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\section{Introduction} Many extensions of the standard model(SM) which place quarks and leptons on a symmetric footing predict the existence of leptoquarks, which are spin-0 or 1, color-triplet objects that couple to a $q\ell$ or $\bar q\ell$ pair{\cite {bigref}}. While these particles may be sought indirectly through their influence on low energy processes{\cite {sacha}}, the most promising approach is via direct production at colliders. In particular, searches for leptoquarks at LEP{\cite {lep}}, HERA{\cite {hera}}, and the Tevatron{\cite {tev}} have already been performed, in most cases concentrating on the specific scenario of scalar leptoquarks. Based on both the direct and indirect searches we might expect that if leptoquarks exist their masses must be above a few hundred GeV, and possibly up in the TeV range. In this paper we will examine the search reach for both scalar and vector leptoquarks at future hadron colliders. The production rates for leptoquarks at such colliders will be shown to be sufficiently large so that particles of this type in the TeV mass range and above become accessible. In addition, we will see that the size of the production cross section alone is sufficient to distinguish scalar from vector leptoquark types. \section{Leptoquark Pair Production} Leptoquarks can be produced either singly or in pairs in hadronic collisions. The cross section for single production, however, relies on the size of the {\it a priori} unknown Yukawa couplings of the leptoquark and is therefore model dependent. Pair production, on the otherhand, proceeds through QCD interactions and thus depends {\it only} on the leptoquark spin and the fact that it is a color triplet field. It has been shown in Ref.{\cite {scalar}} that this mechanism will be dominant unless the Yukawa couplings, which are governed by the electroweak interactions, are rather large, {\it i.e.}, of order electromagnetic strength or greater. This is an important result in that the production of both scalar and vector leptoquarks at hadron colliders is not dependent upon the electroweak properties of these particles. Of course, the converse is also true, {\it i.e.}, the production properties cannot be used to probe the detailed nature of the leptoquark type. The production cross section for pairs of scalar leptoquarks($S$) in either $gg$ or $q\bar q$ collisions is easily calculated and has been available for some time; we use the results of Hewett {\it et al.} ~in Ref.{\cite {scalar}} in what follows. Given the mass of $S$ there are no real ambiguities in this calculation except for the possible inclusion of a $K-$factor (which we omit) to account for higher order QCD corrections and leads to a slight enhancement in the rate. \vspace*{-0.5cm} \noindent \begin{figure}[htbp] \centerline{ \psfig{figure=compare.ps,height=14cm,width=14cm,angle=-90}} \vspace*{-0.6cm} \caption{Production cross section for a pair of 1 TeV vector leptoquarks at the LHC as a function of $\kappa$. The dotted(dashed) curve corresponds to the $gg$($q\bar q$) production subprocess whereas the solid curve is their sum.} \label{compare} \end{figure} \vspace*{0.1mm} The vector leptoquark($V$) case is not as straightforward. In order to calculate the $gg \rightarrow VV$ cross section we need to determine both the trilinear $gVV$ and quartic $ggVV$ couplings, which may naively at first glance appear to be unknown. (For the $q \bar q$ subprocess, only the $gVV$ coupling is required.) However, in any realistic model wherein vector leptoquarks appear and are {\it fundamental} objects, they will be the gauge bosons of an extended gauge group like $SU(5)$. In this case the $gVV$ and $ggVV$ couplings are {\it completely} fixed by extended gauge invariance. These particular couplings will also insure that the subprocess cross sections obey tree-level unitarity, as is the hallmark of all gauge theories. Of course, it might be that the appearance of vector leptoquarks is simply some low energy manifestation of a more fundamental theory at a higher scale and that these particles may even be composite. In this case so-called `anomalous' couplings in both the $gVV$ and $ggVV$ vertices can appear. One possible coupling of this type is an `anomalous chromomagnetic moment', usually described in the literature by the parameter $\kappa$, which takes the value of unity in the more realistic gauge theory case. Among these `anomalous couplings', the term which induces $\kappa$ is quite special in that it is the only one that conserves $CP$ and is of dimension 4. \vspace*{-0.5cm} \noindent \begin{figure}[htbp] \centerline{ \psfig{figure=slepto60tev.ps,height=9.1cm,width=9.1cm,angle=-90} \hspace*{-5mm} \psfig{figure=slepto60tevb.ps,height=9.1cm,width=9.1cm,angle=-90}} \vspace*{-0.6cm} \caption{Scalar leptoquark pair production cross section as a function of mass at a 60 TeV $pp$(left) or $p\bar p$(right) LSGNA collider. The dotted(dashed) curve corresponds to the $gg$($q\bar q$) production subprocess whereas the solid curve is their sum. MRSA$'$ parton densities are employed.} \label{s60} \end{figure} \vspace*{0.1mm} The Feynman rules for the vector leptoquark-gluon interactions can then be derived from the following effective Lagrangian which includes the most general set of $SU(3)_c$ gauge invariant, $CP$-conserving operators of dimension 4 (or less) \begin{equation} {\cal L}_V =-{1\over 2} F^\dagger_{\mu\nu}F^{\mu\nu}+M_V^2V^\dagger_\mu V^\mu -ig_s\kappa V^\dagger_\mu G^{\mu\nu}V_\nu \,. \end{equation} Here, $G_{\mu\nu}$ is the usual gluon field strength tensor, $V_\mu$ is the vector leptoquark field and $F_{\mu\nu}=D_\mu V_\nu-D_\nu V_\mu$, where $D_\mu=\partial_\mu+ig_sT^a G^a_\mu$ is the gauge covariant derivative (with respect to $SU(3)$ color), $G^a_\mu$ is the gluon field and the $SU(3)$ generator $T^a$ is taken in the triplet representation. In most of the numerical results that follow we will assume $\kappa=1$, {\it i.e.}, we will assume that $V$ is indeed a gauge boson and use the results of Hewett {\it et al.} ~in Ref.{\cite {vlq}} for the evaluation of production rates. The cross sections for other nearby values of $\kappa$ are generally qualitatively comparable as is demonstrated by the results shown in Fig.~\ref{compare} for the case of the pair production of 1 TeV vector leptoquarks at the LHC. If the vector leptoquark is {\it not} a gauge boson then we might, {\it e.g.}, expect it to be minimally coupled to the gluon field, as discussed by Bl\"umlein and R\"uckl{\cite {vlq}}. In this case we have instead that $\kappa=0$. The cross section in this case, as can be seen from the figure, is somewhat smaller than in the situation where $V$ is a vector boson with $\kappa=1$. We remind the reader that changes in $\kappa$ will also lead to modifications in the distributions associated with vector leptoquark pair production but these are subjects are beyond the scope of the present analysis and will be discussed elsewhere. \vspace*{-0.5cm} \noindent \begin{figure}[htbp] \centerline{ \psfig{figure=vlepto60tev.ps,height=9.1cm,width=9.1cm,angle=-90} \hspace*{-5mm} \psfig{figure=vlepto60tevb.ps,height=9.1cm,width=9.1cm,angle=-90}} \vspace*{-0.6cm} \caption{Same as the previous figure but now for a spin-1 vector leptoquark with $\kappa=1$. } \label{v60} \end{figure} \vspace*{0.1mm} \section{Results} We now turn to some numerical results. We will consider the production of only a single type of leptoquark at a time and ignore the possibility of a degenerate multiplet of leptoquarks being produced simultaneously. Here we concentrate on cross sections for $S$ and $V$ pair production at the $\sqrt s$=60 (LSGNA) and 200 (PIPETRON) TeV machines, which are displayed in Figures~\ref{s60},~\ref{v60},~\ref{s200} and~\ref{v200}, since the corresponding results for the Tevatron and LHC can be found in, {\it e.g.}, Ref.{\cite {dj}}. In these figures, the contributions of the two distinct subprocesses $gg\rightarrow SS,VV$ and $q\bar q \rightarrow SS,VV$ are separately displayed together with their sum. From Figures~\ref{s60} and~\ref{v60} several conclusions are immediately obvious for leptoquark production at the $\sqrt s=60$ TeV collider: ($i$) The vector leptoquark cross section is substantially larger than that for scalars in both $pp$ and $p\bar p$ collisions since the rates for both $gg\rightarrow VV$ and $q\bar q\rightarrow VV$ are larger than their scalar counterparts. ($ii$) Due to the contribution of the $q\bar q$ production mode, $p\bar p$ colliders have larger leptoquark cross sections than do $pp$ colliders. For example, the ratio of $p\bar p$ to $pp$ cross sections for a 4(6) TeV scalar(vector) leptoquark is approximately 2(6) at $\sqrt s=60$ TeV. At $pp$ machines, for both vector and scalar leptoquarks, the cross sections are dominated by the $gg$ process out to the machine's anticipated mass reach. In the $\sqrt s=60$ TeV $p\bar p $ case, the $q\bar q$ process dominates over $gg$ for masses greater than about 3.0(1.8) TeV for scalar(vector) leptoquarks. The mass reaches for the 60 TeV machine can be found in Table I. \vspace*{-0.5cm} \noindent \begin{figure}[htbp] \centerline{ \psfig{figure=slepto200tev.ps,height=9.1cm,width=9.1cm,angle=-90} \hspace*{-5mm} \psfig{figure=slepto200tevb.ps,height=9.1cm,width=9.1cm,angle=-90}} \vspace*{-0.6cm} \caption{Same as Fig.2 but now at the 200 TeV PIPETRON collider.} \label{s200} \end{figure} \vspace*{0.1mm} At $\sqrt s$=200 TeV, the patterns observed at 60 TeV are repeated. For example, the ratio of $p\bar p$ to $pp$ cross sections for a 10(15) TeV scalar(vector) leptoquark is approximately 1.5(3.5). In the $p\bar p $ collider mode, the $q\bar q$ process dominates over $gg$ for masses greater than about 10(6) TeV for scalar(vector) leptoquarks. \vspace*{-0.5cm} \noindent \begin{figure}[htbp] \centerline{ \psfig{figure=vlepto200tev.ps,height=9.1cm,width=9.1cm,angle=-90} \hspace*{-5mm} \psfig{figure=vlepto200tevb.ps,height=9.1cm,width=9.1cm,angle=-90}} \vspace*{-0.6cm} \caption{Same as Fig.3 but now for the 200 TeV PIPETRON collider.} \label{v200} \end{figure} \vspace*{0.1mm} Table~\ref{leptos} summarizes and compares the search reaches for both scalar and vector leptoquarks at the Tevatron and LHC as well as the hypothetical 60 and 200 TeV $pp$ and $p\bar p$ colliders. Our results for the Tevatron confirm the expectations of the TeV2000 Study Group {\cite {tev2000}}, who also assume the 10 event discovery limit, while those obtained for the LHC are somewhat smaller{\cite {wrochna}} than that given by a fast CMS detector simulation. As discussed above, the larger cross sections for vector leptoquarks results in higher search reaches at all machines. Similarly, the larger $q\bar q$ subprocess contribution to the total cross section at $p\bar p$ machines leads to a greater reach for both scalar and vector leptoquarks in this collision mode. It is clear from this table that future hadron colliders will be able to significantly extend the present search reaches for scalar and vector leptoquarks. \begin{table}[htbp] \centering \begin{tabular}{lccc} \hline \hline Machine & ${\cal L}(fb^{-1})$ & $S$ & $V$ \\ \hline LHC & 100 & 1.34(1.27) & 2.1(2.0) \\ 60 TeV($pp$) & 100 & 4.9(4.4) & 7.6(7.0) \\ 60 TeV($p\bar p$) & 100 & 5.7(5.2) & 9.6(9.0) \\ 200 TeV($pp$) & 1000 & 15.4(14.1) & 24.2(23.3) \\ 200 TeV($p\bar p$) & 1000 & 18.1(16.2) & 31.1(29.0) \\ TeV33 & 30 & $\simeq 0.35$ & $\simeq 0.58$\\ \hline \hline \end{tabular} \caption{Search reaches in TeV for scalar($S$) and vector($V$) leptoquarks at future hadron colliders assuming a branching fraction into a charged lepton plus a jet of unity($1/2$). For vector leptoquarks, $\kappa=1$ has been assumed and in both cases the MRSA$'$ parton densities have been employed. These results are based on the assumption of 10 signal events.} \label{leptos} \end{table} \section{Acknowledgements} The author would like to thanks S. Godfrey, J. Hewett, G. Wrochna and W. Merritt for discussions related to this work. \def\MPL #1 #2 #3 {Mod.~Phys.~Lett.~{\bf#1},\ #2 (#3)} \def\NPB #1 #2 #3 {Nucl.~Phys.~{\bf#1},\ #2 (#3)} \def\PLB #1 #2 #3 {Phys.~Lett.~{\bf#1},\ #2 (#3)} \def\PR #1 #2 #3 {Phys.~Rep.~{\bf#1},\ #2 (#3)} \def\PRD #1 #2 #3 {Phys.~Rev.~{\bf#1},\ #2 (#3)} \def\PRL #1 #2 #3 {Phys.~Rev.~Lett.~{\bf#1},\ #2 (#3)} \def\RMP #1 #2 #3 {Rev.~Mod.~Phys.~{\bf#1},\ #2 (#3)} \def\ZP #1 #2 #3 {Z.~Phys.~{\bf#1},\ #2 (#3)} \def\IJMP #1 #2 #3 {Int.~J.~Mod.~Phys.~{\bf#1},\ #2 (#3)}

proofpile-arXiv_065-500

\section*{Acknowledgments} I am grateful to Stephan Narison for inviting me to QCD-96.

proofpile-arXiv_065-501

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proofpile-arXiv_065-502

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\newcommand{\overline d_\omega}{\overline d_\omega} \newcommand{{\cal A}}{{\cal A}} \newcommand{{\cal B}}{{\cal B}} \newcommand{{\widetilde{\cal A}}}{{\widetilde{\cal A}}} \newcommand{{\widetilde{\cal B}}}{{\widetilde{\cal B}}} \newcommand{{\cal O}}{{\cal O}} \newcommand{{\cal P}}{{\cal P}} \newcommand{{\cal S}}{{\cal S}} \newcommand{\sigma^\omega}{\sigma^\omega} \newcommand{\Omega^\omega}{\Omega^\omega} \newcommand{\overline\varphi}{\overline\varphi} \newcommand{{\cal T}}{{\cal T}} \newcommand{{\cal U}}{{\cal U}} \newcommand{{\cal V}}{{\cal V}} \newcommand{{\cal M}}{{\cal M}} \newcommand{{\gamma^{(\Sigma_K,K,\xnot)}}}{{\gamma^{(\Sigma_K,K,{x_0})}}} \newcommand{\gammatotp}[1]{{\gamma^{(\Sigma_K,K,{x_0})}_{#1}}} \newcommand{\int_{C_2(K\backslash\xnot)}}{\int_{C_2(K\backslash{x_0})}} \newcommand{\int_{C_3(K\backslash\xnot)}}{\int_{C_3(K\backslash{x_0})}} \newcommand{\int_{\Sigma_K}}{\int_{\Sigma_K}} \newcommand{{\rm g.f.}}{{\rm g.f.}} \newcommand{\Ctilde}[2]{{\widetilde{C}_{#1}^{#2}}} \newcommand{{\rm s.f.}}{{\rm s.f.}} \newcommand{{\gamma^{\Sigma_K}}}{{\gamma^{\Sigma_K}}} \newcommand{{\gamma^{(K,\xnot)}}}{{\gamma^{(K,{x_0})}}} \newcommand{{\gamma^{(\Sigma_K,\xnot)}}}{{\gamma^{(\Sigma_K,{x_0})}}} \newcommand{\avel}[1]{{\ave{#1}_\lambda}} \newcommand{{\widetilde U}}{{\widetilde U}} \newcommand{{\Phi'_\bot}}{{\Phi'_\bot}} \newcommand{\univec}[1]{{\frac{{#1}}{|{#1}|}}} \newcommand{\phiphi}[3]{{\univec{{#2}\univec{\dot\Phi({#1})} +{#3}\univec{{\Phi'_\bot}({#1})}}}} \newcommand{\hatphi}[4]{{\hat\phi_{{#1}{#2}}\hat\phi_{{#3}{#4}} }} \newcommand{\der}[1]{{\frac\de{\de{#1}} }} \newcommand\qq{\rm} \newcommand\cmp[1]{{\qq Commun.\ Math.\ Phys.\ \bf #1}} \newcommand\jmp[1]{{\qq J.\ Math.\ Phys.\ \bf #1}} \newcommand\pl[1]{{\qq Phys.\ Lett.\ \bf #1}} \newcommand\np[1]{{\qq Nucl.\ Phys.\ \bf #1}} \newcommand\mpl[1]{{\qq Mod.\ Phys.\ Lett.\ \bf #1}} \newcommand\pr[1]{{\qq Phys.\ Rev.\ \bf #1}} \newcommand\prl[1]{{\qq Phys.\ Rev.\ Lett.\ \bf #1}} \newcommand\lmp[1]{{\qq Lett.\ Math.\ Phys.\ \bf #1}} \newcommand\jsp[1]{{\qq J. Stat.\ Phys.\ \bf #1}} \newcommand\ijmp[1]{{\qq Int.\ J. Mod.\ Phys.\ \bf #1}} \newcommand\cqg[1]{{\qq Class.\ Quant.\ Grav.\ \bf #1}} \newcommand\prept[1]{{\qq Phys.\ Rept.\ \bf #1}} \newcommand\tmp[1]{{\qq Theor.\ Math.\ Phys.\ \bf #1}} \newcommand\anp[1]{{\qq Ann.\ Phys.\ \bf #1}} \newcommand\anm[1]{{\qq Ann.\ Math.\ \bf #1}} \newcommand\adm[1]{{\qq Adv.\ Math.\ \bf #1}} \newcommand\rms[1]{{\qq Russ.\ Math.\ Surveys \bf #1}} \newcommand\jp[1]{{\qq J.\ Phys.\ \bf #1}} \newcommand\bAMS[1]{{\qq Bull.\ Amer.\ Math.\ Soc.\ \bf #1}} \newcommand\tAMS[1]{{\qq Trans.\ Amer.\ Math.\ Soc.\ \bf #1}} \begin{document} \begin{titlepage} \title{Abelian $BF$ Theories and Knot Invariants} \author{ Alberto S. Cattaneo\\[10pt] Lyman Laboratory of Physics\\ Harvard University\\ {\sc Cambridge, MA 02138, USA}\\ E-mail: {\tt cattaneo@math.harvard.edu} } \date{September 25, 1996\\[10pt] {\sf HUTMP-96/B355, hep-th/9609205}} \maketitle \begin{abstract} In the context of the Batalin--Vilkovisky formalism, a new observable for the Abelian $BF$ theory is proposed whose vacuum expectation value is related to the Alexander--Conway polynomial. The three-dimensional case is analyzed explicitly, and it is proved to be anomaly free. Moreover, at the second order in perturbation theory, a new formula for the second coefficient of the Alexander--Conway polynomial is obtained. An account on the higher-dimensional generalizations is also given. \end{abstract} \end{titlepage} \section{Introduction} In recent years, the study of three-dimensional {\em topological quantum field theories} (TQFT) has shed new light on knot invariants. The non perturbative analysis of the Chern--Simons theory in \cite{WittCS} has given an intrinsically three-dimensional definition of the Jones \cite{Jon} and HOMFLY \cite{HOMFLY} polynomials, while the approach of \cite{FK} has shown a more direct connection with the $(2+1)$-dimensional formulation. Later, the perturbative expansion in the covariant gauge \cite{GMM,BN-th} has shown that numerical knot invariants can be obtained in terms of integrals over copies of the knot times copies of ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$, and the second-order invariant has been computed explicitly. A rigorous mathematical formulation of these integrals has been given in \cite{BT} where some subtleties (``anomalies'') that arise in this framework have also been pointed out. Another three-dimensional TQFT is the so-called $BF$ theory \cite{Schw}. The version with a cosmological term gives results that are equivalent to those obtained in Chern--Simons theory \cite{C,CCFM}. The version without cosmological term (or {\em pure}) admits an observable \cite{CCM,Cat,Lon} whose {\em vacuum expectation value} (v.e.v.) is related to the Alexander--Conway polynomial \cite{AC}. This is a ``classical'' knot invariant that can be defined in any odd dimension; yet it is quite difficult to find a good generalization of the three-dimensional observable of the pure $BF$ theory in higher dimensions (s. \cite{CoM} for an attempt). However, pure $BF$ theory is essentially an Abelian theory, as one can see by rescaling $A\to\epsilon A$ and $B\to B/\epsilon$ since, in the limit $\epsilon\to0$, the non-Abelian perturbation $B\wedge A\wedge A$ gets killed. Therefore, we expected to get the same invariants studying the simpler Abelian version of $BF$ theory. In this framework, we have got to define a new non-trivial observable which, though rather involved, has a natural generalization in any odd dimension. In the three-dimensional case, we show that this observable is not ``anomalous" both in the usual field-theoretical and in the Bott and Taubes's \cite{BT} meaning; i.e., we prove both that a quantum observable corresponding to the classical one exists and that the topological nature of its v.e.v.\ is not spoiled by the collapse of more than two points together ({\em hidden faces}). Moreover, we show that this v.e.v.\ yields, at the second order in perturbation theory, a new integral expression for the second coefficient of the Alexander--Conway polynomial (our conjecture is that the whole v.e.v.\ is the inverse of the Alexander--Conway polynomial). More generally, our approach suggests a new way of defining knot invariants (or even higher-degree forms on the space of imbeddings) as integrals over copies of the knot and a cobounding surface that turn out not to depend on the choice of the surface. Eventually, we recall that the Abelian $BF$ theory has a physical application \cite{FGM} as a tool for studying the bosonization of many-body systems. It would be interesting to see if our observable has a physical interpretation as well. \subsection{Plan of the paper} Explicitly, in the three-dimensional case, we want to consider the classical action \[ S_{\rm cl} = \int_{{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3} B\wedge dA, \] where $A$ and $B$ are one-forms. This theory is invariant under adding an exact form to either $A$ or $B$ . A classical observable for this theory is then given by \[ \gammatotp{\rm cl} = \int_{\Sigma_K} B\wedge A +\frac12 \oint_{x<y\in K} [A(x)\,B(y)-B(x)\,A(y)], \] where $K=\de\Sigma_K$ is a knot. This observable can be shown to be {\em on-shell} invariant under the symmetries of the action, i.e., to be invariant up to terms containing the equations of motion (which state that $A$ and $B$ are closed forms). Observables that are invariant only on shell can be considered in the {\em Batalin--Vilkovisky} (BV) formalism \cite{BV}, which we briefly review in Sec.\ \ref{sec-BV}. The application of the BV formalism to the three-dimensional Abelian $BF$ theory is given in Sec.\ \ref{tdabft}. (In App.\ \ref{app-zm}, we discuss the generalization to the case where there exist harmonic zero modes.) In Sec.\ \ref{sec-obs}, we discuss the BV formulation of our observable and prove that its exponential is not anomalous, i.e., it is possible to make it an observable by adding higher-order corrections in the Planck constant. (In App.\ \ref{app-corr}, we find the lowest-order correction explicitly.) In Sec.\ \ref{sec-comp}, we describe the Feynman diagrams of the theory, discuss the generalization to our case of the ``regularization" framework developed in \cite{BT}, and obtain the simplest---but non-trivial---numerical knot invariant that our theory produces, viz., the second coefficient of the Alexander--Conway polynomial. (In App.\ \ref{app-hf}, we prove that this v.e.v.\ is not anomalous in the sense of \cite{BT}, and generalize this result to any order in perturbation theory.) Finally, in Sec.\ \ref{sec-glim}, we describe the steps to be taken to define the higher-dimensional generalizations of our theory. \section{The BV formalism} \lab{sec-BV} The BV formalism \cite{BV} is a generalization of the BRST formalism \cite{BRST} which is applicable to theories whose symmetry closes only on shell. Moreover, even in the case of off-shell closed symmetries, the BV formalism allows dealing with observables that are invariant only on shell, as the ones we are interested in. In this section we give only a brief introduction. We refer to Ref.\ \cite{Ans}, whose notations we follow, for a thorough exposition of the BV formalism, as well as for a clear-cut discussion of the renormalization issue (which we have not to deal with in the present paper since the theory we consider is topological as well as Gaussian). \subsection{Preliminaries} We denote by $\Phi^i$ the {\em fields} one needs in a theory (i.e., the physical fields, the ghosts, the antighosts, the Lagrange multipliers and, if necessary, the ghosts for ghosts and so on); the space of fields is called the {\em configuration space}. We denote by $\epsilon(\Phi^i)$, or simply by $\epsilon_i$, the ghost number of the field $\Phi^i$. By simplicity, we consider only the case of a theory whose physical fields are bosonic, so the Grassmann parity of $\Phi^i$ is given by $(-1)^{\epsilon_i}$. In the BV formalism, along with every field $\Phi^i$ one introduces an {\em antifield} $\Phi^{{\dag}}_i$ with the same characteristics of its partner but the ghost number, which instead is given by \beq \epsilon(\Phi_i^{{\dag}}) = -\epsilon_i - 1; \lab{ghn} \eeq this also implies that the Grassmann parity is reversed. The space of fields and antifields is called the {\em phase space} \cite{SchwBV}. In the next sections we will also use the antifields $\Phi^*_i$ satisfying \beq \Phi^{{\dag}}_i := *\Phi^*_i, \lab{stardagger} \eeq where $*$ is the Hodge operator. Over the phase space one introduces a supersymplectic structure \cite{SchwBV} which allows the definition of the BV {\em antibracket} \beq \antib XY := X \braket{\dphib i}{\dphid i} Y - X \braket{\dphidb i}{\dphi i} Y \lab{defantib} \eeq and the BV {\em Laplacian} \beq \Delta X := \sum_i (-1)^{\epsilon_i+1} X \braket{\dphib i}{\dphidb i}. \lab{deflap} \eeq Here $X$ and $Y$ are functionals over the phase space and $\braket\cdot\cdot$ denotes the scalar product \beq \braket\alpha\beta := \int_M \alpha\wedge*\beta \eeq and $M$ is the manifold over which the theory is defined. In Ref.\ \cite{WittBV}, the BV phase space is interpreted as the tangent space $T{\cal M}$ over the space of fields ${\cal M}$. In the finite-dimensional case, this is locally isomorphic to the cotangent space $T^*{\cal M}$ by using a volume form on ${\cal M}$. Then the Laplacian $\Delta$ on $T{\cal M}$ is in correspondence with the exterior derivative on $T^*{\cal M}$. However, the product of two functionals on $T{\cal M}$ does not correspond to the wedge product on $T^*{\cal M}$, and the antibracket measures this failure since \beq \antib XY = (-1)^{\epsilon(Y)} [\Delta(XY)-X\,\Delta Y - (-1)^{\epsilon(Y)} \Delta X\ Y] \eeq (notice that the Laplacian as defined in \rf{deflap} acts from the right). Of course, in the infinite-dimensional case (in which we are interested), this description is only formal. In particular, the Laplacian depends on the regularization we use to define the functional integral. \subsection{BV cohomologies} One can define some cohomologies on the phase space, with respect to the gradation provided by the ghost number. Each cohomology is defined by a coboundary operator, i.e., a nilpotent operator of ghost number one. The simplest coboundary operator is the Laplacian itself. The second interesting coboundary operator is \beq \Omega X := \antib X\Sigma - i\hbar \Delta X, \lab{defOmega} \eeq where $\Sigma$, the {\em quantum action}, is a bosonic functional that has to satisfy the {\em quantum master equation} \beq \antib\Sigma\Sigma -2i\hbar\Delta\Sigma = 0 \lab{qme} \eeq for $\Omega$ to be nilpotent. Notice that \rf{qme} is equivalent to asking the Gibbs weight $\exp(i\Sigma/\hbar)$ to be $\Delta$-closed. The third coboundary operator we consider is \beq \sigma X := \antib XS \lab{defsigma} \eeq where $S$, the {\em action}, is a bosonic functional that has to satisfy the {\em master equation} \beq \antib SS =0 \lab{me} \eeq for $\sigma$ to be nilpotent. A particular case, which we encounter in this paper, is provided by a $\Delta$-closed action $S$. In this case \beq \Delta\sigma + \sigma\Delta = 0, \lab{Deltasigma} \eeq so $\Delta$ and $\sigma$ define a double complex. Moreover, by \rf{qme}, $S$ is also a quantum action for any $\hbar$ and, as such, it defines an $\Omega$-cohomology. The restriction of the operator $\sigma$ to the configuration space defines a new operator, \beq sX := (\sigma X)_{\big|_{\Phi^{{\dag}}=0}}, \lab{defs} \eeq which can be shown to be nilpotent {\em on shell}, i.e., modulo the solutions of \beq S_{\rm cl}\,\frac{\overleftarrow\delta}{\delta\Phi^i} = 0, \lab{onshell} \eeq where \beq S_{\rm cl} = S_{\big|_{\Phi^{{\dag}}=0}} \lab{Scl} \eeq is the {\em classical action}. Thus, $s$ defines a cohomology on the configuration space on shell. If $s$ is nilpotent also off shell, one says that the symmetry closes. In this case the BRST approach is available and $s$ is actually the BRST operator. Notice, however, that by \rf{defs} $s$ is defined to act from the right; the usual BRST operator $s_l$ is the corresponding operator acting from the left, and one has \beq s_l X = (-1)^{\epsilon(X)} s X. \lab{defBRST} \eeq \subsection{BV quantization} The interest in the $\Omega$-cohomology relies on the fact that one can formally show that the class of observables whose v.e.v.'s are gauge-fixing independent is given by the $\Omega$-closed bosonic functionals modulo $\Omega$-exact terms. More precisely, one introduces the {\em partition function} \beq Z_\Psi := \int_{ {\cal L}_\Psi } e^{\frac i\hbar\Sigma}, \lab{defZ} \eeq where the {\em Lagrangian submanifold} ${ {\cal L}_\Psi }$ is defined by the equations \beq \Phi^{{\dag}}_i = \dphi i\,\Psi(\Phi), \lab{deflpsi} \eeq and $\Psi$, the {\em gauge-fixing fermion}, is a functional on the configuration space that has ghost number $-1$. In Ref.\ \cite{WittBV}, prescription \rf{deflpsi} is shown to amount to selecting the top form in the functional on $T^*{\cal M}$ that, under the isomorphism between $T{\cal M}$ and $T^*{\cal M}$, corresponds to the Gibbs weight $\exp(i\Sigma/\hbar)$. The v.e.v.\ of a functional $X$ over the phase space is then defined as \beq \ave X_\Psi = \frac1{Z_\Psi} \int_{ {\cal L}_\Psi } e^{\frac i\hbar\Sigma}\, X. \lab{defvev} \eeq By using the formal properties of the functional integration, one has then the following \cite{AD,SchwBV} \begin{Stat} If $\Sigma$ satisfies the quantum master equation \rf{qme}, then \begin{enumerate} \item the partition function $Z_\Psi$ and the expectation values of $\Omega$-closed functionals do not change under infinitesimal variations of the gauge-fixing fermion $\Psi$, and \item the expectation value of an $\Omega$-exact functional vanishes. \end{enumerate} \lab{statBV} \end{Stat} In the finite-dimensional case, Statement \ref{statBV} becomes a rigorous theorem. One can also show that the definitions \rf{defZ} and \rf{defvev} correspond to the usual ones in the BRST formalism whenever applicable. The $\sigma$-cohomology is useful since it is given by the $\Omega$-cohomology in the limit $\hbar\to0$ and is much easier to study. The idea is to solve the quantum master equation and to study the $\Omega$-cohomology by an expansion in powers of $\hbar$. Notice, however, that from an action satisfying \rf{me} is not always possible to obtain a quantum action satisfying \rf{qme} that, in the limit $\hbar\to0$, yields the starting action; if this does not happen, one calls the theory {\em anomalous}. Moreover, even if the theory is not anomalous, a $\sigma$-closed functional of ghost number zero does not always produce an $\Omega$-closed functional that, in the limit $\hbar\to0$, yields the starting one. A sufficient condition for both to happen is that the one-ghost-number $\sigma$-cohomology be trivial. As explained before, the $s$-cohomology is the restriction of the $\sigma$-co\-hom\-ol\-ogy to the configuration space on shell. Since it is easier to study than the $\sigma$-cohomology, one can study the latter by an expansion in antifields. Under some mild assumptions \cite{VBF}, one can prove that the extension from a classical action $S_{\rm cl}$ to an action $S$ satisfying \rf{me} and \rf{Scl} exists and is unique modulo {\em canonical transformations} (i.e., transformations on the phase space that preserve the supersymplectic structure). \section{The three-dimensional Abelian $BF$ theory} \lab{tdabft} In this section we apply the BV formalism to the theory defined by the classical action \beq S_{\rm cl}^\omega = \int_M B\wedge d_\omega A, \lab{Sclomega} \eeq where \begin{itemize} \item $M$ is a three-manifold; \item $A$ and $B$ are fields taking values in $\Omega^1(M)$; \item $d_\omega = d + i\omega$, and \item $\omega$ is an external $d$-closed source in $\Omega^1(M)$ (thus, $d_\omega^2=0$). \end{itemize} In the particular case $\omega=0$, we will simply write $S_{\rm cl}$ and speak of the {\em pure} theory. We can also split $S_{\rm cl}^\omega$ as \beq S_{\rm cl}^\omega = S_{\rm cl} -i \gamma_{\rm cl}^\omega, \lab{Sclomegasplit} \eeq with \beq \gamma_{\rm cl}^\omega = -\int_M B\wedge \omega\wedge A= \int_M \omega\wedge B\wedge A, \lab{defgammaomegacl} \eeq and see $\gamma_{\rm cl}^\omega$ as a perturbation of the pure classical action. If $H^1(M,d_\omega)$ is trivial (for the general case s.\ App. \ref{app-zm}), the symmetries of this theory are simply given by \beq \begin{array}{cc} s^\omega A = d_\omega c, & s^\omega c =0,\\ s^\omega B = \overline d_\omega \psi & s^\omega\psi=0, \end{array} \lab{defso} \eeq where $\overline d_\omega = d - i\omega$, and $c$ and $\psi$ are the ghosts, which take values in $\Omega^0(M)$ and have ghost number one. Notice that $\gamma^\omega_{\rm cl}$ is on-shell invariant under the symmetry \rf{defso} of the pure theory. \subsection{The BV action} The BV action corresponding to \rf{Sclomega} is given by \beq S^\omega = \int_M B\wedge d_\omega A + A^*\wedge d_\omega c + B^*\wedge \overline d_\omega \psi + \bar c^* h_c + \bar\psi^* h_\psi, \lab{Somega} \eeq where $\bar c$ and $\bar \psi$ are the antighosts, and $h_c$ and $h_\psi$ are the Lagrange multipliers. The antighosts and the Lagrange multipliers take values in $\Omega^0(M)$; the former have ghost number minus one, the latter have ghost number zero. The additional terms in the antighosts and Lagrange multipliers are necessary to gauge fix the theory. Notice that we have used here the antifields $*$ instead of the antifields ${{\dag}}$, s.\ \rf{stardagger}. If the Laplacian $d_\omega^*d_\omega+d_\omega\domega^*$ has zero modes, additional terms are required; for simplicity we suppose that there are no zero modes, i.e., we suppose that the cohomology $H^*(M,d_\omega)$ is trivial (s.\ App.\ \ref{app-zm} for the case when $H^1(M,d_\omega)$ is not trivial). It is not difficult to see that $S^\omega$ satisfies the master equation \rf{me} for any closed one-form $\omega$. Notice moreover that $\Delta S=0$, so $S$ also satisfies the quantum master equation \rf{qme}. Thus, we can quantize the theory with Gibbs weight $\exp(iS^\omega/\hbar)$ for any $\hbar$. In the following we will set $\hbar=1$. Notice moreover that the action $S^\omega$ does not require the choice of a metric on $M$, so we expect its partition function to be a topological invariant of $M$. The $\sigma^\omega$ operator \rf{defsigma} acts on the fields and antifields as follows \beq \begin{array}{cccc} \sigma^\omega\psi^*=-d_\omega B^*, & \sigma^\omega B^*=-d_\omega A, & \sigma^\omega A = d_\omega c, & \sigma^\omega c=0,\\ \sigma^\omega c^*=-\overline d_\omega A^*, & \sigma^\omega A^*=-\overline d_\omega B, & \sigma^\omega B = \overline d_\omega\psi, & \sigma^\omega\psi =0, \end{array} \lab{sigmaAB} \eeq \beq \begin{array}{cccc} \sigma^\omega h_c^*=-\bar c^*, & \sigma^\omega \bar c^* =0, & \sigma^\omega\bar c=h_c, & \sigma^\omega h_c=0,\\ \sigma^\omega h_\psi^*=-\bar\psi^*, & \sigma^\omega\bar\psi^*=0, & \sigma^\omega\bar\psi=h_\psi, & \sigma^\omega h_\psi=0. \end{array} \lab{sigmaanti} \eeq It is very useful to consider the following linear combinations \beq \begin{array}{lcccccccc} && (3,-2) && (2,-1) && (1,0) && (0,1)\\ {\cal A} &=& -\psi^* &+& B^* &+& A &+& c,\\ {\cal B} &=& -c^* &+& A^* &+& B &+& \psi; \end{array} \lab{defcalAB} \eeq where by $(i,j)$ we denote an $i$-form of ghost number $j$. Notice that ${\cal A}$ and ${\cal B}$ have an overall (i.e., form plus ghost) degree equal to one. By \rf{defcalAB}, we can rewrite the action as \beq S^\omega = \int_M {\cal B}\wedged_\omega{\cal A} + \bar c^* h_c + \bar\psi^* h_\psi. \lab{Somegacal} \eeq Moreover, we can rewrite \rf{sigmaAB} as \beq \begin{array}{lcr} \sigma^\omega_l{\cal A} &=& d_\omega{\cal A},\\ \sigma^\omega_l{\cal B} &=& \overline d_\omega{\cal B}, \end{array} \lab{sigmacalAB} \eeq where $\sigma^\omega_l$ is the operator corresponding to $\sigma^\omega$ but acting from the left (as the exterior derivative); notice that \beq \sigma^\omega_l X = (-1)^{\epsilon(X)} \sigma^\omega X. \eeq Following \rf{Sclomegasplit}, we can split $S^\omega$ as \beq S^\omega = S -i \gamma^\omega, \lab{Somegasplit} \eeq where \beq \gamma^\omega = -\int_M{\cal B}\wedge\omega\wedge{\cal A}= \int_M\omega\wedge{\widetilde{\cal B}}\wedge{\cal A}, \lab{defgammaomega} \eeq where the operator $\tilde{}$ acts by changing sign to odd-ghost-number terms. The splitting \rf{Somegasplit} is very convenient since not only do both $S^\omega$ and $S$ satisfy the quantum master equation \rf{qme}, but we also have \beqy \antib{\gamma^\omega}S &=& 0,\lab{goS}\\ \antib{\gamma^\omega}{\gamma^\omega}&=&0,\lab{gogo}\\ \Delta\gamma^\omega&=&0\lab{Deltago} \eeqy [notice that, because of \rf{sigmaAB+}, \rf{goS} does not hold if $H^1(M,d)$ contains nontrivial elements besides $\omega$]. We can also split the Gibbs weight $\exp(iS^\omega)$ into the Gibbs weight $\exp(iS)$ times the observable \beq \Gamma[\omega] = \exp\gamma^\omega, \lab{defOomega} \eeq which we can prove to be $\Omega$-closed as a consequence of \rf{goS}, \rf{gogo} and \rf{Deltago}. Notice that, if $\omega$ is not trivial, the action $S$ as to be modified as in App.\ \ref{app-zm} [with $b_1=1$ and $\varphi_1=\overline\varphi_1=v\,\omega'/\braket{\omega'}{\omega'}$, where $\omega'=\omega+d\alpha$ and $d^*\omega'=0$]. By using these notations, it is easy to prove that the theory really depends only on the cohomology class of $\omega$. In fact, if we substitute $\omega$ with $\omega+df$, the action $S^\omega$ gets an extra contribution \beq S^\omega \longrightarrow S^\omega + \Omega^\omega T^f \eeq with \beq T^f=\int_M f{\widetilde{\cal B}}\wedge{\cal A}. \eeq By noticing that, $\Delta T^f=0$, $\antib{\Omega^\omega T^f}{\Omega^\omega T^f}=0$ and $\antib{\Omega^\omega T^f}{T^f}=0$, we can show that \beq \exp(i\Omega^\omega T^f) = 1 + \Omega^\omega U^f, \eeq with \beq U^f = \sum_{n=1}^{\infty}\frac{i^n}{n!}\, (\Omega^\omega T^f)^{n-1}\,T^f. \eeq Since, by Statement \ref{statBV}, the v.e.v.\ of an $\Omega^\omega$-exact functional vanishes, we conclude the (formal) proof that the partition function of the theory depends only on the cohomology class of $\omega$. \subsection{The quantization} To quantize the theory, we have to choose a gauge-fixing fermion. A convenient choice is \beq \Psi = \braket{d_\omega\bar c}A + \braket{\overline d_\omega\bar \psi}B. \lab{defPsi} \eeq Notice that to gauge fix the theory we need to choose a metric on $M$, but, by Statement \ref{statBV}, the partition function will not depend on it. By \rf{defZ} and \rf{deflpsi}, we have \beq Z[M,\omega] = \int[DA\,DB\,Dc\,D\bar c\,D\psi\,D\bar\psi\, Dh_c\,Dh_\psi] \exp{(i S^\omega_{\rm g.f.})}, \lab{Zomega} \eeq where \beq S^\omega_{\rm g.f.} = \int_M B\wedge d_\omega A + \braket{d_\omega\bar c}{d_\omega c} + \braket{\overline d_\omega\bar\psi}{\overline d_\omega \psi} + \braket{h_c}{d_\omega^* A} + \braket{h_\psi}{\overline d_\omega^*B}. \lab{Somegagf} \eeq The partition function \rf{Zomega} can then be computed by using the zeta-function regularization of the determinants and yields \cite{Schw} \beq Z[M,\omega] = {\cal T}(M,d_\omega), \lab{ZomegaT} \eeq where ${\cal T}$ is the Ray--Singer torsion, which is a topological invariant of the manifold $M$ and depends only on the cohomology class of the closed one-form $\omega$. This explicit result confirms the previously discussed formal arguments. Notice that any multiple of $\omega$ is still a closed one-form; thus, we can consider $Z[M,\lambda\omega]$ as well. Now we recall that the Ray--Singer torsion is equal to the Reidemeister torsion \cite{CM} and the Reidemeister torsion of the complement of a knot is proportional to the inverse of the Alexander--Conway polynomial of the knot itself \cite{MT}; thus, we can see the inverse of the Alexander--Conway polynomial as the partition function \rf{ZomegaT} of an Abelian $BF$ theory. More precisely, we have the following (cfr.\ \cite{BG} and \cite{Roz}) \begin{Th} If $M={\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3\backslash{\rm Tub}(K)$, where ${\rm Tub}(K)$ is a tubular neighborhood of the knot $K\in{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$, and $\omega\in H^1(M)$ is such that \beq \oint_{K_1}\omega=1, \lab{c1omega} \eeq with $K_1$ a closed circle wrapping around $K$ only once, then \beq \frac{Z[M,\lambda\omega]}{Z[M]} = \frac1{i\lambda}\frac{z(\lambda)}{\Delta(K;z(\lambda))}, \quad z(\lambda) = 2i\sin(\lambda/2), \lab{aveOomega} \eeq where $\Delta(K;z)$ is the Alexander--Conway polynomial satisfying the skein relation \[ \Delta(K_+;z) - \Delta(K_-;z) = z\,\Delta(K_0;z) \] and normalized to one on the unknot. \lab{thmOomega} \end{Th} \section{Observables for the pure theory} \lab{sec-obs} From now on, we will consider only the pure theory defined by the action $S$ in \rf{Somega} with $\omega=0$. We will look for observables (i.e., $\Omega$-closed zero-ghost-number functionals modulo $\Omega$-exact terms) that are metric independent. By Statement \ref{statBV}, their v.e.v.'s will give topological invariants (up to framing) since the action is metric independent as well. Our survey is not exhaustive; i.e., there could exist other more involved, metric-independent observables that could lead to other topological invariants. \subsection{Loop observables} The simplest observables one can build are \beq \gamma_A^K = \oint_K A,\quad \gamma_B^K = \oint_K B, \eeq where $K$ is an exact one-cycle [if $K$ were only closed, these functionals would not be closed under \rf{sigmaAB+}]. These observables are always $\Omega$-exact: \beq \gamma_A^K = -\Omega\beta^*_\Sigma,\quad \gamma_B^K = -\Omega\alpha^*_\Sigma, \lab{alphaO} \eeq where \beq \alpha^*_\Sigma = \int_\Sigma A^*,\quad \beta^*_\Sigma = \int_\Sigma B^*, \eeq and $\Sigma$ is a surface cobounding $K$. Any function of $\gamma_A$ or $\gamma_B$ separately will be $\Omega$-exact, too. To get a nontrivial observable, we have to pair them; e.g., we can consider the observable \beq \tau[K_1,K_2] = \gamma_A^{K_1}\,\gamma_B^{K_2}. \lab{deftau} \eeq By \rf{alphaO}, we can show that \beq \tau[K_1,K_2] = -i\Delta(\gamma_A^{K_1}\,\alpha^*_{\Sigma_2}) -\Omega(\gamma_A^{K_1}\,\alpha^*_{\Sigma_2}). \eeq Since \beq \Delta(\gamma_A^{K_1}\,\alpha^*_{\Sigma_2}) = -\int_{\Sigma_2}\omega_{K_1} = -\#(K_1,\Sigma_2)= -{\rm lk}(K_1,K_2) \eeq (where $\omega_{K_1}$ is the Poincar\'e dual of $K_1$, $\#$ denotes the intersection number and ${\rm lk}$ the linking number), we have \beq \ave{\tau[K_1,K_2]} = i\,{\rm lk}(K_1,K_2). \lab{avetau} \eeq An explicit computation of the l.h.s.\ with the gauge-fixing \rf{defPsi} actually gives Gauss's formula. \subsection{Surface observables} As we have seen, the loop observables are rather trivial. A more interesting observable can be built if $\dim H^1(M)=\dim H_2(M,\de M)=1$; viz., define \beq \gamma^\Sigma = \int_\Sigma{\widetilde{\cal B}}\wedge{\cal A}= \int_\Sigma(B\wedge A + B^*\psi + cA^*), \lab{defgammaSigma} \eeq with $\Sigma\in H_2(M,\de M)$. This observable is essentially the same as in \rf{defgammaomega} with $\Sigma$ the Poincar\'e dual of $\omega$, so we know that its exponential \beq \Gamma[\Sigma,\lambda] = \exp(\lambda\gamma^\Sigma), \lab{expGS} \eeq is an observable as well which, up to $\Omega$-exact terms, depends only on the homology class of $\Sigma$. Moreover, the splitting \rf{Somegasplit}, shows us that \beq \ave{\Gamma[\Sigma,\lambda]}_M = \ave{\exp(\lambda\gamma^\omega)}_M = \frac{Z[M,\lambda\omega]}{Z[M]}. \lab{avegammaomega} \eeq In particular, this holds when $M$ is as in the hypotheses of Thm.\ \ref{thmOomega}. Thus, from the r.h.s.\ of \rf{aveOomega} we can read the v.e.v.\ of $\Gamma[\Sigma,\lambda]$. Notice that condition \rf{c1omega} on $\omega$ requires its Poincar\'e dual $\Sigma$ to satisfy $\#(\Sigma,K_1)=1$. Since any surface $\Sigma_K$ {\em spanning} the knot $K$ (i.e., any oriented surface $\Sigma_K$ imbedded in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$ such that $K$ is identical with the boundary of $\Sigma_K$, and the orientation on $\Sigma_K$ induces the given orientation on $K$) satisfies this property, we have the following \begin{Th} If $M={\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3\backslash{\rm Tub}(K)$ and $\Sigma=\Sigma_K\bigcap M\in H_2(M,\de M)$, then \beq \ave{\Gamma[\Sigma,\lambda]}_M = \frac1{i\lambda}\frac{z(\lambda)}{\Delta(K;z(\lambda))}, \quad z(\lambda) = 2i\sin(\lambda/2). \lab{aveOSigma} \eeq \lab{thmOSigma} \end{Th} Notice that a spanning surface $\Sigma_K$ always exists; e.g., we can take the Seifert surface. An expansion in powers of $\lambda$ of the l.h.s.\ of \rf{aveOSigma} would give a representation of the coefficients of the inverse of the Alexander--Conway polynomial as Feynman diagrams involving only bivalent vertices on $\Sigma$. However, the problem of finding the propagators in a manifold like the one described above is very difficult. In the next subsection, we will see how to recast the problem as the computation of a v.e.v.\ in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$. \subsection{Surface-plus-knot observables} \lab{ssec-spko} From now on we work in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$ and consider a knot $K$ together with a spanning surface $\Sigma_K$. Thm.\ \ref{thmOSigma} suggests to consider the v.e.v.\ of the exponential of $\gamma^{\Sigma_K}$. However, since $\Sigma_K$ has a boundary, $\gamma^{\Sigma_K}$ is not $\sigma$-closed anymore; actually, \beq \sigma\gamma^{\Sigma_K} = \oint_K(\psi A-Bc). \lab{sigmagammaSigma} \eeq Therefore, we have to find another functional depending on $K$ (so that it vanishes when $\Sigma_K$ is closed) such that its $\sigma$-variation cancels \rf{sigmagammaSigma}. We first consider \beq \gamma^{(K,{x_0})} = \frac12\int_{x<y\in K} [A(x)\, B(y)-B(x)\, A(y)], \lab{defgammaK} \eeq where ${x_0}$ is a base point on $K$. Notice that $K\backslash{x_0}$ is diffeomorphic to ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}$, so its configuration spaces $C_n(K\backslash{x_0})$ are diffeomorphic to the configuration spaces $C_n({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}})$ described in \cite{FM} (s.\ also subsection \fullref{ssec-reg}). On these spaces it is possible to introduce the {\em tautological forms} \beq \eta_{ij} = \phi^*_{ij}\,\omega, \quad i,j=1,\ldots,n,\quad i\not= j, \eeq where $\phi^*_{ij}$ denotes the pullback via the map \beq \phi_{ij}(\vec x) = {\rm sgn}(x_i-x_j), \quad \vec x\in C_n({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}), \eeq and $\omega=1/2$ is the the unit volume element on $S^0$. With these notations, we can rewrite \rf{defgammaK} as \beq \gamma^{(K,{x_0})} = \int_{C_2(K\backslash{x_0})}A_1\wedge\eta_{12}\wedge B_2. \lab{AetaB} \eeq Now a simple computation shows that \beq \sigma\gamma^{(K,{x_0})} = -\oint_K(\psi A-Bc)+ \psi({x_0})\,\oint_KA -\oint_KB\,c({x_0}); \lab{sigmagammaK} \eeq so the first term cancels \rf{sigmagammaK}. We have then to find another functional (vanishing when $\Sigma_K$ has no boundary) whose variation cancels the second and third terms. It is not difficult to see that \beq \gamma^{(\Sigma_K,{x_0})}=\psi({x_0})\,\int_{\Sigma_K}B^*+ \int_{\Sigma_K}A^*\ c({x_0}) \lab{defgamma0} \eeq does the job. Thus, we can define the following $\sigma$-closed (actually, $\Omega$-closed) functional \beq {\gamma^{(\Sigma_K,K,\xnot)}} = \gamma^{\Sigma_K} + \gamma^{(K,{x_0})} + \gamma^{(\Sigma_K,{x_0})}. \lab{gammatot} \eeq In the case of links---which we will not consider anymore in the following---the observable has to be modified as \beq \gamma^{(\Sigma_K,K,\{\Sigma_{K_i},x_{0i}\})} = \gamma^{\Sigma_K} + \sum_i\left[{ \gamma^{(K_i,x_{0i})} + \gamma^{(\Sigma_{K_i},x_{0i})} }\right], \eeq where $\Sigma_K$ is a spanning surface for the link $K$ while each $\Sigma_{K_i}$ is a spanning surface only for the component $K_i$, whose base point is denoted by $x_{0i}$. Then, recalling \rf{expGS}, we want to consider the exponential of ${\gamma^{(\Sigma_K,K,\xnot)}}$, \beq {\cal O}_0[K,\lambda]=\exp(\lambda{\gamma^{(\Sigma_K,K,\xnot)}}), \lab{calO0} \eeq which is $\sigma$-closed and hence a candidate to be an observable. Actually, \beq \Delta{\cal O}_0[K,\lambda]=\frac{\lambda^2}2\,{\cal O}_0[K,\lambda]\, \antib{\gamma^{(\Sigma_K,K,\xnot)}}\gammatot \lab{DeltaO0} \eeq vanishes if we are working in {\em standard framing}, i.e., if \beq {\rm slk}(K)=\int_{\Sigma_K}\omega_K=0, \lab{defsf} \eeq where $\omega_K$ is the Poincar\'e dual of $K$ and ${\rm slk}$ denotes the self-linking number [whose definition via \rf{defsf} relies on a choice of regularization]. With this hypothesis, we expect the v.e.v.\ of ${\cal O}_0$ not to depend on the gauge fixing and, as a consequence, to be metric independent. By essentially the same proof that led to the invariance (modulo $\Omega$-exact terms) of $\Gamma[\omega]$, s.\ \rf{defOomega}, under $\omega\rightarrow\omega+d\eta$, we can prove that ${\cal O}_0$ is invariant (modulo $\Omega$-exact terms) under $\Sigma_K\rightarrow\Sigma_K+ \de T$ with $T\in\Omega_3({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3)$. From \rf{aveOSigma} we expect the v.e.v.\ of ${\cal O}_0[K,\lambda]$ to be proportional to the inverse of the Alexander--Conway polynomial. The proportionality constant, which depends on $\lambda$, could be spoiled when we send ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3\backslash{\rm Tub}(K)$ to ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$; thus, we can make only the weaker statement that \beq \frac{\ave{{\cal O}_0[K_{\rm s.f.},\lambda]}} {\ave{{\cal O}_0[\bigcirc_{\rm s.f.},\lambda]}} = \frac1{\Delta(K;z(\lambda))}, \quad z(\lambda) = 2i\sin(\lambda/2), \lab{aveOK0} \eeq where $K_{\rm s.f.}$ and $\bigcirc_{\rm s.f.}$ are, respectively, a generic knot and the unknot in standard framing. This result has to be compared with the similar formulae obtained in the context of the {\em non-Abelian} pure $BF$ theory \cite{CCM,Cat}. It should not amaze that the Abelian and non-Abelian pure $BF$ theories are under this respect equivalent, for, as observed in \cite{Cat}, the v.e.v.'s of the latter can be computed exactly in saddle-point approximation (s.\ also the observation in the Introduction). We want to point out the precise meaning of \rf{aveOK0}: We are not claiming that the coefficients of the $\lambda$-expansion of $\ave{{\cal O}_0}$ are a sum of numerical knot invariants up to factors containing the self-linking number. We are saying that these numerical knot invariants are well defined only if the knot is in standard framing; otherwise ${\cal O}_0$ is not an observable, and we are not guaranteed that its v.e.v.\ is a topological invariant. This means that, to compute this v.e.v., we have to choose a particular presentation of the knot, viz., one in which the self-linking number vanishes, and that this v.e.v.\ should be invariant only under deformations that do not change the self-linking number. In the next section, we will discuss how to drop this cumbersome condition. \subsection{The corrected surface-plus-knot observable} \lab{ssec-cspko} For the purposes of this subsection, it is convenient to rescale \beq {\cal B}\longrightarrow {\cal B}/\lambda, \lab{rescale} \eeq so the Gibbs weight becomes $\exp(iS/\lambda)$ and we recognize $\lambda$ as the Planck constant of the theory. Under this rescaling we also have \beq {\cal O}_0[K,\lambda]\longrightarrow {\cal O}_0[K]=\exp{\gamma^{(\Sigma_K,K,\xnot)}}; \eeq thus, ${\cal O}_0$ is a classical observable, i.e., it is $\sigma$-closed and does not depend on the Planck constant. Now we look for a quantum generalization \beq {\cal O}[K,\lambda] = \sum_{n=0}^\infty (i\lambda)^n\,{\cal O}_n[K] \lab{defOK} \eeq satisfying \beq \Omega{\cal O}=(\sigma-i\lambda\Delta){\cal O}=0, \lab{OmegaO} \eeq and hence \beq \sigma{\cal O}_{n} = \Delta{\cal O}_{n-1},\quad\mbox{$n=1,2,\ldots$}. \lab{Onn} \eeq Notice that a solution $O_{n}$, if it exists, is defined only up to $\sigma$-closed terms. However, if we send $O_n$ into $O_n+\sigma Y_n$, the solution $O_{n+1}$ will be sent into $O_{n+1}-\Delta Y_n$ because of \rf{Deltasigma}. Thus, ${\cal O}$ will be changed by an $\Omega$-exact term. On the other hand, if we add a nontrivial $\sigma$-closed term to $O_n$, then this---together with the extra contribution $O_{n+1}$ receives by \rf{Onn}---lets ${\cal O}$ get an extra $\Omega$-closed term. This means that the solution of \rf{OmegaO}, if it exists, is unique only up to $\Omega$-closed terms, i.e., up to other observables. It is not difficult to see, by \rf{Deltasigma}, that, if \rf{Onn} holds up to a fixed $n-1$, then \beq \sigma\Delta{\cal O}_{n-1}=0. \eeq Thus, the r.h.s.\ of \rf{Onn} is $\sigma$-closed; however, to solve \rf{Onn}, we want it to be $\sigma$-exact, which is not guaranteed. If this happens, then the observable ${\cal O}$ exists and we say that ${\cal O}_0$ is not anomalous. Among the possible solutions of \rf{OmegaO}, we are interested in the ones that depend only on the triple $\Sigma_K,K,{x_0}$ and that reduce to $\exp\gamma^{\Sigma_K}$ when $\Sigma_K$ has no boundary. We call these solutions {\em proper}. Now notice that the action is invariant under the change of variables $({\cal A},{\cal B})\to({\cal B},{\cal A})$ while the observable ${\gamma^{(\Sigma_K,K,\xnot)}}$ is odd under it. This means that the corrections can be chosen to have a well-defined parity. By induction, one can see that they can be written as integrals of products of ${\cal B}\wedge{\cal A}$ and ${\widetilde{\cal B}}\wedge{\cal A}$ over submanifolds of products of configuration spaces of ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$. Moreover, $\Delta{\cal O}_n$ will have the same structure. Since ${\widetilde{\cal B}}\wedge{\cal A}$ and ${\cal B}\wedge{\cal A}$ are overall two-forms (i.e., each component has form degree plus ghost number equal to two), a product of them will be an overall even form. As a consequence, the zero-ghost-number part will be an even form, while the one-ghost-number part will be an odd form. However, only the even homology spaces of the configuration spaces of ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$ are nontrivial. Therefore, since ${\cal O}_n$ has ghost number zero, it can be a non-trivial element of the $\sigma$-cohomology, whereas $\Delta{\cal O}_n$, which has ghost number one and is $\sigma$-closed, must be $\sigma$-exact. Thus, we have proved the following \begin{Th} The classical observable ${\cal O}_0[K]$ is not anomalous and admits a proper extension. \lab{th-notan} \end{Th} By Statement \ref{statBV}, we expect the v.e.v.\ of ${\cal O}$ to be a topological invariant (up to framing) of the triple $\Sigma_K,K,{x_0}$. If the argument proving the invariance (modulo $\Omega$-exact terms) under a deformation $\Sigma_K\rightarrow\Sigma_K+\de T$ with $T\in\Omega_3({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3)$ goes through, we arrive at the following \begin{Conj} The v.e.v.\ of a proper solution ${\cal O}[K,\lambda]$ is a regular-isotopy invariant of the knot $K$. \lab{conj-Kinv} \end{Conj} Now write this invariant as the sum of an ambient-isotopy invariant and a regular-isotopy invariant that vanishes in standard framing. If the second contribution can be written as the v.e.v.\ of an $\Omega$-closed observable, then we can redefine the proper solution ${\cal O}$ by subtracting this term; so we arrive to the following \begin{Conj} There exists a proper solution whose v.e.v.\ is an ambient-isotopy invariant. \lab{conj-amb} \end{Conj} Eventually, since ${\cal O}$ is a quantum generalization of ${\cal O}_0$ to which it reduces in standard framing, and the v.e.v.\ of ${\cal O}_0$ is expected to satisfy \rf{aveOK0}, we have our last \begin{Conj} The proper solution ${\cal O}[K,\lambda]$ of Conjecture \ref{conj-amb} satisfies \beq \frac{\ave{{\cal O}[K,\lambda]}_\lambda} {\ave{{\cal O}[\bigcirc,\lambda]}_\lambda}= \frac1{\Delta(K;z(\lambda))}, \quad z(\lambda) = 2i\sin(\lambda/2), \lab{aveOK} \eeq where $\bigcirc$ denotes the unknot. \lab{conj-KAC} \end{Conj} If only Conjecture \ref{conj-Kinv} holds, we still expect \rf{aveOK} to hold but only if the standard framing is chosen. \subsubsection{The first correction} In App.\ \ref{app-corr}, we discuss how to find the first correction to ${\cal O}_0$. In particular, we show that a proper solution is given by \beq {\cal O}_1 = {\cal O}_0\ U_1, \lab{O1} \eeq with \beq U_1 = \Delta u_1, \lab{Uu} \eeq and \beq u_1 = \gamma_{ABB}\,\int_{\Sigma_K} B^* + \gamma_{BAA}\,\int_{\Sigma_K} A^*, \lab{u1} \eeq where \beqy \gamma_{ABB} &=& \int_{C_3(K\backslash{x_0})}A_1\eta_{12} B_2 \eta_{23}B_3 =-\frac12\int_{x<y<z\in K} B(x)\, A(y)\, B(z), \lab{gammaABB}\\ \gamma_{BAA} &=& \int_{C_3(K\backslash{x_0})}B_1\eta_{12} A_2 \eta_{23}A_3 =-\frac12\int_{x<y<z\in K} A(x)\, B(y)\, A(z). \lab{gammaBAA} \eeqy Remember that ${\cal O}_1$ is defined up to a $\sigma$-closed term. Our choice---\rf{O1}, \rf{Uu} and \rf{u1}---is particularly convenient since it gives the correct v.e.v.\ of ${\cal O}$ to the second order in $\lambda$. To see this, we first observe that any Wick contraction in the computation of a v.e.v.\ carries a factor $\lambda$. Thus, at order $\lambda^2$, the v.e.v.\ of ${\cal O}_0+i\lambda {\cal O}_1$ will contain the v.e.v.'s of $1/2({\gamma^{(\Sigma_K,K,\xnot)}})^2$ and of $i\lambda U_1$. Since $\Delta U_1=0$, we have \beq \Omega O_2=0, \eeq where \beq O_2=\left({ \frac12({\gamma^{(\Sigma_K,K,\xnot)}})^2+i\lambda U_1 }\right); \lab{O2} \eeq therefore, no other correction is needed to make this second-order term an observable. Notice that any redefinition of $U_1$ obtained by adding a $\Delta$-closed term will have the same property. By \rf{Onn}, this additional term must also be $\sigma$-closed, so it will be $\Omega$-closed as well. Thus, as expected, $O_2$ is defined up to $\Omega$-closed terms whose v.e.v.'s are of order $\lambda^2$. There exist only a few of such terms, i.e., $\lambda^2$, $\lambda\tau[K]$ and $\tau[K]^2$, where [cfr.\ \rf{deftau}] \beq \tau[K] = \gamma_A^K\,\gamma_B^K. \lab{deftauK} \eeq By \rf{avetau}---and remembering \rf{rescale}---we see that \beq \ave{O_2+k\tau^2+i\lambda l\tau - \lambda^2 m}_\lambda= \ave{O_2}_\lambda - \lambda^2 [2k({\rm slk} K)^2+l{\rm slk} K + m]; \lab{aveO2tau} \eeq i.e., the second order of $\ave{{\cal O}[K,\lambda]}_\lambda$ is defined up to a quadratic function of the self-linking number of $K$. We will see in the next section that, choosing $k=-3/16$ and $l=m=0$, Conjectures \ref{conj-Kinv}, \ref{conj-amb} and \ref{conj-KAC} hold at this order. \paragraph{Remark} Notice that we are allowed to add to $O_2$ only contributions of the form $\lambda$ times an observable, for $\lambda$-independent contributions would change the classical part of the observable. Thus, $\tau^2$ would not be an allowed contribution. However, as shown at the end of App.\ \ref{app-corr}, adding $\tau^2$ to $O_2$ is equivalent to adding a $\lambda$-independent correction to $U_1$, s.\ \rf{deltaU1}. \section{Computation of v.e.v.'s} \lab{sec-comp} In this section we will describe the perturbative expansion of $\ave{{\cal O}_0}_\lambda$ with the gauge fixing \rf{defPsi}. We will also explicitly compute the second order term of $\ave{{\cal O}_0}$ and $\ave{{\cal O}}$. We will see that the latter satisfies, at this order, Conjectures \ref{conj-Kinv} and \ref{conj-KAC}. \subsection{Gauge fixing and propagators} We will work in the covariant gauge fixing; i.e., we will choose the gauge-fixing fermion as in \rf{defPsi} with $\omega=0$. By \rf{deflpsi} and \rf{stardagger}, this amounts to setting \beq \begin{array}{lr} A^*=*d\bar c, & \bar c^*=*d^*A,\\ B^*=*d\bar\psi, & \bar\psi^*=*d^*B, \lab{setstar} \end{array} \eeq while all the other antifields are set to zero. Thus, the gauge-fixed action reads [cfr. \rf{Somegagf}] \beq S_{\rm g.f.} = \int_{{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3} B\wedge d A + \braket{d\bar c}{d c} + \braket{d\bar\psi}{d\psi} + \braket{h_c}{d^* A} + \braket{h_\psi}{d^*B}, \lab{Sgf} \eeq from which we can read the propagators (we write only the ones we are interested in) \beq \begin{array}{lcccl} \ave{A_\mu(x)\,B_\nu(y)}_\lambda &=& \ave{B_\mu(x)\,A_\nu(y)}_\lambda &=& i\lambda\,\frac1{4\pi}\,\epsilon_{\mu\nu\rho}\, \frac{(x-y)^\rho}{|x-y|^3},\\ \ave{c(x)\,\bar c(y)}_\lambda &=& -\ave{\bar c(x)\,c(y)}_\lambda &=& i\lambda\,\frac1{4\pi}\,\frac1{|x-y|},\\ \ave{\psi(x)\,\bar\psi(y)}_\lambda &=& -\ave{\bar\psi(x)\,\psi(y)}_\lambda &=& i\lambda\,\frac1{4\pi}\,\frac1{|x-y|}.\\ \end{array} \lab{props} \eeq Notice that the propagators are exactly the same as in Chern--Simons theory \cite{GMM,BN-th}. \subsubsection{Parity} We have already observed that, under $({\cal A},{\cal B})\to({\cal B},{\cal A})$, the action is left unchanged while ${\gamma^{(\Sigma_K,K,\xnot)}}$ changes sign. As a consequence, in the gauge-fixing defined by \rf{setstar}, the propagators \rf{props} are invariant under \beq (A,B,c,\bar c,\psi,\bar\psi,h_c,h_\psi)\rightarrow (B,A,\psi,\bar\psi,c,\bar c,h_\psi,h_c). \lab{parity} \eeq Thus, all the terms in perturbation theory that are odd under \rf{parity} [like, e.g., the v.e.v.\ of $({\gamma^{(\Sigma_K,K,\xnot)}})^{2n+1}$] will vanish. \subsubsection{Supersymmetry} Another observation that will simplify the discussion of the perturbation theory concerns the supersymmetry of the action \rf{Sgf} (which is the same that holds in its non-Abelian generalization \cite{MS}); viz., we can define a fermionic vector operator $Q$, i.e., an operator satisfying \beq [Q_\alpha,Q_\beta]_+ = Q_\alpha\,Q_\beta+Q_\beta\,Q_\alpha=0, \eeq which annihilates the action: \beq Q S = 0. \eeq Actually, there exist two such operators (which, moreover, anticommute with each other); the first one acts as \beq \begin{array}{lcl} {}(QA)_{\alpha\beta} &=& \epsilon_{\alpha\beta\gamma}\de^\gamma\bar\psi,\\ {}(QB)_{\alpha\beta} &=& \epsilon_{\alpha\beta\gamma}\de^\gamma\bar c,\\ {}(Qc)_\alpha &=& -A_\alpha,\\ {}(Q\psi)_\alpha &=& -B_\alpha,\\ {}(Q\bar c)_\alpha &=& 0,\\ {}(Q\bar\psi)_\alpha &=& 0,\\ {}(Qh_c)_\alpha &=& \de_\alpha\bar c,\\ {}(Qh_\psi)_\alpha &=& \de_\alpha\bar\psi. \end{array} \lab{defQ} \eeq The second operator is obtained by exchanging $(c,\bar c,h_c)$ with $(\psi,\bar\psi,h_\psi)$. A consequence of this supersymmetry is that the v.e.v.\ of a $Q$-exact functional vanishes. We want to point out that this supersymmetry is peculiar of ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$, but holds (with a different $Q$-operator) also for other gauge fixings. \subsubsection{Regularization} \lab{ssec-reg} The propagators \rf{props} diverge as the two points where the fields are evaluated approach to each other. The non-regularized v.e.v.'s of our observables are integrals of these propagators over products of $C_k(K\backslash{x_0})$ and $\Sigma_K$. To avoid divergences we have to give a prescription to split the points in these integrals. Our choice will essentially follow the approach of \cite{BT} with some important modifications due to the presence of the surface $\Sigma_K$ (s.\ also \cite{Lon}). The idea is to start defining the Fulton--MacPherson \cite{FM} compactification $C_n({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3)$ of the configuration space of $n$ points in $\overline{{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3}$, where the latter is the compactification of ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$ obtained by replacing the infinity with its blow up. Then, denoting by $B^2$ a two-dimensional surface whose boundary is diffeomorphic to $S^1$, we can consider the following imbeddings of compact spaces \beq {\rm pt}\hookrightarrow S^1 \hookrightarrow \overline{B^2} \hookrightarrow\overline{{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3}, \eeq where $\rm pt$ is a base point on the sphere $S^1$ which is mapped to the boundary of $B^2$. This allows us to define the configuration space $C_n^t$ of $n$ points on the knot distinct from the base point and $t$ points on its spanning surface. Notice that the points on the knot can be ordered, so $C_n^t$ has $n!$ connected components. We will denote by $\Ctilde nt$ the identity component (i.e., the component with points on $S^1$ ordered as $0,1,2,\ldots,n$). Our regularization prescription to compute the v.e.v.'s will be to replace $C_k(K\backslash{x_0})^n\times(\Sigma_K)^t$ with $C_{kn}^t$. Moreover, we will rewrite the propagators \rf{props} as \beq \begin{array}{lcccl} \ave{A_i\wedge B_j}_\lambda &=& \ave{B_i\wedge A_j}_\lambda &=& i\lambda\,\theta_{ij},\\ \ave{c_i\,(*d\bar c)_j}_\lambda &=& \ave{(*d\bar c)_i\,c_j}_\lambda &=& -i\lambda\,\theta_{ij},\\ \ave{\psi_i\,(*d\bar\psi)_j}_\lambda &=& \ave{(*d\bar \psi)_i\,\psi_j}_\lambda &=& -i\lambda\,\theta_{ij}, \end{array} \lab{propstheta} \eeq where $\theta_{ij}$ is the tautological form on ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$ defined as the pullback of the $SO(3)$-invariant unit volume element on $S^2$ by the map \beq \phi_{ij}(\vec x) = \frac{x_j-x_i}{|x_j-x_i|}, \quad\vec x\in C_n({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3). \eeq Notice that $\theta_{ij}$ satisfies \beq d\theta_{ij}=0,\quad \theta_{ij}^2 = 0,\quad \theta_{ji} = -\theta_{ij}. \eeq To compute our v.e.v.'s, we will have to integrate these two-forms as well as the tautological zero-forms $\eta_{ij}$ [appearing in \rf{AetaB}] over some $C_n^t$. If we choose the identity component $\Ctilde nt$, we can eliminate the zero-forms $\eta$. Thus, we can represent the contributions to our v.e.v.'s graphically as follows: we represent the knot as a horizontal line (which we suppose directed from left to right) with the base point on its boundary and the spanning surface as the portion of plane above it, and the two-forms $\theta_{ij}$ as arrows connecting the point $i$ to the point $j$ (s.\ fig.\ \ref{fig-examples}). \begin{figure} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(140.00,55.00) \put(0.00,5.00){\line(1,0){40.00}} \put(50.00,5.00){\line(1,0){40.00}} \put(100.00,5.00){\line(1,0){40.00}} \put(35.00,5.00){\vector(1,-3){0.2}} \bezier{224}(15.00,5.00)(25.00,31.00)(35.00,5.00) \put(5.00,5.00){\vector(1,2){4.50}} \put(9.50,14.00){\vector(3,2){10.50}} \put(20.00,21.00){\vector(1,-3){5.33}} \put(75.00,5.00){\vector(1,-4){0.2}} \bezier{408}(55.00,5.00)(65.00,55.00)(75.00,5.00) \put(70.00,5.00){\vector(1,-4){0.2}} \bezier{244}(60.00,5.00)(65.00,35.00)(70.00,5.00) \put(65.00,5.00){\vector(1,2){10.00}} \put(75.00,25.00){\vector(1,-2){10.00}} \put(110.00,5.00){\vector(1,3){5.00}} \put(115.00,20.00){\vector(1,-3){5.00}} \put(125.00,5.00){\vector(-1,2){6.00}} \put(119.00,17.00){\vector(4,-3){16.00}} \put(100.00,5.00){\vector(3,1){15.00}} \put(100.00,5.00){\vector(1,3){3.67}} \put(103.67,16.00){\vector(4,-1){16.33}} \put(120.00,11.92){\vector(3,4){7.56}} \put(50.00,5.00){\vector(1,4){6.50}} \put(56.50,31.00){\vector(3,-2){29.50}} \end{picture} \caption{Some examples of diagrams with nonvanishing v.e.v.}\label{fig-examples} \end{figure} We can give even a better description---cfr.\ \cite{BT}---if we introduce the bundles \beq \begin{array}{c} \Ctilde nt(S^1\times{\cal P}\times{\cal S})\\ \Big\downarrow\vcenter{\rlap{$p$}}\\ S^1\times{\cal P}\times{\cal S} \end{array}, \eeq where ${\cal P}$ is the space of the maps $S^1\to S^1$ homotopic to the identity and ${\cal S}$ is the space of imbeddings of ${B^2}$ in ${{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3}$, and the fiber is $\Ctilde nt$. Now we can see a generic v.e.v.\ as the sum of contributions which read \beq I = p_* [I], \eeq where $p_*$ denotes the push forward along the fiber, and $[I]$ is a form on the bundle. A locally constant function on the base space---i.e., a function whose differential vanishes)---will be a topological invariant on ${\cal S}$, for a locally constant function is a constant on $S^1\times{\cal P}$ which are connected. If, moreover, this topological invariant does not depend on cohomological deformations of the surface, it will eventually be an invariant of its boundary. As in \cite{BT}, the differential of $I$ can be written, by Stokes's theorem, as \beq dI = p_*d[I] + p_*^\de[I] = p_*^\de[I], \lab{dI} \eeq where $p_*^\de$ denotes the push forward along the boundary of the fiber. It is useful to distinguish on this boundary between {\em principal} and {\em hidden} faces. \label{simple} The principal faces are essentially of four types: \begin{enumerate} \item two points on the knot collapse together; \item one point on the knot collapses to the base point; \item two point on the surface collapse together; \item one point on the surface collapses on the knot where either {\em (a)} there are no points, or {\em (b)} there is one point, or {\em (c)} there is the base point. \end{enumerate} All the other components of the boundary (viz., when more points come together) are referred to as the hidden faces. Among the principal faces, we will call {\em simple} the ones of type 1, 2 and 3a. The principal-face contribution $\delta I$ to $dI$ can be evaluated ``graphically'' just by looking at the diagrams. What is not immediate is seeing wether the push forward along the hidden faces vanishes. However, in App.\ \ref{app-hf} we prove a vanishing theorem for the push forward along all faces but the simple principal faces (s. Thm.\ \ref{th-vt}). \subsection{The perturbative expansion of $\ave{{\cal O}_0}$} By \rf{setstar}, the gauge-fixed observable ${\gamma^{(\Sigma_K,K,\xnot)}}$ \rf{gammatot} now reads \beq \gammatotp{\rm g.f.} = \gamma^{\Sigma_K}_{\rm g.f.} + \gamma^{(K,{x_0})}_{\rm g.f.} + \gamma^{(\Sigma_K,{x_0})}_{\rm g.f.}, \lab{gammatotgf} \eeq with \beqy \gamma^{\Sigma_K}_{\rm g.f.} &=& \int_{\Sigma_K} [B\wedge A + (*d\bar\psi)\psi + c\,(*d\bar c), \lab{gammasigmagf}\\ \gamma^{(K,{x_0})}_{\rm g.f.} &=& \int_{C_2(K\backslash\xnot)} A_1\eta_{12}\wedge B_2,\lab{gammaKgf}\\ \gamma^{(\Sigma_K,{x_0})}_{\rm g.f.} &=& \psi({x_0})\,\int_{\Sigma_K} (*d\bar\psi) + \int_{\Sigma_K} (*d\bar c)\,c({x_0}).\lab{gamma0gf} \eeqy \subsubsection{The general structure of the perturbative expansion} \lab{ssec-gspe} The first thing we notice is that all these functionals are odd under \rf{parity}. Thus, only an even product of them will have a nonvanishing v.e.v. This, by the way, proves that $\ave{{\cal O}_0}_\lambda$ is even in $\lambda$, in according with \rf{aveOK0}. The second observation is that, by \rf{defQ}, the functional $\gamma^{\Sigma_K}_{\rm g.f.}$, is $Q$-exact, viz., \beq \gamma^{\Sigma_K}_{\rm g.f.} = \int_{\Sigma_K} dx^\alpha\wedge dx^\beta \,[Q(\psi A - Bc)]_{\alpha\beta}; \eeq thus, the v.e.v.\ of any of its powers vanishes. This implies that no loops appear among the v.e.v.'s we are computing. In fig.\ \ref{fig-ele}.a , we show one of these loops. Notice that, if we were working on a less trivial manifold, \rf{Sgf} would not be supersymmetric, so such loops would exist. As a matter of fact, the v.e.v.\ considered in \rf{aveOSigma} consists entirely of such loop diagrams. \begin{figure} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(140.00,51.01) \put(0.00,20.00){\line(1,0){40.00}} \put(5.00,25.00){\vector(1,4){4.25}} \put(9.25,42.00){\vector(1,-1){5.75}} \put(15.00,36.25){\vector(-1,4){3.69}} \put(11.31,51.00){\vector(4,-1){23.69}} \put(35.00,45.08){\vector(-2,-3){6.72}} \put(28.28,35.00){\vector(-2,-1){23.28}} \put(50.00,20.00){\line(1,0){40.00}} \put(100.00,20.00){\line(1,0){40.00}} \put(60.00,20.00){\vector(1,1){11.00}} \put(71.00,31.00){\vector(4,1){15.00}} \put(86.00,34.75){\vector(-3,4){7.69}} \put(78.31,45.00){\vector(-1,-3){2.00}} \put(76.31,39.00){\vector(1,-4){4.75}} \put(100.00,20.00){\vector(1,1){8.00}} \put(108.00,28.00){\vector(2,-1){9.00}} \put(117.00,23.50){\vector(1,2){5.75}} \put(122.75,35.00){\vector(-3,1){17.75}} \put(20.00,10.00){\makebox(0,0)[cc]{{\em a)} A loop}} \put(70.00,10.00){\makebox(0,0)[cc]{{\em b)} A $5$-chord}} \put(120.00,10.00){\makebox(0,0)[cc]{{\em c)} A $4$-flagellum}} \end{picture} \caption{Elements that appear in the diagrams}\label{fig-ele} \end{figure} The third observation is that $\gamma^{(\Sigma_K,{x_0})}_{\rm g.f.}$ is linear in the Grassmann variables $c({x_0})$ and $\psi({x_0})$. Thus, its square simply reads \beq (\gamma^{(\Sigma_K,{x_0})}_{\rm g.f.})^2= \psi({x_0})\,\int_{\Sigma_K} (*d\bar\psi)\ \int_{\Sigma_K} (*d\bar c)\,c({x_0}), \eeq while all higher powers vanish. We can now describe the features of the perturbative expansion of $\ave{{\cal O}_0}_\lambda$. In the v.e.v.'s where no $\gamma^{(\Sigma_K,{x_0})}_{\rm g.f.}$ appears, we have to Wick contract the fields $A$ and $B$ on $K$ and on $\Sigma_K$ together in all possible ways discarding all the diagrams that contain a loop. Thus, we are left with chains that connect two points on $K$ through a certain number of bivalent vertices on $\Sigma_K$. We will call an {\em $n$-chord} such a chain, where $n>0$ is the number of links (s.\ fig.\ \ref{fig-ele}.b). If the v.e.v.\ contains $\gamma^{(\Sigma_K,{x_0})}_{\rm g.f.}$, besides the chords described above, we will have a chain connecting ${x_0}$ with a point on $\Sigma_K$ through a certain number of bivalent vertices on $\Sigma_K$. We will call an {\em $n$-flagellum} such a chain (s.\ fig.\ \ref{fig-ele}.c). Eventually, we have v.e.v.'s containing $(\gamma^{(\Sigma_K,{x_0})}_{\rm g.f.})^2$. They contain some chords and two flagella. For a v.e.v. not to vanish, the total number of links in all the chords must be even; moreover, the total number of links in the flagella must be even. In figs.\ \ref{fig-examples} and \ref{fig-MCVIJ}, some diagrams of nonvanishing v.e.v.'s are shown. At order $2n$ in $\lambda$, we will have a sum of diagrams of this kind with a total number of links equal to $2n$. \begin{figure} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(140.00,60.00) \put(20.00,40.00){\line(1,0){40.00}} \put(80.00,40.00){\line(1,0){40.00}} \put(35.00,40.00){\vector(1,-4){0.2}} \bezier{164}(25.00,40.00)(30.00,60.00)(35.00,40.00) \put(55.00,40.00){\vector(1,-4){0.2}} \bezier{164}(45.00,40.00)(50.00,60.00)(55.00,40.00) \put(115.00,40.00){\vector(3,-4){0.2}} \bezier{200}(85.00,40.00)(100.00,60.00)(115.00,40.00) \put(110.00,40.00){\vector(1,-1){0.2}} \bezier{112}(90.00,40.00)(100.00,50.00)(110.00,40.00) \put(40.00,35.00){\makebox(0,0)[cc]{$M$}} \put(100.00,35.00){\makebox(0,0)[cc]{$C$}} \put(0.00,10.00){\line(1,0){40.00}} \put(50.00,10.00){\line(1,0){40.00}} \put(100.00,10.00){\line(1,0){40.00}} \put(10.00,10.00){\vector(2,3){10.00}} \put(30.00,10.00){\vector(-2,3){10.00}} \put(50.00,10.00){\vector(1,1){15.00}} \put(50.00,10.00){\vector(3,1){30.00}} \put(100.00,10.00){\vector(2,1){12.00}} \put(112.00,16.00){\vector(3,4){6.75}} \put(20.00,5.00){\makebox(0,0)[cc]{$V$}} \put(70.00,5.00){\makebox(0,0)[cc]{$I_0$}} \put(120.00,5.00){\makebox(0,0)[cc]{$J_0$}} \put(25.00,38.00){\makebox(0,0)[cc]{$1$}} \put(35.00,38.00){\makebox(0,0)[cc]{$2$}} \put(45.00,38.00){\makebox(0,0)[cc]{$3$}} \put(55.00,38.00){\makebox(0,0)[cc]{$4$}} \put(85.00,38.00){\makebox(0,0)[cc]{$1$}} \put(90.00,38.00){\makebox(0,0)[cc]{$2$}} \put(110.00,38.00){\makebox(0,0)[cc]{$3$}} \put(115.00,38.00){\makebox(0,0)[cc]{$4$}} \put(10.00,8.00){\makebox(0,0)[cc]{$1$}} \put(30.00,8.00){\makebox(0,0)[cc]{$2$}} \put(23.00,25.00){\makebox(0,0)[cc]{$3$}} \put(68.00,25.00){\makebox(0,0)[cc]{$1$}} \put(83.00,20.00){\makebox(0,0)[cc]{$2$}} \put(113.00,15.00){\makebox(0,0)[cc]{$1$}} \put(120.00,24.00){\makebox(0,0)[cc]{$2$}} \end{picture} \caption{The second-order diagrams}\label{fig-MCVIJ} \end{figure} From the structure of the perturbative expansion, it is easy to see that, if $\avel{{\cal O}_0}$ is a topological invariant, then it is invariant under $\Sigma_K\to \Sigma_K+\de T$ with $T\in\Omega_3({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3)$. In fact, if we have a topological invariant, we can move the region where this deformation occurs to infinity. Since all the vertices on $\Sigma_K$ are connected through a finite number of links to a point on $K$, this region at infinity does not contribute. This, of course, would not be true if loops on $\Sigma_K$ were allowed. As a final remark, we notice that, if we represent the unknot in standard framing as a planar curve and choose its spanning surface to belong to the same plane, then \beq \ave{{\cal O}_0[\bigcirc_{\rm s.f.},\lambda]}=1. \eeq \subsubsection{The second order} Now we want to compute explictly the v.e.v. of $\frac12({\gamma^{(\Sigma_K,K,\xnot)}})^2$. By the supersymmetry argument, we know that the v.e.v.\ of $({\gamma^{\Sigma_K}})^2$ vanishes. Moreover, the v.e.v.\ of ${\gamma^{(K,\xnot)}}{\gamma^{(\Sigma_K,\xnot)}}$ is equal to the product of the v.e.v.'s of ${\gamma^{(K,\xnot)}}$ and ${\gamma^{(\Sigma_K,\xnot)}}$, which vanish. Thus, we are left with only four contributions, which, after some computations, can be written as \beq \begin{array}{lcl} \avel{\frac12 ({\gamma^{(K,\xnot)}})^2} &=& \lambda^2\,(M-C),\\ \avel{{\gamma^{(K,\xnot)}}{\gamma^{\Sigma_K}}} &=& \lambda^2\, V,\\ \avel{\frac12({\gamma^{(\Sigma_K,\xnot)}})^2} &=& \lambda^2\, I_0,\\ \avel{{\gamma^{\Sigma_K}}{\gamma^{(\Sigma_K,\xnot)}}} &=& -2\lambda^2 J_0, \end{array} \eeq where the diagrams $M$, $C$, $V$, $I_0$ and $J_0$ are shown in fig.\ \ref{fig-MCVIJ}. Explicitly they read \beq \begin{array}{lcl} M &=& \int_{\Ctilde 40}\theta_{12} \wedge \theta_{34},\\ C &=& \int_{\Ctilde 40}\theta_{14} \wedge \theta_{23},\\ V &=& \int_{\Ctilde 21}\theta_{13} \wedge \theta_{23},\\ I_0 &=& \int_{\Ctilde 02}\theta_{01} \wedge \theta_{02},\\ J_0 &=& \int_{\Ctilde 02}\theta_{01} \wedge \theta_{12}. \end{array} \eeq Thus, the second order of $\avel{{\cal O}_0}$ reads \beq \avel{\frac12({\gamma^{(\Sigma_K,K,\xnot)}})^2} = \lambda^2\,(M-C+V+I_0-2J_0). \lab{avegamma2} \eeq \subsection{The v.e.v.\ of the corrected observable} As explained in subsection \ref{ssec-spko}, we do not expect $\avel{{\cal O}_0}$ to be a knot invariant (at least not with a general framing), since, in general, ${\cal O}_0$ is {\em not} an observable. In subsection \ref{ssec-cspko}, we have seen that there is a procedure that leads to an observable ${\cal O}$ starting from ${\cal O}_0$. We have computed the first correction \rf{O1} explicitly and have shown that the corrected second-order v.e.v.\ is given by the v.e.v.\ of the observable $O_2$ defined in \rf{O2}. In this section, we will compute this v.e.v.\ explicitly and show that it is a knot invariant. \subsubsection{The v.e.v. of $O_2$} To evaluate the v.e.v.\ of the correction $i\lambda U_1$, we notice that, by \rf{Uu} and Statement \ref{statBV}, \beq \avel{i\lambda U_1} = \avel{i\lambda\Delta u_1} = \avel{\sigma u_1}. \eeq By \rf{u1}, we have then \beq \avel{i\lambda U_1} = {\widetilde U}_1 + {\widetilde U}_2, \eeq with \beq \begin{array}{lcl} {\widetilde U}_1 &=& \avel{\gamma_{ABB}\,\sigma\int_{\Sigma_K} B^* +\gamma_{BAA}\,\sigma\int_{\Sigma_K} A^*},\\ {\widetilde U}_2 &=& - \avel{(\sigma\gamma_{ABB})\,\int_{\Sigma_K} B^* +(\sigma\gamma_{BAA})\,\int_{\Sigma_K} A^*}. \end{array} \eeq Finally, an explicit evaluation of this v.e.v.'s yields \beq \begin{array}{lcl} {\widetilde U}_1 &=& 2\lambda^2\,(X-2M-C),\\ {\widetilde U}_2 &=& \lambda^2\,(H_l+H_r), \end{array} \eeq where the new diagrams $X$, $H_l$ and $H_r$ are shown in fig.\ \ref{fig-XHH}. \begin{figure} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(140.00,40.00) \put(50.00,10.00){\line(1,0){40.00}} \put(80.00,10.00){\vector(1,-2){0.2}} \bezier{180}(60.00,10.00)(70.00,30.00)(80.00,10.00) \put(60.00,10.00){\vector(1,3){5.00}} \put(70.00,5.00){\makebox(0,0)[cc]{$H_l$}} \put(140.00,10.00){\line(-1,0){40.00}} \put(110.00,10.00){\vector(-1,-2){0.2}} \bezier{180}(130.00,10.00)(120.00,30.00)(110.00,10.00) \put(130.00,10.00){\vector(-1,3){5.00}} \put(120.00,5.00){\makebox(0,0)[cc]{$H_r$}} \put(0.00,10.00){\line(1,0){40.00}} \put(25.00,10.00){\vector(1,-3){0.2}} \bezier{252}(5.00,10.00)(15.00,40.00)(25.00,10.00) \put(35.00,10.00){\vector(1,-3){0.2}} \bezier{252}(15.00,10.00)(25.00,40.00)(35.00,10.00) \put(20.00,5.00){\makebox(0,0)[cc]{$X$}} \put(5.00,8.00){\makebox(0,0)[cc]{$1$}} \put(15.00,8.00){\makebox(0,0)[cc]{$2$}} \put(25.00,8.00){\makebox(0,0)[cc]{$3$}} \put(35.00,8.00){\makebox(0,0)[cc]{$4$}} \put(60.00,8.00){\makebox(0,0)[cc]{$1$}} \put(80.00,8.00){\makebox(0,0)[cc]{$2$}} \put(110.00,8.00){\makebox(0,0)[cc]{$1$}} \put(130.00,8.00){\makebox(0,0)[cc]{$2$}} \put(67.00,25.00){\makebox(0,0)[cc]{$3$}} \put(123.00,25.00){\makebox(0,0)[cc]{$3$}} \end{picture} \caption{The second-order correction diagrams}\label{fig-XHH} \end{figure} Explicitly they read \beq \begin{array}{lcl} X &=& \int_{\Ctilde 40} \theta_{13}\wedge\theta_{24},\\ H_l &=& \int_{\Ctilde 21} \theta_{12}\wedge\theta_{13},\\ H_r &=& \int_{\Ctilde 21} \theta_{21}\wedge\theta_{23}. \end{array} \eeq Therefore, the correction to $\avel{{\cal O}_0}$ reads \beq \avel{i\lambda U_1}=\lambda^2\,(2X-4M-2C+H_l+H_r). \lab{aveU1} \eeq To get the complete v.e.v.\ of $O_2$, we have to add \rf{aveU1} to \rf{avegamma2}. First, however, it is useful to notice that the square of the self-linking number \beq {\rm slk} K = 2\int_{\Ctilde 20} \theta_{12} \eeq can be written as \beq ({\rm slk} K)^2 = 8(M+C-X); \eeq thus, \beq \avel{O_2} = -\frac38\,\lambda^2 ({\rm slk} K)^2 + \lambda^2\, w_2, \lab{aveO2} \eeq with \beq w_2 = -X+V+H_l+H_r+I_0-2J_0. \lab{defw2} \eeq We conclude this subsection by noticing that, if we take the unknot as a planar curve and choose its spanning surface to lie in the same surface, it is immediately proved that \beq w_2(\bigcirc)=0. \lab{w200} \eeq Shortly we will prove that $w_2$ is a knot invariant; thus, \rf{w200} actually holds for any presentation of the unknot. \subsubsection{The invariance of the second-order term} \lab{ssec-isot} Now we will show that the principal-face variation of $w_2$ vanishes; we will follow the approach of Ref.\ \cite{BT}, which we have recalled in subsection \ref{ssec-reg}. Referring to figs.\ \ref{fig-MCVIJ}, \ref{fig-XHH} and \ref{fig-pf}, \begin{figure} \unitlength 1.00mm \linethickness{0.4pt} \begin{picture}(127.00,60.00) \put(0.00,10.00){\line(1,0){30.00}} \put(92.00,10.00){\line(1,0){30.00}} \put(25.00,10.00){\vector(1,-4){0.2}} \bezier{164}(15.00,10.00)(20.00,30.00)(25.00,10.00) \bezier{56}(15.00,10.00)(6.00,13.00)(9.00,16.00) \put(15.00,10.00){\vector(0,-1){0.2}} \bezier{88}(9.00,16.00)(14.00,23.00)(15.00,10.00) \put(76.00,10.00){\line(-1,0){30.00}} \put(51.00,10.00){\vector(-1,-4){0.2}} \bezier{164}(61.00,10.00)(56.00,30.00)(51.00,10.00) \bezier{56}(61.00,10.00)(70.00,13.00)(67.00,16.00) \put(61.00,10.00){\vector(0,-1){0.2}} \bezier{88}(67.00,16.00)(62.00,23.00)(61.00,10.00) \bezier{56}(92.00,10.00)(83.00,13.00)(86.00,16.00) \put(92.00,10.00){\vector(0,-1){0.2}} \bezier{88}(86.00,16.00)(91.00,23.00)(92.00,10.00) \put(92.00,10.00){\vector(3,1){20.00}} \put(35.00,10.00){\makebox(0,0)[cc]{$\ell_l$}} \put(81.00,10.00){\makebox(0,0)[cc]{$\ell_r$}} \put(127.00,10.00){\makebox(0,0)[cc]{$\ell_0$}} \put(0.00,25.00){\line(1,0){30.00}} \put(46.00,25.00){\line(1,0){30.00}} \put(0.00,25.00){\vector(1,1){10.00}} \put(20.00,25.00){\vector(-1,1){10.00}} \put(66.00,25.00){\vector(1,-1){0.2}} \bezier{112}(46.00,25.00)(56.00,35.00)(66.00,25.00) \put(46.00,25.00){\vector(1,2){5.00}} \put(92.00,25.00){\vector(-1,-1){0.2}} \bezier{112}(112.00,25.00)(102.00,35.00)(92.00,25.00) \put(112.00,25.00){\vector(-1,2){5.00}} \put(92.00,25.00){\line(1,0){30.00}} \put(35.00,25.00){\makebox(0,0)[cc]{$v$}} \put(81.00,25.00){\makebox(0,0)[cc]{$h_l$}} \put(127.00,25.00){\makebox(0,0)[cc]{$h_r$}} \put(0.00,40.00){\line(1,0){20.00}} \put(11.00,40.00){\vector(1,-2){0.2}} \bezier{84}(5.00,40.00)(8.00,50.00)(11.00,40.00) \put(17.00,40.00){\vector(1,-3){0.2}} \bezier{168}(5.00,40.00)(11.00,60.00)(17.00,40.00) \put(88.00,40.00){\line(-1,0){20.00}} \put(77.00,40.00){\vector(-1,-2){0.2}} \bezier{84}(83.00,40.00)(80.00,50.00)(77.00,40.00) \put(71.00,40.00){\vector(-1,-3){0.2}} \bezier{168}(83.00,40.00)(77.00,60.00)(71.00,40.00) \put(34.00,40.00){\line(1,0){20.00}} \put(44.00,40.00){\vector(1,-4){0.2}} \bezier{160}(38.00,40.00)(41.00,60.00)(44.00,40.00) \put(50.00,40.00){\vector(1,-4){0.2}} \bezier{160}(44.00,40.00)(47.00,60.00)(50.00,40.00) \bezier{56}(112.00,40.00)(103.00,43.00)(106.00,46.00) \put(112.00,40.00){\vector(0,-1){0.2}} \bezier{88}(106.00,46.00)(111.00,53.00)(112.00,40.00) \put(102.00,40.00){\line(1,0){20.00}} \put(112.00,40.00){\vector(2,3){6.67}} \put(25.00,40.00){\makebox(0,0)[cc]{$c_l$}} \put(93.00,40.00){\makebox(0,0)[cc]{$c_r$}} \put(59.00,40.00){\makebox(0,0)[cc]{$m$}} \put(127.00,40.00){\makebox(0,0)[cc]{$h$}} \end{picture} \caption{The principal-face diagrams}\label{fig-pf} \end{figure} we start considering the principal-face contributions \beq \begin{array}{lcl} \delta X &=& c_l - m + c_r,\\ \delta V &=& c_l + m + c_r - 2v + \ell_l + \ell_r,\\ \delta H_l &=& -m + h - h_l + h_r -\ell_l,\\ \delta H_r &=& -m - h - h_l + h_r -\ell_r,\\ \delta I_0 &=& 2h_l + 2\ell_0,\\ \delta J_0 &=& -v + h_r + \ell_0, \end{array} \lab{delta...} \eeq which by \rf{defw2} imply \beq \delta w_2 = 0. \lab{deltaw2} \eeq It should be clear from fig.\ \ref{fig-pf} what $c_l$, $m$, $c_r$, $v$, $h_l$ and $h_r$ mean. To write the diagrams $h$, $\ell_l$, $\ell_0$ and $\ell_r$, we need to introduce explicitly the map \beq \Phi:B^2\longrightarrow{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3 \eeq that defines the surface $\Sigma_K$. The diagram $h$ is given by \beq h = \int_{\Ctilde 11} \theta_{12}\wedge\theta_{11}, \eeq where $\theta_{11}$ is the pull back of the volume form $\omega$ through the map \beq \phi_{11}(\vec x) = \frac{\dot\Phi(x_1)}{|\dot\Phi(x_1)|}, \quad \vec x\in\Ctilde 11. \eeq and $\dot\Phi$ denotes the derivative of $\Phi$ in the direction tangent to the knot (notice that $\Phi(x_1)$ is on the knot). To describe the remaining diagrams, we have also to introduce $\Phi'$, i.e., the derivative of $\Phi$ w.r.t.\ the other coordinate in the parametrization of the surface. In general the vector $\Phi'$ will {\em not} be orthogonal to $\dot\Phi$. To obtain an orthogonal vector we define \beq {\Phi'_\bot} = \Phi' - \frac{\Phi'\cdot\dot\Phi}{|\dot\Phi|^2}\dot\Phi. \lab{defPhiortho} \eeq Then the diagrams $\ell_l$ and $\ell_r$ read \beq \begin{array}{lcl} \ell_l = \int_{\Ctilde 20} [\theta_{12}\wedge \int_{{\cal U}_2}\theta_2],\\ \ell_r = \int_{\Ctilde 20} [\theta_{12}\wedge \int_{{\cal U}_1}\theta_1], \end{array} \eeq where the one-dimensional manifolds ${\cal U}_i$ are defined as \beq \begin{array}{ll} {\cal U}_i=\{(u_1,u_2^1,u_2^2)\in{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}\times{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}\times{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^+/ &[(u_1)^2 + (u_2^1)^2]|\dot\Phi(x_i)|^2+(u_2^2)^2|{\Phi'_\bot}(x_i)|^2=1,\\ & u_1 + u_2^1 = 0\}, \end{array} \eeq and $\theta_i$ is the pull back of $\omega$ to ${\cal U}_i$ through the map \beq \phi_i(u_1,u_2^1,u_2^2) = \frac {(u_2^1-u_1)\dot\Phi(x_i)+u_2^2{\Phi'_\bot}(x_i)} {|(u_2^1-u_1)\dot\Phi(x_i)+u_2^2{\Phi'_\bot}(x_i)|}. \eeq Finally, \beq \ell_0 = \int_{\Ctilde 01} [\theta_{01}\wedge \int_{\widetilde{\cal U}_0}\widetilde\theta_0], \eeq where the one-dimensional manifold $\widetilde{\cal U}_0$ is defined as \beq \widetilde{\cal U}_0 = \{(u^1,u^2)\in{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}\times{\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^+/ (u^1)^2|\dot\Phi({x_0})|^2 + (u^2)^2|{\Phi'_\bot}({x_0})|^2 = 1\}, \eeq and $\widetilde\theta_0$ is the pull back of $\omega$ to $\widetilde{\cal U}_0$ through the map \beq \widetilde\phi_0(u_1,u_2)=\frac {u^1\dot\Phi({x_0}) + u^2{\Phi'_\bot}({x_0})} {|u^1\dot\Phi({x_0}) + u^2{\Phi'_\bot}({x_0})|}. \eeq Notice that in \rf{delta...} we have written only the non vanishing contributions. (Actually, more sophisticated arguments, s.\ App.\ \ref{app-hf}, show that also $\ell_l$, $\ell_r$ and $\ell_0$ vanish.) \ \ All other possible terms vanish for one of the following reasons: \begin{enumerate} \item we have to integrate a form on a space of lower dimension; \item a factor $\theta_{ij}^2$ appears, or \item the push forward vanish because of a symmetry. \end{enumerate} An example of the first case is the push forward of $[V]$ along the face obtained by sending $3$ to $0$ which gives \[ \int_{\widetilde{\cal U}_0}[\int_{\Ctilde 20} \theta_{10}\wedge\theta_{20}]. \] The second case happens, e.g., when we push forward $[V]$ along the face where we send $1$ to $2$. The third case occurs in the push forward of $[J_0]$ when we send $1$ to $2$, the symmetry being the exchange of $1$ with $2$ which does not reverse the orientation of the manifold we are integrating over but changes the sign of the form to be integrated. For the same reasons, the push forwards of $[X]$, $[I_0]$, $[J_0]$ and $[H_l]+[H_r]$ along the hidden faces vanish. The only non-trivial case is the push forward of $[V]$ along the hidden face where $1$, $2$ and $3$ come together. This case is analyzed in App.\ \ref{app-hf}, and a vanishing theorem is proved. These results together with \rf{deltaw2} prove the following \begin{Th} The corrected second-order term $w_2$ is a topological invariant of the imbedding $\Sigma_K$ of $B^2$ in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$. \lab{thm-Sigmainv} \end{Th} As a consequence, if we deform the imbedding $\Sigma_K$ by adding to it the boundary of a three-cycle, we can always move this deformation to infinity. Since all the vertices on $\Sigma_K$ are connected through at most two $\theta$'s to a point living on $K$, the deformation at infinity will not contribute. Therefore, $w_2$ actually depends only on $K$, and we have the following \begin{Th} The corrected second-order term $w_2$ is a knot invariant. \lab{thm-Kinv} \end{Th} The chord-diagram contribution $-X$ to $w_2$ is exactly the same that appears in the invariant studied in \cite{GMM,BN-th}. This invariant is known to be equal to the second coefficient $a_2$ of the Alexander--Conway polynomial plus a constant term (viz., the value it takes on the unknot). Notice that the chord diagram $-X$ alone is {\em not} a knot invariant. To get a knot invariant we have to add to it either the other terms that define $w_2$---let call $W$ their sum---or the diagram $Y$ considered in \cite{GMM,BN-th,BT} (viz., a diagram with a trivalent vertex in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^3$). Since both $-X+W$ and $-X+Y$ are knot invariants, also $T=W-Y$ is a knot invariant. Our claim is that $T$ is trivial (i.e., it is the same for all knots). To prove it, it is enough to check that $T$ takes the same value on two knots $K_+$ and $K_-$ that differ only around a chosen crossing. We notice that the difference $T(K_+)-T(K_-)$ comes from a singularity at the crossing point where the flip occurs---as in \cite{BN-th}---or along the line where the two spanning surfaces get to intersect. However, it is not difficult to check that such singularities do not arise; so $T$ is a constant. Therefore, $w_2$ is equal to $a_2$ plus a constant. However, since by definition $a_2(\bigcirc)=0$, \rf{w200} implies \beq w_2 = a_2, \eeq and Conjecture \fullref{conj-KAC} is satisfied at this order. As a concluding remark, we notice that in passing from $Y$ to $W$ one of the integrations on the knot is replaced by an integration on the spanning surface; so it should be possible to relate $W$ and $Y$ directly via Stokes's theorem. \subsubsection{Higher orders} Thm.\ \ref{th-notan} ensures that a quantum observable ${\cal O}$ extending ${\cal O}_0$ exists. Its v.e.v.\ at order $\lambda^n$ will be given by diagrams containing $n$ propagators connecting points on the knot and/or on the spanning surface. Of course, the restrictions given in subsection \ref{ssec-gspe} for the v.e.v.\ of ${\cal O}_0$ do not hold anymore; in particular, the vertices on the knot will not necessarily be univalent and the vertices on the spanning surface will not necessarily be bivalent (we have already seen a counterexample at the second order). However, no loops on the surface will appear (since the corrections must vanish when the spanning surface is boundariless). Moreover, since the v.e.v.\ of ${\cal O}_0$ vanishes at odd order, we do not need odd-order corrections. The combinatorics of these diagrams will be dictated by the specific form of the corrected observable ${\cal O}$. What we expect, by field-theoretical arguments, is that these combinations of diagrams will be metric independent, i.e., will be the sum of invariants possibly times powers of the self-linking number, i.e., ``isolated chords" in the diagrams. The true invariants will then be obtained by factorizing the isolated chords. A rigorous mathematical proof that they are actually knot invariants will simply require checking that the principal-face contributions of the diagrams that sum up cancel each other, for Thm.\ \fullref{th-vt} ensures that the push forwards along hidden faces always vanish. Notice that now we could also throw away the $BF$ field theory and directly study $\delta$-closed combinations of diagrams with vertices on the knot and/or on the spanning surface. By Thm.\ \ref{th-vt}, these will yield knot invariants as well as higher-degree cohomology classes on the space of imbeddings (the degree being given by $2l-n-2t$ where $l$ is the number of propagators, $n$ the number of points on the knot and $t$ the number of points on the surface). \section{A glimpse to higher dimensions} \lab{sec-glim} There is no problem in defining the Abelian $BF$ theory in any dimension: just take $A$ and $B$ to be fields taking values in $\Omega^p(M)$ and $\Omega^q(M)$ respectively, with $p+q+1=d$ and $d=\dim M$. The classical action \rf{Sclomega} can easily be extended to a BV action. The partition function then is known to be equal to the Ray--Singer torsion or to its inverse (depending on $p$) \cite{Schw,BlT}. M oreover, it is not difficult to generalize the observable \rf{defgammaSigma}, where now $\Sigma\in H_{d-1}(M,\de M)$. The classical part of this observable reads \beq \gamma^\Sigma_{\rm cl} = \int_\Sigma B\wedge A, \eeq and satisfies \beq s\gamma^\Sigma_{\rm cl} = 0,\quad\mbox{on shell}, \eeq where {\em on shell} means modulo the classical equations of motion, \beq dA=0,\quad dB=0, \eeq and $s$ is the BRST operator \beq sA=dc,\quad sB=d\psi \eeq (now $c$ and $\psi$ are a $(p-1)$- and a $(q-1)$-form respectively). If $\Sigma_K$ is a spanning surface for a $(d-2)$-knot $K$ (i.e., an imbedding of $S^{d-2}$ in ${\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^d$), then \beq s\gamma^{\Sigma_K}_{\rm cl} = \oint_K[\psi\wedge A + (-1)^q\, B\wedge c], \quad\mbox{on shell}. \lab{sgSd} \eeq To get an on-shell $s$-closed functional, we have to add to $\gamma^{\Sigma_K}$ another term canceling the r.h.s.\ of \rf{sgSd}. We first notice that $\gamma^{(K,{x_0})}$ as in \rf{AetaB} can be generalized in any dimension, where now $\eta$ is the tautological $(d-3)$-form on the configuration space of $K\backslash{x_0}$. An explicit computation shows that \beq s\gamma^{(K,{x_0})}_{\rm cl} = (-1)^{d+1+p}\, \oint_K [\psi\wedge A + (-1)^{q+d+1}\, B\wedge c], \quad\mbox{on shell}. \eeq Therefore, {\em in odd dimension} we can define the following on-shell $s$-closed functional \beq \gammatotp{\rm cl} = \gamma^{\Sigma_K}_{\rm cl} + (-1)^{p+1}\,\gamma^{(K,{x_0})}_{\rm cl}. \lab{defgtcld} \eeq Then, starting from \rf{defgtcld}, the BV procedure will yield a $\sigma$-closed observable. Finally, we would like to consider an object like ${\cal O}_0$ in \rf{calO0}, for its v.e.v.\ should be related to the Alexander--Conway polynomial (or its inverse, depending on $p$). Of course, we do not expect ${\cal O}_0$ to be an observable, so we should look for corrections as explained in subsection \ref{ssec-cspko}. Notice that in any dimension it is possible to define linear combinations ${\cal A}$ and ${\cal B}$ generalizing \rf{defcalAB}, \rf{Somegacal} and \rf{sigmacalAB} (and including the whole set of ghosts for ghosts). Moreover, in odd dimension the classical action is invariant under $(A,B)\to (B,A)$ while $\gammatotp{\rm cl}$ is odd under it. Thus, their BV extensions will share the same property under $({\cal A},{\cal B})\to ({\cal B},{\cal A})$. This leads to proving a generalization of Thm.\ \fullref{th-notan} stating that ${\cal O}_0$ is never anomalous. We have only to check that the form degree of the one-ghost-number component of any form with well-defined parity under the above transformation never matches with the dimension of a nontrivial homology space of $C_n({\hbox{{\rm I}\kern-.2em\hbox{\rm R}}}^d)$. As a matter of fact, these dimensions are multiples of $(d-1)$. On the other hand, forms with well-defined parity are obtained by products of ${\cal B}\wedge{\cal A}$ and $\widetilde{\cal B} \wedge{\cal A}$, both of which are overall $(d-1)$-forms, times a certain number $r$ of tautological $(d-3)$-forms $\eta$; thus, the form degree of the one-ghost-number component will be conguent to $-1-2r$ mod $(d-1)$. Then our claim follows from the fact that \[ 2r+1\equiv0\ \rm{mod}(d-1) \] has no solutions if $d$ is odd. The v.e.v.\ of ${\cal O}$ should then yield metric-independent functionals of the knot and its spanning surface. Eventually, if a vanishing theorem holds, these functionals will be knot invariants. Hence, we could compute numerical knot invariants (presumably the coefficients of the Alexander--Conway polynomial or its inverse) in any odd dimension in terms of integrals over the configuration spaces of points on the knot and on its spanning surface. Via Stokes's theorem, the second-order invariants should correspond, up to a constant term, to the ones proposed in \cite{Bott-un}. As a final remark, we notice that in three dimensions we could have chosen $A$ to be a zero-form and $B$ a two-form. In this case, the v.e.v.\ of ${\cal O}$ should give directly the Alexander--Conway polynomial instead of its inverse. \section{Conclusions} In this paper we have considered a new way of obtaining knot invariants from a TQFT. The nice feature of our theory is that it is Abelian. What makes things non-trivial is a rather involved observable, which can be defined only in the context of BV formalism; yet, as observed in the last section, it can be generalized in any odd dimension. In the three-dimensional case, we have shown that at the second order the theory actually produces a numerical knot invariant which, despite the fact it is not new, comes out written in an entirely new way. The next task is to find the other corrections to the observable and to evaluate higher-order v.e.v.'s, in three dimensions as well as in any odd dimension. Of course, an alternative way would be working directly on the space of surface-plus-knot diagrams, as described in subsection \ref{ssec-reg}, and try to find combinations whose differential vanishes. This would allow studying higher-degree forms on the space of imbeddings as well. Notice that while the Chern-Simons and the $BF$ theory with a cosmological term produce the whole set of HOMFLY polynomials and their ``colored" generalizations, pure $BF$ theories, both Abelian and non-Abelian, give only the Alexander--Conway polynomial. However, it is possible that even more involved observables exist whose v.e.v.\ is a more general knot invariant. As a matter of fact, pure $BF$ theory comes out naturally as a particular limit of the v.e.v.\ of a cabled Wilson loop in the theory with a cosmological term \cite{Cat}. This limit corresponds to the first diagonal in the $(h,d)$ expansion of the colored Jones function \cite{MM}. A generalization of the computation done in \cite{Cat} should give the observables whose v.e.v.'s correspond to the upper diagonals in this expansion. Then a careful study of the ``Abelianizing limit" described in the Introduction should yield the corresponding observables for the Abelian theory. \section*{Acknowledgements} I thank D.~Anselmi, P.~Cotta-Ramusino and R.~Longoni for helpful conversations. I am especially thankful to R.~Bott for a number of very useful discussions. This work was supported by INFN Grant No.\ 5077/94.

proofpile-arXiv_065-503

\chapter{Introduction} \label{intro_chapt} \proverb{No time like the present.} \section{Subject of this thesis} \label{thesis_subject} Over the past thirty years, there have been significant advances in the area of natural language interfaces to databases (\textsc{Nlidb}\xspace{s}). \textsc{Nlidb}\xspace{s} allow users to access information stored in databases by typing requests expressed in natural language (e.g.\ English). (The reader is referred to \cite{Perrault}, \cite{Copestake}, and \cite{Androutsopoulos1995} for surveys of \textsc{Nlidb}\xspace{s}.\footnote{The project described in this thesis began with an extensive survey of \textsc{Nlidb}\xspace{s}. The results of this survey were reported in \cite{Androutsopoulos1995}.}) Most of the existing \textsc{Nlidb}\xspace{s} were designed to interface to ``snapshot'' database systems, that provide very limited facilities for manipulating time-dependent data. Consequently, most \textsc{Nlidb}\xspace{s} also provide very limited support for the notion of time. In particular, they were designed to answer questions that refer mainly to the present (e.g.\ \pref{intro:1} -- \pref{intro:3}), and do not support adequately the mechanisms that natural language uses to express time. For example, very few (if any) temporal adverbials (\qit{in 1991}, \qit{after 5:00pm}, etc.) and verb forms (simple past, past continuous, past perfect, etc.) are typically allowed, and their semantics are usually over-simplified or ignored. \begin{examps} \item What is the salary of each engineer? \label{intro:1} \item Who is at site 4? \label{intro:2} \item Which generators are in operation? \label{intro:3} \end{examps} The database community is becoming increasingly interested in \emph{temporal} database systems. These are intended to store and manipulate in a principled manner information not only about the present, but also about the past and the future (see \cite{Tansel3} and \cite{tdbsglossary} for an introduction, and \cite{Bolour1983}, \cite{McKenzie2}, \cite{Stam}, \cite{Soo}, \cite{Kline1993}, and \cite{Tsotras1996} for bibliographies). When interfacing to temporal databases, it becomes crucial for \textsc{Nlidb}\xspace{s} to interpret correctly the temporal linguistic mechanisms (verb tenses, temporal adverbials, temporal subordinate clauses, etc.) of questions like \pref{intro:4} -- \pref{intro:6}. \begin{examps} \item What was the salary of each engineer while ScotCorp was building bridge 5? \label{intro:4} \item Did anybody leave site 4 before the chief engineer had inspected the control room? \label{intro:5} \item Which systems did the chief engineer inspect on Monday after the auxiliary generator was in operation? \label{intro:6} \end{examps} In chapter \ref{comp_chapt}, I argue that previous approaches to natural language interfaces for \emph{temporal} databases (\textsc{Nlitdb}\xspace{s}) are problematic, mainly because they ignore important time-related linguistic phenomena, and/or they assume idiosyncratic temporal database systems. This thesis develops a principled framework for constructing English \textsc{Nlitdb}\xspace{s}, drawing on research in linguistic theories of time, temporal logics, and temporal databases. \section{Some background} This section introduces some ideas from \textsc{Nlidb}\xspace{s}, linguistic theories of time, temporal logics, and temporal databases. Ideas from the four areas will be discussed further in following chapters. \subsection{Natural language interfaces to databases} \label{domain_config} Past work on \textsc{Nlidb}\xspace{s} has shown the benefits of using the abstract architecture of figure \ref{pipeline_fig}. The natural language question is first parsed and analysed semantically by a linguistic front-end, which translates the question into an intermediate meaning representation language (typically, some form of logic). The generated intermediate language expression captures formally what the system understands to be the meaning of the natural language question, without referring to particular database constructs. The intermediate language expression is then translated into a database language (usually \textsc{Sql}\xspace \cite{Ullman} \cite{Melton1993}) that is supported by the underlying \emph{database management system} (\textsc{Dbms}\xspace; this is the part of the database system that manipulates the information in the database). The resulting database language expression specifies what information needs to be retrieved in terms of database constructs. The \textsc{Dbms}\xspace retrieves this information by evaluating the database language expression, and the obtained information is reported back to the user. \begin{figure} \hrule \medskip \begin{center} \includegraphics[scale=.7]{pipeline_arch} \caption{Abstract architecture of many modern NLIDBs} \label{pipeline_fig} \end{center} \hrule \end{figure} Most \textsc{Nlidb}\xspace{s} can only handle questions referring to a particular knowledge-domain (e.g.\ questions about train departures, or about the employees of a company), and need to be configured before they can be used in a new domain. The configuration typically includes ``teaching'' the \textsc{Nlidb}\xspace words that can be used in the new domain, and linking basic expressions of the formal intermediate language to database constructs (see section 6 of \cite{Androutsopoulos1995}). The architecture of figure \ref{pipeline_fig} has proven to have several advantages (see sections 5.4 and 6 of \cite{Androutsopoulos1995}), like modularity (e.g.\ the linguistic-front end is shielded from database-level issues), and \textsc{Dbms}\xspace portability (the same linguistic front-end can be used with \textsc{Dbms}\xspace{s} that support different database languages). This thesis examines how this architecture can be used to construct \textsc{Nlitdb}\xspace{s}. \subsection{Tense and aspect theories} \label{tense_aspect_intro} In English, temporal information can be conveyed by verb forms (simple past, past continuous, present perfect, etc.), nouns (\qit{beginning}, \qit{predecessor}, \qit{day}), adjectives (\qit{earliest}, \qit{next}, \qit{annual}), adverbs (\qit{yesterday}, \qit{twice}), prepositional phrases (\qit{at 5:00pm}, \qit{for two hours}), and subordinate clauses (\qit{while tank 4 was empty}), to mention just some of the available temporal mechanisms. A linguistic theory of time must account for the ways in which these mechanisms are used (e.g.\ specify what is the temporal content of each verb form, how temporal adverbials or subordinate clauses affect the meaning of the overall sentences, etc.). The term ``tense and aspect theories'' is often used in the literature to refer to theories of this kind. (The precise meanings of ``tense'' and ``aspect'' vary from one theory to the other; see \cite{Comrie} and \cite{Comrie2} for some discussion. Consult chapter 5 of \cite{Kamp1993} for an extensive introduction to tense and aspect phenomena.) It is common practice in tense and aspect theories to classify natural language expressions or situations described by natural language expressions into \emph{aspectual classes}. (The term \qit{Aktionsarten} is often used to refer to these classes.) Many aspectual classifications are similar to Vendler's taxonomy \cite{Vendler}, that distinguishes between \emph{state} verbs, \emph{activity} verbs, \emph{accomplishment} verbs, and \emph{achievement} verbs.\footnote{According to Mourelatos \cite{Mourelatos1978}, a similar taxonomy was developed independently in \cite{Kenny1963}, where Kenny notes that his classification is similar to the distinction between \emph{kineseis} and \emph{energiai} introduced by Aristotle in \textit{Metaphysics}, $\Theta$.1048\textit{b}, 18--36.} For example, \qit{to run} (as in \qit{John ran.}) is said to be an activity verb, \qit{to know} (as in \qit{John knows the answer.}) a state verb, \qit{to build} (as in \qit{John built a house.}) an accomplishment verb, and \qit{to find} (as in \qit{Mary found the treasure.}) an achievement verb. Vendler's intuition seems to be that activity verbs describe actions or changes in the world. For example, in \qit{John ran.} there is a running action in the world. In contrast, state verbs do not refer to any actions or changes. In \qit{John knows the answer.} there is no change or action in the world. Accomplishment verbs are similar to activity verbs, in that they denote changes or actions. In the case of accomplishment verbs, however, the action or change has an inherent ``climax'', a point that has to be reached for the action or change to be considered complete. In \qit{build a house} the climax is the point where the whole of the house has been built. If the building stops before the whole of the house has been built, the building action is incomplete. In contrast, the action of the activity verb \qit{to run} (with no object, as in \qit{John ran.}) does not seem to have any climax. The runner can stop at any time without the running being any more or less complete. If, however, \qit{to run} is used with an object denoting a precise distance (e.g.\ \qit{to run a mile}), then the action \emph{does} have a climax: the point where the runner completes the distance. In this case, \qit{to run} is an accomplishment verb. Finally, achievement verbs, like \qit{to find}, describe instantaneous events. In \qit{Mary found the treasure.} the actual finding is instantaneous (according to Vendler, the time during which Mary was searching for the treasure is not part of the actual finding). In contrast, in \qit{John built a house.} (accomplishment verb) the actual building action may have lasted many years. Aspectual taxonomies are invoked to account for semantic differences in similar sentences. The so-called ``imperfective paradox'' \cite{Dowty1977} \cite{Lascarides} is a well-known example (various versions of the imperfective paradox have been proposed; see \cite{Kent}). The paradox is that if the answer to a question like \pref{taintro:1} is affirmative, then the answer to the non-progressive \pref{taintro:2} must also be affirmative. In contrast, an affirmative answer to \pref{taintro:3} does not necessarily imply an affirmative answer to \pref{taintro:4} (John may have abandoned the repair before completing it). The \textsc{Nlitdb}\xspace must incorporate some account for this phenomenon. If the \textsc{Nlitdb}\xspace generates an affirmative response to \pref{taintro:1}, there must be some mechanism to guarantee that the \textsc{Nlitdb}\xspace's answer to \pref{taintro:2} will also be affirmative. No such mechanism is needed in \pref{taintro:3} and \pref{taintro:4}. \begin{examps} \item Was IBI ever advertising a new computer? \label{taintro:1} \item Did IBI ever advertise a new computer? \label{taintro:2} \item Was J.Adams ever repairing engine 2? \label{taintro:3} \item Did J.Adams ever repair engine 2? \label{taintro:4} \end{examps} The difference between \pref{taintro:1} -- \pref{taintro:2} and \pref{taintro:3} -- \pref{taintro:4} can be accounted for by classifying \qit{to advertise} as an activity, \qit{to repair} as an accomplishment, and by stipulating that: (i) the simple past of an accomplishment requires the climax to have been reached; (ii) the past continuous of an accomplishment or activity, and the simple past of an activity impose no such requirement. Then, the fact that an affirmative answer to \pref{taintro:3} does not necessarily imply an affirmative answer to \pref{taintro:4} is accounted for by the fact that \pref{taintro:4} requires the repair to have been completed, while \pref{taintro:3} merely requires the repair to have been ongoing at some past time. In contrast \pref{taintro:2} does not require any climax to have been reached; like \pref{taintro:1}, it simply requires the advertising to have been ongoing at some past time. Hence, an affirmative answer to \pref{taintro:1} implies an affirmative answer to \pref{taintro:2}. It will become clear in chapter \ref{linguistic_data} that aspectual taxonomies pertain to the semantics of almost all temporal linguistic mechanisms. \subsection{Temporal logics} \label{temp_log_intro} Time is an important research topic in logic, and many formal languages have been proposed to express temporal information \cite{VanBenthem} \cite{Gabbay1994b}. One of the simplest approaches is to use the traditional first-order predicate logic, introducing time as an extra argument of each predicate. \pref{tlogi:1} would be represented as \pref{tlogi:2}, where $t$ is a time-denoting variable, $\prec$ stands for temporal precedence, $\sqsubseteq$ for temporal inclusion, and $now$ is a special term denoting the present moment. The answer to \pref{tlogi:1} would be affirmative iff \pref{tlogi:2} evaluates to true, i.e.\ iff there is a time $t$, such that $t$ precedes the present moment, $t$ falls within 1/10/95, and tank 2 contained water at $t$. (Throughout this thesis, I use ``iff'' as a shorthand for ``if and only if''.) \begin{examps} \item Did tank 2 contain water (some time) on 1/10/95? \label{tlogi:1} \item $\exists t \; contain(tank2, water, t) \land t \prec now \land t \sqsubseteq \mathit{1/10/95}$ \label{tlogi:2} \end{examps} An alternative approach is to employ \emph{temporal operators}, like Prior's $P$ (past) and $F$ (future) \cite{Prior}. In that approach, formulae are evaluated with respect to particular times. For example, $contain(tank2, water)$ would be true at a time $t$ iff tank 2 contained water at $t$. Assuming that $\phi$ is a formula, $P\phi$ is true at a time $t$ iff there is a time $t'$, such that $t'$ precedes $t$, and $\phi$ is true at $t'$. Similarly, $F\phi$ is true at $t$ iff there is a $t'$, such that $t'$ follows $t$, and $\phi$ is true at $t'$. \qit{Tank 2 contains water.} can be expressed as $contain(tank2, water)$, \qit{Tank 2 contained water.} as $P\, contain(tank2, water)$, \qit{Tank 2 will contain water.} as $F \, contain(tank2, water)$, and \qit{Tank 2 will have contained water.} as $F \, P \, contain(tank2, water)$. Additional operators can be introduced, to capture the semantics of temporal adverbials, temporal subordinate clauses, etc. For example, an $On$ operator could be introduced, with the following semantics: if $\phi$ is a formula and $\kappa$ specifies a day (e.g.\ the day 1/10/95), then $On[\kappa, \phi]$ is true at a time $t$ iff $t$ falls within the day specified by $\kappa$, and $\phi$ is true at $t$. Then, \pref{tlogi:1} could be represented as \pref{tlogi:4}. \begin{examps} \item $P \;\; On[\mathit{1/10/95}, contains(tank2, water)]$ \label{tlogi:4} \end{examps} The intermediate representation language of this thesis, called \textsc{Top}\xspace, adopts the operators approach (\textsc{Top}\xspace stands for ``language with Temporal OPerators''). Temporal operators have also been used in \cite{Dowty1982}, \cite{Lascarides}, \cite{Richards}, \cite{Kent}, \cite{Crouch2}, \cite{Pratt1995}, and elsewhere. Unlike logics designed to be used in systems that reason about what changes or remains the same over time, what can or will happen, what could or would have happened, or how newly arrived information fits within already known facts or assumptions (e.g.\ the situation calculus of \cite{McCarthy1969}, the event calculus of \cite{Kowalski1986}, and the logics of \cite{Allen1983}, \cite{Allen1984}, and \cite{McDermott1982} -- see \cite{Vila1994} for a survey), \textsc{Top}\xspace is not intended to be used in reasoning. I provide no inference rules for \textsc{Top}\xspace, and this is why I avoid calling \textsc{Top}\xspace a logic. \textsc{Top}\xspace is only a formal language, designed to facilitate the systematic mapping of temporal English questions to formal expressions (this mapping is not a primary consideration in the above mentioned logics). The answers to the English questions are not generated by carrying out reasoning in \textsc{Top}\xspace, but by translating the \textsc{Top}\xspace expressions to database language expressions, which are then evaluated by the underlying \textsc{Dbms}\xspace. The definition of \textsc{Top}\xspace will be given in chapter \ref{TOP_chapter}, where other ideas from temporal logics will also be discussed. \subsection{Temporal databases} \label{tdbs_general} In the \emph{relational model} \cite{Codd1970}, currently the dominant database model, information is stored in \emph{relations}. Intuitively, relations can be thought of as tables, consisting of rows (called \emph{tuples}) and columns (called \emph{attributes}). For example, the $salaries$ relation below shows the present salaries of the current employees of a company. In the case of $salaries$, whenever the salary of an employee is changed, or whenever an employee leaves the company, the corresponding tuple is modified or deleted. Hence, the database ``forgets'' past facts, and does not contain enough information to answer questions like \qit{What was the salary of T.Smith on 1/1/1992?}. \adbtable{2}{|c|c|}{$salaries$} {$employee$ & $salary$ } {$J.Adams$ & $17000$ \\ $T.Smith$ & $19000$ \\ \ \dots & \ \dots } It is certainly true that traditional database models and languages \emph{can} and \emph{have} been used to store temporal information. (This has led several researchers to question the need for special temporal support in database systems; see \cite{Davies1995} for some discussion.) For example, two extra attributes ($\mathit{from}$ and $\mathit{to}$) could be added to $salaries$ (as in $salaries2$) to \emph{time-stamp} its tuples, i.e.\ to show when each employee had the corresponding salary. \adbtable{4}{|c|c|c|c|}{$salaries2$} {$employee$ & $salary$ & $from$ & $to$ } {$J.Adams$ & $17000$ & $1/1/88$ & $5/5/90$ \\ $J.Adams$ & $18000$ & $6/5/90$ & $9/8/91$ \\ $J.Adams$ & $21000$ & $10/8/91$ & $27/3/93$ \\ \ \dots & \ \dots & \ \dots & \ \dots \\ $T.Smith$ & $17000$ & $1/1/89$ & $1/10/90$ \\ $T.Smith$ & $21000$ & $2/10/90$ & $23/5/92$ \\ \ \dots & \ \dots & \ \dots & \ \dots } The lack of special temporal support in traditional database models and languages, however, complicates the task of expressing in database language time-related data manipulations. We may want, for example, to compute from $salaries2$ a new relation $same\_salaries$ that shows the times when J.Adams and T.Smith had the same salary, along with their common salary: \adbtable{3}{|c|c|c|}{$same\_salaries$} {$salary$ & $from$ & $to$ } {$17000$ & $1/1/89$ & $5/5/90$ \\ $21000$ & $10/8/91$ & $23/5/92$ \\ \ \dots & \ \dots & \ \dots } That is, for every tuple of J.Adams in $salaries2$, we need to check if the period specified by the $\mathit{from}$ and $\mathit{to}$ values of that tuple overlaps the period specified by the $\mathit{from}$ and $\mathit{to}$ values of a tuple for T.Smith which has the same $salary$ value. If they overlap, we need to compute the intersection of the two periods. This cannot be achieved easily in the present version of \textsc{Sql}\xspace (the dominant database language for relational databases \cite{Ullman} \cite{Melton1993}), because \textsc{Sql}\xspace currently does not have any special commands to check if two periods overlap, or to compute the intersection of two periods (in fact, it does not even have a period datatype). As a further example, the approach of adding a $\mathit{from}$ and a $\mathit{to}$ attribute to every relation allows relations like $rel1$ and $rel2$ to be formed. Although $rel1$ and $rel2$ contain different tuples, they represent the same information. \vspace{-9mm} \begin{center} \begin{tabular}{lr} \dbtable{4}{|c|c|c|c|}{$rel1$} {$employee$ & $salary$ & $from$ & $to$} {$G.Foot$ & $17000$ & $1/1/88$ & $9/5/88$ \\ $G.Foot$ & $17000$ & $10/5/88$ & $9/5/93$ \\ $G.Foot$ & $18000$ & $10/5/93$ & $1/3/94$ \\ $G.Foot$ & $18000$ & $2/3/94$ & $11/2/95$ \\ $G.Foot$ & $17000$ & $12/2/95$ & \ $31/3/96$ } & \dbtable{4}{|c|c|c|c|}{$rel2$} {$employee$ & $salary$ & $from$ & $to$} {$G.Foot$ & $17000$ & $1/1/88$ & $31/5/89$ \\ $G.Foot$ & $17000$ & $1/6/89$ & $10/8/92$ \\ $G.Foot$ & $17000$ & $11/8/92$ & $9/5/93$ \\ $G.Foot$ & $18000$ & $10/5/93$ & $11/2/95$ \\ $G.Foot$ & $17000$ & $12/2/95$ & \ $31/3/96$ } \end{tabular} \end{center} Checking if the two relations represent the same information is not easy in the current \textsc{Sql}\xspace version. This task would be greatly simplified if \textsc{Sql}\xspace provided some mechanism to ``normalise'' relations, by merging tuples that apart from their $from$ and $to$ values are identical (tuples of this kind are called \emph{value-equivalent}). In our example, that mechanism would turn both $rel1$ and $rel2$ into $rel3$. To check that $rel1$ and $rel2$ contain the same information, one would check that the normalised forms of the two relations are the same. \adbtable{4}{|c|c|c|c|}{$rel3$} {$employee$ & $salary$ & $from$ & $to$} {$G.Foot$ & $17000$ & $1/1/88$ & $9/5/93$ \\ $G.Foot$ & $18000$ & $10/5/93$ & $11/2/96$ \\ $G.Foot$ & $17000$ & $12/2/95$ & \ $31/3/96$ } Numerous temporal versions of \textsc{Sql}\xspace and the relational model have been proposed (e.g.\ \cite{Clifford2}, \cite{Ariav1986}, \cite{Tansel},\cite{Snodgrass}, \cite{Navathe1988}, \cite{Gadia1988}, \cite{Lorentzos1988}; see \cite{McKenzie} for a summary of some of the proposals). These add special temporal facilities to \textsc{Sql}\xspace (e.g.\ predicates to check if two periods overlap, functions to compute intersections of periods, etc.), and often special types of relations to store time-varying information (e.g.\ relations that force value-equivalent tuples to be merged automatically). Until recently there was little consensus on how temporal support should be added to \textsc{Sql}\xspace and the relational model (or other database languages and models), with every researcher in the field adopting his/her own temporal database language and model. Perhaps as a result of this, very few temporal \textsc{Dbms}\xspace{s} have been implemented (these are mostly early prototypes; see \cite{Boehlen1995c}). This thesis adopts \textsc{Tsql2}\xspace, a recently proposed temporal extension of \textsc{Sql-92}\xspace that was designed by a committee comprising most leading temporal database researchers. (\textsc{Sql-92}\xspace is the latest \textsc{Sql}\xspace standard \cite{Melton1993}. \textsc{Tsql2}\xspace is defined in \cite{TSQL2book}. An earlier definition of \textsc{Tsql2}\xspace can be found in \cite{Tsql2Sigmod}.) \textsc{Tsql2}\xspace and the version of the relational model on which \textsc{Tsql2}\xspace is based will be presented in chapter \ref{tdb_chapter}, along with some modifications that were introduced to them for the purposes of this thesis. Until recently, there was no implemented \textsc{Dbms}\xspace supporting \textsc{Tsql2}\xspace. A prototype system, however, which is capable of evaluating \textsc{Tsql2}\xspace queries now exists. (This system is called \textsc{TimeDB}. See \cite{Boehlen1995c} for a brief technical description of \textsc{TimeDB}. \textsc{TimeDB} actually supports \textsc{Atsql2}, a variant of \textsc{Tsql2}\xspace. See \cite{Boehlen1996} for some information on \textsc{Atsql2}.) Researchers in temporal databases distinguish between \emph{valid time} and \emph{transaction time}.\footnote{I adopt the consensus terminology of \cite{tdbsglossary}. A third term, \emph{user-defined time}, is also employed in the literature to refer to temporal information that is stored in the database without the \textsc{Dbms}\xspace treating it in any special way.} The valid time of some information is the time when that information was true in the \emph{world}. The transaction time of some information is the time when the \emph{database} ``believed'' some piece of information. In this thesis, I ignore the transaction-time dimension. I assume that the natural language questions will always refer to the information that the database currently believes to be true. Questions like \pref{dbi:5}, where \qit{on 2/1/95} specifies a transaction time other than the present, will not be considered. \begin{examps} \item According to what the database believed on 2/1/95, what was the salary of J.Adams on 1/1/89? \label{dbi:5} \end{examps} \section{Contribution of this thesis} \label{contribution} As mentioned in section \ref{thesis_subject}, most existing \textsc{Nlidb}\xspace{s} were designed to interface to snapshot database systems. Although there have been some proposals on how to build \textsc{Nlidb}\xspace{s} for temporal databases, in chapter \ref{comp_chapt} I argue that these proposals suffer from one or more of the following: (i) they ignore important English temporal mechanisms, or assign to them over-simplified semantics, (ii) they lack clearly defined meaning representation languages, (iii) they do not provide complete descriptions of the mappings from natural language to meaning representation language, or (iv) from meaning representation language to database language, (v) they adopt idiosyncratic and often not well-defined temporal database models or languages, (vi) they do not demonstrate that their ideas are implementable. In this thesis, I develop a principled framework for constructing English \textsc{Nlitdb}\xspace{s}, attempting to avoid pitfalls (i) -- (vi). Building on the architecture of figure \ref{pipeline_fig}: \begin{itemize} \item I explore temporal linguistic phenomena that are likely to appear in English questions to \textsc{Nlitdb}\xspace{s}. Drawing on existing linguistic theories of time, I formulate an account for many of these phenomena that is simple enough to be embodied in practical \textsc{Nlitdb}\xspace{s}. \item Exploiting ideas from temporal logics, I define a temporal meaning representation language (\textsc{Top}\xspace), which I use to represent the semantics of English questions. \item I show how \textsc{Hpsg}\xspace \cite{Pollard1} \cite{Pollard2}, currently a highly regarded linguistic theory, can be modified to incorporate the tense and aspect account of this thesis, and to map a wide range of English questions involving time to appropriate \textsc{Top}\xspace expressions. \item I present and prove the correctness of a mapping that translates \textsc{Top}\xspace expressions to \textsc{Tsql2}\xspace queries. \end{itemize} This way, I establish a sound route from English questions involving time to a general-purpose temporal database language, that can act as a principled framework for constructing \textsc{Nlitdb}\xspace{s}. To ensure that this framework is workable: \begin{itemize} \item I demonstrate how it can be employed to implement a prototype \textsc{Nlitdb}\xspace, using the \textsc{Ale}\xspace grammar development system \cite{Carpenter1992} \cite{Carpenter1994} and Prolog \cite{Clocksin1994} \cite{Sterling1994}. I configure the prototype \textsc{Nlitdb}\xspace for a hypothetical air traffic control domain, similar to that of \cite{Sripada1994}. \end{itemize} Unfortunately, during most of the work of this thesis no \textsc{Dbms}\xspace supported \textsc{Tsql2}\xspace. As mentioned in section \ref{tdbs_general}, a prototype \textsc{Dbms}\xspace (\textsc{TimeDB}) that supports a version of \textsc{Tsql2}\xspace (\textsc{Atsql2}) was announced recently. Although it would be obviously very interesting to link the \textsc{Nlitdb}\xspace of this thesis to \textsc{TimeDB}, there is currently very little documentation on \textsc{TimeDB}. The task of linking the two systems is further complicated by the fact that both adopt their own versions of \textsc{Tsql2}\xspace (\textsc{TimeDB} supports \textsc{Atsql2}, and the \textsc{Nlitdb}\xspace of this thesis adopts a slightly modified version of \textsc{Tsql2}\xspace, to be discussed in chapter \ref{tdb_chapter}). One would have to bridge the differences between the two \textsc{Tsql2}\xspace versions. Due to shortage of time, I made no attempt to link the \textsc{Nlitdb}\xspace of this thesis to \textsc{TimeDB}. The \textsc{Tsql2}\xspace queries generated by the \textsc{Nlitdb}\xspace are currently not executed, and hence no answers are produced. Although several issues (summarised in section \ref{to_do}) remain to be addressed, I am confident that this thesis will prove valuable to both those wishing to implement \textsc{Nlitdb}\xspace{s} for practical applications, and those wishing to carry out further research on \textsc{Nlitdb}\xspace{s}, because: (a) it is essentially the first in-depth exploration of time-related problems the \textsc{Nlitdb}\xspace designer has to face, from the linguistic level down to the database level, (b) it proposes a clearly defined framework for building \textsc{Nlitdb}\xspace{s} that addresses a great number of these problems, and (c) it shows how this framework was used to implement a prototype \textsc{Nlitdb}\xspace on which more elaborate \textsc{Nlitdb}\xspace{s} can be based. Finally, I note that: (i) the work of this thesis is one of the first to use \textsc{Tsql2}\xspace, and one of the first to generate feedback to the \textsc{Tsql2}\xspace designers (a number of obscure points and possible improvements in the definition of \textsc{Tsql2}\xspace were revealed during this project; these were reported in \cite{Androutsopoulos1995b}); (ii) the prototype \textsc{Nlitdb}\xspace of this thesis is currently one of the very few \textsc{Nlidb}\xspace{s} (at least among \textsc{Nlidb}\xspace{s} whose grammar is publicly documented) that adopt \textsc{Hpsg}\xspace.\footnote{See also \cite{Cercone1993}. A version of the \textsc{Hpsg}\xspace grammar of this thesis, stripped of its temporal mechanisms, was used in \cite{Seldrup1995} to construct a \textsc{Nlidb}\xspace for snapshot databases.} \section{Issues that will not be addressed} \label{no_issues} To allow the work of this thesis to be completed within the available time, the following issues were not considered. \paragraph{Updates:} This thesis focuses on \emph{questions}. Natural language requests to \emph{update} the database (e.g.\ \pref{noiss:1}) are not considered (see \cite{Davidson1983} for work on natural language updates.) \begin{examps} \item Replace the salary of T.Smith for the period 1/1/88 to 5/5/90 by 17000. \label{noiss:1} \end{examps} Assertions like \pref{noiss:2} will be treated as yes/no questions, i.e.\ \pref{noiss:2} will be treated in the same way as \pref{noiss:3}. \begin{examps} \item On 1/1/89 the salary of T.Smith was 17000. \label{noiss:2} \item Was the salary of T.Smith 17000 on 1/1/89? \label{noiss:3} \end{examps} \paragraph{Schema evolution:} This term refers to cases where the \emph{structure}, not only the \emph{contents}, of the database change over time (new relations are created, old deleted, attributes are added or removed from relations, etc.; see \cite{McKenzie1990}). Schema evolution is not considered in this thesis. The structure of the database is assumed to be static, although the information in the database may change over time. \paragraph{Modal questions:} Modal questions ask if something could have happened, or could never have happened, or will necessarily happen, or can possibly happen. For example, \qit{Could T.Smith have been an employee of IBI in 1985?} does not ask if T.Smith was an IBI employee in 1985, but if it would have been possible for T.Smith to be an IBI employee at that time. Modal questions are not examined in this thesis (see \cite{Mays1986} and \cite{Lowden1991} for related work). \paragraph{Future questions:} A temporal database may contain predictions about the future. At some company, for example, it may have been decided that T.Smith will retire two years from the present, and that J.Adams will replace him. These decisions may have been recorded in the company's database. In that context, one may want to submit questions referring to the future, like \qit{When will T.Smith retire?} or \qit{Who will replace T.Smith?}. To simplify the linguistic data that the work of this thesis had to address, future questions were not considered. The database may contain information about the future, but the framework of this thesis does not currently allow this information to be accessed through natural language. Further work could extend the framework of this thesis to handle future questions as well (see section \ref{to_do}). \paragraph{Cooperative responses:} In many cases, it is helpful for the user if the \textsc{Nlidb}\xspace reports more information than what the question literally asks for. In the dialogue below (from \cite{Johnson1985}), for example, the system has reasoned that the user would be interested to know about the United flight, and has included information about that flight in its answer although this was not requested. \begin{examps} \item Do American Airlines have a night flight to Dallas? \label{noiss:4} \item \sys{No, but United have flight 655.} \label{noiss:5} \end{examps} In other cases, the user's requests may be based on false presumptions. \pref{noiss:4a}, for example, presumes that there is a flight called BA737. If this is not true, it would be useful if the \textsc{Nlidb}\xspace could generate a response like \pref{noiss:4b}. \begin{examps} \item Does flight BA737 depart at 5:00pm? \label{noiss:4a} \item \sys{Flight BA737 does not exist.} \label{noiss:4b} \end{examps} The term \emph{cooperative responses} \cite{Kaplan1982} is used to refer to responses like \pref{noiss:5} and \pref{noiss:4b}. The framework of this thesis includes no mechanism to generate cooperative responses. During the work of this thesis, however, it became clear that such a mechanism is particularly important in questions to \textsc{Nlitdb}\xspace{s}, and hence a mechanism of this kind should be added (this will be discussed further in section \ref{to_do}). \paragraph{Anaphora:} Pronouns (\qit{she}, \qit{they}, etc.), possessive determiners (\qit{his}, \qit{their}), and some noun phrases (\qit{the project}, \qit{these people}) are used anaphorically, to refer to contextually salient entities. The term \emph{nominal anaphora} is frequently used to refer to this phenomenon (see \cite{Hirst1981} for an overview of nominal anaphora, and \cite{Hobbs1986} for methods that can be used to resolve pronoun anaphora). Verb tenses and other temporal expressions (e.g.\ \qit{on Monday}) are often used in a similar anaphoric manner to refer to contextually salient times (this will be discussed in section \ref{temporal_anaphora}). The term \emph{temporal anaphora} \cite{Partee1984} is used in that case. Apart from a temporal anaphoric phenomenon related to noun phrases like \qit{the sales manager} (to be discussed in section \ref{noun_anaphora}), for which support is provided, the framework of this thesis currently provides no mechanism to resolve anaphoric expressions (i.e.\ to determine the entities or times these expressions refer to). Words introducing nominal anaphora (e.g.\ pronouns) are not allowed, and (excluding the phenomenon of section \ref{noun_anaphora}) temporal anaphoric expressions are treated as denoting any possible referent (e.g.\ \qit{on Monday} is taken to refer to any Monday). \paragraph{Elliptical sentences:} Some \textsc{Nlidb}\xspace{s} allow elliptical questions to be submitted as follow-ups to previous questions (e.g.\ \qit{What is the salary of J.Adams?}, followed by \qit{His address?}; see section 4.6 of \cite{Androutsopoulos1995} for more examples). Elliptical questions are not considered in this thesis. \section{Outline of the remainder of this thesis} The remainder of this thesis is organised as follows: Chapter 2 explores English temporal mechanisms, delineating the set of linguistic phenomena that this thesis attempts to support. Drawing on existing ideas from tense and aspect theories, an account for these phenomena is formulated that is suitable to the purposes of this thesis. Chapter 3 defines formally \textsc{Top}\xspace, discussing how it can be used to represent the semantics of temporal English expressions, and how it relates to other existing temporal representation languages. Chapter 4 provides a brief introduction to \textsc{Hpsg}\xspace, and discusses how \textsc{Hpsg}\xspace can be modified to incorporate the tense and aspect account of this thesis, and to map English questions involving time to appropriate \textsc{Top}\xspace expressions. Chapter 5 defines the mapping from \textsc{Top}\xspace to \textsc{Tsql2}\xspace, and proves its correctness (parts of this proof are given in appendix \ref{trans_proofs}). It also discusses the modifications to \textsc{Tsql2}\xspace that are adopted in this thesis. Chapter 6 describes the architecture of the prototype \textsc{Nlitdb}\xspace, provides information about its implementation, and explains which additional modules would have to be added if the system were to be used in real-life applications. Several sample English questions directed to a hypothetical temporal database of an airport are shown, discussing the corresponding output of the prototype \textsc{Nlitdb}\xspace. Chapter 7 discusses previous proposals in the area of \textsc{Nlitdb}\xspace{s}, comparing them to the framework of this thesis. Chapter 8 summarises and proposes directions for further research. \chapter{The Linguistic Data and an Informal Account} \label{linguistic_data} \proverb{There is a time for everything.} \section{Introduction} This chapter explores how temporal information is conveyed in English, focusing on phenomena that are relevant to \textsc{Nlitdb}\xspace{s}. There is a wealth of temporal English mechanisms (e.g.\ verb tenses, temporal adverbials, temporal adjectives, etc.), and it would be impossible to consider all of those in this thesis. Hence, several English temporal mechanisms will be ignored, and simplifying assumptions will be introduced in some of the mechanisms that will be considered. One of the goals of this chapter is to specify exactly which linguistic phenomena this thesis attempts to support. For the phenomena that will be supported, a further goal is to provide an informal account of how they will be treated. Although this chapter draws on existing tense and aspect theories, I stress that it is in no way an attempt to formulate an improved tense and aspect theory. The aim is more practical: to explore how ideas from existing tense and aspect theories can be integrated into \textsc{Nlitdb}\xspace{s}, in a way that leads to provably implementable systems. \section{Aspectual taxonomies} \label{asp_taxes} As mentioned in section \ref{tense_aspect_intro}, many tense and aspect theories employ aspectual classifications, which are often similar to Vendler's distinction between states (e.g.\ \qit{to know}, as in \qit{John knows the answer.}), activities (e.g.\ \qit{to run}, as in \qit{John ran.}), accomplishments (e.g.\ \qit{to build}, as in \qit{John built a house.}), and achievements (e.g.\ \qit{to find}, as in \qit{Mary found the treasure.}). Vendler proposes a number of linguistic tests to determine the aspectual classes of verbs. For example, according to Vendler, activity and accomplishment verbs can appear in the progressive (e.g.\ \qit{John is running}, \qit{John is building a house}), while state and achievement verbs cannot (*\qit{John is knowing the answer.}, *\qit{Mary is finding the treasure}). Activity verbs are said to combine felicitously with \qit{for~\dots} adverbials specifying duration (\qit{John ran for two minutes.}), but sound odd with \qit{in~\dots} duration adverbials (?\qit{John ran in two minutes.}). Accomplishment verbs, in contrast, combine felicitously with \qit{in~\dots} adverbials (\qit{John built a house in two weeks.}), but sound odd with \qit{for~\dots} adverbials (?\qit{John built a house for two weeks.}). Finally, according to Vendler state verbs combine felicitously with \qit{for~\dots} adverbials (e.g.\ \qit{John knew the answer for ten minutes (but then forgot it).}), while achievement verbs sound odd with \qit{for~\dots} adverbials (?\qit{Mary found the treasure for two hours.}). The exact nature of the objects classified by Vendler is unclear. In most cases, Vendler's wording suggests that his taxonomy classifies verbs. However, some of his examples (e.g.\ the fact that \qit{to run} with no object is said to be an activity, while \qit{to run a mile} is said to be an accomplishment) suggest that the natural language expressions being classified are not always verbs, but sometimes larger syntactic constituents (perhaps verb phrases). In other cases, Vendler's arguments suggest that the objects being classified are not natural language expressions (e.g.\ verbs, verb phrases), but world situations denoted by natural language expressions. According to Vendler, \qit{Are you smoking?} ``asks about an activity'', while \qit{Do you smoke?} ``asks about a state''. In this case, the terms ``activity'' and ``state'' seem to refer to types of situations in the world, rather than types of natural language expressions. (The first question probably asks if somebody is actually smoking at the present moment. The second one has a \emph{habitual} meaning: it asks if somebody has the habit of smoking. Vendler concludes that habits ``are also states in our sense''.) Numerous variants of Vendler's taxonomy have been proposed. These differ in the number of aspectual classes they assume, the names of the classes, the nature of the objects being classified, and the properties assigned to each class. Vlach \cite{Vlach1993} distinguishes four aspectual classes of sentences, and assumes that there is a parallel fourfold taxonomy of world situations. Moens \cite{Moens} distinguishes between ``states'', ``processes'', ``culminated processes'', ``culminations'', and ``points'', commenting that his taxonomy does not classify real world situations, but ways people use to describe world situations. Parsons \cite{Parsons1989} distinguishes three kinds of ``eventualities'' (``states'', ``activities'', and ``events''), treating eventualities as entities in the world. Lascarides \cite{Lascarides} classifies propositions (functions from time-periods to truth values), distinguishing between ``state'', ``process'', and ``event'' propositions. \section{The aspectual taxonomy of this thesis} \label{aspectual_classes} Four aspectual classes are employed in this thesis: \emph{states}, \emph{activities}, \emph{culminating activities}, and \emph{points}. (Culminating activities and points correspond to Vendler's ``accomplishments'' and ``achievements'' respectively. Similar terms are used in \cite{Moens} and \cite{Blackburn1994}.) These aspectual classes correspond to ways of \emph{viewing world situations} that people seem to use: a situation can be viewed as involving no change or action (state view), as an instantaneous change or action (point view), as a change or action with no climax (activity view), or as a change or action with a climax (culminating activity view). (Throughout this thesis, I use ``situation'' to refer collectively to elements of the world that other authors call ``events'', ``processes'', ``states'', etc.) Determining which view the speaker has in mind is important to understand what the speaker means. For example, \qit{Which tanks contained oil?} is typically uttered with a state view. When an \qit{at \dots} temporal adverbial (e.g.\ \qit{at 5:00pm}) is attached to a clause uttered with a state view, the speaker typically means that the situation of the clause simply holds at the time of the adverbial. There is normally no implication that the situation starts or stops holding at the time of the adverbial. For example, in \qit{Which tanks contained oil at 5:00pm?} there is normally no implication that the tanks must have started or stopped containing oil at 5:00pm. In contrast, \qit{Who ran to the station?} is typically uttered with a culminating activity view. In this case, an \qit{at \dots} adverbial usually specifies the time when the situation starts or is completed. \qit{Who ran to the station at 5:00pm?}, for example, probably asks for somebody who started running to the station or reached it at 5:00pm. Some linguistic markers seem to signal which view the speaker has in mind. For example, the progressive usually signals a state view (e.g.\ unlike \qit{Who ran to the station at 5:00pm?}, \qit{Who was running to the station at 5:00pm?} is typically uttered with a state view; in this case, the running is simply ongoing at 5:00pm, it does not start or finish at 5:00pm). Often, however, there are no such explicit markers. The processes employed in those cases by hearers to determine the speaker's view are not yet fully understood. In an \textsc{Nlitdb}\xspace, however, where questions refer to a restricted domain, reasonable guesses can be made by observing that in each domain, each verb tends to be associated mainly with one particular view. Certain agreements about how situations are to be viewed (e.g.\ that some situations are to be treated as instantaneous -- point view) will also have been made during the design of the database. These agreements provide additional information about how the situations of the various verbs are viewed in each domain. More precisely, the following approach is adopted in this thesis. Whenever the \textsc{Nlitdb}\xspace is configured for a new application domain, the base form of each verb is assigned to one of the four aspectual classes, using criteria to be discussed in section \ref{aspect_criteria}. These criteria are intended to detect the view that is mainly associated with each verb in the particular domain that is being examined. Following \cite{Dowty1986}, \cite{Moens}, \cite{Vlach1993}, and others, aspectual class is treated as a property of not only verbs, but also verb phrases, clauses, and sentences. Normally, all verb forms will inherit the aspectual classes of the corresponding base forms. Verb phrases, clauses, or sentences will normally inherit the aspectual classes of their main verb forms. Some linguistic mechanisms (e.g.\ the progressive or some temporal adverbials), however, may cause the aspectual class of a verb form to differ from that of the base form, or the aspectual class of a verb phrase, clause, or sentence to differ from that of its main verb form. The aspectual class of each verb phrase, clause, or sentence is intended to reflect the view that users typically have in mind when using that expression in the particular domain. In the case of a verb like \qit{to run}, that typically involves a culminating activity view when used with an expression that specifies a destination or specific distance (e.g.\ \qit{to run to the station/five miles}), but an activity view when used on its own, it will be assumed that there are two different homonymous verbs \qit{to run}. One has a culminating activity base form, and requires a complement that specifies a destination or specific distance. The other has an activity base form, and requires no such complement. A similar distinction would be introduced in the case of verbs whose aspectual class depends on whether or not the verb's object denotes a countable or mass entity (e.g.\ \qit{to drink a bottle of wine} vs.\ \qit{to drink wine}; see \cite{Mourelatos1978}). Similarly, when a verb can be used in a domain with both habitual and non-habitual meanings (e.g. \qit{BA737 (habitually) departs from Gatwick.} vs.\ \qit{BA737 (actually) departed from Gatwick five minutes ago.}), a distinction will be made between a homonym with a habitual meaning, and a homonym with a non-habitual meaning.\footnote{When discussing sentences with multiple readings, I often use parenthesised words (e.g.\ \qit{(habitually)}) to indicate which reading is being considered.} The base forms of habitual homonyms are classified as states. (This agrees with Vendler, Vlach \cite{Vlach1993}, and Moens and Steedman \cite{Moens2}, who all classify habituals as states.) The aspectual classes of non-habitual homonyms depend on the verb and the application domain. Approaches that do not postulate homonyms are also possible (e.g.\ claiming that \qit{to run} is an activity which is transformed into a culminating activity by \qit{the station}). The homonyms method, however, leads to a more straight forward treatment in the \textsc{Hpsg}\xspace grammar of chapter \ref{English_to_TOP} (where the base form of each homonym is mapped to a different sign). In the rest of this thesis, I refer to verbs whose base forms are classified as states, activities, culminating activities, or points as \emph{state verbs}, \emph{activity verbs}, \emph{culminating activity verbs}, and \emph{point verbs}. \section{Criteria for classifying base verb forms} \label{aspect_criteria} This section discusses the criteria that determine the aspectual class of a verb's base form in a particular \textsc{Nlitdb}\xspace domain. Three criteria are employed, and they are applied in the order of figure \ref{decision_tree}. \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.5]{decision_tree} \caption{Determining the aspectual class of a verb's base form} \label{decision_tree} \end{center} \hrule \end{figure} \subsection{The simple present criterion} \label{simple_present_criterion} The first criterion distinguishes state verbs (verbs whose base forms are states) from point, activity, and culminating activity verbs. If the simple present of a verb can be used (in the particular domain) in single-clause questions with non-futurate meanings, the verb is a state one; otherwise it is a point, activity, or culminating activity verb. For example, in domains where \pref{crit:1} and \pref{crit:2} are possible, \qit{to contain} and \qit{to own} are state verbs. \begin{examps} \item Does any tank contain oil? \label{crit:1} \item Which employees own a car? \label{crit:2} \end{examps} Some clarifications are needed. First, the simple present sometimes refers to something that is scheduled to happen. For example, \pref{crit:2.7} could refer to a scheduled assembling (in that case, \pref{crit:2.7} is very similar to \pref{crit:2.7.2}). I consider this meaning of \pref{crit:2.7} futurate. Hence, this use of \pref{crit:2.7} does not constitute evidence that \qit{to assemble} is a state verb. \begin{examps} \item When does J.Adams assemble engine 5? \label{crit:2.7} \item When will J.Adams assemble engine 5? \label{crit:2.7.2} \end{examps} In reporting contexts, the simple present of verbs whose base forms I would not want to be classified as states can be used with a non-futurate meaning. For example, in a context where the speaker reports events as they happen, \pref{crit:2.8} is possible. (This use of the simple present is unlikely in \textsc{Nlitdb}\xspace questions.) \begin{examps} \item J.Adams arrives. He moves the container. He fixes the engine. \label{crit:2.8} \end{examps} The simple present criterion examines questions directed to a \textsc{Nlitdb}\xspace, not sentences from other contexts. Hence, \pref{crit:2.8} does not constitute evidence that \qit{to arrive}, \qit{to move}, and \qit{to fix} are state verbs. The reader is reminded that when verbs have both habitual and non-habitual meanings, I distinguish between habitual and non-habitual homonyms (section \ref{aspectual_classes}). Ignoring scheduled-to-happen meanings (that do not count for the simple present criterion), \pref{crit:3} and \pref{crit:4} can only have habitual meanings. \begin{examps} \item Which flight lands on runway 2? \label{crit:3} \item Does any doctor smoke? \label{crit:4} \end{examps} \pref{crit:3} asks for a flight that habitually lands on runway 2, and \pref{crit:4} for doctors that are smokers. That is, \pref{crit:3} and \pref{crit:4} can only be understood as involving the habitual homonyms of \qit{to land} and \qit{to smoke}. (In contrast, \pref{crit:5} and \pref{crit:6} can be understood with non-habitual meanings, i.e.\ as involving the non-habitual homonyms.) \begin{examps} \item Which flight is landing on runway 2? \label{crit:5} \item Is any doctor smoking? \label{crit:6} \end{examps} Therefore, in domains where \pref{crit:3} and \pref{crit:4} are possible, the habitual \qit{to land} and \qit{to smoke} are state verbs. \pref{crit:3} and \pref{crit:4} do not constitute evidence that the non-habitual \qit{to land} and \qit{to smoke} are state verbs. \subsection{The point criterion} \label{point_criterion} The second criterion, the \emph{point criterion}, distinguishes point verbs from activity and culminating activity ones (state verbs will have already been separated by the simple present criterion; see figure \ref{decision_tree}). The point criterion is based on the fact that some verbs will be used to describe kinds of world situations that are modelled in the database as being always instantaneous. If a verb describes situations of this kind, its base form should be classified as point; otherwise, it should be classified as activity or culminating activity. In section \ref{aspect_examples}, for example, I consider a hypothetical airport database. That database does not distinguish between the times at which a flight starts or stops entering an airspace sector. Entering a sector is modelled as instantaneous. Also, in the airport domain \qit{to enter} is only used to refer to flights entering sectors. Consequently, in that domain \qit{to enter} is a point verb. If \qit{to enter} were also used to refer to, for example, groups of passengers entering planes, and if situations of this kind were modelled in the database as non-instantaneous, one would have to distinguish between two homonyms \qit{to enter}, one used with flights entering sectors, and one with passengers entering planes. The first would be a point verb; the second would not. The person applying the criterion will often have to decide exactly what is or is not part of the situations described by the verbs. The database may store, for example, the time-points at which a flight starts to board, finishes boarding, starts to taxi to a runway, arrives at the runway, and leaves the ground. Before classifying the non-habitual \qit{to depart}, one has to decide exactly what is or is not part of departing. Is boarding part of departing, i.e.\ is a flight departing when it is boarding? Is taxiing to a runway part of departing? Or does departing include only the time at which the flight actually leaves the ground? If a flight starts to depart when it starts to board, and finishes departing when it leaves the ground, then the base form of \qit{to depart} should not be classified as point, because the database does not treat departures as instantaneous (it distinguishes between the beginning of the boarding and the time when the flight leaves the ground). If, however, departing starts when the front wheels of the aircraft leave the ground and finishes when the rear wheels leave the ground, the base form of \qit{to depart} \emph{should} be classified as point, because the database does not distinguish the two times. In any case, the user should be aware of what \qit{to depart} is taken to mean. The point criterion is similar to claims in \cite{Vendler}, \cite{Singh}, \cite{Vlach1993}, and elsewhere that achievement (point) verbs denote instantaneous situations. \subsection{The imperfective paradox criterion} \label{ip_criterion} The third criterion distinguishes activity from culminating activity verbs (state and point verbs will have already been separated by the point and simple present criteria). The criterion is based on the imperfective paradox (section \ref{tense_aspect_intro}). Assertions containing the past continuous and simple past of the verbs, like \pref{crit:20} -- \pref{crit:23}, are considered. \begin{examps} \item John was running. \label{crit:20} \item John ran. \label{crit:21} \item John was building a house. \label{crit:22} \item John built a house. \label{crit:23} \end{examps} The reader is reminded that assertions are treated as yes/no questions (section \ref{no_issues}). If an affirmative answer to the past continuous assertion implies an affirmative answer to the simple past assertion (as in \pref{crit:20} -- \pref{crit:21}), the verb is an activity one; otherwise (e.g.\ \pref{crit:22} -- \pref{crit:23}), it is a culminating activity one. As will be discussed in section \ref{progressives}, the past continuous sometimes has a futurate meaning. Under this reading, \pref{crit:20} means \qit{John was going to run.}, and an affirmative answer to \pref{crit:20} does not necessarily imply an affirmative answer to \pref{crit:21}. When applying the imperfective paradox criterion, the past continuous must not have its futurate meaning. In various forms, the imperfective paradox criterion has been used in \cite{Vendler}, \cite{Vlach1993}, \cite{Kent}, and elsewhere. \subsection{Other criteria} \label{other_aspect_criteria} The three criteria above are not the only ones that could be used. The behaviour of verbs when appearing in various forms or when combining with some temporal adverbials varies depending on their aspectual classes. Alternative criteria can be formulated by observing this behaviour. For example, some authors classify verbs (or situations denoted by verbs) by observing how easily they appear in progressive forms (to be discussed in section \ref{progressives}), how easily they combine with \qit{for~\dots} and \qit{in~\dots} duration adverbials (sections \ref{for_adverbials} and \ref{in_adverbials} below), or what the verbs entail about the start or the end of the described situation when they combine with \qit{at \dots} temporal adverbials (section \ref{point_adverbials} below). In some cases, the person classifying the base verb forms may be confronted with a verb for which the three criteria of sections \ref{simple_present_criterion} -- \ref{ip_criterion} do not yield a clear verdict. In such cases, additional evidence for or against classifying a base verb form into a particular class can be found by referring to following sections, where the typical behaviour of each class is examined. \subsection{Classifying base verb forms in the airport domain} \label{aspect_examples} To illustrate the use of the criteria of sections \ref{simple_present_criterion} -- \ref{ip_criterion}, I now consider a hypothetical \textsc{Nlitdb}\xspace to a temporal database that contains information about the air-traffic of an airport. (I borrow some terminology from \cite{Sripada1994}. The airport domain will be used in examples throughout this thesis.) The airport database shows the times when flights arrived at, or departed from, the airport, the times flights spent circling around the airport while waiting for permission to land, the runways they landed on or took off from, the gates where the flights boarded, etc. The database is occasionally queried using the \textsc{Nlitdb}\xspace to determine the causes of accidents, and to collect data that are used to optimise the airport's traffic-handling strategies. The airport's airspace is divided into sectors. Flights approaching or leaving the airport cross the boundaries of sectors, each time \emph{leaving} a sector and \emph{entering} another one. The airport is very busy, and some of its runways may also \emph{be closed} for maintenance. Hence, approaching flights are often instructed to \emph{circle} around the airport until a runway \emph{becomes} free. When a runway is freed, flights \emph{start} to \emph{land}. Landing involves following a specific procedure. In some cases, the pilot may abort the landing procedure before completing it. Otherwise, the flight lands on a runway, and it then \emph{taxies} to a gate that \emph{is free}. The moment at which the flight \emph{reaches} the gate is considered the time at which the flight \emph{arrived} (reaching a location and arriving are modelled as instantaneous). Normally (habitually) each flight arrives at the same gate and time every day. Due to traffic congestion, however, a flight may sometimes arrive at a gate or time other than its normal ones. Before \emph{taking off}, each flight is \emph{serviced} by a service company. This involves carrying out a specific set of tasks. Unless all tasks have been carried out, the service is incomplete. Each service company normally (habitually) services particular flights. Sometimes, however, a company may be asked to service a flight that it does not normally service. After being serviced, a flight may be \emph{inspected}. Apart from flights, inspectors also inspect gates and runways. In all cases, there are particular tasks to be carried out for the inspections to be considered complete. Shortly before taking off, flights start to \emph{board}. Unless all the passengers that have checked in enter the aircraft, the boarding is not complete, and the flight cannot depart. (There are special arrangements for cases where passengers are too late.) The flight then \emph{leaves} the gate, and that moment is considered the time at which the flight \emph{departed} (leaving a location and departing are modelled as instantaneous). Normally (habitually) each flight departs from the same gate at the same time every day. Sometimes, however, flights depart from gates, or at times, other than their normal ones. After leaving its gate, a flight may be told to \emph{queue} for a particular runway, until that runway becomes free. When the runway is free, the flight starts to \emph{take off}, which involves following a specific procedure. As with landings, the pilot may abort the taking off procedure before completing it. The database also records the states of parts of the airport's emergency system. There are, for example, emergency tanks, used by the fire-brigade. Some of those may \emph{contain} water, others may contain foam, and others may \emph{be empty} for maintenance. \begin{table} \begin{center} {\small \begin{tabular}{|l|l|l|l|} \hline state verbs & activity verbs & culm.\ activity verbs & point verbs \\ \hline service (habitually) & circle & land & cross \\ arrive (habitually) & taxi (no destination) & take off & enter \\ depart (habitually) & queue & service (actually) & become \\ contain & & inspect & start/begin \\ be (non-auxiliary) & & board & stop/finish \\ & & taxi (to destination) & reach \\ &&& leave \\ &&& arrive (actually) \\ &&& depart (actually) \\ \hline \end{tabular} } \end{center} \caption{Verbs of the airport domain} \label{airport_verbs} \end{table} Table \ref{airport_verbs} shows some of the verbs that are used in the airport domain. \qit{To depart}, \qit{to arrive}, and \qit{to service} are used with both habitual and non-habitual meanings. \pref{criteg:1.3} and \pref{criteg:1.4}, for example, can have habitual meanings. In \pref{criteg:1.3.2} and \pref{criteg:1.4.1}, the verbs are probably used with their non-habitual meanings. I distinguish between habitual and non-habitual homonyms of \qit{to depart}, \qit{to arrive}, and \qit{to service} (section \ref{aspectual_classes}). \begin{examps} \item Which flights depart/arrive at 8:00am? \label{criteg:1.3} \item Which flight departed/arrived at 8:00am yesterday? \label{criteg:1.3.2} \item Which company services BA737? \label{criteg:1.4} \item Which company serviced BA737 yesterday? \label{criteg:1.4.1} \end{examps} I also distinguish between two homonyms of \qit{to taxi}, one that requires a destination-denoting complement (as in \qit{BA737 was taxiing to gate 2.}), and one that requires no such complement (as in \qit{BA737 was taxiing.}). The simple present criterion and sentences like \pref{criteg:1.1}, \pref{criteg:1.2}, \pref{criteg:1.3}, and \pref{criteg:1.4} imply that the non-auxiliary \qit{to be}, \qit{to contain}, and the habitual \qit{to depart}, \qit{to arrive}, and \qit{to service} are state verbs. \begin{examps} \item Which gates are free? \label{criteg:1.1} \item Does any tank contain foam? \label{criteg:1.2} \end{examps} All other verbs of table \ref{airport_verbs} are not state verbs. For example, (excluding habitual and futurate meanings) \pref{criteg:4} -- \pref{criteg:12} sound unlikely or odd in the airport domain. \pref{criteg:19} -- \pref{criteg:19.1} would be used instead. \begin{examps} \item \odd Which flights circle? \label{criteg:4} \item \odd Which flight taxies to gate 2? \label{criteg:9} \item \odd Which flight departs? \label{criteg:12} \item Which flights are circling? \label{criteg:19} \item Which flight is taxiing to gate 2? \label{criteg:20} \item Which flight is departing? \label{criteg:19.1} \end{examps} The verbs in the rightmost column of table \ref{airport_verbs} are used in the airport domain to refer to situations which I assume are modelled as instantaneous in the database. Consequently, by the point criterion these are all point verbs. In contrast, I assume that the situations of the verbs in the two middle columns are not modelled as instantaneous. Therefore, those are activity or culminating activity verbs. In the airport domain, a sentence like \pref{criteg:31} means that BA737 spent some time circling around the airport. It does not imply that BA737 completed any circle around the airport. Hence, an affirmative answer to \pref{criteg:30} implies an affirmative answer to \pref{criteg:31}. By the imperfective paradox criterion, \qit{to circle} is an activity verb. \begin{examps} \item BA737 was circling. \label{criteg:30} \item BA737 circled. \label{criteg:31} \end{examps} Similar assertions and the imperfective paradox criterion imply that \qit{to taxi} (no destination) and \qit{to queue} are also activity verbs. In contrast, the verbs in the third column of table \ref{airport_verbs} are culminating activity verbs. For example, in the airport domain an affirmative answer to \pref{criteg:43.2} does not imply an affirmative answer to \pref{criteg:43.3}: J.Adams may have aborted the inspection before completing all the inspection tasks, in which case the inspection is incomplete. \begin{examps} \item J.Adams was inspecting runway 5. \label{criteg:43.2} \item J.Adams inspected runway 5. \label{criteg:43.3} \end{examps} \section{Verb tenses} \label{verb_tenses} I now turn to verb tenses. I use ``tense'' with the meaning it has in traditional English grammar textbooks (e.g.\ \cite{Thomson}). For example, \qit{John sings.} and \qit{John is singing.} will be said to be in the simple present and present continuous tenses respectively. In linguistics, ``tense'' is not always used in this way. According to \cite{Comrie2}, for example, \qit{John sings.} and \qit{John is singing.} are in the same tense, but differ aspectually. Future questions are not examined in this thesis (section \ref{no_issues}). Hence, future tenses and futurate meanings of other tenses (e.g.\ the scheduled-to-happen meaning of the simple present; section \ref{simple_present_criterion}) will be ignored. To simplify further the linguistic data, the present perfect continuous and past perfect continuous (e.g.\ \qit{has/had been inspecting}) were also not considered: these tenses combine problems from both continuous and perfect tenses. This leaves six tenses to be discussed: simple present, simple past, present continuous, past continuous, present perfect, and past perfect. \subsection{Simple present} \label{simple_present} The framework of this thesis allows the simple present to be used only with state verbs, to refer to a situation that holds at the present (e.g.\ \pref{sp:1}, \pref{sp:2}). \begin{examps} \item Which runways are closed? \label{sp:1} \item Does any tank contain water? \label{sp:2} \end{examps} Excluding the scheduled-to-happen meaning (which is ignored in this thesis), \pref{sp:3} can only be understood as asking for the current normal (habitual) servicer of BA737. Similarly, \pref{sp:4} can only be asking for the current normal departure gate of BA737. \pref{sp:3} would not be used to refer to a company that is actually servicing BA737 at the present moment (similar comments apply to \pref{sp:4}). That is, \pref{sp:3} and \pref{sp:4} can only involve the habitual homonyms of \qit{to service} and \qit{to depart} (which are state verbs), not the non-habitual ones (which are culminating activity and point verbs respectively; see table \ref{airport_verbs}). This is consistent with the assumption that the simple present can only be used with state verbs. \begin{examps} \item Which company services BA737? \label{sp:3} \item Which flights depart from gate 2? \label{sp:4} \end{examps} In the airport domain, \qit{to circle} is an activity verb (there is no state habitual homonym). Hence, \pref{sp:3.7} is rejected. This is as it should be, because \pref{sp:3.7} can only be understood with a habitual meaning, a meaning which is not available in the airport domain (there are no circling habits). \begin{examps} \item Does BA737 circle? \label{sp:3.7} \end{examps} The simple present can also be used with non-state verbs to describe events as they happen (section \ref{simple_present_criterion}), or with a historic meaning (e.g.\ \qit{In 1673 a fire destroys the palace.}), but these uses are extremely unlikely in \textsc{Nlitdb}\xspace questions. \subsection{Simple past} \label{simple_past} Like the simple present, the simple past can be used with verbs from all four classes (e.g.\ \pref{spa:2} -- \pref{spa:7}). \begin{examps} \item Which tanks contained water on 1/1/95? \label{spa:2} \item Did BA737 circle on 1/1/95? \label{spa:5} \item Which flights (actually) departed from gate 2 on 1/1/95? \label{spa:3} \item Which flights (habitually) departed from gate 2 in 1994? \label{spa:1} \item Which company (actually) serviced BA737 yesterday? \label{spa:6} \item Which company (habitually) serviced BA737 last year? \label{spa:7} \end{examps} \pref{spa:3} -- \pref{spa:7} show that both the habitual and the non-habitual homonyms of verbs like \qit{to depart} or \qit{to service} are generally possible in the simple past. \pref{spa:8} is ambiguous. It may refer either to flights that actually departed (perhaps only once) from gate 2 in 1994, or to flights that normally (habitually) departed from gate 2 in 1994. \begin{examps} \item Which flights departed from gate 2 in 1994? \label{spa:8} \end{examps} The simple past of culminating activity verbs normally implies that the situation of the verb reached its climax. For example, in \pref{spa:13} the service must have been completed, and in \pref{spa:12} BA737 must have reached gate 2 for the answer to be affirmative. \begin{examps} \item Did Airserve service BA737? \label{spa:13} \item Did BA737 taxi to gate 2? \label{spa:12} \item BA737 was taxiing to gate 2 but never reached it. \label{spa:12.5} \end{examps} Some native English speakers consider simple negative answers to \pref{spa:13} and \pref{spa:12} unsatisfactory, if for example BA737 was taxiing to gate 2 but never reached it. Although they agree that strictly speaking the answer should be negative, they consider \pref{spa:12.5} a much more appropriate answer. To generate answers like \pref{spa:12.5}, a mechanism for \emph{cooperative responses} is needed, an issue not addressed in this thesis (section \ref{no_issues}). The simple past (and other tenses) often has an \emph{anaphoric} nature. For example, \pref{spa:13} probably does not ask if Airserve serviced BA737 at \emph{any} time in the past. \pref{spa:13} would typically be used with a particular time in mind (e.g.\ the present day), to ask if Airserve serviced BA737 during that time. As will be discussed in section \ref{temporal_anaphora}, a temporal anaphora resolution mechanism is needed to determine the time the speaker has in mind. The framework of this thesis currently provides no such mechanism, and \pref{spa:13} is taken to refer to any past time. (The same approach is adopted with all other tenses that refer to past situations.) \subsection{Present continuous and past continuous} \label{progressives} \paragraph{Futurate meanings:} The present and past continuous can be used with futurate meanings. In that case, \pref{prog:13} is similar to \pref{prog:14}. \begin{examps} \item Who is/was inspecting BA737? \label{prog:13} \item Who will/would inspect BA737? \label{prog:14} \end{examps} Futurate meanings of tenses are not examined in this thesis. Hence, this use of the present and past continuous will be ignored. \paragraph{Activity and culminating activity verbs:} The present and past continuous can be used with activity and culminating activity verbs to refer to a situation that is or was in progress (e.g.\ \pref{prog:1} -- \pref{prog:7} from the airport domain). \begin{examps} \item Are/Were any flights circling? \label{prog:1} \item Is/Was BA737 taxiing to gate 2? \label{prog:7} \end{examps} In the case of culminating activity verbs, there is no requirement for the climax to be reached at any time. The past continuous version of \pref{prog:7}, for example, does not require BA737 to have reached gate 2 (cf.\ \pref{spa:12}). \paragraph{Point verbs:} The present and past continuous of point verbs often refers to a preparatory process that is or was ongoing, and that normally leads to the instantaneous situation of the verb. For example, in the airport domain where \qit{to depart} is a point verb and departing includes only the moment where the flight leaves its gate, one could utter \pref{prog:19.110} when the checking-in is ongoing or when the flight is boarding. \begin{examps} \item BA737 is departing. \label{prog:19.110} \end{examps} The framework of this thesis provides no mechanism for determining exactly which preparatory process is asserted to be in progress. (Should the checking-in be in progress for the response to \pref{prog:19.110} to be affirmative? Should the boarding be ongoing?) Hence, this use of the present and past continuous of point verbs is not allowed. The response to \pref{prog:19.110} will be affirmative only at the time-point where BA737 is leaving its gate (as will be discussed in section \ref{tsql2_time}, the database may model time-points as corresponding to minutes or even whole days). To avoid misunderstandings, the \textsc{Nlitdb}\xspace should warn the user that \pref{prog:19.110} is taken to refer to the actual (instantaneous) departure, not to any preparatory process. This is again a case for cooperative responses, an issue not examined in this thesis. \paragraph{State verbs:} It has often been observed (e.g.\ Vendler's tests in section \ref{asp_taxes}) that state verbs are not normally used in progressive forms. For example, \pref{prog:28} and \pref{prog:30} are easily rejected by most native speakers. (I assume that \qit{to own} and \qit{to consist} would be classified as state verbs, on the basis that simple present questions like \pref{prog:29} and \pref{prog:31} are possible.) \begin{examps} \item \bad Who is owning five cars? \label{prog:28} \item Who owns five cars? \label{prog:29} \item \bad Which engine is consisting of 34 parts? \label{prog:30} \item Which engine consists of 34 parts? \label{prog:31} \end{examps} The claim that state verbs do not appear in the progressive is challenged by sentences like \pref{prog:35} (from \cite{Smith1986}, cited in \cite{Passonneau}; \cite{Kent} and \cite{Vlach1993} provide similar examples). \pref{prog:35} shows that the non-auxiliary \qit{to be}, which is typically classified as state verb, can be used in the progressive. \begin{examps} \item My daughter is being very naughty. \label{prog:35} \end{examps} Also, some native English speakers find \pref{prog:32} and \pref{prog:36} acceptable (though they would use the non-progressive forms instead). (I assume that \qit{to border} would be classified as state verb, on the basis that \qit{Which countries border France?} is possible.) \begin{examps} \item \odd Tank 4 was containing water when the bomb exploded. \label{prog:32} \item \odd Which countries were bordering France in 1937? \label{prog:36} \end{examps} Not allowing progressive forms of state verbs also seems problematic in questions like \pref{prog:40}. \pref{prog:40} has a reading which is very similar to the habitual reading of \pref{prog:41} (habitually serviced BA737 in 1994). \begin{examps} \item Which company was servicing BA737 in 1994? \label{prog:40} \item Which company serviced (habitually) BA737 in 1994? \label{prog:41} \end{examps} The reader is reminded that in the airport domain I distinguish between a habitual and a non-habitual homonym of \qit{to service}. The habitual homonym is a state verb, while the non-habitual one is a culminating activity verb. If progressive forms of state verbs are not allowed, then only the non-habitual homonym (actually servicing) is possible in \pref{prog:40}. This does not account for the apparently habitual meaning of \pref{prog:40}. One could argue that the reading of \pref{prog:40} is not really habitual but \emph{iterative} (servicing many times, as opposed to having a servicing habit). As pointed out in \cite{Comrie} (p.~27), the mere repetition of a situation does not suffice for the situation to be considered a habit. \pref{prog:44}, for example, can be used when John is banging his hand on the table repeatedly. In this case, it seems odd to claim that \pref{prog:44} asserts that John has the habit of banging his hand on the table, i.e.\ \pref{prog:44} does not seem to be equivalent to the habitual \pref{prog:45}. \begin{examps} \item John is banging his hand on the table. \label{prog:44} \item John (habitually) bangs his hand on the table. \label{prog:45} \end{examps} In sentences like \pref{prog:40} -- \pref{prog:41}, however, the difference between habitual and iterative meaning is hard to define. For simplicity, I do not distinguish between habitual and iterative readings, and I allow state verbs to be used in progressive forms (with the same meanings as the non-progressive forms). This causes \pref{prog:40} to receive two readings: one involving the habitual \qit{to service} (servicing habitually in 1994), and one involving the non-habitual \qit{to service} (actually servicing at some time in 1994; this reading is more likely without the \qit{in 1994}). \pref{prog:28} and \pref{prog:30} are treated as equivalent to \pref{prog:29} and \pref{prog:31}. As will be discussed in section \ref{temporal_adverbials}, I assume that progressive tenses cause an aspectual shift from activities and culminating activities to states. In the airport domain, for example, although the base form of \qit{to inspect} is a culminating activity, \qit{was inspecting} is a state. \subsection{Present perfect} \label{present_perfect} Like the simple past, the present perfect can be used with verbs of all four classes to refer to past situations (e.g.\ \pref{prep:5} -- \pref{prep:10}). With culminating activity verbs, the situation must have reached its climax (e.g.\ in \pref{prep:10} the service must have been completed). \begin{examps} \item Has BA737 (ever) been at gate 2? \label{prep:5} \item Which flights have circled today? \label{prep:8} \item Has BA737 reached gate 2? \label{prep:7} \item Which company has (habitually) serviced BA737 this year? \label{prep:9} \item Has Airserve (actually) serviced BA737? \label{prep:10} \end{examps} It has often been claimed (e.g.\ \cite{Moens}, \cite{Vlach1993}, \cite{Blackburn1994}) that the English present perfect asserts that some consequence of the past situation holds at the present. For example, \pref{prep:11} seems to imply that there is a consequence of the fact that engine 5 caught fire that still holds (e.g.\ engine 5 is still on fire, or it was damaged by the fire and has not been repaired). In contrast, \pref{prep:12} does not seem to imply (at least not as strongly) that some consequence still holds. \begin{examps} \item Engine 5 has caught fire. \label{prep:11} \item Engine 5 caught fire. \label{prep:12} \end{examps} Although these claims are intuitively appealing, it is difficult to see how they could be used in a \textsc{Nlitdb}\xspace. Perhaps in \pref{prep:15} the \textsc{Nlitdb}\xspace should check not only that the landing was completed, but also that some consequence of the landing still holds. \begin{examps} \item Has BA737 landed? \label{prep:15} \end{examps} It is unclear, however, what this consequence should be. Should the \textsc{Nlitdb}\xspace check that the plane is still at the airport? And should the answer be negative if the plane has departed since the landing? Should the \textsc{Nlitdb}\xspace check that the passengers of BA737 are still at the airport? And should the answer be negative if the passengers have left the airport? Given this uncertainty, the framework of this thesis does not require the past situation to have present consequences. When the present perfect combines with \qit{for~\dots} duration adverbials (to be discussed in section \ref{for_adverbials}), there is often an implication that the past situation still holds at the present (this seems related to claims that the past situation must have present consequences). For example, there is a reading of \pref{prep:16.10} where J.Adams is still a manager. (\pref{prep:16.10} can also mean that J.Adams was simply a manager for two years, without the two years ending at the present moment.) In contrast, \pref{prep:16.12} carries no implication that J.Adams is still a manager (in fact, it seems to imply that he is no longer a manager). \begin{examps} \item J.Adams has been a manager for two years. \label{prep:16.10} \item J.Adams was a manager for two years. \label{prep:16.12} \end{examps} Representing in \textsc{Top}\xspace the still-holding reading of sentences like \pref{prep:16.10} has proven difficult. Hence, I ignore the possible implication that the past situation still holds, and I treat \pref{prep:16.10} as equivalent to \pref{prep:16.12}. The present perfect does not combine felicitously with some temporal adverbials. For example, \pref{prep:16} and \pref{prep:19} sound at least odd to most native English speakers (they would use \pref{prep:16.1} and \pref{prep:19.1} instead). In contrast, \pref{prep:17} and \pref{prep:20} are acceptable. \begin{examps} \item \odd Which flights have landed yesterday? \label{prep:16} \item Which flights landed yesterday? \label{prep:16.1} \item Which flights have landed today? \label{prep:17} \item \odd Which flights has J.Adams inspected last week? \label{prep:19} \item Which flights did J.Adams inspect last week? \label{prep:19.1} \item Which flights has J.Adams inspected this week? \label{prep:20} \end{examps} \pref{prep:16} -- \pref{prep:20} suggest that the present perfect can only be used if the time of the adverbial contains not only the time where the past situation occurred, but also the speech time, the time when the sentence was uttered. (A similar explanation is given on p.~167 of \cite{Thomson}.) \pref{prep:17} is felicitous, because \qit{today} contains the speech time. In contrast, \pref{prep:16} is unacceptable, because \qit{yesterday} cannot contain the speech time. The hypothesis, however, that the time of the adverbial must include the speech time does not account for the fact that \pref{prep:22} is acceptable by most native English speakers (especially if the \qit{ever} is added), even if the question is not uttered on a Sunday. \begin{examps} \item Has J.Adams (ever) inspected BA737 on a Sunday? \label{prep:22} \end{examps} As pointed out in \cite{Moens2}, a superstitious person could also utter \pref{prep:23} on a day other than Friday the 13th. \begin{examps} \item They have married on Friday 13th! \label{prep:23} \end{examps} One could attempt to formulate more elaborate restrictions, to predict exactly when temporal adverbials can be used with the present perfect. In the case of a \textsc{Nlitdb}\xspace, however, it is difficult to see why this would be worth the effort, as opposed to simply accepting questions like \pref{prep:16} as equivalent to \pref{prep:16.1}. I adopt the latter approach. Given that the framework of this thesis does not associate present consequences with the present perfect, that the still-holding reading of sentences like \pref{prep:16.10} is not supported, and that questions like \pref{prep:16} are allowed, there is not much left to distinguish the present perfect from the simple past. Hence, I treat the present perfect as equivalent to the simple past. \subsection{Past perfect} \label{past_perfect} The past perfect is often used to refer to a situation that occurred at some past time before some other past time. Following Reichenbach \cite{Reichenbach} and many others, let us call the latter time the \emph{reference time}. \pref{pap:1} and \pref{pap:4} have readings where \qit{at 5:00pm} specifies the reference time. In that case, \pref{pap:1} asks for flights that Airserve serviced before 5:00pm, and \pref{pap:4} asks if BA737 was at gate 2 some time before 5:00pm. (When expressing these meanings, \qit{by 5:00pm} is probably preferable. I claim, however, that \qit{at 5:00pm} can also be used in this way. \qit{By~\dots} adverbials are not examined in this thesis.) \begin{examps} \item Which flights had Airserve serviced at 5:00pm? \label{pap:1} \item Had BA737 been at gate 2 at 5:00pm? \label{pap:4} \end{examps} With culminating activity verbs, the climax must have been reached before (or possibly at) the reference time. For example, in \pref{pap:1} the services must have been completed up to 5:00pm. Perhaps some consequence of the past situation must still hold at the reference time (see similar comments about the present perfect in section \ref{present_perfect}). As with the present perfect, however, I ignore such consequential links. When the past perfect combines with temporal adverbials, it is often unclear if the adverbial is intended to specify the reference time or directly the time of the past situation. For example, \pref{pap:6} could mean that BA737 had already reached gate 2 at 5:00pm, or that it reached it at 5:00pm. In the latter case, \pref{pap:6} is similar to the simple past \pref{pap:6.1}, except that \pref{pap:6} creates the impression of a longer distance between the time of the reaching and the speech time. \begin{examps} \item BA737 had reached gate 2 at 5:00pm. \label{pap:6} \item BA737 reached gate 2 at 5:00pm. \label{pap:6.1} \end{examps} When the past perfect combines with \qit{for~\dots} duration adverbials and a reference time is specified, there is often an implication that the past situation still held at the reference time. (A similar implication arises in the case of the present perfect; see section \ref{present_perfect}.) For example, \pref{pap:8} seems to imply that J.Adams was still a manager on 1/1/94. As in the case of the present perfect, I ignore this implication, for reasons related to the difficulty of representing it in \textsc{Top}\xspace. \begin{examps} \item J.Adams had been a manager for two years on 1/1/94. \label{pap:8} \end{examps} As will be discussed in section \ref{temporal_adverbials}, I assume that the past perfect triggers an aspectual shift to state (e.g.\ the base form of \qit{to inspect} is a culminating activity, but \qit{had inspected} is a state). This shift seems to be a property of all perfect tenses. For reasons, however, related to the fact that I treat the present perfect as equivalent to the simple past (section \ref{present_perfect}), I do not postulate any shift in the case of the present perfect. \section{Special temporal verbs} \label{special_verbs} Through their tenses, all verbs can convey temporal information. Some verbs, however, like \qit{to begin} or \qit{to precede}, are of a more inherently temporal nature. These verbs differ from ordinary ones (e.g.\ \qit{to build}, \qit{to contain}) in that they do not describe directly situations, but rather refer to situations introduced by other verbs or nouns. (A similar observation is made in \cite{Passonneau}.) In \pref{spv:1}, for example, \qit{to begin} refers to the situation of \qit{to build}. \qit{To start}, \qit{to end}, \qit{to finish}, \qit{to follow}, \qit{to continue}, and \qit{to happen} all belong to this category of special temporal verbs. \begin{examps} \item They began to build terminal 9 in 1985. \label{spv:1} \end{examps} From all the special temporal verbs, I have considered only \qit{to start}, \qit{to begin}, \qit{to stop}, and \qit{to finish}. I allow \qit{to start}, \qit{to begin}, \qit{to end}, and \qit{to finish} to be used with state and activity verbs, even though with state verbs \qit{to begin} and \qit{to finish} usually sound unnatural (e.g.\ \pref{spv:11}), and with activity verbs (e.g.\ \pref{spv:13}) it could be argued that the use of \qit{to begin} or \qit{to finish} signals that the speaker has in mind a culminating activity (not activity) view of the situation. \begin{examps} \item \odd Which tank began to contain/finished containing water on 27/7/95? \label{spv:11} \item Which flight began/finished circling at 5:00pm? \label{spv:13} \end{examps} When combining with culminating activity verbs, \qit{to start} and \qit{to begin} have the same meanings. \qit{To stop} and \qit{to finish}, however, differ: \qit{to finish} requires the climax to be reached, while \qit{to stop} requires the action or change to simply stop (possibly without being completed). For example, in \pref{spv:20} the service must have simply stopped, while in \pref{spv:21} it must have been completed. \begin{examps} \item Which company stopped servicing (actually) BA737 at 5:00pm? \label{spv:20} \item Which company finished servicing (actually) BA737 at 5:00pm? \label{spv:21} \end{examps} With point verbs (like \qit{to enter} and \qit{to leave} in the airport domain), the use of \qit{to start}, \qit{to begin}, \qit{to stop}, and \qit{to finish} (e.g.\ \pref{spv:22}, \pref{spv:23}) typically signals that the person submitting the question is unaware that the situation of the point verb is taken to be instantaneous. In these cases, I ignore the temporal verbs (e.g.\ \pref{spv:22} is treated as \pref{spv:24}). Ideally, the \textsc{Nlitdb}\xspace would also warn the user that the temporal verb is ignored, and that the situation is modelled as instantaneous (another case for cooperative responses; see section \ref{no_issues}). The framework of this thesis, however, provides no mechanism for generating such warnings. \begin{examps} \item Which flight started to enter sector 2 at 5:00pm? \label{spv:22} \item Which flight finished leaving gate 2 at 5:00pm? \label{spv:23} \item Which flight entered sector 2 at 5:00pm? \label{spv:24} \end{examps} \section{Temporal nouns} \label{temporal_nouns} Some nouns have a special temporal nature. Nouns like \qit{development} or \qit{inspection}, for example, are similar to verbs, in that they introduce world situations that occur in time. The role of \qit{development} in \pref{tn:0} is very similar to that of \qit{to develop} in \pref{tn:0.1}. \begin{examps} \item When did the development of \textsc{Masque} start? \label{tn:0} \item When did they start to develop \textsc{Masque}? \label{tn:0.1} \end{examps} Other nouns indicate temporal order (e.g.\ \qit{predecessor}, \qit{successor}), or refer to start or end-points (e.g.\ \qit{beginning}, \qit{end}). Finally, many nouns (and proper names) refer to time periods, points, or generally entities of the temporal ontology (e.g.\ \qit{minute}, \qit{July}, \qit{event}). From all the temporal nouns (and proper names), this thesis examines only nouns like \qit{year}, \qit{month}, \qit{week}, \qit{day}, \qit{minute}, \qit{second}, \qit{1993}, \qit{July}, \qit{1/1/85}, \qit{Monday}, \qit{3:05pm}. Temporal nouns of a more abstract nature (e.g.\ \qit{period}, \qit{point}, \qit{interval}, \qit{event}, \qit{time}, \qit{duration}), nouns referring to start or end-points, nouns introducing situations, and nouns of temporal precedence are not considered. \section{Temporal adjectives} \label{temporal_adjectives} There are also adjectives of a special temporal nature. Some refer to a temporal order (e.g.\ \qit{current}, \qit{previous}, \qit{earliest}), others refer to durations (e.g.\ \qit{brief}, \qit{longer}), and others specify frequencies (e.g.\ \qit{annual}, \qit{daily}). Adjectives of this kind are not examined in this thesis, with the exception of \qit{current} which is supported to illustrate some points related to temporal anaphora (these points will be discussed in section \ref{noun_anaphora}). (``Normal'' adjectives, like \qit{open} and \qit{free}, are also supported.) \section{Temporal adverbials} \label{temporal_adverbials} This section discusses adverbials that convey temporal information. \subsection{Punctual adverbials} \label{point_adverbials} Some adverbials are understood as specifying time-points. Following \cite{Vlach1993}, I call these \emph{punctual adverbials}. In English, punctual adverbials are usually prepositional phrases introduced by \qit{at} (e.g.\ \qit{at 5:00pm}, \qit{at the end of the inspection}). In this thesis, only punctual adverbials consisting of \qit{at} and clock-time expressions (e.g.\ \qit{at 5:00pm}) are considered. \paragraph{With states:} When combining with state expressions, punctual adverbials specify a time where the situation of the state expression holds. There is usually no implication that the situation of the state starts or stops at the time of the adverbial. \pref{pa:1}, for example, asks if tank 5 was empty at 5:00pm. There is no requirement that the tank must have started or stopped being empty at 5:00pm. Similar comments apply to \pref{pa:2}. \begin{examps} \item Was tank 5 empty at 5:00pm? \label{pa:1} \item Which flight was at gate 2 at 5:00pm? \label{pa:2} \end{examps} In other words, with states punctual adverbials normally have an \emph{interjacent} meaning, not an \emph{inchoative} or \emph{terminal} one. (``Interjacent'', ``inchoative'', and ``terminal'' are borrowed from \cite{Kent}. Kent explores the behaviour of \qit{at}, \qit{for}, and \qit{in} adverbials, and arrives at conclusions similar to the ones presented here.) In narrative contexts, punctual adverbials combining with states sometimes have inchoative meanings. For example, the \qit{at 8:10am} in \pref{pa:4} most probably specifies the time when J.Adams arrived (started being) in Glasgow. In \textsc{Nlitdb}\xspace questions, however, this inchoative meaning seems unlikely. For example, it seems unlikely that \pref{pa:5} would be used to enquire about persons that \emph{arrived} in Glasgow at 8:10am. Hence, for the purposes of this thesis, it seems reasonable to assume that punctual adverbials combining with states always have interjacent meanings. \begin{examps} \item J.Adams left Edinburgh early in the morning, and at 8:10am he was in Glasgow. \label{pa:4} \item Who was in Glasgow at 8:10am? \label{pa:5} \end{examps} \paragraph{With points:} With point expressions, punctual adverbials specify the time where the instantaneous situation of the point expression takes place (e.g.\ \pref{pa:8}, \pref{pa:9}; \qit{to enter} and \qit{to reach} are point verbs in the airport domain). \begin{examps} \item Which flight entered sector 2 at 23:02? \label{pa:8} \item Which flight reached gate 5 at 23:02? \label{pa:9} \end{examps} \pref{pa:10} is ambiguous. It may either involve the non-habitual homonym of \qit{to depart} (this homonym is a point verb in the airport domain), in which case 5:00pm is the time of the actual departure, or the state habitual homonym (to depart habitually at some time), in which case 5:00pm is the habitual departure time. In the latter case, I treat \qit{at 5:00pm} as a prepositional phrase complement of the habitual \qit{to depart}, not as a temporal adverbial. This reflects the fact that the \qit{at 5:00pm} does not specify a time when the habit holds, but it forms part of the description of the habit, i.e.\ it is used in a way very similar to how \qit{from gate 2} is used in \pref{pa:3}. \begin{examps} \item Which flight departed at 5:00pm? \label{pa:10} \item Which flight departed (habitually) from gate 2 (last year)? \label{pa:3} \end{examps} \paragraph{With activities:} With activities, punctual adverbials usually have an inchoative meaning, but an interjacent one is also possible in some cases. \pref{pa:11}, for example, could refer to a flight that either joined the queue of runway 2 at 5:00pm or was simply in the queue at 5:00pm. (In the airport domain, \qit{to queue} and \qit{to taxi} (no destination) are activity verbs.) The inchoative meaning seems the preferred one in \pref{pa:11}. It also seems the preferred one in \pref{pa:13}, though an interjacent meaning is (arguably) also possible. The interjacent meaning is easier to accept in \pref{pa:14}. \begin{examps} \item Which flight queued for runway 2 at 5:00pm? \label{pa:11} \item BA737 taxied at 5:00pm. \label{pa:13} \item Which flights circled at 5:00pm? \label{pa:14} \end{examps} With past continuous forms of activity verbs, however, punctual adverbials normally have only interjacent meanings (compare \pref{pa:11} and \pref{pa:13} to \pref{pa:17} and \pref{pa:19}). (One would not normally use punctual adverbials with present continuous forms, since in that case the situation is known to take place at the present.) \begin{examps} \item Which flight was queueing for runway 2 at 5:00pm? \label{pa:17} \item BA737 was taxiing at 5:00pm. \label{pa:19} \end{examps} To account for sentences like \pref{pa:17} and \pref{pa:19} (and other phenomena to be discussed in following sections), I classify the progressive tenses (present and past continuous) of activity and culminating activity verbs as states. For example, in the airport domain, the base form of \qit{to queue} is an activity. Normally, all other forms of the verb (e.g.\ the simple past) inherit the aspectual class of the base form. The progressive tenses (e.g.\ \qit{is queueing}, \qit{was queueing}) of the verb, however, are states. (The progressive can be seen as forcing an aspectual shift from activities or culminating activities to states. No such aspectual shift is needed in the case of point verbs.) This arrangement, along with the assumption that punctual adverbials combining with states have only interjacent meanings, accounts for the fact that \pref{pa:17} and \pref{pa:19} have only interjacent meanings. In various forms, assumptions that progressive tenses cause aspectual shifts to states have also been used in \cite{Dowty1986}, \cite{Moens}, \cite{Vlach1993}, \cite{Kent}, and elsewhere. \paragraph{With culminating activities:} When combining with culminating activities, punctual adverbials usually have inchoative or terminal meanings (when they have terminal meanings, they specify the time when the climax was reached). The terminal reading is the preferred one in \pref{pa:23}. In \pref{pa:25} both readings seem possible. In \pref{pa:24} the inchoative meaning seems the preferred one. (In the airport domain, \qit{to land}, \qit{to taxi} (to destination), and \qit{to inspect} are culminating activity verbs.) \begin{examps} \item Which flight landed at 5:00pm? \label{pa:23} \item Which flight taxied to gate 4 at 5:00pm? \label{pa:25} \item Who inspected BA737 at 5:00pm? \label{pa:24} \end{examps} Perhaps, as with activities, an interjacent meaning is sometimes also possible (e.g.\ \pref{pa:25} would refer to a flight that was on its way to gate 4 at 5:00pm). This may be true, but with culminating activities the inchoative or terminal reading is usually much more dominant. For simplicity, I ignore the possible interjacent meaning in the case of culminating activities. With past continuous forms of culminating activity verbs, punctual adverbials normally have only interjacent meanings. Compare, for example, \pref{pa:23} -- \pref{pa:24} to \pref{pa:28} -- \pref{pa:30}. This is in accordance with the assumption that the progressive tenses of activity and culminating activity verbs are states. \begin{examps} \item Which flight was landing at 5:00pm? \label{pa:28} \item Which flight was taxiing to gate 4 at 5:00pm? \label{pa:29} \item Who was inspecting BA737 at 5:00pm? \label{pa:30} \end{examps} \paragraph{With past perfect:} As discussed in section \ref{past_perfect}, in sentences like \pref{pa:31} the adverbial can be taken to refer either directly to the taxiing (the taxiing started or ended at 5:00pm) or to the reference time (the taxiing had already finished at 5:00pm). \begin{examps} \item BA737 had taxied to gate 2 at 5:00pm. \label{pa:31} \item BA737 had [taxied to gate 2 at 5:00pm]. \label{pa:32} \item BA737 [had taxied to gate 2] at 5:00pm. \label{pa:33} \end{examps} The way in which sentences like \pref{pa:31} are treated in this thesis will become clearer in chapter \ref{TOP_chapter}. A rough description, however, can be given here. \pref{pa:31} is treated as syntactically ambiguous between \pref{pa:32} (where the adverbial applies to the past participle \qit{taxied}) and \pref{pa:33} (where the adverbial applies to the past perfect \qit{had taxied}). The past participle (\qit{taxied}) inherits the aspectual class of the base form, and refers directly to the situation of the verb (the taxiing). In contrast, the past perfect (\qit{had taxied}) is always classified as state (the past perfect can be seen as causing an aspectual shift to state), and refers to a time-period that starts immediately after the end of the situation of the past participle (the end of the taxiing), and extends up to the end of time. Let us call this period the \emph{consequent period}. In the airport domain, the base form of \qit{to taxi} (to destination) is a culminating activity. Hence, the past participle \qit{taxied} (which refers directly to the taxiing) is also a culminating activity. In \pref{pa:32}, a punctual adverbial combines with the (culminating activity) past participle. According to the discussion above, two readings arise: an inchoative (the taxiing started at 5:00pm) and a terminal one (the taxiing finished at 5:00pm). In contrast, in \pref{pa:33} the punctual adverbial combines with the past perfect \qit{had taxied}, which is a state expression that refers to the consequent period. Hence, only an interjacent reading arises: the time of the adverbial must simply be within the consequent period (there is no need for the adverbial's time to be the beginning or end of the consequent period). This requires the taxiing to have been completed at the time of the adverbial. A similar arrangement is used when the past perfect combines with period adverbials, duration \qit{for~\dots} and \qit{in~\dots} adverbials, or temporal subordinate clauses (to be discussed in following sections). The assumption that the past perfect causes an aspectual shift to state is similar to claims in \cite{Moens}, \cite{Vlach1993}, and elsewhere, that English perfect forms are (or refer to) states. \paragraph{Lexical, consequent, and progressive states:} There is sometimes a need to distinguish between expressions that are states because they have inherited the state aspectual class of a base verb form, and expressions that are states because of an aspectual shift introduced by a past perfect or a progressive tense. Following \cite{Moens}, I use the terms \emph{lexical state}, \emph{consequent state}, and \emph{progressive state} to distinguish the three genres. In the airport domain, for example, the base form of \qit{to queue} is a lexical state. The simple past \qit{queued} and the past participle \qit{queued} are also lexical states. The past perfect \qit{had queued} is a consequent state, while the present continuous form \qit{is queueing} is a progressive state. Finally, for reasons that will be discussed in section \ref{hpsg:mult_mods}, I assume that punctual adverbials cause the aspectual class of the syntactic constituent they modify to become point. In \pref{pa:33}, for example, the \qit{taxied to gate 2} inherits the culminating activity aspectual class of the base form. The past perfect causes the aspectual class of \qit{had taxied to gate 2} to become consequent state. Finally, the \qit{at 5:00pm} causes the aspectual class of \qit{had departed at 5:00pm} to become point. Table \ref{punctual_adverbials_table} summarises the main points of this section. \begin{table} \begin{center} {\small \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{meanings of punctual adverbials} \\ \hline \hline with state & interjacent \\ \hline with activity & inchoative or interjacent \\ \hline with culm.\ activity & inchoative or terminal \\ \hline with point & specifies time of instantaneous situation \\ \hline \hline \multicolumn{2}{|l|}{The resulting aspectual class is point.}\\ \hline \end{tabular} } \end{center} \caption{Punctual adverbials in the framework of this thesis} \label{punctual_adverbials_table} \end{table} \subsection{Period adverbials} \label{period_adverbials} Unlike punctual adverbials, which are understood as specifying points in time, adverbials like \qit{in 1991}, \qit{on Monday}, \qit{yesterday} (e.g.\ \pref{padv:1} -- \pref{padv:2}) are usually understood as specifying longer, non-instantaneous periods of time. In \pref{padv:1}, for example, the period of \qit{in 1991} covers the whole 1991. I call adverbials of this kind \emph{period adverbials}. \begin{examps} \item Who was the sales manager in 1991? \label{padv:1} \item Did BA737 circle on Monday? \label{padv:3} \item Which flights did J.Adams inspect yesterday? \label{padv:2} \end{examps} \qit{Before~\dots} and \qit{after~\dots} adverbials (e.g.\ \pref{padv:3.2}) can also be considered period adverbials, except that in this case one of the boundaries of the period is left unspecified. (I model time as bounded; see section \ref{temporal_ontology} below. In the absence of other constraints, I treat the unspecified boundary as the beginning or end of time.) In \pref{padv:3.2}, for example, the end of the period is the beginning of 2/5/95; the beginning of the period is left unspecified. \qit{Before} and \qit{after} can also introduce temporal subordinate clauses; this will be discussed in section \ref{before_after_clauses}. \begin{examps} \item Which company serviced BA737 before 2/5/95? \label{padv:3.2} \end{examps} This thesis examines only period adverbials introduced by \qit{in}, \qit{on}, \qit{before}, and \qit{after}, as well as \qit{today} and \qit{yesterday}. (\qit{In~\dots} adverbials can also specify durations, e.g.\ \qit{in two hours}; this will be discussed in section \ref{in_adverbials}.) Other period adverbials, like \qit{from 1989 to 1990}, \qit{since 1990}, \qit{last week}, or \qit{two days ago}, are not considered. Extending the framework of this thesis to support more period adverbials should not be difficult. \paragraph{With states:} When period adverbials combine with state expressions, the situation of the state expression must hold for at least some time during the period of the adverbial. In \pref{padv:10}, for example, the person must have been a manager for at least some time in 1995. Similarly, in \pref{padv:11}, the person must have been at gate 2 for at least some time on the previous day. \begin{examps} \item Who was a manager in 1995? \label{padv:10} \item Who was at gate 2 yesterday? \label{padv:11} \end{examps} There is often, however, an implication that the situation holds \emph{throughout} the period of the adverbial. \pref{padv:13}, for example, could mean that the tank was empty throughout January, not at simply some part of January. Similarly, in \pref{padv:12} the user could be referring to tanks that were empty \emph{throughout} January. In that case, if a tank was empty only some days in January and the \textsc{Nlitdb}\xspace included that tank in the answer, the user would be misled to believe that the tank was empty throughout January. Similar comments can be made for \pref{padv:13.9} and \pref{padv:15}. \begin{examps} \item Tank 4 was empty in January. \label{padv:13} \item Which tanks were empty in January? \label{padv:12} \item Was runway 2 open on 6/7/95? \label{padv:13.9} \item Which flights departed (habitually) from gate 2 in 1993? \label{padv:15} \end{examps} The same implication is possible in sentences with \qit{before~\dots} or \qit{after~\dots} adverbials. \pref{padv:20.1}, for example, could mean that the runway was open all the time from some unspecified time up to immediately before 5:00pm (and possibly longer). \begin{examps} \item Runway 2 was open before 5:00pm. \label{padv:20.1} \end{examps} One way to deal with such implications is to treat sentences where period adverbials combine with states as ambiguous. That is, to distinguish between a reading where the situation holds throughout the adverbial's period, and a reading where the situation holds at simply some part of the adverbial's period. \cite{Vlach1993} (p.~256) uses the terms \emph{durative} and \emph{inclusive} to refer to the two readings. (A \textsc{Nlitdb}\xspace could paraphrase both readings and ask the user to select one, or it could provide answers to both readings, indicating which answer corresponds to which reading.) This approach has the disadvantage of always generating two readings, even in cases where the durative reading is clearly impossible. For example, when the state expression combines not only with a period adverbial but also with a \qit{for~\dots} duration adverbial, the meaning can never be that the situation must necessarily hold all the time of the adverbial's period. For example, \pref{padv:29} can never mean that the tank must have been empty throughout January (cf.\ \pref{padv:12}). \begin{examps} \item Which tank was empty for two days in January? \label{padv:29} \item When on 6/7/95 was tank 5 empty? \label{padv:27} \end{examps} Similarly, in time-asking questions like \pref{padv:27}, the durative reading is impossible. \pref{padv:27} can never mean that the tank must have been empty throughout 6/7/95 (cf.\ \pref{padv:13.9}). Formulating an account of exactly when the durative reading is possible is a task which I have not undertaken. Although in chapter \ref{TOP_chapter} I discuss how the distinction between durative and inclusive readings could be captured in \textsc{Top}\xspace, for simplicity in the rest of this thesis I consider only the inclusive readings, ignoring the durative ones. \paragraph{With points:} When period adverbials combine with point expressions, the period of the adverbial must contain the time where the instantaneous situation of the point expression occurs (e.g.\ \pref{padv:62}). \begin{examps} \item Did BA737 enter sector 5 on Monday? \label{padv:62} \end{examps} \paragraph{With culminating activities:} When period adverbials combine with culminating activity expressions, I allow two possible readings: (a) that the situation of the culminating activity expression both starts and reaches its completion within the adverbial's period, or (b) that the situation simply reaches its completion within the adverbial's period. In the second reading, I treat the culminating activity expression as referring to only the completion of the situation it would normally describe, and the aspectual class is changed to point. The first reading is the preferred one in \pref{padv:42} which is most naturally understood as referring to a runner who both started and finished running the 40 miles on Wednesday (\qit{to run} (a distance) is typically classified as culminating activity verb). \begin{examps} \item Who ran 40 miles on Wednesday? \label{padv:42} \end{examps} In the airport domain, the first reading is the preferred one in \pref{padv:47.1} (the inspection both started and was completed on Monday). \begin{examps} \item J.Adams inspected BA737 on Monday. \label{padv:47.1} \end{examps} The second reading (the situation simply reaches its completion within the adverbial's period) is needed in questions like \pref{padv:48} and \pref{padv:49}. In the airport domain, \qit{to land} and \qit{to take off} are culminating activity verbs (landings and taking offs involve following particular procedures; the landing or taking off starts when the pilot starts the corresponding procedure, and is completed when that procedure is completed). If only the first reading were available (both start and completion within the adverbial's period), in \pref{padv:48} the \textsc{Nlitdb}\xspace would report only flights that both started and finished landing on Monday. If a flight started the landing procedure at 23:55 on Sunday and finished it at 00:05 on Monday, that flight would not be reported. This seems over-restrictive. In \pref{padv:48} the most natural reading is that the flights must have simply touched down on Monday, i.e.\ the landing must have simply been completed within Monday. Similar comments can be made for \pref{padv:49} and \pref{padv:52} (in domains where \qit{to fix} is a culminating activity verb). \begin{examps} \item Which flights landed on Monday? \label{padv:48} \item Which flights took off after 5:00pm? \label{padv:49} \item Did J.Adams fix any faults yesterday? \label{padv:52} \end{examps} The problem in these cases is that \qit{to land}, \qit{to take off}, and \qit{to fix} need to be treated as point verbs (referring to only the time-points where the corresponding situations are completed), even though they have been classified as culminating activity verbs (section \ref{aspect_examples}). The second reading allows exactly this. The culminating activity expression is taken to refer to only the completion point of the situation it would normally describe, its aspectual class is changed to point, and the completion point is required to fall within the adverbial's period. The fact that two readings are allowed when period adverbials combine with culminating activities means that sentences like \pref{padv:47.1} -- \pref{padv:52} are treated as ambiguous. In all ambiguous sentences, I assume that a \textsc{Nlitdb}\xspace would present all readings to the user asking them to choose one, or that it would provide answers to all readings, showing which answer corresponds to which reading. (The prototype \textsc{Nlitdb}\xspace of this thesis adopts the second strategy, though the mechanism for explaining which answer corresponds to which reading is primitive: the readings are shown as \textsc{Top}\xspace formulae.) In the case of \qit{before~\dots} adverbials (e.g.\ \pref{padv:49b}), the two readings are semantically equivalent: requiring the situation to simply reach its completion before some time is equivalent to requiring the situation to both start and reach its completion before that time. To avoid generating two equivalent readings, I allow only the reading where the situation both starts and reaches its completion within the adverbial's period. \begin{examps} \item Which flights took off before 5:00pm? \label{padv:49b} \end{examps} Even with the second reading, the answers of the \textsc{Nlitdb}\xspace may not always be satisfactory. Let us assume, for example, that J.Adams started inspecting a flight late on Monday, and finished the inspection early on Tuesday. None of the two readings would include that flight in the answer to \pref{padv:47}, because both require the completion point to fall on Monday. While strictly speaking this seems correct, it would be better if the \textsc{Nlitdb}\xspace could also include in the answer inspections that \emph{partially} overlap the adverbial's period, warning the user about the fact that these inspections are not completely contained in the adverbial's period. This is another case where cooperative responses (section \ref{no_issues}) are needed. \begin{examps} \item Which flights did J.Adams inspect on Monday? \label{padv:47} \end{examps} Finally, I note that although in the airport domain \qit{to taxi} (to destination) is a culminating activity verb, in \pref{padv:61.7} the verb form is a (progressive) state. Hence, the \textsc{Nlitdb}\xspace's answer would be affirmative if BA737 was taxiing to gate 2 some time within the adverbial's period (before 5:00pm), even if BA737 did not reach the gate during that period. This captures correctly the most natural reading of \pref{padv:61.7}. \begin{examps} \item Was BA737 taxiing to gate 2 before 5:00pm? \label{padv:61.7} \end{examps} \paragraph{With activities:} When period adverbials combine with activities, I require the situation of the verb to hold for at least some time within the adverbial's period (same meaning as with states). In \pref{padv:66}, for example, the flight must have circled for at least some time on Monday, and in \pref{padv:67} the flights must have taxied for at least some time after 5:00pm. \begin{examps} \item Did BA737 circle on Monday? \label{padv:66} \item Which flights taxied after 5:00pm? \label{padv:67} \end{examps} Another stricter reading is sometimes possible (especially with \qit{before} and \qit{after}): that the situation does not extend past the boundaries of the adverbial's period. For example, \pref{padv:67} would refer to flights that \emph{started} to taxi after 5:00pm (a flight that started to taxi at 4:55pm and continued to taxi until 5:05pm would not be reported). This reading is perhaps also possible with states (e.g.\ \pref{padv:20.1}), though with activities it seems easier to accept. As a simplification, such readings are ignored in this thesis. \paragraph{Elliptical forms:} \qit{Before} and \qit{after} are sometimes followed by noun phrases that do not denote entities of the temporal ontology (e.g.\ \pref{padv:71}). \begin{examps} \item Did J.Adams inspect BA737 before/after UK160? \label{padv:71} \item Did J.Adams inspect BA737 before/after he inspected UK160? \label{padv:71.1} \end{examps} Questions like \pref{padv:71} can be considered elliptical forms of \pref{padv:71.1}, i.e.\ in these cases \qit{before} and \qit{after} could be treated as when they introduce subordinate clauses (section \ref{before_after_clauses} below). Questions like \pref{padv:71} are currently not supported by the framework of this thesis. Table \ref{period_adverbials_table} summarises the main points of this section. \begin{table} \begin{center} {\small \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{meanings of period adverbials} \\ \hline \hline with state or activity & situation holds for at least part of adverbial's period \\ \hline with culm.\ activity & situation starts and is completed within adverbial's period,\\ & or situation is simply completed within adverbial's period$^{*\dagger}$\\ \hline with point & instantaneous situation occurs within adverbial's period \\ \hline \hline \multicolumn{2}{|l|}{$^*$Not with \qit{before~\dots} adverbials.} \\ \multicolumn{2}{|l|}{$^{\dagger}$The resulting aspectual class is point. (In all other cases the aspectual class} \\ \multicolumn{2}{|l|}{\ \ remains the same.)}\\ \hline \end{tabular} } \end{center} \caption{Period adverbials in the framework of this thesis} \label{period_adverbials_table} \end{table} \subsection{Duration \qit{for~\dots} adverbials} \label{for_adverbials} This section discusses \qit{for~\dots} adverbials that specify durations (e.g.\ \pref{dura:1}). \begin{examps} \item Runway 2 was open for five days. \label{dura:1} \end{examps} \paragraph{With states and activities:} When \qit{for~\dots} adverbials combine with states or activities, one reading is that there must be a period with the duration of the \qit{for~\dots} adverbial, such that the situation of the state or activity holds throughout that period. According to this reading, in \pref{dura:3} there must be a five-year period, throughout which the person was a manager, and in \pref{dura:3} a twenty-minute period throughout which the flight was circling. If J.Adams was a manager for six consecutive years (e.g.\ 1981 -- 1986), he would be included in the answer to \pref{dura:3}, because there is a five-year period (e.g.\ 1981 -- 1985) throughout which he was a manager. \begin{examps} \item Who was a manager for five years? \label{dura:3} \item Did BA737 circle for twenty minutes? \label{dura:4} \end{examps} In some cases, however, \qit{for~\dots} adverbials are used with a stricter meaning: they specify the duration of a \emph{maximal} period where a situation held. In that case, if J.Adams started to be a manager at the beginning of 1981 and stopped being a manager at the end of 1986 (six consecutive years), he would \emph{not} be included in the answer to \pref{dura:3}. For simplicity, this stricter reading is ignored in this thesis . In other cases, a \qit{for~\dots} adverbial does not necessarily specify the duration of a \emph{single} period, but a \emph{total duration}. According to this reading, if J.Adams was a manager during several non-overlapping periods, and the total duration of these periods is five years, he would be included in the answer to \pref{dura:3}, even if he was never a manager for a continuous five-year period. This reading of \qit{for} adverbials is also not supported in this thesis. There is a problem if \qit{for~\dots} adverbials are allowed to combine with consequent states (section \ref{point_adverbials}). This problem will be discussed in section \ref{duration_adverbials}, once some formal apparatus has been established. For the moment, I note that the solution involves disallowing \qit{for~\dots} adverbials to be used with consequent states. \paragraph{With points:} \qit{For~\dots} adverbials sometimes specify the duration of a situation that \emph{follows} the situation of the verb. This is particularly common when \qit{for~\dots} adverbials combine with point expressions. For instance, \pref{dura:11} (based on an example from \cite{Hwang1994}), probably does not mean that J.Adams was actually leaving his office for fifteen minutes. It means that he stayed (or intended to stay) out of his office for fifteen minutes. (I assume here that \qit{to leave} is a point verb, as in the airport domain.) This use of \qit{for~\dots} adverbials is not supported in this thesis. \begin{examps} \item J.Adams left his office for fifteen minutes. \label{dura:11} \end{examps} \qit{For~\dots} adverbials also give rise to iterative readings (section \ref{progressives}). This is again particularly common with point expressions. \pref{dura:12} (from \cite{Hwang1994}) probably means that Mary won several times (\qit{to win} is typically classified as point verb). Such iterative uses of \qit{for~\dots} adverbials are not supported in this thesis. \begin{examps} \item Mary won the competition for four years. \label{dura:12} \end{examps} Excluding iterative readings and readings where \qit{for~\dots} adverbials refer to consequent situations (both are not supported in this thesis), sentences where \qit{for~\dots} adverbials combine with point expressions either sound odd or signal that the user is unaware that the situation of the point expression is modelled as instantaneous (an explanatory message to the user is needed in the latter case; this thesis, however, provides no mechanism to generate such messages). Hence, for the purposes of this thesis it seems reasonable not to allow \qit{for~\dots} adverbials to combine with point expressions. \paragraph{With culminating activities:} When \qit{for~\dots} adverbials combine with culminating activities, the resulting sentences sometimes sound odd or even unacceptable. For example, \pref{dura:38} (based on an example from \cite{Moens}) sounds odd or unacceptable to most native English speakers (\qit{to build} is typically classified as culminating activity verb). In contrast, \pref{dura:37} where the adverbial combines with a (progressive) state is easily acceptable. \begin{examps} \item \odd Housecorp built a shopping centre for two years. \label{dura:38} \item Housecorp was building a shopping centre for two years. \label{dura:37} \end{examps} Based on similar examples, Vendler (section \ref{asp_taxes}) concludes that accomplishments (culminating activities) do not combine with \qit{for~\dots} adverbials. This, however, seems over-restrictive. \pref{dura:40} and \pref{dura:42}, for example, seem acceptable. \begin{examps} \item BA737 taxied to gate 2 for two minutes. \label{dura:40} \item Did J.Adams inspect BA737 for ten minutes? \label{dura:42} \end{examps} Unlike \pref{dura:43}, in \pref{dura:40} there is no requirement that the taxiing must have been completed, i.e.\ that BA737 must have reached the gate. Similar comments can be made for \pref{dura:42} and \pref{dura:45}. \qit{For~\dots} adverbials seem to cancel any requirement that the climax must have been reached. (Similar observations are made in \cite{Dowty1986}, \cite{Moens2}, and \cite{Kent}.) \begin{examps} \item BA737 taxied to gate 2. \label{dura:43} \item Did J.Adams inspect BA737? \label{dura:45} \end{examps} In the framework of this thesis, I allow \qit{for~\dots} adverbials to combine with culminating activities, with the same meaning that I adopted in the case of states and activities, and with the proviso that any requirement that the climax must have been reached should be cancelled. That is, in \pref{dura:42} there must be a ten-minute period throughout which J.Adams was inspecting BA737. Table \ref{for_adverbials_table} summarises the main points of this section. \begin{table} \begin{center} {\small \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{meanings of duration \qit{for~\dots} adverbials} \\ \hline \hline with lexical or progressive state & situation holds continuously for at least that long\\ \hline with consequent state & (not allowed in the framework of this thesis) \\ \hline with activity & situation holds continuously for at least that long \\ \hline with culminating activity & situation holds continuously for at least that long \\ & (no need for climax to be reached)\\ \hline with point & (not allowed in the framework of this thesis) \\ \hline \end{tabular} } \end{center} \caption{Duration \qit{for~\dots} adverbials in the framework of this thesis} \label{for_adverbials_table} \end{table} \subsection{Duration \qit{in~\dots} adverbials} \label{in_adverbials} This section discusses \qit{in~\dots} adverbials that specify durations (e.g.\ \pref{inad:1}, \pref{inad:2}). \qit{In} can also introduce period adverbials (e.g.\ \qit{in 1995}; see section section \ref{period_adverbials}). \begin{examps} \item Airserve serviced BA737 in two hours. \label{inad:1} \item Which flight did J.Adams inspect in one hour? \label{inad:2} \end{examps} \paragraph{With culminating activities:} With culminating activity expressions, \qit{in~\dots} adverbials usually specify the length of a period that ends at the time-point where the situation of the culminating activity expression is completed. In \pref{inad:1}, for example, two hours is probably the length of a period that ends at the time-point where the service was completed. \pref{inad:2} is similar. The period whose length is specified by the \qit{in~\dots} adverbial usually starts at the time-point where the situation of the culminating activity expression begins. In \pref{inad:1}, for example, the two hours probably start at the time-point where the service began. The period of the adverbial, however, may sometimes not start at the beginning of the situation of the culminating activity expression, but at some other earlier time. In \pref{inad:1}, the start of the two hours could be the time-point where Airserve was asked to service BA737, not the beginning of the actual service. The framework of this thesis supports only the case where the period of the adverbial starts at the beginning of the situation described by the culminating activity expression. \paragraph{With points:} With point expressions, the period of the \qit{in~\dots} adverbial starts before the (instantaneous) situation of the point expression, and ends at the time-point where the situation of the point expression occurs. In \pref{inad:10} the ten minutes end at the point where BA737 arrived at gate 2, and start at some earlier time-point (e.g.\ when BA737 started to taxi to gate 2). \pref{inad:11} is similar. \begin{examps} \item BA737 reached gate 2 in ten minutes. \label{inad:10} \item BA737 entered sector 2 in five minutes. \label{inad:11} \end{examps} Determining exactly when the period of the adverbial starts is often difficult. It is not clear, for example, when the five minutes of \pref{inad:11} start. As a simplification, I do not allow duration \qit{in~\dots} adverbials to combine with point expressions. \paragraph{With states and activities:} \qit{In~\dots} adverbials are sometimes used with activity expressions, with the \qit{in~\dots} duration adverbial intended to specify the duration of the situation described by the activity expression. Typically, in these cases the speaker has a culminating activity view in mind. For example, \pref{inad:17} can be used in this way if the speaker has a particular destination (say gate 2) in mind. In that case, \pref{inad:17} can be thought as an elliptical form of \pref{inad:19}. The framework of this thesis does not support this use of \pref{inad:17}. \begin{examps} \item BA737 taxied in ten minutes. \label{inad:17} \item BA737 taxied to gate 2 in ten minutes. \label{inad:19} \end{examps} With state and activity expressions, \qit{in~\dots} adverbials can also specify the duration of a period that ends at the beginning of the situation of the state or activity expression. In \pref{inad:5}, for example, the two hours probably end at the time-point where tank 5 started to be empty. The beginning of the two hours could be, for example, a time-point where a pump started to empty the tank, or a time-point where a decision to empty the tank was taken. Similar comments apply to \pref{inad:17}. \begin{examps} \item Tank 5 was empty in two hours. \label{inad:5} \end{examps} As with point expressions, determining exactly when the period of the adverbial starts is often difficult. As a simplification, I do not allow duration \qit{in~\dots} adverbials to combine with state or activity expressions. Table \ref{in_adverbials_table} summarises the main points of this section. \begin{table} \begin{center} {\small \begin{tabular}{|l|l|} \hline \multicolumn{2}{|c|}{meanings of duration ``in \dots'' adverbials} \\ \hline \hline with state, activity, or point & (not allowed in the framework of this thesis) \\ \hline with culminating activity & distance from the start to the completion of the situation \\ \hline \end{tabular} } \end{center} \caption{Duration \qit{in~\dots} adverbials in the framework of this thesis} \label{in_adverbials_table} \end{table} \subsection{Other temporal adverbials} \label{other_adverbials} Other temporal adverbials, that are not supported by the framework of this thesis, include some adverbials that specify boundaries (e.g.\ \qit{until 1/5/95}, \qit{since 1987}, \qit{by Monday}), frequency adverbials (\qit{always}, \qit{twice}, \qit{every Monday}), and adverbials of temporal order (\qit{for the second time}, \qit{earlier}). \section{Temporal subordinate clauses} \label{subordinate_clauses} Three kinds of temporal subordinate clauses are examined in this thesis: clauses introduced by \qit{while}, \qit{before}, and \qit{after} (e.g.\ clauses introduced by \qit{since}, \qit{until}, or \qit{when} are not examined). From the temporal subordinate clauses that are not examined, \qit{when~\dots} clauses are generally considered the most difficult to support (see \cite{Ritchie}, \cite{Yip1985}, \cite{Hinrichs1986}, \cite{Moens}, \cite{Moens2}, and \cite{Lascarides1993} for explorations of \qit{when~\dots} clauses). \subsection{\qit{While~\dots} clauses} \label{while_clauses} \paragraph{Subordinate clause:} As with period adverbials (section \ref{period_adverbials}), each \qit{while~\dots} clause is understood as specifying a time period. This is a maximal period throughout which the situation of the \qit{while~\dots} clause holds. Let us assume, for example, that J.Adams was a manager only from 1/1/1980 to 31/12/1983, and from 1/1/1987 to 31/12/1990. Then, in \pref{whc:1} the period of the \qit{while~\dots} clause can be either one of these two periods. The user may have in mind a particular one of the two periods. In that case, a temporal anaphora resolution mechanism is needed to determine that period (temporal anaphora is discussed in section \ref{temporal_anaphora}). The framework of this thesis, however, provides no such mechanism (the answer to \pref{whc:1} includes anybody who was fired during any of the two periods). \begin{examps} \item Who was fired while J.Adams was a manager? \label{whc:1} \end{examps} Sentences where the aspectual class of the \qit{while~\dots} clause is point (e.g.\ \pref{whc:4} in the airport domain) typically signal that the user is unaware that the situation of the \qit{while~\dots} clause is modelled as instantaneous. In the framework of this thesis, the answer to \pref{whc:4} includes any flight that was circling at the time-point where BA737 entered sector 2. Ideally, a message would also be generated to warn the user that entering a sector is modelled as instantaneous (no warning is currently generated). This is another case where cooperative responses (section \ref{no_issues}) are needed. \begin{examps} \item Which flights were circling while BA737 entered sector 2? \label{whc:4} \end{examps} Sentences containing \qit{while~\dots} clauses whose aspectual class is consequent state (section \ref{point_adverbials}) usually sound unnatural or unacceptable. For example, \pref{whc:5.2} -- \pref{whc:5.8} sound at least unnatural (e.g.\ instead of \pref{whc:5.2} one would normally use \pref{whc:5.10} or \pref{whc:5.11}). Hence, I do not allow \qit{while~\dots} clauses whose aspectual class is consequent state. This also avoids some complications in the English to \textsc{Top}\xspace mapping. \begin{examps} \item \odd Did any flight depart while BA737 had landed? \label{whc:5.2} \item \odd Did ABM fire anybody while J.Adams had been the manager? \label{whc:5.5} \item \odd Had any flight departed while J.Adams had inspected BA737? \label{whc:5.8} \item Did any flight depart while BA737 was landing? \label{whc:5.10} \item Did any flight depart after BA737 had landed? \label{whc:5.11} \label{whc:5.12} \end{examps} When the aspectual class of the \qit{while~\dots} clause is culminating activity, there is no requirement that the climax of the situation of the \qit{while~\dots} clause must have been reached, even if the tense of that clause normally requires this. In \pref{whc:8} and \pref{whc:6}, for example, there does not seem to be any requirement that the service or the boarding must have been completed (cf.\ \pref{whc:10} and \pref{whc:11}). \pref{whc:8} and \pref{whc:6} appear to have the same meanings as \pref{whc:9} and \pref{whc:7} (in progressive tenses, there is no requirement for the climax to be reached; see section \ref{progressives}). Table \ref{while_clauses_table} summarises the main points about \qit{while~\dots} clauses so far. \begin{examps} \item Did Airserve service BA737? \label{whc:10} \item Which flights departed while Airserve serviced BA737? \label{whc:8} \item Which flights departed while Airserve was servicing BA737? \label{whc:9} \item Did BA737 board? \label{whc:11} \item Which flights departed while BA737 boarded? \label{whc:6} \item Which flights departed while BA737 was boarding? \label{whc:7} \end{examps} \begin{table} \begin{center} {\small \begin{tabular}{|l|l|} \hline aspectual class of & \\ \qit{while~\dots} clause & period specified by \qit{while~\dots} clause \\ \hline \hline consequent state & (not allowed in the framework of this thesis) \\ \hline lexical/progressive state & maximal period where situation of \qit{while~\dots} clause holds \\ or activity & \\ \hline culminating activity & maximal period where situation of \qit{while~\dots} clause holds \\ & (no need for climax of \qit{while~\dots} clause to be reached) \\ \hline point & instant.\ period where situation of \qit{where~\dots} clause occurs \\ \hline \end{tabular} } \end{center} \caption{Periods of \qit{while~\dots} clauses in the framework of this thesis} \label{while_clauses_table} \end{table} \paragraph{Main clause:} Once the periods of the \qit{while~\dots} clauses have been established (following table \ref{while_clauses_table}), the behaviour of \qit{while~\dots} clauses appears to be the same as that of period adverbials (i.e.\ it follows table \vref{period_adverbials_table}). With main clauses whose aspectual class is point, the instantaneous situation of the main clause must occur within the period of the \qit{while~\dots} clause (e.g.\ in \pref{whc:30} the departures must have occurred during a maximal period where runway 5 was closed; \qit{to depart} is a point verb in the airport domain). \begin{examps} \item Did any flight depart from gate 2 while runway 5 was closed? \label{whc:30} \end{examps} With activity main clauses, the situation of the main clause must be ongoing some time during the period of the \qit{while~\dots} clause. In \pref{whc:33}, for example, the flights must have taxied some time during a maximal period where BA737 was circling. As with period adverbials, stricter readings are sometimes possible with activity main clauses. \pref{whc:33}, for example, could refer to flights that both started and stopped taxiing during a maximal period where BA737 was circling. As with period adverbials, I ignore such stricter readings. \begin{examps} \item Which flights taxied while BA737 circled? \label{whc:33} \end{examps} As in the case of period adverbials, with culminating activity main clauses I allow two readings: (a) that the situation of the main clause both starts and reaches its completion within the period of the \qit{while~\dots} clause, or (b) that the situation of the main clause simply reaches its completion within the period of the \qit{while~\dots} clause. In the second reading, the main clause is taken to refer to only the completion point of the situation it would normally describe, and its aspectual is changed to point. In the airport domain, the first reading is the preferred one in \pref{whc:34}. The second reading allows the answer to \pref{whc:35} to contain flights that simply touched down during the service, even if their landing procedures did not start during the service. \begin{examps} \item J.Adams inspected BA737 while Airserve was servicing UK160. \label{whc:34} \item Which flights landed while Airserve was servicing UK160? \label{whc:35} \end{examps} With state main clauses, I require the situation of the main clause to hold some time during the period of the \qit{while~\dots} clause (inclusive reading; see section \ref{period_adverbials}). For example, the answer to \pref{whc:20} must contain anybody who was a lecturer some time during a maximal period where J.Adams was a professor (the non-auxiliary \qit{to be} is typically classified as state verb). As with period adverbials, there is often an implication that the situation of the main clause holds \emph{throughout} the period of the \qit{while~\dots} clause (durative reading). The durative reading is unlikely in \pref{whc:20}, but seems the preferred one in \pref{whc:21} (progressive state main clause). According to the durative reading, \pref{whc:21} refers to a flight that was circling \emph{throughout} a maximal period where runway 2 was closed. \begin{examps} \item Who was a lecturer while J.Adams was a professor? \label{whc:20} \item Which flight was circling while runway 2 was closed? \label{whc:21} \end{examps} The treatment of \qit{while~\dots} clauses of this thesis is similar to that of \cite{Ritchie}. Ritchie also views \qit{while~\dots} clauses as establishing periods, with the exact relations between these periods and the situations of the main clauses depending on the aspectual classes of the main clauses. Ritchie uses only two aspectual classes (``continuing'' and ``completed''), which makes presenting a direct comparison between his treatment of \qit{while~\dots} clauses and the treatment of this thesis difficult. Both approaches, however, lead to similar results, with the following two main exceptions. (a) In \pref{whc:20} and \pref{whc:21} (state main clause), Ritchie's treatment admits only durative readings. In contrast, the framework of this thesis admits only inclusive ones. (b) In \pref{whc:35} (culminating activity main clause), Ritchie's arrangements allow only one reading, where the landings must have both started and been completed during the service. The framework of this thesis allows an additional reading, whereby it is enough if the landings were simply completed during the service. \subsection{\qit{Before~\dots} and \qit{after~\dots} clauses} \label{before_after_clauses} I treat \qit{before~\dots} and \qit{after~\dots} clauses as establishing periods, as in the case of the \qit{before~\dots} and \qit{after~\dots} adverbials of section \ref{period_adverbials}. In \qit{before~\dots} clauses, the period starts at some unspecified time-point (in the absence of other constraints, the beginning of time), and ends at a time-point provided by the \qit{before~\dots} clause. In \qit{after~\dots} clauses, the period starts at a time-point provided by the \qit{after~\dots} clause, and ends at some unspecified time-point (the end of time, in the absence of other constraints). I use the terms \emph{before-point} and \emph{after-point} to refer to the time-points provided by \qit{before~\dots} and \qit{after~\dots} clauses respectively. Once the periods of the \qit{before~\dots} and \qit{after~\dots} clauses have been established, the behaviour of the clauses appears to be the same as that of period adverbials (i.e.\ it follows table \vref{period_adverbials_table}). \paragraph{State \qit{before/after~\dots} clause:} Let us first examine sentences where the aspectual class of the \qit{before~\dots} or \qit{after~\dots} clause is state. With \qit{before~\dots} clauses, the before-point is a time-point where the situation of the \qit{before~\dots} clause starts (table \ref{before_clauses_table}). In \pref{bac:1}, for example, the before-point is a time-point where runway 2 started to be open. The aspectual class of the main clause is point (\qit{to depart} is a point verb in the airport domain). Hence, according to table \vref{period_adverbials_table}, the departures must have occurred within the period of the \qit{before~\dots} clause, i.e.\ before the time-point where runway 2 started to be open. Similar comments apply to \pref{bac:1.1}, \pref{bac:2} (progressive \qit{before~\dots} clause), and \pref{bac:3} (consequent state \qit{before~\dots} clause). In \pref{bac:3}, the before-point is the beginning of the consequent period of the inspection (the period that contains all the time after the completion of the inspection; see section \ref{point_adverbials}), i.e.\ the departures must have happened before the inspection was completed. \begin{examps} \item Which flights departed before runway 2 was open? \label{bac:1} \item Which flights departed before the emergency system was in operation? \label{bac:1.1} \item Which flights departed before BA737 was circling? \label{bac:2} \item Which flights departed before J.Adams had inspected BA737? \label{bac:3} \end{examps} \begin{table}[t] \begin{center} {\small \begin{tabular}{|l|l|} \hline aspectual class of & before-point \\ \qit{before~\dots} clause & (right boundary of period specified by \qit{before~\dots} clause) \\ \hline \hline state & time-point where situation of \qit{before~\dots} clause starts \\ \hline activity & time-point where situation of \qit{before~\dots} clause starts \\ \hline culm.\ activity & time-point where situation of \qit{before~\dots} clause \\ & starts or is completed \\ \hline point & time-point where situation of \qit{before~\dots} clause occurs \\ \hline \end{tabular} } \end{center} \caption{Boundaries of \qit{before~\dots} clauses in the framework of this thesis} \label{before_clauses_table} \end{table} According to table \vref{period_adverbials_table}, in \pref{bac:12} where the main clause is a state, the flight must have been at gate 2 some time during the period of the \qit{before~\dots} clause, i.e.\ for some time before runway 2 started to be open. In \pref{bac:10} (activity main clause), the flight must have circled for some time before runway 2 started to be open, and in \pref{bac:11} (culminating activity main clause) the inspections must have both started and been completed before runway 2 started to be open. (As with the \qit{before~\dots} adverbials of section \ref{period_adverbials}, in \pref{bac:11} it would be better if the \textsc{Nlitdb}\xspace could also report inspections that started but were not completed before runway 2 opened, warning the user that these inspections were not completed before runway 2 opened.) \begin{examps} \item Was any flight at gate 2 before runway 2 was open? \label{bac:12} \item Did any flight circle before runway 2 was open? \label{bac:10} \item Which flights did J.Adams inspect before runway 2 was open? \label{bac:11} \end{examps} In the case of \qit{after~\dots} clauses, when the aspectual class of the \qit{after~\dots} clause is state, the after-point is a time-point where the situation of the \qit{after~\dots} clause either starts or ends. \pref{bac:1a}, for example, has two readings: that the flights must have departed after runway 2 \emph{started} to be open, or that the flights must have departed after runway 2 \emph{stopped} being open. Similar comments apply to \pref{bac:1.1a} and \pref{bac:2a}. \begin{examps} \item Which flights departed after runway 2 was open? \label{bac:1a} \item Which flights departed after the emergency system was in operation? \label{bac:1.1a} \item Which flights departed after BA737 was circling? \label{bac:2a} \end{examps} In sentences like \pref{bac:3a}, where the aspectual class of the \qit{after~\dots} clause is consequent state, the after-point can only be the beginning of the consequent period (the first time-point after the completion of the inspection). It cannot be the end of the consequent period: the end of the consequent period is the end of time; if the after-point were the end of the consequent period, the departures of \pref{bac:3a} would have to occur after the end of time, which is impossible. This explains the distinction between lexical/progressive and consequent states in table \ref{after_clauses_table}. \begin{examps} \item Which flights departed after J.Adams had inspected BA737? \label{bac:3a} \end{examps} \begin{table}[t] \begin{center} {\small \begin{tabular}{|l|l|} \hline aspectual class of & after-point \\ \qit{before~\dots} clause & (left boundary of period specified by \qit{after~\dots} clause) \\ \hline \hline lexical/progressive state & time-point where situation of \qit{after~\dots} clause starts or ends \\ \hline consequent state & time-point where consequent period of \qit{after~\dots} clause starts \\ \hline activity & time-point where situation of \qit{after~\dots} clause ends \\ \hline culm.\ activity & time-point where situation of \qit{after~\dots} clause is completed \\ \hline point & time-point where situation of \qit{before~\dots} clause occurs \\ \hline \end{tabular} } \end{center} \caption{Boundaries of \qit{after~\dots} clauses in the framework of this thesis} \label{after_clauses_table} \end{table} \paragraph{Point \qit{before/after~\dots} clause:} If the aspectual class of the \qit{before~\dots} or \qit{after~\dots} clause is point, the before/after-point is the time-point where the instantaneous situation of the subordinate clause occurs. In \pref{bac:20}, for example, the before/after-point is the point where BA737 reached gate2. \begin{examps} \item Which flights departed before/after BA737 reached gate 2? \label{bac:20} \end{examps} \paragraph{Activity \qit{before/after~\dots} clause:} With activity \qit{before/after~\dots} clauses, I consider the before-point to be a time-point where the situation of the \qit{before~\dots} clause starts, and the after-point to be a point where the situation of the \qit{after~\dots} clause ends. In \pref{bac:23} and \pref{bac:24}, for example, the departures must have occurred before BA737 \emph{started} to taxi or circle. In \pref{bac:25} and \pref{bac:26}, the departures must have occurred after BA737 \emph{stopped} taxiing or circling. \begin{examps} \item Which flights departed before BA737 taxied? \label{bac:23} \item Which flights departed before BA737 circled? \label{bac:24} \item Which flights departed after BA737 taxied? \label{bac:25} \item Which flights departed after BA737 circled? \label{bac:26} \end{examps} Perhaps another reading is sometimes possible with \qit{after~\dots} clauses: that the after-point is a time-point where the situation of the \qit{after~\dots} clause \emph{starts} (e.g.\ \pref{bac:26} would refer to departures that occurred after BA737 \emph{started} to circle). This reading, however, does not seem very likely, and for simplicity I ignore it. \paragraph{Culminating activity \qit{before/after~\dots} clause:} With \qit{after~\dots} clauses whose aspectual class is culminating activity, I consider the after-point to be a time-point where the situation of the \qit{after~\dots} clause reaches its completion. In \pref{bac:27}, the departures must have occurred after the completion of the inspection, and in \pref{bac:28} they must have occurred after the time-point where BA737 reached gate 2. \begin{examps} \item Which flights departed after J.Adams inspected BA737? \label{bac:27} \item Which flights departed after BA737 taxied to gate 2? \label{bac:28} \end{examps} With culminating activity \qit{before~\dots} clauses, I allow the before-point to be a time-point where the situation of the \qit{before~\dots} clause either starts or reaches its completion. In the airport domain, the first reading seems the preferred one in \pref{bac:30} (the flights must have departed before the \emph{beginning} of the inspection). The second reading seems the preferred one in \pref{bac:31} (the flights must have departed before the \emph{completion} of the landing). Both readings seems possible in \pref{bac:31}. \begin{examps} \item Which flights departed before J.Adams inspected BA737? \label{bac:30} \item Which flights departed before BA737 landed? \label{bac:33} \item Which flights departed before BA737 taxied to gate 2? \label{bac:31} \end{examps} If the first reading is adopted (the situation of the \qit{before~\dots} clause \emph{starts} at the before-point) and the \qit{before~\dots} clause is in the simple past, it is unclear if the situation of the \qit{before~\dots} clause must have necessarily reached its climax (the simple past of culminating activity verbs normally requires this; see section \ref{simple_past}). For example, let us assume that the first reading is adopted in \pref{bac:30}. Should the before-point be the beginning of an inspection that was necessarily completed, or can it also be the beginning of an inspection that was never completed? The framework of this thesis currently adopts the first approach, but this is perhaps over-restrictive. It would probably be better if the \textsc{Nlitdb}\xspace allowed the before-point to be the beginning of both inspections that were and were not completed, warning the user about inspections that were not completed. This is another case for cooperative responses (section \ref{no_issues}). \paragraph{Other uses:} \qit{Before} and \qit{after} can be preceded by expressions specifying durations (e.g.\ \pref{bac:40}). This use of \qit{before} and \qit{after} is not considered in this thesis. \begin{examps} \item BA737 reached gate 2 five minutes after UK160 departed. \label{bac:40} \end{examps} \qit{Before~\dots} clauses also have counter-factual uses. For example, in \pref{bac:43} (from \cite{Crouch}) the situation where the car runs into the tree never takes place. This use of \qit{before} is not considered in this thesis. \begin{examps} \item Smith stopped the car before it ran into the tree. \label{bac:43} \end{examps} The treatment of \qit{before~\dots} and \qit{after~\dots} clauses of this thesis is similar to that of \cite{Ritchie}. Ritchie also views \qit{before~\dots} and \qit{after~\dots} clauses as providing before and after-points. As noted in section \ref{while_clauses}, however, Ritchie uses only two aspectual classes. According to Ritchie, in the case of \qit{before~\dots} clauses, the before-point is a time-point where the situation of the \qit{before~\dots} clause starts, and the situation of the main clause must simply start before that point. In \pref{bac:11}, this requires the inspections to have simply \emph{started} before the time-point where runway 2 started to be open. In contrast, the framework of this thesis requires the inspections to have been \emph{completed} before that time-point. In the case of \qit{after~\dots} clauses, the main difference between Ritchie's treatment and the treatment of this thesis concerns state \qit{after~\dots} clauses. In that case, Ritchie allows the after-point to be only the beginning of the situation of the \qit{after~\dots} clause. In \pref{bac:1.1a}, this requires the flights to have departed after the time-point where the system \emph{started} to be in operation. The framework of this thesis allows an additional reading, where the flights must have departed after the time-point where the system \emph{stopped} being in operation. \subsection{Tense coordination} \label{tense_coordination} Some combinations of tenses in the main and subordinate clauses are unacceptable (e.g.\ \pref{coo:0}, \pref{coo:2}). This thesis makes no attempt to account for the unacceptability of such combinations. The reader is referred to \cite{Harper} and \cite{Brent1990} for methods that could be used to detect and reject sentences like \pref{coo:0} and \pref{coo:2}. \begin{examps} \item \bad BA737 left gate 2 before runway 2 is free. \label{coo:0} \item \bad Which runways are closed while runway 2 was circling? \label{coo:2} \end{examps} \section{Noun phrases and temporal reference} \label{noun_anaphora} A question like \pref{nana:1} can refer either to the present sales manager (asking the 1991 salary of the present sales manager) or to the 1991 sales manager (asking the 1991 salary of the 1991 sales manager). Similarly, \pref{nana:2} may refer either to present students or last year's students. In \pref{nana:3.1}, \qit{which closed runway} probably refers to a runway that is \emph{currently} closed, while in \pref{nana:3.5} \qit{a closed runway} probably refers to a runway that was closed at the time of the landing. \begin{examps} \item What was the salary of the sales manager in 1991? \label{nana:1} \item Which students failed in physics last year? \label{nana:2} \item Which closed runway was open yesterday? \label{nana:3.1} \item Did BA737 ever land on a closed runway in 1991? \label{nana:3.5} \end{examps} It seems that noun phrases (e.g.\ \qit{the sales manager}, \qit{which students}, \qit{a closed runway}) generally refer either to the present or to the time of the verb tense (if this time is different than the present). In \pref{nana:1}, the simple past tense refers to some time in 1991. Therefore, there are two options: \qit{the sales manager} can refer either to the present sales manager or to somebody who was the sales manager in 1991. Similar comments apply to \pref{nana:2}. In contrast, in \pref{nana:3} the verb tense refers to the present. Hence, there is only one possibility: \qit{the sales manager} refers to the present sales manager. \begin{examps} \item What is the salary of the sales manager? \label{nana:3} \end{examps} In \pref{nana:3.1}, the verb tense refers to a time (within the previous day) where the runway was open. There should be two readings: it should be possible for \qit{which closed runway} to refer either to a currently closed runway, or to a runway that was closed at the time it was open. Since a runway cannot be closed at the same time where it is open, the second reading is ruled out. (This clash, however, cannot be spotted easily by a \textsc{Nlitdb}\xspace without some inferential capability.) The hypothesis that noun phrases refer either to the present or to the time of the verb tense is not always adequate. For example, a person submitting \pref{nana:5} to the \textsc{Nlitdb}\xspace of a university most probably refers to \emph{previous} students of the university. In contrast, the hypothesis predicts that the question can refer only to \emph{current} students. (Similar examples can be found in \cite{Enc1986}.) \begin{examps} \item How many of our students are now professors? \label{nana:5} \end{examps} The hypothesis also predicts that \pref{nana:6} can refer only to current Prime Ministers or to persons that were Prime Ministers at the time they were born (an extremely unlikely reading). There is, however, a reading where the question refers to all past and present Prime Ministers. This reading is incorrectly ruled out by the hypothesis. \begin{examps} \item Which Prime Ministers were born in Scotland? \label{nana:6} \end{examps} Hinrichs \cite{Hinrichs} argues that determining the times to which noun phrases refer is part of a more general problem of determining the entities to which noun phrases refer. According to Hinrichs, a noun phrase like \qit{every admiral} generally refers to anybody who was, is, or will be an admiral of any fleet in the world at any time. If, however, \pref{nana:8} is uttered in a context where the current personnel of the U.S.\ Pacific fleet is being discussed, the temporal scope of \qit{every admiral} is restricted to current admirals, in the same way that the scope of \qit{every admiral} is restricted to admirals of the U.S.\ Pacific fleet (e.g.\ \pref{nana:8} does not mean that all Russian admirals also graduated from Annapolis). \begin{examps} \item Every admiral graduated from Annapolis. \label{nana:8} \end{examps} The fact that Hinrichs does not limit the times of the noun phrases to the present and the time of the verb tense is in accordance with the fact that \qit{our students} in \pref{nana:5} is not limited to present students, and the fact that \qit{which Prime Ministers} in \pref{nana:6} may refer to all past and present Prime Ministers. Hinrichs' approach, however, requires some mechanism to restrict the scope of noun phrases as the discourse evolves. Hinrichs offers only a very limited sketch of how such a mechanism could be constructed. Also, in the absence of previous discourse, Hinrichs' treatment suggests that \pref{nana:1} refers to the sales managers of all times, an unlikely interpretation. The hypothesis that noun phrases refer either to the present or to the time of the verb tense performs better in this case. Given these deficiencies of Hinrichs' approach, I adopt the initial hypothesis that noun phrases refer to the present or the time of the verb tense. (An alternative approach would be to attempt to merge this hypothesis with Hinrichs' method. \cite{Dalrymple1988} goes towards this direction.) A further improvement can be made to the hypothesis that noun phrases refer to the present or the time of the verb tense. When a noun phrase is the complement of the predicative \qit{to be}, it seems that the noun phrase can refer only to the time of the verb tense. \pref{nana:11}, for example, can only be a request to report the 1991 sales manager, not the current sales manager. Similarly, \pref{nana:11.5} cannot mean that J.Adams is the current sales manager. This also accounts for the fact that in \pref{nana:1}, unlike \qit{the sales manager} which can refer either to the present or 1991, \qit{the salary of the sales manager} (the complement of \qit{was}) can refer only to a 1991 salary, not to a present salary. (I assume that the restriction that the complement of the predicative \qit{to be} must refer to the time of the verb tense does not extend to noun phrases that are subconstituents of that complement, like \qit{the sales manager} in \pref{nana:1}.) The same restriction applies to bare adjectives used as complements of the predicative \qit{to be}. In \pref{nana:3.1}, for example, \qit{open} can only refer to runways that were open on the previous day. It cannot refer to currently open runways. \begin{examps} \item Who was the sales manager in 1991? \label{nana:11} \item J.Adams was the sales manager in 1991. \label{nana:11.5} \end{examps} The hypothesis that noun phrases refer to the present or the time of the verb tense does not apply when a temporal adjective (e.g.\ \qit{current}) specifies explicitly the time of the noun phrase (e.g.\ \pref{nana:9}). (Although temporal adjectives are not considered in this thesis, I support \qit{current} to be able to illustrate this point.) \begin{examps} \item Which current students failed in Physics last year? \label{nana:9} \end{examps} In chapter \ref{English_to_TOP}, an additional mechanism will be introduced, that allows the person configuring the \textsc{Nlitdb}\xspace to force some noun phrases to be treated as always referring to the time of the verb tense, or as always referring to the present. \section{Temporal anaphora} \label{temporal_anaphora} There are several English expressions (e.g.\ \qit{that time}, \qit{the following day}, \qit{then}, \qit{later}) that refer implicitly to contextually salient times, in a way that is similar to how pronouns, possessive determiners, etc.\ refer to contextually salient world entities (the terms \emph{temporal} and \emph{nominal anaphora} were used in section \ref{no_issues} to refer to these two phenomena; the parallels between temporal and nominal anaphora are discussed in \cite{Partee1984}). For example, the user of a \textsc{Nlitdb}\xspace may submit \pref{tan:1}, followed by \pref{tan:2}. In \pref{tan:2}, \qit{at that time} refers to the time when John became manager (temporal anaphora). In a similar manner, \qit{he} refers to John (nominal anaphora). \begin{examps} \item When did John become manager? \label{tan:1} \item Was he married at that time? \label{tan:2} \end{examps} Names of months, days, etc.\ often have a similar temporal anaphoric nature. For example, in a context where several questions about the 1990 status of a company have just been asked, \pref{tan:6} most probably refers to the January of 1990, not any other January. In the absence of previous questions, \pref{tan:6} most probably refers to the January of the current year. (See section 5.5.1 of \cite{Kamp1993} for related discussion.) \begin{examps} \item Who was the sales manager in January? \label{tan:6} \end{examps} Verb tenses also seem to have a temporal anaphoric nature (the term \emph{tense anaphora} is often used in this case). For example, the user may ask \pref{tan:7} (let us assume that the response is ``\sys{no}''), followed by \pref{tan:8}. In that case, the simple past \qit{was} of \pref{tan:8} does not refer to an arbitrary past time, it refers to the past time of the previous question, i.e.\ 1993. \begin{examps} \item Was Mary the personnel manager in 1993? \label{tan:7} \item Who was the personnel manager? \label{tan:8} \end{examps} The anaphoric nature of verb tenses is clearer in multi-sentence text (see \cite{Hinrichs1986}, \cite{Webber1988}, \cite{Kamp1993}, \cite{Kameyama1993} for related work). In \pref{tan:9}, for example, the simple past \qit{landed} refers to a landing that happened immediately after the permission of the first sentence was given. It does not refer to an arbitrary past time where BA737 landed on runway 2. Similar comments apply to the \qit{taxied}. \begin{examps} \item BA737 was given permission to land at 5:00pm. It landed on runway 2, and taxied to gate 4. \label{tan:9} \end{examps} In dialogues like the one in \pref{tan:7} -- \pref{tan:8}, a simplistic treatment of tense anaphora is to store the time of the adverbial of \pref{tan:7}, and to require the simple past of \pref{tan:8} to refer to that time. (A more elaborate version of this approach will be discussed in section \ref{lt_anaphora}.) The behaviour of noun phrases like \qit{the sales manager} of section \ref{noun_anaphora} can be seen as a case of temporal anaphora. This is the only type of temporal anaphora that is supported by the framework of this thesis. Expressions like \qit{at that time}, \qit{the following day}, etc.\ are not supported, and tenses referring to the past are taken to refer to \emph{any} past time. For example, \pref{tan:8} is taken to refer to anybody who was the personnel manager at any past time. The reader is also reminded (section \ref{no_issues}) that nominal anaphora is not considered in this thesis. \section{Other phenomena that are not supported} \label{ling_not_supported} This section discusses some further phenomena that are not supported by the framework of this thesis. \paragraph{Cardinality and duration questions:} Questions about the cardinality of a set or the duration of a situation (e.g.\ \pref{oiss:1}, \pref{oiss:2}) are not supported. (\textsc{Top}\xspace is currently not powerful enough to express the meanings of these questions.) \begin{examps} \item How many flights have landed today? \label{oiss:1} \item For how long was tank 2 empty? \label{oiss:2} \end{examps} \paragraph{Cardinalities and plurals:} Expressions specifying cardinalities of sets (e.g.\ \qit{eight passengers}, \qit{two airplanes}) are not supported (this does not include duration expressions like \qit{five hours}, which are supported). Expressions of this kind give rise to a distinction between \emph{distributive} and \emph{collective} readings \cite{Stirling1985} \cite{Crouch2}. \pref{oiss:10}, for example, has a collective reading where the eight passengers arrive at the same time, and a distributive one where there are eight separate arrivals. This distinction was not explored during the work of this thesis. \textsc{Top}\xspace is also currently not powerful enough to express cardinalities of sets. \begin{examps} \item Eight passengers arrived. \label{oiss:10} \end{examps} The framework of this thesis accepts plural noun phrases introduced by \qit{some} and \qit{which} (e.g.\ \qit{some flights}, \qit{which passengers}), but it treats them semantically as singular. For example, \pref{oiss:11} and \pref{oiss:12} are treated as having the same meanings as \pref{oiss:11.1} and \pref{oiss:12.1} respectively. \begin{examps} \item Which flights landed? \label{oiss:11} \item Which flight landed? \label{oiss:11.1} \item Some flights entered sector 2. \label{oiss:12} \item A flight entered sector 2. \label{oiss:12.1} \end{examps} \paragraph{Quantifiers:} Expressions introducing universal quantifiers at the logical level (e.g.\ \qit{every}, \qit{all}) are not supported. This leaves only existential quantifiers (and an interrogative version of them, to be discussed in chapter \ref{TOP_chapter}) at the logical level, avoiding issues related to quantifier scoping (see also section \ref{quantif_scoping}). It also simplifies the semantics of \textsc{Top}\xspace and the mapping from \textsc{Top}\xspace to \textsc{Tsql2}\xspace. \paragraph{Conjunction, disjunction, and negation:} Conjunctions of words or phrases are not supported. Among other things, this avoids phenomena related to sequencing of events. For example, \pref{oiss:15} is understood as saying that the patient died \emph{after} (and probably as a result of) being given Qdrug (cf.\ \pref{oiss:16} which sounds odd). In contrast, in \pref{oiss:17} the patient was given Qdrug \emph{while} he had high fever. (See, for example, \cite{Hinrichs1986}, \cite{Hinrichs}, \cite{Webber1988}, \cite{Kamp1993}, \cite{terMeulen1994}, and \cite{Hwang1994} for related work.) \begin{examps} \item Which patient was given Qdrug and died? \label{oiss:15} \item \odd Which patient died and was given Qdrug? \label{oiss:16} \item Which patient had high fever and was given Qdrug? \label{oiss:17} \end{examps} Expressions introducing disjunction or negation (e.g.\ \qit{or}, \qit{either}, \qit{not}, \qit{never}) are also not supported. This simplifies the semantics of \textsc{Top}\xspace and the \textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping. Not supporting negation also avoids various temporal phenomena related to negation (see section 5.2.5 of \cite{Kamp1993}), and claims that negation causes aspectual shifts (see, for example, \cite{Dowty1986} and \cite{Moens}). \paragraph{Relative clauses:} Relative clauses are also not supported. Relative clauses require special temporal treatment. \pref{oiss:20}, for example, most probably does not refer to a runway that was closed at an \emph{arbitrary} past time; it probably refers to a runway that was closed at the time of the landing. The relation between the time of the relative clause and that of the main clause can vary. In \pref{oiss:21} (from \cite{Dowty1986}), for example, the woman may have seen John during, before, or even after the stealing. \begin{examps} \item Which flight landed on a runway that was closed? \label{oiss:20} \item The woman that stole the book saw John. \label{oiss:21} \end{examps} Relative clauses can also be used with nouns that refer to the temporal ontology (e.g.\ \qit{period} in \pref{oiss:22}). Additional temporal phenomena involving relative clauses are discussed in section 5.5.4.2 of \cite{Kamp1993}. \begin{examps} \item Who was fired during the period that J.Adams was personnel manager? \label{oiss:22} \end{examps} \paragraph{Passives:} Finally, I have concentrated on active voice verb forms. This simplifies the \textsc{Hpsg}\xspace grammar of chapter 4. It should be easy to extend the framework of this thesis to cover passive forms as well. \section{Summary} The framework of this thesis uses an aspectual taxonomy of four classes (states, points, activities, and culminating activities). This taxonomy classifies verb forms, verb phrases, clauses, and sentences. Whenever the \textsc{Nlitdb}\xspace is configured for a new application, the base form of each verb is assigned to one of the four aspectual classes. All other verb forms normally inherit the aspectual class of the base form. Verb phrases, clauses, and sentences normally inherit the aspectual classes of their main verb forms. Some linguistic mechanisms (e.g.\ progressive tenses, or some temporal adverbials), however, may cause the aspectual class of a verb form to differ from that of the base form, or the aspectual class of a verb phrase, clause, or sentence to differ from that of its main verb form. The aspectual taxonomy plays an important role in most time-related linguistic phenomena. Six tenses (simple present, simple past, present continuous, past continuous, present perfect, and past perfect) are supported, with various simplifications introduced in their meanings. Some special temporal verbs were identified (e.g.\ \qit{to happen}, \qit{to start}); from these only \qit{to start}, \qit{to begin}, \qit{to stop}, and \qit{to finish} are supported. Some nouns have a special temporal nature. For example, some introduce situations (e.g.\ \qit{inspection}), others specify temporal order (e.g.\ \qit{predecessor}), and others refer to entities of the temporal ontology (e.g.\ \qit{day}, \qit{period}, \qit{event}). From all these, only nouns like \qit{year}, \qit{month}, \qit{day}, etc.\ (and proper names like \qit{Monday}, \qit{January}, and \qit{1/5/92}) are supported. Nouns referring to more abstract temporal entities (e.g.\ \qit{period}, \qit{event}) are not supported. No temporal adjectives (e.g.\ \qit{first}, \qit{earliest}) are handled, with the only exception of \qit{current} which is supported to demonstrate the anaphoric behaviour of some noun phrases. Among temporal adverbials, only punctual adverbials (e.g.\ \qit{at 5:00pm}), \qit{for~\dots} and \qit{in~\dots} duration adverbials, and period adverbials introduced by \qit{on}, \qit{in}, \qit{before}, or \qit{after}, as well as \qit{today} and \qit{yesterday} are handled. Frequency, order, or other adverbials that specify boundaries (e.g.\ \qit{twice}, \qit{for the second time}, \qit{since 1992}) are not supported. Only subordinate clauses introduced by \qit{while}, \qit{before}, and \qit{after} are handled (e.g.\ clauses introduced by \qit{when} or \qit{since} and relative clauses are not supported). The issue of tense coordination between main and subordinate clauses is ignored. Among temporal anaphora phenomena, only the temporal anaphoric nature of noun phrases like \qit{the sales manager} is supported. Proper names like \qit{May} or \qit{Monday} are taken to refer to \emph{any} May or Monday. Similarly, past tenses are treated as referring to \emph{any} past time. Temporal anaphoric expressions like \qit{that time} or \qit{the following day} are not allowed. (Nominal anaphoric expressions, e.g.\ \qit{he}, \qit{her salary}, are also not allowed.) The framework of this thesis does not support cardinality or duration queries (\qit{How many~\dots?}, \qit{How long~\dots?}) and cardinality expressions (e.g.\ \qit{five flights}). Plurals introduced by \qit{which} and \qit{some} (e.g.\ \qit{which flights}, \qit{some gates}) are treated semantically as singular. Conjunctions of words or phrases, and expressions introducing universal quantifiers, disjunction, or negation are also not supported. Finally, only active voice verb forms have been considered, though it should be easy to extend the mechanisms of this thesis to support passive voice as well. Table \ref{coverage_table} summarises the linguistic coverage of the framework of this thesis. \begin{table} \begin{tabular}{|l|l|} \hline verb tenses & \supp simple present (excluding scheduled meaning) \\ & \supp simple past \\ & \supp present continuous (excluding futurate meaning) \\ & \supp past continuous (excluding futurate meaning) \\ & \supp present perfect (treated as simple past) \\ & \supp past perfect \\ & \nosupp other tenses \\ \hline temporal verbs & \supp \qit{to start}, \qit{to begin}, \qit{to stop}, \qit{to finish} \\ & \nosupp other temporal verbs (e.g.\ \qit{to happen}, \qit{to follow}) \\ \hline temporal nouns & \supp \qit{year}, \qit{month}, \qit{day}, etc.\ \\ & \nosupp \qit{period}, \qit{event}, \qit{time}, etc.\ \\ & \nosupp nouns introducing situations (e.g.\ \qit{inspection}) \\ & \nosupp nouns of temporal order (e.g.\ \qit{predecessor}) \\ \hline temporal adjectives & \nosupp (only \qit{current}) \\ \hline temporal adverbials & \supp punctual adverbials (e.g.\ \qit{at 5:00pm})\\ & \supp period adverbials (only those introduced by \qit{on}, \qit{in}, \\ & \ \ \ \ \qit{before}, or \qit{after}, and \qit{today}, \qit{yesterday}) \\ & \supp \qit{for~\dots} adverbials \\ & \supp \qit{in~\dots} duration adverbials (only with culm.\ act.\ verbs) \\ & \nosupp frequency adverbials (e.g.\ \qit{twice}) \\ & \nosupp order adverbials (e.g.\ \qit{for the second time}) \\ & \nosupp other boundary adverbials (e.g.\ \qit{since 1987}) \\ \hline subordinate clauses & \supp \qit{while~\dots} clauses \\ & \supp \qit{before~\dots} clauses \\ & \supp \qit{after~\dots} clauses \\ & \nosupp relative clauses \\ & \nosupp other subordinate clauses (e.g.\ introduced by \qit{when}) \\ & \nosupp tense coordination between main-subordinate clauses \\ \hline anaphora & \supp noun phrases and temporal reference \\ & \nosupp \qit{January}, \qit{August}, etc.\ \\ & \ \ \ \ (taken to refer to any January, August, etc.) \\ & \nosupp tense anaphora \\ & \ \ \ \ (past tenses taken to refer to any past time) \\ & \nosupp \qit{that time}, \qit{the following day}, etc.\ \\ & \nosupp nominal anaphora (e.g.\ \qit{he}, \qit{her salary}) \\ \hline other phenomena & \nosupp cardinality and duration queries \\ & \ \ \ \ (\qit{How many~\dots?}, \qit{How long~\dots?}) \\ & \nosupp cardinality expressions (e.g.\ \qit{five flights})\\ & \nosupp plurals (treated as singulars) \\ & \nosupp conjunctions of words or phrases \\ & \nosupp expressions introducing universal quantifiers, \\ & \ \ \ \ disjunction, negation \\ & \nosupp passive voice \\ \hline \end{tabular} \caption{The linguistic coverage of the framework of this thesis} \label{coverage_table} \end{table} \chapter{The TOP Language} \label{TOP_chapter} \proverb{Time will tell.} \section{Introduction} \label{top_intro} This chapter defines \textsc{Top}\xspace, the intermediate representation language of this thesis. As noted in section \ref{temp_log_intro}, \textsc{Top}\xspace employs temporal operators. \pref{tintro:1}, for example, is represented in \textsc{Top}\xspace as \pref{tintro:2}. Roughly speaking, the \ensuremath{\mathit{Past}}\xspace operator requires $contain(tank2, water)$ to be true at some past time $e^v$, and the \ensuremath{\mathit{At}}\xspace operator requires that time to fall within 1/10/95. The answer to \pref{tintro:1} is affirmative iff \pref{tintro:2} evaluates to true. \begin{examps} \item Did tank 2 contain water (some time) on 1/10/95? \label{tintro:1} \item $\ensuremath{\mathit{At}}\xspace[\mathit{1/10/95}, \ensuremath{\mathit{Past}}\xspace[e^v, contain(tank2, water)]]$ \label{tintro:2} \end{examps} An alternative operator-less approach is to introduce time as an extra argument of each predicate (section \ref{temp_log_intro}). I use temporal operators because they lead to more compact formulae, and because they make the semantic contribution of each linguistic mechanism easier to see (in \pref{tintro:2}, the simple past tense contributes the \ensuremath{\mathit{Past}}\xspace operator, while the \qit{on~\dots} adverbial contributes the \ensuremath{\mathit{At}}\xspace operator). \textsc{Top}\xspace is period-based, in the sense that the truth of a \textsc{Top}\xspace formula is checked with respect to a time-period (a segment of the time-axis) rather than an individual time-point. (The term ``period'' is used here to refer to what other authors call ``intervals''; see section \ref{temporal_ontology} below.) A \textsc{Top}\xspace formula may be true at a time-period without being true at the subperiods of that period. Actually, following the Reichenbachian tradition \cite{Reichenbach}, \textsc{Top}\xspace formulae are evaluated with respect to more than one times: \emph{speech time} (time at which the question is submitted), \emph{event time} (time where the situation described by the formula holds), and \emph{localisation time} (a temporal window within which the event time must be placed; this is different from Reichenbach's reference time, and similar to the ``location time'' of \cite{Kamp1993}). While speech time is always a time-point, the event and localisation times are generally periods, and this is why I consider \textsc{Top}\xspace period-based. Period-based languages have been used in \cite{Dowty1982}, \cite{Allen1984}, \cite{Lascarides}, \cite{Richards}, \cite{Pratt1995}, and elsewhere. Multiple temporal parameters have been used by several researchers (e.g.\ \cite{Dowty1982}, \cite{Hinrichs}, \cite{Brent1990}, \cite{Crouch2}). The term ``localisation time'' is borrowed from \cite{Crouch2}, where $lt$ is a temporal window for $et$ as in \textsc{Top}\xspace. Although the aspectual classes of linguistic expressions affect how these expressions are represented in \textsc{Top}\xspace, it is not always possible to tell the aspectual class of a linguistic expression by examining the corresponding \textsc{Top}\xspace formula. The approach here is different from those of \cite{Dowty1977}, \cite{Dowty1986}, \cite{Lascarides}, and \cite{Kent}, where aspectual class is a property of formulae (or denotations of formulae). \textsc{Top}\xspace was greatly influenced by the representation language of Pirie et al.\ \cite{Pirie1990} \cite{Crouch} \cite{Crouch2}, that was used in a natural language front-end to a planner. \textsc{Top}\xspace, however, differs in numerous ways from the language of Pirie et al.\ (several of these differences will be mentioned in following sections). \section{Syntax of TOP} \label{top_syntax} This section defines the syntax of \textsc{Top}\xspace. Some informal comments about the semantics of the language are also given to make the syntax definition easier to follow. The semantics of \textsc{Top}\xspace will be defined formally in following sections. \paragraph{Terms:} Two disjoint sets of strings, \ensuremath{\mathit{CONS}}\xspace \index{cons@\ensuremath{\mathit{CONS}}\xspace (set of all \textsc{Top}\xspace constants)} (constants) and \ensuremath{\mathit{VARS}}\xspace \index{vars@\ensuremath{\mathit{VARS}}\xspace (set of all \textsc{Top}\xspace variables)} (variables), are assumed. I use the suffix ``$^v$'' to distinguish variables from constants. For example, $\mathit{runway^v}, \mathit{gate1^v} \in \ensuremath{\mathit{VARS}}\xspace$, while $\mathit{ba737}, \mathit{1/5/94} \in \ensuremath{\mathit{CONS}}\xspace$. \ensuremath{\mathit{TERMS}}\xspace \index{terms@\ensuremath{\mathit{TERMS}}\xspace (set of all \textsc{Top}\xspace terms)} (\textsc{Top}\xspace terms) is the set $\ensuremath{\mathit{CONS}}\xspace \union \ensuremath{\mathit{VARS}}\xspace$. (\textsc{Top}\xspace has no function symbols.) \paragraph{Predicate functors:} A set of strings \ensuremath{\mathit{PFUNS}}\xspace \index{pfuns@\ensuremath{\mathit{PFUNS}}\xspace (set of all \textsc{Top}\xspace predicate functors)} is assumed. These strings are used as predicate functors (see atomic formulae below). \paragraph{Complete partitioning names:} A set of strings \ensuremath{\mathit{CPARTS}}\xspace \index{cparts@\ensuremath{\mathit{CPARTS}}\xspace (set of all \textsc{Top}\xspace complete partitioning names)} is assumed. These strings represent \emph{complete partitionings} of the time-axis. A complete partitioning of the time-axis is a set of consecutive non-overlapping periods, such that the union of all the periods covers the whole time-axis. (A formal definition will be given in section \ref{top_model}.) For example, the word \qit{day} corresponds to the complete partitioning that contains the period that covers exactly the day 13/10/94, the period that covers exactly 14/10/94, etc. No day-period overlaps another one, and together all the day-periods cover the whole time-axis. Similarly, \qit{month} corresponds to the partitioning that contains the period for October 1994, the period for November 1994, etc. I use the suffix ``$^c$'' for elements of \ensuremath{\mathit{CPARTS}}\xspace. For example, $\mathit{day}^c$ could represent the partitioning of day-periods, and $\mathit{month}^c$ the partitioning of month-periods. \paragraph{Gappy partitioning names:} A set of strings \ensuremath{\mathit{GPARTS}}\xspace \index{gparts@\ensuremath{\mathit{GPARTS}}\xspace (set of all \textsc{Top}\xspace gappy partitioning names)} is assumed. These strings represent \emph{gappy partitionings} of the time-axis. A gappy partitioning of the time-axis is a set of non-overlapping periods, such that the union of all the periods does \emph{not} cover the whole time-axis. For example, \qit{Monday} corresponds to the gappy partitioning that contains the period which covers exactly the Monday 17/10/94, the period that covers exactly the Monday 24/10/94, etc. No Monday-period overlaps another Monday-pariod, and all the Monday-periods together do not cover the whole time-axis. I use the suffix ``$^g$'' for elements of \ensuremath{\mathit{GPARTS}}\xspace. For example, $\mathit{monday}^g$ could represent the partitioning of Monday-periods, and $\text{\textit{5:00pm}}^g$ the partitioning of all 5:00pm-periods (the period that covers exactly the 5:00pm minute of 24/10/94, the period that covers the 5:00pm minute of 25/10/94, etc.). \paragraph{Partitioning names:} \index{parts@\ensuremath{\mathit{PARTS}}\xspace (set of all \textsc{Top}\xspace partitioning names)} \ensuremath{\mathit{PARTS}}\xspace (partitioning names) is the set $\ensuremath{\mathit{CPARTS}}\xspace \union \ensuremath{\mathit{GPARTS}}\xspace$. \paragraph{Atomic formulae:} \index{aforms@\ensuremath{\mathit{AFORMS}}\xspace (set of all \textsc{Top}\xspace atomic formulae)} \ensuremath{\mathit{AFORMS}}\xspace (atomic formulae) is the smallest possible set, such that: \begin{itemize} \item If $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, and $\tau_1, \tau_2, \dots, \tau_n \in \ensuremath{\mathit{TERMS}}\xspace$, then $\pi(\tau_1, \tau_2, \dots, \tau_n) \in \ensuremath{\mathit{AFORMS}}\xspace$. $\pi(\tau_1, \tau_2, \dots, \tau_n)$ is called a \emph{predicate}. $\tau_1, \tau_2, \dots, \tau_n$ are the \emph{arguments} of the predicate. \item \index{part@$\ensuremath{\mathit{Part}}\xspace[\;]$ (used to select periods from partitionings)} If $\sigma \in \ensuremath{\mathit{PARTS}}\xspace$, $\beta \in \ensuremath{\mathit{VARS}}\xspace$, and $\nu_{ord} \in \{\dots, -3, -2, -1, 0\}$, then $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}] \in \ensuremath{\mathit{AFORMS}}\xspace$ and $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta] \in \ensuremath{\mathit{AFORMS}}\xspace$. \end{itemize} Greek letters are used as meta-variables, i.e.\ they stand for expressions of \textsc{Top}\xspace. Predicates (e.g.\ $be\_at(ba737, gate^v)$) describe situations in the world. $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]$ means that $\beta$ is a period in the partitioning $\sigma$. The $\nu_{ord}$ is used to select a particular period from the partitioning. If $\nu_{ord} = 0$, then $\beta$ is the current period of the partitioning (the one that contains the present moment). If $\nu_{ord} < 0$, then $\beta$ is the $-\nu_{ord}$-th period of the partitioning before the current one. When there is no need to select a particular period from a partitioning, the $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ form is used. \paragraph{Yes/no formulae:} Yes/no formulae represent questions that are to be answered with a \qit{yes} or \qit{no} (e.g.\ \qit{Is BA737 circling?}). \ensuremath{\mathit{YNFORMS}}\xspace \index{ynforms@\ensuremath{\mathit{YNFORMS}}\xspace (set of all \textsc{Top}\xspace yes/no formulae)} is the set of all yes/no formulae. It is the smallest possible set, such that if $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, $\tau_1, \dots, \tau_n \in \ensuremath{\mathit{TERMS}}\xspace$, $\phi, \phi_1, \phi_2 \in \ensuremath{\mathit{FORMS}}\xspace$, $\sigma_c \in \ensuremath{\mathit{CPARTS}}\xspace$, $\nu_{qty} \in \{1,2,3,\dots\}$, $\beta$ is a \textsc{Top}\xspace variable that does not occur in $\phi$, and $\tau$ is a \textsc{Top}\xspace variable that does not occur in $\phi$ or a \textsc{Top}\xspace constant, all the following hold. (The restriction that $\beta$ and $\tau$ must not be variables that occur in $\phi$ is needed in the translation from \textsc{Top}\xspace to \textsc{Tsql2}\xspace of chapter \ref{tdb_chapter}.) \begin{itemize} \item $\ensuremath{\mathit{AFORMS}}\xspace \subseteq \ensuremath{\mathit{YNFORMS}}\xspace$ \item $\phi_1 \land \phi_2 \in \ensuremath{\mathit{YNFORMS}}\xspace$ \index{^@$\land$ (\textsc{Top}\xspace's conjunction)} \item \index{pres@$\ensuremath{\mathit{Pres}}\xspace[\;]$ (used to refer to the present)} \index{past@$\ensuremath{\mathit{Past}}\xspace[\;]$ (used to refer to the past)} \index{perf@$\ensuremath{\mathit{Perf}}\xspace[\;]$ (used to express the past perfect)} $\ensuremath{\mathit{Pres}}\xspace[\phi]$, $\ensuremath{\mathit{Past}}\xspace[\beta, \phi]$, $\ensuremath{\mathit{Perf}}\xspace[\beta, \phi] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{at@$\ensuremath{\mathit{At}}\xspace[\;]$ (narrows the localisation time)} $\ensuremath{\mathit{At}}\xspace[\tau, \phi]$, $\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{before@$\ensuremath{\mathit{Before}}\xspace[\;]$ (used to express \qit{before})} \index{after@$\ensuremath{\mathit{After}}\xspace[\;]$ (used to express \qit{after})} $\ensuremath{\mathit{Before}}\xspace[\tau, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\phi_1, \phi_2]$, $\ensuremath{\mathit{After}}\xspace[\tau, \phi]$, $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{ntense@$\ensuremath{\mathit{Ntense}}\xspace[\;]$ (used when expressing nouns or adjectives)} $\ensuremath{\mathit{Ntense}}\xspace[\beta, \phi]$, $\ensuremath{\mathit{Ntense}}\xspace[\mathit{now}^*, \phi] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{for@$\ensuremath{\mathit{For}}\xspace[\;]$ (used to express durations)} \index{fills@$\ensuremath{\mathit{Fills}}\xspace[\;]$ (requires $et = lt$)} $\ensuremath{\mathit{For}}\xspace[\sigma_c, \nu_{qty}, \phi]$, $\ensuremath{\mathit{Fills}}\xspace[\phi] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{begin@$\ensuremath{\mathit{Begin}}\xspace[\;]$ (used to refer to start-points of situations)} \index{end@$\ensuremath{\mathit{End}}\xspace[\;]$ (used to refer to end-points of situations)} $\ensuremath{\mathit{Begin}}\xspace[\phi]$, $\ensuremath{\mathit{End}}\xspace[\phi] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \item \index{culm@$\ensuremath{\mathit{Culm}}\xspace[\;]$ (used to express non-progressives of culminating activity verbs)} $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)] \in \ensuremath{\mathit{YNFORMS}}\xspace$ \end{itemize} No negation and disjunction connectives are defined, because English expressions introducing these connectives are not considered (section \ref{ling_not_supported}). For the same reason no universal quantifiers are defined. All variables can be thought of as existentially quantified. Hence, no explicit existential quantifier is needed. An informal explanation of \textsc{Top}\xspace's operators follows (\textsc{Top}\xspace's semantics will be defined formally in following sections). $\ensuremath{\mathit{Pres}}\xspace[\phi]$ means that $\phi$ is true at the present. For example, \qit{Runway 2 is open.} is represented as $\ensuremath{\mathit{Pres}}\xspace[open(runway2)]$. $\ensuremath{\mathit{Past}}\xspace[\beta, \phi]$ means that $\phi$ is true at some past time $\beta$. The \ensuremath{\mathit{Perf}}\xspace operator is used along with the \ensuremath{\mathit{Past}}\xspace operator to express the past perfect. For example, \qit{Runway 2 was open.} is represented as $\ensuremath{\mathit{Past}}\xspace[e^v, open(runway2)]$, and \qit{Runway 2 had been open.} as: \[\ensuremath{\mathit{Past}}\xspace[e1^v,\ensuremath{\mathit{Perf}}\xspace[e2^v, open(runway2)]] \] $\ensuremath{\mathit{At}}\xspace[\tau, \phi]$ means that $\phi$ holds some time within a period $\tau$, and $\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2]$ means that $\phi_2$ holds at some time where $\phi_1$ holds. For example, \qit{Runway 2 was open (some time) on 1/1/94.} is represented as $\ensuremath{\mathit{At}}\xspace[\mathit{1/1/94}, \ensuremath{\mathit{Past}}\xspace[e^v, open(runway2)]]$, and \qit{Runway 2 was open (some time) while BA737 was circling.} as: \[\ensuremath{\mathit{At}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, circling(ba737)], \ensuremath{\mathit{Past}}\xspace[e2^v, open(runway2)]]\] $\ensuremath{\mathit{Before}}\xspace[\tau, \phi]$ means that $\phi$ is true at some time before a period $\tau$, and $\ensuremath{\mathit{Before}}\xspace[\phi_1, \phi_2]$ means that $\phi_2$ is true at some time before a time where $\phi_1$ is true. $\ensuremath{\mathit{After}}\xspace[\tau, \phi]$ and $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]$ have similar meanings. For example, \qit{Tank 2 was empty (some time) after 1/1/92.} is represented as $\ensuremath{\mathit{After}}\xspace[\mathit{1/1/92}, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$, and \qit{Tank 2 was empty (some time) before the bomb exploded.} as: \[ \ensuremath{\mathit{Before}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, explode(bomb)], \ensuremath{\mathit{Past}}\xspace[e2^v, empty(tank2)]] \] \ensuremath{\mathit{Ntense}}\xspace is used when expressing noun phrases (see section \ref{noun_anaphora}). $\ensuremath{\mathit{Ntense}}\xspace[\beta, \phi]$ means that at a time $\beta$ something has the property specified by $\phi$. $\ensuremath{\mathit{Ntense}}\xspace[\mathit{now}^*, \phi]$ means that something has the property specified by $\phi$ at the present. The reading of \qit{The president was visiting Edinburgh.} that refers to the person who was the president during the visit is represented as $\ensuremath{\mathit{Ntense}}\xspace[e1^v, president(p^v)] \land \ensuremath{\mathit{Past}}\xspace[e1^v, visiting(p^v, edinburgh)]$. In contrast, the reading that refers to the current president is represented as: \[\ensuremath{\mathit{Ntense}}\xspace[\mathit{now}^*,president(p^v)] \land \ensuremath{\mathit{Past}}\xspace[e1^v, visiting(p^v, edinburgh)] \] $\ensuremath{\mathit{For}}\xspace[\sigma_c, \nu_{qty}, \phi]$ means that $\phi$ holds throughout a period that is $\nu_{qty}$ $\sigma_c$-periods long. \qit{Runway 2 was open for two days.} is represented as: \[\ensuremath{\mathit{For}}\xspace[day^c,2, \ensuremath{\mathit{Past}}\xspace[e^v, open(runway2)]] \] The \ensuremath{\mathit{Fills}}\xspace operator is currently not used in the framework of this thesis, but it could be used to capture readings of sentences like \qit{Tank 2 was empty in 1992.} whereby the situation of the verb holds \emph{throughout} the period of the adverbial (see section \ref{period_adverbials}). $\ensuremath{\mathit{At}}\xspace[1992, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Fills}}\xspace[empty(tank2)]]]$ means that the tank was empty \emph{throughout} 1992, while $\ensuremath{\mathit{At}}\xspace[1992, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ means that the tank was empty some time in 1992, but not necessarily throughout 1992. $\ensuremath{\mathit{Begin}}\xspace[\phi]$ means that $\phi$ starts to hold, and $\ensuremath{\mathit{End}}\xspace[\phi]$ means that $\phi$ stops holding. For example, \qit{BA737 started to land.} can be represented as $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Begin}}\xspace[landing(ba737)]]$, and \qit{Tank 2 stopped being empty.} as $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{End}}\xspace[empty(tank2)]]$. Finally, \ensuremath{\mathit{Culm}}\xspace is used to represent sentences where verbs whose base forms are culminating activities appear in tenses that require some inherent climax to have been reached. The \ensuremath{\mathit{Culm}}\xspace operator will be discussed in section \ref{culm_op}. \paragraph{Wh-formulae:} \emph{Wh-formulae} are used to represent questions that contain interrogatives (e.g.\ \qit{Which~\dots?}, \qit{Who~\dots?}, \qit{When~\dots}?). \ensuremath{\mathit{WHFORMS}}\xspace is the set of all wh-formulae. $\ensuremath{\mathit{WHFORMS}}\xspace \defeq \ensuremath{\mathit{WHFORMS}}\xspace_1 \union \ensuremath{\mathit{WHFORMS}}\xspace_2$, \index{whforms@\ensuremath{\mathit{WHFORMS}}\xspace (set of all \textsc{Top}\xspace wh-formulae)} \index{whforms1@$\ensuremath{\mathit{WHFORMS}}\xspace_1$ (set of all \textsc{Top}\xspace wh-formulae with no $?_{mxl}$)} \index{whforms2@$\ensuremath{\mathit{WHFORMS}}\xspace_2$ (set of all \textsc{Top}\xspace wh-formulae with a $?_{mxl}$)} where: \begin{itemize} \item \index{?@$?$ (\textsc{Top}\xspace's interrogative quantifier)} $\ensuremath{\mathit{WHFORMS}}\xspace_1$ is the set of all expressions of the form $?\beta_1 \; ?\beta_2 \; \dots \; ?\beta_n \; \phi$, where $\beta_1, \beta_2, \dots, \beta_n \in \ensuremath{\mathit{VARS}}\xspace$, $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, and each one of $\beta_1, \beta_2, \dots, \beta_n$ occurs at least once within $\phi$. \item \index{?@$?$ (\textsc{Top}\xspace's interrogative quantifier)} \index{?mxl@$?_{mxl}$ (\textsc{Top}\xspace's interrogative-maximal quantifier)} $\ensuremath{\mathit{WHFORMS}}\xspace_2$ is the set of all expressions of the form $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi$, where $\beta_1, \beta_2, \beta_3, \dots, \beta_n \in \ensuremath{\mathit{VARS}}\xspace$, $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, each one of $\beta_2$, $\beta_3$, \dots, $\beta_n$ occurs at least once within $\phi$, and $\beta_1$ occurs at least once within $\phi$ as the first argument of a \ensuremath{\mathit{Past}}\xspace, \ensuremath{\mathit{Perf}}\xspace, \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, \ensuremath{\mathit{After}}\xspace, or \ensuremath{\mathit{Ntense}}\xspace operator, or as the second argument of a \ensuremath{\mathit{Part}}\xspace operator. \end{itemize} ``$?$'' is the \emph{interrogative quantifier}, and $?_{mxl}$ the \emph{interrogative-maximal quantifier}. The interrogative quantifier is similar to an explicit existential quantifier, but it has the additional effect of reporting the values of its variables that satisfy its scope. Intuitively, $?\beta_1 \; ?\beta_2 \; ?\beta_n \; \phi$ means \qit{report all $\beta_1, \beta_2, \dots, \beta_n$ such that $\phi$}. For example, \qit{Which runways are open?} is represented as $?r^v \; \ensuremath{\mathit{Ntense}}\xspace[\mathit{now}^*, runway(r^v)] \land \ensuremath{\mathit{Pres}}\xspace[open(r^v)]$. The constraint that each one of $\beta_1, \dots, \beta_n$ must occur at least once within $\phi$ rules out meaningless formulae like $?o^v \; \ensuremath{\mathit{Past}}\xspace[manager(john)]$, where the $o^v$ does not have any relation to the rest of the formula. This constraint is similar to the notion of \emph{safety} in \textsc{Datalog}\xspace \cite{Ullman}, and it is needed in the translation from \textsc{Top}\xspace to \textsc{Tsql2}\xspace of chapter \ref{tdb_chapter}. The interrogative-maximal quantifier is similar, except that it reports only \emph{maximal periods}. $?_{mxl}$ is intended to be used only with variables that denote periods, and this is why in the case of $?_{mxl}$, $\beta_1$ is required to occur within $\phi$ as the first argument of a \ensuremath{\mathit{Past}}\xspace, \ensuremath{\mathit{Perf}}\xspace, \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, \ensuremath{\mathit{After}}\xspace, or \ensuremath{\mathit{Ntense}}\xspace operator, or as the second argument of a \ensuremath{\mathit{Part}}\xspace operator (the semantics of these operators ensure that variables occurring at these potitions denote periods). Intuitively, $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_n \; \phi$ means \qit{report all the maximal periods $\beta_1$, and all $\beta_2$, \dots, $\beta_n$, such that $\phi$}. The interrogative-maximal quantifier is used in \qit{When \dots?} questions, where we want the answer to contain only the \emph{maximal} periods during which a situation held, not all the periods during which the situation held. If, for example, gate 2 was open from 9:00am to 11:00am and from 3:00pm to 5:00pm, we want the answer to \qit{When was gate 2 open?} to contain only the two maximal periods 9:00am to 11:00am and 3:00pm to 5:00pm; we do not want the answer to contain any subperiods of these two maximal periods (e.g.\ 9:30am to 10:30am). To achieve this, the question is represented as $?_{mxl}e^v \; \ensuremath{\mathit{Past}}\xspace[e^v, open(gate2)]$. \paragraph{Formulae:} \index{forms@\ensuremath{\mathit{FORMS}}\xspace (set of all \textsc{Top}\xspace formulae)} \ensuremath{\mathit{FORMS}}\xspace is the set of all \textsc{Top}\xspace formulae. $\ensuremath{\mathit{FORMS}}\xspace \defeq \ensuremath{\mathit{YNFORMS}}\xspace \union \ensuremath{\mathit{WHFORMS}}\xspace$. \section{The temporal ontology} \label{temporal_ontology} \paragraph{Point structure:} A \emph{point structure} for \textsc{Top}\xspace is an ordered pair $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$, such that \ensuremath{\mathit{PTS}}\xspace \index{pts@\ensuremath{\mathit{PTS}}\xspace (set of all time-points)} is a non-empty set, $\prec$ \index{<@$\prec$ (precedes)} is a binary relation over $\ensuremath{\mathit{PTS}}\xspace \times \ensuremath{\mathit{PTS}}\xspace$, and $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ has the following five properties: \begin{description} \item[transitivity:] If $t_1, t_2, t_3 \in \ensuremath{\mathit{PTS}}\xspace$, $t_1 \prec t_2$, and $t_2 \prec t_3$, then $t_1 \prec t_3$. \item[irreflexivity:] If $t \in \ensuremath{\mathit{PTS}}\xspace$, then $t \prec t$ does not hold. \item[linearity:] If $t_1, t_2 \in \ensuremath{\mathit{PTS}}\xspace$ and $t_1 \not= t_2$, then exactly one of the following holds: $t_1 \prec t_2$ or $t_2 \prec t_1$. \item[left and right boundedness:] There is a $t_{first} \in \ensuremath{\mathit{PTS}}\xspace$, \index{tfirst@$t_{first}$ (earliest time-point)} such that for all $t \in \ensuremath{\mathit{PTS}}\xspace$, $t_{first} \preceq t$. Similarly, there is a $t_{last} \in \ensuremath{\mathit{PTS}}\xspace$, \index{tlast@$t_{last}$ (latest time-point)} such that for all $t \in \ensuremath{\mathit{PTS}}\xspace$, $t \preceq t_{last}$. \item[discreteness:] For every $t_1, t_2 \in \ensuremath{\mathit{PTS}}\xspace$, with $t_1 \not= t_2$, there is at most a finite number of $t_3 \in \ensuremath{\mathit{PTS}}\xspace$, such that $t_1 \prec t_3 \prec t_2$. \end{description} Intuitively, a point structure $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ for \textsc{Top}\xspace is a model of time. \textsc{Top}\xspace models time as being discrete, linear, bounded, and consisting of time-points (see \cite{VanBenthem} for other time models.) \ensuremath{\mathit{PTS}}\xspace is the set of all time-points, and $p_1 \prec p_2$ means that the time-point $p_1$ precedes the time-point $p_2$. \paragraph{prev(t) and next(t):} \index{prev@$prev()$ (previous time-point)} \index{next@$next()$ (next time-point)} If $t_1 \in \ensuremath{\mathit{PTS}}\xspace - \{t_{last}\}$, then $next(t_1)$ denotes a $t_2 \in \ensuremath{\mathit{PTS}}\xspace$, such that $t_1 \prec t_2$ and for no $t_3 \in \ensuremath{\mathit{PTS}}\xspace$ is it true that $t_1 \prec t_3 \prec t_2$. Similarly, if $t_1 \in \ensuremath{\mathit{PTS}}\xspace - \{t_{first}\}$, then $prev(t_1)$ denotes a $t_2 \in \ensuremath{\mathit{PTS}}\xspace$, such that $t_2 \prec t_1$ and for no $t_3 \in \ensuremath{\mathit{PTS}}\xspace$ is it true that $t_2 \prec t_3 \prec t_1$. In the rest of this thesis, whenever $next(t)$ is used, it is assumed that $t \not= t_{last}$. Similarly, whenever $prev(t)$ is used, it is assumed that $t \not= t_{first}$. \paragraph{Periods and instantaneous periods:} A \emph{period} $p$ over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ is a non-empty subset of \ensuremath{\mathit{PTS}}\xspace with the following property: \begin{description} \item[convexity:] If $t_1, t_2 \in p$, $t_3 \in \ensuremath{\mathit{PTS}}\xspace$, and $t_1 \prec t_3 \prec t_2$, then $t_3 \in p$. \end{description} The term ``interval'' is often used in the literature instead of ``period''. Unfortunately, \textsc{Tsql2}\xspace uses ``interval'' to refer to a duration (see chapter \ref{tdb_chapter}). To avoid confusing the reader when \textsc{Tsql2}\xspace will be discussed, I follow the \textsc{Tsql2}\xspace terminology and use the term ``period'' to refer to convex sets of time-points. \index{periods1@$\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ (set of all periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$)} $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ is the set of all periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$. If $p \in \ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ and $p$ contains only one time-point, then $p$ is an \emph{instantaneous period over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$}. $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ \index{instants1@$\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ (set of all instantaneous periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$)} is the set of all instantaneous periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$. For simplicity, I often write \ensuremath{\mathit{PERIODS}}\xspace \index{periods@$\ensuremath{\mathit{PERIODS}}\xspace$ (set of all periods)} and \ensuremath{\mathit{INSTANTS}}\xspace \index{instants@$\ensuremath{\mathit{INSTANTS}}\xspace$ (set of all instantaneous periods)} instead of $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ and $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$, and I often refer to simply ``periods'' and ``instantaneous periods'' instead of ``periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$'' and ``instantaneous periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$''. \index{periods*@$\ensuremath{\mathit{PERIODS}}\xspace^*$ ($\ensuremath{\mathit{PERIODS}}\xspace \union \emptyset$)} $\ensuremath{\mathit{PERIODS}}\xspace^*_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ (or simply $\ensuremath{\mathit{PERIODS}}\xspace^*$) is the set $\ensuremath{\mathit{PERIODS}}\xspace \union \{\emptyset\}$, i.e.\ $\ensuremath{\mathit{PERIODS}}\xspace^*$ is the same as $\ensuremath{\mathit{PERIODS}}\xspace$, except that it also contains the empty set. (The reader is reminded that periods are non-empty sets.) \paragraph{Subperiods:} $p_1$ is a \emph{subperiod} of $p_2$, iff $p_1, p_2 \in \ensuremath{\mathit{PERIODS}}\xspace$ and $p_1 \subseteq p_2$. In this case I write $p_1 \subper p_2$. \index{<sq@$\subper$ (subperiod)} ($p_1 \subseteq p_2$ is weaker than $p_1 \subper p_2$, because it does not guarantee that $p_1, p_2 \in \ensuremath{\mathit{PERIODS}}\xspace$.) Similarly, $p_1$ is a \emph{proper subperiod} of $p_2$, iff $p_1, p_2 \in \ensuremath{\mathit{PERIODS}}\xspace$ and $p_1 \subset p_2$. In this case I write $p_1 \propsubper p_2$. \index{<sq@$\propsubper$ (proper subperiod)} \paragraph{Maximal periods:} \index{mxlpers@$mxlpers()$ (maximal periods of a set or temporal element)} If $S$ is a set of periods, then $\ensuremath{\mathit{mxlpers}}\xspace(S)$ is the set of \emph{maximal periods} of $S$. $\ensuremath{\mathit{mxlpers}}\xspace(S) \defeq \{p \in S \mid \text{for no } p' \in S \text{ is it true that } p \propsubper p'\}$. \paragraph{minpt(S) and maxpt(S):} \index{minpt@$minpt()$ (earliest time-point in a set)} \index{maxpt@$maxpt()$ (latest time-point in a set)} If $S \subseteq \ensuremath{\mathit{PTS}}\xspace$, $minpt(S)$ denotes the time-point $t \in S$, such that for every $t' \in S$, $t \preceq t'$. Similarly, if $S \subseteq \ensuremath{\mathit{PTS}}\xspace$, $maxpt(S)$ denotes the time-point $t \in S$, such that for every $t' \in S$, $t' \preceq t$. \paragraph{Notation:} Following standard conventions, $[t_1, t_2]$ denotes the set $\{t \in \ensuremath{\mathit{PTS}}\xspace \mid t_1 \preceq t \preceq t_2 \}$. (This is not always a period. If $t_2 \prec t_1$, then $[t_1, t_2]$ is the empty set, which is not a period.) Similarly, $(t_1, t_2]$ denotes the set $\{t \in \ensuremath{\mathit{PTS}}\xspace \mid t_1 \prec t \preceq t_2 \}$. $[t_1, t_2)$ and $(t_1,t_2)$ are defined similarly. \section{TOP model} \label{top_model} A \textsc{Top}\xspace model $M$ is an ordered 7-tuple: \[ M = \tup{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}, \ensuremath{\mathit{OBJS}}\xspace, \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{gparts}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace} \] such that $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ is a point structure for \textsc{Top}\xspace (section \ref{temporal_ontology}), $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}} \subseteq \ensuremath{\mathit{OBJS}}\xspace$, and \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{gparts}}}\xspace, and \ensuremath{\mathit{f_{cparts}}}\xspace are as specified below. \paragraph{$\mathbf{OBJS}$:} \index{objs@\ensuremath{\mathit{OBJS}}\xspace (\textsc{Top}\xspace's world objects)} \ensuremath{\mathit{OBJS}}\xspace is a set containing all the objects in the modelled world that can be denoted by \textsc{Top}\xspace terms. For example, in the airport domain \ensuremath{\mathit{OBJS}}\xspace contains all the gates and runways of the airport, the inspectors, the flights, etc. The constraint $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}} \subseteq \ensuremath{\mathit{OBJS}}\xspace$ ensures that all periods are treated as world objects. This simplifies the semantics of \textsc{Top}\xspace. \paragraph{$\mathbf{f_{cons}}$:} \index{fcons@$\ensuremath{\mathit{f_{cons}}}\xspace()$ (maps \textsc{Top}\xspace constants to world objects)} \ensuremath{\mathit{f_{cons}}}\xspace is a function $\ensuremath{\mathit{CONS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace$. (I use the notation $D \mapsto R$ to refer to a function whose domain and range are $D$ and $R$ respectively.) \ensuremath{\mathit{f_{cons}}}\xspace specifies which world object each constant denotes. In the airport domain, for example, \ensuremath{\mathit{f_{cons}}}\xspace may map the constants $gate2$ and $ba737$ to some gate of the airport and some flight respectively. \paragraph{$\mathbf{f_{pfuns}}$:} \index{fpfuns@$\ensuremath{\mathit{f_{pfuns}}}\xspace()$ (returns the maximal periods where predicates hold)} \ensuremath{\mathit{f_{pfuns}}}\xspace is a function that maps each pair $\tup{\pi, n}$, where $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, to another function $(\ensuremath{\mathit{OBJS}}\xspace)^n \mapsto \ensuremath{\mathit{pow}}\xspace(\ensuremath{\mathit{PERIODS}}\xspace)$. ($\ensuremath{\mathit{pow}}\xspace(S)$\/ \index{pow@$\ensuremath{\mathit{pow}}\xspace()$ (powerset)} denotes the powerset of $S$, i.e.\ the set of all subsets of $S$. $(\ensuremath{\mathit{OBJS}}\xspace)^n$ \index{objsn@$(\ensuremath{\mathit{OBJS}}\xspace)^n$ ($\ensuremath{\mathit{OBJS}}\xspace \times \dots \times \ensuremath{\mathit{OBJS}}\xspace$)} is the $n$-ary cartesian product $\ensuremath{\mathit{OBJS}}\xspace \times \ensuremath{\mathit{OBJS}}\xspace \times \dots \times \ensuremath{\mathit{OBJS}}\xspace$.) That is, for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and each $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi,n)$ is a function that maps each $n$-tuple of elements of $\ensuremath{\mathit{OBJS}}\xspace$ to a set of periods (an element of $\ensuremath{\mathit{pow}}\xspace(\ensuremath{\mathit{PERIODS}}\xspace)$). Intuitively, if $\tau_1, \tau_2, \dots, \tau_n$ are \textsc{Top}\xspace terms denoting the world objects $o_1, o_2, \dots, o_n$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(o_1, o_2, \dots, o_n)$ is the set of the maximal periods throughout which the situation described by $\pi(\tau_1, \tau_2, \dots, \tau_n)$ is true. For example, if the constant $ba737$ denotes a flight-object $o_1$, $gate2$ denotes a gate-object $o_2$, and $be\_at(ba737,gate2)$ describes the situation whereby the flight $o_1$ is located at the gate $o_2$, then $\ensuremath{\mathit{f_{pfuns}}}\xspace(be\_at, 2)(o_1, o_2)$ will be the set that contains all the maximal periods throughout which the flight $o_1$ is located at the gate $o_2$. For every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)$ must have the following property: for every $\tup{o_1,o_2,\dots,o_n} \in (\ensuremath{\mathit{OBJS}}\xspace)^n$, it must be the case that: \[ \text{if } p_1, p_2 \in \ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(o_1,o_2,\dots,o_n) \text{ and } p_1 \union p_2 \in \ensuremath{\mathit{PERIODS}}\xspace, \text{ then } p_1 = p_2 \] This ensures that no two different periods $p_1, p_2$ in $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(o_1,\dots,o_n)$ overlap or are adjacent (because if they overlap or they are adjacent, then their union is also a period, and then it must be true that $p_1 = p_2$). Intuitively, if $p_1$ and $p_2$ overlap or are adjacent, we want $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(o_1,o_2,\dots,o_n)$ to contain their union $p_1 \union p_2$ instead of $p_1$ and $p_2$. \paragraph{$\mathbf{f_{culms}}$:} \index{fculms@$\ensuremath{\mathit{f_{culms}}}\xspace()$ (shows if the situation of a predicate reaches its climax)} \ensuremath{\mathit{f_{culms}}}\xspace is a function that maps each pair $\tup{\pi, n}$, where $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, to another function $(\ensuremath{\mathit{OBJS}}\xspace)^n \mapsto \{T,F\}$ ($T,F$ are the two truth values). That is, for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and each $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{f_{culms}}}\xspace(\pi,n)$ is a function that maps each $n$-tuple of elements of \ensuremath{\mathit{OBJS}}\xspace to $T$ or $F$. \ensuremath{\mathit{f_{culms}}}\xspace is only consulted in the case of predicates that represent actions or changes that have inherent climaxes. If $\pi(\tau_1, \tau_2, \dots, \tau_n)$ represents such an action or change, and $\tau_1, \tau_2, \dots, \tau_n$ denote the world objects $o_1, o_2, \dots, o_n$, then $\ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(o_1, o_2, \dots, o_n)$ is the set of all maximal periods throughout which the action or change is ongoing. $\ensuremath{\mathit{f_{culms}}}\xspace(\pi, n)(o_1, o_2, \dots, o_n)$ shows whether or not the change or action reaches its climax at the latest time-point at which the change or action is ongoing. For example, if the constant $j\_adams$ denotes a person $o_1$ in the world, $bridge2$ denotes an object $o_2$, and $building(j\_adams, ba737)$ describes the situation whereby $o_1$ is building $o_2$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(building,2)(o_1,o_2)$ will be the set of all maximal periods where $o_1$ is building $o_2$. $\ensuremath{\mathit{f_{culms}}}\xspace(building,2)(o_1,o_2)$ will be $T$ if the building is completed at the end-point of the latest maximal period in $\ensuremath{\mathit{f_{pfuns}}}\xspace(building,2)(o_1,o_2)$, and $F$ otherwise. The role of \ensuremath{\mathit{f_{culms}}}\xspace will become clearer in section \ref{culm_op}. \paragraph{$\mathbf{f_{gparts}}$:} \index{fgparts@$\ensuremath{\mathit{f_{gparts}}}\xspace()$ (assigns gappy partitionings to elements of \ensuremath{\mathit{GPARTS}}\xspace)} \ensuremath{\mathit{f_{gparts}}}\xspace is a function that maps each element of \ensuremath{\mathit{GPARTS}}\xspace to a \emph{gappy partitioning}. A gappy partitioning is a subset $S$ of \ensuremath{\mathit{PERIODS}}\xspace, such that for every $p_1, p_2 \in S$, $p_1 \intersect p_2 = \emptyset$, and $\bigcup_{p \in S}p \not= \ensuremath{\mathit{PTS}}\xspace$. For example, $\ensuremath{\mathit{f_{gparts}}}\xspace(monday^g)$ could be the gappy partitioning of all Monday-periods. \paragraph{$\mathbf{f_{cparts}}$:} \index{fcparts@$\ensuremath{\mathit{f_{cparts}}}\xspace()$ (assigns complete partitionings to elements of \ensuremath{\mathit{CPARTS}}\xspace)} \ensuremath{\mathit{f_{cparts}}}\xspace is a function that maps each element of \ensuremath{\mathit{CPARTS}}\xspace to a \emph{complete partitioning}. A complete partitioning is a subset $S$ of \ensuremath{\mathit{PERIODS}}\xspace, such that for every $p_1, p_2 \in S$, $p_1 \intersect p_2 = \emptyset$, and $\bigcup_{p \in S}p = \ensuremath{\mathit{PTS}}\xspace$. For example, $\ensuremath{\mathit{f_{cparts}}}\xspace(day^c)$ could be the complete partitioning of all day-periods. \section{Variable assignment} \label{var_assign} A variable assignment with respect to (w.r.t.) a \textsc{Top}\xspace model $M$ is a function $g: \ensuremath{\mathit{VARS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace$ \index{g@$g()$, $g^\beta_o()$ (variable assignment)} ($g$ assigns to each variable an element of \ensuremath{\mathit{OBJS}}\xspace). $G_M$, or simply $G$, \index{G@$G$, $G_M$ (set of all variable assignments)} is the set of all possible variable assignments w.r.t.\ $M$, i.e.\ $G$ is the set of all functions $\ensuremath{\mathit{VARS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace$. If $g \in G$, $\beta \in \ensuremath{\mathit{VARS}}\xspace$, and $o \in \ensuremath{\mathit{OBJS}}\xspace$, then $g^\beta_o$ \index{g@$g()$, $g^\beta_o()$ (variable assignment)} is the variable assignment defined as follows: $g^\beta_o(\beta) = o$, and for every $\beta' \in \ensuremath{\mathit{VARS}}\xspace$ with $\beta' \not= \beta$, $g^\beta_o(\beta') = g(\beta)$. \section{Denotation of a TOP expression} \label{denotation} \paragraph{Index of evaluation:} An index of evaluation is an ordered 3-tuple $\tup{st,et,lt}$, such that $st \in \ensuremath{\mathit{PTS}}\xspace$, $et \in \ensuremath{\mathit{PERIODS}}\xspace$, and $lt \in \ensuremath{\mathit{PERIODS}}\xspace^*$. $st$ \index{st@$st$ (speech time)} (\emph{speech time}) is the time-point at which the English question is submitted to the \textsc{Nlitdb}\xspace. $et$ \index{et@$et$ (event time)} (\emph{event time}) is a period where the situation described by a \textsc{Top}\xspace expression takes place. $lt$ \index{lt@$lt$ (localisation time)} (\emph{localisation time}) can be thought of as a temporal window, within which $et$ must be located. When computing the denotation of a \textsc{Top}\xspace formula that corresponds to an English question, $lt$ is initially set to \ensuremath{\mathit{PTS}}\xspace. That is, the temporal window covers the whole time-axis, and $et$ is allowed to be located anywhere. Various operators, however, may narrow down $lt$, imposing constraints on where $et$ can be placed. \paragraph{Denotation w.r.t.\ M, st, et, lt, g:} The denotation of a \textsc{Top}\xspace expression $\xi$ w.r.t.\ a model $M$, an index of evaluation $\tup{st,et,lt}$, and a variable assignment $g$, is written $\denot{M,st,et,lt,g}{\xi}$ or simply $\denot{st,et,lt,g}{\xi}$. When the denotation of $\xi$ does not depend on $st$, $et$, and $lt$, I often write $\denot{M,g}{\xi}$ or simply $\denot{g}{\xi}$. The denotations w.r.t.\ $M,st,et,lt,g$ of \textsc{Top}\xspace expressions are defined recursively, starting with the denotations of terms and atomic formulae which are defined below. \begin{itemize} \item If $\kappa \in \ensuremath{\mathit{CONS}}\xspace$, then $\denot{g}{\kappa} = \ensuremath{\mathit{f_{cons}}}\xspace(\kappa)$. \item If $\beta \in \ensuremath{\mathit{VARS}}\xspace$, then $\denot{g}{\beta} = g(\beta)$. \item If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, then $\denot{st,et,lt,g}{\phi} \in \{T,F\}$. \end{itemize} The general rule above means that in the case of yes/no formulae, we only need to define when the denotation is $T$. In all other cases the denotation is $F$. \begin{itemize} \item If $\phi_1, \phi_2 \in \ensuremath{\mathit{YNFORMS}}\xspace$, then \index{^@$\land$ (\textsc{Top}\xspace's conjunction)} $\denot{st,et,lt,g}{\phi_1 \land \phi_2} = T$ iff $\denot{st,et,lt,g}{\phi_1} = T$ and $\denot{st,et,lt,g}{\phi_2} = T$. \item \index{part@$\ensuremath{\mathit{Part}}\xspace[\;]$ (used to select periods from partitionings)} If $\sigma \in \ensuremath{\mathit{PARTS}}\xspace$, $\beta \in \ensuremath{\mathit{VARS}}\xspace$, and $\nu_{ord} \in \{\dots, -3, -2, -1, 0\}$, then $\denot{g}{\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]}$ is $T$, iff all the following hold (below $f = \ensuremath{\mathit{f_{cparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{CPARTS}}\xspace$, and $f = \ensuremath{\mathit{f_{gparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{GPARTS}}\xspace$): \begin{itemize} \item $g(\beta) \in f(\sigma)$, \item if $\nu_{ord} = 0$, then $st \in g(\beta)$, \item if $\nu_{ord} \leq -1$, then the following set contains exactly $-\nu_{ord} - 1$ elements: \[ \{ p \in f(\sigma) \mid maxpt(g(\beta)) \prec minpt(p) \text{ and } maxpt(p) \prec st \} \] \end{itemize} \end{itemize} Intuitively, if $\nu_{ord} = 0$, then $\beta$ must denote a period in the partitioning that contains $st$. If $\nu_{ord} \leq -1$, $\beta$ must denote the $-\nu_{ord}$-th period of the partitioning that is completely situated before the speech time (e.g.\ if $\nu_{ord} = -4$, $\beta$ must denote the 4th period which is completely situated before $st$); that is, there must be $-\nu_{ord} - 1$ periods in the partitioning that fall completely between the end of the period denoted by $\beta$ and $st$ ($-(-4) - 1 = 3$ periods if $\nu_{ord} = -4$). For example, if $\ensuremath{\mathit{f_{cparts}}}\xspace(day^c)$ is the partitioning of all day-periods, then $\denot{g}{\ensuremath{\mathit{Part}}\xspace[day^c, \beta, 0]}$ is $T$ iff $g(\beta)$ covers exactly the whole current day. Similarly, $\denot{g}{\ensuremath{\mathit{Part}}\xspace[day^c, \beta, -1]}$ is $T$ iff $g(\beta)$ covers exactly the whole previous day. ($\ensuremath{\mathit{Part}}\xspace[day^c, \beta, 0]$ and $\ensuremath{\mathit{Part}}\xspace[day^c, \beta, -1]$ can be used to represent the meanings of \qit{today} and \qit{yesterday}; see section \ref{at_before_after_op}.) The definition of \ensuremath{\mathit{Part}}\xspace could be extended to allow positive values as its third argument. This would allow expressing \qit{tomorrow}, \qit{next January}, etc. \begin{itemize} \index{part@$\ensuremath{\mathit{Part}}\xspace[\;]$ (used to select periods from partitionings)} \item If $\sigma \in \ensuremath{\mathit{PARTS}}\xspace$ and $\beta \in \ensuremath{\mathit{VARS}}\xspace$, then $\denot{g}{\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]} = T$ iff $g(\beta) \in f(\sigma)$ (where $f = \ensuremath{\mathit{f_{cparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{CPARTS}}\xspace$, and $f = \ensuremath{\mathit{f_{gparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{GPARTS}}\xspace$). \end{itemize} $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ is a simplified version of $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]$, used when we want to ensure that $g(\beta)$ is simply a period in the partitioning of $\sigma$. \begin{itemize} \item If $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $\tau_1, \tau_2, \dots, \tau_n \in \ensuremath{\mathit{TERMS}}\xspace$, then $\denot{st,et,lt,g}{\pi(\tau_1, \tau_2, \dots, \tau_n)}$ is $T$ iff $et \subper lt$ and for some $p_{mxl} \in \ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(\denot{g}{\tau_1}, \denot{g}{\tau_2}, \dots, \denot{g}{\tau_n})$, $et \subper p_{mxl}$. \end{itemize} Intuitively, for the denotation of a predicate to be $T$, $et$ must fall within $lt$, and $et$ must be a subperiod of a maximal period where the situation described by the predicate holds. It is trivial to prove that the definition above causes all \textsc{Top}\xspace predicates to have the following property: \paragraph{Homogeneity:} A \textsc{Top}\xspace formula $\phi$ is \emph{homogeneous}, iff for every $st \in \ensuremath{\mathit{PTS}}\xspace$, $et \in \ensuremath{\mathit{PERIODS}}\xspace$, $lt \in \ensuremath{\mathit{PERIODS}}\xspace^*$, and $g \in G$, the following implication holds:\footnote{The term ``homogeneity'' is also used in the temporal databases literature, but with a completely different meaning; see \cite{tdbsglossary}.} \[ \text{if } et' \subper et \text{ and } \denot{st,et,lt,g}{\phi} = T, \text{ then } \denot{st,et',lt,g}{\phi} = T \] Intuitively, if a predicate is true at some $et$, then it is also true at any subperiod $et'$ of $et$. Although \textsc{Top}\xspace predicates are homogeneous, more complex formulae are not always homogeneous. Various versions of homogeneity have been used in \cite{Allen1984}, \cite{Lascarides}, \cite{Richards}, \cite{Kent}, \cite{Pratt1995}, and elsewhere. The denotation of a wh-formula w.r.t.\ $st$, $et$, $lt$, and $g$ is defined below. It is assumed that $\beta_1, \beta_2, \beta_3, \dots, \beta_n \in \ensuremath{\mathit{VARS}}\xspace$ and $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$. \begin{itemize} \item \index{?@$?$ (\textsc{Top}\xspace's interrogative quantifier)} $\denot{st,et,lt,g}{?\beta_1 \; ?\beta_2 \; \dots \; ?\beta_n \; \phi} = \{\tup{g(\beta_1), g(\beta_2), \dots, g(\beta_n)} \mid \denot{st,et,lt,g}{\phi} = T\}$ \end{itemize} That is, if $\denot{st,et,lt,g}{\phi} = T$, then $\denot{st,et,lt,g}{?\beta_1 \; ?\beta_2 \; \dots \; ?\beta_n \; \phi}$ is a one-element set; it contains one tuple that holds the world-objects assigned to $\beta_1, \beta_2, \dots, \beta_n$ by $g$. Otherwise, $\denot{st,et,lt,g}{?\beta_1 \; ?\beta_2 \; \dots \; ?\beta_n \; \phi}$ is the empty set. \begin{itemize} \item \index{?@$?$ (\textsc{Top}\xspace's interrogative quantifier)} \index{?mxl@$?_{mxl}$ (\textsc{Top}\xspace's interrogative-maximal quantifier)} $ \denot{st,et,lt,g}{?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi} = $ \\ $ \{\tup{g(\beta_1), g(\beta_2), g(\beta_3), \dots, g(\beta_n)} \mid \denot{st,et,lt,g}{\phi} = T \text{, and } $ \\ $ \text{ for no } et' \in \ensuremath{\mathit{PERIODS}}\xspace \text{ and } g' \in G \text{ is it true that } \denot{st,et',lt,g'}{\phi} = T, $ \\ $ g(\beta_1) \propsubper g'(\beta_1), \; g(\beta_2) = g'(\beta_2), \; g(\beta_3) = g'(\beta_3), \; \dots, \; g(\beta_n) = g'(\beta_n)\} $ \end{itemize} The denotation $\denot{st,et,lt,g}{?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi}$ is either a one-element set that contains a tuple holding the world-objects $g(\beta_1), g(\beta_2), \dots, g(\beta_n)$, or the empty set. Intuitively, the denotation of $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi$ contains the values assigned to $\beta_1, \beta_2, \beta_3, \dots, \beta_n$ by $g$, if these values satisfy $\phi$, and there is no other variable assignment $g'$ that assigns the same values to $\beta_2, \beta_3, \dots, \beta_n$, a superperiod of $g(\beta_1)$ to $\beta_1$, and that satisfies $\phi$ (for any $et' \in \ensuremath{\mathit{PERIODS}}\xspace$). That is, it must not be possible to extend any further the period assigned to $\beta_1$ by $g$, preserving at the same time the values assigned to $\beta_2, \beta_3, \dots, \beta_n$, and satisfying $\phi$. Otherwise, the denotation of $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi$ is the empty set. The syntax of \textsc{Top}\xspace (section \ref{top_syntax}) requires $\beta_1$ to appear at least once within $\phi$ as the first argument of a \ensuremath{\mathit{Past}}\xspace, \ensuremath{\mathit{Perf}}\xspace, \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, \ensuremath{\mathit{After}}\xspace, or \ensuremath{\mathit{Ntense}}\xspace operator, or as the second argument of a \ensuremath{\mathit{Part}}\xspace operator. The semantics of these operators require variables occurring at these positions to denote periods. Hence, variable assignments $g$ that do not assign a period to $\beta_1$ will never satisfy $\phi$, and no tuples for these variable assignments will be included in $\denot{st,et,lt,g}{?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_n \; \phi}$. The rules for computing the denotations w.r.t.\ $M,st,et,lt,g$ of other \textsc{Top}\xspace expressions will be given in following sections. \paragraph{Denotation w.r.t.\ M, st:} I now define the denotation of a \textsc{Top}\xspace expression with respect to only $M$ and $st$. The denotation w.r.t.\ $M, st$ is similar to the denotation w.r.t.\ $M, st, et, lt, g$, except that there is an implicit existential quantification over all $g \in G$ and all $et \in \ensuremath{\mathit{PERIODS}}\xspace$, and $lt$ is set to \ensuremath{\mathit{PTS}}\xspace (the whole time-axis). The denotation of $\phi$ w.r.t.\ $M, st$, written $\denot{M,st}{\phi}$ or simply $\denot{st}{\phi}$, is defined only for \textsc{Top}\xspace formulae: \begin{itemize} \item If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, then $\denot{st}{\phi} =$ \begin{itemize} \item $T$, if for some $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$, $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T$, \item $F$, otherwise \end{itemize} \item If $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$, then $\denot{st}{\phi} = \bigcup_{g \in G, \; et \in \ensuremath{\mathit{PERIODS}}\xspace}\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\phi}$. \end{itemize} Each question will be mapped to a \textsc{Top}\xspace formula $\phi$ (if the question is ambiguous, multiple formulae will be generated, one for each reading). $\denot{st}{\phi}$ specifies what the \textsc{Nlitdb}\xspace's answer should report. When $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, $\denot{st}{\phi} = T$ (i.e.\ the answer should be \qit{yes}) if for some assignment to the variables of $\phi$ and for some event time, $\phi$ is satisfied; otherwise $\denot{st}{\phi} = F$ (the answer should be \qit{no}). The localisation time is set to \ensuremath{\mathit{PTS}}\xspace (the whole time-axis) to reflect the fact that initially there is no restriction on where $et$ may be located. As mentioned in section \ref{denotation}, however, when computing the denotations of the subformulae of $\phi$, temporal operators may narrow down the localisation time, placing restrictions on $et$. In the case where $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ (i.e $\phi = ?\beta_1 \; \dots \; ?\beta_n \; \phi'$ or $\phi = ?_{mxl}\beta_1 \; \dots \; ?\beta_n \; \phi'$ with $\phi' \in \ensuremath{\mathit{YNFORMS}}\xspace$), $\denot{st}{\phi}$ is the union of all $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\phi}$, for any $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$. For each $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$, $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\phi}$ is either an empty set or a one-element set containing a tuple that holds values of $\beta_1, \beta_2, \beta_3, \dots, \beta_n$ that satisfy $\phi'$ ($\beta_1$ must be maximal if $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace_2$). Hence, $\denot{st}{\phi}$ (the union of all $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\phi}$) is the set of all tuples that hold values of $\beta_1, \beta_2, \beta_3, \dots, \beta_n$ that satisfy $\phi'$. The answer should report these tuples to the user (or be a message like \qit{No answer found.}, if $\denot{st}{\phi} = \emptyset$). \section{The Pres operator} \label{pres_op} The \ensuremath{\mathit{Pres}}\xspace operator is used to express the simple present and present continuous tenses. For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$: \begin{itemize} \item \index{pres@$\ensuremath{\mathit{Pres}}\xspace[\;]$ (used to refer to the present)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Pres}}\xspace[\phi]} = T$, iff $st \in et$ and $\denot{st,et,lt,g}{\phi} = T$. \end{itemize} \pref{presop:1}, for example, is represented as \pref{presop:2}. \begin{examps} \item Is BA737 at gate 2? \label{presop:1} \item $\ensuremath{\mathit{Pres}}\xspace[be\_at(ba737, gate2)]$ \label{presop:2} \end{examps} Let us assume that the only maximal periods where BA737 was/is/will be at gate 2 are $p_{mxl_1}$ and $p_{mxl_2}$ (i.e.\ \pref{presop:3} holds; see section \ref{top_model}), and that \pref{presop:1} is submitted at a time-point $st_1$, such that \pref{presop:4} holds (figure \ref{pres_op_fig}). \begin{gather} \ensuremath{\mathit{f_{pfuns}}}\xspace(be\_at, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ba737), \ensuremath{\mathit{f_{cons}}}\xspace(gate2)) = \{p_{mxl_1}, p_{mxl_2}\} \label{presop:3} \\ st_1 \in p_{mxl_2} \label{presop:4} \end{gather} \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{pres_op} \caption{\qit{Is BA737 at gate 2?}} \label{pres_op_fig} \end{center} \hrule \end{figure} The answer to \pref{presop:1} will be affirmative iff \pref{presop:5} is $T$. \begin{equation} \denot{st_1}{\ensuremath{\mathit{Pres}}\xspace[be\_at(ba737, gate2)]} \label{presop:5} \end{equation} According to section \ref{denotation}, \pref{presop:5} is $T$ iff for some $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$, \pref{presop:6} holds. \begin{equation} \denot{st_1, et, \ensuremath{\mathit{PTS}}\xspace, g}{\ensuremath{\mathit{Pres}}\xspace[be\_at(ba737, gate2)]} = T \label{presop:6} \end{equation} By the definition of \ensuremath{\mathit{Pres}}\xspace, \pref{presop:6} holds iff both \pref{presop:7} and \pref{presop:8} hold. \begin{gather} st_1 \in et \label{presop:7} \\ \denot{st_1,et,\ensuremath{\mathit{PTS}}\xspace,g}{be\_at(ba737, gate2)} = T \label{presop:8} \end{gather} By the definitions of $\denot{st,et,lt,g}{\pi(\tau_1, \dots, \tau_n)}$ and $\denot{g}{\kappa}$ (section \ref{denotation}), \pref{presop:8} holds iff for some $p_{mxl}$, \pref{presop:9} -- \pref{presop:11} hold. \begin{gather} et \subper \ensuremath{\mathit{PTS}}\xspace \label{presop:9} \\ p_{mxl} \in \ensuremath{\mathit{f_{pfuns}}}\xspace(be\_at, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ba737), \ensuremath{\mathit{f_{cons}}}\xspace(gate2)) \label{presop:10} \\ et \subper p_{mxl} \label{presop:11} \end{gather} By \pref{presop:3}, \pref{presop:10} is equivalent to \pref{presop:13}. \begin{gather} p_{mxl} \in \{p_{mxl_1}, p_{mxl_2}\} \label{presop:13} \end{gather} The answer to \pref{presop:1} will be affirmative iff for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and some $p_{mxl}$, \pref{presop:7}, \pref{presop:9}, \pref{presop:11}, and \pref{presop:13} hold. For $p_{mxl} = p_{mxl_2}$, and $et$ any subperiod of $p_{mxl_2}$ that contains $st_1$ (figure \ref{pres_op_fig}), \pref{presop:7}, \pref{presop:9}, \pref{presop:11}, and \pref{presop:13} hold. Hence, the answer to \pref{presop:1} will be affirmative, as one would expect. If the question is submitted at an $st_2$ that falls outside $p_{mxl_1}$ and $p_{mxl_2}$ (figure \ref{pres_op_fig}), then the answer will be negative, because in that case there is no subperiod $et$ of $p_{mxl_1}$ or $p_{mxl_2}$ that contains $st_2$. The present continuous is expressed similarly. For example, the reading of \pref{presop:14} where Airserve is actually servicing BA737 at the present moment is expressed as \pref{presop:15}. Unlike \cite{Dowty1977}, \cite{Lascarides}, \cite{Pirie1990}, and \cite{Crouch2}, in \textsc{Top}\xspace progressive tenses do not introduce any special progressive operator. This will be discussed in section \ref{culm_op}. \begin{examps} \item Airserve is (actually) servicing BA737. \label{presop:14} \item $\ensuremath{\mathit{Pres}}\xspace[servicing(airserve, ba737)]$ \label{presop:15} \end{examps} The habitual \pref{presop:16} is represented using a different predicate functor from that of \pref{presop:14}, as in \pref{presop:17}. As will be explained in chapter \ref{English_to_TOP}, \pref{presop:14} is taken to involve a non-habitual homonym of \qit{to service}, while \pref{presop:16} is taken to involve a habitual homonym. The two homonyms introduce different predicate functors. \begin{examps} \item Airserve (habitually) services BA737. \label{presop:16} \item $\ensuremath{\mathit{Pres}}\xspace[hab\_server\_of(airserve, ba737)]$ \label{presop:17} \end{examps} \textsc{Top}\xspace's \ensuremath{\mathit{Pres}}\xspace operator is similar to that of \cite{Pirie1990}. The main difference is that the \ensuremath{\mathit{Pres}}\xspace of Pirie et al.\ does not require $st$ to fall within $et$. Instead, it narrows $lt$ to start at or after $st$. This, in combination with the requirement $et \subper lt$, requires $et$ to start at or after $st$. Using this version of \ensuremath{\mathit{Pres}}\xspace in \pref{presop:2} would cause the answer to be affirmative if \pref{presop:1} is submitted at $st_2$ (figure \ref{pres_op_fig}), i.e.\ at a point where BA737 is not at gate 2, because there is an $et$ at which BA737 is at gate 2 (e.g.\ the $et$ of figure \ref{pres_op_fig}), and this $et$ starts after $st_2$. This version of \ensuremath{\mathit{Pres}}\xspace was adopted by Pirie et al.\ to cope with sentences like \qit{J.Adams inspects BA737 tomorrow.}, where the simple present refers to a future inspection (section \ref{simple_present}). In this case, $et$ (inspection time) must be allowed to start after $st$. The \ensuremath{\mathit{Pres}}\xspace of Pirie et al.\ is often over-permissive (e.g.\ it causes the answer to be affirmative if \pref{presop:1} is submitted at $st_2$). Pirie et al.\ employ a post-processing mechanism, which is invoked after the English sentence is translated into logic, and which attempts to restrict the semantics of \ensuremath{\mathit{Pres}}\xspace in cases where it is over-permissive. In effect, this mechanism introduces modifications in only one case: if the \ensuremath{\mathit{Pres}}\xspace is introduced by a state verb (excluding progressive states) which is not modified by a temporal adverbial, then $et$ is set to $\{st\}$. For example, in \qit{J.Adams is at site 2.} where the verb is a state, the mechanism causes $et$ to be set to $\{st\}$, which correctly requires J.Adams to be at gate 2 at $st$. In \qit{J.Adams is at site 2 tomorrow.}, where the state verb is modified by a temporal adverbial, the post-processing has no effect, and $et$ (the time where J.Adams is at site 2) is allowed to start at or after $st$. This is again correct, since in this case $et$ must be located within the following day, i.e.\ after $st$. In \qit{J.Adams is inspecting site 2.}, where the verb is a progressive state, the post-processing has again no effect, and $et$ (inspection time) can start at or after $st$. The rationale in this case is that $et$ cannot be set to $\{st\}$, because there is a reading where the present continuous refers to a future inspection (section \ref{progressives}). For the purposes of this project, where the futurate readings of the simple present and the present continuous are ignored, \textsc{Top}\xspace's \ensuremath{\mathit{Pres}}\xspace is adequate. If, however, these futurate readings were to be supported, a more permissive \ensuremath{\mathit{Pres}}\xspace operator, like that of Pirie et al., might have to be adopted. \section{The Past operator} \label{past_op} The \ensuremath{\mathit{Past}}\xspace operator is used when expressing the simple past, the past continuous, the past perfect, and the present perfect (the latter is treated as equivalent to the simple past; section \ref{present_perfect}). For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $\beta \in \ensuremath{\mathit{VARS}}\xspace$: \begin{itemize} \item \index{past@$\ensuremath{\mathit{Past}}\xspace[\;]$ (used to refer to the past)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Past}}\xspace[\beta, \phi]} = T$, iff $g(\beta) = et$ and $\denot{st,et, lt \intersect [t_{first}, st), g}{\phi} = T$. \end{itemize} The \ensuremath{\mathit{Past}}\xspace operator narrows the localisation time, so that the latter ends before $st$. $et$ will eventually be required to be a subperiod of the localisation time (this requirement will be introduced by the rules that compute the denotation of $\phi$). Hence, $et$ will be required to end before $st$. $\beta$ is used as a pointer to $et$ (the definition of $\ensuremath{\mathit{Past}}\xspace[\beta, \phi]$ makes sure that the value of $\beta$ is $et$). $\beta$ is useful in formulae that contain \ensuremath{\mathit{Ntense}}\xspace{s} (to be discussed in section \ref{ntense_op}). It is also useful in time-asking questions, where $et$ has to be reported. For example, \qit{When was gate 2 open?} is represented as $?_{mxl}e^v \; \ensuremath{\mathit{Past}}\xspace[e^v, open(gate2)]$, which reports the maximal $et$s that end before $st$, such that gate 2 is open throughout $et$. \textsc{Top}\xspace's \ensuremath{\mathit{Past}}\xspace operator is essentially the same as that of \cite{Pirie1990}. (A slightly different \ensuremath{\mathit{Past}}\xspace operator is adopted in \cite{Crouch2}.) \section{Progressives, non-progressives, and the Culm operator} \label{culm_op} Let us now examine in more detail how \textsc{Top}\xspace represents the simple past and the past continuous. Let us start from verbs whose base forms are culminating activities, like \qit{to inspect} in the airport domain. The past continuous \pref{culmop:1} is represented as \pref{culmop:2}. \begin{examps} \item Was J.Adams inspecting BA737? \label{culmop:1} \item $\ensuremath{\mathit{Past}}\xspace[e^v, inspecting(j\_adams, ba737)]$ \label{culmop:2} \end{examps} Let us assume that the inspection of BA737 by J.Adams started at the beginning of $p_{mxl_1}$ (figure \ref{culm_op_fig}), that it stopped temporarily at the end of $p_{mxl_1}$, that it was resumed at the beginning of $p_{mxl_2}$, and that it was completed at the end of $p_{mxl_2}$. Let us also assume that there is no other time at which J.Adams was/is/will be inspecting BA737. Then, \pref{culmop:3} and \pref{culmop:3.2} hold. \begin{gather} \ensuremath{\mathit{f_{pfuns}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(j\_adams), \ensuremath{\mathit{f_{cons}}}\xspace(ba737)) = \{p_{mxl_1},p_{mxl_2}\} \label{culmop:3} \\ \ensuremath{\mathit{f_{culms}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(j\_adams), \ensuremath{\mathit{f_{cons}}}\xspace(ba737)) = T \label{culmop:3.2} \end{gather} \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{culm_op} \caption{\qit{Was J.Adams inspecting BA737?} vs.\ \qit{Did J.Adams inspect BA737?}} \label{culm_op_fig} \end{center} \hrule \end{figure} The reader can check that \pref{culmop:4} is $T$ iff there is an $et$ that is a subperiod of $p_{mxl_1}$ or $p_{mxl_2}$, and that ends before $st$. \begin{equation} \label{culmop:4} \denot{st}{\ensuremath{\mathit{Past}}\xspace[e^v, inspecting(j\_adams, ba737)]} \end{equation} If \pref{culmop:1} is submitted at $st_1$ or $st_2$ (figure \ref{culm_op_fig}), then \pref{culmop:4} is $T$ (the answer to \pref{culmop:1} will be \qit{yes}), because in both cases there is an $et$ (e.g.\ the $et_1$ of figure \ref{culm_op_fig}) that ends before $st_1$ and $st_2$, and that is a subperiod of $p_{mxl_1}$. In contrast, if the question is submitted at $st_3$, \pref{culmop:4} is $F$ (the answer will be negative), because in this case there is no subperiod of $p_{mxl_1}$ or $p_{mxl_2}$ that ends before $st_3$. This is what one would expect: at $st_1$ and $st_2$ the answer to \pref{culmop:1} should be affirmative, because J.Adams has already spent some time inspecting BA737. In contrast, at $st_3$ J.Adams has not yet spent any time inspecting BA737, and the answer should be negative. Let us now consider the simple past \pref{culmop:5}. We want the answer to be affirmative if \pref{culmop:5} is submitted at $st_1$ (or any other time-point after the end of $p_{mxl_2}$), but not if it is submitted at $st_2$ (or any other time-point before the end of $p_{mxl_2}$), because at $st_2$ J.Adams has not yet completed the inspection (section \ref{simple_past}). \begin{examps} \item Did J.Adams inspect BA737? \label{culmop:5} \end{examps} \pref{culmop:5} cannot be represented as \pref{culmop:2}, because this would cause the answer to \pref{culmop:5} to be affirmative if the question is submitted at $st_2$. Instead, \pref{culmop:5} is represented as \pref{culmop:6}. The same predicate $inspecting(j\_adams, ba737)$ of \pref{culmop:2} is used, but an additional \ensuremath{\mathit{Culm}}\xspace operator is inserted. \begin{equation} \label{culmop:6} \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]] \end{equation} Intuitively, the \ensuremath{\mathit{Culm}}\xspace requires the event time to be the $et_2$ of figure \ref{culm_op_fig}, i.e.\ to cover the whole time from the point where the inspection starts to the point where the inspection is completed. (If the inspection is never completed, \ensuremath{\mathit{Culm}}\xspace causes the denotation of \pref{culmop:6} to be $F$.) Combined with the \ensuremath{\mathit{Past}}\xspace operator, the \ensuremath{\mathit{Culm}}\xspace causes the answer to be affirmative if \pref{culmop:5} is submitted at $st_1$ (because $et_2$ ends before $st_1$), and negative if the question is submitted at $st_2$ (because $et_2$ does not end before $st_2$). More formally, for $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $\tau_1, \dots, \tau_n \in \ensuremath{\mathit{TERMS}}\xspace$: \begin{itemize} \item \index{culm@$\ensuremath{\mathit{Culm}}\xspace[\;]$ (used to express non-progressives of culminating activity verbs)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]} = T$, iff $et \subper lt$, $\ensuremath{\mathit{f_{culms}}}\xspace(\pi,n)(\denot{g}{\tau_1}, \dots, \denot{g}{\tau_n}) = T$, $S \not= \emptyset$, and $et = [minpt(S), maxpt(S)]$, where: \[ S = \bigcup_{p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(\pi, n)(\denot{g}{\tau_1}, \dots, \denot{g}{\tau_n})}p \] \end{itemize} The $et = [minpt(S), maxpt(S)]$ requires $et$ to start at the first time-point where the change or action of $\pi(\tau_1, \dots, \tau_n)$ is ongoing, and to end at the latest time-point where the change or action is ongoing. The $\ensuremath{\mathit{f_{culms}}}\xspace(\pi)(\denot{g}{\tau_1}, \dots, \denot{g}{\tau_n}) = T$ means that the change or action must reach its climax at the latest time-point where it is ongoing. Let us now check formally that the denotation \pref{culmop:10} of \pref{culmop:6} is in order. \begin{equation} \label{culmop:10} \denot{st}{\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]]} \end{equation} \pref{culmop:10} is $T$ iff for some $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$, \pref{culmop:11} holds. \begin{equation} \label{culmop:11} \denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g}{\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]]} = T \end{equation} By the definition of \ensuremath{\mathit{Past}}\xspace, \pref{culmop:11} holds iff \pref{culmop:12} and \pref{culmop:13} hold ($\ensuremath{\mathit{PTS}}\xspace \intersect [t_{first}, st) = [t_{first}, st)$). \begin{gather} g(e^v) = et \label{culmop:12} \\ \denot{st,et, [t_{first}, st), g}{\ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]} = T \label{culmop:13} \end{gather} By the definition of \ensuremath{\mathit{Culm}}\xspace, \pref{culmop:13} holds iff \pref{culmop:14} -- \pref{culmop:18} hold. \begin{gather} et \subper [t_{first}, st) \label{culmop:14} \\ \ensuremath{\mathit{f_{culms}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(j\_adams), \ensuremath{\mathit{f_{cons}}}\xspace(ba737)) = T \label{culmop:15} \\ S \not= \emptyset \label{culmop:17} \\ et = [minpt(S), maxpt(S)] \label{culmop:16} \\ S = \bigcup_{p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(j\_adams), \ensuremath{\mathit{f_{cons}}}\xspace(ba737))}p \label{culmop:18} \end{gather} By \pref{culmop:3}, and assuming that $maxpt(p_{mxl_1}) \prec minpt(p_{mxl_2})$ (as in figure \ref{culm_op_fig}), \pref{culmop:17} -- \pref{culmop:18} are equivalent to \pref{culmop:19} -- \pref{culmop:20}. \pref{culmop:19} holds (the union of two periods is never the empty set), and \pref{culmop:15} is the same as \pref{culmop:3.2}, which was assumed to hold. \begin{gather} p_{mxl_1} \union p_{mxl_2} \not= \emptyset \label{culmop:19} \\ et = [minpt(p_{mxl_1}), maxpt(p_{mxl_2})] \label{culmop:20} \end{gather} Hence, \pref{culmop:10} is $T$ (i.e.\ the answer to \pref{culmop:5} is affirmative) iff for some $g \in G$ and $et \in \ensuremath{\mathit{PTS}}\xspace$, \pref{culmop:12}, \pref{culmop:14}, and \pref{culmop:20} hold. Let $et_2 = [minpt(p_{mxl_1}), maxpt(p_{mxl_2})]$ (as in figure \ref{culm_op_fig}). Let us assume that \pref{culmop:5} is submitted at an $st$ that follows the end of $et_2$ (e.g.\ $st_1$ in figure \ref{culm_op_fig}). For $et = et_2$, \pref{culmop:14} and \pref{culmop:20} are satisfied. \pref{culmop:12} is also satisfied by choosing $g = g_1$, where $g_1$ as below. Hence, the answer to \pref{culmop:5} will be affirmative, as required. \[g_1(\beta) = \begin{cases} et_2 & \text{if } \beta = e^v \\ o & \text{otherwise ($o$ is an arbitrary element of \ensuremath{\mathit{OBJS}}\xspace)} \end{cases} \] In contrast, if the question is submitted before the end of $et_2$ (e.g.\ $st_2$ or $st_3$ in figure \ref{culm_op_fig}), then the answer to \pref{culmop:5} will be negative, because there is no $et$ that satisfies \pref{culmop:14} and \pref{culmop:20}. In the case of verbs whose base forms are processes, states, or points, the simple past does not introduce a \ensuremath{\mathit{Culm}}\xspace operator. In this case, when both the simple past and the past continuous are possible, they are represented using the same \textsc{Top}\xspace formula. (A similar approach is adopted in \cite{Parsons1989}.) For example, in the airport domain where \qit{to circle} is classified as process, both \pref{culmop:28} and \pref{culmop:29} are represented as \pref{culmop:30}. \begin{examps} \item Was BA737 circling? \label{culmop:28} \item Did BA737 circle? \label{culmop:29} \item $\ensuremath{\mathit{Past}}\xspace[e^v, circling(ba737)]$ \label{culmop:30} \end{examps} The reader can check that the denotation of \pref{culmop:30} w.r.t.\ $st$ is $T$ (i.e.\ the answer to \pref{culmop:28} and \pref{culmop:29} is affirmative) iff there is an $et$ which is a subperiod of a maximal period where BA737 was circling, and $et$ ends before $st$. That is, the answer is affirmative iff BA737 was circling at some time before $st$. There is no requirement that any climax must have been reached. The reader will have noticed that in the case of verbs whose base forms are culminating activities, the (non-progressive) simple past is represented by adding a \ensuremath{\mathit{Culm}}\xspace operator to the expression that represents the (progressive) past continuous. For example, assuming that \qit{to build (something)} is a culminating activity, \pref{culmop:36} is represented as \pref{culmop:37}, and \pref{culmop:38} as \pref{culmop:39}). \begin{examps} \item Housecorp was building bridge 2. \label{culmop:36} \item $\ensuremath{\mathit{Past}}\xspace[e^v, building(housecorp, bridge2)]$ \label{culmop:37} \item Housecorp built bridge 2. \label{culmop:38} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[building(housecorp, bridge2)]]$ \label{culmop:39} \end{examps} In contrast, in \cite{Dowty1977}, \cite{Lascarides}, \cite{Pirie1990}, \cite{Crouch2}, and \cite{Kamp1993}, progressive tenses are represented by adding a progressive operator to the expressions that represent the non-progressive tenses. For example, ignoring some details, Pirie et al.\ represent \pref{culmop:36} and \pref{culmop:38} as \pref{culmop:44} and \pref{culmop:46} respectively. \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Prog}}\xspace[build(housecorp, bridge2)]]$ \label{culmop:44} \item $\ensuremath{\mathit{Past}}\xspace[e^v, build(housecorp, bridge2)]$ \label{culmop:46} \end{examps} In \pref{culmop:46}, the semantics that Pirie et al.\ assign to $build(housecorp, bridge2)$ require $et$ to cover the whole building of the bridge by Housecorp, from its beginning to the point where the building is complete. (The semantics of \textsc{Top}\xspace's $building(housecorp, bridge2)$ in \pref{culmop:37} require $et$ to be simply a period throughout which Housecorp is building bridge 2.) The \ensuremath{\mathit{Past}}\xspace of \pref{culmop:46} requires $et$ (start to completion of inspection) to end before $st$. Hence, the answer to \pref{culmop:38} is affirmative iff the building was completed before $st$. In \pref{culmop:44}, the semantics that Pirie et al.\ assign to \ensuremath{\mathit{Prog}}\xspace require $et$ to be a subperiod of another period $et'$ that covers the whole building (from start to completion; figure \ref{prog_op_fig}). The \ensuremath{\mathit{Past}}\xspace of \pref{culmop:44} requires $et$ to end before $st$. If, for example, \pref{culmop:36} is submitted at an $st$ that falls between the end of $et$ and the end of $et'$ (figure \ref{prog_op_fig}), the answer will be affirmative. This is correct, because at that $st$ Housecorp has already been building the bridge for some time (although the bridge is not yet complete). \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{prog_op} \caption{A flawed Prog operator} \label{prog_op_fig} \end{center} \hrule \end{figure} The \ensuremath{\mathit{Prog}}\xspace of Pirie et al., however, has a flaw (acknowledged in \cite{Crouch2}): \pref{culmop:44} implies that there is a period $et'$, such that the building is completed at the end of $et'$; i.e.\ according to \pref{culmop:44} the building was or will be necessarily completed at some time-point. This does not capture correctly the semantics of \pref{culmop:36}. \pref{culmop:36} carries no implication that the building was or will ever be completed. (\textsc{Top}\xspace's representation of \pref{culmop:36}, i.e.\ \pref{culmop:37}, does not suffer from this problem: it contains no assumption that the building is ever completed.) To overcome similar problems with \ensuremath{\mathit{Prog}}\xspace operators, ``branching'' models of time or ``possible worlds'' have been employed (see, for example, \cite{Dowty1977}, \cite{McDermott1982}, \cite{Mays1986}, \cite{Kent}; see also \cite{Lascarides} for criticism of possible-worlds approaches to progressives.) Approaches based on branching time and possible worlds, however, seem unnecessarily complicated for the purposes of this thesis. \section{The At, Before, and After operators} \label{at_before_after_op} The \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, and \ensuremath{\mathit{After}}\xspace operators are used to express punctual adverbials, period adverbials, and \qit{while~\dots}, \qit{before~\dots}, and \qit{after~\dots} subordinate clauses (sections \ref{temporal_adverbials} and \ref{subordinate_clauses}). For $\phi, \phi_1, \phi_2 \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $\tau \in \ensuremath{\mathit{TERMS}}\xspace$: \begin{itemize} \item \index{at@$\ensuremath{\mathit{At}}\xspace[\;]$ (narrows the localisation time)} $\denot{st,et,lt,g}{\ensuremath{\mathit{At}}\xspace[\tau, \phi]} = T$, iff $\denot{g}{\tau} \in \ensuremath{\mathit{PERIODS}}\xspace$ and $\denot{st,et,lt \intersect \denot{g}{\tau},g}{\phi} = T$. \item \index{at@$\ensuremath{\mathit{At}}\xspace[\;]$ (narrows the localisation time)} $\denot{st,et,lt,g}{\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2]} = T$, iff for some $et'$ \\ $et' \in mxlpers(\{e \in \ensuremath{\mathit{PERIODS}}\xspace \mid \denot{st,e,\ensuremath{\mathit{PTS}}\xspace,g}{\phi_1} = T\})$ and $\denot{st,et,lt \intersect et',g}{\phi_2} = T$. \end{itemize} If the first argument of \ensuremath{\mathit{At}}\xspace is a term $\tau$, then $\tau$ must denote a period. The localisation time is narrowed to the intersection of the original $lt$ with the period of $\tau$. If the first argument of \ensuremath{\mathit{At}}\xspace is a formula $\phi_1$, the localisation time of $\phi_2$ is narrowed to the intersection of the original $lt$ with a maximal event time period $et'$ at which $\phi_1$ holds. For example, \pref{atop:1} is represented as \pref{atop:2}. \begin{examps} \item Was tank 2 empty (some time) on 25/9/95? \label{atop:1} \item $\ensuremath{\mathit{At}}\xspace[\mathit{25/9/95}, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{atop:2} \end{examps} In \pref{atop:2}, $lt$ initially covers the whole time-axis. The \ensuremath{\mathit{At}}\xspace operator causes $lt$ to become the 25/9/95 period (I assume that the constant $\mathit{25/9/95}$ denotes the obvious period), and the \ensuremath{\mathit{Past}}\xspace operator narrows $lt$ to end before $st$ (if 25/9/95 is entirely in the past, the \ensuremath{\mathit{Past}}\xspace operator has not effect). The answer to \pref{atop:1} is affirmative iff it is possible to find an $et$ that is a subperiod of the narrowed $lt$, such that tank 2 was empty during $et$. If \pref{atop:1} is submitted before 25/9/95 (i.e.\ 25/9/95 starts after $st$), the \textsc{Nlitdb}\xspace's answer will be negative, because the \ensuremath{\mathit{At}}\xspace and \ensuremath{\mathit{Past}}\xspace operators cause $lt$ to become the empty set, and hence it is impossible to find a subperiod $et$ of $lt$ where tank 2 is empty. A simple negative response is unsatisfactory in this case: \pref{atop:1} is unacceptable if uttered before 25/9/95, and the system should warn the user about this. The unacceptability of \pref{atop:1} in this case seems related to the unacceptability of \pref{atop:20}, which would be represented as \pref{atop:21} (the definition of \ensuremath{\mathit{Part}}\xspace would have to be extended to allow positive values of its third argument; see section \ref{denotation}). \begin{examps} \item \bad Was tank 2 empty tomorrow? \label{atop:20} \item $\ensuremath{\mathit{Part}}\xspace[day^c, tom^v, 1] \land \ensuremath{\mathit{At}}\xspace[tom^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{atop:21} \end{examps} In both cases, the combination of the simple past and the adverbial causes $lt$ to become the empty set. In \pref{atop:20}, $tom^v$ denotes the period that covers exactly the day after $st$. The \ensuremath{\mathit{At}}\xspace and \ensuremath{\mathit{Past}}\xspace operators set $lt$ to the intersection of that period with $[t_{first}, st)$. The two periods do not overlap, and hence $lt = \emptyset$, and it is impossible to find a subperiod $et$ of $lt$. This causes the answer to be always negative, no matter what happens in the world (i.e.\ regardless of when tank 2 is empty). Perhaps the questions sound unacceptable because people, using a concept similar to \textsc{Top}\xspace's $lt$, realise that the answers can never be affirmative. This suggests that the \textsc{Nlitdb}\xspace should check if $lt = \emptyset$, and if this is the case, generate a cooperative response (section \ref{no_issues}) explaining that the question is problematic (this is similar to the ``overlap rule'' of \cite{Harper} and the ``non-triviality constraint'' on p.~653 of \cite{Kamp1993}). The framework of this thesis currently provides no such mechanism. Moving to further examples, \pref{atop:3} and \pref{atop:22} are represented as \pref{atop:4} and \pref{atop:23}. Unlike the \qit{on 25/9/95} of \pref{atop:1}, which is represented using a constant ($\mathit{25/9/95}$), the \qit{on Monday} of \pref{atop:3} is represented using a variable ($mon^v$) that ranges over the periods of the partitioning of Monday-periods. Similarly, the \qit{at 5:00pm} of \pref{atop:22} is represented using a variable ($fv^v$) that ranges over the 5:00pm minute-periods. \begin{examps} \item Was tank 2 empty on Monday? \label{atop:3} \item Was tank 2 empty on a Monday? \label{atop:24} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, mon^v] \land \ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{atop:4} \item Was tank 2 empty on Monday at 5:00pm? \label{atop:22} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, mon^v] \land \ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]]$ \label{atop:23} \end{examps} \pref{atop:4} requires tank 2 to have been empty at some past $et$ that falls within some Monday. No attempt is made to determine exactly which Monday the user has in mind in \pref{atop:3} (\pref{atop:3} is treated as equivalent to \pref{atop:24}; section \ref{temporal_anaphora}). Similarly, \pref{atop:23} requires tank 2 to have been empty at some past $et$ that falls within the intersection of some 5:00pm-period with some Monday-period. Assuming that \qit{to inspect} is a culminating activity (as in the airport application), the reading of \pref{atop:28} that requires the inspection to have both started and been completed within the previous day (section \ref{period_adverbials}) is represented as \pref{atop:29}. The \ensuremath{\mathit{Culm}}\xspace requires $et$ to cover exactly the whole inspection, from its beginning to its completion. The \ensuremath{\mathit{Past}}\xspace requires $et$ to end before $st$, and the \ensuremath{\mathit{At}}\xspace requires $et$ to fall within the day before $st$. \begin{examps} \item Did J.Adams inspect BA737 yesterday? \label{atop:28} \item $\ensuremath{\mathit{Part}}\xspace[day^c, y^v, -1] \land \ensuremath{\mathit{At}}\xspace[y^v, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]]]$ \label{atop:29} \end{examps} In contrast, \pref{atop:26} is represented as \pref{atop:27}. In this case, $et$ must be simply a subperiod of a maximal period where J.Adams was inspecting BA737, and also be located within the previous day. \begin{examps} \item Was J.Adams inspecting BA737 yesterday? \label{atop:26} \item $\ensuremath{\mathit{Part}}\xspace[day^c, y^v, -1] \land \ensuremath{\mathit{At}}\xspace[y^v, \ensuremath{\mathit{Past}}\xspace[e^v, inspecting(j\_adams, ba737)]]$ \label{atop:27} \end{examps} Finally, \pref{atop:30} is represented as \pref{atop:31}, which intuitively requires BA737 to have been circling at some past period $e2^v$, that falls within some past maximal period $e1^v$ where gate 2 was open. \begin{examps} \item Did BA737 circle while gate 2 was open? \label{atop:30} \item $\ensuremath{\mathit{At}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, open(gate2)], \ensuremath{\mathit{Past}}\xspace[e2^v, circling(ba737)]]$ \label{atop:31} \end{examps} The \ensuremath{\mathit{Before}}\xspace and \ensuremath{\mathit{After}}\xspace operators are similar. They are used to express adverbials and subordinate clauses introduced by \qit{before} and \qit{after}. For $\phi, \phi_1, \phi_2 \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $\tau \in \ensuremath{\mathit{TERMS}}\xspace$: \begin{itemize} \item \index{before@$\ensuremath{\mathit{Before}}\xspace[\;]$ (used to express \qit{before})} $\denot{st,et,lt,g}{\ensuremath{\mathit{Before}}\xspace[\tau, \phi]} = T$, iff $\denot{g}{\tau} \in \ensuremath{\mathit{PERIODS}}\xspace$ and \\ $\denot{st,et, lt \intersect [t_{first}, minpt(\denot{g}{\tau})), g}{\phi} = T$. \item \index{before@$\ensuremath{\mathit{Before}}\xspace[\;]$ (used to express \qit{before})} $\denot{st,et,lt,g}{\ensuremath{\mathit{Before}}\xspace[\phi_1, \phi_2]} = T$, iff for some $et'$ \\ $et' \in mxlpers(\{e \in \ensuremath{\mathit{PERIODS}}\xspace \mid \denot{st,e,\ensuremath{\mathit{PTS}}\xspace,g}{\phi_1} = T\})$, and \\ $\denot{st,et, lt \intersect [t_{first}, minpt(et')), g}{\phi_2} = T$. \item \index{after@$\ensuremath{\mathit{After}}\xspace[\;]$ (used to express \qit{after})} $\denot{st,et,lt,g}{\ensuremath{\mathit{After}}\xspace[\tau, \phi]} = T$, iff $\denot{g}{\tau} \in \ensuremath{\mathit{PERIODS}}\xspace$ and \\ $\denot{st,et, lt \intersect (maxpt(\denot{g}{\tau}), t_{last}], g}{\phi} = T$. \item \index{after@$\ensuremath{\mathit{After}}\xspace[\;]$ (used to express \qit{after})} $\denot{st,et,lt,g}{\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]} = T$, iff for some $et'$ \\ $et' \in mxlpers(\{e \mid \denot{st,e,\ensuremath{\mathit{PTS}}\xspace,g}{\phi_1} = T\})$ and \\ $\denot{st,et, lt \intersect (maxpt(et'), t_{last}], g}{\phi_2} = T$. \end{itemize} If the first argument of \ensuremath{\mathit{Before}}\xspace is a term $\tau$, $\tau$ must denote a period. The localisation time is required to end before the beginning of $\tau$'s period. If the first argument of \ensuremath{\mathit{Before}}\xspace is a formula $\phi_1$, the localisation time of $\phi_2$ is required to end before the beginning of a maximal event time period $et'$ where $\phi_1$ holds. The \ensuremath{\mathit{After}}\xspace operator is similar. For example, \pref{atop:35} is expressed as \pref{atop:36}, and the reading of \pref{atop:37} that requires BA737 to have departed after the \emph{end} of a maximal period where the emergency system was in operation is expressed as \pref{atop:38}. (I assume here that \qit{to depart} is a point, as in the airport application.) \begin{examps} \item Was tank 2 empty before 25/9/95? \label{atop:35} \item $\ensuremath{\mathit{Before}}\xspace[\mathit{25/9/95}, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{atop:36} \item BA737 departed after the emergency system was in operation. \label{atop:37} \item $\ensuremath{\mathit{After}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, in\_operation(emerg\_sys)], \ensuremath{\mathit{Past}}\xspace[e2^v, depart(ba737)]]$ \label{atop:38} \end{examps} \pref{atop:37} also has a reading where BA737 must have departed after the emergency system \emph{started} to be in operation (section \ref{before_after_clauses}). To express this reading, we need the \ensuremath{\mathit{Begin}}\xspace operator of section \ref{begin_end_op} below. \textsc{Top}\xspace's \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, and \ensuremath{\mathit{After}}\xspace operators are similar to those of \cite{Pirie1990}. The operators of Pirie et al., however, do not narrow $lt$ as in \textsc{Top}\xspace. Instead, they place directly restrictions on $et$. For example, ignoring some details, the $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]$ of Pirie et al.\ requires $\phi_2$ to hold at an event time $et_2$ that follows an $et_1$ where $\phi_1$ holds (both $et_1$ and $et_2$ must fall within $lt$). Instead, \textsc{Top}\xspace's $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]$ requires $et_1$ to be a maximal period where $\phi_1$ holds ($et_1$ does not need to fall within the original $lt$), and evaluates $\phi_2$ with respect to a narrowed $lt$, which is the intersection of the original $lt$ with $et_1$. In most cases, both approaches lead to similar results. \textsc{Top}\xspace's approach, however, is advantageous in sentences like \pref{atop:60}, where one may want to express the reading whereby the tank was empty \emph{throughout} 26/9/95 (section \ref{period_adverbials}). \begin{examps} \item Tank 2 was empty on 26/9/95. \label{atop:60} \item $\ensuremath{\mathit{At}}\xspace[\mathit{26/9/95}, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{atop:61} \end{examps} In these cases one wants $et$ (time where the tank was empty) to cover all the available time, where by ``available time'' I mean the part of the time-axis where the tense and the adverbial allow $et$ to be placed. This notion of ``available time'' is captured by \textsc{Top}\xspace's $lt$: the simple past and the \qit{on 26/9/95} of \pref{atop:60} introduce \ensuremath{\mathit{At}}\xspace and \ensuremath{\mathit{Past}}\xspace operators that, assuming that \pref{atop:60} is submitted after 26/9/95, cause $lt$ to become the period that covers exactly the day 26/9/95. The intended reading can be expressed easily in \textsc{Top}\xspace by including an additional operator that forces $et$ to cover the whole $lt$ (this operator will be discussed in section \ref{fills_op}). This method cannot be used in the language of Pirie et al. Their \ensuremath{\mathit{Past}}\xspace operator narrows the $lt$ to the part of the time-axis up to $st$, but their \ensuremath{\mathit{At}}\xspace does not narrow $lt$ any further; instead, it imposes a direct restriction on $et$ (the semantics of Pirie et al.'s \ensuremath{\mathit{At}}\xspace is not very clear, but it seems that this restriction requires $et$ to be a subperiod of 26/9/95). Hence, $lt$ is left to be the time-axis up to $st$, and one cannot require $et$ to cover the whole $lt$, because this would require the tank to be empty all the time from $t_{first}$ to $st$. The \ensuremath{\mathit{At}}\xspace operator of Pirie et al.\ also does not allow its first argument to be a formula, and it is unclear how they represent \qit{while~\dots} clauses. Finally, Pirie et al.'s \ensuremath{\mathit{Before}}\xspace allows counter-factual uses of \qit{before} to be expressed (section \ref{before_after_clauses}). Counter-factuals are not considered in this thesis, and hence Pirie et al.'s \ensuremath{\mathit{Before}}\xspace will not be discussed any further. \section{The Fills operator} \label{fills_op} As discussed in section \ref{period_adverbials}, when states combine with period adverbials, there is often a reading where the situation of the verb holds \emph{throughout} the adverbial's period. For example, there is a reading of \pref{fop:1} where tank 2 was empty throughout 26/9/95, not at simply some part of that day. \begin{examps} \item Tank 2 was empty on 26/9/95. \label{fop:1} \end{examps} Similar behaviour was observed in cases where states combine with \qit{while~\dots} subordinate clauses (section \ref{while_clauses}). For example, there is a reading of \pref{fop:3} whereby BA737 was at gate 2 throughout the entire inspection of UK160 by J.Adams, not at simply some time during the inspection. \begin{examps} \item BA737 was at gate 2 while J.Adams was inspecting UK160. \label{fop:3} \end{examps} It is also interesting that \pref{fop:5} cannot be understood as saying that tank 2 was empty throughout the period of \qit{last summer}. There is, however, a reading of \pref{fop:5} where tank 2 was empty throughout the August of the previous summer. \begin{examps} \item Tank 2 was empty in August last summer. \label{fop:5} \end{examps} It seems that states give rise to readings where the situation of the verb covers the whole available localisation time. \pref{fop:2} -- \pref{fop:10} would express the readings of \pref{fop:1} -- \pref{fop:5} that are under discussion, if there were some way to force the event times of the predicates $empty(tank2)$, $be\_at(ba737, gate2)$, and $empty(tank2)$ to cover their whole localisation times. \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[\mathit{26/9/95}, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{fop:2} \item $\ensuremath{\mathit{At}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, inspecting(j\_adams, uk160)], \ensuremath{\mathit{Past}}\xspace[e2^v, be\_at(ba737, gate2)]]$ \label{fop:4} \item $\ensuremath{\mathit{Part}}\xspace[august^g, aug^v] \land \ensuremath{\mathit{Part}}\xspace[summer^g, sum^v, -1] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[aug^v, \ensuremath{\mathit{At}}\xspace[sum^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]]$ \label{fop:10} \end{examps} The \ensuremath{\mathit{Fills}}\xspace operator achieves exactly this: it sets $et$ to the whole of $lt$. For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$: \begin{itemize} \item \index{fills@$\ensuremath{\mathit{Fills}}\xspace[\;]$ (requires $et = lt$)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Fills}}\xspace[\phi]} = T$, iff $et = lt$ and $\denot{st,et,lt,g}{\phi} = T$. \end{itemize} The readings of \pref{fop:1} -- \pref{fop:5} that are under discussion can be expressed as \pref{fop:2b} -- \pref{fop:11} respectively. \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[\mathit{26/9/95}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Fills}}\xspace[empty(tank2)]]]$ \label{fop:2b} \item $\ensuremath{\mathit{At}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, inspecting(j\_adams, uk160)],$ \\ $ \ensuremath{\mathit{Past}}\xspace[e2^v, \ensuremath{\mathit{Fills}}\xspace[be\_at(ba737, gate2)]]]$ \label{fop:4b} \item $\ensuremath{\mathit{Part}}\xspace[august^g, aug^v] \land \ensuremath{\mathit{Part}}\xspace[summer^g, sum^v, -1]$ \\ $\ensuremath{\mathit{At}}\xspace[aug^v, \ensuremath{\mathit{At}}\xspace[sum^v, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Fills}}\xspace[empty(tank2)]]]]$ \label{fop:11} \end{examps} This suggests that when state expressions combine with period-specifying subordinate clauses or adverbials, the \textsc{Nlitdb}\xspace could generate two formulae, one with and one without a \ensuremath{\mathit{Fills}}\xspace, to capture the readings where $et$ covers the whole or just part of $lt$. As mentioned in section \ref{period_adverbials}, this approach (which was tested in one version of the prototype \textsc{Nlitdb}\xspace) has the disadvantage that it generates a formula for the reading where $et$ covers the whole $lt$ even in cases where this reading is impossible. In time-asking questions like \pref{fop:35}, for example, the reading where $et$ covers the whole $lt$ (the whole 1994) is impossible, and hence the corresponding formula should not be generated. \begin{examps} \item When was tank 5 empty in 1994? \label{fop:35} \end{examps} Devising an algorithm to decide when the formulae that contain \ensuremath{\mathit{Fills}}\xspace should or should not be generated is a task which I have not addressed. For simplicity, the prototype \textsc{Nlitdb}\xspace and the rest of this thesis ignore the readings that require $et$ to cover the whole $lt$, and hence the \ensuremath{\mathit{Fills}}\xspace operator is not used. The \ensuremath{\mathit{Fills}}\xspace operator, however, may prove useful to other researchers who may attempt to explore further the topic of this section. \section{The Begin and End operators} \label{begin_end_op} The \ensuremath{\mathit{Begin}}\xspace and \ensuremath{\mathit{End}}\xspace operators are used to refer to the time-points where a situation starts or ends. For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$: \begin{itemize} \item \index{begin@$\ensuremath{\mathit{Begin}}\xspace[\;]$ (used to refer to start-points of situations)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Begin}}\xspace[\phi]} = T$, iff $et \subper lt$\\ $et' \in mxlpers(\{e \in \ensuremath{\mathit{PERIODS}}\xspace \mid \denot{st,e,\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T\})$ and $et = \{minpt(et')\}$. \item \index{end@$\ensuremath{\mathit{End}}\xspace[\;]$ (used to refer to end-points of situations)} $\denot{st,et,lt,g}{\ensuremath{\mathit{End}}\xspace[\phi]} = T$, iff $et \subper lt$ \\ $et' \in mxlpers(\{e \in \ensuremath{\mathit{PERIODS}}\xspace \mid \denot{st,e,\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T\})$ and $et = \{maxpt(et')\}$. \end{itemize} $\ensuremath{\mathit{Begin}}\xspace[\phi]$ is true only at instantaneous event times $et$ that are beginnings of maximal event times $et'$ where $\phi$ holds. The \ensuremath{\mathit{End}}\xspace operator is similar. The \ensuremath{\mathit{Begin}}\xspace and \ensuremath{\mathit{End}}\xspace operators can be used to express \qit{to start}, \qit{to stop}, \qit{to begin}, and \qit{to finish} (section \ref{special_verbs}). For example, \pref{beop:3} is expressed as \pref{beop:4}. Intuitively, in \pref{beop:4} the $\ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, uk160)]$ refers to an event-time period that covers exactly a complete inspection of UK160 by J.Adams (from start to completion). $\ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, uk160)]]$ refers to the end of that period, i.e.\ the completion point of J.Adams' inspection. $\ensuremath{\mathit{Begin}}\xspace[inspecting(t\_smith, ba737)]$ refers to the beginning of an inspection of BA737 by T.Smith. The beginning of T.Smith's inspection must precede the completion point of J.Adams' inspection, and both points must be in the past. \begin{examps} \item Did T.Smith start to inspect BA737 before J.Adams finished inspecting UK160? \label{beop:3} \item $\begin{aligned}[t] \ensuremath{\mathit{Before}}\xspace[&\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, uk160)]]], \\ &\ensuremath{\mathit{Past}}\xspace[e2^v, \ensuremath{\mathit{Begin}}\xspace[inspecting(t\_smith, ba737)]]] \end{aligned}$ \label{beop:4} \end{examps} The reading of \pref{atop:37} (section \ref{at_before_after_op}) that requires BA737 to have departed after the emergency system \emph{started} to be in operation can be expressed as \pref{beop:6}. (The reading of \pref{atop:37} where BA737 must have departed after the system \emph{stopped} being in operation is expressed as \pref{atop:38}.) \begin{examps} \item $\ensuremath{\mathit{After}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Begin}}\xspace[in\_operation(emerg\_sys)]], \ensuremath{\mathit{Past}}\xspace[e2^v, depart(ba737)]]$ \label{beop:6} \end{examps} \section{The Ntense operator} \label{ntense_op} The framework of this thesis (section \ref{noun_anaphora}) allows noun phrases like \qit{the sales manager} in \pref{ntop:1} to refer either to the present (current sales manager) or the time of the verb tense (1991 sales manager). The \ensuremath{\mathit{Ntense}}\xspace operator is used to represent these two possible readings. \begin{examps} \item What was the salary of the sales manager in 1991? \label{ntop:1} \item $?slr^v \; \ensuremath{\mathit{Ntense}}\xspace[now^*, manager\_of(mgr^v, sales)] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[1991, \ensuremath{\mathit{Past}}\xspace[e^v, salary\_of(mgr^v, slr^v)]]$ \label{ntop:1.1} \item $?slr^v \; \ensuremath{\mathit{Ntense}}\xspace[e^v, manager\_of(mgr^v, sales)] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[1991, \ensuremath{\mathit{Past}}\xspace[e^v, salary\_of(mgr^v, slr^v)]]$ \label{ntop:1.2} \end{examps} The reading of \pref{ntop:1} where \qit{the sales manager} refers to the present is represented as \pref{ntop:1.1}, while the reading where it refers to the time of the verb tense is represented as \pref{ntop:1.2}. Intuitively, \pref{ntop:1.1} reports any $slr^v$, such that $slr^v$ was the salary of $mgr^v$ at some past time $e^v$ that falls within 1991, and $mgr^v$ is the manager of the sales department \emph{at the present}. In contrast, \pref{ntop:1.2} reports any $slr^v$, such that $slr^v$ was the salary of $mgr^v$ at some past time $e^v$ that falls within 1991, and $mgr^v$ was the manager of the sales department \emph{at} $e^v$. Notice that in \pref{ntop:1.2} the first argument of the \ensuremath{\mathit{Ntense}}\xspace is the same as the first argument of the \ensuremath{\mathit{Past}}\xspace, which is a pointer to the past event time where $salary\_of(mgr^v, slr^v)$ is true (see the semantics of \ensuremath{\mathit{Past}}\xspace in section \ref{past_op}). For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $\beta \in \ensuremath{\mathit{VARS}}\xspace$ : \begin{itemize} \item \index{ntense@$\ensuremath{\mathit{Ntense}}\xspace[\;]$ (used when expressing nouns or adjectives)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Ntense}}\xspace[\beta, \phi]} = T$, iff for some $et' \in \ensuremath{\mathit{PERIODS}}\xspace$, it is true that $g(\beta)= et'$ and $\denot{st,et',\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T$. \item \index{ntense@$\ensuremath{\mathit{Ntense}}\xspace[\;]$ (used when expressing nouns or adjectives)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Ntense}}\xspace[now^*, \phi]} = T$, iff $\denot{st,\{st\},\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T$. \end{itemize} \ensuremath{\mathit{Ntense}}\xspace evaluates $\phi$ with respect to a new event time $et'$, which may be different from the original event time $et$ that is used to evaluate the part of the formula outside the \ensuremath{\mathit{Ntense}}\xspace. Within the \ensuremath{\mathit{Ntense}}\xspace, the localisation time is reset to \ensuremath{\mathit{PTS}}\xspace (whole time-axis) freeing $et'$ from restrictions imposed on the original $et$. If the first argument of \ensuremath{\mathit{Ntense}}\xspace is $now^*$, the new event time is the instantaneous period that contains only $st$, i.e.\ the object to which the noun phrase refers must have at $st$ the property described by $\phi$. If the first argument of \ensuremath{\mathit{Ntense}}\xspace is a variable $\beta$, the new event time $et'$ can generally be any period, and $\beta$ denotes $et'$. In \pref{ntop:1.2}, however, $\beta$ is the same as the first argument of the \ensuremath{\mathit{Past}}\xspace, which denotes the original $et$ that the \ensuremath{\mathit{Past}}\xspace requires to be placed before $st$. This means that $manager\_of(mgr^v, sales)$ must hold at the same event time where $salary\_of(mgr^v, slr^v)$ holds, i.e.\ the person $mgr^v$ must be the sales manager at the same time where the salary of $mgr^v$ is $slr^v$. If the first argument of the \ensuremath{\mathit{Ntense}}\xspace in \pref{ntop:1.2} and the first argument of the \ensuremath{\mathit{Past}}\xspace were different variables, the answer would contain any 1991 salary of anybody who was, is, or will be the sales manager at any time. This would be useful in \pref{ntop:13}, where one may want to allow \qit{Prime Minister} to refer to the Prime Ministers of all times, a reading that can be expressed as \pref{ntop:16}. \begin{examps} \item Which Prime Ministers were born in Scotland? \label{ntop:13} \item $?pm^v \; \ensuremath{\mathit{Ntense}}\xspace[e1^v, pminister(pm^v)] \land \ensuremath{\mathit{Past}}\xspace[e2^v, birth\_in(pm^v,scotland)]$ \label{ntop:16} \end{examps} The framework of this thesis, however, does not currently generate \pref{ntop:16}. \pref{ntop:13} would receive only two formulae, one for current Prime Ministers, and one for persons that were Prime Ministers at the time they were born (the latter reading is, of course, unlikely). Questions like \pref{ntop:5} and \pref{ntop:7}, where temporal adjectives specify explicitly the times to which the noun phrases refer, can be represented as \pref{ntop:6} and \pref{ntop:8}. (The framework of this thesis, however, does not support temporal adjectives other than \qit{current}; see section \ref{temporal_adjectives}.) \begin{examps} \item What was the salary of the current sales manager in 1991? \label{ntop:5} \item $?slr^v \; \ensuremath{\mathit{Ntense}}\xspace[now^*, manager\_of(mgr^v, sales)] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[1991, \ensuremath{\mathit{Past}}\xspace[e^v, salary\_of(mgr^v, slr^v)]]$ \label{ntop:6} \item What was the salary of the 1988 sales manager in 1991? \label{ntop:7} \item $?slr^v \; \ensuremath{\mathit{Ntense}}\xspace[e1^v, \ensuremath{\mathit{At}}\xspace[1988, manager\_of(mgr^v, sales)]] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[1991, \ensuremath{\mathit{Past}}\xspace[e^v, salary\_of(mgr^v, slr^v)]]$ \label{ntop:8} \end{examps} The \ensuremath{\mathit{Ntense}}\xspace operator of \textsc{Top}\xspace is the same as the \ensuremath{\mathit{Ntense}}\xspace operator of \cite{Crouch} and \cite{Crouch2}. \section{The For operator} \label{for_op} The \ensuremath{\mathit{For}}\xspace operator is used to express \qit{for~\dots} and duration \qit{in~\dots} adverbials (sections \ref{for_adverbials} and \ref{in_adverbials}). For $\sigma_c \in \ensuremath{\mathit{CPARTS}}\xspace$, $\nu_{qty} \in \{1,2,3,\dots\}$, and $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$: \begin{itemize} \item \index{for@$\ensuremath{\mathit{For}}\xspace[\;]$ (used to express durations)} $\denot{st,et,lt,g}{\ensuremath{\mathit{For}}\xspace[\sigma_c, \nu_{qty}, \phi]} = T$, iff $\denot{st,et,lt,g}{\phi} = T$, and for some $p_1,p_2,\dots,p_{\nu_{qty}} \in \ensuremath{\mathit{f_{cparts}}}\xspace(\sigma_c)$, it is true that $minpt(p_1) = minpt(et)$, $next(maxpt(p_1)) = minpt(p_2)$, $next(maxpt(p_2)) = minpt(p_3)$, \dots, $next(maxpt(p_{\nu_{qty} - 1})) = minpt(p_{\nu_{qty}})$, and $maxpt(p_{\nu_{qty}}) = maxpt(et)$. \end{itemize} $\ensuremath{\mathit{For}}\xspace[\sigma_c, \nu_{qty}, \phi]$ requires $\phi$ to be true at an event time period that is $\nu_{qty}$ $\sigma_c$-periods long. For example, assuming that $month^c$ denotes the partitioning of month-periods (the period that covers exactly the August of 1995, the period for September of 1995, etc.), \pref{forop:3} can be expressed as \pref{forop:4}. \begin{examps} \item Was tank 2 empty for three months? \label{forop:3} \item $\ensuremath{\mathit{For}}\xspace[month^c, 3, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{forop:4} \end{examps} \pref{forop:4} requires an event time $et$ to exist, such that $et$ covers exactly three continuous months, and tank 2 was empty throughout $et$. As noted in section \ref{for_adverbials}, \qit{for~\dots} adverbials are sometimes used to specify the duration of a \emph{maximal} period where a situation holds, or to refer to the \emph{total duration} of possibly non-overlapping periods where some situation holds. The current version of \textsc{Top}\xspace cannot express such readings. Expressions like \qit{one week}, \qit{three months}, \qit{two years}, \qit{two hours}, etc., are often used to specify a duration of seven days, $3 \times 30$ days, $2 \times 365$ days, $2 \times 60$ minutes, etc. \pref{forop:4} expresses \pref{forop:3} if \qit{three months} refers to \emph{calendar} months (e.g.\ from the beginning of a June to the end of the following August). If \qit{three months} means $3 \times 30$ days, \pref{forop:10} has to be used instead. (I assume that $day^c$ denotes the partitioning of day-periods: the period that covers exactly 26/9/95, the period for 27/9/95, etc.) \begin{examps} \item $\ensuremath{\mathit{For}}\xspace[day^c, 90, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{forop:10} \end{examps} Assuming that \qit{to inspect} is a culminating activity (as in the airport application), \pref{forop:13} represents the reading of \pref{forop:12} where 42 minutes is the duration from the beginning of the inspection to the inspection's completion (section \ref{in_adverbials}). \pref{forop:13} requires $et$ to cover the whole inspection (from beginning to completion), $et$ to be in the past, and the duration of $et$ to be 42 minutes. \begin{examps} \item J.Adams inspected BA737 in 42 minutes. \label{forop:12} \item $\ensuremath{\mathit{For}}\xspace[minute^c, 42, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]]]$ \label{forop:13} \end{examps} Unlike \pref{forop:12}, \pref{forop:14} does not require the inspection to have been completed (section \ref{for_adverbials}). \pref{forop:14} is represented as \pref{forop:15}, which contains no \ensuremath{\mathit{Culm}}\xspace. In this case, $et$ must simply be a period throughout which J.Adams was inspecting BA737, it must be located in the past, and it must be 42 minutes long. \begin{examps} \item J.Adams inspected BA737 for 42 minutes. \label{forop:14} \item $\ensuremath{\mathit{For}}\xspace[minute^c, 42, \ensuremath{\mathit{Past}}\xspace[e^v, inspecting(j\_adams, ba737)]]$ \label{forop:15} \end{examps} \section{The Perf operator} \label{perf_op} The \ensuremath{\mathit{Perf}}\xspace operator is used when expressing the past perfect. For example, \pref{perfop:3} is expressed as \pref{perfop:4}. \ensuremath{\mathit{Perf}}\xspace could also be used to express the present perfect (e.g.\ \pref{perfop:1} could be represented as \pref{perfop:2}). This thesis, however, treats the present perfect in the same way as the simple past (section \ref{present_perfect}), and \pref{perfop:1} is mapped to \pref{perfop:6}, the same formula that expresses \pref{perfop:5}. \begin{examps} \item BA737 had departed. \label{perfop:3} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, depart(ba737)]]$ \label{perfop:4} \item BA737 has departed. \label{perfop:1} \item $\ensuremath{\mathit{Pres}}\xspace[\ensuremath{\mathit{Perf}}\xspace[e^v, depart(ba737)]]$ \label{perfop:2} \item BA737 departed. \label{perfop:5} \item $\ensuremath{\mathit{Past}}\xspace[e^v, depart(ba737)]$ \label{perfop:6} \end{examps} For $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $\beta \in \ensuremath{\mathit{VARS}}\xspace$: \begin{itemize} \item \index{perf@$\ensuremath{\mathit{Perf}}\xspace[\;]$ (used to express the past perfect)} $\denot{st,et,lt,g}{\ensuremath{\mathit{Perf}}\xspace[\beta, \phi]} = T$, iff $et \subper lt$, and for some $et' \in \ensuremath{\mathit{PERIODS}}\xspace$, it is true that $g(\beta) = et'$, $maxpt(et') \prec minpt(et)$, and $\denot{st,et',\ensuremath{\mathit{PTS}}\xspace,g}{\phi} = T$. \end{itemize} $\ensuremath{\mathit{Perf}}\xspace[\beta, \phi]$ holds at the event time $et$, only if $et$ is preceded by a new event time $et'$ where $\phi$ holds (figure \ref{perf_op_fig}). The original $et$ must be a subperiod of $lt$. In contrast $et'$ does not need to be a subperiod of $lt$ (the localisation time in $\denot{st,et',\ensuremath{\mathit{PTS}}\xspace,g}{\phi}$ is reset to \ensuremath{\mathit{PTS}}\xspace, the whole time-axis). The $\beta$ of $\ensuremath{\mathit{Perf}}\xspace[\beta, \phi]$ is a pointer to $et'$, similar to the $\beta$ of $\ensuremath{\mathit{Past}}\xspace[\beta,\phi]$. \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{perf_op} \caption{The Perf operator} \label{perf_op_fig} \end{center} \hrule \end{figure} Ignoring constraints imposed by $lt$, the event time $et$ where $\ensuremath{\mathit{Perf}}\xspace[\beta, \phi]$ is true can be placed anywhere within the period that starts immediately after the end of $et'$ ($et'$ is where $\phi$ is true) and that extends up to $t_{last}$. The informal term ``consequent period'' was used in section \ref{point_adverbials} to refer to this period. Using the \ensuremath{\mathit{Perf}}\xspace operator, the reading of \pref{perfop:7} where the inspection happens at some time before (or possibly on 27/9/95) is expressed as \pref{perfop:8} (in this case, \qit{on 27/9/95} provides a ``reference time''; see section \ref{past_perfect}). In contrast, the reading of \pref{perfop:7} where the inspection happens on 27/9/95 is expressed as \pref{perfop:9}. \begin{examps} \item J.Adams had inspected gate 2 on 27/9/95. \label{perfop:7} \item $\ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]]$ \label{perfop:8} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]]$ \label{perfop:9} \end{examps} Let us explore formally the denotations of \pref{perfop:8} and \pref{perfop:9}. The denotation of \pref{perfop:8} w.r.t.\ $st$ is $T$ iff for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and $g \in G$, \pref{perfop:10} holds. \begin{examps} \item $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g} {\ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]]} = T$ \label{perfop:10} \end{examps} Assuming that $\mathit{27/9/95}$ denotes the obvious period, by the definition of \ensuremath{\mathit{At}}\xspace, \pref{perfop:10} holds iff \pref{perfop:11} is true ($\ensuremath{\mathit{PTS}}\xspace \intersect \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}) = \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95})$). \begin{examps} \item $\denot{st,et,\ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}),g} {\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]} = T$ \label{perfop:11} \end{examps} By the definition of \ensuremath{\mathit{Past}}\xspace, ignoring $e1^v$ which does not play any interesting role here, and assuming that $st$ follows 27/9/95, \pref{perfop:11} is true iff \pref{perfop:12} holds. \begin{examps} \item $\denot{st, et, \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}), g} {\ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]} = T$ \label{perfop:12} \end{examps} By the definition of \ensuremath{\mathit{Perf}}\xspace (ignoring $e2^v$), \pref{perfop:12} holds iff for some $et' \in \ensuremath{\mathit{PERIODS}}\xspace$, \pref{perfop:14}, \pref{perfop:15}, and \pref{perfop:16} hold. \begin{gather} et \subper \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}) \label{perfop:14} \\ maxpt(et') \prec minpt(et) \label{perfop:15} \\ \denot{st,et',\ensuremath{\mathit{PTS}}\xspace,g}{\ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]} = T \label{perfop:16} \end{gather} By the definition of \ensuremath{\mathit{Culm}}\xspace, \pref{perfop:16} holds iff \pref{perfop:17} -- \pref{perfop:21} hold. \begin{gather} et' \subper \ensuremath{\mathit{PTS}}\xspace \label{perfop:17} \\ \ensuremath{\mathit{f_{culms}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ja), \ensuremath{\mathit{f_{cons}}}\xspace(g2)) = T \label{perfop:18} \\ S = \bigcup_{p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ja), \ensuremath{\mathit{f_{cons}}}\xspace(g2))}p \label{perfop:19} \\ S \not= \emptyset \label{perfop:20} \\ et' = [minpt(S), maxpt(S)] \label{perfop:21} \end{gather} Let us assume that there is only one maximal period where J.Adams is inspecting BA737, and that the inspection is completed at the end of that period. Then, the $S$ of \pref{perfop:19} is the maximal period, and \pref{perfop:18} and \pref{perfop:20} hold. \pref{perfop:21} requires $et'$ to be the same period as $S$, in which case \pref{perfop:17} is trivially satisfied. The denotation of \pref{perfop:8} w.r.t.\ $st$ is $T$ (i.e.\ the answer to \pref{perfop:7} is affirmative) iff for some $et$, $et' = S$, and \pref{perfop:14} and \pref{perfop:15} hold, i.e.\ iff there is an $et$ within 27/9/95, such that $et$ follows $S$ ($S = et'$ is the period that covers the whole inspection). The situation is depicted in figure \ref{perf_op2_fig}. In other words, 27/9/95 must contain an $et$ where the inspection has already been completed. \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{perf_op2} \caption{First reading of \qit{J.Adams had inspected gate 2 on 27/9/95}} \label{perf_op2_fig} \end{center} \hrule \end{figure} Let us now consider \pref{perfop:9}. Its denotation w.r.t.\ $st$ will be true iff for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and $g \in G$, \pref{perfop:22} holds. \begin{examps} \item $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g} {\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]]} = T$ \label{perfop:22} \end{examps} By the definition of \ensuremath{\mathit{Past}}\xspace, \pref{perfop:22} holds iff \pref{perfop:23} is true. (For simplicity, I ignore again $e1^v$ and $e2^v$.) \begin{examps} \item $\denot{st,et,[t_{first}, st),g} {\ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]}$ \label{perfop:23} \end{examps} By the definition of \ensuremath{\mathit{Perf}}\xspace, \pref{perfop:23} is true iff for some $et' \in \ensuremath{\mathit{PERIODS}}\xspace$, \pref{perfop:24}, \pref{perfop:25}, and \pref{perfop:26} hold. \begin{gather} et \subper [t_{first}, st) \label{perfop:24} \\ maxpt(et') \prec minpt(et) \label{perfop:25} \\ \denot{st,et',\ensuremath{\mathit{PTS}}\xspace,g}{\ensuremath{\mathit{At}}\xspace[\mathit{27/9/95}, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]} = T \label{perfop:26} \end{gather} By the definition of the \ensuremath{\mathit{At}}\xspace operator, \pref{perfop:26} holds iff \pref{perfop:27} holds. (I assume again that $\mathit{27/9/95}$ denotes the obvious period.) \begin{equation} \denot{st,et',\ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}),g}{\ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]} = T \label{perfop:27} \end{equation} By the definition of \ensuremath{\mathit{Culm}}\xspace, \pref{perfop:27} holds iff \pref{perfop:28} -- \pref{perfop:32} are true. \begin{gather} et' \subper \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{27/9/95}) \label{perfop:28} \\ \ensuremath{\mathit{f_{culms}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ja), \ensuremath{\mathit{f_{cons}}}\xspace(g2)) = T \label{perfop:29} \\ S = \bigcup_{p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(inspecting, 2)(\ensuremath{\mathit{f_{cons}}}\xspace(ja), \ensuremath{\mathit{f_{cons}}}\xspace(g2))}p \label{perfop:30} \\ S \not= \emptyset \label{perfop:31} \\ et' = [minpt(S), maxpt(S)] \label{perfop:32} \end{gather} Assuming again that there is only one maximal period where J.Adams is inspecting BA737, and that the inspection is completed at the end of that period, the $S$ of \pref{perfop:30} is the maximal period, and \pref{perfop:29} and \pref{perfop:31} hold. \pref{perfop:32} requires $et'$ to be the same as $S$. The denotation of \pref{perfop:9} w.r.t.\ $st$ is $T$ (i.e.\ the answer to \pref{perfop:7} is affirmative) iff for some $et$, $et' = S$, and \pref{perfop:24}, \pref{perfop:25}, and \pref{perfop:28} hold. That is there must be some past $et$ that follows $S$ ($S = et'$ is the period that covers the whole inspection), with $S$ falling within 27/9/95 (figure \ref{perf_op2_fig}). The inspection must have been completed within 27/9/95. \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.6]{perf_op3} \caption{Second reading of \qit{J.Adams had inspected gate 2 on 27/9/95}} \label{perf_op3_fig} \end{center} \hrule \end{figure} In \pref{perfop:33}, where there are no temporal adverbials, the corresponding formula \pref{perfop:34} requires some past $et$ (pointed to by $e1^v$) to exist, such that $et$ follows an $et'$ (pointed to by $e2^v$) that covers exactly the whole (from start to completion) inspection of gate 2 by J.Adams. The net effect is that the inspection must have been completed in the past. \begin{examps} \item J.Adams had inspected gate 2 \label{perfop:33} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(ja, g2)]]]$ \label{perfop:34} \end{examps} As noted in section \ref{past_perfect}, there is a reading of \pref{perfop:35} (probably the preferred one) whereby the two-year period ends on 1/1/94, i.e.\ J.Adams was still a manager on 1/1/94. Similarly, there is a reading of \pref{perfop:37}, whereby the two-year period ends at $st$, i.e.\ J.Adams is still a manager (section \ref{present_perfect}). These readings cannot be captured in \textsc{Top}\xspace. \begin{examps} \item On 1/1/94, J.Adams had been a manager for two years. \label{perfop:35} \item J.Adams has been a manager for two years. \label{perfop:37} \end{examps} For example, \pref{perfop:36} requires some past $et$ (pointed to by $e1^v$) to exist, such that $et$ falls within 1/1/94, $et$ follows a period $et'$ (pointed to by $e2^v$), $et'$ is a period where J.Adams is a manager, and the duration of $et'$ is two years. If, for example, J.Adams was a manager only from 1/1/88 to 31/12/89, \pref{perfop:36} causes the answer to \pref{perfop:35} to be affirmative. \pref{perfop:36} does not require the two-year period to end on 1/1/94. \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[\mathit{1/1/94}, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{For}}\xspace[year^c, 2, be(ja, manager)]]]]$ \label{perfop:36} \end{examps} Various versions of \ensuremath{\mathit{Perf}}\xspace operators have been used in \cite{Dowty1982}, \cite{Richards}, \cite{Pirie1990}, \cite{Crouch2}, and elsewhere. \section{Occurrence identifiers} \label{occurrence_ids} Predicates introduced by verbs whose base forms are culminating activities often have an extra argument that acts as an \emph{occurrence identifier}. Let us consider a scenario involving an engineer, John, who worked on engine 2 repairing faults of the engine at several past times (figure \ref{episodes_fig}). John started repairing a fault of engine 2 on 1/6/92 at 9:00am. He continued to work on this fault up to 1:00pm on the same day, at which point he temporarily abandoned the repair without completing it. He resumed the repair at 3:00pm on 25/6/92, and completed it at 5:00pm on the same day. \begin{figure}[tb] \hrule \medskip \begin{center} \includegraphics[scale=.58]{episodes} \caption{Occurrence identifiers} \label{episodes_fig} \end{center} \hrule \end{figure} In 1993, John was asked to repair another fault of engine 2. He started the repair on 1/7/93 at 9:00am, and continued to work on that fault up to 1:00pm on the same day without completing the repair. He then abandoned the repair for ever (John was not qualified to fix that fault, and the repair was assigned to another engineer). Finally, in 1994 John was asked to repair a third fault of engine 2. He started to repair the third fault on 1/6/94 at 9:00am, and continued to work on that fault up to 1:00pm on the same day, without completing the repair. He resumed the repair at 3:00pm, and completed it at 5:00pm on the same day. There is a problem if \pref{epid:1} is represented as \pref{epid:2}. Let us assume that the question is submitted after 1/6/94. One would expect the answer to be affirmative, since a complete past repair of engine 2 by John is situated within 1/6/94. In contrast, \pref{epid:2} causes the answer to be negative. The semantics of \ensuremath{\mathit{Culm}}\xspace (section \ref{culm_op}) requires $et$ to start at the beginning of the earliest maximal period where $repairing(john, eng2)$ holds (i.e.\ at the beginning of $p_1$ in figure \ref{episodes_fig}) and to end at the end of the latest maximal period where $repairing(john, eng2)$ holds (i.e.\ at the end of $p_5$ in figure \ref{episodes_fig}). That is, $et$ must be $p_8$ of figure \ref{episodes_fig}. The \ensuremath{\mathit{At}}\xspace requires $et$ ($p_8$) to be also a subperiod of 1/6/94. Since this is not the case, the answer is negative. \begin{examps} \item Did John repair engine 2 on 1/6/94? \label{epid:1} \item $\ensuremath{\mathit{At}}\xspace[\mathit{1/6/94}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(john,eng2)]]]$ \label{epid:2} \end{examps} The problem is that although John was repairing engine 2 during all five periods ($p_1$, $p_2$, $p_3$, $p_4$, and $p_5$), the five periods intuitively belong to different occurrences of the situation where John is repairing engine 2. The first two periods have to do with the repair of the first fault (occurrence 1), the third period has to do with the repair of the second fault (occurrence 2), and the last two periods relate to the repair of the third fault (occurrence 3). The $\ensuremath{\mathit{Culm}}\xspace[repairing(john,eng2)]$ of \pref{epid:2}, however, does not distinguish between the three occurrences, and forces $et$ to start at the beginning of $p_1$ and to end at the end of $p_5$. Instead, we would like $\ensuremath{\mathit{Culm}}\xspace[repairing(john,eng2)]$ to distinguish between the three occurrences: to require $et$ to start at the beginning of $p_1$ (beginning of the first repair) and to end at the end of $p_2$ (completion of the first repair), or to require $et$ to start at the beginning of $p_4$ (beginning of the third repair) and to end at the end of $p_5$ (completion of the third repair). ($\ensuremath{\mathit{Culm}}\xspace[repairing(john,eng2)]$ should not allow $et$ to be $p_3$, because the second repair does not reach its completion at the end of $p_3$.) To achieve this, an occurrence-identifying argument is added to $fixing(john,eng2)$. If $occ1$, $occ2$, and $occ3$ denote the three repairing-occurrences, $fixing(occ1,john,eng2)$ will be true only at $et$s that are subperiods of $p_1$ or $p_2$, $fixing(occ2, john, eng2)$ only at $et$s that are subperiods of $p_3$, and $fixing(occ3, john, eng2)$ only at $et$s that are subperiods of $p_4$ or $p_5$. In practice, the occurrence-identifying argument is always a variable. For example, \pref{epid:1} is now represented as \pref{epid:3} instead of \pref{epid:2}. \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[\mathit{1/6/94}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occ^v,john,eng2)]]]$ \label{epid:3} \end{examps} Intuitively, according to \pref{epid:3} the answer should be affirmative if there is an $et$ and a particular occurrence $occ^v$ of the situation where John is repairing engine 2, such that $et$ starts at the beginning of the first period where $occ^v$ is ongoing, $et$ ends at the end of the last period where $occ^v$ is ongoing, $occ^v$ reaches its completion at the end of $et$, and $et$ falls within the past and 1/6/94. Now if \pref{epid:1} is submitted after 1/6/94, the answer is affirmative. To see that \pref{epid:3} generates the correct result, let us examine the denotation of \pref{epid:3}. The denotation of \pref{epid:3} w.r.t.\ $st$ is affirmative if for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and $g \in G$, \pref{epid:4} holds. \begin{examps} \item $\denot{st,et,\ensuremath{\mathit{PTS}}\xspace,g} {\ensuremath{\mathit{At}}\xspace[\mathit{1/6/94}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occ^v,john,eng2)]]]} = T$ \label{epid:4} \end{examps} Assuming that the question is submitted after 1/6/94, and that $\mathit{1/6/94}$ denotes the obvious period, by the definitions of \ensuremath{\mathit{At}}\xspace and \ensuremath{\mathit{Past}}\xspace, \pref{epid:4} holds iff \pref{epid:5} and \pref{epid:6} hold. \begin{gather} g(e^v) = et \label{epid:5} \\ \denot{st,et,\ensuremath{\mathit{f_{cons}}}\xspace(\mathit{1/6/94}),g}{\ensuremath{\mathit{Culm}}\xspace[repairing(occ^v,john,eng2)]} = T \label{epid:6} \end{gather} By the definition of \ensuremath{\mathit{Culm}}\xspace, \pref{epid:6} holds iff \pref{epid:7} -- \pref{epid:10} hold, where $S$ is as in \pref{epid:11}. \begin{gather} et \subper \ensuremath{\mathit{f_{cons}}}\xspace(\mathit{1/6/94}) \label{epid:7} \\ \ensuremath{\mathit{f_{culms}}}\xspace(repairing,2)(g(occ^v), \ensuremath{\mathit{f_{cons}}}\xspace(john), \ensuremath{\mathit{f_{cons}}}\xspace(eng2)) = T \label{epid:8} \\ S \not= \emptyset \label{epid:9} \\ et = [minpt(S), maxpt(S)] \label{epid:10} \\ S = \bigcup_{p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(repairing, 2)(g(occ^v), \ensuremath{\mathit{f_{cons}}}\xspace(john), \ensuremath{\mathit{f_{cons}}}\xspace(eng2))}p \label{epid:11} \end{gather} The denotation of \pref{epid:3} w.r.t.\ $st$ is $T$ (i.e.\ the answer to \pref{epid:1} is affirmative), iff for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and $g \in G$, \pref{epid:5} and \pref{epid:7} -- \pref{epid:10} hold. For $et$ as in \pref{epid:21} and $g$ the variable assignment of \pref{epid:20}, \pref{epid:5} and \pref{epid:7} hold. \pref{epid:11} becomes \pref{epid:13}, and \pref{epid:9} holds. \pref{epid:10} becomes \pref{epid:21}, which holds ($et$ was chosen to satisfy it). \pref{epid:8} also holds, because the third repair is completed at the end of $p_5$. \begin{gather} et = [minpt(p_4), maxpt(p_5)] = p_7 \label{epid:21} \\ g(\beta) = \begin{cases} et & \text{if } \beta = e^v \\ \ensuremath{\mathit{f_{cons}}}\xspace(occ3) & \text{if } \beta = occ^v \\ o & \text{otherwise ($o$ is an arbitrary element of \ensuremath{\mathit{OBJS}}\xspace)} \end{cases} \label{epid:20} \\ S = p_4 \union p_5 \label{epid:13} \end{gather} Hence, there is some $et \in \ensuremath{\mathit{PERIODS}}\xspace$ and $g \in G$ for which \pref{epid:5} and \pref{epid:7} -- \pref{epid:10} hold, i.e.\ the answer to \pref{epid:1} will be affirmative as wanted. Occurrence identifiers are a step towards formalisms that treat occurrences of situations (or ``events'' or ``episodes'') as objects in the modelled world (e.g.\ \cite{Parsons1990}, \cite{Kamp1993}, \cite{Blackburn1994}, \cite{Hwang1994}). In \textsc{Top}\xspace all terms (constants and variables) denote elements of \ensuremath{\mathit{OBJS}}\xspace, i.e.\ objects of the modelled world. Thus, allowing occurrence-identifying terms (like $occ^v$ in \pref{epid:3}) implies that occurrences of situations are also world objects. Unlike other formalisms (e.g.\ those mentioned above), however, \textsc{Top}\xspace does not treat these occurrence-identifying terms in any special way, and there is nothing in the definition of \textsc{Top}\xspace to distinguish objects denoted by occurrence-identifiers from objects denoted by other terms. \section{Tense anaphora and localisation time} \label{lt_anaphora} Although tense anaphora (section \ref{temporal_anaphora}) was not considered during the work of this thesis, it seems that \textsc{Top}\xspace's localisation time could prove useful if this phenomenon were to be supported. As noted in section \ref{temporal_anaphora}, some cases of tense anaphora can be handled by storing the temporal window established by adverbials and tenses of previous questions, and by requiring the situations of follow-up questions to fall within that window. \textsc{Top}\xspace's $lt$ can capture this notion of previous window. Assuming that \pref{ltan:1} is submitted after 1993, the \ensuremath{\mathit{At}}\xspace and \ensuremath{\mathit{Past}}\xspace operators of the corresponding formula \pref{ltan:2} narrow $lt$ to the period that covers exactly 1993. This period could be stored, and used as the initial value of $lt$ in \pref{ltan:4}, that expresses the follow-up question \pref{ltan:3}. In effect, \pref{ltan:3} would be taken to mean \pref{ltan:5}. \begin{examps} \item Was Mary the personnel manager in 1993? \label{ltan:1} \item $\ensuremath{\mathit{At}}\xspace[1993, \ensuremath{\mathit{Past}}\xspace[e^v, manager\_of(mary, personnel)]]$ \label{ltan:2} \item Who was the personnel manager? \label{ltan:3} \item $?wh^v \; \ensuremath{\mathit{Past}}\xspace[e^v, manager\_of(wh^v, personnel)]$ \label{ltan:4} \item Who was the personnel manager in 1993? \label{ltan:5} \end{examps} Substantial improvements are needed to make these ideas workable. For example, if \pref{ltan:1} and \pref{ltan:3} are followed by \pref{ltan:6} (expressed as \pref{ltan:7}), and the dialogue takes place after 1993, the \textsc{Nlitdb}\xspace must be intelligent enough to reset $lt$ to the whole time axis. Otherwise, no person will ever be reported, because the \ensuremath{\mathit{Pres}}\xspace of \pref{ltan:7} requires $et$ to contain $st$, and an $et$ that contains $st$ can never fall within the past year 1993 (the $lt$ of the previous question). \begin{examps} \item Who is (now) the personnel manager? \label{ltan:6} \item $?wh^v \; \ensuremath{\mathit{Pres}}\xspace[manager\_of(wh^v, personnel)]$ \label{ltan:7} \end{examps} \section{Expressing habituals} \label{hab_problems} As noted in section \ref{pres_op}, habitual readings of sentences are taken to involve habitual homonyms of verbs. Habitual homonyms introduce different predicates than the corresponding non-habitual ones. For example, \pref{habp:1} and \pref{habp:3} would be expressed as \pref{habp:2} and \pref{habp:4} respectively. Different predicates would used in the two cases. \begin{examps} \item Last month BA737 (habitually) departed from gate 2. \label{habp:1} \item $\ensuremath{\mathit{Part}}\xspace[month^c, mon^v, -1] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, hab\_depart\_from(ba737, gate2)]]$ \label{habp:2} \item Yesterday BA737 (actually) departed from gate 2. \label{habp:3} \item $\ensuremath{\mathit{Part}}\xspace[day^c, y^v, -1] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[y^v, \ensuremath{\mathit{Past}}\xspace[e^v, actl\_depart\_from(ba737, gate2)]]$ \label{habp:4} \end{examps} $hab\_depart\_from(ba737, gate2)$ is intended to hold at $et$s that fall within periods where BA737 has the habit of departing from gate 2. If BA737 departed habitually from gate 2 throughout 1994, $hab\_depart\_from(ba737, gate2)$ would be true at any $et$ that is a subperiod of 1994. In contrast, $actl\_depart\_from(ba737, gate2)$ is intended to hold only at $et$s where BA737 actually departs from gate 2. If departures are modelled as instantaneous (as in the airport application), $actl\_depart\_from(ba737, gate2)$ is true only at instantaneous $et$s where BA737 leaves gate 2. One would expect that if BA737 had the habit of departing from gate 2 during some period, it would also have actually departed from gate 2 at least some times during that period: if $hab\_depart\_from(ba737, gate2)$ is true at an $et$, $actl\_depart\_from(ba737, gate2)$ would also be true at some subperiods $et'$ of $et$. There is nothing in the definition of \textsc{Top}\xspace, however, to guarantee that this implication holds. The event times where $hab\_depart\_from(ba737, gate2)$ and $actl\_depart\_from(ba737, gate2)$ hold are ultimately determined by \ensuremath{\mathit{f_{pfuns}}}\xspace (that specifies the maximal periods where the two predicates hold; see section \ref{top_model}). There is no restriction in the definition of \textsc{Top}\xspace to prohibit whoever defines \ensuremath{\mathit{f_{pfuns}}}\xspace from specifying that $hab\_depart\_from(ba737, gate2)$ is true at some $et$ that does not contain any $et'$ where $actl\_depart\_from(ba737, gate2)$ is true. Another issue is how to represent \pref{habp:5}. \pref{habp:5} cannot be represented as \pref{habp:6}. \pref{habp:6} says that at 5:00pm on some day in the previous month BA737 had the habit of departing. I have found no elegant solution to this problem. \pref{habp:5} is mapped to \pref{habp:7}, where the constant $\text{\textit{5:00pm}}$ is intended to denote a generic representative of 5:00pm-periods. This generic representative is taken to be an entity in the world. \begin{examps} \item Last month BA737 (habitually) departed at 5:00pm. \label{habp:5} \item $\ensuremath{\mathit{Part}}\xspace[month^c, mon^v, -1] \land \ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, hab\_depart(ba737)]]$ \label{habp:6} \item $\ensuremath{\mathit{Part}}\xspace[month^c, mon^v, -1] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, hab\_depart\_time(ba737, \text{\textit{5:00pm}})]]$ \label{habp:7} \end{examps} Unlike \pref{habp:5}, where \qit{at 5:00pm} introduces a constant ($\text{\textit{5:00pm}}$) as a predicate-argument in \pref{habp:7}, the \qit{at 5:00pm} of \pref{habp:8} introduces an \ensuremath{\mathit{At}}\xspace operator in \pref{habp:9}. \begin{examps} \item Yesterday BA737 (actually) departed at 5:00pm. \label{habp:8} \item $\ensuremath{\mathit{Part}}\xspace[day^c, y^v, -1] \land \ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[y^v, \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, actl\_depart(ba737)]]]$ \label{habp:9} \end{examps} The fact that \qit{at 5:00pm} is treated in such different ways in the two cases is admittedly counter-intuitive, and it also complicates the translation from English to \textsc{Top}\xspace (to be discussed in chapter \ref{English_to_TOP}). \section{Summary} \textsc{Top}\xspace is a formal language, used to represent the meanings of the English questions that are submitted to the \textsc{Nlitdb}\xspace. The denotation with respect to $st$ of a \textsc{Top}\xspace formula specifies what the answer to the corresponding English question should report ($st$ is the time-point where the question is submitted to the \textsc{Nlitdb}\xspace). The denotations with respect to $st$ of \textsc{Top}\xspace formulae are defined in terms of the denotations of \textsc{Top}\xspace formulae with respect to $st$, $et$, and $lt$. $et$ (event time) is a time period where the situation described by the formula holds, and $lt$ (localisation time) is a temporal window within which $et$ must be placed. Temporal linguistic mechanisms are expressed in \textsc{Top}\xspace using temporal operators that manipulate $st$, $et$, and $lt$. There are thirteen operators in total. \ensuremath{\mathit{Part}}\xspace picks a period from a partitioning. \ensuremath{\mathit{Pres}}\xspace and \ensuremath{\mathit{Past}}\xspace are used when expressing present and past tenses. \ensuremath{\mathit{Perf}}\xspace is used in combination with \ensuremath{\mathit{Past}}\xspace to express the past perfect. \ensuremath{\mathit{Culm}}\xspace is used to represent non-progressive forms of verbs whose base forms are culminating activities. \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, and \ensuremath{\mathit{After}}\xspace are employed when expressing punctual and period adverbials, and when expressing \qit{while~\dots}, \qit{before~\dots}, and \qit{after~\dots} subordinate clauses. Duration \qit{in~\dots} and \qit{for~\dots} adverbials are expressed using \ensuremath{\mathit{For}}\xspace. \ensuremath{\mathit{Fills}}\xspace can be used to represent readings of sentences where the situation of the verb covers the whole localisation time; \ensuremath{\mathit{Fills}}\xspace, however, is not used in the rest of this thesis, nor in the prototype \textsc{Nlitdb}\xspace. \ensuremath{\mathit{Begin}}\xspace and \ensuremath{\mathit{End}}\xspace are used to refer to time-points where situations start or stop. Finally, \ensuremath{\mathit{Ntense}}\xspace allows noun phrases to refer either to $st$ or to the time of the verb's tense. \chapter{From English to TOP} \label{English_to_TOP} \proverb{One step at a time.} \section{Introduction} This chapter shows how \textsc{Hpsg}\xspace \cite{Pollard1} \cite{Pollard2} was modified to map English questions directed to a \textsc{Nlitdb}\xspace to appropriate \textsc{Top}\xspace formulae.\footnote{The \textsc{Hpsg}\xspace version of this thesis is based on the revised \textsc{Hpsg}\xspace version of chapter 9 of \cite{Pollard2}.} Although several modifications to \textsc{Hpsg}\xspace were introduced, the \textsc{Hpsg}\xspace version of this thesis remains very close to \cite{Pollard2}. The main differences from \cite{Pollard2} are that: (a) \textsc{Hpsg}\xspace mechanisms for phenomena not examined in this thesis (e.g.\ pronouns, relative clauses) were removed, and (b) the situation-theoretic semantic constructs of \textsc{Hpsg}\xspace were replaced by feature structures that represent \textsc{Top}\xspace expressions. Readers with a rudimentary grasp of modern unification-based grammars \cite{Shieber} should be able to follow most of the discussion in this chapter. Some of the details, however, may be unclear to readers not familiar with \textsc{Hpsg}\xspace. The \textsc{Hpsg}\xspace version of this thesis was implemented as a grammar for the \textsc{Ale} system (see chapter \ref{implementation}). \section{HPSG basics} \label{HPSG_basics} In \textsc{Hpsg}\xspace, each word and syntactic constituent is mapped to a \emph{sign}, a feature structure of a particular form, that provides information about the word or syntactic constituent. An \textsc{Hpsg}\xspace grammar consists of signs for words (I call these \emph{lexical signs}), \emph{lexical rules}, \emph{schemata}, \emph{principles}, and a \emph{sort hierarchy}, all discussed below. \subsection{Lexical signs and sort hierarchy} Lexical signs provide information about individual words. (Words with multiple uses may receive more than one lexical sign.) \pref{lentr:1} shows a lexical sign for the base form of \qit{to land} in the airport domain. \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval land\>} \\ synsem & [loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$ ]} \\ aspect & culmact \\ spr & \<\> \\ subj & \< \feat np[-prd]$_{@1}$ \> \\ comps & \< \feat pp[-prd, pform {\fval on}]$_{@2}$ \> ]\\ cont & \sort{landing\_on}{ [arg1 & occr\_var \\ arg2 & @1 \\ arg3 & @2]} ]]] \end{avm} \label{lentr:1} \end{examps} The $<$ and $>$ delimiters denote lists. The {\feat phon} feature shows the list of words to which the sign corresponds (\pref{lentr:1} corresponds to the single word \qit{land}). Apart from {\feat phon}, every sign has a {\feat synsem} feature (as well as other features not shown in \pref{lentr:1}; I often omit features that are not relevant to the discussion). The value of {\feat synsem} in \pref{lentr:1} is a feature structure that has a feature {\feat loc}. The value of {\feat loc} is in turn a feature structure that has the features {\feat cat} (intuitively, syntactic category) and {\feat cont} (intuitively, semantic content). Each \textsc{Hpsg}\xspace feature structure belongs to a particular sort. The sort hierarchy of \textsc{Hpsg}\xspace shows the available sorts, as well as which sort is a subsort of which other sort. It also specifies which features the members of each sort must have, and the sorts to which the values of these features must belong. (Some modifications were made to the sort hierarchy of \cite{Pollard2}. These will be discussed in sections \ref{TOP_FS} and \ref{more_ind}.) In \pref{lentr:1}, for example, the value of {\feat head} is a feature structure of sort {\srt verb}. The value of {\feat head} signals that the word is the base form ({\feat vform} {\fval bse}) of a non-auxiliary ({\feat aux}~{\fval $-$}) verb. The sort hierarchy of \cite{Pollard2} specifies that the value of {\feat head} must be of sort {\srt head}, and that {\srt verb}\/ is a subsort of {\srt head}. This allows feature structures of sort {\srt verb}\/ to be used as values of {\feat head}. The value of {\feat vform} in \pref{lentr:1} is an \emph{atomic feature structure} (a feature structure of no features) of sort {\srt bse}. For simplicity, when showing feature structures I often omit uninteresting sort names. {\feat aspect} \index{aspect@{\feat aspect} (\textsc{Hpsg}\xspace feature)} is the only new \textsc{Hpsg}\xspace feature of this thesis. It is a feature of feature structures of sort {\srt cat}\/ (feature structures that can be used as values of {\feat cat}), and its values are feature structures of sort {\srt aspect}. {\srt aspect}\/ contains only atomic feature structures, and has the subsorts: {\srt state}, \index{state@{\srt state}\/ (\textsc{Hpsg}\xspace sort, state aspectual class)} {\srt activity}, \index{activity@{\srt activity}\/ (\textsc{Hpsg}\xspace sort, activity aspectual class)} {\srt culmact}\/ \index{culmact@{\srt culmact}\/ (\textsc{Hpsg}\xspace sort, culminating activity)} (culminating activity), and {\srt point}. \index{point@{\srt point}\/ (\textsc{Hpsg}\xspace sort, point aspectual class)} {\srt state}\/ is in turn partitioned into: {\srt lex\_state}\/ \index{lexstate@{\srt lex\_state}\/ (\textsc{Hpsg}\xspace sort, lexical state)} (lexical state), {\srt progressive}\/ \index{progressive@{\srt progressive}\/ (\textsc{Hpsg}\xspace sort, progressive state)} (progressive state), and {\srt cnsq\_state}\/ \index{cnsqstate@{\srt cnsq\_state}\/ (\textsc{Hpsg}\xspace sort, consequent state)} (consequent state). This agrees with the aspectual taxonomy of chapter \ref{linguistic_data}. Following table \vref{airport_verbs}, \pref{lentr:1} classifies the base form of \qit{to land} as culminating activity. The {\feat spr}, {\feat subj}, and {\feat comps} features of \pref{lentr:1} provide information about the specifier, subject, and complements with which the verb has to combine. Specifiers are determiners (e.g.\ \qit{a}, \qit{the}), and words like \qit{much} (as in \qit{much more}) and \qit{too} (as in \qit{too late}). Verbs do not admit specifiers, and hence the value of {\feat spr} in \pref{lentr:1} is the empty list. The {\feat subj} value of \pref{lentr:1} means that the verb requires a noun-phrase as its subject. The {\feat np[-prd]$_{\avmbox{1}}$} in \pref{lentr:1} has the same meaning as in \cite{Pollard2}. Roughly speaking, it is an abbreviation for a sign that corresponds to a noun phrase. The {\feat -prd} means that the noun phrase must be non-predicative (see section \ref{hpsg:nouns} below). The \avmbox{1} is intuitively a pointer to the world entity described by the noun phrase. Similarly, the {\feat comps} value of \pref{lentr:1} means that the verb requires as its complement a non-predicative prepositional phrase (section \ref{hpsg:pps} below), introduced by \qit{on}. The \avmbox{2} is intuitively a pointer to the world entity of the prepositional phrase (e.g.\ if the prepositional phrase is \qit{on a runway}, the \avmbox{2} is a pointer to the runway). The value of {\feat cont} in \pref{lentr:1} represents the \textsc{Top}\xspace predicate $landing\_on(\beta, \tau_1, \tau_2)$, where $\tau_1$ and $\tau_2$ are \textsc{Top}\xspace terms corresponding to \avmbox{1} and \avmbox{2}, and $\beta$ is a \textsc{Top}\xspace variable acting as an occurrence identifier (section \ref{occurrence_ids}).\footnote{I follow the approach of section 8.5.1 of \cite{Pollard2}, whereby the {\feat relation} feature is dropped, and its role is taken up by the sort of the feature structure.} The exact relation between \textsc{Hpsg}\xspace feature structures and \textsc{Top}\xspace expressions will be discussed in the following sections. \subsection{Lexical rules} Lexical rules generate new lexical signs from existing ones. In section \ref{hpsg:verb_forms}, for example, I introduce lexical rules that generate automatically lexical signs for (single-word) non-base verb forms (e.g.\ a sign for the simple past \qit{landed}) from signs for base forms (e.g.\ \pref{lentr:1}). This reduces the number of lexical signs that need to be listed in the grammar. \subsection{Schemata and principles} \label{schemata_principles} \textsc{Hpsg}\xspace schemata specify basic patterns that are used when words or syntactic constituents combine to form larger constituents. For example, the \emph{head-complement schema} is the pattern that is used when a verb combines with its complements (e.g.\ when \qit{landed} combines with its complement \qit{on runway 2}; in this case, the verb is the ``head-daughter'' of the constituent \qit{landed on runway 2}). The \emph{head-subject schema} is the one used when a verb phrase (a verb that has combined with its complements but not its subject) combines with its subject (e.g.\ when \qit{landed on runway 2} combines with \qit{BA737}; in this case, the verb phrase is the head-daughter of \qit{BA737 landed on runway 2}). No modifications to the schemata of \cite{Pollard2} are introduced in this thesis, and hence schemata will not be discussed further. \textsc{Hpsg}\xspace principles control the propagation of feature values from the signs of words or syntactic constituents to the signs of their super-constituents. The \emph{head feature principle}, for example, specifies that the sign of the super-constituent inherits the {\feat head} value of the head-daughter's sign. This causes the sign of \qit{landed on runway 2} to inherit the {\feat head} value of the sign of \qit{landed}, and the same value to be inherited by the sign of \qit{BA737 landed on runway 2}. This thesis uses simplified versions of Pollard and Sag's semantics principle and constituent ordering principle (to be discussed in sections \ref{non_pred_nps} and \ref{fronted}), and introduces one new principle (the aspect principle, to be discussed in section \ref{hpsg:punc_adv}). All other principles are as in \cite{Pollard2}. \section{Representing TOP yes/no formulae in HPSG} \label{TOP_FS} According to \cite{Pollard2}, the {\feat cont} value of \pref{lentr:1} should actually be \pref{lentr:2}. \begin{examps} \item \avmoptions{active} \begin{avm} \sort{psoa}{ [quants & \<\> \\ nucleus & \sort{landing\_on}{ [arg1 & occr\_var \\ arg2 & @1 \\ arg3 & @2]} ]} \end{avm} \label{lentr:2} \end{examps} In \cite{Pollard2}, feature structures of sort {\srt psoa} have two features: {\feat quants} and {\feat nucleus}.\footnote{``{\srt Psoa}'' stands for ``parameterised state of affairs'', a term from situation theory \cite{Cooper1990}. The semantic analysis here is not situation-theoretic, but the term ``psoa'' is still used for compatibility with \cite{Pollard2}.} {\feat quants}, which is part of \textsc{Hpsg}\xspace's quantifier storage mechanism, is not used in this thesis. This leaves only one feature ({\feat nucleus}) in {\srt psoa}s. For simplicity, {\feat nucleus} was also dropped, and the {\srt psoa}\/ sort was taken to contain the feature structures that would be values of {\feat nucleus} in \cite{Pollard2}. More precisely, in this thesis {\srt psoa}\/ has two subsorts: {\srt predicate}\/ \index{predicate@{\srt predicate}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace predicates)} and {\srt operator}\/ \index{operator@{\srt operator}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace operators)} (figure \ref{psoa_fig}). {\srt predicate}\/ contains feature structures that represent \textsc{Top}\xspace predicates, while {\srt operator}\/ contains feature structures that represent all other \textsc{Top}\xspace yes/no formulae. (Hence, {\srt psoa}\/ corresponds to all yes/no formulae.) {\srt predicate}\/ has domain-specific subsorts, corresponding to predicate functors used in the domain for which the \textsc{Nlitdb}\xspace is configured. In the airport domain, for example, {\srt landing\_on}\/ is a subsort of {\srt predicate}. The feature structures in the subsorts of {\srt predicate}\/ have features named {\feat arg1}, {\feat arg2}, {\feat arg3}, \index{arg123@{\feat arg1}, {\feat 2}, {\feat 3} (new \textsc{Hpsg}\xspace features, correspond to \textsc{Top}\xspace predicate arguments)} etc. These represent the first, second, third, etc.\ arguments of the predicates. The values of {\feat arg1}, {\feat arg2}, etc.\ are of sort {\srt ind}\/ ({\srt occr\_var}\/ \index{occrvar@{\srt occr\_var}\/ (\textsc{Hpsg}\xspace sort, represents occurrence identifiers)} is a subsort of {\srt ind}). {\srt ind}\/ will be discussed further below. \begin{figure} \avmoptions{} \setlength{\GapWidth}{5mm} \hrule \begin{center} \begin{bundle}{{\srt psoa}} \chunk{ \begin{bundle}{{\srt predicate}} \chunk{\begin{avm} \sort{circling}{ \[arg1 & ind\]} \end{avm}} \chunk{\begin{avm} \osort{landing\_on}{ \[arg1 & occr\_var \\ arg2 & ind \\ arg3 & ind\]} \end{avm}} \chunk{\dots} \end{bundle}} \chunk{ \begin{bundle}{{\srt operator}} \chunk{\begin{avm} \osort{pres}{ \[main\_psoa & psoa\]} \end{avm}} \chunk{\dots} \end{bundle}} \end{bundle} \caption{Subsorts of {\srt psoa}} \label{psoa_fig} \index{pres2@{\srt pres}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Pres}}\xspace)} \index{past2@{\srt past}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Past}}\xspace)} \index{perf2@{\srt perf}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Perf}}\xspace)} \index{atop2@{\srt at\_op}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{At}}\xspace)} \index{beforeop2@{\srt before\_op} (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Before}}\xspace)} \index{afterop2@{\srt after\_op} (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{After}}\xspace)} \index{part2@{\srt part}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Part}}\xspace)} \index{culm2@{\srt culm}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Culm}}\xspace)} \index{end2@{\srt end}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{End}}\xspace)} \index{begin2@{\srt begin}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Begin}}\xspace)} \index{and2@{\srt and}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's conjunction)} \index{ntense2@{\srt ntense}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{Ntense}}\xspace)} \index{forop2@{\srt for\_op}\/ (\textsc{Hpsg}\xspace sort, corresponds to \textsc{Top}\xspace's \ensuremath{\mathit{For}}\xspace)} \index{mainpsoa@{\feat main\_psoa} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{ethandle@{\feat et\_handle} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{timespec@{\feat time\_spec} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{partng@{\feat partng} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{partvar@{\feat part\_var} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{conjunct12@{\feat conjunct1}, {\feat 2} (\textsc{Hpsg}\xspace features, used in the representation of \textsc{Top}\xspace formulae)} \index{durunit@{\feat dur\_unit} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \index{duration@{\feat duration} (\textsc{Hpsg}\xspace feature, used in the representation of \textsc{Top}\xspace formulae)} \end{center} \hrule \end{figure} The {\srt operator}\/ sort has thirteen subsorts, shown in figure \ref{operator_sorts}. These correspond to the twelve \textsc{Top}\xspace operators (\ensuremath{\mathit{Fills}}\xspace is ignored), plus one sort for conjunction.\footnote{The sorts that correspond to the \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, \ensuremath{\mathit{After}}\xspace, and \ensuremath{\mathit{For}}\xspace operators are called {\srt at\_op}, {\srt before\_op}, {\srt after\_op}, and {\srt for\_op}\/ to avoid name clashes with existing \textsc{Hpsg}\xspace sorts.} The order of the features in figure \ref{operator_sorts} corresponds to the order of the arguments of the \textsc{Top}\xspace operators. For example, the {\feat et\_handle} and {\feat main\_psoa} features of the {\srt past}\/ sort correspond to the first and second arguments respectively of \textsc{Top}\xspace's $\ensuremath{\mathit{Past}}\xspace[\beta, \phi]$. For simplicity, in the rest of this thesis I drop the $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]$ version of \ensuremath{\mathit{Part}}\xspace (section \ref{denotation}), and I represent words like \qit{yesterday} using \textsc{Top}\xspace constants (e.g.\ $yesterday$) rather than expressions like $\ensuremath{\mathit{Part}}\xspace[day^c, \beta, -1]$. This is why there is no sort for $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]$ in figure \ref{operator_sorts}. \begin{figure} \avmoptions{active} \hrule \medskip \hspace*{9mm} \begin{tabular}{lll} \begin{avm} \osort{pres}{ [main\_psoa & psoa]} \end{avm} && \hspace*{7mm} \begin{avm} \osort{culm}{ [main\_psoa & predicate]} \end{avm} \\ \begin{avm} \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & psoa]} \end{avm} && \hspace*{7mm} \begin{avm} \osort{and}{ [conjunct1 & psoa \\ conjunct2 & psoa]} \end{avm} \\ \begin{avm} \osort{perf}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & psoa]} \end{avm} && \hspace*{7mm} \begin{avm} \osort{begin}{ [main\_psoa & psoa]} \end{avm} \\ \begin{avm} \osort{at\_op}{ [time\_spec & temp\_ent $\lor$ psoa \\ main\_psoa & psoa]} \end{avm} && \hspace*{7mm} \begin{avm} \osort{end}{ [main\_psoa & psoa]} \end{avm} \\ \begin{avm} \osort{before\_op}{ [time\_spec & temp\_ent $\lor$ psoa \\ main\_psoa & psoa]} \end{avm} && \begin{avm} \osort{ntense}{ [et\_handle & now $\lor$ \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & psoa]} \end{avm} \\ \begin{avm} \osort{after\_op}{ [time\_spec & temp\_ent $\lor$ psoa \\ main\_psoa & psoa]} \end{avm} && \begin{avm} \osort{for\_op}{ [dur\_unit & compl\_partng \\ duration & \sort{sem\_num}{ [tvar $-$]} \\ main\_psoa & psoa]} \end{avm} \\ \multicolumn{3}{l}{\avmoptions{}\begin{avm} \osort{part}{ \[partng & compl\_partng $\lor$ \sort{gappy\_partng}{ \[tvar $-$ \]} \\ part\_var & \sort{temp\_ent}{ \[tvar $+$ \]} \]} \end{avm} } \end{tabular} \caption{Subsorts of {\srt operator}} \label{operator_sorts} \medskip \hrule \end{figure} In \cite{Pollard2}, feature structures of sort {\srt ind}\/ (called indices) have the features {\srt person}, {\srt number}, and {\srt gender}, which are used to enforce person, number, and gender agreement. For simplicity, these features are ignored here, and no agreement checks are made. Pollard and Sag's subsorts of {\srt ind}\/ ({\srt ref}, {\srt there}, {\srt it}\/), which are used in \textsc{Hpsg}\xspace's binding theory, are also ignored here. In this thesis, indices represent \textsc{Top}\xspace terms (they also represent gappy partitioning names, but let us ignore this temporarily). The situation is roughly speaking as in figure \ref{simple_ind_hierarchy}. For each \textsc{Top}\xspace constant (e.g.\ $ba737$, $gate2$), there is a subsort of {\srt ind}\/ that represents that constant. There is also a subsort {\srt var}\/ of {\srt ind}, whose indices represent \textsc{Top}\xspace variables. A {\feat tvar} \index{tvar@{\feat tvar} (\textsc{Hpsg}\xspace feature, shows if an index represents a \textsc{Top}\xspace variable)} feature is used to distinguish indices that represent constants from indices that represent variables. All indices of constant-representing sorts (e.g.\ {\srt ba737}, {\srt uk160}\/) have their {\feat tvar} set to $-$. Indices of {\srt var}\/ have their {\feat tvar} set to $+$. \begin{figure} \avmoptions{} \hrule \begin{center} \begin{bundle}{{\srt ind}} \chunk{\begin{avm} \sort{ba737}{ \[tvar & $-$\]} \end{avm}} \chunk{\begin{avm} \sort{uk160}{ \[tvar & $-$\]} \end{avm}} \chunk{\begin{avm} \sort{gate2}{ \[tvar & $-$\]} \end{avm}} \chunk{\dots} \chunk{\begin{avm} \sort{var}{ \[tvar & $+$\]} \end{avm}} \end{bundle} \caption{{\srt ind} and its subsorts -- simplified version} \label{simple_ind_hierarchy} \end{center} \hrule \end{figure} The fact that there is only one subsort ({\srt var}\/) for \textsc{Top}\xspace variables in figure \ref{simple_ind_hierarchy} does not mean that only one \textsc{Top}\xspace variable can be represented. {\srt var} is a \emph{sort} of feature structures, containing infinitely many feature-structure members. Although the members of {\srt var} cannot be distinguished by their feature values (they all have {\feat tvar} set to $+$), they are still different; i.e.\ they are ``structurally identical'' but not ``token-identical'' (see chapter 1 of \cite{Pollard2}). Each one of the feature-structure members of {\srt var}\/ represents a different \textsc{Top}\xspace variable. The subsorts that correspond to \textsc{Top}\xspace constants also contain infinitely many different feature-structure members. In this case, however, all members of the same subsort are taken to represent the same constant. For example, any feature structure of sort {\srt gate2} represents the \textsc{Top}\xspace constant $gate2$. \section{More on the subsorts of ind} \label{more_ind} The subsorts of {\srt ind}\/ are actually more complicated than in figure \ref{simple_ind_hierarchy}. Natural language front-ends (e.g.\ \textsc{Masque} \cite{Auxerre2}, \textsc{Team} \cite{Grosz1987}, \textsc{Cle} \cite{Alshawi}, \textsc{SystemX} \cite{Cercone1993}) often employ a domain-dependent hierarchy of types of world entities. This hierarchy is typically used in disambiguation, and to detect semantically anomalous sentences like \qit{Gate 2 departed from runway 1}. Here, a hierarchy of this kind is mounted under the {\srt ind}\/ sort. (Examples illustrating the use of this hierarchy are given in following sections.) In the airport domain, there are temporal world entities (the Monday 16/10/95, the year 1995, etc.), and non-temporal world entities (flight BA737, gate 2, etc.). Indices representing temporal entities are classified into a {\srt temp\_ent}\/ \index{tempent@{\srt temp\_ent}\/ (\textsc{Hpsg}\xspace sort, represents temporal entities)} subsort of {\srt ind}, while indices representing non-temporal entities are classified into {\srt non\_temp\_ent}\/ \index{nontempent@{\srt non\_temp\_ent} (\textsc{Hpsg}\xspace sort, represents non-temporal entities)} (see figure \ref{ind_hierarchy}; ignore {\srt partng}\/ and its subsorts for the moment). {\srt non\_temp\_ent}\/ has in turn subsorts like {\srt mass}\/ (indices representing mass entities, e.g.\ foam or water), {\srt flight\_ent}\/ (indices representing flights, e.g.\ BA737), etc. {\srt flight\_ent}\/ has one subsort for each \textsc{Top}\xspace constant that denotes a flight (e.g.\ {\srt ba737}, {\srt uk160}\/), plus one sort ({\srt flight\_ent\_var}\/) whose indices represent \textsc{Top}\xspace variables that denote flights. The other children-sorts of {\srt non\_temp\_ent}\/ have similar subsorts. \begin{figure} \begin{center} \includegraphics[scale=.6]{handle_sorts} \caption{{\srt partng}, {\srt ind}, and their subsorts} \label{ind_hierarchy} \end{center} \end{figure} {\srt temp\_ent}\/ has subsorts like {\srt minute\_ent}\/ (indices representing particular minutes, e.g.\ the 5:00pm minute of 1/1/91), {\srt day\_ent}\/ (indices representing particular days), etc. {\srt minute\_ent}\/ has one subsort for each \textsc{Top}\xspace constant that denotes a particular minute (e.g.\ {\srt 5:00pm1/1/91}\/), plus one sort ({\srt minute\_ent\_var}\/) whose indices represent \textsc{Top}\xspace variables that denote particular minutes. The other children-sorts of {\srt temp\_ent}\/ have similar subsorts. The indices of {\srt other\_temp\_ent\_var}\/ \index{othertempentvar@{\srt other\_temp\_ent\_var}\/ (\textsc{Hpsg}\xspace sort, represents some time-denoting \textsc{Top}\xspace variables)} (figure \ref{ind_hierarchy}) represent \textsc{Top}\xspace variables that denote temporal-entities which do not correspond to sister-sorts of {\srt other\_temp\_ent\_var}\/ ({\srt minute\_ent}, {\srt day\_ent}, etc.). This is needed because not all \textsc{Top}\xspace variables denote particular minutes, days, months, etc. In \pref{lentr:3}, for example, $e^v$ denotes a past period that covers exactly a taxiing of UK160 to gate 1 (from start to completion). The taxiing may have started at 5:00pm on 1/1/95, and it may have been completed at 5:05pm on the same day. In that case, $e^v$ denotes a period that is neither a minute-period, nor a day-period, nor a month-period, etc. \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[taxiing\_to(occr^v,uk160, gate1)]]$ \label{lentr:3} \end{examps} {\srt occr\_var}\/ contains indices that represent \textsc{Top}\xspace variables used as occurrence identifiers (section \ref{occurrence_ids}). Indices of sorts that represent \textsc{Top}\xspace constants (e.g.\ {\srt foam}, {\srt 5:00pm1/1/91}, {\srt Jun93}\/ in figure \ref{ind_hierarchy}) have their {\feat tvar}\/ \index{tvar@{\feat tvar} (\textsc{Hpsg}\xspace feature, shows if an index represent a \textsc{Top}\xspace variable or not)} set to $-$. Indices of sorts that represent \textsc{Top}\xspace variables (e.g.\ {\srt flight\_ent\_var}, {\srt minute\_ent\_var}, {\srt other\_temp\_ent\_var}, {\srt occr\_var}\/) have their {\feat tvar}\/ set to $+$. There is also a special sort {\srt now}\/ (not shown in figure \ref{ind_hierarchy}) that is used to represent the \textsc{Top}\xspace expression $now^*$ \index{now2@{\srt now}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace's $now^*$)} (section \ref{ntense_op}). The sorts of figure \ref{operator_sorts} mirror the definitions of \textsc{Top}\xspace's operators. For example, the {\srt ntense}\/ sort reflects that fact that the first argument of an \ensuremath{\mathit{Ntense}}\xspace operator must be $now^*$ or a variable ({\feat tvar}~$+$) denoting a period ({\srt temp\_ent}), while the second argument must be a yes/no formula ({\srt psoa}). (The {\srt sem\_num}\/ \index{semnum@{\srt sem\_num}\/ (\textsc{Hpsg}\xspace sort, represents numbers)} sort in {\srt for\_op} is a child-sort of {\srt non\_temp\_ent}, with subsorts that represent the numbers 1, 2, 3, etc. The {\srt compl\_partng}\/ and {\srt gappy\_partng}\/ sorts in {\srt for\_op}\/ and {\srt part}\/ are discussed below.) The hierarchy under {\srt ind}\/ is domain-dependent. For example, in an application where the database contains information about a company, the subsorts of {\srt non\_temp\_ent}\/ would correspond to departments, managers, etc.\ I assume, however, that in all application domains, {\srt ind} would have the children-sorts {\srt temp\_ent}, {\srt non\_temp\_ent}, {\srt occr\_var}, {\srt gappy\_partng} (to be discussed below), and possibly more. I also assume that the subsorts of {\srt partng} (see below) and {\srt temp\_ent}\/ would have the general form of figure \ref{ind_hierarchy}, though they would have to be adjusted to reflect the partitionings and temporal entities used in the particular application. I now turn to the {\srt partng}\/ \index{partng2@{\srt partng}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace partitioning names)} sort of figure \ref{ind_hierarchy}, which has the subsorts {\srt compl\_partng}\/ \index{complpartng@{\srt compl\_partng}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace complete partitioning names)} and {\srt gappy\_partng}\/ \index{gappypartng@{\srt gappy\_partng}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace gappy part.\ names and some terms)} (these three sorts do not exist in \cite{Pollard2}). For each \textsc{Top}\xspace complete or gappy partitioning name (e.g.\ $minute^c$, $day^c$, $\text{\textit{5:00pm}}^g$, $monday^g$) there is a leaf-subsort of {\srt compl\_partng}\/ or {\srt gappy\_partng}\/ respectively that represents that name. (The leaf-subsorts of {\srt gappy\_partng} are also used to represent some \textsc{Top}\xspace terms; this is discussed below.) In figure \ref{ind_hierarchy}, the sorts {\srt 5:00pm}, {\srt 9:00am}, etc.\ are grouped under {\srt minute\_gappy}\/ to reflect the fact that the corresponding partitionings contain minute-periods. (I assume here that these partitioning names denote the obvious partitionings.) Similarly, {\srt monday}, {\srt tuesday}, etc.\ are grouped under {\srt day\_gappy}\/ to reflect the fact that the corresponding partitionings contain day-periods. Section \ref{habituals} provides examples where sorts like {\srt minute\_gappy} and {\srt day\_gappy} prove useful. Apart from gappy partitioning names, the subsorts of {\srt gappy\_partng}\/ are also used to represent \textsc{Top}\xspace terms that denote generic representatives of partitionings (section \ref{hab_problems}). (To allow the subsorts of {\srt gappy\_partng}\/ to represent \textsc{Top}\xspace terms, {\srt gappy\_partng}\/ is not only a subsort of {\srt partng}, but also of {\srt ind}; see figure \ref{ind_hierarchy}.) For example, \pref{lentr:6} (that expresses the habitual reading of \pref{lentr:5}) is represented as \pref{lentr:7}. In this case, the subsort {\srt 5:00pm}\/ of {\srt gappy\_partng}\/ represents the \textsc{Top}\xspace constant \textit{5:00pm}. \avmoptions{active} \begin{examps} \item BA737 departs (habitually) at 5:00pm \label{lentr:5} \item $\ensuremath{\mathit{Pres}}\xspace[hab\_departs\_at(ba737, \text{\textit{5:00pm}})]$ \label{lentr:6} \item \begin{avm} \osort{pres}{ [main\_psoa & \sort{hab\_departs\_at}{ [arg1 & \sort{ba737}{ [tvar $-$]} \\ arg2 & \sort{5:00pm}{ [tvar $-$]}]}]} \end{avm} \label{lentr:7} \end{examps} In contrast, \pref{lentr:9} (that expresses the non-habitual reading of \pref{lentr:8}) is represented as \pref{lentr:10}. In this case, the subsort {\srt 5:00pm}\/ of {\srt gappy\_partng}\/ represents the \textsc{Top}\xspace gappy partitioning name $\text{\textit{5:00pm}}^g$. (It cannot represent a \textsc{Top}\xspace term, because \textsc{Top}\xspace terms cannot be used as first arguments of \ensuremath{\mathit{Part}}\xspace operators.) The \avmbox{1}s in \pref{lentr:10} mean that the values of {\feat part\_var} and {\feat time\_spec} must be token-identical, i.e.\ they must represent the same \textsc{Top}\xspace variable. \avmoptions{active} \begin{examps} \item BA737 departed (actually) at 5:00pm. \label{lentr:8} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, depart(ba737)]]$ \label{lentr:9} \item \begin{avm} \osort{and}{ [conjunct1 & \osort{part}{ [partng & \sort{5:00pm}{ [tvar $-$]} \\ part\_var & \sort{minute\_ent\_var}{ [tvar $+$]}@1 ]} \\ conjunct2 & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \sort{depart}{ [arg1 & \sort{ba737}{ [tvar $-$]} ]}]}]}]} \end{avm} \label{lentr:10} \end{examps} The {\srt minute\_gappy}\/ and {\srt gappy\_var}\/ sorts of figure \ref{ind_hierarchy} are used only to represent \textsc{Top}\xspace variables that denote generic representatives of unknown {\srt minute\_gappy}\/ or {\srt day\_gappy}\/ partitionings. The $t^v$ variable of \pref{lentr:16}, for example, denotes the generic representative of an unknown {\srt minute\_gappy}\/ partitioning. (If BA737 departs habitually at 5:00pm, $t^v$ denotes the generic representative of the $\text{\textit{5:00pm}}^g$ partitioning, the same generic representative that the \textit{5:00pm} constant of \pref{lentr:6} denotes.) The $\ensuremath{\mathit{Pres}}\xspace[hab\_departs\_at(ba737, t^v)]$ part of \pref{lentr:16} is represented as \pref{lentr:17}. (The feature-structure representation of quantifiers will be discussed in section \ref{TOP_FS_WH}.) \begin{examps} \item When does BA737 (habitually) depart? \label{lentr:15} \item $?t^v \; \ensuremath{\mathit{Pres}}\xspace[hab\_departs\_at(ba737, t^v)]$ \label{lentr:16} \item \begin{avm} \osort{pres}{ [main\_psoa & \sort{hab\_departs\_at}{ [arg1 & \sort{ba737}{ [tvar $-$]} \\ arg2 & \sort{minute\_gappy\_var}{ [tvar $+$]} ]} ]} \end{avm} \label{lentr:17} \end{examps} The indices of sorts like {\srt minute\_gappy\_var}\/ and {\srt day\_gappy\_var}\/ have their {\feat tvar} \index{tvar@{\feat tvar} (\textsc{Hpsg}\xspace feature, shows if an index represent a \textsc{Top}\xspace variable or not)} set to $+$. The indices of all other leaf-subsorts of {\srt gappy\_partng}\/ (e.g.\ {\srt 5:00pm}, {\srt monday}\/) have their {\feat tvar} set to $-$. \section{Representing TOP quantifiers in HPSG} \label{TOP_FS_WH} \textsc{Top}\xspace yes/no formulae are represented in the \textsc{Hpsg}\xspace version of this thesis as feature-structures of sort {\srt psoa} (figure \ref{psoa_fig}). To represent \textsc{Top}\xspace wh-formulae (formulae with interrogative or interrogative-maximal quantifiers) additional feature-structure sorts are needed. I discuss these below. Feature structures of sort {\srt quant}\/ represent unresolved quantifiers (quantifiers whose scope is not known yet). They have two features: {\feat det} and {\feat restind} (restricted index), as shown in \pref{nps:5}. The {\feat det} feature shows the type of the quantifier. In this thesis, {\feat det} can have the values {\srt exists}\/ (existential quantifier), {\srt interrog}\/ \index{interrog5@{\srt interrog}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace interrogative quantifiers)} (interrogative quantifier), and {\srt interrog\_mxl}\/ \index{interrogmxl5@{\srt interrog\_mxl}\/ (\textsc{Hpsg}\xspace sort, represents \textsc{Top}\xspace interrogative-maximal quantifiers)} (interrogative-maximal quantifier). (Apart from the values of {\feat det}, {\srt quant}\/ is as in \cite{Pollard2}.) \begin{examps} \item \avmoptions{active} \begin{avm} \sort{quant}{ [det & exists $\lor$ interrog $\lor$ interrog\_mxl \\ restind & \osort{nom\_obj}{ [index & \sort{ind}{ [tvar & $+$]} \\ restr & set\(psoa\)]}]} \end{avm} \label{nps:5} \end{examps} The values of {\feat restind} are feature structures of sort {\srt nom\_obj}\/ (nominal object).\footnote{In \cite{Pollard2}, {\srt nom\_obj}\/ has the subsorts {\srt npro}\/ (non-pronoun) and {\srt pron}\/ (pronoun). These subsorts are not used in this thesis.} These have the features {\feat index} (whose values are of sort {\srt ind}\/) and {\feat restr} (whose values are sets of {\srt psoa}s). When a {\srt nom\_obj}\/ feature structure is the value of {\feat restind}, the {\feat index} corresponds to the \textsc{Top}\xspace variable being quantified, and the {\feat restr} corresponds to the restriction of the quantifier. (If the {\feat restr} set contains more than one {\srt psoa}s, the {\srt psoa}-elements of the set are treated as forming a conjunction.) For example, \pref{nps:6} represents \pref{nps:7}. \begin{examps} \item \avmoptions{active} \begin{avm} \sort{quant}{ [det & interrog \\ restind & \osort{nom\_obj}{ [index & \sort{ind}{ [tvar & $+$]}@1 \\ restr & \{\sort{flight}{ [arg1 & @1]} \}]}]} \end{avm} \label{nps:6} \item $?f^v \; flight(f^v)$ \label{nps:7} \end{examps} Although \textsc{Top}\xspace does not use explicit existential quantifiers (universal quantification is not supported, and \textsc{Top}\xspace variables can be thought of as existentially quantified), the \textsc{Hpsg}\xspace version of this thesis employs explicit existential quantifiers ({\srt quant}s whose {\feat det} is {\srt exists}\/) for compatibility with \cite{Pollard2}. These explicit existential quantifiers are removed when extracting \textsc{Top}\xspace formulae from signs (this is discussed in section \ref{extraction_hpsg} below). \section{Extracting TOP formulae from HPSG signs} \label{extraction_hpsg} The parser maps each question to a sign. (Multiple signs are generated when the parser understands a question to be ambiguous.) For example, \qit{Which inspector was at gate 2?} is mapped to \pref{nps:14b} (exactly how \pref{nps:14b} is generated will become clearer in the following sections; see also the comments about \ensuremath{\mathit{Ntense}}\xspace{s} in section \ref{non_pred_nps} below). \begin{examps} \setbox\avmboxa=\hbox{\begin{avm} \sort{inspector}{ [arg1 & @1]} \end{avm}} \avmoptions{active,center} \item \begin{avm} [\avmspan{phon \; \<\fval which, inspector, was, at, gate2\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\>] \\ cont & \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{located\_at}{ [arg1 & @1 \\ arg2 & gate2]}]}] \\ \avmspan{qstore \; \{\sort{quant}{ [det & interrog \\ restind & \osort{nom\_obj}{ [index & \sort{person\_ent}{ [tvar & $+$]}@1 \\ restr & \{\box\avmboxa\} ]} ]}\} } ] \end{avm} \label{nps:14b} \end{examps} Apart from the features that were discussed in section \ref{HPSG_basics}, signs also have the feature {\feat qstore}, whose values are sets of {\srt quant}s (section \ref{TOP_FS_WH}). The {\feat cont} value of signs that correspond to questions is of sort {\srt psoa}\/, i.e.\ it represents a \textsc{Top}\xspace yes/no formula. In the \textsc{Hpsg}\xspace version of this thesis, the {\feat qstore} value represents quantifiers that must be ``inserted'' in front of the formula of {\feat cont}. In the prototype \textsc{Nlitdb}\xspace (to be discussed in chapter \ref{implementation}), there is an ``extractor'' of \textsc{Top}\xspace formulae that examines the {\feat cont} and {\feat qstore} features of the question's sign, and generates the corresponding \textsc{Top}\xspace formula. This is a trivial process, which I discuss only at an abstract level: the extractor first examines recursively the features and feature values of {\feat cont}, rewriting them in term notation (in \pref{nps:14b}, this generates \pref{nps:15}); then, for each element of {\feat qstore}, the extractor adds a suitable quantifier in front of the formula of {\feat cont} (in \pref{nps:14b}, this transforms \pref{nps:15} into \pref{nps:17}). \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, located\_at(p^v, gate2)]$ \label{nps:15} \item $?p^v \; inspector(p^v) \land \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(p^v, gate2)]$ \label{nps:17} \end{examps} In the case of elements of {\feat qstore} that correspond to existential quantifiers, no explicit existential quantifier is added to the formula of {\feat cont} (only the expression that corresponds to the {\feat restr} of the {\srt quant}-element is added). For example, if the {\feat det} of \pref{nps:14b} were {\srt exists}, \pref{nps:17} would be \pref{nps:17b}. \begin{examps} \item $inspector(p^v) \land \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(p^v, gate2)]$ \label{nps:17b} \end{examps} The extracted formula then undergoes an additional post-processing phase (to be discussed in section \ref{post_processing}). This is a collection of transformations that need to be applied to some of the extracted formulae. (In \pref{nps:17} and \pref{nps:17b}, the post-processing has no effect.) \section{Verb forms} \label{hpsg:verb_forms} I now present the treatment of the various linguistic constructs, starting from verb forms (simple present, past continuous, etc.). (Pollard and Sag do not discuss temporal linguistic mechanisms.) \subsection{Single-word verb forms} \label{single_word_forms} Let us first examine the lexical rules that generate signs for (single-word) non-base verb forms from signs for base forms. The signs for simple present forms are generated by \pref{vforms:1}. \begin{examps} \item \lexrule{Simple Present Lexical Rule:} \begin{center} \avmoptions{active} \begin{avm} [\avmspan{phon \; \<$\lambda$\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$]} \\ aspect & lex\_state] \\ cont & @1]] \end{avm} \\ $\Downarrow$ \\ \begin{avm} [\avmspan{phon \; \<\fval morph\($\lambda$, simple\_present\)\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$]} \\ aspect & lex\_state] \\ cont & \sort{pres}{ [main\_psoa & @1]} ]] \end{avm} \end{center} \label{vforms:1} \end{examps} \pref{vforms:1} means that for each lexical sign that matches the first feature structure (the ``left hand side'', LHS) of the rule, a new lexical sign should be generated as shown in the second feature structure (the ``right hand side'', RHS) of the rule. (Following standard \textsc{Hpsg}\xspace notation, I write {\feat synsem$\mid$loc} to refer to the {\feat loc} feature of the value of {\feat synsem}.) The {\feat head}s of the LHS and RHS mean that the original sign must correspond to the base form of a non-auxiliary verb (auxiliary verbs are treated separately), and that the resulting sign corresponds to a finite verb form (a form that does not need to combine with an auxiliary verb). The {\feat cont} of the new sign is the same as the {\feat cont} of the original one, except that it contains an additional \ensuremath{\mathit{Pres}}\xspace operator. Features of the original sign not shown in the LHS (e.g.\ {\feat subj}, {\feat comps}) have the same values in the generated sign. \pref{vforms:1} requires the original sign to correspond to a (lexical) state base form. No simple present signs are generated for verbs whose base forms are not states. This is in accordance with the assumption of section \ref{simple_present} that the simple present can be used only with state verbs. $morph(\lambda, simple\_present)$ denotes a morphological transformation that generates the simple present form (e.g.\ \qit{contains}) from the base form (e.g.\ \qit{contain}). The prototype \textsc{Nlitdb}\xspace actually employs two different simple present lexical rules. These generate signs for singular and plural simple present forms respectively. As mentioned in sections \ref{ling_not_supported} and \ref{TOP_FS}, plurals are treated semantically as singulars, and no number-agreement checks are made. Hence, the two lexical rules differ only in the {\feat phon} values of the generated signs. \pref{vforms:2} shows the base form sign of \qit{to contain} in the airport domain. From \pref{vforms:2}, \pref{vforms:1} generates \pref{vforms:3}. The {\srt tank\_ent}\/ and {\srt mass\_ent}\/ in \pref{vforms:2} and \pref{vforms:3} mean that the indices introduced by the subject and the object must be of sort {\srt tank\_ent} and {\srt mass\_ent}\/ respectively ({\srt tank\_ent}\/ is a sister of {\srt flight\_ent}\/ in figure \ref{ind_hierarchy}). Hence, the semantically anomalous \qit{Gate 2 contains water.} (where the subject introduces an index of sort {\srt gate2}, which is not a subsort of {\srt tank\_ent}\/) would be rejected. All lexical signs of verb forms have their {\feat qstore} set to $\{\}$. For simplicity, I do not show the {\feat qstore} feature here. \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval contain\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{tank\_ent@1}$\> \\ comps & \<\feat np[-prd]$_{mass\_ent@2}$\> ]\\ cont & \sort{contains}{ [arg1 & @1 \\ arg2 & @2]}]] \end{avm} \label{vforms:2} \item \begin{avm} [\avmspan{phon \; \<\fval contains\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{tank\_ent@1}$\> \\ comps & \<\feat np[-prd]$_{mass\_ent@2}$\> ]\\ cont & \osort{pres}{ [main\_psoa & \osort{contains}{ [arg1 & @1 \\ arg2 & @2]}]}]] \end{avm} \label{vforms:3} \end{examps} The simple past signs of culminating activity verbs are generated by \pref{vforms:4}, shown below. The simple past signs of non-culminating activity verbs are generated by a lexical rule that is similar to \pref{vforms:4}, except that it does not introduce a \ensuremath{\mathit{Culm}}\xspace operator in the resulting sign. The signs of past participles (e.g.\ \qit{inspected} in \qit{Who had inspected BA737?}) are generated by two lexical rules which are similar to the simple past ones. There is a rule for culminating activity verbs (which introduces a \ensuremath{\mathit{Culm}}\xspace in the past participle sign), and a rule for non-culminating activity verbs (that introduces no \ensuremath{\mathit{Culm}}\xspace). Both rules do not introduce \ensuremath{\mathit{Past}}\xspace operators. The generated signs have their {\feat vform} set to {\fval psp}\/ (past participle), and the same {\feat aspect} as the base signs, i.e.\ their {\feat aspect} is not changed to {\fval cnsq\_state} (consequent state). The shift to consequent state takes place when the auxiliary \qit{had} combines with the past participle (this will be discussed in section \ref{multi_forms}). \newpage \begin{examps} \item \lexrule{Simple Past Lexical Rule (Culminating Activity Base Form):} \avmoptions{active} \begin{center} \begin{avm} [\avmspan{phon \; \<$\lambda$\>} \\ synsem|loc & [cat & [head & \sort{verb}{ [vform & bse \\ aux & $-$]} \\ aspect & culmact] \\ cont & @1]] \end{avm} \\ $\Downarrow$ \\ \begin{avm} [\avmspan{phon \; \<\fval morph\($\lambda$, simple\_past\)\>} \\ synsem|loc & [cat & [head & \sort{verb}{ [vform & fin \\ aux & $-$]} \\ aspect & culmact] \\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \sort{culm}{ [main\_psoa & @1]} ]} ]] \end{avm} \end{center} \label{vforms:4} \end{examps} The signs for present participles (e.g.\ \qit{servicing} in \qit{Which company is servicing BA737?}) are generated by \pref{vforms:10}. The present participle signs are the same as the base ones, except that their {\feat vform} is {\fval prp}\/ (present participle), and their {\feat aspect} is {\srt progressive}\/ (progressive state). \begin{examps} \item \lexrule{Present Participle Lexical Rule:} \avmoptions{active} \begin{center} \begin{avm} [\avmspan{phon \; \<$\lambda$\>} \\ synsem|loc & [cat & [head & \sort{verb}{ [vform & bse \\ aux & $-$]} \\ aspect & aspect] \\ cont & @1]] \end{avm} \\ $\Downarrow$ \\ \begin{avm} [\avmspan{phon \; \<\fval morph\($\lambda$, present\_participle\)\>} \\ synsem|loc & [cat & [head & \sort{verb}{ [vform & prp \\ aux & $-$]} \\ aspect & progressive] \\ cont & @1]] \end{avm} \end{center} \label{vforms:10} \end{examps} Gerund signs are generated by a lexical rule that is similar to \pref{vforms:10}, except that the generated signs retain the {\feat aspect} of the original ones, and have their {\feat vform} set to {\srt ger}. In English, there is no morphological distinction between gerunds and present participles. \textsc{Hpsg}\xspace and most traditional grammars (e.g.\ \cite{Thomson}), however, distinguish between the two. In \pref{vforms:10x1}, the \qit{inspecting} is the gerund of \qit{to inspect}, while in \pref{vforms:10x2}, the \qit{inspecting} is the present participle. \begin{examps} \item J.Adams finished inspecting BA737. \label{vforms:10x1} \item J.Adams was inspecting BA737. \label{vforms:10x2} \end{examps} The fact that gerund signs retain the {\feat aspect} of the base signs is used in the treatment of \qit{to finish} (section \ref{special_verbs}). The simple past \qit{finished} receives multiple signs. (These are generated from corresponding base form signs by the simple past lexical rules.) \pref{vforms:14} is used when \qit{finished} combines with a culminating activity verb phrase, and \pref{vforms:23} when it combines with a state or activity verb phrase. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval finished\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$ ]} \\ aspect & point \\ spr & \<\> \\ subj & \<@1\> \\ comps & \<\feat vp[subj \<@1\>, vform {\fval ger}, aspect {\fval culmact}]:@2 \> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \osort{end}{ [main\_psoa & \sort{culm}{ [main\_psoa & @2]}]}]}]] \end{avm} \label{vforms:14} \item \begin{avm} [\avmspan{phon \; \<\fval finished\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$ ]} \\ aspect & point \\ spr & \<\> \\ subj & \<@1\> \\ comps & \<\feat vp[subj \<@1\>, vform {\fval ger}, \\ aspect {\fval state $\lor$ activity}]:@2 \> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \sort{end}{ [main\_psoa & @2]}]}]] \end{avm} \label{vforms:23} \end{examps} In \pref{vforms:14}, the {\feat vp[subj $<$\avmbox{1}$>$, vform {\fval ger}, aspect {\fval culmact}]:\avmbox{2}} means that \qit{finished} requires as its complement a gerund verb phrase (a gerund that has combined with its complements but not its subject) whose aspect is culminating activity. The \avmbox{1} of {\feat comps} points to a description of the required subject of the gerund verb phrase, and the \avmbox{2} is a pointer to the {\feat cont} value of the sign of the gerund verb phrase. The two \avmbox{1}s in \pref{vforms:14} have the effect that \qit{finished} requires as its subject whatever the gerund verb phrase requires as its subject. The two \avmbox{2}s cause the sign of \qit{finished} to inherit the {\feat cont} value of the sign of the gerund verb phrase, but with additional \ensuremath{\mathit{Past}}\xspace, \ensuremath{\mathit{End}}\xspace, and \ensuremath{\mathit{Culm}}\xspace operators. \pref{vforms:23} is similar, but it introduces no \ensuremath{\mathit{Culm}}\xspace. In \pref{vforms:10x1}, the sign of the gerund \qit{inspecting} retains the {\feat aspect} of the base sign, which in the airport domain is {\srt culmact}. The sign of the gerund verb phrase \qit{inspecting BA737} inherits the {\srt culmact}\/ {\feat aspect} of the gerund sign (following the aspect principle, to be discussed in section \ref{hpsg:punc_adv}). Hence, \pref{vforms:14} is used. This causes \pref{vforms:10x1} to receive a sign whose {\feat cont} represents \pref{vforms:24x1}, which requires the inspection to have been completed. \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, jadams, ba737)]]]$ \label{vforms:24x1} \end{examps} In \pref{vforms:24}, the sign of \qit{circling} inherits the {\srt activity}\/ {\feat aspect} of the base sign, causing \pref{vforms:23} to be used. This leads to \pref{vforms:25}, which does not require any completion to have been reached. \begin{examps} \item BA737 finished circling. \label{vforms:24} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{End}}\xspace[circling(ba737)]$ \label{vforms:25} \end{examps} There is also a sign of the simple past \qit{finished} for the case where the gerund verb phrase is a point. In that case, the {\feat cont} of the sign of \qit{finished} is identical to the the {\feat cont} of the sign of the gerund verb phrase, i.e.\ the \qit{finished} has no semantic contribution. This is in accordance with the arrangements of section \ref{special_verbs}. The signs of \qit{started}, \qit{stopped}, and \qit{began} are similar, except that they introduce \ensuremath{\mathit{Begin}}\xspace operators instead of \ensuremath{\mathit{End}}\xspace ones. Unlike \qit{finished}, the signs of \qit{stopped} do not introduce \ensuremath{\mathit{Culm}}\xspace operators when \qit{stopped} combines with culminating activities, reflecting the fact that there is no need for a completion to have been reached. \subsection{Auxiliary verbs and multi-word verb forms} \label{multi_forms} I now move on to auxiliary verbs and multi-word verb forms (e.g.\ \qit{had departed}, \qit{is inspecting}). \pref{vforms:30} shows the sign of the simple past auxiliary \qit{had}. According to \pref{vforms:30}, \qit{had} requires as its complement a past participle verb phrase. The \avmbox{1}s mean that \qit{had} requires as its subject whatever the past participle verb phrase requires as its subject. The \avmbox{2}s mean that the {\feat main\_psoa} value of the {\srt perf}\/ is the {\feat cont} value of the sign of the past participle verb phrase. \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval had\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & cnsq\_state \\ spr & \<\> \\ subj & \<@1\> \\ comps & \<\feat vp[subj \<@1\>, vform {\fval psp}]:@2 \> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \osort{perf}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & @2]}]}]] \end{avm} \label{vforms:30} \end{examps} In the airport domain, the past participle \qit{departed} receives multiple signs (for various habitual and non-habitual uses; these signs are generated from the corresponding base form signs by the lexical rules of section \ref{single_word_forms}). The sign of \pref{vforms:32} is used in \pref{vforms:31}. \begin{examps} \avmoptions{active} \item BA737 had departed. \label{vforms:31} \item \begin{avm} [\avmspan{phon \; \<\fval departed\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & psp \\ aux & $-$ ]} \\ aspect & point \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{flight\_ent@3}$\> \\ comps & \<\> ]\\ cont & \sort{actl\_depart}{ [arg1 & @3]}]] \end{avm} \label{vforms:32} \end{examps} According to \pref{vforms:32}, \qit{departed} requires no complements, i.e.\ it counts as a verb phrase, and can be used as the complement of \qit{had}. When \qit{had} combines with \qit{departed}, the {\feat subj} of \pref{vforms:30} becomes the same as the {\feat subj} of \pref{vforms:32} (because of the \avmbox{1}s in \pref{vforms:30}), and the {\feat main\_psoa} of the {\srt perf}\/ in \pref{vforms:30} becomes the same as the {\feat cont} of \pref{vforms:32} (because of the \avmbox{2}s in \pref{vforms:30}). The resulting constituent \qit{had departed} receives \pref{vforms:33}. \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval had, departed\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & cnsq\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{flight\_ent@3}$\> \\ comps & \<\> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \osort{perf}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \sort{actl\_depart}{ [arg1 & @3]}]}]}]] \end{avm} \label{vforms:33} \end{examps} The \textsc{Hpsg}\xspace principles (including the semantics and aspect principles that will be discussed in sections \ref{non_pred_nps} and \ref{hpsg:punc_adv}) cause \pref{vforms:33} to inherit the {\feat head}, {\feat aspect}, {\feat spr}, {\feat subj}, and {\feat cont} values of \pref{vforms:30}. Notice that this causes the aspect of \qit{had departed} to become consequent state (\qit{departed} was a point). As will be discussed in section \ref{hpsg:nouns}, the proper name \qit{BA737} contributes an index that represents the flight BA737. When \qit{had departed} combines with its subject \qit{BA737}, the index of \qit{BA737} becomes the {\feat arg1} value of \pref{vforms:33} (because of the \avmbox{3}s of \pref{vforms:33}). This causes \pref{vforms:31} to receive a sign whose {\feat cont} represents \pref{vforms:35}. \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, actl\_depart(ba737)]]$ \label{vforms:35} \end{examps} As mentioned in sections \ref{present_perfect} and \ref{perf_op}, present perfect forms are treated semantically as simple past forms. This is why, unlike the sign of \qit{had}, the sign of \qit{has} (shown in \pref{vforms:36}) does not introduce a \ensuremath{\mathit{Perf}}\xspace operator, and preserves the aspect of the past participle. This causes \qit{BA737 has departed.} to receive the same \textsc{Top}\xspace formula as \qit{BA737 departed.}. \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval has\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & @1 \\ spr & \<\> \\ subj & \<@2\> \\ comps & \<\feat vp[subj \<@2\>, vform {\fval psp}, aspect @1]:@3 \> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & @3]}]] \end{avm} \label{vforms:36} \end{examps} \qit{Does} receives the sign of \pref{vforms:40.1}, which indicates that it requires as its complement a base verb phrase. The verb phrase must be a (lexical) state. (This is in accordance with the assumption of section \ref{simple_present} that the simple present can be used only with state verbs.) \pref{vforms:40.1} and the (habitual) base sign of \pref{vforms:41.1} cause \pref{vforms:41} to receive \pref{vforms:42}. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval does\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state @1 \\ spr & \<\> \\ subj & \<@2\> \\ comps & \<\feat vp[subj \<@2\>, vform {\fval bse}, aspect @1]:@3 \> ]\\ cont & \sort{pres}{ [main\_psoa & @3]}]] \end{avm} \label{vforms:40.1} \item \begin{avm} [\avmspan{phon \; \<\fval service\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{company\_ent@4}$\> \\ comps & \<\feat np[-prd]$_{flight\_ent@5}$\> ]\\ cont & \sort{hab\_servicer\_of}{ [arg1 & @4 \\ arg2 & @5]} ]] \end{avm} \label{vforms:41.1} \item Does Airserve service BA737? \label{vforms:41} \item \begin{avm} [\avmspan{phon \; \<\fval Does, Airserve, service, BA737\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{pres}{ [main\_psoa & \sort{hab\_servicer\_of}{ [arg1 & airserve \\ arg2 & ba737]}]}]] \end{avm} \label{vforms:42} \end{examps} In the airport domain, the base form of \qit{to service} receives also a sign that corresponds to the non-habitual homonym. This is similar to \pref{vforms:41.1}, but it introduces the predicate functor $actl\_servicing$, and its {\feat aspect} is {\srt culmact}. This sign cannot be used in \pref{vforms:41}, because \pref{vforms:40.1} requires the verb-phrase complement to be a state not a culminating activity. This correctly predicts that \pref{vforms:41} cannot be asking if Airserve is actually servicing BA737 at the present moment. \qit{Did} receives two signs: one for culminating-activity verb-phrase complements (shown in \pref{vforms:43}), and one for state, activity, or point verb-phrase complements (this is similar to \pref{vforms:43}, but introduces no \ensuremath{\mathit{Culm}}\xspace). In both cases, a \ensuremath{\mathit{Past}}\xspace operator is added. In the case of culminating-activity complements, a \ensuremath{\mathit{Culm}}\xspace operator is added as well. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval did\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & culmact @1 \\ spr & \<\> \\ subj & \<@2\> \\ comps & \<\feat vp[subj \<@2\>, vform {\fval bse}, aspect @1]:@3 \> ]\\ cont & \osort{past}{ [et\_handle & \sort{temp\_ent}{ [tvar $+$]} \\ main\_psoa & \sort{culm}{ [main\_psoa & @3]}]}]] \end{avm} \label{vforms:43} \end{examps} The non-habitual sign of \qit{service} and \pref{vforms:43} cause \pref{vforms:45} to be mapped to \pref{vforms:46}, which requires Airserve to have actually serviced BA737 in the past. The habitual sign of \pref{vforms:41.1} and the \qit{did} sign for non-culminating activity complements cause \pref{vforms:45} to be mapped to \pref{vforms:46x1}, which requires Airserve to have been a past habitual servicer of BA737. \begin{examps} \item Did Airserve service BA737? \label{vforms:45} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[actl\_servicing(occr^v, airserve, ba737)]]$ \label{vforms:46} \item $\ensuremath{\mathit{Past}}\xspace[e^v, hab\_servicer\_of(airserve, ba737)]$ \label{vforms:46x1} \end{examps} The sign for the auxiliary \qit{is} is shown in \pref{vforms:50}. The present participle \qit{servicing} receives two signs, a non-habitual one (shown in \pref{vforms:51}) and a habitual one. The latter is similar to \pref{vforms:51}, but it introduces the functor $hab\_servicer\_of$, and its {\feat aspect} is {\srt lex\_state}. (The two present participle signs are generated from the base ones by the present participle lexical rule of section \ref{single_word_forms}.) \pref{vforms:50} and \pref{vforms:51} cause \pref{vforms:52} to be mapped to \pref{vforms:53}, which requires Airserve to be actually servicing BA737 at the present. \pref{vforms:50} and the habitual present participle sign cause \pref{vforms:52} to be mapped to \pref{vforms:53x1}, which requires Airserve to be the current habitual servicer of BA737. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval is\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & progressive \\ spr & \<\> \\ subj & \<@1\> \\ comps & \<\feat vp[subj \<@1\>, vform {\fval prp}]:@2 \> ]\\ cont & \sort{pres}{ [main\_psoa & @2]}]] \end{avm} \label{vforms:50} \item \begin{avm} [\avmspan{phon \; \<\fval servicing\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & prp \\ aux & $-$ ]} \\ aspect & culmact \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{company\_ent@1}$\> \\ comps & \<\feat np[-prd]$_{flight\_ent@2}$\> ]\\ cont & \sort{actl\_servicing}{ [arg1 & occr\_var \\ arg2 & @1 \\ arg3 & @2]} ]] \end{avm} \label{vforms:51} \item Airserve is servicing BA737. \label{vforms:52} \item $\ensuremath{\mathit{Pres}}\xspace[actl\_servicing(occr^v, airserve, ba737)]$ \label{vforms:53} \item $\ensuremath{\mathit{Pres}}\xspace[hab\_servicer\_of(airserve, ba737)]$ \label{vforms:53x1} \end{examps} The sign for the auxiliary \qit{was} is similar to \pref{vforms:50}, except that it introduces a \ensuremath{\mathit{Past}}\xspace operator instead of a \ensuremath{\mathit{Pres}}\xspace one. \section{Predicative and non-predicative prepositions} \label{hpsg:pps} \avmoptions{active} Following Pollard and Sag (\cite{Pollard1}, p.65), prepositions receive separate signs for their predicative and non-predicative uses. In sentences like \pref{pps:3} and \pref{pps:4}, where the prepositions introduce complements of \qit{to be}, the prepositions are said to be predicative. In \pref{pps:1} and \pref{pps:2}, where they introduce complements of other verbs, the prepositions are non-predicative. \begin{examps} \item BA737 is at gate 2. \label{pps:3} \item BA737 was on runway 3. \label{pps:4} \item BA737 (habitually) arrives at gate 2. \label{pps:1} \item BA737 landed on runway 3. \label{pps:2} \end{examps} Predicative prepositions introduce their own \textsc{Top}\xspace predicates, while non-predicative prepositions have no semantic contribution. \subsection{Predicative prepositions} \pref{pps:5} shows the predicative sign of \qit{at}. (The predicative signs of other prepositions are similar.) The {\feat prd}~$+$ shows that the sign is predicative. ({\feat prd} is also used to distinguish predicative adjectives and nouns; this will be discussed in sections \ref{hpsg:nouns} and \ref{hpsg:adjectives}.) {\feat pform} reflects the preposition to which the sign corresponds. Signs for prepositional phrases inherit the {\feat pform} of the preposition's sign. This is useful in verbs that require prepositional phrases introduced by particular prepositions. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval at\>} \\ synsem|loc & [cat & [head & \osort{prep}{ [pform & at \\ prd & $+$]} \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\feat np[-prd]$_{@2}$\> ]\\ cont & \sort{located\_at}{ [arg1 & @1 \\ arg2 & non\_temp\_ent@2 ]}]] \end{avm} \label{pps:5} \end{examps} According to \pref{pps:5}, \qit{at} requires a (non-predicative) noun-phrase (\qit{BA737} in \pref{pps:3}) as its subject, and another one (\qit{gate 2} in \pref{pps:3}) as its complement. As will be discussed in section \ref{hpsg:nouns}, \qit{BA737} and \qit{gate 2} contribute indices that represent the corresponding world entities. The \avmbox{2} of \pref{pps:5} denotes the index of \qit{gate 2}. \pref{pps:5} causes \qit{at gate 2} to receive \pref{pps:6}. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval at, gate2\>} \\ synsem|loc & [cat & [head & \osort{prep}{ [pform & at \\ prd & $+$]} \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \sort{located\_at}{ [arg1 & @1 \\ arg2 & gate2 ]}]] \end{avm} \label{pps:6} \end{examps} Apart from \pref{vforms:50} (which is used when \qit{is} combines with a present-participle complement), \qit{is} also receives \pref{pps:7} (which is used when \qit{is} combines with predicative prepositional-phrases). \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval is\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<@3\> \\ comps & \<\feat pp[subj \<@3\>, prd $+$]:@4 \> ]\\ cont & \sort{pres}{ [main\_psoa & @4]}]] \end{avm} \label{pps:7} \end{examps} According to \pref{pps:7}, \qit{is} requires as its complement a predicative prepositional phrase (a predicative preposition that has combined with its complements but not its subject), like the \qit{at gate 2} of \pref{pps:6}. \pref{pps:6} and \pref{pps:7} cause \pref{pps:3} to receive \pref{pps:10}. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval BA737, is, at, gate2\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\>] \\ cont & \osort{pres}{ [main\_psoa & \osort{located\_at}{ [arg1 & ba737 \\ arg2 & gate2]}]}]] \end{avm} \label{pps:10} \end{examps} Like \qit{is}, \qit{was} receives two signs: one for present-participle complements (as in \qit{BA737 was circling.}), and one for predicative prepositional-phrase complements (as in \pref{pps:4}). These are similar to the signs of \qit{was}, but they introduce \ensuremath{\mathit{Past}}\xspace operators rather than \ensuremath{\mathit{Pres}}\xspace ones. \subsection{Non-predicative prepositions} The non-predicative sign of \qit{at} is shown in \pref{pps:12}. (The non-predicative signs of other prepositions are similar.) The \avmbox{1} is a pointer to the {\feat cont} value of the sign that corresponds to the noun-phrase complement of \qit{at}. Notice that in this case the \qit{at} has no semantic contribution (the \qit{at} sign simply copies the {\feat cont} of the noun-phrase complement). \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval at\>} \\ synsem|loc & [cat & [head & \osort{prep}{ [pform & at \\ prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\feat np[-prd]:@1\> ]\\ cont & @1]] \end{avm} \label{pps:12} \end{examps} \pref{pps:12} and the habitual sign of \qit{arrives} of \pref{pps:13} cause \pref{pps:1} to receive \pref{pps:16}. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval arrives\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{flight\_ent@1}$\> \\ comps & \<\feat pp[-prd, pform {\fval at}]$_{gate\_ent@2}$ \> ]\\ cont & \osort{pres}{ [main\_psoa & \osort{hab\_arrive\_at}{ [arg1 & @1 \\ arg2 & @2]}]}]] \end{avm} \label{pps:13} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval BA737, arrives, at, gate2\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{pres}{ [main\_psoa & \osort{hab\_arrive\_at}{ [arg1 & ba737 \\ arg2 & gate2]}]}]] \end{avm} \label{pps:16} \end{examps} The (predicative and non-predicative) prepositional signs of this section are not used when prepositions introduce temporal adverbials (e.g.\ \qit{BA737 departed at 5:00pm.}). There are additional prepositional signs for these cases (see section \ref{hpsg:pupe_adv} below). \section{Nouns} \label{hpsg:nouns} \avmoptions{active} Like prepositions, nouns receive different signs for their predicative and non-predicative uses. Nouns used in noun-phrase complements of \qit{to be} (more precisely, the lexical heads of such noun-phrase complements), like the \qit{president} of \pref{nps:3}, are \emph{predicative}. The corresponding noun phrases (e.g.\ \qit{the president} of \pref{nps:3}) are also said to be predicative. In all other cases (e.g.\ \qit{the president} of \pref{nps:1}), the nouns and noun phrases are \emph{non-predicative}. \begin{examps} \item J.Adams is the president. \label{nps:3} \item The president was at gate 2. \label{nps:1} \end{examps} \subsection{Non-predicative nouns} \label{non_pred_nps} Let us first examine non-predicative nouns. \pref{nps:2} shows the sign of \qit{president} that would be used in \pref{nps:1}. The {\feat prd} value shows that the sign corresponds to a non-predicative use of the noun. The {\feat spr} value means that the noun requires as its specifier a determiner (e.g.\ \qit{a}, \qit{the}). \begin{examps} \item \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{president}{ [arg1 & @1]}]} \end{avm}} \avmoptions{active,center} \begin{avm} [\avmspan{phon \; \<\fval president\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<[loc|cat|head & {\fval det}] \> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & person\_ent@1 \\ restr & \{\box\avmboxa\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:2} \end{examps} The {\feat cont} values of signs that correspond to non-predicative nouns are of sort {\srt nom\_obj}\/ (section \ref{TOP_FS_WH}). The {\feat index} value stands for the world entity described by the noun, and the {\feat restr} value represents \textsc{Top}\xspace expressions that are introduced by the noun. \pref{nps:4} shows the sign of \qit{the} that is used in \pref{nps:1}. (In this thesis, \qit{the} is treated semantically as \qit{a}. This is of course an over-simplification.) \begin{examps} \setbox\avmboxa=\hbox{\begin{avm} \osort{det}{ [spec|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\_\> \\ subj & \<\> \\ comps & \<\> ] \\ cont & @2]]} \end{avm}} \avmoptions{active,center} \item \begin{avm} [\avmspan{phon \; \<\fval the\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{quant}{ [det & exists \\ restind & [index & [tvar $+$]]@2 ]}@3 ] \\ \avmspan{qstore \; \{@3\}} ] \end{avm} \label{nps:4} \end{examps} The {\feat spec} feature of \pref{nps:4} means that \qit{the} must be used as the specifier of a non-predicative $\bar{N}$, i.e.\ as the specifier of a non-predicative noun that has combined with its complements and that requires a specifier. The \avmbox{3}s of \pref{nps:4} cause an existential quantifier to be inserted into the quantifier store, and the \avmbox{2}s cause the {\feat restind} of that quantifier to be unified with the {\feat cont} of the $\bar{N}$'s sign. According to \pref{nps:2}, \qit{president} is non-predicative, it does not need to combine with any complements, and it requires a specifier. Hence, it satisfies the {\feat spec} restrictions of \pref{nps:4}, and \qit{the} can be used as the specifier of \qit{president}. When \qit{the} combines with \qit{president}, the {\feat restind} of \pref{nps:4} is unified with the {\feat cont} of \pref{nps:2} (because of the \avmbox{2}s in \pref{nps:4}), and the {\feat qstore} of \pref{nps:4} becomes \pref{nps:7.1} (because of the \avmbox{3}s in \pref{nps:4}). The resulting noun phrase receives \pref{nps:8}. \begin{examps} \item \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{president}{ [arg1 & @1]}]} \end{avm}} \avmoptions{active,center} \begin{avm} \{\sort{quant}{ [det & exists \\ restind & [index & \sort{person\_ent}{ [tvar & $+$]}@1 \\ restr & \{\box\avmboxa\} ]@2 ]}@3\} \end{avm} \label{nps:7.1} \item \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{president}{ [arg1 & @1]}]} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval the, president\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & \sort{person\_ent}{ [tvar & $+$]}@1 \\ restr & \{\box\avmboxa\}]}@2] \\ \avmspan{qstore \; \{\sort{quant}{ [det & exists \\ restind & @2]}\}}] \end{avm} \label{nps:8} \end{examps} According to the head feature principle (section \ref{schemata_principles}), \pref{nps:8} inherits the {\feat head} of \pref{nps:2} (which is the sign of the ``head daughter'' in this case). The propagation of {\feat cont} and {\feat qstore} is controlled by the semantics principle, which in this thesis has the simplified form of \pref{nps:9}. (\pref{nps:9} uses the terminology of \cite{Pollard2}. I explain below what \pref{nps:9} means for readers not familiar with \cite{Pollard2}.) \begin{examps} \item \principle{Semantics Principle (simplified version of this thesis):}\\ In a headed phrase, (a) the {\feat qstore} value is the union of the {\feat qstore} values of the daughters, and (b) the {\feat synsem$\mid$loc$\mid$cont} value is token-identical with that of the semantic head. (In a headed phrase, the \emph{semantic head} is the {\feat adjunct-daughter} if any, and the {\feat head-daughter} otherwise.) \label{nps:9} \end{examps} Part (a) means that the {\feat qstore} of each (non-lexical) syntactic constituent is the union of the {\feat qstore}s of its subconstituents. Part (b) means that each syntactic constituent inherits the {\feat cont} of its head-daughter (the noun in noun-phrases, the verb in verb phrases, the preposition in prepositional phrases), except for cases where the head-daughter combines with an adjunct-daughter (a modifier). In the latter case, the mother syntactic constituent inherits the {\feat cont} of the adjunct-daughter. (This will be discussed further in section \ref{hpsg:pupe_adv}.) Readers familiar with \cite{Pollard2} will have noticed that \pref{nps:9} does not allow quantifiers to be unstored from {\feat qstore}. Apart from this, \pref{nps:9} is the same as in \cite{Pollard2}. \pref{nps:9} causes the {\feat qstore} of \pref{nps:8} to become the union of the {\feat qstore}s of \pref{nps:2} (the empty set) and \pref{nps:4} (which has become \pref{nps:7.1}). Since \qit{the president} involves no adjuncts, the ``semantic head'' is the ``head-daughter'' (i.e.\ \qit{president}), and \pref{nps:8} inherits the {\feat cont} of \pref{nps:2} (which is now the {\feat restind} of \pref{nps:7.1}). The \qit{gate 2} of \pref{nps:1} is treated as a one-word proper name. (In the prototype \textsc{Nlitdb}\xspace, the user has to type \qit{terminal 2} as a single word; the same is true for \qit{J.Adams} of \pref{nps:3}. This will be discussed in section \ref{preprocessor}.) Proper names are mapped to signs whose {\feat cont} is a {\srt nom\_obj}\/ with an empty-set {\feat restr}.\footnote{In \cite{Pollard2}, the signs of proper names involve {\srt naming}\/ relations, and {\feat context} and {\feat background} features. These are not used in this thesis.} \qit{Gate 2}, for example, receives \pref{nps:10}. \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval gate2\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & gate2 \\ restr & \{\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:10} \end{examps} The predicative sign of \qit{at} of \pref{pps:5}, the predicative sign of \qit{was} (which is similar to \pref{pps:7}, except that it introduces a \ensuremath{\mathit{Past}}\xspace), and \pref{nps:10} cause the \qit{was at gate 2} of \pref{nps:1} to receive \pref{nps:13}. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval was, at, gate2\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{located\_at}{ [arg1 & @1 \\ arg2 & gate2]}]}] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{nps:13} \end{examps} When \qit{was at gate 2} combines with \qit{the president}, \pref{nps:1} receives \pref{nps:14}. According to the semantics principle, the {\feat qstore} of \pref{nps:14} is the union of the {\feat qstore}s of \pref{nps:13} and \pref{nps:8}, and the {\feat cont} of \pref{nps:14} is the same as the {\feat cont} of \pref{nps:13}. \begin{examps} \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{president}{ [arg1 & @1]} ]} \end{avm}} \avmoptions{active,center} \item \begin{avm} [\avmspan{phon \; \<\fval the, president, was, at, gate2\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\>] \\ cont & \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{located\_at}{ [arg1 & @1 \\ arg2 & gate2]}]}] \\ \avmspan{qstore \; \{[det & exists \\ restind & \osort{nom\_obj}{ [index & \sort{person\_ent}{ [tvar & $+$]}@1 \\ restr & \{\box\avmboxa\} ]} ]\} } ] \end{avm} \label{nps:14} \end{examps} \pref{nps:18} is then extracted from \pref{nps:14}, as discussed in section \ref{extraction_hpsg}. Whenever an \ensuremath{\mathit{Ntense}}\xspace operator is encountered during the extraction of the \textsc{Top}\xspace formulae, if there is no definite information showing that the first argument of the \ensuremath{\mathit{Ntense}}\xspace should be $now^*$, the first argument is taken to be a variable. \pref{nps:14}, for example, shows that the first argument of the \ensuremath{\mathit{Ntense}}\xspace could be either a \textsc{Top}\xspace variable or $now^*$. Hence, in \pref{nps:18} the first argument of the \ensuremath{\mathit{Ntense}}\xspace has become a variable ($t^v$). During the post-processing phase (section \ref{post_processing} below), the \ensuremath{\mathit{Ntense}}\xspace of \pref{nps:18} would give rise to two separate formulae: one where the first argument of the \ensuremath{\mathit{Ntense}}\xspace has been replaced by $now^*$ (current president), and one where the first argument of the \ensuremath{\mathit{Ntense}}\xspace has been replaced by the $e^v$ of the \ensuremath{\mathit{Past}}\xspace operator (president when at gate 2). In contrast, if the sign shows that the first argument of the \ensuremath{\mathit{Ntense}}\xspace is definitely $now^*$, the first argument of the \ensuremath{\mathit{Ntense}}\xspace in the extracted formula is $now^*$, and the post-processing has no effect on this argument. \begin{examps} \item $\ensuremath{\mathit{Ntense}}\xspace[t^v, president(p^v)] \land \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(p^v, gate2)]$ \label{nps:18} \end{examps} It is possible to force a (non-predicative) noun to be interpreted as referring always to the speech time, or always to the time of the verb tense. (This also applies to the non-predicative adjectives of section \ref{hpsg:adjectives} below.) To force a noun to refer always to the speech time, one sets the {\feat et\_handle} of the {\srt ntense}\/ in the noun's sign to simply {\srt now}\/ (instead of allowing it to be either {\srt now}\/ or a variable-representing index as in \pref{nps:2}). This way, the {\feat et\_handle} of the {\srt ntense}\/ in \pref{nps:14} would be {\srt now}. \pref{nps:18} would contain $now^*$ instead of $t^v$ (because in this case the sign shows that the first argument of the \ensuremath{\mathit{Ntense}}\xspace should definitely be $now^*$), and the post-processing mechanism would have no effect. To force a noun to refer always to the time of the verb tense, one simply omits the \ensuremath{\mathit{Ntense}}\xspace from the noun's sign. This would cause the formula extracted from the sign of \pref{nps:1} to be \pref{nps:25}. \begin{examps} \item $president(p^v) \land \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(p^v, gate2)]$ \label{nps:25} \end{examps} The semantics of \textsc{Top}\xspace's conjunction (section \ref{denotation}) and of the \ensuremath{\mathit{Past}}\xspace operator (section \ref{past_op}) require $president(p^v)$ and $located\_at(p^v, gate2)$ to be true at the same (past) event time. Hence, \pref{nps:25} expresses the reading where the person at gate 2 was the president of that time. There are however, two complications when (non-predicative) noun signs do not introduce \ensuremath{\mathit{Ntense}}\xspace{s}. (These also apply to adjective signs, to be discussed in section \ref{hpsg:adjectives}.) First, a past perfect sentence like \pref{nps:26} receives only \pref{nps:27}, which requires $president(p^v)$ to be true at the event time pointed to by $e1^v$ (the ``reference time'', which is required to fall within 1/1/95). That is, \qit{the president} is taken to refer to somebody who was the president on 1/1/95, and who may not have been the president during the visit. \begin{examps} \item The president had visited Rome on 1/1/95. \label{nps:26} \item $president(p^v) \land \ensuremath{\mathit{At}}\xspace[\text{\textit{1/1/95}}, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, visiting(p^v, rome)]]]$ \label{nps:27} \end{examps} In contrast, if the sign of \qit{president} introduces an \ensuremath{\mathit{Ntense}}\xspace, the formula extracted from the sign of \pref{nps:26} is \pref{nps:28}. The post-processing generates three different formulae from \pref{nps:28}. These correspond to readings where \qit{president} refers to the time of the visit ($t^v$ replaced by $e2^v$), the reference time ($t^v$ replaced by $e1^v$, equivalent to \pref{nps:27}), or the speech time ($t^v$ replaced by $now^*$). \begin{examps} \item $\ensuremath{\mathit{Ntense}}\xspace[t^v, president(p^v)] \land$ \\ $\ensuremath{\mathit{At}}\xspace[\text{\textit{1/1/95}}, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, visiting(p^v, rome)]]]$ \label{nps:28} \end{examps} The second complication is that (non-predicative) nouns that do not introduce \ensuremath{\mathit{Ntense}}\xspace{s} are taken to refer to the time of the \emph{main clause's} tense, even if the nouns appear in subordinate clauses (subordinate clauses will be discussed in section \ref{hpsg:subordinates}). For example, if \qit{president} does not introduce an \ensuremath{\mathit{Ntense}}\xspace, \pref{nps:31.1} is mapped to \pref{nps:31.2}. The semantics of \pref{nps:31.2} requires the visitor to have been president during the building of terminal 2 (the visitor is not required to have been president during the visit to terminal 3). \begin{examps} \item Housecorp built terminal 2 before the president visited terminal 3. \label{nps:31.1} \item $\begin{aligned}[t] president(p^v) \land \ensuremath{\mathit{Before}}\xspace[&\ensuremath{\mathit{Past}}\xspace[e1^v, visiting(p^v, term3)],\\ & \ensuremath{\mathit{Past}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[building(housecorp, term2)]]] \end{aligned}$ \label{nps:31.2} \end{examps} In contrast, if \qit{president} introduces an \ensuremath{\mathit{Ntense}}\xspace, the post-processing (section \ref{post_processing} below) generates three readings, where \qit{president} refers to the speech time, the time of the building, or the time of the visit. \medskip The non-predicative signs of nouns like \qit{day} or \qit{summer}, that refer to members of partitionings (section \ref{top_model}) are similar to the non-predicative signs of ``ordinary'' nouns like \qit{president}, except that they introduce \ensuremath{\mathit{Part}}\xspace operators, and they do not introduce \ensuremath{\mathit{Ntense}}\xspace{s}. \pref{nps:32}, for example, shows the non-predicative sign of \qit{day}. (The {\srt day}\/ and {\srt day\_ent\_var}\/ sorts are as in figure \vref{ind_hierarchy}.) \begin{examps} \item \setbox\avmboxa=\hbox{\begin{avm} \sort{part}{ [partng & day \\ part\_var & @1]} \end{avm}} \avmoptions{active,center} \begin{avm} [\avmspan{phon \; \<\fval day\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<[loc|cat|head & {\fval det}] \> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & day\_ent\_var@1 \\ restr & \{\box\avmboxa\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:32} \end{examps} Names of months and days (e.g.\ \qit{Monday}, \qit{January}) that can be used both with and without determiners (e.g.\ \qit{on a Monday}, \qit{on Monday}) receive two non-predicative signs each: one that requires a determiner, and one that does not. Finally, proper names that refer to particular time-periods (e.g.\ the year-name \qit{1991}, the date \qit{25/10/95}) receive non-predicative signs that are similar to those of ``normal'' proper names (e.g.\ \qit{gate 2}), except that their {\feat index} values are subsorts of {\srt temp\_ent}\/ rather than {\srt non\_temp\_ent}. I demonstrate in following sections how the signs of temporal nouns and proper names (e.g.\ \qit{day}, \qit{25/10/95}) are used to form the signs of temporal adverbials (e.g.\ \qit{for two days}, \qit{before 25/10/95}). \subsection{Predicative nouns} \label{pred_nps} I now turn to predicative nouns, like the \qit{president} of \pref{nps:3}. \pref{nps:41} shows the predicative sign of \qit{president}. Unlike non-predicative noun-signs, whose {\feat cont} values are of sort {\srt nom\_obj}, the {\feat cont} values of predicative noun-signs are of sort {\srt psoa}. The {\srt president}\/ in \pref{nps:41} is a subsort of {\srt psoa}. \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval president\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ spr & \<[loc|cat|head & {\fval det}]\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \sort{president}{ [arg1 & person\_ent@1]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:41} \end{examps} Unlike non-predicative nouns that do not require subjects (e.g.\ \pref{nps:2}), predicative nouns do require subjects. In \pref{nps:41}, \qit{president} requires a non-predicative noun phrase as its subject. The \avmbox{1} denotes the index of that noun phrase. In the \textsc{Hpsg}\xspace version of this thesis, the predicative signs of nouns are generated automatically from the non-predicative ones by \pref{nps:42}.\footnote{Apart from the $remove\_ntense$, \pref{nps:42} is essentially the same as Borsley's ``predicative NP lexical rule'', discussed in the footnote of p.360 of \cite{Pollard2}.} \begin{examps} \item \lexrule{Predicative Nouns Lexical Rule:} \avmoptions{active} \begin{center} \begin{avm} [synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ subj & \<\>]\\ cont & \osort{nom\_obj}{ [index & @1 \\ restr & \{@2\}]}]] \end{avm} \\ $\Downarrow$ \\ \begin{avm} [synsem|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ subj & \<\feat np[-prd]$_{@1}$\>]\\ cont & remove\_ntense\(@2\)]] \end{avm} \end{center} \label{nps:42} \end{examps} The $remove\_ntense($\avmbox{2}$)$ in \pref{nps:42} means that if \avmbox{2} (the single element of the {\feat restr} set of the non-predicative sign) is of sort {\srt ntense}, then the {\feat cont} of the predicative sign should be the {\feat main\_psoa} of \avmbox{2} (see also \pref{nps:2}). Otherwise, the {\feat cont} of the predicative sign should be \avmbox{2}. In other words, if the non-predicative sign introduces an \ensuremath{\mathit{Ntense}}\xspace, the \ensuremath{\mathit{Ntense}}\xspace is removed in the predicative sign. This is related to the observation in section \ref{noun_anaphora}, that noun phrases that are complements of \qit{to be} always refer to the time of the verb tense. For example, \pref{nps:43} means that J.Adams was the president in 1992, not at the speech time. \pref{nps:43} is represented correctly by \pref{nps:44} which contains no \ensuremath{\mathit{Ntense}}\xspace{s}. \begin{examps} \item J.Adams was the president in 1992. \label{nps:43} \item $\ensuremath{\mathit{At}}\xspace[1992, \ensuremath{\mathit{Past}}\xspace[e^v, president(j\_adams)]]$ \label{nps:44} \end{examps} \textsc{Top}\xspace predicates introduced by predicative nouns (e.g.\ $president(j\_adams)$ in \pref{nps:44}) end up within the operators of the tenses of \qit{to be} (e.g.\ the \ensuremath{\mathit{Past}}\xspace of \pref{nps:44}). This requires the predicates to hold at the times of the tenses. As with previous lexical rules, features not shown in \pref{nps:42} (e.g.\ {\feat spr}, {\feat comps}) have the same values in both the original and the generated signs. For example, \pref{nps:42} generates \pref{nps:41} from \pref{nps:2}. In this thesis, determiners also receive different signs for their uses in predicative and non-predicative noun phrases. (Pollard and Sag do not provide much information on determiners of predicative noun phrases. The footnote of p.360 of \cite{Pollard2}, however, seems to acknowledge that determiners of predicative noun phrases have to be treated differently from determiners of non-predicative noun phrases.) For example, apart from \pref{nps:4}, \qit{the} is also given \pref{nps:45}. The {\feat spec} of \pref{nps:45} shows that \pref{nps:45} can only be used with predicative nouns (cf.\ \pref{nps:4}). Unlike determiners of non-predicative noun phrases, determiners of predicative noun-phrases have no semantic contribution (the {\feat synsem$\mid$loc$\mid$cont} of \pref{nps:45} is simply a copy of the {\feat cont} of the noun, and no quantifier is introduced in {\feat qstore}; cf.\ \pref{nps:4}). \begin{examps} \setbox\avmboxa=\hbox{\begin{avm} \osort{det}{ [spec|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ spr & \<\_\> \\ subj & \<\_\> \\ comps & \<\> ] \\ cont & @2]]} \end{avm}} \avmoptions{active,center} \item \begin{avm} [\avmspan{phon \; \<\fval the\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & @2 ] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:45} \end{examps} In \pref{nps:3}, when \qit{the} combines with \qit{president}, the resulting noun phrase receives \pref{nps:46}. (\textsc{Hpsg}\xspace's principles, including the semantics principle of \pref{nps:9}, cause \pref{nps:46} to inherit the {\feat head}, {\feat subj}, and {\feat cont} of \pref{nps:41}.) \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval the, president\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \sort{president}{ [arg1 & person\_ent@1]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:46} \end{examps} Apart from \pref{vforms:50} and \pref{pps:7}, \qit{is} also receives \pref{nps:47}, which allows the complement of \qit{is} to be a predicative noun phrase. (There is also a sign of \qit{is} for adjectival complements, as in \qit{Runway 2 is closed.}; this will be discussed in section \ref{hpsg:adjectives}. \qit{Was} receives similar signs.) The \avmbox{4}s in \pref{nps:47} denote the {\feat cont} of the predicative noun-phrase. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval is\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<@3\> \\ comps & \<\feat np[subj \<@3\>, prd $+$]:@4 \> ]\\ cont & \osort{pres}{ [main\_psoa & @4]}] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{nps:47} \end{examps} \pref{nps:47} and \pref{nps:46} cause the \qit{is the president} of \pref{nps:3} to receive \pref{nps:48}. Finally, when \qit{is the president} combines with \qit{J.Adams}, \pref{nps:3} receives a sign with an empty {\feat qstore}, whose {\feat cont} represents \pref{nps:50}. \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval is, the, president\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \osort{pres}{ [main\_psoa & \osort{president}{ [arg1 & person\_ent@1]}]}] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{nps:48} \item $\ensuremath{\mathit{Pres}}\xspace[president(j\_adams)]$ \label{nps:50} \end{examps} There are currently two complications with predicative noun phrases. The first is that in the non-predicative signs of proper names like \qit{gate2}, the value of {\feat restr} is the empty set (see \pref{nps:10}). Hence, \pref{nps:42} does not generate the corresponding predicative signs, because the non-predicative signs do not match the {\feat restr} description of the LHS of \pref{nps:42} (which requires the {\feat restr} value to be a one-element set). This causes \pref{nps:51} to be rejected, because there is no predicative sign for \qit{J.Adams}. \begin{examps} \item The inspector is J.Adams. \label{nps:51} \end{examps} One way to solve this problem is to employ the additional rule of \pref{nps:52}. \begin{examps} \item \lexrule{Additional Predicative Nouns Lexical Rule:} \avmoptions{active} \begin{center} \begin{avm} [synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ subj & \<\>]\\ cont & \osort{nom\_obj}{ [index & @1 \\ restr & \{\}]}]] \end{avm} \\ $\Downarrow$ \\ \begin{avm} [synsem|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ subj & \<\feat np[-prd]$_{@2}$\>]\\ cont & \sort{identity}{ [arg1 & @1 \\ arg2 & @2]}]] \end{avm} \end{center} \label{nps:52} \end{examps} This would generate \pref{nps:53} from the non-predicative sign of \qit{J.Adams} (which is similar to \pref{nps:10}). \pref{nps:47} and \pref{nps:53} would cause \pref{nps:51} to be mapped to \pref{nps:54}. (I assume here that the non-predicative \qit{inspector} does not introduce an \ensuremath{\mathit{Ntense}}\xspace.) \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval J.Adams\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $+$]} \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@2}$\> \\ comps & \<\> ]\\ cont & \sort{identity}{ [arg1 & j\_adams \\ arg2 & @2]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{nps:53} \item $inspector(insp^v) \land \ensuremath{\mathit{Pres}}\xspace[identity(j\_adams, insp^v)]$ \label{nps:54} \end{examps} $identity(\tau_1,\tau_2)$ is intended to be true at event times where its two arguments denote the same entity. This calls for a special domain-independent semantics for $identity(\tau_1,\tau_2)$. I have not explored this issue any further, however, and \pref{nps:52} is not used in the prototype \textsc{Nlitdb}\xspace. A second complication is that the non-predicative sign of \qit{Monday} (which is similar to \pref{nps:32}) and the treatment of predicative noun phrases above lead to an attempt to map \pref{nps:55} to \pref{nps:56}. \begin{examps} \item 23/10/95 was a Monday. \label{nps:55} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Part}}\xspace[monday^g, \text{\textit{23/10/95}}]]$ \label{nps:56} \end{examps} \pref{nps:56} is problematic for two reasons. (a) The past tense of \pref{nps:55} is in effect ignored, because the denotation of $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ does not depend on $lt$, which is what the \ensuremath{\mathit{Past}}\xspace operator affects (see sections \ref{denotation} and \ref{past_op}). Hence, the implication of \pref{nps:55} that 23/10/95 is a past day is not captured. This problem could be solved by adding the constraint $g(\beta) \subper lt$ in the semantics of $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ (section \ref{denotation}). (b) \pref{nps:56} violates the syntax of \textsc{Top}\xspace (section \ref{top_syntax}), which does not allow the second argument of a \ensuremath{\mathit{Part}}\xspace operator to be a constant. This problem could be solved by modifying \textsc{Top}\xspace to allow the second argument of \ensuremath{\mathit{Part}}\xspace to be a constant. \section{Adjectives} \label{hpsg:adjectives} Following Pollard and Sag (\cite{Pollard1}, pp.\ 64 -- 65), adjectives also receive different signs for their predicative and non-predicative uses. When used as complements of \qit{to be} (e.g.\ \qit{closed} in \pref{adj:1}) adjectives are said to be predicative. In all other cases (e.g.\ \qit{closed} in \pref{adj:2}), adjectives are non-predicative. (\pref{adj:1} is actually ambiguous. The \qit{closed} may be a predicative adjective, or the passive form of \qit{to close}. As noted in section \ref{ling_not_supported}, however, passives are ignored in this thesis. Hence, I ignore the passive reading of \pref{adj:1}.) \begin{examps} \item Runway 2 was closed. \label{adj:1} \item BA737 landed on a closed runway. \label{adj:2} \end{examps} In the airport domain, the predicative sign of the adjective \qit{closed} is \pref{adj:3}. \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval closed\>} \\ synsem|loc & [cat & [head & \osort{adj}{ [prd & $+$]} \\ spr & \<\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\> ]\\ cont & \sort{closed}{ [arg1 & \(gate\_ent $\lor$ runway\_ent\)@1]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{adj:3} \end{examps} As noted in section \ref{hpsg:nouns}, \qit{is} and \qit{was} receive four signs each. One for progressive forms (see \pref{vforms:50}), one for prepositional phrase complements (see \pref{pps:7}), one for noun-phrase complements (see \pref{nps:47}), and one for adjectival complements (\pref{adj:4} below). \pref{adj:4} and \pref{adj:3} cause \pref{adj:1} to be mapped to \pref{adj:6}. \begin{examps} \avmoptions{active,center} \setbox\avmboxa=\hbox{\begin{avm} [loc & [cat & [head & \osort{adj}{ [prd $+$]} \\ subj & \<@3\> \\ comps & \<\>] \\ cont & @2]] \end{avm}} \item \begin{avm} [\avmspan{phon \; \<\fval was\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<@3\> \\ comps & \<\box\avmboxa\> ]\\ cont & \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & @2]}] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{adj:4} \item $\ensuremath{\mathit{Past}}\xspace[e^v, closed(runway2)]$ \label{adj:6} \end{examps} \pref{adj:7} shows the non-predicative sign of \qit{closed}. The \qit{closed} in \pref{adj:2} is a modifier (adjunct) of \qit{runway}. The {\feat mod} in \pref{adj:7} refers to the {\feat synsem} of the sign of the noun that the adjective modifies. The {\feat synsem$\mid$loc$\mid$cont} of \pref{adj:7} is the same as the one of the noun-sign, except that an \ensuremath{\mathit{Ntense}}\xspace is added to the {\feat restr} of the noun-sign (i.e.\ to the set denoted by \avmbox{2}). The additional \ensuremath{\mathit{Ntense}}\xspace requires the entity described by the noun (the entity represented by \avmbox{1}) to be closed at some unspecified time. The {\feat index} of the noun's sign is also required to represent a gate or runway. \begin{examps} \item \avmoptions{active,center} \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{closed}{ [arg1 & @1]}]} \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} [cat & [head & noun \\ spr & \<\_\> \\ comps & \<\>] \\ cont & \osort{nom\_obj}{ [index & @1 \\ restr & @2]}] \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval closed\>} \\ synsem|loc & [cat & [head & \osort{adj}{ [\avmspan{\feat prd \; $-$} \\ mod|loc & \box\avmboxb]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & \(gate\_ent $\lor$ runway\_ent\)@1 \\ restr & @2 $\union$ \{\box\avmboxa\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{adj:7} \end{examps} In the airport domain, the non-predicative sign of \qit{runway} is \pref{adj:8}. (I assume that \qit{runway} does not introduce an \ensuremath{\mathit{Ntense}}\xspace.) \begin{examps} \item \avmoptions{active,center} \setbox\avmboxa=\hbox{\begin{avm} \sort{runway}{ [arg1 & @1]} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval runway\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<[loc|cat|head & {\fval det}] \> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & runway\_ent@1 \\ restr & \{\box\avmboxa\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{adj:8} \end{examps} In \pref{adj:2}, \qit{closed} combines with \qit{runway} according to \textsc{Hpsg}\xspace's head-adjunct schema (see \cite{Pollard2}). \qit{Closed runway} receives the sign of \pref{adj:9}, where \avmbox{3} is the set of \pref{adj:9.2}. (Sets of {\srt psoa}s are treated as conjunctions.) \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval closed, runway\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<[loc|cat|head & {\fval det}] \> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & runway\_ent@1 \\ restr & @3]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{adj:9} \item \begin{avm} \{ \sort{runway}{ [arg1 & @1]}, \osort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & \osort{closed}{ [arg1 @1]}]} \}@3 \end{avm} \label{adj:9.2} \end{examps} The principles of \textsc{Hpsg}\xspace cause \pref{adj:9} to inherit the {\feat head} and {\feat spr} of \pref{adj:8}. \pref{adj:9} inherits the {\feat cont} of \pref{adj:7} according to the semantics principle of \pref{nps:9} (in this case, the ``semantic head'' is the adjunct \qit{closed}). \pref{adj:9}, the sign of \qit{landed} (which is the same as \pref{lentr:1}, except that it also introduces \ensuremath{\mathit{Past}}\xspace and \ensuremath{\mathit{Culm}}\xspace operators), and the non-predicative sign of \qit{on} (which is similar to \pref{pps:12}), cause \pref{adj:2} to be mapped to the \pref{adj:10}. During the post-processing (section \ref{post_processing} below), \pref{adj:10} gives rise to two different formulae, one where $t^v$ is replaced by $now^*$ (currently closed runway), and one where $t^v$ is replaced by $e^v$ (closed during the landing). \begin{examps} \item $runway(r^v) \land \ensuremath{\mathit{Ntense}}\xspace[t^v, closed(r^v)] \; \land$\\ $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[landing\_on(occr^v, ba737, r^v)]]$ \label{adj:10} \end{examps} An additional sign is needed for each non-predicative adjective to allow sentences like \pref{adj:13}, where a non-predicative adjective (\qit{closed}) combines with a predicative noun (\qit{runway}). \begin{examps} \item Runway 2 is a closed runway. \label{adj:13} \end{examps} \pref{adj:7} cannot be used in \pref{adj:13}, because here \qit{runway} is predicative, and hence the {\feat cont} of its sign is a {\srt psoa}\/ (the predicative sign of \qit{runway} is similar to \pref{nps:41}). In contrast, \pref{adj:7} assumes that the {\feat cont} of the noun is a {\srt nom\_obj}. One has to use the additional sign of \pref{adj:14}.\footnote{It is unclear how \pref{adj:14} could be written in the \textsc{Hpsg}\xspace version of \cite{Pollard2}. In \cite{Pollard2}, the {\srt and}\/ sort does not exist, and conjunctions of {\srt psoa}s can only be expressed using sets of {\srt psoa}s, as in \pref{adj:9.2}. In \pref{adj:14}, however, the value of {\feat sysnsem$\mid$loc$\mid$cont} cannot be a set of {\srt psoa}s, because {\feat cont} accepts only values whose sort is {\srt psoa}, {\srt nom\_obj}, or {\srt quant}.} Using \pref{adj:14}, \pref{adj:13} is mapped to \pref{adj:15}, which requires runway 2 to be closed at the speech time. \begin{examps} \item \avmoptions{active,center} \setbox\avmboxa=\hbox{\begin{avm} \sort{ntense}{ [et\_handle & \osort{temp\_ent}{ [tvar $+$]} $\lor$ now \\ main\_psoa & ]} \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} [cat & [head & \osort{noun}{ [prd \; $+$]} \\ spr & \<\_\> \\ subj & \<\feat np[-prd]$_{@1}$\> \\ comps & \<\>] \\ cont & @2] \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval closed\>} \\ synsem|loc & [cat & [head & \osort{adj}{ [\avmspan{\feat prd \; $-$} \\ mod|loc & \box\avmboxb]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{and}{ [conjunct1 & @2 \\ conjunct2 & \sort{closed}{ [arg1 & @1]}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{adj:14} \item $\ensuremath{\mathit{Pres}}\xspace[runway(runway2) \land closed(runway2)]$ \label{adj:15} \end{examps} As discussed in section \ref{temporal_adjectives}, temporal adjectives (e.g.\ \qit{former}, \qit{annual}) are not considered in this thesis. The prototype \textsc{Nlitdb}\xspace allows only non-predicative uses of the temporal adjective \qit{current} (as in \pref{adj:20}), by mapping \qit{current} to a sign that sets the first argument of the noun's \ensuremath{\mathit{Ntense}}\xspace to $now^*$. (This does not allow \qit{current} to be used with nouns that do not introduce \ensuremath{\mathit{Ntense}}\xspace{s}; see section \ref{non_pred_nps}.) \begin{examps} \item The current president was at terminal 2. \label{adj:20} \end{examps} \section{Temporal adverbials} \label{hpsg:pupe_adv} I now discuss temporal adverbials, starting from punctual adverbials (section \ref{point_adverbials}). \subsection{Punctual adverbials} \label{hpsg:punc_adv} Apart from \pref{pps:5} and \pref{pps:12} (which are used in sentences like \qit{BA737 is at gate 2.} and \qit{BA737 (habitually) arrives at gate 2.}), \qit{at} also receives signs that are used when it introduces punctual adverbials, as in \pref{pupe:1}. \pref{pupe:2} shows one of these signs. \begin{examps} \item Tank 2 was empty at 5:00pm. \label{pupe:1} \avmoptions{active, center} \setbox\avmboxa=\hbox{\begin{avm} \osort{prep}{ [\avmspan{prd \; $-$} \\ \avmspan{mod \; \feat s[vform {\fval fin}]:@2 $\lor$ \feat vp[vform {\fval psp}]:@2} \\ mod|loc|cat|aspect & {\fval state $\lor$ activity} \\ & {\fval $\lor$ point}]} \end{avm}} \item \begin{avm} [\avmspan{phon \; \<\fval at\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\feat np[-prd]$_{minute\_ent@1}$\> \\ aspect & point]\\ cont & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & @2]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{pupe:2} \end{examps} The {\feat mod} feature refers to the {\feat synsem} of the sign of the constituent modified by \qit{at}. {\feat s[vform {\fval fin}]:\avmbox{2}} is an abbreviation for a finite sentence (a finite verb form that has combined with its subject and complements). The \avmbox{2} refers to the {\feat cont} of the sign of the finite sentence. Similarly, {\feat vp[vform {\fval psp}]:\avmbox{2}} stands for a past participle verb phrase (a past participle that has combined with its complements but not its subject). The {\feat mod} of \pref{pupe:2} means that \pref{pupe:2} can be used when \qit{at} modifies finite sentences or past participle verb phrases, whose aspect is state, activity, or point. Generally, in this thesis temporal adverbials (punctual adverbials, period adverbials, duration adverbials) and temporal subordinate clauses (to be discussed in section \ref{hpsg:subordinates}) are allowed to modify only finite sentences and past participle verb phrases. \pref{pupe:2} and the sign of \qit{5:00pm} (shown in \pref{pupe:3}) cause \qit{at 5:00pm} to receive \pref{pupe:4} (\qit{5:00pm} acts as the noun-phrase complement of \qit{at}). \begin{examps} \avmoptions{active,center} \setbox\avmboxa=\hbox{\begin{avm} \sort{part}{ [partng & 5:00pm \\ part\_var & @1]} \end{avm}} \item \begin{avm} [\avmspan{phon \; \<\fval 5:00pm\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & minute\_ent\_var@1 \\ restr & \{\box\avmboxa\}]}@3 ] \\ \avmspan{qstore \; \{[det & exists \\ restind & @3]\}}] \end{avm} \label{pupe:3} \item \setbox\avmboxa=\hbox{\begin{avm} \osort{prep}{ [\avmspan{prd \; $-$} \\ \avmspan{mod \; \feat s[vform {\fval fin}]:@2 $\lor$ \feat vp[vform {\fval psp}]:@2} \\ mod|loc|cat|aspect & {\fval state $\lor$ activity}\\ & {\fval $\lor$ point}]} \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} \{ \sort{part}{ [partng & 5:00pm \\ part\_var & @1]} \} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval at, 5:00pm\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> \\ aspect & point]\\ cont & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & @2]}] \\ \avmspan{qstore \; \{[det & exists \\ restind & [index & minute\_ent\_var@1 \\ restr & \box\avmboxb]]\} } ] \end{avm} \label{pupe:4} \end{examps} According to \textsc{Hpsg}\xspace's head feature principle (section \ref{schemata_principles}), \pref{pupe:4} inherits the {\feat head} of \pref{pupe:2} (\qit{at} is the ``head-daughter'' of \qit{at 5:00pm}, and \qit{5:00pm} is the ``complement-daughter''). Following the semantics principle of \pref{nps:9}, the {\feat qstore} of \pref{pupe:4} is the union of the {\feat qstore}s of \pref{pupe:2} and \pref{pupe:3}, and the {\feat cont} of \pref{pupe:4} is the same as the {\feat cont} of \pref{pupe:2} (in this case, the ``semantic head'' is the head-daughter, i.e.\ \qit{at}). The propagation of {\feat aspect} is controlled by \pref{pupe:5}, a new principle of this thesis. (As with \pref{nps:9}, in \pref{pupe:5} I use the terminology of \cite{Pollard2}.) \begin{examps} \item \principle{Aspect Principle:} \index{aspect@{\feat aspect} (\textsc{Hpsg}\xspace feature)} \\ In a headed-phrase, the {\feat synsem$\mid$loc$\mid$cat$\mid$aspect} value is token-identical with that of the semantic head. (In a headed phrase, the \emph{semantic head} is the {\feat adjunct-daughter} if any, and the {\feat head-daughter} otherwise.) \label{pupe:5} \end{examps} \pref{pupe:5} means that each syntactic constituent inherits the {\feat aspect} of its head-daughter (the noun in noun phrases, the verb in verb phrases, the preposition in prepositional phrases), except for cases where the head-daughter combines with an adjunct-daughter (a modifier). In the latter case, the mother syntactic constituent inherits the {\feat cont} of the adjunct-daughter. \pref{pupe:5} causes \pref{pupe:4} to inherit the {\feat aspect} value of the semantic head \qit{at}. The \qit{tank 2 was empty} of \pref{pupe:1} receives \pref{pupe:6}. \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval tank2, was, empty\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \<\> \\ comps & \<\>] \\ cont & \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{empty}{ [arg1 & tank2]}]}] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{pupe:6} \setbox\avmboxa=\hbox{\begin{avm} \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{empty}{ [arg1 & tank2]}]} \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} \{ \sort{part}{ [partng & 5:00pm \\ part\_var & @1]} \} \end{avm}} \end{examps} When \qit{tank 2 was empty} combines with \qit{at 5:00pm}, \pref{pupe:1} receives \pref{pupe:7}. In this case, \qit{tank 2 was empty} is the head-daughter, and \qit{at 5:00pm} is an adjunct-daughter (a modifier). Hence, according to \pref{pupe:5}, \pref{pupe:7} inherits the {\feat aspect} of \pref{pupe:4} (i.e.\ {\srt point}\/; in contrast, the {\feat aspect} of \pref{pupe:6} was {\srt lex\_state}.) This is in accordance with the arrangements of section \ref{point_adverbials}, whereby punctual adverbials trigger an aspectual shift to point. \begin{examps} \item \setbox\avmboxa=\hbox{\begin{avm} \osort{past}{ [et\_handle & \osort{temp\_ent}{ [tvar & $+$]} \\ main\_psoa & \osort{empty}{ [arg1 & tank2]}]} \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} \{ \sort{part}{ [partng & 5:00pm \\ part\_var & @1]} \} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval tank2, was, empty, at, 5:00pm\>} \\ synsem|loc & [cat & [head & \osort{verb}{ [vform & fin \\ aux & $+$]} \\ aspect & point \\ spr & \<\> \\ subj & \<\> \\ comps & \<\>] \\ cont & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & \box\avmboxa]}] \\ \avmspan{qstore \; \{[det & exists \\ restind & [index & minute\_ent\_var@1 \\ restr & \box\avmboxb]]\}}] \end{avm} \label{pupe:7} \end{examps} According to the semantics principle, \pref{pupe:7} also inherits the {\feat cont} of \pref{pupe:4} (the sign of the modifier), and the {\feat qstore} of \pref{pupe:7} is the union of the {\feat qstore}s of \pref{pupe:4} and \pref{pupe:6}. Finally, according to the head feature principle (section \ref{schemata_principles}), \pref{pupe:7} inherits the {\feat head} of \pref{pupe:6} (the sign of the head-daughter). The {\feat qstore} and {\feat cont} of \pref{pupe:7} represent \pref{pupe:7.1}. \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{pupe:7.1} \end{examps} The reader may wonder why temporal adverbials (e.g.\ \qit{at 5:00pm} in \pref{pupe:1}) are taken to modify whole finite sentences (\qit{tank 2 was empty}), rather than finite verb phrases (\qit{was empty}). The latter approach leads to problems in questions like \qit{Was tank 2 empty at 5:00pm?}, where \qit{was} combines in one step with both its subject \qit{tank 2} and its complement \qit{empty}, following the head-subject-complement schema of \cite{Pollard2}. In this case, there is no verb phrase constituent (verb that has combined with its complements but not its subject) to be modified by \qit{at 5:00pm}. Apart from finite sentences, temporal adverbials are also allowed to modify past participle verb phrases (see the {\feat mod} of \pref{pupe:2}). This is needed in past perfect sentences like \pref{pupe:9}. \begin{examps} \item BA737 had entered sector 2 at 5:00pm. \label{pupe:9} \end{examps} As discussed in section \ref{past_perfect}, \pref{pupe:9} has two readings: one where the entrance occurs at 5:00pm, and one where 5:00pm is a ``reference time'', a time where the entrance has already occurred. The two readings are expressed by \pref{pupe:10} and \pref{pupe:11} respectively (see also section \ref{perf_op}). \pref{pupe:9} is taken to be syntactically ambiguous with two possible parses, sketched in \pref{pupe:12} and \pref{pupe:13}. These give rise to \pref{pupe:10} and \pref{pupe:11} respectively. \begin{examps} \item BA737 had [[entered sector 2] at 5:00pm]. \label{pupe:12} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[fv^v, enter(ba737, sector2)]]]$ \label{pupe:10} \item {[}BA737 had [entered sector 2]] at 5:00pm. \label{pupe:13} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, enter(ba737, sector2)]]]$ \label{pupe:11} \end{examps} One complication of this approach is that it generates two equivalent formulae for the present perfect \pref{pupe:17}, shown in \pref{pupe:18} and \pref{pupe:19}. (\qit{Has} does not introduce a \ensuremath{\mathit{Perf}}\xspace; see \pref{vforms:36}.) These correspond to the parses of \pref{pupe:17a} and \pref{pupe:17b} respectively. \begin{examps} \item BA737 has entered sector 2 at 5:00pm. \label{pupe:17} \item {[}BA737 has [entered sector 2]] at 5:00pm. \label{pupe:17a} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, enter(ba737, sector2)]]$ \label{pupe:18} \item BA737 has [[entered sector 2] at 5:00pm]. \label{pupe:17b} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{At}}\xspace[fv^v, enter(ba737, sector2)]]$ \label{pupe:19} \end{examps} In the prototype \textsc{Nlitdb}\xspace, the sign of \qit{has} is slightly more complex than \pref{vforms:36}. It requires the {\feat cont} of the verb-phrase complement of \qit{has} to be of sort {\srt predicate}. This blocks \pref{pupe:17b} and \pref{pupe:19}, because in \pref{pupe:17b} the \qit{at 5:00pm} causes the {\feat cont} of \qit{entered sector 2 at 5:00pm} to become of sort {\srt at\_op}\/ (it inserts an \ensuremath{\mathit{At}}\xspace operator), which is not a subsort of {\srt predicate}. \pref{pupe:2} corresponds to the interjacent meaning of punctual adverbials, which according to table \vref{punctual_adverbials_table} is possible only with states and activities. \pref{pupe:2} also covers cases where punctual adverbials combine with points. There are also other \qit{at} signs, that are similar to \pref{pupe:2} but that introduce additional \ensuremath{\mathit{Begin}}\xspace or \ensuremath{\mathit{End}}\xspace operators. These correspond to the inchoative (with activities and culminating activities) and terminal (with culminating activities) meanings of punctual adverbials. \subsection{Period adverbials} \label{hpsg:per_advs} I now turn to period adverbials (section \ref{period_adverbials}). \pref{pupe:29} shows one of the signs of \qit{on} that are used when \qit{on} introduces period adverbials. \begin{examps} \avmoptions{active, center} \item \setbox\avmboxa=\hbox{\begin{avm} \osort{prep}{ [prd & $-$ \\ mod & \feat s[vform {\fval fin}]:@2 $\lor$ \feat vp[vform {\fval psp}]:@2 \\ \avmspan{mod|loc|cat|aspect \; {\fval culmact}}]} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval on\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\feat np[-prd]$_{day\_ent@1}$\> \\ aspect & point]\\ cont & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & \osort{end}{ [main\_psoa & @2]}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{pupe:29} \end{examps} \pref{pupe:29}, which can be used only when the \qit{on~\dots} adverbial modifies a culminating activity, corresponds to the reading where the situation of the culminating activity simply reaches its completion within the adverbial's period (table \vref{period_adverbials_table}). (\pref{pupe:29} causes the aspectual class of the culminating activity to become point. This agrees with table \ref{period_adverbials_table}.) For example, \pref{pupe:29} causes \pref{pupe:31} to be mapped to \pref{pupe:32}. (I assume here that \qit{to repair} is classified as culminating activity verb.) Intuitively, \pref{pupe:32} requires a past period $e^v$ to exist, such that $e^v$ covers a whole repair of engine 2 by J.Adams (from start to completion), and the end-point of $e^v$ falls within some Monday. That is, the repair must have been completed on Monday, but it may have started before Monday. \begin{examps} \item J.Adams repaired engine 2 on Monday. \label{pupe:31} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, m^v] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[m^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, [\ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, eng2)]]]]$ \label{pupe:32} \end{examps} There is also an \qit{on} sign that is similar to \pref{pupe:29}, but that does not introduce an \ensuremath{\mathit{End}}\xspace operator, preserves the {\feat aspect} of the modified expression, and can be used when \qit{on~\dots} adverbials modify expressions from all four aspectual classes. This sign causes \pref{pupe:31} to be mapped to \pref{pupe:33} (the prototype \textsc{Nlitdb}\xspace would generate both \pref{pupe:32} and \pref{pupe:33}). \pref{pupe:33} corresponds to the reading where the repair must have both started and been completed within a (the same) Monday. The \qit{on} sign that does not introduce an \ensuremath{\mathit{End}}\xspace also gives rise to appropriate formulae when \qit{on~\dots} adverbials modify state, activity, or point expressions. \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, m^v] \; \land$\\ $\ensuremath{\mathit{At}}\xspace[m^v, \ensuremath{\mathit{Past}}\xspace[e^v, [\ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, eng2)]]]$ \label{pupe:33} \end{examps} Both \pref{pupe:29} and the \qit{on} sign that does not introduce an \ensuremath{\mathit{End}}\xspace require the noun-phrase complement of \qit{on} to introduce an index of sort {\srt day\_ent}. The signs of \qit{1/1/91} and \qit{Monday} introduce indices of sorts {\srt 1/1/91}\/ and {\srt day\_ent\_var}\/ respectively, which are subsorts of {\srt day\_ent}\/ (see figure \vref{ind_hierarchy}). Hence, \qit{1/1/91} and \qit{Monday} are legitimate complements of \qit{on} in period adverbials. In contrast, \qit{5:00pm} introduces an index of sort {\srt minute\_ent\_var}\/ (see \pref{pupe:3}), which is not a subsort of {\srt day\_ent}. Hence, \pref{pupe:40} is correctly rejected. \begin{examps} \item \bad Tank 2 was empty on 5:00pm. \label{pupe:40} \end{examps} The signs of other prepositions that introduce period adverbials (e.g.\ \qit{\underline{in} 1991}, \qit{\underline{before} 29/10/95}, \qit{\underline{after} 5:00pm}) and the signs of \qit{yesterday} and \qit{today} are similar to the signs of \qit{on}, except that \qit{before} and \qit{after} introduce \ensuremath{\mathit{Before}}\xspace and \ensuremath{\mathit{After}}\xspace operators instead of \ensuremath{\mathit{At}}\xspace{s}. Also, \qit{before} is given only one sign, that does not introduce an \ensuremath{\mathit{End}}\xspace (there is no \qit{before} sign for culminating activities analogous to \pref{pupe:29}, that introduces an \ensuremath{\mathit{End}}\xspace). This is related to comments in section \ref{period_adverbials} that in the case of \qit{before~\dots} adverbials, requiring the situation of a culminating activity to simply reach its completion before some time (reading with \ensuremath{\mathit{End}}\xspace) is equivalent to requiring the situation to both start and reach its completion before that time (reading without \ensuremath{\mathit{End}}\xspace). \subsection{Duration adverbials} \label{duration_adverbials} The treatment of \qit{for~\dots} duration adverbials is rather ad hoc from a syntax point of view. In an adverbial like \qit{for two days}, both \qit{two} and \qit{days} are taken to be complements of \qit{for}, instead of treating \qit{two} as the determiner of \qit{days}, and \qit{two days} as a noun-phrase complement of \qit{for}. Number-words like \qit{one}, \qit{two}, \qit{three}, etc.\ are mapped to signs of the form of \pref{dadv:4}. Their {\feat restr}s are empty, and their indices represent the corresponding numbers. (The {\srt 2}\/ of \pref{dadv:4} is a subsort of {\srt sem\_num}\/; see section \ref{more_ind}.) \begin{examps} \item \avmoptions{active} \begin{avm} [\avmspan{phon \; \<\fval two\>} \\ synsem|loc & [cat & [head & \sort{det}{ [spec & none]}\\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & 2 \\ restr & \{\}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{dadv:4} \end{examps} Although words like \qit{one}, \qit{two}, \qit{three}, etc.\ are classified as determiners (the {\feat head} of \pref{dadv:4} is of sort {\srt det}), the {\srt none}\/ value of their {\feat spec} does not allow them to be used as determiners of any noun. (Determiners combining with nouns are the specifiers of the nouns. The {\srt none}\/ means that the word of the sign cannot be the specifier of any constituent, and hence cannot be used as the determiner of any noun.) \pref{dadv:1} shows the sign of \qit{for} that is used in duration adverbials (for typesetting reasons, I show the feature structures that correspond to \avmbox{5} and \avmbox{6} separately, in \pref{dadv:2} and \pref{dadv:3} respectively). \begin{examps} \avmoptions{active, center} \item \setbox\avmboxa=\hbox{\begin{avm} \osort{prep}{ [\avmspan{prd \; $-$} \\ \avmspan{mod \; \feat s[vform {\fval fin}]:@4 $\lor$ \feat vp[vform {\fval psp}]:@4} \\ mod|loc|cat|aspect & \({\fval lex\_state} \\ & $\lor$ {\fval progressive} \\ & $\lor$ {\fval activity}\)@1]} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval for\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<@5, @6\> \\ aspect & @1]\\ cont & \osort{for\_op}{ [dur\_unit & @2 \\ duration & @3 \\ main\_psoa & @4]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{dadv:1} \item \begin{avm} [loc & [cat|head & det \\ cont|index & sem\_num@3]]@5 \end{avm} \label{dadv:2} \item \begin{avm} [loc & [cat & [head & noun \\ spr & \<\_\> \\ subj & \<\> \\ comps & \<\>] \\ cont|restr & \{\sort{part}{ [partng & compl\_partng@2]}\}]]@6 \end{avm} \label{dadv:3} \end{examps} The {\feat comps} of \pref{dadv:1} means that \qit{for} requires two complements: a determiner that introduces a number-denoting ({\srt sem\_num}\/) index (like the \qit{two} of \pref{dadv:4}), and a noun that introduces a \ensuremath{\mathit{Part}}\xspace operator whose first argument is a complete partitioning name (like the \qit{day} of \pref{nps:32}). In \pref{dadv:6}, \qit{for two days} receives \pref{dadv:7}. (As already mentioned, no number-agreement checks are made, and plural nouns are treated semantically as singular ones. Apart from {\feat phon}, the sign of \qit{days} is the same as \pref{nps:32}.) \begin{examps} \item Tank 2 was empty for two days. \label{dadv:6} \item \avmoptions{active, center} \setbox\avmboxa=\hbox{\begin{avm} \osort{prep}{ [\avmspan{prd \; $-$} \\ \avmspan{mod \; \feat s[vform {\fval fin}]:@4 $\lor$ \feat vp[vform {\fval psp}]:@4} \\ mod|loc|cat|aspect & \({\fval lex\_state}\\ & $\lor$ {\fval progressive} \\ & $\lor$ {\fval activity}\)@1]} \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval for, two, days\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> \\ aspect & @1]\\ cont & \osort{for\_op}{ [dur\_unit & day \\ duration & 2 \\ main\_psoa & @4]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{dadv:7} \end{examps} When \qit{tank 2 was empty} combines with its temporal-adverbial modifier \qit{for two days}, the \avmbox{4} of \pref{dadv:7} becomes a feature structure that represents the \textsc{Top}\xspace formula for \qit{tank 2 was empty}, i.e.\ \pref{dadv:8}. According to the semantics principle of \pref{nps:9}, the sign of \pref{dadv:6} inherits the {\feat cont} of \pref{dadv:7} (where \avmbox{4} now represents \pref{dadv:8}). Hence, \pref{dadv:6} is mapped to \pref{dadv:9}. \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]$ \label{dadv:8} \item $\ensuremath{\mathit{For}}\xspace[day^c, 2, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{dadv:9} \end{examps} Following table \vref{for_adverbials_table}, \pref{dadv:1} does not allow \qit{for~\dots} adverbials to modify point expressions (the {\feat mod$\mid$loc$\mid$cat$\mid$aspect} of \pref{dadv:1} cannot be {\srt point}\/). It also does not allow \qit{for~\dots} adverbials to modify consequent states. If \qit{for~\dots} adverbials were allowed to modify consequent states, \pref{dadv:10} would receive \pref{dadv:11} and \pref{dadv:12}. \begin{examps} \item BA737 had circled for two hours. \label{dadv:10} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{For}}\xspace[hour^c, 2, circling(ba737)]]]$ \label{dadv:11} \item $\ensuremath{\mathit{For}}\xspace[hour^c, 2, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, circling(ba737)]]]$ \label{dadv:12} \end{examps} \pref{dadv:11} corresponds to the parse of \pref{dadv:10} where \qit{for two hours} modifies the past participle \qit{circled} before \qit{circled} combines with \qit{had}. In that case, the \qit{for~\dots} adverbial modifies an activity, because past participles retain the aspectual class of the base form (\qit{to circle} is an activity verb in the airport domain). \pref{dadv:12} corresponds to the parse where \qit{for two hours} modifies the whole sentence \qit{BA737 had circled}. In that case, the \qit{for~\dots} adverbial modifies a consequent state, because the \qit{had} has caused the aspectual class of \qit{BA737 had circled} to become consequent state. By not allowing \qit{for~\dots} adverbials to modify consequent states, \pref{dadv:12} is blocked. This is needed, because in \pref{dadv:12} two hours is the duration of a period (pointed to by $e1^v$) that follows a period (pointed to by $e2^v$) where BA737 was circling. This reading is never possible when \qit{for~\dots} adverbials are used in past perfect sentences. The \qit{for~\dots} adverbial of \pref{dadv:10} can only specify the duration of the circling (a reading captured by \pref{dadv:11}). (A similar observation is made on p.~587 of \cite{Kamp1993}.) The present treatment of \qit{for~\dots} duration adverbials causes \pref{dadv:13} to receive \pref{dadv:14}. \pref{dadv:14} does not capture correctly the meaning of \pref{dadv:13}, because it requires the taxiing to have been completed, i.e.\ BA737 to have reached gate 2. In contrast, as discussed in section \ref{for_adverbials}, the \qit{for~\dots} adverbial of \pref{dadv:13} cancels the normal implication of \qit{BA737 taxied to gate 2.} that the taxiing was completed. The post-processing (section \ref{post_processing} below) removes the \ensuremath{\mathit{Culm}}\xspace of \pref{dadv:14}, generating a formula that does not require the taxiing to have been completed. \begin{examps} \item BA737 taxied to gate 2 for five minutes. \label{dadv:13} \item $\ensuremath{\mathit{For}}\xspace[minute^c, 5, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[taxiing\_to(ba737, gate2)]]]$ \label{dadv:14} \end{examps} Duration adverbials introduced by \qit{in} (e.g.\ \pref{dadv:15}) are treated by mapping \qit{in} to a sign that is the same as \pref{dadv:1}, except that it allows the adverbial to modify only culminating activities. (The framework of this thesis does not allow \qit{in~\dots} duration adverbials to modify states, activities, or points; see section \ref{in_adverbials}.) \begin{examps} \item BA737 taxied to gate 2 in five minutes. \label{dadv:15} \end{examps} This causes \pref{dadv:15} to be mapped to \pref{dadv:14}, which correctly requires the taxiing to have been completed, and the duration of the taxiing (from start to completion) to be five minutes. (In this case, the post-processing does not remove the \ensuremath{\mathit{Culm}}\xspace.) \section{Temporal complements of habituals} \label{habituals} Let us now examine more closely the status of temporal prepositional-phrases, like \qit{at 5:00pm} and \qit{on Monday} in \pref{hab:1} -- \pref{hab:4}. \begin{examps} \item BA737 departed at 5:00pm. \label{hab:1} \item BA737 departs at 5:00pm. \label{hab:2} \item J.Adams inspected gate 2 on Monday. \label{hab:3} \item J.Adams inspects gate 2 on Monday. \label{hab:4} \end{examps} \pref{hab:1} has both a habitual and a non-habitual reading. Under the non-habitual reading, it refers to an actual departure that took place at 5:00pm. Under the habitual reading, it means that BA737 had the habit of departing at 5:00pm (this reading is easier to accept if an adverbial like \qit{in 1992} is added). In \pref{hab:2}, only the habitual reading is possible, i.e.\ BA737 currently has the habit of departing at 5:00pm. (A scheduled-to-happen reading is also possible, but as discussed in section \ref{simple_present} this is ignored in this thesis.) Similar comments apply to \pref{hab:3} and \pref{hab:4}. To account for the habitual and non-habitual readings of \qit{to depart} in \pref{hab:1} and \pref{hab:2}, the base form of \qit{to depart} is given the signs of \pref{hab:7} and \pref{hab:8}. These correspond to what chapter \ref{linguistic_data} called informally the habitual and non-habitual homonyms of \qit{to depart}. \pref{hab:7} classifies the habitual homonym as (lexical) state, while \pref{hab:8} classifies the non-habitual homonym as point (this agrees with table \vref{airport_verbs}). According to \pref{hab:7}, the habitual homonym requires an \qit{at~\dots} prepositional phrase that specifies the habitual departure time (this is discussed further below). In contrast, the non-habitual homonym of \pref{hab:8} requires no complement. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval depart\>} \\ synsem & [loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$ ]} \\ aspect & lex\_state \\ spr & \<\> \\ subj & \< \feat np[-prd]$_{flight\_ent@1}$ \> \\ comps & \< \feat pp[-prd, pform {\fval at}]$_{minute\_gappy@2}$ \> ]\\ cont & \sort{hab\_departs\_at}{ [arg1 & @1 \\ arg2 & @2]} ]] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{hab:7} \item \begin{avm} [\avmspan{phon \; \<\fval depart\>} \\ synsem & [loc & [cat & [head & \osort{verb}{ [vform & bse \\ aux & $-$ ]} \\ aspect & point \\ spr & \<\> \\ subj & \< \feat np[-prd]$_{flight\_ent@1}$ \> \\ comps & \<\> ]\\ cont & \sort{actl\_depart}{ [arg1 & @1]} ]] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{hab:8} \end{examps} In the airport domain, there are actually two habitual signs for \qit{to depart}, one where \qit{to depart} requires an \qit{at~\dots} prepositional-phrase complement (as in \pref{hab:7}), and one where \qit{to depart} requires a \qit{from~\dots} prepositional-phrase complement (this is needed in \pref{hab:8.1}). There are also two non-habitual signs of \qit{to depart}, one where \qit{to depart} requires no complement (as in \pref{hab:8}), and one where \qit{to depart} requires a \qit{from~\dots} prepositional-phrase complement (needed in \pref{hab:8.2}). For simplicity, here I ignore these extra signs. \begin{examps} \item BA737 (habitually) departs from gate 2. \label{hab:8.1} \item BA737 (actually) departed from gate 2. \label{hab:8.2} \end{examps} \pref{hab:7}, \pref{hab:8}, and the simple-past lexical rules of section \ref{single_word_forms} give rise to two signs (a habitual and a non-habitual one) for the simple past \qit{departed}. These are the same as \pref{hab:7} and \pref{hab:8}, except that they contain additional \ensuremath{\mathit{Past}}\xspace operators. In contrast, the simple-present lexical rule of section \ref{single_word_forms} generates only one sign for the simple present \qit{departs}. This is the same as \pref{hab:7}, except that it contains an additional \ensuremath{\mathit{Pres}}\xspace operator. No simple-present sign is generated from \pref{hab:8}, because the simple-present lexical rule requires the aspect of the base sign to be state. The non-habitual simple-past sign of \qit{departed}, the \qit{at} sign of \pref{pupe:2}, and the \qit{5:00pm} sign of \pref{pupe:3}, cause \pref{hab:1} to be mapped to \pref{hab:13}, which expresses the non-habitual reading of \pref{hab:1}. In this case, \qit{at 5:00pm} is treated as a temporal-adverbial modifier of \qit{BA737 departed}, as discussed in section \ref{hpsg:punc_adv}. \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, act\_depart(ba737)]]$ \label{hab:13} \end{examps} In the habitual reading of \pref{hab:1}, where the habitual sign of \qit{departed} (derived from \pref{hab:7}) is used, \qit{at 5:00pm} is treated as a prepositional-phrase complement of \qit{departed}. In this case, the sign of \qit{at} that introduces non-predicative prepositional-phrase complements (i.e.\ \pref{pps:12}) is used. The intention is to map \pref{hab:1} to \pref{hab:14}, where \textit{5:00pm} is a constant acting as a ``generic representative'' of 5:00pm minutes (section \ref{hab_problems}). \begin{examps} \item $\ensuremath{\mathit{Past}}\xspace[e^v, hab\_departs\_at(ba737, \text{\textit{5:00pm}})]$ \label{hab:14} \end{examps} The problem is that in this case the \qit{5:00pm} sign of \pref{pupe:3} cannot be used, because it inserts a \ensuremath{\mathit{Part}}\xspace operator in {\feat qstore}. The semantics principle would cause this \ensuremath{\mathit{Part}}\xspace operator to be inherited by the sign of the overall \pref{hab:1}, and thus the \ensuremath{\mathit{Part}}\xspace operator would appear in the resulting formula. In contrast, \pref{hab:14} (the intended formula for \pref{hab:1}) contains no \ensuremath{\mathit{Part}}\xspace operators. To solve this problem, one has to allow an extra sign for \qit{5:00pm}, shown in \pref{hab:15}, which does not introduce a \ensuremath{\mathit{Part}}\xspace. Similarly, an extra \qit{Monday} sign is needed in \pref{hab:3}. (The fact that these extra signs have to be introduced is admittedly inelegant. This is caused by the fact that \qit{at 5:00pm} is treated differently in \pref{hab:13} and \pref{hab:14}; see also the discussion in section \ref{hab_problems}.) The \qit{5:00pm} sign of \pref{hab:15}, the \qit{at} sign that is used when \qit{at} introduces non-predicative prepositional-phrase complements (i.e.\ \pref{pps:12}), and the habitual \qit{departed} sign (derived from \pref{hab:7}) cause \pref{hab:1} to be mapped to \pref{hab:14}. \avmoptions{active} \begin{examps} \item \begin{avm} [\avmspan{phon \; \<\fval 5:00pm\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & 5:00pm \\ restr & \{\}]} ] \\ \avmspan{qstore \; \{\}}] \end{avm} \label{hab:15} \end{examps} The habitual \qit{departed} sign (which derives from \pref{hab:7}) requires the index of the prepositional-phrase complement to be of sort {\srt minute\_gappy}. As wanted, this does not allow the \qit{5:00pm} sign of \pref{pupe:3} (the one that introduces a \ensuremath{\mathit{Part}}\xspace) to be used in the prepositional-phrase complement of the habitual \qit{departed}, because if \pref{pupe:3} is used, the index of the prepositional phrase will be of sort {\srt minute\_ent\_var}, which is not a subsort of {\srt minute\_gappy}\/ (see figure \vref{ind_hierarchy}). In contrast, \pref{hab:15} introduces an index of sort {\srt 5:00pm}, which is a subsort of {\srt minute\_gappy}, and hence that sign can be used in the complement of the habitual \qit{departed}. The treatment of the simple present \pref{hab:2} is similar. In this case, the habitual simple present sign (that is derived from \pref{hab:7}) is used, and \pref{hab:2} is mapped to \pref{hab:16}. No \textsc{Top}\xspace formula is generated for the (impossible) non-habitual reading of \pref{hab:2}, because there is no non-habitual sign for the simple present \qit{departs} (see comments above about the simple present lexical rule). \begin{examps} \item $\ensuremath{\mathit{Pres}}\xspace[hab\_departs\_at(ba737, \text{\textit{5:00pm}})]$ \label{hab:16} \end{examps} \section{Fronted temporal modifiers} \label{fronted} As discussed in section \ref{hpsg:pupe_adv}, in this thesis temporal-adverbial modifiers (e.g.\ \qit{at 5:00pm} in \pref{front:1} -- \pref{front:2}, \qit{on Monday} in \pref{front:3} -- \pref{front:4}) can modify either whole finite sentences or past participle verb phrases. \begin{examps} \item BA737 entered sector 2 at 5:00pm. \label{front:1} \item At 5:00pm BA737 entered sector 2. \label{front:2} \item Tank 2 was empty on Monday. \label{front:3} \item On Monday tank 2 was empty. \label{front:4} \end{examps} In \textsc{Hpsg}\xspace, the order in which a modifier and the constituent to which the modifier attaches can appear in a sentence is controlled by the ``constituent-ordering principle'' (\textsc{Cop}). This is a general (and not fully developed) principle that controls the order in which the various constituents can appear in a sentence (see chapter 7 of \cite{Pollard1}). This thesis uses an over-simplified version of \textsc{Cop}, that places no restriction on the order between temporal modifiers and modified constituents when the modified constituents are sentences. This allows \qit{at 5:00pm} to either follow \qit{BA737 entered sector 2} (as in \pref{front:1}), or to precede it (as in \pref{front:2}). Similarly, \qit{on Monday} may either follow \qit{tank 2 was empty} (as in \pref{front:3}), or precede it (as in \pref{front:4}).\footnote{An alternative approach is to allow temporal modifiers to participate in unbounded dependency constructions; see pp.~176 -- 181 of \cite{Pollard2}.} When temporal modifiers attach to past-participle verb phrases, however, I require the modifiers to follow the verb phrases, as in \pref{front:6}. This rules out unacceptable sentences like \pref{front:5}, where \qit{at 5:00pm} precedes the \qit{entered sector 2}.\footnote{Constituent-ordering restrictions are enforced in the \textsc{Ale}\xspace grammar of the prototype \textsc{Nlitdb}\xspace in a rather ad hoc manner, which involves partitioning the {\srt synsem}\/ sort into {\srt pre\_mod\_synsem}\/ and {\srt post\_mod\_synsem}, and using feature structures from the two subsorts as values of {\feat mod} to signal that the modifier can only precede or follow the modified constituent. This idea was borrowed from a grammar written by Suresh Manandhar.} \begin{examps} \item BA737 had [[entered sector 2] at 5:00pm]. \label{front:6} \item \bad BA737 had [at 5:00pm [entered sector 2]]. \label{front:5} \end{examps} This approach causes \pref{front:7} to receive only \pref{front:8}, because in \pref{front:7} \qit{at 5:00pm} can modify only the whole \qit{BA737 had entered sector 2} (it cannot modify just \qit{entered sector 2} because of the intervening \qit{BA737 had}). In \pref{front:8}, 5:00pm is a reference time, a time where the entrance had already occurred. In contrast, \pref{front:8.5} receives both \pref{front:8} and \pref{front:11}, because in that case \qit{at 5:00pm} can modify either the whole \qit{BA737 had entered sector 2} or only \qit{entered sector 2}. In \pref{front:11}, 5:00pm is the time where the entrance occurred. \begin{examps} \item At 5:00pm [BA737 had entered sector 2]. \label{front:7} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, enter(ba737, sector2)]]]$ \label{front:8} \item BA737 had entered sector 2 at 5:00pm. \label{front:8.5} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}, fv^v] \land \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[fv^v, enter(ba737, sector2)]]]$ \label{front:11} \end{examps} The fact that \pref{front:7} does not receive \pref{front:11} does not seem to be a disadvantage, because in \pref{front:7} the reading where \qit{at 5:00pm} specifies the time of the entrance seems unlikely (or at least much more unlikely than in \pref{front:8.5}). \section{Temporal subordinate clauses} \label{hpsg:subordinates} I now discuss temporal subordinate clauses (section \ref{subordinate_clauses}), focusing on \qit{while~\dots} clauses. The treatment of \qit{before~\dots} and \qit{after~\dots} clauses is very similar. As with period adverbials, \qit{while~\dots} clauses are treated as temporal modifiers of finite sentences or past participle verb phrases. As with prepositions introducing period adverbials, \qit{while} is given two signs. The first one, shown in \pref{subs:1}, introduces an \ensuremath{\mathit{End}}\xspace operator, causes an aspectual shift to point, and can be used only with culminating activity main clauses (\pref{subs:1} is similar to \pref{pupe:29}.) The second one is the same as \pref{subs:1}, except that it does not introduce an \ensuremath{\mathit{End}}\xspace, it preserves the aspectual class of the main clause, and it can be used with main clauses of any aspectual class. In both cases, \qit{while} requires as its complement a finite sentence whose aspect must not be consequent state (this agrees with table \vref{while_clauses_table}, which does not allow the aspectual class of the \qit{while}-clause to be consequent state). \begin{examps} \avmoptions{active, center} \item \setbox\avmboxa=\hbox{\begin{avm} [mod & \feat s[vform {\fval fin}]:@2 $\lor$ \feat vp[vform {\fval psp}]:@2 \\ \avmspan{mod|loc|cat|aspect \; {\fval culmact}}] \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval while\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\feat s[vform & {\fval fin}\\ aspect & $\neg$cnsq\_state]:@1\> \\ aspect & point]\\ cont & \osort{at\_op}{ [time\_spec & @1 \\ main\_psoa & \osort{end}{ [main\_psoa & @2]}]}] \\ \avmspan{qstore \; \{\}} ] \end{avm} \label{subs:1} \end{examps} The \avmbox{1} in \pref{subs:1} denotes the {\feat cont} of the sign of the complement of \qit{while} (the subordinate clause). The two \qit{while} signs cause \pref{subs:3} to receive \pref{subs:4} and \pref{subs:5}. (\qit{To land} is a culminating activity verb in the airport domain). \pref{subs:4} requires the landing to have simply been completed during the inspection, while \pref{subs:5} requires the landing to have both started and been completed during the inspection. \begin{examps} \item UK160 landed while J.Adams was inspecting BA737. \label{subs:3} \item $\begin{aligned}[t] \ensuremath{\mathit{At}}\xspace[&\ensuremath{\mathit{Past}}\xspace[e1^v, inspecting(j\_adams, ba737)], \\ &\ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[landing(occr^v, uk160)]]]] \end{aligned}$ \label{subs:4} \item $\begin{aligned}[t] \ensuremath{\mathit{At}}\xspace[&\ensuremath{\mathit{Past}}\xspace[e1^v, inspecting(j\_adams, ba737)], \\ &\ensuremath{\mathit{Past}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[landing(occr^v, uk160)]]] \end{aligned}$ \label{subs:5} \end{examps} Since \qit{while~\dots} clauses are treated as temporal modifiers, the ordering arrangements of section \ref{fronted} apply to \qit{while~\dots} clauses as well. Hence, \qit{while~\dots} clauses can either precede or follow finite sentences (e.g.\ \pref{subs:8.1}, \pref{subs:9}). \begin{examps} \item UK160 arrived while J.Adams was inspecting BA737. \label{subs:8.1} \item While J.Adams was inspecting BA737, UK160 arrived. \label{subs:9} \end{examps} One problem with the present treatment of \qit{while~\dots} clauses is that it maps \pref{subs:10} to \pref{subs:11}, which requires the inspection to have been completed. This does not agree with table \vref{while_clauses_table}, according to which any requirement that the situation of a culminating activity sentence must have been reached is cancelled when the sentence is used as a \qit{while~\dots} clause. To overcome this problem, the post-processing (section \ref{post_processing} below) removes any \ensuremath{\mathit{Culm}}\xspace operators that are within first arguments of \ensuremath{\mathit{At}}\xspace operators. This removes the \ensuremath{\mathit{Culm}}\xspace of \pref{subs:11}, generating a formula that no longer requires the inspection to have been completed. \begin{examps} \item UK160 departed while J.Adams inspected BA737. \label{subs:10} \item $\begin{aligned}[t] \ensuremath{\mathit{At}}\xspace[&\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(j\_adams, ba737)]], \\ &\ensuremath{\mathit{Past}}\xspace[e2^v, [actl\_depart(uk160)]]] \end{aligned}$ \label{subs:11} \end{examps} \section{Interrogatives} \label{unb_dep} So far, this chapter has considered mainly assertions (e.g.\ \pref{unb:1}). (The reader is reminded that assertions are treated as yes/no questions; e.g.\ \pref{unb:1} is treated as \pref{unb:3}.) I now explain how the \textsc{Hpsg}\xspace version of this thesis copes with questions (e.g.\ \pref{unb:3} -- \pref{unb:8}). \begin{examps} \item Tank 2 was empty. \label{unb:1} \item Was tank 2 empty? \label{unb:3} \item Did J.Adams inspect BA737? \label{unb:4} \item Which tank was empty? \label{unb:5} \item Who inspected BA737? \label{unb:6} \item What did J.Adams inspect? \label{unb:7} \item When did J.Adams inspect BA737? \label{unb:8} \end{examps} Yes/no questions (e.g.\ \pref{unb:3}, \pref{unb:4}) constitute no particular problem. \textsc{Hpsg}\xspace's schemata allow auxiliary verbs to be used in sentence-initial positions, and cause \pref{unb:3} to receive the same formula (shown in \pref{unb:9.1}) as \pref{unb:9}. In both cases, the same lexical signs are used. Similar comments apply to \pref{unb:4} and \pref{unb:10}, which are mapped to \pref{unb:11}. \begin{examps} \item Tank 2 was empty. \label{unb:9} \item $\ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]$ \label{unb:9.1} \item J.Adams did inspect BA737. \label{unb:10} \item $\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, j\_adams, ba737)]]$ \label{unb:11} \end{examps} The interrogative \qit{which} is treated syntactically as a determiner of (non-predicative) noun phrases. The sign of \qit{which} is the same as the sign of \qit{the} of \pref{nps:4}, except that it introduces an interrogative quantifier rather than an existential one. For example, \pref{unb:5} is analysed syntactically in the same way as \pref{unb:12} (punctuation is ignored). However, the formula of \pref{unb:5} (shown in \pref{unb:14}) contains an additional interrogative quantifier (cf.\ the formula of \pref{unb:12}, shown in \pref{unb:13}). (I assume here that \qit{tank} does not introduce an \ensuremath{\mathit{Ntense}}\xspace. The \qit{a} of \pref{unb:12} introduces an existential quantifier which is removed during the extraction of \pref{unb:13} from the sign of \pref{unb:12}, as discussed in section \ref{extraction_hpsg}.) \begin{examps} \item A tank was empty. \label{unb:12} \item $tank(tk^v) \land \ensuremath{\mathit{Past}}\xspace[e^v, empty(tk^v)]$ \label{unb:13} \item $?tk^v \; tank(tk^v) \land \ensuremath{\mathit{Past}}\xspace[e^v, empty(tk^v)]$ \label{unb:14} \end{examps} The interrogative \qit{who} is treated syntactically as a non-predicative noun-phrase. Its sign, shown in \pref{unb:15}, introduces an interrogative quantifier. \begin{examps} \avmoptions{active} \item \begin{avm} [\avmspan{phon \; \<\fval who\>} \\ synsem|loc & [cat & [head & \osort{noun}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & \sort{person\_ent}{ [tvar & $+$]} \\ restr & \{\}]}@1] \\ \avmspan{qstore \; \{\sort{quant}{ [det & interrog \\ restind & @1]}\}}] \end{avm} \label{unb:15} \end{examps} \pref{unb:6} is analysed syntactically in the same way as \pref{unb:16}. The sign of \qit{who}, however, gives rise to an interrogative quantifier in the formula of \pref{unb:6} (shown in \pref{unb:18}), which is not present in the formula of \pref{unb:16} (shown in \pref{unb:17}). The interrogative \qit{what} is treated similarly. \begin{examps} \item J.Adams inspected BA737. \label{unb:16} \item $\ensuremath{\mathit{Past}}\xspace[\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, j\_adams, ba737)]]$ \label{unb:17} \item $?wh^v \; \ensuremath{\mathit{Past}}\xspace[\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, wh^v, ba737)]]$ \label{unb:18} \end{examps} The \textsc{Hpsg}\xspace version of this thesis admits questions like \pref{unb:19}, which are unacceptable in most contexts. \pref{unb:19} is licensed by the same syntactic analysis that allows \pref{unb:20}, and receives the same formula as \pref{unb:21}. \begin{examps} \item \odd Did J.Adams inspect which flight? \label{unb:19} \item Did J.Adams inspect a flight? \label{unb:20} \item Which flight did J.Adams inspect? \label{unb:21} \end{examps} Questions like \pref{unb:21}, where the interrogative refers to the object of the verb, are treated using \textsc{Hpsg}\xspace's unbounded-dependencies mechanisms (more precisely, using the {\feat slash} feature; see chapter 4 of \cite{Pollard2}).\footnote{Pollard and Sag also reserve a {\feat que} feature, which is supposed to be used in the treatment of interrogatives. They provide virtually no information on the role of {\feat que}, however, pointing to \cite{Ginzburg1992} where {\feat que} is used in a general theory of interrogatives. Ginzburg's theory is intended to address issues well beyond the scope of this thesis (e.g.\ the relation between a question and the facts that can be said to \emph{resolve} that question; see also \cite{Ginzburg1995}, \cite{Ginzburg1995b}). {\feat que} is not used in this thesis.} Roughly speaking, \pref{unb:21} is analysed as being a form of \pref{unb:19}, where the object \qit{which flight} has moved to the beginning of the question. \textsc{Hpsg}\xspace's unbounded-dependencies mechanisms will not be discussed here (see \cite{Pollard2}; the prototype \textsc{Nlitdb}\xspace uses the traceless analysis of unbounded dependencies, presented in chapter 9 of \cite{Pollard2}). The present treatment of interrogatives allows questions with multiple interrogatives, like \pref{unb:24} which receives \pref{unb:24.1}. (\pref{unb:24} is parsed in the same way as \pref{unb:25}.) Unfortunately, it also allows ungrammatical questions like \pref{unb:26}, which is treated as a version of \pref{unb:24} where the \qit{what} complement has moved to the beginning of the sentence. (\pref{unb:26} receives \pref{unb:24}.) \begin{examps} \item Who inspected what. \label{unb:24} \item $?w1^v \; ?w2^v \; \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w1^v, w2^v)]]$ \label{unb:24.1} \item J.Adams inspected BA737. \label{unb:25} \item \bad What who inspected. \label{unb:26} \end{examps} The interrogative \qit{when} of \pref{unb:27} is treated as a temporal-adverbial modifier of finite sentences. \pref{unb:28} shows the sign of \qit{when} that is used in \pref{unb:27}. \pref{unb:28} causes \pref{unb:27} to receive \pref{unb:27.1}. \avmoptions{active, center} \begin{examps} \item When was tank 2 empty? \label{unb:27} \item $?_{mxl}w^v \; \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]$ \label{unb:27.1} \item \setbox\avmboxa=\hbox{\begin{avm} [mod & \feat s[vform {\fval fin}]:@1 \\ \avmspan{mod|loc|cat|aspect \; @2}] \end{avm}} \setbox\avmboxb=\hbox{\begin{avm} [det & interrog\_mxl \\ restind & [index & \sort{temp\_ent}{ [tvar & $+$]} \\ restr & \{\}]] \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval when\>} \\ synsem|loc & [cat & [head & \box\avmboxa \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> \\ aspect & @2]\\ cont & @1] \\ \avmspan{qstore \; \{\box\avmboxb\}} ] \end{avm} \label{unb:28} \end{examps} \pref{unb:28} introduces interrogative-maximal quantifiers whose variables ($w^v$ in \pref{unb:27.1}) do not appear elsewhere in the formula. The post-processing (to be discussed in section \ref{post_processing}) replaces the variables of interrogative-maximal quantifiers by variables that appear as first arguments of \ensuremath{\mathit{Past}}\xspace or \ensuremath{\mathit{Perf}}\xspace operators. In \pref{unb:27.1}, this would replace $w^v$ by $e^v$, generating a formula that asks for the maximal past periods where tank 2 was empty. There is also a second sign for the interrogative \qit{when} (shown in \pref{unb:32}), that is used in habitual questions like \pref{unb:29}. In \pref{unb:29}, \qit{when} is taken to play the same role as \qit{at 5:00pm} in \pref{unb:30}, i.e.\ it is treated as the prepositional-phrase complement of the habitual \qit{depart} (see section \ref{habituals}), which has moved to the beginning of the sentence via the unbounded-dependencies mechanisms. \avmoptions{active, center} \begin{examps} \item When does BA737 depart (habitually)? \label{unb:29} \item Does BA737 depart (habitually) at 5:00pm? \label{unb:30} \item \setbox\avmboxa=\hbox{\begin{avm} [det & interrog \\ restind & @1] \end{avm}} \begin{avm} [\avmspan{phon \; \<\fval when\>} \\ synsem|loc & [cat & [head & \osort{prep}{ [prd & $-$]} \\ spr & \<\> \\ subj & \<\> \\ comps & \<\> ]\\ cont & \osort{nom\_obj}{ [index & \sort{gappy\_partng}{ [tvar & $+$]} \\ restr & \{\}]}@1] \\ \avmspan{qstore \; \{\box\avmboxa\}}] \end{avm} \label{unb:32} \end{examps} In the simple past \pref{unb:33}, both the (state) habitual homonym of \qit{to depart} (that of \pref{hab:7}, which requires a prepositional phrase complement) and the (point) non-habitual homonym (that of \pref{hab:8}, which requires no complement) can be used. Hence, \qit{when} can be either a prepositional-phrase complement of the habitual \qit{depart} (using \pref{unb:32}), or a temporal modifier of the non-habitual sentence \qit{did BA737 depart} (using \pref{unb:28}). This gives rise to \pref{unb:34} and \pref{unb:35}, which correspond to the habitual and non-habitual readings of \pref{unb:33} (the $w^v$ of \pref{unb:35} would be replaced by $e^v$ during the post-processing). \begin{examps} \item When did BA737 depart? \label{unb:33} \item $?w^v \; \ensuremath{\mathit{Past}}\xspace[e^v, hab\_departs\_at(ba737, w^v)]$ \label{unb:34} \item $?_{mxl}w^v \; \ensuremath{\mathit{Past}}\xspace[e^v, actl\_depart(ba737)]$ \label{unb:35} \end{examps} \section{Multiple temporal modifiers} \label{hpsg:mult_mods} The framework of this thesis currently runs into several problems in sentences with multiple temporal modifiers. This section discusses these problems. \paragraph{Both preceding and trailing temporal modifiers:} Temporal modifiers are allowed to either precede or follow finite sentences (section \ref{fronted}). When a finite sentence is modified by both a preceding and a trailing temporal modifier (as in \pref{mults:1}), two parses are generated: one where the trailing modifier attaches first to the sentence (as in \pref{mults:2}), and one where the preceding modifier attaches first (as in \pref{mults:4}). In most cases, this generates two semantically equivalent formulae (\pref{mults:3} and \pref{mults:5} in the case of \pref{mults:1}). A mechanism is needed to eliminate one of the two formulae. \begin{examps} \item Yesterday BA737 was at gate 2 for two hours. \label{mults:1} \item Yesterday [[BA737 was at gate 2] for two hours.] \label{mults:2} \item $\ensuremath{\mathit{At}}\xspace[yesterday, \ensuremath{\mathit{For}}\xspace[hour^c, 2, \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(ba737, gate2)]]]$ \label{mults:3} \item {[}Yesterday [BA737 was at gate 2]] for two hours. \label{mults:4} \item $\ensuremath{\mathit{For}}\xspace[hour^c, 2, \ensuremath{\mathit{At}}\xspace[yesterday, \ensuremath{\mathit{Past}}\xspace[e^v, located\_at(ba737, gate2)]]]$ \label{mults:5} \end{examps} \paragraph{Multiple temporal modifiers and anaphora:} Another problem is that a question like \pref{mults:10} is mapped to \pref{mults:11}. (I assume here that \qit{flight} does not introduce an \ensuremath{\mathit{Ntense}}\xspace.) The problem with \pref{mults:11} is that it does not require $fv^v$ to be the particular 5:00pm-minute of 2/11/95. \pref{mults:11} requires the flight to have arrived on 2/11/95 and after an arbitrary 5:00pm-minute (e.g.\ the 5:00pm-minute of 1/11/95). In effect, this causes the \qit{after 5:00pm} to be ignored. \begin{examps} \item Which flight arrived after 5:00pm on 2/11/95? \label{mults:10} \item $?fl^v \; flight(fl^v) \land \ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land$\\ $\ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{After}}\xspace[fv^v, \ensuremath{\mathit{Past}}\xspace[e^v, arrive(fl^v)]]]$ \label{mults:11} \end{examps} This problem seems related to the need for temporal anaphora resolution mechanisms (section \ref{temporal_anaphora}). In \pref{mults:16}, for example, the user most probably has a particular (contextually-salient) 5:00pm-minute in mind, and an anaphora resolution mechanism is needed to determine that minute. A similar mechanism could be responsible for reasoning that in \pref{mults:10} the most obvious contextually salient 5:00pm-minute is that of 2/11/95. \begin{examps} \item Which tanks were empty before/at/after 5:00pm? \label{mults:16} \end{examps} \paragraph{Culminating activity with both punctual and period adverbial:} A further problem appears when a culminating activity is modified by both a punctual and a period adverbial.\footnote{The problems of this section that involve period adverbials also arise when temporal subordinate clauses are used instead of period adverbials.} The problem is that, unlike what one would expect, \pref{mults:18} and \pref{mults:19} do not receive equivalent \textsc{Top}\xspace formulae. (I assume here that \qit{to repair} is classified as culminating activity verb.) \begin{examps} \item J.Adams repaired fault 2 at 5:00pm on 2/11/95. \label{mults:18} \item J.Adams repaired fault 2 on 2/11/95 at 5:00pm. \label{mults:19} \end{examps} In \pref{mults:18}, the punctual adverbial \qit{at 5:00pm} modifies the culminating activity sentence \qit{J.Adams repaired fault 2}. The punctual adverbial causes \qit{J.Adams repaired fault 2 at 5:00pm} to become a point (see table \vref{punctual_adverbials_table}). Two formulae are generated: one that requires the repair to have started at 5:00pm, and one that requires the repair to have been completed at 5:00pm. \qit{On 2/11/95} then modifies the point expression \qit{J.Adams repaired fault 2 at 5:00pm}. This leads to \pref{mults:22} and \pref{mults:23}. In \pref{mults:22} the repair \emph{starts} at the 5:00pm-minute of 2/11/95, while in \pref{mults:23} the repair is \emph{completed} at the 5:00pm-minute of 2/11/95. (The first reading is easier to accept in \qit{J.Adams inspected BA737 at 5:00pm on 2/11/95}.) \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}},$\\ $\ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Begin}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:22} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}},$\\ $\ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:23} \end{examps} (A digression: this example also demonstrates why punctual adverbials are taken to trigger an aspectual shift to point; see section \ref{point_adverbials}. Without this shift, the aspectual class of \qit{J.Adams repaired fault 2 at 5:00pm} would be culminating activity, and the \qit{on} signs of section \ref{hpsg:per_advs} would lead to the additional formulae of \pref{mults:22f} and \pref{mults:23f}. These are equivalent to \pref{mults:22} and \pref{mults:23} respectively.) \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, $\\ $\ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{Begin}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]]$ \label{mults:22f} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}},$\\ $\ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{At}}\xspace[fv^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]]$ \label{mults:23f} \end{examps} In \pref{mults:19}, \qit{J.Adams repaired fault 2} is first modified by the period adverbial \qit{on 2/11/95}. Two formulae (shown in \pref{mults:24} and \pref{mults:25}) are generated. \pref{mults:24} requires the repair to simply reach its completion on 2/11/95, while \pref{mults:25} requires the repair to both start and reach its completion on 2/11/95. In the first case (where \pref{mults:24} is generated), the aspectual class of \qit{J.Adams repaired fault 2 on 2/11/95} becomes point, while in the other case the aspectual class remains culminating activity (see also table \vref{period_adverbials_table}). \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]$ \label{mults:24} \item $\ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]$ \label{mults:25} \end{examps} In the case of \pref{mults:24}, where the aspectual class of \qit{J.Adams repaired fault 2 on 2/11/95} is point, the signs of section \ref{hpsg:punc_adv} lead to \pref{mults:26}, while in the case of \pref{mults:25}, they lead to \pref{mults:27} and \pref{mults:28}. \begin{examps} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g, fv^v] \land \ensuremath{\mathit{At}}\xspace[fv^v,$\\ $\ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:26} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g] \land \ensuremath{\mathit{At}}\xspace[fv^v,$\\ $\ensuremath{\mathit{Begin}}\xspace[ \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:27} \item $\ensuremath{\mathit{Part}}\xspace[\text{\textit{5:00pm}}^g] \land \ensuremath{\mathit{At}}\xspace[fv^v,$\\ $\ensuremath{\mathit{End}}\xspace[ \ensuremath{\mathit{At}}\xspace[\text{\textit{2/11/95}}, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:28} \end{examps} Hence, \pref{mults:18} receives two formulae (\pref{mults:22} and \pref{mults:23}), while \pref{mults:19} receives three (\pref{mults:26} -- \pref{mults:28}). \pref{mults:26} is equivalent to \pref{mults:23}. They both require the repair to reach its completion within the 5:00pm-minute of 2/11/95. Unlike what one might expect, however, \pref{mults:27} is not equivalent to \pref{mults:22}. \pref{mults:27} requires a past period that covers exactly the whole repair (from start to completion) to fall within 2/11/95, and the beginning of that period to fall within some 5:00pm-minute. This means that the repair must start at the 5:00pm-minute of 2/11/95 (as in \pref{mults:22}), but it also means that the repair must reach its completion within 2/11/95 (this is not a requirement in \pref{mults:22}). Also, unlike what one might expect, \pref{mults:28} is not equivalent to \pref{mults:23} and \pref{mults:26}. \pref{mults:28} requires the repair to reach its completion within the 5:00pm-minute of 2/11/95 (as in \pref{mults:23} and \pref{mults:26}), but it also requires the repair to start within 2/11/95 (which is not a requirement in \pref{mults:23} and \pref{mults:26}). The differences in the number and semantics of the generated formulae in \pref{mults:18} and \pref{mults:19} lead to differences in the behaviour of the \textsc{Nlitdb}\xspace that are difficult to explain to the user. A tentative solution is to adopt some mechanism that would reorder the temporal modifiers, so that the punctual adverbial attaches before the period one. This would reverse the order of \qit{on 2/11/95} and \qit{at 5:00pm} in \pref{mults:19}, and would cause \pref{mults:19} to be treated in the same way as \pref{mults:18} (i.e.\ to be mapped to \pref{mults:22} and \pref{mults:23}; these seem to capture the most natural readings of \pref{mults:18} and \pref{mults:19}). \paragraph{Culminating activity and multiple period adverbials:} A further problem is that a sentence like \pref{mults:30}, where a culminating activity is modified by two period adverbials, receives three formulae, shown in \pref{mults:32} -- \pref{mults:31}. It turns out that \pref{mults:33} is equivalent to \pref{mults:31}, and hence one of the two should be eliminated. \begin{examps} \item J.Adams repaired fault 2 in June in 1992. \label{mults:30} \item $\ensuremath{\mathit{Part}}\xspace[june^g, j^v] \land \ensuremath{\mathit{At}}\xspace[1992,$\\ $\ensuremath{\mathit{At}}\xspace[j^v, \ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:32} \item $\ensuremath{\mathit{Part}}\xspace[june^g, j^v] \land \ensuremath{\mathit{At}}\xspace[1992,$\\ $\ensuremath{\mathit{End}}\xspace[\ensuremath{\mathit{At}}\xspace[j^v, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]]$ \label{mults:33} \item $\ensuremath{\mathit{Part}}\xspace[june^g, j^v] \land \ensuremath{\mathit{At}}\xspace[1992,$\\ $\ensuremath{\mathit{At}}\xspace[j^v, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[repairing(occr^v, j\_adams, fault2)]]]]$ \label{mults:31} \end{examps} A period adverbial combining with a culminating activity can either insert an \ensuremath{\mathit{End}}\xspace operator and cause an aspectual shift to point, or insert no \ensuremath{\mathit{End}}\xspace and leave the aspectual class unchanged (see section \ref{hpsg:per_advs}). In the case where \pref{mults:32} is generated, \qit{in June} inserts an \ensuremath{\mathit{End}}\xspace and changes the aspectual class to point. This does not allow \qit{in 1992} (which attaches after \qit{in June}) to insert an \ensuremath{\mathit{End}}\xspace, because period adverbials combining with points are not allowed to insert \ensuremath{\mathit{End}}\xspace{s} (the \qit{on} sign of \pref{pupe:29} cannot be used with points). In the cases where \pref{mults:33} or \pref{mults:31} are generated, \qit{in June} does not insert an \ensuremath{\mathit{End}}\xspace, and the aspectual class remains culminating activity. \qit{In 1992} can then insert an \ensuremath{\mathit{End}}\xspace (as in \pref{mults:33}) or not (as in \pref{mults:31}). \pref{mults:31} requires the whole repair to be located within a June and 1992 (i.e.\ within the June of 1992). \pref{mults:32} is weaker: it requires only the completion point of the repair to be located within the June of 1992. Finally, \pref{mults:33} requires the whole of the repair to be located within a June, and the completion point of the repair to fall within 1992. This is equivalent to requiring the whole of the repair to fall within the June of 1992, i.e.\ \pref{mults:33} is equivalent to \pref{mults:31}, and one of the two should be eliminated. \section{Post-processing} \label{post_processing} The parsing maps each English question to an \textsc{Hpsg}\xspace sign (or multiple signs, if the parser understands the question to be ambiguous). From that sign, a \textsc{Top}\xspace formula is extracted as discussed in section \ref{extraction_hpsg}. The extracted formula then undergoes an additional post-processing phase. This is a collection of minor transformations, discussed below, that cannot be carried out easily during the parsing. \paragraph{Removing Culms:} \pref{post:2} shows the \textsc{Top}\xspace formula that is extracted from the sign of \pref{post:1}. As discussed in section \ref{duration_adverbials}, \pref{post:2} does not represent correctly \pref{post:1}, because \pref{post:2} requires the taxiing to have been completed. In contrast, as discussed in section \ref{for_adverbials}, the \qit{for~\dots} adverbial of \pref{post:1} cancels the normal implication of \qit{BA737 taxied to gate 2} that the taxiing must have been completed. To express correctly \pref{post:1}, the \ensuremath{\mathit{Culm}}\xspace of \pref{post:2} has to be removed. \begin{examps} \item BA737 taxied to gate 2 for five minutes. \label{post:1} \item $\ensuremath{\mathit{For}}\xspace[minute^c, 5, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[taxiing\_to(ba737, gate2)]]]$ \label{post:2} \end{examps} A first solution would be to remove during the post-processing any \ensuremath{\mathit{Culm}}\xspace operator that is within the scope of a \ensuremath{\mathit{For}}\xspace operator. The problem with this approach is that duration \qit{in~\dots} adverbials also introduce \ensuremath{\mathit{For}}\xspace operators (see section \ref{duration_adverbials}), but unlike \qit{for~\dots} adverbials they do not cancel the implication that the completion must have been reached. For example, the formula extracted from the sign of \pref{post:5} is \pref{post:2}. In this case, \pref{post:2} is a correct rendering of \pref{post:5} (because \pref{post:5} \emph{does} imply that BA737 reached gate 2), and hence the \ensuremath{\mathit{Culm}}\xspace operator should not be removed. To overcome this problem, the prototype \textsc{Nlitdb}\xspace attaches to each \ensuremath{\mathit{For}}\xspace operator a flag showing whether it was introduced by a \qit{for~\dots} or an \qit{in~\dots} adverbial. Only \ensuremath{\mathit{For}}\xspace operators introduced by \qit{for~\dots} adverbials cause \ensuremath{\mathit{Culm}}\xspace operators within their scope to be removed. \begin{examps} \item BA737 taxied to gate 2 in five minutes. \label{post:5} \end{examps} The post-processing also removes any \ensuremath{\mathit{Culm}}\xspace operator from within the first argument of an \ensuremath{\mathit{At}}\xspace operator. As explained in section \ref{hpsg:subordinates}, this is needed to express correctly \qit{while~\dots} clauses. \paragraph{$\mathbf{?_{mxl}}$ quantifiers:} As noted in section \ref{unb_dep}, before the post-processing the variables of interrogative-maximal quantifiers introduced by \qit{when} do not occur elsewhere in their formulae. For example, \pref{post:9} and \pref{post:6} are extracted from the signs of \pref{post:8} and \pref{post:6}. In both formulae, $w^v$ occurs only immediately after the $?_{mxl}$. \begin{examps} \item When was J.Adams a manager? \label{post:8} \item $?_{mxl}w^v \; \ensuremath{\mathit{Past}}\xspace[e^v, manager(j\_adams)]$ \label{post:9} \item When while BA737 was circling was runway 2 open? \label{post:6} \item $?_{mxl}w^v \; \ensuremath{\mathit{At}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, circling(ba737)], \ensuremath{\mathit{Past}}\xspace[e2^v, open(runway2)]]$ \label{post:7} \end{examps} During the post-processing, the variables of interrogative-maximal quantifiers are replaced by variables that appear as first arguments of \ensuremath{\mathit{Past}}\xspace or \ensuremath{\mathit{Perf}}\xspace operators, excluding \ensuremath{\mathit{Past}}\xspace and \ensuremath{\mathit{Perf}}\xspace operators that are within first arguments of \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace operators. In \pref{post:9}, this causes $w^v$ to be replaced by $e^v$. The resulting formula asks for the maximal past periods where J.Adams was a manager. Similarly, the $w^v$ of \pref{post:7} is replaced by $e2^v$. The resulting formula asks for the maximal past periods $e2^v$, such that runway 2 was open at $e2^v$, and $e2^v$ is a subperiod of a period $e1^v$ where BA737 was circling. In \pref{post:7}, $w^v$ cannot be replaced by $e1^v$, because $\ensuremath{\mathit{Past}}\xspace[e1^v, circling(ba737)]$ is within the first argument of an \ensuremath{\mathit{At}}\xspace. \ensuremath{\mathit{Past}}\xspace and \ensuremath{\mathit{Perf}}\xspace operators located within first arguments of \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace operators are excluded, to avoid interpreting \qit{when} as referring to the time where the situation of a subordinate clause held (formulae that express subordinate clauses end-up within first arguments of \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace operators). The interrogative \qit{when} always refers to the situation of the main clause. For example, \pref{post:6} cannot be asking for maximal periods where BA737 was circling that subsume periods where runway 2 was open (this would be the meaning of \pref{post:7} if $w^v$ were replaced by $e1^v$). When the main clause is in the past perfect, this arrangement allows the variable of $?_{mxl}$ to be replaced by either the first argument of the main-clause's \ensuremath{\mathit{Past}}\xspace operator, or the first argument of the main-clause's \ensuremath{\mathit{Perf}}\xspace operator. \pref{post:11}, for example, shows the formula extracted from the sign of \pref{post:10}. The post-processing generates two formulae: one where $w^v$ is replaced by $e1^v$, and one where $w^v$ is replaced by $e2^v$. The first one asks for what section \ref{point_adverbials} called the ``consequent period'' of the inspection (the period from the end of the inspection to the end of time). The second one asks for the time of the actual inspection. \begin{examps} \item When had J.Adams inspected BA737? \label{post:10} \item $?_{mxl}w^v \; \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, j\_adams, ba737)]]]$ \label{post:11} \end{examps} \paragraph{Ntense operators:} As noted in section \ref{non_pred_nps}, when extracting \textsc{Top}\xspace formulae from signs, if an \ensuremath{\mathit{Ntense}}\xspace operator is encountered and the sign contains no definite indication that the first argument of the \ensuremath{\mathit{Ntense}}\xspace should be $now^*$, in the extracted formula the first argument of the \ensuremath{\mathit{Ntense}}\xspace becomes a variable. That variable does not occur elsewhere in the extracted formula. Assuming, for example, that the (non-predicative) \qit{queen} introduces an \ensuremath{\mathit{Ntense}}\xspace, the formula extracted from the sign of \pref{post:14} is \pref{post:15}. The $t^v$ of the \ensuremath{\mathit{Ntense}}\xspace does not occur elsewhere in \pref{post:15}. \begin{examps} \item The queen was in Rome. \label{post:14} \item $\ensuremath{\mathit{Ntense}}\xspace[t^v, queen(q^v)] \land \ensuremath{\mathit{Past}}\xspace[e1^v, located\_at(q^v, rome)]$ \label{post:15} \end{examps} During the post-processing, variables appearing as first arguments of \ensuremath{\mathit{Ntense}}\xspace{s} give rise to multiple formulae, where the first arguments of the \ensuremath{\mathit{Ntense}}\xspace{s} are replaced by $now^*$ or by first arguments of \ensuremath{\mathit{Past}}\xspace or \ensuremath{\mathit{Perf}}\xspace operators. In \pref{post:15}, for example, the post-processing generates two formulae: one where $t^v$ is replaced by $now^*$ (queen at the speech time), and one where $t^v$ is replaced by $e^v$ (queen when in Rome). In \pref{post:17} (the formula extracted from the sign of \pref{post:16}), there is no \ensuremath{\mathit{Past}}\xspace or \ensuremath{\mathit{Perf}}\xspace operator, and hence $t^v$ can only become $now^*$. This captures the fact that the \qit{queen} in \pref{post:16} most probably refers to the queen of the speech time. \begin{examps} \item The queen is in Rome. \label{post:16} \item $\ensuremath{\mathit{Ntense}}\xspace[t^v, queen(q^v)] \land \ensuremath{\mathit{Pres}}\xspace[located\_at(q^v, gate2)]$ \label{post:17} \end{examps} In \pref{post:19} (the formula extracted from the sign of \pref{post:18}), the post-processing leads to three formulae, where $t^v$ is replaced by $now^*$ (queen at speech time), $e2^v$ (queen during the visit), or $e1^v$ (queen at a ``reference time'' after the visit). \begin{examps} \item The queen had visited Rome. \label{post:18} \item $\ensuremath{\mathit{Ntense}}\xspace[t^v, queen(q^v)] \land \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, visiting(q^v, rome)]]$ \label{post:19} \end{examps} \section{Summary} This chapter has shown how \textsc{Hpsg}\xspace can be used to translate English questions directed to a \textsc{Nlitdb}\xspace to appropriate \textsc{Top}\xspace formulae. During the parsing, each question receives one or more \textsc{Hpsg}\xspace signs, from which \textsc{Top}\xspace formulae are extracted. The extracted formulae then undergo an additional post-processing phase, which leads to formulae that capture the semantics of the original English questions. Several modifications were made to \textsc{Hpsg}\xspace. The main modifications were: (i) \textsc{Hpsg}\xspace features and sorts that are intended to account for phenomena not examined in this thesis (e.g.\ pronouns, relative clauses, number agreement) were dropped. (ii) The quantifier storage mechanism of \textsc{Hpsg}\xspace was replaced by a more primitive one, that does not allow quantifiers to be unstored during the parsing; the semantics principle was modified accordingly. (iii) An {\feat aspect} feature was added, along with a principle that controls its propagation. (iv) The possible values of {\feat cont} and {\feat qstore} were modified, to represent \textsc{Top}\xspace expressions rather than situation-theory constructs. (v) A hierarchy of world-entity types was mounted under the {\srt ind}\/ sort; this is used to disambiguate sentences, and to block semantically ill-formed ones. (vi) New lexical signs and lexical rules were introduced to cope with temporal linguistic mechanisms (verb tenses, temporal adverbials, temporal subordinate clauses, etc.). Apart from these modifications, the \textsc{Hpsg}\xspace version of this thesis follows closely \cite{Pollard2}. \chapter{From TOP to TSQL2} \label{tdb_chapter} \proverb{Time is money.} \section{Introduction} This chapter describes the translation from \textsc{Top}\xspace to \textsc{Tsql2}\xspace. The discussion starts with an introduction to \textsc{Tsql2}\xspace and the version of the relational model on which \textsc{Tsql2}\xspace is based. This thesis adopts some modifications to \textsc{Tsql2}\xspace. These are described next, along with some minor alterations in the \textsc{Top}\xspace definition of chapter \ref{TOP_chapter}. The translation from \textsc{Top}\xspace to \textsc{Tsql2}\xspace requires \textsc{Top}\xspace's model to be linked to the database; this is explained next. The translation is carried out by a set of rules. I explore formally the properties that these rules must possess for the translation to be correct, and I describe the intuitions behind the design of the rules. An illustration of how some of the rules work is also given. The full set of the translation rules, along with a proof that they possess the necessary properties, is given in appendix \ref{trans_proofs}. The chapter ends with a discussion of related work and reflections on how the generated \textsc{Tsql2}\xspace code could be optimised. \section{An introduction to TSQL2} \label{TSQL2_intro} This section introduces \textsc{Tsql2}\xspace and the version of the relational model on which \textsc{Tsql2}\xspace is based. Some definitions that are not part of the \textsc{Tsql2}\xspace documentation are also given; these will be used in following sections. I note that although \cite{TSQL2book} defines \textsc{Tsql2}\xspace's syntax rigorously, the semantics of the language is defined very informally, with parts of the semantics left to the intuition of the reader. There are also some inconsistencies in the \textsc{Tsql2}\xspace definition (several of these were pointed out in \cite{Androutsopoulos1995b}). \subsection{The traditional relational model} \label{relational} As explained in section \ref{tdbs_general}, the traditional relational model stores information in relations, which can be thought of as tables. For example, $salaries$ below is a relation showing the current salaries of a company's employees. $salaries$ has two \emph{attributes} (intuitively, columns), $employee$ and $salary$. The \emph{tuples of the relation} are intuitively the rows of the table ($salaries$ has three tuples). \adbtable{2}{|l|l|}{$salaries$} {$employee$ & $salary$ } {$J.Adams$ & $17000$ \\ $T.Smith$ & $19000$ \\ $G.Papas$ & $14500$ } I adopt a set-theoretic definition of relations (see section 2.3 of \cite{Ullman} for alternative approaches). A set of attributes ${\cal D}_A$ \index{da@${\cal D}_A$ (set of all attributes)} is assumed (e.g.\ $employee$ and $salary$ are elements of ${\cal D}_A$). A \emph{relation schema} is an ordered tuple of one or more attributes (e.g.\ $\tup{employee, salary}$). A set of \emph{domains} ${\cal D}_D = \{D_1, D_2, \dots, D_{n_D}\}$ \index{dd@${\cal D}_D$ (set of all domains)} is also assumed. Each element $D_i$ of ${\cal D}_D$ is itself a set. For example, $D_1$ may contain all strings, $D_2$ all positive integers, etc. Each attribute (element of ${\cal D}_A$) is assigned a domain (element of ${\cal D}_D$). $D(A)$ \index{d()@$D(A)$ (domain of the attribute $A$)} denotes the domain of attribute $A$. $D$ \index{d@$D$ (universal domain)} on its own refers to the \emph{universal domain}, the union of all $D_i \in {\cal D}_D$. A \emph{relation} over a relation schema $R = \tup{A_1, A_2, \dots, A_n}$ is a subset of $D(A_1) \times D(A_2) \times \dots \times D(A_n)$, where $\times$ denotes the cartesian product, and $D(A_1)$, $D(A_2)$,~\dots, $D(A_n)$ are the domains of the attributes $A_1$, $A_2$,~\dots, $A_n$ respectively. That is, a relation over $R$ is a set of tuples of the form $\tup{v_1, v_2, \dots, v_n}$, where $v_1 \in D(A_1)$, $v_2 \in D(A_2)$, \dots, $v_n \in D(A_n)$. In each tuple $\tup{v_1, v_2, \dots, v_n}$, $v_1$ is the \emph{attribute value} of $A_1$, $v_2$ is the attribute value of $A_2$, etc. The universal domain $D$ is the set of all possible attribute values. Assuming, for example, that $employee, salary \in {\cal D}_A$, that $D_1$ and $D_2$ are as in the previous paragraph, and that $employee$ and $salary$ are assigned $D_1$ and $D_2$, $r$ below is a relation over $\tup{employee, salary}$. ($r$ is a mathematical representation of $salaries$ above.) On its own, ``relation'' will be used to refer to a relation over any relation schema. \[ r = \{\tup{J.Adams, 17000}, \tup{T.Smith, 19000}, \tup{G.Papas, 14500}\} \] The \emph{arity} of a relation over $R$ is the number of attributes in $R$ (e.g.\ the arity of $r$ is 2). The \emph{cardinality} of a relation is the number of tuples it contains (the cardinality of $r$ is 3). A relational \emph{database} is a set of relations (more elaborate definitions are possible, but this is sufficient for our purposes). I assume that every element of $D$ (universal domain) denotes an object in the modelled world. (``Object in the world'' is used here with a very loose meaning, that covers qualifications of employees, salaries, etc.) \ensuremath{\mathit{OBJS^{db}}}\xspace \index{objsdb@\ensuremath{\mathit{OBJS^{db}}}\xspace (\textsc{Bcdm}\xspace's world objects)} is the set of all the world objects that are each denoted by a single element of $D$. (Some world objects may be represented in the database as collections of elements of $D$, e.g.\ as whole tuples. \ensuremath{\mathit{OBJS^{db}}}\xspace contains only world objects that are denoted by \emph{single} elements of $D$.) I also assume that a function $f_D : D \mapsto \ensuremath{\mathit{OBJS^{db}}}\xspace$ \index{fd@$f_D()$ (maps attribute values to world objects)} is available, that maps each element $v$ of $D$ to the world object denoted by $v$. $f_D$ reflects the semantics assigned to the attribute values by the people who use the database. In practice, an element of $D$ may denote different world objects when used as the value of different attributes. For example, $15700$ may denote a salary when used as the value of $salary$, and a part of an engine when used as the value of an attribute $part\_no$. Hence, the value of $f_D$ should also depend on the attribute where the element of $D$ is used, i.e.\ it should be a function $f_D : D \times {\cal D}_A \mapsto \ensuremath{\mathit{OBJS^{db}}}\xspace$. For simplicity, I overlook this detail. I also assume that $f_D$ is 1-1 (injective), i.e.\ that every element of $D$ denotes a different world object. In practice, $f_D$ may not be 1-1: the database may use two different attribute values (e.g.\ $dpt3$ and $sales\_dpt$) to refer to the same world object. The \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation could be formulated without assuming that $f_D$ is 1-1. This assumption, however, bypasses uninteresting details. By the definition of \ensuremath{\mathit{OBJS^{db}}}\xspace, any element of \ensuremath{\mathit{OBJS^{db}}}\xspace is a world object denoted by some element of $D$. That is, for every $o \in \ensuremath{\mathit{OBJS^{db}}}\xspace$, there is a $v \in D$, such that $f_D(v) = o$, i.e.\ $f_D$ is also surjective. Since $f_D$ is both 1-1 and surjective, the inverse mapping \ensuremath{f_D^{-1}}\xspace is a function, and \ensuremath{f_D^{-1}}\xspace is also 1-1 and surjective. \subsection{TSQL2's model of time} \label{tsql2_time} Like \textsc{Top}\xspace, \textsc{Tsql2}\xspace assumes that time is discrete, linear, and bounded. In effect, \textsc{Tsql2}\xspace models time as consisting of \emph{chronons}. Chronons are the shortest representable units of time, and correspond to \textsc{Top}\xspace's time-points.\footnote{\textsc{Tsql2}\xspace distinguishes between \emph{valid-time chronons}, \emph{transaction-time chronons}, and \emph{bitemporal chronons} (pairs each comprising a valid-time and a transaction-time chronon; see chapter 10 of \cite{TSQL2book}). As noted in section \ref{tdbs_general}, transaction-time is ignored in this thesis. Hence, transaction-time and bitemporal chronons are not used, and ``chronon'' refers to valid-time chronons.} Depending on the \textsc{Tsql2}\xspace implementation, a chronon may represent a nanosecond, a day, or a whole century. Let us call the (implementation-specific) set of chronons \ensuremath{\mathit{CHRONS}}\xspace. \index{chrons@\ensuremath{\mathit{CHRONS}}\xspace (set of all chronons)} Although not stated explicitly, it is clear from the discussion in chapter 6 of \cite{TSQL2book} that $\ensuremath{\mathit{CHRONS}}\xspace \not= \emptyset$, that chronons are ordered by a binary precedence relation (let us call it $\prec^{db}$), and that $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ has the properties of transitivity, irreflexivity, linearity, left and right boundedness, and discreteness (section \ref{temporal_ontology}). I define periods over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ in the same way as periods over $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ (section \ref{temporal_ontology}). A period over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ is a non-empty and convex set of chronons. An instantaneous period over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ is a set that contains a single chronon. $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ \index{periods2@$\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ (set of all periods over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$} and $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ \index{instants2@$\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ (set of all instantaneous periods over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$)} are the sets of all periods and all instantaneous periods over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ respectively. In section \ref{resulting_model}, I set the point structure $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}$ of \textsc{Top}\xspace's model to $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$. Hence, $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ and $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}}$ become $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ and $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$. As in chapter \ref{TOP_chapter}, I write \ensuremath{\mathit{PERIODS}}\xspace \index{periods@$\ensuremath{\mathit{PERIODS}}\xspace$ (set of all periods)} and \ensuremath{\mathit{INSTANTS}}\xspace \index{instants@$\ensuremath{\mathit{INSTANTS}}\xspace$ (set of all instantaneous periods)} to refer to these sets, and $\ensuremath{\mathit{PERIODS}}\xspace^*$ \index{periods*@$\ensuremath{\mathit{PERIODS}}\xspace^*$ ($\ensuremath{\mathit{PERIODS}}\xspace \union \emptyset$)} to refer to $\ensuremath{\mathit{PERIODS}}\xspace \union \{\emptyset\}$. A \emph{temporal element} over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ is a non-empty (but not necessarily convex) set of chronons. $\ensuremath{\mathit{TELEMS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ \index{telems2@$\ensuremath{\mathit{TELEMS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}}$ (set of all temporal elements over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$)} (or simply \ensuremath{\mathit{TELEMS}}\xspace) \index{telems@\ensuremath{\mathit{TELEMS}}\xspace (set of all temporal elements)} is the set of all temporal elements over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$. Obviously, $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{TELEMS}}\xspace$. For every $l \in \ensuremath{\mathit{TELEMS}}\xspace$, $mxlpers(l)$ \index{mxlpers@$mxlpers()$ (maximal periods of a set or temporal element)} is the set of the \emph{maximal periods} of $l$, defined as follows: \begin{align*} mxlpers(l) \defeq \{p \subseteq l \mid & \; p \in \ensuremath{\mathit{PERIODS}}\xspace \text{ and for no } p' \in \ensuremath{\mathit{PERIODS}}\xspace \text{ is it true that } \\ & \; p' \subseteq l \text{ and } p \propsubper p'\} \end{align*} The $mxlpers$ symbol is overloaded. When $l \in \ensuremath{\mathit{TELEMS}}\xspace$, $mxlpers(l)$ is defined as above. When $S$ is a set of periods, $mxlpers(S)$ is defined as in section \ref{temporal_ontology}. \textsc{Tsql2}\xspace supports multiple \emph{granularities}. These correspond to \textsc{Top}\xspace complete partitionings. A granularity can be thought of as a set of periods over $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{db}}$ (called \emph{granules}), such that no two periods overlap, and the union of all the periods is \ensuremath{\mathit{CHRONS}}\xspace. A lattice is used to capture relations between granularities (e.g.\ a year-granule contains twelve month-granules, etc; see chapter 19 of \cite{TSQL2book}). \ensuremath{\mathit{INSTANTS}}\xspace, also called the \emph{granularity of chronons}, is the finest available granularity. \textsc{Tsql2}\xspace allows periods and temporal elements to be specified at any granularity. For example, one may specify that the first day of a period is 25/11/95, and the last day is 28/11/95. If the granularity of chronons is finer than the granularity of days, the exact chronons within 25/11/95 and 28/11/95 where the period starts and ends are unknown. Similarly, if a temporal element is specified at a granularity coarser than \ensuremath{\mathit{INSTANTS}}\xspace, the exact chronon-boundaries of its maximal periods are unknown.\footnote{To bypass this problem, in \cite{Androutsopoulos1995b} periods and temporal elements are defined as sets of granules (of any granularity) rather than sets of chronons.} These are examples of \emph{indeterminate temporal information} (see chapter 18 of \cite{TSQL2book}). Information of this kind is ignored in this thesis. I assume that all periods and temporal elements are specified at the granularity of chronons, and that we know exactly which chronons are or are not included in periods and temporal elements. Granularities other than \ensuremath{\mathit{INSTANTS}}\xspace will be used only to express durations (see below). Finally, \textsc{Tsql2}\xspace uses the term \emph{interval} to refer to a duration (see comments in section \ref{top_intro}). An interval is a number of consecutive granules of some particular granularity (e.g.\ two day-granules, five minute-granules). \subsection{BCDM} \label{bcdm} As noted in section \ref{tdbs_general}, numerous temporal versions of the relational model have been proposed. \textsc{Tsql2}\xspace is based on a version called \textsc{Bcdm}\xspace. Apart from the relations of the traditional relational model (section \ref{relational}), which are called \emph{snapshot relations} in \textsc{Tsql2}\xspace, \textsc{Bcdm}\xspace provides \emph{valid-time relations}, \emph{transaction-time relations}, and \emph{bitemporal relations}. Transaction-time and bitemporal relations are not used in this thesis (see chapter 10 of \cite{TSQL2book}). Valid-time relations are similar to snapshot relations, except that they have a special extra attribute (the \emph{implicit attribute}) that shows when the information of each tuple was/is/will be true. A special domain $D_T \in {\cal D}_D$ \index{dt@$D_T$ (set of all attribute values that denote temporal elements)} is assumed, whose elements denote the elements of \ensuremath{\mathit{TELEMS}}\xspace (temporal elements). For every $v_t \in D_T$, $f_D(v_t) \in \ensuremath{\mathit{TELEMS}}\xspace$; and for every $l \in \ensuremath{\mathit{TELEMS}}\xspace$, $\ensuremath{f_D^{-1}}\xspace(l) \in D_T$. $D_T$ is the domain of the implicit attribute. Since $D_T \in {\cal D}_D$, $D_T \subseteq D$ ($D$ is the union of all the domains in ${\cal D}_D$). The assumptions of section \ref{relational} about $f_D$ still hold: I assume that $f_D$ is an injective and surjective function from $D$ (which now includes $D_T$) to \ensuremath{\mathit{OBJS^{db}}}\xspace. Since the elements of $D_T$ denote all the elements of \ensuremath{\mathit{TELEMS}}\xspace, $D_T \subseteq D$, and \ensuremath{\mathit{OBJS^{db}}}\xspace contains all the objects denoted by elements of $D$, it must be the case that $\ensuremath{\mathit{TELEMS}}\xspace \subseteq \ensuremath{\mathit{OBJS^{db}}}\xspace$. Then, the fact that $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{TELEMS}}\xspace$ (section \ref{tsql2_time}) implies that $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{OBJS^{db}}}\xspace$. A \emph{valid-time relation} $r$ over a relation-schema $R = \tup{A_1, A_2, \dots, A_n}$ is a subset of $D(A_1) \times D(A_2) \times \dots \times D(A_n) \times D_T$, where $D(A_1)$, $D(A_2)$,~\dots, $D(A_n)$ are the domains of $A_1$, $A_2$,~\dots, $A_n$. $A_1$, $A_2$,~\dots, $A_n$ are the \emph{explicit attributes} of $r$. I use the notation $\tup{v_1, v_2, \dots, v_n; v_t}$ to refer to tuples of valid-time relations. If $r$ is as above and $\tup{v_1, v_2, \dots, v_n; v_t} \in r$, then $v_1 \in D(A_1)$, $v_2 \in D(A_2)$,~\dots, $v_n \in D(A_n)$, and $v_t \in D_T$. $v_1$, $v_2$,~\dots, $v_n$ are the \emph{values of the explicit attributes}, while $v_t$ is the \emph{value of the implicit attribute} and the \emph{time-stamp} of the tuple. In snapshot relations, all attributes count as explicit. In the rest of this thesis, ``valid-time relation'' on its own refers to a valid-time relation over any relation-schema. \textsc{Tsql2}\xspace actually distinguishes between \emph{state valid-time relations} and \emph{event valid-time relations} (see chapter 16 of \cite{TSQL2book}). These are intended to model situations that have duration or are instantaneous respectively. This distinction seems particularly interesting, because it appears to capture some facets of the aspectual taxonomy of chapter \ref{linguistic_data}. Unfortunately, it is also one of the most unclear and problematically defined features of \textsc{Tsql2}\xspace. The time-stamps of state and event valid-time relations are supposed to denote ``temporal elements'' and ``instant sets'' respectively. ``Temporal elements'' are said to be unions of periods, while ``instant sets'' simply sets of chronons (see p.314 of \cite{TSQL2book}). This distinction between ``temporal elements'' and ``instant sets'' is problematic. A union of periods is a union of convex sets of chronons, i.e.\ simply a set of chronons. (The union of two convex sets of chronons is not necessarily convex.) Hence, one cannot distinguish between unions of periods and sets of chronons (see also section 2 of \cite{Androutsopoulos1995b}). In section 3.3 of \cite{Androutsopoulos1995b} we also argue that \textsc{Tsql2}\xspace does not allow specifying whether a computed valid-time relation should be state or event. Given these problems, I chose to drop the distinction between state and event valid-time relations. I assume that the time-stamps of all valid-time relations denote temporal elements, with temporal elements being sets of chronons. For example, assuming that the domains of $employee$ and $salary$ are as in section \ref{relational}, $val\_salaries$ below is a valid-time relation over $\tup{employee, salary}$, shown in its tabular form (the double vertical line separates the explicit attributes from the implicit one). According to chapter 10 of \cite{TSQL2book}, the elements of $D_T$ are non-atomic. Each element $v_t$ of $D_T$ is in turn a set, whose elements denote the chronons that belong to the temporal element represented by $v_t$. \adbtable{3}{|l|l||l|}{$val\_salaries$} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $\{c^1_1, c^1_2, c^1_3, \dots, c^1_{n_1}\}$ \\ $J.Adams$ & $18000$ & $\{c^2_1, c^2_2, c^2_3, \dots, c^2_{n_2}\}$ \\ $J.Adams$ & $18500$ & $\{c^3_1, c^3_2, c^3_3, \dots, c^3_{n_3}\}$ \\ $T.Smith$ & $19000$ & $\{c^4_1, c^4_2, c^4_3, \dots, c^4_{n_4}\}$ \\ $T.Smith$ & $21000$ & $\{c^5_1, c^5_2, c^5_3, \dots, c^5_{n_5}\}$ } For example, $c^1_1, c^1_2, c^1_3, \dots, c^1_{n_1}$ in the first tuple for J.Adams above represent all the chronons where the salary of J.Adams was/is/will be 17000. $\{c^1_1, c^1_2, c^1_3, \dots, c^1_{n_1}\}$ is an element of $D_T$. For simplicity, when depicting valid-time relations I often show (in an informal manner) the temporal elements denoted by the time-stamps rather the time-stamps themselves. $val\_salaries$ would be shown as below, meaning that the time-stamp of the first tuple represents a temporal element of two maximal periods, 1/1/92 to 12/6/92 and 8/5/94 to 30/10/94. (I assume here that chronons correspond to days. $now$ refers to the current chronon.) \begin{examps} \item \label{tlang:4} \dbtable{3}{|l|l||l|}{$val\_salaries$} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1/1/92, \; 12/6/92] \union [8/5/94, \; 30/10/94]$ \\ $J.Adams$ & $18000$ & $[13/6/92, \; 7/5/94] \union [31/10/94, \; now]$ \\ $T.Smith$ & $21000$ & $[15/6/92, \; now]$ } \end{examps} Two tuples $\tup{v_1^1, \dots, v_n^1; v_t^1}$ and $\tup{v^2_1, \dots, v_n^2; v_t^2}$ are \emph{value-equivalent} iff if $v^1_1 = v^2_1$, \dots, $v^1_n = v^2_n$. A valid-time relation is \emph{coalesced} iff it contains no value-equivalent tuples. \textsc{Bcdm}\xspace requires all valid-time relations to be coalesced (see p.188 of \cite{TSQL2book}). For example, \pref{bcdm:1} is not allowed (its first and third tuples are value-equivalent). In this thesis, this \textsc{Bcdm}\xspace restriction is dropped, and \pref{bcdm:1} is allowed. \begin{examps} \item \label{bcdm:1} \dbtablec{|l|l||l|} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1/1/92, \; 12/6/92]$ \\ $J.Adams$ & $18000$ & $[13/6/92, \; 7/5/94]$ \\ $J.Adams$ & $17000$ & $[8/5/94, \; 30/10/94]$ \\ $J.Adams$ & $18000$ & $[31/10/94, \; now]$ \\ $T.Smith$ & $21000$ & $[15/6/92, \; now]$ } \end{examps} By the definition of $D_T$, the elements of $D_T$ denote all the elements of $\ensuremath{\mathit{TELEMS}}\xspace$ (temporal elements). Since $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{TELEMS}}\xspace$, some of the elements of $D_T$ denote periods. $D_P$ \index{dp@$D_P$ (set of all attribute values that denote periods)} is the subset of all elements of $D_T$ that denote periods.\footnote{\cite{TSQL2book} seems to adopt a different approach, where $D_P \intersect D_T = \emptyset$.} I also assume that there is a special value $\ensuremath{v_\varepsilon}\xspace \in D$, \index{ve@$\ensuremath{v_\varepsilon}\xspace$ (attribute value denoting $\emptyset$)} that is used to denote the empty set (of chronons). For example, a \textsc{Tsql2}\xspace expression that computes the intersection of two non-overlapping periods evaluates to \ensuremath{v_\varepsilon}\xspace.\footnote{Table 8.3 of \cite{TSQL2book} implies that \ensuremath{v_\varepsilon}\xspace is the special ``null'' value. In \textsc{Sql}\xspace, null has several roles. Here, I assume that there is a special value \ensuremath{v_\varepsilon}\xspace whose only role is to denote the empty set.} I use $D_P^*$ \index{dp*@$D_P^*$ ($D_P \union \emptyset$)} to refer to $D_P \union \{\ensuremath{v_\varepsilon}\xspace\}$. The following notation will prove useful: \begin{itemize} \item \ensuremath{\mathit{VREL}_P}\xspace \index{vrelp@\ensuremath{\mathit{VREL}_P}\xspace (set of all valid time relations time-stamped by elements of $D_P$)} is the set of all valid-time relations whose time-stamps are all elements of $D_P$ (all the time-stamps denote periods). \item \ensuremath{\mathit{NVREL}_P}\xspace \index{nvrelp@\ensuremath{\mathit{NVREL}_P}\xspace (``normalised'' elements of \ensuremath{\mathit{VREL}_P}\xspace)} is the set of all the (intuitively, ``normalised'') relations $r \in \ensuremath{\mathit{VREL}_P}\xspace$ with the following property: if $\tup{v_1, \dots, v_n; v^1_t} \in r$, $\tup{v_1, \dots, v_n; v^2_t} \in r$, and $f_D(v^1_t) \union f_D(v^2_t) \in \ensuremath{\mathit{PERIODS}}\xspace$, then $v^1_t = v^2_t$. This definition ensures that in any $r \in \ensuremath{\mathit{NVREL}_P}\xspace$, there is no pair of different value-equivalent tuples whose time-stamps $v^1_t$ and $v^2_t$ denote overlapping or adjacent periods (because if the periods of $v^1_t$ and $v^2_t$ overlap or they are adjacent, their union is also a period, and then it must be true that $v^1_t = v^2_t$, i.e.\ the value-equivalent tuples are not different). \item \ensuremath{\mathit{SREL}}\xspace \index{srel@\ensuremath{\mathit{SREL}}\xspace (set of all snapshot relations)} is the set of all snapshot relations. \item For every $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{VREL}_P}\xspace(n)$ \index{vrelpn@$\ensuremath{\mathit{VREL}_P}\xspace(n)$ (relations in \ensuremath{\mathit{VREL}_P}\xspace with $n$ explicit attributes)} contains all the relations of \ensuremath{\mathit{VREL}_P}\xspace that have $n$ explicit attributes. Similarly, $\ensuremath{\mathit{NVREL}_P}\xspace(n)$ \index{nvrelpn@$\ensuremath{\mathit{NVREL}_P}\xspace(n)$ (set of all relations in \ensuremath{\mathit{VREL}_P}\xspace with $n$ explicit attributes)} and $\ensuremath{\mathit{SREL}}\xspace(n)$ \index{sreln@$\ensuremath{\mathit{SREL}}\xspace(n)$ (set of all snapshot relations of $n$ attributes)} contain all the relations of \ensuremath{\mathit{NVREL}_P}\xspace and \ensuremath{\mathit{SREL}}\xspace respectively that have $n$ explicit attributes. \end{itemize} To simplify the proofs in the rest of this chapter, I include the empty relation in all $\ensuremath{\mathit{VREL}_P}\xspace(n)$, $\ensuremath{\mathit{NVREL}_P}\xspace(n)$, $\ensuremath{\mathit{SREL}}\xspace(n)$, for $n= 1,2,3,\dots$. \subsection{The TSQL2 language} \label{tsql2_lang} This section is an introduction to the features of \textsc{Tsql2}\xspace that are used in this thesis. \subsubsection*{SELECT statements} As noted in section \ref{tdbs_general}, \textsc{Tsql2}\xspace is an extension of \textsc{Sql-92}\xspace. Roughly speaking, \textsc{Sql-92}\xspace queries (e.g.\ \ref{tlang:1}) consist of three clauses: a \sql{SELECT}, \index{select@\sql{SELECT} (\textsc{Tsql2}\xspace keyword, introduces a \textsc{Tsql2}\xspace query)} a \sql{FROM}, \index{from@\sql{FROM} (\textsc{Tsql2}\xspace keyword, shows the relations on which a \sql{SELECT} operates)} and a \sql{WHERE} \index{where@\sql{WHERE} (\textsc{Tsql2}\xspace keyword, introduces restrictions)} clause. (The term \emph{\sql{SELECT} statement} will be used to refer to the whole of a \textsc{Sql-92}\xspace or \textsc{Tsql2}\xspace query.) \begin{examps} \item \index{as@\sql{AS} (\textsc{Tsql2}\xspace keyword, introduces correlation names)} \index{and@\sql{AND} (\textsc{Tsql2}\xspace's conjunction)} \label{tlang:1} \select{SELECT DISTINCT sal.salary \\ FROM salaries AS sal, managers AS mgr \\ WHERE mgr.manager = 'J.Adams' AND sal.employee = mgr.managed} \end{examps} Assuming that $salaries$ and $managers$ are as below, \pref{tlang:1} generates the third relation below. \begin{examps} \item[] \dbtable{2}{|l|l|}{$salaries$} {$employee$ & $salary$ } {$J.Adams$ & $17000$ \\ $T.Smith$ & $18000$ \\ $G.Papas$ & $14500$ \\ $B.Hunter$ & $17000$ \\ $K.Kofen$ & $16000$ } \ \ \dbtable{2}{|l|l|}{$managers$} {$manager$ & $managed$ } {$J.Adams$ & $G.Papas$ \\ $J.Adams$ & $B.Hunter$ \\ $J.Adams$ & $J.Adams$ \\ $T.Smith$ & $K.Kofen$ \\ $T.Smith$ & $T.Smith$ } \ \ \dbtable{1}{|l|}{$(result)$} {$salary$} {$17000$ \\ $14500$ } \end{examps} \pref{tlang:1} generates a snapshot one-attribute relation that contains the salaries of all employees managed by J.Adams. The \sql{FROM} clause of \pref{tlang:1} shows that the query operates on the $salaries$ and $managers$ relations. \sql{sal} and \sql{mgr} are \emph{correlation names}. They can be thought of as tuple-variables ranging over the tuples of $salaries$ and $managers$ respectively. The (optional) \sql{WHERE} clause imposes restrictions on the possible combinations of tuple-values of \sql{sal} and \sql{mgr}. In every combination, the $manager$ value of \sql{mgr} must be $J.Adams$, and the $managed$ value of \sql{mgr} must be the same as the $employee$ value of \sql{sal}. For example, $\tup{J.Adams, G.Papas}$ and $\tup{G.Papas, 14500}$ is an acceptable combination of \sql{mgr} and \sql{sal} values respectively, while $\tup{J.Adams, G.Papas}$ and $\tup{B.Hunter, 17000}$ is not. In \textsc{Sql-92}\xspace (and \textsc{Tsql2}\xspace), correlation names are optional, and relation names can be used to refer to attribute values. In \pref{tlang:1}, for example, one could omit \sql{AS mgr}, and replace \sql{mgr.manager} and \sql{mgr.managed} by \sql{managers.manager} and \sql{managers.managed}. To simplify the definitions of section \ref{additional_tsql2} below, I treat correlation names as mandatory, and I do not allow relation names to be used to refer to attribute values. The \sql{SELECT} clause specifies the contents of the resulting relation. In \pref{tlang:1}, it specifies that the resulting relation should have only one attribute, $salary$, and that for each acceptable combination of \sql{sal} and \sql{mgr} values, the corresponding tuple of the resulting relation should contain the $salary$ value of \sql{sal}'s tuple. The \sql{DISTINCT} \index{distinct@\sql{DISTINCT} (\textsc{Tsql2}\xspace keyword, removes duplicate tuples)} in \pref{tlang:1} causes duplicates of tuples to be removed from the resulting relation. Without the \sql{DISTINCT} duplicates are not removed. The result of \pref{tlang:1} would contain two identical tuples $\tup{17000}$, deriving from the tuples for J.Adams and B.Hunter in $salaries$. This is against the set-theoretic definition of relations of sections \ref{relational} and \ref{bcdm} (relations were defined to be \emph{sets} of tuples, and hence cannot contain duplicates.) To ensure that relations contain no duplicates, in this thesis \sql{SELECT} statements always have a \sql{DISTINCT} in their \sql{SELECT} clauses. \textsc{Tsql2}\xspace allows \sql{SELECT} statements to operate on valid-time relations as well. A \sql{SNAPSHOT} \index{snapshot@\sql{SNAPSHOT} (\textsc{Tsql2}\xspace keyword, signals that a snapshot relation is to be created)} keyword in the \sql{SELECT} statement indicates that the resulting relation is snapshot. When the resulting relation is valid-time, an additional \sql{VALID} clause is present. In the latter case, the \sql{SELECT} clause specifies the values of the explicit attributes of the resulting relation, while the \sql{VALID} clause specifies the time-stamps of the resulting tuples. Assuming, for example, that $val\_salaries$ is as in \pref{tlang:4}, \pref{tlang:7} returns \pref{tlang:8}. \begin{examps} \item \label{tlang:7} \sql{SELECT DISTINCT sal.employee, sal.salary \\ VALID PERIOD(BEGIN(VALID(sal)), END(VALID(sal))) \\ FROM val\_salaries AS sal} \item \label{tlang:8} \dbtablec{|l|l||l|} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1/1/92, \; 30/10/94]$ \\ $J.Adams$ & $18000$ & $[13/6/92, \; now]$ \\ $T.Smith$ & $21000$ & $[15/6/92, \; now]$ } \end{examps} The \sql{VALID} \index{valid@\sql{VALID} (\textsc{Tsql2}\xspace keyword, refers to time-stamps of tuples)} keyword is used both to start a \sql{VALID}-clause (a clause that specifies the time-stamps of the resulting relation) and to refer to the time-stamp of the tuple-value of a correlation name. In \pref{tlang:7}, \sql{VALID(sal)} refers to the time-stamp of \sql{sal}'s tuple (i.e.\ to the time-stamp of a tuple from $val\_salaries$). \sql{BEGIN(VALID(sal))} \index{begin2@\sql{BEGIN} (\textsc{Tsql2}\xspace keyword, returns the start-point of a temporal element)} refers to the first chronon of the temporal element represented by that time-stamp, and \sql{END(VALID(sal))} \index{end2@\sql{END} (\textsc{Tsql2}\xspace keyword, returns the end-point of a temporal element)} to the last chronon of that temporal element.\footnote{Section 30.5 of \cite{TSQL2book} allows \sql{BEGIN} and \sql{END} to be used only with periods. I see no reason for this limitation. I allow \sql{BEGIN} and \sql{END} to be used with any temporal element.} The \sql{PERIOD} \index{period@\sql{PERIOD} (\textsc{Tsql2}\xspace keyword, constructs periods or introduces period literals)} function generates a period that starts at the chronon of its argument, and ends at the chronon of its second argument. Hence, each time-stamp of \pref{tlang:8} represents a period that starts/ends at the earliest/latest chronon of the temporal element of the corresponding time-stamp of $val\_salaries$. \subsubsection*{Literals} \textsc{Tsql2}\xspace provides \sql{PERIOD} \index{period@\sql{PERIOD} (\textsc{Tsql2}\xspace keyword, constructs periods or introduces period literals)} literals, \sql{INTERVAL} \index{interval@\sql{INTERVAL} (\textsc{Tsql2}\xspace keyword, returns intervals or introduces interval literals)} literals, and \sql{TIMESTAMP} \index{timestamp@\sql{TIMESTAMP} (\textsc{Tsql2}\xspace keyword, introduces chronon-denoting literals)} literals (the use of ``\sql{TIMESTAMP}'' in this case is unfortunate; these literals specify time-points, not time-stamps of valid-time relations, which denote temporal-elements). For example, \sql{PERIOD '[March 3, 1995 - March 20, 1995]'} is a literal that specifies a period at the granularity of days. If chronons are finer than days, the assumption in \textsc{Tsql2}\xspace is that the exact chronons within March 3 and March 20 where the period starts and ends are unknown (section \ref{tsql2_time}). In this thesis, \sql{PERIOD} literals that refer to granularities other than that of chronons are abbreviations for literals that refer to the granularity of chronons. The denoted period contains all the chronons that fall within the granules specified by the literal. For example, if chronons correspond to minutes, \sql{PERIOD '[March 3, 1995 - March 20, 1995]'} is an abbreviation for \sql{PERIOD '[00:00 March 3, 1995 - 23:59 March 20, 1995]'}. \textsc{Tsql2}\xspace supports multiple calendars (e.g.\ Gregorian, Julian, lunar calendar; see chapter 7 of \cite{TSQL2book}). The strings that can appear between the quotes of \sql{PERIOD} literals (e.g.\ \sql{'[March 3, 1995 - March 20, 1995]'}, \sql{'(3/4/95 - 20/4/95]'}) depend on the available calendars and the selected formatting options (see chapter 7 of \cite{TSQL2book}). The convention seems to be that the boundaries are separated by a dash, and that the first and last characters of the quoted string are square or round brackets, depending on whether the boundaries are to be included or not. I also assume that \sql{PERIOD 'today'} can be used (provided that chronons are at least as fine as days) to refer to the period that covers all the chronons of the present day. (There are other \textsc{Tsql2}\xspace expressions that can be used to refer to the current day, but I would have to discuss \textsc{Tsql2}\xspace granularity-conversion commands to explain these. Assuming that \sql{PERIOD 'today'} is available allows me to avoid these commands.) \sql{TIMESTAMP} literals specify chronons. Only the following special \sql{TIMESTAMP} literals are used in this thesis: \sql{TIMESTAMP 'beginning'}, \sql{TIMESTAMP 'forever'}, \sql{TIMESTAMP 'now'}. These refer to the beginning of time, the end of time, and the present chronon. An example of an \sql{INTERVAL} literal is \sql{INTERVAL '5' DAY}, which specifies a duration of five consecutive day-granules. The available granularities depend on the calendars that are active. The granularities of years, months, days, hours, minutes, and seconds are supported by default. Intervals can also be used to shift periods or chronons towards the past or the future. For example, \sql{PERIOD '[1991 - 1995]' + INTERVAL '1' YEAR} is the same as \sql{PERIOD '[1992 - 1996]'}. If chronons correspond to minutes, \sql{PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE)} specifies the period that covers all the chronons from the beginning of time up to (but not including) the current chronon. \subsubsection*{Other TSQL2 functions and predicates} The \sql{INTERSECT} \index{intersect@\sql{INTERSECT} (\textsc{Tsql2}\xspace keyword, computes the intersection of two sets of chronons)} function computes the intersection of two sets of chronons.\footnote{Section 8.3.3 of \cite{TSQL2book} requires both arguments of \sql{INTERSECT} to denote periods, but section 30.14 allows the arguments of \sql{INTERSECT} to denote temporal elements. I follow the latter. I also allow the arguments of \sql{INTERSECT} to denote the empty set.} For example, \sql{INTERSECT(PERIOD '[May 1, 1995 - May 10, 1995]', PERIOD '[May 3, 1995 - May 15, 1995]')} is the same as \sql{PERIOD '[May 3, 1995 - May 10, 1995]'}. If the intersection is the empty set, \sql{INTERSECT} returns \ensuremath{v_\varepsilon}\xspace (section \ref{bcdm}). The \sql{CONTAINS} \index{contains@\sql{CONTAINS} (\textsc{Tsql2}\xspace keyword, checks if a chronon belongs to a set of chronons)} predicate checks if a chronon belongs to a set of chronons. For example, if $val\_salaries$ is as in \pref{tlang:4}, \pref{tlang:9} generates a snapshot relation showing the current salary of each employee. \sql{CONTAINS} can also be used to check if a set of chronons is a subset of another set of chronons.\footnote{Table 8.7 in section 8.3.6 and additional syntax rule 3 in section 32.4 of \cite{TSQL2book} allow the arguments of \sql{CONTAINS} to denote periods but not generally temporal elements. Table 32.1 in section 32.4 of \cite{TSQL2book}, however, allows the arguments of \sql{CONTAINS} to denote temporal elements. I follow the latter. I also allow the arguments of \sql{CONTAINS} to denote the empty set. The same comments apply in the case of \sql{PRECEDES}.} \begin{examps} \item \label{tlang:9} \select{SELECT DISTINCT SNAPSHOT sal.employee, sal.salary \\ FROM val\_salaries AS sal \\ WHERE VALID(sal) CONTAINS TIMESTAMP 'now'} \end{examps} The \sql{PRECEDES} \index{precedes@\sql{PRECEDES} (\textsc{Tsql2}\xspace keyword, checks temporal precedence)} predicate checks if a chronon or set of chronons strictly precedes another chronon or set of chronons. Section 8.3.6 of \cite{TSQL2book} specifies the semantics of \sql{PRECEDES} only in cases where its arguments are chronons or periods. I assume that $expr_1$ \sql{PRECEDES} $expr_2$ is true, iff the chronon of $expr_1$ (if $expr_1$ specifies a single chronon) or all the chronons of $expr_1$ (if $expr_1$ specifies a set of chronons) strictly precede the chronon of $expr_2$ (if $expr_2$ specifies a single chronon) or all the chronons of $expr_2$ (if $expr_2$ specifies a set of chronons). For example, \sql{PERIOD '[1/6/95 - 21/6/95]' PRECEDES PERIOD '[24/6/95 - 30/6/95]'} is true, but \sql{PERIOD '[1/6/95 - 21/6/95]' PRECEDES PERIOD '[19/6/95 - 30/6/95]'} is not. \subsubsection*{Embedded SELECT statements} \textsc{Tsql2}\xspace (and \textsc{Sql-92}\xspace) allow embedded \sql{SELECT} statements to be used in the \sql{FROM} clause, in the same way that relation names are used (e.g.\ \pref{tlang:10}). \begin{examps} \item \label{tlang:10} \select{SELECT DISTINCT SNAPSHOT sal2.salary \\ FROM (\select{SELECT DISTINCT sal1.salary \\ VALID VALID(sal1) \\ FROM val\_salaries AS sal1} \\ \ \ \ \ \ ) AS sal2 \\ WHERE sal2.salary > 17500} \end{examps} Assuming that $val\_salaries$ is as in \pref{tlang:4}, the embedded \sql{SELECT} statement above simply drops the $employee$ attribute of $val\_salaries$, generating \pref{tlang:11}. \sql{sal2} ranges over the tuples of \pref{tlang:11}. \pref{tlang:10} generates a relation that is the same as \pref{tlang:11}, except that tuples whose $salary$ values are not greater than 17500 are dropped. \begin{examps} \item \label{tlang:11} \dbtablec{|l||l|} {$salary$ &} {$17000$ & $[1/1/92, \; 12/6/92] \union [8/5/94, \; 30/10/94]$ \\ $18000$ & $[13/6/92, \; 7/5/94] \union [31/10/94, \; now]$ \\ $21000$ & $[15/6/92, \; now]$ } \end{examps} \subsubsection*{Partitioning units} In \textsc{Tsql2}\xspace, relation names and embedded \sql{SELECT} statements in the \sql{FROM} clause can be followed by \emph{partitioning units}.\footnote{Section 30.3 of \cite{TSQL2book} allows relation names but not embedded \sql{SELECT} statements to be followed by partitioning units in \sql{FROM} clauses. \cite{Snodgrass1994d} (queries Q.1.2.2, Q.1.2.5, Q.1.7.6), however, shows \sql{SELECT} statements embedded in \sql{FROM} clauses and followed by partitioning units. I follow \cite{Snodgrass1994d}.} \textsc{Tsql2}\xspace currently provides two partitioning units: \sql{(PERIOD)} and \sql{(INSTANT)} \index{instant@\sql{(INSTANT)} (\textsc{Tsql2}\xspace partitioning unit)} (see section 30.3 and chapter 12 of \cite{TSQL2book}). \sql{(INSTANT)} is not used in this thesis. Previous \textsc{Tsql2}\xspace versions (e.g.\ the September 1994 version of chapter 12 of \cite{TSQL2book}) provided an additional \sql{(ELEMENT)}. For reasons explained below, \sql{(ELEMENT)} is still used in this thesis. \sql{(ELEMENT)} \index{element@\sql{(ELEMENT)} (\textsc{Tsql2}\xspace partitioning unit)} merges value-equivalent tuples.\footnote{The semantics of \sql{(ELEMENT)} was never clear. The discussion here reflects my understanding of the September 1994 \textsc{Tsql2}\xspace documentation, and the semantics that is assigned to \sql{(ELEMENT)} in this thesis.} For example, if $rel1$ is the relation of \pref{pus:1}, \pref{pus:2} generates the coalesced relation of \pref{pus:3}. \begin{examps} \item \label{pus:1} \dbtable{3}{|l|l||l|}{$rel1$} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1986, \; 1988]$ \\ $J.Adams$ & $17000$ & $[1987, \; 1990]$ \\ $J.Adams$ & $17000$ & $[1992, \; 1994]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1990]$ \\ $G.Papas$ & $14500$ & $[1990, \; 1992]$ } \item \label{pus:2} \select{SELECT DISTINCT r1.employee, r1.salary \\ VALID VALID(r1) \\ FROM rel1(ELEMENT) AS r1} \item \label{pus:3} \dbtablec{|l|l||l|} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1986, \; 1990] \union [1992, \; 1994]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1992]$ } \end{examps} The effect of \sql{(ELEMENT)} on a valid-time relation $r$ is captured by the $coalesce$ function: \index{coalesce@$coalesce()$ (effect of \sql{(ELEMENT)})} \[ \begin{aligned} coalesce(r) \defeq \{&\tup{v_1, \dots, v_n; v_t} \mid \tup{v_1, \dots, v_n; v_t'} \in r \text{ and} \\ &f_D(v_t) = \bigcup_{\tup{v_1, \dots, v_n; v_t''} \in r}f_D(v_t'') \} \\ \end{aligned} \] \sql{(ELEMENT)} has no effect on already coalesced valid-time relations. Hence, in the \textsc{Bcdm}\xspace version of \cite{TSQL2book}, where all valid-time relations are coalesced, \sql{(ELEMENT)} is redundant (and this is probably why it was dropped). In this thesis, valid-time relations are not necessarily coalesced (section \ref{bcdm}), and \sql{(ELEMENT)} plays an important role. \sql{(PERIOD)} \index{period2@\sql{(PERIOD)} (\textsc{Tsql2}\xspace partitioning unit)} intuitively breaks each tuple of a valid-time relation into value-equivalent tuples, each corresponding to a maximal period of the temporal element of the original time-stamp. Assuming, for example, that $rel2$ is the relation of \pref{pus:3}, \pref{pus:4} generates \pref{pus:5}. \begin{examps} \item \label{pus:4} \select{SELECT DISTINCT r2.employee, r2.salary \\ VALID VALID(r2) \\ FROM rel2(PERIOD) AS r2} \item \label{pus:5} \dbtablec{|l|l||l|} {$employee$ & $salary$ &} {$J.Adams$ & $17000$ & $[1986, \; 1990]$ \\ $J.Adams$ & $17000$ & $[1992, \; 1994]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1992]$ } \end{examps} As the example shows, \sql{(PERIOD)} may generate non-coalesced relations. This is mysterious in the \textsc{Bcdm}\xspace version of \cite{TSQL2book}, where non-coalesced valid-time relations are not allowed. The assumption seems to be that although non-coalesced valid-time relations are not allowed, during the execution of \sql{SELECT} statements temporary non-coalesced valid-time relations may be generated. Any resulting valid-time relations, however, are coalesced automatically at the end of the statement's execution. \pref{pus:5} would be coalesced automatically at the end of the execution of \pref{pus:4} (cancelling, in this particular example, the effect of \sql{(PERIOD)}). In this thesis, no automatic coalescing takes place, and the result of \pref{pus:4} is \pref{pus:5}. To preserve the spirit of \sql{(PERIOD)} in the \textsc{Bcdm}\xspace version of this thesis where valid-time relations are not necessarily coalesced, I assume that \sql{(PERIOD)} operates on a coalesced copy of the original relation. Intuitively, \sql{(PERIOD)} first causes \pref{pus:1} to become \pref{pus:3}, and then generates \pref{pus:5}. The effect of \sql{(PERIOD)} on a valid-time relation $r$ is captured by the $pcoalesce$ function: \index{pcoalesce@$pcoalesce()$ (effect of \sql{(PERIOD)})} \[ \begin{aligned} pcoalesce(r) \defeq \{&\tup{v_1, \dots, v_n; v_t} \mid \tup{v_1, \dots, v_n; v_t'} \in coalesce(r) \text { and} \\ &f_D(v_t) \in mxlpers(f_D(v_t'))\} \end{aligned} \] \section{Modifications of TSQL2} \label{TSQL2_mods} This thesis adopts some modifications of \textsc{Tsql2}\xspace. Some of the modifications were mentioned in section \ref{TSQL2_intro}. The main of those were: \begin{itemize} \item The requirement that all valid-time relations must be coalesced was dropped. \item The distinction between state and event valid-time relations was abandoned. \item \sql{(ELEMENT)} was re-introduced. \item The semantics of \sql{(PERIOD)} was enhanced, to reflect the fact that in this thesis valid-time relations are not necessarily coalesced. \item All periods and temporal elements are specified at the granularity of chronons. Literals referring to other granularities are used as abbreviations for literals that refer to the granularity of chronons. \end{itemize} This section describes the remaining \textsc{Tsql2}\xspace modifications of this thesis. \subsection{Referring to attributes by number} \label{by_num} In \textsc{Tsql2}\xspace (and \textsc{Sql-92}\xspace) explicit attributes are referred to by their names. In \pref{tlang:1b}, for example, \sql{sal.salary} refers to the $salary$ attribute of $val\_salaries$. \begin{examps} \item \label{tlang:1b} \select{SELECT DISTINCT sal.salary \\ VALID VALID(sal) \\ FROM val\_salaries AS sal} \end{examps} In the \textsc{Tsql2}\xspace version of this thesis, explicit attributes are referred to by number, with numbers corresponding to the order in which the attributes appear in the relation schema (section \ref{relational}). For example, if the relation schema of $val\_salaries$ is $\tup{employee, salary}$, $employee$ is the first explicit attribute and $salary$ the second one. \pref{tlang:1c} would be used instead of \pref{tlang:1b}. To refer to the implicit attribute, one still uses \sql{VALID} (e.g.\ \sql{VALID(sal)}). \begin{examps} \item \label{tlang:1c} \select{SELECT DISTINCT sal.2 \\ VALID VALID(sal) \\ FROM salaries AS sal} \end{examps} Referring to explicit attributes by number simplifies the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation, because this way there is no need to keep track of the attribute names of the various relations. \subsection{(SUBPERIOD) and (NOSUBPERIOD)} \label{new_pus} Two new partitioning units, \sql{(SUBPERIOD)} and \sql{(NOSUBPERIOD)}, were introduced for the purposes of this thesis. \sql{(SUBPERIOD)} is designed to be used with relations from \ensuremath{\mathit{VREL}_P}\xspace (section \ref{bcdm}). The effect of \sql{(SUBPERIOD)} \index{subperiod2@\sql{(SUBPERIOD)} (\textsc{Tsql2}\xspace partitioning unit)} on a relation $r$ is captured by the $subperiod$ function: \index{subperiod@$subperiod()$ (effect of \sql{(SUBPERIOD)})} \[ subperiod(r) \defeq \{ \tup{v_1, \dots, v_n; v_t} \mid \tup{v_1, \dots, v_n; v_t'} \in r \text{ and } f_D(v_t) \subper f_D(v_t') \} \] For each tuple $\tup{v_1, \dots, v_n; v_t'} \in r$, the resulting relation contains many value-equivalent tuples of the form $\tup{v_1, \dots, v_n; v_t}$, one for each period $f_D(v_t)$ that is a subperiod of $f_D(v_t')$. Assuming, for example, that chronons correspond to years, and that $rel$ is the relation of \pref{subper:0}, \pref{subper:1} returns the relation of \pref{subper:2}. \begin{examps} \item \label{subper:0} \dbtableb{|l|l||l|} {$J.Adams$ & $17000$ & $[1992, \; 1993]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1990]$ \\ $G.Papas$ & $14500$ & $[1990, \; 1991]$ } \item \label{subper:1} \sql{SELECT DISTINCT r.1, r.2 \\ VALID VALID(r) \\ FROM rel(SUBPERIOD) AS r} \item \label{subper:2} \dbtableb{|l|l||l|} {$J.Adams$ & $17000$ & $[1992, \; 1993]$ \\ $J.Adams$ & $17000$ & $[1992, \; 1992]$ \\ $J.Adams$ & $17000$ & $[1993, \; 1993]$ \\ & & \\ $G.Papas$ & $14500$ & $[1988, \; 1990]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1988]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1989]$ \\ $G.Papas$ & $14500$ & $[1989, \; 1989]$ \\ $G.Papas$ & $14500$ & $[1989, \; 1990]$ \\ $G.Papas$ & $14500$ & $[1990, \; 1990]$ \\ & & \\ $G.Papas$ & $14500$ & $[1990, \; 1991]$ \\ $G.Papas$ & $14500$ & $[1991, \; 1991]$ } \end{examps} The first three tuples of \pref{subper:2} correspond to the first tuple of \pref{subper:0}. The following six tuples correspond to the first tuple of $G.Papas$ in \pref{subper:0}. The remaining tuples of \pref{subper:2} derive from the second tuple of $G.Papas$ in \pref{subper:0} (the tuple for the subperiod $[1990, \; 1990]$ has already been included in \pref{subper:2}). Notice that \sql{(SUBPERIOD)} does not coalesce the original relation before generating the result (this is why there is no tuple for G.Papas time-stamped by $[1988, \; 1991]$ in \pref{subper:2}). Obviously, the cardinality of the resulting relations can be very large (especially if chronons are very fine, e.g.\ seconds). The cardinality, however, is never infinite (assuming that the cardinality of the original relation is finite): given that time is discrete, linear, and bounded, any period $p$ is a finite set of chronons, and there is at most a finite number of periods (convex sets of chronons) that are subperiods (subsets) of $p$\/; hence, for any tuple in the original relation whose time-stamp represents a period $p$, there will be at most a finite number of tuples in the resulting relation whose time-stamps represent subperiods of $p$. It remains, of course, to be examined if \sql{(SUBPERIOD)} can be supported efficiently in \textsc{Dbms}\xspace{s}. It is obviously very inefficient to store (or print) individually all the tuples of the resulting relation. A more space-efficient encoding of the resulting relation is needed. I have not explored this issue. Roughly speaking, \sql{(SUBPERIOD)} is needed because during the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation every \textsc{Top}\xspace formula is mapped to a valid-time relation whose time-stamps denote the event-time periods where the formula is true. Some (but not all) formulae are homogeneous (section \ref{denotation}). For these formulae we need to ensure that if the valid-time relation contains a tuple for an event-time $et$, it also contains tuples for all the subperiods of $et$. This will become clearer in section \ref{trans_rules}. \sql{(NOSUBPERIOD)} \index{nosubperiod2@\sql{(NOSUBPERIOD)} (\textsc{Tsql2}\xspace partitioning unit)} is roughly speaking used when the effect of \sql{(SUBPERIOD)} needs to be cancelled. \sql{(NOSUBPERIOD)} is designed to be used with relations from \ensuremath{\mathit{VREL}_P}\xspace. It eliminates any tuple $\tup{v_1, \dots, v_n; v_t}$, for which there is a value-equivalent tuple $\tup{v_1, \dots, v_n; v_t'}$, such that $f_D(v_t) \propsubper f_D(v_t')$. The effect of \sql{(NOSUBPERIOD)} on a valid-time relation $r$ is captured by the $nosubperiod$ function: \index{nosubperiod@$nosubperiod()$ (effect of \sql{(NOSUBPERIOD)})} \[ \begin{aligned} nosubperiod(r) \defeq \{ \tup{v_1, \dots, v_n; v_t} \in r \mid &\text{ there is no } \tup{v_1, \dots, v_n; v_t'} \in r \\ &\text{ such that } f_D(v_t) \propsubper f_D(v_t') \} \end{aligned} \] Applying \sql{(NOSUBPERIOD)} to \pref{subper:2} generates \pref{subper:3}. \begin{examps} \item \label{subper:3} \dbtableb{|l|l||l|} {$J.Adams$ & $17000$ & $[1992, \; 1993]$ \\ $G.Papas$ & $14500$ & $[1988, \; 1990]$ \\ $G.Papas$ & $14500$ & $[1990, \; 1991]$ } \end{examps} Although \sql{(SUBPERIOD)} and \sql{(NOSUBPERIOD)} are designed to be used (and in practice will always be used) with relations from \ensuremath{\mathit{VREL}_P}\xspace, I allow \sql{(SUBPERIOD)} and \sql{(NOSUBPERIOD)} to be used with any valid-time relation. In the proofs of appendix \ref{trans_proofs}, this saves me having to prove that the original relation is an element of \ensuremath{\mathit{VREL}_P}\xspace whenever \sql{(SUBPERIOD)} and \sql{(NOSUBPERIOD)} are used. \subsection{Calendric relations} \label{calrels} As mentioned in section \ref{tsql2_lang}, \textsc{Tsql2}\xspace supports multiple calendars. Roughly speaking, a \textsc{Tsql2}\xspace calendar describes a system that people use to measure time (Gregorian calendar, Julian calendar, etc.). \textsc{Tsql2}\xspace calendars also specify the meanings of strings within the quotes of temporal literals, and the available granularities. According to section 3.2 of \cite{TSQL2book}, \textsc{Tsql2}\xspace calendars are defined by the database administrator, the \textsc{Dbms}\xspace vendor, or third parties. In this thesis, I assume that \textsc{Tsql2}\xspace calendars can also provide \emph{calendric relations}. Calendric relations behave like ordinary relations in the database, except that they are defined by the creator of the \textsc{Tsql2}\xspace calendar, and cannot be updated. The exact purpose and contents of each calendric relation are left to the calendar creator. I assume, however, that a calendric relation provides information about the time-measuring system of the corresponding \textsc{Tsql2}\xspace calendar.\footnote{Future work could establish a more systematic link between calendric relations and \textsc{Tsql2}\xspace calendars. For example, calendric relations could be required to reflect (as a minimum) the lattice that shows how the granularities of the calendar relate to each other (section \ref{tsql2_time}).} The Gregorian \textsc{Tsql2}\xspace calendar could, for example, provide the calendric valid-time relation $gregorian$ below. (I assume here that chronons are finer than minutes.) \adbtable{7}{|c|c|c|c|c|c||c|}{$gregorian$} {$year$ & $month$ & $dnum$ & $dname$ & $hour$ & $minute$ &} { \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots \\ $1994$ & $Sept$ & $4$ & $Sun$ & $00$ & $00$ & $\{c_{n_1}, \dots, c_{n_2}\}$ \\ $1994$ & $Sept$ & $4$ & $Sun$ & $00$ & $01$ & $\{c_{n_3}, \dots, c_{n_4}\}$ \\ \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots \\ $1995$ & $Dec$ & $5$ & $Tue$ & $21$ & $35$ & $\{c_{n_5}, \dots, c_{n_6}\}$ \\ \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots & \ \dots } The relation above means that the first minute (00:00) of September 4th 1994 (which was a Sunday) covers exactly the period that starts at the chronon $c_{n_1}$ and ends at the chronon $c_{n_2}$. Similarly, the period that starts at $c_{n_3}$ and ends at $c_{n_4}$ is the second minute (00:01) of September 4th 1994. Of course, the cardinality of $gregorian$ is very large, though not infinite (time in \textsc{Tsql2}\xspace is bounded, and hence there is at most a finite number of minute-granules). It is important, however, to realise that although $gregorian$ behaves like a normal relation in the database, it does not need to be physically present in the database. Its tuples could be computed dynamically, whenever they are needed, using some algorithm specified by the \textsc{Tsql2}\xspace calendar. Other calendric relations may list the periods that correspond to seasons (spring-periods, summer-periods, etc.), special days (e.g.\ Easter days), etc. Calendric relations like $gregorian$ can be used to construct relations that represent the periods of partitionings. \pref{calrels:6}, for example, constructs a one-attribute snapshot relation, that contains all the time-stamps of $gregorian$ that correspond to 21:36-minutes. The resulting relation represents all the periods of the partitioning of 21:36-minutes. \begin{examps} \item \select{SELECT DISTINCT SNAPSHOT VALID(greg) \\ FROM gregorian AS greg \\ WHERE greg.5 = 21 AND greg.6 = 36} \label{calrels:6} \end{examps} Similarly, \pref{calrels:5} generates a one-attribute snapshot relation that represents the periods of the partitioning of Sunday-periods. The embedded \sql{SELECT} statement generates a valid-time relation of one explicit attribute (whose value is $\mathit{Sun}$ in all tuples). The time-stamps of this relation are all the time-stamps of $gregorian$ that correspond to Sundays (there are many tuples for each Sunday). The \sql{(PERIOD)} coalesces tuples that correspond to the same Sunday, leading to a single period-denoting time-stamp for each Sunday. These time-stamps become the attribute values of the relation generated by the overall \pref{calrels:5}. \begin{examps} \item \select{SELECT DISTINCT SNAPSHOT VALID(greg2) \\ FROM (\select{SELECT DISTINCT greg1.4 \\ VALID VALID(greg1) \\ FROM gregorian AS greg1 \\ WHERE greg1.4 = 'Sun'} \\ \ \ \ \ \ )(PERIOD) AS greg2} \label{calrels:5} \end{examps} In \cite{Androutsopoulos1995b} we argue that calendric relations constitute a generally useful addition to \textsc{Tsql2}\xspace, and that unless appropriate calendric relations are available, it is not possible to formulate \textsc{Tsql2}\xspace queries for questions involving existential or universal quantification or counts over day-names, month names, season-names, etc.\ (e.g.\ \pref{calrels:1} -- \pref{calrels:3}). \begin{examps} \item Which technicians were at some site on a Sunday? \label{calrels:1} \item Which technician was at Glasgow Central on every Monday in 1994? \label{calrels:2} \item On how many Sundays was J.Adams at Glasgow Central in 1994? \label{calrels:3} \end{examps} \subsection{The INTERVAL function} \label{interv_fun} \index{interval@\sql{INTERVAL} (\textsc{Tsql2}\xspace keyword, returns intervals or introduces interval literals)} \textsc{Tsql2}\xspace provides a function \sql{INTERVAL} that accepts a period-denoting expression as its argument, and returns an interval reflecting the duration of the period. The assumption seems to be that the resulting interval is specified at whatever granularity the period is specified. For example, \sql{INTERVAL(PERIOD '[1/12/95 - 3/12/95]')} is the same as \sql{INTERVAL '3' DAY}. In this thesis, all periods are specified at the granularity of chronons, and if chronons correspond to minutes, \sql{PERIOD '[1/12/95 - 3/12/95]'} is an abbreviation for \sql{PERIOD '[00:00 1/12/95 - 23:59 3/12/95]'} (sections \ref{tsql2_time} and \ref{tsql2_lang}). Hence, the results of \sql{INTERVAL} are always specified at the granularity of chronons. When translating from \textsc{Top}\xspace to \textsc{Tsql2}\xspace, however, there are cases where we want the results of \sql{INTERVAL} to be specified at other granularities. This could be achieved by converting the results of \sql{INTERVAL} to the desired granularities. The \textsc{Tsql2}\xspace mechanisms for converting intervals from one granularity to another, however, are very obscure (see section 19.4.6 of \cite{TSQL2book}). To avoid these mechanisms, I introduce an additional version of the \sql{INTERVAL} function. If $expr_1$ is a \textsc{Tsql2}\xspace expression that specifies a period $p$, and $expr_2$ is the \textsc{Tsql2}\xspace name (e.g.\ \sql{DAY}, \sql{MONTH}) of a granularity $G$, then \sql{INTERVAL(}$expr_1$, $expr_2$\sql{)} specifies an interval of $n$ granules (periods) of $G$, where $n$ is as follows. If there are $k$ consecutive granules $g_1, g_2, g_3, \dots, g_k$ in $G$ such that $g_1 \union g_2 \union g_3 \union \dots \union g_k = p$, then $n = k$. Otherwise, $n = 0$. For example, \sql{INTERVAL(PERIOD '[May 5, 1995 - May 6, 1995]', DAY)} is the same as \sql{INTERVAL '2' DAY}, because the period covers exactly 2 consecutive day-granules. Similarly, \sql{INTERVAL(PERIOD '[May 1, 1995 - June 30, 1995]', MONTH)} is the same as \sql{INTERVAL '2' MONTH}, because the period covers exactly two consecutive month-granules. In contrast, \sql{INTERVAL(PERIOD '[May 1, 1995 - June 15, 1995]', MONTH)} is the same as \sql{INTERVAL '0' MONTH} (zero duration), because there is no union of consecutive month-granules that covers exactly the period of \sql{PERIOD '[May 1, 1995 - June 15, 1995]'}. \subsection{Correlation names used in the same FROM clause where they are defined} \label{same_FROM} The syntax of \textsc{Tsql2}\xspace (and \textsc{Sql-92}\xspace) does not allow a correlation name to be used in a \sql{SELECT} statement that is embedded in the same \sql{FROM} clause that defines the correlation name. For example, \pref{sfrom:1} is not allowed, because the embedded \sql{SELECT} statement uses \sql{r1}, which is defined by the same \sql{FROM} clause that contains the embedded \sql{SELECT} statement. \begin{examps} \item \label{sfrom:1} \select{SELECT \dots \\ VALID VALID(r1) \\ FROM rel1 AS r1, \\ \ \ \ \ \ (\select{SELECT \dots \\ VALID VALID(r2) \\ FROM rel2 AS r2 \\ WHERE VALID(r1) CONTAINS VALID(r2)} \\ \ \ \ \ \ ) AS r3 \\ WHERE \dots} \end{examps} By \emph{definition of a correlation name} $\alpha$, I mean the expression ``\sql{AS $\alpha$}'' that associates $\alpha$ with a relation. For example, in \pref{sfrom:1} the definition of \sql{r1} is the ``\sql{AS r1}''.\footnote{In \sql{SELECT} statements that contain other embedded \sql{SELECT} statements, multiple definitions of the same correlation name may be present (there are rules that determine the scope of each definition). We do not need to worry about such cases, however, because the generated \textsc{Tsql2}\xspace code of this chapter never contains multiple definitions of the same correlation name.} A correlation name $\alpha$ is \emph{defined by a \sql{FROM} clause} $\xi$, if $\xi$ contains the definition of $\alpha$, and this definition is not within a \sql{SELECT} statement which is embedded in $\xi$. For example, in \pref{sfrom:1} the \sql{r2} is defined by the ``\sql{FROM rel2 AS r2}'' clause, not by the ``\sql{FROM rel1 AS r1, (\dots) AS r3}'' clause. In this thesis, I allow a correlation name to be used in a \sql{SELECT} statement that is embedded in the same \sql{FROM} clause that defines the correlation name, provided that the definition of the correlation name precedes the embedded \sql{SELECT} statement. \pref{sfrom:1} is acceptable, because the definition of \sql{r1} precedes the embedded \sql{SELECT} statement where \sql{r1} is used. In contrast, \pref{sfrom:2} is not acceptable, because the definition of \sql{r1} follows the embedded \sql{SELECT} statement where \sql{r1} is used. \begin{examps} \item \label{sfrom:2} \select{SELECT \dots \\ VALID VALID(r1) \\ FROM (\select{SELECT \dots \\ VALID VALID(r2) \\ FROM rel2 AS r2 \\ WHERE VALID(r1) CONTAINS VALID(r2)} \\ \ \ \ \ \ ) AS r3, \\ \ \ \ \ \ rel1 AS r1 \\ WHERE \dots} \end{examps} The intended semantics of statements like \pref{sfrom:1} should be easy to see: when evaluating the embedded \sql{SELECT} statement, \sql{VALID(r1)} should represent the time-stamp of a tuple from $rel1$. The restriction that the definition of the correlation name must precede the embedded \sql{SELECT} is imposed to make this modification easier to implement. The modification of this section is used in the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation rules for $\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2]$, $\ensuremath{\mathit{Before}}\xspace[\phi_1, \phi_2]$, and $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]$ (section \ref{trans_rules} below and appendix \ref{trans_proofs}). \subsection{Equality checks and different domains} \label{eq_checks} Using the equality predicate (\sql{=}) with expressions that refer to values from different domains often causes the \textsc{Tsql2}\xspace (or \textsc{Sql-92}\xspace) interpreter to report an error. If, for example, the domain of the first explicit attribute of $rel$ is the set of all integers, \sql{r.1} in \pref{eqs:1} stands for an integer. \textsc{Tsql2}\xspace (and \textsc{Sql-92}\xspace) does not allow integers to be compared to strings (e.g.\ ``J.Adams''). Consequently, \pref{eqs:1} would be rejected, and an error message would be generated. \begin{examps} \item \select{SELECT DISTINCT SNAPSHOT r.2 \\ FROM rel AS r \\ WHERE r.1 = 'J.Adams'} \label{eqs:1} \end{examps} In other cases (e.g.\ if a real number is compared to an integer), type conversions take place before the comparison. To by-pass uninteresting details, in this thesis I assume that no type conversions occur when ``\sql{=}'' is used. The equality predicate is satisfied iff both of its arguments refer to the same element of $D$ (universal domain). No error occurs if the arguments refer to values from different domains. In the example of \pref{eqs:1}, \sql{r.1 = 'J.Adams'} is not satisfied, because \sql{r.1} refers to an integer in $D$, \sql{'J.Adams'} to a string in $D$, and integers are different from strings. Consequently, in the \textsc{Tsql2}\xspace version of this thesis \pref{eqs:1} generates the empty relation (no errors occur). \subsection{Other minor changes} \textsc{Tsql2}\xspace does not allow partitioning units to follow \sql{SELECT} statements that are not embedded into other \sql{SELECT} statements. For example, \pref{pus:10} on its own is not acceptable. \begin{examps} \item \label{pus:10} \sql{(}\select{SELECT DISTINCT r1.1, r1.2 \\ VALID VALID(r1) \\ FROM rel AS r1}\\ \sql{)(PERIOD)} \end{examps} \sql{SELECT} statements like \pref{pus:10} can be easily made acceptable by embedding them into another \sql{SELECT} statement (e.g.\ \pref{pus:11}). \begin{examps} \item \label{pus:11} \select{SELECT DISTINCT r2.1, r2.2 \\ VALID VALID(r2) \\ FROM (\select{SELECT DISTINCT r1.1, r1.2 \\ VALID VALID(r1) \\ FROM rel AS r1} \\ \ \ \ \ \ )(PERIOD) AS r2} \end{examps} For simplicity, I allow stand-alone statements like \pref{pus:10}. I assume that \pref{pus:10} generates the same relation as \pref{pus:11}. I also allow stand-alone \sql{SELECT} statements enclosed in brackets (e.g.\ \pref{pus:12}). I assume that the enclosing brackets are simply ignored. \begin{examps} \item \label{pus:12} \sql{(}\select{SELECT DISTINCT r1.1, r1.2 \\ VALID VALID(r1) \\ FROM rel AS r1}\\ \sql{)} \end{examps} \section{Additional TSQL2 terminology} \label{additional_tsql2} This section defines some additional terminology, that is used to formulate and prove the correctness of the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation. \paragraph{Column reference:} A \emph{column reference} is an expression of the form $\alpha.i$ or \sql{VALID(}$\alpha$\sql{)}, where $\alpha$ is a correlation name and $i \in \{1,2,3,\dots\}$ (e.g.\ \sql{sal.2}, \sql{VALID(sal)}). \paragraph{Binding context:} A \sql{SELECT} statement $\Sigma$ is a \emph{binding context} for a column reference $\alpha.i$ or \sql{VALID(}$\alpha$\sql{)} iff: \begin{itemize} \item the column reference is part of $\Sigma$, \item $\alpha$ is defined (in the sense of section \ref{same_FROM}) by the topmost \sql{FROM} clause of $\Sigma$, and \item the column reference is not in the topmost \sql{FROM} clause of $\Sigma$; or it is in the topmost \sql{FROM} clause of $\Sigma$, but the definition of $\alpha$ precedes the column reference. \end{itemize} By \emph{topmost \sql{FROM} clause of $\Sigma$} I mean the (single) \sql{FROM} clause of $\Sigma$ that does not appear in any \sql{SELECT} statement embedded in $\Sigma$ (e.g.\ the topmost \sql{FROM} clause of \pref{fcn:1} is the ``\sql{FROM tab1 AS r1, (\dots) AS r3}''). We will often have to distinguish between individual \emph{occurrences} of column references. For example, \pref{fcn:2} is a binding context for the occurrence of \sql{VALID(r1)} in the \sql{VALID} clause, because that occurrence is part of \pref{fcn:2}, \sql{r1} is defined by the topmost \sql{FROM} clause of \pref{fcn:2}, and the occurrence of \sql{VALID(r1)} is not in the topmost \sql{FROM} clause of \pref{fcn:2}. \pref{fcn:2}, however, is \emph{not} a binding context for the occurrence of \sql{VALID(r1)} in the embedded \sql{SELECT} statement of \pref{fcn:2}, because that occurrence is in the topmost \sql{FROM} clause, and it does not follow the definition of \sql{r1}. \begin{examps} \item \label{fcn:2} \select{SELECT DISTINCT r1.1, r3.2 \\ VALID VALID(r1) \\ FROM (\select{SELECT DISTINCT SNAPSHOT r2.1, r2.2 \\ FROM tab2 AS r2 \\ WHERE VALID(r2) CONTAINS VALID(r1)} \\ \ \ \ \ \ ) AS r3, \\ \ \ \ \ \ tab1 AS r1 \\ WHERE r1.1 = 'J.Adams'} \end{examps} In contrast, \pref{fcn:1} \emph{is} a binding context for the \sql{VALID(r1)} in the embedded \sql{SELECT}, because the definition of \sql{r1} precedes that occurrence of \sql{VALID(r1)}. \begin{examps} \item \label{fcn:1} \select{SELECT DISTINCT r1.1, r3.2 \\ VALID VALID(r1) \\ FROM tab1 AS r1, \\ \ \ \ \ \ (\select{SELECT DISTINCT SNAPSHOT r2.1, r2.2 \\ FROM tab2 AS r2 \\ WHERE VALID(r2) CONTAINS VALID(r1)} \\ \ \ \ \ \ ) AS r3 \\ WHERE r1.1 = 'J.Adams'} \end{examps} In both \pref{fcn:2} and \pref{fcn:1}, the overall \sql{SELECT} statement is not a binding context for \sql{r2.1}, \sql{r2.2}, and \sql{VALID(r2)}, because \sql{r2} is not defined by the topmost \sql{FROM} clause of the overall \sql{SELECT} statement. The embedded \sql{SELECT} statement of \pref{fcn:2} and \pref{fcn:1}, however, \emph{is} a binding context for \sql{r2.1}, \sql{r2.2}, and \sql{VALID(r2)}. \paragraph{Free column reference:} A column reference $\alpha.i$ or \sql{VALID($\alpha$)} is a \emph{free column reference} in a \textsc{Tsql2}\xspace expression $\xi$, iff: \begin{itemize} \item the column reference is part of $\xi$, and \item there is no \sql{SELECT} statement in $\xi$ (possibly being the whole $\xi$) that is a binding context for the column reference. \end{itemize} The \sql{VALID(r1)} in the embedded \sql{SELECT} statement of \pref{fcn:2} is free in \pref{fcn:2}, because there is no binding context for that occurrence in \pref{fcn:2}. In contrast, the \sql{VALID(r1)} in the \sql{VALID} clause of \pref{fcn:2} is not free in \pref{fcn:2}, because \pref{fcn:2} is a binding context for that occurrence. The \sql{VALID(r2)} of \pref{fcn:2} is not free in \pref{fcn:2}, because the embedded \sql{SELECT} statement is a binding context for \sql{VALID(r2)}. A correlation name $\alpha$ \emph{has a free column reference in} a \textsc{Tsql2}\xspace expression $\xi$, iff there is a free column reference $\alpha.i$ or \sql{VALID($\alpha$)} in $\xi$. For every \textsc{Tsql2}\xspace expression $\xi$, $\ensuremath{\mathit{FCN}}\xspace(\xi)$ \index{fcn@$\ensuremath{\mathit{FCN}}\xspace(\xi)$ (set of all correlation names with free column references in $\xi$)} is the set of all correlation names that have a free column reference in $\xi$. For example, if $\xi$ is \pref{fcn:2}, $\ensuremath{\mathit{FCN}}\xspace(\xi) = \{$\sql{r1}$\}$ (the \sql{VALID(r1)} of the embedded \sql{SELECT} statement is free in \pref{fcn:2}). There must be no free column references in the overall \sql{SELECT} statements that are submitted to the \textsc{Tsql2}\xspace (or \textsc{Sql-92}\xspace) interpreter (though there may be free column references in their embedded \sql{SELECT} statements). Hence, it is important to prove that there are no free column references in the overall \sql{SELECT} statements generated by the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation. \paragraph{Value expression:} In \textsc{Tsql2}\xspace (and \textsc{Sql-92}\xspace), \emph{value expression} refers to expressions that normally evaluate to elements of $D$ (universal domain). (The meaning of ``normally'' will be explained in following paragraphs.) For example, \sql{'J.Adams'}, \sql{VALID(sal)}, and \sql{INTERSECT(PERIOD '[1993 - 1995]', PERIOD '[1994 - 1996]')} are all value expressions. \paragraph{Assignment to correlation names:} An \emph{assignment to correlation names} is a function $g^{db}$ \index{gdb@$g^{db}()$, $(g^{db})^{\alpha}_{\tup{v_1, v_2, \dots}}()$ (assignment to correlation names)} that maps every \textsc{Tsql2}\xspace correlation name to a possible tuple of a snapshot or valid-time relation. $G^{db}$ \index{Gdb@$G^{db}$ (set of all assignments to correlation names)} is the set of all assignments to correlation names If $\alpha$ is a (particular) correlation name, $\tup{v_1, v_2, \dots}$ is a (particular) tuple of a snapshot or valid-time relation, and $g^{db} \in G^{db}$, $(g^{db})^{\alpha}_{\tup{v_1, v_2, \dots}}$ \index{gdb@$g^{db}()$, $(g^{db})^{\alpha}_{\tup{v_1, v_2, \dots}}()$ (assignment to correlation names)} is the same as $g^{db}$, except that it assigns $\tup{v_1, v_2, \dots}$ to $\alpha$. (For every other correlation name, the values of $g^{db}$ and $(g^{db})^{\alpha}_{\tup{v_1, v_2, \dots}}$ are identical.) \paragraph{eval:} \index{eval@$eval()$ (evaluates \textsc{Tsql2}\xspace expressions)} For every \textsc{Tsql2}\xspace \sql{SELECT} statement or value expression $\xi$, and every $st \in \ensuremath{\mathit{CHRONS}}\xspace$ and $g^{db} \in G^{db}$, $eval(st, \xi, g^{db})$ is the relation (if $\xi$ is a \sql{SELECT} statement) or the element of $D$ (if $\xi$ is a value expression) that is generated when the \textsc{Tsql2}\xspace interpreter evaluates $\xi$ in the following way: \begin{itemize} \item $st$ is taken to be the current chronon. \item Every free column reference of the form $\alpha.i$ is treated as a value expression that evaluates to $v_i$, where $v_i$ is the $i$-th attribute value in the tuple $g^{db}(\alpha)$. \item Every free column reference of the form \sql{VALID($\alpha$)} is treated as a value expression that evaluates to $v_t$, where $v_t$ is the time-stamp of $g^{db}(\alpha)$. \end{itemize} If $\xi$ cannot be evaluated in this way (e.g.\ $\xi$ contains a free column reference of the form $\alpha.4$, and $g^{db}(\alpha) = \tup{v_1, v_2, v_3}$), $eval(st, \xi, g^{db})$ returns the special value $error$. \index{error@$error$ (signals evaluation error)} (I assume that $error \not\in D$.) A value expression $\xi$ \emph{normally} (but not always) evaluates to an element of $D$, because when errors arise $eval(st, \xi, g^{db}) = error \not\in D$. If, however, $eval(st, \xi, g^{db}) \not= error$, $eval(st, \xi, g^{db}) \in D$. Strictly speaking, $eval$ should also have as its argument the database against which $\xi$ is evaluated. For simplicity, I overlook this detail. Finally, if $\ensuremath{\mathit{FCN}}\xspace(\xi) = \emptyset$ ($\xi$ contains no free column references), $eval(st, \xi, g^{db})$ does not depend on $g^{db}$. In this case, I write simply $eval(st, \xi)$. \section{Modifications in TOP and additional TOP terminology} \label{TOP_mods} In the formulae generated by the English to \textsc{Top}\xspace translation, each $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ is conjoined with a subformula that is (or contains another subformula) of the form $\ensuremath{\mathit{At}}\xspace[\beta, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\beta, \phi]$, or $\ensuremath{\mathit{After}}\xspace[\beta, \phi]$ ($\sigma \in \ensuremath{\mathit{PARTS}}\xspace$, $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, $\beta \in \ensuremath{\mathit{VARS}}\xspace$, and the $\beta$ of \ensuremath{\mathit{Part}}\xspace is the same as that of \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace). For example, \pref{tmods:1} and \pref{tmods:3} are mapped to \pref{tmods:2} and \pref{tmods:4}. Also the reading of \pref{tmods:5} where Monday is the time when the tank was empty (rather than a reference time; section \ref{past_perfect}) is mapped to \pref{tmods:6}. \begin{examps} \item Tank 2 was empty on a Monday. \label{tmods:1} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, mon^v] \land \ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{tmods:2} \item On which Monday was tank 2 empty? \label{tmods:3} \item $?mon^v \; \ensuremath{\mathit{Part}}\xspace[monday^g, mon^v] \land \ensuremath{\mathit{At}}\xspace[mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{tmods:4} \item Tank 2 had been empty on a Monday. \label{tmods:5} \item $\ensuremath{\mathit{Part}}\xspace[monday^g, mon^v] \land \ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[mon^v, empty(tank2)]]]$ \label{tmods:6} \end{examps} In this chapter, I use a slightly different version of \textsc{Top}\xspace, where the \ensuremath{\mathit{Part}}\xspace is merged with the corresponding \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace. For example, \pref{tmods:2}, \pref{tmods:4}, and \pref{tmods:6} become \pref{tmods:7}, \pref{tmods:8}, and \pref{tmods:9} respectively. \begin{examps} \item $\ensuremath{\mathit{At}}\xspace[monday^g, mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{tmods:7} \item $?mon^v \; \ensuremath{\mathit{At}}\xspace[monday^g, mon^v, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{tmods:8} \item $\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Perf}}\xspace[e2^v, \ensuremath{\mathit{At}}\xspace[monday^g, mon^v, empty(tank2)]]]$ \label{tmods:9} \end{examps} The semantics of $\ensuremath{\mathit{At}}\xspace[\sigma, \beta, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\sigma, \beta, \phi]$, and $\ensuremath{\mathit{After}}\xspace[\sigma, \beta, \phi]$ follow ($f$ is $\ensuremath{\mathit{f_{gparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{GPARTS}}\xspace$, and $\ensuremath{\mathit{f_{cparts}}}\xspace$ if $\sigma \in \ensuremath{\mathit{CPARTS}}\xspace$.) \begin{itemize} \item $\denot{st,et,lt,g}{\ensuremath{\mathit{At}}\xspace[\sigma, \beta, \phi]} = T$ iff $g(\beta) \in f(\sigma)$ and $\denot{st, et, lt \intersect g(\beta), g}{\phi} = T$. \item $\denot{st,et,lt,g}{\ensuremath{\mathit{Before}}\xspace[\sigma, \beta, \phi]} = T$ iff $g(\beta) \in f(\sigma)$ and $\denot{st, et, lt \intersect [t_{first}, minpt(\denot{g}{\beta})), g} {\phi} = T$. \item $\denot{st,et,lt,g}{\ensuremath{\mathit{After}}\xspace[\sigma, \beta, \phi]} = T$ iff $g(\beta) \in f(\sigma)$ and $\denot{st, et, lt \intersect (maxpt(\denot{g}{\beta}), t_{last}], g} {\phi} = T$. \end{itemize} In the \textsc{Top}\xspace version of this chapter, $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$, $\ensuremath{\mathit{At}}\xspace[\beta, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\beta, \phi]$, and $\ensuremath{\mathit{After}}\xspace[\beta, \phi]$ ($\beta \in \ensuremath{\mathit{VARS}}\xspace$) are no longer yes/no formulae. $\ensuremath{\mathit{At}}\xspace[\kappa, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\kappa, \phi]$, and $\ensuremath{\mathit{After}}\xspace[\kappa, \phi]$ ($\kappa \in \ensuremath{\mathit{CONS}}\xspace$), however, are still yes/no formulae. The \textsc{Top}\xspace version of chapter \ref{TOP_chapter} is more convenient for the English to \textsc{Top}\xspace mapping, while the version of this chapter simplifies the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation. In the prototype \textsc{Nlitdb}\xspace, there is a converter between the module that translates from English to \textsc{Top}\xspace and the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. The module that translates from English to \textsc{Top}\xspace maps \pref{tmods:1}, \pref{tmods:3}, and \pref{tmods:5} to \pref{tmods:2}, \pref{tmods:4}, and \pref{tmods:6} respectively. The converter turns \pref{tmods:2}, \pref{tmods:4}, and \pref{tmods:6} into \pref{tmods:7}, \pref{tmods:8}, and \pref{tmods:9}, which are then passed to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. The reader is reminded that the $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta, \nu_{ord}]$ version of \ensuremath{\mathit{Part}}\xspace is not used in the translation from English to \textsc{Top}\xspace (section \ref{TOP_FS}). Hence, only the $\ensuremath{\mathit{Part}}\xspace[\sigma, \beta]$ form of \ensuremath{\mathit{Part}}\xspace is possible in formulae generated by the English to \textsc{Top}\xspace translation. In the \textsc{Top}\xspace version of this chapter, \ensuremath{\mathit{Part}}\xspace operators of this form are merged with \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace operators. Therefore, no \ensuremath{\mathit{Part}}\xspace operators occur in the formulae that are passed to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. As with the \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, and \ensuremath{\mathit{After}}\xspace of chapter \ref{TOP_chapter} (section \ref{top_syntax}), in every $\ensuremath{\mathit{At}}\xspace[\sigma, \beta, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\sigma, \beta, \phi]$, and $\ensuremath{\mathit{After}}\xspace[\sigma, \beta, \phi]$, I require $\beta$ not to occur within $\phi$. This is needed to prove the correctness of the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation. To avoid complications in the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation, I require that in any $\ensuremath{\mathit{At}}\xspace[\kappa, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\kappa, \phi]$, or $\ensuremath{\mathit{After}}\xspace[\kappa, \phi]$ ($\kappa \in \ensuremath{\mathit{CONS}}\xspace$, $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$) that is passed to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator, $\ensuremath{\mathit{f_{cons}}}\xspace(\kappa) \in \ensuremath{\mathit{PERIODS}}\xspace$. (The definitions of section \ref{at_before_after_op} are more liberal: they allow $\ensuremath{\mathit{f_{cons}}}\xspace(\kappa)$ not to belong to \ensuremath{\mathit{PERIODS}}\xspace, though if $\ensuremath{\mathit{f_{cons}}}\xspace(\kappa) \not\in \ensuremath{\mathit{PERIODS}}\xspace$, the denotation of $\ensuremath{\mathit{At}}\xspace[\kappa, \phi]$, $\ensuremath{\mathit{Before}}\xspace[\kappa, \phi]$, or $\ensuremath{\mathit{After}}\xspace[\kappa, \phi]$ is always $F$.) In practice, formulae generated by the English to \textsc{Top}\xspace mapping never violate this constraint. For every $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, $\corn{\phi}$ \index{'`@$\corn{}$ (corners)} (pronounced ``corners $\phi$'') is the tuple $\tup{\tau_1, \tau_2, \tau_3, \dots, \tau_n}$, where $\tau_1, \dots, \tau_n$ are all the constants that are used as arguments of predicates in $\phi$, and all the variables that occur in $\phi$, in the same order (from left to right) they appear in $\phi$. If a constant occurs more than once as a predicate argument in $\phi$, or if a variable occurs more than once in $\phi$, there are multiple $\tau_i$s in $\corn{\phi}$ for that constant or variable. If $\corn{\phi} = \tup{\tau_1, \tau_2, \tau_3, \dots, \tau_n}$, the \emph{length} of $\corn{\phi}$ is $n$. For example, if: \[ \phi = \ensuremath{\mathit{Ntense}}\xspace[t^v, woman(p^v)] \land \ensuremath{\mathit{At}}\xspace[1991, \ensuremath{\mathit{Past}}\xspace[e^v, manager\_of(p^v, sales)]] \] then $\corn{\phi} = \tup{t^v, p^v, e^v, p^v, sales}$, and the length of $\corn{\phi}$ is 5. \section{Linking the TOP model to the database} \label{linking_model} As discussed in section \ref{denotation}, the answer to an English question submitted at $st$ must report the denotation $\denot{M,st}{\phi}$ of the corresponding \textsc{Top}\xspace formula $\phi$. $\denot{M,st}{\phi}$ follows from the semantics of \textsc{Top}\xspace, provided that the model $M$, which intuitively provides all the necessary information about the modelled world, has been defined. In a \textsc{Nlidb}\xspace, the only source of information about the world is the database.\footnote{This is not entirely true in the framework of this thesis, as there is also a type-hierarchy of world-entities in the \textsc{Hpsg}\xspace grammar (section \ref{HPSG_basics}).} Hence, $M$ has to be defined in terms of the information in the database. This mainly involves defining \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace (which are parts of $M$) in terms of database concepts. \begin{figure} \hrule \begin{center} \medskip \includegraphics[scale=.6]{link_paths} \caption{Paths from basic \textsc{Top}\xspace expressions to the modelled world} \label{link_paths_fig} \end{center} \hrule \end{figure} \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace show how certain basic \textsc{Top}\xspace expressions (constants, predicates, and partitioning names) relate to the modelled world. These functions will be defined in terms of the functions \ensuremath{\mathit{h_{cons}}}\xspace, \ensuremath{\mathit{h_{pfuns}}}\xspace, \ensuremath{\mathit{h_{culms}}}\xspace, \ensuremath{\mathit{h_{cparts}}}\xspace, and \ensuremath{\mathit{h_{gparts}}}\xspace (to be discussed in section \ref{h_funs}), and $f_D$ (section \ref{relational}). Roughly speaking, the $h$ functions map basic \textsc{Top}\xspace expressions to database constructs (attribute values or relations), and $f_D$ maps the attribute values of these constructs to world objects (figure \ref{link_paths_fig}). \ensuremath{\mathit{h_{cons}}}\xspace, \ensuremath{\mathit{h_{pfuns}}}\xspace, \ensuremath{\mathit{h_{culms}}}\xspace, \ensuremath{\mathit{h_{cparts}}}\xspace, and \ensuremath{\mathit{h_{gparts}}}\xspace will in turn be defined in terms of the functions \ensuremath{\mathit{h'_{cons}}}\xspace, \ensuremath{\mathit{h'_{pfuns}}}\xspace, \ensuremath{\mathit{h'_{culms}}}\xspace, \ensuremath{\mathit{h'_{cparts}}}\xspace, and \ensuremath{\mathit{h'_{gparts}}}\xspace (to be discussed in section \ref{via_TSQL2}), and $eval$ (section \ref{additional_tsql2}). The $h'$ functions map basic \textsc{Top}\xspace expressions to \textsc{Tsql2}\xspace expressions, and $eval$ maps \textsc{Tsql2}\xspace expressions to database constructs. After defining the $h'$ functions, one could compute $\denot{M,st}{\phi}$ using a reasoning system, that would contain rules encoding the semantics of \textsc{Top}\xspace, and that would use the path basic \textsc{Top}\xspace expressions $\rightarrow$ \textsc{Tsql2}\xspace expressions $\rightarrow$ database constructs $\rightarrow$ modelled world (figure \ref{link_paths_fig}) to compute any necessary values of \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace. That is, only basic \textsc{Top}\xspace expressions would be translated into \textsc{Tsql2}\xspace, and the \textsc{Dbms}\xspace would be used only to evaluate the \textsc{Tsql2}\xspace translations of these expressions. The rest of the processing to compute $\denot{M,st}{\phi}$ would be carried out by the reasoning system. This thesis adopts an alternative approach that exploits the capabilities of the \textsc{Dbms}\xspace to a larger extent, and that requires no reasoning system. Based on the $h'$ functions (that map only basic \textsc{Top}\xspace expressions to \textsc{Tsql2}\xspace expressions), a method to translate \emph{any} \textsc{Top}\xspace formula into \textsc{Tsql2}\xspace will be developed. Each \textsc{Top}\xspace formula $\phi$ will be mapped to a single \textsc{Tsql2}\xspace query (figure \ref{trans_paths_fig}). This will be executed by the \textsc{Dbms}\xspace, generating a relation that represents (via an interpretation function) $\denot{M,st}{\phi}$. It will be proven formally that this approach generates indeed $\denot{M,st}{\phi}$ (i.e.\, that paths 1 and 2 of figure \ref{trans_paths_fig} lead to the same result). \begin{figure} \hrule \begin{center} \medskip \includegraphics[scale=.6]{trans_paths} \caption{Paths from TOP formulae to their denotations} \label{trans_paths_fig} \end{center} \hrule \end{figure} There is one further complication: the values of \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace will ultimately be obtained by evaluating \textsc{Tsql2}\xspace expressions returned by \ensuremath{\mathit{h'_{cons}}}\xspace, \ensuremath{\mathit{h'_{pfuns}}}\xspace, \ensuremath{\mathit{h'_{culms}}}\xspace, \ensuremath{\mathit{h'_{cparts}}}\xspace, and \ensuremath{\mathit{h'_{gparts}}}\xspace. A \textsc{Tsql2}\xspace expression, however, may generate different results when evaluated at different times (e.g.\ a \sql{SELECT} statement may return different results after a database relation on which the statement operates has been updated). This causes the values of \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace to become sensitive to the time where the \textsc{Tsql2}\xspace expressions of the $h'$ functions are evaluated. We want this time to be $st$, so that the \textsc{Tsql2}\xspace expressions of the $h'$ functions will operate on the information that is in the database when the question is submitted, and so that a \textsc{Tsql2}\xspace literal like \sql{PERIOD 'today'} (section \ref{tsql2_lang}) in the expressions of the $h'$ functions will be correctly taken to refer to the day that contains $st$. To accommodate this, \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace must be made sensitive to $st$: \ensuremath{\mathit{f_{cons}}}\xspace becomes a function $\ensuremath{\mathit{PTS}}\xspace \mapsto (\ensuremath{\mathit{CONS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace)$ instead of $\ensuremath{\mathit{CONS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace$. This allows the world objects that are assigned to \textsc{Top}\xspace constants via \ensuremath{\mathit{f_{cons}}}\xspace to be different at different $st$s. Similarly, \ensuremath{\mathit{f_{pfuns}}}\xspace is now a function over \ensuremath{\mathit{PTS}}\xspace. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)$ is in turn a function that maps each pair $\tup{\pi,n}$, where $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, to another function $(\ensuremath{\mathit{OBJS}}\xspace)^n \mapsto pow(\ensuremath{\mathit{PERIODS}}\xspace)$ (cf.\ the definition of \ensuremath{\mathit{f_{pfuns}}}\xspace in section \ref{top_model}). The definitions of \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace are modified accordingly. Whatever restrictions applied to \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace, now apply to $\ensuremath{\mathit{f_{cons}}}\xspace(st)$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)$, $\ensuremath{\mathit{f_{culms}}}\xspace(st)$, $\ensuremath{\mathit{f_{cparts}}}\xspace(st)$, and $\ensuremath{\mathit{f_{gparts}}}\xspace(st)$, for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$. Also, wherever \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{gparts}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace were used in the semantics of \textsc{Top}\xspace, $\ensuremath{\mathit{f_{cons}}}\xspace(st)$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)$, $\ensuremath{\mathit{f_{culms}}}\xspace(st)$, and $\ensuremath{\mathit{f_{gparts}}}\xspace(st)$ should now be used. The \textsc{Top}\xspace model also becomes sensitive to $st$, and is now defined as follows: \[ M(st) = \tup{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}, \ensuremath{\mathit{OBJS}}\xspace, \ensuremath{\mathit{f_{cons}}}\xspace(st), \ensuremath{\mathit{f_{pfuns}}}\xspace(st), \ensuremath{\mathit{f_{culms}}}\xspace(st), \ensuremath{\mathit{f_{gparts}}}\xspace(st), \ensuremath{\mathit{f_{cparts}}}\xspace(st)} \] Intuitively, $M(st)$ reflects the history of the world as recorded in the database at $st$. (If the database supports both valid and transaction time, $M(st)$ reflects the ``beliefs'' of the database at $st$; see section \ref{tdbs_general}.) The answer to an English question submitted at $st$ must now report the denotation $\denot{M(st),st}{\phi}$ of the corresponding \textsc{Top}\xspace formula $\phi$. \section{The $h$ functions} \label{h_funs} I first discuss \ensuremath{\mathit{h_{cons}}}\xspace, \ensuremath{\mathit{h_{pfuns}}}\xspace, \ensuremath{\mathit{h_{culms}}}\xspace, \ensuremath{\mathit{h_{cparts}}}\xspace, and \ensuremath{\mathit{h_{gparts}}}\xspace, the functions that -- roughly speaking -- map basic \textsc{Top}\xspace expressions to database constructs. As with \ensuremath{\mathit{f_{cons}}}\xspace, \ensuremath{\mathit{f_{pfuns}}}\xspace, \ensuremath{\mathit{f_{culms}}}\xspace, \ensuremath{\mathit{f_{cparts}}}\xspace, and \ensuremath{\mathit{f_{gparts}}}\xspace, the values of \ensuremath{\mathit{h_{cons}}}\xspace, \ensuremath{\mathit{h_{pfuns}}}\xspace, \ensuremath{\mathit{h_{culms}}}\xspace, \ensuremath{\mathit{h_{cparts}}}\xspace, and \ensuremath{\mathit{h_{gparts}}}\xspace will ultimately be obtained by evaluating \textsc{Tsql2}\xspace expressions at $st$. The results of these evaluations can be different at different $st$s, and hence the definitions of the $h$ functions must be sensitive to $st$. \paragraph{$\mathbf{h_{cons}}$:} \index{hcons@$\ensuremath{\mathit{h_{cons}}}\xspace()$ (\textsc{Top}\xspace constants to attribute values)} \ensuremath{\mathit{h_{cons}}}\xspace is a function $\ensuremath{\mathit{PTS}}\xspace \mapsto (\ensuremath{\mathit{CONS}}\xspace \mapsto D)$. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{h_{cons}}}\xspace(st)$ is in turn a function that maps each \textsc{Top}\xspace constant to an attribute value that represents the same world-entity. For example, $\ensuremath{\mathit{h_{cons}}}\xspace(st)$ could map the \textsc{Top}\xspace constant $sales\_department$ to the string attribute value $Sales \; Department$, and the constant $\mathit{today}$ to the element of $D_P$ ($D_P \subseteq D$) which denotes the day-period that contains $st$. \paragraph{$\mathbf{h_{pfuns}}$:} \index{hpfuns@$\ensuremath{\mathit{h_{pfuns}}}\xspace()$ (predicates to relations showing maximal periods of situations)} \ensuremath{\mathit{h_{pfuns}}}\xspace is a function over \ensuremath{\mathit{PTS}}\xspace. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)$ is in turn a function over $\ensuremath{\mathit{PFUNS}}\xspace \times \{1,2,3,\dots\}$, such that for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$ (section \ref{bcdm}). $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)$ is intended to map every \textsc{Top}\xspace predicate of functor $\pi$ and arity $n$ to a relation that shows for which arguments of the predicate and at which maximal periods the situation represented by the predicate is true, according to the ``beliefs'' of the database at $st$. For example, if $circling(ba737)$ represents the situation where BA737 is circling, and according to the ``beliefs'' of the database at $st$, $p$ is a maximal period where BA737 was/is/will be circling, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(circling, 1)$ must contain a tuple $\tup{v;v_t}$, where $f_D(v) = \ensuremath{\mathit{f_{cons}}}\xspace(ba737)$ ($v$ denotes the flight BA737), and $f_D(v_t) = p$. Similarly, if $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(circling, 1)$ contains a tuple $\tup{v;v_t}$, where $f_D(v) = \ensuremath{\mathit{f_{cons}}}\xspace(ba737)$ and $f_D(v_t) = p$, $p$ is a maximal period where BA737 was/is/will be circling, according to the ``beliefs'' of the database at $st$. \paragraph{$\mathbf{h_{culms}}$:} \index{hculms@$\ensuremath{\mathit{h_{culms}}}\xspace()$ (predicates to relations showing if situations reach their climaxes)} \ensuremath{\mathit{h_{culms}}}\xspace is a function over \ensuremath{\mathit{PTS}}\xspace. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{h_{culms}}}\xspace(st)$ is in turn a function over $\ensuremath{\mathit{PFUNS}}\xspace \times \{1,2,3,\dots\}$, such that for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n) \in \ensuremath{\mathit{SREL}}\xspace(n)$. Intuitively, \ensuremath{\mathit{h_{culms}}}\xspace plays the same role as \ensuremath{\mathit{f_{culms}}}\xspace (section \ref{top_model}). In practice, \ensuremath{\mathit{h_{culms}}}\xspace is consulted only for predicates that describe situations with inherent climaxes. $\ensuremath{\mathit{h_{culms}}}\xspace(st)$ maps each \textsc{Top}\xspace predicate of functor $\pi$ and arity $n$ to a relation that shows for which predicate arguments the situation of the predicate reaches its climax at the latest time-point where the situation is ongoing, according to the ``beliefs'' of the database at $st$. If, for example, $inspecting(j\_adams, ba737)$ represents the situation where J.Adams is inspecting BA737, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(inspecting, 2)$ is a relation in $\ensuremath{\mathit{NVREL}_P}\xspace(2)$ and $\ensuremath{\mathit{h_{culms}}}\xspace(st)(inspecting, 2)$ a relation in $\ensuremath{\mathit{SREL}}\xspace(2)$. If, according to the ``beliefs'' of the database at $st$, the maximal periods where J.Adams was/is/will be inspecting BA737 are $p_1, p_2, \dots, p_j$, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(inspecting, 2)$ contains the tuples $\tup{v_1, v_2; v^1_t}$, $\tup{v_1, v_2; v^2_t}$, \dots, $\tup{v_1, v_2; v^j_t}$, where $f_D(v_1) = \ensuremath{\mathit{f_{cons}}}\xspace(j\_adams)$, $f_D(v_2) = \ensuremath{\mathit{f_{cons}}}\xspace(ba737)$, and $f_D(v^1_t) = p_1$, $f_D(v^2_t) = p_2$, \dots, $f_D(v^j_t) = p_j$. Let us assume that $p$ is the latest maximal period among $p_1, \dots, p_j$. $\ensuremath{\mathit{h_{culms}}}\xspace(st)(inspecting, 2)$ contains $\tup{v_1, v_2}$ iff according to the ``beliefs'' of the database at $st$, the inspection of BA737 by J.Adams reaches its completion at the end of $p$. \paragraph{$\mathbf{h_{gparts}}$:} \index{hgparts@$\ensuremath{\mathit{h_{gparts}}}\xspace()$ (gappy part.\ names to relations representing gappy partitionings)} \ensuremath{\mathit{h_{gparts}}}\xspace is a function over \ensuremath{\mathit{PTS}}\xspace. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{h_{gparts}}}\xspace(st)$ is in turn a function that maps every element of \ensuremath{\mathit{GPARTS}}\xspace to an $r \in \ensuremath{\mathit{SREL}}\xspace(1)$, such that the set $S = \{ f_D(v) \mid \tup{v} \in r \}$ is a gappy partitioning. $\ensuremath{\mathit{h_{gparts}}}\xspace(st)$ is intended to map each \textsc{Top}\xspace gappy partitioning name $\sigma_g$ to a one-attribute snapshot relation $r$, whose attribute values represent the periods of the gappy partitioning $S$ that is assigned to $\sigma_g$. For example, $\ensuremath{\mathit{h_{gparts}}}\xspace(st)$ could map $monday^g$ to a one-attribute snapshot relation whose attribute values denote all the Monday-periods. As with the other $h$ functions, the values of \ensuremath{\mathit{h_{gparts}}}\xspace will ultimately be obtained by evaluating \textsc{Tsql2}\xspace expressions at $st$ (see section \ref{via_TSQL2} below). The results of these evaluations can in principle be different at different $st$s, and this is why \ensuremath{\mathit{h_{gparts}}}\xspace is defined to be sensitive to $st$. In practice, however, the \textsc{Tsql2}\xspace expressions that are evaluated to obtain the values of \ensuremath{\mathit{h_{gparts}}}\xspace will be insensitive to their evaluation time, and hence the values of \ensuremath{\mathit{h_{gparts}}}\xspace will not depend on $st$. Similar comments apply to \ensuremath{\mathit{h_{cparts}}}\xspace below. \paragraph{$\mathbf{h_{cparts}}$:} \index{hcparts@$\ensuremath{\mathit{h_{cparts}}}\xspace()$ (compl.\ part.\ names to relations representing compl.\ partitionings)} \ensuremath{\mathit{h_{cparts}}}\xspace is a function over \ensuremath{\mathit{PTS}}\xspace. For every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{h_{cparts}}}\xspace(st)$ is in turn a function that maps every element of \ensuremath{\mathit{CPARTS}}\xspace to an $r \in \ensuremath{\mathit{SREL}}\xspace(1)$, such that the set $S = \{f_D(v) \mid \tup{v} \in r \}$ is a complete partitioning. $\ensuremath{\mathit{h_{cparts}}}\xspace(st)$ is intended to map each \textsc{Top}\xspace complete partitioning name $\sigma_c$ to a one-attribute snapshot relation $r$, whose attribute values represent the periods of the complete partitioning $S$ that is assigned to $\sigma_c$. For example, $\ensuremath{\mathit{h_{cparts}}}\xspace(st)$ could map $day^c$ to a one-attribute snapshot relation whose attribute values denote all the day-periods. \section{The TOP model in terms of database concepts} \label{resulting_model} The \textsc{Top}\xspace model (see section \ref{top_model} and the revisions of section \ref{linking_model}) can now be defined in terms of database concepts as follows. \paragraph{Point structure:} $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec} \defeq \tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}$ \\ As mentioned in section \ref{tsql2_time}, $\ensuremath{\mathit{CHRONS}}\xspace \not= \emptyset$, and $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}$ has the properties of transitivity, irreflexivity, linearity, left and right boundedness, and discreteness. Hence, $\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}$ qualifies as a point structure for \textsc{Top}\xspace (section \ref{temporal_ontology}). Since $\tup{\ensuremath{\mathit{PTS}}\xspace, \prec} = \tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}$, $\ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}} = \ensuremath{\mathit{PERIODS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}}$, and $\ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{PTS}}\xspace, \prec}} = \ensuremath{\mathit{INSTANTS}}\xspace_{\tup{\ensuremath{\mathit{CHRONS}}\xspace, \prec^{chrons}}}$. I write simply \ensuremath{\mathit{PERIODS}}\xspace and \ensuremath{\mathit{INSTANTS}}\xspace to refer to these sets. \paragraph{$\mathbf{OBJS}$:} $\ensuremath{\mathit{OBJS}}\xspace \defeq \ensuremath{\mathit{OBJS^{db}}}\xspace$ \\ Since $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{OBJS^{db}}}\xspace$ (section \ref{bcdm}) and $\ensuremath{\mathit{OBJS}}\xspace = \ensuremath{\mathit{OBJS}}\xspace^{db}$, $\ensuremath{\mathit{PERIODS}}\xspace \subseteq \ensuremath{\mathit{OBJS}}\xspace$, as required by section \ref{top_model}. \paragraph{$\mathbf{f_{cons}}$:} \index{fcons@$\ensuremath{\mathit{f_{cons}}}\xspace()$ (maps \textsc{Top}\xspace constants to world objects)} For every $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\kappa \in \ensuremath{\mathit{CONS}}\xspace$, I define $\ensuremath{\mathit{f_{cons}}}\xspace(st)(\kappa) \defeq f_D(\ensuremath{\mathit{h_{cons}}}\xspace(st)(\kappa))$. Since $\ensuremath{\mathit{h_{cons}}}\xspace(st)$ is a function $\ensuremath{\mathit{CONS}}\xspace \mapsto D$, and $f_D$ is a function $D \mapsto \ensuremath{\mathit{OBJS^{db}}}\xspace$, and $\ensuremath{\mathit{OBJS}}\xspace = \ensuremath{\mathit{OBJS^{db}}}\xspace$, $\ensuremath{\mathit{f_{cons}}}\xspace(st)$ is a function $\ensuremath{\mathit{CONS}}\xspace \mapsto \ensuremath{\mathit{OBJS}}\xspace$, as required by section \ref{top_model} and the revisions of section \ref{linking_model}. \paragraph{$\mathbf{f_{pfuns}}$:} \index{fpfuns@$\ensuremath{\mathit{f_{pfuns}}}\xspace()$ (returns the maximal periods where predicates hold)} According to section \ref{top_model} and the revisions of section \ref{linking_model}, for every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)$ must be a function: \[ \ensuremath{\mathit{PFUNS}}\xspace \times \{1,2,3,\dots\} \mapsto ((\ensuremath{\mathit{OBJS}}\xspace)^n \mapsto \ensuremath{\mathit{pow}}\xspace(\ensuremath{\mathit{PERIODS}}\xspace)) \] That is, for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, every $n \in \{1,2,3,\dots\}$, and every $o_1, \dots, o_n \in \ensuremath{\mathit{OBJS}}\xspace$, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi,n)(o_1, \dots, o_n)$ must be a set of periods. I define $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi,n)(o_1, \dots, o_n)$ as follows: \[ \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1, \dots, o_n) \defeq \{ f_D(v_t) \mid \tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n); v_t} \in \ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n) \} \] The restrictions of section \ref{h_funs} guarantee that $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi,n) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$, which implies that for every $\tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n); v_t} \in \ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n)$, $f_D(v_t) \in \ensuremath{\mathit{PERIODS}}\xspace$. Hence, $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi,n)(o_1, \dots, o_n)$ is a set of periods as wanted. As discussed in section \ref{h_funs}, if $\pi(\tau_1, \dots, \tau_n)$ represents some situation, and $\tau_1, \dots, \tau_n$ denote $o_1, \dots, o_n$, then $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n)$ contains $\tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n); v_t}$ iff $f_D(v_t)$ is a maximal period where the situation of $\pi(\tau_1, \dots, \tau_n)$ holds. $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1, \dots, o_n)$ is supposed to be the set of the maximal periods where the situation of $\pi(\tau_1, \dots, \tau_n)$ holds. The definition of \ensuremath{\mathit{f_{pfuns}}}\xspace above achieves this. According to section \ref{top_model} and the revisions of section \ref{linking_model}, it must also be the case that: \[ \text{if } p_1, p_2 \in \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1,\dots,o_n) \text{ and } p_1 \union p_2 \in \ensuremath{\mathit{PERIODS}}\xspace, \text{ then } p_1 = p_2 \] \ensuremath{\mathit{f_{pfuns}}}\xspace, as defined above, has this property. The proof follows. Let us assume that $p_1$ and $p_2$ are as above, but $p_1 \not= p_2$. As discussed above, the assumption that $p_1, p_2 \in \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1,\dots,o_n)$ implies that $p_1, p_2 \in \ensuremath{\mathit{PERIODS}}\xspace$. Let $v_t^1 = \ensuremath{f_D^{-1}}\xspace(p_1)$ and $v_t^2 = \ensuremath{f_D^{-1}}\xspace(p_2)$ (i.e.\ $p_1 = f_D(v_t^1)$ and $p_2 = f_D(v_t^2)$). Since, $p_1 \not= p_2$ and \ensuremath{f_D^{-1}}\xspace is 1-1 (section \ref{relational}), $\ensuremath{f_D^{-1}}\xspace(p_1) \not= \ensuremath{f_D^{-1}}\xspace(p_2)$, i.e. $v_t^1 \not= v_t^2$. The definition of $\ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1,\dots,o_n)$, the assumptions that $p_1, p_2 \in \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(\pi, n)(o_1,\dots,o_n)$ and that $p_1 \union p_2 \in \ensuremath{\mathit{PERIODS}}\xspace$, and the fact that $p_1 = f_D(v_t^1)$ and $p_2 = f_D(v_t^2)$ imply that $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi,n)$ contains the value-equivalent tuples $\tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n); v_t^1}$ and $\tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n); v_t^2}$, where $f_D(v_t^1) \union f_D(v_t^2) \in \ensuremath{\mathit{PERIODS}}\xspace$. This conclusion, the fact that $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi,n) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$ (see previous paragraphs), and the definition of $\ensuremath{\mathit{NVREL}_P}\xspace(n)$ (section \ref{bcdm}) imply that $v_t^1 = v^t_2$, which is against the hypothesis. Hence, it cannot be the case that $p_1 \not= p_2$, i.e.\ $p_1 = p_2$. {\small Q.E.D.}\xspace \paragraph{$\mathbf{f_{culms}}$:} \index{fculms@$\ensuremath{\mathit{f_{culms}}}\xspace()$ (shows if the situation of a predicate reaches its climax)} According to section \ref{top_model} and the revisions of section \ref{linking_model}, for every $st \in \ensuremath{\mathit{PTS}}\xspace$, $\ensuremath{\mathit{f_{culms}}}\xspace(st)$ must be a function: \[ \ensuremath{\mathit{PFUNS}}\xspace \times \{1,2,3,\dots\} \mapsto ((\ensuremath{\mathit{OBJS}}\xspace)^n \mapsto \{T, F\}) \] For every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, $n \in \{1,2,3,\dots\}$, and $o_1, \dots, o_n \in \ensuremath{\mathit{OBJS}}\xspace$, I define: \[ \ensuremath{\mathit{f_{culms}}}\xspace(\pi, n)(o_1, \dots, o_n) \defeq \begin{cases} T, & \text{if } \tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n)} \in \ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n) \\ F, & \text{otherwise} \end{cases} \] The restrictions of section \ref{h_funs}, guarantee that $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n) \in \ensuremath{\mathit{SREL}}\xspace(n)$. As discussed in section \ref{h_funs}, if a predicate $\pi(\tau_1, \dots, \tau_n)$ represents some situation with an inherent climax, and $\tau_1$, \dots, $\tau_n$ denote $o_1$, \dots, $o_n$, then $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n)$ contains $\tup{\ensuremath{f_D^{-1}}\xspace(o_1), \dots, \ensuremath{f_D^{-1}}\xspace(o_n)}$ iff the situation reaches its climax at the end of the latest maximal period where the situation is ongoing. $\ensuremath{\mathit{f_{culms}}}\xspace(st)(\pi, n)(o_1, \dots, o_n)$ is supposed to be $T$ iff the situation of $\pi(\tau_1, \dots, \tau_n)$ reaches its climax at the end of the latest maximal period where it is ongoing. The definition of \ensuremath{\mathit{f_{culms}}}\xspace above achieves this. \paragraph{$\mathbf{f_{gparts}}$:} \index{fgparts@$\ensuremath{\mathit{f_{gparts}}}\xspace()$ (assigns gappy partitionings to elements of \ensuremath{\mathit{GPARTS}}\xspace)} For every $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\sigma_g \in \ensuremath{\mathit{GPARTS}}\xspace$, $\ensuremath{\mathit{f_{gparts}}}\xspace(st)(\sigma_g) \defeq \{f_D(v) \mid \tup{v} \in \ensuremath{\mathit{h_{gparts}}}\xspace(st)(\sigma_g)\}$. The restrictions on \ensuremath{\mathit{h_{gparts}}}\xspace of section \ref{h_funs} guarantee that $\ensuremath{\mathit{f_{gparts}}}\xspace(st)(\sigma_g)$ is always a gappy partitioning, as required by section \ref{top_model} and the revisions of section \ref{linking_model}. \paragraph{$\mathbf{f_{cparts}}$:} \index{fcparts@$\ensuremath{\mathit{f_{cparts}}}\xspace()$ (assigns complete partitionings to elements of \ensuremath{\mathit{CPARTS}}\xspace)} For every $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\sigma_c \in \ensuremath{\mathit{CPARTS}}\xspace$, $\ensuremath{\mathit{f_{cparts}}}\xspace(st)(\sigma_c) \defeq \{f_D(v) \mid \tup{v} \in \ensuremath{\mathit{h_{cparts}}}\xspace(st)(\sigma_c)\}$. The restrictions on \ensuremath{\mathit{h_{cparts}}}\xspace of section \ref{h_funs} guarantee that $\ensuremath{\mathit{f_{cparts}}}\xspace(st)(\sigma_c)$ is always a complete partitioning, as required by section \ref{top_model} and the revisions of section \ref{linking_model}. \section{The $h'$ functions} \label{via_TSQL2} I now discuss $\ensuremath{\mathit{h'_{cons}}}\xspace$, $\ensuremath{\mathit{h'_{pfuns}}}\xspace$, $\ensuremath{\mathit{h'_{culms}}}\xspace$, $\ensuremath{\mathit{h'_{gparts}}}\xspace$, and $\ensuremath{\mathit{h'_{cparts}}}\xspace$, the functions that map basic \textsc{Top}\xspace expressions (constants, predicates, etc.) to \textsc{Tsql2}\xspace expressions. I assume that these functions are defined by the configurer of the \textsc{Nlitdb}\xspace (section \ref{domain_config}). \paragraph{$\mathbf{h_{cons}'}$:} \index{hconsp@$\ensuremath{\mathit{h'_{cons}}}\xspace()$ (similar to \ensuremath{\mathit{h_{cons}}}\xspace but returns \textsc{Tsql2}\xspace expressions)} $\ensuremath{\mathit{h'_{cons}}}\xspace$ maps every \textsc{Top}\xspace constant $\kappa$ to a \textsc{Tsql2}\xspace value expression $\xi$, such that $\ensuremath{\mathit{FCN}}\xspace(\xi) = \emptyset$, and for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $eval(st, \xi) \in D$. (The latter guarantees that $eval(st, \xi) \not= error$.) $\xi$ is intended to represent the same world object as $\kappa$. For example, $\ensuremath{\mathit{h'_{cons}}}\xspace$ could map the \textsc{Top}\xspace constant $sales\_department$ to the \textsc{Tsql2}\xspace value expression \sql{'Sales Department'}, and the \textsc{Top}\xspace constant $yesterday$ to \sql{PERIOD 'today' - INTERVAL '1' DAY}. In practice, the values of \ensuremath{\mathit{h'_{cons}}}\xspace need to be defined only for \textsc{Top}\xspace constants that are used in the particular application domain. The values of \ensuremath{\mathit{h'_{cons}}}\xspace for other constants are not used, and can be chosen arbitrarily. Similar comments apply to \ensuremath{\mathit{h'_{pfuns}}}\xspace, \ensuremath{\mathit{h'_{culms}}}\xspace, \ensuremath{\mathit{h'_{gparts}}}\xspace, and \ensuremath{\mathit{h'_{cparts}}}\xspace. $\ensuremath{\mathit{h_{cons}}}\xspace$ is defined in terms of $\ensuremath{\mathit{h'_{cons}}}\xspace$. For every $st \in \ensuremath{\mathit{CHRONS}}\xspace$ and $\kappa \in \ensuremath{\mathit{CONS}}\xspace$: \index{hcons@$\ensuremath{\mathit{h_{cons}}}\xspace()$ (\textsc{Top}\xspace constants to attribute values)} \[ \ensuremath{\mathit{h_{cons}}}\xspace(st)(\kappa) \defeq eval(st, \ensuremath{\mathit{h'_{cons}}}\xspace(\kappa)) \] The restrictions above guarantee that $eval(st, \ensuremath{\mathit{h'_{cons}}}\xspace(\kappa)) \in D$. Hence, $\ensuremath{\mathit{h_{cons}}}\xspace(st)$ is a function $\ensuremath{\mathit{CONS}}\xspace \mapsto D$, as required by section \ref{h_funs}. \paragraph{$\mathbf{h_{pfuns}'}$:} \index{hpfunsp@$\ensuremath{\mathit{h'_{pfuns}}}\xspace()$ (similar to \ensuremath{\mathit{h_{pfuns}}}\xspace but returns \textsc{Tsql2}\xspace expressions)} $\ensuremath{\mathit{h'_{pfuns}}}\xspace$ is a function that maps every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$ to a \textsc{Tsql2}\xspace \sql{SELECT} statement $\Sigma$, such that $\ensuremath{\mathit{FCN}}\xspace(\Sigma) = \emptyset$, and for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $eval(st, \Sigma) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$. $\ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi, n)$ is intended to be a \textsc{Tsql2}\xspace \sql{SELECT} statement that generates the relation to which $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)$ maps $\pi$ and $n$ (the relation that shows for which arguments and at which maximal periods the situation described by $\pi(\tau_1, \dots, \tau_n)$ is true). \ensuremath{\mathit{h_{pfuns}}}\xspace is defined in terms of $\ensuremath{\mathit{h'_{pfuns}}}\xspace$. For every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, and $n \in \{1,2,3,\dots\}$: \index{hpfuns@$\ensuremath{\mathit{h_{pfuns}}}\xspace()$ (predicates to relations showing maximal periods of situations)} \[ \ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n) \defeq eval(st, \ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi, n)) \] The restrictions on $\ensuremath{\mathit{h'_{pfuns}}}\xspace$ above guarantee that $eval(st, \ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi, n)) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$. Hence, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n) \in \ensuremath{\mathit{NVREL}_P}\xspace(n)$, as required by section \ref{h_funs}. Let us assume, for example, that $manager(\tau)$ means that $\tau$ is a manager, and that $manager\_of$ is the relation of $\ensuremath{\mathit{NVREL}_P}\xspace(2)$ in \pref{hpfuns:99a} that shows the maximal periods where somebody is the manager of a department. (To save space, I often omit the names of the explicit attributes. These are not needed, since explicit attributes are referred to by number.) \begin{examps} \item \label{hpfuns:99a} \dbtableb{|l|l||l|} {$J.Adams$ & $sales$ & $[1/5/93, \; 31/12/94]$ \\ $J.Adams$ & $personnel$ & $[1/1/95, \; 31/3/95]$ \\ $J.Adams$ & $research$ & $[5/9/95, \; 31/12/95]$ \\ $T.Smith$ & $sales$ & $[1/1/95, \; 7/5/95]$ \\ \ \dots & \ \dots & \ \dots } \end{examps} $\ensuremath{\mathit{h'_{pfuns}}}\xspace(manager, 1)$ could be defined to be \pref{hpfuns:2}, which generates \pref{hpfuns:3} (\pref{hpfuns:3} is an element of $\ensuremath{\mathit{NVREL}_P}\xspace(1)$, as required by the definition of $\ensuremath{\mathit{h'_{pfuns}}}\xspace$). The embedded \sql{SELECT} statement of \pref{hpfuns:2} discards the second explicit attribute of $manager\_of$. The \sql{(PERIOD)} coalesces tuples that correspond to the same employees (e.g.\ the three periods for J.Adams), generating one tuple for each maximal period. \begin{examps} \item \select{SELECT DISTINCT mgr2.1 \\ VALID VALID(mgr2) \\ FROM (\select{SELECT DISTINCT mgr1.1 \\ VALID VALID(mgr1) \\ FROM manager\_of AS mgr1} \\ \ \ \ \ \ )(PERIOD) AS mgr2} \label{hpfuns:2} \item \dbtableb{|l||l|} {$J.Adams$ & $[1/5/93, \; 31/3/95]$ \\ $J.Adams$ & $[5/9/95, \; 31/12/95]$ \\ $T.Smith$ & $[1/1/95, \; 7/5/95]$ \\ \ \dots & \ \dots } \label{hpfuns:3} \end{examps} \paragraph{$\mathbf{h_{culms}'}$:} \index{hculmsp@$\ensuremath{\mathit{h'_{culms}}}\xspace()$ (similar to \ensuremath{\mathit{h'_{culms}}}\xspace but returns \textsc{Top}\xspace expressions)} \ensuremath{\mathit{h'_{culms}}}\xspace is a function that maps every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$ to a \textsc{Tsql2}\xspace \sql{SELECT} statement $\Sigma$, such that $\ensuremath{\mathit{FCN}}\xspace(\Sigma) = \emptyset$, and for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $eval(st, \Sigma) \in \ensuremath{\mathit{SREL}}\xspace(n)$. $\ensuremath{\mathit{h'_{culms}}}\xspace(\pi, n)$ is intended to be a \textsc{Tsql2}\xspace \sql{SELECT} statement that generates the relation to which $\ensuremath{\mathit{h_{culms}}}\xspace(st)$ maps $\pi$ and $n$ (the relation that shows for which arguments of $\pi(\tau_1, \dots, \tau_n)$ the situation of the predicate reaches its climax at the end of the latest maximal period where it is ongoing). \ensuremath{\mathit{h_{culms}}}\xspace is defined in terms of $\ensuremath{\mathit{h'_{culms}}}\xspace$. For every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$, and $n \in \{1,2,3,\dots\}$: \index{hculms@$\ensuremath{\mathit{h_{culms}}}\xspace()$ (predicates to relations showing if situations reach their climaxes)} \[ \ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n) \defeq eval(st, \ensuremath{\mathit{h'_{culms}}}\xspace(\pi, n)) \] The restrictions on $\ensuremath{\mathit{h'_{culms}}}\xspace$ above guarantee that $eval(st, \ensuremath{\mathit{h'_{culms}}}\xspace(\pi, n)) \in \ensuremath{\mathit{SREL}}\xspace(n)$. Hence, for every $\pi \in \ensuremath{\mathit{PFUNS}}\xspace$ and $n \in \{1,2,3,\dots\}$, $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n) \in \ensuremath{\mathit{SREL}}\xspace(n)$, as required by section \ref{h_funs}. In the airport application, for example, $inspecting(\tau_1, \tau_2, \tau_3)$ means that an occurrence $\tau_1$ of an inspection of $\tau_3$ by $\tau_2$ is ongoing. $inspections$ is a relation of the following form: \adbtable{5}{|l|l|l|l||l|}{$inspections$} {$code$ & $inspector$ & $inspected$ & $status$ &} {$i158$ & $J.Adams$ & $UK160$ & $complete$ & $[9\text{:}00am \; 1/5/95 - 9\text{:}45am \; 1/5/95]$ \\ &&&& $\;\; \union \; [10\text{:}10am \; 1/5/95 - 10\text{:}25am \; 1/5/95]$ \\ $i160$ & $J.Adams$ & $UK160$ & $incomplete$ & $[11\text{:}00pm \; 2/7/95 - 1\text{:}00am \; 3/7/95]$ \\ &&&& $\;\; \union \; [6\text{:}00am \; 3/7/95 - 6\text{:}20am \; 3/7/95]$ \\ $i205$ & $T.Smith$ & $BA737$ & $complete$ & $[8\text{:}00am \; 16/11/95 - 8\text{:}20am \; 16/11/95]$ \\ $i214$ & $T.Smith$ & $BA737$ & $incomplete$ & $[8\text{:}10am \; 14/2/96 - now]$ } The first tuple above shows that J.Adams started to inspect UK160 at 9:00am on 1/5/95, and continued the inspection up to 9:45am. He resumed the inspection at 10:10am, and completed the inspection at 10:25am on the same day. $status$ shows whether or not the inspection reaches its completion at the last time-point of the time-stamp. In the first tuple, its value is $complete$, signaling that the inspection was completed at 10:25am on 1/5/95. The inspection of the second tuple was ongoing from 11:00pm on 2/7/95 to 1:00am on 3/7/95, and from 6:00am to 6:20am on 3/7/95. It did not reach its completion at 6:20am on 3/7/95 (perhaps it was aborted for ever). The inspection of the last tuple started at 8:10am on 14/2/96 and is still ongoing. Each inspection is assigned a unique inspection code, stored as the value of the $code$ attribute. The inspection codes are useful to distinguish, for example, J.Adams' inspection of UK160 on 1/5/95 from that on 2-3/7/95 (section \ref{occurrence_ids}). $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting, 3)$ and $\ensuremath{\mathit{h'_{culms}}}\xspace(inspecting, 3)$ are defined to be \pref{hpfuns:4} and \pref{hpfuns:5} respectively. \begin{examps} \item \select{SELECT DISTINCT insp.1, insp.2, insp.3 \\ VALID VALID(insp) \\ FROM inspections(PERIOD) AS insp} \label{hpfuns:4} \item \select{SELECT DISTINCT SNAPSHOT inspcmpl.1, inspcmpl.2, inspcmpl.3\\ FROM inspections AS inspcmpl \\ WHERE inspcmpl.4 = 'complete'}\label{hpfuns:5} \end{examps} This causes $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(inspecting, 2)$ and $\ensuremath{\mathit{h_{culms}}}\xspace(st)(inspecting, 2)$ to be \pref{hpfuns:6} and \pref{hpfuns:7} respectively. \begin{examps} \item \dbtableb{|l|l|l||l|} {$i158$ & $J.Adams$ & $UK160$ & $[9\text{:}00am \; 1/5/95, \; 9\text{:}45am \; 1/5/95]$ \\ $i158$ & $J.Adams$ & $UK160$ & $[10\text{:}10am \; 1/5/95, \; 10\text{:}25am \; 1/5/95]$ \\ $i160$ & $J.Adams$ & $UK160$ & $[11\text{:}00pm \; 2/7/95, \; 1\text{:}00am \; 3/7/95]$ \\ $i160$ & $J.Adams$ & $UK160$ & $[6\text{:}00am \; 3/7/95, \; 6\text{:}20am \; 3/7/95]$ \\ $i205$ & $T.Smith$ & $BA737$ & $[8\text{:}00am \; 16/11/95, \; 8\text{:}20am \; 16/11/95]$ \\ $i214$ & $T.Smith$ & $BA737$ & $[8\text{:}10am \; 14/2/96, \; now]$ } \label{hpfuns:6} \item \dbtableb{|l|l|l|} {$i158$ & $J.Adams$ & $UK160$ \\ $i205$ & $T.Smith$ & $BA737$ } \label{hpfuns:7} \end{examps} \paragraph{$\mathbf{h_{gparts}'}$:} \index{hgpartsp@$\ensuremath{\mathit{h'_{gparts}}}\xspace()$ (similar to \ensuremath{\mathit{h_{gparts}}}\xspace but returns \textsc{Tsql2}\xspace expressions)} \ensuremath{\mathit{h'_{gparts}}}\xspace is a function that maps every \textsc{Top}\xspace gappy partitioning name $\sigma_g$ to a \textsc{Tsql2}\xspace \sql{SELECT} statement $\Sigma$, such that $\ensuremath{\mathit{FCN}}\xspace(\Sigma) = \emptyset$, and for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, it is true that $eval(st, \Sigma) \in \ensuremath{\mathit{SREL}}\xspace(1)$ and $\{f_D(v) \mid \tup{v} \in eval(st, \Sigma)\}$ is a gappy partitioning. $\ensuremath{\mathit{h'_{gparts}}}\xspace(\sigma_g)$ is intended to generate the relation to which $\ensuremath{\mathit{h_{gparts}}}\xspace(st)$ maps $\sigma_g$ (the relation that represents the members of the gappy partitioning). Assuming, for example, that the $gregorian$ calendric relation of section \ref{calrels} is available, $\ensuremath{\mathit{h'_{gparts}}}\xspace(sunday^g)$ could be \pref{calrels:5} of page \pageref{calrels:5}. \ensuremath{\mathit{h_{gparts}}}\xspace is defined in terms of \ensuremath{\mathit{h'_{gparts}}}\xspace. For every $st \in \ensuremath{\mathit{CHRONS}}\xspace$ and $\sigma_g \in \ensuremath{\mathit{GPARTS}}\xspace$: \index{hgparts@$\ensuremath{\mathit{h_{gparts}}}\xspace()$ (gappy part.\ names to relations representing gappy partitionings)} \[ \ensuremath{\mathit{h_{gparts}}}\xspace(st)(\sigma_g) \defeq eval(st, \ensuremath{\mathit{h'_{gparts}}}\xspace(\sigma_g)) \] The restrictions on \ensuremath{\mathit{h'_{gparts}}}\xspace and the definition of $\ensuremath{\mathit{h_{gparts}}}\xspace(st)$ above satisfy the requirements on \ensuremath{\mathit{h_{gparts}}}\xspace of section \ref{h_funs}. \paragraph{$\mathbf{h_{cparts}'}$:} \index{hcpartsp@$\ensuremath{\mathit{h'_{cparts}}}\xspace()$ (similar to \ensuremath{\mathit{h_{cparts}}}\xspace but returns \textsc{Tsql2}\xspace expressions)} I assume that for each complete partitioning used in the \textsc{Top}\xspace formulae, there is a corresponding \textsc{Tsql2}\xspace granularity (section \ref{tsql2_time}). \ensuremath{\mathit{h'_{cparts}}}\xspace is a function that maps each \textsc{Top}\xspace complete partitioning name to an ordered pair $\tup{\gamma, \Sigma}$, where $\gamma$ is the name of the corresponding \textsc{Tsql2}\xspace granularity, and $\Sigma$ is a \sql{SELECT} statement that returns a relation representing the periods of the partitioning. More precisely, it must be the case that $\ensuremath{\mathit{FCN}}\xspace(\Sigma) = \emptyset$, and for every $st \in \ensuremath{\mathit{CHRONS}}\xspace$, $eval(st, \Sigma) \in \ensuremath{\mathit{SREL}}\xspace(1)$ and $\{f_D(v) \mid \tup{v} \in eval(st, \Sigma)\}$ is a complete partitioning. For example, if the $gregorian$ relation of section \ref{calrels} is available, \ensuremath{\mathit{h'_{cparts}}}\xspace could map $day^c$ to $\langle$\sql{DAY}$,\Sigma\rangle$, where $\Sigma$ is \pref{hpfuns:8.87}. \pref{hpfuns:8.87} returns a one-attribute snapshot relation whose attribute values denote all the day-periods. \begin{examps} \item \label{hpfuns:8.87} \select{SELECT DISTINCT SNAPSHOT VALID(greg2) \\ FROM (\select{SELECT DISTINCT greg1.4 \\ VALID VALID(greg1) \\ FROM gregorian AS greg1} \\ \ \ \ \ \ )(PERIOD) AS greg2} \end{examps} \ensuremath{\mathit{h_{cparts}}}\xspace is defined in terms of \ensuremath{\mathit{h'_{cparts}}}\xspace. For every $st \in \ensuremath{\mathit{CHRONS}}\xspace$ and $\sigma_c \in \ensuremath{\mathit{CPARTS}}\xspace$, if $\ensuremath{\mathit{h'_{cparts}}}\xspace(\sigma_c) = \tup{\gamma, \Sigma}$, then: \index{hcparts@$\ensuremath{\mathit{h_{cparts}}}\xspace()$ (compl.\ part.\ names to relations representing compl.\ partitionings)} \[ \ensuremath{\mathit{h_{cparts}}}\xspace(st)(\sigma_c) = eval(st, \Sigma) \] The restrictions on \ensuremath{\mathit{h'_{cparts}}}\xspace and the definition of $\ensuremath{\mathit{h_{cparts}}}\xspace(st)$ above satisfy the requirements on \ensuremath{\mathit{h_{cparts}}}\xspace of section \ref{h_funs}. The $\gamma$ is used in the translation rule for $\ensuremath{\mathit{For}}\xspace[\sigma_c, \nu_{qty}, \phi]$ (appendix \ref{trans_proofs}). \section{Formulation of the translation problem} \label{formulation} Let us now specify formally what we want the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation to achieve. I first define $interp$ (interpretation of a resulting relation). For every $\phi \in \ensuremath{\mathit{FORMS}}\xspace$ and every relation $r$\/: \index{interp@$interp()$ (interpretation of resulting relation)} \begin{equation} interp(r, \phi) \defeq \begin{cases} T, \text{ if } \phi \in \ensuremath{\mathit{YNFORMS}}\xspace \text{ and } r \not= \emptyset \\ F, \text{ if } \phi \in \ensuremath{\mathit{YNFORMS}}\xspace \text{ and } r = \emptyset \\ \{\tup{f_D(v_1), \dots, f_D(v_n)} \mid \tup{v_1, \dots, v_n} \in r \}, \\ \text{\ \ \ \ if } \phi \in \ensuremath{\mathit{WHFORMS}}\xspace \end{cases} \label{formulation:2} \end{equation} Intuitively, if $\phi$ was translated to a \sql{SELECT} statement that generated $r$, $interp(r, \phi)$ shows how to interpret $r$. If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ (yes/no English question) and $r \not= \emptyset$, the answer should be affirmative. If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and $r = \emptyset$, the answer should be negative. Otherwise, if $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ (the English question contains interrogatives, e.g.\ \qit{Who~\dots?}, \qit{When~\dots?}), the answer should report all the tuples of world objects $\tup{f_D(v_1), \dots, f_D(v_n)}$ represented by tuples $\tup{v_1, \dots, v_n} \in r$. A translation function $tr$ is needed, that maps every $\phi \in \ensuremath{\mathit{FORMS}}\xspace$ to a \textsc{Tsql2}\xspace \sql{SELECT} statement $\mathit{tr(\phi)}$, \index{tr@$tr()$ (\textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping)} such that for every $st \in \ensuremath{\mathit{PTS}}\xspace$, \pref{formulation:1sq} and \pref{formulation:1} hold. \begin{gather} \ensuremath{\mathit{FCN}}\xspace(tr(\phi)) = \emptyset \label{formulation:1sq} \\ interp(eval(st, tr(\phi)), \phi) = \denot{M(st), st}{\phi} \label{formulation:1} \end{gather} $M(st)$ must be as in section \ref{linking_model}. As discussed in section \ref{denotation}, each (reading of an) English question is mapped to a \textsc{Top}\xspace formula $\phi$. The answer must report $\denot{M(st), st}{\phi}$. If $\mathit{tr}$ satisfies \pref{formulation:1}, $\denot{M(st), st}{\phi}$ can be computed as $interp(eval(st, tr(\phi)), \phi)$, by letting the \textsc{Dbms}\xspace execute $tr(\phi)$ (i.e.\ compute $eval(st, tr(\phi))$). $\mathit{tr}$ will be defined in terms of an auxiliary function $\mathit{trans}$. $\mathit{trans}$ is a function of two arguments: \index{trans@$trans()$ (auxiliary \textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping)} \[ trans(\phi, \lambda) = \Sigma \] where $\phi \in \ensuremath{\mathit{FORMS}}\xspace$, $\lambda$ is a \textsc{Tsql2}\xspace value expression, and $\Sigma$ a \textsc{Tsql2}\xspace \sql{SELECT} statement. A set of ``translation rules'' (to be discussed in section \ref{trans_rules}) specifies the $\Sigma$-values of $\mathit{trans}$. In practice, $\lambda$ always represents a period. Intuitively, $\lambda$ corresponds to \textsc{Top}\xspace's $lt$. When $trans$ is first invoked (by calling $tr$, discussed below) to translate a formula $\phi$, $\lambda$ is set to \sql{PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever')} to reflect the fact that \textsc{Top}\xspace's $lt$ is initially set to \ensuremath{\mathit{PTS}}\xspace (see the definition of $\denot{M,st}{\phi}$ in section \ref{denotation}). $trans$ may call itself recursively to translate subformulae of $\phi$ (this will become clearer in following sections). When calling $trans$ recursively, $\lambda$ may represent a period that does not cover the whole time-axis, to reflect the fact that already encountered \textsc{Top}\xspace operators may have narrowed $lt$. I define $\mathit{tr}$ as follows: \begin{equation} tr(\phi) \defeq trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace) \label{formulation:3} \end{equation} where $\ensuremath{\mathit{\lambda_{init}}}\xspace \defeq$ \sql{PERIOD (TIMESTAMP 'beginning', TIMESTAMP 'forever')}. Obviously, \ensuremath{\mathit{\lambda_{init}}}\xspace contains no correlation names, and hence $\ensuremath{\mathit{FCN}}\xspace(\ensuremath{\mathit{\lambda_{init}}}\xspace) = \emptyset$. This implies that $eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace, g^{db})$ does not depend on $g^{db}$. \ensuremath{\mathit{\lambda_{init}}}\xspace evaluates to the element of $D_P$ that represents the period that covers the whole time-axis, i.e.\ for every $st \in \ensuremath{\mathit{PTS}}\xspace$, it is true that $eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace) \in D_P$ and $f_D(eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace)) = \ensuremath{\mathit{PTS}}\xspace$. Therefore, lemma \ref{linit_lemma} holds. \begin{lemma} \label{linit_lemma} {\rm $\ensuremath{\mathit{FCN}}\xspace(\ensuremath{\mathit{\lambda_{init}}}\xspace) = \emptyset$, and for every $st \in \ensuremath{\mathit{PTS}}\xspace$, $eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace) \in D_P$ and $f_{D}(eval(st,\ensuremath{\mathit{\lambda_{init}}}\xspace)) = \ensuremath{\mathit{PTS}}\xspace$.} \end{lemma} Using \pref{formulation:3}, \pref{formulation:1sq} and \pref{formulation:1} become \pref{formulation:6} and \pref{formulation:4} respectively. The translation rules (that specify the values of $\mathit{trans}$ for each $\phi$ and $\lambda$) must be defined so that for every $\phi \in \ensuremath{\mathit{FORMS}}\xspace$ and $st \in \ensuremath{\mathit{PTS}}\xspace$, \pref{formulation:6} and \pref{formulation:4} hold. \begin{gather} \ensuremath{\mathit{FCN}}\xspace(trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) = \emptyset \label{formulation:6} \\ interp(eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)), \phi) = \denot{M(st), st}{\phi} \label{formulation:4} \end{gather} Appendix \ref{trans_proofs} proves that theorems \ref{wh_theorem} and \ref{yn_theorem} hold for the translation rules of this thesis. \begin{theorem} \label{wh_theorem} {\rm If $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$, $st \in \ensuremath{\mathit{PTS}}\xspace$, $trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace) = \Sigma$, and the total number of interrogative and interrogative-maximal quantifiers in $\phi$ is $n$, then: \begin{enumerate} \item $\ensuremath{\mathit{FCN}}\xspace(\Sigma) = \emptyset$ \item $eval(st, \Sigma) \in \ensuremath{\mathit{SREL}}\xspace(n)$ \item $\{\tup{f_D(v_1), \dots, f_D(v_n)} \mid \tup{v_1, \dots, v_n} \in eval(st, \Sigma)\} = \denot{M(st), st}{\phi}$ \end{enumerate} } \end{theorem} That is, the translation $\Sigma$ of $\phi$ contains no free column references, and it evaluates to a snapshot relation of $n$ attributes, whose tuples represent $\denot{M(st), st}{\phi}$. \begin{theorem} \label{yn_theorem} {\rm If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, $st \in \ensuremath{\mathit{PTS}}\xspace$, $\lambda$ is a \textsc{Tsql2}\xspace expression, $g^{db} \in G^{db}$, $eval(st, \lambda, g^{db}) \in D_P^*$, $\corn{\phi} = \tup{\tau_1, \dots, \tau_n}$, and $\Sigma = trans(\phi, \lambda)$, then: \begin{enumerate} \item $\ensuremath{\mathit{FCN}}\xspace(\Sigma) \subseteq \ensuremath{\mathit{FCN}}\xspace(\lambda)$ \item $eval(st, \Sigma, g^{db}) \in \ensuremath{\mathit{VREL}_P}\xspace(n)$ \item $\tup{v_1, \dots, v_n; v_t} \in eval(st, \Sigma, g^{db})$ iff for some $g \in G$: \\ $\denot{M(st), g}{\tau_1} = f_D(v_1)$, \dots, $\denot{M(st), g}{\tau_n} = f_D(v_n)$, and \\ $\denot{M(st), st, f_D(v_t), f_D(eval(st, \lambda, g^{db})), g}{\phi} = T$ \end{enumerate} } \end{theorem} $\tau_1, \tau_2, \dots, \tau_n$ are all the constants in predicate argument positions and all the variables in $\phi$ (section \ref{TOP_mods}). Clause 3 intuitively means that the tuples of $eval(st, \Sigma, g^{db})$ represent all the possible combinations of values of $\tau_1, \dots, \tau_n$ and event times $et$, such that $\denot{M(st), st, et, lt, g}{\phi} = T$, where $lt$ is the element of $\ensuremath{\mathit{PERIODS}}\xspace^*$ represented by $\lambda$. I now prove that theorems \ref{wh_theorem} and \ref{yn_theorem} imply that \pref{formulation:6} and \pref{formulation:4} hold for every $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\phi \in \ensuremath{\mathit{FORMS}}\xspace$, i.e.\ that $trans$ has the desired properties. \textbf{Proof of \pref{formulation:6}:} Let $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\phi \in \ensuremath{\mathit{FORMS}}\xspace$. We need to show that \pref{formulation:6} holds. Since $\ensuremath{\mathit{FORMS}}\xspace = \ensuremath{\mathit{WHFORMS}}\xspace \union \ensuremath{\mathit{YNFORMS}}\xspace$, the hypothesis that $\phi \in \ensuremath{\mathit{FORMS}}\xspace$ implies that $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ or $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$. In both cases \pref{formulation:6} holds: \begin{itemize} \item If $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$, then by theorem \ref{wh_theorem}, $\ensuremath{\mathit{FCN}}\xspace(trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) = \emptyset$, i.e.\ \pref{formulation:6} holds. \item If $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, then by theorem \ref{yn_theorem} and lemma \ref{linit_lemma}, the following holds, which implies that \pref{formulation:6} also holds. \[ \ensuremath{\mathit{FCN}}\xspace(trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \subseteq \ensuremath{\mathit{FCN}}\xspace(\ensuremath{\mathit{\lambda_{init}}}\xspace) = \emptyset \] \end{itemize} \textbf{Proof of \pref{formulation:4}:} Let $st \in \ensuremath{\mathit{PTS}}\xspace$ and $\phi \in \ensuremath{\mathit{FORMS}}\xspace$. Again, it will either be the case that $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ or $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$. If $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$, then by theorem \ref{wh_theorem} the following is true: \[ \{\tup{f_D(v_1), \dots, f_D(v_n)} \mid \tup{v_1, \dots, v_n} \in eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace))\} = \denot{M(st), st}{\phi} \] The definition of $interp$, the hypothesis that $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$, and the equation above imply \pref{formulation:4}. It remains to prove \pref{formulation:4} for $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$. Let $\corn{\phi} = \tup{\tau_1, \dots,\tau_n}$. By lemma \ref{linit_lemma}, for every $g^{db} \in G^{db}$, $eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace, g^{db}) = eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace) \in D_P$ and $f_D(eval(st, \ensuremath{\mathit{\lambda_{init}}}\xspace)) = \ensuremath{\mathit{PTS}}\xspace$. Also, \pref{formulation:6} (proven above) implies that $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace), g^{db})$ does not depend on $g^{db}$. Then, from theorem \ref{yn_theorem} we get \pref{formulation:10} and \pref{formulation:12}. \begin{examples} \item \label{formulation:10} $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \in \ensuremath{\mathit{VREL}_P}\xspace(n)$ \item \label{formulation:12} $\tup{v_1, \dots, v_n; v_t} \in eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace))$ iff for some $g \in G$: \\ $\denot{M(st), g}{\tau_1} = f_D(v_1), \dots, \denot{M(st), g}{\tau_n} = f_D(v_n)$, and \\ $\denot{M(st), st, f_D(v_t), \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T$ \notag \end{examples} The hypothesis that $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and the definition of $\mathit{interp}$ imply that the left-hand side of \pref{formulation:4} has the following values: \[ \begin{cases} T, & \text{if } eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \not= \emptyset \\ F, & \text{if } eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) = \emptyset \end{cases} \] The hypothesis that $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$ and the definition of $\denot{M(st), st}{\phi}$ (section \ref{denotation}) imply that the right-hand side of \pref{formulation:4} has the following values: \[ \begin{cases} T, & \text{if for some } g \in G \text{ and } et \in \ensuremath{\mathit{PERIODS}}\xspace, \; \denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T \\ F, & \text{otherwise} \end{cases} \] Hence, to prove \pref{formulation:4} it is enough to prove \pref{formulation:15}. \begin{examples} \item \label{formulation:15} $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \not= \emptyset$ iff \\ for some $g \in G$ and $et \in \ensuremath{\mathit{PERIODS}}\xspace$, $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T$ \end{examples} I first prove the forward direction of \pref{formulation:15}. If it is true that $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \not= \emptyset$, by \pref{formulation:10} $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace))$ contains at least a tuple of the form $\tup{v_1, \dots, v_n; v_t}$, i.e.\ \pref{formulation:20} is true. \begin{equation} \label{formulation:20} \tup{v_1, \dots, v_n; v_t} \in eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \end{equation} \pref{formulation:20} and \pref{formulation:12} imply that for some $g \in G$, \pref{formulation:21} holds. \begin{equation} \label{formulation:21} \denot{M(st), st, f_D(v_t), \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T \end{equation} \pref{formulation:10} and \pref{formulation:20} imply that $v_t$ is the time-stamp of a tuple in a relation of \ensuremath{\mathit{VREL}_P}\xspace, which implies that $f_D(v_t) \in \ensuremath{\mathit{PERIODS}}\xspace$. Let $et = f_D(v_t)$. Then, \pref{formulation:21} becomes \pref{formulation:22}, where $g \in G$ and $et = f_D(v_t) \in \ensuremath{\mathit{PERIODS}}\xspace$. The forward direction of \pref{formulation:15} has been proven. \begin{equation} \label{formulation:22} \denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T \end{equation} I now prove the backwards direction of \pref{formulation:15}. I assume that $g \in G$, $et \in \ensuremath{\mathit{PERIODS}}\xspace$, and $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T$. Let $v_t = \ensuremath{f_D^{-1}}\xspace(et)$, which implies that $et = f_D(v_t)$. Then \pref{formulation:23} holds. \begin{equation} \label{formulation:23} \denot{M(st), st, f_D(v_t), \ensuremath{\mathit{PTS}}\xspace, g}{\phi} = T \end{equation} Let $v_1 = \ensuremath{f_D^{-1}}\xspace(\denot{M(st),g}{\tau_1})$, \dots, $v_n = \ensuremath{f_D^{-1}}\xspace(\denot{M(st),g}{\tau_n})$. This implies that \pref{formulation:24} also holds. \begin{equation} \label{formulation:24} \denot{M(st), g}{\tau_1} = f_D(v_1), \; \dots, \; \denot{M(st), g}{\tau_n} = f_D(v_n) \end{equation} \pref{formulation:24}, \pref{formulation:23}, the hypothesis that $g \in G$, and \pref{formulation:12} imply \pref{formulation:25}, which in turn implies that $eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \not= \emptyset$. The backwards direction of \pref{formulation:15} has been proven. \begin{equation} \label{formulation:25} \tup{v_1, \dots, v_n; v_t} \in eval(st, trans(\phi, \ensuremath{\mathit{\lambda_{init}}}\xspace)) \end{equation} This concludes the proof of \pref{formulation:4}. I have proven that $trans$ satisfies \pref{formulation:6} and \pref{formulation:4} for every $\phi \in \ensuremath{\mathit{FORMS}}\xspace$ and $st \in \ensuremath{\mathit{PTS}}\xspace$, i.e.\ that $trans$ has all the desired properties. \section{The translation rules} \label{trans_rules} The values (\sql{SELECT} statements) of $trans$ are specified by a set of ``translation rules''. These rules are of two kinds: (a) base (non-recursive) rules that specify $trans(\phi, \lambda)$ when $\phi$ is an atomic formula or a formula of the form $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$; and (b) recursive rules that specify $trans(\phi, \lambda)$ in all other cases, by recursively calling other translation rules to translate subformulae of $\phi$. In this section, I attempt to convey the intuitions behind the design of the translation rules, and to illustrate the functionality of some representative rules. In the case of a yes/no formula $\phi$, the aim is for the resulting \sql{SELECT} statement to return a relation of $\ensuremath{\mathit{VREL}_P}\xspace(n)$ that shows all the combinations of event-times $et$ and values of $\tau_1, \dots, \tau_n$ ($\tup{\tau_1, \dots, \tau_n} = \corn{\phi}$) for which $\phi$ is satisfied. More precisely, the tuples of the relation must represent all the combinations of event times $et$ and world objects assigned (by $\ensuremath{\mathit{f_{cons}}}\xspace(st)$ and some variable assignment $g$) to $\tau_1, \dots, \tau_n$, for which $\denot{M(st), st, et, lt, g}{\phi} = T$, where $lt$ is the element of $\ensuremath{\mathit{PERIODS}}\xspace^*$ represented by $\lambda$. In each tuple $\tup{v_1, \dots, v_n; v_t}$, $v_t$ represents $et$, while $v_1, \dots, v_n$ represent the world objects of $\tau_1, \dots, \tau_n$. For example, the rule for predicates is as follows: \textbf{Translation rule for predicates:} \\ $trans(\pi(\tau_1, \dots, \tau_n), \lambda) \defeq$\\ \sql{(}\select{SELECT DISTINCT $\alpha.1$, $\alpha.2$, \dots, $\alpha.n$ \\ VALID VALID($\alpha$) \\ FROM ($\ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi, n)$)(SUBPERIOD) AS $\alpha$ \\ WHERE \dots \\ \ \ AND \dots \\ \ \ \vdots \\ \ \ AND \dots \\ \ \ AND $\lambda$ CONTAINS VALID($\alpha$))} where the ``\dots''s in the \sql{WHERE} clause stand for all the strings in $S_1 \union S_2$, and: \begin{gather*} S_1 = \{\text{``}\alpha.i = \ensuremath{\mathit{h'_{cons}}}\xspace(\tau_i)\text{''} \mid i \in \{1,2,3,\dots,n\} \text{ and } \tau_i \in \ensuremath{\mathit{CONS}}\xspace\} \\ S_2 = \{\text{``}\alpha.i = \alpha.j\text{''} \mid i,j \in \{1,2,3,\dots,n\}, \; i < j, \; \tau_i = \tau_j, \text{ and } \tau_i, \tau_j \in \ensuremath{\mathit{VARS}}\xspace\} \end{gather*} I assume that whenever the translation rule is invoked, a new correlation name $\alpha$ is used, that is obtained by calling a \emph{generator of correlation names}. Whenever called, the generator returns a new correlation name that has never been generated before. I assume that the correlation names of the generator are of some distinctive form (e.g.\ \sql{t1}, \sql{t2}, \sql{t3},~\dots), and that the correlation names in the \sql{SELECT} statements returned by \ensuremath{\mathit{h'_{pfuns}}}\xspace, \ensuremath{\mathit{h'_{culms}}}\xspace, \ensuremath{\mathit{h'_{cparts}}}\xspace, and \ensuremath{\mathit{h'_{gparts}}}\xspace are not of this distinctive form. I also assume that some mechanism is in place to ensure that no correlation name of the distinctive form of the generator can be used before it has been generated. The use of the generator means that $\mathit{trans}$ is strictly speaking not a pure function, since the same $\pi$ and $\tau_1, \dots, \tau_n$ lead to slightly different \sql{SELECT} statements whenever $trans(\pi(\tau_1, \dots, \tau_n), \lambda)$ is computed: each time the resulting statement contains a different $\alpha$ (similar comments apply to other translation rules). There are ways to make $\mathit{trans}$ a pure function, but these complicate the translation rules and the proof of their correctness, without offering any practical advantage. Let us consider, for example, the predicate $inspecting(i158, j\_adams, uk160)$. According to section \ref{denotation}, $\denot{M(st), st, et, lt, g}{inspecting(i158, j\_adams, uk160)} = T$ iff $et \subper lt$ and $et \subper p$, where: \[ p \in \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(inspecting, 3)(\denot{M(st),g}{i158}, \denot{M(st),g}{j\_adams}, \denot{M(st),g}{uk160}) \] Let us assume that $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting, 3)$ and $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(inspecting, 3)$ are \pref{hpfuns:4} and \pref{hpfuns:6} respectively (p.~\pageref{hpfuns:4}), that $i158$, $j\_adams$, and $uk160$ correspond to the obvious attribute values of \pref{hpfuns:6}, and that $\lambda$ is \sql{PERIOD '[9:00am 1/5/95 - 9:30pm 1/5/95]'}. $lt$ is the period represented by $\lambda$. By the definition of \ensuremath{\mathit{f_{pfuns}}}\xspace of section \ref{resulting_model}: \[ \ensuremath{\mathit{f_{pfuns}}}\xspace(st)(inspecting,3)(\denot{M(st),g}{i158}, \denot{M(st),g}{j\_adams}, \denot{M(st),g}{uk160}) = \{p_1, p_2\} \] where $p_1$ and $p_2$ are the periods of the first two tuples of \pref{hpfuns:6}. The denotation of $inspecting(i158, j\_adams, uk160)$ is $T$ for all the $et$s that are subperiods of $p_1$ or $p_2$ and also subperiods of $lt$. The translation rule above maps $inspecting(i158, j\_adams, uk160)$ to \pref{trans:1}, where $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting, 3)$ is the \sql{SELECT} statement of \pref{hpfuns:4} (that returns \pref{hpfuns:6}). \begin{examps} \item \label{trans:1} \sql{(}\select{SELECT DISTINCT t1.1, t1.2, t1.3 \\ VALID VALID(t1) \\ FROM ($\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting, 3)$)(SUBPERIOD) AS t1 \\ WHERE t1.1 = 'i158' \\ \ \ AND t1.2 = 'J.Adams' \\ \ \ AND t1.3 = 'UK160' \\ \ \ AND PERIOD '[9:00am 1/5/95 - 9:30pm 1/5/95]' CONTAINS VALID(t1))} \end{examps} \pref{trans:1} returns \pref{trans:2}, where the time-stamps correspond to all the subperiods of $p_1$ and $p_2$ ($p_1$ and $p_2$ are the periods of the first two time-stamps of \pref{hpfuns:6}) that are also subperiods of $lt$ (the period represented by $\lambda$). \begin{examps} \item \dbtableb{|l|l|l||l|} {$i158$ & $J.Adams$ & $UK160$ & $[9\text{:}00am \; 1/5/95, \; 9\text{:}30pm \; 1/5/95]$ \\ $i158$ & $J.Adams$ & $UK160$ & $[9\text{:}10am \; 1/5/95, \; 9\text{:}15pm \; 1/5/95]$ \\ $i158$ & $J.Adams$ & $UK160$ & $[9\text{:}20am \; 1/5/95, \; 9\text{:}25pm \; 1/5/95]$ \\ \ \dots & \ \dots & \ \dots & \ \dots } \label{trans:2} \end{examps} In other words, the time-stamps of \pref{trans:2} represent correctly all the $et$s where the denotation of $inspecting(i158, j\_adams, uk160)$ is $T$. In this example, all the predicate arguments are constants. Hence, there can be no variation in the values of the arguments, and the values of the explicit attributes in \pref{trans:2} are the same in all the tuples. When some of the predicate arguments are variables, the values of the corresponding explicit attributes are not necessarily fixed. The $S_2$ constraints in the \sql{WHERE} clause of the translation rule are needed when the predicate contains the same variable in more than one argument positions. In those cases, $S_2$ requires the attributes that correspond to the argument positions where the variable appears to have the same values. $S_2$ contains redundant constraints when some variable appears in more than two argument positions. For example, in $\pi(\beta, \beta, \beta)$ ($\beta \in \ensuremath{\mathit{VARS}}\xspace$), $S_2$ requires the tuples $\tup{v_1, v_2, v_3; v_t}$ of the resulting relation to satisfy: $v_1 = v_2$, $v_1 = v_3$, and $v_2 = v_3$. The third constraint is redundant, because it follows from the others. The prototype \textsc{Nlitdb}\xspace employs a slightly more complex definition of $S_2$ that does not generate the third constraint. Similar comments apply to the rule for $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$ below, and the rules for conjunction, $\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2]$, $\ensuremath{\mathit{Before}}\xspace[\phi_1, \phi_2]$, and $\ensuremath{\mathit{After}}\xspace[\phi_1, \phi_2]$ (appendix \ref{trans_proofs}). \textbf{Translation rule for $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$:}\\ $trans(\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)], \lambda) \defeq$\\ \sql{(}\select{SELECT DISTINCT $\alpha_1.1$, $\alpha_1.2$, \dots, $\alpha_1.n$ \\ VALID PERIOD(BEGIN(VALID($\alpha_1$)), END(VALID($\alpha_1$))) \\ FROM ($\ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi, n)$)(ELEMENT) AS $\alpha_1$, \\ \ \ \ \ \ ($\ensuremath{\mathit{h'_{culms}}}\xspace(\pi, n)$) AS $\alpha_2$ \\ WHERE $\alpha_1.1 = \alpha_2.1$ \\ \ \ AND $\alpha_1.2 = \alpha_2.2$ \\ \ \ \ \ \vdots \\ \ \ AND $\alpha_1.n = \alpha_2.n$ \\ \ \ AND \dots \\ \ \ \ \ \vdots \\ \ \ AND \dots \\ \ \ AND $\lambda$ CONTAINS PERIOD(BEGIN(VALID($\alpha_1$)), END(VALID($\alpha_1$)))} Whenever the rule is used, $\alpha_1$ and $\alpha_2$ are two new different correlation names, obtained by calling the correlation names generator after $\lambda$ has been supplied. The ``\dots'' in the \sql{WHERE} clause stand for all the strings in $S_1 \union S_2$, where $S_1$ and $S_2$ are as in the translation rule for predicates, except that $\alpha$ is now $\alpha_1$. The rule for $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$ is similar to that for $\pi(\tau_1, \dots, \tau_n)$. The resulting \sql{SELECT} statement returns an element of $\ensuremath{\mathit{VREL}_P}\xspace(n)$ that shows the $et$s and the values of the predicate arguments for which the denotation of $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$ is $T$. In the case of $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$, however, the generated relation contains only tuples $\tup{v_1, \dots, v_n; v_t}$, for which $\tup{v_1, \dots, v_n}$ appears in $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n)$ (the relation returned by $\ensuremath{\mathit{h'_{culms}}}\xspace(\pi, n)$). That is, the situation of $\pi(\tau_1, \dots, \tau_n)$ must reach its climax at the latest time-point where it is ongoing. Also, $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n)$ (the relation returned by $\ensuremath{\mathit{h'_{pfuns}}}\xspace(\pi,n)$) is coalesced using \sql{(ELEMENT)}. This causes all tuples of $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n)$ that refer to the same situation to be merged into one tuple, time-stamped by a temporal element that is the union of all the periods where the situation is ongoing. Let us refer to this coalesced version of $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(\pi, n)$ as $r$. $\alpha_1$ ranges over the tuples of $r$, while $\alpha_2$ over the tuples of $\ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi,n)$. The relation returned by $trans(\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)], \lambda)$ contains all tuples $\tup{v_1, \dots, v_n; v_t}$, such that $\tup{v_1, \dots, v_n; v_t'} \in r$, $v_t$ represents the period that starts at the beginning of the temporal element of $v_t'$ and ends at the end of the temporal element of $v_t'$, $\tup{v_1, \dots, v_n} \in \ensuremath{\mathit{h_{culms}}}\xspace(st)(\pi, n)$, and $v_t$'s period (i.e.\ $et$) is a subperiod of $\lambda$'s period (i.e.\ $lt$). $S_1$ and $S_2$ play the same role as in the translation rule for predicates. Let us assume that $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting,3)$ and $\ensuremath{\mathit{h'_{culms}}}\xspace(inspecting,3)$ are \pref{hpfuns:4} and \pref{hpfuns:5} respectively, that $\ensuremath{\mathit{h_{pfuns}}}\xspace(st)(inspecting,3)$ and $\ensuremath{\mathit{h_{culms}}}\xspace(st)(inspecting,3)$ are \pref{hpfuns:6} and \pref{hpfuns:7}, and that $\lambda =$ \sql{PERIOD '[1/5/95 - 18/11/95]'}. The translation rule above maps $\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, person^v, flight^v)]$ to \pref{trans:5}. \begin{examps} \item \label{trans:5} \sql{(}\select{SELECT DISTINCT t1.1, t1.2, t1.3 \\ VALID PERIOD(BEGIN(VALID(t1)), END(VALID(t1))) \\ FROM ($\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting,3)$)(ELEMENT) AS t1, \\ \ \ \ \ \ ($\ensuremath{\mathit{h'_{culms}}}\xspace(inspecting,3)$) AS t2 \\ WHERE t1.1 = t2.1 \\ \ \ AND t1.2 = t2.2 \\ \ \ AND t1.3 = t2.3 \\ \ \ AND PERIOD '[1/5/95 - 18/11/95]' CONTAINS \\ \ \ \ \ \ \ PERIOD(BEGIN(VALID(t1)), END(VALID(t1))))} \end{examps} \pref{trans:5} returns \pref{trans:6}. There is (correctly) no tuple for inspection $i160$: the semantics of \ensuremath{\mathit{Culm}}\xspace (section \ref{culm_op}) requires the inspection to reach its completion at the latest time-point where it is ongoing; according to \pref{hpfuns:7}, this is not the case for $i160$. There is also (correctly) no tuple for $i214$: the semantics of \ensuremath{\mathit{Culm}}\xspace requires $et$ (the time of the inspection) to be a subperiod of $lt$ ($\lambda$'s period), but $i214$ does not occur within $lt$. Finally, \pref{trans:6} does not contain tuples for the subperiods of [9:00am 1/5/95, 10:25am 1/5/95] and [8:00am 16/11/95, 8:20am 16/11/95]. This is in accordance with the semantics of \ensuremath{\mathit{Culm}}\xspace, that allows $\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, j\_adams, ba737)]$ to be true only at $et$s that cover entire inspections (from start to completion). \begin{examps} \item \dbtableb{|l|l|l||l|} {$i158$ & $J.Adams$ & $UK160$ & $[9\text{:}00am \; 1/5/95, \; 10\text{:}25am \; 1/5/95]$ \\ $i205$ & $T.Smith$ & $BA737$ & $[8\text{:}00am \; 16/11/95, \; 8\text{:}20am \; 16/11/95]$ } \label{trans:6} \end{examps} All the other translation rules for yes/no formulae are recursive. For example, $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$ is translated using the following: \textbf{Translation rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$:}\\ \label{past_trans_discuss} $trans(\ensuremath{\mathit{Past}}\xspace[\beta, \phi'], \lambda) \defeq$\\ \sql{(}\select{SELECT DISTINCT VALID($\alpha$), $\alpha$.1, $\alpha$.2, \dots, $\alpha$.$n$ \\ VALID VALID($\alpha$) \\ FROM $trans(\phi', \lambda')$ AS $\alpha$)} $\lambda'$ is the expression \sql{INTERSECT($\lambda$, PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' $\chi$))}, $\chi$ stands for the \textsc{Tsql2}\xspace name of the granularity of chronons (e.g.\ \sql{DAY}), and $n$ is the length of $\corn{\phi'}$. Whenever the rule is used, $\alpha$ is a new correlation name obtained by calling the correlation names generator. The rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$ calls recursively $\mathit{trans}$ to translate $\phi'$. $\phi'$ is translated with respect to $\lambda'$, which represents the intersection of the period of the original $\lambda$ with the period that covers all the time up to (but not including) the present chronon. This reflects the semantics of \ensuremath{\mathit{Past}}\xspace (section \ref{past_op}), that narrows $lt$ to $lt \intersect [t_{first}, st)$. The relation returned by $trans(\ensuremath{\mathit{Past}}\xspace[\beta, \phi'], \lambda)$ is the same as that of $trans(\phi', \lambda')$, except that the relation of $trans(\ensuremath{\mathit{Past}}\xspace[\beta, \phi'], \lambda)$ contains an additional explicit attribute, that corresponds to the $\beta$ of $\ensuremath{\mathit{Past}}\xspace[\beta,\phi']$. The values of that attribute are the same as the corresponding time-stamps (that represent $et$). This reflects the semantics of $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$, that requires the value of $\beta$ to be $et$. As a further example, $\ensuremath{\mathit{At}}\xspace[\kappa, \phi']$ ($\kappa \in \ensuremath{\mathit{CONS}}\xspace$) is translated using the following: \textbf{Translation rule for $\ensuremath{\mathit{At}}\xspace[\kappa, \phi']$:}\\ $trans(\ensuremath{\mathit{At}}\xspace[\kappa, \phi'], \lambda) \defeq trans(\phi', \lambda')$, where $\lambda'$ is \sql{INTERSECT($\lambda$, $\ensuremath{\mathit{h'_{cons}}}\xspace(\kappa)$)}. The translation of $\ensuremath{\mathit{At}}\xspace[\kappa, \phi']$ is the same as the translation of $\phi'$, but $\phi'$ is translated with respect to $\lambda'$, which represents the intersection of $\lambda$'s period with that of $\kappa$. This reflects the fact that in $\ensuremath{\mathit{At}}\xspace[\kappa, \phi']$, the \ensuremath{\mathit{At}}\xspace narrows $lt$ to the intersection of the original $lt$ with $\kappa$'s period. There are separate translation rules for $\ensuremath{\mathit{At}}\xspace[\sigma_c, \beta, \phi']$, $\ensuremath{\mathit{At}}\xspace[\sigma_g, \beta, \phi']$, and $\ensuremath{\mathit{At}}\xspace[\phi_1, \phi_2]$ ($\sigma_c \in \ensuremath{\mathit{CPARTS}}\xspace$, $\sigma_g \in \ensuremath{\mathit{GPARTS}}\xspace$, and $\phi', \phi_1, \phi_2 \in \ensuremath{\mathit{YNFORMS}}\xspace$). The complete set of translation rules for yes/no formulae is given in appendix \ref{trans_proofs}, along with a formal proof that $trans(\phi, \lambda)$ satisfies theorem \ref{yn_theorem}. Theorem \ref{yn_theorem} is proven by induction on the syntactic complexity of $\phi$. I first prove that theorem \ref{yn_theorem} holds if $\phi$ is a predicate or $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$. For all other $\phi \in \ensuremath{\mathit{YNFORMS}}\xspace$, $\phi$ is non-atomic. In those cases, I prove that theorem \ref{yn_theorem} holds if it holds for the subformulae of $\phi$. Let us now consider wh-formulae. These have the form $?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'$ or $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_k \; \phi'$, where $\phi' \in \ensuremath{\mathit{YNFORMS}}\xspace$ (section \ref{top_syntax}). The first case is covered by the following rule. (The rules for wh-formulae define $trans(\phi, \lambda)$ only for $\lambda = \ensuremath{\mathit{\lambda_{init}}}\xspace$. The values of $\mathit{trans}$ for $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ and $\lambda \not= \ensuremath{\mathit{\lambda_{init}}}\xspace$ are not used anywhere and can be chosen arbitrarily. Intuitively, for $\phi \in \ensuremath{\mathit{WHFORMS}}\xspace$ the goal is to define $trans(\phi, \lambda)$ so that it satisfies theorem \ref{wh_theorem}. That theorem is indifferent to the values of $\mathit{trans}$ for $\lambda \not= \ensuremath{\mathit{\lambda_{init}}}\xspace$.) \textbf{Translation rule for $?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'$:} \\ $trans(?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi', \ensuremath{\mathit{\lambda_{init}}}\xspace) \defeq$ \\ \sql{(}\select{SELECT DISTINCT SNAPSHOT $\alpha.\omega_1$, $\alpha.\omega_2$, \dots, $\alpha.\omega_k$ \\ FROM $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ AS $\alpha$)} Whenever the rule is used, $\alpha$ is a new correlation name, obtained by calling the correlation names generator. Assuming that $\corn{\phi'} = \tup{\tau_1, \dots, \tau_n}$, for every $i \in \{1,2,3, \dots, \kappa\}$: \[ \omega_i = min(\{j \mid j \in \{1,2,3,\dots,n\} \text{ and } \tau_j = \beta_j\}) \] That is, the first position (from left to right) where $\beta_i$ appears in $\tup{\tau_1, \dots, \tau_n}$ is the $\omega_i$-th one. Intuitively, we want $?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'$ to be translated to a \sql{SELECT} statement that returns a snapshot relation, whose tuples represent $\denot{M(st), st}{?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'}$. According to section \ref{denotation}, $\denot{M(st), st}{?\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'}$ is the set of all tuples that represent combinations of values assigned to $\beta_1, \dots, \beta_k$ by some $g \in G$, such that for some $et \in \ensuremath{\mathit{PERIODS}}\xspace$, $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi'} = T$. By theorem \ref{yn_theorem}, the relation returned by $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ (see the translation rule) is a valid-time relation, whose tuples show all the possible combinations of $et$s and values assigned (by $\ensuremath{\mathit{f_{cons}}}\xspace(st)$ and some $g \in G$) to $\tau_1, \dots, \tau_n$, for which $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi'} = T$. The syntax of \textsc{Top}\xspace (section \ref{top_syntax}) guarantees that $\beta_1, \dots, \beta_k$ appear within $\phi'$. This in turn guarantees that $\beta_1, \dots, \beta_k$ appear among $\tau_1, \dots, \tau_n$, i.e.\ the relation of $trans(\phi',\ensuremath{\mathit{\lambda_{init}}}\xspace)$ contains attributes for $\beta_1,\dots,\beta_k$. To find all the possible combinations of values of $\beta_1, \dots, \beta_k$ for which (for some $et$) $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi'} = T$, we simply need to pick (to ``project'' in relational terms) from the relation of $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ the attributes that correspond to $\beta_1, \dots, \beta_k$. For $i \in \{1,2,3,\dots,k\}$, $\beta_i$ may appear more than once in $\phi'$. In this case, the relation of $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ contains more than one attributes for $\beta_i$ (these attributes have the same values in each tuple). We only need to project one of the attributes that correspond to $\beta_i$. The translation rule projects only the first one; this is the $\omega_i$-th attribute of $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$, the attribute that corresponds to the first (from left to right) $\tau_j$ in $\tup{\tau_1, \dots, \tau_n}$ that is equal to $\beta_i$. Let us consider, for example, the following wh-formula (\qit{Who inspected what?}): \begin{equation} \label{trans:10.1} ?w1^v \; ?w2^v \; \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w1^v, w2^v)]] \end{equation} Here, $\phi' = \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w1^v, w2^v)]]$ and $\corn{\phi'} = \tup{e^v, occr^v, w1^v, w2^v}$. Let us assume that $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ returns \pref{trans:10}. \pref{trans:10} shows all the possible combinations of $et$s and values that can be assigned by some $g \in G$ to $e^v$, $occr^v$, $w1^v$, and $w2^v$, such that $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\phi'} = T$. In every tuple, the time-stamp is the same as the value of the first explicit attribute, because the semantics of \ensuremath{\mathit{Past}}\xspace requires the value of $e^v$ (represented by the first explicit attribute) to be $et$ (represented by the time-stamp). To save space, I omit the time-stamps of \pref{trans:10}. \begin{examps} \item \label{trans:10} \dbtableb{|l|l|l|l||l|} {$[9\text{:}00am \; 1/5/95, \; 3\text{:}00pm \; 1/5/95]$ & $i158$ & $J.Adams$ & $UK160$ & \dots \\ $[10\text{:}00am \; 4/5/95, \; 11\text{:}30am \; 4/5/95]$ & $i165$ & $J.Adams$ & $BA737$ & \dots \\ $[7\text{:}00am \; 16/11/95, \; 7\text{:}30am \; 16/11/95]$ & $i204$ & $T.Smith$ & $UK160$ & \dots } \end{examps} To generate the snapshot relation that represents $\denot{M(st), st}{?w1^v \; ?w2^v \; \phi'}$, i.e.\ the relation that shows the combinations of values of $w1^v$ and $w2^v$ for which (for some $et$ and $g$) $\denot{M(st), st, et, \ensuremath{\mathit{PTS}}\xspace, g}{\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w1^v, w2^v)]]} = T$, we simply need to project the explicit attributes of \pref{trans:10} that correspond to $w1^v$ and $w2^v$. The first positions where $w1^v$ and $w2^v$ appear in $\corn{\phi'} = \tup{e^v, occr^v, w1^v, w2^v}$ are the third and fourth (i.e.\ $\omega_1 = 3$ and $\omega_2 = 4$). Hence, we need to project the third and fourth explicit attributes of \pref{trans:10}. The translation rule for $?\beta_1 \; \dots \; ?\beta_k \; \phi'$ maps \pref{trans:10.1} to \pref{trans:11}, which achieves exactly that (it returns \pref{trans:12}). \begin{examps} \item \label{trans:11} \sql{(}\select{SELECT DISTINCT SNAPSHOT t1.3, t1.4 \\ FROM $trans(\ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w1^v, w2^v)]], \ensuremath{\mathit{\lambda_{init}}}\xspace)$ AS t1)} \item \label{trans:12} \dbtableb{|l|l|} {$J.Adams$ & $UK160$ \\ $J.Adams$ & $BA737$ \\ $T.Smith$ & $UK160$ } \end{examps} Wh-formulae of the form $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_k \; \phi'$ ($\phi' \in \ensuremath{\mathit{YNFORMS}}\xspace$) are translated using the following: \textbf{Translation rule for $?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi'$:} \\ $trans(?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi', \ensuremath{\mathit{\lambda_{init}}}\xspace) \defeq$ \\ \sql{(}\select{SELECT DISTINCT SNAPSHOT VALID($\alpha_2$), $\alpha_2$.2, $\alpha_2$.3, \dots, $\alpha_2$.$k$ \\ FROM (\select{SELECT DISTINCT 'dummy', $\alpha_1.\omega_2$, $\alpha_1.\omega_3$, \dots, $\alpha_1.\omega_k$ \\ VALID $\alpha_1.\omega_1$ \\ FROM $trans(\phi', \ensuremath{\mathit{\lambda_{init}}}\xspace) $ AS $\alpha_1$}\\ \ \ \ \ \ )(NOSUBPERIOD) AS $\alpha_2$)} Whenever the rule is used, $\alpha_1$ and $\alpha_2$ are two different new correlation names, obtained by calling the correlation names generator. Assuming that $\corn{\phi'} = \tup{\tau_1, \dots, \tau_n}$, $\omega_1, \dots, \omega_k$ are as in the rule for $?\beta_1 \; \dots \; ?\beta_k \; \phi$. That is, the first position (from left to right) where $\beta_i$ appears in $\tup{\tau_1, \dots, \tau_n}$ is the $\omega_i$-th one. Let us consider, for example, \pref{trans:14} (\qit{What circled when.}). \begin{equation} ?_{mxl}e^v \; ?w^v \; \ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)] \label{trans:14} \end{equation} Let us also assume that $trans(\ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)], \ensuremath{\mathit{\lambda_{init}}}\xspace)$ returns \pref{trans:15}. In this case, $\phi' = \ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)]$ and $\corn{\phi'} = \tup{e^v, w^v}$. \pref{trans:15} shows all the combinations of $et$s and values of $e^v$ and $w^v$, for which the denotation of $\ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)]$ is $T$. In each tuple, the value of the first explicit attribute (that corresponds to $e^v$) is the same as the time-stamp, because the semantics of \ensuremath{\mathit{Past}}\xspace requires the value of $e^v$ to be the same as $et$ (represented by the time-stamp). To save space, I omit the time-stamps. \begin{examps} \item \label{trans:15} \dbtableb{|l|l||l|} { $[5\text{:}02pm \; 22/11/95, \; 5\text{:}17pm \; 22/11/95]$ & $BA737$ & \ \dots \\ $[5\text{:}05pm \; 22/11/95, \; 5\text{:}15pm \; 22/11/95]$ & $BA737$ & \ \dots \\ $[5\text{:}07pm \; 22/11/95, \; 5\text{:}13pm \; 22/11/95]$ & $BA737$ & \ \dots \\ \ \dots & \ \dots & \ \dots \\ $[4\text{:}57pm \; 23/11/95, \; 5\text{:}08pm \; 23/11/95]$ & $BA737$ & \ \dots \\ $[4\text{:}59pm \; 23/11/95, \; 5\text{:}06pm \; 23/11/95]$ & $BA737$ & \ \dots \\ $[5\text{:}01pm \; 23/11/95, \; 5\text{:}04pm \; 23/11/95]$ & $BA737$ & \ \dots \\ \ \dots & \ \dots & \ \dots \\ $[8\text{:}07am \; 22/11/95, \; 8\text{:}19am \; 22/11/95]$ & $UK160$ & \ \dots \\ $[8\text{:}08am \; 22/11/95, \; 8\text{:}12am \; 22/11/95]$ & $UK160$ & \ \dots \\ $[8\text{:}09am \; 22/11/95, \; 8\text{:}10am \; 22/11/95]$ & $UK160$ & \ \dots \\ \ \dots & \ \dots & \ \dots } \end{examps} BA737 was circling from 5:02pm to 5:17pm on 22/11/95, and from 4:57pm to 5:08pm on 23/11/95. UK160 was circling from 8:07am to 8:19am on 22/11/95. \pref{trans:15} also contains tuples for the subperiods of these periods, because $circling(w^v)$ (like all \textsc{Top}\xspace predicates) is homogeneous (section \ref{denotation}). $\ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)]$ is true at all these subperiods that end before the present chronon. In our example, the embedded \sql{SELECT} statement of $trans(?_{mxl}\beta_1 \; ?\beta_2 \; ?\beta_3 \dots \; ?\beta_k \; \phi', \ensuremath{\mathit{\lambda_{init}}}\xspace)$ is: \begin{examps} \item \label{trans:16} \sql{(}\select{SELECT DISTINCT 'dummy', t1.2 \\ VALID t1.1 \\ FROM $trans(\ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)], \ensuremath{\mathit{\lambda_{init}}}\xspace)$ AS t1)} \end{examps} \pref{trans:16} generates \pref{trans:17}, where the time-stamps are the values of the first explicit attribute of \pref{trans:15} (i.e.\ they correspond to $e^v$). The \sql{'dummy'} in the embedded \sql{SELECT} statement (\pref{trans:16} in our example) means that the first explicit attribute of that statement's resulting relation should have the string ``$dummy$'' as its value in all tuples. This is needed when $k = 1$. If, for example, \pref{trans:14} were $?_{mxl}e^v \; \ensuremath{\mathit{Past}}\xspace[e^v, circling(ba737)]$, without the \sql{'dummy'} the \sql{SELECT} clause of \pref{trans:16} would contain nothing after \sql{DISTINCT} (this is not allowed in \textsc{Tsql2}\xspace). \begin{examps} \item \label{trans:17} \dbtableb{|l|l||l|} { $dummy$ & $BA737$ & $[5\text{:}02pm \; 22/11/95, \; 5\text{:}17pm \; 22/11/95]$ \\ $dummy$ & $BA737$ & $[5\text{:}05pm \; 22/11/95, \; 5\text{:}15pm \; 22/11/95]$ \\ $dummy$ & $BA737$ & $[5\text{:}07pm \; 22/11/95, \; 5\text{:}13pm \; 22/11/95]$ \\ \ \dots & \ \dots & \ \dots \\ $dummy$ & $BA737$ & $[4\text{:}57pm \; 23/11/95, \; 5\text{:}08pm \; 23/11/95]$ \\ $dummy$ & $BA737$ & $[4\text{:}59pm \; 23/11/95, \; 5\text{:}06pm \; 23/11/95]$ \\ $dummy$ & $BA737$ & $[4\text{:}59pm \; 23/11/95, \; 5\text{:}06pm \; 23/11/95]$ \\ \ \dots & \ \dots & \ \dots \\ $dummy$ & $UK160$ & $[8\text{:}07am \; 22/11/95, \; 8\text{:}19am \; 22/11/95]$ \\ $dummy$ & $UK160$ & $[8\text{:}08am \; 22/11/95, \; 8\text{:}12am \; 22/11/95]$ \\ $dummy$ & $UK160$ & $[8\text{:}09am \; 22/11/95, \; 8\text{:}10am \; 22/11/95]$ \\ \ \dots & \ \dots & \ \dots } \end{examps} The \sql{(NOSUBPERIOD)} of the translation rule removes from \pref{trans:17} any tuples that do not correspond to maximal periods. That is \pref{trans:17} becomes \pref{trans:18}. \begin{examps} \item \label{trans:18} \dbtableb{|l|l||l|} { $dummy$ & $BA737$ & $[5\text{:}02pm \; 22/11/95, \; 5\text{:}17pm \; 22/11/95]$ \\ $dummy$ & $BA737$ & $[4\text{:}57pm \; 23/11/95, \; 5\text{:}08pm \; 23/11/95]$ \\ $dummy$ & $UK160$ & $[8\text{:}07am \; 22/11/95, \; 8\text{:}19am \; 22/11/95]$ } \end{examps} The overall \pref{trans:14} is mapped to \pref{trans:19}, which generates \pref{trans:20}. \pref{trans:20} represents the denotation of \pref{trans:14} w.r.t.\ $M(st)$ and $st$ (pairs of maximal circling periods and the corresponding flights). \begin{examps} \item \label{trans:19} \sql{(}\select{SELECT DISTINCT SNAPSHOT VALID(t2), t2.2 \\ FROM (\select{SELECT DISTINCT 'dummy', t1.2 \\ VALID t1.1 \\ FROM $trans(\ensuremath{\mathit{Past}}\xspace[e^v, circling(w^v)], \ensuremath{\mathit{\lambda_{init}}}\xspace)$ AS t1} \\ \ \ \ \ \ )(NOSUBPERIOD) AS t2)} \item \label{trans:20} \dbtableb{|l|l|} { $[5\text{:}02pm \; 22/11/95, \; 5\text{:}17pm \; 22/11/95]$ & $BA737$ \\ $[4\text{:}57pm \; 23/11/95, \; 5\text{:}08pm \; 23/11/95]$ & $BA737$ \\ $[8\text{:}07am \; 22/11/95, \; 8\text{:}19am \; 22/11/95]$ & $UK160$ } \end{examps} Appendix \ref{trans_proofs} proves that the translation rules for wh-formulae satisfy theorem \ref{wh_theorem}. \section{Optimising the generated TSQL2 code} \label{tsql2_opt} The generated \textsc{Tsql2}\xspace code is often verbose. There are usually ways in which it could be shortened and still return the same results. Figure \ref{optimise_code}, for example, shows the code that is generated by the translation of \pref{opt:1}, if chronons correspond to minutes. (\pref{opt:1} expresses the reading of \qit{Who inspected UK160 yesterday?} where the inspection must have both started and been completed on the previous day.) \begin{eqnarray} &&?w^v \; \ensuremath{\mathit{At}}\xspace[yesterday, \ensuremath{\mathit{Past}}\xspace[e^v, \ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w^v, uk160)]]] \label{opt:1} \end{eqnarray} \begin{figure} \hrule \medskip {\small \begin{verbatim} (SELECT DISTINCT SNAPSHOT t4.3 FROM (SELECT DISTINCT VALID(t3), t3.1, t3.2, t3.3 VALID VALID(t3) FROM (SELECT DISTINCT t1.1, t1.2, t1.3 VALID PERIOD(BEGIN(VALID(t1)), END(VALID(t1))) FROM (SELECT DISTINCT insp.1, insp.2, insp.3 VALID VALID(insp) FROM inspections(PERIOD) AS insp)(ELEMENT) AS t1, (SELECT DISTINCT SNAPSHOT inspcmpl.1, inspcmpl.2, inspcmpl.3 FROM inspections AS inspcmpl WHERE inspcmpl.4 = 'complete') AS t2 WHERE t1.1 = t2.1 AND t1.2 = t2.2 AND t1.3 = t2.3 AND t1.3 = 'UK160' AND INTERSECT( INTERSECT( PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever'), PERIOD 'today' - INTERVAL '1' DAY), PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE)) CONTAINS PERIOD(BEGIN(VALID(t1)), END(VALID(t1))) ) AS t3 ) AS t4) \end{verbatim} } \vspace*{-5mm} \caption{Example of generated \textsc{Tsql2}\xspace code} \label{optimise_code} \medskip \hrule \end{figure} I assume here that $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting, 3)$ and $\ensuremath{\mathit{h'_{culms}}}\xspace(inspecting, 3)$ are \pref{hpfuns:4} and \pref{hpfuns:5} respectively. The embedded \sql{SELECT} statements of figure \ref{optimise_code} that are associated with \sql{t1} and \sql{t2} are \pref{hpfuns:4} and \pref{hpfuns:5}. The embedded \sql{SELECT} statement that is associated with \sql{t3} corresponds to $\ensuremath{\mathit{Culm}}\xspace[inspecting(occr^v, w^v, uk160)]$ (see the rule for $\ensuremath{\mathit{Culm}}\xspace[\pi(\tau_1, \dots, \tau_n)]$ in section \ref{trans_rules}). It generates a relation whose explicit attributes show all the combinations of codes, inspectors, and inspected objects that correspond to complete inspections. The time-stamps of this relation represent periods that cover whole inspections (from start to completion). The last constraint in the \sql{WHERE} clause (the one with \sql{CONTAINS}) admits only tuples whose time-stamps (whole inspections) are subperiods of $lt$. The two nested \sql{INTERSECT}s before \sql{CONTAINS} represent $lt$. The \ensuremath{\mathit{At}}\xspace has narrowed $lt$ to the intersection of its original value (whole time-axis) with the previous day (\sql{PERIOD 'today' - INTERVAL '1' DAY)}). The \ensuremath{\mathit{Past}}\xspace has narrowed $lt$ further to the intersection with $[t_{first}, st)$ (\sql{PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE)}). The embedded \sql{SELECT} statement that is associated with \sql{t4} is generated by the translation rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$ (section \ref{trans_rules}). It returns the same relation as the statement that is associated with \sql{t3}, except that the relation of \sql{t4}'s statement has an additional explicit attribute that corresponds to the first argument of \ensuremath{\mathit{Past}}\xspace. In each tuple, the value of this extra attribute is the same as the time-stamp ($et$). The topmost \sql{SELECT} clause projects only the third explicit attribute of the relation returned by \sql{t4}'s statement (this attribute corresponds to $w^v$ of \pref{opt:1}). The code of figure \ref{optimise_code} could be shortened in several ways. \sql{t4}'s statement, for example, simply adds an extra attribute for the first argument of \ensuremath{\mathit{Past}}\xspace. In this particular case, this extra attribute is not used, because \pref{opt:1} contains no interrogative quantifier for the first argument of \ensuremath{\mathit{Past}}\xspace. Hence, \sql{t4}'s statement could be replaced by \sql{t3}'s (the topmost \sql{SELECT} clause would have to become \sql{SELECT DISTINCT SNAPSHOT t3.2}). One could also drop the top-level \sql{SELECT} statement, and replace the \sql{SELECT} clause of \sql{t3}'s statement with \sql{SELECT DISTINCT SNAPSHOT t1.2}. Furthermore, the intersection of the whole time-axis (\sql{PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever')}) with any period $p$ is simply $p$. Hence, the second \sql{INTERSECT(\dots, \dots)} could be replaced by its second argument. The resulting code is shown in figure \ref{optimise_code2}. Further simplifications are possible. \begin{figure} \hrule \medskip {\small \begin{verbatim} (SELECT DISTINCT SNAPSHOT t1.2 FROM (SELECT DISTINCT insp.1, insp.2, insp.3 VALID VALID(insp) FROM inspections(PERIOD) AS insp)(ELEMENT) AS t1, (SELECT SNAPSHOT inspcmpl.1, inspcmpl.2, inspcmpl.3 FROM inspections AS inspcmpl WHERE inspcmpl.4 = 'complete') AS t2 WHERE t1.1 = t2.1 AND t1.2 = t2.2 AND t1.3 = t2.3 AND t1.3 = 'UK160' AND INTERSECT(PERIOD 'today' - INTERVAL '1' DAY, PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE)) CONTAINS PERIOD(BEGIN(VALID(t1)), END(VALID(t1))) \end{verbatim} } \vspace*{-5mm} \caption{Shortened \textsc{Tsql2}\xspace code} \label{optimise_code2} \medskip \hrule \end{figure} Most \textsc{Dbms}\xspace{s} employ optimisation techniques. A commercial \textsc{Dbms}\xspace supporting \textsc{Tsql2}\xspace would probably be able to carry out at least some of the above simplifications. Hence, the reader may wonder why should the \textsc{Nlitdb}\xspace attempt to optimise the \textsc{Tsql2}\xspace code, rather than delegate the optimisation to the \textsc{Dbms}\xspace. First, as mentioned in section \ref{tdbs_general}, only a prototype \textsc{Dbms}\xspace currently supports \textsc{Tsql2}\xspace. Full-scale \textsc{Tsql2}\xspace \textsc{Dbms}\xspace{s} with optimisers may not appear in the near future. Second, long database language queries (like the ones generated by the framework of this thesis) can often confuse generic \textsc{Dbms}\xspace optimisers, causing them to produce inefficient code. Hence, shortening the \textsc{Tsql2}\xspace code before submitting it to the \textsc{Dbms}\xspace is again important. It would be interesting to examine if optimisations like the ones discussed above could be automated, and integrated into the framework of this thesis as an additional layer between the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator and the \textsc{Dbms}\xspace. I have not explored this issue. \section{Related work} Various mappings from forms of logic to and from relational algebra (e.g.\ \cite{Ullman}, \cite{VanGelder1991}), from logic programming languages to \textsc{Sql}\xspace (e.g.\ \cite{Lucas1988}, \cite{Draxler1992}), and from logic formulae generated by \textsc{Nlidb}\xspace{s} to \textsc{Sql}\xspace (\cite{Lowden1}, \cite{Androutsopoulos}, \cite{Androutsopoulos3}, \cite{Rayner93}) have been discussed in the past. The mapping which is most relevant to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation of this chapter is that of \cite{Boehlen1996}. Boehlen et al.\ study the relation between \textsc{Tsql2}\xspace and an extended version of first order predicate logic (henceforth called \textsc{Sul}\xspace), that provides the additional temporal operators $\mathbf{\bullet}\xspace$ (previous), $\mathbf{\circ}\xspace$ (next), $\ensuremath{\mathbf{since}\xspace}$, and $\ensuremath{\mathbf{until}\xspace}$. \textsc{Sul}\xspace is point-based, in the sense that \textsc{Sul}\xspace formulae are evaluated with respect to single time-points. \textsc{Sul}\xspace assumes that time is discrete. Roughly speaking, $\mathbf{\bullet}\xspace \phi$ \index{.@$\mathbf{\bullet}\xspace$ (\textsc{Sul}\xspace operator, previous time-point)} is true at a time-point $t$ iff $\phi$ is true at the time-point immediately before $t$. Similarly, $\mathbf{\circ}\xspace \phi$ \index{..@$\mathbf{\circ}\xspace$ (\textsc{Sul}\xspace operator, next time-point)} is true at $t$ iff $\phi$ is true at the time-point immediately after $t$. $\phi_1 \; \ensuremath{\mathbf{since}\xspace} \; \phi_2$ \index{since@$\ensuremath{\mathbf{since}\xspace}$ (\textsc{Sul}\xspace operator)} is true at $t$ iff there is some $t'$ before $t$, such that $\phi_2$ is true at $t'$, and for every $t''$ between $t'$ and $t$, $\phi_1$ is true at $t''$. Similarly, $\phi_1 \; \ensuremath{\mathbf{until}\xspace} \; \phi_2$ \index{until@$\ensuremath{\mathbf{until}\xspace}$ (\textsc{Sul}\xspace operator)} is true at $t$ iff there is some $t'$ after $t$, such that $\phi_2$ is true at $t'$, and for every $t''$ between $t$ and $t'$, $\phi_1$ is true at $t''$. Various other operators are also defined, but these are all definable in terms of $\mathbf{\bullet}\xspace$, $\mathbf{\circ}\xspace$, $\ensuremath{\mathbf{since}\xspace}$, and $\ensuremath{\mathbf{until}\xspace}$. For example, $\lozenge\xspace \phi$ \index{<>@$\lozenge\xspace$ (\textsc{Sul}\xspace's past operator)} is equivalent to $true \; \ensuremath{\mathbf{since}\xspace} \; \phi$ ($true$ is a special formula that is true at all time-points). In effect, $\lozenge\xspace \phi$ is true at $t$ if there is a $t'$ before $t$, and $\phi$ is true at $t'$. For example, \pref{sul:1a} and \pref{sul:2a} can be expressed as \pref{sul:1} and \pref{sul:2} respectively. \begin{examps} \item BA737 departed (at some past time). \label{sul:1a} \item $\lozenge\xspace depart(ba737)$ \label{sul:1} \item Tank 2 has been empty (all the time) since BA737 departed. \label{sul:2a} \item $empty(tank2) \; \ensuremath{\mathbf{since}\xspace} \; depart(ba737)$ \label{sul:2} \end{examps} Boehlen et al.\ provide rules that translate from \textsc{Sul}\xspace to \textsc{Tsql2}\xspace. (They also show how to translate from a fragment of \textsc{Tsql2}\xspace back to \textsc{Sul}\xspace, but this direction is irrelevant here.) The underlying ideas are very similar to those of this chapter. Roughly speaking, there are non-recursive rules for atomic formulae, and recursive rules for non-atomic formulae. For example, the translation rule for $\phi_1 \; \ensuremath{\mathbf{since}\xspace} \; \phi_2$ calls recursively the translation algorithm to translate $\phi_1$ and $\phi_2$. The result is a \sql{SELECT} statement, that contains two embedded \sql{SELECT} statements corresponding to $\phi_1$ and $\phi_2$. Devising rules to map from \textsc{Sul}\xspace to \textsc{Tsql2}\xspace is much easier than in the case of \textsc{Top}\xspace, mainly because \textsc{Sul}\xspace formulae are evaluated with respect to only one time-parameter (\textsc{Top}\xspace formulae are evaluated with respect to three parameters, $st$, $et$, and $lt$), \textsc{Sul}\xspace is point-based (\textsc{Top}\xspace is period-based; section \ref{top_intro}), and \textsc{Sul}\xspace provides only four temporal operators whose semantics are very simple (the \textsc{Top}\xspace version of this chapter has eleven operators, whose semantics are more complex). Consequently, proving the correctness of the \textsc{Sul}\xspace to \textsc{Tsql2}\xspace mapping is much simpler than in the case of \textsc{Top}\xspace. It has to be stressed, however, that \textsc{Top}\xspace and \textsc{Sul}\xspace were designed for very different purposes. \textsc{Sul}\xspace is interesting from a theoretical temporal-logic point of view. Roughly speaking, it has been proven that whatever can be expressed in traditional first-order predicate logic with a temporal precedence connective by treating time as an extra predicate argument (e.g.\ \pref{tlogi:2} of page \pageref{tlogi:2}) can also be expressed in first-order predicate logic enhanced with only a $\ensuremath{\mathbf{since}\xspace}$ and an $\ensuremath{\mathbf{until}\xspace}$ operator, subject to some continuity conditions (the reverse is not true; see chapter II.2 of \cite{VanBenthem}). The mapping from \textsc{Sul}\xspace to \textsc{Tsql2}\xspace (and the reverse mapping from a fragment of \textsc{Tsql2}\xspace to \textsc{Sul}\xspace) is part of a study of the relative expressiveness of \textsc{Sul}\xspace and \textsc{Tsql2}\xspace. The existence of a mapping from \textsc{Sul}\xspace to \textsc{Tsql2}\xspace shows that \textsc{Tsql2}\xspace is at least as expressive as \textsc{Sul}\xspace. (The reverse is not true. Full \textsc{Tsql2}\xspace is more expressive than \textsc{Sul}\xspace; see \cite{Boehlen1996}.) In contrast, \textsc{Top}\xspace was not designed to study expressiveness issues, but to facilitate the mapping from (a fragment of) English to logical form. Chapter \ref{English_to_TOP} showed how to translate systematically from a non-trivial fragment of English temporal questions into \textsc{Top}\xspace. No such systematic translation has been shown to exist in the case of \textsc{Sul}\xspace, and it is not at all obvious how temporal English questions (e.g.\ containing progressive and perfect tenses, temporal adverbials, temporal subordinate clauses) could be mapped systematically to appropriate \textsc{Sul}\xspace formulae. Although the study of expressiveness issues is not a goal of this thesis, I note that the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation of this chapter implies that \textsc{Tsql2}\xspace is at least as expressive as \textsc{Top}\xspace (every \textsc{Top}\xspace formula can be mapped to an appropriate \textsc{Tsql2}\xspace query). The reverse is not true: it is easy to think of \textsc{Tsql2}\xspace queries (e.g.\ queries that report cardinalities of sets) that cannot be expressed in (the current version of) \textsc{Top}\xspace. Finally, neither \textsc{Top}\xspace nor \textsc{Sul}\xspace can be said to be more expressive than the other, as there are English sentences that can be expressed in \textsc{Sul}\xspace but not \textsc{Top}\xspace, and vice-versa. For example, the \textsc{Sul}\xspace formula \pref{sul:2} expresses \pref{sul:2a}, a sentence that cannot be expressed in \textsc{Top}\xspace. Also, the \textsc{Top}\xspace formula \pref{sul:11} expresses \pref{sul:10}. There does not seem to be any way to express \pref{sul:10} in \textsc{Sul}\xspace. \begin{examps} \item Tank 2 was empty for two hours. \label{sul:10} \item $\ensuremath{\mathit{For}}\xspace[hour^c, 2, \ensuremath{\mathit{Past}}\xspace[e^v, empty(tank2)]]$ \label{sul:11} \end{examps} \section{Summary} \textsc{Tsql2}\xspace is an extension of \textsc{Sql-92}\xspace that provides special facilities for manipulating temporal information. Some modifications of \textsc{Tsql2}\xspace were adopted in this chapter. Some of these are minor, and were introduced to bypass uninteresting details (e.g.\ referring to explicit attributes by number) or obscure points in the \textsc{Tsql2}\xspace definition (e.g.\ the new version of the \sql{INTERVAL} function). Other modifications are more significant, and were introduced to facilitate the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation (e.g.\ \sql{(SUBPERIOD)} and \sql{(NOSUBPERIOD)}). One of these more significant modifications (calendric relations) is generally useful. Some minor modifications of \textsc{Top}\xspace were also adopted in this chapter. A method to translate from \textsc{Top}\xspace to \textsc{Tsql2}\xspace was framed. Each \textsc{Top}\xspace formula $\phi$ is mapped to a \textsc{Tsql2}\xspace query. This is executed by the \textsc{Dbms}\xspace, generating a relation that represents (via an interpretation function) $\denot{M(st),st}{\phi}$. Before the translation method can be used, the configurer of the \textsc{Nlitdb}\xspace must specify some functions (\ensuremath{\mathit{h'_{cons}}}\xspace, \ensuremath{\mathit{h'_{pfuns}}}\xspace, \ensuremath{\mathit{h'_{culms}}}\xspace, \ensuremath{\mathit{h'_{gparts}}}\xspace, \ensuremath{\mathit{h'_{cparts}}}\xspace) that link certain basic \textsc{Top}\xspace expressions to \textsc{Tsql2}\xspace expressions. The \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation is then carried out by a set of translation rules. The rules have to satisfy two theorems (\ref{wh_theorem} and \ref{yn_theorem}) for the translation to be correct (i.e.\ for the \textsc{Tsql2}\xspace query to generate a relation that represents $\denot{M(st),st}{\phi}$). An informal description of the functionality of some of the rules was given. The full set of the translation rules, along with a proof that they satisfy theorems \ref{wh_theorem} and \ref{yn_theorem}, is given in appendix \ref{trans_proofs}. Further work could explore how to optimise the generated \textsc{Tsql2}\xspace code. The \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation is in principle similar to the \textsc{Sul}\xspace to \textsc{Tsql2}\xspace translation of \cite{Boehlen1996}. \textsc{Top}\xspace and \textsc{Sul}\xspace, however, were designed for very different purposes, and the \textsc{Sul}\xspace to \textsc{Tsql2}\xspace translation is much simpler than the \textsc{Top}\xspace to \textsc{Tsql2}\xspace one. \chapter{The prototype NLITDB} \label{implementation} \proverb{Time works wonders.} \section{Introduction} This chapter discusses the architecture of the prototype \textsc{Nlitdb}\xspace, provides some information on how the modules of the system were implemented, and explains which modules would have to be added if the prototype \textsc{Nlitdb}\xspace were to be used in real-life applications. A description of the hypothetical airport database is also given, followed by sample questions from the airport domain and the corresponding output of the \textsc{Nlitdb}\xspace. The chapter ends with information on the speed of the system. \section{Architecture of the prototype NLITDB} \label{prototype_arch} Figure \ref{simple_arch_fig} shows the architecture of the prototype \textsc{Nlitdb}\xspace. Each English question is first parsed using the \textsc{Hpsg}\xspace grammar of chapter \ref{English_to_TOP}, generating an \textsc{Hpsg}\xspace sign. Multiple signs are generated for questions that the parser understands to be ambiguous. A \textsc{Top}\xspace formula is then extracted from each sign, as discussed in section \ref{extraction_hpsg}. Each extracted formula subsequently undergoes the post-processing of section \ref{post_processing}. (The post-processor also converts the formulae from the \textsc{Top}\xspace version of chapters \ref{TOP_chapter} and \ref{English_to_TOP} to the version of \ref{tdb_chapter}; see section \ref{TOP_mods}.) As discussed in section \ref{post_processing}, the post-processing sometimes generates multiple formulae from the same original formula. \begin{figure} \hrule \medskip \begin{center} \includegraphics[scale=.6]{simple_archit} \caption{Architecture of the prototype NLITDB} \label{simple_arch_fig} \end{center} \hrule \end{figure} Each one of the formulae that are generated at the end of the post-processing captures what the \textsc{Nlitdb}\xspace understands to be a possible reading of the English question. Many fully-fledged \textsc{Nlidb}\xspace{s} use preference measures to guess which reading among the possible ones the user had in mind, or generate ``unambiguous'' English paraphrases of the possible readings, asking the user to select one (see \cite{Alshawi}, \cite{Alshawi2}, \cite{DeRoeck1986} and \cite{Lowden1986}). No such mechanism is currently present in the prototype \textsc{Nlitdb}\xspace. All the formulae that are generated at the end of the post-processing are translated into \textsc{Tsql2}\xspace. The \textsc{Nlitdb}\xspace prints all the resulting \textsc{Tsql2}\xspace queries along with the corresponding \textsc{Top}\xspace formulae. The \textsc{Tsql2}\xspace queries would be executed by the \textsc{Dbms}\xspace to retrieve the information requested by the user. As mentioned in section \ref{tdbs_general}, however, the prototype \textsc{Nlitdb}\xspace has not been linked to a \textsc{Dbms}\xspace. Hence, the \textsc{Tsql2}\xspace queries are currently not executed, and no answers are produced. The following sections provide more information about the grammar and parser, the module that extracts \textsc{Top}\xspace formulae from \textsc{Hpsg}\xspace signs, the post-processor, and the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. \section{The grammar and parser} \label{parser} The \textsc{Hpsg}\xspace version of chapter \ref{English_to_TOP} was coded in the formalism of \textsc{Ale}\xspace \cite{Carpenter1992} \cite{Carpenter1994}, building on previous \textsc{Ale}\xspace encodings of \textsc{Hpsg}\xspace fragments by Gerald Penn, Bob Carpenter, Suresh Manandhar, and Claire Grover.\footnote{The prototype \textsc{Nlitdb}\xspace was implemented using \textsc{Ale}\xspace version 2.0.2 and Sicstus Prolog version 2.1.9. Chris Brew provided additional \textsc{Ale}\xspace code for displaying feature structures. The software of the prototype \textsc{Nlitdb}\xspace, including the \textsc{Ale}\xspace grammar, is available from \texttt{http://www.dai.ed.ac.uk/groups/nlp/NLP\_home\_page.html}. An earlier version of the prototype \textsc{Nlitdb}\xspace was implemented using the \textsc{Hpsg-Pl} and \textsc{Pleuk} systems \cite{Popowich} \cite{Calder}.} \textsc{Ale}\xspace can be thought of as a grammar-development environment. It provides a chart parser (which is the one used in the prototype \textsc{Nlitdb}\xspace; see \cite{Gazdar1989} for an introduction to chart parsers) and a formalism that can be used to write unification-grammars based on feature structures. Coding the \textsc{Hpsg}\xspace version of chapter \ref{English_to_TOP} in \textsc{Ale}\xspace's formalism proved straight-forward. \textsc{Ale}\xspace's formalism allows one to specify grammar rules, definite constraints (these are similar to Prolog rules, except that predicate arguments are feature structures), lexical entries, lexical rules, and a hierarchy of sorts of feature structures. The schemata and principles of the \textsc{Hpsg}\xspace version of chapter \ref{English_to_TOP} were coded using \textsc{Ale}\xspace grammar rules and definite constraints. \textsc{Ale}\xspace's lexical entries, lexical rules, and sort hierarchy were used to code \textsc{Hpsg}\xspace's lexical signs, lexical rules, and sort hierarchy respectively. The \textsc{Ale}\xspace grammar rules and definite constraints that encode the \textsc{Hpsg}\xspace schemata and principles are domain-independent, i.e.\ they require no modifications when the \textsc{Nlitdb}\xspace is configured for a new application domain. The lexical rules of the prototype \textsc{Nlitdb}\xspace are also intended to be domain-independent, though their morphology parts need to be extended (e.g.\ more information about the morphology of irregular verbs is needed) before they can be used in arbitrary application domains. The lexical entries of the system that correspond to determiners (e.g.\ \qit{a}, \qit{some}), auxiliary verbs, interrogative words (e.g.\ \qit{who}, \qit{when}), prepositions, temporal subordinators (e.g.\ \qit{while}, \qit{before}), month names, day names, etc.\ (e.g.\ \qit{January}, \qit{Monday}) are also domain-independent. The person configuring the \textsc{Nlitdb}\xspace, however, needs to provide lexical entries for the nouns, adjectives, and (non-auxiliary) verbs that are used in the particular application domain (e.g.\ \qit{flight}, \qit{open}, \qit{to land}). The largest part of the \textsc{Nlitdb}\xspace's sort hierarchy is also domain independent. Two parts of it need to be modified when the system is configured for a new domain: the hierarchy of world entities that is mounted under {\srt ind}\/ (section \ref{more_ind} and figure \vref{ind_hierarchy}), and the subsorts of {\srt predicate}\/ that correspond to \textsc{Top}\xspace predicates used in the domain (section \ref{TOP_FS} and figure \vref{psoa_fig}). As will be discussed in section \ref{modules_to_add}, tools could be added to help the configurer modify the domain-dependent modules. \section{The extractor of TOP formulae and the post-processor} \label{extraction_impl} The module that extracts \textsc{Top}\xspace formulae from \textsc{Hpsg}\xspace signs actually generates \textsc{Top}\xspace formulae written as Prolog terms. For example, it generates \pref{extr:5} instead of \pref{extr:3}. The correspondence between the two notations should be obvious. The Prolog-like notation of \pref{extr:5} is also used in the formulae that are passed to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator, and in the output of the \textsc{Nlitdb}\xspace (see section \ref{samples} below). \begin{examps} \item $?x1^v \; \ensuremath{\mathit{Ntense}}\xspace[x3^v, president(x1^v)] \land \ensuremath{\mathit{Past}}\xspace[x2^v, located\_at(x1^v, terminal2)]$ \label{extr:3} \item \texttt{\small interrog(x1\^{}v, and(\hspace*{-2mm}\begin{tabular}[t]{l} ntense(x3\^{}v, president(x1\^{}v)),\\ past(x2\^{}v, located\_at(x1\^{}v, terminal2)))) \end{tabular}} \label{extr:5} \end{examps} The extractor of the \textsc{Top}\xspace formulae is implemented using Prolog rules and \textsc{Ale}\xspace definite constraints (Prolog-like rules whose predicate-arguments are feature structures). Although the functionality of the extractor's code is simple, the code itself is rather complicated (it has to manipulate the internal data structures that \textsc{Ale}\xspace uses to represent feature structures) and will not be discussed. As mentioned in section \ref{prototype_arch}, the post-processor of figure \ref{simple_arch_fig} implements the post-processing phase of section \ref{post_processing}. The post-processor also eliminates \ensuremath{\mathit{Part}}\xspace operators by merging them with the corresponding \ensuremath{\mathit{At}}\xspace, \ensuremath{\mathit{Before}}\xspace, or \ensuremath{\mathit{After}}\xspace operators, to convert the formulae into the \textsc{Top}\xspace version of the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator (section \ref{TOP_mods}). The post-processor's code, which is written in Prolog, presents no particular interest and will not be discussed. \section{The TOP to TSQL2 translator} \label{translator_module} Implementing in Prolog the \textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping of chapter \ref{tdb_chapter} proved easy. The code of the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator of figure \ref{simple_arch_fig} is basically a collection of Prolog rules for the predicate \texttt{trans}. Each one of these rules implements one of the translation rules of section \ref{trans_rules} and appendix \ref{trans_proofs}. For example, the following Prolog rule implements the translation rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$ (p.~\pageref{past_trans_discuss}). (I omit some uninteresting details of the actual Prolog rule.) \singlespace\small \begin{verbatim} trans(past(_^v, PhiPrime), Lambda, Sigma):- chronons(Chronon), multiappend([ "INTERSECT(", Lambda, ", ", "PERIOD(TIMESTAMP 'beginning', ", "TIMESTAMP 'now' - INTERVAL '1' ", Chronon, "))" ], LambdaPrime), trans(PhiPrime, LambdaPrime, SigmaPrime), new_cn(Alpha), corners(PhiPrime, CList), length(CList, N), generate_select_list(Alpha, N, SelectList), multiappend([ "(SELECT DISTINCT VALID(", Alpha, "), ", SelectList, "VALID VALID(", Alpha, ")", "FROM ", SigmaPrime, " AS ", Alpha, ")", ], Sigma). \end{verbatim} \normalsize\doublespace The first argument of \texttt{trans} is the \textsc{Top}\xspace formula to be translated (in the notation of \pref{extr:5}). \texttt{Lambda} is a string standing for the $\lambda$ argument of the $trans$ function of section \ref{formulation} (initially \texttt{"PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever')"}). The generated \textsc{Tsql2}\xspace code is returned as a string in \texttt{Sigma}. The \texttt{chronons(Chronon)} causes \texttt{Chronon} to become a string holding the \textsc{Tsql2}\xspace name of the granularity of chronons (e.g.\ \sql{"MINUTE"}). The \texttt{chronons} predicate is supplied by the configurer of the \textsc{Nlitdb}\xspace, along with Prolog predicates that define the $h'$ functions of section \ref{via_TSQL2}. For example, the following predicate defines $\ensuremath{\mathit{h'_{pfuns}}}\xspace(inspecting,3)$ to be the \sql{SELECT} statement of \pref{hpfuns:4} on page \pageref{hpfuns:4}. The \texttt{chronons} predicate and the predicates that define the $h'$ functions are the only domain-dependent parts of the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. \singlespace\small \begin{verbatim} h_prime_pfuns_map(inspecting, 3, ["SELECT DISTINCT insp.1, insp.2, insp.3", "VALID VALID(insp)", "FROM inspections(PERIOD) AS insp"]). \end{verbatim} \normalsize\doublespace The first \texttt{multiappend} in the $trans$ rule above generates the $\lambda'$ string of the translation rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$ (p.~\pageref{past_trans_discuss}). It concatenates the string-elements of the list provided as first argument to \texttt{multiappend}, and the resulting string ($\lambda'$) is returned in \texttt{LambdaPrime}. As in the translation rule for $\ensuremath{\mathit{Past}}\xspace[\beta, \phi']$, the translation mapping is then invoked recursively to translate $\phi'$ (\texttt{PhiPrime}). The result of this translation is stored in \texttt{SigmaPrime}. \texttt{new\_cn(Alpha)} returns in \texttt{Alpha} a string holding a new correlation name (\texttt{new\_cn} implements the correlation names generator of section \ref{trans_rules}). The \texttt{corners(PhiPrime, CList)} causes \texttt{CList} to become $\corn{\phi'}$, and \texttt{length(CList, N)} returns in \texttt{N} the length of $\corn{\phi'}$. The \texttt{generate\_select\_list(Alpha, N, SelectList)} returns in \texttt{SelectList} a string of the form \texttt{Alpha.1, Alpha.2, \dots, Alpha.N}. Finally, the second \texttt{multiappend} returns in \texttt{Sigma} a string that holds the overall \textsc{Tsql2}\xspace code. \section{Modules to be added} \label{modules_to_add} \begin{figure} \hrule \medskip \begin{center} \includegraphics[scale=.6]{architecture} \caption{Extended architecture of the prototype NLITDB} \label{arch_fig} \end{center} \hrule \end{figure} The prototype \textsc{Nlitdb}\xspace is intended to demonstrate that the mappings from English to \textsc{Top}\xspace and from \textsc{Top}\xspace to \textsc{Tsql2}\xspace are implementable. Consequently, the architecture of the prototype \textsc{Nlitdb}\xspace is minimal. Several modules, to be sketched in the following sections, would have to be added if the system were to be used in real-life applications. Figure \ref{arch_fig} shows how these modules would fit into the existing system architecture. (Modules drawn with dashed lines are currently not present.) \subsection{Preprocessor} \label{preprocessor} The \textsc{Ale}\xspace parser requires its input sentence to by provided as a Prolog list of symbols (e.g.\ \pref{prepro:2}). \begin{examps} \item \texttt{[was,ba737,circling,at,pm5\_00]} \label{prepro:2} \end{examps} As there is essentially no interface between the user and the \textsc{Ale}\xspace parser in the prototype \textsc{Nlitdb}\xspace, English questions have to be typed in this form. This does not allow words to start with capital letters or numbers, or to contain characters like ``\texttt{/}'' and ``\texttt{:}'' (Prolog symbols cannot contain these characters and must start with lower case letters). To bypass these constraints, proper names, dates, and times currently need to be typed in unnatural formats (e.g.\ ``\texttt{london}'', ``\texttt{d1\_5\_92}'', ``\texttt{pm5\_00}'', ``\texttt{y1991}'' instead of ``\texttt{London}'', ``\texttt{1/5/92}'' ``\texttt{5:00pm}'', ``\texttt{1991}''). A preprocessing module is needed, that would allow English questions to be typed in more natural formats (e.g.\ \pref{prepro:4}), and would transform the questions into the format required by the parser (e.g.\ \pref{prepro:2}). \begin{examps} \item Was BA737 circling at 5:00pm? \label{prepro:4} \end{examps} Similar preprocessing modules are used in several natural language front-ends (e.g.\ \textsc{Cle} \cite{Alshawi} and \textsc{Masque} \cite{Lindop}). These modules typically also merge parts of the input sentence that need to be processed as single words. For example, the lexicon of the airport domain has a single lexical entry for \qit{gate 2}. The preprocessor would merge the two words of \qit{gate 2} in \pref{prepro:7}, generating \pref{prepro:8}. (Currently, \qit{gate 2} has to be typed as a single word.) \begin{examps} \item Which flights departed from gate 2 yesterday? \label{prepro:7} \item \texttt{[which,flights,departed,from,gate2,yesterday]} \label{prepro:8} \end{examps} The preprocessing modules typically also handle proper names that cannot be included in the lexicon because they are too many, or because they are not known when creating the lexicon. In a large airport, for example, there would be hundreds of flight names (\qit{BA737}, \qit{UK1751}, etc.). Having a different lexical entry for each flight name is impractical, as it would require hundreds of entries to be added into the lexicon. Also, new flights (and hence flight names) are created occasionally, which means that the lexicon would have to be updated whenever a new flight is created. Instead, the lexicon could contain entries for a small number of pseudo-flight names (e.g.\ \qit{flight\_name1}, \qit{flight\_name2}, \dots, \qit{flight\_nameN}; N is the maximum number of flight names that may appear in a question, e.g.\ 5). Each one of these lexical entries would map a pseudo-flight name to a \textsc{Top}\xspace constant (e.g.\ $flight1$, $flight2$, \dots, $flightN$).\footnote{In the \textsc{Hpsg}\xspace grammar of chapter \ref{English_to_TOP}, these constants would be represented using sorts like {\srt flight1}, {\srt flight2}, \dots, {\srt flightN}, which would be daughters of {\srt flight\_ent}\/ and sisters of {\srt flight\_ent\_var}\/ in figure \vref{ind_hierarchy}.} The preprocessor would use domain-dependent formatting conventions to identify flight names in the English question (e.g.\ that any word that starts with two or three capital letters and is followed by three or four digits is a flight name). Each flight name in the question would be replaced by a pseudo-flight name. For example, the preprocessor would turn \pref{prepro:4.1} into \pref{prepro:4.2}. \begin{examps} \item Did BA737 depart before UK160 started to land? \label{prepro:4.1} \item \texttt{[did,flight\_name1,depart,before,flight\_name2,started,to,land]} \label{prepro:4.2} \end{examps} \pref{prepro:4.2} would then be parsed, giving rise to \pref{prepro:4.3} ($flight1$ and $flight2$ are \textsc{Top}\xspace constants introduced by the lexical entries of \qit{flight\_name1} and \qit{flight\_name2}). An extra step would be added to the post-processing phase of section \ref{post_processing}, to substitute $flight1$ and $flight2$ with \textsc{Top}\xspace constants that reflect the original flight names. For example, the preprocessor could pass to the post-processor the original flight names (\qit{BA737} and \qit{UK160}), and the post-processor could replace $flight1$ and $flight2$ by the original flight names in lower case. This would cause \pref{prepro:4.3} to become \pref{prepro:4.4}. Similar problems arise in the case of dates, times, numbers, etc. \begin{examps} \item $\ensuremath{\mathit{Before}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Begin}}\xspace[landing(flight2)]], \ensuremath{\mathit{Past}}\xspace[e2^v, depart(flight1)]]$ \label{prepro:4.3} \item $\ensuremath{\mathit{Before}}\xspace[\ensuremath{\mathit{Past}}\xspace[e1^v, \ensuremath{\mathit{Begin}}\xspace[landing(uk160)]], \ensuremath{\mathit{Past}}\xspace[e2^v, depart(ba737)]]$ \label{prepro:4.4} \end{examps} No preprocessing mechanism is currently present in the prototype \textsc{Nlitdb}\xspace. The lexicon contains (for demonstration purposes) only a few entries for particular (not pseudo-) flight names, times, dates, and numbers (e.g.\ \qit{BA737}, \qit{9:00am}). For example, there is no entry for \qit{9:10am}. This causes the parsing of \qit{Which tanks were empty at 9:10am?} to fail. In contrast the parsing of \qit{Which tanks were empty at 9:00am?} succeeds, because there \emph{is} a lexical entry for \qit{9:00am}. \subsection{Quantifier scoping} \label{quantif_scoping} When both words that introduce existential quantification (e.g.\ \qit{a}, \qit{some}) and words that introduce universal quantification (e.g.\ \qit{every}, \qit{each}) are allowed, it is often difficult to decide which quantifiers should be given scope over which other quantifiers. For example, \pref{qsco:1} has two possible readings. Ignoring the temporal information of \pref{qsco:1}, these readings would be expressed in the traditional first-order predicate logic (\textsc{Fopl}) using formulae like \pref{qsco:2} and \pref{qsco:3}. \begin{examps} \item A guard inspected every gate. \label{qsco:1} \item $\exists x \; (guard(x) \land \forall y \; (gate(y) \rightarrow inspect(x, y)))$ \label{qsco:2} \item $\forall y \; (gate(y) \rightarrow \exists x \; (guard(x) \land inspect(x, y)))$ \label{qsco:3} \end{examps} In \pref{qsco:2}, the existential quantifier (introduced by \qit{a}) is given wider scope over the universal one (introduced by \qit{every}). According to \pref{qsco:2}, all the gates were inspected by the same guard. In contrast, in \pref{qsco:3} where the universal quantifier is given wider scope over the existential one, each gate was inspected by a possibly different guard. In \pref{qsco:1}, both scopings seem possible (at least in the absence of previous context). In many cases, however, one of the possible scopings is the preferred one, and there are heuristics to determine this scoping (see chapter 8 of \cite{Alshawi}). For example, universal quantifiers introduced by \qit{each} tend to have wider scope over existential quantifiers (e.g. if the \qit{every} of \pref{qsco:1} is replaced by \qit{each}, the scoping of \pref{qsco:3} becomes more likely than that of \pref{qsco:2}). In this thesis, words that introduce universal quantifiers were deliberately excluded from the linguistic coverage (section \ref{ling_not_supported}). This leaves only existential quantification and by-passes the quantifier scoping problem, because if all quantifiers are existential ones, the relative scoping of the quantifiers does not matter. (In \textsc{Top}\xspace, existential quantification is expressed using free variables. There are also interrogative and interrogative-maximal quantifiers, but these are in effect existential quantifiers that have the additional side-effect of including the values of their variables in the answer.) If the linguistic coverage of the prototype \textsc{Nlitdb}\xspace were to be extended to support words that introduce universal quantification, an additional scoping module would have to be added (figure \ref{arch_fig}). The input to that module would be an ``underspecified'' \textsc{Top}\xspace formula (see the discussion in chapter 2 of \cite{Alshawi}), a formula that would not specify the exact scope of each quantifier. In a \textsc{Fopl}-like formalism, an underspecified formula for \pref{qsco:1} could look like \pref{qsco:10}. \begin{examps} \item $inspect((\exists x \; guard(x)), (\forall y \; gate(y)))$ \label{qsco:10} \end{examps} The scoping module would generate all the possible scopings, and determine the most probable ones, producing formulae where the scope of each quantifier is explicit (e.g.\ \pref{qsco:2} or \pref{qsco:3}). Alternatively, one could attempt to use the the \textsc{Hpsg}\xspace quantifier scoping mechanism (see \cite{Pollard2} and relevant comments in section \ref{TOP_FS}) to reason about the possible scopings during the parsing. That mechanism, however, is a not yet fully developed part of \textsc{Hpsg}\xspace. \subsection{Anaphora resolution} \label{anaphora_module} As discussed in sections \ref{no_issues} and \ref{temporal_anaphora}, nominal anaphora (e.g.\ \qit{she}, \qit{his salary}) and most cases of temporal anaphora (e.g.\ \qit{in January}, tense anaphora) are currently not supported. An anaphora resolution module would be needed if phenomena of this kind were to be supported. As in the case of quantifier scoping, I envisage a module that would accept ``underspecified'' \textsc{Top}\xspace formulae, formulae that would not name explicitly the entities or times to which anaphoric expressions refer (figure \ref{arch_fig}). The module would determine the most probable referents of these expressions, using a discourse model that would contain descriptions of previously mentioned entities and times, information showing in which segments of the previous discourse the entities or times were mentioned, etc.\ (see \cite{Barros1994} for a description of a similar module). The output of this module would be a formula that names explicitly the referents of anaphoric expressions. \subsection{Equivalential translator} \label{equiv_translat} The translation from \textsc{Top}\xspace to \textsc{Tsql2}\xspace of chapter \ref{tdb_chapter} assumes that each pair $\tup{\pi,n}$ of a \textsc{Top}\xspace predicate functor $\pi$ and an arity $n$ can be mapped to a valid-time relation (stored directly in the database, or computed from information in the database) that shows the event times where $\pi(\tau_1, \dots, \tau_n)$ is true, for all the possible world entities denoted by the \textsc{Top}\xspace terms $\tau_1, \dots, \tau_n$. The configurer of the \textsc{Nlitdb}\xspace specifies this mapping when defining \ensuremath{\mathit{h'_{pfuns}}}\xspace (section \ref{via_TSQL2}). Although the assumption that each $\tup{\pi,n}$ can be mapped to a suitable relation is valid in most situations, there are cases where this assumption does not hold. The ``doctor on board'' problem \cite{Rayner93} is a well-known example of such a case. Let us imagine a database that contains only the following coalesced valid-time relation, that shows the times when a (any) doctor was on board each ship of a fleet. \adbtable{2}{|l||l|}{$doctor\_on\_board$} {$ship$ & } {$Vincent$ & $[8\text{:}30am \; 22/1/96 - 11\text{:}45am \; 22/1/96]$ \\ & $\;\;\union \; [3\text{:}10pm \; 23/1/96 - 5\text{:}50pm \; 23/1/96]$ \\ & $\;\; \union \; [9\text{:}20am \; 24/1/96 - 2\text{:}10pm \; 24/1/96]$ \\ $Invincible$ & $[8\text{:}20am \; 22/1/96 - 10\text{:}15am \; 22/1/96]$ \\ & $\;\; \union \; [1\text{:}25pm \; 23/1/96 - 3\text{:}50pm \; 23/1/96]$ \\ \; \dots & \; \dots } Let us also consider a question like \pref{doct:1}, which would be mapped to the \textsc{Top}\xspace formula \pref{doct:2}. I assume here that \qit{doctor} and \qit{ship} introduce predicates of the form $doctor(\tau_1)$ and $ship(\tau_2)$, and that the predicative preposition \qit{on} introduces a predicate of the form $located\_on(\tau_3, \tau_4)$ ($\tau_1, \dots, \tau_4 \in \ensuremath{\mathit{TERMS}}\xspace$). For simplicity, I assume that \qit{doctor} and \qit{ship} do not introduce \ensuremath{\mathit{Ntense}}\xspace operators (section \ref{non_pred_nps}). \begin{examps} \item Is there a doctor on some ship? \label{doct:1} \item $doctor(d^v) \land ship(s^v) \land \ensuremath{\mathit{Pres}}\xspace[located\_on(d^v, s^v)]$ \label{doct:2} \end{examps} To apply the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation method of chapter \ref{tdb_chapter}, one needs to map $\tup{doctor,1}$ to a valid-time relation (computed from information in the database) that shows the event times where $doctor(\tau_1)$ is true, i.e.\ when the entity denoted by $\tau_1$ was a doctor. Unfortunately, the database (which contains only $doctor\_on\_board$) does not show when particular entities were doctors, and hence such a relation cannot be computed. In the same manner, $\tup{ship,1}$ has to be mapped to a relation that shows the ships that existed at each time. This relation cannot be computed: $doctor\_on\_board$ does not list all the ships that existed at each time; it shows only ships that had a doctor on board at each time. Similarly, $\tup{located\_on,2}$ has to be mapped to a relation that shows when $located\_on(\tau_3,\tau_4)$ is true, i.e.\ when the entity denoted by $\tau_3$ was on the entity denoted by $\tau_4$. Again, this relation cannot be computed. If, for example, $\tau_3$ denotes a doctor (e.g.\ Dr.\ Adams) and $\tau_4$ denotes Vincent, there is no way to find out when that particular doctor was on Vincent: $doctor\_on\_board$ shows only when some (any) doctor was on each ship; it does not show when particular doctors (e.g.\ Dr.\ Adams) were on each ship. Hence, the translation method of chapter \ref{tdb_chapter} cannot be used. It should be easy to see, however, that \pref{doct:2} is equivalent to \pref{doct:3}, if $doctor\_on\_ship(\tau_5)$ is true at event times where the entity denoted by $\tau_5$ is a ship, and a doctor of that time is on that ship. What is interesting about \pref{doct:3} is that there \emph{is} enough information in the database to map $\tup{doctor\_on\_ship,1}$ to a relation that shows the event times where $doctor\_on\_ship(\tau_5)$ holds. Roughly speaking, one simply needs to map $\tup{doctor\_on\_ship,1}$ to the $doctor\_on\_board$ relation. Hence, the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translation method of chapter \ref{tdb_chapter} \emph{can} be applied to \pref{doct:3}, and the answer to \pref{doct:1} can be found by evaluating the resulting \textsc{Tsql2}\xspace code. \begin{examps} \item $\ensuremath{\mathit{Pres}}\xspace[doctor\_on\_ship(s^v)]$ \label{doct:3} \end{examps} The problem is that \pref{doct:1} cannot be mapped directly to \pref{doct:3}: the English to \textsc{Top}\xspace mapping of chapter \ref{English_to_TOP} generates \pref{doct:2}. We need to convert \pref{doct:2} (whose predicates are introduced by the lexical entries of nouns, prepositions, etc.) to \pref{doct:3} (whose predicates are chosen to be mappable to relations computed from information in the database). An ``equivalential translator'' similar to the ``abductive equivalential translator'' of \cite{Rayner93} and \cite{Alshawi2} could be used to carry out this conversion. Roughly speaking, this would be an inference module that would use domain-dependent conversion rules, like \pref{doct:4} which allows any formula of the form $doctor(\tau_1) \land ship(\tau_2) \land \ensuremath{\mathit{Pres}}\xspace[located\_on(\tau_1,\tau_2)]$ ($\tau_1, \tau_2 \in \ensuremath{\mathit{TERMS}}\xspace$) to be replaced by $\ensuremath{\mathit{Pres}}\xspace[doctor\_on\_ship(\tau_2)]$. \pref{doct:4} would license the conversion of \pref{doct:2} into \pref{doct:3}. \begin{examps} \item $doctor(\tau_1) \land ship(\tau_2) \land \ensuremath{\mathit{Pres}}\xspace[located\_on(\tau_1, \tau_2)] \equiv \ensuremath{\mathit{Pres}}\xspace[doctor\_on\_ship(\tau_2)]$ \label{doct:4} \end{examps} There would be two kinds of pairs $\tup{\pi,n}$ ($\pi$ is a predicate functor and $n$ an arity): pairs that are mapped to relations, and pairs for which this mapping is impossible (the value of \ensuremath{\mathit{h'_{pfuns}}}\xspace would be undefined for the latter). The formula generated after the scoping and anaphora resolution would be passed to the equivalential translator (figure \ref{arch_fig}). If all the predicate functors and arities in the formula are among the pairs that are mapped to relations, the equivalential translator would have no effect. Otherwise, the equivalential translator would attempt to convert the formula into another one that contains only predicate functors and arities that are mapped to relations (an error would be reported if the conversion is impossible). The new formula would then be passed to the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator. \subsection{Response generator} \label{response_generator} The execution of the \textsc{Tsql2}\xspace code produces the information that is needed to answer the user's question. A response generator is needed to report this information to the user. In the simplest case, if the question is a yes/no one, the response generator would simply print a \qit{yes} or \qit{no}, depending on whether or not the \textsc{Tsql2}\xspace code retrieved at least one tuple (section \ref{formulation}). Otherwise, the response generator would print the tuples retrieved by the \textsc{Tsql2}\xspace code. Ideally, the response generator would also attempt to provide \emph{cooperative responses} (section \ref{no_issues}; see also section \ref{to_do} below). In \pref{respgen:1}, for example, if BA737 is at gate 4, the response generator would produce \pref{respgen:2} rather than a simple \qit{no}. That is, it would report the answer to \pref{respgen:1} along with the answer to \pref{respgen:3}. \begin{examps} \item Is BA737 at gate 2? \label{respgen:1} \item \sys{No, BA737 is at gate 4.} \label{respgen:2} \item Which gate is BA737 at? \label{respgen:3} \end{examps} In that case, the architecture of the \textsc{Nlitdb}\xspace would have to be more elaborate than that of figure \ref{arch_fig}, as the response generator would have to submit questions (e.g.\ \pref{respgen:3}) on its own, in order to collect the additional information that is needed for the cooperative responses. \subsection{Configuration tools} As already noted, there are several parts of the prototype \textsc{Nlitdb}\xspace that need to be modified whenever the \textsc{Nlitdb}\xspace is configured for a new application. Most large-scale \textsc{Nlidb}\xspace{s} provide tools that automate these modifications, ideally allowing people that are not aware of the details of the \textsc{Nlitdb}\xspace's code to configure the system (see section 6 of \cite{Androutsopoulos1995}, and chapter 11 of \cite{Alshawi}). A similar tool is needed in the prototype \textsc{Nlitdb}\xspace of this thesis. Figure \ref{arch_fig} shows how this tool would fit into the \textsc{Nlitdb}\xspace's architecture.\footnote{Some of the heuristics of the quantifier scoping module and parts of the anaphora resolution module may in practice be also domain-dependent. In that case, parts of these modules would also have to be modified during the configuration. For simplicity, this is not shown in figure \ref{arch_fig}.} \section{The airport database} This section provides more information about the hypothetical airport database, for which the prototype \textsc{Nlitdb}\xspace was configured. \begin{figure} \hrule \medskip \begin{center} $\begin{array}{l} gates(gate, availability) \\ runways(runway, availability) \\ queues(queue, runway) \\ servicers(servicer) \\ inspectors(inspector) \\ sectors(sector) \\ flights(flight) \\ tanks(tank, content) \\ norm\_departures(flight, norm\_dep\_time, norm\_dep\_gate) \\ norm\_arrivals(flight, norm\_arr\_time, norm\_arr\_gate) \\ norm\_servicer(flight, servicer) \\ flight\_locations(flight, location) \\ circling(flight) \\ inspections(code, inspector, inspected, status) \\ services(code, servicer, flight, status) \\ boardings(code, flight, gate, status) \\ landings(code, flight, runway, status) \\ \mathit{takeoffs}(code, flight, runway, status) \\ taxiings(code, flight, origin, destination, status) \end{array}$ \caption[Relations of the airport database]{Relations of the airport database} \label{db_relations} \end{center} \hrule \end{figure} The airport database contains nineteen relations, all valid-time and coalesced (section \ref{bcdm}). Figure \ref{db_relations} shows the names and explicit attributes of the relations. For simplicity, I assume that the values of all the explicit attributes are strings. I also assume that chronons correspond to minutes, and that the $gregorian$ calendric relation of section \ref{calrels} is available. The $runways$ relation has the following form: \adbtable{3}{|l|l||l|}{$runways$} {$runway$ & $\mathit{availability}$ & } {$runway1$ & $open$ & $[8\text{:}00am \; 1/1/96, \; 7\text{:}30pm \; 3/1/96]$ \\ & & $\;\; \union \; [6\text{:}00am \; 4/1/96, \; 2\text{:}05pm \; 8/1/96] \; \union \; \dots$ \\ $runway1$ & $closed$ & $[7\text{:}31pm \; 3/1/96, \; 5\text{:}59am \; 4/1/96]$ \\ & & $\;\; \union \; [2\text{:}06pm \; 8/1/96, \; 5\text{:}45pm \; 8/1/96]$ \\ $runway2$ & $open$ & $[5\text{:}00am \; 1/1/96, \; 9\text{:}30pm \; 9/1/96] \; \union \; \dots$ \\ $runway2$ & $closed$ & $[9\text{:}31pm \; 9/1/96, \; 10\text{:}59am \; 10/1/96] \; \union \; \dots$ } The $\mathit{availability}$ values are always $open$ or $closed$. There are two tuples for each runway: one showing the times when the runway was open, and one showing the times when it was closed. If a runway did not exist at some time (e.g.\ a runway may not have been constructed yet at that time), both tuples of that runway exclude this time from their time-stamps. The $gates$ relation is similar. Its $availability$ values are always $open$ or $closed$, and there are two tuples for each gate, showing the times when the gate was open (available) or closed (unavailable) respectively. Runways that are used for landings or take-offs have queues, where flights wait until they are given permission to enter the runway. The $queues$ relation lists the names of the queues that exist at various times, along with the runways where the queues lead to. The $servicers$ relation shows the names of the servicing companies that existed at any time. The $inspectors$, $sectors$, and $flights$ relations are similar. The $tanks$ relation shows the contents ($water$, $foam$, etc., or $empty$ if the tank was empty) of each tank at every time where the tank existed. Each outgoing flight is assigned a normal departure time and gate (see also section \ref{aspect_examples}). The $norm\_departures$ relation shows these times and gates. For example, if $norm\_departures$ were as follows, this would mean that from 9:00am on 1/1/92 to 5:30pm on 31/11/95 BA737 normally departed each day from gate 2 at 2:05pm. (For simplicity, I assume that all flights are daily.) At 5:31pm on 31/11/95, the normal departure time of BA737 was changed to 2:20pm, while the normal departure gate remained gate 2. No further change to the normal departure time or gate of BA737 was made since then. \adbtable{4}{|l|c|c||l|}{$norm\_departures$} {$flight$ & $norm\_dep\_time$ & $norm\_dep\_gate$ &} {$BA737$ & $2\text{:}05pm$ & $gate2$ & $[9\text{:}00am \; 1/1/92, \; 5\text{:}30pm \; 31/11/95]$ \\ $BA737$ & $2\text{:}20pm$ & $gate2$ & $[5\text{:}31pm \; 31/11/95, \; now]$ } Similarly, each incoming flight is assigned a normal arrival time and a gate, listed in $norm\_arrivals$. Flights are also assigned normal servicers, servicing companies that over a period of time normally service the flights whenever they arrive or depart. This information is stored in $norm\_servicer$. The $flight\_locations$ relation shows the location of each flight over the time. Possible $location$ values are the names of airspace sectors, gates, runways, or queues of runways. The $circling$ relation shows the flights that were circling at each time. As discussed in section \ref{aspect_examples}, flights, gates, and runways are occasionally inspected. The $inspections$ relation was discussed in section \ref{via_TSQL2}. It shows the inspection code, inspector, inspected object, status (completed or not), and time of each inspection. The $services$, $boardings$, $landings$, $\mathit{takeoffs}$, and $taxiings$ relations are very similar. They provide information about actual services, boardings, landings, take-offs, and taxiings from one location ($origin$) to another ($destination$). Each service, boarding, landing, take-off, or taxiing is assigned a unique code, stored in the $code$ attribute. The $status$ attribute shows if the climax is reached at the latest time-point of the time-stamp. The values of the $origin$ and $destination$ attributes of $taxiings$ are names of gates, runways, and queues. Apart from relations, a database would in practice also contain \emph{integrity constraints} (see \cite{Gertz1995} and \cite{Wijsen1995}). There would be, for example, a constraint saying that if the $circling$ relation shows a flight as circling at some time, the $flights$ relation must show that flight as existing at the same time. I do not discuss integrity constraints, as they are not directly relevant to this thesis. \section{Sample questions and output} \label{samples} This section presents sample questions from the airport domain, along with the corresponding output of the prototype \textsc{Nlitdb}\xspace. The questions are chosen to demonstrate that the \textsc{Nlitdb}\xspace behaves according to the specifications of the previous chapters. The questions are \emph{not} intended to be (and are probably not) a representative sample of questions that a real user might want to submit in the airport domain (see comments about Wizard of Oz experiments in section \ref{wizard} below). The user submits questions using the \texttt{nli} Prolog predicate: \singlespace\small \begin{verbatim} | ?- nli([which,flight,left,sector3,at,pm5_00,yesterday]). \end{verbatim} \normalsize\doublespace The system parses the question and reports the generated \textsc{Hpsg}\xspace sign. \singlespace\small \begin{verbatim} HPSG Sign: (phrase, qstore:(ne_set_quant, elt:(det:exists, restind:(index:(_10148, minute_ent, tvar:plus), restr:(ne_set_psoa, elt:(part, part_var:_10148, partng:pm5_00), elts:e_set))), elts:(ne_set_quant, elt:(det:interrog, restind:(index:(_10486, flight_ent, tvar:plus), restr:(ne_set_psoa, elt:(flight, arg1:_10486), elts:e_set))), elts:e_set)), synsem:(synsem, loc:(cat:(aspect:point, comps:e_list, head:(aux:minus, inv:minus, mod:none, prd:minus, vform:fin), spr:e_list, subj:e_list), cont:(at_op, main_psoa:(at_op, main_psoa:(past, et_handle:(temp_ent, tvar:plus), main_psoa:(leave_something, arg1:_10486, arg2:sector3)), time_spec:_10148), time_spec:yesterday)), nonloc:(inherited:slash:e_set, to_bind:slash:e_set))) \end{verbatim} \normalsize\doublespace The sign above is written in \textsc{Ale}\xspace's notation. The sign is of sort {\srt phrase}\/ (it corresponds to a phrase rather than a single word), and it has the features {\feat qstore} and {\feat synsem}. The {\feat qstore} value represents a non-empty set of quantifiers ({\srt ne\_set\_quant}). Its {\feat elt} feature describes the first element of that set, which is an existential quantifier. The quantifier ranges over a \textsc{Top}\xspace variable, represented by an \textsc{Hpsg}\xspace index of sort {\srt minute\_ent} (see figure \vref{ind_hierarchy}) whose {\feat tvar} is $+$ (the index represents a \textsc{Top}\xspace variable rather than a constant). The {\feat elt} value represents the \textsc{Top}\xspace-like expression $\exists \, x2^v \; \ensuremath{\mathit{Part}}\xspace[pm5\_00^g, x2^v]$. The Prolog variable \texttt{\_10148} is a pointer to the index of the quantifier, i.e.\ it plays the same role as the boxed numbers (e.g.\ \avmbox{1}, \avmbox{2}) in the \textsc{Hpsg}\xspace formalism of chapter \ref{English_to_TOP}. The {\feat elts} value describes the rest of the set of quantifiers, using in turn an {\feat elt} feature (second element of the overall set), and an {\feat elts} feature (remainder of the set, in this case the empty set). The second element of the overall set represents the \textsc{Top}\xspace expression $?x1^v \; flight(x1^v)$. In the airport application, the lexical entries of non-predicative nouns do not introduce \ensuremath{\mathit{Ntense}}\xspace operators (this generates appropriate readings in most cases; see the discussion in section \ref{non_pred_nps}). This is why no \ensuremath{\mathit{Ntense}}\xspace operator is present in the second quantifier of the sign. (The effect of \ensuremath{\mathit{Ntense}}\xspace{s} can still be seen in the airport application in the case of non-predicative adjectives, that do introduce \ensuremath{\mathit{Ntense}}\xspace{s}.) The features of the {\feat synsem} value are as in chapter \ref{English_to_TOP}. The {\feat cont} value represents the \textsc{Top}\xspace expression $\ensuremath{\mathit{At}}\xspace[yesterday, \ensuremath{\mathit{At}}\xspace[x2^v, \ensuremath{\mathit{Past}}\xspace[x3^v, leave\_something(x1^v, sector3)]]]$. The extractor of section \ref{extraction_impl} extracts a \textsc{Top}\xspace formula from the sign, and prints it as a Prolog term. \singlespace\small \begin{verbatim} TOP formula extracted from HPSG sign: interrog(x1^v, and(part(pm5_00^g, x2^v), and(flight(x1^v), at(yesterday, at(x2^v, past(x3^v, leave_something(x1^v, sector3))))))) \end{verbatim} \normalsize\doublespace The Prolog term above stands for: \begin{examps} \item[] $?x1^v \; \ensuremath{\mathit{Part}}\xspace[pm5\_00^g, x2^v] \land flight(x1^v) \; \land$ \\ $\ensuremath{\mathit{At}}\xspace[yesterday,\ensuremath{\mathit{At}}\xspace[x2^v, \ensuremath{\mathit{Past}}\xspace[x3^v, leave\_something(x1^v, sector3)]]]$ \end{examps} The extracted formula then goes through the post-processor of section \ref{extraction_impl}. The post-processor eliminates the \ensuremath{\mathit{Part}}\xspace operator, adding the $pm5\_00^g$ as an extra argument to the corresponding \ensuremath{\mathit{At}}\xspace operator: \singlespace\small \begin{verbatim} TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), at(yesterday, at(pm5_00^g, x2^v, past(x3^v, leave_something(x1^v, sector3)))))) \end{verbatim} \normalsize\doublespace The Prolog term above stands for: \begin{examps} \item $?x1^v \; flight(x1^v) \land \ensuremath{\mathit{At}}\xspace[yesterday,$ \\ $\ensuremath{\mathit{At}}\xspace[pm5\_00^g, x2^v, \ensuremath{\mathit{Past}}\xspace[x3^v, leave\_something(x1^v, sector3)]]]$ \label{log:1} \end{examps} \nspace{1.4} The post-processed formula is then translated into \textsc{Tsql2}\xspace: \singlespace\small \begin{verbatim} TSQL2 query: (SELECT DISTINCT SNAPSHOT t8.1 FROM (SELECT DISTINCT t6.1, t7.1, t7.2, t7.3, t7.4 VALID VALID(t6) FROM (SELECT DISTINCT t1.1 VALID VALID(t1) FROM (SELECT DISTINCT fl.1 VALID VALID(fl) FROM flights(PERIOD) AS fl )(SUBPERIOD) AS t1 WHERE PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever') CONTAINS VALID(t1) ) AS t6, (SELECT DISTINCT t2.1, t5.1, t5.2, t5.3 VALID VALID(t5) FROM (SELECT DISTINCT SNAPSHOT VALID(cp2) FROM gregorian AS cp2 WHERE cp2.5 = '17' AND cp2.6 = '00' ) AS t2, (SELECT DISTINCT VALID(t4), t4.1, t4.2 VALID VALID(t4) FROM (SELECT DISTINCT t3.1, t3.2 VALID VALID(t3) FROM (SELECT DISTINCT flocs.1, flocs.2 VALID PERIOD(END(VALID(flocs)), END(VALID(flocs))) FROM flight_locations(PERIOD) AS flocs )(SUBPERIOD) AS t3 WHERE t3.2 = 'sector3' AND INTERSECT( INTERSECT(t2.1, INTERSECT( PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever'), PERIOD 'today' - INTERVAL '1' DAY)), PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE)) CONTAINS VALID(t3) ) AS t4 ) AS t5 ) AS t7 WHERE t6.1 = t7.3 AND VALID(t6) = VALID(t7) ) AS t8 ) \end{verbatim} \normalsize\doublespace The ``\sql{SELECT DISTINCT fl.1}~\dots \sql{FROM flights(PERIOD) AS fl}'' that starts at the sixth line of the \textsc{Tsql2}\xspace code is the \sql{SELECT} statement to which \ensuremath{\mathit{h'_{pfuns}}}\xspace maps predicates of the form $flight(\tau_1)$. This statement returns a relation that shows the flights that existed at each time. The embedded \sql{SELECT} statement that is associated with the correlation name \sql{t6} is the result of applying the translation rule for predicates (section \ref{trans_rules}) to the $flight(x1^v)$ of \pref{log:1}. The ``\sql{WHERE PERIOD(TIMESTAMP 'beginning', TIMESTAMP 'forever') CONTAINS VALID(t1)}'' corresponds to the restriction that $et$ must fall within $lt$. (At this point, no constraint has been imposed on $lt$, and hence $lt$ covers the whole time-axis.) This \sql{WHERE} clause has no effect and could be removed during an optimisation phase (section \ref{tsql2_opt}). The ``\sql{SELECT DISTINCT flocs.1}~\dots \sql{flight\_locations(PERIOD) AS flocs}'' that starts at the 23rd line of the \textsc{Tsql2}\xspace code is the \sql{SELECT} statement to which \ensuremath{\mathit{h'_{pfuns}}}\xspace maps predicates of the form $leave\_something(\tau_1, \tau_2)$. This statement generates a relation that for each flight and location, shows the end-points of maximal periods where the flight was at that location. The embedded \sql{SELECT} statement that is associated with \sql{t4} is the result of applying the translation rule for predicates to the $leave\_something(x1^v, sector3)$ of \pref{log:1}. \sql{VALID(t3)} is the leaving-time, which has to fall within $lt$. The three nested \sql{INTERSECT}s represent constraints that have been imposed on $lt$: the \ensuremath{\mathit{Past}}\xspace operator requires $lt$ to be a subperiod of $[p_{first}, st)$ (i.e.\ a subperiod of \sql{TIMESTAMP 'beginning', TIMESTAMP 'now' - INTERVAL '1' MINUTE}), the $\ensuremath{\mathit{At}}\xspace[pm5\_00^g, \dots]$ requires $lt$ to be a subperiod of a 5:00pm-period (\sql{t2.1} ranges over 5:00pm-periods), and the $\ensuremath{\mathit{At}}\xspace[yesterday,\dots]$ requires the localisation time to be a subperiod of the previous day (\sql{PERIOD 'today' - INTERVAL '1' DAY}). The \sql{SELECT} statement that is associated with \sql{t5} is generated by the translation rule for \ensuremath{\mathit{Past}}\xspace (section \ref{trans_rules}), and the \sql{SELECT} statement that is associated with \sql{t7} is introduced by the translation rule for $\ensuremath{\mathit{At}}\xspace[\sigma_g, \beta, \phi']$ (section \ref{atsg_rule}). (The $\ensuremath{\mathit{At}}\xspace[yesterday, \dots]$ of \pref{log:1} does not introduce its own \sql{SELECT} statement, it only restricts $lt$; see the translation rule for $\ensuremath{\mathit{At}}\xspace[\kappa, \phi']$ in section \ref{trans_rules}.) The \sql{SELECT} statement that is associated with \sql{t8} is introduced by the translation rule for conjunction (section \ref{conj_rule}). It requires the attribute values that correspond to the $x1^v$ arguments of $flight(x1^v)$ and $leave\_something(x1^v, sector3)$, and the event times where the two predicates are true to be identical. Finally, the top-level \sql{SELECT} statement is introduced by the translation rule for $?\beta_1 \; ?\beta_2 \; ?\beta_3 \; \dots \; ?\beta_k \; \phi'$ (section \ref{wh1_rule}). It returns a snapshot relation that contains the attribute values that correspond to $x1^v$ (the flights). No further comments need to be made about the generated \textsc{Hpsg}\xspace signs and \textsc{Tsql2}\xspace queries. To save space, I do not show these in the rest of the examples. I also do not show the \textsc{Top}\xspace formulae before the post-processing, unless some point needs to be made about them. As noted in section \ref{progressives}, no attempt is made to block progressive forms of state verbs. The progressive forms of these verbs are taken to have the same meanings as the corresponding non-progressive ones. This causes the two questions below to receive the same \textsc{Top}\xspace formula. \singlespace\small \begin{verbatim} | ?- nli([which,tanks,contain,water]). TOP formula after post-processing: interrog(x1^v, and(tank(x1^v), pres(contains(x1^v, water)))) | ?- nli([which,tanks,are,containing,water]). TOP formula after post-processing: [same formula as above] \end{verbatim} \normalsize\doublespace There are two lexical entries for the base form of \qit{to service}, one for the habitual homonym, and one for the non-habitual one. The habitual entry introduces the predicate functor $hab\_servicer\_of$ and classifies the base form as state. The non-habitual entry introduces the functor $actl\_servicing$ and classifies the base form as culminating activity. The simple present lexical rule (section \ref{single_word_forms}) generates a simple present lexical entry for only the habitual homonym (whose base form is state). Hence, the \qit{services} below is treated as the simple present of the habitual homonym (not as the simple present of the non-habitual homonym), and only a formula that contains the $hab\_servicer\_of$ functor is generated. This captures the fact that the question can only have a habitual meaning (it cannot refer to a servicer that is actually servicing BA737 at the present; the reader is reminded that the scheduled-to-happen reading of the simple present is ignored in this thesis -- see section \ref{simple_present}). \singlespace\small \begin{verbatim} | ?- nli([which,servicer,services,ba737]). TOP formula after post-processing: interrog(x1^v, and(servicer(x1^v), pres(hab_servicer_of(x1^v, ba737)))) \end{verbatim} \normalsize\doublespace In contrast, the present participle lexical rule (section \ref{single_word_forms}) generates progressive entries for both the non-habitual (culminating activity base form) and the habitual (state base form) homonyms. This causes the question below to receive two parses, one where the \qit{is servicing} is the present continuous of the non-habitual homonym, and one where it is the present continuous of the habitual homonym. This gives rise to two formulae, one involving the $actl\_servicing$ functor (the servicer must be servicing BA737 at the present), and one involving the $hab\_servicer\_of$ functor (the servicer must be the current normal servicer of BA737). (The $x2^v$ in the first formula is an occurrence identifier; see section \ref{occurrence_ids}.) The habitual reading of the second formula seems rather unlikely in this case. \singlespace\small \begin{verbatim} | ?- nli([which,servicer,is,servicing,ba737]). TOP formula after post-processing: interrog(x1^v, and(servicer(x1^v), pres(actl_servicing(x2^v, x1^v, ba737)))) TOP formula after post-processing: interrog(x1^v, and(servicer(x1^v), pres(hab_servicer_of(x1^v, ba737)))) \end{verbatim} \normalsize\doublespace There are also different lexical entries for the actual \qit{to depart} and the habitual \qit{to depart} (at some time). The habitual entry introduces the functor $hab\_dep\_time$, requires an \qit{at~\dots} complement, and classifies the base form as state. The non-habitual entry introduces the functor $actl\_depart$, requires no complement, and classifies the base form as point. When \qit{BA737 departed at 5:00pm.} is taken to involve the habitual homonym, \qit{at 5:00pm} is treated as the complement that specifies the habitual departure time (the second argument of $hab\_dep\_time(\tau_1, \tau_2)$). When the sentence is taken to involve the non-habitual homonym, \qit{at 5:00pm} is treated as a temporal modifier, and it introduces an \ensuremath{\mathit{At}}\xspace operator (section \ref{hpsg:punc_adv}). In the following question, this analysis leads to two formulae: one where each reported flight must have actually departed at 5:00pm at least once in 1993, and one where the habitual departure time of each reported flight must have been 5:00pm some time in 1993. The second reading seems the preferred one in this example. \singlespace\small \begin{verbatim} | ?- nli([which,flights,departed,at,pm5_00,in,y1993]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), at(y1993, at(pm5_00^g, x2^v, past(x3^v, actl_depart(x1^v)))))) TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), at(y1993, past(x2^v, hab_dep_time(x1^v, pm5_00))))) \end{verbatim} \normalsize\doublespace The first question below receives no parse, because \qit{to circle} is classified as activity verb (there is no habitual state homonym in this case), and the simple present lexical rule does not generate simple present lexical entries for activity verbs. In contrast, the present participle lexical rule does generate progressive entries for activity verbs. This causes the second question below to be mapped to the formula one would expect. The failure to parse the first question is justified, in the sense that the question seems to be asking about flights that have some circling habit, and the \textsc{Nlitdb}\xspace has no access to information on circling habits. A more cooperative response, however, is needed to explain this to the user. \singlespace\small \begin{verbatim} | ?- nli([does,ba737,circle]). **No (more) parses. | ?- nli([is,ba737,circling]). TOP formula after post-processing: pres(circling(ba737)) \end{verbatim} \normalsize\doublespace Following the arrangements of section \ref{hpsg:per_advs}, in the following question where a culminating activity combines with a period adverbial, two formulae are generated: one where the inspection must have simply been completed on 1/5/92, and one where the whole inspection (from start to completion) must have been carried out on 1/5/92. The first reading seems unlikely in this example, though as discussed in section \ref{period_adverbials}, there are sentences where the first reading is the intended one. \singlespace\small \begin{verbatim} | ?- nli([who,inspected,uk160,on,d1_5_92]). TOP formula after post-processing: interrog(x1^v, at(d1_5_92, end(past(x2^v, culm(inspecting(x3^v, x1^v, uk160)))))) TOP formula after post-processing: interrog(x1^v, at(d1_5_92, past(x2^v, culm(inspecting(x3^v, x1^v, uk160))))) \end{verbatim} \normalsize\doublespace In the following question, the punctual adverbial \qit{at 5:00pm} combines with a culminating activity. According to section \ref{point_adverbials}, two readings arise: one where the taxiing starts at 5:00pm, and one where it finishes at 5:00pm. In both cases, the punctual adverbial causes the aspect of \qit{which flight taxied to gate 2 at 5:00pm} to become point. That point sentence then combines with the period adverbial \qit{yesterday}. According to section \ref{period_adverbials}, the instantaneous situation of the point phrase (the start or end of the taxiing) must occur within the period of the adverbial. This analysis leads to two formulae: one where the taxiing starts at 5:00pm on the previous day, and one where the taxiing finishes at 5:00pm on the previous day. These formulae capture the most likely readings of the question. Unfortunately, if the order of \qit{at 5:00pm} and \qit{yesterday} is reversed, the generated formulae are not equivalent to the ones below (see the discussion in section \ref{hpsg:mult_mods}) \singlespace\small \begin{verbatim} | ?- nli([which,flight,taxied,to,gate2,at,pm5_00,yesterday]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), at(yesterday, at(pm5_00^g, x2^v, end(past(x3^v, culm(taxiing_to(x4^v, x1^v, gate2)))))))) \end{verbatim} \newpage \begin{verbatim} TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), at(yesterday, at(pm5_00^g, x2^v, begin(past(x3^v, culm(taxiing_to(x4^v, x1^v, gate2)))))))) \end{verbatim} \normalsize\doublespace In the sentence below (which is treated as a yes/no question), the treatment of past perfects and punctual adverbials of section \ref{hpsg:punc_adv} allows \qit{at 5:00pm} to modify either the verb phrase \qit{left gate 2}, or the entire \qit{BA737 had left gate 2}. This gives rise to two \textsc{Top}\xspace formulae: one where 5:00pm is the time at which BA737 left gate 2, and one where 5:00pm is a reference time at which BA737 had already left gate 2. The two formulae capture the two most likely readings of the sentence. \singlespace\small \begin{verbatim} | ?- nli([ba737,had,left,gate2,at,pm5_00]). TOP formula after post-processing: past(x2^v, perf(x3^v, at(pm5_00^g, x1^v, leave_something(ba737, gate2)))) TOP formula after post-processing: at(pm5_00^g, x1^v, past(x2^v, perf(x3^v, leave_something(ba737, gate2)))) \end{verbatim} \normalsize\doublespace Similarly, in the following question, the \qit{at 5:00pm} is allowed to modify either the verb phrase \qit{taken off}, or the entire \qit{BA737 had taken off}. In the first case, the verb phrase still has the aspectual class of the base form, i.e.\ culminating activity. According to section \ref{point_adverbials}, 5:00pm is the time where the taking off was completed or started. These two readings are captured by the first and second formulae below. (The second reading seems unlikely in this example.) In the case where \qit{at 5:00pm} modifies the entire \qit{BA737 had taken off}, the \qit{had} has already caused the aspect of \qit{BA737 had taken off} to become (consequent) state. According to section \ref{point_adverbials}, in that case 5:00pm is simply a time-point where the situation of the sentence (having departed) holds. This reading is captured by the third formula. \singlespace\small \begin{verbatim} | ?- nli([ba737,had,taken,off,at,pm5_00]). TOP formula after post-processing: past(x2^v, perf(x3^v, at(pm5_00^g, x1^v, end(culm(taking_off(x4^v, ba737)))))) TOP formula after post-processing: past(x2^v, perf(x3^v, at(pm5_00^g, x1^v, begin(culm(taking_off(x4^v, ba737)))))) TOP formula after post-processing: at(pm5_00^g, x1^v, past(x2^v, perf(x3^v, culm(taking_off(x4^v, ba737))))) \end{verbatim} \normalsize\doublespace The first question below receives the formula one would expect. As discussed in section \ref{hpsg:mult_mods}, in the second question below the grammar of chapter \ref{English_to_TOP} allows two parses: one where \qit{yesterday} attaches to \qit{BA737 was circling}, and one where \qit{yesterday} attaches to \qit{BA737 was circling for two hours}. These two parses give rise to two different but logically equivalent formulae. \singlespace\small \begin{verbatim} | ?- nli([ba737,was,circling,for,two,hours,yesterday]). TOP formula after post-processing: at(yesterday, for(hour^c, 2, past(x1^v, circling(ba737)))) | ?- nli([yesterday,ba737,was,circling,for,two,hours]). TOP formula after post-processing: for(hour^c, 2, at(yesterday, past(x1^v, circling(ba737)))) TOP formula after post-processing: at(yesterday, for(hour^c, 2, past(x1^v, circling(ba737)))) \end{verbatim} \normalsize\doublespace The following example reveals a problem in the current treatment of temporal modifiers. The \textsc{Hpsg}\xspace version of this thesis (section \ref{hpsg:pupe_adv}) allows temporal modifiers to attach only to finite sentences (finite verb forms that have already combined with their subjects and complements) or past participle verb phrases (past participles that have combined with all their complements but not their subjects). In both cases, the temporal modifier attaches after the verb has combined with all its complements. English temporal modifiers typically appear either at the beginning or the end of the sentence (not between the verb and its complements), and hence requiring temporal modifiers to attach after the verb has combined with its complements is in most cases not a problem. However, in the following question (which most native English speakers find acceptable) the temporal modifier (\qit{for two hours}) is between the verb (\qit{queued}) and its complement (\qit{for runway2}). Therefore, the temporal modifier cannot attach to the verb after the verb has combined with its complement, and the system fails to parse the sentence. (In contrast, \qit{UK160 queued for runway 2 for two hours.}, where the temporal modifier follows the complement, is parsed without problems.) \singlespace\small \begin{verbatim} | ?- nli([uk160,queued,for,two,hours,for,runway2]). **No (more) parses. \end{verbatim} \normalsize\doublespace As explained in section \ref{post_processing}, the post-processing removes \ensuremath{\mathit{Culm}}\xspace operators that are within \ensuremath{\mathit{For}}\xspace operators introduced by \qit{for~\dots} adverbials. This is demonstrated in the following example. The \qit{for~\dots} adverbial introduces a \texttt{for\_remove\_culm} pseudo-operator, which can be thought of as a \ensuremath{\mathit{For}}\xspace operator with a flag attached to it, that signals that \ensuremath{\mathit{Culm}}\xspace{s} within the \ensuremath{\mathit{For}}\xspace operator must be removed. The post-processor removes the \ensuremath{\mathit{Culm}}\xspace, and replaces the \texttt{for\_remove\_culm} with an ordinary \ensuremath{\mathit{For}}\xspace operator. \singlespace\small \begin{verbatim} | ?- nli([which,flight,boarded,for,two,hours]). TOP formula extracted from HPSG sign: interrog(x1^v, and(flight(x1^v), for_remove_culm(hour^c, 2, past(x2^v, culm(boarding(x3^v, x1^v)))))) TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), for(hour^c, 2, past(x2^v, boarding(x3^v, x1^v))))) \end{verbatim} \normalsize\doublespace Duration \qit{in~\dots} adverbials introduce \ensuremath{\mathit{For}}\xspace operators that carry no flag to remove enclosed \ensuremath{\mathit{Culm}}\xspace{s}. In the following question, this leads to a formula that (correctly) requires the boarding to have been completed. \singlespace\small \begin{verbatim} | ?- nli([which,flight,boarded,in,two,hours]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), for(hour^c, 2, past(x2^v, culm(boarding(x3^v, x1^v)))))) \end{verbatim} \normalsize\doublespace As explained in section \ref{present_perfect}, the present perfect is treated in exactly the same way as the simple past. This causes the two questions below to receive the same formula. \singlespace\small \begin{verbatim} | ?- nli([which,flight,has,been,at,gate2,for,two,hours]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), for(hour^c, 2, past(x2^v, located_at(x1^v, gate2))))) | ?- nli([which,flight,was,at,gate2,for,two,hours]). TOP formula after post-processing: [same formula as above] \end{verbatim} \normalsize\doublespace As discussed in section \ref{special_verbs}, when \qit{finished} combines with a culminating activity, the situation must have reached its completion. In contrast, when \qit{stopped} combines with a culminating activity, the situation must have simply stopped, without necessarily reaching its completion. This difference is captured in the two formulae below by the existence or absence of a \ensuremath{\mathit{Culm}}\xspace. \singlespace\small \begin{verbatim} | ?- nli([j_adams,finished,inspecting,uk160,at,pm5_00]). TOP formula after post-processing: at(pm5_00^g, x1^v, past(x2^v, end(culm(inspecting(x3^v, j_adams, uk160))))) | ?- nli([j_adams,stopped,inspecting,uk160,at,pm5_00]). TOP formula after post-processing: at(pm5_00^g, x1^v, past(x2^v, end(inspecting(x3^v, j_adams, uk160)))) \end{verbatim} \normalsize\doublespace In the airport domain, non-predicative adjectives (like \qit{closed} below) introduce \ensuremath{\mathit{Ntense}}\xspace operators. In the question below, the formula that is extracted from the \textsc{Hpsg}\xspace sign contains an \ensuremath{\mathit{Ntense}}\xspace whose first argument is a variable. As explained in section \ref{post_processing}, this leads to two different formulae after the post-processing, one where \qit{closed} refers to the present, and one where \qit{closed} refers to the time of the verb tense. \singlespace\small \begin{verbatim} | ?- nli([was,any,flight,on,a,closed,runway,yesterday]). TOP formula extracted from HPSG sign: and(flight(x1^v), and(and(ntense(x2^v, closed(x3^v)), runway(x3^v)), at(yesterday, past(x4^v, located_at(x1^v, x3^v))))) **Post processing of TOP formula generated 2 different formulae. TOP formula after post-processing: and(flight(x1^v), and(and(ntense(now, closed(x3^v)), runway(x3^v)), at(yesterday, past(x4^v, located_at(x1^v, x3^v))))) TOP formula after post-processing: and(flight(x1^v), and(and(ntense(x4^v, closed(x3^v)), runway(x3^v)), at(yesterday, past(x4^v, located_at(x1^v, x3^v))))) \end{verbatim} \normalsize\doublespace In the following question, the \qit{currently} clarifies that \qit{closed} refers to the present. The \ensuremath{\mathit{Ntense}}\xspace in the formula extracted from the \textsc{Hpsg}\xspace sign has $now^*$ as its first argument. The post-processing has no effect. \singlespace\small \begin{verbatim} | ?- nli([was,any,flight,on,a,currently,closed,runway,yesterday]). TOP formula extracted from HPSG sign: and(flight(x1^v), and(and(ntense(now, closed(x2^v)), runway(x2^v)), at(yesterday, past(x3^v, located_at(x1^v, x2^v))))) \end{verbatim} \normalsize\doublespace In the following question, the verb tense refers to the present, and hence \qit{closed} can only refer to a currently closed runway. The post-processor generates only one formula, where the first argument of \ensuremath{\mathit{Ntense}}\xspace is $now^*$. \singlespace\small \begin{verbatim} | ?- nli([is,any,flight,on,a,closed,runway]). TOP formula extracted from HPSG sign: and(flight(x1^v), and(and(ntense(x2^v, closed(x3^v)), runway(x3^v)), pres(located_at(x1^v, x3^v)))) TOP formula after post-processing: and(flight(x1^v), and(and(ntense(now, closed(x3^v)), runway(x3^v)), pres(located_at(x1^v, x3^v)))) \end{verbatim} \normalsize\doublespace Predicative adjectives do not introduce \ensuremath{\mathit{Ntense}}\xspace{s} (section \ref{hpsg:adjectives}), and \textsc{Top}\xspace predicates introduced by these adjectives always end up within the operator(s) of the verb tense. This captures the fact that predicative adjectives always refer to the time of the verb tense. \singlespace\small \begin{verbatim} | ?- nli([was,gate2,open,on,monday]). TOP formula after post-processing: at(monday^g, x1^v past(x2^v, open(gate2))) \end{verbatim} \normalsize\doublespace For reasons explained in section \ref{pred_nps}, the system fails to parse sentences that contain proper names or names of days, months, etc.\ when these are used as predicative noun phrases (e.g.\ the first two questions below). Other predicative noun phrases pose no problem (e.g.\ the third question below). \singlespace\small \begin{verbatim} | ?- nli([d1_1_91,was,a,monday]). **No (more) parses. | ?- nli([ba737,is,uk160]). **No (more) parses. | ?- nli([ba737,is,a,flight]). TOP formula after post-processing: pres(flight(ba737)) \end{verbatim} \normalsize\doublespace Multiple interrogative words can be handled, as demonstrated below. \singlespace\small \begin{verbatim} | ?- nli([which,flight,is,at,which,gate]). TOP formula after post-processing: interrog(x1^v, interrog(x2^v, and(gate(x1^v), and(flight(x2^v), pres(located_at(x2^v, x1^v)))))) \end{verbatim} \normalsize\doublespace In the first question below, the grammar of chapter \ref{English_to_TOP} allows \qit{yesterday} to attach to either \qit{BA737 was circling} or to the whole \qit{did any flight leave a gate while BA737 was circling}. Two \textsc{Hpsg}\xspace signs are generated as a result of this, from which two different but logically equivalent formulae are extracted. In contrast, in the second question below, the \qit{yesterday} cannot attach to \qit{BA737 was circling}, because of the intervening \qit{while} (\qit{while BA737 was circling} is treated as an adverbial, and \qit{yesterday} cannot attach to another adverbial). Consequently, only one formula is generated. \singlespace\small \begin{verbatim} | ?- nli([did,any,flight,leave,a,gate,while,ba737,was,circling,yesterday]). TOP formula after post-processing: and(flight(x1^v), and(gate(x2^v), at(at(yesterday, past(x3^v, circling(ba737))), past(x4^v, leave_something(x1^v, x2^v))))) TOP formula after post-processing: and(flight(x1^v), and(gate(x2^v), at(yesterday, at(past(x3^v, circling(ba737)), past(x4^v, leave_something(x1^v, x2^v)))))) | ?- nli([did,any,flight,leave,a,gate,yesterday,while,ba737,was,circling]). TOP formula after post-processing: and(flight(x1^v), and(gate(x2^v), at(past(x3^v, circling(ba737)), at(yesterday, past(x5^v, leave_something(x1^v, x2^v)))))) \end{verbatim} \normalsize\doublespace In the questions below, the subordinate clause is a (progressive) state. According to section \ref{before_after_clauses}, in the first question the flights must have arrived before a time-point where BA737 started to board (\qit{to arrive} is a point verb in the airport domain). In the second question, section \ref{before_after_clauses} allows two readings: the flights must have arrived after a time-point where BA737 started or stopped boarding. The generated formulae capture these readings. \singlespace\small \begin{verbatim} | ?- nli([which,flights,arrived,before,ba737,was,boarding]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), before(past(x2^v, boarding(x3^v, ba737)), past(x4^v, actl_arrive(x1^v))))) | ?- nli([which,flights,arrived,after,ba737,was,boarding]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), after(begin(past(x2^v, boarding(x3^v, ba737))), past(x4^v, actl_arrive(x1^v))))) TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), after(past(x2^v, boarding(x3^v, ba737)), past(x4^v, actl_arrive(x1^v))))) \end{verbatim} \normalsize\doublespace Below, the subordinate clause is a culminating activity. In the first question, according to section \ref{before_after_clauses} the flights must have arrived before a time-point where BA737 finished or started to board. In the second question, the flights must have arrived after a time-point where BA737 finished boarding. These readings are captured by the generated formulae. \singlespace\small \begin{verbatim} | ?- nli([which,flights,arrived,before,ba737,boarded]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), before(end(past(x2^v, culm(boarding(x3^v, ba737)))), past(x4^v, actl_arrive(x1^v))))) TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), before(past(x2^v, culm(boarding(x3^v, ba737))), past(x4^v, actl_arrive(x1^v))))) | ?- nli([which,flights,arrived,after,ba737,boarded]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), after(past(x2^v, culm(boarding(x3^v, ba737))), past(x4^v, actl_arrive(x1^v))))) \end{verbatim} \normalsize\doublespace In the next two questions, the subordinate clause is a consequent state. According to section \ref{before_after_clauses}, in the first question the flights must have arrived before the situation of the subordinate clause (having boarded) began, i.e.\ before BA737 finished boarding. In the second question, the flights must have arrived after the situation of the subordinate clause (having boarded) began, i.e.\ after BA737 finished boarding. These readings are captured by the generated \textsc{Top}\xspace formulae. \singlespace\small \begin{verbatim} | ?- nli([which,flights,arrived,before,ba737,had,boarded]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), before(past(x2^v, perf(x3^v, culm(boarding(x4^v, ba737)))), past(x5^v, actl_arrive(x1^v)))))) | ?- nli([which,flights,arrived,after,ba737,had,boarded]). TOP formula after post-processing: interrog(x1^v, and(flight(x1^v), after(begin(past(x2^v, perf(x3^v, culm(boarding(x4^v, ba737))))), past(x5^v, actl_arrive(x1^v)))))) \end{verbatim} \normalsize\doublespace The question below combines a \qit{when} interrogative and a \qit{while~\dots} clause. The generated formula asks for maximal past circling-periods of BA737 that fall within maximal past periods where UK160 was located at gate 2. \singlespace\small \begin{verbatim} | ?- nli([when,while,uk160,was,at,gate2,was,ba737,circling]). TOP formula after post-processing: interrog_mxl(x3^v, at(past(x2^v, located_at(uk160, gate2)), past(x3^v, circling(ba737)))) \end{verbatim} \normalsize\doublespace Finally, the question below receives two formulae: the first one asks for times of past actual departures; the second one asks for past normal departure times. (The latter reading is easier to accept if an adverbial like \qit{in 1992} is attached.) In the second question, only a formula for the habitual reading is generated, because the simple present lexical rule (section \ref{single_word_forms}) does not generate a simple present lexical entry for the non-habitual \qit{to depart} (which is a point verb). \singlespace\small \begin{verbatim} | ?- nli([when,did,ba737,depart]). TOP formula after post-processing: interrog_mxl(x2^v, past(x2^v, actl_depart(ba737))) TOP formula after post-processing: interrog(x1^v, past(x2^v, hab_dep_time(ba737, x1^v))) | ?- nli([when,does,ba737,depart]). TOP formula after post-processing: interrog(x1^v, pres(hab_dep_time(ba737, x1^v))) \end{verbatim} \normalsize\doublespace \section{Speed issues} As already noted, the prototype \textsc{Nlitdb}\xspace was developed simply to demonstrate that the mappings from English to \textsc{Top}\xspace and from \textsc{Top}\xspace to \textsc{Tsql2}\xspace are implementable. Execution speed was not a priority, and the \textsc{Nlitdb}\xspace code is by no means optimised for fast execution. On a lightly loaded Sun \textsc{Sparc}station 5, single-clause questions with single parses are typically mapped to \textsc{Tsql2}\xspace queries in about 15--30 seconds. Longer questions with subordinate clauses and multiple parses usually take 1--2 minutes to process. (These times include the printing of all the \textsc{Hpsg}\xspace signs, \textsc{Top}\xspace formulae, and \textsc{Tsql2}\xspace queries.) The system's speed seems acceptable for a research prototype, but it is unsatisfactory for real-life applications. Whenever a modification is made in the software, the code of the affected modules has to be recompiled. This takes only a few seconds in the case of modules that are written in Prolog (the post-processor and the \textsc{Top}\xspace to \textsc{Tsql2}\xspace translator), but it is very time-consuming in the case of modules that are written in \textsc{Ale}\xspace's formalism (the components of the \textsc{Hpsg}\xspace grammar and the extractor of \textsc{Top}\xspace formulae). This becomes particularly annoying when experimenting with the grammar, as in many cases after modifying the grammar all its components (sort hierarchy, lexical rules, etc.) have to be recompiled, and this recompilation takes approximately 8 minutes on the above machine. \section{Summary} The framework of this thesis was tested by developing a prototype \textsc{Nlitdb}\xspace, implemented using Prolog and \textsc{Ale}\xspace. The prototype was configured for the hypothetical airport application. A number of sample questions were used to demonstrate that the system behaves according to the specifications of the previous chapters. The architecture of the prototype is currently minimal. A preprocessor, mechanisms for quantifier-scoping and anaphora resolution, an equivalential translator, a response generator, and configuration tools would have to be added if the system were to be used in real-life applications. Execution speed would also have to be improved. \chapter{Comparison with Previous Work on NLITDBs} \label{comp_chapt} \proverb{Other times other manners.} This chapter begins with a discussion of previous work on \textsc{Nlitdb}\xspace{s}. The discussion identifies six problems from which previous proposals on \textsc{Nlitdb}\xspace{s} suffer. I then examine if the framework of this thesis overcomes these problems. \section{Previous work on NLITDBs} \label{previous_nlitdbs} This section discusses previous work on \textsc{Nlitdb}\xspace{s}. Clifford's work, which is the most significant and directly relevant to this thesis, is presented first. \subsection{Clifford} \label{Clifford_prev} Clifford \cite{Clifford} defined a temporal version of the relational model. He also showed how a fragment of English questions involving time can be mapped systematically to logical expressions whose semantics are defined in terms of a database structured according to his model.\footnote{Parts of \cite{Clifford} can be found in \cite{Clifford4}, \cite{Clifford5}, and \cite{Clifford3}. The database model of this section is that of \cite{Clifford}. A previous version of this model appears in \cite{Clifford2}.} Clifford's approach is notable in that both the semantics of the English fragment and of the temporal database are defined within a common model-theoretic framework, based on Montague semantics \cite{Dowty}. Clifford extended the syntactic coverage of Montague's \textsc{Ptq}\xspace grammar, to allow past, present, and future verb forms, some temporal connectives and adverbials (e.g.\ \qit{while}, \qit{during}, \qit{in 1978}, \qit{yesterday}), and questions. \pref{cliff:2} -- \pref{cliff:7} are all within Clifford's syntactic coverage. (Assertions like \pref{cliff:4} are treated as yes/no questions.) \begin{examps} \item Is it the case that Peter earned 25K in 1978? \label{cliff:2} \item Does Rachel manage an employee such that he earned 30K? \label{cliff:3} \item John worked before Mary worked. \label{cliff:4} \item Who manages which employees? \label{cliff:6} \item When did Liz manage Peter? \label{cliff:7} \end{examps} Clifford does not allow progressive verb forms. He also claims that no distinction between progressive and non-progressive forms is necessary in the context of \textsc{Nlitdb}\xspace{s} (see p.12 of \cite{Clifford4}). According to Clifford's view, \pref{cliff:1a} can be treated in exactly the same manner as \pref{cliff:1b}. This ignores the fact that \pref{cliff:1b} most probably refers to a company that habitually or normally services BA737, or to a company that will service BA737 according to some plan, not to a company that is actually servicing BA737 at the present. In contrast, \pref{cliff:1a} most probably refers to a company that is actually servicing BA737 at the present, or to a company that is going to service BA737. Therefore, the \textsc{Nlitdb}\xspace should not treat the two questions as identical, if its responses are to be appropriate to the meanings users have in mind. \begin{examps} \item Which company is servicing flight BA737? \label{cliff:1a} \item Which company services flight BA737? \label{cliff:1b} \end{examps} Clifford also does not discuss perfect tenses (present perfect, past perfect, etc.), which do not seem to be allowed in his framework. Finally, he employs no aspectual taxonomy (this will be discussed in section \ref{sem_assess}). Following the Montague tradition, Clifford employs an intensional higher order language (called \textsc{Il}$_s$\xspace) to represent the meanings of English questions. There is a set of syntactic rules that determine the syntactic structure of each sentence, and a set of semantic rules that map syntactic structures to expressions of \textsc{Il}$_s$\xspace. For example, \pref{cliff:7} is mapped to the \textsc{Il}$_s$\xspace expression of \pref{cliff:8}. \begin{examps} \item $\begin{aligned}[t] \lambda i_1 [[i_1 < i] \land \exists y [&EMP'_*(i_1)(Peter) \land \\ &MGR'(i_1)(y) \land y(i_1) = Liz \land AS\_1(Peter, y)]] \end{aligned}$ \label{cliff:8} \end{examps} Roughly speaking, \pref{cliff:8} has the following meaning. $EMP'_*(i_1)(Peter)$ means that Peter must be an employee at the time-point $i_1$. $MGR'(i_1)(y)$ means that $y$ must be a partial function from time-points to managers (an \emph{intension} in Montague semantics terminology) which is defined for (at least) the time-point $i_1$. $AS\_1(Peter, y)$ requires $y$ to represent the history of Peter's managers (i.e.\ the value $y(i_1)$ of $y$ at each time-point $i_1$ must be the manager of Peter at that time-point). The $y(i_1) = Liz$ requires the manager of Peter at $i_1$ to be Liz. Finally, $i$ is the present time-point, and $i_1 < i$ means that $i_1$ must precede $i$. \pref{cliff:8} requires all time-points $i_1$ to be reported, such that $i_1$ precedes the present time-point, Peter is an employee at $i_1$, and Peter's manager at $i_1$ is Liz. The following (from \cite{Clifford}) is a relation in Clifford's database model (called \textsc{Hrdm}\xspace\ -- Historical Relational Database Model). \nspace{1.0} \begin{center} {\small \begin{tabular}{|l|l|l|l|l|} \hline \multicolumn{5}{|l|}{$emprel$} \\ \hline \hline $EMP$ & $MGR$ & $DEPT$ & $SAL$ & $lifespan$ \\ \hline &&&& \\ $Peter$ & $ \left[ \begin{array}{l} S2 \rightarrow Elsie\\ S3 \rightarrow Liz \end{array} \right] $ & $ \left[ \begin{array}{l} S2 \rightarrow Hardware\\ S3 \rightarrow Linen \end{array} \right] $ & $ \left[ \begin{array}{l} S2 \rightarrow 30K \\ S3 \rightarrow 35K \end{array} \right] $ & $\{S2,S3\}$\\ &&&& \\ \hline &&&& \\ $Liz$ & $ \left[ \begin{array}{l} S2 \rightarrow Elsie\\ S3 \rightarrow Liz \end{array} \right] $ & $ \left[ \begin{array}{l} S2 \rightarrow Toy\\ S3 \rightarrow Hardware \end{array} \right] $ & $ \left[ \begin{array}{l} S2 \rightarrow 35K \\ S3 \rightarrow 50K \end{array} \right] $ & $\{S2,S3\}$\\ &&&& \\ \hline &&&& \\ $Elsie$ & $ \left[ \begin{array}{l} S1 \rightarrow Elsie\\ S2 \rightarrow Elsie \end{array} \right] $ & $ \left[ \begin{array}{l} S1 \rightarrow Toy\\ S2 \rightarrow Toy \end{array} \right] $ & $ \left[ \begin{array}{l} S1 \rightarrow 50K \\ S2 \rightarrow 50K \end{array} \right] $ & $\{S1,S2\}$\\ &&&& \\ \hline \end{tabular} } \end{center} \nspace{1.4} The $\mathit{lifespan}$ of each tuple shows the time-points (``states'' in Clifford's terminology) for which the tuple carries information. In \textsc{Hrdm}\xspace, attribute values are not necessarily atomic. They can also be sets of time-point denoting symbols (as in the case of $\mathit{lifespan}$), or partial functions from time-point denoting symbols to atomic values. The relation above means that at the time-point $S2$ the manager of Peter was Elsie, and that at $S3$ the manager of Peter was Liz. \textsc{Hrdm}\xspace uses additional time-stamps to cope with schema-evolution (section \ref{no_issues}). I do not discuss these here. Clifford shows how the semantics of \textsc{Il}$_s$\xspace expressions can be defined in terms of an \textsc{Hrdm}\xspace database (e.g.\ how the semantics of \pref{cliff:8} can be defined in terms of information in $emprel$). He also defines an algebra for \textsc{Hrdm}\xspace, similar to the relational algebra of the traditional relational model \cite{Ullman}. (Relational algebra is a theoretical database query language. Most \textsc{Dbms}\xspace{s} do not support it directly. \textsc{Dbms}\xspace users typically specify their requests in more user-friendly languages, like \textsc{Sql}\xspace. \textsc{Dbms}\xspace{s}, however, often use relational algebra internally, to represent operations that need to be carried out to satisfy the users' requests.) The answer to \pref{cliff:7} can be found using \pref{cliff:9}, which is an expression in Clifford's algebra. \begin{examps} \item $\mathit{\omega(\sigma\text{-}WHEN_{EMP = Peter, MGR = Liz}(emprel))}$ \label{cliff:9} \end{examps} $\sigma\text{-}WHEN_{EMP = Peter, MGR = Liz}(emprel)$ \index{swhen@$\sigma\text{-}WHEN$ (operator of Clifford's algebra)} generates a single-tuple relation (shown below as $emprel2$) that carries the information of Peter's tuple from $emprel$, restricted to when his manager was Liz. The $\omega$ \index{o@$\omega$ (operator of Clifford's algebra)} operator returns a set of time-point denoting symbols, that represents all the time-points for which there is information in the relation-argument of $\omega$. In our example, \pref{cliff:9} returns $\{S3\}$. \nspace{1.0} \begin{center} {\small \begin{tabular}{|l|l|l|l|l|} \hline \multicolumn{5}{|l|}{$emprel2$} \\ \hline \hline $EMP$ & $MGR$ & $DEPT$ & $SAL$ & $lifespan$ \\ \hline &&&& \\ $Peter$ & $[S3 \rightarrow Liz] $ & $[S3 \rightarrow Linen]$ & $[S3 \rightarrow 35K] $ & $\{S3\}$\\ &&&& \\ \hline \end{tabular} } \end{center} \nspace{1.4} Clifford outlines an algorithm for mapping \textsc{Il}$_s$\xspace expressions to appropriate algebraic expressions (e.g.\ mapping \pref{cliff:8} to \pref{cliff:9}; see p.170 of \cite{Clifford}). The description of this algorithm, however, is very sketchy and informal. According to Clifford (\cite{Clifford4}, p.16), a parser for his version of the \textsc{Ptq}\xspace grammar (that presumably also maps English questions to \textsc{Il}$_s$\xspace expressions) has been developed. Clifford, however, does not provide any information on whether or not a translator from \textsc{Il}$_s$\xspace to his algebra was ever implemented (as noted above, this mapping is not fully defined), and there is no indication that Clifford's framework was ever used to implement an actual \textsc{Nlitdb}\xspace. \subsection{Bruce} \label{Bruce_prev} Bruce's \textsc{Chronos} \cite{Bruce1972} is probably the first natural language question-answering system that attempted to address specifically time-related issues. \textsc{Chronos} is not really an interface to a stand-alone database system. When invoked, it has no information about the world. The user ``teaches'' \textsc{Chronos} various facts (using statements like \pref{Bruce:1} and \pref{Bruce:2}), which are stored internally as expressions of a Lisp-like representation language. Questions about the stored facts can then be asked (e.g.\ \pref{Bruce:3}, \pref{Bruce:4}). \begin{examps} \item The American war for independence began in 1775. \label{Bruce:1} \item The articles of confederation period was from 1777 to 1789. \label{Bruce:2} \item Does the American war for independence coincide with the time from 1775 to 1781? \label{Bruce:3} \item Did the time of the American war for independence overlap the articles of confederation period? \label{Bruce:4} \end{examps} Bruce defines formally a model of time, and explores how relations between time-segments of that model can represent the semantics of some English temporal mechanisms (mainly verb tenses). Bruce's time-model and temporal relations seem to underlie \textsc{Chronos}' Lisp-like representation language. Bruce, however, provides no information about the representation language itself. With the exception of verb tenses, there is very little information on the linguistic coverage of the system and the linguistic assumptions on which the system is based, and scarcely any information on the mapping from English to representation language. (The discussion in \cite{Bruce1972} suggests that the latter mapping may be based on simplistic pattern-matching techniques.) Finally, Bruce does not discuss exactly how the stored facts are used to answer questions like \pref{Bruce:3} and \pref{Bruce:4}. \subsection{De, Pan, and Whinston} De, Pan, and Whinston \cite{De} \cite{De2} describe a question-answering system that can handle a fragment of English questions involving time. The ``temporal database'' in this case is a rather ad hoc collection of facts and inference rules (that can be used to infer new information from the facts), rather than a principled database built on a well-defined database model. Both the grammar of the linguistic processor and the facts and rules of the database are specified in ``equational logic'' (a kind of logic-programming language). There is no clear intermediate representation language, and it is very difficult to distinguish the part of the system that is responsible for the linguistic processing from the part of the system that is responsible for retrieving information from the ``database''. De et al.\ consider this an advantage, but it clearly sacrifices modularity and portability. For example, it is very hard to see which parts of the software would have to be modified if the natural language processor were to be used with a commercial \textsc{Dbms}\xspace. The system of De et al.\ does not seem to be based on any clear linguistic analysis. There is also very little information in \cite{De} and \cite{De2} on exactly which temporal linguistic mechanisms are supported, and which semantics are assigned to these mechanisms. Furthermore, no aspectual classes are used (see related comments in section \ref{sem_assess}). \subsection{Moens} \label{Moens_prev} Moens' work on temporal linguistic phenomena \cite{Moens} \cite{Moens2} has been highly influential in the area of tense and aspect theories (some ideas from Moens' work were mentioned in chapter \ref{linguistic_data}). In the last part of \cite{Moens} (see also \cite{Moens3}), Moens develops a simplistic \textsc{Nlitdb}\xspace. This has a very limited linguistic coverage, and is mainly intended to illustrate Moens' tense and aspect theory, rather than to constitute a detailed exploration of issues related to \textsc{Nlitdb}\xspace{s}. As in the case of Bruce and De et al., Moens' ``database'' is not a stand-alone system built according to an established (e.g.\ relational) database model. Instead, it is a collection of Prolog facts of particular forms, that record information according to an idiosyncratic and unclearly defined database model. Apart from purely temporal information (that shows when various events took place), Moens' database model also stores information about \emph{episodes}. According to Moens, an episode is a sequence of ``contingently'' related events. Moens uses the term ``contingency'' in a rather vague manner: in some cases it denotes a consequence relation (event A was a consequence of event B); in other cases it is used to refer to events that constitute steps towards the satisfaction of a common goal. The intention is, for example, to be able to store an event where John writes chapter 1 of his thesis together with an event where John writes chapter 2 of his thesis as constituting parts of an episode where John writes his thesis. Some episodes may be parts of larger episodes (e.g.\ the episode where John writes his thesis could be part of a larger episode where John earns his PhD). Moens claims that episodic information of this kind is necessary if certain time-related linguistic mechanisms (e.g.\ \qit{when~\dots} clauses, present perfect) are to be handled appropriately. Although I agree that episodic information seems to play an important role in how people perceive temporal information, it is often difficult to see how Moens' episodic information (especially when events in an episode are linked with consequence relations) can be used in a practical \textsc{Nlitdb}\xspace (e.g.\ in section \ref{present_perfect}, I discussed common claims that the English present perfect involves a consequence relation, and I explained why an analysis of the present perfect that posits a consequence relation is impractical in \textsc{Nlitdb}\xspace{s}). By assuming that the database contains episodic information, one also moves away from current proposals in temporal databases, that do not consider information of this kind. For these reasons, I chose not to assume that the database provides episodic information. As was demonstrated in the previous chapters, even in the absence of such information reasonable responses can be generated in a large number of cases. Moen's database model is also interesting in that it provides some support for \emph{imprecise temporal information}. One may know, for example, that two events A and B occurred, and that B was a consequence of A, without knowing the precise times where A and B occurred. Information of this kind can be stored in Moens' database, because in his model events are not necessarily associated with times. One can store events A and B as a sequence of contingently related events (here contingency would have its consequence meaning) without assigning them specific times. (If, however, there is no contingency relation between the two events and their exact times are unknown, Moens' model does not allow the relative order of A and B to be stored.) Although there has been research on imprecise temporal information in databases (e.g.\ \cite{Brusoni1995}, \cite{Koubarakis1995}), most of the work on temporal databases assumes that events are assigned specific times. To remain compatible with this work, I adopted the same assumption. Moens' system uses a subset of Prolog as its meaning representation language. English questions are translated into expressions of this subset using a \textsc{Dcg} grammar \cite{Pereira1980}, and there are Prolog rules that evaluate the resulting expressions against the database. Moens provides no information about the \textsc{Dcg} grammar. Also, the definition of the meaning representation language is unclear. It is difficult to see exactly which Prolog expressions are part of the representation language, and the semantics of the language is defined in a rather informal way (by listing Prolog code that evaluates some of the possible expressions of the representation language against the database). \subsection{Spenceley} \label{Spenceley_prev} Spenceley \cite{Spenceley1989} developed a version of the \textsc{Masque} natural language front-end \cite{Auxerre2} that can cope with certain kinds of imperatives and temporal questions. The front-end was used to interface to a Prolog database that modelled a blocks-world similar to that of Winograd's \textsc{Shrdlu} \cite{Winograd1973}. The dialogue in \pref{Spenc:1} -- \pref{Spenc:5.5} illustrates the capabilities of Spenceley's system. The user can type imperatives, like \pref{Spenc:1} and \pref{Spenc:2}, that cause the database to be updated to reflect the new state of the blocks-world. At any point, questions like \pref{Spenc:3} and \pref{Spenc:5} can be issued, to ask about previous actions or about the current state of the world. \begin{examps} \item Take Cube1. \label{Spenc:1} \item Put Cube1 on Cube2. \label{Spenc:2} \item What was put on Cube2? \label{Spenc:3} \item \sys{Cube1.} \label{Spenc:4} \item Is Cube2 on Cube1? \label{Spenc:5} \item \sys{No.} \label{Spenc:5.5} \end{examps} A simplistic aspectual taxonomy is adopted, that distinguishes between \emph{states} and \emph{actions} (the latter containing Vendler's activities, accomplishments, and achievements; see section \ref{asp_taxes}). The linguistic coverage is severely restricted. For example, the user can ask about past actions and present states (e.g.\ \pref{Spenc:3}, \pref{Spenc:5}), but not about past states (\pref{Spenc:6} is rejected). Only \qit{while~\dots}, \qit{before~\dots}, and \qit{after~\dots} subordinate clauses can be used to specify past times, and subordinate clauses can refer only to actions, not states (e.g.\ \pref{Spenc:7} is allowed, but \pref{Spenc:8} is not). Temporal adverbials, like \qit{at 5:00pm} in \pref{Spenc:9}, are not supported. Spenceley also attempts to provide some support for \emph{tense anaphora} (section \ref{temporal_anaphora}), but her tense anaphora mechanism is very rudimentary. \begin{examps} \item \rej Where was Cube1? \label{Spenc:6} \item What was taken before Cube1 was put on Cube2? \label{Spenc:7} \item \rej What was taken before Cube1 was on Cube2? \label{Spenc:8} \item \rej What was taken at 5:00pm? \label{Spenc:9} \end{examps} The English requests are parsed using an ``extraposition grammar'' \cite{Pereira}, and they are translated into a subset of Prolog that acts as a meaning representation language.\footnote{The syntax and semantics of a similar Prolog subset, that is used as the meaning representation language of another version of \textsc{Masque}, are defined in \cite{Androutsopoulos}.} The resulting Prolog expressions are then executed by the Prolog interpreter to update the database or to retrieve the requested information. The ``database'' is a collection of ad hoc Prolog facts (and in that respect similar to the ``databases'' of Bruce, De et al., and Moens). It stores information about past actions, but not states (this is probably why questions like \pref{Spenc:8} are not allowed). Also, the database records temporal relations between actions (which action followed which action, which action happened during some other action), but not the specific times where the actions happened. Hence, there is no information in the database to answer questions like \pref{Spenc:9}, that require the specific times where the actions happened to be known. \subsection{Brown} Brown \cite{Brown1994} describes a question-answering system that can handle some temporal linguistic phenomena. As in Bruce's system, the user first ``teaches'' the system various facts (e.g.\ \qit{Pedro is beating Chiquita.}), and he/she can then ask questions about these facts (e.g.\ \qit{Is he beating her?}). Brown's system is interesting in that it is based on Discourse Representation Theory (\textsc{Drt}\xspace), a theory in which tense and aspect have received particular attention \cite{Kamp1993}. Brown's system, however, seems to implement the tense and aspect mechanisms of \textsc{Drt}\xspace to a very limited extent. Brown shows only how simple present, simple past, present continuous, and past continuous verb forms can be handled. Other tenses, temporal adverbials, temporal subordinate clauses, etc.\ do not seem to be supported. Brown's system transforms the English sentences into \textsc{Drt}\xspace discourse representation structures, using a grammar written in an extended \textsc{Dcg} version \cite{Covington1993}. Brown provides very little information about this grammar. The relation of Brown's grammar to that sketched in \cite{Kamp1993} is also unclear. The discourse representation structures are then translated into Prolog facts (this turns out to be relatively straight-forward). As in Moens' and Spenceley's systems, the ``database'' is a collection of Prolog facts, rather than a principled stand-alone system. \subsection{Other related work} \label{other_related_prev} Hafner \cite{Hafner} considers the inability of existing \textsc{Nlidb}\xspace{s} to handle questions involving time a major weakness. Observing that there is no consensus among database researchers on how the notion of time should be supported in databases (this was true when \cite{Hafner} was written), Hafner concludes that \textsc{Nlidb}\xspace designers who wish their systems to handle questions involving time cannot look to the underlying \textsc{Dbms}\xspace for special temporal support. She therefore proposes a temporal reasoning model (consisting of a temporal ontology, a Prolog-like representation language, and inference rules written in Prolog), intended to be incorporated into a hypothetical \textsc{Nlidb}\xspace to compensate for the lack of temporal support from the \textsc{Dbms}\xspace. Hafner, however, does not describe exactly how her reasoning model would be embedded into a \textsc{Nlidb}\xspace (e.g.\ how the semantics of verb tenses, temporal adverbials, etc.\ could be captured in her representation language, how English questions could be translated systematically into her representation language, and exactly how her inference rules would interact with the \textsc{Dbms}\xspace). Also, although when \cite{Hafner} was written it was true that there was no consensus among temporal database researchers, and that the \textsc{Nlidb}\xspace designer could not expect special temporal support from the \textsc{Dbms}\xspace, this is (at least to some extent) not true at the present. A temporal database query language (\textsc{Tsql2}\xspace) that was designed by a committee comprising most leading temporal database researchers now exists, and a prototype \textsc{Dbms}\xspace (\textsc{TimeDB}; section \ref{tdbs_general}) that supports \textsc{Tsql2}\xspace has already appeared. Instead of including into the \textsc{Nlitdb}\xspace a temporal reasoning module (as sketched by Hafner), in this thesis I assumed that a \textsc{Dbms}\xspace supporting \textsc{Tsql2}\xspace is available, and I exploited \textsc{Tsql2}\xspace's temporal facilities. Mays \cite{Mays1986} defines a modal logic which can be used to reason about possible or necessary states of the world (what may or will become true, what was or could have been true; see also the discussion on modal questions in section \ref{no_issues}). Mays envisages a reasoning module based on his logic that would be used when a \textsc{Nlidb}\xspace attempts to generate cooperative responses. In \pref{Mays:2}, for example, the system has offered to monitor the database, and to inform the user when Kitty Hawk reaches Norfolk. In order to avoid responses like \pref{Mays:4}, the system must be able to reason that the distance between the two cities will never change. Mays, however, does not discuss exactly how that reasoning module would be embedded into a \textsc{Nlidb}\xspace (e.g.\ how English questions would be mapped to expressions of his logic, and how the reasoning module would interact with the database). \begin{examps} \item Is the Kitty Hawk in Norfolk? \label{Mays:1} \item \sys{No, shall I let you know when she is?} \label{Mays:2} \item Is New York less than 50 miles from Philadelphia? \label{Mays:3} \item \sys{No, shall I let you know when it is?} \label{Mays:4} \end{examps} Hinrichs \cite{Hinrichs} proposes methods to address some time-related linguistic phenomena, reporting on experience from a natural language understanding system that, among other things, allows the user to access time-dependent information stored in a database. Although Hinrichs' methods are interesting (some of them were discussed in section \ref{noun_anaphora}), \cite{Hinrichs} provides little information on the actual natural language understanding system, and essentially no information on the underlying \textsc{Dbms}\xspace and how the intermediate representation language expressions are evaluated against the database. There is also no indication that any aspectual taxonomy is used, and the system uses a version of Montague's \textsc{Ptq}\xspace grammar (see related comments in section \ref{eval_Grammar} below). Finally, in \textsc{Cle} (a generic natural language front-end \cite{Alshawi}) verb tenses introduce into the logical expressions temporal operators, and variables that are intended to represent states or events. The semantics of these operators and variables, however, are left undefined. In \textsc{Clare} (roughly speaking, a \textsc{Nlidb}\xspace based on \textsc{Cle}; see \cite{Alshawi2}) the temporal operators are dropped, and verb tenses are expressed using predications over event or state variables. The precise semantic status of these variables remains obscure. Both \cite{Alshawi} and \cite{Alshawi2} do not discuss temporal linguistic phenomena in any detail. \section{Assessment} \label{evaluation} It follows from the discussion in section \ref{previous_nlitdbs} that previous approaches to \textsc{Nlitdb}\xspace{s} suffer from one or more of the following: (i) they ignore important English temporal mechanisms, or assign to them over-simplified semantics (e.g.\ Clifford, Spenceley, Brown), (ii) they lack clearly defined meaning representation languages (e.g.\ Bruce, De et al., Moens), (iii) they do not provide complete descriptions of the mappings from natural language to meaning representation language (e.g.\ Bruce, Moens, Brown), or (iv) from meaning representation language to database language (e.g.\ Clifford), (v) they adopt idiosyncratic and often not well-defined database models or languages (e.g.\ Bruce, De et al., Moens, Spenceley, Brown), (vi) they do not demonstrate that their ideas are implementable (e.g.\ Clifford, Hafner, Mayes). In this section I assess the work of this thesis with respect to (i) -- (vi), comparing mainly to Clifford's work, which constitutes the most significant previous exploration of \textsc{Nlitdb}\xspace{s}. \subsection{English temporal mechanisms and their semantics} \label{sem_assess} In section \ref{Clifford_prev}, I criticised Clifford's lack of aspectual taxonomy. It should be clear from the discussion in chapter \ref{linguistic_data} that the distinction between aspectual classes pertains to the semantics of most temporal linguistic mechanisms, and that without an aspectual taxonomy important semantic distinctions cannot be captured (e.g.\ the fact that the simple past of a culminating activity verb normally implies that the climax was reached, while the simple past of a point, state, or activity verb carries no such implication; the fact that an \qit{at~\dots} adverbial typically has an inchoative or terminal meaning with a culminating activity, but an interjacent meaning with a state, etc.) The aspectual taxonomy of this thesis allowed me to capture many distinctions of this kind, which cannot be accounted for in Clifford's framework. Generally, this thesis examined the semantics of English temporal mechanisms at a much more detailed level compared to Clifford's work. Particular care was also taken to explain clearly which temporal linguistic mechanisms this thesis attempts to support, which simplifications were introduced in the semantics of these mechanisms, and which phenomena remain to be considered (see table \vref{coverage_table} for a summary). This information is difficult to obtain in the case of Clifford's work. In terms of syntactic coverage of time-related phenomena, the grammar of this thesis is similar to Clifford's. Both grammars, for example, support only three kinds of temporal subordinate clauses: \qit{while~\dots}, \qit{before~\dots}, and \qit{after~\dots} clauses. Clifford's grammar allows simple-future verb forms (these are not supported by the grammar of this thesis), but it does not allow progressive or perfect forms (which are partially supported by the grammar of this thesis). The two grammars allow similar temporal adverbials (e.g.\ \qit{in~1991}, \qit{before 3/5/90}, \qit{yesterday}), though there are adverbials that are supported by Clifford's grammar but not by the grammar of this thesis (e.g.\ \qit{never}, \qit{always}), and adverbials that are supported by the grammar of this thesis but not by Clifford's (e.g.\ \qit{for five hours}, \qit{in two days}). Both grammars support yes/no questions, \qit{Who/What/Which~\dots?} and \qit{When~\dots?} questions, multiple interrogatives (e.g.\ \qit{Who inspected what on 1/1/91?}), and assertions (which are treated as yes/no questions). The reader is reminded, however, that Clifford assigns to temporal linguistic mechanisms semantics which are typically much shallower than the semantics of this thesis. Although the framework of this thesis can cope with an interesting set of temporal linguistic phenomena, there are still many English temporal mechanisms that are not covered (e.g.\ \qit{since~\dots} adverbials, \qit{when~\dots} clauses, tense anaphora). Hence, the criticism about previous approaches, that important temporal linguistic mechanisms are not supported, applies to the work of this thesis as well. (It also applies to Clifford's framework, where most of these mechanisms are also not covered.) I claim, however, that the temporal mechanisms that are currently supported are assigned sufficiently elaborate semantics, to the extent that the other criticism about previous approaches, that they use over-simplified semantics, does not apply to the work of this thesis. I hope that further work on the framework of this thesis will extend its coverage of temporal phenomena (see section \ref{to_do} below). \subsection{Intermediate representation language} From the discussion in section \ref{previous_nlitdbs}, it follows that some previous proposals on \textsc{Nlitdb}\xspace{s} (e.g.\ Bruce, De et al., Moens) use meaning representation languages that are not clearly defined. (Clifford's work does not suffer from this problem; his \textsc{Il}$_s$\xspace language is defined rigorously.) This is a severe problem. Without a detailed description of the syntax of the representation language, it is very difficult to design a mapping from the representation language to a new database language (one may want to use the linguistic front-end with a new \textsc{Dbms}\xspace that supports another database language), and to check that existing mappings to database languages cover all the possible expressions of the representation language. Also, without a rigorously defined semantics of the representation language, it is difficult to see the exact semantics that the linguistic front-end assigns to natural language expressions, and it is impossible to prove formally that the mapping from representation language to database language preserves the semantics of the representation language expressions. This pitfall was avoided in this thesis: both the syntax and the semantics of \textsc{Top}\xspace are completely and formally defined (chapter \ref{TOP_chapter}). \subsection{Mapping from English to representation language} \label{eval_Grammar} In section \ref{previous_nlitdbs}, I noted that some previous \textsc{Nlitdb}\xspace proposals (e.g.\ Bruce, Moens) provide very little or no information on the mapping from English to meaning representation language. (Again, this criticism does not apply to Clifford's work; his mapping from English to \textsc{Il}$_s$\xspace is well-documented.) In this thesis, this pitfall was avoided: I adopted \textsc{Hpsg}\xspace, a well-documented and currently widely-used grammar theory, and I explained in detail (in chapter \ref{English_to_TOP}) all the modifications that were introduced to \textsc{Hpsg}\xspace, and how \textsc{Hpsg}\xspace is used to map from English to \textsc{Top}\xspace. I consider the fact that this thesis adopts \textsc{Hpsg}\xspace to be an improvement over Clifford's framework, which is based on Montague's ageing \textsc{Ptq}\xspace grammar, and certainly a major improvement over other previous \textsc{Nlitdb}\xspace proposals (e.g.\ Bruce, Spenceley, De et al., Moens) that employ ad hoc grammars which are not built on any principled grammar theory. \subsection{Mapping from representation language to database language} As mentioned in section \ref{Clifford_prev}, Clifford outlines an algorithm for translating from \textsc{Il}$_s$\xspace (his intermediate representation language) to a version of relational algebra. This algorithm, however, is described in a very sketchy manner, and there is no proof that the algorithm is correct (i.e.\ that the generated algebraic expressions preserve the semantics of the \textsc{Il}$_s$\xspace expressions). In contrast, the \textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping of this thesis is defined rigorously, and I have proven formally that it generates appropriate \textsc{Tsql2}\xspace queries (chapter \ref{tdb_chapter} and appendix \ref{trans_proofs}). \subsection{Temporal database model and language} Several previous proposals on \textsc{Nlitdb}\xspace{s} (e.g.\ De et al., Spenceley, Moens) adopt temporal database models and languages that are idiosyncratic (not based on established database models and languages) and often not well-defined. Although Clifford's database model and algebra are well-defined temporal versions of the traditional relational database model and algebra, they constitute just one of numerous similar proposals in temporal databases, and it is unlikely that \textsc{Dbms}\xspace{s} supporting Clifford's model and algebra will ever appear. This thesis adopted \textsc{Tsql2}\xspace and its underlying \textsc{Bcdm}\xspace model. As already noted, \textsc{Tsql2}\xspace was designed by a committee comprising most leading temporal database researchers, and hence it has much better chances of being supported by forthcoming temporal \textsc{Dbms}\xspace{s}, or at least of influencing the models and languages that these \textsc{Dbms}\xspace{s} will support. As mentioned in section \ref{tdbs_general}, a prototype \textsc{Dbms}\xspace for a version of \textsc{Tsql2}\xspace has already appeared. Although I had to introduce some modifications to \textsc{Tsql2}\xspace and \textsc{Bcdm}\xspace (and hence the database language and model of this thesis diverge from the committee's proposal), these modifications are relatively few and well-documented (chapter \ref{tdb_chapter}). \subsection{Implementation} As mentioned in section \ref{Clifford_prev}, although a parser for Clifford's \textsc{Ptq}\xspace version has been implemented, there is no indication that a translator from \textsc{Il}$_s$\xspace to his relational algebra was ever constructed, or that his framework was ever used to build an actual \textsc{Nlitdb}\xspace. (Similar comments apply to the work of Hafner and Mays of section \ref{other_related_prev}.) In contrast, the framework of this thesis was used to implement a prototype \textsc{Nlitdb}\xspace. Although several modules need to be added to the prototype \textsc{Nlitdb}\xspace (section \ref{modules_to_add}), the existence of this prototype constitutes an improvement over Clifford's work. Unfortunately, the \textsc{Nlitdb}\xspace of this thesis still suffers from the fact that it has never been linked to a \textsc{Dbms}\xspace (section \ref{prototype_arch}). I hope that this will be achieved in future (see section \ref{to_do} below). \section{Summary} In terms of syntactic coverage of temporal linguistic mechanisms, the framework of this thesis is similar to Clifford's. The semantics that Clifford assigns to these mechanisms, however, are much shallower than those of this thesis. In both frameworks, there are several time-related phenomena that remain to be covered. Unlike some of the previous \textsc{Nlitdb}\xspace proposals, the intermediate representation language of this thesis (\textsc{Top}\xspace) is defined rigorously, and the mapping from English to \textsc{Top}\xspace is fully documented. Unlike Clifford's and other previous proposals, this thesis adopts a temporal database model and language (\textsc{Tsql2}\xspace) that were designed by a committee comprising most leading temporal database researchers, and that are more likely to be supported by (or at least influencing) forthcoming temporal \textsc{Dbms}\xspace{s}. The mapping from \textsc{Top}\xspace to \textsc{Tsql2}\xspace is fully defined and formally proven. In contrast, Clifford's corresponding mapping is specified in a sketchy way, with no proof of its correctness. Also, unlike Clifford's and other previous proposals, the framework of this thesis was used to implement a prototype \textsc{Nlitdb}\xspace. The implementation of this thesis still suffers from the fact that the prototype \textsc{Nlitdb}\xspace has not been linked to a \textsc{Dbms}\xspace. I hope, however, that this will be achieved in future. \chapter{Conclusions} \label{conclusions_chapt} \proverb{Times change and we with time.} \section{Summary of this thesis} This thesis has proposed a principled framework for constructing natural language interfaces to temporal databases (\textsc{Nlitdb}\xspace{s}). This framework consists of: \begin{itemize} \item a formal meaning representation language (\textsc{Top}\xspace), used to represent the semantics of English questions involving time, \item an \textsc{Hpsg}\xspace version that maps a wide range of English temporal questions to appropriate \textsc{Top}\xspace expressions, \item a set of translation rules that turn \textsc{Top}\xspace expressions into suitable \textsc{Tsql2}\xspace queries. \end{itemize} The framework of this thesis is principled, in the sense that it is clearly defined and based on current ideas from tense and aspect theories, grammar theories, temporal logics, and temporal databases. To demonstrate that it is also workable, it was employed to construct a prototype \textsc{Nlitdb}\xspace, implemented using \textsc{Ale}\xspace and Prolog. Although several issues remain to be addressed (these are discussed in section \ref{to_do} below), the work of this thesis constitutes an improvement over previous work on \textsc{Nlitdb}\xspace{s}, in that: (i) the semantics of English temporal mechanisms are generally examined at a more detailed level, (ii) the meaning representation language is completely and formally defined, (iii) the mapping from English to meaning representation language is well-documented and based on a widely-used grammar theory, (iv) a temporal database language and model that were designed by a committee comprising most leading temporal database researchers are adopted, (v) the mapping from meaning representation language to database language is clearly defined and formally proven, (vi) it was demonstrated that the theoretical framework of this thesis is implementable, by constructing a prototype \textsc{Nlitdb}\xspace on which more elaborate systems can be based. \section{Further work} \label{to_do} There are several ways in which the work of this thesis could be extended: \paragraph{Extending the linguistic coverage:} \label{wizard} In section \ref{evaluation}, I noted that although the framework of this thesis can handle an interesting set of temporal linguistic mechanisms, there are still many time-related linguistic phenomena that are not supported (see table \vref{coverage_table}). One could explore how some of these phenomena could be handled. The temporal anaphoric phenomena of section \ref{temporal_anaphora} are among those that seem most interesting to investigate: several researchers have examined temporal anaphoric phenomena, e.g.\ \cite{Partee1984}, \cite{Hinrichs1986}, \cite{Webber1988}, \cite{Eberle1989}, and it would be interesting to explore the applicability of their proposals to \textsc{Nlitdb}\xspace{s}. A Wizard of Oz experiment could also be carried out to determine which temporal phenomena most urgently need to be added to the linguistic coverage, and to collect sample questions that could be used as a test suite for \textsc{Nlitdb}\xspace{s} \cite{King1996}. (In a Wizard of Oz experiment, users interact through terminals with a person that pretends to be a natural language front-end; see \cite{Diaper1986}.) \paragraph{Cooperative responses:} In section \ref{no_issues}, I noted that the framework of this thesis provides no mechanism for cooperative responses. It became evident during the work of this thesis that such a mechanism is particularly important in \textsc{Nlitdb}\xspace{s} and should be added (cases where cooperative responses are needed were encountered in sections \ref{simple_past}, \ref{progressives}, \ref{special_verbs}, \ref{period_adverbials}, \ref{while_clauses}, \ref{before_after_clauses}, \ref{at_before_after_op}, and \ref{samples}). To use an example from section \ref{simple_past}, \pref{coop:2} is assigned a \textsc{Top}\xspace formula that requires BA737 to have \emph{reached} gate 2 for the answer to be affirmative. This causes a negative response to be generated if BA737 was taxiing to gate 2 but never reached it. While a simple negative response is strictly speaking correct, it is hardly satisfactory in this case. A more cooperative response like \pref{coop:4} is needed. \begin{examps} \item Did BA737 taxi to gate 2? \label{coop:2} \item \sys{BA737 was taxiing to gate 2 but never reached it.} \label{coop:4} \end{examps} In other cases, the use of certain English expressions reveals a misunderstanding of how situations are modelled in the database and the \textsc{Nlitdb}\xspace. In \pref{coop:3}, for example, the \qit{for~\dots} adverbial shows that the user considers departures to have durations (perhaps because he/she considers the boarding part of the departure; see section \ref{point_criterion}). In the airport application, however, departures are treated as instantaneous (they include only the time-points where the flights leave the gates), and \qit{to taxi} is classified as a point verb. The \qit{for~\dots} adverbial combines with a point expression, which is not allowed in the framework of this thesis (see table \vref{for_adverbials_table}). This causes \pref{coop:3} to be rejected without any explanation to the user. It would be better if a message like \pref{coop:3a} could be generated. \begin{examps} \item Which flight was departing for twenty minutes? \label{coop:3} \item \sys{Departures of flights are modelled as instantaneous.} \label{coop:3a} \end{examps} \paragraph{Paraphrases:} As explained in section \ref{prototype_arch}, a mechanism is needed to generate English paraphrases of possible readings in cases where the \textsc{Nlitdb}\xspace understands a question to be ambiguous. \paragraph{Optimising the TSQL2 queries:} As discussed in section \ref{tsql2_opt}, there are ways in which the generated \textsc{Tsql2}\xspace queries could be optimised before submitting them to the \textsc{Dbms}\xspace. One could examine exactly how these optimisations would be carried out. \paragraph{Additional modules in the prototype NLITDB:} Section \ref{modules_to_add} identified several modules that would have to be added to the prototype \textsc{Nlitdb}\xspace if this were to be used in real-life applications: a preprocessor, modules to handle quantifier scoping and anaphora resolution, an equivalential translator, and a response generator. Adding a preprocessor and a simplistic response generator (as described at the beginning of section \ref{response_generator}) should be easy, though developing a response generator that would produce cooperative responses is more complicated (see the discussion above and section \ref{response_generator}). It should also be possible to add an equivalential translator without introducing major revisions in the work of this thesis. In contrast, adding modules to handle quantifier scoping and anaphora requires extending first the theoretical framework of this thesis: one has to modify \textsc{Top}\xspace to represent universal quantification, unresolved quantifiers, and unresolved anaphoric expressions (sections \ref{quantif_scoping} and \ref{anaphora_module}), and to decide how to determine the scopes or referents of unresolved quantifiers and anaphoric expressions. \paragraph{Linking to a DBMS:} As explained in sections \ref{tdbs_general} and \ref{contribution}, a prototype \textsc{Dbms}\xspace (\textsc{TimeDb}) that supports a version of \textsc{Tsql2}\xspace was released recently, but the prototype \textsc{Nlitdb}\xspace of this thesis has not been linked to that system (or any other \textsc{Dbms}\xspace). Obviously, it would be particularly interesting to connect the \textsc{Nlitdb}\xspace of this thesis to \textsc{TimeDb}. This requires bridging the differences between the versions of \textsc{Tsql2}\xspace that the two systems adopt (section \ref{contribution}). \paragraph{Embedding ideas from this thesis into existing NLIDBs:} Finally, one could explore if ideas from this thesis can be used in existing natural language front-ends. In section \ref{other_related_prev}, for example, I noted that \textsc{Cle}'s formulae contain temporal operators whose semantics are undefined. One could examine if \textsc{Top}\xspace operators (whose semantics are formally defined) could be used instead. Ideas from the \textsc{Top}\xspace to \textsc{Tsql2}\xspace mapping of chapter \ref{tdb_chapter} could then be employed to translate the resulting \textsc{Cle} formulae into \textsc{Tsql2}\xspace. \newpage \nspace{1.0} \addcontentsline{toc}{chapter}{Bibliography}

proofpile-arXiv_065-504

\section{Introduction} It is widely held that Quasi-Monte Carlo\ integration, in which the integration points are distributed more uniformly than in classical Monte Carlo\ integration which uses truly (or approximately) random points, can lead to potentially much smaller integration errors for the same amount of effort ({\it i.e.\/} the same number of integrand evaluations). A number of theorems are known that relate information on the fluctuating behaviour of the integrand (such as variation, modulus of continuity, etc.) and information on the degree of uniformity of the point set employed (in terms of some quantitative notion of discrepancy) to the integration error \cite{niederreiter}. These results, however, do not easily lend themselves to practical error estimates and moreover, being usually upper limits, may be too pessimistic in many applications. This situation is to be contrasted to that in classical Monte Carlo\ integration: there, one settles for a {\it probabilistic\/} error estimate, which on the one hand does not give perfectly certain information but only confidence levels, but on the other hand can be easily computed by estimating not only the integral but at the same time the variance of the integrand. The essential point in this procedure is the existence of the Central Limit Theorem, which states that for a large number $N$ of randomly chosen integration points, the integration error has an approximately Gaussian distribution with zero mean and a standard deviation related to the integrand's variance. The estimation of this single parameter therefore suffices to determine the shape of the error distribution. In this paper, we attempt to derive results similar to the Central Limit Theorem, for the case of Quasi-Monte Carlo. In previous publications \cite{first,second} we have argued that such considerations require a definition of what constitutes an ensemble of $N$-point quasi-random\ point sets. For truly random points, this is an easy problem since we may simply assume the points to be iid uniformly over the integration region. For quasi-random\ points the situation is somewhat more subtle. We propose to use the fact that such more evenly distributed point sets are generally characterized by a low value of {\it discrepancy\/}: given {\it some\/} definition of discrepancy (we shall specify one later on), we restrict ourselves to the set of $N$-point point sets in which the points are all uniformly iid, but with the additional condition that the discrepancy has a particular value $s$ (by suitable integration over $s$, we shall of course obtain again the classical results for truly random points). We can then study the distribution of the integration error over this ensemble of point sets. The lay-out of this paper is as follows. In section 2 we establish some notation and define our point set ensemble. In section 3 we derive our main result on the distribution of the integration error, in terms of a single complex integral. In section 4, we present explicit results for a particular, simple definition of discrepancy. In section 5 we attempt to do the same for what we believe constitutes a realistic discrepancy. In each case we aim at arriving at an error distribution that depends on only a single parameter (so that confidence levels for the integration result can easily be computed), and ultimately, of course, the ideal Gaussian error distribution. \section{Notation and definitions} Our integration region will always be the $D$-dimensional hypercube $K=[0,1)^D$, containing the point set $X_N=\{x_1,x_2,\ldots,x_N\}$. Where necessary, we shall denote the individual components of the vector $x_k$ with Greek indices: so, $x_k=x_k^{\mu}=(x_k^1,x_k^2,\ldots,x_k^D)$. Let the integrand be denoted by $f(x)$; we assume, for simplicity, that the moments \begin{equation} J_p = \intl_K dx\;f(x)^p \end{equation} exist at least for the first few values of $p$. The numerical integral estimate is given by \begin{equation} S = {1\over N}\sum\limits_{k=1}^N\;f(x_k)\;\;, \end{equation} and the integration error $\eta$ is then, of course, \begin{equation} \eta = S - J_1\;\;. \end{equation} It is the probability distribution of $\eta$ over the ensemble of point sets $X_N$ which is our object of concern here. We now turn to the definition of a discrepancy. We introduce the Fourier base of orthonormal function as follows. Starting with $D=1$, we define \begin{equation} u_{2n-1}(x) = \sqrt{2}\sin(2\pi nx)\;\;\;,\;\;\; u_{2n}(x) = \sqrt{2}\cos(2\pi nx)\;\;, \end{equation} for $n=1,2,3,\ldots$, and $u_0(x)=1$. In more dimensions, we define vectors $\vec{n}=n^{\mu}=(n^1,n^2,\ldots,n^D)$ with integer, non-negative components, and write \begin{equation} u\svn(x) = \prod\limits_{\mu=1}^Du_{n^{\mu}}(x^{\mu})\;\;. \end{equation} We assume that the integrand $f$ can be decomposed into its various Fourier modes as follows: \begin{equation} f(x) = \sum\limits_{\vec{n}}v_{\vec{n}}u\svn(x)\;\;, \end{equation} from which it immediately follows that \relax \begin{equation} J_1 = v_{(0,0,\ldots,0)}\;\;\;,\;\;\; V \equiv \sum\limits_{\vec{n}>0}v_{\vec{n}}^2 = J_2-J_1^2 \;\;. \end{equation} \relax Here and in the following, the notation $\vec{n}>0$ means a sum over all vectors $\vec{n}$ except the null vector $(0,0,\ldots,0)$. Quadratic integrability of the integrand requires that the variance $V$, i.e. the sum of the $v_{\vec{n}}^2$, converges. \relax To each mode with wave vector $\vec{n}$ we associate a {\it strength\/} $\si\svn$. In \cite{first} and \cite{second} we relate these strengths to a definition of an ensemble of {\it integrands}, by letting every $v_{\vec{n}}$ be normally distributed with zero mean and width $\si\svn$, but here we do not have to assume a particular such ensemble. The definition of (quadratic) discrepancy that we propose to use is \begin{equation} D_N(X_N) = {1\over N}\sum\limits_{k,l=1}^N\beta(x_k,x_l)\;\;\;,\;\;\; \beta(x_k,x_l) = \sum\limits_{\vec{n}>0}\si\svn^2u\svn(x_k)u\svn(x_l)\;\;. \end{equation} An essential property is that \begin{equation} \intl_K dx_k\;\beta(x_k,x_l) = \intl_K dx_l\;\beta(x_k,x_l) = 0\;\;. \label{betaprop} \end{equation} Another important assumption is that of {\it translational invariance}, by which the sines and cosines of each particular wave component have equal strength: \begin{eqnarray} \sigma_{(2n^1,2n^2,\ldots,2n^D)} & = & \sigma_{(2n^1-1,2n^2,\ldots,2n^D)} = \sigma_{(2n^1,2n^2-1,\ldots,2n^D)} =\nonumber \\ =\;\;\cdots & = & \sigma_{(2n^1-1,2n^2-1,\ldots,2n^D-1)} \;\;. \end{eqnarray} One of the consequences of this choice is that $\beta(x_k,x_l)$ only depends on the difference $x_k-x_l$, and therefore \begin{equation} \beta(0) = \intl_K dx\;\beta(x,x) = \sum\limits_{\vec{n}>0}\si\svn^2\;\;. \end{equation} Hence, for truly random points the expected value of the discrepancy is \begin{equation} \avg{D_N(X_N)} = \sum\limits_{\vec{n}>0}\si\svn^2\;\;, \end{equation} and of course we assume this sum to converge. For the particular point set $X_N$ we are employing, we assume the discrepancy $D_N(X_N)$ to have a known value $s$, non-negative by construction. Super-uniform, or quasi-random, point sets are distinguished by the fact that $s$ is small compared to its expectation for random point sets. We now come to the definition of an ensemble of quasi-random\ point sets. We consider it to consist of all point sets $X_N$ that have a value $s$ of the above discrepancy, but are otherwise unrestricted. The combined probability density $P_N$ for the $N$ points $x_k$ is then given by \relax \begin{eqnarray} P_N(s;x_1,x_2,\ldots,x_N) & = & {\delta\left(D_N(x_1,x_2,\ldots,x_N)-s\right)\over H_0(s)}\;\;,\nonumber \\ H_0(s) & = & \intl_K dx_1\cdots dx_N\; \delta\left(D_N(x_1,x_2,\ldots,x_N)-s\right)\;. \label{pndef} \end{eqnarray} \relax The number $H_0(s)$ serves to normalize the probability density $P_N$: it is nothing but the probability for a set of truly random points to attain the value $s$ for its discrepancy. Indeed, we trivially have \begin{equation} \int\limits_0^{\infty}ds\;H_0(s)P_N(s;x_1,x_2,\ldots,x_N) = 1\;\;. \end{equation} \section{The error distribution} We now start to work our way towards a Central Limit Theorem\ for Quasi-Monte Carlo, assuming the point set $X_N$ to be a member of the ensemble constructed above. Let $P(s;\eta)$ be the probability density of the integration error $\eta$ over the ensemble of possible point sets $X_N$. We may write \begin{equation} P(s;\eta) = \intl_K dx_1\cdots dx_N \;P_N(s;x_1,\ldots,x_N)\; \delta\left(\eta - S + J_1\right)\;\;. \label{petadef} \end{equation} Using the definition of the Dirac delta distributions in Eqs.(\ref{pndef},\ref{petadef}) as Fourier integrals, we may write this as \relax \begin{eqnarray} P(s;\eta) & = & {1\over H_0(s)} \intl_{-i\infty}^{i\infty} {dt\over2\pi i}\;{dz\over2\pi i}\;e^{ - z\eta - zJ_1 - ts } M(z,t)\;\;,\nonumber \\ M(z,t) & = & \intl_K dx_1\cdots dx_N\; \exp\left( {z\over N}\sum\limits_{k=1}^Nf(x_k) + {t\over N}\sum\limits_{k,l=1}^N\beta(x_k,x_l)\right)\nonumber \\ & = & \sum\limits_{m\ge0} {t^m\over m!}\;M_m(z)\;\;, \end{eqnarray} \relax where the integration contours for $t$ and $z$ run to the left of any singularities. In the spirit of the classical Central Limit Theorem, we must now proceed to take the asymptotic limit $N\to\infty$ in a careful manner, taking into account that the dominant part of the $z$ integral comes from the region where $z$ is of order $\order{\sqrt{N}}$. The procedure is most easily illustrated by considering the first few powers of $t$. To start, we have \relax \begin{eqnarray} M_0(z) & = & \intl_K dx_1\cdots dx_N\;e^{{z}\sum\limits_kf(x_k)/N} \;\;=\;\; \left\langle e^{zf(x)/N}\right\rangle ^N \vphantom{\intl_K dx_1\cdots dx_N\;e^{{z}\sum\limits_kf(x_k)/N} \left\langle e^{zf(x)/N}\right\rangle ^N} \nonumber \\ & = & \left(1 + {z\over N}J_1 + {z^2\over2N^2}J_2 + \order{{z^3\over N^3}}\right)^N \vphantom{\intl_K dx_1\cdots dx_N\;e^{{z}\sum\limits_kf(x_k)/N} \left\langle e^{zf(x)/N}\right\rangle ^N} \nonumber \\ & = & \exp\left(zJ_1 + {z^2\over2N}(J_2-J_1^2) + \order{{z^3\over N^2}}\right)\;\;. \vphantom{\intl_K dx_1\cdots dx_N\;e^{{z}\sum\limits_kf(x_k)/N} \left\langle e^{zf(x)/N}\right\rangle ^N} \end{eqnarray} \relax Due to \eqn{betaprop} the next contribution evaluates as follows: \relax \begin{eqnarray} M_1(z) & = & \intl_K dx_1\cdots dx_N\;e^{{z}\sum\limits_kf(x_k)/N} {1\over N}\sum\limits_{k,l}\beta(x_k,x_l)\nonumber \\ & = & \left\langle e^{zf(x)/N}\right\rangle^{N-2}\;{N(N-1)\over2N}\; \intl_K dx_1dx_2\;e^{{z}(f(x_1)+f(x_2))/N}\beta(x_1,x_2)\nonumber \\ & & + \left\langle e^{zf(x)/N}\right\rangle^{N-1}\;{N\over N}\; \intl_K dx\;e^{{z}f(x)/N}\beta(x,x)\nonumber \\ & \sim & M_0(z)\left(\intl_K dx\;\beta(x,x) \vphantom{{Z^2\over N}}\;+\right.\nonumber \\ & & \hphantom{M_0(z)XX}\left. +\; {z^2\over2N} \intl_K dx_1dx_2\;f(x_1)\beta(x_1,x_2)f(x_2)\right)\;\;, \end{eqnarray} \relax where we have suppressed all subleading terms. The higher-order terms can easily be worked out: the only combinations that survive in the limit $N\to\infty$ are \begin{eqnarray} C_k & = & \intl_K dx_1dx_2\cdots dx_k\; \beta(x_1,x_2)\beta(x_2,x_3)\cdots\beta(x_{k-1},x_k)\beta(x_k,x_1)\nonumber \\ & = & \sum\limits_{\vec{n}>0}\si\svn^{2k}\;\;, \end{eqnarray} and \begin{eqnarray} F_k & = & \intl_K dx_1dx_2\cdots dx_kdx_{k+1}\; f(x_1)\beta(x_1,x_2)\cdots\beta(x_k,x_{k+1})f(x_{k+1})\nonumber \\ & = & \sum\limits_{\vec{n}>0}v_{\vec{n}}^2\si\svn^{2k}\;\;. \end{eqnarray} These objects come with topological symmetry factors of $2^k/(2k)$ and $2^k/2$, respectively \cite{first}. To leading order in $N$, we can therefore write \relax \begin{eqnarray} M(z,t) & \sim & M_0(z)\exp\left( \sum\limits_{k\ge0}C_k{(2t)^k\over2k} + {z^2\over N}\sum\limits_{k>0}F_k{(2t)^k\over2}\right)\nonumber \\ & = & M_0(z)\exp\left( \sum\limits_{k\ge0}\sum\limits_{\vec{n}>0}{(2t\si\svn^2)^k\over2k} + {z^2\over N}\sum\limits_{k>0}\sum\limits_{\vec{n}>0}{(2t\si\svn^2)^kv_{\vec{n}}^2\over2}\right)\nonumber \\ & = & M_0(z)\exp\left( - {1\over2}\sum\limits_{\vec{n}>0}\log(1-2t\si\svn^2) + {z^2\over2N}\sum\limits_{\vec{n}>0}{2t\si\svn^2v_{\vec{n}}^2\over1-2t\si\svn^2} \right) \end{eqnarray} \relax Combining everything, we have \relax \begin{eqnarray} P(s;\eta) & = & {1\over H_0(s)}\intl_{-i\infty}^{i\infty} {dz\over2\pi i}\;{dt\over2\pi i}\nonumber \\ & & \hphantom{{1\over2\pi i}}\times \exp\left(-z\eta - ts - {1\over2}\sum\limits_{\vec{n}>0}\log\left(1-2t\si\svn^2\right) + {z^2\over2N}B(t)\right)\;\;,\nonumber \\ B(t) & = & \sum\limits_{\vec{n}>0}{v_{\vec{n}}^2\over1-2t\si\svn^2}\;\;. \end{eqnarray} \relax The $z$ integral converges provided $\mbox{Re}B(t)>0$, which certainly holds if $1-2\si\svn^2\mbox{Re}t >0$ for all $\vec{n}$. Performing the $z$ integration, we arrive at our master formula: \relax \begin{eqnarray} P(s;\eta) & = & {1\over H_0(s)} \intl_{-i\infty}^{i\infty} {dt\over2\pi i}\;\sqrt{N\over2\pi B(t)}\nonumber \\ & & \hphantom{\sqrt{N\over2\pi} {1\over H}}\times \exp\left(-ts-{1\over2}\sum\limits_{\vec{n}>0}\log\left(1-2t\si\svn^2\right) -{\eta^2N\over2B(t)}\right)\;\;. \label{master} \end{eqnarray} \relax We see that, for the types of discrepancy discussed here, the error distribution is symmetric around $\eta=0$. Its precise form, however, will depend on our choice for the $\si\svn$. As we have said, a particular such choice reflects our belief about which kind of function class our actual integrand is a typical member of: but it must be realized that we are, in fact, allowed to take {\it any\/} choice for the $\si\svn$ that satisfies $\sum\si\svn^2<\infty$. A choice that does not `fit' the behaviour of $f(x)$ too well will just result in a somewhat worse error estimate: but the error distribution itself is only based on our assumption on the ensemble of point sets $X_N$, and not on any assumption about the integrand apart from its quadratic integrability. From \eqn{master} a number of results immediately follow. In the first place, we can recover the case of truly random point sets by simply averaging over all possible values of $s$, with the appropriate probability distribution\relax $H_0(s)$: this immediately leads to \relax \begin{equation} \int\limits^\infty_0 ds\;H_0(s)\;P(s;\eta) = \sqrt{{N\over2\pi V}} \exp\left(-{\eta^2N\over2V}\right)\;\;\;, \end{equation} \relax which is the standard Central Limit Theorem. Another result comes from the normalization of $P(s;\eta)$: upon integrating over $\eta$ we find \begin{equation} H_0(s) = \intl_{-i\infty}^{i\infty} {dt\over2\pi i}\; \exp\left( - ts - {1\over2}\sum\limits_{\vec{n}>0}\log\left(1-2t\si\svn^2\right) \right)\;\;, \label{h0def} \end{equation} in accordance with Ref.\cite{first}. A final observation to be made is that the error $\eta$ only occurs in the combination $\eta^2N$. From this it immediately follows that, all other things being equal, the error will only decrease as $1/\sqrt{N}$. Any improved rate of convergence is therefore {\it solely\/} due to a decrease of the discrepancy value $s$ with $N$. \section{A simple model: uniform strengths} The first, and simplest, model that we shall consider is that where $2M$ of the $\si\svn^2$ are equal to $1/2M$, and all the other ones vanish. It is natural to take for the nonzero modes the ones with the lowest frequencies ({\it i.e.\/} small values of the components of $\vec{n}$), but this is not necessary. As mentioned above, the choice of $\si\svn$ only establishes which modes are {\it covered}, that is, enter in the computation of the discrepancy: a general integrand will, of course have modes with different frequencies, which are not covered. We therefore write \begin{equation} V = \sum\limits_{\vec{n}>0}v_{\vn}^2 = V_1 + V_2\;\;, \end{equation} where $V_1$ contains the $2M$ covered modes, for which $\si\svn\ne0$, and $V_2$ contains all the other, uncovered, ones. The larger $V_1$ is with respect to $V_2$, the better our discrepancy model `fits' the integrand. We immediately have \relax \begin{eqnarray} {1\over2}\sum\limits_{\vec{n}>0}\log\left(1-2t\si\svn^2\right) & = & M\log\left(1-{t\over M}\right)\;\;, \vphantom{\sqrt{M\over2\pi}\exp\left(-{M(s-1)^2\over2}\right)} \nonumber \\ B(t) & = & V_1/\left(1-{t\over M}\right) + V_2\;\;, \vphantom{\sqrt{M\over2\pi}\exp\left(-{M(s-1)^2\over2}\right)} \nonumber \\ H_0(s) & = & {M^M\over\Gamma(M)}s^{M-1}e^{-Ms} \vphantom{\sqrt{M\over2\pi}\exp\left(-{M(s-1)^2\over2}\right)} \nonumber \\ & \sim & \sqrt{M\over2\pi}\exp\left(-{M(s-1)^2\over2}\right) \;\;, \end{eqnarray} \relax where the last line holds for large $M$. Both the form of $H_0(s)$ and that of $\beta(x_k,x_l)$ for this model are given in \cite{second}; by construction, the expected discrepancy for truly random points is $\avg{s}=1$. The master formula now becomes \relax \begin{eqnarray} P(s;\eta) & = & {\Gamma(M)\over M^{M-1}} \intl_{-i\infty}^{i\infty} {dx\over2\pi i}\;\sqrt{N\over2\pi(V_2 + sV_1/x)}\nonumber \\ & & \hphantom{{\Gamma(M)\over M^{M-1}}} \times \exp\left( Mx - M\log x - {\eta^2N\over2(V_2+sV_1/x)}\right)\;\;, \end{eqnarray} \relax where we have written $x\equiv s(1-t/M)$. \relax Consequently the integration contour must cross the positive real axis. Two special cases can immediately be derived from this. In the first place, suppose that we had chosen the nonzero $\si\svn$ in a very bad way, such that $V_1=0$: that is, the integrand consists only of uncovered modes. It then follows immediately that \relax \begin{equation} \left.P(s;\eta)\right|_{V_1=0} = \sqrt{N\over2\pi V_2}\exp\left(-{\eta^2N\over2V_2}\right)\;\;, \end{equation} \relax which is the standard Central Limit Theorem. In this case, nothing is really lost, and the error estimate is just as good (or bad) as in classical Monte Carlo. On the other hand, if the integrand consists only of covered modes, so that $V_2=0$, we find after some straightforward manipulations: \relax \begin{eqnarray} P(s;\eta) & = & \xi(M)\;\sqrt{N\over2\pi V_1s}\; \left(1-{\eta^2N\over2V_1sM}\right)^{M-3/2}\;\;,\nonumber \\ \xi(M) & = & {4^{M-1}\over\sqrt{M\pi}}{\Gamma(M)^2\over\Gamma(2M-1)} \;\;{=}\;\;1+\order{{1\over M}}\;\;, \label{v2zerocase} \end{eqnarray} \relax with the strict constraint $\eta^2N<2V_1sM$. This follows from the fact that, if this inequality is violated, the complex integration contour for $x$ can be closed to the right, where the integrand has no singularities; for the same reason \cite{second}, $H_0(s)$ vanishes for $s<0$. Note that, for this particular discrepancy, $s$ can actually vanish: this happens in one dimension, if the point set is equidistant and $N>M$. In that case, $\eta$ is always zero, so that the function is integrated exactly. This is just another instance of the Nyqvist theorem \cite{nyq}. For general $V_1$ and $V_2$, we may consider the case where $M$ becomes large. The integral can then be approximated by the saddle-point method. The saddle point is located at $x=1+\order{1/M}$, and we find \begin{equation} P(s;\eta) \sim \sqrt{{N\over2\pi(V_1s+V_2)}} \exp\left(-{\eta^2N\over2(V_1s+V_2)}\right)\;\;. \label{simplemodelclt} \end{equation} Again, we recover a Gaussian limiting distribution; its width is no longer parameterized by $V=V_1+V_2$ but rather by $V_1s+V_2$: the information we have gathered by computing the discrepancy $s$ is seen to result in a reduced error, depending on how much of the fluctuating behaviour of the integrand is actually covered by the modes entering in the discrepancy. The limit of large $M$ is actually justified by a self-consistency argument: the error distribution (\ref{simplemodelclt}) heavily suppresses the region $\eta^2N\gg2(V_1s+V_2)$, so that (as can also be gleaned from \eqn{v2zerocase}) $M$ does not actually have to be a huge number for the saddle-point approximation to work. Note, moreover, that if we only allow the lowest frequency mode in each dimension, that is, only $n^{\mu}=0,1,2$ for each component of $\vec{n}$, $M$ already equals $(3^D-1)/2$ which grows very rapidly with increasing $D$. The upshot of this (admittedly simple-minded, but nevertheless possible) model is: first, that we may hope for an error distribution which tends to a Gaussian (especially in high dimension), and, secondly, the width of this distribution depends on the discrepancy $s$ in a manner which depends on the degree in which the relevant modes of the integrand correspond to those used in the evaluation of the discrepancy. We conjecture that these two conclusions will persist in more realistic models of discrepancy. \section{A more realistic model: one dimension} The model of discrepancy discussed above has the advantages both of simplicity and dimensionality-independence: but it may not be altogether too realistic, in particular because covered modes with high frequency are assumed to have the same strength as those with low frequency. An alternative, which we discuss now, covers all modes, but with strengths that decrease with increasing frequency. For simplicity, we start with $D=1$. We shall take \relax \begin{equation} \sigma_{2n} = \sigma_{2n-1} = {1\over n}\;\;\;,\;\;\;n=1,2,3,\ldots, \end{equation} \relax just the same as in \cite{second}. For truly random points we have, then, $\avg{s}=\pi^2/3$, and we shall assume that we have at our disposal a point set with a discrepancy value $s$ much lower than this average. First of all, we compute $H_0(s)$ for this small $s$. In Ref.\cite{second}, we performed an exact calculation, but here we shall settle for a more simple-minded saddle-point approximation. We assume that the $t$ integral in \eqn{h0def} is saturated by a saddle-point lying at $t=-a^2/2$, that is, \relax \begin{eqnarray} H_0(s) & = & \intl_{-i\infty}^{i\infty} {dt\over2\pi i}\;e^{\phi(t)}\nonumber \\ & \sim & {\exp\left({\phi(-a^2/2)}\right)\over\sqrt{2\pi\phi''(-a^2/2)}}\;\;, \vphantom{{1\over2\pi i}\intl_{-i\infty}^{i\infty} dt\;e^{\phi(t)}} \nonumber \\ \phi(t) & = & -st - \sum\limits_{n>0}\log\left(1-{2t\over n^2}\right)\;\;, \vphantom{{1\over2\pi i}\intl_{-i\infty}^{i\infty} dt\;e^{\phi(t)}} \nonumber \\ \phi(-a^2/2) & = & {sa^2\over2} - \pi a + \log(2\pi a) + \order{{1\over a}}\;\;, \vphantom{{1\over2\pi i}\intl_{-i\infty}^{i\infty} dt\;e^{\phi(t)}} \nonumber \\[-3mm] \phi'(-a^2/2) & = & -s + {\pi\over a} + \order{{1\over a^2}}\;\;\equiv0\;\;, \vphantom{{1\over2\pi i}\intl_{-i\infty}^{i\infty} dt\;e^{\phi(t)}} \nonumber \\[-3mm] \phi''(-a^2/2) & = & {\pi\over a^3} + \order{{1\over a^4}}\;\;. \vphantom{{1\over2\pi i}\intl_{-i\infty}^{i\infty} dt\;e^{\phi(t)}} \end{eqnarray} \relax The saddle point is seen to correspond to $a\sim\pi/s$ which is large for small values of $s$, thus justifying the neglect of higher orders in $1/a$. The resulting form for $H_0$ is ($s\ll{\pi^2/3}$) \relax \begin{equation} H_0(s) \,\sim\, {\pi^2\sqrt{2\pi}\over s^{5/2}}\exp\left(-{\pi^2\over2s}\right)\;\;, \end{equation} \relax in agreement with the corresponding limit of the exact result from \cite{second}. For the evaluation of the error distribution $P(s;\eta)$ we must now also compute $B(t)$, which involves the unknown coefficients $v_n$ of the integrand. It is certainly too crude, but nonetheless instructive, to study the simple case where $$ v_n^2 = \sigma_n^2\;\;\;,\;\;\;n=1,2,3,\ldots. $$ In that case, we have \begin{equation} B(t) = 2\sum\limits_{n>0}{1\over n^2-2t} = 2\sum\limits_{n>0}{1\over n^2+a^2} = {\pi\over a} + \order{{1\over a^2}}\;\;, \label{btavg} \end{equation} where $a$ has now to be determined anew for the saddle point in the $t$ integration of \eqn{master}. It is seen to be equal to \begin{equation} a \sim {\pi\gamma\over s}\;\;\;,\;\;\; \gamma = 1 + {\eta^2N\over2\pi^2}\;\;. \end{equation} Performing the saddle integral we arrive at \begin{eqnarray} P(s;\eta) & \sim & \sqrt{N\over2\pi s}\;\gamma^{5/2}\; \exp\left(-{\pi^2\over2s}(\gamma^2-1)\right)\nonumber \\ & \sim & \sqrt{N\over2\pi s}\exp\left(-{\eta^2N\over2s}\right)\;\;. \end{eqnarray} This last, Gaussian, central limit is self-consistently justified from the fact that it implies $\eta^2N=\order{s}$ which is indeed small by assumption. \relax Note that we may write for this case (see \eqn{simplemodelclt}): $$ s = V{s\over\left<s\right>}\;\;. $$ \relax What, now, happens for more general $v_n$? One answer is to assume that, since the integrand must be quadratically integrable, the sum $\sum v_n^2$ must converge; if we also assume that it has no exceptionally strong higher modes, it is reasonable to write $$ v_{2n-1}^2 + v_{2n}^2 = {C\omega_n\over n^2}\;\;, $$ where $C$ is a constant, and the $\omega_n$ are numbers that are not too different from unity. Not rigorously, but at least reasonably, we may then write \begin{equation} B(t) = \sum\limits_{n>0}{C\omega_n\over n^2+a^2} \sim {C\pi\over a} + \order{{1\over a^2}}\;\;, \end{equation} leading to \begin{equation} P(s;\eta) \sim \sqrt{N\over2\pi sC}\; \exp\left(-{\eta^2N\over2sC}\right)\;\;. \end{equation} The essential point here is that the deviations of the individual $\omega_n$ from unity can give rise, in $B(t)$ to contributions that are of order $\order{1/a^2}$, and not of order $\order{1/a}$. Another argument leading to the same conclusion is to compute the moments of $B(t)$ over the ensemble of integrands described in Refs.\cite{first,second}: the $v_n$ are assumed to be normally distributed around zero with standard deviation $\sigma_n$. The expectation of $B(-a^2/2)$ is then, of course, just the result of \eqn{btavg}, but its {\it variance\/} goes as $\order{1/a^3}$. If $a$ increases (for decreasing $s$), the probable values for $B(t)$ therefore cluster together more and more closely around the expectation value, again justifying our approximations. A last example in this context is that of an integrand that has only a single mode, with frequency $k$, so that only $v_{2k}$ and $v_{2k-1}$ are non-vanishing, and we have \relax \begin{equation} B(t) = {Vk^2\over k^2-2t}\;\;. \end{equation} \relax We immediately find that \begin{equation} \hat{s} \equiv s - {\eta^2N\over Vk^2} > 0\;\;, \end{equation} by the same arguments as above\relax; and, for $\eta$ values smaller than this limit, we may again apply the saddle-point method to find \begin{eqnarray} P(s;\eta) & \sim & \sqrt{{N\over2\pi V}}\; \left({s\over\hat{s}}\right)^{5/2}\sqrt{1+{\pi^2\over\hat{s}^2k^2}} \exp\left(-{\eta^2N\over2V}+{\pi^2\over2s}-{\pi^2\over2\hat{s}}\right)\nonumber \\ & \sim & \sqrt{{N\over2\pi V}\left(1+{\pi^2\over k^2s^2}\right)} \exp\left(-{\eta^2N\over2V}\left(1+{\pi^2\over k^2s^2}\right)\right)\;\;, \end{eqnarray} where the last line holds if $\hat{s}$ and $s$ are close in value. In this limit, again a Gaussian distribution is obtained, with variance $V/(1+\pi^2/k^2s^2)$. Note that the error improvement now not only depends on the smallness of $s$ but also on the number $k$; this is reasonable because the mode with frequency $k$ enters in this particular discrepancy with a factor $1/k^2$ so that, when $k$ is large, a small value of $s$ does not tell us too much about how well the $k$th mode is integrated by the point set. \relax \section{Conclusions and outlook} We have shown that we can define a Central Limit for the case of Quasi-Monte Carlo using a suitable definition of the discrepancy. A master-formula was derived for the error-distribution density over point sets with a fixed discrepancy. We have given two simple examples of problem classes and their error-distribution densities. These results indicate that the expected error will improve if low-discrepancy point sets are used to evaluate integrals. We would like to extend these results to more realistic and more dimensional cases. An explicit result seems to be too far-fectched, at the moment, but it might be possible to use saddle-point methods to derive similar results for more realistic cases. \relax \newpage

proofpile-arXiv_065-505

\section{Introduction} The discovery of the Higgs particle is of utmost importance in particle physics. Over the years, various theoretical bounds have been made \cite{Cabibbo,Lindner,SherRep,HunterGuide,Schrempp,Sher,AI,CEQ1}, and most recently an experimental {\em lower} bound of 65 GeV was set \cite{LowerB}. But the Higgs boson still remains elusive. Its nature --mass and couplings-- would reveal the most fundamental aspects of the kind of mechanism that governs the spontaneous symmetry breakdown of the Standard Model (SM). In particular, one would like to know whether or not such a discovery, if and when it will be made, will be accompanied by ``new physics'' at some energy scale $\Lambda$. Of equal importance is the following question: at roughly what mass scale will the Higgs boson be considered elementary or composite? Can one make some meaningful statement concerning its nature once it is discovered? These are the issues we would like to explore in this paper. A first step in this direction has been recently achieved by detailed analyses of the Higgs potential \cite{Sher,AI,CEQ1}. Indeed, with the discovery of the top quark with mass $m_t = 175 \pm 9$ GeV \cite{CDF}, the Higgs mass ($m_H$) is severely constrained by the requirement of vacuum stability. In particular, two interesting conclusions have been drawn: \begin{enumerate} \item[{\bf i.}] If a Higgs will be discovered at LEP200, i.e. with with $m_H\leq m_Z$, then some new physics must appear at very low scales: $\Lambda \stackrel{<}{_\sim} 10$ TeV \cite{AI,CEQ1,Hung}. \item[{\bf ii.}] The Standard Model with an high cut--off (without new particles below $10^{15}$ GeV) requires $m_H \stackrel{>}{_\sim} 130$ GeV and is incompatible with the Minimal Supersymmetric Standard Model (MSSM), where the mass of the lightest Higgs boson is expected below $130$ GeV \cite{CEQ1}. \end{enumerate} We shall re--analyze the above statements, trying to clarify the stability of the physical conclusions with respect to the theoretical errors, and we shall extend the discussion studying the implications of a Higgs discovery up to to approximately 700 GeV. The plan of the paper is as follows. In section II and III we shall review what is known about the Higgs sector of the SM and of the MSSM. We then divide our analysis into three separate mass regions: the region below $m_Z$, between $m_Z$ and $2m_Z$, above $2m_Z$. \section{The Higgs boson in the minimal Standard Model} The symmetry breaking sector of the SM is highly unstable and make sense only in presence of a cut--off scale $\Lambda$. The instability of the scalar sector implies that upper and lower limits on $m_H$, imposed by the requirement of no Landau pole and vacuum stability, both below $\Lambda$, tend to shrink together as the cut--off increases \cite{Cabibbo,Lindner}. The instability of the scalar potential is generated by the quantum loop corrections to the classical expression \begin{equation} V^{tree}(\Phi) =-m^2\Phi^\dagger\Phi+{\lambda \over 6}(\Phi^\dagger\Phi)^2, \qquad\qquad \Phi=\left(\begin{array}{c} \phi^+ \\ \phi^0 \end{array} \right) \label{treepot1} \end{equation} where $v^2 = 2 \langle \phi^0 \rangle^2 =1/(\sqrt{2}G_F)\simeq (246~\mbox{GeV})^2$ and $\phi = (\sqrt{2} \Re e \phi^0 -v)$ is the physical Higgs field. As already noticed in Ref.~\cite{Cabibbo}, and successively confirmed by detailed analysis of the renormalization group (RG) improved potential \cite{SherRep,Ford}, the issue of vacuum stability for $\phi\sim\Lambda \gg m_Z$ practically coincides with the requirement that the running coupling $\lambda(\Lambda)$ never becomes negative. On the other hand, the requirement that no Landau pole appears before $\Lambda$ is equivalent to the condition that $\lambda(\Lambda)$ always remains in the perturbative region. The evolution of $\lambda$ as a function of $\Lambda$ is ruled by a set of coupled differential equations \begin{eqnarray} \begin{array}{ccc} \mbox{\rm d}\lambda(t)/\mbox{\rm d}t &=&\beta_\lambda(\lambda,g_i), \\ \mbox{\rm d}g^2_i(t)/\mbox{\rm d}t &=& \beta_i(\lambda,g_i), \end{array} \qquad\qquad t=\ln\left(\Lambda/\mu\right) \label{evol1} \end{eqnarray} with the corresponding set of initial conditions which relate $\lambda(\mu)$ and $g^2_i(\mu)$ to physical observables ($g_3$, $g_2$ and $g_1$ denote $SU(3)_C\times SU(2)_L\times U(1)_Y$ gauge couplings and $g_{t}$ the top--quark Yukawa coupling, all couplings are understood in the $\overline{MS}$ scheme). The $\beta$--functions of Eq.~(\ref{evol1}) are known in perturbation theory up to two loops (see Ref.~\cite{Ford,Schrempp} for the complete expressions), i.e. up to the third order in the expansion around zero in terms of $\lambda$ and $g_i^2$, whereas the finite parts of the initial conditions around $\mu=m_Z$ (threshold corrections) are known up to one--loop accuracy \cite{Sirlin,Arason,Hempfling}. This knowledge enable us to re--sum all the next--to--leading logs in the evolution of the coupling constants and thus to calculate them with high accuracy in the perturbative region. Nevertheless, the instable character of $\lambda(t)$ can be simply read--off by the one--loop expression \begin{equation} \beta_\lambda ={1\over 16 \pi^2} \left[ 4\lambda^2 +12 \lambda g_t^2 -36 g_t^4 + O(g_1^2,g_2^2) \right], \label{lambda1} \end{equation} together with the tree--level relations \begin{equation} \lambda(m_H) = {3 m_H^2 \over v^2 } \qquad\quad \mbox{and} \qquad\quad g_t^2(m_t) = {2 m_t^2 \over v^2 }. \label{thres0} \end{equation} For small values of $m_H$ the $g_t^4$ term in Eq.~(\ref{lambda1}) drives $\lambda$ to negative values, whereas if $m_H$ is large enough the Higgs self--interaction dominates and eventually $\lambda$ ``blow--up''. The situation is summarized in fig.~1 where we plot the evolution of $\lambda$ as obtained by integrating two--loop beta functions. For $m_t=175$ (pole mass) and $\alpha_S(m_Z)=0.118$, if we impose the condition \begin{equation} 0 < \lambda(\Lambda) <10 \label{cond1} \end{equation} at the Planck scale, then $m_H$ is confined in a very narrow range (full lines in fig.~1):\footnote{~With respect to Ref.~\cite{AI} we have removed a small error in the threshold correction of $g_t$ (the correct expression is given in Ref.~\cite{Hempfling}) obtaining a $\sim 1$ GeV decrease of the lower limit.} \begin{equation} 136~\mbox{GeV} \leq m_H \leq 174~\mbox{GeV}.\label{stablerange} \end{equation} The lower limit on $m_H$ depends strongly on the values of $m_t$ and $\alpha_S(m_Z)$ \cite{Sher,AI,CEQ1} whereas the upper one is more or less independent from them, both are weakened if the condition (\ref{cond1}) is imposed at scales $\Lambda$ below the Planck mass (dashed lines in fig.~1). What happens if $m_H$ is outside the range (\ref{stablerange})? For what concerns the problem of vacuum instability, the usual wisdom asserted that new physics must show up at or before the scale $\Lambda$ where $\lambda(\Lambda)$ becomes negative. However, as shown recently in Ref.~\cite{Hung}, the physical meaning of the previous statement is not trivial. In particular, there are models where the masses of the new particles could be substantially larger than $\Lambda$ and still stabilize the vacuum. We shall come back to this in section IV. The upper bound in Eq.~(\ref{cond1}) can be considered as an upper limit for the applicability of perturbation theory ($\lambda/4\pi$ is the expansion parameter) and indeed below this value the difference between one-- and two--loop beta functions is not large (dotted curve in fig.~1). However, for $m_H\simeq 180$ GeV, i.e. just above the upper limit imposed by Eq.~(\ref{cond1}), the integration of one--loop beta functions originates a singularity at $\Lambda_L < M_{Planck}$. As the Higgs mass increases $\Lambda_L$ decreases and approaches $10^{5}$ GeV for $m_H \approx$ 300 GeV. What is the physical meaning of the singularity scale $\Lambda_L$? If one believes that the Landau pole is not an artifact of perturbation theory but a non--perturbative feature of the model, as suggested by lattice simulations (see e.g. Refs.~\cite{lattice,Schrempp}), then is tempting to think that some new physics must occur around $\Lambda_L$. If that is so, this kind of new physics must be {\em very} different from the one needed to stabilize the vacuum, since one is now dealing with a strong coupling domain. One is then tempted to attribute this behaviour (strong coupling) to the nature of the Higgs boson. In particular, one might think that the Higgs boson is a composite particle which acts like an elementary field below the scale $\Lambda_L$. How one can tell if this is the case is the subject of our discussion in section VI. \section{The Higgs sector of the MSSM} The Higgs sector of the MSSM (see Refs.~\cite{Susy} for excellent reviews) contains two Higgs doublets, one is responsible for charged--lepton and down--type--quark masses ($H_1$), the other for up--type--quark masses ($H_2$). Of the eight degrees of freedom, two charged, one $CP$--odd and two $CP$--even neutral scalars correspond to physical particles after the $SU(2)_L\times U(1)_Y$ breaking. The tree--level potential for the two neutral components $H_1^0$ and $H_2^0$ is given by \begin{equation} V^{tree}(H^0_1,H^0_2) = { g_1^2 + g_2^2 \over 8} (|H_1^0|^2 - |H_2^0|^2)^2 + m^2_1 |H_1^0|^2 + m^2_2 |H_2^0|^2 + [m^2_{12} H_1^0H_2^0 + \mbox{h.c.}]. \label{treepot2} \end{equation} The sum of the two vacuum expectation values squared is fixed by the gauge boson masses: $v^2_1+v^2_2=v^2$ ($v_{1,2}=\sqrt{2}\langle H^0_{1,2} \rangle$), while the ratio $\tan(\beta)=v_2/v_1$ is a free parameter. The remarkable feature of the potential~(\ref{treepot2}) is that the coefficient of the dimension four operator is completely fixed in terms of the gauge couplings $g_1$ and $g_2$. This property leads to the tree--level relation \begin{equation} m^2_{H,H'} = {1\over 2}\left( m_A^2 + m^2_Z \mp \sqrt{ ( m_A^2 + m_Z^2 )^2 -4m_A^2 m_Z^2 \cos^2 2\beta } \right), \end{equation} where $m_{H,H'}$ are the two $CP$--even Higgs boson masses and $m_A$ is the $CP$--odd one, which implies a strict upper bound \begin{equation} m_H \leq m_Z \cos 2\beta \leq m_Z \label{treeboud} \end{equation} on the lightest Higgs boson mass. As it is well--known~\cite{SusyLim,SusyLim2,CEQ1}, the bound~(\ref{treeboud}) receive large radiative corrections if SUSY particles, and in particular the ${\widetilde t}$ squark, are heavy. This can be easily understood by means of the SM evolution of $\lambda$ previously discussed. Indeed, if all SUSY particles (including additional Higgs bosons) have a mass of the order of $M_S$ ($M_S^2 \gg m^2_Z$), the lightest Higgs boson decouples below $M_S$ and mimics the SM Higgs. Then, the evolution of the scalar self-coupling $\lambda(\Lambda)$ is dictated by SM beta functions up to $\Lambda=M_S$, where SUSY is restored and, according to the potential~(\ref{treepot2}), the following relation must old: \begin{equation} \lambda(M_S)={3\over 4}\left[ g_1(M_S)^2 +g_2(M_S)^2 \right]\cos^2 2\beta. \label{Susycond1} \end{equation} Eq.~(\ref{Susycond1}) saturates the bound~(\ref{treeboud}) for $M_S\sim m_Z$ but, due to the rapidly decreasing behaviour of $\lambda(\Lambda)$ (see fig.~2), implies $(20\div 30)\%$ violations of the tree--level bound for $M_S\sim 1$ TeV \cite{SusyLim}. Analogously to the tree--level relations (\ref{thres0}), the boundary condition~(\ref{Susycond1}) is not differentiable with respect to the scale of $\lambda$: in order to calculate precise bounds on the Higgs mass is necessary to include threshold effects in both cases. The most important correction to Eq.~(\ref{Susycond1}) is the one generated by stop loops, that is proportional to $g_t^4$. If we include this effect Eq.~(\ref{Susycond1}) is modified in \begin{equation} \lambda(\Lambda)={3\over 4}\left[ g_1(\Lambda)^2 +g_2(\Lambda)^2 \right] \cos^2 2\beta +\Delta \lambda(\Lambda), \label{Susycond2} \end{equation} where \begin{equation} {\mbox{d} \Delta \lambda(\Lambda) \over \mbox{d}\ln (\Lambda/\mu)} = -{36 \over 16 \pi^2} g_t^4 +..., \label{delta00} \end{equation} by this way the leading term in the derivative of both sides of Eq.~(\ref{Susycond2}) is the same. The explicit expression of $\Delta \lambda(\Lambda)$, obtained by the one--loop stop contribution to the potential~(\ref{treepot2}), is given by \cite{CEQ1} \begin{eqnarray} \Delta \lambda(\Lambda) = &\displaystyle{9 g_t^4 \over 16 \pi^2}& \left\{ { (m_t+X_t)^2 \over {\widetilde m_+}^2 } \left[ 1 - { (m_t+X_t)^2 \over 12{\widetilde m_+}^2 } \right] \right. \nonumber \\ && \left. + { (m_t-X_t)^2 \over {\widetilde m_-}^2 } \left[ 1 - { (m_t-X_t)^2 \over 12{\widetilde m_-}^2 } \right] +\ln\left( {{\widetilde m_-}^2 {\widetilde m_+}^2 \over \Lambda^4} \right) \right\}, \label{delta01} \end{eqnarray} where ${\widetilde m_\pm} = M_S^2 +m_t \pm m_tX_t$ are the eigenvalues of the stop mass matrix and $X_t$ is the usual stop--mixing parameter \cite{CEQ1,SusyLim}. As noticed in Ref.~\cite{CEQ1}, $\Delta \lambda(M_S)$ has a maximum for $X_t^2= 6M_S^2 +O(m_t^2)$. Imposing the boundary condition (\ref{delta01}) at $\Lambda\sim M_S$, using two--loop SM beta functions to evolve down at $\mu\sim m_Z$ and finally using SM one--loop matching conditions to relate $m_H$ and $m_t$ to $\lambda$ and $g_t$, we find (masses are in units of GeV): \begin{eqnarray} M_H^{MSSM} < 127 +0.9\left[m_t-175\right] -0.8\left[\displaystyle{\alpha_S(m_Z)-.118 \over .006} \right] +7\cdot\log_{10} \left( {M_S \over 10^3 }\right) \pm 4 \label{Susy127} \end{eqnarray} in good agreement with the detailed analysis of Ref.~\cite{CEQ1}. The error in Eq.~(\ref{Susy127}) has been estimated by varying low and high energy matching scales in the following intervals: $\Lambda \in [M_S, 2M_S]$ and $\mu \in [m_Z, 2 m_t]$. Obviously the upper limit is very sensitive to $M_S$, defined as the soft stop mass, and is valid for $M_S$ near $1$~TeV; on the other hand, the dependence form other SUSY masses is within the quoted error. As can be noticed from fig.~2, for $m_t=175$~GeV and $\alpha_S=0.118$, the SM with $\Lambda=M_{Plank}$ is compatible with the MSSM only for unnatural large values of $M_S$. \section{Physics of the ``low'' mass Higgs boson: $m_H \leq m_Z$ GeV} As we have discussed in section II, the SM becomes unstable when $m_H \leq 136$ GeV. Moreover, if the Higgs mass is below the $Z$ mass, the SM breaks down at a scale $\Lambda$ situated in the TeV region \cite{AI,CEQ1,Hung}. Recently, it has been pointed out \cite{CEQ2} that for small values of $\Lambda$ the lower limit on $m_H$ imposed by the condition \begin{equation} \left. {\mbox{d} V^{1-RG}(\phi) \over \mbox{d} \phi}\right\vert_{\phi=\Lambda} >0, \label{cond2} \end{equation} where $V^{1-RG}(\phi)$ denotes the one--loop RG--improved potential, do not coincide with the one imposed by $\lambda(\Lambda)>0$. We agree with the above statement, however it must be stressed that the two conditions lead to equivalent results up to a small re--definition of $\Lambda$ \cite{CEQ2}. As an example, the lower limit on $m_H$ imposed by Eq.~(\ref{cond1}) with $\Lambda=1$~TeV, namely $m_H> 72$ GeV, is equivalent to the one imposed by Eq.~(\ref{cond1}) with $\Lambda \simeq 3.4$~TeV. On the other hand, the two conditions coincide for large values of the cut--off, where the corresponding $\lambda(\Lambda)$ curves are almost flat (fig.~1). Since the exact relation between $\Lambda$, understood as the scale where the evolution of $\lambda$ is no more ruled by Standard Model beta functions, and the masses of hypothetical new particles depends on the details of the new--physics model \cite{Hung}, in our opinion is meaningless to fix $\Lambda$ with great accuracy. In other words, for a given value of $m_H$, the scale $\Lambda$ where Eq.~(\ref{cond1}) or Eq.~(\ref{cond2}) are no more satisfied can give only an indication of the order of magnitude below which new physics must appear, and within this interpretation the two conditions are completely equivalent and consistent with the statement {\bf i} of sect.~I. To stabilize the SM vacuum, one has to add more scalar degrees of freedom which couple to the SM Higgs, a well-known fact from studies of the effective potential or from studies of the RG equations. The most natural new--physics candidate in this case is the MSSM. There there is a plethora of scalar fields: the supersymmetric partners of quarks and leptons, and the additional Higgses. However, as we have seen in the previous section, the ``stabilizing scalar'' is the stop which cancel the $g_t$ dependence in the evolution of $\lambda$. More light is the Higgs and more light must be the stop. What happens if the Higgs mass is very light, say 70 GeV, and the top is not found in the TeV region? It could mean several things. Either the MSSM is not correct and a more complicated version is needed or something other than SUSY enters the picture. In Ref.~\cite{Hung} this question has been studied using a toy model with electroweak singlet scalars, with multiplicity $N$ and with a coupling $\delta$ to the standard Higgs field. It was found that the mass of the new singlet scalars could be as high as ten times the scale $\Lambda$ where $\lambda(\Lambda)$ becomes negative. In the above discussion, there was never any need for the Higgs boson to be composite. In fact, it appears to be more natural for the Higgs boson to be {\em elementary} in this case. Although there are models for a ``light'' Higgs boson where an elementary Higgs field is mixed with a top condensate \cite{Gerard}, it does not appear to be possible to construct a model where the Higgs boson is entirely composite. It is in this sense that we say that the Higgs boson is elementary if its mass is $m_Z$ or below. The main conclusion of this section is the following: if $m_H \leq m_Z$, the Higgs boson is most likely elementary and there should be new physics, within the 10 TeV scale, either in terms of SUSY particles or in terms of new scalar degrees of freedom. \section{Physics of the Higgs boson with $m_Z \leq m_H \leq 2 m_Z$} This is a region where it will be extremely hard to detect the Higgs boson \cite{HunterGuide}. Theoretically, this is a region where one can still presume that the Higgs boson is an elementary particle. Indeed, a look at fig.~1 will convince us that $\lambda$ blows up below the Planck scale only when $m_H \stackrel{>}{_\sim} 2 m_Z$. Furthermore, there is no known mechanism which can give rise to a composite Higgs boson that light (without additional scalars). As we have seen in section III, if $m_H \stackrel{<}{_\sim} 130$ GeV the most natural candidate is still the MSSM. On the other hand, for $m_H \stackrel{>}{_\sim} 130$ GeV the MSSM it is unnatural because the SUSY scale is too high. Above 130 GeV natural candidates are SUSY extensions of the SM with a non--minimal scalar sector \cite{Espinosa2}. In this region also the SM itself can be considered a good candidate. Indeed, a part from the problem of quadratic divergences, new--physics can be pushed up to the Planck scale if $m_H \stackrel{>}{_\sim} 130$. In this framework, an interesting scenario is the one proposed in Ref.~\cite{FN}. \section{Physics of the Higgs boson with $m_H \geq 2 m_Z$} We finally come to the question of which kind of new physics is expected if $m_H$ if found above $\sim 180$ GeV, i.e. in the region where $\lambda(\Lambda)$ develops a singularity at $\Lambda_L < M_{Planck}$. As we have already said in sect. I, the Landau pole might just be an artifact of perturbation theory. However we believe this is not the case. Following the indications of lattice simulations \cite{lattice}, we believe that the presence of such singularity is at least qualitatively correct and that indicates the {\em composite nature} of the Higgs boson. The physics below the compositeness scale can be described in terms an effective field theory whose couplings are constrained by the boundary conditions at the compositeness scale. In this framework, a class of models which is particularly attractive, relevant to the present discussion and quite general is the class of the top--condensate models \cite{Topcond1,BHL}. There the relevant boundary conditions are \cite{BHL,BLP}: \begin{eqnarray} && \lambda(\mu),~g_t(\mu) \Lto \infty \label{boundc1} \\ && \lambda(\mu)/g^2_t(\mu) \Lto \mbox{const.} \label{boundc2} \end{eqnarray} Thus the Landau singularity of the Higgs self-couplings naturally fits into this scheme. The only problem is the requirement of a pole also in the evolution of the top Yukawa coupling. As can be noticed in fig.~3, the top Yukawa coupling itself is not large enough to develop a singularity since its evolution is ``softened'' by QCD. However, as we will show in the following, if we include additional heavy fermions with a mass $m_f$ above a critical value, both $g_t$ and $g_f$ can ``blow up'' at a scale $\Lambda_f$. To analyze better the model, let us consider the Lagrangian of a single degenerate quark doublet $q = (u, d)$ coupled to the Higgs field. If we re--scale the Higgs field in the following way \begin{equation} \Phi \longrightarrow \Phi_{0} / g_f, \end{equation} the Lagrangian becomes \begin{equation} {\cal L}= {\cal L}_{kinetic}(u,d) + Z_{\Phi} D_{\mu}\Phi^{\dagger}_{0} D^{\mu}\Phi_{0} +{\widetilde{m}}^2 \Phi^{\dagger}_{0} \Phi_{0} -\frac{\widetilde{\lambda}}{6} (\Phi^{\dagger}_{0} \Phi_{0})^2 + {\bar q}_{L}\Phi_{0} d_R +{\bar q}_{L}{\Phi^C_{0}} u_R + \mbox{h.c.}, \end{equation} where \begin{equation} \Phi^C_0 = i\sigma_2 \Phi_0^{\ast} , \qquad Z_{\Phi} = 1/ g^2_f, \qquad \widetilde{m}^2 = Z_{\Phi} m^2 , \qquad\mbox{and}\qquad \widetilde{\lambda} = Z_{\Phi}^2\lambda. \end{equation} If $\lambda$ and $g^2_f$ develop a singularity at the same scale $\Lambda_f = \Lambda_L =\Lambda_C$, so that the boundary conditions (\ref{boundc1}-\ref{boundc2}) are satisfied, then with an appropriate tuning of the quadratic divergences we can have \cite{BLP} \begin{equation} \widetilde{m}^2/\Lambda_C^2 \Lto \mbox{const} <0. \end{equation} Thus at the compositeness scale the above Lagrangian becomes \begin{equation} {\cal L}= {\cal L}_{kinetic}(u,d) + {\bar q}_{L}\Phi_{0} d_R +{\bar q}_{L} \Phi^C_{0} u_R +{\widetilde{m}}^2 \Phi^{\dagger}_{0} \Phi_{0} +\mbox{h.c.} \end{equation} and $\Phi_0$, which is now just an auxiliary field, can can be integrated out to obtain a four--fermion Nambu--Jona--Lasinio Lagrangian \cite{NambuY} \begin{equation} {\cal L} = {\cal L}_{kinetic}(u,d) + G_0 {\bar q}_L( u_R {\bar u}_R + d_R {\bar d}_R) q_L , \end{equation} with $G_0 = -1/\widetilde{m}^2.$ Viewed in this way, the Higgs boson becomes a dynamical fermion--condensate below the scale $\Lambda_C$, in other words the Higgs boson becomes a composite particle. The necessary conditions for the above view to hold are the constraints (\ref{boundc1}-\ref{boundc2}). In order to understand if these conditions can be satisfied, it is useful to examine the RG equation of the ratio $x=\lambda/g^2_f$. At one loop, it is given by \begin{equation} 16 \pi^2 \frac{dx}{dt} = 4 g^2_f (x - x_{+})(x - x_{-}) , \end{equation} where $x_{\pm} = (-3 \pm 9)/2$, if both members of the quark doublet are degenerate in mass, or $x_{\pm} = \frac{3}{8} (-1 \pm \sqrt{65})$, if one member is much heavier than the other one (e.g. the 3rd generation case). The only possibility to have $\Lambda_f = \Lambda_L $ is that the initial value of $x$ is one of the two fixed points. Since $x_{-}$ is always negative, the solution $x= x_{-}$ is ruled out by vacuum stability. Thus the boundary conditions can be satisfied only if $x=x_{+}$ and this implies a precise relation between $m_H$ and $m_f$. Using the tree--level relations (\ref{thres0}) we find \begin{equation} m^2_H = \frac{2}{3} m^2_f x_{+}.\label{fixedp} \end{equation} Since $x_{+}$ is always greater than $3/2$, the fixed point scenario implies \begin{equation} m_H > m_f. \label{mhgtmf} \end{equation} It is easy to see that, in the large $N_c$ limit, the fixed point takes on the value $x= 6$, giving $m_H = 2 m_f$, a familiar result found in the Nambu-Jona-Lasinio model. For finite $N_c$ one finds in general \begin{equation} m_f < m_H < 2 m_f. \end{equation} As we have mentioned earlier, the top quark is not heavy enough to solely fit into this scenario. This is because the growth of the top Yukawa coupling is dampened by QCD. The minimum top mass for which there will be a Landau singularity at the Planck mass is $m_t \approx 216$ GeV, a value which is way outside the experimental range. Let us then assume that there is an extra doublet of degenerate quarks, $Q= (U, D)$, whose mass is arbitrary. As we shall see below, the addition of this extra doublet changes dramatically the behaviour of the couplings at high energy. To see this let us write the RG equations for $\lambda$, $g_f$ (the new--doublet Yukawa coupling), and $g_t$ (the top Yukawa coupling): \begin{eqnarray} && 16 \pi^2 \frac{d\lambda}{dt} = 4 \lambda^2 + 12 \lambda (g^2_t + 2 g^2_f) - 36 (g^4_t + 2 g^4_f) + O( g_1^2, g_2^2) \label{gg1} \\ && 16 \pi^2 \frac{dg^2_f}{dt} = g^2_f [12 g^2_f + 6 g^2_t -16 g^2_3] + O( g_1^2, g_2^2) \label{gg2} \\ && 16 \pi^2 \frac{dg^2_t}{dt} = g^2_t [9 g^2_t + 12 g^2_f -16 g^2_3 ] + O(g_1^2, g_2^2) . \label{gg3} \end{eqnarray} In the absence of the extra quark doublet, one can easily see from the above equations that the growth of $g_t$ is dampened by the gauge couplings (mainly by $g_3$)\footnote{~The $O( g_1^2, g_2^2)$ terms in Eqs.~(\ref{gg2}-\ref{gg3}), which tend to split the evolution of $U$ and $D$ Yukawa couplings, cannot be neglected if we are interested in a precise determination of the critical value of $g_f$ (see the discussion below). }. On the other hand, in presence of the extra doublet $g_t$ is no longer dampened provided $g_f$ exceeds some critical value. In addition, $g_t$ and $g_f$ tend to ``drag'' each other. If we allow the possibility --not withstanding experimental constraints-- that there could be an extra doublet of degenerate quarks with mass less than the top quark,\footnote{Note that electroweak precision data put severe constraints on possible new--fermion mass splitting but there is still room for an additional degenerate fourth family of quarks and leptons \protect\cite{Langa}.} then a numerical integration of the above equations shows that there is a minimum mass for the new fermions for which $g_t$ and $g_f$ develop a singularity around the Planck scale. As shown in fig.~3, this minimum mass is $m_f\simeq 160$ GeV. The corresponding Higgs mass, determined by the condition that $\lambda$ develops a singularity at the same scale as $g_t$ and $g_f$, is $m_H \simeq 190$ GeV. As $m_f$ increases, the compositeness scale $\Lambda_C$ decreases and the relation between $m_H$ and $m_f$ approaches the fixed point prediction (\ref{fixedp}) with $x_+ \simeq 3$ (see fig.~3). The above scenario cannot be considered as a realistic model. Indeed, if the scale $\Lambda_C$ is high there is clearly a ``fine tuning'' problem related to the large disparity between $\Lambda_C$ and the electroweak scale. However, it is beyond the scope of this paper to try to construct an underlying theory around $\Lambda_C$ and thus we will ignore it. Our purpose is just to show some general features of a wide class of models. In particular, if there are no new bosons (scalars or gauge bosons) below the compositeness scale, the following features hold independently of the multiplicity of the new fermions: \begin{enumerate} \item[{\bf i.}] The compositeness scale $\Lambda_C$, the heavy--fermion mass $m_f$, and the effective Higgs mass $m_H$, are tied together so that $m_H$ and $m_f$ increase as $\Lambda_C$ decrease. \item[{\bf ii.}] As shown in Eq.~(\ref{mhgtmf}), one typically finds $m_H > m_f$. Thus if $m_H$ is not found below $2m_Z$ it should be ``easier'' to search for new fermions instead of searching for the Higgs boson itself. \item[{\bf iii.}] For $\Lambda_C \approx 1$ TeV both $m_H$ and $m_f$ are $O(\Lambda_C)$ and the Higgs effective theory becomes meaningless. In this sense we agree with the more precise and well--defined lattice bound: $m_H \stackrel{<}{_\sim} 700$ GeV \cite{lattice}. \end{enumerate} \section{Conclusions} In this paper we have analyzed the consequences of a Higgs discovery up to approximately $700$ GeV, dividing the mass region into three parts: the region below $m_Z$, between $m_Z$ and $2m_Z$, above $2m_Z$. Regarding the first two regions we have confirmed and refined the results stated in the introduction, namely the SM lower bound due to vacuum stability and the MSSM upper bound. Regarding the last region ($m_H \stackrel{>}{_\sim} 2 m_Z$) we have shown, by means of a simple heavy--fermion condensate model, how the Landau pole of the Higgs self coupling can be related to the compositeness of the Higgs particle. We have analyzed the general features of such scenarios. In particular, we have shown that there exists a precise relationship between the effective Higgs mass, the new--fermion mass and the compositeness scale, which should hold in a wide class of models. \bigskip {\bf Acknowledgments} \noindent G.I. wishes to thank G. Altarelli and L. Maiani for interesting discussions, and the warm hospitality of the University of Virginia where part of this work was done. P.Q.H. wishes to thank the warm hospitality of the Theory Groups at the University of Rome ``La Sapienza'' and at Fermilab where part of this work was done. P.Q.H. is supported in parts by the U.S. Department of Energy under grant No. DE-A505-89ER40518. \section*{References}

proofpile-arXiv_065-506

\section{Introduction} Supersymmetry (SUSY) predicts the existence of a spin 1/2 partner of the gluon $g$, the so-called { gluino} $\mathaccent"7E g$. Like the QCD gauge boson, it is neutral, it is its own anti-particle (i.e., it is a Majorana fermion) and its coupling to ordinary matter is precisely determined in terms of the usual QCD colour matrices and the strong coupling constant $\alpha_s$ \cite{Nilles}. However, in contrast to the gluon, whose mass is predicted to be zero by the theory, that of the gluino $m_{\mathaccent"7E g}$ is a priori an arbitrary parameter, and so is its lifetime $\tau_{\mathaccent"7E g}$. Many searches have been carried out in order to detect or rule out such a particle. A detailed survey, including the description of various experiments, can be found in Ref.~\cite{Farrar-review}. In particular, light gluinos should be directly produced in 4jet events at LEP~I. Motivated by the advances in 4jet analyses based on heavy flavour identification, we further elaborate the study of Ref.~\cite{physlett} to cater for a wider range of light gluino masses and lifetimes. The use of $\mu$-vertex devices \cite{deangelis,Squarcia} provides an independent procedure to settle the ongoing controversy around the light gluino scenario, if one considers that long-lived gluinos might produce 4jet events with detectable secondary vertices \cite{vertex,Cuypers}. We assume them to be long-lived enough to be tagged by present experimental vertex tagging methods. With simple invariant mass cuts based on the different kinematics of the partons in the final state, one can obtain a signal identifiable as a clear excess in the total number of vertex tagged 4jet events, in percentage well beyond the uncertainties related to non-perturbative as well as to higher order perturbative effects. We also consider the possibility of `loopholes' in recent analysis carried out by the ALEPH collaboration~\cite{aleph} and by de Gouv\^ea and Murayama~\cite{Murayama}, which could undermine the validity of the results obtained there. We are especially concerned with the fact that gluinos might decay mainly into missing energy, so that the $q \bar q \tilde g \tilde g$ events are not recognised as 4jet events. As a matter of fact, the ALEPH analysis explicitly assumes that this is not the case whereas Ref.~\cite{Murayama} only considers the case in which the gluino does not decay inside the LEP detectors. The plan of the paper is as follows. In next Section we review the status of the light gluino window and put our work into context. Next we describe our calculations, and in Section 4 we devote some space to discuss possible tagging procedures of SUSY events. In Section 5 we present our results and Section 6 contains some brief conclusions. \section{The light gluino window} Our present knowledge about gluinos is summarised in Fig.~1, which shows the excluded regions in mass and lifetime of the SUSY fermion as they stood in 1993-1994\footnote{We would like to thank the authors of Ref.~\cite{Kileng-Osland1} for their kind permission of exploiting here one of the figures of their paper.}. At that time, it was clear that relatively long-lived and light gluinos (decaying into a `photino', or more correctly, into the lightest neutralino) were not yet excluded by the experiment. In the theory it is natural for gluinos to be much lighter than squarks if their mass is induced radiatively \cite{Masiero}. Furthermore, gluino and lightest neutralino masses are naturally less than few GeV if dimension-3 SUSY-breaking operators are absent from the low energy theory \cite{Farrar-Quarks94}, although the compatibility with a light gluino window in Supergravity models is strongly dependent on the dynamics of the breaking mechanism of the electroweak symmetry \cite{Diaz,Burjassot}. Since these values of $m_{\mathaccent"7E g}$ and $\tau_{\mathaccent"7E g}$ were within the reach of already operating accelerators \cite{PDG}, the regions identified by the white areas in Fig.~1 started receiving some attention in those years \cite{window}. In particular, it was noted that if the gluino is so light, it should be directly produced at LEP I: either in 2jet \cite{Kileng-Osland2} or in 4jet events \cite{epem}. An immediate interest in this possibility raised. This was also motivated by the `historical' discrepancy between the value of $\alpha_s$ determined by low energy deep-inelastic lepton-nucleon scattering and that measured by the $e^+e^-$ CERN experiments \cite{alphas}. In this respect, although the discrepancy between the two values of $\alpha_s$ was statistically small \cite{cpl} and also has slightly decreased in the latest measurements \cite{last_as}, it was speculated that the evolution of the strong coupling can be slowed down by a contribution to the $\beta$ function of a new, coloured, neutral fermion: indeed, a light gluino. The search for SUSY-signals intensified then at LEP I. Studies in other contexts, such as the influence in the Altarelli-Parisi evolution of the structure functions \cite{GLAP} or the so-called `3+1' jet events at HERA \cite{Ramon}, were also pursued. However, the effects are there too small to be tested using the present experimental data. More recently, extensive searches for light gluino signals have been carried out at the Tevatron \cite{CMF}. As for LEP I, the strategy adopted was to search for light gluinos in the context of the so-called QCD colour factor analyses in 4jet samples \cite{colour}. The basic idea is to measure the fundamental colour factors of QCD, that is, $C_A$, $C_F$ (the Casimir operators of the fundamental and adjoint representations of the gauge group $SU(N_C)$) and $T_F$ (the normalisation of the generators of the fundamental representation). In QCD (i.e., $N_C=3$), one gets $C_A=3$ and $C_F=4/3$. The factors $C_A$, $C_F$ and $T_F$ represent the relative strength of the couplings of the processes $q\rightarrow qg$, $g\rightarrow gg$ and $g\rightarrow q\bar q$, respectively (see, e.g., Ref.~\cite{Quigg}). The analytical formulae of the cross section $\sigma(e^+e^-\rightarrow 4\mbox{jet})$ for massless particles were computed long ago in Ref.~\cite{ERT}. The strategy is to compare the theoretical predictions to the data, by leaving the colour factors as free parameters to be determined by the fit. In practice, one of these, e.g., $C_F$, is absorbed in the normalisation of the cross sections leaving two independent ratios $C_A/C_F$ and $T_R/C_F$, with $T_R=N_FT_F$, being $N_F$ the number of active flavours. The experimental analyses are preferentially based on angular correlations between jets \cite{angles}, as they are sensitive to differences between the $2q2g$ (Fig.~2a--c) and the $4q$ (Fig.~2d) component of 4jet events. Light gluinos would enter in 4jet events via diagrams of the type depicted in Fig.~3, in the process $e^+e^-\rightarrow q\bar q {\mathaccent"7E g}{\mathaccent"7E g}$ (through a $g^*\rightarrow {\mathaccent"7E g}{\mathaccent"7E g}$ splitting). Note that gluino production in 4jet events via squarks splitting into quark-gluino pairs is very suppressed due to the large value of the lower limits on the squark masses, so is also the case in 2jet production through squark loops \cite{Kileng-Osland2}. As gluinos are coloured fermions, their contribution would enhance the part of the 4jet cross section with angular structure similar to that of $4q$ events. Naively then, one could say that the total number of flavours $N_F$ of the theory is apparently increased, such that, a SUSY-signal can be revealed in the form of an enhancement of $T_R$, with respect to the predictions of pure QCD. The results of those analyses were that, although the experimental measurements were in good agreement with QCD, it is was not possible to exclude the existence of a light ${\mathaccent"7E g}$ (see, e.g., Ref.~\cite{OPAL}). In particular, gluinos with a mass of at least 2 GeV yield an expectation value for $T_F/C_F$ that was within one standard deviation of the measured one. Even the extreme case of a massless gluino (for which $T_F/C_F\approx0.6$) would have brought the predictions only slightly beyond the upper experimental region of $68\%$ confidence level (CL) given in Ref.~\cite{OPAL}. Therefore, after those studies, the experimental constraints on the gluino mass and lifetime could still be summarised by the plot in Fig.~1. The reason why the LEP analyses showed a limitation in putting stringent bounds on the existence of light gluinos was that contributions to the total cross section of SUSY events are small and further reduced with respect to the ordinary QCD rates when mass suppression is taken into account. In particular, gluino effects on the total number of 4jet events were comparable in percentage to the systematic uncertainties related to jet hadronisation process and the uncalculated next-to-leading order (NLO) corrections through the order ${\cal O}(\alpha_s^3)$. To overcome these systematic limitations, it was recently proposed in Ref.~\cite{Murayama} to consider the colour factor $C_A/C_F$ sufficiently well known as to be taken for its QCD value, 9/4. The other factor is also partially constrained, by the fact that there are clearly five quark flavours which are active in the di-jet $Z$ decays. Therefore, if one wants to pin down possible gluino effects, one should allow variations of $T_F/C_F$ only above the value 3/8. Armed with these two new constraints, Ref.~\cite{Murayama} obtained an improved bound on $m_{\mathaccent"7E g}$. In particular, it was shown that a gluino mass ${\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}~1.5$ GeV is apparently excluded at more than $90\%$ CL by the 1991-92 OPAL data \cite{OPAL}. A new experimental analysis carried out by ALEPH in Ref.~\cite{aleph} obtains results along the same lines. They give new measurements of the QCD colour factors using all the data collected from 1992 to 1995 and obtain excellent agreement with ordinary QCD along with a new 95\% CL constraint $m_{\mathaccent"7E g}>6.3$ GeV on the gluino mass\footnote{Further indications towards the exclusion of somewhat lighter gluino masses come from studies of the running of $\alpha_s$ at higher orders \cite{as3}.}. This stringent limit was achieved thanks to a dedicated treatment to reduce Monte Carlo (MC) uncertainties related to the hadronisation process of the partons and to the fact that meanwhile preliminary results of the NLO corrections to the 4jet rate had become available (the complete calculation has been presented very recently \cite{a3complete}). On the one hand, several different models of parton fragmentation were compared to each other with different parameter settings, on the other hand, it was clear that NLO results have a strong impact on the 4jet rate, but very small influence on the shape of the angular distributions used in Ref.~\cite{aleph}. Although these results represent a clear improvement, the analyses of Refs.~\cite{aleph,Murayama} are still based on the traditional method \cite{colour} of ordering the jets in energy and identifying the two most energetic ones with those originated by the quarks produced in the $Z$ decay. In fact, the angles which are generally used (see Ref.~\cite{angles} for the exact definitions) in measuring the colour factors of QCD require in principle the identification of primary and secondary partons. In practice, the above assumption is often incorrect and the sensitivity of the experimental distributions to the QCD colour factors is considerably reduced. In this respect, it is worth recalling that, e.g., in $Z\rightarrow q\bar qgg$ events the percentage of events in which the two lowest energy partons are both gluons is only $\approx53\%$ \cite{nonAbel}. A possible improvement of the `energy ordering' procedure was advocated in Ref.~\cite{ioebas}, where samples of 4jets with two jets tagged as heavy flavour jets (i.e., $c$- and especially $b$-quarks) \cite{deangelis} are considered. In this way, one gets a greater discrimination power between $q\bar qgg$ and $q\bar q q'\bar q'$ events, for two reasons. First, one is able to distinguish between (heavy) quark and gluon jets, thus assigning the momenta of the final states to the various particles in a more correct way, as heavy quarks are mainly produced as primary partons. Second, $q\bar qgg$ event rates are reduced by the heavy flavour selection by a factor of 3/5 with respect to the $q\bar q q'\bar q'$ ones. Therefore, the $4q$ signal is enhanced by almost a factor of two and the differences between the quark and the gluon components can be more easily studied. The DELPHI collaboration is the only one to date (to our knowledge) that has resorted to flavour identification techniques to analyse 4jet events \cite{ana} (similar studies in the case of 3jet events are performed in Ref.~\cite{fuster}). They examined the data collected in the years 1991-1994 , from which a total of 11,000 4jet events with at least two heavy quark jets were selected. The typical efficiency was 12\% with a purity of events where all jets are correctly assigned of about 70\%. Note that neural networks were employed to combine the information on high transverse momentum leptons, large impact parameters and energy ordering of the jets (see, e.g., Ref.~\cite{deangelis} for a review about techniques of heavy flavour identification). Their results have been presented recently \cite{done}. The important outcome is that with the new selection strategy the errors are substantially reduced compared to previous analyses \cite{colour,OPAL}, especially for $T_R/C_F$. In general, the result was found to be in good agreement with the QCD expectations, but no new constraint on the mass of a possible light gluino was given at that time. Further analyses along the same lines are currently in progress \cite{progress}. As we shall see later on, our findings further support the relevance of such approaches. \section{Calculation} The Feynman diagrams describing at tree-level the reactions \begin{equation}\label{proc1} e^+ + e^-\rightarrow q + \bar q + g + g, \end{equation} \begin{equation}\label{proc2} e^+ + e^-\rightarrow q + \bar q + {q'} + \bar q', \end{equation} \begin{equation}\label{proc3} e^+ + e^-\rightarrow q + \bar q + {\mathaccent"7E g} + {\mathaccent"7E g}, \end{equation} are shown in Figs.~2--3. In the present analysis we have computed the matrix elements of processes (\ref{proc1})--(\ref{proc3}) with the same {\tt FORTRAN} generator used in Refs.~\cite{ioebas,MEs}, which takes exactly into account all masses and both the $\gamma^*$ and $Z$ intermediate contributions. Mass effects in 4jet events are important, as repeatedly recalled in the literature \cite{ioebas,MEs,masses,Mainz}, especially if heavy flavour selection is performed. For this study, the above program has been also checked against the one used in Ref.~\cite{masses} in the appropriate limit (i.e., when the masses along the fermion lines attached to the $\gamma^*,Z$ propagator are neglected). For the details of the numerical computation as well as the explicit helicity amplitude formulae, see Ref. \cite{MEs}. We have analysed processes (\ref{proc1})--(\ref{proc3}) adopting four different jet-resolution criteria for resolvable partons. We have done so in order to investigate the independence of our conclusions from the actual criteria employed and to check whether any of these shows better features for the analysis of light gluino contributions. The jet algorithms are identified through their clustering variable $y_{ij}$. They are ($\sqrt s=M_Z$): the JADE scheme (J) \cite{JADE} based on the `measure' \begin{equation}\label{JADE} y^J_{ij} = {{2E_i E_j(1-\cos\theta_{ij})}\over{s}}, \end{equation} and its `E' variation (E) \begin{equation}\label{E} y^E_{ij} = \frac{(p_i+p_j)\cdot(p_i+p_j)}{s}, \end{equation} the Durham scheme (D) \cite{DURHAM} \begin{equation}\label{DURHAM} y^D_{ij} = {{2\min (E^2_i, E^2_j)(1-\cos\theta_{ij})} \over{s}} \end{equation} and the Geneva algorithm (G) \cite{GENEVA} \begin{equation}\label{GENEVA} y^G_{ij} = \frac{8}{9} {{E_iE_j(1-\cos\theta_{ij})}\over{(E_i+E_j)^2}}. \end{equation} For all of them the two (pseudo)particles $i$ and $j$ (with energy $E_i$ and $E_j$, respectively) for which $y_{ij}$ is minimum are combined into a single (pseudo)particle $k$ of momentum $P_k$ given by the formula \begin{equation} P_k=P_i+P_j. \end{equation} The procedure is iterated until all (pseudo)particle pairs satisfy $y_{ij}\ge y_{\mathrm{cut}}$. The various characteristics of these algorithms are summarised in Ref.~\cite{GENEVA}. In our lowest order calculation, the 4jet cross section for a given algorithm is simply equal to the four parton cross section with a cut $y_{ij}\ge y_{\mathrm{cut}}$ on all pairs of partons $(i,j)$. It is worth noticing that, in the recently calculated NLO corrections to the 4jet rates \cite{a3complete}, the much forgotten Geneva scheme has been shown to be particularly sensitive to the number of light flavours as well as to exhibit a small scale dependence. In this respect, the G scheme may be more suitable than others in enlightening possible gluino contributions in the experimental sample. Concerning the numerical part of our work, we have taken $\alpha_{em}= 1/128$ and $\sin^2\theta_W=0.23$, while for the $Z$ boson mass and width we have adopted the values $M_{Z}=91.1$ GeV and $\Gamma_{Z}=2.5$ GeV, respectively. For the quarks we have: $m_c=1.7$ GeV and $m_b=5.0$ GeV while the flavours $u$, $d$ and $s$ have been considered massless. We have varied the gluino mass $m_{\mathaccent"7E g}$ in the range between 0 and 20 GeV. Finally, the strong coupling constant has been set equal to $0.115$\footnote{Our results will not be affected by the actual value of $\alpha_s$, as we will be interested in the end in relative differences between ordinary QCD and QCD+SUSY event rates.}. \section{Tagging procedure} In this Section we describe possible signatures of long-lived gluinos in 4jet events at LEP I. We will resort to the fact that gluinos should produce displaced vertices and offer a complementary tool to the ALEPH study \cite{aleph}. We also single out those combinations of SUSY parameters which could induce gluino decays into a large amount of missing energy. In this situation, the contribution of SUSY events to the 4jet rate as selected in the ALEPH analysis would be considerably reduced. Since we are implying that gluinos decay inside the LEP detectors, our considerations will only apply to the case of lifetimes less than $10^{-9}$ sec or so. In terms of mass, we will focus our attention on the two regions: (i) 0~${\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}~ m_{\mathaccent"7E g}~{\buildrel{\scriptscriptstyle <}\over {\scriptscriptstyle\sim}}$~1.5 GeV; (ii) $m_{\mathaccent"7E g}~{\buildrel{\scriptscriptstyle >} \over{\scriptscriptstyle\sim}}$~3.5 GeV. Long-lived gluinos can be operationally defined as those which hadronise before decaying. An inevitable consequence is that they live confined into bound states, generically called $R$-hadrons \cite{window}. The lightest of these would probably be the neutral, flavour singlet $({\tilde g}g)$ and $({\tilde g}uds)$ hadrons. If the gluino mass falls in the range (i), decays into the minimum hadronic mass, i.e., $(\tilde gg) \to \tilde\gamma \pi\pi$ and/or $(\tilde g uds) \to \tilde\gamma \Lambda$, maximize the missing energy and, therefore, the SUSY signal in the ALEPH analysis can be well attenuated. If $\tau_{\tilde g}\gg \tau_b$, SUSY events should show displaced vertices at a distance $d$ significantly larger than the decay length produced by a $b$-quark (i.e., 300 $\mu$m or so). If $\tau_{\tilde g}\approx\tau_b$ then the `degeneracy' discussed in Ref.~\cite{physlett} would occur between heavy quarks and gluinos, so that the double vertex tagging procedure combined with appropriate kinematic cuts (see Section 4) should help to disentangle the gluino contribution. If $\tau_{\tilde g}\ll\tau_b$, then it may not be useful to look for detached vertices, since these would not be detectable for $(\tilde g uds) \to \tilde\gamma \Lambda$ and would be too close to the interaction region. In general, for a gluino with mass below 1.5 GeV, the fragmentation function should be similar to that for the charm quark, i.e., with $\langle z \rangle < 0.5$, or even softer if the mass is very light. Hence, there is a maximum missing energy possible, and it may be that this whole region is excluded by the ALEPH analysis. If not, there must be significant missing energy correlated with the directions of the soft visible gluino jets. A combination of a suitable missing energy distribution with the ALEPH analysis should be able to find or exclude such a light gluino. In the second regime (ii), the gluino would presumably fragment into a $(\tilde gg)$ or $(\tilde g uds)$ hadron with a fragmentation function perhaps similar to a $b$-hadron, i.e., with $\langle z \rangle \sim 0.75$. This hadron would then decay into a $\tilde \gamma$ plus hadrons with a distribution similar to that for $\tilde g \to \tilde \gamma q \bar q$. The missing energy would be maximized if the mass of the $\tilde \gamma$ is close to that of the $\tilde g$, but a limit to the mass difference is set by the requirements that the decay occurs inside the tracking volume and that the squark masses be reasonable. The charged multiplicity distribution should be similar to that for $e^+e^- \to q \bar q$ at the same $q \bar q$ mass, and events occurring inside $R \sim 0.5$~m with non-zero charged multiplicity should be observed with high probability. Such events would have two hard jets, two soft jets, missing energy, and two largely detached vertices with $d\gg0.3$ mm in the directions of the soft jets. If the rate corresponding to these events is rather poor, then it is conceivable that the LEP collaborations might not have noticed them so far. \section{Results} \subsection{Production rates} In this Section we compare the production rates of SUSY events, as a function of the gluino mass in the range between 0 and 20 GeV, to the yield of ordinary QCD events. We do this with and without assuming vertex tagging (the titles $e^+e^-\rightarrow$~VVjj and $e^+e^-\rightarrow$~jjjj in the forthcoming plots will refer to these two cases, respectively). In the results of the cross sections for the untagged case a summation over all the quark flavours (massless and massive) is implicit, whereas for the tagged case we will consider the detached vertices as produced by gluinos and $b$-quarks only, thus neglecting the case of $c$-decays\footnote{We assume the rate due to misidentification of gluons and massless quarks as heavy partons negligible \cite{deangelis}.}, and sum over the remaining quark flavours. In principle, one should also retain $c$-quark events among those producing a detached vertex, eventually combining the corresponding rates with those for $b$-quarks, according to the values of efficiency and purity of the experimental analyses. In fact, the lifetime of $c$-quarks is finite (around 1/3 of that of the $b$'s) and is thus responsible for secondary vertices. Therefore, one should expect that, for values of the gluino lifetime around $1/3\tau_b$ (and below), charmed hadrons can represent an additional background from ordinary QCD to the SUSY signal and the actual positions of the decay vertices of $c$- and $b$-quarks can partially overlap. This is the reason why the algorithms determining the efficiency/purity of flavour tagging used by the LEP~I collaborations contain a multiplicity selection rule (the number of tracks produced being higher for $b$'s than for $c$'s) \cite{revb}. The values of purity obtained in a single $b$-tag at LEP~I, around $95\%$ or more \cite{deangelis,Squarcia,revb}, imply that above the $b$-selection cuts (in multiplicity and decay distance) the ordinary QCD contribution is indeed almost entirely due to decaying bottom hadrons, whereas below those cuts charmed hadrons are mainly responsible for secondary vertices. In other terms, the $c$- and $b$-contributions would enter in our analysis `separately' from each other into the QCD background to SUSY signals, the relative small contamination being eventually established by the experimental tagging strategy. For reasons of space, in the following we will illustrate the interplay between gluinos and bottom quarks only. However, it must always be intended that in presence of a short decay distance and/or a low secondary vertex multiplicity the actual rates from ordinary QCD events will in the end need the inclusion of the mentioned corrections due to the differences between $c$- and $b$-quarks. In Fig.~4 we study the effect of a non--zero gluino mass in the total cross section of the process $e^+e^- \to qq\mathaccent"7E g\mathaccent"7E g$ for the schemes described in Section 2 and for different values of the $y_{\mathrm{cut}}$ parameter. For $m_{\mathaccent"7E g} \buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim} 5$ GeV the cross section falls exponentially in the J and D schemes. For the G scheme the exponential behaviour starts somewhat earlier, at $m_{\mathaccent"7E g}\buildrel{\scriptscriptstyle >} \over{\scriptscriptstyle\sim} 2$ GeV, and for the E scheme later, when $m_{\mathaccent"7E g}\buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim}~7$ GeV. We already know that SUSY rates certainly compare rather poorly to both the $q\bar qgg$ and $q\bar q q'\bar q'$ contributions, if all quark flavours are retained and energy ordering is adopted \cite{masses}. Nonetheless, one of the salient features in Fig.~4 is that for $m_{\mathaccent"7E g}~{\buildrel{\scriptscriptstyle <} \over{\scriptscriptstyle\sim}}~10$ GeV the mass suppression on the SUSY rates is always less than one order of magnitude. This is true independently of jet algorithm and for three typical values of the resolution parameter $y_{\mathrm{cut}}$. In any case the contribution for $m_{\mathaccent"7E g}>10$ GeV begins to be very small. Therefore, in the remainder of the paper we will confine ourselves to gluino masses up to 10 GeV only. At this point, one should recall that the ordinary QCD production rates are much larger than the SUSY ones displayed in Fig.~4. For example, when no vertex tagging is exploited and all flavours are retained in the sample, at the minimum of the $y_{\mathrm{cut}}$'s used there, one gets $\sigma(q\bar qgg)=4153(4498)[5862]\{9312\}$ pb and $\sigma(q\bar qQ\bar Q)=187(217)[300]\{548\}$ pb, in correspondence of the J(E)[D]\{G\} scheme. A similar pattern in the relative composition of 4jet events persists also at larger values of the resolution parameter. The cross sections as a function of $y_{\mathrm{cut}}$ for the different subprocesses yielding two displaced vertices are presented in Fig.~5. The ratio between SUSY and pure QCD events is clearly improved, so that $b\bar bq\bar q$ and gluino rates now compare to each other. The largest contribution still comes from the subprocess $e^+e^- \to b\bar{b}gg$, which is almost one order of magnitude larger than the other two in the whole range of $y_{\mathrm{cut}}$. How to ameliorate this situation will be discussed below. The gluino rates are shown for three reference masses, $m_{\mathaccent"7E g} = 1,5, 10$ GeV. The pattern recognised in Fig.~4 as a function of the gluino mass is also visible in Fig.~5 for the $y_{\mathrm{cut}}$ dependence. That is, as the gluino mass increases the production rates get smaller, however still remaining within the same order of magnitude if $m_{\mathaccent"7E g}~ {\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}~10$ GeV. In practice, gluinos in the mass range up to 10 GeV have all sizeable production rates at LEP I in the jet schemes considered for usual values of the jet resolution parameters. This is indeed encouraging, as this means that the 4jet sample could well be sensible to values of $m_{\mathaccent"7E g}$ larger than those usually considered (i.e., of the order of the $b$-mass or below). Fig.~6 shows the improvements that can be achieved with heavy flavour tagging combined with the typical kinematic behaviour of gluino events, see Ref.~\cite{physlett}. For reference, the gluino mass has been fixed at 5 GeV, though the main features of the plots do not depend on $m_{\mathaccent"7E g}$ as these are connected only to the fact that gluinos are always secondary products. Only the D scheme is shown, for the other schemes exhibit very similar behaviour. The variable $Y_{ij}$ is the invariant scaled mass \begin{equation} Y_{ij}= \frac{(p_i+p_j)^2}{s} \label{Yij}, \end{equation} where $s$ is the center of mass energy ($s=M_Z^2$) and the indices $ij$ label the jets as follows: (12) refer to the two vertex tagged jets, and to the most energetic jets in the case of energy ordering; (34) corresponds to the two remaining jets. The distributions are normalised to one. Note that in Fig.~6a the $2\mathaccent"7E g 2q$ and $2b2q$ events are peaked at low $Y_{12}$ while the $2b2g$ events are evenly distributed. The peak in the first two cases is easily understood as it comes from the propagator $g^*\to b\bar{b}/\mathaccent"7E g\mathaccent"7E g$, which is not present in the third case (the tagged jets there come always from the $Z$ decay). The long tail of the $2b2q$ spectra comes from the fact that there can be `mis--tags' of $b$'s coming from the $Z$ propagator. The peak for $2q2\mathaccent"7E g$ is even narrower, as the two gluinos are always produced through gluon splitting, apart from a small contamination of mis-tags coming from $2b2\mathaccent"7E g$. The strategy is now clear: for $Y_{12}<0.2$ most of the SUSY signal is retained while $2b2q(2b2g)$ events are reduced roughly by a factor of two(four). In contrast, when energy ordering is performed (Fig.~6b) this effect is washed out, as all the distributions have a similar shape and no useful cut can be devised. Note that the distributions are finite due to the masses of the tagged jets so that loop corrections will not change significantly the behaviour presented here. The situation is even better if we look at Fig.~6c: the distribution for $2q2\mathaccent"7E g$ is flat and the other two distributions are peaked at $Y_{34}=0$. This effect is just the complementary of Fig.~6a: the (34) jets come from the $Z$ propagator in the $2q2\mathaccent"7E g$ events while show the peak of the gluon splitting for the other two cases. Again, when energy ordering is performed, Fig.~6d, the effect is wiped off. Therefore, we adopt the following requirements to optimise the SUSY signal over the ordinary QCD background: $Y_{12}<0.2$ and $Y_{34}>0.1$. Note that the use of the tagging procedure has been crucial for such an achievement. These simple invariant mass distributions serve the purpose of reducing the ordinary QCD rates in case of $\tau_{\tilde g}~ {\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}~\tau_b$, as it can happen when $m_{\tilde g}~{\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}} ~1.5$ GeV. For $m_{\tilde g}~{\buildrel{\scriptscriptstyle >} \over{\scriptscriptstyle\sim}}~3.5$ GeV, where $\tau_{\tilde g}\gg\tau_b$ and the gluino and quark vertices are in principle well distinguishable, the kinematic distributions would clearly help to elucidate the underlying SUSY dynamics. In Fig.~7 we show the different contributions to the total cross section in our tagging procedure like in Fig.~5, but with the improved sample. The $2b2g$ event rates, which were one order of magnitude larger than those of the other two partonic components, have been greatly reduced. All contributions are now comparable (at least for $m_{\mathaccent"7E g}= 1\div5$ GeV). For $m_{\mathaccent"7E g}~ {\buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim}}~5$ GeV the ordinary QCD events can be most likely eliminated from the sample already on a displaced vertex basis, by asking, e.g., that the decay length is much longer than 0.3 mm. However, for completeness we report the rates for large gluino masses too, as the tagging procedure could be complicated by the fact that a large part of the vertex tagged hadronic sample at LEP I has been collected via a bi-dimensional tagging \cite{bidimensional}. Therefore, projections of different decay lengths $d$ could well appear the same on the reproduced event plane. It is also worth recalling that the fact that gluinos are electrically neutral whereas quarks are charged can hardly be useful in 4jet analyses as there is extremely low efficiency in measuring the total jet charge, especially in multijet events. That explains why, for instance, this difference is not used to discriminate partonic compositions in ordinary 4jet events (as gluons too are neutral). In summary, we have shown that it is feasible to significantly enhance the signal of possible light gluino species over the QCD background using tagged samples with the help of elementary kinematical distributions. \subsection{Missing energy distributions} In this section we study the decays rates of SUSY events, for three representative values of the gluino mass which yield sizeable production rates. In particular, we will investigate the spectrum in missing energy inside the gluino jets, trying to establish the quantitative relevance in the total SUSY sample of hadronic events carrying an energetic imbalance that does not meet the usual trigger thresholds of the LEP I detectors. In discussing the possible decay modes of the gluino, two assumptions need to be made. The first is the condition of $R$-parity conservation. The second is the choice of the lowest mass Supersymmetric particle. $R$-parity, defined to be even for ordinary particles and odd for their Supersymmetric counterparts, needs to be preserved if lepton and baryon numbers are exactly conserved. This implies that the lightest Supersymmetric particle is exactly stable. In this paper we shall take it for granted that the neutralino (photino) is the lowest mass Supersymmetric particle. Failing this condition, the next likely choice would be the case where the scalar neutrinos are lower in mass. However, very light doublet sneutrinos are excluded by the $Z$ width constraints \cite{PDG}. The choice of the scalar quark masses ${\mathaccent"7E M}_L$ and ${\mathaccent"7E M}_R$ \cite{Nilles} affects the gluino branching ratios. In particular, considering only one flavour of massless quarks and assuming that the photino is massless, the ratio between the widths of the two dominant gluino decay modes is given by: \begin{equation}\label{BR} \frac{\Gamma({\tilde{g}\rightarrow g\tilde{\gamma}})} {\Gamma({\tilde{g}\rightarrow q\bar{q}\tilde{\gamma}})}= \frac{3\alpha_s}{4\pi} \frac{({\mathaccent"7E M}_R^2-{\mathaccent"7E M}_L^2)^2} {({\mathaccent"7E M}_L^4+{\mathaccent"7E M}_R^4)}. \end{equation} Therefore, the quark-antiquark-neutralino decay channel is dominant over the gluon-neutralino one. However, in some SUSY models the $L$ and $R$ mass eigenstates may differ by a factor of two or even more, such that ${({\mathaccent"7E M}_R^2-{\mathaccent"7E M}_L^2)^2}/ {({\mathaccent"7E M}_L^4+{\mathaccent"7E M}_R^4)}\buildrel {\scriptscriptstyle >}\over{\scriptscriptstyle\sim}1/2$ \cite{Haber-Kane}. Furthermore, as the photino mass approaches that of the gluino, $m_{\mathaccent"7E \gamma}/m_{\mathaccent"7E g}\rightarrow1$, the three-body decay mode suffers a further suppression, which goes as $(1-m_{\mathaccent"7E \gamma}/m_{\mathaccent"7E g})^2$. A more extensive review of the gluino decay channels can be found, for example, in Section 3.4 of Haber and Kane \cite{Haber-Kane} (see also references therein). We have computed the relevant decay currents by using helicity amplitude techniques, and incorporated these into a complete matrix element for gluino production and decay, over the appropriate phase space. In doing so, two different formalisms were employed: the usual helicity projector method \cite{IZ} and the techniques of Ref.~\cite{KS}. The results obtained with the two methods agree for any polarisation state if in the latter formalism one modifies the helicity projections to coincide with the physical choice along the direction of the final partons. Before studying the decay spectra, a few comments are in order concerning the fragmentation of a gluino. As already mentioned, a gluino would appear at the end of a hadronisation process confined into a bound state. Now, the decay kinematics of $R$-hadrons is in principle different from that of free ${\mathaccent"7E g}$'s. However, if the gluino is sufficiently heavy (say, $m_{\mathaccent"7E g}~ {\buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim}}~3.5$ GeV \cite{Haber-Kane}), the phenomenology of the decay products of such $R$-hadrons would be similar to that of unbounded gluinos. In particular, the basic result is that the $\mathaccent"7E \gamma$ energy spectrum roughly agrees with that produced by a freely decaying $\mathaccent"7E g$ as long as $m_{\mathaccent"7E \gamma}/m_{\mathaccent"7E g}$ is not too close to one \cite{ACCMM}. For lighter gluinos the analysis is less straightforward. However, according to Ref.~\cite{Franco}, it is reasonable to expect that these SUSY hadrons would again decay similarly to free gluinos, provided that $m_{\mathaccent"7E g}$ is replaced by an `effective' $R$-hadron mass equal to $\approx 0.75~m_{\mathaccent"7E g}$. For our purposes, we assume that the mass appearing in the decay spectra is in fact that of the SUSY parton in the mass range (ii), whereas in the interval (i) it represents the mentioned effective mass. Furthermore, on the one hand, we confine ourselves to values of $m_{\mathaccent"7E \gamma}$ strictly smaller than $m_{\mathaccent"7E g}$ in order to maintain valid our approximation over the range $m_{\mathaccent"7E g}~ {\buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}}~1.5$ GeV; on the other hand, we will push the ratio $m_{\mathaccent"7E \gamma}/m_{\mathaccent"7E g}$ up to 3/4 in order to maximise the amount of missing energy carried away by the undetected photino. The results we have obtained for the energy distribution of the missing energy after the two decays are displayed in Figs.~8a--c and Figs.~9a--c (in correspondence of the two possible decays). The crucial point is that the amount of missing energies produced could be so large that SUSY events of the type $q\bar q {\tilde g}{\tilde g}$ are not recognised as 4jet events. In fact, experimental analyses have a minimal hadronic energy cut on each of the four jets, in order to reduce the background due to poorly reconstructed events. The $E_{\mathrm{miss}}$ spectra are shown for four kinematical decay configurations: a massless photino, and a massive one with $m_{\mathaccent"7E \gamma} =n/4m_{\mathaccent"7E g}$, with $n=1,2,3$, and for three gluino masses $m_{\mathaccent"7E g} =1,5,10$ GeV. It is clear from both Fig.~8a--c and 9a--c that the missing energy spectrum gets harder as $m_{\tilde g}$ and $m_{\tilde\gamma}$ increase in both decay channels considered. The effect is common to all algorithms. For $m_{\tilde g}=1$ GeV the mean value of the missing energy is always below 10 GeV in both decay channels, and it can grow up to more than 15 GeV if $m_{\tilde g}=10$ GeV and $m_{\tilde \gamma}=7.5$ GeV. Under such conditions, it could be argued that $q\bar q {\tilde g}{\tilde g}$ events can pass unobserved as actual 4jet events if tight constraints are implemented on the missing mass energy of the hadronic event sample. \section{Summary and conclusions} In this paper we have studied the production and decay rates of $e^+e^-\to q\bar q{\tilde g}{\tilde g}$ events at LEP~I, where ${\tilde g}$ represents a relatively light (up to 10 GeV in mass) and long-lived (up to $10^{-9}$ sec in lifetime) gluino, and compared these to the yield of ordinary QCD events of the type $e^+e^-\to q\bar qgg$ and $e^+e^-\to q\bar qq'\bar q'$, involving quarks $q$ and gluons $g$. The presence of such SUSY events in 4jet samples at LEP~I has been advocated in the past years to explain the disagreement between the values of the strong coupling constant $\alpha_s$ as measured from the deep-inelastic scattering and the $Z$-peak data. This was further motivated by the initial discrepancy between the QCD predictions for the colour factors $C_A$, $C_F$ and $T_F$ and their actual measurements obtained in earlier analyses \cite{colour} by the LEP collaborations, as these constants are sensitive to additional SUSY contributions. The claim about the possible existence of gluinos in LEP I data has apparently become less convincing during the recent two or three years, as the experimental and theoretical analyses of the data have reached a higher level of sophistication and precision. Very recent studies seem to exclude gluinos with masses up to 6.3 GeV. Although such results represent clear progress towards settling the ongoing dispute about the existence of SUSY signals at LEP I, we have outlined here a complementary approach guided by two considerations. First, in all the mentioned analyses no vertex tagging was exploited in assigning the momenta of the jets to the corresponding partons from which the former originate. The study of 4jet events showing two secondary vertices produced in the decay of $c$- and $b$-quarks has in fact been proved to be successful in reducing the error on the QCD colour factor which is most sensitive to the possible presence of light gluinos (that is, $T_R=N_FT_F$). Furthermore, we have also shown that simple kinematic distributions (such as the invariant masses of the two vertex tagged jets and of the remaining two) can effectively help to enrich significantly the 4jet samples of gluino events (if existing). In fact, the latter, on the one hand, should yield displaced vertices and, on the other hand, are always produced as secondary partons (contrary to heavy quarks). Second, the validity of the result quoted by the ALEPH collaboration (the most constraining one) could be undermined by the fact that, in their procedure of selecting candidate 4jet samples, events carrying a large fraction of missing energy were not included. As a matter of fact, gluinos (or better, $R$-hadrons, in which the SUSY partner of the gluon is confined) should predominantly decay into `photinos', which escape detection. Indeed, there are kinematic configurations in which the ratio between the gluino and photino mass is such that the missing energy is rather large and, conversely, the left-over one for the hadronic system arising from the SUSY decay is rather small, such that these events might not pass the experimental 4jet resolution and selection criteria. In the above context, we believe to have obtained interesting results for future studies. In fact, we have shown that, in the vertex tagged sample of 4jets, SUSY events become comparable to the rates of ordinary QCD events for gluino masses up to about 10 GeV, thus well beyond the bounds presently set on this quantity. Furthermore, we have indicated that the latter cannot be reliable if the photino mass is not negligible compared to that of the gluino. Therefore, we conclude that experimental analyses based on our approach should help in clarifying the present debate, either contradicting the present bounds on the gluino mass or improving these by extending the experimental coverage of the so-called `light gluino window'. For example, over the mass region $m_b < m_{\tilde g}~ \buildrel{\scriptscriptstyle <}\over{\scriptscriptstyle\sim}~10$ GeV, SUSY rates should still be sizable and yield an unmistakable signature with two hard jets, two soft ones, large missing energy, two detached vertices with $c\tau~\buildrel{\scriptscriptstyle >}\over{\scriptscriptstyle\sim}~0.3$ mm, a neat peak in the invariant mass of the vertex tagged dijet system and a very flat distribution in the mass of the other two jets. In carrying out our study we have resorted to parton level calculations, which include all masses of primary and secondary partons exactly. Although we have not implemented a full Monte Carlo procedure including the fragmentation of the gluinos into hadrons or the decay of the latter into jets and missing particles, we have used analytic approximations which should mimic well the actual gluino phenomenology to a degree of accuracy compatible with that of the current experimental analyses. In this respect, we have indicated possible signatures of gluinos decaying inside the LEP detectors, as a function of both the mass and the lifetime of the SUSY particle. Before closing, we would like to point out a few crucial aspects of our work. First, contrary to many previous studies (which did not exploit vertex tagging and/or kinematical cuts) in which the gluino component represents an effect of just a few percent (thereby being of the same order as next-to-leading and/or hadronisation corrections), we have been concerned with SUSY rates that are always comparable or even larger than those produced by pure QCD events. Therefore, the inclusion of the mentioned corrections will not spoil our results. Second, for values of gluino masses up to 10 GeV or so, our conclusions are essentially the same {independently} of the jet algorithm and of the value used for $y_{\mathrm{cut}}$ (although the actual cross sections and the behaviour of the distributions do certainly depend on them). In the end, the magnitude of higher order and hadronisation effects as well as experimental considerations will determine which algorithm and which resolution parameter are the most suitable to use, though the Geneva algorithm seems to be slightly favoured due to its special sensitivity to the actual number of active flavours and a smaller scale dependence in NLO corrections. Third, by adopting the current LEP I values of vertex tagging efficiency and luminosity, we should expect a statistically significant analysis, based on several thousands of doubly tagged 4jet events. Fourth, since those presented here are theoretical results from parton level calculations, they will necessarily have to be folded with detailed experimental simulations (including both fragmentation/hadronization and detector effects), such that one could even improve at that stage our procedure: for example, by exploiting various differences (in charge, mass, lifetime) that occur between heavy quarks and gluinos. We finally remark that in the long term our arguments could well be of interest also to the SLC experiment at SLAC, as microvertex devices are installed there and they are known to have achieved by now a considerable tagging efficiency, so to hopefully compensate for the present lack of statistics of their data with respect to the LEP ones. \section*{Acknowledgements} We thank Bas Tausk and Val Gibson for valuable discussions. We are also grateful to Ben Bullock for carefully reading the manuscript. This work is supported in part by the Ministero dell' Universit\`a e della Ricerca Scientifica, the UK PPARC, the Spanish CICYT project AEN 94-0936, and the EC Programme ``Human Capital and Mobility'', Network ``Physics at High Energy Colliders'', contracts CHRX-CT93-0357 DG 12 COMA (SM) and ERBCHBICHT (RMT). KO is grateful to Trinity College and the Committee of Vice-Chancellors and Principals of the Universities of the United Kingdom for financial support. \goodbreak

proofpile-arXiv_065-507

\section{Introduction} \noindent The NRQCD factorization approach\cite{B-B-L} is a systematic framework for analyzing the inclusive cross sections and annihilation decay rates of heavy quarkonium. The cross section (or decay rate) is factored into short-distance coefficients that are computable using perturbation theory and long-distance NRQCD matrix elements. The matrix elements scale as a definite power of $v$, the typical relative velocity of the heavy quark in quarkonium. Thus the cross section can be organized into a double expansion in powers of $v$ and powers of $\alpha_s(m_c)$, where $m_c$ is the heavy quark mass. The NRQCD factorization framework makes definite, and in some cases rather dramatic, predictions for the dependence of the cross section on the polarization of the quarkonium. Below, I outline the NRQCD factorization approach as it applies to polarized quarkonium states and then summarize the applications that have been carried out thus far. \section{NRQCD Factorization} \noindent The general expression for the cross section for the inclusive production of a quarkonium state $H$ with four-momentum $P$ is \begin{eqnarray} \sum_X d \sigma(12 \to H(P) + X) &=& {1 \over 4 E_1 E_2 v_{12}}\; {d^3P \over (2 \pi)^3 2 E_P} \nonumber \\ && \hspace{-1in} \times \sum_X \; (2 \pi)^4 \delta^4(k_1 + k_2 - P - k_X) \; |{\cal T}_{1 2 \to H(P) + X}|^2 \,, \label{dsig} \end{eqnarray} where ${\cal T}_{1 2 \to H(P) + X}$ is the T-matrix element and the sum on the right side includes integration over the phase space of the additional particles. This cross section involves both ``short distances'' of order $1/m_c$ or smaller and ``long distances'' of order $1/(m_c v)$ or larger. The production of the $c \bar c$ pair involves short distances, because the parton processes that produce a $c \bar c$ pair always involve particles that are off their mass shells by amounts of order $m_c$ and can therefore propagate only over short distances. The binding of the $c$ and $\bar c$ into the state $H$ involves long distances, because gluons whose wavelengths are comparable to or larger than the size of the bound state, which is of order $1/(m_c v)$, play a large role in the binding. If the cross section in (\ref{dsig}) is sufficiently inclusive, short-distance and long-distance effects can be factored\cite{B-B-L}. (If there are hadrons in the initial state, it is also necessary to restrict the four-momentum of the quarkonium to be significantly different from that of the initial hadrons in order to avoid contributions from diffractive scattering.) Like many other factorization ``theorems'' of perturbative QCD, the factorization of quarkonium cross sections has not been proven with complete rigor, but it is a plausible generalization of the factorization theorems for the Drell-Yan production of muon pairs and for heavy quark production. The factorization relies on cancellations between soft partons that are emitted by the $c \bar c$ pair and soft partons that are exchanged between the $c \bar c$ pair and other jet-like collections of collinear partons. After taking into account these cancellations, the cross section can be factored into short-distance and long-distance parts. The short-distance parts involve the production of a $c \bar c$ pair with small relative momentum plus hard partons. The $c \bar c$ pair that emerges from the short-distance part is essentially pointlike on the scale of the quarkonium wavefunction. The long-distance parts involve the formation of $H$ plus soft partons from the pointlike $c \bar c$ pair. The standard factorization methods of perturbative QCD produce an expression for the cross section that involves an integral over the relative momentum of the $c$ and $\bar c$ that form the quarkonium. Long-distance and short-distance effects can be further untangled by expanding the short-distance factors in powers of the relative momentum ${\bf q}$ and absorbing the integration over ${\bf q}$ into the long-distance parts. The resulting long-distance factors can be expressed as matrix elements in an effective field theory called {\it nonrelativistic QCD} (NRQCD). The T-matrix elements in (\ref{dsig}) that survive after soft-parton cancellations can be expressed in the form \begin{equation} {\cal T}_{1 2 \to H(P) + X_H + X_S} \;\approx\; \sum_n \hat{\cal T}_{1 2 \to c \bar c(P,n) + X_H} \langle H + X_S | \psi^\dagger {\cal K}_n \chi (x=0) | 0 \rangle \,, \label{T-fact} \end{equation} where the sum includes all possible color and angular-momentum states of the $c \bar c$ pair. The factor $\hat{\cal T}_{1 2 \to c \bar c(P,n) + X_H}$ can be interpreted as a T-matrix element for producing a $c \bar c$ pair in the state $n$ plus the hard partons $X_H$. The second factor on the right is a matrix element in NRQCD between the vacuum state and a state that in the asymptotic future consists of the quarkonium $H$ at rest plus the soft partons $X_S$. The local operator $\psi^\dagger {\cal K}_n \chi$ creates a pointlike $c \bar c$ pair in the state $n$. After inserting the expression (\ref{T-fact}) for the T-matrix elements into (\ref{dsig}), we can rearrange the cross section into a form in which the short-distance and long-distance contributions are factored: \begin{eqnarray} \sum_X \; (2 \pi)^4 \delta^4(k_1 + k_2 - P - k_X) \; | {\cal T}_{1 2 \to H(P) + X}|^2 && \nonumber \\ && \hspace{-2.5in} \;\approx\; \sum_{m n} \left( \sum_{X_H} (\hat{\cal T}_{1 2 \to c \bar c(P,m) + X_H})^* \hat{\cal T}_{1 2 \to c \bar c(P,n) + X_H} \right) \nonumber \\ && \hspace{-2in} \times \left( \sum_{X_S} \langle 0 | \chi^\dagger {\cal K}^\dagger_m \psi | H + X_S \rangle \langle H + X_S | \psi^\dagger {\cal K}_n \chi | 0 \rangle \right) \,. \label{TT-fact} \end{eqnarray} The short-distance factor is \begin{equation} C_{mn}(k_1,k_2, P) \;=\; \sum_{X_H} (\hat{\cal T}_{1 2 \to c \bar c(P,m) + X_H})^* \hat{\cal T}_{1 2 \to c \bar c(P,n) + X_H} \,. \label{Cmn} \end{equation} The long-distance factor is \begin{equation} \langle {\cal O}^H_{mn} \rangle \;=\; \langle 0 | \chi^\dagger {\cal K}^\dagger_m \psi \; {\cal P}_H \; \psi^\dagger {\cal K}_n \chi | 0 \rangle \,, \label{O-H} \end{equation} where ${\cal P}_H$ projects onto states that in the asymptotic future contain the quarkonium state $H$ plus soft partons: \begin{equation} {\cal P}_H \;=\; \sum_{X_S} | H + X_S \rangle \langle H + X_S | \,. \label{P-H} \end{equation} Inserting (\ref{TT-fact}) into (\ref{dsig}), we obtain the NRQCD factorization formula for the inclusive cross section: \begin{equation} \sum_X d \sigma(12 \to H(P) + X) \;=\; {1 \over 4 E_1 E_2 v_{12}}\; {d^3P \over (2 \pi)^3 2 E_P} \sum_{mn} C_{mn}(k_1,k_2, P) \; \langle {\cal O}^H_{mn} \rangle \,. \label{dsig-fact} \end{equation} We can write a similar factorization formula for the inclusive decay rate of the quarkonium $H$ via the annihilation of the $c \bar c$ pair. The general formula for the decay rate is \begin{equation} \sum_X d \Gamma(H \to X) \;=\; {1 \over 2 M_H} \; \sum_X \; (2 \pi)^4 \delta^4(P - k_X) \; |{\cal T}_{H(P) \to X}|^2 \,, \label{dGam} \end{equation} where $P = (M_H,{\bf 0})$. This can be expressed in the factored form \begin{equation} \sum_X d \Gamma(H \to X) \;=\; {1 \over 2 M_H} \sum_{mn} C_{mn} \; \langle H | {\cal O}_{mn} | H \rangle \,, \label{dGam-fact} \end{equation} where the $C_{mn}$'s are short-distance coefficients. The long-distance factors are expectation values in the quarkonium state of local four-fermion operators of the form ${\cal O}_{mn} = \psi^\dagger {\cal K}^\dagger_m \chi \chi^\dagger {\cal K}_n \psi$. \section{Short-distance Coefficients} \noindent Since the coefficients $C_{mn}$ in (\ref{dsig-fact}) and (\ref{dGam-fact}) involve only short distances of order $1/m_c$ or larger, they can be expressed as perturbation series in $\alpha_s(m_c)$. These coefficients are known at tree level for many processes, and in a few cases they are known at the one-loop level. Most of these coefficients have been obtained by calculating the perturbative cross section for producing a $c \bar c$ pair in a state with a prescribed nonrelativistic wavefunction. The inclusive cross section is sensitive only to the behavior of the wavefunction near the origin. The analogs of the NRQCD matrix elements (\ref{O-H}) for the $c \bar c$ state can also be calculated using perturbation theory in terms of the prescribed wavefunction. Knowing the cross section and the matrix elements, we can read off the short-distance coefficients. Unfortunately, this method is not sufficiently general to determine all the short-distance coefficients, especially for polarized quarkonium states. The {\it threshold expansion method}, developed recently by Braaten and Chen\cite{Braaten-Chen}, is a general prescription for calculating the short-distance coefficients. It is based directly on the NRQCD factorization approach, and can be readily applied to polarized quarkonium. \subsection{Threshold expansion method} \noindent The threshold expansion method relies on the fact that the short-distance coefficients in (\ref{TT-fact}) are insensitive to the long-distance effects that bind the $c \bar c$ pairs into the quarkonium state $H$. Thus the factorization formula will hold with the same short-distance coefficients if we replace $H$ by asymptotic perturbative states $c \bar c = c \bar c({\bf q},\xi,\eta)$ that consist of a $c$ and a $\bar c$ with relative momentum ${\bf q}$ and spin/color state that is represented by the Pauli spinors $\xi$ and $\eta$. To completely determine the short-distance coefficients, we need to use different states $c \bar c$ and $c \bar c'$ in the T-matrix element and in its complex conjugate. The resulting matching prescription is \begin{eqnarray} \sum_X \; (2 \pi)^4 \delta^4(k_1 + k_2 - P - k_X) \; ({\cal T}_{1 2 \to c \bar c'(P) + X})^* {\cal T}_{1 2 \to c \bar c(P) + X} \Big|_{pQCD} && \nonumber \\ && \hspace{-3.5in} \;\approx\; \sum_{m n} C_{mn}(k_1,k_2,P) \langle 0 | \chi^\dagger {\cal K}_m \psi \; {\cal P}_{c \bar c',c \bar c} \; \psi^\dagger {\cal K}_n \chi | 0 \rangle \Big|_{pNRQCD} \,, \label{TT-match} \end{eqnarray} where the projection operator in the matrix element is \begin{equation} {\cal P}_{c \bar c',c \bar c} \;=\; \sum_{X_S} | c \bar c' + X_S \rangle \langle c \bar c + X_S | \,. \label{P-ccbar} \end{equation} The left side of (\ref{TT-match}) is to be calculated using perturbative QCD, and then expanded in powers of the relative momenta ${\bf q}$ and ${\bf q}'$. The matrix elements on the right side are to be calculated using perturbative NRQCD, and then expanded in powers of ${\bf q}$ and ${\bf q}'$. The coefficients $C_{mn}$ are then determined by matching these expansions order by order in $\alpha_s$. \subsection{Example} \noindent We illustrate the threshold expansion method by carrying out one of the simplest matching calculations. It gives the short-distance coefficient corresponding to the parton process $q \bar q \to c \bar c$. The T-matrix element for this process is \begin{equation} {\cal T}_{1 2 \to c \bar c} \;=\; g^2 {1 \over P^2} \; {\bar v}(k_2) \gamma_\mu T^a u(k_1) \; \bar u(p) \gamma^\mu T^a v({\bar p}) . \label{T-qqbar} \end{equation} Making a nonrelativistic expansion of the spinors of the $c$ and $\bar c$, this reduces to \begin{equation} {\cal T}_{1 2 \to c \bar c} \;=\; {g^2 \over 2 m_c} \; {\bar v}(k_2) \gamma_\mu T^a u(k_1) \; L^\mu_{\ i} \; \xi^\dagger \sigma^i T^a \eta , \label{T-qqbar0} \end{equation} where $L^\mu_{\ i}$ are elements of the boost matrix that transforms from the rest frame of the $c \bar c$ pair to the frame in which it has total four-momentum $P$. Multiplying by the complex conjugate of ${\cal T}_{1 2 \to c \bar c'}$ and averaging over initial spins and colors, we obtain \begin{equation} ({\cal T}_{1 2 \to c \bar c'})^* \; {\cal T}_{1 2 \to c \bar c} \;=\; {4 \pi^2 \alpha_s^2 \over 9} \left[ \delta^{ji} - {\hat z}^j {\hat z}^i \right] {\eta'}^\dagger \sigma^j T^a \xi' \xi^\dagger \sigma^i T^a \eta , \label{TT-qqbar} \end{equation} where $\bf {\hat z}$ is a unit vector in the direction of the momenta of the colliding $c$ and $\bar c$. The spinor factor can be expressed in terms of an NRQCD matrix element: \begin{equation} \langle \chi^\dagger \sigma^j T^a \psi | c \bar c' \rangle \langle c \bar c | \psi^\dagger \sigma^i T^a \chi \rangle \Big|_{pNRQCD} \;=\; 4 m_c^2 \; {\eta'}^\dagger \sigma^j T^a \xi' \xi^\dagger \sigma^i T^a \eta \,. \label{M-qqbar} \end{equation} Using the matching prescription (\ref{TT-match}), the short-distance coefficient of the matrix element $\langle \chi^\dagger \sigma^j T^a \psi {\cal P}_{c \bar c',c \bar c} \psi^\dagger \sigma^i T^a \chi \rangle$ is \begin{equation} C_{ij} \;=\; (2 \pi)^4 \delta^4(k_1 + k_2 - P) \; { \pi^2 \alpha_s^2 \over 9 m_c^2} \left[ \delta^{ji} - {\hat z}^j {\hat z}^i \right] \,. \label{C-qqbar} \end{equation} Inserting the short-distance coefficient (\ref{C-qqbar}) into the factorization formula (\ref{dsig-fact}) and integrating over the phase space of the quarkonium, we get an expression for the inclusive cross section: \begin{eqnarray} \sum_X \sigma(q \bar q \to H + X) && \nonumber \\ && \hspace{-1in} \;=\; \delta(s - 4 m_c^2) \; {\pi^3 \alpha_s^2 \over 36 m_c^4} \left[ \delta^{ji} - {\hat z}^j {\hat z}^i \right] \langle \chi^\dagger \sigma^j T^a \psi \; {\cal P}_H \; \psi^\dagger \sigma^i T^a \chi \rangle \,. \label{sig-qqbar} \end{eqnarray} This is a term in the factorization formula for the inclusive cross section of any quarkonium state $H$, whether polarized or unpolarized. There are additional terms of order $\alpha_s^2$ from the process $gg \to c \bar c$. All other parton processes give terms with short-distance coefficients of order $\alpha_s^3$ or higher. \section{NRQCD Matrix Elements} \noindent The long-distance factors in the NRQCD factorization formulas are expressed as matrix elements of local four-fermion operators in NRQCD. Since long-distance effects in QCD are inherently nonperturbative, the NRQCD matrix elements can only be calculated using nonperturbative methods like lattice gauge theory. There are effective lattice prescriptions for calculating the matrix elements $\langle H | {\cal O}_{mn} | H \rangle$ that appear in quarkonium decay rates\cite{B-K-S}. Unfortunately, these methods cannot be used to calculate directly the production matrix elements $\langle {\cal O}^H_{mn} \rangle$. The problem lies in implementing on the lattice the projection defined by (\ref{P-H}). In the absence of nonperturbative calculations, the only alternative is to treat the NRQCD matrix elements as phenomenological parameters to be determined by experiment. \subsection{Model-independent framework} \noindent The factorization formula (\ref{dsig-fact}) provides a model-independent framework for analyzing quarkonium production. In any reasonable model for the production of quarkonium through short-distance parton processes, the inclusive cross section can be expressed in the factored form (\ref{dsig-fact}). The model can therefore be reduced to a set of assumptions about the NRQCD matrix elements. Until recent years, most calculations of quarkonium production were carried out using either the {\it color-singlet model} or the {\it color-evaporation model}\cite{Schuler}. In the color-singlet model, the quarkonium is assumed to be simply a color-singlet $c \bar c$ pair in an appropriate angular-momentum state. Only one NRQCD matrix element is assumed to be important, and it can be expressed in terms of the $c \bar c$ wavefunction, or one of its derivatives, evaluated at the origin. In the color-evaporation model, the color and angular-momentum quantum numbers of the quarkonium are simply ignored. The NRQCD matrix elements are assumed to be calculable in perturbation theory up to an overall normalization constant that depends on the state $H$. This model implies that the matrix elements scale like $v^{3+D}$, where $v$ is a small parameter and $D$ is the number of covariant derivatives ${\bf D}$ in the operator ${\cal O}^H_{mn}$. NRQCD predicts a much more intricate hierarchy among the matrix elements. The matrix element $\langle {\cal O}^H_{mn} \rangle$ defined in (\ref{O-H}) scales like $v^{3+D+E+2M}$, where $D$ is the number of covariant derivatives ${\bf D}$ that appear in the operator and $E$ and $M$ are the number of chromoelectric and chromomagnetic transitions that are required for $c \bar c$ pairs in the states created by $\psi^\dagger {\cal K}_m \chi$ and $\psi^\dagger {\cal K}_n \chi$ to reach the dominant Fock state of $H$. These velocity-scaling rules determine the approximate magnitudes of NRQCD matrix elements. By keeping only those matrix elements that scale with the fewest powers of $v$, we can reduce their number sufficiently that a phenomenological approach becomes tractable. One should keep in mind, however, that the importance of a term in the cross section is determined not only by the magnitude of the matrix element but also by the magnitude of its short-distance coefficient. \subsection{Reducing the matrix elements} \noindent The matrix elements can be further simplified by using symmetries of NRQCD. To illustrate the simplifications, we will use matrix elements of the $J/\psi(\lambda)$, where the polarization state is specified by the helicity $\lambda$. \subsubsection{Rotational symmetry} \noindent Rotational symmetry is an exact symmetry of NRQCD. It implies, for example, that the matrix element for $H = \psi(\lambda)$ in (\ref{sig-qqbar}) must be a linear combination of the tensors $\delta_{ij}$, $U_{\lambda j} U^\dagger_{i \lambda}$, and $U_{\lambda j} U^\dagger_{i \lambda}$, where $U_{i \lambda}$ is the unitary matrix that transforms vectors from the spherical basis to the Cartesian basis. If that matrix element is summed over the polarizations of the $\psi$, the only possible tensor is $\delta_{ij}$. The matrix element must therefore satisfy \begin{equation} \sum_\lambda \langle \chi^\dagger \sigma^j T^a \psi \; {\cal P}_{\psi(\lambda)} \psi^\dagger \sigma^i T^a \chi \rangle \;=\; {\delta_{ij} \over 3} \langle \chi^\dagger \sigma^k T^a \psi \; {\cal P}_\psi \; \psi^\dagger \sigma^k T^a \chi \rangle . \label{rot-sym} \end{equation} \subsubsection{Heavy-quark spin symmetry} \noindent Heavy-quark spin symmetry is an approximate symmetry of NRQCD that holds up to corrections of order $v^2$. The symmetry follows from the fact that the spin of the heavy quark is conserved at leading order in $v^2$ in NRQCD. It implies, for example, that the matrix element for $H = \psi(\lambda)$ in (\ref{sig-qqbar}) must be proportional to $U_{\lambda j} U^\dagger_{i \lambda}$: \begin{equation} \langle \chi^\dagger \sigma^j T^a \psi \; {\cal P}_{\psi(\lambda)} \psi^\dagger \sigma^i T^a \chi \rangle \;\approx\; {U_{\lambda j} U^\dagger_{i \lambda} \over 3} \langle \chi^\dagger \sigma^k T^a \psi \; {\cal P}_\psi \; \psi^\dagger \sigma^k T^a \chi \rangle . \label{hqs-sym} \end{equation} Summing over helicities, we recover (\ref{rot-sym}). \subsubsection{Vacuum-saturation approximation} \noindent The vacuum-saturation approximation can only be applied to specific color-singlet matrix elements. If the matrix element is expressed in the form (\ref{O-H}), the operators $\psi^\dagger {\cal K}_m \chi$ and $\psi^\dagger {\cal K}_n \chi$ must create pointlike $c \bar c$ pairs in the dominant Fock state of the quarkonium. In the vacuum-saturation approximation, the projection operator $P_H$ defined in (\ref{P-H}) is replaced by $ | H \rangle \langle H|$, which corresponds to keeping only the vacuum term in the sum over soft states $X_S$. This is a controlled approximation in NRQCD, holding up to corrections that are of order $v^4$. The vacuum saturation approximation can be illustrated by the following matrix element for $J/\psi$ production: \begin{equation} \langle 0 | \chi^\dagger \sigma^i \psi \; {\cal P}_{\psi(\lambda)} \; \psi^\dagger \sigma^j \chi | 0 \rangle \;\approx\; \langle 0 | \chi^\dagger \sigma^i \psi | \psi(\lambda) \rangle \; \langle \psi(\lambda) | \psi^\dagger \sigma^j \chi | 0 \rangle \,. \label{vsa-prod} \end{equation} The vacuum-saturation approximation can also be used for decay matrix elements: \begin{equation} \langle \psi | \chi^\dagger \mbox{\boldmath{$\sigma$}} \psi \cdot \psi^\dagger \mbox{\boldmath{$\sigma$}} \chi | \psi \rangle \;\approx\; \sum_\lambda \big| \langle \psi(\lambda) | \psi^\dagger \mbox{\boldmath{$\sigma$}} \chi | 0 \rangle \big|^2 \,. \label{vsa-decay} \end{equation} The vacuum-to-quarkonium matrix element $\langle \psi | \psi^\dagger \mbox{\boldmath{$\sigma$}} \chi | 0 \rangle$ that appears on the right sides of (\ref{vsa-prod}) and (\ref{vsa-decay}) is proportional to the wavefunction at the origin. This matrix element can be easily calculated using lattice simulations of NRQCD. Thus the vacuum-saturation approximation provides a way to calculate certain production matrix elements on the lattice. \section{Polarization Predictions} \noindent The NRQCD factorization formulas apply equally well to any quarkonium state $H$, including a polarized state. Since the short-distance coefficients are independent of $H$, the dependence on $H$ enters only through the NRQCD matrix elements. In particular, the dependence on the polarization comes only from the matrix elements. In many cases, it is completely determined by the symmetries of NRQCD. The color evaporation model predicts that quarkonium states are always produced unpolarized. Both the color-singlet model and the NRQCD factorization approach give nontrivial predictions for polarized quarkonium. There have been many calculations of polarization effects in the color-singlet model. In most cases, the resulting predictions are not particularly dramatic. Below, we discuss several examples for which the calculations have have been extended to include color-octet mechanisms predicted by the NRQCD factorization formalism. In one case, we find a very dramatic prediction. \subsection{Spin alignment at the Tevatron} \noindent The NRQCD factorization framework has led to a dramatic change in our understanding of the production of charmonium at large transverse momentum in $p \bar p$ collisions\cite{B-F-Y}. As pointed out by Braaten and Yuan\cite{Braaten-Yuan} in 1993, the cross section for $p \bar p \to \psi + X$ at sufficiently large transverse momentum $p_T$ is dominated by gluon fragmentation. It can be factored into the cross section for producing a gluon with large transverse momentum and a fragmentation function: \begin{equation} d \sigma (p \bar p \to \psi(P) + X) \;=\; \int_0^1 dz \; d \hat{\sigma} (p \bar p \to g(P/z) + X) \; D_{g \to \psi}(z) \,. \label{dsig-frag} \end{equation} The fragmentation function $D_{g \to \psi}(z)$ gives the probability that the jet initiated by the gluon includes a $\psi$ carrying a fraction $z$ of the gluon momentum. Using the NRQCD factorization approach, the fragmentation function can be expressed in the form \begin{equation} D_{g \to \psi}(z) \;=\; \sum_{mn} d_{mn}(z) \langle {\cal O}_{mn}^H \rangle \,. \label{frag-fact} \end{equation} The matrix element that is leading order in $v$ is $|\langle \psi | \psi^\dagger \mbox{\boldmath{$\sigma$}} \chi | 0 \rangle|^2$, which scales like $v^3$. It has a short-distance coefficient of order $\alpha_s^3$. Using this term in the fragmentation function (\ref{frag-fact}), the cross section predicted by (\ref{dsig-frag}) is about a factor of 30 below recent data on prompt $\psi$ production at the Tevatron from the CDF detector\cite{Sansoni}. In 1995, Braaten and Fleming\cite{Braaten-Fleming} suggested that the gluon fragmentation function for the $\psi$ might actually be dominated by a term that represents a color-octet production mechanism. The matrix element is $\langle \chi^\dagger \sigma^k T^a \psi \; {\cal P}_\psi \; \psi^\dagger \sigma^k T^a \chi \rangle$, which is of order $v^7$ and measures the probability of producing a $\psi$ from a pointlike $c \bar c$ pair in a color-singlet $^3S_1$ state. The reason this matrix element might be important is that its short-distance coefficient is of order $\alpha_s$. The enhancement from the two fewer powers of $\alpha_s$ can overcome the suppression by $v^4$. The leading-order expression for this term in the fragmentation function is \begin{equation} D_{g \to \psi}(z) \;=\; {\pi \alpha_s \over 96 m_c^4} \delta(1-z) \; \langle \chi^\dagger \sigma^k T^a \psi \; {\cal P}_\psi \; \psi^\dagger \sigma^k T^a \chi \rangle \,. \label{D-psi} \end{equation} The $p_T$-dependence predicted by this mechanism is in agreement with the CDF data. The normalization depends on the unknown matrix element in (\ref{D-psi}). The value of the matrix element required to fit the CDF data is consistent with suppression by a factor of $v^4$ relative to the corresponding color-singlet matrix element $|\langle \psi | \psi^\dagger \mbox{\boldmath{$\sigma$}} \chi| 0 \rangle|^2$. Cho and Wise\cite{Cho-Wise} pointed out in 1995 that this production mechanism has dramatic implications for the polarization of the $\psi$. At leading order in $\alpha_s$, the $\psi$'s produced by gluon fragmentation will be 100\% transversely polarized. The radiative corrections were examined by Beneke and Rothstein\cite{Beneke-Rothstein}, who concluded that the spin alignment at large $p_T$ will remain greater than 90\%. The dominant corrections to the spin alignment at the values of $p_T$ measured at the Tevatron come from nonfragmentation contributions to the cross section\cite{Cho-Leibovich}. At large $p_T$, they fall like $1/p_T^2$ relative to gluon fragmentation. Unfortunately, the calculations required to obtain the prediction for the spin alignment as a function of $p_T$ have not yet been carried out. The predictions for the spin alignment of the $\psi'$ are identical to those for the $\psi$. The spin alignment for the $\psi'$ is easier to measure, because, in the case of the $\psi$, one has to take into account the effects of the radiative decays $\chi_{cJ} \to \psi \gamma$. It should be possible to make at least a crude measurement of the spin alignment of the $\psi'$ from existing CDF data. \subsection{Spin alignment in $Z^0$ decay} \noindent The color-singlet model predicts that the cross sections for producing charmonium in $Z^0$ decay are too small to be observed at LEP. As pointed out by Cheung, Keung, and Yuan and by Cho\cite{C-K-Y}, the dominant contribution comes instead from a color-octet production mechanism involving the same matrix element that appears in the fragmentation function (\ref{D-psi}). Using the value of this matrix element obtained by fitting the CDF data, they found that the rate is almost an order of magnitude larger than predicted by the color-singlet model and thus large enough to be observed. The spin alignment of the $\psi$ in $Z^0$ decay was also calculated by Cheung, Keung, and Yuan. Unfortunately, the alignment is predicted to be small and is completely unobservable in the present data sample from LEP. \subsection{Spin alignment in $B$ decay} \noindent In the production of $\psi$ from $B$ decay, color-singlet contributions are suppressed by a near cancellation between Wilson coefficients in the effective weak hamiltonian. Color-octet production mechanisms are therefore important, in spite of the $v^4$ suppression of the matrix elements\cite{K-L-S}. The spin alignment of the $\psi$'s that are produced in the decay $B \to \psi + X$ were recently calculated by Fleming et al.\cite{F-H-M-N}. It depends sensitively on the values of 3 independent color-octet matrix elements, including the one that appears in (\ref{D-psi}). A measurement of the spin alignment would therefore place strong constraints on these matrix elements. \section{Conclusions} \noindent The factorization formulas (\ref{dsig-fact}) and (\ref{dGam-fact}) provide a model-independent framework for analyzing heavy quarkonium production and annihilation rates. All the short-distance factors can be calculated systematically using the threshold expansion method. The long-distance factors are defined in terms of NRQCD matrix elements. The decay matrix elements can be computed using lattice simulations of NRQCD, but most of the production matrix elements must be treated as phenomenological parameters. The relative magnitudes of the matrix elements are predicted by the velocity-scaling rules of NRQCD. These magnitudes have a pattern that is very different from that assumed in the color-singlet model or in the color-evaporation model. The NRQCD factorization approach gives unambiguous predictions for the polarization of heavy quarkonium states. These predictions are inescapable consequences of this framework. The polarization predictions from NRQCD factorization can be dramatic. An example is the spin alignment of prompt $\psi$ and $\psi'$ at the Tevatron, which is predicted to be greater than 90\% at the largest values of $p_T$. An experimental measurement of this spin alignment would be a crucial test of the NRQCD factorization framework. \nonumsection{Acknowledgements} \noindent This work was supported in part by the United States Department of Energy, Division of High Energy Physics, under grant DE-FG02-91-ER40690. \nonumsection{References}

proofpile-arXiv_065-508

\section{Introduction} One of the most promising candidates for physics beyond the so-called Standard Model (SM) is that of supersymmetry (SUSY). In this paper we shall be concerned with the implications of a particular possible feature of SUSY, namely that of $R$-parity violation (RPV).\cite{rpv,barbm,suzuki} $R$-parity is a $Z\!\!\! Z_2$ symmetry of both the SM and its minimal SUSY extension, the MSSM, under which all of the SM particles have charge 0, while all their SUSY partners have charge 1. Its implications include the stability of the lightest supersymmetric particle (LSP), and hence the typical SUSY collider signatures of missing $E_T$ and the existence of a source of dark matter. Its violation changes both the implied cosmology and the expected collider signatures, allowing such effects as LSPs decaying inside the detector and leptoquarks. In addition to these, further constraints on RPV can be derived by considering experimental limits on rare decays.\cite{bgh,bpw,bgnn} $R$-parity is violated by the superpotential and soft potential terms \begin{eqnarray} W^{RPV}&=& \frac{1}{2}\l{ijk}L_iL_je_k + \lp{ijk}L_iQ_jd_k \cr && \quad + \frac{1}{2}\lpp{ijk}u_id_jd_k + \mu_iL_iH_2 \cr V^{RPV}_{\rm soft}&=& \frac{1}{2}\C{ijk}L_iL_je_k + \Cp{ijk}L_iQ_jd_k \cr && \quad + \frac{1}{2}\Cpp{ijk}u_id_jd_k + D_iL_iH_2 \cr && \qquad + m^2_{L_ih_1}L_iH_1^* + h.c. \end{eqnarray} From the point of view of deriving constraints on the $R$-parity violating couplings in the model, the most extensively studied couplings are the dimensionless couplings $\l{}$, $\lp{}$, and $\lpp{}$, which directly generate many effects which can be experimentally limited. The extra soft terms by definition mostly couple only heavy SUSY particles and hence are relevant mostly because of their impact on the RGEs, although they can have significant effects on the neutrino-neutralino and Higgs-sneutrino sectors.\cite{suzuki,rv,hemp,rpar} In SUSY models, flavour changing effects may be caused by the existence of off-diagonal terms in the sfermion mass matrices in the basis in which the fermion masses are diagonal. Such flavour-violating soft masses can be generated either from the high energy theory such as a GUT directly, or else through the RGEs by couplings which violate flavour symmetries, such as Yukawa couplings mediated by the CKM matrix or here RPV couplings. This paper is a short summary of work contained in two previous papers,\cite{rpar,qfv} in which we presented the renormalisation group equations (RGEs) for the couplings of the full $R$-parity violating sector of the model, and investigated the implications of typical scenarios at the GUT scale for the generation of neutrino masses and other flavour violation processes. \section{Effects of the RGEs} We begin with a brief discussion of the dimensionless couplings, whose RGEs have recently been presented in a number of papers.\cite{rpar,dimless} Here we present simple analytic solutions to the RGEs in the limit where the Yukawa couplings are much smaller than the gauge couplings \cite{rpar} leading to \begin{eqnarray} \lambda(M_Z) & = & 1.5\ \lambda(M_{GUT}) \nonumber \\ \lambda^{\prime}(M_Z) & = & 3.4 - 3.7\ \lambda^{\prime}(M_{GUT}) \label{LEHE} \\ \lambda^{\prime\prime}(M_Z) & = & 4.0 - 4.7\ \lambda^{\prime\prime}(M_{GUT}) \nonumber \end{eqnarray} where the ranges are caused by the error on $\alpha_3(M_Z)$. In addition to running themselves, the RPV couplings can alter the mass spectrum, giving quite a tight constraint on the $\lp{}$ couplings from the requirement that the sneutrino mass should be above its experimental limit,\cite{qfv} and can generate patterns of soft masses which violate flavour and lepton number symmetries. \section{Sneutrino VEVs} Sneutrino VEVs are an important signature of $R$-parity violation which arise because of the existence of $\mu_i$, $D_i$, and $m^2_{L_iH_1}$ terms which explicitly cause the effective potential to contain terms linear in the sneutrino field, either from explicit \cite{suzuki,hemp,lee} or spontaneous $R$-parity violation. They can also be caused by one loop effects involving dimensionless $R$-parity violating couplings,\cite{bgmt,enqvist} and by generation from the RGEs \cite{rpar} as discussed below. Once sneutrinos have acquired VEVs, neutrinos and neutralinos mix, so that we may derive bounds on $R$-parity violating terms by imposing experimental limits on neutrino masses, since in general we find that with $l_i$ as the VEV of $L_i$, a neutrino mass is generated is of order $(g_1^2+g_2^2)l_i^2/2M$, where $M$ is some typical neutralino mass. If we assume universal soft masses, then at the GUT scale we have only $R$-parity violation dimensionless and trilinear terms. The dangerous terms for generating sneutrino VEVs are then generated by the following terms mixing $L_i$ and $H_1$. \begin{equation} \begin{array}{cccl} \l{i33}h_\tau & \hbox{or} & \lp{i33}h_b & \hbox{generating $\mu_i$, $D_i$, $m^2_{H_1L_i}$} \\ \C{i33}h_\tau & \hbox{or} & \Cp{i33}h_b & \hbox{generating $D_i$} \\ \C{i33}\eta_\tau & \hbox{or} & \Cp{i33}\eta_b & \hbox{generating $m^2_{H_1L_i}$} \end{array} \end{equation} The effects are largest when $\tan\beta$ is large, but the dependence is rather complicated. The sneutrino VEV will in general be proportional to the $R$-parity violating coupling, and hence the neutrino mass to the coupling squared. We have performed a GUT scale analysis, setting universal parameters at the unification scale, together with some choice of GUT scale $R$-parity violating Yukawas, then running masses and couplings to low energy to give output. Unfortunately, the behaviour is a sufficiently complicated function of the many different parameters that it is not really possible to derive useful bounds on the couplings, but it is nonetheless possible to get an idea of the order of magnitude of the neutrino mass which we expect. For example we find that $\l{133}$ and $\lp{133}$ of order $10^{-3}$ and $10^{-4}$ are still large enough to be inconsistent with present experimental limits over much of parameter space.\cite{rpar} \section{Rare and Forbidden Processes} \subsection{$\mu\to e\gamma$} One of the most tightly bounded experimental constraints on flavour changing neutral currents is through the rare decay $\mu\to e\gamma$, forbidden in the SM. In SUSY models, a non-zero rate can be generated through non-diagonal slepton mass matrices and also through the direct effects of $R$-parity violating couplings.\cite{suzuki,lee,bgmt} However, as noted above, $R$-parity violation induces flavour violation through soft terms, and so here we shall consider the two effects together. We shall set only two $R$-parity violating dimensionless couplings non-zero at $M_{GUT}$ and see what effects they generate. We will be mainly interested in comparing the relatively simple ``direct'' contributions from diagrams which have $R$-parity violating couplings at the vertices, with the ``indirect'' contributions where the flavour violation is driven by off-diagonal mass insertions $\Delta m^2$ generated through the RGEs. As an example we consider the impact of $\lp{111}(M_{GUT})=\lp{211}(M_{GUT})=0.001$. We show the resulting contributions to the amplitude as a function of $M_{1/2}$ in Figure~\ref{oldfig9}. Here we have set $\tan\beta=10$, $m_0=100$GeV, $A_0=0$, $\mu_4>0$. What is remarkable about this figure is that it is clear that in this case the direct contributions are completely negligible relative to those from the chargino and neutralino mediated diagrams. \begin{figure*}[htb] \center \psfig{figure=oldfig9.ps,height=10cm} \caption{ Absolute values of amplitudes for $\mu\to e\gamma$ from direct $R$-parity violation diagrams (dashed lines), neutralino mediated diagrams (dot-dashed lines), and chargino mediated diagrams (dotted lines) plotted against $M_{1/2}$. We also show the total amplitude (solid line) and the experimental bound on the amplitude (horizontal solid line). Parameters are $m_t=175$GeV, $\alpha_3(M_Z)=0.12$, $\tan\beta=10$, $m_0=100$GeV, $A_0=0$, $\mu_4>0$, and $\lp{111}(M_{GUT})=\lp{211}(M_{GUT})=0.001$. } \label{oldfig9} \end{figure*} We conclude this section by summarising our results. Firstly we find rather different behaviour for the three different scenarios of non-zero $\lp{}$, $LH$ $\l{}$ (where the flavour violation occurs in the left handed slepton sector through $\l{1ij}\l{2ij}$) and $RH$ $\l{}$ (where it occurs in the right handed slepton sector through $\l{ij1}\l{ij2}$). For the $\lp{}$ case, the chargino and neutralino mediated diagrams with flavour violation through soft mass insertions dominate completely the direct contributions, giving very much tighter constraints, particularly for large $\tan\beta$. For the $LH$ effects due to $\l{}$ couplings we find again that the chargino contribution dominates, but not overwhelmingly, and there can be large cancellations. For the $RH$ case there are no chargino contributions, and the neutralino and direct effects are usually of comparable size and opposite sign. However, since there are so many possible cancellations between terms, it is essentially impossible to derive concrete bounds. The strongest reasonable statement is that, for the values we have considered for pairs of couplings at $M_{GUT}$ of $\lambda\lambda\simeq 10^{-4}$ and $\lp{}\lp{}\simeq 10^{-6}$ we expect contributions of order the experimental limit for a very light spectrum, with the branching ratio scaling as $\lambda^4$ or $\lp{}^4$ respectively. \subsection{$b\to s\gamma$} Another process which has been studied in the context of constraining flavour violation in SUSY theories is that of $b\to s\gamma$. Here we find that the indirect effects again often dominate the direct ones. However, the bounds on couplings derived here are quite weak, since $b\to s\gamma$ is much harder to constrain as the large SM contribution complicates matters, and indeed we find that the bounds on $\lp{}$ are weaker than those derived from requiring the sneutrino mass to be above its experimental limit, while those on $\lpp{}$ are only of order 0.2 for the relevant product with a very light spectrum.\cite{qfv} \subsection{$K^0-\bar K^0$ Mixing} The final process which we shall consider is that of $K^0-\bar K^0$ mixing, the direct contributions to which have been extensively studied.\cite{barbm,crs} Here we find a complete contrast to the situation for the other processes, in that even when the large tree level contribution is neglected the indirect contributions are always smaller than the direct ones.\cite{qfv} \section{Conclusion} The full RGEs for the MSSM with $R$-parity violation with the inclusion of all soft terms as well as all dimensionless couplings lead to some interesting physics. The most important effects of including $R$-parity violating couplings at the unification scale are those associated with flavour violation, both through ``direct'' terms where these couplings appear at the vertices of the diagrams, and ``indirect'' terms where they generate off-diagonal soft masses through the RGEs which then generate effects through one loop diagrams. The inclusion of $R$-parity violation in our superpotential through dimensionless terms allows the generation of lepton-Higgs mixing which leads to sneutrino VEVs and hence neutrino masses. We have shown that the indirect generation of sneutrino VEVs through the running of the RGEs for the soft terms often leads to larger effects than those derived directly from one loop diagrams. Typically we find that values of $\l{i33}$ and $\lp{i33}$ of order $10^{-2}$ and $10^{-3}$ respectively at the GUT scale give masses to the corresponding neutrino of order hundreds to thousands of eV, although the exact value is quite dependent on the unification scale parameters, and these form the tightest constraints on these couplings which have been derived. Similarly, we have studied the process $\mu\to e\gamma$, which we have shown to be very strongly affected by chargino and neutralino mediated diagrams. These typically dominate the direct contributions which had already been calculated, often by several orders of magnitude for the case of the $\lp{}$ couplings, but there are strong cancellations so that it is not possible to give precise bounds on the couplings from such processes. However, unless we invoke arbitrary cancellations, the typical size of such indirect effects on FCNC are likely to be the dominant constraint on the building of a model with non-zero $R$-parity violating couplings. In comparison, bounds derived from $b\to s\gamma$ are extremely weak except where the spectrum is already experimentally ruled out, while for $K^0-\bar K^0$ mixing the direct contributions dominate. Our main conclusion from these calculations is that $R$-parity violation can generate large flavour violating effects through the running of the dimensionful RGEs, and that these effects are often much larger than those which are generated directly by the couplings themselves, so that merely studying diagrams with $R$-Parity violating vertices can be very misleading. \section*{References}

proofpile-arXiv_065-509

\section{Introduction} Borel transforms provide a convenient and insightful method of describing the large-order behaviour of QCD perturbation series. An observable \begin{equation} \label{series} R(a) = \sum_{n=0}^{\infty} r_n a^{n+1}, \qquad a \equiv \alpha_s /\pi \end{equation} is represented as the integral \begin{equation} \label{transform} R(a) = \int_0^{\infty} F(z) e^{-z/a} dz \end{equation} where $F(z)$ is the transform. (If necessary, principal values are used to define the integral -- see Appendix). A singularity of the form \begin{equation} \label{renorm} F(z) \sim {A \over (1- \alpha z)^{\beta}}, \qquad z \sim 1/\alpha \end{equation} gives rise to a factorial divergence \begin{equation} \label{fac} r_n \sim n ! A {\alpha}^n {\beta (\beta+1) \ldots (\beta+n-1) \over n!} \end{equation} in the perturbation expansion of $R(a)$. Much attention has been devoted to characterising the $F(z)$ of QCD observables, both in general and for particular cases. Arguments related to power corrections show that a typical observable has an infinite chain of singularities of the form (\ref{renorm}), equally spaced along the positive real axis of the Borel $z$-plane \cite{poles}\cite{mueller}. These are called IR renormalons and are crucial is assessing the practical reliability of perturbation theory. Although the positions and general form of the renormalons are fixed for any $R(a)$ by these arguments, the particular details (especially the $A$ and $\beta$ of a singularity) have to be established for each observable separately. Such work has both reproduced the predicted general features and provided specific information about $F(z)$ in cases of interest, see e.g. \cite{broadhurst}-\cite{stein}. A notable feature of these results is that $F(z)$ is invariably a simpler function than $R(a)$. However to fully exploit the potential of these results it is often necessary to consider functions related to $R(a)$ rather than the calculated $R(a)$ itself. For example, $R(a)$ and its Borel transform may have been calculated using the $\overline{MS}$ renormalisation scheme, but it is now the corresponding results in the effective charge scheme \cite{EC} that are wanted. The function of interest is now the effective charge $\beta$-function \begin{equation} \label{beta} \rho(R) \equiv {\beta(a(R)) \over da/dR} \end{equation} where $\beta(a)$ is the $\overline{MS}$ $\beta$-function. Calculating this involves doing a series of operations to the function $R(a)$: inverting it to give $a(R)$, finding the function-of-a-function $\beta(a(R))$, differentiation and division. In terms of the function these are straightforward enough operations, though possibly difficult to do exactly in practice. A natural question -- particularly if, as noted, the observable is simpler as a transform than as a function of $a$ -- is what effect these operations have in the Borel plane ? If we know the Borel transform of $R(a)$, what does this tell us about the transform of $a(R)$ ? Because such operations are commonplace, questions like this arise continually when applying the results of renormalon calculations. It is this practical need that the method outlined here seeks to meet. \section{The Operations} The operations that this paper will discuss are \begin{enumerate} \item Multiplication $R_1(a) R_2 (a)$ \item Differentiation $dR/da$ \item Function-of-Function $R_1 (R_2 (a))$ \item Division $R_1 (a)/ R_2 (a)$ \item Inversion $a(R)$ \end{enumerate} Some of these are easy and some are hard. The first two are the straightforward ones. Like with any simple integral transform, the Borel transform of a product is given by a convolution integral over the transforms of the factors. If $R_1 (a)$ and $R_2 (a)$ have transforms $F_1 (z)$ and $F_2 (z)$, then $R_1 (a) R_2 (a)$ has a transform \begin{equation} \label{conv} F_3 (z) = \int_0^z F_1 (u) F_2 (z-u) du. \end{equation} This will be true even if the transforms contain renormalon singularities (see Appendix). As noted by 't Hooft \cite{thooft}, the presence of such singularities in the factors leads to similar singularities in the product as a result of (\ref{conv}). This aspect is discussed in more detail below. Indeed much of this paper can be considered a generalization of this observation about the convolution integral. The other straightforward operation is differentiation, since the transform of $a^n d^n R/ d a^n$ is \begin{equation} \label{diff} {d \over dz} \biggl( z^n {d^{n-1} F \over d z^{n-1}}\biggr). \end{equation} It is possible to remove powers of $a$ from $a^n d^n R/da^n$ by using \begin{equation} \label{arg} {R(a) \over a} = F(0) + \int_0^{\infty} e^{-z/a} F'(z) dz. \end{equation} The $F(0)$ piece here corresponds to a delta function term in the transform which we will seek to avoid. Thus (\ref{arg}) will only be used in this paper on functions for which $F(0)=0$. The remaining operations 3-5 are all intrinsically difficult ones. By this we mean that it is unlikely that closed-form results will ever be found for their effect on transforms, in the way that (\ref{conv}) and (\ref{diff}) exist for multiplication and differentiation. Even with a simple example like $F(z) = 1/(1-z)$, it appears that they can produce a transform that is much more complicated in its fine detail than the input one. Our simple and natural question does not have a simple and natural full answer. However the issue of practicality remains and for this a complete answer to the problem is not necessary. For the effective use of current and foreseeable QCD calculations it will be sufficient to extract only the more important effects that the use of these operations will have on the transforms. Furthermore, by concentrating on the main effects, a simple and natural picture does emerge. The general question of the effect of the operations 3-5 on Borel summable functions has been addressed before, in a pair of papers by Auberson and Mennessier \cite{AM}. Their main conclusion is that the functions thus produced are also Borel summable. This result is clearly a necessary starting point for our investigation in that it establishes that the transforms we will be trying to describe actually exist. In general \cite{AM} provides a firm foundation of rigour that will be relied upon implicitly throughout what follows. As a consequence of the emphasis on extracting information that can be used in practical QCD contexts, the approach and language used here is however very different than in \cite{AM}. \section{The $\lambda$-Expansion} As a concrete example, consider operation 5, inverting a function $R(a)$ to $a(R)$. The central idea of our approach is to introduce the following split in the initial function \begin{equation} \label{split} R(a) = a + \lambda \tilde R (a). \end{equation} The parameter $\lambda$ is a bookkeeping device which will be set equal to one at the end. Thus \begin{equation} \label{tilde} \tilde R (a) = \sum_{n=1}^{\infty} r_n a^{n+1} \end{equation} is simply all of $R(a)$ except for its lowest order term. Of course (\ref{split}) really defines a function $R(a,\lambda)$ such that $R(a)=R(a,1)$. And if we are starting from a QCD observable, $R(a)$ will be renormalisation scheme invariant, but $R(a,\lambda \neq 1)$ will not be. But these nuances are unproblematical and the $\lambda$-dependence of $R(a,\lambda)$ will be suppressed in the notation. It is also possible to think of $a(R)$ as now having a $\lambda$-dependence, though one more complicated than that in (\ref{split}). Expand $a(R)$ as an expansion in $\lambda$. \begin{equation} \label{taylor} a(x) = x + \lambda {\partial a \over \partial \lambda}\bigg|_{\lambda=0} + {\lambda^2 \over 2} {\partial^2 a \over \partial \lambda^2} \bigg|_{\lambda=0} + \ldots \end{equation} The coefficients are functions of $x$ and can be found by repeatedly differentiating (\ref{split}) rewritten as \begin{equation} \label{split2} x= a(x) + \lambda \tilde R (a(x)) \end{equation} with respect to $\lambda$. Thus the first differentiation gives \begin{equation} \label{diff2} 0= {\partial a \over \partial \lambda} + \tilde R (a) + \lambda {\partial \tilde R \over \partial a} {\partial a \over \partial \lambda} \end{equation} and so \begin{equation} \label{first} {\partial a \over \partial \lambda}\bigg|_{\lambda=0} = - \tilde R (x). \end{equation} The first few terms of (\ref{taylor}) are found to be \begin{eqnarray} a(x) = x & - & \lambda \tilde R (x) + \lambda^2 \tilde R (x) {d \tilde R \over dx} \nonumber \\ & - & {\lambda^3 \over 2} \biggl( {\tilde R}^2 {d^2 \tilde R \over dx^2} + 2 \tilde R \biggl( {d \tilde R \over dx} \biggr)^2 \biggr) + \ldots \label{expan} \end{eqnarray} This will be called the $\lambda$-expansion of $a(R)$. Given that all the other perturbative expansions in this paper are divergent, it is important to emphasize that the $\lambda$-expansion is usually convergent as a series in $\lambda$. A detailed discussion of this issue is deferred \cite{BM}, but heuristic arguments can be offered for (\ref{expan}) being convergent. First, note that so far nothing has been assumed about the convergence properties of (\ref{tilde}). $\tilde R (a)$ can be as favourable and well-behaved a function as required and (\ref{expan}) will still be derived. As a relation between $\tilde R(x)$ and $a (x)$ as functions, the structure of the $\lambda$-expansion is independent of their behaviour as expansions in $x$. In this light, there is no {\it a priori} reason to expect (\ref{expan}) to be especially problematic for the cases of interest. Secondly, when $\tilde R (x)$ is divergent, the resulting divergence of $a(x)$ as a series in $x$ can be thought of as having been absorbed into the $\tilde R(x)$ that appear in (\ref{expan}) to leave a convergent expression. Thirdly, consider the function \begin{equation} \label{yfun} \lambda(y) = \lambda(y,x) \equiv {y \over \tilde R (x-y)} \end{equation} and its inverse $y(\lambda) = y(\lambda,x)$ for a fixed $x$. $\tilde R (a)$ is expected to have one cut in the complex $a$-plane, along the negative real axis. Thus $\tilde R (x-y)$ is analytic about $y=0$ for all $x > 0$ and $\lambda(y)$ will converge for $|y| < x$. Hence its inverse, $\lambda(y)$, has a non-zero radius of convergence \cite{Bromwich}, but since \begin{equation} \label{aasy} a(x) = x - y(\lambda,x), \end{equation} so does (\ref{expan}). Clearly none of these arguments is sufficient to prove that the $\lambda$-expansion is convergent for $\lambda=1$, as required. Pending the detailed discussion \cite{BM}, the issue is set aside. For present purposes, we merely note that in the results presented here the expansions actually have infinite radii of convergence. Turning to the use of (\ref{expan}), for a given $R(a)$ and hence $\tilde R (a)$, setting $\lambda=1$ in (\ref{expan}) provides a systematic means of calculating $a(R)$ from $\tilde R (a)$. It is a method of inverting functions. Its importance as such is that it serves to reduce the hard operation of inversion to a sequence (albeit infinite) of easy ones, namely differentiation and multiplication. Given the Borel transform of $R(a)$, (\ref{conv}) - (\ref{arg}) can be used to calculate the Borel transform of any particular term in the $\lambda$-expansion and then these contributions summed to obtain the Borel transform of $a(R)$. \section{Calculating the Transform} The simplest case is where the transform of $R(a)$ (and thus $\tilde R (a)$) has a single singularity at $z=1/\alpha$ such that \begin{equation} \label{input} F(z) \sim {A \over (1-\alpha z)^{\beta}}, \qquad z \sim 1/\alpha. \end{equation} The generalization to multiple poles, as required for realistic QCD examples, will be straightforward. Neglecting numerical factors, the general term in (\ref{expan}) is \begin{equation} \label{general} \lambda^n {d^{q_1} \tilde R \over dx^{q_1}} \ldots {d^{q_n} \tilde R \over dx^{q_n}} \end{equation} \begin{equation} \label{constraint} q_1+ \ldots q_n = n-1. \end{equation} What is the transform of this, given an $\tilde R$ implied by (\ref{input}) ? Using (\ref{diff})-(\ref{arg}) this is easily found in principle, but for any particular $F(z)$ that accords with (\ref{input}) the convolution integrals quickly become impossible to evaluate exactly. It becomes necessary to follow only the more important features of the transforms through the calculation. Note that if $F(z)$ in (\ref{input}) has another singularity at $z=1/\alpha$ with $\beta'=\beta-1$, then according to (\ref{fac}) the additional contribution to the factorial divergence of the coefficient $r_n$ is suppressed by $1/n$ and so can be neglected at large orders. All singularities at $z=1/\alpha$ with smaller $\beta$ are similarily sub-leading in $r_n$. The singularity with largest $\beta$ at $z=1/\alpha$ will be called the dominant singularity, the others sub-dominant. It is the dominant singularities (possibly at different $z=1/\alpha$) that will be most important in practice. Furthermore, for the operations 1-5 it turns out that sub-dominant terms in the initial transform only give rise to sub-dominant terms in the result. From now on the sub-dominant singularities will be neglected. The features of the transforms that are to be tracked through the calculation will be their behaviour close to $z=0$, the positions of the poles and the dominant behaviour there. For any transform of interest, these can be summarised thus \begin{equation} \label{form} F_i (z) \sim \left\{ \begin{array}{cl} z^{m_i}, & z \sim 0 \\ A_i (1-\alpha z)^{-\beta_i}, & z \sim 1/\alpha \end{array} \right. \end{equation} for an input transform like (\ref{input}). Note that all of these are also only singular at $z=1/\alpha$. Why is this ? Firstly, differentiation of $\tilde R$ and the use of (\ref{diff}) cannot cause the transform to become non-analytic at any other point. Sub-dominant terms also remain sub-dominant. Secondly, multiplication and the use of the convolution integral (\ref{conv}) have much the same effect. Consider where $F_1(z)$ and $F_2(z)$ in (\ref{conv}) are of the form (\ref{form}). How does $F_3(z)$ behave ? For $z < 1/\alpha$ the integrand is finite and so is $F_3(z)$. But as $z \rightarrow 1/\alpha$ the integrand begins to diverge at both ends of its interval and hence so can $F_3(z)$ as $z \rightarrow 1/\alpha$. For $z > 1/\alpha$, a principal value is taken in the integral where necessary (see Appendix) and $F_3(z)$ is finite. The only point where $F_3(z)$ is non-analytic is $z=1/\alpha$. This is essentially the observation made by 't Hooft \cite{thooft}. Furthermore, if $F_1(z)$ and $F_2(z)$ are as in (\ref{form}), so is $F_3(z)$, but with \begin{equation} \label{mmm} m_3 = m_1 + m_2 + 1 \end{equation} \begin{equation} \label{bbb} \beta_3 = \left\{ \begin{array}{lll} -\beta_1+m_2+1 & \hbox{if} & -\beta_1+m_2+1<-\beta_2 +m_1+1 \\ -\beta_2+m_1+1 & \hbox{if} & -\beta_2+m_1+1<-\beta_1 +m_2+1 \end{array} \right. \end{equation} One has to know the behaviours near $z=0$ because when one of the transforms in the integrand of (\ref{conv}) is diverging to give the divergence in $F_3(z)$, the other transform's argument is tending towards $z=0$. All this is neglecting the sub-dominant terms. However, such terms in either $F_1(z)$ or $F_2(z)$ do not produce dominant terms in $F_3(z)$ as a result of this convolution; they are safely neglected. Since differentiation and multiplication are the only operations involved in finding (\ref{general}), its transform and any intermediate ones involved in finding it will indeed thus behave as (\ref{form}). In addition, one can use (\ref{diff}), (\ref{mmm}) and (\ref{bbb}) to find the $A_i$, $\beta_i$ and $m_i$ of these transforms. Most of the terms (\ref{general}) turn out not to contribute to the dominant singularity in the transform of $a(R)$. At each order in $\lambda$ there is one term of the form \begin{equation} \label{main} \lambda^n {\tilde R}^{n-1} {d^{n-1} \tilde R \over dx^{n-1}} \end{equation} and it is only these that contribute to this singularity. Working through the (tedious) details and summing the $\lambda$-expansion one finds that, for $\tilde R(a)$ given by (\ref{input}), the dominant part of the transform for $a(R)$ is \begin{equation} \label{answer} F(z) \sim {- \lambda A e^{-\lambda r_1 /\alpha} \over (1 - \alpha z)^{\beta} }, \qquad z \sim 1/\alpha, \end{equation} where $r_1$ is the one-loop coefficient of $R(a)$. Finally, $\lambda$ is set equal to one. The main effect that inversion has had on the transform is thus to change the overall constant (the $A_i$ in (\ref{form})). The $\beta_i$ doesn't change; this need not be the case for other operations. Most importantly, the position of the singularity hasn't changed. In retrospect this is an obvious consequence of the non-obvious fact that inversion can be reduced to multiplication and differentiation. Additional singularities at other positions do not change this basic picture. If (\ref{input}) is generalised to \begin{equation} \label{string} F(z) \sim \sum_m {A_{m} \over (1- \alpha_m z)^{\beta_m}}, \qquad z \sim 1/\alpha_m, \end{equation} (\ref{answer}) becomes \begin{equation} \label{strans} F(z) \sim \sum_m {-A_{m} e^{-r_1/\alpha_m} \over (1-\alpha_m z)^{\beta_m}}, \qquad z \sim 1/\alpha_m. \end{equation} The additional complications (\ref{string}) introduces are all sub-dominant in (\ref{strans}). The idea of a split (\ref{split}) in one of the functions leading to a $\lambda$-expansion also renders operations 3 and 4 tractable. Indeed in these cases the $\lambda$-expansion appears much more familiar. \begin{equation} \label{funfun} R_1(x + \lambda \tilde R_2(x)) = \sum_{n=0}^{\infty} {\lambda^n \over n!} {\tilde R_2 (x)}^n {d^n R_1 \over dx^n} \end{equation} \begin{equation} \label{quot} {R_1(x) \over x+\lambda \tilde R_2(x)} = {R_1(x) \over x} \sum_{n=0}^{\infty} \biggl( - {\lambda \tilde R_2(x) \over x} \biggr)^n. \end{equation} However since these operations involve two functions, the details and the results are contingent on the specifics of two input transforms and a discussion is again deferred \cite{BM}. A full set of results covers the situations one is liable to be confronted with in practice. A general feature is however clear. Because all these operations can be reduced to multiplication and differentiation, if the initial transforms have the renormalon structure predicted by QCD \cite{poles}\cite{mueller}, so do the transforms produced by the operations. The universality of that structure is further confirmed: all QCD observables look the same in the Borel plane. $$ $$ {\bf Acknowledgements} $$ $$ Chris Maxwell is thanked for useful comments and invaluable encouragement. \section*{Appendix} Although it is a standard theorem \cite{Hardy} that the Cauchy product of two Borel summable series is also Borel summable, the result for products required here is that the Borel transform of the product is given by the convolution (\ref{conv}), even when singularities are present. Since we know of no proof of this in the literature, one is outlined here. Consider two transform integrals \begin{equation} \label{defs} f(a) = \int_0^{\infty} F(z) e^{-z/a} dz, \qquad g(a) = P \int_0^{\infty} G(z) e^{-z/a} dz, \end{equation} where $F(z)$ is analytic, but $G(z)$ has a singularity at $z=\eta$. The $P$ indicates a principal value. The convolution integral will thus be \begin{equation} \label{cint} H(z) = P \int_0^z G(w)F(z-w) dw. \end{equation} Now \begin{eqnarray} h(a) & = & g(a)f(a) \label{m1} \\ & = & \lim_{\epsilon \rightarrow 0} \biggl( \int_0^{\eta-\epsilon} G(z)e^{-z/a} dz + \int_{\eta+\epsilon}^{\infty} G(z)e^{-z/a} dz \biggr) f(a) \label{m2} \\ & = & {\overline h}(a) + \lim_{\epsilon \rightarrow 0} \int_0^{\eta-\epsilon} dz G(z) e^{-z/a} \int_{\eta-z-\epsilon}^{\eta-z+\epsilon} dw F(w) e^{-z/a}. \label{m3} \end{eqnarray} where \begin{equation} \label{m4} {\overline h}(a) \equiv P \int_0^{\infty} H(z) e^{-z/a} dz. \end{equation} It is the final term in (\ref{m3}) that is at issue. The details of whether or not it tends to zero depend on the specific $G(z)$. It suffices to consider whether \begin{equation} \label{eps} \epsilon \int_0^{\eta-\epsilon} G(z) dz \end{equation} vanishes as $\epsilon \rightarrow 0$. If it does, then the natural generalisation of the convolution theorem, namely that $h(a) = {\overline h}(a)$, is true. If, as if in QCD, \begin{equation} \label{type} G(z) \sim (\eta - z)^{-\beta}, \qquad z \sim \eta, \end{equation} then (\ref{eps}) vanishes only if $\beta<2$. However the naive principal value definition in (\ref{defs}) only holds for $\beta \leq 1$ anyway. For $\beta >1$, one can define the transform via \begin{equation} \label{altdef} g(a) = a^{-n} P \int_0^{\infty} {\overline G}(z) e^{-z/a}dz \end{equation} for some $n$ such that ${\overline G}(z)$ has a singularity with $\beta'=\beta-n<1$. In this paper, the transforms are implicitly defined like this. For clarity, the $a^{-n}$ factors are suppressed and $\beta>1$ ia allowed, but with care it can be arranged such that $\beta<1$ in all transforms. The convolution theorem thus holds for all products considered here. The generalization to both $F$ and $G$ having multiple renormalon singularities is straightforward. Principal values have been used to define the Borel integrals because the generalization of the theorem, (\ref{cint}) and (\ref{m4}), is then particularly natural. However it is more common in QCD to define Borel integrals using contours that detour around singularities. These versions have the disadvantage here that the contours required for the equivalents of (\ref{cint}) and (\ref{m4}) are then not obvious. \newpage

proofpile-arXiv_065-510

\section{MAX} The Millimeter wave Anisotropy eXperiment (MAX) was a balloon borne experiment that measured the cosmic microwave background anisotropy (CMBA) on half degree angular scale from 1989 to 1994. It was a collaboration between groups in the University of California, Santa Barbara and Berkeley. Between 1989 and 1994 the instrument was launched 5 times and scanned 9 regions of the sky for CMBA. First detection of CMBA signals was reported by Alsop {\it et al.\ } \cite{alsopetal92}. Table 1 summarizes the flights, the regions scanned, flat band power results, and CMBA papers published. \subsection{Overview of the MAX experiment} \label{The MAX Instrument, Observing Strategy and Window Function} Several papers describe the MAX instrument \cite{alsopetal92,fischeretal92,meinhodletal93b}, here we only summarize selected aspects of the experiment. Particular details of the experiment, e.g. exact beam size, frequency bands, and bolometer temperature, where modified during the duration of the program. The numbers that will be quoted here refer to the last flight of the program, MAX-5, unless otherwise noted. MAX had a single pixel photometer at the focal plane of an off-axis Gregorian telescope. The telescope provided a beam of 0.5 degree FWHM. The beam was split inside the photometer to 4 frequency bands by means of dichroic mesh-filters. During an observation the beam was modulated on the sky with two frequencies. The secondary mirror modulated the beam sinusoidally, in cross-elevation direction, at a frequency of 5.4 Hz and amplitude of 1.4 deg. Simultaneously the entire gondola was scanned in azimuth at constant velocity at an amplitude of 4 degrees and a frequency of 0.0075 Hz. The center of the gondola scan tracked the location of a bright star for the duration of the observation. The fast secondary mirror chop provided effective discrimination against low frequency electronic noise which had a $1/f$ knee at $\sim 3$ Hz. The slow scan enabled the subtraction of temporal variations in the bolometer temperature and the atmosphere brightness. The MAX detectors were composite bolometers operating at 300 mK for the first three flights and at 85 mK for the last two. The MAX-5 bolometer for the 450 GHz band was background limited. At the lower frequency bands phonon and Johnson noise was dominant. During the duration of the program sensitivity to CMB temperature differences improved significantly. For example, the sensitivity of the 180 GHz channel improved by a factor of 10 from $\sim 2$ mK$\sqrt{\mbox{sec}}$ \cite{fischeretal92} to 0.24 mK$\sqrt{\mbox{sec}}$ \cite{limetal96}. The optical chop at $\sim 5$ Hz, which was chosen as an optimum in the trade-off between the bolometer time constants and the onset of low frequency noise, determined MAX's $l$ space coverage to a single window function. The window function peaked at $l = 150$ and had half power points at $l= 72$ and $l=248$. MAX was calibrated by observing a planet once during the flight and then using a partially reflecting membrane as a transfer standard for additional periodic calibrations. The typical absolute calibration error was 10\% which was dominated by uncertainties in the brightness temperature of the planets observed. \subsection{MAX results} MAX provided seven detections of CMBA signals at an angular scale of $\sim 0.5$ degrees. For these seven detections the wide frequency coverage, up to 4 channels between 90 GHz\ and 450 GHz, enabled unambiguous spectral discrimination against emission from galactic dust. Extrapolation of the fluctuations observed in the 408 MHz Haslam map to the MAX frequency bands, using the expected spectral dependence of either synchrotron or Bremstrahlung radiation, yields a fluctuations' amplitude of typically less than 10\% of the amplitude observed. Thus it is unlikely that synchrotron or Bremstrahlung are the dominant source of the detected fluctuations. Searches in available catalogs found no sufficiently intense radio sources in the regions observed. The treatment of potential temporal variations in the signal due to atmosphere variability, beam motion relative to the balloon or earth, moon location, etc. are discussed in the references mentioned in Table 1. Two measurements near the star $\mu$-Pegasi, where 100$\mu$ IRAS maps indicate significant dust contrast, were expected to reveal dust signals. Indeed, the dust signature detected was morphologically consistent with IRAS. Only upper limits on the CMB fluctuation power were derived in these regions. Most cosmological models predict an increase in the power spectrum of the CMB fluctuations near the peak of MAX's window function. The MAX results are suggestive of a combined flat band power larger than that detected by the COBE/DMR experiment. Statistical analysis to combine the seven detections to a single estimate of the CMB fluctuation power within MAX's window function is in progress. \begin{table}[htb] \label{table max summary} \centerline{ \begin{tabular}{|| l | c | l ||} \hline Flight/Year/ Region Observed & $(\Delta T/T)_{flat}$ & Publication \\ \hline \hline MAX--1 / 1989 & & Fischer {\it et al.\ } 1992 \cite{fischeretal92} \\ \hline MAX--2 / 1990 / GUM$^{*}$ & $2.9^{+4.3}_{-1.8}$ & Alsop {\it et al.\ } 1992 \cite{alsopetal92} \\ \hline MAX--3 / 1991 / GUM$^{*}$ & $2.7^{+1.1}_{-0.7}$ & Gundersen {\it et al.\ } 1993 \cite{gundersenetal93} \\ \hline MAX--3 / 1991 /$\mu$-Pegasi & $<$ 1.6 & Meinhold {\it et al.\ } 1993 \cite{meinholdetal93} \\ \hline MAX--4 / 1993 / GUM$^{*}$ & $2.0^{+0.6}_{-0.4}$$^{**}$ & Devlin {\it et al.\ } 1994 \cite{devlinetal94} \\ \hline MAX--4 / 1993 / $\sigma$-Herculis & $1.8^{+0.8}_{-0.6}$$^{**}$ & Clapp {\it et al.\ } 1994 \cite{clappetal94} \\ \hline MAX--4 / 1993 / $\iota$-Draconis & $1.9^{+0.7}_{-0.4}$$^{**}$ & Clapp {\it et al.\ } 1994 \cite{clappetal94} \\ \hline MAX--5 / 1994 / HR5127 & $1.2^{+0.4}_{-0.3}$ & Tanaka {\it et al.\ } 1996 \cite{tanakaetal96} \\ \hline MAX--5 / 1994 / $\phi$-Herculis & $1.9^{+0.7}_{-0.4}$ & Tanaka {\it et al.\ } 1996 \cite{tanakaetal96} \\ \hline MAX--5 / 1994 / $\mu$-Pegasi & $ < 1.3 $ & Lim {\it et al.\ } 1996 \cite{limetal96} \\ \hline \end{tabular} } \caption{Summary of MAX results. Values of $\Delta T /T$ are for flat band $\langle {l(l+1) C_{l} \over 2\pi} \rangle^{1/2} $, 95\% confidence interval. ($^{*}$) GUM stands for the region near the star Gamma Ursa Minoris. ($^{**}$) Original results were revised as described by Tanaka {\it et al.\ } (1996). } \end{table} \section{MAXIMA} The goal of next generation experiments is to make precise measurements of the CMBA power spectrum. Theoretical work within the last several years has demonstrated that the optimal observing strategy to constrain the power spectrum, in the absence of systematic errors or foregrounds, is to observe as many sky pixels as possible with modest ($\sim 1$) signal to noise per pixel \cite{knox95}. It has also been argued that small to intermediate scale measurements, at $100 \mathrel{\mathpalette\mathrelfun <} l \mathrel{\mathpalette\mathrelfun <} 1500$ covering the region where CDM models predict adiabatic peaks, could discriminate between various cosmological models \cite{whitehumoriond,joaoalbrecht96} and provide information about the cosmological parameters independent of the underlying cosmological model \cite{huwhitemoriond}. The Millimeter wave Anisotropy eXperiment Imaging Array (MAXIMA) was designed to address these scientific requirements by scanning many pixels on the sky within a single flight, providing large $l$ space coverage and high $l$ resolution, while improving on the systematic-error rejection achieved for MAX. MAXIMA\ is a balloon borne program designed to constrain the CMBA power spectrum on a range of angular scales between $l \sim 60$ and $l \sim 650$. It is a collaboration between groups at the University of California, Berkeley, University of Rome, IROE -- Florence, Queen Mary and Westfield College -- London, and the California Institute of Technology. \subsection{Experimental Configuration} MAXIMA will observe 14 sky pixels simultaneously with 0.18 degree FWHM beams. The attached Figure shows the experiment, the focal plane and its orientation on the sky. The 14 single frequency photometers detect radiation in three frequency bands centered around 150 GHz, 240 GHz, and 420 GHz. The bolometers will be maintained at $100$ mK to provide high sensitivity and short time constants. The experiment is designed for up to 24 hour of observations and it will fly in north America. \begin{figure} \centerline{\psfig{file=cryosta4.plt,height=7.5in}} \label{figure maxima experiment} \end{figure} \subsubsection{optics} The optical system is a three mirror off-axis f/1.8 Gregorian telescope. The primary mirror is a $1.3$ meter diameter off axis section of a parabola. The secondary and tertiary mirrors (21 cm and 18 cm in diameter respectively) are conic sections with aspheric components which compensate the aberrations introduced by the primary mirror. The secondary and tertiary mirrors and a Lyot stop are mounted in a well baffled box inside the cryostat and are cooled to liquid helium temperature. The cold Lyot stop provides excellent sidelobe rejection and cooling the secondary optics reduces the optical loading on the bolometers. A three mirror system was designed to provide for a diffraction limited $\sim 1 \times 1$ deg$^2$ field of view at 150 GHz. The secondary and tertiary mirrors are fixed and the light, 11 kg, primary mirror can be modulated around the optical axis of the telescope (the line connecting the center of the primary and the center of the secondary). \subsubsection{Cryogenics, Detectors and Electronics} The cryostat was designed for a north-American flight of up to 24 hours. The bolometers will be cooled to 100 mK by means of an adiabatic demagnetization refrigerator (ADR). The heat of magnetization generated during the ADR cycle will be sunk into a $^{3}$He\ refrigerator operating at 300 mK with a cooling capacity of 25 Joules. The resulting cooling capacity of the ADR is 93 mJoules. With expected heat loads both the ADR and the $^{3}$He\ refrigerator will maintain cooling capacity much longer than the cryostat. Spider-web bolometers \cite{bock94} operating at 100 mK will be used to detect the incoming radiation\footnote{See also a paper by Debernardis in these proceedings. The bolometers, readout electronics, and attitude control system are shared technology between BOOMERanG \ and MAXIMA.}. Extrapolation from measurements at 300 mK, and preliminary measurements at 100 mK, indicate that the detectors will be background limited and will have time constants $\mathrel{\mathpalette\mathrelfun <} 10$ msec. We expect a detector NET of 60 $\mu$K$\sqrt{\mbox{sec}}$ (90 $\mu$K$\sqrt{\mbox{sec}}$ ) at the 150 GHz (240 GHz) frequency band. The detectors will be AC-biased at a frequency of several hundred Hz$^{1}$. The post lock-in noise of the readout electronics was measured to be less than 10 $n$V$\sqrt{\mbox{Hz}}$, down to frequencies smaller than 100 mHz. \subsubsection{Gondola and Attitude Control} The gondola provides for pointing in azimuth and elevation. Pointing control is achieved with a 5 Hz feedback loop control relying on a two axis magnetometer for coarse pointing ($\pm 2$ degrees) and on a CCD camera as a fine sensor. The CCD camera and its associated f/0.7 lens provide a field of view of 7.4 degrees in azimuth and 5.5 degrees in elevation, and pixel resolution of 0.8 arcmin/pixel and 0.9 arcmin/pixel, respectively. The on board image processing is expected to provide sub-pixel resolution. Overall pointing stability is expected to be 1 arcminute RMS or better. \subsection{Observing Strategy and $l$ Space Coverage} \label{Observing Strategy and $l$ Space Coverage} MAXIMA's beam will be scanned in azimuth with two frequencies. The primary mirror will modulate the beam in a triangular wave with frequency $f_{1}$ and amplitude $A_{1}$, while the gondola will be simultaneously scanned in azimuth at a slower rate. Here we discuss the choice of $f_{1}$ and $A_{1}$. A bolometer with time constant $\tau$ acts as a single pole low pass filter on the detected optical signals. The -3 dB point of this filter is used to set a criterion on the maximum speed that the beam can be scanned across the sky. The signal detected by a fast bolometer ($\tau \simeq 0$) when a Gaussian beam with width $\sigma = 0.425 \times \mbox{FWHM}$ crosses a point source at constant speed $\dot{\theta}$ has a Gaussian frequency distribution with width $\tilde{\sigma}=\dot{\theta}/(2 \pi \sigma)$. If we require that the -3 dB roll-off of a real bolometer will be larger than $3\tilde{\sigma}$ we obtain a relation between the maximum scan speed and the bolometer time constant \begin{equation} {1 \over 2 \pi \tau} \geq { 3 \dot{\theta} \over 2 \pi \sigma} \; \; \Rightarrow \; \; \dot{\theta} \leq {\sigma \over 3 \tau} = {\mbox{beam FWHM} \over 7 \tau}. \end{equation} For a 0.18 degrees FWHM beam width and $\tau=10$ msec, $\dot{\theta} \leq 2.6$ degrees/sec. By moving the beams across the sky at this (or somewhat lower) speed the bolometers remain sufficiently sensitive to all spatial frequencies up to $\sim$1/beam size. The amplitude $A_{1}$ is determined by requiring that the scan frequency $f_{1}$ be higher than the knee of the $1/f$ noise. preliminary measurements during the first flight of MAXIMA\ suggest that $f_{1} \simeq 0.5$ Hz is appropriate. For a triangular wave \begin{equation} 4 A_{1} f_{1} = \dot{\theta} \leq 2.6 \; \mbox{deg/sec}, \end{equation} so that $A_{1} = 1.3$ degrees. Larger amplitudes are possible with shorter bolometer time constants. This scan strategy is efficient and enables the synthesis of multiple window functions in a single scan. In combination with the 11 arcminute beams we expect an $l$ space coverage between $l=60$ and $l=650$. (see also a companion paper in this proceedings \cite{adrianmoriond}.) \subsection{Status and Flight Program} The MAXIMA\ set of measurements is being comissioned in stages. In the first flight, which was launched from Palestine, Texas, on Sept. 2, 1995, we flew the single beam receiver used on MAX-4, and MAX-5. Most other flight systems, including the gondola, the pointing system, AC-bias electronics, and chopping primary mirror, were new. The flight goals were to test all new flight systems, scan regions of the sky for CMBA signals, and test new scan strategies. All of these have been successfully accomplished. Data analysis is in progress. The 14-beam array is presently under construction and is scheduled to be launched as MAXIMA-2 in the spring of 1997.

proofpile-arXiv_065-511

\section{Introduction} \setcounter{equation}{0} It is well known that the ground state of one-dimensional interacting electron systems is non-Fermi liquid. The state, which is called as ``Luttinger liquid'', is characterized by a separation of spin and charge degrees of freedom, and anomalous exponents of correlation functions which depend on the interaction\cite{Solyom-review,Emery-review,Haldane-review,Fukuyama-Takayama}. The parallel chains of the interacting electrons coupled by the interchain hopping are basic models of quasi-one-dimensional electron systems. Theoretical understanding of these systems is fundamental for studying electronic properties of organic conductors \cite{Ishiguro-Yamaji}, high temperature superconductors \cite{Andersonconjecture} and interacting electron systems applied to strong magnetic field (Fractional Quantum Hall Effects) \cite{WenFQHE}. In addition, the problem is important as a first step toward studying two-dimensional interacting electron systems. Consequently, there has been a growing interest in the system of the coupled chains, in particular two chains. Several work have been devoted to investigating the two chains coupled by the interchain hopping for the case of spinless Fermion\cite{Wen,Kusmartsev-Luther-Nersesyan,Yakovenko,Yoshioka-Suzumura-1,Yoshioka-Suzumura-2} and for the case including spin degree of freedom\cite{Castellani-Dicastro-Metzner,Fabrizio,Finkelstein-Larkin,Clarke-Strong-Anderson,Yamaji-Shimoi,Shimoi-Yamaji-Yanagisawa,Nagaosa-Oshikawa,Schulz,Balents-Fisher,Yoshioka-Suzumura-IV}. In addition, Anderson localization in such a system has been studied based on the above investigation\cite{Kimura-Kuroki-Aoki,Orignac-Giamarchi}. However, interchain interaction also plays a role of the interchain coupling\cite{Nersesyan-Luther-Kusmartsev,Yoshioka-Suzumura-3}. Therefore, by extending the work by Finkel'stein and Larkin\cite{Finkelstein-Larkin} and that by Schulz\cite{Schulz}, we study the two chains of Luttinger liquids in the presence of both the interchain hopping and the interchain interaction. To our knowledge, much is not known about such a problem in the case including spin degree of freedom. We clarify the electronic properties originated from the interchain hopping by calculating excitation spectrum, phase diagram and susceptibilities. The gap appears in the excitation spectrum of the transverse fluctuations. This leads to splitting of the degenerated states in the absence of the hopping, {i.e.,} ``in phase'' ordering states and ``out of phase'' ordering states between the chains. From the calculation of charge and spin susceptibilities with $q_x$ and $q_y$ being the longitudinal and transverse wavenumber, it is shown that both susceptibilities with $q_y = 0$ are the same as those in the absence of the hopping, and those with $q_y = \pi$ show remarkable dependence on $q_x$. The plan of the paper is as follows. In II, the Hamiltonian of the model is given and is expressed by use of the phase variables. In III, we study the excitation spectrum and the phase diagram at $T=0$ with $T$ being temperature. The charge and the spin susceptibilities are also calculated. IV is devoted to discussion on the present results. \section{Model and Phase Representation} \setcounter{equation}{0} \subsection{Model Hamiltonian} We investigate the system where two chains of Luttinger liquids are coupled by both the interchain hopping and the interchain interaction. The Hamiltonian is given by \begin{equation} \label{eqn:Hamiltonian} {\cal H} = {\cal H_{\rm k}} + {\cal H_{\rm int}} + {\cal H'_{\rm int}} \;\;, \end{equation} where \begin{eqnarray} {\cal H_{\rm k}} &=& \sum_{k,p,\sigma} \epsilon_{kp} \left\{ a^{\dagger}_{k,p,\sigma,1} a_{k,p,\sigma,1} + ( 1 \to 2 ) \right\} - t \sum_{k,p,\sigma} \left\{ a^{\dagger}_{k,p,\sigma,1} a_{k,p,\sigma,2} + ( 1 \leftrightarrow 2 ) \right\}, \label{eqn:2.1}\\ {\cal H_{\rm int}} &=& { {\pi v_F g_{2}} \over {L} } \sum_{p,\sigma,\sigma'} \sum_{k_1, k_2, q} \left\{ a^{\dagger}_{k_1,p,\sigma,1} a^{\dagger}_{k_2,-p,\sigma',1} a_{k_2+q,-p,\sigma',1} a_{k_1-q,p,\sigma,1} + ( 1 \to 2 ) \right\}, \label{eqn:2.2}\\ {\cal H'_{\rm int}} &=& { {\pi v_F g'_{2}} \over {L} } \sum_{p,\sigma,\sigma'} \sum_{k_1, k_2, q} \left\{ a^{\dagger}_{k_1,p,\sigma,1} a^{\dagger}_{k_2,-p,\sigma',2} a_{k_2+q,-p,\sigma',2} a_{k_1-q,p,\sigma,1} + ( 1 \leftrightarrow 2 ) \right\}. \label{eqn:2.3} \end{eqnarray} Equations (\ref{eqn:2.1}), (\ref{eqn:2.2}) and (\ref{eqn:2.3}) express the kinetic energy, the intrachain interaction and the interchain interaction, respectively. Quantities $k$ and $\epsilon_{kp} ( = v_F(pk-k_F)) $ denote the momentum and the kinetic energy of a Fermion where $v_F$, $p = +(-)$ and $k_F$ are the Fermi velocity, the right-going (left-going) state of a Fermion and the Fermi momentum. The operator $a^{\dagger}_{k,p,\sigma,i}$ expresses creation of the Fermion with $k$, $p$, $\sigma$ and $i$ where $\sigma = +(-)$ and $i(=1,2) $ are the spin $\uparrow(\downarrow)$ state and the index of the chains. The interchain hopping and the length of the chain are defined by $t$ and $L$, respectively. The normalized quantities, $g_{2}$ ($g'_{2}$) denotes the matrix element of the interaction for the intrachain (interchain) forward scattering between particles moving oppositely. Note that the conventional definition of the elements are given by $g \to g/(2\pi v_F)$\cite{Solyom-review}. \subsection{Phase Representation} We represent the above Hamiltonian, Eqs.(\ref{eqn:2.1}) $\sim$ (\ref{eqn:2.3}) and the field operators of Fermions in terms of phase variables based on bosonization method. The separation of the Fermi wavenumber due to the hopping is taken into account by use of the unitary transformation, $c_{k,p,\sigma,\mu} = ( - \mu a_{k,p,\sigma,1} + a_{k,p,\sigma,2} ) / \sqrt{2}$ ($\mu = \pm$). Then the kinetic term, ${\cal H}_{\rm k}$ is rewritten as \begin{equation} {\cal H_{\rm k}} = \sum_{k,p,\sigma,\mu} v_F(pk-k_{F\mu}) c^{\dagger}_{k,p,\sigma,\mu} c_{k,p,\sigma,\mu}, \label{eqn:2.4} \end{equation} where $k_{F\mu} = k_F - \mu t/v_F$. The interaction terms are rewritten as follows, \begin{eqnarray} {\cal H}_{\rm int} &+& {\cal H}'_{\rm int} = {\cal H}^1_{\rm int} + {\cal H}^2_{\rm int} + {\cal H}^3_{\rm int}, \label{eqn:2.5}\\ {\cal H}^1_{\rm int} &=& \frac{\pi v_F}{2L} (g_2 + g_2') \sum_{p,\sigma,\sigma',\mu,\mu'} \sum_{q} \rho_{p,\sigma,\mu}(q) \rho_{-p,\sigma',\mu'}(-q) , \label{eqn:2.6}\\ {\cal H}^2_{\rm int} &=& \frac{\pi v_F}{2} (g_2 - g_2') \sum_{p,\sigma,\sigma',\mu} \int {\rm d} x \left\{ \psi^\dagger_{p,\sigma,\mu} \psi^\dagger_{-p,\sigma',\mu} \psi_{-p,\sigma',-\mu} \psi_{p,\sigma,-\mu} \right\}, \label{eqn:2.7}\\ {\cal H}^3_{\rm int} &=& \frac{\pi v_F}{2} (g_2 - g'_2) \sum_{p,\sigma,\sigma',\mu} \int {\rm d} x \left\{ \psi^\dagger_{p,\sigma,\mu} \psi^\dagger_{-p,\sigma',-\mu} \psi_{-p,\sigma',\mu} \psi_{p,\sigma,-\mu} \right\}, \label{eqn:2.8} \end{eqnarray} where $\psi_{p,\sigma,\mu} \equiv (1\sqrt{L}) \sum_k {\rm e}^{{\rm i} k x} c_{k,p,\sigma,\mu}$. The operator $\rho_{p,\sigma,\mu}(q) (= \sum_k c^\dagger_{k+q,p,\sigma,\mu} c_{k,p,\sigma,\mu})$ expresses the density fluctuation around the new Fermi point $k_{F\mu}$ and satisfies the commutation relations, $\left[ \rho_{p,\sigma,\mu}(-q), \rho_{p',\sigma',\mu'}(q') \right] = \delta_{\sigma \sigma'} \delta_{p p'}\delta_{q q'} \delta_{\mu,\mu'} pqL/(2\pi)$. The Hamiltonian ${\cal H}^1_{\rm int}$ denotes the interaction described in terms of the quadratic form of the density fluctuation. For the processes of Eqs.(\ref{eqn:2.7}) and (\ref{eqn:2.8}), the band index is not conserved in ${\cal H}^2_{\rm int}$ and the index is exchanged in ${\cal H}^3_{\rm int}$, respectively. Here we define phase variables $\theta_{\pm}(x)$, $\phi_{\pm}(x)$, $\tilde{\theta}_{\pm}(x)$ and $\tilde{\phi}_{\pm}(x)$ as\cite{Yoshioka-Suzumura-3,Suzumura_P,correspondence} \begin{eqnarray} \theta_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\pi {\rm i}}{\sqrt{2} q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \sum_{\sigma,\mu} \left( \rho_{+,\sigma,\mu}(-q) \pm \rho_{-,\sigma,\mu}(-q) \right), \label{eqn:2.9} \\ \phi_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\pi {\rm i}}{\sqrt{2} q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \sum_{\sigma,\mu} \sigma \left( \rho_{+,\sigma,\mu}(-q) \pm \rho_{-,\sigma,\mu}(-q) \right), \label{eqn:2.10}\\ \tilde{\theta}_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\pi {\rm i}}{\sqrt{2} q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \sum_{\sigma,\mu} \mu \left( \rho_{+,\sigma,\mu}(-q) \pm \rho_{-,\sigma,\mu}(-q) \right), \label{eqn:2.11}\\ \tilde{\phi}_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\pi {\rm i}}{\sqrt{2} q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \sum_{\sigma,\mu} \sigma \mu \left( \rho_{+,\sigma,\mu}(-q) \pm \rho_{-,\sigma,\mu}(-q) \right) , \label{eqn:2.12} \end{eqnarray} where $\alpha^{-1}$ is a cutoff of the large momentum corresponding to the band width $v_F\alpha^{-1}$. Equations (\ref{eqn:2.9}) $\sim$ (\ref{eqn:2.12}) satisfy commutation relations given by $[ \theta_+ (x), \theta_- (x') ] = [ \phi_+ (x), \phi_- (x') ] = [ \tilde{\theta}_+ (x), \tilde{\theta}_- (x') ] = [ \tilde{\phi}_+ (x), \tilde{\phi}_- (x') ] = {\rm ln} \left\{1 + {\rm i}(x-x')\alpha^{-1}\right\} - {\rm ln} \left\{1 - {\rm i}(x-x')\alpha^{-1}\right\} \simeq {\rm i} \pi {\rm sgn} ( x - x' )$ and zero for the others. The variables, $\theta_{\pm}$ and $\phi_{\pm}$ express the fluctuation of the total charge density and that of the total spin density, respectively, while $\tilde{\theta}_{\pm}$ and $\tilde{\phi}_{\pm}$ express the transverse fluctuation of the charge density and that of the spin density, respectively. Actually these properties are understood from the facts that $\partial_x \theta_+ = ( \pi / \sqrt{2} ) \sum_{p,\sigma, i} \psi^\dagger_{p,\sigma,i} \psi_{p,\sigma,i} $, $\partial_x \phi_+ = ( \pi / \sqrt{2} ) \sum_{p,\sigma, i} \sigma \psi^\dagger_{p,\sigma,i} \psi_{p,\sigma,i} $, $ \partial_x \tilde{\theta}_+ = ( - \pi / \sqrt{2} ) \sum_{p,\sigma} \{ \psi^\dagger_{p,\sigma,1} \psi_{p,\sigma,2} + h.c. \} $ and $\partial_x \tilde{\phi}_+ = ( - \pi / \sqrt{2} ) \sum_{p,\sigma} \sigma \{ \psi^\dagger_{p,\sigma,1} \psi_{p,\sigma,2} + h.c. \} $ where $\psi_{p,\sigma,i} = (1/\sqrt{L}) \sum_k {\rm e}^{{\rm i} k x} a_{k,p,\sigma,i}$. In terms of the phase variables, the field operator of Fermions, $\psi_{p,\sigma,\mu} ( x )$, is expressed as \cite{Luther-Peschel,Luther-Emery}, \begin{eqnarray} \psi_{p,\sigma,\mu} ( x ) &=& { 1 \over \sqrt{2 \pi \alpha} } \exp \left[ {\rm i} p k_{F\mu} x + {\rm i} \Theta_{p,\sigma,\mu} + {\rm i} \pi \Xi_{p, \sigma, \mu} \right] \nonumber \\ &\equiv& \psi'_{p,\sigma,\mu} ( x ) \exp \left( {\rm i} \pi \Xi_{p, \sigma, \mu} \right), \label{eqn:2.13} \end{eqnarray} with \begin{equation} \Theta_{p,\sigma,\mu} = \frac{1}{2\sqrt{2}} \Big\{ p \theta_+ + \theta_- + \mu ( p \tilde{\theta}_+ + \tilde{\theta}_- ) + \sigma ( p \phi_+ + \phi_- ) + \sigma \mu ( p \tilde{\phi}_+ + \tilde{\phi}_- ) \Big\}. \label{eqn:2.14} \end{equation} In Eq.(\ref{eqn:2.13}), the phase factor, $\pi \Xi_{p, \sigma, \mu}$, is added so that the Fermion operators with different indices satisfy the anticommutation relation \cite{Solyom-review}. The factor $ \Xi_{p, \sigma, \mu}$ is given by $\Xi_1 =0$ and $\Xi_i = \sum_{j=1}^{i-1} \hat N_j$, $(i = 2 \sim 8)$ with ${\hat N}_i$ being the number operator of the Fermions with indices $i$ where the index $(p,\sigma,\mu)$ corresponds to $(+,+,+) = 1$, $(+,-,+) = 2$, $(+,+,-) = 3$, $(+,-,-) = 4$, $(-,+,+) = 5$, $(-,-,+) = 6$, $(-,+,-) = 7$ and $(-,-,-) = 8$, respectively. Note that the above choice of $ \Xi_{p, \sigma, \mu}$ is not unique. By substituting Eqs.(\ref{eqn:2.13}) and (\ref{eqn:2.14}) into Eqs.(\ref{eqn:2.7}) and (\ref{eqn:2.8}), and by using the fact that ${\cal H}_{\rm k} = (\pi v_F / L ) \sum_{p,\sigma,\mu} \sum_{q} \rho_{p,\sigma,\mu}(q) \rho_{p,\sigma,\mu}(-q)$ \cite{Luther-Peschel,Luther-Emery,Mattis-Lieb}, the Hamiltonian ${\cal H} = {\cal H}_T + {\cal H}_R$ can be expressed as, \begin{eqnarray} {\cal H}_T & = & \frac{v_\rho}{4 \pi} \int {\rm d} x \left\{ \frac{1}{\eta_\rho}(\partial_x \theta_+)^2 + {\eta_\rho}(\partial_x \theta_-)^2 \right\} + \frac{v_F}{4 \pi} \int {\rm d} x \left\{ (\partial_x \phi_+)^2 + (\partial_x \phi_-)^2 \right\}, \label{eqn:2.15}\\ {\cal H}_R & = & \frac{v_F}{4 \pi} \int {\rm d} x \left\{ \tilde{A}_{\theta,+}(\partial_x \tilde{\theta}_+)^2 + \tilde{A}_{\theta,-}(\partial_x \tilde{\theta}_-)^2 + \tilde{A}_{\phi,+}(\partial_x \tilde{\phi}_+)^2 + \tilde{A}_{\phi,-}(\partial_x \tilde{\phi}_-)^2\right\} \nonumber \\ & + & { { v_{F} (g_{2}-g'_{2}) } \over {\pi\alpha^2} } \int {\rm d}x \left\{ \cos \sqrt{2} \tilde{\theta}_- - \cos (2 q_0 x -\sqrt{2} \tilde{\theta}_+) \right\} \left\{ \cos \sqrt{2} \tilde{\phi}_- + \cos \sqrt{2} \tilde{\phi}_+ \right\}, \label{eqn:2.16} \end{eqnarray} where $v_\rho = v_F \sqrt{(1 + 2 g_2 + 2 g'_2)(1 - 2 g_2 - 2 g'_2)}$, $\eta_\rho = \sqrt{(1 - 2 g_2 - 2 g'_2)/(1 + 2 g_2 + 2 g'_2)}$, $\tilde{A}_{\theta,\pm} = \tilde{A}_{\phi,\pm} = 1$ and $q_0 = 2t/v_F$. Here ${\cal H}_T$ expresses fluctuations of total charge and spin densities, whose excitations are given by $v_\rho |k|$ and $v_F |k|$, respectively. On the other hand, ${\cal H}_R$, which expresses the transverse fluctuation, includes the complex nonlinear terms. The nonlinear terms including $\cos \sqrt{2} \tilde{\theta}_-$ result from ${\cal H}^2_{\rm int}$ and those with $\cos (2 q_0 x -\sqrt{2} \tilde{\theta}_+)$ are due to ${\cal H}^3_{\rm int}$. In deriving Eq.(\ref{eqn:2.16}), we chose a Hilbert space with the even integers for respective numbers $N_1 + N_3$, $N_2 + N_4$, $N_5 + N_7$, $N_6 + N_8$, $N_1 + N_5$, $N_1 + N_6$ and $N_1 + N_2$ where $N_i$ is the eigenvalue of ${\hat N}_{i}$. The negative sign of $\cos (2 q_0 x -\sqrt{2} \tilde{\theta}_+)$ is due to such a choice of the phase factor, $\pi \Xi_{p,\sigma,\mu}$. It is noticed that the interchain interaction does not change the structure of the Hamiltonian, {i.e.,} the parameters in the presence of $g_2'$ are obtained by rewriting as $g_2 \to g_2 + g_2'$ in ${\cal H}_T$ and $g_2 \to g_2 - g_2'$ in ${\cal H}_R$. The Hamiltonian, ${\cal H}_R$ can be rewritten as \begin{eqnarray} {\cal H}_R & = & v_F \int {\rm d} x \left\{ \bar{\psi}_1^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_1 - \bar{\psi}_2^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_2 + \bar{\psi}_3^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_3 - \bar{\psi}_4^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_4 \right\} \nonumber \\ & + & \pi v_F (g_2 - g_2') \int {\rm d} x \left\{ {\rm i} \bar{\psi}^\dagger_3 \bar{\psi}^\dagger_4 - {\rm i} \bar{\psi}_4 \bar{\psi}_3 - \bar{\psi}^\dagger_4 \bar{\psi}_3 {\rm e}^{- {\rm i} 2 q_0 x} - \bar{\psi}^\dagger_3 \bar{\psi}_4 {\rm e}^{{\rm i} 2 q_0 x} \right\} \nonumber \\ & & \hspace{3.4cm} \times \left\{ {\rm i} \bar{\psi}^\dagger_1 \bar{\psi}^\dagger_2 - {\rm i} \bar{\psi}_2 \bar{\psi}_1 + \bar{\psi}^\dagger_2 \bar{\psi}_1 + \bar{\psi}^\dagger_1 \bar{\psi}_2 \right\}, \label{eqn:2.17} \end{eqnarray} where $\bar{\psi}_{i}$ ($i = 1 \sim 4$) are new Fermion fields defined by \begin{eqnarray} \bar{\psi}_1 & = & \frac{1}{\sqrt{ 2 \pi \alpha }} {\rm e}^{{\rm i} \frac{1}{\sqrt{2}}( \tilde{\phi}_+ + \tilde{\phi}_-)} {\rm e}^{{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_1} + \hat{\bar{N}_2} )} \equiv \bar{\psi}_1' {\rm e}^{{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_1} + \hat{\bar{N}_2} )}, \label{eqn:2.18}\\ \bar{\psi}_2 & = & \frac{1}{\sqrt{ 2 \pi \alpha }} {\rm e}^{-{\rm i} \frac{1}{\sqrt{2}}( \tilde{\phi}_+ - \tilde{\phi}_-)} {\rm e}^{-{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_1} + \hat{\bar{N}_2} )} \equiv \bar{\psi}_2' {\rm e}^{-{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_1} + \hat{\bar{N}_2} )}, \label{eqn:2.19}\\ \bar{\psi}_3 & = & \frac{1}{\sqrt{ 2 \pi \alpha }} {\rm e}^{{\rm i} \frac{1}{\sqrt{2}}( \tilde{\theta}_+ + \tilde{\theta}_-)} {\rm e}^{{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_3} + \hat{\bar{N}_4} ) + {\rm i} \pi ( \hat{\bar{N}_1} + \hat{\bar{N}_2} ) } \equiv \bar{\psi}_3' {\rm e}^{{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_3} + \hat{\bar{N}_4} ) + {\rm i} \pi ( \hat{\bar{N}_1} + \hat{\bar{N}_2} ) }, \label{eqn:2.20}\\ \bar{\psi}_4 & = & \frac{1}{\sqrt{ 2 \pi \alpha }} {\rm e}^{-{\rm i} \frac{1}{\sqrt{2}}( \tilde{\theta}_+ - \tilde{\theta}_-)} {\rm e}^{-{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_3} + \hat{\bar{N}_4} ) + {\rm i} \pi ( \hat{\bar{N}_1} + \hat{\bar{N}_2} ) } \equiv \bar{\psi}_4' {\rm e}^{-{\rm i} \frac{\pi}{2} ( \hat{\bar{N}_3} + \hat{\bar{N}_4} ) + {\rm i} \pi ( \hat{\bar{N}_1} + \hat{\bar{N}_2} ) } . \label{eqn:2.21} \end{eqnarray} In Eqs.(\ref{eqn:2.18})$\sim$(\ref{eqn:2.21}), the phase factors with the number operator are introduced again to satisfy the anticommutation relation. Equation (\ref{eqn:2.17}) is derived by choosing the Hilbert space with both $\bar{N}_1 + \bar{N}_2 $ and $\bar{N}_3 + \bar{N}_4 $ being even integers. It is noted that the phase variables in Eqs.(\ref{eqn:2.18}) $\sim$ (\ref{eqn:2.21}) are expressed as \begin{eqnarray} \tilde{\theta}_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\sqrt{2} \pi {\rm i}}{q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \left( \bar{\rho}_3 (-q) \pm \bar{\rho}_4 (-q) \right) \;\;, \label{eqn:2.22}\\ \tilde{\phi}_{\pm}(x) &=& - \sum_{q \not= 0 } \frac{\sqrt{2} \pi {\rm i}}{q L} {\rm e}^{(- \alpha |q|/2 + {\rm i} q x)} \left( \bar{\rho}_1(-q) \pm \bar{\rho}_2 (-q) \right) \;\;, \label{eqn:2.23} \end{eqnarray} where $\bar{\rho}_j (q) \equiv \sum_k \bar{\psi}^\dagger_j (k+q) \bar{\psi}_j (k)$ with $\bar{\psi}_j = 1/\sqrt{L} \sum_k \bar{\psi}_j (k) {\rm e}^{{\rm i} k x}$ ($j = 1 \sim 4$). \section{Properties at Low Temperatures} \setcounter{equation}{0} By using renormalization group method, Finkel'stein and Larkin \cite{Finkelstein-Larkin} showed that the nonlinear terms in Eq.(\ref{eqn:2.16}) with $g_2 \neq g_2'$ tend to the strong coupling in the limit of low energy and the terms without (with) the misfit parameter, $2 q_0$ become relevant (irrelevant). Schulz insisted that the transverse charge excitation is completely gapful and that of spin excitation has two kinds of excitation with gap and gapless from the symmetry of the Hamiltonian. Here we explicitly show the results by use of the mean field approximation in which the terms including the misfit parameter are neglected. This method is expected to be effective in the limit of strong coupling and has an advantage of the straightforward calculation of the several quantities. It should be noted that the break of the balance between $\tilde{\theta}_+$ and $\tilde{\theta}_-$ due to the misfit parameter may lead to the renormalization of $\tilde{\eta}_\rho$ defined by $(\tilde{A}_{\theta,-}/\tilde{A}_{\theta,+})^{1/2}$. We neglect such an effect as zeroth-approximation since the system tends to strong coupling and the gap appears. On the other hand, $\tilde{\eta}_\sigma ( \equiv (\tilde{A}_{\phi,-}/\tilde{A}_{\phi,+})^{1/2} )$ remains unity due to the balance between $\tilde{\phi}_+$ and $\tilde{\phi}_-$. The present method is effective for the energy lower than the hopping, and then the large momentum cutoff $\alpha^{-1}$ must be read as $t/v_F$. \subsection{Excitation Spectrum} By making use of the mean-field approximation, ${\cal H}_R$ with $g_2 \neq g_2'$ is rewritten as \begin{eqnarray} {\cal H}_R = {\cal H}^{12}_{\rm MF} + {\cal H}^{34}_{\rm MF} - \Delta \Delta' L /(2 \pi v_F (g_2-g'_2)) \;\;, \end{eqnarray} where \begin{eqnarray} {\cal H}^{12}_{\rm MF} & = & v_F \int {\rm d} x \left\{ \bar{\psi}_1^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_1 - \bar{\psi}_2^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_2 \right\} \nonumber \\ &+& \frac{\Delta}{2} \int {\rm d} x \left\{ {\rm i} \bar{\psi}^\dagger_1 \bar{\psi}^\dagger_2 - {\rm i} \bar{\psi}_2 \bar{\psi}_1 + \bar{\psi}^\dagger_2 \bar{\psi}_1 + \bar{\psi}^\dagger_1 \bar{\psi}_2 \right\}, \label{eqn:3.1}\\ {\cal H}^{34}_{\rm MF} & = & v_F \int {\rm d} x \left\{ \bar{\psi}_3^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_3 - \bar{\psi}_4^\dagger ( -{\rm i} \partial_x ) \bar{\psi}_4 \right\} + \Delta' \int {\rm d} x \left\{ {\rm i} \bar{\psi}^\dagger_3 \bar{\psi}^\dagger_4 - {\rm i} \bar{\psi}_4 \bar{\psi}_3 \right\}. \label{eqn:3.2} \end{eqnarray} The quantities, $\Delta$ and $\Delta'$ in Eqs.(\ref{eqn:3.1}) and (\ref{eqn:3.2}) are gap parameters determined by the following self-consistent equations, \begin{eqnarray} \frac{\Delta}{2} & = & \pi v_F ( g_2 - g'_2 ) \left\{ {\rm i} \left\langle \bar{\psi}^\dagger_3 \bar{\psi}^\dagger_4 \right\rangle - {\rm i} \left\langle \bar{\psi}_4 \bar{\psi}_3 \right\rangle \right\}, \label{eqn:3.3} \\ \Delta' & = & \pi v_F ( g_2 - g'_2) \left\{ {\rm i} \left\langle \bar{\psi}^\dagger_1 \bar{\psi}^\dagger_2 \right\rangle - {\rm i} \left\langle \bar{\psi}_2 \bar{\psi}_1 \right\rangle + \left\langle \bar{\psi}^\dagger_2 \bar{\psi}_1 \right\rangle + \left\langle \bar{\psi}^\dagger_1 \bar{\psi}_2 \right\rangle \right\}. \label{eqn:3.4} \end{eqnarray} The eigenvalues, $\omega_{12}$ of Eq.(\ref{eqn:3.1}) and $\omega_{34}$ of Eq.(\ref{eqn:3.2}) are calculated as \begin{eqnarray} \omega_{12} &=& \left\{ \begin{array}{l} \pm E_k \equiv \pm \sqrt{ (v_F k)^2 + \Delta^2 } \;\;, \\ \pm v_F k \;\;, \end{array} \right. \label{eqn:n3.5} \\ \omega_{34} & =& \pm E'_k \equiv \pm \sqrt{ (v_F k)^2 + \Delta'^2 } \;\,. \label{eqn:n3.6} \end{eqnarray} The excitations of Eqs.(\ref{eqn:n3.5}), whose spectral weights are 1/2, are obtained by ${\cal H}^{12}_{\rm MF}$ in terms of Majorana Fermion as \begin{eqnarray} {\cal H}_{\rm MF}^{12} &=& \frac{v_F}{2} \int {\rm d} x \left\{ C_1(-{\rm i} \partial_x)C_1 - C_2(-{\rm i} \partial_x)C_2 \right\} + {\rm i} \Delta \int {\rm d} x C_1 C_2 \nonumber \\ &+& \frac{v_F}{2} \int {\rm d} x \left\{ C_0(-{\rm i} \partial_x)C_0 - C_3(-{\rm i} \partial_x)C_3 \right\} \;\;, \label{eqn:3.5} \end{eqnarray} where $C_0 = {\rm i} / \sqrt{2} ( {\rm e}^{{\rm i} \pi /4} \bar{\psi}_1 - {\rm e}^{-{\rm i} \pi /4} \bar{\psi}_1^\dagger)$, $C_1 = 1 / \sqrt{2} ( {\rm e}^{{\rm i} \pi /4} \bar{\psi}_1 + {\rm e}^{-{\rm i} \pi /4} \bar{\psi}_1^\dagger)$, $C_2 = 1 / \sqrt{2} ( {\rm e}^{- {\rm i} \pi /4} \bar{\psi}_2 + {\rm e}^{{\rm i} \pi /4} \bar{\psi}_2^\dagger)$ and $C_3 = {\rm i} / \sqrt{2} ( {\rm e}^{-{\rm i} \pi /4} \bar{\psi}_2 - {\rm e}^{{\rm i} \pi /4} \bar{\psi}_2^\dagger)$. Finkel'stein and Larkin\cite{Finkelstein-Larkin} have already obtained the same form as the first line in Eq. (\ref{eqn:3.5}) by replacing $ \cos \sqrt{2} \tilde{\theta}_-$ in Eq.(\ref{eqn:2.16}) with a value of a fixed point. However they did not show the gapless excitation which contributes to low energy properties {\it e.g.,} specific heat\cite{Schulz}. The gap equations of Eqs.(\ref{eqn:3.3}) and (\ref{eqn:3.4}) are calculated (see Appendix A) as \begin{eqnarray} \Delta & = & - 2 \Delta' ( g_2 - g'_2 ) \log \frac{\xi_c + \sqrt{\xi_c^2 + \Delta'^2}}{|\Delta'|}, \label{eqn:3.6} \\ \Delta' & = & - \Delta ( g_2 - g'_2 ) \log \frac{\xi_c + \sqrt{\xi_c^2 + \Delta^2}}{|\Delta|}, \label{eqn:3.7} \end{eqnarray} where $\xi_c$ is cut-off energy of the order of $t$. From the the gap equations, it is found that both $(-\Delta,-\Delta')$ and $(\Delta,\Delta')$ are solutions, and that ${\rm sgn} ( \Delta \Delta') = -1$ for $g_2 - g_2' > 0$ and ${\rm sgn} ( \Delta \Delta' ) = 1$ for $g_2 - g_2' < 0$. The solutions of Eqs.(\ref{eqn:3.6}) and (\ref{eqn:3.7}) in the case of $\Delta' > 0$ is shown in Fig.\ref{fig:1}. \subsection{Possible States} We examine phase diagram which shows the most divergent state. Since the dominant contribution in the low energy limit is given by ${\cal H}^2_{\rm int}$ in Eq.(\ref{eqn:2.7}) \cite{Finkelstein-Larkin}, we rewrite the term as \begin{eqnarray} {\cal H}^2_{\rm int} & = & \frac{\pi v_F}{4} ( g_2 - g'_2 ) \sum_{p,\sigma,\sigma'} \int {\rm d} x \Big\{ \big(\sum_{\mu'} \psi^\dagger_{-p,\sigma,\mu'} \psi^\dagger_{p,\sigma',\mu'} \big) \big(\sum_\mu \psi_{p,\sigma',\mu} \psi_{-p,\sigma,\mu} \big) \nonumber \\ & & \hspace{4.5cm} - \big(\sum_{\mu'} \mu' \psi^\dagger_{-p,\sigma,\mu'} \psi^\dagger_{p,\sigma',\mu'} \big) \big(\sum_\mu \mu \psi_{p,\sigma',\mu} \psi_{-p,\sigma,\mu} \big) \Big\} \nonumber \\ & = & - \frac{\pi v_F}{4} ( g_2 - g'_2 ) \sum_{p,\sigma,\sigma'} \int {\rm d} x \Big\{ \big(\sum_{\mu'} \psi^\dagger_{-p,\sigma,-\mu'} \psi_{p,\sigma',\mu'} \big) \big(\sum_\mu \psi^\dagger_{p,\sigma',\mu} \psi_{-p,\sigma,-\mu} \big) \nonumber \\ & & \hspace{4.5cm} - \big(\sum_{\mu'} \mu' \psi^\dagger_{-p,\sigma,-\mu'} \psi_{p,\sigma',\mu'} \big) \big(\sum_\mu \mu \psi_{p,\sigma',\mu} \psi_{-p,\sigma,-\mu} \big) \Big\} \;\,. \label{eqn:3.8} \end{eqnarray} Since the states should be selected so as to gain the energy from Eq.(\ref{eqn:3.8}), the possible states in the case from $g_2 - g'_2 > 0$ are given by \begin{eqnarray} S_{-}^{\sigma,\sigma'} &\equiv& \sum_{\mu} \mu \psi_{p,\sigma,\mu} \psi_{-p,\sigma',\mu} = - \left\{ \psi_{p,\sigma,1} \psi_{-p,\sigma',2} + (1 \leftrightarrow 2) \right\} \nonumber \\ & \sim & \frac{{\rm i}}{\pi \alpha} {\rm e}^{\frac{{\rm i}}{\sqrt{2}}\theta_-} {\rm e}^{\frac{{\rm i} p}{2\sqrt{2}}(\sigma- \sigma')\phi_+} {\rm e}^{\frac{{\rm i}}{2\sqrt{2}}(\sigma + \sigma')\phi_-} \sin \left\{ \frac{\tilde{\theta}_-}{\sqrt{2}} + p \frac{\sigma-\sigma'}{2\sqrt{2}}\tilde{\phi}_+ + \frac{\sigma +\sigma'}{2\sqrt{2}}\tilde{\phi}_- \right\}, \label{eqn:3.9} \\ DW_{+}^{\sigma,\sigma'} &\equiv& \sum_{\mu} \psi^\dagger_{p,\sigma,\mu} \psi_{-p,\sigma',-\mu} = - \left\{ \psi^\dagger_{p,\sigma,1} \psi_{-p,\sigma',1} - (1 \to 2) \right\} \nonumber \\ & \sim & \frac{-{\rm i}}{\pi \alpha} {\rm e}^{-{\rm i} 2 p k_F x} {\rm e}^{\frac{-{\rm i} p}{\sqrt{2}}\theta_+} {\rm e}^{\frac{-{\rm i} p}{2\sqrt{2}}(\sigma + \sigma')\phi_+} {\rm e}^{\frac{-{\rm i}}{2\sqrt{2}}(\sigma - \sigma')\phi_-} \sin \left\{ \frac{\tilde{\theta}_-}{\sqrt{2}} + p \frac{\sigma-\sigma'}{2\sqrt{2}}\tilde{\phi}_+ + \frac{\sigma +\sigma'}{2\sqrt{2}}\tilde{\phi}_- \right\}, \nonumber \\ & & \label{eqn:3.10} \end{eqnarray} and those in the case of $g_2 - g'_2 < 0$ are given by \begin{eqnarray} S_{+}^{\sigma,\sigma'} &\equiv& \sum_{\mu} \psi_{p,\sigma,\mu} \psi_{-p,\sigma',\mu} = \psi_{p,\sigma,1} \psi_{-p,\sigma',1} + (1 \to 2) \nonumber \\ & \sim & \frac{1}{\pi \alpha} {\rm e}^{\frac{{\rm i}}{\sqrt{2}}\theta_-} {\rm e}^{\frac{{\rm i} p}{2\sqrt{2}}(\sigma- \sigma')\phi_+} {\rm e}^{\frac{{\rm i}}{2\sqrt{2}}(\sigma + \sigma')\phi_-} \cos \left\{ \frac{\tilde{\theta}_-}{\sqrt{2}} + p \frac{\sigma-\sigma'}{2\sqrt{2}}\tilde{\phi}_+ + \frac{\sigma +\sigma'}{2\sqrt{2}}\tilde{\phi}_- \right\}, \label{eqn:3.11} \\ DW_{-}^{\sigma,\sigma'} &\equiv& \sum_{\mu} \mu \psi^\dagger_{p,\sigma,\mu} \psi_{-p,\sigma',-\mu} = - \left\{ \psi^\dagger_{p,\sigma,1} \psi_{-p,\sigma',2} - (1 \leftrightarrow 2) \right\} \nonumber \\ & \sim & \frac{1}{\pi \alpha} {\rm e}^{-{\rm i} 2 p k_F x} {\rm e}^{\frac{-{\rm i} p}{\sqrt{2}}\theta_+} {\rm e}^{\frac{-{\rm i} p}{2\sqrt{2}}(\sigma + \sigma')\phi_+} {\rm e}^{\frac{-{\rm i}}{2\sqrt{2}}(\sigma - \sigma')\phi_-} \cos \left\{ \frac{\tilde{\theta}_-}{\sqrt{2}} + p \frac{\sigma-\sigma'}{2\sqrt{2}}\tilde{\phi}_+ + \frac{\sigma +\sigma'}{2\sqrt{2}}\tilde{\phi}_- \right\}. \nonumber \\ & & \label{eqn:3.12} \end{eqnarray} Here $DW^{\sigma,\sigma'}_{-}$ ( $DW^{\sigma,\sigma'}_{+}$ ) expresses density wave with interchain and out of phase ordering ( with intrachain and out of phase ordering), while $S^{\sigma,\sigma'}_{-}$ ( $S^{\sigma,\sigma'}_{+}$) expresses superconductivity with interchain and in phase ordering ( with intrachain and in phase ordering). The most dominant state between $S_-^{\sigma,\sigma'}$ and $DW_+^{\sigma,\sigma'}$ for $g_2 > g_2'$ ($S_+^{\sigma,\sigma'}$ and $DW_-^{\sigma,\sigma'}$ for $g_2 < g_2'$ ) is determined by the total charge and spin fluctuations. By noting that the correlation functions for the total fluctuations are calculated as, \begin{eqnarray} \left\langle {\rm e}^{\frac{-{\rm i}}{\sqrt{2}}\theta_-(x)} {\rm e}^{\frac{-{\rm i} p}{2\sqrt{2}}(\sigma- \sigma')\phi_+(x)} {\rm e}^{\frac{-{\rm i}}{2\sqrt{2}}(\sigma + \sigma')\phi_-(x)} {\rm e}^{\frac{{\rm i}}{\sqrt{2}}\theta_-(0)} {\rm e}^{\frac{{\rm i} p}{2\sqrt{2}}(\sigma- \sigma')\phi_+(0)} {\rm e}^{\frac{{\rm i}}{2\sqrt{2}}(\sigma + \sigma')\phi_-(0)} \right\rangle &\sim& \left( \frac{\alpha}{|x|} \right)^{\frac{1}{2} + \frac{1}{2\eta_\rho} }, \nonumber \\ & & \label{eqn:3.13} \\ \left\langle {\rm e}^{\frac{{\rm i} p}{\sqrt{2}}\theta_+(x)} {\rm e}^{\frac{{\rm i} p}{2\sqrt{2}}(\sigma + \sigma')\phi_+(x)} {\rm e}^{\frac{{\rm i}}{2\sqrt{2}}(\sigma - \sigma')\phi_-(x)} {\rm e}^{\frac{-{\rm i} p}{\sqrt{2}}\theta_+(0)} {\rm e}^{\frac{-{\rm i} p}{2\sqrt{2}}(\sigma + \sigma')\phi_+(0)} {\rm e}^{\frac{-{\rm i}}{2\sqrt{2}}(\sigma - \sigma')\phi_-(0)} \right\rangle &\sim& \left( \frac{\alpha}{|x|} \right)^{\frac{1}{2} + \frac{\eta_\rho}{2} }, \nonumber \\ \label{eqn:3.14} \end{eqnarray} we obtain the phase diagram in Fig.\ref{fig:2}. The exponent of the correlation function of CDW is the same as that of SDW, and the exponent of singlet superconductivity is the same as that of triplet superconductivity, respectively. This comes from the symmetry of the interaction for the spin degree of freedom, as is also seen in spin dependent Tomonaga model with the isotropic interaction \cite{Fukuyama-Takayama}. Therefore the phase diagram shown in Fig.\ref{fig:2} is essentially the same as that of spinless Fermion studied previously\cite{Yoshioka-Suzumura-3}. In the repulsive case of $g_2 + g_2' >0$, the most dominant state is the density wave. However, the interchain hopping leads to the superconducting state being subdominant even in such a region. \subsection{Susceptibilities} In this subsection, we calculate the charge susceptibilities, $ \chi_\rho(q_x, q_y; {\rm i} {\omega_n})$, and the spin susceptibilities, $\chi_\sigma (q_x, q_y; {\rm i} {\omega_n})$, in the case of $q_x \ll 2k_F$ and $q_y = 0$ or $\pi$ which are defined as \begin{equation} \chi_\nu (q_x, q_y; {\rm i} {\omega_n}) = \int^\beta_0 {\rm d} \tau \int {\rm d} (x - x') {\rm e}^{{\rm i} {\omega_n} \tau} {\rm e}^{-{\rm i} q_x (x - x') } \chi_\nu (x-x', q_y; \tau), \label{eqn:3.15} \end{equation} and $\nu$ = $\rho$ or $\sigma$. In Eq.(\ref{eqn:3.15}), \begin{eqnarray} \chi_\rho(x-x', q_y; \tau) & = & \frac{1}{2}\left\langle T_\tau \left\{ \rho(x, 1 ; \tau) + {\rm e}^{{\rm i} q_y} \rho(x, 2 ; \tau) \right\} \left\{ \rho(x', 1 ; 0) + {\rm e}^{{\rm i} q_y} \rho(x', 2 ; 0) \right\} \right\rangle, \label{eqn:3.16} \\ \chi_\sigma (x-x', q_y; \tau) & = & \frac{1}{2}\left\langle T_\tau \left\{ m(x, 1 ; \tau) + {\rm e}^{{\rm i} q_y} m(x, 2 ; \tau) \right\} \left\{ m(x', 1 ; 0) + {\rm e}^{{\rm i} q_y} m(x', 2 ; 0) \right\} \right\rangle, \label{eqn:3.17} \end{eqnarray} where $\rho(x,i;\tau) = \sum_{p,\sigma} \psi^\dagger_{p,\sigma,i}(x;\tau) \psi_{p,\sigma,i}(x;\tau)$ and $m(x,i;\tau) = \sum_{p,\sigma} \sigma \psi^\dagger_{p,\sigma,i}(x;\tau) \psi_{p,\sigma,i}(x;\tau)$ denote operators of the charge and spin densities at the $i$-th chain, respectively. At first we consider the case of $q_y = 0$. From Eq.(\ref{eqn:2.15}), both $\chi_\rho(x-x', 0; \tau)$ and $\chi_\sigma(x-x', 0; \tau)$ are calculated as \begin{eqnarray} \chi_\rho(x-x', 0; \tau) &=& \frac{1}{\pi^2} \left\langle T_\tau \partial_x \theta_+(x,\tau) \partial_{x'} \theta_+(x',0) \right\rangle \nonumber \\ &=& \frac{2 \eta_\rho}{\pi v_\rho} \frac{1}{\beta L} \sum_{q_x} \frac{(v_\rho q_x)^2}{{\omega_n}^2 + (v_\rho q_x)^2} {\rm e}^{{\rm i} q_x(x-x') - {\rm i} {\omega_n} \tau}, \label{eqn:3.18} \\ \chi_\sigma(x-x', 0; \tau) &=& \frac{1}{\pi^2} \left\langle T_\tau \partial_x \phi_+(x,\tau) \partial_{x'} \phi_+(x',0) \right\rangle \nonumber \\ &=& \frac{2}{\pi v_F} \frac{1}{\beta L} \sum_{q_x} \frac{(v_F q_x)^2}{{\omega_n}^2 + (v_F q_x)^2} {\rm e}^{{\rm i} q_x(x-x') - {\rm i} {\omega_n} \tau} . \label{eqn:3.19} \end{eqnarray} Therefore (${\rm i} \omega_n \to \omega$), one obtains \begin{eqnarray} {\rm Re} \chi_\rho(q_x, 0;\omega) & = & \frac{2 \eta_\rho}{\pi v_\rho} \frac{(v_\rho q_x)^2}{(v_\rho q_x)^2 - \omega^2} \;\;, \label{eqn:3.20} \\ {\rm Re} \chi_\sigma(q_x, 0;\omega) & = & \frac{2}{\pi v_F} \frac{(v_F q_x)^2}{(v_F q_x)^2 - \omega^2} \;\;, \label{eqn:3.21} \end{eqnarray} which are familiar to Luttinger liquid\cite{Solyom-review}. Next we examine the case of $q_y = \pi$, for which Eqs.(\ref{eqn:3.16}) and (\ref{eqn:3.17}) are expressed as \begin{eqnarray} & & \chi_\rho (x-x', \pi; \tau) \nonumber \\ & = & \frac{1}{2}\left\langle T_\tau \Bigg( \sum_{p',\sigma',\mu'} \psi^\dagger_{p',\sigma',-\mu'}(x;\tau) \psi_{p',\sigma',\mu'}(x;\tau) \Bigg) \Bigg( \sum_{p,\sigma,\mu} \psi^\dagger_{p,\sigma,\mu}(x') \psi_{p,\sigma,-\mu}(x') \Bigg) \right\rangle, \label{eqn:3.22} \\ & & \chi_\sigma (x-x', \pi; \tau) \nonumber \\ & = & \frac{1}{2} \left\langle T_\tau \Bigg( \sum_{p',\sigma',\mu'} \sigma' \psi^\dagger_{p',\sigma',-\mu'}(x;\tau) \psi_{p',\sigma',\mu'}(x;\tau) \Bigg) \Bigg( \sum_{p,\sigma,\mu} \sigma \psi^\dagger_{p,\sigma,\mu}(x') \psi_{p,\sigma,-\mu}(x') \Bigg) \right\rangle. \nonumber \\ & & \label{eqn:3.23} \end{eqnarray} After some manipulations (Appendix B), the static susceptibilities, ${\rm Re} \chi_\rho(q_x,\pi;0)$ and ${\rm Re} \chi_\sigma(q_x,\pi;0)$ at $T=0$ are respectively obtained as \begin{eqnarray} {\rm Re} \chi_\rho(q_x,\pi;0) & = & \frac{1}{2\pi} \int {\rm d} k \Bigg\{ \frac{1}{E_k + E'_{k - q_x - q_0}} \Big( 1 - \frac{\xi_k}{E_k} \frac{\xi_{k - q_x -q_0}}{E'_{k - q_x -q_0}} + \frac{\Delta}{E_k} \frac{\Delta'}{E'_{k - q_x -q_0}} \Big) \nonumber \\ & & \hspace{1cm} + \frac{1}{E_k + E'_{k + q_x - q_0}} \Big( 1 - \frac{\xi_k}{E_k} \frac{\xi_{k + q_x -q_0}}{E'_{k + q_x -q_0}} + \frac{\Delta}{E_k} \frac{\Delta'}{E'_{k + q_x -q_0}} \Big) \Bigg\}, \label{eqn:3.24} \\ {\rm Re} \chi_\sigma(q_x,\pi,0) &= & \frac{1}{2\pi} \Bigg\{ P \int^{q_x + q_0} {\rm d} k \left( 1 + \frac{\xi_k}{E'_k} \right) \frac{1}{E'_k - \xi_k + v_F(q_x + q_0)} \nonumber \\ & &\hspace{0.5cm} + P \int_{q_x + q_0} {\rm d} k \left( 1 - \frac{\xi_k}{E'_k} \right) \frac{1}{E'_k + \xi_k - v_F(q_x + q_0)} \nonumber \\ & &\hspace{0.5cm} + P \int^{-q_x + q_0} {\rm d} k \left( 1 + \frac{\xi_k}{E'_k} \right) \frac{1}{E'_k - \xi_k + v_F(-q_x + q_0)} \nonumber \\ & &\hspace{0.5cm} + P \int_{-q_x + q_0} {\rm d} k \left( 1 - \frac{\xi_k}{E'_k} \right) \frac{1}{E'_k + \xi_k - v_F(-q_x + q_0)} \Bigg\} \nonumber \\ &= & \frac{1}{\pi v_F} \left\{ 2 - \frac{\Delta'^2}{2 \epsilon_+^2} \ln \left( 1 + \frac{\epsilon_+^2}{\Delta'^2} \right) - \frac{\Delta'^2}{2 \epsilon_-^2} \ln \left( 1 + \frac{\epsilon_-^2}{\Delta'^2} \right) \right\}, \label{eqn:3.25} \end{eqnarray} where $P$ denotes the principal value, $ \xi_k = v_{\rm F}k$ and $\epsilon_\pm = v_F (\pm q_x + q_0)$. In Fig.\ref{fig:3} and Fig.\ref{fig:4}, we show the normalized quantities ${\bar \chi}_\rho(q_x,\pi;0)$ and ${\bar \chi}_\rho(q_x,\pi;0)$ which are defined by ${\rm Re} \chi_\rho (q_x,\pi;0) / (2/\pi v_F)$ and ${\rm Re} \chi_\sigma (q_x,\pi,0)/(2/\pi v_F)$, respectively. The cutoff energy, $\xi_c$, defined in Eqs.(\ref{eqn:3.6}) and (\ref{eqn:3.7}) is taken as $2t$. Note that Eqs.(\ref{eqn:3.24}) and (\ref{eqn:3.25}) are valid in the case of $|q_x \pm q_0| \lower -0.3ex \hbox{$<$} \kern -0.75em \lower 0.7ex \hbox{$\sim$} q_0$. Equation (\ref{eqn:3.24}) shows that the value of ${\rm Re}\chi_\rho(q_x,\pi;0)$ in the case of $g_2 - g_2' < 0$ is larger than that in the case of $g_2 - g_2' > 0$ because $g_2 - g_2' < 0$ ($> 0$) leads to ${\rm sgn}( \Delta \Delta') = 1$ ($-1$). Such dependence on $g_2 - g_2'$ is also found in the absence of the hopping (see Eq.(\ref{eqn:D15})). On the other hand, ${\rm Re} \chi_\sigma (q_x,\pi;0)$ is independent of sign of the relative interaction. This result seems to correspond to the fact that ${\rm Re} \chi_\sigma (q_x,\pi;0)$ in the absence of the hopping is independent of $g_2 - g_2'$ (see Eq.(\ref{eqn:D16})). In Fig.\ref{fig:3}, ${\rm Re} \chi_\rho(q_x,\pi;0)$ takes a minimum around $q_x = q_0$ in the case of $g_2 - g_2'>0$ and has a small dependence on $q_x$ and $g_2-g_2'$, {i.e.,} being nearly unity for $g_2 - g_2'<0$. On the other hand, ${\rm Re} \chi_{\sigma}(q_x,\pi;0)$ in Fig.\ref{fig:4} takes the minimum for both $g_2 - g_2'>0$ and $g_2 - g_2'<0$. The characteristic dependence is due to the separation of Fermi wavenumber {i.e.,} $k_{F-} - k_{F+} = q_0$ and the gap in the transverse fluctuation, both of which result from the interchain hopping. Note that the $q_x$-dependence of these ${\rm Re} \chi_\rho(q_x,\pi;0)$ and ${\rm Re} \chi_\sigma(q_x,\pi;0)$ does not change qualitatively by the choice of $\xi_c/t$. \section{Discussion} \setcounter{equation}{0} In the present paper, we studied the low temperature properties of two chains coupled by the interchain hopping and the interchain interaction where the interactions of only the forward scattering between oppositely moving particles were taken into account as a simplest model of Luttinger liquid. There are four kinds of excitations originated from the fluctuations of total charge, total spin, transverse charge and transverse spin respectively. The total fluctuations which show the gapless excitation are the same as those in the absence of the interchain hopping (Appendix C). On the other hand, the transverse fluctuations of both charge density and spin density which are expressed by the complicated non-linear terms are crucial in the presence of the interchain hopping. By utilizing the mean field approximation, it was shown that the transverse fluctuation of the charge is completely gapful and that of the spin has the two kinds of excitations with and without gap. The most dominant states are obtained as $DW^{\sigma,\sigma'}_+$ for $g_2 > |g_2'|$, $DW^{\sigma,\sigma'}_-$ for $g_2' > |g_2|$, $S^{\sigma,\sigma'}_+$ for $g_2 < -|g_2'|$ and $S^{\sigma,\sigma'}_-$ for $g_2' <- |g_2|$, respectively where density wave (superconductivity) belongs to out of phase (in phase) ordering between the chains, {i.e.,} the transverse wavenumber being $\pi$($0$). We note that in the case of quasi one-dimensional electron system with only the hopping of pairs\cite{Suzumura-Fukuyama}, the density wave with the transverse wave number $(\pi,\pi)$ or superconductivity with $(0,0)$ has maximum critical temperature. It is worth while noting that the states of the superconductivity remain subdominant even for the repulsive interaction, {i.e.,} $g_2 + g'_2 > 0$. Such a result may be the important point toward understanding of the competition of superconductivity and SDW observed in quasi one-dimensional conductors, $\rm{(TMTSF)_2 X}$\cite{Yoshioka-Suzumura-IV}. Possible states in the absence of the interchain hopping are obtained by calculating correlation functions for $S^{\sigma,\sigma'}_{||}$ $(= \psi_{p,\sigma,1} \psi_{-p,\sigma',1}$ or $\psi_{p,\sigma,2} \psi_{-p,\sigma',2})$, $S^{\sigma,\sigma'}_{\bot}$ $( = \psi_{p,\sigma,1} \psi_{-p,\sigma',2}$ or $\psi_{p,\sigma,2} \psi_{-p,\sigma',1})$, $DW^{\sigma,\sigma'}_{||}$ $( = \psi^\dagger_{p,\sigma,1} \psi_{-p,\sigma',1}$ or $\psi^\dagger_{p,\sigma,2} \psi_{-p,\sigma',2})$ and $DW^{\sigma,\sigma'}_{\bot}$ $(= \psi^\dagger_{p,\sigma,1} \psi_{-p,\sigma',2}$ or $\psi^\dagger_{p,\sigma,2} \psi_{-p,\sigma',1}) $ which express the order parameters of the intrachain superconductivity, the interchain superconductivity, the intrachain density wave and the interchain density wave, respectively (Appendix C). The phase diagram in the absence of the interchain hopping is shown in Fig.\ref{fig:5}. By comparing the phase diagram in Fig.\ref{fig:2} and that in Fig.\ref{fig:5}, it is found that the energy gain due to the interchain hopping removes the degeneracy of ``in phase'' and ``out of phase'' ordering. The $q_x$-dependence of charge and spin susceptibilities were calculated for both $q_y = 0$ and $q_y = \pi$. The susceptibilities in the case of $q_y = 0$ are the same as those in the absence of the hopping since the total dynamics is not affected by the hopping. On the other hand, the static susceptibilities, ${\rm Re} \chi_\rho(q_x,\pi;0)$ for $g_2 - g_2'>0$ and ${\rm Re} \chi_\sigma (q_x,\pi;0)$ in the case of $q_y = \pi$ show the minimum around $q_x = q_0$ which is ascribed to the separation of the Fermi wavenumber and the excitation gaps of the transverse fluctuation in the presence of the interchain hopping. The fact that ${\rm Re} \chi_\rho(q_x,\pi;0)$ in the case of $g_2 -g'_2<0 $ is larger than that in the case of $g_2 -g'_2>0 $ is found also in the absence of the interchain hopping. We treated the system with the interaction processes of the forward scattering between the oppositely moving particles as the simplest model of Luttinger liquid. However, the two chains coupled by the interchain hopping have been known to show the various properties depending on the parameters of the system. The repulsive backward scattering becomes relevant in the low energy limit and opens the gap in the excitations of the total spin, and thus modifies the electronic properties\cite{Schulz}. This fact is different from the strictly one-dimensional case, where the repulsive backward scattering is renormalized to zero. The two chains of Hubbard Model shows the richer phases depending on the magnitude of the intrachain interaction, the interchain hopping and the filling\cite{Fabrizio,Balents-Fisher}. In addition, it has been reported that the two chains coupled by both the Coulomb repulsion and the exchange interaction show the superconductivity \cite{Shelton-Tsvelik}. Therefore further investigations are needed to identify the ground state of two coupled chains as a crossover from one dimension to higher dimension. \section*{Acknowledgment} The authors would like to thank A. M. Finkel'stein for discussion. This work was financially supported by the Grant-in-Aid for Scientific Research on the priority area, Novel Electronic States in Molecular Conductors, from the Ministry of Education. \newpage

proofpile-arXiv_065-512

\section{Introduction} \renewcommand{\theequation}{\thesection.\arabic{equation}} It is well known that the renormalization group (RG) equations \cite{rg} have a peculiar power to improve the global nature of functions obtained in the perturbation theory in quantum field theory (QFT)\cite{JZ}: The RG equations may be interpreted as representing the fact that the physical quantities ${\cal O}(p,\alpha, \mu)$ should not depend on the renormalization point $\mu$ having any arbitrary value, \begin{eqnarray} \frac{\partial {\cal O}(p, \alpha;\mu)}{\d \mu}=0. \end{eqnarray} Such a floating renormalization point was first introduced by Gell-Mann and Low in the celebrated paper\cite{rg}. It is Goldenfeld, Oono and their collaborators ( to be abbreviated to GO) \cite{goldenfeld1,goldenfeld2} who first showed that the RG equation can be used for purely mathematical problems as to improving the global nature of the solutions of differential equations obtained in the perturbation theory. One might say, however, that their presentation of the method is rather heuristic, heavily relied on the RG prescription in QFT and statistical physics; it seems that they were not so eager to give a mathematical reasoning to the method so that their method may be understandable even for those who are not familiar with the RG.\footnote{In Appendix A, we give a brief account of the Goldenfeld et al's prescription.} In fact, the reason why the RG equations even in QFT ``improve'' naive perturbation had not been elucidated. One may say that when GO successfully applied the RG method to purely mathematical problems such as solving differential equations, it had shaped a clear problem to reveal the mathematical reason of the powefullness of the RG method, at least, a la Stuckelberg-Peterman and Gell-Mann-Low. Quite recently, the present author has formulated the method and given the reasoning of GO's method on the basis of the classical theory of envelopes\cite{kunihiro,kunihiro2}: It was demonstrated that owing to the very RG equation, the functions consturcted from the solutions in the perturbation theory certainly satisfies the differential equation in question uniformly up to the order with which local solutions around $t=t_0$ is constructed. It was also indicated in a generic way that the RG equation may be regarded as the envelope equation. In fact, if a family of curves $\{{\rm C}_{\mu}\}_{\mu}$ in the $x$-$y$ plane is represented by $y=f(x; \mu)$, the function $g(x)$ representing the envelope E is given by eliminating the parameter $\mu$ from the equation \begin{eqnarray} \frac{\partial f(x; \mu)}{\partial \mu}=0. \end{eqnarray} One can readily recognize the similarity of the envelope equation Eq.(1.2) with the RG equation Eq.(1.1). In Ref.'s\cite{kunihiro,kunihiro2}, a simplified prescription of the RG method is also presented. For instance, the perturbative expansion is made with respect to a small parameter and independent functions\footnote{Such an asmptotic series is called {\em generalizes asymptotic series}. The author is gratefull to T. Hatusda for telling him this fact and making him recognize its significance.}, and the procedure of the ''renormalization" has been shown unnecessary. However, the work given in \cite{kunihiro,kunihiro2} may be said to be incomplete in the following sense: To give the proof mentioned above, the scalar equation in question was converted to a system of {\em first order} equations, which describe a vetor field. But the theory of envelopes for vetor fields, i.e.,envelopes of trajectories, has not been presented in \cite{kunihiro,kunihiro2}. The theory should have been formulated for vector equations to make the discussion self-contained and complete. One of the purposes of the present paper is therefore to reformulate geometrically the RG method for vector equations, i.e., systems of ODE's and PDE's and to complete the discussion given in \cite{kunihiro,kunihiro2}. Another drawback of the previous work is that a reasoning given to a procedure to setting $t_0=t$ in the RG method\footnote{See Appendix A.} of Goldenfeld et al was not fully persuasive.\footnote{The author is grateful to Y. Oono for his criticism on this point.} In this paper, we present a more convincing reasoning for the procedure. Once the RG method is formulated for vector fields, the applicability of the RG method developed by Goldenfeld, Oono and their collaborators is found to be wider than one might have imagined: The RG method is applicable also to, say,$n$-dimensional vector equations that are not simply converted to a scalar equation of the $n$-th order; needless to say, it is not necessarily possible to convert a system of ordinary differential equations (or dynamical system) to a scalar equation of a high order with a simple structure, although the converse is always possible trivially. For partial differential equations, it is not always possible to convert a system to a scalar equation of a high order\cite{curant}. Moreover, interesting equations in science including physics and applied mathematics are often given as a system. Therefore, it is of interest and importance to show that the RG method can be extended and applied to vector equations. To demonstrate the powefulness of the method, we shall work out some specific examples of vector equations. We shall emphasize that the RG method provides a general method for the reduction of the dynamics as the reductive perturbation method (abbreviated to RP method)\cite{kuramoto} does. It should be mentioned that Chen, Goldenfeld and Oono\cite{goldenfeld2} already indicated that it is a rule that the RG equation gives equations for slow motions which the RP method may also describe. In this paper, we shall confirm their observation in a most general setting for vector equations. Furthermore, one can show \cite{kunihiro3} that the natural extension of the RG method also applies to {\em difference} equations or maps, and an extended envelope equation leads to a reduction of the dynamics even for discrete maps. Thus one sees that the RG method is truly a most promising candidate for a general theory of the reduction of the dynamics, although actual computation is often tediuous in such a general and mechanical method. This paper is organized as follows: In the next section, we desribe the theory of envelopes for curves (or trajectories) in parameter representation. In section 3, the way how to construct envelope surfaces is given when a family of surfaces in three-dimensional space are parametrized with two parameters. In section 4, we give the basic mathematical theorem for the RG method applied to vector fields. This section is partially a recapitulation of a part of Ref.\cite{kunihiro}, although some clarifications are made here. In section 5, some examples are examined in this method, such as the forced Duffing\cite{holmes}, the Lotka-Volterra\cite{lotka} and the Lorenz\cite{lorenz,holmes} equations. The Duffing equation is also an example of non-autonomous one, containing an external force. In section 6, we treat generic equations with a bifurcation; the Landau-Stuart\cite{stuart} and the Ginzburg-Landau equation will be derived in the RG method. The final section is devoted to a brief summary and concluding remarks. In Appendix A, a critical review of the Goldenfeld et al's method is given. In Appendix B, the Duffing equation is solved as a scalar equation in the RG method. \section{Envelopes of trajectories} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} To give a geometrical meaning to the RG equation for systems, one needs to formulate a theory of envelopes of curves which are given in a parameter representation: For example, if the equation is for ${\bf u} (t)=\ ^t(x(t), y(t))$, the solution forms a trajectory or curve in the $x$-$y$ plane with $t$ being a parameter. In this section, we give a brief account of the classical theory of envelopes for curves in the $n$-dimensional space, given in a parameter representation. Let a family of curves $\{{\rm C}_{\alpha}\}_{\alpha}$ in an $n$-dimensional space be given by \begin{eqnarray} {\bf X} (t; \alpha)=\ ^t(X_1(t; \alpha), X_2(t; \alpha), ... , X_n(t;\alpha)), \end{eqnarray} where the point $(X_1, X_2,... X_n)$ moves in the $n$-dimensional space when $t$ is varied. Curves in the family is parametrized by $\alpha$. We suppose that the family of curves $\{{\rm C}_{\alpha}\}_{\alpha}$ has the envelope E: \begin{eqnarray} {\bf X} _E(t)=\ ^t(X_{E1}(t; \alpha), X_{E2}(t; \alpha), ... ,X_{En} (t;\alpha)). \end{eqnarray} The functions ${\bf X} _E(t)$ may be obtained from ${\bf X}(t;\alpha)$ as follows. If the contact point of C$_{\alpha}$ and E is given by $t=t_{\alpha}$, we have \begin{eqnarray} {\bf X}(t_\alpha; \alpha)={\bf X}_E(t_{\alpha}). \end{eqnarray} For each point in E, there exists a parameter $\alpha=\alpha(t)$: Thus the envelope function is given by \begin{eqnarray} {\bf X}_E(t_{\alpha})={\bf X}(t_\alpha; \alpha(t_{\alpha})). \end{eqnarray} Then the problem is to get the function $\alpha(t)$, which is achieved as follows. The condition that E and C$_{\alpha}$ has the common tangent line at ${\bf X}(t_\alpha; \alpha)={\bf X}_E(t_{\alpha})$ reads \begin{eqnarray} \frac{d{\bf X}}{dt}\biggl\vert_{t=t_{\alpha}}= \frac{d{\bf X}_E}{dt}\biggl\vert_{t=t_{\alpha}}. \end{eqnarray} On the other hand, differentiating Eq.(2.4), one has \begin{eqnarray} \frac{d{\bf X}_E}{dt}\biggl\vert_{t=t_{\alpha}}=\frac{\d {\bf X}}{\d t} \biggl\vert_{t=t_{\alpha}}+ \frac{\d {\bf X}}{\d \alpha}\frac{d \alpha}{dt} \biggl\vert_{t=t_{\alpha}} . \end{eqnarray} From the last two equations, we get \begin{eqnarray} \frac{\d {\bf X}}{\d \alpha}={\bf 0}. \end{eqnarray} From this equation, the function $\alpha=\alpha(t)$ is obtained. This is of the same form as the RG equation. Thus one may call the envelope equation the RG/E equation, too. In the application of the envelope theory for constructing global solutions of differential equations, the parameter is the initial time $t_0$, i.e., $\alpha =t_0$. Actually, apart from $t_0$, we have unknown functions given as initial values in the applications. We use the above condition to determine the $t_0$ dependence of the initial values by imposing that $t_0=t$. In section 4, we shall show that the resultant function obtained as the envelope of the local solutions in the perturbation theory becomes an approximate but uniformly valid solution. \section{Envelope Surfaces} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} This section is devoted to give the condition for constructing the envelope surface of a family of surfaces with two parameters in the three-dimensional space. The generalization to the $n$-dimensional case is straightforward. Let $\{ {\rm S}_{\tau_1 \tau_2}\}_{\tau_1\tau_2}$ be a family of surfaces given by \begin{eqnarray} F({\bf r}; \tau_1, \tau_2)=0, \end{eqnarray} and E the envelope surface of it given by \begin{eqnarray} G({\bf r})=0, \end{eqnarray} with ${\bf r}=(x, y, z)$. The fact that E contacts with S$_{\tau_1\tau_2}$ at $(x, y,z)$ implies \begin{eqnarray} G({\bf r})=F({\bf r};\tau_1({\bf r}), \tau_2({\bf r}))=0. \end{eqnarray} Let $({\bf r}+d{\bf r}, \tau_1+d\tau_1, \tau_2+d\tau_2)$ gives another point in E, then \begin{eqnarray} G({\bf r}+d{\bf r})=F({\bf r}+d{\bf r};\tau_1+d\tau_1, \tau_2+d\tau_2)=0. \end{eqnarray} Taking the difference of the two equations, we have \begin{eqnarray} \nabla F\cdot d{\bf r}+\frac{\d F}{\d \tau_1}d\tau_1+ \frac{\d F}{\d \tau_2}d\tau_2=0. \end{eqnarray} On the other hand, the fact that E and S$_{\tau_1\tau_2}$ have a common tangent plane at ${\bf r}$ implies that \begin{eqnarray} \nabla F\cdot d{\bf r}=0. \end{eqnarray} Combining the last two equations, one has \begin{eqnarray} \frac{\d F}{\d \tau_1}d\tau_1+\frac{\d F}{\d \tau_2}d\tau_2=0. \end{eqnarray} Since $d\tau_1$ and $d\tau_2$ may be varied independently, we have \begin{eqnarray} \frac{\d F}{\d \tau_1}=0,\ \ \ \ \frac{\d F}{\d \tau_2}=0. \end{eqnarray} From these equations, we get $\tau_i$ as a function of ${\bf r}$; $\tau _i=\tau_i({\bf r})$. As an example, let \begin{eqnarray} F(x, y, z; \tau_1, \tau_2)={\rm e} ^{-\tau_1y}\{1-y(x-\tau_1)\}+{\rm e} ^{-\tau_2x} \{1-x(y-\tau_2)\}-z. \end{eqnarray} The conditions ${\d F}/{\d \tau_1}=0$ and ${\d F}/{\d \tau_2}=0$ give \begin{eqnarray} \tau_1=x,\ \ \ \tau_2=y, \end{eqnarray} respectively. Hence one finds that the envelope is given by \begin{eqnarray} G(x, y, z)=F(x, y, z; \tau_1=x, \tau_2=y)=2{\rm e} ^{-xy}-z=0, \end{eqnarray} or $z=2{\rm exp}(-xy)$. It is obvious that the discussion can be extended to higher dimensional cases. In Ref.\cite{kunihiro2}, envelope surfaces were constructed in multi steps when the RG method was applied to PDE's. However, as has been shown in this section, the construction can be performed by single step. \setcounter{section}{3} \setcounter{equation}{0} \section{The basis of the RG method for systems } \renewcommand{\theequation}{\thesection.\arabic{equation}} \subsection{ODE's} Let ${\bf X}=\, ^t(X_1, X_2, \cdots , X_n)$ and ${\bf F}({\bf X}, t; \epsilon) =\, ^t(F _1({\bf X}, t; \epsilon)$, $F _2({\bf X}, t; \epsilon),\cdots , F _n({\bf X}, t; \epsilon))$, and ${\bf X}$ satisfy the equation \begin{eqnarray} \frac{d{\bf X}}{dt} = {\bf F}({\bf X} , t; \epsilon). \end{eqnarray} Let us try to have the perturbation solution of Eq.(4.1) around $t=t_0$ by expanding \begin{eqnarray} {\bf X} (t; t_0)= {\bf X} _0(t; t_0) + \epsilon {\bf X} _1(t; t_0) + \epsilon^2{\bf X} _2(t; t_0) \cdots. \end{eqnarray} We suppose that an approximate solution $\tilde{{\bf X}}=\tilde{\bf X} (t; t_0, {\bf W}(t_0))$ to the equation up to $O(\epsilon ^p)$ is obtained, \begin{eqnarray} \frac{d\tilde{\bf X} (t; t_0, {\bf W}(t_0))}{dt}= {\bf F} (\tilde{\bf X} (t), t; \epsilon) + O(\epsilon^p), \end{eqnarray} where the $n$-dimensional vector ${\bf W}(t_0)$ denotes the initial values assigned at the initial time $t=t_0$. Here notice that $t_0$ is arbitrary. Let us construct the envelope function ${\bf X} _E(t)$ of the family of trajectories given by the functions $\tilde{\bf X}(t; t_0, {\bf W}(t_0))$ with $t_0$ parameterizing the trajectories. The construction is performed as follows: First we impose the RG/E equation, which now reads \begin{eqnarray} \frac{d\tilde{\bf X}}{d t_0}={\bf 0}. \end{eqnarray} Notice that $\tilde{\bf X}$ contains the unknown function ${\bf W}(t_0)$ of $t_0$.\footnote{ This means that Eq.(4.4) is a total derivative w.r.t. $t_0$; \begin{eqnarray} \frac{d\tilde{\bf X}}{d t_0}=\frac{\d\tilde{\bf X}}{\d t_0}+ \frac{d{\bf W}}{d t_0}\cdot\frac{\d\tilde{\bf X}}{\d {\bf W}}={\bf 0}.\nonumber \end{eqnarray} } In the usual theory of envelopes, as given in section 2, this equation gives $t_0$ as a function of $t$. However, since we are now constructing the perturbation solution that is as close as possible to the exact one around $t=t_0$, we demand that the RG/E equation should give the solution $t_0=t$, i.e., the parameter should coincide with the point of tangency. It means that the RG/E equation should determine the $n$-components of the initial vector ${\bf W}(t_0)$ so that $t_0=t$. In fact, Eq.(4.4) may give equations as many as $n$ which are independent of each other.\footnote{In the applications given below, the equation is, however, reduced to a scalar equation.} Thus the envelope function is given by \begin{eqnarray} {\bf X} _E(t)=\tilde{\bf X} ( t; t, {\bf W}(t)). \end{eqnarray} Then the fundamental theorem for the RG method is the following:\\ {\bf Theorem:}\ \ {\em ${\bf X}_E(t)$ satisfies the original equation uniformly up to $O(\epsilon ^p)$.} {\bf Proof} \ \ The proof is already given in Eq.(3.21) of Ref.\cite{kunihiro}. Here we recapitulate it for completeness. $\forall t_0$, owing to the RG/E equation one has \begin{eqnarray} \frac{d{\bf X}_E}{dt}\Biggl\vert _{t=t_0} &=& \frac{d\tilde{\bf X}(t; t_0, {\bf W}(t_0))}{d t}\Biggl\vert _{t=t_0}+ \frac{d\tilde{\bf X}(t; t_0, {\bf W}(t_0))}{d t_0}\Biggl\vert _{t=t_0}, \nonumber \\ \ \ &=& \frac{d\tilde{\bf X}(t; t_0, {\bf W}(t_0))}{d t}\Biggl\vert _{t=t_0}, \nonumber \\ \ \ &=& {\bf F} ({\bf X} _E(t_0), t_0; \epsilon) + O(\epsilon^p), \end{eqnarray} where Eq.(4.4) has been used in the last equality. This concludes the proof. \subsection{PDE's} It is desirable to develop a general theory for systems of PDE's as has been done for ODE's. But such a general theorem is not available yet. Nevertheless it {\em is} known that the simple generalization of Eq. (4.4) to envelope surfaces works. Let $\tilde{\bf X} (t, {\bf x} : t_0, {\bf x} _0; {\bf W} (t_0, {\bf x} _0))$ is an approximate solution given in the perturbation theory up to $O(\epsilon^p)$ of a system of PDE's with respect to $t$ and ${\bf x} =(x_1, x_2, \dots , x_n)$. Here we have made explicit that the solution has an initial and boundary value ${\bf W} (t_0, {\bf x} _0)$ dependent on $t_0$ and ${\bf x} _0= (x_{10}, x_{20}, \dots , x_{n0})$. As has been shown in section 3, the RG/E equation now reads \begin{eqnarray} \frac{d \tilde{\bf X}}{d t_0}={\bf 0}, \ \ \ \frac{d \tilde{\bf X}}{d x_{i0}}={\bf 0}, \ \ (i=1, 2, \dots , n). \end{eqnarray} Notice again that $\tilde{\bf X}$ contains the unknown function ${\bf W}(t_0, {{\bf x} _0})$ dependent on $t_0$ and ${\bf x} _0$, hence the derivatives are total derivatives. As the generalization of the case for ODE's, we demand that the RG/E equation should be compatible with the condition that the coordinate of the point of tangency becomes the parameter of the family of the surfaces;i.e., \begin{eqnarray} t_0=t, \ \ \ {\bf x} _0 ={\bf x}. \end{eqnarray} Then the RG/E equation is now reduced to equations for the unknown function ${\bf W}$, which will be shown to be the amplitude equations such as time-dependent Ginzburg-Landau equation. Here we remark that although Eq.(4.7) is a vector equation, the equation to appear below will be reduced to a scalar one; see subsection 6.2. It can be shown, at least for equations treated so far and here, that the resultant envelope functions satisfy the original equations uniformly up to $O(\epsilon^p)$; see also Ref.\cite{kunihiro2}. \section{Simple examples} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} In this section, we treat a few of simple examples of systems of ODE's to show the how the RG method works. The examples are the Duffing\cite{holmes} equation of non-autonomous nature, the Lotka-Volterra\cite{lotka} and the Lorenz\cite{lorenz} equation. The first one may be treated as a scalar equation. Actually, the equation is easier to calculate when treated as a scalar one. We give such a treatment in Appendix B. We shall work out to derive the time dependence of the solution to the Lotka-Volterra equation explicitly. The last one is an example with three degrees of freedom, which shows a bifurcation\cite{holmes}. We shall give the center manifolds to this equation around the first bifurcation of the Lorenz model. A general treatment for equations with a bifurcation will be treated in section 6. \subsection{Forced Duffing equation} The forced Duffing equations are reduced to \begin{eqnarray} \ddot {x}+ 2\epsilon \gamma \dot{x}+ (1+\epsilon \sigma)x + \epsilon hx^3&=& \epsilon f\cos t, \nonumber \\ \ddot {y}+ 2\epsilon \gamma \dot{y}+ (1+\epsilon \sigma)y + \epsilon hy^3 &=& \epsilon f\sin t. \end{eqnarray} Defining a complex variable $z=x+i y$, one has \begin{eqnarray} \ddot {z}+ 2\epsilon \gamma \dot{z}+ (1+\epsilon \sigma)z + \frac{\epsilon h}{2}(3\vert z\vert^2z +{z^{\ast}}^3)= \epsilon f{\rm e}^{it}. \end{eqnarray} We suppose that $\epsilon$ is small. We convert the equation to the system \begin{eqnarray} \biggl(\frac{d}{dt} -L_0\biggl){\bf u} = -\epsilon F(\xi, \eta; t) \pmatrix{0\cr 1}, \end{eqnarray} where \begin{eqnarray} {\bf u} &=& \pmatrix{\xi \cr \eta}, \ \ \ \xi= z, \ \ \eta = \dot{z},\nonumber \\ L_0 &=& \pmatrix{\ 0 & 1\cr -1& 0}, \end{eqnarray} and \begin{eqnarray} F(\xi, \eta; t)=\sigma \xi + 2\gamma \eta \frac{h}{2}(3\vert \xi\vert^2 + {\xi ^{\ast}}^3) - f{\rm e}^{it}. \end{eqnarray} Let us first solve the equation in the perturbation theory by expanding \begin{eqnarray} {\bf u} = {\bf u} _0 + \epsilon {\bf u} _1 + \dots, \end{eqnarray} with ${\bf u} _i=\ ^t(\xi _i, \eta _i)$\, $(i=0, 1, \dots)$. We only have to solve the following equations successively; \begin{eqnarray} \biggl(\frac{d}{dt} -L_0\biggl){\bf u} _0&=&{\bf 0}, \nonumber \\ \biggl(\frac{d}{dt} -L_0\biggl){\bf u} _1&=& - F(\xi _0, \eta _0; t)\pmatrix{0\cr 1}, \end{eqnarray} and so on. The solution of the zero-th order equation is found to be \begin{eqnarray} {\bf u} _0(t; t_0)= W(t_0){\bf U} {\rm e}^{it}, \end{eqnarray} where ${\bf U} $ is an eigenvector belonging to an eigen value $i$ of $L_0$, \begin{eqnarray} L_0{\bf U} = i{\bf U}, \ \ \ {\bf U}=\pmatrix{1\cr i}. \end{eqnarray} The other eigenvector is given by the complex conjugate ${\bf U}^{\ast}$, which belongs to the other eigenvalue $-i$. We have made it explicit that the constant $W$ may be dependent on the initial time $t_0$. In terms of the component, \begin{eqnarray} \xi_0(t;t_0)= W(t_0){\rm e} ^{it}, \ \ \ \eta _0(t;t_0) = iW(t_0){\rm e}^{it}. \end{eqnarray} Inserting these into $F(\xi _0, \eta _0;t)$, one has \begin{eqnarray} F(\xi _0, \eta _0;t)={\cal W}(t_0){\rm e}^{it} + \frac{h}{2}{W^{\ast}}^3{\rm e}^{-3it}, \end{eqnarray} with \begin{eqnarray} {\cal W}(t_0)\equiv (\sigma +2i\gamma)W + \frac{3h}{2}\vert W\vert ^2W -f \end{eqnarray} We remark that the inhomogeneous term includes a term proportional to the zero-th order solution. Thus ${\bf u} _1$ contains a resonance or a secular term as follows; \begin{eqnarray} {\bf u} _1(t; t_0)&=& -\frac{1}{2i}{\cal W}{\rm e}^{it}\{(t-t_0 +\frac{1}{2i}) {\bf U} -\frac{1}{2i}{\bf U} ^{\ast}\} -\frac{h}{16}{W^{\ast}}^3{\rm e}^{-3it}({\bf U} -2{\bf U} ^{\ast}). \end{eqnarray} In terms of the components \begin{eqnarray} \xi _1(t;t_0)&=& \frac{i}{2}{\cal W}{\rm e} ^{it}(t-t_0)+\frac{h}{16}{W^{\ast}}^3 {\rm e}^{-3it}, \nonumber \\ \eta _1(t; t_0)&=& -\frac{{\cal W}}{2}{\rm e}^{it}(t-t_0 - i) -\frac{3i}{16}h{W^{\ast}}^3{\rm e}^{-3it}. \end{eqnarray} Adding the terms, we have \begin{eqnarray} {\bf u}(t)&\simeq& {\bf u}_0(t;t_0) + \epsilon {\bf u}_1(t;t_0), \nonumber \\ \ \ \ &=& W(t_0){\bf U} {\rm e}^{it}- \epsilon \frac{1}{2i}{\cal W}{\rm e}^{it}\{(t-t_0 +\frac{1}{2i}) {\bf U} -\frac{1}{2i}{\bf U} ^{\ast}\} -\epsilon \frac{h}{16}{W^{\ast}}^3{\rm e}^{-3it}({\bf U} -2{\bf U} ^{\ast}), \nonumber \\ \ \ \ &\equiv& \tilde{{\bf u}}(t;t_0). \end{eqnarray} In terms of the components, \begin{eqnarray} \xi(t;t_0)&\simeq&W(t_0){\rm e}^{it} +\epsilon \frac{i}{2}{\cal W}(t_0){\rm e}^{it}(t-t_0) +\epsilon \frac{h}{16}{W^{\ast}}^3{\rm e}^{-3it}\equiv \tilde{\xi},\nonumber \\ \eta(t;t_0)&\simeq&iW(t_0){\rm e}^{it}-\epsilon\frac{{\cal W}}{2}{\rm e}^{it}(t-t_0 -i) -\epsilon \frac{3i}{16}h{W^{\ast}}^3{\rm e}^{-3it}\equiv\tilde{\eta}. \end{eqnarray} Now let us construct the envelope ${\bf u}_E(t)$ of the family of trajectories or curves $\tilde{{\bf u}}(t; t_0)=(\tilde{\xi}(t;t_0), \tilde{\eta}(t;t_0))$ which is parametrized with $t_0$; ${\bf u}_E(t)$ will be found to be an approximate solution to Eq. (5.3) in the global domain. According to section 2, the envelope may be obtained from the equation \begin{eqnarray} \frac{d\tilde{{\bf u}}(t;t_0)}{d t_0}=0. \end{eqnarray} In the usual procedure for constructing the envelopes, the above equation is used for obtaining $t_0$ as a function of $t$, and the resulting $t_0=t_0(t)$ is inserted in $\tilde{{\bf u}}(t;t_0)$ to make the envelope function ${\bf u} _E(t)=\tilde{{\bf u}}(t; t_0(t))$. In our case, we are constructing the envelope around $t=t_0$, so we rather impose that \begin{eqnarray} t_0=t, \end{eqnarray} and Eq.(5.17) is used to obtain the initial value $W(t_0)$ as a function of $t_0$. That is, we have \begin{eqnarray} 0&=&\frac{d\tilde{{\bf u}}(t;t_0)}{d t_0}\biggl\vert _{t_0=t},\nonumber \\ \ &=& \frac{dW}{dt}{\bf U} {\rm e}^{it} +\epsilon\frac{{\cal W}}{2i}{\rm e}^{it}{\bf U} + \epsilon\frac{i}{2}\frac{d{\cal W}}{dt}{\rm e}^{it}\frac{1}{2i}({\bf U} -{\bf U} ^{\ast}) -\frac{3\epsilon h}{16}\frac{dW^{\ast}}{dt}{\rm e}^{-3it}({\bf U} -2{\bf U} ^{\ast}). \end{eqnarray} Noting that the equation is consistent with $dW/dt=O(\epsilon)$, one has \begin{eqnarray} \frac{dW}{dt}&=& i\frac{\epsilon}{2}{\cal W}(t),\nonumber \\ \ \ \ &= & i\frac{\epsilon}{2}\{(\sigma +2i\gamma)W(t)+ \frac{3h}{2} \vert W(t)\vert ^2 W(t) -f\}. \end{eqnarray} This is the amplitude equation called Landau-Stuart equation, which may be also given by the RP method\cite{kuramoto} as a reduction of the dynamics. With this equation, the envelope trajectory is given by \begin{eqnarray} \xi_E(t)&=& W(t){\rm e}^{it} + \epsilon \frac{h}{16}{W^{\ast}}^3{\rm e}^{-3it}, \nonumber \\ \eta _E(t)&=& i(W(t)+\epsilon \frac{1}{2}{\cal W}(t)){\rm e}^{it} -\epsilon \frac{3i}{16}h{W^{\ast}}^3{\rm e}^{-3it}. \end{eqnarray} For completeness, let us examine the stationary solution of the Landau-Stuart equation, briefly; \begin{eqnarray} {\cal W}=(\sigma +2i\gamma)W + \frac{3}{2}\epsilon h\vert W\vert ^2W-f=0. \end{eqnarray} Writing $W$ as \begin{eqnarray} W=A{\rm e} ^{-i\theta}, \end{eqnarray} we have \begin{eqnarray} A^2\biggl[(\frac{3}{2}hA^2+\sigma)^2+4\gamma^2\biggl]=f^2, \end{eqnarray} which describes the jumping phenomena of the Duffing oscillator. \subsection{Lotka-Volterra equation} As another simple example, we take the Lotka-Volterra equation\cite{lotka}; \begin{eqnarray} \dot{x}= ax -\epsilon xy, \ \ \ \ \dot{y}=-by+\epsilon'xy, \end{eqnarray} where the constants $a, b, \epsilon$ and $\epsilon'$ are assumed to be positive. It is well known that the equation has the conserved quantity, i.e., \begin{eqnarray} b\ln\vert x\vert + a\ln \vert y\vert -(\epsilon' x+\epsilon y)={\rm const.}. \end{eqnarray} The fixed points are given by $(x=0, y=0)$ and $(x=b/\epsilon', y=a/\epsilon)$. Shifting and scaling the variables by \begin{eqnarray} x=(b+ \epsilon\xi)/\epsilon', \ \ \ \ y=a/\epsilon + \eta, \end{eqnarray} we get the reduced equation given by the system \begin{eqnarray} \biggl(\frac{d}{dt}- L_0\biggl){\bf u}= -\epsilon\xi\eta\pmatrix{\ 1\cr -1}, \ \ \ \ \end{eqnarray} where \begin{eqnarray} {\bf u} = \pmatrix{\xi\cr \eta},\ \ \ \ L_0=\pmatrix{0 & -b\cr a & \ 0}. \end{eqnarray} The eigen value equation \begin{eqnarray} L_0{\bf U}=\lambda _0{\bf U} \end{eqnarray} has the solution \begin{eqnarray} \lambda _0=\pm i\sqrt{ab}\equiv \pm i\omega, \ \ \ \ {\bf U} =\pmatrix{\, 1\cr \mp i\frac{\omega}{b}}. \end{eqnarray} Let us try to apply the perturbation theory to solve the equation by expanding the variable in a Taylor series of $\epsilon$; \begin{eqnarray} {\bf u}={\bf u}_0+\epsilon{\bf u}_1 +\epsilon^2{\bf u}_2+\cdots, \end{eqnarray} with ${\bf u} _i=\ ^t(\xi _i, \eta_i)$. The lowest term satisfies the equation \begin{eqnarray} \biggl(\frac{d}{dt}- L_0\biggl){{\bf u}}_0={\bf 0}, \end{eqnarray} which yields the solution \begin{eqnarray} {\bf u} _0(t;t_0)=W(t_0){{\rm e}}^{i\omega t}{\bf U} + {\rm c.c.}, \end{eqnarray} or \begin{eqnarray} \xi _0= W(t_0){\rm e} ^{i\omega t} + {\rm c.c.}, \ \ \ \ \eta _0=-\frac{\omega}{b}\big(iW(t_0){\rm e} ^{i\omega t} + {\rm c.c.}\big). \end{eqnarray} Here we have supposed that the initial value $W$ depends on the initial time $t_0$. Noting that \begin{eqnarray} \pmatrix{\ 1\cr -1}=\alpha {\bf U} + {\rm c.c.}, \end{eqnarray} with $\alpha=(1- ib/\omega)/2$, one finds that the first order term satisfies the equation \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _1= \frac{\omega}{b}\biggl[iW^2 {\rm e} ^{2i\omega t} (\alpha {\bf U} + {\rm c.c.}) + {\rm c. c.}\biggl], \end{eqnarray} the solution to which is found to be \begin{eqnarray} {\bf u} _1=\frac{1}{b}\biggl[W^2(\alpha {\bf U} + \frac{\alpha ^{\ast}}{3} {\bf U} ^{\ast}) {\rm e}^{2i\omega t} + {\rm c.c.}\biggl], \end{eqnarray} or \begin{eqnarray} \xi _1 &=&\frac{1}{b}\bigl( \frac{2\omega - ib}{3\omega}W^2{\rm e} ^{2i\omega t} + {\rm c.c.}\bigl), \nonumber \\ \eta _1 &=& -\frac{\omega}{3b^2} \bigl( \frac{2b+ i\omega }{\omega}W^2{\rm e} ^{2i\omega t} +{\rm c.c.}\bigl). \end{eqnarray} The second order equation now reads \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _2 = \frac{1}{3b^2}\biggl[\{ (b-i\omega)\vert W\vert ^2W{\rm e}^{i\omega t} + 3(b+i\omega)W^3{\rm e} ^{3i\omega t}\} + {\rm c.c.}\biggl]\pmatrix{\ 1\cr -1}. \end{eqnarray} We remark that the inhomogeneous term has a part proportional to the zero-th-order solution, which gives rise to a resonance. Hence the solution necessarily includes secular terms as follows; \begin{eqnarray} {\bf u} _2&=& \Biggl[\frac{b-i\omega}{3b^2}\vert W\vert ^2W \biggl\{ \alpha (t-t_0 +i\frac{\alpha^{\ast}}{2\omega}) {\bf U} + \frac{\alpha ^{\ast}}{2i\omega}{\bf U} ^{\ast}\biggl\} {\rm e} ^{i\omega t } \nonumber \\ \ \ \ & & + \frac{b+i\omega}{4b^2i\omega}W^3(2\alpha {\bf U} + \alpha^{\ast}{\bf U} ^{\ast}){\rm e}^{3i\omega t}\Biggl] + {\rm c.c.} . \end{eqnarray} In terms of the components, one finds \begin{eqnarray} \xi _2 &=& \Biggl[ \frac{-i}{6\omega}\frac{b^2+\omega^2}{b^2}\vert W\vert ^2W(t-t_0) {\rm e} ^{i\omega t}\nonumber + \frac{W^3}{8b^2\omega ^2}\{ (3\omega ^2 -b^2) - 4ib\omega \}{\rm e} ^{3i\omega t}\Biggl] + {\rm c.c.} \nonumber \\ \eta _2 &=& \frac{\vert W\vert ^2W}{6b^3} \Biggl[ -(b^2 +\omega^2)(t-t_0) +\frac{1}{\omega} \{2b\omega +i (b^2 -\omega ^2)\}\Biggl]{\rm e}^{i\omega t}\nonumber \\ \ \ \ \ & \ & + \frac{W^3}{8b^3}\{ -4b + \frac{i}{\omega}(3b^2 -\omega ^2)\} {\rm e}^{3i\omega t} + {\rm c.c.} . \end{eqnarray} The RG/E equation reads \begin{eqnarray} \frac{d {\bf u}}{d t_0}={\bf 0}, \end{eqnarray} with $t_0=t$, which gives the equation for $W(t)$ as \begin{eqnarray} \frac{d W}{dt}= - i\epsilon^2 \frac{\omega ^2+b^2}{6\omega b^2}\vert W\vert ^2 W. \end{eqnarray} If we define $A(t)$ and $\theta (t)$ by $W(t)=(A(t)/2i) {\rm exp} i\theta(t)$, the equation gives \begin{eqnarray} A(t)= {\rm const.}, \ \ \ \ \theta (t) = - \frac{\epsilon^2A^2}{24}(1+ \frac{b^2}{\omega ^2})\omega t + \bar{\theta }_0, \end{eqnarray} with $\bar{\theta }_0$ being a constant. Owing to the prefactor $i$ in r.h.s. of Eq. (5.44), the absolute value of the amplitude $A$ becomes independent of $t$, while the phase $\theta$ has a $t$-dependence. The envelope function is given by \begin{eqnarray} {\bf u} _E(t)=\pmatrix{\xi _E(t)\cr \eta _E(t)}= {\bf u} (t, t_0)\Biggl\vert_{t_0=t, \d {\bf u}/\d t_0=0}. \end{eqnarray} In terms of the components, one has \begin{eqnarray} \xi _{_E}&= & A\sin \Theta (t) - \epsilon \frac{A^2}{6\omega}(\sin 2\Theta (t) + \frac{2\omega }{b}\cos 2\Theta (t))\nonumber \\ \ \ \ & \ & -\frac{\epsilon^2 A^3}{32}\frac{3\omega ^2 -b^2}{\omega ^2b^2} (\sin 3\Theta (t) - \frac{4\omega b}{3\omega ^2 -b^2}\cos 3\Theta (t) ), \nonumber \\ \eta _{_E} &=& -\frac{\omega}{b}\Biggl[ \biggl(A - \frac{\epsilon^2A^3}{24}\frac{b^2-\omega ^2}{b^2\omega ^2}\biggl) \cos \Theta (t) - \frac{\epsilon ^2 A^3}{12b\omega}\sin \Theta (t) \nonumber \\ \ \ \ \ & \ & + \epsilon \frac{A^2}{2b}\biggl(\sin 2\Theta (t) - \frac{2b}{3\omega}\cos 2\Theta (t)\biggl) - \frac{\epsilon^2A^3}{8b\omega}\biggl( \sin 3\Theta (t) - \frac{3b^2 -\omega ^2}{4b^2\omega ^2}\cos 3\Theta (t)\biggl)\Biggl], \end{eqnarray} where \begin{eqnarray} \Theta (t) \equiv \tilde {\omega} t + \bar{\theta}_0, \ \ \ \ \tilde {\omega} \equiv \{ 1- \frac{\epsilon^2A^2}{24}(1+ \frac{b^2}{\omega ^2})\}\omega . \end{eqnarray} One sees that the angular frequency is shifted. We identify ${\bf u}_E(t)= (\xi _E(t), \eta _E(t))$ as an approximate solution to Eq.(5.28). According to the basic theorem presented in section 4, ${\bf u}_E(t)$ is an approximate but uniformly valid solution to the equation up to $O(\epsilon^3)$. We remark that the resultant trajectory is closed in conformity with the conservation law given in Eq. (5.26). ``Explicit solutions'' of two-pieces of Lotka-Volterra equation were considered by Frame \cite{frame}; however, his main conceren was on extracting the period of the solutions in an average method. Comparing the Frame's method, the RG method is simpler, more transparent and explicit. The present author is not aware of any other work which gives an explicit form of the solution as given in Eq. (5.47,48). \subsection{The Lorenz model} The Lorenz model\cite{lorenz} for the thermal convection is given by \begin{eqnarray} \dot{\xi}&=&\sigma(-\xi+\eta),\nonumber \\ \dot{\eta}&=& r\xi -\eta -\xi\zeta,\nonumber \\ \dot{\zeta}&=& \xi\eta - b \zeta. \end{eqnarray} The steady states are give by \begin{eqnarray} {\rm (A)}\ \ (\xi, \eta, \zeta)=(0, 0, 0),\ \ \ {\rm (B)}\ \ (\xi, \eta, \zeta)= (\pm \sqrt{b(r-1)},\pm \sqrt{b(r-1)},r-1). \end{eqnarray} The linear stability analysis\cite{holmes} shows that the origin is stable for $0<r<1$ but unstable for $r>1$, while the latter steady states (B) are stable for $1<r<\sigma(\sigma+b+3)/(\sigma -b-1)\equiv r_c$ but unstable for $r>r_c$. In this paper, we examine the non-linear stability around the origin for $r\sim 1$; we put \begin{eqnarray} r=1+\mu \ \ \ {\rm and}\ \ \ \mu =\chi \epsilon^2, \ \ \ \chi={\rm sgn}\mu. \end{eqnarray} We expand the quantities as Taylor series of $\epsilon$: \begin{eqnarray} {\bf u}\equiv \pmatrix{\xi\cr \eta\cr \zeta} = \epsilon {\bf u}_1+\epsilon^2{\bf u}_2 + \epsilon ^3{\bf u}_3 + \cdots, \end{eqnarray} where ${\bf u} _i=\ ^t(\xi_i, \eta_i, \zeta_i) $\ $(i=1, 2, 3, \dots)$. The first order equation reads \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u}_1={\bf 0}, \end{eqnarray} where \begin{eqnarray} L_0=\pmatrix{-\sigma & \sigma & 0\cr 1 & -1 & 0\cr 0 & 0 & -b}, \end{eqnarray} the eigenvalues of which are found to be \begin{eqnarray} \lambda _1=0, \ \ \ \lambda _2= - \sigma -1,\ \ \ \lambda _3= -b. \end{eqnarray} The respective eigenvectors are \begin{eqnarray} {\bf U} _1=\pmatrix{1\cr 1\cr 0}, \ \ \ {\bf U} _2=\pmatrix{\sigma\cr -1\cr 0}, \ \ \ {\bf U} _3=\pmatrix{0\cr 0\cr 1}. \end{eqnarray} When we are interested in the asymptotic state as $t\rightarrow \infty$, one may take the neutrally stable solution \begin{eqnarray} {\bf u} _1(t; t_0)=W(t_0){\bf U}_1, \end{eqnarray} where we have made it explicit that the solution may depend on the initial time $t_0$, which is supposed to be close to $t$. In terms of the components, \begin{eqnarray} \xi_1(t)=W(t_0), \ \ \ \eta_1(t)=W(t_0), \ \ \ \zeta _1(t) =0. \end{eqnarray} The second order equation now reads \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u}_2=\pmatrix{\ \ 0\cr -\xi_1\zeta_1\cr \xi_1\eta_1} = W^2{\bf U}_3, \end{eqnarray} which yields \begin{eqnarray} {\bf u}_2(t)=\frac{W^2}{b}{\bf U}_3, \end{eqnarray} or in terms of the components \begin{eqnarray} \xi_2=\eta_2=0, \ \ \ \zeta_2=\frac{W^2}{b}. \end{eqnarray} Then the third order equation is given by \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u}_3= \pmatrix{\ \ \ 0\cr -\chi\xi_1-\xi_2\zeta_1-\xi_1\zeta_2\cr \xi_2\eta_1+\xi_1\eta_2} = \frac{1}{1+\sigma}(\chi W-\frac{1}{b}W^3)(\sigma{\bf U}_1 -{\bf U}_2), \end{eqnarray} which yields \begin{eqnarray} {\bf u}_3=\frac{1}{1+\sigma}(\chi W-\frac{1}{b}W^3) \{\sigma(t-t_0 + \frac{1}{1+\sigma}){\bf U}_1 - \frac{1}{1+\sigma}{\bf U}_2\}. \end{eqnarray} Thus gathering all the terms, one has \begin{eqnarray} {\bf u} (t;t_0)&=& \epsilon W(t_0){\bf U}_1 + \frac{\epsilon^2}{b}W(t_0)^2{\bf U}_3 \nonumber \\ \ \ \ \ &\ & \ \ \ + \frac{\epsilon ^3}{1+\sigma}(\chi W(t_0) -\frac{1}{b}W(t_0)^3) \{\sigma(t-t_0 + \frac{1}{1+\sigma}){\bf U}_1 - \frac{1}{1+\sigma}{\bf U}_2\}, \end{eqnarray} up to $O(\epsilon ^4)$. The RG/E equation now reads \begin{eqnarray} {\bf 0}&=&\frac{d {\bf u}}{d t_0}\biggl\vert_{t_0=t},\nonumber \\ \ &=& \epsilon \frac{dW}{dt}{\bf U}_1+ 2 \frac{\epsilon^2}{b}W\frac{dW}{dt}{\bf U}_3 -\frac{\sigma}{1+\sigma}\epsilon^3(\chi W - \frac{1}{b}W^3){\bf U}_1, \end{eqnarray} up to $O(\epsilon^4)$. Noting that one may self-consistently assume that $dW/dt=O(\epsilon^2)$, we have the amplitude equation \begin{eqnarray} \frac{dW}{dt}=\epsilon^2\frac{\sigma}{1+\sigma}(\chi W(t) - \frac{1}{b}W(t)^3). \end{eqnarray} With this $W(t)$, the envelope function is given by \begin{eqnarray} {\bf u}_E(t)&=&{\bf u} (t; t_0=t),\nonumber \\ \ \ \ &=& \epsilon W(t){\bf U}_1 + \frac{\epsilon^2}{b}W(t)^2{\bf U}_3 + \frac{\epsilon ^3}{(1+\sigma)^2}(\chi W(t) -\frac{1}{b}W(t)^3) (\sigma {\bf U}_1 -{\bf U}_2), \end{eqnarray} or \begin{eqnarray} \xi_E(t)&=&\epsilon W(t),\nonumber \\ \eta_E(t)&=& \epsilon W(t) +\frac{\epsilon^3}{1+\sigma} (\chi W(t)-\frac{1}{b}W(t)^3),\nonumber \\ \zeta_E(t)&=& \frac{\epsilon^2}{b}W(t)^2. \end{eqnarray} We may identify the envelope functions thus constructed as a global solution to the Lorenz model; according to the general theorem given in section 4, the envelope functions satisfy Eq.(5.49) approximately but uniformly for $\forall t$ up to $O(\epsilon ^4)$. A remark is in order here; Eq.(5.68) shows that the slow manifold which may be identified with a center manifold\cite{holmes} is given by \begin{eqnarray} \eta=(1+ \epsilon^2\frac{\chi}{1+\sigma})\xi - \frac{1}{b(1+\sigma)}\xi^3, \ \ \ \zeta= \frac{1}{b}\xi^2. \end{eqnarray} Notice here that the RG method is also a powefull tool to extract center manifolds in a concrete form. It is worth mentioning that since the RG method utilizes neutrally stable solutions as the unperturbed ones, it is rather natural that the RG method can extract center manifolds when exist. The applicability of the RG method was discussed in \cite{goldenfeld2} using a generic model having a center manifold, although the relation between the exitence of center manifolds and neutrally stable solutions is not so transparent in their general approach. \setcounter{equation}{0} \section{Bifurcation Theory} \renewcommand{\theequation}{\thesection.\arabic{equation}} In this section, we take generic equations with a bifurcation. We shall derive the Landau-Stuart and Ginzburg-Landau equations in the RG method. In this section, we shall follow Kuramoto's monograph\cite{kuramoto} for notations to clarify the correspondence between the RG method and the reductive perturbation (RP) method. \subsection{Landau-Stuart equation} We start with the $n$-dimensional equation \begin{eqnarray} \frac{d{\bf X}}{dt} = {\bf F}({\bf X} , t; \mu ). \end{eqnarray} Let ${\bf X}_0(\mu)$ is a steady solution \begin{eqnarray} {\bf F}({\bf X}_0(\mu) ; \mu)=0. \end{eqnarray} Shifting the variable as ${\bf X} = {\bf X}_0 + {\bf u}$, we have a Taylor series \begin{eqnarray} \frac{d{\bf u}}{dt} = L{\bf u} + M{\bf u} {\bf u} + N{\bf u} {\bf u} {\bf u}+\cdots , \end{eqnarray} where we have used the diadic and triadic notations\cite{kuramoto}; \begin{eqnarray} L_{ij}&=&\frac{\d F_i}{\d X_j}\biggl\vert _{{\bf X} ={\bf X} _0}, \ \ \ (M{\bf u} {\bf u})_i=\sum _{j, k} {1\over 2} \frac{\d ^2 F_i}{\d X_j\d X_k}\biggl\vert _{{\bf X} ={\bf X} _0}u_ju_k, \nonumber \\ (N{\bf u} {\bf u} {\bf u})_i&=& \sum _{j, k, l} \frac{1}{6}\frac{\d ^3 F_i}{\d X_j\d X_k\d X_l} \biggl\vert _{{\bf X} ={\bf X} _0}u_ju_ku_l. \end{eqnarray} We suppose that when $\mu<0$, ${\bf X}_0$ is stable for sufficiently small perturbations, while when $\mu >0$, otherwise. We also confine ourselves to the case where a Hopf bifurcation occurs. We expand $L, M$ and $N$ as \begin{eqnarray} L=L_0 + \mu L_1 + \cdots , \ \ M=M_0 + \mu M_1 + \cdots , \ \ N=N_0 + \mu N_1 + \cdots . \end{eqnarray} The eigenvalues $\lambda^{\alpha}\, (\alpha=1, 2, \dots , n)$ of $L$ are also expanded as \begin{eqnarray} \lambda^{\alpha}=\lambda^{\alpha}_0 + \mu \lambda^{\alpha}_1 + \cdots, \end{eqnarray} with \begin{eqnarray} L_0{\bf U} _{\alpha}= \lambda^{\alpha}_0{\bf U} _{\alpha}. \end{eqnarray} We assume that $\lambda^{1}_0=-\lambda^{2}_0$ are pure imaginary, i.e., $\lambda^{1} _0 = i\omega_0$, and $\Re \lambda^{\alpha}_0<0$ for $\alpha=3, 4, \dots$. Defining $\epsilon$ and $\chi$ by $\epsilon = \sqrt{\vert \mu\vert}$ and $\chi={\rm sgn}\mu$, we expand as \begin{eqnarray} {\bf u} = \epsilon {\bf u} _1 + \epsilon^2 {\bf u}_2 + \epsilon ^3{\bf u}_3 +\cdots. \end{eqnarray} The ${\bf u} _i$ $(i=1, 2, 3, ...)$ satisfies \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _1&=& {\bf 0}, \nonumber \\ \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _2&=& M_0{\bf u} _1{\bf u} _1, \nonumber \\ \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _3&=& \chi L_1{\bf u} _1 + 2M_0{\bf u} _1 {\bf u} _2 + N_0 {\bf u} _1{\bf u} _1 {\bf u} _1, \end{eqnarray} etc. To see the asymptotic behavior as $t\rightarrow \infty$, we take the neutrally stable solution as the lowest one around $t= t_0$; \begin{eqnarray} {\bf u} _1 (t; t_0)=W(t_0){\bf U} {\rm e}^{i\omega_0t} + {\rm c.c.}, \end{eqnarray} where c.c. stands for the complex conjugate. With this choice, we have only two degrees of freedom for the initial value $W(t_0)$. The second order equation is solved easily to yield \begin{eqnarray} {\bf u} _2(t;t_0)= \bigl({\bf V}_{+}W(t_0)^2 {\rm e}^{2i\omega_0 t} + {\rm c. c.} \bigl) + {\bf V}_0\vert W(t_0)\vert ^2, \end{eqnarray} where \begin{eqnarray} {\bf V}_{+}= - (L_0-2i\omega _0)^{-1} M_0{\bf U} {\bf U},\ \ \ {\bf V}_{0}= - 2L_0^{-1} M_0{\bf U} \bar{{\bf U}}, \end{eqnarray} with $\bar{{\bf U}}$ being the complex conjugate of ${\bf U}$.\footnote{ In other sections, we use the notation $a^{\ast}$ for the complex conjugate of $a$. In this section, $^{\ast}$ is used for a different meaning, following ref.\cite{kuramoto}; see Eq. (6.16).} Inserting ${\bf u} _1$ and ${\bf u} _2$ into the r.h.s of Eq. (6.9), we get \begin{eqnarray} \biggl(\frac{d}{dt} - L_0\biggl){\bf u} _3 &=&\bigl\{\chi L_1 W{\bf U} + (2M_0\bar{{\bf U}}{\bf V}_{+} + 3N_0{\bf U} {\bf U}\bar{{\bf U}}) \vert W\vert ^2W\bigl\}{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.}, \nonumber \\ \ \ \ & \equiv & {\bf A}{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.}, \end{eqnarray} where h.h. stands for higher harmonics. So far, the discussion is a simple perturbation theory and has proceeded in the same way as given in the RP method except for not having introduced multiple times. Now we expand ${\bf A}$ by the eigenvectors ${\bf U} _{\alpha}$ of $L_0$ as \begin{eqnarray} {\bf A}=\sum _{\alpha}A_{\alpha}{\bf U} _{\alpha}, \end{eqnarray} where \begin{eqnarray} A_{\alpha}= {\bf U} ^{\ast}_{\alpha}{\bf A}. \end{eqnarray} Here ${\bf U} ^{\ast}_{\alpha}$ satisfies \begin{eqnarray} {\bf U} ^{\ast}_{\alpha}L_0=\lambda^{\alpha}_0L_0 , \end{eqnarray} and is normalized as ${\bf U}^{\ast}_{\alpha}{\bf U}_{\alpha} =1$. Then we get for ${\bf u}_3$ \begin{eqnarray} {\bf u}_3(t;t_0)=\{A_1(t-t_0+\delta){\bf U} + \sum _{\alpha\not= 1}\frac{A_{\alpha}}{i\omega_0 - \lambda_0^{\alpha}} {\bf U}_{\alpha}\}{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.}. \end{eqnarray} The constant $\delta$ is chosen so that the coefficient of the secular term of the first component vanishes at $t=t_0$. Note the appearance of the secular term which was to be avoided in the RP method: The condition for the secular terms to vanish is called the solvability condition which plays the central role in the RP method\cite{kuramoto}. Thus we finally get \begin{eqnarray} {\bf u}(t;t_0)=\{\epsilon W(t_0){\bf U} + \epsilon ^3 \bigl(A_1(t-t_0+ \delta){\bf U} + \sum _{\alpha\not= 1}\frac{A_{\alpha}}{i\omega_0 - \lambda_0^{\alpha}} {\bf U}_{\alpha}\bigl)\}{\rm e}^{i\omega_0t}+ {\rm c.c.} + {\rm h.h.}. \end{eqnarray} The RG/E equation \begin{eqnarray} \frac{d {{\bf u}}}{d t_0}\Biggl\vert_{t_0=t}={\bf 0}, \end{eqnarray} yields \begin{eqnarray} \frac{dW}{dt}&=&\epsilon ^2A_1, \nonumber \\ \ \ &=& \epsilon^2\bigl[ \chi {\bf U} ^{\ast}L_1{\bf U} W+ \{ 2{\bf U}^{\ast}M_0\bar{{\bf U}}{\bf V}_{+} +3{\bf U}^{\ast}N_0{\bf U}\bfU\bar{{\bf U}}\}\vert W\vert^2W\bigl] , \end{eqnarray} up to $O(\epsilon^3)$. Here note that the terms coming from h.h. do not contribute to this order because $dW/dt_0$ is $O(\epsilon ^2)$. The resultant equation is so called the Landau-Stuart equation and coincides with the result derived in the RP method\cite{kuramoto}. \subsection{The Ginzburg-Landau equation} We add the diffusion term to Eq.(6.1); \begin{eqnarray} \frac{d{\bf X}}{dt} = {\bf F}({\bf X} )+ D\nabla ^2 {\bf X}, \end{eqnarray} where $D$ is a diagonal matrix. Let ${\bf X} _0$ be a uniform and steady solution. Shifting the variable ${\bf X} = {\bf X} _0 +{\bf u}$ as before, we have \begin{eqnarray} \frac{d{\bf u}}{dt} = \hat{L}{\bf u} + M{\bf u} {\bf u} + N{\bf u} {\bf u} {\bf u}+\cdots , \end{eqnarray} with \begin{eqnarray} \hat{L} = L +D\nabla ^2. \end{eqnarray} Then using the same expansion as before, we have the same equation for ${\bf u} _1, {\bf u}_2$ and ${\bf u}_3$ as given in Eq.(6.9) with $L_0$ being replaced with $\hat{L}_0\equiv L_0 + D\nabla ^2$. To see the asymptotic behavior as $t\rightarrow \infty$, we take the neutrally stable uniform solution as the lowest one around $t= t_0$ and ${\bf r}={\bf r}_0$; \begin{eqnarray} {\bf u} _1 (t, {\bf r}; t_0, {\bf r}_0) = W(t_0, {\bf r}_0){\bf U} {\rm e}^{i\omega_0t} + {\rm c.c.}. \end{eqnarray} With this choice, we have only two degrees of freedom for the initial value $W(t_0, {\bf r}_0)$. The second order equation is solved easily to yield the same form as that given in Eq.(6.11). Inserting ${\bf u} _1$ and ${\bf u} _2$ into the r.h.s of Eq. (6.9) with $L_0$ replaced with $\hat{L}_0$, we have \begin{eqnarray} \biggl(\frac{\d}{\d t} - \hat{L}_0\biggl){\bf u} _3 &=&\bigl\{\chi L_1 W{\bf U} + (2M_0\bar{{\bf U}}{\bf V}_{+} + 3N_0{\bf U} {\bf U}\bar{{\bf U}}) \vert W\vert ^2W\bigl\}{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.}, \nonumber \\ \ \ \ & \equiv & {\bf A}{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.} . \end{eqnarray} Then we get for ${\bf u}_3$ in the spatially 1-dimensional case, \begin{eqnarray} {\bf u}_3(t;t_0)&=&\biggl[A_1\{c_1(t-t_0+\delta) -\frac{c_2}{2} D^{-1}(x^2 -x_0^2+\delta')\}{\bf U} + \sum _{\alpha\not= 1}\frac{A_{\alpha}}{i\omega_0 - \lambda_0^{\alpha}} {\bf U}_{\alpha}\biggl]{\rm e}^{i\omega_0t} \nonumber \\ \ \ \ &\ & + {\rm c.c.} + {\rm h.h.}, \end{eqnarray} with $c_1+c_2=1$. We have introduced constants $\delta$ and $\delta'$ so that the secular terms of the first component of ${\bf u} _3$ vanish at $t=t_0$ and $x=x_0$. Note the appearance of the secular terms both $t$- and $x$-directions; these terms were to be avoided in the RP method with the use of the solvability condition. Adding all the terms, we finally get \begin{eqnarray} {\bf u}(t;t_0)&=&\biggl[(\epsilon W(t_0,x_0){\bf U} + \epsilon ^3 \{A_1\Big(c_1(t-t_0+\delta)- \frac{c_2}{2} D^{-1}(x^2 -x_0^2+\delta')\Big){\bf U} \nonumber \\ \ \ \ &\ & \ \ \ + \sum _{\alpha\not=1}\frac{A_{\alpha}}{i\omega_0 - \lambda_0^{\alpha}} {\bf U}_{\alpha}\}\biggl]{\rm e}^{i\omega_0t} + {\rm c.c.} + {\rm h.h.}, \end{eqnarray} up to $O(\epsilon ^4)$. The RG/E equation\footnote{See section 3.} \begin{eqnarray} \frac{d {{\bf u}}}{d t_0}\Biggl\vert_{t_0=t}={\bf 0}, \ \ \ \frac{d {{\bf u}}}{d x_0}\Biggl\vert_{x_0=x}={\bf 0}, \ \ \ \end{eqnarray} yields \begin{eqnarray} \frac{\d W}{\d t}=\epsilon ^2c_1A_1 + O(\epsilon^3), \ \ \ \ D\frac{\d W}{\d x}=-\epsilon ^2xc_2A_1 +O(\epsilon^3). \end{eqnarray} We remark that the seemingly vector equation is reduced to a scalar one. Differentiating the second equation once again, we have \begin{eqnarray} D\frac{\d ^2W}{\d x^2}=-\epsilon ^2c_2A_1 +O(\epsilon^3). \end{eqnarray} Here we have utilized the fact that $\d W/\d x= O(\epsilon^2)$. Noting that $c_1+c_2=1$, we finally reach \begin{eqnarray} \frac{\d W}{\d t}- D\frac{\d ^2W}{\d x^2}&=&\epsilon ^2A_1, \nonumber \\ \ \ &=& \epsilon^2\bigl[ \chi {\bf U} ^{\ast}L_1{\bf U} W+ \{ 2{\bf U}^{\ast}M_0\bar{{\bf U}}{\bf V}_{+} +3{\bf U}^{\ast}N_0{\bf U}\bfU\bar{{\bf U}}\}\vert W\vert^2W\bigl] , \end{eqnarray} up to $O(\epsilon^3)$. This is so called the time-dependent Ginzburg-Landau (TDGL) equation and coincides with the amplitude equation derived in the RP method\cite{kuramoto}. We have seen that the RG method can reduce the dynamics of a class of non-linear equations as the RP method can. Therefore it is needless to say that our method can be applied to the Brusselators\cite{brussel}, for instance, and leads to the same amplitude equations as the RP method\cite{kuramoto} does\cite{kunihiro4}. \section{A brief summary and concluding remarks} In this paper, we have shown that the RG method of Goldenfeld, Oono and their collaborators can be equally applied to vector equations, i.e., systems of ODE's and PDE's, as to scalar equations.\cite{goldenfeld1,goldenfeld2,kunihiro,kunihiro2} We have formulated the method on the basis of the classical thoery of envelopes, thereby completed the argument given in \cite{kunihiro,kunihiro2}. We have worked out for some examples of systems of ODE's, i.e., the forced Duffing, the Lotka-Volterra and the Lorenz equation. It has been also shown in a generic way that the method applied to equations with a bifurcation leads to the amplitude equations, such as the Landau-Stuart and the (time-dependent) Ginzburg-Landau equation. Then how about the phase equations\cite{kuramoto}?: The phase equations describe another reduced dynamics. The basis of the reduction of the dynamics by the phase equations lies in the fact that when a symmetry is broken, there appears a slow motion which is a classical counter part of the Nambu-Goldstone boson in quantum field theory. We believe that if the phase equations are related to slow motions of the system at all, the RG method should also leads to the phase equations. It is an interesting task to be done to show that it is the case. There is another class of dynamics than those described by differential equations, i.e., difference equations or discrete maps. It is interesting that a natural extension of the RG/E equation to difference equations leads to a reduction of the dynamics.\cite{kunihiro3} This fact suggests that the RG method pioneered by Goldenfeld , Oono and their collaborators provides one of the most promising candidate for a general theory of the reduction of dynamics, although it is certain that such a mechanical and general method is often tedious in the actual calculations.\footnote{ It should be mentioned that there are other methods \cite{other1,other2} for the dynamical reduction as promising as the RG and RP method are.} As an application of the reduction of difference equations, it will be interesting to see whether the coupled map lattice equations as systems of non-linear difference equations\cite{cml} can be reduced to simpler equations by the RG method. We hope that we can report about it in the near future. \vspace{2.5cm} \centerline{\large{\bf Acknowledgements}} This work is partly a reply to some people who asked if the RG method could be applied to vector equations. The author acknowledges M. Davis and Y. Kuramoto for questions and comments for the previous papers\cite{kunihiro,kunihiro2}. He also thanks J. Matsukidaira and T. Ikeda for indicating the significance of examining vector equations. J Matsukidaira is gratefully acknowledged for useful comments in the earlist stage of this work. He thanks M. Yamaguti and Y. Yamagishi for discussions on difference equations. He is indebted to R. Hirota, H. Matano, Y. Nishiura, J. Satsuma and M. Yamaguti for their interest in this work. He expresses his sincere thanks to M. Yamaguti for his encouragement. {\large {\bf Note added}} After submitting the paper, the author was informed that S. Sasa applied the RG method to derive phase equations in a formal way. The author is grateful to S. Sasa for sending me the TEX file({\tt patt-sol/9608008}) of the paper before its publication. \newpage \setcounter{equation}{0} \centerline{\bf {\large Appendix A}} \renewcommand{\theequation}{A.\arabic{equation}} In this Appendix, we give a quick review of Goldenfeld, Oono and their collaborators' prescription for the RG method. Then we summarize the problems to which a mthematical reasoning is needed in the author's point of view. We take the following simplest example to show their prescription: \begin{eqnarray} \frac{d^2 x}{dt^2}\ +\ \epsilon \frac{dx}{dt}\ +\ x\ =\ 0, \end{eqnarray} where $\epsilon$ is supposed to be small. The exact solution reads \begin{eqnarray} x(t)= A \exp (-\frac{\epsilon}{2} t)\sin( \sqrt{1-\frac{\epsilon^2}{4}} t + \theta), \end{eqnarray} where $A$ and $\theta$ are constant to be determined by an initial condition. Now, let us blindly apply the perturbation theory expanding $x$ as \begin{eqnarray} x(t) = x_0(t) \ +\ \epsilon x_1(t)\ +\ \epsilon ^2 x_2(t)\ +\ ... . \end{eqnarray} The result is found to be\cite{kunihiro} \begin{eqnarray} x(t; t_0)&=& A_0\sin (t +\theta_0) -\epsilon\frac{A_0}{2} (t -t_0)\sin(t+\theta_0) \nonumber \\ \ \ \ & \ \ \ & +\epsilon^2\frac{A_0}{8} \{ (t-t_0)^2\sin(t +\theta_0) - (t-t_0)\cos(t+\theta_0)\} + O(\epsilon^3). \end{eqnarray} Now here come the crucial steps of the Goldenfeld et al's prescription: \begin{description} \item{(i)} First they introduce a dummy time $\tau$ which is close to $t$, and ``renormalize" $x(t; t_0)$ by writing $t - t_0 = t-\tau +\tau - t_0$; \begin{eqnarray} x(t, \tau)&=& A(\tau)\sin (t +\theta(\tau)) -\epsilon\frac{A(\tau)}{2} (t -\tau)\sin(t+\theta(\tau)) \nonumber \\ \ \ \ & \ \ \ & + \epsilon^2\frac{A(\tau)}{8} \{ (t-\tau)^2\sin(t +\theta(\tau)) - (t-\tau)\cos(t+\theta(\tau))\} + O(\epsilon^3), \end{eqnarray} with \begin{eqnarray} x(\tau, \tau)= A(\tau)\sin (\tau +\theta(\tau)). \end{eqnarray} Here $A_0$ and $\theta_0$ have been multiplicatively renormalized to $A(\tau)$ and $\theta(\tau)$. \item{(ii)} They observe that $\tau $ is an arbitrary constant introduced by hand, thus they claim that the solution $x(t, \tau)$ should not depend on $\tau$; namely, $x(t, \tau)$ should satisfy the equation \begin{eqnarray} \frac{d x(t, \tau)}{d \tau}=0. \end{eqnarray} This is similar to the RG equation in the field theory where $\tau$ corresponds to the renormalization point $\tau$; hence the name of the RG method. \item{(iii)} Finally they impose another important but a mysterious condition that \begin{eqnarray} \tau=t. \end{eqnarray} \end{description} From (ii) and (iii), one has \begin{eqnarray} \frac{dA}{d\tau} + \frac{\epsilon}{2} A=0, \ \ \ \frac{d\theta}{d\tau}+\frac{\epsilon^2}{8}=0, \end{eqnarray} which gives \begin{eqnarray} A(\tau)= \bar{A}{\rm e}^{-\epsilon\tau/2}, \ \ \ \theta (\tau)= -\frac{\epsilon^2}{8}\tau + \bar{\theta}, \end{eqnarray} where $\bar{A}$ and $\bar{\theta}$ are constant numbers. Thus, rewriting $\tau$ to $t$ in $x(\tau)$, one gets \begin{eqnarray} x(t,t)= \bar{A}\exp(-\frac{\epsilon}{2} t)\sin((1-\frac{\epsilon ^2}{8})t + \bar{\theta}). \end{eqnarray} They identify $x(t,t)$ with the desired solution $x(t)$. Then one finds that the resultant $x(t)$ is an approximate but uniformly valid solution to Eq.(A.1). In short, the solution obtained in the perturbation theory with the local nature has been ``improved'' by the RG equation Eq.(A.7) to become a global solution. But what have we done mathematically? what is a mathematical meaning of the "renormalization'' replacing $t_0$ with the extra dummy time $\tau$? Can't we avoid the "renormalization'' procedure to solve a purely mathematical problem? Why can we identify $x(t,t)$ with the desired solution?; with $\tau $ being a constant, $x(t, \tau)$ can be a(n) (approximate) solution to Eq. (A.1), can't it? In other words, when the operator $d/dt$ hits the second argument of $x(t, t)$, what happens? In Ref.\cite{kunihiro}, it was shown that the ``renormalization" procedure to introduce the extra dummy time $\tau$ is not necessary. Furthermore, it was clarified that the conditions (ii) and (iii) are the ones to construct the {\em envelope} of the family of the local solutions obtained in the perturbation theory; $x(t; t)$ is the envelope function of the family of curves given by $x(t; t_0)$ where $t_0$ parametrizes the curves in the family. Furthermore, it was shown that the envelope function $x(t,t)$ satisfies the orginal equations approximately but uniformly; the hitting of $d/dt$ on the second argument of $x(t, t)$ does not harm anything. In short, the prescription given by Goldenfeld, Oono and their collaborators is not incorrect, but the reasoning for the prescription is given in \cite{kunihiro,kunihiro2} and will be more refined in the present paper. In Ref.\cite{kunihiro2}, a simplification of the prescription and its mathematical foundation is given for PDE's. \newpage \centerline{\bf {\large Appendix B}} \setcounter{equation}{0} \renewcommand{\theequation}{B.\arabic{equation}} In this Appendix, we solve the forced Duffing equation without converting it to a system. It is easier to solve it in this way than in the way shown in the text. We start with Eq. (2.6) \begin{eqnarray} \ddot {z}+ 2\epsilon \gamma \dot{z}+ (1+\epsilon \sigma)z + \frac{\epsilon h}{2}(3\vert z\vert^2z +{z^{\ast}}^3)= \epsilon f{\rm e}^{it}, \end{eqnarray} where $\epsilon$ is small. Expanding $z$ as \begin{eqnarray} z=z_0+\epsilon z_1 +\epsilon ^2z_2 + \cdots, \end{eqnarray} one gets for $z$ in the perturbation theory \begin{eqnarray} z(t; t_0)= W(t_0){\rm e} ^{it}+ \epsilon (t-t_0)\{f- W(\sigma + 2i\gamma) - \frac{3h}{2}\vert W\vert^2W\} {\rm e} ^{it} + \epsilon \frac{1}{16}{W^{\ast}(t)}^3{\rm e}^{3it} + O(\epsilon^2). \end{eqnarray} Note that there exists a secular term in the first order term. The RG/E equation reads\cite{kunihiro} \begin{eqnarray} \frac{d z}{d t_0}=0 \end{eqnarray} with $t_0=t$, which leads to \begin{eqnarray} \dot{W}= -\epsilon(\sigma +2i\gamma)W - \frac{3}{2}\epsilon h\vert W\vert ^2W+ \epsilon f \end{eqnarray} up to $O(\epsilon ^2)$. Here we have discarded terms such as $\epsilon dW/dt$, which is $O(\epsilon ^2)$ because $dW/dt=O(\epsilon)$. The resultant equation for the amplitude is the Landau-Stuart equation for the Duffing equation. The envelope is given \begin{eqnarray} z_E(t)=z(t; t_0=t)= W(t){\rm e} ^{it} + \frac{\epsilon}{16} {W^{\ast}}^3{\rm e}^{3it} + O(\epsilon^2). \end{eqnarray} We identify $z_E(t)$ with a global solution of Eq.(B.2), and $x(t)={\rm Re}[z_E]$ and $y(t)={\rm Im}[z_E]$ are solutions to Eq.(B.1). As shown in the text, $\forall t$, $z_E(t)$ satisfies Eq.(B.2) uniformly up to $O(\epsilon ^2)$. \newpage \newcommand{N. \ Goldenfeld}{N. \ Goldenfeld} \newcommand{Y.\ Oono}{Y.\ Oono}

proofpile-arXiv_065-513

\section{Introduction} Quantum groups or q-deformed Lie algebra implies some specific deformations of classical Lie algebras. From a mathematical point of view, it is a non-commutative associative Hopf algebra. The structure and representation theory of quantum groups have been developed extensively by Jimbo [1] and Drinfeld [2]. The q-deformation of Heisenberg algebra was made by Arik and Coon [3], Macfarlane [4] and Biedenharn [5]. Recently there has been some interest in more general deformations involving an arbitrary real functions of weight generators and including q-deformed algebras as a special case [6-10]. In the mean time some theoretical physicists studied the q-deformation of quantum mechanic in one dimension [11-16]. The purpose of this paper is to use the $gl_q(n)$-covariant oscillator algebra to construct the q-analogue of the quantum mechanics with harmonic potential in n dimensions. \section{Coherent states of $gl_q(n)$-covariant oscillator algebra} $gl_q(n)$-covariant oscillator algebra is defined as [17] \begin{displaymath} a^{\dagger}_ia^{\dagger}_j =\sqrt{q} a^{\dagger}_j a^{\dagger}_i,~~~(i<j) \end{displaymath} \begin{displaymath} a_ia_j=\frac{1}{\sqrt{q}}a_j a_i,~~~(i<j) \end{displaymath} \begin{displaymath} a_ia^{\dagger}_j=\sqrt{q} a^{\dagger}_ja_i,~~~(i \neq j) \end{displaymath} \begin{displaymath} a_ia^{\dagger}_i =1+q a^{\dagger}_ia_i +(q-1) \Sigma_{k=i+1}^na^{\dagger}_k a_k,~~~(i=1,2,\cdots,n-1) \end{displaymath} \begin{displaymath} a_n a^{\dagger}_n =1+q a^{\dagger}_n a_n, \end{displaymath} \begin{equation} [N_i, a_j]=-\delta_{ij}a_j,~~~[N_i, a^{\dagger}_j]=\delta_{ij}a^{\dagger}_j,~~~(i,j=1,2,\cdots, n ) \end{equation} where we restrict our concern to the case that $q$ is real and $0<q<1$. Here $N_i$ plays a role of number operator and $a_i(a^{\dagger}_i)$ plays a role of annihilation(creation) operator. From the above algebra one can obtain the relation between the number operators and mode opeartors as follows \begin{equation} a^{\dagger}_ia_i=q^{\Sigma_{k=i+1}^nN_k}[N_i], \end{equation} where $[x]$ is called a q-number and is defined as \begin{displaymath} [x]=\frac{q^{x}-1}{q-1}. \end{displaymath} \def|n_1,n_2,\cdots,n_n>{|n_1,n_2,\cdots,n_n>} Let us introduce the Fock space basis $|n_1,n_2,\cdots,n_n>$ for the number operators $N_1,N_2,\cdots, N_n$ satisfying \begin{equation} N_i|n_1,n_2,\cdots,n_n>=n_i|n_1,n_2,\cdots,n_n>,~~~(n_1,n_2,\cdots,n_n=0,1,2\cdots) \end{equation} Then we have the following representation \begin{displaymath} a_i|n_1,n_2,\cdots,n_n>=\sqrt{q^{\Sigma_{k=i+1}^nn_k}[n_i]}|n_1,\cdots, n_i-1,\cdots,n_n> \end{displaymath} \begin{equation} a^{\dagger}_i|n_1,n_2,\cdots,n_n>=\sqrt{q^{\Sigma_{k=i+1}^nn_k}[n_i+1]}|n_1,\cdots, n_i+1,\cdots,n_n>. \end{equation} From the above representation we know that there exists the ground state $|0,0,\cdots,0>$ satisfying $a_i|0,0>=0$ for all $i=1,2,\cdots,n$. Thus the state $|n_1,n_2,\cdots,n_n>$ is obtatind by applying the creation operators to the ground state $|0,0,\cdots,0>$ \begin{equation} |n_1,n_2,\cdots,n_n>=\frac{(a^{\dagger}_n)^{n_n}\cdots(a^{\dagger}_1)^{n_1}}{\sqrt{[n_1]!\cdots [n_n]!}}|0,0,\cdots,0>. \end{equation} If we introduce the scale operators as follows \begin{equation} Q_i=q^{N_i},~~(i=1,2,\cdots,n), \end{equation} we have from the algebra (1) \begin{equation} [a_i,a^{\dagger}_i]=Q_iQ_{i+1}\cdots Q_n. \end{equation} Acting the operators $Q_i$'s on the basis $|n_1,n_2,\cdots,n_n>$ produces \begin{equation} Q_i|n_1,n_2,\cdots,n_n>=q^{n_i}|n_1,n_2,\cdots,n_n> . \end{equation} From the relation $a_i a_j =\frac{1}{\sqrt{q}}a_j a_i,~~(i<j)$, the coherent states for $gl_q(n)$ algebra is defined as \begin{equation} a_i|z_1,\cdots,z_i,\cdots,z_n>=z_i|z_1,\cdots, z_{i},\sqrt{q}z_{i+1},\cdots,\sqrt{q}z_n>. \end{equation} Solving the eq.(9) we obtain \begin{equation} |z_1,z_2,\cdots,z_n>=c(z_1,\cdots,z_n)\Sigma_{n_1,n_2,\cdots,n_n=0}^{\infty} \frac{z_1^{n_1}z_2^{n_2}\cdots z_n^{n_n}}{\sqrt{[n_1]![n_2]!\cdots [n_n]!}}|n_1,n_2,\cdots,n_n> . \end{equation} Using eq.(5) we can rewrite eq.(10) as \begin{equation} |z_1,z_2,\cdots,z_n>=c(z_1,\cdots,z_n) \exp_q(z_na^{\dagger}_n)\cdots\exp_q(z_2a^{\dagger}_2)\exp_q(z_1a^{\dagger}_1)|0,0,\cdots,0>. \end{equation} where q-exponential function is defined as \begin{displaymath} \exp_q(x)=\Sigma_{n=0}^{\infty}\frac{x^n}{[n]!}. \end{displaymath} The q-exponential function satisfies the following recurrence relation \begin{equation} \exp_q(q x)=[1-(1-q)x]\exp_q(x) \end{equation} Using the above relation and the fact that $0<q<1$, we obtain the formula \begin{equation} \exp_q(x) =\Pi_{n=0}^{\infty}\frac{1}{1-(1-q)q^{n}x} \end{equation} Using the normalization of the coherent state , we have \begin{equation} c(z_1,z_2,\cdots,z_n)=\exp_q(|z_1|^2)\exp_q(|z_2|^2)\cdots \exp_q(|z_n|^2). \end{equation} The coherent state satisfies the completeness relation \begin{equation} \int\cdots \int |z_1,z_2,\cdots,z_n><z_1,z_2,\cdots,z_n|\mu(z_1,z_2,\cdots,z_n) d^2z_1 d^2z_2\cdots d^2 z_n=I, \end{equation} where the weighting function $\mu(z_1,z_2,\cdots,z_n)$ is defined as \begin{equation} \mu(z_1,z_2,\cdots,z_n)=\frac{1}{\pi^2}\Pi_{i=1}^n\frac{\exp_q(|z_i|^2)} {\exp_q(q|z_i|^2)}. \end{equation} In deriving eq.(15) we used the formula \begin{equation} \int_0^{1/(1-q)}x^n \exp_q(q x)^{-1} d_q x=[n]! \end{equation} \section{q-Deformed Weyl-Heisenberg Group} The purpose of this section is to explain what is the q-analogue of the q-deformed Weyl-Heisenberg group. From the algebra (1) we obtain \begin{displaymath} a_i f(a^{\dagger}_i)=f(q a^{\dagger}_i) a_i +(Df)(a^{\dagger}_i) Q_{i+1}\cdots Q_n \end{displaymath} \begin{equation} a_n f(a^{\dagger}_n)=f(q a^{\dagger}_n) a_n +(Df)(a^{\dagger}_n) , \end{equation} where $D$ is called q-derivative and defined as \begin{displaymath} DF(x)=\frac{F(x)-F(qx)}{x(1-q)}. \end{displaymath} Putting $f(x)=\exp_q(tx)$ we have \begin{equation} a_i \exp_q(ta^{\dagger}_i) =\exp_q(qta^{\dagger}_i)a_i +t\exp_q(ta^{\dagger}_i)Q_{i+1}\cdots Q_n. \end{equation} Using the formula (12) we have \begin{equation} a_i^n\exp_q(ta^{\dagger}_i) =\exp_q(ta^{\dagger}_i)(a_i+tQ_iQ_{i+1}\cdots Q_n)^n \end{equation} and thus \begin{equation} \exp_q(s_ia_i)\exp_q(t_ia^{\dagger}_i)=\exp_q(t_ia^{\dagger}_i)\exp_q(s_ia_i+s_it_iQ_iQ_{i+1}\cdots Q_n). \end{equation} Taking account of $[a_i,Q_i]_q=a_iQ_i -q Q_i a_i=0$,we have \begin{equation} \exp_q(s_ia_i)\exp_q(t_ia^{\dagger}_i)=\exp_q(t_ia^{\dagger}_i)\exp_q(s_it_iQ_iQ_{i+1}\cdots Q_n)\exp_q(s_ia_i). \end{equation} If we muliply above equations from $i=1$ to $n$, we obtain the q-deformed Weyl-Heisenberg relation \begin{equation} \Pi_{i=1}^n\exp_q(s_ia_i)\exp_q(t_ia^{\dagger}_i)= \Pi_{i=1}^n\exp_q(t_ia^{\dagger}_i)\exp_q(s_it_iQ_iQ_{i+1}\cdots Q_n)\exp_q(s_ia_i). \end{equation} \section{q-deformed quantum mechanics in n dimensions} It is intersting to study the q-deformed harmonic oscillator system in n dimensions. In order to formulate it we define the position and momentum operators \begin{displaymath} X_i=\frac{1}{\sqrt{2}}(a_i +a^{\dagger}_i) \end{displaymath} \begin{equation} P_i=-\frac{i}{\sqrt{2}}(a_i -a^{\dagger}_i). \end{equation} Then the Hamiltonian of q-deformed harmonic oscillator in n dimensions is given by \begin{equation} H=\Sigma_{i=1}^n H_i, \end{equation} where \begin{equation} H_i=\frac{1}{2}(P_i^2 +X_i^2) =\frac{1}{2} (a_ia^{\dagger}_i+a^{\dagger}_i a_i). \end{equation} Now, the q-cannonical commutation relation can be expressed by \begin{equation} X_iP_i-P_iX_i =i(\frac{q+1}{2})^{i-n-1} +i(q-1)\Sigma_{k=i}^n (\frac{q+1}{2})^{i-k-1}H_k. \end{equation} Expressing $H_i$'s in terms of $Q_i$'s operators, we get \begin{displaymath} H_i=\frac{q+1}{2(q-1)}Q_iQ_{i+1}\cdots Q_n -\frac{1}{q-1}Q_{i+1}Q_{i+2}\cdots Q_n, ~~(i=1,2,\cdots, n-1) \end{displaymath} \begin{equation} H_n=\frac{q+1}{2(q-1)}Q_n -\frac{1}{q-1}. \end{equation} Thus the Hamiltonian is given by \begin{equation} H=\frac{Q-1}{q-1}+\frac{1}{2}\Sigma_{i=1}^nQ_iQ_{i+1}\cdots Q_n \end{equation} where \begin{displaymath} Q=Q_1Q_2 \cdots Q_n \end{displaymath} Thus we have \begin{equation} H|n_1,\cdots,n_n>=E(n_1,\cdots,n_n)|n_1,\cdots,n_n> \end{equation} where the energy spectrum is given by \begin{equation} E(n_1,\cdots,n_2)=[n_1+\cdots+n_n]+\frac{1}{2}\Sigma_{i=1}^nq^{n_1+\cdots +n_n} \end{equation} \section{Concluding Remark} In this paper we used $gl_q(n)$-covariant oscillator algebra to obtain its coherent state and showed the completeness relation. Moreover we construct the q-deformed quantum mechanical hamiltonian in n dimensions by using $gl_q(n)$-covariant oscillators. In conclusion, it was known that we can obtain the q-analogue of n-dimensional Schroedinger equation with harmonic potential by using $gl_q(n)$-covariant oscillator system. \section*{Acknowledgement} This paper was supported by the KOSEF (961-0201-004-2) and the present studies were supported by Basic Science Research Program, Ministry of Education, 1995 (BSRI-95-2413).

proofpile-arXiv_065-514

proofpile-arXiv_065-515

\section{INTRODUCTION} Simulations of full QCD with Wilson fermions at zero temperature so far have been carried out on lattices of size $\le 16^{3}\times 32$, physical volumes $<$ 1.5 (fm)$^{3}$ and ratios of $\frac{m_{\pi}}{m_{\rho}} > 0.6$ \cite{SESAM1}. The latter quantity is a monitor for the closeness to the chiral point. It has been demonstrated \cite{GERO} that a statistically significant full QCD (reference) sample can be generated in one year's runtime on a 256-node APE computer, at $\frac{m_{\pi}}{m_{\rho}}=0.71$. This however corresponds still to a rather heavy quark mass $m_q$: by use of chiral perturbation theory we can mock a fictitious "physical" pseudoscalar meson containing two strange quarks, with mass ratio of the size quoted, $ \frac{m_{ps}}{m_{\phi}}\approx \frac{\sqrt{2m^{2}_{K}}}{m_{\phi}}=0.69$. Thus, in order to quantify light sea quark effects in full QCD, one would rather prefer to work on larger volumes that accommodate a large $\pi$-correlation length both in physical and lattice units. This clearly asks for simulations on lattices $>16^3$ and $\beta\ge 5.6$. In this note, we describe the {{\color{grey3\ project, which is geared to push QCD simulations with standard Wilson fermions further {\em towards} the {\em chiral} {\em limit}, i.e. beyond $\frac{m_{\pi}}{m_{\rho}}<0.6$ and at appropriate volumes. The 512-node APE Tower offers sufficient memory to handle a $24^3\times40$ lattice. With its CPU-power it can drive an optimized HMC at sufficient speed ({\em i}) to increase the lattice size by more than a factor of 4 compared to the previous standards described above, ({\em ii}) to go more chiral, i.e., cope with worse conditioned fermion matrices. In our exploratory study, we are guided by the experiences described in \cite{SESAM1,SESAM2}, taking advantage of algorithmic achievements such as improved inverters \cite{FROMMER1} and new parallel preconditioning techniques \cite{FROMMER2}. We shall report on a Hybrid Monte Carlo simulation on a $24^3 \times 40$ lattice at $\beta=5.6$ and two $\kappa$-values, 0.1575 and 0.158. \section{HOW CHIRAL?} The $24^3\times 40$ lattice allows to increase $\xi_{\pi}$ by a factor of 1.5 compared to Ref.~\cite{SESAM3}, which should suffice to target for $\frac{m_{\pi}}{m_{\rho}}$ in the range of $.5$ -- $.6$. Concerning physical volumes and scales we benefit from the increasing $\Delta\beta$-shift \cite{SESAM2} as we go to smaller bare quark masses and choose $\beta=5.6$. \begin{figure}[htb] \tinypicture{find.eps} \tinypictur{kappa_ex.eps} \caption{ Fixing $\kappa_{\mbox{sea}}$ and estimate for $\kappa_{\rm c,v}$ compared to the results from Ref~\cite{SESAM3}. \label{EXTRA}} \end{figure} For the determination of $\kappa_{\rm sea}$, we extrapolated the relation $m_qa=\frac{1}{2}\Big( \frac{1}{\kappa}- \frac{1}{\kappa_{\rm c}} \Big)$ on the data set of Ref.~\cite{SESAM3} to $m_qa=0.023$, cf.\ \fig{EXTRA}a, where $\frac{\xi_{\pi}}{24a}$ is estimated from the mass trajectory to be about $.23$ of the spatial extension. This value is small enough to protect us from finite size effects. We thus will work at $\kappa=0.1580$. To put this parameter choice into perspective, we sketched in \fig{EXTRA}b the approach of the critical $\kappa$ at fixed $\kappa_{\rm sea}$, $\kappa_{\rm c,v}$, \cite{SESAM3}, towards $\kappa_{\rm c}$, the locus of which is $\kappa_{\rm sea}=\kappa_{\rm c}$ (the diagonal line). The cross marks the current estimate for our working point. Notice that our parameter choice, $\kappa_{\rm sea}=0.158$, appears to be reasonably positioned within the `chirality gap'. In the lattice discretization of the fermionic action, we switched from the usual o/e representation to the full representation of $M={\bf 1}-\kappa D$, employing a new SSOR preconditioning scheme \cite{FROMMER2,FROMMER3,SESAM4}. In particular on the APE machine, this method offers an overall gain of 100 \% in execution time, as seen from \tab{TIMES}. \begin{table}[htb] \footnotesize \begin{tabular}{lllll} \hline Algo & & $\kappa=0.1575$ & $\kappa=0.1580$ \\ \hline o/e &t/s & 8200 & $-$ \\ SSOR &t/s & 3800 & 9100 \\ \hline \vspace{8pt} \end{tabular} \caption{Average time to generate 1 trajectory on the APE100 Tower. \label{TIMES}} \end{table} \section{SIMULATION} We have tuned the HMC timestep to achieve acceptance rates larger than $60$ \%. For the SSOR scheme with twice as many degrees of freedom as in the o/e case, we chose $T=0.5$. For this trajectory length, in the production runs, the 32 bit machine precision induces a reversibility error $\delta(\Delta S)$ in the range of just 2 \% of the average $\Delta S$ for an inversion residue of $r= 10^{-8}$. This is due to {\em local} computations, {\em global} summations being carried out in emulated double precision arithmetic. It should be said that the impact of this error of $\Delta S$ onto the canonical distribution deserves further attention. The chosen HMC run parameters are given in \tab{PARAMETER}. \begin{table}[htb] \footnotesize \begin{tabular}{llllllll} \hline Algo&$\kappa$&$T$&$dt$ &acc/{\footnotesize\%} &$r$ \\ \hline o/e &$0.1575$&1 &0.008&70 &$10^{-8}$\\ SSOR &$0.1575$&0.5&0.004&72 &$10^{-8}$\\ \hline SSOR &$0.1580$&0.5&0.004&66 &$10^{-8}$\\ \hline \vspace{8pt} \end{tabular} \caption{Parameters of the HMC simulation. \label{PARAMETER}} \end{table} During the thermalization phase we carefully approached the lowest quark mass in a near adiabatic fashion, to protect the system from oscillating through the shielding transition. We forked the run into two $\kappa$-branches after an initial thermalization of 480 trajectories. The production status reported is given by 830 trajectories (out of which 350 are thermalized) for $\kappa=0.1575$ and 1500 trajectories (out of which 750 are thermalized) for $\kappa=0.1580$. \section{PRELIMINARY RESULTS} We have performed first tentative measurements of the autocorrelation $C(t)$ of the plaquette. In \fig{AUTO}, we plot the autocorrelation function for the plaquette at $\kappa=0.158$. On the large lattice volume we can profit from a substantial self averaging effect suppressing the fluctuations of the plaquette as well as of other intensive quantities. The plot indicates that the autocorrelation times come out surprisingly small and might settle well below $\tau_{\mbox{int}}=50$ for the plaquette. \begin{figure}[htb] \smallpicture{c_plaq.eps} \caption{ Autocorrelation function of plaquette. \label{AUTO}} \end{figure} We mention that the autocorrelation function for light meson masses looks similar. Using 22 configurations drawn from a sample of 600 trajectories we have computed the potential following Ref.~\cite{SESAM2}. We performed a 30 step APE smearing and evaluated the potential for a time extension of 5 where a plateau in the local mass is emerging. We have fitted for a "string tension" in the range up to 1 fm. In \tab{RESULTS} we quote a preliminary estimate for the ensueing scale and physical lattice volume $V_{s}$. \begin{table}[htb] \footnotesize \begin{tabular}{llllllll} \hline $a^{-1}$ & $V_{s}$ &$m_{\pi}$ SL &$m_{\rho}$ SL &$\frac{m_{\pi}}{m_{\rho}}$ \\ \hline $2.37(1)$ GeV & 2 fm & 0.178(5) &0.32(2) &0.56(4) \\ \hline \vspace{8pt} \end{tabular} \caption{Results for the lattice scale from potential, and masses in lattice units. \label{RESULTS}} \end{table} We monitored local meson masses to position the run with respect to chirality: on a sample of 19 configurations we retrieve a rough first guesstimate of $m_{\pi}a$ and $m_{\rho}a$, see \tab{RESULTS}. Our findings suggest $\frac{m_{\pi}}{m_{\rho}}$ to be 0.56(4). \section{CONCLUSIONS AND OUTLOOK} In the {{\color{grey3-feasibility study, we find that $\frac{m_{\pi}}{m_{\rho}}$-ratio appears to reach the target region indeed, where $\frac{\xi_{\pi}}{a}=5.6<0.25 \times V_s^{\frac{1}{3}}$. The lattice resolution is increased (with respect to the small lattice results) to $a^{-1}=2.37$ GeV. We are encouraged by the observed autocorelation times and expect $>$ 50 independent configurations from 8 months future runtime on APE Towers. \section*{Acknowledgements} We thank Prof. Mathis at ENEA/Italy and his staff for kind support. We thank the Caspur group of La Sapienzia/Roma for help. Th. L. and K. S. acknowledge the DFG-grant Schi 257/5-1.

proofpile-arXiv_065-516

\section{Introduction} The minimal supersymmetric standard model (MSSM) is one of the strongest candidate for physics beyond the standard model (SM). Since the MSSM naturally contains two Higgs doublets, $CP$ could be violated in the Higgs sector both spontaneously and explicitly. The possibility of spontaneous $CP$ violation in the MSSM, which is caused by the non-trivial phase of vacuum expectation values (VEVs), was first discussed in Refs.\cite{MAEKAWA}\cite{POMAROL}. They have shown the following; \\ (i) Spontaneous $CP$ violation is caused essentially by the chargino and the neutralino radiative corrections. \\ (ii) This scenario is excluded from the experiment since the "pseudoscalar" mass is of $O(\sqrt{\lambda_5} v)$ which is about 5 GeV. \par In this paper we reconsider (i) and (ii), and results are the following; \\ \par \noindent {\bf (a)} $CP$ violating vacuum discussed in Ref.\cite{MAEKAWA} and \cite{POMAROL} is not stable. It becomes stable when top and stop contributions are added, provided that the stop mass is larger than 180 GeV in the limit of small stop left-right mixing. \\ \par \noindent {\bf (b)} Result (ii) was obtained in the case where there is no left-right mixing of stop. If the effect of the stop left-right mixing is included, Higgs masses depend on new parameters $A_t$ and $\mu$. Thus, there might be the experimental allowed region for spontaneous $CP$ violation scenario in the MSSM. Unfortunately, however, the numerical analysis shows that $CP$ violating vacuum is unstable in the parameter region of $A_t /m_{\tilde t}$ and/or $\mu /m_{\tilde t} \geq O(1/3)$, and the situation is almost same as the small left-right mixing limit case when $A_t /m_{\tilde t}$ and $\mu /m_{\tilde t} \leq O(1/3)$. We find that the result (ii) does not change even when we consider the possibility of left-right mixing. \\ \par \noindent {\bf (c)} In this paper we consider only the region $\tan \beta \geq 1$ \footnote{Parameter space $\tan \beta < 1$ is strongly disfavored in low energy SUSY models\cite{NOJIRI} as pointed out in Ref.\cite{POMAROL}.}. As for the experimental constraints, since spontaneous $CP$ violation makes scalars and a pseudoscalar mix, it is not accurate to compare the lightest Higgs mass predicted in this scenario to the lower limit of pseudoscalar mass 24.3 GeV in Ref.\cite{PDG} in which they assume that scalar and pseudoscalar do not mix. Therefore, we need to consider the precise experimental constraints from (A): $Z \rightarrow h_1h_2$ and (B): $Z \rightarrow h_i l^+l^-$ $(i=1,2)$. \\ \par \noindent {\bf (d)} We also discuss briefly explicit $CP$ violation in the Higgs sector of the MSSM in this paper. Since it is also caused by the radiative correction, its effect is too small to influence the phenomenology. \\ \par We therefore conclude that this scenario is excluded and the Higgs sector can not, by itself, trigger $CP$ violation. \par In Section 2, we discuss spontaneous $CP$ violation. Section 3 gives summary and discussion. In Appendix, we show explicit $CP$ violation in the Higgs sector. \section{Reanalysis of Spontaneous $CP$ Violation in the MSSM} The most general two Higgs doublet model potential\cite{THDM} is given by \begin{eqnarray} \label{TDLEE} V(H_1, H_2) &=& m_1^2 |H_1|^2 + m_2^2 |H_2|^2 - (m_{12}^2 H_1 H_2 + {\rm h.c.}) + \lambda_1 |H_1|^4 + \lambda_2 |H_2|^4 \nonumber \\ &+& \lambda_3 |H_1|^2|H_2|^2 + \lambda_4 |H_1 H_2|^2 + {1\over2}[\lambda_5 (H_1 H_2)^2 + {\rm h.c.}] \\ &+& {1\over2}[\lambda_6 (H_1 H_2)|H_1|^2 + {\rm h.c.}] + {1\over2}[\lambda_7 (H_1 H_2)|H_1|^2 + {\rm h.c.}] . \nonumber \end{eqnarray} $H_1$ and $H_2$ are Higgs doublet fields denoted as \begin{equation} H_1=\left( \begin{array}{c} H_1^{0} \\ H_1^- \\ \end{array} \right) , \qquad H_2=\left( \begin{array}{c} H_2^+ \\ H_2^0 \\ \end{array} \right), \end{equation} with \begin{equation} H_1H_2=H_1^0H_2^0-H_1^-H_2^+. \end{equation} Quartic couplings $\lambda_i$s $(i = 1 \sim 4)$ are written by gauge couplings in the MSSM as \begin{equation} \label{lambda1234} \lambda_1 = \lambda_2 = {1\over8}(g^2+g'^2), \;\; \lambda_3 = {1\over4}(g^2-g'^2), \;\; \lambda_4 = -{1\over2}g^2 , \end{equation} where $g$ and $g'$ are gauge couplings of $SU(2)_L$ and $U(1)_Y$, respectively. Parameters $m_1$, $m_2$, and $m_{12}$ are arbitrary determined by supersymmetric Higgs mass $\mu$ and soft SUSY breaking parameters. Coupling $\lambda_{5}$ get non-zero positive value by radiative corrections of the chargino and the neutralino\cite{MAEKAWA}\cite{POMAROL}. The value of $\lambda_5$ is \begin{equation} \label{lambda5} \lambda_5 = {g^4 \over 32 \pi^2} \sim 5 \times 10^{-4}, \end{equation} in the limit of small squark left-right mixings and SUSY breaking mass parameter $B$, and equal mass limit of charginos and neutralinos\cite{POMAROL}. Couplings $\lambda_{6}$ and $\lambda_{7}$ are expected to be the same order of $\lambda_5$. Parameters $m_{12}^2$ and $\lambda_{5 \sim 7}$ are all complex in general. In the case that the Higgs sector has $CP$ symmetry, \begin{equation} \label{lambdarelation} {\rm Im}(\lambda_5^* m_{12}^4) = {\rm Im}(\lambda_5^* \lambda_6^2) = {\rm Im}(\lambda_5^* \lambda_7^2) =0 \end{equation} are satisfied, and all these parameters can be real by the redefinition of Higgs fields. Then we can set all parameters to be real in spontaneous $CP$ violation scenario. As for the explicit $CP$ violation, Eq.(\ref{lambdarelation}) is not held as shown in Appendix. Eq.(\ref{lambdarelation}) shows that the Higgs potential of the MSSM is automatically $CP$ invariant in the tree level because $\lambda_{5 \sim 7}^{\rm (tree)}=0$ . \par Assuming that the charged Higgs does not get VEV, we denote VEVs of neutral components as \begin{equation} \label{VEVs} \langle H_1^0 \rangle = v_1 , \;\;\;\;\; \langle H_2^0 \rangle = v_2 e^{i \phi} , \end{equation} where $v_1$ and $v_2$ are real and positive parameters which satisfy $v \equiv \sqrt{v_1^2+v_2^2}=174$ GeV. We define fields around this vacuum as \begin{eqnarray} \label{fielddifinition0} H_1^0 &=& v_1 + {1 \over \sqrt{2}} (S_1+i \sin \beta A), \nonumber \\ \label{fielddifinition1} H_2^0 &=& v_2 e^{i \phi} + {1 \over \sqrt{2}}e^{i \phi} (S_2+i \cos \beta A), \end{eqnarray} where $S_1$ and $S_2$ are scalar fields and $A$ is a pseudoscalar field, and $\tan \beta = v_2/v_1$. \par The stationary condition of the phase \begin{equation} \label{stationaryconditionp} \left.{\partial V \over \partial \phi}\right| = 0 \end{equation} induces \begin{equation} \label{theta} \sin \phi = 0 \;\; {\rm or}\;\; \cos \phi = {2 m_{12}^2 - \lambda_6 v_1^2 - \lambda_7 v_2^2 \over 4 \lambda_5 v_1 v_2}. \end{equation} The solution which has non-vanishing phase is derived from the second equation of Eq.(\ref{theta}) and we denote $\phi = \phi_0$ {}for this case. The necessary condition for spontaneous $CP$ violation is \begin{equation} \left. \langle V \rangle \right|^{\langle H_1 \rangle = v_1}_{ \langle H_2 \rangle = v_2 (-v_2)}\; > \; \left. \langle V \rangle \right|^{\langle H_1 \rangle = v_1}_{ \langle H_2 \rangle = v_2 e^{i \phi_0}} \end{equation} {}for $\phi_0 \neq 0, \pi$, which derives \begin{equation} \label{Ncondition} \lambda_5 > 0, \;\;\;\;\;\; \left| {2 m_{12}^2 - \lambda_6 v_1^2 - \lambda_7 v_2^2 \over 4 \lambda_5 v_1 v_2} \right| < 1. \end{equation} It means that $m_{12}^2$ must be small of $O(\lambda_{5} v^2)$ in order to get spontaneous $CP$ violation. Eq.(\ref{Ncondition}) is just a necessary and not the sufficient condition for spontaneous $CP$ violation. We must not forget that there exist another stationary point with vanishing phase corresponding to the first equation of Eq.(\ref{theta}). \par By the use of stationary conditions of VEVs \begin{equation} \label{stationarycondition} \left.{\partial V \over \partial v_i}\right| = 0 \quad (i=1,2) , \\ \end{equation} we eliminate $m_1^2$ and $m_2^2$ as \begin{eqnarray} \label{m12} m_1^2 &=& {\overline{g^2} \over 2}(v_2^2 - v_1^2) + \lambda_5 v_2^2 - \left({3 \lambda_6 v_1 v_2 \over 2} + {\lambda_7 v_2^3 \over 2 v_1} \right) \cos \phi_0 , \\ m_2^2 &=& {\overline{g^2} \over 2}(v_1^2 - v_2^2) + \lambda_5 v_1^2 - \left({\lambda_6 v_1^3 \over 2 v_2} + {3 \lambda_7 v_1 v_2 \over 2} \right) \cos \phi_0 , \end{eqnarray} where $\overline{g}^2 \equiv (g^2+g'^2)/2$. Now we can decide specific Higgs potential with definite values of $m_1^2, m_2^2$, and $m_{12}^2 = 2 \lambda_5 v_1 v_2 \cos \phi_0 + (\lambda_6 v_1^2 + \lambda_7 v_2^2)/2$, which should have the stationary point at the non-trivial phase $\phi = \phi_0$. Expanding fields around this point as Eq.(\ref{fielddifinition0}), mass spectra become \begin{eqnarray} \label{mat2} M_{S1-S1}^2 &=& \overline{g^2} v_1^2 + 2 ( \lambda_5 v_2^2 \cos^2 \phi_0 + \lambda_6 v_1 v_2 \cos \phi_0), \\ M_{S2-S2}^2 &=& (\overline{g^2} + \Delta) v_2^2 + 2 ( \lambda_5 v_1^2 \cos^2 \phi_0 + \lambda_7 v_1 v_2 \cos \phi_0), \\ M_{S1-S2}^2 &=& - {\overline{g^2}\over 2} v_1 v_2 - 2 \lambda_5 v_1 v_2 \sin^2 \phi_0 + \lambda_6 v_1^2 \cos \phi_0 + \lambda_7 v_2^2 \cos \phi_0, \\ M_{S1-A}^2 &=& -(2 \lambda_5 \cos \phi_0 v_2 + \lambda_6 v_1) v \sin \phi_0 , \\ M_{S2-A}^2 &=& -(2 \lambda_5 \cos \phi_0 v_1 + \lambda_7 v_2) v \sin \phi_0 , \\ \label{mat222} M_{A-A}^2 &=& 2 \lambda_5 v^2 \sin^2 \phi_0 . \end{eqnarray} $\Delta$ represents the top and stop effects \begin{equation} \label{DELTA} \Delta \equiv {3 h_t^4 \over 4 \pi^2} \; {\rm ln}\: {m_t^2 + m_{\tilde t}{}^2 \over m_t^2} \;, \end{equation} where $m_{\tilde t}$ is the soft breaking stop mass parameter. Eq.(\ref{DELTA}) is derived from the one loop effective potential\cite{COLMAN} including only top and stop contributions, that is \begin{equation} \label{topeffect} V_{\rm top} = {3 \over 16 \pi^2} \left[ (h_t^2 |H_2|^2 + m_{\tilde t}^2)^2 {\rm ln} {(h_t^2 |H_2|^2 + m_{\tilde t}^2) \over Q^2} - h_t^4 |H_2|^4 {\rm ln} {h_t^2 |H_2|^2 \over Q^2} \right] \ , \end{equation} where stop left-right mixing are neglected. The values of $M_{S1-A}^2$, $M_{S2-A}^2$ and $M_{A-A}^2$ are same as calculated by Pomarol\cite{POMAROL}. \par Next we show that if top and stop radiative corrections are not included, $CP$ violating vacuum with non-vanishing phase can not be a global minimum. We expand the determinant by small parameters $\lambda_{5 \sim 7}$ as \begin{equation} \label{detsp} {\rm Det} M_{ij}^2 = {\rm Det^{(0)}} M_{ij}^2 + {\rm Det^{(1)}} M_{ij}^2 + {\rm Det^{(2)}} M_{ij}^2 + ..... \;\; . \end{equation} As for the order $O(\lambda_{5 \sim 7}^0)$, ${\rm Det^{(0)}} M_{ij}^2 =0$. It is the result from so-called Georgi-Pais theorem\cite{GP}, which says that the radiative symmetry breaking can be possible only when massless particle exists in the tree level. As for $O(\lambda_{5 \sim 7}^1)$, \begin{equation} \label{detsp1} {\rm Det^{(1)}} M_{ij}^2 = 2 \lambda_5 \overline{g^2} \Delta v^2 v_1^2 v_2^2 \sin^2 \phi_0 . \end{equation} And for the next order $O(\lambda_{5 \sim 7}^2)$, \begin{equation} \label{detsp2} {\rm Det^{(2)}}M_{ij}^2 = - \overline{g^2} [ 8 \lambda_5^2 + (\lambda_6 + \lambda_7)^2 ] v^2 v_1^2 v_2^2 \sin^2 \phi_0 . \end{equation} ${\rm Det^{(1)}} M_{ij}^2$ is positive definite and ${\rm Det^{(2)}} M_{ij}^2$ is negative definite. In order for $CP$ violating vacuum to be stable, where is a global minimum in fact, the relation \begin{equation} {\rm Det^{(1)}} M_{ij}^2 > |{\rm Det^{(2)}} M_{ij}^2| \end{equation} must be satisfied. {}For this inequality to be satisfied, top and stop contributions are essential, and stop mass must be larger than 178 GeV at $\tan \beta = 1$ (188 GeV at $\tan \beta = \infty$) when $m_t = 174$ GeV. Otherwise the determinant of this neutral Higgs mass matrix becomes negative. Without top and stop contributions, the stationary point which break $CP$ symmetry is not the true vacuum and $CP$ conserving point corresponding to the first equation of Eq.(\ref{theta}) becomes the true vacuum. {}For example, in the case of $\phi_0 = \pi/2$, we can really show \begin{eqnarray} & & \left. \langle V \rangle \right|^{ \langle H_1 \rangle = v_1}_{ \langle H_2 \rangle = v_2 e^{i \phi_0}} - \left. \langle V \rangle \right|^{ \langle H_1 \rangle = v_1^2 - v_2^2}_{ \langle H_2 \rangle = 0} = \lambda_5 v_2^4 + O(\lambda_5^2) \; > 0 \;\;\;\; (v_1^2 > v_2^2) , \\ & & \left. \langle V \rangle \right|^{ \langle H_1 \rangle = v_1}_{ \langle H_2 \rangle = v_2 e^{i \phi_0}} - \left. \langle V \rangle \right|^{ \langle H_1 \rangle = 0}_{ \langle H_2 \rangle = v_2^2 - v_1^2} = \lambda_5 v_1^4 + O(\lambda_5^2) \; > 0 \;\;\;\; (v_2^2 > v_1^2) , \end{eqnarray} where we neglect $\lambda_{6,7}$ for simplicity. We stress that spontaneous $CP$ violation can not occur only by one loop diagram of the chargino and the neutralino contrary to Refs.\cite{MAEKAWA}\cite{POMAROL}. Top and stop effects are essentially needed. However these effects do not influence to the $M_{A-A}^2$ component at all. Then the lightest Higgs mass has little dependence of $m_{\tilde t}$, and its mass becomes smaller than about 5.5 GeV. \par There is no allowed region in $\tan \beta \geq 1$ which satisfies following experimental constraints; (A): the branching ratio $B(Z \rightarrow h_1h_2)$ should be less than $10^{-7}$\cite{PDG}, (B): $B(Z \rightarrow h_i l^+l^-)$ should be smaller than $1.3 \times 10^{-7}$\cite{PDG}\cite{ALEPH}, where $h_1$ and $h_2$ are lightest and second lightest physical Higgs states, respectively. However in $\tan \beta < 1$, there is allowed region, {}for example, \begin{equation} \label{agree} \tan \beta = 0.2, \;\;\; \phi_0 = \pi/2, \;\;\;m_{\tilde t} = 3 \:{\rm TeV} . \end{equation} But in this case, $h_t/4 \pi^2 \simeq 1.35$, so we can not trust the loop expansion of Eq.(\ref{topeffect}). \par How does the situation change if the stop left-right mixing is included? Are there possibilities that there appears experimentally allowed region in $\tan \beta \geq 1$ by additional parameters $A_t$ and $\mu$ appeared in Eqs.(\ref{mat2})$\sim$(\ref{mat222})? Here $A_t$ is the SUSY breaking parameter of stop-stop-Higgs interaction. In order for $A_t$ and/or $\mu$ to have large effects on Higgs masses, they must be of $O(m_{\tilde t})$, since stop left-right mixing is proportional to $m_t \: A_t$ and $m_t \: \mu$. However it is shown that $CP$ violating vacuum becomes unstable in the parameter region of $A_t /m_{\tilde t}$ and/or $\mu /m_{\tilde t} \geq O(1/3)$ from the numerical analysis. And in the region of $A_t /m_{\tilde t}$ and $\mu /m_{\tilde t} \leq O(1/3)$, the situation is almost same as the limit case of small left-right mixing. In addition, the magnitude of $\lambda_5$, which is proportional to $M_{A-A}^2$, itself becomes small if stop left-right mixing exists. Thus, there is no experimentally allowed region even if parameters $A_t$ and $\mu$ take any values. Therefore we can conclude that spontaneous $CP$ violation in the MSSM is excluded from experimental constraints. \section{Summary and Discussion} We show that $CP$ violating vacuum can not be the true vacuum only by the chargino and the neutralino contributions. Top and stop contributions are crucially needed for spontaneous $CP$ violation in the MSSM. In the limit of small stop left-right mixing, the stop mass must be larger than about 180 GeV for the vacuum stability, however, there is no experimentally allowed region in $\tan \beta \geq 1$. If we include the stop left-right mixing, additional parameters $A_t$ and $\mu$ appear in Higgs masses. However numerical analysis shows that both $A_t$ and $\mu$ should be smaller than $O(m_{\tilde t}/3)$ {}for the vacuum stability, and the situation is not so changed as the limit case of small stop left-right mixing. Thus, there is no experimental allowed region for spontaneous $CP$ violation in the MSSM in $\tan \beta \geq 1$. In order to obtain experimentally consistent spontaneous $CP$ violation scenario in the SUSY model, we should extend the MSSM to, for example, the next-to-minimal supersymmetric standard model (NMSSM)\cite{NMSSM} which contains an additional gauge singlet field $N$. In Refs.\cite{SPNMSSM}\cite{SPNMSSM2}\cite{SPNMSSM3}, they discuss spontaneous $CP$ violation in the NMSSM. Especially for the NMSSM with the scale invariant superpotential\cite{SPNMSSM2}\cite{SPNMSSM3}, spontaneous $CP$ violation occurs radiatively, so we can not avoid Georgi-Pais theorem. However the large VEV of $N$ can lift up the lightest Higgs mass, which is relatively light compared to $\langle N \rangle$ in actual, and spontaneous $CP$ violation in the NMSSM can be consistent with the experimental constraints (A) and (B)\cite{SPNMSSM3}. \par As for explicit $CP$ violation in the Higgs sector of the MSSM, the mixing with scalars and a pseudoscalar appears also at the loop level. In this case $m_{12}^2$ does not need to be small of $O(\lambda_{5} v^2)$ as spontaneous $CP$ violation scenario. Angles of $CP$ mixings are negligibly small of $O(\lambda_{5 \sim 7})$\cite{POMAROL} as shown in Appendix, which are too small to influence the phenomenology \vskip 1 cm \noindent {\bf Acknowledgements}\par I would like to thank Professor A. I. Sanda for useful discussions and careful reading of manuscripts.

proofpile-arXiv_065-517

\section{Soft $vs.$ hard dynamics in QCD } {\it QCD and virtual Compton scattering.} The kinematics of the amplitude of the process $\gamma^* p \to \gamma p'$ can be specified by the initial nucleon momentum $p$, the momentum transfer $r=p-p'$ and the momentum $q$ of the initial virtual photon, $q^2 \equiv -Q^2$. The final photon momentum $q'$ is then given by $q'= q +r$ with $q'^2=0$. Other important momentum invariants are $t \equiv (p'-p)^2=(q-q')^2$ and $s \equiv (q+p)^2$. Taking $Q^2$ large, $i.e.,$ at least above $1 \, GeV^2$, one can hope to enter the region where the amplitude is dominated by short distances between the two photon vertices and pQCD may be applicable. In this situation, it is tempting to speak about the ``virtual Compton scattering on a single quark'' implying that the large-$Q^2$ behaviour is given just by the quark propagator (see Fig.$1a$), while the long-distance information is accumulated in a distribution function $F(X,t)$ described by the matrix element of $\langle p' | \bar q \ldots q | p \rangle$ type. However, the factorization formula \begin{equation} M(Q^2,s,t) \to \int m(s/Q^2,X) F(X,t) dX \label{fact} \end{equation} only makes sense if $|t| \ll Q^2$. Otherwise, if $|t| \sim Q^2$, large momentum enters into the hadron wave function and one deals with the scattering process on the hadron as a whole. \begin{figure}[htb] \mbox{ \epsfxsize=8cm \epsfysize=3cm \hspace{2.0cm} \epsffile{fig1.eps} } \vspace{-0.5cm} {\caption{\label{fig:1} $a)$ Handbag diagram contributing to the DVCS amplitude at $t=0$. The lower blob corresponds to the double quark distribution $F(x,y)$. $b)$ One of the two-gluon-exchange diagrams dominating the VCS amplitude for asymptotically large $Q^2$ and $t$. $c)$ Lowest-order diagram for the $\gamma^* \gamma \to \pi^0$ transition form factor. The blob corresponds to the pion distribution amplitude $\varphi_{\pi}(x)$. } } \end{figure} For $|t| \ll Q^2$, the function $F(X,t)$ in eq.(\ref{fact}) looks like a parton distribution function $f(x)$ with an additional form-factor-type dependence on $t$. To make analogy with deep inelastic scattering, it is instructive to recall that the imaginary part of the virtual forward Compton amplitude (for which $q'=q$ and $p'=p$) in the limit of large $Q^2$ and fixed Bjorken variable $\zeta \equiv Q^2/2(pq)$ can be written as \begin{eqnarray} \int_0^1 f(x) \delta((q+xp)^2) \cdot 2 (qp) dx = \int_0^1 f(x) \delta(x-Q^2/ 2 (qp) ) dx = f(\zeta)\, , \end{eqnarray} where $xp$ is the fraction of the initial hadron momentum carried by the interacting quark. The usual parton distribution functions $f(\zeta)$ correspond to exactly forward matrix elements, with $r \equiv p' -p =0$, while the kinematics of VCS requires that $r \neq 0$ and $t \equiv r^2 \neq 0$. Hence, we need a new type of parton functions $F(X,t)$ \cite{ji}. In the limit $t \to 0$, they reduce to the ``asymmetric distribution functions'' $F(X)$ \cite{compton,gluon}. Hence, the studies of deeply virtual Compton scattering (DVCS) are related to a new field of pQCD applications. As shown in refs.\cite{compton,gluon}, the asymmetric distributions $F(X)$ have features of both the distribution functions and distribution amplitudes (wave functions). A more detailed discussion of DVCS at small $t$ will be given in Section 3. Another situation in which pQCD is applicable is when both $Q^2$ and $|t|$ are asymptotically large. Then the virtual Compton scattering amplitude factorizes into a convolution of the short-distance amplitude $m(\{x_i\},\{y_j\},Q^2,s,t)$ and two distribution amplitudes $\varphi(x_1,x_2,x_3)$, $\varphi(y_1,y_2,y_3)$ describing the proton in the initial and final state, respectively (see Fig.$1b$). They are related to matrix elements of $\langle 0|q\ldots q \ldots q |p \rangle$ type and give the probability amplitude, that $e.g.,$ the initial proton can be treated as three collinear quarks with the momentum $p$ divided into fractions $x_1 p,x_2 p, x_3 p$ with $x_1+x_2+x_3 =1$. The short-distance amplitude $m(\{x_i\},\{y_j\},Q^2,s,t)$ is given by Feynman diagrams involving two hard gluon exchanges, suppressed by a factor $(\alpha_s/\pi)^2 \sim 1/100$ compared to the ``soft contribution'' produced by a simple overlap of soft wave functions, without any gluon exchanges. The soft term, however, has an extra power of $1/Q^2$ for large $Q^2$. As a result, the hard term asymptotically dominates, though the soft term may be much larger than the hard one for accessible $Q^2$. {\it Quark-hadron duality and $\gamma^* p \to \Delta^+$ transition.} In particular, a purely soft contribution to $G_M^p(Q^2)$ calculated within the local quark-hadron duality approach \cite{nr83} is in good agreement with experimental data up to $Q^2 \sim \, 20 \, GeV^2$. The same approach was used recently \cite{del} to get the estimates of the soft term for the $\gamma^* p \to \Delta^+$ transition. For the magnetic form factor $G_M^*(Q^2)$ these estimates are rather close to the results of the analysis of inclusive SLAC data \cite{stoler,keppel}. A small value for the ratio $G_E^*(Q^2)/G_M^*(Q^2)$ obtained in ref.\cite{del} is also in agreement with available data \cite{burkert}, in contrast to the pQCD prediction \cite{carlson} which gives $G_E^*(Q^2)/G_M^*(Q^2) \to \, -1$ for the ratio of hard contributions. \begin{figure}[htb] \epsfxsize=6cm \epsfysize=4cm \hspace{-0.5cm} \epsffile{pf2.eps} \epsfxsize=6cm \epsfysize=4cm \hspace{0.1cm} \epsffile{pf3.eps} \vspace{-1cm} {\caption{\label{fig:2} Quark-hadron duality estimates for the $\gamma^* p \to \Delta^+$ transition. Left: form factor $G_M^*(Q^2)$. Right: ratio of form factors $G_E^*(Q^2)$ and $G_M^*(Q^2)$. }} \end{figure} Within different nonperturbative approaches \cite{nr82,nr83,chizhit,kroll96}, it was observed that soft terms are sufficiently large to describe the data or that the hard terms are too small compared to the data. Hence, there is a growing evidence that soft terms dominate the exclusive amplitudes at accessible energies. Of course, the magnitude of the hard contribution depends on the shape of distribution amplitudes (DA's). The latter are usually integrated with the weights like $1/x_1 x_2$, so the humpy DA's of Chernyak-Zhitnitsky (CZ) type \cite{cz82,cz84} produce contributions which are much larger than those obtained with smooth DA's close to the ``asymptotic'' forms. The CZ wave functions were originally motivated by QCD sum rule analysis \cite{cz82}. However, the results of the QCD sum rule calculations of the DA's are extremely model-dependent and unreliable. Furthermore, for the theoretically most clean case of the $\gamma^* \gamma \to \pi^0$ form factor, both a direct QCD sum rule calculation of this form factor and available experimental data show no enhancement compared to the pQCD result obtained with the asymptotic DA for the pion. \section{$\gamma^* \gamma \to \pi^0$ form factor} {\it pQCD analysis.} The transition $\gamma^* \gamma^* \to \pi^0$ of two virtual photons $\gamma^*$ into a neutral pion provides an exceptional opportunity to test QCD predictions for exclusive processes. In the lowest order of perturbative QCD, its asymptotic behaviour is due to the subprocess $\gamma^*(q) + \gamma^*(q^{\prime}) \to \bar q(\bar xp) + q (xp) $ with $x$ ($\bar x$) being the fraction of the pion momentum $p$ carried by the quark produced at the $q$ ($q')$ photon vertex (see Fig.1$c$). The relevant diagram is similar to the handbag diagram for deep inelastic scattering, with the main difference that one should use the pion distribution amplitude $\varphi_{\pi}(x)$ instead of parton densities. For large $Q^2$, the perturbative QCD prediction is given by \cite{bl80}: \begin{equation} F_{\gamma^* \gamma^* \pi^0 }^{as}(Q^2, q^{\prime 2}) = \frac{4\pi}{3} \int_0^1 {{\varphi_{\pi}(x)}\over{xQ^2 - \bar x q^{\prime 2} }} \, dx \stackrel{q^{\prime 2}=0}{\longrightarrow} \frac{4\pi}{3} \int_0^1 {{\varphi_{\pi}(x)}\over{xQ^2}} \, dx \equiv \frac{4\pi}{3Q^2} I \, . \label{eq:gg*pipqcd} \end{equation} Experimentally, the most important situation is when the lower virtuality photon is (almost) real $q^{\prime 2} \approx 0$. In this case, necessary nonperturbative information is accumulated in the same integral $I$ (see eq.(\ref{eq:gg*pipqcd})) that appears in the one-gluon-exchange diagram for the pion electromagnetic form factor \cite{cz84,pl80,blpi79}. The value of $I$ depends on the shape of the pion distribution amplitude $\varphi_{\pi}(x)$. In particular, using the asymptotic form $ \varphi_{\pi}^{as}(x) = 6 f_{\pi} x \bar x $ \cite{pl80,blpi79} gives $F_{\gamma \gamma^* \pi^0 }^{as}(Q^2) = 4 \pi f_{\pi}/Q^2 $ for the asymptotic behaviour \cite{bl80}. If one takes the Chernyak-Zhitnitsky form \cite{cz82} $\varphi_{\pi}^{CZ}(x) = 30 f_{\pi} x(1-x)(1-2x)^2$, the integral $I$ increases by a sizable factor of 5/3, and this difference can be used for experimental discrimination between the two forms. Note, that the pQCD hard scattering term for $\gamma \gamma^* \to \pi^0$ has the zeroth order in the QCD coupling constant $\alpha_s$, just like in deep inelastic scattering. Hence, there are good reasons to expect that pQCD for $F_{\gamma \gamma^* \pi^0 }(Q^2)$ may work at rather low $Q^2$. The $Q^2=0$ limit of $F_{\gamma \gamma^* \pi^0 }(Q^2)$ is known from $\pi^0 \to \gamma \gamma$ decay rate. Using PCAC and ABJ anomaly \cite{ABJ}, one can calculate $F_{\gamma \gamma^* \pi^0 }(0)$ theoretically: $ F_{\gamma \gamma^* \pi^0 }(0) =1/ \pi f_{\pi} . $ It is natural to expect that a complete QCD result does not strongly deviate from a simple interpolation \cite{blin} $ \pi f_{\pi} F_{\gamma \gamma^* \pi^0 }(Q^2) = 1/(1+ Q^2/4 \pi^2 f_{\pi}^2) $ between the $Q^2=0$ value and the large-$Q^2$ asymptotics \footnote{In particular, such an interpolation agrees with the results of a constituent quark model calculation \cite{hiroshi}}. This interpolation implies the asymptotic form of the distribution amplitude for the large-$Q^2$ limit and agrees with CELLO experimental data \cite{CELLO}. It was also claimed \cite{CLEO} that the new CLEO data available up to $8 \, GeV^2$ also agree with the interpolation formula. Comparing the data with theoretical predictions, one should take into account the one-loop pQCD radiative corrections to the hard scattering amplitude calculated in ref.\cite{braaten}. Effectively, the correction decreases the leading-order result by about $20 \%$, still leaving a sizable gap between the prediction based on the CZ amplitude and the phenomenologically successful Brodsky-Lepage interpolation formula\cite{blin}. Hence, the new preliminary data\cite{CLEO} seem to indicate that the magnitude of $I$ is close to that corresponding to the asymptotic form of the pion distribution amplitude. Because of the far-reaching consequences of this conclusion, it is desirable to have a direct calculation of the $\gamma\gamma^* \to \pi^0$ form factor in the intermediate region of moderately large momentum transfers $Q^2 \raisebox{-.2ex}{$\stackrel{\textstyle> 1 \, GeV^2$. As we will see below, the QCD sum rules allow one to calculate $F_{\gamma \gamma^* \pi^0 }(Q^2)$ for large $Q^2$ without any assumptions about the shape of the pion distribution amplitude, and the result can be used to get information about $\varphi_{\pi}(x)$. {\it QCD sum rules and pion distribution amplitude.} The CZ sum rule written directly for the pion distribution amplitude $\varphi_\pi(x)$ is \begin{eqnarray} f_\pi\varphi_\pi(x)&=&\frac{3M^2}{2\pi^2}(1-e^{-s_0/M^2})x(1-x) +\frac{\alpha_s\langle GG\rangle}{24\pi M^2}[\delta(x)+\delta(1-x)] \nonumber \\ &+ & \frac{8}{81}\frac{\pi\alpha_s\langle\bar qq\rangle^2}{M^4} \{11[\delta(x)+\delta(1-x)]+2[\delta^\prime(x)+\delta^\prime(1-x)]\}. \label{eq:wfsr} \end{eqnarray} Here, $M$ is the auxiliary Borel parameter which must be taken in the region where the r.h.s. is least sensitive to its variations, and $s_0$ is the effective onset of the continuum fitted to maximize the $M^2$-stability region. From the QCD sum rule for $f_{\pi}$, \cite{svz} $s_0 \approx 0.7 \, GeV^2$. As emphasized in ref.\cite{MR}, the lowest condensates $\langle GG\rangle$ and $\langle \bar qq\rangle^2$ taken into account in eq.(\ref{eq:wfsr}) do not provide all the information necessary for a reliable determination of $\varphi_\pi(x)$. The humpy CZ shape is, in fact, a compromise between the $\delta(x)$, $\delta(1-x)$ condensate peaks and the smooth $x(1-x)$ behaviour of the perturbative term. Adding higher condensates, $e.g.,$ $\langle \bar q D^2 q\rangle$, one would get even higher derivatives of $\delta(x)$ and $\delta(1-x)$. The sum of such singular terms can be treated as an expansion of some finite-width function $\delta \varphi (x)$: \begin{equation} \delta \varphi (x) = a_0 \delta(x) + a_1 \delta^{\prime} (x) + a_2 \delta^{\prime \prime}(x) + \ldots + \{x \to 1-x \}. \end{equation} Of course, the knowledge of $a_0$ alone is not sufficient for a reliable reconstruction of $\delta \varphi (x)$. On the other hand, the higher coefficients $a_1, a_2, etc.$ are given by a sum of several higher condensates whose magnitudes are completely unknown. Hence, no strict conclusions can be made. The CZ procedure is equivalent to assuming that $a_1, a_2, \ldots \, \sim 0$, though other choices ($e.g.,$ nonlocal condensate model \cite{MR}) may look more realistic. {\it QCD sum rule for doubly virtual form factor $F_{\gamma^* \gamma^* \pi^0} (q^2, q^{\prime 2})$.} Instead of following the steps dictated by the old logic: $1)$ pQCD factorization for $F_{\gamma^* \gamma \pi^0} (Q^2)$; $2)$ QCD sum rules for the moments of $\varphi_{\pi} (x)$ (which are unreliable); $3)$ calculation of $I= \int_0^1 \varphi_{\pi} (x)/x \, dx$, we developed in ref. \cite{rr} the approach which $1)$ starts with the QCD sum rule for $ F_{\gamma^* \gamma^* \pi^0} (q^2, q^{\prime 2})$ in the $q^{\prime 2} \to 0$ limit; $2)$ information about $I$ is extracted from this sum rule and $3)$ then used to make conclusions about the shape of $\varphi_{\pi} (x)$. When both virtualities of the photons are large, we have the following QCD sum rule: \begin{eqnarray} \pi f_{\pi} \mbox{$F_{\gamma^*\gamma^*\pi^\circ}$}(Q^2, q^{\prime 2})= 2\int_0^{s_o} ds \, e^{-s/{M^2}} \int_0^1 \frac{x\bar{x}(xQ^2 - \bar x q^{\prime 2})^2} {[s{x}\bar{x}+xQ^2 - \bar x q^{\prime 2}]^3} \,dx \, \nonumber \\ +\frac{\pi^2}{9} {\langle \frac{\alpha_s}{\pi}GG \rangle} \left(\frac{1}{2M^2 Q^2} - \frac{1}{2M^2 q^{\prime 2}} + \frac1{Q^2 q^{\prime 2}}\right) \nonumber\\ + \frac{64}{243}\pi^3\alpha_s{\langle \bar{q}q\rangle}^2 \left( \frac1{M^4} \left [ \frac{Q^2}{q^{\prime 4}} - \frac9{2q^{\prime 2}}+\frac9{2Q^2}-\frac{q^{\prime 2}}{Q^4} \right ] + \frac9{Q^2 q^{\prime 4}} -\frac9{Q^4q^{\prime 2} } \right ) . \label{eq:SR1} \end{eqnarray} In this situation, the pQCD approach is also expected to work. Indeed, neglecting the $s{x}\bar{x}$-term compared to $xQ^2 - \bar x q^{\prime 2}$ and keeping only the leading $O(1/Q^2)$ and $O(1/q^{\prime 2})$ terms in the condensates, we can write eq.(\ref{eq:SR1}) as \begin{eqnarray} \mbox{$F_{\gamma^*\gamma^*\pi^\circ}$}(,Q^2) = \frac{4\pi}{3f_{\pi}} \int_0^1 \frac{dx}{ xQ^2 - \bar x q^{\prime 2}} \, \left \{ \frac{3M^2}{2\pi^2}(1-e^{-s_0/M^2}) x\bar{x} \right. \nonumber \\ \left. + \frac{1}{24M^2} \langle \frac{\alpha_s}{\pi}GG\rangle [\delta(x) + \delta (\bar{x})] \right. \nonumber \\ + \left. \frac{8}{81M^4}\pi\alpha_s{\langle \bar{q}q\rangle}^2 \biggl ( 11[\delta(x) + \delta (\bar{x})] + 2[\delta^{\prime}(x) + \delta ^{\prime}(\bar{x})] \biggr ) \right \} \label{eq:SRlargeQ2wf}. \end{eqnarray} The expression in curly brackets coincides with the QCD sum rule (\ref{eq:wfsr}) for the pion distribution amplitude $f_{\pi} \varphi_{\pi}(x)$. Hence, when both $Q^2$ and $q^{\prime 2}$ are large, the QCD sum rule (\ref{eq:SR1}) exactly reproduces the pQCD result (\ref{eq:gg*pipqcd}). One may be tempted to get a QCD sum rule for the integral $I$ by taking $ q^{\prime 2}=0$ in eq.(\ref{eq:SR1}). Such an attempt, however, fails immediately because of the power singularities $1/q^{\prime 2}$, $1/ q^{\prime 4}$, $etc.$ in the condensate terms. It is easy to see that these singularities are produced by the $\delta(x)$ and $\delta'(x)$ terms in eq.(\ref{eq:SRlargeQ2wf}). In fact, it is precisely these terms that generate the two-hump form for $\varphi_{\pi}(x)$ in the CZ-approach \cite{cz82}. The advantage of having a direct sum rule for $F_{\gamma \gamma^* \pi^0}(Q^2,q^{\prime 2})$ is that the small-$q^{\prime 2}$ behavior of $F_{\gamma \gamma^* \pi^0}(Q^2,q^{\prime 2})$ is determined by the position of the closest resonances in the $q^{\prime }$ channel, which is known. Eventually, $1/q^{\prime 2}$ is substituted for small $q^{\prime 2}$ by something like $1/m_{\rho}^2$ and the QCD sum rule in the $q^{\prime 2}$ limit is \begin{eqnarray} & \,& \pi f_{\pi} F_{\gamma \gamma^* \pi^0}(Q^2) = \int_0^{s_0} \left \{ 1 - 2 \frac{Q^2-2s}{(s+Q^2)^2} \left (s_{\rho} - \frac{s_{\rho}^2}{2 m_{\rho}^2} \right ) \right. \nonumber \\ &+& \left. 2\frac{Q^2-6s+3s^2/Q^2}{(s+Q^2)^4} \left (\frac{s_{\rho}^2}{2} - \frac{s_{\rho}^3}{3 m_{\rho}^2} \right ) \right \} e^{-s/M^2} \frac{Q^2 ds }{(s+Q^2)^2} \nonumber \\ &+&\frac{\pi^2}{9} {\langle \frac{\alpha_s}{\pi}GG \rangle} \left \{ \frac{1}{2 Q^2 M^2} + \frac{1}{Q^4} - 2 \int_0^{s_0} e^{-s/M^2} \frac{ds }{(s+Q^2)^3} \right \} \nonumber \\ &+&\frac{64}{27}\pi^3\alpha_s{\langle \bar{q}q\rangle}^2 \lim_{\lambda^2 \to 0} \left \{ \frac1{2Q^2 M^4} + \frac{12}{Q^4 m_{\rho}^2 } \left [ \log \frac{Q^2}{\lambda ^2} -2 \right. \right. \nonumber \\ &+& \left. \left. \int_0^{s_0} e^{-s/M^2} \left ( \frac{s^2+3sQ^2+4Q^4} {(s+Q^2)^3} - \frac1{s+\lambda ^2} \right) ds \right] \right. \label{eq:finsr} \\ &-& \left. \frac4{Q^6} \left [ \log \frac{Q^2}{\lambda^2} -3+ \int_0^{s_0} e^{-s/M^2} \left ( \frac{s^2+3sQ^2+6Q^4} {(s+Q^2)^3} - \frac1{s+\lambda ^2} \right) ds \right] \right \} \nonumber \end{eqnarray} In Fig.\ref{fig:3}, we present a curve for $Q^2F_{\gamma \gamma^* \pi^0}(Q^2)/4\pi f_{\pi}$ calculated from eq.(\ref{eq:finsr}) for standard values of the condensates, $\rho$- and $\pi$-meson duality intervals $s_{\rho} = 1.5 \, GeV^2$, \cite{svz}, $s_0 = 0.7 \, GeV^2$ and $M^2 = 0.8\, GeV^2$. It is rather close to the curve corresponding to the Brodsky-Lepage interpolation formula $\pi f_{\pi} F_{\gamma \gamma^* \pi^0}(Q^2) = 1/(1+Q^2/4\pi^2 f_{\pi}^2)$ and to that based on the $\rho$-pole approximation $\pi f_{\pi} F(Q^2) = 1/(1+Q^2/m_{\rho}^2)$. Hence, our result favors a pion distribution amplitude which is close to the asymptotic form. It should be noted, that the $\rho$-pole behaviour in the $Q^2$-channel has nothing to do with the explicit use of the $\rho$-contributions in our models for the correlators in the $q^{\prime 2}$-channel: the $Q^2$-dependence of the $\rho$-pole type emerges due to the fact that the pion duality interval $s_0 \approx 0.7 \, GeV^2$ is numerically close to $m_{\rho}^2\approx 0.6\,GeV^2$. Taking the lowest-order perturbative spectral density $\rho^{quark}(s,q^{\prime 2}=0 ,Q^2) = {{Q^2}/{(s+Q^2)^2}}$ and assuming the local quark-hadron duality, we obtain the result \begin{equation} f_{\pi} F_{\gamma \gamma^* \pi^0}^{LD}(Q^2) = \frac1{\pi } \int_0^{s_0} \rho^{quark}(s,0,Q^2) \, ds = \frac1{\pi (1+Q^2/s_0)} \label{eq:FLDgg} \end{equation} coinciding, for $s_0=4 \pi^2 f_{\pi}^2 \approx 0.67 \, GeV^2$ with the BL-interpolation formula. \begin{figure}[htb] \mbox{ \hspace{1.5cm} \epsfxsize=5cm \epsfysize=7.5cm \epsffile{ruas2.eps} } \vspace{-3.5cm} {\caption{\label{fig:3} Combination $Q^2 F_{\gamma \gamma^* \pi^0}(Q^2)/4\pi f_{\pi}$ as calculated from the QCD sum rule (solid line), $\rho$-pole model (short-dashed line) and Brodsky-Lepage interpolation (long-dashed line). }} \end{figure} {\it Lessons.} $1)$ CZ sum rule is an unreliable source of information about the pion DA; $2)$ Pion DA is narrow; $3)$ Since the diagrams for the nucleon DA's have the structure as in the pion case, we should expect that the nucleon DA's are also close to asymptotic and , hence, the two-gluon-exchange hard terms are very small for accessible $Q^2$ and $t$. Thus, it is very important to get the estimates for the soft contributions to the virtual Compton scattering amplitude (see, $e.g.,$ refs. \cite{crs}, where the high-$t$ real Compton scattering on the pion was considered). \section{Small-$t$, large-$Q^2$ limit of VCS: a new pQCD area} Recently, X. Ji \cite{ji} suggested to use the deeply virtual Compton scattering (DVCS) to get information about some parton distribution functions inaccessible in standard inclusive measurements. He also emphasized that the DVCS amplitude has a scaling behavior in the region of small $t$ and fixed $x_{Bj}$ which makes it a very interesting object on its own ground. {\it Double distributions.} In the scaling limit, the square of the proton mass $m_p^2=p^2$ can be neglected compared to the virtuality $Q^2 \equiv -q^2$ of the initial photon and the energy invariant $p \cdot q \equiv m_p \nu$. Thus, we set $p^2=0$ and, for small $t$, we also have $r^2 = 0$. Then the requirement $p'^2 \equiv (p-r)^2=p^2$ reduces to the condition $p\cdot r = 0$ which can be satisfied only if $r$ is proportional to $p$: $r= \zeta p$, where $\zeta$ coincides with the Bjorken variable $x_{Bj} \equiv Q^2/2(p \cdot q)$, $0 \leq x_{Bj} \leq 1$. Naturally, the light-like limit of 4-momenta $p$, $r$ is more convenient to visualize in a frame where the initial proton is moving fast, rather than in its rest frame. Though the momenta $p$ and $r$ are proportional to each other, one should make a clear distinction between them since $p$ and $r$ specify the momentum flow in two different channels. Since the initial quark momentum originates both from $p$ and $r$, we write it as $xp +y r$. In more formal terms, the relevant light-cone matrix elements are parameterized as \begin{eqnarray} && \hspace{-6mm} \langle p-r\, | \, \bar \psi_a(0) \hat z E(0,z;A) \psi_a(z) \, | \, p \rangle |_{z^2=0} = \bar u(p-r) \hat z u(p) \label{eq:vec} \\ && \hspace{-6mm} \int_0^1 \int_0^1 \, \left ( e^{-ix(pz)-iy(r z)}F_a(x,y) \right - \left. e^{ix(pz)-i\bar y(r z)}F_{\bar a}(x,y) \right ) \, \theta( x+y \leq 1) dy \, dx , \nonumber \end{eqnarray} $etc.,$ where $\hat z \equiv \gamma_{\mu} z^{\mu}$ and $\bar u(p-r), u(p)$ are the Dirac spinors for the nucleon. Taking the limit $r =0$ gives the matrix element defining the parton distribution functions $f_a(x)$, $f_{\bar a}(x)$. This leads to the reduction formula: \begin{equation} \int_0^{1-x} \, F_a(x,y)\, dy= f_a(x) . \label{eq:redf} \end{equation} {\it Asymmetric distribution functions.} Since $r = \zeta p$, the variable $y$ appears in eq.(\ref{eq:vec}) only in $x+y\zeta \equiv X$ and $x- \bar y\zeta \equiv X - \zeta$ combinations, where $X$ and $(X - \zeta)$ are the total fractions of the initial hadron momentum $p$ carried by the quarks. Integrating the double distribution $F(X-y \zeta,y)$ over $y$ we get the asymmetric distribution function \begin{equation} {\cal F}_{\zeta}^a (X) = \int_0^{{\rm min} \{ X/\zeta, \bar X / \bar \zeta \}} F_a(X-y \zeta,y) \, dy, \label{eq:asdf} \end{equation} where $\bar \zeta \equiv 1- \zeta$. Since $\zeta \leq 1$ and $x+y \leq 1$, the variable $X$ satisfies a natural constraint $0\leq X \leq 1$. In the region $X > \zeta$ (Fig.4$a$), the initial quark momentum $Xp$ is larger than the momentum transfer $r = \zeta p$, and we can treat ${\cal F}_{\zeta}^a (X)$ as a generalization of the usual distribution function $f_a(x)$. In this case, the quark goes out of the hadron with a positive fraction $Xp$ of the original hadron momentum and then comes back into the hadron with a changed (but still positive) fraction $(X - \zeta)p$. The Bjorken ratio $\zeta$ specifies the momentum asymmetry of the matrix element. Hence, one deals now with a family of asymmetric distribution functions ${\cal F}_{\zeta}^a (X)$ whose shape changes when $\zeta$ is changed. The basic distinction between the double distributions $F(x,y)$ and the asymmetric distribution functions ${\cal F}_{\zeta} (X)$ is that the former do not depend on the momentum asymmetry parameter $\zeta$, while the latter are explicitly labelled by it. \begin{figure}[htb] \mbox{ \epsfxsize=5.5cm \epsfysize=2.5cm \hspace{2.1cm} \epsffile{fig2.eps} } {\caption{\label{fig:4} Momentum flow corresponding to the asymmetric distribution function in two regimes: {\it a) } $X \geq \zeta$ and {\it b)} $X \equiv Y \zeta \leq \zeta $. } } \end{figure} When $\zeta \to 0$, the limiting curve for ${\cal F}_{ \zeta}(X)$ reproduces the usual distribution function: \begin{equation} {\cal F}^a_{\zeta=0} \, (X) = f_a(X) \ . \label{eq:Fzeta0} \end{equation} Another region is $X < \zeta$ (Fig.4$b$), in which the ``returning'' quark has a negative fraction $(X- \zeta)$ of the light-cone momentum $p$. Hence, it is more appropriate to treat it as an antiquark going out of the hadron and propagating together with the original quark. Writing $X$ as $X = Y \zeta$, we see that the quarks carry now positive fractions $Y \zeta p \equiv Y r$ and $\bar Y r \equiv (1-Y)r $ of the momentum transfer $r$, and the asymmetric distribution function in the region $X= Y \zeta < \zeta$ looks like a distribution amplitude $\Psi_{\zeta}(Y)$ for a $\bar q q $ state with the total momentum $r= \zeta p$: \begin{equation} \Psi_{\zeta}(Y) = \int_0^Y F((Y-y) \zeta , y ) \, dy . \label{eq:Psi} \end{equation} {\it Leading-order contribution.} Using the parameterization for the matrix elements given above, we get a parton-type representation for the handbag contribution at $t=0$: $$ T^{\mu \nu}_{symm} (p,q,r) = \left (g^{\mu \nu} -\frac1{p \cdot q } (p^{\mu}q^{\nu} +p^{\nu}q^{\mu}) \right ) \, \sum_a e_a^2\, \sqrt{1- \zeta} \ ( T_V^a(\zeta ) + T_V^{\bar a}(\zeta ) ) , $$ where only the $\{\mu \leftrightarrow \nu \}$-symmetric part is shown explicitly and $T_V^a(\zeta )$ is the invariant amplitude depending on the scaling variable $\zeta $: \begin{equation} T_V^a(\zeta ) = \int_0^{1} \left ( \frac1{X-\zeta +i\epsilon} + \frac1{X} \right ) {\cal F}_{\zeta}^a (X) \, dX \, . \label{eq:tv} \end{equation} The term containing $1/(X-\zeta +i\epsilon)$ generates the imaginary part: \begin{equation} - \frac1{\pi}\, {\rm Im} \, T_V^a(\zeta ) = {\cal F}^a_{\zeta} \, (\zeta)\, . \label{eq:imtv} \end{equation} Though ${\cal F}_{\zeta = 0}^a(X) = f_a(X)$, in the general case when $\zeta \neq 0$, these two functions differ. Furthermore, the imaginary part appears for $X= \zeta$, $i.e.,$ in a highly asymmetric configuration in which the second quark carries a vanishing fraction of the original hadron momentum, in contrast to the usual distribution $f_a(\zeta)$ which corresponds to a symmetric configuration with the final quark having the momentum equal to that of the initial one. A characteristic feature of the asymmetric distribution functions ${\cal F}_{\zeta}^a(X)$ is that they rapidly vary in the region $X \raisebox{-.2ex}{$\stackrel{\textstyle< \, \zeta$ and vanish for $X=0$. However, the limiting curve ${\cal F}_{\zeta=0}(X)$ does not necessarily vanish for $X=0$, $i.e.,$ the limits $\zeta \to 0$ and $X \to 0$ do not commute. For this reason, if $\zeta$ is small, the substitution of ${\cal F}_{\zeta}^a(X)$ by $f_a(X)$ may be a good approximation for all $X$-values except for the region $X \raisebox{-.2ex}{$\stackrel{\textstyle< \, \zeta$, and it is not clear {\it a priori} how close are the functions ${\cal F}_{\zeta}^a(\zeta)$ and $ f_a(\zeta)$. { \it Evolution of the double distributions.} The purely scaling behavior of the DVCS amplitude is violated by the logarithmic $Q^2$-dependence of $F_{NS}(x,y;Q^2)$ governed by the evolution equation \begin{equation} Q \frac{d}{d Q} F_{NS}(x,y;Q^2) = \int_0^1 d \xi \int_0^1 R_{NS} (x,y; \xi, \eta;g) F_{NS}( \xi, \eta;Q^2) d \eta \label{eq:nfwdev} \end{equation} (the flavor-nonsinglet (NS) component is taken for simplicity). Since integration over $y$ converts $F_{NS} (x,y)$ into the parton distribution function $f_{NS} (x)$, whose evolution is governed by the GLAPD equation \cite{gl,ap,d}, our kernel has the property \begin{equation} \int_0^ {1-x} R_{NS} (x,y; \xi, \eta;g) d y = \frac1{\xi} P_{NS} (x/\xi). \label{eq:rtop} \end{equation} For a similar reason, integrating $R_{NS}(x,y; \xi, \eta;g)$ over $x$ one should get the evolution kernel $V(y,\eta;g)$ \cite{blpi79,pl80} for the pion distribution amplitude \begin{equation} \int_0^{1-y} R_{NS}(x,y; \xi, \eta;g) d x = V(y,\eta;g). \label{eq:rtov} \end{equation} In the formal $Q^2 \to \infty$ limit, $F(x,y; Q^2\to \infty) \sim \delta(x) y \bar y,$ $i.e.,$ in each of its variables $x,y$, the double distribution tends to the characteristic asymptotic form: $\delta(x)$ is specific for the distribution functions, while the $y \bar y$-form is the asymptotic shape for the lowest-twist two-body distribution amplitudes \cite{pl80,blpi79}. { \it Evolution of asymmetric distribution functions.} As a result, the evolution of the asymmetric distribution functions ${\cal F}_{\zeta}^a(X)$ proceeds in the following way. Due to the GLAP-type evolution, the momenta of the partons decrease and distributions become peaked in the regions of smaller and smaller $X$. However, when the parton momentum degrades to values smaller than the momentum transfer $r = \zeta p$, the further evolution is like that for a distribution amplitude: it tends to make the distribution symmetric with respect to the central point $X= \zeta/2$ of the $(0, \zeta)$ segment. {\it Conclusions.} DVCS opens a new class of scaling phenomena characterized by absolutely new nonperturbative functions describing the structure of the proton. The continuous electron beam accelerators like TJNAF and ELFE may be an ideal place to study DVCS \cite{afanas}. The asymmetric distributions can also be studied in the processes of large-$Q^2$ meson electroproduction \cite{gluon}, $etc.$ {\it Acknowledgement.} This work was supported by the US Department of Energy under contract DE-AC05-84ER40150.

proofpile-arXiv_065-518

\subsection*{1. Introduction} As is well known, the Standard Model \cite{Glashow} predictions for the electroweak observables are in perfect agreement with the current data \cite{Schaile2,Renton}. But frankly speaking, the SM is also a complicated theory with many free parameters and other open questions. Very recently the discovery of the top quark has been announced by CDF with $M_t=176 \pm 8 \pm 10\;GeV$ \cite{CDF}, which we interprete as $M_t=176 \pm 13 \;GeV$, and by the D0 Collaboration with $M_t=199 ^{+19}_{-21} \pm 22\;GeV$\cite{D0}). This direct measurement of top quark mass is in very good agreement with the prediction based on the SM electroweak fits of the LEP and other data, $M_t=178 \pm 8 ^{+17}_{-18}\;GeV$\cite{Renton}, where the central value and the first error refer to $M_H=300\;GeV$. This direct measurement of $M_t$, while still not very precise, should help in reducing the present uncertainties on almost all electroweak observables. And consequently, the knowledge of $M_t$ will be very important for one to look for the hints of new physics. Technicolor(TC) \cite{Farhi} is one of the important candidates for the mechanism of electroweak symmetry breaking. The comparison of theoretical predictions based on the TC theories and the precision electroweak measurements is very specialized and rapidly changing, as new data becomes available as well as new theoretical variables with which theory can be compared with experiment. This subject is of immense importance to TC theory, because it has been widely reported that the data disfavor TC theories, a claim that has been disputed by several authors (for a recent review see ref.\cite{King}). Very recently, Burgess et al., \cite{Burgess} extended the (S,T,U) parametrization\cite{Peskin} by introducing three additional parameters (V,W,X) to describe the lowest non-trivial momentum dependence in oblique diagrams. The inclusion of (V, W, X) in the fit weakens the bounds on S, T strongly: $S < 2.5$, $T < 1.3$ \cite{King,Burgess}. In this paper we define a parameter $\Delta_b^{new}$ which only measures the non-oblique corrections on $Zb\overline{b}$ vertex from new physics, especially that from the ETC dynamics and the charged PGBs appeared in QCD-like TC theories. By the comparison of the theoretical prediction for $\Delta_b^{new}$ in TC theory with the experimentally determined $\Delta_{b,exp}^{new}$ one can obtain some constraints on the Clebsch-Gordon coefficient $\xi$ and put new lower limits on the masses of charged PGBs. This paper is organized as follows: In Sec.2 we at first present the standard model predictions for $R_b$ and other observables and then define the new parameter $\Delta_b^{new}$. In Sec.3 are collected the relevant calculations and the constraints for the parameter $\xi$ and the masses $m_{p1}$ and $m_{p2}$ for QCD-like TC theories. We also list and comment on several new TC models proposed very recently in the sense of avoiding the existed constraints imposed by the precision data. The conclusions and the related discussions are in Sec.4. \subsection*{2. $Zb\overline{b}$ veretx, the SM predictions and the data} For LEP processes there are two types of radiative corrections: the corrections to the gauge boson self-energies and the corrections to the $Zb\overline{b}$ vertex. In the evaluation of self-energy corrections the error due to our ignorance of the Higgs mass is substantial after the direct measurement of $m_t$ at Fermilab\cite{CDF,D0}. On the other hand, in the corrections to the $Zb\overline{b}$ vertex, where the leading contribution due to the large top quark mass is produced by the exchange of the W bosons, there is no dependence on the unknown Higgs mass. Moreover, the possible new physics contributions to the $Zb\overline{b}$ vertex are much more restricted. Any non-standard behavior most possibly means the existence of new physics! The Z-pole observables considered in this paper include $\Gamma_b$, $\Gamma_h$, $\Gamma_Z$, $R_b$, $R_c$ and $R_l$ (in which $\Gamma_l=(\Gamma_e + \Gamma_\mu + \Gamma_\tau)/3$), they are well determined theoretically and experimentally. Because the asymmetry $A_{FB}^b$ is almost unaffected by the $Zb\overline{b}$ vertex correction \cite{Altarelli} we will not include this quantity in our analysis. Calculations of the one-loop corrections to the $Zb\overline{b}$ vertex has been performed by several groups \cite{Akhundov}. The partial decay width $\Gamma (Z\rightarrow f \overline{f})$ has been calculated in the $\overline{MS}$ renormalization scheme \cite{Degrassi} and has been expressed in a compact form \cite{Pich}, \begin{eqnarray} \Gamma(Z\rightarrow f\overline{f})&=& \frac{N_c^f}{48}\frac{\hat{\alpha}} {\hat{s}^2_w \hat{c}_w^2}\,m_Z [\hat{a}_f^2 + \hat{v}_f^2](1+\delta^{(0)}_f)(1+\delta_{QED}^f)\nonumber \\ &&\cdot (1+\delta_{QCD}) (1+\delta_\mu^f)(1+\delta_{tQCD}^f)(1+\delta_b), \end{eqnarray} where $N_c^f=3(1)$ for quarks (leptons) is the color factor. The partial decay widths in eq.(1) has included the genuine electroweak corrections, the QED and QCD corrections, as well as the corrections to $Zb\overline{b}$ vertex due to the large top quark mass. The definitions and the explicit expressions for all functions and factors appeared in eq.(1) can be found in refs.\cite{Degrassi,Pich}. In ref.\cite{Fleischer}, J.Fleischer et al. calculated the two-loop $0(\alpha\alpha_s)$ QCD corrections to the partial decay width $\Gamma_b$, and they found a screening of the leading one-loop top mass effects by $m_t\rightarrow $ $m_t\,[1-\frac{1}{3}(\pi^2-3)\alpha_s/\pi]$. In this paper we will include this two-loop QCD corrections. For more details about the calculations of $\Gamma_b$ and other relevant quantities in the SM one can see the refs.\cite{Akhundov,Degrassi} and a more recent paper\cite{Xiao1}. In our analysis, the measured values \cite{Schaile2,pdg,BES,CDF} $m_Z=91.1888$ $\pm 0.0044$ $\;GeV$, $G_\mu=1.16639\times 10^{-5}(GeV)^{-2}$ , $\alpha^{-1}=137.0359895$, $\alpha_s(m_Z)=0.125\pm0.005$, $m_e=0.511\;MeV$, $m_\mu=105.6584\;MeV$ and $m_\tau = 1776.9\;MeV$, together with $m_t=176 \pm 13 \;GeV$ and the assumed value $M_H=300^{+700}_{-240}\;GeV$ are used as the input parameters. In the numerical calculations we conservatively take the ``on-shell'' mass of the b-quark the value $m_b=4.6\pm 0.3\;GeV$ (in ref.\cite{Pich}, the authors used $m_b=4.6\pm 0.1\;GeV$), and use the known relation\cite{ckg}between the ``on-shell'' and the $\overline{MS}$ schemes to compute the running mass $\overline{m}_b(m_Z)$ at the Z scale: $\overline{m}_b(m_Z)=3 \pm 0.2\;GeV$ for $m_Z=91.1888\; GeV$. We also use the same treatment for the c-quark, $\overline{m}_c(m_Z)=1\,GeV$ if we take $m_c=1.6\,GeV$ as its ``on-shell'' mass. For other three light quarks we simply assume that $\overline{m}_i(m_Z)=0.1\,GeV\;(i=u,d,s)$. All these input parameters will be referred to as the {\em Standard Input Parameters} (SIP). Among the electroweak observables the ratio $R_b=\Gamma_b/\Gamma_h$ is the special one. For this ratio most of the vacuum polarization corrections depending on the $m_t$ and $m_h$ cancel out, while the experimental uncertainties in the detector response to hadronic events also basically cancel. Furthermore, this ratio is also insensitive to extensions of the SM which would only contribute to vacuum polarizations. In Table 1 we list the SM predictions for the Z boson decay widths (in MeV) and the ratios $R_b$, $R_c=\Gamma(Z\rightarrow c\overline{c})/ \Gamma_h$ and $R_l=\Gamma_h/\Gamma(Z\rightarrow l\overline{l})$, the corresponding measured values at LEP are also listed. It is easy to see that the $R_b$ predicted by the SM is smaller than that measured. The deviation reaches 2.2-$\sigma$ (or 2.5-$\sigma$ at one-loop order) for $m_t=176\;GeV$. The precision data can be used to set limits on TC theory as well as other kinds of possible new physics. Besides the $m_t$ dependence the $Zb\overline{b}$ vertex is also sensitive to a number of types of new physics. One can parametrize such effects by \begin{eqnarray} \Gamma_b=\Gamma_b^{SM}(1+\Delta_b^{new}) \end{eqnarray} where the term $\Delta_b^{new}$ represents the pure non-oblique corrections to the $Zb\overline{b}$ veretx from new physics. The partial decay width $\Gamma_b^{SM}$ can be determined theoretically by eq.(1), and consequently other five observables studied in this paper can be written as the form of \begin{eqnarray} \Gamma_h&=&\Gamma_h^{SM}+\Gamma_b^{SM}\cdot \Delta_b^{new}, \ \ \ \ \Gamma_Z=\Gamma_Z^{SM}+\Gamma_b^{SM}\cdot \Delta_b^{new},\nonumber\\ R_b&=&R_b^{SM}+R_b^{SM}(1-R_b^{SM})\cdot \Delta_b^{new},\ \ \ \ R_c=R_c^{SM}-R_b^{SM}R_c^{SM}\cdot \Delta_b^{new},\nonumber\\ R_l&=&R_l^{SM}+\frac{\Gamma_b^{SM}}{\Gamma_l^{SM}}\cdot \Delta_b^{new}. \end{eqnarray} Obviously, the oblique corrections and the heavy top quark vertex effect have been absorbed into the evaluations for the observables $X_i^{SM}$ in the SM. This definition of $\Delta_b^{new}$ in eq.(2) is different from that of $\epsilon_b$ \cite{Altarelli}(as well as the parameter $\Delta_b$ in refs.\cite{Blondel,Cornet}). In the SM the parameters $\epsilon_b$\cite{Altarelli} and $\Delta_b$\cite{Cornet} are closely related to the quantity $-Re\{\delta_{b-vertex} \}$ defined in ref.\cite{Pich} and are dominated by quadratic terms in $m_t$ of order $G_F m_t^2$. While the parameter $\Delta_b^{new}$ only measures the new physics effects on the $Zb\overline{b}$ vertex, and $\Delta_b^{new} \equiv 0$ in the SM. We think that this definition of $\Delta_b^{new}$ is more convenient than other similar definitions to measure the new physics effects on the $Zb\overline{b}$ vertex, since new physics can be disentangled if not masked by large $m_t$ effects. In order to extract the vertex factor $\Delta_b^{new}$ from the data set $(\Gamma_b, \Gamma_h, \Gamma_Z, R_b, R_c, R_l)$ as listed in Table 1 more quantitatively, we construct the likelihood function of $\Delta_b^{new}$ as the form of \begin{eqnarray} {\cal L}(x_{exp}, \Delta_b^{new})= N\,Exp[-\sum_x \frac{1}{2} (\frac{x_{exp}-x( \Delta_b^{new})}{\sigma_x})^2] \end{eqnarray} where the $\sigma_x$ is the experimental error of the observable $ x_{exp}$, and N is the normalization factor. With the SIP, the point which maximizes ${\cal L}(x_{exp}, \Delta_b^{new})$ is found to be $\Delta_b^{new} = 0.001$ for $m_t=176\; GeV$. And we also have \begin{eqnarray} \Delta_b^{new} = 0.001 \pm 0.005 \end{eqnarray} at $1-\sigma$ level for $m_t=176\;GeV$ and $M_H=300\;GeV$, while the remainder uncertainties of $\Delta_b^{new}$ are $\pm 0.002$ and $ ^{+0.004}_{-0.002}$ corresponding to $\delta m_t=13\,GeV$ and $M_H=300^{+700}_{-240}$ respectively. It is easy to see that $\Delta_b^{new}$ is now consistent with zero at $1-\sigma$ level. By its own definition the parameter $\Delta_b^{new}$ has no dependence on $M_H$, the present weak dependence is coming from the standard model calculations for the six observables. In the following analysis we always use $(m_t=176\,GeV$, $M_H=300\,GeV$) as reference point and don't discuss the variation of $M_H$. If we interpret the quantity \begin{eqnarray} P(\Delta_b^{new} > A) = \int_A^{+\infty} d \Delta_b^{new} {\cal L}(x_{exp}, \Delta_b^{new}) \end{eqnarray} and \begin{eqnarray} P(\Delta_b^{new} < B) = \int^B_{-\infty} d \Delta_b^{new} {\cal L}(x_{exp}, \Delta_b^{new}) \end{eqnarray} as the probability that $\Delta_b^{new} > A$ ($\Delta_b^{new} < B$ ), then one can obtain the $95\%$ one-sided upper (lower) confidence limits on $\Delta_b^{new}$: \begin{eqnarray} \Delta_{b,exp}^{New} > -0.010,\ \ and \ \ \Delta_{b,exp}^{New} < 0.012 \end{eqnarray} for $m_t=176\pm 13\;GeV$. For any kinds of new physics which may contribute to the $Zb\overline{b}$ vertex, they should satisfy this constraint from $Zb\overline{b}$ vertex as well as those from the (S, T, U, V, W, X) oblique parameters simultaneously. \subsection*{3. Updated constraints on $\xi$ and masses of charged PGBs} In the TC models \cite{Farhi,King,Weinberg}, the larger top quark mass is presumably the result of ETC \cite{Susskind} dynamics at relatively low energy scales. There are two sources of corrections to this $Zb\overline{b}$ vertex in TC models, namely from ETC gauge boson exchange \cite{Simmons,Chivukula} and from charged PGB exchange \cite{Xiao2,Xiao3} For the One-Doublet Technicolor Model(ODTM)\cite{Dimopoulos}, no Pseudo-Goldstone bosons can be survived when the chiral symmetry was broken by the condensate $<T\overline{T}> \neq 0$, but the ETC gauge boson exchange can produce typically large and negative contributions to the $Zb\overline{b}$ vertex, as described in ref.\cite{Simmons}, \begin{eqnarray} \Delta_1^{ETC} \approx -6.5\%\times \xi^2\cdot [\frac{m_t}{176GeV}] \end{eqnarray} where the constant $\xi$ is an ETC-gauge-group-dependent Clebsch-Gordon coefficient and expected to be of order 1 \cite{Simmons}. Theoretically, the exact value of $\xi$ will be determined by the choice of ETC gauge group and by the assignments of the technifermions. As shown in eq.(9), the non-oblique correction on the $Zb\overline{b}$ vertex from the ETC dynamics is quadratic in $\xi$. The variation of $\xi$ will strongly affect the size of $\Delta_1^{ETC}$. Naturally the experimental limits on the vertex factor $\Delta_b^{new}$ cab be interpreted as the bounds on $\xi$. For $m_t=189\;GeV$ one can have, \begin{eqnarray} \xi < 0.4, \ \ at\ \ 95\%\,C.L. \end{eqnarray} For lighter top quark this bound will be loosened slightly. In the most frequently studied Farhi-Susskind One Generation Technicolor Model (OGTM) \cite{Dimopoulos}, the global flavor symmetry $SU(8)_L\times SU(8)_R$ will break down to the $SU(8)_V$ by technifermion condensate $<\overline{T}T>\neq 0$. And consequently 63 massless (Pseudo)-Goldstone bosons will be produced from this breaking. Besides the nonoblique corrections $\Delta_2^{ETC}$ from the ETC gauge boson exchange, the charged PGBs in the OGTM also contribute a negative correction to the $Zb\overline{b}$ vertex as estimated in ref.\cite{Xiao2,Xiao3}. In short, \begin{eqnarray} \Delta_b^{new}(OGTM) = \Delta_2^{ETC} + \Delta_b^{P^\pm} + \Delta_b^{P_8^\pm}. \end{eqnarray} where the terms $\Delta_b^{P^\pm}$ and $\Delta_b^{P_8^\pm}$ represent the contributions from the color singlet charged PGBs $P^{\pm}$ and the color octets $P_8^{\pm}$. Specifically, all three terms in the right-hand side of this equation are negative. For simplicity, we assume that the ETC part of the OGTM studied here are the same or very similar with the ODTM studied in ref.\cite{Simmons} except for the difference in the value of $F_\pi$ (in the OGTM, $F_\pi= 123\;GeV$), and then we can write \begin{eqnarray} \Delta_2^{ETC} \approx -12.9\%\times \xi^2\cdot [\frac{m_t}{176GeV}] \end{eqnarray} Typically, $\Delta_2^{ETC}\approx -6.5\%$ for $m_t=176\;GeV$ and $\xi=1/\sqrt{2}$, which is consistent with the result as shown in the Fig.3 of ref.\cite{Chivukula} for the $SU(4)_{ETC} \rightarrow$ $SU(3)_{TC}$ model with a full family of technifermions. In ref.\cite{Xiao2,Xiao3}, we have calculated the non-oblique corrections on the $Zb\overline{b}$ vertex from the color singlet PGBs $P^\pm$ and the color octet PGBs $P_8^{\pm}$ respectively. The size of the vertex factor $\Delta_{b}^{P^\pm}$ ( $\Delta_{b}^{P_8^\pm}$) depends on $m_t$ and $m_{p1}$ ($m_{p2}$). Using the SIP, one can estimate the ranges of the term $\Delta_b^{P^\pm}$ and $\Delta_b^{P_8^\pm}$: \begin{eqnarray} \Delta_b^{P^\pm} &=& (-0.013 \sim -0.002),\; for \ \ m_{p1}=50 - 400 \;GeV, \\ \Delta_b^{P_8^\pm} &=& (-0.050 \sim -0.003),\; for \ \ m_{p2}=200 - 650 \;GeV, \end{eqnarray} where $m_{p1}$ is the mass of $P^\pm$, and $m_{p2}$ is the mass of $P_8^\pm$. The contributions from the charged PGBs are always negative and will push the OGTM prediction for $\Delta_b^{new}$ away from the measured $\Delta_{b,exp}^{New}$ to a high degree. These negative corrections are clearly disfavored by the current data. But fortunately, the charged PGBs show a clear decoupling behavior as listed in eqs.(13, 14). In the OGTM, the size of vertex factor $\Delta_b$ generally depend on three " free" parameters, the Clebsch-Gordon coefficient $\xi$, the masses $m_{p1}$ and $m_{p2}$ if we use $m_t=176\pm 13$ GeV as input. In order to study the nonoblique corrections on the $Zb\overline{b}$ vertex more quantitatively, we consider the following two ultimate cases: (a). Under the limit $\xi \rightarrow 0$, to extract the possible bounds on the masses of $m_{p1}$ and $m_{p2}$; (b). Under the limits $\Delta_b^{P^{\pm}}\rightarrow 0$ and $\Delta_b^{P_8^{\pm}}\rightarrow 0$( e.g. the charged PGBs are heavy enough and decoupled from the low energy physics), to extract the bounds on the parameter $\xi$. At first if we set $\xi \rightarrow 0$ the current data will permit us to exclude large part of the ranges of $m_{p1}$ and $m_{p2}$ in the $m_{p1}-m_{p2}$ plan, the updated bounds on the masses of charged PGBs are the following: \begin{eqnarray} m_{p1} > 200\;GeV\ \ at\ \ 95\%\;C.L., \ \ for\ \ ``free''\ \ m_{p2} \end{eqnarray} and \begin{eqnarray} m_{p2} > 600\;GeV\ \ at \ \ 95\%\;C.L., \ \ for \ \ m_{p1} \leq \;400\; GeV. \end{eqnarray} while the uncertainties of $m_t$, $\delta m_t=13\;GeV$, almost don't affect the constraints. These limits are much stronger than that has been given before in ref.\cite{Xiao3}. Of cause, the inclusion of the negative corrections from ETC dynamics in the OGTM will strengthen the bounds on $m_{p1}$ and $m_{p2}$. Secondly, if we set the limits $\Delta_b^{P^{\pm}}\rightarrow 0$ and $\Delta_b^{P_8^{\pm}}\rightarrow 0$, the current data means a stringent bound on the size of $\xi$ in the OGTM: $\xi < 0.28$ at $95\%\;C.L.$ for $m_t = 189\;GeV$. If the charged PGBs are heavy and decoupled and, at the same time, the coefficient $\xi$ in QCD-like TC models can be reduced to $0.28$ instead of the popular size $1/\sqrt{2}$ as used in ref.\cite{Chivukula}, the magnitude of both the $\Delta_{b}^{new}(ODTM)$ and $\Delta_b^{new}(OGTM)$ will be consistent with the present constraints on $\Delta_b^{new}$. \subsection*{4. Conclusions} As mentioned at the beginning, TC theory can provide a natural, dynamical explanation for electroweak symmetry breaking. But, as is well known, this theory (including the ETC) also encountered many problems as discussed in detail in refs.\cite{King}. At present, the situation becomes better than 3 years ago\cite{Lane}. The experimentally determined parameters $S_{exp}$ and $\Delta_{b,exp}^{new}$ are all close to zero with small errors, and therefore the former strong constraints are now weakened. In ref.\cite{Chivukula}, the authors have shown that a slowly running technicolor coupling will affect the size of non-oblique corrections to the $Zb\overline{b}$ vertex from ETC dynamics. Numerically, the ``Walking TC'' \cite{Holdom} reduces the magnitude of the corrections at about $20\%$ level. Although this decrease is helpful to reduce the discrepancy between the TC models and the current precision data, however, this improvement is not large enough to resolve this problem. More recently, N.Evans\cite{Evans} points out that the constraints from $Zb\overline{b}$ veretx may be avoided if the ETC scale $M_{ETC}$ can be boosted by strong ETC effects. For standard ETC dynamics\cite{Susskind,King} the ETC gauge bosons are the $SU(2)_w$ singlets, and the exchanges of such kinds of ETC gauge bosons will produce large negative corrections to the $Zb\overline{b}$ vertex as described in refs.\cite{Simmons,Chivukula}. In ``Non-commuting'' theories ( i.e., in which the ETC gauge boson which generates the top quark mass does carry weak SU(2) charge), as noted in refs.\cite{Simmons,Chivukula2}, the contributions on the $Zb\overline{b}$ vertex come from the physics of top-quark mass generation and from weak gauge boson mixing (the signs of the two effects are opposite)\cite{Chivukula2}, and therefore both the size and the sign of the corrections are model dependent and the overall effect may be small and may even increase the $Zb\overline{b}$ branching ratio. It is important to explore this class of models further, since the experiments favor a larger $R_b$\cite{Xiao1}. Besides the new TC models just mentioned above several TC models with novel ideas have also been constructed since 1993, such as the ``Low-scale technicolor''\cite{King2}, the ``Technicolor model with a scaler'' \cite{Georgi2}, the `` Topcolor assisted technicolor'' \cite{Hill}, ``Chiral technicolor''\cite{Terning2} and other models. The main motivation for constructing these new models is evident: Generating the larger top quark mass and at the same time being consistent with the precision data. In summary we defined a parameter $\Delta_b^{new}$ which measures the non-oblique corrections on the $Zb\overline{b}$ vertex from the new physics, such as the ETC dynamics and the charged PGBs appeared in QCD-like TC theories. By its own definition the parameter $\Delta_b^{new}$ is different from the $\epsilon_b$ and the $\Delta_b$ as defined in refs. \cite{Altarelli,Blondel}, and this parameter can be determined experimentally from the data set ($\Gamma_b$, $\Gamma_h$, $\Gamma_Z$, $R_b$, $R_c$, $R_l$). By the comparison of the theoretical prediction for $\Delta_b^{new}$ in QCD-like TC theories with the experimentally determined $\Delta_{b,exp}^{new}$ one can obtain some constraints on the Clebsch-Gordon coefficient $\xi$ and put more stringent lower limits on the masses of charged PGBs. From the numerical calculations and the phenomenological analysis we found that: (a). The charged Pseudo-Goldstone bosons must be heavier than that estimated before in Ref.\cite{Xiao2}. At present for $m_t=176\pm 13\;GeV$, we have $m_{p1} > 200\;GeV$ at $95\% C.L$ for ``free'' $m_{p2}$, and $m_{p2} > 600\;GeV$ at $95\% C.L$ for $m_{p1}\leq 400\;GeV$; (b). If the charged PGBs are indeed very heavy and decoupled and, at the same time, the coefficient $\xi$ in the new QCD-like TC models can be smaller than 0.28, such kinds of QCD-like TC models still be allowed. (c). There is definite discrepancy about the value of $R_b$ between the SM and the experiment. But at present it is hard to explain this deviation as a signal of new physics. From the data set of $(\Gamma_b, \Gamma_h, \Gamma_Z, R_b, R_c, R_l)$, one can determine the size of the nonoblique corrections on the $Zb\overline{b}$ vertex from the new physics experimentally: $\Delta_{b,exp}^{new}=0.001\pm 0.005\pm 0.002(m_t)$, which is close to zero with small errors. \newpage Table 1. The SM predictions for the observables $(\Gamma_b,\; \Gamma_h,\; \Gamma_Z,\;R_b,\;R_c,\;R_l)$, compared with the measured Z parameters at LEP. \begin{center} \vspace{0.2cm} \begin{tabular}{c|l|l} \hline\hline & SM Predictions& LEP Values \\ \hline $\Gamma_b$& $377.7 \pm 0.2(m_t) ^{+0.2}_{-0.9}(m_h) \pm 0.5(\alpha_s) \pm 0.4(\hat{\alpha}) \pm 0.3(\overline{m}_b)$ & $382.7\pm 3.1$, \cite{Altarelli} \\ \hline $\Gamma_h$& $1749.3\; \pm 3.2(m_t)\; ^{+1.4}_{-4.5}(m_h)\; \pm 2.9(\alpha_s) \; \pm 1.7(\hat{\alpha})\; \pm 0.3(\overline{m}_b)$ & $1745.9\pm 4.0$, \cite{Schaile2} \\ \hline $\Gamma_Z$& $2503.9\; \pm 4.3(m_t)\; ^{+1.2}_{-5.9}(m_h)\; \pm 2.9(\alpha_s) \; \pm 2.4(\hat{\alpha})\; \pm 0.3(\overline{m}_b)$ & $2497.4\pm 3.8$, \cite{Schaile2} \\ \hline $R_b$& $ 0.2159 \pm 0.0005(m_t) \pm 0.00003(m_h) \pm 0.00004(\alpha_s)$ $ \pm 0.0001(\overline{m}_b)$, & $0.2202\pm 0.0020$, \cite{Schaile2} \\ \hline $R_c$& $0.1721\; \pm 0.0002(m_t)\; \pm 0.00004(m_h)\; \pm 0.0001(\alpha_s)$$ \pm 0.00003(\overline{m}_b)$, & $0.1583\pm 0.0098$,\cite{Schaile2}\\ \hline $R_l$& $20.820 \pm 0.002(m_t) \pm 0.015(m_h) \pm 0.034(\alpha_s) \pm 0.003(\overline{m}_b) $ & $20.795\pm 0.040$, \cite{Schaile2}\\ \hline \hline \end{tabular} \end{center} \newpage

proofpile-arXiv_065-519

\section{Introduction} The last stage of coalescing binary neutron stars(BNS's) is one of the most promising sources for kilometer size interferometric gravitational wave detectors, LIGO\cite{ligo} and VIRGO\cite{virgo}. When the orbital separation of BNS's becomes $\sim 700$km as a result of the emission of gravitational waves, it is observed that the frequency of gravitational waves from them becomes $\sim 10$Hz. After then, the orbit of BNS's shrinks owing to the radiation reaction toward merging in a few minutes\cite{cult}. In such a phase, BNS's are the strongly self-gravitating bound systems, and gravitational waves from them will have various general relativistic(GR) imformations. In particular, in the last few milliseconds before merging, BNS's are in a very strong GR gravitational field because the orbital separation is less than ten times of the Schwarzschild radius of the system. Thus, if we could detect the signal of gravitational waves radiated in the last few milliseconds, we would be able to observe directly the phenomena in the GR gravitational field. To interpret the implication of the signal of gravitational waves, we need to understand the theoretical mechanism of merging in detail. The little knowledge we have about the very last phase of BNS's is as follows: When the orbital separation of BNS's is $\raisebox{0.3mm}{\em $\, <$} \hspace{-3.3mm 10 GM/c^2$, where $M$ is the total mass of BNS's, they move approximately in circular orbits because the timescale of the energy loss due to gravitational radiation $t_{GW}$ is much longer than the orbital period $P$ as \begin{equation} {t_{GW} \over P} \sim 15 \biggl({dc^2 \over 10GM}\biggr)^{5/2} \biggl({M \over 4\mu}\biggr), \label{time} \end{equation} where $\mu$ and $d$ are the reduced mass and the separation of BNS's. Thus, BNS's adiabatically evolve radiating gravitational waves. However, when the orbital separation becomes $6-10GM/c^2$, they cannot maintain the circular orbit because of instabilities due to the GR gravity\cite{kidder} or the tidal field\cite{lai}. As a result of such instabilities, the circular orbit of BNS's changes into the plunging orbit to merge. This means that the nature of the signal of gravitational waves changes around the transition between the circular orbit and plunging one. Gravitational waves emitted at this transition region may bring us an important information about the structure of NS's because the location where the instability occurs will depend on the equation of state(EOS) of NS sensitively\cite{lai,joan}. Thus, it is very important to investigate the location of the innermost stable circular orbit(ISCO) of BNS's. As mentioned above, the ISCO is determined not only by the GR effects, but also by the hydrodynamic one. We emphasize that the tidal effects depend strongly on the structure of NS. Here, NS is a GR object because of its compactness, $Gm/c^2R\sim 0.2$, where $m$ and $R$ are the mass and radius of NS. Thus, in order to know the location of the ISCO accurately, we need to solve the GR hydrodynamic equations in general. A strategy to search the ISCO in GR manner is as follows; since the timescale of the energy loss is much longer than the orbital period according to Eq.$(\ref{time})$, we may suppose that the motion of BNS's is composed of the stationary part and the small radiation reaction part. From this physical point of view, we may consider that BNS's evolve quasi-stationally, and we can take the following procedure; first, neglecting the evolution due to gravitational radiation, equilibrium configurations are constructed, and then the radiation reaction is taken into account as a correction to the equilibrium configurations. The ISCO is determined from the point, where the dynamical instability for the equilibrium configurations occurs. It may be a grand challenge, however, to distinguish the stationary part from the nonstationary one in general relativity. As Detweiler has pointed out\cite{detweiler}, a stationary solution of the Einstein equation with standing gravitational waves, which will be constructed by adding the incoming waves from infinity, may be a valuable approximation to physically realistic solutions. However, these solutions are not asymptotically flat\cite{detweiler} because GWs contribute to the total energy of the system and the total energy of GWs inside a radius $r$ grows linearly with $r$. The lack of asymptotic flatness forces us to consider only a bounded space and impose boundary conditions in the near zone. Careful consideration will be necessary to find out an appropriate boundary condition for describing the physically realistic system in the near zone. Recently, Wilson and his collaborators\cite{wilson} proposed a simirelativistic approximation method in order to calculate the equilibrium configuration of BNS's just before merging. In their method, they assume the line element as \begin{equation} ds^2=-(\alpha^2-\beta_i \beta^i)c^2 dt^2+2 \beta_i c dt dx^i+\psi^4 dx^3, \end{equation} i.e., three metric $\gamma_{ij}$ is chosen as the conformal flat(i.e., $\gamma_{ij}=\psi^4\delta_{ij}$), and solve only the constraint equations in the Einstein equation. In their approach, they claim that they ignore only the contribution of gravitational waves, but it is not correct at all; as shown in previous post-Newtonian(PN) analyses\cite{shafer,asada,rieth}, the tensor potential term exists in the three metric even if we ignore the radiation reaction of gravitational waves(i.e., $\psi^{-4}\gamma_{ij}\not=\delta_{ij}$). Since such a term appears from the second PN order in the PN approximation, the accuracy of their results is less than the 2PN order: In reality, from results by Cook et al.\cite{cook} in which they obtain equilibrium configurations of the axisymmetric NS using both the Einstein equation and Wilson's method, we can see that some quantities obtained from Wilson's scheme, such as the lapse function, the three metric, the angular velocity, and so on, deviate from the exact solution by about $O((Gm/Rc^2)^2)$. This seems to indicate that their approach for the system of BNS's is valid only at the 1PN level from the PN point of view. Furthermore, the meaning of their approximation is obscure: It is not clear at all how to estimate errors due to such an approximation scheme and in which situation but the spherical symmetric system, the scheme based on the assumption of the conformal flatness is justified. In contrast with Wilson's method, the meaning of the PN approximation is fairly clear: In the PN approximation, the metric is formally expanded with respect to $c^{-1}$ assuming the slow motion and weak self-gravity of matter. If we will take into account the next PN order, the accuracy of approximate solutions will be improved. This means that we can estimate the order of magnitude of the error due to the ignorance of higher PN terms. Also, in the PN approximation, we can distinguish the radiation reaction terms, which begin at the 2.5PN order\cite{esposito}, from other terms in the metric. Thus, it is possible to construct the equilibrium configuration of BNS's without the radiation reaction terms in the 2PN approximation. We schematically describe two approaches in Tables 1(a) and 1(b). As mentioned above, in close binary of NS's, it is important to take into account GR effects to orbital motion as well as to the internal structure of each NS. As for the orbital motion, there exist two parameters; one is the PN parameter $v/c$ and the other is the mass ratio $\eta$ of the reduced mass $\mu$ to the total mass $M$, and both parameters are less than unity. Thus, the physical quantities such as the orbital frequency are expanded with respect to them. In Table 1(a), we show schematically various levels of approximations in terms of $v/c$ and $\eta$. If all terms in a level are taken into account in the 2PN approximation, we mark $P^2N$, while $W$ means that all terms in the marked level are taken into account in Wilson's approach. From Table 1(a), we see that the 2PN approximation can include all corrections in $\eta$ up to the 2PN order in contrast with Wilson's approach. On the other hand, Wilson's approach will hold completely in the test particle limit, i.e., at $O(\eta^0)$, whereas even in this limit the 2PN approximation is not valid at higher PN orders. As for the internal structure of each NS, there also exist two small parameters; one is the compactness $Gm/c^2R$ and the other is the deformation parameter from its spherical shape, such as an ellipticity $e$. In this case, the PN approximation becomes an expansion in terms of $Gm/c^2R$. In Table 1(b), we also show various levels of approximation in terms of these parameters. Although Wilson's approach is exact for spherical NS's, it is not valid in nonspherical cases even at the 2PN order. On the other hand, in the 2PN approximation, the spherical compact star cannot be obtained correctly in contrast with Wilson's approach. In this way, the 2PN approximation has a week point: Although it can take into account all effects up to the 2PN order, it is inferior to Wilson's approach when we take a test-particle limit, $\eta\to 0$, or we describe an exactly spherical NS. However, as shown below, the error due to the ignorance of higher PN terms in those cases is not so large . To estimate the error due to the ignorance of the higher PN terms, let us compare the GR exact solutions with their PN approximations. First, we consider a small star of mass $\mu$ orbiting a Schwarzschild black hole of mass $m_{bh} \gg \mu$. In this case, we may consider that the small star moves on the geodesic around the Schwarzschild black hole, and the orbital angular velocity becomes\cite{kidder} \begin{equation} \Omega=\sqrt{ {Gm_{bh} \over (\bar r+Gm_{bh}c^{-2})^3 } },\label{bheq} \end{equation} where $\bar r$ is the coordinate radius of the orbit in the harmonic coordinate. In the PN approximation, Eq.$(\ref{bheq})$ becomes \begin{equation} \Omega=\sqrt{ {Gm_{bh} \over \bar r^3} }\biggl\{1- {3Gm_{bh} \over 2\bar r c^2}+{15 \over 8} \Bigl({Gm_{bh} \over \bar r c^2}\Bigr)^2+O(c^{-6}) \biggr\}. \label{bhbheq} \end{equation} Comparing Eq.$(\ref{bheq})$ with Eq.$(\ref{bhbheq})$, it is found that the error size of the 2PN angular velocity is $\sim 0.3\%$ at $\bar r=9Gm_{bh}c^{-2}$, and $\sim 1\%$ at $\bar r=6Gm_{bh}c^{-2}$. Thus, the 2PN approximation seems fairly good to describe the motion of relativistic binary stars just before coalescence. Next, we consider a spherical NS of a uniform density in order to investigate the applicability of the PN approximation for determination of the internal structure of NS's. In this model, the pressure, $P$, and the density, $\rho=$const., are related with each other\cite{compact}: \begin{eqnarray} {P \over \rho c^2}&=&{ (1-2Gmr_s^2/c^2R^3)^{1/2}-(1-2Gm/c^2R)^{1/2} \over 3(1-2Gm/c^2R)^{1/2}-(1-2Gmr_s^2/c^2R^3)^{1/2} } \nonumber\\ &=&{1 \over 2}{Gm \over c^2R}\Bigl(1-{r_s^2 \over R^2}\Bigr) +{G^2 m^2 \over c^4 R^2}\Bigl(1-{r_s^2 \over R^2}\Bigr) +{G^3 m^3 \over c^6 R^3}\Bigl({17 \over 8}-{19 r_s^2 \over 8 R^2} +{3 r_s^4 \over 8 R^4}-{r_s^6 \over 8 R^6}\Bigr) +O(c^{-8}), \label{ppppeq} \end{eqnarray} where $r_s$ is the coordinate radius in the Schwarzschild coordinate and terms of order $c^{-2}$, $c^{-4}$ and $c^{-6}$ denote Newtonian, 1PN and 2PN terms respectively. In the second line in Eq.$(\ref{ppppeq})$, we expand the equation in power of $Gm/c^2R$ regarding it as a small quantity. In fig.1, we shows the error, $1-\tilde P/P$, in Newtonian, 1PN and 2PN cases as a function of $r_s$ for $R=5Gm/c^2$(solid lines) and $8Gm/c^2$(dotted lines), where $\tilde P$ denotes the PN approximate pressure. It is found that the discrepancy in the Newtonian treatment is very large, while in the 2PN approximation the error is less than 10$\%$. In this way, we can estimate rigidly the typical error size in the 2PN approximation. Furthermore, the accuracy is fairly good if the NS is not extremely compact; the 2PN approximation will be fairly accurate if the radius of NS is larger than $\sim$10km. Thus, in the present paper, we develop a formalism to obtain equilibrium configurations of uniformly rotating fluid in the 2PN order as a first step. In section 2, we review the basic equations up to the 2PN order. In section 3, we rewrite the Poisson equation for potential functions, which are described in section 2, into useful forms in which the source terms of the Poisson equations decrease rapidly enough($O(r^{-4})$). In section 4, we show a formulation to obtain numerically equilibrium solutions of uniformly rotating fluid in the 2PN approximation: Taking into account the formulation in the first PN approximation\cite{shibata}, we further rewrite potentials defined in section 3 into a polynomial form in the angular velocity, $\Omega$. Then, we transform the integrated Euler equation into the polynomial form in $\Omega^2$ so that the convergence property in iteration procedures may be much improved. For the sake of analysis for numerical results, we describe the 2PN expression of the conserved quantities, such as the conserved mass, the ADM mass, the total energy and the total angular momentum in section 5. Section 6 is devoted to summary. Throughout this paper, $G$ and $c$ denote the gravitational constant and the speed of light. Hereafter, we use units of $G=1$. \section{Formulation} We write the line element in the following form; \begin{equation} ds^2=-(\alpha^2-\beta_i \beta^i)c^2dt^2+2\beta_i c dt dx^i+ \psi^4 \tilde \gamma_{ij}dx^i dx^j, \end{equation} where we define ${\rm det}(\tilde \gamma_{ij})=1$. To fix the gauge condition in the time coordinate, we use the maximal slice condition $K_i^{~i}=0$, where $K_i^{~i}$ is the trace part of the extrinsic curvature, $K_{ij}$. As the spatial gauge condition, we adopt the transverse gauge $\tilde \gamma_{ij,j}=0$ in order to remove the gauge modes from $\tilde \gamma_{ij}$. In this case, up to the 2 PN approximation, each metric variable is expanded as\cite{asada} \begin{eqnarray} \psi&=&1+{1 \over c^2}{U \over 2}+{1 \over c^4}\hbox{$_{(4)}$}\psi+O(c^{-6}),\\ \alpha&=&1-{1 \over c^2}U+{1 \over c^4}\Bigl({U^2 \over 2}+X\Bigr) +{1 \over c^6}\hbox{$_{(6)}$}\alpha+O(c^{-7}), \\ \beta^i&=&{1 \over c^3}\hbox{$_{(3)}$}\beta_i+{1 \over c^5}\hbox{$_{(5)}$}\beta_i+ O(c^{-7}),\\ \tilde \gamma_{ij}&=&\delta_{ij}+{1 \over c^4}h_{ij}+O(c^{-5}). \end{eqnarray} As for the energy-momentum tensor of the Einstein equation, we consider the perfect fluid as \begin{equation} T_{\mu\nu}=\Bigl(\rho c^2+\rho \varepsilon+P \Bigr)u_{\mu}u_{\nu} +P g_{\mu\nu}. \end{equation} For simplicity, we assume that the matter obeys the polytropic equation of state(EOS); \begin{equation} P=(\Gamma-1)\rho \varepsilon=K \rho^{\Gamma}, \end{equation} where $\Gamma$ and $K$ are the polytropic exponent and polytropic constant, respectively. Up to the 2PN order, the four velocity is expanded as\cite{chandra,asada} \begin{eqnarray} u^0&=&1+{1 \over c^2}\Bigl({1 \over 2}v^2+U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{5 \over 2}v^2U +{1 \over 2}U^2+\hbox{$_{(3)}$}\beta_i v^i-X \Bigr)+O(c^{-6}),\nonumber\\ u_0&=&-\biggl[ 1+{1 \over c^2}\Bigl({1 \over 2}v^2-U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{3 \over 2}v^2U +{1 \over 2}U^2+X \Bigr)\biggr]+O(c^{-6}),\nonumber\\ u^i&=&{v^i \over c} \biggl[1+{1 \over c^2}\Bigl({1 \over 2}v^2+U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{5 \over 2}v^2U +{1 \over 2}U^2+\hbox{$_{(3)}$}\beta_i v^i-X \Bigr)\biggr]+O(c^{-7}),\nonumber\\ u_i&=&{v^i \over c}+{1 \over c^3}\Bigl\{\hbox{$_{(3)}$}\beta_i +v^i\Bigl({1 \over 2}v^2+3 U\Bigr)\Bigr\} +{1 \over c^5}\Bigl\{\hbox{$_{(5)}$}\beta_i+\hbox{$_{(3)}$}\beta_i\Bigl({1 \over 2}v^2+3U\Bigr) +h_{ij}v^j \nonumber\\ &&~~~~~~~~~~~~~~+v^i\Bigl({3 \over 8}v^4+{7 \over 2}v^2 U+4U^2-X+4\hbox{$_{(4)}$}\psi +\hbox{$_{(3)}$}\beta_j v^j\Bigr)\Bigr\}+O(c^{-6}), \label{uueq} \end{eqnarray} where $v^i=u^i/u^0$ and $v^2=v^i v^i$. Since we need $u^0$ up to 3PN order to obtain the 2PN equations of motion, we derive it here. Using Eq.$(\ref{uueq})$, we can calculate $(\alpha u^0)^2$ up to 3PN order as \begin{eqnarray} (\alpha u^0)^2&=&1 + \psi^{-4}\tilde \gamma^{ij}u_i u_j \nonumber\\ &=&1+{v^2 \over c^2}+{1 \over c^4}\Bigl(2\hbox{$_{(3)}$}\beta_j v^j+4Uv^2+v^4\Bigr) +{1 \over c^6}\Bigl\{\hbox{$_{(3)}$}\beta_j\hbox{$_{(3)}$}\beta_j+8\hbox{$_{(3)}$}\beta_j v^j U +h_{ij}v^i v^j \nonumber\\ &&~~~~~~~~~+2\hbox{$_{(5)}$}\beta_iv^i +\Bigl(4\hbox{$_{(3)}$}\beta_j v^j+4\hbox{$_{(4)}$}\psi+{15 \over 2}U^2-2X\Bigr)v^2 +8Uv^4+v^6\Bigr\}+O(c^{-7}), \end{eqnarray} where we use $\tilde \gamma^{ij}=\delta_{ij}-c^{-4}h_{ij}+O(c^{-5})$. Thus, we obtain $u^0$ up to the 3PN order as \begin{eqnarray} u^0&=&1+{1 \over c^2}\Bigl({1 \over 2}v^2+U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{5 \over 2}v^2U +{1 \over 2}U^2+\hbox{$_{(3)}$}\beta_i v^i-X \Bigr) \nonumber\\ &&+{1 \over c^6}\Bigl\{-\hbox{$_{(6)}$}\alpha +{1 \over 2}\Bigl( \hbox{$_{(3)}$}\beta_j\hbox{$_{(3)}$}\beta_j+h_{ij}v^iv^j\Bigr) +\hbox{$_{(5)}$} \beta_j v^j + 5\hbox{$_{(3)}$}\beta_j v^j U -2UX \nonumber\\ &&~~~~~+\Bigl({3 \over 2} \hbox{$_{(3)}$}\beta_j v^j+2\hbox{$_{(4)}$}\psi+6U^2-{3 \over 2} X\Bigr)v^2 +{27 \over 8}Uv^4+{5 \over 16}v^6\Bigr\}+O(c^{-7}). \label{utime} \end{eqnarray} Substituting PN expansions of metric and matter variables into the Einstein equation, and using the polytropic EOS, we find that the metric variables obey the following Poisson equations\cite{asada}; \begin{eqnarray} &&\Delta U=-4\pi \rho, \\ &&\Delta X=4\pi\rho\Bigl(2 v^2+2U+(3\Gamma-2)\varepsilon \Bigr), \\ &&\Delta \hbox{$_{(4)}$}\psi=-2\pi\rho\Bigl(v^2+\varepsilon+{5 \over 2}U \Bigr), \\ &&\Delta \hbox{$_{(3)}$}\beta_i=16\pi \rho v^i-\dot U_{,i}, \\ &&\Delta \hbox{$_{(5)}$}\beta_i =16\pi\rho\biggl[v^i \Bigl(v^2+2U+\Gamma\varepsilon \Bigr)+\hbox{$_{(3)}$}\beta_i\biggr]-4U_{,j} \Bigl(\hbox{$_{(3)}$}\beta_{i,j}+\hbox{$_{(3)}$}\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\beta_{k,k}\Bigr) \nonumber\\ &&{\hskip 2cm} -2\hbox{$_{(4)}$}\dot\psi_{,i} +{1 \over 2}(U \dot U)_{,i}+(\hbox{$_{(3)}$}\beta_l U_{,l})_{,i}, \\ &&\Delta h_{ij}=\Bigl(U U_{,ij}-{1 \over 3}\delta_{ij}U \Delta_{flat}U -3U_{,i} U_{,j}+\delta_{ij}U_{,k} U_{,k} \Bigr) -16\pi \Bigl(\rho v^i v^j -{1 \over 3}\delta_{ij}\rho v^2 \Bigr) \nonumber\\ &&\hskip 1cm -\Bigl(\hbox{$_{(3)}$}\dot\beta_{i,j}+\hbox{$_{(3)}$}\dot\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\dot\beta_{k,k} \Bigr) -2\Bigl( (X+2\hbox{$_{(4)}$}\psi)_{,ij}-{1 \over 3}\delta_{ij} \Delta (X+2\hbox{$_{(4)}$}\psi) \Bigr), \\ &&\Delta \hbox{$_{(6)}$}\alpha=4\pi\rho\biggl[2v^4 +2v^2\Bigl(5U+\Gamma\varepsilon\Bigr)+(3\Gamma-2)\varepsilon U +4\hbox{$_{(4)}$}\psi+X+4\hbox{$_{(3)}$}\beta_i v^i \biggr] \nonumber\\ &&{\hskip 3cm}-h_{ij}U_{,ij} -{3 \over 2}U U_{,l}U_{,l}+U_{,l}(2 \hbox{$_{(4)}$}\psi -X)_{,l} \nonumber\\ &&{\hskip 3cm} +{1 \over 2} \hbox{$_{(3)}$}\beta_{i,j}\Bigl(\hbox{$_{(3)}$}\beta_{i,j}+\hbox{$_{(3)}$}\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\beta_{k,k}\Bigr), \end{eqnarray} where $\Delta$ is the flat Laplacian, and $~\cdot~$ denotes $\partial/\partial t$. Equations of motion for fluid are derived from \begin{equation} \nabla_{\mu} T^{\mu}_{~\nu}=0. \label{eom} \end{equation} In this paper, we consider the uniformly rotating fluid around $z$-axis with the angular velocity $\Omega$, i.e., \begin{equation} v^i=\epsilon_{ijk}\Omega^j x^k=(-y\Omega, x\Omega, 0), \end{equation} where we choose $\Omega^j=(0,0,\Omega)$ and $\epsilon_{ijk}$ is the completely anti-symmetric unit tensor. In this case, the following relations hold; \begin{equation} \Bigl({\partial \over \partial t}+\Omega{\partial \over \partial \varphi} \Bigr) Q =\Bigl({\partial \over \partial t}+\Omega{\partial \over \partial \varphi} \Bigr) Q_i =\Bigl({\partial \over \partial t}+\Omega{\partial \over \partial \varphi} \Bigr)Q_{ij}=0,\label{killeq} \end{equation} where $Q$, $Q_i$ and $Q_{ij}$ are arbitrary scalars, vectors, and tensors, respectively. Then, Eq.$(\ref{eom})$ can be integrated as\cite{lightman} \begin{equation} \int {dP \over \rho c^2+\rho\varepsilon+P}=\ln u^0+C,\label{euler} \end{equation} where $C$ is a constant. For the polytropic EOS, Eq.$(\ref{euler})$ becomes \begin{equation} \ln \biggl[1+{\Gamma K \over c^2(\Gamma-1)} \rho^{\Gamma-1} \biggr] = \ln u^0+C, \label{eulerf} \end{equation} or \begin{equation} 1+{\Gamma K \over c^2(\Gamma-1)} \rho^{\Gamma-1}=u^0 \exp(C). \label{euleri} \end{equation} Using Eq.$(\ref{utime})$, the 2PN approximation of Eq.$(\ref{eulerf})$ is written as \begin{eqnarray} H-{H^2 \over 2c^2}+{H^3 \over 3c^4}&=&{v^2 \over 2}+U +{1 \over c^2}\biggl(2Uv^2+{v^4 \over 4}-X+\hbox{$_{(3)}$}\beta_i v^i \biggl) \nonumber\\ &&+{1 \over c^4}\biggl(-\hbox{$_{(6)}$}\alpha +{1 \over 2}\hbox{$_{(3)}$}\beta_i\hbox{$_{(3)}$}\beta_i +4\hbox{$_{(3)}$}\beta_i v^i U-{U^3 \over 6}+\hbox{$_{(3)}$}\beta_i v^i v^2+2 \hbox{$_{(4)}$} \psi v^2 \nonumber\\ &&{\hskip 1.2cm} +{15\over 4} U^2 v^2+2Uv^4 +{1 \over 6} v^6-UX-v^2 X+\hbox{$_{(5)}$}\beta_i v^i +{1 \over 2}h_{ij}v^iv^j\biggr)+C, \label{bern} \end{eqnarray} where $H=\Gamma K \rho^{\Gamma-1}/(\Gamma-1)$, $v^2=R^2\Omega^2$ and $R^2=x^2+y^2$. Note that Eq.$(\ref{bern})$ can be also obtained from the 2PN Euler equation like in the first PN case\cite{jcb,shibata}. If we solve the coupled equations (2.11-17) and $(\ref{bern})$, we can obtain equilibrium configurations of the non-axisymmetric uniformly rotating body. \section{Derivation of the Poisson equation of compact sources for \lowercase{$h_{ij}$, $\hbox{$_{(3)}$}\beta_i$ and $\hbox{$_{(5)}$}\beta_i$}} In section 2, we derive the Poisson equations for metric variables. However, the source terms in the Poisson equations for $\hbox{$_{(3)}$} \beta_i$, $\hbox{$_{(5)}$} \beta_i$, and $h_{ij}$ fall off slowly as $r \rightarrow \infty$ because these terms behave as $O(r^{-3})$ at $r \rightarrow \infty$. These Poisson equations do not take convenient forms when we try to solve them as the boundary value problem in numerical calculation. Hence in the following, we rewrite them into other convenient forms in numerical calculation. As for $h_{ij}$, first of all, we split the equation into three parts as\cite{asada} \begin{eqnarray} \Delta h^{(U)}_{ij}&=&U\Bigl(U_{,ij}-{1 \over 3}\delta_{ij}\Delta U\Bigr) -3U_{,i}U_{,j}+\delta_{ij}U_{,k}U_{,k} \equiv -4\pi S^{(U)}_{ij}, \\ \Delta h^{(S)}_{ij}&=& -16\pi\Bigl(\rho v^iv^j-{1 \over 3}\delta_{ij}\rho v^2\Bigr), \\ \Delta h^{(G)}_{ij}&=&-\Bigl(\hbox{$_{(3)}$}\dot\beta_{i,j}+\hbox{$_{(3)}$}\dot\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\dot\beta_{k,k} \Bigr) \nonumber\\ &&{\hskip 1.6cm}-2\Bigl( (X+2\hbox{$_{(4)}$}\psi)_{,ij}-{1 \over 3}\delta_{ij} \Delta (X+2\hbox{$_{(4)}$}\psi) \Bigr). \label{hijG} \end{eqnarray} The equation for $h^{(S)}_{ij}$ has a compact source, and also the source term of $h^{(U)}_{ij}$ behaves as $O(r^{-6})$ at $r \rightarrow \infty$, so that Poisson equations for them are solved easily as the boundary value problem. On the other hand, the source term of $h^{(G)}_{ij}$ behaves as $O(r^{-3})$ at $r \rightarrow \infty$, so that it seems troublesome to solve the equation for it as the boundary value problem. In order to solve the equation for $h^{(G)}_{ij}$ as the boundary value problem, we had better rewrite the equation into useful forms. As shown in a previous paper\cite{asada}, Eq.$(\ref{hijG})$ is integrated to give \begin{eqnarray} h_{ij}^{(G)}&=& 2{\partial \over \partial x^i}\int(\rho v^j)^{\cdot}\vert {\bf x}-{\bf y} \vert d^3y +2{\partial \over \partial x^j}\int(\rho v^i)^{\cdot}\vert {\bf x}-{\bf y} \vert d^3y +\delta_{ij}\int\ddot\rho \vert {\bf x}-{\bf y} \vert d^3y \nonumber\\ &&+{1 \over 12}{\partial^2 \over \partial x^i \partial x^j}\int\ddot\rho \vert {\bf x}-{\bf y} \vert^3 d^3y +{\partial^2 \over \partial x^i \partial x^j}\int\Bigl(\rho v^2+3P-{\rho U \over 2} \Bigr) \vert {\bf x}-{\bf y} \vert d^3y \nonumber\\ &&~~~~~~~~~~~~~~-{2 \over 3}\delta_{ij}\int{\Bigl(\rho v^2+3P-\rho U / 2 \Bigr) \over \vert {\bf x}-{\bf y} \vert} d^3y. \label{haeq} \end{eqnarray} Using the relations \begin{eqnarray} &&\ddot \rho=-(\rho v^j)_{,j}^{\cdot}+O(c^{-2}),\nonumber\\ &&\dot v^i=0,\nonumber\\ &&v^i x^i=0, \end{eqnarray} Eq.$(\ref{haeq})$ is rewritten as \begin{eqnarray} h_{ij}^{(G)}&=& {7 \over 4}\biggl[\int(\rho v^j)^{\cdot} {x^i-y^i \over |{\bf x}-{\bf y}|} d^3y +\int(\rho v^i)^{\cdot}{x^j-y^j \over |{\bf x}-{\bf y}|} d^3y\biggr] -\delta_{ij}x^k \int {(\rho v^k)^{\cdot} \over |{\bf x}-{\bf y}|}d^3y \nonumber\\ &&-{1 \over 8}x^k \biggl[ {\partial \over \partial x^i}\int (\rho v^k)^{\cdot} {x^j-y^j \over |{\bf x}-{\bf y}|} d^3y +{\partial \over \partial x^j}\int (\rho v^k)^{\cdot} {x^i-y^i \over |{\bf x}-{\bf y}|} d^3y \biggr] \nonumber\\ &&+{1 \over 2}\biggl[ {\partial \over \partial x^i}\int\Bigl(\rho v^2+3P-{\rho U \over 2} \Bigr) {x^j-y^j \over |{\bf x}-{\bf y}|}d^3y +{\partial \over \partial x^j}\int\Bigl(\rho v^2+3P-{\rho U \over 2} \Bigr) {x^i-y^i \over |{\bf x}-{\bf y}|}d^3y \biggr] \nonumber\\ &&{\hskip 3cm}-{2 \over 3}\delta_{ij}\int{\Bigl(\rho v^2+3P-\rho U / 2 \Bigr) \over \vert {\bf x}-{\bf y} \vert} d^3y. \label{hbeq} \end{eqnarray} From Eq.$(\ref{hbeq})$, it is found that $h_{ij}^{(G)}$ is written as \begin{eqnarray} h_{ij}^{(G)}&=&{7 \over 4} \Bigl(x^i \hbox{$_{(3)}$} \dot P_j+x^j \hbox{$_{(3)}$} \dot P_i-\dot Q^{(T)}_{ij} -\dot Q^{(T)}_{ji}\Bigr) -\delta_{ij}x^k \hbox{$_{(3)}$} \dot P_k \nonumber\\ &&-{1 \over 8} x^k \biggl[{\partial \over \partial x^i}\Bigl(x^j \hbox{$_{(3)}$} \dot P_k -\dot Q^{(T)}_{kj} \Bigr) +{\partial \over \partial x^j}\Bigl(x^i \hbox{$_{(3)}$} \dot P_k -\dot Q^{(T)}_{ki} \Bigr)\biggr] \nonumber\\ &&+{1 \over 2} \biggl[{\partial \over \partial x^i}\Bigl(x^j Q^{(I)}- Q^{(I)}_j \Bigr) +{\partial \over \partial x^j}\Bigl(x^i Q^{(I)}- Q^{(I)}_i \Bigr) \biggr] -{2 \over 3}\delta_{ij} Q^{(I)}, \end{eqnarray} where \begin{eqnarray} &&\Delta \hbox{$_{(3)}$} P_i=-4\pi \rho v^i, \\ &&\Delta Q^{(T)}_{ij}=-4\pi \rho v^i x^j, \\ &&\Delta Q^{(I)}=-4\pi \Bigl(\rho v^2+3P-{1 \over 2}\rho U\Bigr), \\ &&\Delta Q^{(I)}_i =-4\pi \Bigl(\rho v^2+3P-{1 \over 2}\rho U \Bigr)x^i. \end{eqnarray} Therefore, $h_{ij}^{(G)}$ can be deduced from variables which satisfy the Poisson equations with compact sources. The source terms in the Poisson equations for $\hbox{$_{(3)}$}\beta_i$ and $\hbox{$_{(5)}$}\beta_i$ also fall off slowly. However, if we rewrite them as\cite{asada} \begin{eqnarray} \hbox{$_{(3)}$}\beta_i&=&-4\hbox{$_{(3)}$} P_i-{1 \over 2}\Bigl(x^i \dot U-\dot q_i\Bigr), \\ \hbox{$_{(5)}$} \beta_i&=&-4\hbox{$_{(5)}$} P_i-{1 \over 2}\Bigl(2x^i \hbox{$_{(4)}$} \dot \psi -\dot \eta_i \Bigr), \end{eqnarray} where \begin{eqnarray} &&\Delta q_i=-4\pi\rho x^i , \\ &&\Delta \hbox{$_{(5)}$} P_i=-4\pi \rho\biggl[v^i\Bigl(v^2+2U+\Gamma\varepsilon \Bigr) +\hbox{$_{(3)}$} \beta_i\biggr]+U_{,j}\Bigl(\hbox{$_{(3)}$}\beta_{i,j}+\hbox{$_{(3)}$}\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\beta_{k,k}\Bigr) \nonumber\\ &&{\hskip 4cm}-{1 \over 8}(\dot U U)_{,i} -{1 \over 4}(\hbox{$_{(3)}$} \beta_l U_{,l})_{,i}, \\ &&\Delta \eta_i=-4\pi\rho\Bigl(v^2+\varepsilon+{5 \over 2}U\Bigr)x^i, \end{eqnarray} then $\hbox{$_{(3)}$}\beta_i$ and $\hbox{$_{(5)}$}\beta_i$ can be obtained by solving the Poisson equations in which the fall-off of the source terms is fast enough, $O(r^{-5})$, for numerical calculation. Note that, using the relation $\hbox{$_{(3)}$} P_i=\epsilon_{izk}q_k \Omega$ and Eqs.$(\ref{killeq})$, $\hbox{$_{(3)}$} \beta_i$ and $\hbox{$_{(5)}$}\beta_i$ may be written as \begin{eqnarray} &&\hbox{$_{(3)}$} \beta_i = \Omega\Bigl\{-4\epsilon_{izk} q_k+{1 \over 2}\Bigl(x^i U_{,\varphi} -q_{i,\varphi}\Bigr)\Bigr\}\equiv \Omega\hbox{$_{(3)}$} \hat \beta_i, \\ &&\hbox{$_{(5)}$} \beta_i = \Omega\Bigl\{-4\hbox{$_{(5)}$} \hat P_i+{1 \over 2}\Bigl(2x^i \hbox{$_{(4)}$}\psi_{,\varphi} -\eta_{i,\varphi}\Bigr)\Bigr\}, \end{eqnarray} where \begin{eqnarray} \Delta \hbox{$_{(5)}$} \hat P_i &=&-4\pi\rho\biggl[ \epsilon_{izk}x^k\Bigl(v^2+2U +\Gamma \varepsilon\Bigr)+\hbox{$_{(3)}$} \hat \beta_i\biggr] +U_{,j}\Bigl(\hbox{$_{(3)}$}\hat \beta_{i,j}+\hbox{$_{(3)}$}\hat \beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\hat \beta_{k,k}\Bigr) \nonumber\\ &&{\hskip 4cm} +{1 \over 8}(UU_{,\varphi})_{,i}-{1 \over 4} (\hbox{$_{(3)}$}\hat \beta_k U_{,k})_{,i}. \end{eqnarray} \section{Derivation of basic equations} In this section, we derive the basic equation which has a suitable form to construct equilibrium configurations of uniformly rotating body in numerical calculation: Although equilibrium configurations can be formally obtained by solving Eq.$(\ref{bern})$ as well as metric potentials, $U$, $X$, $\hbox{$_{(4)}$}\psi$, $\hbox{$_{(6)}$}\alpha$, $\hbox{$_{(3)}$} \beta_i$, $\hbox{$_{(5)}$}\beta_i$ and $h_{ij}$, they do not take convenient forms for numerical calculation. Thus, we here change Eq.$(\ref{bern})$ into other forms appropriate to obtain numerically equilibrium configurations. In numerical calculation, the standard method to obtain equilibrium configurations is as follows\cite{hachisu,oohara,shibata}; \noindent (1) We give a trial density configuration for $\rho$. \noindent (2) We solve the Poisson equations. \noindent (3) Using Eq.$(\ref{bern})$, we give a new density configuration. \noindent These procedures are repeated until a sufficient convergence is achieved. Here, at (3), we need to specify unknown constants, $\Omega$ and $C$. In standard numerical methods\cite{hachisu,oohara}, these are calculated during iteration fixing densities at two points; i.e., if we put $\rho_1$ and $\rho_2$ at $x_1$ and $x_2$ into Eq.$(\ref{bern})$, they become two simultaneous equations for $\Omega$ and $C$. Hence, we can calculate them. However, the procedure is not so simple in the PN case: $\Omega$ is included in the source of the Poisson equations for the variables such as $X$, $\hbox{$_{(4)}$} \psi$, $\hbox{$_{(6)}$}\alpha$, $\eta_i$, $\hbox{$_{(5)}$} \hat P_i$, $h_{ij}^{(S)}$, $Q^{(T)}_{ij}$, $Q^{(I)}$ and $Q^{(I)}_i$. Thus, if we use Eq.$(\ref{bern})$ as it is, equations for $\Omega$ and $C$ become implicit equations for $\Omega$. As found in a previous paper\cite{shibata}, in such a situation, the convergence to a solution is very slow. Therefore, we transform those equations into other forms in which the potentials as well as Eq.$(\ref{bern})$ become explicit polynomial equations in $\Omega$. First of all, we define $q_2$, $q_{2i}$, $q_4$, $q_u$, $q_e$ and $q_{ij}$ which satisfy \begin{eqnarray} \Delta q_2 &=&-4\pi\rho R^2, \\ \Delta q_{2i} &=&-4\pi\rho R^2 x^i, \\ \Delta q_4 &=&-4\pi\rho R^4, \\ \Delta q_u &=&-4\pi\rho U, \\ \Delta q_e &=&-4\pi\rho \varepsilon, \\ \Delta q_{ij} &=&-4\pi\rho x^i x^j. \end{eqnarray} Then, $X$, $\hbox{$_{(4)}$}\psi$, $Q^{(I)}$, $Q_i^{(I)}$, $\eta_i$, $\hbox{$_{(5)}$} \hat P_i$, $Q_{ij}^{(T)}$, and $h^{(S)}_{ij}$ are written as \begin{eqnarray} X&=&-2q_2\Omega^2-2 q_u-(3\Gamma-2)q_e, \\ \hbox{$_{(4)}$}\psi&=&{1 \over 2}\Bigl(q_2\Omega^2+q_e+{5 \over 2}q_u\Bigr), \\ Q^{(I)}&=&q_2 \Omega^2+3(\Gamma-1)q_e-{1 \over 2}q_u \equiv q_2\Omega^2+Q^{(I)}_0, \\ Q_i^{(I)}&=&q_{2i}\Omega^2+Q_{0i}^{(I)}, \\ \eta_i&=&q_{2i}\Omega^2+\eta_{0i}, \\ \hbox{$_{(5)}$} \hat P_i&=&\epsilon_{izk}q_{2k}\Omega^2+\hbox{$_{(5)}$} P_{0i}, \\ Q_{ij}^{(T)}&=&\epsilon_{izl}q_{lj} \Omega, \\ h^{(S)}_{ij}&=&4\Omega^2\Bigl(\epsilon_{izk}\epsilon_{jzl}q_{kl} -{1 \over 3}\delta_{ij}q_2\Bigr), \end{eqnarray} where $Q_{0i}^{(I)}$, $\eta_{0i}$ and $\hbox{$_{(5)}$} P_{0i}$ satisfy \begin{eqnarray} \Delta Q_{0i}^{(I)} &=&-4\pi\Bigl( 3P- { 1 \over 2} \rho U\Bigr)x^i =-4\pi\rho\Bigl( 3(\Gamma-1)\varepsilon-{1 \over 2}U \Bigr)x^i, \\ \Delta \eta_{0i} &=&-4\pi\rho\Bigl(\varepsilon+{5 \over 2}U\Bigr)x^i, \\ \Delta \hbox{$_{(5)}$} P_{0i} &=&-4\pi \rho \biggl[\epsilon_{izk} x^k \Bigl(2U +\Gamma \varepsilon \Bigr)+\hbox{$_{(3)}$} \hat \beta_i\biggr] +U_{,j} \Bigl( \hbox{$_{(3)}$} \hat \beta_{i,j}+ \hbox{$_{(3)}$} \hat \beta_{j,i}-{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\hat \beta_{k,k}\Bigr) \nonumber\\ &&{\hskip 3cm} +{1 \over 8}(UU_{,\varphi})_{,i}-{1 \over 4} (\hbox{$_{(3)}$} \hat \beta_k U_{,k})_{,i} \equiv -4\pi S^{(P)}_i. \end{eqnarray} Note that $\hbox{$_{(5)}$} \beta_i$ and $h_{ij}^{(G)}$ are the cubic and quadratic equations in $\Omega$, respectively, as \begin{eqnarray} \hbox{$_{(5)}$} \beta_i &=&\Omega\biggl[-4\hbox{$_{(5)}$} P_{0i}+{1 \over 2}\Bigl\{x^i \Bigl(q_e+{5 \over 2}q_u\Bigr)_{,\varphi}-\eta_{0i,\varphi}\Bigr\}\biggr] +\Omega^3\biggl[ -4\epsilon_{izk}q_{2k} +{1 \over 2}\Bigl(x^iq_{2,\varphi}-q_{2i,\varphi}\Bigr)\biggr] \nonumber\\ && \equiv \hbox{$_{(5)}$} \beta^{(A)}_i\Omega +\hbox{$_{(5)}$} \beta_i^{(B)}\Omega^3, \\ h_{ij}^{(G)} &=&{1 \over 2}\biggl[ {\partial \over \partial x^j}\Bigl(x^i Q^{(I)}_0-Q^{(I)}_{0i}\Bigr)+ {\partial \over \partial x^i}\Bigl(x^j Q^{(I)}_0-Q^{(I)}_{0j}\Bigr) -{4 \over 3}\delta_{ij}Q^{(I)}_0\biggr] \nonumber\\ &&+\Omega^2 \biggl[{1 \over 2}\biggl\{ {\partial \over \partial x^j}\Bigl(x^i q_2-q_{2i}\Bigr)+ {\partial \over \partial x^i}\Bigl(x^j q_2-q_{2j}\Bigr) -{4 \over 3}\delta_{ij}q_2 \biggr\} \nonumber\\ &&\hskip 2cm -{7 \over 4}\Bigl(x^i\epsilon_{jzk}q_{k,\varphi}+x^j\epsilon_{izk}q_{k,\varphi} -\epsilon_{izk}q_{kj,\varphi}-\epsilon_{jzk}q_{ki,\varphi}\Bigr) +\delta_{ij}x^k \epsilon_{kzl} q_l \nonumber\\ &&\hskip 2cm +{1 \over 8}x^k\biggl\{ {\partial \over \partial x^i} \Bigl(x^j\epsilon_{kzl}q_{l,\varphi}-\epsilon_{kzl}q_{lj,\varphi}\Bigr) +{\partial \over \partial x^j} \Bigl(x^i\epsilon_{kzl}q_{l,\varphi}-\epsilon_{kzl}q_{li,\varphi}\Bigr) \biggr\} \biggr] \nonumber\\ && \equiv h_{ij}^{(A)}+h_{ij}^{(B)} \Omega^2 . \end{eqnarray} Finally, we write $\hbox{$_{(6)}$}\alpha$ as \begin{equation} \hbox{$_{(6)}$}\alpha=\hbox{$_{(6)}$} \alpha_0+\hbox{$_{(6)}$} \alpha_2 \Omega^2-2q_4\Omega^4, \end{equation} where $\hbox{$_{(6)}$} \alpha_0$ and $\hbox{$_{(6)}$} \alpha_2$ satisfy \begin{eqnarray} \Delta \hbox{$_{(6)}$} \alpha_0&=&4\pi\rho\biggl[\Bigl(3\Gamma-2\Bigr) \varepsilon U -\Bigl(3\Gamma-4\Bigr)q_e+3q_u\biggr] \nonumber\\ &&-\Bigl(h_{ij}^{(U)}+h_{ij}^{(A)}\Bigr)U_{,ij}-{3 \over 2}UU_{,l}U_{,l} +U_{,l}{\partial \over \partial x^l}\Bigl({9 \over 2}q_u+(3\Gamma+1)q_e\Bigr) \nonumber\\ &\equiv& -4\pi S^{(\alpha_0)}, \\ \Delta \hbox{$_{(6)}$} \alpha_2&=&8\pi\rho R^2\Bigl(5U+\Gamma\varepsilon +2\hbox{$_{(3)}$}\hat\beta_{\varphi} \Bigr) -\Bigl(4\epsilon_{izk}\epsilon_{jzl}q_{kl}-{4 \over 3}\delta_{ij}q_2 +h_{ij}^{(B)}\Bigr)U_{,ij}+3q_{2,l}U_{,l} \nonumber\\ && \hskip 2cm +{1 \over 2} \hbox{$_{(3)}$}\hat\beta_{i,j} \Bigl(\hbox{$_{(3)}$}\hat\beta_{i,j}+\hbox{$_{(3)}$}\hat\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\hat\beta_{k,k}\Bigr) \nonumber\\ &\equiv& -4\pi S^{(\alpha_2)} . \end{eqnarray} Using the above quantities, Eq.$(\ref{bern})$ is rewritten as \begin{equation} H-{H^2 \over 2c^2}+{H^3 \over 3c^4}=A+B\Omega^2+D\Omega^4 +{R^6 \over 6c^4}\Omega^6+C, \label{berneq} \end{equation} where \begin{eqnarray} A&=&U+{1 \over c^2}\Bigl(2q_u+(3\Gamma-2)q_e\Bigr)+{1 \over c^4}\Bigl\{ -\hbox{$_{(6)}$}\alpha_0-{U^3 \over 6}+U\Bigl(2q_u+(3\Gamma-2)q_e\Bigr)\Bigr\},\nonumber\\ B&=&{R^2 \over 2}+{1 \over c^2}\Bigl(2R^2U+2q_2+\hbox{$_{(3)}$}\hat\beta_{\varphi}\Bigr) +{1 \over c^4}\Bigl\{ -\hbox{$_{(6)}$}\alpha_2+{1 \over 2}\hbox{$_{(3)}$}\hat\beta_i\hbox{$_{(3)}$}\hat\beta_i +4\hbox{$_{(3)}$}\hat\beta_{\varphi}U \nonumber\\ &&+(3\Gamma-1)q_eR^2+{9 \over 2}q_uR^2+{15 \over 4}U^2R^2+2q_2U+ \hbox{$_{(5)}$}\beta_{\varphi}^{(A)} +{1 \over 2}\Bigl(h_{\varphi\varphi}^{(U)}+h_{\varphi\varphi}^{(A)}\Bigr) \Bigr\}, \nonumber\\ D&=&{R^4 \over 4c^2}+{1 \over c^4}\Bigl\{ 2q_4+\hbox{$_{(3)}$}\hat\beta_{\varphi}R^2+{7 \over 3}q_2R^2+2UR^4+ \hbox{$_{(5)}$}\beta_{\varphi}^{(B)}+{1 \over 2}\Bigl(h_{\varphi\varphi}^{(B)} +4R^2q_{RR}\Bigr)\Bigr\}. \end{eqnarray} Note that in the above, we use the following relations which hold for arbitrary vector $Q_i$ and symmetric tensor $Q_{ij}$, \begin{eqnarray} Q_{\varphi}&=&-yQ_{x}+xQ_{y}, \nonumber\\ Q_{\varphi\varphi}&=&y^2Q_{xx}-2xyQ_{xy}+x^2Q_{yy}, \nonumber\\ R^2Q_{RR}&=&x^2Q_{xx}+2xyQ_{xy}+y^2Q_{yy}. \end{eqnarray} We also note that source terms of Poisson equations for variables which appear in $A$, $B$ and $D$ do not depend on $\Omega$ explicitly. Thus, Eq.$(\ref{berneq})$ takes the desired form for numerical calculation. In this formalism, we need to solve 29 Poisson equations for $U$, $q_x$, $q_y$, $q_z$, $\hbox{$_{(5)}$} P_{0x}$, $\hbox{$_{(5)}$} P_{0y}$, $\eta_{0x}$, $\eta_{0y}$, $Q^{(I)}_{0x}$, $Q^{(I)}_{0y}$, $Q^{(I)}_{0z}$, $q_2$, $q_{2x}$, $q_{2y}$, $q_{2z}$, $q_u$, $q_e$, $h_{xx}^{(U)}$, $h_{xy}^{(U)}$, $h_{xz}^{(U)}$, $h_{yy}^{(U)}$, $h_{yz}^{(U)}$, $q_{xx}$, $q_{xy}$, $q_{xz}$, $q_{yz}$, $\hbox{$_{(6)}$}\alpha_0$, $\hbox{$_{(6)}$}\alpha_2$ and $q_4$. In Table 2, we show the list of the Poisson equations to be solved. In Table 3, we also summarize what variables are needed to calculate the metric variables $U$, $X$, $\hbox{$_{(4)}$}\psi$, $\hbox{$_{(6)}$} \alpha$, $\hbox{$_{(3)}$} \beta_i$, $\hbox{$_{(5)}$}\beta_i$, $h_{ij}^{(U)}$, $h_{ij}^{(S)}$, $h_{ij}^{(A)}$ and $h_{ij}^{(B)}$. Note that we do not need $\hbox{$_{(5)}$} P_{0z}$, $\eta_{0z}$, and $q_{zz}$ because they do not appear in any equation. Also, we do not have to solve the Poisson equations for $h^{(U)}_{zz}$ and $q_{yy}$ because they can be calculated from $h^{(U)}_{zz}=-h^{(U)}_{xx}-h^{(U)}_{yy}$ and $q_{yy}=q_2-q_{xx}$. In order to derive $U$, $q_i$, $q_2$, $q_{2i}$, $q_4$, $q_e$ and $q_{ij}$, we do not need any other potential because only matter variables appear in the source terms of their Poisson equations. On the other hand, for $q_u$, $Q_{0i}^{(I)}$, $\eta_{0i}$ and $h_{ij}^{(U)}$, we need the Newtonian potential $U$, and for $\hbox{$_{(5)}$} P_{0i}$, $\hbox{$_{(6)}$}\alpha_0$ and $\hbox{$_{(6)}$}\alpha_2$, we need the Newtonian as well as PN potentials. Thus, $U$, $q_i$, $q_2$, $q_{2i}$, $q_4$, $q_e$ and $q_{ij}$ must be solved first, and then $q_u$, $Q_{0i}^{(I)}$, $\eta_{0i}$, $h_{ij}^{(U)}$, $\hbox{$_{(5)}$} P_{0i}$ and $\hbox{$_{(6)}$}\alpha_2$ should be solved. $\hbox{$_{(6)}$}\alpha_0$ must be solved after we obtain $q_u$ because its Poisson equation involves $q_u$ in the source term. In Table 2, we also list potentials which are included in the source terms of the Poisson equations for other potentials. The configuration which we are most interested in and would like to obtain is the equilibrium state for BNS's of equal mass. Hence, we show the boundary condition at $r \rightarrow \infty$ for this problem. When we consider equilibrium configurations for BNS's where the center of mass for each NS is on the $x$-axis, boundary conditions for potentials at $r \rightarrow \infty$ become \begin{eqnarray} U &=&{1 \over r}\int \rho dV+O(r^{-3}), \hskip 2cm~~~ q_{x}={n^x \over r^2}\int \rho x^2 dV+O(r^{-4}), \nonumber\\ q_2 &=&{1 \over r}\int \rho R^2 dV+O(r^{-3}),\hskip 2cm q_{y}={n^y \over r^2}\int \rho y^2 dV+O(r^{-4}), \nonumber\\ q_e &=&{1 \over r}\int \rho \varepsilon dV+O(r^{-3}),\hskip 2cm~~ q_{z}={n^z \over r^2}\int \rho z^2 dV+O(r^{-4}), \nonumber\\ q_u&=&{1 \over r}\int \rho U dV+O(r^{-3}),\hskip 2cm~ q_4={1 \over r}\int \rho R^4 dV+O(r^{-3}), \end{eqnarray} \begin{eqnarray} \hbox{$_{(5)}$} P_{0x}&=&{n^x \over r^2}\int S^{(P)}_x x dV +{n^y \over r^2}\int S^{(P)}_y y dV +O(r^{-3}), \nonumber\\ \hbox{$_{(5)}$} P_{0y}&=&{n^x \over r^2}\int S^{(P)}_y x dV +{n^y \over r^2}\int S^{(P)}_y y dV +O(r^{-3}), \end{eqnarray} \begin{eqnarray} \eta_{0x}&=&{n^x \over r^2}\int \rho x^2\Bigl(\varepsilon+{5 \over 2}U\Bigr) dV+O(r^{-4}), \nonumber\\ \eta_{0y}&=&{n^y \over r^2}\int \rho y^2\Bigl(\varepsilon+{5 \over 2}U\Bigr) dV+O(r^{-4}), \end{eqnarray} \begin{eqnarray} Q^{(I)}_{0x}&=&{n^x \over r^2}\int \rho x^2\Bigl(3(\Gamma-1)\varepsilon -{1 \over 2}U\Bigr) dV+O(r^{-4}),\hskip 1cm q_{2x}={n^x \over r^2}\int \rho R^2 x^2 dV+O(r^{-4}), \nonumber\\ Q^{(I)}_{0y}&=&{n^y \over r^2}\int \rho y^2\Bigl(3(\Gamma-1)\varepsilon -{1 \over 2}U\Bigr) dV+O(r^{-4}),\hskip 1cm q_{2y}={n^y \over r^2}\int \rho R^2 y^2 dV+O(r^{-4}), \nonumber\\ Q^{(I)}_{0z}&=&{n^y \over r^2}\int \rho z^2\Bigl(3(\Gamma-1)\varepsilon -{1 \over 2}U\Bigr) dV+O(r^{-4}),\hskip 1cm q_{2z}={n^z \over r^2}\int \rho R^2 z^2 dV+O(r^{-4}), \end{eqnarray} \begin{eqnarray} h^{(U)}_{xx}&=&{1 \over r}\int S^{(U)}_{xx} dV+O(r^{-3}),\hskip 3cm h^{(U)}_{xy}={3n^x n^y \over r^3}\int S^{(U)}_{xy}xy dV+O(r^{-5}), \nonumber\\ h^{(U)}_{yy}&=&{1 \over r}\int S^{(U)}_{yy} dV+O(r^{-3}),\hskip 3cm h^{(U)}_{xz}={3n^x n^z \over r^3}\int S^{(U)}_{xz}xz dV+O(r^{-5}), \\ h^{(U)}_{yz}&=&{3n^y n^z \over r^3}\int S^{(U)}_{yz}yz dV+O(r^{-5}), \nonumber \end{eqnarray} \begin{eqnarray} q_{xx}&=&{1 \over r}\int \rho x^2 dV+O(r^{-3}),\hskip 3cm q_{xy}={3n^x n^y \over r^3}\int \rho x^2y^2 dV+O(r^{-5}), \nonumber\\ q_{xz}&=&{3n^x n^z \over r^3}\int \rho x^2 z^2 dV+O(r^{-5}),\hskip 2cm q_{yz}={3n^y n^z \over r^3}\int \rho y^2z^2 dV+O(r^{-5}), \end{eqnarray} \begin{eqnarray} \hbox{$_{(6)}$}\alpha_0&=&{1 \over r}\int S^{(\alpha_0)} dV+O(r^{-3}),\hskip 2cm \hbox{$_{(6)}$}\alpha_2 ={1 \over r}\int S^{(\alpha_2)} dV+O(r^{-3}), \end{eqnarray} where $dV=d^3x$, and \begin{equation} n^i={x^i \over r}. \end{equation} We note that at $r \rightarrow \infty$, $S_i^{(P)} \rightarrow O(r^{-5})$, $S_{ij}^{(U)} \rightarrow O(r^{-6})$, $S^{(\alpha_0)} \rightarrow O(r^{-4})$ and $S^{(\alpha_2)} \rightarrow O(r^{-4})$, so that all the above integrals are well defined. \section{Conserved quantities} In this section, we show the conserved quantities in the 2PN approximation because they will be useful to investigate the stability property of equilibrium solutions obtained in numerical calculations. \noindent (1)Conserved mass\cite{asada}; \begin{equation} M_{\ast} \equiv \int \rho_{\ast}d^3x, \end{equation} where \begin{eqnarray} \rho_{\ast}&=&\rho \alpha u^0 \psi^6 \nonumber\\ &=&\rho \biggl[1+{1 \over c^2}\Bigl({1 \over 2}v^2+3 U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{7 \over 2}v^2 U +{15 \over 4}U^2+6 \hbox{$_{(4)}$}\psi+\hbox{$_{(3)}$}\beta_i v^i\Bigr)+O(c^{-6})\biggr]. \label{stareq} \end{eqnarray} Equation $(\ref{stareq})$ may be written as \begin{equation} \rho_{\ast}=\rho \biggl[1+{1 \over c^2}\Bigl({1 \over 2}v^2+3 U\Bigr) +{1 \over c^4}\Bigl({3 \over 8}v^4+{13 \over 2}v^2 U +{45 \over 4}U^2+3U\varepsilon+\hbox{$_{(3)}$}\beta_i v^i\Bigr)+O(c^{-6})\biggr]. \end{equation} \noindent (2)ADM mass\cite{wald,asada}; \begin{equation} M_{ADM}=-{1 \over 2\pi}\int \Delta \psi d^3 x \equiv \int \rho_{ADM}d^3x, \end{equation} where \begin{eqnarray} \rho_{ADM}&=&\rho \biggl[1+{1 \over c^2} \Bigl(v^2+\varepsilon+{5 \over 2}U\Bigr) +{1 \over c^4}\biggl\{v^4+{13 \over 2}v^2 U+\Gamma\varepsilon v^2 +{5 \over 2}U\varepsilon+{5 \over 2}U^2+5\hbox{$_{(4)}$}\psi \nonumber\\ &&~~~~~~~~~~~~~+2\hbox{$_{(3)}$}\beta_i v^i +{1 \over 32\pi\rho}\hbox{$_{(3)}$}\beta_{i,j} \Bigl(\hbox{$_{(3)}$}\beta_{i,j}+\hbox{$_{(3)}$}\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\beta_{k,k}\Bigr) \biggr\} +O(c^{-6}) \biggr], \end{eqnarray} or \begin{eqnarray} \rho_{ADM}&=&\rho \biggl[1+{1 \over c^2} \Bigl(v^2+\varepsilon+{5 \over 2}U\Bigr) +{1 \over c^4}\Bigl(v^4+9 v^2 U+\Gamma\varepsilon v^2+5 U\varepsilon +{35 \over 4}U^2 +{3 \over 2}\hbox{$_{(3)}$}\beta_i v^i \Bigr) \nonumber\\ && \hskip 9cm +O(c^{-6}) \biggr]. \end{eqnarray} \noindent (3)Total energy, which is calculated from $M_{ADM}-M_*$ in the third PN order\cite{asada}; \begin{equation} E \equiv \int \rho_E d^3x, \end{equation} where \begin{eqnarray} \rho_E &=&\rho\biggl[ \biggl( {1 \over 2}v^2+\varepsilon-{1 \over 2}U \biggr) +{1 \over c^2}\biggl({5 \over 8}v^4+{5 \over 2}v^2 U +\Gamma v^2\varepsilon+2 U\varepsilon-{5 \over 2}U^2 +{1 \over 2}\hbox{$_{(3)}$}\beta_i v^i \biggr) \nonumber\\ &&\hskip 1.5cm +{1 \over c^4}\biggl\{ {11 \over 16}v^6+v^4\Bigl(\Gamma \varepsilon +{47 \over 8} U\Bigr) +v^2 \Bigl( 4\hbox{$_{(4)}$}\psi+6\Gamma \varepsilon U +{41 \over 8}U^2+{5 \over 2}\hbox{$_{(3)}$}\beta_i v^i-X \Bigr) \nonumber\\ &&\hskip 2cm -{5 \over 2}U^3+2\Gamma \hbox{$_{(3)}$}\beta_i v^i \varepsilon+5 \varepsilon\hbox{$_{(4)}$}\psi +5U \hbox{$_{(3)}$}\beta_i v^i-{15 \over 2}U\hbox{$_{(4)}$}\psi+{5 \over 4}U^2\varepsilon \nonumber\\ &&\hskip 2cm +{1 \over 2}h_{ij}v^iv^j +{1 \over 2}\hbox{$_{(3)}$}\beta_i\hbox{$_{(3)}$}\beta_i \nonumber\\ &&\hskip 2cm +{U \over 16\pi \rho}\biggl(2h_{ij}U_{,ij}+ \hbox{$_{(3)}$}\beta_{i,j} \Bigl(\hbox{$_{(3)}$}\beta_{i,j}+\hbox{$_{(3)}$}\beta_{j,i} -{2 \over 3}\delta_{ij}\hbox{$_{(3)}$}\beta_{k,k}\Bigr) \biggr)\biggr\}+O(c^{-6}) \biggr]. \end{eqnarray} \noindent (4)Total linear and angular momenta: In the case $K_i^{~i}=0$, these are calculated from\cite{wald} \begin{eqnarray} P_i&=&{1 \over 8\pi}\lim_{r \to \infty}\oint K_{ij}n^j dS \nonumber\\ &=&{1 \over 8\pi}\lim_{r \to \infty}\oint \psi^6 K_{ij} n^j dS \nonumber\\ &=&{1 \over 8\pi} \int (\psi^6 K_i^{~j})_{,j} d^3x \nonumber\\ &=& \int \Bigl( J_i +{1 \over 16\pi} \psi^4 \tilde \gamma_{jk,i} K^{jk} \Bigr) \psi^6 d^3x, \label{jjjeq} \end{eqnarray} where $J_i=(\rho c^2+\rho \varepsilon+P) \alpha u^0u_i$. Up to the 2PN order, the second term in the last line of Eq.$(\ref{jjjeq})$ becomes \begin{eqnarray} &&{1 \over 16\pi} \int h_{jk,i} \hbox{$_{(3)}$} \beta_{j,k} d^3x, \nonumber\\ &=&{1 \over 16\pi} \int \biggl[ \Bigl(h_{jk,i} \hbox{$_{(3)}$} \beta_{j} \Bigr)_{,k}- h_{jk,ik} \hbox{$_{(3)}$} \beta_{j} \biggr] d^3x, \nonumber\\ &=&{1 \over 16\pi} \lim_{r \to \infty}\oint h_{jk,i} \hbox{$_{(3)}$} \beta_{j} n^k dS=0, \end{eqnarray} where we use $h_{jk} \rightarrow O(r^{-1})$ and $\hbox{$_{(3)}$} \beta_{j} \rightarrow O(r^{-2})$ at $r \rightarrow \infty$, and the gauge condition $h_{jk,k}=0$. Thus, in the 2PN approximation, $P_i$ becomes \begin{equation} P_i \equiv \int p_{i} d^3x , \end{equation} where \begin{eqnarray} p_i&=&\rho \biggl[v^i+{1 \over c^2}\biggl\{ v^i\Bigl(v^2+\Gamma\varepsilon+6U\Bigr)+\hbox{$_{(3)}$}\beta_i \biggr\} +{1 \over c^4}\biggl\{h_{ij}v^j+\hbox{$_{(5)}$} \beta_i+\hbox{$_{(3)}$}\beta_i\Bigl(v ^2+6U+\Gamma \varepsilon\Bigr) \nonumber\\ &&\hskip 1.5cm+v^i\Bigl(2\hbox{$_{(3)}$}\beta_i v^i+10\hbox{$_{(4)}$}\psi+6\Gamma \varepsilon U+ {67 \over 4}U^2+\Gamma \varepsilon v^2+10Uv^2+v^4-X\Bigr)\biggr\} +O(c^{-5}) \biggr]. \end{eqnarray} The total angular momentum $J$ becomes \begin{equation} J= \int p_{\varphi} d^3x , \end{equation} where $p_{\varphi}=-yp_x+xp_y$. \section{Summary} It is generally expected that there exists no Killing vector in the spacetime of coalescing BNS's because such a spacetime is filled with gravitational radiation which propagates to null infinity. However, we may consider coalescing BNS's as the almost stationary object from physical point of view as described in section 1. Motivated by this idea, in this paper, we have developed a formalism to obtain equilibrium configurations of uniformly rotating fluid up to the 2PN order using the PN approximation. The concept of being ``almost'' stationary becomes clear in the framework of the PN approximation and, in particular, the stationary rotating objects can exist exactly at the 2PN order, since the energy loss due to the gravitational radiation does occur from the 2.5PN order. There appear, at the 2PN order, tensor potentials $h_{ij}$ which were completely ignored in Wilson's approach\cite{wilson}. It should be noted that these tensor potentials play an important role at the 2PN order: This is because they appear in the equations to determine equilibrium configurations as shown in previous sections and they also contribute to the total energy and angular momentum of systems. This means that if we performed the stability analysis ignoring the tensor potentials, we might reach an incorrect conclusion. In our formalism, we extract terms depending on the angular velocity $\Omega$ from the integrated Euler equation and Poisson equations for potentials, and rewrite the integrated Euler equation as an explicit equation in $\Omega$. This reduction will improve the convergence in numerical iteration procedure. As a result, the number of Poisson equations we need to solve in each step of iteration reaches 29. However, source terms of the Poisson equations decrease rapidly enough, at worst $O(r^{-4})$, in the region far from the source, so that we can solve accurately these equations as the boundary value problem like in the case of the first PN calculations\cite{shibata}. Thus, the present formalism will be useful to obtain equilibrium configurations for synchronized BNS's or the Jacobi ellipsoid. These configurations will be obtained in future work. \vskip 5mm {\bf Acknowledgments} For helpful discussions, we would like to thank T. Nakamura, M. Sasaki and T. Tanaka. H. A. would like to thank Professor S. Ikeuchi, Professor M. Sasaki and Professor Futamase for their encouragement. This work was in part supported by the Japanese Grant-in-Aid on Scientific Research of the Ministry of Education, Science, and Culture, No. 07740355. \vspace{5mm}

proofpile-arXiv_065-520

\section{Introduction} The quest for $\Omega_0$, the current ratio of the mean mass density of the universe to the critical density required for closure, has been a focus of the research efforts of many astrophysicists involving a variety of different techniques. At present, most observational evidence suggests a universe with sub-critical matter density, perhaps with a cosmological constant making up the difference required for a critical universe (e.g., Coles \& Ellis 1994; Ostriker \& Steinhardt 1995). The possibility of measuring $\Omega_0$ using the amount of ``substructure'' in galaxy clusters has thus generated some interest, ``This is a critical area for further research, as it directly tests for $\Omega$ in dense lumps, so both observational and theoretical studies on a careful quantitative level would be well rewarded.'' (Ostriker 1993). Early analytical work (e.g., Richstone, Loeb, \& Turner 1992) and simulations (Evrard et al. 1993; Mohr et al. 1995) found that the morphologies of X-ray clusters strongly favored $\Omega_0\sim 1$ over low-density universes. Along with $POTENT$ analysis of cosmic velocity fields (e.g., Dekel 1994), these substructure analyses were the only indicators in support of a critical value of $\Omega_0$. However, the analytical results (e.g., Kauffmann \& White 1993; Nakamura, Hattori, \& Mineshige 1995), simulations (e.g., Jing et al. 1994), and morphological statistics (e.g., Buote \& Tsai 1995b) have been criticized rendering the previous conclusions about $\Omega_0$ uncertain. Buote \& Tsai (1995b, hereafter BTa) introduced the power ratios (PRs) for quantifying the spatial morphologies of clusters in terms of their dynamical states. The PRs essentially measure the square of the ratio of a higher order moment of the two-dimensional gravitational potential to the monopole term computed within a circular aperture, where the radius is specified by a metric scale (e.g., 1 Mpc). Buote \& Tsai (1996, hereafter BTb) computed PRs of $ROSAT$ X-ray images for a sample of 59 clusters and discovered that the clusters are strongly correlated in PR space, obeying an ``evolutionary track'' which describes the dynamical evolution of the clusters (in projection). Tsai \& Buote (1996, hereafter TB) studied the PRs of a small sample of clusters formed in the hydrodynamical simulation of Navarro, Frenk, \& White (1995) and verified the interpretation of the ``evolutionary track''. In contrast to the previous studies (e.g., Richstone et al. 1992; Mohr et al. 1995), TB concluded that their small cluster sample, formed in a standard $\Omega_0=1$, CDM simulation, possessed too much substructure (as quantified by the PRs) with respect to the $ROSAT$ clusters, and thus favored a lower value of $\Omega_0$. However, a statistically large sample of clusters is important for studies of cluster morphologies. The PRs are most effective at categorizing clusters into different broad morphological types; i.e. the distinction between equal-sized bimodals and single-component clusters is more easily quantified than are small deviations in ellipticities and core radii between single-component clusters (see BTa). The efficiency of the PRs at classifying clusters into a broad range of morphological types is illustrated by their success at quantitatively discriminating the $ROSAT$ clusters along the lines of the morphological classes of Jones \& Forman (1992) (see BTb). There is a lower frequency of nearly equal-sized bimodals in the $ROSAT$ sample than clusters with more regular morphologies. Hence, to make most effective use of the PRs the models need to be adequately sampled (i.e. simulations have enough clusters) to ensure that relatively rare regions of PR-space are sufficiently populated. In this paper we build on the previous studies and investigate the ability of the PRs to distinguish between models having different values of $\Omega_0$. Unlike the previous theoretical studies of cluster morphologies mentioned above, we also consider models having different power spectra, $P(k)$, since $P(k)$ should affect the structures of clusters as well. At the time we began this project it was too computationally costly to use hydrodynamical simulations to generate for several cosmological models a large, statistically robust, number of clusters with sufficient resolution. To satisfy the above criteria and computational feasibility we instead used pure N-body simulations. The organization of the paper is as follows. We discuss the selection of cosmological models in \S \ref{models}; the specifications of the N-body simulations in \S \ref{sample}; the validity of using dark-matter-only simulations to generate X-ray images and the construction of the images in \S \ref{xray}; and computation of the PRs in \S \ref{prs}. We analyze the models having different values of $\Omega_0$ and a cosmological constant in \S \ref{omega}, and models with different spectral slopes and $\sigma_8$ in \S \ref{spectrum}. The implications of the results for all of the models and comparison of the simulations to the $ROSAT$ sample of BTb is discussed in \S \ref{disc}. Finally, in \S \ref{conc} we present our conclusions. \section{Simulations\label{sim}} \subsection{Cosmological Models\label{models}} To test the sensitivity of cluster morphologies to the cosmological density parameter due to matter, $\Omega_0$, and the power spectrum of density fluctuations, $P(k)$, we examined several variants of the standard Cold Dark Matter (CDM) model (e.g., Ostriker 1993). In Table \ref{table.models} we list the models and their relevant parameters: $\Omega_0$; $\lambda_0=\Lambda/3H^2_0$, where $\Lambda$ is a cosmological constant and $H_0$ is the present value of the Hubble parameter; the spectral index, $n$, of the scale-free power spectrum of density fluctuations, $P(k)\propto k^n$; and $\sigma_8$, the present rms density fluctuations in spheres of radius $8h^{-1}$ Mpc, where $h$ is defined by $H_0=100h$ km s$^{-1}$ Mpc$^{-1}$. The parameters of the open CDM model (OCDM) and low-density, flat model (LCDM) were chosen to be consistent with current observations (e.g., Ostriker \& Steinhardt 1995). Their normalizations were set according to the $\sigma_8 - \Omega_0$ relationship of Eke, Cole, \& Frenk (1996) to agree with the observed abundance of X-ray clusters. The biased CDM model (BCDM) was also normalized in this way. However, the BCDM simulation, because it has $\Omega_0=1$, necessarily has poorer resolution (i.e. fewer particles per cluster) than the OCDM and LCDM models due to the fixed box size of our simulations (see \S \ref{xray}). For the purposes of our investigation of cluster morphologies it is paramount to compare simulations having similar resolution. Hence we use the SCDM model (with $\sigma_8=1$) as our primary $\Omega_0=1$ simulation for analysis, which has resolution equivalent to the OCDM and LCDM simulations. (We show in \S \ref{svsb} that the means of the PR distributions for BCDM and SCDM are very similar which turns out to be most important for examining the effects of $\Omega_0$.) Hence, the SCDM, OCDM, and LCDM models allow us to explore the effects of $\Omega_0$ and $\lambda_0$ on the cluster morphologies; comparing SCDM and BCDM provides information on the influence of $\sigma_8$. We explore the effects of different $P(k)$ on the PRs using the scale-free models, which have different $n$ from SCDM. For the scale-free models we normalized each to the same characteristic mass, $M_{\star}$, defined to be (Cole \& Lacey 1996) the mass scale when the linear rms density fluctuation is equal to $\delta_c$, the critical density for a uniform spherically symmetric perturbation to collapse to a singularity. For $\Omega_0=1$, the linear theory predicts $\delta_c \approx 1.686$ (e.g., Padmanabhan 1993). We take the SCDM model with $\sigma_8=1$ as a reference for these scale-free models which gives a characteristic mass of $10^{14}M_{\sun}$. This procedure allows a consistent means to normalize the scale-free models relative to each other on the mass scales of clusters. Unfortunately, as a result of this normalization procedure, at earlier times the models have different large-scale power and thus the cluster mass functions are different for each of the models. The scale-free model with $n=-1.5$ is similar to the SCDM model and will be used to ``calibrate'' the scale-free models with respect to the other models (see Table \ref{table.models}). \subsection{N-body Cluster Sample\label{sample}} We use the Tree-Particle-Mesh (TPM) N-body code (Xu 1995b) to simulate the dissipationless formation of structure in a universe filled with cold dark matter. The simulations consist of $128^3$ particles in a square box of width $200h^{-1}$ Mpc. The gravitational softening length is $25h^{-1}$ kpc which translates to a nominal resolution of $\sim 50h^{-1}$ kpc. This resolution is sufficient for exploring the structure of clusters with PRs in apertures of radii $R_{ap}\gtrsim 0.5$ Mpc; for a discussion of the related effects of resolution on the performance of PRs on $ROSAT$ X-ray images see Buote \& Tsai (1995b, \S 4). All of the realizations have the same initial random phase. For each simulation we located the 39 most massive clusters using a version of the DENMAX algorithm (Bertschinger \& Gelb 1991) modified by Xu (1995a). This convenient selection criterion yields well defined samples for each simulation and allows consistent statistical comparison between different simulations which is the principal goal of our present investigation. For the various cosmological models we explore (see Table \ref{table.models}) these clusters generally have masses ranging from $(0.3-3)\times h^{-1}10^{15}M_{\sun}$, which correspond to typical cluster masses observed in X-ray (e.g., Edge et al. 1990; David et. al. 1993) and optically (e.g., Carlberg et. al. 1995) selected samples. \subsection{X-ray Images\label{xray}} \subsubsection{Motivation for $j_{g} \propto \rho^2_{DM}$} By letting the gas density trace the dark matter density $(\rho_{gas}\propto\rho_{DM})$ and by assuming that the plasma emissivity of the gas is constant, we computed the X-ray emissivity of the clusters, $j_{g} \propto \rho^2_{DM}$. Given its importance on the results presented in this paper, here we discuss at some length the suitability of this approximation. (Cooling of the gas is discussed in \S \ref{rosat}.) For clusters in the process of formation or merging, the gas can have hot spots appearing where gas is being shock heated (e.g., Frenk, Evrard, Summers, \& White 1996). One effect of such temperature fluctuations on the intrinsic X-ray emissivity is that the intrinsic plasma emissivity will vary substantially over the cluster thus rendering $j_{g} \propto \rho^2_{DM}$ a poor approximation. However, the intrinsic X-ray emissivity is not observed, but rather that which is convolved with the spectral response of the detector. For $ROSAT$ observations of clusters with the PSPC, the plasma emissivity is nearly constant over the relevant ranges of temperatures (NRA 91-OSSA-3, Appendix F, $ROSAT$ Mission Description), and thus temperature fluctuations contribute negligibly to variations in the emissivity; for previous discussions of this issue for the PRs see BTa and TB. A more serious issue is whether the shocking gas invalidates the $\rho_{gas}\propto\rho_{DM}$ approximation, in which case the dynamical state inferred from the gas would not reflect that of the underlying mass. TB, who analyzed the hydrodynamical simulation of Navarro et al. (1995a), showed that the PRs computed for both the gas and the dark matter gave similar indications of the dynamical states of the simulated clusters (see \S 4 of TB). In particular, this applied at early times when the clusters underwent mergers with massive subclusters. \footnote{See Buote \& Tsai (1995a) for a related discussion of the evolution of the shape of the gas and dark matter in the Katz \& White (1993) simulation.} Hence, our approximation for the X-ray emissivity should be reasonable even during the early, formative stages of clusters. Another possible concern with setting $\rho_{gas}\propto\rho_{DM}$ is that a gas in hydrostatic equilibrium, which should be a more appropriate description for clusters in the later stages of their evolution, traces the shape of the potential of the gravitating matter which is necessarily rounder than the underlying mass; if the gas is rounder, then the PRs will be {\it smaller}. However, the core radius (or scale length) of the radial profile of the gas also influences the PRs. In fact, clusters with larger core radii have {\it larger} PRs; see BTa who computed PRs for toy X-ray cluster models having a variety of ellipticities and core radii. When isothermal gas, which is a good approximation for a nearly relaxed cluster, is added to the potential generated by an average cluster formed in a $\Omega_0=1$, CDM simulation, the gas necessarily has a {\it larger} core radius than that of the dark matter (see Figure 14 of Navarro, Frenk, \& White 1995b). Hence, a gas in hydrostatic equilibrium will have a larger core radius than that of the dark matter, at least in the context of the CDM models we are studying. Considering the competing effects of smaller ellipticity and larger core radii (factors of 2-3 in each), and from consulting Table 6 of BTa, we conclude that no clear bias in the PRs is to be expected by assuming that the gas follows the dark matter. In further support of this conclusion are the similarities of the morphologies of clusters in the N-body study of Jing et al. (1995). Using centroid-shifts and axial ratios they find similar results when $\rho_{gas}\propto\rho_{DM}$ and when the gas is in hydrostatic equilibrium (see their Figures 5, 6, and 8). \subsubsection{Construction of the Images} Having chosen our representation for the X-ray emissivity, we then generated two-dimensional ``images'' for each cluster. A rectangular box of dimensions $4\times 4\times 10$ $h^{-3}$ Mpc$^3$ with random orientation was constructed about each cluster. We converted the particle distribution for each cluster to a mass density field using the interpolation technique employed in Smoothed Particle Hydrodynamics (SPH) (e.g., Hernquist \& Katz 1989), from which the X-ray emissivity was generated, $j_{g} \propto \rho^2_{DM}$. The SPH interpolation calculates the density at a grid point by searching for the nearest neighbors and is thus more robust and physical than other linear interpolation schemes like Cloud-in-Cell. For our SPH interpolation we use 20 neighbors and the spline kernel described in Hernquist \& Katz. The interpolation result is independent of the cell we choose for the X-ray emissivity calculations. Typically, the boxes contained $\sim 2500$ particles for a cluster; e.g., SCDM (1799-3965), OCDM (1359-3656), LCDM (1439-3877), and BCDM (603-1740). We projected the emissivity along the long edge of the box into a square $4\times 4$ $h^{-2}$ Mpc$^2$ ``image'' consisting of ``pixels'' of width $20h^{-1}$ kpc. This pixel width was chosen sufficiently small so as not to inhibit reliable computation the PRs. We do not add statistical noise or other effects associated with real observations to the X-ray images since our principal objective is to examine the intrinsic response of cluster morphologies to different cosmological parameters. However, the investigation of observational effects on the PRs by BTa, and the derived error bars on the PRs from $ROSAT$ clusters by BTb, do not show any large systematic biases; comparison of the simulations to the $ROSAT$ cluster sample is discussed in \S \ref{disc}. \subsection{Power Ratios\label{prs}} The PRs are derived from the multipole expansion of the two-dimensional gravitational potential, $\Psi(R,\phi)$, generated by the mass density, $\Sigma(R,\phi)$, interior to $R$, \begin{equation} \Psi(R,\phi) = -2Ga_0\ln\left({1 \over R}\right) -2G \sum^{\infty}_{m=1} {1\over m R^m}\left(a_m\cos m\phi + b_m\sin m\phi\right), \label{eqn.multipole} \end{equation} where $\phi$ is the azimuthal angle, $G$ is the gravitational constant and, \begin{eqnarray} a_m(R) & = & \int_{R^{\prime}\le R} \Sigma(\vec x^{\prime}) \left(R^{\prime}\right)^m \cos m\phi^{\prime} d^2x^{\prime}\label{eqn.mom1},\\ b_m(R) & = & \int_{R^{\prime}\le R} \Sigma(\vec x^{\prime}) \left(R^{\prime}\right)^m \sin m\phi^{\prime} d^2x^{\prime}\label{eqn.mom2}. \end{eqnarray} Because of various advantageous properties of X-ray images of clusters, we associate the surface mass density, $\Sigma$, with X-ray surface brightness, $\Sigma_X$ (which is derived from the projection of $\rho^2_{DM}$ -- see previous section); for more complete discussions of this association see BTa and TB. The square of each term on the right hand side of eq. (\ref{eqn.multipole}) integrated over the boundary of a circular aperture of radius $R_{ap}$ is given by (ignoring factors of $2G$), \begin{equation} P_m={1\over 2m^2 R^{2m}_{ap}}\left( a^2_m + b^2_m\right),\label{eqn.powerm} \end{equation} for $m>0$ and, \begin{equation} P_0=\left[a_0\ln\left(R_{ap}\right)\right]^2,\label{eqn.power0} \end{equation} for $m=0$. It is more useful for studies of cluster structure to consider the ratios of the higher order terms to the monopole term, $P_m/P_0$, which we call ``power ratios'' (PRs). By dividing each term by $P_0$ we normalize to the flux within $R_{ap}$. Because for clusters, $P_m/P_0\ll 1$ for $m>0$ (e.g., BTb), it is preferable to take the logarithm of the PRs, \begin{equation} PR_m \equiv \log_{10}{P_m\over P_0}, \end{equation} which we shall henceforward analyze in this paper. Since the $P_m$ depend on the origin of the chosen coordinate system, we consider two choices for the origin. First, we take the aperture to lie at the centroid of $\Sigma_X$; i.e. where $P_1$ vanishes. Of these centroided $PR_m$, $PR_2$, $PR_3$, and $PR_4$ prove to be the most useful for studying cluster morphologies (see BTa). In order to extract information from the dipole term, we also consider the origin located at the peak of $\Sigma_X$. We denote this dipole ratio by $P^{(pk)}_1/P^{(pk)}_0$, and its logarithm $PR^{(pk)}_1$, to distinguish it from the centroided power ratios. To obtain the centroid of $\Sigma_X$ in a consistent manner for all clusters we adopted the following procedure. First, when projecting the cluster (see \S \ref{xray}), the cluster was roughly centered on the X-ray image by eye. For each image we computed the centroid in a circular aperture with $R_{ap}=1.5$ $h^{-1}$ Mpc located about the field center. This centroid was then used as our initial center for each cluster; see BTa for a description of how the peak of $\Sigma_X$ is located. In addition to considering the $PR_m$ individually, we also analyze the cluster distributions along the ``evolutionary tracks'' in the $(PR_2, PR_4)$ and $(PR_2, PR_3)$ planes obeyed by the $ROSAT$ clusters of BTb. We refer the reader to TB for a detailed discussion of the cluster properties along the evolutionary tracks. Using the augmented Edge et al. (1990) sample of BTb we recomputed the lines defining the evolutionary tracks of the $ROSAT$ data for the $1h^{-1}_{80}$ Mpc apertures. This was done since TB selected a subset of the clusters based on $PR_m$ measurement uncertainty rather than flux. Following TB we fit $PR_4 = a + bPR_2$ considering the uncertainties in both axes ; a similar fit was done for the $(PR_2, PR_3)$ plane. We obtained $a=-0.92$, $b=1.18$ for the $(PR_2, PR_4)$ track which we denote by $PR_{2-4}$. Similarly, for the $(PR_2, PR_3)$ track, which we denote by $PR_{2-3}$, we obtained $a=-0.49$, $b=1.16$. These results are nearly the same found by TB for the slightly different sample. To facilitate comparison to a previous study of the $PR_m$ of $ROSAT$ clusters (BTb), we compute $PR_m$ of the simulated clusters in apertures ranging in radius from $0.5h^{-1}_{80}$ Mpc to $1.5h^{-1}_{80}$ Mpc ($H_0=80h_{80}$ km s$^{-1}$ Mpc$^{-1}$) in steps of $0.25h^{-1}_{80}$ Mpc; i.e. $(0.4-1.2)h^{-1}$ Mpc in steps of $0.2h^{-1}$ Mpc. We refer the reader to \S 2 of BTa and \S 2 of TB for discussions of the advantages of using a series of fixed metric aperture sizes to study cluster morphologies. \section{$PR_m$ for Models with Different $\Omega_0$ and $\lambda_0$\label{omega}} First we consider clusters formed in the SCDM, OCDM, and LCDM models. Contour plots for 16 of the clusters formed in each of the models are displayed in Figures \ref{fig.scdm}, \ref{fig.ocdm}, and \ref{fig.lcdm}. In Figure \ref{fig.p2p4cdm} we show the $(PR_2, PR_4)$ plane for the $0.5h^{-1}_{80}$ Mpc and $1.0h^{-1}_{80}$ Mpc apertures; the SCDM model appears in each plot for comparison. The clusters in each of the models exhibit tight correlations very similar to the evolutionary tracks of the $ROSAT$ clusters (BTb) and the simulated hydrodynamic clusters $(\Omega_0=1)$ of Navarro et al. (1995a) studied by TB. Along the evolutionary tracks a shift in the means of the $PR_m$ is easily noticeable in the $0.5h^{-1}_{80}$ Mpc aperture, being most apparent for the SCDM-OCDM models. The spread of the $PR_m$ along the track in the $1.0h^{-1}_{80}$ Mpc aperture for SCDM-OCDM also appears to be different. The distributions perpendicular to $PR_{2-4}$ do not show discrepancies obvious to the eye. At this time we shift our focus away from the evolutionary tracks and instead analyze the individual $PR_m$ distributions, which prove to be more powerful for distinguishing between cosmological models as we show below. We give the individual $PR_m$ distributions of clusters in the three models for the $0.5h^{-1}_{80}$ Mpc and $1.0h^{-1}_{80}$ Mpc apertures in Figures \ref{fig.prhist.05mpc} and \ref{fig.prhist.10mpc}. We found it most useful to compare these distributions in terms of their means, variances, and Kolmogorov-Smirnov (KS) statistics. For the number of clusters in each of our simulations (39) higher order statistics like the skewness and kurtosis are unreliable ``high variance'' distribution shape estimates (e.g., Bird \& Beers 1993). We did consider more robust statistics like the ``Asymmetry Index'' (AI), which measures a quantity similar to the skewness, and the ``Tail Index'' (TI), which is similar to the Kurtosis (Bird \& Beers 1993). However, we found that they did not clearly provide useful information in addition to the lower order statistics and KS test, and thus we do not discuss them further. \footnote{Actually, the KS test turns out not to provide much additional information over the t-test and F-test, but we include it for ease of comparison to previous studies; e.g., Jing et al. (1995); Mohr et al. (1995); TB.} The means and standard deviations for the $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc apertures are listed in Table \ref{table.avgpr}. As a possible aid to understanding the relationships between the values in Table \ref{fig.avgpr.omega}, we plot in Figure \ref{fig.avgpr.omega} the standard deviation vs the mean for the $0.75h^{-1}_{80}$ Mpc aperture. We do not present the results for the larger apertures because they did not significantly improve the ability to distinguish between the models. Moreover, we found that $PR_4$ and $PR^{(pk)}_1$ do not provide much useful information in addition to $PR_2$ and $PR_3$. Generally $PR_4$ tracks the behavior of $PR_2$, though showing less power to discriminate between models; the similarity to $PR_2$ is understandable given the strong correlation shown in Figure \ref{fig.p2p4cdm}. Likewise, $PR^{(pk)}_1$ is similar to, but not quite so effective as, $PR_3$. For compactness, thus, we shall henceforward mostly restrict our discussion to results for $PR_2$ and $PR_3$ in the $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc apertures. We compare the means, standard deviations, and total distributions of the models in Table \ref{table.test} using standard non-parametric tests as described in Press et al. (1994). The Student's t-test, which compares the means of two distributions, computes a value, $p_t$, indicating the probability that the distributions have significantly different means. Similarly, the F-test, which compares the variances of two distributions, computes a value, $p_F$, indicating the probability that the distributions have significantly different variances. Finally, the KS test, which compares the overall shape of two distributions, computes a value, $p_{KS}$, indicating the probability that the distributions originate from the same parent population; the probabilities listed in Table \ref{table.test} are given as percents; i.e. decimal probability times 100. Note that for the cases where the F-test gives a probability less than 5\% we use the variant of the t-test appropriate for distributions with significantly different variances (i.e. program {\it tutest} in Press et. al.). \subsection{SCDM vs. OCDM} As is clear from inspection of Figures \ref{fig.p2p4cdm} - \ref{fig.avgpr.omega}, and Tables \ref{table.avgpr} - \ref{table.test}, the means of the $PR_m$ of the SCDM model exceed those of OCDM. In terms of the t-test the significance of the differences is very high. Of all the $PR_m$, generally the means of $PR_2$ and $PR_3$ exhibit the largest significant differences; the most significant differences are seen for $PR_3$ in the $0.5h^{-1}_{80}$ Mpc aperture, $p_t=0.02\%$, and for $PR_2$ in the $0.75h^{-1}_{80}$ Mpc aperture, $p_t=0.06\%$. Hence, though different in all the apertures, the discrepancy in the means is most significant for the smallest apertures, $(0.5,0.75)h^{-1}_{80}$ Mpc. The variances of $PR_3$ in the SCDM model are essentially consistent at all radii with their corresponding values in OCDM. However, for $PR_2$ the variances are consistent at $0.75h^{-1}_{80}$ Mpc, but marginally inconsistent at $(0.5,1.0)h^{-1}_{80}$ Mpc (and inconsistent at $(1.25,1.5)h^{-1}_{80}$ Mpc). The KS test generally indicates a significant difference in SCDM and OCDM when also indicated by the t-test, or the t-test and F-test together. The level of discrepancy is usually not as significant as given by the t-test, except when $p_F$ is small as well. Since the KS test does not indicate discrepancy when both the t-test and F-test indicate similarity, we conclude that higher order properties of the PR distributions are probably not very important for the SCDM and OCDM models (at least for the samples of 39 clusters in our simulations). Since this qualitative behavior holds for the other model comparisons, we shall not emphasize the KS tests henceforward. Finally, in terms of the various significance tests we find that the $PR_{2-4}$ distribution essentially gives a weighted probability of the individual $PR_2$ and $PR_4$ distributions; i.e. it does not enhance the discrepancy in the individual distributions. Perpendicular to $PR_{2-4}$ the distributions are consistent. The same behavior is seen for $PR_{2-3}$ as well. This behavior is seen for the remaining model comparisons in this section so we will not discuss the joint distributions further. \subsection{SCDM vs. LCDM} The $PR_m$ means for the SCDM clusters also systematically exceed those in the LCDM model, however the significance of the difference is not as large as with the OCDM clusters. The largest discrepancy is observed for $PR_2$ in the $(0.75,1.0)h^{-1}_{80}$ Mpc apertures for which $p_t=(0.7\%,0.8\%)$. The other $PR_m$ show only a marginal discrepancy in the means. For apertures $(0.5,0.75)h^{-1}_{80}$ Mpc, $PR_3$ has $p_t=(4\%,3\%)$, but is quite consistent at larger radii. The variances for the SCDM and LCDM models are consistent for essentially all radii and all $PR_m$. \subsection{OCDM vs. LCDM} The $PR_m$ means for the LCDM clusters appear to systematically exceed those in the OCDM model, however the formal significances of the differences are quite low. The means are entirely consistent at all radii for $PR_2$. However, $PR_3$ shows a marginal difference in the $0.5h^{-1}_{80}$ Mpc aperture ($p_t=9\%$). The variances of the $PR_m$ of the OCDM and LCDM models behave similarly as with the SCDM and OCDM comparison above, as expected since the SCDM-LCDM variances are essentially identical. However, the degree of discrepancy is not as pronounced. \subsection{Performance Evaluation I.} The means of the individual $PR_m$ distributions generally exhibit the most significant differences between the SCDM, OCDM, and LCDM models; the variances are much less sensitive to the models, with $PR_3$ showing no significant variance differences. The larger means for the $PR_m$ in the SCDM models are expected from the arguments of, e.g., Richstone et al. (1992). That is, in a sub-critical universe the growth of density fluctuations ceased at an early epoch and so present-day clusters should show less ``substructure'' than in an $\Omega_0=1$ universe where formation continues to the present. Clusters with more structure will have systematically larger values of the $PR_m$. The $PR_m$ whose means show the most significant differences between the models are $PR_2$ and $PR_3$, where $PR_2$ typically performs best for apertures $(0.75,1.0)h^{-1}_{80}$ Mpc and $PR_3$ is most effective for $(0.5,0.75)h^{-1}_{80}$ Mpc. Although useful, $PR^{(pk)}_1$ is often the least effective $PR_m$ for differentiating models in terms of its mean; this relatively weak performance of the dipole ratio with respect to other moments is echoed in the results of Jing et al. (1995) who found that their measure of an axial ratio performed better than a centroid shift for discriminating between models (see their tables 3-6). \section{$PR_m$ for Models with Different $n$ and $\sigma_8$\label{spectrum}} In this section we investigate the effects of different power spectra for models otherwise conforming to the specifications of the SCDM model. First, we examine models with different spectral indices of the scale-free power spectrum $(P(k)\propto k^n)$, $n=0,-1,-1.5,-2$. Then we examine the BCDM model which has a lower power-spectrum normalization as expressed by $\sigma_8$. As in the previous section, we find the $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc apertures to be more useful than the larger apertures, and that $PR_4$ and $PR^{(pk)}_1$ do not provide much useful information in addition to that provided by $PR_2$ and $PR_3$. Hence, for compactness we again mostly restrict the discussion to $PR_2$ and $PR_3$ in the smaller apertures. \subsection{$n=-1.5$ vs. SCDM\label{n15}} Before analyzing the $PR_m$ of models with different $n$ we calibrate the scale-free models by comparing the $n=-1.5$ scale-free model to the SCDM model since they should have similar properties (see \S \ref{models}). We find that the means, variances, and KS statistics of the centroided $PR_m$ for the SCDM and $n=-1.5$ models are entirely consistent for all aperture radii with only one possible exception. The variances of $PR_3$ exhibit a marginal $(p_F=5\%)$ discrepancy in the $0.5h^{-1}_{80}$ Mpc aperture. The significance of this variance discrepancy should be treated with caution given the complete consistency of the means $(p_t=31\%)$ and KS (32\%) test at this radius as well as the consistency of all the tests at all the other radii investigated. Hence, the cluster morphologies of the SCDM and $n=-1.5$ models are very consistent expressed in terms of the centroided $PR_m$ ($m=2,3,4$). \subsection{$n=0$ vs. $n=-2$\label{n0n2}} In Figure \ref{fig.ets.sf} we plot for the $n=0,-2$ models the PR correlations for $m=(2,3)$ and $m=(2,4)$ in the $(0.5,1.0)h^{-1}_{80}$ Mpc apertures. Histograms for the individual $PR_m$ in these apertures are displayed in Figures \ref{fig.sfhist.05mpc} and \ref{fig.sfhist.10mpc}. Table \ref{table.avgpr} lists the means, Figure \ref{fig.avgpr.pk} plots the standard deviations versus the averages of the $PR_m$ in the $0.75h^{-1}_{80}$ Mpc aperture, and Table \ref{table.test} gives the results of the significance tests. The means of $PR_3$ are very consistent for the $n=0,-2$ models at all radii examined. Those for $PR_2$ may show some differences in their means, with the $n=-2$ models perhaps having systematically smaller values. The significance of the different means for $PR_2$ is only formally marginal, with $p_t=(11\%,4\%,10\%)\%$ for aperture radii $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc. However, the possible small differences in the means of $PR_2$ are dwarfed by the corresponding highly significant differences in its variances. Generally the variances for all the $PR_m$ in all the apertures are smaller for the $n=-2$ clusters. The most significant variance differences are observed for $PR_2$ which has $p_F<1\%$ in $(0.75,1.0)h^{-1}_{80}$ Mpc apertures and $p_F\sim 5\%$ for $0.5h^{-1}_{80}$ Mpc . The variances for $PR_3$ show differences but at a lower level of significance and only in the $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc apertures; i.e. $p_t=(3\%,1\%,4\%)$. Similar to what we found in \S \ref{omega}, the differences implied by the KS test generally follow the significances implied by the t-test and F-test; i.e. higher order effects in the distributions are probably not overly important (at least for our sample sizes of 39 clusters). Moreover, again we find that analysis of the $PR_m$ in terms of the evolutionary tracks does not add useful information to the previous results. The mean and variance effects for the individual $PR_m$ translate to very similar behavior along $PR_{2-4}$. The direction perpendicular to $PR_{2-4}$ is essentially consistent for all of the tests. As a result, we do not emphasize the KS tests or the evolutionary tracks further. \subsection{Intermediate $n$} The behavior for other $n$ is similar, but depends to some extent on the range examined. We find that the range of $n$ which accentuates differences in the $PR_m$ is between $n=0,-1$. Over the range $n=0,-1$ the discrepancy of means for $PR_2$ essentially follows that of the full $n=0,-2$ discussed in \S \ref{n0n2}. However, the variances are not so highly discrepant as before, with $p_F=3\%$ for $PR_2$ for aperture radii $(0.5,0.75)h^{-1}_{80}$ Mpc; elsewhere the variances of $PR_2$ are consistent between the $n=0,-1$ models. Over the $n=0,-1$ range $PR_3$ is consistent for all statistics at all radii examined. The $PR_m$ exhibit very few differences over the range of indices $n=-1,-2$. For all radii examined the means and KS statistics are consistent for all the $PR_m$. However, the variances do show some marginal differences. The $1.0h^{-1}_{80}$ Mpc apertures has the most significance difference where $p_F=1.5\%$ for both $PR_2$ and $PR_3$. Also in the $0.75h^{-1}_{80}$ Mpc aperture $PR_2$ has $p_F=7\%$. Otherwise the variances of these $PR_m$ are consistent. (We mention that the variance of $PR_2$ for the $n=-1.5$ model in the $0.75h^{-1}_{80}$ Mpc aperture lies above that for the $n=-1$ model in Figure \ref{fig.avgpr.pk}, but the difference is not statistically significant.) \subsection{SCDM vs. BCDM\label{svsb}} Now we consider the $\Omega_0=1$, CDM model with a lower normalization, $\sigma_8=0.51$, which we refer to as the biased CDM model, BCDM. The means and variances for the BCDM model are listed in Table \ref{table.avgpr}, the standard deviation versus the average $PR_m$ in the $0.75h^{-1}_{80}$ Mpc aperture are plotted in Figure \ref{fig.avgpr.pk}, and the results for the significance tests in comparison to SCDM are given in Table \ref{table.test}. The means of all the $PR_m$ at all aperture radii are consistent for the SCDM and BCDM models. The $PR_m$ variances of the SCDM clusters generally exceed those of the BCDM clusters. The significance levels of the differences are only marginal $(p_F\sim 3\%)$ and appear to be most important in the $0.75h^{-1}_{80}$ Mpc aperture. It is possible that the slight variance differences between the SCDM and BCDM models are due to the difference in resolution between the two simulations; i.e. the clusters in the BCDM simulations contain about half the number of particles of the SCDM clusters. We would expect that the effects of resolution would be most important in the smallest apertures (which we do observe), although we would probably expect that the means as well as the variances would be affected (which we do not observe). We mention that the BCDM model performs virtually identically to the SCDM model when compared to the OCDM and LCDM models. \subsection{Performance Evaluation II.} The variances of the $PR_m$ show the most significant differences between models with different power spectra; $PR_2$ generally has the most sensitive variances over the parameter ranges explored. Decreasing $n$ and $\sigma_8$ both decrease the $PR_m$ variances, the differences being of similar magnitude for the $n=0,-1$ models and the SCDM and BCDM models. The means of the $PR_m$ are much less sensitive to the models with different $n$ and $\sigma_8$, with $PR_2$ showing the largest significant differences which are always less than differences in the variances. No significant differences in the means are observed for $PR_3$ over the range of power spectra studied. The predominant effect of the power spectrum on the variances of the $PR_m$ is intriguing. It is reasonable that when the amount of small-scale structures is reduced (smaller $n$) or the population of cluster-sized structures is made more uniform (smaller $\sigma_8$) that the $PR_m$ distributions would also be more uniform. The observed low sensitivity of the $PR_m$ means to the power spectra is also reasonable since on average the $PR_m$ means should only be affected by the rate of mass accretion through the aperture of radius $R_{ap}$, not by the sizes of the individual accreting clumps. \section{Discussion\label{disc}} In the previous sections we have seen that differences in $\Omega_0$ and $P(k)$ in CDM models are reflected in the spatial morphologies of clusters when expressed in terms of the $PR_m$. For the purposes of probing $\Omega_0$, our analysis indicates that $PR_3$ is the best PR since its mean is quite sensitive to $\Omega_0$ but very insensitive to $P(k)$. It is advantageous to also consider $PR_2$ when a cosmological constant is introduced since its means differ for the SCDM and LCDM models by $\sim 3\sigma$ whereas $PR_3$ only distinguishes the models at the $\sim 2\sigma$ level. The marginal dependence of the mean of $PR_2$ on $P(k)$ is not overly serious for studying differences in $\Omega_0$ because the differences in means due to $P(k)$ are always accompanied by larger, more significant differences in the variances; i.e. different means for $PR_2$ but consistent variances should reflect differences only in $\Omega_0$. The best apertures for segregating models are generally $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc. A few previous studies have examined the influence of $\Omega_0$ and $\lambda_0$ on the morphologies of galaxy clusters. Perhaps the most thorough investigation is that of Jing et. al. (1995) who used N-body simulations of a variety of CDM models, including versions similar to our SCDM, OCDM, and LCDM, to study variations of center-shifts and axial ratios. Jing et. al. reached the same qualitative conclusions as we do; i.e. the SCDM model is easily distinguished from OCDM and LCDM because it produces clusters with much more irregular morphologies than than the others. However, Jing et al. obtained infinitesimal KS probabilities for the axial ratio when comparing SCDM to OCDM and LCDM, a level of significance orders of magnitude different from that found in this paper. The source of this discrepancy is unclear given the qualitative similarities of their axial ratio and our $PR_2$. The disagreement may arise from differences in numerical modeling between the simulations; i.e. the results of Jing et al. are derived from simulations with a larger force resolution ($0.1h^{-1}$ Mpc), and smaller particle number for the non-SCDM models ($64^3$) than in our simulations, and have clusters which visually do not show the rich structures seen in our simulations (Figures \ref{fig.scdm}, \ref{fig.ocdm}, and \ref{fig.lcdm}). The qualitative results of Jing et al. agree with the hydrodynamic simulations of Mohr et al. (1995) who also used center shifts and axial ratios as diagnostics for ``substructure''. If we visually estimate the means of the center shifts and axial ratios from Figures 6 and 8 of Jing et al. for their SCDM, OCDM, and LCDM models (actually OCDM with $\Omega_0=0.2$ and LCDM with $\Omega_0=0.2, \lambda_0=0.8$), we find that they agree quite well with the corresponding values in Table 3 of Mohr et al.; i.e. the results from the N-body and hydrodynamic simulations are very similar, despite the many other differences between the simulations (e.g., large number of baryons in OCDM clusters for Mohr et al.). We can make a similar comparison of the $PR_m$ derived in this paper with the results from TB who analyzed the small sample of SCDM clusters formed in the hydrodynamic simulation of Navarro et al. (1995a). We find that the means (and variances) of the $PR_m$ computed in this paper are very similar to those of the hydrodynamic clusters; e.g., the mean for $PR_{2-4}$ for $1h^{-1}_{80}$ Mpc may be read off Figure 7 of TB which shows excellent agreement with the SCDM value we obtain from the N-body simulations (average $PR_{2-4}=3.76$). The quantitative similarity between the results, particularly between the means of the morphological statistics, for the N-body and hydrodynamic simulations of (Jing et al.,Mohr et al.) and (this paper,TB) suggest that it is useful to compare the $PR_m$ derived from N-body simulations directly to the X-ray data. \subsection{Comparison to $ROSAT$ Clusters\label{rosat}} Among the biases that need to be considered in such a comparison are the effects of cooling flows (e.g., Fabian 1994), selection, and noise. Cooling flows increase the X-ray emission in the cluster center, which has the effect on the $PR_m$ of essentially decreasing the core size of the cluster. Judging by the observed core radii of ``regular'' X-ray clusters we would expect at most a factor of $\sim 2$ difference in core radii (e.g., A401 vs. A2029 in Buote \& Canizares 1996; also see Jones \& Forman 1984). \footnote{Large cooling flows only appear in clusters with regular morphologies (e.g., Jones \& Forman 1992; Fabian 1994; BTb).} Changing the core radius by a factor of 2 typically changes $PR_2$ (for example) by a small fraction of a decade (see Table 6 of BTa); this behavior, as we show below, is confirmed using a more thorough treatment. The issue of biases between X-ray-selected and mass-selected samples needs to be addressed with hydrodynamical simulations. The estimated uncertainties of the $PR_m$ for the $ROSAT$ cluster sample of BTb, which take into account noise and unresolved sources, do not show any clear biases. In Figure \ref{fig.ets.ros} we display the correlations of the centroided $PR_m$ for the $ROSAT$ sample of BTb in the $(0.5,1.0)h^{-1}_{80}$ Mpc apertures; the SCDM clusters are also plotted for a comparison. $PR_m$ histograms for these apertures are shown in Figure \ref{fig.roshist.05mpc} and \ref{fig.roshist.10mpc}, along with those for the SCDM and OCDM models. We list the means and variances for the $ROSAT$ clusters in Table \ref{table.ros.avg}; we plot in Figure \ref{fig.avgpr.rosat} the standard deviations versus the means for the $ROSAT$ clusters and models in the $0.5h^{-1}_{80}$ Mpc aperture; the results of the significance tests between the $ROSAT$ clusters and model clusters are given in Table \ref{table.ros.test}. We analyze the $ROSAT$ clusters corresponding to the ``updated Edge et. al. (1990)'' flux-limited sample in BTb which gives 37 and 27 clusters respectively for the $(0.5,1.0)h^{-1}_{80}$ Mpc apertures; note that all the qualitative features of the results we obtain below are reproduced when all of the clusters studied in BTb are used (i.e. 59 and 44 clusters respectively). The means of the SCDM clusters exceed those of the $ROSAT$ sample to a high level of significance, with the differences being most pronounced in the $0.5h^{-1}_{80}$ Mpc aperture. The most significant discrepancy is for $PR_3$ in the $0.5h^{-1}_{80}$ Mpc aperture for which $p_t=1.5\times 10^{-4}\%$. The variances for all the $PR_m$ except $PR_3$ are also significantly different, with the variances of the SCDM clusters exceeding those of the $ROSAT$ clusters. The SCDM model has $\sigma_8=1$ which is too high to fit other observations (e.g., Ostriker \& Steinhardt 1995). The BCDM model, which has $\sigma_8=0.51$, does have $PR_m$ variances in better agreement with the $ROSAT$ sample. However, the means are in essentially the same level of disagreement. In fact, $PR_2$ has a much more significant mean discrepancy $(p_t=1.0\times 10^{-4}\%)$ in the $0.5h^{-1}_{80}$ Mpc aperture. In contrast, the $PR_m$ have means that are entirely consistent for the OCDM and $ROSAT$ clusters in both apertures. The variances of the centroided $PR_m$ are significantly discrepant, particularly in the $1.0h^{-1}_{80}$ Mpc aperture, where the OCDM variances exceed the $ROSAT$ variances. This suggests a lower $\sigma_8$ or $n$ is needed to bring the variances of the OCDM models into agreement with the $ROSAT$ sample. The $PR_m$ means of the LCDM clusters systematically exceed the $ROSAT$ means, but at a lower level of significance than does SCDM. The discrepancies are only significant in the $0.5h^{-1}_{80}$ Mpc aperture, where $PR_3$ $(p_t=0.4\%)$ and $PR^{(pk)}_1$ $(p_t=0.2\%)$ show the most significant discrepancies; the even $PR_m$ show at best a marginal discrepancy in their means $(p_t=10\%-15\%)$. The variances for the even $PR_m$ are also significantly different, though only in the $0.5h^{-1}_{80}$ Mpc aperture as well. As the LCDM and SCDM variances are very similar, we expect that the variance differences can be largely obviated with a lower value of $\sigma_8$. The difference in the means of $PR_3$ for the LCDM and $ROSAT$ clusters in the $0.5h^{-1}_{80}$ Mpc aperture, though formally significant at better than the $3\sigma$ level, represents a shift of about one-half a decade in $PR_3$; also, when using all 59 clusters of BTb the significance is only $p_t=4\%$ ($\sim 2\sigma$). As we have discussed earlier, it is difficult to completely account for a discrepancy of this magnitude by invoking, e.g., the unsuitability of the $\rho_{gas}\propto \rho_{DM}$ approximation, observational noise, or cooling flows. We may make a more precise estimate of the effects of cooling flows on the $PR_m$. The ROSAT clusters in the augmented Edge sample all have estimated mass-flow rates (Fabian 1994) from which we may compute a luminosity (bolometric) due to the cooling flow following Edge (1989), $L_{cool} = 3.0\times 10^{41}h^{-2}_{50}\dot{M}T$ erg/s, where $\dot{M}$ is in $M_{\sun}$/year and $T$ is in keV. Comparing this cooling luminosity to the total cluster luminosity, $L_{bol}$, using the results of David et al. (1993) allows us to in effect remove the cooling gas from the ROSAT $PR_m$. To a first approximation the cooling flow affects only $P_0$ because the cooling emission is weighted heavily towards the aperture center. Hence, to approximately remove the effects of the cooling flows from the ROSAT clusters we reduce $P_0$ for each cluster by $(1-L_{cool}/L_{bol})$. We find that the $PR_m$ of the ROSAT clusters are modified minimally, the effect being that the means of the $PR_m$ are increased by $1/10$ of a decade: means for $PR_2$ and $PR_3$ are -5.60 and -7.52 respectively in the $0.5h^{-1}_{80}$ aperture; the variances show no significant systematic effect. These small mean shifts do reduce the significance of the LCDM-ROSAT discrepancy, but the discrepancy is still significant at the $\sim 3\sigma$ level; e.g., $p_t=1.6\%$ for $PR_3$ and $p_t=1\%$ for $PR^{(pk)}_1$ in the $0.5h^{-1}_{80}$ aperture, and $p_t=34\%$ for the even $PR_m$. Although cooling flows alone cannot completely account for the differences in the ROSAT clusters and the LCDM model, it is very possible that when combined with the the other effects mentioned above a sizeable fraction of the half-decade difference could be made up which would in any event reduce the significance level of the difference. As a result, we believe the discrepancy of the LCDM-$ROSAT$ means must be considered preliminary and await confirmation from appropriate hydrodynamical simulations.\footnote{This would not necessarily rule out low-density, flat models having $\Omega_0<0.35$.} On the other hand, the means of $PR_3$ for the SCDM and BCDM models exceed the $ROSAT$ means by almost a full decade to a higher formal significance level $(\sim 4\sigma)$, which in light of the previous discussion should be considered robust. {\it We conclude that the $\Omega_0=1$, CDM models cannot produce the observed $PR_m$ of the $ROSAT$ clusters, and that the discrepancy in $PR_m$ means is due to $\Omega_0$ being too large.} This agrees with our conclusions obtained in TB for the small sample of clusters drawn from the hydrodynamic simulation of Navarro et al. (1995a). \footnote{If the small sample of clusters in the Navarro et al. simulation are in fact biased towards more relaxed configurations at the present day, then the agreement discussed above between the $PR_m$ computed for the $\Omega_0=1$ N-body simulations in this paper and the $PR_m$ that TB computed for the Navarro et al. simulation further strengthens the SCDM-$ROSAT$ discrepancy.} Our conclusions are opposite those of Mohr et al. (1995) who instead concluded that their {\it Einstein} cluster sample favored SCDM over both OCDM and LCDM. Given the qualitative agreement discussed above between the Jing et al. and Mohr et al. simulations, as well as between our present simulations and TB, it would seem that the discrepancy lies not in the details of the individual simulations. Moreover, since the centroid shift is qualitatively similar to our $P^{(pk)}_1/P^{(pk)}_0$, and the axial ratio is qualitatively related to our $P_2/P_0$, it would seem unlikely that we would reach entirely opposite conclusions. The other plausible variable is to consider how BTb and Mohr et al. computed their statistics on the real cluster data. The $ROSAT$ data analyzed by BTb have better spatial resolution and sensitivity than the {\it Einstein} data analyzed by Mohr et al.. This implies that the Mohr et al. data should be biased in the direction of less ``substructure'' with respect to BTb, which is the opposite of what is found. Another important difference between the two investigations is that the $PR_m$ are computed within apertures of fixed metric size, whereas Mohr et. al. use a $S/N$ criterion to define the aperture size. The fixed metric radius used by the $PR_m$ ensures that cluster structure on the scale $\sim R_{ap}$ is compared consistently which is not true for the $S/N$ criterion (see BTa); e.g., Mohr et al. use aperture sizes of $0.38h^{-1}_{80}$ Mpc for Coma and of $0.81h^{-1}_{80}$ Mpc for A2256. It is not obvious, however, how this confusion of cluster scales would explain the discrepancy of our results with Mohr et al..\footnote{This issue could be addressed by computing $PR_m$ on the {\it Einstein} sample of Mohr et al., however such a task is beyond the scope of the present paper.} \section{Conclusions\label{conc}} Using the power ratios ($PR_m=\log_{10}(P_m/P_0)$) of Buote \& Tsai (1995 -- BTa; 1996 -- BTb; Tsai \& Buote 1996 -- TB) we have examined the sensitivity of galaxy cluster morphologies to $\Omega_0$ and $P(k)$ using large, high-resolution N-body simulations. X-ray images are generated from the dark matter by letting the gas density trace the dark matter. We argue that the $PR_m$ should not be seriously biased by this approximation because a real gas in hydrostatic equilibrium with potentials of CDM clusters is rounder, but also has a larger core radius, the effects of which partially cancel. We also argue that the approximation should be reasonable during mergers because of the agreement shown between the evolution of the dark matter and gas found by TB who analyzed the hydrodynamical simulation of Navarro et al. (1995a). Finally, The $PR_m$ generated from the N-body simulations in this paper agree with results from the Navarro et al. hydrodynamical simulation (TB). Similar agreement is seen between the results of the N-body simulations of Jing et al. (1995) and the hydrodynamical simulations of Mohr et al. (1995). From analysis of several variants of the standard Cold Dark Matter model, we have shown that the $PR_m$ can distinguish between models with different $\Omega_0$ and $P(k)$. Generally, $\Omega_0$ influences the means of the $PR_m$ distributions such that larger values of $\Omega_0$ primarily imply larger average PR values. The slope of the power spectrum and $\sigma_8$ primarily influence the variances of the $PR_m$; smaller $n$ and $\sigma_8$ generally imply smaller $PR_m$ variances. For examining $\Omega_0$, our analysis indicates that $PR_3$ is the best $PR_m$ since its mean is quite sensitive to $\Omega_0$ but very insensitive to $P(k)$. It is advantageous also to consider $PR_2$ when a cosmological constant is introduced since its means differ for the SCDM and LCDM models by $\sim 3\sigma$ whereas $PR_3$ only distinguishes the models at the $\sim 2\sigma$ level. The dependence of the mean of $PR_2$ on $P(k)$ is not overly serious for studying differences in $\Omega_0$ because the differences in means due to $P(k)$ are always accompanied by larger differences in the variances; i.e. different means but consistent variances mostly reflect differences in $\Omega_0$ for $PR_2$. Typically, the best apertures for segregating models are $(0.5,0.75,1.0)h^{-1}_{80}$ Mpc. We did not find it advantageous to compare the distributions along and perpendicular to the ``evolutionary tracks'' in the $(PR_2,PR_4)$ and $(PR_2,PR_3)$ planes (see BTb and TB). The distributions along the tracks performed essentially as a weighted sum of the constituent $PR_m$. The distributions perpendicular to the tracks were in almost all cases consistent for the models. Hence, although the evolutionary tracks are useful for categorizing the dynamical states of clusters, they do not allow more interesting constraints on $\Omega_0$ and $P(k)$ to be obtained over the individual $PR_m$. The consistency of the distributions perpendicular to the evolutionary tracks seems to be a generic feature of the CDM models. We compared the $PR_m$ of the CDM models to the $ROSAT$ cluster sample of Buote \& Tsai (1996). We find that the means of the $\Omega_0=0.35$ OCDM and $ROSAT$ clusters are consistent, but the means of $PR_3$ for the LCDM and $ROSAT$ clusters are formally inconsistent at the $\sim 3\sigma$ level. We assert that this discrepancy should be considered marginal due to various issues associated with the simulation -- observation comparison. However, the means of $PR_3$ for the SCDM and BCDM models (with $\Omega_0=1$) exceed the $ROSAT$ means by almost a full decade with a high level of significance $(\sim 4\sigma)$. Though the formal significance level of this $\rho^2_{DM}$ / X-ray comparison should be considered only an approximation, we argue that taking into account the hydrodynamics and cooling will not reconcile a discrepancy this large. {\it We conclude that the $\Omega_0=1$ CDM models cannot produce the observed $PR_m$ of the $ROSAT$ clusters, and that the discrepancy in $PR_m$ means is due to $\Omega_0$ being too large.} This agrees with our conclusions obtained in TB for the small sample of clusters drawn from the hydrodynamic simulation of Navarro et al. (1995a). These conclusions are also consistent with other indicators of a low value of $\Omega_0$ such as the dynamical analyses of clusters (e.g., Carlberg et al. 1995), the large baryon fractions in clusters (e.g., White et al. 1993), and the heating of galactic disks (Toth \& Ostriker 1992). Our conclusions are inconsistent with those of Mohr et al. (1995) who instead concluded that their {\it Einstein} cluster sample favored $\Omega_0=1$, CDM over equivalents of our low-density models, OCDM and LCDM. We argue that this type of discrepancy is unlikely due to numerical differences between our simulations. We discuss possible differences due to how BTb and Mohr et. al. computed their statistics on the real cluster data. Large hydrodynamical simulations are necessary to render the comparison to the $ROSAT$ data more robust. In addition, the effects of combining data at different redshifts needs to be explored since cluster formation rates should behave differently as a function of $z$ in different models (e.g., Richstone et. al. 1992). It may also prove useful to apply $PR_m$ to mass maps of clusters obtained from weak lensing (Kaiser \& Squires 1993)\footnote{See Wilson, Cole, \& Frenk (1996), who have recently studied weak-lensing maps obtained from N-body simulations, and concluded that a ``global quadrupole statistic'' ($\sim\sqrt{P_2/P_0}$) can distinguish between low-density and critical density models.}, though for cosmological purposes it is not clear whether $\rho_{mass}$ will be as responsive as $\rho^2_{gas}$ to different $\Omega_0$ and $P(k)$. \acknowledgements We gratefully acknowledge J. Tsai for his role in facilitating this collaboration. We thank E. Bertschinger for suggesting to us the scheme to normalize the scale-free models using $M_{\star}$, T. Beers for providing his ROSTAT programs to compute AI and TI, and A. Edge for providing the expression for $L_{cool}$. We appreciated the anonymous referee's prompt reviewing and comments that helped improve the presentation of the paper. DAB was supported by grants NASGW-2681 (through subcontract SVSV2-62002 from the Smithsonian Astrophysical Observatory) and NAG5-2921, and acknowledges the hospitality of the Institute of Astronomy where the final stages of this work were carried out. DAB also expresses gratitude to several senior scientists who offered encouragement during the early stages of this project. GX acknowledges support from NFS HPCC grant ASC93-18185 and thanks the Pittsburgh Supercomputer Center for use of the CRAY-T3D machine. \clearpage \vfill\eject \begin{table}[p] \caption{Cosmological Models \label{table.models}} \begin{tabular}{lccccccc} \tableline\tableline \\[-1pt] Name & $\Omega_0$ & $\lambda_0$ & $n$ & $\sigma_8$ & $h$ & $z_i$\\ \tableline SCDM & 1 & 0 & 1 & 1.00 & 0.5 & 20 \\ OCDM & 0.35 & 0 & 1 & 0.79 & 0.7 & 25 \\ LCDM & 0.35 & 0.65 & 1 & 0.83 & 0.7 & 39 \\ BCDM & 1 & 0 & 1 & 0.51 & 0.5 & 20 \\ SF00 & 1 & 0 & 0 & $\ldots$ & $\ldots$ & $\ldots$ \\ SF10 & 1 & 0 & -1.0 & $\ldots$ & $\ldots$ & $\ldots$ \\ SF15 & 1 & 0 & -1.5 & $\ldots$ & $\ldots$ & $\ldots$ \\ SF20 & 1 & 0 & -2.0 & $\ldots$ & $\ldots$ & $\ldots$ \\ \tableline \end{tabular} \tablecomments{$z_i$ is the redshift where the simulations started. The scale-free models (SF) are normalized to have the same value of $M_{\star}$ as SCDM (see \S \ref{models}).} \end{table} \renewcommand{\arraystretch}{0.75} { \begin{table}[p] \small \caption{Average Power Ratios \label{table.avgpr}} \begin{tabular}{lrc|rc|rc|rc|rc|rc} \tableline\tableline \\[-1pt] & \multicolumn{6}{c}{\large $PR_2$} & \multicolumn{6}{c}{\large $PR_3$}\\ \\ & \multicolumn{2}{c}{0.5 Mpc} & \multicolumn{2}{c}{0.75 Mpc} & \multicolumn{2}{c}{1.0 Mpc} & \multicolumn{2}{c}{0.5 Mpc} & \multicolumn{2}{c}{0.75 Mpc} & \multicolumn{2}{c}{1.0 Mpc} \\ & avg & $\sigma$ & avg & $\sigma$ & avg & $\sigma$ & avg & $\sigma$ & avg & $\sigma$ & avg & $\sigma$ \\ \\[-5pt] \tableline\\[-5pt] SCDM & -5.14 & 0.86 & -5.16& 0.83& -5.38 & 0.76 & -6.72 & 0.73 & -6.82 & 1.01 & -7.06 & 0.98\\ OCDM & -5.55 & 0.61 & -5.82 & 0.78 & -5.93 & 1.03 & -7.40 & 0.81 & -7.56 & 1.13 & -7.59 & 1.28\\ LCDM & -5.45 & 0.83 & -5.69 & 0.85 & -5.87 & 0.83 &-7.08 & 0.83 & -7.32 & 1.00 & -7.38 & 1.02\\ BCDM & -5.01 & 0.57 & -5.24 & 0.60 & -5.41 & 0.63 & -6.83 & 0.65 & -6.98 & 0.67 & -7.07 & 0.68 \\ SF00 & -5.34 & 0.82 & -5.64 & 1.00 & -5.75 & 0.97 & -7.12 & 1.02 & -7.28 & 1.14 & -7.35 & 1.10 \\ SF10 & -5.02 & 0.57 & -5.22 & 0.71 & -5.47 & 0.85 & -6.89 & 0.87 & -7.09 & 1.02 & -7.36 & 1.18 \\ SF15 & -5.20 & 0.85 & -5.22 & 0.94 & -5.40 & 0.91& -6.92 & 0.99 & -6.93 & 1.02 & -7.05 & 0.96 \\ SF20 & -5.07 & 0.60 & -5.24 & 0.62 & -5.45 & 0.57 & -7.01 & 0.71 & -7.13 & 0.75 & -7.20 & 0.79 \\ \tableline \end{tabular} \tablecomments{Aperture sizes assume h=0.8.} \end{table} } \renewcommand{\arraystretch}{1.0} \renewcommand{\arraystretch}{0.75} {\scriptsize \begin{table}[p] \small \caption{Significance Tests for Power Ratios \label{table.test}} \begin{tabular}{ccc|rrr|rrr|rrr} \tableline\tableline \\[-1pt] &&& \multicolumn{3}{c}{0.5 Mpc} & \multicolumn{3}{c}{0.75 Mpc} & \multicolumn{3}{c}{1.0 Mpc}\\[5pt] &&& \multicolumn{1}{c}{$p_t$} & \multicolumn{1}{c}{$p_F$} & \multicolumn{1}{c}{$p_{KS}$} & \multicolumn{1}{c}{$p_t$} & \multicolumn{1}{c}{$p_F$} & \multicolumn{1}{c}{$p_{KS}$} & \multicolumn{1}{c}{$p_t$} & \multicolumn{1}{c}{$p_F$} & \multicolumn{1}{c}{$p_{KS}$}\\ \multicolumn{3}{c}{Models} & (\%) & (\%) & (\%) & (\%) & (\%) & (\%) & (\%) & (\%) & (\%)\\ \tableline \tableline \\ \multicolumn{11}{c}{\large $PR_2$}\\ \\ \tableline \\[-5pt] SCDM &vs. &OCDM & 1.63 & 4.09 & 0.94 & 0.06 & 71.71 & 0.07 & 0.89 & 7.18 & 0.94 \\ SCDM &vs. &LCDM & 10.47 & 87.90 & 21.79 & 0.74 & 89.05 & 1.97 & 0.83 & 58.78 & 3.90 \\ OCDM &vs. &LCDM & 54.64 & 5.79 & 51.42 & 47.52 & 61.72 & 21.79 & 78.26 & 20.53 & 34.56 \\ SCDM &vs. &BCDM & 43.06 & 1.51 & 98.09 & 65.19 & 4.65 & 21.79 & 83.52 & 24.87 & 34.56 \\ SF00 &vs. &SF20 & 10.76 & 6.35 & 7.31 & 4.19 & 0.38 & 0.94 & 10.44 & 0.13 & 3.90 \\ \tableline \\ \multicolumn{11}{c}{\large $PR_3$}\\ \\ \tableline \\[-5pt] SCDM &vs. &OCDM & 0.02 & 52.99 & 0.18 & 0.31 & 46.71 & 0.94 & 4.43 & 10.64 & 7.31 \\ SCDM &vs. &LCDM & 4.44 & 43.91 & 21.79 & 3.26 & 97.10 & 7.31 & 15.88 & 81.58 & 51.42 \\ OCDM &vs. &LCDM & 9.10 & 88.39 & 21.79 & 30.99 & 44.52 & 51.42 & 43.51 & 16.61 & 51.42 \\ SCDM &vs. &BCDM & 49.58 & 47.26 & 70.85 & 40.62 & 1.39 & 34.56 & 97.46 & 2.48 & 70.85 \\ SF00 &vs. &SF20 & 57.65 & 2.89 & 21.79 & 49.89 & 1.28 & 21.79 & 50.92 & 4.31 & 12.97 \\ \tableline \end{tabular} \tablecomments{Aperture sizes assume h=0.8.} \end{table} } \renewcommand{\arraystretch}{1.0} \begin{table}[p] \caption{PR Statistics for $ROSAT$ Clusters \label{table.ros.avg}} \begin{tabular}{lrr|rr} \tableline\tableline \\[-1pt] & \multicolumn{2}{c}{0.5 Mpc} & \multicolumn{2}{c}{1.0 Mpc} \\[5pt] & avg & $\sigma$ & avg & $\sigma$\\ \tableline $PR_2$ & -5.70& 0.44& -6.00& 0.50\\ $PR_3$ & -7.62& 0.77& -7.61& 0.77\\ \\[-10pt] \tableline \end{tabular} \tablecomments{Aperture sizes assume h=0.8.} \end{table} {\small \begin{table}[p] \small \caption{Significance Tests for $ROSAT$ Clusters \label{table.ros.test}} \begin{tabular}{llll|lll} \tableline\tableline \\[-1pt] & \multicolumn{3}{c}{0.5 Mpc} & \multicolumn{3}{c}{1.0 Mpc} \\[5pt] Models & $p_t(\%)$ & $p_F(\%)$ & $p_{KS}(\%)$ & $p_t(\%)$ & $p_F(\%)$ & $p_{KS}(\%)$\\ \tableline \tableline \\ \multicolumn{7}{c}{\large $PR_2$}\\ \\ \tableline \\[-5pt] SCDM & 0.60E-01& 0.12E-01& 0.68E-03& 0.14E-01& 0.24E+01& 0.33E-01\\ BCDM & 0.10E-04& 0.12E+02& 0.56E-04& 0.12E-01& 0.20E+02& 0.12E-01\\ OCDM & 0.23E+02& 0.52E+01& 0.16E+02& 0.68E+02& 0.23E-01& 0.43E+02\\ LCDM & 0.11E+02& 0.21E-01& 0.13E+01& 0.41E+02& 0.69E+00& 0.23E+02\\ \tableline \\ \multicolumn{7}{c}{\large $PR_3$}\\ \\ \tableline \\[-5pt] SCDM& 0.15E-03& 0.71E+02& 0.69E-01& 0.17E+01& 0.20E+02& 0.10E+02\\ BCDM& 0.64E-03& 0.28E+02& 0.32E-02& 0.35E+00& 0.46E+02& 0.12E+02\\ OCDM& 0.22E+02& 0.80E+02& 0.42E+02& 0.93E+02& 0.83E+00& 0.29E+02\\ LCDM& 0.44E+00& 0.69E+02& 0.97E+00& 0.33E+02& 0.14E+02& 0.32E+02\\ \tableline \end{tabular} \tablecomments{Aperture sizes assume h=0.8.} \end{table} } \clearpage

proofpile-arXiv_065-521

\section{Introduction} In Heavy Fermion (HF) materials \cite{Grewe91}, especially in Uranium based compounds, a simple Anderson-lattice model with single $N$=fold degenerate ionic ground-state, cannot explain the rich variety of transport measurements in the whole accessibly temperature range. The additional maxima found experimentally in the specific heat, the resistivity and the thermo-power are related to higher Crystal-Electric-Field (CEF) levels and have been used to propose CEF level schemes for many different HF compounds using high temperature impurity approaches. Many theoretical attempts have been put forward to include CEF effects in a many body description of HF materials \cite{Maekawa86}. For magnetic impurities the high temperature spin-disorder resistance calculations \cite{CornutCoqblin72} have been extended to describe anisotropy of transport of CePt$_2$Si$_2$ \cite{Bhatta89} in third order perturbation theory neglecting lattice coherence effects, for the low temperature phase it can included within some limits in a non-crossing approximation (NCA) calculation \cite{Bickers87,Strong94}. Nevertheless a theoretical approach for the lattice problem is missing. Additionally, the low temperature scale $T^*$ will reflect higher excited multiplets due to the remaining virtual fluctuations in the ground state. Since in Uranium based compounds the bare spectroscopic CEF structure is still unsettled and a CEF singlet cannot be ruled out as ground state a model for competition between CEF and Kondo singlet was put forward only recently \cite{Kuramoto93}. But for the Anderson lattice, approaches for including CEF effects exist only within the slave boson mean field theory \cite{Evens92}. The aim of this paper is to present a formalism to incorporate CEF-levels in a dynamical mean field (local) approximation \cite{Georges96} which is independent of the method used to solve the effective site problem. Even though we will restrict ourselves to one unoccupied state, and singly occupied CEF-multiplets transforming according to different irreducible representations $\Gamma_i$ of the point group of the lattice, we can extend our method to address the question how to generate a multi-channel Kondo lattice model from first principles. \section{Theory} It is assumed that a spin degenerate conduction electron band couples to localized ionic states via hybridization matrix elements. The matrix elements can be derived by expanding the conduction band locally in the irreducible representations of the point group \cite{CoqblinSchrieffer69}. First we introduce the Hamiltonian and the notation used throughout the paper. \begin{equation} H = \begin{array}[t]{l} \displaystyle \sum_{{\kk\sigma}} \varepsilon_{{\kk\sigma}} c^\dagger_{\kk\sigma} c_{\kk\sigma} + \sum_{\nu{\Gamma\alpha}}E_{\Gamma\alpha} X_{{\Gamma\alpha},{\Gamma\alpha}}^{\nu} \\ \displaystyle + \sum_{\nu,{\Gamma\alpha},{\kk\sigma}} V_{{\Gamma\alpha}}({\kk\sigma})e^{i\ul{k}\ul{R}_\nu} c^\dagger_{{\kk\sigma}} X_{0,{\Gamma\alpha}}^\nu + h.c. \end{array} \label{eqn-1} \end{equation} The ionic states are labeled by their corresponding irreducible representation of the point group and an index $\alpha$ which is a shorthand notation for all other quantum numbers, like occupation number $n$, spin, or orbit quantum number, and $ X_{{\Gamma'\alpha'},{\Gamma\alpha}}^\nu = |{\Gamma'\alpha'}><{\Gamma\alpha}|$ at the site $\nu$. It is the most general formulation of the periodic Anderson model which takes into account only local hybridization. In this paper we concentrate on local fluctuations between singly and empty states only ($U\rightarrow \infty$). While propagating through the lattice, the conduction electrons are scattered by the CEF states via the local $T$-matrix $V_{{\Gamma\alpha}}({\kk\sigma}) F_{{\Gamma\alpha},{\Gamma'\alpha'}}(z)V_{{\Gamma'\alpha'}}({\kk\sigma})$, where $F_{{\Gamma\alpha},{\Gamma'\alpha'}}(z) \equiv \ll X_{0,{\Gamma\alpha}}(\tau) X_{{\Gamma'\alpha'},0}\gg $ is the local Green's function. In the summation of all scattering processes of the conduction electrons, terms of the structure $$ \displaystyle \sum_{{\Gamma\alpha}} \sum_\sigma V^\star_{\gapara{1}{1}}({\kk\sigma}) \underbrace{\frac{1}{z-\varepsilon_{{\kk\sigma}}}}_{Band} \underbrace{V_{{\Gamma\alpha}}({\kk\sigma}) F_{{\Gamma\alpha},\gapara{2}{2}}(z)V_{\gapara{2}{2}}({\kk\sigma})}_{T-matrix} $$ occur which can be identified as a component of a matrix product $ \left[ \mat{d}(\ul{k},z) \cdot \mat{F}(z) \right]_{\gapara{1}{1},\gapara{2}{2}} $ where we have defined the conduction electron matrix $$ \mat{d}(\ul{k},z) = \sum_\sigma \mat{d}(\ul{k},\sigma,z) = \sum_\sigma \ul{V}({\kk\sigma}) \frac{1}{z-\varepsilon_{{\kk\sigma}}} \ul{V}^T({\kk\sigma}) $$ and $ \ul{V}^T({\kk\sigma}) = \left( V^\star_{\gapara{1}{1}}({\kk\sigma}),V^\star_{\gapara{1}{2}}({\kk\sigma}), \cdots, V^\star_{\gapara{n}{\Gamma_n}}({\kk\sigma}) \right) $. The f-Green's function $\mat{F}(z)$ is a $N\times N$ matrix, where $N$ is the number of singly occupied states included in (\ref{eqn-1}). {\bf The local ("$d\rightarrow\infty$") approximation:}\\ The local approximation \cite{Kuramoto87}, which is equivalent to the limit $d\rightarrow\infty$ with an appropriate rescaling of the effective hopping \cite{Georges96}, choose one $f$-site as an effective site which is embedded in an effective medium Green's function generated self-consistently by the rest of lattice. While in a single impurity problem the bare medium GF $\mat{\Delta}_0(z) = \frac{1}{N_s}\sum_{\ul{k}}\mat{d}(\ul{k},z)$ enters, the condition that the local $f$-Green's function has to be equal to the $\ul{k}$-summed lattice Green's function \begin{equation} \label{eqn-scc-doo} \frac{1}{N_s} \sum_{\ul{k}} \frac{1}{\mat{1}- \mat{\tilde F}(z)\left(\mat{d}(k,z) - \mat{\tilde\Delta}(z)\right)} = 1 \;\; . \end{equation} determined self-consistently the renormalized media $\mat{\tilde\Delta}(z)$. The effective Anderson width $\mat{\tilde\Gamma}(z) = \frac{\Im m}{\pi}\mat{\tilde\Delta}(z)$ and the Green's function of the effective site $\mat{\tilde F}(z)$ are block-diagonal in the irreducible representations of the point group, since they are local quantities. {\bf The lattice Green's-functions}\\ The $f$-Green's-function (GF) matrix $\mat{F}(k,z)$ is obtain by summing over all possible intermediate scattering events taking an electron from site $i$ to $j$ and Fourier transforming the result in the reciprocal lattice space is \begin{equation} \mat{F}(k,z) = \frac{1}{\mat{\tilde F}^{-1}(z) - \left(\mat{d}(\ul{k},z) -\mat{\tilde\Delta}(z) \right)} \;\;. \label{eqn-lnca-f} \end{equation} While the self-consistency condition (SCC) of the so-called lattice-NCA, another well established local approximation for the Anderson lattice, is derived from a different philosophy \cite{Grewe87}, the structure of the lattice Green's function can be obtained from (\ref{eqn-lnca-f}) by replacing $\mat{\tilde\Delta}(z)$ by the bare $\mat{\Delta}_0(z)$. Even though $\mat{\tilde F}(z)$ and $\mat{\tilde\Delta}(z)$ transform according to the irreducible representations of the point group, the term $\mat{d}(k,z)$ mixes different representations for an arbitrary $k$-point destroying the point-group symmetry in $k$-space. Via the {\em exact} equation of motion \begin{equation} \begin{array}{rcl} G_{{\kk\sigma}}(z) &=&\displaystyle G_{{\kk\sigma}}^{(0)}(z) + G_{{\kk\sigma}}^{(0)}(z) T_{{\kk\sigma}}(z) G_{{\kk\sigma}}^{(0)}(z) \\ &\equiv &\displaystyle \frac{1}{ [G_{{\kk\sigma}}^{(0)}]^{-1}(z) - \Sigma_{{\kk\sigma}}(z)} \end{array} \end{equation} and \begin{equation} T_{{\kk\sigma}}(z) = \begin{array}[t]{l} \displaystyle \sum_{{\Gamma\alpha},{\Gamma'\alpha'}} V_{{\Gamma\alpha}}({\kk\sigma}) F_{{\Gamma'\alpha'},{\Gamma\alpha}}(\ul{k},z)V^\star_{{\Gamma'\alpha'}}({\kk\sigma}) \\ = \ul{V}^{T}({\kk\sigma}) \mat{F}(\ul{k},z) \ul{V}({\kk\sigma}) \end{array} \end{equation} a compact equation for the $k$-dependent self-energy \begin{equation} \Sigma_{{\kk\sigma}}(z) \begin{array}[t]{l} = \displaystyle \frac{ \ul{V}^{T}({\kk\sigma}) \mat{F}(\ul{k},z) \ul{V}({\kk\sigma})} {1 + G_{{\kk\sigma}}^{(0)}(z) \ul{V}^{T}({\kk\sigma}) \mat{F}(\ul{k},z) \ul{V}({\kk\sigma})} \\[10pt] \displaystyle = \ul{V}^{T}({\kk\sigma}) \frac{1}{\mat{\tilde F}^{-1}(z) + \mat{\tilde\Delta}(z) - \mat{d}(\ul{k},-\sigma,z)} \ul{V}({\kk\sigma}) \end{array} \label{eqn-sigma-c} \end{equation} is obtained from with the $k$-dependent inverse relaxation time $\tau^{-1}({\kk\sigma},\omega) = 2\Im m \Sigma_{{\kk\sigma}}(\omega-i\delta)$ is calculate entering Boltzmann transport theory. Even though a local approximation has been used in derivating Eqn.(\ref{eqn-sigma-c}) the self-energy is anisotropic due the $k$-dependence of the hybridization. {\bf Example: Two Kramers Doubles:}\\ The $4\times 4$ problem separates in two identical $2\times 2$ matrices for each pseudo-spin. Using the Faddeeva-function $w(z)$, the diagonal elements $\tilde F_1(z)$ and $\tilde F_2(z)$ of the Green's function matrix $\mat{F}(z)$ and angular averaged hybridization, we obtain two SCC \begin{equation} \begin{array}{l} \label{eqn-ceff-d1} \displaystyle \frac{1}{1 + \tilde F_\alpha(z)\tilde\Delta_\alpha(z)} \left[1 - i\, w\left(\sqrt{\pi}\rho_0(z-\Sigma_{\sigma}(z) -\varepsilon_0)\right) \right . \\ \displaystyle \hspace{20mm} \left. \cdot \frac{\pi V_\alpha^2\rho_0 \tilde F_\alpha(z)}{1 + \tilde F_\alpha(z)\tilde \Delta_\alpha(z)} \right] = 1 \hspace{5mm} \alpha=1,2 \end{array} \end{equation} which are only coupled via the self-energy of the conduction electrons \begin{equation} \Sigma_{\sigma}(z)= \sum_\alpha \frac{V_\alpha^2 \tilde F_\alpha(z)}{1 + \tilde F_\alpha(z)\tilde\Delta_\alpha(z)} \;\; . \label{equ-c-self} \end{equation} found from Eqn.(\ref{eqn-sigma-c}). The averaging of anisotropy effects is justified in the SCC, since only the effective density of states enters the local approximation. For the transport calculation, however, the full angular dependence of the hybridization gives rise to the anisotropy of the transport properties even though a local approximation has been used. From Eqn.~(\ref{equ-c-self}) it is clearly seen that the different position of the Abricosov-Suhl resonaces in the two $\tilde F_\alpha(z)$ will produced two maxima in the resistivity, as demonstrated in \cite{Huth95}. {\bf Vertex corrections in the Transport Theory:}\\ Normally a vertex function enters the calculation of transport properties: \begin{equation} \begin{array}{rcl} \displaystyle \Gamma_{\ul{k}}(z, z+\nu)& =&\displaystyle \partial_{\ul{k}}\varepsilon_{\ul{k}} + \sum_{\ul{k'}} \partial_{\ul{k'}}\varepsilon_{\ul{k'}}G_{\ul{k'}}(z) \\ && \cdot G_{\ul{k'}}(z+\nu) W_{\ul{k'},\ul{k}}(z,z+\nu) \;\; \end{array} \end{equation} $W_{\ul{k'},\ul{k}}(z,z+\nu)$ being the irreducible two particle propagator. If we restrict ourself to CEF-levels arising from the same Hund's rule multiplet, the hybridization matrix element coupling both conduction electron Green's function $G_{\ul{k'}}(z)$ to a local $f$ site will have the same parity. The total parity of the $\ul{k'}$-momentum loop is given by $\partial_{\ul{k}}\varepsilon_{\ul{k}}$. Therefore the vertex correction will be exactly zero in a lattice with inversion symmetry and $\Gamma_{\ul{k}}(z, z+\nu) = \partial_{\ul{k}}\varepsilon_{\ul{k}}$. Neverlethess, the total current has contributions from conduction and $f$-electrons when a $\ul{k}$-dependent hybridization is present \cite{Leder81}. \section{Conclusion} We have developed a formalism to include local CEF-levels in a local approximation. The lattice coherence effects, which have been neglected in previous approaches have been taken into account. Eqn.(\ref{eqn-lnca-f}) and Eqn.(\ref{eqn-sigma-c}) show how the point group symmetry, which is present in all local quantities is destroyed for a general $\ul{k}$-point. Vertex corrections to transport properties vanish identically on symmetry grounds if we restrict ourselves to the lowest lying Hund's rule multiplet. Anisotropy effects enter via angular dependent hybridization matrix elements in the conduction electron self-energy, Eqn.~(\ref{eqn-sigma-c}). In the case of two doublets we can interpret the derived conduction electron self-energy (\ref{equ-c-self}) as a superposition of contributions of individual symmetry channels which recovers the proposed semi-phenomenological extension of the LNCA \cite{Huth95}. Additional maxima in the resistivity arise naturally from contributions of the different Abricosov-Suhl resonances from additional CEF levels. This work has been supported by US Department of Energy, Office of Basic Energy Science, Division of Material Research and the Deutsche Forschungsgemeinschaft and in parts by the National Science Foundation under Grant No.~PHY94-07194. We like to thank the ITP, Santa Barbara for its hospitality.

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\chapter*{} \lecturename{Bibliography} \lecturestar{BIBLIOGRAPHY} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Bibliography} \bibliographystyle{amsplain} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi \addvspace\linespacing \noindent{\large\bfseries Books}\par \addvspace{.5\linespacing} \chapter*{Mathematical Aspects of Mirror~Symmetry} \auth{David R. Morrison} \lecturename{Introduction} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Mathematical Aspects of Mirror Symmetry} \addcontentsline{toc}{chapter}{Introduction} \addvspace\linespacing \noindent{\large\bfseries Introduction}\par \addvspace{.5\linespacing} \noindent {\em Mirror symmetry}\/ is the remarkable discovery in string theory that certain ``mirror pairs'' of Calabi--Yau manifolds apparently produce isomorphic physical theories---related by an isomorphism which reverses the sign of a certain quantum number---when used as backgrounds for string propagation. The sign reversal in the isomorphism has profound effects on the geometric interpretation of the pair of physical theories. It leads to startling predictions that certain geometric invariants of one Calabi--Yau manifold (essentially the numbers of rational curves of various degrees) should be related to a completely different set of geometric invariants of the mirror partner (period integrals of holomorphic forms). The period integrals are much easier to calculate than the numbers of rational curves, so this idea has been used to make very specific predictions about numbers of curves on certain Calabi--Yau manifolds; hundreds of these predictions have now been explicitly verified. Why either the pair of manifolds, or these different invariants, should have anything to do with each other is a great mathematical mystery. The focus in these lectures will be on giving a precise mathematical description of two string-theoretic quantities which play a primary r\^ole in mirror symmetry: the so-called $A$-model and $B$-model correlation functions on a Calabi--Yau manifold. The first of these is related to the problem of counting rational curves while the second is related to period integrals and variations of Hodge structure. A natural mathematical consequence of mirror symmetry is the assertion that Calabi--Yau manifolds often come in pairs with the property that the $A$-model correlation function of the first manifold coincides with the $B$-model correlation function of the second, and {\em vice versa}. Our goal will be to formulate this statement as a precise mathematical conjecture. There are other recent mathematical expositions of mirror symmetry, by Voisin \cite{voisin} and by Cox and Katz \cite{coxkatz}, which concentrate on other aspects of the subject; the reader may wish to consult those as well in order to obtain a complete picture. I have only briefly touched on the physics which inspired mirror symmetry (in lectures one and eight), since there are a number of good places to read about some of the physics background: I recommend Witten's address at the International Congress in Berkeley \cite{witten:physgeom}, a book on ``Differential Topology and Quantum Field Theory'' by Nash \cite{nash}, and the first chapter of H\"ubsch's ``Calabi--Yau Manifolds: A Bestiary for Physicists'' \cite{hubsch}. There are, in addition, three collections of papers related to string theory and mirror symmetry which contain some very accessible expository material: ``Mathematical Aspects of String Theory'' (from a 1986 conference at U.C. San Diego) \cite{stringbook}, ``Essays on Mirror Manifolds'' (from a 1991 conference at MSRI) \cite{mirrorbook}, and its successor volume ``Mirror Symmetry II'' \cite{MSII}. I particularly recommend the paper by Greene and Plesser ``An introduction to mirror manifolds'' \cite{GP:intro} and the paper by Witten ``Mirror manifolds and topological field theory'' \cite{witten:mirror}, both in the MSRI volume. This is a revised version of the lecture notes which I prepared in conjunction with my July, 1993 Park City lectures, and which I supplemented when delivering a similar lecture series in Trento during June, 1994. The field of mirror symmetry is a rapidly developing one, and in finalizing these notes for publication I have elected to let them remain as a ``snapshot'' of the field as it was in 1993 or 1994, making only minor modifications to the main text to accommodate subsequent developments. I have, however, added a postscript that sketches the progress which has been made in a number of different directions since then. \addvspace\linespacing \noindent{\large\bfseries Acknowledgments}\par \addvspace{.5\linespacing} \noindent The ideas presented here concerning the mathematical aspects of mirror symmetry were largely shaped through conversations and collaborations I have had with Paul Aspinwall, Brian Greene, Sheldon Katz, Ronen Plesser, and Edward Witten. It is a pleasure to thank them all for their contributions. I am grateful to Antonella Grassi and Yiannis Vlassopoulos for providing me with copies of the notes they took during the lectures. I am also grateful to Grassi, Katz, Plesser, and Vlassopoulos, as well as to Michael Johnson, Lisa Traynor, and the referee of \cite{compact}, for pointing out errors in the first drafts of these notes. This research was supported in part by the National Science Foundation under grant number DMS-9103827. \chapter*{} \lecturename{Some Ideas From String Theory} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 1. Some Ideas From String Theory} \label{stringtheory} \section{String theory and quantum field theory} The origins of the startling calculations which have led to tremendous interest among mathematicians in the phenomenon of ``mirror symmetry'' lie in string theory. String theory is a proposed model of the physical world which idealizes its fundamental constituent particles as one-dimensional mathematical objects (``strings'') rather than zero-dimensional objects (``points''). In theories such as general relativity, one has traditionally imagined a point as tracing out what is known as a ``worldline'' in spacetime; the corresponding notion in string theory is of a ``worldsheet'' which will describe the trace of a {\em string}\/ in spacetime. We will consider here only ``closed string theory'' in which the string is a closed loop; the worldsheets are then (locally) closed surfaces: if we look at a portion of a worldsheet which represents the history of several interacting particles over a finite time interval, we will see a closed surface which has a boundary consisting of a finite number of closed loops. An early version of string theory was proposed as a model for nuclear processes in the 1960's. Those early investigations revealed a somewhat disturbing property: in order to get a sensible physical theory, the spacetime $M$ in which the string is propagating must have dimension twenty-six. Obviously, when we look around us, we do not see twenty-six dimensions. A later variant which incorporates supersymmetry\footnote{I shall not attempt to explain supersymmetry in these lectures.} is sensible exactly when the spacetime has dimension ten---again a bit larger than the four-dimensional spacetime which we observe. Partly for this reason, but primarily because a better model for nuclear processes was found, the original research activity in string theory largely died out in the early 1970's. String theory was subsequently revived in the 1980's when it was shown \cite{quantum:gravity} that if the ten-dimensional string theory were used to model things at much smaller distance scales, an apparently consistent quantum theory of gravity could be produced. (In fact, gravity is predicted as an essential ingredient of this theory.) This ``anomaly cancellation'' result explained how certain potential inconsistencies in the quantum theory are avoided through an interaction between gravity and the other forces present. Tremendous optimism and excitement pervaded this period, particularly since the new model contains a rich spectrum of elementary particles at low energies and exhibits many features one would expect of a ``grand unified field theory'' which could describe in a single theory all of the forces observed in nature. The ``problem'' of ten dimensions in this context can be resolved by assuming that the ten-dimensional spacetime is locally a product $M=M^{1,3}\times M^6$ of a macroscopic four-dimensional spacetime and a compact six-dimensional space whose size is on the order of the Planck length ($10^{-33}$ cm). Because this is so small compared to macroscopic lengths, one wouldn't expect to observe the compact space directly, but its effect on four-dimensional physics could be detectable in various indirect ways. The next step was even more remarkable for mathematicians---a group of string theorists \cite{CHSW} calculated that the compact six-dimensional space must have a Ricci-flat metric on it. (The physically relevant metric is actually a perturbation of this Ricci-flat one.) This is a very restrictive property---it implies, for example, that the six-manifold is a complex K\"ahler manifold of complex dimension three which has trivial canonical bundle; conversely, such K\"ahler manifolds always admit Ricci-flat metrics. (This had been conjectured by Calabi \cite{calabi} in the late 1950's and proved by Yau \cite{yau} in the mid 1970's.) These manifolds have since been named ``Calabi--Yau manifolds;'' finding and studying them become problems in algebraic geometry, thanks to Yau's theorem. The model being described here of a string propagating in a spacetime (with a specified metric) is generally regarded as a woefully inadequate description of the ``true'' string theory, a good formulation of which is as yet unknown. Indeed, if string theory is truly a theory of gravity as we observe it, then the theory should approximate general relativity when the distance scale approaches macroscopic levels. Since the metric on spacetime is part of general relativity, it should be a part of that ``approximation'' which is somehow to be deduced from a solution to the ultimate ``string equations,'' rather than being something which is put in by hand in advance. Even the {\em topology}\/ of spacetime should be dictated by the string theory. However, neither these ``string equations'' nor their exact solutions are known at present. The Calabi--Yau manifolds and their connections with string theory have been studied intensively for more than a decade. In the earliest period, these manifolds were analyzed using standard mathematical techniques, and the results were applied in a string-theoretic context. However, at the same time, other advances were being made in string theory which suggested other ways of looking at certain aspects of the theory of Calabi--Yau manifolds. This eventually led to the discovery of a surprising new phenomenon known as ``mirror symmetry,'' in which it was observed that different Calabi--Yau manifolds could lead to identical physical theories in a way that implied surprising connections between certain geometric features of the manifolds. To explain this mirror symmetry observation in more detail, we must first describe a few aspects of quantum field theory and its relationship to string theory. In classical mechanics, the worldline representing a particle is required to minimize the ``action'' (which is the energy integrated with respect to time), or more precisely, to be a stationary path for the action functional. Due to this ``stationary action principle,'' the location of the path in spacetime is completely determined by a knowledge of boundary conditions. Other physically measurable quantities associated to the particle (which are often represented as some kind of ``internal variables'') will also evolve from their boundary states in a completely predictable manner, again minimizing the action. In quantum field theory, however, this changes. Only the probability of various possible outcomes can be predicted with certainty, and {\em all}\/ trajectories---not just the action-minimizing ones---contribute to the measurement of this probability. The probability is calculated from an integral over the space of all possible paths with these initial and final states,\footnote{There are enormous mathematical difficulties in dealing with these ``path integrals'' or ``functional integrals,'' and they do not in general have a rigorous mathematical formulation. Nevertheless, in the hands of skilled practitioners they can be used to make predictions which agree with laboratory experiments to a remarkable degree of precision.} and the classical trajectory is recovered as the leading term in a stationary phase approximation to the path integral. Relativistic quantum field theories are frequently studied by treating the theory as a small perturbation of a simple type of theory---called a free field theory---whose functional integrals are well-understood. For example, the path integral describing the interaction of two charged particles can be expanded in a perturbative series whose terms are described by ``Feynman diagrams.'' The zeroth order term is the diagram \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn0.eps}} }$$ \else \vskip1in \noindent \fi which represents two particles which do not interact at all, the leading perturbative correction is described by the diagram \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn1.eps}} }$$ \else \input fakepic1.tex \noindent \fi which represents a transfer of momentum from one particle to the other via the emission and absorption of a third particle carrying the force, and higher order corrections involve diagrams with more complicated topologies---loops are allowed, for example. Such diagrams can be cut into simpler pieces, at the expense of performing an integral over all possible intermediate states. For example, the interacting Feynman diagram illustrated above can be decomposed into two more primitive pieces, \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn3.eps}} }$$ \else \input fakepic2.tex \noindent \fi each of which represents a fundamental ``interaction'' vertex. In string theory, the paths are replaced by surfaces: the interacting diagram might be represented as a sphere with four disks removed (or perhaps as something with more complicated topology), \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn4.eps}} }$$ \fi and this could also be decomposed into more primitive pieces \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{feyn5.eps}} }$$ \fi (called ``pair of pants'' surfaces). One of the advantages of string theory is that this fundamental piece, the ``pair of pants'' surface, is a smooth surface, in contrast to the interaction vertex which introduced a singularity into the worldline. The methods of quantum field theory are applied to string theory in a rather interesting way. If we fix the spacetime and consider a string propagating through it, the location of the worldsheet can be viewed as a map from the worldsheet to the spacetime, and we can regard the coordinate functions on the spacetime as functions on the worldsheet. These spacetime coordinate functions are then treated as the ``internal variables'' of a two-dimensional quantum field theory---formulated on the worldsheet itself---which captures many of the important physical features of the string theory. (The functional integral in this theory involves an integration over all possible metrics on the worldsheet as well as all possible maps from the worldsheet to the spacetime.) The two-dimensional quantum field theories arising from string theory are of a particular type known as a {\em conformal field theory}; this means that a conformal change of metric on the worldsheet will act as an automorphism of the theory (typically acting linearly on various spaces of ``internal fields'' of the theory). The formulation in terms of conformal field theory has turned out to be a very fruitful viewpoint for the study of string theory. \section{Correlation functions and pseudo-holomorphic curves} The basic quantities which one needs to evaluate in any quantum field theory are the {\em correlation functions}\/ which determine the probabilities for a specified final state, given an initial state. Specifying the initial and final states means not only specifying positions, but also the values of any ``internal variables'' which form a part of the theory. The possible initial or final states in a conformal field theory can be represented as operators $\O_P$ on some fixed Hilbert space $\mathcal{H}$, often referred to as ``vertex operators.''\footnote{Generally, in quantum field theories states are represented as elements of a Hilbert space $\mathcal{H}$ but in conformal field theories there is also an operator interpretation.} (The label $P$ indicates the position; we should in principle be specifying initial conditions on an entire boundary circle, but in fact it suffices to consider a limit---within the conformal class of the given metric---in which the circles have been shrunk to zero size and the vertex operators are located at points.) The conjugate transpose of an initial state is a final state, so we often don't distinguish between those in our notation; with these conventions, the correlation function of a number of vertex operators is denoted by \[\langle\O_{P_1}\O_{P_2}\dots\O_{P_k}\rangle.\] (Note that the correlation functions are complex-valued and do not directly calculate probabilities, but also include the phase of the quantum-mechanical wavefunction.) If we fix the topology of the worldsheet we must in general integrate over the choice of metric on that worldsheet. A conformal change of metric leaves the correlation functions invariant, so we only need to integrate over the set of conformal classes of metrics, i.e., over the (finite-dimensional) moduli space ${\mathcal{M}}_{g,k}$ of $k$-punctured Riemann surfaces of genus $g$. The two-dimensional quantum field theories which are related to mirror symmetry have a subset of their correlation functions whose values do not depend on the position of the points $P_j$ on the worldsheet; these are called {\em topological}\/ correlation functions. (We don't need to consider an integral over ${\mathcal{M}}_{g,k}$ in this case.) They will be the primary objects of interest for us. In fact, due to the possibility of decomposing the worldsheet into more primitive pieces, the main case to consider is the case of three vertex operators on surfaces of genus zero, i.e., we take $\Sigma$ to be the ``pair of pants'' surface $\C\P^1-\{P_1,P_2,P_3\}$. To evaluate a correlation function \[\langle\O_{P_1}\,\O_{P_2}\,\O_{P_3}\rangle,\] however, we must still integrate over the infinite-dimensional space $\operatorname{Maps}(\Sigma,M)$ of maps from $\Sigma$ to the spacetime $M$. To proceed further, we need to introduce the ``action'' functional on the space of maps. We fix Riemannian metrics\footnote{This is a ``Euclidean'' version of the theory, whose correlation functions are related by analytic continuation to those of the ``Lorentzian'' version in which the worldsheet metric has signature $(1,1)$.} on both the worldsheet $\Sigma$ and the spacetime $M$ and define for any sufficiently smooth $\varphi\in\operatorname{Maps}(\Sigma,M)$ \[\mathcal{S}[\varphi]=\int_\Sigma\|d\varphi\|^2\,d\mu\] using the metrics to define the norm. (In practice, we take $M$ to be the compact six-dimensional manifold rather than the entire space.) The properties of the action functional are easier to analyze if we assume that $M$ has some additional structure---the minimal structure needed is a symplectic form $\omega$ and an almost-complex structure $J$ which is $\omega$-tamed. (We will review the definitions of these in lecture three.) When these have been chosen, there is a ``d-bar'' operator $\overline{\partial}_J$ on maps, and an alternate formula for the action \[\mathcal{S}[\varphi]=\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu +\int_\Sigma\varphi^*(\omega).\] A lower bound for $\mathcal{S}[\varphi]$ in any homotopy class of maps is thus given by $\int_\Sigma\varphi^*(\omega)$; this bound will be achieved by the so-called pseudo-holomorphic maps---the ones for which $\overline{\partial}_J\varphi\equiv0$. These have been extensively studied by Gromov \cite{gromov} and others as a natural generalization of complex curves on K\"ahler manifolds. This action functional now appears in an integrand which is supposed to be integrated over the infinite-dimensional space of all maps. We will outline the standard manipulations which are made with these functional integrals in physics in order to express the correlation function as an infinite sum of finite-dimensional integrals. We will subsequently use the outcome of those manipulations to make mathematical definitions of the corresponding quantities in the form of a formal sum of these finite-dimensional integrals. The topological correlation functions we are studying are to be evaluated by a functional integral of the form \begin{equation}\label{eq:path} \begin{split} \langle\O_{P_1}\dots\O_{P_k}\rangle&= \int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi \,{\mathcal{S}}[\varphi]}\\ &=e^{-2\pi\,\int_\Sigma\varphi^*(\omega)} \int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi \,\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu}. \end{split} \end{equation} (We are suppressing the ``fermionic'' part of this functional integral, which is actually very important, but explaining it would take us too far afield.) The ``topological'' property of these correlation functions turns out to imply \cite{tsm,witten:mirror} that if we introduce a parameter $t$ into the exponent of the last functional integral in eq.~\eqref{eq:path} to produce \[\int {\mathcal{D}}\varphi\,\O_{P_1}\dots\O_{P_k}\,e^{-2\pi t \,\int_\Sigma\|\overline{\partial}_J\varphi\|^2\,d\mu}, \] then the resulting expression is independent of $t$ and can be evaluated in a limit in which $t\to\infty$. In such a limit, the only contributions to the functional integral are the maps $\varphi$ for which $\overline{\partial}_J\varphi\equiv0$, i.e., the pseudo-holomorphic maps. (This trick for reducing to a finite-dimensional integral is known as the ``method of stationary phase.'') The space of pseudo-holomorphic maps in a given homotopy class is finite-dimensional, so we have reduced the evaluation of our correlation function to an infinite sum of finite-dimensional integrals, of the form \begin{equation}\label{eq:reduced} \langle\O_{P_1}\dots\O_{P_k}\rangle=\sum_{\text{homotopy classes}} e^{-2\pi\,\int_\Sigma\varphi^*(\omega)} \int_{\mathcal{M}}\mathcal{D}\varphi\,\O_{P_1}\dots\O_{P_k}, \end{equation} where $\mathcal{M}$ denotes the moduli space of pseudo-holomorphic maps in a fixed homotopy class. It is these finite-dimensional integrals on which we shall eventually base our definitions. The convergence of the infinite sum will remain an issue in our approach, and will lead us to (in some cases) assign a provisional interpretation to this formula as being a formal power series only. From the physics one expects convergence whenever the volume of the corresponding metric is sufficiently large. \section{A glimpse of mirror symmetry} If the target space $M$ for our maps is a Calabi--Yau manifold (equipped with a Ricci-flat metric), all of the vertex operators which participate in a given topological correlation functions must be of one of two distinct types. Correlation functions involving vertex operators of the first type are called {\em $A$-model correlation functions}\/ while those involving vertex operators of the second type are known as {\em $B$-model correlation functions}\/ \cite{witten:mirror}. (These are actually the correlation functions in two ``topological field theories'' \cite{tsm} which are closely related to the original quantum field theories.) For each type, the vertex operators $\O_P$ in the quantum field theory or topological field theory have a geometric interpretation; we will treat the correlation functions as functions of these geometric objects. The $A$-model correlation functions can be defined in a much broader context than Calabi--Yau manifolds: they can be defined for any semipositive symplectic manifold $M$ (where semipositive roughly means that $-c_1(M)$ is nonnegative---we will give the precise definition in lecture three). The vertex operators $\O_{P_j}$ in the topological field theory correspond to harmonic differential forms $\alpha_j$ on $M$, and the correlation functions $\langle\alpha_1\,\alpha_2\,\alpha_3\rangle$ take the form of an infinite series whose constant term---corresponding to homotopically trivial maps from $\Sigma$ to $M$---is the familiar trilinear function $\int_M\alpha_1\wedge\alpha_2\wedge\alpha_3$. To evaluate the non-constant terms we need an integral over the moduli space of pseudo-holomorphic two-spheres. In the Calabi--Yau case, those two-spheres are expected to be discrete (based on a formal dimension count), so there should be invariants which count the number of rational curves in a given homology class. (There are certain technical difficulties with this, as we shall see in lecture two.) More generally, the non-constant terms in the $A$-model correlation functions will be related to certain kinds of counting problems for pseudo-holomorphic curves on a semipositive symplectic manifold. The $B$-model correlation functions, on the other hand, require a choice of nonvanishing holomorphic $n$-form $\Omega$ on $M$ for their definition, so they are restricted to the Calabi--Yau case. The vertex operators in the topological correlation functions correspond to elements in the space $H^q(\Lambda^pT^{(1,0)}_M)$, where we use $T^{(1,0)}$ to denote the holomorphic tangent bundle of an almost-complex manifold. (More precisely, we should use Dolbeault cohomology to describe $H^q(\Lambda^pT^{(1,0)}_M)$, and take harmonic representatives to get the vertex operators in the topological field theory.) The ``first term'' in the correlation function is then defined as a composition of the standard map on cohomology groups \[H^{q_1}(\Lambda^{p_1}T^{(1,0)}_M)\times H^{q_2}(\Lambda^{p_2}T^{(1,0)}_M)\times H^{q_3}(\Lambda^{p_3}T^{(1,0)}_M) \to H^{n}(\Lambda^{n}T^{(1,0)}_M)\] (for $p_1+p_2+p_3=q_1+q_2+q_3=n$) with some isomorphisms depending on the choice of $\Omega^{\otimes2}$ \[H^n(\Lambda^n(T^{(1,0)}_M)) \overset{\lhk\,\Omega}{\longrightarrow} H^n(\O_M)\cong\left(H^0(K_M)\right)^* \overset{\otimes\Omega}{\longrightarrow} \mathbb{C},\] where the middle isomorphism is Serre duality. (This can be written as an integral over $M$, and so can be thought of as coming from integrating over the moduli space of homotopically trivial maps from $\Sigma$ to $M$---this is why we identify it with the first term in an expansion like eq.~\eqref{eq:reduced}.) Remarkably, all of the other terms in the expansion \eqref{eq:reduced} of a $B$-model correlation function are known to vanish on physical grounds \cite{DG:exact,witten:mirror}, so we can calculate these correlation functions exactly using geometry, and even use the geometric version of the correlation function as a mathematical definition. In brief, the idea of mirror symmetry is this. There could be pairs of complex manifolds $M$, $W$ (each with trivial canonical bundle) which produce identical physics when used for string compactification, except that the r\^oles of the $A$-model and $B$-model correlation functions are reversed. In particular, this would imply the existence of isomorphisms \[H^q(\Lambda^p(T_M^{(1,0)})^*)\cong H^q(\Lambda^p(T_W^{(1,0)}))\] (and vice versa), as well as formulas relating the $A$-model correlation functions on $M$ (which count the number of rational curves) to the $B$-model correlation functions on $W$ (which are related to period integrals of $\Omega$). \chapter*{} \lecturename{Counting Rational Curves} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 2. Counting Rational Curves} \noindent In this lecture we begin the discussion of the problem of counting rational curves on a complex threefold with trivial canonical bundle (a ``Calabi--Yau threefold''). These curve-counting invariants will eventually be used to formulate a mathematical version of the $A$-model correlation functions. In the present lecture, we focus on the problems one encounters in formulating these invariants purely algebraically; we give a number of examples. Consider the deformation theory of holomorphic maps from $\C\P^1\to M$, where $M$ is a complex projective variety. If we are given such a map $\varphi:\C\P^1\to M$, then a first order variation of that map can be described by specifying in which direction (and at what rate) each point of the image moves. That is, we need to specify a holomorphic tangent vector of $M$ for every point on $\C\P^1$, or in other words, a section of $H^0(\C\P^1,\varphi^*(T^{(1,0)}_M))$. As might be expected from other deformation problems, the obstruction group for these deformations is $H^1(\C\P^1,\varphi^*(T^{(1,0)}_M))$. The moduli problem for such maps will be best-behaved if the obstruction group vanishes, that is, if $h^1(\C\P^1,\varphi^*(T^{(1,0)}_M))=0$. When that is true, the moduli space will be a smooth complex manifold of complex dimension $h^0(\C\P^1,\varphi^*(T^{(1,0)}_M))$. More generally, the Euler--Poincar\'e characteristic \[\chi(\varphi^*(T^{(1,0)}_M))= h^0(\C\P^1,\varphi^*(T^{(1,0)}_M))-h^1(\C\P^1,\varphi^*(T^{(1,0)}_M))\] can be regarded as the ``expected complex dimension'' of the moduli space. Although the Euler--Poincar\'e characteristic can be easily computed from the Riemann--Roch theorem for vector bundles, we shall make a more elementary calculation, based on a structure theorem for bundles on $\C\P^1$. \begin{theorem}[Grothendieck] Every vector bundle ${\mathcal{E}}$ on $\C\P^1$ can be written as a direct sum of line bundles: \[{\mathcal{E}}\cong\O(a_1)\oplus\cdots\oplus\O(a_n).\] \end{theorem} Using a Grothendieck decomposition for $\varphi^*(T^{(1,0)}_M)$, we can calculate the cohomology directly. For if \[\varphi^*(T^{(1,0)}_M)\cong\O(a_1)\oplus\cdots\oplus\O(a_n)\] then using the fact the $h^0(\O(a))=1+a$ we find \[h^0(\varphi^*(T^{(1,0)}_M))=\sum_j \begin{cases} 1+a_j&\text{if } a_j\ge-1\\ 0&\text{if } a_j<-1 \end{cases} \] while since $H^1(\O(a_1)\oplus\cdots\oplus\O(a_m))\cong H^0(\O(-2-a_1)\oplus\cdots\oplus\O(-2-a_m))^*$ we have \[h^1(\varphi^*(T^{(1,0)}_M))=\sum_j \begin{cases} -(1+a_j)&\text{if } -2-a_j\ge0\\ 0&\text{if } -2-a_j<0 \end{cases} \] since $1+(-2-a_j)=-(1+a_j)$. Taking the difference, we find \[ \chi(\varphi^*(T^{(1,0)}_M)) =\sum_j(1+a_j)=n+\sum_ja_j=\dim_\mathbb{C} M+\deg\varphi^*(-K_M).\] Thus, the ``expected dimension'' is independent of the decomposition. The same result can be obtained from Riemann--Roch. But our calculation shows more---to ensure vanishing of the obstruction group, we must have $a_j\ge-1$ for all $j$. In addition to this condition on the $a_j$'s, they must also satisfy $\max\{a_j{-}2\}\ge0$, which is seen as follows. From the exact sequence \[0\toT^{(1,0)}_{\C\P^1}\to\varphi^*(T^{(1,0)}_M)\to N_\varphi\to0\] (where $N_\varphi$ denotes the normal bundle) and the fact that $T^{(1,0)}_{\C\P^1}\cong\O(2)$, we see that there must be a nontrivial homomorphism \[\O(2)\to\O(a_1)\oplus\cdots\oplus\O(a_m),\] which implies that $\max\{a_j{-}2\}\ge0$ as claimed. Without loss of generality, we may therefore assume that $a_1\ge2.$ In the case relevant to string theory ($K_M=0$, $\dim_\mathbb{C} M=3$) we then find that in order to have vanishing obstruction group we need \[0=a_1+a_2+a_3\ge2-1-1=0\] and so $a_1=2$, $a_2=a_3=-1$. In this case, the moduli space of holomorphic maps will be smooth of dimension three; if we mod out by the automorphism group $\operatorname{PGL}(2,\mathbb{C})$, the moduli space of unparameterized maps will be smooth of dimension $0$. The points in that space are what we would like to ``count.'' We discuss some examples, drawn largely from \cite{katz:mirror}, to which we refer the reader for more details. \begin{example} \label{exampleone} {\it Lines on the Fermat quintic threefold.} All of the lines on the Fermat quintic threefold \[\{x_0^5+x_1^5+x_2^5+x_3^5+x_4^5=0\}\subset\C\P^4\] can be described as follows.\footnote{We use the Fermat quintic because it is an easily-described nonsingular hypersurface, and because it will be related to a mirror symmetry construction later on, {\em not}\/ because Wiles announced a proof of Fermat's Last Theorem while the 1993 Park City Institute was underway!} \medskip \noindent {\it First type}\/ (375 lines): The line described by $x_0+x_1=x_2+x_3=x_4=0$, and others whose equations are obtained from these by permutations and multiplication by fifth roots of unity. \medskip \noindent {\it Second type}\/ (50 one-parameter families of lines): The lines described parametrically by \[(u,v)\mapsto(u,-u,av,bv,cv)\] for fixed constants $a$, $b$, $c$ satisfying $a^5+b^5+c^5=0$, and others whose parameterizations are obtained from these by permutations and multiplication by fifth roots of unity. \medskip \noindent So we see that the lines are not always finite in number, even for smooth hypersurfaces. (One might have suspected such a ``universal finiteness for smooth hypersurfaces'' based on experience with cubic surfaces---every smooth cubic surface in $\C\P^3$ has precisely twenty-seven lines.) \end{example} \begin{example} {\it Lines on the general quintic threefold.} However, if we deform from the Fermat quintic threefold to a general one, it is possible to show that the number of lines is finite. The generic number of lines can then be computed as follows. Start from the Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^4)$ of lines in $\C\P^4$. Consider the universal bundle $U$ whose fiber at a line $L$ is the two-dimensional subspace $U_L\subset\mathbb{C}^5$ such that $\P(U_L)=L$. We define a bundle ${\mathcal{B}}=\operatorname{Sym}^5(U^*)$ whose fibers describe the quintic forms on the lines $L$. Then every quintic threefold $M$ determines a section $s_M\in\Gamma({\mathcal{B}})$: the equation of $M$ is restricted to $L$ to give a homogeneous quintic there. Clearly, the lines contained in $M$ are precisely those whose corresponding points in the Grassmannian are zeros of the section $s_M$. The Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^4)$ has complex dimension six, and the bundle ${\mathcal{B}}$ has rank six; when things are generic, the section $s_M$ will have finitely many zeros, which can be counted by calculating \[\#\{L\ |\ s_M(L)=0\}=c_6({\mathcal{B}})=2875.\] \end{example} \begin{exampleonebis} Katz \cite{katz:mirror} has found a way to assign multiplicities to each of the isolated lines, and one-parameter families of lines, on the Fermat quintic threefold. His multiplicity assignment for each of the 375 isolated lines is ``5,'' and that for each of the 50 one-parameter families is ``20.'' Thus, the total count is \[5\cdot375+20\cdot50=2875.\] Katz's methods of assigning multiplicities are not yet completely general,\footnote{See the ``Postscript: Recent Developments'' section for the current status.} but they do hold out the hope that a ``count'' of rational curves might be made even in cases when the actual number of curves is not finite. \end{exampleonebis} \begin{example} {\it Conics on the general quintic threefold.} We can make a similar calculation for conics on the general quintic threefold. The key observation is that every conic spans a $\C\P^2$, so the starting point for describing them is the Grassmannian $\operatorname{Gr}(\C\P^2,\C\P^4)$. We need the bundle over the Grassmannian whose fiber is the set of conics in the $\C\P^2$ in question: this is described by $\P(\operatorname{Sym}^2(U^*))$, where $U$ is the universal subbundle as before. The space $\P(\operatorname{Sym}^2(U^*))$ contains degenerate conics (pairs of lines, and double lines) as well as smooth conics. However, if $M$ is sufficiently general, then the actual locus of conics which lie in $M$ will be finite in number, and contain only smooth conics. The vector bundle which will get a section $s_M$ for every quintic $M$ is the bundle ${\mathcal{B}}:=\operatorname{Sym}^5(U^*)/(\operatorname{Sym}^3(U^*)\oplus\O_\P(-1))$. This describes the effect of restricting the quintic equation to the conic: one gets a quintic equation on the $\C\P^2$, but must mod out by those quintics which can be written as the product of a cubic (the $\operatorname{Sym}^3(U^*)$ factor) and the given conic. We have $\dim_\mathbb{C}\P(\operatorname{Sym}^2(U^*))=\operatorname{rank}{\mathcal{B}}=11$, so the computation is made by calculating: \[\#\{C\ |\ s_M(C)=0\}=c_{11}({\mathcal{B}})=609250.\] \end{example} \begin{example} {\it Twisted cubics on the general quintic threefold.} The problem gets more difficult for twisted cubics. Again, we can look at the linear span (a $\C\P^3$) and begin by considering a Grassmannian $\operatorname{Gr}(\C\P^3,\C\P^4)$. But this time we must use a bundle ${\mathcal{H}}\to\operatorname{Gr}(\C\P^3,\C\P^4)$ whose fibers are isomorphic to the Hilbert scheme of twisted cubics in $\C\P^3$. That scheme contains limits which are quite complicated. (For example, there is a limit which is a nodal plane curve with an embedded point at the node which points out of the plane: \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{cubics.eps}} }$$ \fi see Hartshorne \cite{Hartshorne}, pp.~259--260.) Although the bundle ${\mathcal{B}}$ and the section $s_M$ can be defined and understood at points representing smooth twisted cubics, their extension to the locus of degenerate cubics is by no means easy. Ellingsrud and Str\o mme \cite{ES} have, however, carried this out, and they find that the number of twisted cubics on the general quintic threefold is 317206375. \end{example} Clemens \cite{Clemens:AJ} has conjectured that the general quintic threefold will have only a finite number of rational curves of each degree, and that all of them will satisfy $\varphi^*(T^{(1,0)}_M)=\O(2)\oplus\O(-1)\oplus\O(-1)$. This has been verified up through degree nine by Katz \cite{katz:degree7}, Johnsen--Kleiman \cite{JK:nine} and Nijsse \cite{nijsse}, and the prospects are good for degrees as high as twenty-four \cite{JK:high}. However, as we have seen, making the calculation of the number becomes very difficult past degree two. In fact, for degree greater than three, effective techniques for calculating this number are not presently known. We now turn to another example which demonstrates that we cannot always expect finiteness, even for the generic deformation of a given threefold with trivial canonical bundle. \begin{example} {\it Rational curves on double solids.} We let $M$ be the double cover of $\C\P^3$, branched along a general surface $S$ of degree eight in $\C\P^3$; the double cover map is denoted by $\pi:M\to\C\P^3$. We let $\pi^*(H)$ be the pullback of a hyperplane $H$ from $\C\P^3$; the {\em degree}\/ of a rational curve $C$ will mean $\pi^*(H)\cdot C$. To find ``lines'' on $M$, that is, curves $L$ with $\pi^*(H)\cdot L=1$ we consider their images $\pi(L)$. Since $\pi^*(H)$ meets $L$ in a single point $P$, $H$ meets $\pi(L)$ in the single point $\pi(P)$. Thus, $\pi(L)$ must itself be a line. But its inverse image on $M$ will necessarily have two components: $\pi^{-1}(\pi(L))=L+L'$. In order to have this splitting into two components, the line $\pi(L)$ must be tangent to $S$ at every point of intersection with $S$, i.e., it must be four-times tangent to $S$. Now the Grassmannian $\operatorname{Gr}(\C\P^1,\C\P^3)$ has dimension four, and it is one condition to be tangent to a surface, so the dimension of the set of four-tangent lines is nonnegative, and can be expected to be equal to zero. (In fact, it turns out to equal zero as expected, when $S$ is general.) The number of such lines in the Grassmannian can be calculated with the Schubert calculus; it turns out to be 14752. The corresponding count of lines on $M$ is 29504. Finding ``conics'' on $M$ is a different story, as has been observed by Katz and by Koll\'ar. Given a curve $C$ with $\pi^*(H)\cdot L=2$, there are two possibilities for $\pi(C)$: it could be a line, or it could be a conic. In the latter case, the conic $\pi(C)$ must be eight-times tangent to $S$. But in the former case, in order to have an irreducible double cover with a rational normalization, the line $\pi(C)$ must be three-times tangent to $S$. By our previous dimension count, there is at least a one-parameter family of such lines for any choice of $S$. \end{example} So we won't always have a finite number of things to ``count,'' even if we perturb to a general member of a particular family. And there is an additional difficulty if we wish to count maps from $\C\P^1$ to $M$ when multiple covers are allowed, as the next example shows. \begin{example}\label{ex:multiple} {\it Multiple covers.} Suppose that $\varphi:\C\P^1\to M$ is generically one-to-one, but that we consider a map $\varphi':=u\circ\varphi$, where $u:\C\P^1\to\C\P^1$ is a covering of degree $\mu$. Even if $\varphi^*(T^{(1,0)}_M)=\O(2)\oplus\O(-1)\oplus\O(-1)$, we will get a bad splitting of the pullback via the new map: \[\varphi'{}^*(T^{(1,0)}_M)=\O(2\mu)\oplus\O(-\mu)\oplus\O(-\mu).\] Furthermore, the dimension of the moduli space can be calculated: the moduli space of maps $u:\C\P^1\to\C\P^1$ of degree $\mu$ has dimension $2\mu{+}1$. So we see that the dimension of the space of maps will go up and up. \end{example} To handle cases such as multiple covers, ``virtual'' numbers of curves must be introduced; Katz's approach to this is to use excess intersection theory \cite{fulton:intersection}. However, this introduction of ``virtual'' numbers leads to another complication, as our final example shows. \begin{example}\label{ex:negative}{\it Negative numbers of curves}\/ (see \cite{2param2}, section 8). There are cases in which the ``virtual'' number of curves is negative. In general, when the parameter space $B$ for a family of curves is smooth of dimension $b$, the virtual number of curves should be the top Chern class of the holomorphic cotangent bundle $c_b((T^{(1,0)}_B)^*)$. If $M$ is a complex threefold with $K_M=0$ which contains $\C\P^2$ as a submanifold (which can arise from resolving a $\mathbb{Z}/3\mathbb{Z}$-quotient singularity, for example), then the lines on $\C\P^2$ are parameterized by $\C\P^2$ and have virtual number $c_2((T^{(1,0)}_{\C\P^2})^*)=3$, but the conics on $\C\P^2$, being parameterized by $\C\P^5$, have virtual number $c_5((T^{(1,0)}_{\C\P^5})^*)=-6$. This negative value actually agrees with the predictions of mirror symmetry as shown in \cite{2param2}. \end{example} \chapter*{} \lecturename{Gromov--Witten Invariants} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 3. Gromov--Witten Invariants} \section{Counting curves via symplectic geometry} The difficulties we encountered in trying to count rational curves on a Calabi--Yau threefold can be avoided by enlarging the category we are considering, and using Gromov's theory of pseudo-holomorphic spheres in symplectic manifolds \cite{gromov}. This approach has the advantage that the almost-complex structure can be slightly perturbed to make the number of such spheres finite, and the finite number so obtained is independent of the choice of small perturbation. Let $(M,\omega)$ be a compact {\em symplectic manifold}\/ of dimension $2n$. This means that $M$ is a compact oriented differentiable manifold of (real) dimension $2n$ and $\omega$ is a closed real two-form on $M$ which is nondegenerate in the sense that its $n^{\text{th}}$ exterior power $\omega^{\wedge n}$ is nonzero at every point. An {\it almost complex structure}\/ on a manifold $M$ is a map $J:T_M\to T_M$ whose square is $-1$. If we complexify the tangent spaces, we get $T_{M,p}\otimes\mathbb{C}= T_{M,p}^{(1,0)}\oplus T_{M,p}^{(0,1)}$, the decomposition into $+i$ and $-i$ eigenspaces for $J$. If these subspaces are closed under Lie bracket, we say that the almost-complex structure is {\em integrable}; in this case, these subspaces give $M$ the structure of a complex manifold. If $(M,\omega)$ is a symplectic manifold, an almost-complex structure $J$ on $M$ is said to be {\em $\omega$-tamed}\/ if $\omega(\xi,J\xi)>0$ for all nonzero $\xi\in T_pM$. If we have fixed an ($\omega$-tamed) almost-complex structure $J$ on $M$, and $\varphi$ is a differentiable map from $S^2$ to $M$, we define \[\overline{\partial}_J\varphi=\frac12(d\varphi+J\,d\varphi\,J_0),\] where $J_0$ is the standard almost-complex structure on $S^2$. The main example we have in mind is this: $M$ is a compact K\"ahler manifold, $\omega$ is the K\"ahler form, and $J$ is an $\omega$-tamed perturbation of the original complex structure on $M$. \begin{definition}[McDuff \cite{McD:contact}] $(M,\omega)$ is {\em semipositive}\/ if there is no map $\varphi:S^2\to M$ satisfying \[\int_{S^2} \varphi^*(\omega)>0, \quad \text{and} \quad 3-n\le\int_{S^2}\varphi^*(-K_M)<0,\] where we are writing $-K_M$ as in algebraic geometry to indicate the first Chern class $c_1(M)$, which may be represented as a two-form. \end{definition} \begin{examples} Here are three ways of producing semipositive symplectic manifolds. \begin{enumerate} \item If $K_M=0$ (the Calabi--Yau case) then $(M,\omega)$ is semipositive for any $\omega$. \item If $M$ is a complex projective manifold with $|{-}K_M|$ ample (a ``Fano variety''), then we can take $\omega=-K_M$ to produce a semipositive $(M,\omega)$. \item If $n\le3$ then $M$ is automatically semipositive. \end{enumerate} \end{examples} Because it is sometimes difficult to check whether a homology class $\eta$ is represented as the image of a map $\varphi:S^2\to M$ we also introduce a variant of this property. \begin{definition} $(M,\omega)$ is {\em strongly semipositive}\/ if there is no class $\eta\in H_2(M,\mathbb{Z})$ satisfying \[\omega\cdot \eta>0, \quad \text{and} \quad 3-n\le(-K_M)\cdot \eta<0.\] All three of our examples satisfy this stronger property as well. \end{definition} Fix a homology class $\eta\in H_2(M,\mathbb{Z})$. As we saw in example \ref{ex:multiple}, there are technical problems caused by ``multiple-covered'' maps---maps whose degree onto the image is greater than one. Let us call a map {\em simple}\/ if its degree onto its image is one. We let $\operatorname{Maps}^*_\eta(S^2,M)\subset \operatorname{Maps}_\eta(S^2,M)$ denote the subset of simple maps with fundamental class $\eta$. We also let $\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ be the set of simple differentiable maps $S^2\to M$ with fundamental class $\eta$ whose derivative lies in $L_p$. Using an appropriate Sobolev norm, $\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ can be given the structure of a Banach manifold. We can then regard $\overline{\partial}_J$ as a section of the bundle $\mathcal{W}\to\operatorname{Maps}^*_\eta(S^2,M)_{(p)}$ whose fibers are \[\mathcal{W}_\varphi:=H^0_{(p)}(S^2,{\mathcal{A}}^{(0,1)}_{S^2} \otimes\varphi^*(T^{(1,0)}_M)),\] where the subscript $(p)$ denotes $L_p$-cohomology, and ${\mathcal{A}}^{(0,1)}_{S^2}$ denotes the sheaf of $(0,1)$-forms on $S^2$ (with respect to the complex structure $J_0$). The key technical properties we need are summarized in the following two theorems. \begin{theorem}[McDuff \cite{McD:examples}] If $J$ is generic, then \[\MMhol\eta:= \{\varphi\in\operatorname{Maps}^*_\eta(S^2,M)_{(p)}\ |\ \overline{\partial}_J\varphi=0\}\] is a smooth manifold of dimension \[\dim_\mathbb{R}\MMhol\eta=2\,\chi(\varphi^*(T^{(1,0)}_M)).\] (The dimension is calculated using the Atiyah--Singer index theorem, which yields the same result as the Riemann--Roch theorem did in algebraic geometry.) \end{theorem} \noindent (This theorem would have failed if we had allowed multiple-covered maps to be included.) The next theorem is due to Gromov \cite{gromov}, based on some techniques of Sacks--Uhlenbeck \cite{SacksUhlenbeck} and with further improvements by several authors \cite{PW,wolfson,rye}. (We refer the reader to those papers for a more precise statement.) \begin{theorem} $\MMhol\eta$ can be compactified by using limits of graphs of maps; this compactification has good properties. \end{theorem} In the case relevant to string theory in which $M$ is a projective manifold with $K_M=0$ of complex dimension three, we find that for generic $J$, the (real) dimension of $\MMhol\eta$ is six, and the dimension of \[\mathcal{M}^*_{(\eta,J)}:=\MMhol\eta/\operatorname{PGL}(2,\mathbb{C})\] is zero. The space $\mathcal{M}^*_{(\eta,J)}$ itself is already compact in this case; the number of points in that space is our desired invariant. (These points may need to be counted with multiplicity, or with signs.) This invariant counts the number of rational curves (of fixed topological type) on $M$ with respect to its original complex structure, if that number is finite; it can be used as a substitute for that count in the general case.\footnote{It has not yet been verified that Katz's method of assigning multiplicities to positive-dimensional components in the algebro-geometric context produces the same results as this method from symplectic geometry. Because of the need to include signs in certain circumstances, this invariant can even accommodate the ``negative virtual numbers'' which occurred in example \ref{ex:negative}.} To describe the invariants in situations more general than complex threefolds with trivial canonical bundle, we must introduce the oriented bordism group $\Omega_*(M)$. The elements of $\Omega_k(M)$ are equivalence classes of pairs $(B^k,F)$ consisting of a compact oriented differentiable manifold $B$ of dimension $k$ (but not necessarily connected), together with a differentiable map $F:B^k\to M$. We say that $(B^k,F)\sim0$ if there exists an {\em oriented bordism}\/ $(C^{k+1},H)$: i.e., a differentiable manifold $C$ of dimension $k{+}1$ and a differentiable map $H:C^{k+1}\to M$ with $\partial C^{k+1}=B^k$ and $H|_{B^k}=F$. We add elements of $\Omega_k(M)$ by means of disjoint union: $(B^k_1,F_1)+(B^k_2,F_2)=(B^k_1\cup B^k_2,F_1\cup F_2)$; the additive inverse is given by reversing orientation. The oriented bordism group $\Omega_*(M)$ is a module over the Thom bordism ring $\Omega$ (consisting of oriented manifolds modulo oriented bordisms, with no maps to target spaces) via \[N^j\cdot(B^k,F)=(N^j\times B^k,G)\] where $G(x,y)=F(y)$. \begin{theorem}[Thom \cite{Thom}, Conner--Floyd \cite{CF}]\quad \begin{enumerate} \item $\Omega_*(M)\otimes\mathbb{Q}\cong H_*(M,\mathbb{Q})\otimes\Omega$. \item If $H_*(M,\mathbb{Z})$ is torsion-free, then $\Omega_*(M)\cong H_*(M,\mathbb{Z})\otimes\Omega$. \end{enumerate} \end{theorem} To describe our basic invariants, we choose three classes $\alpha_1$, $\alpha_2$, $\alpha_3$ in $\Omega_*(M)$ represented by elements $(B^{k_1}_1,F_1)$, $(B^{k_2}_2,F_2)$, $(B^{k_3}_3,F_3)$, and let $Z_j=\operatorname{Image}(F_j)$. We call the invariants defined below the {\em Gromov--Witten invariants}, since it was Witten \cite{tsm} who pointed out how Gromov's study of $\MMhol\eta$ could be used in principle to describe invariants relevant in topological quantum field theory. The detailed construction of these invariants was recently carried out by Ruan \cite{ruan}. There are two cases to consider, with one being more technically challenging than the other.\footnote{To simplify the exposition, we have altered Ruan's description of the second case, ignored the necessity of passing to the inhomogeneous $\overline{\partial}$ equation (introduced already by Gromov \cite{gromov}), and built into our definition the so-called ``multiple cover formula'' expected from the physics \cite{CDGP,aspmor}. (This latter step is now justified thanks to a theorem of Voisin \cite{voisin:multiple}; there is also a related result of Manin \cite{Manin}.) We are also abusing notation somewhat by using $\Phi_\eta$ in both cases, since the second case is actually related to Ruan's $\widetilde\Phi_\eta$ invariant.} \begin{construction}[Ruan] Let $\eta\in H_2(M,\mathbb{Z})$, let $\alpha_j=(B^{k_j},F_j)$ be a bordism class, and let $Z_j=\operatorname{Image}(F_j)$, for $j=1,2,3$. Suppose that $\sum_{j=1}^3(2n-k_j)=2n-2K_M\cdot \eta$, where $\eta$ is the class of the image of $\varphi$, and suppose that the almost-complex structure $J$ is generic. \begin{itemize} \item[(a)] If $-K_M\cdot\eta>0$, then \[\{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] is a finite set. Let $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ denote the signed number of points in this set, with signs assigned according to orientations at the specified points of intersection. \item[(b)] If $-K_M\cdot\eta=0$, then there exists an integer $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ which agrees with \[\#\{\text{generically injective}\ \varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] (counted with signs) whenever the latter makes sense. (The signs are all positive if the almost-complex structure is integrable.) \end{itemize} These invariants $\Phi_{\eta}(\alpha_1,\alpha_2,\alpha_3)$ depend only on the bordism classes $\alpha_1$, $\alpha_2$, $\alpha_3$, and do not change under small variation of $J$. \end{construction} \section{Simple properties of Gromov--Witten invariants} In spite of the fact that we needed to pass to bordism to ensure that the Gromov--Witten invariants are well-defined, their dependence on bordism-related phenomena is minimal. In fact, Ruan checks that the invariants are trivial with respect to the $\Omega$-module structure on $\Omega_*(M)$, and so it follows from the theorem of Thom and Conner--Floyd that we get a well-defined $\mathbb{Q}$-valued invariant on rational homology $H_*(M,\mathbb{Q})$. If $M$ has no torsion in homology, we even get an integer-valued invariant on integral homology. The Gromov--Witten invariants $\Phi_\eta(\alpha_1,\alpha_2,\alpha_3)$ will vanish if $\alpha_1$ corresponds to a class of real codimension zero or one. This is easy to see---if there are any elements in the set \[\{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1, \varphi(1)\in Z_2, \varphi(\infty)\in Z_3\}\] then the intersection of the image of $\varphi$ with the image $Z_1$ of $F_1$ has real dimension two or one. By varying the location of $\varphi(0)$, we will produce a two-{} or one-parameter family of maps. This contradicts the set being finite; thus, the set must be empty and the invariant vanishes. Note what happens to the Gromov--Witten invariants in the case of interest to string theory ($\dim_\mathbb{C} M=3$, $K_M=0$): the only relevant invariants are those with $k_1=k_2=k_3=4$. (This is because $k_j\le 4$ to get a nonzero invariant, so that $6=\sum (6-k_j)\ge\sum_{j=1}^3 2=6$, which implies that each $k_j$ is $4$.) The possible location of $0$ under a generically injective map is easy to spot: the image curve $\varphi(S^2)$ is some rational curve on $M$, and meets the four-manifold $Z_1$ in precisely $\#(Z_1\cap \eta)=\alpha_1\cdot\eta$ points; we can choose any of these for the image of $0$. Similar remarks about the images of $1$ and $\infty$ lead to the calculation: \[\Phi_\eta(Z_1,Z_2,Z_3) =(\alpha_1\cdot\eta)(\alpha_1\cdot\eta)(\alpha_1\cdot\eta)\, \#(\mathcal{M}^*_{(\eta,J)}).\] \section{The $A$-model correlation functions} Although we have defined the Gromov--Witten invariants for oriented bordism classes, we will now use them in cohomology instead. As previously remarked, thanks to the triviality of the invariants under the $\Omega$-module structure, if we tensor with $\mathbb{Q}$ we can move the invariants to homology (and then by Poincar\'e duality, to cohomology). This is at the expense of possibly allowing them to become $\mathbb{Q}$-valued on integral classes. One hopes that they will remain integer valued on integer classes, but this has not yet been established. Therefore, we will give a presentation using $\mathbb{Q}$-coefficients, but the reader should bear in mind that most of the formulas are expected to be valid with integer coefficients if one uses integer cohomology classes. In brief, the bordism class of $\alpha=(B^k,F)$ gives rise to a homology class $[Z]\in H_k(M,\mathbb{Z})$ (using the image $Z$ of $F$ to represent the class), and by duality to a cohomology class $\zeta=[Z]^\vee\in H^{2n-k}(M,\mathbb{Q})$. (Our retreat to $\mathbb{Q}$-coefficients will be in part because we do not know that every integer cohomology class can be so represented.) We extend the definition of Gromov--Witten invariants to cohomology by defining \[\Phi_\eta(\zeta_1,\zeta_2,\zeta_3):=\Phi_\eta(\alpha_1,\alpha_2,\alpha_3)\] when $\alpha_j=(B^{k_j}_j,F_j)$ and $\zeta_j=[\operatorname{Image}(F_j)]^\vee$; then extend by linearity to all of $H^*(M,\mathbb{Q})$. Our ``$A$-model correlation functions'' are then built from the Gromov--Witten invariants, following a calculation from the physics literature \cite{strominger,DSWW,CDGP,aspmor}. There is a certain danger in using the {\em outcome}\/ of a physics calculation as a {\em definition}---later, the physicists may become interested in a slightly different problem, whose outcome is radically different from the original one, and we mathematicians will find that our definitions are inadequate. Nevertheless, we will go ahead and define the $A$-model correlation functions. These are trilinear functions on the cohomology $H^*(M,\mathbb{Q})$ defined by: \begin{equation}\label{A:correlation} \begin{split} \langle\zeta_1\,\zeta_2\,\zeta_3\rangle:= (\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,q^{\eta}\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,\sum_{m=1}^\infty q^{m\eta} \end{split}\end{equation} It is sometimes convenient to formally sum the geometric series in the final term, and write $q^\eta/(1-q^\eta)$ in place of $\sum_{m=1}^\infty q^{m\eta}$, in which case eq.~\eqref{A:correlation} becomes \begin{equation}\label{A:correlationbis} \begin{split} \langle\zeta_1\,\zeta_2\,\zeta_3\rangle:= (\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\,q^{\eta}\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \Phi_\eta(\zeta_1,\zeta_2,\zeta_3)\, \frac{q^{\eta}}{1-q^{\eta}} \end{split}\end{equation} The terms with $K_M\cdot\eta=0$ have been separated out because they are where the multiple-covered maps cause the greatest difficulty. Heuristically, the coefficients in these functions (as we have defined them) are expected to count the simple maps only. The symbol $q^\eta$ which appears in these formulas has not yet been defined. In fact, there are two natural interpretations of eq.~\eqref{A:correlation}, one algebraic and one geometric, and we consider them in turn in the next two lectures. \chapter*{} \lecturename{The Quantum Cohomology Ring} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 4. The Quantum Cohomology Ring} \section{Coefficient rings} There are several possible ways to interpret the ``$A$-model correlation functions'' defined by eq.~\eqref{A:correlation}. In this lecture, we will focus on the algebraic interpretation, in which the symbol $q^\eta$ can be regarded as an element of a group ring or semigroup ring.\footnote{I am grateful to A.~Givental for pointing out the relevance of group rings.} Recall that for any commutative semigroup ${\mathcal{S}}$ and any commutative ring $R$, the {\em semigroup ring of ${\mathcal{S}}$ with coefficients in $R$}\/ is the ring \[R[q;{\mathcal{S}}]:=\left\{\sum_{\eta\in{\mathcal{S}}}a_\eta q^\eta\ |\ a_\eta\in R\text{ and } \{\eta\ |\ a_\eta\ne0\}\text{ is finite}\right\}.\] The symbol $q$ serves as a placeholder, translating the semigroup operation (usually written additively) into a multiplicative structure on a set of monomials. If ${\mathcal{S}}$ is a group, this coincides with the usual ``group ring'' construction. In the case of a Fano variety, the sum in eq.~\eqref{A:correlation} defining the $A$-model correlation function is finite, and we can regard it as taking values in the rational group ring\footnote{If we knew that the Gromov--Witten invariants were integers, we could use the integral group ring $\mathbb{Z}[q;H_2(M,\mathbb{Z})]$. But when we passed from bordism to cohomology we lost control of the integer structure.}\ \ $\mathbb{Q}[q;H_2(M,\mathbb{Z})]$. To be more concrete, if we assume for simplicity that $H_2(M,\mathbb{Z})$ has no torsion, and choose a basis $e_1$,\dots,$e_r$ of $H_2(M,\mathbb{Z})$, then writing $\eta=\sum a^je_j$ we may associate to $\eta$ the rational monomial $q^{\eta}\in\mathbb{Q}(q_1,\dots,q_r)$ defined by \[\log q^{\eta}=\sum a^j\log q_j .\] (One can also write this multiplicatively: \[q^{\eta}=\prod (q_j)^{(a^j)}\] but then great care is required in distinguishing exponents from superscripts.) If we choose our basis so that the coefficients $a^j$ are nonnegative for all classes $\eta$ which have nonvanishing Gromov--Witten invariants $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$ for some $\zeta_1$, $\zeta_2$, $\zeta_3$, then each $q^{\eta}$ occurring in eq.~\eqref{A:correlation} is a {\em regular}\/ monomial, i.e., $q^{\eta}$ belongs to the polynomial ring $\mathbb{Q}[q_1,\dots,q_r]$, and we can calculate eq.~\eqref{A:correlation} in that ring. In the Calabi--Yau case in which $K_M=0$, the sum in eq.~\eqref{A:correlation} is not finite and we must work harder. The simplest interpretation would be to simply allow infinite sums $\sum a_\eta q^\eta$ as formal expressions. However, in order to construct quantum cohomology (which we shall do in the next section) we need the values of the correlation function to lie in a {\em ring}. In the definition of semigroup rings one restricts to finite sums in order to ensure that the partial sums which occur in the expansion of a product will be finite. That finiteness can still be guaranteed for products of infinite sums if the semigroup satisfies a special property, given below. We say that a semigroup ${\mathcal{S}}$ has the {\em finite partition property}\/ if for every $\eta\in\mathcal{S}$ there are only finitely many pairs $(\eta_1,\eta_2)\in{\mathcal{S}}\times{\mathcal{S}}$ such that $\eta=\eta_1+\eta_2$. For such semigroups, any expression of the form \[\sum_{\substack{(\eta_1,\eta_2)\text{ s.t.}\\ \eta_1+\eta_2=\eta}} a_{\eta_1}a_{\eta_2}\] (for fixed $\eta$) will be finite. Thus, infinite sums can be multiplied. So if ${\mathcal{S}}$ is a semigroup with the finite partition property, we define the {\em formal semigroup ring of ${\mathcal{S}}$ with coefficients in $R$}\/ to be \[R[[q;{\mathcal{S}}]]:=\{\sum_{\eta\in{\mathcal{S}}}a_\eta q^\eta\},\] with the product defined by \[(\sum_{\eta_1\in{\mathcal{S}}}a_{\eta_1} q^{\eta_1})\cdot (\sum_{\eta_2\in{\mathcal{S}}}a_{\eta_2} q^{\eta_2})= \sum_{\eta\in{\mathcal{S}}} (\sum_{\substack{(\eta_1,\eta_2)\text{ s.t.}\\ \eta_1+\eta_2=\eta}} a_{\eta_1}a_{\eta_2}) q^\eta.\] The semigroup $H_2(M,\mathbb{Z})$ of interest to us is actually a {\em group}\/ with a nontrivial free abelian part, and so does not satisfy the finite partition property. However, in eq.~\eqref{A:correlation} we are only required to sum over classes which can be realized by pseudo-holomorphic curves---these generate a smaller semigroup. If we are using an integrable almost-complex structure $J$ on $M$ for which $M$ is a K\"ahler manifold, this smaller semigroup is the {\em integral Mori semigroup}\/ defined (in the case $h^{2,0}=0$, for simplicity) as \[\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z}):=\{\eta\in H_2(M,\mathbb{Z})\ |\ (\omega,\eta)\ge0\ \forall\ \omega\in\overline{\mathcal{K}}_J\},\] where $\mathcal{K}_J$ is the K\"ahler cone and $\overline{\mathcal{K}}_J$ is its closure. The Mori semigroup has the finite partition property (the free part lies in a strongly convex cone, and the torsion part is finite), so we can form the formal semigroup ring $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]$. Presumably, by using the symplectic version of the K\"ahler cone, we would find a similar property for the analogous semigroup in the almost-complex case and could form a similar ring in that case. There is an important variant which we will have occasion to consider. Let $\operatorname{Aut}_J(M)$ be the image in $\operatorname{Aut} H_2(M,\mathbb{Z})$ of the group of diffeomorphisms of $M$ compatible with the almost-complex structure $J$. This group acts on the pseudo-holomorphic curves and so permutes the Gromov--Witten invariants. The values of the $A$-model correlation function are preserved by the group action, and can be regarded as lying in the ring of invariants \[R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]^{\operatorname{Aut}_J(M)}.\] As in the Fano variety case, if we choose an appropriate basis (and assume $H_2(M,\mathbb{Z})$ is torsion-free) then we can regard the correlation function defined in eq.~\eqref{A:correlation} as taking values in a formal power series ring $\mathbb{Q}[[q_1,\dots,q_r]]$. Note that if we set all $q_j$'s to $0$, we simply recover the topological trilinear function $(\zeta_1\cup\zeta_2\cup\zeta_3)|_{[M]}$. But although the formal series in eq.~\eqref{A:correlation} is expected by the physicists to converge near $q_j=0$, no convergence properties of the series (as we have defined it) are known at present. There is an alternative to using the semigroup rings: we could instead use the Novikov rings \cite{Novikov} which have played a r\^ole elsewhere in symplectic geometry \cite{HS}. For each K\"ahler class $\omega$, the {\em Novikov ring}\/ $\Lambda_\omega$ consists of all formal power series \[\sum_{\eta\in H_2(M,\mathbb{Z})} a_\eta q^\eta\] such that the set \[\{\eta\ |\ a_\eta\ne0 \ \text{and}\ (\omega,\eta) <c\}\] is finite for all $c\in\mathbb{R}$. (If it is necessary to specify the ring $R$ in which the coefficients $a_\eta$ take their values, the notation $\Lambda(\omega,R)$ is used.) The product of two elements of $\Lambda_\omega$ is well-defined, and also belongs to $\Lambda_\omega$. In the case $H_2(M,\mathbb{Z})=\mathbb{Z}^r$, $\Lambda_\omega$ is the ring of {\em generalized Laurent series}\/ \[\{\sum a_{\vec{k}} q^{\vec{k}}\ |\ \ \text{there are only finitely many terms with $\omega\cdot\vec{k}<c$ for any $c\in\mathbb{R}$}\}.\] \section{A new algebra structure} The correlation functions defined in the previous lecture can be used to describe a new algebra structure on the cohomology of $M$, in the following way. Let $R$ be an integral domain (usually we use $R=\mathbb{Z}$ or $R=\mathbb{Q}$), and choose a coefficient ring $\mathcal{R}$ from among \begin{enumerate} \item the group ring $R[q;H_2(M,\mathbb{Z})]$ (in the case of a Fano variety), \item the formal semigroup ring with coefficients in $R$ for the Mori semigroup $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]$ (when this is well-defined, such as in the case of a K\"ahler manifold), \item the subring $R[[q;\mathop{\overline{\text{NE}}}\nolimits(M,\mathbb{Z})]]^{\operatorname{Aut}_J(M)}$ of $\operatorname{Aut}_J(M)$-invariants, or \item one of the Novikov rings $\Lambda(\omega,R)$. \end{enumerate} We introduce a binary operation $\zeta_1\star\zeta_2$ on $H^*(M,\mathcal{R})$ defined by the requirement \[((\zeta_1\star\zeta_2) \cup \zeta_3)|_{[M]} =\langle\zeta_1\,\zeta_2\,\zeta_3\rangle.\] (This is well-defined since the cup product is a perfect pairing.) The class $\mbox{\rm 1\kern-2.7pt l}:=[M]^\vee\in H^0(M)$ which is dual to the fundamental class $[M]\in H_{2n}(M)$ has the property that the Gromov--Witten invariants $\Phi_\eta(\mbox{\rm 1\kern-2.7pt l},\zeta_2,\zeta_3)$ all vanish, hence \[\langle\mbox{\rm 1\kern-2.7pt l}\,\zeta_1\,\zeta_2\rangle=(\zeta_2\cup\zeta_3)|_{[M]};\] it follows that $\mbox{\rm 1\kern-2.7pt l}$ serves as the identity element for the binary operation $\star$. This interpretation of the correlation function as a binary operation also comes from physics \cite{MooreSeiberg,topgrav}. Let us return to the picture we had of the ``pair of pants'' surface \iffigs $$\vbox{\centerline{\epsfysize=2cm\epsfbox{assoc1.eps}} }$$ \else \vglue2in\noindent \fi as describing a possible evolution between an initial state with two ``incoming'' vertex operators $\zeta_1$, $\zeta_2$ on the left and a final state with one ``outgoing'' vertex operator $\zeta_1\star\zeta_2$ on the right. This point of view leads to the remarkable expectation that the binary operation should be associative! A heuristic argument for this runs as follows: the product $(\zeta_1\star\zeta_2)\star\zeta_3$ is evaluated by means of the surface \iffigs $$\vbox{\centerline{\epsfysize=3.5cm\epsfbox{assoc2.eps}} }$$ \else \vglue2in\noindent \fi (with an outgoing vertex operator of one piece attached to an incoming vertex operator of the other) while the product $\zeta_1\star(\zeta_2\star\zeta_3)$ is evaluated by means of the surface \iffigs $$\vbox{\centerline{\epsfysize=3.5cm\epsfbox{assoc3.eps}} }$$ \else \vglue2in\noindent \fi which is a deformation of the first one. So long as the values of the resulting quadrilinear function do not depend on the location of the four points in $\C\P^1$ used in defining it, these two products will agree. In fact, as we pointed out in the introduction, the correlation functions we are studying are expected from the physics to be precisely of this ``topological'' nature which makes them independent of the location of the points \cite{tsm}. This associativity property of the binary operation $\star$ can be rewritten as a set of relations which must be satisfied among the Gromov--Witten invariants themselves. This turns out to be a very deep property, which had not been proved at the time these lectures were delivered (although proofs were given not too long thereafter \cite{RuanTian,Liu,MS}). We have formulated the Gromov--Witten invariants and the binary operation at this level of generality primarily because this associativity property is such an interesting one. However, as we will see in more detail below, for the case of primary interest in mirror symmetry---that of Calabi--Yau threefolds---the associativity is automatic, and there is nothing to prove. (Associativity {\it does}\/ say something interesting for Calabi--Yau manifolds of higher dimension.) The $\mathcal{R}$-module $H^*(M,\mathcal{R})$ equipped with the binary operation $\star$ is called the {\em quantum cohomology ring}\/ of $M$, or the {\em quantum cohomology algebra}\/ when we wish to emphasize the $\mathcal{R}$-module structure. We can give a more geometric description of the new binary operation, by turning each Gromov--Witten invariant itself into a kind of binary operation. Here is a heuristic description of what this construction should look like. We want a cohomology class $Q_\eta(\zeta_1,\zeta_2)$ with the property that \[(Q_\eta(\zeta_1,\zeta_2)\cup \zeta_3)|_{[M]}= \Phi_\eta(\zeta_1,\zeta_2,\zeta_3).\] Consider the set of pseudo-holomorphic curves which satisfy the conditions imposed by $\zeta_1$ and $\zeta_2$ only: \[\mathcal{M}_\eta(\zeta_1,\zeta_2):= \{\varphi\in\MMhol\eta\ |\ \varphi(0)\in Z_1,\varphi(1)\in Z_2\},\] where $\zeta_j=[Z_j]^\vee$. To count the maps contributing to $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$, we must look for all maps in $\mathcal{M}_\eta(\zeta_1,\zeta_2)$ which also send $\infty$ into $Z_3$. What subset of $M$ has the property that its intersection with $Z_3$ is in one-to-one correspondence with such maps? It is the subset consisting of {\em all possible}\/ points $\varphi(\infty)$ which might be mapping to $Z_3$. In other words, we can write $Q_\eta(\zeta_1,\zeta_2)=[T_\eta(Z_1,Z_2)]^\vee$, where $T_\eta(Z_1,Z_2)$ is the cycle defined by \begin{align*} T_\eta(Z_1,Z_2)&:= \{P\in M\ |\ P=\varphi(\infty) \text{ for some }\varphi\in\mathcal{M}_\eta(\zeta_1,\zeta_2)\} \\&= \bigcup_{\varphi\in\mathcal{M}_\eta(\zeta_1,\zeta_2)}\operatorname{Image}(\varphi) .\end{align*} Then $T_\eta(Z_1,Z_2)\cap Z_3$ will correspond to the maps counted by $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$, where $\zeta_3=[Z_3]^\vee$. Note that for this heuristic description to work, we need the set $T_\eta(Z_1,Z_2)$ to be of the expected dimension. A better formal definition of $Q_\eta(\zeta_1,\zeta_2)$ would be the pushforward under evaluation at $\infty$ of the pullback of $\mathcal{M}_\eta(\zeta_1,\zeta_2)$ to the universal family of maps. Expressed in these terms, then, the binary operation can be written: \begin{equation}\label{A:binary} \begin{split} \zeta_1\star\zeta_2:= \zeta_1\cup\zeta_2\ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} q^{\eta}\,Q_\eta(\zeta_1,\zeta_2)\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \frac{q^{\eta}}{1-q^{\eta}}\,Q_\eta(\zeta_1,\zeta_2) \end{split}\end{equation} Recall that the Gromov--Witten invariant $\Phi_\eta(\zeta_1,\zeta_2,\zeta_3)$ with $\zeta_j\in H^{\ell_j}(M,\mathbb{Q})$ is zero unless \[\ell_1+\ell_2+\ell_3=2n+2({-}K_M\cdot\eta), \quad\text{and\ \ }\ell_j\ge2.\] It follows that if the cycle $Q_\eta(\zeta_1,\zeta_2)$ is nonzero, we have \[Q_\eta(\zeta_1,\zeta_2)\in H^{2n-\ell_3}(M,\mathbb{Q})= H^{\ell_1+\ell_2-2({-}K_M\cdot\eta)}(M,\mathbb{Q}).\] Thus, if $K_M=0$, then the binary operation $\star$ preserves the grading on cohomology, while if $-K_M\cdot\eta>0$ the grading is shifted down by $2({-}K_M\cdot\eta)$. But note that in any case, the $\mathbb{Z}/2\mathbb{Z}$-grading on cohomology is preserved. Note also that $\ell_j\le2n$ implies $\ell_1+\ell_2+\ell_3\le6n$ and hence $-K_M\cdot\eta\le2n$. \begin{exercise} Show that the semipositivity condition $3-n<-K_M\cdot \eta$ implies that the grading cannot shift up, it can only shift down. \end{exercise} \begin{example} (cf.\ \cite{example:pm,vafa}) We now compute an example of the quantum cohomology ring. Let $M=\C\P^n$ (with $\omega$ induced from the Fubini--Study metric, which will ensure semipositivity). The formal semigroup ring in this case can be written as $\mathcal{R}=\mathbb{Q}[[q]]$ (or we could use $\mathcal{R}=\mathbb{Q}[q]$ since we know the sums are finite, this being a Fano variety). If $C$ is any complex curve on $M$, then $-K_M\cdot C=d(n+1)$, where $d$ is the degree of the curve. Since $-K_M\cdot C\le2n$, we must have $d=1$. So only lines (and constant maps) will contribute to our correlation function. Now the predicted real dimension of the space of maps $\C\P^1\to M$ whose image $L$ has degree one is \[2n+2(-K_M\cdot L)=2n+2(n+1)=4n+2\] while the actual dimension is \[\dim_\mathbb{R}\operatorname{PGL}(2,\mathbb{C})+\dim_\mathbb{R}\operatorname{Gr}(\C\P^1,\C\P^n) =6+2\cdot2(n-1)=4n+2\] so we should be able to use the given complex structure to compute the invariants. The Gromov--Witten invariants are evaluated as follows. A basis for $H^*(M,\mathbb{Q})$ is given by the classes $\zeta^k\in H^{2k}(M,\mathbb{Q})$ where $\zeta$ is the class of a hyperplane. We choose $k_1$, $k_2$, $k_3$, satisfying \[2k_1+2k_2+2k_3=4n+2\] and find that there is a {\em unique}\/ line in $\C\P^m$ meeting three fixed linear spaces of codimensions $k_1$, $k_2$ and $k_3$. And there is a unique map sending $0$, $1$, $\infty$ to the intersection points with the three linear spaces. Thus, \[\Phi_L(\zeta^{k_1},\zeta^{k_2},\zeta^{k_3})=1.\] Expressed in terms of the binary operation, we find that \[\zeta^{k_1}\star\zeta^{k_2}= \begin{cases} \zeta^{k_1+k_2}&\text{if }k_1+k_2\le n\\ \zeta^{k_1+k_2-n-1}\,q&\text{if }k_1+k_2\ge n+1\\ \end{cases}.\] It follows that the quantum cohomology ring can be described as: \[\mathcal{R}[\zeta]/(\zeta^{\star (n+1)}-q).\] \end{example} \begin{example} If we consider the case relevant to string theory ($\dim_\mathbb{C}(M)=3$, $K_M=0$), we find that the only products which differ from the cup product are products $\zeta_1\star \zeta_2$, with $\zeta_1, \zeta_2\in H^2(M)$, and these are given by \begin{equation*} \zeta_1\star\zeta_2:= \zeta_1\cup\zeta_2\ \ + \sum_{0\ne\eta\in H_2(M,\mathbb{Z})} \left( \zeta_1(\eta)\cdot\zeta_2(\eta)\cdot \#(\mathcal{M}^*_{(\eta,J)})\right)\,\frac{q^{\eta}}{1-q^{\eta}}\,\eta \end{equation*} Here, $\#(\mathcal{M}^*_{(\eta,J)})$ denotes the number of curves in class $\eta$ (counted with appropriate multiplicity). \end{example} \begin{remark} Note that the associativity of the binary operation $\star$ is automatically satisfied by threefolds with trivial canonical bundle, since only one of the products being associated can be different from the cup product. \end{remark} \begin{example} \label{example43} Let $\lambda\in H^2(M)$ be represented by $L$, a submanifold of real codimension two. If we define \[\mathcal{M}_{\eta}(\zeta):= \{\varphi\in\MMhol\eta\ |\ \varphi(1)\in \zeta\},\] and \[\Gamma_\eta(\zeta):=[\{P\in M\ |\ P=\varphi(\infty)\text{ for some } \varphi\in\mathcal{M}_\eta(\zeta)\}]^\vee,\] then we can expect that \[Q_\eta(\lambda,\zeta)=\lambda(\eta)\cdot\Gamma_\eta(\zeta).\] This is because the image of each $\varphi$ should meet $L$ in precisely $\lambda(\eta)$ points, any of which may be chosen as $\varphi(0)$. The binary operation can then be written: \begin{equation*} \begin{split} \lambda \star\zeta := \lambda \cup\zeta \ \ &+\sum_{\substack{\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta>0}} \lambda(\eta)\,q^{\eta}\,\Gamma_\eta(\zeta)\\ &+\sum_{\substack{0\ne\eta\in H_2(M,\mathbb{Z}),\\{-}K_M\cdot\eta=0}} \lambda(\eta)\, \frac{q^{\eta}}{1-q^{\eta}}\,\Gamma_\eta(\zeta) \end{split}\end{equation*} We regard $\Gamma_\eta$ as a map on cohomology, and call it the {\em Gromov--Witten map}. \end{example} \section{Algebraic properties of the correlation functions} Let $K$ be the field of fractions of our coefficient ring $\mathcal{R}$; tensoring the quantum cohomology ring with $K$ makes it into a $K$-algebra. This quantum cohomology algebra carries some additional structure which makes it into what is known as a {\em Frobenius algebra}.\footnote{We follow standard mathematical usage \cite{CR,Karp} and do not require a Frobenius algebra to be commutative; our definition therefore differs slightly from that in \cite{Dubrov}. However, we will primarily be interested in the even part $H^{ev}(M)$ of the cohomology of $M$, on which the quantum product will in fact be commutative.} By definition this is a $K$-algebra $A$ with a multiplicative identity element $\mbox{\rm 1\kern-2.7pt l}$, such that there exists a linear functional $\varepsilon:A\to K$ for which the induced bilinear pairing $(x,y)\mapsto\varepsilon(x\star y)$ is nondegenerate. There does not seem to be a standard name for such a functional; we call it an {\em expectation function}\/ (cf.~\cite{summing}). If an expectation function exists at all, then most linear functionals on $A$ can serve as expectation functions. If $A$ is $\mathbb{Z}$-graded, we call $\varepsilon$ a {\em graded expectation function}\/ when $\ker(\varepsilon)$ is a graded subalgebra of $A$ (and we call $A$ a {\em graded Frobenius algebra}\/ when such a function exists). There is much less freedom to choose graded expectation functions. The cohomology of a compact manifold $M$ has the structure of a graded Frobenius algebra, with multiplication given by cup product, $\mbox{\rm 1\kern-2.7pt l}$ given by the standard generator of $H^0(M)$, and a graded expectation function given by ``evaluation on the fundamental class.'' The quantum cohomology algebra is a deformation of this algebra, with the expectation function given by \[\varepsilon(\zeta)= \langle\zeta\,\mbox{\rm 1\kern-2.7pt l}\,\mbox{\rm 1\kern-2.7pt l}\rangle,\] which again can be interpreted as evaluation on the fundamental class. The induced bilinear pairing \[(\zeta_1,\zeta_1)\mapsto \varepsilon(\zeta_1\star\zeta_2) =\langle\zeta_1\,\zeta_2\,\mbox{\rm 1\kern-2.7pt l}\rangle\] coincides with the usual cup product pairing. Note that the correlation function is also determined by $\varepsilon$ and $\star$, via \[\langle\zeta_1\,\zeta_2\,\zeta_3\rangle= \varepsilon(\zeta_1\star\zeta_2\star\zeta_3).\] That is, rather than specifying the correlation function first and using it to determine the quantum product, we can simply work with the quantum product and the expectation function. For most symplectic manifolds, the Frobenius algebra structure on quantum cohomology is not graded; however, in the Calabi--Yau case we get the structure of a graded Frobenius algebra. Generally, given any associative $K$-algebra $A$ with multiplicative identity, and any linear functional $\varphi$ on $A$, the kernel of the bilinear form $(x,y)\mapsto\varphi(x*y)$ is an ideal ${\mathcal{J}_\varphi}$, and the quotient ring $A/{\mathcal{J}_\varphi}$ is a Frobenius algebra with expectation function induced by $\varphi$. If $A$ is itself a Frobenius algebra with an expectation function $\varepsilon$, then by a theorem of Nakayama \cite{Nakayama} (see \cite{Karp} for a modern discussion), $\varphi$ takes the form $\varphi(x)=\varepsilon(\alpha*x)$ for some fixed element $\alpha\in A$, and $\mathcal{J}_\varphi$ coincides with the annihilator of $\alpha$. Although the correlation functions determine the ring structure, the opposite does not hold in general---there can be many expectation functions on a given algebra. However, if $A$ is a graded Frobenius algebra of finite length as a $K$-module and all elements of $A$ have nonnegative degree, then the graded expectation functions on $A$ are in one-to-one correspondence with degree $0$ elements of $A$ which are not zero-divisors. (This is because they must all be of the form $\varphi(x)=\varepsilon(\alpha*x)$ for some $\alpha$ which is not a zero-divisor, but every element of degree ${}>0$ must be a zero-divisor.) In particular, in the case of the quantum cohomology algebra of a Calabi--Yau manifold $M$ (equipped with a symplectic structure), we have a graded Frobenius algebra of finite length in which the degree $0$ elements are just the one-dimensional vector space $H^0(M)$. This means that the graded expectation function is unique up to multiplication by an element of $K$, and that the ring structure determines the correlation functions up to this overall factor. (It is not hard to see in the Calabi--Yau case that the graded expectation function is nonzero precisely on the top degree piece $H^{2n}(M)$, where $n=\dim_{\mathbb{C}}M$, and that $H^{2n}(M)$ must also be one-dimensional.) We will see this structure again when we study the $B$-model correlation functions in lecture six. \chapter*{} \lecturename{Moduli Spaces of $\sigma$-Models} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 5. Moduli Spaces of $\sigma$-Models} \section{Calabi--Yau manifolds and nonlinear $\sigma$-models}\label{sec:51} In this lecture, we wish to give a more geometric interpretation to the $A$-model correlation functions as defined by eq.~\eqref{A:correlation}. This geometric interpretation is motivated in part by a study of the moduli spaces of the conformal field theories associated to Calabi--Yau manifolds, so we begin with a description of those moduli spaces. Let $M$ be a K\"ahler manifold with $K_M=0$. Underlying $M$ is a differentiable manifold $X$ of real dimension $2n$. We can regard $M$ as consisting of $X$ together with a chosen integrable almost-complex structure $J$ and a K\"ahler metric $g_{ij}$, such that $K_M=0$. (The complex manifold specified by $J$ will then be denoted $X_J$.) If $\omega$ denotes the K\"ahler form of the metric, then by a theorem of Calabi \cite{calabi} there is at most one Ricci-flat metric whose K\"ahler form is cohomologous to $\omega$; by a theorem of Yau \cite{yau} such a Ricci-flat metric always exists. The global holonomy of such a metric is necessarily contained in $\operatorname{SU}(n)$. (The metric being K\"ahler implies that its holonomy is contained in $\operatorname{U}(n)\subset SO(2n)$; the Ricci-flatness further restricts the holonomy to $\operatorname{SU}(n)$, and also implies that the canonical bundle is trivial. See \cite{beauville} for an account of these holonomy conditions.) We use the term {\em Calabi--Yau manifold}\/ to mean a compact connected orientable manifold $X$ of dimension $2n$ which admits Riemannian metrics whose (global) holonomy is contained in $\operatorname{SU}(n)$. You should be aware that there are some places in the literature (including papers of mine \cite{guide}) where ``Calabi--Yau manifold'' is used in the more restrictive sense of a Riemannian manifold with holonomy precisely $\operatorname{SU}(n)$. These alternate definitions will often also insist that a complex structure has been chosen on $X$. Given a Calabi--Yau manifold $X$ (in our sense) and a metric on it whose holonomy lies in $\operatorname{SU}(n)$, there always exist complex structures on $X$ for which the given metric is K\"ahler. If $h^{2,0}=0$, then there are only a finite number of such complex structures. (If the universal cover is a written as a product of indecomposable pieces, one may apply conjugation on the various factors to obtain other complex structures.) When $h^{2,0}>0$, however, the complex structures depend on parameters. There are some very interesting cases with $h^{2,0}>0$, including the famous K3 surfaces, but lack of time in these lectures forces us to assume---with regret---that $h^{2,0}=0$ henceforth. The physical model discussed in lecture one which considers maps from surfaces to a six-dimensional target space is a special case of a class of physical theories called ``nonlinear $\sigma$-models.'' One regards these as quantum field theories on the surfaces themselves, with various vertex operators and correlation functions derived from the space of maps from the surface to the target. The target should be a fixed Riemannian manifold, usually assumed to be compact. When the Riemannian metric on the target is (a particular perturbation of) one which has holonomy in $\operatorname{SU}(n)$, the resulting ``nonlinear $\sigma$-model'' is believed to be invariant under conformal transformations of the surface. It thus is a type of ``conformal field theory''---an even broader class of physical models which have a rich literature devoted to their study (see \cite{ginsparg} for an introduction and further references). Conformal field theories typically depend on finitely many parameters, and in the case of a nonlinear $\sigma$-model those parameters have a direct geometric interpretation. In the Lagrangian formulation of the theory, one must specify the metric $g_{ij}$ on the target $X$ together with an auxiliary harmonic two-form $B$ on $X$ called the ``$B$-field.'' (To simplify matters, we take our metrics to have holonomy in $\operatorname{SU}(n)$, even though the true metrics of interest in physics will be perturbations of those; we also assume that $H_2(X,\mathbb{Z})$ has no torsion.\footnote{The correct description of the moduli space will be slightly different if torsion is included---see section \ref{sec:53} below.}) The data consisting of the pair $(g_{ij},B)$ accounts for all local parameters in the conformal field theory moduli space, so we get at least a good local description of moduli if we specify such a pair. More details about these moduli spaces can be found in \cite{ICM}. Two pairs $(g_{ij},B)$ and $(g_{ij}',B')$ will determine isomorphic conformal field theories if there is a diffeomorphism $\varphi:X\to X$ such that $\varphi^*(g_{ij}')=g_{ij}$, and $\varphi^*(B')-B\in H^2_{\text{DR}}(X,\mathbb{Z})$. (We use the notation $H^k_{\text{DR}}(X,\mathbb{Z})$ to denote the image of integral cohomology in de Rham cohomology.) This second condition arises because the appearance of $B$ in the Lagrangian is always in the form $\int_\Sigma B$, and the Lagrangian is exponentiated (with an appropriate factor of $2\pi i$) in every physically observable quantity. We call the set of all isomorphism classes of such pairs the {\em semiclassical nonlinear $\sigma$-model moduli space}, or simply the {\em $\sigma$-model moduli space}\/ (for short). This may differ from the actual {\em conformal field theory moduli space}\/ for three reasons. \begin{enumerate} \item It may happen that the physical theory does not exist for all values of $g_{ij}$ and $B$. Most of the study of these theories uses perturbative methods, valid near a limit of ``large volume'' of the metric, but it may be that the theory breaks down when the volume (either of $X$, or of images of holomorphic maps into $X$) becomes too small. \item On the other hand, there may be a sort of analytic continuation of the theory beyond the region where the $\sigma$-model description is valid. (This was shown to occur in \cite{mmm,phases}.) It was only claimed above that the specification of $(g_{ij},B)$ gave good {\em local}\/ parameters for the moduli. \item Furthermore, there could be subtle isomorphisms between conformal field theories which do not show up in the $\sigma$-model interpretation. This is known to happen in the K3 surface case \cite{AM:K3}, for example (which we have no time to discuss here)---mirror symmetry provides a new identification of conformal field theories. \end{enumerate} We will ignore these phenomena for the present, and concentrate on the ``$\sigma$-model moduli space'' which parameterizes pairs $(g_{ij},B)$ modulo equivalence. To study this moduli space using the tools of algebraic geometry, we must choose a complex structure on $X$. In fact, if we consider the set of triples $(g_{ij},B,J)$ modulo equivalence, with $J$ being an integrable almost-complex structure for which the metric $g_{ij}$ is a Ricci-flat K\"ahler metric, then the map from the set of equivalence classes of triples to that of pairs is a finite map. (It is a map of degree two if the holonomy is precisely $SU(n)$.) On the other hand, we can map the set of triples $(g_{ij},B,J)$ to the moduli space $\MM_{\text{cx}}$ of complex structures on $X$. That moduli space is quite well-behaved, both locally and globally. The local structure is given by the theorem of Bogomolov--Tian--Todorov \cite{bogomolov,tian,todorov}, which says that all first-order deformations are unobstructed. (I recommend Bob Friedman's paper \cite{Friedman:threefolds} for a very readable account of this theorem.) Thus, the moduli space $\MM_{\text{cx}}$ will be smooth, and the tangent space at $[J]$ can be canonically identified with $H^1(T^{(1,0)}_{X_J})$. Globally, $\MM_{\text{cx}}$ is known to be a quasi-projective variety (if one specifies a ``polarization'') by a theorem of Viehweg \cite{viehweg}. We will study the moduli space $\MM_{\text{cx}}$ in more detail (using variations of Hodge structure) in the next section. The fibers of the map \begin{equation}\label{fibrebundle} \{(g_{ij},B,J)\}/{\sim}\ \to\ \MM_{\text{cx}}\end{equation} (from the set of equivalence classes of triples to the moduli space) are spaces of the form $\mathcal{D}/\Gamma$, with \begin{align*}\mathcal{D}&=H^2(X,\mathbb{R})+i\,\mathcal{K}_J\\ \Gamma&=H^2_{\text{DR}}(X,\mathbb{Z})\rtimes \operatorname{Aut}_J(X). \end{align*} One hopes that the map \eqref{fibrebundle} is some kind of fiber bundle (at least generically); this would require that both the family of K\"ahler cones and the family of automorphism groups are generically locally constant. This has been shown for the K\"ahler cones in the case of complex dimension three by Wilson \cite{wilson}. The tangent spaces to the fibers of the map \eqref{fibrebundle} can be canonically identified with $H^1((T^{(1,0)}_{X_J})^*)$. Mirror symmetry predicts that $X$ should have a mirror partner $Y$, such that the moduli spaces of conformal field theories on $X$ and $Y$ should be isomorphic, but with a reversal of r\^oles of $H^1(T^{(1,0)}_{X_J})$ and $H^1((T^{(1,0)}_{X_J})^*)$. That is, under the isomorphism between the conformal field theory moduli spaces, the part of the tangent space corresponding to $H^1((T^{(1,0)}_{X_J})^*)$ on $X$ should map to the part corresponding to $H^1(T^{(1,0)}_{Y_{J'}})$ on $Y$, and vice versa. In particular, the r\^oles of base and fiber in \eqref{fibrebundle} should be reversed. This is at first sight a rather peculiar statement, since the base and the fiber do not look much alike: the base $\MM_{\text{cx}}$ is a quasi-projective variety, whereas the fiber $\mathcal{D}/\Gamma$ looks much more like a Zariski open subset of a bounded domain---a typical model for the space is $(\Delta^*)^r$, where $\Delta^*$ is the punctured disk. This is in fact one of the indicators that the conformal field theory moduli space must be analytically continued beyond the realm of $\sigma$-models, as suggested in point 2 above. We will see further evidence of this at the end of lecture seven. \section{Geometric interpretation of the $A$-model correlation functions} We turn now to a geometric interpretation of the $A$-model correlation functions, which in the case of Calabi--Yau manifolds will turn out to be closely related to the spaces $\mathcal{D}/\Gamma$ described above. In the previous lecture, the symbols $q^\eta$ were treated purely formally, which allowed us to discuss some algebraic aspects of quantum cohomology. Now, however, we would like to make the new product more geometric by giving specific values to the $q^\eta$'s, thereby making the quantum cohomology ring into a deformation of the usual cohomology ring. Turning algebraic parameters into geometric data is a familiar task for algebraic geometers; however here, we only have formal parameters. We will describe a natural parameter space as a formal completion of a certain geometric space---if some day someone proves that the series \eqref{A:correlation} and \eqref{A:binary} are convergent power series, then the true parameter space will be a neighborhood (in the classical topology) of the completion point within the geometric space which we will construct. Let $\mathcal{R}=\mathbb{Q}[[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]]$ be the formal semigroup ring of the integral Mori semigroup. If $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$ is finitely generated, then we can take as the geometric space $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ (the spectrum of the semigroup ring), and as its completion the formal scheme $\operatorname{Spf} {\mathcal{R}}_{\mathbb{C}}$, where ${\mathcal{R}}_{\mathbb{C}}$ denotes ${\mathcal{R}}\otimes_{\mathbb{Q}}\mathbb{C}$ and $\operatorname{Spf}$ is the formal spectrum. More generally, if the ring of $\operatorname{Aut}_J(X)$-invariants ${\mathcal{R}}^{\operatorname{Aut}_J(X)}$ is the formal completion of a ring of finite type over $\mathbb{Q}$, we take our completed parameter space to be $\operatorname{Spf} ({\mathcal{R}}_{\mathbb{C}}^{\operatorname{Aut}_J(X)})$. In the finitely generated case, this geometric space $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ is in a natural way an affine toric variety, and as such admits a rather concrete description: the geometric points are in one-to-one correspondence with the set of semigroup homomorphisms $\operatorname{Hom}_{\text{sg}}(\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z}),\mathbb{C})$, where $\mathbb{C}$ is given the structure of a {\em multiplicative}\/ semigroup. Any geometric point $\xi$ in the parameter space---regarded as a semigroup homomorphism---specifies compatible values $\xi(q^\eta)$ for the symbols $q^\eta$. An important open problem is to decide for which $\xi$ the series expressions \eqref{A:correlation} for the correlation functions converge. If convergent, the correlation functions would become actual $\mathbb{C}$-valued functions on a parameter space (as expected by the physicists), which would be an open subset of $\operatorname{Spec}\mathbb{C}[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]$ in the classical topology. To make this even more concrete, consider the case in which the Mori semigroup is freely generated by elements $e_1$, \dots, $e_r$ which also serve as a basis of the lattice $H_2(X,\mathbb{Z})$. In this case, we can define $q_j:=q^{e_j}$, and write the ring $\mathcal{R}$ as a formal power series ring ${\mathcal{R}}=\mathbb{Q}[[q_1,\dots,q_r]]$. The geometric space $\operatorname{Spec}\mathbb{C}[q_1,\dots,q_r]$ can then be identified as $\mathbb{C}^r$ with coordinates $q_1,\dots q_r$. One natural candidate for the open set on which the correlation functions might converge is \[\{(q_1,\dots,q_r)\in\mathbb{C}^r\ |\ 0\le|q_j|<1\}.\] More generally, still assuming that $H_2(X,\mathbb{Z})$ is torsion-free, suppose we choose a basis $e_1$, \dots, $e_r$ whose span as a semigroup {\em contains}\/ $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$. Then the corresponding formal power series ring $\mathbb{Q}[[q_1,\dots,q_r]]$ contains our coefficient ring $\mathcal{R}$. If we let $\sigma$ denote the open real cone generated by the dual basis $e^1$, \dots, $e^r$, then that formal power series ring can be more canonically described as the formal semigroup ring $\mathcal{R}_\sigma:=\mathbb{Q}[[q;\check\sigma\cap H_2(X,\mathbb{Z})]]$. The same cone $\sigma$ can be used to give a canonical description of the open set specified by $0<|q_j|<1$ in the form \[(H^2(X,\mathbb{R})+i\sigma)/H^2(X,\mathbb{Z}).\] (To see this, write a general element of $H^2(X,\mathbb{C})$ modulo $H^2(X,\mathbb{Z})$ in the form \[\frac1{2\pi i}\sum(\log q_j)e^j,\] and note that the condition $0<|q_j|<1$ is equivalent to $\Im(\frac1{2\pi i}\log q_j)>0$.) The Mori semigroup $\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})$ will be contained in the semigroup spanned by $\{e_j\}$ precisely when the cone $\sigma$ is contained in the K\"ahler cone of $X_J$. We will treat such a choice of cone $\sigma$ as specifying a coordinate chart on the geometric space we are trying to construct. For any such cone, we define \[\mathcal{D}_\sigma=H^2(X,\mathbb{R})+i\,\sigma\subset H^2(X,\mathbb{C})\] In terms of local coordinates, as pointed out above we have \[\mathcal{D}_\sigma/H^2(X,\mathbb{Z})=\{(q_1,\dots,q_r)\ |\ 0<|q_j|<1\}.\] The open subset of our desired geometric space will be a partial compactification of this, defined by \[(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-=\{(q_1,\dots,q_r)\ |\ 0\le|q_j|<1\}.\] We call the origin $0\in(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$ the {\em distinguished limit point}\/ in this space. It is hoped that the expressions for the $A$-model correlation functions, or for the binary operation $\zeta_1\star\zeta_2$, will converge in a neighborhood of the distinguished limit point $0$ in $(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$. The different possible choices of $\sigma$ will correspond to operations---such as blowing up the boundary---which change the compactification without changing the underlying space. Intrinsically, we can describe $\mathcal{R}_\sigma\otimes\mathbb{C}$ as the formal completion of the local ring of $(\mathcal{D}_\sigma/H^2(X,\mathbb{Z}))^-$ at its distinguished limit point $0$. The geometric space which is emerging from this discussion is very closely related to the space $\mathcal{D}/\Gamma$ which formed part of the nonlinear $\sigma$-model moduli space in the case of a Calabi--Yau manifold with $h^{2,0}=0$. In fact, if $\mathcal{K}_J$ is the K\"ahler cone of such a Calabi--Yau manifold which can be partitioned into cones $\sigma_\alpha$ which are spanned by various bases of $H^2(X,\mathbb{Z})$, then $\mathcal{D}/H^2(X,\mathbb{Z})$ is the interior of the closure of the union of the sets $\mathcal{D}_{\sigma_\alpha}/H^2(X,\mathbb{Z})$. Ideally, one could make such a partition in an $\operatorname{Aut}_J(X)$-equivariant way. This would be guaranteed by the following conjecture. \begin{ConeConjecture} Let $X$ be a Calabi--Yau manifold on which a complex structure $J$ has been chosen, and suppose that $h^{2,0}(X)=0$. Let $\mathcal{K}_J$ be the K\"ahler cone of $X$, let $(\mathcal{K}_J)_+$ be the convex hull of $\overline{\mathcal{K}}_J\cap H^2(X,\mathbb{Q})$, and let $\operatorname{Aut}_J(X)$ be the group of holomorphic automorphisms of $X$. Then there exists a rational polyhedral cone $\Pi\subset(\mathcal{K}_J)_+$ such that $\operatorname{Aut}_J(X).\Pi=(\mathcal{K}_J)_+$. \end{ConeConjecture} A nontrivial case of this conjecture---Calabi--Yau threefolds which are fiber products of generic rational elliptic surfaces with section (as studied by Schoen \cite{schoen})---has been checked by Grassi and the author \cite{GM}. There are some other pieces of supporting evidence in examples worked out by Borcea \cite{borcea} and Oguiso \cite{oguiso}. When this conjecture holds, there is a partial compactification of $\mathcal{D}/\Gamma$ constructed in \cite{compact} by gluing together the spaces $(\mathcal{D}_{\sigma_\alpha}/H^2(X,\mathbb{Z}))^-$ for an $\operatorname{Aut}_J(X)$-equivariant partitioning of $\mathcal{K}_J$, and modding out by $\operatorname{Aut}_J(X)$. This produces a ``semi-toric'' partial compactification of the type introduced by Looijenga \cite{Looijenga}. Because it is covered by explicit coordinate charts, this is a convenient type of compactification for making comparisons of correlation functions. There is also a ``minimal'' semi-tori compactification determined from the same data, which partially compactifies $\mathcal{D}/\Gamma$ more directly, adding several new strata but only a single stratum of maximal codimension (the analogue of the ``distinguished limit points''). When the cone conjecture holds, the ring of invariants $\mathcal{R}^{\operatorname{Aut}_J(X)}$ is the formal completion of a ring of finite type over $\mathbb{Q}$, and the completion of the local ring of the minimal semi-toric compactification at its distinguished point $P$ coincides with $\operatorname{Spf}(\mathcal{R}_{\mathbb{C}}^{\operatorname{Aut}_J(X)})$. On such a compactification, we will expect \begin{equation}\label{eq:limA} \lim_{Q\to P}\langle\zeta_1\,\zeta_2\,\zeta_3\rangle_Q =(\zeta_1\cup\zeta_2\cup\zeta_3)|_{[X]} \end{equation} (the ``$q_j=0$ values'' in coordinate charts). Such a point is called a ``semiclassical limit'' in the physics literature \cite{AL}. \section{The r\^ole of torsion in the moduli space}\label{sec:53} Up to this point, we have not considered the effects of possible torsion in $H_2(X,\mathbb{Z})$ and in fact we have explicitly assumed at several points that there was no torsion. If torsion is present, we can define the formal semigroup ring $\mathcal{R}=\mathbb{Q}[[q;\mathop{\overline{\text{NE}}}\nolimits(X_J,\mathbb{Z})]]$ as before, and it will have a torsion part ${\mathcal{R}}_{\text{torsion}}$ whose spectrum is a finite set of geometric points. This can be identified with the set of connected components of our parameter space. It can also be seen in the following description of the $\sigma$-model moduli space. The complete description of the $\sigma$-model moduli space (with the torsion included) considers the quantity $e^{2\pi i(B+i\omega)}$ to lie in $\operatorname{Hom}(H_2(X,\mathbb{Z}),\mathbb{C}^*)$. This can be thought of concretely as having a torsion part, together with a free part which lies in the space \[\operatorname{Hom}(H_2(X,\mathbb{Z})/\text{torsion},\mathbb{C}^*)\cong H^2(X,\mathbb{C}^*)\cong H^2(X,\mathbb{C})/H^2_{\text{DR}}(X,\mathbb{Z})\] where (as in section \ref{sec:51}) $H^2_{\text{DR}}(X,\mathbb{Z})$ is the image of $H^2(X,\mathbb{Z})$ in de Rham cohomology, isomorphic to $H^2(X,\mathbb{Z})/\text{torsion}$. A representative of the free part can be written as $B_{\text{free}}+i\omega\in H^2(X,\mathbb{C})$, where $\omega$ is the K\"ahler form and $B_{\text{free}}$ is the real two-form which appeared in section \ref{sec:51}. The torsion part of $e^{2\pi i(B+i\omega)}\in\operatorname{Hom}(H_2(X,\mathbb{Z}),\mathbb{C}^*)$ can be identified with the torsion part of our coefficient ring ${\mathcal{R}}_{\text{torsion}}$ from the algebraic interpretation. One way to interpret this ``$B$-field with torsion included'' is to regard it as an element of $H^2(X,\mathbb{R}/\mathbb{Z})$. \chapter*{} \lecturename{Variations of Hodge Structure} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 6. Variations of Hodge Structure} \section{The $B$-model correlation functions} Our goal in this lecture is to describe the $B$-model correlation functions and how they are related to variations of Hodge structure. We work with Calabi--Yau manifolds on which complex structures have been chosen. That is, we let $W$ be a complex manifold with $K_W=0$. The assumption of trivial canonical bundle is needed in order to define the $B$-model correlation functions. Let us define \[H^{-p,q}(W):=H^q(\Lambda^p(T^{(1,0)}_W)),\] and consider all of these groups together: \[H^{-*}(W):=\bigoplus_{p,q}H^{-p,q}(W).\] There is a natural ring structure on $H^{-*}(W)$ which can be thought of as a sheaf cohomology version of the cup product pairing: \[H^q(\Lambda^p(T^{(1,0)}_W))\otimes H^{q'}(\Lambda^{p'}(T^{(1,0)}_W)) \to H^{q+q'}(\Lambda^{p+p'}(T^{(1,0)}_W)).\] Note that since these are sheaf cohomology groups, this ring structure is not ``topological'' in nature; in fact, it depends heavily on the choice of complex structure on $W$. Recall that in the case of the $A$-model correlation functions on a symplectic manifold $M$, the expectation function which determined the Frobenius algebra structure was a very familiar object, given by evaluating a cohomology class on the fundamental class of $M$ (which determines a canonical map $H^{n,n}(M)\to\mathbb{C}$). By contrast, the ring structure on quantum cohomology was unusual. In this new ``$B$-model'' case, however, the ring structure is straightforward but the expectation function is more elusive. To define it, we must choose a nonvanishing global section $\Omega^{\otimes2}$ of $(K_W)^{\otimes2}$. This is then used in two steps to specify the expectation function: \[H^{-n,n}(W)=H^n(\Lambda^n(T^{(1,0)}_W)) \overset{\lhk\,\Omega}{\longrightarrow} H^n(\O_W)\cong\left(H^0(K_W)\right)^* \overset{\otimes\Omega}{\longrightarrow} \mathbb{C},\] where the middle isomorphism is Serre duality. Using this expectation function and the ``sheaf cup product'' binary operation, we define the $B$-model correlation functions (in the standard way from the Frobenius algebra structure): \[\langle\beta_1\,\beta_2\,\beta_3\rangle= ((\beta_1\cup\beta_2\cup\beta_3)\lhk\,\Omega)\otimes\Omega.\] (Once again we have a definition which is inspired by the outcome of a calculation in the physics literature \cite{strwit}.) This gives a map \[H^{-p,q}(W)\times H^{-p',q'}(W)\times H^{-(n-p-p'),n-q-q'}(W)\to\mathbb{C}.\] Note that as in the $A$-model case, we actually have a graded Frobenius algebra of finite length, so the expectation function is uniquely defined up to a scalar multiple (which can be absorbed in the choice of $\Omega^{\otimes 2}$.) In order to relate this correlation function to a more familiar mathematical object, we can proceed as follows: first use the two $\Omega$'s to transform two of the arguments, and then use the cup product: \[\langle\beta_1\,\beta_2\,\beta_3\rangle= ((\beta_1\lhk\,\Omega)\cup\beta_2\cup(\beta_3\lhk\,\Omega)).\] This variant of the correlation function can be regarded as a map \[H^{n-p,q}(W)\times H^{-p',q'}(W)\times H^{p+p',n-q-q'}(W)\to\mathbb{C},\] or, if we treat it as a modified ``binary operation,'' as a map \[H^{n-p,q}(W)\times H^{-p',q'}(W)\to H^{n-p-p',q+q'}(W).\] This version of the ``binary operation'' expresses the cohomology $H^*(W)$ as a module over the ring $H^{-*}(W)$. As we shall see, this variant has the pleasant property that it can be directly interpreted in terms of variations of Hodge structure and the differential of the period map. Of course, the original version of the correlation function can be recovered from this, once we have specified $\Omega^{\otimes 2}$. \section{Variations of Hodge structure} We now briefly review the theory of variations of Hodge structure, in order to explain the mathematical origin of the $B$-model correlation functions. Variations of Hodge structure were introduced as a tool for measuring how the complex structure on a differentiable manifold can vary. Good general references for this are Griffiths et al.~\cite{transcendental}, and Schmid \cite{schmid}. There are two primary ways one can view deformations of complex structure. In the first viewpoint, we fix a compact differentiable manifold $Y$, and consider various integrable almost-complex structures $J$ on $Y$. Then the set of such, modulo diffeomorphism, is known to be a finite-dimensional space. In the second viewpoint, we consider proper holomorphic maps $\pi:\mathcal{W}\to S$ with $W_s=\pi^{-1}(s)$ diffeomorphic to $Y$. Each fiber $W_s$ has an induced structure of a complex manifold. If $S$ is contractable, then $\pi$ can be trivialized in the $C^\infty$ category, and we can regard $\pi$ as specifying a family of complex structures. One wants to represent the functor \[S\mapsto\{\pi:\mathcal{W}\to S\}/(\text{isomorphism}),\] by maps to a moduli space which has a ``universal family.'' This is generally too much to hope for, but there are often ``coarse moduli spaces'' whose points are in one-to-one correspondence with the possible complex structures. (The appendices in \cite{MumfordFogarty} provide good background for moduli problems in general.) We will study complex structures on $Y$ by studying the Hodge decomposition induced on cohomology by each choice of complex structure. In general, if $W_s$ is a K\"ahler manifold there is a {\em Hodge decomposition}\/ of the cohomology: \begin{equation}\label{eq:hodge} H^k(W_s,\mathbb{C})\cong \bigoplus_{p+q=k}H^{p,q}(W_s). \end{equation} Now in a family over a contractable base, the bundle of $H^k(W_s,\mathbb{C})$'s may be canonically trivialized. Over more general bases $S$ (assumed to be connected), it is convenient to consider $R^k\pi_*\mathbb{C}_\mathcal{W}$, which is simply the sheaf whose local sections are topologically constant families of cohomology classes. This sheaf has the structure of a {\em local system}: it can be characterized by its fiber $H^k(W_s,\mathbb{C})$ at a particular point $s\in S$ together with a representation of the fundamental group \[\rho:\pi_1(S,s)\to\operatorname{Aut}(H^k(W_s,\mathbb{C}))\] which specifies what happens when the locally constant sections are followed around loops. There is useful dictionary \cite{rsp} between local systems and pairs $({\mathcal{H}},\nabla)$ consisting of a holomorphic vector bundle ${\mathcal{H}}$ on $S$ and a flat holomorphic connection \[\nabla:\mathcal{H}\to(T^{(1,0)}_S)^*\otimes\mathcal{H}.\] The way the dictionary works is this: given a local system $\mathbb{H}$, define $\mathcal{H}=\O_S\otimes\mathbb{H}$, and $\nabla(\sum \varphi_j h^j)=\sum d\varphi_j\otimes h^j$ for $\{h^j\}$ a local basis of sections of $\mathbb{H}$. Conversely, given $(\mathcal{H},\nabla)$, define $\Gamma(U,\mathbb{H})=\{h\in\Gamma(U,\mathcal{H})\ |\ \nabla(h)=0\}$ for every open set $U$. In the case of the cohomology local system $R^k\pi_*{\mathbb{C}}_{\mathcal{W}}$, the associated connection $\nabla$ on $\mathcal{H}^k$ is called the {\em Gauss--Manin connection}. An explicit version of this Gauss--Manin connection goes like this: if we choose a local basis $\alpha^1,\dots,\alpha^r$ for the space of sections $\Gamma(U,R^k\pi_*{\mathbb{C}}_{\mathcal{W}})$, then any $\beta(s)\in\Gamma(U,{\mathcal{H}}^k)$ can be written $\beta(s)=\sum f_j(s)\alpha^j$ for some coefficient functions $f_j\in\Gamma(U,{\O}_S)$. Then \[\nabla(\beta)=\sum df_j\otimes \alpha^j \in\Gamma(U,(T^{(1,0)}_S)^*\otimes{\mathcal{H}^k}).\] This can be given an interpretation in terms of classical ``period integrals'' as follows. The basis $\alpha^1,\dots,\alpha^r$ is dual to some basis $\gamma_1,\dots,\gamma_r\in H_k(W_{s_0},{\mathbb{C}})$. Then the coefficient functions are the period integrals $f_j(s)=\int_{\gamma_j}\beta(s)$. (We use integration to denote the pairing between homology and cohomology.) The great advantage of expressing everything in terms of the Gauss--Manin connection is that the Gauss--Manin connection can be computed algebraically, without knowing the topological cycles in advance. Although the sheaf $R^k\pi_*\mathbb{C}_{\mathcal{W}}$ of cohomology groups can be locally trivialized over the base $S$, the Hodge decomposition \eqref{eq:hodge} will vary as we vary the complex structure. The properties of this variation are more conveniently expressed using the {\em Hodge filtration}: \[F^p(W_s):=\bigoplus_{p'\ge p}H^{p',k-p'}(W_s)\subset H^k(W,\mathbb{C})\] rather than the Hodge groups $H^{p,q}(W_s)$ directly. The spaces $F^p(W_s)$ in the Hodge filtration vary holomorphically with parameters, fitting together to form a holomorphic subbundle $\mathcal{F}^p\subset\mathcal{H}^k$. One might also try to construct a bundle of $H^{p,q}$'s by the simple procedure \[{\mathcal{H}}^{p,q}_{C^\infty}:=\bigcup_{s\in S} H^{p,q}(W_s)\subset{\mathcal{H}^k}.\] As the notation indicates, this defines a $C^\infty$ bundle, but it is not in general holomorphic. There is a holomorphic bundle ${\mathcal{H}}^{p,k-p}$ defined by the exact sequence \begin{equation}\label{nonsplit} 0\to{\mathcal{F}}^{p+1}\to{\mathcal{F}}^p\to {\mathcal{H}}^{p,k-p}\to0, \end{equation} but this exact sequence {\em has no canonical splitting}, and $\mathcal{H}^k$ cannot in general be written as a direct sum of these holomorphic $\mathcal{H}^{p,k-p}$ bundles. The key property satisfied by the Hodge bundles is known as {\em Griffiths transversality}: when we differentiate with respect to parameters by using the Gauss--Manin connection, the Hodge filtration only shifts by one, i.e., \[\nabla(\mathcal{F}^p)\subset(T^{(1,0)}_W)^*\otimes \mathcal{F}^{p-1}.\] To study the totality of complex structures on $W$, we can map the moduli space, or any parameter space $S$ for a family, to the classifying space for Hodge structures. Each Hodge structure on a fixed vector space $H^k$ determines a point in a flag variety \[\operatorname{Flags}_{(f_j)}:=\{\{0\}\subset F^k\subset\cdots\subset F^0= H^k\ |\ \dim F^j=f_j\},\] with the $f_j$'s specifying the dimensions of the spaces making up the filtration. The group $\operatorname{GL}(f_0,\mathbb{C})$ acts transitively on such flags, and if we fix a reference flag $F_0^{\scriptscriptstyle\bullet}$, then the flag variety can be described as $\operatorname{GL}(f_0,\mathbb{C})/\operatorname{Stab}(F_0^{\scriptscriptstyle\bullet})$. (The stabilizer $\operatorname{Stab}(F_0^{\scriptscriptstyle\bullet})$ is the group of block lower triangular matrices.) There are some additional conditions which should be imposed to get a good Hodge structure (cf.~\cite{transcendental,schmid}); these restrict us to an open subset $\mathcal{U}$ of a subvariety\footnote{We must pass to a subvariety to restrict to the so-called {\it polarized}\/ Hodge structures---see \cite{transcendental} or \cite{deligne} for an explanation of this.} of the flag variety on which a discrete group $\Gamma$ acts, and the desired classifying space for Hodge structures is $\mathcal{U}/\Gamma$. The classifying map $S\to\mathcal{U}/\Gamma$ for a family is often referred to as the {\em period map}. The tangent space to the flag variety can be described as \[\bigoplus_j \operatorname{Hom}(F^j/F^{j+1},H^k/F^j).\] So another way of stating Griffiths transversality is to say that the differential of the period map $S\to\mathcal{U}/\Gamma$ sends $T^{(1,0)}_S$ to the subspace \[\bigoplus_j \operatorname{Hom}(F^j/F^{j+1},F^{j-1}/F^j) =\bigoplus_j \operatorname{Hom}(H^{j,k-j}(W_s),H^{j-1,k+1}(W_s))\] of the tangent space. The differential of the map $S\to\operatorname{Flags}_{(f_j)}$ factors through a map $T^{(1,0)}_S\to H^1(T^{(1,0)}_W)$ which describes the first-order deformations represented by $S$ at $[W]$. The map which then induces the differential is the map \begin{equation}\label{eq:differential} H^1(T^{(1,0)}_W)\to \bigoplus_j \operatorname{Hom}(H^{j,k-j}(W),H^{j-1,k+1}(W)) \end{equation} given by sheaf cup product. The success of this approach to studying the moduli of complex structures derives from the {\em local Torelli theorem}\/ for Calabi--Yau manifolds, which states that the map \eqref{eq:differential} is injective. This means that at least locally, the moduli space can be accurately described by using variations of Hodge structure. However, that same map can now be given a new interpretation, as a $B$-model correlation function. That is, {\em the $B$-model correlation function \[H^1(T^{(1,0)}_W)\times H^{j,k-j}(W)\to H^{j-1,k+1}(W)\] coincides with the differential of the period map!} We now restrict our attention to the middle-dimensional cohomology $H^n(W,\mathbb{C})$. Stated in terms of the Gauss--Manin connection, we find the following ``bundle version'' of our correlation function \cite{guide}: given a vector field $\theta$ on the moduli space and sections $\alpha\in\mathcal{F}^j$, $\beta\in\mathcal{F}^{j-1}$, the correlation function is \[\langle\theta\,\alpha\,\beta\rangle=\int_W\nabla_\theta(\alpha)\wedge\beta\] (where $\nabla_\theta=\theta\lhk\nabla$ denotes the directional derivative in direction $\theta$). However, as used in physics the correlation function is a specific function rather than a map between bundles. To find this interpretation, we will need to choose specific sections of these bundles on which to evaluate the map. It is this issue to which we now turn. \section{Splitting the Hodge filtration} Our method for specifying sections of the Hodge bundles will be given in terms of a choice of splitting for the Hodge filtration on the middle-dimensional cohomology $H^n(W,\mathbb{C})$, i.e., a set of splittings of the exact sequences (\ref{nonsplit}) (but defined only locally in the parameter space). We determine such a splitting by means of a filtration on {\em homology}, which we think of as specifying ``which periods to calculate.'' Let ${\mathbb{S}}_{\scriptscriptstyle\bullet}$ be a filtration of the homology local system $\operatorname{Hom}(R^n\pi_*{\mathbb{C}}_{\mathcal{W}},{\mathbb{C}}_S)$ by sub-local systems, and let \[{\mathbb{S}}^\ell:=\operatorname{Ann}({\mathbb{S}}_{\ell-1}):= \{\alpha\in \mathcal{H}^n\ |\ \int_\gamma\alpha=0\ \forall\ \gamma\in{\mathbb{S}}_{\ell-1}\}.\] be the associated filtration of annihilators of $\mathbb{S}_{\scriptscriptstyle\bullet}$ in cohomology. We say that $\mathbb{S}_{\scriptscriptstyle\bullet}$ is a {\em splitting filtration for $\mathcal{F}^{\scriptscriptstyle\bullet}$}\/ if $({\mathcal{H}}^n)_s \cong ({\mathcal{F}}^p)_s\oplus({\mathbb{S}}^{n-p+1})_s$ for every $s\in S$ and for every $0\le p\le n$. (In this case, $\mathbb{S}^{\scriptscriptstyle\bullet}$ and $\mathcal{F}^{\scriptscriptstyle\bullet}$ are called {\em opposite filtrations of weight $n$}\/ \cite{deligne}.) One way of producing examples of splitting filtrations is as follows: fix a point $s\in S$, and consider the conjugate of the Hodge filtration at $s_0$, namely, $\overline{F^q}_{s_0}$. The ``opposite'' property for these filtrations is easy to check: by definition \begin{align*}({\mathcal{F}}^p)_s&=H^{n,0}(W_s)\oplus\cdots\oplus H^{p,n-p}(W_s)\\ \intertext{and so} (\overline{{\mathcal{F}}^{n-p+1}})_s&= \overline{H^{n,0}(W_s)\oplus\cdots\oplus H^{n-p+1,p-1}(W_s)}\\ &=H^{0,n}(W_s)\oplus\cdots\oplus H^{p-1,n-p+1}(W_s), \end{align*} where we have used the fact that $\overline{H^{p,q}(W_s)}=H^{q,p}(W_s)$. The Gauss--Manin connection can be used to extend this from a filtration at one point to a filtration of the local system. Although this filtration only coincides with the conjugate of the Hodge filtration at one point in the parameter space, it remains opposite to the Hodge filtration at all points nearby. Given a splitting filtration ${\mathbb{S}}_{\scriptscriptstyle\bullet}$, we define \[{\mathcal{H}}^{p,q}_{\mathbb{S}}:={\mathcal{F}}^p\cap\operatorname{Ann}({\mathbb{S}}_{q-1}),\] on any open set on which $\mathbb{S}_{\scriptscriptstyle\bullet}$ is single-valued. Then \[{\mathcal{H}}=\bigoplus_{p=0}^n {\mathcal{H}}^{p,q}_{\mathbb{S}} \quad \text{and} \quad \mathcal{F}^p=\bigoplus_{p'\ge p}\mathcal{H}^{p',n-p'}_\mathbb{S}.\] (This is the promised splitting of the Hodge filtration.) More concretely, this space can be described in terms of conditions on the periods as follows. The sections of ${\mathcal{H}}^{p,q}_{\mathbb{S}}$ over $U$ are \[\Gamma(U,\mathcal{H}^{p,q}_{\mathbb{S}}):= \{\beta\in\Gamma(U,{\mathcal{F}}^p)\ |\ \int_\gamma\beta=0\ \forall\ \gamma\in{\mathbb{S}}_{q-1}\}.\] We also define a space of {\em distinguished sections}\/ of ${\mathcal{H}}^{p,q}_{\mathbb{S}}$ by \[\Gamma(U,{\mathcal{H}}^{p,q}_{\mathbb{S}})_{\text{dist}}:= \{\beta\in\Gamma(U,{\mathcal{H}}^{p,q}_{\mathbb{S}})\ |\ d\left(\int_\gamma\beta\right) =0\ \forall\ \gamma\in{\mathbb{S}}_{q}\}.\] (That is, the period integrals $\int_\gamma\beta$ are constant for all $\gamma\in{\mathbb{S}}_{q}$, and vanish for all $\gamma\in\mathbb{S}_{q-1}$.) For each ${\mathbb{S}}_{\scriptscriptstyle\bullet}$, then, we can define specific $B$-model correlation functions, using the $\Omega$ coming from the distinguished section of ${\mathcal{H}}^{n,0}_{\mathbb{S}}$ (which is well defined up to a complex scalar multiple). This has the advantage that the correlation functions have been turned into actual functions on a parameter space (in accord with the physicists' interpretation) rather than sections of a bundle. The disadvantage is that further parameters---in the form of a choice of splitting---have been introduced. However, the necessity of considering further parameters such as these, on which the correlation functions will depend anti-holomorphically rather than holomorphically, was recently realized in the physics literature \cite{t:tstar}. In addition to the distinguished $n$-form $\Omega$, our choice of splitting determines a family of distinguished vector fields which when contracted with $\Omega$ yield the distinguished sections of $\mathcal{H}^{n-1,1}_{\mathbb{S}}$. These vector fields can be integrated into {\em canonical coordinates}, well-defined up to a $\operatorname{GL}(r,{\mathbb{C}})$ transformation. (The flexibility of that final $\operatorname{GL}(r,\mathbb{C})$ choice comes from the constants of integration, which must also be specified in order to completely determine a set of canonical coordinates.) A bit more explicitly, if $\gamma_0$ spans ${\mathbb{S}}_0$ and $\gamma_0, \gamma_1,\dots,\gamma_r$ span ${\mathbb{S}}_1$, then the distinguished $\Omega$ satisfies $\int_{\gamma_0}\Omega=\text{constant}$, and the coordinates are given by \[\int_{\gamma_1}\Omega,\dots,\int_{\gamma_r}\Omega.\] If we start with an arbitrary $n$-form $\widetilde\Omega$, we can write the distinguished $n$-form as \[\Omega:=\frac{\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}\] and the canonical coordinates as \[\frac{\int_{\gamma_1}\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}, \dots, \frac{\int_{\gamma_r}\widetilde\Omega}{\int_{\gamma_0}\widetilde\Omega}.\] This is the most general possible form for canonical coordinates (and a distinguished $n$-form) needed for the physical theory, according to recent work in physics \cite{BCOV:KS}. Let us fix a splitting filtration ${\mathbb{S}_{\scriptscriptstyle\bullet}}$. Consider a basis $\{\beta^i\}$ of ${\mathcal{H}^n}$ consisting of distinguished sections of the bundles ${\mathcal{H}}_{\mathbb{S}}^{p,q}$ (ordered so that the basis is also adapted to the Hodge filtration ${\mathcal{F}}^{{\scriptscriptstyle\bullet}}$), and a multi-valued basis $\{\gamma_j\}$ of the homology local system $\operatorname{Hom}(R^n\pi_*{\mathbb{C}}_{\mathcal{W}},{\mathbb{C}}_S)$, adapted to the splitting filtration ${\mathbb{S}_{\scriptscriptstyle\bullet}}$. Then the period matrix $(\int_{\gamma_j}\beta^i)$ (which has multi-valued entries) will take a block upper triangular form with constant diagonal blocks. And if we calculate the connection matrix in the basis $\{\beta^i\}$, it takes the special form \[\begin{pmatrix} 0&A^1_0&0&&\cdots&0\\ &0&A^1_1&0&\cdots&0\\ &&\ddots&\ddots&&\vdots\\ \vdots&\vdots&&&0&A^1_{n-1}\\ 0&0&\cdots&&&0 \end{pmatrix}\] in which the only nonzero entries are in the first block superdiagonal of the matrix. The entries $A^1_j$ precisely contain the data for the $B$-model correlation functions, calculated in our distinguished basis. \chapter*{} \lecturename{The $A$-Variation of Hodge Structure} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 7. The $A$-Variation of Hodge Structure} \section{Variations of Hodge structure near the boundary of moduli} In this lecture, we begin by reviewing the asymptotic behavior of a variation of Hodge structure near the boundary of moduli space, and the behavior of the $B$-model correlation functions there. Comparing to the $A$-model correlation functions will reveal some similarities---this is one of the hints of mirror symmetry. We make the similarities even more apparent by using the $A$-model correlation functions to construct a new variation of Hodge structure, which we call the $A$-variation of Hodge structure. Let $S=(\Delta^*)^r\subset \overline{S}=\Delta^r$, and suppose we are given a family $\pi:\mathcal{W}\to S$ of complex manifolds. We will assume that there is a way to complete this to a family $\bar\pi:\overline{\mathcal{W}}\to\overline{S}$ in which $\bar\pi$ is still proper (but no longer smooth). Thus, $0\in\overline{S}$ is a boundary point in the parameter space. Pick a basepoint $s\in S$; then the fundamental group $\pi_1(S,s)$ is generated by loops $\gamma^{(1)}$,\dots,$\gamma^{(r)}$ with $\gamma^{(j)}$ homotopic to the standard generator of $\pi_1(\Delta^*_j)$, where $\Delta^*_j$ is the $j^{\text{th}}$ factor in $(\Delta^*)^r$. \begin{MonodromyTheorem}[Landman \cite{monodromy}] The action of each generator $\gamma^{(j)}$ gives a quasi-unipotent automorphism $T^{(j)}$ of $H^k(W_s,\mathbb{Q})$, i.e., $(((T^{(j)})^{b_j}-I)^{r_j}=0$. (This is called {\em unipotent}\/ if $b_j=1$.) \end{MonodromyTheorem} We will restrict attention to the unipotent case. This is partially for technical convenience, but in fact, in the examples which have been calculated for mirror symmetry purposes, only unipotent monodromy transformations have played a r\^ole. When $T^{(j)}$ is unipotent, its logarithm can be defined by the following sum (which is finite). \[N^{(j)}:=\log T^{(j)}:= (T^{(j)}-I) - \frac12\,(T^{(j)}-I)^2+\cdots.\] (Note that the $T^{(j)}$'s and $N^{(j)}$'s all commute.) Let $z_1,\dots,z_r$ be coordinates on $\overline{S}$, with $z_j$ a coordinate on the $j^{\text{th}}$ disk. Consider the operator \begin{multline*} \mathcal{N}:=\exp\left(-\frac1{2\pi i}\sum\,\log z_j\,N^{(j)}\right)=\\ I + \left(-\frac1{2\pi i}\sum\, \log z_j\,N^{(j)}\right) + \frac1{2!}\left(-\frac1{2\pi i}\sum\,\log z_j\,N^{(j)}\right)^2+\cdots \end{multline*} (also a finite sum). For any section $e$ of the local system $R^k\pi_*(\mathbb{C}_\mathcal{W})$, a simple calculation shows that \begin{equation}\label{eq:GMext} \nabla(\mathcal{N} (e)) = -\frac1{2\pi i}\sum\frac{dz_j}{z_j}\,N^{(j)}(e). \end{equation} The key facts about the asymptotic behavior are as follows. \begin{NilpotentOrbitTheorem}[Schmid \cite{schmid}] Assume that each monodromy transformation $T^{(j)}$ is unipotent. Let $e_1(s),\dots,e_r(s)$ be a multi-valued basis of $R^k\pi_*(\mathbb{C}_\mathcal{W})$, and let $\eta_\ell:=\mathcal{N}(e_\ell)$. Then each $\eta_\ell$ is a single-valued section of $\mathcal{H}^k$ on $S$, and together they can be used to generate an extension $\overline{\mathcal{H}}^k$ of $\mathcal{H}^k$ to $\overline{S}$. By eq.~\eqref{eq:GMext}, the Gauss--Manin connection extends to a connection on $\overline{\mathcal{H}}^k$ (again denoted by $\nabla$) with {\em regular singular points}, i.e., the extended connection is a map \[\nabla:\overline{\mathcal{H}}^k\to (T^{(1,0)}_{\overline{S}})^*(\log B)\otimes \overline{\mathcal{H}}^k\] where $(T^{(1,0)}_{\overline{S}})^*(\log B)$ is the free $\O_{\overline{S}}$-module generated by $\frac{dz_j}{z_j}$, $j=1,\dots,r$. Moreover, the Hodge bundles $\mathcal{F}^p$ have locally free extensions to subbundles $\overline{\mathcal{F}}^p\subset\overline{\mathcal{H}}^k$ such that \[\nabla(\overline{\mathcal{F}}^p)\subset (T^{(1,0)}_{\overline{S}})^*(\log B)\otimes \overline{\mathcal{F}}^{p-1}.\] \end{NilpotentOrbitTheorem} The asymptotic behavior as $z_j\to0$ of the $B$-model correlation functions \[\langle\theta\,\alpha\,\beta\rangle=\int_W\nabla_\theta(\alpha)\wedge\beta\] can be deduced from this theorem. If we let $\theta_j=2\pi i\,z_j\,\frac{d}{dz_j}$ (chosen to remove poles in the asymptotic expression for the correlation function) then the leading term in $\langle\theta_j\,\eta_\ell\,\beta\rangle$ is given by the monodromy: \begin{equation}\label{eq:limB} \lim_{z_j\to0}\langle\theta_j\,\eta_\ell\,\beta\rangle =-\int_W N^{(j)}(e_\ell)\wedge\beta. \end{equation} The essential properties of the monodromy are captured by the {\em monodromy weight filtration}\/ ${\mathbb{W}}_{{\scriptscriptstyle\bullet}}$ on the cohomology, which has the properties that $N^{(j)}{\mathbb{W}}_\ell\subset {\mathbb{W}}_{\ell-2}$, and that for any positive real numbers $a_1$, \dots, $a_r$, the operator $N:=\sum a_j N^{(j)}$ induces isomorphisms $N^\ell:\operatorname{Gr}^{\mathbb{W}}_{n+\ell}\to\operatorname{Gr}^{\mathbb{W}}_{n-\ell}$. Any splitting filtration which we use to make calculations of $B$-model correlation functions must be somehow compatible with this monodromy weight filtration, if those calculations are to make sense near the boundary. If mirror symmetry is going to hold, there must be a correspondence between the limiting behaviors described in eqs.~\eqref{eq:limA} and \eqref{eq:limB}. In fact, the first thing to notice is that the natural flat coordinates on the $A$-model moduli space are multiple-valued, with the ambiguity precisely specified by $H_{\text{DR}}^2(M,\mathbb{Z})$. So there must be some part of the monodromy weight filtration which matches that behavior. This motivated the following definition, first given in \cite{guide,compact} (cf.~also \cite{deligne}). We say that a boundary point is {\em maximally unipotent}\/ if \[{\mathcal{H}}_s=({\mathcal{F}}^n)_s\oplus({\mathbb{W}}_{2n-2})_s\] and \[{\mathcal{H}}_s=({\mathcal{F}}^{n-1})_s\oplus({\mathbb{W}}_{2n-4})_s\] for all $s$ near the point. With this definition, the distinguished holomorphic $n$-form and the canonical coordinates can be defined as in lecture six. There is an alternate version of this ``maximally unipotent monodromy'' condition, which agrees with the original one for Calabi--Yau threefolds, but is more restrictive in higher dimension. We say that a boundary point is {\em strongly maximally unipotent}\/ if the weight filtration ${\mathbb{W}}_{{\scriptscriptstyle\bullet}}$ has nontrivial graded pieces in even degree only, and if the induced filtration on homology defined by \[\mathbb{S}_\ell:=\operatorname{Ann}(\mathbb{W}_{2n-2\ell+2})\] is a splitting filtration. (Note that the corresponding filtration on cohomology is then \[\mathbb{S}^\ell:=\operatorname{Ann}(\mathbb{S}_{\ell-1})=\mathbb{W}_{2n-2\ell};\] this is the filtration which should be opposite to the Hodge filtration.) In this case, we will be able to use distinguished sections to calculate $B$-model correlation functions, as explained earlier. At the moment, only the original version of the definition has been justified to the satisfaction of physicists as an appropriate characterization of points which should be useful for mirror symmetry. To completely carry out a mirror symmetry type calculation, though, the second version would seem to be necessary. And as we shall see, that version has been extremely successful in examples. Actually, even just at the level of the monodromy action, the parallels between the structure of the Lefschetz operators on the cohomology and the action of monodromy are rather striking, as was first observed by Cattani, Kaplan and Schmid \cite{CKS}. The operators $\operatorname{ad}(e^j)$ describe Lefschetz decompositions of the cohomology of $M$, which have many structural parallels to the monodromy weight filtration at a maximally unipotent point. \section{Reinterpreting the $A$-model correlation functions} Let $M$ be a Calabi--Yau manifold on which a complex structure and K\"ahler metric have been fixed. Inspired by some of the similarities between the two different types of correlation functions, we wish to improve the analogy by translating the $A$-model correlation functions into data describing a variation of Hodge structure. Consider the moduli space $\mathcal{D}/\Gamma$ for $A$-model correlation functions, and a coordinate chart specified by a cone $\sigma$: \[\begin{array}{ccccc} \mathcal{D}/\Gamma&\leftarrow&\mathcal{D}_\sigma/H^2(M,\mathbb{Z})&\cong&(\Delta^*)^r\\ &&\cap\raise1pt\hbox{$\scriptstyle|$}&&\cap\raise1pt\hbox{$\scriptstyle|$}\\ &&(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-&\cong&\Delta^r \end{array}\] We assume that the cone $\sigma$ (which we call a {\em framing}\/) is generated by a basis $e^1$, \dots, $e^r$ of $H^2(M,\mathbb{Z})$. Let $t_1$, \dots, $t_r$ be coordinates on $H^2(M,\mathbb{C})$ dual to this basis (so that elements of $H^2(M,\mathbb{C})$ take the form $\sum t_j e^j$). The natural vector fields for making calculations of correlation functions which involve a term from the tangent space $H^2(M,\mathbb{C})$ are the vector fields $\partial/\partial t_j$. These are the analogues of the distinguished vector fields which we had on the $B$-model side. On the other hand, natural coordinates on $\mathcal{D}_\sigma/H^2(M,\mathbb{Z})$ are furnished by $q_j=\exp(2\pi i\,t_j)$. Then \[\frac{\partial}{\partial t_j}=2\pi i\,q_j\,\frac{\partial}{\partial q_j},\] from which we conclude that those correlation functions should naturally be evaluated on the basis $2\pi i\,q_j\,\partial/\partial q_j$ of the sheaf of logarithmic vector fields on the space $(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-$. We identify $\partial/\partial t_j$ with the operation of taking the quantum product with the basis element $e^j\in H^2(M,\mathbb{Q})$. The resulting map is determined by the correlation functions of the form $\langle e^j\, \alpha\,\beta\rangle$. We had a particularly simple form for these correlation functions, given in example \ref{example43}, in terms of the Gromov--Witten maps $\Gamma_\eta$. We now wish to reinterpret that formula in the following way. We will describe a holomorphic bundle\footnote{There are a few variants to this construction, in which one uses slightly different bundles. Essentially, one can restrict to any subbundle of $\bigoplus H^{\ell,\ell}(M)$ which is preserved by cup products with the part of $H^{1,1}(M)$ which it contains.} $\mathcal{E}:=\left(\bigoplus H^{\ell,\ell}(M)\right)\otimes \O_{(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-}$ with a connection\footnote{I am indebted to P. Deligne for advice which led to this form of the formula (cf.~\cite{deligne}).} (with regular singular points) \[\nabla:=\frac1{2\pi i}\,\left( \sum d\mskip0.5mu\log q_j\otimes\operatorname{ad}(e^j)+ \sum_{0\ne\eta\in H_2(M,\mathbb{Z})} d\mskip0.5mu\log\left(\frac1{1-q^{\eta}}\right)\otimes\Gamma_\eta \right)\] which was derived from the formulas for $e^j{\star}$, where $\operatorname{ad}(e^j):H^k(M)\to H^{k+2}(M)$ is defined by $\operatorname{ad}(e^j)(A)=e^j\cup A$. We also define a ``Hodge filtration'' \[\mathcal{E}^p:=\left(\bigoplus_{0\le\ell\le m-p} H^{\ell,\ell}(M)\right)\otimes\O_{(\mathcal{D}_\sigma/H^2(M,\mathbb{Z}))^-}.\] This describes a structure we call the {\em framed $A$-variation of Hodge structure with framing $\sigma$}. To be a bit more precise, we should study ``formally degenerating variations of Hodge structure,'' since the series used to define $\nabla$ is only formal. (We won't formulate that theory in detail here.) The connection $\nabla$ which we defined from the Gromov--Witten invariants is in fact a {\em flat}\/ holomorphic connection \cite{topgrav}. The flatness follows from the associativity (and commutativity) of the binary operation. In fact, since the directional derivatives with respect to $\nabla$ corresponded to binary products $e^j\star\zeta$ (where $e^j$ describes the direction of the derivative), iterated directional derivatives have the form $e^k\star(e^j\star\zeta)$. We would simply need to know that reversing the order of $j$ and $k$ produces the same result, and this is guaranteed by the commutativity and associativity. In particular, the flatness is automatic when $\dim_\mathbb{C} M=3$, a case in which there is no issue of associativity. The recent proofs of associativity of quantum cohomology \cite{RuanTian,Liu,MS} guarantee that this connection is flat in arbitrary dimension. As in the geometric case, there is an additional structure associated to this variation of Hodge structure: a local system. The local system on homology takes the simple form \[\mathbb{S}_\ell:=H_{0,0}\oplus H_{1,1}\oplus\dots\oplus H_{\ell,\ell},\] and the corresponding local system on cohomology then becomes \[\mathbb{S}^\ell=H^{\ell,\ell}\oplus H^{\ell+1,\ell+1}\oplus\dots\oplus H^{n,n}.\] The logarithms of the monodromy actions which define these local systems are specified by the topological pairings, and coincide with the cup-product maps \[H^2(M,\mathbb{Z})\otimes \mathbb{S}^\ell\to \mathbb{S}^{\ell+1}.\] In the next lecture, we will formulate a precise conjecture which equates this $A$-variation of Hodge structure with the geometric variation of Hodge structure on a mirror partner. \section{Beyond the K\"ahler cone}\label{sec73} We indicated in lecture five that the conformal field theory moduli space is actually {\em larger}\/ than the nonlinear $\sigma$-model moduli space. We can now explain how this comes about---it is due to an analysis of the effect of flops on the conformal field theory. Flops are birational transformations among Calabi--Yau threefolds which have been studied extensively as part of the minimal model program (see for example \cite{CKM}). The effect of flops on the K\"ahler cone of a Calabi--Yau threefold is as follows. Given a Calabi--Yau threefold $X$ with a complex structure $J$, and a linear system $|L|$ inducing a flopping contraction from $X_J$ to $\widehat X_{\widehat J}$, the K\"ahler cones $\mathcal{K}_J$ and $\widehat{\mathcal{K}}_{\widehat J}$ share a common wall, which contains the class of $|L|$, as depicted in figure \ref{fig0}. The K\"ahler cone has already occurred in our discussion of the moduli spaces of $\sigma$-models. The natural question arises: suppose we attempt to ``attach'' the moduli spaces $\mathcal{D}/\Gamma$ and $\widehat{\mathcal{D}}/\widehat{\Gamma}$ along (the images of) their common wall? In fact, it now appears likely that the conformal field theory moduli spaces of $X$ and $\widehat X$ are analytic continuations of each other, and that this ``attached space'' is a part of the full conformal field theory moduli space \cite{mmm,phases}. (This at least seems to happen in examples---the arguments for this rely on mirror symmetry, and involve finding regions in the mirror's moduli space which correspond to the $X_J$ and $\widehat X_{\widehat J}$ theories, respectively.) One of the consequences of this would be an analytic continuation of correlation functions from $\mathcal{D}/\Gamma$ to $\widehat{\mathcal{D}}/\widehat{\Gamma}$. \begin{figure} \iffigs $$\vbox{\centerline{\epsfysize=3cm\epsfbox{cones.eps}} }$$ \else \vglue2in\noindent \fi \caption{Adjacent K\"ahler cones} \label{fig0} \end{figure} Here is a formal calculation from \cite{phases,small} which supports this analytic continuation idea (see also \cite{beyond} for a more mathematical treatment). The union of all of the K\"ahler cones of birational models of $X_J$ is known as the {\em movable cone}\/ $\operatorname{Mov}{X_J}$ \cite{kawamata}. We compute in the formal semigroup ring $\mathbb{Q}[q;\operatorname{Mov}(X_J)^\vee]$ (which we identify canonically with the same ring for $\widehat X_{\widehat J}$), and so the computation is purely formal. Consider the simplest flop: the flop based on a collection of disjoint holomorphic rational curves $\Gamma_i\subset X_J$ (in a common homology class $[\Gamma]$) such that the normal bundle is $N_{\Gamma_i/X_J}=\O(-1)\oplus \O(-1)$. (These curves must be flopped simultaneously in order to ensure that the flopped variety is K\"ahler.) A reasonable genericity assumption about the {\em other}\/ rational curves on $X_J$ is this: all (pseudo-)holomorphic curves in classes $\eta\not\in\mathbb{R}_{>0}[\Gamma]$ are disjoint from the $\Gamma_i$'s. Since there is a proper transform map on divisors, the Gromov--Witten invariants (which in this case are determined entirely by intersection properties of $\eta$ and the number of elements in $\mathcal{M}^*_{(\eta,J)}$) do not change when passing from $X_J$ to $\widehat X_{\widehat J}$, except for the invariants $\Phi_{[\Gamma]}$ themselves. The cup product can also change. The $A$-model correlation functions on $X_J$ can be written in the form \begin{align*} \langle A\,B\,C\rangle=A\cdot B\cdot C &+\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ &+\sum_{\substack{\eta\in H_2(X,\mathbb{Z})\\ \eta\ne\lambda\Gamma}} \frac{q^\eta}{1-q^\eta}\,\Phi_\eta(A,B,C). \end{align*} Only the first terms change when passing to $\widehat X_{\widehat J}$ and in fact we claim that \begin{align*} A\cdot B\cdot C &+\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ = & {\widehat A}\cdot {\widehat B}\cdot {\widehat C} +\frac{q^{[\widehat \Gamma]}}{1-q^{[\widehat \Gamma]}}\, (\widehat A\cdot\widehat \Gamma)(\widehat B\cdot\widehat \Gamma) (\widehat C\cdot\widehat \Gamma)\,n_{\widehat \Gamma}, \end{align*} where $\widehat A$, $\widehat B$, and $\widehat C$ are the proper transforms of $A$, $B$, and $C$. (In other words, the change in the topological term is precisely compensated for by the change in the $q^{[\Gamma]}$ term.) We will check this formula in the case in which $A$ and $B$ meet one of the curves $\Gamma$ transversally at $a$ and $b$ points, respectively, and $\widehat C$ meets $\widehat\Gamma$ transversally at $c$ points. (The general case can be deduced from this one.) Then $C$ must contain $\Gamma$ with multiplicity $c$, and the configuration of divisors is as in figure \ref{fig1} (which illustrates the case $a=b=c=1$ for simplicity). $A$ and $B$ have no intersection points along $\Gamma$, but both $\widehat A$ and $\widehat B$ contain $\widehat \Gamma$, and they meet $\widehat C$. The total number of intersection points of $\widehat A$, $\widehat B$ and $\widehat C$ (counted with multiplicity) which lie in $\widehat \Gamma$ is thus $abc$. \refstepcounter{figure}\label{fig1} \begin{figure} \iffigs $$ \matrix\epsfxsize=2in\epsfbox{fig1a.eps} & \qquad & \epsfxsize=2in\epsfbox{fig1b.eps} \cr \quad & & \cr \hbox{{\footnotesize\bfseries Figure \ref{fig1}a}{.\footnotesize\mdseries\upshape\enspace Before the flop.}} & & \hbox{{\footnotesize\bfseries Figure \ref{fig1}b}{.\footnotesize\mdseries\upshape\enspace After the flop.}} \cr\endmatrix $$ \else \vglue3in\noindent \fi \end{figure} Since a similar thing happens for each curve $\Gamma_i$ in the numerical equivalence class, we see that \begin{equation} \widehat A\cdot\widehat B\cdot\widehat C -A\cdot B\cdot C = abc\,n_\Gamma =-(A\cdot\Gamma)(B\cdot\Gamma)(C\cdot\Gamma)\,n_\Gamma \label{eq:three} \end{equation} (using $A\cdot\Gamma=a$, $B\cdot\Gamma=b$, $C\cdot\Gamma=-c$). On the other hand, since $[\widehat\Gamma]=-[\Gamma]$ and $n_{\widehat\Gamma}=n_\Gamma$, we can compute: \begin{equation}\label{eq:four}\begin{split} \frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\,&(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma - \frac{q^{[\widehat \Gamma]}}{1-q^{[\widehat \Gamma]}}\, (\widehat A\cdot\widehat \Gamma)(\widehat B\cdot\widehat \Gamma) (\widehat C\cdot\widehat \Gamma)\,n_{\widehat \Gamma}\\ =&\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}}\, (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma +\frac{q^{-[\Gamma]}}{1-q^{-[\Gamma]}} (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ =&\left(\frac{q^{[\Gamma]}}{1-q^{[\Gamma]}} +\frac{1}{q^{[\Gamma]}-1}\right) (A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma \\ =&-(A\cdot\Gamma)(B\cdot\Gamma) (C\cdot\Gamma)\,n_\Gamma. \end{split}\end{equation} Adding eqs.~\eqref{eq:three} and \eqref{eq:four} proves the desired formula. The conclusion from all of this should be that the mirror symmetry phenomenon is really about birational equivalence classes. For if there is any analytic continuation of the correlation function from the region associated to $\mathcal{K}_J$ out into the next cone $\widehat{\mathcal{K}}_{\widehat J}$, the calculation above shows that this analytic continuation must in fact reproduce the correlation function of the flopped model $\widehat X_{\widehat J}$. It is tempting to think that if we combined the $\sigma$-model moduli spaces for all birational models of $X$ we would fill out the entire conformal field theory moduli space. However, some examples that have been worked out by Witten \cite{phases} and by Aspinwall, Greene and the author \cite{catp} show that this is not the case. In those examples, there are other regions in the moduli space which correspond to rather different kinds of physical model, including some called {\em Landau--Ginzburg theories}\/ which will play a r\^ole again in the next lecture. \chapter*{} \lecturename{Mirror Symmetry} \lecture \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Lecture 8. Mirror Symmetry} \section{Mirror manifold constructions} The original speculations about mirror symmetry were based on the appearance of arbitrariness of a choice that was made in identifying certain constituents of the conformal field theory associated to a Calabi--Yau manifold with geometric objects on the manifold. The distinction between vertex operators which appear in the $A$-model and $B$-model correlation functions is simply a difference in sign of a certain quantum number; if that sign is changed, the geometric interpretation is altered dramatically. This led Dixon \cite{Dixon} and Lerche--Vafa--Warner \cite{LVW} to propose that there might be a second Calabi--Yau manifold producing essentially the same physical theory as the first, but implementing this change of sign. Some time later,\footnote{At about the same time, another important piece of evidence for mirror symmetry was given by Candelas, Lynker, and Schimmrigk \cite{CLS}, who found an almost perfect symmetry under the exchange $h^{1,1}\leftrightarrow h^{2,1}$ on the set of Hodge numbers coming from Calabi--Yau threefolds which can be realized as weighted projective hypersurfaces.} an explicit construction was made by Greene and Plesser \cite{GreenePlesser} which showed that this phenomenon does indeed occur in physics. The construction rests on a chain of equivalences which are believed to hold among different physical models, as follows. \begin{enumerate} \item Certain $\sigma$-models on Calabi--Yau manifolds are believed to correspond to so-called Landau--Ginzburg theories \cite{GVW}. (It has recently been recognized \cite{phases} that this correspondence is not direct, but involves analytic continuation on the moduli space.) Roughly speaking, the class of Calabi--Yau manifolds for which this correspondence can be made is the class of ample anti-canonical hypersurfaces in toric varieties. Such a hypersurface will have an equation of the form $\Phi(x_1,\dots,x_{n+1})=0$ (in some appropriate coordinates on the torus), and this same polynomial is used as a ``superpotential'' in constructing the Landau--Ginzburg theory. \item Certain Landau--Ginzburg theories---quotients of the ones for which the superpotential is of ``Fermat type'' \[\Phi(x_1,\dots,x_{n+1})=x_1^{d_1}+\dots+x_{n+1}^{d_{n+1}}\] by certain finite groups $\Gamma$---are believed to correspond to yet another type of conformal field theory. This other theory is described in terms of discrete series representations $V^{(k)}$ of the ``$N{=}2$ superconformal algebra,'' and it takes the form \[\left(\bigotimes_j V^{(d_j+2)}\right)/G\] where $G$ is a slight enlargement of the group $\Gamma$. (Note that the case of $\Gamma$ being trivial is allowed, but then $G$ is not trivial.) The representation theory of the $N{=}2$ superconformal algebra is related to these things by analyzing the conformal field theory on an infinite cylinder. (The superconformal algebra can be described in terms of automorphisms of the cylinder.) \item By studying the representation theory, Greene and Plesser find a kind of duality among the finite groups $G$: there is a dual group $\widehat{G}$ and an isomorphism \[\left(\bigotimes_j V^{(d_j+2)}\right)/G\cong \left(\bigotimes_j V^{(d_j+2)}\right)/\widehat{G}\] which has the ``sign-reversing property'' of mirror symmetry. \item The duality can be extended to the groups $\Gamma$, and the mirror Landau--Ginzburg theory of $\Phi/\Gamma$ is $\Phi/\widehat{\Gamma}$. This looks a bit asymmetric, since for example the case $\Gamma$ trivial leads to a rather large group $\widehat{\Gamma}$. But the group $\Gamma$ continues to act as a group of ``quantum symmetries'' on the quotient theory, in a way that restores symmetry to this construction. \item Finally, the Calabi--Yau which is the quotient of the Fermat hypersurface by $\Gamma$ should have as its mirror the one which is the quotient by $\widehat{\Gamma}$. \end{enumerate} This is called the {\em Greene--Plesser orbifolding construction}. \medskip There is a conjectural generalization of this construction, which as of yet has no basis in conformal field theory---it is simply a mathematician's guess. This generalization would work for an arbitrary family of Calabi--Yau hypersurfaces in toric varieties. The construction is due to V.~Batyrev \cite{batyrev1}.\footnote{We restrict ourselves to the hypersurface case here; further generalizations---to complete intersections---were subsequently given by Borisov and Batyrev--Borisov \cite{borisov,BB:dual}.} Take an ample anticanonical hypersurface $M$ in a toric variety $V$, and let $\{M_t\}$ be the family of such. This family is determined by the Newton polygon of the corresponding equations---that is a polygon $P\subset L_\mathbb{R}:=L\otimes \mathbb{R}$, where $L$ is the {\em monomial lattice}\/ of the torus $T$ (of which $V$ is a compactification). Batyrev shows that the Calabi--Yau condition admits a particularly simple characterization in terms of $P$: the polyhedron $P$ is {\em reflexive}, which means that each hyperplane $H$ which supports a face of codimension one of $P$ can be written in the form \[H=\{y\in L_\mathbb{R}\ |\ (\ell,y)=-1\}\] for some appropriate vector $v\in \operatorname{Hom}(L,\mathbb{Z})$. (The key property here is the {\em integrality}\/ of the vector $v$---there would always be some $v\in \operatorname{Hom}(L,\mathbb{R})$ to define $H$.) \begin{lemma}[Batyrev] If $P$ is reflexive, then the {\em polar polyhedron}\/ \[P^o:=\{x\in \operatorname{Hom}(L,\mathbb{R})\ |\ (x,y)\ge-1\text{ for all }y\in P\}\] is also reflexive. \end{lemma} The conjectured generalization is that the mirror of the family $\{M_t\}$ of hypersurfaces determined by $P$ should be the family $\{W_s\}$ of hypersurfaces (in a compactification of the dual torus of $T$) determined by the polar polyhedron $P^o$. One of the pieces of evidence for this conjecture is \begin{theorem}[Batyrev]\label{batthm} \[\dim H^{\pm1,1}(\widehat{M})=\dim H^{\mp1,1}(\widehat{W}),\] where $\widehat{M}$ and $\widehat{W}$ are $\mathbb{Q}$-factorial terminalizations of $M$ and $W$ respectively. \end{theorem} \noindent Batyrev and collaborators have also explored the Hodge structures of these hypersurfaces in considerable detail \cite{Bat:vmhs,BvS,BatCox}. A refinement of Batyrev's theorem called the {\em monomial-divisor mirror map}\/ was introduced in \cite{mondiv}. This map gives an explicit combinatorial correspondence between (appropriate subspaces of) $H^{\pm1,1}(\widehat{M})$ and $H^{\mp1,1}(\widehat{W})$, and is expected to correctly determine the derivative of the mirror map near the large radius limit point. That derivative data is precisely what one needs in order to evaluate the ``constants of integration'' in finding the canonical coordinates $q_j$. \medskip There is another mirror manifold construction for a class of threefolds which has been proposed by Voisin \cite{Voisin:K3} and Borcea \cite{Borcea:K3}. Let $S$ be a K3 surface with an involution $\iota$ such that $\iota^*(\Omega)=-\Omega$ for any holomorphic two-form $\Omega$ on $S$, and let $E$ be an elliptic curve. The quotient $\overline{M}=(S\times E)/(\iota\times(-1))$ has singularities along the fixed curves of the involution $\iota\times(-1)$, but they can be resolved by a simple blowing up to produce a Calabi--Yau threefold $M$. Involutions of this type on K3 surfaces have been classified by Nikulin \cite{Nikulin:involutions}, who found that they fall into a pattern with a remarkable symmetry; when the Hodge numbers of the associated Calabi--Yau threefold are calculated, this symmetry becomes the expected mirror relation among Hodge numbers. The detailed knowledge which is available concerning the variations of Hodge structure on K3 surfaces can be used to study the correlation functions in detail for these models \cite{Voisin:K3}, which provides further evidence that mirror partners have been correctly identified. In fact, there is also a physics argument explaining why these pairs of conformal field theories are actually mirror to each other \cite{AM:K3}, based on the physics of mirror symmetry for K3 surfaces. \section{Hodge-theoretic mirror conjectures} We can now formulate the main conjecture in the mathematical study of mirror symmetry. \begin{HTmirrorconjecture} \quad Given a boundary point $P\in\overline{\mathcal{M}}_W$ with maximally unipotent monodromy (or perhaps with strongly maximally unipotent monodromy), there should exist a mirror partner $M$ of $W$, a framing $\sigma$ of $M$, a neighborhood $U$ of $P$ in $\mathcal{M}_W$, and a ``mirror map'' \[\mu:U \to (\mathcal{D}_\sigma/L)^-\] which is determined up to constants of integration by the property that \[\mu^*(d\mskip0.5mu\log q_j)= d\left(\frac{\int_{\gamma_j}\Omega}{\int_{\gamma_0}\Omega}\right),\] such that $\mu$ induces an isomorphism between appropriate sub-variations of Hodge structure of \begin{enumerate} \item the formal completion of the geometric variation of Hodge structure at $P$, and \item the framed $A$-variation of Hodge structure with framing $\sigma$. \end{enumerate} (The sub-variations of Hodge structure should contain the entire first two terms of the Hodge filtration on both sides.) \end{HTmirrorconjecture} There are additional conjectures one wants to make about the relationship between $M$ and $W$: there should also be isomorphisms \[H^{p,q}(W)\cong H^{-p,q}(M)\quad p\ge0,\] and these should preserve all correlation functions. (In particular, the ``reverse'' mirror isomorphism should hold, and there should also be isomorphisms between correlation functions which do not come from variations of Hodge structure.) Of course, such isomorphisms only make sense if we have specified the constants of integration. In fact, one wants to conjecture that the entire conformal field theory moduli spaces are isomorphic, but this is a difficult conjecture to make precisely at present since we do not have a complete mathematical understanding of conformal field theory moduli spaces. If we start with the $A$-variation of Hodge structure, there is another conjecture we can make. \begin{converse} Conversely, given $(M,\sigma)$, the corresponding $A$-variation of Hodge structure comes from geometry, in the sense that there is a family $\mathcal{Z}\to\overline{S}$ of varieties degenerating at $0\in\overline{S}$ such that the framed $A$-variation of Hodge structure is isomorphic to the formal completion at $0$ of a (Tate-twisted) sub-variation of Hodge structures of the variation of Hodge structures on some cohomology of $Z_s$. \end{converse} Due to the phenomenon of rigid Calabi--Yau manifolds, we can't assume any stronger properties about $Z_s$: Calabi--Yau threefolds with $h^{2,1}=0$ cannot have mirror partners in the usual sense, since such a mirror partner would satisfy $h^{1,1}=0$, which is absurd. However, there is an example in the physics literature of a rigid Calabi--Yau manifold, known as the ``$Z$-orbifold,'' which has a mirror physical theory that was worked out recently by Candelas, Derrick and Parkes \cite{CDP} (see also \cite{AspGr}). In this example, the variation of Hodge structure associated to the mirror theory can be described by the family of cubic sevenfolds in $\P^8$ (with a suitable Tate twist). \section{Some computations} We explain some of the evidence in favor of the mirror symmetry conjectures which has been accumulated through specific computations.\footnote{The computations presented here are taken from the original paper of Candelas, de la Ossa, Green and Parkes \cite{CDGP} on the quintic threefold, and a paper of Greene, Plesser and the present author \cite{GMP} on higher dimensional mirror manifolds. A survey of other calculations of this type (and the methods for making them) can be found in \cite{predictions}.} We will compute with Calabi--Yau hypersurfaces of dimension $n\ge3$ in ordinary projective space $\C\P^{n+1}$; the degree of the hypersurface must be $n+2$. The family of such hypersurface includes a Fermat hypersurface, which is part of the ``Dwork pencil'' with defining equation: \[x_0^{n+2}+\cdots+x_{n+1}^{n+2}-(n+2)\psi\,x_0{\cdots}x_{n+1}=0,\] where $\psi$ is a parameter. The group \[\Gamma:=\{(\alpha_0,\dots,\alpha_{n+1})\ |\ \alpha_j\in\mmu_{n+2}, \prod\alpha_j=1\}/\{(\alpha,\dots,\alpha)\}\] acts on the fibers of this family by componentwise multiplication on the coordinates. Using either the Greene--Plesser orbifolding construction, or Batyrev's polar polyhedron construction, one sees that the family $\{M\}$ of hypersurfaces of degree $n+2$ in $\C\P^{n+1}$ has as its predicted mirror the family $\{W\}$ described as the Dwork pencil modulo $\Gamma$ (living in the quotient space $\C\P^{n+1}/\Gamma$). In fact, we can describe the moduli space of this mirrored family in terms of the parameter $\psi^{n+2}$---the reason for passing to a power is the existence of an additional automorphism, acting on the family as a whole, generated by componentwise multiplication in the $x$'s by $(\alpha,1,\dots,1)$ while simultaneously multiplying $\psi$ by $\alpha^{-1}$. It is not difficult to compute where this family becomes singular. The partial derivatives of the defining equation are all of the form \[(n+2)\left(x_j^{n+1}-x_j^{-1}\psi\,x_0{\cdots}x_{n+1}\right)\] and for these to vanish simultaneously we must have $\psi^{n+2}=1$. Moreover, the additional automorphism of the family fixed the fiber $\psi=0$, and so causes additional singularities there. Thus, we can describe the moduli space as $\C\P^1-\{0,1,\infty\}$, with its natural compactification being $\C\P^1$. What is the monodromy behavior at the boundary points? (We label the monodromy transformations according to the point.) At $\psi^{n+2}=0$, we find that the monodromy has finite order, at $\psi^{n+2}=1$ it is unipotent but $(T_1-I)^2=0$ so the order is not maximal (since $n\ne1$), and at $\psi^{n+2}=\infty$ we find maximal order of unipotency $(T_\infty -I)^n\ne0$. In fact, this point is maximally unipotent, and even strongly maximally unipotent, in the terminology established earlier. To compute canonical coordinates and correlation functions near $\psi^{n+2}=\infty$ we need to know the period functions there. These can be found by studying the differential equations which they satisfy. In this case of toric hypersurfaces, we have a special method available---the representation of cohomology by means of residues of differential forms on the ambient space with poles along the hypersurface. A basis for the primitive cohomology can be written (in the affine chart $x_0=1$, say) as \[\beta_j:=\operatorname{Res}\left( \frac{\psi^{j+1}\,(x_1{\cdots}x_{n+1})^j\,dx_1\wedge\cdots\wedge dx_{n+1}} {\left(1+x_1^{n+2}+\cdots+x_{n+1}^{n+2} -(n+2)\psi\,x_1{\cdots}x_{n+1}\right)^{j+1}} \right)\] The connection matrix in this basis can then be found using Griffiths' ``reduction of pole order'' lemma \cite{Griffiths} to calculate coefficients $\theta_{ij}$ such that \[\nabla(\beta_i)=\sum\theta_{ij}\beta_j.\] To find the period matrix from the connection matrix, one must solve some differential equations. For if $\{e_k\}$ is a basis for the local system and we write $e_k=\sum\eta_{ki}\beta_i$ then \[0=\nabla(e_k)=\sum d\eta_{ki}\, \beta_i + \sum\eta_{ki}\theta_{ij}\beta_j\] gives differential equations for the unknown coefficient functions $\eta_{ki}$: \[d\eta_{ki}=-\sum\eta_{k\ell}\theta_{\ell i}.\] The flatness of $\nabla$ is equivalent to the integrability of these equations, which can therefore be solved. \begin{table} \begin{center} \begin{tabular}{|l|l|} \hline $n$&$n$-point function\\ \hline $3$& $5+2875\,q+4876875\,q^2+8564575000\,q^3+15517926796875\,q^4 $\\ &$\phantom{5} +28663236110956000\, q^5+53621944306062201000\,q^6 $\\ &$\phantom{5} +101216230345800061125625\,q^7+ 192323666400003538944396875\,q^8 $\\ &$\phantom{5} +367299732093982242625847031250\,q^9 $\\ &$\phantom{5} +704288164978454714776724365580000\,q^{10} $\\ &$\phantom{5} +1354842473951260627644461070753075500\,q^{11} $\\ &$\phantom{5} +2613295702542192770504516764304958585000\,q^{12} $\\ &$\phantom{5} +5051976384195377826370376750184667397150000\,q^{13} $\\ &$\phantom{5} +9784992122065556293839548184561593434114765625\,q^{14} $\\ &$\phantom{5} +18983216783256131050355758292004110332155634496875\,q^{15} $\\ &$\phantom{5} +36880398908911843175757970052077286676680907186572875\,q^{16} $\\ &$\phantom{5} +71739993072775923425756947313710004388338109828244718125\,q^{17} $\\ &$\phantom{5} +139702324572802672116486725324237666156179096139345867681250\,q^{18} $\\ &$\phantom{5} +\dots$\\[6pt] $4$& $6 + 120960 \,q \!+\! 4136832000 \,q^2 \!+\! 148146924602880 \,q^3 \!+ 5420219848911544320 \,q^4 $\\ &$\phantom{6} + 200623934537137119778560 \,q^5 + 7478994517395643259712737280 \,q^6 $\\ &$\phantom{6} + 280135301818357004749298146851840 \,q^7 $\\ &$\phantom{6} + 10528167289356385699173014219946393600 \,q^8 $\\ &$\phantom{6} + 396658819202496234945300681212382224722560 \,q^9 $\\ &$\phantom{6} + 14972930462574202465673643937107499992165427200 \,q^{10} $\\ &$\phantom{6} + 566037069767251121484562070892662863943365345190400 \,q^{11} $\\ &$\phantom{6} + 21424151141341932048068067497996096856724987411324108800 \,q^{12} \!+\! \dots$\\[6pt] $5$& $7 + 3727381 \,q + 2637885990187 \,q^2 + 1927092954108108787 \,q^3 $\\ &$\phantom{7} +1425153551321014327663291 \,q^4 + 1060347883438857662557634869906 \,q^5 $\\ &$\phantom{7} + 791661306374088776109692880989252173 \,q^6 $\\ &$\phantom{7} + 592348256908461616176898022359492565546566 \,q^7 $\\ &$\phantom{7} + 443865568545713063761643598030194801299861575595 \,q^8 $\\ &$\phantom{7} + 332947403131697202086626568381790256001850741509664373 \,q^9 +\dots$\\[6pt] $6$& $8 + 106975232 \,q + 1672023727001600 \,q^2 + 26611692333081695092736 \,q^3 $\\ &$\phantom{8} + 426129121674687823674948571136 \,q^4 $\\ &$\phantom{8} + 6842148599241293047857339542861643776 \,q^5 $\\ &$\phantom{8} + 110018992594692024449889564415904439556898816 \,q^6 $\\ &$\phantom{8} + 1770551943055574073245974844490813198478975912902656 \,q^7 $\\ &$\phantom{8} + 28508925683951911989843155602330000507452539542539447947264 \,q^8 $\\ &$\phantom{8} +\cdots$\\ \hline \end{tabular} \end{center} \medskip \caption{$n$-point functions in dimension $n$} \end{table} If we work in a local coordinate $z=\psi^{-n-2}$ near $\psi^{n+2}=\infty$, we find that a basis $e_0(z)$, \dots, $e_{n+1}(z)$ of local solutions can be found such that $e_0(z)$ is single-valued near $z=0$, and \[e_{j+1}(z) = (\log z)\, e_j(z) + \text{single-valued function}.\] (This is a consequence of the maximally unipotent monodromy.) The vectors $e_j(z)$ form the columns of the period matrix. One can then use row operations to put the period matrix in upper triangular form, with constant diagonal elements. (Let us choose the diagonal elements to all be $n+2$.) This implements the change of basis to a basis consisting of distinguished sections of $\mathcal{H}^{p,q}_\mathbb{S}$. The nonzero entries $A_j^1$ in the connection matrix are then calculated by differentiating rows of the period matrix, and writing the result as a multiple of a subsequent row. Each such entry takes the form \[A^1_j=Y^1_j\,\frac{dq}q,\] and the functions $Y^1_j$ represent correlation functions $\langle (\partial/\partial t)\,\beta_j\,\beta_{n-j-1}\rangle$. This can all be done very explicitly, using power series expansions of the unknown single-valued functions, in these examples. (I advise using {\sc maple} or {\sc mathematica} if you would like to try it for yourself.) We show two kinds of calculations in the tables. For the first, only the ``maximally unipotent'' assumption is required, since the calculation requires only the distinguished $n$-form and the canonical coordinates. What is computed in table 1 is the ``$n$-point function,'' which iterates the differential of the period map $n$ times. (This was introduced some years ago in the variation of Hodge structures context by Carlson, Green, Griffiths and Harris \cite{CGGH}.) \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1= 5+2875\,\cuone3+609250\,\cu23+317206375\,\cu33+242467530000\,\cu43 $\\$\phantom{Y_1^1=5} +229305888887625\,\cu53+ 248249742118022000\,\cu63 $\\$\phantom{Y_1^1=5} +295091050570845659250\,\cu73+375632160937476603550000\,\cu83 $\\$\phantom{Y_1^1=5} +503840510416985243645106250\,\cu93 $\\$\phantom{Y_1^1=5} +704288164978454686113488249750\,\cu{10}3 $\\$\phantom{Y_1^1=5} +1017913203569692432490203659468875\,\cu{11}3 $\\$\phantom{Y_1^1=5} +1512323901934139334751675234074638000 \,\cu{12}3 $\\$\phantom{Y_1^1=5} +2299488568136266648325160104772265542625\,\cu{13}3 $\\$\phantom{Y_1^1=5} +3565959228158001564810294084668822024070250\,\cu{14}3 $\\$\phantom{Y_1^1=5} +5624656824668483274179483938371579753751395250\,\cu{15}3 $\\$\phantom{Y_1^1=5} +9004003639871055462831535610291411200360685606000\,\cu{16}3+\dots $\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point function in dimension three} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=6+60480\,\cuone2+440884080\,\cu22+6255156277440\,\cu32$\\ $\phantom{Y_1^1=6}+117715791990353760\,\cu42 +2591176156368821985600\cdot5^2\,\cu52 +\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point function in dimension four} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=7+1009792\,\cuone2+122239786088\,\cu22 +30528671745480104\,\cu32$\\ $\phantom{Y_1^1=7}+10378199509395886153216\,\cu42 +\dots$\\[6pt] $Y_2^1=7+1707797\,\cuo1+510787745643\,\cuo2 +222548537108926490\,\cuo3$\\ $\phantom{Y_2^1=7}+113635631482486991647224\,\cuo4 +\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point functions in dimension five} \end{table} \begin{table} \begin{center} \begin{tabular}{|l|} \hline $Y_1^1=8+15984640\,\cuone2+33397159706624\,\cu22 +154090254047541417984\,\cu32 $\\$\phantom{Y_1^1=8} +1000674891265872131899670528\,\cu42+\dots$\\[6pt] $Y_2^1=8+\!37502976\,\cuo1\!+\!224340704157696\,\cuo2 \!+\!2000750410187341381632\,\cuo3$\\ $\phantom{Y_2^1=8} +21122119007324663457380794368\,\cuo4+\dots$\\[6pt] $Y_2^2=8+\!59021312\,\cue{}\!+\!821654025830400\,\cue2 \!+\!\!12197109744970010814464\,\cue3$\\ $\phantom{Y_2^2=8} +186083410628492378226388631552\,\cue4+\dots$\\ \hline \end{tabular} \end{center} \medskip \caption{Three-point functions in dimension six} \end{table} The other computations, displayed in tables 2--5, are of three-point functions $Y^a_b$, read off of the connection matrix in a distinguished basis. (There is a symmetry $Y^a_b=Y^a_{n-a-b}=Y^b_{n-a-b}$ so we only show some of these.) The coefficients in the series expansions are the predicted values of the Gromov--Witten invariants. The three-point function $Y^1_0$ has the value $n+2$ (a constant, due to the definition of canonical coordinates) and is not shown in the tables. The other functions $Y^1_j$ come directly from the connection matrix. In dimension six, there is also a ``secondary'' function, which (by the $B$-model version of the associativity, which is simply the associativity of the ``sheaf cup product'' pairing) can be calculated as $Y^2_2 =(Y^1_2)^2/Y^1_1$. There is a relation between the computations in table 1, and those in tables 2--5, which can be explicitly verified from these tables: it is \[\text{$n$-point function } = \frac{Y^1_0\cdot Y^1_1\cdot {} \dotsm {} \cdot Y^1_{n-1}}{(n+2)^{n}}.\] The functions $Y^a_b$ are predicted to agree with quantum products on the mirror manifolds \[\zeta^a\star\zeta^b\star\zeta^{n-a-b},\] where $\zeta^j$ is the class of a linear space (in $\C\P^{n+1}$) of complex codimension $j$. In fact, we have displayed things in tables 2--5 with this in mind, writing series in terms of $q^k/(1-q^k)$. Also in tables 2--5, we have pulled out some factors of the degree of the rational curve. If there are $\ell$ occurrences of ``1'' among $\{a,b,n-a-b\}$, then there will be $\ell$ of the linear spaces of codimension one, and each meets a given rational curve $\Gamma$ in $\deg(\Gamma)$ points, giving rise to a factor of $(\deg(\Gamma))^{\ell}$ in the Gromov--Witten invariants. Pulling out those factors makes the comparison with ``counting'' problems more transparent. All of the predicted Gromov--Witten invariants in degrees one and two in these tables have been verified by Katz \cite{katz:verifying}; most of the invariants in degree three have been verified by Ellingsrud and Str{\o}mme \cite{ES,ESii}. \chapter*{} \lecturename{Postscript: Recent Developments} \lectureoptionstar{POSTSCRIPT:}{Recent Developments} \markboth{D. R. Morrison, Mathematical Aspects of Mirror Symmetry}{Postscript: Recent Developments} As mentioned in the introduction, the subject of mirror symmetry is a rapidly developing one, and much has happened since the lectures on which these notes are based were delivered. We will briefly sketch some of these developments in this postscript. The Gromov--Witten invariants and their generalizations have been studied particularly intensively. The definition of Ruan \cite{ruan} which we presented in the lectures has been supplanted by other definitions drawn from symplectic geometry (cf.~\cite{MS,RuanTian}) which work directly in cohomology (avoiding the bordism technicalities) and are also more general. In full generality these extended Gromov--Witten invariants are not only associated to curves of genus zero with three vertex operators, but also to curves of arbitrary genus $g$ with $k$ vertex operators (provided that $2g-2+k>0$) and even to some non-topological correlation functions.\footnote{There have also been investigations into the physical interpretation of these higher genus invariants, and how they should transform under mirror symmetry (in the case of Calabi--Yau threefolds) \cite{BCOV:anom,BCOV:KS}. At one time, it had been expected that for Calabi--Yau threefolds the genus zero topological correlation functions would completely determine the conformal field theory, but now it is known that higher genus invariants are needed as well \cite{chiral}.} There are at least three proofs of the associativity relations for these symplectic Gromov--Witten invariants \cite{RuanTian,Liu,MS}, including proofs of a stronger form of associativity known as the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equations \cite{topgrav,DVV,Wit:twoDgrav,Dubrov} which are relevant in the case of higher genus. As in the genus zero case, these higher genus invariants can be used to encode a kind of quantum cohomology ring (somewhat larger than the one we studied here); it is also possible to interpret the WDVV associativity relation as the flatness of a certain connection \cite{Dubrov}. A very accessible exposition of this circle of ideas has been written by McDuff and Salamon \cite{MS}. Parallel to this development, Gromov--Witten invariants have also been defined purely within algebraic geometry. The methods of Katz described in the lectures were developed further (see \cite{katz:GW} and the appendix to \cite{BCOV:anom}), and similar methods based on the construction of a ``virtual moduli cycle'' were developed independently by Li and Tian \cite{LiTian}. The foundations for an algebraic theory of Gromov--Witten invariants were carefully laid by Kontsevich and Manin \cite{KM} (again, the higher genus invariants and the WDVV equations play an important r\^ole), and the program they initiated was ultimately carried out \cite{BehMan,BehFant,Beh}, producing a definition of Gromov--Witten invariants based on stable maps. (The work of Li--Tian mentioned above \cite{LiTian} is also closely related to this program.) Even before this program was complete, Kontsevich had applied it to obtain some spectacular results in enumerative geometry, including a verification of the predicted number $242467530000$ of rational quartics on the general quintic threefold \cite{Kontsevich}. The stable map theory is nicely explained, with further references, in \cite{FulPan}. Kontsevich has also formulated a ``homological'' version of the mirror conjecture \cite{kont:icm} involving what are known as $A^\infty$-categories (cf.~\cite{stasheff}), which is related to the ``extended moduli space'' introduced by Witten \cite{witten:mirror}. By a construction of Fukaya \cite{fukaya}, to every compact symplectic manifold $(Y,\omega)$ with vanishing first Chern class, one can associate an $A^\infty$-category whose objects are essentially the Lagrangian submanifolds of $Y$, and whose morphisms are determined by the intersections of pairs of submanifolds. Kontsevich's conjecture relates the bounded derived category of the Fukaya category of $Y$ (playing the r\^ole of the $B$-model) to the bounded derived category of the category of coherent sheaves on a mirror partner $X$ (playing the r\^ole of the $A$-model). I must refer the reader to \cite{kont:icm} for further details concerning this fascinating conjecture. The art of making predictions about enumerative geometry from calculations with the variation of Hodge structure on a candidate mirror partner has been considerably refined: see \cite{predictions} for a survey and references to the literature. The era of numerical experiments in mirror symmetry seems to be largely over, and has been supplanted by a more analytical period. Witten's analysis of the physics related to Calabi--Yau manifolds which are hypersurfaces in toric varieties \cite{phases} was further developed in \cite{summing}, where techniques were found---somewhat related to methods introduced by Batyrev \cite{Bat:qcoho} for the study of quantum cohomology of toric varieties---for precisely calculating a variant of the quantum cohomology ring of the Calabi--Yau manifold. (The variant is derived from enumerative problems on the ambient space rather than directly on the Calabi--Yau manifold.) There is a physics argument, but not a complete mathematics argument, which explains why this variant should coincide with the usual quantum cohomology ring after a change of coordinates in the coefficient ring. This variant {\em can}\/ be rigorously shown to agree with the correlation functions of the mirror Calabi--Yau manifold, again calculated in the ``wrong'' coordinates. In this way, the results of \cite{summing} provided the first analytical proof that some kind of enumerative problem on one side of the mirror could be related to a variation of Hodge structure calculation on the other side. Further development of these ideas in \cite{towards-duality} led to a preliminary argument to the effect that the physical theories associated to a Batyrev--Borisov pair should actually be mirror to each other. In a striking recent development, Givental has proved \cite{Givental:homological,Givental:ICM,Givental:equivariant} that for Calabi--Yau complete intersections in projective spaces, the ``predicted'' enumerative formulas which one calculates by using a Batyrev--Borisov candidate mirror partner are in fact correct evaluations of the Gromov--Witten invariants. This establishes, for example, the accuracy of {\em all}\/ of the predictions about the general quintic threefold made by Candelas et al. \cite{CDGP} (and which we listed in table 2). Givental's remarkable proof actually has very little to do with mirror symmetry {\em per se}: in studying an equivariant version of quantum cohomology, he finds enough structure to enable a calculation which is formally similar to (and certainly inspired by) the variation of Hodge structure calculations on the candidate mirror partner. The last several years have also been a period of dramatic developments in string theory. There are new techniques which go by the names of ``duality'' and ``nonperturbative methods,'' and a number of the recent results have been closely related to Calabi--Yau manifolds and mirror symmetry. One of the earliest nonperturbative results \cite{Str:,bhole} was the discovery\footnote{This had been anticipated some time earlier in the physics literature \cite{CDLS,GreenHubsch,CGH,Cd:con} based on the discovery of and speculations about conifold transitions in the mathematics literature \cite{Clemens:double,Friedman:simult,Hirzebruch:examples,tianyau,% reid,Friedman:threefolds}, but an understanding of the physical mechanism behind the attachment of the moduli spaces was lacking.} that the string theory moduli spaces associated to Calabi--Yau manifolds should be attached along loci corresponding to ``conifold transitions''---a process in which a collection of rational curves is contracted to ordinary double points and the resulting space is then smoothed to produce another Calabi--Yau manifold. This new attaching procedure supplements, but is rather different from, the gluing of K\"ahler cones which we discussed in section \ref{sec73}. In the new procedure, a moduli space of a different dimension (corresponding to a Calabi--Yau manifold with different Hodge numbers than the original) is cemented on at the same point where the two like-dimensional pieces (K\"ahler cones differing by a flop) have been glued together. The ``cement'' which holds these two spaces together (i.e., the physical process responsible) is a phase transition between charged black holes on one component of the moduli space and elementary particles on the other. The string theory moduli spaces mentioned above are actually somewhat larger than the conformal field theory moduli spaces which were one of the primary subjects of these lectures. There are two variants of string theory which are relevant, called type IIA and type IIB string theories, and the additional parameters which must be added to the conformal field theory moduli space differs between the two. In the case of type IIA, the extra parameters are a choice of holomorphic $3$-form and the choice of an element in the intermediate Jacobian of the Calabi--Yau threefold. (Some of the mathematical structure of these spaces related to the intermediate Jacobians was anticipated in work of Donagi and Markman \cite{DonMark}.) In the case of type IIB, the new parameters are similar, but related to the even cohomology of the manifold. These two types of parameters should be mapped to each other under mirror symmetry \cite{udual,mirrorII}. In fact, a large number of other related structures called ``D-brane moduli spaces'' should also correspond under mirror symmetry---the precise implications of this correspondence (which appears to be connected to Kontsevich's homological mirror symmetry conjecture) are still being worked out. Finally, in a very exciting recent development, a completely new geometric aspect of mirror symmetry has been discovered by Strominger, Yau and Zaslow \cite{SYZ}. A Calabi--Yau manifold $X$ of real dimension $2n$ on which a complex structure $J$ and K\"ahler form $\omega$ have been fixed has a natural class of $n$-dimensional submanifolds $M$ defined by the property that $\omega|_M\equiv0$ and $\Im(\Omega)|_M\equiv0$ for some choice of holomorphic $n$-form $\Omega$. These {\em special Lagrangian submanifolds}\/ were introduced by Harvey and Lawson \cite{HL} as a natural class of volume-minimizing submanifolds; they have many other interesting properties, including an exceptionally well-behaved deformation theory \cite{mclean}. Strominger, Yau and Zaslow argue on physical grounds (using the correspondence of D-brane moduli spaces mentioned above) that whenever $X$ has a mirror partner, then $X$ must admit a map $\rho:X^{2n}\to B^n$ whose generic fiber is a special Lagrangian $n$-torus, and which has a section $\sigma:B\to X$ whose image is itself a special Lagrangian submanifold. Given this structure, the mirror partner of $X$ is then predicted to be a compactification of the family of dual tori of the fibers of $\rho$. (The section specifies a point $p_b:=\sigma(b)$ on each torus $T_b:=\rho^{-1}(b)$; the dual torus is then $\operatorname{Hom}(\pi_1(T_b,p_b),\operatorname{U}(1))$.) There is also an argument---quite similar in nature to \cite{towards-duality}---that such a structure should suffice for producing a mirror isomorphism between the corresponding physical theories. A mathematical account of this construction can be found in \cite{underlying}, which attempts to make the mathematical implications of this story precise: given a ``special Lagrangian $m$-torus fibration,'' all of the structure we have seen relating the quantum cohomology and the variation of Hodge structure should (conjecturally) follow as a consequence. For the Voisin--Borcea threefolds, the structure of these special Lagrangian torus fibrations (using a mildly degenerate metric) has been worked out in complete detail by Gross and Wilson \cite{GrossWilson}, who find compatibility with the previously observed mirror phenomena in a beautiful geometric form.

proofpile-arXiv_065-523

\section*{Acknowledgments} We are grateful to V.A. Kuzmin and M.A. Shifman for useful discussions. We also wish to thank R. Ball for the clarification of the current status of the QCD analysis of the HERA data. It is the pleasure to thank the members of the University of Minnesota for hospitality during this interesting meeting. The participation at the DPF-96 Meeting of APS was partly supported by the Russian Fund for Fundamental Research, Grant N 96-02-18897. The work on this report is done within the framework of the Grant N 96-01-01860, supported by the Russian Fund for Fundamental Research. \newpage

proofpile-arXiv_065-524

\subsection*{1. Introduction} Supersymmetric theories (SUSY) \cite{R0,R9} are the best motivated extensions of the Standard Model (SM) of the electroweak and strong interactions. They provide an elegant way to stabilize the huge hierarchy between the Grand Unification or Planck scale and the Fermi scale, and its minimal version, the Minimal Supersymmetric Standard Model (MSSM) allows for a consistent unification of the gauge coupling constants and a natural solution of the Dark Matter problem \cite{R1a}. \\ \vspace*{-3mm} Supersymmetry predicts the existence of a left-- and right--handed scalar partner to each Standard Model (SM) quark. The current eigenstates, $\tilde{q}_L$ and $\tilde{q}_R$, mix to give the mass eigenstates $\tilde{q}_1$ and $\tilde{q}_2$; the mixing angle is proportional to the quark mass and is therefore important only in the case of the third generation squarks \cite{R1}. In particular, due to the large value of the top mass $m_t$, the mixing between the left-- and right--handed scalar partners of the top quark, $\tilde{t}_L$ and $\tilde{t}_R$, is very large and after diagonalization of the mass matrix, the lightest scalar top quark mass eigenstate $\tilde{t}_1$ can be much lighter than the top quark and all the scalar partners of the light quarks \cite{R1}. \\ \vspace*{-3mm} If the gluinos [the spin $1/2$ superpartners of the gluons] are heavy enough, scalar quarks will mainly decay into quarks and charginos and/or neutralinos [mixtures of the SUSY partners of the electroweak gauge bosons and Higgs bosons]. These are in general tree--level two--body decays, except in the case of the lightest top squark which could decay into a charm quark and a neutralino through loop diagrams if the decay into a chargino and a bottom quark is not overwhelming \cite{R2}. These decays have been extensively discussed in the Born approximation \cite{R3}. In this paper we will extend these analyses by including the ${\cal O}(\alpha_s)$ corrections, which due to the relatively large value of the strong coupling constant, might be large and might affect significantly the decay rates and the branching ratios\footnote{If the gluinos are lighter than squarks, then squarks will mainly decay into quarks plus gluinos; the QCD corrections to these processes have been recently discussed in Refs.\cite{R5,R5a}.}. \\ \vspace*{-3mm} The particular case of the QCD corrections to scalar quark decays into massless quarks and photinos has been discussed in Refs.~\cite{R4,R5}. In the general case that we will address here, there are three [related] features which complicate the analysis, the common denominator of all these features being the finite value of quark masses: (i) In the case of the decays of top and bottom squarks, one needs to take into account the finite value of the top quark mass in the phase space as well as in the loop diagrams. (ii) Scalar quark mixing will introduce a new parameter which will induce additional contributions; since the mixing angle appears in the Born approximation, it needs to be renormalized. (iii) The finite quark mass [which enters the coupling between scalar quarks, quarks and the neutralino/chargino states] needs also to be renormalized. \\ \vspace*{-3mm} The QCD corrections to the reaction $\tilde{q} \rightarrow q \chi$ analyzed in the present paper are very similar to the case of the reverse process, $t \rightarrow \tilde{t} \chi^0$ and $t \rightarrow \tilde{b} \chi^+$ recently discussed in Ref.~\cite{R6} (see also Ref.~\cite{R7}). During the preparation of this paper, we received a report by Kraml et al. \cite{R8}, where a similar analysis has been conducted. Our analytical results agree with those given in this paper\footnote{We thank the Vienna group and in particular S. Kraml for their cooperation in resolving some discrepancies with some of the formulae and plots given in the early version of the paper Ref.~\cite{R8}. We also thank T. Plehn for checking independently the results.}. We extend their numerical analysis, which focused on the decay of the lightest top squark into the lightest charginos and neutralinos, by discussing the decays into the heavier charginos and neutralinos and by studying the case of bottom squarks and the SUSY partners of light squarks. \subsection*{2. Born Approximation} In the Minimal Supersymmetric Standard Model \cite{R0,R9}, there are two charginos $\chi_i^+ [i=1,2$] and four neutralinos $\chi_{i}^0$ [$i=1$--4]. Their masses and their couplings to squarks and quarks are given in terms of the Higgs--higgsino mass parameter $\mu$, the ratio of the vacuum expectation values $\tan\beta$ of the two Higgs doublet MSSM fields needed to break the electroweak symmetry, and the wino mass parameter $M_2$. The bino and gluino masses are related to the parameter $M_2$ [$M_1 \sim M_2/2$ and $m_{\tilde{g}} \sim 3.5 M_2$] when the gaugino masses and the three coupling constants of SU(3)$\times$SU(2)$\times$U(1) are unified at the Grand Unification scale. \\ \vspace*{-3mm} The squark masses are given in terms of the parameters $\mu$ and $\tan\beta$, as well as the left-- and right--handed scalar masses $M_{\tilde{q}_L}$ and $M_{\tilde{q}_R}$ [which in general are taken to be equal] and the soft--SUSY breaking trilinear coupling $A_q$. The top and bottom squark mass eigenstates, and their mixing angles, are determined by diagonalizing the following mass matrices \begin{equation} {\cal M}^2_{\tilde{t}} = \left( \begin{array}{cc} M_{\tilde{t}_L}^2 + m_t^2 + \cos 2 \beta (\frac{1}{2} - \frac{2}{3}s_W^2) \, M_Z^2 & m_t \, M^{LR}_t \\ m_t \, M^{LR}_t & M_{\tilde{t}_R}^2 + m_t^2 + \frac{2}{3}\cos 2 \beta \; s_W^2 \, M_Z^2 \end{array} \right) \end{equation} \begin{equation} {\cal M}^2_{\tilde{b}} = \left( \begin{array}{cc} M_{\tilde{t}_L}^2 + m_b^2 + \cos 2 \beta (-\frac{1}{2} +\frac{1}{3}s_W^2) \, M_Z^2 & m_b \, M^{LR}_b \\ m_b \, M^{LR}_b & M_{\tilde{b}_R}^2 + m_b^2 - \frac{1}{3}\cos 2 \beta \; s_W^2 \, M_Z^2 \end{array} \right) \end{equation} where $M^{LR}_{t,b}$ in the off--diagonal terms read: $M^{LR}_t = A_t - \mu \, \cot \beta$ and $M^{LR}_b = A_b - \mu \tan\beta$. \\ \vspace*{-3mm} In the Born approximation, the partial widths for the decays $\tilde{t}_i \rightarrow t\chi^0_j$, $\tilde{t}_i \rightarrow b\chi^+_j$ can be written as $[q\equiv t$ or $b$, and we drop the indices of the neutralino/chargino states] \begin{eqnarray} \Gamma_0( \tilde{t}_i \rightarrow q \chi) = \frac{\alpha}{4\,m_{\tilde{t}_i}^3} \bigg[ ( {c_L^i}^2 + {c_R^i}^2 ) \, ( m_{\tilde{t}_i}^2 - m_{q}^2 - m_{\chi}^2 ) - 4 \, c_L^i\,c_R^i \, m_{q}\,m_{\chi}\, \epsilon_\chi \bigg] \,\lambda^{1/2}(m_{\tilde{t}_i}^2,m_{q}^2,m_{\chi}^2) \end{eqnarray} where $\lambda (x,y,z)=x^2+y^2+z^2-2\,(xy+xz+yz)$ is the usual two--body phase space function and $\epsilon_\chi$ is the sign of the eigenvalue of the neutralino $\chi$. The couplings $c_{L,R}^i$ for the neutral current process, $\tilde{t}_i \rightarrow t \chi^0$, are given by \begin{eqnarray} \left\{ \begin{array}{c} c_R^1 \\ c_R^2 \end{array} \right\} &=& b\,m_t\, \left\{ \begin{array}{c} \st{t} \\ \ct{t} \end{array} \right\} + f_L\, \left\{ \begin{array}{c} \ct{t} \\ -\st{t} \end{array} \right\} \nonumber \\ \left\{ \begin{array}{c} c_L^1 \\ c_L^2 \end{array} \right\} &=& b\,m_t\, \left\{ \begin{array}{c} \ct{t} \\ -\st{t} \end{array} \right\} + f_R\, \left\{ \begin{array}{c} \st{t} \\ \ct{t} \end{array} \right\} \end{eqnarray} \begin{eqnarray} b & = & \frac{1}{\sqrt{2}\, M_W \sin\beta\,s_W} \; N_{j4} \nonumber \\ f_L & = & \sqrt{2}\left[ \frac{2}{3} \; N_{j1}' + \left(\frac{1}{2} - \frac{2}{3}\, s_W^2 \right) \frac{1}{c_W s_W}\;N_{j2}' \right] \nonumber \\ f_R & = &-\sqrt{2}\left[ \frac{2}{3} \; N_{j1}' - \frac{2}{3} \frac{s_W}{c_W} \; N_{j2}' \right] \ , \end{eqnarray} and for the charged current process, $\tilde{t}_i \rightarrow b\chi^+$, \begin{eqnarray} \left\{ \begin{array}{c} c_L^1 \\ c_L^2 \end{array} \right\} & = & \frac{m_b\,U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta} \left\{ \begin{array}{c} -\ct{t} \\ \st{t} \end{array} \right\} \nonumber \\ \left\{ \begin{array}{c} c_R^1 \\ c_R^2 \end{array} \right\} & = & \frac{V_{j1}}{s_W} \, \left\{ \begin{array}{c} \ct{t} \\ -\st{t} \end{array} \right\} - \frac{m_t\,V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta} \left\{ \begin{array}{c} \st{t} \\ \ct{t} \end{array} \right\} \ . \end{eqnarray} In these equations, $\theta_t$ is the $\tilde{t}$ mixing angle [which as discussed previously can be expressed in terms of the Higgs--higgsino SUSY mass parameter $\mu$, $\tan\beta$ and the soft--SUSY breaking trilinear coupling $A_t$] with $s_{\theta}=\sin\theta$, $c_{\theta}=\cos\theta$ etc.; $s_W^2=1-c_W^2\equiv \sin^2\theta_W$ and $N, U/V$ are the diagonalizing matrices for the neutralino and chargino states \cite{R10} with \begin{eqnarray} N'_{j1}= c_W N_{j1} +s_W N_{j2} \ \ \ , \ \ \ N'_{j2}= -s_W N_{j1} +c_W N_{j2} \ . \end{eqnarray} A similar expression eq.~(3) can be obtained for the neutral and charged decays of bottom squarks, $\tilde{b}_i \rightarrow b \chi_j^0$ and $ \tilde{b} \rightarrow t \chi_j^-$ \begin{eqnarray} \Gamma_0( \tilde{b}_i \rightarrow q \chi) = \frac{\alpha}{4\,m_{\tilde{b}_i}^3} \bigg[ ( {c_L^i}^2 + {c_R^i}^2 ) \, ( m_{\tilde{b}_i}^2 - m_{q}^2 - m_{\chi}^2 ) - 4 \, c_L^i\,c_R^i \, m_{q}\,m_{\chi}\, \epsilon_\chi \bigg] \,\lambda^{1/2}(m_{\tilde{b}_i}^2,m_{q}^2,m_{\chi}^2) \end{eqnarray} with the couplings $c_{L,R}^i$ in the neutral decay $\tilde{b} \rightarrow b \chi^0$ given by [$\theta_b$ is the $\tilde{b}$ mixing angle] \begin{eqnarray} \left\{ \begin{array}{c} c_R^1 \\ c_R^2 \end{array} \right\} &=& b\,m_b\, \left\{ \begin{array}{c} \st{b} \\ \ct{b} \end{array} \right\} + f_L\, \left\{ \begin{array}{c} \ct{b} \\ -\st{b} \end{array} \right\} \nonumber \\ \left\{ \begin{array}{c} c_L^1 \\ c_L^2 \end{array} \right\} &=& b\,m_b\, \left\{ \begin{array}{c} \ct{b} \\ -\st{b} \end{array} \right\} + f_R\, \left\{ \begin{array}{c} \st{b} \\ \ct{b} \end{array} \right\} \end{eqnarray} \begin{eqnarray} b & = & \frac{1}{\sqrt{2}\, M_W \cos\beta\,s_W} \; N_{j3} \nonumber \\ f_L & = & \sqrt{2}\left[ -\frac{1}{3} \; N_{j1}' + \left(-\frac{1}{2} +\frac{1}{3}\, s_W^2 \right) \frac{1}{c_W s_W}\;N_{j2}' \right] \nonumber \\ f_R & = &-\sqrt{2}\left[ -\frac{1}{3} \; N_{j1}' + \frac{1}{3} \frac{s_W}{c_W} \; N_{j2}' \right] \ , \end{eqnarray} and for the charged current process, $\tilde{b}_i \rightarrow t\chi^-$, \begin{eqnarray} \left\{ \begin{array}{c} c_L^1 \\ c_L^2 \end{array} \right\} & = & \frac{m_t\,V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta} \left\{ \begin{array}{c} -\ct{b} \\ \st{b} \end{array} \right\} \nonumber \\ \left\{ \begin{array}{c} c_R^1 \\ c_R^2 \end{array} \right\} & = & \frac{U_{j1}}{s_W} \, \left\{ \begin{array}{c} \ct{b} \\ -\st{b} \end{array} \right\} - \frac{m_b\,U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta} \left\{ \begin{array}{c} \st{b} \\ \ct{b} \end{array} \right\} \ . \end{eqnarray} In the case where the mass of the final quark and the squark mixing angle are neglected [as it is the case for the first and second generation squarks], the decay widths simplify to \begin{eqnarray} \Gamma_0(\tilde{q}_i \rightarrow q \chi) = \frac{\alpha}{4}\,m_{\tilde{q}_i}\, \left( 1- \frac{m_{\chi}^2}{m_{\tilde{q}_i}^2} \right)^2 f_i^2 \end{eqnarray} where the $f_i$'s [with now $i=L,R$ since there is no squark mixing] in the case of the neutral decays, $\tilde{q} \rightarrow q \chi^0$, are given in terms of the quark isospin $I_{3L}^q$ and charge $e_q$, by \begin{eqnarray} f_L & = & \sqrt{2}\left[ e_q \; N_{j1}' + \left(I_{3L}^q - e_q s_W^2 \right) \frac{1}{c_W s_W}\;N_{j2}' \right] \nonumber \\ f_R & = &-\sqrt{2}\left[ e_q \; N_{j1}' -e_q\, \frac{s_W}{c_W} \; N_{j2}' \right] \ , \end{eqnarray} while for the charged decays, $\tilde{q} \rightarrow q' \chi^+$ one has for up--type (down--type) squarks: \begin{eqnarray} f_L= V_{j1}/s_W \ (U_{j1}/s_W) \ \ , \ \ f_R=0 \ . \end{eqnarray} \subsection*{3. QCD corrections to Top Squark Decays} The QCD corrections to the top squark decay width, eq.~(3), consist of virtual corrections Figs.1a--d, and real corrections with an additional gluon emitted off the initial $\tilde{t}$ or final $t$ [for the neutral decay] or $b$ [for the charged decay] quark states, Fig.~1e. The ${\cal O}(\alpha_s)$ virtual contributions can be split into gluon and gluino exchange in the $q$--$\tilde{t}$--$\chi$ [$q=t,b$] vertex as well as mixing diagrams and the $\tilde{t}$ and $t/b$ wave function renormalization constants. The renormalization of the $q$--$\tilde{t}$--$\chi$ coupling is achieved by renormalizing the top/bottom quark masses and the $\tilde{t}$ mixing angle. We will use the dimensional reduction scheme\footnote{The quark mass and wave-function counterterms will be different in the dimensional regularization \cite{R11a} and dimensional reduction schemes \cite{R11}. Since dimensional reduction is the scheme which preserves supersymmetry, we will present our results in this scheme.} to regularize the ultraviolet divergencies, and a fictitious gluon mass $\lambda$ is introduced to regularize the infrared divergencies. \subsubsection*{3.1 Virtual Corrections} The QCD virtual corrections to the $\tilde{t}_i$--$\chi$--$q$ interaction vertex can be cast into the form \begin{eqnarray} \delta \Gamma^i = ie \ \frac{\alpha_s}{3\pi} \, \sum_{j=g, \tilde{g}, {\rm mix}, {\rm ct} } \left[ G_{j,L}^i P_L + G_{j,R}^i P_R \right] \end{eqnarray} where $G^i_{g}, \,G^i_{\tilde{g}}, \,G^i_{\rm mix}$ and $G^i_{\rm ct}$ denote the gluon and gluino exchanges in the vertex, and the mixing and counterterm contributions, respectively. \\ \vspace*{-3mm} The contribution of the gluonic exchange [Fig.~1a] can be written as \begin{eqnarray} G^i_{g,L,R} = c_{L,R}^i \, F_1^i + c_{R,L}^i \,F_2^i \end{eqnarray} with the form factors $F^i_{1,2}$ given by \begin{eqnarray} F_1^i & = & B_0 + 2 \, m_{q}^2 \, C_0 - 2 \, m_{\tilde{t}_i}^2 \, (C_{11}-C_{12}) + 2 \, m_{\chi}^2 \, C_{11} \nonumber \\ F_2^i & = & -2 \, m_{q} \, m_{\chi} \, (C_0+C_{11}) \end{eqnarray} with $q \equiv t$ for the neutral and $q \equiv b$ for the charged decays; the two and three--point Passarino--Veltman functions, $B_0 \equiv B_0(m_{\tilde{t}_i}^2,\lambda,m_{\tilde{t}_i})$ and $C_{..} \equiv C_{..}(m_{q}^2, m_{\tilde{t}_i}^2, m_{\chi}^2,$ $m_{q}^2, \lambda^2,m_{\tilde{t}_i}^2)$ can be found in Ref.~\cite{R12}. \\ \vspace*{-3mm} The gluino exchange contributions [Fig.~1b], are given by \begin{eqnarray} G_{\tilde{g},L,R}^i & = & -2 \sum_{k=1,2} \, d_{L,R}^k \bigg[ (v_{\tilde{q}}^k v_{\tilde{t}}^i+a_{\tilde{q}}^k a_{\tilde{t}}^i) F_4^{ik} \mp (a_{\tilde{q}}^k v_{\tilde{t}}^i+v_{\tilde{q}}^k a_{\tilde{t}}^i) F_5^{ik} \nonumber \\ & & \hspace{1.5cm} + (v_{\tilde{q}}^k v_{\tilde{t}}^i-a_{\tilde{q}}^k a_{\tilde{t}}^i) F_6^{ik} \mp (a_{\tilde{q}}^k v_{\tilde{t}}^i-v_{\tilde{q}}^k a_{\tilde{t}}^i) F_7^{ik} \bigg] \nonumber \\ & & \hspace{1.5cm} + d_{R,L}^k \bigg[ (v_{\tilde{q}}^k v_{\tilde{t}}^i+a_{\tilde{q}}^k a_{\tilde{t}}^i) F_1^{ik} \mp (a_{\tilde{q}}^k v_{\tilde{t}}^i+v_{\tilde{q}}^k a_{\tilde{t}}^i) F_1^{ik} \nonumber \\ & & \hspace{1.5cm} + (v_{\tilde{q}}^k v_{\tilde{t}}^i-a_{\tilde{q}}^k a_{\tilde{t}}^i) F_2^{ik} \mp (a_{\tilde{q}}^k v_{\tilde{t}}^i-v_{\tilde{q}}^k a_{\tilde{t}}^i) F_3^{ik} \bigg] \end{eqnarray} with again $q=t$ for the neutral decay and $q=b$ for the charged one; the form factors $F^{ik}_{1,..,7}$ read \begin{eqnarray} F_1^{ik} & = & m_{\tilde{g}}\,m_{\chi}\, [C_0+C_{12}] \nonumber \\ F_{2,3}^{ik} & = & m_{\chi}\, [\pm m_{q}\, (C_0+C_{11})+m_t\,C_{12}] \nonumber \\ F_{4,5}^{ik} & = & m_{\tilde{g}}\, [m_t \,C_0 \pm m_{q}\,(C_{11}-C_{12})] \nonumber \\ F_{6,7}^{ik} & = & m_{\tilde{q}_k}^2\,C_0 \pm m_t \,m_{q}\, [C_0+C_{11}-C_{12}] +m_{q}^2\, [C_{11}-C_{12}] +m_{\chi}^2\,C_{12}+B_0 \end{eqnarray} with the two-- and three--point functions $B_0\equiv B_0(m_{\tilde{t}_i}^2,m_{\tilde{g}},m_t)$ and $C_{..} \equiv C_{..} (m_{q}^2,m_{\tilde{t}_i}^2,m_{\chi}^2,m_{\tilde{q}}^2,$ $m_{\tilde{g}}^2, m_t^2)$. The couplings $d_{R,L}^k$ are given by \begin{eqnarray} d_{L,R}^k \hspace{0.3cm} = c_{R,L}^k \ \end{eqnarray} for neutralinos, while for the charginos one has \begin{eqnarray} \left\{\begin{array}{c} d_L^1 \\ d_L^2 \end{array} \right\} & = & \frac{U_{j1}}{s_W}\, \left\{\begin{array}{c} \ct{b} \\ -\st{b} \end{array} \right\} -\frac{m_b\,U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta}\, \left\{\begin{array}{c} \st{b} \\ \ct{b} \end{array} \right\} \nonumber \\ \left\{\begin{array}{c} d_R^1 \\ d_R^2 \end{array} \right\} & = & \frac{m_t\,V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta}\, \left\{\begin{array}{c} -\ct{b} \\ \st{b} \end{array} \right\} \ . \end{eqnarray} The $v_{\tilde{q}}^i$ and $a_{\tilde{q}}^i$ couplings read \begin{eqnarray} v_{\tilde{q}}^1 & = & {\textstyle\frac{1}{2}}\,( \ct{q}-\st{q} ) \ , \hspace{1.cm} v_{\tilde{q}}^2 \; = \; {\textstyle - \frac{1}{2}}\,( \ct{q}+\st{q} ) \ , \nonumber \\ a_{\tilde{q}}^1 & = & {\textstyle\frac{1}{2}}\,( \ct{q}+\st{q} ) \ , \hspace{1.cm} a_{\tilde{q}}^2 \; = \; {\textstyle\frac{1}{2}}\,( \ct{q}-\st{q} ) \ . \end{eqnarray} \vspace*{3mm} Finally, the mixing contributions due to the diagrams Fig.~1c, yield the expressions \begin{eqnarray} G_{\rm mix,L,R}^i & = & \frac{(-1)^i\,(\delta_{1i}\,c_{L,R}^2 + \delta_{2i}\,c_{L,R}^1)} {m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2} \, \bigg[ 4 m_t \,m_{\tilde{g}}\, c_{2 \theta_t}\,B_0(m_{\tilde{t}_i}^2, m_t,m_{\tilde{g}}) \nonumber \\ & & \hspace{4.5cm} + \, c_{2 \theta_t} s_{2\theta_t} ( A_0(m_{\tilde{t}_2}^2)- A_0(m_{\tilde{t}_1}^2) ) \bigg] \ . \end{eqnarray} Therein, $A_0$ is the Passarino--Veltman one--point function. Note that all these contributions are the same in both the dimensional reduction and dimensional regularization schemes. \subsubsection*{3.2 Counterterms} The counterterm contributions in eq.~(15) are due to the $\tilde{t}$ and $t/b$ wave function renormalizations [Fig.~1d] as well as the renormalization of the quark mass $m_t$ or $m_b$ and the mixing angle $\theta_t$, which appear in the Born couplings. \\ \vspace*{-3mm} For the neutral decay process, $\tilde{t}_i \rightarrow t\chi^0_j$, the counterterm contribution is given by \begin{eqnarray} G^{1,2}_{\rm ct,L} & = & \frac{1}{2}\,c^{1,2}_L\,( \delta Z^t_R + \delta Z_{\tilde{t}_{1,2}}) + b \, \{\ct{t},-\st{t}\} \, \delta m_t - b\,m_t \, \{\st{t},\ct{t}\} \, \delta \theta_t + f_R \, \{\ct{t},-\st{t}\} \, \delta \theta_t \nonumber \\ G^{1,2}_{\rm ct,R} & = & \frac{1}{2}\,c^{1,2}_R\,( \delta Z^t_L + \delta Z_{\tilde{t}_{1,2}}) + b \, \{\st{t},\ct{t}\} \, \delta m_t + b\,m_t \, \{\ct{t},-\st{t}\} \, \delta \theta_t - f_L \, \{\st{t},\ct{t}\} \, \delta \theta_t \ , \nonumber \\ && \end{eqnarray} whereas for the charged current process, $\tilde{t}_i \rightarrow b\chi^+_j$, one obtains, \begin{eqnarray} G^{1,2}_{\rm ct,L} & = & \frac{1}{2}\,c^{1,2}_L\, \left[ \delta Z^b_R + \delta Z_{\tilde{t}_{1,2}} + 2\,\frac{\delta m_b}{m_b} \right] + \frac{ m_b U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta} \,\{\st{t},\ct{t}\}\,\delta \theta_t \nonumber \\ G^{1,2}_{\rm ct,R} & = & \frac{1}{2}\,c^{1,2}_R\, \left[ \delta Z^b_L + \delta Z_{\tilde{t}_{1,2}} \right] - \,\frac{ \delta m_t \, V_{j2}}{\sqrt{2}\,s_W\,M_W\, \sin\beta} \,\{\st{t},\ct{t}\} \nonumber \\ & & \vspace{0.5cm} - \frac{V_{j1}}{s_W}\,\{\st{t},\ct{t}\}\,\delta \theta_t - \frac{m_t V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta} \,\{\ct{t},-\st{t}\}\,\delta \theta_t \ . \end{eqnarray} In the on--shell scheme, the quark and squark masses are defined as the poles of the propagators and the wave--function renormalization constants follow from the residues at the poles; the corresponding counterterms are given by (see also Refs.~\cite{R6,R8}) \begin{eqnarray} \frac{\delta m_q}{m_q} & = & \frac{1}{2} \bigg[ \Sigma^q_R(m_q^2)+\Sigma^q_L(m_q^2)\bigg] + \Sigma^q_S(m_q^2) \nonumber \\ \delta Z^q_L & = & - \Sigma^q_L(m_q^2) - m_q^2 \bigg[ {\Sigma^q_L}^{\prime}(m_q^2) +{\Sigma^q_R}^{\prime}(m_q^2)+2\,{\Sigma^q_S}^{\prime}(m_q^2) \bigg] \nonumber \\ \delta Z^q_R & = & - \Sigma^q_R(m_q^2) - m_q^2 \bigg[ {\Sigma^q_L}^{\prime}(m_q^2) +{\Sigma^q_R}^{\prime}(m_q^2)+2\,{\Sigma^q_S}^{\prime}(m_q^2) \bigg] \nonumber \\ \delta Z_{\tilde{t}_i} & = & - \left(\Sigma_{\tilde{t}}^{ii} \right)'(m_{\tilde{t}_i}^2) \end{eqnarray} In the dimensional reduction scheme, the self--energies $\Sigma$ and their derivatives $\Sigma'$, up to a factor $\alpha_s /3\pi$ which has been factorized out, are given by \cite{R6,R8} \begin{eqnarray} \Sigma^q_L(k^2) & = & - \bigg[ 2 \,B_1(k^2,m_q,\lambda) + (1+c_{2 \theta_q}) B_1(k^2,m_{\tilde{g}},m_{\tilde{q}_1}) + (1-c_{2 \theta_q}) B_1(k^2,m_{\tilde{g}},m_{\tilde{q}_2}) \bigg] \nonumber \\ \Sigma^q_R(k^2) & = & - \bigg[ 2 \,B_1(k^2,m_q,\lambda) + (1-c_{2 \theta_q}) B_1(k^2,m_{\tilde{g}},m_{\tilde{q}_1}) + (1+c_{2 \theta_q}) B_1(k^2,m_{\tilde{g}},m_{\tilde{q}_2}) \bigg] \nonumber \\ \Sigma^q_S(k^2) & = & - \bigg[ 4 \,B_0(k^2,m_q,\lambda) + \frac{m_{\tilde{g}}}{m_q}\,s_{2 \theta_q}\, ( B_0(k^2,m_{\tilde{g}},m_{\tilde{q}_1}) - B_0(k^2,m_{\tilde{g}},m_{\tilde{q}_2}) ) \bigg] \nonumber \\ (\Sigma_{\tilde{t}}^{ii})'(k^2) & = & - 2 \bigg[ - 2\,B_1(k^2,m_{\tilde{t}_i},\lambda) - 2\,k^2 \,B_1'(k^2,m_{\tilde{t}_i},\lambda) + (m_t^2+m_{\tilde{g}}^2-k^2)\,B_0'(k^2,m_t,m_{\tilde{g}}) \nonumber \\ & & \hspace{0.8cm} - \,B_0(k^2,m_t,m_{\tilde{g}}) + (-1)^i\,2\,s_{2 \theta}\,m_t\,m_{\tilde{g}} B_0'(k^2,m_t,m_{\tilde{g}}) \bigg] \ . \end{eqnarray} Using dimensional regularization, the quark self--energies differ from the previous expressions by a constant; in terms of the their values in the dimensional reduction scheme, they are given by \begin{eqnarray} \left. \Sigma_{L,R}^q \right|_{\rm dim.~reg.} = \Sigma_{L,R}^q -2 \ \ , \ \ \left. \Sigma_{S}^q \right|_{\rm dim.~reg.} = \Sigma_{S}^q + 2 \ . \end{eqnarray} Finally, we need a prescription to renormalize the $\tilde{t}$ mixing angle $\theta_t$. Following Ref.~\cite{R13}, we choose this condition in such a way that it cancels exactly the mixing contributions eq.~(23) for the decay $\tilde{t_2} \rightarrow t \chi^0$ \begin{eqnarray} \delta\theta_t & = & \frac{1} {m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2} \left[4 \, m_t \,m_{\tilde{g}} \,c_{2 \theta_t} \,B_0(m_{\tilde{t}_2}^2,m_t, m_{\tilde{g}}) + c_{2 \theta_t} s_{2\theta_t} (A_0(m_{\tilde{t}_2}^2)- A_0(m_{\tilde{t}_1}^2) ) \right] \ . \end{eqnarray} Alternatively, since the lightest top squark $\tilde{t}_1$ can be lighter than the top quark and then is more likely to be discovered first in the top decays $t \rightarrow \tilde{t}_1 \chi_0$, one can choose the renormalization condition such that the mixing contributions are cancelled in the latter process; this leads to a counterterm similar to eq.~(29) but with $B_0(m_{\tilde{t}_2}^2,m_t, m_{\tilde{g}})$ replaced by $B_0(m_{\tilde{t}_1}^2,m_t, m_{\tilde{g}})$. The difference between the two renormalization conditions, \begin{eqnarray} \Delta \delta\theta_t = \frac{4 m_t \,m_{\tilde{g}} \,c_{2 \theta_t}} {m_{\tilde{t}_1}^2-m_{\tilde{t}_2}^2} \left[ B_0(m_{\tilde{t}_1}^2,m_t,m_{\tilde{g}}) - B_0(m_{\tilde{t}_2}^2,m_t,m_{\tilde{g}}) \right] \end{eqnarray} is, however, very small numerically. Indeed, if $m_{\tilde{t}_1}$ is a few GeV away from $m_{\tilde{t}_2}$, one has $\theta_t \simeq -\pi/4$ and therefore $c_{2 \theta_t} \sim 0$, leading to a difference which is less than one permille for the scenario of Figs.~2a/b. For degenerate top squarks, one has $\Delta \delta \theta =4m_t m_{\tilde{g}} c_{2 \theta_t} B_0' (m_{\tilde{t}_2}^2,m_t,m_{\tilde{g}})$ which is also very small numerically [less than $\sim 1\% $ for the scenarios of Fig.~2.] \\ \vspace*{-3mm} The complete virtual corrections to the $\tilde{t}_i \rightarrow q \chi$ decay width is then given by \begin{eqnarray} \Gamma^V(\tilde{t}_i \rightarrow q \chi) & = & \frac{\alpha}{6 \, m_{\tilde{t}_i}^3} \frac{\alpha_s}{\pi} \; \mbox{Re} \; \bigg\{ (c_L^i \, G_L^i + c_R^i \, G_R^i)\, ( m_{\tilde{t}_i}^2 - m_{q}^2 - m_{\chi}^2 ) \nonumber \\ & & \hspace{1.6cm} - \; 2 \; ( c_L^i \, G_R^i + c_R^i \, G_L^i ) \, m_{q} \, m_{\chi} \epsilon_\chi \, \bigg\} \, \lambda^{1/2}(m_{\tilde{t}_i}^2,m_{q}^2,m_{\chi}^2) \ . \end{eqnarray} The sum of all virtual contributions including the counterterms are ultraviolet finite as it should be, but they are still infrared divergent; the infrared divergencies will be cancelled after adding the real corrections. \subsubsection*{3.3 Real Corrections} The contributions to the squark decay widths from the real corrections, with an additional gluon emitted from the initial $\tilde{t}$ or final $t/b$ states, can be cast into the form \begin{eqnarray} \Gamma_{\rm real}^i & = & \frac{2\,\alpha}{3 \, m_{\tilde{t}_i}} \frac{\alpha_s}{\pi} \bigg\{ 8 \; c_L^i \, c_R^i \; m_{q} \, m_{\chi} \epsilon_\chi \, \big[\; ( m_{\tilde{t}_i}^2 + m_{q}^2 - m_{\chi}^2) \, I_{01} + m_{\tilde{t}_i}^2 \, I_{00} + m_{q}^2 \, I_{11} + I_0 + I_1 \big] \nonumber \\ & & \hspace{1.6cm} +\; ({c_L^i}^2+{c_R^i}^2) \, \big[\; 2 \, ( m_{q}^2 + m_{\chi}^2 - m_{\tilde{t}_i}^2 ) \, ( m_{\tilde{t}_i}^2 \, I_{00} + m_{q}^2 \, I_{11} + I_0 + I_1 ) \nonumber \\ & & \hspace{4.1cm} + 2 \, ( m_{q}^4 - \; ( m_{\chi}^2 - m_{\tilde{t}_i}^2 )^2 ) \, I_{01} - I - I_1^0 \big] \bigg\} \end{eqnarray} where the phase space integrals $ I(m_{\tilde{t}_i},m_{q},m_{\chi}) \equiv I $ are given by \cite{R14} \begin{eqnarray} I_{00} & = & \frac{1}{4\,m_{\tilde{t}_i}^4}\bigg[ \kappa \, \ln \bigg( \frac{\kappa^2}{\lambda\,m_{\tilde{t}_i}\,m_{q}\,m_{\chi}}\bigg) -\kappa-(m_{q}^2-m_{\chi}^2) \ln \bigg(\frac{\beta_1}{\beta_2}\bigg)-m_{\tilde{t}_i}^2\,\ln (\beta_0) \bigg] \nonumber \\ I_{11} & = & \frac{1}{4\,m_{q}^2\,m_{\tilde{t}_i}^2}\bigg[ \kappa \, \ln \bigg( \frac{\kappa^2}{\lambda\,m_{\tilde{t}_i}\,m_{q}\,m_{\chi}}\bigg) -\kappa-(m_{\tilde{t}_i}^2-m_{\chi}^2)\ln \bigg(\frac{\beta_0}{\beta_2}\bigg)-m_{q}^2\,\ln (\beta_1) \bigg] \nonumber \\ I_{01} & = & \frac{1}{4\,m_{\tilde{t}_i}^2} \bigg[ -2\,\ln \bigg(\frac{\lambda\,m_{\tilde{t}_i}\, m_{q}\,m_{\chi}}{\kappa^2} \bigg)\,\ln (\beta_2) + 2\,\ln^2(\beta_2) - \ln^2(\beta_0) - \ln^2(\beta_1) \nonumber \\ & & + 2\,\mbox{Li}_2\,(1-\beta_2^2) - \mbox{Li}_2 \,(1-\beta_0^2) - \mbox{Li}_2\,(1-\beta_1^2) \bigg] \nonumber \\ I & = & \frac{1}{4\,m_{\tilde{t}_i}^2} \bigg[ \frac{\kappa}{2}(m_{\tilde{t}_i}^2 +m_{q}^2+m_{\chi}^2) +2\,m_{\tilde{t}_i}^2\,m_{q}^2\,\ln (\beta_2) +2\,m_{\tilde{t}_i}^2\,m_{\chi}^2\,\ln (\beta_1) +2\,m_{q}^2\,m_{\chi}^2\,\ln (\beta_0) \bigg] \nonumber \\ I_0 & = & \frac{1}{4\,m_{\tilde{t}_i}^2} \bigg[ -2\,m_{q}^2\,\ln (\beta_2) -2\,m_{\chi}^2\,\ln (\beta_1)-\kappa \bigg] \nonumber \\ I_1 & = & \frac{1}{4\,m_{\tilde{t}_i}^2}\bigg[ -2\,m_{\tilde{t}_i}^2\, \ln (\beta_2) -2\,m_{\chi}^2\,\ln (\beta_0)-\kappa \bigg] \nonumber \\ I_1^0 & = & \frac{1}{4\,m_{\tilde{t}_i}^2}\bigg[ m_{\tilde{t}_i}^4 \, \ln (\beta_2) -m_{\chi}^2 \,(2\,m_{q}^2-2\,m_{\tilde{t}_i}^2 +m_{\chi}^2) \, \ln (\beta_0) -\frac{\kappa}{4}\,(m_{q}^2-3\,m_{\tilde{t}_i}^2+5\,m_{\chi}^2) \bigg] \ . \end{eqnarray} with $\kappa = \lambda^{1/2}(m_{\tilde{t}_i}^2,m_{q},m_{\chi})$ and \begin{equation} \beta_0 = \frac{m_{\tilde{t}_i}^2-m_{q}^2-m_{\chi}^2+\kappa} {2\,m_{q}\,m_{\chi}},\;\; \beta_1 = \frac{m_{\tilde{t}_i}^2-m_{q}^2+m_{\chi}^2-\kappa} {2\,m_{\tilde{t}_i}\,m_{\chi}},\;\; \beta_2 = \frac{m_{\tilde{t}_i}^2+m_{q}^2-m_{\chi}^2-\kappa} {2\,m_{\tilde{t}_i}\,m_{q}} \ . \end{equation} \bigskip \noindent Our analytical results agree with the results obtained recently in Ref.~\cite{R8}. \subsection*{4. QCD corrections to other squark decays} \subsubsection*{4.1 Bottom Squark Decays} In the case of the bottom squark decays, $\tilde{b}_i \rightarrow b \chi^0$ and $\tilde{b}_i \rightarrow t \chi^-$, the analytical expressions of the QCD corrections are just the same as in the previous section once the proper changes of the squark [$m_{\tilde{t}_i} \rightarrow m_{\tilde{b}_i}$], the quark $[q\equiv b$ and $q\equiv t$ for the neutral and charged decays] masses and the mixing angles $[\theta_t \rightarrow \theta_b$] are performed. The couplings for $\tilde{b}$ decays are as given in section 2: for the $d^k_{L,R}$ couplings, one has in the case of the neutral decay $\tilde{b}_i \rightarrow b \chi^0$ \begin{eqnarray} d_{L,R}^k \hspace{0.3cm} = c_{R,L}^k \ , \end{eqnarray} with $c_{L,R}^k$ of eq.~(11), while in the charged decay $\tilde{b}_i \rightarrow t \chi^-$, they read \begin{eqnarray} \left\{\begin{array}{c} d_L^1 \\ d_L^2 \end{array} \right\} & = & \frac{V_{j1}}{s_W}\, \left\{\begin{array}{c} \ct{t} \\ -\st{t} \end{array} \right\} -\frac{m_t\,V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta}\, \left\{\begin{array}{c} \st{t} \\ \ct{t} \end{array} \right\} \nonumber \\ \left\{\begin{array}{c} d_R^1 \\ d_R^2 \end{array} \right\} & = & \frac{m_b\,U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta}\, \left\{\begin{array}{c} -\ct{t} \\ \st{t} \end{array} \right\} \ . \end{eqnarray} The counterterm contributions are the same as in eq.~(24) with the change $(t, \tilde{t}) \rightarrow (b, \tilde{b})$ in the neutral decay; in the charged decay mode they are different due to different couplings (see also Refs.~\cite{R6,R8}): \begin{eqnarray} G^{1,2}_{\rm ct,L} & = & \frac{1}{2}\,c^{1,2}_L\, \left[ \delta Z^t_R + \delta Z_{\tilde{b}_{1,2}} + 2\,\frac{\delta m_t}{m_t} \right] + \frac{ m_t V_{j2}}{\sqrt{2}\,s_W\,M_W\,\sin\beta} \,\{\st{b},\ct{b}\}\,\delta \theta_b \nonumber \\ G^{1,2}_{\rm ct,R} & = & \frac{1}{2}\,c^{1,2}_R\, \left[ \delta Z^t_L + \delta Z_{\tilde{b}_{1,2}} \right] - \,\frac{ \delta m_b \, U_{j2}}{\sqrt{2}\,s_W\,M_W\, \cos\beta} \,\{\st{b},\ct{b}\} \nonumber \\ & & \vspace{0.5cm} - \frac{U_{j1}}{s_W}\,\{\st{b},\ct{b}\}\,\delta \theta_b - \frac{m_b U_{j2}}{\sqrt{2}\,s_W\,M_W\,\cos\beta} \,\{\ct{b},-\st{b}\}\,\delta \theta_b \ . \end{eqnarray} where again the $c_{L,R}^k$ are given by eq.~(11). Except for very large values of $\tan\beta$, the $\tilde{b}$ mixing angle [as well as the bottom quark mass] can be set to zero and the analytical expressions simplify considerably\footnote{In the absence of mixing, the left-- and right--handed bottom squarks are, to a very good approximation, degenerate if $M_{\tilde{q}_L} = M_{\tilde{q}_R}$. In the rest of the discussion, ${\tilde{b}_L}$ and ${\tilde{b}_R}$ [and a {\it fortiori} the partners of the light quarks ${\tilde{q}_L}$ and ${\tilde{q}_R}$] will be considered as degenerate.}. The case of the neutral decay $\tilde{b} \rightarrow b \chi^0$ is even simpler since one can also neglect the mass of the final $b$ quark. In fact, the latter situation corresponds to the case of decays of first and second generation squarks into light quarks and charginos/neutralinos, which will be discussed now. \subsubsection*{4.2 Light Quark Partners Decays} Neglecting the squark mixing angle as well as the mass of the final quarks, the virtual corrections of the processes $\tilde{q}_i \rightarrow q \chi$ [where the subscript $i$ stands now for the chirality of the squark, since in the absence of squark mixing one has $\tilde{q}_{L,R} =\tilde{q}_{1,2}$] are given by the sum of the gluon and gluino exchange vertices and the wave--function counterterm, plus the real correction. The total width can then be written as \begin{eqnarray} \Gamma^i = \Gamma^i_0 \bigg[ 1 \,+ \, \frac{4}{3} \frac{\alpha_s}{\pi} \, \left( F_{\rm g}+ F_{\rm \tilde{g}}+ F_{\rm ct} + F_{\rm r} \right) \bigg] \end{eqnarray} where the decay width in the Born approximation $\Gamma^i_0$ has been given in eq.~(12). In terms of the ratio $\kappa= m_{\chi}^2/m_{\tilde{q}}^2$, the gluon exchange corrections are given by [$\Delta =1/(4-n)$ with $n$ the space-time dimension, and $\mu$ is the renormalization scale] \begin{eqnarray} F_{\rm g} &=& \frac{\Delta}{2} + 1 - \frac{1}{2} \ln \frac{ m_ {\tilde{q}}^2 } {\mu^2} -\frac{1}{4} \ln^2 \frac{ \lambda^2/ m_ {\tilde{q}}^2 } {(1-\kappa)^2 } - \ln \frac{ \lambda^2/ m_ {\tilde{q}}^2 } {1-\kappa} -{\rm Li_{2}}(\kappa) \ . \end{eqnarray} The gluino exchange contribution, with $\gamma= m_{\tilde{g}}^2 /m_{\tilde{q}}^2$, is given by \begin{eqnarray} F_{\rm \tilde{g}} = \sqrt{ \kappa \gamma} \left[ \frac{1}{ \kappa} \ln (1-\kappa)+ \frac{1}{1-\kappa} \left[ \gamma \ln \gamma -(\gamma-1) \ln (\gamma-1) \right] + \frac{ \kappa +\gamma -2}{(1-\kappa)^2} \, I \, \right] \end{eqnarray} with \begin{eqnarray} I & \equiv & \frac{1}{m_{\tilde{q}_i}^2\,(1-\kappa)}\, C_0 (0,m_{\tilde{q}}^2, m_{\chi}^2,m_{\tilde{q}}^2,m_{\tilde{g}}^2, 0) \ . \end{eqnarray} In terms of dilogarithms, the function $I$ is given for $\kappa \gamma <1$ by \begin{eqnarray} I= {\rm Li_{2}} \left( \frac{\gamma-1}{\gamma \kappa-1} \right) - {\rm Li_{2}} \left( \kappa \frac{\gamma-1}{\gamma \kappa-1} \right) - {\rm Li_{2}} \left( \frac{\gamma+\kappa-2}{\gamma \kappa-1} \right) + {\rm Li_{2}} \left( \kappa \frac{\gamma+\kappa-2}{\gamma \kappa-1} \right) \end{eqnarray} and for $\kappa \gamma > 1$ one has \begin{eqnarray} I & = &-{\rm Li_{2}}\left( \frac{\gamma \kappa-1}{\gamma-1} \right) +{\rm Li_{2}}\left( \frac{\gamma \kappa-1}{\gamma+\kappa-2} \right) +{\rm Li_{2}}\left( \frac{\gamma \kappa-1}{\kappa(\gamma-1)} \right) -{\rm Li_{2}}\left( \frac{\gamma \kappa-1}{\kappa(\gamma+\kappa-2)} \right) \nonumber \\ & & -\ln (\kappa)\,\ln \frac{\gamma+\kappa-2}{\gamma-1} \ . \end{eqnarray} The counterterm contribution, consisting of the sum of the squark and quark wave--function renormalization constants, reads \begin{eqnarray} F_{\rm ct} &=& - \frac{\Delta}{2} + \frac{\gamma}{4\,(1-\gamma)} - \frac{\gamma}{2} - \frac{15}{8} + \frac{1}{2} \ln \frac{ m_ {\tilde{q}}^2}{\mu^2} - \frac{1}{4} \ln \frac{ \lambda^2}{ m_ {\tilde{q}}^2 } \nonumber \\ & & - \frac{1}{2}(\gamma^2-1) \ln \frac{\gamma -1}{\gamma} + \frac{1}{4}\left[ \frac{2\,\gamma-1}{(1-\gamma)^2}+3 \right] \ln \gamma \ . \end{eqnarray} Finally, the real corrections with massless quarks in the final state contribute \begin{eqnarray} F_{\rm r} &=& \frac{1}{4} \ln^2 \frac{ \lambda^2/ m_{\tilde{q}}^2 }{(1-\kappa)^2} + \frac{5}{4} \ln \frac{ \lambda^2/ m_{\tilde{q}}^2 }{(1-\kappa)^2} - \frac{\kappa\,(4-3 \kappa)}{4\,(1-\kappa)^2}\ln \kappa \nonumber \\ & & - {\rm Li_{2}}(\kappa) -\ln \kappa \ln (1-\kappa) - \frac{3 \kappa-5}{8\,(\kappa-1)} - \frac{\pi^2}{3} + 4 \ . \end{eqnarray} We see explicitly that the ultraviolet divergences $\Delta/2$ and the scale $\mu$ cancel when $F^i_g$ and $F^i_{\rm ct}$ are added, and that the infrared divergences $\ln^2(\lambda^2/ m_{\tilde{q}}^2)$ and $\ln (\lambda^2/ m_{\tilde{q}}^2)$ disappear when $F_g$, $F_{\rm ct}$ and $F_{\rm r}$ are summed. The gluino exchange contribution eq.~(40) does not contain any ultraviolet or infrared divergences. The total correction in eq.~(38) then reads \begin{eqnarray} F_{\rm tot} & = & F_{\rm g}+ F_{\rm \tilde{g}}+ F_{\rm ct} + F_{\rm r} \nonumber \\ & = & - \frac{1}{8}\left( \frac{4\,\gamma^2-27\,\gamma+25}{\gamma-1} + \frac{3\,\kappa-5}{\kappa-1} \right) - \frac{\pi^2}{3} - 2\, {\rm Li_2}(\kappa) - \frac{1}{2}\,(\gamma^2-1)\,\ln \frac{\gamma-1}{\gamma} \nonumber \\ & & + \frac{3\,\gamma^2-4\,\gamma+2}{4\,(1-\gamma)^2}\,\ln \gamma - \frac{3}{2}\,\ln (1- \kappa) + \frac{3\,\kappa^2-4\,\kappa}{4\,(\kappa-1)^2}\,\ln \kappa - \ln \kappa\,\ln (1-\kappa) \nonumber \\ & & +\sqrt{ \kappa \gamma} \left[ \frac{1}{ \kappa} \ln (1-\kappa)+ \frac{1}{1-\kappa} \left[ \gamma \ln \gamma -(\gamma-1) \ln (\gamma-1) \right] + \frac{ \kappa +\gamma -2}{(1-\kappa)^2} \, I \, \right] . \end{eqnarray} \smallskip In the limit where the mass of the final neutralino or chargino is much smaller than the mass of the initial squark, the analytical expression of the QCD correction further simplifies: \begin{eqnarray} F_{\rm tot}= \frac{3 \gamma^2-4\gamma+2}{4(\gamma-1)^2} \ln \gamma - \frac{1}{2} (\gamma^2-1) \ln \frac{\gamma-1}{\gamma} -\frac{2 \gamma^2-11 \gamma +10}{4(\gamma-1)} -\frac{\pi^2}{3} \ . \end{eqnarray} Note the explicit logarithmic dependence on the gluino mass in the correction. This logarithmic behaviour, leading to a non-decoupling of the gluinos for very large masses, \begin{eqnarray} F_{\rm tot} = \frac{3}{4} \ln \frac{ m_{\tilde{g}}^2} {m_{\tilde{q}}^2} +\frac{5}{2} -\frac{\pi^2}{3} \ \ \ {\rm for} \ \ m_{\tilde{g}} \gg m_{\tilde{q}} \end{eqnarray} is due to the wave function renormalization and is a consequence of the breakdown of SUSY as discussed in Ref.~\cite{R4}. Had we chosen the $\overline{\rm MS}$ scheme when renormalizing the squark/quark wave functions [i.e. subtracting only the poles and the related constants in the expression eq.~(23)] we would have been left with contributions which increase linearly with the gluino mass. \\ \vspace*{-3mm} Our analytical results in the case of massless final quarks agree with the corresponding results obtained in Refs.\cite{R4,R5}, where the QCD corrections to the decay of a squark into a massless quark and a photino have been derived, after correcting the sign of $F_{\tilde{g}}$ in Ref.~\cite{R4}; see also the discussion given in Ref.~\cite{R5}. \subsection*{5. Numerical Analysis and Discussion} In the numerical analysis of the QCD corrections to squark decays, we will choose $m_t=180$ GeV (consistent with \cite{R17}) and $m_b=5$ GeV for the top and bottom quark masses and a constant value for the strong coupling constant $\alpha_s =0.12$ [the value of $\alpha_s$ in the running from a scale of 0.1 to 1 TeV does not change significantly]; the other fixed input parameters are $\alpha=1/128$, $M_Z=91.187$ GeV and $s_W^2=0.23$ \cite{R18}. For the SUSY parameters, we will take into account the experimental bounds from the Tevatron and LEP1.5 data \cite{R16}, and in some cases use the values favored by fits of the electroweak precision data from LEP1 \cite{R15}. \\ \vspace*{-3mm} Fig.~2 shows the partial widths for the decays of the lightest top squark into the two charginos $\chi_{1,2}^+$ and a bottom quark [2a] and into the lightest neutralino $\chi_1^0$ and the sum of all neutralinos [the opening of the neutralino thresholds can be seen in the curves] and a top quark [2b]. In these figures, $\tan\beta$ is fixed to $\tan\beta=1.6$, a value favored by $b$--$\tau$ Yukawa coupling unification \cite{R19}. The solid, dashed and dot--dashed curves correspond to the $(M_2, \mu$) values [in GeV]: $(70, -500), (70, -70)$ and $(300,-70)$ in Fig.~2a [which give approximately the same value for the lightest chargino mass, $m_{\chi_1^+} \simeq 70$ GeV] and $(100, -500), (100, -100)$ and $(250,-50)$ in Fig.~2b [giving an LSP mass of $m_{\chi_1^0} \sim 50$ GeV]. These values correspond to the scenarios $M_2 \ll |\mu|$, $M_2 \simeq \mu$ and $M_2 \gg |\mu|$, and have been chosen to allow for a comparison with the numerical analysis given in \cite{R8}. The parameters in the $\tilde{t}$ mass matrix are fixed by requiring $m_{\tilde{t}_2} =600$ GeV and varying $M_{\tilde{t}_L}$. The mixing angle is then completely fixed assuming $M_{\tilde{t}_R}= M_{\tilde{t}_L}$ ($\theta_{\tilde{t}}\approx -\pi/4$ except for $m_{\tilde{t}_1}$ very close to $m_{\tilde{t}_2}$); in the bottom squark sector we have $m_{\tilde{b}_1}= 220$ GeV, $m_{\tilde{b}_2} \sim 230$ GeV and $\theta_{\tilde{b}} \simeq 0$. \\ \vspace*{-3mm} Fig.~3 shows the magnitude of the QCD corrections relative to the Born width to the decays of the lightest top squark into charginos+bottom [3a/b] and neutralinos+stop [3c/d] for the scenarios described in Fig.~2a [for Figs.3a/b] and Fig.~2b [for Figs.3c/d]. For both the neutral and charged decays, the QCD corrections can be rather large and vary in a wide margin: from $\sim \pm 10\%$ for light top squarks up to $\sim -40\%$ for $m_{\tilde{t}_1} \sim m_{\tilde{t}_2}$ and some $(M_2, \mu)$ values. \\ \vspace*{-3mm} The small spikes near $m_{\tilde{t}_1} \sim 425$ (530) GeV for $\chi^+ b$ $(\chi^0 t$) decays are due to thresholds in the top squark wave function renormalization constants from the channel $\tilde{t}_1 \rightarrow \tilde{g} t$. For the depicted $m_{\tilde{t}_1}$ range, this happens only for the value $M_2 =70$ (100) GeV which leads to $m_{\tilde{g}} \simeq 3.5 M_2 \sim 245 (350)$ GeV. Note, however, that when this occurs, the channel $\tilde{t}_1 \rightarrow \tilde{g} t$ becomes by far the main decay mode, and the chargino/neutralino modes are very rare. \\ \vspace*{-3mm} In Fig.~4 the variation of the QCD corrections for the decay $\tilde{t}_1 \rightarrow b \chi_1^+$ [4a] and $\tilde{t}_1 \rightarrow t \chi_1^0$ [4b] is displayed as a function of the gluino mass, for two values of $\mu=-50$ and $-500$ GeV and $\tan\beta=1.6$ and $20$. The top squark masses are fixed to $m_{\tilde{t}_1} =300$ and $m_{\tilde{t}_2}=600$ GeV ($\theta_{\tilde{t}}= -\pi/4$) and the $\tilde{b}$ masses are as in Fig.~2. $M_2$ and hence the chargino and neutralino masses are fixed by $m_{\tilde{g}}$. The figure exhibits a slight dependence of the QCD correction on the gluino mass. For the chosen set of squark mass parameters, the variation of the QCD correction with $\mu$ is rather pronounced, while the variation with $\tan\beta$ is milder. \\ \vspace*{-3mm} Fig.~5 shows the partial decay widths for the decays of the lightest bottom squark [which in our convention is denoted by $\tilde{b}_1$ and is almost left--handed] into the lightest chargino $\chi_{1}^-$ and a top quark [5a] and into the lightest neutralino $\chi_1^0$ and a bottom quark [5b]. As in Fig.~2, $\tan\beta$ is fixed to $\tan\beta=1.6$ and $m_{\tilde{t}_1}=600$ GeV; the mass difference between the two squarks is $\simeq 10$ GeV and we have for the mixing angle $\theta_{\tilde{b}} \simeq 0$. The solid, dashed and dot--dashed curves correspond to the $(M_2, \mu$) values [in GeV]: $(60, -500), (70, -60)$ and $(300,-60)$ in Fig.~5a and $(100, -500), (100, -100)$ and $(250,-50)$ in Fig.~5b. The decay $\tilde{b}_1 \rightarrow t \chi_1^-$ is by far dominant when the channel $\tilde{b}_1 \rightarrow \tilde{g}b$ is closed, since its decay width is almost two orders of magnitude larger than the $\tilde{b}_1 \rightarrow$ LSP+bottom decay width. \\ \vspace*{-3mm} Fig.~6 presents the magnitude of the relative QCD corrections to the decays $\tilde{b}_1 \rightarrow t \chi_1^-$ [6a] and $\tilde{b}_1 \rightarrow b \chi_1^0$ [6b] as a function of the bottom squark mass, for the same scenarios as in Fig.~5. Again, depending on the values of $\mu, M_2$ and $m_{\tilde{b}_1}$, the QCD corrections vary from ($\pm$) a few percent up to $-50\%$. \\ \vspace*{-3mm} Finally, Fig.~7 displays the QCD corrections to the decays of the SUSY partners of massless quarks into neutralinos, $\tilde{q} \rightarrow q \chi_0$, as a function of the ratio $\kappa=m_{\chi}^2/ m_{\tilde{q}}^2$ for several values of the ratio $\gamma=m_{\tilde{g}}^2 /m_{\tilde{q}}^2, \gamma=1.2, 1.5$ and 2 [7a] and as a function of $\gamma$ for several values of $\kappa, \kappa=0.2, 0.5$ and $0.8$ [7b]. The quark mass and the squark mixing angle are set to zero and all squarks are taken to be degenerate. The corrections then depend only on the two parameters, $\kappa$ and $\gamma$ since the dependence on the other SUSY parameters factorizes in the Born term. The QCD corrections vary from small [most of the time negative] values for small $\kappa$ values and small gluino masses, up to $\sim 20\%$ near threshold. \\ \vspace*{-3mm} For the decays $\tilde{q}_L \rightarrow q' \chi^\pm_j$ [the right--handed squark does not decay into charginos], the matrix elements in the chargino mass matrix do not factorize in the Born expressions and the QCD corrections further depend on the ratios $U_{j1}/V_{j1}$ through the contribution $F_{\tilde{g}}$. This dependence is, however, rather mild since first the ratio $U_{j1}/V_{j1}$ is of order unity in most of the relevant SUSY parameter space [in particular for $|\mu| > M_2$] and second the contribution $F_{\tilde{g}}$ is small compared to the other contributions for gluino masses below 1 TeV. The QCD corrections for the decays $\tilde{q}_L \rightarrow q' \chi^\pm$ are thus approximately the same as in the case of the decays into neutralinos. \\ \vspace*{-3mm} In conclusion: we have calculated the ${\cal O}(\alpha_s)$ QCD corrections to decay modes of scalar squarks into quarks plus charginos or neutralinos in the Minimal Supersymmetric Standard Model. We have paid a particular attention to the case of $\tilde{t}$ [and also $\tilde{b}$] squarks, where mixing effects are important. In the case of top squark decays, the QCD corrections can reach values of the order of a few ten percent depending on the various SUSY parameters. They can be either positive or negative and increase logarithmically with the gluino mass. For the scalar partners of light quarks, the corrections do not exceed the level of ten to twenty percent for gluino masses less than 1 TeV. \vspace*{2cm} \noindent {\bf Acknowledgements}: \\ \vspace*{-3mm} \noindent We thank Tilman Plehn and Peter Zerwas for discussions and for the comparison between their results Ref.~\cite{R5a} and ours, and the Vienna group, in particular Sabine Kraml, for discussions about Ref.~\cite{R8}. \newpage

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\section{Theory} \paragraph*{Theory.} The two channel Anderson lattice Hamiltonian under investigation reads \begin{eqnarray} \hat H &= &\displaystyle \sum_{\alpha <i,j>} \frac{t^*}{\sqrt{d}} c^\dagger_{i\alpha\sigma}c_{j\alpha\sigma} +\sum_{i\sigma} E_\sigma X_{\sigma,\sigma}^{(i)} \nonumber\\ & &\displaystyle + \sum_{i\sigma\alpha} V\left\{ c^\dagger_{i\alpha\sigma} X_{\alpha,\sigma}^{(i)} + h.c \right\} \;\; . \end{eqnarray} $X$ are the usual Hubbard operators, $d$ being the spatial dimension, $i$ the lattice site, $t^*$ the reduced hopping matrix element of the conduction electron between nearest neighbours which carry a spin $\sigma$ and a channel index $\alpha=(1,2)$. The conduction electrons couple via hybridization matrix element $V$ to the ionic many-body states on each lattice site. The symmetry breaking magnetic field enters by a Zeeman term in $E_\sigma = \varepsilon_f + g |\mu_b|\sigma H$. The Zeeman splitting of the conduction electrons only result in a shift of the band centers and turn out to be a small correction. When $|E_\alpha- E_\sigma|$ is much larger than the hybridization width $\Gamma_0 = \pi\rho V^2$, the model can be mapped onto an two channel Kondo model \cite{Cox87} via the Schrieffer-Wolff transformation \cite{SchriefferWol66}. In the local approximation for the Anderson Lattice \cite{KimKurKas87}, which is equivalent to the limit $d\rightarrow\infty$ \cite{Georges96} with an appropriate rescaling of the effective hopping \cite{MetznerVoll89}, we choose one $f$-site as an effective "'impurity"' site which is self-consistently embedded in an effective medium reflection the rest of lattice. While in a single impurity problem only the bare medium $\Delta_{\alpha\sigma}^{0}(z) = N_s^{-1}\sum_{\ul{k}}d_{\alpha\sigma}(\ul{k},z) =N_s^{-1} \sum_{\ul{k}}{V^2}(z-\varepsilon_{\ul{k} \alpha\sigma})^{-1}$ enters, in the lattice the condition that the local $f$-Green's function (GF) has to be equal to the $\ul{k}$-summed lattice GF \begin{equation} \label{eqn-scc-doo} \frac{1}{N_s} \sum_{\ul{k}} \frac{1}{1- \tilde F_{\alpha\sigma}(z)\left(d_{\alpha\sigma}(\ul{k},z) - \tilde\Delta_{\alpha \sigma}(z)\right)} = 1 \;\; . \end{equation} where $\tilde\Delta_{\alpha\sigma}(z)$ is the self-consistent one body self energy of the local "impurity" propagator $\tilde F_{\alpha\sigma}(z) =\ll\hat X_{\alpha,\sigma}^{(i)}|\hat X_{\sigma,\alpha}^{(i)}\gg (z) $. The effective hybridization width $\tilde\Gamma_{\alpha\sigma}(\omega) = \frac{\Im m}{\pi}\tilde\Delta_{\alpha\sigma}(\omega-i\delta)$ enters then the local two channel impurity problem. The self-energy of the conduction electrons is given by \begin{equation} \Sigma_{\alpha\sigma}(z)= \frac{V^2 \tilde F_{\alpha\sigma}(z)}{1 + \tilde F_{\alpha\sigma}(z)\tilde\Delta_{\alpha\sigma}(z)} \;\; . \label{equ-c-self} \end{equation} In the single channel case, the unitarity limit of the $T$-matrix $\tilde T(z)=V^2\tilde F(z)$ leads to the Fermi-liquid behaviour of the conduction band self-energy $\Sigma_{\sigma}(z) \propto a T^2 + b w^2$. Since the value of the $T$-matrix at the chemical potential and $T\rightarrow 0$ is smaller than the unitarity limit in the two(multi)-channel case Eqn.(\ref{equ-c-self}) tells us immediately, that the corresponding conduction band self-energy for the exact solution in the paramagnetic phase of the lattice has to be finite. This has been recently called an {\em incoherent metal} \cite{Cox96,JarrellPanCoxLuk96}. The physical origin is the following: the local spin is over-compensated by two conduction electron spins. On each lattice site a residual free thermodynamically fluctuating degree of freedom (DOF) acts as a scatterer for conduction electrons. An residual entropy of $1/2\log(2)$ per site is associated with this DOF freedom which has been interpreted as a free Majorana fermion \cite{EmeryKiv92}. The finite self-energy yields a finite value for $\rho(T\rightarrow 0,H=0)$. Since in translational invariant system a vanishing dc resistivity is expected this indicates that the paramagnetic state is {\em not} the ground state of the two-channel Anderson lattice. In the absence of a magnetic field the NCA equations of the effective impurity \begin{eqnarray} \Sigma_\alpha(z) & = & \frac{1}{\pi} \sum_\sigma \intl_{-\infty}^{\infty} \tilde\Gamma_{\alpha\sigma}(\varepsilon) f(\varepsilon)\tilde P_\sigma(z+\varepsilon) \label{eqn-sig-a} \\ \Sigma_\sigma(z) & = & \frac{1}{\pi} \sum_\alpha \intl_{-\infty}^{\infty} \tilde\Gamma_{\alpha\sigma}(\varepsilon) f(-\varepsilon)\tilde P_\alpha (z-\varepsilon) \label{eqn-sig-s} \end{eqnarray} are equivalent to a resonant level system with an effective Anderson width $\tilde\Gamma_0= 2\Gamma_0$. The NCA pathology in the local GF becomes the physical Abrikosov-Suhl-resonance (ASR) in the two channel case \cite{Cox87,KimCox95,Cox93}. The NCA threshold exponents of the effective ionic propagators $\tilde P$ \cite{MuellerHartman84} have the exact value of $1/2$ calculated within a conformal field theory approach\cite{AffleckLud93}. In limit of infinite spin $N$ and channel $M$ degeneracy with a fix ratio $N/M$ the NCA becomes exact \cite{CoxRuck93}. The effective local GF is given by the convolution \begin{equation} \tilde F_{\alpha\sigma}(i \omega_n) = \frac{1}{\tilde Z_f}\oin\frac{dz}{2\pi i}e^{-\beta z} \tilde P_\alpha(z)\tilde P_\sigma(z+i \omega_n) , \end{equation} where $\tilde Z_f$ is the effective local partition function. Even though higher order vertex corrections \cite{Anders95} will modify the spectral distribution, the leading physical effect and the correct thermodynamics is captured correctly within the Eqns.(\ref{eqn-sig-a}) and (\ref{eqn-sig-s}). Since the saturation value of the effective site $T$-matrix is half the unitary limit no pseudo-gap develops in the quasi-particle spectrum as in the onc-channel lattice \cite{GrewePruKei88}. In infinite spatial dimensions the vertex corrections in the two-particle propagators vanish \cite{MuellerHartman89} and the conductivity itself is a $1/d$ correction which can be calculated by evaluating the lowest order bubble diagram \cite{PruschkeCoxJar93}, given by \begin{eqnarray} \sigma_{\alpha}(\omega) & = & \displaystyle A \intl_{-\infty}^{\infty} d\omega'\frac{[f(w')-f(w+w')]}{\omega} \intl_{-\infty}^{\infty} d\varepsilon \rho_0(\varepsilon) \nonumber \\ && \sum_{\sigma} \displaystyle \Im m G^{(c)}_{\alpha\sigma}(\omega'-i\delta,\varepsilon)\Im m G^{(c)}_{\alpha\sigma}(\omega '+ \omega-i\delta,\varepsilon) \;\; , \label{equ-opti-cond} \end{eqnarray} which can be written as an integral over four complex error function; $A = \pi e^2 a^2 t^{*^2}N (h d Vol)^{-1} =t^* \omega_p^2/(4\pi)$, the Gaussian density of states $\rho_0(\varepsilon)$ \cite{MetznerVoll89}, $G^{(c)}_{\alpha\sigma}(z)$ the conduction electron GF, and $a$ the lattice constant of the $d$-dimensional hypercube. The $f$-electrons do not contribute to the conductivity since the hybridization is $\ul{k}$ independent. The dc-conductivity is obtained by the limit $\sigma_{dc}(T) = \lim\limits_{\omega\rightarrow 0}\sigma(\omega,T)$. \paragraph*{Results.} To obtain a self-consistent solution of the lattice problem, {\em (i)} the effective hybridization with $\tilde\Gamma(\varepsilon)$ has been treated with the same accuracy as the threshold singularity of the ionic propagators in Eqns.(\ref{eqn-sig-a}) and (\ref{eqn-sig-s}), and {\em (ii)} only 10\% of the calculated change of $\tilde\Gamma(\varepsilon)$ is added in each lattice iteration step, i.e.~the lattice is switched on adiabatically. The error in norm of the $\tilde P(z)$ reaches $0.01\%$, the sum rule for the self-energy is obeyed within 0.02\% and the maximum norm $max\{|\tilde\Gamma_{n}(\varepsilon)-\tilde\Gamma_{n-1}(\varepsilon)|\} < 10^{-8}$. All energies, if not otherwise stated, are measured in the original Anderson-width $\Gamma_0$. We chose $\varepsilon_f= E_\sigma-E_\alpha = -3\Gamma_0$ in the absence of $H$ and $t^*=10\Gamma_0$ with a band center at $\omega=0$. $T_K= 0.016\Gamma_0$ \cite{tkondo}. In Fig.~\ref{fig-1} $\rho(T,H))$ normalized to the estimated $T\rightarrow 0$ value of the QMC data \cite{JarrellPanCoxLuk96} is shown for different values of the an applied magnetic field measured in units of $H_K = k_BT_K/(g\mu_B)$. We have fix the lattice scale $T_0 = 1.3T_K$ \cite{tkondo} by matching the QMC resistivity data. The agreement with the higher temperature data for the Kondo lattice - the open symbols - is excellent. Nevertheless, the resistivity has a maximum and slowly decreases with decreasing temperature, as expected from a lattice calculation. The calculations are done slightly below half-filling and with fixed chemical potential. At half-filling in the Kondo-regime an the analytical solution obtained with an artificial Lorentzian density of states \cite{Cox96} predicts a resistivity of $\rho(T) = \rho(0)(1-a\sqrt{T})$ at low temperature. While for $H=0$ still a positive intercept at $T=0$ is expected consistent with an infinitly degenerated ground state, clearly a crossover to Fermi-liquid is found within an applied field. The NCA pathology \cite{MuellerHartman89} prevents access to the Fermi-liquid regime when $T\rightarrow 0$ \cite{KimKurKas87}. Evaluation the constants for $A$ in Eqn.~(\ref{equ-opti-cond}), assuming a lattice constant of 5\AA\ and 2 electrons per unit cell in a three dimensional lattice gives a resistivity prefactor of $\approx 12.6 \mu\Omega cm/\Gamma_0^2$ which leads to a resistivity maximum of $\approx 250\mu\Omega cm$ using our absolute maximum of $20\Gamma_0^2$. This is very close to the experimentally found value of $\approx 190 \mu\Omega cm$ for UBe$_{13}$ \cite{WillisThoSmiFis87}. Motivated by the experimental data for $\rho(T,H)$ for UBe$_{13}$ \cite{AndrakaSte94}, we have attempted to scale our $\rho(T,H)$ data with the {\em Ansatz} $\Delta\rho/ \rho = [\rho(T,H)-\rho(T,0)]/ \rho(T,0) \propto f(H/(T+ T^*)^{\beta})$. While for the impurity model, we expect $T^* =0., \beta = 1/2$, we find approximate scaling for $T^*=0.006T_K, \beta=0.39$, as ploted in Fig.~\ref{fig-2}. The inset of the figure shows the imaginary part of the conduction electron self-energy $\Sigma_c(\omega-i\delta)$ (\ref{eqn-sig-s}) for $H=0$ is plotted for four different temperatures in Fig.~\ref{fig-2}. It shows a shift of the maxima away from the chemical potential in this metallic regime. Very close to $\omega=0$ a very small onset of coherence is observed for $T\rightarrow 0$, but the relaxation rate remains of the order of $2\Gamma_0$. Now we focus on the optical conductivity, displayed in figure \ref{fig-opti}. The large peak at $\approx 0.9\Gamma_0$ results from high energy charge excitations. With decreasing temperatures the optical conductivity develops a pseudo-gap. The $f$-sum rule relates the integrated optical conductivity \begin{equation} \int\limits_0^\infty \sigma_{\alpha}(\omega) d\omega = \frac{\pi^2e^2 a^2 t^{*^2}}{h}\frac{1}{Vol} \sum\limits_\sigma \ll \hat T_x\gg \;\; \;\propto \frac{1}{d} \end{equation} to the average kinetic energy in the direction of the current flow \cite{Maldague77}, which is checked numerically. It indicates a shift of small amount of spectral weight to higher frequencies. This can be seen clearly in the figure by comparing the $T=10T_K$ and the $T=T_K$ curve (note that the logarithmic plot overemphazies the area of the gap). At low temperatures a small increase in $\sigma(\omega)$ can be observed when $\omega\rightarrow 0$. Nevertheless, no clear Drude peak is seen even for $T=0.01T_K$, one decade lower than the observed maximum in $1/\sigma(0,T)$. However, in a magnetic field of $H=H_K$ a low frequency "Drude"-peak develops again, consistent with the return to Fermi-liquid behaviour suggested in $\rho(T,H)$. Note that in the single channel Anderson lattice a clear Drude peak develops already a little below the maximum in $1/\sigma(0,T)$. Using the Kramers-Kronig relation the imaginary part of the optical conductivity has been calculated. With the phenomenological {\em Ansatz} \begin{equation} \sigma_{opt}(\omega) = \sigma(\omega) + i\sigma''(\omega) = \frac{\omega_p^2}{4\pi} \frac{1}{\Gamma_{opt}(\omega) -i\omega (1+\lambda(\omega))} \end{equation} the dynamical optical relaxation rate $\Gamma_{opt}(\omega)$ and the dimensionless mass enhancement factor $\lambda(\omega)$ have been determined. In Fig.~\ref{fig-gamma}(a) $\Gamma_{opt}(\omega)$ is plotted for the same parameters as in Fig.~\ref{fig-opti}. While for temperatures $T>0.1T_K$ $\Gamma_{opt}(\omega)$ is nearly frequency independent for low frequencies, at the lowest temperature $\Gamma_{opt}(0)$ has decreased reflecting the decrease of the dc-resistivity. The low frequency behaviour has an exponent slightly lower then $n=1$. We emphasize the {\em difference} between $\Im m\Sigma_c(\omega)$ and $\Gamma_{opt}(w)$: the first is a true {\em one-particle} relaxation rate, the second, however, reflects the two-particle nature of the energy absorption process associated with electrical charge transport. Generally, only for a Fermi-liquid at very low temperatures and frequencies should $\Gamma_{opt}=-2\Im m\Sigma_c(\omega)$. In an applied magnetic field $H=H_K$ and low temperatures $T=0.01T_K$ the optical relaxation rate $\Gamma_{opt}(0)$ shows the expected trend to Fermi-liquid behaviour. Fig.~\ref{fig-gamma}(b) shows the quantum Monte Carlo (QMC) calculation of $\Gamma_{opt}(\omega)$ for the two-channel Kondo lattice at particle hole symmetry. A strict quantitative calculation is not possible because: (i) of the overlap of $T_0$ and the Kondo interaction $J$ to within an order of magnitude in the QMC calculations (in the Anderson model calculations of this paper $T_0$ is well separated from high energy scales), and (ii) because the particle-hole symmetry removes the non-monotonicity experienced in the NCA calculations. Modulo these concerns, the separate calculations agree qualitatively in the overlapping temperature and frequency regions. {\em Comparison to Experiment:} As mentioned earlier, $\rho(T,H=0)$ is reminiscent in form and magnitude to UBe$_{13}$ \cite{WillisThoSmiFis87,AndrakaSte94}. The magneto-resistance also resembles that of UBe$_{13}$, though our scaling form in detail is different. However, a strict comparison is not possible, since assuming a quadrupolar Kondo model applies to UBe$_{13}$, we should rather split $|\alpha i>$ states (order H) and quadratically split $|\sigma i>$ stated (van Vleck processes). Additionally, recent experiments suggest a possible U$^{3+}$-U$^{4+}$ configuration degeneracy with is lifted with Th substitution \cite{Aliev95}. We refer the necessary intermediate valance calculation to a future work. In this case, a crossover from NFL to Fermi-liquid physics is still expected. Our $\sigma(T,\omega),\Gamma_{opt}(\omega)$ calculations are very compatible with data for the alloys Y$_{0.8}$U$_{0.2}$Pd$_{3}$ and Th$_{1-x}$U$_{x}$Pd$_{2}$Al$_{3}$ \cite{Degiorgi95}, as well as the compound UBe$_{13}$\cite{Bonn87}. Because of the incoherent normal metal phase, we expect little qualitative difference between these more dilute alloys and the lattice. For UBe$_{13}$, the existing optical data only go to 50 cm$^{-1}\simeq 5-6k_BT_K$ in frequency\cite{Bonn87}, and it is clearly desireable to extend these measurements to lower frequencies. We remark that for Th$_{1-x}$U$_{x}$Pd$_{2}$Al$_{3}$, if a hexagonal quadrupolar Kondo picture applies, a $c$-axis magnetic field will split the $|\sigma i>$ levels \cite{Cox93}, permitting comparison to our calculations. In addition, it should be interesting to test whether a magnetic two-channel lattice picture applies to CeCu$_2$Si$_2$\cite{KimCox95} and thus have detailed optical conductivity measurements carried out in applied field for this system. We would also like to thank P.~Coleman, A.~Millis and M.B.~Maple for organizing a stimulating workshop on {\em NFL behaviour in solid} at the Institute for Theoretical Physics, where part of the work was performed. Especially, we thank P. Coleman for suggesting a careful analysis of the magnetotransport. One of us (FBA) also like to acknowledge encouraging comments by H.~Castella, D.~Vollhardt and N.~Grewe. This~work~was~supported~by the Deutsche Forschungsgemeinschaft in part by the National Science Foundation under Grant No. PHY94-07194, and the US Department of Energy, Office of Basic Energy Science, Division of Materials Research (FBA and DLC), and by NSF grants DMR-9406678 and DMR-9357199 (MJ). Quantum Monte Carlo calculations were carried out with a grant of supercomputer time from the Ohio Supercomputer Center. The NCA calculations were performed partially on a Pentium driven LINUX laptop.

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\section{Gap survival for $\gamma p$ case} In order to study soft interactions which accompany a hard scattering, Bjorken \cite{Bjorken} suggested to investigate the ratio of the cross sections of the high $p_t$ dijet production with a large rapidity gap (LRG) to that of dijets without a rapidity gap: \begin{equation} f_{ac}={\sigma(a+c\rightarrow \left(jet(p_t)+X\right) +LRG + \left(jet(-p_t)+Y\right)) \over \sigma(a+c\rightarrow jet(p_t)+jet(-p_t)+Z)}= \kappa P_{LRG} \end{equation} Here $c$ can be a proton or a nuclear target. To account for the difference between scales of hard and soft processes quantify the role of soft physics Bjorken evaluated $f_{ac}$ as the product of 2 factors: \begin{equation} f_{ac}\equiv \kappa P_{RGS}. \end{equation} Factor $\kappa$ is the probability of producing a rapidity gap in hard subprocess, while $P_{RGS}$ characterizes probability of gap survival due to soft interactions of constituents which do not participate in the hard collision. Natural mechanism for the colorless hard collision is the exchange by 2 gluons. At first sight this contribution should be 0. Really it follows from the QCD factorization theorem that the exchange by an extra gluon between the partons involved in a hard collision is canceled out for the total cross section of dijet production. However for diffractive processes the presence of the LRG trigger in the final state destroys the cancelation between different terms, leading to the factorization theorem breaking\cite{CFS}. In perturbative QCD $\kappa$, can be estimated as the ratio of cross sections of hard collisions of partons due to a double gluon color singlet exchange to that due to a single gluon exchange \cite{Bjorken,MT,DT}, give $\kappa \sim 0.15$ cf.discussion in \cite{Zeppenfeld} which depends rather weakly on $p_t$ of the jets. Account for the leading $\alpha_s\ln x$ corrections may lead to a certain increase of $\kappa$ with the length of rapidity gap. $\kappa$ is different for the hard collisions of partons belonging to the different representations of $SU(3)_{color}$. This leads to a certain dependence of $\kappa$ on the kinematics and to a weak dependence on a projectile. Within the framework of conventional soft dynamics $P_{RGS}$ should be approximately independent of the projectile. This is because of the different geometry of collisions characteristic for soft and for hard collisions. Hard collisions are concentrated at small impact parameters which are characterized by the average slope of the diffractive cross section: $a +b \rightarrow X_1+X_2$, where $X_1$, $X_2$ are diffractive states. On the contrary, soft interactions are predominantly peripheral, at impact parameters increasing with energy. This has been established experimentally via the observation of the diffractive cone shrinkage with increase of the energy. Thus a reasonable approximation is that $P_{RGS}$ is determined by collisions at zero impact parameters. Within the eikonal approximation used by Bjorken \cite{Bjorken} the eikonal phase at zero impact parameters is a function of the dimensionless ratio $\sigma_{tot}(ac)/B_{ac}$, where $B_{ac}$ is the slope of the differential cross section for the soft $ac$ scattering. We observe that this ratio is practically the same for proton and photon projectiles. Here for a photon projectile we use as a guide the vector dominance model where $B_{\gamma c} \approx B_{\pi c}$ and $\sigma_{inel} \approx \sigma_{\pi c}$. Hence in the eikonal approximation: \begin{equation} P_{RGS}(p\bar p)=P_{RGS}(\gamma p). \end{equation} This projectile independence is because a collision at central impact parameters is almost black. A second possible source of filling the gap between the jets can be radiation from the two gluon exchange. This radiation should be a small effect since both gluons are located at the same parameter. In this case radiation of gluons with transverse momenta $\ll p_t$ is cancelled out because such a gluon can not resolve colorless exchange, cf.\cite{Gribov}. Radiation of hard gluon is suppressed by the smallness of the coupling constant. Besides, this radiation is projectile independent since it is determined by the properties of the 2 gluon exchange. Very recently photoproduction events which have two or more jets have been observed in the $W_{\gamma p}$ range $135 <W_{\gamma p}<~280 GeV$ with the ZEUS detector at HERA \cite{ZEUS}. A class of the events is observed with little hadronic activity between the jets. The value of $f_{\gamma p}=0.07 \pm 0.03$ is reported based on the last bin: $\Delta \eta \ge 3$. This value is rather close to the estimates in perturbative QCD \cite{Bjorken,MT,DT} neglecting absorptive effects due to interactions of spectator partons in colliding particles, i.e.assuming $P_{RGS}\sim 1$. It is significantly larger that the values reported by D0 \cite{D0} and CDF \cite{CDF} at $\sqrt{s}$=1.8 TeV: $f_{p\bar p}=0.0107\pm 0.0010(stat.)^{+0.0025}_{-0.0013}(sys.)$ \cite{D0}, and $0.0086 \pm 0.0012$ \cite{CDF}. The difference in the gap survival probability is another manifestation of the lack of factorization in the hard processes when extra constraints are imposed on the event selection, see review in \cite{AFS}. We thus conclude that the probability of gap survival seems to be an effective probe of soft interactions which accompany hard interactions. Specifics of the photon projectile is that its wave function contains a significant $q\bar q$ component with large transverse momenta where color is screened. For such configurations, CT would lead to significant enhancement of $P_{RGS}$. In the ZEUS experiment the requirement of observing two high $p_t$ jets in the acceptance of the detector have led to an effective selection of jets carrying a fraction of more than 0.7 of the photon momentum. This component of the wave function is dominated by the small size $q\bar q$ component of the photon wave function since the soft component is suppressed at least by a factor $1-z$. Hence the larger value of $f_{\gamma p}$ observed in this experiments as compared to $f_{pp}$ maybe a manifestation of CT. In other words, kinematics of of the ZEUS experiment may {\bf effectively suppress the soft component in the parton wave function of photon}. One of the ways to check this interpretation is to investigate the dependence of $P_{RGS}$ as a function of the fraction of the photon momentum carried by the jet. The prediction is a significant depletion of $f_{\gamma p}$ when this fraction decreases to values below 0.5. One should also try to introduce a cut for the jet fraction larger that 0.7, but to avoid kinematics when the jet from accompanying quark would fill the gap. This may increase the color transparency effect. \begin{figure} \centerline{ \epsfig{file=nucgap.eps,width=10.0cm,height=10.0cm}} \vspace{-2.25cm} \caption{$A$ dependence of the rapidity gap survival probability on $\sigma_{eff}$.} \end{figure} \section{A-dependence of gap survival} Another way to check the color transparency interpretation of the ZEUS data would be to study the $A$-dependence of $P_{RGS}$. One can address here in {\bf a quantitative way } the key question of {\it how large is the effective cross section for the interaction of the photon in the configuration which leads to the production of events with rapidity gaps between jets?} Is it close to the average value of $\sigma_{eff} \sim $20 mb or maybe much smaller, as the CT interpretation of the ZEUS data suggests. Let us define \begin{equation} R(A) = { f_{\gamma A}(\Delta \eta) \over f_{\gamma p}(\Delta \eta)}, \end{equation} for $\Delta \eta \ge 3$ where $f_p(\Delta \eta)$ flattens out. It is easy to calculate the $A$-dependence of $R(A)$ using the eikonal approximation \cite{BT}: \begin{equation} R(A)=\int d^2B {\tilde T}(B)\exp(-\sigma_{eff} {\tilde T}(B)). \end{equation} Here $ {\tilde T}(B)$ is the standard nuclear thickness function: $ {\tilde T}(B)=\int_{-\infty}^{\infty} d z \rho_A(\sqrt{B^2+z^2})$, where the nuclear density $\rho_A(r)$ is normalized according to $\int \rho_A(r) d^3r=1$. $\sigma_{eff}$ is the cross section of inelastic soft interaction of the hadronic component of the photon wave function, excluding diffractive cross section. The results of the calculation of $R(A$) are presented in Fig.1 as a function $A$ for several values of $\sigma_{eff}$. One can see that measurements with nuclear targets could provide a quantitative measurement of $\sigma_{eff}$ and hence shed a new light on the dynamics of strongly interacting color singlet object responsible for the jet events with rapidity gaps. If one would observe $\sigma_{eff} \leq 10 mb$ this would provide a clear evidence for CT in the production of dijets with LRG. It seems that the optimal range of the targets is $A \le 40$ since for larger $A$, $R(A)$ depends rather weakly on $A$. \section{Acknowledgments} We would like to thank A.Levy for useful comments. This work was supported in part by U.S.Department of Energy and by BSF.

proofpile-arXiv_065-527

\section*{References\@mkboth {REFERENCES}{REFERENCES}}\list {[\arabic{enumi}]}{ \settowidth\labelwidth{[ #1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\hskip .11em plus .33em minus -.07em{\hskip .11em plus .33em minus -.07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi \def\@citea{}\@cite{\@for\@citeb:=#2\do {\@citea\def\@citea{,\penalty\@m\ }\@ifundefined {b@\@citeb}{{\bf ?}\@warning {Citation `\@citeb' on page \thepage \space undefined}}% \hbox{\csname b@\@citeb\endcsname}}}{#1}} \def\@cite#1#2{[{#1 \if@tempswa , #2\fi}]} \def[{[} \def]{]} \def{} \def[{[} \def]{]} \def\CITE#1{$^{\hbox{\tiny \cite{#1}}}$} \catcode`\@=12 \def{\cal A}{{\cal A}} \def{\cal B}{{\cal B}} \def{\cal C}{{\cal C}} \def{\cal D}{{\cal D}} \def{\cal E}{{\cal E}} \def{\cal F}{{\cal F}} \def{\cal G}{{\cal G}} \def{\cal H}{{\cal H}} \def{\cal I}{{\cal I}} \def{\cal J}{{\cal J}} \def{\cal K}{{\cal K}} \def{\cal L}{{\cal L}} \def{\cal M}{{\cal M}} \def{\cal N}{{\cal N}} \def{\cal O}{{\cal O}} \def{\cal P}{{\cal P}} \def{\cal Q}{{\cal Q}} \def{\cal R}{{\cal R}} \def{\cal S}{{\cal S}} \def{\cal T}{{\cal T}} \def{\cal U}{{\cal U}} \def{\cal V}{{\cal V}} \def{\cal W}{{\cal W}} \def{\cal X}{{\cal X}} \def{\cal Y}{{\cal Y}} \def{\cal Z}{{\cal Z}} \def{\mit \Gamma}{{\mit \Gamma}} \def{\mit \Delta}{{\mit \Delta}} \def{\mit \Theta}{{\mit \Theta}} \def{\mit \Lambda}{{\mit \Lambda}} \def{\mit \Xi}{{\mit \Xi}} \def{\mit \Pi}{{\mit \Pi}} \def{\mit \Sigma}{{\mit \Sigma}} \def{\mit \Upsilon}{{\mit \Upsilon}} \def{\mit \Phi}{{\mit \Phi}} \def{\mit \Psi}{{\mit \Psi}} \def{\mit \Omega}{{\mit \Omega}} \def\vol#1{{\bf #1}} \def\Eq#1{Eq.(\ref{#1})} \def\Eqs#1#2{Eqs.(\ref{#1})-(\ref{#2})} \def\rightarrow{\rightarrow} \def\r#1{{\rm #1}} \def\e#1{{10^{#1}}} \def\bm#1{\mbox{\boldmath $#1$}} \let\bg=\bm \def\Frac(#1/#2){\left(\frac{#1}{#2}\right)} \def\r{Tr}{\r{Tr}} \def\Tp#1{\,{}^t\! #1} \def\stackrel{<}{\sim}{\stackrel{<}{\sim}} \def\stackrel{>}{\sim}{\stackrel{>}{\sim}} \def\stackrel{\rm def}{=}{\stackrel{\rm def}{=}} \def{\bm{N}}{{\bm{N}}} \def{\bm{Z}}{{\bm{Z}}} \def{\bm{Q}}{{\bm{Q}}} \def{\bm{R}}{{\bm{R}}} \def{\bm{C}}{{\bm{C}}} \def{\bm{H}}{{\bm{H}}} \def{\bm{F}}{{\bm{F}}} \def{\bm{\scriptstyle Z}}{{\bm{\scriptstyle Z}}} \def{\bm{\scriptstyle Q}}{{\bm{\scriptstyle Q}}} \def{\bm{\scriptstyle R}}{{\bm{\scriptstyle R}}} \def{\bm{\scriptstyle C}}{{\bm{\scriptstyle C}}} \def{\bm{\scriptstyle H}}{{\bm{\scriptstyle H}}} \def{\bm{\scriptstyle F}}{{\bm{\scriptstyle F}}} \def\Tdot#1{{{#1}^{\hbox{.}}}} \def\Tddot#1{{{#1}^{\hbox{..}}}} \def\Tdddot#1{{{#1}^{\hbox{...}}}} \def\dddot#1{\stackrel{...}{#1}{}\!\!} \def\Order#1{\r{O}\!\left(#1\right)} \def\begin{equation}{\begin{equation}} \def\end{equation}{\end{equation}} \def\begin{eqnarray}{\begin{eqnarray}} \def\end{eqnarray}{\end{eqnarray}} \def\begin{eqnarray*}{\begin{eqnarray*}} \def\end{eqnarray*}{\end{eqnarray*}} \def\begin{itemize}{\begin{itemize}} \def\end{itemize}{\end{itemize}} \font\elevenmib=cmmib10 scaled\magstephalf \skewchar\elevenmib='177 \def\YUKAWAmark{\hbox{\elevenmib Yukawa\hskip0.05cm Institute\hskip0.05cm Kyoto \hfill}} \addtolength{\voffset}{-2cm} \addtolength{\oddsidemargin}{-1.5cm} \addtolength{\textwidth}{2.5cm} \addtolength{\textheight}{4cm} \begin{document} \def\bm{x}{\bm{x}} \def\bm{y}{\bm{y}} \def\bm{k}{\bm{k}} \thispagestyle{empty} \begin{titlepage} \hbox to \hsize{\YUKAWAmark \hfill YITP-96-29} \rightline{KUNS 1381} \rightline{September 1996} \vspace{2cm} \begin{center}\large\bf Evolution of Cosmological Perturbations \\ during Reheating \end{center} \bigskip \begin{center} Takashi Hamazaki\footnote{email address: hamazaki@murasaki.scphys.kyoto-u.ac.jp} \end{center} \begin{center}\it Department of Physics, Faculty of Science, Kyoto University,\\ Kyoto 606-01, Japan \end{center} \begin{center} and \end{center} \begin{center} Hideo Kodama\footnote{email address: kodama@yukawa.kyoto-u.ac.jp} \end{center} \begin{center}\it Yukawa Institute for Theoretical Physics, Kyoto University, \\ Kyoto 606-01, Japan\\ \end{center} \bigskip \bigskip \begin{center}\bf Abstract\end{center} The behavior of scalar perturbations on superhorizon scales during the reheating stage is investigated by replacing the rapidly oscillating inflaton field by a perfect fluid obtained by spacetime averaging and the WKB approximation. The influence of the energy transfer from the inflaton to radiation on the evolution of the Bardeen parameter is examined for realistic reheating processes. It is shown that the entropy perturbation generated by the energy transfer is negligibly small, and therefore the Bardeen parameter is conserved in a good accuracy during reheating. This justifies the conventional prescription relating the amplitudes of quantum fluctuations during inflation and those of adiabatic perturbations at horizon crossing in the post-Friedmann stage. \end{titlepage} \section{Introduction} In the inflationary paradigm cosmological large scale structures such as galaxies and their distribution are thought to be formed from seed density perturbations produced by quantum fluctuations of an inflaton field\cite{r:qfluct,Bardeen.J&Steinhardt&Turner1983}. In this scenario one can in principle determine the statistical properties of the present large scale structure of the universe by calculating the amplitude and the spectrum of the seed perturbations and tracing their evolution, if the fundamental laws of nature are specified. The former task is relatively easy and the result can be put into a simple formula which is valid for a wide variety of the inflaton potential. For simplified evolutionary universe models, it is also the case for the second task as far as the linear evolutionary stage of perturbations is concerned. This simplification is brought about by the fact that a gauge invariant variable called the Bardeen parameter is conserved with a good accuracy for growing modes in Friedmann stages and the inflationary stage% \cite{Bardeen.J&Steinhardt&Turner1983,Friemann.J&Turner1984,Brandenberger.R&Kahn1984,% Kodama.H&Sasaki1984}. For example, if the reheating at the end of the inflationary stage is instantaneous, one can determine the amplitudes of perturbations when they reenter the horizon in the post-Friedmann stage from those of the quantum fluctuations on the Hubble horizon scales during inflation simply by matching the values of the Bardeen parameter. This powerful conservation law of the Bardeen parameter still holds for smooth transition from the inflationary to the Friedmann stage, provided that the equation of state of the cosmic matter changes slowly with the cosmic expansion and that entropy perturbations can be neglected\cite{Bardeen.J&Steinhardt&Turner1983}. In realistic models of reheating, however, it is not clear whether these conditions are satisfied or not because matters with different dynamical properties such as the inflaton field and radiation coexist possibly for a long while. Further, according to the perturbation theory of multi-component systems, the energy transfer processes, for example, from the inflaton to radiation, themselves may produce additional entropy perturbations if the energy transfer rate depends on the energy densities or the cosmic expansion rate. For example, it is proposed recently that a parametric resonance of the coherent inflaton field and other massless scalar fields may contribute as the dominant energy transfer process in the early phase of reheating\cite{Traschen.J&Brendenberger1990,Kofman.L&Linde&Starobinsky1994,% Shtanov.Y&Traschen&Brandenberger1995,Yoshimura.M1995}. In this case large entropy perturbations can be produced by the perturbations of the energy transfer rate since this rate for the parametric resonance is very sensitive to the amplitude of the inflaton oscillation and the cosmic expansion rate. In the present paper we investigate this problem in detail and examine whether the Bardeen parameter is well conserved or not during the reheating phase by evaluating the entropy perturbations produced by realistic reheating processes with the help of the gauge invariant formalism for multi-component systems\cite{Kodama.H&Sasaki1984}. On the basis of the result in our previous work\cite{Kodama.H&Hamazaki1996} that the behavior of perturbations in the stage dominated by an oscillatory inflaton $\phi$ coincides with that of a perfect fluid obtained by a spacetime averaging and the WKB approximation except for a sequence of negligibly short intervals around the zero points of $\dot\phi$, we consider a system consisting of gravity, radiation and a perfect fluid corresponding to the inflaton. The paper is organized as follows. First in the next section we give the basic assumptions and explain their motivations. In particular the behavior of a scalar field in the flat spacetime which decays to a massless scalar particles via the standard one particle decay process(Born decay) is briefly analyzed to prove the preservation of coherence of the decaying inflaton field. This analysis is also used to determine the structure of the energy transfer term. In \S3 the evolution equations for perturbations during reheating are given by applying the gauge-invariant formalism for the perturbations of a multi-component system to our system. Then in \S4 the amplitudes of entropy perturbations produced during reheating are estimated with the helps of these evolution equations to show that the Bardeen parameter is conserved with a good accuracy for perturbations corresponding to the present large scale structures. \S5 is devoted to conclusion and discussion. The gauge-invariant perturbation theory of a multi-component system used in \S3 is reviewed in Appendix A. The definitions and formulae are given in a generic form in order to correct errors in the corresponding equations in the original article\cite{Kodama.H&Sasaki1984}. In Appendix B a general method to get upper bounds on solutions to a first-order differential equation system is explained. It is used to find upper bounds on the entropy perturbations in \S4. Throughout this paper, the natural units $c=\hbar=1$ are adopted and $8 \pi G$ is denoted as $\kappa^2$. Further the notations for the perturbation variables adopted in the article\cite{Kodama.H&Sasaki1984} are used and their definitions are sometimes omitted except for those newly defined in this paper. \section{Fundamental assumptions and preliminary considerations} In order to investigate the evolution of perturbations during the reheating phase, we must treat a coupled system of the inflaton, matter and gravitational field. In this paper, as the cosmic matter, we only consider radiation whose dynamics is described by the energy-momentum tensor of a perfect fluid \begin{eqnarray} &&\tilde T_{(r)\mu\nu}=(\tilde \rho_r + \tilde P_r) \tilde u_{(r)\mu} \tilde u_{(r)\nu}+\tilde g_{\mu\nu}\tilde P_r,\\ && \tilde P_r={1\over 3}\tilde\rho_r, \end{eqnarray} where and in the followings the tilde denotes perturbed quantities. Further we assume that the inflaton is well described by a classical and coherent real scalar field $\tilde \phi$ minimally coupled with gravity, and its energy-momentum tensor is given by \begin{equation} \tilde T_{(\phi)\mu\nu}=\partial_\mu\tilde\phi\partial_\nu\tilde\phi -{1\over2}\tilde g_{\mu\nu}\left[(\tilde\nabla\tilde\phi)^2+U(\tilde\phi) \right]. \end{equation} Due to the decay of the inflaton these energy-momentum tensors are not conserved separately: \begin{eqnarray} && \tilde\nabla_\nu \tilde T_{ (\phi)\mu }^\nu=\tilde Q_{(\phi)\mu} \equiv -\tilde Q_\mu,\\ && \tilde\nabla_\nu \tilde T_{ (r)\mu }^\nu=\tilde Q_{(r)\mu} \equiv \tilde Q_\mu. \end{eqnarray} \subsection{Coherence of the inflaton} One subtle point in the assumptions above is the requirement of coherence on the inflaton field. If the interaction of the inflaton field and radiation is neglected, no problem arises when we assume that it is described by a coherent classical field. On the other hand, when the interaction is taken into account, the consistency of this assumption is no longer obvious because the inflaton field by itself is a quantum field and its decay into radiation is a quantum process in general. Further such quantum nature should be properly taken into account when one determines the structure of the energy transfer term $\tilde Q_\mu$. Though we cannot justify this assumption for a generic case, it seems reasonable at least in the case in which the potential of the scalar field is quadratic in $\phi$ from the following observation. Let us consider a massive scalar field $\phi$ on a flat background interacting with a massless scalar field $\chi$ which plays the role of radiation in the realistic situations. For simplicity we assume that the interaction of these fields is cubic and their Lagrangian density is given by \begin{equation} {\cal L}=-{1\over2}\left[(\partial\phi)^2+m^2\phi^2+(\partial\chi)^2 +2\mu\phi\chi^2\right]. \end{equation} Let us quantize these scalar fields by the standard canonical quantization and introduce the creation and annihilation operators, $(A_{\bm{p}},A_{\bm{p}}^\dagger)$ and $(B_{\bm{k}},B_{\bm{k}}^\dagger)$, in the Schr\"{o}dinger picture by \begin{eqnarray} &&\phi(\bm{x})=\sum_{\bm{p}}{1\over\sqrt{2V\omega_p}} \left(A_{\bm{p}}+A_{-\bm{p}}^\dagger\right)e^{i\bm{p}\cdot\bm{x}},\\ &&\pi_\phi(\bm{x})=-i\sum_{\bm{p}}\sqrt{\omega_p\over 2V} \left(A_{\bm{p}}-A_{-\bm{p}}^\dagger\right)e^{i\bm{p}\cdot\bm{x}},\\ &&\chi(\bm{x})=\sum_{\bm{k}}{1\over\sqrt{2V|\bm{k}|}} \left(A_{\bm{k}}+A_{-\bm{k}}^\dagger\right)e^{i\bm{k}\cdot\bm{x}},\\ &&\pi_\chi(\bm{x})=-i\sum_{\bm{k}}\sqrt{|\bm{k}|\over 2V} \left(A_{\bm{k}}+A_{-\bm{k}}^\dagger\right)e^{i\bm{k}\cdot\bm{x}}, \end{eqnarray} where $\pi_\phi$ and $\pi_\chi$ are the conjugate momentums for $\phi$ and $\chi$, respectively, and \begin{equation} \omega_p:=\sqrt{|\bm{p}|^2+m^2}. \end{equation} Then these canonical variables are represented on the Fock space ${\cal H}={\cal H}_\phi\otimes {\cal H}_\chi$ spanned by the basis vectors \begin{equation} |\bm{n},\bm{m}>=\left(\sum_{\bm{n}}{1\over\sqrt{\bm{n}!}} (A^\dagger)^{\bm{n}}\right)\Omega_\phi \otimes\left(\sum_{\bm{m}}{1\over\sqrt{\bm{m}!}} (B^\dagger)^{\bm{m}}\right)\Omega_\chi, \end{equation} where $\Omega=\Omega_\phi\otimes\Omega_\chi$ is the Fock vacuum defined by \begin{equation} A_{\bm{p}}\Omega_\phi=0, \quad B_{\bm{k}}\Omega_\chi=0. \end{equation} In this representation the free part of the Hamiltonian for each component is written in the standard diagonal form: \begin{equation} :H_\phi: =\sum_{\bm{p}}\omega_p A^\dagger_{\bm{p}}A_{\bm{p}}, \qquad :H_\chi: =\sum_{\bm{k}}|\bm{k}| B^\dagger_{\bm{k}}B_{\bm{k}}. \end{equation} Hence $\Omega$ is the ground state of the free part of the Hamiltonian, but not of the total Hamiltonian. Though this implies that $\Omega$ is unstable against the time evolution, it is a convenient cyclic vector for constructing coherent states. Let us consider a state $\Phi(t)$ which is represented at the initial time $t=0$ as \begin{equation} \Phi(0)=\Phi_\phi\otimes \Omega_\chi. \end{equation} Then by the perturbation theory, taking account of the mass renormalization, the expectation value of the annihilation operator for $\phi$ is calculated as \begin{equation} <\Phi(t)|A_{\bm{p}}|\Phi(t)>=e^{- {t\over\tau_p} } <A_{\bm{p}}(t)>_0 + \Order{\mu^3}, \label{EV:A}\end{equation} where $A_{\bm{p}}(t)$ is the operator defined by \begin{equation} A_{\bm{p}}(t):=A_{\bm{p}}e^{-i\omega_p t} -{i\over \omega_p\tau_p} A_{-\bm{p}}^\dagger \cos \omega_pt, \end{equation} $<X>_0$ denotes the expectation value of $X$ in the state $\Phi_\phi$, and $\tau_p$ is the life of $\phi$-particles with a momentum $\bm{p}$ given by \begin{equation} \tau_p={\mu^2\over 16\pi\omega_p}. \end{equation} The expectation value of the creation operator is just given by the complex conjugate of the above expression. Next we consider the expectation values of products of creation and annihilation operators. It is convenient to define $ << X >>_t $ by \begin{equation} << X >>_t := <\Phi(t)| X |\Phi(t)> -<\Omega(t)| X |\Omega(t)> , \end{equation} where $\Omega(t)$ is the solution to the Schr\"{o}dinger equation with the initial condition $\Omega(0)=\Omega$. This subtraction of the expectation with respect to $\Omega(t)$ is to eliminate the effect of the instability of the perturbative vacuum. In this notation, the expectation values of products of creation and annihilation operators are given by \begin{eqnarray} && << A_{\bm{p}}A_{\bm{q}} >>_t =e^{-t({1\over \tau_p}+{1\over\tau_q})} <:A_{\bm{p}}(t)A_{\bm{q}}(t):>_0 + \Order{\mu^3}, \label{EV:AA}\\ && << A_{\bm{p}}^\dagger A_{\bm{q}} >>_t =e^{-t({1\over \tau_p}+{1\over\tau_q})} <:A_{\bm{p}}^\dagger(t)A_{\bm{q}}(t):>_0 + \Order{\mu^3}. \label{EV:AA*}\end{eqnarray} In particular if we take the initial state $\Phi_\phi$ as a coherent state given by \begin{equation} \Phi_\phi=\exp\left(-{1\over2}\sum_{\bm{p}}|c_p|^2\right) \exp\left(\sum_{\bm{p}}c_p A_{\bm{p}}^\dagger\right)\Omega_\phi, \end{equation} the expectation values of $A_{\bm{p}}$ and $A_{\bm{p}}^\dagger$, and their products with respect to this initial state are simply given by \begin{eqnarray} && <A_{\bm{p}}>_0 =c_p, \quad <A_{\bm{p}}^\dagger>_0=\bar c_p,\\ && <A_{\bm{p}}A_{\bm{q}}>_0=c_p c_q, \quad <A_{\bm{p}}^\dagger A_{\bm{q}}>_0=\bar c_p c_q. \end{eqnarray} Hence from \Eq{EV:A}, \Eq{EV:AA} and \Eq{EV:AA*} we obtain \begin{equation} << X(\bm{x})Y(\bm{y}) >>_t \simeq <\Phi(t)|X(\bm{x})|\Phi(t)><\Phi(t)|Y(\bm{y})|\Phi(t)>, \end{equation} up to $\Order{\mu^2}$ where $X$ and $Y$ are any of $\phi$ and $\pi_\phi$. This result shows that the coherence of the scalar field is preserved by its Born decay at least within a time shorter than the decay life $\tau$. Though the above argument does not apply to the energy transfer by the parametric resonance, it is reasonable to assume the coherence of the inflaton in that case as well because the parametric resonance occurs only when the inflaton field has a good coherence. \subsection{Energy-momentum transfer term} The simplified model analysis in the previous subsection can be used to determine the structure of the energy-momentum transfer term $\tilde Q_\mu$ as well. In that model the divergence of the energy-momentum tensor of the quantum $\phi$-field is given by \begin{equation} \partial_\nu T_{(\phi) \mu}^\nu=\mu \chi^2\partial_\mu\phi-\delta m^2 \phi\partial_\mu\phi, \end{equation} where the second term on the right-hand side is the mass counter term. Calculation of the expectation value with respect to $\Phi(t)$ in the order $\mu^2$ yields \begin{eqnarray} &Q_{(\phi) \mu}&=<< \partial_\nu T_{ (\phi)\mu }^\nu >>_t \nonumber\\ &&=-{\mu^2\over 16\pi} <:\partial_\mu \hat\phi(t,\bm{x}) \sum_{\bm{p}}{1\over \omega_p\sqrt{V}} \hat\pi_p(t)e^{i\bm{p}\cdot\bm{x}}:>_0, \end{eqnarray} where \begin{eqnarray} &&\hat \phi(t,\bm{x}):=\sum_{\bm{p}}{1\over\sqrt{2V\omega_p}} \left(A_{\bm{p}}e^{-i\omega_p t}+A_{-\bm{p}}^\dagger e^{i\omega_p t} \right)e^{i\bm{p}\cdot\bm{x}},\\ &&\hat\pi_p(t):=-i\sqrt{2\omega_p}\left(A_{\bm{p}}e^{-i\omega_p t} -A_{-\bm{p}}e^{i\omega_p t}\right). \end{eqnarray} In the case in which the initial coherent state $\Phi_\phi$ contains only particles with small momentums, as in the case of inflaton, this expression is approximately written as \begin{eqnarray} &Q_{(\phi) \mu} & = -\gamma <:\partial_\mu\hat\phi(t,\bm{x})\hat\pi(t,\bm{x}):>_0 \nonumber\\ && \simeq -\gamma \partial_\mu <\hat\phi(t,\bm{x})>_0 <\hat\pi(t,\bm{x})>_0 \equiv -\gamma \partial_\mu\phi \pi, \end{eqnarray} where \begin{equation} \gamma := {\mu^2\over 16\pi m}. \end{equation} This result suggests that in curved spacetimes in general the energy transfer term for the Born decay has the form \begin{equation} \tilde Q_\mu = - \tilde Q_{(\phi) \mu} = \partial_\mu \tilde \phi \tilde \Gamma(\tilde \phi, \tilde n^\nu \partial_\nu\tilde\phi), \label{EnegyTranserTermByphi}\end{equation} where $\tilde n^\mu$ is some timelike unit vector which coincides with the unit normal to the constant time hypersurfaces in the spatially homogeneous case. Though we cannot determine this unit vector, its choice has no effect in the framework of the linear perturbation theory for the following reason. In the linear perturbation, since $\tilde n^j$ is a first-order quantity, the perturbation of $\tilde n^\mu \partial_\mu\tilde\phi$ depends only on $\delta n^0$ as \begin{equation} \delta(\tilde n^\mu \partial_\mu\tilde\phi)=\partial_0 \delta\phi +\delta n^0\partial_0 \phi. \end{equation} On the other hand, for the same reason, $\delta n^0$ is determined only by the perturbation of $\tilde g_{00}$ as \begin{equation} \delta n^0=-{1\over 2}{\delta g_{00}\over g_{00}}. \end{equation} Hence the freedom of $\tilde n^\mu$ has no effect. The above argument on the structure of $\tilde Q_\mu$ may not apply to the energy transfer term for other processes such as the parametric resonance. However, since $\partial_\mu \tilde \phi$ is the unique vector field constructed from $\phi$ in the inflaton dominated stage, it is reasonable to assume that ${\tilde Q}_\mu$ has the same structure as that given in Eq.(\ref {EnegyTranserTermByphi}) for such cases as well, though $\tilde \Gamma$ may depend on higher derivatives of $\tilde \phi$ \subsection{WKB approximation and replacement of the scalar field by a perfect fluid} From the discussion so far we can formulate the evolution equation for perturbations during the reheating stage by applying the gauge-invariant formalism for a general multi-component system to the system consisting of the classical scalar field, radiation and the gravitational field. However, the equations obtained by this procedure is rather difficult to analyze because some of the terms in the equations becomes very large periodically when $\dot\phi$ vanishes. In order to avoid this difficulty and make the problem tractable, we utilize the result of our previous paper\cite{Kodama.H&Hamazaki1996}. It is shown there that during the stage in which the rapidly oscillating classical scalar field dominates the energy density of the universe and its behavior is well described by the WKB form \begin{equation} \tilde \phi = F(\tilde S, x), \end{equation} where $\tilde S$ is a rapidly oscillating phase function, the behavior of superhorizon scale perturbations coincides with that for a perfect fluid system obtained by a spacetime averaging of the energy-momentum tensor of $T_{(\phi)}^\mu{}_\nu$ over Hubble horizon scales except for a sequence of negligibly short intervals around the zero points of $\dot\phi$. This implies that we can replace the rapidly oscillating scalar field by a perfect fluid in the investigation of the behavior of superhorizon perturbations as far as the perturbation variables averaged over the oscillation period are concerned. On the basis of this result we consider a perfect fluid instead of treating the classical scalar field directly. The energy-momentum tensor of the perfect fluid corresponding to the classical scalar field is given by\cite{Kodama.H&Hamazaki1996} \begin{eqnarray} &&\tilde T_{(f)}^\mu{}_\nu=(\tilde \rho_f + \tilde P_f)\tilde u_{(f)}^\mu \tilde u_{(f)\nu} +\tilde P \delta^\mu_\nu,\\ && \tilde P_f=w_f \tilde \rho_f; \quad w_f={n-2\over n+2}, \end{eqnarray} where we have assumed that the potential of the scalar field is given by a simple power-law function \begin{equation} U={\lambda\over n}|\phi|^n. \end{equation} We assume this power law form throughout this paper. $\tilde \rho_f$ and $\tilde u_{(f)}^\mu$ is represented in terms of the original scalar field as \begin{eqnarray} &&\tilde \rho_f={n+2\over 2n}<(\nabla\tilde\phi)^2> \simeq {n+2\over 2n}<(\partial_{\tilde S} F)^2>(\tilde \nabla<\tilde S>)^2,\\ && \tilde u_{(f)\mu}= {\partial_\mu<\tilde S>\over [ (\tilde\nabla<\tilde S>)^2 ]^{1 \over 2} }, \end{eqnarray} where $<X>$ represents a spacetime average of $X$ over the Hubble horizon scales. This approximation is good if the parameter defined by \begin{equation} \epsilon^2:={<(\nabla F)^2>\over <|\partial_S F|^2(\nabla S)^2>} \end{equation} is much smaller than unity. Since $\epsilon$ represents the ratio of the cosmic expansion rate $H$ to the oscillation frequency of the scalar field, this condition is satisfied in the rapidly oscillating phase. In order to formulate the perturbation equations for this perfect fluid and radiation, we must rewrite the spacetime average of the energy-momentum transfer term (\ref{EnegyTranserTermByphi}) in terms of the fluid variables. If we write $<\tilde Q_\mu>$ as $\tilde Q_\mu$ for simplicity, it must have the form \begin{equation} \tilde Q_\mu=\partial_\mu <\tilde S> R(<\tilde\phi>^2, \tilde n^\nu \partial_\nu <\tilde S>). \end{equation} From the argument in the previous subsection we can take $\partial^\mu <\tilde S>$ as $\tilde n^\mu$. Hence, from the relativistic virial theorem\cite{Kodama.H&Hamazaki1996} \begin{equation} <U(\tilde\phi)>=-{1\over n}<(\tilde \nabla\tilde\phi)^2> \left[1+\Order{\epsilon}\right], \end{equation} it is simply written as \begin{equation} \tilde Q_\mu=\tilde u_{(f)\mu}{\tilde Q} , \hspace{0.5cm} {\tilde Q}= G (\tilde\rho_f) . \end{equation} For example, for the Born decay in the model considered in the previous subsection, $G$ is simply given by \begin{equation} G (\tilde\rho_f) ={2n\over n+2}\gamma \tilde \rho_f. \end{equation} On the other hand, for the energy transfer by the parametric resonance, $\tilde Q$ should be modified as \begin{equation} {\tilde Q}= G({\tilde \rho}_f, {1 \over 3}{\tilde \nabla}_\mu {\tilde u}^\mu_f) , \end{equation} where ${\tilde \nabla}_\mu {\tilde u}^\mu_f /3$ represents the expansion rate of the $\phi={\rm const}$ hypersurface, and coincides with the Hubble parameter $H$ in the unperturbed background. This dependence arises because the duration of the parametric resonance for each mode of massless fields coupled with $\phi$ depends on the cosmic expansion rate. Though ${\tilde \nabla}_\mu {\tilde u}^\mu_f$ is of order $\epsilon$ in the WKB approximation scheme, it may not be neglected because ${\tilde Q}$ has a strong dependence on it for the parametric resonance decay. \section{Evolution equations for perturbations} Under the assumptions given in the previous section, we can easily write down the evolution equation of perturbations during reheating by applying the gauge invariant perturbation theory of a multicomponent system to the current system. Basically a perturbation of this system is described by the gauge invariant density contrasts and the shears of the 4-velocity for the inflaton fluid and radiation. But in order to investigate the dynamical behavior of the Bardeen parameter, it is more convenient to choose variables which respect the decomposition of perturbations into the adiabatic modes and the entropy modes. Hence, as the basic variables, we adopt the curvature perturbation $\Phi$, the total shear velocity $V$, and the quantities representing the difference of the density contrasts and the shear velocities between the inflaton and radiation, $Y$ and $Z$, defined by \begin{equation} Y = {\rho_f \rho_r \over \rho^2}S_{f r}, \hspace{0.5cm} Z = {\rho_f \rho_r \over \rho^2}{aH \over k} V_{f r}. \label {e:defent} \end{equation} Note that $Y$ and $Z$ become zero at the beginning and at the end of the reheating stage when the energy density of radiation or the inflaton is negligible. With the help of the general formulae given in Appendix A, we can easily write down the evolution equations for the basic variables by specializing the space dimension to 3. First, from Eq.(\ref {e:Ds}), (\ref {e:Vs}), (\ref {e:gamcom}), and (\ref {e:gamrel}), the evolution equations for $\Phi$ and $\Upsilon$ are written in terms of these variables as \begin{eqnarray} && {\cal D} \Phi + \Phi = -{3 \over 2} I_K (1+w) \Upsilon , \label {e:evophi} \\ && {\cal D} \Upsilon + {3 \over 2} I_K (1+w) \Upsilon -{K \over a^2 H^2} \Upsilon \nonumber \\ && \quad \quad = \left[ -1+{2 \over 3} \left(k \over a H\right)^2 {C_K \over I_K} {1 \over 1+w} {1 \over h} \left(w_f h_f +{1 \over 3}h_r\right) \right] \Phi \nonumber \\ && \quad \quad \quad + {4 \over 3}{1+w_f \over (1+w)^2}\left(w_f-{1 \over 3}\right)Y , \label {e:evoups} \end{eqnarray} where ${\cal D}= a ( d / da )=d / d({\log a})$. On the other hand, by taking account of Eqs. (\ref {e:Fcalp}), (\ref {e:EMconp}), and (\ref {e:momdif}), Eq.(\ref {e:ED}) and (\ref {e:EV}) reduce to the following evolution equations for $Y$, $Z$: \begin{eqnarray} &&{\cal D}Y + 3\left({1 \over 3}+w_f-2w\right)Y = - \left({k \over aH}\right)^2 Z -{3 \over 4}{Q \over H \rho}{1+w \over 1+w_f}E_{c f} \nonumber \\ && \quad \quad +{2Q \over 3H \rho}{1 \over 1+w_f}\left( 1+{3 \over 4}w_f \right) {C_K \over I_K}\left({k \over a H}\right)^2 \Phi, \label {e:evoY} \\ && {\cal D}Z +\left[ {3 \over 2}(1+w)I_K-{K \over a^2 H^2} +3(w_f-2w)+(1-3 w_f){h_r \over h}\right. \nonumber \\ && \quad \quad \left. +{Q \over Hh}\left( w_f + { 2 \over 3 } +{4 \over 3}{1 \over 1+w_f}{\rho_r \over \rho_f} \right) \right]Z \nonumber \\ && \quad \quad = {1 \over h}\left[ {4 \over 3} w_f \rho_r +{1 \over 3}(1+w_f) \rho_f \right] Y \nonumber \\ && \quad \quad +{2 \over 3}\left(w_f -{1 \over 3}\right){C_K \over I_K} \left({k \over a H }\right)^2 {\rho_f \rho_r \over \rho^2}{\Phi \over 1+w} . \label {e:evoZ} \end{eqnarray} In particular, subtracting Eq.(\ref {e:evoups}) from Eq.(\ref {e:evophi}), we obtain \begin{eqnarray} && {\cal D} (\Phi - \Upsilon) = - {K \over a^2 H^2} \Upsilon - {2 \over 3} \left({k \over a H}\right)^2 {C_K \over I_K} {1 \over 1+w} {1 \over h} \left(w_f h_f +{1 \over 3}h_r \right) \Phi \nonumber \\ && \quad \quad - {4 \over 3}{1+w_f \over (1+w)^2}\left(w_f-{1 \over 3}\right)Y, \label {e:Bard} \end{eqnarray} where $\Phi - \Upsilon$ denoted as $\zeta$ in the article\cite{r:Mukh} is referred to as the Bardeen parameter from now on\cite{Bardeen.J&Steinhardt&Turner1983}. Taking into account that $\Phi - \Upsilon$ and $\Phi$ are of the same order, and that the spatial curvature $K$ is practically zero because of the inflationary expansion, we immediately confirm from this equation that the Bardeen parameter $\Phi - \Upsilon$ is conserved on superhorizon scales, if the entropy perturbation $Y$ is negligibly small. In the next section, we will investigate in how good accuracy the Bardeen parameter is conserved by evaluating the amplitude of the entropy perturbation generated in realistic reheating processes. In the evolution equations of the entropy perturbation, the perturbation of the energy transfer rate works as the source term. When the energy transfer rate ${\tilde Q}$ depends only on ${\tilde \rho}_f$, the perturbation of the energy transfer rate is expressed in terms of the basic variables as \begin{eqnarray} E_{c f} &=& { G_{\rho_f} (\rho_f) \rho_f \over G(\rho_f) } \Delta_{c f} \nonumber \\ &=& { G_{\rho_f} (\rho_f) \rho_f \over G(\rho_f) } {1 + w_f \over 1+w} \left[ {2 \over 3}{C_K \over I_K} \left({k \over aH}\right)^2 \Phi + {4 \over 3}{\rho \over \rho_f}Y \right], \label {e:enerho} \end{eqnarray} where $G_{\rho_f}$ denotes the partial derivative of $G$ with respect to ${\rho_f}$. If the energy transfer rate depends also on the cosmic expansion rate ${\tilde \nabla}_\mu {\tilde u}^\mu_f$ as \begin{equation} {\tilde Q}= G({\tilde \rho}_f, {1 \over 3}{\tilde \nabla}_\mu {\tilde u}^\mu_f), \end{equation} the following term should be added to the right hand side of Eq.({\ref {e:enerho}}): \begin{eqnarray} && {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left( {1 \over 3}{k \over a H }V_f - {K \over a^2 H^2}\Upsilon \right) \nonumber \\ && \quad \quad = {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left[ {1 \over 3}\left({k \over a H }\right)^2 \Upsilon + {4 \over 9} {1 \over 1+w}\left({k \over a H }\right)^2 {\rho \over \rho_f }Z - {K \over a^2 H^2}\Upsilon \right]. \label {e:traHub1} \end{eqnarray} On the other hand, if ${\tilde \nabla}_\mu {\tilde u}^\mu$, or ${\tilde \nabla}_\mu {\tilde u}^\mu_r$ are adopted as the local Hubble constant, the terms to be added are given by \begin{eqnarray} && {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left( {1 \over 3}{k \over a H }V - {K \over a^2 H^2}\Upsilon \right) \nonumber \\ && \quad \quad = {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left[ {1 \over 3}\left({k \over a H }\right)^2 \Upsilon - {K \over a^2 H^2}\Upsilon \right], \end{eqnarray} and \begin{eqnarray} && {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left( {1 \over 3}{k \over a H }V_r - {K \over a^2 H^2}\Upsilon \right) \nonumber \\ && \quad \quad = {G_H (\rho_f, H) ~ H \over G (\rho_f, H) } \left[ {1 \over 3}\left({k \over a H }\right)^2 \Upsilon - {1 \over 3} {1+w_f \over 1+w} \left({k \over a H }\right)^2 {\rho \over \rho_r }Z - {K \over a^2 H^2}\Upsilon \right], \label {e:traHub2} \end{eqnarray} respectively. In all of these expressions for $E_{c f}$, all the terms in proportion to $\Phi$ or $\Upsilon$ are multiplied by coefficients of order $ ({k / a H})^2 $. In the next section, we will show that this suppression factor makes the contribution of the entropy perturbations negligibly small, even if we take into account a possible dynamical growth of $Y$ and $Z$. \section{Conservation of the Bardeen parameter during the reheating stage} From Eq.(\ref {e:Bard}), it follows that the entropy perturbation does not affect the conservation of the Bardeen parameter if the condition \begin{equation} \int_{a_s}^{a_e} { da \over a } |Y| \ll |\Phi| \label {e:purpo} \end{equation} is satisfied, where $a_s$ and $a_e$ are the values of the scale factor at the start and at the end of the reheating stage. In this section we show that this condition are satisfied in the realistic chaotic inflation scenario whose dominant reheating processes are the parametric resonance and/or the Born decay. First we define the index $g_{\rho_f}$ and $g_H$ by \begin{equation} g_{\rho_f} := {G_{\rho_f} {\rho_f} \over G} , \hspace{0.5cm} g_H := {G_H H \over G} . \end{equation} Though $g_{\rho_f}$ is bounded by unit from below, \begin{equation} g_{\rho_f} \ge 1, \end{equation} for realistic reheating processes, it may become much larger than unity in the stage in which the parametric resonance is effective. It is difficult to evaluate the upper bound on $g_{\rho_f}$ in this stage because we have poor knowledge on the strong parametric resonance. However, since the analysis in the weak parametric resonance\cite{Shtanov.Y&Traschen&Brandenberger1995} shows that $G$ behaves as $G \sim \exp ( {\rm O}(g_{\rho_f}) )$ when $g_{\rho_f} \gg 1$, it is expected that $g_{\rho_f}$ does not exceed unity by many orders of magnitude. For example, according to the recent numerical investigation taking account of rescattering of produced particles\cite {r:param4}, the effective value of $g_{\rho_f}$ does not exceed $100$. In contrast to $g_{\rho_f}$, $g_H$ does not have a definite sign and may becomes negative. However, from the analysis of weak parametric resonance, it is expected that its absolute value $|g_H|$ is at most of the same order as $g_{\rho_f}$. Hence, when the distinction of these is not important, we use \begin{equation} g := \max \{ g_{\rho_f}, |g_H| \} . \end{equation} Note that as the Born decay dominates in the energy transfer, $g_H$ vanishes and $g_{\rho_f}=g$ approaches unity. In order to evaluate the amplitude of entropy perturbations, it is convenient to decompose the reheating stage into the following four substages: \begin{eqnarray} && 1)~ a_s \le a < a_1 \quad {G \over H \rho_f} \gg 1 , \nonumber \\ && 2)~ a_1 \le a < a_2 \quad {G \over H \rho_f} g \gg 1 \stackrel{>}{\sim} {G \over H \rho_f} , \nonumber \\ && 3)~ a_2 \le a < a_3 \quad 1 \stackrel{>}{\sim} {G \over H \rho_f} g , \quad 1\stackrel{>}{\sim} {G\over H\rho_f}, \nonumber \\ && 4)~ a_3 \le a < a_e \quad {G \over H \rho_f} \gg 1 . \end{eqnarray} The first substage corresponds to the period during which an explosive energy transfer occurs by the parametric resonance. As the amplitude of the inflaton oscillation decreases and the parametric resonance gets less effective, $G/H\rho_f$ becomes less than unity. If $g$ is much greater than unity in this phase, the second substage appears. On the other hand, if the Born decay already dominates at that time and $g=1$, this stage does not appear and the system goes directly to the third substage, during which the Born decay is the main process of reheating but it is still slower than the cosmic expansion. Finally as the cosmic expansion rate decreases with time, $G/H\rho_f$ becomes greater than unity again and the reheating completes. This is the last substage. We evaluate the upper bound of the amplitude of the entropy perturbation in each stage by utilizing the technique explained in Appendix B. We assume $K=0$ henceforth. First we put Eqs. (\ref {e:evoY}) and (\ref {e:evoZ}) into the matrix form \begin{equation} {\cal D}\bm{X}= \bg{\Omega}\bm{X} + \bm{S} , \label {e:2times2} \end{equation} where 2-column vectors $\bm{X}$ and $\bm{S}$, and $2 \times 2$ matrix $\bg{\Omega}$ are defined by \begin{eqnarray} &&\bm{X} := \left ( \begin{array}{c} Y \\ Z \end{array} \right ) , \nonumber \\ && \bg{\Omega} := \left ( \begin{array}{cc} \begin{array}{l} -1-3 w_f +6w \\ -{G \over H \rho_f}{g_{\rho_f}} \end{array} & ({k \over a H})^2 ( -1-{G \over 3 H \rho_f} g_H {1 \over 1+w_f} ) \\ \quad & \quad \\ {4 \over 3}w_f{\rho_r \over h}+{1 \over 3}(1+w_f){\rho_f \over h} & \begin{array}{c} -{3 \over 2}+{9 \over 2}w-3 w_f-(1-3 w_f){h_r \over h} \\ -{G \over H \rho_f}\{ (w_f+ {2 \over 3}) {\rho_f \over h} +{4 \over 3}{1 \over 1+w_f}{\rho_r \over h} \} \end{array} \end{array} \right ) , \nonumber \\ && \bm{S} := \left ( \begin{array}{c} -{1 \over 2} {G \over H \rho} g_{\rho_f} + {1 \over 6}{G \over H \rho} { 4+3 w_f \over 1+ w_f } \\ \quad \\ -{2 \over 9}{ 1 - 3 w_f \over 1+w } {\rho_f \rho_r \over \rho^2} \end{array} \right ) \left({k \over a H}\right)^2 \Phi \nonumber \\ && \quad \quad \quad \quad + \left ( \begin{array}{c} -{1 \over 4}{G \over H \rho} g_H { 1+w \over 1+w_f } \\ \quad \\ 0 \end{array} \right ) \left({k \over a H}\right)^2 \Upsilon . \nonumber \\ && \quad \end{eqnarray} In order to estimate the upper bound of $\bg{\Omega}_H := \bg{\Omega} +\bg{\Omega}^\dagger$, we decompose it into a sum of three matrices as \begin{eqnarray} && \bg{\Omega}_{H1} = \left( \begin{array} {cc} -2-6 w_f +12w & 0 \\ 0 & -3+9w-6 w_f -(2-6 w_f){h_r \over h} \end{array} \right) , \nonumber \\ && \bg{\Omega}_{H2} = \left( \begin{array} {cc} 0 & \begin{array}{c} -({k \over a H})^2 ( 1+{G \over 3 H \rho_f} g_H {1 \over 1+w_f} ) \\ + {4 \over 3}w_f{\rho_r \over h} +{1 \over 3}(1+w_f){\rho_f \over h} \end{array} \\ \quad & \quad \\ \begin{array}{c} -({k \over a H})^2 ( 1+{G \over 3 H \rho_f} g_H {1 \over 1+w_f} ) \\ + {4 \over 3}w_f{\rho_r \over h} +{1 \over 3}(1+w_f){\rho_f \over h} \end{array} & 0 \end{array} \right) , \nonumber \\ && \bg{\Omega}_{H3} = \left( \begin{array} {cc} -2{G \over H \rho_f}{g_{\rho_f}} & 0 \\ 0 & -{G \over H \rho_f}\{ (2 w_f+ {4 \over 3}) {\rho_f \over h} +{8 \over 3}{1 \over 1+w_f}{\rho_r \over h} \} \end{array} \right) . \end{eqnarray} Then the maximum eigenvalue of these matrices are given by \begin{eqnarray} && \lambda_{m1} = \max \left\{ -2-6 w_f +12w , -3+9w-6 w_f -(2-6 w_f){h_r \over h} \right\} , \\ && \lambda_{m2} = -\left({k \over a H}\right)^2 \left( 1+{G \over 3 H \rho_f} g_H {1 \over 1+w_f} \right) + {4 \over 3}w_f{\rho_r \over h} +{1 \over 3}(1+w_f){\rho_f \over h} , \label {e:2nd} \\ && \lambda_{m3} = - \min \left\{ 2{G \over H \rho_f}{g_{\rho_f}} , {G \over H \rho_f}\left[ \left(2 w_f+ {4 \over 3}\right) {\rho_f \over h} +{8 \over 3}{1 \over 1+w_f}{\rho_r \over h}\right] \right\}, \end{eqnarray} respectively. Hence the maximum eigenvalue of $\bg{\Omega}_H$ is bounded by the sum of them, \begin{eqnarray} \lambda_m &:=& \lambda_{m1} + \lambda_{m2} + \lambda_{m3} \nonumber \\ &\le& N -\left({k \over a H}\right)^2 \left( 1+{G \over 3 H \rho_f} g_H {1 \over 1+w_f}\right) \nonumber \\ && \quad \quad - \min \left\{ 2{G \over H \rho_f}{g_{\rho_f}} , {G \over H \rho_f}\left[ \left(2 w_f+ {4 \over 3}\right) {\rho_f \over h} +{8 \over 3}{1 \over 1+w_f}{\rho_r \over h} \right] \right\}, \label {e:lambup} \end{eqnarray} Here $N$ is the maximum value of the sum of $\lambda_{m1}$ and the two last terms on the right hand side of (\ref {e:2nd}). Since this sum becomes maximum at $\rho_r=0$ or $\rho_f=0$ for fixed $w_f$, $N$ is given by \begin{equation} N := \max \{ 2-5 w_f, 6 w_f - {5 \over 3} \} . \end{equation} The source term is evaluated as \begin{equation} \| \bm{S} \| \sim \Order{ \left(g { G \over H \rho } +1\right) \left( { k \over a H } \right)^2 |\Phi| }, \end{equation} taking account of the fact that $\Phi$ is of the same order as $\Upsilon$. First in the first stage, since $\lambda_m$ is negative and its absolute value$\sim G/H\rho_f$ is much larger than unity, Eqs.(\ref {e:uplim}) and (\ref {e:uplimp}) in the Appendix B yields \begin{equation} \| \bm{X}(a) \| \le \Order{g \left( { k \over a H } \right)^2 |\Phi|}. \end{equation} Next in the second stage, $\lambda_m$ is bounded as $\lambda_m\le N$ since $(k/aH)^2g$ is practically much smaller than unity. Hence, by using Eq.(\ref {e:uplim}) in the Appendix B and by taking account of the order of magnitude of $\|\bm{X}\|$ at the end of the previous stage, $\| \bm{X}(a_1) \|$, we obtain \begin{equation} \| \bm{X}(a) \| \le \left( { a \over a_1 } \right)^{N / 2} \Order{ g_1 \left( { k \over a H } \right)^2 |\Phi| }, \end{equation} where $g_1$ is the value of $g$ at $a=a_1$. In the third stage, in the same way we obtain \begin{equation} \| \bm{X}(a) \| \le \left( { a \over a_1 } \right)^{N / 2} \Order{ g_1 \left( { k \over a H } \right)^2 |\Phi| }. \end{equation} Finally in the fourth stage, since the eigenvalue of $\bg{\Omega}$ is negative, and its absolute value increases exponentially in time, the influence of the previous stage on $\bm{X}(a)$, corresponding to the first-term in Eq.(\ref{e:uplim}), is rapidly erased, and $\|\bm{X}(a)\|$ settles down to a value determined locally by $\|\bm{S}\|$ as \begin{equation} \| \bm{X}(a) \| \le \Order{ \left( { k \over a H } \right)^2 |\Phi|}, \end{equation} as seen from Eqs.(\ref {e:uplim}) and (\ref {e:uplimp}). This estimate is actually a quite weak one. The actual value of $\|\bm{X}\|$ decreases exponentially in time in this stage due to the suppression factor $\rho_f/\rho$ in the definition (\ref{e:defent}). Hence $Y$ and $Z$ take nonnegligible values only in stages when the inflaton and radiation coexists, and vanish as soon as the energy transfer completes effectively. These estimates on the upper bound of $\| \bm{X} \|$ can be used to obtain stronger bounds on $Y$ and $Z$. To see this, let us write the second row of Eq.(\ref{e:2times2}) as \begin{eqnarray} && {\cal D} Z = \Omega_Z Z + S_Z, \\ && \Omega_Z := -{3 \over 2}+{9 \over 2}w-3 w_f - (1-3 w_f){h_r \over h} - \Order{ {G \over H \rho} }, \\ && S_Z := \Order{Y} + \Order{\left({ k \over a H }\right)^2 \Phi }, \end{eqnarray} and regard this as the evolution equation for $Z$. Then since $\Omega_Z$ is bounded as \begin{eqnarray} &&\Omega_Z \le N_Z - \Order{{G \over H \rho} },\\ && N_Z := \max \left\{ -3 w_f, {3 \over 2}(w_f-1) \right\}, \end{eqnarray} using the values of the upper bound obtained in the previous analysis on $|Y| ( \le \| \bm{X} \| )$ and applying Eqs.(\ref {e:uplim}) and (\ref {e:uplimp}) to the above equation on $Z$, we obtain the following stronger upper bound on $|Y|$ in the first and the fourth stage: \begin{equation} |Z| \le \Order{{ H \rho \over G } g \left({ k \over a H }\right)^2 |\Phi|}. \end{equation} We can apply the same method to the evolution equation for $Y$ obtained from the first row of \Eq{e:2times2}, \begin{eqnarray} && {\cal D} Y = \Omega_Y Y + S_Y, \\ && \Omega_Y := -1-3 w_f +6 w -{G \over H \rho_f}{ g_{\rho_f} },\\ && S_Y := \Order{\left(1+ {G \over H \rho_f} g \right) \left({ k \over a H }\right)^2 Z } + \Order{ {G \over H \rho} g \left({ k \over a H }\right)^2 \Phi}. \end{eqnarray} Now $\Omega_Y$ is bounded as \begin{eqnarray} && \Omega_Y \le N_Y -{G \over H \rho_f}{ g_{\rho_f} },\\ && N_Y := |1-3 w_f|. \end{eqnarray} Hence we obtain in the first stage \begin{equation} |Y(a)| \le \Order{\left({ k \over a H }\right)^2 \Phi }, \end{equation} assuming that \begin{equation} g \left({ k \over a H }\right)^2 < 1, \end{equation} in the second stage \begin{equation} |Y(a)| \le \Order{ \max \left\{ 1, g_1 \left({ a \over a_1 }\right)^{N/2} \left({ k \over a H }\right)^2 \right\} } \left({ k \over a H }\right)^2 \Phi, \end{equation} and in the third stage \begin{equation} |Y| \le \left({ a \over a_2 }\right)^{N_Y} \Order{ \max \left\{ 1, g_1 \left({ a \over a_1 }\right)^{N/2} \left({ k \over a H }\right)^2 \right\} } \left({ k \over a H }\right)^2 |\Phi| . \end{equation} Putting these estimates together, we finally obtain \begin{eqnarray} \int_{a_s}^{a_e} { da \over a } |Y| &\cong& \int_{a_1}^{a_3} { da \over a } |Y| \nonumber \\ &\le& \left({ a_3 \over a_1 }\right)^{N_Y} \Order{ \max \left\{ 1, g_1 \left({ a_3 \over a_1 }\right)^{N/2} \left({ k \over a H }\right)^2_3 \right\} } \left({ k \over a H }\right)^2_3 |\Phi|. \end{eqnarray} Taking account of $N/2 \ge N_Y$ when $n \ge 2$, we can conclude that the entropy perturbation $Y$ does not affect the conservation of the Bardeen parameter if \begin{equation} g_1 \left( { a_e \over a_s } \right)^{N_Y/2+N/4} \left( { k \over a H } \right)^2_e \ll 1 . \label {e:cnsrvcnd} \end{equation} is satisfied. Since $g_1$ is a monotonic function of $\rho_f$ and it is expected that it does not exceed unity much, we drop it henceforth. The condition (\ref {e:cnsrvcnd}) gives a lower bound on the energy density $\rho(a_e)$ at the end of reheating. Let us evaluate its order of magnitude for realistic values of the physical parameters. First we introduce the parameter $y$ defined by $\rho(a_e)^{ 1 /4 } = 10^{y} $GeV. This parameter is related to the value of $k/aH$ at the end of reheating for perturbations corresponding to the present cosmological structures of $10$ Mpc scales by \begin{equation} \left( { k \over a H } \right)^2_e \sim 10^{-2y-15} . \end{equation} Here we have assumed $a(t_{eq}) / a(t_0) \sim 10^{-4}$ where $t_{eq}$ is the equality time. In order to generate density perturbations consistent with the observed anisotropy of the cosmic microwave background ${ \delta T / T } \sim 10^{-5}$, the energy density at the time when the relevant perturbations cross the Hubble horizon in the inflationary stage should be given by \begin{equation} {\rho}^{ 1 /4 } \sim 10^{16} \r{GeV}, \end{equation} which is approximately equal to $\rho(a_s)$. Hence from the inequality \begin{eqnarray} { a_e \over a_s } \le \left( { \rho(a_s) \over \rho (a_e) } \right)^{ 1 / [ 3(1+w_{min}) ] } \nonumber \\ w_{min}:=\min \left\{ w_f, {1 \over 3} \right\}, \end{eqnarray} we can see that the condition (\ref {e:cnsrvcnd}) holds, if \begin{equation} y> \left\{\begin{array}{ll} \displaystyle {576w_f-356 \over 36w_f+13} & \r{for}\quad w_f\ge{1\over3},\\ \displaystyle {221w_f-19\over 5w_f-10} & \r{for}\quad w_f\le{1\over3} \end{array}\right. \label {e:solid} \end{equation} is satisfied. This condition on $y$ is a strict one from a mathematical point of view. However, it seems to be too strong from a physical point of view because $Y$ takes nonnegligible value only around $x:=\rho_r / \rho_f \sim 1 $, while the above estimate is based on the upper bounds on $\lambda_m$ and $\Omega_Y$ that correspond to the values at $x=0$ or $x=\infty$. Hence a practical condition is obtained by replacing $N$ and $N_Y$ in Eq.(\ref {e:cnsrvcnd}) by the values of $\lambda_m$ and $\Omega_Y$ at $x=1$. It is expressed as \begin{equation} y> \left\{\begin{array}{ll} \displaystyle -{100w_f+404\over29w_f+57} & \r{for}\quad w_f\ge{1\over3},\\ \displaystyle -{135w_f^2+370w_f+299\over18w_f^2+65w_f+43} & \r{for}\quad w_f\le{1\over3}. \end{array}\right. \label {e:dotted} \end{equation} These conditions are depicted in Figure 1. \begin{figure}[htb] \centerline{\epsfysize=8.5cm \epsfbox{bound.eps}} \vspace {5mm} \caption{ The energy density at the end of reheating ($\rho^{1/4}=10^y {\rm GeV}$) vs power of the inflaton potential $n$ } \protect\label{f:fig1} \end{figure} \noindent In this figure the solid line and the dotted line correspond to (\ref {e:solid}) and (\ref {e:dotted}), respectively, and the Bardeen parameter is well conserved in the region above these lines. This figure shows that reheating does not affect the conservation of the Bardeen parameter if it terminates before the primordial nucleosynthesis. Hence we can conclude that in all the realistic models based on chaotic inflation, the Bardeen parameter stays constant in a good accuracy during reheating. This justifies the conventional prescription relating the amplitude of adiabatic perturbations at horizon crossing in the post-Friedman stage to the value of the Bardeen parameter in the inflationary stage. \section{Discussion} In this paper we have investigated the evolution of perturbations during reheating taking account of the effect of the energy transfer from the inflaton to radiation, by replacing the inflaton field by a perfect fluid obtained by the spacetime averaging and the WKB approximation. By evaluating the amplitudes of entropy perturbations generated during reheating and their influence on the adiabatic component of perturbations, we have shown that the Bardeen parameter is well conserved during the reheating stage as well as in the inflationary stage and the post-Friedmann stage for realistic models. Though we have considered only the parametric resonance and the Born decay as the dominant reheating processes for definiteness, the conclusion holds rather generally since the arguments are insensitive to the details of the models. Of course our analysis does not exhaust all the possible models of inflation. In particular we have only considered the case in which the inflaton is described by a single component field. Though a simple multi-component extension does not seem to change the conclusion as far as the fluid replacement of the inflaton fields during the reheating stage gives a good approximation, some subtlety may occur if scalar fields with very tiny masses coexist with the inflaton fields. For example, if there exists a scalar field which affects physical parameters controlling the reheating processes such as particle masses or coupling constants of physical particles but has no dynamical effect by itself during reheating, it may produce large entropy perturbations and affect the Bardeen parameter. In particular, as our analysis suggests, this possibility may become important if such a field affects the parameters controlling the parametric resonance processes. In such multi-component systems, entropy perturbations produced during reheating may survive after reheating and have important effects on the present universe, as discussed by Yokoyama et al. as a mechanism to generate the baryon isocurvature perturbation\cite{r:Yokoya}. These problems in the multi-component extension are under investigation. \section*{Acknowledgments} T. H. would like to thank Prof. H. Sato for continuous encouragements. He would like to thank Y. Nambu, M. Sasaki, E. Stewart, and J. Yokoyama for fruitful discussions. This work was partly supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan(H.K.:40161947).

proofpile-arXiv_065-528

\section{Introduction} The study of the density matrices of identical particles (bosons or fermions) moving freely in a box \cite{FeynStat} is generalized in this paper to the case of identical particles in a parabolic confinement potential with either harmonic interactions between the particles, or with an anisotropy induced by a homogeneous magnetic field on top of the parabolic confinement. This model, giving rise to repetitive Gaussian integrals, allows to derive an analytical expression for the generating function of the partition function. For an ideal gas of non-interacting particles in a parabolic well, this generating function coincides with the grand-canonical partition function. With interactions, the calculation of this generating function circumvents the constraints on the summation over the cycles of the permutation group. Moreover, it allows one to calculate the canonical partition function recursively for the system with harmonic two-body interactions. The theory is developed both for fermions and for bosons. In view of the recent interest in Bose-Einstein condensation in a trap \cite{BEC1,BEC2,BEC3}, more attention has been payed to the boson case in the discussion of the results. The model system discussed here has already been studied in the context of quantum dots with operator techniques, and the eigenvalues and eigenstates were calculated including the effect of harmonic two-body interactions and in the presence of a magnetic field \cite{Johnson}. However, to the best of our knowledge neither the boson case nor the thermodynamics seems to have been analyzed previously. It should also be mentioned that the idea of first expanding the Hilbert space to the configuration space and then projecting onto the appropriate subspace by group theoretical means has been used recently \cite {Haase,Hausler} in the context of quantum dots to study the ground state correlations for fermions and bosons. Another motivation to perform the present analytical calculations lies in our path integral formulation of the density matrices for $N$ particles\cite {LBD,BDL,LBDPR}. Indeed, particles in a parabolic potential are a favorite testing ground for the path integral method\cite{FeynHibbs,FeynKlei,GiachTog}% . It should be noted that for our formulation the existence of a positive measure over a well defined domain in the $R^{3N}$ configuration space is essential in view of any algorithmic approach to the problem. In the present paper, which is in essence analytical, integrations over the configuration space are performed. The reason is that the extension of the state space to the configuration space makes the Gaussian integrals tractable. The permutation symmetry leads to summations over the cycles which are performed using the generating function technique, which is one of the main results of the present paper. The model of $N$ identical particles in a parabolic well, in the presence of a magnetic field and with harmonic repulsive or attractive two-body interactions has its intrinsic value, since it constitutes an exactly soluble idealization of atoms in a magnetic trap. It should be stressed that the association of identical particles with each three oscillator degrees of freedom makes the model $3D$. Without Bose-Einstein or Fermi-Dirac statistics, i.e. for ``distinguishable'' particles, the model is equivalent with $3N$ one dimensional oscillators, because each degree of freedom decouples in such a way that there is no difference in statistical behavior between $3N$ $1D$ oscillators and $N$ $3D$ oscillators \cite{Ford}. This paper is organized as follows: the calculation technique is explained in the next section. In the subsequent section we repeat the same calculation for the model with a homogeneous magnetic field. In section IV the results for $1D$ bosons and fermions are given. In section V bosons in $% 3D$ are analyzed in some detail, and in the last section the conclusions are given. \section{Harmonically interacting identical particles in a parabolic well.} In this section we calculate the partition function of $N$ identical particles with the following Lagrangian including one-body and two-body potentials \begin{equation} L=\frac 12\sum_{j=1}^N{\bf \dot{r}}_j^2-V_1-V_2;\quad V_1=\frac{\Omega ^2}2% \sum_{j=1}^N\left. {\bf r}_j\right. ^2;\quad V_2=-\frac{\omega ^2}4% \sum_{j,l=1}^N\left( {\bf r}_j-{\bf r}_l\right) ^2. \end{equation} (Atomic units are used.) The potentials can be rewritten in terms of the center-of-mass coordinate ${\bf R}$ and coordinates ${\bf u}_j$ describing the coordinates of the particles measured from the center of mass \begin{equation} {\bf R}=\frac 1N\sum_{j=1}^N{\bf r}_j;\quad {\bf u}_j={\bf r}_j-{\bf R}, \end{equation} from which \begin{equation} V_1+V_2=V_{CM}+V;\quad V_{CM}=\frac 12N\Omega ^2{\bf R}^2;\quad V=w^2\sum_{j=1}^N\left. {\bf u}_j\right. ^2, \end{equation} with \begin{equation} w=\sqrt{\Omega ^2-N\omega ^2}. \label{eqeigenw} \end{equation} The requirement that $w$ has to be positive expresses the stability condition that the confining potential has to be strong enough to overcome the repulsion between the particles. If an harmonic interparticle {\bf % attraction} is considered, the eigenfrequency $w$ would become $w=\sqrt{% \Omega ^2+N\omega ^2},$ and no stability condition has to be imposed on the confining potential. Notice that these transformations do neither diagonalize the Lagrangian nor the Hamiltonian, because the coordinates $% {\bf u}_j$ are {\sl not} independent of the center-of-mass coordinate. Since the system consists in each direction of 1 degree of freedom with frequency $\Omega $ and $\left( N-1\right) $ degrees of freedom with frequency $w,$ the propagator \begin{equation} K_D\left( {\bf r}_1^{\prime \prime }\cdots {\bf r}_N^{\prime \prime },\beta |% {\bf r}_1^{\prime }\cdots {\bf r}_N^{\prime },0\right) \equiv \left\langle {\bf r}_1^{\prime \prime }\cdots {\bf r}_N^{\prime \prime }\left| e^{-\beta H}\right| {\bf r}_1^{\prime }\cdots {\bf r}_N^{\prime }\right\rangle _D \label{eqKDdef} \end{equation} for {\bf distinguishable} particles (indicated by the subscript $D$ for 3 dimensions and $d$ in 1 dimension) can be calculated from the action expressed in the imaginary time variable, and it is of course a product of the propagators $K_d$ per component: \begin{equation} K_D\left( {\bf r}_1^{\prime \prime }\cdots {\bf r}_N^{\prime \prime },\beta |% {\bf r}_1^{\prime }\cdots {\bf r}_N^{\prime },0\right) =K_d\left( \bar{x}% ^{\prime \prime },\beta |\bar{x}^{\prime },0\right) K_d\left( \bar{y}% ^{\prime \prime },\beta |\bar{y}^{\prime },0\right) K_d\left( \bar{z}% ^{\prime \prime },\beta |\bar{z}^{\prime },0\right) , \label{eqKDseparate} \end{equation} where the column vector $\bar{x}$ contains the $x$-components of the particles, i.e. $\bar{x}^T=\left( x_1,\cdots ,x_N\right) $ and similarly for $\bar{y}$ and $\bar{z}.$ Knowing the propagator $K\left( x^{\prime \prime },\beta |x^{\prime },0\right) _\varpi $ of a single harmonic oscillator with frequency $\varpi $% \begin{equation} K\left( x_\beta ,\beta |x_0,0\right) _\varpi =\sqrt{\frac \varpi {2\pi \sinh \varpi \beta }}\exp \left\{ -\frac \varpi 2\frac{\left( x_\beta ^2+x_0^2\right) \cosh \varpi \beta -2x_\beta x_0}{\sinh \varpi \beta }% \right\} , \end{equation} one finds for the 1-dimensional propagator $K_d$ of the $N$ distinguishable oscillators in the interacting system: \begin{equation} K_d\left( \bar{x}^{\prime \prime },\beta |\bar{x}^{\prime },0\right) =\frac{% K\left( \sqrt{N}X^{\prime \prime },\beta |\sqrt{N}X^{\prime },0\right) _\Omega }{K\left( \sqrt{N}X^{\prime \prime },\beta |\sqrt{N}X^{\prime },0\right) _w}\prod_{j=1}^NK\left( x_j^{\prime \prime },\beta |x_j^{\prime },0\right) _w, \label{eqKdexplicit} \end{equation} where the factor $\sqrt{N}$ in $\sqrt{N}X^{\prime \prime }$ accounts for the mass $N$ (in atomic units) of the center. The denominator in (\ref {eqKdexplicit}) compensates for the fact that $\left( N-1\right) $ instead of $N$ degrees of freedom of frequency $w$ are available. The 3-dimensional propagator $K_D$ (\ref{eqKDdef}) for $N$ distinguishable oscillators of the interacting system is, according to (\ref{eqKDseparate}) and (\ref {eqKdexplicit}), given by \begin{equation} K_D\left( {\bf \bar{r}}^{\prime \prime },\beta |{\bf \bar{r}}^{\prime },0\right) =\frac{K\left( \sqrt{N}{\bf R}^{\prime \prime },\beta |\sqrt{N}% {\bf R}^{\prime },0\right) _\Omega }{K\left( \sqrt{N}{\bf R}^{\prime \prime },\beta |\sqrt{N}{\bf R}^{\prime },0\right) _w}\prod_{j=1}^NK\left( {\bf r}% _j^{\prime \prime },\beta |{\bf r}_j^{\prime },0\right) _w, \label{eqKDproduct} \end{equation} \begin{equation} K\left( {\bf r}_j^{\prime \prime },\beta |{\bf r}_j^{\prime },0\right) _w=K\left( x_j^{\prime \prime },\beta |x_j^{\prime },0\right) _wK\left( y_j^{\prime \prime },\beta |y_j^{\prime },0\right) _wK\left( z_j^{\prime \prime },\beta |z_j^{\prime },0\right) _w \end{equation} where ${\bf \bar{r}}$ denotes a point in the configuration space $R^{3N},$ i.e. ${\bf \bar{r}}^T\equiv \left( \begin{array}{lll} \left( x_1,y_1,z_1\right) , & \cdots , & \left( x_N,y_N,z_N\right) \end{array} \right) $. The symmetrized density matrix $K_I$ for 3D identical particles (indicated by the subscript $I)$ can be obtained by using the following projection, with $P$ denoting the permutation matrix: \begin{equation} K_I\left( {\bf \bar{r}}^{\prime \prime },\beta |{\bf \bar{r}}^{\prime },0\right) =\frac 1{N!}\sum_p\xi ^pK_D\left( P{\bf \bar{r}}^{\prime \prime },\beta |{\bf \bar{r}}^{\prime },0\right) , \end{equation} where $\xi =+1$ for bosons and $\xi =-1$ for fermions. It should be emphasized that $P$ acts on the particle indices, not on the components of $% {\bf r}$ separately. The partition function is then readily obtained by integrating over the configuration space \begin{equation} Z_I=\int d{\bf \bar{r}}K_I\left( {\bf \bar{r}},\beta |{\bf \bar{r}},0\right) =\int d{\bf \bar{r}}\frac 1{N!}\sum_p\xi ^pK_D\left( P{\bf \bar{r}},\beta |% {\bf \bar{r}},0\right) . \end{equation} The remaining part of this section will be devoted to the explicit evaluation of this integral for the partition function. The integration proceeds in 3 stages: the first stage deals with the center-of-mass treatment, the second one concerns the cyclic decomposition, and in the third step the summation over the cycles will be performed. \subsection{The center of mass} The center-of-mass coordinate ${\bf R}$ does not only depend on the coordinates of all the particles, but it also has its own propagator. Therefore substituting ${\bf R}$ by its expression in terms of the particle positions and then performing the integration seems not to be the most adequate way to deal with the integration over the configuration space. Instead, the following identity is used for the formal treatment of ${\bf R}$ as an independent coordinate, at the expense of additional integrations: \begin{equation} \int d{\bf \bar{r}}f\left( {\bf \bar{r},}\frac 1N\sum_{j=1}^N{\bf r}% _j\right) =\int d{\bf R}\int d{\bf \bar{r}}f\left( {\bf \bar{r},R}\right) \delta \left( {\bf R}-\frac 1N\sum_{j=1}^N{\bf r}_j\right) . \end{equation} Fourier transformation of the $\delta $-function then leads to \begin{equation} \int d{\bf \bar{r}}f\left( {\bf \bar{r},}\frac 1N\sum_{j=1}^N{\bf r}% _j\right) =\int d{\bf R}\int \frac{d{\bf k}}{\left( 2\pi \right) ^3}e^{i{\bf % k}\cdot {\bf R}}\int d{\bf \bar{r}}f\left( {\bf \bar{r},R}\right) e^{-i{\bf \bar{k}\cdot \bar{r}}}, \end{equation} where ${\bf \bar{k}}^T=\frac kN\left( \begin{array}{lll} \left( 1,1,1\right) , & \cdots , & \left( 1,1,1\right) \end{array} \right) $ is a $3N$ dimensional row vector. Applying this transformation to the partition function $Z_I$ and rearranging the factors one obtains \begin{equation} Z_I=\int d{\bf R}\int \frac{d{\bf k}}{\left( 2\pi \right) ^3}e^{i{\bf k}% \cdot {\bf R}}\frac{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _\Omega }{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _w}\int d% {\bf \bar{r}}\frac 1{N!}\sum_p\xi ^p\prod_{j=1}^NK\left( \left( P{\bf r}% \right) _j,\beta |{\bf r}_j,0\right) _we^{-i\vec{k}{\bf \cdot }\vec{r}_j/N}. \end{equation} This transformation makes ${\bf R}$ independent of the particle positions relative to the center of mass. The real dependence on the relative positions is reintroduced by the Fourier transform. It should be noted that the explicit dependence of the propagator (\ref{eqKDproduct}) on ${\bf R,}$ and the presence of the factor $e^{-i\vec{k}{\bf \cdot }\vec{r}_j/N},$ are consequences of the two-body interactions. The next step is to rewrite the sum over the permutations as a sum over all possible cycles. This will be done in the next subsection. An excellent example of such a decomposition into cycles has been given by Feynman \cite {FeynStat} for a system of non-interacting particles in a box. \subsection{Cyclic decomposition} A permutation can be broken up into cycles. Suppose that a particular permutation contains $M_\ell $ cycles of length$~\ell .$ The positive integers $M_\ell $ and $\ell $ then have to satisfy the constraint \begin{equation} \sum_\ell \ell M_\ell =N. \label{eqMlsum} \end{equation} Furthermore, the number $M\left( M_1,\cdots M_N\right) $ of cyclic decompositions with $M_1$ cycles of length $1,$ $\cdots ,$ $M_\ell $ cycles of length $\ell ,$ $\cdots $ is known to be \begin{equation} M\left( M_1,\cdots M_N\right) =\frac{N!}{\prod_\ell M_\ell !\ell ^{M_\ell }}. \end{equation} A cycle of length $\ell $ will be obtained from $\left( \ell -1\right) $ permutations. Therefore the sign factor $\xi ^p$ can be decomposed as \begin{equation} \xi ^p=\prod_\ell \xi ^{\left( \ell -1\right) M_\ell }. \end{equation} Combining these result originating from the permutation symmetry one obtains \begin{equation} Z_I=\int d{\bf R}\int \frac{d{\bf k}}{\left( 2\pi \right) ^3}e^{i{\bf k}% \cdot {\bf R}}\frac{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _\Omega }{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _w}% \sum_{M_1\cdots M_N}\prod_\ell \frac{\xi ^{\left( \ell -1\right) M_\ell }}{% M_\ell !\ell ^{M_\ell }}\left( {\cal K}_\ell \left( {\bf k}\right) \right) ^{M_\ell }, \end{equation} \begin{equation} {\cal K}_\ell \left( {\bf k}\right) =\int d{\bf r}_{\ell +1}\int d{\bf r}% _\ell \cdots \int d{\bf r}_1\delta \left( {\bf r}_{\ell +1}-{\bf r}_1\right) \prod_{j=1}^NK\left( {\bf r}_{j+1},\beta |{\bf r}_j,0\right) _we^{-i{\bf % k\cdot r}_j/N}. \end{equation} The $\delta $-function expresses that the decomposition is cyclic. It is obvious that \begin{equation} {\cal K}_\ell \left( {\bf k}\right) ={\cal K}_\ell ^{\left( 1D\right) }\left( k_x\right) {\cal K}_\ell ^{\left( 1D\right) }\left( k_y\right) {\cal % K}_\ell ^{\left( 1D\right) }\left( k_z\right) , \end{equation} which allows to analyze ${\cal K}_\ell \left( {\bf k}\right) $ from its 1-dimensional constituents: \begin{equation} {\cal K}_\ell ^{\left( 1D\right) }\left( k_x\right) =\int dx_{\ell +1}\int dx_\ell \cdots \int dx_1\delta \left( x_{\ell +1}-x_1\right) \prod_{j=1}^NK\left( x_{j+1},\beta |x_j,0\right) _we^{-ik_xx_j/N}. \end{equation} Using the semigroup property \cite{Roep} of the harmonic oscillator propagator $K\left( x_{j+1},\beta |x_j,0\right) _w,$ all integrations but one can be performed \begin{equation} {\cal K}_\ell ^{\left( 1D\right) }\left( k_x\right) =\int dxK\left( x,\ell \beta |x,0\right) _we^{-\int_0^{\ell \beta }d\tau f_x\left( \tau \right) x\left( \tau \right) } \label{eqKl1D} \end{equation} where \begin{equation} f_x\left( \tau \right) =i\frac{k_x}N\sum_{j=0}^{\ell -1}\delta \left( \tau -j\beta \right) . \label{eqdriving} \end{equation} The integral (\ref{eqKl1D}) is the propagator $K_{w,f}$ of a driven harmonic oscillator with the Lagrangian \begin{equation} L_{w,f_x}=\frac 12\dot{x}^2-\frac 12w^2x^2+f_x\left( \tau \right) x. \end{equation} studied in \cite{FeynHibbs,FeynOperator}. It should be noted again that without two-body interactions the driving force (\ref{eqdriving}) is lacking. Taking over the result from \cite{FeynOperator} and integrating over the configuration space one obtains \begin{eqnarray} Z_{w,f_x}\left( \beta \right) &=&\int dxK_{w,f_x}\left( x,\beta |x,0\right) \nonumber \\ &=&\frac 1{2\sinh \frac 12\beta w}\exp \left( \frac 12\int_0^\beta d\tau \int_0^\beta d\sigma \frac{f_x\left( \tau \right) f_x\left( \sigma \right) }{% 2w}\frac{\cosh \left( \left( \frac \beta 2-\left| \tau -\sigma \right| \right) w\right) }{\sinh \frac 12\beta w}\right) . \end{eqnarray} After straightforward algebra one obtains for the 1D function ${\cal K}_\ell ^{\left( 1D\right) }\left( k_x\right) $: \begin{equation} {\cal K}_\ell ^{(1D)}\left( k_x\right) =\frac 1{2\sinh \frac 12\ell \beta w}% \exp \left( -\frac \ell {4N^2}\frac{k_x^2}w\frac{1+e^{-\beta w}}{1-e^{-\beta w}}\right) ; \end{equation} and for its 3D extension: \begin{equation} {\cal K}_\ell \left( \vec{k}\right) =\left( \frac 1{2\sinh \frac 12\ell \beta w}\right) ^3\exp \left( -\frac \ell {4N^2}\frac{k^2}w\frac{1+e^{-\beta w}}{1-e^{-\beta w}}\right) . \end{equation} Using (\ref{eqMlsum}) one then is left with a sixfold integral for the partition function \begin{equation} Z_I= \begin{array}[t]{l} \int d{\bf R}\int \frac{d{\bf k}}{\left( 2\pi \right) ^3}e^{i{\bf k}\cdot {\bf R}}\frac{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _\Omega }{K\left( \sqrt{N}{\bf R},\beta |\sqrt{N}{\bf R},0\right) _w}\exp \left( -\frac 1{4N}\frac{k^2}w\frac{1+e^{-\beta w}}{1-e^{-\beta w}}\right) \\ \times \sum_{M_1\cdots M_N}\prod_\ell \frac{\xi ^{\left( \ell -1\right) M_\ell }}{M_\ell !\ell ^{M_\ell }}\left( \frac 1{2\sinh \frac 12\ell \beta w}% \right) ^{3M_\ell } \end{array} . \end{equation} Both the integrations over ${\bf k}$ and ${\bf R}$ are Gaussian, leading to the following series for $Z_I$: \begin{equation} Z_I=\left( \frac{\sinh \frac 12\beta w}{\sinh \frac 12\beta \Omega }\right) ^3{\Bbb Z}_I\left( N\right) ;{\Bbb \quad Z}_I\left( N\right) \equiv \sum_{M_1\cdots M_N}\prod_\ell \frac{\xi ^{\left( \ell -1\right) M_\ell }}{% M_\ell !\ell ^{M_\ell }}\left( \frac{e^{-\frac 12\ell \beta w}}{1-e^{-\ell \beta w}}\right) ^{3M_\ell }. \label{eqZgeneral} \end{equation} Without two-body interactions ($w=\Omega $), ${\Bbb Z}_I\left( N\right) $ is the partition function of a set of identical oscillators. The partition function $Z_I$ only differs from it by a center-of-mass correction and the actual values of $w.$ The remaining summation over the cycles involves the constraint (\ref {eqMlsum}), which however can be removed by the use of the generating function technique, which will be considered in the next subsection. \subsection{The generating function} Concentrating on the explicit dependence of ${\Bbb Z}_I\left( N\right) $ on $% N$ (with $w$ considered as a parameter), one can construct the following generating function \begin{equation} \Xi \left( u\right) =\sum_{N=0}^\infty {\Bbb Z}_I\left( N\right) u^N, \end{equation} with ${\Bbb Z}_I\left( 0\right) =1$ by definition. The partition function $% {\Bbb Z}_I\left( N\right) $ can then be obtained by taking the appropriate derivatives of $\Xi \left( u\right) $ with respect to $u,$ assuming that the series for $\Xi \left( u\right) $ is convergent near $u=0$: \begin{equation} {\Bbb Z}_I\left( N\right) =\frac 1{N!}\left. \frac{d^N}{du^N}\Xi \left( u\right) \right| _{u=0}. \end{equation} The summation over the number of cycles with length $\ell $ is now unrestricted, and can easily be performed: \begin{equation} \Xi _I\left( u\right) =\exp \left( \sum_{\ell =1}^\infty \xi ^{\ell -1}\frac{% e^{-\frac 32\ell \beta w}u^\ell }{\ell \left( 1-e^{-\ell \beta w}\right) ^3}% \right) . \end{equation} This series can be rewritten into the more familiar form \begin{equation} \Xi _I\left( u\right) =\exp \left( -\xi \sum_{\nu =0}^\infty \frac 12\left( \nu +1\right) \left( \nu +2\right) \ln \left( 1-\xi ue^{-\beta w\left( \frac % 32+\nu \right) }\right) \right) . \label{eqKsinomagn} \end{equation} It should be noted in view of the remarks at the end of the preceding section concerning ${\Bbb Z}_I\left( N\right) $, that $\Xi _I\left( u\right) $ is also the generating function of a model without two-body interactions. In that case $w$ equals $\Omega ,$ and $\Xi _I\left( u\right) $ coincides with the well known \cite{BDLSSCpress,Grossmann,Kirsten} grand canonical partition function of a set of identical particles in a parabolic well. \subsection{Recurrence relations for the partition function} Starting from the expression for $\Xi _I\left( u\right) $ derived in the previous subsection for the interacting model, a recursion relation can be obtained for ${\Bbb Z}_I\left( N\right) $. Introducing: \begin{equation} b=e^{-\beta w} \end{equation} for brevity in the notations, we observe that \begin{equation} \frac d{du}\Xi _I\left( u\right) =\Xi _I\left( u\right) \sum_{\nu =0}^\infty \frac 12\left( \nu +1\right) \left( \nu +2\right) \frac{b^{\frac 32+\nu }}{% 1-\xi ub^{\frac 32+\nu }} \end{equation} Considering next ${\Bbb Z}_I\left( N\right) =\frac 1{N!}\frac{d^{N-1}}{% du^{N-1}}\left. \frac d{du}\Xi \left( u\right) \right| _{u=0},$ the product rule and an elementary binomial expansion can be used to find \begin{equation} {\Bbb Z}_I\left( N\right) =\frac 1N\sum_{m=0}^{N-1}\xi ^{N-m-1}\left( \frac{% b^{\frac 12\left( N-m\right) }}{1-b^{N-m}}\right) ^3{\Bbb Z}_I\left( m\right) . \label{ZIrecur} \end{equation} The corresponding 1-dimensional version of this recurrence relation (indicated with the subscript $i$ to distinguish it from the 3D case with capital subscript) becomes \begin{equation} {\Bbb Z}_i\left( N\right) =\frac 1N\sum_{m=0}^{N-1}\xi ^{N-m-1}\frac{b^{% \frac 12\left( N-m\right) }}{1-b^{N-m}}{\Bbb Z}_i\left( m\right) , \end{equation} leading to the following partition functions in closed form for one-dimensional bosons and one-dimensional fermions \begin{equation} {\Bbb Z}_b=\frac{b^{\frac 12N}}{\prod_{j=1}^N\left( 1-b^j\right) };\quad {\Bbb Z}_f=\frac{b^{\frac 12N^2}}{\prod_{j=1}^N\left( 1-b^j\right) }. \end{equation} It is easy to check that these partition functions are the solution of the recurrence relation for ${\Bbb Z}_i\left( N\right) $ with $\xi =1$ for bosons and $\xi =-1$ for fermions. However we did not find a systematic method to obtain analytical solutions of this type of recurrence relations. E.g. for the 3D case we had to rely on numerical schemes, as will be discussed below. But at this stage, it is worthwhile first to consider the presence of an homogeneous magnetic field, as the origin of anisotropy in our model of $N$ identical oscillators. \section{$N$ identical oscillators in a magnetic field} The Lagrangian of $N$ particles in a confining parabolic potential in the presence of a magnetic field is \begin{equation} L_{\omega _c}=\frac 12\sum_{j=1}^N\left( {\bf \dot{r}}_j-2\omega _cx_j\dot{y}% _j\right) ^2-\frac 12\Omega ^2\sum_{j=1}^N{\bf r}_j^2, \label{eqLmagn} \end{equation} where $\omega _c$ is the cyclotron frequency. For this model, the calculations of the preceding section can in essence be repeated. First the propagator for distinguishable particles is calculated. The next step will be the projection on the irreducible representation of the permutation group, and performing the cyclic decomposition. Then the generating function is introduced to circumvent the constraints on the partition in cycles, and finally the summation over the cycles is performed. The fact that the energy spectrum and the wavefunction can be calculated when harmonic interparticle interactions are included \cite{Johnson} indicates that the propagator and the partition function for the model in a magnetic field with two-body interaction can be obtained using our methods. The calculation technique is very demanding, therefore it seemed appropriate to illustrate the method at the hand of the simple model (\ref{eqLmagn}). \subsection{The propagator for distinguishable particles} For distinguishable particles, the many-particle propagator is a product of one-particle propagators. The one-particle propagator $K_{\omega _c}^{\left( 1\right) }\left( {\bf r},\beta |{\bf r}^{\prime }\right) $ of this model can be calculated by stochastic techniques\cite{Simon} or by path integral techniques\cite{FeynHibbs}. The evaluation is somewhat lengthy but straightforward, resulting eventually in \begin{equation} K_{\omega _c}^{\left( 1\right) }\left( {\bf r},\beta |{\bf r}^{\prime }\right) = \begin{array}[t]{l} \sqrt{\frac \Omega {2\pi \sinh \beta \Omega }}\frac s{2\pi \sinh \beta s}% \exp \left\{ -\frac{\Omega \left( \left( z^2+\left( z^{\prime }\right) ^2\right) \cosh \beta \Omega -2zz^{\prime }\right) }{2\sinh \beta \Omega }% \right\} \\ \times \exp \left\{ -\frac s2\frac{\left( x^2+y^2+(x^{\prime })^2+(y^{\prime })^2\right) \cosh \beta s-2\left( xx^{\prime }+yy^{\prime }\right) \cosh \frac 12\beta \omega }{\sinh \beta s}\right\} \\ \times \exp \left\{ -i\left( \frac 12\omega _c\left( xy-x^{\prime }y^{\prime }\right) -s\frac{\sinh \frac 12\beta \omega }{\sinh \beta s}\left( y^{\prime }x-yx^{\prime }\right) \right) \right\} , \end{array} \end{equation} with the eigenfrequency $s$ given by \begin{equation} s=\sqrt{\Omega ^2+\frac 14\omega _c^2}. \end{equation} The partition function corresponding to this single-particle propagator is obtained by integrating over the configuration space: \begin{equation} {\Bbb Z}_{\omega _c}^{\left( 1\right) }\left( \beta \right) =\int d{\bf r}% K_{\omega _c}^{\left( 1\right) }\left( {\bf r},\beta |{\bf r}\right) =\left[ 8\sinh \beta \frac \Omega 2\sinh \beta \left( \frac s2+\frac{\omega _c}4% \right) \sinh \beta \left( \frac s2-\frac{\omega _c}4\right) \right] ^{-1} \label{eqZmagn} \end{equation} which coincides with the partition function of a 3D harmonic oscillator if $% \omega _c=0.$ \subsection{The partition function and generating function for identical particles} For $N$ identical particles, the propagator becomes \begin{equation} K_{I,\omega _c}\left( {\bf \bar{r}},\beta |{\bf \bar{r}}^{\prime }\right) =% \frac 1{N!}\sum_p\xi ^p\prod_{j=1}^NK_{\omega _c}^{\left( 1\right) }\left( \left( P{\bf r}\right) _j,\beta |{\bf r}_j^{\prime }\right) , \end{equation} with the corresponding partition function given by \begin{equation} {\Bbb Z}_{I,\omega _c}\left( N\right) =\int d{\bf \bar{r}}K_{I,\omega _c}\left( {\bf \bar{r}},\beta |{\bf \bar{r}}\right) . \end{equation} Using the cyclic decomposition and the semigroup property as before, followed by the relaxation of the constraint (\ref{eqMlsum}) on the number of cycles, one finds for the generating function \begin{equation} \Xi _{\omega _c}\left( u\right) =\exp \left( \sum_{\ell =1}^\infty \frac{\xi ^{\ell -1}}\ell {\Bbb Z}_{\omega _c}^{\left( 1\right) }\left( \ell \beta \right) u^\ell \right) . \label{eqKsimagn} \end{equation} Because in this model with a magnetic field no two-body interactions were taken into account, this expression equals the grand-canonical partition function of the model if $u=e^{\beta \mu }$ is interpreted as the fugacity, with $\mu $ the chemical potential. We have thus obtained an expression for the grand-canonical partition function in the absence of two-body interactions for $N$ identical harmonic oscillators, both with and without a magnetic field. For a system with harmonic two-body interactions, repulsive or attractive, expressions for the canonical partition function are obtained (in the absence of a magnetic field). The corresponding thermodynamic properties (grand-canonical ground state occupancy, canonical and grand-canonical specific heat and average square radius) of the confined system will be calculated in the next section. \section{Identical particles in 1D} In the preceding sections we have obtained the grand canonical partition function of identical harmonic oscillators in a confining parabolic potential well without a magnetic field [equation (\ref{eqKsinomagn})] and in the presence of a magnetic field [equation (\ref{eqKsimagn})]. For the same system the canonical partition function is obtained for an attractive as well as for a repulsive harmonic two-body interaction [equation (\ref {eqZgeneral})]. The expressions are given for bosons and for fermions. In this section we will analyze the thermodynamical properties of the confined system in the case of motion in 1D. It should be noted that for the non-interacting case in 1D, we recover the results of Ref. \onlinecite{Takahashi}% . To the best of our knowledge, the interacting case has not been analyzed up to now for identical particles. For distinguishable particles it has been studied before \cite{Lieb}. In 1D, the explicit expressions for $Z$ largely facilitate the analysis: the free energy $F=-\frac 1\beta \ln Z,$ the internal energy $U=\frac{d\beta F}{% d\beta }$ and the specific heat $C=\frac{dU}{dT}$ can be readily obtained. The results are summarized in Table I. The subscript $CM\;$refers to the center-of-mass contribution. The center-of-mass correction $C_{CM}$ to the specific heat --which is an excess specific heat and therefore may be negative-- is shown in Fig. 1. In Fig. 2 we plot the contribution $\Delta C$ to the specific heat, due to the internal degrees of freedom. Note that $\Delta C$ is the same for 1D fermions and bosons; bosonization implies this result \cite{Tomonaga}. The spatial extension of the cloud of interacting identical particles can be obtained by applying the Hellman-Feynman theorem to the free energy: \begin{equation} \frac{\partial F}{\partial \left( \Omega ^2\right) }=\frac 1{2Z}\text{Tr}% \left( e^{-\beta H}\sum_{j=1}^Nx_j^2\right) =\frac N2\left\langle x^2\right\rangle . \end{equation} The difference $\left\langle x^2\right\rangle _f-\left\langle x^2\right\rangle _b$ between the mean square length of the 1D fermion and boson system turns out to be: \begin{equation} \left\langle x^2\right\rangle _f-\left\langle x^2\right\rangle _b=\frac{N-1}{% 2w}. \end{equation} It is proportional to the number of particles, and inversely proportional to the frequency $w$ of the internal degrees of freedom. \section{Bosons in 3D} The calculation of the thermodynamic properties or the mean square radius of the 3D cloud of identical particles is substantially complicated by the lack of an explicit expression in closed form for the partition function. Numerical methods have therefore to be used. We concentrate this analysis on the boson case. For fermions a different scheme will have to be developed and will be published later. \subsection{In the absence of a magnetic field} The recurrence relation (\ref{ZIrecur}) is not directly accessible for numerical computation, as can easily be seen by evaluating the expected dominant factor $b^{3N/2}/\prod_{j=1}^N\left( 1-b^j\right) ^3$ for bosons. For a relatively low temperature and a moderate number of particles, say $% b=0.75$ and $N=1000,$ this factor is as small as $1.0402\times 10^{-182}.$ We therefore isolate this factor using the following scaling for the partition function \begin{equation} {\Bbb Z}_B\left( N\right) =\sigma _N\frac{b^{\frac 32N}}{\prod_{j=1}^N\left( 1-b^j\right) ^3}, \end{equation} and rewrite the recurrence relation (\ref{ZIrecur}) in terms of the activity $\rho _N$ (see e.g. \cite{Lieb}) defined as \begin{equation} \sigma _N=\rho _N\sigma _{N-1}\Longrightarrow \sigma _N=\sigma _0\prod_{j=0}^N\rho _j \end{equation} where $\sigma _0=\rho _0\equiv 1$ have been introduced for convenience. The recurrence relation (\ref{ZIrecur}) then becomes after some manipulations \begin{equation} \rho _N=\frac 1N\left( \frac{1-b^N}{1-b}\right) ^3\left( 1+\sum_{m=0}^{N-2}\left( \frac{1-b}{1-b^{N-m}}\right) ^3\prod_{j=m+1}^{N-1}% \frac{\left( 1-b^j\right) ^3}{\rho _j}\right) . \label{eqrhorecur} \end{equation} The corresponding recurrence relation for the internal energy of the internal degrees of freedom becomes \begin{equation} \frac{{\Bbb U}_B\left( N\right) }{\hbar w}=\frac 1N\frac 1{\rho _N}\left( \frac{1-b^N}{1-b}\right) ^3\left( \begin{array}{l} \frac 32\frac{1+b}{1-b}+\frac{{\Bbb U}_B\left( N-1\right) }{\hbar w} \\ +\sum_{m=0}^{N-2}\left( \left( \frac 32\left( N-m\right) \frac{1+b^{N-m}}{% 1-b^{N-m}}+\frac{{\Bbb U}_B\left( m\right) }{\hbar w}\right) \left( \frac{1-b% }{1-b^{N-m}}\right) ^3\prod_{j=m+1}^{N-1}\frac{\left( 1-b^j\right) ^3}{\rho _j}\right) \end{array} \right) . \label{eqUrecur} \end{equation} The temperature scale used to express $b=e^{-w/kT}$ is \begin{equation} t=\left( \frac N{\zeta \left( 3\right) }\right) ^{-1/3}\frac{kT}w\equiv \frac T{T_c}\text{ where }\zeta \left( 3\right) =1.\,2021. \end{equation} The routine used to calculate the values $\rho _1\cdots \rho _N$ and ${\Bbb U% }_B\left( 1\right) \cdots {\Bbb U}_B\left( N\right) $ from these recurrence relations (\ref{eqrhorecur}) and (\ref{eqUrecur}) is displayed in Table II. The values of $\rho _1\cdots \rho _N$ for $N=100$ are shown in Fig. 3 for the three temperatures $T/T_c=0.5,$ 1 and 2. For $T=2T_c$ it turns out that $% \rho _j$ is about 8 times larger than $\rho _{j-1}$ for $j$ approaching $N,$ which makes it extremely difficult to deal numerically with the values of the partition function rather than with the proportionality factors. Once $\rho _N$ is known, the internal energy ${\Bbb U}_B\left( N\right) $ is readily obtained from (\ref{eqUrecur}). The results for $N=10,$ 100 and 1000 are shown in Fig. 4. The specific heat in the canonical ensemble can also be calculated from these parameters. This is shown in Fig. 5, clearly illustrating the effect of condensation for a finite number of particles. Anticipating the next subsection, it should be noted that the specific heat in the canonical ensemble for $N$ particles and without repulsive two-body interactions is identical to the specific heat in the grand canonical ensemble with the average number of particles given by the same value of $N.$ The reason why we compare both ensembles without repulsive interactions is the requirement that in the grand canonical ensemble the system should be stable for any large number of particles, which is not the case in the Gaussian model with repulsion, because the confining potential can only accommodate a finite number of particles as a consequence of (\ref{eqeigenw}% ). Another interesting consequence of the repulsive interactions follows from the dependence of the condensation temperature $T_c$ on the number of particles, which obeys the following scaling law taking (\ref{eqeigenw}) into account: \begin{equation} T_c=\frac{\sqrt{\Omega ^2-N\omega ^2}}k\left( \frac N{\zeta \left( 3\right) }% \right) ^{1/3}\Longrightarrow \frac{kT_c}\Omega \left( \frac{\omega ^2}{% \Omega ^2}\zeta \left( 3\right) \right) ^{1/3}=\sqrt{1-\frac{N\omega ^2}{% \Omega ^2}}\left( \frac{N\omega ^2}{\Omega ^2}\right) ^{1/3} \end{equation} For the case of attractive interactions and no confinement potential, $T_c$ is proportional to $N^{4/3}.$ The condensation temperature for both cases is plotted in Fig. 6. For the Gaussian model, the case of harmonic attraction does not pose any problem because in this model there is no sign of an ``extra'' collapse due to the nature of the interaction. The only consequence seems to be that the condensation occurs at a much higher temperature than would be the case without two-body interactions or with a repulsive harmonic interaction. \subsection{In the presence of a magnetic field} The thermodynamic properties in the grand canonical ensemble of an ideal Bose gas in a parabolic well and in the presence of a magnetic field can be obtained directly from (\ref{eqZmagn}) and (\ref{eqKsimagn}). Substituting $% u $ by the fugacity $e^{\beta \mu },$ the Gibbs free energy $G_{\omega _c}$ for the boson case ($\xi =+1$) becomes: \begin{equation} G_{\omega _c}=-\frac 1\beta \sum_{\ell =1}^\infty \frac 1\ell \frac{e^{\ell \beta \left( \mu -\frac 12\Omega -s\right) }}{\left( 1-e^{-\ell \beta \Omega }\right) \left( 1-e^{-\ell \beta \left( s+\frac 12\omega _c\right) }\right) \left( 1-e^{-\ell \beta \left( s-\frac 12\omega _c\right) }\right) }. \end{equation} After a power series expansion of the denominators, the summation over the cycle lengths $\ell $ can be performed: \begin{equation} G_{\omega _c}=\frac 1\beta \sum_{j,k,l=0}^\infty \ln \left( 1-e^{\beta \mu }e^{-\beta \Omega \left( \frac 12+j\right) }e^{-\beta s\left( 1+k+l\right) }e^{-\frac 12\beta \omega _c\left( k-l\right) }\right) , \end{equation} as expected from a Bose-Einstein distribution with energy levels \begin{equation} \epsilon _{j,k,l}=\frac 12\Omega +s+j\Omega +k\left( s+\frac 12\omega _c\right) +l\left( s-\frac 12\omega _c\right) . \end{equation} The average number of particles $N=-\frac{\partial G_{\omega _c}}{\partial \mu }$ is given by \begin{equation} N=\sum_{j,k,l=0}^\infty n_{j,k,l};\quad n_{j,k,l}=\frac{e^{\beta \mu }e^{-\beta \Omega \left( \frac 12+j\right) }e^{-\beta s\left( 1+k+l\right) }e^{-\frac 12\beta \omega _c\left( k-l\right) }}{1-e^{\beta \mu }e^{-\beta \Omega \left( \frac 12+j\right) }e^{-\beta s\left( 1+k+l\right) }e^{-\frac 12% \beta \omega _c\left( k-l\right) }}, \end{equation} and the fugacity can be eliminated in favor of the ground state occupancy $% n_0$: \begin{equation} n_0=\frac{e^{\beta \mu }e^{-\frac 12\beta \Omega }e^{-\beta s}}{1-e^{\beta \mu }e^{-\frac 12\beta \Omega }e^{-\beta s}}\Longrightarrow e^{\beta \mu }=\alpha e^{\beta \left( \frac 12\Omega +s\right) }\text{ with }\alpha \equiv \frac{n_0}{n_0+1} \end{equation} Restoring the cyclic summation for reasons of numerical convergence, one obtains for the average number of particles \begin{equation} N=\sum_{\ell =1}^\infty \frac{\alpha ^\ell }{D_\ell };\quad D_\ell \equiv \left( 1-e^{-\ell \beta \Omega }\right) \left( 1-e^{-\ell \beta \left( s+% \frac 12\omega _c\right) }\right) \left( 1-e^{-\ell \beta \left( s-\frac 12% \omega _c\right) }\right) , \end{equation} which can be cast in the form of the following numerically tractable series \begin{equation} N=\frac \alpha {1-\alpha }+\sum_{\ell =1}^\infty \alpha ^\ell \left( \frac 1{% D_\ell }-1\right) . \end{equation} For given $N,$ standard numerical techniques can be used to determine $% \alpha \in \left[ 0,1\right] ,$ and hence the ground state occupancy $n_0,$ which is shown in Fig. 7 for $\omega _c=0$ and in Fig. 8 for $\omega _c/\Omega =5$ as a function of $T/T_c$ for several values of $N.$ It turns out that the parabolic well is more important than the anisotropy due to the presence of the magnetic field. The anisotropy only moderately influences the temperature dependence of the ground state occupancy. The internal energy $U_{\omega _c}=\frac{\partial \left( \beta G_{\omega _c}\right) }{\partial \beta }-\mu N$ becomes \begin{equation} U_{\omega _c}=\sum_{\ell =1}^\infty \frac{\alpha ^\ell }{D_\ell }\left( \Omega \frac{e^{-\ell \beta \Omega }}{1-e^{-\ell \beta \Omega }}+\left( s+% \frac 12\omega _c\right) \frac{e^{-\ell \beta \left( s+\frac 12\omega _c\right) }}{1-e^{-\ell \beta \left( s+\frac 12\omega _c\right) }}+\left( s-% \frac 12\omega _c\right) \frac{e^{-\ell \beta \left( s-\frac 12\omega _c\right) }}{1-e^{-\ell \beta \left( s-\frac 12\omega _c\right) }}\right) \end{equation} and its numerical evaluation once $\alpha $ is determined presents no numerical difficulties. The resulting specific heat is shown in Fig. 9 for $% \omega _c/\Omega =5.$ Comparison with Fig. 5 reveals that the anisotropy due to the magnetic field essentially broadens the peak in the specific heat near the condensation temperature, but does not substantially alter the structure. \section{Discussion and conclusion} In this paper we applied the method of symmetrical density matrices, developed by Feynman for a system of non-interacting particles in a box, to a system of harmonically interacting particles in a confining parabolic potential. The interaction could be taken into account, due to the Gaussian nature of the propagators, allowing integration over the configuration space. The symmetrization resulting from the projection of the propagators for distinguishable particles on the appropriate representation of the permutation group gives rise to a series which could be summed using the generating function technique. Without these generating functions the calculation has to be restricted to a limited number of particles \cite {Haase,Hausler,Hammermesh}. Using them not only the grand canonical partition function $\Xi \left( u\right) $ could be obtained in the parameter range of the model where $\Xi \left( u\right) $ is well defined, but also the canonical partition functions $Z\left( N\right) $ for a given number $N$ of identical particles could be obtained as a recursion of partition functions of a smaller number of particles for the interacting system. The recurrence relation for the activity $\rho _N,$ i.e. the proportionality factor between $Z\left( N\right) $ and $Z\left( N-1\right) ,$ allows for an accurate numerical treatment of the thermodynamical quantities of the model, such as the internal energy and the specific heat. Also the thermodynamic properties of the same model in the presence of a homogeneous magnetic field could be investigated along these lines. A detailed analysis of the ideal gas in a confining parabolic potential with anisotropy due to a magnetic field was presented; special attention was payed to the relation between the number of condensed atoms, the magnetic field, the strength of the confinement potential and the total number of particles. The relationship between the parameters of our model and the characteristics of atomic traps \cite{BEC1,BEC2,BEC3} lies beyond the scope of the present paper. It should be mentioned that in the absence of two-body interactions, the generating function can be identified as the grand canonical partition function. In that case the specific heat as obtained from the grand canonical ensemble for an average number of $\left\langle N\right\rangle $ particles calculated using the chemical potential equals the specific heat obtained from the canonical ensemble with the number of particles $N$ given by $\left\langle N\right\rangle .$ For a more elaborate discussion we refer to \cite{BDLSSCpress}. It should also be mentioned that in this the model it is assumed that the spin degrees of freedom are fixed. This simplifying assumption is imposed by the symmetrization method, which becomes more involved if the spin degrees of freedom depend on the configuration of the particles. In summary, the partition function of a general Gaussian model for bosons and fermions, with or without a magnetic field, has been calculated analytically, and the thermodynamical properties of this model have been studied. Particular attention has been given to the Bose-Einstein condensation in the presence of two-body interactions, repulsive and attractive, and in the presence of a magnetic field. \acknowledgments Part of this work is performed in the framework of the NFWO projects No. 2.0093.91, 2.0110.91, G. 0287.95 and WO.073.94N (Wetenschappelijke Onderzoeksgemeenschap, Scientific Research Community of the NFWO on ``Low-Dimensional Systems''), and in the framework of the European Community Program Human Capital and Mobility through contracts no. CHRX-CT93-0337 and CHRX-CT93-0124. One of the authors (F.B.) acknowledges the National Fund for Scientific Research for financial support.

proofpile-arXiv_065-529

\section{Introduction} \setcounter{equation}{0} In this paper we are concerned with symmetry reductions of the nonlinear third order partial differential equation given by \begin{equation} \Delta \equiv u_t -\epsilon u_{xxt} +2\kappa u_x-u u_{xxx} -\alpha u u_x-\beta u_x u_{xx}=0, \label{fulleqn} \end{equation} where $\epsilon$, $\kappa$, $\alpha$ and $\beta$ are arbitrary constants. Three special cases of (\ref{fulleqn})\ have appeared recently in the literature. Up to some rescalings, these are: (i), the Fornberg-Whitham equation \cite{\refFW,\refWp,\refWbk}, for the parameters $\epsilon=1$, $\alpha=-1$, $\beta=3$ and $\kappa=\tfr12$, (ii), the Rosenau-Hyman equation \cite{\refRH} for the parameters $\epsilon=0$, $\alpha=1$, $\beta=3$ and $\kappa=0$, and (iii), the Fuchssteiner-Fokas-Camassa-Holm equation \cite{\refCH,\refCHH,\refFuch,\refFFb} for the parameters $\epsilon=1$, $\alpha=-1$ and $\beta=2$. The Fornberg-Whitham (FW) equation \begin{equation} u_t -u_{xxt} +u_x= u u_{xxx} -u u_x +3u_x u_{xx} \label{fw} \end{equation} was used to look at qualitative behaviours of wave-breaking \cite{\refWp}. It admits a wave of greatest height, as a peaked limiting form of the travelling wave solution \cite{\refFW}, $$ u(x,t) = A \exp\left(-\tfr12 \vert x-\tfr43 t \vert\right),$$ where $A$ is an arbitrary constant. The Rosenau-Hyman (RH) equation \begin{equation} u_t =u u_{xxx}+u u_x +3u_x u_{xx}. \label{rh} \end{equation} models the effect of nonlinear dispersion in the formation of patterns in liquid drops \cite{\refRH}. It also has an unusual solitary wave solution, known as a ``compacton'', $$ u(x,t) = \left \{ \begin{array}{cc} -\tfr83 c \cos^2\{\tfr14 (x-ct)\},& \mbox{if } \,|x-ct|\le 2\pi, \\[1.0ex] 0, & \mbox{if } \,|x-ct|> 2\pi. \end{array} \right . $$ These waves interact producing a ripple of low amplitude compacton-anticompacton pairs. The Fuchssteiner-Fokas-Camassa-Holm (FFCH) equation \begin{equation} u_t -u_{xxt} +2\kappa u_x= u u_{xxx} -3u u_x + 2u_x u_{xx}, \label{ch} \end{equation} first arose in the work of Fuchssteiner and Fokas \cite{\refFuch,\refFFb} using a bi-Hamiltonian approach; we remark that it is only implicitly written in \cite{\refFFb} --- see equations (26e) and (30) in this paper --- though is explicitly written down in \cite{\refFuch}. It has recently been rederived by Camassa and Holm \cite{\refCH} from physical considerations as a model for dispersive shallow water waves. In the case $\kappa=0$, it admits an unusual solitary wave solution $$ u(x,t) = A\exp\left(-\vert x-ct \vert\right), $$ where $A$ and $c$ are arbitrary constants, which is called a ``peakon''. A Lax-pair \cite{\refCH} and bi-Hamiltonian structure \cite{\refFFb} have been found for the FFCH equation (\ref{ch}) and so it appears to be completely integrable. Recently the FFCH equation (\ref{ch}) has attracted considerable attention. In addition to the aforementioned, other studies include \cite{\refCHH,\refCS,\refFoka,\refFokb,\refFokS,\refFuchb,\refGP,\refMB,\refOR}. The FFCH equation (\ref{ch}) may be thought of as an integrable modification of the regularized long wave (RLW) equation \cite{\refBBM,\refPer} \begin{equation} u_{xxt} + u u_{x} - u_{t} - u_{x} = 0,\label{eqrlw} \end{equation} sometimes known as the Benjamin-Bona-Mahoney equation. However, in contrast to (\ref{ch}), the RLW equation (\ref{eqrlw}) is thought \underbar{\it not} to be solvable by inverse scattering (cf., \cite{\refMcLO}); its solitary wave solutions interact inelastically (cf., \cite{\refMak}) and only has finitely many local conservation laws \cite{\refOlverb}. However physically it has more desirable properties than the celebrated Korteweg-de Vries (KdV) equation \begin{equation} u_{t} + u_{xxx} + 6uu_x=0,\label{eqkdv} \end{equation} which was the first equation to be solved by inverse scattering \cite{\refGGKM}. We remark that two other integrable variants of the RLW equation (\ref{eqrlw}) are \begin{equation} u_{xxt} + 2 u u_{t} - u_{x}\partial_x^{-1}u_t - u_{t} - u_{x} = 0,\label{eqswwa} \end{equation} where $\left(\partial_x^{-1} f\right)(x) = \int_x^\infty f(y)\,\mbox{d} y$, which was introduced by Ablowitz, Kaup, Newell and Segur \cite{\refAKNS}, and \begin{equation} u_{xxt} + u u_{t} - u_{x}\partial_x^{-1}u_t - u_{t} - u_{x} = 0,\label{eqswwb} \end{equation} which was discussed by Hirota and Satsuma \cite{\refHS}. We also note that (\ref{ch}), with $\kappa=\tfr12$, (\ref{eqrlw}), (\ref{eqswwa}) and (\ref{eqswwb}) all have the same linear dispersion relation $\omega(k)=-k/(1+k^2)$ for the complex exponential $u(x,t)\sim\exp\{\mbox{i}[kx+\omega(k)t]\}$. Recently, Gilson and Pickering \cite{\refGP} have shown that no equation in the entire class of equations (\ref{fulleqn})\ will satisfy the necessary conditions of either the Painlev\'e PDE test due to Weiss, Tabor and Carnevale \cite{\refWTC} or the Painlev\'e ODE test due to Ablowitz, Ramani and Segur \cite{\refARS,\refARSi} to be solvable by inverse scattering. However, the integrable FFCH equation (\ref{ch}) does possess the ``weak Painlev\'e'' property (cf., \cite{\refRDG,\refRRDG}), as does the FW equation (\ref{fw}). All these special travelling wave solutions are essentially exponential solutions, or sums of exponential solutions, and thus would suggest some sort of linearity in the differential equation. This is discussed by Gilson and Pickering \cite{\refGP}, who show that (\ref{fulleqn}), with $\alpha\not=0$ and $\beta(1+\beta)\not=0$, can be written as \begin{equation} \left( \beta u_x +u {\partial_x} +\epsilon {\partial_t}\right) ( u_{xx} -\mu^2 u -2\kappa/\beta) =0, \label{ufactor} \end{equation} where $\partial_x\equiv{\partial/\partial x}$, $\partial_t\equiv{\partial/\partial t}$ and $\mu^2=-\alpha/(1+\beta)$, provided that $\epsilon\alpha+\beta +1=0$, which includes the FFCH equation (\ref{ch}). For the travelling wave reduction, $$u=w(z),\qquad z=x-ct,$$ the resulting ordinary differential equation is \begin{equation} (2\kappa -c) w'+\epsilon c w'''-ww'''-\alpha ww'- \beta w'w''=0, \label{travwav} \end{equation} where $'\equiv{\mbox{d}/ \mbox{d} z}$, which also may be factorised as \begin{equation} \left [ \beta w'+(w-\epsilon c ){ \mbox{d} \over \mbox{d} z} \right ] (w''-\mu^2 w+\gamma)=0, \label{wfactor} \end{equation} provided that $$\mu^2=-{\alpha\over 1+\beta},\qquad \beta(1+\beta)\gamma -2\kappa(1+\beta) + c(1+\beta+\alpha\epsilon)=0.$$ This includes all three special cases (\ref{fw})--(\ref{ch}); since $\beta(1+\beta)$ is strictly non-zero in these three cases then a suitable $\gamma$ can always be found. Furthermore, if $1+\beta+\alpha\epsilon=0$ and $\epsilon\not=0$, then (\ref{fulleqn}) with $\kappa=0$ possesses the ``peakon'' solution $$ u(x,t) = A \exp\left(-\epsilon^{-1/2}\vert x-ct \vert\right),$$ where $A$ and $c$ are arbitrary constants. More generally, if $\alpha/(1+\beta)<0$, $1+\beta+\alpha\epsilon\not=0$ and $\kappa\not=0$, then (\ref{fulleqn})\ possesses the solution $$ u(x,t) = A \exp\left\{-\left(-\,{\alpha\over1+\beta}\right)^{1/2}\vert x-ct \vert\right\},\qquad c={2(1+\beta)\kappa\over 1+\beta+\alpha\epsilon},$$ where $A$ is an arbitrary constant. If $\alpha/(1+\beta)>0$, $\beta\not=-1$ and $\alpha\beta\not=0$, then (\ref{fulleqn})\ possesses the ``compacton'' solution $$ u(x,t) = {2[2(1+\beta)\kappa-(1+\beta+\alpha\epsilon)c]\over\alpha\beta} \cos^2\left\{\tfr12\left(\alpha\over1+\beta\right)^{1/2}(x-ct)\right\},$$ where $c$ is an arbitrary constant. The classical method for finding symmetry reductions of partial differential equations is the Lie group method of infinitesimal transformations. As this method is entirely algorithmic, though often both tedious and virtually unmanageable manually, symbolic manipulation programs have been developed to aid the calculations. An excellent survey of the different packages available and a description of their strengths and applications is given by Hereman \cite{\refH} (see also his contribution in this volume). In this paper we use the {\rm MACSYMA} package {\tt symmgrp.max} \cite{\refCHW} to calculate the determining equations. In recent years the nonclassical method due to Bluman and Cole \cite{\refBC} (in the sequel referred to as the ``nonclassical method''), sometimes referred to as the ``method of partial symmetries of the first type'' \cite{\refV}, or the ``method of conditional symmetries'' \cite{\refLW}, and the direct method due to Clarkson and Kruskal \cite{\refCK} have been used to generate many new symmetry reductions and exact solutions for several physically significant partial differential equations\ that are not obtainable using the classical Lie method (cf., \cite{\refPAC} and the references therein). The nonclassical method is a generalization of the classical Lie method, whereas the direct method is an ansatz-based approach which involves no group theoretic techniques. Nucci and Clarkson \cite{\refNC} showed that for the Fitzhugh-Nagumo equation the nonclassical method is more general than the direct method, since they demonstrated the existence of a solution of the Fitzhugh-Nagumo equation, obtainable using the nonclassical method but not using the direct method. Subsequently Olver [\refOlverb] (see also [\refABH,\refPucci]) has proved the general result that for a scalar equation, every reduction obtainable using the direct method is also obtainable using the nonclassical method. Consequently we use the nonclassical method in this paper rather than the direct method. Symmetry reductions and exact solutions have several different important applications in the context of differential equations. Since solutions of partial differential equations\ asymptotically tend to solutions of lower-dimensional equations obtained by symmetry reduction, some of these special solutions will illustrate important physical phenomena. In particular, exact solutions arising from symmetry methods can often be used effectively to study properties such as asymptotics and ``blow-up'' (cf., \cite{\refG,\refGDEKS}). Furthermore, explicit solutions (such as those found by symmetry methods) can play an important role in the design and testing of numerical integrators; these solutions provide an important practical check on the accuracy and reliability of such integrators (cf., \cite{\refA,\refS}). Classical symmetries of differential equations are found in practice by a two-step process. The first involves finding the determining equations for the infinitesimals of the group action. These determining equations form an overdetermined, linear system of {partial differential equations}. The second step involves integrating this system. The first step is entirely algorithmic, and has been implemented in all the commercial symbolic manipulation languages (cf., \cite{\refH}). The second step involves heuristic integration procedures which have been implemented in some symbolic manipulation programs and are largely successful, though not infallible. Commonly, the overdetermined systems to be solved are simple, and heuristic integration is both fast and effective. However, there are three areas where heuristics can break down (cf., \cite{\refMC} for further details and examples). \begin{itemize} \item[1.]{\it Arbitrary parameters and functions}. If the partial differential equation\ whose symmetries are sought involves arbitrary parameters, such as (1.1) or more generally, arbitrary functions, heuristics yield usually the general solution, and miss those special cases of the parameters and arbitrary functions where additional symmetries exist. \item[2.]{\it Termination}. Heuristic algorithms are not guaranteed to terminate, and may become trapped in infinite loops for some examples. \item[3.]{\it Too difficult to solve}. The system may not be solvable by the heuristic. The heuristic will then attempt to represent the general solution in terms of functions satisfying certain conditions, but may give up before a useful representation is obtained. \end{itemize} These problems are addressed by use differential Gr\"obner bases (DGBs) which we describe below. The method used to find solutions of the determining equations in the nonclassical method is that of DGBs, defined to be a basis \ss\ of the differential ideal generated by the system such that every member of the ideal pseudo-reduces to zero with respect to \ss. This method provides a systematic framework for finding integrability and compatibility conditions of an overdetermined system of partial differential equations. It avoids the problems of infinite loops in reduction processes and yields, as far as is currently possible, a ``triangulation'' of the system from which the solution set can be derived more easily \cite{\refCMi,\refMF,\refR,\refRi}. In a sense, a DGB provides the maximum amount of information possible using elementary differential and algebraic processes in finite time. In pseudo-reduction, one must, if necessary, multiply the expression being reduced by differential (non-constant) coefficients of the highest derivative terms of the reducing equation, so that the algorithms used will terminate \cite{\refMF}. In practice, such coefficients are assumed to be non-zero, and one needs to deal with the possibility of them being zero separately. These are called singular cases. The triangulations of the systems of determining equations\ for infinitesimals\ arising in the nonclassical method in this paper were all performed using the {\rm MAPLE} package {\tt diffgrob2} \cite{\refM}. This package was written specifically to handle nonlinear equations of polynomial type. All calculations are strictly `polynomial', that is, there is no division. Implemented there are the Kolchin-Ritt algorithm using pseudo-reduction instead of reduction, and extra algorithms needed to calculate a DGB (as far as possible using the current theory), for those cases where the Kolchin-Ritt algorithm is not sufficient \cite{\refMF}. The package was designed to be used interactively as well as algorithmically, and much use is made of this fact here. It has proved useful for solving many fully nonlinear systems \cite{\refCMi--\refCMiv}. In the following sections we shall consider the cases $\epsilon=0$ and $\epsilon\not=0$, when we set $\epsilon=1$ without loss of generality, separately because the presence or lack of the corresponding third order term is significant. In \S 2 we find the classical Lie group of symmetries and associated reductions of (\ref{fulleqn}). In \S3 we discuss the nonclassical symmetries and reductions of (\ref{fulleqn})\ in the generic case. In \S4 we consider special cases of the the nonclassical method in the so-called $\tau=0$; in full generality this case generates a single equation which is considerably more complex than our original equation! In \S5 we discuss our results. \section{Classical symmetries} \setcounter{equation}{0} To apply the classical method we consider the one-parameter Lie group of infinitesimal transformations in ($x,t,u$) given by \begin{equation} \label{trans} \begin{array}{ccl} x^*&=&x+\varepsilon \xi (x,t,u) + O (\varepsilon^2), \\[1.0ex] t^*&=&t+\varepsilon \tau (x,t,u) + O (\varepsilon^2), \\[1.0ex] u^*&=&u+\varepsilon \phi (x,t,u) + O (\varepsilon^2), \end{array} \end{equation} where $\varepsilon$ is the group parameter. Then one requires that this transformation leaves invariant the set \begin{equation} S_{\Delta} \equiv \{ u(x,t) : \Delta =0 \} \label{sdel} \end{equation} of solutions of (\ref{fulleqn}). This yields an overdetermined, linear system of equations for the infinitesimals $\xi (x,t,u),\tau (x,t,u),\phi (x,t,u)$. The associated Lie algebra is realised by vector fields of the form \begin{equation} {\bf v} = \xi (x,t,u) {\partial_x} +\tau (x,t,u) {\partial_t} + \phi (x,t,u) {\partial_u}. \label{vecfi} \end{equation} Having determined the infinitesimals, the symmetry variables are found by solving the characteristic equation \begin{equation} {\mbox{d} x \over \xi (x,t,u)} = {\mbox{d} t \over \tau (x,t,u)}= {\mbox{d} u \over \phi (x,t,u)}, \label{chareq} \end{equation} which is equivalent to solving the invariant surface condition\ \begin{equation} \psi \equiv \xi (x,t,u) u_x+\tau (x,t,u) u_t - \phi (x,t,u) =0. \label{invsc} \end{equation} The set $S_{\Delta}$ is invariant under the transformation (\ref{trans}) provided that $ {\rm pr}^{(3)} {\bf v} (\Delta) |_{\Delta \equiv 0} =0$ where ${\rm pr}^{(3)}{\bf v}$ is the third prolongation of the vector field (\ref{vecfi}), which is given explicitly in terms of $\xi,\tau$ and $\phi$ (cf.\ \cite{\refO}). This procedure yields the determining equations. There are two cases to consider. \subsection{$\epsilon=0$} In this case using the {\rm MACSYMA} package {\tt symmgrp.max} we obtain the following system of ten determining equations \begin{equation} \label{clez} \begin{array}{@{\hspace{0.0ex}}l@{\hspace{0.0ex}}} \tau_{u}=0, \quad \tau_{x}=0, \quad \xi_{u} =0, \quad u\phi_{uuu}+\beta\phi_{uu}=0, \quad 3u^2\phi_{uu}+\beta u\phi_{u}-\beta\phi=0,\\[1.0ex] 3u\phi_{xu}-3u\xi_{xx}+\beta\phi_{x}=0, \quad 3u\phi_{xuu}+2\beta\phi_{xu}-\beta\xi_{xx}=0,\\[1.0ex] \tau_{t}u-3\xi_{x}u+\phi=0,\quad \phi_{xxx}u+(\alpha u-2\kappa)\phi_{x}-\phi_{t}=0,\\[1.0ex] 3u^2\phi_{xxu}+\beta u\phi_{xx}+2\kappa\phi -u^2\xi_{xxx} +(2\alpha u^2-4\kappa u)\xi_{x}+u\xi_{t}=0. \end{array} \end{equation} Next applying the {\tt reduceall} algorithm in the {\rm MAPLE} package {\tt diffgrob2} to this system yields $$ \begin{array}{l} (2+\beta)\xi_{xx}=0,\quad (2+\beta)[\alpha u\xi_{xt}+\xi_{tt} -2\kappa\xi_{xt}]=0, \\[1.0ex] \xi_{u}=0, \quad \tau_{x}=0, \quad \tau_{u}=0,\\[1.0ex] 2\alpha u\xi_{x}+2\kappa\xi_{x}+\xi_{t} -2\kappa\tau_{t}=0,\quad (2+\beta)[2\kappa\phi+(2\alpha u^2-4\kappa u)\xi_x+u\xi_t]=0. \end{array} $$ This is simple enough to solve; there is no need to do the full Kochin-Ritt algorithm in this case. The output shows that there are three special values of the parameters, namely $\alpha=0$, $\beta=-2$ and $\kappa=0$, and combinations thereof. It transpires that the special case $\beta=-2$ is purely an artefact. For the three special cases (a) $\alpha=0$, $\kappa\not=0$, (b) $\alpha\not=0$, $\kappa=0$ and (c) $\alpha=\kappa=0$, applying the {\tt reduceall} algorithm of {\tt diffgrob2} to (\ref{clez}) yields\\ \noindent \begin{tabular}{cl@{\hspace{5.0ex}}l} (a)&$\alpha=0$,\quad $\kappa\not=0$ &$\xi_{xx}=0,\quad \xi_{tt} -2\kappa\xi_{xt}=0,\quad\xi_{u}=0,$\\[1.0ex] &&$\tau_{x}=0,\quad 2\kappa\xi_{x}+\xi_{t}-2\kappa\tau_{t}=0, \quad\tau_{u}=0,$\\[1.0ex] &&$2\kappa\phi+(2\alpha u^2-4\kappa u)\xi_x+u\xi_t=0.$\\[2.0ex] \end{tabular} \begin{tabular}{cl@{\hspace{5.0ex}}l} (b)&$\alpha\not=0$,\quad $\kappa=0$ &$2\alpha u\xi_{x} +\xi_{t}=0,\quad \xi_{tt}=0,\quad\xi_{u}=0,$\\[1.0ex] &&$\tau_{x}=0,\quad\tau_{tt}=0,\quad \tau_{u}=0,$\\[1.0ex] &&$2\kappa\phi+(2\alpha u^2-4\kappa u)\xi_x+u\xi_t=0.$\\[2.0ex] \end{tabular}\\ \begin{tabular}{cl@{\hspace{5.0ex}}l} (c)&$\alpha=\kappa=0$ &$ \xi_{xx}=0,\quad\xi_{t}=0,\quad\xi_{u}=0,$\\[1.0ex] &&$\tau_{x}=0,\quad\tau_{tt}=0,\quad\tau_{u}=0,$\\[1.0ex] &&$\phi-3u\xi_x+u\tau_t=0$.\\[1.0ex] \end{tabular}\\ Hence we obtain the following infinitesimals:\\ \noindent {\sc Case} 2.1(i) $\alpha\not=0$ and $\kappa\not=0$ \begin{equation} \xi= 2\kappa c_3 t + c_1,\qquad\tau= c_3 t+ c_2,\qquad\phi=-c_3 u.\label{kzii} \end{equation} \noindent {\sc Case} 2.1(ii) $\alpha=0$ and $\kappa\not=0$ \begin{equation} \xi= c_3x+2\kappa(c_4-c_3) t + c_1,\qquad\tau=c_4 t+ c_2,\qquad\phi=(3c_3-c_4) u.\label{kziii} \end{equation} \noindent {\sc Case} 2.1(iii) $\alpha\not=0$ and $\kappa=0$ \begin{equation} \xi= c_1,\qquad\tau=c_3 t+ c_2,\qquad\phi=-c_3 u.\label{knii} \end{equation} \noindent {\sc Case} 2.1(iv) $\alpha=0$ and $\kappa=0$ \begin{equation} \xi= c_3x+c_1,\qquad\tau=c_4 t+c_2,\qquad\phi=(3c_3-c_4) u,\label{kniii} \end{equation} where $c_1,c_2,c_3,c_4$ are arbitrary constants. Solving the invariant surface condition (\ref{invsc})\ yields four different canonical reductions:\\ \noindent{\bf Reduction 2.1} $\alpha$ and $\kappa$ arbitrary. If $c_3=c_4=0$ in (\ref{kzii})--(\ref{kniii}) we may set $c_1=c$ and $c_2=1$ and we obtain the travelling wave reduction $$ u(x,t)= w(z),\qquad z=x-c t,$$ where $w(z)$ satisfies $$ w w''' +\alpha w w' +\beta w'w''+ (c-2\kappa)w'=0. $$ This can be integrated to yield $$ w w''+\tfr12 (\beta-1) (w')^2 +\tfr12 \alpha w^2 + (c-2\kappa)w=A, $$ where $A$ is an arbitrary constant. Multiplying this by $w^{\beta-2} w'$ and integrating again yields \begin{equation} (w')^2 +{\alpha\over1+\beta} w^2 + {2(2\kappa -c)\over\beta} w={2A\over\beta-1}+B w^{1-\beta},\label{eqeztw} \end{equation} where $B$ is an arbitrary constant, for $\beta\not=-1,0,1$. Generally if $\beta\not=-1,0,1$, then (\ref{eqeztw}) is solvable using quadratures, though for certain special values of the parameters there are explicit solutions. For example (i), if $\beta=-2$ or $\beta=-3$, then (\ref{eqeztw}) is solvable in terms of Weierstrass or Jacobi elliptic functions, respectively, (ii), if $B=0$, then (\ref{eqeztw}) is solvable in term of trigonometric functions, and (iii), if $c=2\kappa$ and $\beta=3$, then $w(z)$ can be expressed in terms of trigonometric functions via the transformation $w(z)=v^{1/2}$. In the special cases $\beta=-1,0,1$ we obtain the equations $$ \begin{array}{l} (w')^2 +\alpha w^2\ln w+2(2\kappa -c) w= B w^2 - A, \\[1.0ex] (w')^2 +\alpha w^2+2(2\kappa -c) w\ln w= B w - 2A, \\[1.0ex] (w')^2 + \alpha w^2 +2(2\kappa -c) w= B- A \ln w, \end{array} $$ respectively, with $A$ and $B$ arbitrary functions. If the coefficient of $\ln w$ in these equations is zero, then $w(z)$ is expressible in terms of elementary functions, otherwise in terms of quadratures.\\ \noindent{\bf Reduction 2.2} $\alpha\not=0$, $\kappa$ arbitrary. If $c_3\not=0$ in (\ref{kzii}) and (\ref{knii}) we may set $c_3=1$, $c_1=c$ and $c_2=0$, without loss of generality, and obtain the reduction \begin{equation} u(x,t) = w(z) t^{-1},\qquad z= x -c\ln t -2\kappa t,\label{redIIiiw} \end{equation} where $w(z)$ satisfies $$ w w'''+\beta w' w''+\alpha w w'+c w' + w =0.$$ Also if $c_3=0$ and $c_4\not=0$ in (\ref{kziii}) we may set $c_4=1$, $c_1=c$ and $c_2=0$, without loss of generality, and obtain the same reduction (\ref{redIIiiw}).\\ \noindent{\bf Reduction 2.3} $\alpha=0$, $\kappa$ arbitrary. If $c_3\not=0$ and $c_4\not=0$ in (\ref{kziii}) and (\ref{kniii}), we may set $c_3=m+\tfr13$, $c_4=1$ and $c_1=c_2=0$, without loss of generality, and obtain the reduction $$ u(x,t) = w(z) t^{3m},\qquad z=(x-2\kappa t)t^{-m-1/3},$$ where $w(z)$ satisfies $$ w w''' +\beta w'w''+(m+\tfr13) zw'-3m w=0.$$ \vspace{1.0ex} \noindent{\bf Reduction 2.4} $\alpha=0$, $\kappa$ arbitrary. If $c_3\not=0$ and $c_4=0$ in (\ref{kziii}) and (\ref{kniii}), we may set $c_3=m$, $c_1=2\kappa$ and $c_2=1$, without loss of generality, and obtain the reduction $$ u(x,t) = w(z) \mbox{e}^{3m t}, \qquad z= (x-2\kappa t) \mbox{e}^{-m t}, $$ where $w(z)$ satisfies $$ w w''' +\beta w' w'' +m z w' -3m w =0.$$ \subsection{$\epsilon=1$} In this case we obtain the following system of eleven determining equations: \begin{equation} \label{clen} \begin{array}{l} \tau_{u}=0, \quad \tau_{x}=0, \quad \xi_u=0, \quad \phi_{uu}=0,\quad 2 \phi_{xu} - \xi_{xx}=0, \\[1.0ex] \beta(u\phi_{u} - \phi + \xi_{t})=0, \quad \phi + u\tau_{t} -u \xi_{x} - \xi_{t}=0,\\[1.0ex] 3u \phi_{xu} + \phi_{tu}+ \beta \phi_{x} - 3u \xi_{xx}- 2 \xi_{xt} =0, \\[1.0ex] u\phi_{xxu} + \phi + u\tau_{t} - 3 \xi_{x} u - \xi_{t} =0, \\[1.0ex] u\phi_{xxx} + \phi_{xxt} -\phi_{t} +(\alpha u -2\kappa) \phi_{x}=0, \\[1.0ex] 3 u^{2}\phi_{xxu} + 2 u\phi_{xtu} +\beta u\phi_{xx} + 2 \kappa \phi - u^{2} \xi_{xxx} - u\xi_{xxt} \\[1.0ex] {~}\hfill +(2\alpha u^2-4\kappa)\xi_{x} +[(\alpha+1) u-2\kappa]\xi_{t} =0. \end{array} \end{equation} As in the previous case, we apply the {\tt reduceall} algorithm in the {\rm MAPLE} package {\tt diffgrob2}, to this system, which yields $$ \begin{array}{l} \xi_x=0,\quad (\alpha+1)\xi_{tt}=0,\quad \xi_u=0,\\[1.0ex] \tau_x=0,\quad (\alpha+1)\tau_{tt}=0,\quad \tau_u=0,\\[1.0ex] 2\kappa\phi = [2\kappa - (\alpha+1) u ]\xi_t. \end{array} $$ This shows that there are two special values of the parameters, namely $\alpha=-1$ and $\kappa=0$. For the three special cases (a) $\alpha=-1$, $\kappa\not=0$, (b) $\alpha\not=-1$, $\kappa=0$ and (c) $\alpha=-1$, $\kappa=0$, applying the {\tt reduceall} algorithm of {\tt diffgrob2} to (\ref{clen}) yields\\ \begin{tabular}{cl@{\hspace{5.0ex}}l} (a)& $\alpha=-1$,\quad $\kappa\not=0$ & $\xi_x=0,\quad \xi_{tt}=0,\quad \xi_u=0,$\\[1.0ex] && $\tau_x=0,\quad \tau_{t}=0,\quad \tau_u=0,$\\[1.0ex] && $\phi = \xi_t.$\\[2.0ex] (b)& $\alpha\not=-1$,\quad $\kappa=0$ & $\xi_x=0,\quad \xi_{t}=0,\quad \xi_u=0,$\\[1.0ex] && $\tau_x=0,\quad \tau_{tt}=0,\quad \tau_u=0,$\\[1.0ex] && $\phi = -u\tau_t.$\\[2.0ex] (c)& $\alpha=-1$, $\kappa=0$ & $\xi_{x}=0,\quad\xi_{tt}=0,\quad\xi_{u}=0,$\\[1.0ex] && $\tau_{x}=0,\quad\tau_{tt}=0,\quad\tau_{u}=0,$\\[1.0ex] && $\phi-u\xi_t+u\tau_t=0.$\\[1.0ex] \end{tabular}\\ Hence we obtain the following infinitesimals:\\ \noindent {\sc Case} 2.2(i) $\alpha\not=-1$, $\kappa\not=0$ \begin{equation} \xi=c_3 t + c_1,\quad \tau= {(1+\alpha)c_3 t\over2\kappa} + c_2,\quad \phi=c_3\left[1-{(1+\alpha)u\over2\kappa}\right].\label{kzi} \end{equation} \noindent {\sc Case} 2.2(ii) $\alpha=-1$, $\kappa\not=0$ \begin{equation} \xi=c_3 t + c_1,\quad \tau=c_2,\quad \phi=c_3.\label{Kzii}\end{equation} \noindent {\sc Case} 2.2(iii) $\alpha\not=-1$, $\kappa=0$ \begin{equation} \xi=c_1,\quad \tau=c_3 t + c_2,\quad \phi=-c_3u. \label{Kziii}\end{equation} \noindent {\sc Case} 2.2(iv) $\alpha=-1$, $\kappa=0$ \begin{equation} \xi=c_3 t + c_1,\quad \tau=c_4 t + c_2,\quad \phi=c_3-c_4u,\label{kziv}\end{equation} where $c_1,c_2,c_3,c_4$ are arbitrary constants.\\ There are four canonical reductions.\\ \noindent{\bf Reduction 2.5} $\alpha$ and $\kappa$ arbitrary. If in (\ref{kzi})--(\ref{kziv}) $c_3=c_4=0$, we may set $c_1=c$ and $c_2=1$ without loss of generality. Thus we obtain the reduction $$ u(x,t)=w(z)+c, \quad z=x-c t, $$ where $w(z)$ satisfies $$ w w'''+\beta w'w''+\alpha ww'=[2\kappa-(1+\alpha)c]w'. $$ This can be integrated to yield $$ w w''+\tfr12(\beta+1)(w')^2+\tfr12\alpha w^2=[2\kappa-(1+\alpha)c]w + A, $$ where $A$ is an arbitrary constant. Then multiplying through by $w^{\beta-2} w'$ and integrating again yields \begin{equation} (w')^2+{2\alpha w^2\over \beta+1} ={2[2\kappa-(1+\alpha)c]w\over\beta} + {2A\over \beta-1} + Bw^{1-\beta},\label{eqentw} \end{equation} provided that $\beta\not=-1,0,-1$. Generally if $\beta\not=-1,0,1$, then (\ref{eqentw}) is solvable using quadratures, though for certain special values of the parameters, there are explicit solutions. For example (i), if $\beta=-2$ or $\beta=-3$, then (\ref{eqentw}) is solvable in terms of Weierstrass or Jacobi elliptic functions, respectively, (ii) if $B=0$, then (\ref{eqentw}) is solvable in term of trigonometric functions, and (iii) if $(1+\alpha)c=2\kappa$ and $\beta=3$, then $w(z)$ can be expressed in terms of trigonometric functions via the transformation $w(z)=v^{1/2}$. In the special cases $\beta=-1,0,1$ we obtain the following equations, $$ \begin{array}{l} (w')^2+2\alpha w^2\ln w= -2[2 \kappa -(1+\alpha)c]w-A+B w^2,\\[1.0ex] (w')^2+2\alpha w^2 = -2[2 \kappa -(1+\alpha)c]w\ln w-2A+B w^2,\\[1.0ex] (w')^2+2\alpha w^2 = -2[2 \kappa -(1+\alpha)c]w+2A\ln w+B w^2, \end{array} $$ respectively, where $A$ and $B$ are arbitrary constants. If the coefficient of $\ln w$ in these equations is zero, then $w(z)$ is expressible in terms of elementary functions, otherwise in terms of quadratures.\\ \noindent{\bf Reduction 2.6} $\alpha\not=-1$, $\kappa$ arbitrary. If $c_3\not=0$ in (\ref{kzi}), we may set $c_3=1$, $c_2=0$ and $c_1=2\kappa c/(1+\alpha)$, without loss of generality. Thus we obtain the reduction \begin{equation} u(x,t) = {w(z)+c\over t} +{2\kappa\over1+\alpha},\quad z=x-{2\kappa t\over1+\alpha} -c \ln t, \label{redIIIiiw} \end{equation} where $w(z)$ satisfies \begin{equation} ww'''+\beta w'w''- w''+\alpha ww'+(\alpha +1)c w'+w+c=0. \label{redIIIiieq} \end{equation} If $c_3\not=0$ in (\ref{Kziii}) we may set $c_3=1$, $c_1=c$ and $c_2=0$ to obtain the reduction (\ref{redIIIiiw}) with $\kappa=0$.\\ \noindent{\bf Reduction 2.7} $\alpha=-1$, $\kappa\not=0$. If $c_3\not=0$ in (\ref{Kzii}) then we set $c_3=m$, $c_1=0$ and $c_2=1$, without loss of generality. Thus we obtain the reduction \begin{equation} u(x,t)=w(z) +m t,\quad z=x-\tfr12m t^2, \label{redIIIiiiw} \end{equation} where $w(z)$ satisfies $$ ww'''+\beta w'w''- ww'-2\kappa w'-m=0,$$ which may be integrated to yield \begin{equation} w w'' +\tfr12 (\beta-1) (w')^2 -\tfr12 w^2 -2\kappa w -m z=A,\label{redIIIiiieq} \end{equation} where $A$ is an arbitrary constant.\\ \noindent{\bf Reduction 2.8} $\alpha=-1$, $\kappa=0$. If $c_3\not=0$ and $c_4\not=0$ in (\ref{kziv}) we may set $c_3=m$, $c_4=1$, $c_1=c$ and $c_2=0$, without loss of generality. Thus we obtain the reduction \begin{equation} u(x,t)={w(z)+c\over t} +m,\quad z=x-m t - c\ln t, \label{redIIIivw} \end{equation} where $w(z)$ satisfies \begin{equation} ww''' +\beta w'w''-w''- ww' +w+c =0.\label{redIIIiveq} \end{equation} \section{Nonclassical symmetries ($\tau\not=0$)} \setcounter{equation}{0} In the nonclassical method one requires only the subset of $S_{\Delta}$ given by \begin{equation} S_{\Delta,\psi} = \{ u(x,t) : \Delta (u) =0, \psi (u) =0 \}, \label{sdelpsi} \end{equation} where $S_{\Delta}$ is defined in (\ref{sdel})\ and $\psi=0$ is the invariant surface condition\ (\ref{invsc}), to be invariant under the transformation (\ref{trans}). The usual method of applying the nonclassical method (e.g. as described in \cite{\refLW}), involves applying the prolongation ${\rm pr}^{(3)} {\bf v}$ to the system composed of (\ref{fulleqn})\ and the invariant surface condition\ (\ref{invsc})\ and requiring that the resulting expressions vanish for $u\in S_{\Delta,\psi}$, i.e. \begin{equation} {\rm pr}^{(3)} {\bf v} (\Delta) |_{\Delta=0,\psi=0}=0, \quad {\rm pr}^{(1)} {\bf v} (\psi) \vert_{\Delta=0,\psi=0}=0. \label{twoprol} \end{equation} It can well known that the latter vanishes identically when $\psi=0$ without imposing any conditions upon $\xi$, $\tau$ and $\phi$. To apply the method in practice we advocate the algorithm described in \cite{\refCMiii} for calculating the determining equations, which avoids difficulties arising from using differential consequences of the invariant surface condition\ (\ref{invsc}). In the canonical case when $\tau\not=0$ we set $\tau=1$ without loss of generality. We proceed by eliminating $u_t$ and $u_{xxt}$ in (\ref{fulleqn})\ using the invariant surface condition\ (\ref{invsc})\ which yields \begin{equation} \label{delstar} \begin{array}{l} \epsilon\xi u_{xxx} -uu_{xxx} +3\epsilon\xi_u u_x u_{xx} -\beta u_x u_{xx}-\epsilon\phi_u u_{xx} +2\epsilon\xi_x u_{xx} \\[0.7ex] +\epsilon\xi_{uu} u_x^3 -\epsilon\phi_{uu} u_x^2 +2\epsilon\xi_{xu} u_x^2-\alpha u u_x -2\epsilon\phi_{xu} u_x +2 \kappa u_x \\[0.7ex] +\epsilon\xi_{xx} u_x -\epsilon\phi_{xx} +\phi-\xi u_x =0. \end{array} \end{equation} We note that this equation now involves the infinitesimals $\xi$ and $\phi$ that are to be determined. Then we apply the classical Lie algorithm to (\ref{delstar}) using the third prolongation $ {\rm pr}^{(3)} {\bf v}$ and eliminating $u_{xxx}$ using (\ref{delstar}). It should be noted that the coefficient of $u_{xxx}$ is ($\xi-\epsilon u$). Therefore, if this is zero the removal of $u_{xxx}$ using (\ref{delstar}) is invalid and so the next highest derivative term, $u_{xx}$, should be used instead. We note again that this has a coefficient, $\beta-3$, and so that in the case $\xi=u$ one needs to calculate the determining equations\ for the cases $\beta\ne 3$ and $\beta=3$ separately. Continuing in this fashion, there is a cascade of cases to be considered. In the remainder of this section, we consider these cases in turn. First, however, we discuss the case given by $\epsilon=0$. \subsection{$\epsilon=0$} The first determining equation gives $\xi_u=0$, and substituting this into the other seven determining equations yields \begin{equation} \label{nez} \begin{array}{l} \phi_{uuu} u + \beta \phi_{uu}=0, \quad 3 \phi_{xuu} u + 2 \beta \phi_{xu} - \beta \xi_{xx}=0,\\[2.0ex] 3 \phi_{uu} u^{2} + \beta \phi_{u} u - \beta \phi=0,\quad 3 \phi_{xu} u - 3 \xi_{xx} u + \beta \phi_{x}=0,\\[2.0ex] \phi_{t} u-\phi_{xxx}u^{2}-\alpha\phi_{x}u^{2} + 2 \kappa \phi_{x} u + 3 \xi_{x} \phi u - \phi^{2}=0,\\[2.0ex] 3 \phi_{xxu} u^{2} - \xi_{xxx} u^{2} + 2 \alpha \xi_{x} u^{2} + \beta \phi_{xx} u - 4 \xi_{x} \kappa u + 3 \xi \xi_{x} u \\[1.0ex] + \xi_{t} u + 2 \kappa \phi - \xi \phi =0. \end{array} \end{equation} It is quite straightforward to solve these equations and so we obtain the following infinitesimals: (a), if $\alpha\not=0$ $$ \begin{array}{l} {\hbox to 30pt{(i)\hfill}} \displaystyle \quad \xi=2\kappa +{c_1\over t+c_2}, \quad \phi={- u \over t+c_2},\\[3.0ex] {\hbox to 30pt{(ii)\hfill}} \displaystyle \quad \xi=c_1, \quad \phi=0, \end{array} $$ and (b), if $\alpha=0$ $$ \begin{array}{l} {\hbox to 30pt{(i)\hfill}} \displaystyle \quad \xi= { (c_1+1) x+2\kappa (2c_1-1) t+c_2 \over 3(c_1 t+ c_3)}, \quad \phi={u\over c_1 t+ c_3},\\[3.0ex] {\hbox to 30pt{(ii)\hfill}} \displaystyle \quad \xi={x+4 \kappa t + c_1 \over 3t+c_2}, \quad \phi=0. \end{array} $$ These are all equivalent to classical infinitesimals. Hence in this case there are no new nonclassical symmetries. \subsection{$\epsilon=1$} As discussed in the preamble to this section, we must consider, in addition to the general case of the determining equations, each of the singular cases of the determining equations. \\ \noindent{\sc Case 3.2.1} {$\xi\not=u$.} We can remove factors of ($\xi-u$) from the determining equations, and we have then that $\xi_u=0$. Reducing the remaining eight determining equations with respect to\ this, only the last six are non-zero: $$ \begin{array}{l} 3 \phi_{uu} u^{2} - 6 \xi \phi_{uu} u + \beta \phi_{u} u + 3 \xi^{2} \phi_{uu} - \beta \xi \phi_{u} - \beta \phi + \beta \xi \xi_{x} + \beta \xi_{t}=0,\\[2.0ex] \phi_{uuu} u - \xi \phi_{uuu} + \beta \phi_{uu}=0,\\[2.0ex] \xi_{x} \phi_{u} u - \beta \xi \phi_{x} + \phi \phi_{uu} u + \beta \phi_{x} u - \xi \phi \phi_{uu} - 5 \xi \phi_{xu} u+4\xi\xi_{xx}u+\phi_{tu} u \\[0.7ex] -\phi \phi_{u} + \xi_{t} \phi_{u} - \xi \phi_{tu} - \xi^{2} \xi_{xx} + 3 \phi_{xu} u^{2} - 3 \xi_{xx} u^{2}-2 \xi_{x}^{2}u-2\xi_{xt} u \\[0.7ex] + 2 \xi^{2} \phi_{xu} + 2 \xi_{x} \phi - 2 \xi_{t} \xi_{x} + 2 \xi \xi_{xt} =0,\\[2.0ex] 2 \xi \kappa \phi_{x} - \phi_{t} u + \alpha \phi_{x} u^{2} - 2 \kappa \phi_{x} u - \alpha \xi \phi_{x} u + 2 \phi_{xu} \phi_{x} u + \phi \phi_{xxu} u - 2 \xi \phi_{xu} \phi_{x} \\[0.7ex] - \xi_{xx} \phi_{x} u - \xi \phi \phi_{xxu} - 3 \xi_{x} \phi u + 2 \xi \xi_{x} \phi + \xi \xi_{xx} \phi_{x} + \xi_{x} \phi_{xx} u - \xi \phi_{xxx} u \\[0.7ex] - \xi_{t} \phi + \phi^{2} - \phi \phi_{xx} + \phi_{xxx} u^{2} + \phi_{xxt} u - \xi \phi_{xxt} + \xi_{t} \phi_{xx} + \xi \phi_{t}=0,\\[2.0ex] 2 \beta \phi_{xu} u - \xi_{x} \phi_{uu} u - \beta \xi_{xx} u + 2 \phi_{u} \phi_{uu} u + \beta \xi \xi_{xx} - 5 \xi \phi_{xuu} u - \xi \phi \phi_{uuu} \\[0.7ex] + \phi \phi_{uuu} u + \phi_{tuu} u - \phi \phi_{uu} + \xi_{t} \phi_{uu} - \xi \phi_{tuu} + 3 \phi_{xuu} u^{2} + 2 \xi^{2} \phi_{xuu} \\[0.7ex] - 2 \xi \phi_{u} \phi_{uu} + 2 \xi \xi_{x} \phi_{uu} - 2 \beta \xi \phi_{xu} =0,\\[2.0ex] 4 \xi_{x} \kappa u - 2 \phi_{uu} \phi_{x} u - \beta \phi_{xx} u - \xi \xi_{xx} \phi_{u} - 2 \kappa \phi - 2 \phi \phi_{xuu}u -2 \phi_{xtu} u +\xi\phi \\[0.7ex] +\xi_{xxt} u + \xi_{xxx} u^{2} + \xi_{t} \xi_{xx} - 3 \phi_{xxu} u^{2} + 2 \phi \phi_{xu} - \xi \xi_{xxt} + 2 \xi^{2} \xi_{x} -\xi^{2}\phi_{xxu} \\[0.7ex] -2\xi_{t}\phi_{xu} - \alpha \xi_{t} u + \alpha \xi \phi + \xi_{xx} \phi_{u} u - 2 \xi \xi_{x} \kappa - \xi \xi_{xxx} u - 3 \xi \xi_{x} u + 2 \xi \phi_{uu} \phi_{x} \\[0.7ex] +\xi \xi_{x} \xi_{xx} + 4 \xi \phi_{xxu} u + \beta \xi \phi_{xx} + 2 \xi \phi \phi_{xuu} + 2 \xi \phi_{u} \phi_{xu} - 2 \xi \xi_{x} \phi_{xu} -\alpha \xi_{x} u^{2}\\[0.7ex] + \alpha \xi \xi_{x} u + 2 \xi \phi_{xtu} + 2 \xi_{t} \kappa - \xi_{t} u - \xi_{xx} \phi - 2 \phi_{u} \phi_{xu} u =0. \end{array} $$ Reducing the fifth of these equations with respect to\ the fourth yields $$ (\beta-3)\left[(u-\xi) \phi_u -\phi +\xi \xi_x+\xi_t\right]=0. $$ If $\beta=3$, then one easily finds via another route that the expression in the second bracket is necessarily zero. The equation for $\phi$ can be solved to give $$ \phi= F(x,t)(u-\xi) +\xi \xi_x+\xi_t. $$ When this is substituted into the remaining equations we can then take coefficients of powers of $u$ to be zero, and our problem is then easily solved. As in the $\epsilon=0$ case discussed in \S3.1 above, it is quite straightforward to solve the resulting equations. The complete solution set is\\ \noindent (a), if $\alpha\not=-1$ \begin{equation} \label{enqarbi} \begin{array}{l} {\hbox to 30pt{(i)\hfill}}\displaystyle \qquad \xi=c_1, \quad \phi=0, \\[1.0ex] {\hbox to 30pt{(ii)\hfill}}\displaystyle \qquad \xi ={2\kappa\over (1+\alpha)}-{c_1\over t+c_2}, \quad \phi={2\kappa-(1+\alpha)u \over (1+\alpha)(t+c_2)}, \end{array} \end{equation} (b), if $\alpha=-1$ \begin{equation} \xi=c_1 t+c_2, \quad \phi=c_1,\label{enqnnegi} \end{equation} (c), if $\alpha=-1$ and $\kappa=0$ \begin{equation} \xi=c_1 - {c_3\over t+c_2}, \quad\phi={c_1-u \over t+c_2}, \label{enqnnegii} \end{equation} (d), if $\beta=-1$ and $\alpha=0$ \begin{equation} \xi= c_1 x-2c_1 \kappa t +c_2, \quad \phi=3c_1 u-2c_1^2 x+4c_1^2\kappa t -2c_1c_2 -2c_1 \kappa, \quad \beta\not=0.\label{enqneg} \end{equation} The infinitesimals (\ref{enqarbi})--(\ref{enqnnegii}) give rise to classical reductions, but (\ref{enqneg}) gives the following new nonclassical reduction.\\ \noindent{\bf Reduction 3.1} If in (\ref{enqneg}), we set $c_1\not=0$ and $c_2=0$, without loss of generality, then we obtain $$ u(x,t)=w(z) \exp\left(3c_1 t\right) + c_1 z \exp\left(c_1 t\right)+2\kappa, \quad z=(x-2\kappa t-2\kappa/c_1)\exp\left(-c_1t\right),$$ where $w(z)$ satisfies $$ w w''' -w'w'' +c_1 z w' -3c_1 w =0.$$ \noindent{\sc Case 3.2.2} {$\xi=u$, $\beta\not=3$, $\beta\not=1$.} We generate five determining equations, the first of which is $\phi_{uu} =0$. Thus $\phi$ is a linear function of $u$, and substituting this into the remaining four determining equations, we take coefficients of powers of $u$ to be zero. These equations are easily solved to give $ \phi=0$ provided that $\kappa=0$ and $\alpha=-1$. The invariant surface condition\ and (\ref{fulleqn})\ are then solved to give the simple exact solution $$ u(x,t) = {x +c_1\over t+c_2},$$ where $c_1$ and $c_2$ are arbitrary constants.\\ \noindent{\sc Case 3.2.3} {$\xi=u$, $\beta=1$.} We consider here the case $\phi_{uu}\ne0$, since taking $\phi_{uu}=0$ yields the same solution as in Case 3.2.2 above. In this instance the remaining four determining equations\ are $$ \begin{array}{l} 12\kappa- 2 \phi_{xuu} u - 6\alpha u - 6 u - 2 \phi \phi_{uuu} - 3 \phi_{u} \phi_{uu} - 4 \phi_{xu} - 2 \phi_{tuu}=0,\\[2.0ex] \phi_{xu} \phi_{xx} u- \phi_{u} \phi_{xxx} u -\alpha \phi_{u} \phi_{x} u - \phi \phi_{xu} u - 2 \phi_{x} \phi_{xx} + \phi \phi_{uu} \phi_{xx} \\[0.7ex] + \phi_{tu} \phi_{xx} - 2 \phi_{u} \phi_{xu} \phi_{x} + 2 \kappa \phi_{u} \phi_{x} + 2 \phi \phi_{x} - \phi^{2} \phi_{uu} - \phi \phi_{u} \phi_{xxu} \\[0.7ex] - \phi_{xxt} \phi_{u} + \phi_{t} \phi_{u} - \phi \phi_{tu}=0,\\[2.0ex] \phi_{u} \phi_{xuu} u +4\alpha\phi_{u} u - \phi \phi_{uu}^{2} - \phi_{tu} \phi_{uu} + \phi \phi_{u} \phi_{uuu}+6\phi -4\phi_{xx} \\[0.7ex] + \phi_{tuu} \phi_{u} - 4 \phi_{xxu} u + 4 \phi_{u} u - 2 \phi_{uu} \phi_{x} - 4 \phi \phi_{xuu} + 2 \phi_{u}^{2} \phi_{uu} - 8 \kappa \phi_{u} \\[0.7ex] -2\alpha \phi - \phi_{xu} \phi_{uu} u - 4 \phi_{xtu}=0,\\[2.0ex] \phi \phi_{uu} u +\alpha\phi_{xu} u^{2} - 2 \phi_{u} \phi_{uu} \phi_{x} - 2 \phi \phi_{u} \phi_{xuu} - 2 \phi_{u} \phi_{xxu} u +\alpha\phi_{tu} u \\[0.7ex] + 2 \phi \phi_{xu} \phi_{uu} -2 \kappa\phi\phi_{uu} + 2 \phi_{tu} \phi_{xu} +\alpha \phi \phi_{uu} u + 2 \phi_{xxx} u - 2 \phi_{x} u \\[0.7ex] + 2 \phi_{xu}^{2} u - 3 \phi_{u} \phi_{xx} + 2 \phi \phi_{xxu} - 2 \phi_{u}^{2} \phi_{xu} - 2 \phi_{xtu} \phi_{u} + 4 \phi \phi_{u} - 2 \kappa \phi_{tu} \\[0.7ex] -\alpha\phi \phi_{u} - 2 \kappa \phi_{xu} u + \phi_{tu} u + \phi_{xu} u^{2} + 2 \phi_{xxt} - 2 \phi_{t}=0. \end{array} $$ Using the procedures in the package {\tt diffgrob2} with an ordering designed to eliminate first derivatives with respect to $t$, then derivatives with respect to $x$, one can obtain several equations for derivatives of $\phi$ with respect to $u$ only. One can then continue to produce lower order and lower degree equations in the $u$-derivatives of $\phi$, using repeated cross-differentiation and reductions. For example, the ``Direct Search" procedure in the {\tt diffgrob2} manual, \cite{\refM} may be used. This process suffers from expression swell. No termination of this process was observed by us within the computer memory available, and the expressions obtained contained thousands of summands! One of three results appear likely. Firstly, the process terminates with the highest derivative term being $\phi$ itself, yielding $\phi$ to be a function of $u$ alone (note that $x$ and $t$ do not appear explicitly in any of the determining equations). Inserting this into the determining equations, one must have that $\phi$ is constant, a contradiction to our standing assumption in this subcase. Secondly, the process may terminate with an inconsistency, and thirdly, the process may terminate but with such a large expression that the result is useless.\\ \noindent{\sc Case 3.2.4} {$\xi=u$, $\beta=3$, $\phi_u\not=0$.} Four determining equations were obtained, the first of which is $\phi_{uu}=0$, so we substitute $\phi=F(x,t)u+G(x,t)$ into the remaining three and require $F(x,t)\not=0$. We find that there are no such solutions.\\ \noindent{\sc Case 3.2.5} {$\xi=u$, $\beta=3$, $\phi_u=0$ and not both $\kappa$ and $\alpha+1$ are zero.} One determining equation\ was obtained which was a polynomial in $u$ of degree two whose coefficients are functions of $x,t$ only, so the coefficients of powers of $u$ must be zero. These equations were easily simplified using the procedures in {\tt diffgrob2} to yield, \begin{eqnarray} && \kappa\not=0,\quad \alpha=-1,\quad \phi= 0 , \label{xiuini}\\ && \kappa\not=0,\quad \alpha=-1,\quad \phi= {-2\kappa \over t+c_1}, \label{xiuinii} \\[1.0ex] && \begin{array}{@{\hspace{0.0ex}}ll} \kappa\hbox{ arbitrary},\quad\alpha\not=-1, &\phi=c_1\exp(\zeta)+c_2\exp(-\zeta),\\[1.0ex] &\displaystyle \zeta=\mbox{i}\sqrt{\alpha}\left(x-{2\kappa t\over1+\alpha}\right). \end{array} \label{xiuiniii} \end{eqnarray} In (\ref{xiuini}) if we solve (\ref{fulleqn})\ and the invariant surface condition\ as a system of equations we find that the only solution is $u(x,t)=c$, a constant. In (\ref{xiuinii}) we can solve (\ref{fulleqn})\ and the invariant surface condition\ to give the exact (canonical) solution $$ u(x,t)= -2\kappa+{x/t},$$ which cannot be realised by any of the previously found reductions, though it would not appear to be a particularly interesting solution. It is interesting to note that performing the {\tt KolRitt} algorithm of {\tt diffgrob2} on the system comprising the original equation with the invariant surface condition\ led to a simple calculation for $u$. By contrast, the usual procedure of solving the invariant surface condition\ using the method of characteristics and inserting the result into the original equation to obtain the reduction was considerably more difficult due to the implicit nature of the reduction. In (\ref{xiuiniii}) we can again solve our problem to yield the exact (canonical) solution $$ u(x,t)={-2\kappa \over 1+\alpha} \pm (c_0 +c_1 \mbox{e}^{\zeta} +c_2 \mbox{e}^{-\zeta})^{1/2}, \qquad \zeta=\mbox{i}\sqrt{\alpha}\left(x- {2\kappa t \over 1+\alpha}\right),$$ which is a special case of the travelling wave reduction 2.5.\\ \noindent{\sc Case 3.2.6} {$\xi=u$, $\beta=3$, $\phi_u=0$, $\kappa=0$, $\alpha=-1$.} We are left simply with the determining equation $\phi_{xx}-\phi=0$, which produces the following infinitesimal, \begin{equation} \phi= g(t) \mbox{e}^x +h(t) \mbox{e}^{-x}, \label{xiuiv} \end{equation} where $g$ and $h$ are arbitrary functions. Hence we have to solve the invariant surface condition \begin{equation} u u_x +u_t = g(t) \mbox{e}^x +h(t) \mbox{e}^{-x}. \label{eqIIIxv} \end{equation} It is straightforward to show that every solution of this equation is also a solution of (\ref{fulleqn}). \section{Nonclassical ($\tau=0$) and Direct Methods} \setcounter{equation}{0} \def\redIIi{\redIIi} \def\redIIii{\redIIii} \def\redIIIi{\redIIIi} In the canonical case of the nonclassical method when $\tau=0$ we set $\xi=1$ without loss of generality. We proceed by eliminating $u_x,u_{xx},u_{xxx}$ and $u_{xxt}$ in (\ref{fulleqn})\ using the invariant surface condition\ (\ref{invsc})\ which yields \begin{equation} \label{delstard} \begin{array}{l} u_t -\epsilon \phi \phi_{uu} u_t -\epsilon \phi_{xu} u_t -\epsilon \phi_{u}^2 u_t -\phi_{xx} u -\phi_{u} \phi_x u -\phi^2 \phi_{uu} u -2 \phi \phi_{xu} u \\[1.0ex] -\phi \phi_u^2 u -\alpha\phi u-\beta \phi \phi_x -\epsilon \phi_t \phi_u -\beta \phi^2 \phi_u -\epsilon \phi_{xt} -\epsilon \phi \phi_{tu}+2 \kappa \phi =0, \end{array} \end{equation} which involves the infinitesimal $\phi$ that is to be determined. As in the $\tau\not=0$ case we apply the classical Lie algorithm to this equation using the first prolongation $ {\rm pr}^{(1)} {\bf v}$ and eliminate $u_t$ using (\ref{delstard}). The equivalent approach using the direct method of Clarkson and Kruskal \cite{\refCK} is to consider the ansatz $u=U(x,t,w(t))$ and require that the result be ordinary differential equation\ for $w(t)$; see also \cite{\refPACiii,\refLouii}. It is straightforward to show that this yields the equivalent reductions.\\ {\bf Case 4.1} {$\epsilon=0$.} The nonclassical method generates a single equation of 25 terms, without any singular solutions. Since this is difficult to solve explicitly, we seek polynomial solutions in $u$.\\ \noindent {\it Ansatz 1}.\quad$\phi=F(x,t)$.\quad In this case we obtain the following three exact solutions for (\ref{fulleqn}) with $\epsilon=0$: \begin{equation} u(x,t)= \mu_2\left[x-(2\kappa-\beta\mu)t\right]^2 + \mu_0, \label{IVsoli} \end{equation} where $\mu_2$ and $\mu_0$ are arbitrary constants, provided that $\alpha=0$, \begin{equation} u(x,t)={(x-2 \kappa t)^3\over 12 t}+ \mu(x-2\kappa t) + \delta t^{1/2}, \label{IVsolii} \end{equation} where $\delta$ is an arbitrary constant, provided that $\alpha=0$ and $\beta=-1$, and \begin{equation} u(x,t)=-\,{x-2\kappa t \over \alpha t},\label{IVsoliii} \end{equation} provided that $\alpha\not=0$.\\ \hide{\noindent {\it Ansatz 2}.\quad$\phi=F(x,t)u+G(x,t)$. \quad The coefficients of powers of $u$ set to zero yield three equations, one a third order ordinary differential equation\ in $F$, the other two both have both dependent variables present. It doesn't seem possible to integrate the ordinary differential equation\ (in order to simplify the problem), and there is large expression swell when trying to use {\tt diffgrob2}.} \noindent {\it Ansatz 2}.\quad$\phi=F(x,t)u^2+G(x,t)u+H(x,t)$. \quad In this case we obtain the following three exact solutions for (\ref{fulleqn}) with $\epsilon=0$; \begin{equation} u(x,t)=A\tan\left[\tfr12\sqrt\alpha(x-2\kappa t) \right], \label{IVbsoli} \end{equation} where $\mu$ is an arbitrary constant, provided that $\beta=-3$, \begin{equation} u(x,t)= A \exp\{\mu(x-2\kappa t)\},\qquad \mu^2= -\,{\alpha\over1+\beta},\label{IVbsolii} \end{equation} provided that $\beta\not=-1$, and \begin{equation} u(x,t)=A\,\mbox{sech}\{\tfr12\sqrt\alpha(x-2\kappa t)\},\label{IVbsoliii} \end{equation} provided that $\beta=-3$.\\ {\bf Case 4.2} {$\epsilon=1$.} In this case the nonclassical method generates a single equation of 150 terms, which has a singular solution if and only if $$ \phi\phi_u +\phi_x -u - {2\kappa/\beta} =0, $$ provided that $ \alpha-\beta-1 =0$. We again seek polynomial solutions of $\phi$ using one ansatz.\\ \noindent {\it Ansatz 1}.\quad$\phi=F(x,t)$.\quad In this case we obtain three following three exact solutions for (\ref{fulleqn}) with $\epsilon=0$: \begin{equation} u(x,t)= \mu_2\left[x-(2\kappa-\beta\mu)t\right]^2 + \mu_1\left[x-(2\kappa-\beta\mu)t\right]+\mu_0, \label{IVcsoli} \end{equation} where $\mu_2$, $\mu_1$ and $\mu_0$ are arbitrary constants, provided that $\alpha=0$, \begin{equation} \label{IVcsolii} \begin{array}{r} \displaystyle u(x,t)={(x-2 \kappa t)^3\over 12 t}+{\mu_2(x-2 \kappa t)^2\over t} + \left({1+8\mu_2^2\over2t}+\mu_1\right)(x-2\kappa t) + \delta t^{1/2}\\[2.5ex] \displaystyle +{\mu_2(6+16\mu_2^2)\over3t}+2\kappa + \mu_1+\mu_2, \end{array} \end{equation} where $\mu_2$, $\mu_1$ and $\delta$ are arbitrary constants, provided that $\alpha=0$ and $\beta=-1$, and \begin{equation} u(x,t)=-\,{x-2\kappa t \over \alpha t},\label{IVcsoliii} \end{equation} provided that $\alpha\not=0$. \section{Discussion} \setcounter{equation}{0} In this paper we have classified symmetry reductions of the nonlinear third order partial differential equation\ (1.1), which contains three special cases that have attracted considerable interest recently, using the classical Lie method and the nonclassical method due to Bluman and Cole \cite{\refBC}. The use of the MAPLE package {\tt diffgrob2} was crucial in this classification procedure. In the classical case it identified the special cases of the parameters for which additional symmetries might occur whilst in the nonclassical case, the use of {\tt diffgrob2} rendered a daunting calculation tractable and thus solvable. In their recent paper, Gilson and Pickering \cite{\refGP} discuss the application of the Painlev\'e tests for integrability due to Ablowitz, Ramani and Segur \cite{\refARS,\refARSi} and Weiss, Tabor and Carnevale \cite{\refWTC} to equation (1.1). In particular, they investigate the integrability of the ordinary differential equations\ arising from the travelling-wave reductions 2.1 and 2.5 above. It would be interesting to investigate the integrability of some of the ordinary differential equations\ arising from the other reductions derived in this paper using standard Painlev\'e analysis, ``weak Painlev\'e analysis'' \cite{\refRDG,\refRRDG} and ``perturbative Painlev\'e analysis'' \cite{\refCFP}, though we shall not pursue this further here. Marinakis and Bountis \cite{\refMB} have also applied Painlev\'e analysis to the FFCH equation (\ref{ch}); an interesting aspect of their analysis is the use of a hodograph transformation. To conclude we remark that the RH equation (\ref{rh}) is a quasilinear partial differential equation\ of the form discussed by Clarkson, Fokas and Ablowitz \cite{\refCFA}. It is routine to apply their algorithm, which involves a hodograph transformation, for applying the Painlev\'e PDE test to such quasilinear partial differential equations\ and show that (\ref{rh}) does not satisfy the necessary conditions to be solvable by inverse scattering. \vspace{3.0ex} \begin{center} \large \bf Acknowledgments \end{center} We thank the editors for inviting us to write an article. We also thank the Program in Applied Mathematics, University of Colorado at Boulder, for their hospitality during our visit whilst some of this work was done. The research of PAC and ELM is supported by EPSRC (grant GR/H39420) and that of TJP by an EPSRC Postgraduate Research Studentship, which are gratefully acknowledged. \def\refbk#1#2#3#4#5 {\bibitem{#1}{\frenchspacing#2},{\frenchspacing\sl#3}, #4\ (#5).} \def\refpp#1#2#3#4 {\bibitem{#1}{\frenchspacing#2}, #3, #4.} \def\refjl#1#2#3#4#5#6{\bibitem{#1}{\frenchspacing#2}, {\frenchspacing\it#3},\ {\bf#4}, #5\ (#6).} \def\refeb#1#2#3#4#5#6#7{\bibitem{#1}{\frenchspacing#2}, {\frenchspacing\it#3},\ {\bf#4}\ no.#5, #6\ (#7).} \def\reftoap#1#2#3#4#5{\bibitem{#1}{\frenchspacing#2}, {\frenchspacing\it#3}, #4\ (#5).}

proofpile-arXiv_065-530

\section{Introduction}\par Among the several quantities that can be measured in the process of electron-positron annihilation into a fermion-antifermion couple, the longitudinal polarization asymmetry $A_{LR} \equiv {\sigma_{L}-\sigma_{R}\over \sigma_{L}+\sigma_{R}}$ has represented in the last few years an example of, least to say, remarkable theoretical interest. This is due to the known fact that, as it was stressed in a number of dedicated papers \cite{1},\cite{2}, \cite{3},\cite{4}, the properties of this observable on top of Z resonance are indeed special. In particular one can stress two main facts i.e. that $A_{LR}$ is independent of the final produced state (this was shown in particular detail in Ref.\cite{3}), and that it is particularly sensitive to possible virtual effects of a large number of models of new physics (this was exhaustively discussed in Refs.\cite{2} and \cite{4}). These features, that appear essentially unique, have deeply motivated the tough experimental effort at SLC \cite{5} where $A_{LR}$ has been (in fact, it is still being) measured to an extremely high precision \cite{6}, fully exploiting the fact that at a linear electron-positron collider it is "relatively" easy to produce longitudinally polarized electron beams with a high and accurately known polarization degree\cite{7}. This is not the case of a circular accelerator, and for this reason neither at LEP1 (in spite of the several impressive experimental studies and efforts of recent years \cite{8}) nor at LEP2 a measurement of $A_{LR}$ has been, or will be predicticably performed.\par The possibility that a linear electron-positron collider of an overall c.m. energy not far from 500 GeV is eventually built in a not too distant future has been very seriously investigated in the last few years, and the results of a remarkable combined experimental and theoretical effort have been published in several dedicated Proceedings\cite{8bis}. At such a kind of machine it would be, again, "relatively" easy to produce longitudinally polarized electron beams, which implies the possibility of measuring $A_{LR}$, for various possible final states. One might therefore wonder whether the special theoretical properties valid on top of Z resonance will still be true and, if not, how they would be modified at about 500 GeV.\par The purpose of this paper is precisely that of investigating the general features of $A_{LR}$ at such a future linear collider (NLC) and to show that, from a theoretical point of view, this quantity still retains beautiful and interesting features, that make it particularly promising as a tool for investigating virtual effects of models of new physics. This will be shown in some detail in the following Section 2. Section 3 will be devoted to an illustrative example, i.e. to the case of a model with anomalous gauge couplings, for which the benefits of a measurement of $A_{LR}$ will be explicitely shown in a quantitative way. A short final discussion will then be given in Section 4, valid for a more general class of theoretical models. \newpage \section{Longitudinal polarization asymmetries at one loop.} \vspace{0.25cm} {\bf 2a. General features} \vspace{0.25cm} The purpose of this section is that of deriving relatively simple and compact expressions for longitudinal polarization asymmetries in the process of electron-positron annihilation into two final fermions at arbitrary c.m. energy at one loop. With this aim, we shall follow a procedure that has been fully illustrated in two recent papers \cite{9},\cite{10} and has been called "$Z$-peak subtracted" representation. In order, though, to make this paper, at least reasonably, self-contained, we shall sketch a quick derivation of all the fundamental formulae, defering to refs.\cite{9},\cite{10} for a more complete discussion of several technical details.\par The starting point of our derivation will be the expression of the longitudinally polarized cross sections $\sigma_{L,f}$ and $\sigma_{R,f}$ (left-handed and right-handed initial electrons) in Born approximation, where $f$ denotes the final fermion (in the case that we shall consider, charged lepton or quark). In practice, though, it will be more useful to consider from the very beginning the difference and the sum of such cross sections, that appear directly as the numerator and the denominator of the longitudinal polarization asymmetry. Denoting by $\sigma_{LR,f}$ and $\sigma_f$ these quantities, one easily finds that \begin{eqnarray} && \sigma^{(0)}_{LRf}(q^2)=\sigma^{(0)}_{Lf}(q^2)-\sigma^{(0)}_{Rf}(q^2) =N_f({4\pi q^2\over3})\times \nonumber\\ &&\{[{\sqrt2 G^{(0)}_{\mu} M^2_{Z0}\over4\pi}]^2{2g^{(0)}_{Vl}g^{(0}_{Al}) (g^{(0)\, 2}_{Vf}+g^{(0)\, 2}_{Af})\over(q^2-M^2_{Z0})^2}- 2{\alpha_0 Q_f\over q^2}[{\sqrt2 G^{(0)}_{\mu} M^2_{Z0}\over4\pi}] {g^{(0)}_{Al}g^{(0)}_{Vf}\over q^2-M^2_{Z0}}\} \end{eqnarray} \begin{eqnarray} && \sigma^{(0)}_{f}(q^2)=\sigma^{(0)}_{Lf}(q^2)+\sigma^{(0)}_{Rf}(q^2) =N_f({4\pi q^2\over3})\{\{{\alpha^2_0 Q^2_f\over q^4}+ \nonumber\\ && [{\sqrt2 G^{(0)}_{\mu} M^2_{Z0}\over4\pi}]^2{(g^{(0)\, 2}_{Vl}+g^{(0\, 2}_{Al}) (g^{(0)\, 2}_{Vf}+g^{(0)\, 2}_{Af})\over(q^2-M^2_{Z0})^2}- 2{\alpha_0 Q_f\over q^2}[{\sqrt2 G^{(0)}_{\mu} M^2_{Z0}\over4\pi}] {g^{(0)}_{Vl}g^{(0)}_{Vf}\over q^2-M^2_{Z0}}\} \end{eqnarray} In the previous formulae, $q^2$ is the total c.m. squared energy, $N_f$ is the colour factor and the various couplings are defined in the conventional way, i.e $g^{(0)}_{Al,f} \equiv I^{3L}_{l,f}$ ; $g^{(0)}_{Vl,f}\equiv I^{3L}_{l,f} -2Q_{l,f}s^{2}_{0}$, with $Q_{l,f}$ the charge of the lepton $l$ or fermion $f$. Note that all the couplings and the $Z$ mass (with index (0)) are, by definition, 'bare' ones.\par From eqs.(1),(2) it is straightforward to derive the expression at Born level of the longitudinal polarization asymmetry $A^{(0)}_{LR,f}(q^2)$ defined as $\sigma^{(0)}_{LR,f}/\sigma_f$. From a glance to eqs.(1),(2) one can derive a rather important and well-known fact. On top of $Z$ resonance, where the pure $Z$ exchange term largely dominates, the dependence on the final state completely disappears, so that $A^{(0)}_{LR,f}$ becomes only dependent on the initial electron-$Z$ couplings. But when one moves away from the $Z$ peak this peculiar feature disappears, and other terms become competitive. As a result of this, $A^{(0)}_{LR,f}$ will now effectively depend on products of $Z$ couplings to the initial and to the final considered fermion, and several different observables will therefore become potentially relevant.\par Concerning the final fermion, we shall be limited in this paper to the case of "light" charged ones ($f=l,u,d,s,c,b$). Moreover, the considered $q^2$ values will always be (much) larger than $M^2_Z$. In terms of the final masses, this means that they will be safely negligible, $m_f\simeq 0$. For what concerns calculations within the Standard Model framework, this will have the consequence that at the one loop level the independent Lorentz structures of the invariant scattering amplitude will be of only four types corresponding to initial and final axial and vector "currents". Equivalently, one shall have, following the definitions of ref.\cite{10}, a "$\gamma\gamma$", a "$ZZ$", a "$\gamma Z$" and a "$Z\gamma$" structure, that will appear as the four independent combinations of the elementary $\gamma, Z$ "currents" defined as \begin{equation} v_{\mu f}^{(\gamma)}= e_0Q_f\bar u_f\gamma_{\mu}v_f \end{equation} \begin{equation} v_{\mu f}^{(Z)}= {e_0\over 2c_0s_0} \bar u_f\gamma_{\mu}(g^{(0)}_{Vf}-\gamma^5 g^{(0)}_{Af})v_f \end{equation} For instance, the "$\gamma Z$" structure will correspond in our notations to the product of $v^{(\gamma)}_{\mu,l}v^{\mu(Z)}_{f}$, while the "$Z\gamma$" structure will correspond to $v^{(Z)}_{\mu,l}v^{\mu(\gamma)}_{f}$.\par In this paper, we shall focuse our attention on three cases that we consider to be realistic at a future $500~GeV$ electron-positron collider, i.e. those of production of two final charged leptons ($A_{LR,l}$), of a final $b\bar b$ couple ($A_{LR,b}$), and of production of all possible light final quark couples ($A_{LR,5}$). This should cover all the meaningful possibilities for two final light fermions production.\par The previous equations (1),(2) were strictly valid at Born level. To make more rigorous statements, one has now to move to the one loop expressions. This implies a redefinition of the various bare quantities and also a consideration of the potentially dangerous QED radiation. For what concerns the latter point, a rigorous treatment of $A_{LR,f}$ at NLC (on $Z$ resonance an exhaustive discussion is available \cite{11}), does not yet exist to our knowledge (and is, in fact, under examination). We shall assume that, as it happens in all other cases, a proper apparatus-dependent calculation allows to eliminate the unwanted difficulties and we proceed from now on to the treatment of the purely "non QED" component. In the latter one we shall leave aside, and consider it as a separate and fixed component, the contribution to the considered observables originated by standard strong interactions that, in the conventional treatment, will be denoted as the "QCD" term $\simeq \alpha_s(q^2)$. Our interest will be concentrated on the purely electroweak components of the various $A_{LR,f}$, computed at one loop. The color factor of all these quantities will consequently continue to cancel exactly in the ratio, as it did at pure Born level.\par To illustrate the philosophy and the main features of our approach with the simplest example, we shall consider the modifications at one loop of the "pure $Z$" Born exchange term $\simeq{1\over (q^2-M^2_Z)^2}$ in the denominator of a general $A^{(0)}_{LR,f} \equiv \sigma^{(0)}_{LR,f}/\sigma^{(0)}_{f}$. As it has been shown in full detail in ref.\cite{10}, sect.2, in the discussion leading to eq.(38), and as one can easily derive, the Born expression becomes at one loop: \begin{eqnarray} && \sigma^{(Z)}_{lf}(q^2)=N_f({4\pi q^2\over3}){[{3\Gamma_l\over M_Z}][{3\Gamma_f\over N^{QCD}_f M_Z}] \over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}[ 1-2R^{(lf)}(q^2)\nonumber\\ && -8s_lc_l\{{\tilde{v}_l\over1+\tilde{v}^2_l}V^{(lf)}_{\gamma Z}(q^2)+{\tilde{v}_f |Q_f|\over1+\tilde{v}^2_f}V^{(lf)}_{Z\gamma}(q^2)\}] \end{eqnarray} We want to stress now again the main features of this equation. As one sees, the squared Fermi coupling $G^{(0)2}_{\mu}$ has been replaced by the product of the two $Z$ widths $\Gamma_l\Gamma_f$, for which one is supposed to take the experimental values measured on top of $Z$ resonance (in fact, for $f\neq l$, $\Gamma_f$ appears divided by the quantity $N^{QCD}_f\simeq 3(1+\alpha_s (M^2_Z)/\pi)$, where also $\alpha_s(M^2_Z)$ is supposed to be measured on top of $Z$ resonance). As a consequence of this bargain, the one loop "form factors" $R^{lf}(q^2)$, $V^{lf}(q^2)$ are \underline{subtracted} at $q^2=M^2_Z$. More precisely, they will be given by integrations over the angular variable of the following expressions: \begin{equation} R^{(lf)}(q^2,\theta) \equiv \tilde{I}^{(lf)}_Z (q^{2}, \theta) - \tilde{I}^{(lf)}_Z (M^2_Z, \theta) \end{equation} \begin{equation} V^{(lf)}_{\gamma Z}(q^2,\theta) \equiv \tilde{F}^{(lf)} _{\gamma Z} (q^{2}, \theta) - \tilde{F}^{(lf)}_{\gamma Z} (M^2_Z, \theta) \end{equation} \begin{equation} V^{(lf)}_{Z\gamma}(q^2,\theta) \equiv \tilde{F}^{(lf)} _{Z\gamma} (q^{2}, \theta) - \tilde{F}^{(lf)}_{Z\gamma} (M^2_Z, \theta) \end{equation} \noindent where the "auxiliary" quantity $\tilde{I}_Z$ is defined as \begin{equation} \tilde{I}^{(lf)}_Z(q^2,\theta) = {q^2\over q^2-M^2_Z}[\tilde{F}^{(lf)}_Z (q^2, \theta)- \tilde{F}^{(lf)}_{Z} (M^2_Z, \theta)] \end{equation} \noindent and all the quantities denoted as $\tilde{F}^{(ij)}$ in the previous equations are conventional, \underline{gauge invariant} combinations of self-energies, vertices and boxes (defined following the conventions of Degrassi and Sirlin \cite{12}) that belong to the previously defined "$ZZ$", "$\gamma Z$" and "$Z\gamma$" Lorentz structures. To fix the normalization, the \underline{self-energy} ($cos\theta$ independent) component of $\tilde{F}^{ij}$ is the one appearing in the usual definition of the transverse self-energies: \begin{equation} A_{i}(q^{2}) \equiv A_{i}(0) + q^{2} F_{i} (q^{2}) \end{equation} The generalization of the given example to the complete expressions of the asymmetries will now be a trivial one. In the final \underline{"$\gamma Z$" component}, for instance, new quantities measured on $Z$ resonance will appear. One will be the longitudinal polarization asymmetry itself, defined as ; \begin{equation} A_{LR}(M^2_Z)={2\tilde{v}_{l}(M^2_Z)\over 1 + \tilde{v}^2_{l}(M^2_Z)} \end{equation} \noindent where $\tilde{v}_l(M^2_Z)=1-4s^2_l(M^2_Z)$ and $s^2_l(M^2_Z)$ is the effective (leptonic) Weinberg-Salam angle measured at $M^2_Z$. Also, the corresponding hadronic variables $\tilde{v}_f(M^2_Z)$ will enter, whose \underline{exact} definition is provided by the so called polarized forward-backward asymmetries originally \cite{13} defined as: \begin{equation} A_{b,c} = {2\tilde{v}_{b,c}(M^2_Z)\over 1 + \tilde{v}^2_{b,c}(M^2_Z)} \end{equation} \noindent (In practice, $\tilde{v}_f\simeq 1-4|Q_f|\tilde{s}^2_l(M^2_Z)$.) One should also say at this point that, for what concerns the photon contribution, the treatment is of strictly conventional type, with the bare $\alpha^{(0)}$ replaced by the physical coupling computed at zero momentum transfer $\alpha_{QED}\equiv \alpha(0)$ and the form factor \begin{equation} \tilde{\Delta}^{(lf)}\alpha(q^2,\theta) \equiv \tilde{F}^{(lf)}_{\gamma} (0, \theta) - \tilde{F}^{(lf)}_{\gamma} (q^{2}, \theta) \end{equation} \noindent where $\tilde{F}_{\gamma}$ is a proper projection on the "$\gamma\gamma$" structure of the usual photon self-energy with corresponding vertices and boxes.\par After this long but, we hope, useful discussion we shall now write the required one loop expressions of the electroweak component of the considered asymmetries. Using the previous notations, we have that: \begin{eqnarray} &&\sigma^{(1)}_{LR,f}= N_f({4\pi q^2\over3})\{{[{3\Gamma_l\over M_Z}][{3\Gamma_f\over N^{QCD}_f M_Z}]\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}[ 1-2R^{(lf)}(q^2)\nonumber\\ && -{4s_lc_l\over \tilde{v}_l}V^{(lf)}_{\gamma Z}(q^2)-{8s_lc_l\tilde{v_f} |Q_f|\over1+\tilde{v}^2_f}V^{(lf)}_{Z\gamma}(q^2)] +2\alpha(0)|Q_f|{q^2-M^2_Z\over q^2((q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z)}\nonumber\\ && [{3\Gamma_l\over M_Z}]^{1/2}[{3\Gamma_f\over N^{QCD}_f M_Z}]^{1/2}{\tilde{v}_f\over(1+\tilde{v}^2_l)^{1/2}(1+ \tilde{v}^2_f)^{1/2}}[1+ \tilde{\Delta}^{(lf)}\alpha(q^2) -R^{(lf)}(q^2)\nonumber\\ &&-{4s_lc_l|Q_f|\over \tilde{v}_f}V^{(lf)}_{Z\gamma}(q^2)]\} \end{eqnarray} \begin{eqnarray} &&\sigma^{(1)}_f=N_f({4\pi q^2\over3})\{ Q^2_f {\alpha^2(0)\over q^4}[1+2\tilde{\Delta}^{(lf)}\alpha(q^2)] +{[{3\Gamma_l\over M_Z}][{3\Gamma_f\over N^{QCD}_f M_Z}]\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}[ 1-2R^{(lf)}(q^2)\nonumber\\ && -8s_lc_l\{{\tilde{v}_l\over1+\tilde{v}^2_l}V^{(lf)}_{\gamma Z}(q^2)+{\tilde{v}_f |Q_f|\over1+\tilde{v}^2_f}V^{(lf)}_{Z\gamma}(q^2)\}] +2\alpha(0)|Q_f|{q^2-M^2_Z\over q^2((q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z)}\nonumber\\ && [{3\Gamma_l\over M_Z}]^{1/2}[{3\Gamma_f\over N^{QCD}_f M_Z}]^{1/2}{\tilde{v}_l\tilde{v}_f\over(1+ \tilde{v}^2_l)^{1/2}(1+\tilde{v}^2_f)^{1/2}}[1+ \tilde{\Delta}^{(lf)}\alpha(q^2) -R^{(lf)}(q^2) \nonumber\\ && -{4s_lc_l\over\tilde{v}_l}V^{(lf)}_{\gamma Z}(q^2)+ {|Q_f|\over\tilde{v}_f}V^{(lf)}_{Z\gamma}(q^2)]\} \end{eqnarray} Eqs.(14) and (15) conclude our introductory discussion. In the forthcoming part of this section we shall consider in more detail the various cases corresponding to the three chosen different final states. \newpage {\bf 2b. Discussion of different final states} \vspace{0.5cm} We begin with the (simplest) case of two final charged leptons. Since $f=l$, only three independent form factors will remain ($V_{\gamma Z}\equiv V_{Z\gamma}$). To maintain the notations of refs.\cite{9}, \cite{10} we shall put no fermion index on them, so that they will be labelled as $\tilde{\Delta}\alpha$, $R$ and $V$. Also, we shall use here the quantity \begin{equation} \kappa\equiv {\alpha(0)\over [3\Gamma_l/M_Z]} \end{equation} From eqs.(14).(15) one is then led to the desired expression. Here for simplicity we shall write it in an "effective" way i.e. throwing away terms that are numerically irrelevant and only retaining the meaningful contributions. In this way we obtain the (relatively simple) formula: \begin{eqnarray} &&A^{(1)}_{LR,l}(q^2)={q^2[\kappa(q^2-M^2_Z)+q^2] \over \kappa^2(q^2-M^2_Z)^2+q^4}A_{LR}(M^2_Z)\times \nonumber \\ &&\bigm\{1+[{\kappa(q^2-M^2_Z) \over\kappa(q^2-M^2_Z)+q^2}-{2\kappa^2(q^2-M^2_Z)^2\over \kappa^2(q^2-M^2_Z)^2+q^4}] [\tilde{\Delta}\alpha(q^2)+R(q^2)] -{4c_ls_l\over \tilde{v}_l}V(q^2) \bigm\} \end{eqnarray} A few comments are, at this point, appropriate. First of all, one sees that numerically the value of eq.(17) (more precisely, of its leading term, the first one in the r.h.s. of the equation) decreases when $q^2$ becomes larger than $M^2_Z$, pointing to an asymptotic value of about ${1\over 2}A_{LR}(M^2_Z)\simeq 0.07$. The one loop modifications to the leading term contain two quantities, the combination $[\tilde{\Delta}\alpha + R]$ and the "$\gamma Z$" term $V$. The fact that the sum $[\tilde{\Delta}\alpha + R]$ appears in eq.(17) is not accidental: it will be a general feature for the new physics effects in any ratio of cross sections. We shall return on this point in the next section. The point that deserves attention is the fact that the coefficient of $V$ is relatively enhanced with respect to the coefficient of $[\tilde{\Delta}\alpha + R]$ by the factor ${1\over \tilde{v}_l}$, which makes it one order of magnitude larger. Note that this fact comes from the (accidental) smallness of the quantity $\tilde{v}_l(M^2_Z)$ and is generated by the contribution to the "$\gamma Z$" structure in the "pure $Z$" exchange component of $\sigma_{LR,l}$ \underline{at one loop}, that is strongly reminiscent of the situation met on top of $Z$ resonance. As a consequence of this, one expects a relative enhancement of the virtual effects for those models of new physics where the contribution to $V(q^2)$ is not accidentally depressed. We shall provide one specific example in section 3.\par The next case that we shall consider is that of a final $b\bar b$ couple. From the relevant expressions, making the same numerical approximations as in the previous case i.e. only retaining the dominant contributions to the various coefficients, we obtain in this case: \begin{equation} A^{(1)}_{LR,b}(q^2)=\bar A_{LR,b} (q^2)[1+a_b (q^2)[\tilde{\Delta}^{lb}\alpha(q^2) +R^{lb}(q^2)]+b_b (q^2)V^{lf}_{\gamma Z}(q^2)+ c_b (q^2)V^{lf}_{Z\gamma}(q^2)] \end{equation} \noindent where \begin{equation} \bar A_{LR,b} (q^2)= {C_{LR,b}(q^2)\over C_b(q^2)} \end{equation} \begin{equation} a_b (q^2)=C^{\gamma Z}_{LR,b}-2C^{\gamma\gamma}_{b}-C^{\gamma Z}_{b} =-2C^{ZZ}_{LR,b}-C^{\gamma Z}_{LR,b}+ 2C^{ZZ}_{b}+C^{\gamma Z}_{b} \end{equation} \begin{equation} b_b (q^2)=-{4s_lc_l\over \tilde{v}_l}[C^{ZZ}_{LR,b}-C^{\gamma Z}_{b}] +{8s_lc_l\tilde{v}_l\over1+\tilde{v}^2_l}C^{ZZ}_{b} \end{equation} \begin{equation} c_b (q^2)=-{4s_lc_l\over 3\tilde{v}_b}[C^{\gamma Z}_{LR,b} -C^{\gamma Z}_{b}] +{8s_lc_l\tilde{v}_b\over3(1+\tilde{v}^2_b)}[C^{ZZ}_{b}-C^{ZZ}_{LR,b}] \end{equation} \noindent and \begin{equation} C_{LR,b}=N^{ZZ}_{LR,b}+N^{\gamma Z}_{LR,b} \end{equation} \begin{equation} N^{ZZ}_{LR,b}={18\tilde{v}_l\over1+\tilde{v}^2_l}({3\Gamma_l/M_Z\over \alpha})({3\Gamma_b/N^{QCD}_b M_Z\over\alpha})({q^4 \over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}) \end{equation} \begin{equation} N^{\gamma Z}_{LR,b}={6\tilde{v}_b\over(1+\tilde{v}^2_l)^{1/2} (1+\tilde{v}^2_b)^{1/2}} ({{3\Gamma_l/M_Z}^{1/2}\over \alpha})([3\Gamma_b/N^{QCD}_b M_Z]^{1/2})({q^2(q^2-M^2_Z) \over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}) \end{equation} \begin{equation} C^{ZZ}_{LR,b}= N^{ZZ}_{LR,b}/C_{LR,b} \ \ \ \ \ C^{\gamma Z}_{LR,b}=N^{\gamma Z}_{LR,b}/C_{LR,b} \ \ \ \ \ C_{b}=1+N^{ZZ}_{b}+N^{\gamma Z}_{b} \end{equation} \begin{equation} N^{ZZ}_{b}=9({3\Gamma_l/M_Z\over \alpha})({3\Gamma_b/N^{QCD}_b M_Z\over\alpha})({q^4 \over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}) \end{equation} \begin{equation} N^{\gamma Z}_{b}={6\tilde{v}_l\tilde{v}_b\over(1+\tilde{v}^2_l)^{1/2} (1+\tilde{v}^2_b)^{1/2}} ({{3\Gamma_l/M_Z}^{1/2}\over \alpha})([3\Gamma_b/N^{QCD}_b M_Z]^{1/2})({q^2(q^2-M^2_Z) \over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z}) \end{equation} \begin{equation} C^{\gamma\gamma}_{b}=1/C_{b} \ \ \ \ \ C^{ZZ}_{b}= N^{ZZ}_{b}/C_{b} \ \ \ \ \ C^{\gamma Z}_{b}=N^{\gamma Z}_{b}/C_{b} \end{equation} Comparing eq.(18) with the previous eq.(17), we notice the following facts. The numerical value of the leading term becomes asymptotically, as one easily sees, much larger than that of the corresponding leptonic quantity. For large $q^2$, using the experimental inputs for the various widths and asymmetries, it approaches a value of approximately $0.65$. In the one loop corrections, the largely dominant coefficient is that of $V_{\gamma Z}$, approximately one order of magnitude larger than that of $[\tilde{\Delta}\alpha + R]$, again as a consequence of the $\simeq{1\over \tilde{v}_l}$ factor. Note that the enhanced coefficient, i.e. that of $V_{\gamma Z}$, comes from the \underline{pure Z} exchange contribution to $\sigma_{LR,b}$, where $\tilde{v}_l$ appears at Born level. This feature, that is not valid for the crossed term $V_{Z\gamma}$ where $\tilde{v}_l$ is replaced by the much bigger $\tilde{v}_b$, is the same that has been already met in the case of the leptonic asymmetry.\par The final case to be considered is that of the full longitudinal asymmetry for production of the five light quarks $A_{LR,5}=\sigma_{LR,5}/\sigma_5$. To derive its expression is straightforward once the prescriptions of our approach have been made clear. In pratice, the only new experimental quantities that will enter in the theoretical formulae will be the $c$ asymmetry on $Z$ resonance and the overall Z width $\Gamma_5$. The various relevant expressions have all been given and computed in Ref.\cite{10}, where it has also been shown that the related experimental error would not produce any consequence in the theoretical formulae for unpolarized quantities that contain them as an input. We shall return on this point at the end of the paper. For the moment we write the final expression for the asymmetry introducing a separation of the new physics effects that will be useful for our next analysis. More precisely, we define systematically, for any model of new physics and final state $f$: \begin{equation} \tilde{\Delta}^{(lf)}\alpha(q^2) = \tilde{\Delta}\alpha(q^2) + \delta\tilde{\Delta}^{(lf)}\alpha(q^2) \end{equation} \begin{equation} R^{(lf)}(q^2) = R(q^2) + \delta R^{(lf)}(q^2) \end{equation} \begin{equation} V^{(lf)}_{\gamma Z}(q^2) = V(q^2) + \delta V^{(lf)}_{\gamma Z}(q^2) \end{equation} \begin{equation} V^{(lf)}_{Z\gamma}(q^2) = V(q^2) + \delta V^{(lf)}_{Z\gamma}(q^2) \end{equation} \noindent where the first bracket contains the "universal" (without index) effects, i.e. those that would be exactly the same for final leptons or quarks.\par Using the previous definitions, it is relatively easy to derive the expression of $A_{LR,5}$ for models of new physics that are of \underline{universal type}. Working in the usual spirit of only retaining the important contributions we would obtain the following formula \begin{equation} A^{(1)}_{LR,5}(q^2)= \bar A_{LR,5}(q^2)\{1+a_5 (q^2)[\tilde{\Delta}\alpha(q^2) +R(q^2)]+[b_5 (q^2)+c_5 (q^2)]V(q^2)\} \end{equation} \noindent where \begin{equation} \bar A_{LR,5} (q^2)= {C_{LR,5}(q^2)\over C_5(q^2)} \end{equation} \begin{equation} a_5 (q^2)=C^{\gamma Z}_{LR,5}-2C^{\gamma\gamma}_{5}-C^{\gamma Z}_{5} =-2C^{ZZ}_{LR,5}-C^{\gamma Z}_{LR,5}+ 2C^{ZZ}_{5}+C^{\gamma Z}_{5} \end{equation} \begin{equation} b_5 (q^2)+c_5 (q^2)=-4s_lc_l\{[p_{LR,5}C^{ZZ}_{LR,5} +p'_{LR,5}C^{\gamma Z}_{LR,5}] -[p_5 C^{ZZ}_{5}+p'_5 C^{\gamma Z}_{LR,5}]\} \end{equation} \noindent and \begin{equation} C_{LR,5}=N^{ZZ}_{LR,5}+N^{\gamma Z}_{LR,5} \end{equation} \begin{equation} N^{ZZ}_{LR,5}=({2\tilde{v}_l\over1+\tilde{v}^2_l}) {[{3\Gamma_l\over M_Z}] [{3\Gamma_5\over M_Z}]\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z} \end{equation} \begin{equation} N^{\gamma Z}_{LR,5}={2\alpha\over3(1+\tilde{v}^2_l)^{1/2}}[{3\Gamma_l\over M_Z}]^{1/2} \Sigma_5{ q^2(q^2-M^2_Z)\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z} \end{equation} \begin{equation} C^{ZZ}_{LR,5}= N^{ZZ}_{LR,5}/C_{LR,5} \ \ \ \ \ C^{\gamma Z}_{LR,5}=N^{\gamma Z}_{LR,5}/C_{LR,5} \ \ \ \ \ C_{5}=1+N^{ZZ}_{5}+N^{\gamma Z}_{5} \end{equation} \begin{equation} N^{ZZ}_{5}=(\frac{9}{33})[{3\Gamma_l/ M_Z\over\alpha}] [{3\Gamma_5/ M_Z\over\alpha}]{q^4\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z} \end{equation} \begin{equation} N^{\gamma Z}_{5}=(\frac{2}{11}){\tilde{v}_l\over(1+\tilde{v}^2_l)^{1/2}} {[3\Gamma_l/ M_Z]^{1/2}\over\alpha} \Sigma_5 {q^2(q^2-M^2_Z)\over(q^2-M^2_Z)^2+M^2_Z\Gamma^2_Z} \end{equation} \begin{equation} \Sigma_5= \sum_q{3|Q_q|\tilde{v}_q\over(1+\tilde{v}^2_q)^{1/2}}({3N_q\Gamma_q \over M_Z})^{1/2} \end{equation} \begin{equation} C^{\gamma\gamma}_{5}=1/C_{5} \ \ \ \ \ C^{ZZ}_{5}= N^{ZZ}_{5}/C_{5} \ \ \ \ \ C^{\gamma Z}_{5}=N^{\gamma Z}_{5}/C_{5} \end{equation} \begin{equation} p_5= {\tilde{v}_l\over1+\tilde{v}^2_l}+ \sum_q({\tilde{v}_q\over1+\tilde{v}^2_q}){|Q_q|\Gamma_q\over\Gamma_5} \ \ \ \ \ \ p'_5= {1\over \tilde{v}_l}+ p'_{LR,5} \end{equation} \begin{equation} p'_{LR,5}=\sum_q({3|Q_q|^2\over\Sigma_5(1+\tilde{v}^2_q)^{1/2}}) ({3N_q\Gamma_q\over M_Z})^{1/2} \ \ \ \ \ \ p_{LR,5}= {1\over\tilde{v}_l}+\sum_q({\tilde{v}_q\over1 +\tilde{v}^2_q}) {2|Q_q|\Gamma_q\over3\Gamma_5} \end{equation} The coefficient $\bar A_{LR,5}(q^2)$, in this particular notation, contains both the leading ("effective" Born) terms and the one-loop corrections $\delta \bar A^{SM}_{LR,5}$ of the pure SM. The latter ones will not be, in general, of universal type, since they involve vertices and boxes. Neglecting their numerical value for a first estimate of the leading term gives us the expected large $q^2$ value of the asymmetry, that is approximately $\bar A_{LR,5}(q^2)\simeq 0.50$. For what concerns the remaining coefficients, one easily sees that, once again, that of $[\tilde{\Delta}\alpha + R]$ is more than one order of magnitude smaller than that of $V$. The latter one, in turn, comes mostly from the \underline{universal} component of $V_{\gamma Z}$ reproducing the situation that we have already met in the two previous examples.\par This recurrent feature of "$V_{\gamma Z}$" dominance of the one loop effects of new physics survives, in the last considered asymmetry, even in the most general case of non universal type of effects, as one can see if one writes the full expression that generalizes eq.(34) to this case. This can be done in a straightforward way, and leads to the rather lengthy expression that we write here for completeness: \begin{eqnarray} && A^{(1)}_{LR,5}(q^2)= \bar A_{LR,5}(q^2)\{1+a_5 (q^2)[\tilde{\Delta}\alpha(q^2) +R(q^2)]+[b_5 (q^2)+c_5 (q^2)]V(q^2)\nonumber\\ &&+\sum_q\delta\tilde{\Delta}^{(lq)}\alpha(q^2) [(C^{\gamma Z}_{LR,5}(q^2)-C^{\gamma Z}_{5}(q^2)){3|Q_q|\tilde{v}_q \over\Sigma_5(1+\tilde{v}^2_q)^{1/2}} ({3N_q\Gamma_q\over M_Z})^{1/2} -C^{\gamma\gamma}_5(q^2)\frac{18}{11}|Q_q|^2]\nonumber\\ &&+\sum_q\delta R^{(lq)}(q^2)[2(C^{ZZ}_{5}(q^2)-C^{ZZ}_{LR,5} (q^2)){\Gamma_q\over \Gamma_5} +(C^{\gamma Z}_{5}(q^2)-C^{\gamma Z}_{LR,5}(q^2)){3|Q_q|\tilde{v}_q \over\Sigma_5(1+\tilde{v}^2_q)^{1/2}} ({3N_q\Gamma_q\over M_Z})^{1/2}]\nonumber\\ &&+\sum_q\delta V^{(lq)}_{\gamma Z}(q^2)[ ({8s_lc_l\tilde{v}_l\over1+\tilde{v}^2_l}) C^{ZZ}_{5}(q^2){\Gamma_q\over \Gamma_5}-{4s_lc_l\over \tilde{v}_l} (C^{ZZ}_{LR,5}(q^2)2{\Gamma_q\over \Gamma_5} -C^{\gamma Z}_{5}{3|Q_q|\tilde{v}_q \over\Sigma_5(1+\tilde{v}^2_q)^{1/2})} ({3N_q\Gamma_q \over M_Z})^{1/2})]\nonumber\\ &&+\sum_q\delta V^{(lq)}_{Z\gamma}(q^2)][ {|Q_q|4s_lc_l\over\tilde{v}_q}(C^{\gamma Z}_{5}(q^2) -C^{\gamma Z}_{LR,5}(q^2)) {3|Q_q|\tilde{v}_q \over\Sigma_5(1+\tilde{v}^2_q)^{1/2})} ({3N_q\Gamma_q \over M_Z})^{1/2}\nonumber\\ &&+({8s_lc_l|Q_q|\tilde{v}_q\over1+\tilde{v}^2_q}) (C^{ZZ}_{5}(q^2)-C^{ZZ}_{LR,5}(q^2)){\Gamma_q\over \Gamma_5}]\} \end{eqnarray} \noindent and, this time, the one-loop corrections contain both the SM and the new physics effects. Note that the $[\tilde{\Delta}\alpha(q^2) +R(q^2)]$ combination only appears for the universal term. Leaving aside a more quantitative discussion in this non-universal case, we only remark that, as we said previously, the weight of both the universal and the non-universal $V^{lf}_{\gamma Z}$ components remain essentially enhanced by the typical $1/\tilde{v}_l$ effect, that remains in conclusion the relevant feature of all the considered longitudinal polarization asymmetries.\par This characteristic feature should be compared now with those of other specific unpolarized observables. We have done this for the following relevant quantities, exploiting their theoretical expressions in our approach that can be found in refs.\cite{9},\cite{10}: I) $A_{FB,\mu}$, the muon forward-backward asymmetry. Here the size of the coefficient of the sum $[\tilde{\Delta}\alpha + R]$ that still appears as a unique block is approximately three times bigger than that of $V$. II) $\sigma_{\mu}$, the muon cross-section. Here the dominant effect is by far (one order of magnitude) concentrated in the correction $\tilde{\Delta}\alpha$ (that now is no more related to $R$ as in the previous asymmetries). III) $\sigma_5$, the five light quark cross section. For the case of \underline{universal} effects, the coefficients of all the three form factors $\tilde{\Delta}\alpha$, $R$ and $V$ are now roughly equal (this remains qualitatively true for general non universal effects). IV) $\sigma_b$, the $b\bar b$ cross section. Here, the leading coefficients of nearly equal size are those of $R$ and $V$.\par This short analysis shows that, indeed, longitudinal polarization asymmetries are much more sensitive to one specific one-loop effect $\simeq V_{\gamma Z}$ and therefore to all those models that contribute this quantity in a sensible way. One the contrary, unpolarized leptonic observables are a better place for looking at effects generated by either the combination $[\tilde{\Delta}\alpha + R]$ (e.g. $A_{FB,\mu}$) or by the separate quantity $\tilde{\Delta}\alpha$ (e.g. $\sigma_{\mu}$). In unpolarized hadronic quantities, the three form factors $\tilde{\Delta}\alpha$, $R$ and $V$ all appear with coefficients of similar size.\par We still have to discuss three specific points. The first one is that, as previously stressed, it is the combination $[\tilde{\Delta}\alpha + R]$ that appears systematically in ratios of cross sections. This can be understood from the general ($\gamma$, $Z$) structure if we write the general expression of any cross section in the following way \begin{equation} \sigma^{lf}_1\equiv c^{\gamma\gamma}_1(1+2\tilde{\Delta}^{lf}\alpha) +c^{ZZ}_1(1-2R^{lf})+c^{\gamma Z}_1(1+\Delta^{lf}\alpha-R^{lf}) +\ \ V~ terms \end{equation} \noindent defining $c_1\equiv c^{\gamma\gamma}_1+c^{ZZ}_1+c^{\gamma Z}_1$, one can write \begin{equation} \sigma^{lf}_1\equiv c_1[1+\tilde{\Delta}^{lf}\alpha -R^{lf}+{ c^{\gamma\gamma}_1 -c^{ZZ}_1\over c_1}(\tilde{\Delta}^{lf}\alpha)+R^{lf})] +\ \ V~terms \end{equation} Keeping only first order terms in $\tilde{\Delta}^{lf}\alpha$, $R^{lf}$ and $V$, the ratio of two such cross sections $\sigma^{lf}_1$ and $\sigma^{lf}_2$, is given by: \begin{equation} {\sigma^{lf}_1\over \sigma^{lf}_2} = {c_1\over c_2}[1+ ({ c^{\gamma\gamma}_1-c^{ZZ}_1\over c_1}-{ c^{\gamma\gamma}_2-c^{ZZ}_2\over c_2})(\tilde{\Delta}^{lf}\alpha +R^{lf})] +\ \ V~terms \end{equation} \noindent in which only the combination $[\tilde{\Delta}^{lf}\alpha +R^{lf}]$ appears. Note that this property in general disappears if one considers ratios of sums of different flavors $(\sum_f\sigma^{lf}_1)/(\sum_f\sigma^{lf}_2)$, as we have seen in the case of $A^{(1)}_{LR,5}$, eq.(48).\par The second point is the statement that, in order to exploit the properties of $A_{LR,f}$, the contribution of the model of new physics to $V$ must not be accidently depressed with respect to that of $[\tilde{\Delta}\alpha + R]$. Although we cannot prove this fact in general, we shall provide now in the next section a specific example of a model where this is actually the case, and for which the role of $A_{LR,f}$ will consequently be very useful.\par The final point is that of whether the bargain introduced in our approach by the replacement of $G_{\mu}$ with $Z$-peak quantities does not generate a dangerous theoretical input error (in the case of unpolarized observables, this was shown not to be the case for future $e^+e^-$ colliders at their realistically expected accuracy in refs.\cite{9},\cite{10}. Let us start with the leptonic asymmetry eq.(17). In our approach, its new theoretical expression at the "effective" Born level is the first member on the r.h.s. of eq.(17). This contains the $Z$ leptonic width $\Gamma_l$ and $A_{LR}$ measured on top of $Z$ resonance. With the available errors on these quantities, one computes a theoretical error in eq.(17) of approximately $0.0018$ which is mostly coming from $A_{LR}$. Assuming an (optimistic) experimental error on $A_{LR}$ at a $500~GeV$ NLC \cite{NLC} of $0.007$ (purely statistical) one sees that the induced theoretical error is completely negligible. This statement will also be made more accurate by future improvements on the measurement of $A_{LR}$ at SLD \cite{7},\cite{SLDLecce}.\par In the case of eq.(18), one easily sees that the major source of error in the expression of the "effective" Born terms comes from the quantity ${\tilde{v}_b\over \sqrt{1+\tilde{v}^2_b}}$. To compute this error, we have used the definition eq.(12), from which we obtain: \begin{equation} \delta \tilde{v}_b={(1+\tilde{v}^2_b)^2\over2(1-\tilde{v}^2_b)}\delta A_b \end{equation} \noindent Using the experimental LEP+SLD value \cite{7}, \cite{lastres} \begin{equation} A_b= 0.867\pm0.022 \end{equation} \noindent we derive $\delta\tilde{v}_b\simeq 0.04$. A standard calculation then gives for the theoretical input error: \begin{equation} \delta^{(th)} \bar A_{LR,b}(q^2)\simeq 0.02 \end{equation} Note that this numerical result is directly proportional to the experimental error on $A_b$, and will be correspondingly reduced by future improved measurements of this quantity. This final error should be compared to the expected experimental precision on $A_{LR,b}$. Although a detailed study does not exist yet to our knowledge, we can reasonably foresee a picture for $b\bar b$ detection similar to that found for previous LEP2 studies \cite{LEP2}, that would lead to an overall error of at least a few percent, sufficiently larger than our theoretical input error.\par To conclude, we have considered the case of $A_{LR,5}$. This case can be treated in a reasonably simple way since in the theoretical expression of the leading term $\bar A_{LR,5}(q^2)$ the only relevant theoretical uncertainty affects the "$\gamma Z$" component of the numerator (for the denominator, a previous discussion given in ref.\cite{10} has shown that the main error is coming from $\Gamma_h$, the $Z$ hadronic width measured on $Z$ peak, and is completely negligible i.e. much smaller than a relative one percent). The $\gamma Z$ component contains the $Z$ peak quantities $\Gamma_{u,d,s,c,b}$ and the related quantities $A_{u,d,s,c,b}$ defined by a generalization of eq.(12). In fact, no experimental information is available on the ($u,d,s$) variables. A reasonable attitude seems to us to be that of assuming a universality property, i.e. \begin{equation} \Gamma_u=\Gamma_c \ \ \ \ \ \Gamma_d=\Gamma_s=\Gamma_b(m_t=0) \end{equation} \noindent and to derive $\Gamma_b(m_t=0)$ from its knowns experimental value where the theoretical top quark contribution has been subtracted. Analogously, we shall assume that $A_u=A_c$ and $A_d=A_s=A_b(m_t=0)$ and for the latter quantity we have again subtracted the known (and relatively small) top quark contribution. With these assumptions, one easily sees that the major theoretical error is coming from that of $\tilde{v}_c$ and $\tilde{v}_b$ (the induced error by the widths is much smaller). Using the experimental SLD results for $A_{b,c}$ then leads to an error of $A_{LR,5}$: \begin{equation} \delta^{th} \bar A_{LR,5}(q^2)\simeq 0.02 \end{equation} This is not a very comfortable result, since one would expect an experimental error on $A_{LR,5}$ at NLC not far from the purely statistical one, that is around one percent. In order to reduce the theoretical error of our input to such values, an extra effort from SLD that reduces to the one percent level the error on $A_b$ and to the three percent level that on $A_c$ would be requested. Such a desirable goal seems to be reachable in future SLD measurements \cite{7}. In the rest of this paper, we shall illustrate as an application the consequences of having been able to reduce the overall error on $A_{LR,5}$ to the one percent level. This will be done immediately in the next Section 3.\par \section{A model with anomalous gauge couplings} To illustrate the previous considerations with a concrete example, we shall now consider the case of a model where anomalous gauge couplings (AGC) are present. To be more precise, we shall discuss the consequences of our approach for the study of a model proposed by Hagiwara et al \cite{Hag}, to whose paper we defer for a full discussion of various theoretical aspects. Briefly, the model assumes that physics below a scale $\Lambda$ of supposed order $1~TeV$ can be described by an "effective" Lagrangian obtained by adding to the conventional SM component an extra $SU(2)\times U(1)$ invariant, C and CP conserving, dimension six piece. The latter contains, a priori, eleven parameters of which four enter at the one loop level for production of two final \underline{massless} fermions from electron-positron annihilation. In the notation of ref.\cite{Hag} these are called $f_{DW}$, $f_{DB}$, $f_{BW}$ and $f_{\Phi,1}$. In a conventional treatment that does not use our $Z$-peak representation they would all contribute this process at one loop. The treatment of this model in our approach turns out to be particularly convenient. As it has already been shown in ref.\cite{9}, the number of effective parameters that appear in the subtracted form factors is reduced to two ($f_{DW}$ and $f_{DB}$) since $f_{BW}$ and $f_{\Phi,1}$ are fully reabsorbed in the used input parameters $\Gamma_l$ and $s^2_l(M^2_Z)$. Another welcome feature of this model is that its effects for massless fermions are of universal type, so that the \underline{same} two parameters will enter both leptonic and quark observables. This allows to determine informations on bounds on these parameters in a greatly simplified way, using several measurements of different experimental quantities. This was done in a very recent paper \cite{clean} where the bounds that would be obtained from negative searches both at LEP2 and at NLC without polarization were derived. In Fig.1 the results of that investigation are presented showing the region within which the two parameters ($f_{DW}$, $f_{DB}$) would be constrained by negative searches in the unpolarized case. Numerically, we would find in this case: \begin{equation} \Delta f_{DB}=\pm 0.16 \end{equation} \begin{equation} \Delta f_{DW}=\pm 0.025 \end{equation} In practice, the determination of such bounds in Fig.1 is mostly provided by two quantities i.e. the muon cross section and the five light hadron production cross section $\sigma_5$ (the forward-backward asymmetry $A_{FB,\mu}$ plays a negligible role because of a weaker sensitivity). Their expression in the considered model are provided in ref.\cite{clean} and are fixed by the (AGC) content of the three form factors $\tilde{\Delta}\alpha$, $R$ and $V$, that read respectively: \begin{equation} \tilde{\Delta}^{(AGC)}\alpha(q^2)= -q^2({2e^2\over\Lambda^2}) (f^r_{DW}+f^r_{DB}) \end{equation} \begin{equation} R^{(AGC)}(q^2)= (q^2-M^2_Z)({2e^2\over s^2_lc^2_l\Lambda^2}) (f^r_{DW}c^4_l+f^r_{DB}s^4_l) \end{equation} \begin{equation} V^{(AGC)}(q^2) = (q^2-M^2_Z)({2e^2\over s_lc_l\Lambda^2}) (f^r_{DW}c^2_l-f^r_{DB}s^2_l) \end{equation} We have now added to the previous unpolarized information that derivable from longitudinal polarization asymmetries. To avoid problems related to $b$ quark identification and to stick more rigorously to the massless quark configuration, we have only considered the leptonic and the full light hadronic asymmetry (where the weight of the $b$ contribution is sufficiently depressed). For the latter ones we have assumed, following our previous discussion (and an optimistic attitude), an experimental error $\delta A_{LR,l}=\pm 0.007$ and $\delta A_{LR,5}=\pm 0.01$. This is based on an integrated luminosity of $20~fb^{-1}$ leading at $\sqrt{q^2}=500~GeV$ to about $5\times10^{4}$ hadronic events and $1.7\times10^{4}$ (muon + tau events). To give a hint of how the "$V$ enhancement" mechanism works, we write the two corresponding theoretical expressions in the chosen configuration $\sqrt{q^2}=500~GeV$, that numerically read:\\ from $A_{LR,5}$ \begin{equation} |{32\pi\alpha(0)M^2_Z\over\Lambda^2}[-53.36f_{DW} + 14.43f_{DB}]|=|{\delta A_{LR,5}\over A_{LR,5}}|\gsim0.02 \end{equation} \noindent and from $A_{LR,l}$ \begin{equation} |{32\pi\alpha(0)M^2_Z\over\Lambda^2}[-342.65f_{DW} + 92.55f_{DB}]|= |{\delta A_{LR,l}\over A_{LR,l}}|\gsim0.1 \end{equation} \noindent (For $\Lambda=1~TeV$, the coefficient $32\pi\alpha(0)M^2_Z\over\Lambda^2$ is equal to 0.0061).\par In eqs.(62) and (63) the last numbers on the r.h.s. represent the visibility threshold for the effect. Note that both equations involve the same type of combination of $f_{DW}$ and $f_{DB}$ couplings. With the expected accuracies, eq.(63) due to $A_{LR,l}$ is slightly more stringent than eq.(62) due to $A_{LR,5}$. This is fortunate because of the uncertainty on the final accuracy that will be reachable on $A_{LR,5}$. In the following numerical analysis we shall combine quadratically the informations coming from these two constraints and this reduces somewhat the importance of $A_{LR,5}$. These expressions should be compared with those provided by the unpolarized observables. Taking for simplicity the two most sensitive quantities i.e. the muon and the hadron cross sections, the corresponding equations would be:\\ from $\sigma_{\mu}$ \begin{equation} |{32\pi\alpha(0)M^2_Z\over\Lambda^2}[-22.02f_{DW} -13.07f_{DB}]|=|{\delta\sigma_{\mu}\over \sigma_{\mu}}|\gsim0.01 \end{equation} \noindent and from $\sigma_{5}$ \begin{equation} |{32\pi\alpha(0)M^2_Z\over\Lambda^2}[-49.53f_{DW} -5.45f_{DB}]|=|{\delta\sigma_{5}\over \sigma_{5}}|\gsim0.005 \end{equation} Comparing eqs.(64)(65) with eqs.(62),(63) one actually sees that the combination of $f_{DW}$, $f_{DB}$ that appear in the two sets are almost orthogonal. This corresponds indeed, as we discussed in Section 2, to the fact that different form factors are selected in the two cases.\par From a practical point of view, the additional improvements for future negative bounds derivable from the addition of the two extra asymmetries is shown in Fig.1 . As one sees, the final limits would be: \begin{equation} \Delta f_{DB}=\pm 0.08 \end{equation} \begin{equation} \Delta f_{DW}=\pm 0.014 \end{equation} In other words, the additional constraint provided by longitudinal polarization would lead, in this example, to an improvement in the bounds equal to, roughly, a factor of two. \section{Conclusions} We have shown in this paper that longitudinal polarization asymmetries of electron-positron annihilation into pairs of light fermion-antifermion at energies larger than $M_Z$ exhibit interesting theoretical features that might be useful for detection of a certain type of virtual effects of new physics at one loop, and that are due to a special enhancement of the subtracted $V$ form factor. This feature is analogous to that found on top of $Z$ resonance, showing that $A_{LR}$ continues to be a relevant observable even far from that privilegded kinematical configuration.\par We have presented in this paper only one concrete example of how this enhancement mechanism works, for the special case of one model of universal type. Other similar cases could be examined. For instance, general models of technicolour type (already qualitatively considered in ref.\cite{9}) would probably benefit from a more detailed numerical calculation. This will be done in a separate work. Also, the more complicated case of models of non universal type would deserve consideration. An interesting case would be that of general supersymmetric models. Here the virtual effects are usually depressed on $Z$ resonance. Away from $Z$ resonance, there might be, though, unconventional effects of non universal type (we have in mind e.g. boxes, that are kinematically depressed on $Z$ peak but resuscitate when $(q^2-M^2_Z)$ is sufficiently large). These would enter in our subtracted form factors at large energies since they would \underline{not} be reabsorbed, by definition, in the $Z$ peak observables that are the new inputs of our procedure. The study of this possibility is by now in progress. \vspace{0.5cm} \leftline{\Large \bf Acknowledgments} One of us (C.V.) wishes to thank the Department of Physique Math\'ematique et Th\'eorique of Montpellier, where this work was written, for the warm and friendly hospitality. \newpage

proofpile-arXiv_065-531

\section{Introduction} \markright{Gauss-Bonnet type identity in Weyl-Cartan space} In the modern gravitational theory quadratic Lagrangians are used widly that is stimulated by the gauge treatment of gravitation and the renormalization problems in quantum gravity.$^{1}$ In this connection the Gauss-Bonnet type identity becomes the object of a considerable amount of attention. The generalization of the Gauss-Bonnet formula to a 4-dimensional Riemann space $V_{4}$ was performed by Bach $^{2}$ and Lanczos $^{3}$ and on the basis of the variational method by Ray.$^{4}$ The Bach-Lanczos identity in Riemann spaces imlpies the one-loop renormalizability of pure gravitation.$^{5}$ The generalization of the Bach-Lanczos identity to a Riemann-Cartan space $U_{4}$ was performed in. $^{6-8}$ We shall obtain the Gauss-Bonnet type identity in a Weyl-Cartan space $Y_{4}$ that can be essential for the dilatonic gravitational theory with quadratic Lagrangians. On the preliminary version of our results see Ref. 9. \section{Preliminaries to the variational procedure} \markright{Gauss-Bonnet type identity in Weyl-Cartan space} We shall consider a Weyl-Cartan space $Y_{4}$ that is a connected 4-di\-men\-si\-o\-nal oriented differentiable manifold ${\cal M}$ equipped with a linear connection $\Gamma$ and a metric $g$ (with the Lorenzian signature) which obey the constraints, \begin{equation} Q_{\lambda}\!^{\alpha\beta} = \frac{1}{4}Q_{\lambda}g^{\alpha\beta}\; , \quad Q_{\lambda}\!^{\alpha\beta}:= \nabla_{\lambda} g^{\alpha\beta}\; , \quad Q_{\lambda}:= Q_{\lambda}\!^{\alpha\beta}g_{\alpha\beta}\;. \label{eq:1a} \end{equation} The tensor $Q_{\lambda}\!^{\alpha\beta}$ is a nonmetricity tensor.$^{10,11}$ \par We shall use a holonomic local vector frame $\vec{e}_{\mu}=\vec{\partial} _{\mu}$ ($\mu=1,2,3,4$) with $\Gamma_{\sigma\rho }\!^{\lambda}$ as a connection coefficients. A space $Y_{4}$ contains a nonvanishing torsion tensor, $T_{\sigma\rho}\!^{\lambda}:= 2\Gamma_{[\sigma\rho ]}\!^{\lambda}$, in general. A curvature tensor of $Y_{4}$ and its various contractions read, $R_{\mu}\!^{\nu} = R_{\sigma\mu}\!^{\nu\sigma}$, $\tilde{R}_{\mu}\!^{\nu} = R_{\mu\sigma}\!^{\sigma\nu}$, $R = R_{\sigma}\!^{\sigma}$ and \begin{equation} R_{\alpha\beta\sigma}\!^{\lambda} = 2\partial_{[\alpha}\Gamma_{\beta]\sigma} \!^{\lambda} + 2\Gamma_{[\alpha\vert\rho\vert}\!^{\lambda}\Gamma_ {\beta]\sigma}\!^{\rho}\; . \label{eq:2} \end{equation} \par Let us consider the Lagrangian density, \begin{equation} {\cal L}_{0} = \sqrt{-g}L_{0}\; , \quad L_{0} = R^{2} - (R_{\alpha\beta} + \tilde{R}_{\alpha\beta})(R^{\beta\alpha} + \tilde{R}^{\beta\alpha}) + R_{\alpha\beta\mu\nu}R^{\mu\nu\alpha\beta}\; . \label{eq:1} \end{equation} The variation of (\ref{eq:1}) with respect to the metric $g^{\sigma\rho}$ and the connection $\Gamma_{\lambda \nu}\!^{\sigma }$ reads, \begin{equation} \delta{\cal L}_{0} = -\frac{1}{2}\sqrt{-g}H_{\sigma\rho}\delta g^{\sigma \rho} - \sqrt{-g}H_{\sigma}\!^{\nu\lambda}\delta\Gamma_{\lambda\nu}\!^ {\sigma} + \mbox{total divergence} \; , \label{eq:3} \end{equation} where \begin{eqnarray} \sqrt{-g}H_{\sigma\rho}: = -2\left[ \frac{\delta{\cal L}_{0}} {\delta g^{\sigma\rho}}\right]_{\Gamma=const}\;,\quad \sqrt{-g}H_{\sigma} \!^{\nu\lambda}:= -\left[\frac{\delta{\cal L}_{0}}{\delta\Gamma_{\lambda\nu} \!^{\sigma}}\right]_{g^{\sigma\rho}=const}\;, \label{eq:4} \\ H_{\sigma\rho} = g_{\sigma\rho}L_{0} - 4R_{\alpha\beta(\sigma}\!^{\tau} R_{\rho )\tau}\!^{\alpha\beta} - 4R_{\tau (\sigma\rho )}\!^{\kappa}(R_{\kappa} \!^{\tau} + \tilde{R}_{\kappa}\!^{\tau}) \nonumber \\ - 4 R_{\tau (\sigma}(R_{\rho )}\!^{\tau} + \tilde{R}_{\rho )}\!^{\tau}) - 4RR_{(\sigma\rho )}\; , \label{eq:5}\\ \sqrt{-g}H_{\sigma}\!^{\nu\lambda}= 4\nabla_{\mu} \{ \sqrt{-g} [R_{\sigma}\!^ {\nu [\lambda\mu ]} + (R_{\sigma }\!^{[\lambda} +\tilde{R}_{\sigma }\!^ {[\lambda})g^{\mu ]\nu} \nonumber \\ - (R^{\nu [\lambda} + \tilde{R}^{\nu [\lambda})\delta_{\sigma}\! ^{\mu]} - R\delta_{\sigma}^{[\lambda}g^{\mu ]\nu} ]\} \nonumber\\ + 2\sqrt{-g} [ R_{\sigma}\!^{\nu\alpha\beta} + (R_{\sigma}\!^{\alpha} + \tilde{R}_{\sigma}\!^{\alpha})g^{\beta\nu} - (R^{\nu\alpha} + \tilde{R}^ {\nu\alpha})\delta_{\sigma}^{\beta} + Rg^{\nu\alpha}\delta_{\sigma}^{\beta} ] M_{\alpha\beta}\!^{\lambda}\; . \label {eq:6} \end{eqnarray} Here the modified torsion tensor is introduced, $M_{\sigma\rho}\!^{\lambda}:= T_{\sigma\rho}\!^{\lambda} + 2\delta_{[\sigma}^{\lambda}T_{\rho]}$, where $T_{\rho} := T_{\rho\tau}\!^{\tau}$ is the torsion trace. \section{The Gauss-Bonnet type Teorem \newline in Weyl-Cartan space} \markright{Gauss-Bonnet type identity in Weyl-Cartan space} The following Gauss-Bonnet type Teorem generalized to $Y_{4}$ is valid. \par {\em Theorem}: The integral quantity, \begin{equation} \int_{{\cal M}} \, \sqrt{-g} [R^{2} - (R_{\alpha\beta} + \tilde{R}_{\alpha \beta})(R^{\beta\alpha} + \tilde{R}^{\beta\alpha}) + R_{\alpha\beta\mu\nu} R^{\mu\nu\alpha\beta}] \, d^{4}x \; , \label{eq:18} \end{equation} over the oriented 4-dimensional manifold ${\cal M}$ without boundary equipped with the Weyl-Cartan differential-geometric structure does not depend on the choice of a metric and a connection of the manifold and is a topological invariant. \par {\em Proof}. The main idea of the proof consists in the demonstration that the variation of the integrand of (\ref{eq:18}) with respect to a metric and a connection in a Weyl-Cartan space $Y_{4}$ is equal identically to a total divergence. In $Y_{4}$ the metric and the connection are not independent because of the constraints (\ref{eq:1a}). Therefore one can vary the modified integrand expression, \begin{equation} {\cal L} = {\cal L}_{0} + \frac{1}{2} \sqrt{-g}H_{\alpha\beta}\!^{\lambda} \left (Q_{\lambda}\!^{\alpha\beta} - \frac{1}{4}Q_{\lambda}g^{\alpha\beta} \right )\; , \label{eq:9} \end{equation} which in $Y_{4}$ coinsides with the integrand of (\ref{eq:18}). \par The variation of (\ref{eq:9}) has the form, \begin{eqnarray} &\delta{\cal L} = -\frac{1}{2}\left (\sqrt{-g}H_{\sigma}\!^{\nu} + \stackrel{*}{\nabla}_{\lambda}\left [\sqrt{-g}(H_{\sigma}\!^{\nu\lambda} - \frac{1}{4}\delta_{\sigma}^{\nu} H_{\tau}\!^{\tau\lambda})\right ] \right ) g_{\nu\rho}\delta g^{\sigma\rho}\nonumber \\ &+\frac{1}{2}\sqrt{-g} \left ( H_{\sigma}\!^{\nu\lambda}(Q_{\lambda\nu\rho}- \frac{1}{4}Q_{\lambda}g_{\nu\rho}) - \frac{1}{2}g_{\sigma\rho} H_{\alpha\beta}\!^{\lambda}(Q_{\lambda}\!^{\alpha\beta}-\frac{1}{4} Q_{\lambda}g^{\alpha\beta}) \right )\delta g^{\sigma\rho} \nonumber \\ & - \sqrt{-g}\left (H_{[\sigma\rho ]}\!^{\lambda} + \frac{1}{4}g_{\sigma\rho} H_{\tau}\!^{\tau\lambda}\right )g^{\rho\nu}\delta\Gamma_{\lambda\nu}\!^ {\sigma} \nonumber \\ & + \frac{1}{2} \sqrt{-g} \left (Q_{\lambda}\!^{\alpha\beta} -\frac{1}{4} Q_{\lambda}g^{\alpha\beta}\right )\delta H_{\alpha\beta}\!^{\lambda} + \mbox{total divergence} \; , \label{eq:90} \end{eqnarray} where $\stackrel{*}{\nabla}_{\lambda} = \nabla_{\lambda} + T_{\lambda}$. If the constraints (\ref{eq:1a}) are taken into account, then the hypothesis of the Theorem is the consequence of the Lemma. \par {\em Lemma}: In a Weyl-Cartan space $Y_{4}$ the following identities are valid, \begin{equation} (a)\,\;\sqrt{-g}H_{\sigma}\!^{\rho} + \stackrel{*}{\nabla}_{\lambda} (\sqrt{-g}H_{\sigma}\!^{\rho\lambda})= 0\;,\quad (b)\,\;H_{[\sigma\rho ]} \!^{\lambda}= 0\;,\quad (c)\,\;H_{\tau}\!^{\tau\lambda}=0\;. \label{eq:13} \end{equation} \par {\em Proof}. The statement (c) follows immediately from (\ref{eq:6}). Using (\ref{eq:6}) and the Bianchi identities in $Y_{4}$ one gets, \begin{eqnarray} H_{[\sigma\nu ]}\!^{\lambda} = [ 8R_{\tau [\sigma}\!^{[\lambda\alpha ]} \delta_{\nu ]}^{\beta} + 4R_{\tau [\sigma}\!^{\alpha\beta}\delta_{\nu]}^{\lambda} + 4(R_{\tau}\!^{\alpha} + \tilde{R}_{\tau}\!^{\alpha})\delta_{[\sigma}^{\beta} \delta_{\nu ]}^{\lambda} \nonumber \\ + 2(R_{\tau}\!^{\lambda} + \tilde{R}_{\tau}\! ^{\lambda})\delta_{\sigma}^{\alpha}\delta_{\nu}^{\beta}] T_{\alpha \beta}\!^{\tau} \nonumber \\ + [ 2R_{\sigma\nu}\!^{\alpha\beta} + 4(R_{[\sigma}\! ^{\alpha} + \tilde{R}_{[\sigma}\!^{\alpha})\delta_{\nu ]}^{\beta} - 2R\delta _{\sigma}^{\alpha}\delta_{\nu}^{\beta}] M_{\alpha\beta}\!^{\lambda}\; . \label{eq:14} \end{eqnarray} Let us consider the expression, \begin{equation} B_{\sigma\nu}\!^{\lambda} = \frac{1}{2}\eta_{\alpha\beta\kappa\omega} \eta^{\mu\tau\kappa\epsilon}\eta^{\phi\rho\omega\lambda} \eta_{\delta\sigma\epsilon\nu}R_{\phi\rho}\!^{\alpha\beta} T_{\mu\tau}\!^{\delta}\; , \label{eq:15} \end{equation} where $\eta_{\alpha\beta\gamma\delta}=\sqrt{-g}e_{[\alpha\beta\gamma\delta]}$ is the Levi-Civita tensor ($e_{[1234]} = -1$), and calculate this expression in two ways: the first way consists in combining the first factor with the second one and the third factor with the forth one, while the second way consists in combining the first factor with the third one and the second factor with the forth one. By equating the two results one gets that the statement (b) of the Lemma is valid. \par With the help of the Bianchi identities one can find, \begin{eqnarray} \sqrt{-g}H_{\sigma}\!^{\nu} + \stackrel{*}{\nabla}_{\lambda}(\sqrt{-g}H_ {\sigma}\!^{\nu\lambda}) = \sqrt{-g} [\delta_{\sigma}^{\nu}L_{0} + 4R_{\sigma\tau}\!^{\alpha\beta}R_{\alpha\beta}\!^{[\tau\nu ]} \nonumber \\ + 4R_{\sigma}\!^{\kappa [\tau\nu ]}(R_{\tau\kappa} + \tilde{R}_{\tau \kappa}) + 2(R_{\sigma}\!^{\tau} +\tilde{R}_{\sigma}\!^{\tau}) (R_{\tau}\!^{\nu} + \tilde{R}_{\tau}\!^{\nu}) -2R(R_{\sigma}\!^{\nu} + \tilde{R}_{\sigma}\!^{\nu})]\; . \label{eq:16} \end{eqnarray} Let us consider the expression, \begin{equation} B_{\sigma\rho} = \frac{1}{4}\eta_{\delta\tau\phi}\!^{\psi}\eta_{\mu\nu}\!^ {\phi\omega}\eta_{\alpha\beta\psi\rho}\eta_{\kappa\epsilon\omega\sigma} R^{\alpha\beta\delta\tau}R^{\mu\nu\kappa\epsilon}\; , \label{eq:17} \end{equation} and calculate it in two ways, as before. After equating the two results one can be convinced that the statement (a) of the Lemma is valid. The proof of the Lemma is finished. \par Using the Lemma one can see that in $Y_{4}$ the variation of (\ref{eq:90}) and therefore the variation of (\ref{eq:18}) is equal to a total divergence. Q.E.D. \vskip 0.6cm \par {\large\bf References} \vskip 0.4cm \begin{description} \item 1. K.S. Stelle, {\em Phys. Rev.} {\bf D16}, 953 (1977). \item 2. R. Bach, {\em Math. Z.} {\bf 9}, 110 (1921). \item 3. C. Lanczos, {\em Ann. Math.(N.Y.)} {\bf 39}, 842 (1938). \item 4. J.R. Ray, {\em J. Math. Phys.} {\bf 19}, 100 (1978). \item 5. G. 't Hooft and M. Veltman, {\em Ann. Inst. H. Poincar\'{e}} {\bf 20}, 69 (1974). \item 6. V.N. Tunjak, {\em Izvestija vyssh. uch. zaved. (Fizika)} N9, 74 (1979) [in Russ.]. \item 7. H.T. Nieh, {\em J. Math. Phys.} {\bf 21}, 1439 (1980). \item 8. K. Hayashi and T. Shirafuji, {\em Prog. Theor. Phys.} {\bf 65}, 525 (1981). \item 9. O.V. Babourova, S.B. Levina and B.N. Frolov, in {\em Teoreticheskie i eksperimental\'{}nye problemy gravitacii} (Abstracts of Contr. Pap. 9 Russ. Grav. Conf., Novgorod) (Moscow, 1996), Part 1, p. 43 (in Russian). \item 10. J.A. Schouten, {\em Ricci-Calculus} (Springer, Berlin, 1954). \item 11. F.W. Hehl, J.D. McCrea, E.W. Mielke and Yu. Ne'eman, {\em Phys. Reports} {\bf 258}, 1 (1995). \end{description} \end{document}

proofpile-arXiv_065-532

\section{Introduction} The set of ideas enclosed in the Renormalization Group (RG, hereafter) has led to a variety of developments in many fields, as this Conference has made apparent. From a particle theorist point of view they englobe a bunch of ideas form which we may understand what a quantum field theory is. Moreover, it provides a framework for nonperturbative calculations. In recent years, there has been an intensive development of the field mainly for scalar theories.\cite{yuri} It can probably be said that we thoroughly understand all the subtleties the RG reserves for us in this case. Nevertheless what would ultimately justify the whole approach, as applied for Particle Physics, is the construction of Lorentz and gauge invariant nonperturbative equations, manageable for reliable approximations. The hope of RG practitioners is that we are really not that far from there.\footnote{We have to mention that although the search for nonperturbative gauge invariant equations is still open, there are a variety of results for perturbative definitions of a gauge invariant theory based on these grounds.\cite{marisa}} Particle physicists may imagine themselves, thus, applying in the nearby future the powerfulness of the approach to atack long-standing nonperturbative problems for, say, quantum gluodynamics. And the next step would probably be the introduction of matter to have the full physical theory. At this point it will for sure be helpful to have already understood the peculiarities associated to fermions on their own, both conceptual and technical. This is the reason we believe that we should try, as it has been done for bosons, to master as deeply as we can fermionic theories, even with no additional fields. We do not want to suggest that the features of fermionic equations have to be significantly different from the ones for bosons. On the contrary, we believe that they have to be ultimately a direct translation of ideas from one subject to the other. What we do mean is that these features may not be noticeable in any obvious way. On the other hand, new technicalities may also appear. And before embarking ourselves in a more ambitious project these peculiarities must be previously worked out and clearly understood. We thus want to present some recent work in that direction.\cite{ours} Namely, the study of a two dimensional sample model (the so-called Gross-Neveu model$\,$\cite{gn}) which we will define below. This model is sufficiently simple in order to be able to carry out the algebra as far as we need but that it still captures the essentials of the approach for the Grassman case. Let us briefly review the main ideas involved in bosonic theories as they are studied by Polchinski.\cite{polchinski} We first have to choose a regulator adequated for our purposes. It is done by simply modifying the propagator \( P(p) \) \begin{equation} P(p)=\frac{1}{p^2} \end{equation} to \begin{equation} P_{\Lambda}(p)=\frac{K(p^2/\Lambda^{2})}{p^2} \end{equation} where \( \Lambda \) is a momentum-space cutoff and \( K(x) \) a regulating function which decays sufficiently rapid to zero when \( x\rightarrow\infty \). With this kind of regulator, a quick (and somehow sloppy) argument that leads to an appropriate RG equation is to identify all the \( \Lambda \)-dependences in a partly integrated action by signaling all possible occurrences of the propagator and multiplying by the \( \Lambda \)-derivative of it. In this manner we immediately obtain \begin{equation} \label{RGeq:bos} -\Lambda\frac{d}{d\Lambda}S_{int} \equiv\dot{S}_{int} =\frac{1}{2}\frac{\delta S_{int}}{\delta \phi}\cdot\dot{P}_{\Lambda} \cdot\frac{\delta S_{int}}{\delta \phi} -\frac{1}{2}\mbox{tr} \left( \dot{P}_{\Lambda}\cdot\frac{\delta^{2}S_{int}}{\delta \phi\delta\phi} \right) \end{equation} with \begin{equation} S_{int}=S-\frac{1}{2}\phi\cdot P_{\Lambda}^{-1}\cdot\phi \end{equation} and $S$ the full action. The first term takes into account tree-type propagators and the second one loop-type propagators. We are using a compact notation regarding the propagator as a matrix with a dot standing for matrix multiplication. The equation for a pure fermionic theory can be written in a similar form. From the propagator \begin{equation} P_{\Lambda}=i\not\!{p}\,\frac{K(p^{2}/\Lambda^{2})}{p^{2}} \end{equation} we obtain the RG equation \begin{equation} \dot{S}_{int} =\frac{\delta S_{int}}{\delta \psi}\cdot\dot{P}_{\Lambda}\cdot \frac{\delta S_{int}}{\delta\bar{\psi}} -\mbox{tr} \left( \frac{\delta}{\delta\psi}\cdot\dot{P}_{\Lambda}\cdot \frac{\delta S_{int}}{\delta\bar{\psi}} \right) \end{equation} Returning to \( S \) and expressing the resultant equation in dimensionless variables it is obtained \begin{eqnarray} \label{RGeq:fer} \dot{S} &=&2K'(p^{2})\frac{\delta S}{\delta\psi}\cdot i\not\!{p} \cdot\frac{\delta S}{\delta\bar{\psi}} -\mbox{tr} \left( 2K'(p^{2})\frac{\delta}{\delta\psi}\cdot i\not\!{p}\, \frac{\delta S}{\delta\bar{\psi}} \right)\nonumber \\&-&2p^{2}\frac{K'(p^{2})}{K(p^{2})} \left( \bar{\psi}\cdot\frac{\delta S}{\delta \bar{\psi}} +\psi\frac{\delta S}{\delta \psi} \right) \\&+&dS\nonumber \\&+&\frac{1-d+\eta(t)}{2} \left(\bar{\psi}\frac{\delta S}{\delta\bar{\psi}} +\psi\frac{\delta S}{\delta\psi} \right) -\left( \bar{\psi}\cdot p^{\mu}\frac{\partial'}{\partial p^{\mu}} \frac{\delta S}{\delta\bar{\psi}} +\psi\cdot p^{\mu}\frac{\partial'}{\partial p^{\mu}} \frac{\delta S}{\delta \psi} \right)\nonumber \end{eqnarray} where we work on a $d$-dimensional Euclidean space; \( \eta \) is the anomalous dimension (needed to obtain a physically interesting fixed point); \( t\equiv-\ln \Lambda \); and the prime in \( \frac{\partial'}{\partial p^{\mu}} \) means that the derivative does not act on the momentum conservation delta functions and thus only serves to count powers of momenta. Note the first difference between bosons and fermions. Due to the different structure of the propagators, the fermionic equation presents an explicit factor of $p$ in the first two terms of the right hand side of the RG equation~\ref{RGeq:fer} while this is not the case for bosons (Eq.~\ref{RGeq:bos}). This may just look like a technical remark without any relevance. However it turns out that the seemingly most powerful approximations to these equations are based on the so-called derivative expansion\cite{tim} whose first order term is obtained by restricting the action to be a kinetic term plus a general potential term with no derivatives. In a fermionic theory this approximation will not be feasible, because we will be left only with a fairly simple linear equation. The derivative expansion should nevertheless be applicable, but it would lead to more complicated structures even at first non-trivial order. \section{The model} Let us now apply the RG equation~\ref{RGeq:fer} to the so-called chiral Gross-Neveu model.\cite{gn} As any other field theory, it is best defined through its symmetries. We will consider thus an Euclidean invariant $N$-flavoured model, with an \( U(N)\times U(N) \) internal symmetry group. It is also chosen to obey the discrete symmetries of parity, charge conjugation and reflection hermiticity.\cite{zinn-justin} When one imposes these restictions and further use Fierz reorderings, it is easily sown that there appear only three basic structures, \begin{eqnarray} V_{12}^{j}&\equiv& \bar{\psi}^{a}(p_{1})\gamma^{j}\psi^{a}(p_{2})\nonumber\\ S_{12}S_{34}-P_{12}P_{34}&\equiv& \bar{\psi}^{a}(p_1)\psi^{a}(p_2)\bar{\psi}^{b}(p_3)\psi^{b}(p_4) \nonumber\\ &-& \bar{\psi}^{a}(p_1)\gamma_{s}\psi^{a}(p_2) \bar{\psi}^{b}(p_3)\gamma_{s}\psi^{b}(p_4)\\ S_{12}P_{34}-P_{12}S_{34}&\equiv& \bar{\psi}^{a}(p_1)\psi^{a}(p_2) \bar{\psi}^{b}(p_3)\gamma_{s}\psi^{b}(p_4) \nonumber\\ &-& \bar{\psi}^{a}(p_1)\gamma_{s}\psi^{a}(p_2) \bar{\psi}^{b}(p_3)\psi^{b}(p_4)\nonumber \end{eqnarray} which have to be combined in an arbitrary way with combinations of momenta. The next step is to define a reasonable approximation to handle the above functional-derivative equation. We would like to choose one that closely resembles the bosonic derivative approximation. Nevertheless, due to the number of different structures it is not that easy to parametrize the general action up to, say, two derivatives while maintaining arbitrary the number of fields. Moreover, we should keep in mind that the allowed action is, as long as we are working with a finite number of different species, composed by a finite number of operators:\ the Grassman character of our variables constraints the number of fields allowed at one point of space. We have already commented that derivative terms should also be included. In fact, this is an important point because in \( d=2 \) the anomalous dimension \( \eta \) usually plays an important role: we would probably be too naive if we try to obtain numbers without letting it to be nonzero. In fact two derivatives may seem to do the job. However, once one goes through the calculations, it turns out to be quite clear that \( \eta =0 \) is the only consistent value. This implies that we need at least three derivatives. The maximum number of fields was chosen to be six. This seems a number both sufficiently low in order to keep the action relatively simple and sufficiently high to let non-trivial results appear. The action thus obtained has the usual kinetic term; one term with three derivatives and only two fields; two derivative-free four-fields operators \begin{equation} g_{1}(S_{12}S_{34}-P_{12}P_{34})\ ,\ \ \ \ \ g_{2}V_{12}^{j}V_{34}^{j} \end{equation} with coupling constants \( g_{1} \) and \( g_{2} \); eleven operators with also four fields but two derivatives; and ninety-two six-fermions operators, five of them with only one derivative and the rest with three derivatives. After some algebra we can now obtain the set of beta functions. The fixed points are the solutions for these functions to vanish. They are a set of 106 non-linear algebraic equations. Up to this point, the function \( K(p^{2}) \) can be mainted arbitrary, thus keeping some freedom of chosing a scheme. The fixed point solutions in our approximation will in general depend on two parameters which serve as a scheme parametrization. In principle, this should not worry us, because it is well known that the actual expression of the fixed point action has no intrinsic physical meaning. For Particle Physics it is specially interesting the value of the relevant directions from the fixed points. That is, we linearize the RG transformations, \begin{equation} \dot{g}_{i}={\cal R}_{i}(g_{j}) \end{equation} to \begin{equation} \dot{g}_{i}={\cal R}_{ij}\cdot\delta g_{j}\ ,\ \ \ \ \ {\cal R}_{ij}\equiv \left.\frac{\partial{\cal R}_{i}}{\partial g_{j}}\right|_{g^{0}} \end{equation} where \( g^{0} \) is the fixed-point solution and \( \delta g_{j} \) are the deviations from it. The number of positive eigenvalues of the matrix \( {\cal R}_{ij} \) coincide with the number of possible parameters we can fine-tune in the corresponding cutoff-free theory$\,$\cite{wilson} and the actual value of these eigenvalues gives the speed of departure from the fixed point. These eigenvalues are directly related to the so-called critical exponents in the terminology of second-order phase transitions. They are universal and, therefore, they should be free from schemes dependences. In our approximation, however, this is not so, as often happens with truncations. The scheme ambiguities are solved by a translation of the principle of minimal sensitivity used in perturbative calculations.\cite{stevenson} \section{Results} We now sketch the main results. The equations simplify enormously when \( N\rightarrow\infty. \) Two fixed points can be clearly identified. One of them with vanishing anomalous dimension (it is of order \( N^{-1} \)) and with the most relevant eigenvalue \( \sqrt{17} - 3 \) in this approximation. Moreover the coupling constant \( g_{2} \) which corresponds to \( U(1) \) Thirring-like excitations becomes free (we have, in fact, a line of fixed points) and \( g_{1} \) is also of order \( N^{-1} \). All these features but the anomalous dimension remind the fixed point solution found by Dashen and Frishman.\cite{df} The other solution, which corresponds to a different definition of the large $N$ limit (different assumed $N$ dependences of the coupling constants) has a non-vanishing anomalous dimension. It is scheme dependent with a range of variation of 1.11--1.14 for most of the schemes. The most relevant eigenvalue is also scheme-dependent with a range of 2.1--2.3 and, unlike the previous case, there are no free parameters. Before going on we must comment on a quite unpleasant feature of this kind of approximations. By now it is generally believe that any approximation based on truncations leads to a system of fixed-point equations with many spurious solutions.\cite{spur} It seems that a pure derivative expansion (that is, one with a truncation in the number of derivatives but without any further truncation in the number of fields) cures all this kind of problems. We work with a truncated action and thus we expect on general grounds that this unwanted peculiarity appears and, actually, it does. The solution of the puzzle is not always simple. One usually tries to discriminate among solutions by checking the stability of the obtained ones either going one step beyond in the approximation or else tuning some parameters. In our case we are lucky to have nitid results in the large $N$ limit. Therefore, we take as reliable solutions only those whose limit when \( N\rightarrow\infty \) concides with one of the solutions found above. This procedure will probably not be available in all cases and one should wonder if there is any systematic procedure to deal with the problem without relying on technical details of the studied model. Of course, one can always try to perform a true derivative expansion instead of mutilating each term as we have done. It should eliminate at once the spurious results. In fact we have a special case, which we will refer later on that suggests that this is true. Nevertheless the expansion proposed is not that simple, specially for $N$ moderately large. Moreover, it seems to be difficult to deal with different values of $N$ simultaneously and still preserving each term in the expansion without truncating it at some arbitrary point. One solution for finite $N$ matches the first one discussed above, with the most relevant eigenvalue smoothly decreasing to \( \sqrt{17} - 3 \) and with \( N\cdot\eta \) increasing with N to 4.87\ldots\@\ Unlike the strict large $N$ limit, we do not find, nevertheless, a line of fixed points but an isolated one. We blame this feature to the crudeness of the approximation. The solution that matches the second one above has a much more conspicuous behaviour. In fact it is valid only for \( N>142.8 \). At this value it matches another branch of solutions, which exists even when \( N\rightarrow\infty \) although the couplings do not scale with $N$ as integer powers but as noninteger ones. Both the anomalous dimension and the most relevant eigenvalue present strong scheme dependences, quite difficult to disentangle. Finally, we can consider separately the \( N=1 \) case. It is worth going through it because it is a simple case where we can treat the equations in a purely derivative approximation: due to Fermi statistics, we cannot have more than six fields if we consider terms up to three derivatives. Our action is thus exact in this sense. Also, we have to work out further relations imposed by Fierz reorderings, not present for \( N\not=1 \). Our action is, therefore, even shorter than that. The results are nevertheless very messy and, probably, not reliable. When considering terms up to two derivatives, we find a line of fixed points (as it is expected in the Thirring model) with \( \eta=0 \) as stated previously. But once we go to three derivatives, \( \eta\not=0 \) but, unexpectedly, the line of fixed points disappears and we obtain only an isolated one. Nevertheless one piece of nice news comes out: the spurious solutions disappear, as expected. A final comment is in order. We have found \( \eta \) by imposing that the normalization of the kinetic term is fixed at some standard value. This is surely not the most general way to proceed. If an exact computation is performed we know that this normalization does not matter and we will be able to fix it safely to whatever value we want: we will find a whole line of physically equivalent fixed points. It is generally known that this kind of symmetry is broken for most of the approximations$\,$\cite{bell} (in particular it is broken by the derivative expansion). Moreover, we generally expect that the true physical fixed point mixes with non-local ones with similar behaviour for the truncated action but with different anomalous dimensions. To pick up the local solution among the non-local ones, one should try to find the reminiscence of the line of fixed points:\ a marginal redundant operator. Its presence would signal that our scheme is truly approximating the local fixed point behaviour and not something else. This would probably fixed some, or perhaps even all, of the scheme dependences. This analysis has not been performed. Summarizing, we have presented a fermionic RG equation and an example of its application. It seems that, although technically harder to work with, there emerges the same patterns as in the bosonic case. In particular the annoying issue of spurious solutions is also present. However, it does seem that with sufficiently accurate work and restricting oneself to a derivative expansion without further truncations reliable non-trivial results should come out. \section*{Acknowledgments} I would like to acknowledge J.I. Latorre and Tim R. Morris for discussions on this and related subjects and A. Travesset for reading the manuscript. This work has been supported by funds form MEC under contract AEN95-0590. \section*{References}

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\section{Introduction} The strong, $\sim$10~Jy, flat-spectrum radio source PKS~1830$-$211 was first suggested to be a gravitational lens by Rao and Subrahmanyan (1988). Three years later the source was identified as an Einstein ring/gravitational lens (\cite{jau91}) and remains the brightest such object found in the radio sky by almost two orders of magnitude. While the interpretation of the source as a gravitational lens beyond the Galaxy (\cite{sk92}) is secure, it lies in a crowded and heavily obscured field close to the Galactic Center and so far all efforts to identify optical or infra-red counterparts either for the lensing galaxy or the lensed source have been unsuccessful (\cite{djo92}; \cite{jau93}). In particular, the failure of optical measurements to furnish any redshifts has driven the search for these critical parameters into the radio spectrum. The symmetric morphology of the source, comprising two compact, flat-spectrum components of similar brightness located on opposite sides of a 1 arcsec ring, immediately suggests a close alignment of the lensed source behind the lensing mass. Moreover, there is evidence of unusually high rotation measures in some parts of the source which argues that the lensing galaxy is probably a gas-rich spiral (\cite{nnr93}), and suggests the possibility of detecting molecular absorption. \section{Observations and Interpretation} Accordingly, we undertook a survey on 1995 June 10 and 11 for redshifted \hi and OH absorption with the Parkes 64~m radio telescope, as part of a cooperative observation programme with the Project Phoenix group (\cite{tar96}). The Project Phoenix receiver and signal processing equipment were used to cover a frequency range of 995 to 1675~MHz, nicely complementing a previous absorption search over the frequency range 400$-$1000~MHz at Green Bank, which yielded a negative result (\cite{mcm93}). Our observations excluded the two intervals 1535-1635~MHz and 1165-1175~MHz because of excessive interference. Although interference of both terrestrial and satellite origin was profuse over most of the remaining band, it was generally of very narrow bandwidth and easily recognizable, and did not greatly impede the search. We detected only a single absorption feature with two sub-components of similar amplitude, centered at 1191.1~MHz with an overall line width of approximately 50~km/s (Figure 1a). The absorption feature was detected with comparable strength on two consecutive days. On the second day, observations of PKS~1830$-$211 were bracketed with those of PKS~1921$-$293, a nearby source of similar flux density. No evidence of the absorption feature was seen in this comparison source. VLBI observations were made on 1995 September 18 with four telescopes of the Australian Long Baseline Array (LBA) (\cite{pre93}; \cite{jau94}); Hobart, Coonabarabran, Parkes and five antennas of the Australia Telescope Compact Array (ATCA) acting as a phased array. S2 recorders (\cite{wei91}) were used, operating in dual polarization (LCP and RCP) with 4~MHz bandwidth centered on 1191.0~MHz. Correlated visibilities between all six antennas of the ATCA were also recorded to produce an improved total-power spectrum (Figure 1b). PKS~1830$-$211 was observed over an 8 hour period interleaved with a bandpass calibrator. The VLBI data were correlated at the ATNF VLBI correlator (\cite{wil96}). No correlated flux was detected on the long ($>4\mbox{~M}\lambda$) baselines to Hobart owing to interstellar scattering at this frequency (\cite{jon96}). Our VLBI image in Figure 2 is similar to an earlier 2.3 GHz VLBI image (\cite{jau91}) and shows the two compact components of PKS~1830$-$211, but little of the low brightness ring which is heavily resolved at this resolution. Figure 2 also shows the absorption spectrum of each component, clearly demonstrating that the two velocity components of the absorption system obscure different parts of the source. The low velocity component is also heavily resolved and therefore must be obscuring only the extended ring while the high velocity component is resolved only partially and covers the NE component but is weak or absent in the SW component. This, together with a comparison of the relative optical depths in Figures 1 and 2, also allows us to infer that the angular size of the absorbing features must be greater than a few tenths of an arcsecond. A molecular absorption system at a redshift of 0.88582 has already been found in this source (\cite{wik96}), and is argued to arise in an intervening galaxy rather than in the lensed object. This has been confirmed by observations with the BIMA array which have spatially resolved the $z = 0.89$ absorption (\cite{fry96}). We believe that our detection constitutes a second absorption system in PKS~1830$-$211 and is probably \hi absorption in an intervening galaxy at a redshift of $0.1926 \pm 0.0001$. In support of this we note that there are no catalogued lines near 2246~MHz, the rest frequency of our observed absorption if it belongs to the $z = 0.89$ system. Further, we observed the bright Galactic sources Sgr~B2, Ori~KL and IRC~10216 with the 26~m telescope at Hobart to search for a possible unlisted transition at this frequency, and detected nothing above an rms noise level of $\sim$0.1\% of the continuum. Moreover, our detection appears to lie mainly in front of the NE component whilst the $z = 0.89$ absorption is confined to the SW component (\cite{wik96}; Frye {\em et al.} 1996). Our absorption feature also displays velocity structure not seen in the $z = 0.89$ absorption profiles. The interpretation of the feature in Figure 1 as OH is unlikely for two reasons. First, the spectra show none of the `satellite' profiles typical of OH absorption and second, we see no evidence of absorption at 1016~MHz corresponding to \hi at the same redshift, which we would expect to be clearly visible. Neither can this be an hydrogen recombination line as we would have detected many such lines across the band and did not. T. Wiklind and F. Combes (1996b) report no evidence of molecular absorption in PKS~1830$-$211 at $z = 0.19$ in their SEST observations. However this is not totally unexpected as the total solid angle subtended by the source at these high frequencies is small ($\sim$1mas$^2$) and hence the probability of intersecting a dense molecular cloud along the line of sight is presumably small. The absorption feature seen in Figure 1 is very similar in both width and column density to that found in the lens system 0218$+$357 (\cite{cry93}), which is convincingly argued to arise from \hi absorption in the lensing galaxy, probably a spiral galaxy seen nearly edge-on. We favour a similar interpretation for the absorption in PKS~1830$-$211, with the line of sight intercepting several \hi clouds in a gas-rich galaxy at $z = 0.19$. The two features seen in the absorption profile may well correspond to two spiral arms of $\sim$kpc scale seen nearly superposed, both of which partially obscure the ring and one of which obscures the NE compact component. Such a picture is entirely consistent with the properties of \hi clouds and galaxy dynamics observed within our own Galaxy. Independent evidence for a considerable amount of material along the line of sight is suggested by the unusually high rotation measure seen in the NE component (Nair {\em et al.} 1993; \cite{sub90}). Furthermore, the observed downturn in total flux density of PKS~1830$-$211 below 1~GHz (\cite{rao88}) implies significant free-free absorbing material obscuring the non-compact structure, which has a steep spectrum and cannot be synchrotron self-absorbed. \section{PKS~1830$-$211\ as a Compound Gravitational Lens} While the \hi absorption toward PKS~1830$-$211 indicates the presence of a galaxy at $z = 0.19$, this does not infallibly imply {\it a priori}\/ that gravitational lensing is taking place at this redshift, any more than it does for the $z = 0.89$ system. That lensing of some kind is taking place is beyond dispute given, for example, the near-simultaneous variation seen in the flux density variations of the two compact components (\cite{van95}), apparently separated by more than $\sim10$kpc: a most improbable effect in any non-lensed interpretation. It seems almost certain therefore, that at least one of the two redshift systems at $z = 0.19$ and $z = 0.89$ is partaking in the lensing and there is further evidence that both systems may be involved. Firstly, the lens evinces a quite paradoxical appearance at high resolution. VLBI images made over a 3 year period (\cite{jon93}; \cite{gar96}) show the SW component to be unresolved at $\sim$milliarcsecond resolution while the NE component shows a well resolved linear structure. This difference in morphology of the two components is too long-lived to be due to the difference in propagation times to the two components, estimated to be no more than a few tens of days (Nair {\em et al.} 1993; \cite{van95}). The results presented here suggest that the second absorption system at $z = 0.19$ may be responsible for this striking disparity in the two images by causing additional lensing distortion of the NE image, thus forming a compound gravitational lens. The alternative explanation that scattering by a high gas concentration in front of the NE component is causing the additional linear structure is ruled out as there is no $\lambda^2$ dependence on the length of this feature in the 5 and 15~GHz images (\cite{jon93} and \cite{gar96} respectively). Secondly, attempts to model PKS~1830$-$211 with a single, simple gravitational potential have achieved only modest success, and have produced markedly different models. The models developed by Nair {\em et~al.} (1993) and by Kochanek {\em et~al.} (1992), for example, differ in a number of material respects, not least in the time delay between the two compact components, which has the opposite sign in the Nair {\it et al.} model to that inferred from the Kochanek {\it et al.} model. The presence of a second lensing galaxy in this system adds an extra layer of complexity to any lensing model. This galaxy is likely to influence the light travel time through the NE component and so must be considered before ${\rm H}_{0}$ can be determined from the time delay. It is important therefore to estimate the mass and position of the $z = 0.19$ system. We are unable to obtain this information from our data. However polarization images of PKS~1830$-$211 at two or more frequencies to map rotation measure would help delineate the intervening material. \acknowledgments The Australia Telescope is operated as a national facility by CSIRO\@. J.E.J.L and C.J.P. are supported by Australian Postgraduate Research Awards. This research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under contract to the NASA. \newpage

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\section{Introduction} \label{s1} Corrections are utterances such as (\ref{ex:e1}b) where a discourse participant corrects the utterance of some other discourse participant\footnote{Here and in what follows, we use capital letters to indicate prosodic prominence.}. \bex \label{ex:e1} \bsex A: Jon likes Mary. \\ B: No, PETER likes Mary. \esex \eex Although there is much literature on corrections (e.g. \cite{ScJeSa:tpfscitooric77,norrick:otooceic91,ringlebruce:sfnlp82}), a thorough investigation of their linguistics is still outstanding. In this paper, we build up on \cite{Leusen:tioc94} and examine some of the requirements corrections place on context or in other words, the relationship between correction (the correcting utterance) and correctee (the utterance being corrected). For instance, it is clear that the pair of utterances in (\ref{ex:e2}) does not form a well--formed dialog. \bex \label{ex:e2} \bsex A: Jon likes Mary. \\ $\star$ B: No, PETER likes Sarah. \esex \eex On the other hand, it is also clear that a simple equality requirement between the semantic representation of the deaccented part of the correction and that of its parallel counterpart in the source is not appropriate either: \bex \label{ex:e3} \bsex A: Jon likes [the woman with the red hat]$_1$ \\ B: No, PETER likes Sarah$_1$ \esex \eex Here the correction contains an NP {\it Sarah} whose semantic representation is not identical with that of its source parallel element {\it the woman with the red hat}. In other words, a requirement such as \cite{Sag:dalf76}'s alphabetical variant constraint would fail\footnote{Sag proposes an analysis of VP ellipsis which requires that the semantic representation of a VP ellipsis be an alphabetical variant of the semantic representation of its antecedent. The basic assumption is that semantic representations are $\lambda$--terms. Two terms are alphabetical variants of each other iff they are identical up to renaming of bound variables.}. At this stage one could be tempted to conclude that the equality requirement is a semantic one: the deaccented part of the correction must be semantically equivalent with its parallel correlate in the source utterance. However, this is also incorrect. Thus in (\ref{ex:e4}), the property denoted by the VP in (\ref{ex:e4}b) need not be the same as the property denoted by its parallel counterpart in (\ref{ex:e4}a): whereas the VP in (\ref{ex:e4}a) denotes the property of loving Jon's wife, the VP in (\ref{ex:e4}b) may denote the property of loving Peter's wife\footnote{This is of course similar to the sloppy/strict ambiguity characteristic of VP ellipsis. Indeed, as we shall later see, our treatment is very similar to \cite{DaShPe:eahou91}'s treatment of VPE.}. \bex \label{ex:e4} \bsex A: Jon$_1$ loves his$_1$ wife. \\ B: No, PETER loves his wife. \esex \eex In short, it is clear that some identity requirement is needed to appropriately characterise the relation between correctum and correction (cf. example \ref{ex:e2}). On the other hand, it is less clear what this identity requirement should be (cf. examples \ref{ex:e3},\ref{ex:e4}). In this paper, we contend that the correct notion of identity is given by Higher--Order Unification with equivalences, a form of Unification which takes into account not only syntactic identity, but also denotational equivalence. We show that the HOUE--based analysis of corrections we propose, not only captures some of the contextual requirements of corrections, but also makes appropriate predictions about the interaction of corrections with both pronominal anaphora and (in)definiteness. \section{HOU with Equivalences} \label{s2} Now we will briefly review higher-order unification and its properties, for details we refer the reader to~\cite{Snyder91}. Higher-order unification solves the problem of finding substitutions $\sigma$ that for a given equation $A=B$ make both sides equal in the theory of $\beta\eta$-equality ($\sigma(A)=_{\beta\eta}\sigma(B)$). Huet's well-known algorithm~\cite{Huet75} solves the problem by recursively decomposing formulae and binding Function variables to most general formulae of a given type and given head. However, even though HOU considers $\beta\eta$-equality of formulae, it does not take into account the semantics of the logical connectives and quantifiers contained in the logical representation of natural language utterances. For this we need a unification algorithm for $\beta\eta$-equality augmented by logical equivalence. Obviously, such an algorithm has to generalize theorem proving methods for higher-order logic, since the task of unifying an equation $(A\vee\neg A) = T$, where $T$ is a sentence, is equivalent to proving the validity of the theorem $T$\footnote{The formula $(A\vee\neg A)$ must be true in all models, so $T$ can only be equivalent to it, if it is a theorem.}. An algorithm that solves this problem is described in~\cite{Kohlhase:hot95}. It is a generalization of the first-order Tableau method~\cite{Fitting:fotp} for automated theorem proving, which refutes a negated theorem by analyzing the connectives in an and/or tree and finding instantiations that close each branch of the tree by finding elementary contradictions on it. Instead of a formal recapitulation of the tableau method, we discuss the example of the logical theorem $(p(a)\vee p(b)\Rightarrow \exists x.p(x))$. The negation of this is equivalent\footnote{In addition to the de Morgan laws we use the identity $\exists x.A=\neg\forall x.\neg A$.} to the formula at the root of the following tableau. \[\begin{array}{c} p(a)\vee p(b) \wedge \forall x.\neg p(x) \\ p(a)\vee p(b) \\ \forall x.\neg p(x) \\ \left. \begin{array}{c} p(a) \\{} \neg p(y) \\{} *[y=a] \end{array}\right| \begin{array}{c} p(b) \\{} \neg p(z) \\{} *[z=b] \end{array} \end{array}\] Here we see that conjuncts are simply added to the branch, whereas disjuncts are analyzed in separate branches of the tree. The scopes of universal quantifications (with new variables) can be inserted at the end of branches, the same is possible with the scopes of existential quantifications (with the bound variables replaced by Skolem\footnote{Skolem terms serve as witnesses for the objects whose existence is claimed by the existential formula $A$. Since this object may depend on the values of free variables $x_1,\ldots,x_n$ occurring in $A$, they have the form $f(x_1,\ldots,x_n)$ where $f$ is a new function.} terms). Finally, both branches of the tableau are closed, i.e. the last formula can be instantiated (by the substitution in brackets) so that it contradicts a formula in the branch above. These instantiations are computed by unification, and in the case of higher-order logic by HOU. The distinguishing feature of the HOUE algorithm~\cite{Kohlhase:hot95} is that intermediate equations $(A = B)$ of type $t$ (generated either by unifying two formulae on the branch to make them contradictory or by processing other unification problems) can be transformed into negated equivalences (which can then be treated by the theorem proving component). Actually, tableau development for the negated equivalence $\neg(A\Leftrightarrow B)$ contains trivial branches, so we use the following (optimized) rule, which splits an equation of type $t$ into two tableau branches \[ \begin{array}{c} A=B \\ \begin{array}{c|c} A & B \\ \neg B & \neg A \end{array}\end{array}\] This way, HOU and tableau theorem proving recursively call each other in HOUE, until a refutation is found (all branches of the tableau are closed). \section{The basic analysis} \label{s3} Typically, a correction partially or completely repeats a previous utterance and one of its characteristic properties is that the repeated material is deaccented, that is, it is characterised by an important reduction in pitch, amplitude and duration (cf. \cite{Bartels:sot95}). Our proposal is to analyse corrections as involving a deaccented anaphor which consists of the repeated material. Furthermore, we require that the semantic representation of a deaccented anaphor unify with the semantic representation of its antecedent. More precisely, let $SSem$ and $TSem$ be the semantic representations of the source (i.e. antecedent) and target (i.e. anaphoric) clause respectively, and $TP^1 \dots TP^n, SP^1 \dots SP^n$ be the target and source parallel elements\footnote{As in \cite{DaShPe:eahou91}, we take the identification of parallel elements as given.}, then the interpretation of an SOE must respect the following equations: \\ $ \ba{l} An(SP^1,\dots,SP^n) = SSem \\ An(TP^1,\dots,TP^n) = TSem \\ \\ \ea $ Intuitively, these two equations require that target and source clause share a common semantics: $An$, the semantics of the deaccented anaphor. We illustrate the workings of the analysis by a simple example. Given the dialog in (\ref{ex:e1}), the equations to be solved are: \\ $ \ba{l} An(j) = like(j,m) \\ An(p) = like(p,m) \\ \\ \ea $ Given these equations, HOU yields a unique solution $An = \lambda x. like(x,m)$. In contrast, the equations required for the analysis of example (\ref{ex:e2}) are: \\ $ \ba{l} An(j) = like(j,m) \\ An(p) = like(p,s) \\ \\ \ea $ Since there is no substitution of values for free variables which simultaneously makes $An(j)$ $\alpha\beta\eta$--identical with $like(j,m)$ and $An(p)$ $\alpha\beta\eta$--identical with $like(p,s)$, unification fails thereby indicating the ill--formedness of (\ref{ex:e2}). \section{Corrections and pronominal anaphora} \label{s4} The resolution of pronouns occurring in the destressed part of a correction appears to be subject to very strong parallelism constraints. For instance in (\ref{ex:e10}b), the pronoun {\it her} can only be understood as referring to its source parallel element {\it Sarah} -- else it must be stressed. \bex \label{ex:e10} \bsex Jon loves Sarah$_1$ . \\ No, PETER loves her. \esex \eex Intuitively, there is a simple explanation for this: if the destressed part of a correction is a repeat of its parallel element in the source utterance, then pronouns occurring in it must necessarily resolve to their parallel counterpart in the source expression. As we shall see, the picture is somewhat more complex however. In some cases, a destressed pronoun in the correction may be ambiguous. In other cases, it functions as a paycheck pronoun. Finally, extraneous factors such as scope constraints and world knowledge interact with the semantics of corrections in determining the resolution of destressed pronouns. In what follows, we show how HOUE allows us to correctly predict this array of empirical facts. \subsection{Pronouns} Let us start with example (\ref{ex:e10}) above. Given the analysis of corrections described in section \ref{s3}, the equations to be resolved are\footnote{Unresolved pronouns are represented by free variables i.e. variables whose value is determined by unification. Alternatively, pronouns could be resolved first and unification would then function as a filter on admissible resolutions.}: \\ $ \begin{array}{ll} An(j) & = \txt{love}(j, s) \\ An(p) & = \txt{love}(p, x) \\ & \end{array} $ By unification, the only possible values for $An$ and $x$ are $\lambda y\txt{love}(y, s)$ and $s$ respectively. That is, the destressed pronoun is resolved by unification to its parallel element in the source utterance, {\it Sarah}. As required. In some cases however, a destressed pronoun in the correction is ambiguous. For instance in (\ref{ex:e6}b), the pronoun {\it his} may resolve either to {\it Jon} or to {\it Peter}. \bex \label{ex:e6} \bsex Jon$_1$ loves his$_1$ wife. \\ No, PETER $_2$ loves his$_{1,2}$ wife. \esex \eex Interestingly, such cases are similar to the sloppy/strict ambiguity\footnote{The terminology {\it sloppy/strict} originated with \cite{Ross:covis67}. Intuitively, a pronoun has a strict interpretation if it denotes as its antecedent. By contrast, a pronoun which denotes differently from its antecedent is said to have a sloppy interpretation.} characteristic of VP ellipsis and as \cite{DaShPe:eahou91} have shown, HOU straightforwardly captures such cases because of its ability to yield several solutions. In the case of (\ref{ex:e6}), the analysis proceeds as follows. First, the following equations must be resolved: \\ $ \begin{array}{ll} An(j) & = \txt{love}(j, \txt{wof }(j)) \\ An(p) & = \txt{love}(p, \txt{wof }(x)) \\ & \end{array} $ Resolution of the first equation yields two values for $An$\footnote{Unification yields a third value for $An$, namely $\lambda y\txt{love}(j, \txt{wof}(y))$. This solution however is ruled out by the second equation. More generally, we assume a restriction similar to \cite{DaShPe:eahou91}'s {\bf Primary Occurrence Restriction} (POR): the occurrences directly associated with the contrastive elements are primary occurrences and any solution containing a primary occurrence is discarded as linguistically invalid. For instance, in $An(j) = \txt{love}(j, \txt{wof }(j))$, the first occurrence of $j$ is a primary occurrence so that the solution $An = \lambda y$ {\it love(j, wof(y))} is ruled out. For a proposal of how the POR can be formally modelled, see \cite{GaKo:hocuanls96}.}, $\lambda y \txt{love}(y, \txt{wof}(j))$ and $\lambda y \txt{love}(y, \txt{wof}(y))$. By applying $An$ to $p$, we then get two possible values for $An(p)$: $\txt{love}(p, \txt{wof}(j))$ and $\txt{love}(p, \txt{wof}(p))$. As a side effect, the pronoun {\it his} represented by $x$ is resolved either to {\it Jon} or to {\it Peter}. In short, for such cases, the multiple solutions delivered by HOU match the ambiguity of natural language. \subsection{Paycheck pronouns} Destressed pronouns whose source parallel element is a pronominal possessive NP are particularly interesting. At first sight, they seem to behave just like any other destressed pronouns occurring in a correction, that is, they seem to resolve unambiguously to their parallel source element. For instance, in (\ref{ex:e14}b), the most likely resolution of {\it her} is {\it Jon's wife}. \bex \label{ex:e14} \bsex Jon$_1$ likes his$_1$ wife. \\ No, PETER likes her (= his$_1$ wife) \esex \eex However, a closer investigation of the data suggests that this reading is a kind of default reading which is preferred out of a pair of two grammatically possible interpretations. To see this, consider examples (\ref{ex:e15}) and (\ref{ex:e16}). \bex \label{ex:e15} \bsex Jon$_1$ broke his$_1$ arm yesterday. \\ No, PETER$_2$ broke it (= his$_{1,2}$ arm) yesterday. \esex \eex \bex \label{ex:e16} \bsex Jon$_1$ had his$_1$ nose remodelled in Paris. \\ No, PETER$_2$ had it (= his$_{2}$ nose) remodelled in Paris. \esex \eex Although these examples are structurally identical with (\ref{ex:e14}), they differ in the interpretation of the destressed pronoun occurring in the correction. Whereas (\ref{ex:e14}) only allows for a strict interpretation of this pronoun, (\ref{ex:e15}) permits both a strict and a sloppy interpretation whilst (\ref{ex:e16}) only admits of a sloppy reading. Our contention is that a destressed pronoun in the correction whose source parallel element is a possessive definite, is systematically ambiguous between a strict and a sloppy interpretation. However extraneous factors may have the effect that only one reading is available. For instance, in (\ref{ex:e16}) the strict reading is ruled out by our world knowledge that one can only have one's own nose remodelled. As for (\ref{ex:e14}), the absence of sloppy reading can be explained if we assume that the interpretation of a destressed anaphor follows a default strategy geared toward maximal semantic identity between the destressed anaphor and its antecedent. Under this assumption, the strict reading is the most natural since it establishes a strict denotational identity between the antecedent VP {\it likes Jon's wife} and the destressed anaphor {\it likes her}. The behaviour of these pronouns is simply explained once they are viewed as {\it paycheck pronouns} as illustrated by Karttunen's famous example (cf. \cite{karttunen:pav69}): \bex \label{ex:e30} The man who gave his paycheck to his wife was wiser than the man who gave it to his mistress \eex Paycheck pronouns differ from other pronouns in that they can neither be seen as coreferential constants nor as bound variables -- instead they pick up the definite description introduced by their antecedent and reanchor its possessive pronoun in its immediate context. For instance in (\ref{ex:e30}) above, the paycheck pronoun {\it it} picks up the description {\it his paycheck} and reanchors its possessive pronoun {\it his} to the second occurrence of {\it the man}. There are various ways in which paycheck pronouns can be accounted for but essentially, the idea is that their denotation is fixed by a definite description containing either an unresolved pronoun or an unresolved property. As \cite{cooper:tiop79} convincingly argues, the second solution is methodologically more satisfactory. We will therefore assume that paycheck pronouns are definite NPs whose representation includes a free variable of type $(e \rightarrow t)$ i.e. a property. More specifically, we assume that a paycheck pronoun is assigned the following representation: \[ \lambda Q. \exists x[P(x) \wedge \forall y[P(y) \leftrightarrow y = x] \wedge Q(x)] \] \noindent where $P\in\txt{wff}_{(e \rightarrow t)}$. Given this, the analysis of (\ref{ex:e14}) runs as follows. The equations to be resolved to check the well-formedness of the destressed anaphor {\it likes her} are:\footnote{In what follows, we abbreviate $\lambda Q. \exists x[P(x) \wedge \forall y[P(y) \leftrightarrow y = x] \wedge Q(x)]$ to $\lambda Q. \exists x[P(x) \wedge unique(x) \wedge Q(x)]$.}. $ \begin{array}{ll} An(j) & = \exists x [\txt{wof}(x, j) \wedge unique(x)\wedge love(j,x)] \\ An(p) & = \exists x [P(x) \wedge unique(x) \wedge love(p,x)] \end{array} $ Resolution of the first equation yields the two values $\lambda y. \exists x [\txt{wof}(x, j) \wedge unique(x) \wedge \txt{love}(y,x)]$ and $\lambda y. \exists x [\txt{wof}(x, y) \wedge unique(x)\wedge \txt{love}(y,x)]$ for $An$ and thus, the values $\exists x [\txt{wof}(x, j) \wedge unique(x)\wedge\txt{love}(p,x)]$, and $\exists x [\txt{wof}(x, p)\wedge unique(x) \wedge \txt{love}(p,x)]$ for $An(p)$. The first result yields the strict reading (Peter loves Jon's wife) whereas the second yields the sloppy reading (Peter loves Peter's wife). \section{Corrections and definiteness} \label{s5} So far, we have only considered cases where the semantic representation of the destressed anaphor could syntactically unify with that of its antecedent. That is, in each case it was possible to find a substitution of values for free variables which made the two semantic representations $\alpha\beta\eta$--identical. In this section, we turn to more semantic cases, cases in which the relation between destressed anaphor and source parallel element is one of denotational -- rather than syntactical -- identity. Definites are a primary example of such a phenomenon: since one and the same individual can be referred to by several, distinct definite descriptions, it often happens that the definite description used in the destressed part of a correction is not structurally identical with the description used in its source parallel element. This is illustrated in example (\ref{ex:e20}) where the source utterance contains the definite {\it the woman with the red hat}. As illustrated by (\ref{ex:e20}a--d), the parallel element in the correction can be {\it his wife, her, the neighbour's daughter} or {\it Sarah}. In each case, the description does not syntactically unify with the source description {\it the woman with the red hat}. Note however that the correction is only well--formed when the parallel descriptions are interpreted as referring to one and the same individual (cf. the ill--formedness of (\ref{ex:e20}e--g)). That is, when they are semantically equivalent. \bex \label{ex:e20} Jon$_2$ likes [the woman with the red hat]$_1$ \bsex No, PETER$_3$ likes his wife (= NP$_1$) \\ No, PETER likes her$_1$.\\ No, PETER likes [the neighbour's daughter]$_1$. \\ No, PETER likes Sarah$_1$. \\ $\star$ No, PETER likes her$_4$.\\ $\star$ No, PETER likes Mary$_4$. \\ $\star$ No, PETER likes him. \esex \eex How does HOUE account for such examples? To show this, we now sketch the main steps of the unification process for example (\ref{ex:e20}d) with equations: \[\begin{array}{ll} An(p) & = like(p,s) \\ An(j) & = \exists x (w(x) \wedge wrh(x) \wedge unique(x) \wedge like(j,x)) \end{array}\] These are solved in a context, where Sarah is the only woman with a red hat. The HOUE method is given access to the hypotheses $unique(s)$, $w(s)$ and $wrh(s)$ by adding them to the initial tableau. In a first step, we solve the first equation to $An=\lambda z.like(z,s)$ and obtain the following tableau: \[\begin{array}{c} unique(s)\\ w(s)\\ wrh(s)\\ An(p) = like(p,s) \\ \vdots\\ like(j,s)=\exists x (w(x)\wedge wrh(x)\wedge unique(x) \wedge like(j,x)) \end{array}\] The HOUE rule discussed in section~\ref{s2} now splits the initial equation into two branches. The first one has the form \[\begin{array}[t]{c} like(j,s) \\ \neg\exists x (w(x)\wedge\ldots \wedge like(j,x))\\ \begin{array}{c|c|c|c} \neg w(z) & \neg wrh(z) &\neg unique(z) &\neg like(j,z) \\{} *[z=s] & *[z=s] & *[z=s] & *[z=s] \end{array} \end{array}\] and contains the formulae $like(j,s)$ and $(\neg\exists x (w(x)\wedge wrh(x)\wedge unique(x) \wedge like(j,x)))$. The latter is universally quantified\footnote{We use that $\neg\exists x.A$ is equivalent to $\forall x.\neg A$ here.} and can therefore be developed into four branches $\neg w(z)$, $\neg wrh(z)$, $\neg unique(z)$, and $\neg like(j,z)$. The first three branches can be closed using the hypotheses on Sarah and the last one with the first formula, all by binding the new variable $z$ to $s$. The second branch has the form \[\begin{array}[t]{c} \neg like(j,s)\\ \exists x (\ldots \wedge unique(x)\wedge like(j,x))\\ unique(c)\\ like(j,c))\\ \vdots \\ c=s \\ like(j,s)\\{} *[] \end{array}\] and consists of the formulae $\neg like(j,s)$ and $\exists x (w(x)\wedge wrh(x)\wedge unique(x) \wedge like(j,x))$, which is developed into the single branch containing the conjuncts $w(c)$, $wrh(c)$, $unique(c)$, and $like(j,c))$, where $c$ is a Skolem constant for $x$. Here an expansion of the definition of uniqueness \[unique(x)\Leftrightarrow\forall z(w(z)\wedge wrh(z)\leftrightarrow x=z)\] closes the branch (if Sarah and $c$ are unique, then $s=c$). By now, it should be clear that our treatment will also encounter no particular problem in dealing with examples such as (\ref{ex:e23}) and (\ref{ex:e24}) below. The first example relies on the world-knowledge that marrying is a symmetric relation (both partners have to say ``yes I do''), whereas the second relies on the fact that getting wounded is synonymous to being hurt by someone/thing. Once these equivalences are taken into account, the HOUE analysis of corrections will correctly predict that these examples are well--formed. \bex \label{ex:e23} \bsex A: Jon married Sarah \\ B: No, Sarah married PETER \esex \eex \bex \label{ex:e24} \bsex A: Sarah hurt Paul. \\ B: No, PETER was wounded. \esex \eex We have seen that a deaccented anaphor must either have a semantic representation which syntactically unifies with that of its antecedent, or be semantically equivalent to this antecedent. To show that this is a necessary condition, we need to provide some ill--formed examples in which neither condition holds. Such examples are given when the correction contains a destressed pronoun whose source parallel element is either an indefinite (\ref{ex:e21}) or a quantifier (\ref{ex:e22}). \bex \label{ex:e21} \bsex Jon eats an$_1$ apple. \\ $\ast$ No, PETER eats it$_1$. \esex \eex \bex \label{ex:e22} \bsex Jon kissed most$_1$ women at the party yesterday. \\ $\ast$ No, PETER kissed them$_1$. \esex \eex In both cases, the semantic representation of the pronoun in the correction fails to syntactically unify with the semantic representation of its antecedent. Neither can it be proved that {\it it} and {\it them} are semantically equivalent to {\it an apple} and {\it most women at the party } respectively. Therefore, unification fails correctly ruling out (\ref{ex:e21}) and (\ref{ex:e22}). The logical reason for this e.g. in (\ref{ex:e21}), is that while the second equation $An(p)=eat(p,y)$ can be solved to $An=\lambda x.eat(x,y)$ yielding the negated $\neg(eat(j,y)\Leftrightarrow\exists x (ap(x)\wedge eat(j,x))$, this cannot be refuted \footnote{Example (\ref{ex:e21}) is in fact ambiguous between a specific reading of the indefinite {\it an apple} and a non-specific one. In the first case, the indefinite denote uniquely so that {\it it} in (\ref{ex:e21}b) refers to this unique apple. Since it is denotationally equivalent with its antecedent, HOUE will succeed. In the second case, there is no unique apple salient in the context, hence {\it it} and {\it an apple} cannot be denotationally equivalent. Therefore HOUE fails. The above discussion focuses on this second possibility.}. \section{Conclusion} \label{s6} In a sense, it would be much more natural to express the proposed analysis in a dynamic setting (cf. \cite{Kamp:atotasr81}). The data discussed in section \ref{s5} clearly shows that definite, indefinites and quantifiers behave differently wrt. corrections. The intuition is that whereas, a definite can bind a pronoun in the correction (cf. example \ref{ex:e20}), indefinites and quantifiers cannot (cf. examples \ref{ex:e21},\ref{ex:e22}). These are of course precisely the sort of facts dynamic semantics was designed to deal with: if we assume that the correctee--correction pair is semantically represented by a disjunction $(\Phi \vee \Psi)$, then a definite in the correctee will be able to bind an anaphor in the correction (because definites have global scope) whereas indefinites and quantifiers won't (because traditionally disjunction is static and the discourse referents introduced by one disjunct are not accessible to the other disjunct). In this paper, we've shown that such facts could be modelled by means of HOUE on static semantic representations; it would be interesting to see how the analysis would transpose to a more dynamic setting. This however must await the development of Higher--Order Unification for a dynamic lambda--calculus. Another question worth investigating is whether the interleaving of anaphora resolution and quantification proposed in \cite{DaShPe:eahou91} could account for the data considered here. The approach has the advantage that it does not resort to equivalences, thus permitting better computational properties. However, unless definites are treated in a special way, it is unlikely that the approach will be able to capture examples such as (\ref{ex:e20}) where denotational equivalence, rather than strict unification, is required. Finally, an interesting issue concerns the relationship between HOUE and accommodation. A simple way to model accomodation would be to posit that, as theorem proving hits a dead-end, accomodation can be used to close off a branch: the accomodated fact is the fact needed to derive a contradiction and close off this tableau branch. Naturally, this idea is too simplistic in that some model must be defined which constrains accomodation. This we leave as an open research issue.

proofpile-arXiv_065-535

\section{Introduction} It took decades until physicists understood that all known fundamental interactions can be described in terms of gauge theories. My historical account begins with Einstein's general theory of relativity (GR), which is a non-Abelian gauge theory of a special type (see Secs. 3,4). The other gauge theories emerged in a slow and complicated process gradually from GR, and their common geometrical structure --- best expressed in terms of connections of fiber bundles --- is now widely recognized. Thus, also in this respect, H. Weyl was right when he wrote in the preface to the first edition of Space -- Time -- Matter (RZM) early in 1918: ``Wider expanses and greater depths are now exposed to the searching eye of knowledge, regions of which we had not even a presentiment. It has brought us much nearer to grasping the plan that underlies all physical happening'' \cite{1}. It was Weyl himself who made in 1918 the first attempt to extend GR in order to describe gravitation and electromagnetism within a unifying geometrical framework \cite{2}. This brilliant proposal contains all mathematical aspects of a non-Abelian gauge theory, as I will make clear in $\S 2$. The words gauge (Eich--) transformation and gauge invariance appear the first time in this paper, but in the everyday meaning of change of length or change of calibration. Einstein admired Weyl's theory as ``a coup of genius of the first rate \ldots '', but immediately realized that it was physically untenable: ``Although your idea is so beautiful, I have to declare frankly that, in my opinion, it is impossible that the theory corresponds to nature.'' This led to an intense exchange of letters between Einstein (in Berlin) and Weyl (at the ETH in Z\"urich), which will hopefully soon be published in {\sl The Collected Papers} of Einstein. (In my article \cite{3} I gave an account of this correspondence which is preserved in the Archives of the ETH.) No agreement was reached, but Einstein's intuition proved to be right. Although Weyl's attempt was a failure as a physical theory it paved the way for the correct understanding of gauge invariance. Weyl himself re-interpreted his original theory after the advent of quantum theory in a seminal paper \cite{4} which I will discuss at length in $\S 3$. Parallel developments by other workers and interconnections are indicated in Fig.1. At the time Weyl's contributions to theoretical physics were not appreciated very much, since they did not really add new physics. The attitude of the leading theoreticians is expressed in familiar distinctness in a letter by Pauli to Weyl from July 1, 1929, after he had seen a preliminary account of Weyl's work: \begin{quotation} {\sl Before me lies the April edition of the Proc.Nat.Acad. (US). Not only does it contain an article from you under ``Physics'' but shows that you are now in a `Physical Laboratory': from what I hear you have even been given a chair in `Physics' in America. I admire your courage; since the conclusion is inevitable that you wish to be judged, not for success in pure mathematics, but for your true but unhappy love for physics \cite{5}.} \end{quotation} Weyl's reinterpretation of his earlier speculative proposal had actually been suggested before by London, but it was Weyl who emphasized the role of gauge invariance as a {\em symmetry principle} from which electromagnetism can be {\em derived}. It took several decades until the importance of this symmetry principle --- in its generalized form to non-Abelian gauge groups developed by Yang, Mills, and others --- became also fruitful for a description of the weak and strong interactions. The mathematics of the non-Abelian generalization of Weyl's 1929 paper would have been an easy task for a mathematician of his rank, but at the time there was no motivation for this from the physics side. The known properties of the weak and strong nuclear interactions, in particular their short range, did not point to a gauge theoretical description. We all know that the gauge symmetries of the Standard Model are very hidden and it is, therefore, not astonishing that progress was very slow indeed. Today, the younger generation, who learned the Standard Model from polished textbook presentations, complains with good reasons about many of its imperfections. It is one of the aims of this talk to make it obvious that it was extremely difficult to reach our present understanding of the fundamental interactions. The Standard Model, with all its success, is a great achievement, and one should not be too discouraged when major further progress is not coming rapidly. Because of limitations of time and personal knowledge, I will discuss in the rest of my talk mainly the two important papers by Weyl from 1918 and 1929. The latter contains also his two-component theory of massless spin $1/2$ fermions. In this context I will make in $\S 5$ a few remarks about the developments which led in 1958 to the phenomenological $V-A$ current-current Lagrangian for the weak interactions. My historical account of the non-Abelian generalizations by Klein, Pauli and others, culminating in the paper by Yang and Mills, will also be much abbreviated. This is not too bad, since there will soon be a book by Lochlain O'Raifeartaigh that is devoted entirely to the early history of gauge theories \cite{6}. Those who do not know German will find there also English translations of the most important papers of the first period (1918--1929). The book contains in addition the astonishing paper by Klein (1938) \cite{7}, Pauli's letters to Pais on non-Abelian Kaluza-Klein reductions \cite{8}, parts of Shaw's dissertation, in which he develops a non-Abelian $SU(2)$ gauge theory \cite{9}, and Utiyama's generalization of Yang-Mills theory to arbitrary gauge groups \cite{10}. These works are behind the diagram in Fig.1. This talk covers mostly material contained in the papers \cite{3}, \cite{11}, and \cite{12}, which I have published some time ago in German, partly because all early publications and letters related to our subject are written in this language. \begin{figure} \begin{center} \epsfig{file=fig1.eps,height=20cm} \caption{Key papers in the development of gauge theories.} \end{center} \end{figure} \clearpage \section{Weyl's attempt to unify gravitation \newline and electromagnetism} On the 1st of March 1918 Weyl writes in a letter to Einstein: ``These days I succeeded, as I believe, to derive electricity and gravitation from a common source \ldots ''. Einstein's prompt reaction by postcard indicates already a physical objection which he explained in detail shortly afterwards. Before I come to this I have to describe Weyl's theory of 1918. \subsection{Weyl's generalization of Riemannian geometry} Weyl's starting point was purely mathematical. He felt a certain uneasiness about Riemannian geometry, as is clearly expressed by the following sentences early in his paper\footnote{I am using here and at other places the English translation of L. O'Raifeartaigh \cite{6}.}: \begin{quotation} {\sl But in Riemannian geometry described above there is contained a last element of geometry ``at a distance'' (ferngeometrisches Element) --- with no good reason, as far as I can see; it is due only to the accidental development of Riemannian geometry from Euclidean geometry. The metric allows the two magnitudes of two vectors to be compared, not only at the same point, but at any arbitrarily separated points.} {\it A true infinitesimal geometry should, however, recognize only a principle for transferring the magnitude of a vector to an infinitesimally close point} {\sl and then, on transfer to an arbitrary distant point, the integrability of the magnitude of a vector is no more to be expected that the integrability of its direction.} \end{quotation} After these remarks Weyl turns to physical speculation and continues as follows: \begin{quotation} {\sl On the removal of this inconsistency there appears a geometry that, surprisingly, when applied to the world,} {\it explains not only the gravitational phenomena but also the electrical.} {\sl According to the resultant theory both spring from the same source, indeed} {\it in general one cannot separate gravitation and electromagnetism in a unique manner}. {\sl In this theory} {\it all physical quantities have a world geometrical meaning; the action appears from the beginning as a pure number. It leads to an essentially unique universal law; it even allows us to understand in a certain sense why the world is four-dimensional}. \end{quotation} In brief, Weyl's geometry can be described as follows. First, the spacetime manifold $M$ is equipped with a conformal structure, i.e., with a class $[g]$ of conformally equivalent Lorentz metrics $g$ (and not a definite metric as in GR). This corresponds to the requirement that it should only be possible to compare lengths at one and the same world point. Second, it is assumed, as in Riemannian geometry, that there is an affine (linear) torsion-free connection which defines a covariant derivative $\nabla$, and respects the conformal structure. Differentially this means that for any $g\in[g]$ the covariant derivative $\nabla g$ should be proportional to $g$: \begin{equation} \label{2.1} \nabla g =-2A\otimes g\ \ \ \ \ \ \ (\nabla_{\lambda}g_{\mu\nu}=-2A_{\lambda}g_{\mu\nu}), \end{equation} where $A=A_{\mu}dx^{\mu}$ is a differential 1-form. Consider now a curve $\gamma: [0,1]\rightarrow M$ and a parallel-transported vector field $X$ along $\gamma$. If $l$ is the length of $X$, measured with a representative $g\in[g]$, we obtain from (\ref{2.1}) the following relation between $l(p)$ for the initial point $p=\gamma(0)$ and $l(q)$ for the end point $q=\gamma(1)$: \begin{equation} \label{2.2} l(q)=\exp\left(-\int_{\gamma}A\right)\ l(p). \end{equation} Thus, the ratio of lengths in $q$ and $p$ (measured with $g\in[g]$) {\it depends in general on the connecting path $\gamma$} (see Fig.2). The length is only independent of $\gamma$ if the curl of $A$, \begin{equation} \label{2.3} F=dA\ \ \ \ \ \ \ \ (F_{\mu\nu}=\partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu}), \end{equation} vanishes. \begin{figure} \begin{center} \epsfig{file=fig2.eps,width=10cm} \caption{Path dependence of parallel displacement and transport of length in Weyl space.} \end{center} \end{figure} The compatibility requirement (\ref{2.1}) leads to the following expression for the Christoffel symbols in Weyl's geometry: \begin{equation} \label{2.4} \Gamma^{\mu}_{\nu\lambda}=\frac{1}{2}g^{\mu\sigma}( g_{\lambda\sigma,\nu}+g_{\sigma\nu,\lambda}-g_{\nu\lambda,\sigma}) +g^{\mu\sigma}(g_{\lambda\sigma}A_{\nu}+g_{\sigma\nu}A_{\lambda}- g_{\nu\lambda}A_{\sigma}). \end{equation} The second $A$-dependent term is a characteristic new piece in Weyl's geometry which has to be added to the Christoffel symbols of Riemannian geometry. Until now we have chosen a fixed, but arbitrary metric in the conformal class $[g]$. This corresponds to a choice of calibration (or gauge). Passing to another calibration with metric $\bar{g}$, related to $g$ by \begin{equation} \label{2.5} \bar{g}=e^{2\lambda}g, \end{equation} the potential $A$ in (\ref{2.1}) will also change to $\bar{A}$, say. Since the covariant derivative has an absolute meaning, $\bar{A}$ can easily be worked out: On the one hand we have by definition \begin{equation} \nabla \bar{g} =-2\bar{A}\otimes\bar{g}, \end{equation} and on the other hand we find for the left side with (\ref{2.1}) \begin{equation} \nabla\bar{g}=\nabla(e^{2\lambda}g)= 2d\lambda\otimes\bar{g}+e^{2\lambda}\nabla g= 2d\lambda\otimes\bar{g}-2A\otimes\bar{g}. \end{equation} Thus \begin{equation} \label{2.6} \bar{A}=A- d\lambda\ \ \ \ \ \ (\bar{A}_{\mu}=A_{\mu}-\partial_{\mu}\lambda). \end{equation} This shows that a change of calibration of the metric induces a {\it ``gauge transformation''} for $A$: \begin{equation} \label{2.7} g\rightarrow e^{2\lambda}g,\ \ \ \ A\rightarrow A-d\lambda. \end{equation} Only gauge classes have an absolute meaning. (The Weyl connection is, however, gauge-invariant.) \subsection{Electromagnetism and Gravitation} Turning to physics, Weyl assumes that his ``purely infinitesimal geometry'' describes the structure of spacetime and consequently he requires that physical laws should satisfy a double-invariance: 1. They must be invariant with respect to arbitrary smooth coordinate transformations. 2. They must be {\it gauge invariant}, i.e., invariant with respect to substitutions (\ref{2.7}) for an arbitrary smooth function $\lambda$. Nothing is more natural to Weyl, than identifying $A_{\mu}$ with the vector potential and $F_{\mu\nu}$ in eq.(\ref{2.3}) with the field strength of electromagnetism. In the absence of electromagnetic fields ($F_{\mu\nu}=0$) the scale factor $\exp(-\int_{\gamma}A)$ in (\ref{2.2}) for length transport becomes path independent (integrable) and one can find a gauge such that $A_{\mu}$ vanishes. In this special case one is in the same situation as in GR. Weyl proceeds to find an action which is generally invariant as well as gauge invariant and which would give the coupled field equations for $g$ and $A$. I do not want to enter into this, except for the following remark. In his first paper \cite{2} Weyl proposes what we call nowadays the Yang-Mills action \begin{equation} \label{2.8} S(g,A)=-\frac{1}{4}\int Tr(\Omega\wedge\ast\Omega). \end{equation} Here $\Omega$ denotes the curvature form and $\ast\Omega$ its Hodge dual\footnote{The integrand in (\ref{2.8}) is in local coordinates indeed just the expression $R_{\alpha\beta\gamma\delta} R^{\alpha\beta\gamma\delta} \sqrt{-g}dx^{0}\wedge\ldots\wedge dx^{3}$ which is used by Weyl ($R_{\alpha\beta\gamma\delta}$ $=$ curvature tensor of the Weyl connection).}. Note that the latter is gauge invariant, i.e., independent of the choice of $g\in[g]$. In Weyl's geometry the curvature form splits as $\Omega=\hat{\Omega}+F$, where $\hat{\Omega}$ is the metric piece \cite{13}. Correspondingly, the action also splits, \begin{equation} \label{2.9} Tr (\Omega\wedge\ast\Omega) = Tr (\hat{\Omega}\wedge\ast\hat{\Omega}) +F\wedge\ast F. \end{equation} The second term is just the Maxwell action. Weyl's theory thus contains formally all aspects of a non-Abelian gauge theory. Weyl emphasizes, of course, that the Einstein-Hilbert action is not gauge invariant. Later work by Pauli \cite{14} and by Weyl himself \cite{1,2} led soon to the conclusion that the action (\ref{2.8}) could not be the correct one, and other possibilities were investigated (see the later editions of RZM). Independent of the precise form of the action Weyl shows that in his theory gauge invariance implies the {\it conservation of electric charge} in much the same way as general coordinate invariance leads to the conservation of energy and momentum\footnote{I adopt here the somewhat naive interpretation of energy-momentum conservation for generally invariant theories of the older literature.}. This beautiful connection pleased him particularly: ``\ldots [it] seems to me to be the strongest general argument in favour of the present theory --- insofar as it is permissible to talk of justification in the context of pure speculation.'' The invariance principles imply five `Bianchi type' identities. Correspondingly, the five conservation laws follow in two independent ways from the coupled field equations and may be ``termed the eliminants'' of the latter. These structural connections hold also in modern gauge theories. \subsection{Einstein's objection and reactions of other physicists} After this sketch of Weyl's theory I come to Einstein's striking counterargument which he first communicated to Weyl by postcard (see Fig.3). The problem is that if the idea of a nonintegrable length connection (scale factor) is correct, then the behavior of clocks would depend on their history. Consider two identical atomic clocks in adjacent world points and bring them along different world trajectories which meet again in adjacent world points. According to (\ref{2.2}) their frequencies would then generally differ. This is in clear contradiction with empirical evidence, in particular with the existence of stable atomic spectra. Einstein therefore concludes (see \cite{3}): \begin{quotation} {\sl \ldots (if) one drops the connection of the $ds$ to the measurement of distance and time, then relativity looses all its empirical basis.} \end{quotation} Nernst shared Einstein's objection and demanded on behalf of the Berlin Academy that it should be printed in a short amendment to Weyl's article, and Weyl had to cope with it. I have described the intense and instructive subsequent correspondence between Weyl and Einstein elsewhere \cite{3}. As an example, let me quote from one of the last letters of Weyl to Einstein: \begin{quotation} {\sl This [insistence] irritates me of course, because experience has proven that one can rely on your intuition; so little convincing your counterarguments seem to me, as I have to admit \ldots} \end{quotation} \begin{quotation} {\sl By the way, you should not believe that I was driven to introduce the linear differential form in addition to the quadratic one by physical reasons. I wanted, just to the contrary, to get rid of this `methodological inconsistency {\it (Inkonsequenz)}' which has been a stone of contention to me already much earlier. And then, to my surprise, I realized that it looks as if it might explain electricity. You clap your hands above your head and shout: But physics is not made this way ! (Weyl to Einstein 10.12.1918).} \end{quotation} Weyl's reply to Einstein's criticism was, generally speaking, this: The real behavior of measuring rods and clocks (atoms and atomic systems) in arbitrary electromagnetic and gravitational fields can be deduced only from a dynamical theory of matter. Not all leading physicists reacted negatively. Einstein transmitted a very positive first reaction by Planck, and Sommerfeld wrote enthusiastically to Weyl that there was ``\ldots hardly doubt, that you are on the correct path and not on the wrong one.'' In his encyclopedia article on relativity \cite{15} Pauli gave a lucid and precise presentation of Weyl's theory, but commented Weyl's point of view very critically. At the end he states: \begin{quotation} {\sl \ldots Resuming one may say that Weyl's theory has not yet contributed to get closer to the solution of the problem of matter.} \end{quotation} Also Eddington's reaction was first very positive but he changed his mind soon and denied the physical relevance of Weyl's geometry. The situation was later appropriately summarized by F.London in his 1927 paper \cite{16} as follows: \begin{quotation} {\sl In the face of such elementary experimental evidence, it must have been an unusually strong metaphysical conviction that prevented Weyl from abandoning the idea that Nature would have to make use of the beautiful geometrical possibility that was offered. He stuck to his conviction and evaded discussion of the above-mentioned contradictions through a rather unclear re-interpretation of the concept of ``real state'', which, however, robbed his theory of its immediate physical meaning and attraction.} \end{quotation} \begin{figure} \epsfig{file=fig3.eps,width=16cm} \caption{Postcard of Einstein to Weyl 15.4.1918 (Archives of ETH).} \end{figure} \clearpage \section{Weyl's 1929 Classic: ``Electron and Gravitation''} Shortly before his death late in 1955, Weyl wrote for his {\it Selecta} \cite{17} a postscript to his early attempt in 1918 to construct a `unified field theory'. There he expressed his deep attachment to the gauge idea and adds (p.192): \begin{quotation} {\sl Later the quantum-theory introduced the Schr\"odinger-Dirac potential $\psi$ of the electron-positron field; it carried with it an experimentally-based principle of gauge-invariance which guaranteed the conservation of charge, and connected the $\psi$ with the electromagnetic potentials $\phi_{i}$ in the same way that my speculative theory had connected the gravitational potentials $g_{ik}$ with the $\phi_{i}$, and measured the $\phi_{i}$ in known atomic, rather than unknown cosmological units. I have no doubt but that the correct context for the principle of gauge-invariance is here and not, as I believed in 1918, in the intertwining of electromagnetism and gravity.} \end{quotation} This re-interpretation was developed by Weyl in one of the great papers of this century \cite{4}. Weyl's classic does not only give a very clear formulation of the gauge principle, but contains, in addition, several other important concepts and results --- in particular his two-component theory. The richness of the paper is clearly visible from the following table of contents: \begin{quotation} {\sl Introduction. Relationship of General Relativity to the quantum-theoretical field equations of the spinning electron: mass, gauge-invariance, distant-parallelism. Expected modifications of the Dirac theory. -I. Two-component theory: the wave function $\psi$ has only two components. -$\S 1$. Connection between the transformation of the $\psi$ and the transformation of a normal tetrad in four-dimensional space. Asymmetry of past and future, of left and right. -$\S 2$. In General Relativity the metric at a given point is determined by a normal tetrad. Components of vectors relative to the tetrad and coordinates. Covariant differentiation of $\psi$. -$\S 3$. Generally invariant form of the Dirac action, characteristic for the wave-field of matter. -$\S 4$. The differential conservation law of energy and momentum and the symmetry of the energy-momentum tensor as a consequence of the double-invariance (1) with respect to coordinate transformations (2) with respect to rotation of the tetrad. Momentum and moment of momentum for matter. -$\S 5$. Einstein's classical theory of gravitation in the new analytic formulation. Gravitational energy. -$\S 6$. The electromagnetic field. From the arbitrariness of the gauge-factor in $\psi$ appears the necessity of introducing the electromagnetic potential. Gauge invariance and charge conservation. The space-integral of charge. The introduction of mass. Discussion and rejection of another possibility in which electromagnetism appears, not as an accompanying phenomenon of matter, but of gravitation.} \end{quotation} The modern version of the gauge principle is already spelled out in the introduction: \begin{quotation} {\sl The Dirac field-equations for $\psi$ together with the Maxwell equations for the four potentials $f_{p}$ of the electromagnetic field have an invariance property which is formally similar to the one which I called gauge-invariance in my 1918 theory of gravitation and electromagnetism; the equations remain invariant when one makes the simultaneous substitutions $$\psi\ \ \ {\rm by}\ \ \ e^{i\lambda}\psi\ \ \ \ {\rm and}\ \ \ f_{p}\ \ \ {\rm by}\ \ \ f_{p}-\frac{\partial\lambda}{\partial x^{p}}, $$ where $\lambda$ is understood to be an arbitrary function of position in four-space. Here the factor $\frac{e}{ch}$, where $-e$ is the charge of the electron, $c$ is the speed of light, and $\frac{h}{2\pi}$ is the quantum of action, has been absorbed in $f_{p}$. The connection of this ``gauge invariance'' to the conservation of electric charge remains untouched. But a fundamental difference, which is important to obtain agreement with observation, is that the exponent of the factor multiplying $\psi$ is not real but pure imaginary. $\psi$ now plays the role that Einstein's $ds$ played before. It seems to me that this new principle of gauge-invariance, which follows not from speculation but from experiment, tells us that the electromagnetic field is a necessary accompanying phenomenon, not of gravitation, but of the material wave-field represented by $\psi$. Since gauge-invariance involves an arbitrary function $\lambda$ it has the character of ``general'' relativity and can naturally only be understood in that context.} \end{quotation} We shall soon enter into Weyl's justification which is, not surprisingly, strongly associated with general relativity. Before this I have to describe his incorporation of the Dirac theory into GR which he achieved with the help of the tetrad formalism. One of the reasons for adapting the Dirac theory of the spinning electron to gravitation had to do with Einstein's recent unified theory which invoked a distant parallelism with torsion. E.Wigner \cite{18} and others had noticed a connection of this theory and the spin theory of the electron. Weyl did not like this and wanted to dispense with teleparallelism. In the introduction he says: \begin{quotation} {\sl I prefer not to believe in distant parallelism for a number of reasons. First my mathematical intuition objects to accepting such an artificial geometry; I find it difficult to understand the force that would keep the local tetrads at different points and in rotated positions in a rigid relationship. There are, I believe, two important physical reasons as well. The loosening of the rigid relationship between the tetrads at different points converts the gauge-factor $e^{i\lambda}$, which remains arbitrary with respect to $\psi$, from a constant to an arbitrary function of space-time. In other words, only through the loosening the rigidity does the established gauge-invariance become understandable. } \end{quotation} This thought is carried out in detail after Weyl has set up his two-component theory in special relativity, including a discussion of $P$ and $T$ invariance. He emphasizes thereby that the two-component theory excludes a linear implementation of parity and remarks: ``It is only the fact that the left-right symmetry actually appears in Nature that forces us to introduce a second pair of $\psi$-components.'' To Weyl the mass-problem is thus not relevant for this. Indeed he says: ``Mass, however, is a gravitational effect; thus there is hope of finding a substitute in the theory of gravitation that would produce the required corrections.'' We shall return to the two-component theory in $\S 5$ in connection with parity violation and the $V-A$ interaction. \subsection{Tetrad formalism} The method of Weyl for incorporating his two-component spinors into general relativity makes use of local tetrads (Vierbeins). In the tetrad formalism the metric is described by an arbitrary basis of orthonormal vector fields $\{e_{\alpha}(x);\alpha=0,1,2,3\}$. If $\{e^{\alpha}(x)\}$ denotes the dual basis of 1-forms, the metric is given by \begin{equation} \label{3.1} g=\eta_{\mu\nu}e^{\nu}(x)\otimes e^{\nu}(x),\ \ \ \ (\eta_{\mu\nu})=diag(1,-1,-1,-1). \end{equation} Weyl emphasizes, of course, that only a class of such local tetrads is determined by the metric: the metric is not changed if the tetrad fields are subject to spacetime-dependent Lorentz transformations: \begin{equation} \label{3.2} e^{\alpha}(x)\rightarrow\Lambda^{\alpha}_{\ \beta}(x)e^{\beta}(x). \end{equation} With respect to a tetrad, the connection forms\footnote{I am using more modern notations; for details see \cite{18}.} $\omega=(\omega^{\alpha}_{\ \beta})$ have values in the Lie algebra of the homogeneous Lorentz group: \begin{equation} \label{3.3} \omega_{\alpha\beta}+\omega_{\beta\alpha}=0. \end{equation} (Indices are raised and lowered with $\eta^{\alpha\beta}$ and $\eta_{\alpha\beta}$, respectively.) They are determined (in terms of the tetrad) by the first structure equation of Cartan: \begin{equation} \label{3.4} de^{\alpha}+\omega^{\alpha}_{\ \beta}\wedge e^{\beta}=0. \end{equation} Under local Lorentz transformations (\ref{3.2}) the connection forms transform in the same way as the gauge potential of a non-Abelian gauge theory: \begin{equation} \label{3.5} \omega(x)\rightarrow \Lambda(x)\omega(x)\Lambda^{-1}(x)- d\Lambda(x)\Lambda^{-1}(x). \end{equation} The curvature forms $\Omega=(\Omega^{\mu}_{\ \nu})$ are obtained from $\omega$ in exactly the same way as the Yang-Mills field strength from the gauge potential: \begin{equation} \label{3.6} \Omega=d\omega+\omega\wedge\omega \end{equation} (second structure equation). For a vector field $V$, with components $V^{\alpha}$ relative to $\{e_{\alpha}\}$, the covariant derivative $DV$ is given by \begin{equation} \label{3.7} DV^{\alpha}=dV^{\alpha}+\omega^{\alpha}_{\ \beta}V^{\beta}. \end{equation} Weyl generalizes this in a unique manner to spinor fields $\psi$: \begin{equation} \label{3.8} D\psi=d\psi+\frac{1}{4}\omega_{\alpha\beta}\sigma^{\alpha\beta}\psi. \end{equation} Here, the $\sigma^{\alpha\beta}$ describe infinitesimal Lorentz transformations (in the representation of $\psi$). For a Dirac field these are the familiar matrices \begin{equation} \label{3.9} \sigma^{\alpha\beta}=\frac{1}{2}[\gamma^{\alpha},\gamma^{\beta}]. \end{equation} (For 2-component Weyl fields one has similar expressions in terms of the Pauli matrices.) With these tools the action principle for the coupled Einstein-Dirac system can be set up. In the massless case the Lagrangian is \begin{equation} \label{3.10} {\cal L}=\frac{1}{16\pi G}R-i\bar{\psi}\gamma^{\mu}D_{\mu}\psi, \end{equation} where the first term is just the Einstein-Hilbert Lagrangian (which is linear in $\Omega$). Weyl discusses, of course, immediately the consequences of the following two symmetries: (i) local Lorentz invariance, (ii) general coordinate invariance. \subsection{The new form of the gauge-principle} All this is kind of a preparation for the final section of Weyl's paper, which has the title ``electric field''. Weyl says: \begin{quotation} {\sl We come now to the critical part of the theory. In my opinion the origin and necessity for the electromagnetic field is in the following. The components $\psi_{1}$ $\psi_{2}$ are, in fact, not uniquely determined by the tetrad but only to the extent that they can still be multiplied by an arbitrary ``gauge-factor'' $e^{i\lambda}$. The transformation of the $\psi$ induced by a rotation of the tetrad is determined only up to such a factor. In special relativity one must regard this gauge-factor as a constant because here we have only a single point-independent tetrad. Not so in general relativity; every point has its own tetrad and hence its own arbitrary gauge-factor; because by the removal of the rigid connection between tetrads at different points the gauge-factor necessarily becomes an arbitrary function of position.} \end{quotation} In this manner Weyl arrives at the gauge-principle in its modern form and emphasizes: ``From the arbitrariness of the gauge-factor in $\psi$ appears the necessity of introducing the electromagnetic potential.'' The first term $d\psi$ in (\ref{3.8}) has now to be replaced by the covariant gauge derivative $(d-ieA)\psi$ and the nonintegrable scale factor (\ref{2.1}) of the old theory is now replaced by a phase factor: $$ \exp\left(-\int_{\gamma}A\right)\rightarrow \exp\left(-i\int_{\gamma}A\right), $$ which corresponds to the replacement of the original gauge group {\bf R} by the compact group $U(1)$. Accordingly, the original Gedankenexperiment of Einstein translates now to the Aharonov-Bohm effect. The close connection between gauge invariance and conservation of charge is again uncovered. The current conservation follows, as in the original theory, in two independent ways: On the one hand it is a consequence of the field equations for matter plus gauge invariance, at the same time, however, also of the field equations for the electromagnetic field plus gauge invariance. This corresponds to an identity in the coupled system of field equations which has to exist as a result of gauge invariance. All this is nowadays familiar to students of physics and needs not to be explained in more detail. Much of Weyl's paper penetrated also into his classic book ``The Theory of Groups and Quantum Mechanics'' \cite{19}. There he mentions also the transformation of his early gauge-theoretic ideas: ``This principle of gauge invariance is quite analogous to that previously set up by the author, on speculative grounds, in order to arrive at a unified theory of gravitation and electricity. But I now believe that this gauge invariance does not tie together electricity and gravitation, but rather electricity and matter.'' When Pauli saw the full version of Weyl's paper he became more friendly and wrote \cite{20}: \begin{quotation} {\sl In contrast to the nasty things I said, the essential part of my last letter has since been overtaken, particularly by your paper in Z. f. Physik. For this reason I have afterward even regretted that I wrote to you. After studying your paper I believe that I have really understood what you wanted to do (this was not the case in respect of the little note in the Proc.Nat.Acad.). First let me emphasize that side of the matter concerning which I am in full agreement with you: your incorporation of spinor theory into gravitational theory. I am as dissatisfied as you are with distant parallelism and your proposal to let the tetrads rotate independently at different space-points is a true solution.} \end{quotation} In brackets Pauli adds: \begin{quotation} {\sl Here I must admit your ability in Physics. Your earlier theory with $g'_{ik}=\lambda g_{ik}$ was pure mathematics and unphysical. Einstein was justified in criticizing and scolding. Now the hour of your revenge has arrived.} \end{quotation} Then he remarks in connection with the mass-problem: \begin{quotation} {\sl Your method is valid even for the massive {\rm [Dirac]} case. I thereby come to the other side of the matter, namely the unsolved difficulties of the Dirac theory (two signs of $m_{0}$) and the question of the 2-component theory. In my opinion these problems will not be solved by gravitation \ldots the gravitational effects will always be much too small.} \end{quotation} Many years later, Weyl summarized this early tortuous history of gauge theory in an instructive letter to the Swiss writer and Einstein biographer C.Seelig, which I reproduce in the German original \cite{21}. \begin{quotation} {\sl Aus dem Jahre 1918 datiert der von mir unternommene erste Versuch, eine einheitliche Feldtheorie von Gravitation und Elektromagnetismus zu entwickeln, und zwar auf Grund des Prinzips der Eichinvarianz, das ich neben dasjenige der Koordinaten-Invarianz stellte. Ich habe diese Theorie selber l\"angst aufgegeben, nachdem ihr richtiger Kern: die Eichinvarianz, in die Quantentheorie her\"uberge- rettet ist als ein Prinzip, das nicht die Gravitation, sondern das Wellenfeld des Elektrons mit dem elektromagnetischen verkn\"upft. --- Einstein war von Anfang dagegen, und das gab zu mancher Diskussion Anlass. Seinen konkreten Einw\"anden glaubte ich begegnen zu k\"onen. Schliesslich sagte er dann: ``Na, Weyl, lassen wir das! So --- das heisst auf so spekulative Weise, ohne ein leitendes, anschauliches physikalisches Prinzip --- macht man keine Physik!'' Heute haben wir in dieser Hinsicht unsere Standpunkte wohl vertauscht. Einstein glaubt, dass auf diesem Gebiet die Kluft zwischen Idee und Erfahrung so gross ist, dass nur der Weg der mathematischen Spekulation, deren Konsequenzen nat\"urlich entwichelt und mit den Tatsachen konfrontiert werden m\"ussen, Aussicht auf Erfolg hat, w\"ahrend mein Vertrauen in die reine Spekulation gesunken ist und mir ein engerer Anschluss an die quanten-physikalischen Erfahrungen geboten scheint, zumal es nach meiner Ansicht nicht genug ist, Gravitation und Elektromagnetismus zu einer Einheit zu verschmelzen. Die Wellenfelder des Elektrons und was es sonst noch an unreduzierbaren Elementarteilchen geben mag, m\"ussen mit eigeschlossen werden.} \end{quotation} \section{Yang-Mills Theory} In his Hermann Weyl Centenary Lecture at the ETH \cite{22}, C.N. Yang commented on Weyl's remark ``The principle of gauge-invariance has the character of general relativity since it contains an arbitrary function $\lambda$, and can certainly only be understood in terms of it'' \cite{23} as follows: \begin{quotation} {\sl The quote above from Weyl's paper also contains something which is very revealing, namely, his strong association of gauge invariance with general relativity. That was, of course, natural since the idea had originated in the first place with Weyl's attempt in 1918 to unify electromagnetism with gravity. Twenty years later, when Mills and I worked on non-Abelian gauge fields, our motivation was completely divorced from general relativity and we did not appreciate that gauge fields and general relativity are somehow related. Only in the late 1960's did I recognize the structural similarity mathematically of non-Abelian gauge fields with general relativity and understand that they both were connections mathematically.} \end{quotation} Later, in connection with Weyl's strong emphasis of the relation between gauge invariance and conservation of electric charge, Yang continues with the following instructive remarks: \begin{quotation} {\sl Weyl's reason, it turns out, was also one of the melodies of gauge theory that had very much appealed to me when as a graduate student I studied field theory by reading Pauli's articles. I made a number of unsuccessful attempts to generalize gauge theory beyond electromagnetism, leading finally in 1954 to a collaboration with Mills in which we developed a non-Abelian gauge theory. In [\ldots ] we stated our motivation as follows: The conservation of isotopic spin points to the existence of a fundamental invariance law similar to the conservation of electric charge. In the latter case, the electric charge serves as a source of electromagnetic field; an important concept in this case is gauge invariance which is closely connected with (1) the equation of motion of the electro-magnetic field, (2) the existence of a current density, and (3) the possible interactions between a charged field and the electromagnetic field. We have tried to generalize this concept of gauge invariance to apply to isotopic spin conservation. It turns out that a very natural generalization is possible. Item (2) is the melody referred to above. The other two melodies, (1) and (3), where what had become pressing in the early 1950's when so many new particles had been discovered and physicists had to understand now they interact which each other. I had met Weyl in 1949 when I went to the Institute for Advanced Study in Princeton as a young ``member''. I saw him from time to time in the next years, 1949--1955. He was very approachable, but I don't remember having discussed physics or mathematics with him at any time. His continued interest in the idea of gauge fields was not known among the physicists. Neither Oppenheimer nor Pauli ever mentioned it. I suspect they also did not tell Weyl of the 1954 papers of Mills' and mine. Had they done that, or had Weyl somehow came across our paper, I imagine he would have been pleased and excited, for we had put together two things that were very close to his heart: gauge invariance and non-Abelian Lie groups.} \end{quotation} It is indeed astonishing that during those late years Pauli never talked with Weyl on non-Abelian generalizations of gauge-invariance, since he himself had worked on this --- even before the work of Yang and Mills. During a discussion following a talk by Pais at the 1953 Lorentz Conference \cite{24} in Leiden, Pauli said: \begin{quotation} {\sl \ldots I would like to ask in this connection whether the transformation group [isospin] with constant phases can be amplified in a way analogous to the gauge group for electromagnetic potentials in such a way that the meson-nucleon interaction is connected with the amplified group \ldots } \end{quotation} Stimulated by this discussion, Pauli worked on this problem and drafted a manuscript to Pais that begins with \cite{8}: \begin{quotation} {\sl Written down July 22-25, 1953, in order to see how it looks.} {\it Meson-Nucleon Interaction and Differential Geometry.} \end{quotation} Unaware of Klein's earlier contribution \cite{7}, Pauli generalizes in this manuscript the Kaluza-Klein theory to a sixdimensional space, and arrives through dimensional reduction at the essentials of an $SU(2)$ gauge theory. The extra-dimensions are two-spheres with spacetime dependent metrics on which $SU(2)$ operates in a spacetime dependent manner. Pauli develops first in ``local language'' the geometry of what we now call a fiber bundle with a homogeneous space as typical fiber (in his case $S^{2}\cong SU(2)/U(1)$). Studying the curvature of the higher dimensional space, Pauli automatically finds for the first time the correct expression for the non-Abelian field strengths. Afterwards, Pauli sets up the 6-dimensional Dirac equation and writes it out in an explicit manner which is adapted to the fibration. Later, in December 1953, he sends a ``Mathematical Appendix'' to Pais and determines --- among other things --- the mass spectrum implied by this equation. The final sentence reads: ``So this leads to some rather unphysical `shadow particles'.'' Pauli did not write down a Lagrangian for the gauge fields, but as we shall see shortly, it was clear to him that the gauge bosons had to be massless. This, beside the curious fermion spectrum, must have been the reason why he did not publish anything. With this background, the following story of spring 1954 becomes more understandable. In late February, Yang was invited by Oppenheimer to return to Princeton for a few days and to give a seminar on his joint work with Mills. Here, Yang's report \cite{25}: \begin{quotation} {\sl Pauli was spending the year in Princeton, and was deeply interested in symmetries and interactions. (He had written in German a rough outline of some thoughts, which he had sent to A. Pais. Years later F.J. Dyson translated this outline into English. It started with the remark, ``Written down July 22-25, 1953, in order to see how it looks,'' and had the title ``Meson-Nucleon Interaction and Differential Geometry.'') Soon after my seminar began, when I had written down on the blackboard, $$ (\partial_{\mu}-i\epsilon B_{\mu})\psi, $$ Pauli asked, ``What is the mass of this field $B_{\mu}$?'' I said we did not know. Then I resumed my presentation, but soon Pauli asked the same question again. I said something to the effect that that was a very complicated problem, we had worked on it and had come to no definite conclusions. I still remember his repartee: ``That is not sufficient excuse.'' I was so taken aback that I decided, after a few moments' hesitation to sit down. There was general embarrassment. Finally Oppenheimer said, ``We should let Frank proceed.'' I then resumed, and Pauli did not ask any more questions during the seminar. I don't remember what happened at the end of the seminar. But the next day I found the following message: February 24, Dear Yang, I regret that you made it almost impossible for me to talk with you after the seminar. All good wishes. Sincerely yours, W.Pauli. I went to talk to Pauli. He said I should look up a paper by E. Schr\"odinger, in which there were similar mathematics\footnote{E. Schr\"odinger, Sitzungsberichte der Preussischen (Akademie der Wissenschaften, 1932), p. 105.}. After I went back to Brookhaven, I looked for the paper and finally obtained a copy. It was a discussion of space-time-dependent representations of the $\gamma_{\mu}$ matrices for a Dirac electron in a gravitational field. Equations in it were, on the one hand, related to equations in Riemannian geometry and, on the other, similar to the equations that Mills and I were working on. But it was many years later when I understood that these were all different cases of the mathematical theory of connections on fiber bundles.} \end{quotation} Later Yang adds: \begin{quotation} {\sl I often wondered what he [Pauli] would say about the subject if he had lived into the sixties and seventies.} \end{quotation} At another occasion \cite{22} he remarked: \begin{quotation} {\sl I venture to say that if Weyl were to come back today, he would find that amidst the very exciting, complicated and detailed developments in both physics and mathematics, there are fundamental things that he would feel very much at home with. He had helped to create them.} \end{quotation} Having quoted earlier letters from Pauli to Weyl, I add what Weyl said about Pauli in 1946 \cite{26}: \begin{quotation} {\sl The mathematicians feel near to Pauli since he is distinguished among physicists by his highly developed organ for mathematics. Even so, he is a physicist; for he has to a high degree what makes the physicist; the genuine interest in the experimental facts in all their puzzling complexity. His accurate, instructive estimate of the relative weight of relevant experimental facts has been an unfailing guide for him in his theoretical investigations. Pauli combines in an exemplary way physical insight and mathematical skill.} \end{quotation} To conclude this section, let me emphasize the main differences of GR and Yang-Mills theories. Mathematically, the $so(1,3)$-valued connection forms $\omega$ in $\S 3.1$ and the Liealgebra-valued gauge potential $A$ are on the same footing; they are both representatives of connections in (principle) fiber bundles over the spacetime manifold. Eq.(\ref{3.6}) translates into the formula for the Yang-Mills field strength $F$, \begin{equation} \label{4:1} F=dA+A\wedge A. \end{equation} In GR one has, however, additional geometrical structure, since the connection derives from a metric, or the tetrad fields $e^{\alpha}(x)$, through the first structure equation (\ref{3.4}). Schematically, we have: \begin{figure} \begin{center} \epsfig{file=fig4.eps,width=6cm} \caption{General Relativity versus Yang-Mills theory.} \end{center} \end{figure} (In bundle theoretical language one can express this as follows: The principle bundle of GR, i.e., the orthonormal frame bundle, is constructed from the base manifold and its metric, and has therefore additional structure, implying in particular the existence of a canonical 1-form (soldering form), whose local representative are the tetrad fields; see, e.g. \cite{Bleecker}.) Another important difference is that the gravitational Lagrangian $\ast R=\frac{1}{2}\Omega_{\alpha\beta}\wedge\ast (e^{\alpha}\wedge e^{\beta})$ is linear in the field strengths, whereas the Yang-Mills Lagrangian $F\wedge\ast F$ is quadratic. \section{Parity Violation and 2-Component Neutrino} The two-component spinor theory was only briefly mentioned in my discussion of Weyl's great 1929 paper. Since this massless spin $1/2$ equation became very important after the discovery of parity violation I would now like to add a few remarks. Due to the fact that there exist two inequivalent irreducible (projective) representations of the one-component of the homogeneous Lorentz group, $L^{\uparrow}_{+}$ (with $SL(2,C)$ as universal covering group), there are two types of fundamental Weyl spinors, $\phi_{\alpha}$ and $\chi{\dot{\beta}}$, for which the free Weyl equations read as follows: \begin{equation} \label{5.1} \hat{\sigma}^{\mu}\partial_{\mu}\phi=0,\ \ \ \ \sigma^{\mu}\partial_{\mu}\chi=0. \end{equation} Here, $(\sigma^{\mu})=(1 \! {\rm I},-\vec{\sigma})$, $(\hat{\sigma}^{\mu})=(1 \! {\rm I},\vec{\sigma})$ ($\vec{\sigma}$: Pauli matrices). In spinor calculus these equations become \begin{equation} \label{5.2} \partial^{\alpha\dot{\beta}}\phi_{\alpha}=0,\ \ \ \partial_{\alpha\dot{\beta}}\chi^{\dot{\beta}}=0. \end{equation} In his ``New Testament'' from 1933 \cite{27}, Pauli rejected these equations: ``Indessen sind diese Wellengleichungen, wie ja aus ihrer Herleitung hervorgeht, nicht invariant gegen\"uber Spiegelungen (Vertauschlung von links und recht) und infolgedessen sind sie auf die physikalische Wirklichkeit nicht anwendbar.'' However, as long as no interactions are taken into account, this statement is not correct. To make this evident one only has to note that both equations in (\ref{5.1}) are equivalent to the Majorana formulation: Consider, for instance, the $\phi$-field and set \begin{equation} \label{5.3} \psi=\left(\begin{array}{c} \phi \\ \varepsilon\phi^{\ast} \end{array}\right),\ \ \ \ \ \ \varepsilon=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), \end{equation} then the first equation in (\ref{5.1}) is equivalent to the massless Dirac equation, \begin{equation} \label{5.4} \gamma^{\mu}\partial_{\mu}\psi=0. \end{equation} Furthermore, $\psi$ is self-conjugate: A general Dirac spinor $\left(\begin{array}{c} \phi_{\alpha} \\ \chi^{\dot{\beta}} \end{array}\right)$ transforms under charge conjugation $C$ according to \begin{equation} \label{5.5} C: \left(\begin{array}{c} \phi \\ \chi \end{array}\right) \rightarrow \left(\begin{array}{c} -\varepsilon\chi^{\ast} \\ \varepsilon\phi^{\ast} \end{array}\right) \end{equation} and, for (\ref{5.3}), this reduces to the Majorana condition $C:\psi\rightarrow\psi$. Nobody would say that the Majorana theory is not reflection invariant. Note in this connection also the following: A Dirac field transforms under $P$ as \begin{equation} \label{5.6} P:\psi\rightarrow\psi'(x)=\gamma^{0}\psi(Px). \end{equation} For the Majorana field (\ref{5.3}) this translates into an antilinear transformation for $\phi$, \begin{equation} \label{5.7} P:\phi\rightarrow\phi'(x)=\varepsilon\phi^{\ast}(Px), \end{equation} which leaves the Weyl equation invariant. Usually this operation is interpreted as $CP$, but without interactions this is a matter of semantics. Before I will return to history, let me also remind you of the formulation of Lee and Yang \cite{28}. These authors introduce in the Weyl representation of the $\gamma$-matrices the Dirac spinor $\psi=\left(\begin{array}{c} \phi \\ 0 \end{array}\right)$, whence $(1-\gamma^{5})\psi=0$. The first Weyl equation in (\ref{5.1}) is then again equivalent to the massless Dirac equation (\ref{5.4}). In the Lee-Yang formulation one thus has \begin{equation} \label{5.8} \gamma^{\mu}\partial_{\mu}\psi=0,\ \ \ \ \ (1-\gamma^{5})\psi=0. \end{equation} These equations are, of course, independent of the representation of the $\gamma$-algebra. Thus, the three formulations of Weyl, Majorana, and Lee-Yang are entirely equivalent. This was noticed by several authors \cite{28} shortly after the discovery of parity violation, but had been worked out by J.Serpe \cite{29} already in 1952. Today, because of the chiral nature of the fundamental fermions, the use of Weyl spinors has become common practice. The discovery of parity violation early in 1957 in several experiments suggested by Lee and Yang \cite{30} was one of the most exciting events in the fifties. Its impact was enormous, as is illustrated by the following letter from Pauli to Weisskopf \cite{31}: \begin{quotation} {\sl \noindent Dear Weisskopf, Now the first shock is over and I begin to collect myself again (as one says in Munich). Yes, it was very dramatic. On Monday 21st at 8:15 p.m. I was supposed to give a talk about ``past and recent history of the neutrino''. At 5 p.m. the mail brought me three experimental papers: C.S. Wu, Lederman and Telegdi; the latter was so kind to send them to me. The same morning I received two theoretical papers, one by Yang, Lee and Oehme, the second by Yang and Lee about the two-component spinor theory. The latter was essentially identical with the paper by Salam, which I received as a preprint already six to eight weeks ago and to which I referred in my last short letter to you. (Was this paper known in the USA?) ( At the same time came a letter from Geneva by Villars with the New York Times article.) Now, where shall I start? It is good that I did not make a bet. I would have resulted in a heavy loss of money (which I cannot afford); I did make a fool of myself, however (which I think I can afford to do)--- incidentally, only in letters or orally and not in anything that was printed. But the others now have the right to laugh at me. What shocks me is not the fact that ``God is just left-handed'' but the fact that in spite of this He exhibits Himself as left/right symmetric when He expresses Himself strongly. In short, the real problem now is why the strong interaction are left/right symmetric. How can the strength of an interaction produce or create symmetry groups, invariances or conservation laws? This question prompted me to my premature and wrong prognosis. I don't know any good answer to that question but one should consider that already there exists a precedent: the rotation group in isotopic spin-space, which is not valid for the electromagnetic field. One does not understand why it is valid at all. It seems that there is a certain analogy here! In my lecture I described how Bohr (Faraday lecture, 1932, Solvay Conference, 1932), as my main opponent in regard to the neutrino, considered plausible the violation of the energy law in the beta-decay (what one calls today ``weak interaction''), how his opposition then became weaker and how he said in a more general way (1933) that one must be ``prepared for surprises'' not anywhere but specifically with the beta-decay. Then I said spontaneously (on the spur of the moment) that at the end of my talk I would come back to the surprises which Professor Bohr had foreseen here \ldots Many questions, no answers !} \end{quotation} Let me say a bit more about the paper of Salam which is mentioned in Pauli's letter. In September 1956 Salam had heard Yang's talk at the Seattle Conference on his and Lee's famous solution of the $\vartheta -\tau$ puzzle by abandoning left/right symmetry in weak interactions. In his Nobel Price lecture Salam recollects \cite{32}: \begin{quotation} {\sl I remember travelling back to London on an American Air Force (MATS) transport flight. Although I had been granted, for that night, the status of a Brigadier or a Field Marshal --- I don't quite remember which --- the plane was very uncomfortable, full of crying servicemen's children --- that is, the children were crying, not the servicemen. I could not sleep. I kept reflecting on why Nature should violate left/right symmetry in weak interactions. Now the hallmark of most weak interactions was the involvement in radioactivity phenomena of Pauli's neutrino. While crossing over the Atlantic came back to me a deeply perceptive question about the neutrino which Professor Rudolf Peierls had asked when he was examining me for a Ph.D. a few years before. Peierls' question was: ``The photon mass is zero because of Maxwell's principle of a gauge symmetry for electromagnetism; tell me, why is the neutrino mass zero?'' } \end{quotation} During that comfortless night he realized that Weyl's two-component equation for the neutrino would account for both parity violation and the masslessness of the neutrino. Soon afterwards he presented the idea to Peierls, who replied: ``I do not believe left/right symmetry is violated in weak forces at all.'' After that, Salam was hoping to find more resonance at CERN. There he communicated the idea to Pauli, through Villars, who ``returned the next day with a message of the Oracle: Give my regards to my friend Salam and tell him to think of something better.'' Meanwhile parity violation was discovered and Salam got a kind, apologetic letter from Pauli. But this changed again soon afterwards. I quote: \begin{quotation} {\sl Thinking that Pauli's spirit should by now be suitably crushed, I sent him two short notes (Salam, 1957b) I had written in the meantime. These contained suggestions to extend chiral symmetry to electrons and muons, assuming that their masses were a consequence of what has come to be known as dynamical spontaneous symmetry breaking. With chiral symmetry for electrons, muons, and neutrinos, the only mesons that could mediate weak decays of the muons would have to carry spin one. Reviving thus the notion of charged intermediate spin-one bosons, one could then postulate for these a type of gauge invariance which I called the ``neutrino gauge''. Pauli's reaction was swift and terrible. He wrote on 30 January 1957, then on 18 February and later on 11, 12 and 13 March: ``I am reading (along the shores of Lake Zurich) in bright sunshine quietly your paper \ldots'' ``I am very much startled on the title of your paper `Universal Fermi Interaction' \ldots For quite a while I have for myself the rule if a theoretician says universal it just means pure nonsense. This holds particularly in connection with the Fermi interaction, but otherwise too, and now you too, Brutus, my son, come with this word \ldots'' Earlier, on 30 January, he had written: ``There is a similarity between this type of gauge invariance and that which was published by Yang and Mills... In the latter, of course, no $\gamma_{5}$ was used in the exponent,'' and he gave me the full reference of Yang and Mills' paper \cite{18}. I quote from this letter: ``However, there are dark points in your paper regarding the vector field $B_{\mu}$. If the rest mass is infinite (or very large), how can this be compatible with the gauge transformation $B_{\mu}\rightarrow B_{\mu}-\partial_{\mu}\Lambda $ ?'' and he concludes his letter with the remark: ``Every reader will realize that you deliberately conceal here something and will ask you the same questions.} \end{quotation} \section{Chiral Invariance and Universal $V$--$A$ Interaction} These recollections bring me to the last subject of my lecture. The two-component model of the neutrino paved also the way for a successful phenomenological description of weak interaction processes at low energies. In his masterly written review article ``On the earlier and more recent history of the Neutrino'' \cite{33}, Pauli remarks: \begin{quotation} {\sl For some time I faced this particular model with a certain skepticism [42], since it seemed to me that the special role of the neutrino was emphasized too strongly. It turned out, however, that by further developing the ideas of Stech and Jensen (see $\S 3$ above) the model allowed an interesting generalization for the form of the interaction energy for all weak interactions.} \end{quotation} After an inventory of the experimental situation, mentioning in particular the new recoil experiments on ${\rm ^{6}He}$, Pauli continues with: \begin{quotation} {\sl Based on the Stech-Jensen transformation and the two-component model of the neutrino the following postulate suggests itself for the theoretical interpretation:} {\it The Hamiltonian of each weak 4-fermion interaction shall ``universally'' contain either only R or only L components of the involved fermions.} {\sl Equivalent to this postulate is the formulation that in the transformation $\psi'=\gamma_{5}\psi$ the density of the interaction energy for each particle separately should ``universally'' remain unchanged or change its sign.} \end{quotation} At this point the classical papers \cite{34} are quoted, followed by the statement: \begin{quotation} {\sl The Stech-Jensen transformation referred to a pair of the particles simultaneously while the two-component model of the neutrino is equivalent to the validity of the result of the transformation for the neutrino alone. The} {\it postulate of the extended Stech-Jensen transformation now under discussion is therefore a generalization of the two-component model of the neutrino.} \end{quotation} As we all know this postulate leads uniquely to the universal $V$--$A$ interaction. At the time it was disturbing that the $V$ and $A$ interaction strengths for nucleons in beta decay are empirically not equal. Today we know that the equality does hold on the level of the quark fields. It is, unfortunately, not generally known that W. Theis proposed independently the parity violating V-A interaction in a paper submitted on 20 December 1957 to the {\it Zeitschrift f\"ur Physik} \cite{35}. Theis emphasized that in the spinor calculus a Dirac spinor can be expressed in terms of a single two-component Weyl spinor \begin{equation} \label{6.1} \psi=\left(\begin{array}{c} \phi_{\alpha} \\ \frac{i}{m}\partial^{\alpha\dot{\beta}}\phi_{\alpha} \end{array}\right), \end{equation} and that the Dirac equation is then equivalent to the Klein-Gordon equation for $\phi_{\alpha}$. Since in this representation $\psi$ contains derivatives, the author finds Fermi's requirement of a derivative-free coupling not so convincing and requires instead a derivative-free four-Fermi interaction for the Weyl spinors. This allows for only one possibility, namely \begin{equation} \label{6.2} p^{\ast}_{\alpha}n_{\dot{\beta}}e^{\ast\alpha}\nu^{\dot{\beta}} +{\rm h.c.}, \end{equation} which is just the $V$--$A$ coupling. This formal argument is similar to the one in the classic paper by Feynman and Gell-Mann \cite{34}. The latter goes, however, beyond the $V$--$A$ interaction and advocates a current-current interaction Lagrangian, containing also hypothetical self-terms. These imply processes like neutrino-electron scattering or the annihilation process $e^{-}+e^{+}\rightarrow\nu+\bar{\nu}$, which was soon recognized to be very important in the later evolutionary stages of massive stars \cite{36}. (We have heard a lot about this during the school.) It may also not be known to the young generation that various experiments\footnote{For a description of the classic experiments, I refer to an excellent paper by Telegdi \cite{37}.} were in conflict with chiral invariance at the time when Feynman and Gell-Mann wrote their paper. They had the courage to question the correctness of these experiments: \begin{quotation} {\sl These theoretical arguments seem to the authors to be strong enough to suggest that the disagreement with the $^{6}He$ recoil experiment and with some other less accurate experiments indicates that these experiments are wrong. The $\pi\rightarrow e+\bar{\nu}$ problem may have a more subtle solution.} \end{quotation} The later verification of the prediction for the ratio $\Gamma(\pi\rightarrow e\nu)/ \Gamma(\pi\rightarrow\mu\nu)$ was one of the triumphs of the universal $V$--$A$ interaction. We will certainly hear more from J. Steinberger about the experimental side of the story. \section{Epilogue} The developments after 1958 consisted in the gradual recognition that --- contrary to phenomenological appearances --- Yang-Mills gauge theory can describe weak and strong interactions. This important step was again very difficult, with many hurdles to overcome. One of them was the mass problem which was solved, perhaps in a preliminary way, through spontaneous symmetry breaking. Of critical significance was the recognition that spontaneously broken gauge theories are renormalizable. On the experimental side the discovery and intensive investigation of the neutral current was, of course, extremely crucial. For the gauge description of the strong interactions, the discovery of asymptotic freedom was decisive . That the $SU(3)$ color group should be gauged was also not at all obvious. And then there was the confinement idea which explains why quarks and gluons do not exist as free particles. All this is described in numerous modern text books and does not have to be repeated. The next step of creating a more unified theory of the basic interactions will probably be much more difficult. All major theoretical developments of the last twenty years, such as grand unification, supergravity and supersymmetric string theory are almost completely separated from experience. There is a great danger that theoreticians get lost in pure speculations. Like in the first unification proposal of Hermann Weyl they may create beautiful and highly relevant mathematics which does, however, not describe nature. Remember what Weyl wrote to C. Seelig in his late years: \begin{quotation} {\sl Einstein glaubt, dass auf diesem Gebiet die Kluft zwischen Idee und Erfahrung so gross ist, dass nur der Weg der mathematischen Spekulation (\ldots) Aussicht auf Erfolg hat, w\"ahrend mein Vertrauen in die reine Spekulation gesunken ist \ldots } \end{quotation}

proofpile-arXiv_065-536

\section*{Acknowledgments} This work was supported in part by National Science Foundation Grants PHY9507313 (BKB) and PHY9408439 (DG) to Oakland University (OU). DG was also supported in part by a Cottrell College Science Award of Research Corporation to OU. BKB would like to thank the Institute for Geophysics and Planetary Physics at Lawrence Livermore National Laboratory and the Department of Astronomy at the University of Michigan for hospitality. Part of this work was performed by ES in partial fulfillment of the requirements for the M.S.~in Physics degree at OU.

proofpile-arXiv_065-537

\section{Introduction} One of the most promising candidates for physics beyond the so-called Standard Model (SM) is that of supersymmetry (SUSY) \cite{revs}. SUSY has the highly attractive properties of giving a natural explanation to the hierarchy problem of how it is possible to have a low energy theory containing light scalars (the Higgs) when the ultimate theory must include states with masses of order the Planck mass. This allows the ultimate hope of constructing a theory of gauge unification and fermion (and superpartner) masses defined at a scale near the Planck mass yet whose structure and parameters may be deduced from physics at accessible scales. In this paper we shall be concerned with the implications of a particular possible feature of SUSY, namely that of $R$-Parity violation (RPV) \cite{rpv,barbm,suzuki}. $R$-Parity is a $Z\!\!\! Z_2$ symmetry of both the SM and its minimal SUSY extension, the MSSM, under which all of the SM particles have charge 0, while all their SUSY partners have charge 1. Its implications include the stability of the lightest supersymmetric particle (LSP), and hence the typical SUSY collider signatures of missing $E_T$ and the existence of a source of dark matter. Its violation changes both the implied cosmology and the expected collider signatures, allowing such effects as LSPs decaying inside the detector and leptoquarks \cite{bhat}. In addition to these, further constraints on RPV can be derived by considering experimental limits on rare decays \cite{bgh,bpw,bgnn}, and by the demands of proton stability. In practice it is usual to evade the problems of proton decay by considering either baryon number or lepton number violation but not both simultaneously. $R$-Parity is violated by the superpotential and soft potential \begin{eqnarray} W&=& h^u_{ij}Q_iH_2u_j + h^d_{ij} Q_iH_1d_j + h^e_{ij} L_iH_1e_j \cr &&+ \frac{1}{2}\l{ijk}L_iL_je_k + \lp{ijk}L_iQ_jd_k + \frac{1}{2}\lpp{ijk}u_id_jd_k \cr &&+ \mu_4 H_1H_2 + \mu_iL_iH_2 \cr V_{\rm soft}&=& \eta^u_{ij}Q_iH_2u_j + \eta^d_{ij} Q_iH_1d_j + \eta^e_{ij}L_iH_1e_j + {\rm h.c.} \cr &&+ \frac{1}{2}\C{ijk}L_iL_je_k + \Cp{ijk}L_iQ_jd_k + \frac{1}{2}\Cpp{ijk}u_id_jd_k + {\rm h.c.} \cr &&+ \frac{1}{2}M_a\lambda_a^c\lambda_a + \sum_{a,b} m_{ab}^2 \varphi_a\bar\varphi_b \cr &&+ D_4 H_1H_2 + D_iL_iH_2 + {\rm h.c.} \end{eqnarray} From the point of view of deriving constraints on the $R$-Parity violating couplings in the model, the most extensively studied couplings are the dimensionless couplings $\l{}$, $\lp{}$, and $\lpp{}$, which directly generate many effects which can be experimentally limited. The extra soft terms by definition mostly couple only heavy SUSY particles and hence are relevant mostly because of their impact on the Renormalisation Group Equations (RGEs), although they can have significant effects on the neutrino-neutralino and Higgs boson-sneutrino sectors \cite{suzuki,rv,paper1}. In SUSY models, flavour-changing effects may be caused by the existence of off-diagonal terms in the sfermion mass matrices in the basis in which the fermion masses are diagonal, in which there are no tree level SM flavour-changing neutral currents (FCNC). Such flavour-violating soft masses can be generated either from the high energy theory such a GUT directly, or else through the RGEs by couplings which violate flavour symmetries, such as Yukawa couplings mediated by the CKM matrix and both possibilities have been studied extensively \cite{hagelin,otherfcnc}. However, the inclusion of the RPV couplings in the RGEs allows many extra violations of quark and lepton flavour. In a previous analysis \cite{paper1}, we presented the renormalisation group equations (RGEs) for the couplings of the full $R$-Parity violating sector of the model, and investigated the implications of typical scenarios at the GUT scale for the generation of neutrino masses and the decay $\mu\to e\gamma$. Contributions can conveniently be split into two categories which we term ``direct'' (where the flavour violation occurs directly through $R$-Parity violating vertices in the diagrams) and ``indirect'' (where $R$-Parity violation induces flavour violation through the RGEs on the soft masses). We found that the indirect effects were typically large, often orders of magnitude larger than the direct ones. However, the extremely complicated dependence on the spectrum and the existence of many possible cancellations in the amplitude render the process of deriving bounds impossible except on an order of magnitude basis. The intention of this paper is to extend the work of reference \cite{paper1} to consider quark flavour violation (QFV) effects (for recent work on the subject see ref.~\cite{rel}) . Following this introduction, we give a short discussion on a new bound from requiring the sneutrino masses to be above their experimental limits in section 2, while section 3 is devoted to the effects of RPV on $b\to s\gamma$, and section 4 to the $K^0-\bar K^0$ mixing term $\Delta m_K$. Many useful formulae for our RGEs and definitions are relegated to the appendices, while section 5 contains our conclusions. \section{Bounds from Sneutrino Masses} We now comment on a new simple bound on certain of the Yukawa couplings, which is associated with our assumptions of unification of couplings at a high scale. If we add a new Yukawa coupling associated with the sneutrinos (here $\lp{}$), the mass of the sneutrino is driven down by its effect on the RGEs, and so increasing the value of the coupling will ultimately drive the mass of the sneutrino below the experimental limit. The low energy values of the lepton masses and the corresponding soft masses are given by \begin{eqnarray} m^2_{L_i}&=& m_0^2 + 0.51 M_{1/2}^2 - \sum_{jk}\lp{ijk}^2(M_{GUT}) \bigl [ 13m_0^2 + 49M_{1/2}^2 - 1.5 M_{1/2}A_0 - 12 A_0^2\bigr ] \cr m^2_{\tilde\nu_i}&=& m^2_{L_i} + \frac{1}{2}M_Z^2\cos2\beta \end{eqnarray} where we have assumed universal masses at the unification scale and solved the RGEs numerically. The experimental limit on sneutrinos is 37 GeV \cite{pdb} (assuming that the sneutrinos are not degenerate), so that we conclude that the Yukawa couplings $\lp{}$ must be bounded by \begin{equation} \sum_{jk}\lp{ijk}^2(M_{GUT}) < \frac {m_0^2 + 0.51 M_{1/2}^2 + \frac{1}{2}M_Z^2\cos2\beta - (37\hbox{GeV})^2} {13m_0^2 + 49M_{1/2}^2 - 1.5 M_{1/2}A_0 - 12 A_0^2} \label{sneutmassbound} \end{equation} Given that masses for squarks and sleptons as low as 100 GeV are barely tenable in a unified framework with present collider limits, this constraint is the tightest available on certain $\lp{}$. For example, if we set $A_0\simeq 0$, $M_{1/2}=m_0=200$ GeV and $\tan\beta=10$ then this gives a bound on all of the $\lp{}$ of 0.15, while if two of the $\lp{}$ are equal then they must be less than 0.11, bounds which become only very slightly weaker with increasing soft masses, while the corresponding numbers with $m_0$ and $M_{1/2}$ both 100 GeV are 0.12 and 0.09. Note that these couplings are at the GUT scale, and the electroweak scale values are then around three times larger, while there is an error of at least 10 to 20\% from the uncertainty in the value of the strong coupling. Although we have neglected the effect of the tau Yukawa coupling, this in fact only makes the bound for the third generation rather tighter by reducing the sneutrino mass for that generation still further. We can also derive bounds on the $\l{}$ and $\lpp{}$ couplings from similar arguments, but the results are not very restrictive. \section{$b\to s\gamma$} \subsection{Contributions} The rare decay process $b\to s\gamma$ has a branching rate which can be deduced from the decay $B\to K^*\gamma$ which has been measured \cite{CLEO} to obtain a value \begin{equation} \hbox{B}(b\to s\gamma)=(2.32\pm 0.57 \pm 0.35) \times 10^{-4} \end{equation} consistent with the standard model result, and hence a 95\% confidence level limit of \begin{equation} 1.0\times 10^{-4} < \hbox{B}(b\to s\gamma) < 4.2 \times 10^{-4} \label{bsglimit} \end{equation} This gives a very strong test of new physics such as supersymmetry, since SUSY models with light spectra can give large contributions to this process through charged Higgs and chargino diagrams \cite{bsgsusy}. The total branching ratio for the process $b \rightarrow s \gamma$ can be written (in units of the BR for the semileptonic $b$ decay) as: \begin{equation} \frac{\hbox{B}(b \rightarrow s \gamma)}{\hbox{B}(b \rightarrow c e \bar{\nu})} = \frac{3 \pi \alpha}{G_F^2 |K_{cb}|^2 I(z)} \left(|\tilde{A}_{LR}|^2 + |\tilde{A}_{RL}|^2 \right) F \label{br} \end{equation} where $G_F$ is the Fermi constant, $z=m_c/m_b$, $I(z) = 1- 8z^2+ 8z^6- 24z^4\log(z)$ is the phase space factor, $K_{ij}$ will be the different CKM matrix elements, $\tilde{A}_{LR}$, $\tilde{A}_{RL}$ are the total amplitudes to the LR and RL transitions respectively, and \begin{equation} F \sim \left( 1 - \frac{8}{3} \frac{\alpha_s(m_b)}{\pi} \right) \frac{1}{\kappa(z)} \end{equation} contains NLO effects ($\kappa(z)$ being the NLO correction to the semileptonic decay). Here \begin{equation} \tilde{A}_i = \eta^{16/23} \tilde{A}_i^{\gamma} + \frac{8}{3} (\eta^{14/23}-\eta^{16/23}) \tilde{A}_i^{g} + C \tilde{A}_i^0 \end{equation} with $i=LR,RL$, $\eta=\alpha_s(M_W)/\alpha_s(m_b)$ and the different terms are as follows: $\tilde A_{LR,RL}^{\gamma,g}$ are the coefficients of the effective operators for the $bs\gamma$ and $bsg$ interactions \begin{eqnarray} C^{\gamma}_{LR,RL}&=& \frac{e}{4\pi}m_b(\bar s\sigma^{\mu\nu}P_{R,L}b)F_{\mu\nu} \cr C^g_{LR,RL}&=& \frac{g_3}{4\pi}m_b(\bar s^i\sigma^{\mu\nu}P_{R,L}b_j) G^a_{\mu\nu}T^a_i{}^j \end{eqnarray} $\tilde A_i^0=-\alpha_W K^*_{ts} K_{tb}/M_W^2$ is from the coefficient of an operator \begin{equation} C^0_i= [\bar sP_Lc] [\bar cP_Lb] \;\;, \end{equation} and $C$ stands for the leading logarithmic QCD corrections (for a complete list of references see \cite{bsgQCD}). The LR amplitudes can be divided into an $R$-Parity conserving part plus an RPV one; the former has been calculated in ref.~\cite{bert}, and corresponds mainly to the contributions coming from the SM diagram plus those with top quark and charged Higgs, and stops/scharms and charginos running in the loop, plus smaller contributions from loops with neutralinos or gluinos and d--type squarks, which are generated due to QFV explicitly through the CKM matrices. Their expressions are \begin{eqnarray} \tilde{A}^{\gamma,g}_{SM} & = & \frac{\alpha_W}{2} K^*_{ts} K_{tb} \frac{3}{M_W^2} \frac{m_t^2}{M_W^2} f_{\gamma,g}^{(1)} \left( \frac{m_t^2}{M_W^2} \right) \\ \tilde{A}^{\gamma,g}_{H^-} & = & \frac{\alpha_W}{2} K^*_{ts} K_{tb} \frac{1}{M_W^2} \frac{m_t^2}{m_{H^-}^2} \left[ \frac{1}{\tan^{2} \beta} f_{\gamma,g}^{(1)} \left( \frac{m_t^2}{m_{H^-}^2} \right) + f_{\gamma,g}^{(2)} \left( \frac{m_t^2}{m_{H^-}^2} \right) \right] \\ \tilde{A}^{\gamma,g}_{\chi^{-}} & = & - \alpha_W K^*_{ts} K_{tb} \sum_{j=1}^{2} \left\{ \frac{-1}{m^2_{\tilde{c}_1}} |V_{j1}|^{2} f_{\gamma,g}^{(3)} \left( \frac{M^2_{\chi_j^-}}{m_{\tilde{c}_1}^2} \right) \right. \cr && \qquad\qquad + \sum_{k=1}^{2} \frac{1}{m^2_{\tilde{t}_k}} \left| V_{j1} T_{k1} - \frac{m_t V_{j2} T_{k2}}{\sqrt{2} M_W \sin\beta} \right|^{2} f_{\gamma,g}^{(3)} \left( \frac{M^2_{\chi_j^-}}{m^2_{\tilde{t}_k}} \right) \cr && - \frac{U_{j2}}{\sqrt{2}\cos\beta} \frac{M_{\chi_j^-}}{M_W} \left[ \frac{-1}{m^2_{\tilde{c}_1}} V_{j1} f_{\gamma,g}^{(4)} \left( \frac{M_{\chi_j^-}^2}{m^2_{\tilde{c}_1}} \right) \right. \cr && \qquad \left. \left. + \sum_{k=1}^{2} \frac{1}{m^2_{\tilde{t}_k}} \left( V_{j1} T_{k1} - \frac{m_tV_{j2} T_{k2}}{\sqrt{2}M_W\sin\beta} \right) T_{k1} f_{\gamma,g}^{(4)} \left( \frac{M_{\chi_j^-}^2}{m^2_{\tilde{t}_k}} \right) \right] \right\} \;\; , \label{amps} \end{eqnarray} where our notation is as in ref.~\cite{paper1}. $T$, $B$ are the orthogonal matrices that diagonalise the stop and sbottom mass matrices respectively through $T M^2_{\tilde{t}-weak} T^{\dagger} = M^2_{\tilde{t}-diag}$ where $T_{11}=T_{22}=\cos\theta_t$, $T_{12}=-T_{21}=\sin\theta_t$, so that we may write the mass eigenstates $|\tilde t^{(1)}>$ and $|\tilde t^{(2)}>$ as $\cos\theta_t|\tilde t_L>+\sin\theta_t|\tilde t_R>$ and $-\sin\theta_t|\tilde t_L>+\cos\theta_t|\tilde t_R>$ respectively, and similarly for other flavours. For the first and second generations with small left-right mixing we shall take $\cos\theta_i=1$, so that for example $\tilde d^{(1)}_i=\tilde d_L^i$ and $\tilde d^{(2)}_i=\tilde c_R^i$. For simplicity, we shall use the notation $\tilde s_1$ rather than $\tilde d_2^{(1)}$, except where confusion might arise. The different functions $f_{\gamma,g}^{(i)}$ are defined in Appendix C. We generate new contributions to both the LR and RL amplitudes from $R$-parity violating couplings ($\lp{}$ and $\lpp{}$) directly and also indirectly from the induced QFV soft terms. Therefore we can write \begin{equation} \tilde{A}_{LR}^{RPV} = \tilde{A}_{LR}^{\lp{}} + \tilde{A}_{LR}^{\Delta m_{\chi^-}} + \tilde{A}_{LR}^{\Delta m_{\chi^0}} + \tilde{A}_{LR}^{\Delta m_{\tilde{g}}} \;\; . \end{equation} The direct amplitude is given by \begin{equation} \tilde{A}_{LR}^{\gamma\lp{}} = Q_d \tilde{A}_{LR}^{g\lp{}} = - Q_d \sum_{i,j=1}^{3} \frac{\tlp{i2j} \tlp{i3j}}{4\pi} \left( \frac{1}{12} \left[ \frac{\sin^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(1)}}} + \frac{\cos^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(2)}}} \right] - \frac{1}{m^2_{\tilde{\nu_i}}} F_1(x_{ji}) \right) \; , \label{alplr} \end{equation} where $\tlp{}$ are the different RPV couplings in the fermion mass eigenstate basis which, in this case, is related to the weak one by: \begin{equation} \tlp{ijk} = \lp{imk} K_{mj} \; , \end{equation} where a sum over $m$ is understood. Note that this still leaves the possibility of generating effects even with only one non-zero RPV coupling in the weak basis \cite{ag}. Here $x_{ji}={m^2_{d_j}}/{m^2_{\tilde{\nu_i}}}$. The other amplitudes are: \begin{eqnarray} \tilde{A}_{LR}^{\gamma,g\Delta m_{\chi^-}} & = & -\alpha_W K^*_{cs} K_{tb} \sum_{n=1}^{2} \frac{\Delta m^2_{\tilde{c}_1 \tilde{t}_{n}}}{m^2_{\tilde{t}_n}-m^2_{\tilde{c}_1}} \sum_{j=1}^{2} V_{j1} \left\{ \left( V^{*}_{j1} T^{*}_{n1} - \frac{m_t V^{*}_{j2}T^{*}_{n2}}{\sqrt{2} M_W \sin\beta} \right) \right. \\ & \times & \left. \left( \frac{f_{\gamma,g}^{(3)}(x^t_{jn})}{m^2_{\tilde{t}_n}} - \frac{f_{\gamma,g}^{(3)}(x^c_{j1})}{m^2_{\tilde{c}_1}} \right) - \frac{U_{j2}T^{*}_{n1}}{\sqrt{2} \cos \beta} \frac{M_{\chi_j^-}}{M_W} \left( \frac{f_{\gamma,g}^{(4)}(x^t_{jn})}{m^2_{\tilde{t}_n}} - \frac{f_{\gamma,g}^{(4)}(x^c_{j1})}{m^2_{\tilde{c}_1}} \right) \right\} \cr \tilde{A}_{LR}^{\gamma\Delta m_{\chi^0}} & = & Q_d \tilde{A}_{LR}^{g\Delta m_{\chi^0}} = - 2\alpha_W Q_d \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_1 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_1}} \sum_{j=1}^4 \Biggl\{ \cr && \left|s_W Q_d N'_{j1} - \frac{1}{c_W}(1/2+ Q_d s_W^2) N'_{j2} \right|^2 B^*_{n1} \left[ \frac{F_2(x_{jn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{j1}^s)}{m^2_{\tilde{s}_1}} \right] \nonumber \\ & - & \left( s_W Q_d N'_{j1} - \frac{1}{c_W} (1/2 + Q_d s_W^2) N'_{j2} \right) \left[ \left( s_W Q_d N'_{j1}-\frac{s_W^2}{c_W} Q_d N'_{j2} \right) B^*_{n2} \right. \nonumber \\ && \qquad\qquad\left.\left. - \frac{m_b}{2M_W\cos\beta} N_{j3} B^*_{n1} \right] \frac{M_{\chi_j^0}}{m_b} \left[ \frac{F_4(x_{jn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{j1}^s)}{m^2_{\tilde{s}_1}} \right] \right\} \label{admlr}\\ \tilde{A}_{LR}^{\gamma\Delta m_{\tilde{g}}} & = & - 2 \alpha_s Q_d C(R) \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_1 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_1}} \nonumber \\ & \times & \left[ B_{n1}^* \left( \frac{F_2(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) - B_{n2}^* \frac{M_{\tilde{g}}}{m_b} \left( \frac{F_4(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) \right] \;\; \\ \tilde{A}_{LR}^{g\Delta m_{\tilde{g}}} & = & - \alpha_s \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_1 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_1}} \left\{ -B_{n1}^*C(G) \left( \frac{F_1(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_1(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) \right. \nonumber \\ && +B_{n1}^*(2C(R)-C(G))\left( \frac{F_2(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) \cr && +B_{n2}^* \frac{M_{\tilde{g}}}{m_b} C(G) \left( \frac{F_3(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_3(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) \cr && \left. - B_{n2}^* \frac{M_{\tilde{g}}}{m_b} (2C(R)-C(G)) \left( \frac{F_4(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{\tilde g1}^s)}{m^2_{\tilde{s}_1}} \right) \right\} \;\; , \end{eqnarray} where $\alpha_s$ is the strong gauge coupling constant. Also, \begin{equation} x^c_{j1}=\frac{M^2_{\chi_j^-}}{m_{\tilde{c}_1}^2}, \quad x^s_{j1}=\frac{M^2_{\chi_j^0}}{m_{\tilde{s}_1}^2}, \quad x^s_{\tilde g1}=\frac{M^2_{\tilde g}}{m_{\tilde{s}_1}^2} \end{equation} \begin{equation} x^t_{jn}=\frac{M^2_{\chi_j^-}}{m_{\tilde{t}_n}^2}, n=1,2 \quad x^b_{jn}=\frac{M^2_{\chi_j^0}}{m_{\tilde{b}_n}^2}, n=1,2, \quad x^b_{\tilde gn}=\frac{M^2_{\tilde g}}{m_{\tilde{b}_n}^2}, \end{equation} and the QCD factors are $C(R)=4/3$, $C(G)=3$. Now we turn to the RPV contributions to $\tilde{A}_{RL}$. These are given by: \begin{equation} \tilde{A}^{RPV}_{RL} = \tilde{A}_{RL}^{\lp{}} + \tilde{A}_{RL}^{\lpp{}} + \tilde{A}_{RL}^{\Delta m_{\chi^0}} + \tilde{A}_{RL}^{\Delta m_{\tilde{g}}} \; , \end{equation} with: \begin{eqnarray} \tilde{A}_{RL}^{\gamma,g\lp{}} & = & - \sum_{i,j=1}^3 \left[ \frac{\tlp{ij2}\tlp{ij3}}{4\pi} \{ Q_d,1 \} \left( \frac{1}{12} \frac{\cos^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(1)}}} + \frac{1}{12} \frac{\sin^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(2)}}} - \frac{1}{m^2_{\tilde{\nu_i}}} F_1(x_{ji}) \right) \right. \nonumber \\ && +\frac{\lp{ij2}\lp{ij3}}{4\pi} \left( -\frac{\cos^2 \theta_{e_i}}{m^2_{\tilde{e}_i^{(1)}}} f_{\gamma,g}^{(1)} \biggl(\frac{m^2_{u_j}}{m^2_{\tilde{e}_i^{(1)}}}\biggr) - \frac{\sin^2 \theta_{e_i}}{m^2_{\tilde{e}_i^{(2)}}} f_{\gamma,g}^{(1)} \biggl(\frac{m^2_{u_j}}{m^2_{\tilde{e}_i^{(2)}}}\biggr) \right. \nonumber \\ && \left. \left. \qquad\qquad + \frac{\cos^2 \theta_{u_j}}{m^2_{\tilde{u}_j^{(1)}}} f_{\gamma,g}^{(3)} \biggl(\frac{m^2_{e_i}}{m^2_{\tilde{u}_j^{(1)}}}\biggr)+ \frac{\sin^2 \theta_{u_j}}{m^2_{\tilde{u}_j^{(2)}}} f_{\gamma,g}^{(3)} \biggl(\frac{m^2_{e_i}}{m^2_{\tilde{u}_j^{(2)}}}\biggr) \right) \right] \label{alprl} \end{eqnarray} where both $\tlp{}$ and $x_{ji}$ are defined after eq.~(\ref{alplr}). \begin{eqnarray} \tilde{A}_{RL}^{\gamma,g\lpp{}} & = & -2 \sum_{i,j=1}^3 \frac{\lpp{ij2}\lpp{ij3}}{4\pi} \left[ \frac{\sin^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(1)}}} f_{\gamma,g}^{(5)} \biggl(\frac{ m^2_{u_i} }{ m^2_{\tilde d_j^{(1)}} } \biggr ) +\frac{\cos^2 \theta_{d_j}}{m^2_{\tilde{d}_j^{(2)}}} f_{\gamma,g}^{(5)} \biggl(\frac{ m^2_{u_i} }{ m^2_{\tilde d_j^{(2)}} } \biggr ) \right. \nonumber \\ && \qquad\qquad\left. + \frac{\sin^2 \theta_{u_i}}{m^2_{\tilde{u}_i^{(1)}}} f_{\gamma,g}^{(6)} \biggl(\frac{ m^2_{d_j} }{ m^2_{\tilde u_i^{(1)}} } \biggr ) (x_{j1}^i) + \frac{\cos^2 \theta_{u_i}}{m^2_{\tilde{u}_i^{(2)}}} f_{\gamma,g}^{(6)} \biggl(\frac{ m^2_{d_j} }{ m^2_{\tilde u_i^{(2)}} } \biggr ) \right] \label{alpprl} \\ \tilde{A}_{RL}^{\gamma\Delta m_{\chi^0}} & = & Q_d \tilde{A}_{RL}^{g\Delta m_{\chi^0}} = -2 \alpha_W Q_d \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_2 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_2}} \sum_{j=1}^4 \left\{ \left|s_W Q_d N'_{j1} - \frac{s_W^2}{c_W} Q_d N'_{j2} \right|^2 \right. \nonumber \\ & \times & B^*_{n2} \left[ \frac{F_2(x_{jn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{j2}^s)}{m^2_{\tilde{s}_2}} \right] - \left( s_W Q_d N'^{*}_{j1} - \frac{s_W^2}{c_W} Q_d N'^{*}_{j2} \right) \nonumber \\ & \times & \left[ \left( s_W Q_d N'^{*}_{j1}-\frac{1}{c_W}(1/2+ Q_d s_W^2) N'^{*}_{j2} \right) B^*_{n1} + \frac{m_b}{2M_W\cos\beta} N^*_{j3} B^*_{n2} \right] \nonumber \\ & \times & \left. \frac{M_{\chi_j^0}}{m_b} \left[ \frac{F_4(x_{jn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{j2}^s)}{m^2_{\tilde{s}_2}} \right] \right\} \\ \tilde{A}_{RL}^{\gamma\Delta m_{\tilde{g}}} & = & -2\alpha_s Q_d C(R) \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_2 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_2}} \nonumber \\ & \times & \left[ B^*_{n2} \left( \frac{F_2(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) - B_{n1}^* \frac{M_{\tilde{g}}}{m_b} \left( \frac{F_4(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) \right] \\ \tilde{A}_{RL}^{g\Delta m_{\tilde{g}}} & = & -\alpha_s \sum_{n=1}^2 \frac{\Delta m^2_{\tilde{s}_2 \tilde{b}_n}}{m^2_{\tilde{b}_n} - m^2_{\tilde{s}_2}} \left\{ -B^*_{n2}C(G) \left( \frac{F_1(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_1(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) \right. \nonumber \\ && \qquad\qquad + B_{n2}^*(2C(R)-C(G)) \left( \frac{F_2(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_2(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) \nonumber \\ && \qquad\qquad + B_{n1}^* \frac{M_{\tilde{g}}}{m_b} C(G) \left( \frac{F_3(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_3(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) \cr && \qquad\qquad \left. - B_{n1}^* \frac{M_{\tilde{g}}}{m_b} (2C(R)-C(G)) \left( \frac{F_4(x_{\tilde gn}^b)}{m^2_{\tilde{b}_n}} - \frac{F_4(x_{\tilde g2}^s)}{m^2_{\tilde{s}_2}} \right) \right\} \end{eqnarray} \subsection{Analytical Discussion} In general the results for the implications of the MSSM for $b\to s\gamma$ are well known \cite{bsgsusy}. The SM contribution to the branching rate is large and in good agreement with the data at around $3\times10^{-4}$. The MSSM charged Higgs contribution is always of the same sign as that from the $W$ and can easily push the total over the experimental limit, hence giving quite a tight constraint on the charged Higgs mass or equivalently $m_A$. In practice, however, the situation changes when the full spectrum is considered, because the SUSY partners can give contributions to the amplitude of both signs with the dominant contribution being that of the chargino which can be of opposite sign to that of the SM and charged Higgs \cite{bsgsusy}. There are thus two separate scenarios according to the sign of $\mu$. For $\mu_4<0$, the structure of the chargino mass matrix and mixings makes its contribution comparable in size to the other two with the opposite sign giving rise to a strong cancellation, and hence the constraints on SUSY parameters are rather weak. For $\mu_4>0$ the chargino amplitude is not so big and moreover its sign is not always opposed to that of SM and charged Higgs, and therefore cannot cancel them off completely so we must have a fairly heavy charged Higgs, which given our unification and radiative electroweak symmetry breaking assumptions implies fairly large soft masses. If we now turn to the implications of $R$-parity violating couplings, there are three distinct possibilities. The first of these is where the product $\lp{i2j}\lp{i3j}$ is non-zero. Here the main effects will be to change $\tilde A_{LR}$, since mixing is generated in left handed superfields only. This will give an additional term which is added directly to the amplitude from the usual MSSM terms, but may be of opposite sign and hence less restricted. By contrast, the cases of non-zero $\lp{ij2}\lp{ij3}$ or $\lpp{ij2}\lpp{ij3}$ generate contributions to $\tilde A_{RL}$ (negligible in the MSSM), which cannot therefore have interference with the MSSM effects but which are only added to them in quadrature. Beginning with the direct $R$--Parity violating contributions, the SM contribution to $\tilde A_{LR}$ is $-6.7\times 10^{-8}\hbox{GeV}^{-2}$, with an experimental limit on $\sqrt{(\tilde A_{LR}^{\gamma})^2+(\tilde A_{RL}^{\gamma})^2}$ derived from equation~(\ref{bsglimit}) of 4.1 to 8.5$\times 10^{-8}\hbox{GeV}^{-2}$. Since the QCD corrections typically give a contribution to $\tilde A_{i}$ of around $0.65\tilde A_i^{\gamma}$, we shall consider what values of the couplings give a contribution of $-2\times 10^{-8}\hbox{GeV}^{-2}$ to $\tilde A_{LR}^{\gamma}$ or $7\times 10^{-8}\hbox{GeV}^{-2}$ to $\tilde A_{RL}^{\gamma}$ (recall that the sign of the former matters, since it is being added to a negative SM contribution, while the latter is added in quadrature). It is then straightforward to derive the bounds \begin{eqnarray} \tlp{i2j}\tlp{i3j}(M_Z)&\stackrel{<}{\sim}& 0.09 \Biggl [ 2 \left ( \frac{100\hbox{GeV}}{m_{\tilde \nu^i}} \right )^2 - \left ( \frac{100\hbox{GeV}}{m_{\tilde d_R^j}} \right )^2 \Biggr ]^{-1} \\ \vert \lp{ij2}\lp{ij3}(M_Z)\vert &\stackrel{<}{\sim}& 0.035 \Biggl [ \left ( \frac{100\hbox{GeV}}{m_{\tilde e_L^i}} \right )^2 - \left ( \frac{100\hbox{GeV}}{m_{\tilde d_L^j}} \right )^2 \Biggr ]^{-1} \\ \vert \lpp{i2j}\lpp{i3j}(M_Z)\vert &\stackrel{<}{\sim}& 0.16 \left ( \frac{m_{\tilde q_R}}{100\hbox{GeV}} \right )^2 \label{bsglppRRdirect} \end{eqnarray} where we have neglected the distinction between $\tlp{}$ and $\lp{}$ in deriving the second of these. The indirect contributions are far more complicated, but it is instructive to consider their size in the case where we include only the diagrams with a helicity flip on the gaugino line (and hence an overall factor of the gaugino mass), set all sparticle masses approximately degenerate at $\tilde m$ and ignore the various mixing factors. This then gives an approximate contribution for each chargino of \begin{equation} \tilde{A}_{LR}^{\gamma\Delta m_{\chi^-}} \sim \lp{i2j}\lp{i3j}(M_{\rm GUT}) \frac{10^{-7}\hbox{GeV}^{-2}}{\cos\beta} \left ( \frac{100\hbox{GeV}}{\tilde m} \right )^2 \frac{M_{\chi_j^-}}{M_W} \end{equation} where we have used equation~(\ref{dmapp}) and have evaluated the functions with all masses equal. Since the product of two of these couplings at the GUT scale is an order of magnitude smaller than at $M_Z$ \cite{paper1}, for $\lp{i2j}\lp{i3j}$ we expect a significant chargino contribution relative to the direct contributions for large $\tan\beta$ unless the gaugino-higgsino mixing is small. A simple way of understanding how the chargino contribution must be relevant is to consider this in terms of results from the MSSM. The chargino contribution here is driven by the CKM matrix element $K_{23}\simeq 0.04$, while the $R$--Parity Violating contributions are driven by a similar mixing $\Delta m^2/m_{\tilde q}^2$, which is roughly equal to the product of $\lp{i2j}\lp{i3j}(M_{GUT})$ and hence will be similarly important when this product of couplings at the GUT scale is around $10^{-2}$. The gluino contribution is simpler, and neglecting mixing between sbottom squarks and the mass difference between squarks and gluinos we find a bound on each of the pairs of couplings \begin{eqnarray} \lp{ij2}\lp{ij3}(M_Z)&\stackrel{<}{\sim}& 0.003 \left ( \frac{\tilde m}{100\hbox{GeV}} \right )^2 \\ \lp{ij2}\lp{ij3}(M_Z)&\stackrel{<}{\sim}& 0.006 \left ( \frac{\tilde m}{100\hbox{GeV}} \right )^2 \\ \lpp{i2j}\lpp{i3j}(M_Z)&\stackrel{<}{\sim}& 0.006 \left ( \frac{\tilde m}{100\hbox{GeV}} \right )^2 \end{eqnarray} where the bound is in fact on the sum of these indirect terms plus the direct contributions quoted above. Unfortunately, we might expect the suppression from the mixing in the sbottom sector to weaken these by an order of magnitude or so. We can conclude however that it is likely that the $RL$ contributions are dominated by the gluino, especially in the case of non-zero $\lpp{ij2}\lpp{ij3}$, while for $\lp{i2j}\lp{i3j}$ the chargino is likely to dominate, results which are confirmed by our numerical studies. \subsection{Numerical Results} Given the discussion above, we are now in a position to discuss the effects of setting these products non-zero with a realistic spectrum from universality at the GUT scale. The products of couplings which we are considering have not been constrained before, but we find $\lp{i2j}\lp{i3j}<0.06(100\hbox{GeV}/\tilde m)^2$ using the bounds on each coupling independently from $K^+$ decays \cite{ag}. Other such bounds can always be evaded by selecting the indices in the products appropriately; for example the tight bounds on $\lpp{ijk}$ from neutron anti-neutron oscillation \cite{barbm,gs} only apply if $i=1$. In Figure~\ref{bsglpLL} we show a simple example of mixing involving the product $\lp{i2j}\lp{i3j}$, where we choose both couplings 0.05 at the GUT scale, hence making the product around $0.023$ at low energy, and with other parameters $\tan\beta=10$, $\mu_4<0$, $m_0=100$ GeV, $A_0=0$, and $M_{1/2}$ varying. We see that in fact here the gluino contribution is dominated by the direct one, and this in turn is dominated by the chargino contribution. However, even at very low values of the mass spectrum (for $M_{1/2}\stackrel{<}{\sim} 110$ GeV the chargino mass is actually below the experimental limit) and even though we have taken the product of couplings to be close to the limit of equation~(\ref{sneutmassbound}), the contributions are still too small to greatly affect the total. Hence we conclude that bounds on this product are really not usefully found from $b\to s\gamma$ decays. Similarly, for $\lp{ij2}\lp{ij3}$ we expect that the product at the GUT scale will be $\stackrel{<}{\sim} 0.01$ from equation~(\ref{sneutmassbound}), which in turn leads to the product at the weak scale being around $\stackrel{<}{\sim} 0.1$, which can only give a useful bound if the spectrum is extremely light, and in practice it again appears that constructing a spectrum so light as to have interesting consequences for $b\to s\gamma$ through $R$-Parity Violation requires one or other of the masses to become lighter than its experimental bound. For the case of $\lpp{ij2}\lpp{ij3}$ we do not have such tight bounds from the sneutrino mass and can increase the coupling to quite a large value. In Figure~\ref{bsglppRR}, we show a plot with the same input parameters as in Figure~\ref{bsglpLL}, but with $\lpp{ij2}\lpp{ij3}(M_{GUT})=0.01$. Here again the effects on $b\to s\gamma$ are only really beginning to be significant for a very light spectrum. However, it is interesting that the contribution is very much larger than might be expected from equation~(\ref{bsglppRRdirect}), since the gluino term is overwhelmingly dominant. In conclusion, while the $R$--Parity violating couplings give a contribution to $b\to s\gamma$ which is enhanced typically by an order of magnitude when the indirect contributions are included, and although this in principle gives a tighter bound than others in the literature, in fact these constraints are very weak in the context of universal mass spectrum at the GUT scale. This is because, for example, it is virtually impossible to arrange that squarks should have masses as light as 100 GeV without violating one or another of the experimental limits. In the case of non-zero $\lp{}$ couplings the bounds from $b\to s\gamma$ are weaker than those from the sneutrino mass limits given in equation~(\ref{sneutmassbound}). For the $\lpp{}$ couplings, the indirect gluino contribution is far larger than the direct contribution, and we find that for a light spectrum with $m_0$ and $M_{1/2}$ of order 100 GeV, the product of $\lpp{ij2}\lpp{ij3}(M_Z)$ should be $\stackrel{<}{\sim} 0.2$. This bound scales very roughly as $M_{1/2}$, but with some complicated dependence on the sbottom squark mixing as well as on the mass spectrum. \section{$K^0-\bar K^0$ mixing} \subsection{SUSY contributions} The $K^0-\bar K^0$ sector has long been a probe of physics beyond the standard model, starting with the original motivation of the study of CP violation and recently also as a way of investigating flavour changing neutral currents in SUSY. These occur even in the absence of $R$-Parity violation, since in the MSSM (and SM) quark flavour is not conserved. Hence we shall find that the RPV contribution is complementary to the already existing CKM contribution. We shall ask the question of how much it is possible to constrain RPV couplings through their impact on $\Delta m_K$. Here we shall consider the relative sizes of three effects, the MSSM $\Delta m^2$ contribution and the direct and indirect RPV contributions. The first has already been investigated \cite{hagelin,otherfcnc}, as has the last \cite{barbm,bgnn,ag,crs,chr}, but the question of whether the simple direct effects are in fact dominant has never been studied. The upper bound on the $K_L-K_S$ mass difference has been measured \cite{pdb} as \begin{equation} \Delta m_K = (3.491 \pm 0.009) \times 10^{-15} \hbox{GeV} \end{equation} $K^0-\bar K^0$ mixing is generated by the effective Lagrangian \begin{eqnarray} \Delta{\cal L}^{\Delta S=2}&=& c_{LL}[\bar d_i P_L s^i][\bar d_j P_L s^j] +c^{\prime}_{LL}[\bar d_i P_L s^j][\bar d_j P_L s^i]\cr &+& c_{RR}[\bar d_i P_R s^i][\bar d_j P_R s^j] +c^{\prime}_{RR}[\bar d_i P_R s^j][\bar d_j P_R s^i]\cr &+& c_{LR}[\bar d_i P_L s^i][\bar d_j P_R s^j] +c^{\prime}_{LR}[\bar d_i P_L s^j][\bar d_j P_R s^i]\cr &+& d_{LL}[\bar d_i \gamma_\mu P_L s^i][\bar d_j \gamma^\mu P_L s^j] +d_{RR}[\bar d_i \gamma_\mu P_R s^i][\bar d_j \gamma^\mu P_R s^j] \end{eqnarray} giving a contribution to the $K_L-K_S$ mass difference \begin{eqnarray} \Delta m_K&=&\frac{1}{12} f_K^2m_K \Bigl ( c_{LR} + 3c^{\prime}_{LR} + 8d_{LL} +8d_{RR} \cr && \qquad + \frac{m_K^2}{(m_d+m_s)^2} (6c_{LR}+2c^{\prime}_{LR}+5c_{LL}-c^{\prime}_{LL} +5c_{RR}-c^{\prime}_{RR}) \Bigr ) \label{dmKeqn} \end{eqnarray} Here the terms involving both left and right handed fields in the effective lagrangian will give a larger contribution to $\Delta m_K$ than those involving purely left or right handed fields because of the factor of $m_K^2/m_s^2$. We shall not try to include QCD corrections to these formulae since these are unlikely to be more significant than the errors implicit in, for example, our choice of $\alpha_3$, $m_s$, and SUSY parameters. Contributions to the parameters in the effective Lagrangian come from a variety of different classes of diagram, some of which are shown in Figure~\ref{KKdiag}. The standard model contribution from Figure~\ref{KKdiag}a is \begin{eqnarray} {d_{LL}}^{SM}&=&\sum_{i,j}\frac{g^4}{64\pi^2} K_{i1}K^*_{i2}K_{j2}K^*_{j1} I_4(m_{u_i}^2,m_{u_j}^2,M_W^2,M_W^2) \cr &\simeq& \frac{g^4}{128\pi^2}\vert K_{cd}\vert^2\frac{m_c^2}{M_W^4} \label{SMKKbar} \end{eqnarray} where for the second line we only consider the charm and up quark contributions. Apart from the standard model contributions, there are contributions from direct $R$-Parity violating diagrams which have been calculated previously \cite{barbm,ag,crs,chr}. The only tree level diagram is that with the interchange of a sneutrino and two $\tlp{}$ couplings shown in Figure~\ref{KKdiag}b \begin{equation} {c_{LR}}^{TL}=-\sum_i\frac{ \tlp{i21}\tlp{i12}^* } { m_{\tilde\nu_i}^2 } \end{equation} where our notation is as in the previous section and our earlier work \cite{paper1}. The $\lp{}$ box diagrams such as Figure~\ref{KKdiag}c give a contribution \begin{eqnarray} {d_{LL}}^{\lp{}}&=&\sum_{i,j,k,m} \frac{1}{64\pi^2}\tlp{i1k}^*\tlp{j2k}\tlp{j1m}^*\tlp{i2m} I_4(m_{\tilde\nu_i}^2,m_{\tilde\nu_j}^2,m_{d_k}^2,m_{d_m}^2) \cr &+&\sum_{i,j,k,m} \frac{1}{64\pi^2}\tlp{i1k}^*\tlp{j2k}\tlp{j1m}^*\tlp{i2m} I_4(m_{\nu_i}^2,m_{\nu_j}^2,m_{\tilde d_R^k}^2,m_{\tilde d_R^m}^2)\\ {d_{RR}}^{\lp{}}&=&\sum_{i,j,k,m} \frac{1}{64\pi^2}\tlp{ik1}\tlp{jk2}^*\tlp{jm1}\tlp{im2}^* I_4(m_{\tilde\nu_i}^2,m_{\tilde\nu_j}^2,m_{d_k}^2,m_{d_m}^2) \cr &+&\sum_{i,j,k,m} \frac{1}{64\pi^2}\tlp{ik1}\tlp{jk2}^*\tlp{jm1}\tlp{im2}^* I_4(m_{\nu_i}^2,m_{\nu_j}^2,m_{\tilde d_L^k}^2,m_{\tilde d_L^m}^2)\cr &+&\sum_{i,j,k,m} \frac{1}{64\pi^2}\lp{ik1}\lp{jk2}^*\lp{jm1}\lp{im2}^* I_4(m_{\tilde e_L^i}^2,m_{\tilde e_L^j}^2,m_{u_k}^2,m_{u_m}^2) \cr &+&\sum_{i,j,k,m} \frac{1}{64\pi^2}\lp{ik1}\lp{jk2}^*\lp{jm1}\lp{im2}^* I_4(m_{e_i}^2,m_{e_j}^2,m_{\tilde u_L^k}^2,m_{\tilde u_L^m}^2)\\ {c^{\prime}_{LR}}^{\lp{}}&=&-\sum_{i,j,k,m} \frac{1}{32\pi^2}\tlp{i1k}^*\tlp{j2k}\tlp{im1}\tlp{jm2}^* I_4(m_{\tilde\nu_i}^2,m_{\tilde\nu_j}^2,m_{d_k}^2,m_{d_m}^2) \cr &&-\sum_{i,j,k,m} \frac{1}{32\pi^2}\tlp{i1k}^*\tlp{j2k}\tlp{im1}\tlp{jm2}^* I_4(m_{\nu_i}^2,m_{\nu_j}^2,m_{\tilde d_R^k}^2,m_{\tilde d_R^m}^2) \end{eqnarray} while $\lpp{}$ box diagrams such as Figure~\ref{KKdiag}d give \begin{eqnarray} {d_{RR}}^{\lpp{}}&=&\sum_{i,j} \frac{1}{32\pi^2}\lpp{i13}\lpp{i23}^*\lpp{j13}\lpp{j23}^* \biggl [ I_4(m_b^2,m_b^2,m_{\tilde u_R^i}^2,m_{\tilde u_R^j}^2) \cr && \qquad\qquad\qquad\qquad\qquad\qquad + I_4(m_{\tilde b_R}^2,m_{\tilde b_R}^2,m_{u_i}^2,m_{u_j}^2) \biggr ] \end{eqnarray} In addition we have diagrams such as those Figure~\ref{KKdiag}e with one internal $W$ boson and one internal squark line, where a helicity flip is needed on the internal fermion line, forcing it to be a top or charm. Hence we have \begin{eqnarray} {c_{LR}}^{\lpp{}W}&=&-\sum_{i,j} \frac{\alpha}{4\pi\sin^2\theta_W}\lpp{i31}\lpp{j32}^*K_{i1}^*K_{j2} m_{u_i}m_{u_j}J_4(m_{\tilde b_R}^2,M_W^2,m_{u_i}^2,m_{u_j}^2) \cr {c^{\prime}_{LR}}^{\lpp{}W}&=&\sum_{i,j} \frac{\alpha}{4\pi\sin^2\theta_W}\lpp{i31}\lpp{j32}^*K_{i1}^*K_{j2} m_{u_i}m_{u_j}J_4(m_{\tilde b_R}^2,M_W^2,m_{u_i}^2,m_{u_j}^2) \end{eqnarray} We neglect similar diagrams with $\lp{}$ vertices, since these are always dominated by the tree level contribution. Although they have not been considered in the context of $R$-Parity, the diagrams involving mass insertions on squark lines have been analysed in the case where the off-diagonal mass terms are generated by the CKM matrix and by boundary conditions at the unification scale \cite{hagelin,otherfcnc,bert}. Most of these analyses include only gluino mediated box diagrams, such as Figure~\ref{KKdiag}f. \begin{eqnarray} {d_{LL}}^{\tilde g}&=& \left( \Delta m^2_{\tilde s_L\tilde d_L}\right)^2 \alpha_3^2 \biggl [ \frac{11}{18} I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde g}^2,M_{\tilde g}^2) \cr &&\qquad\qquad\qquad\qquad -\frac{1}{9} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ {d_{RR}}^{\tilde g}&=& \left( \Delta m^2_{\tilde s_R\tilde d_R}\right)^2 \alpha_3^2 \biggl [ \frac{11}{18} I_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \cr &&\qquad\qquad\qquad\qquad -\frac{1}{9} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ {c_{LR}}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \alpha_3^2 \biggl [ -\frac{2}{3} I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \cr &&\qquad\qquad\qquad\qquad\qquad -\frac{7}{3} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ &+& \left(\Delta m^2_{\tilde s_L\tilde d_R}\right) \left(\Delta m^2_{\tilde s_R\tilde d_L}\right) \alpha_3^2 \biggl [ -\frac{11}{9} I_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \cr {c_{LR}^{\prime}}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \alpha_3^2 \biggl [ \frac{10}{9} I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \cr && \qquad\qquad\qquad\qquad\qquad -\frac{1}{9} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_L/}^2,m_{\tilde d_R}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ &+& \left(\Delta m^2_{\tilde s_L\tilde d_R}\right) \left(\Delta m^2_{\tilde s_R\tilde d_L}\right) \alpha_3^2 \biggl [ -\frac{5}{3} I_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \cr {c_{LL}}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_L\tilde d_R}\right)^2 \alpha_3^2 \biggl [ -\frac{17}{18} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ {c_{LL}^\prime}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_L\tilde d_R}\right)^2 \alpha_3^2 \biggl [ \frac{1}{6} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ {c_{RR}}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_R\tilde d_L}\right)^2 \alpha_3^2 \biggl [ -\frac{17}{18} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \\ {c_{RR}^\prime}^{\tilde g}&=& \left(\Delta m^2_{\tilde s_R\tilde d_L}\right)^2 \alpha_3^2 \biggl [ \frac{1}{6} M_{\tilde g}^2 J_4^{\prime\prime} (m_{\tilde d_{L}}^2,m_{\tilde d_{R}}^2,M_{\tilde g}^2,M_{\tilde g}^2) \biggr ] \;\;, \end{eqnarray} where we have assumed that $m_{\tilde d_L}\simeq m_{\tilde s_L}$. These results and those for the resulting contribution to $\Delta m_K$ in equation~(\ref{dmKeqn}) agree with those presented in Gabbianni {\it et al} \cite{otherfcnc}, where the disagreements between these results and others in references~\cite{hagelin,otherfcnc} are discussed. The chargino mediated box diagrams give \begin{eqnarray} {d_{LL}}^{\tilde\chi^{\pm}}&=& \left( \Delta m^2_{\tilde c_L\tilde u_L}\right)^2 \frac{\alpha^2}{4\sin^4\theta_w} \sum_{i,j} \vert V_{i1}\vert^2 \vert V_{j1}\vert^2 I_4^{\prime\prime} (m_{\tilde u_L}^2,m_{\tilde u_L}^2, M_{\tilde\chi^{\pm}_i}^2,M_{\tilde\chi^{\pm}_j}^2) \end{eqnarray} while from neutralino mediated box diagrams we have \begin{eqnarray} {d_{LL}}^{\tilde\chi^0}&=& \sum_{ij} \frac{1}{16\pi^2} \left( \Delta m^2_{\tilde s_L\tilde d_L}\right)^2 \biggl [ \vert A_{dLi} \vert^2 \vert A_{dLj} \vert^2 I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2) \cr &&\qquad\qquad -(A_{dLi})^2 ( A_{dLj}^* )^2 M_{\tilde\chi_i^0}M_{\tilde\chi_j^0} J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2) \biggr ] \\ {d_{RR}}^{\tilde\chi^0}&=& \sum_{ij} \frac{1}{16\pi^2} \left( \Delta m^2_{\tilde s_R\tilde d_R}\right)^2 \biggl [ \vert A_{dRi} \vert^2 \vert A_{dRj} \vert^2 I_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2, M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2) \cr &&\qquad\qquad -(A_{dRi})^2 ( A_{dRj}^* )^2 M_{\tilde\chi_i^0}M_{\tilde\chi_j^0} J_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2) \biggr ]\\ {c_{LR}}^{\tilde\chi^0}&=& -\sum_{ij} \frac{ A_{dLi}A_{dRi}^*A_{dLj}^*A_{dRj} } {4\pi^2} \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \cr && \qquad\qquad\qquad\qquad \times M_{\tilde\chi_i^0}M_{\tilde\chi_j^0} J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2)\\ {c_{LR}^{\prime}}^{\tilde\chi^0}&=& \sum_{ij} \frac{ A_{dLi}A_{dRi}A_{dLj}^*A_{dRj}^* } {4\pi^2} \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \cr && \qquad\qquad\qquad\qquad \times I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde\chi_j^0}^2) \end{eqnarray} and from those with one gluino and one neutralino : \begin{eqnarray} {d_{LL}}^{\tilde\chi^0\tilde g}&=& -\sum_{i} \frac{\alpha_3}{6\pi} \left( \Delta m^2_{\tilde s_L\tilde d_L}\right)^2 \biggl [ \vert A_{dLi} \vert^2 I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \cr &&\qquad\qquad +\frac{1}{2}(A_{dLi}^{*2} + A_{dLi}^2) M_{\tilde\chi_i^0}M_{\tilde g} J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_L}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \biggr ] \\ {d_{RR}}^{\tilde\chi^0\tilde g}&=& -\sum_{i} \frac{\alpha_3}{6\pi} \left( \Delta m^2_{\tilde s_R\tilde d_R}\right)^2 \biggl [ \vert A_{dRi} \vert^2 I_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \cr &&\qquad\qquad +\frac{1}{2}(A_{dRi}^{*2} + A_{dRi}^2) M_{\tilde\chi_i^0}M_{\tilde g} J_4^{\prime\prime} (m_{\tilde d_R}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \biggr ] \\ \label{dRRglu} {c_{LR}}^{\tilde\chi^0\tilde g}&=& -\sum_{i} \frac{\alpha_3}{2\pi} \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \biggl [ (A_{dRi}A_{dLi} + A_{dRi}^* A_{dLi}^* ) \cr && \qquad\qquad\qquad\qquad\qquad\qquad\qquad \times I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \cr &&\qquad\qquad + (A_{dRi}A_{dLi}^* + A_{dRi}^* A_{dLi} ) M_{\tilde\chi_i^0}M_{\tilde g} J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \biggr ] \\ {c_{LR}^{\prime}}^{\tilde\chi^0\tilde g}&=& \sum_{i} \frac{\alpha_3}{6\pi} \left(\Delta m^2_{\tilde s_L\tilde d_L}\right) \left(\Delta m^2_{\tilde s_R\tilde d_R}\right) \biggl [ (A_{dRi}A_{dLi} + A_{dRi}^* A_{dLi}^* ) \cr && \qquad\qquad\qquad\qquad\qquad\qquad\qquad \times I_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \cr &&\qquad\qquad + (A_{dRi}A_{dLi}^* + A_{dRi}^* A_{dLi} ) M_{\tilde\chi_i^0}M_{\tilde g} J_4^{\prime\prime} (m_{\tilde d_L}^2,m_{\tilde d_R}^2,M_{\tilde\chi_i^0}^2,M_{\tilde g}^2) \biggr ] \qquad \end{eqnarray} Here we use \begin{eqnarray} A_{dLj}&=&eQ_dN_{j1}^{\prime} - \frac{g}{\cos\theta_W} \left( \frac{1}{2}+Q_d\sin^2\theta_W\right)N_{j2}^{\prime}\cr A_{dRj}&=&-eQ_dN_{j1}^{\prime*} + \frac{gQ_d\sin^2\theta_W}{\cos\theta_W}N_{j2}^{\prime*} \end{eqnarray} and we note that \begin{eqnarray} \Delta m^2_{\tilde s_L \tilde d_L}&=&m^2_{Q_1Q_2} \nonumber \\ \Delta m^2_{\tilde s_R \tilde d_R}&=&m^2_{d_1d_2} \\ (\Delta m^2_{\tilde s_L \tilde d_R})^2&=&\nu_1^2\left( \eta^d_{21} \right)^2 +\nu_1^2 \eta^d_{21} \left( \eta^d_{11} m^2_{Q_1Q_2}\frac{\delta}{\delta m^2_{\tilde d_L}} +\eta^d_{22} m^2_{d_1d_2}\frac{\delta}{\delta m^2_{\tilde d_R}} \right) + \ldots \nonumber \\ (\Delta m^2_{\tilde s_R \tilde d_L})^2&=&\nu_1^2 \left( \eta^d_{12} \right)^2 +\nu_1^2 \eta^d_{12} \left( \eta^d_{22} m^2_{Q_1Q_2}\frac{\delta}{\delta m^2_{\tilde d_L}} +\eta^d_{11} m^2_{d_1d_2}\frac{\delta}{\delta m^2_{\tilde d_R}} \right) + \ldots \nonumber \\ \Delta m^2_{\tilde s_L \tilde d_R} \Delta m^2_{\tilde s_R \tilde d_L} &=& \nu_1^2 \left( \eta^d_{21}\eta^d_{12}+ \frac{1}{2} \eta^d_{21} \left(\eta^d_{22} m^2_{Q_1Q_2}\frac{\delta}{\delta m^2_{\tilde d_L}} +\eta^d_{11} m^2_{d_1d_2}\frac{\delta}{\delta m^2_{\tilde d_R}} \right) \right. \nonumber \\ &+& \left. \frac{1}{2} \eta^d_{12} \left( \eta^d_{11} m^2_{Q_1Q_2}\frac{\delta}{\delta m^2_{\tilde d_L}} +\eta^d_{22} m^2_{d_1d_2}\frac{\delta}{\delta m^2_{\tilde d_R}} \right) + \ldots \right) \nonumber \end{eqnarray} where the ellipsis stands for higher order contributions, the derivatives are assumed to act only of one of the arguments of the appropriate $I_4^{\prime\prime}$, and we have only included terms proportional to $\Delta m^2_{\tilde s_L \tilde d_R}$ and $\Delta m^2_{\tilde s_R \tilde d_L}$ in the dominant gluino contributions, not in the neutralino contributions where such terms also appear but are smaller. \subsection{Analytical Discussion} We begin by noting that the standard model term from equation~(\ref{SMKKbar}) gives a contribution to $\Delta m_K$ of around $2\times 10^{-15}$ GeV, which given the large theoretical errors in the input parameters to the calculation is in reasonable agreement. Hence we shall here require that the new contributions from SUSY and RPV do not destroy this agreement by being larger than the experimental limit themselves. However, it is important to note that there are unknown relative signs between the various contributions, and hence that there can be cancellations. The direct contributions have been discussed in refs.~\cite{barbm,bgnn,ag,crs,chr}. The most stringent bound on couplings is that from the tree level diagram of Figure~\ref{KKdiag}b which leads to a constraint \cite{bgnn,chr} \begin{equation} \tlp{i12}\tlp{i21}(M_Z) \equiv\lp{ij2}\lp{ik1}(M_Z)K_{j1}K_{k2} \stackrel{<}{\sim} 1.3\times 10^{-7} \left( \frac{\tilde m_{\tilde\nu_i}}{500\hbox{GeV}}\right )^2 \label{TLbound} \end{equation} The box diagrams of Figures~\ref{KKdiag}c and \ref{KKdiag}d and the competing diagram with an internal W line of Figure~\ref{KKdiag}e lead to bounds on products of two couplings of order $10^{-2}$ to $10^{-3}$ \cite{barbm,crs} for a very light spectrum. Ref.~\cite{ag} was mostly concerned with the case where, in the weak basis, only one $R$-Parity violating coupling is non-zero. This leads to bounds of order 0.1 on $\lp{i1k}$ and $\lp{i2k}$ for a very light spectrum, but scaling more weakly with masses. We now consider how large the expected contributions from the main indirect processes are, beginning with non-zero $\lpp{}$, by deriving approximate bounds from each of the most important direct and indirect contributions in turn. The term which we shall bound will be $\lpp{}^2\equiv\lpp{i31}\lpp{i32}$, and for purposes of the discussion in this section we shall assume that all superpartners are degenerate at $\tilde m\simeq 3M_{1/2}$. The main indirect contributions are the gluino mediated box diagrams, which arise because $\Delta m^2_{\tilde s_R\tilde d_R}$ is non-zero. Using the various functions in Appendix C and Eqs.~(\ref{dmapp}) and (\ref{dRRglu}) we can find $d_{RR}^{\tilde g}$ and hence derive an approximate bound of \begin{equation} \lpp{}^2(M_Z)\stackrel{<}{\sim} 0.07 \left( \frac{\tilde m}{500\hbox{GeV}} \right ) \label{dRRcont} \end{equation} Although other indirect contributions exist, this is the simplest and is sometimes the largest. However, the various contributions from $\Delta m^2_{\tilde s_L \tilde d_R}$ and $\Delta m^2_{\tilde s_R \tilde d_L}$ often in practice give comparably large effects, and so equation~(\ref{dRRcont}) should be treated with caution. The direct contributions are those from the $\lpp{}$ box diagram and the $\lpp{}-W$ diagram \cite{barbm,crs}. The former gives a bound of \begin{equation} \lpp{}^2(M_Z) \stackrel{<}{\sim} 0.01 \left( \frac{\tilde m}{500\hbox{GeV}}\right ) \label{dRRdirectcontn} \end{equation} which may be rather inaccurate if the first index of the non-zero $\lpp{}$ is 3. The second gives \begin{eqnarray} \lpp{213}\lpp{223}(M_Z) &\stackrel{<}{\sim}& 0.05 \left( \frac{\tilde m}{500\hbox{GeV}}\right )^2 \\ \lpp{213}\lpp{323}(M_Z) &\stackrel{<}{\sim}& 0.1 \left( \frac{\tilde m}{500\hbox{GeV}}\right )^2 \\ \lpp{313}\lpp{223}(M_Z) &\stackrel{<}{\sim}& 0.2 \left( \frac{\tilde m}{500\hbox{GeV}}\right )^2 \\ \lpp{313}\lpp{323}(M_Z) &\stackrel{<}{\sim}& 0.1 \left( \frac{\tilde m}{500\hbox{GeV}}\right )^2 \label{toptop} \end{eqnarray} respectively, where the scaling is very approximate for the cases involving stop squarks. These numbers are in reasonable agreement with those in reference~\cite{crs} given the uncertainties in (for example) $m_s$. We conclude that although the indirect contribution appears rather smaller, none of the contributions to $\Delta m_K$ is obviously negligible in the region of interest, and so we must turn to a numerical analysis, noting that we expect $\lpp{}^2(M_{GUT})\sim 10^{-3}$ to give a result comparable with the experimental limit for masses of order a few hundred GeV. We now turn to the case of non-zero $\lp{}$. Here the tree level diagram is inevitably completely dominant where it exists \cite{bgnn,chr}, and even with only one non-zero coupling in the weak basis there can be measurable effects \cite{ag}. Note that here it is possible to arrange the couplings so that the indirect contributions are zero but the direct contribution is not, for example if we arrange for non-zero $\tlp{113}$, $\tlp{223}$, $\tlp{212}$, and $\tlp{122}$. However, such scenarios seem contrived given the need to avoid the extremely tight bounds from the tree level term while simultaneously having non-negligible values for four couplings, requiring remarkable cancellations in $\tlp{i12}\tlp{i21}$. The only scenario which we shall consider is thus non-zero $\lp{i13}$ and $\lp{i23}$. For this case the direct contribution lead to a constraint which is similar to that of Eq.~(\ref{dRRcont}), \begin{equation} \lp{}^2(M_Z)\stackrel{<}{\sim} 0.07 \left( \frac{\tilde m}{500\hbox{GeV}} \right ) \label{dLLcont} \end{equation} where again this result is very approximate given the number of different contributions which can be relevant. There is no $\lp{}-W$ diagram by construction since it only exists when the tree level diagram exists, while the direct contribution from the box diagram then gives \begin{equation} \lp{}^2(M_Z) \stackrel{<}{\sim} 8\times 10^{-3} \left( \frac{\tilde m}{500\hbox{GeV}}\right ) \label{dLLdirectcontn} \end{equation} Since these results are in general only reliable as order of magnitude estimates, we use a numerical study to find out which contributions are most significant. \subsection{Numerical Results} The results for $\Delta m_K$ are fairly straightforward. We find that for each scenario which we have considered the indirect contributions are in fact completely dominated by the direct ones, with the gluino term being between one and two orders of magnitude too small to compete with the direct box diagrams. We illustrate this in Figure~\ref{KKbarlppRR}, which shows a typical situation. Here we have set $\lpp{213}(M_{GUT})=\lpp{223}(M_{GUT})=0.02$, $M_{1/2}=100$ GeV, $A_0=0$, $\tan\beta=10$, $\mu_4<0$, and show the relevant contributions. Apart from the fact that the direct contributions are dominant here, we note that the contribution from diagrams with one neutralino and one gluino line is comparable in magnitude to that from gluinos alone. This clearly will be significant for models involving mass insertions from a GUT theory. In general we find that for flavour violation in the left handed sector the contributions from charginos and mixed gluino-neutralino diagrams are up to half that of the gluino, while for right handed flavour violation the chargino contribution is negligible but the mixed neutralino-gluino contribution is similar. In summary then, we find that in practice this is the only process which has been studied where the indirect contributions are generically overwhelmed by the direct one, and hence we can essentially simply use the results quoted above in equations~(\ref{dRRdirectcontn}) and (\ref{dLLdirectcontn}) to bound the couplings, thus confirming the results of references~\cite{bgnn,ag,crs,chr}. Why this should be so is unclear, since the formulae for the contributions to $\Delta m_K$ are so complicated, but it appears the values of the various four point functions are simply such as to suppress the indirect contributions, while for the three point functions relevant for $\mu\to e\gamma$ and $b\to s\gamma$ the indirect contributions are enhanced. We also find that different parts in the gluino contribution are of different signs and comparable magnitude, and so tend to partially cancel. Finally, we note that the qualitative nature of this result is such that the direct contribution will also dominate in $B^0-\bar B^0$ and $D^0-\bar D^0$ mixing. \section{Conclusion} We now conclude with a brief summary of our results. We have extended our analysis of the RGEs for $R$-Parity violating supersymmetry to include the effects of the CKM matrix, and have studied two well-known flavour changing processes, including both the direct contributions, with $R$-Parity violating couplings at the vertices of diagrams, and the indirect ones, where flavour violation arises through soft masses generated by the Renormalisation Group Equations. We first showed that constraints on $\lp{}$ can be derived by demanding that the sneutrino masses not be driven below their experimental limits, which gives a bound of around 0.3 on the low energy values of the $\lp{}$ couplings, with the particularly interesting feature that it is extremely insensitive to the sparticle spectrum and does not disappear in the limit where the masses go to infinity. Given that the sparticle spectrum is now known to be heavy (with many limits well above 100 GeV if we assume unification) this means that this bound is one of the tightest existing in such a scenario. For the case of $b\to s\gamma$ we find that the indirect contributions caused by the flavour violation are dominant, and enhance the amplitude contribution by up to an order of magnitude, allowing new constraints on both $\lpp{}$ and $\lp{}$. However, these are rather weak, and the $\lp{}$ constraints are weaker than those from sneutrino masses derived earlier in this paper. With regard to $K^0-\bar K^0$ mixing, we have included all the indirect contributions including those from gluino, neutralino, chargino, and mixed neutralino-gluino diagrams. We find that here the chargino and mixed diagrams are sometimes comparable in size to those from gluinos, with consequences for the study of flavour changing from GUT theories. However, in the context of $R$-Parity violation, we find that the indirect contributions are consistently around an order of magnitude or more less than the direct ones, so that for this process and for the similar ones of $B^0-\bar B^0$ and $D^0-\bar D^0$ mixing we cannot improve on results in the literature \cite{bgnn,ag,crs,chr}. \noindent {\bf Acknowledgments \hfil} \\ PLW would like to thank Jonathan Flynn for teaching him something about QCD. BdC is very grateful to the University of Oxford for their hospitality during the early stages of this work. The work of BdC was supported by a PPARC Postdoctoral Fellowship.

proofpile-arXiv_065-538

\section{INTRODUCTION} The idea to use heavy quarkonia as a probe of excited QCD matter produced in relativistic heavy ion collisions was proposed a decade ago \cite{MS}. This suggestion was based on the concept of colour screening of static potential acting between the heavy quark and antiquark, which occurs in hot and/or dense quark-gluon matter. A year later, the $J/\psi$\ suppression in nucleus-nucleus collisions was observed experimentally \cite{NA38}. Since that time, quarkonium suppression has always been a respectable prospective signal of deconfinement. However, the experimentally observed suppression has been consistent not only with the deconfinement scenario, but with a plenty of ``conventional" explanations as well \cite{Blaizot}. Moreover, the differences between various conventional approaches, invoking such different mechanisms as nuclear absorption, gluon shadowing, or comover absorption, but nevertheless all providing more or less reasonable description of the data, created a lively controversy. The problem thus was clearly awaiting a more detailed and systematic analysis, many essential inputs for which were missing. In fact, the lack of precision data on $J/\psi$\ and \cal P\ production in $p-A$ collisions, poor theoretical understanding of $J/\psi$\ production and its interactions with light hadrons made this analysis virtually impossible. Fortunately, the situation has started to change recently: new high precision data on quarkonium production (see \cite{Carlos}-\cite{San} and references therein) have significantly advanced the theory \cite{Braa}, and the operator product expansion techniques have allowed a systematic calculation of quarkonium absorption cross sections \cite{Peskin}-\cite{KS} and dissociation rates in confined matter \cite{KS}-\cite{K1}. The dominance of higher Fock states in quarkonium production, revealed by the new Fermilab data \cite{San} and naturally emerging theoretically, has inspired a new approach to nuclear attenuation of quarkonium production \cite{KS2}. This approach, as I shall discuss in this talk, has enabled a quantitative understanding of quarkonium suppression in both $p-A$ and $A-B$ collisions, giving a credit to the nuclear absorption model \cite{GH}. It has become clear that the existing $S-U$ data in fact do {\it not} exhibit any anomalous behaviour and are {\it not} consistent with a deconfinement scenario, which requires additional strong suppression. This is why the results on $J/\psi$~ suppression in Pb-Pb collisions were so anxiously awaited -- they were our last chance to see something unusual before the advent of future experiments at RHIC and the LHC. These results, presented by the NA50 Collaboration at this Conference \cite{MG}, are striking. The data clearly show a strong $J/\psi$~ suppression, going way beyond the expected. But is this suppression really anomalous? Have we finally reached the border of the long-awaited {\it terra incognita} of deconfined quark-gluon matter? In this talk, I shall attempt to address these questions. \section{QCD ATOMS IN EXTERNAL FIELDS} \subsection{Quarkonium Interactions and the Operator Product Expansion} \vskip0.3cm In the Operator Product Expansion (OPE) approach, the amplitude of heavy quarkonium interaction with light hadrons is represented in the form \begin{eqnarray} F_{\Phi h} = i\int d^4x e^{iqx} \langle h|T\{J(x)J(0)\}|h \rangle = \sum_n c_n(Q,m_Q) \langle O_n \rangle , \label{2.5} \end{eqnarray} where the set $\{O_n\}$ should include all local gauge-invariant operators expressible in terms of gluon fields; the matrix elements $\langle O_n \rangle$ are taken between the initial and final light-hadron states. The coefficients $c_n$ are expected to be computable perturbatively and are process-independent. \vskip0.3cm \begin{minipage}[t]{6cm} \setlength{\unitlength}{0.00800in}% \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(229,207)(83,535) \thinlines \put(270,715){\oval(20,50)} \put(200,570){\oval(110,70)} \put( 85,740){\line( 1, 0){225}} \put( 85,690){\line( 1, 0){225}} \multiput(160,690)(0.00000,-8.26087){12}{\line( 0,-1){ 4.130}} \put(130,715){\oval(20,50)} \multiput(240,740)(0.00000,-7.83784){19}{\line( 0,-1){ 3.919}} \put(255,560){\line( 1, 0){ 55}} \put( 85,580){\line( 1, 0){ 60}} \put( 85,570){\line( 1, 0){ 60}} \put( 85,560){\line( 1, 0){ 60}} \put(255,580){\line( 1, 0){ 55}} \put(255,570){\line( 1, 0){ 55}} \end{picture} \end{minipage} \begin{minipage}[b]{9cm} Figure 1. A sample diagram describing quarkonium interaction with a light hadron in the OPE scheme; dashed lines are the gluon propagators, ovals represent the quarkonium wave function, and the blob stands for the gluon structure function of the hadron. \end{minipage} \vskip0.3cm The Wilson coefficients $c_n$ were computed for S \cite{Peskin} and P \cite{K1}, \cite{KS4} states in the leading order in $1/N^2$ ($N$ is the number of colours). The expectation values $\langle O_n \rangle$ of the operators composed of gluon fields can be expressed as Mellin transforms \cite{Parisi} of the gluon structure function of the light hadron, evaluated at the scale $Q^2=\epsilon_0^2$, \begin{eqnarray} \langle O_n \rangle = \int_0^1 dx\ x^{n-2} g(x, Q^2 = \epsilon_0^2). \label{2.12} \end{eqnarray} \begin{figure}[h] \begin{center} \epsfxsize14cm \mbox{\epsfig{file=qm96f5.ps,bbllx=2.cm,bblly=10.cm,bburx=21.5cm,bbury=26.cm,width=14.cm,angle=0}} \end{center} \noindent Figure 2. $J/\psi$ photoproduction cross section; the curve is the theoretical prediction \cite{KSSZ}. \label{prod} \end{figure} Since the total $\Phi-h$ cross section is proportional to the imaginary part of the amplitude $F_{\Phi h}$, the dispersion integral over the c.m.s. energy $\lambda$ leads to the set of sum rules, relating the cross section to the gluon structure function of the light hadron. This relation, illustrated in Fig.\ 1, has a very important property: the magnitude and energy dependence of the quarkonium dissociation cross section at low energies is entirely determined by the behaviour of the gluon structure function at large $x \sim 1/\lambda$, whereas the cross section at high energy is governed by the small $x$ behaviour of the structure function. Since the gluon structure functions of light hadrons are suppressed at large $x$, the calculated cross section rises very slowly from the threshold. When the hadron momentum in the $J/\psi$ rest frame is $P_h \simeq 5 $ GeV, the cross section is more than an order of magnitude below its asymptotic value. Recently, the calculation sketched above has been refined \cite{KSSZ} by taking into account target mass corrections, the real part of the scattering amplitude restored by dispersion relations, and the use of modern gluon structure functions inferred from the analyses of HERA data. This allows to evaluate the cross section in the entire energy range accessible to present experiments; the results confirm the threshold behaviour of the absorption cross section established previously. Vector meson dominance relates the cross sections of $J/\psi$~ dissociation and photo-production; Fig.\ 2 shows the results compared to the available data. One can see that a strong threshold suppression of the $J/\psi$~ absorption cross section is actually required by the data. \subsection{Quarkonium Interactions with Pions} \vskip0.3cm It can be shown that spontaneously broken chiral and scale symmetries of QCD imply decoupling of low-energy pions from heavy quarkonium \cite{K1}. The proof is based on the application of low-energy QCD theorems \cite{JE} (see \cite{EK} for a recent review and introduction) to the amplitude of quarkonium interactions with light hadrons \cite{VZ}. Qualitatively, the origin of the decoupling can be explained in the following way. At low energies, the amplitude of quarkonium interaction is proportional to the gluon field operator dominating the trace of the energy-momentum tensor of QCD. The appearance of this operator in the trace of the energy--momentum tensor is a reflection of the broken scale invariance of QCD, so the coupling is determined by the scale dimension of the hadron field. Chiral symmetry, however, implies zero scale dimension for the Goldstone boson fields -- otherwise scale transformations would break chiral invariance. \subsection{Quarkonium Production in Hadron Collisions} \vskip0.3cm The perturbative approach to quarkonium production \cite{CS},\cite{Ger} is based on the assumption that the production process is localized at distances $\sim m_{Q}^{-1}$, much shorter than the size of quarkonium $r \sim [\alpha_s(r^{-1})m_{Q}]^{-1}$. This approach is justified if all gluons involved in the production carry a high momentum $q \sim m_{Q}$. However, the hadroproduction of vector states, for example, requires at least three gluons, of which only two must be hard to create the $\bar{Q}Q$ pair. At small $P_T$ (the domain that dominates the integrated quarkonium production cross sections), the third gluon can be very soft, and is emitted (or absorbed) at distances of the order of quarkonium size. This is clearly inconsistent with the factorized form of the amplitude, and may ``explain" the failure of perturbative approach in describing the integrated cross sections of quarkonium production at fixed target energies. At collider energies, the perturbative approach fails even at high $P_T$, since the non-perturbative contribution to the gluon fragmentation becomes important \cite{Braa}. These arguments help to understand the phenomenological success of the colour evaporation model in explaining the data (see \cite{Gav} for a recent study). A consistent solution of this problem emerges if one assigns the soft gluon to the quarkonium wave function introducing the notion of $|\bar{Q}Qg...\rangle$ higher Fock states \cite{Braa}. In fact, such states appear naturally in the OPE scheme described above. Consider, for example, the amplitude of quarkonium interaction with an external gluon field (see Figs.\ 1 and 3): it includes the transformation of the colour-singlet quarkonium into a colour-octet $[\bar{Q}Q]_8$ state. The overall colour neutrality is of course preserved and ensured by the coloured gluon cloud surrounding the $[\bar{Q}Q]_8$ state. The simplest example of such system is provided by the $|[\bar{Q}Q]_8 g\rangle$ state. Since the vacuum of QCD has a complicated structure \cite{SVZ} with $\langle g^2 G^2 \rangle \neq 0$, it induces a significant admixture of the $|[\bar{Q}Q]_8 g\rangle$ component in the wave function of quarkonium \cite{Vol} -- see Fig. 3a. \begin{figure} \vskip0.5cm \begin{center} \setlength{\unitlength}{0.008in}% \begingroup\makeatletter\ifx\SetFigFont\undefined% \gdef\SetFigFont#1#2#3#4#5{% \reset@font\fontsize{#1}{#2pt}% \fontfamily{#3}\fontseries{#4}\fontshape{#5}% \selectfont}% \fi\endgroup% \begin{picture}(589,149)(83,593) \thinlines \put(270,715){\oval(20,50)} \put(465,715){\oval(20,50)} \put(635,715){\oval(20,50)} \put( 85,740){\line( 1, 0){225}} \put( 85,690){\line( 1, 0){225}} \multiput(160,690)(0.00000,-8.26087){12}{\line( 0,-1){ 4.130}} \multiput(230,740)(0.00000,-7.83784){19}{\line( 0,-1){ 3.919}} \multiput(155,605)(0.40000,-0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \multiput(155,595)(0.40000,0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \multiput(225,605)(0.40000,-0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \multiput(225,595)(0.40000,0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \put(425,740){\line( 1, 0){ 90}} \put(130,715){\oval(20,50)} \put(515,740){\line( 0,-1){ 50}} \multiput(610,605)(0.40000,-0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \put(425,690){\line( 1, 0){ 90}} \put(585,740){\line( 0,-1){ 50}} \put(585,740){\line( 1, 0){ 85}} \put(585,690){\line( 1, 0){ 85}} \multiput(515,740)(8.23529,0.00000){9}{\line( 1, 0){ 4.118}} \multiput(515,690)(8.23529,0.00000){9}{\line( 1, 0){ 4.118}} \multiput(485,690)(0.00000,-8.26087){12}{\line( 0,-1){ 4.130}} \multiput(615,690)(0.00000,-8.26087){12}{\line( 0,-1){ 4.130}} \multiput(480,595)(0.40000,0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \multiput(610,595)(0.40000,0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \multiput(480,605)(0.40000,-0.40000){26}{\makebox(0.1111,0.7778){\SetFigFont{5}{6}{\rmdefault}{\mddefault}{\updefault}.}} \end{picture} \end{center} \hskip6.5cm a) \hskip6.8cm b) \vskip0.3cm \noindent Figure 3. Interactions of quarkonium with external gluon fields; solid (dashed) lines are the heavy quark (gluon) propagators, and ovals represent the quarkonium wave function. \end{figure} For a physical $J/\psi$ state, this leads to the following generic decomposition: \begin{eqnarray} |J/\psi\rangle = a_1\ |\bar{c}c\rangle\ + \ a_2\ |[\bar{c}c]_8 g\rangle \ +\ ... \label{fock} \end{eqnarray} Similar decompositions hold for other quarkonium states; for $\chi$ states, for instance, the importance of higher Fock component is implied by the divergence of the perturbative annihilation amplitude in the soft gluon limit \cite{NR}. The magnitude of the $|[\bar{Q}Q]_8 g\rangle$ state admixture is reflected by the magnitude of relativistic corrections in the NRQCD approach \cite{NR} and by the size of power corrections in the QCD sum rule approach \cite{SVZ}. These corrections are generally not very large, making applicable the familiar concept of heavy quarkonium as of a non-relativistic system essentially composed of just $\bar{Q}Q$ state. However in certain processes -- like production and annihilation of quarkonium -- these components can play extremely important role\footnote{Another example is provided by the scattering of quarkonium states at very high energies \cite{Mue}.}. In fact, the leading order production of heavy vector quarkonium proceeds via the gluon fusion producing the $\bar{Q}Q$ pair in a colour-octet state that later neutralizes its colour emitting (or absorbing) an extra gluon. If this extra gluon is soft (as is the case in the small $P_T$ domain), the production process can be visualized as proceeding via the higher Fock state $|[\bar{Q}Q]_8 g\rangle$ (see Fig.3b). Since the colour Coulomb interaction between the heavy quarks in the colour-octet state is repulsive and weak ($\sim 1/(N^2-1)$ with respect to the attraction in the colour-singlet state, where $N$ is the number of colours), the $|[\bar{Q}Q]_8 g\rangle$ state is separated from the basic $|\bar{Q}Q\rangle$ state by the mass gap of $\simeq \epsilon_0$, where $\epsilon_0$ is quarkonium binding energy. This (virtual) state therefore has a proper lifetime of $\tau \simeq 1/\epsilon_0$. In the frame where quarkonium moves with momentum $P$, the superposition (\ref{fock}) will be coherent over a distance $z_c \simeq \tau P/2M_{Q}$. At high energies, this distance is sufficient for a produced $|[\bar{Q}Q]_8 g\rangle$ state to traverse the entire nuclear volume. What will be the effect of the nuclear medium on the propagation of such a state? To answer this question, let us first note that the produced $[\bar{Q}Q]_8$ pair is initially almost pointlike, with the transverse size of $r^{\bar{Q}Q}_{\perp}\sim 1/2m_Q$ (see Fig. 3b). The produced $|[\bar{Q}Q]_8 g\rangle$ state can be thus considered as a colour dipole formed by an almost pointlike colour-octet $\bar{Q}Q$ state and a collinear gluon. The transverse size of the $|\bar{c}c g\rangle$ state can be estimated \cite{KS2} from the characteristic virtualities of the diagram of Fig.3b as $r_{\perp} \simeq (2m_c \Lambda_{{\rm QCD}})^{-1/2} \simeq 0.20-0.25$ fm. An interaction inside nuclear matter will most likely prevent this state from binding, at later stage, to the quarkonium -- the colour octet $\bar{c}c$, with its collinear gluon stripped off, will preferably produce open charm mesons\footnote{Note that the $J/\psi$~ production cross section represents only a tiny part, of the order of 1\%, of the total charm production -- this means that the probability to pick up a collinear gluon for the colour-octet $\bar{Q}Q$ state is in general very small.}. Let us try to estimate the break-up cross section of such $|[\bar{Q}Q]_8 g\rangle$ state in its interaction with nucleons \cite{KS2}. The transverse size of the $|[\bar{c}c]_8 g\rangle$ state estimated above is roughly the same as the size of $J/\psi$. The $|[\bar{c}c]_8 g\rangle$ state however is not bound, so, contrary to the case of $J/\psi$, we do not expect any threshold suppression of the break-up cross section. We can therefore estimate the $|[\bar{c}c]_8 g\rangle$ break-up cross section rescaling the value of the $J/\psi$ break-up cross section at high energy (where the threshold suppression does not affect the cross section) by the colour factor $9/4$, arising from the difference between the couplings of colour dipoles formed by the triplet and octet charges. At the energy range relevant for the fixed target experiments, the $J/\psi$ break-up cross section evaluated in the formalism of section 2.1 is $\sigma_{J/\psi N} \simeq 2.5-3$ mb. We therefore get $\sigma_{(\bar{c}c g) N} \simeq 6 - 7$ mb as an estimate of the $|[\bar{c}c]_8 g\rangle$ absorption cross section. The analogous estimate for bottomonium states yields $\sigma_{(\bar{b}b g) N} \simeq 1.5 - 2$ mb. These estimates are admittedly rough; they show, however, that the nuclear attenuation of quarkonium production is in general quite strong and, in the first approximation, is universal for various quarkonium states. \section{QUARKONIUM AS A PROBE OF DECONFINED MATTER} We have shown in the previous section that the absorption cross sections of tightly bound quarkonium states at low energies are very small due to the softness of gluon fields confined inside light hadrons; this protects $J/\psi$~ in a thermal hadron gas at all meaningful temperatures ($T\leq 0.3$ GeV) \cite{KS}-\cite{K1}. On the other hand, the distribution of gluons in a deconfined medium is directly thermal, so that the deconfined gluons are hard, with the average momentum of $\langle p_g \rangle_{\rm deconf} = 3T$. An immediate consequence of deconfinement is thus a considerable hardening of the gluon momentum distribution \cite{KS,KS1}. Hard deconfined gluons can easily break up the $J/\psi$; the cross section of this ``gluo-effect" is given by \begin{eqnarray} \sigma_{g J/\psi}(k) = {2\pi \over 3} \left( {32 \over 3} \right)^2 \left( {m_c \over \epsilon_0} \right)^{1/2} {1 \over m^2_c} {(k/\epsilon_0 - 1)^{3/2} \over (k/\epsilon_0)^5 }, \label{5.4} \end{eqnarray} where $k$ is the momentum of the gluon incident on a stationary quarkonium with binding energy $\epsilon_0$. We thus see qualitatively how a deconfinement test can be carried out. If we put a $J/\psi$~into matter at a temperature $T=0.2$ GeV, then the $J/\psi$~will survive if the matter is confined, and will disappear if the matter is deconfined, since in the latter case the gluons will be hard enough to break it up. The latter part of this statement is in accordance with the original prediction that the formation of a QGP should lead to a $J/\psi$~suppression \cite{MS,KMeS}. There it was argued that in a QGP, colour screening would prevent any resonance binding between the perturbatively produced $c$ and ${\bar c}$, allowing the heavy quarks to separate. At the hadronization point of the medium, they would then be too far apart to bind to a $J/\psi$~and would therefore form a $D$ and a $\bar D$. Our picture complements this argument by the conclusion that additional suppression of physical $J/\psi$\ in dense matter will occur {\it if and only if} there is deconfinement. The dissociation of $J/\psi$\ (or $\Upsilon$) in both pictures is a consequence of the interaction with strong gluon fields present in deconfined matter. There is, however, a difference between the two mechanisms: the static screening picture takes into account the effect of deconfined fields on the binding potential acting between the heavy quarks, but neglects the energy-momentum transfer between the $J/\psi$\ and the heat bath. The dynamical ``gluo-effect" picture of quarkonium suppression, on the other hand, emphasizes the role of the energy-momentum transfer from deconfined gluons to the $J/\psi$~, but neglects the screening of the binding potential. Both pictures are expected to describe the physics of $J/\psi$\ suppression in their respective domains of applicability; they should emerge as two limits in one unified microscopic approach, that still has to be developed. The parameter that is relevant in this problem is $X(T) \equiv \Delta E(T) / T$, where the binding energy of quarkonium $\Delta E$ depends on the temperature of the system $T$ because of the Debye screening. In the weak coupling limit of $X\ll 1$, the binding energy is negligible compared to the temperature, and the quarkonium will simply fall apart with the rate $R = 4 / L (T / \pi M_Q)^{1/2}$ ($L$ is quarkonium size), which is the classical high temperature limit of thermal activation rate \cite{KMS}. In the strong coupling limit of $X\gg 1$, on the other hand, the system is tightly bound, and the binding energy threshold has to be overcome by the absorption of hard gluons from the heat bath. The rate of dissociation in this case should be computed from the thermal average of the gluon-quarkonium cross section (\ref{5.4}). The actual value of $X$ at different temperatures depends, of course, on the detailed dynamics of screening; lattice calculations can be of significant help here, fixing the temperature dependence of quarkonium mass. It is important to note that the dynamical ``gluo-effect" approach to $J/\psi$~suppression does not require a thermal equilibrium of the gluon fields, so that it will remain applicable even in deconfined pre-equilibrium stages. Quarkonium interactions in an equilibrating parton gas were considered in ref. \cite{XW}. \section{PHENOMENOLOGY OF QUARKONIUM PRODUCTION\\ IN NUCLEAR COLLISIONS$^4$} \addtocounter{footnote}{1}\footnotetext{This section is based on the work \cite{KLNS}.} \subsection{$p-A$ collisions} \vskip0.3cm According to our discussion in section 2.3, in the presently accessible kinematic region of $J/\psi$~production by $p-A$ collisions ($x_F\geq 0$), the target nucleus sees only the passage of the pre-resonance state; physical charmonium states are formed outside the nucleus. The size of the pre-resonance state is determined by the charmed quark mass and confinement scale and is therefore the same for $J/\psi$~and \cal P. The nuclear attenuation of $J/\psi$~ and \cal P~ production in $p-A$ collisions should thus be universal. Indeed, the $J/\psi$~and \cal P~production in $pA$ collisions shows to the same $A$-dependence. Fitting the available data on the \cal P/($J/\psi$) ratio \cite{Carlos} to the form $A^{\alpha}$ leads to \begin{eqnarray} \alpha = 0.0 \pm 0.02, \ \ 95\%\ C.L.;\label{5} \end{eqnarray} this rules out variations of more than 10 \% between $pp$ and $pU$ collisions. The suppression of $J/\psi$~production in $p-A$ collisions should thus be understood as pre-resonance absorption in normal nuclear matter. This accounts naturally for the equal suppression observed for the two states, which would be impossible for physical resonances of such different sizes. We shall now determine the pre-resonance absorption cross section from the NA38/51 $p-A$ data at incident proton beam energies of 200 and 450 GeV \cite{Carlos}. In Glauber theory, the survival probability for a $J/\psi$~produced in a $p-A$ collision is given by \begin{eqnarray} S_{pA}^{Gl} = {\sigma_{pA\to \psi}\over A \sigma_{pN \to \psi}} = \int d^2b~dz \rho_A(b,z) \exp\left\{-(A-1)\int_z^{\infty} dz' \rho_A(b,z') \sigma_{abs} \right\}. \label{2} \end{eqnarray} Here $\rho_A$ is the nuclear density distribution, for which we take the standard three-parameter Woods-Saxon form with parameters as tabulated in Ref.\ \cite{deJager}; it is normalized to unity, with $\int d^2b dz \rho_A(b,z) = 1$. The suppression is thus fully determined by the absorption cross section $\sigma_{abs}$ in nuclear matter. From the NA38/51 data we obtain the best fit for \begin{eqnarray} \sigma_{abs} = 6.3 \pm 0.6~ {\rm mb}, \ \ 95\%\ C.L.; \label{4} \end{eqnarray} the corresponding survival probabilities are plotted in Fig. 4. The agreement is seen to be excellent in all cases. The value (\ref{4}) is consistent with the theoretical estimates of section 2.3, which suggest for the absorption cross section of the ${\bar c}c-g$ on nucleons $\sigma_{abs} \simeq 6 - 7$ mb \cite{KS2}. \begin{figure}[h] \hskip1.4cm {\large{$S_{pA}^{J/\psi}$}} \begin{center} \epsfxsize12cm \mbox{\epsfig{file=qm96f6.ps,bbllx=1.cm,bblly=9.2cm,bburx=21.5cm,bbury=16.5cm,width=12.cm,angle=0}} \end{center} \hskip12cm {\large{A}}\\ \noindent Figure 4. $J/\psi$ suppression in $pA$ collisions; the NA38/50 data (black points) are compared to the Glauber theory calculations (grey points) with $\sigma_{abs}=6.3 \pm 0.6$ mb. \end{figure} We have carried out the same analysis for the 800 GeV E772 data (see \cite{Pat} for a review); here the cross section is slightly larger: $\sigma_{abs} = 7.4 \pm 0.7$ mb , but within errors compatible with the value (\ref{4}) obtained from the NA38/51 data. A slow increase of the absorption cross section with energy can be attributed to the growth of the gluon structure function towards smaller $x$ (the same effect is responsible for the increase of the $J/\psi$ absorption cross section in the relevant energy range, see Fig.\ 2). We thus conclude that $J/\psi$~and \cal P~production in $pA$ collisions is quantitatively well described by absorption of a pre-resonance charmonium state in nuclear matter, with the absorption cross section for both states in the energy range of SPS experiments given by Eq.\ (\ref{4}). We now extend this description to nuclear collisions. \subsection{{\it{\bf S-U}} Collisions} \vskip0.3cm In nucleus-nucleus collisions, charmonium production can be measured as function of the centrality of the collision, and hence we have to calculate the $J/\psi$~survival probability at fixed impact parameter $b$. It is given by \begin{eqnarray} { d S_{AB}^{Gl}(b) \over d^2 b} = {1 \over A B~ \sigma_{NN \to \psi}}\left[{d \sigma_{AB \to \psi} \over d^2 b} \right] = \int d^2s dz dz' \rho_A(\vec{s},z) \rho_B(\vec{b}-\vec{s},z') S_A(z,\vec{s}) S_B(z',\vec{s}), \label{6} \end{eqnarray} where $S_A(z,\vec{s}) = \exp\left\{-(A-1)\int_z^{\infty} dz_A~ \rho_A(\vec{s},z_A)~ \sigma_{abs} \right\}$, and analogously for $S_B(z',\vec{s})$. Here $\vec{s}$ specifies the position of the production point in a plane orthogonal to the collision axis, while $z$ and $z'$ give the position of this point within nucleus $A$ and within nucleus $B$, respectively. The nuclear density distributions $\rho_A$ and $\rho_B$ are defined as above. To obtain normalized survival probability at fixed impact parameter $b$, we have to divide $[dS^{Gl}/d^2b]$ by $[dS^{Gl}(b;\sigma_{abs}=0)/d^2b]$. \par Experimentally, the centrality of the collision is determined by a calorimetric measurement of the associated transverse energy $E_T$; we thus have to establish and test a correpondence between impact parameter $b$ and transverse energy $E_T$. This correlation can be expressed in terms of the number of ``wounded" nucleons \cite{Bialas}. Each wounded nucleon contributes on the average an amount $q$ to the overall transverse energy produced in the collision, so we have the relation \begin{eqnarray} {\bar E}_T(b)~ =~q~{\bar N}_w(b) \label{propw} \end{eqnarray} between the average number ${\bar N}_w$ of nucleons wounded in a collision at fixed impact parameter $b$ and the associated average transverse energy ${\bar E}_T$ produced in that collision. In the analysis of specific experimental results, the proportionality factor $q$ depends on the details of the detector, in particular on the rapidity and transverse momentum range in which the produced secondaries are measured. \par The average number of wounded nucleons in an $AB$ collision at impact parameter $b$ is given by \begin{eqnarray} {\bar N}^w_{AB}(b)~\equiv \int d^2 s\ n^w_{AB}(b,s) = ~& A \int d^2 s~ T_A(\vec{s}) \left\{ 1-[1-\sigma_N T_B(\vec{s}-\vec{b})]^B\right\} + & \nonumber \\ +~ & B \int d^2 s~ T_B(\vec{s}-\vec{b}) \left\{ 1-[1-\sigma_N T_A(\vec{s})]^A\right\}. & \label{9} \end{eqnarray} Here $\sigma_N\simeq 30$ mb denotes the inelastic production cross section, and $T_A(\vec{s}) =\int dz~\rho_A(z,\vec{s})$ the nuclear profile function; the $\vec{s}$-integration runs again over a plane orthogonal to the collision axis. The distribution (\ref{9}) is normalized in the following way: \begin{eqnarray} {\bar N}^w_{AB} = {1 \over \sigma_{AB}} \int d^2 b {\bar N}^w_{AB}(b) = {1 \over \sigma_{AB}}\ (A \sigma_B + B \sigma_A). \end{eqnarray} Since there are fluctuations in the number of wounded nucleons and in the transverse energy of the secondaries that each wounded nucleon produces, there will be corresponding fluctuations in the relation between $E_T$ and $b$. We assume the dispersion $D$ in the produced transeverse energy to be proportional to $\sqrt{N_w}$, $D^2~ =~ a {\bar E}_T(b)$, with a universal physical parameter $a$ to be determined from $pA$ or $AB$ collisions. We choose the $E_T-b$ correlation function $P_{AB}(E_T,b)$ as a conventional (see, e.g., \cite{GV}) Gaussian distribution around the central value (\ref{propw}) with dispersion $D$; it is normalized at fixed $b$: $\int dE_T ~P_{AB}(E_T,b) = 1$. We have checked \cite{KLNS} that both minimum bias \cite{Stock} and Drell-Yan associated \cite{NA38},\cite{MG} transverse energy spectra are very well reproduced in the approach outlined above. With the relation between the measured transverse energy $E_T$ and the impact parameter $b$ of the collision thus determined, we can now calculate the $E_T$ dependence of the charmonium survival probability in nuclear matter. We begin with $J/\psi$~production. The experimentally determined quantity is the ratio $(d\sigma_{AB}^{J/\psi}/dE_T)/(d\sigma_{AB}^{DY})$ of $J/\psi$ to Drell-Yan production, measured in the mass interval $2.9 \leq M_{\mu\mu} \leq 5.5$ GeV. From this we obtain the survival probability at fixed $E_T$ \begin{eqnarray} S^{J/\psi}_{exp}(E_T) = {\sigma^{DY}_{AB} \over \sigma^{J/\psi}_{AB}}\ \left[{ d\sigma_{AB}^{J/\psi} \over dE_T} / { d\sigma_{AB}^{DY} \over dE_T} \right] \label{sexp} \end{eqnarray} by normalizing the measured ratio at fixed $E_T$ by the measured integrated cross sections. The quantity (\ref{sexp}) can be directly computed in the Glauber theory formalism outlined above. Using the value of the pre-resonance absorption cross section (\ref{4}), determined from the analysis of $p-A$ data, we have found a good agreement with the $E_T$-integrated $O-Cu$, $O-U$ and $S-U$ data and with $E_T$ distributions measured in $S-U$ collisions, as we shall shortly show. For \cal P~production, the situation changes. The data for the integrated and the differential survival probabilities are considerably lower than what nuclear absorption predicts, and the additional suppression moreover increases with increasing $E_T$. We therefore need to include the effect of additional \cal P\ suppression on $J/\psi$~production. The branching ratio for the reaction $\psi' \to J/\psi$ is 0.57; therefore the $\psi/\psi'$ ratio measured in $pp$ and $pA$ collisions \cite{Carlos} implies that $8\pm 2$ \% of the observed $J/\psi$'s are due to \cal P~decay. Since the \cal P\ is suppressed in $S-U$ collisions, the corresponding fraction of the observed $J/\psi$'s must be suppressed as well. This correction reduces theoretical predictions on the average by $\simeq 5\%$. We show in Fig.\ 5 (left) the resulting corrected theoretical $E_T$ dependence of $J/\psi$~survival probabilities. The agreement between the data and predictions is seen to be excellent. I wish to stress that we do not need to invoke any additional sources of direct $J/\psi$\ suppression (apart from the nuclear absorption of pre-resonance charmonium state) to describe the data. On the other hand, the additional \cal P\ suppression found in $S-U$ collisions clearly indicates the presence of produced matter at the stage when charmonium states are formed. The agreement of our Glauber calculations with the measured $J/\psi$\ survival probabilities shows, however, that this matter cannot break up $J/\psi$\ states. This can be explained by the smallness of $J/\psi$\ dissociation rate in confined hadronic gas \cite{KS}-\cite{KMS} advocated in sections 2 and 3. \vskip0.2cm \begin{figure}[h] \noindent{\large{$S_{AB}^{J/\psi}$}} \begin{center} \epsfxsize14cm \mbox{\epsfig{file=qm96f8.ps,bbllx=2.5cm,bblly=8.8cm,bburx=21.5cm,bbury=17.cm,width=14.cm,angle=0}} \end{center} \hskip12cm {\large{$E_T$}}\\ \vskip0.3cm \noindent Figure 5. $J/\psi$ suppression in $S-U$ (left) and $Pb-Pb$ (right) collisions; the NA38/50 data \cite{Carlos} (black points) are compared to the Glauber theory predictions \cite{KLNS} (grey points) with $\sigma_{abs}=6.3\pm 0.6$ mb. \label{supp} \end{figure} \subsection{Pb-Pb Collisions} \vskip0.3cm We can further check our approach in $Pb-Pb$ collisions, since the NA50 experiment \cite{MG} is equipped with a zero degree calorimeter (ZDC), which determines at each $E_T$ the associated number of projectile spectators -- those projectile nucleons which reach the ZDC with their full initial energy $E_{in}=158$ GeV/c. This additional information is important, since it uniquely identifies the peripherality of the collision. Denoting the projectile as $A$, the number of projectile spectators is evidently $A - N_w^A$, with $N_w^A$ of the $A$ nucleons in the projectile wounded. We thus have $E_{ZDC}=(A-N_w^A)E_{in}$; using $\bar{E}_T=q~\bar{N}_w$, we predict \cite{KLNS} the $E_T-E_{ZDC}$ correlation which agrees very well with the measured one \cite{MG}, \cite{web}. \par We are now ready to address the $J/\psi$\ production in the NA50 experiment \cite{MG}. The $Pb-Pb$ results, plotted as a function of the average path $L$ of $J/\psi$\ in nuclear matter, clearly show strong additional suppression beyond the expected on the basis of $\sim exp(-\rho_0 \sigma_{abs} L)$ dependence \cite{MG}. The conclusion on the ``anomalous" nature of this suppression, however, crucially depends on the magnitude of $L$, assigned to the $Pb-Pb$ points. We would like therefore first to check the $L$ assignment of the NA50 Collaboration in our approach, which directly gives the $J/\psi$\ survival probability at a given $E_T$. The results are presented in Fig.\ 5 (right). One can see that while the lowest $E_T$ point is still marginally consistent with Glauber theory, the suppression observed at higher $E_T$ indeed goes significantly beyond expected. Equating our calculated survival probability to the form used by the NA50, $S_{Gl}=exp(-\rho_0 \sigma_{abs} L)$, we confirm the NA50 $L$ assignments \cite{MG}. Since, as we have shown, the Glauber theory approach has been extremely successful in reproducing the bulk of $J/\psi$\ production data in $p-A$ and $A-B$ collisions, the suppression observed in $Pb-Pb$ indeed can be called ``anomalous". \section{IS THE QUARK-GLUON PLASMA DISCOVERED?} \noindent Before we address this provocative question, posed at this Conference also by \mbox{J.-P. Blaizot} \cite{JP} and C.-Y. Wong \cite{Wong}, let us consider the possible differences between the collision dynamics in $S-U$ and $Pb-Pb$ systems. The success of Glauber theory in describing the $J/\psi$\ suppression in $S-U$ collisions and its failure in $Pb-Pb$ points to a difference in the properties of matter produced in these two reactions. We shall try to describe this difference in terms of two variables, one of which characterizes the energy density of produced matter, and the other its degree of equilibration. In Glauber theory, the initial energy density achieved in the collision is proportional to the density of wounded nucleons $n_w$ (see Eq.\ (\ref{9})) in the transverse plane. The average energy densities achieved in $S-U$ and $Pb-Pb$ collisions are almost identical; at first glance this suggests that the matter seen by produced $J/\psi$'s should be the same in both cases. This is not so, however, for two reasons. First, the profile of the energy density in the two systems is different: central $Pb-Pb$ collisions produce a ``hot core", inside which the energy density is higher than the highest one attainable in $S-U$ system by about $25\%$. Second, the $J/\psi$'s are produced mostly in this central region (see Eq.\ (\ref{6})), and thus feel the matter which is hotter than average. These two effects combined lead to significant difference in the energy densities of matter seen by $J/\psi$'s in $S-U$ and $Pb-Pb$ collisions. This is illustrated in Fig.\ 6, where we plot the ratio of experimental $J/\psi$\ suppression to the Glauber theory predictions versus the average density of matter seen by $J/\psi$\ (to calculate this latter quantity, we convolute the density distribution with the $J/\psi$\ production profile). \begin{figure}[h!] \hskip1cm {\large{${S_{exp}^{J/\psi} / S_{Gl}^{J/\psi}}$}} \begin{center} \epsfxsize9.5cm \mbox{\epsfig{file=qm96f9.ps,bbllx=2.5cm,bblly=2.cm,bburx=21.5cm,bbury=25.cm,width=9.5cm,angle=0}} \end{center} \hskip10cm {\large{$\bar{n}_w^{J/\psi}, {\rm fm}^{-2}$}}\\ \vskip0.3cm \noindent Figure 6. The ratio of experimental $J/\psi$ suppression in $S-U$ \cite{Carlos} (crosses) and $Pb-Pb$ \cite{MG} (circles) collisions to the Glauber theory predictions versus the average density of wounded nucleons seen by $J/\psi$ (the latter quantity is proportional to the energy density). \label{ratio} \end{figure} Apart from being more dense, the matter seen by $J/\psi$\ is also likely to be more thermalized. We can quantify this statement introducing the variable \begin{eqnarray} \kappa = {\nu + 1 \over w}, \label{kappa} \end{eqnarray} where $\nu$ is the number of inelastic $NN$ collisions and $w$ is the number of wounded nucleons these collisions produce. The value of $\kappa$ tells how many times, on the average, each wounded nucleon was hit. In a $pp$ collision, $\nu=1$ and $w=2$, so that $\kappa=1$. In $pA$ collisions, the number of wounded nucleons in the target is equal to the number of collisions \cite{Bialas}, which, after taking into account the wounded projectile nucleon, again yields $\kappa=1$. In nucleus-nucleus collisions, however, the value of $\kappa$ can exceed unity because the nucleons once wounded can collide again and again. These collisions can break down the independence of fragmentation of wounded nucleons and provide initial conditions for the onset of collective behaviour in the system. Indeed, when $\kappa>1$, the partons from different wounded nucleons interact, which is a necessary initial stage for producing deconfined matter -- at large $\kappa$, partons can no longer be attributed to a particular pair of wounded nucleons and overlap in the transverse plane. It is evident that in nuclear collisions $\kappa$ grows with atomic number and/or energy (since the number of collisions depends on the inelastic cross section). In central $S-U$ collisions at SPS energy, the Glauber theory calculation yields $\bar{\kappa}_{SU}\simeq 1.7$, whereas for a central $Pb-Pb$ collision we find $\bar{\kappa}_{PbPb}\simeq 2.4$. The $Pb-Pb$ value thus is as far from the $S-U$ one as the $S-U$ is from $pA$. Central $Pb-Pb$ collisions therefore are likely to produce not only more dense, but also more thermalized matter. It is important to note that the value of $\kappa$ and its variation with centrality can be determined in a model--independent way directly from the experimental data. Indeed, the number of collisions $\nu$ is proportional to the number of produced Drell-Yan pairs, and the number of wounded nucleons to the produced transverse energy (see Eq.\ (\ref{propw})), so at a given $E_T$ one has \begin{eqnarray} \kappa (E_T) \sim q {N_{DY}(E_T) \over E_T} = {q \over E_T}\ {1 \over \sigma_{DY}\Delta E}\ \int_{\Delta E} {d\sigma_{DY} \over dE_T} dE_T, \label{kapexp} \end{eqnarray} where $N_{DY}(E_T)$ is the number of Drell-Yan pairs associated with a given $E_T$ bin of the width $\Delta E_T$. We thus conclude that the matter seen by $J/\psi$\ in central $Pb-Pb$ collisions is more dense and more thermalized, so the occurence of new phenomena at least cannot be excluded {\it a priori}. If the observed $J/\psi$\ suppression were indeed to be interpreted as a signal of deconfinement phase transition, apart from being ``anomalous", it also has to exhibit a threshold behaviour. This feature seems to be present in the data (see Fig.\ 6): a slight increase in the energy density induces dramatic deviation from the trend established previously. Moreover, if one considers the profile of the density $n_w$ and $\kappa$ in the transverse plane and assumes that {\it all} $J/\psi$'s produced in the region where $n_w > n_{SU}^{max},\ \kappa > \kappa_{SU}^{max}$ are dissociated, the resulting suppression falls below the data points by only $10-15\%$ (similar analysis has been presented at this Conference by J.-P. Blaizot \cite{JP}). This shows that the observed suppression is almost as strong as we could possibly accomodate! Can we still find a ``conventional" explanation of the NA50 effect? It is of course too early to try to answer this question; let us therefore limit ourselves to some preliminary observations. Since Glauber calculations show that nuclear absorption of pre-resonance state cannot explain the $Pb-Pb$ data, a conventional explanation has to invoke additional suppression of $J/\psi$'s in the produced confined matter. This suppression would be characterized by a smooth (basically, exponential) dependence on the density of produced matter. Looking at Fig.\ 6, one realizes that it would not be easy to fit the data by this kind of a smooth dependence -- the fits would most likely overestimate the slope of the $S-U$ data and underestimate it for the $Pb-Pb$ points. The physical reason for this is transparent -- if hadronic comovers do not induce an additional $J/\psi$\ suppression in $S-U$ collisions, it is difficult to make them effective in $Pb-Pb$ system. One may try to assume a larger density variation with $E_T$ (sometimes the density is assumed to be directly proportional to $E_T$ \cite{Sean}). Indeed, a large variation of density is possible in very central collisions due to fluctuations in the number of produced hadrons. However the measured $E_T-E_{ZDC}$ correlation discussed in section 4.3 shows that in the presented $Pb-Pb$ data the variation of $E_T$ results from the variation of the collision centrality. Even the highest $E_T$ point of NA50 is not entirely in the fluctuation domain, and corresponds to the mean impact parameter of $\bar{b}\simeq 2$ fm. In this regime, the $E_T$ measured by the NA50 varies by more than four times, but this leads only to $\sim 30\%$ variation of the initial energy density (see Figs.\ 5 and 6). Additional constraint on a conventional scenario is imposed by the $\psi'/\psi$ ratio: if the density of comovers in $Pb-Pb$ were much higher than the density in $S-U$, this would imply a smaller $\psi / \psi'$ ratio in the former case -- the prediction that would bring us in conflict with the data \cite{MG}. A possible way to describe the $Pb-Pb$ points conventionally would be to decrease artificially the value of the nuclear absorption cross section $\sigma_{abs}$, leaving thus room for comover absorption already in $S-U$ collisions (the need for comover effects to explain the $S-U$ data was advocated at this Conference by S. Gavin \cite{Sean}). This, however, would contradict to the $pA$ data, that fix the value of $\sigma_{abs}$ rather precisely (see section 4.1). Of course, it remains to be seen if a convincing conventional explanation can eventually be found. To summarize, the NA50 $Pb-Pb$ results indeed seem to suggest that a new mechanism of $J/\psi$\ suppression sets in at higher energy densities. The observed effect can be considered as a strong evidence of some kind of deconfinement in nuclear collisions. What can we do to turn this evidence into a proof, or to discard it? \section{WHAT HAS TO BE DONE NEXT?} New precision data on quarkonium production coming from CERN SPS, Fermilab, HERA and elsewhere allow us today to get rid of many uncertainties inherent to the analyses of $J/\psi$\ suppression over the years. A coherent picture, providing a good description of the bulk of existing $pp$, $pA$ and $AB$ data, has started to emerge -- this makes us ready to recognize and study unusual phenomena. It is therefore particularly important to learn more about the onset of anomalous behaviour of $J/\psi$\ suppression seen by the NA50 Collaboration. More statistics and more data points, both in the transition regime of small $E_T$ and in the fluctuation region of the highest $E_T$, are needed to establish the threshold behaviour suggested by the present data. An important information would be also provided by the $J/\psi$\ transverse momentum distributions \cite{PT}. Direct measurement of the low-energy $J/\psi$\ absorption cross section in the proposed inverse kinematics experiment \cite{KS},\cite{KS4} has become possible with the advent of $Pb$ beam at CERN SPS. This experiment would allow us to directly constrain the $J/\psi$\ absorption possible in a hadronic medium, providing important additional check of the deconfinement transition as the cause of ``anomalous" $J/\psi$\ suppression. Heavy quarkonium represents a rare example of a strongly interacting system that is simple enough to be systematically analyzed by the current theoretical methods. It has already proved to be extremely useful for understanding the properties of QCD and its ground state - the vacuum. I believe that quarkonium will tell us much also about the critical behaviour of QCD matter produced by relativistic heavy ion collisions. \vskip0.3cm {\bf Acknowledgements} \vskip0.2cm \noindent The results presented here were obtained together with H. Satz; I wish to thank him for an enjoyable collaboration. I am grateful to L. McLerran for his collaboration, valuable comments and encouragement. Very important contributions to the work presented here have been made by C. Louren\c{c}o, M. Nardi, A. Syamtomov, X.-N. Wang, X.-M. Xu and G.M. Zinoviev. Stimulating and enlightening discussions of various aspects of this work with J.-P. Blaizot, E. Braaten, A. Capella, J. Ellis, K.J. Eskola, S. Gavin, M. Ga\'{z}dzicki, \mbox{C. Gerschel,} \mbox{M. Gonin,} M. Gyulassy, M. Jacob, A.B. Kaidalov, F. Karsch, L. Kluberg, \mbox{T. Matsui,} \mbox{A.H. Mueller,} B. M\"{u}ller, J.-Y. Ollitrault, J. Schukraft, G. Schuler, \mbox{R. Stock,} \mbox{A.I. Vainshtein} and R. Vogt are gratefully acknowledged.

proofpile-arXiv_065-539

\section{Introduction} In this paper we use a new real space renormalization-group map to study renormalization of SUSY $\phi^4$ theories. Symanzik's \cite{Sy} work proved that $\phi^4$ theories can be represented as weakly self-avoiding random walk models. They are in the same universality class as the self-avoiding random walk \cite{La}.\ Our renormalization-group transformation is carried out on the space of probabilities. Real space renormalization-group methods have proved to be useful in the study of a wide class of phenomena and are used here to provide a new stochastic meaning to the parameters involved in the flow of the interacting constant and the mass, as well as the $\beta$ function for SUSY $\phi^4$.\ The hierarchical models introduced by Dyson \cite{Dy} have the feature of having a simple renormalization-group transformation. We use a hierarchical lattice where the points are labeled by elements of a countable, abelian group {\it G} with ultrametric $\delta$; i.e. the metric space $({\it G},\delta)$ is hierarchical. The hierarchical structure of this metric space induces a renormalization-group map that is ``local"; i.e. instead of studying the space of random functions on the whole lattice, we can descend to the study of random functions on L-blocks (cosets of {\it G}) \cite{Ev}. Our method provides a probabilistic meaning to every parameter appearing in the flow of interaction constant and the mass for any $\phi^n$.\ This paper is organized as follows; in Section 2 we present the lattice and the corresponding class of L\'{e}vy processes which are studied here. In Section 3 we define the renormalization-group map and apply this to SUSY $\phi^4$, in random walk representation. In Section 4 we use the results obtained in previous Section to give new ligth into the stochastic meaning of the map and results are presented.\ \section{The lattice with ultrametric and the L\'{e}vy process.} The hierarchical lattice used in this paper was introduced by Brydges, Evans and Imbrie \cite{Ev}.Here, we present a slight variant. Fix an integer $L\geq 2$. The points of the lattice are labeled by elements of the countable, abelian group ${\it G}=\oplus ^{\infty}_{k=0}{\bf Z}_{L^d}$, d being the dimension of the lattice. A one-dimensional example can be found in Brydges et al \cite{Ev} and a two dimensional example in Rodr\'{\i}guez-Romo \cite{Su}. An element $X_i$ in ${\it G}$ is an infinite sequence $$ X_i\equiv (...,y_k,...,y_2, y_1, y_0 )\;\;;\;\;y_i\in {\bf Z}_{L^d}\;\; \mbox{ thus }\;\; X_i\in {\it G}=\oplus^{\infty}_{k=0}{\bf Z}_{L^d}, $$ where only finitely many $y_i$ are non-zero.\ Let us define subgroups \begin{equation} \{0\}={\it G}_0\subset {\it G}_1\subset ...\subset {\it G} \;\;\;\mbox{where } {\it G}_k=\{X_i\in {\it G}| y_i=0, i\geq k \} \end{equation} and the norm $|\cdot|$ as \begin{equation} |X_i|=\left\{\begin{array}{cc} 0 & \mbox{ if $X_i=0$} \\ L^p & \mbox{where $p=\inf\{k| X_i\in {\it G}_k\}$ if $X_i\neq 0$} \end{array} \right. \end{equation} Then, the map $\delta:(X_{i+1}, X_i)\rightarrow |X_{i+1}-X_i|$ defines a metric on ${\it G}$. In this metric the subgroups ${\it G}_k$ are balls $|X_i|\leq L^k$ containing $L^{dk}$ points. Here the operation + (hence - as well) is defined componentwise.\ The metric defined by eq(2) satisfies a stronger condition than the triangle inequality, i.e. \begin{equation} |X_i+X_{i+1}|\leq \mbox{ Max}(|X_i|,|X_{i+1}|). \end{equation} From eq(3), it is clear that the metric introduced is an ultrametric. \ For the purposes of this paper we introduce the L\'{e}vy process as a continuous time random walk $w$. This is the following ordered sequence of sites in {\it G}; \begin{equation} (w(t_0),...,w(t_0+...+t_n))\;\;,w(t_0+...+t_i)=X_i\in {\it G}, \;\;T=\sum ^n_{i=0}t_i ,\; n\geq 0 \end{equation} where $t_i$ is the time spent in $X_i\in {\it G}$ (waiting time at $X_i$) and T, fixed at this point, is the runnig time for the process. For convenience we take $X_0=0$.\ We are not dealing with nearest neighbour random walks on the lattice, provided we mean neighbourhood with respect to the ultrametric distance $\delta$ previously defined. We propose the L\'{e}vy process we are dealing with, having a probability $P(w)=r^ne^{-rT}\prod^{n-1}_{i=0}q(X_{i+1},X_{i})$. Namely the continuous time random walk has a probability $rdt$ (r is the jumping rate) of making a step in time $(t,dt)$ and, given that it jumps, the probability of jumping from $X_i$ to $X_{i+1}$ is $q(X_{i+1},X_i)$, conditioned to a fixed running time $T$ for the process. $q(X_{i+1},X_i)$ is an open function of the initial and final sites of jumps in the lattice ${\it G}$. Here we define $Dw$ by $\int(\cdot)Dw$=\newline $\sum_n\sum_{[X_i]^n_{i=0}}\int^T_0\prod^n_{i=0}dt_i \delta(\sum^n_{i=0}t_i-T)(\cdot)$. From this follows $\int P(w)Dw$=$1$.\ Let the space of simple random walks of length $n$, be $\Lambda_n$, with probability measure $P(w)$, we construct on this space the weakly SARW model that represents SUSY $\phi^4$ (through McKane-Parisi-Sourlas theorem \cite{MKPS}). We take advantage of this feature to provide a better understanding of SUSY $\phi^4$ renormalization in terms of stochastic processes. This method can be straightforward generalized to any SUSY $\phi^n$ with ultrametric. \section{The renormalization-group map on the random walk representation of SUSY $\phi^4$.} We propose a renormalization-group map on the lattice $R(X_i)=LX'_i$ where $X_i\in {\it G}$ and $LX'_i\in {\it G}'$=${\it G}/{\it G}_1\sim {\it G}$; i.e. the renormalized lattice ${\it G}'$ is isomorphic to the original lattice ${\it G}$. Here $LX'_i=(...,y_2,y_1)$.\ Besides we propose the action of the renormalization-group map on the space of random walks $R(w)=w'$, from $w$ above as defined, to $w'$. Here, $w'$ is the following ordered sequence of sites in ${\it G}'= {\it G}/{\it G}_1\approx {\it G}$; \begin{equation} (w'(t'_0),...,w'(t'_0+...+t'_k))\;\;,\mbox{ where} \end{equation} $$ w'(t'_0,...,t'_{i'})=X'_{i'}\in {\it G},\;\;T'=\sum ^k_{i'=0}t'_i , \;0\leq k\leq n, \;\;T=L^{\varphi}T'. $$ $R$ maps $w(t_0)+{\it G}_1,w(t_0+t_1)+{\it G}_1,...,w(t_0+...+t_n)+{\it G}_1$ to cosets $Lw(t_0)$,$Lw(t_0+t_1)$,...,$Lw(t_0+...+t_n)$ respectively. If two or more successive cosets in the image are the same, they are listed only as one site in $w'(t'_0),...,w(t'_0+...+t'_k)$, and the times $t'_j$ are sums of the corresponding $t_i$ for which successive cosets are the same, rescaled by $L^{\varphi}$. For $\varphi=2$, we are dealing with normal diffusion (this is the standard version of SUSY $\phi^4$), in case $\varphi<2$ with superdiffusion, and subdiffusion for $\varphi>2$. In the following this parameter is arbitrary, so we can study general cases.\ We can now work out probability measures at the $(p+1)^{th}$ stage in the renormalization provided only that we know the probabilities at the $p^{th}$ stage. We integrate the probabilities of all the paths $w^{(p)}$ consistent with a fixed path $w^{(p+1)}$ in accordance with the following. Let $R(w)=w'$ be the renormalization-group map above as stated, then $P'(w')$= \newline $L^{\varphi k}\int Dw P(w)\chi (R(w)=w')$. Here $R(w)=w'$ is a renormalization-group transformation that maps an environment $P(w)$ to a new one, $P'(w')$, thereby implementing the scaling.\ Hereafter \begin{equation} m_{j'}=\sum^{j'}_{i'=0}n_{i'}+j' \;\;\mbox{ and} \end{equation} $$ n=\sum^{k}_{i'=0}n_{i'}+k\;\;\;\;\;\;0\leq j'\leq k, \mbox{ being} $$ $n_{i'}$=$max\{i|w(t_0+...+t_j)\in LX_{i'}, \forall j\leq i\}$; i.e. the number of steps (for paths $w$) in the contracting ${\it G}_1$ coset that, once the renormalization-group map is applied, has the image $LX'_{i'}$.\ Concretely $P'(w')$ can be written like \begin{equation} P'(w')= L^{\varphi k}\;\sum_{\left[ n_{i'}\right]^{k}_{i'=0}} \;\sum_{\left[ X_{i}\right]^{n}_{i=0}} \int \prod^{n}_{i=0}dt_{i}\; \prod^{k}_{j'=0}\; \delta (\sum^{m_{j'}}_{{i}=m_{j'-1}+1}\;\;t_{i}- L^{\varphi }t'_{j'})\times \end{equation} $$ \times \prod^{k}_{j'=0}\; \prod^{m_{j'}}_{{i}=m_{j'-1}+1}\; \chi (X_{i}\in LX'_{j'}\;\;)P(w). $$ It is straightforward to prove that the probability $P(w)$ where we substitute $q(X_{i+1},X_i)$ by $c|X_{i+1}-X_i|^{-\alpha}$ $\forall X_i,X_{i+1}\in {\it G}$ ($c$ is a constant fixed up to normalization and $\alpha$ another constant), is a fixed point of the renormalization-group map provided $\varphi$= $\alpha-d$. Even more, if in $P(w)$ we substitute $q(X_{i+1},X_i)$ by $c\left(|X_{i+1}-X_i|^{-\alpha}+|X_{i+1}-X_i|^{-\gamma}\right)$ $\forall X_i,X_{i+1}\in {\it G}$, $\gamma>>\alpha$ ($\gamma$ is an additional constant), then this flows to the very same fixed point of the renormalization-group map that in the first case. This holds provided $log\left(\frac{L^{-\alpha}-L^{-\gamma}-2L^{d-\gamma-\alpha}} {L^{-\alpha}-2L^{d-\gamma}}\right)\rightarrow 0$ and $\varphi$=$\alpha-d$.\ Let us substitute $q(X_{i+1},X_i)$ by $q_1(|X_{i+1}-X_i|)+\epsilon b(X_{i+1},X_i)$ where $q_1(|X_{i+1}-X_i|)$ is any function of the distance between $X_{i+1}$ and $X_i$, both sites in the lattice; $b(X_{i+1},X_i)$ is a random function and $\epsilon$ a small parameter. We can impose on $b(X_{i+1},X_i)$ the following conditions.\ a) $\sum_{X_{i+1}}b(X_{i+1},X_i)$=$0$.\ b) Independence. We take $b(X_{i+1},X_i)$ and $b(X'_{i+1},X'_i)$ to be independent if $X_{i+1}\neq X'_i$.\ c) Isotropy.\ d) Weak randomness.\ In this case, $P(w)$ is still a fixed point of the renormalization-group map provided\newline \begin{equation} \sum_{(n_j)^k_0}\sum_{(X_i)^n_0}\prod^{n-1}_{i=0}b(X_{i+1},X_i) \prod^k_{j=0}\prod^{m_j}_{i=m_{j-1}+1}\chi(X_i\in LX'_j)= \end{equation} $$ \prod^k_{j=0}b(X'_{j+1},X'_j)L^{-\varphi k}(b(1)(L^d-1))^{n_j} $$ where $b(1)=\frac{1-L^{-\varphi}}{L^d-1}$ is the probability of jumping from one specific site to another specific site inside the ${\it G}_1$ coset.\ One formal solution to eq(8) is the following \begin{equation} b(X_{i+1},X_i)=\left\{\begin{array}{cc} \frac{1-L^{\varphi}}{L^d-1} & \mbox{ if $|X_{i+1}-X_i|=L$} \\ \sum_t\left( \begin{array}c d+\varphi\\ t \end{array} \right) f(X_{i+1})^tf(X_i)^{d+\varphi-t} & \mbox{where $|X_{i+1}-X_i|>L$} \end{array} \right. \end{equation} up to proper normalization. Here $f(X_i)$ and $f(X_{i+1})$ are homogeneous function of sites in the lattice, order -1. Besides they add to $...,1)$ and are positive defined. Since in the limit $d+\varphi\rightarrow \infty$ (provided the mean remains finite) binomial probability distribution tends to Poissson distribution; we think a nontrivial SUSY $\phi^4$ theory could be included in this case \cite{Kl}.\ The random walk representation of the SUSY $\phi^4$ is a weakly SARW that penalizes two-body interactions, this is a configurational measure model. Configurational measures are measures on $\Lambda_n$. Let $P_U(w)$ be the probability on this space such that \begin{equation} P_U(w)=\frac{U(w)P(w)}{Z} \end{equation} where $Z=\int U(w)P(w)Dw$ and $P(w)$ is the probability above described, thus a fixed point of the renormalization-group map. Besides $U(w)$ is the energy of the walks. To study the effect of the renormalization-group map on $P_U(w)$ we need to follow the trajectory of $U(w)$ after applying several times the renormalization-group map.\ Therefore, from previous definition of the renormalization-group map follows; $$ P'_{U'}(w')=L^{\varphi k}\int P_{U}(w) \chi (R(w)=w')Dw $$ where $Z'=Z$, thus \begin{equation} U'(w')= \frac{\int Dw P(w)\chi (R(w)=w')U(w)} {\int Dw P(w)\chi (R(w)=w')} \end{equation} Note that eq(11) can be view as the conditional expected value of $U(w)$ given that the renormalization-group map is imposed. Therefore and hereafter, to simplify notation, we write eq(11) as $U'(w')=< U(w) >_{w'}$.\ In the random walk representation of the SUSY $\phi^4$ model with interaction $\lambda$, and mass (killing rate in the stochastic framework) $m$, $U$ is as follows. \begin{equation} U(w)=\prod_{X\in {\it G}} e^{-m\sum_{i\in J_{X}}t_{i}-\lambda \sum_{i<j\in J_{X}} t_{i}t_{j} {\bf 1}_{\left\{w(t_i)=w(t_j)\right\} }}, \end{equation} being $m<0$ and $\lambda\stackrel{>}{\scriptscriptstyle<}0$ (small) constants. Here we set a randomly free running time for the process $T$. The probability $P_U(w)$, where $U(w)$ is defined as in eq(12), flows to a fixed form after the renormalization-group map is applied. This fixed form is characterized by the renormalized energy \begin{equation} U'(w')= \prod_{X'\in{\it G}} e^{ -m' \sum_{i'\in J_{X'}} t'_{i'}- \lambda' \sum_ {\stackrel{i'< j'} {\left\{i',j'\right\}\in J_{X'}}} t'_{i'}t'_{j'} {\bf 1}_{(w(t'_{i'})=w(t'_{j'}))}} \times \end{equation} $$ \left\{1+\eta'_{1} \sum_{ \stackrel{i'< j'< k'} {\left\{i',j', k'\right\}\in J_{X'}}} t'_{i'}t'_{j'} t'_{k'} {\bf 1}_{(w(t'_{i'})=w(t'_{j'}) =w(t'_{k'}))}+ \right. $$ $$ +\left. \eta'_2 \sum_ {\stackrel{i'< j'} {\left\{i',j'\right\}\in J_{X'}}} t'_{i'}t'_{j'} {\bf 1}_{(w(t'_{i'})=w(t'_{j'}))} +\eta'_{3}\sum_{i'\in J_{X'}} t'_{i'}\right\}+r'. $$ Here \begin{equation} m' = L^{\varphi}m+m'_1 \mbox{$\;\;\;where$} \end{equation} \begin{equation} m'_{1}= \gamma_{1}\lambda-\gamma_{2}\lambda^2+r_{m'_1}. \end{equation} \begin{equation} \lambda'=L^{2\varphi-d}\lambda-\chi\lambda^2+r_{\lambda'}. \end{equation} \begin{equation} \eta'_{1}=\eta_{1}L^{3\varphi-2d}+\eta \lambda^2. \end{equation} \begin{equation} \eta'_2=\eta_1A+L^{(2\varphi-d)}\eta_2 \mbox{$\;\;\;and$} \end{equation} \begin{equation} \eta'_3=\eta_1B+\eta_2\gamma_{1}+L^{\varphi}\eta_{3}. \end{equation} All parameters involved in eq(15), eq(16), eq(17), eq(18) and eq(19); namely $\gamma_1$, $\gamma_2$, $\chi$, $\eta$, $A$ and $B$ have precise, well defined formulae \cite{Su}. They are linearized conditional expectations of events inside contracting ${\it G}_1$ cosets which, upon renormalization, maps to a fixed random walk with totally arbitrary topology. Even more, we have precise formulae for all the remainders, also \cite{Su}. Concretely speaking, $\gamma_1$ and $\gamma_2$ are contributions to renormalized local times coming from one and two two-body interactions inside the contracting ${\it G}_1$ cosets, respectively. $\chi$ is the two two-body interaction (inside the contracting ${\it G}_1$ cosets) contribution to renormalized two-body interaction. $\eta$ is the contribution to renormalized three-body interaction coming from two two-body interactions. Finally $A$ and $B$ are the one three-body interaction (inside the contracting ${\it G}_1$ cosets) contribution to renormalized two-body interaction and local time, respectively.\ In the SUSY representation of $\phi^4$ we can say that $\gamma_1$ and $\gamma_2$ are first and second order contributions of SUSY $\phi^4$ to renormalized SUSY mass; $\chi$ is the second order contribution of SUSY $\phi^4$ to renormalized SUSY $\phi^4$; $\eta$ is the second order contribution (the first order contribution is null due to topological restriccions) of SUSY $\phi^4$ to renormalized SUSY $\phi^6$. Finally, $A$ and $B$ are first order contributions of SUSY $\phi^6$ (already generated at this stage by previous renormalization stages) to renormalized SUSY $\phi^4$ and mass, respectively.\ Eq(13) is presented in terms of the product of two factors. The first one (exponential) involves only; a) trivial flow of mass and interacting constant b) $\lambda\phi^4$ contribution (inside contracting ${\it G}_1$ cosets) to renormalized mass and $\lambda'\phi^4$ up to leading order. The second factor involves mixed terms; namely $\lambda\phi^4$ and $\phi^6$ contributions (inside the contracting ${\it G}_1$ cosets) to renormalized mass, $\phi^4$ and $\phi^6$. $\phi^6$ terms come into the scheme because they are produced from $\lambda\phi^4$ due to the fixed topology of the continuous-time random walk on the hierarchical lattice. This arrangment allows us to distinguish the physically meaningful (leading order) magnitudes. From this, we analyze some results in next section.\ We can choose either representation to obtain the final formulae for parameters and remainders. Here and in Rodr\'{\i}guez-Romo S. \cite{Su} we choose the one to provide new stochastic meaning to renormalizing SUSY field theories.\ We claim that this result is the space-time renormalization-group trajectory, for the weakly SARW energy interaction studied by Brydges, Evans and Imbrie \cite{Ev} provided $\varphi=2$ and $d=4$. In their paper the trajectory of a SUSY $\phi^{4}$ was studied (recall that this can be understood in terms of intersection of random walks due to Mc Kane, Parisi, Sourlas theorem) from a SUSY field-theoretical version of the renormalization-group map, on almost the same hierarchical lattice we are studying here. We improve the model by providing exact expressions for $\lambda$ and $m$ for each step the renormalization-group is applied in the stochastic framework, among others.\ To obtain eq(13) we have introduced an initial mass term $m$, $O(\lambda^2)$ (this allows a factorization whose errors are included in the remainder $r'$, automatically). We use the Duplantier's hypotesis \cite{Du} and assume all divergences of the SUSY $\phi^4$ with ultrametric as coming from the vertices or interactions per site of the lattice. This hypothesis has been proved to be correct in dimension 2 by means of conformal field theory. Then, a formal Taylor series expansion is applied which is analyzed for each particular topology in the renormalized field theory (this is done in random walk representation) per site of the new lattice. Putting everything together and by induction, we obtain the final result.\ We can apply the very same method to study any SUSY $\Phi^n$ model on this ultrametric space.\ \section{Renormalized SUSY $\phi^4$ with ultrametric. The stochastic approach.} To start with, we write the physically meaningful (leading order) part of eq(13); namely eq(14), eq(15) and eq(16) in parameter space. Let us define the following vector \begin{equation} {\bf H}=(m,\lambda) \end{equation} Here we have approached up to the most probable events (first order in SUSY representation). The action of the renormalization-group map (RG) is expressed as \begin{equation} {\bf H'}=R({\bf H})=(m',\lambda'). \end{equation} The fixed points in our theory, $(m^*_1,\lambda^*_1)$ and $(m^*_2,\lambda^*_2)$ are as follows.\\ a) The trivial $m^*_1$=$\lambda^*_1$=$0$.\\ b) $\lambda^*_2$=$\frac{L^{2\varphi-d}-1}{\chi}$ ; $m^*_2$=$\frac{\gamma_1(L^{2\varphi-d}-1)}{\chi(1-L^{\varphi})}- \frac{\gamma_2(L^{2\varphi-d}-1)^2}{\chi^2(1-L^{\varphi})}$.\\ The nontrivial fixed point involves a renormalized two-body interaction which is inverse to the conditional expectation of two two-body interactions that renormalizes to a two-body interaction inside the contracting ${\it G}_1$ cosets ($\chi$) given that the RG map is applied. Meanwhile the renormalized mass in this point is given in terms of two ratios. The first one involves the ratio of conditional expectations of one two-body interaction that renormalizes to local times ($\gamma_1$) inside a contracting ${\it G}_1$ coset and $\chi$. The second ratio involves the conditional expectation of two two-body interactions that renormalize to local times ($\gamma_2$) inside a contracting ${\it G}_1$ coset and $\chi^2$. Both; $\lambda^*_2,m^*_2$, are independent of the scaling factor $L$ for large lattices.\ As we come infinitesimally close to a particular fixed point, (called this ${\bf H^*}$), the trajectory is given completely by the single matrix $M$ (its eigenvalues and eigenvectors). Namely \begin{equation} M_{ij}=\left.\frac{\partial R_i({\bf H})} {\partial H_i}\right|_{{\bf H}={\bf H}^*} \end{equation} From the random walk representation of SUSY $\phi^4$ with ultrametric, up to the most probable event approach (leading order in SUSY representation), we obtain \begin{equation} M= \left( \begin{array}{cc} L^{\varphi} & \gamma_1-2\gamma_2\lambda^* \\ 0 & L^{2\varphi-d}-2\chi\lambda^* \end{array} \right), \end{equation} where $\lambda^*$ can be either $\lambda^*_1$ or $\lambda^*_2$.\ The eigenvalues and eigenvectors of this matrix are as follows.\\ a) $l_1=L^{\varphi}$ with eigenvector $(m,0)$.\\ b) $l_2=L^{2\varphi-d}-2\chi\lambda^*$ with eigenvector $\left(m, -\frac{L^{\varphi}-2L^{2\varphi-d}+2\chi\lambda^*} {\gamma_1-2\gamma_2\lambda^*}m\right)$, where $\lambda^*$ can be either $\lambda^*_1$ or $\lambda^*_2$.\\ For $L\geq 2$ and $\varphi> 0$; both fixed points $(m^*_1, \lambda^*_1)$ and $(m^*_2, \lambda^*_2)$ are repulsive in the direction of the eigenvector $(m,0)$, marginal if $\varphi=0$ and attractive if $\varphi<0$. The trivial fixed point $(m^*_1, \lambda^*_1)$ is repulsive in the direction of the eigenvector $\left(m,-\frac{L^{\varphi}-2L^{2\varphi-d}+2\chi}{\gamma_1} m\right)$ provided $\varphi>d/2$, marginal if $\varphi=d/2$ and attractive otherwise. Finally, the fixed point $(m^*_2, \lambda^*_2)$ is repulsive in the direction of the eigenvector $\left(m,-\frac{L^{\varphi}-2L^{2\varphi-d}+2\chi}{\gamma_1} m\right)$ provided $d/2>\varphi$, marginal if $d/2=\varphi$ and attractive otherwise. This means that the only critical line which forms the basin of attraction for both fixed points is given only for $0<\varphi<d/2$ and is locally defined by $g_1=0$. Here $g_1$ is the linear scaling field associated with the eigenvector $(m,0)$.\ The largest eigenvalue defines the critical exponent $\nu$. In the trivial fixed point $(m^*_1,\lambda^*_1)$, $\nu=1/\varphi$ provided $d\geq \varphi$. If $d< \varphi$ than $\nu=\frac{1}{2\varphi-d}$. Here the eigenvalue $l_1=L^{\varphi}$$>1$ provided $2\varphi<d$; i.e. this fixed point is repulsive in the direction of the eigenvector $(m,0)$ if and only if $2\varphi<d$. Although our results are rather general, let us consider the Flory's case as an example \cite{Fl}. For $d\geq 5$ this trivial fixed point, in the Flory's case, is attractive in the direction of the eigenvector $(m,0)$, marginal in dimension four and repulsive otherwise.\ In the fixed point $(m^*_2,\lambda^*_2)$, $\nu=\frac{1}{\varphi}$, provided $\beta>-log_L\left(\frac{1+L^{\beta-d}}{2}\right)$. Back to the example we are considering here (Flory's case) \cite{Fl}; for $d\geq 5$ this fixed point is repulsive, marginal in $d=4$ and attractive otherwise.\ We cannot explain, from this first order approach (the most probable event), logarithmic corrections to the end-to-end distance in the critical dimension. This is correctly explained, although heuristically, elsewhere \cite{Su}.\ Using the spin representation, we find the following.\\ a) For the trivial fixed point $(m^*_1,\lambda^*_1)$.\\ $\alpha$=$2-d/\varphi$ ; $\beta$=$\frac{2(d-\varphi)}{\varphi}$ ; $\gamma$=$\frac{4\varphi-3d}{\varphi}$ ; $\delta$=$\frac{2\varphi-d}{2d-2\varphi}$ ; $\nu$=$\frac{1}{\varphi}$ and finally $\eta$=$2-4\varphi+3d$.\\ b) For the fixed point $(m^*_2, \lambda^*_2)$.\\ $\alpha$=$2-d/\varphi$ ; $\beta$=$\frac{d-log_L(2-L^{2\varphi-d})}{\varphi}$ ; $\gamma$=$\frac{2log_L(2-L^{2\varphi-d})-d}{\varphi}$ ; $\delta$=$\frac{Log_L(2-L^{2\varphi-d})}{d-log_L(2-L^{2\varphi-d})}$ ; $\nu$=$\frac{1}{\varphi}$ and finally $\eta$=$2+d-2Log_L(2-L^{2\varphi-d})$.\\ Besides, if we introduce critically the mass as was done in Brydges et al. \cite{Ev} in $d=4$ and $\varphi=2$, the critical exponents look as follows.\\ $\alpha$=$0$ ; $\beta$=$\frac{1}{2}$ ; $\gamma$=$1$ ; $\delta$=$3$ ; $\nu$=$\frac{1}{2}$ and finally $\eta$=0.\\ On the other hand, we know that for the SUSY $\lambda\phi^4$, $\beta(\lambda')$=$\mu\frac{\partial \lambda'}{\partial \mu}$, where $\mu$ is a parameter with the dimensions of mass; namely $\mu$ is an arbitrary mass parameter.\ Since we know the fixed points for the theory in random walk representation; these must be the zeros of $\beta(\lambda)$ in the SUSY $\lambda\phi^4$ representation. Using this criteria we obtain the following expression for $\beta(\lambda)$; \begin{equation} \beta(\lambda)=\frac{\gamma_2}{1-L^{\varphi}}\lambda^2- \frac{\gamma_1}{1-L^{\varphi}}\lambda+m \end{equation} up to a multiplicative constant.\ An interesting pictorial interpretation of the renormalized group equation was suggested by S. Coleman \cite{Co}. The equation can be viewed as a flow of bacteria in a fluid streaming in a one dimensional channel. Here we provide a new interpretation of the velocity of the fluid at the point $\lambda$, $\beta(\lambda)$, in terms of stochastic events (function of conditional expectations of two-body interactions inside contracting ${\it G}_1$ cosets). For large lattice $\beta(\lambda)$ is independent of the lattice parameter $L$.\ Concretely, $\beta(\lambda)$ (or the velocity of the fluid at the point $\lambda$) is written in terms of one two-body and two two-body contributions to renormalized local times (stochastic approach) or mass (field theory approach). The first contribution is $O(\lambda)$ and the second, $O(\lambda^2)$. Let us call $\beta'(\lambda) =\left(\frac{\partial \beta(\lambda)}{\partial \lambda}\right)_m$, then \begin{equation} \beta'(\lambda)=\frac{2\gamma_2\lambda-\gamma_1}{1-L^{\varphi}} \end{equation} In the trivial fixed point, $\beta'(\lambda^*_1)> 0$ (infrared stable), provided $\varphi> 0$ and $L\geq 2$; besides $\beta'(\lambda^*_1)< 0$ (ultraviolet stable), provided $\varphi< 0$ and $L\geq 2$. In the fixed point $(m^*_2,\lambda^*_2)$, $\beta'(\lambda^*_2)\geq 0$ (infrared stable), provided $\varphi\geq 1/2\left[d+log_L\left(\frac{\gamma_1\chi}{2\gamma_2}+ 1\right)\right]$, and $\beta'(\lambda^*_2)< 0$ (ultraviolet stable) otherwise. Here we define $ d_H=log_L\left(\frac{\gamma_1\chi}{2\gamma_2}+1\right)$ which is given in terms of the ratio for conditional expectations of two-body interactions which renormalizes to local time and two-body interactions. From this, the following estimates are obtained \\ a) d=4; $\beta'(\lambda^*_2)\leq 0$, provided $d_H\geq 0$.\ b) d=3; $\beta'(\lambda^*_2)\leq 0$, provided $d_H\geq 1/3$.\ b) d=2; $\beta'(\lambda^*_2)\leq 0$, provided $d_H\geq 2/3$.\ d) d=1; $\beta'(\lambda^*_2)\leq 0$, provided $d_H\geq 1$.\ \section{Summary} Because of the equivalence between the polymer and SAW problems, functional integration methods were employed in the majority of theoretical approaches to these problems. It should, however, be remarked that the critical exponents for the SAW obtained by this method are only meaningful if the spatial dimensionality $d$ is close to its formal value $d=4$, and it is not yet clear how to get results for real space in this way. There is another method based on the search for a solution to the exact equation for the probability density of the end-to-end distance of the random walk \cite{Al}. By defining the self- consistent field explicitly, the density could be found with the help of the Fokker-Planck equation. In this paper we provide another alternative view where the probability density, as a random function of the random walk, is proposed.\ Discrete random walks approximate to diffusion processes and many of the continuous equations of mathematical physics can be obtained, under suitable limit conditions, from such walks. Besides we can look at this relation the other way around; that is, suppose that the movement of an elementary particle can be described as a random walk on the lattice, which represents the medium it traverses, and that the distance between two neighbouring vertices, though very small, is of a definite size; therefore the continuous equations can be considered as merely approximations which may not hold at very small distances. We show in this paper how the mathematical results are easily derived by standard methods. The main interest lies in the interpretation of the results.\ In our approach the properties of the medium will be described by the lattice and the transition probabilities. We obtain $m'$, the ``mass" of the field as observed on this particular hierarchical lattice. The lattice is characterized by the ultrametric space used to label this.\ We propose to obtain renormalized $n$-body interactions out of a set of stochastic diagrams with a fixed totally arbitrary topology.\ Here we would like to stress that the search for a proper mathematical foundation of a physical theory does not mean only a concession to the quest for aesthetic beauty and clarity but is intended to meet an essential physical requirement. The mathematical control of the theory plays a crucial role to allow estimates on the proposed approximations and neglected terms.\ Usually approximations must be introduced which often have the drawback that, although they can work well, they are uncontrolled: there is no small parameter which allows an estimate of the error.\ Explicit mathematical formulae for all the parameters and remainders in the method can be provided. In sake of brevity we present these elsewhere \cite{Su}. All of them are expressed in terms of conditional expectations of events inside contracting ${\it G}_1$ cosets.\ Once a successful theoretical scheme has been found it is conceivable that it is possible to reformulate its structure in equivalent terms but in different forms in order to isolate, in the most convenient way, some of its aspects and eventually find the road to successive developments.\ Let us remark that we are talking of a particle picture even when we deal with systems containing many particles or even field systems.\ We hope our method and ideas may help in the proper understanding of the association of stochastic processes to the quantum states of a dynamical system; i.e. stochastic quantization.\ Summarizing, in this paper we present an heuristic space-time \newline renormalization-group map, on the space of probabilities, to study SUSY $\phi^4$ in random walk representation, on a hierarchical metric space defined by a countable, abelian group ${\it G}$ and an ultrametric $\delta$. We present the L\'evy process on $\Lambda_n$ that correspond to the random walk representation of SUSY $\phi^4$ which is a configurational measure model from the point of view of a stochastic processes. We apply the renormalization-group map on the random walk representation and work out explicitly the weakly SARW case for double intersecting paths which corresponds to SUSY $\phi^4$, as an example. The generalization to SUSY $\phi^n$, for any $n$, is straightforward. New conclusions are derived from our analysis.\ Our result improves the field-theoretical approach \cite{Ev} by obtaining an exact probabilistic formula for the flow of the interaction constant and the mass under the map. \section{Acknowledgments} This research was partially supported by CONACYT, Ref 4336-E, M\'exico.

proofpile-arXiv_065-540

\section{Introduction} The problem of classification of trivial Lagrangians i.e. Lagrangians leading to trivial Euler-Lagrange equations has a long history and it seems that it is not solved in full generality even today in the framework of classical field theory. The basic input to the rigorous formulation of this problem is a fiber bundle $ \pi: Y \rightarrow X $ where $X$ is usually interpreted as the space-time manifold and the fibres describe the internal degrees of freedom of the problem. To formulate the Lagrangian formalism one has to include the ``velocities" i.e. one must work on some jet-bundle extension of $ \pi: Y \rightarrow X. $ There are two approaches to this problem. The first one uses the infinite jet bundle extension formalism. In this framework one can prove that, quite generally, a trivial Lagrangian is what is usually called, a total derivative \cite{T}, \cite{A}. The second approach, which is more suitable for the needs of specific physical problems is to use a finite jet bundle extension. This corresponds to field theories with a finite order of partial derivatives. The most common physical theories are, at most, of second order. The r\^ole of the finite-order approach has been particulary emphasised by Krupka (see for instance \cite{K1}). In the finite-order approach there is no general solution of the trivial Lagrangian problem. However, there are a number of partial results which deserve mentioning. For instance, the first-order case has been completely elucidated (see \cite{E}-\cite{GP}; needless to say, this rather simple case has been rediscovered independently by many people from time to time). For the second and higher order Lagrangians only partial results are known to the author. More precisely, we refer to Olver's papers \cite{BCO}, \cite{O} where one studies the case of a trivial Lagrangian of order $r$ which is dependent only of the highest order derivatives in a polynomially homogeneous way. In this case one can show that the Lagrangian is a linear combination of some polynomial expressions called hyper-Jacobians. The existence of some linear dependence between the hyper-Jacobians is emphasised but the general problem of finding all possible linear dependencies is not solved. It is worth mentioning that the results of Olver are based on a very complex algebraic machinery of higher dimensional determinants, invariant theory, etc. combined with a generalization of Gel'fand-Dikii transform. In this paper we will present a rather complete analysis of case of the second-order trivial Lagrangian. Without any limitations we will be able to prove that the dependence of the second order derivatives is through the hyper-Jacobian polynomials. In particular the polynomiality emerges naturally from the proof and the homogeneity condition is not needed. Moreover we will elucidate the question of possible linear dependencies between the hyper-Jacobians and it will emerge that only the tracelessness conditions already appearing in \cite{BCO}, \cite{O} are sufficient i.e. no other constraints are independent of these. All these results will be obtained from a long but quite elementary proof. The methods used are essentially those used in \cite{G1}, \cite{G2} for the analysis of the most general form of a Euler-Lagrange expression: they consist in complete induction and some Fock space techniques. We feel that this completely new method may offer a way for the analysis of the general case of $r$-th order Lagrangians and so it deserves some attention. The structure of the paper is the following one. In Section 2 we present the jet-bundle formalism to fix the notations and illustrate our problem on the well-known first-order trivial Lagrangian case. We also present the equations to be solved in the second-order case and comment on the best strategy to solve them. In Section 3 we consider the second-order case for a scalar field theory. The proof is based on induction on the dimension of the space-time manifold and is contained in the proof of the main theorems in \cite{G1}, \cite{G2}. We give it, nevertheless in this Section because we will need further some notations, results and techniques. We also start the analysis for the general case by an obvious corollary of the result obtained for the scalar field and we will formulate the general result we want to prove. In Section 4 the case of a two component classical field is analysed and one is able to perceive the nature of the difficulties involved in this analysis. In Section 5 the case of a field with three or more components is settled completing the analysis. In Section 6 we combine the results from the preceding two Sections in the main theorem and make some final comments. \section{The Jet-Bundle Formalism in the Lagrangian Theory} 2.1 As we have said in the Introduction, the kinematical structure of classical field theory is based on a fibred bundle structure $ \pi: Y \mapsto X $ where $Y$ and $X$ are differentiable manifolds of dimensions $ dim(X) = n, \quad dim(Y) = N + n $ and $\pi$ is the canonical projection of the fibration. Usually $X$ is interpreted as the ``space-time" manifold and the fibres of $Y$ as the field variables. Next, one considers the $r$-th jet bundle $ J^{r}_{n}(Y) \mapsto X \quad (r \in \rm I \mkern -3mu N). $ A $r$-th order jet with source $x \in X$, and target $y \in Y$ is, by definition, an equivalence class of all the smooth maps $ \zeta: X \rightarrow Y $ verifying $\zeta(x) = y$ and having the same partial derivatives in $x$ up to order $r$ (in any chart on $X$ and respectively on $Y$). We denote the equivalence class of $\zeta$ by $ j^{r}\zeta $ and the factor set by $ J^{r}_{x,y}. $ Then the $r$-th order jet bundle extension is, by definition $ J^{r}_{n}(Y) \equiv \cup J^{r}_{x,y}. $ One usually must take $ r \in \rm I \mkern -3mu N $ sufficiently large such that all formulas make sense. Let us consider a local system of coordinates in the chart $ U \subseteq X: \quad (x^{\mu}) \quad (\mu = 1,...,n). $ Then on some chart $ V \subseteq \pi^{-1}(U) \subset Y $ we take a local coordinate system adapted to the fibration structure: $ (x^{\mu},\psi^{A}) \quad (\mu = 1,...,n, \quad A = 1,...,N) $ such that the canonical projection is $ \pi(x^{\mu},\psi^{A}) = (x^{\mu}). $ Then one can extend this system of coordinates to $ J^{r}_{n}(Y) $ as follows: on the open set $ V^{r} \equiv (\pi^{r,0})^{-1}(V) $ we define the coordinates of $ j^{r}_{x}\zeta $ to be $ (x^{\mu},\psi^{A},\psi^{A}_{\mu},...,\psi^{A}_{\mu_{1}.,,,,\mu_{r}}) $ where $ \mu_{1} \leq \cdots \leq \mu_{s} \qquad (s \leq r). $ Explicitly \begin{equation} \psi^{A}_{\mu_{1},...,\mu_{s}}(j^{r}_{x}\zeta) \equiv \prod_{i=1}^{s} {\partial\over \partial x^{\mu_{i}}} \zeta(x) \quad (s=1,...,r). \end{equation} If $ \mu_{1},...,\mu_{s} $ are arbitrary numbers belonging to the set $ \{1,...,n\} $ then by the expression $ \{\mu_{1},...,\mu_{s}\} $ we understand the result of the operation of increasing ordering. Then the notation $ \psi^{A}_{\{\mu_{1},...,\mu_{s}\}} $ becomes meaningful for all set of numbers $ \mu_{1},...,\mu_{s}. $ If $ I = \{\mu_{1},...,\mu_{s}\} $ is an arbitrary set from $ \{1,...,n\}^{\times s} $ then we define \begin{equation} \psi^{A}_{I} = \psi^{A}_{\mu_{1},...,\mu_{s}} \equiv \psi^{A}_{\{\mu_{1},...,\mu_{s}\}}. \end{equation} This notation makes sense whenever the cardinal of $I$ verifies: $ |I| \leq r $ where if $ I = \emptyset $ then we put $ \psi^{A}_{\emptyset} = \psi^{A}. $ With this convention the expression $ \psi^{A}_{I} $ is completely symmetric in the individual $ \mu_{1},...,\mu_{s} $ which make up the multi-index $I$. 2.2 Let us consider $ s \leq r $ and $T$ a $(n + 1)$-form which can be written in the local coordinates introduced above as: \begin{equation} T = {\cal T}_{A} \quad d\psi^{A} \wedge dx^{1} \wedge \cdots \wedge dx^{n} \label{edif} \end{equation} with $ {\cal T}_{A} $ some smooth functions of $ (x^{\mu},\psi^{A}_{I}) \qquad (|I| \leq s). $ Then $T$ is a globally defined object. We call such a $T$ a {\it differential equation of order s}. 2.3 To introduce some special type of differential equations we need some very useful notations \cite{AD}. We define the differential operators: \begin{equation} \partial^{I}_{A} \equiv {r_{1}!...r_{l}! \over |I|!} {\partial \over \partial \psi^{A}_{I}} \label{pdif} \end{equation} where $ r_{i} $ is the number of times the index $i$ appears in $I$. The combinatorial factor in (\ref{pdif}) avoids possible overcounting in the computations which will appear in the following. One has then: \begin{equation} \partial^{\mu_{1},...,\mu_{l}}_{A} \psi^{B}_{\nu_{1},...,\nu_{m}} = \cases{ { 1\over l!} \delta^{A}_{B} perm(\delta^{\mu_{i}}_{\nu_{j}}), & for $l = m$ \cr 0, & for $l \not= m$ \cr} \end{equation} where \begin{equation} perm\left( \delta^{\mu_{i}}_{\nu_{j}} \right) \equiv \sum_{P \in {\cal P}_{l}} \delta^{\mu_{1}}_{\nu_{P(1)}}\cdots \delta^{\mu_{l}}_{\nu_{P(l)}} \end{equation} is a permanent. (In general we denote by $ perm(A) $ the permanent of the matrix $A$). Next, we define the total derivative operators: \begin{equation} D_{\mu} = {\partial\over \partial x^{\mu}} + \sum_{l=0}^{r-1} \psi^{A}_{\nu_{1},...,\nu_{l},\mu} \partial^{\nu_{1},...,\nu_{l}}_{A} = {\partial\over \partial x^{\mu}} + \sum_{|I|\leq r-1} \psi^{A}_{I\mu}~ \partial^{I}_{A} \label{tdif} \end{equation} where we use the convention $ IJ \equiv I \cup J. $ One can check that \begin{equation} D_{\mu}\psi^{A}_{I} = \psi^{A}_{I\mu}, \qquad |I| \leq r-1 \label{der} \end{equation} and \begin{equation} [D_{\mu}, D_{\nu}] = 0. \label{com} \end{equation} Finally we define the differential operators \begin{equation} D_{I} \equiv \prod_{i \in I} D_{\mu_{i}}. \label{tdifs} \end{equation} Because of (\ref{com}) the order of the factors in the right hand side is irrelevant. 2.4 A differential equation $T$ is called {\it locally variational} (or of the {\it Euler-Lagrange type}) {\it iff} there exists a local real function ${\cal L}$ such that the functions $ {\cal T}_{A} $ from (\ref{edif}) are of the form: \begin{equation} {\cal E}_{A}({\cal L}) \equiv \sum_{l=0}^{r} (-1)^{l} D_{\mu_{1},...,\mu_{l}} (\partial^{\mu_{1},...,\mu_{l}}_{A} {\cal L}) \label{Eop} \end{equation} One calls ${\cal L}$ a {\it local Lagrangian} and: \begin{equation} L \equiv {\cal L}~dx^{1}\wedge\cdots \wedge dx^{n} \label{Lform} \end{equation} a {\it local Lagrange form}. If the differential equation $T$ is constructed as above then we denote it by $ E(L). $ A local Lagrangian is called a {\it total divergence} if it is of the form: \begin{equation} {\cal L} = D_{\mu} V^{\mu}. \end{equation} A Lagrangian is called {\it trivial} (or {\it null} in the terminology of \cite{BCO}, \cite{O}) if it satisfies: \begin{equation} E(L) = 0. \label{trEL} \end{equation} One can check that a total divergence Lagrangian is trivial. The converse of this statement has been proved only in the infinite jet bundle approach \cite{T}. 2.5 Let us briefly review the case of trivial first-order Lagrangians. One must take in the equation (\ref{trEL}) above the function ${\cal L}$ depending only on the variables $ (x^{\mu}, \psi^{A}, \psi^{A}_{\mu}). $ Then we obtain the condition of triviality as follows: \begin{equation} \partial_{A} {\cal L} - D_{\mu} \partial_{A}^{\mu} {\cal L} = 0. \label{trEL1} \end{equation} One can easily find out that this equation is equivalent to the following two equations: \begin{equation} \left(\partial^{\mu}_{A}~\partial^{\nu}_{B} + \mu \leftrightarrow \nu \right) {\cal L} = 0 \label{I1} \end{equation} and \begin{equation} \partial_{A} {\cal L} - (\partial_{\mu} + \psi^{B}_{\mu} \partial_{B}) \partial_{A}^{\mu} {\cal L} + (\partial_{\mu} + \psi^{B}_{\mu} \partial_{B}) (\partial_{\nu} + \psi^{C}_{\nu} \partial_{C}) {\cal L} = 0. \label{I2} \end{equation} From (\ref{I1}) one easily discovers that ${\cal L}$ is a polynomial of maximal degree $n$ in the first-order derivatives of the following form: \begin{equation} {\cal L} = \sum_{k=1}^{n} {1 \over k!} L^{\mu_{1},...,\mu_{k}}_{A_{1},...,A_{k}} \prod_{i=1}^{k} \psi^{A_{i}}_{\mu_{i}} \label{TrI} \end{equation} where the functions $ L^{...}_{...} $ depend only on the variables $ (x^{\mu}, \psi^{A}) $ and are completely antisymmetric in the upper indices $ \mu_{1},...,\mu_{k} $ and in the lower indices $ A_{1},...,A_{k}. $ To exploit the equation (\ref{I2}) one defines the form \begin{equation} \Lambda = \varepsilon_{\mu_{1},...,\mu_{n}} \quad \sum_{k=1}^{n} {1 \over k1} L^{\mu_{1},...,\mu_{k}}_{A_{1},...,A_{k}} \quad d\psi^{A_{1}} \wedge \cdots \wedge d\psi^{A_{k}} \wedge dx^{\mu_{k+1}} \wedge \cdots \wedge dx^{\mu_{n}} \label{lam} \end{equation} and shows that (\ref{I2}) is equivalent to \begin{equation} d\Lambda = 0. \end{equation} 2.6 Finally we give the similar details for the second-order case. Let us consider a second-order Lagrangian: $ {\cal L}(x^{\mu} ,\psi^{A},\psi_{\nu}^{A},\psi_{\nu \rho }^{A}) $ and impose the triviality condition (\ref{trEL}). One obtains explicitly: \begin{equation} \partial_{A} {\cal L} - D_{\mu} \partial_{A}^{\mu} {\cal L} + D_{\mu} D_{\nu} \partial_{A}^{\mu\nu} {\cal L} = 0. \label{trEL2} \end{equation} We write in detail this equation and note a linear dependence on the fourth-order derivatives; applying the operator $ \partial^{\zeta_{1}\zeta_{2}\zeta_{3}\zeta_{4}}_{B} $ we obtain after some rearrangements: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \partial_{A}^{\mu\rho_{1}} \partial_{B}^{\rho_{2}\rho_{3}} {\cal L} + (A \leftrightarrow B) = 0. \label{II1} \end{equation} We will use many times from now on a convenient notation, namely by $ \sum_{(\mu,\nu,\rho)} $ we will mean the sum over all the {\it cyclic} permutations of the indices $ \mu, \nu, \rho. $ We take into account this equations in the original equation (\ref{trEL2}) to simplify it a little bit by eliminating the dependence on the fourth order derivatives. What remains is an equations having a quadratic dependence on the third-order derivatives. We differentiate twice this equation with respect to the third-order derivatives and obtain as before: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \sum_{(\zeta_{1},\zeta_{2},\zeta_{3})} \partial^{\zeta_{1}\zeta_{2}}_{D}\partial^{\zeta_{3}\rho_{3}}_{A} \partial^{\rho_{1}\rho_{2}}_{B} {\cal L} = 0. \label{II2} \end{equation} Taking into account (\ref{II1}) and (\ref{II2}) the initial equation becomes linear in the third-order derivatives; differentiating once with respect to the third-order derivatives one gets: \begin{equation} \sum_{(\zeta_{1},\zeta_{2},\zeta_{3})} \left[ \left(\partial_{D}^{\zeta_{1}\zeta_{2}} \partial_{A}^{\zeta_{3}} - \partial_{A}^{\zeta_{1}\zeta_{2}} \partial_{D}^{\zeta_{3}} \right) + 2 \left(\partial_{\mu} + \psi^{B}_{\mu} \partial_{B} + \psi^{B}_{\mu\rho} \partial^{\rho}_{B}\right) \partial^{\zeta_{1}\zeta_{2}}_{D} \partial^{\zeta_{3}\mu}_{A} \right] {\cal L} = 0. \label{II3} \end{equation} From the initial equation what is left is: \begin{equation} \begin{array}{c} \partial_{A} {\cal L} - (\partial_{\mu} + \psi^{B}_{\mu} \partial_{B}+ \psi^{B}_{\mu\rho} \partial_{B}^{\rho}) \partial_{A}^{\mu} {\cal L} + \nonumber \\ (\partial_{\mu} + \psi^{B}_{\mu} \partial_{B}+ \psi^{B}_{\mu\rho} \partial_{B}^{\rho}) (\partial_{\nu} + \psi^{C}_{\nu} \partial_{C}+ \psi^{C}_{\mu\sigma} \partial_{C}^{\sigma}) {\cal L} \equiv 0. \end{array} \label{II4} \end{equation} So, equation (\ref{trEL2}) is equivalent to the identities (\ref{II1})-(\ref{II4}). Our strategy in the following will be to find the most general solution of the equations involving only the second-order derivatives i.e. (\ref{II1}) and (\ref{II2}). Some comments related to the dependence on the first-order derivatives will be made at the end. Inspecting more carefully the equation (\ref{II2}) it becomes clear that it for $ N = 1, 2 $ it follows from (\ref{II1}). Also, (\ref{II1}) for $ A = B $ coincides with the same equation for $ N = 1. $ This is the reason for studying separately the cases $ N = 1, 2 $ and $ N \geq 3. $ \section{Trivial Second-Order Lagrangians in the Case $N = 1$} 3.1 In this case we will omit completely the field indices $ A, B,... $ because they take only one value. The dependence on the second-order derivatives is encoded in the equation (\ref{II1}) which becomes in this case: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \partial^{\mu\rho_{1}} \partial^{\rho_{2}\rho_{3}} {\cal L} = 0. \label{N=1} \end{equation} As we have said in the Introduction, we intend to find the most general solution of this equation using induction over $n$. To be able to formulate the induction hypothesis we introduce the following polynomial expressions, called {\it hyper-Jacobians} \cite{BCO}, \cite{O} (see also \cite{G1}, \cite{G2}) which in this case have the following form: \begin{equation} J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} \equiv \varepsilon^{\rho_{1},...,\rho_{n}} \prod_{i=1}^{r} \psi_{\rho_{i}\sigma_{i}} \qquad (r = 0,...,n) \label{hyperJ} \end{equation} We will use consistently Bourbaki conventions: $ \sum_{\emptyset} \cdots = 0 $ and $ \prod_{\emptyset} \cdots = 1. $ We note the following symmetry properties: \begin{equation} J^{\rho_{Q(r+1)},...,\rho_{Q(n)}}_{\sigma_{P(1)},...,\sigma_{P(r)}} = (-1)^{|P|+|Q|} J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} \quad (r = 0,...,n) \label{antisym} \end{equation} where $P$ is a permutation of the numbers $1,...,r$ and $Q$ is a permutations of the numbers $r+1,...,n$. We also note that the following identities are true (see \cite{BCO}): \begin{equation} J^{\rho_{r+1},...,\rho_{n-1},\zeta}_{\sigma_{1},...,\sigma_{r-1},\zeta} = 0 \quad (r = 1,...,n-1). \label{trace} \end{equation} In other words, the hyper-Jacobians are completely antisymmetric in the upper indices, in the lower indices and are also traceless. In the following we will need the expression for the derivatives of the hyper-Jacobians. On easily finds out the following formula: true: \begin{equation} \begin{array}{c} \partial^{\mu\nu} J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\nu_{r}} = \nonumber \\ {1 \over 2} \quad \sum_{i=1}^{r} (-1)^{n-i} \delta_{\sigma_{i}}^{\nu} J^{\rho_{1},...,\rho_{n},\mu}_{\sigma_{1},...,\hat{\sigma_{i}},...,\sigma_{r}} + (\mu \leftrightarrow \nu) \quad (r = 0,...,n). \end{array} \label{derJ} \end{equation} This formula suggests the use of the Fock space techniques. Let us emphasize this point in detail. We will consider the functions $ J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} $ as the components of a tensor $ \{J_{r}\} \in {\cal H} \equiv {\cal F}^{-}(\rm I \mkern -3mu R^{n}) \otimes {\cal F}^{-}(\rm I \mkern -3mu R^{n}) $ where $ J_{r} $ belongs to the subspace of homogeneous tensors $ {\cal H}_{n-r,r} $ (where $ {\cal H}_{p,q} $ is the subspace of homogeneous tensors of degree $ p, q $ respectively.) We will denote by $ b^{*(\mu)}, c^{*}_{(\mu)}, b_{(\mu)}, c^{(\mu)} $ the fermionic creation and respectively the annihilation operators acting in $ {\cal H}. $ With these notations one can rewrite (\ref{derJ}) in a more compact way, namely: \begin{equation} \partial^{\mu\nu} J_{r} = \alpha_{r} [b^{*(\mu)} c^{(\nu)} + b^{*(\nu)} c^{(\mu)}] J_{r-1}; \qquad \alpha_{r} \equiv (-1)^{r-1} {1 \over 2} \times \sqrt{r \over n-r+1} \qquad (r = 0,...,n). \label{derJF} \end{equation} Also, the identities (\ref{trace}) can be compactly written as follows: \begin{equation} C J_{r} = 0 \qquad (r = 0,...,n) \label{traceF} \end{equation} where we have defined \begin{equation} C \equiv b^{(\mu)} c_{(\mu)}. \label{constraint} \end{equation} We need one more notation for our Fock space machinery, namely $ <\cdot,\cdot> $ which is the duality form between $ {\cal H} $ and $ {\cal H}^{*}. $ 3.2 We prove now the main result. \begin{thm} The general solution of the equations (\ref{N=1}) is of the following form: \begin{equation} {\cal L} = \sum_{r=0}^{n} {\cal L}^{\sigma_{1},...,\sigma_{r}}_{\rho_{r+1},...,\rho_{n}} J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} \label{polyn} \end{equation} where the functions $ {\cal L}^{...}_{...} $ are independent of $ \psi_{\mu\nu} $: \begin{equation} \partial^{\mu\nu} {\cal L}^{\sigma_{1},...,\sigma_{r}}_{\rho_{r+1},...,\rho_{n}} = 0 \quad (r = 0,...,n), \end{equation} and have analogous properties as the hyper-Jacobians, namely the (anti)symmetry property: \begin{equation} {\cal L}^{\sigma_{P(1)},...,\sigma_{P(r)}}_{\rho_{Q(r+1)},...,\rho_{Q(n)}} = (-1)^{|P|+|Q|} {\cal L}^{\sigma_{1},...,\sigma_{r}}_{\rho_{r+1},...,\rho_{n}} \quad (r = 0,...,n) \label{antisym-l} \end{equation} (where $P$ is a permutation of the numbers $1,...,r$ and $Q$ is a permutations of the numbers $r+1,...,n$) and also verify the identities: \begin{equation} {\cal L}_{\rho_{r+1},...,\rho_{n-1},\zeta}^{\sigma_{1},...,\sigma_{r-1},\zeta} = 0 \quad (r = 1,...,n-1) \label{trace-l} \end{equation} (i. e. are traceless). The function coefficients $ {\cal L}^{...}_{...} $ are uniquely determined by $ {\cal L} $ and the properties (\ref{antisym-l}) and (\ref{trace-l}) above. \label{structure} \end{thm} {\bf Proof:} It is a particular case of the main theorem from \cite{G2}. It is convenient to consider that $ {\cal L}^{\sigma_{1},...,\sigma_{r}}_{\rho_{r+1},...,\rho_{n}} $ are the components of a tensor $ \{{\cal L}^{r}\} $ in the dual space $ {\cal H}^{*}; $ explicitly: $ {\cal L}^{r} \in {\cal H}^{*}_{n-r+1,r} $ (where $ {\cal H}^{*}_{p,q} $ is the subspace of homogeneous tensors of degree $ p, q $ respectively.) With this trick, formula (\ref{polyn}) can be written in compact notations as: \begin{equation} {\cal L} = \sum_{r=0}^{n} <{\cal L}^{r},J_{r}>. \label{compact} \end{equation} We will denote by $ b_{*(\mu)}, c^{*(\mu)}, b^{(\mu)}, c_{(\mu)} $ the fermionic creation and respectively the annihilation operators acting in $ {\cal H}^{*}. $ (i) We now prove the uniqueness statement. So we must show that if \begin{equation} \sum_{r=0}^{n} <{\cal L}^{r},J_{r}> = 0 \label{uniqueness} \end{equation} then $ {\cal L}^{r} = 0 \quad r = 0,...,n. $ To prove this, we apply to the equation (\ref{uniqueness}) the operator $ \prod_{i=1}^{p} \partial^{\rho_{i}\sigma_{i}} \quad (p \leq n) $ and then we will make $ \psi_{\mu\nu} \rightarrow 0. $ Using (\ref{derJF}) one easily discovers the following equations: \begin{equation} \prod_{i=1}^{p} \quad \left[ b^{(\rho_{i})} c^{*(\sigma_{i})} + b^{(\sigma_{i})} c^{*(\rho_{i})} \right] \quad {\cal L}^{p} = 0, \qquad (p = 0,...,n). \label{unicity} \end{equation} To analyze this system we first define the operator: \begin{equation} {\cal C} \equiv b^{(\rho)} c_{(\rho)} \end{equation} and prove by elementary computations that the condition (\ref{trace-l}) can be rewritten as: \begin{equation} {\cal C} {\cal L}^{r} = 0 \quad (r = 0,...,n). \label{trace-l-compact} \end{equation} At this point it is convenient to define the dual tensors \begin{equation} \tilde{\cal L}^{\sigma_{1},...,\sigma{r};\rho_{1},...,\rho_{r}} \equiv {(-1)^{r} \over \sqrt{r! (n-r)!}} \varepsilon^{\rho_{1},....\rho_{n}} {\cal L}^{\sigma_{1},...,\sigma{r}}_{\rho_{r+1},...,\rho_{n}}. \label{dual} \end{equation} Because we have \begin{equation} \tilde{\tilde{\cal L}} = (-1)^{n} {\cal L} \end{equation} it is equivalent and more convenient to work in the dual space $ \tilde{\cal H}. $ We will denote by $ b^{*}_{(\mu)}, c^{*}_{(\mu)} $ and $ b^{(\mu)}, c^{(\mu)} $ the fermionic creation and respectively the annihilation operators acting in $ \tilde{\cal H}. $ Then the condition (\ref{trace-l-compact}) rewrites as: \begin{equation} \tilde{\cal C} \tilde{\cal L}^{r} = 0 \qquad (r = 0,...,n) \label{trace-tilde} \end{equation} where \begin{equation} \tilde{\cal C} \equiv b^{(\mu)} c^{*}_{(\mu)} \label{constraint-tilde} \end{equation} and the equation (\ref{unicity}) becomes: \begin{equation} \prod_{i=1}^{p} \quad \left[ b^{(\rho_{i})} c^{(\sigma_{i})} + b^{(\sigma_{i})} c^{(\rho_{i})} \right] \tilde{\cal L}^{p} = 0, \qquad (p = 0,...,n). \label{unicity-tilde} \end{equation} Finally, we need the following number operators: \begin{equation} N_{b} \equiv b^{*(\rho)} b_{(\rho)}; \quad N_{c} \equiv c^{*(\rho)} c_{(\rho)}. \end{equation} Then one knows that: \begin{equation} N_{b} \vert_{\tilde{\cal H}_{p,q}} = p {\bf 1}, \quad N_{c} \vert_{\tilde{\cal H}_{p,q}} = q {\bf 1}. \label{number} \end{equation} We analyze the system (\ref{unicity-tilde}) using some simple lemmas. The proofs are elementary and are omitted. \begin{lemma} The following formula is true: \begin{equation} b^{*}_{(\mu)} c^{*}_{(\nu)} \left[ b^{(\mu)} c^{(\nu)} + b^{(\nu)} c^{(\mu)} \right] = N_{b} (N_{c} + {\bf 1}) - \tilde{\cal C}^{*} \tilde{\cal C}. \end{equation} \label{inverse} \end{lemma} \begin{lemma} The operator $\tilde{\cal C}$ commutes with all the operators of the form $$ b^{(\mu)} c^{(\nu)} + b^{(\nu)} c^{(\mu)}. $$ Explicitly: \begin{equation} \left[ \tilde{\cal C}, b^{(\mu)} c^{(\nu)} + b^{(\nu)} c^{(\mu)} \right] = 0. \end{equation} \label{commute} \end{lemma} \begin{lemma} If the tensor $t$ verifies the identity $ \tilde{\cal C} t = 0 $ the the tensors $$ \left[ b^{(\mu)} c^{(\nu)} + b^{(\nu)} c^{(\mu)} \right] t $$ also verify this identity. \label{iteration} \end{lemma} We now have: \begin{prop} Suppose the tensor $ t \in \tilde{\cal H}_{r,r} \quad (r = 0,...,n) $ verify the system: \begin{equation} \prod_{i=1}^{p} \quad \left[ b^{(\rho_{i})} c^{(\sigma_{i})} + b^{(\sigma_{i})} c^{(\rho_{i})} \right] \quad t = 0. \end{equation} Then we have $ t = 0. $ \end{prop} {\bf Proof:} We apply to this system the operator $ \prod_{i=1}^{p} b^{*}_{(\rho_{i})} c^{*}_{(\rho_{i})} $ and make repeated use of the lemmas above. $\nabla$ The argument involved in the proof above will be called {\it the unicity argument}. In conclusion the system (\ref{unicity-tilde}) has the solution $ \tilde{\cal L}^{p} = 0 \quad (p = 0,...,n). $ (ii) We start to prove the formula (\ref{polyn}) by induction over $n$. For $ n = 1 $ the derivation of (\ref{polyn}) is elementary. We suppose that we have the assertion of the theorem for a given $n$ and we prove it for $ n + 1. $ In this case the indices $ \mu,\nu, ... $ takes values (for notational convenience) $ \mu,\nu, ...= 0,...,n $ and $ i,j,...= 1,...,n. $ If we consider in (\ref{N=1}) that $ \mu,\rho_{1},\rho_{2},\rho_{3} = 1,...,n $ then we can apply the induction hypothesis and we get: \begin{equation} {\cal L} = \sum_{r=0}^{n} l^{i_{1},...,i_{r}}_{j_{r+1},...,j_{n}} I_{i_{1},...,i_{r}}^{j_{r+1},...,j_{n}}. \label{polyn'} \end{equation} Here $ l^{...}_{...} $ have properties of the type (\ref{antisym-l}) and ({\ref{trace-l}) and can depend on $ x, \psi^{A}, \psi^{A}_{\mu} $ {\it and} $ \psi^{A}_{0\mu}. $ The expressions $ I^{...}_{...} $ are constructed from $ \psi_{ij} $ according to the prescription (\ref{hyperJ}). (iii) We still have at our disposal the relations (\ref{N=1}) where at least one index takes the value $0$. The computations are rather easy to do using instead of (\ref{polyn'}) the compact tensor notation (see (\ref{compact})) and the unicity argument. We give the results directly for the dual tensors $ \tilde{l}^{r}. $ \begin{equation} (\partial^{00})^{2} \tilde{l}^{r} = 0 \qquad (r = 0,...,n), \label{eq1} \end{equation} \begin{equation} \partial^{00} \partial^{0i} \tilde{l}^{r} = 0 \qquad (r = 0,...,n), \label{eq2} \end{equation} \begin{equation} \alpha_{r+1} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \partial^{00} \tilde{l}^{r+1} + 2 \partial^{0i} \partial^{0j} \tilde{l}^{r} = 0 \quad (r = 0,...,n-1), \label{eq3} \end{equation} \begin{equation} \partial^{0i} \partial^{0j} \tilde{l}^{n} = 0, \label{eq4} \end{equation} \begin{equation} \sum_{(i,j,k)} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \partial^{0k} \tilde{l}^{r} = 0 \qquad (r = 1,...,n). \label{eq5} \end{equation} The expressions $ \tilde{l}^{r} $ are obviously considered as tensors from $ \tilde{\cal H}_{r,r} $ verifying the restriction: \begin{equation} \tilde{\cal C} \tilde{l}^{r} = 0 \quad (r = 0,...,n). \label{id} \end{equation} As in \cite{G2}, these equations can be solved i.e. one can describe the most general solution. From (\ref{eq1}) we have: \begin{equation} \tilde{l}^{r} = \tilde{l}^{r}_{(0)} + \psi_{00} \tilde{l}^{r}_{(1)} \label{sol} \end{equation} where the functions $ \tilde{l}^{r}_{(\alpha)} \quad (\alpha = 0,1; \quad r = 0,...,n) $ verify: \begin{equation} \partial^{00} \tilde{l}^{r}_{(\alpha)} = 0 \quad (\alpha = 0,1; \quad r = 0,...,n) \end{equation} and also verify identities of the type (\ref{id}): \begin{equation} \tilde{\cal C} \tilde{l}^{r}_{(\alpha)} = 0 \quad (\alpha = 0,1; \quad r = 0,...,n). \label{id-alpha} \end{equation} From (\ref{eq2}) we also get: \begin{equation} \partial^{0i} \tilde{l}^{r}_{(1)} = 0, \quad (r = 0,...,n) \label{restr1} \end{equation} and finally (\ref{eq3}) - (\ref{eq5}) become: \begin{equation} \alpha_{r+1} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \tilde{l}^{r+1}_{(1)} + 2 \partial^{0i} \partial^{0j} \tilde{l}^{r}_{(0)} = 0, \qquad (r = 0,...,n-1) \label{eq3'} \end{equation} \begin{equation} \partial^{0i} \partial^{0j} \tilde{l}^{n}_{(0)} = 0 \label{4'} \end{equation} \begin{equation} \sum_{(i,j,k)} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \partial^{0k} \tilde{l}^{r}_{(0)} = 0 \qquad (r = 0,...,n). \label{eq5'} \end{equation} (iv) We proceed further by applying the operator $ \partial^{0k} $ to (\ref{eq3'}); taking into account (\ref{restr1}) we obtain: \begin{equation} \partial^{0i} \partial^{0j} \partial^{0k} \tilde{l}^{r}_{(0)} = 0 \qquad (r = 0,...,n-1). \label{eq3''} \end{equation} From this relation one obtains a polynomial structure in $ \psi_{0i} $ for $ \tilde{l}^{r}_{(0)} \quad (r = 0,...,n-1): $ \begin{equation} \tilde{l}^{r}_{(0)} = \tilde{l}^{r}_{(00)} + \psi_{0i} \tilde{l}^{r}_{(0i)} + {1 \over 2} \psi_{0i} \psi_{0j} \tilde{l}^{r}_{(0ij)} \quad (r = 0,...,n-1). \label{sol1} \end{equation} From (\ref{eq4}) one a obtains a similar polynomial structure: \begin{equation} \tilde{l}^{n}_{(0)} = \tilde{l}^{n}_{(00)} + \psi_{0i} \tilde{l}^{n}_{(0i)} . \label{sol2} \end{equation} Moreover we have the following restrictions on the various tensors appearing in the preceding two formulae: \begin{equation} \partial^{0i} \tilde{l}^{r}_{(0\mu)} = 0 \quad (r = 0,...,n); \qquad \partial^{0k} \tilde{l}^{r}_{(0ij)} = 0 \quad (r = 0,...,n-1) \label{restr1'} \end{equation} and \begin{equation} \tilde{\cal C} \tilde{l}^{r}_{(0\mu)} = 0 \quad (r = 0,...,n); \qquad \tilde{\cal C} \tilde{l}^{r}_{(0ij)} = 0 \quad (r = 0,...,n-1) \label{id''} \end{equation} and we also can impose \begin{equation} \tilde{l}^{r}_{(0ij)} = \tilde{l}^{r}_{(0ji)} \quad (r = 0,...,n-1). \label{restr2} \end{equation} If we substitute now (\ref{sol1}) into the original equation (\ref{eq3'}) we obtain \begin{equation} \tilde{l}^{r}_{(0ij)} = - 2 \alpha_{r+1} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \tilde{l}^{r+1}_{(1)} \qquad (r = 0,...,n-1). \label{sol3} \end{equation} Finally we substitute the expressions (\ref{sol1}) and (\ref{sol2}) into the equation (\ref{eq5'}) and we obtain: \begin{equation} \sum_{(i,j,k)} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \tilde{l}^{r}_{(0k)} = 0 \qquad (r = 0,...,n) \label{eq6} \end{equation} and \begin{equation} \sum_{(i,j,k)} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] \tilde{l}^{r}_{(0kl)} = 0 \qquad (r = 0,...,n). \label{eq7} \end{equation} One must check that the expression (\ref{sol3}) for $ \tilde{l}^{r}_{(0kl)} $ is compatible with the restrictions (\ref{id''}) by applying by applying the operator $ \tilde{\cal C} $ to this relation. Also one notes that (\ref{sol3}) identically verifies the equation (\ref{eq7}). In conclusion we are left to solve only (\ref{eq6}) together with the restrictions (\ref{restr1'}), and (\ref{id''}). We have the following results: \begin{lemma} The following formula is valid \begin{equation} \left[ \tilde{\cal C}^{*}, \tilde{\cal C} \right] = N_{b} - N_{c}. \end{equation} \end{lemma} \begin{lemma} If $ t \in \tilde{\cal H}_{p,p} $ verifies $ {\cal C} t = 0 $ then it also verifies $ {\cal C}^{*} t = 0 $ and conversely. \label{c-star} \end{lemma} The proofs of these lemmas are elementary and are omitted. Based on them we have \begin{lemma} Let $ t^{k} \in \tilde{\cal H}_{p,p} \quad (k = 1,...,n) $ be tensors verifying the restriction \begin{equation} \tilde{\cal C} t^{k} = 0 \label{Ct} \end{equation} and the system: \begin{equation} \sum_{(i,j,k)} \left[ b^{(i)} c^{(j)} + b^{(j)} c^{(i)} \right] t^{k} = 0. \label{permutation} \end{equation} Then one can write {\it uniquely} $t$ in of the following form: \begin{equation} t^{k} = b^{(k)} U + c^{(k)} V \label{T} \end{equation} with $ U \in \tilde{\cal H}_{p+1,p} $ and $ V \in \tilde{\cal H}_{p,p+1} $ verifying \begin{equation} \tilde{\cal C} U = V \quad \tilde{\cal C}^{*} U = 0 \quad \tilde{\cal C} V = 0 \quad \tilde{\cal C}^{*} V = U. \label{UV} \end{equation} Here we put by convention $ \tilde{\cal H}_{p,q} \equiv \{0\} $ if at least one of the indices $p$ and $q$ is negative or $n+1$. \label{permutation-lemma} \end{lemma} {\bf Proof:} We apply to the equation (\ref{permutation}) the operator $ b^{*}_{(i)} c^{*}_{(j)} $ and we find out (after summation over $i$ and $j$ and taking into account (\ref{Ct}): \begin{equation} (p+2)t^{k} = b^{(k)} b^{*}_{(l)} t^{l} + c^{(k)} c^{*}_{(l)} t^{l}. \end{equation} So we have the formula from the statement with: $ U = (p+2)^{-1} b^{*}_{(l)} t^{l} $ and $ V = (p+2)^{-1} c^{*}_{(l)} t^{l}. $ These expressions verify the identities (\ref{UV}). Conversely, if we have (\ref{T}) and (\ref{UV}) it remains to check that the equations (\ref{permutation}) and (\ref{Ct}) are indeed identically satisfied. $\nabla$ From this lemma one can write down that the most general solution of (\ref{eq6}). Combining with the previous results one obtains the most general expression for the tensors $ \tilde{l}^{r}. $ Reverting to the original tensors $ l^{r} $ one obtains easily that the most general expression for them is: \begin{equation} l^{r} = l^{r}_{(0)} + \psi_{0i} \left[ b^{(i)} U^{r} + c^{*(i)} V^{r} \right] - 2 \alpha_{r+1} \psi_{0i} \psi_{0j} b^{(i)} c^{*(j)} l^{r+1}_{(1)} + \psi_{00} l^{r}_{(1)} \qquad (r = 0,...,n). \label{sol-gen} \end{equation} The tensors $ l^{r}_{(0)}, l^{r}_{(1)} \in \tilde{\cal H}_{r,n-r}, U^{r} \in \tilde{\cal H}_{r+1,n-r}, V^{r} \in \tilde{\cal H}_{r,n-r-1} $ are not completely arbitrary; they must satisfy the following relations: \begin{equation} {\cal C} l^{r}_{(\alpha)} = 0, \quad (\alpha = 0,1; \quad r = 0,...,n), \label{iden1} \end{equation} \begin{equation} {\cal C} U^{r} = V^{r}, \quad {\cal C} V^{r} = 0, \quad {\cal C}^{*} U^{r} = 0, \quad {\cal C}^{*} V^{r} = U^{r} \quad (r = 0,...,n) \label{iden2} \end{equation} and \begin{equation} \partial^{\mu\nu} l^{r}_{(\alpha)} = 0, \quad \partial^{\mu\nu} U^{r} = 0, \quad \partial^{\mu\nu} V^{r} = 0, \qquad (r = 0,...,n; \quad \alpha = 0,1). \label{iden3} \end{equation} The structure of the tensors $ l^{r} \quad (r = 0,...,n) $ is completely elucidated. (v) It remains to introduce these expressions for $ l^{r} $ in (\ref{polyn'}) and regroup the terms. Like in \cite{G1}, \cite{G2} one obtains the desired formula (\ref{polyn}) for $ n+1 $ with the tensors $ {\cal L}^{r} $ expressed in terms of the tensors defined in the proof above. Finally one must check that the tensors $ {\cal L}^{r} $ also verify the induction hypothesis i.e. the identities (\ref{trace-l}). This is done after some computations using (\ref{iden1}) - (\ref{iden3}) and the induction is finished. \vrule height 6pt width 6pt depth 6pt \begin{rem} We make a last comment concerning the unicity statement from the proof. First, the non-uniqueness is easy to explain because if one add to the tensors $ {\cal L}^{...}_{...} $ contributions containing at least a factor $ \delta_{\rho_{j}}^{\sigma_{i}} $ then it immediately follows from the identity (\ref{trace}) that the right hand side of the formula (\ref{polyn}) is not changed. So, the constrain (\ref{trace-l}) is a way of eliminating these type of contributions respecting in the same time the antisymmetry properties of the functions $ {\cal L}^{...}_{...} $ i.e. to obtain the {\it traceless} part of $ {\cal L}^{...}_{...}. $ In this context we mention that such a decomposition of a tensor in a traceless part and a rest containing at least a delta factor is true in extremely general conditions as it is proved in \cite{K3}. \end{rem} 3.2 Let us prepare the ground for the analysis of the more complicated case $ N \geq 2. $ First we note that if in analogy to (\ref{dual}) we define: \begin{equation} \tilde{J}_{\sigma_{1},...,\sigma_{r};\rho_{1},...,\rho_{r}} \equiv {(-1)^{r} \over \sqrt{r! (n-r)!}} \varepsilon_{\rho_{1},....\rho_{n}} J_{\sigma_{1},...,\sigma_{r}}^{\rho_{r+1},...,\rho_{n}}. \label{dual-hyperJ} \end{equation} then can rewrite (\ref{polyn}) as follows: \begin{equation} {\cal L} = \sum_{r=0}^{n} \tilde{\cal L}^{\sigma_{1},...,\sigma_{r};\rho_{1},...,\rho_{r}} \tilde{J}_{\sigma_{1},...,\sigma_{r},\rho_{1},...,\rho_{r}}. \label{polyn-tilde} \end{equation} We intend to use equation (\ref{II1}) first for the case $ A = B = 1,2...,N. $ It is clear that we will be able to apply the theorem above. To do this we define in analogy to (\ref{hyperJ}) and (\ref{dual-hyperJ}) the expressions \begin{equation} J^{(A)\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} \equiv \varepsilon^{\rho_{1},...,\rho_{n}} \prod_{i=1}^{r} \psi_{\rho_{i}\sigma_{i}}^{A} \qquad (r = 0,...,n; \quad A = 1,...,N) \label{hyperJN} \end{equation} and \begin{equation} \tilde{J}^{(A)}_{\sigma_{1},...,\sigma_{r};\rho_{1},...,\rho_{r}} \equiv {(-1)^{r} \over \sqrt{r! (n-r)!}} \varepsilon_{\rho_{1},....\rho_{n}} J_{\sigma_{1},...,\sigma_{r}}^{(A)\rho_{r+1},...,\rho_{n}}. \label{dual-hyperJN} \end{equation} Then the equations (\ref{II1}) for $ A = B $ will produce an expression of the following form: \begin{equation} {\cal L} = \sum_{r,s,...,=0}^{n} {\cal L}^{\sigma_{1},...,\sigma_{r};\mu_{1},...,\mu_{s}; \cdots}_{\rho_{r+1},...,\rho_{n};\nu_{s+1},...,\nu_{n};\cdots} J^{\rho_{r+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{r}} J^{\nu_{s+1},...,\nu_{n}}_{\mu_{1},...,\mu_{s}} \cdots \label{polynN} \end{equation} where the functions $ {\cal L}^{...}_{...} $ are verifying the following properties: \begin{equation} \partial^{\mu\nu}_{A} {\cal L}^{...}_{...} = 0 \end{equation} \begin{equation} {\cal L}^{\cdots;\sigma_{P(1)},...,\sigma_{P(r)};\cdots}_{\cdots; \rho_{Q(r+1)},...,\rho_{Q(n)};\cdots} = (-1)^{|P|+|Q|} {\cal L}^{\cdots;\sigma_{1},...,\sigma_{r};\cdots}_{\cdots; \rho_{r+1},...,\rho_{n};\cdots} \end{equation} (where $P$ is a permutation of the numbers $ 1,...,r $ and $Q$ is a permutation of the numbers $ r+1,...,n $) and \begin{equation} {\cal L}^{\cdots;\sigma_{1},...,\sigma_{r-1},\zeta;\cdots}_{\cdots; \rho_{r+1},...,\rho_{n-1},\zeta;\cdots} = 0. \end{equation} Again the analysis is much simplified if one uses tensor notations. Generalizing in an obvious way the scalar case the functions $ {\cal L}^{...}_{...} $ will become the components of a tensor $ {\cal L} \in ({\cal H}^{*})^{\otimes N} $ and one can write (\ref{polynN}) in a more compact manner: \begin{equation} {\cal L} = \sum_{r_{1},...,r_{N}=1}^{n} <{\cal L}^{r_{1},...,r_{N}}, J^{(1)}_{r_{1}} \otimes \cdots J^{(N)}_{r_{N}}> \label{polynN-compact} \end{equation} where $ J^{(1)}_{r_{1}} \otimes \cdots J^{(N)}_{r_{N}} \in {\cal H}^{\otimes N} $ and $ <\cdot,\cdot> $ is the duality form. Let $ b^{*(\mu)}_{(A)}, c^{*(A)}_{(\mu)}, b^{(A)}_{(\mu)},c_{(A)}^{(\mu)} $ be the creation and the annihilation operators acting in $ {\cal H}^{\otimes N} $ and $ b^{*(A)}_{(\mu)}, c^{*(\mu)}_{(A)}, b^{(\mu)}_{(A)},c_{(\mu)}^{(A)} $ the corresponding operators from $ ({\cal H}^{*})^{\otimes N}. $ Then the constraints of the type (\ref{trace-l-compact}) can be written as follows: \begin{equation} {\cal C}_{A} {\cal L}^{r_{1},...,r_{N}} = 0 \quad (A = 1,...,N) \label{trace-lN} \end{equation} where we have defined: \begin{equation} {\cal C}_{A} \equiv b^{(\mu)}_{(A)} c_{(\mu)}^{(A)} \quad (A = 1,...,N). \end{equation} The expressions of the type (\ref{polynN}) or (\ref{polynN-compact}) are unique in the sense that $ {\cal L} $ uniquely determines the function coefficients $ {\cal L}^{r_{1},...,r_{N}}; $ this follows directly from the uniqueness statement of theorem \ref{structure}. It is convenient to work with the dual tensors $ \tilde{\cal L}^{r_{1},...,r_{N}} \in \tilde{\cal H}^{\otimes N} \quad (A = 1,...,N) $ defined analogously as in (\ref{dual}) which will verify the constraints: \begin{equation} \tilde{\cal C}_{A} \tilde{\cal L}^{r_{1},...,r_{N}} = 0 \quad (A = 1,...,N) \label{C} \end{equation} where \begin{equation} \tilde{\cal C}_{A} \equiv b^{(\mu)}_{(A)} c_{(\mu)}^{*(A)} \quad (A = 1,...,N) \end{equation} are the expressions of the constraints in the dual space. Our goal in the next two sections will be to prove the following result: \begin{thm} The most general solution of the equations (\ref{II1}) and (\ref{II2}) is of the form: \begin{equation} {\cal L} = \sum_{r_{1},...,r_{N}=1}^{n} <\tilde{\cal L}^{r_{1},...,r_{N}}, \tilde{J}^{(1)}_{r_{1}} \otimes \cdots \tilde{J}^{(N)}_{r_{N}}>. \label{polynN-compact-dual} \end{equation} The tensors $ \tilde{\cal L}^{\sigma_{1},...,\sigma_{r};\rho_{1},...,\rho_{r}; \mu_{1},...,\mu_{s};\nu_{1},...,\nu_{s};\cdots} $ verify, the usual antisymmetry and tracelessness properties, but moreover they verify the property of complete antisymmetry in {\it all} the indices $ \sigma_{1},...,\sigma_{r},\mu_{1},...,\mu_{s},... $ i.e. they verify the identities \begin{equation} \left[ b^{(\mu)}_{(A)} b^{(\nu)}_{(B)} + (\mu \leftrightarrow \nu) \right] \tilde{\cal L}^{r_{1},...,r_{N}} = 0. \label{antisymmetry} \end{equation} \label{structureN} \end{thm} To do this we will use the remaining equations i.e. (\ref{II1}) for $ A \not= B $ and (\ref{II2}). Using the compact expression (\ref{polynN-compact}) one obtains from (\ref{II1}): \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left[ b^{(\mu)}_{(A)} c^{(\rho_{1})}_{(A)} + b^{(\rho_{1})}_{(A)} c^{(\mu)}_{(A)} \right] \left[ b^{(\rho_{2})}_{(B)} c^{(\rho_{3})}_{(B)} + b^{(\rho_{3})}_{(B)} c^{(\rho_{2})}_{(B)} \right] \tilde{\cal L}^{r_{1},...,r_{N}} + (A \leftrightarrow B) = 0 \label{II1-Fock} \end{equation} and from (\ref{II2}) it follows: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \sum_{(\zeta_{1},\zeta_{2},\zeta_{3})} \left[ b^{(\zeta_{1})}_{(D)} c^{(\zeta_{2})}_{(D)} + b^{(\zeta_{2})}_{(D)} c^{(\zeta_{1})}_{(D)} \right] \left[ b^{(\rho_{1})}_{(B)} c^{(\rho_{2})}_{(B)} + b^{(\rho_{2})}_{(B)} c^{(\rho_{1})}_{(B)} \right] \left[ b^{(\zeta_{3})}_{(A)} c^{(\rho_{3})}_{(A)} + b^{(\rho_{3})}_{(A)} c^{(\zeta_{3})}_{(A)} \right] \tilde{\cal L}^{r_{1},...,r_{N}} = 0 \label{II2-Fock} \end{equation} For the case $ N = 2 $ to be analysed in the next Section it easily follows that (\ref{II2-Fock}) follows from (\ref{II1-Fock}), so we will have to analyse only the first equation for $ A \not= B. $ For the case $ N \geq 3 $ to be analysed in Section 5 it will be convenient to start with (\ref{II2-Fock}). \section{Trivial Second-Order Lagrangians for the Case $N = 2$} As we have said before we analyse (\ref{II1-Fock}) in the case $ A \not= B. $ It is convenient to redefine $ b^{(\mu)}_{(A)} \rightarrow d^{\mu}, c^{(\mu)}_{(A)} \rightarrow e^{\mu}, b^{(\mu)}_{(B)} \rightarrow b^{\mu}, c^{(\mu)}_{(B)} \rightarrow c^{\mu}. $ Then the equation (\ref{II1-Fock}) takes the generic form: \begin{equation} \sum b^{\mu} c^{\nu} d^{\rho} e^{\sigma} t = 0 \label{II1-compact} \end{equation} where the sum runs over all the permutations of indices $ \mu, \nu, \rho, \sigma $ and $t$ is an arbitrary tensor from $ \tilde{\cal H}_{k,k',r,r}. $ Here $k, k', r, r$ are the eigenvalues of the operators $ N_{b}, N_{c}, N_{d} $ and $ N_{e} $ respectively. We now define the following operators: \begin{equation} A \equiv c^{\rho} d^{*}_{\rho}, \quad B \equiv d^{\rho} e^{*}_{\rho}, \quad C \equiv e^{\rho} c^{*}_{\rho}, \quad Z \equiv b^{\rho} c^{*}_{\rho} \label{ABC} \end{equation} and there are some constraints to be taken into account, namely (see (\ref{C}) and lemma \ref{c-star}): \begin{equation} B t = 0, \quad B^{*} t = 0 \label{B} \end{equation} and \begin{equation} Z t = 0, \quad Z^{*} t = 0. \label{Z} \end{equation} We will use in the following only the constraint (\ref{B}). We start the proof of theorem \ref{structureN} by a series of lemmas and propositions. \begin{lemma} If the tensor $t$ verifies the equation (\ref{II1-compact}) and the constraints (\ref{B}) then it also verifies the equation \begin{equation} (r+2) \left[ (k+1) (r+3) {\bf 1} - M \right] b^{\mu} t = c^{\mu} U_{(0)} + d^{\mu} V_{(0)} + e^{\mu} W_{(0)} \label{II1-contraction} \end{equation} where \begin{equation} M \equiv A^{*} A + C C^{*} \label{M} \end{equation} and $ U_{(0)}, V_{(0)}, W_{(0)} $ are some tensors constructed from $t$. (We will not need in the following their explicit expressions.) \end{lemma} {\bf Proof:} One applies to the equation (\ref{II1-compact}) the operator $ c^{*}_{\nu} d^{*}_{\rho} e^{*}_{\sigma} $ and uses the constraints (\ref{B}). $\nabla$ We want to prove that the operator in the square brackets from (\ref{II1-contraction}) is invertible. To do this we notice that $M$ can be restricted to the subspace of $ \tilde{\cal H}_{k,k',r,r} $ determined by $ Ker{B} \cap Ker(B^{*}). $ We denote this subspace by $h$ and by $M'$ the restriction of $M$ to $h$. Then we have \begin{prop} The spectrum of the operator $M'$ is described by: \begin{equation} {\it Spec}(M') \subset \{ v(v+r-k+2) | v = 0,...,v_{0}\} \label{spectrum} \end{equation} where $ v_{0} \equiv min\{k, 2(n-r)\}. $ In particular we have \begin{equation} {\it Spec}(M') \subset [0, k (r+2)]. \label{S} \end{equation} \label{spectrum-M} \end{prop} {\bf Proof:} One finds out after some tedious but straightforward computations that if the tensor $t$ verifies the eigenvalue equation \begin{equation} M' t = \lambda t \label{X} \end{equation} then one also has \begin{equation} M' A^{s} (C^{*})^{u} t = \lambda_{s,u} A^{s} (C^{*})^{u} t. \label{eigen} \end{equation} Here $s$ and $u$ are natural numbers verifying $ s, u \leq n-r, \quad s+u \leq k $ and the expression for the eigenvalue is \begin{equation} \lambda_{s,u} \equiv \lambda - \Lambda_{s+u} \end{equation} where \begin{equation} \Lambda_{v} \equiv v(v-r-k+2). \end{equation} The proof of the formula above is most easily done by induction: first one considers the case $ s = 0 $ and use the induction over $u$ and next one proves (\ref{eigen}) by induction over $s$. Needless to say, one must make use of the various anticommutation relations between the operators $A, B, C$ and their Hermitian conjugates. Now one supposes that $ \lambda \not\in {\it Spec}(M') $ i.e. $ \lambda \not= \Lambda_{v} \quad (v= 0,1,...,v_{0}). $ In this case it follows that $ \lambda_{s,u} \not= 0 $ and we will be able to prove that one has \begin{equation} A^{s} (C^{*})^{u} t = 0 \quad (s,u \leq n-r, \quad s+u \leq k). \label{recurrence} \end{equation} We analyse separately two cases. If $ 2r + k \leq 2n $ we can take in (\ref{eigen}) $s$ and $u$ such that $ s + u = k. $ One obtains that \begin{equation} \lambda_{s,u} A^{s} (C^{*})^{u} t = 0. \end{equation} Because $ \lambda_{s,u} \not= 0 $ we have (\ref{recurrence}) for $ s + u = k. $ Next, one proves this relation for $ s + u \leq k $ by recurrence (downwards) over $ v = s + u $ using again (\ref{eigen}). In the case $ 2r + k > 2n $ the proof of (\ref{recurrence}) is similar, only one has to start the induction downwards from $ s + u = 2(n-r). $ The relation (\ref{recurrence}) is proved. If we take in this relation $ s = u = 0 $ we get $ t = 0. $ In conclusion, if $ \lambda $ does not belong to the set $ {\it Spec}(M') $ then the equation (\ref{X}) does not have non-trivial solutions. Because the operator $M'$ lives in a finite dimensional Hilbert space the first assertion of the proposition follows. The second assertion follows from the first and from $ M \geq 0. $ $\vrule height 6pt width 6pt depth 6pt$ \begin{cor} The matrix $ (k+1) (r+3) {\bf 1} - M' $ is invertible. \end{cor} {\bf Proof:} $ (k+1) (r+3) $ does not belong to the spectrum of $M'$. $\nabla$ Now we come back to the equation (\ref{II1-contraction}); using the corollary above and the finite dimensional functional calculus, it is not hard to prove that one obtains from this equation the following consequence: \begin{equation} b^{\mu} t = c^{\mu} U + d^{\mu} V + e^{\mu} W \label{UVW} \end{equation} where $ U, V, W $ are some tensors verifying \begin{equation} B U = 0, \quad B V = W, \quad B W = 0, \quad B^{*} U = 0, \quad B^{*} V = 0, \quad B^{*} W = V. \end{equation} A structure formula of the type (\ref{UVW}) is valid for every tensor $ \tilde{\cal L}^{r,k} $ appearing in the structure formula for the trivial Lagrangian. It it important to note that in deriving this result we have used only the constraints (\ref{B}) and not the constraints (\ref{Z}). So, the tensors $ \tilde{\cal L}^{r,k} $ are not uniquely fixed by the unicity argument. We use the possibility of redefining these tensors to our advantage. Indeed, if one inserts formulas of the type (\ref{UVW}) into the expression of the Lagrangian one can show that the contribution following from the first term is null (one must use the tracelessness of the hyper-Jacobians). In other words, one can redefine the tensors $ \tilde{\cal L}^{r,k} $ such that one has: \begin{equation} (r+2) b^{\mu} \tilde{\cal L}^{r,k} = d^{\mu} V + e^{\mu} W. \end{equation} Now one can make this formula more precise is one uses in a clever way lemma \ref{permutation-lemma}, namely the following relation stands true: \begin{equation} (r+2) b^{\mu} \tilde{\cal L}^{r,k} = (d^{\mu} {\cal D} + e^{\nu} {\cal E}) {\cal L}^{r,k} \label{L} \end{equation} where we have introduce new notations: \begin{equation} {\cal D} \equiv b^{\mu} d^{*}_{\mu}, \quad {\cal E} \equiv b^{\mu} e^{*}_{\mu}. \end{equation} Now we have \begin{lemma} The following formula is valid: \begin{equation} {(r+p+1)! \over (r+1)!} b^{\mu_{1}} \cdots b^{\mu_{p}} \tilde{\cal L}^{r,k} = (-1)^{[p/2]} \sum_{s=0}^{p} (-1)^{(p+1)s} C^{s}_{p} {\cal A}_{p} (d^{\mu_{1}} \cdots d^{\mu_{s}} e^{\mu_{s+1}} \cdots e^{\mu_{p}}) {\cal D}^{s} {\cal E}^{p-s} \tilde{\cal L}^{r,k}. \label{bbb} \end{equation} Here $ [m] $ is the integer part of $m$, $ C^{s}_{p} \equiv {p! \over s! (p-s)!} $ and $ {\cal A}_{p} $ is the operator of antisymmetrization in the indices $ \mu_{1},...,\mu_{p}. $ \end{lemma} {\bf Proof:} By very long computations using induction over $p$. Indeed, for $ p = 0 $ the formula is trivial and for $ p = 1 $ we have (\ref{L}). $\nabla$ In particular, if we take in (\ref{bbb}) $ p = k $ one obtains after some prelucrations \begin{equation} {(r+k+1)! \over (r+1)!} b^{\mu_{1}} \cdots b^{\mu_{k}} \tilde{\cal L}^{r,k} = \sum_{s=0}^{k} (-1)^{(k+1)s+[s/2]} {1 \over (k-s)!} {\cal A}_{p} (d^{\mu_{1}} \cdots d^{\mu_{s}} e^{\mu_{s+1}} \cdots e^{\mu_{k}}) B^{k-s} L^{r,k} \label{bcd} \end{equation} where $ L^{r,k} \in \tilde{\cal H}_{r+k,r,0,k} $ is given by \begin{equation} L^{r,k} = {\cal D}^{k} \tilde{\cal L}^{k,r}. \end{equation} Using indices the expression of $ {\cal L} $ becomes \begin{equation} \begin{array}{c} {\cal L} = \sum_{r,k=0}^{n} {(r+1)! \over (r+k+1)!} \sum_{s=0}^{k} (-1)^{(k+1)s+[s/2]} {1 \over (k-s)!} (B^{k-s} L)^{\mu_{1},...\mu_{s}\rho_{1},...,\rho_{r}; \mu_{s+1},...,\mu_{k}\sigma_{1},...,\sigma_{r};\emptyset;\nu_{1},...,\nu_{k}} \nonumber \\ \tilde{J}^{(1)}_{\rho_{1},...,\rho_{r};\sigma_{1},...,\sigma_{r}} \tilde{J}^{(2)}_{\mu_{1},...,\mu_{k};\nu_{1},...,\nu_{k}} \end{array} \end{equation} Now one uses the explicit expression for the operator $B$ and the tracelessness of the hyper-Jacobians to show that one can replace in the formula above $ B^{k-s} $ by a sum of lower powers of $B$. In the end one finds out by recurrence that the sum over $s$ disappears and the formula above is transformed into: \begin{equation} {\cal L} = \sum_{r,k=0}^{n} {(r+1)! \over (r+k+1)!} (-1)^{k(k-1)/2} L^{\mu_{1},...\mu_{k}\rho_{1},...,\rho_{r}; \sigma_{1},...,\sigma_{r};\emptyset;\nu_{1},...,\nu_{k}} \tilde{J}^{(1)}_{\rho_{1},...,\rho_{r};\sigma_{1},...,\sigma_{r}} \tilde{J}^{(2)}_{\mu_{1},...,\mu_{k};\nu_{1},...,\nu_{k}}. \label{structureII} \end{equation} In other words, by redefining $ {\cal L}^{r,k} \rightarrow {\cal L}^{r,k}_{1} $ where: \begin{equation} {\cal L}_{1}^{\rho_{1},...,\rho_{r};\sigma_{1},...,\sigma_{r}; \mu_{1},...\mu_{k};\nu_{1},...,\nu_{k}} \equiv (-1)^{k(k-1)/2} {(r+1)! \over (r+k+1)!} L^{\mu_{1},...\mu_{k}\rho_{1},...,\rho_{r}; \sigma_{1},...,\sigma_{r};\emptyset;\nu_{1},...,\nu_{k}} \label{redefine} \end{equation} one preserves the formula (\ref{structureII}) and has moreover \begin{equation} (b^{\mu} d^{\nu} + b^{\nu} d^{\mu}) {\cal L}^{r,k}_{1} = 0. \end{equation} This observation finishes the proof of the theorem \ref{structureN} for the case $ N = 2. $ \section{Trivial Second-Order Lagrangians in the Case $N \geq 3$} In this case we start with the equation (\ref{II2-Fock}) and note that it gives something non-trivial {\it iff} all the three indices $ A, B, D $ are distinct one of the other. In this case it is convenient to redefine $$ b^{(\mu)}_{(D)} \rightarrow d^{\mu}, c^{(\mu)}_{(D)} \rightarrow e^{\mu}, b^{(\mu)}_{(B)} \rightarrow f^{\mu}, c^{(\mu)}_{(B)} \rightarrow g^{\mu}, b^{(\mu)}_{(A)} \rightarrow b^{\mu}, c^{(\mu)}_{(A)} \rightarrow c^{\mu} $$ and to obtain an equation of the following type: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \sum_{(\zeta_{1},\zeta_{2},\zeta_{3})} \left( d^{\zeta_{1}} e^{\zeta_{2}} + d^{\zeta_{2}} e^{\zeta_{1}} \right) \left( f^{\rho_{1}} g^{\rho_{2}} + f^{\rho_{2}} g^{\rho_{1}} \right) \left( b^{\zeta_{3}} c^{\rho_{3}} + b^{\rho_{3}} c^{\zeta_{3}} \right) t = 0 \label{II2-compact} \end{equation} where $ t \in \tilde{\cal H}_{r,r,t,t,k,k}; $ here $ r_{D} = r, r_{B} = t, r_{A} = k. $ Moreover, the tensor $t$ must satisfy the constraints (\ref{B}) and (\ref{Z}). It is convenient to define \begin{equation} t_{1} \equiv \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left( f^{\rho_{1}} g^{\rho_{2}} + f^{\rho_{2}} g^{\rho_{1}} \right) b^{\rho_{3}} t \label{tensor-X} \end{equation} and \begin{equation} t_{2} \equiv \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left( f^{\rho_{1}} g^{\rho_{2}} + f^{\rho_{2}} g^{\rho_{1}} \right) c^{\rho_{3}} t. \label{tensor-Y} \end{equation} Let us note that $ t_{1} \in \tilde{\cal H}_{r,r,t-1,t-1,k-1,k} $ and $ t_{2} \in \tilde{\cal H}_{r,r,t-1,t-1,k,k-1}. $ Then we can rewrite (\ref{II2-compact}) in equivalent way if we use lemma \ref{permutation-lemma} and the constraint (\ref{B}), namely: \begin{equation} (r+2) (c^{\mu} t_{1} + b^{\mu} t_{2}) = d^{\mu} (A t_{1} + {\cal D} t_{2}) + c^{\mu} (C^{*} t_{1} + {\cal E} t_{2}). \label{II2-XY} \end{equation} We can formulate now the central result of this Section: \begin{prop} The equation (\ref{II2-XY}) above together with the constrains (\ref{B}) and (\ref{Z}) have only the trivial solution $ t_{1} = 0. $ \end{prop} {\bf Proof:} The proof is extremely tedious and we give only a minimal number of details. (i) We apply to the equation (\ref{II2-XY}) the operator $ c^{*}_{\mu}. $ Let us define the following operators \begin{equation} Q \equiv A^{*} {\cal D} + C {\cal E} \label{Q} \end{equation} and \begin{equation} R \equiv M + Q Z^{*}. \label{R} \end{equation} Then one can write the result in a very simple form: \begin{equation} R t_{1} = (r+2) (k+1) t_{1} \label{RX} \end{equation} so one must investigate this eigenvalue problem. (ii) We denote by $h'$ the subspace of $ \tilde{\cal H}_{r,r,t-1,t-1,k-1,k} $ defined by $ Ker(B) \cap Ker(B^{*}) \cap Ker(Z) $ and note that $ t_{1} \in h'. $ Next one shows by simple computations that the operator $R$ commutes with the operators $ B, B^{*}, Z $ so it makes sense to restrict it to the subspace $h'.$ The same assertion is true with respect to the operator $R^{*}. $ We will denote these restrictions by $R'$ and $R'^{*}$ respectively. From this observations it follows in particular that the operator $ R - R^{*} $ leaves the subspace $h'$ invariant. But one computes that \begin{equation} R - R^{*} = Q Z^{*} - Z Q^{*} = Z^{*} Q - Q^{*} Z \label{RQZ} \end{equation} and from the last equality it easily follows that \begin{equation} (t_{1}, (R - R^{*}) t_{1}') = 0 \quad \forall t_{1}, t_{1}' \in h'. \end{equation} Because $ R - R^{*} $ leaves the subspace $h'$ invariant we conclude that \begin{equation} R' - R'^{*} = 0. \end{equation} Combining with (\ref{RQZ}) one discovers that: \begin{equation} (Q Z^{*} - Z Q^{*}) t_{1} = 0 \quad \forall t_{1} \in h'. \end{equation} Applying to this relation the operator $Z$ produces after some computations: \begin{equation} Q t_{1} = 0. \label{QX} \end{equation} Let us introduce the following notation: \begin{equation} N \equiv {\cal D}^{*} {\cal D} + {\cal E}^{*} {\cal E}. \label{N} \end{equation} Then, taking into account (\ref{QX}) one can obtain from (\ref{RX}) the following relation: \begin{equation} (2M - N) t_{1} = (r+2) (k+1) t_{1}. \label{MN} \end{equation} (iii) In the same way one can obtain similar results concerning the tensor $t_{2}$. More precisely one denotes by $h''$ the subspace of $ \tilde{\cal H}_{r,r,t-1,t-1,k,k-1} $ defined by $ Ker(B) \cap Ker(B^{*}) \cap Ker(Z^{*}) $ and notes that $ t_{2} \in h''. $ Then one shows that similarly to (\ref{QX}) and (\ref{MN}) one has: \begin{equation} Q^{*} t_{2} = 0 \label{Q-star} \end{equation} and \begin{equation} (2N - M) t_{2} = (r+2) (k+1) t_{2}. \label{NM} \end{equation} (iv) The relation (\ref{QX}) suggests to investigate if it is possible to restrict the operator $ 2M - N $ to the subspace $ h_{Q} \equiv h' \cap Ker(Q). $ One shows by elementary but rather long computations that this assertion is true for the operators $M, N$ so it makes sense to define by $ M_{Q}, N_{Q} $ the corresponding restrictions. So, indeed the operator $ 2M - N $ leaves the subspace $h_{Q}$ invariant. (v) Moreover one computes the commutator of $M$ and $N$ and shows that: \begin{equation} [M_{Q}, N_{Q}] = 0. \end{equation} It follows that the operators $ M_{Q}, N_{Q} $ can be simultaneously diagonalized. Let us suppose that the equation (\ref{MN}) can have a non-trivial solution. Then it is easy to show that there must exists at least one non-trivial solution $t_{1}$ of this equation such that $t_{1}$ is a simultaneous eigenvalue of the operators $M, N.$ Explicitly: \begin{equation} M t_{1} = \lambda t_{1}, \quad N t_{1} = \mu t_{1} \label{MNX} \end{equation} and \begin{equation} 2 \lambda - \mu = (k+1) (r+2). \end{equation} (vi) Let us notice now that the relation (\ref{Q-star}) can be written as follows: \begin{equation} Q^{*} Z^{*} t_{1} = 0. \end{equation} One applies to this relation, first the operator $Z$ and next the operator $Q$; after some computations one gets: \begin{equation} (M - N)^{2} t_{1} = [Q, Q^{*}] t_{1}. \end{equation} One evaluates explicitly the commutator and uses (\ref{MNX}) to obtain the following restriction on the eigenvalue $ \lambda $: \begin{equation} (\lambda - \lambda_{0})^{2} + (r-k+2) (\lambda - \lambda_{0}) + \lambda = 0 \label{lambda} \end{equation} where we have denoted for simplicity \begin{equation} \lambda_{0} \equiv (k+2) (r+1) \end{equation} i.e. the eigenvalue appearing in the right hand side of the equation (\ref{MN}). Now it is easy to prove that the solutions of the equation (\ref{lambda}) (if they exist at all) must be greater than $ k (r+2). $ But this conflicts with Proposition \ref{spectrum-M} (see formula (\ref{S})) so we conclude that the equation (\ref{MN}) have only the trivial solution in the subspace $ h_{Q}. $ This finishes the proof of the similar assertion from the statement. $\vrule height 6pt width 6pt depth 6pt$ Remembering the definition of the tensor $t_{1}$ we just have find out that we have \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left( f^{\rho_{1}} g^{\rho_{2}} + f^{\rho_{2}} g^{\rho_{1}} \right) b^{\rho_{3}} \tilde{\cal L}^{...} = 0. \label{fgb} \end{equation} Applying $ Z^{*} $ one finds out that also \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left( f^{\rho_{1}} g^{\rho_{2}} + f^{\rho_{2}} g^{\rho_{1}} \right) c^{\rho_{3}} \tilde{\cal L}^{...} = 0. \label{fgc} \end{equation} In particular these relations make the starting point -relation (\ref{II2-compact})- identically satisfied. Because the indices $A, B, D$ in the relation (\ref{II2-Fock}) are arbitrary we have proved in fact that this relation is equivalent to the following two relations: \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left[ b^{(\rho_{1})}_{(B)} c^{(\rho_{2})}_{(B)} + b^{(\rho_{2})}_{(B)} c^{(\rho_{1})}_{(B)} \right] b^{(\rho_{3})}_{(A)} \tilde{\cal L}^{...} = 0 \label{II-final1} \end{equation} and \begin{equation} \sum_{(\rho_{1},\rho_{2},\rho_{3})} \left[ b^{(\rho_{1})}_{(B)} c^{(\rho_{2})}_{(B)} + b^{(\rho_{2})}_{(B)} c^{(\rho_{1})}_{(B)} \right] c^{(\rho_{3})}_{(A)} \tilde{\cal L}^{...} = 0 \label{II-final2} \end{equation} for any $ A \not= B. $ It is easy to see that the preceding two relations make (\ref{II1-Fock}) and (\ref{II2-Fock}) identities. Now, starting for instance from (\ref{II-final1}) one can use recurrsively the redefinition trick from the preceding Section to show the $ \tilde{\cal L}^{...} $ can be changed such that it verifies: \begin{equation} \left[ b^{(\mu)}_{(1)} b^{(\nu)}_{(B)} + b^{(\nu)}_{(1)} b^{(\mu)}_{(B)} \right] \tilde{\cal L}^{...} = 0 \end{equation} for $ B = 2,3,...,N. $ The preceding relation immediately implies the relation (\ref{antisymmetry}) and accordingly theorem \ref{structureN}. This finishes the proof of the main theorem. \section{Main Theorem} 6.1 In this Section we will exhibit the result of theorem \ref{structureN} in a more compact way. To do this we introduce the second-order hyper-Jacobians in full generality: \begin{equation} J^{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}}_{\nu_{1},...,\nu_{k}} \equiv \varepsilon^{\mu_{1},...,\mu_{n}} \prod_{i=1}^{k} \psi^{A_{i}}_{\mu_{i}\nu_{i}} \quad (k = 0,...,n). \label{hJ} \end{equation} One notices that these expressions are the natural generalization of the expressions (\ref{hyperJ}) defined in Section 3 and that they have properties of the same kind; namely a antisymmetry property: \begin{equation} J^{A_{P(1)},...,A_{P(k)};\mu_{Q(k+1)},...,\mu_{Q(n)}}_{\nu_{P(1)},..., \nu_{P(k)}} = (-1)^{|P|+|Q|} J^{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}}_{\nu_{1},...,\nu_{k}} \quad (k = 0,...,n) \label{antisym-hJ} \end{equation} (where $P$ is a permutation of the numbers $1,...,k$ and $Q$ is a permutations of the numbers $k+1,...,n$) and a tracelessness property (see \cite{O}): \begin{equation} J^{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n-1},\zeta}_{\nu_{1},..., \nu_{r-1},\zeta} = 0. \quad (k = 1,...,n-1). \label{trace-hJ} \end{equation} The relations (\ref{antisym}) and (\ref{trace}) are particular cases of these relations (for the case of the scalar field). Then we have \begin{thm} The most general solution of the equations (\ref{II1}) and (\ref{II2}) is of the following form: \begin{equation} {\cal L} = \sum_{k=0}^{n} {\cal L}^{\nu_{1},...,\nu_{k}}_{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}} J^{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}}_{\nu_{1},...,\nu_{k}} \label{polyn-hJ} \end{equation} where the functions $ {\cal L}^{...}_{...} $ are independent of $ \psi_{\mu\nu}^{B} $: \begin{equation} \partial^{\mu\nu}_{B} {\cal L}^{\nu_{1},...,\nu_{k}}_{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}} = 0 \quad (k = 0,...,n), \end{equation} and have analogous properties as the hyper-Jacobians, namely the (anti)symmetry property: \begin{equation} {\cal L}_{A_{P(1)},...,A_{P(k)};\mu_{Q(k+1)},...,\mu_{Q(n)}}^{\nu_{P(1)},..., \nu_{P(k)}} = (-1)^{|P|+|Q|} {\cal L}_{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n}}^{\nu_{1},...,\nu_{k}} \quad (k = 0,...,n) \label{antisym-l-hJ} \end{equation} (where $P$ is a permutation of the numbers $1,...,k$ and $Q$ is a permutations of the numbers $k+1,...,n$) and also verify the identities: \begin{equation} {\cal L}_{A_{1},...,A_{k};\mu_{k+1},...,\mu_{n-1},\zeta}^{\nu_{1},..., \nu_{k-1},\zeta} = 0. \label{trace-l-hJ} \end{equation} (i. e. are traceless). The function coefficients $ {\cal L}^{...}_{...} $ are uniquely determined by $ {\cal L} $ and the properties (\ref{antisym-l-hJ}) and (\ref{trace-l-hJ}) above. \label{structure-hJ} \end{thm} {\bf Proof:} For $ N = 1 $ this result coincides with theorem \ref{structure}. For $ N \geq 2 $ we will prove it using the results from the preceding two sections. Namely, we will show that the expression (\ref{polynN-compact-dual}) can be rearranged such that it coincides with (\ref{polyn-hJ}) above. In fact, it is easier to start with (\ref{polyn-hJ}) and to obtain the previous expression (\ref{polynN-compact-dual}). To this purpose we will make $ N \rightarrow N+1 $ and suppose that the indices $ A, B, ... $ run from $0$ to $N$; the indices $ a, b, ... $ will run from $1$ to $N$. We separate in the expression (\ref{polyn-hJ}) the contributions in which $ n - r $ indices take values from $1$ to $N$ and the rest take the value $0$. One obtains \begin{equation} {\cal L} = \sum_{k=0}^{n} \sum_{r=k}^{n} C^{n-r}_{n-k} {\cal L}^{\mu_{k+1},...,\mu_{n}}_{0,...,0,a_{r+1},...,a_{k}; \nu_{1},...,\nu_{k}} J^{0,...,0,a_{r+1},...,a_{k};\nu_{1},...,\mu_{k}}_{\mu_{k+1},...,\mu_{n}} \label{polyn-doublesum} \end{equation} where it is understood that there are $ r - k $ entries equal to $0$. One can rearrange this expression as follows: \begin{equation} {\cal L} = \sum_{k=0}^{n} l^{\mu_{k+1},...,\mu_{n}}_{a_{k+1},...,a_{k};\nu_{1},...,\nu_{k}} J^{a_{k+1},...,a_{n};\nu_{1},...,\nu_{k}}_{\mu_{k+1},...,\mu_{n}} \label{polyn-a} \end{equation} where \begin{equation} l^{\mu_{k+1},...,\nu_{n}}_{a_{k+1},...,a_{n};\nu_{1},...,\nu_{k}} \equiv \sum_{r=0}^{k} C^{n-r}_{n-k} {\cal A}_{\nu_{1},...,\nu_{k}} \left({\cal L}^{\mu_{r+1},...,\mu_{n}}_{0,...,0,a_{k+1},...,a_{k}; \nu_{1},...,\nu_{r}} \prod_{l=r+1}^{k} \psi^{0}_{\mu_{l}\nu_{l}} \right) \end{equation} where $ {\cal A}_{\nu_{1},,...,\nu_{k}} $ is the projector which antisymmetrizes in the indices $ \nu_{1},...,\nu_{k} $ and there are $k-r$ entries equal to $0$. One defines the dual tensor $ \tilde{l}^{...}_{...} $ by analogy to (\ref{dual}) and discovers after some combinatorics that it is given by the following relation: \begin{equation} \tilde{l}^{\mu_{k+1},...,\nu_{n};\nu_{k+1},...,\nu_{n}}_{a_{k+1},...,a_{n}} = \sum_{s=0}^{k} L^{\mu_{k+1},...,\nu_{n};\nu_{k+1},...,\nu_{n}; \sigma_{1},...,\sigma_{s}}_{a_{k+1},...,a_{n};\rho_{s+1},...,\rho_{n}} J^{(0)\rho_{s+1},...,\rho_{n}}_{\sigma_{1},...,\sigma_{s}} \end{equation} where \begin{equation} L^{\mu_{k+1},...,\mu_{n};\nu_{k+1},...,\nu_{n}; \sigma_{1},...,\sigma_{s}}_{a_{k+1},...,a_{n};\rho_{s+1},...,\rho_{n}} \equiv {\rm const.} {\cal A}_{\rho_{s+1},...,\rho_{n}} \left( {\cal L}^{\mu_{k+1},...,\nu_{n};\nu_{k+1},...,\nu_{n}; \sigma_{1},...,\sigma_{s},\mu_{k+1},...,\mu_{n}}_{0,...,0, a_{k+1},...,a_{n};\rho_{s+1},...,\rho_{k}} \prod_{l=k+1}^{n} \delta^{\nu_{l}}_{\rho_{l}} \right) \end{equation} and we have $s$ entries equal to $0$. If one defines the dual tensor $ \tilde{L}^{....} $ as in (\ref{dual}), one finally finds out that \begin{equation} L^{\mu_{k+1},...,\nu_{n};\nu_{k+1},...,\nu_{n}; \sigma_{1},...,\sigma_{s}}_{a_{k+1},...,a_{n};\rho_{s+1},...,\rho_{n}} = {\rm const} \tilde{\cal L}^{\sigma_{1},...,\sigma_{s}\mu_{k+1},...,\mu_{n}; \rho_{1},...,\rho_{s}\nu_{k+1},...,\nu_{n}}_{0,...,0,a_{k+1},...,a_{n}} \end{equation} and we have $s$ entries equal to $0$. So, finally, after some relabelling we get the following expression from (\ref{polyn-hJ}): \begin{equation} {\cal L} = \sum_{k=0}^{n} \sum_{r=0}^{n-k} {\rm const.} \tilde{\cal L} ^{\sigma_{1},...,\sigma_{r}\mu_{1},...,\mu_{k}; \rho_{1},...,\rho_{r},\nu_{1},...,\nu_{k}}_{0,...,0,a_{1},...,a_{k}} \tilde{J}^{a_{1},...,a_{k}}_{\mu_{1},...,\mu_{k};\nu_{1},...,\nu_{k}} \tilde{J}^{(0)}_{\rho_{1},...,\rho_{r};\sigma_{1},...,\sigma_{r}} \end{equation} It is clear that we can iterate the procedure with the indices $1,...,N$ and we will obtain finally the expression (\ref{polynN-compact-dual}). $\vrule height 6pt width 6pt depth 6pt$ 6.2 It is interesting to define the so-called {\it horizontalisation} operation ${\bf h}$ on the space on differential form on the jet-bundle space (see for instance \cite{K4}). In particular, it is defined by linearity, multiplicity and: \begin{equation} {\bf h} dx^{\mu} \equiv dx^{\mu}, \quad {\bf h} d \psi^{A} \equiv \psi^{A}_{\mu} dx^{\mu}, \quad {\bf h} d \psi^{A}_{\mu} \equiv \psi^{A}_{\mu\nu} d x^{\nu}, \quad {\bf h} d \psi^{A}_{\mu\nu} \equiv \psi^{A}_{\mu\nu\rho} d x^{\rho}. \end{equation} Let us now define the differential form \begin{equation} \Lambda \equiv \sum_{k=0}^{n} (-1)^{k(n+1)} {\cal L}^{\mu_{k+1},...,\mu_{n}}_{A_{k+1},...,A_{n};\nu_{1},...,\nu_{k}} d \psi^{A_{k+1}}_{\mu_{k+1}} \wedge \cdots \wedge d \psi^{A_{n}}_{\mu_{n}} \wedge d x^{\nu_{1}} \wedge \cdots d x^{\nu_{k}}. \end{equation} Then it is elementary to prove \begin{prop} The Euler-Lagrange form associated to the trivial Lagrangian (\ref{polyn-hJ}) is given by \begin{equation} L = {\bf h} \Lambda. \label{hor} \end{equation} \end{prop} In other words, any second-order trivial Lagrangian can be obtained as a result of the horizontalisation operation. \begin{rem} It is elementary to prove the same assertion for the first-order trivial Lagrangians. In fact, relation (\ref{hor}) stands true if instead of the form $\Lambda$ above one uses the expression (\ref{lam}). \end{rem} 6.3 We still did not exploited the other two conditions of triviality, namely (\ref{II3}) and (\ref{II4}). This analysis of these relations proves to be not very conclusive. Indeed it is very hard to obtain some relations involving directly the coefficients $ {\cal L}^{...}_{...} $ from (\ref{polyn-hJ}). The best we could do was to re-express this formula as \begin{equation} {\cal L} = \sum_{k=0}^{n} {1 \over k!} L^{\mu_{1},...,\mu_{k};\nu_{1},...,\nu_{k}}_{A_{1},...,A_{k}} \prod_{i=1}^{k} \psi^{A_{i}}_{\mu_{i}\nu_{i}} \end{equation} where the functions $ L^{...}_{...} $ are completely symmetric in the couples $ (A_{l},\mu_{l},\nu_{l}) $ and are invariant at the interchanges $ \mu_{l} \leftrightarrow \nu_{l} \quad (l = 0,...,n). $ Then one can obtain from (\ref{II3}) and (\ref{II4}) some rather complicated differential equations on the functions $ L^{...}_{...}. $ These equations seems to be rather hard to analyse. In particular it is not clear if a polynomial structure in the first-order derivatives can be established, although this seems not very plausible. \section{Final Comments} It is quite reasonable to expect that in the same way one can prove a formula of the type (\ref{polynN-compact}) for the case of a trivial Lagrangian of arbitrary order. In the $p$-th order case one must have hyper-Jacobians (\ref{dual-hyperJN}) with $p$ groups of indices of the type $ \rho_{1},...,\rho_{r}. $ One must also expect that a trivial Lagrangian, more precisely its dependence on the highest order derivatives, is also a result of the horizontalisation operation. These problems deserve some investigation and we intend to do this in future. \vskip 1cm

proofpile-arXiv_065-541

\section{Introduction} One of the most interesting aspects of electron transport in submicron scale devices is the interplay between quantum coherence and aperiodic but reproducible conductance fluctuations. Over the past decade the phenomenon of universal conductance fluctuations in disordered systems (where impurity scattering dominates) has been understood through the use of stochastic models. More recently, a new generation of experiments \cite{Marcus92} was designed to measure conductance fluctuations in the ballistic regime where the dynamics of the electrons in the device is determined by the geometry of its boundary. The stochastic approach to these systems is justified by the underlying classical chaotic dynamics. This situation is distinct from the diffusive case, where the corresponding classical limit of the quantum problem is not fully understood. In this paper we discuss the conductance fluctuations in quantum dots. \cite{Kastner92,Meirav90,Foxman93,Foxman94} These are semiconductor devices in which the electrons are confined to a two-dimensional region whose typical linear dimension is in the submicron range. In particular we are interested in the Coulomb blockade regime where the leads are weakly coupled to the dot, either because the leads are very narrow, or due to the presence of potential barriers at the lead-dot interface. \cite{Kastner92} The electrons inside the dot are characterized by isolated resonances whose width is smaller than their average spacing, and conductance occurs through resonant tunneling. As a consequence, the conductance peaks when the Fermi energy matches a resonance energy of the electrons inside the dot and an additional electron tunnels into the dot. Such a system resembles the compound nucleus in its region of isolated resonances. \cite{Porter65} The macroscopic charging energy required to add an electron to a dot is determined by its capacitance $C$ and is given by $e^2/C$. Since $C$ is a constant which is determined essentially by the geometry of the dot, the conductance exhibits equally spaced oscillations as a function of the gate voltage (or Fermi energy). At low temperatures $\Gamma \ll kT < \Delta$ the width of the conductance peaks is $\sim kT$, but the heights exhibit order of magnitude variations. \cite{Meirav90,Foxman93,Foxman94} When the electron-impurity mean free path is larger than the size of the dot, the classical dynamics of the electron inside the dot is determined by the scattering from the dot's boundary. Due to small irregularities in the dot's shape, the electron displays chaotic motion, and its quantum transport through the dot can be described by statistical $S$-matrix theory.\cite{Lewenkopf91} Since the Coulomb blockade regime is dominated by resonances, the conductance peaks can be used to probe the chaoticity of the underlying resonance wavefunctions. A statistical theory of the conductance peaks was originally developed in Ref. \onlinecite{Jalabert92}. By using $R$-matrix theory, \cite{Wigner47,Lane58} the conductance peak amplitude was expressed in terms of the electronic resonance wavefunction across the contact region between the dot and the leads. When the dynamics of the electron inside the dot is chaotic, the fluctuations of the wavefunction inside the dot are assumed to be well described by random matrix theory (RMT). In Ref.\onlinecite{Jalabert92} the conductance distribution was derived in closed form for one-channel leads. These results were rederived in Ref. \onlinecite{Prigodin93}, and later extended to the case of two-channel leads in the absence of time-reversal symmetry \cite{Mucciolo95} through the use of the supersymmetry technique.\cite{Efetov83} However, the calculations required by this technique become too complicated to apply in the general case of any number of possibly correlated and/or non-equivalent channels. The conductance distributions for one-channel leads were recently measured \cite{Chang96,Folk96} and found to be in agreement with theory for both cases of conserved and broken time-reversal symmetry. This indicates that the dephasing effect, which plays an important role in open dots, \cite{Chan95,Baranger95} is of little importance for closed dots. In this paper we discuss in detail the width and conductance peak distributions for leads with any number of channels that are in general correlated and non-equivalent. Exact closed expressions for these distributions are derived for both cases of conserved and broken time-reversal symmetry. \cite{Alhassid95} We find that these distributions are entirely characterized by the eigenvalues of the channel correlation matrices $M^l$ and $M^r$ in the left and right leads, respectively. The strength of our approach is in its simplicity, since it relies solely on standard RMT techniques. To test our predictions we compare our analytical findings to numerical simulations of a chaotic dynamical model, the conformal billiard. \cite{Robnik83} Statistical width and conductance distributions of one-channels leads were recently studied in detail in this model. \cite{Bruus94} Although our paper deals mainly with ballistic dots whose classical dynamics is chaotic, our results should also be valid in the diffusive regime of weakly disordered dots, where random matrix theory is applicable. We note that under certain conditions the partial width is analogous to the wavefunction intensity at a given point. Therefore our width distributions can also be tested by microwave cavity experiments,\cite{Alt95,Sridhar91} where the intensities are measured at several points that are spatially correlated. The plan of the paper is as follows: In Section II we briefly review the conductance in quantum dots in their Coulomb blockade regime. In Section III we discuss the statistical model and derive analytic results for the partial and total width distributions in each lead, for the channel correlation matrix and for the conductance distribution. We investigate the variation of these distributions as a function of the number of channels and their sensitivity to the degree of correlations between them. Those findings are compared in Section IV with numerical results obtained for the conformal billiard. Finally, in Section V we discuss the validity of our assumptions in the the context of typical experiments. \section{Conductance in Quantum Dots} In this section we briefly review the formalism and introduce the notation used throughout this paper. In particular, we express the conductance peak heights in terms of the channel and resonance wavefunctions of the dot. For $\Gamma \ll kT \ll \Delta$, which is typical of many experiments, \cite{Meirav90} the observed on resonance conductance peak amplitude is given by \cite{Beenakker91,Zirnbauer93} \begin{eqnarray} \label{condpeak} G_\lambda =\frac{e^2}{h}\, \frac{\pi}{2 kT} \, g_\lambda \qquad \mbox{with} \qquad g_\lambda =\frac{\Gamma^l_\lambda \Gamma_\lambda^{r}} {\Gamma_\lambda^{l}+\Gamma_\lambda^{r}} \;, \end{eqnarray} where $\Gamma_\lambda^{l(r)}$ is the partial decay width of the resonance $\lambda$ into the left (right) lead. Since each lead can support several open channels we have $\Gamma_\lambda^{l(r)}=\sum_c\Gamma_{c\lambda}^{l(r)}$, where $\Gamma_{c\lambda}^{l(r)}$ is the partial width to decay into channel $c$ in the left (right) lead. In the $R$-matrix formalism, \cite{Lane58} the partial widths are related to the resonance wavefunction inside the dot. More specifically, introducing the partial amplitudes $\gamma_{c\lambda}$, such that $\Gamma_{c\lambda}= |\gamma_{c\lambda}|^2$, one can write \begin{eqnarray} \label{gamdefR} \gamma_{c\lambda} = \sqrt{\frac{\hbar^2 k_c P_c}{m} } \int \! dS\,\Phi_c^*(\bbox{r}) \Psi_\lambda(\bbox{r}) \;. \end{eqnarray} Here $\Psi_\lambda(\bbox{r})$ is the $\lambda$--th resonance wavefunction in the dot, $\Phi_c(\bbox{r})$ is the transverse wavefunction in the lead that corresponds to an open channel $c$, and the integral is taken over the contact area between the lead and the dot. $k_c$ and $P_c$ are the longitudinal wavenumber and penetration factor in channel $c$, respectively. Eq.~(\ref{gamdefR}) shows that the contributions to the partial width amplitude from the internal and external regions of the dot factorize. The information from the region external to the dot is contained in $k_c$ and $P_c$. These quantities are determined by the wave dynamics in the leads and are non-universal. They affect the average widths and enter explicitly in the correlation matrix $M$. However, the fluctuation properties of the conductance are generic and depend only on the statistical properties of the electronic wavefunction at the dot-lead boundary inside the barrier region. A different physical modeling of a quantum dot assumes point-like contacts and each lead is composed of several such point contacts. \cite{Prigodin93,Mucciolo95} In this model the conductance is also given by (\ref{gamdefR}) with each point contact $\bbox{r}_c$ considered as one channel. The corresponding partial width is \cite{Prigodin93} \begin{eqnarray} \label{3} \gamma_{c\lambda} = \sqrt{\frac{ \alpha_c {\cal A} \Delta}{\pi}}\, \Psi_\lambda(\bbox{r}_c) \;, \end{eqnarray} where ${\cal A}$ is the area of the dot, $\Delta$ is the mean spacing and $\alpha_c$ is a dot-lead coupling parameter. Both models can be treated by our formalism. This becomes apparent after the following considerations. A resonance eigenfunction with eigenenergy $E=E_\lambda$ can be approximated by an expansion in a fixed basis $\rho_\mu$ of wavefunctions with the given energy $E$ inside the dot \begin{eqnarray} \label{4} \Psi_\lambda({\bf r})= \sum_\mu \psi_{\lambda\mu} \,\rho_\mu(\bbox{r}) \;. \end{eqnarray} The sum over $\mu$ is truncated at $N$ basis states, where $N$ is large and determined by precision requirements. The partial width in channel $c$ can then be expressed by the scalar product \begin{eqnarray} \label{5} \gamma_{c\lambda} = \langle \bbox{\phi}_c | \bbox{\psi}_\lambda \rangle \equiv \sum_\mu \phi_{c\mu}^* \psi_{\lambda\mu} \;, \end{eqnarray} where \begin{eqnarray} \label{5'} \phi_{c\mu} \equiv \sqrt{\frac{\hbar^2 k_c P_c}{m} } \int\! dS \, \Phi_c^*(\bbox{r})\, \rho_\mu (\bbox{r}) \end{eqnarray} for the extended leads model, and \begin{eqnarray} \label{5''} \phi_{c\mu} \equiv \sqrt{{ \alpha_c {\cal A} \Delta \over \pi}} \rho^*_\mu (\bbox{r}_c) \end{eqnarray} for the point contact model. Thus, we are led to similar formulations of both the extended leads and point-like contacts problems; in the corresponding $N$-dimensional space the partial width amplitudes of a level are simply the projections of its corresponding eigenstate vector $\bbox{\psi}_\lambda$ on the channel vectors $\bbox{\phi}_c$. The only difference between the two models is the explicit expression for the channel vector $\bbox{\phi}_c$. We note that the scalar product (\ref{5}) (that will be used throughout this paper) is different from the original scalar product defined in the spatial region extended by the dot. \section{Statistical Model} Due to the irregularity of the dot's shape, the motion of the electron inside the dot is expected to be chaotic. In Ref. \onlinecite{Jalabert92} we have developed a statistical theory of the conductance peaks by assuming that the vectors $\psi_\lambda$ that correspond to the resonance wavefunctions inside the dot have the same statistical properties as the eigenvectors of a random matrix ensemble. Here we study the limits of conserved time-reversal symmetry, corresponding to the Gaussian Orthogonal Ensemble (GOE), and of broken time-reversal symmetry, corresponding to the Gaussian Unitary Ensemble (GUE). The transition from one symmetry to another occurs when an external magnetic field is applied. The width distribution (or equivalently the wavefunction intensity distribution) was derived in the crossover regime between symmetries for the case of one channel leads only. \cite{Sommers94} \subsection{The Joint Distribution of Partial Width Amplitudes} In RMT the eigenvector $\bbox{\psi} \equiv (\psi_1, \psi_2, \ldots, \psi_N)$ (here and in the following we omit the eigenvector label $\lambda$) is distributed randomly \cite{Brody81} on a sphere $P(\bbox{\psi}) \propto \delta (\sum_{\mu=1}^N | \psi_\mu|^2-1)$. The joint distribution of the partial width amplitudes $\bbox{\gamma} = (\gamma_1, \gamma_2, \ldots, \gamma_\Lambda)$ for $\Lambda$ channels is then given by \begin{eqnarray} \label{6} P(\bbox{\gamma}) = \frac{\Gamma(\beta N/2)}{\pi^{\beta N/2}} \int \! D[\bbox{\psi}] \left[\prod_{c=1}^{\Lambda} \delta ( \gamma_c - \langle\bbox{\phi}_c|\bbox{\psi}\rangle) \right] \delta \!\left( \sum_{\mu=1}^N |\psi_\mu |^2 -1 \right) \;, \end{eqnarray} where $D[\bbox{\psi}] \equiv \prod_{\mu=1}^{N}{d\psi_\mu}$ for the GOE and $D[\bbox{\psi}] \equiv \prod_{\mu=1}^{N}{d\psi^*_\mu d\psi_\mu/2\pi i}$ for the GUE. To evaluate (\ref{6}) we transform the $\Lambda$ channels to a new set of orthonormal channels $\hat{\bbox{\phi}}_c$ \begin{eqnarray} \label{newbasis} \bbox{\phi}_c =\sum_{c^\prime} \hat{\bbox{\phi}}_{c^\prime} F_{c^\prime c} \qquad \mbox{with} \qquad \langle \hat{\bbox{\phi}}_c | \hat{\bbox{\phi}}_{c^\prime} \rangle= \delta_{cc^\prime} \;. \end{eqnarray} We then take advantage of the invariance of the corresponding Gaussian ensemble under an orthogonal (unitary) transformation to rotate the eigenvector $\bbox{\psi}$ such that its first $\Lambda$ components are along the new orthonormal channels. Denoting by ${\cal O}$ the orthogonal (unitary) matrix whose first $\Lambda$ rows are the orthonormal vectors $\hat{\phi}_c (c=1,\ldots, \Lambda)$, we change variables in (\ref{5}) to $\hat{\psi}_\mu =\sum_{\nu} {\cal O}_{\mu \nu} \psi_{\nu}$. Using $\hat{\psi}_c=\langle\hat{\bbox{\phi}}_c |\bbox{\psi}\rangle$ we find \begin{eqnarray} \label{8} P(\bbox{\gamma}) = \frac{\Gamma (\beta N/2)}{\pi^{\beta N/2} } \int \! && \left( \prod^\Lambda_{c=1} d \hat{\psi}_c \right) \left( \prod^N_{\mu=\Lambda+1} d \hat{\psi}_\mu \right) \left[ \prod^\Lambda_{c=1} \delta \left( \gamma_c - F_{c^\prime c} \hat{\psi}_{c^\prime} \right) \right] \nonumber \\ && \times \, \delta\!\left( \sum^\Lambda_{c=1} |\hat{\psi}_c|^2 + \sum^N_{\mu=\Lambda+1} |\hat{\psi}_\mu|^2 -1 \right) \;. \end{eqnarray} The integration over these first $\Lambda$ components is now easily done and gives \begin{eqnarray} \label{9} P(\bbox{\gamma}) = \frac{\Gamma(\beta N/2)}{\pi^{\beta N/2}\,|\det F|} \int\! D[\bbox{\hat\psi}]\,\delta \!\left(\bbox{\hat\gamma}^\dagger \bbox{\hat\gamma} + \sum_{\mu=\Lambda+1}^N | {\hat\psi}_\mu |^2 -1 \right) \;, \end{eqnarray} where $\hat{\gamma}_c \equiv \langle\hat{\bbox{\phi}}_c |\bbox{\psi}\rangle = \sum_{c^\prime} \gamma_{c^\prime} F_{c^\prime c}^{-1*}$ are the partial widths to decay to the new channels and the metric is as before but excluding the first $\Lambda$ components of $\bbox{\psi}$. Finally, the latter integral is easily done by introducing spherical coordinates in the $N-\Lambda$ dimensional space. We obtain \begin{eqnarray} \label{10} P(\bbox{\gamma}) = \frac{\Gamma ( \beta N/2)}{\pi^{\beta \Lambda/2} \Gamma \left(\beta (N-\Lambda)/2\right) |\mbox{det} F|}\, \left[ 1 - \bbox{\gamma}^\dagger (F^\dagger F)^{-1} \bbox{\gamma} \right] ^{\beta\frac{N-\Lambda}{2} -1} \;. \end{eqnarray} For $\Lambda \ll N$ and in the limit $N\rightarrow \infty$, we recover a simplified expression \begin{eqnarray} \label{11} P(\bbox{\gamma}) = (\det M)^{-\beta/2} \, \mbox{e}^{- \frac{\beta}{2} \bbox{\gamma}^\dagger M^{-1} \bbox{\gamma} }\;, \end{eqnarray} where the matrix $M \equiv (NF^\dagger F)^{-1}$ is just the metric defined by the original channels \begin{eqnarray} \label{12} M_{cc^\prime} = \frac{1}{N} \langle \bbox{\phi}_c |\bbox{\phi}_{c^\prime}\rangle \;. \end{eqnarray} The distribution (\ref{11}) is normalized with the measure $D[\bbox{\gamma}] \equiv \prod_{c=1}^{\Lambda}{d\gamma_c/2\pi}$ for the GOE and $D[\bbox{\gamma}]\equiv\prod_{c=1}^{\Lambda} {d\gamma^*_c d\gamma_c/2\pi i}$ for the GUE. Note that for both ensembles the joint partial width amplitudes distribution is Gaussian, the main difference being that the partial amplitudes are real for the GOE and complex for the GUE. Such a Gaussian distribution is also obtained by assuming that the distribution is form-invariant under an orthogonal (unitary) transformation. \cite{Krieger63} It follows from (\ref{11}) that the matrix $M$ is just the correlation matrix of the partial widths \begin{eqnarray} \label{13} M_{cc^\prime} = \overline{ \gamma_c^* \gamma_{c^\prime}} \;. \end{eqnarray} In general the channels are correlated and non-equivalent, {\sl i.e.} non-equal average partial widths. According to (\ref{12}) this is equivalent to assuming channels that are non-orthogonal and have non-equal norms. \subsection{The Channel Correlation Matrix $M$} We shall now derive explicit expressions for the correlation matrix $M$ in a chaotic quantum dot. Using Eq.~(\ref{12}) and the definition of the scalar product (\ref{5}) we find \begin{eqnarray} \label{14} M_{cc^\prime}=\frac{\hbar^2}{2m} \sqrt{k_c k_{c^\prime} P_c P_{c^\prime}} \int\!dS\!\int\! dS^\prime \,\Phi^*_c(\bbox{r}) \left[ \frac{1}{N} \sum_\mu \rho_\mu (\bbox{r}) \rho_\mu (\bbox{r}^\prime) \right] \Phi_c(\bbox{r}^\prime)\;. \end{eqnarray} We first discuss the case where there is no magnetic field so that the motion inside the dot is that of a free particle. Therefore, a resonance eigenstate inside the dot at energy $E =\hbar^2k^2/2m$ can be expanded in a basis of free particle states at the given energy $E$. Since RMT is applicable on a local energy scale, this is the fixed basis $\rho_\mu$ for which the eigenvector coefficients $\psi_\mu$ are distributed randomly (on the sphere). Using polar coordinates, such a basis of free waves is given by $\rho_\mu (\bbox{r}) \propto J_\mu(kr)\exp(i\mu\theta)$ with $\mu=0,\pm 1, \pm2,\ldots$, where $J_\mu$ are Bessel functions of the first kind. Denoting by $N$ the number of such waves on the energy shell, we find \begin{eqnarray} \label{15} \frac{1}{N} \sum_\mu \rho^*_\mu(\bbox{r}) \rho_\mu(\bbox{r}^\prime)= \frac{1}{{\cal A}}\sum_\mu J_\mu(kr) J_\mu(kr^\prime) {\rm e}^{i\mu(\theta^\prime - \theta)} = \frac{1}{{\cal A}}J_0(k| \bbox{r} - \bbox{r}^\prime |) \;, \end{eqnarray} where we have used the addition theorem for the Bessel functions. \cite{Lebedev72} A similar relation holds if we choose a plane waves basis $\rho_\mu(\bbox{r}) = {\cal A}^{-1/2} \exp(i\bbox{k}_\mu\cdot\bbox{r})$ at a fixed energy $\hbar^2k^2/2m$ but with random orientation of $\bbox{k}_\mu$ and use the integral representation of $J_0$. With help of Eq.~(\ref{15}) we obtain for the correlation matrix \begin{eqnarray} \label{16} M_{cc^\prime} = \frac{\hbar^2}{2m {\cal A}}\sqrt{k_c k_{c^\prime} P_c P_{c^\prime}} \int \! dS \! \int \! dS^\prime \, \Phi^*_c(\bbox{r}) J_0(k|\bbox{r} - \bbox{r}^\prime|) \Phi_c(\bbox{r}^\prime)\;, \end{eqnarray} for extended leads, while for the point contact model we find \begin{eqnarray} \label{17} M_{cc^\prime} = \frac{ \alpha_c \Delta}{\pi} J_0(k| \bbox{r}_c - \bbox{r}_c^\prime|)\;. \end{eqnarray} We remark that Eq.~(\ref{17}) is equivalent to $C(k|\Delta\bbox{r}|) \equiv \overline{\Psi^*( \bbox{r}) \Psi(\bbox{r}^\prime)}/\overline{|\Psi(\bbox{r})|^2} = J_0(k|\bbox{r}-\bbox{r}^\prime|)$. This result was first derived in Ref. \onlinecite{Berry77} based on the assumption that the Wigner function of a classically chaotic system is microcanonical on the energy surface, and recently studied extensively in the Africa billiard. \cite{Li94} However, in these references the average is taken for a fixed wavefunction over a local region around $(\bbox{r} + \bbox{r}^\prime)/2$. When an external magnetic field $B$ is present, the electronic classical underlying dynamics undergoes a transition from chaotic to integrable as the field gets stronger, regardless the shape of the billiard. In this paper, however, we only discuss the case of weak fields for which the motion is chaotic, and we are interested in the transition from orthogonal to unitary symmetry. While in the unitary case the wavefunctions become complex, the arguments that lead to Eq. (\ref{15}) are still valid and the wavefunction correlator $C(k|\Delta\bbox{r}|)$ is unchanged. The wavefunction correlation $C(k|\Delta\bbox{r}|)$ has been also derived for weakly disordered systems using the supersymmetry technique \cite{Prigodin95} in the unitary and orthogonal symmetries. In addition Ref. \onlinecite{Prigodin95} derives the joint probability distribution for the intensity of an eigenfunction at two different points. We remark that the joint distribution of the wavefunction amplitude at $\Lambda$ points $\bbox{r}_c$ is a special case of (\ref{11}) obtained for $\gamma_{c\lambda} \equiv \Psi_\lambda(\bbox{r}_c)$ (see the point contact case (\ref{5''}) except that the points $r_c$ can be chosen anywhere within the dot and not only on the boundary). We then obtain \begin{eqnarray} \label{17'} P(\Psi_\lambda(\bbox{r}_1), \Psi_\lambda(\bbox{r}_2), \ldots, \Psi_\lambda(\bbox{r}_\Lambda)) = (\det M)^{-\beta/2} \, \exp \left[ - \frac{\beta}{2} \sum\limits_{c c^\prime=1}^\Lambda \Psi^\ast_\lambda(\bbox{r}_c) \left(M^{-1}\right)_{c c^\prime} \Psi_\lambda(\bbox{r}_{c^\prime}) \right]\;, \end{eqnarray} where $M_{cc^\prime} = {\cal A}^{-1}J_0(k| \bbox{r}_c - \bbox{r}_c^\prime|)$. The distributions of Ref. \onlinecite{Prigodin95} are then easily obtained from (\ref{17'}) when $\Lambda=2$.\cite{Srednicki95} \subsection{Total Width Distribution} We calculate next the total width distribution $P(\Gamma)$ in a given lead that supports $\Lambda$ channels and is characterized by a correlation matrix $M$. Although this quantity is not directly measurable in experiments with quantum dots, it appears very often in resonant scattering by complex objects. \cite{Alt95,compnuc} We remark that for a dot with reflection symmetry $\Gamma^l=\Gamma^r \equiv \Gamma$ the conductance peak $g$ in (\ref{condpeak}) is proportional to $\Gamma$. Such dots are, however, difficult to fabricate. Using $\Gamma=\sum_c|\gamma_c |^2=\bbox{\gamma}^\dagger\bbox{\gamma}$, the characteristic function of $P(\Gamma)$ is given by \begin{eqnarray} \label{18} \widetilde{P}(t) = \int^{\infty}_{-\infty} d \Gamma \exp (it\Gamma) P(\Gamma) = \int^{\infty}_{-\infty} D \left[\bbox{\gamma} \right] \exp (it\bbox{\gamma}^\dagger\bbox{\gamma})\, P(\bbox{\gamma}) \;. \end{eqnarray} Since $P(\bbox{\gamma})$ is a Gaussian, we readily obtain $\widetilde{P} (t)=\left[\det (I -2iMt/\beta)\right]^{-\beta/2}$. The distribution itself is then given by an inverse Fourier transform \begin{eqnarray} \label{Pgamma} P(\Gamma) = \frac{1}{2 \pi} \int^{\infty}_{-\infty} dt \,\frac{e^{-i t \Gamma}}{\left[\det (I - 2 i t M/\beta)\right]^{\beta/2}} \; . \end{eqnarray} The matrix $M$ is Hermitean and positive definite (since $\bbox{x} ^\dagger M \bbox{x}= \overline{| \bbox{x}\cdot \bbox{\gamma}|^2} > 0$ for any $\bbox{x} \neq 0$) and therefore its eigenvalues $w_c^2$ are all positive. According to (\ref{Pgamma}), $P(\Gamma)$ depends only on $w_c^2$. This is a consequence of the invariance of $\Gamma$ under a orthogonal (unitary) transformation of the $\Lambda$ partial width amplitudes. We first discuss the simpler GUE case, for which the integrand has poles $-i/w_c^2$ along the negative imaginary axis. Taking a contour integration along the real line and a half-circle that encloses all the poles in the lower half of the plane, we can calculate (\ref{Pgamma}) by residues. Assuming that all eigenvalues of $M$ are non-degenerate, the poles are all simple and we find \begin{eqnarray} \label{PgamGUE} P_{GUE}(\Gamma) = \left(\prod_c\frac{1}{w_c^2}\right) \sum_{c=1}^\Lambda \left[\prod_{c^\prime \neq c} (\frac{1}{w_{c^\prime}^2} - \frac{1}{w_c^2})\right]^{-1} \mbox{e}^{-\Gamma/w_c^2} \;. \end{eqnarray} The distribution $P_{GUE}(\Gamma)$ given by (\ref{PgamGUE}) must be positive, which can be directly verified by using the concavity of the exponential function. For two channels ($\Lambda=2$) which are in general non-equivalent ($M_{11} \neq M_{22}$) and correlated ($M_{12}\neq 0$), the eigenvalues are given by $w_{1,2}^2 = (M_{11}+M_{22})/2 \pm \sqrt{\left((M_{11}-M_{22})/2\right)^2 + | M_{12}|^2}$. Then, Eq. (\ref{PgamGUE}) reduces to \begin{eqnarray} \label{21} P_{GUE}^{\Lambda=2}(\hat\Gamma) = \frac{2a_+}{\sqrt{a_-^2 + | f |^2}} \,\mbox{e}^{-2 a_+^2 \hat\Gamma/(1- |f|^2) } \sinh \left( \frac{2a_+ \sqrt{a_-^2 + |f|^2}} {1 -|f |^2 }\hat\Gamma \right)\;, \end{eqnarray} where ${\hat\Gamma}=\Gamma/\overline{\Gamma}$ is the width in units of its average value, $f=M_{12}/\sqrt{M_{11}M_{22}}$ measures the degree of correlation between the two channels and $a_\pm = 1/2 \left(\sqrt{M_{11}/M_{22}} \pm \sqrt{M_{22}/M_{11}}\right)$ are dimensionless parameters such that for equivalent channels $a_+=1$ and $a_-=0$. In the latter case, we reproduce the result of Ref. \onlinecite{Mucciolo95}. For degenerate eigenvalues, we can calculate (\ref{Pgamma}) by using the residue formula for higher order poles. Alternatively we can slightly break the degeneracy of the eigenvalues by $\eta$ and take the limit $\eta\rightarrow 0$. For example, for two channels Eq. (\ref{PgamGUE}) gives $P(\Gamma)= \left( \mbox{e}^{-\Gamma/w_1^2} - \mbox{e}^{-\Gamma/w_2^2} \right)/ (w_2^2-w_1^2)$. By taking $w_2^2=w_1^2 + \eta$, in the limit $\eta \rightarrow 0$ we recover \begin{eqnarray} \label{23} P(\Gamma) = \frac{\Gamma}{w^4} \, \mbox{e}^{-\Gamma/w^2} \;, \end{eqnarray} which is the $\chi^2$ distribution in four degrees of freedom. More generally, when all $\Lambda$ channels are uncorrelated and equivalent ($M=w^2 I$) we recover the well-known $\chi^2$ distribution in $2\Lambda$ degrees of freedom\cite{Porter65} \begin{eqnarray} \label{chi22L} P_{GUE}^{(0)}(\Gamma) = \frac{1}{w^{2\Lambda} (\Lambda-1) !} \Gamma^{\Lambda -1}\, \mbox{e}^{-\Gamma/w^2} \;. \end{eqnarray} We have denoted this limiting distribution in (\ref{chi22L}) by $P_{GUE}^{(0)}$ as it will serve as our reference distribution against which to compare the distributions in the general case of correlated and/or inequivalent channels. For the GOE case, the integral of Eq.(\ref{Pgamma}) is more difficult to evaluate since the singularities of the integrand along the negative imaginary axis $t=-i\tau$ are of the type $(\tau -1/2w_c^2)^{-1/2}$. In this case the semi-circle part of the contour (in the lower half of the plane) is deformed to go up and then down along the negative imaginary axis so as to exclude all the singularities. When going around a singularity of the above type the function changes sign. Therefore, after sorting the inverse eigenvalues of $M$ in ascending order $w_1^{-2}< w_2^{-2} < \ldots$, we have \begin{eqnarray} \label{PgamGOE} P_{GOE}(\Gamma) = \frac{1}{\pi 2^{\Lambda/2}} \left(\prod_c \frac{1}{w_c} \right) \sum_{m=1} \int_{1/2 w^2_{2m-1}}^{1/2 w^2_{2m}} d \tau \frac{\mbox{e}^{-\Gamma \tau}} {\sqrt{\prod_{r=1}^{2m-1} (\tau - \frac{1}{2 w^2_r}) \prod_{s=2m}^\Lambda (\frac{1}{2 w^2_{s}} - \tau)}} \;, \end{eqnarray} where for an odd number of channels $\Lambda$, we define $1/2 w^2_{\Lambda+1} \rightarrow \infty $. The integrand of each term on the r.h.s. of (\ref{PgamGOE}) is singular at the two endpoints of the integration interval, but this singularity is integrable. For the case of two channels that are in general non-equivalent but correlated, Eq. (\ref{PgamGOE}) reduces to \begin{eqnarray} \label{25} P^{\Lambda=2}_{GOE}(\hat\Gamma) = \frac{a_+}{ \sqrt{1-|f|^2}}\,\mbox{e}^{-{a_+^2\hat\Gamma}/(1-|f|^2)} \; I_0\!\left( \frac{a_+ \sqrt{a_-^2 + |f|^2}} {1 -|f |^2 }{\hat\Gamma} \right) \;, \end{eqnarray} where $f$ and $a_\pm$ are defined as before (see following Eq. (\ref{21})) and $I_0$ is the Bessel function of order zero. The case of equivalent channels is obtained in (\ref{25}) by substituting $a_+=1$ and $a_-=0$. The reference distribution $P_{GOE}^{(0)}$, defined as before for the case where all $\Lambda$ channels are equivalent and uncorrelated, is found directly from (\ref{Pgamma}) to be the $\chi^2$ distribution in $\Lambda$ degrees of freedom\cite{Alhassid86,Jalabert92} \begin{eqnarray} \label{chi2L} P_{GOE}^{(0)}(\Gamma) = \frac{1}{(2w^2) ^{\Lambda/2} (\Lambda/2 -1) !} \Gamma^{\Lambda/2 -1}\, \mbox{e}^{-\Gamma/2w^2} \;. \end{eqnarray} The top panels (a and b) in Fig. \ref{fig1} show the width distributions for a two-channels lead in the GOE statistics. The left panel is for equivalent channels ($M_{22}/M_{11}=1$) and for various degrees of correlations $f=0.25, 0.5,0.75$, and $0.95$. The right panel is for uncorrelated ($f=0$) but non-equivalent channels: $M_{22}/M_{11}= 2, 3, 4$, and $5$. The bottom panels (c and d) in Fig. \ref{fig1} are similar to (a and b) except that they correspond to the GUE case. We note that in all figures we display as $\Gamma$ the normalized total width $\Gamma/\overline{\Gamma}$. The correlation matrix in the point contact model is fully determined by $k|\Delta \bbox{r}|$ and the number of channels $\Lambda$. The left panels in Fig. \ref{fig2} show the GOE width distributions for $k|\Delta \bbox{r}|=0.25,1, 4$ and for different number of channels $\Lambda=2, 4$, and $6$. The right panels of Fig. \ref{fig2} show similar results but for the GUE statistics. The deviation of the width distribution from the reference distribution $P^{(0)}(\Gamma)$ which corresponds to equivalent and uncorrelated channels (dashed lines in Fig. \ref{fig2}) becomes larger as the number of channels increases for a given $k|\Delta \bbox{r}|$. \subsection{Conductance Peaks Distribution} To calculate the conductance distribution $P(g)$ in the general case, we assume that the left and right leads are far enough from each other and thus uncorrelated. \cite{corrections} The left and right leads are characterized by their own correlation matrix $M^{l}$ and $M^{r}$, respectively. Under this assumption \begin{eqnarray} \label{Pg} P(g) = \int\!d\Gamma^l \, d\Gamma^r \, \delta\!\left(g - \frac{\Gamma^l \Gamma^r}{\Gamma^l + \Gamma^r}\right) P(\Gamma^l)\,P(\Gamma^r) \;, \end{eqnarray} where $P(\Gamma)$ is given by (\ref{PgamGUE}) in the unitary case and by (\ref{PgamGOE}) in the orthogonal case. The distribution $P(g)$ can be evaluated by the following identity \begin{eqnarray} \label{29} \int_0^\infty d\Gamma_1\int_0^\infty d\Gamma_2\, &\mbox{e}^{-\Gamma_1/\delta_1} \mbox{e}^{-\Gamma_2/\delta_2} & \,\delta\!\left(g - \frac{\Gamma_1\Gamma_2}{\Gamma_1 + \Gamma_2}\right) \nonumber \\ = & 4g \,\mbox{e}^{-\left( {1\over \delta_1 } + {1\over\delta_2} \right) g} & \left[ K_0\left({2g\over \sqrt{\delta_1\delta_2}} \right) + {1\over 2} \left( \sqrt{{\delta_2\over\delta_1}} + \sqrt{{\delta_1\over\delta_2}}\right) K_1\left( {2g\over \sqrt{\delta_1\delta_2}} \right) \right] \;, \end{eqnarray} provided $\delta_1, \delta_2 > 0$. To obtain this identity we have used the integral representation of the Bessel function $K_\nu(z) = 1/2 (z/2)^\nu \int_0^\infty \! dt\,t^{-\nu-1} e^{-t-z^2/4t}$\,. For the unitary case, Eqs. (\ref{PgamGUE}) and (\ref{29}) give \begin{eqnarray} \label{PgGUE} P_{GUE}&&(g) = 16 g \left(\prod_c \frac{1}{v_c^2}\right) \left(\prod_d \frac{1}{w_d^2}\right) \sum_{c,d}^{\Lambda,\Lambda^\prime} \left[ \prod_{c^\prime \neq c}(\frac{1}{v_{c^\prime}^2}-\frac{1}{v_c^2}) \prod_{d^\prime \neq d}(\frac{1}{w_{d^\prime}^2}-\frac{1}{w_d^2}) \right]^{-1} \nonumber \\ && \times \,\mbox{e}^{-(\frac{1}{v_c^2} + \frac{1}{w_d^2})g} \left[ K_0\!\left(\frac{2g}{v_c w_d}\right) + \frac{1}{2}\left(\frac{ v_c }{ w_d}+\frac{ w_d}{ v_c} \right) K_1\!\left( \frac{2g}{v_c w_d} \right) \right] \;, \end{eqnarray} where $v^2_c$ and $w^2_d$ are the eigenvalues of the left and right lead correlation matrices $M^l$ and $M^r$, respectively. The previous published results\cite{Jalabert92,Prigodin93} are special cases of Eq.~(\ref{16}) for one channel leads with $\overline{\Gamma}^l = \overline{\Gamma}^r$ ({\sl i.e.} $v_1=w_1$), while the distribution of Ref. \onlinecite{Mucciolo95} is obtained for two (equivalent) channels leads whose matrices are related by an overall asymmetry factor $M^r=a M^l$. A similar calculation for the orthogonal limit gives \begin{eqnarray} \label{PgGOE} P_{GOE}&&(g) = \frac{4g}{\pi^2 2^\Lambda} \left(\prod_c \frac{1}{v_c}\right) \left(\prod_d \frac{1}{w_d}\right) \sum_{m,m^\prime} \int_{1/2 v^2_{2m-1} }^{1/2 v^2_{2m} } d\tau \int_{1/2 w^2_{2m^\prime-1}}^{1/2 w^2_{2m^\prime}} d\tau^\prime \,\mbox{e}^{-(\tau - \tau^\prime) g} \nonumber \\ & & \times\frac{ K_0 (2g \sqrt{\tau\tau^\prime}) + \frac{1}{2}\left( \sqrt{\frac{\tau}{\tau^\prime}} + \sqrt{\frac{\tau^\prime}{\tau}} \right) K_1 (2g \sqrt{\tau\tau^\prime}) } {\left[\prod_{r=1}^{2m-1} \left(\tau - \frac{1}{2 v^2_r}\right) \prod_{s=2m}^\Lambda\left(\frac{1}{2 v^2_s} - \tau\right) \prod_{r^\prime=1}^{2m^\prime-1} \left(\tau^\prime-\frac{1}{2 w^2_{r^\prime}}\right) \prod_{s^\prime=2m^\prime}^\Lambda \left(\frac{1}{2 w^2_{s^\prime}}-\tau^\prime\right) \right]^{1/2}}\;. \end{eqnarray} Fig. \ref{fig3} shows the GOE (left) and GUE (right) conductance peak distribution (\ref{PgGOE}) and (\ref{PgGUE}), respectively, for symmetric $\Lambda$-point leads with $k|\Delta \bbox{r}|=0.25, 1, 4$ and for $\Lambda=2, 4$, and 6 (the same cases shown in Fig. \ref{fig2}). In analogy to $P(\Gamma)$, all figures depicting $P(g)$ display the normalized conductance $g$ defined as $g/\overline{g}$. By comparing Fig. \ref{fig3} with Fig. \ref{fig2} we conclude that, as $\Lambda$ increases, the conductance distribution shows stronger deviation from its limiting case of uncorrelated equivalent channels (dashed lines) than the width distribution does. Fig. \ref{fig4} shows the case of asymmetric leads for the asymmetry factor $a=1$ and 10, for four-point leads with $k|\Delta \bbox{r}| = 1$ and for the orthogonal (a) and the unitary (b) limits. $P(g)$ is not very sensitive to the leads asymmetry and a large value of $a$ is needed to see significant variation from the symmetric leads case. In the limit $a \rightarrow \infty$, one can neglect the smaller width in (\ref{condpeak}) and the conductance peak $g$ is proportional to the partial width in the dominating lead. In this limit $P(g)$ is reduced to $P(\Gamma)$ shown by the dashed lines in Fig. \ref{fig4}). The asymmetry effect becomes larger for an increasing number of channels. This effect can be noticed by comparing the GOE and GUE cases, since for the same number of physical channels $\Lambda$ the GUE has a larger number of ``effective'' channels. \section{Dynamical Model} To test the RMT predictions for the statistical distributions, we modeled a quantum dot by a system whose classical dynamics is chaotic. The model is the conformal billiard, \cite{Robnik83,Berry86} whose shape is defined by the image of the unit circle in the complex $z$-plane under the conformal mapping \begin{eqnarray} \label{mapping} w(z) = \frac{z+bz^2+c\mbox{e}^{i\delta}z^3}{\sqrt{1+2b^2+3c^2}} \;. \end{eqnarray} The parameters $b, c$ and $\delta$ control the billiard shape. Eq. (\ref{mapping}) ensures that area ${\cal A}$ enclosed by $w(z)$ is normalized to $\pi$ and is independent of the shape. We analyze the case $b=0.2$, $c=0.2$ and $\delta=\pi/2$, for which the classical phase space is known to be chaotic. \cite{Bruus94} We have verified that the corresponding spectrum exhibits GOE-like spectral fluctuations (we used 300 converged levels by diagonalizing a matrix of order 1000). This is demonstrated in Fig. \ref{fig5}, where the nearest-neighbors level spacing distribution $P(s)$ and the $\Delta_3$ statistics, which measures the spectral rigidity, are shown. To investigate the effect of an external magnetic field, we consider the same billiard threaded by an Aharonov-Bohm flux line, \cite{Berry86,Bruus94} which does not affect the classical dynamics. The flux is parametrized by $\Phi = \alpha \Phi_0$ where $\Phi_0$ is the unit flux. We use the same set of values for $b, c$, and $\delta$ to insure classical chaotic motion, and choose $\alpha=1/4$ for maximal time-reversal symmetry breaking. The statistical tests shown in Fig.~\ref{fig5} confirm that this choice of $\alpha$ corresponds to the unitary limit . We remark that the $\Delta_3$-statistics is a better measure to distinguish between the GOE and GUE cases than the level spacing distribution $P(s)$ (used in Ref. \onlinecite{Bruus94}.) \subsection{Spatial Correlations} The eigenfunction amplitude correlation $C(k |\Delta \bbox{r}|)= \overline{\Psi^*(\bbox{r}) \Psi(\bbox{r}^\prime)}/\overline{|\Psi( \bbox{r})|^2}$ was recently investigated thoroughly for the conformal billiard. \cite{Li94} The results agree fairly well with the theoretical prediction, namely $C(k |\Delta \bbox{r}|)= J_0(k|\bbox{r}-\bbox{r}^\prime|)$, \cite{Berry77} if one averages over the orientation of $\Delta \bbox{r}$. This result is obtained based on semiclassical arguments and the eigenfunctions studied in Ref.\onlinecite{Li94} were chosen accordingly to be highly excited states (deep in the semiclassical region). In order to apply this result to quantum dots, further considerations are in order. First, a typical semiconductor quantum dot in the submicrometer range contains several hundred electrons, and it is therefore not obvious that the eigenstates around the Fermi level are necessarily semiclassical. Second, scars associated with isolated periodic orbits give corrections to $C(k|\Delta \bbox{r}|)$ which depend on the orientation of $\Delta \bbox{r}$ and are of order $O(\hbar^{1/2})$. The fluctuations of the spatial correlation of the billiard eigenfunctions were recently studied\cite{Srednicki96} and found also to be suppressed by $O(\hbar^{1/2})$. These corrections are negligible if one averages over all orientations around a given point $\bbox{r}$, keeping the modulus $|\Delta\bbox{r}|$ fixed, but this is difficult to implement experimentally. At a fixed orientation the fluctuations of the spatial correlations seem to be rather small if $k|\Delta\bbox{r}| {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$} \lower5pt\vbox{\hbox{$\sim$}}}}\ } 3$ so that (\ref{17}) is a good approximation. For larger values of $k|\Delta\bbox{r}|$, there could be significant fluctuations from (\ref{17}) but in this region the width and conductance distributions are closer to their limiting case of independent channels and are not very sensitive to the exact correlations. Our results were obtained by using the billiard eigenfunctions with Neumann boundary conditions where the normal derivative of the wavefunctions vanishes on the boundaries. We analyze eigenfunctions in the vicinity of the 100$^{th}$ excited level which resembles the experimental situation. By moving the points around the circle we generate more statistics and average over orientations. The results are shown in Fig.~\ref{fig6} where the correlations in the model (solid line) compare well with the theoretical result (dashed line) for both cases with and without magnetic flux. The agreement is fair, particularly for $k|\Delta\bbox{r}| {\ \lower-1.2pt\vbox{\hbox{\rlap{$<$}\lower5pt\vbox{\hbox{$\sim$}}}}\ }5$. For $k|\Delta\bbox{r}|\gg 1$, the deviations from the theoretical value of $C(k|\Delta\bbox{r}|)$ are not important since the channels are weakly correlated and the distributions are very close to those describing uncorrelated channels. Thus, corrections to our analytical findings should not be large, as is supported by the numerical evidence presented below. In our model studies we imposed Neumann boundary conditions around the entire billiard and not just at the dot-lead boundary. To mimic the experimental situation we would have to impose mixed boundary conditions, \cite{Bruus94} which makes the calculations much more computationally intensive. However, we now argue that our simplified situation still provides reasonable results. For extended leads, the length of the dot-lead contact region $D$ must satisfy $k D \gg 1$ in order to support open transverse channels (in dots containing several hundred electrons). Therefore, deviations from $C(k|\Delta\bbox{r}|)$ at the edge of the dot-lead contact region (where our boundary conditions are unrealistic) are averaged out. For point-like contacts, the physical picture is that the conductance is probing the wavefunction in the vicinity of the constriction (the region that couples the dot to the external lead). We then need to know the characteristic properties of the wavefunction inside the dot where our model is quite satisfactory. \subsection{Coupling to Leads and Distributions} We first studied the point-like contacts model by describing the lead as a sequence of $\Lambda$ equally spaced points on the boundary of the billiard (in the $w$-plane). According to Eq.~(\ref{17}) the correlation matrix $M$ is then completely determined by $k |\Delta \bbox{r}| \approx k\delta\theta |w^\prime(r=1, \theta)|$ (where $\delta \theta$ is the angle that spans the arc between two neighboring points in the $z$-plane) and $\Lambda$. In this model it is easy to generate strong correlations by choosing the points close enough, unlike the (discretized) Anderson model \cite{Mucciolo95} where the channels are weakly correlated even if the lead is composed of nearest neighboring points. The eigenvalues $w_c^2$ are found by diagonalizing the matrix $M$. In Figs. \ref{fig7} and \ref{fig8} we compare for the unitary and orthogonal limits, respectively, the total width distribution $P(\Gamma)$ obtained by solving the conformal billiard (histograms) with the theoretical predictions (solid lines), for several values of $k |\Delta \bbox{r}|=0.5,1,2$ and $\Lambda=2,4,6$. The distributions $P^{(0)}(\Gamma)$ for equivalent and uncorrelated channels are indicated by the dashed lines, and are just the $\chi^2$ distributions in $\Lambda$ ($2\Lambda$) degrees of freedom for the GOE (GUE). The agreement between the model and the analytic RMT predictions confirms the validity of the statistical model for a chaotic dot. We observe from Figs. \ref{fig7} and \ref{fig8} that for the larger values of $k |\Delta \bbox{r}|$, the distributions get closer to those for uncorrelated channels. This is consistent with the decrease in spatial correlations (see Fig. \ref{fig7}). Another interesting observation is that, for a constant $k |\Delta \bbox{r}|$ ({\sl i.e.} fixed correlations), the deviation from the limiting case of independent channels becomes larger with an increasing number of channels. Figs. \ref{fig9} and \ref{fig10} show a comparison between the theoretical conductance peaks distributions for symmetric leads, as given by Eqs.~(\ref{PgGUE}) and (\ref{PgGOE}) for the unitary and orthogonal cases, respectively, and those calculated for the Africa billiard with symmetric $\Lambda$-point leads ($\Lambda=2,4$, and 6) and for different values of $k|\Delta \bbox{r}|$. The dashed lines are again the limiting case of uncorrelated and equivalent leads. Observations that are similar to the ones made above for the width distributions, can be made with respect to the conductance peaks distributions. Comparing the width and conductance peaks distributions, we note that the conductance distribution shows stronger deviation from its limit for uncorrelated equivalent channels than does the width distribution. We also studied extended leads by taking the contact region of the lead and the dot to have a finite length $D \approx |w^\prime| \Delta \theta$ on the dot's boundary (in the $w$-plane) where $w^\prime$ is evaluated at the corresponding angle where the lead is located. In this case the channels are defined by the allowed quantized transverse momenta $\kappa_c=\pi n_c/D$ with $n_c=1, 2, \ldots, \Lambda$, where $\Lambda=\mbox{int}[kD/\pi]$. To calculate the partial amplitude for the conformal billiard, the integral in Eq.~(\ref{gamdefR}) (defined in the $w$-plane) is mapped into an integral along an arc in the $z$-plane which is spanned by an angle $\Delta \theta$ \begin{eqnarray} \label{27} \gamma_{c\lambda} = \sqrt{\frac{\hbar^2}{2m} } \int_{\Delta \theta} d\theta\,|w^\prime(r=1,\theta)| \,\Phi_c^*(\theta) \Psi_\lambda (r=1,\theta) \;, \end{eqnarray} where $\Phi_c(\theta)= \sqrt{2/D} \sin (\kappa_c |w^\prime| \theta)$ are the transverse channel wavefunction and for simplicity we have set $k_cP_c=1$. The resonance eigenfunction $\Psi_\lambda$ is given in terms of its expansion in $\mbox{e}^{i m \theta}$ (with $m= 0, \pm 1, \pm 2, \ldots)$ \begin{eqnarray} \label{27'} \Psi_\lambda(r=1,\theta) = {\cal N}_\lambda \sum_j \frac{c_j^\lambda} {\sqrt{\pi(\gamma_j^2 - |\ell_j-\alpha|^2) }} \,\mbox{e}^{i\ell_j \theta} \;, \end{eqnarray} where ${\cal N}_\lambda$ is a normalization constant, $\gamma_j$ are the zeros of $J^\prime_{\mid\ell_j-\alpha \mid}$ and $c_j$ are expansion coefficients as in Ref. \onlinecite{Bruus94}. To guarantee that the correlation matrix $M$ in (\ref{8}) is the same for eigenfunctions of the billiard which belong to different energies, we choose $D$ such that $kD=constant$ and scale the partial amplitude (\ref{gamdefR}) by $k^{1/2}$. The resulting matrix is \begin{eqnarray} \label{27''} kM_{cc^\prime} = \frac{\hbar^2}{2m} {2 \over kD} \int_{\Delta \theta} d\theta\, \int_{\Delta \theta} d\theta^\prime\, & &|w^\prime(r=1,\theta)| \, |w^\prime(r=1,\theta^\prime)| \, \nonumber \\ & & \times \sin \left({\pi n_c \over kD} |w^\prime| \theta\right) J_0 \left(|w^\prime| |\theta-\theta^\prime| \right) \sin \left({\pi n_{c^\prime} \over kD} |w^\prime| \theta^\prime \right) \;. \end{eqnarray} This scaling is desirable in order to be consistent with the theoretical approach presented above, but experimentally it is very hard to accomplish. Fortunately, this scaling of $D$ is insignificant for present experiments \cite{Chang96,Folk96} that deal with dots containing several hundred electrons ${\cal N}$. Indeed, from the Weyl formula we have $k_F \propto {\cal N}^{1/2}$ so that $\delta k_F / k_F = \delta {\cal N} / 2{\cal N} \ll 1$. The latter inequality is obtained when we estimate $\delta {\cal N}$ to be the number of observed Coulomb blockade peaks (since each Coulomb blockade peak corresponds to the addition of one electron into the dot). The relative variation of $k_F$ is thus small and can be neglected. We find that the channels in the extended leads model are weakly correlated and that the average partial widths in the various channels exhibit a moderate variation. In such a case the total width distribution is not very different from the case of uncorrelated equivalent channels. Our model calculations for extended leads are shown in Fig. \ref{fig11} and are in agreement with the RMT predictions for uncorrelated channels (dashed lines). An interesting effect is that with an increasing number of channels even small deviations in $P(\Gamma)$ give rise to relatively large deviations in $P(g)$, as can be seen in Fig. \ref{fig11}d. \section{Connection to experiments and conclusions} We have discussed both the cases of orthogonal and unitary symmetries. To relate to actual experimental situations, it is important to estimate the minimal strength of the magnetic field $B_c$ which ensures complete time-reversal symmetry breaking. For a ballistic electron \cite{Berry86,Jalabert92,Bruus94,Frahm95} $B_c {\cal A} \propto \sqrt { \tau_{cr} / \tau} \Phi_0$, where $\tau_{cr}$ and $\tau$ are, respectively, the time it takes the electron to cross the dot and the Heisenberg time $\tau=h / \Delta$. For an electron at the Fermi energy $B_c {\cal A} \propto {\cal N}^{-1/4} \Phi_0$, where ${\cal N}$ is the number of electrons in the dot. The proportionality factor is non-universal and depends on the exact geometry of the dot. In a semiclassical analysis\cite{Bohigas95,Pluhar95} it can be expressed in terms of classical quantities. For the dots used in some recent experiments, \cite{Chang96,Folk96} $B_c$ is of order of a few mT. Such small values of $B_c$ do not alter significantly the classical dynamics of the electron,\cite{Obs1} and our assumption that the correlation $C(k|\Delta \bbox{r}|)$ is unchanged is justified. Nevertheless, these small variations in the magnetic field have appreciable quantum mechanical effects, {\sl i.e.} the crossover from orthogonal to unitary symmetry. In conclusion, we have derived closed expressions for the width and conductance peak distributions in quantum dots in the Coulomb blockade regime. The main assumption is that the electron's dynamics is chaotic for a ballistic dot or weakly diffusive for a disordered dot. For given correlation matrices that characterize the left and right leads, these distributions are universal and distinct for conserved and broken time-reversal symmetry. While recent experiments have measured the conductance distributions in symmetric one-channel leads, it would be interesting to measure and compare with theory the conductance distributions in more general situations of dots with multi-channel leads. \acknowledgements This work was supported in part by the U.S. Department of Energy Grant DE-FG02-91ER40608. C.H.L. acknowledges financial support by the Conselho Nacional de Pesquisas (CNPq -- Brazil). We thank H.~U.~Baranger and A.D. Stone for discussions and H. Bruus for the use of his billiard computer program.

proofpile-arXiv_065-542

\section{Introduction}\label{sec:intro} The CVC hypothesis \cite{FG} arose at the time when only charged current weak interactions were known. The Hamiltonian for semi-leptonic, weak interactions was written as \begin{equation} H^{SL}_{CC} = \frac{G}{\sqrt{2}} \left[ J^+_{\lambda} L^{\lambda} + J^-_{\lambda} L^{\lambda \dagger} \right] , \label{eq:1} \end{equation} with $J^{\pm}_{\lambda}$ the charge changing hadronic current and $L^{\lambda}$ the leptonic current. The hypothesis had two parts: firstly, if we break the hadronic weak current into vector and axial pieces, $J^{\pm}_{\lambda} = V^{\pm}_{\lambda} - A^{\pm}_{\lambda}$, that the vector pieces $V^{\pm}_{\lambda}/cos \theta_C$ and the isovector piece of the electromagnetic current were the three components, $j_{i \lambda}$, of a vector in isospace; secondly, that all 3 components of this current were conserved, $\partial ^{\lambda} j_{i \lambda} = 0$. Viewed in hindsight, with the full Standard Model at hand, it is hard to appreciate the power and insight that it represented. For the purposes of this brief review we begin with the Standard Model. The first part of the CVC hypothesis is then trivial because all interactions involving strongly interacting systems are built from the same vector (and axial vector) quark currents. On the other hand, the second part of the hypothesis is incorrect because the QCD Hamiltonian contains a piece proportional to $(m_u - m_d) (\bar{u}u - \bar{d}d)$. The fact that $m_u \neq m_d$ implies that $\partial ^{\lambda} j_{i \lambda} \propto (m_u - m_d) \neq 0 $ and thus CVC can only be, at best, a good approximation. Nevertheless, it has proven such a successful approximation that it is built into all phenomenological treatments. In the next section we review two of the classic applications of CVC as well as an important illustration of the fact that $m_u$ is not exactly equal to $m_d$. \section{Classic Tests}\label{sec:class} The most famous testing grounds for CVC are $\beta$-decay of the free neutron, bound nucleons and the pion. For the nucleon the most accurate measurements by far are made in finite nuclei and these are traditionally not considered part of particle physics. Therefore, we begin with the case of pion $\beta$-decay, which still gives the most precise, particle physics test. We then recall the potential importance of neutrino scattering where, as we shall see, the errors are still very large. To complete the section, we briefly review the most spectacular example of the effect of having $m_u \neq m_d$, namely $\rho -\omega$ mixing. This phenomenon is also very important in modern tests of CVC (e.g., at LEP) as well as for determinations of $CP$-violation at $B$-factories -- as we shall discuss in the next section. \subsection{Pion Beta Decay}\label{subsec:pion} The decay \begin{equation} \pi^- \rightarrow \pi^0 + e^- + \bar{\nu_e}, \label{eq:2} \end{equation} is severely depressed by the small mass difference between the charged and neutral pions. Only the vector part of the hadronic weak current can play a role in this decay. The CVC hypothesis then naturally relates the isovector, vector matrix element to the electromagnetic form factor of the pion ($F_{\pi}$): \begin{eqnarray} < \pi^0 (k') | J^{\lambda} | \pi^- (k) > & = & - cos \theta_C \sqrt{2} < \pi^- (k') | j_{e-m}^{\lambda} | \pi^- (k) > \nonumber \\ & = & - cos \theta_C \sqrt{2} F_{\pi}(q^2) (k + k')^{\lambda}, \label{eq} \end{eqnarray} with $q = k' - k$. The very small mass difference involved, $\Delta \sim 4$ MeV, means that the difference between $F_{\pi}(0) = 1$ and $F_{\pi}(q^2)$ is negligible. Hence the $\beta$-decay lifetime can be written in terms of essentially kinematic quantities: \begin{equation} \tau^{-1} = G^2 \frac{cos\theta_C^2}{30\pi^3} \left[ 1 - \frac{\Delta}{2m} \right] \Delta^5 \bar{F} (1 + \delta_{\pi}). \end{equation} The phase space factor, $\bar{F}$, is near one and the (electromagnetic) radiative correction ($\delta_{\pi}$) is of order 1\% \cite{sirlin}. This leads to a theoretical branching ratio $1.0482 \pm 0.0048 \times 10^{-8}$ \cite{mcf}. The most recent experiment to accurately determine this ratio was carried out at LAMPF more than 10 years ago. McFarlane et al. obtained 1224 $\pm$ 36 good events for $\pi^+ \beta$-decay after observing more than $10^{11}$ pions \cite{mcf}. Their final branching ratio, $1.026 \pm 0.039 \times 10^{-8}$, dominates the current world average ($1.025 \pm 0.034 \times 10^{-8}$)\cite{pdg}. Until this time the best measurement was from Depommier et al.\cite{dep}, in 1968. Clearly the theoretical and experimental values are completely consistent at the current experimental limit of about 3\%. \subsection{Neutrino Nucleon Scattering}\label{subsec:neutrino} Neutrino nucleon scattering is the primary source of information on the weak axial vector current, notably $G_A(q^2)$. However, in order to extract information on the axial vector current it is more or less compulsory to assume that the vector matrix elements can be taken from electromagnetic interactions using CVC. Because of the difficulties of dealing with a neutrino beam it is preferable to leave a charged nucleon in the final state. Thus one is led to a deuteron target because of the lack of a free neutron target. However, by using the resolving power of a bubble chamber to tag a low momentum, spectator proton \begin{equation} \nu_{\mu} + d \rightarrow \mu^- + p + p_{spectator}, \end{equation} one can reduce the experimental uncertainties. The data is not sufficiently accurate to determine the detailed shape of $G_A(q^2)$, which is usually parametrized as a dipole by analogy with the vector form factors. It is possible to relax the CVC constraint on the vector form factors and search simultaneously for the best-fit dipole masses $M_A$ {\it and} $M_V$. The values obtained in the early 80's by Baker {\it et al.} ($M_V = 0.86 \pm 0.07, M_A = 1.04 \pm 0.14$) \cite{baker} and Miller {\it et al.} ($M_V = 0.96 \pm 0.04, M_A = 0.80 \pm 0.10$) \cite{miller} were consistent with CVC in that $M_V$ measured in electron scattering is 0.84 GeV. As there is now very good agreement on $M_A$, at about 1.02 GeV, the latter value is probably nothing to worry about, even though the apparent discrepancy in $M_V$ is $2\frac{1}{2}$ standard deviations. The latest result from the Brookhaven group (Kitagaki {\it et al.}) is reassuring, yielding ($M_V = 0.89 +0.04 -0.07$ and $M_A = 0.97 +0.14 -0.11$) \cite{kitagaki}. The same data can also be used to search for evidence for the second-class vector current. The most recent limit comes from a measurement of the reaction $\bar{\nu}_{\mu}p \ra \mu^+n$ at Brookhaven \cite{Ahrens}. Unfortunately the sensitivity to the scalar form factor is reduced by a factor $(m_{\mu}/M_N)^2$. If it is parametrized as \begin{equation} F_S(q^2) = \rho \frac{F_V(0)}{1 - \frac{q^2}{M_S^2}}, \end{equation} then the limit on $\rho$ is $\rho < 1.8$, with $M_S = 1$GeV and the axial tensor term set to zero. One can actually set a better limit on the axial tensor, second class current but that is not our concern. \subsection{Neutrino Deep Inelastic Scattering}\label{subsec:neutdis} Although it is not derived from CVC alone, the Adler sum-rule provides a fundamental test of the quark currents within the Standard Model. It relies on the equal-time commutation relation \begin{equation} \delta (x_0) \left[ J^-_0(x),J^+_0(0) \right] = -4\delta^{(4)}(x) \left[ V^3_0 + A^3_0 \right]. \end{equation} Taking the matrix element of this relation between hadronic states and averaging over spin (so that $<A^3_0> = 0$) we find \begin{equation} \int^{\infty}_0 d\nu \left[ W^{\nu}_2(Q^2,\nu) - W^{\bar{\nu}}_2(Q^2,\nu) \right] = -4<I_3 >, \end{equation} and finally, changing the integration variable to Bjorken $x$, and using the fact that $\nu W^{\nu}_2 = F^{\nu}_2(x,Q^2)$ scales, we find \begin{equation} \int^1_0 dx \frac{F^{\nu n}_2 - F^{\nu p}_2}{2x} = 1. \end{equation} Note that this sum-rule is protected against $0(\alpha_s)$ corrections. Even now, the best experimental test of this fundamental sum-rule comes from the long extinct BEBC facility at CERN, with the result $1.01 \pm 0.08 (stat.) \pm 0.18 (syst.)$ \cite{Allas}. While the data is clearly consistent with expectations, an error of more than 20\% is really unacceptable in such a fundamental quantity. \subsection{Rho-Omega Mixing}\label{subsec:rhom} The most spectacular evidence that the vector current is not conserved comes from electromagnetic interactions. In particular, the data for $e^+ e^- \rightarrow \pi^+ \pi^-$, which is dominated by the isovector $\rho$-meson, shows a sharp interference pattern near the mass of the isoscalar $\omega$-meson. The natural explanation of this is that, because $m_u \neq m_d$, the isospin pure $\rho$ and $\omega$ mesons are not eigenstates of the full QCD Hamiltonian. (N.B. One must, of course, include the mixing induced by coupling to the photon, namely $\rho \rightarrow \gamma \rightarrow \omega$, but this is only 10\% of the observed amplitude. Also, the $\rho$ and $\omega$ are resonances and therefore not eigenstates of any Hamiltonian, but one can give the statement some rigour within models, such as the cloudy bag \cite{cbm}, by turning off the coupling to decay channels.) Suppose we make the standard simplification\cite{Renard}, which is that the direct decay of the isospin pure $\omega$ to two pions cancels the imaginary piece of the two pion loop contribution to the mixing self-energy. This means that we can neglect the pure isospin state, $\omega_I$, coupling to two pions ($\M^{\nu}_{\omega_I\ra\pi\pi}=0$) with the understanding that it is the real part of the mixing amplitude that is being extracted. To lowest order in the mixing amplitude, the amplitude for the virtual $\gamma$ to decay to two pions can be written: \begin{equation} \M^{\mu}_{\gamma\ra\pi\pi} = \M^{\mu}_{\rho_I\ra\pi\pi} \frac{1}{s_{\rho}} \M_{\gamma\ra\rho_I} + \M^{\mu}_{\rho_I\ra\pi\pi} \frac{1}{s_{\rho}} \Pi_{\rho\omega} \frac{1}{s_{\omega}} \M_{\gamma\ra\omega_I}, \end{equation} where $1/s_V$ is the vector meson propagator. The couplings that enter this expression, through $\M^{\mu}_{\rho_I\ra\pi\pi}$, $\M_{\gamma\ra\rho_I}$ and $\M_{\gamma\ra\omega_I}$, involve the pure isospin states $\rho_I$ and $\omega_I$. However, we can re-express it in terms of the physical states by first diagonalising the vector meson propagator. This leads to the result \begin{eqnarray} \nonumber \M^{\mu}_{\gamma\ra\pi\pi} &=& \M^{\mu}_{\rho\ra\pi\pi} \frac{1}{s_{\rho}} \M_{\gamma\ra\rho} + \M^{\mu}_{\omega\ra\pi\pi} \frac{1}{s_{\omega}} \M_{\gamma\ra\omega} \\ &=& \M^{\mu}_{\rho\ra\pi\pi} \frac{1}{s_{\rho}} \M_{\gamma\ra\rho} + \M^{\mu}_{\rho\ra\pi\pi} \frac{\Pi_{\rho\omega}}{s_\rho-s_\omega} \frac{1}{s_{\omega}} \M_{\gamma\ra\omega} , \end{eqnarray} which is the form usually seen in older works \cite{CSM,CB,MSC,us}. A recent analysis \cite{Heath} of the world data gave a value for the mixing amplitude of $\Pi^{\rho \omega} = -3800 \pm 370$ MeV$^2$. \section{Recent Tests and Applications}\label{sec:recent} \subsection{Testing CVC in $\tau^-$ Decay}\label{subsec:tau} All modern $e^+e^-$ colliders with sufficient energy (including LEP) have studied the decay \begin{equation} \tau^- \ra \pi^- + \pi^0 + \nu_{\tau}. \end{equation} At present the world average for the branching ratio into this channel is\cite{pdg} $25.24 \pm 0.16$\%. Only the hadronic vector current is involved and according to CVC this will be only the I=1 piece of the vector current. Thus, unlike $e^+ e^- \ra \pi^+ \pi^-$, there will be no $\rho - \omega$ interference. The procedure is therefore to fit the $e^+ e^- \ra \pi^+ \pi- $ data, including $\rho - \omega$ mixing as well as the $\rho' (1450)$ and $\rho''(1700)$, and then use the purely I=1 piece of the amplitude to calculate the decay rate \cite{Sobie,EI}. The most recent estimate of the theoretical branching ratio comes from the analysis of Sobie\cite{Sobie}, namely $24.6 \pm 1.4$\%. This differs from experiment by about $2\frac{1}{2}$\% with a mainly theoretical error of $\sim 5$\%. Once again CVC works but the level of accuracy is relatively modest. In order to improve the accuracy one would need to resolve the differences between the existing data sets for $e^+ e^- \ra \pi^+ \pi^-$. \subsection{An Application to CP Violation}\label{subsec:CP} The fact that CVC is not exact, and in particular the mixing of $\rho$ and $\omega$, can be put to use in quite a spectacular fashion in the study of CP violation. We shall briefly report on the recent analysis of Enomoto and Tanabashi\cite{ET}, following a suggestion of Lipkin\cite{Lip}. Rather than the traditional proposals to study $(B^0,\bar{B}^0)$ decays, these authors aim to {\it use} $\rho - \omega$ interference to generate a large CP violation signal in charged $B$ decays. One can show that the CP-violating difference in the decay rates $B^- \ra \rho^0 \rho^-$ and $B^+ \ra \rho^0 \rho^+$ is proportional to $cos(\delta + \phi) - cos (\delta - \phi)$, where $\phi$ is the CP violation phase and $\delta$ the known, strong phase arising from $\rho - \omega$ mixing. Notice that the signal vanishes if $\delta = 0$. Although the branching ratios for the decay modes $B^- \ra \rho^0 h^-$ (with $h = K, K^*,\rho$, etc.) is very small ($\sim 10^{-8}$), the asymmetry can be as large as 90\% ! This is clearly a very important suggestion to pursue further. In concluding this brief summary we note one question which needs urgent study. At recent analysis of $\rho - \omega$ mixing by Maltman {\it et al.}\cite{Malt} has suggested that, contrary to earlier conclusions, the direct coupling of the I = 0 $\omega$ to two pions leads to a large uncertainty in $\Pi^{\rho \omega}$ and in the relative phase. In view of the need to know the strong phase $\delta$ very well, in order to extract $\phi$, this is a worrying conclusion. We simply note that this ambiguity vanishes identically at $q^2 = \hat{m}^2_{\rho}$, but that the residual ambiguity needs careful study. \section{Beyond the Standard Model}\label{sec:sm} The Standard Model has proven very successful in every area of particle physics, including recent high-energy collider experiments. However, it has three features which are not well understood: the origin of mass, the three fermion generations and the phenomenon of CP violation. The question of mass is usually framed in terms of (fundamental) Higgs fields \cite{Higgs} and why the corresponding Yukawa couplings take particular values. Instead, one might ask whether a formulation of the Standard Model with massless fermions makes sense. For example, it is well known that QED with massless electrons is not well defined at the quantum level \cite{Muta,8}. In a recent paper, Bass and Thomas \cite{BT} considered the pure Standard Model with gauge symmetry $SU(3) \otimes SU(2)_L \otimes U(1)$ and no additional interaction -- i.e., with no grand unification. They examined the physical theory corresponding to the bare Standard Model Lagrangian with no elementary Higgs and just one generation of massless fermions and gauge bosons. At asymptotic scales, where the U(1) coupling is significantly greater than the asymptotically free SU(3) and SU(2)$_L$ couplings, the left and right handed states of any given charged fermion couple to the U(1) gauge boson with different charges. At the Landau scale of this non-asymptotically free theory, it was suggested that there should be three separate phase transitions -- corresponding to each of the right-right, right-left and left-left interactions becoming supercritical. These transitions correspond to three generations of fermions. As one passes through each transition from a higher scale (shorter distance) the corresponding scalar condensate ``melts'', releasing a dynamical fermion into the Dirac phase studied in the laboratory. In this picture the three generations emerge as quasi-particle states built on a ``fundamental fermion'' interacting self-consistently with the condensates. Clearly this proposal differs in a fundamental manner from the conventional approaches to the Standard Model. While the conceptual framework is extremely simple and elegant, the techniques for dealing with non-perturbative physics at the Landau scale are not well developed. In particular, at the present stage it has not yet been possible to present a rigorous, quantitative derivation of all of the features of the Standard Model. Nevertheless, we believe that the potential for understanding so many phenomena, including mass, CP-violation and the generations, not to mention CVC, is so compelling that the ideas merit further study. \section{Conclusion}\label{sec:conc} We have seen clearly that CVC is a very natural approximation within the Standard Model. Certainly the isovector, vector current (modulo the CKM matrix) is exactly the vector current involved in the charged current weak interactions. The hypothesis that the current is conserved is an approximation only because $m_u \neq m_d$ -- and, of course, because of the electromagnetic interaction. Within particle physics the best test of CVC is pion $\beta$-decay, with the present limit being about 3\%. Using the decay of the heavier (1.777 GeV) $\tau^-$ to $\pi^0 \pi^- \nu_{\tau}$ one can set a limit that is only slightly worse, around 5\%. Tests involving neutrino-nucleon scattering are quite imprecise, with the fundamental Adler sum-rule (which, of course, tests current algebra not just CVC) being in desperate need of accurate data. We also saw that $\rho - \omega$ mixing, which is a spectacular example of the non-conservation of the vector current, provides a very beautiful alternative way to study CP violation in $B$-decays. Finally, we briefly reviewed a rather ambitious framework for understanding the origin of many phenomenological features of the Standard Model, including the three generations and their masses. Such an approach may someday provide us with a real understanding of the origins of CVC. \section{Acknowledgements}\label{sec:Ack} I would like to thank a number of colleagues for helpful discussions, especially H. O'Connell and A.G. Williams concerning $\rho - \omega$ mixing, S.V. Gardner concerning the CP violation proposal and S.D. Bass concerning our generations proposal. It is a pleasure to thank Professor Minamisono and his colleagues for the opportunity to participate in a most stimulating symposium. This work was supported by the Australian Research Council. \section{References}\label{sec:ref}

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\section{Introduction}\label{intro} The time-evolution due to $B^0_s-\overline{B^0_s}$ mixing is governed by the $B_s$ mass eigenstates $B_s^{\mbox{{\scriptsize Heavy}}}$ and $B_s^{\mbox{{\scriptsize Light}}}$ which are characterized by their mass eigenvalues $M_H^{(s)}$, $M_L^{(s)}$ and decay widths $\Gamma_H^{(s)}$, $\Gamma_L^{(s)}$. Because of these mixing effects, oscillatory $\Delta M_s t$ terms with $\Delta M_s\equiv M_H^{(s)}- M_L^{(s)}$ show up in the time-dependent transition rates \cite{evol} $\Gamma(B^0_s(t)\to f)$ and $\Gamma(\overline{B^0_s} (t)\to f)$ describing decays of initially present $B^0_s$ and $\overline{B^0_s}$ mesons into a final state $f$, respectively. The ``strength'' of the $B^0_s-\overline{B^0_s}$ oscillations is measured by the mixing parameter $x_s\equiv\Delta M_s/\Gamma_s$, where $\Gamma_s\equiv(\Gamma^{(s)}_H+\Gamma^{(s)}_L)/2$. Within the Standard Model one expects \cite{xs} $x_s={\cal O}(20)$ implying very rapid $B^0_s-\overline{B^0_s}$ oscillations which require an excellent vertex resolution system to keep track of the $\Delta M_s t$ terms. That is obviously a formidable experimental task. However, as pointed out by Dunietz \cite{dunietz}, it may not be necessary to trace the rapid $\Delta M_s t$ oscillations in order to obtain insights into the mechanism of CP violation. This remarkable feature is due to the expected sizable width difference \cite{deltagamma} $\Delta\Gamma_s\equiv \Gamma_H^{(s)}-\Gamma_L^{(s)}$. The major contributions to $\Delta\Gamma_s$, which may be as large as ${\cal O}(20\%)$ of the average decay width $\Gamma_s$, originate from $\bar b\to \bar cc\bar s$ transitions into final states that are common both to $B_s^0$ and $\overline{B_s^0}$. Because of this width difference already {\it untagged} $B_s$ rates, which are defined by \begin{equation}\label{e1} \Gamma[f(t)]\equiv\Gamma(B_s^0(t)\to f)+\Gamma(\overline{B^0_s}(t)\to f), \end{equation} may provide valuable information about the phase structure of the observable \begin{equation}\label{e2} \xi_f^{(s)}=\exp\left(-i\,\Theta_{M_{12}}^{(s)}\right) \frac{A(\overline{B^0_s}\to f)}{A(B^0_s\to f)}, \end{equation} where $\Theta_{M_{12}}^{(s)}$ is the weak $B_s^0$--$\overline{B_s^0}$ mixing phase \cite{evol}. This can be seen nicely by writing Eq.~(\ref{e1}) in a more explicit way as follows: \begin{eqnarray} \lefteqn{\Gamma[f(t)]\propto\left[\left(1+\left|\xi_f^{(s)} \right|^2\right)\left(e^{-\Gamma_L^{(s)} t}+e^{-\Gamma_H^{(s)} t}\right) \right.}\nonumber\\ &&-2\mbox{\,Re\,}\xi_f^{(s)}\left(e^{-\Gamma_L^{(s)} t}- e^{-\Gamma_H^{(s)} t}\right)\biggr].\label{e3} \end{eqnarray} In this expression the rapid oscillatory $\Delta M_s t$ terms, which show up in the {\it tagged} rates, cancel~\cite{dunietz}. Therefore it depends only on the two exponents $e^{-\Gamma_L^{(s)} t}$ and $e^{-\Gamma_H^{(s)} t}$, where $\Gamma_L^{(s)}$ and $\Gamma_H^{(s)}$ can be determined e.g.\ from the angular distribution \cite{ddlr} of the decay $B_s\to J/\psi\,\phi$. From an experimental point of view such {\it untagged} analyses are clearly much more promising than tagged ones in respect of efficiency, acceptance and purity. \section{A Transparent Example}\label{example} In order to illustrate these untagged rates in more detail, let me discuss an estimate of the angle $\gamma$ of the usual ``non-squashed'' unitarity triangle~\cite{ut} of the Cabibbo--Kobayashi--Maskawa matrix \cite{ckm} (CKM matrix) using {\it untagged} $B_s\to K^+K^-$ and $B_s\to K^0\overline{K^0}$ decays. This approach has been proposed very recently by Dunietz and myself~\cite{fd1}. Using the $SU(2)$ isospin symmetry of strong interactions to relate the QCD penguin contributions to these decays (electroweak penguins are color-suppressed in these modes and thus play a minor role), we obtain \begin{eqnarray} \lefteqn{\Gamma[K^+K^-(t)]\propto |P'|^2\Bigl[\bigl(1-2\,|r|\cos\rho\, \cos\gamma} \nonumber\\ &&+|r|^2\cos^2\gamma\bigr)e^{-\Gamma_L^{(s)} t}+|r|^2\sin^2\gamma\, e^{-\Gamma_H^{(s)} t}\Bigr]\label{e4} \end{eqnarray} and \begin{equation}\label{e5} \Gamma[K^0\overline{K^0}(t)]\propto |P'|^2\,e^{-\Gamma_L^{(s)} t}, \end{equation} where \begin{equation}\label{e6} r\equiv|r|e^{i\rho}=\frac{|T'|}{|P'|}e^{i(\delta_{T'}-\delta_{P'})}. \end{equation} Here $P'$ denotes \cite{ghlrsu3} the $\bar b\to\bar s$ QCD penguin amplitude, $T'$ is the color-allowed $\bar b\to\bar uu\bar s$ tree amplitude, and $\delta_{P'}$ and $\delta_{T'}$ are the corresponding CP-conserving strong phases. In order to determine $\gamma$ from the untagged rates Eqs.~(\ref{e4}) and (\ref{e5}) we need an additional input that is provided by the $SU(3)$ flavor symmetry of strong interactions. If we neglect the color-suppressed current-current contributions to $B^+\to\pi^+\pi^0$ we find \cite{ghlrsu3} \begin{equation}\label{e7} |T'|\approx\lambda\,\frac{f_K}{f_\pi}\,\sqrt{2}\,|A(B^+\to\pi^+\pi^0)|, \end{equation} where $\lambda$ is the Wolfenstein parameter \cite{wolf}, $f_K$ and $f_\pi$ are the $K$ and $\pi$ meson decay constants, respectively, and $A(B^+\to\pi^+\pi^0)$ denotes the appropriately normalized $B^+\to\pi^+\pi^0$ decay amplitude. Since $|P'|$ is known from $B_s\to K^0\,\overline{K^0}$, the quantity $|r|=|T'|/|P'|$ can be estimated with the help of Eq.~(\ref{e7}) and allows the extraction of $\gamma$ from the part of Eq.~(\ref{e4}) evolving with the exponent $e^{-\Gamma_H^{(s)} t}$. \section{$B_s$ Decays into Admixtures of CP Eigenstates}\label{admixtures} As we will see in a moment, one can even do better than in the previous section, i.e.\ without using an $SU(3)$ flavor symmetry input, by considering the decays corresponding to $B_s\to K \overline{K}$ where two vector mesons (or higher resonances) are present in the final states \cite{fd1}. \subsection{An Extraction of $\gamma$ using Untagged $B_s\to K^{\ast+} K^{\ast-}$ and $B_s\to K^{\ast0}\overline{K^{\ast0}}$ Decays}\label{kkbar} The untagged angular distributions of these decays, which take the general form \cite{dighe} \begin{equation}\label{e8} [f(\theta,\phi,\psi;t)]=\sum_k\left[\overline{b^{(k)}}(t)+b^{(k)}(t)\right] g^{(k)}(\theta,\phi,\psi), \end{equation} provide many more observables than the untagged modes $B_s\to K^+K^-$ and $B_s\to K^0\overline{K^0}$ discussed in Section~\ref{example}. Here $\theta$, $\phi$ and $\psi$ are generic decay angles describing the kinematics of the decay products arising in the decay chain $B_s\to K^\ast(\to\pi K)\,\overline{K^\ast}(\to \pi\overline{K})$. The observables $\left[\overline{b^{(k)}}(t)+b^{(k)}(t)\right]$ governing the time-evolution of the angular distribution Eq.~(\ref{e8}) are given by real or imaginary parts of bilinear combinations of decay amplitudes that are of the following structure: \begin{eqnarray} \lefteqn{\left[A_{\tilde f}^\ast(t)\,A_f(t)\right]\equiv\left\langle \left(K^\ast\overline{K^\ast}\right)_{\tilde f}\left|{\cal H}_{\mbox{{\scriptsize eff}}}\right|\overline{B_s}(t)\right\rangle^\ast} \label{e9}\\ &&\times\left\langle\left(K^\ast\overline{K^\ast}\right)_{f}\left|{\cal H}_{\mbox{{\scriptsize eff}}}\right|\overline{B_s}(t)\right\rangle+ \left(\overline{B_s}\to B_s\right).\nonumber \end{eqnarray} In this expression $f$ and $\tilde f$ are labels that define the relative polarizations of $K^\ast$ and $\overline{K^\ast}$ in final state configurations $\left(K^\ast\overline{K^\ast}\right)_f$ (e.g.\ linear polarization states \cite{rosner} $\{0,\parallel,\perp\}$) with CP eigenvalues $\eta_{\mbox{{\tiny CP}}}^f$: \begin{equation}\label{e10} ({\cal CP})\left|\left(K^\ast\overline{K^\ast}\right)_f\right\rangle =\eta_{\mbox{{\tiny CP}}}^f\left| \left(K^\ast\overline{K^\ast}\right)_f\right\rangle. \end{equation} An analogous relation holds for $\tilde f$. The observables of the angular distributions for $B_s\to K^{\ast+} K^{\ast-}$ and $B_s\to K^{\ast0}\overline{K^{\ast0}}$ are given explicitly in Ref.~\cite{fd1}. In the case of the latter decay the formulae simplify considerably since it is a penguin-induced $\bar b\to\bar sd\bar d$ mode and receives therefore no tree contributions. Using -- as in Section~\ref{example} -- the $SU(2)$ isospin symmetry of strong interactions, the QCD penguin contributions of these decays can be related to each other. If one takes into account these relations and goes very carefully through the observables of the angular distributions, one finds that they allow the extraction of the CKM angle $\gamma$ {\it without} any additional theoretical input \cite{fd1}. In particluar no $SU(3)$ symmetry arguments as in Section~\ref{example} are needed. The angular distributions provide moreover information about the hadronization dynamics of the corresponding decays, and the formalism \cite{fd1} developed for $B_s\to K^{\ast+} K^{\ast-}$ applies also to $B_s\to\rho^0\phi$ if we perform a suitable replacement of variables. Since that channel is expected to be dominated by electroweak penguins \cite{ewp}, it may allow interesing insights into the physics of these operators. \subsection{The ``Gold-plated'' Transitions to Extract $\eta$}\label{gold} This subsection is devoted to an analysis \cite{fd1} of the {\it untagged} decays $B_s\to D_s^{\ast+}D_s^{\ast-}$ and $B_s\to J/\psi\,\phi$, which is the counterpart of the ``gold-plated'' mode $B_d\to J/\psi\, K_{\mbox{{\tiny S}}}$ to measure the angle $\beta$ of the unitarity triangle. These decays are dominated by a single CKM amplitude. Consequently the hadronic uncertainties cancel in the quantity $\xi_f^{(s)}$ defined by Eq.~(\ref{e2}), which takes in that particular case the form \begin{equation}\label{e11} \xi_f^{(s)}=\exp(i\,\phi_{\mbox{{\tiny CKM}}}), \end{equation} and the observables of the angular distributions simplify considerably. A characteristic feature of these angular distributions is {\it interference} between CP-even and CP-odd final state configurations leading to observables that are proportional to \begin{equation}\label{e12} \left(e^{-\Gamma_L^{(s)}t}-e^{-\Gamma_H^{(s)}t} \right)\sin\phi_{\mbox{{\tiny CKM}}}. \end{equation} Here the CP-violating weak phase is given by \cite{xs} $\phi_{\mbox{{\tiny CKM}}}=2\lambda^2\eta\approx{\cal O}(0.03)$, where the Wolfenstein parameter $\eta$ fixes the height of the unitarity triangle \cite{ut}. The observables of the angular distributions \cite{fd1} for both the color-allowed channel $B_s\to D_s^{\ast+} D_s^{\ast-}$ and the color-suppressed transition $B_s\to J/\psi\,\phi$ each provide sufficient information to determine the CP-violating weak phase $\phi_{\mbox{{\tiny CKM}}}$ from their {\it untagged} data samples thereby fixing the Wolfenstein parameter $\eta$. The extraction of $\phi_{\mbox{{\tiny CKM}}}$ is not as clean as that of $\beta$ from $B_d\to J/\psi\,K_{\mbox{{\tiny S}}}$. This is due to the smallness of $\phi_{\mbox{{\tiny CKM}}}$ with respect to $\beta$ enhancing the importance of the unmixed amplitudes proportional to the CKM factor $V_{ub}^\ast V_{us}$ which are similarly suppressed in both cases. \section{$B_s$ Decays caused by $\bar b\to\bar cu\bar s$}\label{nonCP} The $B_s$ decays discussed in this section are pure tree decays and probe the CKM angle $\gamma$ in a {\it clean} way~\cite{gam}. There are by now well-known strategies on the market using the time evolutions of such modes, e.g.\ $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to\stackrel{{\mbox{\tiny (---)}}}{D^0}\phi$~\cite{gam,glgam} and $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to D_s^\pm K^\mp$ \cite{adk}, to extract $\gamma$. However, in these strategies {\it tagging} is essential and the rapid $\Delta M_s t$ oscillations have to be resolved which is an experimental challenge. The question what can be learned from {\it untagged} data samples of these decays, where the $\Delta M_s t$ terms cancel, has been investigated by Dunietz in Ref.~\cite{dunietz}. In the untagged case the determination of $\gamma$ requires additional inputs: a measurement of the untagged $B_s\to D^0_{\mbox{{\tiny CP}}} \phi$ rate in the case of the color-suppressed modes $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to\stackrel{{\mbox{\tiny (---)}}}{D^0}\phi$, and a theoretical input corresponding to the ratio of the unmixed rates $\Gamma(B^0_s\to D_s^-K^+)/\Gamma(B^0_s\to D_s^-\pi^+)$ in the case of the color-allowed decays $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to D_s^\pm K^\mp$. This ratio can be estimated with the help of the ``factorization'' hypothesis which may work reasonably well for these color-allowed channels. Interestingly the {\it untagged} data samples may exhibit CP-violating effects that are described by observables of the form \begin{equation}\label{e13} \Gamma[f(t)]-\Gamma[\overline{f}(t)]\propto\left(e^{-\Gamma_L^{(s)}t}- e^{-\Gamma_H^{(s)}t}\right)\sin\varrho\,\sin\gamma. \end{equation} Here $\varrho$ is a CP-conserving strong phase shift and $\gamma$ is the usual angle of the unitarity triangle. Because of the $\sin\varrho$ factor, a non-trivial strong phase shift is essential in that case. Consequently the CP-violating observables Eq.~(\ref{e13}) vanish within the factorization approximation predicting $\varrho\in\{0,\pi\}$. Since factorization may be a reasonable working assumption for the color-allowed modes $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to D_s^\pm K^\mp$, the CP-violating effects in their untagged data samples are expected to be very small. On the other hand, the factorization hypothesis is very questionable for the color-suppressed decays $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to \stackrel{{\mbox{\tiny (---)}}}{D^0}\phi$ and sizable CP violation may show up in the corresponding untagged rates \cite{dunietz}. Concerning such CP-violating effects and the extraction of $\gamma$ from {\it untagged} rates, the decays $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to D_s^{\ast\pm} K^{\ast\mp}$ and $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to \stackrel{{\mbox{\tiny(---)}}}{D^{\ast0}}\phi$ are expected to be more promising than the transitions discussed above. As was shown in Ref.~\cite{fd2}, the time-dependences of their untagged angular distributions allow a {\it clean} extraction of the CKM angle $\gamma$ {\it without} any additional input. The final state configurations of these decays are not admixtures of CP eigenstates as in Section~\ref{admixtures}. They can, however, be classified by their parity eigenvalues. A characteristic feature of the angular distributions are interferences between parity-even and parity-odd configurations that may lead to potentially large CP-violating effects in the untagged data samples even when all strong phase shifts vanish. An example of such an {\it untagged} CP-violating observable is the following quantity \cite{fd2}: \begin{eqnarray} \lefteqn{\mbox{Im}\left\{\left[A_f^\ast(t)\,A_\perp(t)\right]\right\}+ \mbox{Im}\left\{\left[A_f^{\mbox{{\tiny C}}\ast}(t)\, A_\perp^{\mbox{{\tiny C}}}(t)\right]\right\}}\nonumber\\ &&\propto\left(e^{-\Gamma_L^{(s)}t}-e^{-\Gamma_H^{(s)}t}\right) \bigl\{|R_f|\cos(\delta_f-\vartheta_\perp)\nonumber\\ &&+|R_\perp|\cos(\delta_\perp-\vartheta_f)\bigr\}\,\sin\gamma.\label{e14} \end{eqnarray} In that expression bilinear combinations of certain decay amplitudes (see Eq.~(\ref{e9})) show up, $f\in\{0,\parallel\}$ denotes a linear polarization state \cite{rosner} and $\delta_f$, $\vartheta_f$ are CP-conserving phase shifts that are induced through strong final state interaction effects. For the details concerning the observable Eq.~(\ref{e14}) -- in particular the definition of the relevant charge-conjugate amplitudes $A_f^{\mbox{{\tiny C}}}$ and the quantities $|R_f|$ -- the reader is referred to Ref.~\cite{fd2}. Here I would like to emphasize only that the strong phase shifts enter in the form of {\it cosine} terms. Therefore non-trivial strong phases are -- in contrast to Eq.~(\ref{e13}) -- not essential for CP violation in the corresponding untagged data samples and one expects, even within the factorization approximation, which may apply to the color-allowed modes $\stackrel{{\mbox{\tiny (---)}}}{B_s}\to D_s^{\ast\pm} K^{\ast\mp}$, potentially large effects. Since the soft photons in the decays $D_s^\ast\to D_s\gamma$, $D^{\ast0} \to D^0\gamma$ are difficult to detect, higher resonances exhibiting significant all-charged final states, e.g.\ $D_{s1}(2536)^+\to D^{\ast+}K^0$, $D_1(2420)^0\to D^{\ast+}\pi^-$ with $D^{\ast+}\to D^0\pi^+$, may be more promising for certain detector configurations. A similar comment applies also to the mode $B_s\to D_s^{\ast+}D_s^{\ast-}$ discussed in Subsection~\ref{gold}. \section{Conclusions} The oscillatory $\Delta M_st$ terms arising from $B_s^0-\overline{B_s^0}$ mixing, which may be too rapid to be resolved with present vertex technology, cancel in {\it untagged} rates of $B_s$ decays that depend therefore only on the two exponents $e^{-\Gamma_L^{(s)}t}$ and $e^{-\Gamma_H^{(s)}t}$. If the width difference $\Delta\Gamma_s$ is sizable -- as is expected from theoretical analyses -- {\it untagged} $B_s$ decays may allow the determination both of the CKM angle $\gamma$ and of the Wolfenstein parameter $\eta$ and may furthermore provide valuable insights into the mechanism of CP violation and the hadronization dynamics of the corresponding decays. To this end certain angular distributions may play a key role. Compared to the tagged case, such untagged measurements are much more promising in view of efficiency, acceptance and purity. A lot of statistics is required, however, and the natural place for these experiments seems to be a hadron collider. Obviously the feasibility of untagged strategies to extract CKM phases depends crucially on a sizable width difference $\Delta\Gamma_s$. Even if it should turn out to be too small for such untagged analyses, once $\Delta\Gamma_s\not=0$ has been established experimentally, the formulae developed in Refs.~\cite{fd1,fd2} have also to be used to determine CKM phases correctly from tagged measurements. In this sense we cannot lose and an exciting future concerning $B_s$ decays may lie ahead of us! \section*{Acknowledgment} I would like to thank Isi Dunietz for the most pleasant and enjoyable collaboration on the \mbox{topics} presented in this talk. \section*{References}

proofpile-arXiv_065-544

\section{\@startsection {section}{1}{\z@}{-3.5ex plus-1ex minus -.2ex}{2.3ex plus.2ex}{\reset@font\center\bf}} \def\thebibliography#1{\section*{\refname\@mkboth {\uppercase{\refname}}{\uppercase{\refname}}}\list {\@biblabel{\arabic{enumiv}}}{\settowidth\labelwidth{\@biblabel{#1}}% \usecounter{enumiv}% \let\p@enumiv\@empty \def\theenumiv{\arabic{enumiv}}}% \def\newblock{\hskip .11em plus.33em minus.07em}% \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \def\ps@myheadings{\let\@mkboth\@gobbletwo \def\@oddhead{\hfil{\sl\rightmark}\hfil}% \def\@oddfoot{\hfil\rm\thepage}\def\@evenhead{\hfil{\sl\leftmark}\hfil}% \def\@evenfoot{\rm\thepage\hfil}\def\sectionmark##1{} \def\subsectionmark##1{}} \@addtoreset{equation}{section} \def\thesection.\arabic{equation}{\thesection.\arabic{equation}} \catcode`@=12 \pagestyle{myheadings} \def\begin{equation}{\begin{equation}} \def\end{equation}{\end{equation}} \def\Comm#1{{\tt [#1]}} \def{\exp(2m\pi i/3)}{{\exp(2m\pi i/3)}} \def{\exp(-2m\pi i/3)}{{\exp(-2m\pi i/3)}} \def{\exp(2\pi i/3)}{{\exp(2\pi i/3)}} \def\exp({2\pi i\over n}){{\exp(-2\pi i/3)}} \def{q\to\q}{{q\to{\exp(2\pi i/3)}}} \deff(\theta){f(\theta)} \def{d\over d\theta}{{d\over d\theta}} \def{\rm id}{{\rm id}} \def{\theta^{n-1}\over[n-1]_q!}{{\theta^{n-1}\over[n-1]_q!}} \def{\partial\over\partial\theta}{{\partial\over\partial\theta}} \def{\partial\over \partial t}{{\partial\over \partial t}} \def{d_R\over d_R\theta}{{d_R\over d_R\theta}} \def{d_L\over d_L\theta}{{d_L\over d_L\theta}} \def\partial_t{\partial_t} \def\lim_{q\to\q}{\lim_{q\to{\exp(2\pi i/3)}}} \def\lim_{q\to\qq}{\lim_{q\to\exp({2\pi i\over n})}} \def{\sum_{n=0}^\infty}{{\sum_{n=0}^\infty}} \def{\sum_{m=0}^\infty}{{\sum_{m=0}^\infty}} \def{\cal D}{{\cal D}} \def{\it i.e.}{{\it i.e.}} \catcode`@=11 \newdimen\z@ \z@=0pt \def\m@th{\mathsurround=\z@} \def\ialign{\everycr{}\tabskip\z@skip\halign} \def\eqalign#1{\null\,\vcenter{\openup\jot\m@th \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} \def\matrix#1{\null\,\vcenter{\normalbaselines\m@th \ialign{\hfil$##$\hfil&&\quad\hfil$##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \catcode`@=12 \begin{document} \begin{center} {\bf BRAIDED STRUCTURE OF FRACTIONAL $Z_3$-SUPERSYMMETRY} \footnote{Presented at $5^{th}$ Colloquium `Quantum groups and integrable systems', Prague, 20-22 June 1996.} \\ \vspace{0.5cm} \setcounter{footnote}{3} {\sc J. A. de Azc\'arraga$^\dagger$, R. S. Dunne$^\ddagger$, \\ A. J. Macfarlane$^\ddagger$ and J. C. P\'erez Bueno$^\dagger$ \footnote{E-mails: azcarrag@evalvx.ific.uv.es, r.s.dunne@damtp.cam.ac.uk, \\ $\phantom{AAAAAAAi}$ a.j.macfarlane@damtp.cam.ac.uk and pbueno@lie.ific.uv.es.} } \begin{small} \\[2mm] {\sl $\dagger$ Departamento de F\'{\i}sica Te\'orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,} {\sl E-46100 Burjassot (Valencia) Spain.} \\[2mm] {\sl $\ddagger$ Department of Applied Mathematics \& Theoretical Physics,} \\ {\sl University of Cambridge, Cambridge CB3 9EW, U.K.} \end{small} \end{center} \vspace{0.5cm} \noindent {\small It is shown that fractional $Z_3$-superspace is isomorphic to the ${q\to\q}$ limit of the braided line. $Z_3$-supersymmetry is identified as translational invariance along this line. The fractional translation generator and its associated covariant derivative emerge as the ${q\to\q}$ limits of the left and right derivatives from the calculus on the braided line. } \section{Brackets and q-grading} \hskip\parindent Our aim here is to reformulate some results of a previous paper \cite{AM}, where the structure of fractional supersymmetry was investigated from a group theoretical point of view, from a braided Hopf algebra approach. We shall not be concerned here with the possible applications of fractional supersymmetry and will refer instead to \cite{AM,DMPA} for references on this aspect. We begin by defining the bracket \begin{equation} [A,B]_{q^r}:=AB-q^r BA \quad,\label{two} \end{equation} where $q$ and $r$ are just arbitrary complex numbers. If we assign an integer grading $g(X)$ to each element $X$ of some algebra, such that $g(1)$=0 and $ g(XY)=g(X)+g(Y),$ for any $X$ and $Y$, we can define a bilinear graded $z$-bracket as follows, \begin{equation} [A,B]_z=AB-q^{-g(A)g(B)}BA\quad,\quad z=q^{-g(A)g(B)}\quad. \label{four} \end{equation} Here $A$ and $B$ are elements of the algebra, and of pure grade. The definition may be extended to mixed grade terms using the bilinearity. We also have \begin{equation} [r]_q:={{1-q^r}\over{1-q}}\quad,\quad [r]_q!:=[r]_q[r-1]_q[r-2]_q...[2]_q[1]_q\quad, \label{six} \end{equation} supplemented by $[0]_q!=1$. When $q$ is $n$-root of unity the previous grading scheme becomes degenerate, so that in effect the grading of an element is only defined modulo $n$. In this case also have $[r]_q=0$ when $r$ modulo $n$ is zero $(r\neq0)$. \section{$q$-calculus and the braided line} \hskip\parindent Consider the braided line \cite{MajI,MajII}, a simple deformation of the ordinary line characterized by a single parameter $q$. Our braided line Hopf algebra will consist of a single variable $\theta$, of grade 1, upon which no additional conditions are placed for generic $q$, by which we mean that $q$ is not a root of unity. The braided Hopf structure of this deformed line is as follows. It has braided a coproduct, \begin{equation} \Delta\theta=\theta\otimes1+1\otimes\theta\quad, \label{seven} \end{equation} \begin{equation} (1\otimes\theta)(\theta\otimes1)=q\theta\otimes\theta\quad, \quad (\theta\otimes1)(1\otimes\theta)=\theta\otimes\theta\quad, \label{nine} \end{equation} the braiding being given by ${\cal B}(\theta\otimes\theta)=q\theta\otimes\theta$. There are also a counit and antipode, \begin{equation} \varepsilon(\theta)=0\quad,\quad S(\theta^r)=q^{r(r-1)\over2}(-\theta)^r\quad, \label{oneone} \end{equation} which satisfy all the usual Hopf algebraic relations, as long as the braiding is remembered. From the braided Hopf algebra perspective, the coproduct generates a shift along the braided line. To bring this out more clearly we use a group-like notation for the coproduct and write \begin{equation} \theta=1\otimes\theta\quad,\quad \delta\theta=\epsilon=\theta\otimes1\quad, \label{onetwo} \end{equation} so that (\ref{nine}) leads to \begin{equation} [\epsilon,\theta]_{q^{-1}}=0\quad. \label{onethree} \end{equation} In this form, the coproduct (\ref{seven}) expresses the additive group law, \begin{equation} \Delta\theta=\epsilon+\theta \quad \label{onefour} \end{equation} which on the fractional Grassmann variable $\theta$ corresponds \cite{AM} to the action of the left translation $L_\epsilon$, $L_\epsilon\theta\equiv\theta'=\epsilon+\theta$. The above expressions provide a basis upon which to construct a differential (and integral \cite{DMPA}) calculus on the braided line. We can introduce an algebraic left differentiation operator ${\cal D}_L$, in analogy with the undeformed case, via the requirement $[\epsilon {\cal D}_L,\theta]=\epsilon,$ which implies that \begin{equation} [{\cal D}_L,\theta]_q=1\quad,\quad ({d\over d\theta}\theta=1)\quad. \label{onesix} \end{equation} This corresponds to defining [cf. (\ref{four})] the left derivative ${\cal D}_L$ by \begin{equation} [{\cal D}_L,\theta]_z:=1\quad,\quad z=q\quad. \label{onesixa} \end{equation} Regarding (\ref{onefour}) as the definition the left translation by $\epsilon$, and identifying ${\cal D}\equiv{\cal D}_L$, we can go on considering right shifts $R_\eta:\theta\mapsto\theta+\eta$ of parameter $\eta$ where $[\theta,\eta]_{q^{-1}}=[\eta,\theta]_q=0$. Reasonings similar to the above lead us to a right derivative operator ${\cal D}_R$, which satisfies \begin{equation} [\theta,{\cal D}_R]_z=[\theta,{\cal D}_R]_q:=1\quad,\quad ([{\cal D}_R,\theta]_{q^{-1}}=-q^{-1})\quad. \label{oneseven} \end{equation} It may be shown that the left and right derivative operators are in general related \cite{DMPA} by \begin{equation} {\cal D}_R=-q^{-(1+N)}{\cal D}_L\quad, \label{oneeight} \end{equation} where $N$ is a number-like operator satisfying, \begin{equation} [N,\theta]=\theta\quad,\quad [N,{\cal D}_L]=-{\cal D}_L\quad, \label{onenine} \end{equation} and consequently $[N,{\cal D}_R]=-{\cal D}_R$. This implies that $ [{\cal D}_L,{\cal D}_R]_{q^{-1}}=0$ or, alternatively, $[\epsilon{\cal D}_L,\eta{\cal D}_R]=0$ (commutation of the left $L_\epsilon$ and right $R_\eta$ shifts). Let $f(\theta)$ be a function of $\theta$ defined by the positive power series expansion, \begin{equation} f(\theta)={\sum_{m=0}^\infty} {C_m\theta^m\over[m]_q!}\quad, \label{twoone} \end{equation} where the $C_m$ are ordinary numbers. The derivative of $f(\theta)$ is generated by the graded bracket (\ref{onesix}) as follows, \begin{equation} \eqalign{{d\over d\theta} f(\theta)&:=[{\cal D},f(\theta)]_z ={\sum_{m=0}^\infty} C_m\left[{\cal D},{\theta^m\over[m]_q!}\right]_z ={\sum_{m=0}^\infty} C_m\left[{\cal D},{\theta^m\over[m]_q!}\right]_{q^m}\cr &={\sum_{m=0}^\infty} C_m{\theta^{m-1}\over[m-1]_q!}\quad.\cr} \label{twotwo} \end{equation} This clearly reduces to the calculus on the undeformed line when $q=1$. The left translation {\it i.e.}, the coproduct (\ref{onefour}) is given \cite{DMPA,AM} by a deformed exponentiation (see \cite{GR,MajIV,McAnally}) of $\epsilon{\cal D}_L$. We shall refer to the differential calculus defined by (\ref{onesix}), (\ref{oneseven}) and (\ref{oneeight}) as $q$-calculus. \section{$q$-calculus in the $q$ root of unity limit and $(Z_3)$ fractional supersymmetry} \hskip\parindent When $q^m=1$, $[m]_q=0\ (m\ne 1)$ and expressions such as ${\theta^m\over [m]_q!}$ can be made meaningful only by assuming that $\theta^m$ is also zero. In that case, we may identify the limit $\lim_{q\to\exp(2\pi i/m)}{\theta^m\over [m]_q!}$ with a degree zero (`bosonic') variable $t$. It was shown in \cite{PLB} that for the $m=2$ case this procedure leads to a braided interpretation of supersymmetry, the $Z_2$-graded group structure of which was discussed in \cite{AldAz}. We shall consider here in detail the case of $Z_3$ fractional supersymmetry, the group analysis of which may be found in \cite{AM} (the general $Z_n$ case will be discussed in \cite{DMPA}). The limit relevant here is the ${q\to\q}$ limit. To take this limit we note that for $q$ not a root of unity we have in general the relationships \begin{equation} \left[{\cal D}_L,{\theta^m\over[m]_q!}\right]_{z}= \left[{\cal D}_L,{\theta^m\over[m]_q!}\right]_{q^m}= {\theta^{m-1}\over[m-1]_q!}= \left[{\theta^m\over[m]_q!},{\cal D}_R\right]_{q^m}= \left[{\theta^m\over[m]_q!},{\cal D}_R\right]_{z}\quad. \label{twofive} \end{equation} In taking the ${q\to\q}$ limit of the above formulae we encounter difficulties when $m=3$ since $[3]_q=0$. But it is possible to retain (\ref{twofive}) by requiring that the ${q\to\q}$ limit of $\theta^3\over[3]_q!$ be finite and nonzero. This in turn requires $\theta^3\to 0$ as ${q\to\q}$. This is preserved under the left shift $\theta\to\epsilon+\theta$, since $(\epsilon+\theta)^3=0$ follows from (\ref{onethree}) when ${q\to\q}$ provided that $\theta^3=0=\epsilon^3.$ We now note that under complex conjugation we have $\overline{[3]_q!}=q^{-3}[3]_q!$ (along the circle of radius 1). Then, in the ${q\to\q}$ limit the $q$-factorial $[3]_{\exp(2\pi i/3)}!$ is real. As a result, we define \begin{equation} t:=-\lim_{q\to\q}{\theta^3\over[3]_q!}\quad, \label{twoseven} \end{equation} where the - sign is introduced to compare easily with \cite{AM}. Since $\theta$ is assumed real, $t$ will also be real. By using the identities \begin{equation} \lim_{q\to\q}\left({[3r]_q\over[3]_q}\right)=\lim_{q\to\q} \left({1-q^{3r}\over1-q^3}\right)=r \quad, \label{twoeight} \end{equation} \begin{equation} \lim_{q\to\q}\left({[3r+1]_q\over[1]_q}\right)=\lim_{q\to\q}\left({1-q^{3r+1}\over1-q} \right)=1\quad, \label{twonine} \end{equation} and \begin{equation} \lim_{q\to\q}\left({[3r+2]_q\over[2]_q}\right)=\lim_{q\to\q}\left({1-q^{3r+2}\over1-q^2} \right)=1\quad, \label{twoninea} \end{equation} we have, for $p=0,1,2,$ \begin{equation} \lim_{q\to\q}\left({\theta^{3r+p}\over[3r+p]_q!}\right)= {\theta^p\over[p]_{\exp(2\pi i/3)}!}{(-t)^r\over r!} \quad. \label{thirty} \end{equation} As mentioned, the finite limit in (\ref{twoseven}) denoted by $t$ was introduced in order to make possible the ${q\to\q}$ limit of (\ref{twofive}) at $m=3$. Similar problems arise for all $m\geq 3$ \cite{DMPA}, and the importance of (\ref{thirty}) is that it shows that these can also be handled in terms of $t$. Thus, any function $f(\theta)$ on the braided line at generic $q$ leads in the ${q\to\q}$ limit to a function of the form $f(t,\theta)$ (or `fractional superfield' on fractional superspace $(t,\theta)$). To investigate further the properties of $t$, and to see how it fits into our $q$-calculus, let us now consider \begin{equation} \left[{\cal D}_L, \left[{\cal D}_L, \left[{\cal D}_L,{\theta^3\over[3]_q!}\right]_{q^3} \right]_{q^2} \right]_q=1= \left[ \left[ \left[ {\theta^3\over[3]_q!},{\cal D}_R \right]_{q^3}, {\cal D}_R\right]_{q^2},{\cal D}_R\right]_q \quad, \label{threeone} \end{equation} (notice the appropriate $q$-factor in each bracket depending on the grading of its components, cf. (\ref{twofive}), (\ref{four})) valid for all $q\neq{\exp(2\pi i/3)}$. Taking the ${q\to\q}$ limit we see that (\ref{threeone}) reduces to $[{\cal D}_L^3,t]=-1=[t,{\cal D}_R^3],$ so that by identifying \begin{equation} \partial_t=-{\cal D}_L^3={\cal D}_R^3\quad, \label{threethree} \end{equation} we have \begin{equation} [\partial_t,t]=1\quad,\label{threefour} \end{equation} which is just the defining relation of the algebra associated with ordinary calculus. Let us consider the {\it left calculus}. Using $\partial_t$ given by (\ref{threethree}) to induce differentiation with respect to $t$, the full $q$ calculus for $q={\exp(2\pi i/3)}$ obtained from (\ref{twotwo}) and (\ref{twoseven}) is given by, \begin{equation} \eqalign{ &\qquad\qquad\qquad\qquad {d\over d\theta}\theta =[{\cal D},\theta]_{\exp(2\pi i/3)}=1 \quad,\cr {d\over d\theta} t&=[{\cal D}_L,t]=\lim_{{q\to\q}}[{\cal D}_L,-{\theta^3\over[3]_q!}]_{q^3} =-{\theta^2\over[2]_{\exp(2\pi i/3)}}={\exp(2\pi i/3)}\theta^2\quad,\cr {\partial\over \partial t} t & =[\partial_t,t]=1\quad,\quad {\partial\over \partial t}\theta=[\partial_t,\theta]=-[{\cal D}_L^3,\theta]= -[{\cal D}_L,[{\cal D}_L,[{\cal D}_L,\theta]_q]_{q^0}]_{q^{-1}}=0\quad.\cr} \label{threefive} \end{equation} Since ${\partial\over \partial t}\theta=0$ and ${d\over d\theta} t\neq0$, we can only avoid a contradiction by interpreting ${\partial\over \partial t}$ as a partial derivative, and ${d\over d\theta}$ as a total derivative, a result which we implicitly took into account when choosing our notation. We can also define partial differentiation with respect to $\theta$. We do this as follows \begin{equation} {\partial\over\partial\theta}\theta:=[\partial_\theta,\theta]_{\exp(2\pi i/3)}=1\quad,\quad {\partial\over\partial\theta} t:=[\partial_\theta,t]=0\quad. \label{threesix} \end{equation} Using this definition we are able to perform a chain rule expansion of the total derivative, so that \begin{equation} {d\over d\theta}={d\theta\over d\theta}{\partial\over\partial\theta}+{d t\over d\theta}{\partial\over \partial t}={\partial\over\partial\theta} -{\theta^2\over[2]_{\exp(2\pi i/3)}}{\partial\over \partial t}={\partial\over\partial\theta}+{\exp(2\pi i/3)}\theta^2{\partial\over \partial t} \quad. \label{threeseven} \end{equation} By substituting (\ref{threeseven}) into the definition (\ref{threethree}) we obtain an additional but expected condition, \begin{equation} {\partial^3\over\partial\theta^3}=0\quad. \label{threeeight} \end{equation} This can all be put into the algebraic form ${\cal D}_L=\partial_\theta+ {\exp(2\pi i/3)}{\theta^2}\partial_t,$ where \begin{equation} [\partial_\theta,\theta]_{\exp(2\pi i/3)}=1 \quad ([\theta,\partial_\theta]_\exp({2\pi i\over n})=-\exp({2\pi i\over n})) \label{leftder} \end{equation} and $\partial_\theta^3=0.$ The {\it right calculus} may be introduced similarly. Besides $\partial_t={\cal D}_R^3$, we have \begin{equation} \eqalign{ {d_R\over d_R\theta}\theta=[\theta,{\cal D}_R]_{\exp(2\pi i/3)}=1\;, &\; {d_R\over d_R\theta}t= -\lim_{q\to\q}[{\theta^3\over [3]_q!},{\cal D}_R]_{q^3}= \exp(2\pi i/3)\theta^2\,, \cr {\partial\over\partial t}t=[\partial_t,t]=1\quad, &\quad {\partial\over\partial t}\theta=[{\cal D}_R^3,\theta]=0\quad. \cr} \label{new} \end{equation} Introducing a partial right derivative $\delta_\theta$ by \begin{equation} [\theta,\delta_\theta]_{\exp(2\pi i/3)}:=1\quad ([\delta_\theta,\theta]_{\exp({2\pi i\over n})}=-\exp({2\pi i\over n}))\quad, \label{rightder} \end{equation} the expression of ${\cal D}_R$ differs by a sign from that of ${\cal D}\equiv{\cal D}_L$, namely \begin{equation} {d_R\over d_R\theta}={\delta\over\delta\theta}+{\theta^2\over[2]_{\exp(2\pi i/3)}}{\partial\over \partial t}= {\delta\over\delta\theta}-{\exp(2\pi i/3)}\theta^2{\partial\over \partial t}\ ,\ {\cal D}_R=\delta_\theta-{\exp(2\pi i/3)}{\theta^2}\partial_t\,. \label{chainrule} \end{equation} The $\delta_\theta$ introduced above differs from that found in \cite{AM} in a $-{\exp(2\pi i/3)}$ factor, $\delta_\theta$(here)$=-\exp(-2\pi i/3)\delta_\theta(\hbox{ref.\ }\cite{AM})$ \section{$Z_3$-fractional supersymmetry from a braided point of view} \hskip\parindent If follows from the above that ${\cal D}_L\equiv Q$ and $-{\exp(2\pi i/3)}{\cal D}_R\equiv D$ are just the supercharge and the corresponding covariant derivative encountered in $(Z_3)$ fractional supersymmetry \cite{AM}. Hence \begin{equation} Q^3=-\partial_t\quad,\quad D^3=-\partial_t\quad. \label{algebra} \end{equation} We may identify $Q$ and $D$ as, respectively, the generators of left and right shifts along the braided line at $q={\exp(2\pi i/3)}$. These were shown in \cite{AM} to correspond to the right- $[Q]$ and left-invariant $[D]$ `fractional translation' generators. To further investigate this point of view we examine the Hopf structure on the braided line in the ${q\to\q}$ limit. When $q={\exp(2\pi i/3)}$, (\ref{nine}) reduces to \begin{equation} \eqalign{(1\otimes\theta)(\theta\otimes1)&={\exp(2\pi i/3)}\theta\otimes\theta\quad,\quad (\theta\otimes1)(1\otimes\theta)=\theta\otimes\theta\quad,\cr} \label{fourseven} \end{equation} so that from (\ref{seven}), we find \begin{equation} \Delta\theta^3=\theta^3\otimes1+1\otimes\theta^3+ (1+{\exp(2\pi i/3)}+[{\exp(2\pi i/3)}]^2)(\theta\otimes\theta^2 + \theta^2\otimes\theta)=0\quad, \label{fournine} \end{equation} as required by the homomorphism property of the coproduct. The counit and antipode take the following form \begin{equation} \varepsilon(\theta)=0\quad,\quad S(\theta)=-\theta\quad. \label{fifty} \end{equation} The braided structure (\ref{seven}),(\ref{fourseven}) is the standard one, see for example \cite{MajIII,MajII} for a discussion of super and anyonic quantum groups. The new structure appears when the variable $t$ defined by the limit (\ref{twoseven}) is introduced \cite{PLB,DMPA}. From the definition (\ref{twoseven}) it follows that $\theta$ and $t$ commute. From (\ref{twoseven}) and (\ref{seven})-(\ref{nine}) we compute $\lim_{q\to\q}\Delta [{-\theta^3\over[3]_q!}]$ using that $[n]_q=1+q+\ldots+q^{n-1}$ for generic $q$. This, together with $[t,\theta]=0$ and (\ref{seven})-(\ref{oneone}), shows that the algebra generated by $(t,\theta)$ has a braided Hopf algebra structure, with \begin{equation} \Delta t=t\otimes1+1\otimes t+{\exp(2\pi i/3)}(\theta\otimes\theta^2 + \theta^2\otimes\theta) \quad,\quad \epsilon(t)=0\quad,\quad S(t)=-t\quad, \label{fiveone} \end{equation} (for instance $\Delta([t,\theta])=0=[\Delta t,\Delta\theta]$). This means that although $t$ and $\partial_t$ satisfy the algebra associated with ordinary (undeformed) calculus, $t$ has non primitive coproduct: the coaddition no longer corresponds to a time translation. Considered along with the chain rule expansion of the $q$-calculus derivative (\ref{threeseven}), it is clear that in the ${q\to\q}$ limit we cannot decompose the $q$-calculus algebra into unrelated $t$ and $\theta$ parts. Indeed, from (\ref{fiveone}) we see that when the braided group is considered, the appearance of $\theta$ in the coproduct of $t$ means that no such decomposition can be performed. The fact that we cannot regard this braided Hopf algebra as a product entity is an essential feature of fractional supersymmetry in general. To see this for the present $Z_3$-case we rewrite the coproducts of $\theta$ and $t$ using the notation (\ref{onetwo}). Using (\ref{seven}), (\ref{fournine}) and definition (\ref{twofive}) we obtain \begin{equation} \eqalign{\theta\to\Delta\theta=\epsilon+\theta\quad,\quad t\to\Delta t=t+\tau+q(\epsilon\theta^2+\epsilon^2\theta)\quad,\cr} \label{fivetwo} \end{equation} where now $q\equiv{\exp(2\pi i/3)}$ (so that $-1/[2]_q=q$) and \begin{equation} \tau=\lim_{q\to{\exp(2\pi i/3)}}=-{\epsilon^3\over [3]_q!} \label{fivethree} \end{equation} is a time translation independent of $t$. This is just the form of the finite $Z_3$-supersymmetry transformation of \cite{AM}; the transformation of $t$ follows from that of $\theta$ via the relationship (\ref{twoseven}) and the coproduct $\Delta$. \section{Final remarks} \hskip\parindent To conclude, let us summarize our results \cite{PLB,DMPA} and outline the new point of view which they provide. For generic $q$ the braided line described in section 3 is well defined. In the ${q\to\q}$ limit the nilpotency of $\theta$ prevents us from having a complete description of the braided line and its associated differential calculus. A convenient way in which we can obtain such a description is to introduce an additional variable $t$, defined as in (\ref{twoseven}). From (\ref{thirty}) this is seen to carry a structure which, for generic $q$, is related to $\theta^3$ and higher powers of $\theta$. So in the ${q\to\q}$ limit the braided line is made up of the two variables $\theta$ and $t$, which span the one-dimensional fractional superspace. Furthermore, under a shift along this braided line $\theta$ and $t$ transform exactly as in $Z_3$ fractional supersymmetry. Thus we are able to identify $Z_3$-superspace with the $q={\exp(2\pi i/3)}$ limit of the braided line, and $Z_3$-supersymmetry as translational invariance along this line. Clearly, (fractional) superspace cannot be regarded as the tensor product of independent $t$ and $\theta$ parts; it is instead a single braided geometric entity. This provides a braided interpretation of the central extension character of the $Z_3$-graded group aspect of fractional supersymmetry discussed in \cite{AM}. To conclude, we wish to stress that the above results are not restricted to the $Z_3$ case. Similar results also hold for supersymmetry \cite{PLB} (cf. \cite{AldAz}) and in the $Z_n$ case \cite{DMPA}. \section*{Acknowledgements} This paper describes research supported in part by E.P.S.R.C and P.P.A.R.C. (UK) and by the C.I.C.Y.T (Spain). J.C.P.B. wishes to acknowledge an FPI grant from the CSIC and the Spanish Ministry of Education and Science. \thebibliography{References} \bibitem{AM} {J.A. de Azc\'arraga and A.J.Macfarlane, J. Math. Phys {\bf 37}, 1115-1127 (1996).} \bibitem{DMPA} {R.S. Dunne, A.J. Macfarlane, J. A. de Azc\'arraga and J.C. P\'erez Bueno, {\it Geometrical foundations of fractional supersymmetry}, forthcoming.} \bibitem{MajI} {S. Majid, {\it Introduction to braided geometry and q-Minkowski space}, Varenna lectures, hep-th/9410241 (1994).} \bibitem{MajII} {S. Majid, {\it Foundations of quantum group theory}, Camb. Univ. Press, (1995).} \bibitem{GR} {G. Gasper and M. Rahman, {\it Basic hypergeometric series}, Camb. Univ. Press, (1995).} \bibitem{MajIV} {S. Majid, J. Math. Phys. {\bf 34}, 4843-4856 (1993).} \bibitem{McAnally} {D.S. McAnally, J. Math. Phys {\bf 36}, 546 (1996).} \bibitem{PLB} {R.S. Dunne, A.J. Macfarlane, J. A. de Azc\'arraga and J.C. P\'erez Bueno, {\it Supersymmetry from a braided point of view}, DAMTP/96-51, FTUV/96-27, IFIC/96-31 (hep-th/9607220), to appear in Phys. Lett. B.} \bibitem{AldAz} {V. Aldaya and J. A. de Azc\'arraga, J. Math. Phys. {\bf 26}, 1818-1821 (1985).} \bibitem{MajIII} {S. Majid, {\it Anyonic Quantum Groups}, in {\it Spinors, Twistors, Clifford Algebras and Quantum Deformations (Proc. of 2nd Max Born Symposium, Wroclaw, Poland, 1992)}, Z. Oziewicz et al, eds., p. 327-336, Kluwer.} \end{document}

proofpile-arXiv_065-545

\section{Introduction: The Spinless Salpeter Equation} One's attitude to the well-known ``spinless Salpeter equation'' may be reflected by either of the following two approaches (or points of view): \begin{itemize} \item On the one hand, this spinless Salpeter equation may be regarded to represent some standard approximation to the Bethe--Salpeter formalism for the description of bound states within a relativistic quantum field theory. It may be derived from the Bethe--Salpeter equation \cite{salpeter51} by two steps: \begin{enumerate} \item Eliminate---in full accordance with the spirit of instantaneous interactions---any dependence on timelike variables to obtain in this way the so-called Salpeter equation \cite{salpeter52}. \item Neglect any reference to all the spin degrees of freedom of the involved bound-state constituents and restrict your formalism exclusively to positive-energy solutions. \end{enumerate} \item On the other hand, this spinless Salpeter equation may be viewed as one of the most straightforward generalizations of the standard nonrelativistic quantum theory towards the reconciliation with all the requirements imposed by special relativity. To be precise, this generalization consists of incorporating the square-root operator of the relativistic expression for the kinetic energy of the involved particles. For the particular case of two particles of equal mass $m$ and relative momentum ${\bf p}$, the kinetic-energy operator $T$ is given by \begin{equation} T({\bf p}) \equiv 2\,\sqrt{{\bf p}^2 + m^2}\ . \label{eq:kinenergy} \end{equation} All the forces operating between the bound-state constituents are tacitly assumed to be described by an arbitrary static interaction potential $V$. For the special case of two particles, this interaction potential should depend only on the relative coordinate ${\bf x}$ of these particles: $V = V({\bf x})$. \end{itemize} In any case, the self-adjoint Hamiltonian $H$ governing the dynamics of any quantum system to be described by the spinless Salpeter equation will be of the form \begin{equation} H = T({\bf p}) + V({\bf x})\ . \label{eq:ham-sseq} \end{equation} The two-particle spinless Salpeter equation to be investigated here is then nothing else but the eigenvalue problem for this Hamiltonian $H$, $$ H|\chi_k\rangle = E_k|\chi_k\rangle\ ,\quad k = 0,1,2,\dots\ , $$ for Hilbert-space eigenvectors $|\chi_k\rangle$ corresponding to energy eigenvalues $$ E_k \equiv \frac{\langle\chi_k|H|\chi_k\rangle}{\langle\chi_k|\chi_k\rangle}\ . $$ For the sake of simplicity, we shall focus our attention to the physically most relevant case of central potentials, i. e., potentials which depend only on the modulus $|{\bf x}|$ of the configuration-space relative coordinate: \begin{equation} V = V(|{\bf x}|)\ . \label{eq:centralpot} \end{equation} In the above form, the spinless Salpeter equation appears to be a very promising candidate for the (semi)relativistic description of hadrons as bound states of (constituent) quarks within the framework of potential models \cite{lucha91,lucha92,lucha95}, or, at least, the first step in the correct direction \cite{gara89,lucha92sign}. However, the presence of the relativistic kinetic-energy operator (\ref{eq:kinenergy}) in (\ref{eq:ham-sseq}) or, to do justice to the spinless Salpeter equation, the nonlocality of this operator $H$, that is, more precisely, of either the kinetic-energy operator $T$ in configuration space or the interaction-potential operator $V$ in momentum space, renders difficult to arrive at rigorous analytical statements about the corresponding energy spectrum. In view of this, numerous attempts to circumvent these problems have been proposed. Some very brief account of the history of these attempts may be found, for instance, in Ref.~\cite{lucha94}. These approaches include, among others, the development of elaborate numerical approximation methods \cite{nickisch84,lucha91num,fulcher93} as well as the construction of effective Hamiltonians which, in spite of their apparently nonrelativistic form, incorporate relativistic effects by sophisticated momentum dependence of the involved parameters \cite{lucha93eff}. A lot of information on the solutions of the spinless Salpeter equation may even be gained by application of a relativistic virial theorem \cite{lucha89rvt}, most easily derived from a rather general ``master virial theorem'' \cite{lucha90rvt}. The (from the physical point of view perhaps most interesting) case of a Coulomb-type static interaction potential, the so-called relativistic Coulomb problem, has been investigated particularly carefully. For the corresponding lowest-lying energy eigenvalues, both lower \cite{herbst77,martin89} and upper \cite{martin89,lucha94varbound,lucha96rcprefer,lucha96rcplowly} bounds have been derived and series expansions \cite{leyaouanc94} in powers of the involved fine structure constant have been given. Here, we intend to pave the way for the calculation of upper bounds on the energy eigenvalues of the spinless Salpeter equation with rather arbitrary interaction potentials. To this end, we apply the famous min--max principle---which controls any such attempt---in a particular basis of our trial space, characterized by generalized Laguerre polynomials. \section{Minimum--Maximum Principle and Rayleigh--Ritz Variational Technique} The derivation of upper bounds on the eigenvalues of some operator $H$ makes, of course, only sense for those operators $H$ which are bounded from below. Accordingly, let us assume from now on that the arbitrary interaction potential (\ref{eq:centralpot}) in our semirelativistic Hamiltonian (\ref{eq:ham-sseq}) is such that this necessary prerequisite holds. For example, for the crucial case of a Coulomb-type static interaction potential, the so-called relativistic Coulomb problem, the demanded semi-boundedness of the spectrum of the Hamiltonian $H$ has been (rigorously) demonstrated by Herbst~\cite{herbst77}. The theoretical basis as well as the primary tool for the derivation of rigorous upper bounds on the eigenvalues of some self-adjoint operator is, beyond doubt, the so-called min--max principle \cite{reed-simon}. An immediate consequence of this min--max principle is the Rayleigh--Ritz technique: Let $H$ be a semi-bounded self-adjoint operator. Let $E_k$, $k = 0,1,2,\dots$, denote the eigenvalues of $H$, ordered according to $E_0\le E_1\le E_2\le\dots$. Let $D_d$ be some $d$-dimensional subspace of the domain of $H$ and let $\widehat E_k$, $k = 0,1,\dots,d-1$, denote all $d$ eigenvalues of this operator $H$ restricted to the space $D_d$, ordered according to $\widehat E_0\le\widehat E_1\le\dots\le\widehat E_{d-1}$. Then the $k$th eigenvalue $E_k$ (counting multiplicity\footnote{\normalsize\ For instance, for a Hamiltonian $H$ depending only on the moduli of momentum ${\bf p}$ and coordinate ${\bf x}$, respectively, states of given orbital angular momentum but different projections of the latter will be degenerate.}) of $H$ satisfies the inequality $$ E_k \le \widehat E_k\ ,\quad k = 0,1,\dots,d-1\ . $$ (For a discussion of the history of inequalities and variational methods for eigenvalue problems, see, e.~g., Ref.~\cite{weinstein72}; for some applications, see, e.~g., Ref.~\cite{flamm82}.) Now, let us assume that this $d$-dimensional subspace $D_d$ is spanned by some set of $d$ orthonormalized (and therefore beyond doubt linearly independent) basis vectors $|\psi_k\rangle$, $k = 0,1,\dots,d-1$: $$ \langle\psi_i|\psi_j\rangle = \delta_{ij}\ ,\quad i,j = 0,1,\dots,d-1\ . $$ Then the set of eigenvalues $\widehat E$ may immediately be determined as the $d$ roots of the characteristic equation \begin{equation} \det\left(\langle\psi_i|H|\psi_j\rangle - \widehat E\,\delta_{ij}\right) = 0\ , \quad i,j = 0,1,\dots,d-1\ , \label{eq:chareq} \end{equation} as becomes clear from an expansion of any eigenvector of the restricted operator $H$ in terms of the set of basis vectors $|\psi_k\rangle$, $k = 0,1,\dots,d-1$, of the subspace $D_d$. \section{Generalized Laguerre Basis} The crucial step in any investigation of the present type is the suitable choice of a basis in the subspace $D_d$. For the case of the semirelativistic Hamiltonian (\ref{eq:ham-sseq}), we find it convenient to work in a basis which involves the so-called generalized Laguerre polynomials. The latter are specific orthogonal polynomials, defined by the power series \cite{abramow} $$ L_k^{(\gamma)}(x) = \sum_{r=0}^k\,(-1)^r \left(\begin{array}{c}k+\gamma\\ k-r \end{array}\right)\frac{x^r}{r!} $$ and normalized according to \cite{abramow} $$ \int\limits_0^\infty{\rm d}x\,x^\gamma\exp(-x)\,L_k^{(\gamma)}(x)\, L_{k'}^{(\gamma)}(x) = \frac{\Gamma(\gamma+k+1)}{k!}\,\delta_{kk'}\ . $$ Consequently, introducing two variational parameters, namely, one, $\mu$, with the dimension of mass as well as a dimensionless one, $\beta$, a generic trial vector $|\psi\rangle$ of the subspace $D_d$, with orbital angular momentum $\ell$ and its projection $m$, will be characterized by the following admittedly very suggestive ansatz for its coordinate-space representation $\psi_{k,\ell m}({\bf x})$: \begin{equation} \psi_{k,\ell m}({\bf x}) = {\cal N}\,|{\bf x}|^{\ell+\beta-1}\exp(-\mu\,|{\bf x}|)\, L_k^{(\gamma)}(2\,\mu\,|{\bf x}|)\,{\cal Y}_{\ell m}(\Omega_{\bf x})\ , \label{eq:ansatz} \end{equation} where normalizability restricts the variational parameter $\mu$ to positive values, $$ \mu>0\ . $$ Here, ${\cal Y}_{\ell m}(\Omega)$ are the spherical harmonics for angular momentum $\ell$ and projection $m$ depending on the solid angle $\Omega$; they are orthonormalized according to \begin{equation} \int{\rm d}\Omega\,{\cal Y}^\ast_{\ell m}(\Omega)\,{\cal Y}_{\ell'm'}(\Omega) = \delta_{\ell\ell'}\,\delta_{mm'}\ . \label{eq:spharorth} \end{equation} The proper orthonormalization of the ansatz (\ref{eq:ansatz}) fixes the parameter $\gamma$ necessarily to the value $\gamma = 2\,\ell+2\,\beta$ and determines the normalization constant ${\cal N}$: \begin{eqnarray*} \psi_{k,\ell m}({\bf x}) &=& \sqrt{\frac{(2\,\mu)^{2\ell+2\beta+1}\,k!} {\Gamma(2\,\ell+2\,\beta+k+1)}}\,|{\bf x}|^{\ell+\beta-1} \exp(-\mu\,|{\bf x}|)\\[1ex] &\times&L_k^{(2\ell+2\beta)}(2\,\mu\,|{\bf x}|)\, {\cal Y}_{\ell m}(\Omega_{\bf x}) \end{eqnarray*} satisfies the normalization condition $$ \int{\rm d}^3x\,\psi_{k,\ell m}^\ast({\bf x)}\,\psi_{k',\ell'm'}({\bf x)} = \delta_{kk'}\,\delta_{\ell\ell'}\,\delta_{mm'}\ . $$ Rather obviously, normalizability constrains the variational parameter $\beta$ too, namely, to a range characterized by $2\,\beta>-1$, i.~e., to the range $$ \beta > -\frac{1}{2}\ . $$ The Fourier transform $\widetilde\psi_{k,\ell m}({\bf p})$ of the above trial function involves the hypergeometric series $F$, defined with the help of the gamma function $\Gamma$ by \cite{abramow} $$ F(u,v;w;z) = \frac{\Gamma(w)}{\Gamma(u)\,\Gamma(v)}\,\sum_{n=0}^\infty\, \frac{\Gamma(u+n)\,\Gamma(v+n)}{\Gamma(w+n)}\,\frac{z^n}{n!}\ ; $$ it reads \begin{eqnarray*} \widetilde\psi_{k,\ell m}({\bf p}) &=& \sqrt{\frac{(2\,\mu)^{2\ell+2\beta+1}\,k!} {\Gamma(2\,\ell+2\,\beta+k+1)}}\,\frac{(-i)^\ell\,|{\bf p}|^\ell}{2^{\ell+1/2} \,\Gamma\left(\ell+\frac{3}{2}\right)}\\[1ex] &\times& \sum_{r=0}^k\,\frac{(-1)^r}{r!} \left(\begin{array}{c}k+2\,\ell+2\,\beta\\ k-r\end{array}\right) \frac{\Gamma(2\,\ell+\beta+r+2)\,(2\,\mu)^r} {({\bf p}^2+\mu^2)^{(2\ell+\beta+r+2)/2}}\\[1ex] &\times& F\left(\frac{2\,\ell+\beta+r+2}{2},-\frac{\beta+r}{2}; \ell+\frac{3}{2};\frac{{\bf p}^2}{{\bf p}^2+\mu^2}\right) {\cal Y}_{\ell m}(\Omega_{\bf p}) \end{eqnarray*} and satisfies the normalization condition $$ \int{\rm d}^3p\,\widetilde\psi_{k,\ell m}^\ast({\bf p)}\, \widetilde\psi_{k',\ell'm'}({\bf p)} = \delta_{kk'}\,\delta_{\ell\ell'}\,\delta_{mm'}\ . $$ In principle, it is straightforward to calculate the expectation values $$ H_{ij}\equiv\langle\psi_i|H|\psi_j\rangle $$ of the Hamiltonian (\ref{eq:ham-sseq}), necessary for applying the min--max principle. Due to the orthonormalization (\ref{eq:spharorth}) of the spherical harmonics ${\cal Y}_{\ell m}(\Omega)$, however, only matrix elements taken between states of identical orbital angular momentum $\ell$ and its projection $m$ will be nonvanishing. \section{Power-Law Potentials} When speculating about the possible shape of a physically meaningful (or phenomenologically acceptable) interaction potential, the very first idea which unavoidably comes to one's mind as a reasonable candidate is an interaction potential of the power-law form, the power being only constrained by requiring that the Hamiltonian is bounded from below: \begin{equation} V(|{\bf x}|) = \sum_n a_n\,|{\bf x}|^{b_n}\ , \label{eq:polapot} \end{equation} with sets of arbitrary real constants $a_n$ and $b_n$, the latter only subject to the constraint $$ b_n\ge -1\quad\mbox{if}\quad a_n<0\ . $$ By close inspection of our ansatz (\ref{eq:ansatz}) it should become clear that we are able to handle even potentials of the type ``power--times--exponential,'' that is, potentials of the form $$ V(|{\bf x}|) = \sum_n a_n\,|{\bf x}|^{b_n}\exp(c_n\,|{\bf x}|)\ ,\quad b_n\ge -1\quad\mbox{if}\quad a_n<0\ . $$ It is a rather simple task to write down the matrix elements for the power-law potential (\ref{eq:polapot}): \begin{eqnarray*} V_{ij} &\equiv& \langle\psi_i|V(|{\bf x}|)|\psi_j\rangle\\[1ex] &=& \sum_n\,a_n\int{\rm d}^3x\,\psi_{i,\ell m}^\ast({\bf x})\,|{\bf x}|^{b_n} \,\psi_{j,\ell m}({\bf x})\\[1ex] &=& \sqrt{\frac{i!\,j!}{\Gamma(2\,\ell+2\,\beta+i+1)\, \Gamma(2\,\ell+2\,\beta+j+1)}}\\[1ex] &\times& \sum_n\,\frac{a_n}{(2\,\mu)^{b_n}}\,\sum_{r=0}^i\,\sum_{s=0}^j\, \frac{(-1)^{r+s}}{r!\,s!} \left(\begin{array}{c}i+2\,\ell+2\,\beta\\ i-r\end{array}\right) \left(\begin{array}{c}j+2\,\ell+2\,\beta\\ j-s\end{array}\right)\\[1ex] &\times& \Gamma(2\,\ell+2\,\beta+b_n+r+s+1)\ . \end{eqnarray*} For instance, considering merely radial excitations by letting $\ell=0$ and choosing, just for the sake of definiteness, for the variational parameter $\beta$ the value $\beta=1$, the explicit form of the potential matrix $V\equiv (V_{ij})$ is $$ V = \frac{1}{6}\,\sum_n\,\frac{a_n}{(2\,\mu)^{b_n}}\,\Gamma(3+b_n) \left(\begin{array}{ccc}3&-\sqrt{3}\,b_n&\cdots\\ -\sqrt{3}\,b_n&3+b_n+b_n^2&\cdots\\ \vdots&\vdots&\ddots\end{array}\right)\ . $$ \section{Analytically Evaluable Special Cases} It should be really no great surprise that the evaluation of the matrix elements of the kinetic-energy operator $T$, \begin{eqnarray*} T_{ij} &\equiv& \langle\psi_i|T({\bf p})|\psi_j\rangle\\[1ex] &=& \int{\rm d}^3p\,\widetilde\psi_{i,\ell m}^\ast({\bf p})\,T({\bf p})\, \widetilde\psi_{j,\ell m}({\bf p})\ , \end{eqnarray*} is somewhat more delicate than the previous calculation of the matrix elements of the power-law potentials $V$. Consequently, let us focus our attention to those situations which allow for a fully analytic evaluation of the above kinetic-energy matrix elements. \subsection{Orbital Excitations} On the one hand, we may restrict our formalism to the case $i=j=0$ but allow, nevertheless, for still arbitrary values of the orbital angular momentum $\ell$ (which means to consider arbitrary orbital excitations), and set $\beta=1$. Then the matrix elements $V_{ij}$ of the power-law potential (\ref{eq:polapot}) reduce to $$ V_{00} = \frac{1}{\Gamma(2\,\ell+3)}\,\sum_n\,\frac{a_n}{(2\,\mu)^{b_n}}\, \Gamma(2\,\ell+b_n+3) $$ whereas for the matrix elements $T_{ij}$ of the kinetic energy (\ref{eq:kinenergy}) we obtain \begin{equation} T_{00} = \frac{4^{\ell+2}\,[\Gamma(\ell+2)]^2} {\sqrt{\pi}\,\Gamma\left(2\,\ell+\frac{7}{2}\right)}\,\mu\, F\left(-\frac{1}{2},\ell+2;2\,\ell+\frac{7}{2};1-\frac{m^2}{\mu^2}\right)\ . \label{eq:t00} \end{equation} At this point, our primary aim must be to get rid of the hypergeometric series $F$ in the above intermediate result. \begin{itemize} \item In the ultrarelativistic limit, realized in the case of vanishing mass $m$ of the involved particles, that is, for $m=0$, the hypergeometric series $F$ in (\ref{eq:t00}) may be simplified with the help of the relation \cite{abramow} $$ F(u,v;w;1) = \frac{\Gamma(w)\,\Gamma(w-u-v)}{\Gamma(w-u)\,\Gamma(w-v)} $$ for $$ w\ne 0,-1,-2,\dots\ ,\quad \Re(w-u-v)>0\ , $$ in order to yield for the kinetic-energy matrix element $T_{00}$, Eq.~(\ref{eq:t00}), the much more innocent expression $$ T_{00} = \frac{2\,[\Gamma(\ell+2)]^2} {\Gamma\left(\ell+\frac{3}{2}\right)\Gamma\left(\ell+\frac{5}{2}\right)}\, \mu \ . $$ The resulting upper bounds, $H_{00}$, can be optimized be minimizing $H_{00}$ with respect to the variational parameter $\mu$. For instance, for a linear potential $V(|{\bf x}|) = a\,|{\bf x}|$, this minimization procedure thus yields $$ \min_{\mu>0}H_{00} = 2\,\Gamma(\ell+2)\,\sqrt{\frac{(2\,\ell+3)\,a} {\Gamma\left(\ell+\frac{3}{2}\right)\Gamma\left(\ell+\frac{5}{2}\right)}}\ . $$ In the limit of large orbital angular momenta $\ell$, that is, for $\ell\to\infty$, this minimal upper bound turns out to be not in conflict with the experimentally well-established linearity of ``Regge trajectories:'' $$ \lim_{\ell\to\infty}\left(\min_{\mu>0}H_{00}\right)^2 = 8\,a\,\ell\ , $$ which is in striking accordance with all previous findings \cite{kang75,lucha91regge}. \item Fixing the variational parameter $\mu$ to the particular value $\mu=m$ allows us to take advantage from the fact that $$ F(u,v;w;0) =1\ , $$ whence the kinetic-energy matrix element $T_{00}$, Eq.~(\ref{eq:t00}), reduces to $$ T_{00} = \frac{4^{\ell+2}\,[\Gamma(\ell+2)]^2} {\sqrt{\pi}\,\Gamma\left(2\,\ell+\frac{7}{2}\right)}\,m\ . $$ \end{itemize} \subsection{Radial Excitations} On the other hand, considering only states of vanishing orbital angular momentum $\ell$, i.~e., only states with $\ell=0$, confines our investigation to the analysis of radial excitations. In this case, we may use the relation \cite{abramow} $$ F\left(u,1-u;\frac{3}{2};\sin^2z\right) = \frac{\sin[(2\,u-1)\,z]}{(2\,u-1)\sin z} $$ in order to recast the hypergeometric series $F$ in the momentum-space representation $\widetilde\psi_{k,00}(|{\bf p}|)$ of our trial states into the form \begin{eqnarray*} F\left(\frac{\beta+r+2}{2},-\frac{\beta+r}{2};\frac{3}{2}; \frac{{\bf p}^2}{{\bf p}^2+\mu^2}\right) &=& \frac{\displaystyle\sqrt{{\bf p}^2+\mu^2}}{(\beta+r+1)\,|{\bf p}|}\\[1ex] &\times& \sin\left[(\beta+r+1)\arctan\frac{|{\bf p}|}{\mu}\right]\ . \end{eqnarray*} Simplifying the momentum-space trial function $\widetilde\psi_{k,00}(|{\bf p}|)$ in this way, \begin{eqnarray*} \widetilde\psi_{k,00}(|{\bf p}|) &=& \sqrt{\frac{k!} {\mu\,\Gamma(2\,\beta+k+1)}}\,\frac{2^\beta}{\pi\,|{\bf p}|}\\[1ex] &\times& \sum_{r=0}^k\,\frac{(-2)^r}{r!}\left(\begin{array}{c}k+2\,\beta\\ k-r\end{array}\right)\Gamma(\beta+r+1)\\[1ex] &\times& \left(1+\frac{{\bf p}^2}{\mu^2}\right)^{-(\beta+r+1)/2} \sin\left[(\beta+r+1)\arctan\frac{|{\bf p}|}{\mu}\right]\ , \end{eqnarray*} the matrix elements $T_{ij}$ of the kinetic energy (\ref{eq:kinenergy}) immediately become \begin{eqnarray*} T_{ij} &=& \sqrt{\frac{i!\,j!}{\Gamma(2\,\beta+i+1)\,\Gamma(2\,\beta+j+1)}}\, \frac{4^{\beta+1}}{\pi}\,\mu\\[1ex] &\times& \sum_{r=0}^i\,\sum_{s=0}^j\,\frac{(-2)^{r+s}}{r!\,s!} \left(\begin{array}{c}i+2\,\beta\\ i-r\end{array}\right) \left(\begin{array}{c}j+2\,\beta\\ j-s\end{array}\right)\\[1ex] &\times& \Gamma(\beta+r+1)\,\Gamma(\beta+s+1)\,I_{rs}\ , \end{eqnarray*} where $I_{rs}$ denotes the only remaining integration, \begin{eqnarray*} I_{rs} &\equiv& \int\limits_0^\infty{\rm d}y\,\sqrt{y^2+\frac{m^2}{\mu^2}}\\[1ex] &\times& \frac{\cos[(r-s)\arctan y] - \cos[(2\,\beta+r+s+2)\arctan y]} {(1+y^2)^{(2\beta+r+s+2)/2}}\ . \end{eqnarray*} This integration may, of course, always be performed by some standard numerical integration procedure. However, for $\mu=m$, the integral $I_{rs}$ simplifies to $$ I_{rs} = \int\limits_0^\infty{\rm d}y\, \frac{\cos[(r-s)\arctan y] - \cos[(2\,\beta+r+s+2)\arctan y]} {(1+y^2)^{(2\beta+r+s+1)/2}}\ , $$ which, for $2\,\beta$ integer and thus, because of the previous normalizability constraint $2\,\beta>-1$, non-negative, i.~e., for the values $2\,\beta=0,1,2,\dots$, may be evaluated with the help of the expansion $$ \begin{array}{r}\cos(N\arctan y) = \displaystyle\frac{1}{(1+y^2)^{N/2}}\,\displaystyle\sum_{n=0}^N \left(\begin{array}{c}N\\n\end{array}\right)\cos\left(\frac{n\,\pi}{2}\right) y^n\\[3ex]\mbox{for}\ N=0,1,2,\dots\ ,\end{array} $$ with the result \begin{eqnarray*} I_{rs} &=& \frac{1}{2} \left[\Gamma\left(\frac{2\,\beta+r+s+|r-s|+1}{2}\right)\right]^{-1} \sum_{n=0}^{|r-s|}\left(\begin{array}{c}|r-s|\\ n\end{array}\right)\\[1ex] &\times& \Gamma\left(\frac{n+1}{2}\right) \Gamma\left(\frac{2\,\beta+r+s+|r-s|-n}{2}\right) \cos\left(\frac{n\,\pi}{2}\right)\\[1ex] &-& \frac{1}{2}\left[\Gamma\left(2\,\beta+r+s+\frac{3}{2}\right)\right]^{-1} \sum_{n=0}^{2\beta+r+s+2} \left(\begin{array}{c}2\,\beta+r+s+2\\ n\end{array}\right)\\[1ex] &\times& \Gamma\left(\frac{n+1}{2}\right) \Gamma\left(2\,\beta+r+s+1-\frac{n}{2}\right) \cos\left(\frac{n\,\pi}{2}\right)\ . \end{eqnarray*} The case $\beta=0$, however, requires special care for the following reason. For $\beta=0$, the integral $I_{00}$ and therefore also the kinetic-energy matrix element $T_{00}$ become singular, as may be read off from the above explicit expression for the integral $I_{rs}$. This singularity may be cancelled by the contribution of a Coulomb-type term $\kappa\,|{\bf x}|^{-1}$ in the power-law potential (\ref{eq:polapot}) if the involved coupling constant $\kappa$ takes some particular, ``critical'' value. This cancellation can then be made manifest by observing that \cite{lucha96rcplowly} $$ \lim_{\beta\to 0}\,\int\limits_0^\infty{\rm d}y\, \frac{1 - \cos[(2+2\,\beta)\arctan y]}{(1+y^2)^{1/2+\beta}} = 2\lim_{\beta\to 0}\,\int\limits_0^\infty{\rm d}y\, \frac{y^2}{(1+y^2)^{3/2+\beta}}\ . $$ Explicitly, for $\beta=1$, the kinetic-energy matrix $T\equiv(T_{ij})$ is given by $$ T = \frac{128}{15\,\pi}\,m\left(\begin{array}{ccc} 1&\displaystyle\frac{\sqrt{3}}{7}&\cdots\\[2ex] \displaystyle\frac{\sqrt{3}}{7}&\displaystyle\frac{11}{9}&\cdots\\[2ex] \vdots&\vdots&\ddots\end{array}\right)\ . $$ In any case, our approach yields analytic expressions for the matrix elements $H_{ij}$ of our semirelativistic Hamiltonian $H$ with an interaction potential out of the rather large class given by the power-law form (\ref{eq:polapot}). In principle, the $d$ (real) roots of the characteristic equation (\ref{eq:chareq}) may be determined algebraically up to and including the case $d=4$, entailing, of course, analytic expressions of rather rapidly increasing complexity. For larger values of the dimension $d$ of our trial space $D_d$, the resulting energy matrix, $(H_{ij})$, may be easily diagonalized numerically, however, without the necessity to apply time-consuming integration procedures. In order to be able to estimate and appreciate the quality of all the upper bounds obtained in this way, we apply the above results to four prototype potentials, namely, to \begin{itemize} \item the harmonic-oscillator potential, $$ V(|{\bf x}|) = \omega\,|{\bf x}|^2\ ,\quad\omega>0\ , $$ \item the Coulomb potential, $$ V(|{\bf x}|) = -\frac{\kappa}{|{\bf x}|}\ ,\quad\kappa>0\ , $$ \item the linear potential, $$ V(|{\bf x}|) = a\,|{\bf x}|\ ,\quad a>0\ , $$ and \item the funnel potential, $$ V(|{\bf x}|) = -\frac{\kappa}{|{\bf x}|} + a\,|{\bf x}|\ , \quad\kappa>0\ ,\quad a>0\ , $$ \end{itemize} for typical values \cite{lucha92sign} of the involved coupling parameters $\omega$, $\kappa$, and $a$. The upper bounds on the energy eigenvalues of the lowest-lying radial excitations (1S, 2S, 3S, and 4S in usual spectroscopic notation) for the harmonic-oscillator, Coulomb, linear, and funnel potentials are shown in Tables~\ref{tab:osci} through \ref{tab:funnel}, respectively; the upper bounds on the respective energy eigenvalues of just the first orbital excitation (1P again in usual spectroscopic notation) for the above potentials are listed in Table~\ref{tab:1pstates}. \normalsize \begin{table}[htb] \caption{Energy eigenvalues of the spinless Salpeter equation with harmonic-oscillator potential $V(|{\bf x}|) = \omega\,|{\bf x}|^2$, for the parameter values $\mu=m=1\;\mbox{GeV}$, $\omega=0.5\;\mbox{GeV}^3$, $\beta=1$, and a size $d\times d$ of the energy matrix $(H_{ij})$. Numbers in italics (for small matrix sizes) indicate analytically obtained results. All eigenvalues are given in units of GeV.} \label{tab:osci} \begin{center} \begin{tabular}{ccccc} \hline\hline\\[-1.5ex] State&$1\times 1$&$2\times 2$&$25\times 25$&Schr\"odinger\\[1ex] \hline\\[-1.5ex] 1S&{\it 4.2162}&{\it 3.9276}&3.8249&3.8249\\ 2S&---&{\it 8.1085}&5.7911&5.7911\\ 3S&---&---&7.4829&7.4823\\ 4S&---&---&9.0215&9.0075\\[1ex] \hline\hline \end{tabular} \end{center} \end{table} \begin{table}[htb] \caption[]{Energy eigenvalues of the spinless Salpeter equation with Coulomb potential $V(|{\bf x}|) = -\kappa/|{\bf x}|$, for the parameter values \cite{lucha92sign} $\mu=m=1\;\mbox{GeV}$, $\beta=1$, $\kappa = 0.456$, and the size $d\times d$ of the energy matrix $(H_{ij})$. Numbers in italics (for small matrix sizes) indicate analytically obtained results. All eigenvalues are given in units of GeV.} \label{tab:coulomb} \begin{center} \begin{tabular}{cccc} \hline\hline\\[-1.5ex] State&$1\times 1$&$2\times 2$&$25\times 25$\\[1ex] \hline\\[-1.5ex] 1S&{\it 2.2602}&{\it 2.0539}&1.9450\\ 2S&---&{\it 3.0702}&1.9868\\ 3S&---&---&2.0015\\ 4S&---&---&2.0238\\[1ex] \hline\hline \end{tabular} \end{center} \end{table} \begin{table}[htb] \caption[]{Energy eigenvalues of the spinless Salpeter equation with the linear potential $V(|{\bf x}|) = a\,|{\bf x}|$, for the parameter values \cite{lucha92sign} $\mu=m=1\;\mbox{GeV}$, $\beta=1$, $a = 0.211\;\mbox{GeV}^2$, and the size $d\times d$ of the energy matrix $(H_{ij})$. Numbers in italics (for small matrix sizes) indicate analytically obtained results. All eigenvalues are given in units of GeV.} \label{tab:linear} \begin{center} \begin{tabular}{cccc} \hline\hline\\[-1.5ex] State&$1\times 1$&$2\times 2$&$20\times 20$\\[1ex] \hline\\[-1.5ex] 1S&{\it 3.0327}&{\it 2.8034}&2.7992\\ 2S&---&{\it 4.0767}&3.3629\\ 3S&---&---&3.8079\\ 4S&---&---&4.1905\\[1ex] \hline\hline \end{tabular} \end{center} \end{table} \begin{table}[htb] \caption[]{Energy eigenvalues of the spinless Salpeter equation with the funnel potential $V(|{\bf x}|) = -\kappa/|{\bf x}| + a\,|{\bf x}|$, for the parameter values \cite{lucha92sign} $\mu=m=1\;\mbox{GeV}$, $\beta=1$, $\kappa = 0.456$, $a = 0.211\mbox{ GeV}^2$, and the size $d\times d$ of the energy matrix $(H_{ij})$. Numbers in italics (for small matrix sizes) indicate analytically obtained results. All eigenvalues are given in units of GeV.} \label{tab:funnel} \begin{center} \begin{tabular}{cccc} \hline\hline\\[-1.5ex] State&$1\times 1$&$2\times 2$&$20\times 20$\\[1ex] \hline\\[-1.5ex] 1S&{\it 2.5767}&{\it 2.5182}&2.5162\\ 2S&---&{\it 3.4499}&3.1570\\ 3S&---&---&3.6337\\ 4S&---&---&4.0348\\[1ex] \hline\hline \end{tabular} \end{center} \end{table} \begin{table}[htb] \caption[]{Energy eigenvalues for the 1P states of the spinless Salpeter equation with the harmonic-oscillator potential $V(|{\bf x}|) = \omega\,|{\bf x}|^2$, the Coulomb potential $V(|{\bf x}|) = -\kappa/|{\bf x}|$, the linear potential $V(|{\bf x}|) = a\,|{\bf x}|$, and the funnel potential $V(|{\bf x}|) = -\kappa/|{\bf x}| + a\,|{\bf x}|$, respectively, for the parameter values \cite{lucha92sign} $\mu=m=1\;\mbox{GeV}$, $\beta=1$, $\omega=0.5\;\mbox{GeV}^3$, $\kappa = 0.456$, $a = 0.211\;\mbox{GeV}^2$, and the size $d\times d$ of the energy matrix $(H_{ij})$. Numbers in italics (for small matrix sizes) indicate analytically obtained results. All eigenvalues are given in units of GeV.} \label{tab:1pstates} \begin{center} \begin{tabular}{ccccc} \hline\hline\\[-1.5ex] Potential&$1\times 1$&$20\times 20$&Schr\"odinger\\[1ex] \hline\\[-1.5ex] Harmonic oscillator&{\it 6.5094}&4.9015&4.9015\\ Coulomb&{\it 2.5314}&1.9875&---\\ Linear&{\it 3.2869}&3.1414&---\\ Funnel&{\it 3.0589}&2.9816&---\\[1ex] \hline\hline \end{tabular} \end{center} \end{table} \large For the case of the harmonic-oscillator potential, the corresponding Hamiltonian $H$ in its momentum-space representation is equivalent to a nonrelativistic Hamiltonian with some effective interaction potential, which clearly is reminiscent of that troublesome square-root operator. In this form, it is then rather easily accessible to numerical procedures for solving a nonrelativistic Schr\"odinger equation \cite{falk85}. For comparison, we quote, in Tables~\ref{tab:osci} and \ref{tab:1pstates}, the eigenvalues obtained along these lines. We find a very encouraging, rapid convergence of the upper bounds. \section{Summary} By application of the well-known min--max principle, which represents the theoretical foundation of any computation of upper bounds on the eigenvalues of self-adjoint operators, to trial spaces spanned by sets of basis states which enable us to handle the square-root operator of the relativistic kinetic energy $T$ in a satisfactory manner, we demonstrated how to derive (for lowest-lying states even analytically!) upper bounds on the energy levels of the spinless Salpeter equation with some (linear combination of) power-law potentials. Interestingly, in the case of the funnel potential, which is the prototype of almost all of the ``realistic,'' that is, phenomenologically acceptable, inter-quark potentials used for the description of hadrons as bound states of (constituent) quarks, the obtained lowest-order approximation to the upper bound on, e.~g., the ground-state energy is merely some 2 \% above the corresponding value. Of course, all the bounds derived here may be improved numerically by a minimization with respect to the variational parameters introduced. \normalsize \clearpage \newpage

proofpile-arXiv_065-546

\section{Introduction} Late-type Low Surface Brightness galaxies (LSBs) are considered to be very young stellar systems, because of their rather blue colors (de Blok, van der Hulst \& Bothun 1995, McGaugh \& Bothun 1996) and very low oxygen abundances (McGaugh, 1994). Based on these observational evidences there have been recently theoretical suggestions that LSBs are formed inside dark matter halos that collapsed very recently, at $z\le 1$, from density fluctuations of small amplitude (Dalcanton, Spergel, \& Summers 1996, Mo, McGaugh, \& Bothun 1994). In this work we study the colors of LSBs from the point of view of synthetic stellar populations (SSP), and show that LSBs could not be as young as claimed in the quoted literature. Recently one of us (PP) has obtained a stellar Initial Mass Function (hereafter P-IMF) starting from high-resolution numerical simulations of the supersonic random motions in the interstellar medium (Nordlund \& Padoan, 1997; Padoan, Jones \& Nordlund,1997). Here we will plug this P-IMF into the latest version of our synthetic stellar population code which is based on Jimenez \& MacDonald (1997) evolutionary tracks and Kurucz atmospheric models (Kurucz 1992). With this we compute synthetic colors and colors gradients for LSBs (section 2) and we show how these can be used to set tight bounds on the ages of their stellar discs (section 3). We also show that the color gradients are well fitted (section 4), and we speculate on the cosmological implications of these results in section 5. \section{Synthetic stellar populations for LSBs} In the following when we will refer to LSBs' we will always mean the sample of late-type disc galaxies observed by de Blok, van der Hulst \& Bothun (1995). For each galaxy of their sample the HI surface density, and the surface brightness profiles in several bands are published. LSBs are found to be rather blue; the color tends to become bluer in the outer regions of their discs. De Blok, van der Hulst \& Bothun (1995) noted that it is difficult to understand the colors of LSBs, if their stellar population is old or forming at a declining rate. McGaugh and Bothun (1996) from the analysis of their sample concluded that the stellar populations in LSBs must be very young, because of the very blue colors and of the very low metallicity. In fact an IMF appropriate to the solar neighbourhood, like the one by Miller and Scalo (1979), has a shape very flat for ${\rm M}\leq 0.1 {\rm M}_{\odot}$ and this results in too red V-I colors when B-V are properly fitted. Since the discs of LSBs are rather quiescent when compared with HSB discs, we suppose that their colors are an excellent probe of their stellar IMF. Although this can at most be taken as first approximation, it gives an excellent fit to many observed relations, as we will show. Moreover, it allows us to probe to which extent our P-IMF can provide a realistic interpretation of observed data. At variance with other IMF, in the P-IMF there are no free parameters, and it is based on a model for the structure and dynamics of molecular clouds, that has strong observational support (Padoan, Jones, \& Nordlund 1997, Padoan \& Nordlund 1997). The P-IMF is designed to model large scale star formation, and contains a dependence on mean density $n$, temperature $T$, and velocity dispersion $\sigma_{v}$ of the star forming gas. The mean stellar mass is given by: \begin{equation} M_{*}=1\rm {M}_{\odot}\left(\frac{T}{10\,K}\right)^2\left(\frac{n}{10\, cm^{-3}}\right)^{-1/2} \left(\frac{\sigma_{v}}{5\, km/s}\right)^{-1} \label{eq1} \end{equation} As a significant example we apply the P-IMF to a simple exponential disc model, with height-scale equal to $100\rm\, {pc}$, length scale equal to $3\, \rm{Kpc}$, and total mass equal to $\rm{M_{D}}=3\times10^9 \rm {M}_{\odot}$, a set of parameters chosen to be representative of the LSBs. Our results about colors depend only slightly on these particular values, however. As a measure of the gas velocity dispersion we use the disc vertical velocity dispersion. We also assume that all stars are formed in a cold gas phase, at T$=10\, K$. Note that the same stellar mass would be obtained if the vertical velocity dispersion, instead of the height-scale, were kept constant along the radius, because of the dependence on velocity dispersion and density in equation (1). Fig.~1 shows the IMF predicted for such a disc at 1$ \rm {kpc}$ and 6$ \rm {kpc}$ from its center. The IMF is more massive than the Miller-Scalo (dashed line), but also less broad. The IMF at 6$ \rm {kpc}$ is also more massive than at 1$ \rm {kpc}$. We then expect that with these properties the stellar populations which will form will be rather blue, and will become bluer at larger distances from the center, as is observed in LSBs. To compute the synthetic colors we used the latest version of our synthetic stellar population code (Jimenez et al. 1996). The code uses the library of stellar tracks computed with JMSTAR9 and the set of atmospheric models calculated by Kurucz (Kurucz 1992). A careful treatment of {\rm all} evolutionary stages has been done following the prescriptions in Jimenez et al. (1995), and Jimenez et al. (1996). Different star formation rates and stellar IMF are incorporated in the code, so a large parameter space can be investigated. We find that the star formation in LSBs can be adequately described with an initial burst, followed by a quiescent evolution up to the present time. It has been already remarked (van der Hulst et al., 1993) that LSBs' gas surface densities are too low to allow efficient star formation according to Kennicut criterion (Kennicut 1989). Therefore it is reasonable to argue that significant star formation is limited to an initial burst. The duration of the burst is almost irrelevant to the colors, because of its rather old age, but it cannot be much longer than a few $10^7$ yr, in order to be consistent with the low metallicity of the synthetic stellar population, and with the low oxygen abundance of the HII regions observed by McGaugh (1994) in LSBs. We find that the colors of LSBs are not difficult to reproduce, as long as stars smaller than $1 \rm{M}_{\odot}$ are not as numerous as in the solar-neighborhood population, which would give a too red V-I color, and as long as a low metallicity is used. Indeed, one can easily see, from the theoretical models by Kurucz (1992), that even a {\it single} star with low metallicity (Z=0.0002) can reproduce the colors of LSBs. As an example, the colors of a typical galaxy from the sample of de Blok, van der Hulst, \& Bothun, namely F568-V1, are: U-B=-0.16, B-V=0.57, B-R=0.91, V-I=0.77 (luminosity weighted); the colors of a Kurucz model with temperature T=5500 K, $\log$(g)=4.5, Z=0.0002 are: U-B=-0.17, B-V=0.56, B-R=0.94, V-I=0.75. This model corresponds to a star of $0.94 \rm{M}_{\odot}$, having a lifetime of 11 Gyr. Obviously, the reason for such a good match does not lie in the fact that the stellar IMF does not contain any star more massive than $1 \rm{M}_{\odot}$, as suggested in the past (Romanishin, Strom, \& Strom 1983, Schombert et al. 1990), but simply in the fact that $0.94 \rm{M}_{\odot}$ is the mass at the turn-off for the stellar population of F568-V1 in our model, which gives an age for this galaxy's disc of about 11 Gyr. \section{The age of LSBs} In Fig.~2 we plot the time evolution of the colors for a very low metallicity ($Z=0.0002$), and in Fig.~3 for a higher metallicity ($Z=0.0040$). In order to compare the theoretical prediction with the observed colors, we have used the mean values of the luminosity-weighted colors listed in Table~4 of de Block, van der Hulst, \& Bothun (1995). Since the color of a stellar population is affected by age and metallicity, we also plot in Fig.~2 the mean of the observed colors, excluding the three galaxies for which U-B is observed and has a positive value. The error bars represent the dispersion around the mean. It is clear that the fit is excellent for an age of $12 \,\rm{Gyr}$, and that an age $\le 9 \,\rm{Gyr}$ is definitely inconsistent with the data. In Fig.~3 we plot the mean of the colors for the three galaxies with positive U-B. These redder galaxies are better fitted by a higher metallicity, $Z=0.0040$, which is one fifth of the solar metallicity, and is one of the highest metallicity estimated by McGaugh (1994) in LSB HII regions. The best fit for the age is $9\, \rm{Gyr}$. The effect of the metallicity on the colors is illustrated in Fig.~4 and 5, where we show the trajectories of the time evolution of our models in color-color diagrams. It is evident that we do not find LSBs younger than $9\,{\rm Gyr}$, for any metallicity consistent with the observations. The spread in colors is a result of the spread in metallicity, as is shown by the remarkable agreement between the trajectories and the observations. For instance, in the (B-V,U-B) diagram, where the trajectories are well separated, also the observed points show a similar spread in U-B. On the other hand, in the (B-R,B-V) diagram, where the theoretical trajectories are almost coincident, also the observational points are nicely aligned around the trajectories. Therefore {\it the ages of LSBs' discs rule out the possibility that they formed from primordial density fluctuations of low amplitude, collapsed at $z\le1$}. Such old ages may seem difficult to reconcile with those of the relatively young stellar populations in normal late-type galaxies, that have U-B and B-V colors comparable to those of LSBs, and B-R and V-I even redder. However, the very blue U-B and B-V colors in LSBs are very well explained by the very low metallicities, rather than by the young stellar ages, and the B-R and V-I colors are explained by the lack of small stars (as the P-IMF predicts), in comparison with a Miller-Scalo IMF. The diagram (B-V,U-B) shown in Fig.~5 is particularly important, because it can be used to estimate the age of single galaxies, without an independent determination of the metallicity of its stellar population. In fact, in that diagram the time evolution is almost horizontal, along B-V, while the metallicity variations are almost vertical, along U-B. In other words, the degeneracy age-metallicity in the colors is broken in such a digram. We can therefore see that galaxies of different metallicities have all about the same age (11-12 Gyr). The horizontal dispersion of the observational points, along B-V, is approximately 0.1 mag, which is comparable to the observational uncertainty. Therefore, the determination of the age of LSB discs with presently available photometry, and without an independent estimate of the metallicity, has an uncertainty of $\pm 2.0$ Gyr (0.1 mag in B-V). \section{Color gradients} An interesting feature of LSBs is their color gradient: LSBs are bluer in the periphery than near the center of their disc (de Blok, van der Hulst, \& Bothun 1995). Our theoretical models predict a color gradient in agreement with the observations. In fact, the exponential disc model has a volume density that decreases with increasing radius, and equation (1) shows that the typical stellar mass in the IMF grows with decreasing gas density, producing increasingly bluer colors. We have computed the color gradients per disc length-scale, for a model with an age of 12 Gyr, and metallicity Z=0.0002. In Table~1 we show the results, compared with the observational data, which are obtained from the mean of the values listed by de Blok, van der Hulst, \& Bothun (1995), in their Table~3. Again, we have excluded from the mean the three galaxies with U-B$>0$, since they require a metallicity significantly larger than Z=0.0002. Together with the mean gradients, we give the mean of the errors listed by the above mentioned authors. The agreement between observational data and theory is striking. Note that the model is just the one that best fits the colors of LSBs, as shown in Fig.~2, rather than being an ad hoc model which fits the color gradients. Therefore {\it the color gradient of LSBs indicates that the stellar IMF is more massive towards the periphery of the discs than near the center}, as predicted by our P-IMF. \section{Conclusions} In this work we have shown that the P-IMF, applied to a simple exponential disc model, allows an excellent description of the colors and color gradients of LSBs. This allows us to draw a few interesting consequences: \begin{itemize} \item The Miller-Scalo IMF produces too red V-I colors, and therefore cannot describe the stellar population of LSB galaxies; \item The P-IMF, applied to a simple exponential disc model with an initial burst of star formation, produces excellent fits of the LSBs' colors and color gradients; \item The metallicity of LSB stellar populations ranges from practically zero to about one fifth solar. \item Although most stars in LSBs are formed in an initial burst, a relation between colors and surface brightness is not expected, because the colors are strongly affected also by the metallicity. \item The age of LSBs, inferred from the UBVRI colors, is between $9$ and $13\, \rm{Gyr}$. These disc populations are therefore about as old as the disc of our Galaxy. \item Since LSBs galaxies are old they cannot be explained as late collapsed objects (low density fluctuations at $z\le1$), therefore their origin remains still unexplained. \end{itemize} \acknowledgements This work has been supported by the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center. RJ and VAD thank TAC for the kind hospitality and support.

proofpile-arXiv_065-547

\section{INTRODUCTION} The availability of the first truly heavy beams at the AGS allows the study of hot and dense nuclear matter over the largest volumes that will effectively be available in the laboratory. Early results on transverse energy production~\cite{AuPRL} indicate that with such heavy beams, energy densities predicted to lead to a quark-gluon plasma phase transition are probably reached. The study of hadron spectra provides a more detailed description of the reaction dynamics and, in particular, of its evolution as a function of the mass of the colliding system. Preliminary results on hadron production at 10.8$\cdot$A$~$GeV/c from the E877 Collaboration were presented at Quark Matter 1995~\cite{qm95}. Au beam data were taken during the 1993, '94 and '95 AGS heavy ion runs. Here, we will present mostly '93 data with some preliminary results from the '94 run. The E877 experimental setup is presented in T.~K.~Hemmick's contribution to this conference~\cite{hemmick}. It can be summarized into two groups of detectors: nearly $4\pi$ calorimetry around the target and a forward spectrometer. The centrality determination is done using the calorimeters surrounding the target~\cite{AuMult} and the particle spectra discussed here were obtained with the spectrometer. \section{PROTON DISTRIBUTIONS} Since at AGS energies creation of baryon-antibaryon pairs is nearly negligible, the rapidity distribution of protons allows the study of the energy deposition in the reaction, i.e. the nuclear stopping power. \begin{figure}[thb] \epsfig{file=p_dndmtdy.ps,height=6in} {\begin{minipage}[c]{6cm} \vspace*{-12.5cm} \caption{\small Proton transverse mass spectra. The data are presented in rapidity bins of 0.1 unit widths successively multiplied by increasing powers of 10 for decreasing rapidity. Errors are statistical. Also shown are exponential fits which for $y\geq 2.9$ exclude $m_t-m_p<0.2$~GeV/c$^2$. \label{fig:pmt}} \end{minipage}} \end{figure} E877's measurement of proton transverse mass spectra in Au+Au collisions for the most central 4\% of $\sigma_{\rm geom}$ is presented in figure~\ref{fig:pmt}. One of the key features of the E877 spectrometer is that its acceptance includes $p_t=0$. In terms of rapidity, the spectrometer's acceptance for protons starts at $y=2.2$ with data extending to $y=3.5$. The vertical scale in figure~\ref{fig:pmt} is $1/m_{t}^{2} \cdot dN/dy dm_{t}$ so that a thermal (Boltzmann type) source would produce an exponential distribution. As in all the figures of this paper, the data are presented with statistical error bars only. Overall, the transverse mass spectra exhibit thermal shapes with increasing slope parameter when approaching mid-rapidity. As expected, deviation from a purely exponential distribution is observed around beam rapidity ($y_{\rm beam}=3.14$) for $m_t-m_p<0.2$~GeV/c$^2$. This colder low transverse momentum component to the proton spectra can be attributed to spectator protons from the projectile~\cite{DeeCentral}. \begin{figure}[thb] \begin{center} \epsfig{file=temps.eps,height=4.5in} \end{center} \caption{\small Proton inverse slope constants as a function of rapidity. The measured temperature parameters for Au+Au (filled circles) are compared to data from the Si+Al (filled stars) system. The data are reflected about mid-rapididity (open symbols). \label{fig:temps}} \end{figure} The values of the inverse slope constants ${\rm T}_{\rm B}$ obtained from fitting each of the proton transverse mass spectra with an exponential (full lines in figure~\ref{fig:pmt}) are plotted in figure~\ref{fig:temps}. The data are compared to values obtained for 14.6$\cdot$A~GeV/c Si+Al ($y_{\rm beam}=3.44$)~\cite{SiBaryons} and are plotted as a function of $y/y_{\rm beam}$. The measured inverse slope constants for the two systems are quite similar around beam rapidity. Systematically increasing differences are observed between the Au+Au and Si+Al systems which reach their largest values closest to the center of mass rapidity. The present result could be interpreted as being due to a larger collective transverse flow component in Au+Au compared to what was observed in Si+Al~\cite{Expansion}. \begin{figure}[thb] \begin{center} \epsfig{file=dndy_over.eps,height=4.5in} \end{center} \caption{\small Rapidity distribution of protons in Au+Au (full circles) compared to data from the Si+Al (full stars) system and to a prediction from RQMD (solid line). The filled symbols are the measured points and the open symbols are obtained from reflecting the measurement about $y_{cm}$. \label{fig:pdndy}} \end{figure} The proton rapidity density distribution shown in figure~\ref{fig:pdndy} is obtained by integrating the transverse mass spectra where data are available and extrapolating to $p_t=\infty$ using the Boltzmann fits (see figure~\ref{fig:pmt}). The measured rapidity distribution for Au+Au (circles) is plotted as a function of $y/y_{\rm beam}$ in order to compare with Si+Al data (squares)~\cite{SiBaryons} and is multiplied with the proper Jacobian ($y_{\rm beam}$). The Si+Al rapidity density distribution was also renormalized by the ratio of the number of nuleons in Au+Au to that in Si+Al so as to better compare the shapes of the two distributions. The open symbols are obtained by reflecting the experimental results about $y_{\rm cm}$. The solid line is the result of an RQMD~1.08~\cite{RQMD} calculation for Au+Au. The RQMD calculation exhibits a structure near $y_{beam}$ originating from projectile protons which have had no or minimal interaction with the target. This structure includes protons originating from the dissociation of the spectators treated as unbound in RQMD. The measured rapidity distribution of protons in Au+Au is significantly narrower than what was observed for Si+Al. The distribution is still wider than predicted by an isotropic thermal distribution, using the temperature deduced from the pion spectra, and is consistent with a longitudinaly expanding thermalized source~\cite{stachel}. Although the stopping was already shown to be high in Si+Al~\cite{SiStopping}, the measured Au+Au rapidity distribution indicates that an even larger amount of stopping is involved. This is understood as being a consequence of the smaller surface to volume ratio and increased average number of rescatterings in the Au+Au system. This interpretation is supported by the RQMD calculation that reproduces both overall widths of the Si+Al~\cite{SiBaryons} and the Au+Au measurements. \section{PION DISTRIBUTIONS} Pions are produced copiously at AGS energies. Because of their light masses and large cross sections for interaction in nuclear matter they are expected to thermalize easily. Furthermore, their spectra are not as affected as those of heavier particles by a given collective flow velocity and thus are good probes for studying thermal properties at freeze-out. \begin{figure}[thb] \begin{center} \epsfig{file=pi_dndmtdy.ps,height=5.55in} \end{center} \caption{\small Pion transverse mass spectra. The data are presented in rapidity bins of 0.1 unit widths successively multiplied by increasing powers of 10. \label{fig:pimt}} \end{figure} The pion spectra in Au+Au collisions for the most central 4\% of $\sigma_{\rm geom}$ are shown in figure~\ref{fig:pimt} as a function of transverse mass and rapidity. The pion data extend to 0.8~GeV/c$^2$ in $m_t-m_{\pi}$ and cover rapidities 2.8 to 4.5. The transverse mass spectra are exponential with slopes decreasing with increasing rapidity. A clear enhancement above a pure exponential is observed for $m_t-m_\pi < $~0.2~GeV/c$^2$. A similar effect was already observed for the Si+Pb system and was explained by the contribution from decay pions of $\Delta$ resonances~\cite{Delta}. To better display the enhancement for the $y=3.2-3.3$ rapidity slice, the data were divided by an exponential fitted to the part of the spectra above $m_t-m_{\pi} >$~0.2~GeV/c$^2$. As shown on the insert, the enhancement reaches values as large as 4 at $m_t-m_{\pi} =$~0~GeV/c$^2$. The enhancement is systematically increasing as one goes toward $y_{\rm cm}$. \begin{figure}[thb] \begin{center} \epsfig{file=pm_ratios_2.ps,width=5in} \end{center} \caption{\small Ratio of the transverse mass spectra of $\pi^+$ and $\pi^-$ in 3 rapidity bins. \label{fig:piratio}} \end{figure} Close inspection of the transverse mass curves of figure~\ref{fig:pimt} reveals that the deviations from a pure exponential emission are systematically larger for $\pi^{-}$ than for $\pi^{+}$. The charge asymmetry of the pion production is better studied by plotting the ratio of $\pi^{+}$ to $\pi^{-}$ as done in figure~\ref{fig:piratio}. In order to improve statistical errors in the ratio, the data were divided into three large rapidity bins. The $\pi^{+}/\pi^{-}$ ratio is consistent with unity at large transverse mass values. A strong charge asymmetry is observed starting at $m_t-m_{\pi}< 0.2$~GeV/c$^2$ with a minimum value of $\pi^{+}/\pi^{-}\simeq 0.6$ at $m_t-m_{\pi}=0$~GeV/c$^2$ for the $y=2.9-3.2$ rapidity slice. The asymmetry is also observed to systematically decrease as a function of rapidity. The observed rapidity dependence is in line with the charge asymmetry of $\pi^{+}/\pi^{-} \simeq 0.5$ measured by the E866 collaboration near $y_{\rm cm}$ at similar centrality~\cite{qm95e866}. The observed pion charge asymmetry and its rapidity dependence could be due to the different Coulomb potentials seen by each charge type at freeze-out. Coulomb interactions have been discussed at this conference in terms of their influence on the interpretation of particle correlations~\cite{dariusz,kadija,baym}. The study of the Coulomb effect on the shape of the particle spectra will be pursued since it provides a different approach to the determination of the spacetime particle distribution at freeze-out. \begin{figure}[thb] \begin{center} \epsfig{file=prot_over.eps,height=5in} \end{center} \caption{\small Rapidity distributions for $\pi^{+}$, $\pi^{-}$ and protons. The data (solid symbols) are reflected about $y_{\rm cm}$ (open symbols). An RQMD calculation is overlaid (solid line) for comparison. \label{fig:pidndy}} \end{figure} The rapidity distribution for pions obtained by integrating the transverse mass spectra is shown in figure~\ref{fig:pidndy}. The distributions are plotted as a function of laboratory rapidity, covering the rapidity range $2.9<y<4.4$. The RQMD calculations agree very well with our data over their entire range. A similar agreement was also observed for the the Si+Pb system~\cite{qm93delta}. The proton rapidity distribution is also included at the bottom of figure~\ref{fig:pidndy}. Contrary to what was observed in the Si+Al~\cite{Expansion}, where the proton distribution was wider than that of the pion, the rapidity distribution widths are similar for Au+Au. The measured narrowing of the proton distribution from the Si+Al is reproduced by the RQMD calculation. These observations are consistent with full stopping and similar values of longitudinal and transverse expansion of $\beta\sim0.5$~\cite{stachel}. \begin{figure}[htb] \epsfig{file=photon.eps,height=3.in} \epsfig{file=photons2.ps,height=2.95in} \caption{\small Photon measurement made with the CsI detector. Shown on the left is the data and an RQMD+GEANT calculation and on the right the decomposition of the different sources that contribute to the calculation. \label{fig:pizero}} \end{figure} The inclusive photon spectra, and indirectly the $\pi^{\circ}$ yield, was measured using a CsI photon detector~\cite{zou}. This detector was placed 35 m from the target at an angle of $\theta = 4.8^{\circ}$ and covered $\delta\theta \sim 0.34^{\circ}$. The photon inclusive transverse momentum measurement is shown on left side of figure~\ref{fig:pizero} (open symbols). Because of the small size of the detector, reconstruction of the $\pi^{\circ}$ using the two decay photons was impossible. A model dependent method was used to derive a limit on $\pi^{\circ}$ production using single photons. An RQMD calculation was performed followed by tracking of the decay photons through the E877 spectrometer using a GEANT (full symbols)~\cite{GEANT}. The RQMD+GEANT calculation is in fair agreement with the shape of the measured photon spectra except for the lowest transverse momentum values $p_t<100$~MeV/c. As the calculation shows on the right side of figure~\ref{fig:pizero}, the single photon spectrum is dominated, above $p_t > 100$~MeV/c, by photons that originate from $\pi^{\circ}$ decays. The calculation also indicates that the dominant contributions at low transverse momentum ($p_t< 100$~MeV/c) are secondary photons. These secondary photons originate from interactions in the spectrometer material located upstream of the CsI detector. The contributions to the measured photon spectrum from neutron contamination and $\eta$ decays are also shown. A careful inspection of the higher end of the inclusive photon spectrum on left side of figure~\ref{fig:pizero} reveals that the calculation systematically overpredicts the yield of photons. Using the photon spectrum decomposition made above, the discrepancy is found to correspond to an overprediction by RQMD of roughly $\sim 20\%$ of the $\pi^{\circ}$ yield~\cite{zou}. RQMD predicts a ratio between the pion yields of $\pi^+/\pi^{\circ}/\pi^- = 1/1.26/1.16$ at a centrality of 12\% of $\sigma_{\rm geom}$. Thus the present result does not support the predicted excess of neutral pions. \section{KAON DISTRIBUTIONS} \begin{figure}[htb] \begin{center} \epsfig{file=k_dndmtdy.ps,height=5in} \end{center} \caption{\small Transverse mass spectra for K$^+$ on the left and K$^-$ on the right. The data are presented in rapidity bins of 0.1 unit widths successively multiplied by increasing powers of 10. \label{fig:kaons}} \end{figure} The study of the production kaons and other strange particles provides information on the level of chemical equilibrium achieved and is also relevant to determine whether chiral restoration is achieved. At Quark Matter 95, preliminary kaon spectra were presented that created a high level of interest~\cite{qm95}. The measured transverse mass spectra of K$^+$ seemed to exhibit an unexpected enhancement at the lowest transverse mass values. In order to validate the shape of these spectra, new measurements of the kaon spectra were performed in 1994 with slightly different experimental conditions. In addition, the preliminary results from the 1993 run were complemented by an independent re-analysis of the data~\cite{thesis}. The results from the re-analysis of the 1993 data for the top 10\% of $\sigma_{\rm geom}$ are plotted in figure~\ref{fig:kaons}. The $m_t$ scales are divided into bins equal to 20~MeV/c in $p_t$ for K$^+$ and 40~MeV/c for K$^-$. The K$^+$ spectra no longer show a statistically significant enhancement at low transverse mass. The origin of the structure was traced to an albedo source located at the spectrometer's collimator edge, closest to the beam trajectory. The present results are confirmed by the preliminary analysis of the 1994 data. Both sets of data show an indication of a dip at very low $p_t$ in the K$^+$ spectra. The fitted exponential inverse slopes have values of 60 to 90~MeV which are consistent with those presented at Quark Matter '95 for K$^+$. These values are lower than the ones obtained from the E866 data (T$_{\rm B}\sim 150$ to $170$~MeV) in the same rapidity range~\cite{stachel,akiba}. The E866 data cover transverse mass values of up to 1~GeV/c$^2$ whereas our measurement covers the first 0.1~GeV/c$^2$. \section{DEUTERON DISTRIBUTIONS} \begin{figure}[th] \begin{center} \epsfig{file=d_dndmtdy.ps,height=5.5in} \end{center} \caption{\small Transverse mass spectra for deuterons are shown on the left. The data are presented in rapidity bins of 0.1 widths successively multiplied by increasing powers of 10. Different RQMD predictions for the rapidity bin $1.4<y<1.8$ are compared to deuteron (circles) and proton (triangles) data on the right. The dashed histograms correspond to predictions of standard RQMD while the full histograms include the effect of mean field. The full line is a Boltzmann distribution fitted to the large $p_t$ region of the RQMD result for deuterons. \label{fig:deuts}} \end{figure} The production of composite particles at the AGS has yet to be studied extensively. Their production is just starting to be considered in the cascade models. Deuterons are, because of their mass, good probes to study expansion and collective transverse flow velocities. Preliminary results from the measurement of deuteron production from the 1994 run are presented in figure~\ref{fig:deuts} for the most central collisions (4\% of $\sigma_{\rm geom}$). The deuteron acceptance covers the rapidity 1.2 to 2.2 and the data extend to 0.25~GeV/c$^2$ in $m_t-m_d$ for $y=2.1-2.2$. The measured deuteron spectra are very flat over the covered transverse mass range and exhibit only a slight rapidity dependence. On the right side of figure~\ref{fig:deuts}, data for deuterons (circles) and protons (triangles) are compared to the corresponding RQMD calculations (lines) for the production of different types of composites (p, d, t, He$^4$) in the rapidity slice $1.4<y<1.8$~\cite{mattiello}. The result from the cascade version of RQMD (dashed histogram) is found to overshoot the data. A modified version of RQMD that simulates the presence of a mean field (full line histogram) reproduces the measured deuterons yield quite well but still overshoots the proton data. A Boltzmann fit to the large $p_t$ region of the RQMD deuteron calculation (full line) shows that even the simple cascade model deviates from a thermal shape and produces much flatter distributions at low $p_t$. The spectral shapes of the deuteron data support the importance of the mean field to describe the measured yields. The model predicts an increasing flattening with mass and thus the measurement of the spectra of more complex composites would serve to better determine the importance of mean field effects in relativistic heavy ions collisions. \section{CONCLUSION} We have presented new data on the particle distributions of hadrons produced in Au+Au collisions at AGS energies. The rapidity distributions show increased stopping relative to lighter systems leading to a center of mass region richer in baryons. The rapidity dependence of the slopes of the particle spectra and the rapidity distributions show evidence of large collective transverse and longitudinal flow. The data are consistent with a larger collective component in the Au+Au final state than what was observed for the Si+Al system. The slope and yield of the measured deuteron spectra at low $p_t$ suggest the presence of sizable mean field effects. The addition of a new vertex detector system to the E877 spectrometer and the better statistical samples from the runs of 1994 and 1995 should soon provide interesting new results, particularly on deuteron, K$^-$ and $\Lambda$ production. The avalaible data on both global observables and particle spectra in Au+Au collisions are consistent with the formation of baryon rich nuclear matter at density and temperature close to that expected for a phase transition. It is particularly interesting to note that heavy systems provide us with new observables, such as collective flow, mean field and Coulomb effects, that will help us to better understand the space-time evolution of the system during the collision and yield additional signatures for new phenomena. Support from US DoE, the NSF, the Canadian NSERC, and CNPq Brazil is gratefully acknowledged.

proofpile-arXiv_065-548

\section{Introduction} Polarimetry is a powerful way of probing asymmetries in supernova explosions. Because polarization produced within the supernova atmosphere may have spectral features that are distinguishable from the featureless interstellar polarization (ISP), which is caused by dust within both the Galaxy and the host galaxy, spectropolarimetry is usually a more powerful tool than broad-band polarimetry in decomposing the intrinsic polarization of a distant supernova from ISP. SN 1987A and SN 1993J are the only two supernovae for which spectropolarimetry has been published and intrinsic polarizations are positively detected. SN 1987A is so far the best observed supernova for which linear polarimetry was obtained from several days to about 260 days after explosion (Mendez et al. 1988; Cropper et al. 1988; Jeffery 1991a). The degree of polarization evolved with time, indicating that the cause of the polarization is related intrinsically to SN 1987A. The data were analyzed (H\"oflich 1987; Jeffery 1991b) in terms of the photospheric scattering model (Brown \& McLean 1977; Shapiro \& Sutherland 1982). Recently, Wang \& Wheeler (1996a) provided a different view in which time delayed scattering by a hypothesized circumstellar dust clump successfully reproduced both the broad-band and spectropolarimetry of the supernova and the early infrared light curve of SN 1987A (Bouchet et al. 1989). Linear polarization indicated by polarization changes across spectral features was also detected in SN 1993J (Trammell, Hines, \& Wheeler 1993). Broad band polarimetry shows also variable polarization before and after the second optical maximum of SN 1993J (Doroshenko, Efimov, \& Shakhovskoi 1995). Only two Type Ia supernovae -- SN 1983G (McCall et al. 1984) and SN 1992A (Spyromilio \& Bailey 1992) have been previously observed by spectropolarimetry. The noise levels of the SN 1983G and SN 1992A data are 0.5\% and 0.3\%, respectively. These data indicate SN 1983G and SN 1992A are not polarized at a level higher than 0.5\%. As part of our program of systematic supernova polarimetry, we have observed 5 supernovae with broad-band polarimetry and compiled a catalog of all the supernovae with polarimetry reported in the literature (Wang et al. 1996). This sample shows that all the Type II supernovae with sufficient data are intrinsically polarized while no intrinsic polarization can be established for any of the Type Ia supernovae. Supernovae, of both Type Ia and II, are now being widely used as standard candles for distance determinations. As a self-consistency test, it is imperative to set some observational constraints on the degree of the polarization. Furthermore, normal spectroscopy provides only part of the information carried by the supernova light. Spectropolarimetry can further constrain various models for spectral line formation and radiative transfer through supernova atmospheres, as shown in recent examples by H\"oflich (1995a), H\"oflich et al. (1996), and H\"oflich, Wheeler, \& Wang (1997). Furthermore, it may be the only way to detect chemical blobs in the ejecta which usually produce little observable effects on the flux spectra of a supernova (H\"oflich, Wheeler, \& Wang 1997). \section{The Observations} SN 1996X in NGC 5061 was discovered independently by several observers (Garradd 1996) on April 12.5 UT at a V magnitude about 13.5. Spectroscopy on April 14 UT showed SN 1996X to be a Type Ia supernova prior to optical maximum (Suntzeff 1996; Wang and Wheeler 1996b; Benetti and Patat 1996). Our spectropolarimetry of SN 1996X was obtained on April 14.3 using the Imaging Grism Polarimeter (IGP) mounted at the Cassegrain focus of the 2.1 meter telescope of the McDonald Observatory. The IGP is a simple, high efficiency, dual beam polarimeter which can be easily switched between imaging and spectropolarimetry mode. A rotatable half waveplate was used as the polarization analyzer. A description of the instrument can be found in Hill \& Trammell (1996) and Trammell (1994). A TK4 ($1024\times1024$) CCD was used as the detector. A broad filter with pass band from 4400 to 7500 was used for the broad-band polarimetry. Such a filter effectively filters out photons with wavelengths beyond the effective range of the waveplate, but allows a large number of photons to be transmitted for polarimetry. The slit width used for the spectropolarimetry was 2$^{\prime\prime}$.1 which gave a spectral resolution of about $14$\AA. Wavelength calibration was obtained by taking exposures of an argon lamp. \begin{figure} \psfig{figure=fig1.eps,width=7.8cm,clip=,angle=0} \caption{ Instrumental polarization from observations of unpolarized standard stars. } \psfig{figure=fig2.eps,width=7.8cm,angle=0} \caption{ (a) The flux spectra of SN 1996X on April 14.3, 1996 UT in arbitrary units; (b) the Stokes parameter $Q$; (c) the Stokes parameter $U$; (d) the noise level of the Stokes parameters. In (b) - (c), the thin lines are the observed data at the original sampling step of 3.8\AA/pixel, the thick lines are the same data resampled with a sample window width of 64.8\AA (17 pixels). } \psfig{figure=fig3.eps,width=7.8cm,clip=,angle=0} \caption{ The wavelength range showing correlations between the flux spectrum (dotted line), the polarization spectrum from the resampled data (thin solid line), and the polarization of the data smoothed by convolving with a Gaussian of FWHM of 64.8\AA (thick solid line). } \end{figure} The data were taken with clear and dark sky. Several unpolarized and polarized standard stars were observed each night. The data reduction process is the same as outlined in Miller, Robinson, \& Goodrich (1988). The instrumental polarization obtained by observing several unpolarized standards is shown in Figure 1 (a) and 1 (b) and is found to be less than 0.1\% in the entire wavelength range. The polarized standards were used to calibrate the polarization angles. In addition, observations were also obtained by inserting a polarizer at the top of the waveplate to characterize the efficiency of the waveplate and the wavelength dependence of the fast axis of the waveplate. The polarization efficiency is typically 95\% at around 6500 \AA\ and decreases to about 90\% at 4500 \AA. The variation of the fast axis with wavelength is less than 2$^\circ$. These effects and also the instrumental polarization were taken into account and corrected for in the data that will be presented below. \section{Data and Results} Although the exact phase of the supernova awaits the publication of an accurate light curve, the supernova was at pre-maximum phase on April 14, as indicated by the flux spectrum shown in Figure 2 (a). The continuum was very blue and the Fe II lines were still not fully developed. The Stokes parameters $Q$ and $U$ derived from the spectropolarimetry are shown in Figures 2 (b) and 2 (c), and their errors in Figure 2 (d). The dominant error in $Q$ and $U$ was from photon statistics, and was the largest in the red portion of the spectra where it reached $1\%$; typical errors were less than 0.6\% in regions around 5500 \AA. Polarizations higher than 1\% can be reliably excluded from the $Q$ and $U$ spectra shown in Figure 2. However, at a lower level of about 0.3\%, some broad features are present. For a simple test, the $Q$ and $U$ spectra were convolved with a Gaussian of FWHM of 65 \AA. As shown in Figure 3, in the wavelength range from 4900 to 5500 \AA, there exists a correlation between the wavelengths of these broad features and the spectral features in the flux spectrum. To investigate the authenticity of this polarization variation with wavelength, the data were resampled to increase the signal to noise ratio. This was done by deriving the Stokes parameters within a certain wavelength window $\Delta\lambda$ centered on a particular wavelength, $\lambda$. This is equivalent to calculating the following weighted mean of the original Stokes parameters: {\small $$ Q(\lambda|\Delta\lambda)\ = \ \int_{\lambda-\Delta\lambda/2}^{\lambda+\Delta\lambda/2}\,N(\lambda^\prime) Q(\lambda-\lambda^\prime) d\lambda^\prime/ \int_{\lambda-\Delta\lambda/2}^{\lambda+\Delta\lambda/2}\,N(\lambda^\prime) d\lambda^\prime, \eqno(1) $$} where $N$ is the number of counts per wavelength interval. The corresponding equation for $U$ can be similarly derived. Such an operation increases the signal to noise ratio of the data, but at the expenses of degrading the spectral resolution. Fortunately, the spectral features in a supernova spectrum are in general as broad as 150 \AA, and useful information can still be obtained even if the resolution is as low as 100 \AA. The Stokes parameters with an integration window of width $2\times\Delta\lambda$ = 64.8 \AA\ are shown in Figure 2 (b) and (c). The noise level ($\sigma$) which is also shown in Figure 2 (d), is now typically around 0.1\%. The degree of polarization $P$ can be constructed from the Stokes parameters $Q$ and $U$. However, it is well known that for relatively high noise level, $P\ = \ \sqrt{Q^2\ +\ U^2}$ is biased toward values larger than the true degree of polarization $P_{\rm o}$\ ; $P$ is usually not the best estimate of $P_{\rm o}$\ . The distribution function for $(P, \theta)$ takes the form (Simmons \& Stewart 1985) {\small $$ f(P, \theta)\ =\ {P\over2\pi\sigma} \exp\{-{1\over2}[( {P\over\sigma})^2+({P_{\rm o}\over\sigma})^2-{P\over\sigma} {P_{\rm o}\over\sigma} \cos(\theta-\theta_{\rm o})]\}, \eqno(2)$$} where $(P_{\rm o}, \theta_{\rm o})$ stands for the true degree of polarization and polarization position angle. Simmons \& Stewart (1985) constructed optimal estimates of degree and position angle of polarization $(P_{\rm o}$ and $\theta_{\rm o})$ from the marginal distribution of polarization obtained by integrating over $\theta$ in equation (2). This method, however, does not usually simultaneously optimize $P_{\rm o}$ and $\theta_{\rm o}$. A simpler method can be constructed by requiring the best estimate of $(P_{\rm o}, \theta_{\rm o})$ to take the values for which $(P,\theta)$ gives the maximum of the distribution $f(P,\theta)$. It is easy to derive from equation (2) that this results in the following simple formula for $P_{\rm o}$\ and $\theta_o$ $$ \begin{array}{lll} P_{\rm o}\ &=\ P-\sigma^2/P & P\ \ge\ \sigma\nl P_{\rm o}\ &=\ 0 & P\ <\ \sigma \nl \theta_{\rm o} &=\ \theta & \nl \end{array} \eqno(3)$$ Both the polarization $P_{\rm o}$\ derived from equation (3) and $P$ are shown in Figure 4(a). They differ mainly in wavelength regions with low signal to noise ratios. The polarization position angles could not be reliably determined for data with large errors, therefore only position angles calculated from the resampled $Q$ and $U$ are shown (Figure 4(b)). An enlarged version of the part of the spectrum with strong correlations between the spectral features in the polarimetry and flux spectrum is shown in Figure 3, where the features are marked by dashed lines for illustrative purposes. The corresponding spectral features in the polarization spectrum which are correlated with the spectral lines in the flux spectrum are blue shifted by about 2200 $\rm{km\,s^{-1}}$. No strong correlation between the flux and polarization spectra could be established at wavelengths longer than 6000 \AA. This may be due to some intrinsic characteristics of the polarization of Type Ia SN (see \S 4 for a more detailed discussion). The position angles change across spectral features. This should not be a surprise when compared with the best observed supernova, SN 1987A, where the intrinsic component shows dramatic variations of both degree and polarization position angle across spectral features (Cropper et al. 1988). \begin{figure}[h] \psfig{figure=fig4.eps,width=8.4cm,clip=,angle=0} \caption{ (a) The number of counts in a single beam of a 20 minute exposure on a logarithmic scale (dotted line); the degree of polarization resampled with a sample window width of 64.8\AA ; the degree of polarization $P_{\rm o}$\ as defined in equation (2) (thick solid line). (b) The polarization position angle of the resampled data. } \end{figure} It is impossible to accurately correct for the effect of ISP. The Galactic extinction to the host galaxy NGC 5061 is $A_B\ = \ 0.25$, which alone can produce polarizations as large as 0.7\% according to the limit derived by Serkowski (1970). The ISP correction is even more complicated by the polarization due to dust in the host galaxy; however, the ISP follows Serkowski's law, and is a smooth function of wavelength; the uncertainties introduced by the ISP will not change the $Q$ and $U$ spectral structures on wavelength scales less than a few hundred \AA. The ISP correction can, however, dramatically change the appearance of the polarization spectra. The spectral features in Figure 4(a) may change from `emission' to `absorption' even if a very small amount of ISP correction $\sim 0.2\%$ is required. This should be borne in mind when interpretating the polarization spectra. In addition to the spectropolarimetry discussed above, we have also obtained broad-band polarimetry using a wide filter with pass band from 4000 -- 7500 \AA. Three and five data sets were obtained on April 14 and May 22, respectively. Each data set consists of four 100 second exposures which gave estimates of the Stokes parameters. The measured $Q$ and $U$ are practically zero for all data sets. The averaged values are $Q\ = \ 0.079\ \pm \ 0.041 \%$ and $U\ = \ 0.056\ \pm\ 0.039\%$ on April 14, and $Q\ = \ 0.090\ \pm\ 0.021\%$ and $U\ = \ 0.063\ \pm\ 0.019\%$ on May 22, where the errors are due to photon statistics. The systematic instrumental uncertainty is around 0.06\% (as indicated by repeated observations of unpolarized standards) and is therefore the dominant error for the broad-band data. The broad-band data are consistent with zero polarization. We do not confirm the short-term temporal variations of polarization reported in early polarimetry of another Type Ia SN 1972E (Lee, Wamsteker, \& Wisniewski 1972, quoting observations made by Serkowski and Wamsteker). It should be pointed out that the null detection in broad-band polarimetry did not conflict with the spectral features detected in the spectropolarimetry data. As a check for consistency, we have weighted the spectropolarimetry data with the transmission curve of the filter and the photon counts at different wavelengths to simulate broad-band polarimetry. The resulting $Q$ and $U$ are in agreement with the broad-band polarimetry data. \section{Models} Besides the attempt to identify features of the polarization with those of the flux spectrum, a more direct approach is a comparison with theory which provides additional information on the spectral patterns to be expected. A simplified approach is suitable to answer the following questions: what does the polarization spectrum look like for a SN~Ia, what asphericity is needed to be consistent with the observation of SN1996X, and how do the predicted spectra compare with the observations ? Delayed detonation models (Khokhlov 1991) have been found to provide a good representation of the observed light curves and spectra of normal bright SN~Ia (H\"oflich 1995b, H\"oflich \& Khokhlov 1996 and references therein) and we use such a model to produce a representative theoretical SN~Ia polarization. The polarization spectrum is given by Monte Carlo calculations based on the following assumptions: a) homologously expanding, ellipsoidal envelopes with density and chemical profiles given by a hydrodynamical model, b) occupation numbers given by local thermodynamical equilibrium (LTE); c) electron scattering, bound-free and free-free for continuum opacities; d) lines treated in a Sobolev approximation with an assumed constant thermalization fraction; e) line transitions result in depolarization; f) the temperature structure is given; g) and elliptical geometry with a axis ratio being independent of radius. Note that discrepancies between predicted and observed wavelengths of polarization features corresponding to 1000 to 2000 $\rm{km\,s^{-1}}$\ may be expected because this difference is well within the observed variation of expansion velocities among normal bright SNe~Ia. Lines of the same ion must, nevertheless, be expected to be shifted consistently. For more details, see H\"oflich et al. (1995). \begin{figure}[h] \vskip -5.1cm \vskip +5.1cm \psfig{figure=fig5.eps,width=8.4cm,rwidth=4.4cm,angle=270} \caption{ Comparison of the theoretical polarization spectrum of the delayed detonatinon model DD200c five days before maximum light (thick line; H\"oflich et al. 1996; axis ratio 0.88 seen at an inclination of 30$^{\rm o}$) with the observations (thin line). The identification of some features are given. Most of the weaker features are due to iron group elements in the second and third ionization stage. The S II and Si II features at about 4900, 5300, 5600 and 5670 \AA\ are too strong and the overall polarization above 6500 \AA\ is too large. Possible solutions are discussed in the text. Note, that the observed spectrum is oversampled although smoothed with a triangle of 64.8 \AA FWHM. } \end{figure} For the density and chemical structure we used model DD200c with T(r) based on light curve calculations corresponding to about 5 days before maximum (H\"oflich et al. 1996). In Figure 5, the polarization spectrum for an oblate ellipsoid with an axis ratio of 0.89 is given which is seen at an inclination of 30$^{\rm o}$\ which fits best. Note that, quantitatively, the spectrum depends critically on the inclination. The depolarization in the ubiquitous metal lines has two main effects. Firstly, the mean polarization is about a factor of 2 to 3 smaller compared to that of a Thomson scattering atmosphere which provides a good approximation to Type II supernovae. Consequently, the same asphericity will produce much less polarization for an SN Ia compared to an SN II. Secondly, $P$ varies strongly with wavelength on a typical scale of about the Doppler width. The variations with wavelength are significantly stronger than those in the flux spectra because it takes only one interaction to depolarize but many to thermalize a photon. A comparison with the observations shows some qualitative similarity of the wavelength distribution of the features. The pattern in the models is mainly influenced by the atomic physics rather than the detailed explosion model. Therefore, the agreement or disagreement may be used to ask if the measurement can be regarded as a real detection. Most of the observed features have their equivalents in the theoretical model, although the predicted polarization above 6500 \AA\ is much higher than observed (see below). The mean size of $P$ implies an asphericity of $\approx $ 11 \% at an inclination angle of 30$^{\rm o}$ for the symmetry axis of the ejecta (Figure 5). There are also some distinct shortcomings in the model polarization spectrum. The features corresponding to the products of partial burning (S II and Si II) are somewhat too strong compared to the observations. In principle, this may be compensated by slightly higher temperatures, small changes in the explosion model (e.g. a slightly larger Ni production) or, as test calculations have shown, by making the outer Si-rich layers almost spherical. A more severe problem is the spectrum above 6500 \AA. Although the spectral features resemble the models, the absolute polarization is much lower. A component due to interstellar polarizaton (U=-0.1 \%, Q = 0.25 \%) helps to bring up $P$ above 6500~\% but completely destroys any reasonable fit at other wavelengths and, thus, does not solve the problem. This ISP is also inconsistent with broad-band data (see above). The increasing predicted polarization is insensitive to the details of the explosion models. It is caused primarily by the decreasing line blending towards longer wavelengths. Lower line blending also implies that the polarization in the red is produced at deeper layers (H\"oflich 1995b). Consequently, a possible solution to the problem is a nearly spherical, inner Ni region corresponding to expansion velocities $\approx 8000$$\rm{km\,s^{-1}}$. This, together with the depolarization in Si and S, imply that asphericities must be mainly attributed to the distribution of chemical elements in the transition region between the Ni and Si layers (H\"oflich et al. 1996). The correlation between flux and polarization spectra depends sensitively on the size of the line blanketing, thermalization parameter of lines and the actual wavelength. For a more detailed discussion see H\"oflich, Wheeler, \& Wang (1997). \section{Discussions and Conclusions} SN 1996X is the first Type Ia supernova for which spectropolarimetry has suggested a polarized component intrinsic to the supernova. The peak to valley spectral variation in the $Q$ and $U$ spectra (Figure 4) is as high as 0.6\% in many cases which, at an error level of 0.15\%, gives a 4$\sigma$ detection of the spectral features. The degree of the intrinsic polarization can be measured in terms of the deviation of the spectral features from the wavelength averaged degree of polarization. For SN 1996X, this is about 0.3\%, which is small compared with the typical degree of polarization in SN 1987A and other Type II SNe (Wang et al. 1996). Previous observations of SN 1983N and SN 1992A have given only upper limits on the degree of polarization of Type Ia supernova (Spyromilio \& Bailey, 1992; McCall et al. 1984). The SN 1996X observations were obtained prior to the optical maximum, and the detected polarization is most likely to be due to a distorted photosphere or element distribution. The correlation of features between the polarization spectra and flux spectra is not perfect in the entire observed wavelength range. This is expected because such a correlation depends sensitively on multiple scattering effects, the ratio between continuum and line optical depths, and thermalization in lines. The 2200 $\rm{km\,s^{-1}}$\ blue shift of the features in the polarization spectrum relative to the flux spectrum (cf. Figure 3) can also be understood as due to multiple scattering. The model calculation shows that a strong feature in the polarization spectra does not require the presence of a strong feature in the flux spectra. Lines which hardly change the flux may produce significant depolarizations. The comparison with theoretical polarization spectra suggests an asphericity of about 11 \% in the distribution of chemical elements. There are evidences that, the strong asphericity is limited to the transition region between nuclear statistical equillibrium (NSE) and partial burning. The distorted nature of the supernova envelope might arise from inhomogenous burning during the deflagration phase of the burning front (Khokhlov 1995). Such a low degree of polarization probably does not seriously endanger the use of Type Ia as calibrated candles for distance indicators, but nonetheless poses a source of uncertainty. We will discuss these effects including the effects of inclination and the relation between flux and polarization spectra quantitatively in a separate study (H\"oflich, Wheeler, \& Wang 1997). We thank Gary Hill, and Paul Shapiro for many discussions and help. We are grateful to the McDonald staff, especially to David Doss and Jerry Martin for their excellent support. This research is supported in part by NSF grant AST 9528110, and NASA grant GO-2563 through the Space Telescope Science Institute and by grand Ho 1177/2-1 of the Deutsche Forschungsgemeinschaft.

proofpile-arXiv_065-549

\section{INTRODUCTION } Magnetic fields in galaxies have strengths of order few $10^{-6} G$, and are coherent on scales of several kpc (cf. Beck et al 1996). How such ordered, large scale fields arise is a problem of considerable interest. They can arise in principle, due to dynamo amplification of a weak but nonzero seed field $\sim 10^{-19} - 10^{-23} G$, if the galactic dynamo can operate efficiently to exponetiate the field by a factor $\sim 30 - 40$ (cf.Zeldovich {\it et al.} 1983). But the origin of even such a small seed field needs some physical explanation. We review here some of the issues relevant to galactic magnetic field generation, in particular, the galactic dynamo theory, its problems and possible solutions. The evolution of the magnetic field is generally described by the induction equation $ (\partial {\bf B}/ \partial t) = {\bf \nabla } \times ( {\bf v} \times {\bf B} - \eta {\bf \nabla } \times {\bf B}), $ provided one assumes the usual form of Ohms law and neglects the displacement current term in Maxwells equation. Here ${\bf B}$ is the magnetic field, ${\bf v}$ the velocity of the fluid and $\eta$ the resistivity. If $\eta \to 0$ the magnetic flux through any area in the fluid can be shown to be conserved during the motion of the fluid. The presence of a finite resistivity allows for a violation of such "flux freezing" and the magnetic reynolds number $R_m = vL/\eta$ measures the relative importance of flux freezing versus resistive diffusion. (Here $v$ and $L$ are typical velocity and length scales of the fluid motions.) In most astrophysical contexts flux freezing greatly dominates over diffusion with $R_m >> 1$. Since ${\bf B} =0$ is a perfectly valid solution of the induction equation, there would be no magnetic fields generated if one were to start with a zero magnetic field initially. It is generally believed that the universe did not start with an initial magnetic field. So one needs some way of violating the induction equation and produce a cosmic battery effect, to drive curents from a state with initially no current. There are a number of such battery mechanisms which have been suggested (see Rees 1994; Subramanian 1995 for reviews). All of them lead to only small fields much smaller than the galactic fields. Therefore, it would be good to find velocity fields, which can act to exponentiate the small seed fields efficiently. Some form of dynamo action is needed to explain the observed magnetic fields. We will discuss below the possiblities of galactic dynamos after touching briefly on one battery mechanism, which appears capable of seeding the whole IGM with ordered fields, albiet with a small value (Subramanian et al 1994). \section{A BATTERY MECHANISM} The basic problem that any battery has to address is how to produce finite currents from zero currents? Most mechanisms use the fact that the positively and negatively charged particles in a charge neutral universe do not have identical properties. For example if one considered a gas of ionised hydrogen, then the electrons have a much smaller mass compared to protons. This means that for a given pressure gradient of the gas the electrons tend to be accelerated much more than the ions. This leads in general to an electric field, which couples back positive and negative charges, of the form ${\bf E}_T = {\bf \nabla}p_e / e n_e$, where $p_e$ and $n_e$ are the electron pressure and number density, respectively. If such a thermally generated electric field has a curl, then by Faradays law of induction a magnetic field can grow. Taking $p_e = n_e kT$ with $T$ the electron temperature we have ${\bf \nabla} \times {\bf E}_T = - (c k/ e) ( {\bf \nabla } n_e / n_e) \times {\bf \nabla } T$. So ${\bf E}_T$ has a curl only if the density and temperature gradients, are not parallel to each other. Biermann (1950) and Mestel and Roxburgh (1962) applied this idea to stars. Subramanian et al (1994) have applied it to cosmic ionisation fronts, which are produced when the first UV sources turn on to ionise the intergalactic medium. The temperature gradient in a cosmic ionisation front is normal to the front. However, a component to the density gradient can arise in a different direction, if the ionisation front is sweeping across arbitrarily laid down density fluctuations, associated with protogalaxies/clusters since these in general have no correlation to the source of the ionising photons. The resulting thermally generated magnetic fields on galactic scales turn out to have a strength $B \sim 3 \times 10^{-20} G$. This field by itself is far short of the observed microgauss strength fields in galaxies, but it can provide a seed field for a dynamo. Recently Kulsrud et al (1996) have suggested that the Biermann battery can also operate in collapsing protogalaxies/clusters. \section{ THE GALACTIC DYNAMO } Spiral galaxies are differentially rotating systems. Also the magnetic flux is to a large extent frozen into the fluid. So any radial component of the magnetic field will be efficiently wound up and amplified to produce a toroidal component of the field. But this results in only a linear amplification of the field and to obtain the observed galactic fields starting from small seed fields one should find a way to generate the radial components of the field in the galaxy from the toroidal one. If this can be done, the field can grow exponentially and one has a dynamo. A mechanism to produce the radial components from the toroidal field was originally invented by Parker (1955) (cf. Zeldovich et al 1983). The essential feature is to invoke the effects of cyclonic turbulence in the galactic gas. The galactic interstellar medium is assumed to be turbulent, due to for example the effect of supernovae randomly going off in different regions. In a rotating, stratified (in density and pressure) medium like a disk galaxy, such turbulence becomes cyclonic and aquires a net helicity. Helical motions of the galactic gas perpendicular to the disk can draw out the toroidal field into a loop which looks like a twisted $\Omega$. Such a loop is connected to a current and because of the twist this current has a component parallel to the original field. Since the gas has a net helicity, in the presence of such motions a toroidal current can be produced from the toroidal field, Hence, poloidal fields can be generated from toroidal ones. In quantitative terms isotropic and homogeneous turbulence with helicity, in the presence of a large scale magnetic field {\bf B}, leads to an extra electromotive force of the form $ {\bf E} = \alpha {\bf B} - \eta_t {\bf \nabla } \times {\bf B}$ where $\alpha$ depends on the helical part of the turbulence and $\eta_t$ called the turbulent diffusion depends on the non helical part of the turbulent velocity correlation function. A physics comment is in order at this stage. When one considers the effect of turbulent fluid motions on say smoke, one only gets a mean diffusion of the smoke particles, associated with the random walking nature of turbulent fluid motions. But for magnetic fields the induction equation has terms which not only imply a body transport due to the random motions of the fluid, but also a term which describes the generation of magnetic fields due to velocity shear. It is this qualitative difference between magnetic fields and smoke that leads to an alpha effect, over and above turbulent diffusion (and also leads to the small scale dynamo action discussed below). Note that both these effects also crucially depend on the diffusive (random walk) property of fluid motion. So if due to some reason (see below) the fluid motion becomes wavelike, then the alpha effect and turbulent diffusion will be suppressed. The induction equation, with the extra turbulent component of the electric field, with a prescribed large scale velocity field, can have exponentially growing solutions for the large scale field. These have been studied extensively in the literature (cf. Beck et al. 1996 for a review). One can even modify it to discuss the possible reasons why large scale magnetic fields in spirals are sometimes bi-symmetric and why these bi-symmetric magnetic spirals are correlated with the optical spirals (cf. Mestel and Subramanian 1991; Chiba and Tosa 1990). We have assumed here that the turbulent velocities do not get affected by the Lorentz forces due to the magnetic field, at least not until the mean large scale field builds up sufficiently. However this does not turn out to be valid due to the more rapid build up of magnetic noise compared to the mean field, a problem to which we now turn. \section{PROBLEM OF MAGNETIC NOISE} Suppose one splits up the magnetic field ${\bf B} = {\bf B}_0 + \delta{\bf B}$, into a mean field $ {\bf B}_0$ and a fluctuating component $\delta{\bf B}$. Here the mean is defined either as a spatial average over scales larger than the turbulent eddy scales or more correctly as an ensemble average. The dynamics of the fluctuating field has been worked out in detail by Kulsrud and Anderson (1992) (KA) in fourier space and by Subramanian (1996) using a complimentary co-ordinate space approach. We summarise some of the results drawing mainly on the later work. This analysis shows firstly that the fluctuating field, tangled on a scale $l$, can grow on the turn over time scale of a turbulent eddy of scale $l$, with a growth rate $\Gamma_l \sim v_l/l$, provided the magnetic reynolds number on that scale $R_m(l) = v_l l/\eta $ is greater than a critical reynolds number $R_c \sim 100$. Here $v_l$ is the velocity associated with eddies of scale $l$. For Kolmogorov turbulence, since $v_l \propto l^{1/3}$, $\Gamma_l \propto l^{-2/3}$. If the magnetic reynolds number associated with eddies at the cut-off scale (inner scale), say $l_c$, of the turbulence is larger than $R_c$, then these eddies will themselves be able to exponentiate the magnetic field first on these scales. Since the time scale for mean field growth is $\sim 10^9 yrs$, of order a few rotation time scales of the disk, is much larger than the turn around time scales of the turbulent eddies in the galaxy, the magnetic field will be rapidly dominated by the fluctuating component. KA argued that as the small scale field builds up it will drain energy from the turbulence, mainly due to the friction of the ionised gas and the neutrals resulting from ambipolar drift. Also once the energy density in the small scale component achieves equipartition with the turbulent energy density, the turbulence will become weak, a more wavelike "alfven" turbulence, than an eddy like fluid turbulence resulting in a suppression of the alpha effect. All this happens much before the mean field has grown appreciably. So KA speculated that the galactic field is primordial in origin. \section{ A POSSIBLE SOLUTION } Before accepting the above conclusions, it is worth re-examining carefully the dynamics of the small scale fields and its back reaction on the turbulence. We had pointed out earlier (Subramanian 1995) that if the small scale field is intermittent in space, then it may saturate without drastically draining the power from the turbulence. This idea has been now investigated more thoroughly (Subramanian 1996), taking into account in a quantitative fashion the effects of ambipolar drift in the galaxy as well. We summarise below some of the relevant results. For this it is useful look at the behaviour of the magnetic correlation function, say $w(r,t) = <\delta{\bf B}({\bf x},t). \delta{\bf B}({\bf y},t)> $, where $r= \vert {\bf x} - {\bf y}\vert$ and the angular brackets $<>$ indicates an ensemble average. Suppose the turbulence was initially isotropic and homogeneous. Then in the kinematic regime, if $R_m(l_c) > R_c$, the fastest growing $w(r,t)$ has a form $f(r)e^{\Gamma t}$ with $f(r)$ strongly peaked within $r= r_1 = l_c / R_m^{1/2}(l_c)$, and a negative tail extending to $r_2 \sim l_c$. Zeldovich et al. interpret such a correlation function as implying that field is concentrated into ropes of thickness $ r_1$ and radius of order $r_2$. There can be slower growing higher order modes $w$ with more complicated structure for the field but all with a ropy structure with rope thickness of order $r_1$. The question arises as to how such ropes evolve in the non-linear regime when the lorentz force due to the generated field reacts back on fluid motions? We show that, in a partially ionised plasma, due to ambipolar drift, the effective diffusivity changes to $\eta_{eff} = \eta + <\delta{\bf B}^2> / ( 6\pi \rho_i \nu_{in})$, where $\rho_i$ is the ion density and $\nu_{in}$ is the neutral-ion collision frequency. So, as the energy density in the fluctuating field increases, the effective magnetic reynolds number, for fluid motion on any scale of the turbulence say $ R_{ambi}(l) = v_l l/ \eta_{eff}$, decreases. Firstly, this makes it easier for the field energy to reach the diffusive scales $r_d \sim l/R_{ambi}^{1/2}$, from a general initial configuration. Subsequent field amplification, leads to a decreasing $R_{ambi}$ and an increase in $r_d$ and hence the thickeness of the flux ropes. However for the galactic turbulence with say an outer scale $L\sim 100pc$ and $v_L \sim 10$ km s$^{-1}$, $R_{ambi}(L) \sim n_if^{-1} 10^6 $, where $n_i$ is the ion density in units of cm$^{-3}$, and $f$ is the ratio of the magnetic to the turbulent energy density $E_T$. The flux ropes then remain relatively thin with a thickness at most a value of order $L/R_{ambi}^{1/2} << L$, even taking account of the ambipolar drift. The volume filling fraction of the field depends not only on the thickness of the rope but also on its total length. In the kinematic regime, a rough estimate of the length of the rope is $\sim N L$, where $N$ is related to the number of higher order modes excited. For $R_{ambi} \sim 10^6$, only a few higher order modes can be exited. In this case the volume filling factor of the field is $\sim N/R_m$. How this changes as the field reaches a stationary state is more difficult to quantify. (cf. Vishniac 1995, in a different context). Since $R_{ambi} > R_c$, ambipolar drift alone cannot lead to a saturation for the small scale dynamo. The field keeps building up until the effects of the magnetic pressure in the ropes acting on the fluid as a whole becomes important. Due to the increasing importance of this pressure, stretching of field lines can lead to a partial decrease in fluid density in the ropes rather than a decrease in the rope cross section and the associated increase in the rope magnetic field. An upper limit to the magnetic pressure in the ropes is given by the external pressure $P_{ext}$. This implies that the field in the rope, say $B_r$ is limited to $B_r < (8\pi P_{ext})^{1/2}$. In the interstellar medium, the ratio of the gas pressure to the turbulent energy density $P_g/E_T \sim 1.7 (T/10^4 K)(v/10 km s^{-1})^{-2}$. If $P_{ext}$ in the galaxy is dominated by the gas pressure, the peak field in the ropes will not be much larger than the equipartition field. [Vishniac (1995) has also discussed the ropy nature of the magnetic field in accretion disks. There are however crucial differences due to the dominance of ambipolar drift in the galaxy. Also in the case considered here, when the field grows from a small seed field, the kinematic evolution naturally leads to an initial ropy structure for the field which can be preserved when the field becomes strong]. The above considerations lead us to conjecture that when the small scale dynamo saturates, 1. The magnetic noise generated by the small scale dynamo is ropy and does not fill the volume of the fluid. 2. The peak fields in the ropes are not much greater than equipartition fields. Given these conjectures one can show that the power dissipated in ambipolar drift is much smaller than the turbulent power. Also, from the conjectures 1 and 2 one can see that the average energy density of the generated small scale field is much smaller than the average energy density in the turbulence. So any wave-like motion induced by the presence of the field will have a period larger than the eddy turn around time. This implies that such tangled small scale fields do not change the diffusive nature of the turbulence. Due to the above reasons the large scale dynamo can still operate to grow the mean field. More details of the above work can be found in Subramanian (1996). In conclusion, it seems that an understanding of how galactic magnetic fields originate, is far from complete and is still a challenging problem. \acknowledgements I thank the SOC/LOC of the Asia-Pacific IAU meeting for support. Periodic discussions with T. Padmanabhan and a brief one with Ethan Vishniac were very helpful.

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\section{Introduction} The ALEPH \cite{ALEPH} and OPAL \cite{OPAL} collaborations have measured the dependence of single-particle inclusive cross sections in $e^+e^-$ annihilation on the scattering angle $\theta$ between the observed hadron $h$ and the incoming electron beam. The angular dependence discriminates between contributions from transversely and longitudinally polarized virtual bosons, and from $Z^0$-photon interference \cite{NAS94} \begin{eqnarray}\label{def:TL} \lefteqn{ \frac{d^2\sigma^h}{dx d\cos\theta}(e^+e^-\to hX) =} \nonumber\\ &\hspace*{-0.5cm} =& {}\!\!\!\!\!\!\frac{3}{8}(1+\cos^2\theta) \frac{d\sigma_T^h}{dx}(x,Q^2) +\frac{3}{4}\sin^2\theta\, \frac{d\sigma_L^h}{dx}(x,Q^2) \nonumber\\&&{}\!\!\!\!\!\!+ \frac{3}{4}\cos\theta\, \frac{d\sigma_A^h}{dx}(x,Q^2) . \end{eqnarray} In the following, dropping the superscript `$h$' implies summation over all hadrons $h$. In this paper we concentrate on the longitudinal cross section. It is given as a convolution of a parton fragmentation function $D^h_p$ ($p=q,\bar{q},g$) with a partonic cross section ${d\hat\sigma_L^p}/{dx}$ \begin{equation} \frac{d\sigma_L^h}{dx}(x,Q^2) =\sum_p\int_x^1\frac{dz}{z} \frac{d\hat\sigma_L^p}{dz}(z) D_p^h(x/z,Q^2) . \end{equation} The perturbative expansion of the longitudinal parton cross section starts at order $\alpha_s$. Summed over all hadrons, the fragmentation functions satisfy the energy conservation sum rule $\sum_h \int_0^1\!dx\,x D^h_p(x,Q^2)=1$. Consequently, the integrated longitudinal cross section is an infrared (IR) safe quantity which is calculable in perturbation theory \begin{eqnarray}\label{sigmaL} \sigma_L &\equiv & \sum_h\frac{1}{2}\int_0^1\!\! dx \, x\frac{d\sigma_L^h}{dx} = \sum_p\frac{1}{2}\int_0^1\!\! dx \,x \frac{d\hat\sigma_L^p}{dx} \nonumber\\ &&\hspace*{-1.4cm}=\,\sigma_0\!\left[ \frac{\alpha_s}{\pi}+(14.583-1.028N_f)\left(\frac{\alpha_s}{\pi}\right)^2 \!+\ldots\right] . \end{eqnarray} Here $\sigma_0$ is the Born total $e^+e^-$ annihilation cross section, $\alpha_s\equiv \alpha_s(Q)$ and $N_f$ is the number of active fermion flavours. The next-to-leading order contribution has been obtained in \cite{RIJ96}. OPAL \cite{OPAL} has measured $\sigma_L$ the $Z_0$ peak: \begin{equation} \sigma_L/\sigma_{\rm tot}(M_Z^2) =0.057\pm 0.005~. \end{equation} One of the main motivations for the present study is to investigate whether measurements of the total longitudinal cross sections can yield a precise determination of the strong coupling. This requires that we control higher-order perturbative corrections and nonperturbative effects, both of which are expected to be much larger for $\sigma_L$ than for the total cross section $\sigma_{tot}$. We address both types of corrections in this report, by studying the structure of IR renormalons, a certain class of higher-order perturbative corrections, for the longitudinal cross section. The study of nonperturbative effects in fragmentation functions is an interesting topic in its own right \cite{BBM}. The light-cone expansions for fragmentation functions and for structure functions in DIS are similar \cite{BB91}, and suggest that nonperturbative effects in both cases are of order $1/Q^2$ and can be described in terms of multi-parton distributions. In contrast to DIS, however, the relevant operator structures for fragmentation are essentially nonlocal and cannot be expanded at small distances. Hence the usual operator product expansion does not apply and the status of the light-cone expansion is less established. Hadronization models generically introduce nonperturbative corrections of order $1/Q$, while current data on scaling violations in fragmentation do not distinguish between $1/Q$ or $1/Q^2$ behaviour. A nonperturbative correction of order $1/Q$ to the total longitudinal cross section was suggested in \cite{WEB94} as a consequence of phase-space reduction in the one-loop diagram calculated with a massive gluon. We address these apparently conflicting statements below. While this work was in writing, Dasgupta and Webber \cite{DAS96} have addressed a similar set of questions with closely related methods. \section{General formalism} There is suggestive evidence from exact low-order results that $\beta_0$ is a large parameter and that keeping corrections of order $(\beta_0\alpha_s)^n$ in higher orders resums important contributions. Moreover, the infrared renormalons encoded in the corresponding series can elucidate the power-behaviour of nonperturbative corrections and, perhaps, even their $x$-dependence, as in the case of $d\sigma_L/d x$. The $(\beta_0\alpha_s)^n$ corrections can be traced by inserting a chain of fermion loops into the gluon propagator, and by restoring the full QCD $\beta$-function coefficient $\beta_0 =-1/(4\pi)[11-2/3N_f]$ from the dependence on $N_f$. For $\sigma_L$ we obtain two contributions, according to whether the registered parton comes from the primary vertex or a fermion loop. The corresponding diagrams are shown in Fig.~1 and Fig.~2, for the contributions of the `primary' and `secondary' quarks, respectively. Note that the secondary quark contribution reduces to the gluon contribution at lowest order in $\alpha_s$. \setlength{\unitlength}{0.7mm} \begin{figure}[t] \vspace{2.3cm} \hspace*{-3cm} \begin{picture}(120,200)(0,1) \mbox{\epsfxsize13.0cm\epsffile{QCD96fig1.eps}} \end{picture} \vspace*{-13.0cm} \caption{The `primary' quark contribution to $\sigma_L$. Sum over all possible insertions of the bubble chain is understood.} \end{figure} \setlength{\unitlength}{0.7mm} \begin{figure}[t] \vspace{1.3cm} \hspace*{-3cm} \begin{picture}(120,200)(0,1) \mbox{\epsfxsize13.0cm\epsffile{QCD96fig2.eps}} \end{picture} \vspace*{-11.0cm} \caption{The `secondary' quark contribution to $\sigma_L$. Sum over all possible insertions of the bubble chain is understood.} \end{figure} The evaluation of the two classes of diagrams, for an arbitrary number of internal fermion loops, and for their sum, is relatively straightforward by means of the dispersion technique developed in \cite{BEN95,NEU95,BBB}, in terms of the distribution function over the invariant mass $k^2$ of the bubble chain: \begin{eqnarray}\label{xi-distributions} \lefteqn{ \frac{d\hat\sigma_L^{\rm{[p,s]}}}{dx}(x,\xi=k^2/Q^2)\equiv } \nonumber\\ &\equiv & \frac{8\pi}{N_c q^2}\int\! d\,{\rm Lips}[p_1,p_2,k]\, \frac{1}{k^2}{\cal M}_{\mu\nu}{\cal M}_{\mu'\nu'}^\ast \nonumber\\ &&\hspace*{-0.4cm}{}\times \int\! d\,{\rm Lips}[k_1,k_2]\, \frac{1}{2}{\rm Tr}[\gamma_\nu\!\not\!k_1\gamma_{\nu'}\!\not\!k_2] \nonumber\\ &&\hspace*{-0.4cm}{}\times \left\{\begin{array}{l} \frac{p_{1,\mu}p_{1,\mu'}}{2|\vec p_1|^2} \delta\left(x-\frac{2p_1 q}{q^2}\right)~~{\mbox{`primary'}} \\ \frac{k_{1,\mu}k_{1,\mu'}}{2|\vec k_1|^2} \delta\left(x-\frac{2k_1 q}{q^2}\right)~~{\mbox{`secondary'}} \end{array}\right . \end{eqnarray} The notation for momenta and Lorenz indices corresponds to Fig.~2. We denote by $[p]$ ($[s]$) the contribution of the `primary' (`secondary') quark, while $\int\!d\,{\rm Lips[\ldots]}$ are Lorentz-invariant phase space integrals with the momentum conservation $\delta$-function included, and ${\cal M}_{\mu\nu}{\cal M}_{\mu'\nu'}^\ast$ is the matrix element for the primary $q\bar{q} g$ amplitude squared. Note that in the case of the `primary' quark contribution the phase space integral over $k_1,k_2$ is proportional to $k^2$, so that the result takes the form of the one-loop diagram calculated with a gluon of mass $k^2$. This equivalence does not hold for the registered `secondary' quark because of the nontrivial longitudinal projector. Because of this inequivalence, the restoration of $\beta_0$ from fermion loops is not unambiguous and the relation of fermion loop chains with running coupling effects is partially lost. In practice, we have found that the numerical differences are small, so that a detailed discussion is deferred to \cite{BBM}. The analytic expressions for the primary and secondary quark contribution are rather lengthy and will also be given there. Given the invariant mass distributions (in $\xi=k^2/Q^2$), finite order results are obtained in terms of the logarithmic integrals \cite{BEN95,BBB} \begin{equation} \int_0^1 d\xi \,\ln^n\xi \,\frac{d}{d\xi}\frac{d\hat\sigma_L}{dx}(x,\xi) . \end{equation} The sum of the series, defined by a principal value prescription for the Borel integral, equals \begin{eqnarray}\label{BS} \frac{d\hat\sigma_L^{(BS)}}{dx} &=& \int_0^1\!d\xi \,\Phi(\xi)\,\frac{d}{d\xi}\frac{d\hat\sigma_L}{dx}(x,\xi) \nonumber\\ &&\hspace*{-1cm}{}+\left[\frac{d\hat\sigma_L}{dx}(x,\xi_L)- \frac{d\hat\sigma_L}{dx}(x,0)\right] , \end{eqnarray} where $\xi_L < 0$ is the position of the Landau pole in the strong coupling and the function $\Phi(\xi)$ is specified in Eq.~(2.25) of \cite{BBB}. Infrared renormalons correspond to nonanalytic terms in the expansion of $d\hat\sigma_L/dx (x,\xi) $ at small $\xi$ \begin{eqnarray}\label{expansion} \frac{d\hat\sigma_L}{dx}(x,\xi) & = & \frac{d\hat\sigma_L}{dx}(x,0) + f_1(x)\sqrt{\xi} \nonumber \\ & + & f_2(x)\,\xi\ln\xi \end{eqnarray} and are interpreted as indications of nonperturbative power corrections of the form \begin{equation}\label{HT} \frac{d\hat\sigma_L}{dx} = \frac{d\hat\sigma_L^{\rm pert}}{dx} -\frac{\mu_{\rm IR}}{Q} f_1(x) - \frac{\mu^2_{\rm IR}}{Q^2} f_2(x) -\ldots \end{equation} Their size can be estimated by the corresponding ambiguity in the summation of the perturbative series, which is of order of the imaginary part (divided by $\pi$) of the sum in (\ref{BS}). Note that identifying the $x$-dependence of the power corrections in (\ref{HT}) with the $x$-dependence of the IR renormalon ambiguity or, equivalently, the coefficients of non-analytic terms in (\ref{expansion}) is an assumption which can not be justified from first principles. Since IR renormalons in short-distance quantities are related to ultraviolet ambiguities in higher-twist matrix elements, we refer to this assumption as the `ultraviolet dominance' of higher-twist corrections. \section{Perturbative series for $\sigma_L$} In this section we consider perturbative corrections to $\sigma_L$, written as \begin{equation} \sigma_L = \sigma_0\,\frac{\alpha_s}{\pi}\,\left[1+ \sum_{n=0}^\infty d_n\,(-\beta_0\alpha_s)^n\right] , \end{equation} where $\sigma_0$ is the Born total $e^+e^-$ cross section. As mentioned earlier, we approximate the exact higher-order coefficient by its value in the `large-$\beta_0$' limit, where $\beta_0$ is restored from the term with the largest power of $N_f$ at each order. This approximation, called `naive nonabelianization' in \cite{BEN95}, reduces to the familiar BLM prescription for $n=1$. To see how it works, we rewrite the exact $\alpha_s^2$ correction in (\ref{sigmaL}) as \begin{equation} d_1=6.17-0.7573/(-\beta_0) . \end{equation} With $-\beta_0=0.61$ for $N_f=5$, neglecting the second term gives an accuracy of about 25\%. We have calculated the coefficients $d_n$ in higher orders, in the $\overline{\mbox{MS}}$ scheme. The `primary' and `secondary' quark contributions, $d_n^{[p]}$ and $d_n^{[s]}$, respectively, add to $d_n$ as $d_n=d_n^{[p]}/3+2 d_n^{[s]}/3$. A few lower order results up to order $\alpha_s^4$ are \begin{equation} \label{eq1} d_1^{[p]} = 11/2 \quad d_2^{[p]} = 29.8 \quad d_3^{[p]} = 164 , \end{equation} \begin{equation} \label{eq2} d_1^{[s]} = 13/2 \quad d_2^{[s]} = 46.0 \quad d_3^{[s]} = 369 . \end{equation} The sum of these contributions to all orders is conveniently written in terms of `enhancement factors' relative to the leading order contribution \cite{BBB} defined by \begin{equation} M^{[p,s]}(\alpha_s) = 1 + \sum_{n=0}^\infty (-\beta_0\alpha_s)^n d_n^{[p,s]}~, \end{equation} so that \begin{equation} \sigma_L^{({\rm BS})} =\sigma_0 \frac{\alpha_s}{\pi} \left[\frac{1}{3}M^{[p]}+\frac{2}{3}M^{[s]}\right]. \end{equation} For various values of $\alpha_s(M_Z)$ we get at $Q=M_Z$ \begin{eqnarray}\label{Mfactors} \lefteqn{\alpha_s =0.110:} \nonumber\\ &M^{[p]}=1.59, & M^{[s]} = 1.92\pm 0.05~. \nonumber\\ \lefteqn{\alpha_s =0.120: } \nonumber\\ &M^{[p]}=1.68, & M^{[s]} = 2.08\pm 0.08~. \nonumber\\ \lefteqn{\alpha_s =0.130:} \nonumber\\ &M^{[p]}=1.79, & M^{[s]} = 2.23\pm 0.12~. \end{eqnarray} The given numbers correspond to a principal value definition of the Borel integral and the uncertainties roughly coincide with the size of the minimal term in the series\footnote{The corresponding uncertainty for $M^{[p]}$ is small in comparison with the one for $M^{[s]}$ and is omitted.}. Let us add the following comments: (i) The perturbative coefficients in (\ref{eq1}), (\ref{eq2}) grow rapidly, especially for the secondary quark contribution. This growth is related to an IR renormalon, that indicates a $1/Q^2$ correction to primary quark fragmentation and a $1/Q$ correction to secondary quark fragmentation, see Sect.~4. (ii) Even though the $1/Q$ power behaviour indicates much larger nonperturbative corrections to $\sigma_L$ as compared to $\sigma_{tot}$, the moderate size of the minimal term of the perturbative series suggests that these corrections are still not large at $Q=M_Z$. The relatively large hadronization correction for $\sigma_L$ within the JETSET model applied in \cite{OPAL} could thus correspond to higher-order perturbative rather than nonperturbative effects. \setlength{\unitlength}{0.7mm} \begin{figure}[t] \vspace{-6.8cm} \begin{picture}(100,200)(0,1) \mbox{\epsfxsize7.5cm\epsffile{QCD96fig3.eps}} \end{picture} \vspace*{-3.5cm} \caption{ Longitudinal fraction in the total $e^+e^-$ cross section: (c) leading order, (b) next-to-leading order and (a) resummation of all orders in $\beta_0^n\alpha_s^{n+1}$ corrected for the exact $O(\alpha_s^2)$ coefficient.} \end{figure} (iii) This suggestion is also supported by Fig.~3, where for $\alpha_s(M_Z)=0.118$ we have plotted the energy dependence of the total longitudinal cross section. Taking into account higher-order perturbative corrections [curve (a)] steepens the energy dependence, such that it is not far from the JETSET prediction, where the steep energy dependence is due to the hadronization correction. It is worth noting that the parton shower Monte Carlo alone does not yield this energy dependence. Experience with similar calculations suggests that the approximation of resumming only $(\beta_0\alpha_s)^n$ contributions overestimates radiative corrections, so that we expect a more realistic estimate in between the curves (a) and (b). An exact $O(\alpha_s^3)$ calculation would reduce the theoretical error considerably. (iv) In \cite{DOK95}, universality of the $1/Q$-power correction was assumed and a corresponding unique phenomenological parameter fitted from the difference between the measured average thrust $\langle 1-T\rangle$ and the theoretical second order prediction. When added to the second order result (\ref{sigmaL}), one obtains a prediction for $\sigma_L$ consistent with data. There is no conflict between the procedure of \cite{DOK95} and the one presented here, if the phenomenological $1/Q$ correction effectively parameterizes the higher-order perturbative contributions added in our approach. If universality of power corrections holds, these perturbative corrections would also be universal, at least asymptotically in large orders. However, from the point of view presented here, the universality assumption is not required, since higher-order corrections are in principle calculable for each observable. \section{Power corrections} Returning to (\ref{expansion}), we quote the expansions for the invariant mass distributions in (\ref{xi-distributions}): \begin{eqnarray} \label{xdist1} \lefteqn{x\frac{d\hat\sigma_L^{[p]}}{dx}(x,\xi) =} \\ \!\!\!\!\!\!&=& \!\!\!\! \frac{C_F\alpha_s}{2\pi} \Big\{2x + \xi \ln\xi[8+4\delta(1-x)] + \ldots\Big\}~, \nonumber \end{eqnarray} \begin{eqnarray} \label{xdist2} \lefteqn{x\frac{d\hat\sigma_L^{[s]}}{dx}(x,\xi)=} \\ \!\!\!\!\!\! &=& \!\!\!\! \frac{C_F\alpha_s}{2\pi} \Big\{ 4(1-x)(2+2x-x^2)+ 12 x\ln x \nonumber\\ & + &\frac{4\xi\ln\xi}{5x^2}\left[ 3+ 30 x - 15 x^3 + 2 x^5+ 15 x^2\ln x\right] \nonumber\\ & + & \ldots\Big\}~. \nonumber \end{eqnarray} Interpreting $\xi$ as $(\Lambda/Q)^2$ where $\Lambda$ is the QCD scale, these expressions are valid for $x>\Lambda/Q$. We note that for such $x$, all power corrections are at most of order $1/Q^2$, in agreement with the result from the light-cone expansion of fragmentation processes in \cite{BB91}. We also see that the power expansion runs in $\Lambda^2/(Q^2 x)$ for the primary quark contribution and $\Lambda^2/(Q^2 x^2)$ for the secondary quark (gluon) contribution. The strong divergence of the second contribution for small $x$ makes it possible for the moments of the $x$-distribution to have parametrically larger power corrections. Indeed, we find for the two contributions to the total longitudinal cross section \begin{eqnarray} \frac{\hat{\sigma}_L^{[p]}}{\sigma_0}\! \!\!&=&\!\!\! \frac{\alpha_s}{\pi} \left[\frac{1}{3} + 0\cdot \sqrt{\xi} + 4\xi\ln \xi + {\cal O}(\xi^2) \right] , \\ \frac{\hat{\sigma}_L^{[s]}}{\sigma_0}\! \!\!&=& \!\!\!\frac{\alpha_s}{\pi} \left[\frac{2}{3} - \frac{5\pi^3}{32}\sqrt{\xi} - 4\xi\ln \xi + {\cal O}(\xi^{\frac{3}{2}}) \right], \end{eqnarray} with a $1/Q$ correction for the secondary quark contribution. Assuming ultraviolet dominance of higher-twist corrections, the $x$-distributions given in (\ref{xdist1}), (\ref{xdist2}) can be used to model the $x$-dependence of power corrections by convoluting the partonic power correction with the leading twist fragmentation function \cite{BBM,DAS96}. Note that the expressions for the secondary quark contribution differs from the gluon contribution to $\sigma_L$ in \cite{DAS96}, because the series of higher-order fermion loop diagrams does not reduce to the massive gluon calculation performed in \cite{DAS96}. The ensuing additional model dependence in the estimate of higher-twist corrections will be discussed in \cite{BBM}. Both the calculation here and the calculation with a massive gluon coincide in the essential aspects --- power corrections of order $\Lambda^2/(Q^2 x^2)$ for finite $x$ and $1/Q$ for the integrated longitudinal cross section. In the remainder of this section, we discuss the origin of the $1/Q$ correction in more detail. Adopting for this purpose the massive gluon approximation, one finds that the Borel transform for the gluon fragmentation contribution factorizes as \begin{equation} B\left[x\frac{d\hat\sigma_L^{[g]}}{dx}\right](x;u) = x^{-2 u} \cdot F(u) , \end{equation} when some terms that can not give rise to a $1/Q$ correction are omitted. Here $u$ is the Borel parameter and analyticity of $F(u)$ for $|u|<1$ corresponds to the statement that only $1/Q^2$ corrections arise at finite $x$. Now, the $x$-dependence can be absorbed completely into a change of scale in the coupling and the Borel integral is given by \begin{equation} x\frac{d\hat\sigma_L^{[g]}}{d x} = \int\limits_0^u \!d u\,\exp\left( -\frac{u}{(-\beta_0\alpha_s(x Q))}\right) F(u). \end{equation} The Borel integral (leaving renormalon poles at finite $u$ aside) does not exist for $x<\Lambda/Q$, because it diverges at infinity. This is a manifestation of the fact that for such small $x$ the power expansion breaks down and that power corrections to integrated distributions depend sensitively on how the small-$x$ region is weighted. Indeed, because of the factorization of the $x$-dependence, the $x$-integration is trivial and we get \begin{equation} B\left[\int_0^1 d x x^{1+\gamma} \frac{d\hat\sigma_L^{[g]}}{d x} \right] = \frac{F(u)}{1+\gamma-2 u} . \end{equation} For the total longitudinal cross section, $\gamma=0$, and the newly generated pole at $u=1/2$ corresponds to the $1/Q$ correction discussed before. Note that effects due to color coherence and angular ordering are expected to change the small-$x$ asymptotic behaviour, which could potentially shift the pole to a different value. Clarifying the impact of resummation of small-$x$ logarithms requires similar efforts to those that have been undertaken to understand the effect of Sudakov resummation on power corrections in Drell-Yan production. Note that the non-uniformity of the power expansion before integration over $x$ does not occur for deep inelastic scattering (DIS) processes. Given the correspondence of IR renormalons with power ultraviolet divergences of higher-twist operators, the difference between fragmentation and DIS must be sought in the renormalization properties of multi-parton correlation functions that appear in the light-cone expansion of \cite{BB91}. For the longitudinal structure function in DIS, we find that the quadratic power divergence at one-loop of the multi-parton operator \begin{equation} g\bar{\psi}(x) \tilde{G}_{\alpha\beta}(v x) x^\alpha \gamma^\beta \gamma_5\psi(-x) \end{equation} takes the form ($\bar{\alpha}\equiv 1-\alpha$) \begin{eqnarray} &&\hspace*{-0.6cm}-\frac{C_F\alpha_s}{4\pi}\frac{\Lambda_{UV}^2}{Q^2} \!\int\limits_0^1\!d\alpha\,(2-\alpha)\Big\{\bar{\psi}(x)\!\not\! x\psi (x[\alpha v-\bar{\alpha}]) \nonumber\\ &&\hspace*{0.5cm}+\,\bar{\psi}(x[\alpha v+\bar{\alpha}]) \!\not\! x\psi(-x)\Big\}, \end{eqnarray} that is, the form of a convolution with the leading twist contribution. The important point to notice is that the operator spreads only a finite distance on the light-cone under renormalization. In contrast, the multi-parton correlations that appear in fragmentation spread over the entire light-cone under renormalization. When the energy fraction $x$ approaches zero, the operator becomes sensitive to very large longitudinal distances and to how fast the gauge fields decrease at infinity. It is this sensitivity to the behaviour at infinity that causes a $1/Q$ correction in the longitudinal cross section upon integration over $x$. We will return to this point in detail in a future publication \cite{BBM}. \section*{Acknowledgements} We are grateful to B. Webber for informing us about his related work \cite{DAS96} prior to publication. M.~B. and V.~B. thank the Institute for Nuclear Theory in Seattle for hospitality during the course of this work. This research is supported in part by the Department of Energy under contract DE-AC03-76SF-00515.

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\chapter{LOCAL QUANTUM STATISTICS AND} \centerline{\Large \bf THERMODYNAMICS OF BOSE GAS} \vspace{24pt} In this chapter we will continue to develop the methods of local statistical thermodynamics \cite{kul1} with application to the ideal bose gas in curved space-time. In chapter VII working with the Schwinger-DeWitt proper time formalism we found that the density of Helmholtz free energy in curved space-time may be found in the form of a series (\ref{i25}) including the geometric structure of space-time. However, as in the previous section, working with this formalism we could have certain difficulties with the introduction of the chemical potential and the computing of the density of the grand thermodynamical potential. To avoid such problems which will appear on the way of construction of local thermodynamics with the Schwinger DeWitt formalism, we turn to the local momentum space method \cite{bunch1}, \cite{panan1} in quantum field theory in an arbitrary curved space time and the imaginary time formalism to introduce the temperature \cite{i15}, \cite{wein1}. For the construction of the local quantum statistics and thermodynamics of a bose gas, we will use the connection between the partition function of ideal quantum systems and finite temperature Green's functions, which may be found by the local momentum space method. \section{Density of grand } \vspace{-4mm} \hspace{13mm}{\Large \bf thermodynamical potential} \vspace{1mm} In this section we will analyze the scalar field model with a conserved charge. The Lagrangian of the model is \newline \vspace{-6mm} \begin{eqnarray} S_m=-(1/2)\int d^4x\sqrt{g(x)}\Phi^{\ast}(x)(-\Box_x+m^2+\xi R)\Phi(x) \label{bos1} \end{eqnarray} \vspace{-6mm} \newline where $\Phi=(\phi_1,\phi_2)$ is a dublet of the real fields. The action written in terms of real fields will be \newline \vspace{-6mm} \begin{eqnarray} S_m=-(1/2)\int d^4x\sqrt{g(x)}\phi^a(x)(-\Box_x+m^2+\xi R)\phi^a(x) \label{bos2} \end{eqnarray} \vspace{-6mm} \newline The total action of the system "matter $+$ gravity" is \newline \vspace{-6mm} \begin{eqnarray} S_{tot}=S_g+S_m \label{bos3} \end{eqnarray} \vspace{-6mm} \newline Now we can write the effective action at finite temperature in analogy with (\ref{i20}) as \newline \vspace{-6mm} \begin{eqnarray} L_{eff}(\beta)=\tilde{L}_g-\omega(\beta, \mu ,R) \label{bos4} \end{eqnarray} \vspace{-6mm} \newline where $\tilde{L}_g$ is (\ref{i21}) and $\omega(\beta, \mu ,R)$ is the density of grand thermodynamic potential. The result (\ref{bos4}) may be obtained with the momentum space representation for the Green's function of a boson (\ref{h33}) \cite{lcs1}. In the momentum space representation, the expression for $L_{eff}$ is split into two parts \newline \vspace{-6mm} \begin{eqnarray} L_{eff}=-(i/2)\int\limits_{m^2}^\infty dm^2 \mbox{tr} G(x,x^{'}) -\omega(\beta, \mu ,R) \label{bos5} \end{eqnarray} \vspace{-6mm} \newline The potential $\omega(\beta, R)$ is \newline \vspace{-6mm} \begin{eqnarray} \omega(\beta, R)= (-1/2) \mbox{tr} \int\limits_{m^2}^\infty dm^2\sum \limits_{j=0}^2 \sum \limits_{n=0}^\infty \gamma_j(x,x^{'})\left(-\frac{\partial}{\partial m^2}\right)^j \times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \int \frac{d^3k}{(2\pi)^3}({\omega_n}^2+\epsilon^2)^{-1} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(1/2)\sum \limits_{j=0}^2 \gamma_j(x^{'})\left(-\frac{\partial}{\partial m^2}\right)^j \mbox{tr} \ln ({\omega_n}^2+\epsilon^2) \label{bos6} \end{eqnarray} \vspace{-6mm} \newline and coincides with the Helmholtz free energy (\ref{i22}) and $\omega _n=2\pi n/\beta$. The symbol $\mbox{tr}$ in (\ref{bos6}) is determined as \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr} (...)=\sum \limits_{n \neq 0} \int \frac{d^3k}{(2\pi)^3}... \nonumber \end{eqnarray} \vspace{-6mm} \newline For introducing the chemical potential, we will change the Matsubara frequencies $\omega_n \to \omega_n+\mu$ and thermodynamic potential will be $\omega(\beta, \mu ,R)$. Since both positive and negative frequencies are summed, we will have \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr} \ln ({\omega_n}^2+\epsilon^2) \to \mbox{tr} \ln [(\omega_n+\mu)^2+\epsilon^2] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{tr} \left\{\ln \left[{\omega_n}^2+(\epsilon-\mu)^2\right] +\ln \left[{\omega_n}^2+(\epsilon+\mu)^2\right]\right\} \label{bos7} \end{eqnarray} \vspace{-6mm} \newline After summation in (\ref{bos7}) we get \newline \vspace{-6mm} \begin{eqnarray} \omega(\beta, \mu ,R)=\omega_{-}(\beta, \mu ,R) +\omega_{+}(\beta, \mu ,R) \label{bos8} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \omega_{-}(\beta, \mu ,R)=(1/\beta)\sum \limits_{j=0}^2 \gamma_j(x^{'})\left(-\frac{\partial}{\partial m^2}\right)^j \ln (1-\exp [-\beta(\epsilon-\mu)]) \label{bos9} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \omega_{+}(\beta, \mu ,R)=(1/\beta)\sum \limits_{j=0}^2 \gamma_j(x^{'})\left(-\frac{\partial}{\partial m^2}\right)^j \ln (1-z \exp [-\beta (\epsilon+\mu)]) \label{bos10} \end{eqnarray} \vspace{-6mm} \newline So, with accordance to (\ref{bos8}) the density of grand thermodynamic potential is the series \newline \vspace{-6mm} \begin{eqnarray} \omega(\beta, \mu ,R)=\sum \limits_{j=0}^2\gamma^{'}_j(x^{'})b_j(\beta m,z) \label{bos11} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} b_0(\beta m,z)=(1/\beta) \ln (1-z \exp (-\beta \epsilon)); \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} b_j(\beta m,z)=\left(-\frac{\partial}{\partial m^2}\right)^j b_0(\beta m,z) \label{bos12} \end{eqnarray} \vspace{-6mm} \newline and the fugacity is $z=\exp (\beta \mu)$. The geometrical coefficients $\gamma^{'}_j(R)$ of the equation (\ref{bos12}) have the form (\ref{h45}). Renormalizations in the total Lagrangian (\ref{bos4}) are the same as in chapter XI. \section{Statistics and thermodynamics of bose gas} \vspace{1mm} We find the bose distribution function as the derivative of the grand thermodynamical potential \newline \vspace{-6mm} \begin{eqnarray} n_{\vec{k}}=-{{\partial \omega _{\bar k}(\beta ,\mu ,R)} \over {\partial \mu }} \nonumber \end{eqnarray} \vspace{-6mm} \newline For occupation numbers with momentum $\vec k$ we get \newline \vspace{-6mm} \begin{eqnarray} n_{\vec{ k}}={1 \over {(z^{-1}e^{\beta \varepsilon _{\vec{ k}}}-1)}}B(\beta ,R),\label{ins1} \end{eqnarray} \vspace{-6mm} \newline where the function $B(\beta ,R)$ is described by the formula \newline \vspace{-6mm} \begin{eqnarray} B(\beta ,R)=\ {1+\gamma _1(R){\beta \over {2\varepsilon _{\vec{k}}}}\left[{1-(1-ze^{-\beta \varepsilon _{\vec{k}}})^{-1}}\right]+\quad .\;.\;.} \label{ins2} \end{eqnarray} \vspace{-6mm} \newline The function $B(\beta ,R)$ depends on the curvature, temperature and energy of the boson. Studying the thermodynamical properties of Bose gases we will start with the equation \newline \vspace{-6mm} \begin{eqnarray} \omega(\beta, \mu ,R)=-(1/\beta)\sum \limits_{j=0}^2 \gamma_j(x^{'})\left(-\frac{\partial}{\partial m^2}\right)^j \ln (1-z \exp [-\beta \epsilon]) \label{bos13} \end{eqnarray} \vspace{-6mm} \newline In the non-relativistic limit for particle energy $\epsilon=\vec{k}^2/2m$ we get from (\ref{bos13}) the equation \newline \vspace{-6mm} \begin{eqnarray} \omega(\beta,\mu,R)=\sum \limits_{j=0}^2 \gamma_j(x^{'})g_{5/2}(z)\left(-\frac{\partial}{\partial m^2}\right)^j \lambda^{-3} \label{bos14} \end{eqnarray} \vspace{-6mm} \newline where $\lambda=(2\pi/mT)^{1/2}$ is a wavelength of the particle, and the function $g_{5/2}(z)$ has the following form \newline \vspace{-6mm} \begin{eqnarray} g_{5/2}(z)=\sum\limits_{l=1}^\infty \frac{z^l}{l^{5/2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-(4/\sqrt{\pi})\int dx x^2\ln \left(1-z \exp (-x^2)\right) \label{bos15} \end{eqnarray} \vspace{-6mm} \newline The average number of particles in a certain momentum state $k$ is obtained as the derivative \newline \vspace{-6mm} \begin{eqnarray} <n_k>=-\frac{\partial}{\partial \mu}\omega(\beta, \mu ,R) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\sum \limits_{j=0}^2 \gamma_j(x^{'})g_{5/2}(z)\left(-\frac{\partial}{\partial m^2}\right)^j \left(z^{-1} \exp (\beta \epsilon)-1\right) \label{bos16} \end{eqnarray} \vspace{-6mm} \newline The density of particles is \newline \vspace{-6mm} \begin{eqnarray} n=\lambda^{-3}\left[1- \gamma_1(R)(3/4m^2)-\gamma_2(R)(3/16m^4)\right]g_{3/2}(z) +n_0 \label{bos17} \end{eqnarray} \vspace{-6mm} \newline where the new function $g_{3/2}(z)$ is \newline \vspace{-6mm} \begin{eqnarray} g_{3/2}(z)=z\frac{\partial}{\partial z}g_{5/2}(z) \nonumber \end{eqnarray} \vspace{-6mm} \newline and $n_0=z/(1-z)$ is the average number of particles with zero momentum. The functions $g_{3/2}(z)$ and $g_{5/2}(z)$ are special cases of a more general class of functions \newline \vspace{-6mm} \begin{eqnarray} g_{n}(z)=\sum \limits_{k=1}^\infty \frac{z^k}{k^n} \label{bos18} \end{eqnarray} \vspace{-6mm} \newline In a more simple form the equation(\ref{bos17}) may be written as \newline \vspace{-6mm} \begin{eqnarray} (n-n_0)\lambda^3=g_{3/2}(z,R) \label{bos19} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} g_{3/2}(z,R)=\left[1- \alpha\frac{R}{m^2}+...\right]g_{3/2}(z) \label{bos20} \end{eqnarray} \vspace{-6mm} \newline is a function which depends on curvature, and $\alpha$ is a numerical parameter. The equation (\ref{bos19}) connects four values: fugacity, temperature, density of the particles and curvature. We can solve it graphically and get the dependence of the (effective) chemical potential on curvature, temperature and density. The function $g_{3/2}(z,R)$ varies with the curvature $R$ as shown in Fig. I-1. The graphical solution of the equation (\ref{bos19}) is presented in Fig. I-2. In this Figure $X$-projection of the point of intersection of the curve $g_{3/2}(z,R)+\lambda^3n_0$ and of the line $\lambda^3n$ shows the dependence of the fugacity on curvature. As follows from the graphical picture, for positive curvature $R<0$ the fugacity $z(R)>z_0$ (or $\mu(R)>\mu_0$), for $R>0$ we have $z(R)<z_0$ (or $\mu(R)>\mu_0$). The fugacity $z_0$ ($\mu_0$-chemical potential) corresponds to statistics in Euclidean space. The behaviour of the effective chemical potential is shown in Fig I-3. \section{Bose-Einstein condensation} \vspace{1mm} At low temperature there is a significal number of particles in the ground state, which can be expressed by the equation \newline \vspace{-6mm} \begin{eqnarray} n_0=n-\lambda^{-3}g_{3/2}(z,R) \label{bos21} \end{eqnarray} \vspace{-6mm} \newline With the rising of the temperature the average number of particles $n_0$ with zero momentum is lowered and for temperatures $T>T_{c}$ it becomes zero. The temperature $T_{c}$ is the critical temperature of a Bose condensation. The critical temperature may be found from the equation with ($z=1$ and $n_0=0$) \newline \vspace{-6mm} \begin{eqnarray} n{\lambda_{c}}^{3}=g_{3/2}(1,R) \label{bos22} \end{eqnarray} \vspace{-6mm} \newline The solution of this equation is \cite{i23} \newline \vspace{-6mm} \begin{eqnarray} {T_{c}(R)}={T_0}\left(1+ \frac{\gamma_1^{'}(R)}{2 m^2}+...\right) \label{bos23} \end{eqnarray} \vspace{-6mm} \newline The temperature \newline \vspace{-6mm} \begin{eqnarray} T_0=T_c(R=0)=\frac{2\pi}{m}\left[ \frac{n}{\zeta(3/2)}\right]^{2/3} \label{bos24} \end{eqnarray} \vspace{-6mm} \newline where ${\zeta(3/2)}=2.612...$ is the Riemann zeta function, $n$ is the density of bosons and $m$ is the mass of boson is degeneracy temperature (condensation temperature) in "flat" space without gravity. The ratio \newline \vspace{-6mm} \begin{eqnarray} \frac{T_{c}(R)}{T_0}=1+ \frac{R}{12 m^2}+... ,\label{bos25} \end{eqnarray} \vspace{-6mm} \newline is not one, but depends on the curvature $R$ of space-time. As we can see, $T_{c}(R)>T_0$, for $R>0$ and $T_{c}(R)<T_0$ for $R<0$. The correction \newline \vspace{-6mm} \begin{eqnarray} \frac{\delta T_c(R)}{T_0}=\frac{R}{12 m^2}+O\left((R/m^2)^2\right) \end{eqnarray} \vspace{-6mm} \newline is small, therefore effects of curvature will be essential for quantum systems in strong gravitational fields. \chapter{ GRAVITATIONAL CHERN-SIMONS } \centerline{\Large \bf MASS TERM } \vspace{3mm} \centerline{\Large \bf AT FINITE TEMPERATURE } \vspace{24pt} In this chapter we will consider fermions interacting with an external gravitational field at finite temperature. In the previous chapter XX we learned that the interaction of fermions with external gauge bosons leads to the induced action of the Chern-Simons type. In the same way the interaction of fermions with external gravitational fields may leads to the effective gravitational Chern-Simons term \cite{vcs1},\\ \cite{gcs1}. Now we will develop a formalism of calculations of the effective gravitational CS action from the action for massive fermions interacting with an external gravitational field. \section{ Induced gravitational Chern-Simons} \vspace{-4mm} \hspace{18mm}{\Large \bf mass term} \vspace{1mm} Let us introduce the action for massive fermions connected with an external gravitational field as: \newline \vspace{-6mm} \begin{eqnarray} S=\int d^3x \sqrt{g}\bar{\psi}(x)(iD+m)\psi(x) \label{gcs1} \end{eqnarray} \vspace{-6mm} \newline where $D=\gamma^\mu D_\mu=\gamma^\mu (\partial_\mu +\omega_{ab\mu}\sigma^{ab})$ with $\gamma^\mu(x)=h^\mu _a(x)\gamma^a $. The function $\omega_{ab\mu}$ is the local Lorentz connection and $\sigma^{ab}=(1/8)[\gamma^a,\gamma^b]$ is the commutator of $\gamma$ matrices. $\gamma$-matrices with Latin indices are constructed from Pauli matrices \newline \vspace{-6mm} \begin{eqnarray} \gamma^a=i\sigma^a ~~,\sigma^a (a=1,2,3) \label{gcs2} \end{eqnarray} \vspace{-6mm} \newline and obey the relations: \newline \vspace{-6mm} \begin{eqnarray} \{\gamma^a,\gamma^b \}=-2\delta^{ab}, ~~~\gamma_a\gamma_b=-\delta_{ab}-\epsilon _{abc}\gamma^c \label{gcs3} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr} (\gamma^a\gamma^b)=-2i\delta^{ab}, ~~~ \mbox{tr} (\gamma^a\gamma^b\gamma^c)=2\epsilon^{abc} \label{gcs4} \end{eqnarray} \vspace{-6mm} \newline We can find the effective action by integrating (\ref{gcs1}) with respect to the fermionic field from the equation \newline \vspace{-6mm} \begin{eqnarray} exp[-S_{eff}]=\int D\bar{\psi}(x)\psi(x) \exp [-S]=Det(-iD-m)\label{gcs5} \end{eqnarray} \vspace{-6mm} \newline To find the one-loop effective action we will rewrite (\ref{gcs5}) in the following form \cite{ojima1}: \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=-ln~ Det\frac{(-iD-m)}{(-iD+i\omega-m)} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\ln~Det\frac{(-iD-m)(-iD-i\omega-m)}{(-iD+i\omega-m)(-iD-i\omega-m)} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\ln~Det(DD-D\omega+im\omega+m^2) +logDet[(D-\omega)^2]+m^2 \label{gcs6} \end{eqnarray} \vspace{-6mm} \newline where we put $\omega=\gamma^\mu \omega_\mu=\gamma^\mu \sigma^{ab}\omega _{ab \mu}$. Now we can use the momentum space formalism we developed in part I of this work for calculation of the first determinant of the expression (\ref{gcs6}). We can simplify the problem by noticing that the second term of (\ref{gcs6}) does not give contributions to the induced CS term. For the calculation of the first term let us introduce normal coordinates and write the metric at the origin ($x^{'}$) of these coordinates. We will use variable $y=x-x^{'}$ for the definition of the point of manifold in tangent space. The tangent space $Y$ will be "flat" with metric $g_{ab}=-\eta_{ab}=diag(-1,-1,-1)$. In the origin of these coordinates the tetrad functions and the metric will be the series: \newline \vspace{-6mm} \begin{eqnarray} h^\mu _a(x)=h^\mu _a(x^{'})-(1/6){R^\mu}_{\nu a \sigma}y^\mu y^\sigma +...\label{gcs7} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} g_{\mu \nu}(x)=g_{\mu \nu}(x^{'})-(1/3)R_{\mu \alpha \nu \beta}y^\mu y^\nu -(1/6){R _{\mu \alpha \nu \beta}}_{;\lambda} y^\mu y^\nu y^\lambda +... \label{gcs8} \end{eqnarray} \vspace{-6mm} \newline The covariant derivative in a tangent space is written as \newline \vspace{-6mm} \begin{eqnarray} D=\gamma_\mu (x)D_\mu (x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\left[\frac{}{}\gamma^a h^\mu _a(x^{'})-(1/6)\gamma^a {R^\mu}_{\nu a \sigma} y^\mu y^\sigma+...\right]\left( \frac{\partial} {\partial y^\mu}+\omega_{ab\mu }(x^{'})\sigma^{ab}\right) \label{gcs9} \end{eqnarray} \vspace{-6mm} \newline The effective action can be represented in the normal coordinates as \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=\int d^3 x\sqrt{g} S_{eff}(x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mathop{\lim}\limits_{x \to x^{'}} \int d^3 x^{'}\sqrt{g( x^{'})} S_{eff}( x^{'})= \mathop{\lim}\limits_{y \to 0}\int d^3 x\sqrt{g} S_{eff}(x,y) \label{gcs10} \end{eqnarray} \vspace{-6mm} \newline In proper time formalism $ln Det$ is \newline \vspace{-6mm} \begin{eqnarray} S_{eff}[\omega]=\ln~Det(DD-D\omega+im\omega+m^2)= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x \to x^{'}} \int d^3x <x|\int \limits_0^\infty \frac{ds}{s}\mbox{tr} \exp \left\{-i\left( D^\mu D_\mu+\frac{R}{4} -D\omega +im\omega +m^2\right)\right\}s|x^{'}> \label{gcs11} \end{eqnarray} \vspace{-6mm} \newline In accordance with (\ref{gcs10}) and (\ref{gcs11}) the density $S_{eff} (x)$ may be written in the momentum-space form \newline \vspace{-6mm} \begin{eqnarray} S_{eff}(x)=\mathop{\lim}\limits_{y \to 0} \int \limits_0^\infty \frac{ds}{s}\int \frac{d^3 k}{(2\pi)^3} \exp [~iky~] \exp [-i(m^2-\Delta_\mu \Delta^\mu )s]\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr} \exp \left\{-i\left[\left( D^\mu D_\mu+\frac{R}{4} -D\omega \right) +(i \Delta_\mu D^\mu +iD_\mu \Delta^ \mu-i\Delta \omega +im\omega ) \right]s \right\} \label{gcs12} \end{eqnarray} \vspace{-6mm} \newline where $\mathop{\lim}\limits_{y \to 0} \Delta_\mu (y)=k_\mu$. Notice that if we like to obtain the contribution of finite terms as $(m \to 0) $, it is sufficient to extract the terms which are proportional to $s^2$ with a factor $m$ or proportional to $s^3$ with a factor $m^3$ (or $k^2$). Thus we get the term in $S_{eff}[\omega]$, which contains the induced CS term. The second and the third orders of the expansion (\ref{gcs12}) give \newline \vspace{-6mm} \begin{eqnarray} S_{eff}(x)=\mathop{\lim}\limits_{y \to 0} \int \frac{d^3 k}{(2\pi)^3} \exp [~iky~]\int \limits_0^\infty \frac{ds}{s} \exp [-i(m^2-\Delta_\mu \Delta^\mu)s](im)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \mbox{tr} \left[\frac{1}{2!}\left( \omega D\omega + D\omega \omega \right)s^2- \right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. - \frac{1}{3!}(\Delta \omega\Delta \omega \omega + \Delta \omega \omega \Delta \omega+\omega\Delta \omega\Delta \omega +m^2\omega\omega\omega)s^3 \right] \label{gcs13} \end{eqnarray} \vspace{-6mm} \newline where $\Delta(y)=\gamma^\mu \Delta_\mu(y)$. In the limit $(y \to 0)$ the equation (\ref{gcs13}) is \newline \vspace{-6mm} \begin{eqnarray} S_{eff}(x)=(im)\int \frac{d^3 k}{(2\pi)^3}\int \limits_0^\infty \frac{ds}{s} \exp [-i(m^2+k^2)s]\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \mbox{tr} \left[\left( \omega \hat{\partial}\omega +\omega \omega\omega\right)~s^2- \right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.- \frac{1}{3!}(\hat{k} \omega\hat{k} \omega \omega + \hat{k}\omega \omega \hat{k}\omega+\omega \hat{k} \omega \hat{k}\omega +m^2\omega\omega\omega)s^3 \right] \label{gcs14} \end{eqnarray} \vspace{-6mm} \newline Noticing, that \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr}(\hat{k} \omega\hat{k} \omega \omega + \hat{k}\omega \omega \hat{k}\omega+\omega \hat{k} \omega \hat{k}\omega +m^2\omega\omega\omega)=(k^2+m^2)\mbox{tr}\omega\omega\omega \label{gcs15} \end{eqnarray} \vspace{-6mm} \newline rewrite (\ref{gcs14}) as \newline \vspace{-6mm} \begin{eqnarray} S_{eff}(x)=(im)\left[\mbox{tr}\omega \hat{\partial}\omega+\frac{2}{3}\mbox{tr}\omega\omega\omega \right] \int \frac{d^3 k}{(2\pi)^3}\int \limits_0^\infty ds(s) \exp [-is(m^2+k^2)] \label{gcs16} \end{eqnarray} \vspace{-6mm} \newline Applying the equation for the $\Gamma(\nu )$ function \newline \vspace{-6mm} \begin{eqnarray} \int \limits_0^\infty ds s^{\nu-1}\exp [-sm^2] =\frac{\Gamma(\nu)}{|m|^{2\nu}} \label{gcs17} \end{eqnarray} \vspace{-6mm} \newline and computing \mbox{tr} of $\gamma$ matrices, find \newline \vspace{-6mm} \begin{eqnarray} S_{eff}^{CS}[\omega]=-\frac{i}{64 \pi}\frac{m}{|m|}\int d^3 x \epsilon^{\mu \nu\rho} \left(\partial_\mu {\omega^b}_{a \nu}{\omega^a}_{b \rho}+ \frac{2}{3}{\omega^a}_{b \nu}{\omega^b}_{c \mu}{\omega^c}_{a \rho}\right) \label{gcs18} \end{eqnarray} \vspace{-6mm} \newline This is the expression for gravitational CS term \cite{ojima1} which we get using momentum-space method. \section{Induced gravitational Chern-Simons} \vspace{-4mm} \hspace{13mm}{ \Large \bf mass term at finite temperature} \vspace{1mm} The calculations of finite temperature gravitational action of CS type are based on the computation of $\ln Det$ (\ref{gcs5}). In Part I we developed the formalism of such calculations with the help of the momentum space methods. For these calculations we will replace the integration procedure over three momenta in the expression (\ref{gcs16}) by the summation over Matsubara frequencies and integration over the two remaining momenta. Let us introduce temperature in the tangent 3-D space to the curved manifold as (part I,(\ref{b47})): \newline \vspace{-6mm} \begin{eqnarray} \int \frac{d^3k}{(2 \pi)^3}F(k^2_1,k^2_2,k^2_3,s) \mathop{\longrightarrow}\limits^{T\neq 0} \frac{1}{\beta}\sum \limits_{n=-\infty}^{\infty} \int \frac{d^2k}{(2 \pi)^2}F(\omega^2_n,k^2_2,k^2_3,s) \label{gcs19} \end{eqnarray} \vspace{-6mm} \newline where $\beta^{-1}=T$ is temperature, and $\omega^2_n=(2\pi /\beta)(n+1/2)$. To introduce temperature in the model, rewrite integral over $(s)$ in (\ref{gcs16}) in the form \newline \vspace{-6mm} \begin{eqnarray} \int \frac{d^3p}{(2\pi)^3}\int ds~s \exp [-is(k^2+m^2)]= -\int \frac{d^3p}{(2\pi)^3}\frac{1}{(k^2+m^2)^2} \label{gcs20} \end{eqnarray} \vspace{-6mm} \newline Using (\ref{gcs19}) and the summation formula (\ref{vcs23}), find the finite temperature CS term \newline \vspace{-6mm} \begin{eqnarray} S_{eff}[\omega]=-\frac{i}{64 \pi}\frac{m}{|m|} tanh\frac{ |m|}{2T} \int d^3 x \epsilon^{\mu \nu\rho} \left(\partial_\mu {\omega^b}_{a \nu}{\omega^a}_{b \rho}+ \frac{2}{3}{\omega^a}_{b \nu}{\omega^b}_{c \mu}{\omega^c}_{a \rho}\right) \label{gcs22} \end{eqnarray} \vspace{-6mm} \newline Comparing (\ref{gcs18}) and (\ref{gcs22}) we get the following result \newline \vspace{-6mm} \begin{eqnarray} \frac{S_{eff}[T \neq 0]}{S_{eff}[T=0]}= tanh\frac{\beta |m|}{2} \label{gcs23} \end{eqnarray} \vspace{-6mm} \newline The expression (\ref{gcs22}) shows that the structure of the induced CS term at finite temperature is exactly the same as at zero temperature, and the functional relations between the CS terms in tensor (\ref{gcs23}) and vector types of interactions (\ref{vcs29}) are the same. \chapter{QUANTUM FIELD METHODS} \centerline{\Large \bf IN STATISTICAL PHYSICS} \vspace{24pt} The customary approach to many-body theory is the method of second quantization \cite{key13,fw1}. This chapter is a short introduction to statistical mechanics of simple quantum systems using this method. \section{Equilibrium statistical mechanics} \vspace{1mm} Statistical mechanics deals with three types of ensembles: 1) $The$ $microcanonical$ $ensemble$ is used to describe an isolated system which has a fixed energy $E$, particle number $N$, and volume $V$. 2) $The$ $canonical$ $ ensemble$ is used to describe a system in contact with a heat reservoir at temperature $T$. The system can freely exchange energy with the reservoir and $T$, $N$, and $V$ are fixed variables 3) $The$ $grand$ $canonical$ $ensemble$ is used to describe a system which can exchange particles as well as energy with reservoir. In this ensemble $T$, $V$ and the chemical potential $\mu$ are fixed variables. The main aim of statistical mechanics is to derive the thermodynamic properties of macroscopic bodies starting from the description of the motion of the microscopic components (atoms, electrons, etc.). To solve the problem it is necessary to find the probability distribution of the microscopic components in thermal equilibrium (after a sufficiently long time), and deduce the macroscopic properties of the system from this microscopic probability distribution. Following this scheme let us consider a classical Hamilton system with $2N$ degrees of freedom in a box of volume $V$. The classical equations of motion are: \newline \vspace{-6mm} \begin{eqnarray} \dot{q}_i=\frac{\partial H}{\partial p_i}, \hspace{1cm}\dot{p}_i=-\frac{\partial H}{\partial q_i} \label{a1} \end{eqnarray} \vspace{-6mm} \newline where $H$ is the Hamiltonian, $q_i$ and $p_i$ $(i=1,2,...,N)$ are the set of coordinates and momenta of the system. Let $A(q,p)$ be an arbitrary measurable quantity (an observable). The equilibrium average of this quantity $\bar{A}$ is defined by \newline \vspace{-6mm} \begin{eqnarray} \bar{A}=\mathop{\lim}\limits_{t \to \infty} \frac{1}{t}\int\limits_o^\infty A(q(\tau),p(\tau))d\tau \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int dqdp A(q,p)\varrho(q,p) \label{a2} \end{eqnarray} \vspace{-6mm} \newline where $\varrho(q,p)$ is the equilibrium probability. This function is never negative and satisfies the normalization condition \newline \vspace{-6mm} \begin{eqnarray} \int dqdp~\varrho(q,p) =1.\label{a3} \end{eqnarray} \vspace{-6mm} \newline The fundamental hypothesis of equilibrium statistical mechanics is that $\varrho$ follows the canonical distribution \newline \vspace{-6mm} \begin{eqnarray} \varrho(q,p) =Z^{-1} \exp[-\beta H(q,p)], \label{a4} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} Z=\int dqdp \exp[-\beta H(q,p)] \label{a5} \end{eqnarray} \vspace{-6mm} \newline is the partition function, and $\beta^{-1}=T$ is the absolute equilibrium temperature. It is useful to find a characterization of the canonical distribution that distinguishes it from all other possible probability distributions. It is convenient to introduce the entropy of a distribution $\varrho$ defined as follows: \newline \vspace{-6mm} \begin{eqnarray} S[\varrho]=-\int dqdp\varrho(q,p) \mbox{ln}\varrho(q,p) =-<\mbox{ln}\varrho> \label{a6} \end{eqnarray} \vspace{-6mm} \newline Entropy has the following properties: the more ordered is the system the smaller is the entropy (i.e., the more concentrated the probability distribution in a restricted region of phase space); the more disordered is the system (i.e., the more uniform the probability distribution), the larger is the entropy. For relativistic quantum statistical systems the equation (\ref{a4}) will be turned into the equation for statistical operator \newline \vspace{-6mm} \begin{eqnarray} \hat{\varrho}=Z^{-1} \exp[-\beta(\hat {H}-\mu\hat{N}] \label{a7} \end{eqnarray} \vspace{-6mm} \newline where $\hat {H}$ is the Hamiltonian and $\hat{N}$ is the number operator. This operator is Hermitian and commutes with Hamiltonian $H$. The parameter $\mu$ is the so called chemical potential. The ensemble average of an operator $\hat{A}$ will be \newline \vspace{-6mm} \begin{eqnarray} <A>=Z^{-1}\mbox{Tr}[\hat{A}\hat{\varrho}] \label{a8} \end{eqnarray} \vspace{-6mm} \newline The factor $Z$ will turn into a so called grand canonical partition function of the form \newline \vspace{-6mm} \begin{eqnarray} Z=\mbox{Tr} \exp[-\beta(\hat {H}-\mu\hat{N}] \label{a9} \end{eqnarray} \vspace{-6mm} \newline This function is the most important in thermodynamics. The average value of the energy $U$ may be written from (\ref{a8}) as \newline \vspace{-6mm} \begin{eqnarray} U=<\hat{H}>=\mbox{Tr}[\hat{H}\hat{\varrho}] \label{a10} \end{eqnarray} \vspace{-6mm} \newline and the entropy as \newline \vspace{-6mm} \begin{eqnarray} S=-\mbox{Tr}[\hat{\varrho}\mbox{ln}\hat{\varrho}] \label{a11} \end{eqnarray} \vspace{-6mm} \newline From the equations (\ref{a9}) and (\ref{a11}) we get a useful relation \newline \vspace{-6mm} \begin{eqnarray} S=-\mbox{Tr}[\hat{\varrho}~\mbox{ln}\hat{\varrho}] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\mbox{Tr}[\hat{\varrho}(-\ln Z-\beta\hat{H}+\mu\hat{N})] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{ln}Z+\beta U-\beta \mu N, \nonumber \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} (1/\beta)\mbox{ln}Z=U-S/\beta-\mu N. \label{a12} \end{eqnarray} \vspace{-6mm} \newline The quantity \newline \vspace{-6mm} \begin{eqnarray} \Omega=-(1/\beta)\mbox{ln} Z \label{a13} \end{eqnarray} \vspace{-6mm} \newline is the grand thermodynamical potential of the grand canonical ensemble. For canonical ensemble ($\mu=0$) we can write the equivalent equation for thermodynamic potential $F$ (free energy). It is easy to find that \newline \vspace{-6mm} \begin{eqnarray} F-\Omega=\mu N. \label{a14} \end{eqnarray} \vspace{-6mm} \newline The grand partition function $Z=Z(V,T,\mu )$ is the most important function in thermodynamics. All other standard termodynamic properties may be determined from this function. For example, the pressure, particle number, entropy, and energy (in the infinite volume limit) are \newline \vspace{-6mm} \begin{eqnarray} P=T\frac{\partial \mbox{ln} Z}{\partial V}, \hspace {2cm} N=T\frac{\partial \mbox{ln }Z}{\partial \mu} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} S=T\frac{\partial ~T\mbox{ln}Z}{\partial T},\hspace {2cm} E=-PV+TS+\mu N \label{a15} \end{eqnarray} \vspace{-6mm} \newline \section{Statistical mechanics of simple systems} \vspace{-4mm} \hspace{15mm}{\Large \bf Formalism of second quantization} \vspace{1mm} Now we can apply the formalism of statistical mechanics developed above to non-interacting many body quantum systems in the frame of the method of second quantization. \vspace{3mm} \centerline{ \large \bf 1. Bosonic quantum system at finite temperature} \vspace{3mm} Let us study a bosonic quantum system. Each quantum state $\epsilon$ of the system may be occupied by bosons. Let $n$ $(n=0,1,2,3,...)$ be number of bosons in this state, and $|n>$ be the wave function of this state. We will call a state $|0>$ a vacuum state. One may introduce creation $a$ and annihilation $a^+$ operators with commutation relation \newline \vspace{-6mm} \begin{eqnarray} [a,a^+]=aa^+-a^+a=1 \label{a16} \end{eqnarray} \vspace{-6mm} \newline The action of these operators on eigenstates is \newline \vspace{-6mm} \begin{eqnarray} a|n>=\sqrt n|n-1>, \hspace{2cm} a^+|n>=\sqrt {n+1}|n+1> \label{a17} \end{eqnarray} \vspace{-6mm} \newline and on vacuum state is \newline \vspace{-6mm} \begin{eqnarray} a|0>=0 \label{a18} \end{eqnarray} \vspace{-6mm} \newline From (\ref{a16}), (\ref{a17}) one can get the number operator $\hat{N}=a^+a$: \newline \vspace{-6mm} \begin{eqnarray} \hat{N}=a^+a|n>=n|n> \label{a19} \end{eqnarray} \vspace{-6mm} \newline We may build all states $|n>$ from vacuum $|0>$ by repeated applications of the creation operator \newline \vspace{-6mm} \begin{eqnarray} |n>=(n!)^{-1/2}(a^+)^n|0> \label{a20} \end{eqnarray} \vspace{-6mm} \newline The Hamiltonian of the system in the state with energy $\epsilon$ may be constructed as a product $\epsilon$ with number operator (up to an additive constant) in the form \newline \vspace{-6mm} \begin{eqnarray} \hat{H}=\epsilon(\hat{N}+1/2)=\epsilon(a^+a+1/2) \label{a21} =\frac{1}{2}\epsilon(a^+a+aa^+) \end{eqnarray} \vspace{-6mm} \newline The additive term $\epsilon/2$ in (\ref{a21}) is the zero-point energy. Usually this term may be ignored and we get $\hat{H}=\epsilon \hat{N}$. The grand partition function according to (\ref{a9}) will be \newline \vspace{-6mm} \begin{eqnarray} Z=\mbox{Tr} \exp[-\beta(\hat{H}-\mu\hat{N})] =\mbox{Tr} \exp[-\beta(\epsilon-\mu)\hat{N}] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_{n=0}^\infty <n|\exp[-\beta(\epsilon-\mu)\hat{N}]|n> =\sum\limits_{n=0}^\infty \exp[-\beta(\epsilon-\mu)n] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(1-\exp[-\beta(\epsilon-\mu)])^{-1} \label{a22} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{a22}) into (\ref{a15}) we get the average number of bosons in the system with energy $\epsilon$ \newline \vspace{-6mm} \begin{eqnarray} N=(\exp[\beta(\epsilon-\mu)]-1)^{-1} \label{a23} \end{eqnarray} \vspace{-6mm} \newline and the average energy in the form $\bar{\epsilon}=\epsilon N$. \vspace{3mm} \centerline{ \large \bf 2. Fermionic quantum system at finite temperature} \vspace{3mm} For a fermionic quantum system there are only two states $|0>$ and $|1>$ with energy $\epsilon$. The action of the fermion creation and annihilation operators on these states is \newline \vspace{-6mm} \begin{eqnarray} \alpha^+|0>=|1>,~~~\alpha|1>=|0> \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \alpha^+|1>=|0>,~~~\alpha|0>=|1> \label{a24} \end{eqnarray} \vspace{-6mm} \newline Thus, these operators have the property that their square is zero \newline \vspace{-6mm} \begin{eqnarray} \alpha^+\alpha^+=\alpha \alpha=0 \label{a25} \end{eqnarray} \vspace{-6mm} \newline The number operator is $\hat{N}=\alpha^+\alpha$. One has \newline \vspace{-6mm} \begin{eqnarray} \hat{N}|0>=\alpha^+\alpha|0>=0;~~~ \hat{N}|1>=\alpha^+\alpha|1>=|1> \label{a26} \end{eqnarray} \vspace{-6mm} \newline Creation and annihilation operators satisfy the anticommutation relation \newline \vspace{-6mm} \begin{eqnarray} \{\alpha,\alpha^+\}=\alpha\alpha^++\alpha \alpha^+=1 \label{a27} \end{eqnarray} \vspace{-6mm} \newline The Hamiltonian for this system may be taken in the form \newline \vspace{-6mm} \begin{eqnarray} \hat{H}=\frac{1}{2}(\alpha^+\alpha-\alpha \alpha^+) =\epsilon(\hat{N}-1/2) \label{a28 } \end{eqnarray} \vspace{-6mm} \newline The partition function for the fermionic system will be \newline \vspace{-6mm} \begin{eqnarray} Z=\mbox{Tr} \exp[-\beta(\hat{H}-\mu \hat{N})] =\sum \limits_{n=0}^{1} <n|\exp [-\beta(\epsilon-\mu)\hat{N}]|n> \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =1+\exp[-\beta(\epsilon-\mu)] \label{a29} \end{eqnarray} \vspace{-6mm} \newline The average number of fermions is \newline \vspace{-6mm} \begin{eqnarray} N=(\exp[\beta(\epsilon-\mu)]+1)^{-1} \label{a30} \end{eqnarray} \vspace{-6mm} \newline The average energy of the system in the state with energy $\epsilon$ is $\bar{\epsilon}=\epsilon N$. The next section describes an alternative approach, the method of functional integrals for studying the behavior of statistical systems. \chapter{ PATH INTEGRALS} \centerline{\Large \bf IN STATISTICAL PHYSICS } \vspace{24pt} Functional integration, introduced several decades ago \cite{fh1}, is one of the most powerful methods of modern theoretical physics. The functional integration approach to systems with an infinite number of degrees of freedom turns out to be very suitable for the introduction and formulation of perturbation theory in quantum field theory and statistical physics. This approach is simpler than the operator method. Using functional integrals in statistical physics allows one to derive numerous interesting results more quickly than other methods. Theories of phase transitions of the second kind, superfluidity, superconductivity, plasma, and the Ising model are examples of the problems for which the functional integration method appears to be very useful. If an exact solution exists, the functional integration method gives a simple way to obtain it. In other cases, when exact knowledge is unobtainable, the application of functional integrals helps to build a qualitative picture of the phenomenon, and to develop an approximate solution scheme. The functional integration method is suitable for obtaining the diagram techniques of the perturbation theory, and also for modifying the perturbative scheme if such a modification is necessary. The extension of functional integral techniques to background curved space-time allows one to take into account the gravitational field by considering the statistical and thermodynamical properties of the systems. The aim of this chapter is to introduce the finite temperature functional integral approach to statistical mechanics and local thermodynamics in curved space-time. \section{Partition function in path integral formalism} \vspace{1mm} This paragraph discusses the scalar field which is described by the Schrodinger field operator $\hat{\varphi}(\vec{x})$ where $\vec{x}$ is the spatial coordinate. We denote eigenstates of $\hat{\varphi}(\vec{x})$ by $|\varphi>$ \newline \vspace{-6mm} \begin{eqnarray} \hat{\varphi}(\vec{x})|\varphi> = \varphi(\vec{x})|\varphi> \label{b1} \end{eqnarray} \vspace{-6mm} \newline where $\varphi(\vec{x})$ is a $c$-number function. Let $\hat{\pi}(\vec{x})$ be its conjugate momentum operator. The usual completeness and orthogonality conditions may be written as: \newline \vspace{-6mm} \begin{eqnarray} \int {d\varphi \left| \varphi \right\rangle }\left\langle \varphi \right|=1 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left\langle {{\varphi _a}} \mathrel {\left | {\vphantom {{\varphi _a} {\varphi _b}}} \right. \kern-\nulldelimiterspace} {{\varphi _b}} \right\rangle =\delta \left[ {\varphi _a(\vec{x})-\varphi _b(\vec{x})} \right] \label{b2} \end{eqnarray} \vspace{-6mm} \newline where $\delta$-is the Dirac delta function. Similarly, the eigenstates of the conjugate momentum field operator are labeled by $|\pi>$ and satisfy the equation \newline \vspace{-6mm} \begin{eqnarray} \hat{\pi}(\vec{x})|\pi> =\pi(\vec{x})|\pi> \label{b3} \end{eqnarray} \vspace{-6mm} \newline with eigenvalies $\pi(\vec{x})$. The completeness and orthogonality conditions are \newline \vspace{-6mm} \begin{eqnarray} \int {d\pi \left| \pi \right\rangle }\left\langle \pi \right|=1 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left\langle {{\pi _a}} \mathrel{\left | {\vphantom {{\pi _a} {\pi _b}}} \right. \kern-\nulldelimiterspace} {{\pi _b}} \right\rangle =\delta \left[ {\pi _a(\vec{x})-\pi _b(\vec{x})} \right] \label{b4} \end{eqnarray} \vspace{-6mm} \newline the overlap between an eigenstate of field operator and an eigenstate of the momentum operator is \newline \vspace{-6mm} \begin{eqnarray} \left\langle {\varphi } \mathrel{\left | {\vphantom {\varphi \pi }} \right. \kern-\nulldelimiterspace} {\pi } \right\rangle =\exp \left[ {i\int {d^3x\pi (\vec{x})\varphi (\vec{x})}} \right] \label{b5} \end{eqnarray} \vspace{-6mm} \newline For dynamics, one requires a Hamiltonian expressed as a functional of the field and its conjugate momentum: \newline \vspace{-6mm} \begin{eqnarray} H=\int {d^3xH(\hat{\pi} ,\hat{\varphi} )} \label{b6} \end{eqnarray} \vspace{-6mm} \newline Suppose that a system is in state $|\varphi_a>$ at a time $t=0$. After time $t_f$ it will evolve into $exp(-iHt_f)|\varphi_a>$. The transition amplitude for going from state $|\varphi_a>$ to some other state $|\varphi_b>$ after time $t_f$ is thus \newline \vspace{-6mm} \begin{eqnarray} <\varphi_b| \exp (-iHt_f)|\varphi_a> \label{b7} \end{eqnarray} \vspace{-6mm} \newline For statistical mechanics purposes, consider the case in which the system returns to its original state after time $t_f$. Divide the time interval $[0,t_f]$ into $N$ equal steps of duration $\Delta t=t_f/N$. Then \newline \vspace{-6mm} \begin{eqnarray} \left\langle {\varphi _a} \right|\exp (-iHt_f)\left| {\varphi _a} \right\rangle =\mathop {\lim }\limits_{N\to \infty } \int {\prod\limits_{i=1}^N {\left( {d\varphi _i{{d\pi _i} \over {2\pi }}} \right)}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left\langle {{\varphi _a}} \mathrel{\left | {\vphantom {{\varphi _a} {\pi _N}}} \right. \kern-\nulldelimiterspace} {{\pi _N}} \right\rangle \left\langle {\pi _N} \right|\exp (-iH \Delta t_f) \left| {\varphi _N} \right\rangle \left\langle {{\varphi _N}} \mathrel{\left | {\vphantom {{\varphi _N} {\pi _{N-1}}}} \right. \kern-\nulldelimiterspace} {{\pi _{N-1}}} \right\rangle \left\langle {\pi _{N-1}} \right|\exp (-iH\Delta t_f) \left| {\varphi _{N-1}} \right\rangle \times ... \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \left\langle {{\varphi _2}} \mathrel{\left | {\vphantom {{\varphi _2} {\pi _1}}} \right. \kern-\nulldelimiterspace} {{\pi _1}} \right\rangle \left\langle {\pi _1} \right|\exp (-iH\Delta t_f)\left| {\varphi _1} \right\rangle \left\langle {{\varphi _1}} \mathrel{\left | {\vphantom {{\varphi _1} {\varphi _a}}} \right. \kern-\nulldelimiterspace} {{\varphi _a}} \right\rangle \label{b8} \end{eqnarray} \vspace{-6mm} \newline It is known, that \newline \vspace{-6mm} \begin{eqnarray} <\varphi_1|\varphi_a>=\delta(\varphi_1-\varphi_a) \label{b9} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} <\varphi_{i+1}|\pi_i>=exp\left[i\int{d^3x} \pi_i(\vec{x})\varphi_{i+1}(\vec{x})\right] \label{b10} \end{eqnarray} \vspace{-6mm} \newline For $\Delta t\rightarrow 0$ one can expand \newline \vspace{-6mm} \begin{eqnarray} \left\langle {\pi _i} \right|\exp (-iH\Delta t) \left| {\varphi _i} \right\rangle \approx \left\langle {\pi _i} \right|(1-iH\Delta t)\left| {\varphi _i} \right\rangle = \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left\langle {{\pi _i}} \mathrel{\left | {\vphantom {{\pi _i} {\varphi _i}}} \right. \kern-\nulldelimiterspace} {{\varphi _i}} \right\rangle (1-iH_i\Delta t) =(1-iH_i\Delta t)\exp \left[ {-i\int {d^3x\pi _i(x)\varphi _i(x)}} \right] \label{b11} \end{eqnarray} \vspace{-6mm} \newline where $H_i=H(\pi_i, \varphi_i)$. Inserting (\ref{b9}), (\ref{b10}) and (\ref{b11}) into (\ref{b8}) gives \newline \vspace{-6mm} \begin{eqnarray} \left\langle {\varphi _a} \right|\exp (-iHt_f)\left| {\varphi _a} \right\rangle =\mathop {\lim}\limits_{N\to \infty } \int {\prod\limits_{i=1}^N {\left( {d\varphi _i{{d\pi _i} \over {2\pi }}} \right)}}\delta (\varphi _1-\varphi _a)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ {-i\Delta t\sum\limits_{j=1}^N {\int {d^3x\left[ {H(\pi _j,\varphi _j)-\pi _j (\varphi _{j+1}-\varphi _j)\/ \Delta t} \right]}}} \right\}\label{b12} \end{eqnarray} \vspace{-6mm} \newline where $\varphi_{N+1}=\varphi_1=\varphi_a$. Taking the continuum limit of (\ref{b12}) , gives the result \newline \vspace{-6mm} \begin{eqnarray} \left\langle {\varphi _a} \right|\exp (-iHt_f)\left| {\varphi _a} \right\rangle =\int {D\pi \int\limits_{\varphi (\vec{x},0) =\varphi _a(\vec{x})}^{\varphi (\vec{x},t_f) =\varphi _a(\vec{x})} {D\varphi }}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ {i\int\limits_0^{t_f} {dt} \int {d^3x\left[ {\pi (\vec{x},t){{\partial \varphi (\vec{x},t)} \over {\partial t}}-H(\pi (\vec{x},t),\varphi (\vec{x},t))} \right]}} \right\} \label{b13} \end{eqnarray} \vspace{-6mm} \newline The integration over $\pi(\vec{x},t)$ is unrestricted, and integration over $\varphi(\vec{x},t)$ is such that the field starts at $\varphi_a(\vec{x})$ at $t=0$ and ends at $\varphi_a(\vec{x})$ at $t=t_f$. Note that all operators are gone and r one can work only with classical variables \cite{key9}. \section {Partition function for bosons.} \vspace{1mm} The partition function in statistical mechanics is expressed as \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =\mbox{Tr}\exp \left\{ {-\beta \left( {H-\mu N } \right)} \right\}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \int {d\varphi _a}\left\langle {\varphi _a} \right|\exp \left\{ {-\beta \left( {H-\mu N } \right)} \right\} \label{b14} \left| {\varphi _a} \right\rangle \end{eqnarray} \vspace{-6mm} \newline where the sum runs over all states. The aim now is to express $Z_\beta$ in terms of a functional integral. First introduce the imaginary time $\tau=it$. The limits of integration on $\tau$ are $[0,\beta]$, then $-it_f=\beta$ and \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =\int {D\pi \int\limits_{periodic} {D\varphi }}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ {\int\limits_0^\beta {d\tau }\int {d^3x\left[ {i\pi (x,t){{\partial \varphi (x,t)} \over {\partial \tau }}-H(\pi ,\varphi ) +\mu N(\pi ,\varphi )} \right]}} \right\} \label{b15} \end{eqnarray} \vspace{-6mm} \newline Integration over the field is constrained so that \newline \vspace{-6mm} \begin{eqnarray} \varphi (\vec{x},0)=\varphi (\vec{x} ,\beta) \label{b16} \end{eqnarray} \vspace{-6mm} \newline This is a consequence of the trace operation. In the equation (\ref{b15}) make the replacement \newline \vspace{-6mm} \begin{eqnarray} H(\pi ,\varphi )\to H(\pi ,\varphi )-\mu N(\pi ,\varphi ) \label{b17} \end{eqnarray} \vspace{-6mm} \newline where $ N(\pi ,\varphi )$ is conserved charge density, if the system admits some conserved charge. Rewrite the equation (\ref{b15}) in a more convenient form. For this purpose, consider a scalar field, which is described by the Lagrangian of the form \newline \vspace{-6mm} \begin{eqnarray} L=-{1 \over 2}(\partial \varphi )^2-{1 \over 2}m^2\varphi ^2-U(\varphi ) \label{b18} \end{eqnarray} \vspace{-6mm} \newline where $U(\varphi)$ is a potential function. The momentum conjugate to the field is \newline \vspace{-6mm} \begin{eqnarray} \pi ={{\partial L} \over {\partial (\partial _0\varphi )}} ={\partial_0 }\varphi \label{b19} \end{eqnarray} \vspace{-6mm} \newline Here the Hamiltonian may be written as \newline \vspace{-6mm} \begin{eqnarray} H=\pi {\partial_0 }\varphi-L= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} {1 \over 2}\pi ^2+{1 \over 2}(\nabla \varphi )^2 +{1 \over 2}m^2\varphi ^2+U(\varphi ) \label{b20} \end{eqnarray} \vspace{-6mm} \newline For evaluating the partition function, return to the discretized version of (\ref{b15}): \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =\mathop {\lim }\limits_{N\to \infty } \left( {\prod\limits_{i=1}^N {{\int\limits_{-\infty}^{\infty} {{d\pi _i} \over {2\pi}}} \int\limits_{periodic} {d\varphi _i}}} \right) \exp \left\{ {\sum\limits_{j=1}^N {\int {d^3x\left[ {i\pi _j} \right.}}} \right.(\varphi _{j+1}-\varphi _j)- \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {\left. {-\Delta \tau \left( {{1 \over 2}\pi _j^2 +{1 \over 2}(\nabla \varphi _j)^2 +{1 \over 2}m^2\varphi _j^2+U(\varphi )} \right)} \right]} \right\} \label{b21} \end{eqnarray} \vspace{-6mm} \newline Using the equation for a Gaussian integral \newline \vspace{-6mm} \begin{eqnarray} {1 \over {\sqrt {2\pi i}}}\int\limits_{-\infty }^\infty {dx\exp \left\{ {{i \over 2}ax^2+ibx} \right\} ={1 \over {\sqrt a}}\exp \left\{ {-{i \over 2}{{b^2} \over a}} \right\}} \label{b23} \end{eqnarray} \vspace{-6mm} \newline one can write \newline \vspace{-6mm} \begin{eqnarray} \int {{{d\pi _j} \over {2\pi }}}\exp \left\{ {i\pi _j(\varphi _{j+1}-\varphi _j) -{{\Delta \tau } \over 2}\pi _j^2} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over {\sqrt {2\pi \Delta \tau }}}\exp \left\{ {-{{(\varphi _{j+1}-\varphi _j)^2} \over {\Delta \tau }}} \right\} \label{b22} \end{eqnarray} \vspace{-6mm} \newline and after momentum integrations in (\ref{b21}) one gets \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\mathop {\lim }\limits_{N\to \infty } \int {\left( {\prod\limits_{i=1}^N {d\varphi _i}} \right)} \exp \left\{ {\Delta \tau \sum\limits_{j=1}^N {\int {d^3x\left[ {-{1 \over 2}\left( {{{(\varphi _{j+1}-\varphi _j)} \over {\Delta \tau }}} \right)^2} \right.}}} \right.- \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {\left. {-{1 \over 2}(\nabla \varphi _j)^2 -{1 \over 2}m^2\varphi _j^2-U(\varphi )} \right]} \right\} \label{b24} \end{eqnarray} \vspace{-6mm} \newline Returning to the continuum limit, one obtains: \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\int\limits_{periodic} {D\varphi \exp \left( {\int\limits_0^\beta {d\tau \int {d^3xL(\varphi ,\partial \varphi )}}} \right)} \label{b25} \end{eqnarray} \vspace{-6mm} \newline Equation (\ref{b25}) expresses $Z_\beta$ as a functional integral in time interval $[0,\beta]$. The normalization constant $N^{'}$ is irrelevant, since it does not change the thermodynamics. Now it is seen how this method works in application to a non-interacting scalar field. Let the Lagrangian of the scalar field be \newline \vspace{-6mm} \begin{eqnarray} L(\varphi ,\partial \varphi )=-{1 \over 2}(\partial \varphi )^2\label{b26} -{1 \over 2}m^2\varphi ^2 \end{eqnarray} \vspace{-6mm} \newline Define the finite temperature action of the Bose system at temperatute $T$ in a large cubic volume $V=L^3$ as \newline \vspace{-6mm} \begin{eqnarray} S_\beta =-{1 \over 2}\int\limits_0^\beta {d\tau \int \limits_{V}{d^3x\varphi \left( {\partial _\tau ^2 +\partial _i^2+m^2} \right)}}\varphi \label{b27} \end{eqnarray} \vspace{-6mm} \newline Due to the constraint of periodicity of the field $\varphi(\vec{x},\tau)$ $(\vec{x}\in V,\tau\in[0,\beta])$ (\ref{b16}), it can be expanded in a Fourier series as \newline \vspace{-6mm} \begin{eqnarray} \varphi (\vec{x},\tau )=(\beta V )^{-1/2}{\sum\limits_{n=-\infty }^\infty } {\sum\limits_{\vec{k}} \varphi _n(\vec{k}) \exp i(\vec{k}\vec{x}+\omega _n\tau )} \label{b28} \end{eqnarray} \vspace{-6mm} \newline where $\varphi _n(\vec{k})$ is the Fourier coefficient, $\omega_n=2\pi n/\beta$, and $\vec{k}=2\pi m/L$ $n$, $m$ are integer numbers. Using the identity \newline \vspace{-6mm} \begin{eqnarray} \int\limits_0^\beta {d\tau }\exp i(\omega _n-\omega _m)\tau =\beta \delta _{n,m} \label{b29} \end{eqnarray} \vspace{-6mm} \newline one obtains the equation for the action in terms of the Fourier coefficients \newline \vspace{-6mm} \begin{eqnarray} S_\beta =-{1 \over {2}}{\sum\limits_{n=-\infty }^\infty} {\sum\limits_{\vec{k}}{\varphi _n^*(\vec{k})} (\omega _n^2+{\vec{k}^2+m^2}})\varphi _n(\vec{k}) \label{b30} \end{eqnarray} \vspace{-6mm} \newline where $\varphi _{-n}(-\vec{k})=\varphi _n^*(\vec{k})$ goes from the reality of the field $\varphi$. Then the partition function (\ref{b25}) may be written as \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\int {\prod\limits_n {\prod\limits_{\vec{k}} {d\varphi _n(\vec{k})}}} \exp \left\{ -{1 \over 2}\varphi _n^*(k)(\omega _n^2+{\vec{k}}^2+m^2) d \varphi _n(k) \right\} \label{b31} \end{eqnarray} \vspace{-6mm} \newline where the explicit form of the measure of the functional integration (\ref{b31}) is \newline \vspace{-6mm} \begin{eqnarray} D\varphi =\prod\limits_n {\prod\limits_{\vec{k}} d{\varphi _n(\vec{k})}} \label{b32} \end{eqnarray} \vspace{-6mm} \newline So, integration over the coefficients $\varphi_n(\vec{k})$ in (\ref{b31}) gives the following equation for the logarithm of the partition function [Bernard 1974] \newline \vspace{-6mm} \begin{eqnarray} \mbox{ln} Z_\beta =-{1 \over 2} \mbox{ln} Det(\omega _n^2+\epsilon _{\vec{k}}^2) =-{1 \over 2}\sum\limits_n {\sum\limits_{\vec{k}}} \mbox{ln}({\omega _n^2}+\epsilon _{\vec{k}}^2) \label{b33} \end{eqnarray} \vspace{-6mm} \newline where $\epsilon _{\vec{k}}^2={\vec{k}}^2+m^2$ is the energy of a one particle state with a certain momentum. Summation in respect with $\omega_n$ in (\ref{b33}) may be done with the help of the basic equation \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_{n=-\infty }^\infty {{z \over {z^2+n^2}}} ={\pi \over 2}\coth \pi z \label{b34} \end{eqnarray} \vspace{-6mm} \newline The derivative of the sum gives \newline \vspace{-6mm} \begin{eqnarray} {\partial \over {\partial \epsilon }} \sum\limits_n {\ln \left( {{{4\pi ^2n^2} \over {\beta ^2}}+\epsilon ^2} \right)} =\sum\limits_n {{{2\epsilon } \over {4\pi ^2n^2/ \beta ^2+\epsilon ^2}}=} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =2\beta \left( {{1 \over 2}+{1 \over {\exp \beta \epsilon -1}}} \right) \label{b35} \end{eqnarray} \vspace{-6mm} \newline After integration with respect to $\epsilon$ we will have \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_n {\ln \left( {{{4\pi ^2n^2} \over {\beta ^2}} +\epsilon ^2} \right)}=2\beta \left[ {{\epsilon \over 2} +{1 \over \beta }\ln (1-\exp[- \beta \epsilon]) } \right] \label{b36} \end{eqnarray} \vspace{-6mm} \newline The expression for the thermodynamic potential will have the standard form \newline \vspace{-6mm} \begin{eqnarray} F=-{1 \over \beta }\ln Z_\beta ={1 \over \beta } \sum\limits_{\vec{k}} {\ln (1-\exp[- \beta \epsilon _{\vec{k}}])} \label{b37} \end{eqnarray} \vspace{-6mm} \newline Here the infinite, temperature independent part of the function $Z_\beta$ is dropped \section {Green's function of boson field} \vspace{1mm} Introduce the two point Green function (or propagator) of boson field at finite temperature as the thermal average of a time-ordered product of two scalar field operators\footnote{In quantum field theory two-points Green's function is introduced as the vacuum expectation value of a time-ordered product of two field operators.}. \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x-y)=<\mbox{T}\varphi(x)\varphi(y)>_\beta \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr}\left[ {\exp (-\beta H)\mbox{T}\varphi (x) \varphi (y)} \right]/ \mbox{Tr}\exp (-\beta H) \label{b38} \end{eqnarray} \vspace{-6mm} \newline One may express the Green's function as the sum \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-x^{'})=\theta(\tau-\tau^{'})G_\beta^+(x-x^{'}) +\theta(\tau^{'}-\tau)G_\beta^-(x-x^{'}) \label{b39} \end{eqnarray} \vspace{-6mm} \newline where $G^{\pm}$ are Wightman functions \newline \vspace{-6mm} \begin{eqnarray} G_\beta^+(x-x^{'})=<\mbox{T}\varphi(x)\varphi(x^{'})>_\beta \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} G_\beta^-(x-x^{'})=<\mbox{T}\varphi(x^{'})\varphi(x)>_\beta \label{b40} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \theta (\tau )=\left\{ \matrix{1,\tau >0,\hfill\cr 0,\tau <0.\hfill\cr} \right. \label{b41} \end{eqnarray} \vspace{-6mm} \newline is the step function. In the interval $[0,\beta]$ \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-y)_{|{x^{4}}=0}=G_\beta^+(x-y)_{|{x^{4}}=0} \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-y)_{|{x^{4}}=\beta}=G_\beta^-(x-y)_{|{x^{4}}=\beta} \label{b42} \end{eqnarray} \vspace{-6mm} \newline Using the fact that $exp(-\beta H)$ and time ordering operation commute and taking into account the cyclic property of the trace, one gets an important property of the thermal Green's function \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-y)_{|{x^{4}}=0}=G_\beta^+(x-y)_{|{x^{4}}=0} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr}\left[ {\exp (-\beta H)\varphi (\vec{x},\tau) \varphi (\vec{y},0)} \right]/ \mbox{Tr}\exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr}\left[ { \varphi (\vec{y},0)\exp (-\beta H)\varphi (\vec{x},\tau) }\right] / \mbox{Tr}\exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr}\left[ { \exp (-\beta H)(\exp (\beta H)\varphi (\vec{y},0) \exp (-\beta H))\varphi (\vec{x},\tau) }\right] / \mbox{Tr}\exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr}\left[ {\exp (-\beta H)\varphi (\vec{y},\beta) \varphi (\vec{x},\tau)} \right]/ \mbox{Tr}\exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =G_\beta^-(x-y)_{|{x^{4}}=\beta}=G_\beta(x-y)_{|{x^{4}}=\beta} \label{b43} \end{eqnarray} \vspace{-6mm} \newline Thus one has a periodicity condition \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-y)_{|{x^{4}}=0}=G_\beta(x-y)_{|{x^{4}}=\beta} \label{b44} \end{eqnarray} \vspace{-6mm} \newline This relation leads to the Green's function in Euclidean quantum field theory. Since the Green's function $G_\beta$ is periodic (\ref{b44}) and depends only on coordinate differences $G_\beta(x^0-y^0, \vec{x}-\vec{y})$, it may be represented by Fourier series and integral as \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x-y)= (1/ \beta ) \sum\limits_{n} {\exp [i\omega _n(x^4-y^4)]} \int{{d^3k} \over {(2\pi )^3}}\exp [i\vec{k}(\vec{x}-\vec{y})] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times G_\beta (\omega _n,\vec{k}) \label{b45} \end{eqnarray} \vspace{-6mm} \newline with $\omega_n=2\pi n/\beta$. In compact form one can write this equation as \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x-y)=\int\limits_k^- {\exp [ik(x-y)] G_\beta (k)} \label{b46} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \int\limits_k ^-{f(k)}={1 \over \beta }\sum\limits_n \int {{{d^3k} \over {(2\pi )^3}}}f(\omega_n,\vec{k}) \label{b47} \end{eqnarray} \vspace{-6mm} \newline The Green's function satisfies the equation \newline \vspace{-6mm} \begin{eqnarray} \left( {-\partial _x^2+m^2} \right)G_\beta (x-y)=\delta (x-y) \label{b48} \end{eqnarray} \vspace{-6mm} \newline with periodic conditions (\ref{b44}) and $\delta$ -function in the form: \newline \vspace{-6mm} \begin{eqnarray} \delta (x)=\sum\limits_{n}\int{d^3k\over{(2\pi)^3}} \exp[i(\omega_nx^4+\vec{k}\vec{x})] \label{b49} \end{eqnarray} \vspace{-6mm} \newline From (\ref{b46}) and (\ref{b48}) it follows that $G_\beta(k)$ is \newline \vspace{-6mm} \begin{eqnarray} G_\beta (k)=\frac{1}{k^2+m^2} \label{b50} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x)=\int\limits_k ^-{\frac{\exp (ikx)}{k^2+m^2}} \label{b51} \end{eqnarray} \vspace{-6mm} \newline \section{ Notation} \vspace{1mm} To work with the standard quantum field theory of signature $(+2)$, return to imaginary time formalism, on the time variable $t=-i\tau$ defined in the interval $[0,-i\beta]$. Therefore the equation (\ref{b44}) may be rewritten as \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x-y)_{|x^0=0}=G_\beta(x-y)_{|x^0=-i\beta} \label{b52} \end{eqnarray} \vspace{-6mm} \newline It leads to the replacement $\omega_n\rightarrow i\omega_n$ and \newline \vspace{-6mm} \begin{eqnarray} \int\limits_k {f(k)}={i \over \beta }\sum\limits_n \int {{{d^3k} \over {(2\pi )^3}}}f(\omega_n,\vec{k}) \label{b53} \end{eqnarray} \vspace{-6mm} \newline with $\omega_n=2\pi in/\beta$. The equation for the Green's function becomes \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x)=-\int\limits_k {\frac{\exp (ikx)}{k^2+m^2}} \label{b54} \end{eqnarray} \vspace{-6mm} \newline This expression looks like the expression for the Green's function in a common field theory. \section{Partition function for fermions} \vspace{1mm} The previous section discussed a quantization scheme for Bose fields in the functional integral formulation. However, the methods of path integral may be applied in the same way to finite temperature Fermi systems. This section develops these methods. Consider a free spinor field which is described by the action \newline \vspace{-6mm} \begin{eqnarray} S=-\int {d^4x}{\bar{\psi}(x) (i\gamma \cdot \partial +m)}\psi(x) \label{c1} \end{eqnarray} \vspace{-6mm} \newline where $\{\gamma_\mu\}$ is a set of Dirac matrices, which are defined by \newline \vspace{-6mm} \begin{eqnarray} \{\gamma_\mu,\gamma_\nu\}=2g_{\mu\nu} \label{c2} \end{eqnarray} \vspace{-6mm} \newline metric $g_{\mu \nu}=diag(-,+,+,+) $and, by notation, $\bar{\psi}=\psi^{+}\gamma^0$ is hermition conjugate. The action (\ref{c1}) has a global $U(1)$ symmetry and that is associated with conserved current \newline \vspace{-6mm} \begin{eqnarray} j^\mu(x)=\bar{\psi}(x)\gamma^\mu\psi(x) \label{c3} \end{eqnarray} \vspace{-6mm} \newline The total conserved charge is \newline \vspace{-6mm} \begin{eqnarray} Q=\int{d^3x}j^0=\int{d^3x}\psi^+\psi \label{c4} \end{eqnarray} \vspace{-6mm} \newline To apply the formalism developed in chapter II.2 treat the field $\psi$ as the basic field, and the momentum congugate to this field \newline \vspace{-6mm} \begin{eqnarray} \pi=\frac{\partial L}{\partial(\partial\psi/\partial t)}=i\psi^+ \label{c5} \end{eqnarray} \vspace{-6mm} \newline Thus consider $\psi$ and $\psi^+$ to be independent entities in the Hamiltonian formulation. The Hamiltonian density is found by the standard procedure as \newline \vspace{-6mm} \begin{eqnarray} H=\pi {{\partial \psi } \over {\partial t}}-L =\psi^+ \left( {i{{\partial \psi } \over {\partial t}}} \right)-L =\bar{\psi}(-i\gamma \partial +m)\psi \label{c6} \end{eqnarray} \vspace{-6mm} \newline Introduce the partition function as \newline \vspace{-6mm} \begin{eqnarray} Z=\mbox{Tr} \exp\left[ -i\beta(H-\mu Q)\right] \label{c7} \end{eqnarray} \vspace{-6mm} \newline Follow the steps leading up to the equation (\ref{b21}). For the partition function the intermediate equation will be: \newline \vspace{-6mm} \begin{eqnarray} Z_\beta \propto \mathop {\lim}\limits_ {N\to \infty }\int {\left( {id\psi ^+{{d\psi } \over {2\pi }}} \right)}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ {-i\Delta t\sum\limits_{j=1}^N {\int {d^3x\left[ {H(\psi ^+,\psi )-\psi _j^ +(\psi _{j+1}-\psi _j)\/ \Delta t} \right]}}} \right\} \label{c8} \end{eqnarray} \vspace{-6mm} \newline Taking into account the limit of this equation, find \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\int {id\psi ^+}\int {d\psi } \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ {i\int\limits_0^{t_f} {dt\int {d^3x\left( {i\psi ^+{{\partial \psi } \over {\partial t}}-H(\psi ^+(x,t),\psi (x,t))} \right)}}} \right\} \label{c9} \end{eqnarray} \vspace{-6mm} \newline For the time variable $\tau$, get \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\int {id\psi ^+}\int {d\psi } \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \left\{ \int\limits_0^\beta {d\tau} \int {d^3x\psi^+} \left( \mu-\frac{\partial} {\partial \tau}+i\gamma ^0\vec{\gamma} \cdot \vec{\nabla} -m\gamma ^0 \right)\psi \right\} \label{c10} \end{eqnarray} \vspace{-6mm} \newline The quantization of a Fermi system can be obtained as a result of integration over the space of anticommuting functions $\psi(\vec{x},\tau)$ $(\vec{x}\in V,\tau\in [0,\beta])$, which are the elements of an infinite Grassman algebra. To obtain the correct result it is neccessary to impose on $\psi$, $\bar\psi$ the antiperiodicity conditions in the variable $\tau$: \newline \vspace{-6mm} \begin{eqnarray} \psi(\vec{x},\beta)=-\psi(\vec{x},0), ~~\bar{\psi}(\vec{x},\beta)=-\bar{\psi}(\vec{x},0) \label{c11} \end{eqnarray} \vspace{-6mm} \newline The Fourier series for $\psi$, $\bar\psi$ in Fermi case (\ref{c11}) are \newline \vspace{-6mm} \begin{eqnarray} \psi (x,\tau )=(\beta V)^{-1/ 2}\sum\limits_n {\sum\limits_{\vec{k}}} {\exp (i[\omega \tau +\vec{k}\vec{x}])\psi _n(\vec{k})} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} {\psi}^+ (x,\tau )=(\beta V)^{-1/ 2}\sum\limits_n {\sum\limits_{\vec{k}}} {\exp (-i[\omega \tau +\vec{k}\vec{x}]){\psi}^+_n(\vec{k})} \label{c12} \end{eqnarray} \vspace{-6mm} \newline where $\psi _n(\vec{k})$ and $\psi^+ _n(\vec{k})$ are the generators of Grassmann algebra. Inserting (\ref{c12}) into (\ref{c10}) get \newline \vspace{-6mm} \begin{eqnarray} Z_\beta =N^{'}\left[ {\prod\limits_i {\prod\limits_n {\int id\psi _n^+d\psi _n}}} \right]\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \exp \sum\limits_n {\sum\limits_{\vec{k}} {\psi _n^+ \left[ {(-i\omega _n+\mu )-\gamma ^0\vec{\gamma} \cdot{\vec{ k}} -m\gamma ^0} \right]\psi _n}} \label{c13} \end{eqnarray} \vspace{-6mm} \newline Calculation of the Gaussian functional integral over the fermi fields in (\ref{c13}) leads to the following equation \newline \vspace{-6mm} \begin{eqnarray} Z_\beta=Det[(-i\omega _n+\mu )-\gamma ^0\vec{\gamma} \cdot{\vec{ k}} -m\gamma ^0] \label{c14} \end{eqnarray} \vspace{-6mm} \newline or using useful relation \newline \vspace{-6mm} \begin{eqnarray} \mbox{ln}DetD=\mbox{Tr}\ln D \label{c15} \end{eqnarray} \vspace{-6mm} \newline find \newline \vspace{-6mm} \begin{eqnarray} \ln Z_\beta=\mbox{Tr}\ln[(-i\omega _n+\mu )-\gamma ^0\vec{\gamma} \cdot{\vec{ k}} -m\gamma ^0]= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =2\sum\limits_n {\sum\limits_{\vec{k}} {\ln \left[ {(\omega _n +i\mu )^2+\epsilon _{\vec{k}}^2} \right]}} \label{c16} \end{eqnarray} \vspace{-6mm} \newline Since both positive and negative frequencies are summed over, (\ref{c16}) can be written in the form \newline \vspace{-6mm} \begin{eqnarray} \mbox{ln} Z_\beta=\sum\limits_n \sum\limits_{\vec{k}} \left\{\ln \left[ {(\omega _n -\mu )^2+\epsilon _{\vec{k}}^2} \right] +\ln \left[ {(\omega _n +\mu )^2+\epsilon _{\vec{k}}^2} \right]\right\} \label{c17} \end{eqnarray} \vspace{-6mm} \newline Summation (\ref{c17}) over $\omega_n$ leads to the folowing equation \newline \vspace{-6mm} \begin{eqnarray} \ln Z_\beta= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} 2V\sum\limits_{\vec{k}} {\left[ \beta {\epsilon_{\vec{k}}} +(1+\exp \{-\beta (\epsilon_{\vec{k}} -\mu )\}) +(1+\exp \{-\beta (\epsilon_{\vec{k}} +\mu )\}) \right]} \label{c18} \end{eqnarray} \vspace{-6mm} \newline The contributions for particles $(\mu)$ and antiparticles $(-\mu)$, and, also, the zero-point energy of vacuum, have now been obtained. [Kapusta 1989] \section{Green's function of fermi field} \vspace{1mm} The finite temperature fermionic Green's function (or fermionic propagator) may be introduced in the similar way to scalar field. Consider that \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta (x-y)=<T\psi(x),\bar{\psi}(y)>_\beta \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr} [\exp (-\beta H)\mbox{T}\psi(x),\bar{\psi}(y)]/\mbox{ Tr}~ \exp (-\beta H) \label{c19} \end{eqnarray} \vspace{-6mm} \newline For fermions the analog of (\ref{b39}) is in the form \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x-y)=\theta(x-y)G^{+F}_\beta(x-y)+\theta(y-x)G^{F}_\beta(x-y) \label{c20} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} G^{+F}_\beta(x-y)=<\psi(x)\bar{\psi}(y)>_\beta,~~x^4>y^4 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} and \newline \vspace{-6mm} \begin{eqnarray} G^{-F}_\beta(x-y)=-<\bar{\psi}(y)\psi(x)>_\beta,~~~~x^4<y^4 \label{c21} \end{eqnarray} \vspace{-6mm} \newline The desired boundary conditions now follow from \newline \vspace{-6mm} \begin{eqnarray} G^{-F}_\beta(x-y)=-\mbox{Tr}[\exp (-\beta H)\bar{\psi}(y^4,\vec{y}) \psi(0,\vec{x})]/\mbox{Tr}~\exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\mbox{Tr}[\exp (-\beta H) \exp (\beta H) \psi(0,\vec{x}) \exp (-\beta H)\bar{\psi}(y^4,\vec{y})]/\mbox{Tr}~ \exp (-\beta H) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\mbox{Tr}[\exp (-\beta H) \psi(\beta,\vec{x})\bar{\psi}(y^4,\vec{y})]/\mbox{Tr} \exp (-\beta H) =-G^{+F}_\beta(x-y) \label{c22} \end{eqnarray} \vspace{-6mm} \newline It leads to antiperiodic boundary conditions \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x-y)_{|x^4=0}=-G^F_\beta(x-y)_{|x^4=\beta} \label{c23} \end{eqnarray} \vspace{-6mm} \newline and this means that the fields $\psi$, $\bar{\psi}$ are antiperiodic \newline \vspace{-6mm} \begin{eqnarray} \psi(\vec{x},0)=-\psi(\vec{x},\beta) \label{c24} \end{eqnarray} \vspace{-6mm} \newline The Fourier series of the fermionic propagator is written as \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x-y)=(1/\beta)\sum\limits_n \exp~[i\omega_n(x^4-y^4)] \int{\frac{d^3k}{2\pi^3}} \exp[i\vec{k}(\vec{x}-\vec{y})] G^F_\beta(\omega_n,\vec{k}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int\limits_k ^- \exp [ik(x-y)] G^F_\beta(k) \label{c25} \end{eqnarray} \vspace{-6mm} \newline with $k^\mu=(\omega_n,\vec{k})$,~~$\omega_n=({2\pi}/{\beta})(n+1/2)$. The Green's function of the fermionic field satisfies the equation \newline \vspace{-6mm} \begin{eqnarray} (i\bar{\gamma}\cdot\partial+m)G^F_\beta(x-y)=\delta(x-y) \label{c26} \end{eqnarray} \vspace{-6mm} \newline and has the following form \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x)=\int\limits_k^-\frac{\bar{\gamma}\cdot k+m}{k^2+m^2} \exp (ikx) \label{c27} \end{eqnarray} \vspace{-6mm} \newline where $\{\bar{\gamma}_\mu\}$ is the set of Euclidean gamma matrices. \section{ Notation } \vspace{1mm} As in the case of the scalar field one can work with a quantum model in Minkowski space-time, restoring the time variable $t=-i\tau$ defined in the interval $[0,-i\beta]$. It gives the following antiperiodic conditions for Green's function \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x-y)_{|x^0=0}=-G^F_\beta(x-y)_{|x^0=-i\beta} \label{c28} \end{eqnarray} \vspace{-6mm} \newline and the rule of integration \newline \vspace{-6mm} \begin{eqnarray} \int\limits_k {f(k)}={i \over \beta }\sum\limits_n \int {{{d^3k} \over {(2\pi )^3}}}f(\omega_n,\vec{k}) \label{c29} \end{eqnarray} \vspace{-6mm} \newline with $\omega_n=(2\pi i/\beta)(n+1/2)$. In the Minkowski metric (signature $(+2)$) the equation for Green's function is written as [Dolan, Jackiw 1974] \newline \vspace{-6mm} \begin{eqnarray} G^F_\beta(x)=\int\limits_k ^{-}\frac{{\gamma}\cdot k+m}{k^2+m^2}\exp (ikx) \label{c30} \end{eqnarray} \vspace{-6mm} \newline \chapter{THERMODYNAMICS OF QUANTUM GASES} \centerline{\Large \bf AND GREEN'S FUNCTIONS} \vspace{24pt} This section develops a method for calculation of thermodynamic potentials directly from finite temperature Green's functions. For this purpose the Schwinger proper time formalism \cite{schw1} has been applied, to write down generating functionals of quantum fields in terms of Green's functions of these fields. The finite temperature generalization of this formalism leads to thermodynamical potentials which are written through finite temperature Green's functions. \section{Thermal bosonic fields} \vspace{1mm} In standard form the generating functional $Z$ of the scalar field $\varphi$ with the Lagrangian (\ref{b26}) is written as : \newline \vspace{-6mm} \begin{eqnarray} Z[J]=\int {D\varphi \exp \left\{ {-(i/ 2)\int {d^4x\varphi (x)K_{xy}\varphi (y)}} +i\int{d^4xJ(x)\varphi(x)}\right\}} \label{d1} \end{eqnarray} \vspace{-6mm} \newline The symmetric operator \newline \vspace{-6mm} \begin{eqnarray} K_{xy}=\left( {-\partial _x^2+m^2} \right)\delta (x-y) \label{d2} \end{eqnarray} \vspace{-6mm} \newline can formally be treated as a symmetric matrix with continuous indices $(x,y)$. It has the following properties: \newline \vspace{-6mm} \begin{eqnarray} \int{d^4y}K^{1/2}_{xy}K^{-1/2}_{yz}=\delta(x-z) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \int{d^4y}K^{1/2}_{xy}K^{1/2}_{yz}=K_{xz} \label{d3} \end{eqnarray} \vspace{-6mm} \newline The functional $Z$ gives the transition amplitude from the initial $|0^->$ and final $|0^+>$ vacuum in the presence of the source $J(x)$, $(Z=<0^+|0^->_J)$ which is producing particles. The Green's function may be treated as the solution of the equation \newline \vspace{-6mm} \begin{eqnarray} \int {d^y}K_{xy}G(y,x^{'})=\delta (x-x^{'}) \label{d4} \end{eqnarray} \vspace{-6mm} \newline with operator $K_{xy}$. Now rewrite (\ref{d1}) in a convenient form changing the integration variable from $\varphi$ to \newline \vspace{-6mm} \begin{eqnarray} \varphi^{'}(x)=\int {d^4 y}K_{xy}\varphi(y) \label{d5} \end{eqnarray} \vspace{-6mm} \newline Then the quadratic form of (\ref{d1}) may be recast as \newline \vspace{-6mm} \begin{eqnarray} -(1/2)\int{d^4x}\left[\varphi^{'}(x)-\int{d^4y}K^{-1/2}_{yx}J(y)\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(1/2)\int{d^4x}{d^4y}\varphi(y)G(x,y)\varphi(y) \label{d6} \end{eqnarray} \vspace{-6mm} \newline Substituting (\ref{d6}) into (\ref{d1}) and performing an integration of the Gaussian type integral, get \newline \vspace{-6mm} \begin{eqnarray} Z[J] \propto (detK^{1/2})^{-1}\exp \left[-(i/2)\int{d^4x}{d^y}\varphi(y)G(x,y)\varphi(y)\right] \label{d7} \end{eqnarray} \vspace{-6mm} \newline where the Jacobian arises from the change of variable (\ref{d5}). The functional determinant in (\ref{d7}) may be written with Green's function $G(x,y)$: \newline \vspace{-6mm} \begin{eqnarray} (DetK^{1/2})^{-1}=(DetG(x,y))^{1/2}=\exp \left[(1/2)\mbox{Tr} \ln G(x,y)\right] \label{d8} \end{eqnarray} \vspace{-6mm} \newline The Green's function is found from functional differentiation of $Z$ with respect to source $J$ \newline \vspace{-6mm} \begin{eqnarray} (i)^2\left(\frac{\delta \ln Z}{\delta J(x)\delta J(y)}\right)_{J=0} =<0|\mbox{T}\varphi(x)\varphi(y)|>=G(x,y) \label{d9} \end{eqnarray} \vspace{-6mm} \newline Now pay attention to the functional determinant in (\ref{d7}) because of its important role in the applications to statistical mechanics. One can introduce the heat kernel for operator $K$ as the solution of the partial differential equation \newline \vspace{-6mm} \begin{eqnarray} i\frac{\partial}{\partial s}\Im(x,x^{'};is) =\int{d^4z}K_{xz}\Im(z,x^{'};is) \label{d10} \end{eqnarray} \vspace{-6mm} \newline with the boundary conditions $\Im(x,x^{'};0)=\delta(x-x^{'})$ The Green's function $G$ may be written with $\Im$ in the form \newline \vspace{-6mm} \begin{eqnarray} G(x,x^{'}) =\int\limits_0^\infty {ids}\Im(z,x^{'};is) \label{d11} \end{eqnarray} \vspace{-6mm} \newline and the logarithm of functional determinant of the operator $K$ as \newline \vspace{-6mm} \begin{eqnarray} \ln Det K=\int \limits_0 ^\infty {ids}(is)^{-1} \mbox{tr} \Im(z,x^{'};is) \label{d12} \end{eqnarray} \vspace{-6mm} \newline To get the equation for the heat kernel $\Im$, find proper time representation of Green's function. Using the useful equation \newline \vspace{-6mm} \begin{eqnarray} (k^2+m^2)^{-1}=i\int\limits_0^\infty ds \exp\{-is(k^2+m^2)\} \label{d13} \end{eqnarray} \vspace{-6mm} \newline write the equation for Green's function in the form \newline \vspace{-6mm} \begin{eqnarray} G(x,x^{'})=\int\frac{d^4k}{(2\pi)^4}\frac{1}{k^2+m^2}\exp(iky) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =i\int\limits_0^\infty ds \int\frac{d^4k}{(2\pi)^4}\exp\{iky-is(k^2+m^2)\} \end{eqnarray} \vspace{-6mm} \newline where $y=x-x^{'}$. After the integration with respect to momentum, we get the Schwinger representation of the Green's function \newline \vspace{-6mm} \begin{eqnarray} G(x,x^{'})=\frac{i}{(4\pi)^2}\int\limits_0^\infty ids(is)^2\exp(-ism^2-\sigma/2is) \label{d14} \end{eqnarray} \vspace{-6mm} \newline The variable $\sigma$ equals half the square of the distance between $x$ and $x^{'}$ \newline \vspace{-6mm} \begin{eqnarray} \sigma=(x-x^{'})^2/2 \nonumber \end{eqnarray} \vspace{-6mm} \newline The equation for the heat kernel directly follows from the comparision (\ref{d11}) and (\ref{d14}): \newline \vspace{-6mm} \begin{eqnarray} \Im(x,x^{'};is)=\frac{i}{(4\pi is)^2} \exp(-ism^2-\sigma/2is) \label{d15} \end{eqnarray} \vspace{-6mm} \newline For future calculations it is better to work with the generating functional of connected Green's functions $W[J]$ which is connected with generating functional $Z$ by means of equation \newline \vspace{-6mm} \begin{eqnarray} Z[J]=\exp\{iW[J]\} \label{d16} \end{eqnarray} \vspace{-6mm} \newline From (\ref{d12}) and (\ref{d16}) we find \newline \vspace{-6mm} \begin{eqnarray} W[0]=(i/2)\ln Det K \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-(i/2)\int\limits_0^\infty ids(is)^{-1}\mbox{tr}\Im (x,x^{'};is) \label{d17} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{d15}) into (\ref{d17}) and taking into account (\ref{d14}) we get an important equation \newline \vspace{-6mm} \begin{eqnarray} W[0]=-(i/2)\int{d^4x}\int\limits_{m^2}^\infty dm^2\mbox{tr}G(x,x^{'}) \label{d18} \end{eqnarray} \vspace{-6mm} \newline Now find finite temperature functional $W_\beta$ inserting Green's function at finite temperature \newline \vspace{-6mm} \begin{eqnarray} W_\beta=-(i/2)\int\limits_\beta{d^4x}\int\limits_{m^2}^\infty dm^2\mbox{tr}G_\beta(x,x^{'}) \label{d19} \end{eqnarray} \vspace{-6mm} \newline Now \newline \vspace{-6mm} \begin{eqnarray} W_\beta=-(\beta/2)\int\limits_V{d^3x}\int\limits_{m^2}^\infty dm^2\int\limits_k(k^2+m^2)^{-1} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(\beta/2)V \int\limits_k \ln (k^2+m^2) \label{d20} \end{eqnarray} \vspace{-6mm} \newline The Helmholtz free energy may be treated as effective potential of the finite part of $W_\beta$, therefore \newline \vspace{-6mm} \begin{eqnarray} F(\beta, V) =-iW_\beta. \label{d21} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} F (\beta, V) =-(i\beta/2)V \int\limits_k \ln (k^2+m^2) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(1/2)V\int\frac{d^3k}{(2\pi)^3}\sum\limits_n \ln(\omega^2_n+\epsilon^2_k) \label{d22} \end{eqnarray} \vspace{-6mm} \newline After making the summation over the frequencies $\omega$, this equation will coincide with (\ref{b37}) (if the infinite, temperature independent term is dropped). So one can conclude that the density of Hemholtz free energy may be written as [Kulikov \& Pronin 1987] \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=(i/2)\int\limits_{m^2}^\infty dm^2\mbox{tr}G(\beta,x-x^{'}), \label{d23} \end{eqnarray} \vspace{-6mm} \newline where $G(\beta,x-x^{'})$ is temperature contribution in $G_\beta(x,x^{'})$. \section{Bosonic finite temperature Green's} \vspace{-4mm} \hspace{20mm}{\Large \bf function in the Schwinger representation} \vspace{1mm} Start from the finite temperature Green's function (\ref{b51}) in the limit of coincidence $x=x^{'}$: \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x\to x^{'}} G_\beta (x,x^{'})=\int \limits _{k}^{-}\frac{d^4k}{(2\pi)^4}\frac{1}{k^2+m^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\int\limits_0^\infty ds (1/\beta)\sum\limits_n\int\frac{d^3k}{(2\pi)^3} \exp\{-is(-\omega^2_n+\vec{k}^2+m^2)\} \end{eqnarray} \vspace{-6mm} \newline After the integration with respect to momentum find \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x\to x^{'}} G_\beta (x,x^{'})=\int\limits _0 ^\infty{(4\pi is)^{3/2}}\exp(-ism^2) (1/\beta)\sum\limits_n \exp(-4\pi^2n^2/\beta^2), \label{d24} \end{eqnarray} \vspace{-6mm} \newline Rewrite the sum in (\ref{d24}) with the help of the equation \cite{dit1} \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_n \exp[-\alpha (n-z)^2]=\sum\limits_n (\pi/\alpha)^{1/2}\exp(-\pi^2n^2/\alpha-2\pi izn) \label{d25} \end{eqnarray} \vspace{-6mm} \newline In bosonic case $(z=0)$ (\ref{d24}) has the form: \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x,x)=\frac{i}{(4\pi)^2}\sum\limits_n \int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is) \label{d26} \end{eqnarray} \vspace{-6mm} \newline Selecting the temperature independent $(n=0)$ part of the equation (\ref{d26}) we find the finite temperature contribution in Green's function of scalar field: \newline \vspace{-6mm} \begin{eqnarray} G_\beta(x,x)=G(x,x)+G(\beta,x-x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{i}{(4\pi)^2}\int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\frac{i}{(4\pi)^2}2\sum\limits_{n=1}^\infty \int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is) \label{d27} \end{eqnarray} \vspace{-6mm} \newline Inserting $G(\beta,x-x)$ of the equation (\ref{d27}) into (\ref{d23}) we find the equation for free energy in the form \newline \vspace{-6mm} \begin{eqnarray} f (\beta)=-(1/(4\pi)^2)\int\limits_{m^2}^\infty dm^2 \sum\limits_{n=1}^\infty \int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is) \label{d28} \end{eqnarray} \vspace{-6mm} \newline The integral over $(s)$ occuring in (\ref{d28}) can be found with \newline \vspace{-6mm} \begin{eqnarray} \int\limits_0^\infty dxx^{\nu-1}\exp\left(-\alpha\frac{1}{x}-\gamma x\right)= 2\left(\alpha/\gamma\right)^{\nu/2}K_\nu(2\sqrt{\alpha\gamma}),\label{d29} \end{eqnarray} \vspace{-6mm} \newline where $K_\nu$ is a modified Bessel function. Then \newline \vspace{-6mm} \begin{eqnarray} \int\limits_0^\infty ids(is)^{j-3}\exp (-ism^2-\beta ^2n^2/ 4is) =2\left( {\beta n/ 2m} \right)^{j-2}K_{j-2}(\beta mn),\label{d30} \end{eqnarray} \vspace{-6mm} \newline and the equation for $f$ will be \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=-(m^2/2\pi^2\beta^2)\sum\limits_{n=1}^\infty(1/n^2)K_2(\beta mn) \label{d31} \end{eqnarray} \vspace{-6mm} \newline The integral representation of the sum of the modified Bessel function (\ref{bb14}) allows the standard equation for the Helmholtz free energy to be written as \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=(1/\beta)\int{\frac{d^3k}{(2\pi)^3}} \ln (1-\exp(-\beta\epsilon_k)) \label{d32} \end{eqnarray} \vspace{-6mm} \newline directly from (\ref{d31}). \section {Thermal fermionic fields} \vspace{1mm} Let us now develop the same formalism for fermionic fields. Add to the fermionic Lagrangian the terms with anticommutative sources $\eta(x)$ and $\bar{\eta}(x)$ \newline \vspace{-6mm} \begin{eqnarray} L\rightarrow L=L_0+\eta(x)\bar{\psi}(x)+\psi(x)\bar{\eta}(x) \label{d33} \end{eqnarray} \vspace{-6mm} \newline and write the generating functional \newline \vspace{-6mm} \begin{eqnarray} Z[\eta,\bar{\eta}]=\int {D\psi}{D\bar{\psi}} \exp \left\{ i \int{d^4x} \left[L_0+\eta(x)\bar{\psi}(x)+\psi(x)\bar{\eta}(x)\right]\right\},\label{d34} \end{eqnarray} \vspace{-6mm} \newline where the functions $\psi$ and $\bar\psi$ are considered as Grassman variables. After integration in respect with these Grassmann variables obtain \newline \vspace{-6mm} \begin{eqnarray} Z[\eta,\bar{\eta}]\propto (Det K) \exp\left[ -i\int{d^4x}\int {d^4y} \bar{\eta}(x)K_{x y}\eta(y)\right] \label{d35} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} K_{xy}=(i\gamma\cdot\partial_x+m)\delta(x-y) \label{d36} \end{eqnarray} \vspace{-6mm} \newline and bi-spinor $G_F$ satisfies the equation which is equivalent to (\ref{d4}). In operator form \cite{schw1} it is written \newline \vspace{-6mm} \begin{eqnarray} KG_F=\hat1 \label{d37} \end{eqnarray} \vspace{-6mm} \newline At zero sources the generating functional $W[0]$ is \newline \vspace{-6mm} \begin{eqnarray} W[0]=-i\ln Z[0]=-i\ln Det (K) \label{d38} \end{eqnarray} \vspace{-6mm} \newline On the other hand it is \newline \vspace{-6mm} \begin{eqnarray} W[0]=-i\ln Det(K) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(i/2)\int\limits_0^\infty ids(is)^{-1}\mbox{tr} \hat{\Im}(x,x^{'};is) \label{d39} \end{eqnarray} \vspace{-6mm} \newline where the kernel $\hat{\Im}(x,x^{'};is)$ is the solution of the equation \newline \vspace{-6mm} \begin{eqnarray} i\frac{\partial}{\partial s}\hat{\Im}(x,x^{'};is) =(K\hat{\Im})(x,x^{'};is), \label{d40} \end{eqnarray} \vspace{-6mm} \newline The equation for the bi-spinor Green's function is written as \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=\int\frac{d^4k}{(2\pi)^4}\frac{\hat1}{k^2+m^2} \exp(iky) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =i\int\limits_0^\infty ds \int\frac{d^4k}{(2\pi)^4} \exp\{iky-is(k^2+m^2)\}\label{aud1} \end{eqnarray} \vspace{-6mm} \newline and after integration over the momentum, we get \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=-\frac{\hat1}{(4\pi)^2}\int\limits_0^\infty ds(is)^2 \exp(-ism^2-\sigma/2is), \label{d41} \end{eqnarray} \vspace{-6mm} \newline As follows from (\ref{d11}) the heart kernel has the form \newline \vspace{-6mm} \begin{eqnarray} \hat{\Im}(x,x^{'};is)=-i\frac{\hat 1}{(4\pi)^2} (is)^2 \exp(-ism^2-\sigma/2is), \label{d42} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{d42}) into (\ref{d39}) and using (\ref{d41}) we find \newline \vspace{-6mm} \begin{eqnarray} W[0]=(i/2)\int{d^4x}\int\limits_{m^2}^\infty dm^2 \mbox{tr} G_F(x,x^{'}) \label{d43} \end{eqnarray} \vspace{-6mm} \newline For non-interacting fermionic field this equation is divergent and does not give any useful information but, in the finite temperature case we can get some interesting results connected with temperature properties of quantum Fermi gas. \section{Fermionic finite temperature Green's} \vspace{-4mm} \hspace{20mm}{\Large \bf function in Schwinger representation} \vspace{1mm} In the limit of coincidence $(x \to x^{'})$ the finite temperature fermionic bi-spinor $G_F$ is written as: \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x\to x^{'}} G_F(x,x^{'})=\int\frac{d^4k}{(2\pi)^4}\frac{\hat{1}}{k^2+m^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\int\limits_0^\infty ds (\hat{1}/\beta)\sum\limits_n\int\frac{d^3k}{(2\pi)^3} \exp\{-is(-\omega^2_n+\vec{k}^2+m^2)\} \end{eqnarray} \vspace{-6mm} \newline where $\omega_n=(2i\pi n/\beta)(n+1/2)$. After the integration in respect with momentum we get \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x\to x^{'}} G_F(x,x^{'})= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \int\limits_0^\infty {(4\pi is)^{3/2}}\exp(-ism^2) (\hat{1}/\beta)\sum\limits_n \exp[-4\pi^2(n+1/2)^2/\beta^2] \label{d44} \end{eqnarray} \vspace{-6mm} \newline Using the equation (\ref{d25}) with $z=-1/2$, we find \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_n \exp[-(2\pi /\beta)^2 (n+1/2)^2]=\sum\limits_n(-1)^n [\beta/(4\pi i s)^{1/2}] \exp(-\pi^2n^2/4is),\label{d45} \end{eqnarray} \vspace{-6mm} \newline Then the equation (\ref{d44}) will be \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x\to x^{'}} G(x,x^{'})=i\frac{\hat 1}{(4\pi)^2}\sum\limits_n (-1)^n\int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is).\label{d46} \end{eqnarray} \vspace{-6mm} \newline Now we can select the temperature independent part of the equation (\ref{d46}): \newline \vspace{-6mm} \begin{eqnarray} G_F^\beta(x,x)=G_F(x,x)+G_F(\beta,x-x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =i\frac{\hat 1}{(4\pi)^2}\int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +i\frac{\hat 1}{(4\pi)^2}2\sum\limits_{n=1}^\infty(-1)^n \int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is) \label{d47} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{d47}) into the equation [Kulikov \& Pronin 1987]: \newline \vspace{-6mm} \begin{eqnarray} f=(-i/2) \int \limits_{m^2}^\infty dm^2 \mbox{tr} G_F(\beta,x-x^{'})) \label{add1} \end{eqnarray} \vspace{-6mm} \newline we find an expression for the free energy of Fermi gas in the form \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=-(1/(4\pi)^2)\mbox{tr} (\hat{1})\int\limits_{m^2}^\infty dm^2 \sum\limits_{n=1}^\infty(-1)^n \int\limits_0^\infty \frac{ids}{(is)^2} \exp(-ism^2-n^2\beta^2/4is) \label{d48} \end{eqnarray} \vspace{-6mm} \newline The integral over $(s)$ occuring in (\ref{d48}) can be written as \newline \vspace{-6mm} \begin{eqnarray} \int\limits_0^\infty ids(is)^{j-3}\exp (-ism^2-\beta ^2n^2/ 4is) =2\left( {\beta n/ 2m} \right)^{j-2}K_{j-2}(\beta mn), \label{d49} \end{eqnarray} \vspace{-6mm} \newline and the equation for density of free energy of fermionic field $f$ will be \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=\mbox{tr} (m^2/2\pi^2\beta^2)(\hat{1}) \sum \limits _{n=1}^\infty \frac{(-1)^n}{n^2}K_2(\beta mn) \label{d50} \end{eqnarray} \vspace{-6mm} \newline The standard equation for Helmholtz free energy follows from (\ref{aa14}) \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=-(4/\beta)\int{\frac{d^3k}{(2\pi)^3}}\ln (1+\exp(-\beta\epsilon_k)) \label{d51} \end{eqnarray} \vspace{-6mm} \newline Now we can describe the thermal behavior of massless vector fields in the Schwinger proper time formalism to prepare the mathematical foundation for further computations in curved space-time. \chapter{FINITE TEMPERATURE GAUGE FIELDS} \vspace{24pt} In the present day progress in quantum field theory is to a great extent due to the development of gauge fields \cite{key18,key19,hooft1}. These fields open up new possibilities for the description of interactions of the elementary particles. Gauge fields are involved in most modern models of strong, weak and electromagnetic interactions. They are extremely attractive for the unification of all interactions into a single universal interaction. On the other hand the functional formulation of the gauge fields helps us to get statistical and thermodynamical results connected with finite temperature properties of the particles which are described by these fields. In this connection we will develop the finite temperature formalism for gauge fields \section{Gauge theories: Pure Yang-Mills theory} \vspace{1mm} The basic idea of gauge field relies on the local gauge invariance principle of the quantum field theory \cite{key18,key19}. Let us consider a gauge tansformation parametrized by the functions $\omega^a(x)$ \newline \vspace{-6mm} \begin{eqnarray} U(x)=\exp[-i\omega^a(x)\tau^a] \label{e1} \end{eqnarray} \vspace{-6mm} \newline where the generators of the Lee algebra obey the equation \newline \vspace{-6mm} \begin{eqnarray} [\tau^a,\tau^b]=if^{abc}\tau^c \label{e2} \end{eqnarray} \vspace{-6mm} \newline Numbers $f^{abc}$ are the structure constants of the group. Let the vector (gauge) fields $A_\mu(x)=A_\mu(x)^a\tau^a$ transform according to \newline \vspace{-6mm} \begin{eqnarray} A_\mu(x) \to A^\omega_\mu(x)=UA_\mu(x)U^{-1}+(i/g)U\partial_\mu U^{-1} \label{e3} \end{eqnarray} \vspace{-6mm} \newline For infinitesimal transformations we have \newline \vspace{-6mm} \begin{eqnarray} (A^{\omega})^a _\mu =A^a_\mu (x)+f^{abc}A^b_\mu(x) \omega^c(x)+(1/g)\partial_\mu \omega^a(x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =A^a_\mu (x)+\delta A^a_ \mu (x) \label{e4} \end{eqnarray} \vspace{-6mm} \newline The transformations (\ref{e1}) preserve the Maxwell type Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L=-(1/2)\mbox{Tr} (F_{\mu\nu}(x))^2=-(1/4) F^a_{\mu \nu} F^{a \mu \nu} \label{e5} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} F_{\mu \nu}=\partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}-ig[A_{\mu} , A_{\nu}] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(\partial_{\mu} A^a_{\nu} -\partial_{\nu} A^a_{\mu} +gf^{abc}A^b_{\mu} A^c_{\nu})\tau^a \label{e6} \end{eqnarray} \vspace{-6mm} \newline is the strength of the vector field $A_\mu$. The path integral over the field $A_\mu$ with the Lagrangian (\ref{e5}) in the form \newline \vspace{-6mm} \begin{eqnarray} \int DA_\mu \exp (iS[A_\mu,\partial_\nu A_\mu]) \label{e7} \end{eqnarray} \vspace{-6mm} \newline is undefined because of the gauge degrees of freedom. Namely, the gauge transformations create the "orbit" of the field $A^\omega_\mu$ in the functional field space and the functional integration (\ref{e7}) over this space overcounts the degrees of freedom of the theory. To improve the situation we have to eliminate all unphysical degrees of freedom which origin from local gauge invariance. So, we have to "slice" the orbit once so that we do not have this infinite overcounting. This is the origin of the gauge fixing problem. To solve this problem one can select the surface in this functional space. The surface is good if it intersects the orbit of any given field under gauge transformation once and only once. The equation of this surface may be written as \newline \vspace{-6mm} \begin{eqnarray} F(A_\mu)=0 \label{e8} \end{eqnarray} \vspace{-6mm} \newline To eliminate the unphysical degrees of freedom we may insert the factor $\delta[F(A _\mu)] $ in the functional integral. We can do it inserting \newline \vspace{-6mm} \begin{eqnarray} 1=\Delta_{FP}(A_\mu)\int D \omega \delta [F(A^\omega _\mu)] \label{e9} \end{eqnarray} \vspace{-6mm} \newline into (\ref{e7}). This expression can not change the measure of integration in (\ref{e7}) so we will have \newline \vspace{-6mm} \begin{eqnarray} \int DA_\mu \exp\left\{i\int d^4x L\right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int DA_\mu \left(\Delta_{FP}(A _\mu) \int D\omega\delta[F(A^\omega _\mu)]\right) \exp\left\{i\int d^4x L\right\}, \label{e10} \end{eqnarray} \vspace{-6mm} \newline where $\delta_{FP}$ is Faddeev-Popov determinant, and $D\omega=\prod\limits_{a}\prod\limits_{x} d\omega^a(x)$ is the invariant group measure \cite{sf1}. One can show that it is gauge invariant \cite{key4} \newline \vspace{-6mm} \begin{eqnarray} \Delta_{FP}(A_\mu)=\Delta_{FP}(A^\omega_\mu) \label{e11} \end{eqnarray} \vspace{-6mm} \newline therefore the equation (\ref{e10}) is \newline \vspace{-6mm} \begin{eqnarray} \int DA_\mu \int D\omega \Delta_{FP}(A^\omega _\mu) \delta[F(A^\omega _\mu)] \exp\left\{i\int d^4x L\right\}. \label{e12} \end{eqnarray} \vspace{-6mm} \newline Making the gauge transformation from $A^\omega _\mu$ to $A_\mu$ and taking into account that action is gauge invariant too, we get, dropping multiplicative divergent factor $\int d\omega$, the expression for generating functional of the vector field $A_\mu$ \newline \vspace{-6mm} \begin{eqnarray} Z[J]=\int DA_\mu \Delta_{FP}(A_\mu) \delta[F(A _\mu)] \exp\left\{i\int d^4x L+i(J,A)\right\} \label{e13} \end{eqnarray} \vspace{-6mm} \newline Our task now is to calculate an explicit result of the Faddeev-Popov determinant $\Delta_{FP}(A_ \mu)$. To do this we denote the numerical value of the function $F(A_\mu)$ at $x$ as \newline \vspace{-6mm} \begin{eqnarray} F(A_\mu)=F^a(x) \label{e14} \end{eqnarray} \vspace{-6mm} \newline Then one can write the equation (\ref{e9}) in the form \newline \vspace{-6mm} \begin{eqnarray} \Delta ^{-1}_{FP}(A)=\prod\limits_x \prod\limits_a \int d\omega^a(x)~ \delta (F^a(x)) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\prod\limits_x \prod\limits_a \int dF^a(x)~ \delta (F^a(x))\frac{\partial(\omega_1(x)...\omega_N(x))} {\partial(F_1(x)...F_N(x))} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\prod\limits_x \left\| {{{\partial \omega ^a(x)} \over {\partial F^b(x)}}} \right\|_{F=0} =Det \left(\frac{\delta \omega}{\delta F}\right)_{|F=0} \label{e15} \end{eqnarray} \vspace{-6mm} \newline The Faddeev-Popov determinant \newline \vspace{-6mm} \begin{eqnarray} \Delta _{FP}(A)=Det \left(\frac{\delta F}{\delta \omega}\right)_{|F=0}\label{e16} \end{eqnarray} \vspace{-6mm} \newline is the functional determinant of the continuous matrix $\parallel \delta F^a(x)/\delta \omega^b(y) \parallel$ with rows labelled by $(a,x)$ and columns by $(b,y)$. The generating functional (\ref{e13}) will have the form \newline \vspace{-6mm} \begin{eqnarray} Z[J]=\int DA_\mu(x)Det\left(\delta F^a/\delta \omega^b\right) \delta[F^a(A_\mu)] \exp \left\{ iS+i(J,A)\right\} \label{e17} \end{eqnarray} \vspace{-6mm} \newline Now we can make the last step to rewrite our expression in a form convenient for practical calculations introducing Faddeev-Popov (ghost) fields. \section{Ghost fields} \vspace{1mm} For the calculation of generating functional (\ref{e17}) we must select a gauge constraint and compute the FP-determinant (\ref{e16}) in this gauge. Let us select the Lorentz gauge constrain \newline \vspace{-6mm} \begin{eqnarray} F^a(A_\mu)=\partial^\mu A^a_\mu(x)=0 \label{e18} \end{eqnarray} \vspace{-6mm} \newline We can find $Det(\delta F^a/\delta \omega^b)$ from the Teylor expansion of the $F^a$ with gauge transformation $\omega ^b$: \newline \vspace{-6mm} \begin{eqnarray} F^a\to (F^ \omega)^a=F^a+Det(\delta F^a/\delta \omega ^b)\omega^b+... \label{e19} \end{eqnarray} \vspace{-6mm} \newline For our gauge conditions (\ref{e18}) it is \newline \vspace{-6mm} \begin{eqnarray} \partial^\mu A^a_\mu(x) \mathop \to \limits^\omega \partial^\mu A^a_\mu(x) +f^{abc}\partial(A^b_\mu (x)\omega^c(x))+(1/g)\partial^2 \omega^a(x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\partial^\mu A^a_\mu (x) +((1/g)\partial ^2 \delta ^{ac}+A^{ac}_\mu (x)\partial ^\mu )\omega ^c(x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\partial^\mu A^a_\mu(x) +\int dy\sum\limits_c \left[(1/g)\partial ^2 \delta ^{ac}\delta (x-y) +A^{ac}_\mu(x)\partial ^\mu \delta (x-y)\right]\omega ^c(y) \label{e20} \end{eqnarray} \vspace{-6mm} \newline where $A^{ac}_\mu=f^{abc}A^b_\mu(x)$. Then we can find, that \newline \vspace{-6mm} \begin{eqnarray} Det\left[\delta F/ \delta \omega\right]=Det \parallel<x,a|\delta F/ \delta \omega |y,c>\parallel \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =Det\left[((1/g)\partial ^2 \delta^{ac} + A^{ac}_\mu (x)\partial^\mu)\delta (x-y)\right] \label{e21} \end{eqnarray} \vspace{-6mm} \newline One can rewrite Faddeev-Popov determinant (\ref{e21}) in terms of a path integral over anti-commuting fields $(\bar{c}, c)$\footnote{The functional determinant of anti-commuting fields $\bar{c}$, $c$ is $$\int D\bar{c}Dc \exp\{i\bar{c}^aM^{ab}c^b\}=Det(M)$$}. They are called Faddeev-Popov ghosts. the fields $\bar{c}$, $c$ are independent anti-commuting scalar fields which obey Fermi statistics. Then the determinant (\ref{e21}) may be written in the form \newline \vspace{-6mm} \begin{eqnarray} Det\left[\delta F/ \delta \omega\right]= \int Dc D\bar{c}\exp\left[i\int d^4x L_{ghost}(x)\right] \label{e22} \end{eqnarray} \vspace{-6mm} \newline Lagrangian $L_{ghost}(x)$ is \newline \vspace{-6mm} \begin{eqnarray} L_{ghost}(x)=\bar{c}^a(x)(\partial ^2 \delta^{ab} + gA^{ab}_\mu (x)\partial^\mu)c^b(x) \label{e23} \end{eqnarray} \vspace{-6mm} \newline The first term in the Lagrangian (\ref{e23}) is the pure ghost part \newline \vspace{-6mm} \begin{eqnarray} L^{(0)}_{ghost}(x)=\bar{c}(x)(\partial ^2 \delta^{ac})c(x) \label{e24} \end{eqnarray} \vspace{-6mm} \newline and the ghost propagator $G_0$ directly follows from (\ref{e24}). The equation for this propagator is \newline \vspace{-6mm} \begin{eqnarray} \partial^2_xG_0(x-y)=\delta (x-y), \label{e25} \end{eqnarray} \vspace{-6mm} \newline In the momentum space $G_0(k)$ is written \newline \vspace{-6mm} \begin{eqnarray} G_0(k)=1/(k^2+i\epsilon). \nonumber \end{eqnarray} \vspace{-6mm} \newline The second term in (\ref{e23}) describes interaction ghosts and the vector field \newline \vspace{-6mm} \begin{eqnarray} igf^{abc}\partial_\mu \label{e26} \end{eqnarray} \vspace{-6mm} \newline For better understanding of the ghost contributions let us study an effective action of the ghost fields. \section{ Effective action} \vspace{1mm} The functional determinant (\ref{e21}) may be rewritten as \newline \vspace{-6mm} \begin{eqnarray} Det (\delta F/\delta \omega) =Det (\partial ^2 \delta^{ac}+gA^{ac}_\mu\partial_\mu) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =Det((1/g)\partial^2)Det\left(1+g(1/\partial^2) A^{ac}_\mu\partial_\mu\right) \label{e27} \end{eqnarray} \vspace{-6mm} \newline Ignoring the first determinant\footnote{This term is important in the finite temperature limit.} \newline \vspace{-6mm} \begin{eqnarray} Det (\partial^2)=Det(G^{-1}_0), \label{e28} \end{eqnarray} \vspace{-6mm} \newline we can write an effective action in the form \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=S-i Tr\ln (1+L), \label{e29} \end{eqnarray} \vspace{-6mm} \newline where the element $L$ is written as \newline \vspace{-6mm} \begin{eqnarray} <x,a|L|y,c>=g<x,b|\left[ (1/\partial^2)A^{ac}_\mu(x)\partial^\mu_x \delta (x-y)\right]|y,c> \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =g\int dzG_0(x-z)A^{ac}_\mu(x)\partial^\mu_z \delta (z-y). \label{e30} \end{eqnarray} \vspace{-6mm} \newline Now we can expand the determinant with \newline \vspace{-6mm} \begin{eqnarray} Det(1+L)=\exp\left\{\mbox{Tr}\ln (1+L)\right\}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\exp\left(\sum\limits_{n=1}^\infty \frac{(-1)^{n-1}}{n}\mbox{Tr} L^n\right) \label{e31} \end{eqnarray} \vspace{-6mm} \newline Write the contributions $\mbox{Tr}(L^n)$ in (\ref{e31}). The first expression of the order $g$ is \newline \vspace{-6mm} \begin{eqnarray} \ln L = \mbox{tr}\int dx \int dz \left\{G_0(x-z)gA^{ac}_\mu (z)\partial_{z}\delta (z-x)\right\} \label{e32} \end{eqnarray} \vspace{-6mm} \newline because $\mbox{tr}(A^{ac}_\mu(x))=\mbox{tr}(f^{abc}A^b_\mu(x))=0$. The second expression of the order $g^2$ will be \newline \vspace{-6mm} \begin{eqnarray} \ln (L^2) =g^2 \mbox{tr}\int dx \int dy\int dz\int dt G_0(x-z)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times A^{ac}_\mu(z) \partial_{z}\delta (z-x)G_0(y-t) A^{ca}_\nu (t) \partial_t \delta (t-x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int dz\int dt \left\{gA^{ca}_\nu(t)\partial_t G_0(t-z)\right\}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \left\{gA^{ac}_\mu(z)\partial _z G_0(t-z)\right\} \label{e33} \end{eqnarray} \vspace{-6mm} \newline As we can see from (\ref{e33}) and (\ref{e31}), for $n>2$ the trace of (\ref{e29}) equals the integral with integrand in the form of the cycle multiplication of the terms \newline \vspace{-6mm} \begin{eqnarray} \left\{gA^{ac}_\mu(z)\partial _z G_0(t-z)\right\} \label{e34} \end{eqnarray} \vspace{-6mm} \newline namely, we can write \newline \vspace{-6mm} \begin{eqnarray} \mbox{Tr} (L^n)=\int dz_1...\int dz_n \left\{gA^{ac_1}_\mu(z_1)\partial _{z_1} G_0(z_1-z_2)\right\}\times... \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times...\left\{gA^{c_na}_\mu(z_n )\partial _{z_n} G_0(z_{n-1}-z_n)\right\} \label{e35} \end{eqnarray} \vspace{-6mm} \newline These terms give non-trivial corrections due to ghosts to the effective action $S_{eff}[A]$ Therefore the determinant (\ref{e31}) may be treated as contributions of the closed loops with internal ghost lines, and external lines of which correspond to vector fields. \section {Propagator of vector field} \vspace{1mm} In the previous section we found out how to compute the Faddeev-Popov determinant in the generating functional (\ref{e17}). Now we must treat the second important term the delta function arising from the gauge constraint. For this we will rewrite the delta function as \cite{vl1} \newline \vspace{-6mm} \begin{eqnarray} \delta(\partial^\mu A^a_\mu) =\mathop {\prod}\limits_{x}\delta(\partial^\mu A^a_\mu(x)) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mathop {\lim}\limits_{\alpha \to 0}\mathop {\prod}\limits_{x} (-2i\pi \alpha)^{-1/2} \exp\left[ -\frac{i}{2\alpha}(\partial^\mu A^a_\mu (x))^2 \right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \sim\mathop {\lim}\limits_{\alpha \to 0} \exp\left\{ -\frac{i} {2\alpha}\int d^4x \left[\partial^\mu A^a_\mu (x)\right]^2 \right\} \label{e36} \end{eqnarray} \vspace{-6mm} \newline The argument of the exponent (\ref{e36}) may be combined with the quadratic part of the vector Lagrangian (\ref{e5}): \newline \vspace{-6mm} \begin{eqnarray} i\int d^4x\left[ -\frac{1}{4}(\partial_\mu A^a_\nu-\partial_\nu A^a_\mu)^2 -\frac{1}{2\alpha} \left(\partial^\mu A^a_\mu (x)\right)^2\right] \label{e37} \end{eqnarray} \vspace{-6mm} \newline The second term of the expression (\ref{e37})fixes the gauge. As a result we have the quadratic part over the vector fields in the form \newline \vspace{-6mm} \begin{eqnarray} -\frac{i}{2}\int d^4xA^a_\mu(x) \left[ (\eta^{\mu\nu}\partial^2 +\left(1-\frac{1}{\alpha}\right)\partial_\mu \partial_\nu) \right]A^a_\nu(x) \label{e38} \end{eqnarray} \vspace{-6mm} \newline The propagator of the vector field may be found from the equation \newline \vspace{-6mm} \begin{eqnarray} \left[ (\eta^{\mu\nu}\partial^2 +\left(1-\frac{1}{\alpha}\right)\partial_\mu \partial_\nu)\right] G_{\nu \lambda}(x-y)=-\delta_{\nu \lambda}\delta(x-y).\label{e39} \end{eqnarray} \vspace{-6mm} \newline The solution of this equation is \newline \vspace{-6mm} \begin{eqnarray} G_{\mu\nu}(x)=\int\frac{d^4k}{(2\pi^4)}\left[\eta_{\mu\nu} +(1-\alpha)\frac{k_\mu k_\nu}{k^2}\right]\frac{\exp(ikx)}{k^2+i\epsilon} \label{e40} \end{eqnarray} \vspace{-6mm} \newline The finite temperature propagator of the vector field in the Feynman gauge $(\alpha =1)$ has the simple form \newline \vspace{-6mm} \begin{eqnarray} G_{\mu\nu}(x)=\int\limits_{k} \frac{\eta_{\mu \nu}} {k^2+i\epsilon} \exp(ikx) \label{e41} \end{eqnarray} \vspace{-6mm} \newline where the definition of the integral $\int \limits_{k}$ is the same as in (\ref{b54}) \section {Partition function for Gauge fields} \vspace{1mm} Let us study thermal gauge fields in the axial gauge $(A_3=0)$. The standard expression for the partition function is \newline \vspace{-6mm} \begin{eqnarray} Z=\mbox{Tr} \exp[-\beta H] \nonumber \end{eqnarray} \vspace{-6mm} \newline where $H=H(\pi^a_i,A^a_i)$ is the Hamiltonian of the Yang-Mills field. According to section II (\ref{b15}) we can write \newline \vspace{-6mm} \begin{eqnarray} Z=(\mbox{Tr}\exp[-\beta H])_{axial} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =N\int \prod\limits_a d{\pi^a_1}d{\pi^a_2} \int \limits_{periodic}d{A^a_1}d{A^a_2}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp\left\{\int \limits_0^\beta d\tau \int d^3x[i\pi_i\dot{A}^a_i-H(\pi^a_i,A^a_i)]\right\} \label{e42} \end{eqnarray} \vspace{-6mm} \newline The Hamiltonian in this gauge has only two degrees of freedom for each vector field: $A^a_{1,2}$. The conjugate momenta for these fields are $\pi^a_{1,2}$. After Gaussian integration over the momenta, we get \newline \vspace{-6mm} \begin{eqnarray} Z=N\int \limits_{periodic}D{A^a}\prod\limits_a\delta[A^a_3] \exp\left\{\int \limits_0^\beta d\tau \int d^3xL(x)\right\} \label{e43} \end{eqnarray} \vspace{-6mm} \newline The result is similar to the expression for the Faddeev-Popov ansatz in the axial gauge. Calculations of (\ref{e43}) give the correct result for the partition function of a photon gas. The same result will also be obtained for Coulomb gauge $(\vec{\nabla} \vec{A}=0)$. For the Feynman gauge the simple calculation of the type (\ref{e42}) leads to an incorrect result. But we may correct the result taking into consideration the ghost contribution and rewrite the partition function (\ref{e42}) in the form of (\ref{e13}). \newline \vspace{-6mm} \begin{eqnarray} Z=N\int \limits_{periodic}D{A^a}Det(\delta F^a/\delta \omega^b) \delta [F^a] \exp\left\{\int \limits_0^\beta d\tau \int d^3xL(x)\right\} \label{e44} \end{eqnarray} \vspace{-6mm} \newline Since under the gauge transformations $\delta A^a_\mu=-\partial_ \mu\omega^a$, we have \newline \vspace{-6mm} \begin{eqnarray} Det(\delta (\partial_\mu A^{a \mu})/\delta \omega ^b)=Det (\partial^2) \label{e45} \end{eqnarray} \vspace{-6mm} \newline that coincides with (\ref{b28}) at order $\sim 0(g)$. For the Feynman gauge $(\alpha=1)$ \newline \vspace{-6mm} \begin{eqnarray} \int \limits_{periodic}D{A^a} \exp\left\{\int \limits_0^\beta d\tau (-1/2)\int d^3x\left[A^a_\mu(x)\partial^2A^{a\mu(x)}\right]\right\} =Det(G_{\mu\nu})^{1/2} \label{e46} \end{eqnarray} \vspace{-6mm} \newline At finite temperature $Det(G_{\mu\nu})$ may be written in the form \newline \vspace{-6mm} \begin{eqnarray} Det(G_{\mu\nu})=\exp(-1/2)\mbox{Tr} \ln (G_{\mu\nu}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\exp(-2)\sum\limits_n\int\frac{d^3k}{(2\pi)^3} \ln (\omega_n^2+\vec{k}^2) \label{e47} \end{eqnarray} \vspace{-6mm} \newline and the temperature contribution of $Det (\partial^2)$ in the form \newline \vspace{-6mm} \begin{eqnarray} Det(\partial^2)=\exp \mbox{Tr}\ln (\partial^2)=\exp \mbox{Tr} \ln (G) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\exp\sum\limits_n\int\frac{d^3k}{(2\pi)^3}\ln (\omega_n^2+\vec{k}^2) \label{e48} \end{eqnarray} \vspace{-6mm} \newline where $G$ is just the propagator of the scalar field. The logarithm of the finite temperature generating functional $Z[0]$, according to previous results, is \newline \vspace{-6mm} \begin{eqnarray} \ln Z[0]=\ln Det(G)-(1/2)\ln Det(G_{\mu\nu}), \label{e49} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \ln Z[0]=-\sum\limits_n\int\frac{d^3k}{(2\pi)^3} \ln (\omega_n^2+\vec{k}^2). \label{e50} \end{eqnarray} \vspace{-6mm} \newline After summation the free energy is found in the form \newline \vspace{-6mm} \begin{eqnarray} f(\beta) =-\frac{1}{\beta}\ln Z_\beta =\left(\frac{2}{\beta} \right) \int\frac{d^3k}{(2\pi)^3} \ln (1-\exp[-\beta \epsilon]), \label{e51} \end{eqnarray} \vspace{-6mm} \newline where $\epsilon^2=(\vec{k})^2$. So, we got the correct answer \cite{key21} by introducing the finite temperature ghost contribution. The index $(2)$ in the equation (\ref{e51}) reflects the two degrees of freedom of the massless vector field. The generalization to Schwinger proper time formalism is the same as in the previous section. \chapter{BOSE FIELDS } \centerline{\Large \bf IN CURVED SPACE-TIME} \vspace{24pt} As we know already the inclusion of the interaction with gravitational field (theory formulated in curved space-time) is accomplished by replacing of the partial derivative by the covariant derivative $\partial_\mu \to \nabla_\mu$. It is necessary to ensure that the Lagrangian is the scalar under the general coordinate transformation. Integration is performed over the invariant volume. This procedure, based on general -coordinate covariance, is called the minimal interaction for gravity and leads to the action (\ref{g23}). However, general coordinate covariance does not forbid adding to the Lagrangian invariant terms which are vanishing in flat space-time. Such terms describe the non-minimal interaction with gravity. Therefore the theory under such consideration can be written in the form \newline \vspace{-6mm} \begin{eqnarray} S=\int d^4x\sqrt{-g}\left(L(\phi,\nabla_\mu \phi) + non-min.~~int.\right) \label{h1} \end{eqnarray} \vspace{-6mm} \newline From dimensional analysis of $R$ and $\phi$ conclude that the term, describing non-minimal interaction of matter field with gravitational field may be written in the form: \newline \vspace{-6mm} \begin{eqnarray} (1/2)\xi R\varphi^2 \label{h2} \end{eqnarray} \vspace{-6mm} \newline where $\xi$ is a dimensionless parameter (non-minimal coupling constant) \cite{key17}. If $m=0$ and $\xi=1/6$ then the action is invariant not only under general-coordinate transformation but also under conformal transformations \\ \cite{key16} \newline \vspace{-6mm} \begin{eqnarray} g_{\mu\nu}^{'}(x)=e^{2\chi(x)}g_{\mu\nu},~~ {\varphi^{'}}(x)=e^{\chi(x)}\varphi(x) \label{h3} \end{eqnarray} \vspace{-6mm} \newline where $\chi(x)$ is an arbitrary scalar field (parameter of the conformal transformation). In the result we will have the action in the form \newline \vspace{-6mm} \begin{eqnarray} S_\varphi=-(1/2)\int d^4x\sqrt{-g}\varphi(x) \left(-\Box_x+m^2+\xi R\right)\varphi(x) \label{h4} \end{eqnarray} \vspace{-6mm} \newline where the d'Alembertian operator is $\Box_x=g_{\mu\nu}(x){\nabla^x}_\mu {\nabla^x}_\nu =\partial_\mu\partial^\mu$. The generating functional will be \newline \vspace{-6mm} \begin{eqnarray} Z[R,J]\propto (Det G)^{-1/2}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp\left\{ -(i/2)\int d^4 x g(x)^{1/2} \int d^4 x^{'} {g(x^{'})}^{1/2}J(x)G(x,x^{'})J(x^{'})\right\} \label{h5} \end{eqnarray} \vspace{-6mm} \newline where $G(x,x^{'})$ is the Green's function of the scalar field which is described by the equation: \newline \vspace{-6mm} \begin{eqnarray} g^{(1/2)}(x)\left(-\Box_x+m^2+\xi R\right)G(x,x^{'})= \delta(x-x^{'}) \label{h6} \end{eqnarray} \vspace{-6mm} \newline So, for computation of the generating functional (\ref{h5}) we have to define the Green's function of the scalar field in curved space-time from the equation (\ref{h6}). \section{Momentum-space representation of} \vspace{-4mm} \hspace{20mm}{\Large \bf the bosonic Green's function } \vspace{1mm} In curved space-time we cannot solve the equation (\ref{h6}) and compute the Green's function $G(x,x^{'})$ for arbitrary points $x$ and $x^{'}$ of the manifold. But we can do so in the particular case of the limit of coincidence $(x \simeq x^{'})$. This limit gives us a possibility to find the $(Det)$ of the Green's functions to get the effective action. Thus we will treat the problem of the Green's functions calculations in the limit of coincidence. Let us select a point of the space-time manifold $x^{'}$ and construct the tangential space at this point as we did in the previous section. Let any point $x$ of the manifold have the normal coordinate $y^\alpha (x)$. This coordinate may be treated as a vector in the tangent space with the origin at the point $x^{'}$. In this tangent space the norm of the vector $y^\alpha (x)$ will be $(y,y)=y^\alpha (x)y^\beta (x)\eta_{\alpha\beta}$, and the metric of the manifold may be written as \cite{petrov1} \newline \vspace{-6mm} \begin{eqnarray} g_{\mu \nu }(x)=\eta _{\mu \nu }-{1 \over 3} R_{\mu \alpha \nu \beta }y^\alpha y^\beta -{1 \over 6}R_{\mu \alpha \nu \beta ;\gamma }y^\alpha y^\beta y^\gamma +...\label{h7} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} g(x)=1-{1 \over 3}R_{\alpha \beta } y^\alpha y^\beta -{1 \over 6} R_{\alpha \beta ;\gamma }y^\alpha y^\beta y^\gamma \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left( {{1 \over {18}}R_{\alpha \beta } R_{\gamma \delta }-{1 \over {90}}R^\nu _{\alpha \beta \lambda } R^\lambda _{\gamma \delta \nu }-{1 \over {20}} R_{\alpha \beta ;\gamma \delta }} \right)y^\alpha y^\beta y^\gamma y^\delta +... \label{h8} \end{eqnarray} \vspace{-6mm} \newline where the coefficients are calculated at the $(y=0)$ origin of the coordinate system. The second derivative for the scalar field is \newline \vspace{-6mm} \begin{eqnarray} \nabla ^{x\mu }\nabla _\mu ^x=\eta ^{\alpha \beta } \partial _\alpha \partial _\beta +{1 \over 3}{{{R_\alpha }^\delta }_\beta }^\gamma y^\alpha y^\beta \partial ^x_\delta\partial ^x_\gamma -{2 \over 3}R_\beta ^\alpha y^\beta \partial ^x_\alpha +... \label{h9} \end{eqnarray} \vspace{-6mm} \newline For our calculations it is convenient to express the Green's function $G(x,x^{'})$ as: \newline \vspace{-6mm} \begin{eqnarray} G(x,x^{'})=g^{-1/4}(x)\Im (x,x^{'})g^{-1/4}(x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =g^{-1/4}(x)\Im (x,x^{'}) \label{h10} \end{eqnarray} \vspace{-6mm} \newline Substituting (\ref{h8}),(\ref{h9}) and (\ref{h10}) into (\ref{h6}), we find \newline \vspace{-6mm} \begin{eqnarray} \eta ^{\alpha \beta }\partial _\alpha \partial _\beta \Im -\left[ {m^2+\left( {\xi -{1 \over 6}} \right)R} \right]\Im -{1 \over 3}{R_\alpha }^\beta y^\alpha \partial _\beta \Im \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{1 \over 3}{{{R_\alpha }^\beta }_\gamma }^\delta y^\alpha y^\gamma \partial _\beta \partial _\delta \Im -\left( \xi -{1 \over 6} \right)R_{;\alpha }y^\alpha \Im \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left( -{1 \over 3}{{R_\alpha}^\beta }_{;\gamma } +{1 \over 6}{{R_\alpha} \gamma }^{;\beta } \right)y^\alpha y^\gamma \partial _\beta \Im \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{1 \over 6}{{{{R_\lambda} ^\gamma} _\alpha }^\zeta}_{;\beta } y^\lambda y^\alpha y^\beta \partial _\gamma \partial _\zeta \Im -{1 \over 2}\left( {\xi -{1 \over 6}} \right) R_{;\alpha \beta }y^\alpha y^\beta \Im + \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left( -{1 \over {30}}{R_\alpha }^\beta R_{\beta \gamma } +{1 \over {60}}{{{R_\alpha }^\beta} _\gamma }^\delta R_{\beta \delta }\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+{1 \over {60}}{R^{\beta \chi \delta }}_\alpha R_{\beta \chi \delta \gamma } -{1 \over {120}}R_{;\alpha \gamma } +{1 \over {40}}\Box R_{\alpha \gamma } \right) \Im \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left( -{3 \over {20}}{R^\delta} _{\alpha ;\beta \gamma } +{1 \over {10}}{{R_{\alpha \beta }}^{;\delta }}_\gamma -{1 \over {60}}{{{R^\chi} _\alpha }^\delta} _\beta R_{\chi \gamma }\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+{1 \over {15}}{R^\chi} _{\alpha \lambda \beta } {{{R_\chi }^\delta} _\gamma }^\lambda \right) y^\alpha y^\beta y^\gamma \partial _\delta \Im \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left( {1 \over {20}}{{{R^\kappa} _\alpha} ^\chi} _{\beta ;\gamma \delta } +{1 \over {15}}{R^\kappa }_{\alpha \lambda \beta } {{{R^\lambda} _\gamma} ^\chi} _\delta \right) y^\alpha y^\beta y^\gamma y^\delta \partial _\kappa \partial _\chi \Im =-\delta (y), \label{h11} \end{eqnarray} \vspace{-6mm} \newline where $y^\alpha$ are the coordinates of the point $x$ and $\partial_\alpha \Im=(\partial/\partial y^\alpha)$. We have retained only terms with coefficients involving four derivatives of the metric. These contributions give the ultraviolet divergences that arise in the course of renormalization. In normal coordinates with origin at $x^{'}$, $\Im (x,x^{'})$ is a function of $y$ and $x^{'}$ \newline \vspace{-6mm} \begin{eqnarray} \Im (x,x^{'})=\Im (y,x^{'}) \label{h12} \end{eqnarray} \vspace{-6mm} \newline where $y$ belongs to the small region around $x^{'}$. In this way the equation (\ref{h6}) may be solved recursively. Namely, we will introduce the momentum space associated with the point $x^{'}$ $(y=0)$ by making the $n$-dimensional Fourier transformation: \newline \vspace{-6mm} \begin{eqnarray} \Im (x,x^{'})=\int \frac{d^n k}{(2\pi)^n}\Im(k)\exp(iky) \label{h13} \end{eqnarray} \vspace{-6mm} \newline where $ky=k_\alpha y^\alpha=\eta^{\alpha \beta}k_\alpha y_\beta$ and expanding $\Im (k)$ in a series: \newline \vspace{-6mm} \begin{eqnarray} \Im (k)=\Im_0 (k)+\Im_1 (k)+\Im _2(k)+...\label{h14} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \Im_i (x,x^{'})=\int \frac{d^n k}{(2\pi)^n}\Im_i(k) \exp(iky), ~~~i=0,1,2,...\label{h15} \end{eqnarray} \vspace{-6mm} \newline where we will assume that the coefficients $\Im (k)$ have geometrical coefficients involving $i$ derivatives of the mertic. On dimensional grounds, $\Im_i(k)$ are the order $k^{-(2+i)}$, so that (\ref{h14}) is an asymptotic expansion of $\Im (k)$ in large $k$ (small $y$). Inserting (\ref{h13}) into (\ref{h11}) we get that the lowest order solution ($\sim O(y^2)$) is \newline \vspace{-6mm} \begin{eqnarray} \Im_0 (k)=(k^2+m^2)^{-1} \label{h16} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \Im _1(k)=0 \label{h17} \end{eqnarray} \vspace{-6mm} \newline The function $\Im _2(k)$ ($\sim O(y^4)$) may be found from \newline \vspace{-6mm} \begin{eqnarray} (\eta^{\alpha \beta}\partial_\alpha \partial_\beta-m^2)\Im _2 -\left( {\xi -{1 \over 6}} \right)R \Im_0 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{1 \over 3}{R_\alpha }^\beta y^\alpha \partial _\beta \Im_0 +{1 \over 3}{{{R_\alpha }^\beta }_\gamma }^\delta y^\alpha y^\gamma \partial _\beta \partial _\delta \Im_0=0 \label{h18} \end{eqnarray} \vspace{-6mm} \newline Using (\ref{h13}) one can get that the last two terms of (\ref{h18}) cancel each other. In another way we can consider that $\Im _0$ is Lorentz invariant of the form \newline \vspace{-6mm} \begin{eqnarray} \Im (y)=y_\alpha y^\alpha=\eta^{\alpha \beta}y_\alpha y_\beta \label{h19} \end{eqnarray} \vspace{-6mm} \newline Then, inserting (\ref{h19}) into (\ref{h18}) we get: \newline \vspace{-6mm} \begin{eqnarray} -{1 \over 3}{R_\alpha }^\beta y^\alpha \partial _\beta \Im_0(y) +{1 \over 3}{{{R_\alpha }^\beta }_\gamma }^\delta y^\alpha y^\gamma \partial _\beta \partial _\delta \Im_0(y)\equiv 0, \label{h20} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} (\eta^{\alpha \beta}\partial_\alpha \partial_\beta-m^2)\Im _2(y) -\left( {\xi -{1 \over 6}} \right)R \Im_0(y)=0 \label{h21} \end{eqnarray} \vspace{-6mm} \newline Therefore \newline \vspace{-6mm} \begin{eqnarray} \Im _2(k)=\left( {1 \over 6}-\xi \right)R (k^2+m^2)^{-2}. \label{h22} \end{eqnarray} \vspace{-6mm} \newline The Lorentz invariance of $\Im_0(y)$ leads to further simplifications of (\ref{h11}). Namely, the contributions \newline \vspace{-6mm} \begin{eqnarray} \left( -{1 \over 3}{{R_\alpha}^\beta }_{;\gamma } +{1 \over 6}{{R_\alpha} \gamma }^{;\beta } \right)y^\alpha y^\gamma \partial _\beta \zeta \Im_0(y) +{1 \over 6}{{{{R_\lambda} ^\gamma} _\alpha }^\zeta}_{;\beta } y^\lambda y^\alpha y^\beta \partial _\gamma \partial _\zeta \Im_0(y) \equiv 0 \label{addh22} \end{eqnarray} \vspace{-6mm} \newline are eliminated. In the same way \newline \vspace{-6mm} \begin{eqnarray} \left( -{3 \over {20}}{R^\delta} _{\alpha ;\beta \gamma } +{1 \over {10}}{{R_{\alpha \beta }}^{;\delta }}_\gamma -{1 \over {60}}{{{R^\chi} _\alpha }^\delta} _\beta R_{\chi \gamma }\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+{1 \over {15}}{R^\chi} _{\alpha \lambda \beta } {{{R_\chi }^\delta} _\gamma }^\lambda \right) y^\alpha y^\beta y^\gamma \partial _\delta \Im_0(y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left( {1 \over {20}}{{{R^\kappa} _\alpha} ^\chi} _{\beta ;\gamma \delta } +{1 \over {15}}{R^\kappa }_{\alpha \lambda \beta } {{{R^\lambda} _\gamma} ^\chi} _\delta \right) y^\alpha y^\beta y^\gamma y^\delta \partial _\kappa \partial _\chi \Im_0(y) \equiv 0 \label{h23} \end{eqnarray} \vspace{-6mm} \newline and we have the following equation for $\Im (y)$ to the fourth order in derivatives of the metric: \newline \vspace{-6mm} \begin{eqnarray} \eta ^{\alpha \beta }\partial _\alpha \partial _\beta \Im (y) -\left[ {m^2+\left( {\xi -{1 \over 6}} \right)R} \right]\Im(y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -\left( \xi -{1 \over 6} \right)R_{;\alpha }y^\alpha \Im(y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(1/2)\left( \xi -{1 \over 6} \right)R_{;\alpha\beta }y^\alpha y^\beta\Im(y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left( -{1 \over {30}}{R_\alpha }^\beta R_{\beta \gamma } +{1 \over {60}}{{{R_\alpha }^\beta} _\gamma }^\delta R_{\beta \delta }\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+{1 \over {60}}{R^{\beta \chi \delta }}_\alpha R_{\beta \chi \delta \gamma } -{1 \over {120}}R_{;\alpha \gamma } +{1 \over {40}}\Box R_{\alpha \gamma } \right) \Im(y)=-\delta(y) \label{h24} \end{eqnarray} \vspace{-6mm} \newline Substitution of $\Im_2(k)$ (\ref{h22}) instead of $\Im_0(k)$ in the identity (\ref{h20}) does not change it, thus we can suggest that $\Im_2(k)$ is Lorentz invariant too and it can be wtitten as $\Im_2(y) \sim (y^\alpha y_\alpha)^2$. The equation (\ref{h11}) is simplified to \newline \vspace{-6mm} \begin{eqnarray} \left[k^2+m^2+\left( \xi -{1 \over 6} \right)R +i\left( \xi -{1 \over 6} \right)R_{;\alpha}\partial^\alpha \right]\Im (k)+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left[-(1/2)\left( \xi -{1 \over 6} \right)R_{;\alpha\beta } -{1 \over {30}}{R_\alpha }^\beta R_{\beta \gamma } +{1 \over {60}}{{{R_\alpha }^\beta} _\gamma }^\delta R_{\beta \delta }\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+{1 \over {60}}{R^{\beta \chi \delta }}_\alpha R_{\beta \chi \delta \gamma } -{1 \over {120}}R_{;\alpha \gamma } +{1 \over {40}}\Box R_{\alpha \gamma } \right] \partial^\alpha \partial^\beta \Im(k)=1 \label{h25} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \partial ^\alpha\Im (k)=\partial\Im (k)/ \partial k_\alpha \nonumber \end{eqnarray} \vspace{-6mm} \newline Making a further recurrent process with this equation we get \newline \vspace{-6mm} \begin{eqnarray} \Im_3(k)=0 \label{h26} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \Im _4 (k)=i\left(\frac{1}{6}-\xi\right)R_{;\alpha}(k^2+m^2)^{-1} \partial^\alpha (k^2+m^2)^{-1}+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left(\frac{1}{6}-\xi\right)^2 R^2 (k^2+m^2)^{-3}+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +a_{\alpha \beta}(k^2+m^2)^{-1} \partial^\alpha \partial^\beta (k^2+m^2)^{-1} \label{h27} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} a_{\alpha \beta}=(1/2)\left( \xi -{1 \over 6} \right)R_{;\alpha\beta } +{1 \over {30}}{R_\alpha }^\beta R_{\beta \gamma } -{1 \over {60}}{{{R_\alpha }^\beta} _\gamma }^\delta R_{\beta \delta } \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{1 \over {60}}{R^{\beta \chi \delta }}_\alpha R_{\beta \chi \delta \gamma } +{1 \over {120}}R_{;\alpha \gamma } -{1 \over {40}}\Box R_{\alpha \gamma } \label{h28} \end{eqnarray} \vspace{-6mm} \newline Now we can write the equation for the Green 's function in a convenient form. Let us introduce useful equations: \newline \vspace{-6mm} \begin{eqnarray} (k^2+m^2)^{-1}\partial^\alpha (k^2+m^2)^{-1}\equiv (1/2)\partial^\alpha (k^2+m^2)^{-2} \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} (k^2+m^2)^{-1}\partial^\alpha \partial^\beta (k^2+m^2)^{-1}\equiv (1/3)\partial^\alpha \partial^\beta (k^2+m^2)^{-2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(2/3)\eta_{\alpha \beta}(k^2+m^2)^{-3} \label{h29} \end{eqnarray} \vspace{-6mm} \newline Using (\ref{h29}) we may write $\Im(k)$ as \newline \vspace{-6mm} \begin{eqnarray} \Im(k) =(k^2+m^2)^{-1}+\left(\frac{1}{6}-\xi\right)R(k^2+m^2)^{-2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} (i/2)\left(\frac{1}{6}-\xi\right)R_{;\alpha}(k^2+m^2)^{-2} \partial^\alpha (k^2+m^2)^{-1} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(1/3)a_{\alpha \beta}\partial^\alpha \partial^\beta (k^2+m^2)^{-2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left[\left(\frac{1}{6}-\xi\right)^2R^2 -(2/3)a^\lambda_\lambda \right](k^2+m^2)^{-3} \label{h30} \end{eqnarray} \vspace{-6mm} \newline Inserting the last equation into (\ref{h13}) we get: \newline \vspace{-6mm} \begin{eqnarray} \Im(x^{'},y)=\int \frac{d^nk}{(2 \pi)^n} \exp (iky) \left[1+\gamma_1(x^{'},y)\left(-\frac{\partial} {\partial m^2}\right)\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+ \gamma_2(x^{'},y)\left(-\frac{\partial} {\partial m^2}\right)^2\right](k^2+m^2)^{-1} \label{h31} \end{eqnarray} \vspace{-6mm} \newline where, to the fourth order in derivatives of the metric, the coefficients are: \newline \vspace{-6mm} \begin{eqnarray} \gamma_1(x^{'},y)=\left(\frac{1}{6}-\xi\right)R +(1/2)\left(\frac{1}{6}-\xi\right)R_{;\alpha}y^\alpha -(1/3)a_{\alpha \beta} y^\alpha y^\beta; \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \gamma_2(x^{'},y)=(1/2)\left(\frac{1}{6}-\xi\right)^2R^2 -(1/3)a^\lambda_\lambda. \label{h32} \end{eqnarray} \vspace{-6mm} \newline The expression for the Green's function (\ref{h10}) will be then \cite{bunch1} \newline \vspace{-6mm} \begin{eqnarray} G(x^{'},y)=g^{-1/2}(y) \int \frac{d^n k}{(2 \pi)^n} \exp(iky) \times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \sum \limits_{j=0}^{2}\gamma_j(x^{'},y)\left(-\frac{\partial} {\partial m^2}\right)^j(k^2+m^2)^{-1} \label{h33} \end{eqnarray} \vspace{-6mm} \newline The equation (\ref{h33}) is very important for further calculations. \section {The Green's function and the } \vspace{-4mm} \hspace{20mm}{\Large \bf Schwinger-DeWitt method} \vspace{1mm} We have found already that the geometrical quantities enter directly into the structure of Green's functions for arbitrary fields through the covariant d'Alembertian operator and non-minimal connection. Now we will treat the equation for the Green's function with the Schwinger-DeWitt method developed for flat space-time in the previous section. Let us multiply the equation (\ref{h6}) on the left side by $g^{(1/4)}(x)$ and on the right by $g^{(1/4)}(x^{'})$, and introducing \newline \vspace{-6mm} \begin{eqnarray} \Im(x,x^{'})=g^{(1/4)}(x)G(x,x^{'})g^{(1/4)}(x^{'}) \label{h34} \end{eqnarray} \vspace{-6mm} \newline we will rewrite this equation in the form \newline \vspace{-6mm} \begin{eqnarray} \left(-\partial_\mu \partial^\mu +m^2+\xi R\right)\Im(x,x^{'})= \delta(x,x^{'}), \label{h35} \end{eqnarray} \vspace{-6mm} \newline where $\delta(x,x^{'})=g^{-1/2}(x)\delta(x-x^{'})$ is scalar with respect to general coordinate transformation, and product of $\delta$ -functon is \newline \vspace{-6mm} \begin{eqnarray} (1,\delta(x,x^{'}))=\int d^4x \delta(x,x^{'})=1 \label{addh35} \end{eqnarray} \vspace{-6mm} \newline For simplicity we put $\xi=0$ and get; \newline \vspace{-6mm} \begin{eqnarray} \left(-\Box_x +m^2)\right)\Im(x,x^{'})=\delta (x,x^{'}), \label{h36} \end{eqnarray} \vspace{-6mm} \newline or, in operator form \newline \vspace{-6mm} \begin{eqnarray} \hat{F}\Im =1 \label{h37} \end{eqnarray} \vspace{-6mm} \newline where $F$ is matrix operator. Let us introduce the representation for $\Im$ in the form \newline \vspace{-6mm} \begin{eqnarray} \Im (x,x^{'};s)=i<x|\int \limits _0 ^\infty ds \exp(is\hat{F})|x^{'}> =i \int\limits_0^\infty ds f(x,x^{'};s) \exp(-im^2 s) \label{h38} \end{eqnarray} \vspace{-6mm} \newline We can get from (\ref{h36}), that the function $f(x,x^{'};s)$ is the solution of the equation \newline \vspace{-6mm} \begin{eqnarray} \Box_x f(x,x^{'};s)=i\frac{\partial}{\partial s}f(x,x^{'};s) \label{h39} \end{eqnarray} \vspace{-6mm} \newline where information about space-time structure is included in the d'Alembertian. We may turn this equation into an elliptic one by rewriting the equation for amplitude with the replacement $x^{(0)}=ix^{(4)}$ and $s=it$. We will have \newline \vspace{-6mm} \begin{eqnarray} \frac{\partial}{\partial t}f(x,x^{'};t)=\tilde{\Box_x }f(x,x^{'};t) \label{h40} \end{eqnarray} \vspace{-6mm} \newline One can write the solution of the (\ref{h38}) as a simple expansion of the solution for flat space-time. This solution is \newline \vspace{-6mm} \begin{eqnarray} f(x,x^{'};\bar{s})=(4\pi \bar{s})^{-n/2} \exp(-|x-x^{'}|^2/4\bar{s}) \label{h41} \end{eqnarray} \vspace{-6mm} \newline Returning to the initial variables we get \newline \vspace{-6mm} \begin{eqnarray} f(x,x^{'};s)=(4\pi is)^{-n/2}\exp(-|x-x^{'}|^2/4is) \label{h42} \end{eqnarray} \vspace{-6mm} \newline In curved space-time we may expand this solution to a local asymptotic expansion (for $x \simeq x^{'}$ and $s\simeq 0$) \newline \vspace{-6mm} \begin{eqnarray} f(x,x^{'};s) \sim (4 \pi is)^{(-n/2)}\exp(-\sigma(x,x^{'})/2is) \sum\limits_{j=0}^\infty \gamma_j(x,x^{'}) (is)^j \label{h43} \end{eqnarray} \vspace{-6mm} \newline where $\sigma(x,x^{'})$ is the so-called geodesic interval (half square of geodesic distance between points $x$ and $x^{'}$). In particular \newline \vspace{-6mm} \begin{eqnarray} f(x,x;s) \sim (4 \pi is)^{(-n/2)} \sum\limits_{j=0}^\infty \gamma_j(x) (is)^j \label{h44} \end{eqnarray} \vspace{-6mm} \newline The explicit form of $f_j(x,x^{'})$ can be calculated recursively \cite{brown1}\\ \cite{dt2}, \cite{key17} In the limit of coincidence $x \to x^{'}$ one finds: \newline \vspace{-6mm} \begin{eqnarray} \gamma_0(x^{'})=1;~~\gamma_1(x^{'})=\left(1/6-\xi\right)R; \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \gamma_2(x^{'})=\left(1/6-\xi\right)^2 R^2-(1/3)a^\lambda_\lambda. \label{h45} \end{eqnarray} \vspace{-6mm} \newline From the equations (\ref{h38}), (\ref{h43}) and (\ref{h34}) we get the explicit expression for Green's function in the Schwinger-DeWitt representation: \newline \vspace{-6mm} \begin{eqnarray} G_{SD}(x,x^{'}) =\frac{i\Delta ^{1/2}(x,x^{'})}{(4 \pi)^{n/2}} \int\limits_0^\infty ids (is)^{(-n/2)}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp\left(-ism^2-\sigma(x,x^{'})/2is\right) \sum\limits_{j=0}^\infty \gamma_j(x,x^{'}) (is)^j \label{h46} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \Delta (x,x^{'})=-g(x)^{-1/2}det[\partial_\alpha \partial_\beta \sigma(x,x^{'})]g(x)^{-1/2} \label{h47} \end{eqnarray} \vspace{-6mm} \newline is the Van Vleck determinant (in normal coordinates about $x^{'}$ this determinant is reduced to $g^{-1/2}(y)$ (\ref{h33})). \section {Connection between the two methods} \vspace{1mm} Let us put \newline \vspace{-6mm} \begin{eqnarray} (k^2+m^2)^{-1}=\int \limits_0 ^\infty ids \exp[-is(k^2+m^2)] \label{h48} \end{eqnarray} \vspace{-6mm} \newline Then integration over the momentum in (\ref{h33}) leads to \newline \vspace{-6mm} \begin{eqnarray} \int \frac{d^n k}{(2 \pi)^n} \exp[-is(k^2+m^2)+iky] =i(4 \pi is)^{n/2}\exp(-ism^2-\sigma/2is) \label{h49} \end{eqnarray} \vspace{-6mm} \newline The resulting equation for the Green's function will be \newline \vspace{-6mm} \begin{eqnarray} G(y,x^{'})=\frac{i}{(4 \pi)^{n/2}}g(y)^{-1/2}\int \limits_0 ^\infty \frac{ds}{(is)^{n/2}}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp\left[ -is(k^2+m^2)+iky \right] F (x^{'},y,is) \label{h50} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} F (x^{'},y,is)=\sum\limits_{j=0}^2 \gamma_j (x^{'},y)(is)^j \nonumber \end{eqnarray} \vspace{-6mm} \newline Comparision with the expression of the Green's function in the form of (\ref{h46}) gives $\Delta(x,x^{'})=g^{-1/2}(y)$, and \newline \vspace{-6mm} \begin{eqnarray} \gamma_j (x^{'},y)=\gamma_j (x,x^{'}) \label{h51} \end{eqnarray} \vspace{-6mm} \newline In this chapter we considered two methods for computing the Green's function of a scalar field in curved space-time and we prepared the basis for future finite temperature calculations. \chapter{QUANTUM FIELDS} \centerline{\Large \bf IN CURVED SPACE-TIME} \vspace{24pt} In this chapter we introduce the basic formalism of quantum fields in curved space-time which was studied in works of DeWitt, Fulling, Bunch et al. \cite{dt2}, \\\cite{key17}, \cite{pr1}, \cite{key16}. \section{Lorentz group and quantum fields} \vspace{1mm} Field theory in Minkowski space-time has been well studied from the group point of view by many authors \cite{key3},\cite{{ramond1}}. Here we will consider the connection of fields of different spin with the Lorentz group. In flat space the spin of the field is classified according to the field's properties under infinitesimal Lorentz transformations \newline \vspace{-6mm} \begin{eqnarray} \bar{x}^\alpha=\Lambda^\alpha_\beta x^\beta =(\delta ^\alpha_\beta+\omega ^\alpha_\beta)x^\beta \label{g1} \end{eqnarray} \vspace{-6mm} \newline with \newline \vspace{-6mm} \begin{eqnarray} \omega _{\alpha\beta}=\omega_{\beta \alpha} \nonumber \end{eqnarray} \vspace{-6mm} \newline which preserve the length of the coordinate vector $x^2$. Under Lorentz transformations the general multicomponent field \newline \vspace{-6mm} \begin{eqnarray} F^{\alpha \beta...\lambda} \nonumber \end{eqnarray} \vspace{-6mm} \newline transforms according to \newline \vspace{-6mm} \begin{eqnarray} F^{\alpha\beta...\lambda} ~\mathop \to \limits^\Lambda~ [D(\Lambda)]^{\alpha\beta...\lambda}_{\alpha^{'}\beta^{'}...\lambda^{'}} F^{\alpha^{'}\beta^{'}...\lambda^{'}} \label{g2} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} [D(\Lambda)]=1+(1/2)\omega^{\alpha\beta}\Sigma_{\alpha\beta}.\label{g3} \end{eqnarray} \vspace{-6mm} \newline In order for the Lorentz transformations to form a group, the antisymmetric $\Sigma_{\alpha\beta}$ group generators are constrained to satisfy \newline \vspace{-6mm} \begin{eqnarray} \left[ {\Sigma _{\alpha \beta },\Sigma _{\gamma \delta }} \right] =\eta _{\gamma \beta }\Sigma _{\alpha \delta } -\eta _{\alpha \gamma }\Sigma _{\beta \delta } +\eta _{\delta \beta }\Sigma _{\gamma \alpha } -\eta _{\delta \alpha }\Sigma _{\gamma \beta } \label{g4} \end{eqnarray} \vspace{-6mm} \newline We may write down $\Sigma _{\alpha \beta }$ for different types of fields: 1) It is easy to see that the scalar field has the following form of $\Sigma$ \newline \vspace{-6mm} \begin{eqnarray} \Sigma _{\alpha \beta }=0 \label{g5} \end{eqnarray} \vspace{-6mm} \newline 2) A vector field $F^\alpha$ transforms as \newline \vspace{-6mm} \begin{eqnarray} F^{\alpha} ~\mathop \to \limits^\Lambda~ \Lambda^{\alpha}_{\alpha^{'}}F^{\alpha^{'}} \label{g6} \end{eqnarray} \vspace{-6mm} \newline so from (\ref{g3}) and (\ref{g1}) we get an expression for $\Sigma$ in vector case in the form \newline \vspace{-6mm} \begin{eqnarray} {[\Sigma_{\alpha\beta}]^\gamma}_\delta= \delta _\alpha ^\gamma \eta _{\beta \delta } -\delta _\beta ^\gamma \eta _{\alpha \delta }.\label{g7} \end{eqnarray} \vspace{-6mm} \newline 3) For a spinor field it may be written as \newline \vspace{-6mm} \begin{eqnarray} {[\Sigma_{\alpha\beta}]^\gamma}_\delta =(1/ 4){[\gamma _\alpha ,\gamma _\beta]^\gamma}_\delta, \label{g8} \end{eqnarray} \vspace{-6mm} \newline where $\gamma$ are Dirac matrices. Now we can generalize this concept to curved space-time. \section{Fields in curved space-time} \vspace{1mm} Let us consider general coordinate transformations. We will define a general coordinate transformation as an arbitrary reparametrization of the coordinate system: \newline \vspace{-6mm} \begin{eqnarray} {x^{'}}^\mu={x^{'}}^\mu (x^\nu) \label{g9} \end{eqnarray} \vspace{-6mm} \newline Unlike Lorentz transformations, which are global space-time transformations, general coordinate transformations are local. Under reparametrizations, a scalar field transforms simply as follows: \newline \vspace{-6mm} \begin{eqnarray} {\varphi^{'}}({x^{'}})=\varphi(x), \label{g10} \end{eqnarray} \vspace{-6mm} \newline and vectors $\partial_\mu$ and $dx^\mu$ as: \newline \vspace{-6mm} \begin{eqnarray} {\partial^{'}}_\mu=\frac{\partial x^\nu}{\partial{x^{'}}^\mu}\partial_\nu \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} d{x^{'}}^\mu=\frac{\partial {x^{'}}^\mu}{\partial x^\nu}dx^\nu \label{g11} \end{eqnarray} \vspace{-6mm} \newline As we can see, the transformation properties of (\ref{g10}) and (\ref{g11}) are based on the set of arbitrary real $(4 \times 4)$ matrices of the group $GL(4)$. Now we can give the abstract definition of covariant and contravariant tensors and write the general form of transformation \newline \vspace{-6mm} \begin{eqnarray} {A^{'}}^{\mu_1...}_{\nu_1...}(x^{'}) =\frac{\partial {x^{'}}^{\mu_1}}{\partial x^{\xi_1}}... \frac{\partial x^{\lambda_1}}{\partial {x^{'}}^{\nu_1}}... A_{\lambda_1...}^{\xi_1...}(x) \label{g12} \end{eqnarray} \vspace{-6mm} \newline Let us introduce a metric tensor $g_{\mu\nu}$ which allows us to write the four interval \newline \vspace{-6mm} \begin{eqnarray} ds^2=g_{\mu\nu}dx^\mu dx^\nu \label{g13} \end{eqnarray} \vspace{-6mm} \newline The tensor $g_{\mu\nu}$ transforms under the general coordinate transformation as genuine tensor. The derivative of the scalar field ${\partial}_\mu\varphi(x)$ is a genuine tensor, but the derivative of a vector is not. To create a vector from ${\partial}_\mu A_\nu (x)$ and $({\partial}_\mu A^\nu (x))$ we introduce new fields, connections, that absorb unwanted terms. The covariant derivatives are written then as \newline \vspace{-6mm} \begin{eqnarray} \nabla_\mu A_\nu=\partial_\mu A_\nu +\Gamma^\lambda _{\mu\nu} A_\lambda \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \nabla_\mu A^\nu=\partial_\mu A_\nu -\Gamma^\nu _{\mu\lambda} A^\lambda \label{g14} \end{eqnarray} \vspace{-6mm} \newline From the restriction \newline \vspace{-6mm} \begin{eqnarray} \nabla g_{\mu\nu}=0 \label{g15} \end{eqnarray} \vspace{-6mm} \newline we get the expression for connection $\Gamma ^\lambda _{\mu \nu }$: \newline \vspace{-6mm} \begin{eqnarray} \Gamma ^\lambda _{\mu \nu }=(1/ 2)g^{\lambda \sigma } (\partial _\mu g_{\nu \sigma } +\partial _\nu g_{\mu \sigma } -\partial _\sigma g_{\mu \nu }) \label{g16} \end{eqnarray} \vspace{-6mm} \newline which are named the Christoffel symbols. Another very important object in curved space-time is the Riemann curvature tensor $R^\xi _{\mu \nu \lambda}$, which arises from the commutator of the covariant derivatives \newline \vspace{-6mm} \begin{eqnarray} [\nabla_\mu,\nabla_\nu] A^\lambda = R^\xi _{\mu \nu \lambda} A_\xi \label{icor1} \end{eqnarray} \vspace{-6mm} \newline The Riemann tensor is written in the form \newline \vspace{-6mm} \begin{eqnarray} R^\xi _{\mu \nu \lambda }=\partial _\lambda \Gamma ^\xi _{\nu \mu } -\partial _\nu \Gamma ^\xi _{\mu \lambda } +\Gamma ^\xi _{\sigma \nu }\Gamma ^\sigma _{\mu \lambda } -\Gamma ^\xi _{\lambda \sigma }\Gamma ^\sigma _{\mu \nu } \label{g17} \end{eqnarray} \vspace{-6mm} \newline Contracting the indicies of the Riemann tensor one can get the Ricci curvature tensor \newline \vspace{-6mm} \begin{eqnarray} R_{\mu \nu}=R^\lambda _{\mu \nu \lambda } \label{g18} \end{eqnarray} \vspace{-6mm} \newline and scalar curvature \newline \vspace{-6mm} \begin{eqnarray} R=g^{\mu\nu}R_{\mu \nu} \label{g19} \end{eqnarray} \vspace{-6mm} \newline The transformation properties of the volume element and the square root of the metric tensor are \newline \vspace{-6mm} \begin{eqnarray} d^4 x^{'}=Det \left(\frac{\partial {x^{'} }^\mu}{\partial x^\nu}\right)d^4x \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \sqrt{g^{'}(x^{'})}=Det \left(\frac{\partial x^\mu}{\partial {x^{'}}^\nu}\right) \sqrt{g(x)} \label{g20} \end{eqnarray} \vspace{-6mm} \newline From (\ref{g20}) we get that the product of these two is invariant \newline \vspace{-6mm} \begin{eqnarray} \sqrt{g(x)}d^4 x=inv. \label{g21} \end{eqnarray} \vspace{-6mm} \newline Now we can construct the Einstein-Hilbert action as an invariant of the form \newline \vspace{-6mm} \begin{eqnarray} S_g=-(1/2k^2)\int d^4 x\sqrt{g(x)}R, \label{g22} \end{eqnarray} \vspace{-6mm} \newline and the action for a scalar field \newline \vspace{-6mm} \begin{eqnarray} S_\varphi=-(1/2)\int \sqrt{g(x)} \left(g^{\mu\nu}\partial_\mu \varphi\partial_\nu\varphi +m^2\varphi^2 \right), \label{g23} \end{eqnarray} \vspace{-6mm} \newline where the scalar matter couples to gravity via the interaction \newline \vspace{-6mm} \begin{eqnarray} \sqrt{g(x)}g^{\mu\nu}\partial_\mu \varphi\partial_\nu\varphi \sim g^{\mu\nu}T_{\mu\nu}. \nonumber \end{eqnarray} \vspace{-6mm} \newline The coupling of the gravitational field to the vector field is also straightforward \newline \vspace{-6mm} \begin{eqnarray} S_v=-(1/4)\int \sqrt{g(x)}g^{\mu\nu}g^{\lambda \xi} F^a_{\mu\lambda}F^a_{\nu\xi} \label{g24} \end{eqnarray} \vspace{-6mm} \newline For the gauge-fixing term the generalization to curved space-time is expressed by \newline \vspace{-6mm} \begin{eqnarray} S_{gf}=-(1/2)\alpha^{-1} \int d^4x\sqrt{g(x)} \left(\nabla_\mu A^\mu \right)^2 \label{g25} \end{eqnarray} \vspace{-6mm} \newline For the ghost term it is of the form: \newline \vspace{-6mm} \begin{eqnarray} S_{gh}= \int d^4x\sqrt{g(x)} \left(-\partial_\mu c^\alpha \partial^\mu \bar{c}^\alpha\right) \label{g26} \end{eqnarray} \vspace{-6mm} \newline However, the coupling of gravity to spinor fields leads to a difficulty, because there are no finite dimensional spinorial representations of $GL(4)$. This prevents a naive incorporation of spinors into general relativity. The method we may use for these constructions involves vierbein formalism. \section{Spinors in General relativity} \vspace{1mm} The vierbein (tetrad) formalism utilizes the fact that we can construct a flat tangent space to the curved manifold and introduce the spinorial representation of the Lorentz group in each point of the manifold in this tangent space. Spinors can then be defined at any point on the curved manifold only if they transform within the flat tangent space. Let us erect normal coordinates ${y^\alpha_{(X)}}$ at each point $X$ of the space-time manifold $M$. To preserve the connection with the previous sections we will label the flat tangent indices with letters $\alpha$, $\beta$, $\gamma$, $\delta$,... from the beginning of the Greek alphabet, as we did it before, and general coordinate transformation indices with letters $\lambda$, $\mu$, $\nu$, $\xi$,... from the end of the Greek alphabet. Introduce the vierbein as the mixed tensor (matrix) \newline \vspace{-6mm} \begin{eqnarray} h^\alpha_\mu(X)=\left(\frac{\partial y^\alpha_{(X)}}{\partial x^\mu}\right)_{x=X} ~~~\alpha=0,1,2,3 \label{g27} \end{eqnarray} \vspace{-6mm} \newline Note that the label $\alpha$ refers to the local inertial frame associated with normal coordinates $y^\alpha_{(X)}$ at the point $X$, while $\mu$ is associated with the general coordinate system $\{x^\mu \}$. The inverse of this matrix is given by $h_\alpha^\mu(X)$. \newline \vspace{-6mm} \begin{eqnarray} h^\alpha_\mu h_\beta^\mu=\delta^{ab} \label{g28} \end{eqnarray} \vspace{-6mm} \newline The vierbein can be viewed as the "square root" of the metric tensor $g_{\mu\nu}$ : \newline \vspace{-6mm} \begin{eqnarray} g_{\mu\nu}=h^\alpha_\mu h^\beta_\nu \eta_{\alpha\beta} \label{g29} \end{eqnarray} \vspace{-6mm} \newline For general coordinate transformations $x^\mu=x^\mu (x^\mu{'})$ we can consider the effect of changing the $x^\mu$ while leaving the $y^\alpha_{(X)}$ fixed. Then the verbein transforms as \newline \vspace{-6mm} \begin{eqnarray} h^\alpha_\mu \to{h^{'}}^\alpha_\mu= \frac{\partial {x^{'}}^\mu}{\partial x^\mu} h^\alpha_\mu \label{g30} \end{eqnarray} \vspace{-6mm} \newline We also can transform the $y^\alpha_{(X)}$ arbitrarily at each point $X$ \newline \vspace{-6mm} \begin{eqnarray} y^\alpha_{(X)} \to {y^{'}}^\alpha_{(X)}= {\Lambda(X)}^\alpha_\beta y^\beta_{(X)} \label{g31} \end{eqnarray} \vspace{-6mm} \newline In this case $h^\alpha_\mu (X)$ transforms as a Lorentz covariant vector \newline \vspace{-6mm} \begin{eqnarray} h^\alpha_\mu (X) \to {h^{'}}^\alpha_\mu (X)= {\Lambda(X)}^\alpha_\beta h^\beta_\mu (X) \label{g32} \end{eqnarray} \vspace{-6mm} \newline which leaves the metric (\ref{g29}) invariant. If a covariant vector $A_\mu$ is contracted into $h_\alpha^\mu$, the resulting object \newline \vspace{-6mm} \begin{eqnarray} A_\alpha=h_\alpha^\mu A_\mu \label{g33} \end{eqnarray} \vspace{-6mm} \newline transforms as a collection of four scalars under a general coordinate transformations, while under local Lorentz transformations (\ref{g1}) it behaves as a vector. Thus, by use of tetrads, one can convert general tensors into local, Lorentz-transforming tensors, shifting the additional space-time dependence into the tetrads. Now we may construct the generally covariant Dirac equation. We introduce a spinor $\psi(x)$ that is defined as a scalar under general coordinate transformations and an ordinary spinor under flat tangent space Lorentz transformation: Coordinate transformations: $\psi \to \psi$ Lorentz transformations: $\psi \to D[\Lambda(x)]\psi$ It is important to note that we have introduced local Lorentz transformations in flat tangent space, so $\omega_{\alpha\beta}$ is a function of the space-time. This means that the derivative of the spinor is no longer a genuine tensor. Therefore we must introduce a connection field $\omega_\mu^{\alpha\beta}$ that allows us to gauge the Lorentz group. The covariant derivative for gauging the Lorentz group may be written as \newline \vspace{-6mm} \begin{eqnarray} \nabla_\mu\psi=(\partial_\mu+(1/2)\Sigma_{\alpha\beta} \omega_\mu^{\alpha\beta})\psi \label{g34} \end{eqnarray} \vspace{-6mm} \newline Let $\{\gamma^\alpha\}$ be a set of Dirac matrices with \newline \vspace{-6mm} \begin{eqnarray} \{\gamma_\alpha,\gamma_\beta\}=2\eta_{\alpha\beta} \label{g35} \end{eqnarray} \vspace{-6mm} \newline in the tangential space-time. The Dirac matrices $\gamma^\alpha$ can be contracted with vierbeins: \newline \vspace{-6mm} \begin{eqnarray} h_\alpha^\mu(x)\gamma_\alpha=\gamma_\mu(x) \label{g36} \end{eqnarray} \vspace{-6mm} \newline Then \newline \vspace{-6mm} \begin{eqnarray} \{\gamma_\mu(x),\gamma_\nu(x)\}=2g_{\mu\nu}(x) \label{g37} \end{eqnarray} \vspace{-6mm} \newline In the result (\ref{g32}) and (\ref{g34}) the generally covariant Dirac equation is given by \newline \vspace{-6mm} \begin{eqnarray} (i\gamma^\mu(x)\nabla_{\mu,x}+m)\psi(x)=0 \label{g38} \end{eqnarray} \vspace{-6mm} \newline and hence the action for Dirac particle interacting with gravity is given by: \newline \vspace{-6mm} \begin{eqnarray} L=\sqrt{g}\bar{\psi}(x)(i\gamma^\mu(x)\nabla_{\mu,x}+m)\psi(x) \label{g39} \end{eqnarray} \vspace{-6mm} \newline where $\sqrt{g}=Det(h^\alpha _\mu)$. We can construct a new, alternative, version of the curvature tensor by taking the commutator of two covariant derivatives: \newline \vspace{-6mm} \begin{eqnarray} \left[ {\nabla _\mu ,\nabla _\nu } \right]\psi = {1 \over 2}R^{\alpha \beta }_{\mu \nu }\Sigma _{\alpha \beta }\psi \label{g40} \end{eqnarray} \vspace{-6mm} \newline Written out, this curvature tensor is generally covariant in $\mu$, $\nu$, but flat in $\alpha$, $\beta$: \newline \vspace{-6mm} \begin{eqnarray} R_{\mu \nu }^{\alpha \beta }=\partial _\mu \omega _\nu ^{\alpha \beta } -\partial _\nu \omega _\mu ^{\alpha \beta } +\omega _{\mu \lambda }^\alpha \omega _\nu ^{\beta \lambda } -\omega _{\nu \lambda }^\alpha \omega _\mu ^{\beta \lambda } \label{g41} \end{eqnarray} \vspace{-6mm} \newline At this point, the spinor connection $\omega _{\mu \lambda }^\alpha $ is still an independent field. Covariant derivative of the object with two different kinds of indicies is written as: \newline \vspace{-6mm} \begin{eqnarray} \nabla _ \mu A^\alpha _\mu = \partial _\mu A^\alpha _\nu+ \Gamma^\xi_{\mu \nu} A^\alpha_ \xi +{\omega ^\alpha}_{\mu \beta} A^ \beta _\nu \label{g42} \end{eqnarray} \vspace{-6mm} \newline The external constraint \newline \vspace{-6mm} \begin{eqnarray} \nabla _ \mu h^\alpha _\mu = \partial _\mu h^\alpha _\nu+ \Gamma^\xi_{\mu \nu} h^\alpha_ \xi +{\omega ^\alpha}_{\mu \beta} h^ \beta _\nu=0 \label{g43} \end{eqnarray} \vspace{-6mm} \newline helps to express the spin connection through veirbeins \newline \vspace{-6mm} \begin{eqnarray} \omega _\mu ^{\alpha \beta }={1 \over 2}h^{\alpha \nu } (\partial _\mu h_\nu ^\beta -\partial _\nu h_\mu ^\beta ) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{1 \over 4}h^{\alpha \nu }h^{\beta \xi } (\partial _\xi h_{\nu \gamma }-\partial _\nu h_{\xi \nu })h_\mu ^\nu -(\alpha \leftrightarrow \beta ) \label{g44} \end{eqnarray} \vspace{-6mm} \newline After this preliminary work we can study properties of bosonic and fermionic fields in an external gravitational field. \chapter{ FINITE TEMPERATURE BOSONS} \centerline{\Large \bf IN CURVED SPACE-TIME} \vspace{24pt} In the previous section we developed a mathematical formalism which is a convenient tool for the description of thermal Bose gas in curved space-time. In this section we can consider the ensemble of bosons interacting with gravity at finite temperature. Let a total system ("matter and gravity") be described by the action \newline \vspace{-6mm} \begin{eqnarray} S_{tot}=S_g+S_m \label{i1} \end{eqnarray} \vspace{-6mm} \newline The gravitational action is \newline \vspace{-6mm} \begin{eqnarray} S_g=\int d^4x\sqrt{g(x)}L_g \label{i2} \end{eqnarray} \vspace{-6mm} \newline with Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L_g={1 \over {16\pi G_0}}(R-2\Lambda _0) +\alpha _0 R^2+\beta _0R_{\alpha \beta }R^{\alpha \beta } +\gamma _0 R_{\alpha \beta \chi \delta }R^{\alpha \beta \chi \delta } \label{i3} \end{eqnarray} \vspace{-6mm} \newline and the action for a matter field is \newline \vspace{-6mm} \begin{eqnarray} S_m=\int d^4x\sqrt{g(x)}L_{eff,m} \label{i4} \end{eqnarray} \vspace{-6mm} \newline To write the action for the matter field we use a generating functional $Z[J]$ \newline \vspace{-6mm} \begin{eqnarray} Z[0]=\int D\varphi \exp\left( -(i/2)\int d^4x \sqrt{g(x)} \varphi(x)(-\Box_x+m^2+\xi R)\varphi(x)\right) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \propto DetG(x,x^{'})^{1/2}\exp\left(-(i/2)(J(x),G(x,y)J(y))\right) \label{i5} \end{eqnarray} \vspace{-6mm} \newline Then the functional $W[0]=-i\ln Z[0]$ will be \newline \vspace{-6mm} \begin{eqnarray} W[0]=-(i/2)\ln Det G(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-(i/2)\int\limits_0^\infty ids(is)^{-1}\mbox{tr}f(x,x^{'},is)\exp (-im^2 s) \label{i6} \end{eqnarray} \vspace{-6mm} \newline We may connect the Green's function and heat kernel with the equation \newline \vspace{-6mm} \begin{eqnarray} G_{SD}(x,x^{'})= \int\limits _0 ^\infty ids f( x, x^{'},is) \exp (-im^2 s)\label{i7} \end{eqnarray} \vspace{-6mm} \newline and write (\ref{i6}) in the following form \newline \vspace{-6mm} \begin{eqnarray} W[0]=-(i/2)\int d^4x \sqrt{g(x)} \int\limits_{m^2}^\infty dm^2 \mbox{tr} G_{SD}(x,x^{'}) \label{i8} \end{eqnarray} \vspace{-6mm} \newline From the equation (\ref{i8}) we find an important expression for the effective Lagrangian of matter field \newline \vspace{-6mm} \begin{eqnarray} L_{eff,m}=(-i/2)\int\limits d^4x\sqrt{g(x)} \int \limits_{m^2}^\infty dm^2\mbox{tr}G(x,x^{'}) \label{i9} \end{eqnarray} \vspace{-6mm} \newline Then the effective action will be: \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=\int d^4x \sqrt{g(x)}L_{eff}(x) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int d^4x \sqrt{g(x)}\left(L_g (x)+ (-i/2)\int \limits_{m^2}^\infty dm^2 \mbox{tr} G(x,x^{'})\right) \label{i10} \end{eqnarray} \vspace{-6mm} \newline In order to apply the usual formalism of finite temperature quantum field theory, we will assume that the space-time manifold $M_{4)}$ is a static manifold with topology $S^{1}\times M_{3)}$ where $S^1$ refers to time coordinate and $M_{3)}$ is the spatial, three dimensional section of $M_{4)}$. We will choose $M_{3)}$ without boundaries \cite{ken1}, then no surface terms will appar in the induced action. The heat kernel may be expressed as the sum of zero-temperature images \\ \cite{i17,dow1,dow2} \newline \vspace{-6mm} \begin{eqnarray} f(x,x,is)=\sum\limits_{n=-\infty}^{\infty} \frac{\exp[-\beta^2 n^2/4is]}{(4\pi is)^{1/2} } f_{3)}(\tilde {x},\tilde{x},is) \label{i11} \end{eqnarray} \vspace{-6mm} \newline where the sum goes from periodic restrictions and $f_{3)}(\tilde {x},\tilde{x},is)$ is the solution of the three-dimensional equation \newline \vspace{-6mm} \begin{eqnarray} \left(i\frac{\partial}{\partial s} -{\tilde \Box }_x+\xi R\right)f_{3)}(\tilde {x},\tilde{x},is)=0 \label{i12} \end{eqnarray} \vspace{-6mm} \newline This solution is written in the form of a series \newline \vspace{-6mm} \begin{eqnarray} f_{3)}(\tilde {x},\tilde{x},is)=\sum \limits_{n=0}^\infty \gamma_j(R)(is)^j \label{i13} \end{eqnarray} \vspace{-6mm} \newline Then from the equation (\ref{i7}) in the limit of coincidence $(x \to x^{'})$ \newline \vspace{-6mm} \begin{eqnarray} \mathop {\lim}\limits_{x \to x^{'}} G(x,x^{'})= \mathop {\lim}\limits_{x \to x^{'}} \int\limits _0 ^\infty ids f( x, x^{'},is)\exp\{-im^2 s\}. \label{i14} \end{eqnarray} \vspace{-6mm} \newline we get the finite temperature Green's function in the Schwinger-DeWitt representation: \newline \vspace{-6mm} \begin{eqnarray} \mathop {\lim}\limits_{x \to x^{'}} {G^ \beta}_{SD}(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{i}{(4\pi)^{3/2}}\sum_{n=-\infty}^\infty \sum_{j=0}^\infty \gamma_j(x^{'})\int_0^\infty ids (is)^{j-3/2} \exp(-ism^2-n^2\beta^2/4is) \label{i15} \end{eqnarray} \vspace{-6mm} \newline Selecting the temperature independent part $(n=0)$ we find \newline \vspace{-6mm} \begin{eqnarray} \mathop {\lim}\limits_{x\to x^{'}}{G^ \beta}_{SD} (x,x^{'})= G_{SD}(x^{'},x^{'})+G_{x^{'}}(\beta) \label{i16} \end{eqnarray} \vspace{-6mm} \newline where the finite temperature contribution $G_{x^{'}}(\beta)$ is \newline \vspace{-6mm} \begin{eqnarray} G_{x^{'}}(\beta)=\frac{i}{(4 \pi)^2}2\sum\limits_{j=0}^\infty \sum \limits_{n=1} ^\infty \gamma_j(x^{'})\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \int _0^\infty ids (is)^{j-2} \exp(-ism^2-n^2\beta^2/4is) \label{i17} \end{eqnarray} \vspace{-6mm} \newline and $G_{SD}(x^{'},x^{'})$ is the limit $(x \to x^{'})$ of the Green's function in the Schwinger-DeWitt representation (\ref{h46}). Summation in (\ref{i17}) may be done with (\ref{d25}) and (\ref{d29}). In the result we find \newline \vspace{-6mm} \begin{eqnarray} G_{x^{'}}(\beta)=\frac{i}{(2\pi)^2} \sum \limits_{n=1} ^\infty \sum \limits_{j=1} ^\infty \gamma_j(x^{'})(\beta n/2m)^{j-3}K_{j-3}(\beta m n) \label{i18} \end{eqnarray} \vspace{-6mm} \newline Total Lagrangian may be written in the form; \newline \vspace{-6mm} \begin{eqnarray} L_{eff}(\beta)=\left(L_g (x)+ (-i/2)\int \limits_{m^2}^\infty dm^2\mbox{tr} G_{SD} (x,x^{'})\right)+ (-i/2)\int \limits_{m^2}^\infty dm^2 G_{x^{'}}(\beta) \label{i19} \end{eqnarray} \vspace{-6mm} \newline The first two terms give the same geometric structure, and after renormalizations we will have the Lagrangian $\tilde{L}_g$, the third one is temperature contribution $f(\beta)$. \newline \vspace{-6mm} \begin{eqnarray} L_{eff}(\beta)=\tilde{L}_g-f(\beta) \label{i20} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \tilde{L}_g=\left(L_g (x)+ (-i/2)\int \limits_{m^2}^\infty dm^2 \mbox{tr} G_{SD}(x,x^{'})\right) \label{i21} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=(i/2)\int \limits_{m^2}^\infty dm^2G_{x^{'}}(\beta) \label{i22} \end{eqnarray} \vspace{-6mm} \newline The last expression can be written as series \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=\sum\limits_{j=0}^3\gamma_j(R)b_j(\beta m) \label{i23} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} b_0 (\beta m)=-\frac{m^2}{2\pi^2 \beta^2}\sum\limits_{n=1}^\infty (1/n^2)K_2 (\beta m n) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} b_1 (\beta m)=-\frac{2m}{4\pi^2 \beta}\sum\limits_{n=1}^\infty (1/n)K_1 (\beta m n) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} b_2 (\beta m)=-\frac{2}{8\pi^2 }\sum\limits_{n=1}^\infty K_0 (\beta m n) \label{i24} \end{eqnarray} \vspace{-6mm} \newline Using the results of Appendix: (\ref{bb10}), (\ref{bb12}), (\ref{bb14}), we may rewrite (\ref{i23}) in the form of series [Kulikov \& Pronin 1993] \newline \vspace{-6mm} \begin{eqnarray} f(\beta)=b_0(\beta m)+{\gamma }_1(R)b_1(\beta m) +{\gamma }_2(R^2)b_2(\beta m) \label{i25} \end{eqnarray} \vspace{-6mm} \newline where ${\gamma}_j(R)$ are (\ref{h32}), coefficient $b_0(\beta m)$ is \newline \vspace{-6mm} \begin{eqnarray} b_0=(1/\beta)\int \frac{d^3k}{(2\pi)^3} \ln (1-\exp(-\beta \epsilon)); \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} b_j(\beta m)=\left(-\frac{\partial}{\partial m^2}\right)^j b_0(\beta m) \label{i26} \end{eqnarray} \vspace{-6mm} \newline The expression $b_0(\beta m)$ in (\ref{i25}) is the density of Helmholtz free energy in flat space and the following ones are created by corrections which are connected with the interaction of the heat bosons with gravity, so the equation (\ref{i23}) describes the Helmholtz free energy of a Bose gas in curved space time. \chapter{FERMI FIELDS} \centerline{\Large \bf IN CURVED SPACE-TIME } \vspace{24pt} \section{Momentum-space representation} \vspace{-4mm} \hspace{25mm}{\Large \bf for the Green's function of a fermion} \vspace{1mm} Now we will get the Green's function of fermions in curved space-time to apply the above results for computation of the effective action of the system "matter field $+$ gravitational field". As we know already, the fermionic action with Lagrangian (\ref{g39}) is written as \newline \vspace{-6mm} \begin{eqnarray} S_\psi=(i/2)\int d^4x \sqrt{g(x)}\bar{\psi}(x) \left(i\gamma^\mu \nabla_{\mu,x}+m\right)\psi(x) \label{fer1} \end{eqnarray} \vspace{-6mm} \newline The generating functional then is \newline \vspace{-6mm} \begin{eqnarray} Z[\bar{\eta},\eta]=\int D \bar{\psi} D \psi \exp \left[ iS_{\psi} +i\left(\bar{\eta},\psi\right)+i\left(\bar{\psi},\eta \right)\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \propto (Det G_F)^{-1/2} \exp\left[-i\left(\bar{\eta}(x),S_F(x,y)\eta \right)\right] \label{fer2} \end{eqnarray} \vspace{-6mm} \newline where the scalar product includes the square root of the metric. Connection between the bi-spinor $G_F$ and Fermionic Green's function $S_F$ is\\ \cite{key16}: \newline \vspace{-6mm} \begin{eqnarray} S_F(x,y)=(i\gamma^\mu D_\mu+m)G_F (x,y) \label{fer3} \end{eqnarray} \vspace{-6mm} \newline and the Green's function satisfies the equation \newline \vspace{-6mm} \begin{eqnarray} (i\gamma^\mu D_\mu +m)S_F (x,y)=-g(x)^{-1/2}\delta (x-y) \hat{1}\label{fer4} \end{eqnarray} \vspace{-6mm} \newline To solve this equation with momentum space methods we introduce Riemann normal coordinates \cite{petrov1} and write the spin connection in the form\\ \cite{panan1} \newline \vspace{-6mm} \begin{eqnarray} \Gamma_\mu(y)=(1/16)[\gamma_\alpha,\gamma_\beta]{R^{\alpha \beta}}_{\mu \nu}y^\nu +O(y^2) \label{fer5} \end{eqnarray} \vspace{-6mm} \newline and the vierbein field as \newline \vspace{-6mm} \begin{eqnarray} h^\alpha_\mu(y)=\delta^\alpha_\mu -(1/6)\eta^{\alpha \nu}R_{\mu \nu\beta\zeta}y^\beta y^\zeta +O(y^3) \label{fer6} \end{eqnarray} \vspace{-6mm} \newline The expression for $\gamma_\mu (x^{'})$ may be written as \newline \vspace{-6mm} \begin{eqnarray} \gamma_\mu(x^{'})=h^\alpha_\mu(y)\gamma_\alpha=\delta^\alpha_\mu\gamma_\alpha -(1/6)\eta^{\alpha \nu}R_{\mu \nu\beta\zeta}y^\beta y^\zeta\gamma_\alpha +O(y^3) \label{fer7} \end{eqnarray} \vspace{-6mm} \newline The spinor derivative appearing in Dirac's equation is written as \newline \vspace{-6mm} \begin{eqnarray} \gamma_\mu D_\mu=\gamma_\mu(x^{'})(\partial_\mu-\Gamma_\mu) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\gamma^\mu\partial_\mu+(1/6) {{{R^\mu}_\beta}^ \nu}_\zeta y^\beta y^\zeta \partial_\mu- \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(1/16)\gamma^\mu[\gamma_\alpha,\gamma_\beta] {R^{\alpha \beta}}_{\mu\nu}y^\nu \label{fer8} \end{eqnarray} \vspace{-6mm} \newline The Fourier transform of $S(x^{'},y)$ is \newline \vspace{-6mm} \begin{eqnarray} S(x^{'},y)=\int\frac{d^nk}{(2 \pi)^n}\exp(iky)S(k) \label{fer9} \end{eqnarray} \vspace{-6mm} \newline Let \newline \vspace{-6mm} \begin{eqnarray} S(k)=S_0(k)+S_1(k)+S_2(k)+... \label{fer10} \end{eqnarray} \vspace{-6mm} \newline be an asymptotic representation of $S(k)$ for large $k$. The values $S_i(k)$ are asymptotic variables of the order $k^{-(1+i)}$. They may be found with recursion procedure from the equation \newline \vspace{-6mm} \begin{eqnarray} \left[(i\gamma^\mu\partial_\mu+m)+(1/6) {{{R^\mu}_\beta}^ \nu}_\zeta {y^\beta}{ y^\zeta }\partial_\mu-\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.-(1/16)\gamma^\mu[\gamma_\alpha,\gamma_\beta] {R^{\alpha \beta}}_{\mu\nu}y^\nu+...\right] S(x^{'},y)=\delta(y) \label{fer11} \end{eqnarray} \vspace{-6mm} \newline In this case the momentum space representation of a propagator of a fermion will be \newline \vspace{-6mm} \begin{eqnarray} S(x^{'},y)=\int\frac{d^nk}{(2\pi)^n}\exp(iky) \left[\frac{(\gamma \cdot k+m)}{k^2+m^2} +(1/4)R\frac{(\gamma \cdot k+m)}{(k^2+m^2)^2}- \right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.-(2/3)R_{\alpha\beta}{ k^\alpha }{k^\beta} \frac{(\gamma \cdot k+m)}{(k^2+m^2)^3}+(i/8){R^{\alpha \beta}}_{\mu\nu} \frac{\gamma^\mu[\gamma_\alpha,\gamma_\beta] k^\nu}{(k^2+m^2)^2}+...\right] \label{fer12} \end{eqnarray} \vspace{-6mm} \newline The momentum space solution of the equation for the bi-spinor \newline \vspace{-6mm} \begin{eqnarray} \left(\Box_x+1/4 R-m^2\right)G_F(x,x^{'})= -g(x)^{-1/2}\delta(x-x^{'})\hat{1} \label{fer13} \end{eqnarray} \vspace{-6mm} \newline where $\Box_x=D^\mu_xD_{\mu,x}$ is the covariant d'Alembertian of spinor field, may be obtained with the same momentum space methods as in chapter VI \cite{bunch1}. The result can be written in the form of the following expression \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=G_F(x^{'},y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =g^{-1/2}(y)\int\frac{d^nk}{(2\pi)^n}\sum\limits_{j=0}^2 \hat{\alpha}_j(x^{'},y) \left(-\frac{\partial}{\partial m^2}\right)^j(k^2+m^2)^{-1} \label{fer14} \end{eqnarray} \vspace{-6mm} \newline where the geometrical coefficients $\hat{\alpha}_j(x^{'},y)$ in the limit of coincidence $(x \to x^{'})$ are \newline \vspace{-6mm} \begin{eqnarray} \hat{\alpha}_0(x^{'},y)=\hat{1}; \nonumber \end{eqnarray} \vspace{-6mm} \newline \newline \vspace{-6mm} \begin{eqnarray} \hat{\alpha}_1(x^{'},y)=(1/12)R\cdot \hat{1}; \nonumber \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \hat{\alpha}_2(x^{'},y)=\left(-(1/120){R_\mu}^{;\mu}+(1/288)R^2-\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.-(1/180)R_{\mu \nu}R^{\mu\nu} +(1/180)R_{\mu\nu\sigma\tau}R^{\mu\nu\sigma\tau}\right)\cdot\hat{1} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(1/48)\Sigma_{[\alpha,\beta]}\Sigma_{[\gamma,\delta]}R^{\alpha\beta\lambda\xi} {{{R\gamma}_\lambda}^\delta}_\xi \label{fer15} \end{eqnarray} \vspace{-6mm} \newline where $\Sigma_{[\alpha,\beta]}=(1/4)[\gamma_\alpha,\gamma_\beta]$ \section{The bi-spinor function in the } \vspace{-4mm} \hspace{27mm}{\Large \bf Schwinger-DeWitt representation} \vspace{1mm} In analogy with the scalar field we can rewrite equation (\ref{fer14}) in the Schwinger-DeWitt representation. Since \newline \vspace{-6mm} \begin{eqnarray} (k^2+m^2)^{-1} =\int \limits _0 ^\infty ids \exp\left[-is(k^2+m^2)\right] \label{fer16} \end{eqnarray} \vspace{-6mm} \newline the equation (\ref{fer14}) will be \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=\frac{i\Delta^{(1/2)}(x,x^{'})}{(4 \pi^{n/2})} \int \limits _0 ^\infty \frac{ids}{(is)^{n/2}} \exp\left[-is(k^2+m^2)\right] F(x,x^{'};s) \label{fer17} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} F(x,x^{'};s)=\sum \limits_{n=0}^\infty \hat{\alpha}_n(x,x^{'})(is)^n \label{fer18} \end{eqnarray} \vspace{-6mm} \newline and coefficients $\hat{\alpha}_n(x,x^{'})$ are defined by (\ref{fer15}) As in the scalar case the determinant $\Delta^{(1/2)}(x,x^{'})$ is defined by the equation \newline \vspace{-6mm} \begin{eqnarray} \Delta^{(1/2)}(x,x^{'})=g^{-1/2}(y) \label{fer19} \end{eqnarray} \vspace{-6mm} \newline \chapter{FINITE TEMPERATURE FERMIONS} \centerline{\Large \bf IN CURVED SPACE-TIME} \vspace{24pt} \section { The Helmholtz free energy of a Fermi gas} \vspace{-4mm} \hspace{22mm}{\Large \bf in curved space-time} \vspace{1mm} After we constructed the free energy for a thermal scalar field we can consider a thermal fermi field. Let the total Lagrangian of the system of "gravity $+$ fermionic matter" be \newline \vspace{-6mm} \begin{eqnarray} S_{tot}=S_g+S_m \label{j1} \end{eqnarray} \vspace{-6mm} \newline where $S_g$ is (\ref{i2}) and \newline \vspace{-6mm} \begin{eqnarray} S_m=\int d^4 x \sqrt{g(x)}L_{eff,\psi} \label{j2} \end{eqnarray} \vspace{-6mm} \newline To write the action for a spinor field we will use the functional method for calculation of the generating functional from (\ref{j2}). Making the same procedure as in (\ref{i6})-(\ref{i8}), write \newline \vspace{-6mm} \begin{eqnarray} W[0]=i\ln Z[0]=(i/2)\ln DetG_F(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int\limits_0^\infty ids (is)^{-1}\mbox{tr}\hat{f}(x,x^{'},is) \exp(-im^2s) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(i/2)\int d^4x \int_{m^2}^\infty dm^2 \mbox{tr}G_F(x,x^{'}) \label{j3} \end{eqnarray} \vspace{-6mm} \newline where the Green's function is expressed by \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=\int_0^\infty ids (is)^{-1}\hat{f}(x,x^{'},is)\exp(-im^2s) \label{j4} \end{eqnarray} \vspace{-6mm} \newline and $\hat{f}(x,x^{'},is)$ is the heat kernel. The kernel $\hat{f}(x,x^{'},is)$ is (in the limit $(x \to x^{'})$) the sum of zero-images \\ \cite{i16} antiperiodic in the imaginary time \newline \vspace{-6mm} \begin{eqnarray} \hat{f}(x,x^{'},is)=\sum\limits_{n=-\infty}^\infty \frac{\exp[-\beta^2(n-1/2)^2/4is]}{(4\pi is)^{1/2}} \hat{f}_{3)}(\tilde{x},\tilde{x},is) \label{j5} \end{eqnarray} \vspace{-6mm} \newline where an equation for $\hat{f}_{3)}(x,x^{'},is)$ is \newline \vspace{-6mm} \begin{eqnarray} \left( i\frac{\partial}{\partial s}-\tilde{\Box}_{3)} -(1/4)R\right)\hat{f}_{3)}(\tilde{x},\tilde{x},is)=0 \label{j6} \end{eqnarray} \vspace{-6mm} \newline and $\tilde{\Box}_{3)}$ is the covariant d'Alembertian on $M_{3)}$. The solution of (\ref{j6}) is the series \newline \vspace{-6mm} \begin{eqnarray} \hat{f}_{3)}(\tilde{x},\tilde{x},is) =\sum\limits_{j=0}^\infty\hat{\alpha}_j(x^{'})(is)^j \label{j7} \end{eqnarray} \vspace{-6mm} \newline where the coefficients $\hat{\alpha}_j(x^{'})$ are determined by (\ref{fer15}). From the equation (\ref{j4}) we get the finite temperature Green's function \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim}\limits_{x \to x^{'}}G^\beta_{F,SD}(x,x^{'}) =G_{F,SD}(x^{'},x^{'})+{G_F}(\beta) \label{j8} \end{eqnarray} \vspace{-6mm} \newline where $G_{F,SD}(x^{'},x^{'})$ is (\ref{fer17}) in the limit $(x=x^{'})$ and \newline \vspace{-6mm} \begin{eqnarray} {G_F}(\beta)=\frac{i}{(4\pi)^3/2}\sum\limits_{j=0}^\infty \sum\limits_{n=-\infty}^\infty \hat{\alpha}_j(R)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \int \limits_0^\infty ids(is)^{j-3/2} \exp\left[-ism^2-(\beta^2/4is)(n+1/2)^2\right] \label{j9} \end{eqnarray} \vspace{-6mm} \newline Summation with respect to (\ref{d25}) with $z=1/2$ gives \newline \vspace{-6mm} \begin{eqnarray} {G_F}(\beta)=\frac{i}{(4\pi)^{3/2}}\sum\limits_{j=0}^\infty \sum\limits_{n=1}^\infty \hat{\alpha}_j(R)\times \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times 2\int \limits_0^\infty ids(is)^{j-2}(-1)^n \exp\left[-ism^2-(n^2\beta^2/4is)\right] \label{j10} \end{eqnarray} \vspace{-6mm} \newline After integration over the proper time $(s)$ (\ref{d29}) we get the following expression for the finite temperature contribution in the Green's function of a fermion \newline \vspace{-6mm} \begin{eqnarray} {G_F}(\beta)=\frac{i}{(4\pi)^2}\sum\limits_{j=0}^\infty \sum\limits_{n=1}^\infty \hat{\alpha}_j(R)(-1)^j(\beta n/2m)^{j-3} K_{j-3}(\beta m n) \label{j11} \end{eqnarray} \vspace{-6mm} \newline The total action of the system will be \newline \vspace{-6mm} \begin{eqnarray} L_{eff}(\beta)=\tilde{ L}_g(R)-f_F(\beta,R) \label{j12} \end{eqnarray} \vspace{-6mm} \newline where $\tilde{ L}(R)_g $ is the temperature independent Lagrangian \newline \vspace{-6mm} \begin{eqnarray} \tilde{ L}(R)_g=L_g+(i/2)\mbox{tr} \int\limits_{m^2}^\infty dm^2 G_{F,SD}(x,x^{'}), \label{j13} \end{eqnarray} \vspace{-6mm} \newline and the finite temperature contribution is expressed in the form of a series: \newline \vspace{-6mm} \begin{eqnarray} f_F(\beta,R)=(-i/2) \mbox{tr} \int\limits_{m^2}^\infty dm^2{G_F}(\beta) =\sum\limits_{j=0}^\infty \alpha_j(R)f_j(\beta m) \label{j14} \end{eqnarray} \vspace{-6mm} \newline with coefficients \newline \vspace{-6mm} \begin{eqnarray} \alpha_j(R)=(1/2s)\mbox{tr}\hat{\alpha_j(R)}, \label{j15} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} f_0(\beta m)=\frac{m^2\cdot 2s}{2\pi^2 \beta^2} \sum\limits_{n=1}^\infty \frac{(-1)^n}{n^2} K_2(\beta m n) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} f_1(\beta m)=\frac{m\cdot 2s}{4\pi^2 \beta} \sum\limits_{n=1}^\infty \frac{(-1)^n}{n }K_1(\beta m n) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} f_2(\beta m)=\frac{ 2s}{8\pi^2} \sum\limits_{n=1}^\infty (-1)^n K_0(\beta m n) \label{j16} \end{eqnarray} \vspace{-6mm} \newline The finite, temperature dependent contribution $f(\beta,R)$ represents the density of Helmholtz free energy in curved space-time. Using the integral representation for the series of modified Bessel functions (\ref{aa10}),(\ref{aa12}),(\ref{aa14}) one can write it as [Kulikov \& Pronin 1995]: \newline \vspace{-6mm} \begin{eqnarray} f_F(\beta,R)=f_0(\beta m)+\alpha _1(R)f_1(\beta m)+\alpha _2(R)f_2(\beta m)+.\;.\;., \label{j17} \end{eqnarray} \vspace{-6mm} \newline where the first term is the standard form of the Helmholtz free energy in Euclidean space \newline \vspace{-6mm} \begin{eqnarray} f_0(\beta m)=-{2s \over \beta }\int {{{d^3k} \over {(2\pi )^3}}}\ln \left( {1+e^{-\beta \varepsilon }} \right) \label{j18} \end{eqnarray} \vspace{-6mm} \newline with energy of particle $\varepsilon =\sqrt {\vec{ k}^{2}+m^{2}}$. The factor $2s=4$ reflects the existence of the four degrees of freedom present in the fermion field: particles and antiparticles, spin up and spin down. The following terms are geometrical corrections of the Riemann space time structure with respect to the Euclidean one with temperature coefficients in the form \newline \vspace{-6mm} \begin{eqnarray} f_j(\beta m)=-{2s \over \beta }\int {{{d^3k} \over {(2\pi )^3}}}\left( {-{\partial \over {\partial m^2}}} \right)^j \ln \left( {1+e^{-\beta \varepsilon }} \right). \label{j19} \end{eqnarray} \vspace{-6mm} \newline The method developed above does not allow us to compute the density of the grand thermodynamical potential. Therefore, in the following calculations we will use the local momentum space formalism as the most convenient for the construction of local thermodynamics. \chapter{TWO-LOOP RENORMALIZATIONS IN} \centerline{\Large \bf $\lambda \phi^4$ MODEL} \vspace{24pt} We will start with the problem of renormalization procedure for the self-interacting scalar model in two-loop approximation of perturbation theory. In our calculations we will use the method of counterterms. Let the Lagrangian of the self-interacting $\lambda\varphi^{4}$ model be \newline \vspace{-6mm} \begin{eqnarray} L={1 \over 2}\left( {\partial \varphi } \right)^2-{1 \over 2} m_B^2\varphi ^2-{{\lambda _B} \over {4!}}\varphi ^4 \label{new1} \end{eqnarray} \vspace{-6mm} \newline We may assume that (\ref{new1}) is \newline \vspace{-6mm} \begin{eqnarray} L=L_0+L_I \label{new2} \end{eqnarray} \vspace{-6mm} \newline where the Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L_0={1 \over 2}\left( {\partial \varphi } \right)^2-{1 \over 2} m_R^2\varphi ^2 \label{new3} \end{eqnarray} \vspace{-6mm} \newline describes propagation of free particles, and the Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L_I=-{1 \over 2}\delta m^2\varphi ^2-{{\lambda _B} \over {4!}} \varphi ^4 \label{new4} \end{eqnarray} \vspace{-6mm} \newline describes interaction. Let the coefficient $\delta m^2$ be the difference of the form \newline \vspace{-6mm} \begin{eqnarray} \delta m^2=m_B^2-m_R^2 \nonumber \end{eqnarray} \vspace{-6mm} \newline and constant $\lambda_B$ is expressed as \newline \vspace{-6mm} \begin{eqnarray} \lambda_B=\mu ^{4-n}(\lambda _R+\delta \lambda ) \label{new5} \end{eqnarray} \vspace{-6mm} \newline We determine $m_B$ and $m_R$ as bare and renormalizable boson masses and $\lambda_B$ and $\lambda_R$ as bare and renormalizable constants of interaction. It is easy to see from (\ref{new3}) and (\ref{new4}) that the free propagator of a scalar field is \begin{picture}(5,2.5) \put(6,1.0){\line(1,0){1}} \put(4,1.0){\makebox(0,0){$G=i/ (p^2-m_R^2)=$}} \end{picture} and its vertex is \begin{picture}(5,2.5) \put(6,1.0){\line(1,1){0.5}} \put(6,1.0){\line(1,-1){0.5}} \put(6,1.0){\circle*{.1}} \put(6,1.0){\line(-1,1){0.5}} \put(6,1.0){\line(-1,-1){0.5}} \put(4,1.0){\makebox(0,0){$-i\lambda _R\mu ^{4-n}=$}} \end{picture} Feynman rules for this model have a standard form \cite{its1}. The Feynman diagrams of the counterterms may be found from the Lagrangian of interaction (\ref{new4}). The two point counterterm diagram is \begin{picture}(8,2.5) \put(5,1){\makebox(0,0){$-i\delta m^2=$}} \put(6,1){\line(1,0){1.5}} \put(6.7,1){\makebox(0,0){$\times$}} \end{picture} and the vertex counterterm is \begin{picture}(5,2.5) \put(6,1.0){\line(1,1){0.5}} \put(6,1.0){\line(1,-1){0.5}} \put(6,1.0){\circle{0.2}} \put(6,1.0){\line(-1,1){0.5}} \put(6,1.0){\line(-1,-1){0.5}} \put(4,1.0){\makebox(0,0){$-i\delta \lambda \mu ^{4-n}=$}} \end{picture} Taking into account these counterterms one can construct contributions of the order $\lambda_B$ to Feynman propagator $G$ \begin{picture}(8,3) \put(1.5,1){\line(1,0){1.5}} \put(3.4,1){\makebox(0,0){+}} \put(2.2,0.3){\makebox(0,0){a)}} \put(4.,1){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1){\circle*{.1}} \put(4.7,0.3){\makebox(0,0){b)}} \put(6.2,1){\makebox(0,0){+}} \put(7,1){\line(1,0){1.5}} \put(7.7,1){\makebox(0,0){$\times$}} \put(7.7,0.3){\makebox(0,0){c)}} \end{picture} Fig. II-1 One loop and counterterm contributions to the self energy of the boson. Propagator Fig. 1a) has a standard form. The self-energy diagram of the first order to $\lambda_B$ for Fig. 1-b) may be constructed from the above mentioned graphs in the form \begin{picture}(8,3) \put(4,1){\makebox(0,0){$Fig.1b~~=~~$}} \put(5,1.){\line(1,0){1.5}} \put(5.7,1.5){\circle{1}} \put(5.7,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {\left( {2\pi } \right)^n}}\int {d^n}k{i \over {k^2-m_R^2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\left( {\lambda _R\mu ^{4-n}} \right)} \over {\left( {2\pi } \right)^n}}\int {d^n}k{1 \over {k^2-m_R^2}} \label{new6} \end{eqnarray} \vspace{-6mm} \newline The factor $(1/2)$ is the symmetry factor and $\mu$ is a parameter with dimension of mass. This parameter is used to absorb the dimension of the coupling constant. Using the expression \cite{key4} for the $n$-dimensional integral \newline \vspace{-6mm} \begin{eqnarray} \int {d^n}k{1 \over {\left( {k^2+2kq-m^2} \right)^\alpha }} =\left( {-1} \right)^\alpha i\pi ^{{n \over 2}}{{\Gamma \left( {\alpha -{n \over 2}} \right)} \over {\Gamma \left( \alpha \right)}}{1 \over {\left( {q^2+m^2} \right)^{\alpha -{n \over 2}}}} \label{new7} \end{eqnarray} \vspace{-6mm} \newline we find \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.1b~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={{\lambda _R} \over 2}{{\mu ^{4-n}} \over {\left( {2\pi } \right)^n}}{{\left( {-i\pi ^{{n \over 2}}} \right)} \over {\left( {m^2} \right)^{1-{n \over 2}}}}\Gamma \left( {1-{n \over 2}} \right)= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={{-i\lambda _R} \over 2}{{m^2} \over {16\pi ^2}} \left( {{{m^2} \over {4\pi \mu ^2}}} \right)^{{n \over 2} -1}\Gamma \left( {1-{n \over 2}} \right) \label{new8} \end{eqnarray} \vspace{-6mm} \newline From the expression for gamma function \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {1-{n \over 2}} \right)={2 \over {n-4}} +\gamma -1 \label{new9} \end{eqnarray} \vspace{-6mm} \newline finally get the divergent part of the self-energy diagram \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.1a~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} =-{{i\lambda _R} \over {16\pi ^2}}{{m^2} \over {n-4}} +\lambda _R\times finite\;term+O\left( {\lambda _R^2} \right) \label{new10} \end{eqnarray} \vspace{-6mm} \newline The renormalization of the first order of $\lambda_R$ may be done with the equation \newline \vspace{-6mm} \begin{eqnarray} i\Gamma ^{(2)}=i\left[ {{{p^2-m_R^2} \over i}+(-i\delta m^2)- {{i\lambda _Rm_R^2} \over {16\pi ^2}}{1 \over {n-4}}+...} \right] \label{new11} \end{eqnarray} \vspace{-6mm} \newline Let us express term $\delta m^2$ in the form of series \cite{col1} \newline \vspace{-6mm} \begin{eqnarray} \delta m^2=m_R^2\sum\limits_{\nu =1}^\infty {\sum\limits_{j=\nu }^\infty {{{b_{\nu j}\lambda _R^j} \over {(n-4)^\nu }}=}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =m_R^2\left\{ {{{b_{11}\lambda _R} \over {(n-4)}}+{{b_{12}\lambda _R^2} \over {(n-4)}}+{{b_{22}\lambda _R^2} \over {(n-4)^2}}+...} \right\} \label{new12} \end{eqnarray} \vspace{-6mm} \newline where the coefficients $b_{\nu j}$ are the numbers. In order for vertex $\Gamma ^{(2)}$ in one loop approximation to be finite \newline \vspace{-6mm} \begin{eqnarray} i\Gamma ^{(2)}=p^2-m_R^2+\left( {\delta m^2+{{\lambda _Rm_R^2} \over {16\pi ^2}}{1 \over {n-4}}} \right)=finite\label{new13} \end{eqnarray} \vspace{-6mm} \newline assume \newline \vspace{-6mm} \begin{eqnarray} \delta m^2=-{{\lambda _Rm_R^2} \over {16\pi ^2}}{1 \over {n-4}} \label{new14} \end{eqnarray} \vspace{-6mm} \newline then \newline \vspace{-6mm} \begin{eqnarray} b_{11}=-{1 \over {16\pi ^2}} \label{new15} \end{eqnarray} \vspace{-6mm} \newline This result completes the one loop calculations. Now we will find the vertex corrections for $\Gamma^{(4)}$. There are four graphs of the order $\lambda^2_R$: \vspace{5mm} \begin{picture}(8,3.2) \put(0,1.5){\line(1,1){0.5}} \put(0,1.5){\line(1,-1){0.5}} \put(0,1.5){\circle{.2}} \put(0,1.5){\line(-1,1){0.5}} \put(0,1.5){\line(-1,-1){0.5}} \put(0,0){\makebox(0,0){a)}} \put(1.3,1.3){\makebox(0.5,0.5){+}} \put(3,1.5){\line(-1,1){0.5}} \put(3,1.5){\line(-1,-1){0.5}} \put(3,1.5){\circle*{.1}} \put(3.5,1.5){\circle{1}} \put(4,1.5){\circle*{.1}} \put(4,1.5){\line(1,1){0.5}} \put(4,1.5){\line(1,-1){0.5}} \put(3.5,0){\makebox(0,0){b)}} \put(5,1.3){\makebox(0.5,0.5){+}} \put(6.5,2){\line(-1,1){0.5}} \put(6.5,1){\line(-1,-1){0.5}} \put(6.5,2){\circle*{.1}} \put(6.5,1.5){\circle{1}} \put(6.5,1){\circle*{.1}} \put(6.5,2){\line(1,1){0.5}} \put(6.5,1){\line(1,-1){0.5}} \put(6.5,0){\makebox(0,0){c)}} \put(8,1.3){\makebox(0.5,0.5){+}} \put(9.5,2){\line(-1,1){0.5}} \put(9.5,1){\line(-1,-1){0.5}} \put(9.5,2){\circle*{.1}} \put(9.5,1.5){\circle{1}} \put(9.5,1){\circle*{.1}} \put(9.5,2){\line(2,-1){1.5}} \put(9.5,1){\line(2,1){1.5}} \put(9.5,0){\makebox(0,0){d)}} \end{picture} \vspace*{0.5cm} \vspace{5mm} Fig. II-2 Feynman diagrams contributing to the vertex correction $\Gamma^{(4)}$. The vertex counterterm Fig. 2a) was introduced before. The loop contribution (Fig. 2b)) of the order $\lambda^2_R$ to the vertex function is \begin{picture}(8,3) \put(5,1){\line(-1,1){0.5}} \put(5,1){\line(-1,-1){0.5}} \put(5,1){\circle*{.1}} \put(5.5,1){\circle{1}} \put(6,1){\circle*{.1}} \put(6,1){\line(1,1){0.5}} \put(6,1){\line(1,-1){0.5}} \put(7,1){\makebox(0,0){=}} \end{picture} \vspace{-3mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2} \over {(2\pi )^{2n}}}\int {d^nk{{(i)^2} \over {\left( {k^2-m^2} \right)\left[ {\left( {p+k} \right)^2-m^2} \right]}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\lambda _R^2\mu ^{8-n}} \over {(2\pi )^{2n}}} \int {d^nk\int\limits_0^1 {dx{1 \over {\left[ {\left( {k^2-m^2} \right)\left( {1-x} \right)+\left[ {\left( {p+k} \right)^2-m^2} \right]x} \right]^2}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\lambda _R^2\mu ^{8-n}} \over {(2\pi )^{2n}}} \int\limits_0^1 {dx\int {d^nk{1 \over {\left[ {k^2+2pkx-\left( {m^2-p^2x} \right)} \right]^2}}}} \label{new16} \end{eqnarray} \vspace{-6mm} \newline Taking into account (\ref{new7}) we get \begin{picture}(8,3) \put(5,1){\line(-1,1){0.5}} \put(5,1){\line(-1,-1){0.5}} \put(5,1){\circle*{.1}} \put(5.5,1){\circle{1}} \put(6,1){\circle*{.1}} \put(6,1){\line(1,1){0.5}} \put(6,1){\line(1,-1){0.5}} \put(7,1){\makebox(0,0){=}} \end{picture} \vspace{-3mm} \begin{eqnarray} ={{\lambda _R^2\mu ^{8-n}i\pi ^{{n \over 2}}} \over {2(2\pi )^{2n}}}\Gamma \left( {2-{n \over 2}} \right)\int\limits_0^1 {dx{1 \over {\left[ {m^2-p^2x(1-x)} \right]^{2-{n \over 2}}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-\frac{i\lambda ^2_R}{16 \pi ^2}\frac{1}{n-4} +\lambda ^2_R \times finite~~term \label{new17} \end{eqnarray} \vspace{-6mm} \newline Here we used the representation for $\Gamma \left( {2-{n / 2}} \right)$ in the form \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {2-{n \over 2}} \right)=-{2 \over {n-4}}-\gamma \label{new18} \end{eqnarray} \vspace{-6mm} \newline Two other loops have the same divergent contributions, therefore $\Gamma^{(4)}$ vertex structure can be expressed in the form \newline \vspace{-6mm} \begin{eqnarray} \Gamma ^{\left( 4 \right)}=-i\lambda _R\mu ^{4-n}-{3 \over {16\pi ^2}} {{\lambda _R^2\mu ^{4-n}} \over {\left( {n-4} \right)}} -i\delta \lambda \mu ^{4-n}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \Gamma ^{\left( 4 \right)}=-i\lambda _R\mu ^{4-n}-i \left[ {{3 \over {16\pi ^2}}{{\lambda _R^2\mu ^{4-n}} \over {\left( {n-4} \right)}}+\delta \lambda } \right]\mu ^{4-n} =finite \label{new19} \end{eqnarray} \vspace{-6mm} \newline From the equation (\ref{new19}) we find, that \newline \vspace{-6mm} \begin{eqnarray} \delta \lambda =-{3 \over {16\pi ^2}}\lambda _R^2 \label{new20} \end{eqnarray} \vspace{-6mm} \newline Putting \newline \vspace{-6mm} \begin{eqnarray} \lambda _B=\mu ^{4-n}(\lambda _R+\delta \lambda ) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mu ^{4-n}\left[ {\lambda _R^2+\sum\limits_{\nu =1}^\infty {\sum\limits_{j=\nu }^\infty {{{a_{\nu j}\lambda _R^j} \over {\left( {n-4} \right)^\nu }}}}} \right] \label{new21} \end{eqnarray} \vspace{-6mm} \newline we find coefficient $a_{12}$: \newline \vspace{-6mm} \begin{eqnarray} a_{12}=-{3 \over {16\pi ^2}} \label{new22} \end{eqnarray} \vspace{-6mm} \newline Therefore \newline \vspace{-6mm} \begin{eqnarray} \lambda _B=\mu ^{4-n}\left[ {\lambda _R-{3 \over {16}}{{\lambda _R^2} \over {(n-4)}}+O\left( {\lambda _R^3} \right)} \right] \label{new23} \end{eqnarray} \vspace{-6mm} \newline Further we will consider two-loop contributions to two-point vertex $\Gamma^{(2)}$ \begin{picture}(8,4) \put(0,1.){\line(1,0){1.5}} \put(0.7,1.5){\circle{1}} \put(0.7,1.){\circle*{.1}} \put(0.7,2.){\makebox(0,0){$\times$}} \put(0.7,0){\makebox(0,0){a)}} \put(1.7,0.9){\makebox(0.5,0.5){+}} \put(3.,1.){\line(1,0){1.5}} \put(3.7,1.5){\circle{1}} \put(3.7,1.){\circle{.2}} \put(3.7,0){\makebox(0,0){b)}} \put(5,0.9){\makebox(0.5,0.5){+}} \put(6,1){\line(1,0){1.5}} \put(6.9,1.5){\circle{1}} \put(6.9,1.){\circle*{.1}} \put(6.9,2.5){\circle{1}} \put(6.9,2){\circle*{.1}} \put(6.7,0){\makebox(0,0){c)}} \put(8,0.9){\makebox(0.3,0.5){+}} \put(8.5,1.){\line(1,0){1.6}} \put(9.3,1.){\circle{1}} \put(8.8,1.){\circle*{.1}} \put(9.8,1.){\circle*{.1}} \put(9.3,0){\makebox(0,0){d)}} \end{picture} \vspace{5mm} Fig. II-3 Counterterms and the loop contributions of the order $\lambda^2_R$ to the self energy. To find counterterms to the order $O(\lambda^3_R)$, we will make loop calculation of all these contributions. Diagram 3(a) gives \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3a~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(6,1){\makebox(0,0){$=$}} \put(4.7,2){\makebox(0,0){$\times$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {\left( {2\pi } \right)^n}}\left( {-i\delta m^2} \right)\int {d^nk{{(i)^2} \over {\left( {k^2-m^2} \right)^2}}=} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\lambda _R\mu ^{4-n}} \over {\left( {2\pi } \right)^n}} \left( {\delta m^2} \right){{i\pi ^{{n \over 2}}\Gamma \left( {2-{n \over 2}} \right)} \over {\left( {m^2} \right)^{2-{n \over 2}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{\lambda_R}{32 \pi^2}(\delta m^2)\Gamma\left(2-\frac{n}{2}\right) \left(\frac{4\pi \mu ^2}{m^2}\right) ^{2-\frac{n}{2}} \label{new24} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{new18}) into (\ref{new24}) we get \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3a~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(4.7,1.){\circle*{.1}} \put(4.7,2){\makebox(0,0){$\times$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={{i\lambda _R} \over {(16\pi ^2)^2}}{{m_R^2} \over {(n-4)^2}} +{{i\gamma \,\lambda _Rm_R^2} \over {2(16\pi ^2)^2}}{{1} \over {(n-4)}}+finite\;terms \label{new25} \end{eqnarray} \vspace{-6mm} \newline Diagram 3(b) gives \begin{picture}(8,2.8) \put(3,1){\makebox(0,0){$Fig.3b~~=~~$}} \put(4.,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle{.2}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\delta \lambda \mu ^{4-n}} \right)} \over {\left( {2\pi } \right)^n}}\int {d^nk{i \over {k^2-m^2}}=} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\delta \lambda \mu ^{4-n}} \over {\left( {2\pi } \right)^n}}\left( -{i\pi ^{{n \over 2}}} \right) \frac{\Gamma \left( {1-{n \over 2}} \right)} {\left( {m^2} \right)^{1-{n \over 2}}}\label{new26} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{new9}) into (\ref{new26}) we find \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3b~~=~~$}} \put(4.,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle{.2}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={{3i} \over {(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)^2}} +{{3i} \over {2(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}} \left( {\gamma -1} \right) \label{new27} \end{eqnarray} \vspace{-6mm} \newline Diagram 3(c) gives: \begin{picture}(8,4) \put(3,1){\makebox(0,0){$Fig.3c~~=~~$}} \put(4,1){\line(1,0){1.5}} \put(4.9,1.5){\circle{1}} \put(4.9,1.){\circle*{.1}} \put(4.9,2.5){\circle{1}} \put(4.9,2){\circle*{.1}} \put(6.2,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={1 \over 4}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2} \over {\left( {2\pi } \right)^{2n}}} \int {d^nk\int {d^nl{{\left( i \right)^3} \over {\left( {k^2-m^2} \right)^2\left( {l^2-m^2} \right)}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={i \over 4}{{\lambda _R^2\mu ^{8-2n}} \over {\left( {2\pi } \right)^{2n}}}\int {d^nk{1 \over {\left( {k^2-m^2} \right)^2}} \int {d^nl{1 \over {\left( {l^2-m^2} \right)}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={i \over 4}{{\lambda _R^2\mu ^{8-2n}} \over {\left( {2\pi } \right)^{2n}}}\left\{ {\left( {i\pi ^{{n \over 2}}} \right){{\Gamma \left( {2-{n \over 2}} \right)} \over {\left( {m^2} \right)^{2-{n \over 2}}}}} \right\}\left\{ {\left( {-i\pi ^{{n \over 2}}} \right){{\Gamma \left( {1-{n \over 2}} \right)} \over {\left( {m^2} \right)^{1-{n \over 2}}}}} \right\}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={i \over 4}{{\lambda _R^2} \over {(16\pi ^2)^2}}\left( {{{4\pi \mu ^2} \over {m^2}}} \right)^{4-n}\Gamma \left( {2-{n \over 2}} \right)\Gamma \left( {1-{n \over 2}} \right)= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={-{i\lambda _R^2} \over {(16\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)^2}} -{{i\lambda _R^2} \over {(32\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)}} \left[ {4\gamma -2} \right]+finite\;terms \label{new28} \end{eqnarray} \vspace{-6mm} \newline Diagram 3(d) is \begin{picture}(8,2.8) \put(3,1){\makebox(0,0){$Fig.3d~~=~~$}} \put(4.5,1.){\line(1,0){1.6}} \put(5.3,1.){\circle{1}} \put(4.8,1.){\circle*{.1}} \put(5.8,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-5mm} \begin{eqnarray} ={1 \over 6}{{\left( {-i\lambda _R\mu ^{4-n}} \right)^2} \over {\left( {2\pi } \right)^{2n}}}\int {d^nk \int {d^nl{{\left( i \right)^3} \over {\left( {k^2-m^2} \right) \left( {l^2-m^2} \right)\left[ {\left( {p+k+l} \right)-m^2} \right]}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={{i\lambda _R^2\mu ^{8-2n}} \over {6\left( {2\pi } \right)^{2n}}}\int {d^nk\int {d^nl{1 \over {\left( {k^2-m^2} \right) \left( {l^2-m^2} \right)\left[ {\left( {p+k+l} \right)-m^2} \right]}}}}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={{i\lambda _R^2\mu ^{8-2n}} \over {6\left( {2\pi } \right)^{2n}}} \pi ^n\Gamma \left( {3-n} \right)\left\{ {-{6 \over {n-4}}(m^2)^{n-3}-{{p^2} \over 2}+3m^2} \right\} \label{new29} \end{eqnarray} \vspace{-6mm} \newline The function $\Gamma(3-n)$ may be written as \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {3-n} \right)={1 \over {n-4}}+\gamma -1 \label{new30} \end{eqnarray} \vspace{-6mm} \newline As the result, the expression for 3(d) will be \begin{picture}(8,2.8) \put(3,1){\makebox(0,0){$Fig.3d~~=~~$}} \put(4.5,1.){\line(1,0){1.6}} \put(5.3,1.){\circle{1}} \put(4.8,1.){\circle*{.1}} \put(5.8,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{-2mm} \begin{eqnarray} =-{{i\lambda _R^2m_R^2} \over {\left( {16\pi ^2} \right)^2}} \left( {{{m^2} \over {4\pi \mu ^2}}} \right)^{n-4}{1 \over {(n-4)^2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{i\lambda _R^2} \over {\left( {16\pi ^2} \right)^2}}{1 \over {(n-4)}} \left\{{ {{p^2} \over {12}}-{{m^2} \over 2}+(\gamma -1)m^2} \right\} \label{new31} \end{eqnarray} \vspace{-6mm} \newline The complete two loop calculations give us the following expression for $\Gamma^{(2)}$ \newline \vspace{-6mm} \begin{eqnarray} \Gamma^{(2)}={{p^2-m^2} \over i}-i\delta m^2-{{i\lambda _Rm_R^2} \over {16\pi ^2(n-4)}}+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{{3i} \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}} +{{3i} \over {2(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)}}\left( {\gamma -1} \right) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{i \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}} +{i \over {2(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)}}\gamma \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{i\lambda _R^2} \over {(16\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)^2}}-{{i\lambda _R^2} \over {(32\pi ^2)^2}}{{m_R^2} \over {\left( {n-4} \right)}} \left[ {4\gamma -2} \right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{i\lambda _R^2m_R^2} \over {\left( {16\pi ^2} \right)^2}} \left( {{{m-R^2} \over {4\pi \mu ^2}}} \right)^{n-4} {1 \over {(n-4)^2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{i\lambda _R^2} \over {\left( {16\pi ^2} \right)^2}} {1 \over {(n-4)}}\left\{ {{{p^2} \over {12}}-{{m-R^2} \over 2} +(\gamma -1)m_R^2} \right\} \label{new32} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} i\Gamma ^{(2)}=p^2-m^2+\delta m^2+{{\lambda _Rm_R^2} \over {16\pi ^2(n-4)}}- \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{2 \over {(16\pi ^2)^2}}{{\lambda _R^2m_R^2} \over {(n-4)^2}} +{1 \over {(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}} \left[ {{{p^2} \over {12}}-{{m_R^2} \over 2}} \right] \label{new33} \end{eqnarray} \vspace{-6mm} \newline To make $\Gamma^{(2)}$ finite in the second order of $\lambda^2_R$, we will put \newline \vspace{-6mm} \begin{eqnarray} \delta m^2=m_R^2\left\{ {{{\lambda _R^{}} \over {(n-4)}}b_{11} +{{\lambda _R^2} \over {(n-4)}}b_{12}+{{\lambda _R^2} \over {(n-4)^2}} b_{22}+O(\lambda _R^3)} \right\} \label{new34} \end{eqnarray} \vspace{-6mm} \newline Then, combining terms in the proper way, we get \newline \vspace{-6mm} \begin{eqnarray} i\Gamma ^{(2)}={{m_R^2} \over {(n-4)}}\left\{ {b_{11} +{1 \over {16\pi ^2}}} \right\}+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}}} \right]\times \left\{ {p^2-m_R^2 \left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}}} \right]^{-1}\times } \right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {\times \left\{ {1+{{\lambda _R^2} \over {(n-4)^2}} \left[ {{1 \over {2(16\pi ^2)^2}}-b_{12}} \right] +{{\lambda _R^2} \over {(n-4)}}\left[ {{2 \over {(16\pi ^2)^2}}-b_{22}} \right]} \right\}} \right\} \label{new35} \end{eqnarray} \vspace{-6mm} \newline It follows from (\ref{new35}) that coefficient $b_{11}$ is exactly (\ref{new15}). Two other coefficients are found from the suggestion that $\Gamma^{(2)}$ is analytic at $n=4$. In the result we get that \newline \vspace{-6mm} \begin{eqnarray} b_{22}={1 \over {2(16\pi ^2)^2}} \label{new36} \end{eqnarray} \vspace{-6mm} \newline and $b_{12}$ is the solution of the equation \newline \vspace{-6mm} \begin{eqnarray} {{\lambda _R^2} \over {(n-4)^2}}\left[ {b_{12} -{5 \over {12(16\pi ^2)^2}}} \right]=0 \label{new37} \end{eqnarray} \vspace{-6mm} \newline It gives \newline \vspace{-6mm} \begin{eqnarray} b_{12}={5 \over {12(16\pi ^2)^2}} \label{new38} \end{eqnarray} \vspace{-6mm} \newline After these calculations we will have \newline \vspace{-6mm} \begin{eqnarray} i\Gamma ^{(2)}=\left[ {1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}}} \right]\times \left( {p^2-m_R^2} \right) \label{new39} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \Gamma ^{(2)}=Z\Gamma ^{(2)}_{reg} \label{new40} \end{eqnarray} \vspace{-6mm} \newline where (wave function) renormalization constant $Z$ is \newline \vspace{-6mm} \begin{eqnarray} Z=1+{1 \over {12(16\pi ^2)^2}}{{\lambda _R^2} \over {(n-4)}} \label{new41} \end{eqnarray} \vspace{-6mm} \newline From the calculations of this section we found that the bare mass $m_B$ and coupling constant $\lambda_B$ for the second order of perturbation theory are \newline \vspace{-6mm} \begin{eqnarray} m_B^2=m_R^2\left\{ {1+{{\lambda _R} \over {(n-4)}} \left[ {-{1 \over {16\pi ^2}}+{5 \over {12}} {{\lambda _R} \over {(16\pi ^2)^2}}} \right]} \right.+ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {+{{2\lambda _R^2} \over {(16\pi ^2)^2}} {1 \over {(n-4)^2}}+O\left( {\lambda _R^3} \right)} \right\} \label{new42} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \lambda _B=\mu ^{4-n}\left\{ {\lambda _R-{3 \over {(16\pi ^2)}} {{\lambda _R^2} \over {(n-4)}}+O\left( {\lambda _R^3} \right)} \right\} \label{new43} \end{eqnarray} \vspace{-6mm} \newline The expressions (\ref{new42}) and (\ref{new43}) connect non-renormalizable and renormalizable parameters of the model, and the model is renormalized in two loop approximation of the perturbative regime. \chapter{GREEN'S FUNCTION OF BOSON} \centerline{\Large \bf IN FINITE TEMPERATURE REGIME } \vspace{24pt} The aim of this chapter is to construct renormalizable Green's function for a boson in a heat bath with a definite temperature. For this purpose we will use the real time representation for the finite temperature propagator that will let us obtain necessary results in a natural and elegant way. We will repeat calculations for the contributions in self energy of the boson in one and two loop approximations based on the scheme, developed in the previous chapter. In contrast to chapter XIV we will consider that all internal lines of Feynman graphs are the finite temperature propagators of the form \newline \vspace{-6mm} \begin{eqnarray} D(k)=D_0(k)+D_\beta (k)={i \over {k^2-m^2}} +{{2\pi \delta (k^2-m^2)} \over {e^{\beta |k^0|}-1}}, \label{n1} \end{eqnarray} \vspace{-6mm} \newline where $\beta^{-1}=T$ is the temperature. Finite temperature calculations don't change Feynmann rules for loop calculations \cite{mor1}, \cite{key21}. Fig 1b) gives \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.1b~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}D(k)= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}D_0(k)+ {1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}D_\beta (k) \nonumber \end{eqnarray} \vspace{-6mm} \newline or in a more compact form \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.1b~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle*{.1}} \put(4.8,0.5){\makebox(0,0){$(T = 0)$}} \put(6,1){\makebox(0,0){$-$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} -{1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}D_{\beta}(k). \label{n2} \end{eqnarray} \vspace{-6mm} \newline \vspace{1cm} The counterterm Fig. 3a) is \begin{picture}(8,2.3) \put(3,1){\makebox(0,0){$Fig.3a~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1){\circle*{.1}} \put(4.7,2){\makebox(0,0){$\times$}} \put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\lambda _R\mu ^{4-n}} \right)} \over {(2\pi )^n}}(-i\delta m^2)\int {d^nk}D_0^2(k) -\lambda _R\delta m^2\int {{d^4k}\over(2\pi)^4}D_\beta (k)D_0(k), \nonumber \end{eqnarray} \vspace{-6mm} \newline or \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3a~~=~~$}} \put(4,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1){\circle*{.1}} \put(4.7,2){\makebox(0,0){$\times$}} \put(4.8,0.5){\makebox(0,0){$(T = 0)$}} \put(6,1){\makebox(0,0){$~~-~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} -\lambda _R\delta m^2\int {{d^4k\over(2\pi)^4}}D_\beta (k)D_0(k). \label{n3} \end{eqnarray} \vspace{-6mm} \newline The contribution of the counterterm Fig. 3b) is \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3b~~=~~$}} \put(4.,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle{.2}} \put(4.8,0.5){\makebox(0,0){$(T \neq 0)$}} \put(6,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\delta\lambda \mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}D(k) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 2}{{\left( {-i\delta \lambda \mu ^{4-n}} \right)} \over {(2\pi )^n}}\int {d^nk}(D_0(k)+D_\beta (k)), \nonumber \end{eqnarray} \vspace{-6mm} \newline and \begin{picture}(8,3) \put(3,1){\makebox(0,0){$Fig.3b~~=~~$}} \put(4.,1.){\line(1,0){1.5}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle{.2}} \put(4.8,0.5){\makebox(0,0){$(T=0)$}} \put(6,1){\makebox(0,0){$~~-~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} -{\left( {i\delta\lambda } \right)}\int {{d^nk}\over{(2\pi)^n}}D_{\beta}(k). \end{eqnarray} \vspace{-6mm} \newline \label{n4} \vspace{1cm} Two loop contribution Fig. 3c) may be written in the form \begin{picture}(8,3.5) \put(3,1){\makebox(0,0){$Fig.3c~~=~~$}} \put(4,1){\line(1,0){1.5}} \put(4.9,1.5){\circle{1}} \put(4.9,1.){\circle*{.1}} \put(4.9,2.5){\circle{1}} \put(4.9,2){\circle*{.1}} \put(5,0.5){\makebox(0,0){$(T \neq 0)$}} \put(6.2,1){\makebox(0,0){$~~=~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} ={1 \over 4}{{(-i\lambda _R\mu ^{4-n})^2} \over {(2\pi )^{2n}}}\int {d^nk}\int {d^nq}(D_0(k)+D_\beta (k))^2 (D_0(q)+D_\beta (q)) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={1 \over 4}{{(-i\lambda _R\mu ^{4-n})^2} \over {(2\pi )^{2n}}}\int {d^nk}\int {d^nq\left[ {D_0^2(k)D_0(q)} \right.+D_0^2(k)D_\beta (q)+} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {+2D_0(k)D_\beta (k)D_0(q)+2D_0(k)D_\beta (k)D_\beta (q) +D_\beta ^2(k)D_0(q)+D_\beta ^2(k)D_\beta (q)} \right]. \nonumber \end{eqnarray} \vspace{-6mm} \newline Then \begin{picture}(8,3.8) \put(3,1){\makebox(0,0){$Fig.3c~~=~~$}} \put(4,1){\line(1,0){1.5}} \put(4.9,1.5){\circle{1}} \put(4.9,1.){\circle*{.1}} \put(4.9,2.5){\circle{1}} \put(4.9,2){\circle*{.1}} \put(5,0.5){\makebox(0,0){$(T = 0)$}} \put(6.2,1){\makebox(0,0){$~~-~~$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} -{{\lambda _R^2} \over 4}\int {{{d^4q} \over {(2\pi )^4}}D_\beta (q)} \left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nk}D_0^2(k)} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{\lambda _R^2} \over 2}\left\{ {\int {{{d^4q} \over {(2\pi )^4}} D_0(k)D_\beta (k)}} \right\}\left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}} \int {d^nq}D_0(q)} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{\lambda _R^2} \over 2}\left\{ {\int {{{d^4q} \over {(2\pi )^4}} D_\beta (q)}} \right\}\left\{ {\int {{{d^4k} \over {(2\pi )^4}}}. D_0(k)D_\beta (k)} \right\} \label{n5} \end{eqnarray} \vspace{-6mm} \newline Finally the contribution Fig. 3d) will be \begin{picture}(8,2.8) \put(3,1){\makebox(0,0){$Fig.3d~~=~~$}} \put(4.5,1.){\line(1,0){1.6}} \put(5.3,1.){\circle{1}} \put(4.8,1.){\circle*{.1}} \put(5.8,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~=~~$}} \put(5.3,0){\makebox(0,0){$(T \neq 0)$}} \end{picture} \newline \vspace{-6mm} \begin{eqnarray} {1 \over 6}{{(-i\lambda _R\mu ^{4-n})^2} \over {(2\pi )^{2n}}}\int {d^nk}D_\beta (k)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times\left[ {\int {d^nq}D_\beta (q)D_0(q-p-k)} \right. +\int {d^nq}D_\beta (q)D_0(k-p-q) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +{\int {d^nq}D_\beta (q)D_0(q+p+k)}+\int {d^nq}D_\beta (q)D_0(q-k+p) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\left. {\int {d^nqD_0(q)D_0(q-p-k)+\int {d^nqD_0(q)D_0(q+p+k)}}} \right] \nonumber \end{eqnarray} \vspace{-6mm} \newline or \begin{picture}(8,2.6) \put(3,1){\makebox(0,0){$Fig.3d~~=~~$}} \put(4.5,1.){\line(1,0){1.6}} \put(5.3,1.){\circle{1}} \put(4.8,1.){\circle*{.1}} \put(5.8,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~+~~$}} \put(5.3,0){\makebox(0,0){$(T = 0)$}} \end{picture} \newline \vspace{-6mm} \begin{eqnarray} +{1 \over 2}{{(-i\lambda _R\mu ^{4-n})^2} \over {(2\pi )^{2n}}}\int {d^nk}D_\beta (k)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray}\times\left[ {\int {d^nq}D_\beta (q)D_0(q+p+k)} +\int {d^nq}D_0 (q)D_0(k+p+q)\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline So, we get \begin{picture}(8,2.8) \put(3,1){\makebox(0,0){$Fig.3d~~=~~$}} \put(4.5,1.){\line(1,0){1.6}} \put(5.3,1.){\circle{1}} \put(4.8,1.){\circle*{.1}} \put(5.8,1.){\circle*{.1}} \put(7,1){\makebox(0,0){$~~-~~$}} \put(5.3,0){\makebox(0,0){$(T = 0)$}} \end{picture} \newline \vspace{-6mm} \begin{eqnarray} -{{\lambda _R^2} \over 2}\int {{{d^4k} \over {(2\pi )^4}}} D_\beta (k)\left\{ {\int {{{d^4q} \over {(2\pi )^4}}}. D_\beta (q)D_0(k+q+p)} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{\lambda _R^2} \over 2}\int {{{d^4k} \over {(2\pi )^4}}}D_\beta (k)\left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nq}D_0(q)D_0(k+q+p)} \right\} \label{n6} \end{eqnarray} \vspace{-6mm} \newline Now we can find counterterms. Assuming the sum of the finite temperature contributions of the equations (\ref{n3}) and (\ref{n5}) zero, \newline \vspace{-6mm} \begin{eqnarray} \left\{ {\lambda _R\delta m^2+{{\lambda _R^2} \over 2} {{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nq}D_0(q)} \right\} \int {{{d^4k} \over {(2\pi )^4}}}D_0(k)D_\beta (k)=0. \nonumber \end{eqnarray} \vspace{-6mm} \newline we find the expression for $\delta m^2$ in the form \newline \vspace{-6mm} \begin{eqnarray} {\delta m^2=-{{\lambda _R} \over 2}{{\mu ^{4-n}} \over {(2\pi )^n}}\int {d^nq}D_0(q)}. \label{n7} \end{eqnarray} \vspace{-6mm} \newline The divergent part of this counterterm will be \newline \vspace{-6mm} \begin{eqnarray} {\delta m_{div}^2=-{{\lambda _R} \over {16\pi ^2}}{{m_R^2} \over {(n-4)}}}. \label{n8} \end{eqnarray} \vspace{-6mm} \newline The following counterterm $\delta\lambda$ may be found by the summation of the finite temperature contributions of the equations (\ref{n4}), (\ref{n5}) and (\ref{n6}) \newline \vspace{-6mm} \begin{eqnarray} \int {{d^4k} \over {(2\pi )^4}}D_\beta (k) \left\{ i\delta \lambda +{{\lambda _R^2} \over 2} \left[ {{\mu ^{n-4}} \over {(2\pi )^n}} \int {d^nqD_0^2(q)} \right]\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+\lambda _R^2 \left[ {{\mu ^{n-4}} \over {(2\pi )^n}} \int {d^nqD_0(q)D_0(k+q+p)} \right] \right\}=0. \nonumber \end{eqnarray} \vspace{-6mm} \newline For zero external momentum $p$ the divergent part of the $\delta\lambda$ will be determined by the divergent part of the integral \newline \vspace{-6mm} \begin{eqnarray} {\delta \lambda ={3 \over 2}(i\lambda _R^2) \left[ {{{\mu ^{n-4}} \over {(2\pi )^n}}\int {d^nqD_0^2(q)}} \right]} \label{n9} \end{eqnarray} \vspace{-6mm} \newline and the divergent contribution will be \newline \vspace{-6mm} \begin{eqnarray} {\delta \lambda_{div} =-{{3\lambda _R^2} \over {16\pi ^2}} {1\over {(n-4)}}}.\label{n10} \end{eqnarray} \vspace{-6mm} \newline The temperature counterterms (\ref{n8}) and (\ref{n10}) have the same structure as the counterterms which annihilate the divergent parts of zero temperature loop contributions. It is easy to see that at $T=0$ the sum of the loop contribution and the counterterm gives zero: \begin{picture}(8,3) \put(3,1.){\line(1,0){1.5}} \put(3.7,1.5){\circle{1}} \put(3.7,1.){\circle*{.1}} \put(5.3,1){\makebox(0,0){+}} \put(6.,1){\line(1,0){1.5}} \put(6.7,1){\makebox(0,0){$\times$}} \put(8,1){\makebox(0,0){$=$}} \end{picture} \vspace{-5mm} \begin{eqnarray} {={{(-i\lambda _R\mu ^{4-n})} \over {2(2\pi )^n}} \int {d^nk}D_0(k)-\delta m^2=0}. \nonumber \end{eqnarray} \vspace{-6mm} \newline It leads us to the equation (\ref{n7}). Loop contributions and counterterm at $T=0$ in $\Gamma^{(4)}$ (Fig.II-2) gives the equation \newline \vspace{-6mm} \begin{eqnarray} {{3 \over 2}{{\lambda _R^2\mu ^{4-n}} \over {(2\pi )^n}} \int {d^nk}D_0(k)D_0(k+p)-i\delta \lambda =0}, \nonumber \end{eqnarray} \vspace{-6mm} \newline which coincides with (\ref{n9}). From the mentioned above analysis we can conclude that the following loops have the same divergent structure: \begin{picture}(8,4) \put(0,1.){\line(1,0){1.5}} \put(0.7,1.5){\circle{1}} \put(0.7,1.){\circle*{.1}} \put(2.3,1){\makebox(0,0){+}} \put(1.7,0.0){\makebox(0,0){$T=0$}} \put(3.,1){\line(1,0){1.5}} \put(3.7,1){\makebox(0,0){$\times$}} \put(5.4,1){\makebox(0,0){$\leftrightarrow$}} \put(6,1){\line(1,0){1.5}} \put(6.9,1.5){\circle{1}} \put(6.9,1.){\circle*{.1}} \put(6.9,2.5){\circle{1}} \put(6.9,2){\circle*{.1}} \put(8,1){\makebox(0,0){+}} \put(8,0){\makebox(0,0){$T \neq 0$}} \put(8.5,1.){\line(1,0){1.6}} \put(9.3,1.){\circle{1}} \put(8.8,1.){\circle*{.1}} \put(9.8,1.){\circle*{.1}} \end{picture} \vspace*{8mm} and \begin{picture}(8,4) \put(0,1){\line(1,1){0.5}} \put(0,1){\line(1,-1){0.5}} \put(0,1){\circle{.2}} \put(0,1){\line(-1,1){0.5}} \put(0,1){\line(-1,-1){0.5}} \put(1,1){\makebox(0,0){+}} \put(1,0){\makebox(0,0){$T=0$}} \put(1.4,1){\makebox(0,0){$3 \times$}} \put(2,1){\line(-1,1){0.5}} \put(2,1){\line(-1,-1){0.5}} \put(2,1){\circle*{.1}} \put(2.5,1){\circle{1}} \put(3,1){\circle*{.1}} \put(3,1){\line(1,1){0.5}} \put(3,1){\line(1,-1){0.5}} \put(3.7,1){\makebox(0,0){$\leftrightarrow$}} \put(4.1,1){\line(1,0){1.2}} \put(4.7,1.5){\circle{1}} \put(4.7,1.){\circle{.2}} \put(6,1){\makebox(0,0){+}} \put(6.3,1){\line(1,0){1.2}} \put(6.9,1.5){\circle{1}} \put(6.9,1.){\circle*{.1}} \put(6.9,2.5){\circle{1}} \put(6.9,2){\circle*{.1}} \put(8,1){\makebox(0,0){+}} \put(7,0){\makebox(0,0){$T \neq 0$}} \put(8.5,1.){\line(1,0){1.6}} \put(9.3,1.){\circle{1}} \put(8.8,1.){\circle*{.1}} \put(9.8,1.){\circle*{.1}} \end{picture} \vspace{10mm} The Green's function $D^{'}(p)$ of the boson is the object which takes into account virtual processes of creation and anihilation of the additional particles when this boson moves through the vacuum. The graph composing $D^{'}(p)$ may be divided into two distinct and unique classes of proper and improper graphs (Fig.II-4)\footnote{The proper graphs cannot be divided into two disjoint parts by the removal of a single line, whereas the improper ones can be disjoint \cite{key6}}. \begin{picture}(8,3) \put(0,1){\makebox(0,0){$D^{'}(p)=$}} \put(1,1){\circle{.2}} \put(1,1){\line(1,0){0.5}} \put(1.5,1){\circle{.2}} \put(1.2,0.2){\makebox(0,0){$D(p)$}} \put(2,1){\makebox(0,0){+}} \put(2.5,1){\line(1,0){0.5}} \put(2.7,0.2){\makebox(0,0){$D(p)$}} \put(2.5,1){\circle{.2}} \put(3.0,1){\circle{.2}} \put(3.5,1){\oval(1,0.7)} \put(4,1){\circle{.2}} \put(4,1){\line(1,0){0.5}} \put(4.3,0.2){\makebox(0,0){$D(p)$}} \put(4.5,1){\circle{.2}} \put(5,1){\makebox(0,0){+}} \put(5.5,1){\circle{.2}} \put(5.5,1){\line(1,0){0.5}} \put(5.7,0.2){\makebox(0,0){$D(p)$}} \put(6,1){\circle{.2}} \put(6.5,1){\oval(1,0.7)} \put(7,1){\circle{.2}} \put(7.5,1){\circle{.2}} \put(7,1){\line(1,0){0.5}} \put(7.1,0.2){\makebox(0,0){$D(p)$}} \put(7.8,0.2){\makebox(0,0){$D(p)$}} \put(7.5,1){\line(1,0){0.5}} \put(8,1){\circle{.2}} \put(8.5,1){\oval(1,0.7)} \put(9,1){\circle{.2}} \put(9,1){\line(1,0){0.5}} \put(9.2,0.2){\makebox(0,0){$D(p)$}} \put(9.5,1){\circle{.2}} \put(10,1){\makebox(0,0){+}} \put(10.8,1){\makebox(0,0){$.~~.~~.$}} \end{picture} \vspace{1cm} Fig.II-4 Green's function $D^{'}(p)$ of the boson as sum of proper self-energy insertions. In accordance with Fig.II-4 the Green's function $D^{'}(p)$ is obtained by the summing of the series: \newline \vspace{-6mm} \begin{eqnarray} D^{'}(p)=D(p)+D(p)\left( {{{\Sigma (p)} \over i}} \right)D(p) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +D(p)\left( {{{\Sigma (p)} \over i}} \right)D(p)D(p) \left( {{{\Sigma (p)} \over i}} \right)D(p)+... \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =D(p){1 \over {1+i\Sigma (p)D(p)}}={i \over {p^2-m^2-\Sigma (p)}} \label{n11} \end{eqnarray} \vspace{-6mm} \newline In this equation $\Sigma(p)$ is the sum of all two point improper graphs. All divergent improper graphs and their counterterm graphs (Fig.II-1,II-3) may be divided into two parts of the first and the second orders with respect to $\lambda_{R}$. We will define them as $\Sigma_1$ and $\Sigma_2$ self-energy graphs. One can write finite contributions in $\Sigma_2$ in the form \begin{picture}(8,4) \put(2,1){\makebox(0,0){$(-i)\Sigma_2=$}} \put(3.3,1){\line(1,0){1.2}} \put(3.9,1.5){\circle{1}} \put(3.9,1.){\circle*{.1}} \put(3.9,2.5){\circle{1}} \put(3.9,2){\circle*{.1}} \put(5,1){\makebox(0,0){+}} \put(5.5,1.){\line(1,0){1.6}} \put(6.3,1.){\circle{1}} \put(5.8,1.){\circle*{.1}} \put(6.8,1.){\circle*{.1}} \put(7.8,1){\makebox(0,0){$=$}} \end{picture} \vspace{0.0mm} \begin{eqnarray} =-{{\lambda _R^2} \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q) \left\{ {\int {d^4k}D_\beta (k)D_0(k+q+p)} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -{{\lambda _R^2} \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q) \left\{ {{{\mu ^{4-n}} \over {(2\pi )^n}} \int {d^4k}D_0(k)D_0(k+q+p)} \right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. {-{{\mu ^{4-n}} \over {(2\pi )^n}} \int {d^4k}D_0(k)D_0(k+p)} \right\}_{p=0} \label{n12} \end{eqnarray} \vspace{-6mm} \newline Let us introduce functions \newline \vspace{-6mm} \begin{eqnarray} F_\beta ={1 \over 2}\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q) \label{n13} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} iI_\beta (p)=\int {{{d^4q} \over {(2\pi )^4}}}D_\beta (q)D_0(q+p). \label{n14} \end{eqnarray} \vspace{-6mm} \newline Then we get \newline \vspace{-6mm} \begin{eqnarray} \Sigma _2=\lambda _R^2F_\beta I_0(0)+{{\lambda _R^2} \over 2}G_\beta \label{n15} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} G_\beta =(F_\beta ,(I_\beta +I_0))_{finite} \label{n16} \end{eqnarray} \vspace{-6mm} \newline The first equation in (\ref{n16}) is a scalar product of the form \newline \vspace{-6mm} \begin{eqnarray} (F_\beta ,I_{\beta})=\int {{{d^4k} \over {(2\pi )^4}}}D_\beta (k)\int {{{d^4q} \over {(2\pi )^4}}}D_0(q)D_\beta (k+q) \label{n117} \end{eqnarray} \vspace{-6mm} \newline and the following one is \newline \vspace{-6mm} \begin{eqnarray} (F_\beta ,I_0)_{finite}=\int {{{d^4k} \over {(2\pi )^4}}}D_\beta (k)\left\{\int {{{d^4q} \over {(2\pi )^4}}}{D_0(q)D_0 (k+q)}\right\}_{finite} \label{n18} \end{eqnarray} \vspace{-6mm} \newline The contribution of the first order in $\Sigma_1$ may be found from (\ref{n2}). This contribution is \newline \vspace{-6mm} \begin{eqnarray} \Sigma _1={{\lambda _R} \over 2}\int {{{d^4q} \over {(2\pi )^4}}} D_\beta (q) ={\lambda _R}F_{\beta}.\label{n19} \end{eqnarray} \vspace{-6mm} \newline The temperature contribution in the boson's mass\footnote{For finite temperature quantum electrodynamics the non-equality of fermionic masses $\delta m_\beta /m \sim\alpha(T/m)^2$ is described by a similar equation \cite{per1}} will have the following form \newline \vspace{-6mm} \begin{eqnarray} m^2(T)=m_R^2+\Sigma _1+\Sigma _2 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =m_R^2+\lambda _{R}F_\beta+\lambda _R^2F_\beta I_\beta (0) +{{\lambda _R^2} \over 2}\lambda _R^2G_\beta \label{n20} \end{eqnarray} \vspace{-6mm} \newline As the result the Green's function will be \newline \vspace{-6mm} \begin{eqnarray} D^{'}(p)=\frac{i}{p^2-m^2(T)} \label{n21} \end{eqnarray} \vspace{-6mm} \newline Thus we have computed the finite temperature Green's function of a boson in two-loop approximation in the form of Feynman propagator with finite temperature dependent mass parameter (\ref{n20}). \chapter{THERMAL PROPERTIES OF BOSON} \vspace{24pt} In chapter XV we showed that the model is renormalizable in each order of the perturbative regime, and found the finite temperature propagator of a boson in a heat bath in two loop approximation. We also got the expression for the finite temperature mass of a boson. These results may help us to get an effective Hamiltonian of the particle and to study its finite temperature behavior in gravitational fields. \section {Effective Hamiltonian of the boson} \vspace{-4mm} \hspace{22mm}{\Large \bf in non-relativistic approximation} \vspace{1mm} After renormalization the pole of boson propagator (\ref{n21}) may be written as \newline \vspace{-6mm} \begin{eqnarray} E=\left[{\vec{p}}^{~2} +m_R^2+\lambda _{R}F_\beta+\lambda _R^2F_\beta I_\beta (0) +{{\lambda _R^2} \over 2}\lambda _R^2G_\beta\right]^{1\over2} \label{n22} \end{eqnarray} \vspace{-6mm} \newline We can rewrite the equation (\ref{n22}) in non-relativistic approximation in the following form \newline \vspace{-6mm} \begin{eqnarray} E=m_R\left\{ {1+\lambda _Rf(\beta m_R)+o(\lambda _R^2)} \right\}^{{1 \over 2}} \left[ {1+{{\vec{p}^{~2}} \over {m_R^2\left\{ {1+\lambda _Rf(\beta m_R) +o(\lambda _R^2)} \right\}}}} \right]^{{1 \over 2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =m_R+{1 \over 2}\lambda _Rm_Rf(\beta m_R) +{\vec{p}^{~2}\over{2m_R}} {\left( 1+{1 \over 2}\lambda _Rf(\beta m_R) \right)}^{-1} +o(\lambda _R^2) \label{n23} \end{eqnarray} \vspace{-6mm} \newline Here the function $f(y)$ is connected with the function $F_\beta (y)$ (\ref{n13}) in the following way \newline \vspace{-6mm} \begin{eqnarray} F_\beta(\beta m_R) ={1 \over 2}\int {{{d^3k} \over {(2\pi )^3}} {1 \over {\varepsilon \left( {e^{\beta \varepsilon }-1} \right)}}} =m_R^2f(\beta m_R) \nonumber \end{eqnarray} \vspace{-6mm} \newline The function $f(y)$ has an asymptotic form (for $(y\ll1)$) \cite{i15}: \newline \vspace{-6mm} \begin{eqnarray} f(y)={1 \over {(2\pi)^2}}\int\limits_1^\infty {dx} {{\sqrt {x^2-1}} \over {e^{xy}-1}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{24y^2} -{1 \over {8\pi y}}+O\left( {y^2\ln y^2} \right), \label{n24} \end{eqnarray} \vspace{-6mm} \newline which is very useful for the analysis of the high temperature behavior of the model. \section{Inertial and gravitational masses of a boson} \vspace{1mm} For our following calculations we will consider that the quantum system interacts with the gravitational field which is described by the metric \newline \vspace{-6mm} \begin{eqnarray} g_{\mu \nu }=\eta _{\mu \nu }+{\Phi\over2}{\delta_{0\mu}}\delta_{0\nu}, \label{n25} \end{eqnarray} \vspace{-6mm} \newline where $\Phi$ is a gravitational potential. The second term in (\ref{n25}) describes a small correction to the Minkowski metric which is connected with the presence of the gravitational field \\ \cite{land2}, \cite{miz1}. In order to write the Hamiltonian of the boson in the presence of gravitational field one can consider that temperature $T$ changes according to Tolmen's law \cite{tol1}. \newline \vspace{-6mm} \begin{eqnarray} T={{T_0} \over {1+\Phi }}, \label{n26} \end{eqnarray} \vspace{-6mm} \newline where $T_0$ is the temperature with $\Phi=0$. The finite temperature Hamiltonian with precision to the first leading term of the series (\ref{n24}) will be \newline \vspace{-6mm} \begin{eqnarray} H={{\vec{p}^{~2}} \over 2}\left( {m_R+{\lambda_{R} \over {48}} {{T_0^2} \over {(1+\Phi )^2m_R}}} \right)^{-1}+m_R+{\lambda_{R} \over {48}}{{T_0^2} \over {m_R(1+\Phi )^2}}+m_R\Phi +... \label{n27} \end{eqnarray} \vspace{-6mm} \newline The last term of the equation (\ref{n27}) describes energy of boson's interaction with gravitational field. Let us rewrite the Hamiltonian (\ref{n27}) as \newline \vspace{-6mm} \begin{eqnarray} H={{\vec{p}^{~2}} \over 2}\left( {m_R+{{\lambda _R} \over {48}}{{T_0^2} \over {(1+\Phi )^2m_R}}} \right)^{-1} +\left( {m_R-{{\lambda _R} \over {24}}{{T_0^2} \over {m_R}}} \right)\Phi +... \label{n28} \end{eqnarray} \vspace{-6mm} \newline The acceleration of the boson in a gravitational field may be found from the quantum mechanical relation \newline \vspace{-6mm} \begin{eqnarray} \vec{a}=-\left[ {H,\left[ H,{\vec{r}} \right]} \right]= -\left( {1-{{\lambda _R} \over {12}}{{T_0^2} \over {m_R^2}}} \right)\nabla \Phi, \label{n29} \end{eqnarray} \vspace{-6mm} \newline and the mass ratio will be \newline \vspace{-6mm} \begin{eqnarray} {{m_g} \over {m_i}}=1-{{\lambda _R} \over {12}}{{T_0^2} \over {m_R^2}} \label{n30} \end{eqnarray} \vspace{-6mm} \newline so the inertial and gravitational masses of the boson in the heat bath are seen to be unequal. One can estimate the value of $\delta m_g/m_i$ for some gravitational source. Let the source of gravitational field be the Sun ( $1.989\times10^{30} kg$) then the relation (\ref{n30}) for the combined boson (Cooper pair with mass $m_b=1Mev$) in the heat bath with temperature $300 K$ gives the following corrections for non-equality between masses \newline \vspace{-6mm} \begin{eqnarray} \frac{\delta m_g}{m_i}=\frac{\lambda _R}{12}\frac{T_0^2}{m_R^2} \sim \lambda \times 10^{-17} \label{n31} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} 10^{-21}< \frac{\delta m_g}{m_i} <10^{-17} \label{n32} \end{eqnarray} \vspace{-6mm} \newline for the range of the coupling constant $10^{-3}<\lambda <10^{-2}$. From the analysis we made in this chapter one may conclude that thermal interaction of the bosons in a gravitational field causes non-equality between inertial and gravitational masses. Non-thermal systems do not demonstrate such properties. The calculations for non-equality between inertial and gravitational mass of electron were made for thermal quantum electrodynamics by Donoghue\\ \cite{don1}. His result for massive fermions has the same functional structure as the equation (\ref{n30}). \chapter{INTRODUCTION} \centerline{\Large \bf TO TOPOLOGICAL FIELD MODELS } \vspace{24pt} As an introduction to odd dimensional topological field models let us consider their origin and topological significance. \vspace{3mm} {\large \it Topological aspect of the model} \vspace{3mm} From gauge-invariant fields in even dimensions we may construct gauge invariant Pontryagin densities: \newline \vspace{-6mm} \begin{eqnarray} P_{(2)}=-(1/2 \pi )\epsilon ^{\mu \nu}{^*F}^{\mu \nu} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} P_{(4)}=-(1/16 \pi ^2) \mbox{tr} ^*F^{\mu \nu}F_{\mu \nu}\label{int1} \end{eqnarray} \vspace{-6mm} \newline whose integrals over the even dimensional space are invariants that measure the topological content of the model. These gauge invariant objects can also be written as total derivatives of gauge invariant quantities \newline \vspace{-6mm} \begin{eqnarray} P_n=\partial_\mu X^\mu _n \label{int2} \end{eqnarray} \vspace{-6mm} \newline The two and four-dimensional expressions are \newline \vspace{-6mm} \begin{eqnarray} X^\mu _2=(1/2\pi)\epsilon^{\mu \nu}A_\nu \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} X^\mu _4=(1/2\pi)\epsilon^{\mu \alpha \nu \beta} \mbox{tr} (A_\alpha F_{\beta \gamma}-(2/3)A_\alpha A_\beta A_\gamma) \label{int3} \end{eqnarray} \vspace{-6mm} \newline The Chern-Simons (CS) secondary characteristic class is obtained by integrating one component of $X^ \mu _n$ over the $(n-1)$ dimensional space which does not include that component. Therefore the 3-D action $S_{CS}$ is proportional to \newline \vspace{-6mm} \begin{eqnarray} S_{CS} \sim \int dx^0dx^1dx^2 X^ 3 _4 \label{int4} \end{eqnarray} \vspace{-6mm} \newline A topological massive term (we will name it CS term) can be added to the fundamental action for a gauge fields, but unlike the ways in which gauge fields are usually given a mass, no gauge symmetry is broken by its introduction. \vspace{3mm} {\large \it Quantum aspects of 3-D field theory} \vspace{3mm} Let us consider non-Abelian quantum model with topological mass term. The Lagrangian of this model is \newline \vspace{-6mm} \begin{eqnarray} L=L_0+L_{CS} +L_{gauge} \label{int5} \end{eqnarray} \vspace{-6mm} \newline where $L_0$ is the usual action for non-Abelian gauge field \newline \vspace{-6mm} \begin{eqnarray} L_0=-(1/2)\mbox{tr}(F_{\mu \nu}F^{\mu \nu}) \label{int6} \end{eqnarray} \vspace{-6mm} \newline with \newline \vspace{-6mm} \begin{eqnarray} F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu +g[A_\mu, A_\nu] \label{int7} \end{eqnarray} \vspace{-6mm} \newline $L_{CS}$ is CS term \newline \vspace{-6mm} \begin{eqnarray} L_{CS}=-im\epsilon ^{\mu \nu \rho} \mbox{tr} (A_\mu \partial_\nu A_\rho-(2/3)A_\mu A_\nu A_\rho) \label{int8} \end{eqnarray} \vspace{-6mm} \newline and $L_{gauge}$ includes the gauge-fixing term \newline \vspace{-6mm} \begin{eqnarray} L_{gauge}= (\partial_\mu\bar{\eta}^a)(\partial^\mu \eta^a) +gf_{abc}(\partial_\mu\bar{\eta}^a)D^\mu \eta \label{int9} \end{eqnarray} \vspace{-6mm} \newline We introduce $SU(N)$ gauge group here with matrix notation: $A_\mu=A^a_\mu\tau^a$, where $\tau^a$ are anti-Hermitian matrices in the fundamental representation: \newline \vspace{-6mm} \begin{eqnarray} [\tau^a, \tau^b]=f^{abc}\tau^c,~~~ \mbox{tr} (\tau^a \tau^b) =-(1/2)\delta^{ab} \label{int10} \end{eqnarray} \vspace{-6mm} \newline and $f^{abc}$ are the structure constants of $SU(N)$. The theory is defined in three space-time dimensions with Euclidean signature $(+~+~+)$. The coupling of the CS term is imaginary in Euclidean space-time and real in Minkowski space-time. \vspace{3mm} {\large \it Properties of the model}: \vspace{3mm} For an odd number of dimensions, the operation of parity, P, can be defined as reflection in all axes: \newline \vspace{-6mm} \begin{eqnarray} x^\mu \mathop{\to}\limits_{P}-x^\mu , ~~~A_\mu \mathop{\to}\limits_{P} -A_\mu \label{int11} \end{eqnarray} \vspace{-6mm} \newline The usual gauge Lagrangian is even under parity, \newline \vspace{-6mm} \begin{eqnarray} L_0 \mathop{\to}\limits_{P}+L_0 \label{int12} \end{eqnarray} \vspace{-6mm} \newline but the CS term is odd \newline \vspace{-6mm} \begin{eqnarray} L_{CS} \mathop{\to}\limits_{P}-L_{CS} \label{int13} \end{eqnarray} \vspace{-6mm} \newline Under gauge transformation \newline \vspace{-6mm} \begin{eqnarray} A_\mu \to \Omega^{-1}\left\{(1/g)\partial_\mu +A_\mu\right\}\Omega \label{int14} \end{eqnarray} \vspace{-6mm} \newline The Lagrangian $L_0$ is invariant but $L_{CS}$ is not: \newline \vspace{-6mm} \begin{eqnarray} \int d^3x L_{CS} \to \int d^3x L_{CS} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(im/g)\int d^3x \epsilon^{\mu \nu \rho}\partial_\mu \mbox{tr} [(\partial_\nu\Omega)\Omega^{-1}A_\rho] +8\pi^2 (m/g^2)i\omega \label{int15} \end{eqnarray} \vspace{-6mm} \newline where: \newline \vspace{-6mm} \begin{eqnarray} \omega=(1/24)\int d^3x \epsilon^{\mu \nu \rho} \mbox{tr}[\Omega^{-1}(\partial_\mu\Omega)\Omega^{-1} (\partial_\nu\Omega)\Omega^{-1}(\partial_\rho\Omega)] \label{int16} \end{eqnarray} \vspace{-6mm} \newline The set of gauge transformations is divided into global gauge rotations, $\partial_\mu \Omega=0$, and all others, for which we assume that $\Omega (x) \to 1$ as $x^\mu \to \infty$. Integrating over global gauge rotations requires the system to have a total color charge equal to zero. In this case, $A_\mu (x)$ falls off faster than $1/|x|$ as $x^\mu \to \infty$, and the second term on the right-hand side of (\ref{int15}), which is a surface integral, vanishes. The last term in (\ref{int15}) does not vanish in general. The $\omega$ of (\ref{int16}) is a winding number, which labels the homotopy class of $\Omega (x)$ \cite{nak1} Changing variables $A \to A^U$, where $A^U$ is a gauge transformation of $A$, implies that the vacuum average of the value $\hat{Q}$ is \newline \vspace{-6mm} \begin{eqnarray} <Q>=\exp |i8\pi^2 (m/g^2)\omega(U)|<Q> \label{int17} \end{eqnarray} \vspace{-6mm} \newline This invariance gives us a quantization condition for the dimensionless ratio \newline \vspace{-6mm} \begin{eqnarray} 4\pi^2 (m/g^2)=n,~~~n=0,\pm 1,\pm 2, \label{int18} \end{eqnarray} \vspace{-6mm} \newline Therefore, for the theory to be invariant under certain large gauge transformations (for a non-Abelian gauge group), which are not continiously deformable to the identity, the ratio of the CS mass $m$ and the gauge coupling $g^2$ must be quantized \cite{int6}. Further we can study the problem of massive excitations. For spinor electrodynamics in three dimensions we have \newline \vspace{-6mm} \begin{eqnarray} L=L_g+L_f +L_{int} \label{int19} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} L_g=-(1/4)F^{\mu \nu}F_{\mu \nu}+ (\mu/4)\epsilon ^{\mu \nu }F_{\mu \nu}A_\alpha, ~~ F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu \label{int20} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} L_f=i\bar{\psi}\hat{\partial}\psi-m\bar{\psi}\psi \label{int21} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} L_{int}=-J^\mu A_\mu,~~ J^\mu=-e\bar{\psi}\gamma^\mu \psi \label{int22} \end{eqnarray} \vspace{-6mm} \newline The coupling constant $e$ has dimension $(mass)^{-1/2}$. The equations of motion will be \newline \vspace{-6mm} \begin{eqnarray} \partial_\mu F^{\mu \nu}+ (\mu /4)\epsilon ^{\nu \mu \alpha}F_{\mu \alpha }=J^\mu \label{int23} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} (i\bar{\partial}+e\bar{A}-m) \psi =0\label{int24} \end{eqnarray} \vspace{-6mm} \newline One can introduce the dual field strength tensor in 3-D space-time \newline \vspace{-6mm} \begin{eqnarray} ^*F^\mu=(1/2)\epsilon ^{\mu \alpha \beta}F_{\alpha \beta}~~ F^{\alpha \beta}={\epsilon ^{ \alpha \beta \mu}}{^*}F_\mu \label{int25} \end{eqnarray} \vspace{-6mm} \newline The Bianchi identity follows from (\ref{int23}): \newline \vspace{-6mm} \begin{eqnarray} {\partial_\mu}{^*F^\mu}=0 \label{int26} \end{eqnarray} \vspace{-6mm} \newline and the equation (\ref{int23}) may be written in a dual form \newline \vspace{-6mm} \begin{eqnarray} {\partial_\alpha}{^*F_\beta}-{\partial_\beta}{^*F_\alpha}- \mu F_{\alpha \beta}=-\epsilon _{\alpha \beta \mu}J^\mu \label{int27} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} (\Box+ \mu^2){^*F^\mu}=\mu\left(\eta^{\mu \nu} -\epsilon^{\mu \nu \alpha} \frac{\partial_\alpha}{\mu}\right)J_\nu \label{int28} \end{eqnarray} \vspace{-6mm} \newline This equation demonstrates that the gauge excitations are massive. \vspace{3mm} {\large \it 3-D gravity. Connection with topology} \vspace{3mm} The results of the topological part of the introduction gives us a hint how to construct the topological term for three dimensional gravity from a four-dimensional $^*RR$ Pontryagin density. \newline \vspace{-6mm} \begin{eqnarray} ^*RR=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu \rho \sigma} {R_{\alpha \beta}}^{\rho \sigma}=\partial_\mu X^\mu \label{int29} \end{eqnarray} \vspace{-6mm} \newline Let us find $X^\mu$ from (\ref{int29}). To do this we will rewrite $^*RR$ in the following way \newline \vspace{-6mm} \begin{eqnarray} ^*RR=(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu \rho \sigma} {R_{\alpha \beta}}^{\rho \sigma} =(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} {R_{\alpha \beta}}^{ab} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(1/2)\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} \left\{ \partial_\alpha {\omega_\beta}^{ab}-\partial_\beta {\omega_\alpha}^{ab} +{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} -{\omega _\beta }^{ac}{\omega _{\alpha c}}^{b} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} \left\{ \partial_\alpha {\omega_\beta}^{ab} +{\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \right\} \label{int30} \end{eqnarray} \vspace{-6mm} \newline In our calculations we used the following expression for the Ricci connection \newline \vspace{-6mm} \begin{eqnarray} R_{\mu \nu \rho \sigma} =\partial_\alpha {\omega_{\beta a b}}-\partial_\beta {\omega_{\alpha a b}} +{\omega_{\alpha a}}^{c}{\omega_{\beta c}}_{b} -{\omega _{\beta a}}^{c}{\omega _{\alpha c}}_{b} \label{int31} \end{eqnarray} \vspace{-6mm} \newline where $\omega_{\mu ab}$ is 3-D spin connection. Then the expression for Pontryagin density will be \newline \vspace{-6mm} \begin{eqnarray} ^*RR=\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} ~\partial_\alpha {\omega_\beta}^{ab} +\epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} ~ {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int32} \end{eqnarray} \vspace{-6mm} \newline Let us find these two contributions separately. The first contribution to (\ref{int32}) gives \newline \vspace{-6mm} \begin{eqnarray} \epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} ~\partial_\alpha {\omega_\beta}^{ab}= \epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} {\omega_\beta}^{ab})- \epsilon^{\mu \nu \alpha \beta} {\omega_\beta}^{ab}\partial_\alpha R_{\mu \nu ab} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} {\omega_\beta}^{ab}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -\epsilon^{\mu \nu \alpha \beta} {\omega_\beta}^{ab}\partial_\alpha \left\{ \partial_\mu \omega_{\nu ab}-\partial_\nu \omega_{\mu ab} +\omega_{\mu a}^c \omega_{\nu cb}\omega_{\nu a}^c \omega_{\mu cb} \right\} \label{int33} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} {\omega_\beta}^{ab})-2\epsilon^{\mu \nu \alpha \beta} {\omega_\beta}^{ab}\partial_\alpha \left\{ \omega_{\mu a}^c \omega_{\nu cb} \right\} \label{int34} \end{eqnarray} \vspace{-6mm} \newline The second contribution to (\ref{int32}) will be \newline \vspace{-6mm} \begin{eqnarray} \epsilon^{\mu \nu \alpha \beta}R_{\mu \nu ab} ~ {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}= \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -\epsilon^{\mu \nu \alpha \beta} \left\{ \partial_\mu \omega_{\nu ab}-\partial_\nu \omega_{\mu ab} +\omega_{\mu a}^c \omega_{\nu cb}-\omega_{\nu a}^c \omega_{\mu cb} \right\} {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =2\epsilon^{\mu \nu \alpha \beta} \partial_\mu \omega_{\nu ab} {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} +2\epsilon^{\mu \nu \alpha \beta}\omega_{\mu a}^c \omega_{\nu cb} {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int35} \end{eqnarray} \vspace{-6mm} \newline On the other hand the second one to (\ref{int35}) is zero, then we have \newline \vspace{-6mm} \begin{eqnarray} ^*RR=\epsilon^{\mu \nu \alpha \beta}\partial_\alpha (R_{\mu \nu ab} {\omega_\beta}^{ab}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -2\epsilon^{\mu \nu \alpha \beta} {\omega_\beta}^{ab}\partial_\alpha \left\{ \omega_{\mu a}^c \omega_{\nu cb} \right\}+2\epsilon^{\mu \nu \alpha \beta} \partial_\mu \omega_{\nu ab} {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b} \label{int36} \end{eqnarray} \vspace{-6mm} \newline One can find that \newline \vspace{-6mm} \begin{eqnarray} -2\epsilon^{\mu \nu \alpha \beta} {\omega_\beta}^{ab}\partial_\alpha \left\{ \omega_{\mu a}^c \omega_{\nu cb} \right\} +2\epsilon^{\mu \nu \alpha \beta} \partial_\mu \omega_{\nu ab} {\omega_\alpha}^{ac}{\omega_{\beta c}}^{b}\ \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =2\epsilon^{\mu \nu \alpha \beta} \partial_\mu \omega_{\nu a}^b {\omega_\alpha b}^{c}{\omega_{\beta c}}^{a} \label{int37} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} 2\epsilon^{\mu \nu \alpha \beta} \partial_\mu \omega_{\nu a}^b {\omega_\alpha b}^{c}{\omega_{\beta c}}^{a} =(2/3)\epsilon^{\mu \nu \alpha \beta} \partial_\mu \left\{ \omega_{\nu a}^b {\omega_\alpha b}^{c}{\omega_{\beta c}}^{a} \right\} \label{int38} \end{eqnarray} \vspace{-6mm} \newline In the result we will have \newline \vspace{-6mm} \begin{eqnarray} ^*RR=\partial_\mu \epsilon^{\mu \nu \alpha \beta} \left\{ {\omega_\nu}^{ab}R_{\alpha \beta ab}+ (2/3)\omega_{\nu a}^b {\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a} \right\} \label{int39} \end{eqnarray} \vspace{-6mm} \newline Therefore \newline \vspace{-6mm} \begin{eqnarray} X^\mu=\epsilon^{\mu \nu \alpha \beta} \left\{ {\omega_\nu}^{ab}R_{\alpha \beta ab}+ (2/3){\omega_{\nu a}}^b {\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a} \right\} \label{int40} \end{eqnarray} \vspace{-6mm} \newline Let parameter $\mu$ be equal to zero, then determining $\epsilon ^{0 \nu \alpha \beta}=\epsilon ^{\nu \alpha \beta}$ with $\nu, \alpha, \beta=1,2,3$ we get the CS action in the form \newline \vspace{-6mm} \begin{eqnarray} S_{CS}\sim \int d^3x X^0=\int d^3x\epsilon^{\nu \alpha \beta} \left\{ {\omega_\nu}^{ab}R_{\alpha \beta ab}+ (2/3){\omega_{\nu a}}^b {\omega_{\alpha b}}^{c}{\omega_{\beta c}}^{a} \right\} \label{int41} \end{eqnarray} \vspace{-6mm} \newline As we can see from (\ref{int41}) the CS term is of the third derivative order in contrast to the first one as in the vector case (\ref{int3}). \vspace{3mm} {\large \it Nonlinear theory of gravity} \vspace{3mm} We can construct total gravitational action in the form \newline \vspace{-6mm} \begin{eqnarray} S_{tot}=(1/k^2) \int d^3x \sqrt{g}R+(1/k^2\mu)S_{CS}\label{int42} \end{eqnarray} \vspace{-6mm} \newline The sign of the Einstein part is opposite to the conventional one in four dimensions. The Einstein part of the action has coefficient $k^{-2}$ with the dimension of mass, while the topological part has a dimensionless coefficient ($\mu$ has a dimension of mass). Now we can find some interesting properties from this action. Varying (\ref{int42}) with respect to the metric, we get field equations \newline \vspace{-6mm} \begin{eqnarray} \Theta^{\mu \nu}\equiv G^{\mu \nu} +(1/\mu)C^{\mu \nu}=0 \label{int43} \end{eqnarray} \vspace{-6mm} \newline where the second rank Weyl tensor $C^{\mu \nu}$ is \newline \vspace{-6mm} \begin{eqnarray} C^{\mu \nu}=(1/\sqrt{g}) \epsilon ^{\mu \alpha \beta} D_\alpha \tilde{R}^\nu _\beta \label{int44} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} \tilde{R}_{\alpha \beta} =R_{\alpha \beta}-(1/4)g_{\alpha \beta}R, ~~ R=R^\alpha_\alpha\label{int45} \end{eqnarray} \vspace{-6mm} \newline The components of the Einstein tensor $G^{\alpha \beta}$ are \newline \vspace{-6mm} \begin{eqnarray} G^{\alpha \beta} =R^{\alpha \beta}-(1/2)g^{\alpha \beta}R, \label{int46} \end{eqnarray} \vspace{-6mm} \newline and the components of the Riemann tensor ${R^\alpha}_{\beta \gamma \delta}$ are \newline \vspace{-6mm} \begin{eqnarray} {R^\alpha}_{\beta \gamma \delta}= \partial _\delta{\Gamma^\alpha}_{\beta \gamma } -\partial _\gamma {\Gamma^\alpha}_{\beta \delta} +{\Gamma^\alpha}_{\mu \gamma }{\Gamma^\mu}_{\beta \delta} -{\Gamma^\alpha}_{\mu \delta}{\Gamma^\mu}_{\beta \gamma} \label{int47} \end{eqnarray} \vspace{-6mm} \newline From (\ref{int43}) we get first order form for field equations \newline \vspace{-6mm} \begin{eqnarray} {K_{\mu \nu}}^{\lambda \sigma} (\mu) R_{\lambda \sigma}=0 \label{int48} \end{eqnarray} \vspace{-6mm} \newline where ${K_{\mu \nu}}^{\lambda \sigma}$ is operator of the form \newline \vspace{-6mm} \begin{eqnarray} {K_{\mu \nu}}^{\lambda \sigma} =(\delta^\lambda _\mu \delta^\lambda _\nu -(1/2)g_{\mu \nu}g^{\lambda \sigma})+ \frac{1}{\mu \sqrt{g}}{\epsilon _\mu}^{\alpha \beta} (\delta^\lambda _\beta \delta^\sigma _\nu -(1/2)g^{\lambda \sigma}g^{\nu \beta}) \label{int49} \end{eqnarray} \vspace{-6mm} \newline Operator ${K_{\mu \nu}}^{\lambda \sigma}(\mu)$ may be multiplied by ${K_{\mu \nu}}^{\lambda \sigma}(- \mu)$ to yield a second-order equation for Ricci tensor. From (\ref{int48}) we find \newline \vspace{-6mm} \begin{eqnarray} {K_{\alpha \beta}}^{\mu \nu}(-\mu){K_{\mu \nu}}^{\lambda \sigma}(\mu) R_{\lambda \sigma}=0 \label{int50} \end{eqnarray} \vspace{-6mm} \newline that is \newline \vspace{-6mm} \begin{eqnarray} (D_\alpha D^\alpha +\mu^2)R_{\mu \nu}=-g_{\mu \nu}R^{\alpha \beta} R_{\alpha \beta}+ 3R^\alpha _\beta R_\alpha ^\beta \label{int51} \end{eqnarray} \vspace{-6mm} \newline This exhibits the massive character of the excitations. After this short review of topological field models we will consider 3-D fermionic models interacting with vector and with tensor fields. These models may give effective induced topological action of the CS type. \chapter{LOCAL STATISTICS AND} \centerline{\Large \bf THERMODYNAMICS OF FERMI GAS } \vspace{24pt} The results of the previous chapter XII show the way to construct thermodynamic potentials of quantum systems with a variable number of particles. In this chapter we will develop this formalism for fermi systems. \section {Grand thermodynamical potential and} \vspace{-4mm} \hspace{20mm}{\Large \bf low temperature properties of Fermi gases} \vspace{1mm} An equivalent to the Schwinger-DeWitt representation, the momentum space representation of the bi-spinor $G_F(x,x^{'})$ has the following form (\ref{fer14}): \newline \vspace{-6mm} \begin{eqnarray} G_F(x,x^{'})=G_F(x,y)=g^{-\frac{1}{ 4}}(y)\sum\limits_{j=0}^2 {\hat \alpha _j(x,y)\left( {-{\partial \over {\partial m^2}}} \right)^j}G_0(y) \label{tf1} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} G_0(y)=\int {{{d^4k} \over {(2\pi )^4}}}\,{{e^{iky}} \over {k^2+m^2}}\label{tf2} \end{eqnarray} \vspace{-6mm} \newline To introduce temperature, we will extend the time $y^{0}$ coordinate of the tangential space $\left\{{y^\mu}\right\}$ to the imaginary interval $[0,-i\beta ]$ and will consider fermionic field to be antiperiodic on that interval. Then, in imaginary time formalism, we can write \newline \vspace{-6mm} \begin{eqnarray} \int {{{d^4k} \over {(2\pi )^4}}}{1 \over {k^2+m^2}}\buildrel \beta \over \longrightarrow {i \over {\beta }}\int {{{d^3k} \over {(2\pi )^3}}}\sum\limits_{n=-\infty }^\infty {{1 \over {\omega _n^2+\varepsilon ^2}}} \label{tf3} \end{eqnarray} \vspace{-6mm} \newline where $\omega _n=\pi T(2n+1),\quad n=0,\;\pm 1,\;\pm 2,\quad .\;.\;.$ are Matsubara frequencies. To take into consideration the chemical potential we will shift the frequencies by $\omega _n \to \omega _n+\mu $ \cite{mor1,kap1} Making the summation in (\ref{tf3}) we will find $G_0(y)$ in the limit of coincidence $x=x'$ in the form \newline \vspace{-6mm} \begin{eqnarray} \mathop{\lim } \limits_{x\to x^{'}} G_0\left( y \right)\mathop = \limits^\beta {i\over \beta } \int {{{d^3k} \over {\left( {2\pi } \right)^3}}}\sum\limits_{n=-\infty }^\infty {{1 \over {-\left( {\omega _n+\mu }\right)^2 + \varepsilon ^2}}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={{i \over \beta }\int {{{d^3k} \over {\left( {2\pi } \right)^3}}\left\{ {{1 \over {2\varepsilon }}\sum\limits_{n=-\infty }^\infty {\left( {{{\varepsilon -\mu } \over {\left( {\varepsilon -\mu } \right)^2-\omega_{n}^2}}+{{\varepsilon +\mu } \over {\left( {\varepsilon +\mu } \right)^2-\omega_{n}^2}}} \right)}} \right\}}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} ={\int {{{d^3k} \over {\left( {2\pi } \right)^3}}}\left\{ {{i \over {2\varepsilon }}\left( {{1 \over 2}-{1 \over {\exp \beta \left( {\varepsilon -\mu } \right)+1}}} \right)+{i \over {2\varepsilon }}\left( {{1 \over 2}-{1 \over {\exp \beta \left( {\varepsilon +\mu } \right)+1}}} \right)} \right\}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mathop {\lim }\limits_{x\to x^{'}}\left[ {G_\beta ^+\left( y \right)+G_\beta ^-\left( y \right)} \right] \label{tf4} \end{eqnarray} \vspace{-6mm} \newline This equation describes the temperature contributions for particles $(\mu)$ and antiparticles $(-\mu)$ separately. As a result one can find the expression for the finite temperature contribution to the Green's function for a fermion with non-zero chemical potential \newline \vspace{-6mm} \begin{eqnarray} G_\beta (x,\mu )=\int {{{d^3k} \over {(2\pi )^3}}}\sum\limits_{j=0}^2 {\hat \alpha _j}(R)\left( {{\partial \over {\partial m^2}}} \right)^j\left( {1+ze^{-\beta \varepsilon }} \right)^{-1} \label{tf5} \end{eqnarray} \vspace{-6mm} \newline Then the density of grand thermodynamical potential for fermions may be written as \newline \vspace{-6mm} \begin{eqnarray} \omega (\beta ,\mu ,R)=-{i \over 2} \mbox{tr} \int\limits_{m^2}^\infty {dm^2G_\beta (x,\mu )}=\sum\limits_{j=0}^2 {\alpha _j}(R)f_j(\beta m;z), \label{tf6} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} f_0(\beta m;z)=-{s \over \beta }\int {{{d^3k} \over {(2\pi )^3}} \ln (1+ze^{-\beta \varepsilon })} \label{tf7} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} f_j(\beta m;z)=\left( {-{\partial \over {\partial m^2}}} \right)^jf_0(\beta m;z),\label{tf8} \end{eqnarray} \vspace{-6mm} \newline where $z=e^{\beta \mu }$ is the fugacity and factor $s=2$ (spin up, down). The coincidence of the finite temperature Schwinger-DeWitt and momentum space methods for calculation of the densities of themodynamical potentials is obvious for $\mu =0$. Using the equations for thermodynamic potentials one can obtain some interesting properties of an ideal fermi gas in an external gravitational field : 1) The Fermi distribution function of the gas in the gravitational field may be found from the expression \newline \vspace{-6mm} \begin{eqnarray} n_{\vec{k}}=-{{\partial \omega _{\bar k}(\beta ,\mu ,R)} \over {\partial \mu }} \nonumber \end{eqnarray} \vspace{-6mm} \newline for occupation numbers with momentum $\vec k$. It has the form \newline \vspace{-6mm} \begin{eqnarray} n_{\vec{ k}}={1 \over {(z^{-1}e^{\beta \varepsilon _{\vec{ k}}}+1)}}F(\beta ,R),\label{tf9} \end{eqnarray} \vspace{-6mm} \newline where the function $F(\beta ,R)$ is described by the formula \newline \vspace{-6mm} \begin{eqnarray} F(\beta ,R)=\ {1+\alpha _1(R){\beta \over {2\varepsilon _{\vec{k}}}}\left[{1-(1+ze^{-\beta \varepsilon _{\vec{k}}})^{-1}}\right]+\quad .\;.\;.} \label{tf10} \end{eqnarray} \vspace{-6mm} \newline and depends on curvature, temperature and energy of the fermion. 2) We can estimate the dependence of the chemical potential on the curvature of space time in non-relativistic approximation. Let \newline \vspace{-6mm} \begin{eqnarray} \varepsilon _{\vec{ k}}=\frac{\vec{k}^{2}}{ 2m} \end{eqnarray} \vspace{-6mm} \newline then from \newline \vspace{-6mm} \begin{eqnarray} n=-{{\partial\omega (\beta,\mu,R)} \over {\partial\mu }} \end{eqnarray} \vspace{-6mm} \newline we find the equation \newline \vspace{-6mm} \begin{eqnarray} \frac{n\lambda ^3(T)}{s}=f_{3\over 2}(z,R),\label{tf11} \end{eqnarray} \vspace{-6mm} \newline where $\lambda =\left( {2\pi/ mT} \right)^{1/ 2}$ is the thermal wave length of the particle, and \newline \vspace{-6mm} \begin{eqnarray} f_{3/ 2}(z,R)=f_{3/2}(z)\left[\alpha _0-{3 \over4}{{\alpha _1(R)} \over {m^2}}-{3 \over {16}}{{\alpha _2(R^2)} \over {m^4}}-...\right]\label{tf12} \end{eqnarray} \vspace{-6mm} \newline is some function with respect to $z$, and $n$ is the density of Femi gas. The function $f_{\frac{3}{2}}(z)$ is \newline \vspace{-6mm} \begin{eqnarray} f_{3/2}(z)=\sum\limits_{n=1}^\infty {{{z^n} \over {n^{3/ 2}}}}\,(-1)^{n+1}={4 \over {\sqrt \pi }}\int\limits_0^\infty {dx{{x^2} \over {z^{-1}\exp (x^2)+1}}}.\label{tf13} \end{eqnarray} \vspace{-6mm} \newline The equation (\ref{tf11}) may be solved with graphical methods. As we can see in Fig. I-4 the fugacity (chemical potential) depends on the curvature R of the space time. 3) The explicit expression for the chemical potential at low temperatures and high densities ($n\lambda ^3>>1$), where quantum effects are essential, may be found by the calculations of (\ref{tf11}) with the following representation of the function $f_{3/2}(z)$ for large $z$ \cite{huang1} \newline \vspace{-6mm} \begin{eqnarray} f_{3/2}(z) = {4 \over {3\sqrt \pi }} \left[ {\left( {\ln z} \right)^{3/2} + {{\pi ^2} \over 8} \left( {\ln z} \right)^{-1/2} + \quad \ldots} \right] + O(z^{-1}).\label{tf14} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{tf12}) into (\ref{tf11}) and taking into account only the first term in (\ref{tf14}) we find the Fermi energy of a gas of fermions in curved space-time \newline \vspace{-6mm} \begin{eqnarray} \left( {3n\sqrt \pi \over 4s} \right)^{2/3}\lambda ^2\left[1+{1 \over {24}}{R \over {m^2}}+\quad .\;.\;.\right]=\beta \varepsilon _F(R),\label{tf15} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \varepsilon _F(R)=\varepsilon _F^{(0)}\left[1+{1 \over {24}}{R \over {m^2}}+\quad .\;.\;.\right],\label{tf16} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \varepsilon _F^{(0)}=\left( {6\pi ^2n\over s} \right)^{2\over 3} \left( {1\over 2m} \right) \end{eqnarray} \vspace{-6mm} \newline is the Fermi energy at the Euclidean space. Taking into account the second term in (\ref{tf14}) we get the expression for chemical potential \newline \vspace{-6mm} \begin{eqnarray} \mu (T,R)=\varepsilon _F(R)\left\{ {1-{{\pi ^2} \over {12}}\left( {{T \over {\varepsilon _F(R)}}} \right)^2+\quad .\;.\;.} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\varepsilon _F^{(0)}\left\{ {1+{1 \over {24}}{R \over {m^2}}-{{\pi ^2} \over {12}}\left( {{T \over {\varepsilon _F^{(0)}}}} \right)^2+\quad .\;.\;.} \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \mu (T,R)=\mu ^{(0)}(T)+{1 \over {24}}{R \over {m^2}}\varepsilon _F^{(0)}+\quad .\;.\;.\; \label{tf17} \end{eqnarray} \vspace{-6mm} \newline which describes the explicit dependence of the chemical potential of the fermionic gas on temperature T and the curvature R of space-time. \vspace{3mm} \centerline{\Large\bf Conclusion} \vspace{3mm} The thermodynamical potentials of quantum bose and fermi gases (as local thermodynamical objects in a curved space-time) were rewritten in terms of the finite temperature Green's functions of bosons and fermions by means of the local momentum space formalism. The phenomenon of Bose condensation was studied and the critical temperature of condensation as a functional of curvature was found (\ref{bos23}). The non-thermal character of Bose and Fermi distribution functions (\ref{ins1}) and (\ref{tf9}) was shown. The dependence of Fermi energy (\ref{tf16}) and the chemical potential (\ref{tf17}) of a fermi gas on the curvature of space-time was computed for low temperatures. The dependence of chemical potential of bose gas on the curvature of space-time was analyzed. It was found that the temperature is a local thermodynamical characteristic of thermal systems in external gravitational fields. \chapter{RENORMALIZATIONS IN } \centerline{\Large \bf LOCAL STATISTICAL MECHANICS } \vspace{24pt} \section{Divergences of finite temperature} \vspace{-4mm} \hspace{22mm}{\Large \bf field models} \vspace{1mm} As one can show by calculations, the contributions $G_{SD}(x,x)$ in the expressions (\ref{i16}) and (\ref{j8}) are divergent \cite{key16}. To have a clear picture of the model under consideration we need to eliminate these divergent contributions. The direct way to eliminate the divergences is to combine the gravitational Lagrangian and the divergent parts of the matter Lagrangian. 1) Renormalization of $a$ $bose$ $field$. As was shown in (\ref{i21}) the effective Lagrangian for gravitational field can be written as \newline \vspace{-6mm} \begin{eqnarray} \tilde{L}_g = L_g(x)+(-i/2)\int \limits_{m^2}^\infty \mbox{tr} G_{SD}(x,x^{'}) \label{l1} \end{eqnarray} \vspace{-6mm} \newline Inserting (\ref{h46}) into the last term of the expression (\ref{l1}), we find \newline \vspace{-6mm} \begin{eqnarray} (-i/2)\int \limits_{m^2}^\infty \mbox{tr} G_{SD}(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{2(4\pi)^2}\sum \limits_{j=0}^\infty \gamma_j(x) \int \limits_0^\infty ids(is)^{j-3} \exp (-ism^2) \label{l2} \end{eqnarray} \vspace{-6mm} \newline We will use the procedure of dimensional regularization \cite{hv1} to select divergent terms of the expression (\ref{l2}). It gives \newline \vspace{-6mm} \begin{eqnarray} (-i/2)\int \limits_{m^2}^\infty \mbox{tr} G_{SD}(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{2}(4\pi)^{-n/2} \sum \limits_{j=0}^\infty \gamma_j(x) \int ids(is)^{(j-1-n/2)} \exp (-ism^2) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{2}(4\pi)^{-n/2}\left(\frac{m}{M}\right)^{n-4} \sum \limits_{j=0}^\infty \gamma_j(x) m^{4-2j}\Gamma\left(j-n/2\right) \label{l3} \end{eqnarray} \vspace{-6mm} \newline where $n$ is a dimension of the space-time and $M$ is an arbitrary mass scale. This parameter is introduced to preserve the dimension of the Lagrangian to $[L]^{-4}$ for the dimensions $n\neq 4$. The functions $\Gamma(z)$ have the poles at $n=4$: \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {-{n \over 2}} \right)= {4 \over {n(n-2)}}\left( {{2 \over {4-n}}-\gamma } \right)+O(n-4) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {1-{n \over 2}} \right) ={2 \over {2-n}}\left( {{2 \over {4-n}}-\gamma } \right)+O(n-4) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \Gamma \left( {2-{n \over 2}} \right)={2 \over {4-n}}-\gamma +O(n-4) \label{l4} \end{eqnarray} \vspace{-6mm} \newline Therefore the first three terms of the expression (\ref{l3}) are divergent. Selecting divergent parts and using the logarithmic expression for \newline \vspace{-6mm} \begin{eqnarray} \left(\frac{m}{M}\right)^{n-4}=1+\frac{1}{2} \ln \frac{m^2}{M^2}+O((n-4)^2) \label{l5} \end{eqnarray} \vspace{-6mm} \newline write the divergent contributions in the effective Lagrangian $\tilde{L}_g$ in the form \newline \vspace{-6mm} \begin{eqnarray} (-i/2)\int \limits_{m^2}^\infty \mbox{tr} G_{SD}(x,x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-(4\pi)^{-n/2}\left\{\frac{1}{n-4} +\frac{1}{2}\left[j+\ln \left(\frac{m^2}{M^2}\right)\right]\right\}\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \left[\frac{4m^2}{n(n-4)}-\frac{2}{n-2}m^2\gamma_1(x^{'}) +\gamma_2(x^{'})+...\right] \label{l6} \end{eqnarray} \vspace{-6mm} \newline Combining (\ref{l6}) with the gravitational Lagrangian (\ref{i3}) we can redetermine the constants of the gravitational Lagrangian (\ref{l1}) as \newline \vspace{-6mm} \begin{eqnarray} {1 \over {8\pi G_R}}\Lambda _R ={1 \over {8\pi G_0}}\Lambda _0-{1 \over {(4\pi )^2}} {1 \over {(n-4)}}{{m^4} \over 2}\vspace{-17mm} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} {1 \over {16\pi G_R}}={1 \over {16\pi G_0}} +{1 \over {(4\pi )^2}}{1 \over {n-4}}m^2 \left( {{1 \over 6}-\xi } \right) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \alpha _R=\alpha _0-{1 \over {(4\pi )^2}} {1 \over {n-4}}\left( {{1 \over 6}-\xi } \right)^2\nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \beta _R=\beta _0+{1 \over {180}}{1 \over {(4\pi )^2}}{1 \over {n-4}}\nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \gamma _R=\gamma _0-{1 \over {180}}{1 \over {(4\pi )^2}}{1 \over {n-4}} \label{l7} \end{eqnarray} \vspace{-6mm} \newline where $G_0$, $\Lambda_0$, $\alpha_0$, $\beta_0$, $\gamma_0$ are bare constants and $G_R$, $\Lambda_R$, $\alpha_R$, $\beta_R$, $\gamma_R$ are the physical (finite) constants. Since all divergences of the matter field can be included in $\tilde{L}_g$, only the finite temperature contributions observed in (\ref{i20}) remain. 2) Renormalization of $a$ $fermi$ $field$. As follows from (\ref{j13}) the effective Lagrangian of the gravitational field is \newline \vspace{-6mm} \begin{eqnarray} \tilde{L}_g=L_g-\frac{1}{2}(4 \pi)^{-n/2}\sum\limits_{j=0}^\infty \mbox{tr} \hat {\alpha}_j(R)\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \int\limits _0 ^\infty ids (is)^{j-n/2-1} \exp (-ism^2) \label{l8} \end{eqnarray} \vspace{-6mm} \newline After dimensional regularization procedure (as in scalar field case), we get \newline \vspace{-6mm} \begin{eqnarray} {1 \over {8\pi G_R}}\Lambda _R ={1 \over {8\pi G_0}}\Lambda _0+{1 \over {(4\pi )^2}}{{2m^4} \over {n-4}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} {1 \over {16\pi G_R}}={1 \over {16\pi G_0}} -{1 \over {(4\pi )^2}}{1 \over {(n-4)}}{{m^2} \over 6} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \alpha _R=\alpha _0+{1 \over {(4\pi )^2}}{1 \over {(n-4)}}{1 \over {144}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \beta _R=\beta _0-{1 \over {(4\pi )^2}}{1 \over {(n-4)}}{1 \over {90}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \gamma _R=\gamma _0-{1 \over {(4\pi )^2}}{1 \over {(n-4)}}{7 \over {720}} \label{l9} \end{eqnarray} \vspace{-6mm} \newline Thus, the Lagrangian (\ref{j12}) does not include the divergencies in the finite temperature loop approximation over the fermi fields. The method of dimensional regularization developed here is not the only one for applications to field models in background curved space-time. The $\zeta$-function method \cite{haw1}, \cite{hurt1} and the method of covariant geodesic point separation also lead to a solution of the problem of regularization of effective Lagrangians of matter fields and calculation of Energy-momentum tensor anomalies \cite{christ1}. \chapter{NON-PERTURBATIVE EFFECTS} \centerline{\Large \bf IN GROSS-NEVEU MODEL } \vspace{24pt} Quantum Field Theory may be essentially symplified in the limit of high interanal symmetries such as O(N), SU(N) and so on. In some appropriate cases field models can be solved strictly in the limit of large flavor numbers N \cite{top10}. Asymptotically free Gross-Neveu model without dimensional parameters in the Lagrangian is one of such models. This model is renormalizable in 3-D dimensions. The solution of this model shows that the phenomenon of dimensional transmutation \cite{top11} has place and there is a gap in mass spectrum of the model. Studying this model one will find the connection between topological characteristics of space time and the behavior of the solution of the mass gap equation and also the behavior of dinamical mass of the model. N-flavor Gross-Neveu model is described by the Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L=\bar \psi _ii\hat \partial \psi _i+{{g^2} \over 2} \left( {\bar \psi _i\psi _i} \right)^2 \label{f1} \end{eqnarray} \vspace{-6mm} \newline This Lagrangian is invariant under the discrete transformations: \newline \vspace{-6mm} \begin{eqnarray} \psi \to \gamma _5\psi,~~\bar{\psi} \to -\bar{\psi}\gamma _5 \label{f2} \end{eqnarray} \vspace{-6mm} \newline The generating functional of the model \newline \vspace{-6mm} \begin{eqnarray} Z[\eta,\bar{\eta}]=\int D\psi D\bar{\psi} \exp \left[i\int d^nx\left(i\bar{\psi}\partial \psi +(1/2) g^2(\bar{\psi}\psi)^2 +\bar{\eta}\psi+\bar{\psi}\eta\right)\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline may be rewritten in the form: \newline \vspace{-6mm} \begin{eqnarray} Z[\eta,\bar{\eta},\sigma]=\int D\psi D\bar{\psi}D\sigma \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp \left[i\int d^nx\left(i\bar{\psi}\partial \psi - g(\bar{\psi}\psi)\sigma-(1/2)\sigma^2 +\bar{\eta}\psi+\bar{\psi}\eta\right)\right] \label{f3} \end{eqnarray} \vspace{-6mm} \newline Here we introduced the new field $\sigma$ and used a useful relation \newline \vspace{-6mm} \begin{eqnarray} \int D\sigma \exp \left[-i(1/2)\left( \sigma, \sigma \right) -i\left(g\bar{\psi}\psi,\sigma\right)\right] \propto \exp \left[(i/2)g^2 \left (\bar{\psi}\psi\right) ^2 \right] \label{f4} \end{eqnarray} \vspace{-6mm} \newline Since \newline \vspace{-6mm} \begin{eqnarray} m\bar{\psi}\psi \to m\left(-\bar{\psi}{\gamma_5}^2\psi \right) =-m\bar{\psi}\psi \label{f5} \end{eqnarray} \vspace{-6mm} \newline we must put $m=0$ for symmetry of the model. Therefore a new Lagrangian may be written in the form \newline \vspace{-6mm} \begin{eqnarray} L_\sigma =\bar \psi _ii\hat \partial \psi _i-\sigma \left( {\bar \psi _i\psi _i} \right)-{1 \over {2g^2}}\sigma ^2 \label{f6} \end{eqnarray} \vspace{-6mm} \newline and the symmetry of (\ref{f6}) is \newline \vspace{-6mm} \begin{eqnarray} \psi \to \gamma _5\psi,~~\bar{\psi} \to-\gamma _5\bar{\psi}, ~~\sigma \to -\sigma \label{f7} \end{eqnarray} \vspace{-6mm} \newline The generating functional of the model after integration over the matter fields will be \newline \vspace{-6mm} \begin{eqnarray} Z[0]=\int {D\bar \psi D\psi D\sigma }\exp \,i\int {L_\sigma \left( x \right)}dx \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\int {D\sigma \cdot }Det(i\hat \partial -\sigma ) \exp [{{-i} \over {2g^2}}(\sigma ,\sigma )] \label{f8} \end{eqnarray} \vspace{-6mm} \newline Then effective action is written as \newline \vspace{-6mm} \begin{eqnarray} \Gamma[\sigma ]=-i\ln \,Det(i\hat \partial -\sigma ) -{1 \over {2g^2}}(\sigma ,\sigma ) \label{f9} \end{eqnarray} \vspace{-6mm} \newline and the effective potential is \newline \vspace{-6mm} \begin{eqnarray} V_{eff}={{iN} \over 2}(\mbox{Tr} \hat 1)\int {{{d^2k} \over {\left( {2\pi } \right)^2}}}\ln \left( {k^2-\sigma ^2} \right) +{1 \over {2g^2}}\sigma ^2 \label{f10} \end{eqnarray} \vspace{-6mm} \newline The conditions for the energy to be minimal are \newline \vspace{-6mm} \begin{eqnarray} \left({{\delta V_{eff}} \over {\delta \sigma }}\right)_{|\sigma =\sigma _c}=0,\quad \left({{\delta ^2V_{eff}} \over {\delta \sigma ^2}}\right)_{|\sigma =\sigma _c}>0 \label{f11} \end{eqnarray} \vspace{-6mm} \newline From these conditions one can get the gap equation for definition of $\sigma _c$ ($\sigma _c$ defines the minimum of the effective potential) in the form \newline \vspace{-6mm} \begin{eqnarray} {1 \over \lambda }=\mbox{Tr} \hat 1\int {{{d^2\bar k} \over {\left( {2\pi } \right)^2}}}{1 \over {\bar k^2+\sigma _c^2}} \label{f12} \end{eqnarray} \vspace{-6mm} \newline where constant $\lambda =g^2N$. \section {Trivial case. Euclidean space time.} \vspace{1mm} Let us find the solution of the gap equation (\ref{f12}). Ultraviolet cut-off of the integral gives: \newline \vspace{-6mm} \begin{eqnarray} \int_{-\Lambda}^{\Lambda} {{{d^2\bar k} \over {\left( {2\pi } \right)^2}}}{1 \over {\bar k^2+\sigma _c^2}} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =(1/2\pi)\int_{0}^{\Lambda} \frac{2\pi dk^2}{\bar k^2+\sigma _c^2} =(1/4\pi) \ln \frac{\Lambda^2}{\sigma^2} \label{f13} \end{eqnarray} \vspace{-6mm} \newline Then, after regularization of (\ref{f13}) we get the equation \newline \vspace{-6mm} \begin{eqnarray} {1 \over {\lambda \left( \Lambda \right)}} ={1 \over {2\pi }}\ln {{\Lambda ^2} \over {\sigma ^2}} \label{f14} \end{eqnarray} \vspace{-6mm} \newline Let the subtraction point be $\mu =\sigma $, then the renormalized coupling constant may be written as \newline \vspace{-6mm} \begin{eqnarray} {1 \over {\lambda \left( \mu \right)}} ={1 \over {2\pi }}\ln {{\mu ^2} \over {\sigma ^2}}\label{f15} \end{eqnarray} \vspace{-6mm} \newline To eliminate the parameter $\mu$ one can use the methods of the renormalization group. The $\beta$-function in one loop approximation is \newline \vspace{-6mm} \begin{eqnarray} \beta \left( {\lambda _R(\mu )} \right)=-{1 \over \pi } \left( {\lambda _R(\mu )} \right)^2 \label{f16} \end{eqnarray} \vspace{-6mm} \newline then Gell-Mann Low equation \newline \vspace{-6mm} \begin{eqnarray} {{d\lambda _R} \over {\lambda _R^2}}=-{1 \over \pi }{{d\xi } \over \xi } \label{f17} \end{eqnarray} \vspace{-6mm} \newline with initial condition $\lambda \left( {\xi _0} \right)=\lambda _0$ determines the behavior of the coupling constant with respect to scaling of the momentum: \newline \vspace{-6mm} \begin{eqnarray} \lambda _R( t )=\frac{\lambda _0}{1+(\lambda _0 /\pi )t} \label{f18} \end{eqnarray} \vspace{-6mm} \newline where $t=\ln ( \xi /\xi _0)$ The dynamical mass can be found from (\ref{f11}) in the form \newline \vspace{-6mm} \begin{eqnarray} \sigma _c(triv.)=\mu \exp (-{\pi \over {\lambda _R\left( \mu \right)}}) =\mu \exp \left( {-{\pi \over {\lambda _0}}} \right)=const.\label{f19} \end{eqnarray} \vspace{-6mm} \newline \section{ Non-trivial topology of space time.} \vspace{1mm} In this case the constant of interaction can be written as \newline \vspace{-6mm} \begin{eqnarray} {1 \over {\lambda \left( {\mu ,L} \right)}} ={1 \over {2\pi }}\left[\ln {{\mu ^2} \over {\sigma _c^2}}+f(L,\sigma _c) \right] \label{f20} \end{eqnarray} \vspace{-6mm} \newline where $f(\sigma _c,L)$ is some function which depends on topological parameter L. The $\beta$ function is \newline \vspace{-6mm} \begin{eqnarray} \beta \left( {\lambda _R(\mu ,L)} \right) =-\frac{1}{\pi }\left( {\lambda _R(\mu ,L)} \right)^2 \label{f21} \end{eqnarray} \vspace{-6mm} \newline and the solution of Gell-Mann-Low equation with $\lambda _R\left( {\xi _0,L} \right)=\lambda _0(L)$ is expressed by the equation: \newline \vspace{-6mm} \begin{eqnarray} \lambda _R\left( {t,L} \right)=\frac{\lambda _0(L)} {1+(\lambda _0(L) /\pi )t} \label{f22} \end{eqnarray} \vspace{-6mm} \newline The dynamical mass is governed by the equation \newline \vspace{-6mm} \begin{eqnarray} \sigma _c=\mu \exp \left( {-{1 \over {\lambda _0(\mu ,L)}} -{{f(\sigma _c,L)} \over 2}} \right) \label{f23} \end{eqnarray} \vspace{-6mm} \newline or \newline \vspace{-6mm} \begin{eqnarray} \sigma _c=\sigma _c(triv)\exp \left( {-{{f(\sigma _c,L)} \over 2}} \right) \label{f24} \end{eqnarray} \vspace{-6mm} \newline One can see that the dynamical mass depends on function $f(\sigma _c,L)$. The explicit expression of the function $f(\sigma _c,L)$ is \newline \vspace{-6mm} \begin{eqnarray} f_{(\pm )}(L,\sigma _c)=\pm {1 \over {\pi ^2}}\int\limits_0^\infty {dx{1 \over {\sqrt {x^2+(L\sigma _c)^2}}}} \left( {\exp \sqrt {x^2+(L\sigma _c)^2}-1} \right)^{-1} \label{f25} \end{eqnarray} \vspace{-6mm} \newline for topologies of the cylinder $(+)$ and the Mobius strip $(-)$. Therefore non-Euclidean structure of space time leads to redefinition of the gap equation (\ref{f12}) for the Gross-Neveu model. That gives us the possibility to define the dependence of the fermionic mass on topology of the space time. Dynamical violation of the $\gamma_5$ symmetry occurs when $\sigma$ is not equal zero. The method developed above is very useful when the number of "flavors" of the fundamental fields is big ($N \to \infty$). This method does not work for a small "flavor" number $N$. In Quantum Field Theory there is another method, based on an analogy with superconductivity. This is a Mean Field Method \cite{km1}, \cite{key14}, which works very well for any number of "flavors" of the particles. The idea of the method is based on effective potential\footnote{Effective potential is the generating functional for (1PI) Green's functions \cite{jak1,rs1}} calculations, that allows us to take into consideration the effects of topology \cite{toms1} and curvature for self-interacting and gauge models \cite{ish1}. In this work the Mean Field Method is used in dynamical modeling of the behavior of elementary particles and is based on the idea that the masses of compound particles (e.g. nucleons) are generated by the self-interaction of some fundamental fermion fields through the same mechanism as superconductivity. Here the combined particles are treated as the quasi-particles excitations. The Mean Field Method also leads to the mass gap equation, and the solution gives the dynamical mass of the particle. In the following section we will treat the problem of non-Euclidean space-time structure in models with a dynamical mass. \chapter{ $(\bar{\psi}\psi)^2$ NON-LINEAR SPINOR MODELS} \vspace{24pt} \section{Dynamical mass and} \vspace{-4mm} \hspace{33mm}{\Large \bf symmetry breaking} \vspace{1mm} In the construction of the unified models of the elementary particles one can admit the possibility that the mass of combined particles appears as the result of self-interaction of certain fundamental fields, for instance, quarks and leptons from preons, or compound fermions in technicolor models \cite{ro1,ta1}, \cite{ito1}, \cite{sd1}. Following this idea we can get the gap equation. Its solution can predict the dynamical mass of the compound particles. As in the previous case of Gross-Neveu model, we will study here the effects of non-trivial topology and, also, geometry of background space-time. In this section we will consider the phenomenon of the dynamical generation of mass of particles in the application to the models in non-trivial space-time. Let us consider Heisenberg-Ivanenko \cite{top12} non-linear spinor model with Lagrangian \newline \vspace{-6mm} \begin{eqnarray} L=\bar \psi i\hat \partial \psi +{{g_0^2} \over {2\mu _0^2}}\left( {\bar \psi \psi } \right)^2 \label{f26} \end{eqnarray} \vspace{-6mm} \newline where $g_0^2$ is a massless parameter and parameter $\mu_0^2$ has dimension which is connected with the dimension of space time. The symmetry of the Lagrangian (\ref{f26}) is \newline \vspace{-6mm} \begin{eqnarray} \psi \to \gamma _5\psi, ~~\bar \psi \to -\bar \psi \gamma _5 \label{f27} \end{eqnarray} \vspace{-6mm} \newline We can rewrite this Lagrangian in the new form \newline \vspace{-6mm} \begin{eqnarray} L_\sigma =\bar \psi i\hat \partial \psi -g_0\sigma \left( {\bar \psi \psi } \right)-{{\mu _0^2} \over 2}\sigma ^2 \label{f28} \end{eqnarray} \vspace{-6mm} \newline The symmetry of the last one is (\ref{f7}). The equation of motion for the $\sigma$ field is \newline \vspace{-6mm} \begin{eqnarray} \sigma =-{{g_0} \over {\mu _0^2}}\left( {\bar \psi \psi } \right) \label{f29} \end{eqnarray} \vspace{-6mm} \newline thus we may assume that the field $\sigma$ is a collective field. Let us consider that $\sigma$ is $\sigma \to \sigma +\tilde \sigma $, where \newline \vspace{-6mm} \begin{eqnarray} \sigma =(g_0 /{\mu _0}^2)<\psi \bar \psi >\ne 0 \nonumber \end{eqnarray} \vspace{-6mm} \newline is the background field and $\tilde \sigma $ is quantum fluctuations of the $\sigma $-field. Then \newline \vspace{-6mm} \begin{eqnarray} L_\sigma =\bar \psi (i\hat \partial -g_0\sigma )\psi -g_0\left( {\bar\psi \tilde{\sigma} \psi } \right) -{{\mu _0^2} \over 2}\sigma ^2 -{{\mu _0^2} \over 2}{\tilde{\sigma }}^2-{\mu_0}^2\sigma\tilde \sigma \label{f30} \end{eqnarray} \vspace{-6mm} \newline The last term of (\ref{f30}) may be eliminated because of the redefinition of the sources of the quantum field $\tilde \sigma$, and the Feynman graphs will be \begin{picture}(8,3.5) \put(2,1){\line(-1,1){1}} \put(2,1){\line(-1,-1){1}} \put(2,0.99){\line(1,0){1}} \put(2,1.01){\line(1,0){1}} \put(1,1.5){\makebox(0,0){$\psi$}} \put(1,0.5){\makebox(0,0){$\bar \psi$}} \put(3.5,1){\makebox(0,0){$\tilde \sigma$}} \put(1.5,1){\makebox(0,0){$g_0$}} \put(2,1){\circle*{.1}} \put(2,0){\makebox(0,0){$a)$}} \put(5,1.01){\line(1,0){1.5}} \put(5.8,1.3){\makebox(0,0){$i\hat k-g_0\sigma_0$}} \put(5.8,0){\makebox(0,0){$b)$}} \put(8,0.99){\line(1,0){1}} \put(8,1.01){\line(1,0){1}} \put(8.5,1.3){\makebox(0,0){$(i/\mu_0^2)$}} \put(8.5,0){\makebox(0,0){$c)$}} \end{picture} \vspace{10mm} Fig. III-1 Feynman graphs including a collective field. Graph $1 a)$ describes the interaction of fermi field with collective field, $1 b)$ is the propagator of fermi field, and $1 c)$ is the propagator of collective field. In tree approximation with respect to $\sigma$-field we can write an effective action: \newline \vspace{-6mm} \begin{eqnarray} \Gamma_{eff} [\sigma]=(-i/2)\ln Det(i\hat k-g_0\sigma) -\frac{\mu_0^2}{2}(\sigma,\sigma) \label{f31} \end{eqnarray} \vspace{-6mm} \newline The effective potential of this model will be \newline \vspace{-6mm} \begin{eqnarray} V_{eff}={i \over 2}(\mbox{Tr} \hat 1)\int {{{d^nk} \over {\left( {2\pi } \right)^n}}}\ln \left( {k^2-(g_0\sigma )^2} \right)+{{\mu _0^2} \over 2}\sigma ^2 \label{f32} \end{eqnarray} \vspace{-6mm} \newline Minimum $V_{eff}$ gives the gap equation \newline \vspace{-6mm} \begin{eqnarray} \sigma _c=(g_0 / \mu _0)^2\sigma _cI\,(g_0\sigma _c) \label{f33} \end{eqnarray} \vspace{-6mm} \newline or, in another form, \newline \vspace{-6mm} \begin{eqnarray} m=s\lambda _0mI\left( m \right) \label{f34} \end{eqnarray} \vspace{-6mm} \newline where $m$ is the dynamical mass of fermionic field $m=g_0\sigma _c$, $s$ is the dimension of $\gamma$- matrices and $\lambda _0=(g_0^2/ \mu _0)^2$ Now one can solve the equation (\ref{f34}) for different space time topologies. \section{ Model with topologies} \vspace{-4mm} \hspace{33mm} {\Large \bf $R_1 \times R_1 \times S_1$ and $R_1 \times Mobius~strip$} \vspace{1mm} The solution of this gap equation is connected with the calculation of the function I(m) of the equation (\ref{f34}). Let us consider two types of topologies: 1) $R_1\times R_1\times S_1$ with $\psi(x,y,0)=\psi(x,y,L)$\\ and 2) $R_1\times Mobius~strip$ with $\psi(x,y,0)=-\psi(x,y,L)$ (Fig.III- 2) We can find for 3-D space time that \newline \vspace{-6mm} \begin{eqnarray} I(m)=\Lambda -F_{(\pm )}(L,m) \label{f35} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} F_{(\pm )}(L,m)={1 \over {\pi L}}\ln (1\;_+^-\;e^{-Lm}) \label{f36} \end{eqnarray} \vspace{-6mm} \newline with (+) for $R_1\times R_1\times S_1$ topology and (-) for $R_1\times Mobius\;strip$ topology. The gap equation will be \newline \vspace{-6mm} \begin{eqnarray} m=m\lambda _0\Lambda \left( {1-{1 \over {\pi L\Lambda }} \ln (1\;_+^-\;e^{-Lm})} \right) \label{f37} \end{eqnarray} \vspace{-6mm} \newline The analysis of this expression can be made by the theory of bifurcations\\ \cite{hk1}. For this purpose let us write the gap equation (\ref{f37}) in the form \newline \vspace{-6mm} \begin{eqnarray} m=f_{(\pm )}(m,\bar \lambda ) \label{f38} \end{eqnarray} \vspace{-6mm} \newline where the functions $f_{(\pm )}(m,\bar \lambda )$ for topologies $(+)$and $(-)$ are \newline \vspace{-6mm} \begin{eqnarray} f_{(\pm )}(m,\bar \lambda )=m\bar \lambda \left( {1-{1 \over {\pi L\Lambda }}\ln (1\;_+^-\;e^{-Lm})} \right) \label{f39} \end{eqnarray} \vspace{-6mm} \newline The equation (\ref{f38}) has stable m=0 solutions, if $f_m(0,\bar \lambda )<1$. For $f_m(0,\bar \lambda )>1$ the equation (\ref{f38}) has no stable trivial solutions. One can see that there is a stable m=0 solution for (-) topology with the critical parameter \newline \vspace{-6mm} \begin{eqnarray} L_c={{\ln 2} \over \pi }{{\lambda _0} \over {\lambda _0\Lambda -1}} \approx {{\ln 2} \over {\pi \Lambda }} \label{f40} \end{eqnarray} \vspace{-6mm} \newline for $\lambda _0\Lambda =\bar \lambda >1$ The gap equation for topology (+) has no stable trivial solutions. The dynamical mass in this case is a smooth function with respect to parameter L: \newline \vspace{-6mm} \begin{eqnarray} m_{(+)}=-{1 \over L}\ln \left( {1-f(\Lambda ) \exp \left( {-{{\pi L} \over {\lambda _0}}} \right)} \right) \label{f41} \end{eqnarray} \vspace{-6mm} \newline for $\lambda _0\Lambda =\bar \lambda <1$ The solution for topology $R_1\times Mobius\;strip$ is \newline \vspace{-6mm} \begin{eqnarray} m_{(-)}=-{1 \over L}\ln \left( {\exp \left( {{L \over {L_c}}\ln 2} \right)-1} \right)\label{f42} \end{eqnarray} \vspace{-6mm} \newline The restoration of the symmetry takes place when $L=L_c$. In this case the condensate function of the bound state equals zero \newline \vspace{-6mm} \begin{eqnarray} \sigma _c={{g_0} \over {\mu _0^2}}<\psi \bar \psi >=0 \label{f43} \end{eqnarray} \vspace{-6mm} \newline Now we can consider the models with the more complicated space-time structures of Klein bottle and Torus topologies. \newpage \section{Torus topology $M_{3)}=R_1 \times S_1 \times S_1$} \vspace{-4mm} \hspace{35mm}{\Large \bf and topology $M_{3)}=R_1\times Klein\;bottle$} \vspace{1mm} We can introduce topologies $R_1\times S_1\times S_1$ and $R_1\times Mobius~strip$ as an identification of space points for wave functions Fig. III-3: \newline \vspace{-6mm} \begin{eqnarray} \psi(x,0,0)=\psi(x,L,L)~and~\psi(x,0,0)=-\psi(x,L,L) \end{eqnarray} \vspace{-6mm} \newline For simplicity consider that these topologies have only one parameter $L=L^{'}$. The gap equations for the topologies will be \newline \vspace{-6mm} \begin{eqnarray} m=m\bar {\lambda} s\left\{1+\sqrt {m /(\Lambda ^2L)} ( \pi)^{-3 /2} [\sum\limits_{n=1}^\infty {1 \over {\sqrt {2n}}}K_{-1/2}(2Lmn)\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+\sum\limits_{n,k=1}^\infty K_{-1/2} (2Lm\sqrt {n^2+k^2})\left( {n^2+k^2} \right)^{-1/4}]\right\} \label{f44} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} m=m\bar {\lambda} s\left\{1+\sqrt {m /(\Lambda ^2L)} ( \pi)^{-3 /2} [\sum\limits_{n=1}^\infty {1 \over {\sqrt {2n}}}K_{-1/2}(2Lmn)\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.+\sum\limits_{n,k=1}^\infty (-1)^k K_{-1/2} (2Lm\sqrt {n^2+k^2})\left( {n^2+k^2} \right)^{-1/4}]\right\} \label{f45} \end{eqnarray} \vspace{-6mm} \newline The solutions of (\ref{f44}) and (\ref{f45}) give dynamical masses for these topologies\\ \cite{top13}. The results of this paragraph show that the application of the Mean Field Method to non-linear models gives non-renormalized solutions, though we can obtain some information about the influence of topology on dynamical mass behavior. The evaporation of condensate and restoration of chiral symmentry proceed in different ways and are ruled by topology. There are topologies in which these phenomena do not take place. We believe that these results are important in the bag models, because the energy of bag is dependent on non-perturbative effects and boundary conditions\\ \cite{guid1}. \section{Non-linear spinor $(\bar \psi \psi)^2$ model in} \vspace{-4mm} \hspace{23mm}{\Large \bf Riemann space-time at finite temperature.} \vspace{1mm} In this section we will treat the problem of dynamical mass generation of the non-linear spinor model in 4-D Riemann space-time at finite temperature. We will get the finite temperature effective potential and find out information about the influence of curvature of the background gravitational field and temperature on the value of dynamical fermionic mass. Let the total Lagrangian of the model be \newline \vspace{-6mm} \begin{eqnarray} L=L_g+L_m \label{f46} \end{eqnarray} \vspace{-6mm} \newline where the gravitational Lagrangian is\footnote{For clear understanding of the problem, we study the fermionic system in a weak gravitational field.} \newline \vspace{-6mm} \begin{eqnarray} L_g={1 \over {16\pi G_0}}(R-2\tilde \Lambda _0) \label{f47} \end{eqnarray} \vspace{-6mm} \newline and the Lagrangian of matter field $L_m$ is \newline \vspace{-6mm} \begin{eqnarray} L_m=\bar \psi i\bar \gamma ^\mu \nabla _\mu \psi +{{g_0^2} \over {2\mu _0^2}}\left( {\bar \psi \psi } \right)^2 \label{f48} \end{eqnarray} \vspace{-6mm} \newline $\nabla _\mu $ is a covariant derivative. The first term of (\ref{f48}) describes kinetics of the fermi field and its interaction with the gravitational field, and the second one describes the interaction of the fields. The effective potential of the model is written from the action \newline \vspace{-6mm} \begin{eqnarray} \Gamma_{eff}[\sigma]=-{i \over 2}\ln Det[\nabla ^2 +{1 \over 4}R-(g_0\sigma )^2]-{{\mu _0^2} \over 2}(\sigma ,\sigma ) \label{f49} \end{eqnarray} \vspace{-6mm} \newline After making ultraviolet regularization and renormalizations of the model we get \newline \vspace{-6mm} \begin{eqnarray} L_{tot}=L_g+L_m=\left( {L_g+L_m(\infty )} \right) +L_m(\beta ) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =L_g^{ren}+L_m(\beta ) \label{f50} \end{eqnarray} \vspace{-6mm} \newline The renormalized gravitational constant $G_R$ and the $\tilde{\Lambda} _R$-term are: \newline \vspace{-6mm} \begin{eqnarray} {1 \over {8\pi G_R}}\tilde \Lambda _R ={1 \over {8\pi G_0}}\tilde \Lambda _0 +{1 \over {16\pi ^2}}\Lambda ^2-{1 \over {16\pi ^2}}m^2\ln \Lambda ^2 \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} {1 \over {16\pi G_R}}={1 \over {16\pi G_0}} +{1 \over {16\pi ^2}}\ln \Lambda ^2 \label{f51} \end{eqnarray} \vspace{-6mm} \newline where $\Lambda$ is a cut-off parameter, and $m$ is a dynamical mass. In the calculations (\ref{f50}) and (\ref{f51}) there was used the cut-off method of regularization of divergent integrals \cite{key7}. The effective potential can be written from (\ref{f49}) in the form \newline \vspace{-6mm} \begin{eqnarray} V_{eff}[\sigma ]=\sum\limits_{j=1}^2 {\hat \alpha _j(R)f^j(\beta g_0\sigma )+{{\mu _0^2} \over 2}}\sigma ^2 \label{f52} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} f^0(\beta g_0\sigma )={{2m^2} \over {\pi ^2\beta ^2}} \sum\limits_{n=1}^\infty {{{\left( {-1} \right)^n} \over {n^2}}}K_{-2}(\beta g_0\sigma ) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =-{4 \over \beta } \int {{{d^3k} \over {\left( {2\pi } \right)^3}} \ln \left( {1+e^{-\beta \varepsilon }} \right)} \label{f53} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} f^j(\beta g_0\sigma_0 )={1 \over {4g_0}} \left( {-{\partial \over {\partial \sigma ^2}}} \right)^jf^0(\beta g_0\sigma ) \label{f54} \end{eqnarray} \vspace{-6mm} \newline The coefficients $\hat \alpha _j(R)$ are \newline \vspace{-6mm} \begin{eqnarray} \hat \alpha _0=1,\quad \hat \alpha _0={1 \over {12}}R,\quad ....\label{f55} \end{eqnarray} \vspace{-6mm} \newline The solution of the gap equation \newline \vspace{-6mm} \begin{eqnarray} \frac{\partial}{\partial \sigma } V_{eff}[\sigma ]_{|\sigma=\sigma_c} =0 \label{f56} \end{eqnarray} \vspace{-6mm} \newline gives, in high temperature approximation (\ref{bb15}), the expression for dynamical mass without redefinition of the coupling constant and temperature \newline \vspace{-6mm} \begin{eqnarray} m^2(R,T)=(g_0 \sigma_c)^2 = A/\lambda + b\cdot T^2 + C\cdot R+...\label{f57} \end{eqnarray} \vspace{-6mm} \newline where the constants \newline \vspace{-6mm} \begin{eqnarray} A=32\pi^2/3.84,~~ B=A/24,~~ C=0.7/12,~~ \lambda=(g_0/\mu_0)^2 \nonumber \end{eqnarray} \vspace{-6mm} \newline are numerical positive coefficients. As we can see from the equation (\ref{f57}), the effective dynamical mass of fernion is a positive function for any temperature and curvature. \chapter{INDUCED CHERN-SIMONS MASS TERM} \centerline{\Large \bf IN TOPOLOGICALLY NON-TRIVIAL SPACE-TIME } \vspace{24pt} In gauge theories with fermions, the a topological mass term is induced by fermionic interactions \cite {vcs1}. If the topology of space-time is not trivial, topological parameters will appear in the effective action and in the CS Lagrangian. In this chapter we show that the CS term depends on topological parameters in such a way that the topological gauge invariance of the total action can be maintained. Let us consider the case of a massive fermion field interacting with external gauge field in 3-D space-time. \section{ Euclidean space-time. Trivial topology} \vspace{1mm} The Euclidean action of this quantum system is given by \newline \vspace{-6mm} \begin{eqnarray} S=\int d^3 x \bar{\psi}(x)(i\hat{\partial}_x+m+g\hat{A}(x))\psi(x) \label{vcs1} \end{eqnarray} \vspace{-6mm} \newline The corresponding generating functional is \newline \vspace{-6mm} \begin{eqnarray} Z=\int D\bar{\psi}(x) D \psi(x)exp\left[-\int d^3x \bar{\psi}(x)(i\hat{\partial}_x+m+g\hat{A}(x))\psi(x)\right] \label{vcs2} \end{eqnarray} \vspace{-6mm} \newline Here $A_\mu(x)=A_\mu^a (x)T^a$, and $T^a$ are generators of the gauge transformations. We will choose to work with two-component Dirac spinors. Euclidean Dirac matrices have the algebra: \newline \vspace{-6mm} \begin{eqnarray} \{\gamma_\mu, \gamma_\nu \}=-\delta_{\mu \nu} \label{vcs3} \end{eqnarray} \vspace{-6mm} \newline This equation is satisfied by the matrices \newline \vspace{-6mm} \begin{eqnarray} \gamma_1=i\sigma_1,~~\gamma_2=i\sigma_2,~~\gamma_3=i\sigma_3 \label{vcs4} \end{eqnarray} \vspace{-6mm} \newline where $\sigma_i$ are the Pauli matrices. Furthermore, they satisfy \newline \vspace{-6mm} \begin{eqnarray} \gamma_\mu \gamma_\nu=-\delta_{\mu \nu}-\epsilon_{\mu \nu \rho}\gamma_\rho, \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mbox{Tr} (\gamma_\mu \gamma_\nu \gamma_\rho)=2 \epsilon_{\mu \nu \rho} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mbox{Tr} (\gamma_\mu \gamma_\nu \gamma_\rho \gamma_\lambda \gamma_\sigma) =-2(\delta _{\mu \nu} \epsilon_{\rho \lambda \sigma} +\delta _{\rho \lambda} \epsilon_{\mu \nu \sigma} -\delta _{ \nu \sigma} \epsilon_{\rho \lambda \mu} +\delta _{\mu \sigma} \epsilon_{\rho \lambda \nu}) \label{vcs5} \end{eqnarray} \vspace{-6mm} \newline The integration in (\ref{vcs2}) over the fermionic fields gives \newline \vspace{-6mm} \begin{eqnarray} Z[A_\mu]=Det(i\hat{\partial}+m+g\hat{A}))=exp(-S_{eff}) \label{vcs6} \end{eqnarray} \vspace{-6mm} \newline Therefore we can write the expression for the effective action \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=-\ln Det(i\hat{\partial}+m+g\hat{A})) =-\mbox{Tr} \ln (i\hat{\partial}+m+g\hat{A}) \label{vcs7} \end{eqnarray} \vspace{-6mm} \newline Now one can split (\ref{vcs7}) in two parts \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=-\mbox{Tr} \ln (i\hat{\partial}_x+m+g\hat{A}(x)) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\mbox{Tr} \ln \hat{S}_f-\mbox{Tr} \ln \left[1+g\hat{S}_f\hat{A}\right] \label{vcs8} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \hat{S}_f=\frac{1}{i\hat{\partial}+m} =\frac{i\hat{\partial}+m}{\partial^2 +m^2} \label{vcs9} \end{eqnarray} \vspace{-6mm} \newline is a fermionic propagator. The first term of the equation (\ref{vcs8}) is a divergent gauge-independent contribution, and the second one may be expanded in a power series \cite{vcs3}: \newline \vspace{-6mm} \begin{eqnarray} S_{eff}=-\mbox{Tr} \ln \left[1+g\hat{S}_f\hat{A}\right] =\mbox{Tr} \sum_{n=1}^{\infty} \frac{1}{n}[g\hat{S}_f\hat{A}] \label{vcs10} \end{eqnarray} \vspace{-6mm} \newline Let us consider the term which is quadratic in $A_\mu (x)$. The corresponding action is given by \newline \vspace{-6mm} \begin{eqnarray} S^{(2)} _{eff}=\frac{1}{2}g^2 \mbox{Tr} \left[ \frac{i\hat{\partial}+m}{\partial^2 +m^2}\hat{A} \frac{i\hat{\partial}+m}{\partial^2 +m^2}\hat{A}\right] \label{vcs11} \end{eqnarray} \vspace{-6mm} \newline The terms with two and four $\gamma$ matrices contribute to the wave function renormalization of the gauge boson and will not be taken into consideration in further calculations. The other two terms are linear with respect to mass $m$ and give \newline \vspace{-6mm} \begin{eqnarray} S^{(2)} _{eff}=\frac{1}{2}g^2 m \mbox{Tr} \left[ \frac{i\hat{\partial}}{\partial^2 +m^2}\hat{A} \frac{1}{\partial^2 +m^2}\hat{A} +\frac{1}{\partial^2 +m^2} \hat{A}\frac{i\hat{\partial}}{\partial^2 +m^2}\hat{A}\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{img^2}{2}\mbox{tr} _D \gamma_\mu \gamma_\nu \gamma_\rho \mbox{tr} \left[\frac{\partial_\mu}{\partial^2 +m^2}A_\nu \frac{1}{\partial^2 +m^2}A_\rho +\frac{1}{\partial^2 +m^2}A_\mu\frac{\partial_\nu}{\partial^2 +m^2}A_\rho \right] \label{vcs12} \end{eqnarray} \vspace{-6mm} \newline where $\mbox{Tr}=\mbox{tr} _D \mbox{tr}$ is the trace of both the $\gamma$ matrices and the vector field, and also includes the integrations. To write $\mbox{tr}$ in the form $\mbox{tr}\left[f(\partial)g(x)\right]$ one can use the relation \newline \vspace{-6mm} \begin{eqnarray} \phi(x)\frac{1}{\partial^2+m^2}=\frac{1}{\partial^2+m^2}\phi(x) +\frac{[\partial^2,\phi(x)]}{(\partial^2+m^2)^2} +\frac{[\partial^2,[\partial^2,\phi(x)]]}{(\partial^2+m^2)^3} +... \label{vcs13} \end{eqnarray} \vspace{-6mm} \newline Then \newline \vspace{-6mm} \begin{eqnarray} S^{(2)}_{eff}=img^2~\epsilon_{\mu \nu \rho} \mbox{Tr}\left[\frac{1}{(k^2 +m^2)^2} (\partial_\nu A_\mu )A_\rho)\right] \nonumber \end{eqnarray} \vspace{-6mm} \newline \newline \vspace{-6mm} \begin{eqnarray} =img^2~\epsilon_{\mu \nu \rho} \int d^3 x <x|\frac{1}{(\partial^2+m^2)^2}|x> tr\{(\partial_\nu A_\mu )A_\rho\} \label{vcs14} \end{eqnarray} \vspace{-6mm} \newline In another form \newline \vspace{-6mm} \begin{eqnarray} S^{(2)}_{eff}=-img^2~\epsilon_{\mu \nu \rho} \int d^3 x \left[\int \frac{d^3k}{(2\pi)^3}\frac{1}{(k^2+m^2)^2}\right] \{(\partial_\mu A_\nu )A_\rho\} \label{vcs15} \end{eqnarray} \vspace{-6mm} \newline The additional contribution to the CS term comes from the terms which are cubic in $A_\mu$'s. We can rewrite them as \newline \vspace{-6mm} \begin{eqnarray} S^{(3)} _{eff}=\frac{1}{3}g^3 m \mbox{Tr} \left[ \frac{i\hat{\partial}+m}{\partial^2 +m^2}\hat{A} \frac{\hat{i\partial}+m}{k^2 +m^2}\hat{A} \frac{\hat{i\partial}+m}{\partial^2 +m^2}\hat{A}\right] \label{vcs17} \end{eqnarray} \vspace{-6mm} \newline Noticing that the terms having even number of $\gamma$ matrices do not contribute to the CS term, we have: \newline \vspace{-6mm} \begin{eqnarray} S^{(3)} _{eff}=-\frac{1}{3}g^3 m Tr\left[ \frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \frac{1}{\partial^2 +m^2}\hat{A}\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \frac{1}{\partial^2 +m^2}\hat{A} \frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +\frac{1}{\partial^2 +m^2}\hat{A} \frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \frac{\hat{\partial}}{\partial^2 +m^2}\hat{A} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.- m^2\frac{1}{\partial^2 +m^2}\hat{A} \frac{1}{\partial^2 +m^2}\hat{A} \frac{1}{\partial^2 +m^2}\hat{A} \right] \label{vcs18} \end{eqnarray} \vspace{-6mm} \newline Using the identity (\ref{vcs13}) find \newline \vspace{-6mm} \begin{eqnarray} S^{(3)}_{eff}=-\frac{mg^3}{3} \left[\frac{}{}\mbox{tr} _D \gamma_\mu \gamma_\nu \gamma_\rho \gamma_\lambda \gamma_\sigma\right. \times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \mbox{tr} \left\{ \frac{\partial_\mu \partial_\rho }{(\partial^2+m^2)^3}A_\nu A_\lambda A_\sigma + \frac{\partial_\mu \partial_\lambda }{(\partial^2+m^2)^3}A_\nu A_\rho A_\sigma + \frac{\partial_\nu \partial_\lambda }{(\partial^2+m^2)^3}A_\mu A_\rho A_\sigma \right\} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left. -m^2 \mbox{tr} _D \gamma_\mu \gamma_\nu \gamma_\rho \mbox{tr} \frac{1} {(\partial^2+m^2)^3}A_\mu A_\nu A_\rho \right] \label{vcs19} \end{eqnarray} \vspace{-6mm} \newline Tracing with respect to $\gamma$ matrices, find \newline \vspace{-6mm} \begin{eqnarray} S^{(3)}_{eff}=-\frac{2mg^3}{3} \left[\int \frac{d^3k}{(2\pi) ^3}\frac{1}{(k^2+m^2)^2}\right] \mbox{tr} \int d^3x \epsilon ^{\mu \nu \rho} A_\mu A_\nu A_\rho \label{vcs20} \end{eqnarray} \vspace{-6mm} \newline Combining (\ref{vcs15}) with (\ref{vcs20}) we get \cite{vcs3} the induced Chern-Simons term in the form \newline \vspace{-6mm} \begin{eqnarray} S^{CS}_{eff}=-img^2\left[\int \frac{d^3k}{(2\pi) ^3}\frac{1}{(k^2+m^2)^2}\right] \mbox{tr} \int d^3x \epsilon _{\mu \nu \rho} \left[\partial_\mu A_\nu A_\rho -\frac{2}{3}ig A_\mu A_\nu A_\rho \right] \label{addcs20} \end{eqnarray} \vspace{-6mm} \newline or, after integration over the momenta \newline \vspace{-6mm} \begin{eqnarray} S^{CS}_{eff}=-\frac{ig^2}{8 \pi}\frac{m}{|m|} \mbox{tr} \int d^3x \epsilon _{\mu \nu \rho} \left[\partial_\mu A_\nu A_\rho -\frac{2}{3}ig A_\mu A_\nu A_\rho \right] \label{vcs21} \end{eqnarray} \vspace{-6mm} \newline Now we can apply the method we have developed for a gauge model in 3-D non-trivial space-time. \section{ Non-trivial topology} \vspace{1mm} a) Model with space-time topology $\Sigma=R^{(2)}\times mobius~strip$. To introduce topology $\Sigma =R^{(2)} \times S^1$ with parameter $\zeta$ we rewrite the expression (\ref{vcs15}) in the following way. Momentum integration in (\ref{vcs15}) will be integration with respect to two dimensions ($(t,x) \to R^{(2)}$),and the third integration will be transformed in the sum, because the fermionic propagator is antisymmetric with respect to the selected axis ($y \to S^1$). We can consider that $k^2=\vec{k}^2+\omega^2_n$ with $\omega_n=(2\pi /\zeta)(n+1/2)$ and $n=0,\pm1,\pm2,...$. Integrating in 2-D momentum space we find that \newline \vspace{-6mm} \begin{eqnarray} \int\limits^\sim \frac{d^3k}{(2\pi)^3}\frac{1}{(k^2 +m^2)^2} =\int \frac{d^2 k}{(2\pi)^2}\frac{1}{\zeta} \sum\limits_{n=-\infty}^\infty \frac{1}{((\vec{k})^2+\omega^2_n +m^2)^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{8\pi |m|}tanh\left\{\frac{|m| \zeta}{2}\right\} \label{vcs22} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} \int\limits^\sim \frac{d^3k}{(2\pi)^3} =\frac{1}{\zeta}\sum \limits_{n=-\infty}^\infty \int \frac{d^2k}{(2\pi)^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline In the process of calculations we used the useful equation \newline \vspace{-6mm} \begin{eqnarray} \sum\limits_{n=-\infty}^\infty \frac{y}{(y^2+(n+1/2)^2)} =\pi tanh (\pi y) \label{vcs23} \end{eqnarray} \vspace{-6mm} \newline Then the expression (\ref{addcs20}) in this topology will be \newline \vspace{-6mm} \begin{eqnarray} S^{(CS)}_{eff}=-img^2\int \frac{d^2 k}{(2\pi)^2}\frac{1}{\zeta} \sum\limits_{n=-\infty}^\infty \frac{1}{((\vec{k})^2+\omega^2_n +m^2)^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr} \int \epsilon_{\mu \nu \rho}\left\{\partial_\nu A_\mu A_\rho -\frac {2i}{3}g A_\nu A_\mu A_\rho \right\}\label{vcs24} \end{eqnarray} \vspace{-6mm} \newline and the induced CS term at non-trivial 3-D space-time is written as \newline \vspace{-6mm} \begin{eqnarray} S^{CS}_{eff}(\zeta)=-\frac{ig^2}{8\pi}\frac{m}{|m|} tanh \left\{\frac{|m|\zeta}{2}\right\} \mbox{tr} \int d^3 x\epsilon_{\mu \nu \rho}(\partial_\mu A_\nu A_\rho-\frac{2}{3}ig A_\mu A_\nu A_\rho ) \label{vcs28} \end{eqnarray} \vspace{-6mm} \newline The relation between CS terms is the function of the form \newline \vspace{-6mm} \begin{eqnarray} \frac{S^{CS}_{eff}(\zeta)}{S^{CS}_{eff}}= tanh \left\{\frac{|m|\zeta}{2}\right\} \label{vcs29} \end{eqnarray} \vspace{-6mm} \newline b) Model with space-time topology $\Sigma=R^{(2)}\times S^1$. For this space-time topology the propagator of the fermionic field will be periodic at the boundary points of interval $[0,\zeta]$, that leads to the modification of equation (\ref{vcs21}) with respect to new frequencies $\omega_n=2\pi n/\zeta$ with $n=0,\pm1,\pm2,...$. Taking into account the summation formula \newline \vspace{-6mm} \begin{eqnarray} \sum \limits_{n=-\infty}^\infty \frac{y}{y^2+n^2} =\pi coth (\pi y) \label{sum2} \end{eqnarray} \vspace{-6mm} \newline we find, that \newline \vspace{-6mm} \begin{eqnarray} \int \limits_\Sigma \frac{d^3k}{(2\pi)^3}\frac{1}{(k^2 +m^2)^2} =\int \frac{d^2 k}{(2\pi)^2}\frac{1}{\zeta} \sum\limits_{n=-\infty}^\infty \frac{1}{((\vec{k})^2+\omega^2_n +m^2)^2} \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =\frac{1}{8\pi |m|}coth\left\{\frac{|m| \zeta}{2}\right\} \label{vcs30} \end{eqnarray} \vspace{-6mm} \newline Using this equation and the equation (\ref{vcs20}), get the induced CS term in the new topology \newline \vspace{-6mm} \begin{eqnarray} S^{CS}_{eff}(\zeta)=-\frac{ig^2}{8\pi}\frac{m}{|m|} coth\left\{\frac{|m|\zeta}{2}\right\} \mbox{tr} \int d^3 x\epsilon_{\mu \nu \rho}(\partial_\mu A_ \nu A_\rho-\frac{2}{3}ig A_\mu A_\nu A_\rho ) \label{vcs31} \end{eqnarray} \vspace{-6mm} \newline The rotio of the CS terms will be the function of the topological parameter $\zeta$: \newline \vspace{-6mm} \begin{eqnarray} \frac{S^{CS}_{eff}(\zeta)}{S^{CS}_{eff}}= coth\left\{\frac{|m|\zeta}{2}\right\} \label{vcs32} \end{eqnarray} \vspace{-6mm} \newline These results show that the relations (\ref{vcs29}) and (\ref{vcs32}) are smooth functions of the topological parameter $\zeta$. \chapter{THERMODYNAMICS OF GAUGE FIELDS} \vspace{24pt} In this chapter we will apply the formalism of gauge fields in curved space-time developed in chapter V to study the properties of thermal photon gas in an external gravitational field. \section{The Green's function of photons} \vspace{1mm} As we know already from chapter IV the total Lagrangian for a vector field in Minkowski space-time is the sum of three contributions: \newline \vspace{-6mm} \begin{eqnarray} L_{tot} = L_m+L_f+L_{gh}, \label{k1} \end{eqnarray} \vspace{-6mm} \newline where \newline \vspace{-6mm} \begin{eqnarray} L_m = -(1/4)F_{\mu \nu}F^{\mu \nu}, \label{k2} \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} L_f = -(1/2\alpha)(\partial_\mu A^\mu)^2 \label{k3} \end{eqnarray} \vspace{-6mm} \newline and \newline \vspace{-6mm} \begin{eqnarray} L_{gh} = g^{\mu \nu}(\partial_\mu c)(\partial_\nu c^{*}) \label{k4} \end{eqnarray} \vspace{-6mm} \newline It can be extended to curved space-time with the transformation \newline \vspace{-6mm} \begin{eqnarray} A_\alpha={h_\alpha}^\mu A_\mu,~~ \partial_\alpha \to {h_\alpha}^\mu\nabla_\mu \label{k5} \end{eqnarray} \vspace{-6mm} \newline where $\nabla_\mu=\partial_\mu+\Gamma_\mu$ and the connection $\Gamma_\mu$ is defined by equation \newline \vspace{-6mm} \begin{eqnarray} \Gamma_\mu=(1/2)\Sigma_{\alpha \beta}{h^\alpha}^\nu(x) \left[\frac{\partial}{\partial x^{\mu}} {{h^\beta}_nu}(x)\right] \label{k6} \end{eqnarray} \vspace{-6mm} \newline where the matrices $\Sigma_{\alpha \beta}$ are (\ref{g7}). The strength tensor of electromagnetic field is \newline \vspace{-6mm} \begin{eqnarray} F_{\mu \nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu \label{k8} \end{eqnarray} \vspace{-6mm} \newline The variation of the action \newline \vspace{-6mm} \begin{eqnarray} S=\int d^4 x \sqrt{g} \left( L+L_f+L_{gh}\right) \label{k9} \end{eqnarray} \vspace{-6mm} \newline gives the equation for the vector field: \newline \vspace{-6mm} \begin{eqnarray} \nabla_\nu F^{\mu \nu}+(1/\alpha)\nabla^\mu (\nabla_\nu A^\nu)=0 \label{k10} \end{eqnarray} \vspace{-6mm} \newline In the Feynman gauge $(\alpha=1)$ this equation has the form: \newline \vspace{-6mm} \begin{eqnarray} \nabla_\nu \nabla^\mu A^\nu-\nabla_\nu \nabla^\nu A^\mu+ \nabla^\mu \nabla_\nu A^\nu=0 \label{k11} \end{eqnarray} \vspace{-6mm} \newline and with the definition of the Riemann tensor (\ref{icor1}) it is \newline \vspace{-6mm} \begin{eqnarray} \nabla_\nu \nabla^\nu A^\mu-{R^\mu }_\nu A^\nu=0 \label{k12} \end{eqnarray} \vspace{-6mm} \newline Based on this equation we write the equation for the Green's function \newline \vspace{-6mm} \begin{eqnarray} \nabla_\nu \nabla^\nu{ D^\mu}_\tau(x,x^{'})-{R^\mu }_\nu{D^\nu}_\tau(x,x^{'}) =-g^{-1/2}(x)\delta(x-x^{'}){\delta^\mu}_\tau \label{k13} \end{eqnarray} \vspace{-6mm} \newline We will be interested in calculations of the Green's function in the limit $(x \to x^{'})$ to find the effective action. We will rewrite the equation (\ref{k13}) in Riemann normal coordinates with origin at the point $x^{'}$. For covenience one may define the Green's function ${\bar{D}^\mu}_\tau(x,x^{'})$ as \newline \vspace{-6mm} \begin{eqnarray} { D^\mu}_\tau(x,x^{'})=g^{-1/4}(x){\bar{D}^\mu}_\tau(x,x^{'})g^{-1/4}(x^{'}) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} =g^{-1/4}(x){\bar{D}^\mu}_\tau(x,x^{'}) \label{k14} \end{eqnarray} \vspace{-6mm} \newline where we used the fact $g(x^{'})=1$ For our calculations we will use the Christoffel symbols which in the Riemann normal coordinates are \newline \vspace{-6mm} \begin{eqnarray} {\Gamma^\sigma}_{\mu \nu}=-(1/3)({R^\sigma}_{\alpha \beta \gamma}+ {R^\sigma}_{\beta \alpha \gamma})y^\gamma \label{k15} \end{eqnarray} \vspace{-6mm} \newline where $y^\gamma$ represents the coordinates of the point $x$ and the Riemann tensor is evaluated at $x^{'}$. The expansion of equation (\ref{k13}) to the second derivative of the metric gives \newline \vspace{-6mm} \begin{eqnarray} \eta ^{\alpha \beta }\partial _\alpha \partial _\beta \bar D_\tau ^\mu (y) +(1/6)R\bar D_\tau ^\mu (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(4/3)R_\nu ^\mu \bar D_\tau ^\nu (y) -(1/3)R_\nu ^\lambda y^\nu \partial _\lambda \bar D_\tau ^\mu (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(1/3){{{R^\alpha}_\gamma} ^\beta }_\delta y^\gamma y^\delta \partial _\alpha \partial _\beta \bar D_\tau ^\mu (y) -(2/3){{{R^\mu}_\gamma} ^\alpha} _\delta y^\delta \partial _\alpha \bar D_\tau ^\gamma (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(2/3){R^{\mu \alpha }}_{\lambda \gamma } y^\gamma \partial _\alpha \bar D_\tau ^\lambda (y) =-\delta (y)\delta _\tau ^\mu \label{k16} \end{eqnarray} \vspace{-6mm} \newline where $\partial_\alpha=\partial/\partial y^\alpha$. The momentum space approximation is defined by introducing the quantity $D_\tau ^\mu (k)$ defined as \newline \vspace{-6mm} \begin{eqnarray} \bar{D}_\tau ^\mu (x,x^{'})=\bar{D}_\tau ^\mu (x^{'},y) =\int \frac{d^n k}{(2 \pi) ^n}\bar{D}_\tau ^\mu (k) \exp[iky] \label{k17} \end{eqnarray} \vspace{-6mm} \newline This quantity is assumed to have the expansion \newline \vspace{-6mm} \begin{eqnarray} \bar{D}_\tau ^\mu (k)=\bar{D}_{0,\tau} ^\mu (k)+\bar{D}_{1,\tau} ^\mu (k) +\bar{D}_{2,\tau} ^\mu (k)+... \label{k18} \end{eqnarray} \vspace{-6mm} \newline where $\bar{D}_{i,\tau} ^\mu (k)$ have a geometric coefficients involving $i$ derivatives of the metric. On dimensional grounds $\bar{D}_{i,\tau} ^\mu (k)$ must be of order $k^{-(2+i)}$ so the equation (\ref{k16}) is an asymptotic expansion in large $k$. As we can see from (\ref{k16}) the first term in (\ref{k18}) is \newline \vspace{-6mm} \begin{eqnarray} \bar{D}_{0,\tau} ^\mu (k)=\delta^\mu_\tau/k^2, \label{k19} \end{eqnarray} \vspace{-6mm} \newline The second one is \newline \vspace{-6mm} \begin{eqnarray} \bar{D}_{1,\tau} ^\mu (k)=0, \label{k20} \end{eqnarray} \vspace{-6mm} \newline The third one is computed from \newline \vspace{-6mm} \begin{eqnarray} \eta ^{\alpha \beta }\partial _\alpha \partial _\beta \bar D_{2,\tau} ^\mu (y) +(1/6)R\bar D_{0,\tau }^\mu (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} -(4/3)R_\nu ^\mu \bar D_{0,\tau} ^\nu (y) -(1/3){R_\nu }^\lambda y^\nu \partial _\lambda \bar D_{0,\tau} ^\mu (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(1/3){{{R^\alpha}_\gamma} ^\beta }_\delta y^\gamma y^\delta \partial _\alpha \partial _\beta \bar D_{0,\tau }^\mu (y) -(2/3){{{R^\mu}_\gamma} ^\alpha} _\delta y^\delta \partial _\alpha \bar D_{0,\tau} ^\gamma (y) \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} +(2/3){R^{\mu \alpha }}_{\lambda \gamma } y^\gamma \partial _\alpha \bar D_{0,\tau }^\lambda (y)+...=0 \label{k21} \end{eqnarray} \vspace{-6mm} \newline To simplify calculations note that $y^\alpha \to (-i\partial /\partial k^\alpha)$, and integrate by parts to find that \newline \vspace{-6mm} \begin{eqnarray} \bar{D}_{2,\tau}^\mu (y) =\int \frac{d^n k}{(2 \pi) ^n}\bar{D}_{2,\tau}^\mu (k) \exp[iky] =\int \frac{d^nk}{(2 \pi)^n} \exp[iky]\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \left[ \left\{(1/6)R\delta^\mu_\tau-(2/3)R^\mu_\nu\delta^\nu_\tau\right\}/k^4- (4/3){R^{\mu\beta}}_{\nu\gamma} k^\gamma k_\beta\delta^\nu_\tau/k^6 \right] \label{k22} \end{eqnarray} \vspace{-6mm} \newline The final expression for the photon propagator will be then: \newline \vspace{-6mm} \begin{eqnarray} D_\tau^\mu (y)={g}^{-1/4} (y)\int \frac{d^nk}{(2 \pi)^n} \exp[iky]\times \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \left[ \delta^\mu_\tau/k^2+ \left\{(1/6)R\delta^\mu_\tau-(2/3)R^\mu_\nu\delta^\nu_\tau\right\}/k^4- (4/3){R^{\mu\beta}}_{\nu\gamma} k^\gamma k_\beta\delta^\nu_\tau/k^6 \right] \label{k23} \end{eqnarray} \vspace{-6mm} \newline The Green's function of the ghost fields obeys the same equation as a scalar field, therefore one can use this equation for our further calculations. \section {The thermodynamic potential of a photon gas} \vspace{1mm} As in the case of flat space-time, one must carry out the calculations of the free energy of a photon gas together with the ghost contributions. The generating functional of an abelian vector field in curved space-time is written as \newline \vspace{-6mm} \begin{eqnarray} Z[J_\mu ,\eta, \bar{\eta}]=\int D A_\mu D \bar{\eta} D \eta \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \times \exp\left[ \int d^4 x \sqrt{g} \left \{ A^{\mu }(D_{\mu \nu})^{-1}A^{\nu} +\bar{c} D^{-1} c+j_\mu A^\mu +\bar{\eta} c +\bar{c}\eta\right\} \right] \label{addk24} \end{eqnarray} \vspace{-6mm} \newline Integration over the fields with zero sources leads to the following result for the logarithm of the generating functional \newline \vspace{-6mm} \begin{eqnarray} \ln Z[0]=\ln Det (G)-(1/2) \ln Det(G{\mu \nu}) \label{k24} \end{eqnarray} \vspace{-6mm} \newline where the Green's functions are defined by (\ref{k23}) and (\ref{h33})\footnote{To find the Feynman propagator of the ghost field from the boson propagator we have to put the mass of boson $m=0$ after calculation. } From this equation and from the definition of the free energy directly follows the expression for density of free energy \newline \vspace{-6mm} \begin{eqnarray} f_{ph} (\beta, R) =(-i/2)\mathop{\lim}\limits_{m\to 0}\int\limits_{m^2}^\infty dm^2 \left\{2 \mbox{tr} G(\beta,x-x^{'})-\mbox{tr} G_{\mu \nu}(\beta,x-x^{'})\right\} \label{k25} \end{eqnarray} \vspace{-6mm} \newline The final result may be found after putting a mass parameter $m^2$ into the expressions for the propagators of the photon and ghost fields and, after calculation of (\ref{k25}) setting the mass equal to zero. After making this procedure we will have \newline \vspace{-6mm} \begin{eqnarray} f_{ph}(\beta, R)=\sum \limits_j g_j(R)\left(-\frac{\partial}{\partial m^2}\right)^j \mbox{tr} \ln (\omega^2_n+\epsilon^2) \label{k26} \end{eqnarray} \vspace{-6mm} \newline where the symbol $\mbox{tr}$ means \newline \vspace{-6mm} \begin{eqnarray} \mbox{tr}=\sum \limits_{n=-\infty}^\infty \int \frac{d^3k}{(2 \pi )^3} \nonumber \end{eqnarray} \vspace{-6mm} \newline and $g_j(R)$ are geometric coefficients. After summation in (\ref{k26}) we will have the final result in the form \newline \vspace{-6mm} \begin{eqnarray} f_{ph}(\beta, R)=\int \frac{d^3k}{(2\pi)^3} \left\{(2/\beta)\ln (1- \exp[\beta \epsilon])\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.-(1/6)(R-2R^\mu_\nu\delta ^\nu_\mu) [\epsilon( \exp[\beta \epsilon]-1)]^{-1}\right\} \label{k27} \end{eqnarray} \vspace{-6mm} \newline This expression is the density of the Helmholtz free energy of a photon gas in an external gravitational field. \section {Internal energy and heat capacity of a photon gas} \vspace{1mm} Thermodynamic properties of the photon gas have been studied very well in flat space-time \cite{key12,huang1}. The formalism developed in this section allows us to get the properties of photon gas in curved space-time. The density of energy of a photon gas is \newline \vspace{-6mm} \begin{eqnarray} u=\frac{\partial}{\partial \beta}(\beta f)= \int \frac{d^3k}{(2\pi)^3} \left\{\frac{2\epsilon}{ \exp[\beta \epsilon]-1}\right. \nonumber \end{eqnarray} \vspace{-6mm} \newline \vspace{-17mm} \newline \vspace{-6mm} \begin{eqnarray} \left.-(1/6)(R-2R^\mu_\nu\delta ^\nu_\mu) \frac{\partial}{\partial \beta}\int \frac{d^3k}{(2\pi)^3} \frac{\beta}{\epsilon (\exp[\beta \epsilon]-1)}\right\} \label{k28} \end{eqnarray} \vspace{-6mm} \newline The first term in (\ref{k28}) corresponds to the results of statistical mechanics, and the second one is the curved space-time correction to the energy of photon gas. Integrating over the momentum \newline \vspace{-6mm} \begin{eqnarray} \int \frac{d^3k}{(2\pi)^3} \frac{1}{\epsilon (\exp[\beta \epsilon]-1)}=\frac{1}{12\beta} \label{k29} \end{eqnarray} \vspace{-6mm} \newline we get the following approximation to $O(R^2)$ \newline \vspace{-6mm} \begin{eqnarray} u(R)=\sigma T^4+(1/72)(R-2R^\mu_\nu\delta ^\nu_\mu)T^2+... \label{k30} \end{eqnarray} \vspace{-6mm} \newline where $\sigma=(\pi ^2/15)$. The heat capacity will be \newline \vspace{-6mm} \begin{eqnarray} c(R)=\frac{\partial}{\partial T}u(R)=4\sigma T^3 +(1/36)(R-2R^\mu_\nu\delta ^\nu_\mu)T^2+... \label{k31} \end{eqnarray} \vspace{-6mm} \newline As was shown in \cite{alt1} the effects of curvature for Einstein space lead to the results for the Planck black body expression which are equivalent to the results (\ref{k30}).

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\section{Introduction} Flavour Changing Neutral Current (FCNC) phenomena represent a major test for any extension of the Standard Model (SM) which is characterized by an energy scale $\Lambda$ close to the electroweak scale. Low-energy supersymmetry (SUSY) is no exception in this way: its prediction of new particles carrying flavour numbers with masses not exceeding a few TeV's (i.e., $\Lambda$ $<$ few TeV's in this case) makes the indirect search of SUSY manifestations through virtual effects in FCNC processes of utmost interest. The potentiality of probing SUSY in FCNC phenomena was readily realized when the era of SUSY phenomenology started in the early 80's \cite{susy2}. In particular, the major implication that the scalar partners of quarks of the same electric charge but belonging to different generations had to share a remarkable high mass degeneracy was emphasized. Throughout the large amount of work in this last decade it became clearer and clearer that generically talking of the implications of low-energy SUSY on FCNC may be rather misleading. We have a minimal SUSY extension of the SM, the so-called Minimal Supersymmetric Standard Model (MSSM) \cite{susy1}, where the FCNC contributions can be computed in terms of a very limited set of unknown new SUSY parameters. Remarkably enough, this minimal model succeeds to pass all the set of FCNC tests unscathed. To be sure, it is possible to severely constrain the SUSY parameter space, for instance using $b \to s \gamma$ in a way which is complementary to what is achieved by direct SUSY searches at colliders. However, the MSSM is by no means equivalent to low-energy SUSY. First, there exists an interesting large class of SUSY realizations where the customary discrete R-parity (which is invoked to suppress proton decay) is replaced by other discrete symmetries which allow either baryon or lepton violating terms in the superpotential. But, even sticking to the more orthodox view of imposing R-parity, we are still left with a large variety of extensions of the MSSM at low energy. The point is that low-energy SUSY ``feels" the new physics at the superlarge scale at which supergravity (i.e., local supersymmetry) broke down. In this last couple of years we have witnessed an increasing interest in supergravity realizations without the so-called flavour universality of the terms which break SUSY explicitly. Another class of low-energy SUSY realizations which differ from the MSSM in the FCNC sector is obtained from SUSY-GUT's. The interactions involving superheavy particles in the energy range between the GUT and the Planck scale bear important implications for the amount and kind of FCNC that we expect at low energy. After an initial effort on the study of FCNC SUSY effects in kaon physics it became clear that $B$ physics represents the new (and, for many aspects, more promising) frontier for probing SUSY through FCNC effects in the hadronic sector. There has already been an intense research activity in the realm of rare FCNC $B$ decays and SUSY. The simultaneous progress on the experimental side and, even more, the prospects that new $B$ facilities open up in these coming years make these studies of enormous interest in our effort to detail the structure (and the existence!) of low-energy SUSY. In this talk I will review some of the most recent work along these lines, in particular distinguishing the situation concerning the MSSM and other low-energy SUSY realizations. \section{FCNC in SUSY without R-Parity} It is well known that in the SM case the imposition of gauge symmetry and the usual gauge assignment of the 15 elementary fermions of each family lead to the automatic conservation of baryon (B) and lepton (L) numbers (this is true at any order in perturbation theory). On the contrary, imposing in addition to the usual $SU(3)\otimes SU(2) \otimes U(1)$ gauge symmetry an N=1 global SUSY does not prevent the appearance of terms which explicitly break B or L \cite{weinb}. Indeed, the superpotential reads: \begin{eqnarray} W&=&h^U Q H_{U}u^c + h^D Q H_{D} d^c + h^L L H_D e^c + \mu H_U H_D \nonumber \\ &+& \mu^\prime H_{U} L + \lambda^{\prime \prime}_{ijk}u^c_{i}d^c_{j}d_{k}^c + \lambda^{\prime}_{ijk}Q_{i}L_{j}d_{k}^c + \lambda_{ijk}L_{i}L_{j}e_{k}^c \, , \label{superp} \end{eqnarray} where the chiral matter superfields $Q$, $u^c$, $d^c$, $L$, $e^c$, $H_{U}$ and $H_{D}$ transform under the above gauge symmetry as: \begin{eqnarray} &\,&Q\equiv (3,2,1/6); \qquad u^c\equiv (\bar{3},1,-2/3);\qquad d^c\equiv (\bar{3},1,1/3);\\ &\,& L\equiv (1,2,-1/2); \; \; e^c \equiv (1,1,1); \;\; H_{U}\equiv (1,2,1/2); \;\; H_{D}\equiv (1,2,-1/2). \nonumber \label{qnumbers} \end{eqnarray} The couplings $h^U$, $h^D$, $h^L$ are $3\times 3$ matrices in the generation space; $i$, $j$ and $k$ are generation indices. Using the product of $\lambda^\prime$ and $\lambda^{\prime \prime}$ couplings it is immediate to construct four-fermion operators leading to proton decay through the exchange of a squark. Even if one allows for the existence of $\lambda^\prime$ and $\lambda^{\prime \prime}$ couplings only involving the heaviest generation, one can show that the bound on the product $\lambda^\prime \times \lambda^{\prime \prime}$ of these couplings is very severe (of $O(10^{-7})$) \cite{smirnov}. A solution is that there exists a discrete symmetry, B-parity \cite{b}, which forbids the B violating terms in eq.~(\ref{superp}) which are proportional to $\lambda^{\prime \prime}$. In that case it is still possible to produce sizeable effects in FC $B$ decays. For instance using the product of $\lambda^\prime_{3jk}\lambda_{ljl^c}$ one can obtain $b \to s \,(d) + l l^c$ taking $k=2 \, (1)$ and through the mediation of the sneutrino of the $j$-th generation. Two general features of these R-parity violating contributions are: \begin{enumerate} \item we completely lose any correlation to the CKM elements. For instance, in the above example, the couplings $\lambda^\prime$ and $\lambda$ have nothing to do with the usual angles $V_{tb}$ and $V_{ts}$ which appear in $b \to s l^+ l^-$ in the SM; \item we also lose correlation among different FCNC processes which are tightly correlated in the SM. For instance, in our example $b \to d l^+ l^-$ would depend on $\lambda^\prime$ and $\lambda$ parameters which are different from those appearing in $B_{d}-\bar{B}_{d}$ mixing. \end{enumerate} In this context it is difficult to make predictions given the arbitrariness of the large number of $\lambda$ and $\lambda^\prime$ parameters. There exist bounds on each individual coupling (i.e. assuming all the other L violating couplings are zero) \cite{barger}. With some exception, they are not very stringent for the third generation (generally of $O(10^{-1})$), hence allowing for conspicuous effects. Indeed, one may think of using the experimental bounds on rare $B$ decays to put severe bounds on products of L violating couplings. Obviously, the most practical way of avoiding any threat of B and L violating operators is to forbid \underline{all} such terms in eq.~(\ref{superp}). This is achieved by imposing the usual R matter parity. This quantum number reads $+1$ over every ordinary particle and $-1$ over SUSY partners. We now turn to rare $B$ decays in the framework of low energy SUSY with R-parity. \section{Model-independent analysis of FCNC processes in SUSY} Given a specific SUSY model it is in principle possible to make a full computation of all the FCNC phenomena in that context. However, given the variety of options for low-energy SUSY which was mentioned in the Introduction (even confining ourselves here to models with R matter parity), it is important to have a way to extract from the whole host of FCNC processes a set of upper limits on quantities which can be readily computed in any chosen SUSY frame. The best model-independent parameterization of FCNC effects is the so-called mass insertion approximation \cite{mins}. It concerns the most peculiar source of FCNC SUSY contributions that do not arise from the mere supersymmetrization of the FCNC in the SM. They originate from the FC couplings of gluinos and neutralinos to fermions and sfermions~\cite{FCNC}. One chooses a basis for the fermion and sfermion states where all the couplings of these particles to neutral gauginos are flavour diagonal, while the FC is exhibited by the non-diagonality of the sfermion propagators. Denoting by $\Delta$ the off-diagonal terms in the sfermion mass matrices (i.e. the mass terms relating sfermion of the same electric charge, but different flavour), the sfermion propagators can be expanded as a series in terms of $\delta = \Delta/ \tilde{m}^2$ where $\tilde{m}$ is the average sfermion mass. As long as $\Delta$ is significantly smaller than $\tilde{m}^2$, we can just take the first term of this expansion and, then, the experimental information concerning FCNC and CP violating phenomena translates into upper bounds on these $\delta$'s \cite{deltas}. Obviously the above mass insertion method presents the major advantage that one does not need the full diagonalization of the sfermion mass matrices to perform a test of the SUSY model under consideration in the FCNC sector. It is enough to compute ratios of the off-diagonal over the diagonal entries of the sfermion mass matrices and compare the results with the general bounds on the $\delta$'s that we provide here from all available experimental information. There exist four different $\Delta$ mass insertions connecting flavours $i$ and $j$ along a sfermion propagator: $\left(\Delta_{ij}\right)_{LL}$, $\left(\Delta_{ij}\right)_{RR}$, $\left(\Delta_{ij}\right)_{LR}$ and $\left(\Delta_{ij}\right)_{RL}$. The indices $L$ and $R$ refer to the helicity of the fermion partners. The size of these $\Delta$'s can be quite different. For instance, it is well known that in the MSSM case, only the $LL$ mass insertion can change flavour, while all the other three above mass insertions are flavour conserving, i.e. they have $i=j$. In this case to realize a $LR$ or $RL$ flavour change one needs a double mass insertion with the flavour changed solely in a $LL$ mass insertion and a subsequent flavour-conserving $LR$ mass insertion. Even worse is the case of a FC $RR$ transition: in the MSSM this can be accomplished only through a laborious set of three mass insertions, two flavour-conserving $LR$ transitions and an $LL$ FC insertion. Instead of the dimensional quantities $\Delta$ it is more useful to provide bounds making use of dimensionless quantities, $\delta$, that are obtained dividing the mass insertions by an average sfermion mass. The FCNC processes in $B$ physics which provide the best bounds on the $\delta_{23}$ and $\delta_{13}$ FC insertions are $b \to s \gamma$ and $B_{d}- \bar{B}_{d}$, respectively. The process $b \to s \gamma$ requires a helicity flip. In the presence of a $\left(\delta^d_{23}\right)_{LR}$ mass insertion we can realize this flip in the gluino running in the loop. On the contrary, the $\left( \delta^d_{23}\right)_{LL}$ insertion requires the helicity flip to occur in the external $b$-quark line. Hence we expect a stronger bound on the $\left(\delta^d_{23}\right)_{LR}$ quantity. Indeed, this is what happens: $\left(\delta^d_{23}\right)_{LL}$ is essentially not bounded, while $\left(\delta^d_{23}\right)_{LR}$ is limited to be $<10^{-3}-10^{-2}$ according to the average squark and gluino masses (see fig.~\ref{bsglr}). Given the upper bound on $\left(\delta^d_{23}\right)_{LR}$ from $b \to s \gamma$, it turns out that the quantity $x_{s}$ of the $B_{s}-\bar{B}_{s}$ mixing receives contributions from this kind of mass insertions which are very tiny. The only chance to obtain large values of $x_s$ is if $\left(\delta^d_{23}\right)_{LL}$ is large, say of $O(1)$. In that case $x_s$ can easily jump up to values of $O (10^{2})$ or even larger. As for the mixing $B_{d}-\bar{B}_{d}$, we obtain \begin{eqnarray} \sqrt{\left\vert {\mbox Re} \left(\delta^d_{13}\right)^{2}_{LL}\right\vert} &<&4.6 \cdot 10^{-2}\, ; \nonumber \\ \sqrt{\left\vert {\mbox Re} \left(\delta^d_{13}\right)^{2}_{LR}\right\vert} &<&5.6 \cdot 10^{-2}\, ; \nonumber \\ \sqrt{\left\vert {\mbox Re} \left(\delta^d_{13}\right)_{LL}\left( \delta^d_{13}\right)_{RR}\right\vert} &<&1.6 \cdot 10^{-2}\, ; \label{limbdbdb} \end{eqnarray} for $x\equiv m^{2}_{\tilde{g}}/m^{2}_{\tilde{q}}=0.3$ with $m_{\tilde{q}}=500$ GeV. The above bounds scale with $m_{\tilde{q}}$(GeV)$/500$ for different values of $m_{\tilde{q}}$ (at fixed $x$).\\ Then, imposing the bounds~(\ref{limbdbdb}), we can obtain the largest possible value for BR($b \to d \gamma$) through gluino exchange. As expected, the $\left( \delta^{d}_{13}\right)_{LL}$ insertion leads to very small values of this BR of $O(10^{-7})$ or so, whilst the $\left( \delta^{d}_{13}\right)_{LR}$ insertion allows for BR($b \to d \gamma$) ranging from few times $10^{-4}$ up to few times $10^{-3}$ for decreasing values of $x=m^{2}_{\tilde{g}}/ m^{2}_{\tilde{q}}$. As reminded by Ali at this meeting, in the SM we expect BR($b \to d \gamma$) to be typically $10-20$ times smaller than BR($b \to s \gamma$), i.e. BR($b \to d \gamma)=(1.7\pm 0.85 )\times 10^{-5}$. Hence a large enhancement in the SUSY case is conceivable if $\left( \delta^{d}_{13}\right)_{LR}$ is in the $10^{-2}$ range. Notice that in the MSSM we expect $\left( \delta^{d}_{13}\right)_{LR}<m^{2}_{b}/ m^{2}_{\tilde{q}}\times V_{td}<10^{-6}$, hence with no hope at all of a sizeable contribution to $b \to d \gamma$. However, as we shall see in Sect.~\ref{sec:rare}, sizeable deviations from the expected values of the $\delta$ quantities in the MSSM are possible in SUSY schemes which are obtained as the low-energy limit of $N=1$ supergravities with a GUT structure and/or non-universal soft breaking terms. \section{Rare $B$ decays in the MSSM and beyond} \label{sec:rare} Although the name seems to indicate a well-defined particle model, actually MSSM denotes at least two quite different classes of low-energy SUSY models. In its most restrictive meaning it denotes the minimal SUSY extension of the SM (i.e. with the smallest needed number of superfields) with R-parity, radiative breaking of electroweak symmetry, universality of the soft breaking terms and simplifying relations at the GUT scale among SUSY parameters. In this ``minimal" version the MSSM exhibits only four free parameters in addition to those of the SM. Moreover, some authors impose specific relations between the two parameters $A$ and $B$ that appear in the trilinear and bilinear scalar terms of the soft breaking sector further reducing the number of SUSY free parameters to three. Then, all SUSY masses are just function of these few independent parameters and, hence, many relations among them exist. Obviously this very minimal version of the MSSM can be very predictive. The most powerful constraint on this minimal model in the FCNC context comes from $b \to s \gamma$. In SUSY there are five classes of one-loop diagrams which contribute to FCNC $B$ processes. They are distinguished according to the virtual particles running in the loop: W and up-quarks, charged Higgs and up-quarks, charginos and up-squarks, neutralinos and down-squarks, gluinos and down-squarks. It turns out that, at least in this ``minimal" version of the MSSM, the charged Higgs and chargino exchanges yield the dominant SUSY contributions. As for $b \to s \gamma$ the situation can be summarized as follows. The CLEO measurement yields BR$(B \to X_{s}\gamma)=(2.32 \pm 0.67)\times 10^{-4}$ \cite{cleo}. On the theoretical side we are going to witness a major breakthrough with the computation of the next-to-leading logarithmic result for the BR. This is achieved thanks to the calculation of the $O(\alpha_{s})$ matrix elements \cite{greub} and of the next-to-leading order Wilson coefficients at $\mu \simeq m_{b}$ \cite{misiak}. The result quoted by Greub and Hurth \cite{hurth} is BR$(B \to X_{s} \gamma)=(3.25 \pm 0.50) \times 10^{-4}$ in the SM with $m_{t}=(170 \pm 15)$ GeV and $m_{b}/2 \le \mu \le 2 m_{b}$. A substantial improvement also on the experimental error is foreseen for the near future. Hence $b \to s \gamma$ is going to constitute the most relevant place in FCNC $B$ physics to constrain SUSY at least before the advent of $B$ factories. So far this process has helped in ruling out regions of the SUSY parameter space which are even larger than those excluded by LEP I and it is certainly going to be complementary to what LEP II is expected to do in probing the SUSY parameter space. After the detailed analysis in 1991 \cite{bertol} for small values of $\tan \beta$, there have been recent analyses \cite{barb} covering the entire range of $\tan \beta$ and including also other technical improvements (for instance radiative corrections in the Higgs potential). It has been shown \cite{vissani} that the exclusion plots are very sensitive also to the relation one chooses between A and B. It should be kept in mind that the ``traditional" relation $B=A-1$ holds true only in some simplified version of the MSSM. A full discussion is beyond the scope of this talk and so we refer the interested readers to the vast literature which exists on the subject. The constraint on the SUSY parameter space of the ``minimal" version of the MSSM greatly affects also the potential departures of this model from the SM expectation for $b \to s l^+ l^-$. The present limits on the exclusive channels BR$(B^{0} \to K^{*0} e^{+} e^{-})_{CLEO}<1.6 \times 10^{-5}$ \cite{cleo2} and BR$( B^{0} \to K^{*0} \mu^{+} \mu^{-})_{CDF}<2.1 \times 10^{-5}$ \cite{cdf} are within an order of magnitude of the SM predictions. On the theoretical side, it has been estimated that the evaluation of $\Gamma (B\to X_{s}l^{+} l^{-})$ in the SM is going to be affected by an error which cannot be reduced to less than $10-20 \%$ due to uncertainties in quark masses and interference effects from excited charmonium states \cite{ligeti}. It turns out that, keeping into account the bound on $b \to s \gamma$, in the MSSM with universal soft breaking terms a $20 \%$ departure from the SM expected BR is kind of largest possible value one can obtain \cite{cho}. Hence the chances to observe a meaningful deviation in this case are quite slim. However, it has been stressed that in view of the fact that three Wilson coefficients play a relevant role in the effective low-energy Hamiltonian involved in $b \to s \gamma$ and $b \to s l^{+} l^{-}$, a third observable in addition to BR$(b \to s \gamma)$ and BR$(b \to s l^{+}l^{-})$ is needed. This has been identified in some asymmetry of the emitted leptons (see refs.~\cite{cho, ali} for two different choices of such asymmetry). This quantity, even in the ``minimal" MSSM, may undergo a conspicuous deviation from its SM expectation and, hence, hopes of some manifestation of SUSY, even in this minimal realization, in $b \to s l^{+} l^{-}$ are still present. Finally, also for the $B_{d}-\bar{B}_{d}$ mixing, in the above-mentioned analysis of rare $B$ physics in the MSSM with universal soft breaking terms \cite{bertol} it was emphasized that, at least in the low $\tan \beta$ regime, one cannot expect an enhancement larger than $20\%-30\%$ over the SM prediction (see also ref.~\cite{kurimoto}). Moreover it was shown that $x_{s}/x_{d}$ is expected to be the same as in the SM. It should be kept in mind that the above stringent results strictly depend not only on the minimality of the model in terms of the superfields that are introduced, but also on the ``boundary" conditions that are chosen. All the low-energy SUSY masses are computed in terms of the $M_{Pl}$ four SUSY parameters through the RGE evolution. If one relaxes this tight constraint on the relation of the low-energy quantities and treats the masses of the SUSY particles as independent parameters, then much more freedom is gained. This holds true even if flavour universality is enforced. For instance, BR$(b \to s \gamma )$ and $\Delta m_{B_{d}}$ may vary a lot from the SM expectation, in particular in regions of moderate SUSY masses \cite{brignole}. Moreover, flavour universality is by no means a prediction of low-energy SUSY. The absence of flavour universality of soft-breaking terms may result from radiative effects at the GUT scale or from effective supergravities derived from string theory. For instance, FCNC contributions in a minimal SUSY SO(10) model might be comparable to some of the upper bounds on the FCNC $\delta$ quantities given above \cite{strumia}. In the non-universal case, BR$(b \to s l^{+} l^{-})$ is strongly affected by this larger freedom in the parameter space. There are points of this parameter space where the nonresonant BR$(B \to X_{s} e^{+} e^{-})$ and BR$(B \to X_{s} \mu^{+}\mu^{-})$ are enhanced by up to $90 \%$ and $110 \%$ while still respecting the constraint coming from $b \to s \gamma$ \cite{cho}. Finally, let us add a short comment on another rare $B$ decay, $b \to s g$, which has attracted some attention in these last years for its potentiality to increase the $b$ hadronic width, hence lowering the $b$ semileptonic branching ratio. It was recently noticed in refs.~\cite{kagan, ciu} that a BR$(b \to s g)$ close to $10 \%$ could simultaneously solve the two famous problems in $B$ physics of the semileptonic branching ratio and of the number of charms per $B$ decay (charm counting). At first sight, one could object that it is unlikely to have such a large BR$(b \to s g)$ given that its close ``friend" BR$(b \to s \gamma)$ is at the $10^{-4}$ level. However it was shown in ref.~\cite{ciu} that there exists an admittedly small region of the SUSY parameter space where, indeed, BR$(b \to s g)$ is as high as $10 \%$ without conflicting with the measured value of BR$(b \to s \gamma)$. However, at this meeting S. Stone and some of his collaborators have emphasized that a BR$(b \to s g)$ at the $10 \%$ level is in difficulty with the results of searches for the $\Phi$ from $B \to X_{s} \Phi$. \section{Conclusions} We summarize the results reported in this talk in the following three points. \begin{enumerate} \item First, a warning: under the commonly used expression of low-energy SUSY there exists actually a large choice of models (with or without R parity, with or without flavour universality of the soft breaking terms, with or without GUT assumptions,\dots) which lead to quite different implications for rare $B$ decays. \item What we called here the ``minimal" version of the MSSM, i.e. its most restrictive version with only four independent parameters, is mainly constrained by $b \to s \gamma$, with little hope of significant departures from SM in other FCNC $b$ physics (however, some exception is possible, like, for instance, lepton asymmetries in $b \to s l^{+} l^{-}$). \item Extensions of the above ``minimal" version of the MSSM with non-universality, or with SUSY-GUT's, have room for conspicuous departures from the SM in $b \to s l^{+} l^{-}$ and $b \to s g$. \end{enumerate} The hope is that the advent of $B$ factories may promote $B$ physics to a ground for ``precision tests" of new physics analogously to what has been done for the LEP Z factory. \section*{Acknowledgements} We thank F. Gabbiani and E. Gabrielli who collaborated with us in the model-independent analyses that we present here. Helpful discussions with A. Ali and G.F. Giudice are kindly acknowledged. Finally, we wish to thank the organizers of BEAUTY '96 for giving us the opportunity of discussing $B$ physics in a stimulating and pleasant atmosphere.

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\section{Introduction.} A new era in the study of string theory and black holes has been opened up by Polchinski's realization~\cite{dbranes} that soliton backgrounds in string theory can be described in a conformally-invariant way, in terms of world sheets with boundaries (thus incorporating open strings), on which Dirichlet boundary conditions for the collective target-space coordinates of the soliton are imposed. One of the most fruitful applications of this $D$-brane technology has been to black holes. In particular, many authors~\cite{counting} have demonstrated that the counting of quantum states of $D$ branes is equivalent to that of black hole states~\cite{hawkbek}. Thus, it is now generally agreed that black-hole entropy may be dissected into string states that are in principle distinguishable, as we conjectured some time ago~\cite{emn,emnhair} on the basis of studies of two-dimensional stringy black holes~\footnote{See also~\cite{KN}: a similar conjecture was made later by Susskind~\cite{susskind} on different, heuristic grounds.}. One of the ways of distinguishing black-hole states is via generalized Aharonov-Bohm measurements~\cite{emnmeas}, as has recently been discussed~\cite{banks} in the context of $D$ branes. \paragraph{} We have shown recently~\cite{emnd} how $D$ branes emerge in a formulation of the string analogue of the field-theoretical path integral, based on a treatment of non-critical string theory~\cite{emn} in which the role of the target time variable $t$ is played by the Liouville field $\phi$~\cite{aben}, treated as a local renormalization scale~\cite{emn,kogan}. We also pointed out in~\cite{emnd} several analogies between studies of $D$ branes and two-dimensional string black holes~\cite{emn}. Our interest in the latter was largely motivated by the black hole information problem and its possible implications for the effective quantum-mechanical description of light-particle degrees of freedom~\cite{emn}. This problem has also been studied extensively in the context of quantum gravity. In particular, we have shown~\cite{emnw} that, at the one-loop level, the entropy of a scalar field in the presence of a four-dimensional black-hole background in conventional general relativity diverges logarithmically with respect to the short-distance cutoff of the model, taken to be the minimum distance from the black-hole horizon at which a `brick-wall' boundary condition is imposed on the wave function of the scalar field. We have interpreted these quantum divergences as reflecting irreversible temporal evolution associated with entropy production as the horizon moves due to black-hole radiation. \paragraph{} The purpose of this paper is to initiate analogous studies in the context of $D$ branes, with an examination of the quantum effects of $D$-brane recoil during the scattering of a light closed-string state, which requires a treatment of $D$-brane excitations. The remarkably simple construction of Polchinski~\cite{dbranes} opens the way to a $\sigma$-model description of such $D$-brane excitations, in which the critical world-sheet string action is perturbed using appropriate boundary terms. We recall the form of the world-sheet boundary operators describing the excitation of a $D$ brane (see ~\cite{callan} and references therein): \begin{equation} {\cal V}_D = \int _{\partial \Sigma}( y_i \partial _n X^i + u_i X^0 \partial_n X^i) \label{dbraneop} \end{equation} where $n$ denotes the normal derivative on the boundary of the world sheet $\partial \Sigma$, which has at tree level the topology of a disk of size $L$, and the $X^i~,i=1,\dots $ denote the collective excitations of the $D$ brane, which satisfy Dirichlet boundary conditions on the world-sheet boundary: \begin{equation} X^i ({\rm boundary}) = 0,~i=1,\dots, \label{fiveb} \end{equation} whilst $X^0$ is the target time variable which satisfies standard Neumann boundary conditions: $\partial _n X^0 ({\rm boundary}) = 0$. For simplicity, we will later be considering the case of a $0$ brane, with the quantity $u_i $ in (\ref{dbraneop}) denoting its velocity, and $y_i$ its initial position~\footnote{We do not discuss here the more general case of curved world volumes, where the simple Dirichlet boundary conditions are known not to be conformally invariant~\cite{dbranes,li}.}. In this case, the operators (\ref{dbraneop}) describe shifts and motion of the $0$ brane, and so can be thought of as generating the action of the Poincar\'e group on the $0$ brane, with the $y_i$ parametrizing translations and the $u_i$ parametrizing boosts. In the general $D$-brane case, these represent translations and boosts acting on the surface ${\cal S}$ of the $D$ brane. \paragraph{} First steps were taken in refs. \cite{recoil} towards the quantum theory of the scattering of string states off a $D$-brane background. This involves considering the motion of a closed string state towards the (initially fixed) surface ${\cal S}$, the latter viewed as a string soliton background, as shown in Fig.~1(a). In the general $D$-brane case, the surface ${\cal S}$ divides the target space-time into two regions. The closed string state is initially far outside the surface of the brane. At a certain moment, say $X^0=0$, the incoming closed string state finds itself lying partly outside and partly inside the $D$-brane surface. There are then two possibilities to be considered: it may be either absorbed (Fig. 1(b)) or rescattered (Fig. 1(c)). We seek below the appropriate conformal field theory description of the latter case. \paragraph{} In general, quantum scattering off the $D$ brane excites an open string state on the surface of the $D$ brane, which in the scattering case of Fig.~1(c) also emits another closed string state. The quantum excitation and emission processes are both described by closed-to-open string amplitudes, which are non-zero in a world-sheet theory with boundaries. The open-string states are excitations on the $D$-brane collective-coordinate surface. As was shown in ref. \cite{cardy}, such processes can be described in terms of data of the bulk theory. As we show in more detail below, tracing over such excitations results in a quantum modification of the Hawking-Bekenstein area law for the entropy, which has been shown to hold in tree-level treatments of $D$ branes. \paragraph{} To discuss the evolution of entropy, it is essential to treat correctly the quantum recoil of the $D$ brane during the scattering process, which we discuss in section 2. The treatment of recoil requires~\cite{recoil,kogmav,periwal,emnd,kogwheat} an operator with non-zero matrix elements between different $D$- (in our case $0$-)brane states. This can be achieved in the impulse approximation by introducing a Heaviside function factor, $\Theta (X^0)$, into the second operator in (\ref{dbraneop}), which describes a $0$ brane that starts moving at time $X^0=0$. The initial position of the $0$ brane at $X^0=0$ is assumed to be given by the $y_i$. To determine the precise form of the recoil operator in our case, we observe that the leading quantum correction to the scattering of a closed string off a $D$ brane is given by an annulus, as shown in Fig.~2(a). This is divergent in the limit where the annulus is pinched, as shown in Fig.~2(b)~\cite{recoil,kogmav,periwal,emnd}. As was done previously for closed strings in the context of two-dimensional black holes~\cite{emn}\footnote{We recall that in the similar analysis~\cite{emn} of two-dimensional stringy black holes, the r\^ole of recoil was played by the back-reaction operator which described a change of state of the black-hole background. In the underlying conformal field-theoretical description of ref. \cite{witt}, the corresponding operator was the world-sheet instanton anti-instanton vertex~\cite{yung}. The instanton vertex induces (target-space infrared) infinities in correlation functions that are identical to those arising at the torus (closed-string) level~\cite{emn}. In this context, the instanton coupling constant is proportional to the string coupling, and thus a one-string loop analysis corresponds to a dilute instanton gas approximation, as adopted in~\cite{emn}.}, we seek a deformation of the $\sigma$-model that reproduces at the tree (disc) level these infinities arising from quantum string loop corrections, which are infrared in target space. This determines the form of the quantum recoil, which is described by a pair of logarithmic operators, and one recovers momentum conservation as a consistency check of the approach\cite{emn}. In the $D$-brane case discussed here, the weakly-coupled string limit: $g_s \rightarrow 0$ corresponds correctly to a one-open-string-loop (annulus) analysis for a semi-classical (heavy) $0$ brane ($D$ particle), since the mass of the latter is $M_D \propto 1/g_s$ in natural string units. \paragraph{} We argue in section 3 that this treatment of the recoil of the $D$ brane using logarithmic operators~\cite{gurarie,tsvelik,kogmav} entails a time evolution of the $D$-brane state that mirrors what we found previously in the cases of non-critical Liouville strings~\cite{emn} and of a scalar field in the presence of a four-dimensional black hole~\cite{emnw}. The interaction between the incident light-string state and the $D$ brane is described by a source term in the world-sheet $\sigma$ model which induces an apparent departure from criticality. This is compensated by non-trivial dynamics of the Liouville field, whose zero mode is identified with target time $t$, i.e., the zero mode of $X^0$. As we demonstrate explicitly in section 4 using the $D$-brane equation of motion based on a world-sheet Wilson-Polchinski exact Renormalization-Group equation~\cite{wilson}, the evolution of the $D$-brane state is of diffusive Fokker-Planck type, which necessarily entails entropy production. This loss of information in the $D$-brane sector, i.e., transition to a mixed quantum-mechanical state, is accompanied, because of quantum entanglement, by a corresponding transition to a mixed state in the truncated external light-particle system, as we discuss in section 5. The time evolution of the scattering closed-string state does not have a simple Hamiltonian description, but provides a new explicit realization of the modified quantum Liouville equation of~\cite{ehns} within the general non-critical string framework of~\cite{emn,kogan}. As we discuss in section 6, if we speculate on the extension of these results to the propagation of a closed string state through ``virtual $D$ brane foam'', the maximum possible rate of entropy growth that we find is similar in magnitude to that we estimated previously in the two-dimensional Liouville-string~\cite{emn} and four-dimensional black-hole contexts~\cite{emnw}. \section{Operator Treatment of $D$-Brane Recoil} \paragraph{} We now review briefly the treatment~\cite{periwal} of the scattering of a closed string off a $D$ brane, restricting ourselves for reasons of simplicity to the case of point-like $0$ branes: the extension to higher-dimensional $D$ branes is straightforward. For this purpose, we need to consider the annulus amplitude of Fig.~2(a), where the crosses denote closed-string vertex operators $V(k)$, which must be integrated over the propagating open string. This computation may be performed using the operator formalism, in which one evaluates ${\rm Tr}V(k_1)\Delta V(k_2)\Delta$, with $\Delta ^{-1} \equiv L_0 -1 $, where $L_0$ is the Virasoro operator~\footnote{Alternatively, one may work in the closed-string channel, in which case one introduces a state $|B>$ into the closed-string Fock space, to impose the appropriate boundary conditions on the end of the closed-string world sheet: we return later to this approach.}. The part of this computation that is relevant for our purposes is that due to the world-sheet zero modes~\cite{periwal}. Writing $\Delta \equiv \int _0^1 dx x^{L_0-2}$, $L_0=2p^2 + N$, where $N$ is the string level number, and picking out the $N=0$ part, we find the following contribution to the annulus amplitude: \begin{equation} {\cal A} = \int dq <q|exp(-ik_{1}^0X^{0})x_1^{-2(p^0)^2} exp(-ik_{2}^0X^{0})x_2^{-2(p^0)^2}|q> \label{annulus} \end{equation} where the superscript $0$ denotes target-time components, and $q$ is the modular parameter of the annulus. The trace over the zero modes yields the generic form~\cite{periwal} \begin{equation} {\cal A} \ni \delta (k_1^0 + k_2^0)\sqrt{\frac{1}{{\rm log}(x_1)}} f(x_2, k_2^0) \label{result} \end{equation} which conserves momentum in the light-state sector, i.e., does not include any $D$-brane recoil momentum. The $\delta (k_1^0 + k_2^0)$ function arises from integrating over the zero modes of $\sigma$-model fields $X^0(z,{\bar z})$ in a standard fashion. There is also a corresponding $\delta (k_1 + k_2)$ over the space components, not written explicitly above, which comes from integrating over the world-sheet zero modes of the $X^i$ fields. The amplitude (\ref{annulus}, \ref{result}) is pathological, in the sense that it is divergent as $x_1 \rightarrow 0$~\cite{periwal}, and requires regularization. It is this regularization that induces recoil effects which modify the $\delta (k_1+k_2)$ term in order to ensure momentum conservation in the presence of recoil. \paragraph{} The pathological behaviour in the limit $x_1 \rightarrow 0$ corresponds to the pinched-annulus configuration shown in Fig.~2(b): \begin{equation} {\cal A} \ni g_{s}\int _{x\sim 0}\frac{dx_1}{x_1\sqrt{8\pi{\rm log}(x_1)}} A_{disk}(k_1,k_2) \label{result2} \end{equation} where $A_{disc}(k_1,k_2) = <V(k_1)V(k_2)V^iV^i>$ is the tree-level disc amplitude, with $\{ V_i \}$ denoting a complete set of eigenstates of the $L-0$ Virasoro operator, which includes $D$-brane Goldstone zero modes leading to dominant divergent contributions~\cite{recoil,kogmav,periwal,emnd}. To cancel this one-loop infrared divergence, one must add the following tree-level closed-string operator counterterm \begin{equation} \delta {\cal A} = \int d^2z \partial _\alpha (f(X^0)\partial ^\alpha X^i) \label{oper} \end{equation} which contributes on the boundary. Its form is determined by general properties of soliton backgrounds in string theory~\cite{kogmav}, and the function $f(X^0)$ in (\ref{oper}) is determined by requiring that the above operator reproduce the infinities of the annulus amplitude (\ref{result2}). We refer the reader to~\cite{periwal} for details. For our purposes, we restrict our attention to the final expression for the `impulse' operator~\cite{periwal}: \begin{equation} V_{imp} \equiv \int d^2z\thinspace \partial _\alpha ([u_i X^0 ]\Theta (X^0) \partial _\alpha X^i) = \int d\tau\thinspace u_i X^0 \Theta (X^0) \partial _n X^i~; \qquad i=1, \dots 9. \label{wrongrecoil} \end{equation} The step-function operator in (\ref{wrongrecoil}) needs to be defined properly, and we adopt the integral representation~\cite{kogwheat}: \begin{equation} \Theta_{\epsilon} (X^0) = -i\int\limits_{-\infty}^\infty \frac{dq}{q - i\epsilon} e^{iq X^0}\quad, \qquad \epsilon \rightarrow 0^+ \label{theta} \end{equation} where $\epsilon$ is an infrared regulator parameter\footnote{The sign of $\epsilon$ is given a physical origin in the next section.}. In (\ref{wrongrecoil}), the coupling $u_i$ denotes a change in the velocity of the brane, which is determined by imposing overall conformal invariance of the annulus and disc amplitudes~\cite{recoil}. The cancellation of tree and annulus divergences requires \begin{equation} u_i =8\sqrt{2}\pi g_s(k_1 + k_2)_i \label{velocity} \end{equation} which expresses momentum conservation. This interpretation of (\ref{velocity}) is consistent with the fact that the soliton mass is proportional to $1/g_s$, confirming the interpretation of $u_i$ as the $D$-brane velocity. \paragraph{} Analysis of the operator product of the operator (\ref{wrongrecoil}) reveals that it is a {\it logarithmic } operator~\cite{gurarie} \begin{equation} V_{imp}(x)V_{imp}(0) \sim {\rm log} x/|x|^2 \label{log} \end{equation} and can be decomposed into one of a {\it pair} of logarithmic operators $C$ and $D$~\cite{gurarie,tsvelik}, namely $D$ in the notation of \cite{kogwheat}. To identify the pair let us concentrate, for convenience, on the $X^0$-dependent parts. The $D_\epsilon$ recoil operator is identified as~\cite{kogwheat} \begin{equation} D_\epsilon \equiv X^0 \Theta (X^0) \label{dopera} \end{equation} The $C_\epsilon $ operator can be found by studying the operator product of $D_\epsilon$ with the stress-energy tensor: \begin{equation} T(w)D_{\epsilon}(z)= ~\frac{-\epsilon^2/2}{(w-z)^2} D_{\epsilon} + \frac{1}{(w-z)^2}\epsilon\Theta(X^0) \label{opetd1} \end{equation} This enables us to identify the $C_\epsilon $ operator, using the general properties of such logarithmic pairs~\cite{gurarie,tsvelik}: \begin{equation} C_{\epsilon} = \epsilon \Theta_{\epsilon} (X^0) \label{cop} \end{equation} Thus, the correct $D$-brane recoil operator is~\cite{kogwheat}: \begin{eqnarray} {\cal V}_{rec} = \int d\tau ~ [y_i C_{\epsilon}(X^0) \partial _nX^i + u_iD_{\epsilon}(X^0) \partial _n X^i ] \label{recoil} \end{eqnarray} The analysis of ref. \cite{kogwheat} showed that the degenerate operators $C_\epsilon$ and $D_\epsilon$ have conformal dimension $\Delta=-{\epsilon^2\over 2}$, which is negative and vanishing in the limit $\epsilon \rightarrow 0^+$. This means that the impulse operators $V_{rec}$ (\ref{recoil}) are {\it relevant} in a renormalization-group sense for any non-zero $\epsilon$, since, due to their $\partial _n X^i$ parts, their anomalous scaling dimension is $\Delta -1 +1 =-{\epsilon ^2\over 2}$. This is intrinsic to the nature of logarithmic operators, which appear on the border line between conformal field theories and general renormalizable two-dimensional field theories \cite{kogmav},\cite{emnd},\cite{gkf}. These relevant deformations in the recoil problem lead to a {\it change} in the background $0$-brane state~\cite{kogmav,emnd}, with the physical consequences discussed in the next section. \paragraph{} Explicit expressions for the one- and two-point functions of the operators $D_{\epsilon}(X^0)$ and $C_{\epsilon}(X^0)$ appearing in (\ref{recoil}) were derived in~\cite{kogwheat}. It is sufficient for our purposes here to quote the results for the two-point functions: \begin{eqnarray} &~&<C_\epsilon (z)C_\epsilon (0)> \sim -\epsilon^2\sqrt{\frac{\pi}{\alpha}} \int\limits_{-\infty}^\infty \frac{dq}{(q^2+\epsilon^2)} e^{-2\eta q^2\log|z/a|^2} \nonumber\\ &~&=-\epsilon^2 \pi\sqrt{\frac{\pi}{\epsilon^2\alpha}} e^{2\eta\epsilon ^2 \log|z/a|^2} \left(1-{\rm erf}\left(\epsilon \sqrt{2\eta \log|z/a|^2}\right)\right)\nonumber\\ &~&\stackrel{\epsilon\to 0}{\sim} 0+O(\epsilon^2) \label{cc3} \end{eqnarray} \newcommand{2\eta\epsilon^2 \log|z/a|^2}{2\eta\epsilon^2 \log|z/a|^2} \begin{eqnarray} &~&<C_\epsilon (z)D_\epsilon (0)> \sim -\frac{\epsilon}{2}\sqrt{\frac{\pi}{\alpha}} \frac{\partial}{\partial \epsilon} \int\limits_{-\infty}^\infty \frac{dq}{q^2 + \epsilon ^2} e^{-2\eta q^2 \log|z/a|^2}\nonumber \\ &~& = {\pi\over2}\sqrt{{\pi\over\epsilon^2\alpha}} \bigg\{e^{2\eta\epsilon^2 \log|z/a|^2}\left(1-4\eta\epsilon^2 \log|z/a|^2\right) \left(1-{\rm erf}\left(\sqrt{2\eta\epsilon^2 \log|z/a|^2}\right)\right)\nonumber\\ &~&\,\qquad\qquad\qquad+2\sqrt{{2\eta\epsilon^2 \log|z/a|^2\over\pi}}\,\bigg\}\nonumber\\ &~&\stackrel{\epsilon\to 0}{\sim}{\pi\over2}\sqrt{{\pi\over\epsilon^2\alpha}} \left(1-2\eta\epsilon^2 \log|z/a|^2\right) \label{cd} \end{eqnarray} \begin{eqnarray} &~&<D_\epsilon (z)D_\epsilon (0)> = \frac{1}{\epsilon^2} <C_\epsilon (z)D_\epsilon (0)> \nonumber \\ &~&\stackrel{\epsilon\to 0}{\sim}{\pi\over2}\sqrt{{\pi\over\epsilon^2\alpha}} \left({1\over\epsilon^2}-2\eta\log|z/a|^2\right) \label{dd} \end{eqnarray} We have denoted by $\eta$ is the signature of the target metric: \begin{equation} \hbox{Lim}_{z \rightarrow 0} < X^0(z) X^0(0) > \simeq \eta \log |a/L|^2 \label{xx} \end{equation} where $a$ is a conventional ultraviolet regulator parameter on the world sheet and $L$ is an infrared regulator parameter which characterizes the size of the world-sheet annulus. We see explicitly that in the limit \begin{equation} \epsilon \rightarrow 0, \qquad \epsilon ^2 \log|L/a|^2 \sim O(1) \label{epsilonlog} \end{equation} we obtain the canonical two-point correlation functions of a pair of logarithmic operators~\cite{gurarie}, with one exception - the singular $1/\epsilon^2$ term in $<DD>$~\footnote{Note that the singularity structure at small $\epsilon$ is the same in connected correlation functions~\cite{kogwheat}.}. The relevance of this singular term will become apparent in the next section, when we discuss the equation of motion of the semiclassical $D$-brane wave function~\cite{wilson,rey,emnd}. \section{Renormalization-Group Rescaling and Time} \paragraph{} We now discuss in more detail renormalization-group rescaling on the world sheet, with the aim of establishing a connection with the target time variable. The value of the numerical constant in (\ref{epsilonlog}) is a free parameter which we may choose it at will, with the differences between different choices being reabsorbed in the redefinition of nonleading $\log z$ terms. It was argued in ref.~\cite{kogwheat} that the most natural choice is (\ref{epsilonlog}), i.e. \begin{equation} {1\over\epsilon^2} = 2 \eta \log|L/a|^2 \label{relation} \end{equation} which leads to the following singularity structure: \begin{eqnarray} <C_{\epsilon}(z) C_{\epsilon}(0) > &\sim& 0 + O[\epsilon^2] \nonumber \\ <C_{\epsilon}(z) D_{\epsilon}(0) > &\sim & 1 \nonumber \\ <D_{\epsilon}(z)D_{\epsilon}(0)> &\sim& -2 \eta \log|z/L|^2 \label{CD} \end{eqnarray} up to an overall normalization factor. \paragraph{} We now consider a finite-size scale transformation \begin{equation} L \rightarrow L' = L e^{t} \label{fsscaling} \end{equation} of the only type which makes physical sense for the open string world-sheet. The relation (\ref{relation}) between $\epsilon$ and $L$ entails the following transformation of $\epsilon$~\footnote{Note that, if $\epsilon$ is infinitesimally small, so also is $\epsilon'$ for any finite $t$.}: \begin{equation} \epsilon^2 \rightarrow \epsilon'^2 = \frac{\epsilon^2}{1 + 4\eta \epsilon^2 t} \label{epsilontransform} \end{equation} We deduce from the scale dependences of the correlation functions (\ref{CD}) that the corresponding transformations of $ C_{\epsilon}$ and $D_{\epsilon}$ are: \begin{eqnarray} D_{\epsilon} &\rightarrow& D_{\epsilon'} = D_{\epsilon} - t C_{\epsilon} \nonumber \\ C_{\epsilon} &\rightarrow& C_{\epsilon'}= C_{\epsilon} \label{correspond} \end{eqnarray} We emphasize that this transformation law is {\it unambiguous}, being in particular {\it independent} of the signature parameter $\eta$. \paragraph{} The corresponding transformation laws for the couplings $y_i$ and $u_i$, which are conjugate to $ D_{\epsilon}$ and $C_{\epsilon}$ in the recoil expression (\ref{recoil}), are \begin{equation} u_i \rightarrow u_i~~,~~y_i \rightarrow y_i + u_i t \label{scale2} \end{equation} This is consistent with the interpretations of $u_i$ as the velocity after the scattering process and $y_i$ as the spatial collective coordinates of the brane, if and only if the scale-change parameter $t$ is interpreted as the {\it target Minkowski time}\footnote{It is apparent from the scale transformation (\ref{correspond}) that even if one started with the $D_\epsilon$ operator alone, a $C_\epsilon$ operator would have been induced, as required if the pair (\ref{recoil}) is to describe correctly $D$-brane recoil, as in (\ref{scale2}).}, and is therefore to be identified with the zero mode of $X^0$. In this analysis we have assumed that the velocity $u_i$ is small, as is appropriate in the weak coupling r\'egime studied here (\ref{velocity}). The $D$-brane $\sigma$-model formalism is completely relativistic, and we anticipate that a complete treatment beyond the one-loop order discussed here will incorporate correctly all relativistic effects, including Lorentz factors wherever appropriate. \paragraph{} This observation that a world-sheet scale transformation leads to the target-time evolution of the $D$ brane, once recoil is taken into account, is in the same spirit as was proposed previously in the context of Liouville strings~\cite{emn,kogan}. There also an identification was made between target time and a world-sheet renormalization scale, provided in that case by the zero mode of the Liouville field. \paragraph{} We now provide technical support for a similar identification in the present model of the zero modes of the target time variable $X^0$ (\ref{dbraneop}) and the Liouville scale field $\phi$~\footnote{In our interpretation~\cite{emn}, $\phi$ is a local renormalization scale on the world sheet, and its zero mode is identified with $t = \log |L/a|$.}, which is defined by $\gamma_{\alpha \beta} = e^{\phi} \delta_{\alpha \beta}$ for the disc topology, where $\gamma_{\alpha \beta}$ is the world-sheet metric. We first rewrite the boundary recoil operators (\ref{recoil}) as world-sheet bulk operators \begin{equation} V_i = \int _\Sigma d^2z \partial _\alpha (y_i (X^0) \partial ^\alpha X^i ) \label{bulk} \end{equation} Since they have non-vanishing anomalous dimension $-\epsilon ^2/2$, the deformed theory is non-critical. Local scale (conformal) invariance may be restored in the usual way by dressing (\ref{bulk}) with the Liouville field $\phi (z,{\bar z})$~\cite{DDK}: \begin{equation} V_i^L \equiv \int _\Sigma d^2z e^{\alpha _i \phi (z,{\bar z})} \partial _\alpha (y_i (X^0) \partial ^\alpha X^i ) \label{bulkliouv} \end{equation} where $\alpha _i (\alpha _i + Q) =-2\delta _i $: $Q$ is the central-charge deficit in the unperturbed theory and $\delta _i=-\epsilon ^2/2$ the anomalous dimension. In our case, $Q=0$, consistent with fixed Dirichlet boundary conditions~\cite{li}, so that $\alpha _i =\epsilon$. Partial integration of (\ref{bulkliouv}) then leads to \begin{equation} V_i^L = \epsilon \int _{\Sigma} d^2z \int \partial _\alpha \phi \partial^\alpha X^i y_i (X^0) + \int _{\partial \Sigma} d\tau e^{\epsilon \phi (\tau)} y_i(X^0) \partial_n X^i \label{expressionliouv} \end{equation} We now use the fact (\ref{scale2}) that $y(X^0) = u_i X^0\Theta (X^0)$, and the integral representation (\ref{theta}), to observe that the second term in (\ref{expressionliouv}) contains a factor $e^{i(q X^0 - i \epsilon \phi)}$. Identifying $\phi = X^0$, we may write \begin{equation} \Theta _\epsilon (X^0) e^{\epsilon X^0} = {1 \over i}\int^{\infty}_{-\infty} \frac{dq}{q-i\epsilon} e^{i(q-i\epsilon)~X^0} = {1 \over i} \int^{\infty - i \epsilon}_{- \infty - i \epsilon} \frac{d\omega}{\omega} e^{i\omega~X^0} \label{thetaliouv} \end{equation} which implies that, as far as the regulated $\Theta _\epsilon (X^0)$ is concerned, the effect of the Liouville dressing and the identification of $X^0$ with $\phi$ is equivalent to setting $\epsilon \rightarrow 0$ in (\ref{theta}), corresponding to the restoration of conformal invariance. Thus the promotion of the local renormalization scale $t$ to a Liouville field and its identification with the time $X^0$ are consistent, if the following target-space metric $G_{MN}$ derived from (\ref{expressionliouv}) is considered: \begin{equation} G_{00}= 1 \qquad ; G_{ij} = \delta _{ij} \qquad G_{0i} = G_{i0} = \epsilon y_i (X^0) = \epsilon u_i X^0 \Theta (X^0) \label{metrictarget} \end{equation} where $0$ indices denote `time' components, and latin indices denote spatial components. In our discussion so far, we have ignored the effects of the $C-D$ mixing discussed previously (\ref{correspond}). Indeed, due to this mixing, the effects of the $C_\epsilon$ recoil operator cannot be ignored~\cite{kogwheat}. Such effects contribute a term $\epsilon y_i \Theta (X^0)$ in the world line of the $0$ brane $y_i(X^0)$, with $y_i$ the initial location of the brane. According to the discussion in ref. \cite{kogwheat} and in the Appendix, such terms express quantum fluctuations in the location of the brane due to the excitation of {\it stringy} modes. Thus (\ref{metrictarget}) becomes \begin{equation} G_{0i}=G_{i0} =\epsilon (\epsilon y_i + u_i X^0)\Theta (X^0) \label{metriccomplete} \end{equation} The off-diagonal metric component $G_{0i}$ appears to have a discontinuity at $X^0 = 0$. As we shall argue below, the $\epsilon y_i$-dependent parts of (\ref{metriccomplete}) lead to a curvature-singularity at $X^0=0$ which will be $crucial$ for the information-loss problem. \paragraph{} The metric (\ref{metriccomplete}) is consistent with conformal invariance of the $\sigma$ model, as can be seen by considering the derivative with respect to the (local scale) $X^0$: \begin{equation} \frac{dG_{0i}}{d X^0}= \epsilon [ \partial _0y_i(X^0) ]= \epsilon [u_i \Theta (X^0) + (\epsilon y_i + u_i X^0) \delta (X^0)] \label{scalecond} \end{equation} The right-hand side of (\ref{scalecond}) may be identified as a generalized coordinate transformation in target space of the form $\nabla_{(0} \chi_{i)} (X^0)$: $\chi_i \sim \epsilon (\epsilon y_i + u_i X^0) \Theta (X^0)$, where the parenthesized index is symmetrized. Equation (\ref{scalecond}) expresses the condition for conformal (Weyl) invariance of the respective $\sigma$ model, in the usual way, and demonstrates the consistency of the identification of $X^0$ with the Liouville scale\footnote{A similar situation occurred in the ($1+1$)-dimensional black hole case~\cite{emn}.}. We note that (\ref{scalecond}) contains a singularity of the $\delta$-function type, corresponding to the discontinuity in the metric (\ref{metrictarget}). There is a corresponding singularity in the Riemann curvature scalar at $X^0=0$: \begin{eqnarray} &~&R \ni -\epsilon ^2 y^i (X^0) \partial _0 y^i (X^0) + \dots \ni -\epsilon ^2 y^i (X^0) y^i (X^0) \delta (X^0) + \dots = \nonumber \\ &~&-\epsilon^2 (\epsilon y_i + u^i X_0)^2 \delta (X^0) \Theta (X^0) + \dots \label{scalrcurv} \end{eqnarray} where the temporal coordinate is $X^0$, and we can regard $\epsilon y_i + u_iX^0 \equiv \xi_i$ as a Galilean-transformed spatial coordinate, as discussed in more detail in section 5, where we construct the $\sigma$ model that describes the low-energy matter degrees of freedom. We stress once more that, as discussed in ref. \cite{kogwheat} and in the Appendix, the singular terms in (\ref{scalrcurv}) are linked to the $C_\epsilon $ (quantum) recoil operator~\cite{kogwheat}, and express quantum fluctuations in the location of the recoiling $D$-brane due to the excitation of the virtual {\it stringy} effects discussed in section 4. \paragraph{} After identifying $X^0$ with a local renormalization scale (zero mode of the Liouville field), we find that $D$-brane recoil {\it induces a back reaction on the space-time geometry which results in a temporal singularity in the curvature scalar}. In our view, this lies at the core of the information-loss problem associated with virtual $D$-brane excitations induced by the recoil at time $X^0$, as we discuss in more detail in the next sections. \paragraph{} \section{Recoil and Information Loss} \paragraph{} We now discuss the implications of the above recoil analysis for the time evolution of the $D$-brane state. We start from the formalism of the Wilson approach to the renormalization group, as refined by Polchinski and applied previously in the a $\sigma$-model approach to $D$ branes~\cite{wilson,rey,emnd}. We denote by $\Psi [X^i]$ the partition function of a $D$ brane with collective coordinates $X^i$, computed at the disc and annulus (tree and one-loop) level, which plays the r\^ole of the semi-classical wavefunction of the $D$ brane~\cite{rey,emnd}. This obeys a renormalization-group equation, which expresses its scaling properties with respect to a generic short-distance cutoff $\Lambda$ on the world-sheet~\cite{rey}: \begin{equation} \Lambda {\partial \over \partial \Lambda} \Psi [X] = \frac{1}{2\!} \int\int _{\partial \Sigma} d\tau _1 d\tau _2 \Lambda {\partial \over \partial \Lambda} G (\tau_1,\tau_2) \frac{\partial ^2}{\partial X^i (\tau _1) \partial X^j (\tau_2)} \Psi [X] + ... \label{wilson} \end{equation} where $\partial \Sigma$ denotes the world-sheet boundary and the two-point Green function $G(\tau _1, \tau_2)=<X^i(\tau _1)~X^j~(\tau _2)>$. The dots denote terms of the form $(\partial \Psi [X]/\partial X)^2$ , which represent interactions among $D$ branes, and are ignored to the order of accuracy of the present work. In our case, we identify $\Lambda$ with $L/a$ and $t = \log |L/a|$ (\ref{fsscaling}), so that $\Lambda \partial/\partial \Lambda = \partial/\partial t$ and the renormalization-group equation~\cite{rey} may be recast in the form \begin{equation} {\partial \over \partial t} \Psi [X] = \frac{1}{2\!} \int\int _{\partial \Sigma} d\tau _1 d\tau _2 {\partial \over \partial t} G (\tau_1,\tau_2) \frac{\partial ^2}{\partial X^i (\tau _1) \partial X^j (\tau_2)} \Psi [X] + ... \label{tform} \end{equation} which has the form of a diffusion equation. In order to evaluate the rate of change of the entropy, we interpret $|\Psi [X]|^2$ in the natural way as the probability density distribution for the $D$ brane to find itself in the position $X^i(t)$ at time $t$. Using this interpretation, (\ref{tform}) may be used to derive a Fokker-Planck equation for the probability distribution ${\cal P} \equiv |\Psi [X]|^2$: \begin{equation} \partial _t {\cal P} = D~\nabla _{X^i}^2 {\cal P} + \dots \label{FP} \end{equation} as we discuss in detail below. In~(\ref{FP}), $D$ denotes a diffusion coefficient, the dots denote terms representing interactions among the $D$ branes, which we will ignore, apart from a few speculative comments in the last section, and the $X^i$ denote the (world-sheet-independent) zero modes of the collective coordinates of the brane. \paragraph{} In order to derive~(\ref{FP}) and demonstrate that it is non-trivial, we show that both of the factors in the integrand of~(\ref{tform}) are non-zero, using the analysis of the pair $C_{\epsilon}, D_{\epsilon}$ discussed in the previous section. We start with $G(\tau_1, \tau_2)$, which depends on $t$ only via its divergent part, which is due to the (world-sheet-independent) zero modes on the annulus. These divergent parts can readily be computed in the $\sigma$-model approach by noting that the collective coordinates of the $D$ brane depend on the cut-off scale, as a result of the shift (\ref{scale2}) due to $C$-$D$ operator mixing (\ref{correspond}). Thus the world-sheet-coordinate-independent logarithmic divergences in $G(\tau _1, \tau_2)$ can be studied~\cite{emnd} by replacing $X^i (\tau _1)$ by the boosted coordinate $u_i X^0(\tau _1) \Theta (X^0)$, i.e. the $D_\epsilon$ recoil operator itself. The calculation then reduces to the evaluation of the free two-point function of this operator, which was given in ref.~\cite{kogwheat}, and quoted above (\ref{dd}): \begin{equation} G(\tau _1, \tau _2) |_{divergence} \sim u_i^2 <D_{\epsilon } (x) D_{\epsilon} (x') >_{divergence} = u_i^2 log|L/a|^2 \label{diver} \end{equation} This leads to a parametrization of $G(\tau _1, \tau _2)$ by an effective diffusion coefficient $D$: \begin{equation} G(\tau_1, \tau_2) \simeq 2D t: \, t \simeq log|L/a| \label{defd} \end{equation} with \begin{equation} D = u_i u^i \label{dformula} \end{equation} which we substitute into~(\ref{tform}) to obtain \begin{equation} \partial _t \Psi [X] = u_i^2 <\int \int _{\partial \Sigma} d\tau _1 d\tau _2 {\partial^2 \over \partial X^i(\tau_1) \partial X^j(\tau_2)} \Psi[X] + \dots > \label{intermediate} \end{equation} We must now verify that the integrand in (\ref{intermediate}) is also non-vanishing. \paragraph{} To see this, we first use the form (\ref{recoil}) of the $D$-brane recoil operator to re-express this integrand as \begin{equation} {\cal A}_2 \equiv <\partial _n X^i (\tau _1) \partial _{n'} X^j (\tau _2) \delta _{ij}>_{annulus} \label{pole} \end{equation} in the absence of any closed-string tachyon perturbation, where the expectation value is taken in the unperturbed $D$-brane $\sigma$ model, without $C,D$ deformations. The two-point function on the annulus in (\ref{pole}) is that of the Goldstone-mode operators $\int_{\partial \Sigma} d\tau \partial _nX^i (\tau) $ and is proportional to the shift in the mass of the Goldstone mode due to loop corrections~\cite{recoil}. However, since the Goldstone mode expresses the spontaneous breaking of an exact global symmetry in target space, namely translation invariance~\cite{kogmav,emnd}, its mass should be strictly zero. Indeed, the contour deformation technique applied in~\cite{recoil} confirms this, since the only contribution to this two-point function comes from the tachyonic pathologies of the bosonic string, which are absent in supersymmetric theories. Thus, in the absence of matter deformations, the Wilson equation (\ref{tform}) has only trivial content. \paragraph{} However, equations (\ref{tform}) and (\ref{FP}) are non-trivial when matter deformations are included. To see this, we employ the effective tree-level approach to recoil, where one deforms the $\sigma$ model by the world-sheet boundary term (\ref{recoil}), and performs a perturbative expansion in powers of the tachyon deformations. Specifically, at first order we replace (\ref{pole}) by the three-point amplitude \begin{equation} {\cal A}_3 \equiv <\int _{\partial \Sigma} d\tau _1 \int _{\partial \Sigma} d\tau _2 \partial _n X^i (\tau _1) \partial _{n'}X^i (\tau _2) \int _{\Sigma} d^2\sigma T(k) e^{ik_M X^M (\sigma) }> \label{3point} \end{equation} evaluated using a free $\sigma$-model action on a world sheet with boundaries, which is the leading-order contribution of matter deformations in the weak-tachyon approximation to the right-hand-side of (\ref{tform}). The non-zero contributions that lead to diffusion arise when the bulk tachyon deformation approaches the boundary of the world sheet. In such a case, we expect the following boundary operator product expansion~\cite{dieh,cardy} to hold: \begin{equation} \Phi (z, {\bar z}, y) \sim \sum _{i} (2y)^{\Delta _i - x_\Phi}C^a _{\Phi\psi _i} \psi _i (y) \label{boundary} \end{equation} provided that the set of boundary conditions $a$ does not break conformal symmetry. In (\ref{boundary}), $z,{\bar z}$ denote world-sheet (bulk) coordinates, the $y$ are world-sheet boundary coordinates, $\Phi (z,{\bar z})=e^{ik_MX^M(z,{\bar z})}$ is the bulk deformation by the closed-string tachyon, $x_\phi$ is its scaling dimension, and $\{ \psi _i (y) \}$ is a (complete) set of boundary operators with scaling dimensions $\Delta _i$. In theories where logarithmic operators are absent, it was shown in~\cite{cardy} that the open-to-closed-string O.P.E. coefficients $C^a_{\Phi\psi} $ could be re-expressed in terms of bulk data, in particular the coefficients $C^i_{jk}$ that arise in the ordinary bulk O.P.E.. An extension of this analysis to general logarithmic operators is an important technical issue that falls beyond the scope of the present work~\cite{kogwheat2}, but we are able to establish that there is a non-vanishing coefficient in our case. \paragraph{} The boundary operator $\psi$ of interest to us is the $D$ operator discussed in the previous sections. To see that the $C^a_{\Phi D}$ O.P.E. coefficient is indeed non-zero, we consider the O.P.E. between two bulk operators \begin{equation} \Phi (z,{\bar z})\Phi (0,0) \sim C_{\Phi\Phi}^I V_I \label{ope} \end{equation} where $\{ V_I \}$ denotes a complete set of bulk (closed-string) operators. In the $D$-brane case, these include the bulk version of the $D$-brane recoil operator (\ref{wrongrecoil}). The one-point function on the disc of this operator is divergent near the boundary of the disk~\cite{periwal}~\footnote{This is cancelled by divergences in the string amplitude on the annulus, as discussed at the beginning of section 2.}. Parametrizing the world-sheet distance from the boundary by $i \zeta / 2$, we have \begin{equation} <X^0 (i \zeta / 2) \Theta (X^0(i\zeta /2))> \sim \int \frac{dq}{2\pi} \frac{1}{q^2} (\zeta)^{-q^2} \sim \sqrt{-{\rm log}\zeta} \label{divoper} \end{equation} Since $\zeta$ is a short-distance scale on the world sheet, we identify $-\log \zeta \sim \log |L/a|$, and then the dominant term in the $D$-brane two-point function of (\ref{ope}) is \begin{equation} <\Phi (z,{\bar z})\Phi (0,0)>_{z \sim 0} \sim C_{\Phi\Phi}^{D} \sqrt{{\rm log}|L/a|} \label{divbulk} \end{equation} Consider now the case where both $\Phi $'s in (\ref{ope}) lie very close to the boundary of the disc, which for convenience we parametrize by the real axis of the upper half of the complex plane. In this case, for each of the $\Phi $ one has the boundary O.P.E. expansion (\ref{boundary}). The separation of the two operators projected on the boundary will also be small, so one may use the O.P.E. on the boundary for the boundary excitations. We see from (\ref{dd}) that the dominant (divergent) contributions in that expansion come from the boundary $D$ operator, whose two-point function is divergent as ${\rm log}|L/a|$: \begin{equation} <\Phi (z,{\bar z})\Phi (0,0)>_{z \sim 0} \sim (C_{\Phi D}^{a})^2 {\rm log}|L/a| \label{boundiv} \end{equation} Equating the divergent contributions of (\ref{boundiv}) and (\ref{divbulk}), one obtains the relation \begin{equation} (C^a_{\Phi D})^2 \sim \sqrt{\frac{1}{{\rm log}|L/a|}}~C^D_{\Phi\Phi} \label{relation2} \end{equation} The O.P.E. coefficient $C_{\Phi\Phi}^D$ on the right-hand side of (\ref{relation2}) encodes the amplitude for an in-state tachyon to scatter off the $D$-brane into an out-state tachyon, including recoil, which is known to be non-zero~\cite{recoil,kogmav,kogwheat,emnd}. Equation (\ref{relation2}) implies that there is a non-trivial `trapping' amplitude, $C_{\Phi D}^a$, which therefore contributes to the diffusion coefficient in (\ref{FP}), and hence to the information loss via quantum recoil operators. The above calculation is incomplete, in that it was based on an analysis of the leading divergences of the recoil operators of the $D$ brane~\cite{kogwheat}, and it remains to calcuate the proportionality coefficients in (\ref{relation2}), as well as the subleading-divergence contributions of the open-to-closed O.P.E. in a theory with logarithmic operators. This programme falls beyond the scope of the present article~\cite{kogwheat2}, but is not needed for our purpose: we have established that both terms in the integrand of (\ref{tform}) are non-vanishing, and hence that the Fokker-Planck equation (\ref{FP}) is non-trivial. \paragraph{} The diffusion interpretation of (\ref{tform}, \ref{FP}) is contingent on the absence of a factor $i$ between the two sides of the equations, and one might wonder whether these equations are correct as written, or whether they should be Wick rotated. We would like to emphasize that the translation laws (\ref{correspond}, \ref{scale2}) were derived independently of the assumed signature $\eta$, and hence that there is no justification for any Wick rotation of the time variable in (\ref{tform}, \ref{FP})~\footnote{Indeed, when $\eta =-1$, in which case $X^0$ has Minkowskian signature, there is a factor of $i$ on the right-hand-side of (\ref{pole}), as a result of the purely imaginary normalization factor $\sqrt{1/\epsilon ^2 \eta Log|L/a|^2}$ of the two-point function of the operator $D$ (\ref{dd}). Such factors are, therefore, crucial. Their presence implies that, in the case $\eta =-1$, the target-space physical time should be identified with $it$, where $t$ denotes the change (\ref{fsscaling}) in the world-sheet scale. This demonstrates that, with respect to the physical time, the equation (\ref{pole}) retains its diffusion form, whichever the signature $\eta$.}. The appearance of a diffusion equation can be traced back to the presence in the recoil operators of a factor $\Theta (X^0)$~\cite{periwal,kogwheat}. In general, one should also take into account the possibility that the $D$ brane is moving initially, in which case one should use the boundary operators (\ref{dbraneop}) {\it as well}. In this case, there are trivial logarithmic operators in the two-point function $G(\tau _1, \tau _2)$ of (\ref{wilson}), coming from the free fields $X^0$ in the expression for the collective coordinates $y(X^0)_i = y_i + v_i X^0$ of a $D$ brane moving with initial velocity $v_i$. In the presence of both sets of operators, the right-hand side of (\ref{wilson}) becomes {\it complex} for Minkowskian-signature $X^0$, as a result of the imaginary normalization factors appearing in the two-point functions of the $\Theta (X^0)$ operators, mentioned above. This leads to an equation of motion which contains a standard Hamiltonian term \`a la Schr\"odinger, corresponding to a freely-moving brane with finite velocity $v^i$, as well as a diffusion term corresponding to the virtual excitations that appear after the time $X^0=0$. We note in passing that the recoil treatment of \cite{recoil} used only eternally-moving $D$ branes, in contrast to the treatments in \cite{periwal,kogwheat}, in which the membrane receives an impulse at the time $X^0$. In this article, our interest is focused on the diffusion aspect of the recoil, and we shall not discuss further the Hamiltonian terms in the wave equation of the moving brane. \paragraph{} In view of the important implications of (\ref{FP}), we now provide an alternative derivation that does not use the above identification of the $\sigma$-model partition function with the semi-classical wavefunction of the $D$ brane. This second derivation is based on the equivalence of the $D$-brane $\sigma$ model with a background gauge-field open string. In this formulation, the boosted collective coordinates $X^i$ are viewed as `couplings' of a non-critical $\sigma$-model theory. The inclusion of higher genera necessitates a summation over pinched handles (cf, the prototypes in Fig.~2(b)), which entails a natural quantization of these couplings~\cite{emnd}, in close analogy to the wormhole calculus familiar from higher-dimensional quantum gravity. As discussed earlier, the spontaneous breaking of translational invariance by the collective coordinates of the $D$ brane, which is a general feature of any string theory involving solitonic structures~\cite{kogmav}, is accompanied by the appearance of Goldstone zero modes. As discussed previously in the literature~\cite{recoil,kogmav,emnd}, the propagation of these zero modes along thin tubes yields logarithmic infinities in loop amplitudes, which manifest themselves in bilocal structures on the world sheet: \begin{equation} {\cal B}_{ii} =\int d^2z V_i (z) \int d^2w V_i (w) \frac{1}{L_0+{\overline L}_0 -2} \label{bilocal} \end{equation} where the last factor represents the string propagator $\Delta _S$ on a degenerate handle, with the symbols $L_0, {\overline L}_0$ denoting Virasoro generators as usual. Inserting a complete set ${\cal E}_{\alpha}$ of intermediate string states, we can rewrite (\ref{bilocal}) as an integral over the parameter $q \equiv e^{2 \pi i \tau}$, where $\tau$ is the complex modular parameter characterizing the world-sheet tube. The string propagator over the world-sheet tube then reads \begin{equation} \Delta _S \,=\, \sum _\alpha \int dq d{\overline q} \frac{1}{q^{1-h_\alpha} {\overline q}^{1-{\overline h}_\alpha}} \{{\cal E}_\alpha (z_1) \otimes (ghosts) \otimes {\cal E}_\alpha (z_2) \}_{\Sigma_1 \oplus \Sigma_2} \label{props} \end{equation} where $h_\alpha, {\overline h}_\alpha $ are the conformal dimensions of the states ${\cal E}_\alpha$. The sum in (\ref{props}) is over all states propagating along the long, thin tube connecting $\Sigma_1$ and $\Sigma_2$, which are both equal to the sphere in the simplest case of a degenerate torus. As indicated in (\ref{props}), the sum over states must include the ghosts, whose central charge cancels that of the world-sheet matter theory in any critical string model. States with $h_\alpha = {\overline h}_\alpha = 0$ may cause extra logarithmic divergences in (\ref{props}) which are not included in the familiar $\beta$-function analysis on $\Sigma$~\cite{recoil}. This is because such states make contributions to the integral of the form $\int dq d{\overline q} / q {\overline q}$ in the limit $q \rightarrow 0$, which represents a long, thin tube. We assume that such states are discrete in the space of states, i.e., they are separated from other states by a gap. In this case, there are factorizable logarithmic divergences in (\ref{props}) which depend on the background surfaces $\Sigma_{1,2}$, e.g., the sphere in the case of the degenerating torus. \paragraph{} The bilocal term (\ref{bilocal}) can be cast in the form of a local contribution to the world-sheet action, if one employs the trick, familiar from the wormhole calculus, of rewriting it as a Gaussian integral~\cite{wormholes,recoil}: \begin{equation} e^{{\cal B}_{ii}} \propto \int d\alpha e^{-\alpha _i^2 + \alpha _i\int V_i } \label{wlocal} \end{equation} where the $\alpha ^i$ are to be viewed as quantum coupling constants/fields of the world-sheet $\sigma$ model. In fact, the factor $e^{-(\alpha _i)^2} $ must be replaced~\cite{wormholes,schmid2,emnd} by a more general Gaussian distribution of width $\Gamma$: \begin{equation} F(\alpha _i) = \frac{1}{\sqrt{2\pi}\Gamma} e^{-\frac{1}{2\Gamma ^2}(\alpha _i)^2} \label{gauss} \end{equation} The extra logarithmic divergences associated with degenerate handles that we mentioned above, which are $\propto \hbox{ln} \delta$ where $q \sim \delta \sim 0$, have the effect of causing the width parameter $\Gamma$ also to depend logarithmically on the cutoff scale $\delta$: \begin{equation} \Gamma \sim \hbox{ln}\delta \label{widthdiv} \end{equation} Using the Fischler-Susskind mechanism~\cite{fiscsussk} for cancelling string loop divergences, one should associate the divergence $\propto\hbox{log}\delta$ with a world-sheet cut-off scale $\hbox{log}\epsilon$. Upon identification of the cut-off scale with the Liouville field, and of the latter with the target-time variable~\cite{emn,kogan}, we infer that the distributions of the couplings $\alpha_i$ become time-dependent~\cite{emn}. \paragraph{} In the particular case of $D$ branes, the couplings which fluctuate are the boosted collective coordinates of the $D$ brane, including the recoil velocities (or momenta). Redefining the `quantum' coordinates by $X^i \equiv y^i + \alpha _i \sqrt{\hbox{log}\delta}$, the above analysis shows that the quantized collective coordinates have a Gaussian `white-noise' distribution of the generic form: \begin{equation} F(\delta X^i_q)=[\frac{1}{2\pi <\delta X^I_q>^2}]^{1/2} {\rm exp}(-\frac{(\delta X^(_q))^2}{2<(\delta X_q^i)^2>}) \label{gaussian} \end{equation} where the $\delta X_q^i$ denote quantum fluctuations of the time-dependent collective coordinates. Due to the above-mentioned divergent structure associated with zero modes, the width of this distribution has the form: \begin{equation} <(\delta X_q^i)^2> \sim~D~\delta t \qquad t =D^{-1}\hbox{log}\delta \label{gaussianshift} \end{equation} where $\delta t$ denotes an infinitesimal fluctuation in target time, and the relative normalization $\hbox{log}\delta = D~t$ of the diffusion coefficient is dictated by analogy with the Liouville theory~\cite{aben,emn}. This structure is very similar to that of an inflationary scenario for string cosmology~\cite{emninfl}, which is understood in the context of Liouville strings~\cite{emn}, to which this theory is equivalent. \paragraph{} We are now in a position complete our second derivation of a Fokker-Planck equation for the probability distribution ${\cal P}(X^i,t)$ in the `space' $X^i$~\cite{infl,emninfl}, using the renormalization-group equation for the coupling $X^i(t)$ \begin{equation} \partial _t X^i(t) = u^i \label{string} \end{equation} which we interpret as an equation of motion for the classical coupling $X^i_c(t)$. We then decompose the value $X_c(t + \delta t)^i $ at a later time $t$ as \begin{equation} X_c(t + \delta t)^i = X_c(t)^i + \partial _t X^i(t)\delta t + \delta X_q^i (t) \label{quantum} \end{equation} where the $\delta X_q^i$ denote the quantum fluctuations that satisfy the white noise distribution (\ref{gaussianshift}). The probability distribution ${\cal P}$ may then be written as~\cite{infl,emninfl}: \begin{equation} {\cal P}(X^i,t) = \int dX_c^i F(\delta X_q^i)P(X_c,t) \label{prob} \end{equation} from which (\ref{FP}) is derived in a straightforward manner, as discussed in \cite{emninfl}~\footnote{We note in passing that {\it if} the Fokker-Planck equation holds also in the presence of $D$-brane interactions, {\it then} target-space diffeomorphism invariance restricts~\cite{goldin} the form of the equation for the temporal evolution of the $D$-brane wave function to a non-linear Schr\"odinger equation: \begin{equation} i \hbar \partial _t \Psi = {\cal H} \Psi + i \hbar D \nabla^2_i \Psi + i D \hbar | \nabla_i {\rm log} \Psi |^2 \Psi \end{equation} This equation might be useful for the quantization of interacting $D$-branes.}. This provides an independent {\it a posteriori} justification for identifying the $D$-brane $\sigma$-model partition function with the semi-classical wavefunction in target space. \paragraph{} Having derived a Fokker-Planck equation for the probability distribution, we now consider the rate of change of the entropy, defined as: \begin{equation} S=-k_B \int [DX_i] {\cal P}(X,t)\hbox{log}{\cal P}(X,t) \label{entr} \end{equation} where $[DX]$ denotes an appropriate functional integral in a $\sigma$-model sense. It is straightforward using (\ref{wilson}) to derive~\cite{emninfl} \begin{equation} \partial _t S = \int [DX] u_ju^j \frac{|\nabla _{X^i}{\cal P} |^2}{{\cal P}} + \dots \label{rate} \end{equation} where we work in a system of units such that $k_B=1$, and the $\dots $ indicate terms due to interacting $D$ branes, that have not been discussed here. The rate of change (\ref{rate}) is not only non-vanishing, but obviously monotonically increasing. \paragraph{} The rate of evolution (\ref{rate}) of the $D$-brane entropy may be estimated for asymptotically-large time $t$, using the analysis of the logarithmic conformal field theory describing the recoil process that we developed in sections 2 and 3. In a semi-classical treatment, we may represent ${\cal P} = |\Psi |^2$, where the $\sigma$-model partition function $\Psi =e^{-F}$, with $F$ denoting the effective $D$-brane action: \begin{equation} \partial _t S = \int [DX] u_iu^i \frac{1}{{\cal P}} | \nabla _{X^i} {\cal P} |^2 = \int[DX] |\Psi |^2 (u_iu^i) |\nabla _{X^i} F|^2 \label{intermsoff} \end{equation} Then, viewing the collective coordinates $X^i$ as couplings of the $D$-brane $\sigma$ model, using the $t$ dependence of $X^i$ expressed in (\ref{string}), and noting that no other couplings of the $D$-brane $\sigma$ model depend on $t$ to this order, we may write \begin{equation} \partial _t S = \int [DX] |\Psi [X]|^2 |\partial _t F|^2 \label{rewrite} \end{equation} Using now the renormalizability of the effective action $F$, and the Zamolodchikov $C$-theorem~\cite{zam} which tells us that \begin{equation} \partial _t F = - \beta^{y^i} G_{CC} \beta^{y^i} \label{zamthis} \end{equation} in this case, since (\ref{scale2}) $\beta^{y^i} = u^i$, $\beta ^{u_i} =0$, and recalling that (\ref{CD}) $G_{CC} \propto \epsilon ^2 \sim 1/t$, we rewrite (\ref{rewrite}) as \begin{equation} \partial _t S \propto (u_iu^i)^2 \frac{1}{t^2} \label{zamol} \end{equation} where we have normalized the wavefunction $\Psi[X^i]$ in the space of the collective coordinates. The result (\ref{zamol}), which is valid for asymptotically-large times $t \rightarrow \infty$, indicates that the change $\Delta S$ in $D$-brane entropy produced over the entire duration of the scattering process $ \Delta \tau \mathrel{\rlap {\raise.5ex\hbox{$ < $} 1/\epsilon$ by the quantum fluctuations of the recoiling $D$ brane is {\it finite}. This is consistent with the $\delta$-function-type temporal singularity of the effective space-time curvature scalar (\ref{scalrcurv}), and the finite range of effective deviations from conventional General Relativity in string theory. \section{Evolution Equation for the Effective Theory of Light States} \paragraph{} We have shown in the previous section that the ``kick" applied to the $D$ brane by the incident light closed-string state provides it with a quantum-mechanical excitation that leaves it with finite entropy $\Delta S$. Since the initial combined (light particle, $D$ brane) state had no entropy, we expect the final light-particle state also to have non-zero entanglement entropy $\Delta S$. In string world-sheet language, the interaction between the light state and the $D$ brane implies that their effective $\sigma$ models receive equal and opposite perturbations. These act as sources in the equations of motion of both subsystems, making them both appear non-critical. Thus one must introduce a renormalization scale also for the light-particle subsystem, which must also acquire non-zero entropy, according to general arguments~\cite{emn}. \paragraph{} To see this explicitly, we first review the general formalism of ref. \cite{emn} for the description of closed strings perturbed away from criticality by quantum-gravitational backgrounds. Criticality is restored by the inclusion of Liouville scaling factors and the target time is re-interpreted as the zero mode of the Liouville field $\phi$, which may then be identified with that of the $D$-brane target-time coordinate $X^0$, as we have shown also in the $D$-brane context in section 3. It follows from the general analysis of~\cite{emn} that the effective time-evolution equation for the light closed-string degrees of freedom takes the form \begin{equation} \partial _t \rho = i [\rho, H] + i\nd{\delta H}\rho \label{ehns} \end{equation} where $H$ is the light-state Hamiltonian, and the non-Hamiltonian term $\nd{\delta H} \rho$ takes the generic form \begin{equation} \nd{\delta H}\rho = \beta^i G_{ij} [g^j, \rho] \label{generalemn} \end{equation} Here the non-zero renormalization coefficients $\beta^i$ parametrize the apparent departure from criticality induced by the couplings $g^j$, which become quantum variables when higher-genus effects are taken into account, and $G_{ij} \propto <V_i V_j>$ is a suitable positive-definite Zamolodchikov metric in the space of the vertex operators $V_{i,j}$ corresponding to these couplings. The presence of the non-Hamiltonian term~(\ref{generalemn}) then induces entropy production in the effective light-particle theory: \begin{equation} {dS^{light} \over dt} = -\beta^i G_{ij} \beta ^j \label{genentropy} \end{equation} causing $S^{light}$ to vary monotonically if any $\beta^i \ne 0$ and $G_{ij}$ has definite sign~\cite{zam}. As we have argued previously~\cite{emn}, the appearance of a term (\ref{generalemn}) is necessarily accompanied by entropy production (decoherence) in the effective light-particle system, at a rate given in general by (\ref{genentropy})~\cite{emn}. \paragraph{} In the $D$-brane case at hand, we specialize to the subspace spanned by the tachyon modes (lowest-lying closed-string states), so that (\ref{genentropy}) reduces to \begin{equation} {dS^{light} \over dt} = -\beta^T G_{TT} \beta ^T \label{genentropy2} \end{equation} where $T$ denotes a tachyon coupling. In this application, the dominant contribution to the conformal anomaly for the tachyon $\beta$ functions comes from the interaction of the tachyon with the Goldstone modes $D_\epsilon,C_\epsilon$. We work in a renormalization scheme~\cite{zam} where \begin{equation} G_{TT} \sim 1 + O(T^2) \label{metriczam} \end{equation} We now use the general expression $\beta ^i \sim C^i_{jk} g^j g^k + \dots$ for the $\beta$ function, where the $\dots$ denote higher-order terms and the $C^i_{jk}=G^{im}C_{mjk}$, where the $C_{mjk}$ are completely symmetric trilinear O.P.E. coefficients, i.e., three-point correlation functions $<V_iV_jV_k>$. In our $D$-brane case the contributions of recoil to the conformal anomaly are of the form \begin{equation} \beta ^T \simeq C^T_{C_gC_g} (u_i)^2 + \dots \label{confan} \end{equation} to leading order in the matter deformations, where $C_g$ denotes a bulk operator that represents the virtual excitations appearing in $D$-brane recoil as discussed in sections 2 and 3, having as couplings the recoil velocities $u_i$, and the $C^T_{C_gC_g}$ denote appropriate O.P..E. coefficients~\footnote{An explicit expression for $C_g$ is given later in (\ref{cgop}).}. In this weak-tachyon perturbative treatment, the dominant contributions to the change in entropy (\ref{genentropy2}) of the light-particle subsystem is of order \begin{equation} dS^{light}/dt \propto \sum _T (u_i u^i)^2 (C^T_{C_gC_g})^2 \qquad t \rightarrow \infty \label{lightentr} \end{equation} Below we shall express the bulk amplitude $C^T_{C_gC_g}$ in (\ref{lightentr}) in terms of open-to-closed scattering amplitudes of the open-string sector. This enables us to show that (\ref{lightentr}) is, to this order in the matter deformations, identical to (\ref{zamol}) but {\it opposite} in sign, thereby providing an explicit consistency check of the overall conformal invariance of the matter-plus-quantum-$D$-brane recoil system. \paragraph{} As a step towards this demonstration, we first develop the proper definition of the light-state $\sigma$ model, so as to represent correctly the motion of low-lying closed-string modes in an `environment' of quantum-fluctuating $D$ branes. It is technically convenient for this purpose first to re-express the $\sigma$-model for the $0$ brane itself in the Liouville-dressed formalism, using a Galilean transformation of the collective coordinates of the $0$ brane. This, as we shall see, facilitates the analysis, as it provides a simple expression for $C_g$ in terms of a compact representation of the $D$-brane boundary excitations (the $C$,$D$ pair). We rewrite the full recoil operator (\ref{recoil}) as \begin{equation} V_{rec}= \epsilon \int _{\partial \Sigma} d\tau \xi _i \partial _n \xi ^i \Theta _\epsilon (X^0) \qquad \xi _i \equiv y_i + \frac{u_i}{\epsilon} X^0 \label{xiform} \end{equation} where the $\xi^i $ {\it do not} satisfy fixed Dirichlet boudnary conditions. Using the analysis of quantum-mechanical uncertainties of the $0$-brane subsystem~\cite{kogwheat}, which are briefly discussed in the Appendix, c.f., eq. (\ref{qmu}) in particular, we see that $\xi _i - y_i $ in (\ref{xiform}) expresses quantum uncertainty in the initial location of the $D$ brane, since $u_i/\epsilon $ is the minimum uncertainty obtained as a result of the action of the $D_\epsilon$ quantum-recoil operator. This interpretation of (\ref{xiform}) may also be expressed in the language of renormalization. We see from eq. (\ref{velocity}) that the effects of $\epsilon$ may be absorbed in an appropriate renormalization of the string coupling constant $g_s$, and a corresponding renormalized recoil velocity $u_i^R \equiv u_i/\epsilon$. Eq. (\ref{qmu}) of the Appendix tells us that these renormalization effects result from quantum fluctuations in the location of the $0$-brane collective coordinates. In this picture, which from a physical point of view corresponds to a Galilean transformation of the collective (spatial) coordinates in a co-moving frame of velocity $u_i^R$, consistency with the renormalization-group approach requires $u_i^R$ to be exactly marginal, whilst $u_i$ will now have an anomalous dimension $du_i/dt=-(1/2t)u_i$, $t \sim 1/\epsilon ^2$, in agreement with the fact that the anomalous scaling dimension of the $C$ and $D$ operators is $-\epsilon ^2/2 \sim -1/(2t)$: $t \sim Log|L/a|^2$. In what follows, unless explicitly stated, we will suppress the superscript when denoting the renormalized recoil velocity: $u_i^R \rightarrow u_i$. In this interpretation, the $\epsilon \xi _i$ factor in (\ref{xiform}) may be considered as the minimum bound on the (quantum) uncertainties in the location of the $0$-brane in a co-moving (Galilean-transformed) frame. According to the analysis in the Appendix (c.f., eq. (\ref{uncertc})), the uncertainty represented by the world-sheet zero mode of $\epsilon \xi _i$ is due to the action of the $C_\epsilon$ quantum-recoil operator, and is thereby related to the {\it stringy modes} of the $0$-brane~\footnote{Note that within our interpetation of $\epsilon ^{-2}$ as the target time $t$, this uncertainty become time dependent. This is important for the correct definition of the light-particle subsystem in this framework, as we discuss later on.}. \paragraph{} Passing to the bulk formalism as in (\ref{bulkliouv}), expressing the coordinate $X^i$ in terms of $\xi ^i$, and identifying $\phi $ with $X^0$, one finds the following Liouville-dressed $\sigma$-model action: \begin{eqnarray} &~& S_\sigma^D = \int _\Sigma d^2z \frac{1}{4\pi \alpha '}[ -(1 - (u_i^R)^2)\partial _\alpha X^0 \partial ^\alpha X^0 + \partial _\alpha \xi ^i \partial ^\alpha \xi ^j \delta _{ij} + \nonumber \\ &~&(-u_R^i + \epsilon ^2 \xi _i \Theta(X^0) ) \partial _\alpha X^0 \partial ^\alpha \xi ^i ] + \int _{\partial \Sigma} d\tau \epsilon \xi _i \partial _n \xi ^i \Theta (X^0) \label{liouvilledressing} \end{eqnarray} where the $\xi _i, X^0$ are independent $\sigma$-model fields, and $\Theta (X^0)$ is the Liouville-dressed $\Theta$ operator (\ref{thetaliouv}), which is conformal~\footnote{One may renormalize the $X^0$ field in (\ref{liouvilledressing}) so that its kinetic term assumes the canonical form, in which case the off-diagonal metric term is multiplied by a factor $1/(1-(u_i^R)^2)$, which is almost unity for slow-moving $0$ branes, or for recoiling branes in a low-energy scattering experiment.}. Notice that in the above approach $\epsilon$ is viewed as a parameter that is independent of the zero mode of the Liouville field $X^0$. The identification of $1/\epsilon ^2$ with $X^0$ should be made only at the very end of the computations. The off-diagonal metric in (\ref{liouvilledressing}) satisfies the generalized Weyl-invariance conditions for a transformation that is singular at $X^0=0$. Moreover, the curvature scalar constructed out of the coordinates $\xi_i, X^0$, exhibits the temporal singularity (\ref{scalrcurv}) at $X^0=0$, $R \ni -\epsilon ^4 \xi _i \xi ^i \delta (X^0) \Theta (X^0)$ due to the effects of $D$-brane recoil, which induces the above-mentioned uncertainty $\epsilon \xi $ in the location of the brane. \paragraph{} In the formalism (\ref{liouvilledressing}), the recoiling boundary quantum state is characterized by a single Goldstone-mode operator $\cal C$ of the $C$ type: \begin{equation} S^D_{\sigma} \ni \int _{\partial \Sigma } d\tau {\cal C} (\tau) \equiv \int _{\partial \Sigma} d\tau \epsilon \xi _i \partial _n \xi ^i \Theta (X^0) \label{cope} \end{equation} which incorporates the $C$,$D$ operator mixing discussed in section 2, by virtue of the transformed $\xi$ variable. On the other hand, a generic macroscopic $D$-brane boundary state $|a>$ is described by an operator of the form (\ref{cope}) without the $\Theta (X^0)$ term and with $\epsilon~\xi$ replaced by a macroscopic (fixed) coordinate $y_i$. This is not exhibited explicitly in (\ref{liouvilledressing}), but it should be understood in the following. We recall that in the modern approach to $D$ branes~\cite{dbranes}, the Dirichlet boundary conditions for the collective coordinates are viewed as reflecting appropriate gauge-field background excitations of ordinary open-string $\sigma$ models, with all the coordinates of the string obeying standard Neumann (closed-string) boundary conditions~\cite{dbranes,bachas}. This will always be implicit in our subsequent manipulations based on the action (\ref{liouvilledressing}). The $\sigma$-model action $S_\sigma ^D$ (\ref{liouvilledressing}) describes the $0$ brane, viewed as a subsystem for which the scattered matter has been integrated out. Hence the Liouville field/time $X^0$ appearing in (\ref{liouvilledressing}) represents collectively the matter effects which induce violations of conformal invariance that are restored by the gravitational (Liouville) dressing. \paragraph{} We now use the above formalism (\ref{liouvilledressing}) to construct the matter-subsystem $\sigma$ model, which enables us to evaluate the right-hand side of (\ref{genentropy2}). In the bulk formalism, our $\sigma$ model describing the matter subsystem is {\it defined} by (\ref{liouvilledressing}), upon {\it dropping} the boundary term (\ref{cope}) which encodes the environment of quantum-recoil fluctuations of the $D$ brane, and {\it adding} the closed-string tachyon deformation: \begin{equation} T(\xi, X^0) = \int d^Dk d\omega e^{ik_i \xi ^i } e^{i\omega X^0} \label{closedtach} \end{equation} where $D$ is the target-space dimension. This definition makes physical sense for the following reason: As we mentioned previously, in the closed-string formalism one may view a $D$ brane located originally at $y_i$ as being created by an appropriate operator which creates the macroscopic boundary state of the brane $|a>$. On the other hand, a {\it low-energy} observer is obliged to `average out' the time-dependent {\it stringy} effects due to quantum uncertainties in the location of the brane associated with the virtual recoil excitations, that are expressed via the boundary term (\ref{cope}). In $\sigma$-model language, such an averaging-out procedure is equivalent to ignoring the corresponding deformations of the world-sheet action. In this framework, the bulk operator $C_g$ appearing in (\ref{confan}), which describes the quantum $D$-brane recoil excitations from a matter-theory point of view, and hence is retained in the action for the light-matter subsystem, may be identified with the following deformation: \begin{equation} S^D_{\sigma} \ni \int _\Sigma d^2 z C_g (z,{\bar z}) \equiv \epsilon ^2 \int _\Sigma d^2z [\xi _i \Theta(X^0) \partial _\alpha X^0 \partial ^\alpha \xi ^i ] \label{cgop} \end{equation} This completes our specification of the matter $\sigma$ model. \paragraph{} We now check the mathematical consistency of this definition for the matter subsystem by reproducing the entropy change (\ref{zamol}) through (\ref{genentropy2}). First we show that the operator $C_g$ (\ref{cgop}) contains the bulk form of the recoil $D$ operator discussed in section 2, by splitting the field $\xi ^i $ into zero-mode and fluctuation parts: \begin{equation} \xi _i = \xi _{0,i} + {\hat \xi }_i (z,{\bar z}) \label{xifluct} \end{equation} where $\xi _{0,i} $ is independent of the world-sheet coordinates. The operator $C_g$ is now seen to be equivalent to \begin{equation} \int C_g \simeq \epsilon^2 \xi _0 \int _{\Sigma} d^2 \partial _\alpha (\int \frac{dq}{iq^2} e^{iq~X^0}) \partial ^\alpha \xi^i + \dots \label{cgdop} \end{equation} where $\dots $ denote the ${\hat \xi }$-dependent parts. Clearly, if the world sheet has a boundary, the operator exhibited in (\ref{cgdop}) becomes the $D$-brane recoil operator $\epsilon ^2 \xi _0 \int _{\partial \Sigma} d\tau D (\tau) + \dots $, where the $\dots $ denote world-sheet terms that vanish on shell. In this spirit, the `coupling' $\epsilon ^2 \xi _0 $ in (\ref{cgdop}), which has the dimensions of velocity, plays the r\^ole of the renormalized recoil velocity $u_i$ of the brane. These $\xi _0$ zero-mode parts of $C_g$ contribute the dominant divergent terms in the conformal anomaly $(\ref{confan})$, and we now concentrate on them. \paragraph{} There is a formal connection between the closed (bulk) and open (boundary) string theories~\cite{cardy}, which leads to a determination of (\ref{lightentr}) in terms of quantities in the open-string formalism, which provides the advertized connection with (\ref{zamol}). We recall that the matrix element $C_{C_gC_g}^T$ describes - to leading order in the matter deformation - the interaction of the bulk gravitational modes of the $D$ brane, expressing quantum recoil, with the light matter in the closed-string channel~\cite{periwal,emnd}. According to ref. \cite{cardy}, such bulk amplitudes may be expressed as `squares' of open-to-closed string amplitudes containing excitations on the (conformal) boundary state $|a>$. The pertinent relation is of the form~\footnote{For a rigorous proof of (\ref{cg}) in the presence of logarithmic operators, see~\cite{kogwheat2}.}: \begin{equation} \sum _{T} C^T _{C_gC_g} \propto (C^{a}_{C_g {\cal C}})^2 \label{cg} \end{equation} with $C_g$ given by (\ref{cgop}) and $\cal C$ by (\ref{cope}). \paragraph{} We now notice that the sum over tachyon modes in (\ref{cg}) actually reduces to a single term in the string case, as a result of the path integration of the zero-mode of $X^0$. This can be seen easily by using (\ref{closedtach}) for the tachyon mode and taking into account the specific form of the dominant $\xi _0$ zero-mode part of the $C_g$ operators (\ref{cgdop}): $C_q \propto \int dq \frac{1}{q^2} e^{iqX^0} $. It is then straightforward to see that the integration over the zero mode of $X^0$ simply imposes energy conservation: $\omega + q + q'=0$ for the three-point amplitudes in (\ref{cg}), thereby reducing the sum over tachyon modes $T$ (i.e., over $\omega$) to a single term which is uniquely specified in terms of the $C_g$. This allows (\ref{cg}) to be inserted in the right-hand side of (\ref{lightentr}). \paragraph{} We now recall the fact, mentioned above, that the dominant contributions to (\ref{confan}) arise from the zero-mode parts of $C_g$. Equivalently, in the open-string formalism, the dominant contributions appear when the operator $C_g$ is near the boundary, and its one-point function diverges~\cite{periwal,kogwheat,emnd}. Using (\ref{cgdop}), we then see easily that the amplitude $C^{a}_{C_g {\cal C}}$ is essentially $<CCD>$. Since the one-point function of the $D$ operator diverges like $1/\epsilon$~\cite{periwal,kogwheat} near the boundary (\ref{divoper}), and the two-point O.P.E. of $CC$ vanishes as~\cite{kogwheat} $\epsilon ^2$ (\ref{cc3}), it follows that the leading short-distance singularity of the above-mentioned three-point function, when the arguments of the three operators are all behaving ${\cal O}(\epsilon)$, is: \begin{equation} C_{C_g {\cal C}}^a \sim \epsilon \label{copeps} \end{equation} which implies that \begin{equation} C^T_{C_gC_g} \sim (C^a_{C_g {\cal C}})^2 \sim \epsilon ^2 \label{ctgg} \end{equation} Identifying $1/\epsilon ^2 \sim t$, where $t$ is the target time/renormalization-group scale, we obtain the following result for the rate of change (\ref{lightentr}) of the entropy of the light subsystem: \begin{equation} dS^{light}/dt \propto -(u_i u^{i} )^2 \frac{1}{t^2} \qquad t \rightarrow \infty \label{fentr} \end{equation} which is {\it identical} to the result (\ref{zamol}), but {\it opposite} in sign. Thus, for the light-particle system, the direction of time $t$ must be taken {\it opposite} to that of the Liouville renormalization-group flow, in order to have increasing entropy. This is consistent with the fact that the total entropy of the brane-plus-matter system remains constant as a consequence of conformal invariance. A similar situation has been argued to hold in the non-critical Liouville analysis of the string black hole. The above analysis completes the consistency check on our definition of the light-state subsystem, to lowest non-trivial order in matter deformations. \paragraph{} The expression (\ref{fentr}) is finite and independent of the area of the $D$-brane collective-coordinate surface. Hence, the Bekenstein-Hawking area law is not valid for the information-theoretic entropy associated with the quantum fluctuations of a recoiling $D$ brane. We recall that the area law is semiclassical, being associated with a tree-level treatment of the gravitational background, which is described in our case by a $D$-brane configuration. The semiclassical treatment ignores recoil, and simply expresses the entropy of the subsystem of tachyons obtained by integrating over the tachyon degrees of freedom behind the collective-coordinate surface of a classical $D$-brane configuration. This is a geometric term which is also present in standard flat-space local field theories~\cite{srednicki}. Our analysis goes beyond this by considering quantum fluctuations in the background induced by recoil during a scattering process. The situation is analogous to that studied in~\cite{emnw}, where the information-theoretic content of quantum fluctuations in a scalar field at the horizon of a four-dimensional black hole in General Relativity also violated the area law. Recent claims~\cite{susskind} to have derived the Bekenstein-Hawking law for the string black-hole entropy, considering the horizon of a stringy black hole as a $D$ brane, apply only at the semiclassical level. \paragraph{} The order of magnitude of the decoherence term (\ref{fentr}) is the same as that of the non-Hamiltonian term $\nd{\delta H}$ in the evolution equation of the density matrix of the light subsystem. This is due to the fact that, for asymptotic $t \rightarrow \infty$ where the non-critical string is near its equilibrium (critical) point, the commutator $i[g^i, \rho] \sim i[g^i, e^{-H}] =O[ \partial _t g^i =\beta ^i ]$, as a result of the canonical formalism for the couplings $g^i$~\cite{emninfl,emnd}. According to the discussion above, the decoherence of the light-state subsystem is determined by the recoil velocity of the collective-coordinate surface of the $D$ brane induced during scattering. This is given by (\ref{velocity}), and can be at most of the same order of magnitude as $g_sE$, where $E$ is a typical energy of the propagating closed-string state, which plays the r\^ole of the low-energy propagating particle treated in~\cite{emn}). Equation (\ref{fentr}) tells us that, for long times $t \rightarrow \infty$, the dominant decoherence effects in the evolution equation of the density matrix are of order $\frac{g_s^4}{M_s^4}E^4$, where $M_s$ is the string scale. To make this estimate, we take into account the fact that the evolution parameter on the world sheet scale is measured in string units $\sqrt{\alpha '}$, where $\alpha ' =1/M_s^2$ is the string Regge slope. \paragraph{} \section{Conclusions} \paragraph{} We have argued in this paper that $D$ branes provide another example of a model of quantum gravity in which the effective theory of the propagating light-particle states obeys a modified quantum Liouville equation of the type (\ref{ehns}), following the previous examples of $1+1$-dimensional string black holes~\cite{emn} and a scalar field outside a four-dimensional black hole with Yang-Mills interactions. As in these previous cases, time is interpreted as a renormalization-group scale parameter, which may be identified in the $D$-brane and string black-hole cases with the zero mode of the Liouville field on the world sheet. In these two string models, there is no question of modifying quantum mechanics on the world sheet. In both cases, the light degrees of freedom are to be regarded as truncated open systems, and the modification (\ref{ehns}) of the quantum Liouville equation can be traced to the need to treat the interaction with the black-hole environment as a source term in the world-sheet equation of motion for the light-particle system, pushing it away from criticality~\cite{emnd}. \paragraph{} We take this opportunity to clarify where we differ from views espoused elsewhere~\cite{polchinskireview,banks} on the possible maintenance of quantum coherence. In our view, the fact that the conventional Hawking-Bekenstein black-hole entropy can be identified with the number of distinguish$able$ $D$-brane states does not mean that these states are necessarily distinguish$ed$ in a realistic experiment: for example, the infinite set of Aharonov-Bohm phase measurements described in~\cite{emnmeas,banks} are $not$ feasible in practice. This is why one must sum over the $un$observed degrees of freedom, which are represented in the present case by the $quantum$ excitations of the $D$ brane, that cause a deviation from the Hawking-Bekenstein area law and were not studied in~\cite{banks}. It is convenient technically to perform the sum over unseen states using the Liouville field, whose zero mode we interpret as time, as confirmed in this $D$-brane analysis. \paragraph{} During the course of this analysis, we have indicated various steps where a more rigorous treatment is desirable and in preparation~\cite{kogwheat2}. There are also some important physical issues that should be addressed in future work on this $D$-brane model. One is the possible r\^ole of infinite-dimensional symmetries such as the $W_{\infty}$ discussed in connection with the $1+1$-dimensional black-hole model. We have argued previously~\cite{emnwhair,emn} that the infinite-dimensional Cartan subalgebra of $W_{\infty}$ provides an infinite set of conserved charges that are available to label black-hole states with $W$ hair. We have further argued that a finite low-energy experiment is unable to measure all the $W$ hair~\cite{emn}, and that the corresponding inevitable leakage of $W$ quantum numbers between the light-particle subsystem and the black-hole background is a measure of the rate of information loss inherent in (\ref{ehns}). Previously, we have presented formal evidence for this leakage in the $1+1$-dimensional black-hole model~\cite{emn}, and exhibited analogues of these $W$ charges in a higher-dimensional $D$-brane model. The possible r\^ole of such charges in the model studied here deserves further investigation. We are optimistic that such $W$ charges do indeed exist, since it is known~\cite{shaf} that a world-sheet $W_{\infty}$ algebra~\cite{pope} is present in any model with logarithmic operators $C,D$, such as this one. Moreover, the singularity (\ref{scalrcurv}) at $X^0 = 0$ in this model is effectively $1+1$ dimensional, leading us to expect that it may have an underlying algebraic structure similar to the $1+1$-dimensional black-hole example. However, these points require further analysis. \paragraph{} Another area worthy of deeper analysis is the creation and annihilation of $D$ branes. In this paper, we have a pre-existing $D$-brane background, and discussed the formalism for quantum excitations on its surface. However, we expect that much of the machinery developed here could also be used to discuss the production, annihilation and decay of $D$ branes. In particular, the $\Theta (X^0)$ and singularity structure of the recoil operator (\ref{recoil}) is well adapted to the creation of a $D$-brane pair. However, an extension in necessary to describe ``$D$-brane foam", i.e., the treatment of virtual $D$-brane fluctuations in the space-time background~\footnote{This was achieved in the $(1+1)$-dimensional-black-hole case using a world-sheet valley approach~\cite{emnval}.}. We have developed previously~\cite{emnval} the corresponding description of $1+1$-dimensional black holes in terms of monopoles on the world sheet. Extending the present analysis to virtual $D$ branes is, however, a non-trivial technical problem. \paragraph{} In the absence of an appropriate treatment, we can nevertheless offer some intriguing speculations. Let us suppose, as was found in this paper, that the information lost in any encounter with a microscopic $D$ brane is $\propto E^4 M_P^{-4}$, where $E$ is a typical energy scale of the light state. Let us further hypothesize that the density of virtual $D$-brane excitations of the space-time background is ${\cal O}(M_P^3)$~\footnote{We do not distinguish for this heuristic argument between $M_P, M_s = 1/\sqrt{\alpha'}$, or any other gravitational scale in string theory.}. On the other hand, the low-energy particle cross section $\sigma$ is assumed to be proportional to $1/E^2$~\cite{emohn}. Thus, if the speculations in the two preceding sentences were correct, we would infer that $\nd{\delta H} \propto 1/M_P$, where by dimensional analysis the numerator would be ${\cal O}(E^2)$, where $E$ is a characteristic energy scale of the light-particle system. This heuristic argument is consistent in order of magnitude with estimates made previously in the contexts of the $1+1$-dimensional black hole and the four-dimensional quantum gravity model. Remarkably, it is also of the same order as the sensitivity of the neutral kaon~\cite{emnkaon} system to any such possible deviations from the canonical quantum Liouville equation. Therefore we cannot yet exclude the possibility that the type of decoherence effect discussed in this paper might have some phenomenological relevance, however far-fetched this may seem. \newpage \paragraph{} {\Large {\bf Acknowledgements}} \paragraph{} It is a pleasure to acknowledge useful discussions with G. Amelino-Camelia, I. Kogan, F. Lizzi and J.F. Wheater. N.E.M. wishes to thank the CERN Theory Division for its hospitality during the final stages of this work. The work of D.V.N. has been supported in part by DOE grant DE-FG05-91-ER-40633. \paragraph{} {\bf Appendix: $D$-brane Decoherence and Position Uncertainty} \paragraph{} Additional support for our picture of information loss and decoherence induced by the recoil of $D$ branes is provided by a closer look at the modified uncertainty principle for $D$ particles which was discussed previously in ref.~\cite{kogwheat}. As was discussed there, the r\^ole of the logarithmic operators describing recoil is crucial in a derivation of an uncertainty principle for $D$ branes. The $D$ operator provides the conventional quantum-mechanical part of the uncertainty, whilst the $C$ operator is associated with intrinsically stringy parts. To be more precise, if one considers the scattering of light closed-string state, playing the r\^ole of a `detector', off a $D$ brane, and assumes that the momentum of the detector is known precisely before and after the scattering, one arrives at the conclusion that there is an uncertainty in the energy of the $D$ brane which is of order $\hbar \epsilon$. \paragraph{} This conclusion was reached~\cite{kogwheat} by first observing that the regulated $\Theta _\epsilon (X^0)$ operator appearing in the definition of the recoil operators (\ref{recoil}) may be written as $\Theta _\epsilon \sim \Theta (X^0)e^{\epsilon X^0}$; from this it follows that the recoil (measurement) takes a finite time $\Delta \tau \sim 1/\epsilon $, thereby implying an uncertainty in the total energy of order $\hbar \epsilon$. The kinematical analysis of ref. \cite{kogwheat} established the following relation: \begin{equation} \Delta P^i \sim \hbar \epsilon/u^i \label{uncert} \end{equation} where $\Delta P^i$ denotes the uncertainty in the $D$-brane momentum. Given that the collective coordinate of the $D$ brane starts growing as $u_i t$ (\ref{scale2}), the uncertainty (\ref{uncert}) implies an uncertainty due to the $D_\epsilon $ recoil operator in the position of the collective coordinate of the $D$ brane of order \begin{equation} \Delta _{D_\epsilon} X^i \sim u_i \Delta t = u_i \frac{1}{\epsilon} = \frac{\hbar}{\Delta P^i} \label{qmu} \end{equation} which is a quantum-mechanical point-like particle uncertainty betweeen coordinates and momenta. \paragraph{} The essentially stringy parts of the uncertainty are obtained from the $C_\epsilon $ operator, which, due to its specific form (\ref{cop}), yields a lower bound on the uncertainty of the coordinate $y^i$ of order $y^i \epsilon $. Using (\ref{uncert}) one obtains \begin{equation} \Delta _{C_\epsilon} X^i \sim \frac{y^i u_i}{\hbar} \Delta P^i \label{uncertc} \end{equation} Thus, the total uncertainty in the collective spatial coordinate of a recoiling $D$ brane, due to the combined action of the $C$ and $D$ logarithmic pair , is \begin{equation} \Delta _{total} X^i \sim \frac{\hbar}{ \Delta P^i} + \frac{y^i u_i}{\hbar} \Delta P^i \label{deltap} \end{equation} It is important to stress once again that, due to the mixing between the operators $C$ and $D$, one cannot evade the influence of the $C$ operator, and hence a `stringy' form of the uncertainty principle for a $D$ brane~\cite{kogwheat}. \paragraph{} For our purposes, it is important to notice the dependence of the second term in (\ref{deltap}) on the coordinate $y^i$. Minimizing the right-hand-side of the uncertainty (\ref{deltap}) with respect to $\Delta P^i$, one obtains a lower bound on $\Delta X^i$ of the form: \begin{equation} (\Delta X^i)_{min} \sim 2 \sqrt{u_iy^i}=4\sqrt{2\sqrt{2} \pi g_s~(k_1 + k_2)_i~y^i} \label{totalunc} \end{equation} The $\sqrt{y^i}$ dependence is reminiscent of the measurability bounds occuring in the non-critical Liouville string approach to target time~\cite{amelino}. As discussed there, the result (\ref{totalunc}) is compatible with the decoherence (\ref{zamol}) implied by the effects of the recoil operator $C_\epsilon$. This similarity should not have come as a surprise, given the discussion above on the marginal perturbation of the recoiling $D$ brane from a conformal point, and the generic connection of soliton backgrounds in string theory with Liouville strings~\cite{emnd}.

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\section{INTRODUCTION} The pseudoscalar spectrum of QCD consists of an octet of mesons which are approximate Goldstone bosons of spontaneously broken $SU(3)$ axial flavor symmetry, plus an anomalously heavy flavor singlet meson, the $\eta'$. The heaviness of the $\eta'$ is attributed to the effects of topology~\cite{REF1}. In an $SU(3)$ symmetric world, $m_{\eta'}$ would obey \begin{equation} m_{\eta'}^2 = m_0^2 + m_8^2 \end{equation} where $m_8^2$ is the average mass squared of the octet mesons which vanishes in the chiral limit, while $m_0^2$ is the topological contribution which does not vanish in the chiral limit. Neglecting $\eta$-$\eta'$ mixing, one plugs in the known meson masses to obtain the ``experimental'' value $m_0(N_f=3)=860{\rm MeV}$. \section{LATTICE CALCULATION OF $m_{\eta'}$} Extraction of $m_{\eta'}$ from first principles has received much attention~\cite{REF4,REF5,REF6,REF7}. The focus in all these studies has been to calculate the ratio $R(t)$ defined below \begin{equation} R(t) = \frac{\langle \eta'(t)\eta'(0){\rangle}_{2-loop}}{\langle \eta'(t)\eta'(0){\rangle}_{1-loop}} \label{eqn2} \end{equation} where $\eta'(t)$ is the operator that creates or destroys an $\eta'$ meson (in terms of quark fields, for staggered fermions, this becomes $\overline{Q}{\gamma}_5\otimes IQ$). Two point correlation function of this operator yields both a disconnected diagram (referred to as 2-loop in eqn~\ref{eqn2}) and a connected diagram (1-loop). On dynamical configurations, when $m_{val} = m_{dyn}$, $R(t)$ takes the following asymptotic form \begin{equation} R(t) = \frac{N_v}{N_f}\,\,\lbrack 1\,\, - \,\,B \exp(-\Delta mt)\rbrack \label{eqn3} \end{equation} where $N_v$ is the number of valence fermions, $N_f$ is the number of dynamical fermions, B is a constant and $\Delta m = m_{\eta'}-m_8$. For the quenched configurations, infinite iteration of the basic double pole vertex does not exist and it can be shown that the ratio is a linear function of time~\cite{REF4}. \begin{equation} R(t) = const. + \frac{m_0^2}{2m_8}\,t \label{eqn5} \end{equation} This style of calculation was employed by the authors of~\cite{REF4} who obtained a result of $m_0(N_f=3)=751(39){\rm MeV}$ using quenched configurations and Wilson fermions. We used staggered fermions and both dynamical and quenched configurations and reported a value of $m_0(N_f=3)=730(250){\rm MeV}$ extracted from dynamical configurations in~\cite{REF5}. \section{SIMULATION DETAILS} The parameters of the ensemble used in the simulation are shown in table~\ref{TAB1}. For all of the configurations listed in table~\ref{TAB1} the inverse lattice spacing is about 2 GeV as obtained from $m_{\rho}$~\cite{CHEN}. $m_{val}$ for the quenched configurations has been chosen 10\% higher than that corresponding to dynamical ($m_{dyn}=0.01$, $N_f=2$) so that $m_8$ is same for both. Propagators were computed using conjugate gradient on the 128 node OSC Cray T3D machine. For details concerning performance, the type of source, the method adopted for calculating the disconnected propagator etc., the reader is referred to ~\cite{REF5}. \begin{table}[tbh] \caption{The Statistical Ensemble} \hspace{5pt} \label{TAB1} \begin{tabular}{|l|llll|} \hline $N_f$ & $m_{dyn}$ & $\beta$ & $N_{samp}$ & $m_{val}$ \\ \hline 0 & $\infty$ & 6.0 & 83 & 0.011 \\ & & & & 0.022 \\ & & & & 0.033 \\ 2 & 0.01 & 5.7 & 79 & 0.01 \\ & & & & 0.02 \\ & & & & 0.03 \\ 2 & 0.015 & 5.7 & 50 & 0.01 \\ & & & & 0.015 \\ 2 & 0.025 & 5.7 & 34 & 0.01 \\ & & & & 0.025 \\ 4 & 0.01 & 5.4 & 70 & 0.01 \\ \hline \end{tabular} \vspace{-0.7cm} \end{table} \subsection{Smearing} The disconnected data is noisy and clean signals exist only for the first few time slices. Smearing helps in reducing excited state contributions from the first few time slices, thus enabling a reliable extraction of $\Delta m$. We use the gauge invariant Wuppertal smearing technique. \begin{figure}[hbt] \vspace{-0.2cm} \centerline{\psfig{figure=osu-eta-fig1.ps,height=4.0cm,width=6.0cm}} \vspace{-0.7cm} \caption{Correlators from Smeared Operators} \label{figcorr} \vspace{-0.2cm} \end{figure} For the connected contraction we computed two propagators, one with a smeared source and point-like sink and the other with a point-like source and smeared sink (see Fig.~\ref{figcorr}). Correspondingly, for the disconnected loops, the sink end was smeared in the same way(SS). \begin{figure}[hbt] \vspace{-0.2cm} \centerline{\psfig{figure=osu-eta-fig2.ps,height=3.3cm,width=6.0cm}} \vspace{-0.8cm} \caption{Ratio from SS and LL correlators at $m_{val}=m_{dyn}=0.01$} \label{fig1} \vspace{-0.4cm} \end{figure} Fig.~\ref{fig1} compares the ratio plot with and without smearing on dynamical configurations. In the initial few time slices both the data are different but they begin to coincide after a few time slices as they should. \section{RESULTS} Fig.~\ref{fig2} shows the form of the ratio on all the configurations listed in Table~\ref{TAB1}. For $N_f=2$ and $N_f=4$, the points shown are for $m_{val} = m_{dyn}$, while for $N_f=0$ we plot $m_{val}=0.011$. It is gratifying to see that this observable clearly distinguishes the number of flavors. \begin{figure}[h] \vspace{-0.3cm} \centerline{\psfig{figure=osu-eta-fig3.ps,height=3.3cm,width=6.0cm}} \vspace{-0.8cm} \caption{$N_f$ dependence of ratio} \label{fig2} \vspace{-0.5cm} \end{figure} One extracts $m_0^2$ from a linear fit to the quenched data. Fig.~\ref{fig3} shows $m_0^2$ versus $m_{val}$ obtained from both local and the smeared data, fit linearly and extrapolated to the chiral limit. As is expected, $m_0^2$ does not vanish in the chiral limit. \begin{figure}[hbt] \vspace{-0.2cm} \centerline{\psfig{figure=osu-eta-fig4.ps,height=3.3cm,width=6.0cm}} \vspace{-0.6cm} \caption{Chiral extrapolation of $m_0^2$} \label{fig3} \vspace{-0.7cm} \end{figure} The $N_f=2$ data shown in Fig.~\ref{fig2} is fit to the form of eqn~\ref{eqn3}. From the fit one extracts $\Delta m$ and hence $m_{\eta'}$ for all the values of $m_{dyn}$ shown. It can be seen in Fig.~\ref{fig4} that $m_{\eta'}$ does not vanish in the chiral limit. \begin{figure}[hbt] \vspace{-0.2cm} \centerline{\psfig{figure=osu-eta-fig5.ps,height=3.3cm,width=6.0cm}} \vspace{-0.6cm} \caption{Chiral extrapolation of $m_{\eta'}^2$} \label{fig4} \vspace{-0.7cm} \end{figure} When $m_{val} \ne m_{dyn}$, the ratio $R(t)$ takes the form \begin{equation} R(t) = \,\,\frac{N_v}{N_f}\,\lbrack\, At\,\, -\,\, B\exp(-\Delta mt)\,\, + \,\,C\rbrack \label{eqn4} \end{equation} Our data are not precise enough to allow a four parameter fit, but using lowest order $PQ\chi PT$ (and neglecting the small momentum dependent self-interaction $\alpha$) we can express $A$, $B$ and $C$ in terms of one unknown parameter, $m_0^2$. For $m_{val} > m_{dyn}$ ($N_f=2, m_{dyn}=0.01$) data we get reasonable fits and find a partially quenched result of $m_{\eta'}(N_f=3)=876 \pm 16$ MeV, remarkably consistent with the fully quenched and fully dynamical data. For $m_{val} \le m_{dyn}$ the $\chi^2$ is not reasonable. On the other hand, one need not venture far from lowest order $PQ\chi PT$ to fit the data. For example, from two parameter fits to the dynamical data, one obtains $Z'/Z$, the ratio of the residues for creating $\eta$ and $\eta'$. While this ratio is set to unity at lowest order, the data prefer values 20-30\% larger, an amount which could easily be accommodated by higher order chiral and $O(1/N_c)$ corrections. \begin{figure}[h] \vspace{-0.3cm} \centerline{\psfig{figure=osu-eta-fig6.ps,height=3.3cm,width=6.0cm}} \vspace{-0.6cm} \caption{$Z'/Z$ versus $m_{dyn}$} \label{fig5} \vspace{-0.5cm} \end{figure} \section{CONCLUSIONS} The value, we obtain for $m_{\eta'}$ in the chiral limit are summarized in the table below. Smeared operators have produced a lower value for $m_{\eta'}$ than that obtained from local operators. \begin{table}[h] \caption{$m_{\eta'}(N_f=3)$ in the chiral limit} \label{TAB2} \begin{tabular}{|ccc|} \hline &Quenched &Dynamical ($m_{dyn}=m_{val}$)\\ &(MeV) &(MeV) \\ \hline LL & 1156(95) & 974(133) \\ SS & 891(101) & 780(187) \\ \hline \end{tabular} \vspace{-0.4cm} \end{table} Within the quoted statistical errors, the values obtained above are consistent with experiment.

proofpile-arXiv_065-556

\section{Introduction} Supernova remnants (SNR) are believed to be responsible for accelerating particles to energies of at least 100 TeV (see e.g. \cite{Drury,BlandfordEichler87,BerezhkoKrymsky88,JonesEllison91}), and a fraction of the accelerated particles would interact within the supernova remnant and produce gamma--rays (\cite{DruryAharonianVolk}). Recent observations above 100 MeV by the EGRET instrument on the Compton Gamma Ray Observatory have found gamma ray signals associated with at least two supernova remnants -- IC~443 and $\gamma$~Cygni (\cite{Esposito96}). Further evidence for acceleration in SNR comes from the recent ASCA observation of non-thermal X--ray emission from SN~1006 (\cite{Koyama95}). Reynolds (1996) and Mastichiadis (1996) interpret the latter as synchrotron emission by electrons accelerated in the remnant up to energies as high as 100 TeV. In the diffuse galactic background the characteristic $\pi^0$ peak at 70 MeV is clearly evident. The power-law spectra, dominated by bremsstrahlung, contribute $\sim 10\%$ at 1 GeV (\cite{Hunter95,Hunter97}). The $\pi^0$ peak is not so clearly visible, however, in the spectra of IC~443 and $\gamma$~Cygni, possibly due to the larger error bars. However, this may instead suggest larger bremsstrahlung and inverse Compton (IC) contributions, and a correspondingly higher electron to proton ratio than in galactic cosmic rays. From detailed modeling of the gamma-ray spectra of IC~443 and $\gamma$~Cygni it should be possible to determine the relative contributions of $\pi^0$ decay, bremsstrahlung, and inverse Compton scattering to these spectra, and thereby to obtain information about the conditions in these sources. This would also allow a better determination of the expected extrapolation of the Egret spectra for these sources to the TeV energy range. Gamma rays are produced by electrons in bremsstrahlung interactions, and by inverse Compton interactions with the cosmic microwave background (\cite{Mastichiadis96}), with the diffuse galactic infrared/optical radiation, and with the radiation fields of the remnant itself. Protons produce gamma rays through the decay of neutral pions produced in proton--nucleus collisions. Drury, Aharonian \& V\"{o}lk (1993) calculated the $\pi^0$ decay gamma ray flux expected from supernova remnants due to interactions of accelerated cosmic ray nuclei with matter in the remnant. They calculated the expected flux as a function of supernova age and showed that a number of remnants should be detectable above 100 MeV. Mastichiadis (1996) has included synchrotron radiation and inverse Compton scattering on the cosmic microwave background, and shown that synchrotron radiation by directly accelerated electrons is probably responsible for the non-thermal X--rays observed in SN~1006. Mastichiadis also showed that SNR could produce a flux at TeV energies by inverse Compton scattering which is comparable to that predicted for pion production by Drury, Aharonian \& V\"{o}lk (1993) . An important point of the papers of both Reynolds (1996) and Mastichiadis (1996) is that electrons should be accelerated up to 100 TeV energies. Mastichiadis also suggests that the observation of TeV gamma rays from supernova remnants would not, in itself, provide direct evidence of acceleration of nuclei at supernova shocks. Pohl (1996) has also recently suggested TeV gamma-rays from supernova remnants may be of leptonic origin. In the present paper we consider gamma-ray production by interactions of accelerated nuclei and electrons in IC~443 and $\gamma$~Cygni. Our approach is to analyze the observed $\gamma$--ray fluxes and attempt to extract the source parameters from the data rather than use theoretical models of particle acceleration in supernova remnants. We assume power law spectra for electrons and protons, possibly different in slope and in maximum momentum on acceleration. We perform a maximum likelihood fit to the gamma ray spectra of IC~443 and $\gamma$~Cygni to determine the ratio of electrons to protons, the power-law spectral index, and the average matter density seen by the accelerated particles. \section{Relative importance of pion production, bremsstrahlung and inverse Compton scattering} Before describing our detailed calculation, we first briefly discuss the general features of the expected $\pi^0$, bremsstrahlung and inverse Compton gamma ray spectra, and make order of magnitude estimates of their relative contributions to the gamma ray flux. The momentum spectrum of particles accelerated by first order Fermi acceleration at a strong shock is expected to be $dN/dp \sim p^{-2}$. Protons accelerated with this spectrum, and interacting within the remnant, will produce at high energies a $\pi^0$ gamma ray number spectrum having approximately the same slope, $\sim E^{-2}$, but with a low-energy turnover at $\sim m_\pi c^2/2$. At high energy, electrons will cool by inverse Compton and synchrotron losses, $dE/dt \propto -E^2$ (neglecting, for the moment, scattering in the Klein-Nishina regime). In an equilibrium situation, the ambient spectrum of electrons will be steepened by one power to $\sim E^{-3}$, giving rise to a spectrum of inverse Compton gamma rays proportional to $\sim E^{-2}$. However, as we shall show below, {\it a priori} it is not obvious that that such an equilibrium exists for the supernova remnants in question. The cooling time for electrons in a magnetic field is \begin{equation} t_{\rm syn} \approx 1.3 \times 10^{10} \left({B \over 1 \; \mu {\rm G}} \right)^{-2} \left({E \over 1 \; {\rm GeV}} \right)^{-1} \hspace{5mm} {\rm years}. \end{equation} So, for typical interstellar magnetic fields ($\sim 3 \, \mu$G), one finds a cooling time at 100 TeV of $\sim 1.4 \times 10^4$ years. This time is longer than, but of the same order of magnitude of the age of the supernova remnants we are considering. Because of this, Mastichiadis (1996) assumed that the ambient electron spectrum is not yet affected by radiative losses. The cooling time for IC scattering on the microwave background is of the same magnitude and so IC losses will have some effect, though possibly minor, on the shape of the electron spectrum. Hence, as a starting point we estimate the $\gamma$--ray luminosities assuming that protons and electrons are accelerated with identical power law momentum spectra extending to the same maximum momentum $p^{\rm max}$, and that these spectra have not significantly evolved. The bremsstrahlung spectrum will have the same power-law as the electron spectrum, extending to low energy so that bremsstrahlung photons eventually dominate below the $\pi^0$ peak even for very small $e/p$ ratios. On the other hand, the inverse Compton spectrum from an $E^{-2}$ electron spectrum will be $\sim E^{-3/2}$, which is much flatter than the $\pi^0$ and bremsstrahlung gamma ray spectra. We next discuss the relative importance of the various processes to the gamma ray flux based on order of magnitude estimates. The crude approximations discussed below are only used for these order of magnitude estimates, and we use exact formulae for the results we shall present later. Furthermore, in section 5 we shall make detailed fits to the gamma ray spectra which allow for the possibility of the ambient electron spectrum being steeper than the proton spectrum, either due to a different spectrum at acceleration or due to evolution of the electron spectrum. For this estimate we approximate the total proton spectrum (integrated over the SNR) by $N(E_p) \equiv dN/dE_p = a_p E_p^{-\alpha}$ protons~GeV$^{-1}$. For $E_\gamma\gg m_\pi c^2/2$, the gamma-ray luminosity from $\pi^0$ production is \begin{equation} L_{\pi^0}(E_\gamma) \approx c\,n\,\left(\sigma_{pp}^{inel}\,{2Z_{N\pi^0}\over\alpha}\right) \times a_p\,E_\gamma^{-\alpha}, \end{equation} where $\sigma_{pp}^{inel}$ is the inelastic proton--proton cross section, $\rho \approx n\,m_p$ is the average matter density sampled by the protons and $Z_{N\pi}$ is a spectrum-weighted moment of the momentum distribution of pions produced in proton-proton collisions (\cite{Gaisser}). For $\alpha =$~2.0, 2.3, and 2.4 respectively, $Z_{N\pi^0} \approx$ 0.16, 0.075 and 0.066. Thus, for $\alpha = 2.0$ \begin{equation} L_{\pi^0}(E_\gamma)\approx 1.5\times 10^{-16}\,a_p\,n\,E_\gamma^{-2} \hspace{5mm} {\rm photons \; GeV^{-1} \; s^{-1}} \end{equation} where $E_\gamma$ is in GeV and $n$ is in cm$^{-3}$. Similarly, we approximate the total electron spectrum by $N(E_e) \equiv dN/dE_e = a_e E_e^{-\alpha}$ electrons GeV$^{-1}$. To obtain the bremsstrahlung luminosity, we assume that after an electron of energy $E_e$ has traveled one radiation length, $X_0$, it is converted into a photon of energy $E_\gamma = E_e$. Hence, \begin{equation} L_{\rm brem}(E_\gamma) \approx N(E_\gamma) \rho c / X_0. \end{equation} Thus, for $\alpha = 2.0$, \begin{equation} L_{\rm brem}(E_\gamma) \approx 7 \times 10^{-16} a_e n E_\gamma^{-2} \hspace{5mm} {\rm photons \; GeV^{-1} \; s^{-1}}. \end{equation} For inverse Compton scattering, we approximate the photon energy after scattering by an electron of energy $E_e = \gamma m_ec^2$ by $\gamma^2 \bar{\varepsilon}$ where $\bar{\varepsilon}$ is the mean photon energy of the radiation field under consideration. Provided the Compton scattering is in the Thomson regime ($\gamma \bar{\varepsilon} \ll m_e c^2$) this gives an inverse Compton luminosity \begin{equation} L_{\rm IC}(E_\gamma) \approx N(\gamma) n_{\rm ph} \sigma_T c / 2 \gamma \bar{\varepsilon} \end{equation} where $N(\gamma) d\gamma = N(E_e) d E_e$, and we obtain \begin{equation} L_{\rm IC}(E_\gamma) \approx a_e {(\bar{\varepsilon})^{1/2} \over E_\gamma^{3/2}} {n_{\rm ph} \sigma_T c \over m_e c^2}. \label{Eq:approxIC} \end{equation} For scattering off the microwave background, we use $n_{\rm ph} = 400$ cm$^{-3}$, $\bar{\varepsilon} = 6.25 \times 10 ^{-4}$ eV, and obtain \begin{equation} L_{\rm IC}(E_\gamma) \approx 1.3 \times 10^{-14} a_e E_\gamma^{-3/2} \hspace{5mm} {\rm photons \; GeV^{-1} \; s^{-1}} \end{equation} where $E_\gamma$ is in GeV. We note that, for an assumed matter density of 1 cm$^{-3}$, at 1 GeV the inverse Compton scattering contribution is an order of magnitude larger than the bremsstrahlung contribution, and the relative importance of the inverse Compton scattering contribution increases with energy. In the next two sections we describe the radiation and matter environments of the two SNR, and our accurate treatment of pion production, bremsstrahlung, and inverse Compton scattering. \section{The environments of IC~443 and $\gamma$~Cygni} In addition to the microwave background radiation, and the galactic infrared/optical background radiation, the radiation produced locally within the SNR will provide target photons for inverse Compton scattering. The two SNR we consider are at galactocentric radii sufficiently close to that of the Sun that we may use the local infrared/optical radiation field for the galactic background. We adopt the spectrum of Mathis et al. (1983) which is the sum of 6 diluted blackbody spectra and is shown in Figure~1. The mean photon energy for this spectrum is $\bar{\varepsilon} \approx 0.08$ eV, and the total photon number density is $n_{\rm ph} \approx 8.6$ cm$^{-3}$, giving an energy density of $U_{\rm ph} \approx 0.66$ eV cm$^{-3}$. Using Equation~\ref{Eq:approxIC} we can estimate the gamma ray luminosity applicable to the Thomson regime \begin{equation} L_{\rm IC}(E_\gamma) \approx 3.1 \times 10^{-15} a_e E_\gamma^{-3/2} \hspace{5mm} {\rm photons \; GeV^{-1} \; s^{-1}.} \end{equation} This is only a factor $\sim 4$ lower than that for scattering off the microwave background, and so it is necessary to include this radiation field in an accurate calculation. Lozinskaya (1991) gives an excellent discussion of the supernova remnants IC~443 and $\gamma$~Cygni. We consider first the remnant IC~443 which is at a distance of $\sim 1.5$ kpc, and has an angular diameter of $\sim 45^\prime$ giving a radius of $\sim 10$ pc. The remnant is probably about 5000 years old and is expanding into a very inhomogeneous medium including dense molecular clouds. Most of the remnant has a temperature of $\sim 1.2 \times 10^7$ K (\cite{Petre88}). Recent measurements by Asaoka and Aschenbach (1994) of the X--ray spectrum indicate two components: $\sim 27$ M$_\odot$ at 1 KeV, and $\sim 6$ M$_\odot$ at 13 KeV. Identifying the swept-up mass with $27$ M$_\odot$ leads to a pre-SNR interstellar density of $\sim 0.3$ cm$^{-3}$. Petre et al. (1988) measured the X--ray flux in the 2--10 keV range, $6.7 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, and a temperature of 1.03 keV. Approximating the spectrum by a thermal bremsstrahlung spectrum, we obtain a total X--ray energy flux of $3 \times 10^2$ eV cm$^{-2}$ s$^{-1}$. Dividing by the solid angle subtended by the SNR, and multiplying by $4 \pi/c$ we get an approximation to the X--ray energy density in the remnant of $3 \times 10^{-3}$ eV cm$^{-3}$. For a mean photon energy of $\bar{\varepsilon} \approx 1$ keV, one obtains a photon number density of $n_{\rm ph} \approx 10^{-6}$ cm$^{-3}$. Using Equation~\ref{Eq:approxIC} we can estimate the gamma ray luminosity applicable to the Thomson regime (in this case for $E_\gamma \ll 1$ GeV) \begin{equation} L_{\rm IC}(E_\gamma) \approx 4.2 \times 10^{-20} a_e E_\gamma^{-3/2} \hspace{5mm} {\rm photons \; GeV^{-1} \; s^{-1}.} \end{equation} At higher energies (where scattering is in the Klein-Nishina regime) the gamma ray luminosity will be much lower. Comparing the flux from inverse Compton scattering on the X--ray emission with that from inverse Compton scattering on the microwave background, we find it to be negligible in IC~443. However, in younger supernova remnants the X--ray emission may present a significant target for IC scattering. The supernova remnant associated with $\gamma$~Cygni, G78.2+2.1, is at a distance of $\sim 1.8$ kpc, and has an angular size of $\sim 1^\circ$ giving a radius of $\sim 16$ pc. The remnant is probably about 7000 years old and is also expanding into a very inhomogeneous medium including dense molecular clouds. A dense cloud occupies $\sim 5$\% of the volume of the remnant and this has been used by Pollock (1985) to predict the gamma ray flux (see also \cite{Aharonian94}). We shall return to a discussion of the effects of an inhomogeneous interstellar medium later when we discuss the electron to proton ratio. The X--ray flux obtained by Higgs, Landecker, \& Seward (1983) in the 0.2--4 keV range is about $6 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$, with a temperature of 1.3 keV and ambient gas density of $\sim 0.2$ cm$^{-3}$. The X-ray intensity is of the same order of magnitude as from IC 443, and so we conclude that inverse Compton scattering on the X--rays in $\gamma$~Cygni, and other old SNR, will not make an important contribution to the gamma ray flux. A major component of the radiation field of SNR is the infrared emission due to shock heated dust. For both sources, Saken, Fesen, \& Shull (1992) provide infrared spectra obtained with the IRAS satellite. We use their two-temperature model fits to represent the radiation field in the infrared. We have used the solid angle subtended by each source to obtain the energy density from the intensity. In IC~443 the mean photon energies, number densities, and energy densities of the two diluted blackbody components are: $\bar{\varepsilon} \approx 0.008$, 0.04 eV; $n_{\rm ph} \approx 18.4$, 11.8 cm$^{-3}$; $U_{\rm ph} \approx 0.15$, 0.51 eV cm$^{-3}$. In $\gamma$~Cygni the mean photon energies, number densities, and energy densities of the two diluted blackbody components are: $\bar{\varepsilon} \approx 0.009$, 0.025 eV; $n_{\rm ph} \approx 37.3$, 6.2 cm$^{-3}$; $U_{\rm ph} \approx 0.32$, 0.15 eV cm$^{-3}$. In both cases the infrared photon densities exceed the diffuse Galactic infrared photon densities, and so inverse Compton scattering of locally produced infrared photons could make a substantial contribution to the total inverse Compton flux. The total radiation field we adopt is shown in Fig.~2(a) for IC~443 and in Fig.~2(b) for $\gamma$~Cygni, and in each case we show the contributions of the diffuse background and the locally produced radiation. The IRAS data (\cite{Saken92,Mufson86}) are also shown. \section{Accurate treatment of pion production, bremsstrahlung and inverse Compton scattering} For interactions with matter we assume standard interstellar composition. For proton interactions we use the event generator TARGET (Gaisser, Protheroe \& Stanev 1983) in its proton target version. TARGET, which has been extensively tested in numerous applications, represents correctly the proton interaction cross sections and the secondary particles spectra starting at the pion production production threshold. At energies above 100 GeV the interaction cross section has a $\ln^2{s}$ behavior for proton--proton interactions. At total proton energy below 2.5 GeV TARGET generates only $\Delta$ resonances. The output of this part of the code was compared with the results of Dermer (1986) and is in good agreement. In the case of bremsstrahlung, yields are calculated for fully ionized matter for which the cross--section is strongly energy dependent. We use the expressions of Koch \& Motz (1959) with form factors for hydrogen and helium adjusted to represent the more precise values of Tsai (1974). The cross-sections for fully ionized matter are calculated with the bremsstrahlung formulae valid in the absence of screening by the atomic electrons. These cross sections are given by Protheroe, Stanev, \& Berezinsky (1995). For inverse Compton scattering we use the Klein-Nishina cross section (\cite{Jauch}) to calculate the mean free path and the distribution of photon energies produced in the IC process. We perform an exact Monte Carlo simulation of interactions as described by Protheroe, Mastichiadis \& Dermer (1992) to build up distributions of secondary electrons and photons arising from interactions of electrons of a given energy with blackbody radiation of given temperature. These distributions are then convoluted with the parent electron spectrum. \section{Results and comparison to experimental data} We assume the momentum spectrum of accelerated particles is of the form \begin{equation} {d N \over d p} = ac \left( {p \over {\rm 1 \, GeV/c}} \right)^{-\alpha} \hspace{5mm} {\rm (GeV/c)^{-1}} \label{Eq:spectrum} \end{equation} so that the energy spectrum is \begin{equation} {d N \over d E} = a \left( {E \over {\rm 1 \,GeV}} \right) \left( {p \over {\rm 1 \, GeV/c}} \right)^{-(1+\alpha)} \hspace{5mm} {\rm GeV^{-1}} \end{equation} where $E=(p^2c^2 + m^2c^4)^{1/2}$. We allow for different normalizations and spectral indices of protons ($a_p$, $\alpha_p$) and electrons ($a_e$, $\alpha_e$), and define the electron to proton ratio as $R_e \equiv a_e/a_p$ which is the ratio of the differential momentum spectra at 1~GeV/c. At high energies ($E \gg m_p c^2$) the energy spectrum of protons is just $d N_p / d E \approx a_p E^{-\alpha_p}$ GeV$^{-1}$, which gives the total number of protons per unit energy at the supernova remnant. Similarly, for electrons at high energies ($E \gg m_e c^2$) the energy spectrum is $d N_e / d E \approx a_e E^{-\alpha_e}$ GeV$^{-1}$. We calculate gamma ray emissivities for $\pi^0$ production, bremsstrahlung, and inverse Compton scattering in the radiation fields appropriate to each SNR for the case where $a_p/V = 1$ GeV$^{-1}$ cm$^{-3}$ ($V$ is the volume of the region where the accelerated particles are located), $R_e = 1$, and a nucleon density of $n=1$ cm$^{-3}$. The emissivities are calculated for each process for a range of $\alpha$ for the case where $\alpha_e =\alpha_p = \alpha$, assuming $n=1$ cm$^{-3}$, $R_e=1$, and a radiation field analogous to Fig.~2a (IC~443). The emissivity, $Q(E_\gamma) \equiv dQ_\gamma/dE_\gamma$ (cm$^{-3}$ s$^{-1}$ GeV$^{-1}$), multiplied by $E_\gamma^2$, is shown in Fig.~3 for $\alpha=2$. With the exception of the local infrared radiation field (short dashes in Fig.~3), and neglecting the small differences of the galactic infrared background as a function of the galactocentric distance, this emissivity spectrum is universal for SNR in the Galaxy (with suitable scaling of $\pi^0$ and bremsstrahlung emissivity with matter density) on the assumption that supernova remnants accelerate charged particles with a $p^{-2}$ spectrum, and that radiative losses have not yet affected the electron spectrum. The main feature of the emissivity spectrum is the dominance of the inverse Compton scattering (as already expected from Eq.~\ref{Eq:approxIC}) with its very flat spectrum. It is instructive to follow the scaling of the emissivities of different processes with the supernova remnant parameters. The relative contribution of IC scales up or down only with the electron to proton ratio. In the vicinity of $E_\gamma$ = 1 GeV the electron to proton ratio has to be of order of 0.01 for the IC scattering on the microwave background not to exceed the $\pi^0$ contribution for a matter density of 1 nucleon cm$^{-3}$ and $\alpha$ = 2. As long as IC is the most important contribution, the gamma-ray spectrum will be harder than the parent electron spectrum. The $\pi^0$ contribution scales only with the matter density. Alternatively, if the matter density is very high, $\sim 100$ nucleons cm$^{-3}$ one can suppress the IC contribution relative to the bremsstrahlung and $\pi^0$ contributions. For such high densities, more than two orders of magnitude higher than the average densities expected from the mass of the ejecta (\cite{Lozinskaya}), the contribution of bremsstrahlung, which is proportional both to matter density and to the electron to proton ratio, would dominate. For a $\pi^0$ `bump' to be visible requires an electron to proton ratio less than 1. We define $Q_0^\pi$, $Q_0^{\rm brem}$, and $Q_0^{\rm IC}$ to be the gamma ray emissivities (cm$^{-3}$ s$^{-1}$ GeV$^{-1}$) for pion production, bremsstrahlung, and inverse Compton scattering calculated for $A \equiv a_p/V = 1$ GeV$^{-1}$ cm$^{-3}$, $R_e=1$, and $n=1$ cm$^{-3}$. Then the gamma ray flux observed at Earth, a distance $d$ from the SNR, is given by \begin{equation} F_\gamma(E_\gamma,\alpha) = {n_1 A_1 V \over 4 \pi d^2} \left[ Q_0^\pi (E_\gamma,\alpha) + R_e Q_0^{\rm brem} (E_\gamma,\alpha) + { R_e \over n_1} Q_0^{\rm IC} (E_\gamma,\alpha) \right] \end{equation} where $n_1 \equiv n/(1$~cm$^{-3}$), $n$ is the nucleon number density in the region of the SNR containing the accelerated particles, and $A_1 \equiv A/(1$~GeV$^{-1}$ cm$^{-3}$). We perform a maximum likelihood fit to the EGRET data (\cite{Esposito96}) for each SNR with the following free parameters: electron to proton ratio $R_e$, particle spectral indices ($\alpha_p$, $\alpha_e$), maximum energy $E^{\rm max}$ (corresponding to an exponential cutoff, i.e. $\exp(-E/E^{\rm max})$), which could also be different for protons and electrons, $R_e/n_1$, and the overall normalization \begin{equation} B \equiv {n_1 A_1 V \over 4 \pi d^2} = \left( {a_p \over 1 \; {\rm GeV^{-1} \; cm^{-3}}} \right) {n_1 \over 4 \pi d^2} \end{equation} which has units cm GeV$^{-1}$. For each SNR, the $\gamma$--rays are binned in 10 energy bins, two of which give upper limits, and so the total number of data points is not sufficiently large to fit all possible parameters simultaneously. Instead we perform four different fits corresponding to the following assumptions. Fit~1. Both protons and electrons are accelerated with power-law spectra with the same index $\alpha$ and to the same maximum energy of 80 TeV. The choice of $E^{\rm max}$=80 TeV is arbitrary but sufficiently high for fitting the EGRET $\gamma$--ray spectra which only extend to $\sim$10 GeV and for extrapolating to the Whipple limits of several hundred GeV. The assumption of a pure power law is an approximation. Non-linear effects in cosmic-ray modified shocks generally lead to some concavity in the spectra (\cite{JonesEllison91,Berezhko96}), though the deviation from a power-law is slight for relativistic particles. For example, the proton spectrum calculated by Ellison (1993) steepens by less than 2 per cent in the spectral index below 1 TeV from its flattest high energy value. Fit~2. Protons and electrons are accelerated with the same spectral index $\alpha$ but the electron spectrum cuts off at an energy $E_e^{\rm max}$ which is a fit parameter. $E_p^{\rm max}$ = 80 TeV. Fit~3. Protons and electrons are accelerated to different power law spectra with indices $\alpha_p$ and $\alpha_e$. The maximum acceleration energy for electrons is 1000 GeV. $E_p^{\rm max}$ = 80 TeV. Fit~4. Protons and electrons are accelerated to identical power law spectra and both cut off at the same maximum energy $E^{\rm max}$. The fitting procedure actually consisted of a tabulation of the maximum likelihood value $L$ for a pre--determined set of parameter values. Except for the spectral indices $\alpha$ which were taken on a 0.05 grid, all other parameters were introduced on logarithmic scales of 10$^{0.1}$, i.e. ten parameter values per decade. This grid for the trial parameters is small enough to produce neighboring spectra well within the uncertainties of the measurements. The parameter set with a maximum $L$ value was taken to be the best fit. \subsection{Results from Fit~1} Fig.~4 shows the fitted spectrum for the supernova remnant IC~443 in Fit~1. The upper limits on the VHE $\gamma$ rays from IC~443 obtained by the Whipple observatory (\cite{Buckley97}) and from the HEGRA array (\cite{Prosch95}) are also shown in Fig.~4, and seen to be consistent with the fit, even though they were not used in the fit. The fit gives $\alpha$ = 2.32$\pm^{0.14}_{0.11}$. The errors are calculated from the values of $\alpha$ at which $\ln L = \ln L_0 - s^2/2$, where $L_0$ is the best fit value of the likelihood function $L$, and $s$ is the number of standard deviations ($s=1$ to give $1\sigma$ errors). The overall normalization gives $B = 4.0 \times 10^8$ cm GeV$^{-1}$. The electron to proton ratio corresponding to this value of $\alpha$ is $R_e=0.16\pm^{0.29}_{0.08}$. The best fit value of $R_e/n_1$ is $\sim 10^{-4}$, with very large error bars because the emission is dominated by $\pi^0$ production and bremsstrahlung. Thus, we can not reliably give the matter density since the fit requires a very high density only to make the IC component negligible. This also prevents us from obtaining a reliable estimate of the cosmic ray density at the source. Fig.~5 shows a similar fit for $\gamma$~Cygni. The best $\alpha$ value is $\alpha = 2.42\pm^{0.09}_{0.07}$, which gives an electron to proton ratio of $R_e = 0.16\pm^{0.14}_{0.08}$, and a similar value of $R_e/n_1$ to IC~443. The overall normalization gives $B = 10^9$ cm GeV$^{-1}$. The VHE limits are from Whipple (\cite{Buckley97}) and from HEGRA (\cite{Prosch96,Willmer95}) and are again consistent with the fit. We can get some useful information from the overall normalization constant $B \equiv n_1 A_1 V / 4 \pi d^2$ which, given that we know $d$, gives us the product of $n$, $N_p(p=1$~GeV/c$)/V$, and $V$. The total energy content of a spectrum of the form of Equation~\ref{Eq:spectrum} with $\alpha=2.32$ and $a_p=1$ is $\sim 3$~GeV. Thus for IC~443 at a distance of 1.5~kpc the normalization $B=4.0\times 10^8$ cm GeV$^{-1}$ requires a total energy in accelerated protons of \begin{equation} U_{\rm CR} \sim 5 \times 10^{50} \left( {n \over {\rm 1 \; cm^{-3}}} \right)^{-1} \hspace{5mm} {\rm erg}, \label{Eq:energy} \end{equation} where $n$ is the number density of gas in the region where the accelerated particles are located. The corresponding value for $\gamma$~Cygni is a factor of 3 higher. Alternatively, we can obtain an estimate of the total mass in the region where the accelerated particles are located as a function of the energy density in accelerated particles \begin{equation} M \sim 3 \times 10^5 \left( {u_{\rm CR} \over {\rm 1 \; eV \; cm^{-3}}} \right)^{-1} \hspace{5mm} {\rm M_\odot} \end{equation} for IC~443 and a factor 3 higher for $\gamma$~Cygni. Fit~1 requires a very high matter density in order to suppress the IC contribution to the $\gamma$--ray flux which is much flatter than the observed spectrum. The SNR G78.2+2.1 associated with $\gamma$~Cygni was tentatively identified with the COS-B source 2CG78+01 (\cite{Pollock85A}). A cloud of density $\sim 300$ cm$^{-3}$ occupies $\sim 5$\% of the volume of the remnant, and this density has been used by Pollock (1985) in predicting GeV gamma ray fluxes. Aharonian, Drury, \& V\"{o}lk (1994) also discussed this association and point out that the emission may extend to TeV energies. At radio frequencies, most of the emission comes from the SE part of the remnant (\cite{Higgs83A}), and this appears to coincide also with a region of exceptionally high 90 $\mu$m intensity, and a molecular cloud near the rim of the remnant, implying that this region may be the source of the gamma ray emission. A density of $\sim 300$ cm$^{-3}$ is comparable with the densities that come from the maximum likelihood fits. However, the fitted density is the density averaged over the region where the accelerated particles are located, and so this value may be unrealistically high unless acceleration is taking place only close to such a massive high-density cloud. This might occur at massive clouds interacting with the SNR shock. Another way to look at the numbers for $\gamma$~Cygni is to note that the energy in cosmic rays required in a region with density $300$~cm$^{-3}$ is $5\times 10^{48}$~erg. If this high density occupies 5\% of the volume of the SNR then the total energy in cosmic rays, assuming them to be uniformly distributed, is $\sim 10^{50}$~erg, approximately 10\% of the initial kinetic energy of the ejecta of a typical supernova. Finally, we check that for the fitted electron spectra the synchrotron X--ray flux predicted for reasonable magnetic fields does not exceed that observed. This is particularly important for the case of Fit~1 where the electron spectrum extends up to 80 TeV and one would expect to generate a significant X--ray flux for any standard value of magnetic field at the shock. Taking the electron spectrum in the source to be \begin{equation} N(E) \equiv {dN \over dE} = a_e E^{-\alpha} \exp(-E/E_e^{\rm max}) \; \; \; \; \rm GeV^{-1}, \end{equation} where $E$ is in GeV, we may obtain the synchrotron X--ray flux at Earth (erg cm$^{-2}$) in the energy range $\epsilon_1$ to $\epsilon_2$ \begin{equation} F_X = {a_e \over 4 \pi d^2} \int_{\epsilon_1/h}^{\epsilon_2/h} d \nu \int dE E^{-\alpha} P(\nu, E, H_\perp). \end{equation} Here $P(\nu, E, H_\perp)$ is the power per unit frequency (erg Hz$^{-1}$) emitted at frequency $\nu$ by an electron of energy $E$ (GeV) in a magnetic field with perpendicular component $H_\perp$ (gauss) and is given by \begin{equation} P(\nu, E, H_\perp) = 4 \pi c_3 H_\perp F(x) \end{equation} where $c_3= 1.87 \times 10^{-23}$, $x=\nu/\nu_c$, $\nu_c \approx c_1 H_\perp (625 E)^2$, $c_1=6.27 \times 10^{18}$, and \begin{equation} F(x) = x \int_x^\infty K_{5/3}(z) dz \end{equation} where $K_{5/3}$ is a Bessel function of the second kind (\cite {Pacholczyk70}). >From Eq.~14 we see that \begin{equation} a_e = {B R_e \over n} 4 \pi d^2, \end{equation} and so we obtain \begin{equation} F_X = 4 \pi c_3 H_\perp {B R_e \over n} \int_{\epsilon_1/h}^{\epsilon_2/h} d \nu \int dE F(x) E^{-\alpha} \exp(-E/E_e^{\rm max}). \end{equation} For the best fit parameters given above, and a perpendicular component of magnetic field of $H_\perp =6 \mu$G, we predict for IC~443 a 2---10~keV flux of $6.8 \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$ which is not in conflict with the observed flux of $6.7 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$. Similarly for $\gamma$~Cygni we predict a 0.2---4~keV flux of $8.2 \times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ which is not in conflict with the observed flux of $6 \times 10^{-11}$ erg cm$^{-2}$ s$^{-1}$. \subsection {Results from Fits~2, 3, and 4} As noted above, the X-ray luminosity calculated using a magnetic field of 6 $\mu$G and the electron spectra from Fit~1 is not in direct contradiction with the observed luminosities of either object. However, for higher magnetic fields, e.g. due to shock compression, the predicted luminosity would exceed that observed if $H_\perp$ is higher than $\sim 54 \mu$G (IC~443) or $\sim 42 \mu$G ($\gamma$~Cygni). Even for fields below these limits, there may be a problem due to the power-law nature of the synchrotron spectrum given that the observed X--ray spectrum appears to be consistent with thermal bremsstrahlung origin. We therefore perform Fits~2, 3, and 4, which tend to suppress the X--ray production either by cutting off the electron spectrum at an arbitrary $E_e^{\rm max}$ or by allowing a steeper electron spectrum. Tables I and II list the results of all four fits for the two supernova remnants. Columns 2 \& 3 give the proton and electron acceleration spectra, column 4 -- the cutoff energy for electrons (and protons in Fit~4), column 5 contains the preferred $R_e$ value, column 6 -- the matter density. Column 7 gives the measure of the fit improvement ($\ln{(L/L_0)}$) and column 8 -- the maximum value of the magnetic field allowed by the observed X--ray luminosity of the source. Note that the synchrotron X--ray flux does not scale simply as a power-law in magnetic field (with exponent related the electron spectral index) because the X--ray emitting electrons are near or above $E_e^{\rm max}$ where the spectrum is affected by the exponential cut-off so that the delta-function approximation to $F(x)$ which is often used breaks down. \begin{table*} \tablenum{1} \caption{ Fit parameters for IC 443. $E^{\rm max}_p$ fixed at 80 TeV for fits 1, 2 \& 3. $E^{\rm max}_e$ fixed at 80 TeV for fit 1 and at 1 TeV in fit 3. \label{tbl1}} \begin{center} \begin{tabular}{rccccccr} \tableline Fit & $\alpha_p$ & $\alpha_e$ & $E_e^{\rm max}$ & $R_e$ & $n$ & ln$(L/L_0)$ & $H_\perp^{\rm max}$ \\ & & & (GeV) & & (cm$^{-3}$) & & (G) \\ \tableline 1 & 2.32$^{+0.14}_{-0.11}$ & $=\alpha_p$ & $8 \times 10^4$ & 0.16$^{+0.29}_{-0.08}$ & $>10^3$ & 0 & $5.4 \times 10^{-5}$ \\ 2 & 2.15$^{+0.15}_{-0.15}$ & $=\alpha_p$ & 25$^{+15}_{-10}$& 0.4$^{+2.0}_{-0.3}$ & 0.6$^{+0.4}_{-0.2}$ & 0.36 & $7.0 \times 10^{-2}$ \\ 3 & 2.25$^{+0.20}_{-0.20}$ & 2.70$^{+0.15}_{-0.35}$ & 10$^3$ & 0.05$^{+0.15}_{-0.02}$ & 6.3$^{+10.}_{-5.0}$ & 0.68 & $5.3 \times 10^{-4}$ \\ 4 & 1.85$^{+0.30}_{-0.15}$ & $=\alpha_p$ & 40$^{+23}_{-24}$& 0.03$^{+0.07}_{-0.02}$ & 0.08$^{+0.12}_{-0.02}$ & 1.22 & $1.2 \times 10^{-2}$ \\ \tableline \end{tabular} \end{center} \end{table*} \begin{table*} \tablenum{2} \caption{ Fit parameters for $\gamma$~Cygni. $E^{\rm max}_p$ fixed at 80 TeV for fits 1, 2 \& 3. $E^{\rm max}_e$ fixed at 80 TeV for fit 1 and at 1 TeV in fit 3. \label{tbl2}} \begin{center} \begin{tabular}{rccccccr} \tableline Fit & $\alpha_p$ & $\alpha_e$ & $E_e^{\rm max}$ & $R_e$ & $n$ & ln$(L/L_0)$ & $H_\perp^{\rm max}$ \\ & & & (GeV) & & (cm$^{-3}$) & & (G) \\ \tableline 1 &2.42$^{+0.09}_{-0.07}$& $=\alpha_p$ & [$8 \times 10^4$] & 0.16$^{+0.29}_{-0.08}$ & $>10^3$ & 0 & $4.2 \times 10^{-5}$ \\ 2 &2.25$^{+0.15}_{-0.25}$& $=\alpha_p$ & 25$^{+15}_{-10}$& 0.50$^{+0.50}_{-0.40}$ & 4.0$^{+2.3}_{-2.7}$ & 1.07 & $6.6 \times 10^{-3}$\\ 3 &2.35$^{+0.10}_{-0.20}$& 2.50$^{+0.20}_{-0.10}$ & [10$^3$] & 0.13$^{+0.20}_{-0.03}$& 12.0$^{+7.}_{-4.}$ & 0.81 & $7.8 \times 10^{-5}$\\ 4 &1.80$^{+0.10}_{-0.15}$& $=\alpha_p$ & 40$^{+23}_{-5}$& 0.04$^{+0.06}_{-0.03}$& 0.06$^{+0.04}_{-0.03}$ & 5.57 & $1.7 \times 10^{-3}$ \\ \tableline \end{tabular} \end{center} \end{table*} Fit~2 ($\alpha_e = \alpha_p$, $E^{\rm max}_e$ fitted) is not very sensitive to the spectral index at acceleration, but is quite sensitive to the cutoff energy of the electrons. Fit prefers a low value of the electron cutoff energy $E_e^{\rm max}$, which eliminates the problem with the flat IC $\gamma$--ray spectrum and thus does not require very large matter density. Within 1$\sigma$ of the best fit values the matter density only varies by a factor of 2. Because of the low electron energy cutoff the fit allows for significantly flatter acceleration spectra. The range of $R_e$ is significantly wider with $R_e$ decreasing from $\sim$1 for flat spectra to $\sim$0.1 for spectral indices 1.35 -- 1.40. The fit quality improves over Fit~1, especially for $\gamma$~Cygni. Adding one more parameter to the fit increases the error bars on all parameters that we evaluate. The best fit spectrum for IC 443 is shown in Fig.~6. Fit~3 ($\alpha_p, \; \alpha_e$ fitted, $E^{\rm max}_e$ fixed) also allows for a wide range of spectral indices. The most important restriction that data requires is an electron acceleration spectrum which is significantly steeper than the proton spectrum. Even these steep electron spectra do not fully compensate for the flat IC production spectrum, so that the matter density required is of order 1--10 cm$^{-3}$. The $R_e$ value is quite stable, $\sim$0.1--0.2 or even less for IC 443. The fit quality improves over Fit~1 by more than 1$\sigma$ for both supernova remnants. Fig.~7 shows the best fit spectrum for IC 443 under the assumptions of Fit~3. Fit~4 ($\alpha_e=\alpha_p,\; E^{\rm max}_p = E^{\rm max}_e$ fitted) produces the greatest improvement for both objects. This is especially true for $\gamma$~Cygni, for which all three previous sets of assumptions give significantly worse fits. Fig.~8 shows the best fit spectrum for $\gamma$~Cygni. It is important to notice that the contribution of the individual processes is now different. The role of bremsstrahlung is negligible. The $\gamma$--ray fluxes above 1 GeV are due to $\pi^0$ production while IC dominates at lower energy. The power law spectra preferred by the fits are very flat ($\alpha<$2), $R_e\;<$0.1, and the matter density is less than 0.1. Both electron and proton spectra cut off at less than 60 GeV. \section{Conclusions} With the current set of assumptions the fits of the $\gamma$--ray spectra of the two supernova remnants identified three possible sets of basic parameters. The first set, represented by Fit~1, consists of a relatively steep acceleration spectra and requires very large matter density in the acceleration region. The second set, Fits~2 \& 3 allows for more reasonable matter densities at the $\gamma$--ray production region. Fit 4, which assumes a cutoff of the electron and proton spectra at the same maximum energy, results in a low matter density but also a very low value of the maximum energy. A fit with $E_p^{max}$ allowed to increase while keeping $E_e^{max}$ low would give a similar fit to the EGRET data, but steepening the electron spectrum through synchrotron radiation at such a low energy would require extreme magnetic fields. Note that recent theoretical calculations, accounting for nonlinear effects (\cite{Berezhko96}; \cite{EllisonRey}), predict compression ratios higher than 4 that could lead to mean slopes at acceleration much flatter than the canonical $\alpha$ = 2 value. Flat electron spectra at low energies would also be more consistent with the radio spectral indices of the two supernova remnants which imply electron spectral indices at low energies of $\alpha_e = 1.72$ for IC443, and $\alpha_e = 2.0$ for $\gamma$~Cyg (\cite{Green91}). Fits 1 to 3 (IC~443) and Fits 1 and 3 ($\gamma$~Cygni) appear to be inconsistent with the radio data. However, if a high magnetic field were present (i.e. close to $H_\perp^{\rm max}$) then the energies of the electrons producing the radio synchrotron emission could be much lower than the energies of the $\gamma$-ray emitting electrons, thereby removing the inconsistency. This would occur, for example in the case of Fit~2, where the synchrotron emitting electrons could have energies as low as 5~MeV (IC~443). The improvement of fit quality, shown in Tables 1 and 2, increases as expected when an additional fit parameter is introduced. We have not employed the $F$--test, which measures the fit improvement with the increase of number of parameters, because our different fits do not correspond to particular models of supernova remnants that we are testing. It is important to notice that $\ln{(L/L_0)}$ value is to a large extent determined by the end points of the data set where the efficiency and the statistical accuracy of the EGRET measurements are the worst. Following the EGRET observation of IC~443 and $\gamma$~Cygni it was anticipated, based on an $E^{-2}$ extrapolation of the observed flux, that these sources should be observable in the TeV range and above with ground based detectors. A simple $E^{-2}$ extrapolation of the Egret data is inconsistent with upper limits from CASA (\cite{Borione95}) and Cygnus (\cite{Allen95}) around 100 TeV, from Hegra above 20 TeV and from Whipple above 300 GeV. In this context, it is interesting to note that, except for Fit 4, which has very low $E^{max}$, our fits to the EGRET data alone seem to indicate steeper source spectra. The upper limits of the Whipple observatory on the TeV emission from the sources is shown on all emission spectra shown in Figs.~4 -- 8. It is important to notice that the fitted spectra are always below, or very close to, the experimental upper limits. In the case of Fit~1 of IC~443 (Fig.~4), for example, the fitted spectrum is $\sim$50\% higher than the upper limit. Half of that difference is due to the contribution of bremsstrahlung. Note however, that if the interactions are taking place with neutral matter, such as that in a molecular cloud, then the cross sections we have used for bremsstrahlung in ionized matter would be inappropriate, and the bremsstrahlung spectral index could be steepened by as much as $\sim 0.1$ which would have the effect of bringing the fitted spectrum below the experimental limit. An electron source spectrum with $\alpha \sim 2.4$ is favored by a comparison of recent cosmic ray electron propagation calculations with direct cosmic ray electron observations above $\sim 10$ GeV (\cite{Strong94A,Strong95A,PorterProtheroe96}). However, source spectra as flat as $E^{-2}$ are possible (\cite{Aharonian95A}) if the distance to the nearest source is $\sim 100$ pc or more. The spectra of the two SNR observed by EGRET show no sign of flattening at, or below, 100 MeV, indicating the presence of a steep spectral component due to bremsstrahlung. There are many different implications of a steep spectrum on acceleration ($\alpha \sim 2.2 - 2.4$), extracted with some of the assumptions of our analysis (Fits~1--3). For example, such a source spectrum for cosmic ray nuclei would require a loss rate in the galaxy, and hence a diffusion coefficient for cosmic ray propagation, proportional to $\sim E^{0.3}$ rather than $\sim E^{0.7}$ which is usually assumed based on cosmic ray secondary to primary ratios (note that a Kolmogorov spectrum actually predicts a diffusion coefficient proportional to $E^{1/3}$, see also Ptuskin et al. 1993, and Biermann 1995). Such a diffusion coefficient is also favored by reacceleration models (\cite{HeinbachSimon95,SeoPtuskin94}). Such spectra, however, would not be consistent with the radio observations and would require a strong spectral steepening above radio emitting energies. In conclusion, we have modeled the gamma ray emission of IC~443 and $\gamma$~Cygni including inverse Compton scattering on all relevant radiation fields, bremsstrahlung, and pion production. We fit the modeled production spectra to the EGRET data assuming a power-law momentum spectrum of electrons and nuclei in the SNR and exponential cutoffs. The results from the fitting show that (a) the dominant contributions come from bremsstrahlung and $\pi^0$ decay provided the spectra of both electrons and nuclei extend above 60 GeV. The IC contribution can only be important with fine tuning, when $E^{\rm max}$ for both electrons and nucleons is $\sim$40 GeV; (b) an electron to proton ratio of $\sim$ 0.05--0.5 is required under all four assumptions that we have explored; (c) a spectral index of $\alpha \sim 2.2$--2.4 is required if electrons and nuclei are accelerated with the same spectral indices unless the acceleration spectra cut off below 60 GeV; (d) if the acceleration spectra of electrons and nuclei are not the same the data requires electron spectra that are significantly steeper than the nuclei spectra, which contribute to the experimentally observed range only the $\pi^0$ feature, which depends mildly on the acceleration spectra of nuclei; (e) a very high, but uncertain, average matter density is required if the electron spectrum has the same spectral index as the proton spectrum and the cut-off energy is very high, e.g. 80 TeV (note, however, that matter densities as high as $\sim 300$ cm$^{-3}$ are present in molecular clouds associated with these SNR). We also note that in this case the bremsstrahlung contribution at 1 GeV is $\sim 40\%$, somewhat higher than the $\sim 10\%$ obtained from a fit to the diffuse galactic gamma ray intensity (\cite{Hunter95,Hunter97}), which is probably lower due to the effects of cosmic ray energy losses during propagation which are different for electrons and nuclei. At the present time and with the current available experimental information we cannot make a choice between the two basic assumptions that we used for fitting the EGRET data. Additional experimental information is necessary for further analysis. One crucial data set would consist of TeV detections (or lower upper limits) of these two and other supernova remnants. A possible observation would allow us the exclude the assumption for a low energy cutoff of the acceleration spectra and to obtain better determined values for the shape of the electron and proton spectra and for the matter density at the source. We also plan to extend our analysis by using theoretically motivated shapes for the spectra of the accelerated particles. It will then be possible to also predict the expected 100 TeV $\gamma$--ray fluxes from supernova remnants in some current models (\cite{Berezhko96}), which extend the accelerated particles spectra to Z$\times$400 TeV.\\ \noindent {\bf Acknowledgements.} We thank Joe Esposito and the EGRET group for providing the measured spectra of IC~443 and $\gamma$~Cygni in numerical form. We are grateful to the referee Steven Reynolds for helping us to improve on the first version of this paper. RJP thanks the Bartol Research Institute for hospitality during part of 1996. We thank Troy Porter for a useful discussion and for reading the manuscript. The research of TKG is supported in part by NASA Grant NAGW--4605. TS is supported in part by NASA grant NAGW--3880. The research of RJP is supported by a grant from the Australian Research Council.

proofpile-arXiv_065-557

\section{Introduction} Contemporary applications of the Einstein, Podolsky and Rosen (EPR) correlations \cite{EPR,Bohm} and the Bell inequality \cite{Bell,Home} range from purely philosophical problems to quantum cryptography, computation and teleportation. In the cryptographic scheme proposed by Ekert \cite{Ekert} Alice and Bob test for eavesdropping by measuring the average of the ``Bell observable" \begin{eqnarray} c(\bbox a,\bbox a',\bbox b,\bbox b')&=& \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b}|\psi\rangle + \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b'}|\psi\rangle\nonumber\\ &\pp =& + \langle \psi|\hat {\bbox a'}\otimes \hat {\bbox b}|\psi\rangle - \langle \psi|\hat {\bbox a'}\otimes \hat {\bbox b'}|\psi\rangle\label{Bell} \end{eqnarray} where $\hat {\bbox a}$ etc. are ``yes--no" observables (say, signs of spin for electrons, or planes of polarization for photons). Quantum mechanics predicts that for some choices of such observables one can obtain $|c(\bbox a,\bbox a',\bbox b,\bbox b')|=2\sqrt{2}$. In an ideal situation a result of the form $|c(\bbox a,\bbox a',\bbox b,\bbox b')|<2\sqrt{2}$ indicates that at least some pairs of particles were not prepared in the singlet state and this indicates an eavesdropper. Practical applicability of quantum cryptographic protocols crucially depends on detector efficiencies. In typical Bell-type photon pair experiments the efficiencies were smaller than $20\%$. The advent of solid state photodiodes provides efficiencies of detection which are much higher \cite{Kwiat} but still far from ideal. An almost ideal experimental scheme has been recently discussed by Fry {\it et al.\/} \cite{Fry} who propose to replace photons with massive particles (pairs of $^{199}$Hg atoms). Detection efficiency is then at least $95\%$ and can be pushed to more than $99\%$. An obvious drawback of such a communication channel is that it is slow. To make it faster one might be tempted to use relativistic velocities. It will be shown below that for high velocities one may expect a surprising effect: The amount of violation of the Bell inequality may decrease with growing velocity of the spin-1/2 particles. Alice and Bob must therefore additionally know the velocity distribution of the particle beam. Otherwise they may be confused and ``detect an eavesdropper" even though the particles remain in a pure zero-helicity singlet state. The effect is related to the old problem described already in 1930 by Schr\"odinger \cite{ES}. As is widely known E.~Schr\"odinger examined the behavior of the coordinate operator $\bbox x$ associated with Dirac's equation and discovered the oscillatory motion he called the {\it Zitterbewegung\/}. The {\it Zitterbewegung\/} takes place with respect to the {\it center-of-mass\/} position operator $\bbox x_A$ and this is the operator which should be used to define a physically meaningful spin operator. The situation is not typical only of the Dirac equation and is not associated with the presence of negative energy solutions as one is sometimes led to believe. The so-called new Dirac equation generalized by Mukunda {\it et al.\/} \cite{Mukunda} admits only positive-energy solutions but the {\it Zitterbewegung\/} is present and the associated center-of-mass operator is algebraically identical to this implied by Schr\"odinger's analysis of the Dirac equation \cite{BZ}. The analysis presented in \cite{Mukunda} shows clearly that in order to obtain a physically consistent model of an extended hadron one has to proceed in the way identical to the one chosen in this paper: First define the center-of-mass operator $\bbox Q$, then introduce the angular momentum $\bbox L=\bbox Q\times\bbox P$, and finally define spin by $\bbox S=\bbox J -\bbox L$. In what follows I use a group representation formulation, elements of which can be found in the 1965 papers by Fleming \cite{Fleming1}. The group theoretic approach has the advantage of being applicable to any physical system whose symmetry group is the Poincar\'e group, or whose symmetry group contains the Poincar\'e group as a subgroup. \section{Relativistic spin operators} Let us begin with generators of the unitary, infinite dimensional irreducible representation of the Poincar\'e group corresponding to a nonzero mass $m$ and spin $j$. Their standard form is \cite{Ohnuki} \begin{eqnarray} \bbox J &=& \frac{\hbar}{i}\bbox p\times \frac{\partial}{\partial \bbox p} +{\bbox s},\\ \bbox K &=& \pm\Bigl( |p_0|\frac{\hbar}{i}\frac{\partial}{\partial \bbox p} - \frac{\bbox p\times{\bbox s}}{mc+|p_0|}\Bigr),\label{46}\\ \bbox P&=& \bbox p,\\ P_0&=&p_0=\pm \sqrt{\bbox p^2 + m^2c^2}. \end{eqnarray} Here {\bbox s} denotes finite dimensional angular momentum matrices corresponding to the $(2j+1)$-dimensional representation $D^j$ of the rotation group. Similar forms are obtained if one uses the hadronic representation introduced in \cite{Mukunda}. The Poincar\'e group has two Casimir operators: The squared mass and the square of the Pauli-Lubanski vector $W^\mu$. The latter operator written in the above representation is \begin{eqnarray} W^\mu&=&(W^0,\bbox W)=(\bbox P\cdot\bbox J,P_0\bbox J -\bbox P\times\bbox K) \\ &=& \bigl(\bbox p\cdot \bbox s, p_0(\bbox n\cdot \bbox s)\bbox n \pm mc\, \bbox s_\perp\bigr), \end{eqnarray} where $\bbox n$ is the unit vector pointing in the momentum direction and $$ \bbox s_\perp=\bbox s-(\bbox n\cdot \bbox s)\bbox n. $$ The center-of-mass position operator which generalizes to any representation the operator $\bbox x_A$ of Schr\"odinger is \begin{eqnarray} {\bbox Q} &=&-\frac{1}{2}\Bigl(P_0^{-1}{\bbox K} +{\bf K}P_0^{-1}\Bigr) \label{Q}\\ &=&i\hbar\frac{\partial}{\partial \bbox p} -i\hbar\frac{\bbox p}{2p_0^2} + \frac{\bbox p\times{\bbox s}}{|p_0|(mc+|p_0|)}.\label{conn} \end{eqnarray} This operator extends naturally also to massless fields and can be shown to be uniquely (up to subtleties with domains of unbounded operators) derived from symmetry considerations in the case of the Maxwell field \cite{JJ,Mourad}. In the Maxwell field case, the formula (\ref{conn}) can be regarded as defining a connection on a light cone. A parallel transport with respect to this connection can be shown to generate a Berry phase \cite{IBB,Pati}. Orbital angular momentum and spin corresponding to $\bbox Q$ were given by Pryce and Fleming \cite{Fleming1,Pryce} \begin{eqnarray} \bbox L &=& \bbox Q\times\bbox P= \frac{\hbar}{i}\bbox p\times \frac{\partial}{\partial \bbox p} +\frac{|p_0|-mc}{|p_0|}\Bigl( {\bbox s} -(\bbox n\cdot{\bbox s})\bbox n\Bigr), \nonumber\\ {\bbox S} &=& \bbox J - \bbox L = \frac{mc}{|p_0|}{\bbox s} + \Bigl(1-\frac{mc}{|p_0|}\Bigr)(\bbox n\cdot{\bbox s})\bbox n \nonumber\\ &=& \sqrt{1-\beta^2}{\bbox s}_\perp+(\bbox n\cdot{\bbox s})\bbox n = \bbox W/p_0.\label{s-w} \end{eqnarray} $\bbox \beta=\bbox n\,|\bbox v|/c$, where $\bbox v=c^2\bbox p/p_0$ is a velocity of the particle. Eq.~(\ref{s-w}) shows that relativistic spin is closely related to the Pauli-Lubanski vector. Projection of spin in a direction given by the unit vector $\bbox a$ commutes with the Hamiltonian $P_0$ and equals \begin{eqnarray} \bbox a\cdot{\bbox S}&=& \Bigl[\frac{mc}{|p_0|}{\bbox a} + \Bigl(1-\frac{mc}{|p_0|}\Bigr)(\bbox n\cdot{\bbox a})\bbox n \Bigr]\cdot{\bbox s}\\ &=& \Bigl[\sqrt{1-\beta^2}\bbox a_\perp +\bbox a_\parallel\Bigr]\cdot \bbox s=: \bbox \alpha(\bbox a,\bbox p)\cdot{\bbox s}. \label{9} \end{eqnarray} The latter equality defines the vector $\bbox \alpha(\bbox a,\bbox p)$ whose length is \begin{eqnarray} |\bbox \alpha(\bbox a,\bbox p)|= \frac{\sqrt{(\bbox p\cdot\bbox a)^2 +m^2c^2}} {|p_0|}=\sqrt{1+(\bbox \beta\cdot\bbox a)^2-\beta^2}.\nonumber \end{eqnarray} The eigenvalues of $\bbox a\cdot{\bbox S}$ are therefore \begin{eqnarray} \lambda_{a}=j_3\hbar \sqrt{1+(\bbox \beta\cdot\bbox a)^2-\beta^2} \label{aS} \end{eqnarray} where $j_3=-j,\dots,+j$. The eigenvalues of the Pauli-Lubanski vector projections are therefore $\omega_a=p_0\lambda_a$. In the infinite momentum/massless limit the eigenvalues of the relativistic spin in a direction perpendicular to $\bbox p$ vanish, which can be regarded as a consequence of the Lorentz flattenning of the moving particle (in these limits $\bbox S=(\bbox n\cdot\bbox s)\bbox n$). Projection of spin on the momentum direction is equal to the helicity, i.e. $\bbox p\cdot\bbox S=\bbox p\cdot\bbox s$ for any $\bbox p$, and $\bbox S=\bbox s$ in the rest frame ($\bbox p=0$). Bacry \cite{Bacry} observed that a nonrelativistic limit of $\bbox Q$ leads to a correct form of the spin-orbit interaction in the Pauli equation if one uses potentials $V(\bbox Q)$ instead of $V(\bbox x)$ \cite{Kaiser}; an analogous effect was described in \cite{BB1} where the internal angular momentum of the {\it Zitterbewegung\/} leads to spin with the correct $g=2$ factor. An algebraic curiosity is the fact that the components of $\bbox S$ satisfy an algebra which is $so(3)$ in the rest frame and formally contracts to the Euclidean $e(2)$ in the infinite momentum/massless limit, and thus provides an interesting alternative explanation of the privileged role played by the Euclidean group in the theory of massless fields \cite{ja,Kim}. In spite of all these facts suggestng that both $\bbox Q$ and $\bbox S$ are natural candidates for physical observables no experimental tests distinguishing them from other definitions of position and spin have been proposed so far. Obviously, it is not easy to test directly $\bbox Q$ which, representing the center of mass, may be expected to couple to the gravitational field. The spin operator, on the other hand, is responsible for the magnetic moment and should couple to the electromagnetic field which is much stronger. Consider now a Stern-Gerlach-type measurement involving spin-1/2 relativistic particles and assume that $\bbox S$ is the physical internal angular momentum which is measured in this experiment \cite{some}. Assume also that we have two spin-1/2 particles in a singlet state (total helicity equals zero) and propagating in the same direction with identical momenta $\bbox p$ (to be more precise one should take wave packets in momentum space, but for simplicity assume that they are sufficiently well localized around momenta $\bbox p$, so that we can approximate them by plane waves): \begin{eqnarray} |\psi\rangle = \frac{1}{\sqrt{2}}\Bigl( |+\frac{1}{2},\bbox p\rangle|-\frac{1}{2},\bbox p\rangle - |-\frac{1}{2},\bbox p\rangle|+\frac{1}{2},\bbox p\rangle \Bigr). \label{st} \end{eqnarray} The kets $|\pm\frac{1}{2},\bbox p\rangle$ form the {\it helicity\/} basis. Consider the binary operators $\hat {\bbox a}=\bbox a\cdot\bbox S/|\lambda_a|$, $\hat {\bbox b}=\bbox b\cdot\bbox S/|\lambda_b|$. Their eigenvalues are $\pm 1$. The relativistic corrections that arise are those resulting from the modification of the spin direction as ``seen" by a measuring device. The average of the relativistic EPR-Bohm-Bell operator is \begin{eqnarray} {}&{}& \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b}|\psi\rangle =\nonumber\\ &{}& - \frac{ \bbox a\cdot\bbox b- \beta^2 \bbox a_\perp\cdot\bbox b_\perp} {\sqrt{1+\beta^2\bigl[(\bbox n\cdot\bbox a)^2 - 1\bigr]} \sqrt{1+\beta^2\bigl[(\bbox n\cdot\bbox b)^2 - 1\bigr]}}\label{ab} \end{eqnarray} There are several interesting particular cases of the formula (\ref{ab}). First, if $\bbox a=\bbox a_\perp$, $\bbox b=\bbox b_\perp$ then \begin{eqnarray} \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b}|\psi\rangle =-\bbox a\cdot\bbox b \end{eqnarray} which is the nonrelativistic result. This case will never occur in a realistic experiment since localization of detectors will lead to a momentum spread. If $\bbox a\cdot\bbox n\neq 0$, $\bbox b\cdot\bbox n\neq 0$ then in the ultrarelativistic case $\beta^2=1$ \begin{eqnarray} \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b}|\psi\rangle =-\frac{(\bbox a\cdot\bbox n)\,(\bbox b\cdot\bbox n)} {|\bbox a\cdot\bbox n|\,|\bbox b\cdot\bbox n|}=\pm 1 \end{eqnarray} independently of the choice of $\bbox a$, $\bbox b$. It is easy to intuitively understand this result: In the ultrarelativistic limit projections of spin in directions perpendicular to the momentum vanish for both particles and spins are (anti)parallel to the momentum. The most striking case occurs if $\bbox a$ and $\bbox b$ are perpendicular and the nonrelativistic average is 0. Let $\bbox a\cdot\bbox b=0$, $\bbox a\cdot\bbox n= \bbox b\cdot\bbox n=1/\sqrt{2}$. Then \begin{eqnarray} \langle \psi|\hat {\bbox a}\otimes \hat {\bbox b}|\psi\rangle = -\frac{\beta^2}{2- \beta^2}.\label{19} \end{eqnarray} This average is 0 in the rest frame ($\beta=0$) and $-1$ for $\beta =1$. Any observable deviation from 0 in an EPR-Bohm type experiment would be an indication that the operators $\bbox S$ and $\bbox Q$ are physically correct observables and that massive spin-1/2 particles are extended in the sense that centers of mass and charge do not coincide. Fig.~\ref{fig3} shows that (\ref{19}) describes a relativistic effect that is even stronger than the Lorentz contraction or the time delay (both are proportional to $\sqrt{1-\beta^2}$). One pecularity of $\bbox Q$ is that its components do not commute for nonzero spins. An uncertainty principle guarantees therefore that such a particle cannot be localized at a point \cite{Kalnay}, or is extended in some nonclassical sense, a property that cannot be without implications for the renormalization and self-energy problems. The definition of $\bbox Q$ implies also that the center of mass does not transform as a spatial component of a four-vector. This apparently counter-intuitive result agrees however with the classical analysis of M{\o}ller \cite{Moller,Fleming1} who showed that the center of mass of a spinning classical body is not a component of a four vector. These interesting properties seem unavoidable and can be proved in various ways at both quantum and classical levels (for their classical derivations see \cite{Mukunda,Zakrzewski}). Consider now the vectors $\bbox a=(1/\sqrt{2},1/\sqrt{2},0)$, $\bbox a'=(-1/\sqrt{2},1/\sqrt{2},0)$, $\bbox b=(0,1,0)$, $\bbox b'=(1,0,0)$ leading to the the maximal violation of the Bell inequality in nonrelativistic domain. Fig.~\ref{fig1} shows the dependence of the average (\ref{Bell}) on $\beta$ and $\phi$ where $\bbox \beta=(\beta\cos\phi,\beta\sin\phi,0)$. Fig.~\ref{fig2} shows the average (\ref{Bell}) for $\bbox \beta= \beta(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$ as a function of the spherical angles and for $\beta=0.99$ and $\beta=0.95$. These results show clearly that the information about the degree of violation of the Bell inequality is not sufficient for determining purity of a massive two-particle zero-helicity state. Additionally one has to know the momentum distribution of the particle beam. \section{Pauli-Lubanski vector vs. spin} \label{PL} The relation between $W^\mu$ and $\bbox S$ is similar to the one between the 4-velocity $u^\mu$ and the 3-velocity $\bbox \beta$. The Casimir operator $W_\mu W^\mu$ equals $-(mc)^2j(j+1)$ if an irreducible representation of the Poincar\'e group is considered. For this reason it is typical to define the spin 4-vector as \begin{eqnarray} w^\mu=W^\mu/(mc)= \Bigl(\bbox u\cdot \bbox s, \frac{p_0}{mc}(\bbox n\cdot \bbox s)\bbox n \pm \bbox s_\perp\Bigr), \end{eqnarray} where $u^\mu$ is the 4-velocity. In the rest frame $p_0=\pm mc$ and $w^\mu= \pm(0,\bbox s)$ which seems to justify this choice. For a moving particle the eigenvalues of $\bbox a\cdot\bbox w$ are $\lambda_a\, p_0/mc$ where $\lambda_a$ denote the respective eigenvalues of $\bbox a\cdot\bbox S$. The eigenvalues of $\bbox a\cdot\bbox w$ therefore tend to $\pm\infty$ in the infinite momentum limit which is unphysical for a spin observable. Nothing of that kind occurs if one divides $\bbox W$ by {\it energy\/} and {\it not by mass\/} which again selects our spin operator as a candidate for a physical observable. Nevertheless, irrespective of this subtlety, the relativistic EPRB average is the same for both $\bbox S$ and $\bbox w$ since we consider a ``yes-no" observable which is obtained by {\it normalization\/} of eigenvalues to $\pm 1$. This is another reason to believe that the discussed suppression of degree of the Bell inequality violation is a physical phenomenon that should be observable in experiments with massive particles. \section{Comparison with the Dirac equation} Just for the sake of completeness let us compare the general formulas to the analogous calculations performed for the Dirac electrons \cite{1984}. The Pauli-Lubanski vector is \begin{eqnarray} W^0&=&\bbox p\cdot \bbox s\\ \bbox W&=&\frac{1}{2}\bigl(\bbox s H + H \bbox s\bigr), \end{eqnarray} where $H$ is the Dirac free Hamiltonian and $$ \bbox s=\frac{\hbar}{2} \left( \begin{array}{cc} \bbox \sigma & 0\\ 0 & \bbox \sigma \end{array} \right) $$ is the spinor part of the generator of rotations. The relativistic spin operator is therefore equal to \begin{eqnarray} \bbox S&=&\bbox W H^{-1}= \frac{1}{2}\bigl(\bbox s + \Lambda \bbox s\Lambda\bigr)\nonumber\\ &=&\Pi_+\bbox s\Pi_++ \Pi_-\bbox s\Pi_-. \end{eqnarray} Here $\Lambda$ is the sign-of-energy operator and $\Pi_\pm$ project on states of given signs of energy. It follows that $\bbox S$ is the so-called even part of the spinor part of the generator of rotations. This operator commutes with $H$ and hence can be used for analyzing the EPRB experiment \cite{wrong}. The explicit form of this observable in units with $\hbar=1$ and $c=1$ is \begin{eqnarray} \bbox S= \frac{m^2}{p^2_0}\bbox s + \frac{\bbox p^2}{p^2_0}(\bbox n\cdot \bbox s)\bbox n + \frac{im}{2p^2_0}\bbox p\times \bbox \gamma. \end{eqnarray} $\bbox \gamma=(\gamma^1,\gamma^2,\gamma^3)$ where $\gamma^k$ are Dirac matrices. The eigenvalues of $\bbox a\cdot \bbox S$ are given by (\ref{aS}) with $j_3=\pm 1/2$. The corresponding positive-energy eigenstates in the standard representation are $$ \Psi^a_\pm= \left( \begin{array}{c} \sqrt{|p_0|+m}\biggl( (|\lambda_a|+\frac{1}{2}\bbox a\cdot \bbox n) w_\pm \pm \frac{m a\cdot t}{2|p_0|}w_\mp\biggr)\\ \sqrt{|p_0|-m}\biggl(\pm (|\lambda_a|+\frac{1}{2}\bbox a\cdot\bbox n) w_\pm -\frac{m a\cdot t}{2|p_0|}w_\mp\biggr) \end{array} \right) $$ where $w_\pm$ satisfies $\bbox n\cdot\bbox \sigma w_\pm =\pm w_\pm$, and $\bbox n\cdot\bbox t=0$. I have remarked that a positive verification of the relativistic center-of-mass concept would indicate that nonzero spin relativistic particles are extended. The example of the Dirac equation illustrates this idea. Consider again the spinor part of the generator of rotations $\bbox s$. It does not commute with $H$ and satisfies in the Heisenberg picture the precession equation \begin{eqnarray} \dot {\bbox s}=\bbox \omega\times \bbox s,\label{prec} \end{eqnarray} where $\bbox \omega=-2c\gamma^5\bbox p/\hbar$. For massive fields $\bbox \omega$ does not commute with $H$ and hence can be decomposed into even and odd parts. The even part is $$ \bbox{\Omega}={c^2+m\,c^3\bbox{\gamma}\cdot\bbox{n}/|\bbox{p}|\over c^2+m^2c^4/\bbox{p}\,^2}\,\bbox{\omega}. $$ $\bbox \Omega$ reduces to $\bbox \omega$ in both massless and infinite-momentum limits. A Hamiltonian of a particle moving with velocity $\bbox{ v}=c\bbox{\beta}$ can now be expressed as \begin{equation} H=\Bigl(1+{m^2c^4\over c^2\bbox{p}^2}\Bigr) \bbox{\Omega}\cdot\bbox{S} = \bbox{\beta}^{-2}\bbox{\Omega}\cdot\bbox{S} = \bbox{\Omega'}\cdot\bbox{ S}\label{NH} \end{equation} where each of the operators appearing in $H$ is even and commuting with $H$. The form (\ref{NH}) is analogous to the one discussed in \cite{Hes}. The limiting form $H=\bbox{\omega}\cdot\bbox{S}$ is characteristic of all massless fields, where for higher spins the equation (\ref{prec}) is still valid, but angular velocities for a given momentum are smaller the greater the helicity. The new form of the Hamiltonian leads to the following observation. Notice that for massless fields the Hamiltonian can be written in either of the following two forms \begin{equation} H=\bbox{\omega}\cdot\bbox{S}\label{H1} \end{equation} or \begin{equation} H=\bbox{c}\cdot\bbox{p}=\bbox{v}\cdot\bbox{p}\label{H2} \end{equation} where $\bbox{v}$ is the velocity operator for a general massless field ($c\bbox{\alpha}$ in case of the Dirac equation) and $\bbox{c}=(\bbox{v}\cdot\bbox{p})\bbox{p}/\bbox{p}\,^2$ is its even part. We recognize here the classical mechanical rule for a transition from a point-like description to the extended-object-like one: linear momentum goes into angular momentum, linear velocity into angular velocity, and vice versa. The third part of this rule (mass--moment of inertia) can be naturally postulated as follows \begin{equation} H=m_k\bbox{c}\,^2=I_k\bbox{\omega}\,^2.\label{H3} \end{equation} where (\ref{H3}) defines the kinetic mass $(m_k)$ and the kinetic moment of inertia $(I_k)$ of the massless field. The explicit form of $I_k$ for massless fields of helicity $\lambda=m-n$ [corresponding to the $(m,n)$ spinor representation of $SL(2,C)$] is, in ordinary units, \begin{equation} I_k=\frac{\lambda\hbar\bbox{p}\cdot\bbox{S}}{c \bbox{p}\,^2}.\label{in} \end{equation} The equation \begin{equation} I_k=m_k r_\lambda^2 \end{equation} characteristic, by the way, of circular strings (here with mass $m_k$) defines some radius which is equal to \begin{equation} r_\lambda=\frac{\hbar \lambda}{|\bbox{p}|}\label{rad} \end{equation} which can be expressed also as a form of the uncertainty principle \begin{equation} {|\bbox{p}|} r_\lambda={\hbar \lambda}.\label{rad'} \end{equation} The center-of-mass commutation relation $$ [Q_k,Q_l]=-i\hbar\epsilon_{klm}S_m/p_0^2 $$ leads in the massless case to the uncertainty relations of the type $$ \Delta Q_1\Delta Q_2\geq \hbar^2|\lambda|/(2 \langle|\bbox p|^2\rangle|)=\langle r_\lambda^2\rangle/|2\lambda|. $$ It is remarkable that the same radius occurs naturally in the twistor formulation of massless fields \cite{PR}. It is known that although spin-0 twistors can be represented geometrically by null straight lines, this does not hold for spin-$\lambda$, $\lambda\neq 0$, twistors \cite{PR}. Instead of the straight line we get a congruence of twisting, null, shear-free world lines, the so-called Robinson congruence. A three-dimensional projection of this congruence consits of {\it circles\/}, whose radii are given exactly by our formula (\ref{rad}) (cf.~the footnote at p.~62 in \cite{PR}). The circles propagate with velocity of light in the momentum direction and rotate in the right- or left-handed sense depending on the sign of helicity. The same construction can be performed for the massive Dirac particle if one uses $\bbox \Omega'$. \section{Summary} The main idea advocated in this paper can be summarized as follows. Consider {\it some\/} procedure leading to a measurement of a nonrelativistic spin \cite{some}. This procedure is based on a black box giving results ``yes" or ``no" for spins equal to, respectively, $+\hbar/2$ and $-\hbar/2$. In the nonrelativistic domain the particles enter the device ``slowly". Imagine now that for some reasons we decide to use faster particles. The measured average may vary with the growing (average) velocity of the particle beam and, obviously, some result will be obtained. The question is how to calculate the result of such an experiment assuming that the procedure measures the spin itself and not the total angular momentum. Many different definitions of relativistic spin exist in literature but all of them are momentum dependent \cite{Bagrov}. Calculations based on the definition which seems the most physical ({\it via\/} the relativistic center-of-mass) show that relativistic corrections are nontrivial. Their strength can be regarded as a combined influence of two independent relativistic phenomena: The Lorentz contraction and the M{\o}ller shift of the center of mass of a spinning body. The same result is obtained if one uses the spin operator defined {\it via\/} the Pauli-Lubanski vector. The effect can be in principle measured and will have to be taken into account in quantum cryptographic tests for eavesdropping if fast massive particles will be used for a key transfer. \acknowledgements I am grateful to Ryszard Horodecki for suggesting the problem, Vasant Natarajan for informations concerning experiments, and Gerald Kaiser for extensive discussions. The paper is a part of the KBN project 2P30B03809.

proofpile-arXiv_065-558

\section{Introduction} \renewcommand{\theequation}{\thesection.\arabic{equation}} The Schr\"odinger operator \begin{equation} H^{(C)} = - \sum_{j=1}^N {\partial^2 \over \partial x_j^2} +\beta (\beta /2 - 1){\left ( \pi \over L \right )^2 } \sum_{1 \le j < k \le N} {1 \over \sin^2 \pi (x_j - x_k)/L}, \quad 0 \le x_j \le L \label{HC} \end{equation} describes quantum particles on a line of length $L$ interacting through a $1/r^2$ pair potential with periodic boundary conditions, or equivalently quantum particles on a circle of circumference length $L$ (hence the superscript $(C)$) with the pair potential proportional to the inverse square of the chord length. It is one of a number of quantum many body systems in one dimension which are of the Calogero-Sutherland type, meaning that the ground state (i.e.~eigenstate with the smallest eigenvalue $E_0$) is of the form $e^{-\beta W/2}$ with $W$ consisting of one and two body terms only. Explicitly, for (\ref{HC}) we have \begin{equation} W = W^{(C)} := - \sum_{1 \le j < k \le N} \log |e^{2 \pi i x_k/L} - e^{2 \pi i x_j/L}|. \label{WC} \end{equation} In studying the integrability properties of (\ref{HC}) it is useful \cite{poly92a} to generalize the Schr\"odinger operator to include the exchange operator $M_{jk}$ for coordinates $x_j$ and $x_k$: \begin{equation} H^{(C,Ex)} = - \sum_{j=1}^N {\partial^2 \over \partial x_j^2} +\beta {\left ( \pi \over L \right )^2 } \sum_{1 \le j < k \le N} {(\beta /2 - M_{jk}) \over \sin^2 \pi (x_j - x_k)/L}, \label{HCE} \end{equation} When acting on functions symmetric in $x_1,\dots, x_N$, (\ref{HCE}) reduces to (\ref{HC}). Conjugation with the ground state of (\ref{HC}) gives the transformed operator \begin{eqnarray}\tilde{H}^{(C,Ex)}& := & \Big ( {L \over 2 \pi} \Big )^2 e^{\beta W^{(C)}/2} (H^{(C,Ex)} - E_0^{(C)}) e^{-\beta W^{(C)}/2} \nonumber \\ & = & \sum_{j=1}^N \Big (z_j {\partial \over \partial z_j} \Big )^2 + { N-1 \over \alpha} \sum_{j=1}^N z_j {\partial \over \partial z_j} + {2 \over \alpha} \sum_{1 \le j < k \le N}{z_j z_k \over z_j - z_k} \nonumber \\ & & \times \left[\Big ({\partial \over \partial z_j} -{\partial \over \partial z_k} \Big ) - {1 - M_{jk} \over z_j - z_k} \right] \label{HCC} \end{eqnarray} where \begin{equation} z_j := e^{2 \pi i x_j / L}, \qquad \alpha := 2/\beta. \end{equation} This operator has non-symmetric eigenfunctions of the form \begin{equation} E_{\eta}(z,\alpha) = z^{\eta} + \sum_{\nu < \eta} b_{\nu \eta} z^\eta. \label{nj} \end{equation} ($z^\eta$ will be referred to as the leading term), where $\eta$ and $\nu$ are $N$-tuples of non-negative integers and the $b_{\nu \eta}$ are coefficients. To define the partial order $<$, let $\kappa$ be a partition and $P$ be the (unique) permutation of minimal length such that \begin{equation} z^{\eta} := z_1^{\eta_1} \dots z_N^{\eta_N} = z_{P(1)}^{\kappa_1} \dots z_{P(N)}^{\kappa_N}, \label{ETA} \end{equation} Equivalently, let $\kappa_i = \eta_{P(i)}$ and similarly define the partition $\mu$ and permutation $Q$ such that $\mu_i = \nu_{Q(i)}$. The partial order $<$ is defined by the statement that $\nu < \eta$ if $\mu < \kappa$ (dominance ordering) or, in the cases $\mu = \kappa$, if the first non-vanishing difference $P(j) - Q(j)$ is positive. An equivalent specification in this later case is that the last nonvanishing difference of $\eta - \nu$ is negative \cite{ug96a}. The eigenfunctions $E_{\eta}(z,\alpha)$ are referred to as non-symmetric Jack polynomials \cite{opdam95a,bern93a}. The eigenvalue of (\ref{HCC}) corresponding to the eigenfunction $E_{\eta}$ is given by \cite{kk95} \begin{equation} \epsilon_{\eta} = \sum_{j=1}^N \kappa_j^2 + {1 \over \alpha} (N + 1 - 2j) \kappa_j \label{KEV} \end{equation} and is thus independent of the permutation $P$ relating $\eta$ to $\kappa$. Note that this implies the linear combination \begin{equation} \sum_{P=1}^{N!} a_P E_{P^{-1}\kappa} (z,\alpha) \label{SNS} \end{equation} is also an eigenfunction of (\ref{HCC}). The exchange operator generalization can also be applied to other Schr\"odinger operators of the Calogero-Sutherland type, in particular when the underlying root system is $A_N$ or $B_N$ and there is an external potential or if the underlying root system is $BC_N$. These operators are given by \cite{poly92a,yamam96a} \begin{equation} H^{(H,Ex)} := - \sum_{j=1}^N {\partial^2 \over \partial x_j^2} +{\beta^2 \over 4} \sum_{j=1}^N x_j^2 +\beta \sum_{1 \le j < k \le N} {(\beta /2 - M_{jk}) \over (x_j - x_k)^2} \label{HHE} \end{equation} \begin{eqnarray} H^{(L,Ex)} & := & - \sum_{j=1}^N {\partial^2 \over \partial x_j^2} +{\beta^2 \over 4} \sum_{j=1}^N x_j^2 + {\beta a' \over 2} \sum_{j=1}^N {\beta a' / 2 - S_j \over x_j^2} \nonumber \\ && + \beta \sum_{1 \le j < k \le N} {\beta / 2 - M_{jk} \over (x_j - x_k)^2} + {\beta / 2 - S_j S_k M_{jk} \over (x_j + x_k)^2} \label{HLE} \end{eqnarray} \begin{eqnarray} H^{(J,Ex)} & := & - \sum_{j=1}^N {\partial^2 \over \partial \phi_j^2} + {\beta a' \over 2} \sum_{j=1}^N {\beta a' / 2 - S_j \over \sin^2 \phi_j} + {\beta b' \over 2} \sum_{j=1}^N {\beta b' / 2 - S_j \over \cos^2 \phi_j} \nonumber \\ & & + \beta \sum_{1 \le j < k \le N} {\beta / 2 - M_{jk} \over \sin^2( \phi_j - \phi_k)} + {\beta / 2 - S_j S_k M_{jk} \over \sin^2(\phi_j + \phi_k)} \label{HJE} \end{eqnarray} respectively, where the operator $S_j$ replaces the coordinate $x_j$ ($\phi_j$) by $-x_j$ $(-\phi_j)$ (the superscripts $(H)$, $(L)$ and $(J)$ stand for Hermite, Laguerre and Jacobi due to the relationship with these classical polynomials in the case $N=1$ \cite{forr96a}). The symmetric ground state eigenfunctions of (\ref{HHE}) and (\ref{HLE}) are of the form $e^{-\beta W}$ with $W$ given by \begin{equation} W^{(H)} := {1 \over 2} \sum_{j=1}^N x_j^2 - \sum_{1 \le j < k \le N} \log |x_k - x_j| \label{WH} \end{equation} \begin{equation} W^{(L)} := {1 \over 2} \sum_{j=1}^N x_j^2 -{a'\over 2} \sum_{j=1}^N \log x^2 - \sum_{1 \le j < k \le N} \log |x_k^2 - x_j^2|, \label{WL} \end{equation} \begin{equation} W^{(J)} := -{a'\over 2} \sum_{j=1}^N \log \sin^2 \phi_j -{b'\over 2} \sum_{j=1}^N \log \cos^2 \phi_j - \sum_{1 \le j < k \le N} \log | \sin^2 \phi_j - \sin^2 \phi_k|. \label{WJ} \end{equation} Also, the transformation analogous to (\ref{HCC}) gives \begin{eqnarray}\tilde{H}^{(H,Ex)} & := & - {2 \over \beta} e^{\beta W^{(H)}/2} (H^{(H,Ex)} - E_0^{(H)}) e^{-\beta W^{(H)}/2} \nonumber \\ & = & \sum_{j=1}^N \Big ( {\partial^2 \over \partial y_j^2 } - 2 y_j {\partial \over \partial y_j } \Big ) + {2 \over \alpha} \sum_{j < k} {1 \over y_j - y_k} \left[ \Big ( {\partial \over \partial y_j } - {\partial \over \partial y_k } \Big ) - {1 - M_{jk} \over y_j - y_k} \right] \label{HTHE} \end{eqnarray} \begin{eqnarray}\tilde{H}^{(L,Ex)} & := & - {1 \over 2 \beta} e^{\beta W^{(L)}/2} (H^{(L,Ex)} - E_0^{(L)}) e^{-\beta W^{(L)}/2} \nonumber \\ & = & \sum_{j=1}^N \Big ( y_j {\partial^2 \over \partial y_j^2 } + (a+1 - y_j) {\partial \over \partial y_j } \nonumber \\ && + {1 \over \alpha} \sum_{j < k} {1 \over y_j - y_k} \left[ 2 \Big ( y_j {\partial \over \partial y_j } - y_k {\partial \over \partial y_k } \Big ) - {y_j + y_k \over y_j - y_k} (1 -M_{jk}) \right] \label{HTLE} \end{eqnarray} \begin{eqnarray}\tilde{H}^{(J,Ex)} & := & -{1 \over 4} e^{\beta W^{(J)}/2} (H^{(J,Ex)} - E_0^{(J)}) e^{-\beta W^{(L)}/2} \nonumber \\ & = & \sum_{j=1}^N \Big ( z_j {\partial \over \partial z_j} \Big )^2 + \Big ( (a + 1/2) {z_j + 1 \over z_j - 1} + (b + 1/2) {z_j - 1 \over z_j + 1} +{2(N-1) \over \alpha} \Big ) z_j {\partial \over \partial z_j} \nonumber \\ && +{2 \over \alpha} \sum_{1 \le j < k \le N}{z_j z_k \over z_j -z_k} \left[\Big ({\partial \over \partial z_j} -{\partial \over \partial z_k} \Big ) - {1 - M_{jk} \over z_j - z_k} \right] \nonumber \\ && + {2 \over \alpha} \sum_{1 \le j < k \le N} {1 \over z_jz_k - 1} \left[\Big (z_j{\partial \over \partial z_j} - z_k{\partial \over \partial z_k} \Big ) - {z_j z_k (1 - M_{jk}) \over z_j z_k - 1} \right] \label{HTJE} \end{eqnarray} where \begin{equation} a:= (a'\beta - 1) / 2, \qquad b:= (b'\beta - 1) / 2. \end{equation} To obtain (\ref{HTHE}) we have made the change of variables $y_j = \sqrt{\beta / 2}\, x_j$, while to obtain (\ref{HTLE}) we have made the change of variables $y_j = \beta x_j^2 / 2$ and imposed the restriction to eigenfunctions which are even in $x_j$, and to obtain (\ref{HTJE}) we have used the variable $z_j = e^{2i \phi_j}$ and imposed the restriction to eigenfunctions unchanged by the mapping $z_j \mapsto 1/z_j$. Our objective is to initiate a study into properties of the polynomial eigenfunctions of the operators (\ref{HTHE})-(\ref{HTJE}), and to supplement some of the results of \cite{forr96c,forr96b} on the polynomial eigenfunctions of (\ref{HCC}) with a prescribed symmetry (i.e.~eigenfunctions which are either symmetric or anti-symmetric with respect to the interchange of specified variables). In Section 2 we consider (\ref{HCC}). We revise the construction of an eigenoperator for the symmetric polynomial eigenfunctions (the Jack polynomials) which separates the eigenvalues, and how it can be used to establish orthogonality. This construction is then generalized to provide an eigenoperator for the Jack polynomials with prescribed symmetry, which is used to establish orthogonality. In Sections 3 and 4 we introduce non-symmetric generalized Hermite and Laguerre polynomials as eigenfunctions of (\ref{HTHE}) and (\ref{HTLE}) respectively. Exponential operator formulas are given relating these polynomials to the non-symmetric Jack polynomials. New commuting operators are identified which are used to prove the orthogonality of these polynomials. Generalized Hermite and Laguerre polynomials with prescribed symmetry are defined, and the commuting operators are used to define an eigenoperator which separates the eigenvalues and establishes orthogonality. In Section 5 we begin by revising known commuting operators which decompose the operator (\ref{HTJE}). Non-symmetric generalized Jacobi polynomials, and Jacobi polynomials with prescribed symmetry are defined as eigenfunctions of these operators, and orthogonality is established. The final subsection of Section 2-5 is devoted to establishing a formula expressing a particular class of the polynomials with prescribed symmetry in a factored form involving the corresponding symmetric polynomials. \section{Eigenfunctions of $\tilde{H}^{(C,Ex)}$} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} \subsection{Revision} The operator $\tilde{H}^{(C,Ex)}$ allows a factorization in terms of so-called Cherednik operators $\hat{D}_j$ \cite{cher91a}. These operators are given in terms of the Dunkl operator \begin{equation} T_j := {\partial \over \partial z_j} + {1 \over \alpha} \sum_{k=1 \atop k \ne j} {1 \over z_j - z_k} (1 - M_{jk}) \label{DU} \end{equation} for the root system $A_{N-1}$ by \begin{eqnarray} \hat{D}_j & := & z_j T_j + {1 \over \alpha} \sum_{k=1}^{j-1} M_{jk} \nonumber \\ & = & z_j {\partial \over \partial z_j} + {1 \over \alpha} \Big ( \sum_{l < j} {z_l \over z_j - z_l} (1 - M_{lj}) + \sum_{l > j} {z_j \over z_j - z_l} (1 - M_{lj}) \Big ) + { (j-1) \over \alpha}. \label{CO} \end{eqnarray} They mutually commute so that \begin{equation} [\hat{D}_j, \hat{D}_k] = 0. \label{DOC1} \end{equation} This can be checked from the fact that the Dunkl operators commute: \begin{equation} [T_j,T_k]=0. \label{DOC} \end{equation} The non-symmetric Jack polynomials are simultaneous eigenfunctions of the $\hat{D}_j$ for each $j=1,\dots,N$, and the corresponding eigenvalues are \begin{equation} e_{j,\eta} = \eta_j + {1 \over \alpha} \Big ( - \sum_{l < j} h(\eta_l - \eta_j) + \sum_{l > j} h(\eta_j - \eta_l) \Big ) + {(j-1) \over \alpha} \label{EV} \end{equation} with \begin{equation} h(x) = \bigg \{ \begin{array}{l} 1,\qquad x>0 \\ 0, \qquad {\rm otherwise.} \end{array} \label{HX} \end{equation} We remark that some authors \cite{sahi96a,knop96c} define non-symmetric Jack polynomials as eigenfunctions of a variant of the Cherednik operators (\ref{CO}): \begin{eqnarray} \xi_j & := & \alpha y_j T_j + \sum_{k=j+1}^N M_{jk} + (1-N) \nonumber \\ & = & \alpha \left[ y_j {\partial \over \partial y_j} + {1 \over \alpha} \Big ( \sum_{l < j} {y_j \over y_j - y_l} (1 - M_{lj}) + \sum_{l > j} {y_l \over y_j - y_l} (1 - M_{lj}) \Big ) - { (j-1) \over \alpha} \right]. \label{COV} \end{eqnarray} (the choice of notation $y_1,\dots,y_N$ for the coordinates here is for later convenience). The operators $\xi_j$ have the same polynomial eigenfunctions as $\hat{D}_j$ except that $z_j$ is replaced by $y_{N+1 - j}$ $(j=1,\dots,N)$ and $\eta_j$ is replaced by $\eta_{N+1-j}$. Following \cite{knop96c,sahi96a} the corresponding eigenvalues are to be denoted $\bar{\eta}_j$ and are explicitly given by \begin{equation} \bar{\eta}_j = \alpha \eta_j - \Big ( \sum_{l < j} h(\eta_l + 1 - \eta_j) + \sum_{l > j} h(\eta_l - \eta_j) \Big ). \label{EVCV} \end{equation} Returning to the relationship to $\tilde{H}^{(C,Ex)}$, a direct calculation shows \cite{bern93a} \begin{equation} \tilde{H}^{(C,Ex)} = \sum_{j=1}^N \left( \hat{D}_j - {N-1 \over 2 \alpha} \right)^2 - \Big ( {L \over 2 \pi} \Big )^2 E_0^{(C)}. \label{FAC} \end{equation} Let us revise \cite{bern93a} how this decomposition can be used to prove that the symmetric polynomial eigenfunctions of (\ref{HCC}) (i.e.~the symmetric Jack polynomials) are orthogonal with respect to the inner product \begin{equation} \langle f | g \rangle^{(C)} := \int_0^L dx_1 \dots \int_0^L dx_N \, \prod_{1 \le j < k \le N} |z_k - z_j|^{2 /\alpha} f(z_1^*,\dots,z_N^*) g(z_1,\dots,z_N), \label{IC} \end{equation} where $ z_j = e^{2 \pi i x_j /L}$ and $^*$ denotes complex conjugate. First we check directly from the definitions (\ref{CO}) and (\ref{IC}) that \begin{equation} \langle f | \hat{D}_j g \rangle^{(C)} = \langle \hat{D}_j f | g \rangle^{(C)} \label{SA} \end{equation} which says that the Cherednik operators are self-adjoint with respect to the inner product (\ref{IC}). In performing this check we use the facts that $$ \prod_{1 \le j < k \le N} |z_k - z_j|^{2 /\alpha} = \psi_0^* \psi_0, \qquad {\rm where} \quad \psi_0 = \prod_{j=1}^N z_j^{-(N-1)/\alpha} \prod_{1 \le j < k \le N} (z_k - z_j)^{1/\alpha} $$ and $$ \psi_0 \hat{D}_j \psi_0^{-1} = z_j {\partial \over \partial z_j} - {1 \over \alpha} \left( \sum_{l < j} {z_l \over z_j - z_l} M_{lj} + \sum_{l > j} {z_j \over z_j - z_l} M_{lj} \right) + { (N-1) \over 2\alpha}. $$ Note that $|\psi_0|^2 = e^{-\beta W^{(C)}}$, where $W^{(C)}$ is given by (\ref{WC}), and is thus the square of the ground state wave function for (\ref{HC}). Next, we compare the eigenvalues $\{ e_{j,\eta} \}$ to $\{ e_{j,\eta'} \}$, where $\eta'$ is obtained from the $N$-tuple $\eta$ by interchanging $\eta_i$ and $\eta_{i'}$. \vspace{.2cm} \noindent {\bf Lemma 2.1} \quad We have $$ e_{i,\eta'}= e_{i',\eta}, \quad e_{i',\eta'}= e_{i,\eta} \quad {\rm and} \quad e_{j,\eta'}= e_{j,\eta}, \quad (j \ne i',i). $$ \vspace{.2cm} \noindent {\bf Proof} \quad These equations are verified directly from (\ref{EV}). \vspace{.2cm} \noindent {\bf Remark} \quad The result analogous to Lemma 2.1 applies for the eigenvalues (\ref{EVCV}). \vspace{.2cm} {}From Lemma 2.1 we see that $\{ e_{j,\eta} \}_{j=1,\dots,N}$ with $\eta = P^{-1} \kappa$ is independent of the permutation $P$. Choosing the permutation $P(j) = N + 1 -j$ $(j=1,\dots,N)$ shows that $\{ e_{j,\eta} \}_{j=1,\dots,N} = \{ \kappa_{N+1-j} + (j-1)/\alpha \}_{j=1,\dots,N}$. This allows an eigenoperator of the $E_{P^{-1}\kappa}$ to be constructed for which the eigenvalues are independent of $P$: \begin{eqnarray}\lefteqn{ \Big ( 1 + u(\hat{D}_1 - (N-1)/2\alpha) \Big ) \dots \Big ( 1 + u(\hat{D}_N - (N-1)/2\alpha) \Big ) E_{P^{-1}\kappa}}\nonumber \\ && \hspace*{2cm} = \prod_{j=1}^N \Big ( 1 + u(\kappa_j + (N+1 - 2j)/2\alpha) \Big ) E_{P^{-1}\kappa} \label{EVC} \end{eqnarray} (the constants $ - (N-1)/2\alpha$ are not essential and could have been omitted). Note that \begin{equation} \kappa_1 -{1 \over \alpha} > \kappa_2 - {2 \over \alpha} > \dots > \kappa_N - {N \over \alpha} \label{OR} \end{equation} so the eigenvalues for different partitions $\kappa$ are distinct. Consider now the symmetric Jack polynomials $J_\kappa^{(\alpha)}$. They can be characterized (up to normalization, which for definiteness we will take to be that adopted by Stanley \cite{stan89a}) as the polynomial eigenfunctions of (\ref{HCC}) with leading term $m_\kappa$. From the fact that the non-symmetric Jack polynomials $E_{P^{-1}\kappa}$ are simultaneous eigenfunctions of all the $\hat{D}_j$ with leading term $z^{P^{-1}\kappa}$ and the triangular structure (\ref{nj}) we must have \begin{equation} J_\kappa^{(\alpha)}(z) = \sum_{P} a_{P^{-1}\kappa} E_{P^{-1}\kappa}(z,\alpha) \label{JNSJ} \end{equation} for some coefficients $a_{P^{-1}\kappa}$ (these coefficients are given explicitly in \cite{sahi96a}). It follows that the symmetric Jack polynomial satisfies the eigenvalue equation (\ref{EVC}). Since by (\ref{SA}) the operator in (\ref{EVC}) is self-adjoint with respect to the inner product (\ref{IC}) and by (\ref{OR}) the eigenvalues are distinct, this implies that the symmetric Jack polynomials are orthogonal with respect to (\ref{IC}). With respect to the eigenvalue equation (\ref{EVC}) with $E_{P^{-1}\kappa}$ replaced by $J_\kappa^{(\alpha)}$, we remark that Macdonald \cite{mac} has constructed an operator $D_N(X;\alpha)$ such that \begin{equation} D_N(X;\alpha) \,J_\kappa^{(\alpha)} = \prod_{i=1}^N (X + N - i + \alpha \kappa_i) \, J_\kappa^{(\alpha)}. \end{equation} Since $\{ J_\kappa^{(\alpha)} \}$ is a basis for symmetric functions, it follows by comparison with (\ref{EVC}) that when acting on symmetric functions \cite{noumi96a,noumi96b} \begin{equation} \prod_{j=1}^N (X + \alpha \hat{D}_j) = D_N(X;\alpha). \end{equation} Another way of establishing the orthogonality of the symmetric Jack polynomials is to use the expansion (2.14) together with the fact that the non-symmetric Jack polynomials form an orthogonal set with respect to (\ref{IC}). This later fact can be established by first noting that \begin{equation} \prod_{l=1}^N(1 + u_l \hat{D}{_l}) \label{PONS} \end{equation} is an eigenoperator of each $E_\eta$ which separates the eigenvalues. The result now follows after using the fact that (\ref{PONS}) is self-adjoint. \subsection{Jack polynomials with prescribed symmetry} As noted above, the fact that the non-symmetric Jack polynomials $E_\eta$ are simultaneous eigenfunctions of $\hat{D}_1, \dots,\hat{D}_N$ implies that the $E_\eta$ are eigenfunctions of (\ref{HCC}). Since (\ref{HCC}) is symmetric in $z_1,\dots,z_N$ it follows that $E_\eta$ with the variables $z_1,\dots,z_N$ permuted is also an eigenfunction of (\ref{HCC}) with the same eigenvalue. Thus, for any permutation $P$, since the leading order term of $J_\kappa^{(\alpha)}(z)$ is proportional to the monomial symmetric function (i.e.~the symmetrization of $z^{P^{-1}\kappa}$) we must have \begin{equation} J_\kappa^{(\alpha)}(z) = A_{P^{-1} \kappa}\,{\rm Sym} \, \Big ( E_{P^{-1}\kappa}(z,\alpha) \Big ). \label{VK} \end{equation} For the case $P^{-1}\kappa=\kappa$, ${\rm Sym} \, \Big ( E_{P^{-1}\kappa}(z,\alpha) \Big )$ has leading term $m_{\kappa}$, so $A_{P^{-1} \kappa} = v_{\kappa \kappa}$ where $ v_{\kappa \kappa}$ is defined and given explicitly in ref.~\cite{stan89a}. Eigenfunctions can be constructed in an analogous way which are symmetric with respect to the interchange of certain sets of variables and antisymmetric with respect to the interchange of other sets of variables. We will refer to such polynomials as having a prescribed symmetry. To facilitate a discussion of this situation, let us rewrite the coordinates $\{z_j \}_{j=1,\dots,N}$ as $$ \Big (\bigcup_{\alpha = 1}^q \{ w_j^{(\alpha)} \}_{j=1,\dots,N_\alpha^{(w)}} \Big ) \Big (\bigcup_{\gamma = 1}^p \{ z_j^{(\gamma)} \}_{j=1,\dots,N_\gamma^{(z)}} \Big ) $$ taken in order so that $w_1^{(1)} = z_1, \dots, z_{N_p}^{(p)} = z_N$ and $ N = \sum_{\mu = 1}^q N_\mu^{(w)} + \sum_{\gamma=1}^p N_\gamma^{(z)}. $ We seek polynomial eigenfunctions of (\ref{HCC}), $S_{P^{-1}\kappa}(z,\alpha)$ say, which are symmetric in $ \{ w_j^{(\mu)} \}_{j=1,\dots,N_\mu^{(w)}}$ and antisymmetric in $\{ z_j^{(\gamma)} \}_{j=1,\dots,N_\gamma^{(z)}}$. We have \begin{equation} S_{P^{-1}\kappa}(z,\alpha) = {\cal O} \Big ( E_{P^{-1}\kappa}(z,\alpha) \Big ) \label{DEFS} \end{equation} where ${\cal O}$ denotes the operation of symmetrization in $\{ w_j^{(1)} \}_{j=1,\dots,N_1^{(w)}}$, antisymmetrization in $\{ z_j^{(\gamma)} \}_{j=1,\dots,N_\gamma^{(z)}}$ and normalization such that the coefficient of $z^{P^{-1}\kappa}$ is unity. Due to the operation ${\cal O }$ the label $P^{-1}\kappa$ in $S_{P^{-1}\kappa}$ can be replaced by $q+p$ partitions $(\rho, \mu) := (\rho^{(1)}, \dots \rho^{(q)}, \mu^{(1)}, \dots, \mu^{(p)}$ where $\rho^{(\mu)}$ consists of $N_\alpha^{(w)}$ parts $(\alpha =1,\dots,q)$ and $\mu^{(\gamma)}$ consists of $N_\gamma^{(z)}$ parts. For the $N$-tuple $\eta = P^{-1}\kappa$ any rearrangements of \begin{equation} \{ \eta_j \}_{j=1,\dots,N_1^{(w)}}, \, \{ \eta_{N_1^{(w)}+j} \}_{j=1,\dots,N_2^{(w)}}, \dots, \{ \eta_{N^{(w)}+\sum_{\gamma=1}^{p-1}N_\gamma^{(z)}+j} \}_{j=1,\dots,N_p^{(z)}}, \label{REA} \end{equation} where $N^{(w)} := \sum_{\mu = 1}^q N_\mu^{(w)}$, give the same partitions $(\rho, \mu)$ and thus the same polynomial with prescribed symmetry. A feature of the polynomials $E_\eta(z,\alpha)$ is that if $\eta_i = \eta_{i+1}$ then $E_\eta$ is symmetric in $z_i$ and $z_{i+1}$ (see (\ref{swap}) below). It follows that if two parts of any $\mu^{(\gamma)}$ are equal the polynomial $S_{(\rho, \mu)}$ vanishes identically due to the antisymmetrization procedure in its construction. Thus each $\mu^{(\gamma)}$ must be restricted to distinct parts. Now we know from \cite[Lemma 2.4 and Proposition 4.3]{knop96c} that with $s_i := M_{i \, i+1}$ and $\delta_i := \bar{\eta}_i - \bar{\eta}_{i+1}$, \begin{equation} s_i E_\eta = \left \{ \begin{array}{ll} {1 \over \delta_i} E_\eta + (1 - {1 \over \delta_i^2}) E_{s_i \eta}, &\quad \eta_i > \eta_{i+1} \\ E_\eta, &\quad \eta_i = \eta_{i+1} \\ {1 \over \delta_i} E_\eta + E_{s_i \eta}, &\quad \eta_i < \eta_{i+1} \end{array} \right. \label{swap} \end{equation} (here $E_\eta$ refers to the eigenfunctions of (\ref{COV}), which as noted below (\ref{COV}) are related to the non-symmetric Jacks defined as eigenfunctions of (\ref{CO}) by relabelling). Also, each permutation can be written as a product of the elementary transpositions $s_i$. Therefore, we conclude that (\ref{DEFS}) can be rewritten as \begin{equation} S_{P^{-1}\kappa}(z,\alpha) = \sum_{\rm rearrangements} b_{Q^{-1}\kappa} E_{Q^{-1}\kappa}(z,\alpha) \label{SSUM} \end{equation} where the sum is over rearrangements $Q^{-1}\kappa$ of $P^{-1}\kappa$ obtained by permuting within the sets (\ref{REA}). Now two distinct sequences of partitions $(\rho, \mu)$ and $(\hat{\rho}, \hat{\mu})$ as defined below (\ref{DEFS}) cannot have any rearrangements of (\ref{REA}) in common, as they wouldn't then be distinct. Hence the expansion (\ref{SSUM}) for $S_{(\rho, \mu)}$ and $S_{(\hat{\rho}, \hat{\mu})}$ does not contain any common $E_\eta$. It follows immediately from the orthogonality of $\{E_\eta \}$ with respect to (\ref{IC}) that $\{S_{(\rho, \mu)} \}$ are also orthogonal with respect to (\ref{IC}). An alternative way to deduce the orthogonality is to note that the operator \begin{equation} \prod_{\mu = 1}^q \prod_{j=1}^{N_\mu^{(w)}} \Big ( 1 + u_\mu \hat{D}_{\sum_{l=1}^{\mu - 1} N_l^{(w)} + j} \Big ) \prod_{\gamma = 1}^p \prod_{j=1}^{N_\gamma^{(z)}} \Big ( 1 + v_\gamma \hat{D}_{N^{(w)}+\sum_{l=1}^{\gamma - 1} N_l^{(z)} + j} \Big ) \label{PCO} \end{equation} is an eigenoperator of $\{ S_{(\rho, \mu)} \}$. To see this, note that this is an eigenoperator of the non-symmetric Jacks, and from Lemma 2.1 the corresponding eigenvalue is independent of the particular rearrangements (\ref{REA}). Furthermore, the eigenvalues of (\ref{PCO}) corresponding to $S_{(\rho, \mu)}(z,\alpha)$ and $S_{(\hat{\rho}, \hat{\mu)}}(z,\alpha)$ are distinct whenever $\rho \mu$ and $\hat{\rho} \hat{\mu}$ are distinct. Since, by (\ref{SA}), (\ref{PCO}) is self-adjoint with respect to (\ref{IC}), the fact that the eigenvalues are distinct implies orthogonality of these functions with respect to the inner product (\ref{IC}). In the case $q=0$, $p=2$, $|\mu^{(1)}| + |\mu^{(2)}| =1,2$ or 3, explicit formulas for $\tilde{S}_{\mu^{(1)}\mu^{(2)}}(z,\alpha)$ have been given in ref.~\cite{yamam96b}, where the $\tilde{S}$ are eigenfunctions of (\ref{HCC}) which are symmetric in $\{w_j^{(1)}\}$ and $\{w_j^{(2)}\}$. However, in general $\tilde{S}$ does not correspond to $S$ as $\tilde{S}$ does not satisfy (\ref{DEFS}), and the $\tilde{S}$ are not orthogonal. \subsection{Some special Jack polynomials with prescribed symmetry} In some previous works \cite{forr96c,forr96b} we have conjectured a formula for certain Jack polynomials with prescribed symmetry in terms of difference products and the symmetric Jack polynomial. In the present notation the conjecture in \cite{forr96c,forr96b} applies to $S_{(\rho,\kappa)}(z,\alpha)$ with $q=1$ and \begin{equation} (\rho,\kappa) = (\rho_1, \rho_2, \dots, \rho_{N_0}, N_1 - 1,N_1-2, \dots, 1,N_2-1, N_2-2, \dots ,1,N_p-1,N_p-2, \dots, 1) \label{PK} \end{equation} (here we have written $N_1^{(w)} =: N_0$, $N_l^{(z)} =: N_l$ to be consistent with refs.~\cite{forr96c,forr96b} and it is assumed $\rho_1 \ge \rho_2 \ge \cdots \ge \rho_{N_1} \ge 0$). The conjecture states that for $(\rho,\mu)$ given by (\ref{PK}) \begin{equation} S_{(\rho,\mu)}(z,\alpha) = A_{(\rho,\mu)} \prod_{\gamma=1}^p \prod_{1 \le j < k \le N_\gamma} (z_k^{(\gamma)} - z_j^{(\gamma)}) J_{\rho}^{(p + \alpha)}(w_1,\dots,w_{N_0}), \label{CON} \end{equation} where $ A_{\rho\mu}$ is some normalization, provided \begin{equation} \rho_1 \le {\rm min} (N_1,\dots,N_p) \label{IN} \end{equation} (A stronger conjecture was also given in \cite{forr96c,forr96b} which replaces $\rho_1$ in (\ref{PK}) by $\rho_1-1$, however we do not consider that extension here.) This conjecture can be verified directly by showing that the r.h.s.~of (\ref{CON}) is an eigenfunction of (\ref{HCC}) (an abbreviated version of the required calculation was given in \cite{forr96c}, however the working there is incomplete and an erroneous conclusion was drawn). We begin by rewriting the variables $z_1,\dots,z_N$ in (\ref{HCC}) as $\{w_j\}_{j=1,\dots,N_0}$ and $\{z_j^{(\gamma)} \}_{j=1,\dots,N_\gamma}$ $(\gamma = 1,\dots, p)$. In terms of these variables, when acting on functions symmetric in $\{w_j\}$ and anti-symmetric in $\{z_j^{(\gamma)}\}$ we have \begin{equation} \tilde{H}^{(C,Ex)} = \tilde{H}^{(C,w)} + \tilde{H}^{(C,z)} + \tilde{H}^{(C,wz)} \label{DEC} \end{equation} where \begin{equation} \tilde{H}^{(C,w)} = \sum_{j=1}^{N_0} \Big (w_j {\partial \over \partial w_j} \Big )^2 + { N-1 \over \alpha} \sum_{j=1}^{N_0} w_j {\partial \over \partial w_j} + {2 \over \alpha} \sum_{ j \ne k }{w_j w_k \over w_j - w_k}{\partial \over \partial w_j} \end{equation} \begin{eqnarray} \tilde{H}^{(C,z)} & = & \sum_{\gamma=1}^p \sum_{j=1}^{N_\gamma} \Big (z_j^{(\gamma)} {\partial \over \partial z_j^{(\gamma)} } \Big )^2 + { N-1 \over \alpha} \sum_{\gamma=1}^p \sum_{j=1}^{N_\gamma} z_j^{(\gamma)} {\partial \over \partial z_j^{(\gamma)}} \nonumber \\ && + {2 \over \alpha} \sum_{\gamma < \nu} \sum_{j=1}^{N_\gamma} \sum_{k=1}^{N_\nu} {z_j^{(\gamma)} z_k^{(\nu)} \over z_j^{(\gamma)} - z_k^{(\nu)}} \left[\Big ({\partial \over \partial z_j^{(\gamma)}} -{\partial \over \partial z_k^{(\nu)}} \Big ) - {1 - M(z_j^{(\gamma)},z_k^{(\nu)}) \over z_j^{(\gamma)} - z_k^{(\nu)}} \right] \nonumber \\ && + {2 \over \alpha} \sum_{\gamma=1}^p \sum_{j<k} {z_j^{(\gamma)} z_k^{(\gamma)} \over z_j^{(\gamma)} - z_k^{(\gamma)}} \left[\Big ({\partial \over \partial z_j^{(\gamma)}} - {\partial \over \partial z_k^{(\gamma)}} \Big ) - {2 \over z_j^{(\gamma)} - z_k^{(\gamma)}} \right] \end{eqnarray} \begin{eqnarray} \tilde{H}^{(C,wz)} = {2 \over \alpha} \sum_{\gamma = 1}^p \sum_{j=1}^{N_\gamma} \sum_{k=1}^{N_0} {z_j^{(\gamma)} w_k \over z_j^{(\gamma)} - w_k} \Big [\Big ({\partial \over \partial z_j^{(\gamma)}} -{\partial \over \partial w_k}\Big ) - {1 - M(z_j^{(\gamma)},w_k) \over z_j^{(\gamma)} - w_k} \Big ] \label{DEFHCWZ} \end{eqnarray} Here we have used the notation $M(x,y)$ to denote the operator which exchanges the coordinates $x$ and $y$. We seek the action of these operators on \begin{equation} \prod_{\gamma = 1}^p \Delta (z^{(\gamma)}) \prod_{l=1}^{N_0} w_l^{\kappa_l}, \qquad \Delta (z^{(\gamma)}) := \prod_{1 \le j < k \le N_\gamma}( z_k^{(\gamma)} - z_j^{(\gamma)}). \label{DEL} \end{equation} For this purpose we require the following result \cite{kk95}, which can be verified directly. \vspace{.2cm} \noindent {\bf Lemma 2.2} \quad Let $$ A(y_j,y_k) := {y_j y_k \over y_j - y_k} \left[\Big ({\partial \over \partial y_j} -{\partial \over \partial y_k}\Big ) - {1 - M(y_j,y_k) \over y_j - y_k} \right] $$ For $\lambda_j \ge \lambda_k$ we have \begin{equation} A(y_j,y_k) \, y_j^{\lambda_j} y_k^{\lambda_k} = -\lambda_k y_j^{\lambda_j} y_k^{\lambda_k} + \left \{ \begin{array}{l}\displaystyle{ \sum_{l=1}^{\lambda_j - \lambda_k-1}} (\lambda_j - \lambda_k - l) y_j^{\lambda_j-l} y_k^{\lambda_k+l}, \quad \lambda_j - \lambda_k \ge 2 \\ 0, \quad {\rm otherwise.} \end{array} \right. \label{AA} \end{equation} \vspace{.2cm} This lemma will first be used to determine the action of $H^{(C,z)}$ on (\ref{DEL}). \vspace{.2cm} \noindent {\bf Lemma 2.3} \quad Let $P_\gamma$ be a permutation of $\{1,2,\dots,N_\gamma\}$, and let \begin{equation} \label{PHI} \Phi(z) := \prod_{\gamma=1}^p \prod_{j=1}^{N_\gamma} (z_j^{(\gamma)})^{P_\gamma(l) - 1}. \end{equation} We have \begin{equation} \tilde{H}^{(C,z)} \Phi(z) = \delta^{(C)} \Phi(z) + \Omega (z). \label{L1} \end{equation} where $\delta^{(C)}$ is independent of the permutations $P_\gamma$ and $\Omega(z)$ is a polynomial such that the exponent of each monomial has at least one repeated part. Hence \begin{equation} \tilde{H}^{(z)}\, \prod_{\gamma=1}^p \Delta (z^{(\gamma)}) = \delta^{(C)} \prod_{\gamma=1}^p \Delta (z^{(\gamma)}). \label{L2} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad The fact that $\delta^{(C)}$ is independent of the permutations $P_\gamma$ follows from the eigenvalue (\ref{KEV}) being independent of the permutation, while each monomial having at least one repeated part is a consequence of the fact that the exponents in $\Phi(z)$ for each set of variables $\{z_j\}_{j=1,\dots,N_\gamma}$ consists of the consecutive integers $0,1,\dots,N_\gamma-1$, and the action of the operator $A(z_j^{(\gamma)},z_k^{(\mu)})$ noted in Lemma 2.2. Anti-symmetrizing both sides of (\ref{L1}) in the variables $\{z_j\}_{j=1,\dots,N_\gamma}$ ($\gamma = 1,\dots,p$) the polynomial $\Omega (z)$ therefore gives zero contribution, and the result (\ref{L2}) follows from the Vandermonde determinant formula \begin{equation} \sum_{P=1}^{N!} \epsilon(P) \prod_{l=1}^N z_l^{P(l) - 1} = \Delta (z), \end{equation} where $\epsilon(P)$ denotes the partity of the permutation $P$. \vspace{.2cm} The crucial point in establishing (\ref{CON}) is the action of $H^{(C,wz)}$ on (\ref{DEL}). \vspace{.2cm} \noindent {\bf Lemma 2.4} \quad Let $F(w,z) = m_\kappa (w) \prod_{\gamma=1}^p \Delta (z^{(\gamma)})$, where $\kappa$ is a partition consisting of $N_0$ parts with the largest part $\kappa_1$ restricted by $\kappa_1 \le {\rm min}(N_1,\dots,N_\gamma)$ and $ m_\kappa (w)$ denotes the monomial symmetric function with exponent $\kappa$. We have \begin{equation} \tilde{H}^{(C,wz)}F(w,z) = \left({p \over \alpha} \sum_{j=1}^{N_0} \Big ( w_j {\partial \over \partial w_j} \Big )^2 - {2 \over \alpha} (N^{(z)} - p/2) \sum_{j=1}^{N_0} w_j {\partial \over \partial w_j} \right) F(w,z). \label{L3} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad Consider first the action of $A(z_j^{(\gamma)},w_k)$ on $\Delta (z^{(\gamma)}) w_k^\lambda$. Expanding $ \Delta (z^{(\gamma)})$ into terms of the form (\ref{PHI}) we see from the argument of the proof of Lemma 2.3 that for $0 \le \lambda \le N_{\gamma}$ only the first term on the r.h.s. of (\ref{AA}) for the action of $A(z_j^{(\gamma)},w_k)$ on $z_j^{(\gamma)\lambda_j} w_k^\lambda$ contributes, and thus $$ \sum_{j=1}^{N_\gamma} A(z_j^{(\gamma)},w_k) \Delta(z^{(\gamma)}) w_k^\lambda = \Big ( - \sum_{j=1}^{N_\gamma} {\rm min} (\lambda, N_\gamma - j) \Big ) \Delta(z^{(\gamma)}) w_k^\lambda. $$ But for $0 \le \lambda \le N_\gamma$ a straightforward calculation gives $$ - \sum_{j=1}^{N_\gamma} {\rm min} (\lambda, N_\gamma - j) = {1 \over 2} \lambda^2 - (N_\gamma - {1 \over 2}) \lambda. $$ Thus $$ \sum_{k=1}^{N_0} \sum_{j=1}^{N_\gamma} A(z_j^{(\gamma)},w_k) \Delta(z^{(\gamma)}) w_1^{\kappa_{Q(1)}} \dots w_{N_0}^{\kappa_{Q(N_0)}} = \Big ( {1 \over 2} |\kappa^2| - (N_\gamma - {1 \over 2})|\kappa| \Big ) w_1^{\kappa_{Q(1)}} \dots w_{N_0}^{\kappa_{Q(N_0)}}, $$ where $|\kappa| := \sum_{j=1}^{N_0} \kappa_j$, $|\kappa^2| := \sum_{j=1}^{N_0} \kappa_j^2$, independent of the permutation $Q$. Summing over $\gamma$ and comparison with the definition of $\tilde{H}^{(C,wz)}$ shows that $$ \tilde{H}^{(C,wz)} F(w,z) = {2 \over \alpha} \Big ( {p \over 2} |\kappa^2| - (N^{(z)} - {p \over 2} |\kappa| \Big ) F(w,z). $$ This equation remains valid with $H^{(C,wz)}$ replaced by the operator on the r.h.s.~of (\ref{L3}), thus verifying the validity of (\ref{L3}). \vspace{.2cm} Substituting the results of Lemmas 2.3 and 2.4 in (\ref{DEC}), assuming the inequality in Lemma 2.4, we have \begin{eqnarray}\lefteqn{ \tilde{H}^{(C,Ex)} \prod_{\gamma=1}^p \Delta (z^{(\gamma)}) \, m_\kappa (w) = } \nonumber \\ & & \bigg ( \delta^{(C)} + {N - 1 -2N^{(z)} + p \over \alpha} \sum_{j=1}^{N_0} w_j {\partial \over \partial w_j} + \Big ( 1 + {p \over \alpha} \Big ) \bigg [ \sum_{j=1}^{N_0} \Big ( w_j {\partial \over \partial w_j} \Big )^2 \nonumber \\ & & + {2 \over \alpha + p} \sum_{j \ne k} {w_jw_k \over w_j - w_k} {\partial \over \partial w_j} \bigg ]\bigg ) \, \Delta (z^{(\gamma)}) \, m_\kappa (w) \label{WW} \end{eqnarray} The first two terms on the r.h.s.~of (\ref{WW}) are eigenoperators of any homogeneous polynomial in $w$, while the terms in the square brackets form the eigenoperator defining the symmetric Jack polynomial $J_\rho^{(p+ \alpha)}(w)$ (recall (\ref{HCC}) with $M_{jk}=1$). Thus by forming an appropriate linear combination of $m_\kappa(w)$ in (\ref{WW}) (with $|\kappa| = |\rho|$ and ${\rm min}(N_1,\dots,N_p) \ge \rho_1 \ge \kappa_1$) we see that indeed (\ref{CON}) is an eigenfunction of (\ref{HCC}), as required. We remark that the above derivation shows that if we replace $J_\rho^{(p+ \alpha)}(w)$ in (\ref{CON}) by $E_\rho(w,\alpha+p)$, then the resulting function is also an eigenfunction of (\ref{HCC}). This is consistent with the construction (\ref{DEFS}) of $S_{(\rho,\mu)}$. In fact this latter eigenfunction must result from antisymmetrizing $E_{(\rho,\mu)}$, with $(\rho,\mu)$ defined by (\ref{PK}), in the variables $\{z_j^{(\gamma)}\}_{ j=1,\dots,N_\gamma}$. Thus, with this operation defined by ${\cal A}$ and assuming the inequality (\ref{IN}), we see from the structure (\ref{nj}) and the fact that antisymmetrization of a monomial with equal exponents vanishes that \begin{eqnarray} {\cal A} E_{(\rho,\mu)} & = & w^\rho {\cal A} z^{\mu} + \sum_{\nu < \rho} c_{\nu \rho} w^{\nu} {\cal A} z^{\mu} \nonumber \\ & = & \prod_{\gamma = 1}^p \Delta(z^{(\gamma)}) \Big ( w^\rho + \sum_{\nu < \rho} c_{\nu \rho} w^{\nu} \Big ). \label{UNI} \end{eqnarray} for some constants $c_{\nu \rho}$. But ${\cal A} E_{(\rho,\mu)}$ must be an eigenfunction of (\ref{HCC}), and the above working gives that the function of $w$ must satisfy a eigenvalue equation in which the eigenoperator is again of the form (\ref{HCC}), which we know has a unique solution of the form required in (\ref{UNI}). In the case $q=0$, $p=1$ we can also provide a formula for $S_{(\rho,\mu)} =: S_\mu$ in terms of the symmetric Jack polynomial: \begin{equation} S_\mu (z,\alpha) = \Delta(z) {1 \over v_{\kappa \kappa}(\alpha/(1+\alpha))} J_\kappa^{(\alpha/(1+\alpha))} (z) \label{ANTIS} \end{equation} where \begin{equation} \kappa := (\mu_1 - N+1, \mu_2 - N+2, \dots, \mu_N) \end{equation} and $v_{\kappa \kappa}$ is as in (\ref{VK}). Note from (\ref{DEFS}) that (\ref{ANTIS}) is equivalent to the statement that \begin{equation} {\cal A} \, E_\mu(z,\alpha) = \Delta(z) {1 \over v_{\kappa \kappa}(\alpha/(1+\alpha))} J_\kappa^{(\alpha/(1+\alpha))} (z), \end{equation} where ${\cal A}$ denotes antisymmetrization in all variables. The easiest way to verify (\ref{ANTIS}) is to try for eigenfunctions of (\ref{HCE}) of the form $|\Delta(z)|^{1/\alpha} \Delta(z) f$ where $f$ is symmetric. A straightforward calculation shows that $f$ must be an eigenfunction of (\ref{HCC}) with $M_{jk}=1$ and $2/\alpha$ replaced by $2/\alpha + 1$. But the unique symmetric eigenfunction of this equation with leading term $m_\kappa$ is $ J_\kappa^{(\alpha/(1+\alpha))}/v_{\kappa \kappa}(\alpha/(1+\alpha))$. The relationship between $\kappa$ and $\mu$, and thus the result follows from (\ref{DEFS}). \section{Eigenfunctions of $\tilde{H}^{(H,Ex)}$} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} \subsection{The non-symmetric generalized Hermite polynomials} The operator (\ref{HTHE}) has unique polynomial eigenfunctions of the form \begin{equation} y^\eta + \sum_{|\nu| < |\eta|} c_{\eta \, \nu} y^\nu \label{3.1} \end{equation} with corresponding eigenvalue $-2|\eta|$. By adding together an appropriate linear combination of these eigenfunctions, we can construct the unique eigenfunction of the form \begin{equation} E_\eta^{(H)}(y,\alpha) := E_\eta(y,\alpha) + \sum_{|\nu| < |\eta|} c_{\eta \, \nu}' E_{\nu}(y;\alpha) \label{3.2} \end{equation} again with eigenvalue $-2|\eta|$. We will refer to the $E_\eta^{(H)}(y,\alpha)$ as the non-symmetric generalized Hermite polynomials (they are related to the symmetric generalized Hermite polynomials defined in \cite{forr96a} by an equation analogous to (\ref{JNSJ}); see eq.~(\ref{A2}) below). In fact by adopting a method due to Sogo \cite{sogo96}, an exponential operator formula can be obtained expressing $E_\eta^{(H)}(y,\alpha)$ in terms of $ E_\eta(y,\alpha)$, which is the analogue of the formula due to Lassalle \cite{lass96a} (see eq.~(\ref{A2}) below) expressing the symmetric generalized Hermite polynomials in terms of the symmetric Jack polynomials. To obtain this formula, we write the eigenvalue equation for the $E_\eta^{(H)}(y,\alpha)$ in the form \begin{equation} ( A + \tilde{D}_0) E_\eta^{(H)}(y,\alpha) = 0 \label{D0} \end{equation} where \begin{equation} A := -2 \sum_{j=1}^N y_j {\partial \over \partial y_j} + 2 |\eta|, \quad \tilde{D}_0 := \sum_{j=1}^N {\partial^2 \over \partial y_j^2 } + {2 \over \alpha} \sum_{j < k} {1 \over y_j - y_k} \left[ \Big ( {\partial \over \partial y_j } - {\partial \over \partial y_k } \Big ) - {1 - M_{jk} \over y_j - y_k} \right] \end{equation} Note that (\ref{D0}) only specifies $ E_\eta^{(H)}$ uniquely after the specification (\ref{3.2}). Since $ E_\eta(y,\alpha)$ is homogeneous of degree $|\eta|$ we also have $A \, E_\eta(y,\alpha) = 0$ which can be equated with (\ref{D0}) and the resulting equation rearranged to give \begin{equation} E_\eta^{(H)}(y,\alpha) = \Big (1 - (A + \tilde{D}_0)^{-1}\tilde{D}_0 \Big ) \, E_\eta(y,\alpha) \label{EX1} \end{equation} Next we make use of the operator identity \begin{equation} (A + \tilde{D}_0)^{-1}\tilde{D}_0 = A^{-1} \tilde{D}_0 - (A^{-1}\tilde{D}_0)^2 + (A^{-1}\tilde{D}_0)^3 + \dots, \label{OID} \end{equation} We note that after $p$ applications of $\tilde{D}_0$, $ E_\eta(y,\alpha)$ is a homogeneous polynomial of degree $|\eta| - 2p$ so we have $$ A^{-1} (\tilde{D}_0)^p \, E_\eta(y,\alpha) = {1 \over 4 p} (\tilde{D}_0)^p \, E_\eta(y,\alpha). $$ Using this in (\ref{EX1}) gives \begin{eqnarray} E_\eta^{(H)}(y,\alpha) & = & \left( 1 - {1 \over 4}\tilde{D}_0 + {1 \over 4^2} {1 \over 2!}(\tilde{D}_0)^2 - {1 \over 4^3} {1 \over 3!} (\tilde{D}_0)^3 + \dots \right) E_\eta(y,\alpha) \nonumber \\ & = & \exp ( - \tilde{D}_0 / 4 ) \, E_\eta(y,\alpha), \label{EX2} \end{eqnarray} which is consistent with (\ref{3.2}) and is thus the sought exponential operator formula. Note that the series in (\ref{EX2}) terminates after the $[|\eta|/2]$ application of $\tilde{D}_0$. To proceed further we note that the operator $ \tilde{D}_0$ can be written in terms of the Dunkl operator (\ref{DU}) with the $z_i$ replaced by $y_i$. We have \cite{dunkl89a} \begin{equation} \tilde{D}_0 = \sum_{j=1}^N T_j^2. \label{DU1} \end{equation} We will use (\ref{EX2}) and (\ref{DU1}) to verify that the $E_\eta^{(H)}$ are simultaneous eigenfunctions of a set of operators more basic than $\tilde{H}^{(H,Ex)}$, which play an analogous role to the Cherednik operators in the theory of the non-symmetric Jack polynomials. \vspace{.2cm} \noindent {\bf Proposition 3.1} \quad The non-symmetric generalized Hermite polynomials $E_\eta^{(H)}$ are eigenfunctions of the operators \begin{equation} h_i := \xi_i - {\alpha \over 2} T_i^2, \qquad (i=1,\dots,N) \label{HI} \end{equation} with corresponding eigenvalue $\bar{\eta}_i$. Here $ \xi_i$ is the Cherednik operator (\ref{COV}) and $\bar{\eta}_i$, which is given explicitly by (\ref{EVCV}), is defined as the eigenvalue in the eigenvalue equation \begin{equation} \xi_i E_\eta = \bar{\eta}_i E_\eta. \label{XEV} \end{equation} \vspace{.2cm} \noindent {\bf Remarks} \\ (i) In \cite[Prop.~3.2 with $j=1$]{forr96a} we noted that $D_N^1 + {1 \over 4} [D_N^1,D_0]$, where \begin{equation} D_N^1 := \alpha \sum_{j=1}^N y_j {\partial \over \partial y_j} +N(N-1)/2, \qquad D_0 := \sum_{j=1}^N {\partial^2 \over \partial y_j^2} + {2 \over \alpha} \sum_{j \ne k} {1 \over y_j - y_k} {\partial \over \partial y_j}, \label{DEFD0} \end{equation} is an eigenoperator for the symmetric generalized Hermite polynomials $H_\kappa(y;\alpha)$. Our construction of (\ref{HI}) was motivated by this result, (\ref{DU1}) and the fact that \begin{equation} \sum_{i=1}^N (\xi_i + (N-1)) = D_N^1. \label{SD} \end{equation} (ii) We could replace $\xi_i$ in (\ref{HI}) by $\hat{D}_{N+1-i}$ (recall the remark below (\ref{COV})). Our use of $\xi_i$ has been influenced by \cite{knop96c,sahi96a}. \vspace{.2cm} In further preparation for proving Proposition 3.1 we will evaluate the commutator $[\xi_j,\tilde{D}_0]$. Due to (\ref{DU1}), we should first consider the commutator $[\xi_j,T_i]$. \vspace{.2cm} \noindent {\bf Lemma 3.1} \quad We have \begin{eqnarray*} {[}\xi_j, T_i] & = & T_i M_{ij}, \qquad i < j \\ {[}\xi_j, T_i] & = & T_j M_{ij}, \qquad i > j \\ {[}\xi_j, T_j] & = & -\alpha T_j - \sum_{p < j} M_{jp} T_j - \sum_{p> j} T_j M_{jp}. \end{eqnarray*} \vspace{.2cm} \noindent {\bf Proof} \quad These formulas are verified by straightforward calculation using the formula (\ref{COV}) relating $\xi_j$ and $T_j$, the commutator formula (\ref{DOC}), and the additional easily verified commutator identities $$ {[}T_i,y_i] = 1 + {1 \over \alpha} \sum_{p \ne i} M_{ip} \qquad {[}T_i,y_j] = - {1 \over \alpha} M_{ij}, \quad i \ne j \qquad {[T}_i, M_{jk}] = 0, \quad i \ne j,k. $$ \vspace{.2cm} Now we can evaluate the commutator $[\xi_i,\tilde{D}_0]$. \vspace{.2cm} \noindent {\bf Lemma 3.2} \quad We have $$ [\xi_i,\tilde{D}_0] = - 2 \alpha T_i^2. $$ \vspace{.2cm} \noindent {\bf Proof} \quad Using (\ref{DU1}) we have $$ [\xi_i,\tilde{D}_0] = \sum_{j=1}^N [\xi_i, T_j^2] = \sum_{j=1}^N \Big ( [\xi_i, T_j] T_j + T_j[\xi_i, T_j] \Big ). $$ The result follows after splitting the sum up into parts $j<i$, $j=i$ and $j > i$, then using Lemma 3.1 and the facts that $$ M_{ij} T_j = T_i M_{ij}, \qquad M_{ij}T_k = T_k M_{ij} \quad (k \ne i,j). $$ \vspace{.2cm} With this preparation we can now provide the verification of the claim of Proposition 3.1. \vspace{.2cm} \noindent {\bf Proof of Proposition 3.1} \quad Using (\ref{EX2}) and (\ref{XEV}) we have \begin{equation} \xi_i \Big ( e^{\tilde{D}_0/4} E_\eta^{(H)} \Big ) = \bar{\eta}_i \Big ( e^{\tilde{D}_0/4} E_\eta^{(H)} \Big ). \label{XHEV} \end{equation} But according to the Baker-Campbell-Hausdorff formula \begin{eqnarray}\xi_i \left( e^{\tilde{D}_0/4} E_\eta^{(H)} \right) & = & e^{\tilde{D}_0/4} \left( \xi_i + {1 \over 4}[\xi_i,\tilde{D}_0] +{1 \over 2!}{1 \over 4^2} [[\xi_i,\tilde{D}_0],,\tilde{D}_0] + \dots \right) E_\eta^{(H)} \nonumber \\ & = & e^{\tilde{D}_0/4} \left( \xi_i -{\alpha \over 2} T_i^2 \right) E_\eta^{(H)} \label{BCH} \end{eqnarray} where to obtain the last line we have used the fact that since $[\xi_i,\tilde{D}_0] = -2 \alpha T_i^2$ (by Lemma 3.2) and $\tilde{D}_0 = \sum_{j=1}^N T_j^2$ (eq.~(\ref{DU1})), the higher order commutators vanish due to (\ref{DOC}). Equating the r.h.s.~of (\ref{BCH}) with the r.h.s.~of (\ref{XHEV}) gives the desired eigenvalue equation. \vspace{.2cm} \noindent {\bf Remark} \quad Since $\{E_\eta^{(H)}\}$ form a basis for analytic functions it follows from Proposition 3.1 that $\{h_i\}$ mutually commute. This fact can also be checked directly using, Proposition 3.1, Lemma 3.1 and (\ref{DOC1}) and (\ref{DOC}). \vspace{.2cm} {}From Proposition 3.1, (\ref{DU1}), (\ref{SD}) and (\ref{HTHE}) we have that \begin{equation} \sum_{i=1}^N h_i = -{\alpha \over 2} \Big ( \tilde{H}^{(H,Ex)} + N(N-1)/\alpha \Big ). \label{A1} \end{equation} Also, by forming the sum (\ref{JNSJ}) in (\ref{EX2}) we have \begin{equation} \sum_{P} a_{P^{-1}\kappa} E_{P^{-1}\kappa}^{(H)}(y;\alpha) = \exp \Big ( -{1 \over 4} \tilde{D}_0 \Big )J_\kappa^{(\alpha)}(y) = 2^{-|\kappa|}J_\kappa^{(\alpha)}(1^N) H_\kappa(y;\alpha) \label{A2} \end{equation} where, after noting that $\tilde{D}_0 = D_0$ as defined in (\ref{DEFD0}) when acting on symmetric functions, the last equality is the exponential operator formula of Lassalle (see ref.~\cite[eq.~(3.21)]{forr96a}). Now from the remark below Lemma 2.1 we have that $ \{ \bar{\eta}_j \}_{j=1,\dots,N} = \{ \alpha \kappa_j - (j-1) \}_{j=1,\dots,N}$ independent of the permutation relating $\eta$ to the partition $\kappa$. Using this fact, (\ref{A1}), (\ref{A2}) and Proposition 3.1 we see by following the argument of the last two paragraphs of Section 2.1 that \begin{equation} \prod_{j=1}^N ( 1+ u h_i ) \label{KAKEI} \end{equation} is an eigenoperator of the symmetric generalized Hermite polynomials $H_\kappa(y;\alpha)$ (an operator with this property equivalent to (\ref{KAKEI}) has recently been identified by Kakei \cite{kakei96}) with corresponding eigenvalue \begin{equation} \prod_{j=1}^N \Big ( 1 + u(\alpha \kappa_j - (j-1)) \Big ). \label{EVH} \end{equation} Note that the inequalities (\ref{OR}) imply that the eigenvalues are distinct. We remark that in ref.~\cite[Prop.~3.2]{forr96a} an operator $\tilde{H}_j^{(H)}$ was constructed such that \begin{equation} \Big ( \sum_{j=0}^N X^{N-j} \tilde{H}_j^{(H)} \Big ) H_\kappa(y;\alpha) = \prod_{i=1}^N ( X+ N - i + \alpha \kappa_i) H_\kappa(y;\alpha). \label{NH1} \end{equation} Comparison with the eigenvalue (\ref{EVH}) corresponding to the operator (\ref{KAKEI}) shows that when acting on symmetric functions \begin{equation} \prod_{i=1}^N ( X+ N + h_i) = \sum_{j=0}^N X^{N-j} \tilde{H}_j^{(H)} \label{NH2} \end{equation} The eigenoperator (\ref{KAKEI}) can be used to establish that $\{ H_\kappa(y;\alpha) \}$ are orthogonal with respect to the inner product \begin{equation} \langle f|g \rangle^{(H)} := \prod_{l=1}^N \int_{-\infty}^\infty dy_l \, e^{-y_l^2} \prod_{1 \le j < k \le N} |y_k - y_j|^{2/\alpha} f g \label{INH} \end{equation} (note that the weight function in (\ref{INH}) is equal to $e^{-\beta W^{(H)}}$, where $ W^{(H)}$ is given by (\ref{WH}), and is thus the square of the ground state wave function of (\ref{HHE})). This is an immediate consequence of the fact that the eigenvalues (\ref{EVH}) are distinct and (\ref{KAKEI}) is self-adjoint with respect to (\ref{INH}) (recall the analogous argument in Section 2.1). The latter result follows from the $h_i$ being self-adjoint with respect to (\ref{INH}), which is to be established in the subsequent lemma. The orthogonality has previously been established in refs.~\cite{forr96a,kakei96}, but the details here are different. \vspace{.2cm} \noindent {\bf Lemma 3.3} \quad We have \begin{equation} \langle f|T_ig \rangle^{(H)} = \langle(2y_i - T_i) f|g \rangle^{(H)} \label{DU4} \end{equation} and thus \begin{equation} \langle f|h_ig \rangle^{(H)} = \langle h_i f|g \rangle^{(H)} \label{DU5} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad The result (\ref{DU4}) is given in \cite[lemma 3.7]{dunkl91a}. It is derived using integration by parts. Using (\ref{DU4}) and the first equation in (\ref{COV}) we find \begin{equation} \langle f|\xi_ig \rangle^{(H)} = \langle (\xi_i + \alpha(2y_i^2 - T_i y_j - y_j T_i) f|g \rangle^{(H)}. \label{DU6} \end{equation} But from Proposition 3.1 $h_i = \xi_i - {(\alpha /2)}T_i^2$. Noting from (\ref{DU4}) that \begin{equation} \langle f|T_i^2g \rangle^{(H)} = \langle(2y_i - T_i)^2 f|g \rangle^{(H)} \label{DU7} \end{equation} the result (\ref{DU5}) follows by subtracting $\alpha/2$ times (\ref{DU7}) from (\ref{DU6}). Analogous to the theory of the symmetric Jack polynomials revised in Section 2.1, the orthogonality of the symmetric generalized Hermite polynomials can be established from the formula (\ref{A2}) and the fact that the non-symmetric generalized Hermite polynomials are orthogonal with respect to (\ref{INH}). This latter fact is established by noting that (\ref{PONS}) with $\hat{D}_l$ replaced by $h_l$ is self-adjoint with respect to (\ref{INH}) and is an eigenoperator of each $E_\eta^{(H)}$ which separates the eigenvalues. \subsection{Generalized Hermite polynomials with prescribed symmetry} Let $S_{(\rho, \mu)}(y,\alpha)$ denote a Jack polynomial with prescribed symmetry. By following the working which led to (\ref{EX2}) we can construct a polynomial eigenfunction of (\ref{HTHE}) according to \begin{equation} S_{(\rho, \mu)}^{(H)}(y,\alpha) = \exp( - \tilde{D}_0/4) \,S_{(\rho, \mu)}(y,\alpha). \label{EXH} \end{equation} Note that $S_{(\rho, \mu)}^{(H)}(y,\alpha)$ has the same symmetry properties as $S_{(\rho, \mu)}(y,\alpha)$. We will refer to $\{ S_{(\rho, \mu)}^{(H)}(y,\alpha) \}$ as the generalized Hermite polynomials with prescribed symmetry. Due to the expansion (\ref{SSUM}), and the formula (\ref{EX2}) we see from (\ref{EXH}) that \begin{equation} S_{(\rho, \mu)}^{(H)}(y,\alpha) = \sum_{\rm rearrangements} b_{Q^{-1}\kappa} E_{Q^{-1}\kappa}^{(H)}(y,\alpha) \label{NH3} \end{equation} {}From this formula we can deduce that the operator (\ref{PCO}) with each operator $\hat{D}_j$ replaced by $h_j$ is an eigenoperator of $S_{(\rho, \mu)}^{(H)}$, and the corresponding eigenvalues are distinct for distinct members of $\{S_{(\rho, \mu)}^{(H)} \}$. This implies $\{S_{(\rho, \mu)}^{(H)} \}$ is an orthogonal set with respect to the inner product (\ref{INH}). This fact can also be deduced from (\ref{NH3}) and the orthogonality of $\{ E_\eta^{(H)} \}$. \subsection{Some special generalized Hermite polynomials with prescribed symmetry} The analogue of the conjecture (\ref{CON}) in the Hermite case is that for $\eta =: P\kappa $ given by (\ref{PK}) \begin{equation} S_\eta^{(H)}(y,\alpha) = A_{\eta}^{(H)} \prod_{\gamma=1}^p \prod_{1 \le j < k \le N_\gamma}\! (y_k^{(\gamma)} - y_j^{(\gamma)}) H_{\rho}(\sqrt{\alpha \over \alpha + p}x_1,\dots,\sqrt{\alpha \over \alpha +p}x_{N_0};p + \alpha), \label{CON1} \end{equation} provided the inequality (\ref{IN}) is satisfied. To verify this conjecture, our task is to show that (\ref{CON1}) is an eigenfunction of (\ref{HTHE}). To do this, we proceed as in (\ref{DEC}) and write (\ref{HTHE}) when acting on functions symmetric in $\{x_j\}_{j=1,\dots,N_0}$ and antisymmetric in $\{y_j^{(\gamma)}\}_{j=1,\dots,N_\gamma}$ $(\gamma = 1,\dots, p)$ as the sum of three terms: \begin{equation} \tilde{H}^{(H,Ex)} = \tilde{H}^{(H,x)} + \tilde{H}^{(H,y)} + \tilde{H}^{(H,xy)} \end{equation} where \begin{equation} \tilde{H}^{(H,x)} = \sum_{j=1}^{N_0} \Big ( {\partial^2 \over \partial x_j^2} - 2 x_j {\partial \over \partial x_j} \Big ) + {2 \over \alpha} \sum_{j \ne k} {1 \over x_j - x_k} {\partial \over \partial x_k} \end{equation} \begin{equation} \tilde{H}^{(H,xy)} = {2 \over \alpha} \sum_{\gamma = 1}^p \sum_{j=1}^{N_\gamma} \sum_{k=1}^{N_0} {1 \over y_j^{(\gamma)} - x_k} \left[\Big ({\partial \over \partial y_j^{(\gamma)}} -{\partial \over \partial x_k}\Big ) - {1 - M(y_j^{(\gamma)},x_k) \over y_j^{(\gamma)} - x_k} \right] \end{equation} and $\tilde{H}^{(H,y)}$ is given by (\ref{HTHE}) with $y_1,y_2\dots,y_N$ replaced by $y_1^{(1)},y_2^{(1)},\dots,y_{N_p}^{(p)}$. The fundamental operator in establishing that (\ref{CON1}) is an eigenfunction of (\ref{HTHE}) is $(y_jy_k)^{-1}A(y_j,y_k)$, where $A(y_j,y_k)$, along with its action on $y_j^{\lambda_j} y_k^{\lambda_k}$ is specified in Lemma 2.2. Using this action, we can repeat the argument of the proof of Lemma 2.3 to conclude that $\prod_{\gamma = 1}^p \Delta(y^{(\gamma)})$ is an eigenfunction of $\tilde{H}^{(H,y)}$. This action can also be used to establish the analogue of Lemma 2.4. \vspace{.2cm} \noindent {\bf Lemma 3.4} \quad Let $F(x,y) = m_\kappa (x) \prod_{\gamma=1}^p \Delta (y^{(\gamma)})$, where $\kappa$ is a partition consisting of $N_0$ parts with the largest part $\kappa_1$ restricted by $\kappa_1 \le {\rm min}(N_1,\dots,N_\gamma)$. We have \begin{equation} \tilde{H}^{(H,xy)}F(x,y) = {p \over \alpha} \sum_{j=1}^{N_0} {\partial^2 \over \partial x_j^2} F(x,y). \label{L34} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad Proceeding as in the proof of Lemma 2.4 we see that for $0 \le \lambda \le N_\gamma$ only the $l=1$ term on the r.h.s.~of (\ref{AA}) contributes to the action of $(y^{(\gamma)}_j x_k)^{-1} A(y^{(\gamma)}_j,x_k) $ on $(y^{(\gamma)}_j)^{\lambda'} x_k^\lambda$, and this requires $\lambda - \lambda' \ge 2$. Thus $$ \sum_{j=1}^{N_\gamma} (y^{(\gamma)}_j x_k)^{-1} A(y^{(\gamma)}_j,x_k) \Delta (y^{(\gamma)}) x_k^\lambda = \left( \sum_{j=0}^{\lambda - 2} (\lambda - j - 1) \right) \Delta (y^{(\gamma)}) x_k^\lambda, $$ which implies $$ \tilde{H}^{(H,xy)}F(x,y) = {p \over \alpha} (|\kappa^2| - |\kappa|) F(x,y), $$ and the result follows. \vspace{.2cm} {}From Lemma 3.4 and the fact that $\tilde{H}^{(H,y)}$ is an eigenoperator for $\prod_{\gamma = 1}^p \Delta (y^{(\gamma)})$ with eigenvalue $\delta^{(H)}$ say we have, assuming the inequality in Lemma 3.4, \begin{eqnarray*} \tilde{H}^{(H,Ex)} \prod_{\gamma = 1}^p \Delta (y^{(\gamma)}) m_\kappa(x) & = & \left(\delta^{(H)} + (1 + {p \over \alpha}) \left[ \sum_{j=1}^{N_0} {\partial^2 \over \partial x_j^2} - {2 \alpha \over \alpha + p} x_j {\partial \over \partial x_j}\right.\right.\\ && \left.\left.+ {2 \over \alpha + p} \sum_{j \ne k}{1 \over x_j - x_k}{\partial \over \partial x_j} \right] \right) \prod_{\gamma = 1}^p \Delta (y^{(\gamma)}) m_\kappa(x). \end{eqnarray*} The fact that (\ref{CON1}) is an eigenfunction of $ \tilde{H}^{(H,Ex)}$ follows from this equation since the operator in square brackets is the defining eigenoperator for $H_\kappa((\alpha/(\alpha + p))^{1/2}x;\alpha)$. As another explicit evaluation of a class of $S_{(\rho, \mu)}^{(H)}$, in the case $p=1$, $q=0$ we have the analogue of (\ref{ANTIS}): \begin{equation} S_\mu^{(H)}(y,\alpha) = \Delta (y) {2^{-|\kappa|} C_\kappa^{(\alpha)}(1^N) \over v_{\kappa \kappa}(\alpha/(1+\alpha))} H_\kappa(y;\alpha/(1+\alpha)), \label{ANTIS1} \end{equation} where the prefactors of $H_\kappa$ are chosen so that the coefficient of $m_\kappa$ in this factor is unity (see ref.~\cite{forr96a}). The derivation of this result is analogous to the derivation of (\ref{ANTIS}), and so will be omitted. \section{Eigenfunctions of $\tilde{H}^{(L,Ex)}$} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} \subsection{The non-symmetric generalized Laguerre polynomials} Analogous to the situation with $\tilde{H}^{(H,Ex)}$, the operator (\ref{HTLE}) has unique polynomial eigenfunctions of the form (\ref{3.1}) with corresponding eigenvalue $-|\eta|$. An appropriate linear combination of these eigenfunctions gives unique eigenfunctions of the form \begin{equation} E_\eta^{(L)}(y,\alpha) := E_\eta(y,\alpha) + \sum_{|\nu| < |\eta|} c_{\eta \, \nu} E_{\nu}(y;\alpha) \label{4.1} \end{equation} which also have eigenvalue $-|\eta|$. As well as depending on the parameter $\alpha$ they also depend on the parameter $a$ in (\ref{HTLE}), however for notational convenience we have suppressed this dependence in (\ref{4.1}). The $E_\eta^{(L)}(y,\alpha)$ will be referred to as the non-symmetric generalized Laguerre polynomials (we recall from ref.~\cite{forr96a} that the symmetric generalized Laguerre polynomials $L_\kappa^a(y;\alpha)$ are the polynomial eigenfunctions of (\ref{HTLE}) with leading term proportional to the symmetric Jack polynomial $J_\kappa^{(\alpha)}(y)$. By repeating the working which led to (\ref{EX2}), starting with the eigenvalue equation for $E_\eta^{(L)}(y,\alpha)$, we obtain the exponential operator formula \begin{equation} E_\eta^{(L)}(y,\alpha) = \exp \bigg ( - \Big ( \tilde{D}_1 + (a+1) \sum_{j=1}^N {\partial \over \partial y_j} \Big ) \bigg ) E_\eta(y,\alpha) \label{EX5} \end{equation} where \begin{equation} \tilde{D}_1 := \sum_{j=1}^N y_j {\partial^2 \over \partial y_j^2} + {1 \over \alpha} \sum_{j<k} {1 \over y_j - y_k} \left[ 2\Big ( y_j {\partial \over \partial y_j} - y_k {\partial \over \partial y_k}\Big ) - {y_j + y_k \over y_j - y_k}(1 - M_{jk}) \right]. \label{WE1} \end{equation} It is convenient to introduce new variables $x_j^2 =: y_j$. It follows from \cite[first eq.~pg.~125]{dunkl93a} that \begin{equation} \tilde{D}_1 + \left.(a+1) \sum_{j=1}^N {\partial \over \partial y_j} \right|_{y_j \mapsto x_j^2} = {1 \over 4} \sum_{i=1}^N \left(T_i^{(B)}\right)^2 \label{WE2} \end{equation} where $T_i^{(B)}$ is the Dunkl operator for the root system $B_N$: \begin{equation} T_i^{(B)} := { \partial \over \partial x_i} + {1 \over \alpha} \sum_{p \ne i} \left( { 1 - M_{ip} \over x_i - x_p} + {1 - S_i S_p M_{ip} \over x_i + x_p} \right) + {a + 1/2 \over x_i}(1 - S_i) \label{DEFTB} \end{equation} (as in (\ref{HLE}) $S_j$ is the operator which replaces the coordinate $x_j$ by $-x_j$). The similarity between (\ref{EX5}) with the substitution (\ref{WE2}), and (\ref{EX2}) with the substitution (\ref{DU1}) suggest we define operators $l_i$ say, analogous to (\ref{HI}): \begin{equation} l_i := \hat{\xi}_i - {\alpha \over 4} (T_i^{(B)})^2 \label{LI} \end{equation} where $ \hat{\xi}_i$ is the Cherednik operator (\ref{COV}) with the change of variables $y_j = x_j^2$ (a literal analogy would have $\alpha/4$ replaced by $\alpha/2$ in (\ref{LI}); the reason for modifying this is connected with the remark accompanying Lemma 4.2 below). We want to show that the $l_i$ are eigenoperators for the $E_\eta^{(L)}(x^2,\alpha)$. To do this we require the analogue of Lemma 3.2, and this in turn requires some preliminary results. \vspace{.2cm} \noindent {\bf Lemma 4.1} \quad When acting on functions even in $x_1,\dots,x_N$ \begin{equation} {1 \over 4} (T_i^{(B)})^2 = x_i^2 \hat{T}_i^2 + (a+1) \hat{T}_i + {1 \over \alpha} \sum_{p \ne i} M_{ip} \hat{T}_i \label{L41} \end{equation} where $ \hat{T}_i$ is the $A$-type Dunkl operator (\ref{DU}) with the change of variables $z_j = x_j^2$ $(j=1,\dots,N)$: \begin{equation} \hat{T}_i = {1 \over 2x_i} { \partial \over \partial x_i} + {1 \over \alpha} \sum_{p \ne i} {1 - M_{ip} \over x_i^2 - x_p^2}. \label{DEFTH} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad Let $f$ be even in $x_1,\dots,x_N$. Then from (\ref{DEFTB}) and (\ref{DEFTH}) we see that $$ T_i^{(B)} f = 2x_i \hat{T}_i \, f. $$ Now $x_i \hat{T}_i \, f$ is odd in $x_i$, and from the definition (\ref{DEFTB}) we see that when acting on a function which is odd in $x_i$, $$ T_i^{(B)} = 2x_i \hat{T}_i + {2a+1 \over x_i} + {1 \over \alpha} \sum_{p \ne i} {1+S_p \over x_i + x_p} M_{ip}. $$ Thus when acting on $f$ \begin{equation} (T_i^{(B)})^2 = 4 (x_i \hat{T}_i)^2 + (4a+2) \hat{T}_i + {4 \over \alpha} \sum_{p \ne i} {1 \over x_i + x_p} M_{ip}x_i \hat{T}_i. \label{PL41} \end{equation} To simplify further, note that \begin{equation} (x_i \hat{T}_i)^2 = x_i ( [ \hat{T}_i,x_i] + x_i \hat{T}_i ) \hat{T}_i \label{PL42} \end{equation} and evaluate the commutator: \begin{equation} [ \hat{T}_i,x_i] = {1 \over 2 x_i} + {1 \over \alpha} \sum_{p \ne i} {1 \over x_i + x_p} M_{ip}. \label{PL43} \end{equation} The stated result (\ref{L41}) follows by substituting (\ref{PL43}) in (\ref{PL42}), substituting the result in (\ref{PL41}) and simplifying. \vspace{.2cm} The analogue of Lemma 3.1 is given by the following result. \vspace{.2cm} \noindent {\bf Lemma 4.2} \quad Let $B_i := {1 \over 4}(T_i^{(B)})^2$ and let \begin{equation} \hat{\xi}_j := \alpha x_j^2 \hat{T}_j + (1 - N) + \sum_{p > j} M_{jp} \label{DEFXH} \end{equation} be the Cherednik operator (\ref{COV}) with the substitution $y_j = x_j^2$ ($j=1,\dots,N$). We have \begin{eqnarray*} [ \hat{\xi}_j,B_i] & = & B_i M_{ij}, \quad i<j \\ {[} \hat{\xi}_j,B_i] & = & B_j M_{ij}, \quad i>j \\ {[} \hat{\xi}_j,B_j]& = &- \alpha B_j - \sum_{p < j} M_{jp} B_j - \sum_{p > j} B_j M_{jp}. \end{eqnarray*} \vspace{.2cm} \noindent {\bf Remark} \quad Although the commutators in Lemma 3.1 and Lemma 4.2 have the same structure, note that in Lemma 3.1 the operator $T_i$ occurs while in Lemma 4.2 it is the operator $(T_i^{(B)})^2$ which occurs. Moreover, it seems that there is no simple formula for $[ \hat{\xi}_j,T_i^{(B)}]$. \vspace{.2cm} \noindent {\bf Proof} \quad First consider the case $i < j$. From Lemma 4.1 we have \begin{eqnarray} [ \hat{\xi}_j,B_i] & = & [ \hat{\xi}_j, x_i^2 \hat{T}_i^2 + (a+1) \hat{T}_i + {1 \over \alpha} \sum_{p \ne i} M_{ip} \hat{T}_i] \nonumber \\ & = & [ \hat{\xi}_j, x_i^2] \hat{T}_i^2 + x_i^2 \Big ( [ \hat{\xi}_j, \hat{T}_i] \hat{T}_i + \hat{T}_i [ \hat{\xi}_j, \hat{T}_i] \Big ) \nonumber \\ & & + (a+1) [ \hat{\xi}_j, \hat{T}_i] + {1 \over \alpha} [ \hat{\xi}_j, \sum_{p \ne i} M_{ip} ] \hat{T}_i + {1 \over \alpha} \sum_{p \ne i} M_{ip} [ \hat{\xi}_j, \hat{T}_i]. \label{MCOM} \end{eqnarray} Now, for $i <j$ a direct calculation gives $ [ \hat{\xi}_j, x_i^2] = -x_j^2 M_{ij}$, while Lemma 3.1 gives $[ \hat{\xi}_j, \hat{T}_i] = \hat{T}_i M_{ij}$. To evaluate the second last commutator in (\ref{MCOM}) we substitute (\ref{DEFXH}), and note that $$ [x_j^2 \hat{T}_j, \sum_{p \ne i} M_{ip}] = (x_j^2 \hat{T}_j - x_i^2 \hat{T}_i) M_{ij}, \qquad [\sum_{q > j} M_{jq}, \sum_{p \ne i} M_{ip}]=0, $$ where to obtain the latter result the formula $M_{ij} M_{jp} = M_{jq}M_{qi}= M_{qi}M_{ij}$ has been used. Substituting these results in (\ref{MCOM}) gives the stated result in the case $i<j$. The cases $i >j$ and $i=j$ are very similar, although the latter case requires more manipulation to simplify the final expression. \vspace{.2cm} Using Lemma 4.2 the required analogue of Lemma 3.2 follows. \vspace{.2cm} \noindent {\bf Lemma 4.3} \quad We have $$ [ \hat{\xi}_j, \sum_{i=1}^N (T_i^{(B)})^2] = -\alpha (T_j^{(B)})^2. $$ \vspace{.2cm} \noindent We can use Lemma 4.3 in the same way as Lemma 3.2 was used in the proof of Proposition 3.1 to prove the analogue of Proposition 3.1 in the Laguerre case. \vspace{.2cm} \noindent {\bf Proposition 4.1} \quad The non-symmetric Laguerre polynomials $E_\eta^{(L)}(x^2,\alpha)$ are simultaneous eigenfunctions of the operators $l_i$ (\ref{LI}) with corresponding eigenvalue $\bar{\eta}_i$. \vspace{.2cm} \noindent {\bf Remark} \quad Since $\{E_\eta^{(L)} \}$ form a basis for analytic functions, it follows from Proposition 4.1 that $\{l_i\}$ mutually commute, a fact that can also be derived directly by using Lemma 4.2 and the fact that the $T_i^{(B)}$ commute, as do the operators $ \hat{\xi}_i$. \vspace{.2cm} {}From (\ref{SD}) and (\ref{WE2}) we see from (\ref{LI}) and (\ref{HTLE}) that \begin{equation} \sum_{i=1}^N l_i =\left. -\alpha \tilde{H}^{(L,Ex)} \right|_{y_j \mapsto x_j^2} - N(N-1)/2. \label{LI2} \end{equation} Also, analogous to (\ref{A2}), forming the sum (\ref{JNSJ}) in (\ref{EX5}) we have \begin{eqnarray} \sum_{P} a_{P^{-1}\kappa} E_{P^{-1}\kappa}^{(L)}(y;\alpha) & = & \exp \bigg ( - \Big ( \tilde{D}_1 + (a+1) \sum_{j=1}^N {\partial \over \partial y_j} \Big ) \bigg ) J_\kappa^{(\alpha)}(y) \nonumber \\ & = & (-1)^{|\kappa|} |\kappa|! J_\kappa^{(\alpha)}(1^N) L_\kappa^a(y;\alpha) \label{NL1} \end{eqnarray} where the last equality follows from \cite[eq.~(4.39)]{forr96a}. From (\ref{LI2}), analogous to (\ref{KAKEI}), we have that \begin{equation} \prod_{j=1}^N ( 1+ u l_i ) \label{KAKEI2} \end{equation} is an eigenoperator of the symmetric generalized Laguerre polynomials $ L_\kappa^a(x^2;\alpha)$ with corresponding eigenvalue (\ref{EVH}). In ref.~\cite[Prop.~4.5]{forr96a} an operator $\tilde{H}_j^{(L)}$ was constructed such that (\ref{NH1}) holds with $\tilde{H}_j^{(H)}$ replaced by $\tilde{H}_j^{(L)}$ and $H_\kappa(y;\alpha)$ replaced by $L_\kappa^a(y;\alpha)$. It follows that (\ref{NH2}) holds with $h_j$ replaced by $l_j$ and $\tilde{H}_j^{(H)}$ replaced by $\tilde{H}_j^{(L)}$. We know from \cite{forr96a} (see also \cite{lass91b}) that $\{L_\kappa^a\}$ are orthogonal with respect to the inner product \begin{eqnarray} \langle f | g \rangle^{(L)} &:=& 2^N \prod_{l=1}^N \int_{-\infty}^\infty dx_l \, e^{-x_l^2} |x_l|^{2a+1} \prod_{1 \le j < k \le N} |x_k^2 - x_j^2|^{2/\alpha} \nonumber \\&& \times f(x_1^2,\dots,x_N^2) g(x_1^2,\dots,x_N^2) \label{INL} \end{eqnarray} Note that the weight function is proportional to $e^{-\beta W^{(L)}}$, where $W^{(L)}$ is given by (\ref{WL}), and is thus proportional to the square of the symmetric ground state wave function of (\ref{HTLE}). The orthogonality can be deduced in the present setting by first checking (see subsequent lemma) that the $l_i$, and thus the eigenoperator (\ref{KAKEI2}), are self adjoint with respect to (\ref{INL}) and recalling that the eigenvalues of (\ref{KAKEI2}) are distinct. \vspace{.2cm} \noindent {\bf Lemma 4.4} \quad We have \begin{equation} \langle f |\hat{T}_i g \rangle^{(L)} = \langle (1 - {a \over x_i^2} - \hat{T}_i )f|g \rangle^{(L)} \label{L441} \end{equation} and \begin{equation} \langle f | l_i g \rangle^{(L)} = \langle l_i f | g \rangle^{(L)}. \label{L442} \end{equation} \vspace{.2cm} \noindent {\bf Proof} \quad The result (\ref{L441}) follows from the explicit formula (\ref{DEFTH}) and integration by parts. From (\ref{L441}), (\ref{PL43}) and (\ref{DEFXH}) we find \begin{equation} \langle f |\hat{\xi}_i g \rangle^{(L)} = \langle (\hat{\xi}_i + \alpha (x_i^2 - a - 2x_i^2 \hat{T}_i - 1 - {1 \over \alpha} \sum_{p \ne i} M_{ip}))f|g \rangle^{(L)}. \label{PL441} \end{equation} Also, analogous to (\ref{DU4}) we have \begin{equation} \langle f|T_i^{(B)}g \rangle^{(L)} = \langle(2x_i - T_i^{(B)}) f|g \rangle^{(L)} \label{PL442} \end{equation} and thus \begin{equation} \langle f|(T_i^{(B)})^2g \rangle^{(L)} = \langle(4x_i^2 - 2x_iT_i^{(B)} -2T_i^{(B)} x_i + (T_i^{(B)})^2) f|g \rangle^{(L)}. \label{PL443} \end{equation} But from the proof of Lemma 4.1 we know that when acting on functions even in $x_1^2, \dots, x_N^2$, $$ x_i T_i^{(B)} = 2x_i^2 \hat{T}_i $$ and from the working in the same proof we can compute that $$ T_i^{(B)} x_i = 2x_i^2\hat{T}_i + 2(a+1) + {2 \over \alpha} \sum_{p \ne i} M_{ip}. $$ Substituting these formulas in (\ref{PL443}) and subtracting $\alpha/4$ times the result from (\ref{PL441}) gives the required result (\ref{L442}). Note that the operator (\ref{PONS}) with the $\hat{D}_j$ replaced by $l_j$ is a self-adjoint (with respect to (\ref{INL})) eigenoperator of the $E_\eta^{(L)}(x^2,\alpha)$ which separates the eigenvalues. It follows that $\{E_\eta^{(L)} \}$ is an orthogonal set with respect to the inner product (\ref{INL}). The orthogonality of $\{ L_\kappa^a \}$ also follows from this fact and the expansion (\ref{NL1}). \subsection{Generalized Laguerre polynomials with prescribed symmetry} The theory here is analogous to that for the generalized Hermite polynomials with prescribed symmetry. Generalized Laguerre polynomials with prescribed symmetry, $S_{(\rho, \mu)}^{(L)}(y,\alpha)$ say, are defined as the eigenfunctions of (\ref{HTLE}) given by the exponential operator formula \begin{equation} S_{(\rho, \mu)}^{(L)}(y,\alpha) = \exp \bigg ( - \Big ( \tilde{D}_1 + (a+1) \sum_{j=1}^N {\partial \over \partial y_j} \Big ) \bigg ) S_{(\rho,\mu)}(y,\alpha). \end{equation} The operator (\ref{PCO}) with each $\hat{D}_j$ replaced by $l_j$ is an eigenoperator for each $S_{(\rho, \mu)}^{(L)}(y,\alpha)$ and the eigenvalues are distinct for distinct members of $\{S_{(\rho, \mu)}^{(L)} \}$. {}From this we see that $\{S_{(\rho, \mu)}^{(L)} \}$ is an orthogonal set with respect to the inner product (\ref{INL}). \subsection{Some special generalized Laguerre polynomials with prescribed symmetry} For $\eta =: P^{-1}\kappa$ given by (\ref{PK}) and the inequality (\ref{IN}) satisfied the analogue of the conjecture (\ref{CON}) in the Laguerre case states \begin{equation} S_\eta^{(L)}(y,\alpha) = A_{\eta}^{(L)} \prod_{\gamma=1}^p \prod_{1 \le j < k \le N_\gamma} (y_k^{(\gamma)} - y_j^{(\gamma)}) L_{\rho}^{\alpha a /(p+\alpha)}\Big ({\alpha \over \alpha + p}x_1,\dots,{\alpha \over \alpha + p}x_{N_0};p + \alpha \Big ), \label{CON2} \end{equation} Here we will verify this statement by showing that the r.h.s.~of (\ref{CON2}) is an eigenfunction of (\ref{HTLE}). Now, when acting on functions symmetric in $\{x_j\}_{j=1,\dots,N_0}$ and antisymmetric in $\{y_j^{(\gamma)}\}_{j=1,\dots,N_\gamma}$ $(\gamma = 1,\dots, p)$ we have \begin{equation} \tilde{H}^{(H,Ex)} = \tilde{H}^{(L,x)} + \tilde{H}^{(L,y)} + \tilde{H}^{(L,xy)} \end{equation} where \begin{equation} \tilde{H}^{(L,x)} = \sum_{j=1}^{N_0} \Big ( x_j{\partial^2 \over \partial x_j^2} +(a+1 - x_j) {\partial \over \partial x_j} \Big ) + {2 \over \alpha} \sum_{j \ne k} {x_j \over x_j - x_k} { \partial \over \partial x_j} \end{equation} \begin{equation} \tilde{H}^{(L,xy)} = {1 \over \alpha} \sum_{\gamma = 1}^p \sum_{j=1}^{N_\gamma} \sum_{k=1}^{N_0} {1 \over y_j^{(\gamma)} - x_k} \left[2\Big ( y_j^{(\gamma)}{\partial \over \partial y_j^{(\gamma)}} - x_k {\partial \over \partial x_k}\Big ) - { y_j^{(\gamma)} + x_k \over y_j^{(\gamma)} - x_k}(1 - M(y_j^{(\gamma)},x_k) \right] \label{HLXY} \end{equation} and $\tilde{H}^{(L,y)}$ is given by (\ref{HTLE}) with $y_1,y_2\dots,y_N$ replaced by $y_1^{(1)},y_2^{(2)},\dots,y_{N_p}^{(p)}$. To compute the action of these operators we require the action of the operator which occurs in the summand of (\ref{HLXY}) on monomials. A direct calculation gives the following result. \vspace{.2cm} \noindent {\bf Lemma 4.4} \quad Let $$ A^{(L)}_{jk} := {1 \over y_j - y_k} \left[ 2 \Big ( y_j{\partial \over \partial y_j} - y_k{\partial \over \partial y_k} \Big ) - {y_j + y_k \over y_j - y_k}(1 - M_{jk}) \right]. $$ For $\lambda_j \ge \lambda_k$ we have $$ A^{(L)}_{jk} y_j^{\lambda_j} y_k^{\lambda_k} = \left\{ \begin{array}{l} {\displaystyle \sum_{l=1}^{\lambda_j - \lambda_k }} ( 2 (\lambda_j - \lambda_k - l) + 1 ) y_j^{\lambda_j - l} y_k^{\lambda_k - l + 1}, \quad \lambda_j - \lambda_k \ge 1 \\[2mm] 0, \quad {\rm otherwise}. \end{array} \right. $$ \vspace{.2cm} Using Lemma 4.4 the argument of the proof of Lemma 2.3 shows that $\prod_{\gamma = 1}^p \Delta(y^{(\gamma)})$ is an eigenfunction of $\tilde{H}^{(L,y)}$. This lemma is also used to determine the action of $\tilde{H}^{(L,xy)}$. The strategy is the same as in the proof of Lemma 2.4, so the details will be omitted. \vspace{.2cm} \noindent {\bf Lemma 4.5} \quad Let $F(x,y)$ be as in Lemma 3.4. We have $$ \tilde{H}^{(L,xy)} F(x,y) = {p \over \alpha} \sum_{j=1}^{N_0} \Big ( x_j {\partial^2 \over \partial x_j^2} + {\partial \over \partial x_j} \Big ) F(x,y). $$ \vspace{.2cm} {}From the above working we see that, assuming the inequality in Lemma 3.4, \begin{eqnarray*}\lefteqn{ \tilde{H}^{(L,Ex)} \prod_{\gamma = 1}^p \Delta (y^{(\gamma)}) m_\kappa(x) = \left( \delta^{(L)} + (1 + {p \over \alpha}) \left[ \sum_{j=1}^{N_0} x_j{\partial^2 \over \partial x_j^2} +\Big ( a\alpha/(p+\alpha) +1 \right.\right. }\\ &&\left.\left. -x_j \alpha/(p+\alpha) \Big ) {\partial \over \partial x_j} + {2 \over \alpha + p} \sum_{j \ne k}{x_j \over x_j - x_k}{\partial \over \partial x_j} \right] \right) \prod_{\gamma = 1}^p \Delta (y^{(\gamma)}) m_\kappa(x). \end{eqnarray*} where $\delta^{(L)}$ is the eigenvalue for the action of $\tilde{H}^{(L,y)}$. Since the operator in square brackets is the defining eigenoperator for $L_\kappa^{\alpha a/(p + \alpha)} (\alpha x/(p + \alpha); \alpha)$, it follows that (\ref{CON2}) is an eigenfunction of $\tilde{H}^{(L,Ex)}$ as required. In the case $p=1$, $q=0$ we have the analogue of (\ref{ANTIS}) and (\ref{ANTIS1}): \begin{equation} S_\mu^{(L)}(y,\alpha) = \Delta (y) {(-1)^{|\kappa|} |\kappa|! C_\kappa^{(\alpha)}(1^N) \over v_{\kappa \kappa}(\alpha/(1+\alpha))} L_\kappa^a(y;\alpha/(1+\alpha)), \end{equation} which is derived according to the same method. \section{Eigenfunctions of $\tilde{H}^{(J,Ex)}$} \setcounter{equation}{0} \renewcommand{\theequation}{\thesection.\arabic{equation}} \subsection{Decomposition} The operators, $\hat{D}_j^{(J)}$ say, which provide a decomposition of $\tilde{H}^{(J,Ex)}$ analogous to the decomposition (\ref{FAC}) of $\tilde{H}^{(C,Ex)}$ have been given by Bernard et al.~\cite{bern95a} and Hikami \cite{hik96a}. We have \begin{equation} \hat{D}_j^{(J)} = \hat{D}_j - {1 \over \alpha} \sum_{k=1 \atop \ne j}^N {1 - S_j S_k M_{jk} \over 1 - z_j z_k} - (a+{1 \over 2}) {1 - S_j \over 1 - z_j} - (b+{1 \over 2}){1 - S_j \over 1 + z_j} + {1 \over 2}(a+b+1) \label{CJ} \end{equation} and \begin{equation} \tilde{H}^{(J,Ex)} = \sum_{j=1}^N \Big ( \hat{D}_j^{(J)} \Big )^2 -{1 \over 4} E_0^{(J)}. \label{HJC} \end{equation} We want to investigate the eigenfunctions of $\hat{D}_j^{(J)}$ and relate them to the eigenfunctions of $\tilde{H}^{(J,Ex)}$. To begin, we note that $\hat{D}_j^{(J)}$ is self-adjoint with respect to the inner product \begin{equation} \langle f | g \rangle^{(J)} := \int_0^{\pi/2} d\phi_1 \dots \int_0^{\pi/2} d\phi_N \, |\psi_0^{(J)}|^2 f(z_1^*,\dots,z_N^*) g(z_1,\dots,z_N) \label{IPJ} \end{equation} where $z_j = e^{2 i \phi_j}$ and \begin{equation} \psi_0^{(J)} := \prod_{j=1}^Nz_j^{-(N-1)/\alpha - (a+b+1)/2} (z_j-1)^{a+1/2} (z_j + 1)^{b+1/2} \prod_{j<k} (z_j - z_k)^{1/\alpha} (1 - z_j z_k)^{1/\alpha}. \end{equation} Note that $|\psi_0^{(J)}|^2$ is proportional to $e^{-\beta W^{(J)}}$ where $W^{(J)}$ is given by (\ref{WJ}), and is thus proportional to the square of the symmetric ground state wave function of $H^{(J,Ex)}$. The self-adjointness is easily checked upon using the operator identity \begin{eqnarray}\lefteqn{ \psi_0^{(J)} \hat{D}_j^{(J)} (\psi_0^{(J)})^{-1} = z_j {\partial \over \partial z_j} - {1 \over \alpha} \Big ( \sum_{l < j} {z_l \over z_j - z_l} M_{lj} + \sum_{l > j} {z_j \over z_j - z_l} M_{lj} \Big )} \hspace{3cm}\nonumber \\ &&+{1 \over \alpha} \sum_{k=1 \atop \ne j} {S_j S_k M_{jk} \over 1 - z_j z_k} + (a+1/2) { S_j \over 1 - z_j} + (b+1/2){ S_j \over 1 + z_j}. \label{REMA} \end{eqnarray} Also, we remarked in a previous study \cite{forr96a} that $\tilde{H}^{(J,Ex)}$ has a complete set of symmetric polynomial eigenfunctions $G_\kappa^{(a,b)}(y;\alpha)$ (the generalized Jacobi polynomials) where $y=\sin^2 \phi = - (z+1/z-2)/4$. To specify the eigenvalues and eigenfunctions of (\ref{CJ}), let $\eta$ be an $N$-tuple of non-negative integers as in (\ref{ETA}), and define the partial order as below (\ref{ETA}). Let $\epsilon$ be an $N$-tuple with each entry $+1$ or $-1$, and define $\epsilon \eta$ as the $N$-tuple formed from $\epsilon$ and $\eta$ by multiplication of the respective parts. A direct calculation shows \begin{equation} \hat{D}_j^{(J)} \, z^{\epsilon \eta} =e_{j,\epsilon\eta}^{(J)} z^{\epsilon \eta} + \sum_{\eta' < \eta} \sum_{\epsilon'} c_{\epsilon \eta, \epsilon' \eta'}z^{\epsilon' \eta'} \label{CHA} \end{equation} where \begin{eqnarray} e_{j,\epsilon\eta}^{(J)} & = & \epsilon_j \eta_j + {1 \over \alpha} \Big ( - \sum_{l<j} h(\epsilon_l \eta_l - \epsilon_j \eta_j) + \sum_{l>j} h(\epsilon_j \eta_j - \epsilon_l \eta_l) + \sum_{k=1 \atop \ne j}^N h(-\epsilon_j \eta_j -\epsilon_k \eta_k) \Big ) \nonumber \\ & & + {1 \over \alpha}(j-1) - (a+b+1) \Big ( h(-\epsilon_j \eta_j) - {1 \over 2} \Big ) \label{CEVJ} \end{eqnarray} with $h(x)$ defined by (\ref{HX}). It follows that $\hat{D}_j^{(J)}$ has a complete set of Laurent polynomial eigenfunctions such that when the highest weight monomial is $z^{\epsilon \eta}$ the corresponding eigenvalue is $e_{j,\epsilon\eta}^{(J)}$. The fact that each operator $\hat{D}_j^{(J)} $ has a unique Laurent polynomial eigenfunction with leading monomial $z^{\epsilon \eta}$ and that the operators $\{ \hat{D}_j^{(J)} \}$ mutually commute imply each eigenfunction is simultaneously an eigenfunction of all the operators $\hat{D}_1^{(J)}, \dots,\hat{D}_N^{(J)}$. Next we note that the eigenvalues $e_{j,\epsilon\eta}^{(J)}$ and $e_{j,\epsilon\eta}^{(J)} \Big |_{\epsilon_p \mapsto - \epsilon_p}$ are simply related. \vspace{.2cm} \noindent {\bf Lemma 5.1} \quad We have $$ e_{j,\epsilon\eta}^{(J)} \Big |_{\epsilon_p \mapsto - \epsilon_p} = e_{j,\epsilon\eta}^{(J)} \quad (j \ne p), \qquad e_{j,\epsilon\eta}^{(J)} \Big |_{\epsilon_p \mapsto - \epsilon_p} = -e_{j,\epsilon\eta}^{(J)} \quad (j = p). $$ \vspace{.2cm} \noindent {\bf Proof} \quad This follows directly from (\ref{CEVJ}). \vspace{.2cm} \noindent A consequence of Lemma 5.1 is that $(\hat{D}_j^{(J)})^2$ permits Laurent polynomial eigenfunctions, $E_\eta^{(J)}(y)$ say ($y=-(z+1/z-2)/4$), with leading term $y^{\eta}$ and corresponding eigenvalue $(e_{j,\epsilon\eta}^{(J)})^2 =(e_{j,\eta}^{(J)})^2 $. Now $e_{j,\eta}^{(J)} = e_{j,\eta} + (a + b +1)/2$ where $e_{j,\eta}$ is given by (\ref{EV}). From Lemma 2.1 we thus have that $\{ (e_{j,\eta}^{(J)})^2 \}$ is independent of the permutation in the equation $\eta = P \kappa$. By following the argument used in the last two paragraphs of Section 2.1 we conclude that \begin{equation} \Big ( 1 + u (\hat{D}_1^{(J)})^2 \Big ) \dots \Big ( 1 + u (\hat{D}_N^{(J)})^2 \Big ) \label{PEJ} \end{equation} is an eigenoperator for the symmetric (Laurent) polynomial eigenfunctions of (\ref{HJC}) with leading term proportional to $y^\kappa$. As remarked below (\ref{REMA}), these polynomials are the generalized Jacobi polynomials $G_\kappa^{(a,b)}(y;\alpha)$. The corresponding eigenvalue is \begin{equation} \prod_{j=1}^N \Big ( 1 + u \Big ( \kappa_j + (N - j)/\alpha + (a + b +1)/2 \Big )^2 \Big ). \label{SEVJ} \end{equation} Now since each operator $\hat{D}_j^{(J)}$ is self-adjoint with respect to the inner product (\ref{IPJ}), the operator (\ref{PEJ}) is also self-adjoint with respect to this inner product. Furthermore, from the inequalities (\ref{OR}) each eigenvalue (\ref{SEVJ}) is distinct. This immediately implies (as has been proved before \cite[in the latter reference an operator equivalent to (\ref{PEJ}) when restricted to acting on symmetric functions of $y$ is also constructed]{beer93a,vandiej95c} that the generalized Jacobi polynomials $\{G_\kappa^{(a,b)}(y;\alpha) \}$ are orthogonal with respect to the inner product (\ref{IPJ}). Note that the operator (\ref{PONS}) with $\hat{D}_j$ replaced by $(\hat{D}_j^{(J)})^2$ can be used to show that $\{ E_\eta^{(J)} \}$ is orthogonal with respect to (\ref{IPJ}). \subsection{Generalized Jacobi polynomials with prescribed symmetry} {}From the decomposition (5.2) and the fact that $(\hat{D}_j^{(J)})^2$ and $\tilde{H}^{(J,Ex)}$ both have unique polynomial eigenfunctions with leading term $y^\eta$, we conclude that $E_{\eta}^{(J)}$ can be specified as the eigenfunction of (\ref{HJC}) with leading term $y^\eta$. To pursue this characterization, it is convenient to introduce the variable $y = \sin^2 \phi$ in (\ref{HJE}) and repeat the computation of (\ref{HTJE}). This gives \begin{eqnarray} - \tilde{H}^{(J,Ex)} & = & \sum_{j=1}^N y_j (1 - y_j) {\partial^2 \over \partial y_j^2} + (a+1) \sum_{j=1}^N {\partial \over \partial y_j} - (a+b + 2 +{2\over\alpha}(N-1)) \sum_{j=1}^N y_j{\partial \over \partial y_j} \nonumber \\ & & + {1 \over \alpha} \sum_{j \ne k}{1 \over y_j - y_k} \Big ( 2y_j(1 - y_k) {\partial \over \partial y_j} - {y_j(1 - y_k) \over y_j - y_k} (1 - M_{jk}) \Big )\nonumber \\& := &U + V \label{RAD} \end{eqnarray} where \begin{equation} U := - \tilde{H}^{(C,Ex)} - (a+b+1+(N-1)/\alpha) \sum_{j=1}^N y_j{\partial \over \partial y_j}, \quad V = \tilde{H}^{(L,Ex)} + \sum_{j=1}^N y_j{\partial \over \partial y_j}. \end{equation} Following ref.~\cite{sogo96} we note that the operator $U$ is an eigenoperator for the non-symmetric Jack polynomials $E_\eta^{(J)}$ while the operator $V$ reduces by one the degree of a homogeneous polynomial (c.f.~(\ref{HTLE})). Using these facts and noting $E_\eta(y,\alpha)$ has leading term $y^\eta$, we see by proceeding as in the derivation of (\ref{EX1}) that \begin{equation} E_\eta^{(J)}(y,\alpha) = \Big (1 - (\hat{U} + V)^{-1}V \Big )E_\eta(y,\alpha). \label{SO2} \end{equation} where with $\eta = P^{-1} \kappa$, \begin{equation} \hat{U} = U + \sum_{j=1}^N \Big ( \kappa_j^2 + 2 \kappa_j (N-j)/\alpha + \kappa_j (a+b+1) \Big ). \end{equation} Note that if this operator is expanded according to (\ref{OID}) the series terminates after $|\eta|$ applications of $V$. The generalized Jacobi polynomials with prescribed symmetry, $S_{(\rho,\mu)}^{(J)}(y,\alpha)$ say, are defined as the eigenfunctions of (\ref{HJC}) given by the formula \begin{equation} S_{(\rho, \mu)}^{(J)}(y,\alpha) =\Big (1 - (\hat{U} + V)^{-1}V \Big ) S_{(\rho, \mu)}^{(J)}(y,\alpha). \label{SO3} \end{equation} (for any polynomial $p$ which is an eigenfunction of $\hat{U}$, $(1 - (\hat{U} + V)^{-1}V)\,p$ is an eigenfunction of (\ref{HJC})). {}From (\ref{SO3}) and (\ref{SSUM}) it follows that \begin{equation} S_{(\rho, \mu)}^{(J)}(y,\alpha) = \sum_{\rm rearrangements} b_{Q^{-1}\kappa} E_{Q^{-1}\kappa}^{(J)}(y,\alpha), \end{equation} which in turn implies (\ref{PCO}) is an eigenoperator for $\{S_{(\rho, \mu)}^{(J)} \}$ with $\hat{D}_j$ therein replaced by $(\hat{D}_j^{(J)})^2$. This operator is self-adjoint with respect to (\ref{IPJ}) and separates the eigenvalues, so we conclude that $\{S_{(\rho, \mu)}^{(J)}\}$ is an orthogonal set with respect to the inner product (\ref{IPJ}), with an appropriate change of variables (see ref.~\cite[eq.~2.17]{forr96a}). \subsection{Some special generalized Jacobi polynomials with prescribed symmetry} Here we want to establish the analogue of (\ref{CON}) in the Jacobi case: \begin{equation} S_\eta^{(J)}(y,\alpha) = A_{\eta}^{(J)} \prod_{\gamma = 1}^p \prod_{1 \le j < k \le N_\gamma} (y_k^{(\gamma)} - y_j^{(\gamma)}) \, G_\rho^{(u,v)}(x_1,\dots,x_{N_0};p+\alpha), \label{CONJJ} \end{equation} where $ A_{\eta}^{(J)}$ is some normalization, $\eta$ is given by (\ref{PK}), $\rho_1$ satisfies (\ref{IN}) and \begin{equation} u := {\alpha \over \alpha + p} \Big ( {p \over \alpha} + a + 1 \Big ) - 1, \qquad v:= {\alpha \over \alpha + p} \Big ( {p \over \alpha} + b + 1 \Big ) - 1 \end{equation} Now set $N= N_0 + \sum_{\gamma=1}^p N_\gamma$ and denote the variables $y_1,\dots,y_N$ as $\{x_j\}_{j=1,\dots,N_0}$ and $\{y_j^{(\gamma)}\}_{j=1, \dots,N_\gamma}$ $(\gamma = 1,\dots,p)$. Analogous to (\ref{DEC}), for $\tilde{H}^{(J,Ex)}$ acting on functions symmetric in $\{x_j\}$ and anti-symmetric in $\{y_j^{(\gamma)} \}$ we make the decomposition \begin{equation} \tilde{H}^{(J,Ex)} = \tilde{H}^{(J,x)} + \tilde{H}^{(J,y)} +\tilde{H}^{(J,xy)} \end{equation} where \begin{eqnarray} \tilde{H}^{(J,x)} & = & \sum_{j=1}^{N_0} \Big ( x_j(1-x_j){\partial^2 \over \partial x_j^2} + (a+1) {\partial \over \partial x_j} - (a+b+2+{2 \over \alpha}(N-1))x_j {\partial \over \partial x_j} \Big ) \nonumber \\ & & +{2 \over \alpha} \sum_{j \ne k} {x_j (1 - x_k) \over x_j - x_k} {\partial \over \partial x_j} \end{eqnarray} \begin{eqnarray} \tilde{H}^{(J,xy)} & = & {1 \over \alpha} \sum_{\gamma=1}^p \sum_{j=1}^{N_\gamma} \sum_{k=1}^{N_0} {1 \over y_j^{(\gamma)} - x_k} \left[ 2 y_j^{(\gamma)} (1 - x_k) {\partial \over \partial y_j^{(\gamma)}} - 2 x_k (1 - y_j^{(\gamma)}) {\partial\over \partial x_k} \nonumber \right.\\ && \left. - { y_j^{(\gamma)} + x_k - 2y_j^{(\gamma)} x_k \over y_j^{(\gamma)} - x_k} (1 - M( y_j^{(\gamma)},x_k)) \right]. \end{eqnarray} and $\tilde{H}^{(J,y)}$ is given by (\ref{HTJE}) with $y_1,y_2\dots,y_N$ replaced by $y_1^{(1)},y_2^{(1)},\dots,y_{N_p}^{(p)}$. In fact we have that $\tilde{H}^{(J,y)}$ is a linear combination of $\tilde{H}^{(C,y)}$ and $\tilde{H}^{(L,y)}$ and hence is an eigenoperator of $\prod_{\gamma = 1}^p \Delta(y^{(\gamma)})$ with eigenvalue $\delta^{(J)}$ say. Also, from (\ref{DEFHCWZ}) and (\ref{HLXY}) we have $$ \tilde{H}^{(J,xy)} = - \tilde{H}^{(C,xy)} + \tilde{H}^{(L,xy)} $$ so when acting on $F(x,y)$ as defined in Lemma 2.4 we see from Lemmas 2.4 and 4.5 that $$ \tilde{H}^{(J,xy)} = {p \over \alpha} \sum_{j=1}^{N_0} x_j (1 - x_j) {\partial^2 \over \partial x_j^2} + {2 \over \alpha} (N^{(z)} - p) \sum_{j=1}^{N_0} x_j{\partial \over \partial x_j} + {p \over \alpha} \sum_{j=1}^{N_0} {\partial \over \partial x_j}. $$ Hence, for $\tilde{H}^{(J,Ex)}$ acting on $F(x,y)$ we have $$ \tilde{H}^{(J,Ex)} = \delta^{(J)} + {p + \alpha \over \alpha} {\cal H} $$ where ${\cal H}$ is the operator (\ref{RAD}) with $N=N_0$, the variables $y_1, \dots,y_N$ replaced by $x_1,\dots,x_N$, $M_{jk} = 1$, $a=u$, $b=v$ and $\alpha$ replaced by $\alpha + p$. The operator ${\cal H}$ is the defining eigenoperator for the generalized Jacobi polynomial in (\ref{CONJJ}), thus establishing the validity of that equation. In the case $p=1$, $q=0$ we have the analogue of (\ref{ANTIS}) and thus another explicit formula: \begin{equation} S_\mu^{(J)}(y,\alpha) = \Delta (y) {1 \over v_{\kappa \kappa}(\alpha/(1+\alpha))} G_\kappa^{(a,b)}(y;\alpha/(1+\alpha)). \end{equation} This result is derived from (\ref{HJE}) in the same manner as (\ref{ANTIS}) is derived from (\ref{HCE}). \section{Conclusion} The Schr\"odinger operators (\ref{HCC}) and (\ref{HTHE})-(\ref{HTJE}) admit polynomial eigenfunctions of the form given on the r.h.s.~of (\ref{nj}). The most basic of these polynomials are the non-symmetric eigenfunctions of (\ref{HCC}), which are referred to as the non-symmetric Jack polynomials. In (\ref{EX2}), (\ref{EX5}), (\ref{SO2}) we give operator formulas which transform the non-symmetric Jack polynomials to the non-symmetric polynomial eigenfunctions of (\ref{HTHE})-(\ref{HTJE}). The operators (\ref{HCC}) and (\ref{HTHE})-(\ref{HTJE}) also admit bases of fully symmetric polynomial eigenfunctions, and polynomial eigenfunctions with a prescribed symmetry. We have established orthogonality of these sets of eigenfunctions with respect to inner products defined as multidimensional integrals, with the corresponding symmetric ground state wave function as the weight function. For the fully symmetric polynomials the orthogonality has previously been established, but for the polynomials with prescribed symmetry this result is new. We expect the polynomials with prescribed symmetry to be relevant to the calculation of correlation functions in the Calogero-Sutherland model with spin (see e.g.~ref.~\cite{kk95} and references therein). For this purpose we require normalization formulas, and an expansion formula expressing the power sum in terms of the Jack polynomials with prescribed symmetry. Regarding the former, we point out that for the special Jack polynomials with prescribed symmetry of Section 2.3, a conjecture for the normalization has been made in ref.~\cite{forr96c}. \vspace{.5cm} \noindent {\bf Acknowledgements} \vspace{.2cm} \noindent We thank K.~Takemura (RIMS) for a useful remark, and T.~Miwa for his hospitality at RIMS where this work was initiated. Also, correspondence with C.~Dunkl is appreciated. \bibliographystyle{unsrt}

proofpile-arXiv_065-559

\section{Introduction} Several years ago, Schwinger wrote a series of papers~\cite{Schwinger0,Schwinger1,Schwinger2} wherein he calculated the Casimir energy released in the collapse of a spherically symmetric bubble or cavity. Using general arguments, he showed that the effect was mostly a volume effect, and derived a simple and elegant formula for the energy release involved in the collapse. He found that (for each polarization state) the {\em ``dielectric energy, relative to the zero energy of the vacuum, [is given] by} \begin{equation} E = - V \int \frac{d^3\vec{k}}{(2 \pi)^3} \frac{1}{2} \,[\hbar c]\, k \left( 1 - \frac{1}{\sqrt{\epsilon}} \right). \label{E-Schwinger-0} \end{equation} \noindent {\em So the Casimir energy of a uniform dielectric is negative"}. {}From the above one finds that a dielectric slab with a spherical vacuum cavity of radius $R$ has a higher Casimir energy than the same slab of material with the cavity re--filled with dielectric. Introducing a wave-number cutoff $K$ into the previous expression,~\footnote{For sonoluminescence, this wave-number cutoff can be related to the wavelength of the electromagnetic radiation emitted in the collapse of the bubble. For generic dielectrics, this wave-number cutoff is related to the high-wave-number asymptotic behaviour of the dispersion relation.} and summing over polarization states, shows that the Casimir energy of a cavity in a dielectric, relative to pure dielectric, is \begin{eqnarray} E_{cavity} &=&+2\frac{4\pi}{3}R^3 \; \int_0^K {4 \pi k^2 dk\over(2\pi)^3} \frac{1}{2} \hbar c k \left( 1 - \frac{1}{\sqrt{\epsilon}} \right) \nonumber\\ &=&+\frac{1}{6 \pi} \hbar c R^3 K^4 \left(1 - \frac{1}{\sqrt{\epsilon}} \right). \end{eqnarray} \noindent In general, this volume term will be the dominant contribution. In view of the elegance and simplicity of this result, it is natural to ask whether it can also be derived by more traditional quantum field theoretic means. Indeed, the existence of such a volume contribution is easy to verify on general physical grounds: \noindent (1) One can view Schwinger's result in elementary terms as simply the difference in zero-point-energies, obtained by integrating the difference in photon dispersion relations over the density of states\footnote{The dots denote finite-volume corrections. We shall develop this density-of-states point of view more fully in a separate publication.} \begin{equation} E_{ cavity} = + 2 V \int \frac{d^3\vec{k}}{(2 \pi)^3} \frac{1}{2} \hbar \left[ c k - \omega(k) \right] + \cdots \end{equation} \noindent At low wave-numbers, we know that the dispersion relation for a dielectric is simply summarized by the zero-frequency refractive index $n$. That is \begin{equation} \omega(k) \to c k/n \qquad {\rm as} \qquad k \to 0. \end{equation} On the other hand, at high enough wave-numbers, the photons propagate freely through the dielectric: They are then simply free photons travelling through the empty vacuum between individual atoms. Thus \begin{equation} \omega(k) \to c k \qquad {\rm as} \qquad k \to \infty. \end{equation} {}From the above we know that the integrand must go to zero at large wave-number. In fact for any real dielectric the integrand must go to zero sufficiently rapidly to make the integral converge, since after all we are talking about a real physical difference in energies. To actually calculate this energy difference one requires a suitable physical model for $\omega(k)$. Schwinger's calculation~\cite{Schwinger0}, is equivalent to picking the particularly simple model \begin{equation}\label{omegamodel} \omega(k) = {c k\over n} \; \Theta(K-k) + ck \; \Theta(k-K). \label{eqn7} \end{equation} \noindent Here $\Theta(x)$ is the Heaviside step function, and $K$ is a wave-number which characterizes the transition from dielectric-like behaviour to vacuum-like behaviour. Note that the cutoff $K$ describes an actual physical situation: It is a surrogate for all of the complicated physics that would be required to make a detailed model for the dielectric to vacuum transition. \noindent (2) We also know that the quantum action in 3+1 dimensions generically contains divergences which range from quartic to logarithmic, in addition to finite contributions. As is well known, this ``cosmological constant" contribution (the quartic divergence) will not vanish unless the theory has very special symmetries (for example---super-symmetry). Thus energy densities that go as $({\rm cutoff})^4$ are {\em generic} in (3+1) dimensions. \noindent (3) Alternatively, one could perform an explicit quantum field theoretic calculation of the Casimir energy in some model problem and thereby verify Schwinger's result. A step in this direction has been provided by Milton {\em et al.}~\cite{Milton80,Milton95,Milton96}, who attempted to compute the Casimir energy associated with a spherical cavity of radius $R$, dielectric constant $\epsilon_1$, and permeability $\mu_1$, embedded in an infinite medium with dielectric constant $\epsilon_2$ and permeability $\mu_2$. They found that the dominant term is not the volume term but a surface term which is proportional to $R^2 K^3 (\epsilon_1 -\epsilon_2)^2$. We, however, have re-analysed these calculations and do find a volume term which dominates except for very small bubbles. We have performed the calculation of the Casimir energy in two different and complementary ways: \noindent (a) We have taken the formalism of Milton {\em et al.}~\cite{Milton80,Milton95,Milton96} and applied it directly to an {\em ab initio} calculation of the Casimir energy. We compute the energy difference between the following configurations: (Case I) an otherwise uniform medium with dielectric constant $\epsilon_2$ and permeability $\mu_2$, containing a spherical cavity of radius $R$ with dielectric constant $\epsilon_1$ and permeability $\mu_1$, and (Case II) a completely uniform medium with dielectric constant $\epsilon_2$ and permeability $\mu_2$. This energy difference is given as a sum over a series of integrals involving Ricatti--Bessel functions. Some of the sums can be evaluated explicitly while others can only be evaluated by using an asymptotic analysis of the type used by Milton {\em et al.} We verify the existence of both volume and sub-dominant surface contributions. \noindent (b) We have analyzed the extant calculations to see exactly where they differ from the present calculation. We find that the subtraction scheme they use to calculate the Casimir energy does not correspond to the physical situation in question. We isolate the difference in energy between these calculations and the correct one. We will explicitly show that this difference is proportional to volume. \section{Physical description of the calculation} Milton {\em et al.}~\cite{Milton80,Milton95,Milton96} explicitly calculated the electromagnetic Green functions for a dielectric ball embedded in an infinite space of (different) dielectric material, and then attempted to calculate the Casimir energy by explicitly integrating these Green functions over ``all space". Note that an important limitation of any such calculation is that any attempt at explicitly calculating Green functions must be restricted to systems of extremely high symmetry---such as half-spaces, slabs, or balls. The basic strategy is to take the classical expression for the energy \begin{eqnarray} E &=& {1\over2} \int_{Geometry} \left[ \epsilon \vec E^2 + {1\over\mu} \vec { B}^2 \right] d^3x, \end{eqnarray} \noindent promote the electric and magnetic fields to be operator quantities, and then calculate the vacuum expectation value \begin{eqnarray} E = {1\over2} \int_{Geometry} \big[ &\epsilon& \langle \vec E(0,x) \cdot \vec E(0,x) \rangle \nonumber\\ + &{1\over\mu}& \langle \vec { B}(0,x) \cdot \vec { B}(0,x) \rangle \big] d^3x. \end{eqnarray} \noindent The geometry is incorporated in the calculation both via the limits of integration and via the boundary conditions satisfied by the fields. Since these two-point functions are of course divergent, they must be rendered finite by some regularization prescription. Milton {\em et al.} use point-splitting in the time direction: \begin{eqnarray} E(\tau) = {1\over2} \int_{Geometry} \big[ &\epsilon& \langle \vec E(\tau,x) \cdot \vec E(0,x) \rangle \nonumber\\ + &{1\over\mu}& \langle \vec { B}(\tau,x) \cdot \vec { B}(0,x) \rangle ] d^3x. \end{eqnarray} All of the technical aspects of the analysis then focus on the calculation of these two-point correlation functions (Green functions) by explicitly solving for the TE and TM modes appropriate for a spherical ball with dielectric boundary conditions; and then explicitly writing down the Green functions as a sum over suitable combinations of Ricatti--Bessel functions and vector spherical harmonics. To avoid unnecessary notational complications, we schematically rewrite the above as \begin{equation} E(\tau) = {1\over2} \int_{Geometry} G_{[\epsilon,\mu]}(\tau,x;0,x) \; d^3x, \end{equation} \noindent where $G_{[\epsilon,\mu]}(t,x;t',x')$ is simply shorthand for the linear combination of Green functions appearing above. We may calculate these Green functions for {\em three} different geometries\footnote{Notice that in Milton {\it et al.} the dielectric properties of these media are taken to be frequency independent, the cutoff being put in ``by hand" via time-splitting.}: \\ {\bf Case I:} A dielectric ball of dielectric constant $\epsilon_1$, permeability $\mu_1$, and radius $R$ embedded in a infinite dielectric of {\em different} dielectric constant $\epsilon_2$ and permeability $\mu_2$. (In applications to sonoluminescence, think of this as an air bubble of radius $R$ in water.) \\ {\bf Case II:} A completely homogeneous space completely filled with dielectric $(\epsilon_2,\mu_2)$. (In applications to sonoluminescence, think of this as pure water.) \\ {\bf Case III:} A completely homogeneous space completely filled with dielectric $(\epsilon_1,\mu_1)$. (In applications to sonoluminescence, think of this as pure air.)\\ We are in complete agreement with the extant calculations and results for these three individual Green functions---where we disagree, as will be shown, is {\em in the way that these three Green functions are inserted into the computation for the Casimir energy.} Milton {\em et al.} calculate an ``energy difference", which we will call $E_{surface}$, and which they define as \begin{equation} E_{surface} = {1\over2} \left\{ \int_{ \small all\ r} G_I - \int_{r>R} G_{II} - \int_{r<R} G_{III} \right\}. \end{equation} The computation of this quantity in~\cite{Milton80,Milton95,Milton96} is mathematically correct. An asymptotic analysis shows that this expression is indeed proportional to the surface area---plus even higher-order terms. However, the energy calculated from the above expression does not correspond to the energy of a physically realizable situation. The physically correct quantity to compute is~\cite{Schwinger0} \begin{equation} \label{E-casimir-define} E_{Casimir} = {1\over2} \left\{ \int_{\small all\ r} G_I - \int_{\small all\ r} G_{II} \right\}. \end{equation} Observe that this quantity is simply the difference in energy between two real physical situations: (Case I) having the dielectric ball present and (Case II) replacing the dielectric ball by the surrounding medium. {\em This is exactly the quantity that Schwinger calculates in reference~\cite{Schwinger0} to describe the Casimir enegy released in the collapse of the bubble: it is the energy released in evolving from bubble to no--bubble.} The difference between the two calculations is \begin{equation} \Delta E = E_{Casimir} - E_{surface} = {1\over2} \int_{r<R} \left\{ G_{III} - G_{II} \right\}. \end{equation} \noindent This difference is easily seen to be proportional to the volume: remember that $G_{III}$ and $G_{II}$ are Green functions corresponding to two spaces that are completely filled with homogeneous dielectrics---therefore they are each individually translation invariant. (When one expresses these Green functions in terms of spherical polar coordinates this is not obvious.) This observation permits one to pull the Green functions outside the integral, so that \begin{equation} \Delta E = {1\over 2} V \left\{ G_{III}(\tau,0;0,0) - G_{II}(\tau,0;0,0) \right\}, \end{equation} \noindent where $V$ is the volume of the ball of radius $R$. We shall now show that this term is in fact exactly in conformity with Schwinger's result. \section{ ``Ab initio" calculation} \subsection{The energy density} We now calculate the Casimir energy for the geometrical configuration previously described. We use techniques developed by Milton {\em et al.}, but use, as Schwinger did, a wave number cutoff and shall present the calculation in as much detail as space permits. We defer many technical details to a forthcoming publication. For each individual geometry, the energy density ${ T}^{ tt}$ can be evaluated by using the dyadic Green function formalism~\cite{Milton80,Milton95,Milton96}. (Henceforth we use natural units.) For Case I one finds \begin{equation}\label{E-energy-density} T^{tt}_I(r) = {\rm Re} \left[{-i\over8\pi} \int_{-\infty}^{+\infty} {d\omega\over 2\pi} e^{-i\omega\tau} \; X_I(k,r)\right], \end{equation} \noindent with identical expressions holding for the other geometries. Here the $\omega$--integral arises from the time-splitting regularization, while we have used the notation $k = |\omega| n$ with $n$ the appropriate position--dependent refractive index { ($n_1$ inside the dielectric sphere, $n_2$ outside)}, and have defined the quantity $X_I(k,r)$ by \begin{eqnarray} X_I(k,r) &\equiv& \sum_{\ell=1}^\infty (2\ell+1) \Bigg\{ \left[ k_{I}^2 + {\ell(\ell+1)\over r^2} \right] F^{I}_\ell(k;r,r) \nonumber\\ &&+ {1\over r^2} {\partial\over\partial r_1} r_1 {\partial\over\partial r_2} r_2 \left[ F^{I}_\ell(k;r_1,r_2) \right]\Big|_{r_1=r_2=r} \Bigg\} \nonumber\\ &&+ \left[ (F^I_\ell) \to (G^I_\ell) \right]. \end{eqnarray} \noindent The functions $F_\ell (r,r')$ and $G_\ell (r,r')$ are the Green functions for the electrical and magnetic fields in the appropriate geometry, and are given below. Similar results, with appropriate substitutions for the momenta, hold when one makes reference to Cases II and III. Note that when making the substitutions $(I)\to(II)$ or $(I)\to(III)$, one should also change the refractive index that implicitly appears in the factor $k$. In addition it should be borne in mind that $k_I$ is a function of position: $k_I = n_1 |\omega|= k_{III}$ inside the dielectric sphere, whereas $k_I = n_2 |\omega| = k_{II}$ outside the dielectric sphere. Because $F_\ell$ and $G_\ell$ depend only on the {\em absolute} value of $\omega$ we can write the energy density as \begin{eqnarray} T^{tt}_I(r) &=& {\rm Re} \left[ {-i\over8\pi} \int_{0}^{\infty} {d\omega\over 2\pi} \left[ e^{-i\omega\tau} + e^{+i\omega\tau} \right] X_I(k,r) \right] \nonumber\\ &=& {\rm Re} \left[{-i\over4\pi} \int_{0}^{\infty} {d\omega\over 2\pi} \cos(\omega\tau) \; X_I(k,r) \right] \nonumber\\ &=& {1\over4\pi} \int_{0}^{\infty} {d\omega\over 2\pi} \cos(\omega\tau) \; {\rm Im}\left[X_I(k,r)\right]. \label{E-density-formula} \end{eqnarray} \noindent The Casimir energy, Eq. (\ref{E-casimir-define}) is obtained by taking the {\em difference} in energy densities and integrating over all space while paying attention to the appropriate index of refraction for each region of space. Thus \begin{eqnarray} \label{E-casimir-energy-1} E_{Casimir} &=& \int_0^\infty r^2 dr \; \int_0^{\infty} {d\omega\over 2\pi} \cos(\omega\tau) \nonumber\\ && \times {\rm Im}\left[X_I(k,r)-X_{II}(k,r)\right]. \end{eqnarray} This expression for the Casimir energy is completely equivalent to equation (41) of~\cite{Milton95}, and equation (4.2b) of~\cite{Milton96} and is also closely related to equations (30a) and (30b) of~\cite{Milton80}. Note that extant calculations use the same time-splitting parameter for the two different media---the physics behind this choice is far from clear, and we shall return to this point in a future publication. It is now clear how one should modify these expressions to replace time-splitting regularization by a wave-number cutoff. For generality we can take an arbitrary wave-number cutoff described by some smooth real function $f(k)$ which goes to zero as $k \rightarrow \infty$ and simply write \begin{eqnarray} \label{E-casimir-energy-2} E_{Casimir} &=& \int_0^\infty r^2 dr \; \int_0^\infty {d\omega\over 2\pi} \nonumber\\ && \times {\rm Im}[f(k_I) X_I(k,r)- f(k_{II}) X_{II}(k,r)]. \end{eqnarray} With due caution, the relevant Green functions can be read off from~\cite{Milton80,Milton95,Milton96}. \subsection{The Green functions} In evaluating the Green functions one must be careful to correctly incorporate the boundary conditions appropriate to the geometry and the physics. This means that they must satisfy appropriate continuity conditions derived from Maxwell's equations. That is \begin{equation} \vec E_\perp, \quad \epsilon \vec E_r, \quad {1\over\mu}\vec B_\perp, \quad {\rm and} \quad \vec B_r, \end{equation} \noindent must be continuous. In terms of $F_{\ell}$ and $G_{\ell}$, one sees that \begin{equation} \mu F_{\ell}, \quad G_{\ell}, \quad {\partial \over \partial r}{r F_{\ell}}, \quad {\rm and} \quad \frac{1}{\epsilon}{\partial \over \partial r}{r G_{\ell}}, \end{equation} \noindent must be continuous. The Green functions are \noindent {\bf Case I:} \\ For $r_1, r_2 < R$: \begin{eqnarray} F^I_\ell, G^I_\ell(r_1,r_2) &=& i k_{III} \; j_\ell(k_{III} r_<) \nonumber\\ &&\times \left[h_\ell(k_{III}r_>) - A^\ell_{F,G} \; j_\ell(k_{III} r_>) \right]. \label{E-i-a} \end{eqnarray} For $r_1, r_2 >R$: \begin{eqnarray} F^I_\ell, G^I_\ell(r_1,r_2) &=& i k_{II} \; h_\ell(k_{II} r_>) \nonumber\\ &&\times \left[j_\ell(k_{II}r_<) - B^\ell_{F,G} \; h_\ell(k_{II} r_<) \right]. \label{E-i-b} \end{eqnarray} \noindent The function $j_\ell(x)$ is the spherical Bessel function of order $\ell$ and $h_\ell(x) \equiv h_\ell^{(1)} (x)$ is the spherical Hankel function of the first kind. See equations (12a) and (12b) of~\cite{Milton80}, equation (16) of~\cite{Milton95}, or equation (2.13) of~\cite{Milton96}. The quantities $A^\ell_{F,G}$ and $B^\ell_{F,G}$ are those given in~\cite{Milton80,Milton95,Milton96}. \noindent {\bf Case II:}\\ For all $r_1,r_2$: \begin{equation} F^{II}_\ell, G^{II}_\ell(r_1,r_2) = i k_{II} \; j_\ell(k_{II} r_<) \; h_\ell(k_{II}r_>). \label{E-ii} \end{equation} \noindent {\bf Case III:}\\ For all $r_1, r_2$: \begin{equation} F^{III}_\ell, G^{III}_\ell(r_1,r_2) = i k_{III} \; j_\ell(k_{III} r_<) \; h_\ell(k_{III}r_>). \label{E-iii} \end{equation} \noindent We are now ready to explicitly compute the Casimir energy. In passing we remark that the object $F^{(0)}_\ell$ defined in equation (29) of~\cite{Milton80}, equation (35) of~\cite{Milton95}, and equation (3.7) of~\cite{Milton96}, which is essential for those calculations, is {\em not} a Green function of {\em any} differential operator. Specifically, $F^{(0)}_\ell$ does not satisfy the dielectric boundary conditions. It is not even continuous, and is merely a potpourri of two different Green functions which does not have any particular physical relevance. \subsection{The Casimir energy} We calculate $E_{Casimir}$ using equation (\ref{E-casimir-energy-2}), [equivalently (\ref{E-casimir-energy-1})] together with equations (\ref{E-i-a}--\ref{E-ii}). When evaluating the imaginary parts of $X$ it is convenient to introduce the Ricatti--Bessel functions $s_\ell(x) = { x j}_\ell(x)$ and $e_\ell(x) = { x h}_\ell(x)$. One also needs the identity \begin{equation} {\ell(\ell+1)\over x^2} s_\ell(x) = s_\ell''(x) + s_\ell(x), \end{equation} \noindent together with an identical equation which holds for $e_\ell(x)$. After some rearrangement (technical details are suppressed and will be relegated to a more detailed forthcoming publication) we find that inside the dielectric sphere \begin{eqnarray} {\rm Im}\{&X_I&(k,r)\}_{in} \nonumber\\ &=& 2\; {k_{III}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) \nonumber\\ &&\times \left\{ 2 [s_\ell(x)]^2 + [s_\ell'(x)]^2 + s_\ell(x) s_\ell''(x) \right\}|_{k_{III}r} \nonumber\\ &-& {k_{III}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) {\rm Re}\{A^\ell_F+A^\ell_G \} \nonumber\\ &&\times \left\{ 2 [s_\ell(x)]^2 + [s_\ell'(x)]^2 + s_\ell(x) s_\ell''(x) \right\}|_{k_{III}r} . \nonumber\\ \end{eqnarray} A remarkable Ricatti--Bessel function identity permits us to perform the first sum over $\ell$ {\em exactly}. Using \begin{equation} \sum_{\ell=1}^{\infty} (2\ell+1) \left[ 2 s_\ell(x)^2 + s_\ell'(x)^2 + s_\ell(x) s_\ell''(x) \right] = 2 x^2, \end{equation} one obtains that \begin{eqnarray} {\rm Im}\{ &X_I&(k,r)\}_{in} =4 k_{III}^3 \nonumber\\ &-& {k_{III}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) {\rm Re}\{A^\ell_F+A^\ell_G \} \nonumber\\ &&\times \left\{ 2 [s_\ell(x)]^2 + [s_\ell'(x)]^2 + s_\ell(x) s_\ell''(x) \right\}\big|_{k_{III}r}. \nonumber\\ \end{eqnarray} Note that for the $A^\ell_{F,G}$--terms we {\em cannot} explicitly perform the $\ell$ summation because of the complicated $\ell$ dependence of these coefficients~\cite{Milton95,Milton96}. It is very important to notice at this point that the $A^\ell_{F,G}$--terms in the above expressions are the {\em only} pieces retained in the currently extant calculations, the other terms unfortunately have been missed there due to the use of the wrong ``Green functions". Taking a cue from the above, the results for the region inside the bubble can be written (using self explanatory notation) as \begin{equation} {\rm Im}\{ X_I(k,r) \}_{in} = 4 n_1^3 |\omega|^3 + Q_{in}^{surface}(k_{III},r). \end{equation} Turning to the region outside the dielectric sphere, one gets that \begin{eqnarray} {\rm Im}\{ &X_I&(k,r)\}_{out} \nonumber\\ &=& 2\; {k_{II}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) \nonumber\\ &&\times \left\{ 2 [s_\ell(x)]^2 + [s_\ell'(x)]^2 + s_\ell(x) s_\ell''(x)\right\}|_{k_{II}r} \nonumber\\ &-& {k_{II}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) {\rm Re}\big[ (B^\ell_F+B^\ell_G) \nonumber\\ &&\times \left\{ 2 [e_\ell(x)]^2 + [e_\ell'(x)]^2 + e_\ell(x) e_\ell''(x) \right\}\big]\big|_{k_{III}r} \nonumber\\ &=& 4 k_{II}^3 - {k_{II}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) {\rm Re}\big[ (B^\ell_F+B^\ell_G) \nonumber\\ &&\times \left\{ 2 [e_\ell(x)]^2 + [e_\ell'(x)]^2 + e_\ell(x) e_\ell''(x) \right\}\big]\big|_{k_{III}r} . \end{eqnarray} Again, in self-explanatory notation \begin{equation} {\rm Im}\{ X_I(k,r) \}_{out} = 4 n_2^3 |\omega|^3 + Q_{out}^{surface}(k_{II},r). \end{equation} For Case II one simply has \begin{eqnarray} {\rm Im}\{ &X_{II}&(k,r) \}_{all~space} = 2\; {k_{II}\over r^2} \; \sum_{\ell=1}^\infty (2\ell+1) \nonumber\\ &&\times \left\{ 2 [s_\ell(x)]^2 + [s_\ell'(x)]^2 + s_\ell(x) s_\ell''(x)\right\}|_{k_{II}r} \nonumber\\ &=& 4 k_{II}^3 = 4 n_2^3 |\omega|^3. \end{eqnarray} Going back to the momentum-space regulated Casimir energy, equation (\ref{E-casimir-energy-2}), we obtain \begin{eqnarray} E_{Casimir} &=& \int_0^R r^2 dr \int_{0}^{\infty} {d\omega \over2\pi} 4 |\omega|^3 \nonumber\\ &&\qquad \times\left[ n_1^3 f(n_1 |\omega|) - n_2^3 f(n_2 |\omega|)\right] \nonumber\\ &+& \int_0^R r^2 dr \int_{0}^{\infty} {d\omega\over2\pi} f(n_1 |\omega|) Q^{surface}_{in}(k,r) \nonumber\\ &+& \int_R^\infty r^2 dr \int_{0}^{\infty} {d\omega\over2\pi} f(n_2 |\omega|) Q^{surface}_{out}(k,r). \nonumber\\ \end{eqnarray} The remaining integrals for the $n^3 |\omega|^3$ term are trivial. Changing the integration variable to $k = n |\omega|$, and explicitly re-inserting the appropriate factors of $\hbar$ and $c$, we get \begin{eqnarray} E_{Casimir} &=& + 2 V \int {d^3\vec k \over (2\pi)^3} {1\over 2} \hbar [ \omega_1(k) - \omega_2(k) ] f(k) \nonumber\\ &+& \int_0^R r^2 dr \int_{0}^{\infty} {d\omega\over2\pi} \hbar f(n_1 |\omega|) Q^{surface}_{in}(k,r) \nonumber\\ &+& \int_R^\infty r^2 dr \int_{0}^{\infty} {d\omega\over2\pi} \hbar f(n_2 |\omega|) Q^{surface}_{out}(k,r). \nonumber\\ \end{eqnarray} \noindent which is the central result of this paper. This is explicitly of the form: \begin{center} (Schwinger's volume term) + (surface term). \end{center} \noindent The ``surface term" corresponds to $E_{surface}$ and is given by the two double integrals in the expression for $E_{Casimir}$ above. For time-splitting regularization and dilute dielectric media, these terms were explicitly shown by Milton {\em et al.} to be proportional to the surface area (plus even higher-order corrections). The first term is the volume term not present in some of the existing calculations. In fact, after approximating air by vacuum (setting $n_1=1$) and using Schwinger's momentum space cutoff, this integral is {\em exactly equal} to Schwinger's result~\cite{Schwinger0}. \section{ Discussion} The main result of this paper can be succinctly stated: in a dielectric medium of dielectric constant $n$ the Casimir energy of a cavity---the difference in zero point energies of a dielectric medium of refractive index $n$ with and without a vacuum cavity of volume $V$---is: \begin{equation} E_{Casimir} = {1\over8\pi^2} \; V \; \hbar c K^4 \; \left[1-{1\over n}\right] + \cdots, \end{equation} with this volume term dominant if the scale of the bubble is {\em larger} than the cutoff wavelength $2 \pi / K$. This result is completely in agreement with Schwinger's calculation in~\cite{Schwinger0}, and Schwinger's argument is now buttressed by our explicit re-assessment of Milton {\em et al.}'s calculation for a spherical dielectric ball. We close with what is perhaps a minor point that we nevertheless feel should be made explicit: the volume contribution to the Casimir energy is always there, and is always physical, but it is {\em sometimes} safe to neglect it. For example, a situation equally physical as the one we have considered here is the following: suppose one is provided with a fixed number of dielectric bodies of fixed shape (in particular, of fixed volume), and suppose that one simply wishes to move the bodies around in space with respect to each other. Then the bulk volume contributions to the Casimir energy, while still present, are constants independent of the relative physical location of the dielectric bodies, and so merely provide a constant offset to the total Casimir energy. {\em If all we are interested in is the energy differences between different spatial configurations of the same bodies then the various volume contributions can be quietly neglected.} On the other hand, the volume contribution is of critical importance whenever one wants to calculate the energy difference between an inhomogeneous dielectric and a homogeneous dielectric wherein the irregularities have been filled in. {\em This is, precisely, the physical situation in the case of bubble collapse in a dielectric medium.} \acknowledgements This work was supported in part by the U.S. Department of Energy, the U.S. National Science Foundation, by the Spanish Ministry of Education and Science and the Spanish Ministry of Defense. Part of this work was carried out at the Laboratory for Space Astrophysics and Fundamental Physics (LAEFF, Madrid), and C.E.C., C.M-P., and M.V. wish to gratefully acknowledge the hospitality shown. Part of this work was carried out at Los Alamos National Laboratory, and J.P-M. wishes to gratefully acknowledge the hospitality shown to him there.

proofpile-arXiv_065-560

\section{Introduction.} Theoretical studies of light hadron spectroscopy have led to the widespread belief that gluonic excitations are present in the spectrum of hadrons, so more resonances should be observed than are predicted by the conventional $q\bar q$ and $qqq$ quark model. The two general categories of gluonic mesons expected are glueballs (dominated by pure glue basis states) and hybrids (dominated by basis states in which a $q\bar q$ is combined with a gluonic excitation). Some of these novel states, notably the light hybrids, are predicted to have exotic quantum numbers (forbidden to $q\bar q$), such as J$^{PC}=1^{-+}$. The confirmation of such a resonance would be proof of the existence of exotic non-$q\bar q$ states, and would be a crucial step towards establishing the spectrum of gluonic states. There are detailed theoretical predictions for the decays of these exotic hybrids \cite{ikp,cp95}, which have motivated several experimental studies of purportedly favored hybrid channels such as $b_1\pi$ and $f_1\pi$. Although one would prefer to find these unambiguously non-$q\bar q$ J$^{PC}$-exotics, glueballs and hybrids with non-exotic quantum numbers are also expected. For example, in the flux tube model the lowest hybrid multiplet, expected at $\approx 1.8$-$1.9$ GeV \cite{paton85,bcs}, contains the non-exotics J$^{PC} = 0^{-+}, 1^{\pm\pm}, 1^{+-}$ and $2^{-+}$ in addition to the exotics $0^{+-}$, $1^{-+}$ and $2^{+-}$. To identify these non-exotic states one needs to distinguish them from the ``background'' of radial and orbital $q\bar q$ excitations in the mass region $\approx 1.5$-$2.5$ GeV, where the first few gluonic levels are anticipated\cite{glueball,wein}. Our point of departure is to calculate the two-body decay modes of all radial and orbital excitations of $n\bar n$ states ($n=u,d$) anticipated up to 2.1 GeV. This includes 2S, 3S, 2P, 1D and 1F multiplets, a total of 32 resonances in the $n\bar n$ sector. We also summarize the experimental status and important decays of candidate members of these multiplets, and compare the predictions for decay rates with experiment. We start by briefly reviewing the established 1S and 1P states that confirm that $^3$P$_0$ pair creation dominates most hadronic decays. SHO wavefunctions are employed for convenience; these lead to analytic results for decay amplitudes and are known to give reasonable empirical approximations. This is sufficient for our main purpose, which is to emphasize selection rules and to isolate major modes to aid in the identification of states. In addition to the 1S and 1P states we also find reasonable agreement between the model and decays of 1D, 2P and 1F states where data exist; this confirms the extended utility of the model and adds confidence to its applications to unknown states. Examples of new results include the following. $\bullet$ The radial 2$^3$P$_1$ $a_{1R} \to \rho \pi$ is strongly suppressed in S-wave, and dominant in D-wave. This contrasts with the expectation for a hybrid $a_1$. The model's prediction of a dominant D-wave has been dramatically confirmed for the $a_1(1700)$\cite{ves951,suchung} and thereby establishes 1.7~GeV as the approximate mass of the $n\bar n$ members of the 2P nonets. This includes the $0^{++}$ nonet whose I=0 members share the quantum numbers of the scalar glueball. $\bullet$ In the scalar glueball sector, we find that the decays of the $f_0(1500)$ and the $f_{{\rm J}}(1710)$ are inconsistent with radially excited quarkonia. $\bullet$ We identify the 2S $0^{-+}$ nonet. The $\eta$ members are predicted to have narrow widths relative to the $\pi$ counterpart. This is consistent with the broad $\pi(1300)$ and the narrower candidates $\eta(1295)$ and $\eta(1440)$. $\bullet$ The vector states $\rho(1465)$ and $\omega(1419)$ are interesting in that the decay branching fractions appear to show anomalous features requiring a hybrid component. We identify the experimental signatures needed to settle this question. $\bullet$ The $\pi(1800)$ has been cited as a likely hybrid candidate\cite{cp95,fec94,ves} on the strength of its decay fractions. The 3S $0^{-+}$ $q\bar q$ $\pi$ is also anticipated in this region. We find that the decays of the hybrid and 3S $0^{-+}$ have characteristic differences which enable them to be distinguished. We identify modes that may enable the separation of these two configurations. Our other results for the many $n\bar n$ states predicted up to 2.1 GeV should be useful in the identification of these higher quarkonia, and in confirming that non-exotic gluonic or molecular states are indeed inconsistent with quarkonium assignments. The order of discussion is 1S and 1P (section 2); 2S and $^3D_1$ (section 3); 3S (section 4); 2P (section 5); 1D (section 6); 1F (section 7). A summary and an outline for experimental strategy is in section 8. \section {1S and 1P Testbed} First we will use the well known decays of light 1S and 1P $n\bar n$ states to motivate and constrain the $^3$P$_0$ decay model. Ackleh, Barnes and Swanson\cite{abs} have carried out a systematic study of $q\bar{q}$ decays in the $^3$P$_0$ and related pair creation decay models: in that work a $^3$P$_0$-type amplitude was established as dominant in most light $n\bar{n}$ decays. (For other discussions of $q \bar q$ decays in the $^3$P$_0$ model see Ref.\cite{3p0}). Fig.1, from Ref.\cite{abs}, shows $^3$P$_0$ model predictions for the decay widths. Large widths are indeed predicted to be large and smaller widths are found to be correspondingly small. If we choose the pair creation strength $\gamma=0.5$ (Eq. A3) to set an approximately correct overall width scale, then $\Gamma (h_1 \rightarrow \rho \pi)$ and $ \Gamma ( a_1 \rightarrow \rho \pi)$ are both $ \approx 0.4$-$0.5$ GeV; $\Gamma(f_2 \rightarrow \pi\pi)$, $\Gamma (\rho \rightarrow \pi\pi)$ and $\Gamma(b_1 \rightarrow \omega \pi)$ are all $\approx 0.1$-$0.2$ GeV, and $\Gamma (a_2 \rightarrow \rho \pi)$ is smallest, $\approx 0.05$ GeV; all are reasonably close to the observed widths. \begin{figure} $$\epsfxsize=5truein\epsffile{bcpsfig1.eps}$$ {Figure~1. Partial widths of light 1S and 1P $q\bar q$ mesons in the $^3$P$_0$ model. The model parameters shown are $\beta=0.2$-$0.6$ GeV (with $\beta\approx 0.4$ GeV preferred) and $\gamma=0.5$. } \end{figure} The optimum parameter values found in a fit to the partial widths of Fig.1\cite{abs} are $\beta=0.40$~GeV (which is actually the length scale most commonly used in light $q\bar q$ decays) and $\gamma=0.51$; with these values the rms relative error for these six decays is $\Delta\Gamma / \Gamma_{expt} = 29 \% $. In this work we have actually found that the pair production amplitude $\gamma=0.5$ is somewhat large for higher-L $q\bar q$ states, so in our discussions of higher quarkonia we will instead use $\gamma=0.4$. In constrained-$\gamma$ fits we find that using $\gamma=0.4$ only moderately decreases the accuracy of the fit to the light 1S and 1P decays, to $\Delta\Gamma / \Gamma_{expt} = 43 \% $, with an optimum $\beta=0.36$~GeV. A more sensitive test of the $^3$P$_0$ model involves amplitude ratios in the decays $b_1\to\omega\pi$ and $a_1\to\rho\pi$. In these decays both S- and D-wave final states are allowed, and the ratio of these decay amplitudes is known to be D/S = $+0.260(35)$ for the $b_1$ and $-0.09(2)$ for the $a_1$ \cite{pdg96}. This ratio is quite sensitive to the quantum numbers of the produced pair; with $^3$P$_0$ quantum numbers and the usual $\beta$ we find reasonable agreement in sign and magnitude, whereas a OGE pair production mechanism gives the wrong sign for D/S \cite{abs}. This ratio test for $b_1\to\omega\pi$ was historically very important in establishing the $^3$P$_0$ decay model \cite{early3p0}. These successes of the $^3$P$_0$ model motivate its use in predicting decays of the less familiar radial and orbital excitations of light quarkonia. \section{2S States} We first consider the decays of the low-lying radially-excited pseudoscalar and vector states. Our general approach will be to review recent data on the state in question and compare these data to predictions for candidate $q\bar q$ and (where appropriate) hybrid states. In each case we will attempt to identify decay modes that distinguish between competing assignments most clearly. \subsection{$0^{-+}$ $2^1$S$_0$: $ \pi$ and $\eta$} $\bullet$ $\pi(1300)$ The $\pi(1300)$ was first reported by Bellini {\it et al.}\cite{bellini} in 1982 but remains rather poorly known. It is seen in $\pi\rho$, $\pi(\pi\pi)_S$ and $\pi f_0(1300)$, with a width of 200-600 MeV; there is however no accurate measurement of the branching fractions \cite{pdg94}. Recently higher statistics have been obtained for the $\pi(1300)$ by VES\cite{ves951,ves} and by E852 at BNL\cite{suchung}. The VES data shows a clear $\pi(1300)$ peak in $3\pi$, with a width of $\Gamma \approx 400$-$500$ MeV in both $\pi (\pi\pi)_S$ and $\rho \pi$; the latter is particularly strong and dominates this channel below 2 GeV. It should be noted, however, that the size of the Deck background in $\pi (\pi \pi)_S$ is uncertain, and it is not clear whether the $\pi(1300)$ reported in $\pi (\pi \pi)_S$ is actually due to the resonance. Fig.1c of Ref.\cite{ves951} suggests that the Deck mechanism could cause {\it all} of the $\pi(1300) \rightarrow \pi (\pi \pi)_S$ enhancement in Fig.4a of that reference. We will assume that this is essentially correct, and that the $\pi(1300)$ resonance decays dominantly to $\rho\pi$. In the $^3$P$_0$ decay model we expect $\rho\pi$ to be the dominant mode of a 2S $q\bar q$ $\pi(1300)$, since this is the only open two-body channel. (We assume that the $f_0(980)$ and $a_0(980)$ are dominantly K$\bar {\rm K}$, so the mode $\pi(1300)\to f_0(980)\pi $ is a more complicated three body or virtual two-body decay.) With our parameter set $\gamma=0.4$ and $\beta=0.4$~GeV we predict a partial width of \begin{equation} \Gamma(\pi(1300) \rightarrow \pi \rho) = 209 \; {\rm MeV} \ . \end{equation} This rate is given in Table B2 of Appendix B. (App.B is a tabulation of all our numerical results for partial widths in the $^3$P$_0$ model.) In Fig.2 we show the dependence of this prediction on the wavefunction length scale $\beta$. Evidently the prediction of a large width, comparable to observation, follows from any plausible choice for $\beta$. Thus the observed $\pi(1300)$ is consistent with expectations for a 2$^1$S$_0$ $q\bar q$ state. \begin{figure} $$\epsfxsize=5truein\epsffile{bcpsfig2.eps}$$ {Figure~2. The $\rho\pi$ partial width of a 2S $\pi(1300)$, with $^3$P$_0$ model parameters $\beta=0.2$-$0.6$ GeV and $\gamma=0.4$. } \end{figure} Although the mode $f_0^{q\bar q}(1300)\pi$ is nominally closed by phase space, the $f_0(1300)$ is a very broad state, so one might anticipate a significant $(\pi\pi)_S\pi$ mode through the low-mass tail of the $f_0(1300)$. This possibility may be tested by varying $M(f_0^{q\bar q})$; the resulting $\Gamma(\pi(1300)\to f_0^{q\bar q}\pi )$ does not exceed $10$ MeV over the range $M(f_0^{q\bar q}) = 400$-$1000$ MeV. Thus, the population of a $\pi (\pi\pi)_S$ mode by $\pi(1300)$ decays through an intermediate $f_0^{q\bar q}\pi $ state is predicted to be a small effect. If there actually is a large $\pi(1300)\to \pi (\pi\pi)_S$ mode, rather than a nonresonant Deck effect, this would be in disagreement with the $^3$P$_0$ model. Thus it would be very interesting to establish the branching fraction for $\pi(1300)\to \pi (\pi\pi)_S$ accurately in future work. $\bullet$ $\eta(1295)$ This state has a width of $\Gamma = 53(6)$ MeV\cite{pdg94}, much narrower than its I=1 2$^1$S$_0$ partner $\pi(1300)$. It has been reported in $a_0(980)\pi $ and $\eta\pi\pi $. This small width is natural if the $\pi(1300)$ does indeed decay dominantly to $\rho\pi $, since G-parity forbids the analogous processes $\eta_{n\bar{n}} \to \rho\pi$ and $\eta_{n\bar{n}}\to \omega\eta$; to the extent that the $a_0(980)$ and $f_0(980)$ are dominantly $K \bar{K}$ there are no quasi-two-body $q \bar q$ modes open to the $\eta(1295)$. Consequently the decays must proceed through the weaker direct three-body and virtual two-body channels such as $a_0^{q\bar q}\pi$ and $f_0^{q\bar q}\eta$. It is interesting to note the r\^ole that the 2S initial wavefunction has played in our discussion. Suppose for illustration that we had instead used 1S wavefunctions for the $\pi(1300)$ and $\eta(1295)$; we would then have predicted partial widths of several hundred MeV into the low-energy tails of the modes $f_0^{q\bar q}\pi $ and $a_0^{q\bar q}\pi $, with consequent broad widths for the $\pi(1300)$ {\it and} the $\eta(1295)$, in contradiction with experiment. $\bullet$ $\eta(1440)$ These successes raise provocative questions regarding the $\eta(1440)$ state(s). This is a purportedly complicated region which may contain more than one resonance\cite{pdg94}. The PDG width of the $\eta(1440)$ is only $\Gamma = 60(30)$ MeV, with signals reported in ${\rm K}^* {\rm K} $, $a_0(980)\pi $, $\eta(\pi\pi)_S$ and $\rho\gamma$. Except for $\rho\gamma$ these modes are not inconsistent with a dominantly $s\bar s$ state. The only two-body strong channel open for a 2$^1$S$_0$ $s\bar s$ $\eta(1440)$ is ${\rm K}^* {\rm K} $, but this could rescatter from KK$\pi$ into the other reported modes $a_0(980)\pi$ and $\eta\pi\pi$. The $^3$P$_0$ model prediction for the partial width $\eta(1440) \to {\rm K}^* {\rm K}$ versus the wavefunction length scale $\beta$ is shown in Fig.3. \begin{figure} $$\epsfxsize=5truein\epsffile{bcpsfig3.eps}$$ {Figure~3. The K$^*\bar {\rm K}$ + h.c. partial width of a 2$^1$S$_0$ $s\bar s$ $\eta(1440)$ in the $^3$P$_0$ model. Other two-body modes are excluded by phase space. } \end{figure} Evidently the predicted ${\rm K}^* {\rm K} $ partial width is comparable to the observed width, so a 2$^1$S$_0$ $s\bar s$ assignment appears possible for this state. Of course the $\rho\gamma$ mode is not expected from $s\bar s$, and if confirmed may imply large $n\bar n \leftrightarrow s\bar s$ mixing in this sector as is observed in the 1S I=0 pseudoscalars. This can be parameterized as \begin{eqnarray} &|\eta(1295)\rangle &= +\cos(\theta)|{n\bar{n}}\rangle + \sin(\theta) |{s\bar{s}}\rangle \\ &|\eta(1440)\rangle &= -\sin(\theta)|{n\bar{n}}\rangle + \cos(\theta) |{s\bar{s}}\rangle \ . \\ \end{eqnarray} A remeasurement of $\eta(1440) \to \rho \gamma$, which should be possible at BEPC and TCF in $\psi \to \gamma\gamma\rho $, would be very useful in clarifying the nature of this state. Ideally we would like to know the invariant mass distributions of $\rho\gamma$, $\omega\gamma$ and $\phi\gamma$ final states, since these are flavor-tagging modes that allow investigation of possible flavor mixing in the parent resonances. Similarly, an accurate measurement of the branching fractions in the flavor-tagging $\psi\to V\eta(1440)$ and $V\eta(1295)$ hadronic decays, with $V=\omega, \phi$, would be useful for the determination of the $n\bar n $-$s\bar s$ mixing angle. In summary, from the total widths alone it is possible to describe the $\eta(1295)$ and $\eta(1440)$ as unmixed $n\bar n$ and $s\bar s$ 2$^1$S$_0$ radial excitations. The report of a large $\eta(1440)\to\rho\gamma$ radiative mode however suggests flavor mixing between these states, and should be remeasured with greater sensitivity together with other $V\gamma$ modes. This mixing could also account for the large $\eta(1440)$ signal seen in $\eta (\pi\pi) $ by GAMS \cite{yup}. \subsection{$1^{--}$: 2$^3$S$_1$ and $^3$D$_1$ $\rho$ and $\omega$} $\bullet${\bf $\rho(1465)$, $\rho(1700)$} If one accepts that the $\pi(1300)$ and $\eta(1295)$ belong to a $2^1$S$_0$ $q\bar q$ nonet, it is then natural to assign the $\rho(1465)$ and the $\omega(1419)$ \cite{pdg94,cd} to $2^3$S$_1$ states. Indeed, one expects the contact hyperfine interaction to raise the mass of the vector nonet with respect to the pseudoscalar nonet by approximately this amount \cite{god}. It is unlikely that the vectors near 1.4-1.5 GeV are dominantly D-waves, since the $^3$D$_1$ $n\bar n$ states should lie close to the other 1D candidates such as the $\pi_2(1670)$, $\rho_3(1691)$ and $\omega_3(1667)$. In the Godfrey-Isgur potential model a mass of 1660 MeV was predicted for the $^3$D$_1$ state, whereas they expect the 2$^3$S$_1$ radial excitation at 1450 MeV \cite{god}. The $\rho(1465)$ also lies well below flux-tube model expectations of M$_H (1^{--}) \approx 1.8$-$1.9$ GeV\cite{paton85,bcs} for vector hybrids, so although the possibility of light vector hybrids has been discussed \cite{cp95,kalash}, these do not appear likely unless the flux tube model for hybrids is misleading. The experimental branching fractions of these $1^{--}$ states are somewhat obscure, because there are at least two broad, overlapping resonances in each flavor sector in this mass region. The status of these vector states as seen in $e^+e^-$ annihilation was reviewed recently by Clegg and Donnachie \cite{cd}. In the $\rho$ sector they find that at least two states are present. The lighter state is assigned a mass of M $ = 1.463(25)$ GeV and a width of $\Gamma = 0.311(62)$ GeV; it couples strongly to $4\pi$ states (including $a_1\pi$ but not $h_1\pi$) and $\omega\pi$, and less strongly to $\pi\pi$. The higher state has M $ = 1.73(3)$ GeV, $\Gamma = 0.40(10)$ GeV, couples most strongly to $4\pi$ ($a_1\pi$ and $h_1\pi$ are not separated) and perhaps $6\pi$; $\pi\pi$ is also important, but the $\omega\pi$ width is found to be small. These states have also been reported recently by Crystal Barrel \cite{cbrhor} in $\pi^-\pi^o$ states in $\bar p d\to \pi^-\pi^o\pi^op$; both vectors appear in $\pi^-\pi^o$, with masses and widths of M $ = 1.411(10)(10)$ GeV, $\Gamma = 0.343(18)(8)$ GeV, and M $ = 1.780{+34\atop -25}(14)$ GeV, $\Gamma = 0.275(42)(17)$ GeV, quite similar to the $e^+e^-$ results. The $^3$P$_0$ model predictions for pure 2$^3$S$_1$ and $^3$D$_1$ $\rho$ states at 1.465 GeV and 1.700~GeV are given in Table I (see also Tables B1, B8), together with flux tube model predictions for a hypothetical 1.5~GeV vector hybrid. Very characteristic differences between the states are evident in their couplings to $4\pi$ final states; 2S couples very weakly to these, 1D couples strongly to both $a_1\pi$ and $h_1\pi$, and the hybrid couples strongly to $a_1\pi$ but not to $h_1\pi$. Both quarkonium states have moderately large couplings to $\pi\pi$ and $\omega\pi$, whereas the hybrid couples strongly only to $a_1\pi$. \begin{table} \caption{ Partial widths of 2S, 1D and hybrid $\rho$ states.} \label{tabrho} \begin{tabular}{lccccccccc} & $\pi\pi$ & $\omega\pi$ & $\rho\eta$ & $\rho\rho$ & KK & K$^*$K & $h_1\pi$ & $a_1\pi$ & total\\ \hline $\rho_{2S}(1465)$ & 74. & 122. & 25. & - & 35. & 19. & 1. & 3. & 279. \\ $\rho_{1D}(1700)$ & 48. & 35. & 16. & 14. & 36. & 26. & 124. & 134. & 435. \\ $\rho_H(1500)$ & 0 & 5 & 1 & 0 & 0 & 0 & 0 & 140 & $\approx 150$ \end{tabular} \end{table} Note that the $|q\bar q\rangle$ components are spin {\it triplet} whereas the hybrid is spin {\it singlet}. This difference in spin underlies the characteristic pattern of branching fractions in Tables I and II. Although there are many similarities between theory and experiment, there are problems in detail. The important couplings of the lighter state to $\pi\pi$ and $\omega\pi$ found by Clegg and Donnachie are consistent with a 2S quarkonium, but we do not expect a significant coupling of a 2$^3$S$_1$ $\rho$ to $4\pi$ final states. The dominant coupling of the heavier state to $4\pi$ is as predicted for the D-wave quarkonium, but the reported absence of $\omega\pi$ is not expected. The presence of two states (2$^3$S$_1$ and $^3$D$_1$) in $\pi\pi$ with comparable strengths, reported by Crystal Barrel \cite{cbrhor}, is expected. Of course it is difficult to distinguish the contributions from two broad states with similar masses, and the $4\pi$ final states themselves have not yet been completely characterized. (The $a_1\pi$ and $h_1\pi$ modes of the $\rho(1700)$ in $e^+e^-$ for example have not been separated.) It appears likely that the states and their branching fractions are still inadequately resolved experimentally in this mass region, so it is not yet appropriate to attempt a detailed fit, using for example linear combinations of the 2S and 1D basis states. It is clear from our $^3$P$_0$ results that in future it will be important to separate the $a_1\pi$ and $h_1\pi$ contributions (which tag 1D and H \cite{cp95,kalash} states), and that the $\pi\pi$ and $\omega\pi$ distributions should also be studied carefully, since these are expected to arise mainly from quarkonia rather than hybrids. $\bullet$ {\bf $\omega(1419)$ and $\omega(1649)$} We anticipate similar problems with at least two broad overlapping resonances in the I=0 sector. Clegg and Donnachie \cite{cd} discuss both one- and two-resonance fits to the $\omega$ sector in the reactions $e^+e^- \to \rho\pi$ and $\omega\pi\pi$. In their two-resonance fit they find a lower state with a mass and width of M$ = 1.44(7)$ GeV, $\Gamma = 0.24(7)$ GeV, and a higher, quite narrow state with M$ = 1.606(9)$ GeV, $\Gamma = 0.113(20)$ GeV. The PDG quote masses and widths of M$ = 1.419(31)$ GeV, $\Gamma = 0.174(59)$ GeV, M$ = 1.649(24)$ GeV, $\Gamma = 0.220(35)$ GeV; the parameters for the lighter state are consistent but the width of the higher-mass $\omega$ state is broader than Clegg and Donnachie estimate. Clegg and Donnachie find that both $\omega$ states couple strongly to $\rho\pi$. Only the second is found to couple to $\omega\pi\pi$, and that coupling is rather weak. A fit with a single resonance finds instead that the $\omega\pi\pi$ branching fraction exceeds $\rho\pi$, so these should be regarded as tentative conclusions. \begin{table} \caption{ Partial widths of 2S, 1D and hybrid $\omega$ states.} \label{tabomega} \begin{tabular}{lcccccc} & $\rho\pi$ & $\omega\eta$ & KK & K$^*$K & $b_1\pi$ & total \\ \hline $\omega_{2S}(1419)$ & 328. & 12. & 31. & 5. & 1. & 378. \\ $\omega_{1D}(1649)$ & 101. & 13. & 35. & 21. & 371. & 542. \\ $\omega_H(1500)$ & 20 & 1 & 0 & 0 & 0 & $\approx 20$ \end{tabular} \end{table} For comparison we again show the numerical predictions of the $^3$P$_0$ model for pure 2S, 1D and H states. The masses assumed are 1996 PDG values (see Tables B1 and B9). The large $\rho\pi$ couplings reported for the vector states are evidently consistent with expectations for both 2S and 1D quarkonia. Again the S+S modes are predicted to be small for a hybrid, so they can be used to tag quarkonia or the $q\bar q$ components of mixed states. Since none of the favored S+P modes is open to an I=0 hybrid at 1.5 GeV, such a state would be quite narrow, as shown in Table II. (The decay $\omega_H\to b_1\pi$ is excluded by the ``singlet selection rule'' \cite{cp95,abs}, which states that $({\rm S}_{q\bar q}=0) \not\to ({\rm S}_{q\bar q}=0) + ({\rm S}_{q\bar q}=0) $ in the $^3$P$_0$ model; the $\omega_H$ hybrid has ${\rm S}_{q\bar q}=0$ in the flux tube model. Interestingly, the singlet selection rule holds for both $^3$P$_0$ and OGE quarkonium decay amplitudes \cite{abs}.) A hybrid in this mass region should be visible as a narrow bump in the $\rho\pi$ invariant mass distribution. (This channel is not favored for a hybrid, but it is allowed at a reduced rate due to different $\rho$ and $\pi$ spatial wavefunctions.) Thus it may be useful to search $\rho\pi$ final states for narrow resonances with improved statistics, although the signal would of course be broadened by the $\rho$ width. The very large $b_1\pi$ mode predicted for the 1D quarkonium is very interesting, because neither 2S nor hybrid vector states are expected to couple significantly to $b_1\pi$. This two-body mode will appear as $\omega\pi\pi$; Clegg and Donnachie do report an $\omega\pi\pi$ mode for their higher $\omega$ state, but the coupling is not as strong as we predict. The total width of their higher-mass state is also much smaller than expected. Since the 1D state is predicted to have a very large width, $\approx 500$ MeV (Table B9), this discrepancy may be due to a distortion of the shape by threshold effects, with resulting inaccuracies in the reported couplings. Assuming that the $^3$P$_0$ model predictions are approximately correct, a study of the $1^{--}$ $\omega\pi\pi$ mass distribution should reveal the $^3$D$_1$ $\omega$ basis state in isolation. (It may be distributed over several resonances.) If the quasi-two-body approximation is correct, the mass distribution of $\omega\pi$ pairs in the resonance contribution to $\omega\pi\pi$ should be consistent with a $b_1(1231)$. \subsection{Mixing in the $1^{--}$ sector.} Although we have considered the decay modes of pure 2S, 1D and H vector states, the physical resonances are certainly linear combinations of these and other basis states. Since the known resonances have similar masses, we should consider the possibility that there is significant mixing and introduce the linear combination \begin{equation} |{\rm V}\rangle = \cos(\theta) \bigg( \cos(\phi) |2^3{\rm S}_1 \rangle + \sin(\phi) |{}^3{\rm D}_1\rangle \bigg) + \sin(\theta) |{\rm H}\rangle \ . \end{equation} The mixing angles for each resonance can be determined from the branching fractions to certain states. The S+S modes identify the $q\bar q$ components of the state (see Tables~I and II). In the I=1 states the $4\pi$ modes $a_1\pi$ and $h_1\pi$ are similarly characteristic; the $h_1\pi$ mode is produced only by the 1D basis state, and $a_1\pi$ comes from both 1D and hybrid states. Similarly in I=0 the mode $b_1\pi$ tags the 1D quarkonium basis state and 2S and 1D states both lead to strong $\rho\pi$ couplings. Determination of the mixing angles in the physical states will be possible given accurate measurements of the branching fractions to these characteristic modes. We have not carried out a fit to determine the mixing angles because the experimental results do not yet appear definitive. However we note that the partial widths reported by Clegg and Donnachie for the $\rho(1465)$, which include a large $\Gamma_{a_1\pi}$ and a small $\Gamma_{h_1\pi}$, are inconsistent with 2S or 1D alone. These widths imply a large H component in this state with the possibility of considerable H-2S mixing. Future experimental work could concentrate on an accurate determination of the $\pi\pi$, $\omega\pi$, $h_1\pi$ and $a_1\pi$ branching fractions of the $\rho$ states. The $h_1\pi$ and $a_1\pi$ modes are especially sensitive to the nature of the initial state. Similarly the $\rho\pi$ and $b_1\pi$ branching fractions of the $\omega$ states are the most interesting experimentally. \section{3S States} \subsection{$0^{-+}: \; 3^1$S$_0\ \pi(1800)$} The same experiments\cite{ves951,ves,bellini,veseta} that see the $\pi(1300)$ in $\rho\pi $ and a possible broad enhancement in $\pi (\pi \pi)_S$ also report a prominent $\pi(1800)$ in $f_0(980)\pi $, $f_0(1300)\pi $, $f_0(1500)\pi $ and K$({\rm K} \pi)_S$. None of these experiments see the $\pi(1800)$ in $\rho\pi$. This is striking, as also is the fact that the total width of $\approx 150$-$200$ MeV is considerably smaller than that of the $\pi(1300)$. Furthermore, the presence of clear signals in both $f_0(1300)\pi $ and $f_0(980)\pi $ is remarkable and was commented upon with some surprise\cite{ves}. The decays into $\pi \rho$ and KK$^*$ are both suppressed; VES quote the limits \cite{ves} \begin{equation} \frac{\pi(1800) \rightarrow \pi^- \rho^0}{\pi(1800) \rightarrow\pi^- f_0(980)_{|\to \pi^+\pi^-}} \; < \; 0.14 \; \ \ (90 \% \ {\rm c.l.}) \end{equation} and \begin{equation} \frac{\pi(1800) \to {\rm K}^- {\rm K}^*}{\pi(1800) \to {\rm K}^-{\rm K}^+ \pi ({\rm S-wave})} \; < \; 0.1 \; \ \ (95 \% \ {\rm c.l.}) \ . \end{equation} A prominent KK$^*_0$ signal is present (observed as ${\rm K}({\rm K}\pi)_S$), so the virtual transition $\pi(1800) \to {\rm KK}^*_0 \to {\rm K}{\rm K}\pi \to f_0(980)\pi$ is probably responsible for the coupling to $f_0(980)\pi$; this mode appears to be stronger than $f_0(1300)\pi$. The mass of this state makes it a candidate for either the radial $3^1$S$_0$ or the ground state hybrid $\pi_H$. The predicted branching fractions for $3^1$S$_0$ (Table B4) and $\pi_H$ hybrid states (from Ref.\cite{cp95}) near this mass are shown in Table III. \begin{table} \caption{ Partial widths of 3S and hybrid $\pi(1800)$ states.} \label{tabpi1800} \begin{tabular}{lccccccc} & $\rho\pi$ & $\rho\omega$ & $\rho(1465)\pi$ & $f_0(1300)\pi $ & $f_2\pi$ & K$^*$K & total \\ \hline $\pi_{3S}(1800)$ & 30. & 74. & 56. & 6. & 29. & 36. & 231. \\ $\pi_H(1800)$ & 30 & 0 & 30 & 170 & 6 & 5 & $\approx 240$ \end{tabular} \end{table} The decay amplitude for $3^1$S$_0 \to {}^3$S$_1 + {}^1$S$_0$ is actually close to a node with these masses, so the weak coupling to $\rho\pi$ is expected for both a 3S quarkonium and a hybrid. The most important differences are in the $\rho\omega$ and $f_0(1300)\pi$ modes: $\rho\omega$ is predicted to be the largest mode of a 3S $\pi(1800)$ state, whereas for a hybrid $\pi_H(1800)\to\rho\omega$ should be very weak (this is the usual selection rule against S+S final states). Conversely, $f_0(1300)\pi$ is predicted to be weak for 3S quarkonium but is expected to be the dominant decay mode of a $\pi_H(1800)$ hybrid. The observation of a large $f_0(1300)\pi$ mode argues in favor of a hybrid assignment for this state. One should note however that the $^3$P$_0$ model also predicts a small branching fraction for $\pi(1300)\to \pi (\pi\pi)_S$; if the observed $\pi (\pi\pi)_S$ signal is really due to the $\pi(1300)$ rather than the Deck effect, the decay model may simply be inaccurate for N$^1$S$_0 \to ^1$S$_0 + ^3$P$_0$ transitions. There may for example be large OGE decay amplitudes in these channels, as was found in the related transition $^3$P$_0\to ^1$S$_0 + ^1$S$_0$ \cite{abs}; this can be checked in a straightforward calculation \cite{bs}. Thus the presence of a strong $\pi(1800)\to f_0(1300)\pi$ mode is indicative of a hybrid {\it assuming} that the $^3$P$_0$ model is accurate. Although the strong $f_0(1300)\pi $ signal in the VES data may well have isolated the $\pi_H(1800)$ hybrid, VES also finds evidence for a large $\rho\omega$ signal at a similar mass\cite{khok}. We expect $\rho\omega$ to arise from the 3S $\pi(1800)$ quarkonium state rather than from a hybrid. These signals may be due to two different resonances; the $\rho \omega$ signal is evident well below 1800 MeV, and persists to higher mass than the $f_0(1300)\pi $ distribution. Similarly the mode $f_2\pi $ is observed (Fig.4d of Ref.\cite{ves951}), but at a mass of $\approx 1700$ MeV, well below the $\pi(1800)$ seen in $f_0(1300)\pi $. This may also indicate a 3S state somewhat below a hybrid $\pi(1800)$. If two $0^{-+}$ $\pi$ resonances were to be isolated in this region, this would be strong evidence through overpopulation for both a hybrid and a 3S $q\bar q$ excitation. Further investigation of the modes $\rho\pi$, $\rho(1465) \pi$, $\rho\omega$, $f_0(1300)\pi$ and $f_2\pi$ could be useful to clarify the resonances in the region of the $\pi(1800)$; establishing the branching fractions to these states is especially important. The most characteristic are $\rho\omega$ and $f_0(1300)\pi$, since the hybrid and 3S quarkonium predictions differ greatly for these modes. Theoretical studies of the stability of the decay amplitudes under variation of parameters and wavefunctions and the assumed decay mechanism \cite{abs} would also be interesting. Searches for the multiplet partners of this state may be useful, since they too have characteristic decay modes. A 3S $n\bar n$ $\eta(1800)$ quarkonium for example (Table B4) is predicted to have large $\rho\rho$ and $\omega\omega$ modes, which should be zero for a hybrid. An $\eta(1760)$ which couples to $\rho\rho$ and $\omega\omega$ was reported by MarkIII \cite{MIIIeta} and by DM2 \cite{DM2eta}. The conclusions regarding the presence of this pseudoscalar signal in the MarkIII $4\pi$ data have since been disputed \cite{bsz}. \subsection{$1^{--}: \; 3^3$S$_1 $} If the $\pi(1800)$ is a 3S quarkonium we should expect to find 3S vector states near 1.9~GeV. No candidates for these states are known at present below 2.1 GeV, however there are possible $\rho$ candidates at 2150 and 2210 MeV\cite{pdg94}. The predictions for decays of 3S vectors are given in Table B3; it is notable that the simple S+S modes have small couplings, with the exception of $\rho(1900)\to\rho\rho$. Unfortunately the relatively obscure 2S+S modes are favored, especially for the $\omega(1900)$. Some S+P modes have sufficiently strong couplings to the 3S vectors to be attractive experimentally, notably $\rho(1900)\to a_2\pi$ and $\omega(1900)\to b_1\pi$. As noted previously, the $b_1\pi$ mode is forbidden to an $\omega$ vector hybrid by the singlet selection rule, since this hybrid decay would have S$_{q\bar q}=0$ for all states. \section{2P States} The 2P states are especially important because the expected mass of this multiplet ($\approx 1700$ MeV) is close to the predicted mass of the lowest hybrid multiplet in the flux tube model, $\approx 1.8$-$1.9$ GeV \cite{paton85,bcs}. Furthermore, the position of the 1P and 2P unmixed $n\bar n$ levels and the 1P $s\bar s$ level are needed for input to quarkonium - glueball mixing studies\cite{cafe} based on the lattice expectations for glueballs in this region\cite{glueball}. Determining the nature of the $f_J(1710)$ will be important in this regard. Since the quantum numbers $1^{++}$ and $1^{+-}$ occur in both the hybrid and 2P multiplets, these states need to be identified to avoid confusion with hybrids. As we shall see, a recently discovered $1^{++}$ state, the $a_1(1700)$, appears to be our first confirmed member of the 2P multiplet, in that it passes a very nontrivial $^3$P$_0$ model amplitude test and thereby for the first time establishes the mass scale of the 2P multiplets. \subsection{$1^{++}: \; 2^3$P$_1 \; a_1(1700)$} A recent experiment at BNL \cite{bnl1} reported a candidate $1^{-+}$ exotic, produced by $\pi \rho$ and decaying to $\pi f_1$. They also see a $1^{++}$ state in this channel at $\approx 1.7$ GeV, with a width of $\approx 0.4$ GeV; the relative phase of the $1^{++}$ and $1^{-+}$ waves was used to support the claim of a resonant $1^{-+}$. A similar $1^{++}$ signal has been reported by VES in $\rho\pi$ \cite{ves951,ves}. The challenge is to establish whether this $1^{++}$ $a_1(1700)$ is a hybrid $a_{1(H)}$ (perhaps a partner of the reported $1^{-+}$ exotic) or a radial $2^3$P$_1$ $n\bar n$ state. The predicted total width of a $1^{++}$ $a_1(1700)$ hybrid in the model of Close and Page \cite{cp95} is $\approx 300$ MeV, comparable to the observed width. However the total width predicted for a $a_1(1700)$ $2^3$P$_1$ $n\bar n$ state is similar, about 250 MeV (see Table B5). Some differences between these assignments are evident when we compare partial widths (see Table IV). \begin{table} \caption{ Partial widths of 2P and hybrid $a_1(1700)$ states.} \label{taba11700} \begin{tabular}{lccccccccc} & $\rho\pi$ & $\rho\omega$ & $\rho(1465)\pi$ & $b_1\pi$ & $f_0(1300)\pi $ & $f_1\pi$ & $f_2\pi$ & K$^*$K & total \\ \hline $a_{1(2P)}(1700)$ & 57. & 15. & 41. & 41. & 2. & 18. & 39. & 33. & 246. \\ $a_{1(H)}(1700)$ & 30 & 0 & 110 & 0 & 6 & 60 & 70 & 20 & $\approx 300$ \end{tabular} \end{table} Clearly the 2P state couples more strongly to S+S modes than does the hybrid, as usual, so an accurate determination of the branching fractions to $\rho\pi$ and $\rho\omega$ would be interesting. The other modes are less characteristic with the exception of $b_1 \pi$, which should come exclusively from the quarkonium state. The absence of the decay $a_{1(H)}\to b_1\pi$ is a special case of the singlet selection rule cited previously as forbidding the transition $\omega_H \to b_1\pi$. We therefore urge that experiments that observe $a_1(1700) \rightarrow \pi f_1$ also seek a signal, or a limit, for $a_1(1700)\to \pi b_1$. A crucial test of 2P versus H assignments for the $a_1(1700)$ arises in the decay amplitudes to $\rho\pi$. From Appendix A, Eqs.(A53,A58,A59), the transition $2 {}^3{\rm P}_1 \to {}^3{\rm S}_1 + {}^1{\rm S}_0 $ has both S and D amplitudes, and the D/S ratio is (where $x \equiv |\vec{p}_f|/\beta$) \begin{equation} {{\rm D}\over {\rm S}}\bigg|_{2 {}^3{\rm P}_1 \to {}^3{\rm S}_1 + {}^1{\rm S}_0} = -{ 2^{1/2} 7 \over 3^2 5} {x^2 (1-{2\over 21} x^2 ) \over (1 - {4\over 9}x^2 + {4\over 135} x^4 )} \ . \end{equation} The inverse of this ratio is shown versus $\beta$ in Fig.4; note that the S-wave amplitude has a zero very close to the preferred value $\beta=0.4$ GeV. This is a striking and unusual result, since in most cases we find that the lower partial waves are dominant. In contrast, for a hybrid one expects S-wave dominance, $a_{1(H)} \to (\rho \pi)_S:(\rho \pi)_D \approx 20:1$. \begin{figure} $$\epsfxsize=5truein\epsffile{bcpsfig4.eps}$$ {Figure~4. The S/D amplitude ratio in the transition 2$^3$P$_1$ $a_1(1700)\to\rho\pi$ predicted by the $^3$P$_0$ model. } \end{figure} Experimentally, VES sees the $a_1(1700)$ prominently in the $\rho\pi$ D-wave (see Fig.2c of Ref.\cite{ves951}); the resonance near 1.7 GeV dominates the entire 1-2 GeV region. In contrast, the $\rho\pi$ S-wave (Fig.2a of \cite{ves951}) is dominated by the $a_1(1230)$ and shows no clear evidence for the $a_1(1700)$. E852 similarly sees this resonance clearly in the $\rho\pi$ D-wave, with a mass and width of M $\approx 1.66$ GeV and $\Gamma\approx 0.22$ GeV \cite{suchung}. This D-wave dominance of the $\rho\pi$ final state appears to be dramatic confirmation that the $a_1(1700)$ is a 2$^3$P$_1$ radial excitation. Furthermore the successful predictions of $a_1 \to \rho \pi$ being in S wave and $a_{1R} \to \rho \pi$ being in D wave supports the extension of the model to radial excitations. With the $a_1(1700)$ established as a 2P $n\bar n$ state, the multiplet partners are expected nearby in mass (multiplet splittings due to spin-orbit and tensor forces appear to be small even at L$_{q\bar q}=1$) and searches for these states should be carried out. In the next sections we will discuss the decay modes predicted for these other 2P states. \subsection{$0^{++},2^{++} \ 2^3$P$_0, 2^3$P$_2 \; $: $a_0(1700), a_2(1700)$ } With the $a_1(1700)$ as the 2$^3$P$_1$ ``$a_{1R}$'' radial state, one may ask why the $a_{0R}$ and $a_{2R}$ partners are not seen in the same experiments. A simple explanation follows from the partial widths shown in Table B5. Since the production mechanism of the $a_1(1700)$ in $\pi p \to \pi f_1 p$ apparently involves natural parity exchange (probably $\rho$ or $f_2$ exchange), the $0^{++}$ scalar state $a_{0R}$ cannot be produced. Although the $2^{++}$ $a_{2R}$ can be produced (note the large $\rho\pi$ coupling), it has a weak coupling to the $\pi f_1$ final state and hence is not readily observable in this channel. There is some very recent evidence for a 2$^3$P$_2$ state from the Crystal Barrel, who report an $a_2(1650)$ in $\eta\pi^o$ final states in $p\bar p \to \eta \eta \pi^o$ \cite{Degener}. Although we expect $\eta\pi$ to be a relatively minor mode, with a branching fraction of 7\%, the mass and reported width of $\Gamma=260(15)$ MeV are consistent with expectations (Table B5). The final states $\rho\pi$ and $\rho\omega$ are predicted to have large couplings to an $a_{2R}$ state, so we expect a large signal in these $ 3 \pi$ and $5\pi$ final states. The prediction of a large coupling to vector meson pairs suggests $\gamma \gamma \to 2^3$P$_{\rm J} \to VV$ as a possible source of the $a_{0R}$ and $a_{2R}$ states. Indeed, ARGUS has evidence that the $\rho\omega$ final state near threshold is mainly in the partial wave J$^{PC}=2^{++}$, J$_z=2$, and the $\gamma\gamma\to \rho^o\omega$ cross section is at maximum near 1.7 GeV \cite{argusa2r}. The J$_z=2$ signal is characteristic of a $2^{++}$ resonance, as there is a selection rule\cite{abc} that $\gamma \gamma \rightarrow ({\rm J}=2^{++}, \lambda=0) = 0$ in the nonrelativistic quark model; hence $\lambda=2$ dominates. A study of $\gamma \gamma \rightarrow 5 \pi$ with improved statistics, perhaps at LEP2, may help to isolate these states. Of course the interpretation of any $\gamma\gamma \rightarrow VV$ reaction should be regarded as tentative until the large $\gamma\gamma\to\rho^o\rho^o$ signal \cite{mpwrev} is understood, as this reaction also is dominated by J$^{PC}=2^{++}$, J$_z=2$, but contains both I=0 and I=2 projections in s-channel and hence cannot come from a single $q\bar q$ resonance. Finally, the reaction $\gamma \gamma \to a_{0R} \to \pi b_1$ may also lead to a significant signal in $5 \pi$ final states, and could be isolated if the $\lambda = 0$ selection rule is used to suppress the $a_{2R}$ signal. \subsection{$2^{++} \ 2^3$P$_2 \; $: $f_2(1600 - 1800)$} Encouraged by the likely confirmation of the radial $1^{++}$ $a_1(1700)$, we now turn our attention to the 2P isoscalar multiplet. First we consider the $f_2(1700)$ 2$^3$P$_2$ $n\bar n$ radial tensor. We predict a large $\rho\rho$ width for the 2$^3$P$_2$ $f_2(1700)$, and the modes $\omega \omega$, $\pi \pi$ and perhaps $\pi a_2$ should also be important (see Table B6). (Note that the simple branching fraction ratio $\rho\rho/\omega\omega \approx 3$ follows trivially from flavor counting.) The total width is predicted to be $\approx 400$ MeV. Although there is no strong evidence for such a state, there are suggestions of its presence in several processes. A large $2^{++}$ enhancement referred to as the $X(1600)$, with $\Gamma=400(200)$ MeV, is well known in $\gamma\gamma\to\rho^o\rho^o$ \cite{pdg96,argus91}. The small charged to neutral $\rho\rho$ ratio however precludes the identification of this signal with a single $f_2(1700)$ resonance. There are also reports of a rather narrow $f_2(1640)$ with a width of $\approx 60$-$120$ MeV in $\omega \omega$\cite{pdg96,argusww,ves92,obelix92}. Although the predicted 2$^3$P$_2$ $f_2(1700)$ width is much larger, it would be reduced somewhat by threshold effects in the $\omega\omega$ channel. Indeed, if the resonance mass is around 1700 MeV and its width is several hundred MeV, as suggested by our analysis, it may decay strongly into $\rho \rho$ (due to the large $\rho$ width leading to a favorable phase space), but the narrowness of the $\omega$ may cause only the upper part of the resonance to feed the $\omega \omega$ channel. Thus the resonance width in $\omega\omega$ may appear smaller than in $\rho\rho$, so both the X(1600) and the $f_2(1640)$ may be aspects of a single state. A recent reanalysis of MarkIII data on $\psi\to\gamma\pi^+\pi^+\pi^-\pi^-$ \cite{bsz} similarly sees evidence of a $2^{++}$ state near M $=1.64$ GeV, with $\Gamma=0.14$ GeV, which couples strongly to $\rho\rho$. (In contrast they observe $0^{++}$ states dominantly in $\sigma\sigma$.) This preference of the tensor state for $\rho\rho$ is consistent with $^3$P$_0$ model expectations for a 2$^3$P$_2$ $f_2(1700)$ state (Table B6). Finally, it is possible that the $f_2(1520)$ or ``AX" state seen in $p\bar{p} \to 3 \pi$ \cite{axstate} may be the low-mass tail of the $f_2(1700)$. \subsection{$0^{++}\ 2^3$P$_0 \; $: $f_0(1500), f_{\rm J}(1710)$} The $0^{++}$ $f_0$ sector in the 1.5 GeV mass region is clearly of interest for glueball searches. It is thus important to identify the $^3$P$_0$ quarkonia in this mass region. We stress that one should not be overly naive in this endeavor since strong recoupling effects, including couplings of quarkonia to nearby glueballs, are expected \cite{cafe}. Nonetheless for initial theoretical guidance it will be useful to consider the predictions of the naive $^3$P$_0$ model for the decays of unmixed $^3$P$_0$ $n\bar n$ quarkonia. The decays predicted for the 2P scalar $f_0(1700)$ state in the $^3$P$_0$ model are given in Table B6. Fortunately they are very characteristic. The dominant modes are $\rho\pi\pi$, with approximately equal contributions from $\pi(1300)\pi$ and $a_1(1230)\pi$. The channels $\rho\rho$ and $\pi\pi$ are also important, and the total width is predicted to be $\approx 400$ MeV. The $\eta\eta$ and KK amplitudes are both close to nodes and are predicted to be quite small. The two well known scalar resonances in this mass region which can be compared to these predictions are the glueball candidate $f_0(1500)$ and the $f_{\rm J}(1710)$. These states have PDG masses and total widths of M $=1503(11)$ MeV, $\Gamma = 120(19)$ MeV and M $=1697(4)$ MeV, $\Gamma = 175(9)$ MeV; both are rather narrow relative to expectations for a 2P $n\bar n$ state. BES has recently reported\cite{besnew} a spin parity analysis of the K$^+$K$^-$ system in $\psi$ radiative decays; they see both J=0 and J=2 states. Both have widths of $\approx 100$~MeV, much narrower than we expect for 2P $n\bar n$ states. The presence of a significant $\eta\eta$ mode for both the $f_0(1500)$ and $f_{\rm J}(1710)$ argues against a 2P $n\bar n$ assignment. The possibility that a node in the 2P decay amplitude is consistent with the observed weakness of $f_{\rm J}(1710)\to\pi\pi$ is found to be unrealistic in practice; although there are actually two nodes, the modes that are strongly suppressed by these in the $^3$P$_0$ model are $\eta\eta$ and KK, not $\pi\pi$. The disagreement of predicted decay modes of 2P $n\bar n$ states with experiment for the $f_0(1500)$ and $f_{\rm J}(1710)$ supports the suggestions that neither of these states is a quarkonium. Amsler and Close \cite{cafe} have noted that the $f_0(1500)$ could be a glueball that is mixed with the nearby $n\bar n$ and $s\bar s$ basis states, which explains the observed branching fractions. Conversely, Weingarten \cite{wein} suggests that the $f_{\rm J}(1710)$ is the scalar glueball, based on its mass and on lattice QCD evidence that flavor symmetry may be inaccurate in glueball decays, together with a different pattern of $q\bar q\leftrightarrow G$ mixing. It may be that the glueball, $n\bar n$ and $s\bar s$ basis states are all strongly mixed in this sector, so that an assumed separation into glueball and quarkonium states is inaccurate \cite{glynnys}. An alternative suggestion is that the $f_{\rm J}(1710)$ may be a vector-vector molecule, analogous to the $f_0(980)$ and $a_0(980)$ K$\bar{\rm K}$ candidates. The two possibilities discussed in the literature are K$^*\bar{\rm K}^*$ \cite{T} and K$^*\bar{\rm K}^*$+$\omega \phi$ \cite{dsb}; these both predict small nonstrange modes and large couplings to KK$\pi\pi$ final states. The weakness of the $\pi\pi$ mode is due to the presence of a hidden $s\bar s$ pair (just as for $f_0(980)\to\pi\pi$), since both models assume that the $f_{\rm J}(1710)$ is dominantly $ns\bar n \bar s$ in flavor. In any case the 2P scalar $n\bar n$ states (or resonances with large $2^3$P$_0$ $n\bar n$ components) should appear in $\rho\pi\pi$ final states, so it would be useful to search for these states, especially in reactions that produce the $f_0(1500)$ or $f_{\rm J}(1710)$. Finally, we should consider the possibility that the $f_{\rm J}(1710)$ is dominantly a 2$^3$P$_2$ $n\bar n$ tensor state (see Table B6), since the quantum numbers have not been determined definitively. Again the quarkonium assignment is inconsistent with experiment; the $\eta\eta$ coupling is predicted to be small, and $\pi\pi$ is predicted to be quite large. The largest mode, $\rho\rho$, has not been reported for the $f_{\rm J}(1710)$. The total width of the $n\bar n$ state is again rather larger than reported for the $f_{\rm J}(1710)$. One must conclude that the $f_{\rm J}(1710)$ does not appear to be consistent with any $n\bar n$ quarkonium assignment. \subsection{$1^{+-}\ 2^1$P$_1 \; $: $b_1(1700), h_1(1700)$} Predictions for the missing spin-singlet 2P states are given in Table B7. These are expected to be only about 250 MeV wide, so they may be easy to detect. Reactions that produce the $h_1(1170)$ and $b_1(1231)$ are obviously the most promising for searches for their radial excitations. The $h_1(1700)$ couples dominantly to $\rho\pi$, so it may be observable for example in $\pi^-p\to \rho\pi n$, in production through natural-parity exchange. Its partner $b_1(1700)$ can be produced similarly in $\omega\pi $ final states, and less characteristically in $\rho\rho$. \section{1D States:} \subsection{$2^{-+}\ ^1$D$_2 \; $} Studies of the decays of hybrids in the flux tube model conclude that a $2^{-+}$ member of the lowest hybrid multiplet may be observably narrow \cite{cp95}. This hybrid multiplet is expected at $\approx 1.8$-$1.9$ GeV \cite{paton85,bcs}, which overlaps the Godfrey-Isgur quark model predictions of 1.68 GeV for the $^1$D$_2$ $n\bar n$, 1.89 GeV for $^1$D$_2 \; s\bar{s}$, and 2.13 GeV for 2$^1$D$_2$ $n\bar n$ \cite{god}. Thus it may be necessary to use characteristic branching fractions to distinguish quarkonia from hybrids in this mass region. Of course the $\pi_2(1670)$ is presumably $n\bar n$ because it has well established 1D multiplet partners such as the $\rho_3(1691)$, but distinguishing the higher-mass $s\bar s$ and 2D quarkonia from hybrids may not be so straightforward. \subsection{$\pi_2$} Experimentally, the $\pi_2(1670)$ couples most strongly to $f_2(1275)\pi$ ($\approx 56\% $) and $\rho\pi$ ($\approx 31\% $), with weaker couplings (at the 5-10$\% $ level) to $f_0(1300)\pi$ and K$^*$K. The 1996 PDG total width is 258(18) MeV \cite{pdg96}. In comparison, the $^3$P$_0$ model predicts a total width of 250 MeV, with branching fractions of $f_2(1275)\pi$ ($\approx 30\% $), $\rho\pi$ ($\approx 47\% $) and K$^*$K ($\approx 12\% $); these are in reasonable qualitative agreement with experiment. There is however disagreement with experiment in that little $f_0(1300)\pi$ is expected; we predict a branching fraction of only $0.2\% $ to this mode, whereas the PDG value is $8.7(3.4)\% $. The largest as yet unreported mode should be $\rho\omega$, predicted to have a branching fraction of $11\% $. In addition to the plausible quarkonium state $\pi_2(1670)$, the ACCMOR Collaboration in 1981 noted a $2^{-+}$ structure near 1.8 GeV, coupled to $f_2\pi $ and weakly to $f_0(1300)\pi$ and $\rho\pi$ \cite{acc}. This is similar to reports of a possible $2^{-+}$ (or even $1^{-+}$) seen in photoproduction of $3\pi$ states near $1.77$~GeV with a width of 100-200 MeV, which couples to $\rho\pi$ and $f_2\pi$ \cite{pi2phot}. The VES Collaboration also claims a peak near 1.8 GeV, which they believe however to be non-resonant \cite{rya}. Lastly, two-photon experiments which see the $\pi_2(1670)$ in $\gamma\gamma\rightarrow\pi_2\rightarrow\pi^0\pi^0\pi^0$ \cite{crystalball} and $\gamma\gamma\rightarrow\pi_2\rightarrow\pi^+\pi^-\pi^0$ \cite{cello} also see indications of a possible contribution around 1.8 GeV. (In both cases the data appear skewed towards the higher masses relative to simple Breit Wigner and PDG values.) This may be expected for $\pi_{2(D)}$ through VMD as its $\rho \omega$ coupling is predicted to be large and thereby provide a further probe for any 2D component in $\pi_2(1800)$ state. It may be possible for LEP2 to clarify this situation. If there is indeed a second $\pi_2$ state near 1.8 GeV, it is much too light to be a radial excitation of the $\pi_2(1670)$, and may instead be a hybrid. To test this possibility we have calculated the branching fractions of a $\pi_2(1800)$ hybrid in the flux tube model, and for comparison we show the partial widths of a hypothetical 1D quarkonium $\pi_2(1800)$. These are given in Table V. (The partial widths to $a_1(1230)\eta$ and K$^*_1(1273)$K are $<1$ MeV in both models, so these modes are not displayed.) \begin{table} \caption{ Partial widths of 1D and hybrid $\pi_2(1800)$ states.} \label{tabpi2} \begin{tabular}{lccccccccc} & $\rho\pi$ & $\omega\rho$ & $\rho_R\pi$ & $b_1\pi$ & $f_0\pi$ & $f_1\pi$ & $ f_2\pi $ & K$^*$K & total \\ \hline $\pi_{2(1D)}(1800)$ & 162. & 69. & 0. & 0. & 1. & 5. & 86. & 49. & 372. \\ $\pi_{2(H)}(1800)$ & 8 & 0 & 5 & 15 & 1 & 0 & 50 & 1 & 80 \end{tabular} \end{table} Evidently there are very characteristic differences between hybrid and 1D ($\pi_2$) branching fractions. First, note that a large $f_2(1275)\pi$ mode is {\it not} distinctive; this is expected from both states. A 1D quarkonium should also couple strongly to $\rho\pi$, $\omega\rho$ and K$^*$K, and the total width should be about 400 MeV. In contrast, these S+S modes are weak for a hybrid; the second largest mode (after $f_2\pi$) should be $b_1\pi$, which is forbidden to quarkonium by the singlet selection rule. Clearly a study of $b_1\pi $ final states in processes that report a $\pi_2(1800)$ would be very useful as a hybrid search. Other modes are quite small, so the hybrid should be a relatively narrow state, with a total width of only about 100 MeV. In summary, the characteristic signature of a $\pi_{2(H)}(1800)$ hybrid is a strong $f_2\pi $ mode and some $b_1\pi$ but weak couplings to $\rho\pi$, $\omega\rho$ and K$^*$K. \subsection{$\eta_2$} A doubling of $2^{-+}$ peaks has also been reported by Crystal Barrel, in the isoscalar sector in $p\bar p \to (\eta\pi^{o}\pi^{o})\pi^{o}$ \cite{bugg}. Masses and widths of M $ = 1645(14)(15)$~MeV, $\Gamma = 180 {+40\atop-21}(25)$~MeV and M $ = 1875(20)(35)$~MeV, $\Gamma = 200(25)(45)$~MeV have been reported for the two $2^{-+}$ states. This $\eta_2(1645)$ is seen in $a_2(1318)\pi$ \cite{eta2cb}, and in view of the approximate degeneracy with the $\pi_2(1670)$ and other 1D candidates is probably the $^1$D$_2$ $n\bar n$ isosinglet partner of $\pi_2(1670)$. The higher-mass state $\eta_2(1875)$ has been seen only in $f_2(1275)\eta$ (only 50 MeV above threshold), and no evidence of it is found in $a_0 (980)\pi$, $f_0(980)\eta$ or $f_0(1300)\eta$. The Crystal Ball Collaboration some time ago reported a $2^{-+}$ (or possibly $0^{-+}$) at 1880 MeV, with a width of 220 MeV, decaying equally to $a_2 (1318)\pi$ and $a_0 (980)\pi$ \cite{crystalball}. These data are also consistent with a contribution from $\eta_2(1645)$. One expects $\gamma\gamma\rightarrow\eta_2 > \gamma\gamma\rightarrow\pi_2$, with the magnitude of the signal in $\gamma\gamma\rightarrow \eta\pi\pi$ depending on BR($\eta_2\rightarrow \eta\pi\pi$). Here again LEP2 may have much to contribute. \begin{table} \caption{ Partial widths of 1D and hybrid $\eta_2(1875)$ states.} \label{tabeta2} \begin{tabular}{lcccccccc} & $\rho\rho$ & $\omega\omega$ & $f_2\eta$ & $a_0(1450)\pi$ & $a_1\pi$ & $ a_2\pi $ & K$^*$K & total \\ \hline $\eta_{2(1D)}(1875)$ & 147. & 46. & 45. & 1. & 43. & 264. & 61. & 607. \\ $\eta_{2(H)}(1875)$ & 0 & 0 & 20 & 2 & 0 & 160 & 10 & $\approx 190$ \end{tabular} \end{table} In Table VI we compare the decay modes expected for a hybrid at 1875 MeV with $^3$P$_0$ model predictions for a hypothetical $^1$D$_2$ $\eta_2(1875)$ quarkonium. Both assignments lead to a significant $f_2\eta$ signal, and both predict a much larger $a_2\pi$ mode. The most characteristic modes are $\rho\rho$ and $\omega\omega$, which should be very weak for a hybrid but large for a 1D quarkonium. Similar results follow for ${\rm K}^*{\rm K}$ and $a_1\pi$. Clearly searches for $a_2\pi$, $\rho\rho$ and $\omega\omega$ would be most useful. The large predicted coupling to $\rho \rho$ for the $\eta_{2(1D)}$ encourages a search in $\gamma \gamma$ for this state. \subsection{$^3$D$_{\rm J}$ states} Here we consider only the $^3$D$_3$ and $^3$D$_2$ states since the $^3$D$_1$ vectors were previously discussed with the 2$^3$S$_1$ states. The $3^{--}$ states $\rho_3(1691)$ and $\omega_3(1667)$ are well established $^3$D$_3$ $n\bar n$ quarkonia, with masses as expected for 1D states and widths of about 200 MeV. The $\rho_3$ (Table B7) is expected to decay mainly to $\rho\rho$ $(41\% )$ and $\pi\pi$ $(34\% )$, with a somewhat weaker $\omega\pi$ mode $(11\% )$. Experimentally the decays to $4\pi$ are about $70\% $, of which $16(6)\% $ is $\omega\pi$. The $\pi\pi$ branching fraction is observed to be $23.6(1.6)\% $. There are also KK and ${\rm K}^*{\rm K}$ modes of a few percent, roughly as predicted. The total width is predicted to be 174 MeV with these parameters, consistent with observation. Thus the $\rho_3(1691)$ appears to decay approximately as predicted by the $^3$P$_0$ model, which supports the application of the model to decays of high-L states. Its isoscalar partner $\omega_3(1667)$ is a more interesting case. Since few modes are open and the couplings are rather weak, we predict a total width of only 69 MeV. Although this appears inconsistent with the PDG width of 168(10) MeV, this observed value is presumably broadened by the hadronic width of the $\rho$ and $b_1$ in the two-body modes $\rho\pi$ and $b_1\pi$. The reported modes are $\rho\pi$ and $\omega\pi\pi$; we expect $\rho\pi$ to be dominant, with $\approx 10\% $ branches to $b_1\pi$ (the source of $\omega\pi\pi$?) and KK. The KK mode affords an opportunity to measure the actual width of the $\omega_3$, which may be much smaller than it appears in $\rho\pi$ and $b_1\pi$ modes. Our results for the $^3$D$_2$ $2^{--}$ states $\rho_2(1670)$ and $\omega_2(1670)$ are especially interesting because these are ``missing mesons" in the quark model. We find that these are rather broad states, with total widths of about 300-400 MeV. The $\rho_2$ is predicted to have a large branching fraction of $54 \% $ to $a_2\pi$, so it should be observable in this final state or in the secondary modes $\omega\pi$ or K$^*$K. The $\omega_2$ is predicted to have an even larger branching fraction of $74 \% $ to $\rho\pi$. It too couples significantly to K$^*$K, and may also be observable in $\omega\eta$. \section{1F States} The 1F states provide us with an opportunity to test the accuracy of the $^3$P$_0$ decay model predictions for higher quarkonium states, since the $4^{++}$ and $3^{+\pm}$ states expected near 2.05~GeV do not have competing assignments as glueballs or hybrids. At present only two of these states are reasonably well established, the $f_4(2044)$ and $a_4(2037)$ \cite{pdg96}. There is also some evidence for an $a_3(2080)$ \cite{pdg94}. We do not yet have experimental branching fractions for the I=1 1F states. The $a_4(2037)$ is seen in KK and $3\pi$, and the $a_3(2080)$ is reported in $3\pi$ and $\rho_3(1691)\pi$, with $\rho_3\pi$ dominant. The branching fractions of the $f_4(2044)$ are known with more accuracy; $\omega\omega$ and $\pi\pi$ are important modes, $26(6) \% $ and $17.0(1.5) \% $. KK and $\eta\eta$ modes are both known, with reported branching fractions of about $0.7\% $ and $0.2 \% $ respectively. $^3$P$_0$ predictions for the decays of these $^3$F$_{\rm J}$ states are given in Tables B11 and B12. The $a_4(2050)$ is indeed expected to appear in $3\pi$ (mainly $\rho\pi$), and the dominant mode is predicted to be $\rho\omega$. This state is predicted to be rather narrower than reported. The $a_3(2080)$ is predicted to decay dominantly to $\rho_3\pi$, as is observed. The $3\pi$ mode is also predicted to be large, and to arise from both $\rho\pi$ and $f_2\pi$. The $f_4(2044)$ $^3$P$_0$ model predictions are also in qualitative agreement with experiment, in that $\pi\pi$ and $\omega\omega$ are expected to be important modes, as observed. The $f_4$ partial widths to pseudoscalar pairs are uniformly too large, for example $\Gamma_{f_4 \to \pi\pi}^{thy.} = 62.$ MeV but $\Gamma_{f_4 \to \pi\pi}^{expt.} = 35(4)$ MeV. This decay however is G-wave, so the rate has a prefactor of $|\vec p_{\pi} / \beta|^9$; this extreme sensitivity means that a small increase of $\beta$ by $\approx 10\%$, halves the decay rate and gives agreement with experiment. Thus this disagreement is quite sensitive to parameters and is probably not significant. The predictions for branching fractions of the five missing I=0,1 1F states suggest that several of them may easily be found by reconstructing the appropriate final states. The total widths of all except the $^3$F$_2$ states are predicted to be $\sim 300$ MeV, so they should be observable experimentally. The $f_3(2050)$ is predicted to couple dominantly to $a_2\pi$. In the spin-singlet $^1$F$_3$ sector, the $h_3(2050)$ should appear in $\rho\pi$ and $\rho_3(1691)\pi$, just as we found for the $a_3(2080)$. The $b_3(2050)$ should be evident in $a_2\pi$, and less strongly in $\omega_3\pi$, $\omega\pi$ and $\rho\rho$. Modes such as $a_2\pi$ are preferable because the two-body mesons are not excessively broad and they are far from threshold, so a resonance can be distinguished from a threshold effect. In some cases the amplitude structure of these final states is also characteristic; these can be determined from the results quoted in App.A. The missing $^3$F$_2$ states may be more difficult to identify, as we predict large total widths of $\approx 600$ MeV for these states. The $a_2(2050)$ couples most strongly to $b_1\pi$; $\eta_2(1645)\pi$ and ${\rm K}^*_1(1273){\rm K}$ are other important modes. Its I=0 partner $f_2(2050)$ should be evident in $\pi_2(1670)\pi$ and will also populate ${\rm K}^*_1(1273){\rm K}$ final states. Identification of these 1F states and determination of their branching fractions and decay amplitudes will be a very useful contribution to the study of resonances, as it will allow detailed tests of the usefulness of the $^3$P$_0$ model as a means for identifying quarkonium states in this crucial 2 GeV region. \section{Summary and Experimental Strategy} We have established that the $a_1(1700)$ is very likely a 2P radial excitation. This follows from the weak S-wave and strong D-wave in $\rho \pi$. This also establishes the natural mass scale for the 2P multiplets as $\approx 1.7$ GeV. We have been unable to identify radial scalars. These are predicted to be broad, and so their non-appearance is not surprising. Conversely it raises interest in the (relatively narrow) $f_0(1500)$ and possible scalar $f_{\rm J}(1710)$. We do identify some (more speculative) potential candidates for $2^{++}$ 2P members. We note that $\gamma \gamma$ production may help identify these radial 2P states and also clarify the nature of $f_0(1500)$ and $f_{\rm J}(1710)$\cite{glynnys}. The $\pi(1300)$ and $\eta(1295)$ appear to be convincing 2S states. This conclusion is based on their relative widths; the large $\rho \pi$ mode of the $\pi(1300)$ has no analog for its $\eta$ counterparts. The status of the $\eta(1440)$ remains open; the mass and width suggest a dominantly $s\bar s$ state, but the $\gamma\rho$ mode argues against it. Studies of $\psi \to \eta(1295,1440) + (\omega, \phi)$ and $\psi \to \gamma + (\gamma\omega,\gamma\rho, \gamma\phi)$ may identify the flavor content of these $\eta$ states. The $\rho(1465)$ and $\omega(1419)$ have masses that are consistent with radial 2S but their decays show characteristics of hybrids, as noted previously \cite{cp95}. We suggest that these states may be 2S-hybrid mixtures analogous to the 3S-hybrid mixing suggested for the $c\bar c$\cite{pagecc}. This can be tested by accurate measurement of the partial widths of these states and their vector partners at 1.6-1.7 GeV to $\pi\pi$, $\omega\pi$, and especially $h_1\pi$ and $a_1\pi$. The 3S $\pi$ is expected in the $1800$ MeV mass region as is a $\pi_H$ hybrid. We find that the decay patterns of these states are very different. A strong $f_0(1300)\pi $ from the hybrid contrasted with a large $\rho \omega$ mode from the 3S quarkonium is the sharpest discriminant. The VES state $\pi(1800)$ clearly exhibits this hybrid signature. It is now necessary to establish the presence of $0^{-+}$ in the $\rho \omega$ channel, and to see if any resonant state is present that is distinct from the $\pi(1800)$ seen in $f_0(1300)\pi $. It is possible that there are two $\pi(\approx 1800)$ states, $q\bar q$ and hybrid, whose production mechanisms and decay fractions differ sufficiently so that they can be separated. We suggest that the possibility of two such $\pi(\approx 1800)$ states be allowed for in data analyses. In the immediate future there are opportunities for $\gamma \gamma$ physics at LEP2 and at B~factories. Possible strategies for isolating some of these higher quarkonia include: $\bullet$ $\gamma \gamma \to 5\pi$ contains (i) $\rho \omega$ which may access the radial $a_{0R}$ and $a_{2R}$ near 1700 MeV and a possible $\pi_{3S}(1800)$. (ii) $\pi b_1$ which can isolate the $a_{0R}$ if the helicity selection rule\cite{abc} is used to suppress the $a_{2R}$. $\bullet$ $\gamma \gamma \to 4 \pi$ may access the radial $f_{2R}$ near 1700 MeV through its decay into $\rho \rho$. The $4 \pi$ channel may also be searched for the $f_0(1500)$ since this state is known to have a significant branching fraction to $4\pi $ but should have a suppressed $\gamma \gamma$ coupling if it is a glueball \cite{glynnys}. $\bullet$ $\gamma \gamma \to 3 \pi$ may be searched for $2^{-+}$ states in order to verify whether the established $\pi_2(1670)$ is accompanied by a higher $\pi_2(1800)$ in $3\pi^o$ and $\pi^+ \pi^- \pi^o$. This $3\pi$ system may also be studied for evidence of one or more $\pi(1800)$ states. $\bullet$ $\gamma \gamma \to \eta \pi \pi$ may access the isoscalar partners of these $\pi_2$ states. In the near future it will be possible to study $e^+ e^-$ annihilation up to $\approx 2$ GeV at DAFNE. The channels $e^+ e^- \to 4\pi$ should be measured and $\pi a_1$ and $\pi h_1$ states separated in order to carry out the analysis of hybrid and radial vector components in section 3B. The isoscalar partners of the vectors also need confirmation, and final states with kaons are needed to investigate possible $\omega$-$\phi$ mixing; a potential weakness of the present data analyses is that such flavor mixing is assumed to be unimportant. In the next century there will be new opportunities at the COMPASS facility at CERN. This will enable further studies of central production and also of diffractive excitation. For the latter one may anticipate improved studies of the $\pi$ excitations (such as the $\pi(1300)$ and $\pi(1800)$ states), possibly including Primakoff excitation. Judicious studies of specific final states as discussed above may help separate 3S and hybrid states. The use of K beams will allow analogous studies of the strange counterparts of these states and may help to clarify the spectrum of quarkonia, glueballs and hybrids. Experiments with $\pi$ beams can access the following interesting channels. $\bullet$ $\pi p \to (\pi f_1) p$, to confirm the D-wave dominance of $a_{1R}(1700)$ and to seek its partner $a_{2R}$. $\bullet$ $\pi p \to (\pi f_2) p$ can access both $\pi_{2(1D)}$ and $\pi_ {2(H)}$. These can be separated in $b_1\pi$; the singlet selection rule forbids this mode for $\pi_{2(1D)}$ but allows it for $\pi_{2(H)}$. $(\pi \rho) p$ can also separate $\pi_{2(1D)}$ from $\pi_{2(H)}$; $\pi_{2(1D)} \to \rho \pi$ is the dominant mode whereas $\pi_{2(H)}$ is much suppressed into S+S hadrons. $\bullet$ $(\pi\pi), (\pi\omega)$, $(a_1\pi) $ and $(h_1\pi) $ are important in the interpretation of the vectors between 1.4 and 1.7 GeV, which may contain large hybrid components. $\bullet$ $(f_0\pi), (f_2\pi)$ and $(\rho\omega) $ can all be searched for evidence of $\pi(1800)$ states. $\bullet$ $\pi^- p \to (\pi \rho)^o n$ or $ (\pi \omega)^o n$ access respectively $h_{1R}$ and $b_{1R}$. Finally, many two-body channels are predicted to couple strongly to specific 2P, 1D and 1F states, as shown in Appendix B. These include ``missing mesons" such as the $^3$F$_2$ and most 2P states, and studies of these two-body final states may reveal the missing resonances. The modes $a_2\pi$, $\rho\rho$ and $b_1\pi$ are important for many of these missing states and merit careful investigation. We reiterate that it is in general a good strategy to study decays into both S+S and S+P meson modes, as the relative couplings of these modes are usually quite distinct for hybrid versus quarkonium assignments. \newpage \acknowledgements We would like to acknowledge useful communications with C.Amsler, D.V.Bugg, S.U.Chung, G.Condo, K.Danyo, A.Dzierba, S.Godfrey, I.Kachaev, Y.Khokhlov, A.Kirk, D.Ryabchikov and A.Zaitsev. This work was supported in part by the United States Department of Energy under contracts DE-FG02-96ER40944 at North Carolina State University and DE-AC05-96OR22464 managed by Lockheed Martin Energy Systems Inc. at Oak Ridge National Laboratory. FEC is supported in part by European Community Human Capital Mobility Programme Eurodafne, Contract CHRX-CT92-0026. \newpage

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\section*{References}

proofpile-arXiv_065-562

\section{Introduction} Since the discovery of quantized vortices in liquid helium II, it has been recognized that they might provide a mechanism for the coupling of the superfluid to the normal fluid. In the two fluid model, this coupling is represented by a mutual drag force with one dissipative and one conservative component, whose respective strengths can be measured investigating the attenuation of second sound at various temperatures \cite{HV,Don1,Don2}. Models for the friction coefficients which successfully fit the data up to 2.1 K have been presented in Refs. \cite{MS,SD}. The phenomenon of vortex mutual friction has been observed as well in rotating superfluid $^{3}$He-B \cite{Bev}; this fact brings support to the conjecture that such a mechanism is indeed a relevant source of dissipative processes in highly degenerate quantum fluids. It is also worthwhile to remind that in the last years, vortices have been seen to play a role in phase transitions taking place either in underpressurized $^{4}$He \cite{Ma} or in supersaturated solutions of $^{3}$He in $^{4}$He \cite{DJ}. The interaction between the velocity field and the density fluctuations of the superfluid, which at nonvanishing temperatures are embodied in the normal fluid, can be accounted for within a Lagrangian description \cite{dem1,dem2}. However, microscopic descriptions of the interaction between the superfluid motion, especially when topological singularities are concerned, and collective excitations, are not conclusive \cite{Don1,Don2}. The investigation of quantum tunneling of vortex lines in superconductors and superfluids \cite{Ao,Ste,Niu} has now improved our comprehension of, for example, the role of the inertial mass of the vortex (see also Refs. \cite{Duan0,Duan1,Duan2} and cited therein) and the influence of either pinning or dissipation on the tunneling rates. Indeed, the value of the vortex inertia is a fundamental parameter in any theoretical description of vortex dynamics and it remains being a controversial issue \cite{Duan2,Niu3}. Different starting points assign to this inertia figures ranging from zero to infinity, the latter arising from a logarithmic divergence with the system size due to the renormalization effect induced by the condensate motion. On the other hand, the vortex mass is known to be finite in superconductors \cite{Su}. Although the phenomenological approaches \cite{Don2} completely disregard inertial effects, the mass enters the description of the dynamics of a free vortex, known to be cyclotron-like \cite{Don1,dem2,Niu3}, through a frequency parameter $\Omega$, which would be a measurable quantity if this time dependent regime were experimentally visualized. In particular, it has been recently shown \cite{dem2} that the cyclotron motion is a natural solution of the nonlinear Schr\"{o}dinger equation applied to a vortex. Dynamical \cite{dem2} and thermodynamical \cite{Mo} methods have been proposed to measure the vortex inertial coefficient; our present purpose is neither to participate in the existing polemics, nor to propose a new model for the calculation of the vortex mass, but rather to assume that it is a numerical parameter and proceed along similar lines as those invoked in the well established cyclotron motion already discussed in textbooks \cite{Don1}. The aim of the present work is to propose a Hamiltonian model for the coupling between a rectilinear vortex immersed into the excitations of the superfluid, as will be discussed in Sec. 2. Due to the translational symmetry of the problem along an axis parallel to the vortex, the problem is a twodimensional one, {\it i. e.}, we consider a point vortex on a plane. We shall show in Sec. 3 that, if one considers the vortex as a quantum particle undergoing Brownian motion \cite{DH} in a heat reservoir, it is possible to establish the irreversible time evolution of its density operator within the Generalized Master Equation (GME) approach \cite{Haa,Cha}. In this way, in Sec. 4 we are able to derive dissipative equations of motion for the canonical position-momentum variables of the vortex and for its velocity. These variables can be seen to evolve under the combined effect of the usual hydrodynamical lift on a rotating cylinder, plus a drag force. If the coupling is linear in the excitation operators, the drag coefficients are governed by the dynamical susceptibility of the liquid. The consequences of the equations of motion thus obtained, the asymptotic velocity of the vortex and the relation of the current description to the phenomenological model are discussed in Sec. 5, where the perspectives of the present approach are also outlined. \section{The Hamiltonian model} Let us first summarize the description of the free motion. The Hamiltonian for a cylindrical vortex parallel to the $z-$axis in liquid helium at zero temperature is in charge of providing the Magnus force \cite{Don1}. It reads \begin{equation} H_v = \frac{1}{2 M} \left [{\bf p} - q {\bf A}({\bf r})\right]^2 + M \, \Omega\, v_s\, y \label{Hfree} \end{equation} where \begin{equation} {\bf A}({\bf r}) = \frac{h\,\rho_s\,l}{2} (y,-x) \end{equation} is the vector potential whose curl yields the vortex-velocity-dependent part of the Magnus force and the potential term $M \Omega v_s y$ gives the superfluid-velocity-dependent part of this force. Here $M$ is the dynamical mass of the vortex, $\rho_s$ denotes the number density of the superfluid, $v_s$ its velocity along the $x-$axis, assumed to be uniform, $h$ is Planck's constant, $l$ the system length along the $z-$ axis and \begin{equation} \Omega= \frac{q \,h\,\rho_s\,l}{M}. \end{equation} The quantity $q = \pm 1$ is the sign of the vorticity according to the right handed convention. Furthermore, at zero temperature, $\rho_s$ coincides with the total density per unit mass $\rho/m$, being $m$ the mass of a helium atom. At this point it is convenient to remember the existing theoretical uncertainty regarding the vortex mass parameter $M$ that should appear in dynamical calculations \cite{Niu,Duan1,Duan2} and keep in mind that in the phenomenological approaches \cite{Don2} the dynamical regime of the vortex is that in which the Magnus force balances the drag plus any applied force \cite{BDV}. As stated in the Introduction, our viewpoint here is identical to that of former authors \cite{Don1,dem2,Ao,Niu} who assume a finite figure for the vortex inertia and consider the cyclotron - like motion of a free vortex as their starting point, with the frequency $\Omega$ as the leading parameter. Since it will be shown in Sec. 4 that the dissipative motion is easily described in terms of the complex position variable $z = x + i y$ and the velocity $d z/d t$, we here write the complex Hamilton equation that stems from (2.1) \begin{equation} \frac{ d^2 z}{d t^2} = i\,\Omega\,\left(\frac{d z}{d t} - v_s\right) \end{equation} with the complex Magnus force at the right hand side. We now assume liquid helium to contain elementary excitations. For nonvanishing temperatures $T$, these excitations can be of thermal origin and thus give rise to the normal fluid, while at zero temperature they must be created by an external probe and yield a vanishing normal density $\rho_n$. If $T$ is above 1 K, the normal component is mainly a gas of rotons, being the phonons the dominant excitations at lower temperatures. Therefore, at any temperature the interaction of the vortex line with these elementary excitations, produces damped motion of the vortex. As stated in the introduction, the main goal of this article is to construct a hamiltonian model which enables us to obtain this dissipative behavior. For this purpose, we shall consider a description of quantum dissipation similar to that recently presented in order to account for the irreversible evolution of solitons \cite{CNCal2}, in which an effective Hamiltonian is constructed for the collective motion coupled to the residual excitations using the Collective Coordinate formalism \cite{Raj}. This model exhibits unexpected features \cite{MD} due to the fact that both the system and the reservoir have the same microscopic origin, which is just the case here discussed. In this spirit, and considering that within a superfluid in its ground state the vortex exhibits a soliton-like behavior, we propose a vortex-plus-reservoir Hamiltonian, that modifies expression (\ref{Hfree}) as follows \begin{equation} H = \frac{1}{2 M} \left [{\bf p} - q {\bf A}({\bf r}) - \lambda\,{\bf B}\right]^2 + M\,\Omega\,v_s\,y + H_B \label{H} \end{equation} with ${\bf B}$ a vector function of operators that represents the elementary excitations of the superfluid and $H_B$ is the Hamiltonian of these excitations. In Eq. (\ref{H}) the interaction term $H_{int} = - \lambda\, {\bf B}\cdot {\bf v}$ couples the reservoir and the vortex through the unperturbed velocity of the latter, being \begin{equation} {\bf v} = \left(\frac{ p_x}{M}- \frac{\Omega}{2} \,y, \frac{ p_y}{M} + \frac{\Omega}{2}\,x\right). \label{vfree} \end{equation} In the present approach, the hermitian operator ${\bf B}$ is associated to the creation of a density fluctuation in the liquid and could then be labelled by a transferred momentum ${\bf q}$. Up to lowest order, one may have for each component of the vector ${\bf B}$ \begin{equation} B_{\bf q} = \frac{\hat{O}^{\dagger}_{\bf q} + \hat{O}_{\bf q}}{\sqrt{2}} \label{B} \end{equation} where $\hat{O}^{\dagger}_{\bf q}\, (\hat{O}_{\bf q})$ is the Feynman-Cohen operator that creates (destroys) a density fluctuation quantum, {\it i.e.}, a phonon or a roton \begin{equation} \hat{O}^{\dagger}_{\bf q} = \rho^{\dagger}_{\bf q} -\frac{1}{N} \sum_{{\bf k} \neq {\bf q}} \frac{{\bf k}\cdot {\bf q}} {k^2}\,\rho^{\dagger}_{\bf k}\,\rho^{\dagger}_{{\bf q} - {\bf k}}. \label{O} \end{equation} being here N the number of atoms in the liquid. Furthermore, we realize that the term $\lambda^2\,B^2/2 M$ appearing in (2.5) can be absorbed into the hamiltonian $H_B$, which is in charge of providing the equilibrium density vector of the reservoir. \section{The Generalized Master Equation} The Hamiltonian (\ref{H}) is of the form system - plus - reservoir - plus - interaction \cite{Haa}. The standard reduction - projection procedure of nonequilibrium statistical mechanics \cite{DH}, enriched with the time convolutionless method developed by Chaturvedi and Shibata\cite{Cha}, has already proven to be useful to derive a generalized master equation (GME), with time dependent coefficients, for the density operator $\sigma$ of a particle interacting with a heat reservoir in the weak coupling - nonmarkovian limit \cite{Mad1,Mad2}. In this case, $\sigma$ is the density operator of the vortex and the generalized master equation reads \cite{MD} \widetext \begin{eqnarray} \label{emg} \frac{d\,\sigma}{d\,t}+\frac{i}{\hbar} [H_v,\sigma] & = & -\frac{\lambda^2}{\hbar^2}\,\int_0^t d \tau\, \left\{[v_x,[v_x(-\tau),\sigma]]+[v_y,[v_y(-\tau),\sigma]]\right\} \phi(\tau) \nonumber \\ && -i \frac{\lambda^2}{\hbar^2}\,\int_0^t d \tau\, \left\{[v_x,[v_x(-\tau),\sigma]_{+}]+[v_y,[v_y(-\tau),\sigma]_{+}]\right\} \psi(\tau) \end{eqnarray} \narrowtext \noindent where $[a,b]_{+}$ denotes an anticommutator. In this expression, the time dependent functions $\phi$ and $\psi$ are the real and imaginary parts, respectively, of the correlation between heat bath operators \cite{esh}, \begin{equation} <B_j(\tau)\,B_j> = \phi(\tau) + i \psi(\tau) \end{equation} for $j = x, y$, assuming an isotropic reservoir. If the hermitian operator $B_j$ is chosen according to Eqs. (\ref{B}), (\ref{O}), the function \begin{equation} S_{\bf q}(\tau) = \phi_{\bf q}(\tau) + i\,\psi_{\bf q}(\tau) \end{equation} is just the Fourier transform of the dynamical structure factor $S({\bf q},\omega)$ of helium II and is experimentally known for a wide range of transferred momenta \cite{Gri,Gly}. Notice that the GME is a differential, rather than an integrodifferential, equation, since the unknown $\sigma$ under the integral sign is taken at time $t$; accordingly, it can be simplified if we define the following time dependent parameters, \begin{equation} \frac{M}{2} \gamma(t) = - \frac{\lambda^2}{\hbar}\,\int_0^t d \tau\,\psi(\tau)\,{\rm sin} \Omega \tau \label{gama} \end{equation} \begin{equation} \frac{M}{2} \mu(t) = \frac{\lambda^2}{\hbar}\,\int_0^t d \tau\,\psi(\tau)\,{\rm cos} \Omega \tau \label{mu} \end{equation} and \begin{equation} C(t) = \frac{\lambda^2}{\hbar^2}\,\int_0^t d \tau\, \phi(\tau)\,{\rm cos} \Omega \tau. \label{C} \end{equation} The velocities appearing in Eq. (\ref{emg}) are those of the free vortex displayed in (\ref{vfree}) and their detailed time dependence is extracted from Hamilton's equations corresponding to the Hamiltonian (\ref{Hfree}), namely \begin{eqnarray} v_x(t)& = &[v_x(0) - v_s]\,{\rm cos} \Omega t - v_y(0)\,{\rm sin} \Omega t+ v_s \nonumber \\ v_y(t) & = & [v_x(0) - v_s]\,{\rm sin} \Omega t + v_y(0) {\rm cos} \Omega t. \label{vt} \end{eqnarray} In terms of these quantities and using Eqs. (3.4) to (3.6), we can write \widetext \begin{eqnarray} \frac{d\,\sigma}{d\,t}+\frac{i}{\hbar} [H_{eff},\sigma] & = & -C(t)\,\left\{[v_x,[v_x,\sigma]]+[v_y,[v_y,\sigma]]\right\} \label{em} \\ && +\frac{i} {\hbar}\,\frac{M \gamma(t)}{2} \left\{[v_x,[v_y,\sigma]_{+}]-[v_y,[v_x,\sigma]_{+}]\right\}. \nonumber \end{eqnarray} \narrowtext The effective Hamiltonian contains a renormalization to the vortex mass, induced by the coupling to the thermal reservoir, plus a drift contribution. Its expression is \widetext \begin{equation} H_{eff} = H_v + \frac{M \mu(t)}{2} \left(v_x^2+v_y^2\right) + M\, v_s\,\omega(t)\, v_x - M\, v_s\,\gamma(t)\, v_y \label{heff} \end{equation} \narrowtext \noindent where $\omega(t) = \left.\mu(t)\right|_{\Omega = 0} - \mu(t)$. It is also worthwhile noticing that, being the system translationally invariant on the $(x,y)$ plane, terms in $H_{eff}$ proportional to $v_x, v_y$ play no role in the dynamics. It is important to observe that the validity of the GME (\ref{em}) is more general than the weak coupling approximation case. Indeed, if one expands the integral, time dependent collisional kernel of the master equation in powers of the coupling parameter $\lambda$, as done, for instance, in Ref. \cite{Cha}, after a lengthy calculation one can realize that the form of the new GME is identical to (\ref{em}), at least up to the fourth order in the expansion parameter, except for the fact that the coefficients (\ref{gama}) to (\ref{C}) become polynomials in $\lambda$. Finally, we should also mention that the most common assumptions considered in many applications are that the reservoir is purely harmonic and/or that the interaction term is linear in its coordinates. If this is not the current case, the nonvanishing mean values $<B_j>$ must be considered \cite{MD} and modify the effective Hamiltonian ({\ref{heff}). However, the equations of motion that we shall derive in the next section remain invariant, since these extra terms can be removed by a Galilean transformation. Moreover, it should be noticed as well that the correlation function $<B_k(\tau)\,B_j> $ for $k\ne j$, which is in general a nonvanishing function, does not enter Eq. (\ref{emg}). \section{The equations of irreversible motion} We are now in a position to derive equations of motion for expectation values $\langle a\rangle$ of arbitrary observables $a$, which can be cast in the form \widetext \begin{eqnarray} \frac{d\,\langle a\rangle}{d\,t}+\frac{i}{\hbar}\, <[a,H_{eff}]> &=& -C(t) \left(<[[a,v_x],v_x]> +<[[a,v_y],v_y]>\right) \nonumber \\ &+& \frac{i}{\hbar}\,\frac{M \,\gamma(t)}{2} \left(<[[a,v_x],v_y]_{+}> -<[[a,v_y],v_x]_{+}>\right). \end{eqnarray} \narrowtext In order to derive equations of motion for the position and momentum components of the vortex we will restrict ourselves to the markovian limit; in other words, we consider that the correlation indicated by $\phi,\psi$ is short lived, within the observational times. The parameters in Eqs. (\ref{gama}) and (\ref{mu}) become then time independent and after some algebra, elimination of the momentum permits us to write a unique complex differential equation for the expectation value of its velocity, that exhibits the effects of the coupling to the reservoir. This equation is \begin{equation} \frac{d^2\,\langle z\rangle}{d\,t^2}= i \Omega\,\beta\, \left(\frac{d\,\langle z\rangle}{d\,t} - v_s \right) \label{z} \end{equation} where the quantity that renormalizes the complex Magnus force is \begin{equation} \beta= 1 + \mu + i \,\gamma. \end{equation} with $\mu$ and $\gamma$ being the asymptotic values of (\ref{mu}) and (\ref{gama}) respectively. We here realize that the reservoir constituted by the excitations of the superfluid provides both a dissipative and a conservative coupling, respectively measured by the parameters $\gamma$ and $\mu$. This is in agreement with the structure of the mutual friction force of the two fluid model \cite{HV,Don1,Don2}. Moreover, keeping in mind that if the density fluctuations of the superfluid carry a definite momentum ${\bf q}$, the heat bath correlation $\langle B_{\bf q} B_{\bf q}(\tau)\rangle$ is, to lowest order, just the Fourier transform of the dynamical structure factor, using standard relations of linear response theory \cite{pin} one can readily show that \begin{equation} \mu + i \gamma = \frac{\lambda^2\,l}{2 \pi \hbar M}\,\chi({\bf q},\Omega) \label{chi} \end{equation} where $\chi({\bf q},\Omega)$ is the dynamical susceptibility or response function of the liquid (per unit length) at momentum ${\bf q}$ and energy $\hbar \Omega$. In the most general situation where thermal excitations cover the whole momentum range, a summation over ${\bf q}$ should be applied on the right hand side of Eq. (\ref{chi}). A consequence of this result is that the temperature dependence of the drag coefficients is provided by the variation of $\chi({\bf q}, \Omega)$ with $T$ \cite{Gly}; it is then worthwhile to keep in mind that the harmonic oscillator heat bath employed in previous investigations of vortex coupling to excitations \cite{Ao,Niu} predicts a temperature independent dissipation strength \cite{esh}. However, if the reservoir operators $B_j$ are described by nonlinear functions of $\hat{O}_{\bf q}^{\dagger},\hat{O}_{\bf q}$ rather than by the Feynman-Cohen operator (\ref{O}), it is possible to show that the drag coefficients vanish at zero temperature \cite{CNCal2,MD}; in such a case, the coupling is uneffective and the vortex moves freely governed by the Hamiltonian (\ref{Hfree}), as expected. Equation (\ref{z}) can be straightforwardly integrated, giving a mean value of the complex velocity operator \begin{equation} \frac {d \langle z\rangle}{d t} = \left[\left.\frac{d \langle z\rangle}{d t}\right|_{t=0} - v_s\right]\,e^{\displaystyle i \Omega (1 + \mu)t}\,e^{\displaystyle - \Omega\gamma t} + v_s. \label{vl} \end{equation} Since $\Omega\,\gamma$ is always a positive quantity, this expression contains exponential decay of the initial conditions, and the limiting value of the vortex velocity is then the superfluid one $v_s$. Let us now examine the situation as described by the phenomenological theory \cite{HV,Don1,Don2} where the drag force is written as \begin{equation} f_D = - (\gamma_0+ i \gamma'_0)\,\left(\frac{d z}{d t} - v_n\right) \label{drag} \end{equation} where $\gamma_0, \gamma'_0$ are the strenghts of the dissipative and conservative components and $v_n$ is the normal fluid velocity. It is also assumed that when equilibrium is reached, $f_D$ can be expressed in the form $ (\alpha - i \alpha')\,\rho_s\,q\,h\, (v_n - v_s)$. If one solves Newton's equation for a point particle with mass $M$ moving under the Magnus and the drag force (\ref{drag}), one finds \widetext \begin{eqnarray} \frac {d z}{d t} &= & \left[\left.\frac{d z}{d t}\right|_{t=0} - v_s-(\alpha'+i\,\alpha)\,(v_n - v_s)\right]\,e^{\displaystyle i (\Omega - \gamma'_0\,l/M) t}\,e^{\displaystyle - \gamma_0\,l t/M} \nonumber \\ & + & (\alpha' + i\,\alpha)\,(v_n - v_s)+ v_s \label{feno} \end{eqnarray} \narrowtext \noindent where $\alpha$ and $\alpha'$ can be written in terms of $\gamma_0$ and $\gamma'_0$ as \begin{eqnarray} \alpha&= &\rho_s\,q\,h\, \frac{\displaystyle \gamma_0}{\displaystyle (\rho_s\,q\,h-\gamma'_0)^2+\gamma_0^2} \nonumber \\ \alpha'&=&\frac{\displaystyle \gamma_0^2 +\gamma'_0\,(\gamma'_0 - \rho_s\,q\,h)}{\displaystyle (\rho_s\,q\,h-\gamma'_0)^2+\gamma_0^2} \label{rela} \end{eqnarray} The inverse relationships giving $\gamma_0, \gamma'_0$ in terms of $\alpha, \alpha'$ can be found in Ref. \cite{Don1}. We see that in this case, the asymptotic velocity contains both the reactive and the resistive coefficients. However, measurements of second sound attenuation in helium II at temperatures below 1.5 K give values for $\alpha, \alpha'$ around 10$^{-2}$, providing thus a negligible correction to the unperturbed velocity $v_s$. On the other hand, it is important to notice that according to the general equation (4.1) derived in this work, the expectation value of the free velocity operator (2.6), whose complex counterpart reads \begin{equation} v = \frac{p}{M} + i \frac{\Omega}{2}\,z \end{equation} satisfies the evolution law \begin{equation} \langle v(t) \rangle= \left[\langle v(0) \rangle - v_s \left( 1- \frac{\mu_0}{\beta}\right)\right] e^{\displaystyle i \Omega \beta t} + v_s\left( 1 - \frac{\mu_ 0}{\beta}\right). \label{mono} \end{equation} with $\mu_0 = \mu_{\Omega=0}$ (cf. Eq. (3.5)). Equation (\ref{mono}) is remarkably close to the above expression (\ref{feno}) for vanishing normal fluid velocity. Indeed, for a normal fluid at rest, Eq. ({\ref{mono}) is of the form (\ref{feno}), with coefficients $\tilde{\gamma}_0,\tilde{\gamma}'_0$ (or $\tilde{\alpha}, \tilde{\alpha}'$), given by \begin{eqnarray} \Omega\,(\mu + i \gamma) &=& - (\tilde{\gamma}'_0 - i \tilde{\gamma}_0)\,\frac{l}{M} \nonumber \\ \frac{\mu_0}{\beta}& =& \tilde{\alpha}'+i\,\tilde{\alpha}. \label{mano} \end{eqnarray} Elimination of $\mu$ and $\gamma$ gives the relationship \begin{eqnarray} \tilde{\alpha}& =& \frac{\displaystyle \rho_s\,q\,h\,\tilde{\gamma}_0}{\displaystyle \tilde{\gamma}_0^2+(\rho_s\,q\,h-\tilde{\gamma}'_0)^2}\,|\mu_0| \nonumber \\ \tilde{\alpha}'& =& {\frac{\displaystyle \rho_s\,q\,h\,( \tilde{\gamma}'_0-\rho_s\,q\,h)}{\displaystyle \tilde{\gamma}_0^2+(\rho_s\,q\,h - \tilde{\gamma}'_0)^2}}\,|\mu_0| \end{eqnarray} The interesting similitude between these relations and those in Eqs. (\ref{rela}) gives support to the conjecture that the present model embodies substantial aspects of the mechanism responsible of damped vortex motion in superfluid helium. The differences between the relationships characterizing the phenomenological model in Eqs. (4.8), and the present ones in Eqs. (4.12), are due to the fact that the structure of the drag force is not identical in both approaches. In fact, a close look at Eqs. (4.2), (4.3) and (4.6) shows that the Hamiltonian description gives rise to an extra component of the force, proportional to the relative two fluid velocity $v_n - v_s$. This supplementary component is not removed by the assumption that the normal fluid lies at rest; however, it is also worthwhile to keep in mind that the assumption that the drag force is proportional to ${\bf v} - {\bf v}_n$ applies under the hypothesis of vanishing vortex mass \cite{Don1}. If one is interested in getting rid of the extra force, a different model should be selected, so as to bring the two fluid dynamics into the picture. Such an improvement does not consist of a simple modification of the Hamiltonian (2.5); instead, a totally different formulation is required stemming from a Hamiltonian description of the two fluids to which a suitable copling is incorporated. This philosophy fits more specifically the spirit of macroscopic, fluiddynamical models and is thus beyond the scope of the present work. As a final remark, we wish to recall that every time dependent quantity here presented owes this dependence to the special model feature that makes room to a finite, although unknown, inertial coefficient of the vortex. This parameter rules the evolution since it appears in both the conservative and the decay time scale (cf. Eqs. (\ref{vl}), ({\ref{feno}) and ({\ref{mono})); we then realize that as pointed out in Ref. \cite{dem2}, experimental detection of the time dependent regime would thus provide a means of measuring the vortex inertia. It should be kept in mind that the present results concerning the dynamics cannot be extrapolated down to $M = 0$; in fact, the free Hamiltonian (2.1), the frequency (2.3), the vortex - reservoir Hamiltonian (2.5) and the velocity (2.6) become meaningless in such a case. However, the asymptotic velocity does not depend upon the mass, since its value causes the Magnus and the drift force to cancel each other in the absence of inertial effects. \section{Discussion and summary} Let us now examine further the characteristics of the model here presented and its relationship to the phenomenological description of dissipation. On the one hand, it is important to keep in mind that the vortex mass is assumed to vanish in the phenomenological two - fluid model of mutual friction, where the velocity arises from the balance between the Magnus and the drag forces; it should be noticed that this regime is also the time - asymptotic form of a Newton - like equation of motion if the vortex mass is finite \cite{Don2}. The precise value of the vortex inertial coefficient is thus not important in the limiting regime, although it influences the dynamics at finite times through the frequency $\Omega$ (cf. Eqs. (2.3) and (\ref{feno})), which is the relevant parameter of the model. In this context, it is important to keep in mind that the coupling to the thermal excitations further renormalizes the vortex mass; in fact, inspection of the effective Hamiltonian (\ref{heff}) shows us that the kinetic energy has been changed into $M (1 + \mu) {\bf v}^2/2$. On the other hand, the phenomenological theories introduce a mutual friction whose drift and dissipative components are proportional to the relative two - fluid velocity ${\bf v}_n - {\bf v}_s$. The vortex velocity, either with respect to the superfluid or to the normal one, is determined by the force balance when inertial effects vanish; consequently, it depends upon the parameters of the drag. Instead, the present model should be regarded as a description in the reference frame of the normal fluid, {\it i.e.}, both $v_s$ and $d \langle z \rangle/d t$ refer to the local velocity ${\bf v}_n$ of the heat reservoir in the neighborhood of the vortex. The coefficients that measure the drag effects thus depend upon the strength of the coupling to the thermal excitations and upon their dynamical response; however the fluid dynamics of the elementary excitations is not explicitly contemplated. We believe that the model here presented covers most aspects of the description of dissipative dynamics of a vortex line in helium II and opens possibilities towards further improvements, among which, a definite one is the introduction of the motion of elementary excitations, to properly account for mutual friction in the sense of phenomenological theories. With respect to previous calculations of the drag coefficients carried, for example, in Refs. \cite{MS,SD}, our model, being quantal in nature, is not subjected to either the low - temperature limitations of a hydrodynamical description as pointed out in \cite{MS}, or to uncertainties associated to a classical approach to the roton - vortex collisions \cite{SD}. It may be also mentioned that the model holds as well for vortex motion in liquid $^3$He; quantum statistics only enters the characterization of the excitations making the heat reservoir, which would consist of the zero sound phonons of the fermion liquid. No special differences with the present results would be expected in that case, except from the fact that the larger core size of vortices in $^3$He could probably enlarge the inertia parameter $M$ with a subsequent decrease in the oscillation frequency $\Omega$. \acknowledgements We are pleased to acknowledge stimulating conversations with Dr. Manuel Barranco. This work has been supported by grants PID 34520092 from Consejo Nacional de Investigaciones Cient\'{\i}ficas y T\'ecnicas, Argentina, and EX100/95 from Universidad of Buenos Aires.

proofpile-arXiv_065-563

\section{INTRODUCTION} Since Quantum Chromodynamics \cite{qcd} and the concept of asymptotic freedom \cite{asymf} were introduced to describe the dynamics of hadronic processes at high momentum transfers, several \char'134 key" predictions of the theory were successfully tested by experiment: `Evidence for jet structure in hadron production by $\rm e^+\rm e^-$ annihilation' was found in 1975 \cite{hanson}, the gluon was explicitly observed at the PETRA $\rm e^+\rm e^-$ storage ring in 1979 \cite{gluon}. The first measurement of the coupling strength $\alpha_{\rm s}$, the basic free parameter of the theory, was reported in that same year \cite{as-mkj}, based on leading order perturbative QCD. The first determination of $\alpha_{\rm s}$ in next-to-leading order (NLO) QCD dates back to 1982 \cite{as-jade}. After these pioneering years, many further tests of QCD were performed in $\rm e^+\rm e^-$ annihilation, in deep inelastic lepton-nucleon scattering and at hadron colliders. In 1988, first evidence for the running $\alpha_{\rm s}$ was obtained from the energy dependence of 3-jet event production rates in $\rm e^+\rm e^-$ annihilation \cite{jadejet2}. An update and a summary of these measurements will be presented in Section~3 of this review. Although many determinations of $\alpha_{\rm s}$, in the energy range of $Q \sim 4$~to~$46$~GeV, were available by 1990, the $running$ of $\alpha_{\rm s}$ could not convincingly be seen from those results \cite{altarelli}. Only in 1992, a compilation of measurements of $\alpha_{\rm s}$ in the energy range from 1.78~GeV (the mass of the $\tau$-lepton) to 91.2~GeV (the mass of the $\rm Z^0$-boson), could demonstrate the characteristic energy dependence of the strong coupling \cite{sb-catani}. The actual evidence for the running coupling strength or, equivalently, for asymptotic freedom is summarised in this review. The results from jet production rates in $\rm e^+\rm e^-$ annihilation are presented in Section~3. Recent studies of jet production and of the proton structure function $F_2$ in deep inelastic electron-proton collisions are discussed in Section~4. New results from jets in $p \overline{p}$ collisions and from hadronic decays of $\tau$-leptons, demonstrating the energy dependence of $\alpha_{\rm s}$ in ranges of very high and very low momentum transfers, respectively, are presented in Sections~5 and~6. An update of the world summary of $\alpha_{\rm s}$ measurements is finally given in Section~7. This report is restricted to results which were published at the time of this conference; preliminary results are not taken into account. \section{QCD AND THE RUNNING $\alpha_{\rm s}$} Within perturbative QCD, the energy dependence of $\alpha_{\rm s}$ is given by the $\beta$-function: \begin{eqnarray} \mu \frac{\partial \alpha_{\rm s}}{\partial \mu} & = & - \frac{\beta_0}{2\pi} \alpha_{\rm s}^2 - \frac{\beta_1}{4 \pi^2} \alpha_{\rm s}^3 - \frac{\beta_2}{64\pi^3} \alpha_{\rm s}^4 - \dots \nonumber \\ \beta_0 & = & 11 - \frac{2}{3} N_f\ , \nonumber \\ \beta_1 & = & 51 - \frac{19}{3} N_f \nonumber \\ \beta_2 & = & 2857 - \frac{5033}{9} N_f + \frac{325}{27} N_f^2\ , \end{eqnarray} where $N_f$ is the number of quark flavours with masses less than the energy scale $\mu$. A solution of Equation 1, in third order expansion, is \cite{pdg96} \begin{eqnarray} \alpha_s(\mu) & = & \frac{4\pi}{\beta_0 \ln(\mu / \Lambda)^2} \big[ 1 - 2\ \frac{\beta_1}{\beta_0^2}\ \frac{\ln\left(\ln( \mu / \Lambda)^2\right)}{\ln(\mu / \Lambda)^2} \nonumber \\ & + & \frac{4 \beta_1^2}{\beta_0^4 \ln^2 (\mu / \Lambda)^2} \big( \big( \ln \left[\ln (\mu / \Lambda)^2\right] - \frac{1}{2}\big)^2 \nonumber \\ &+& \frac{ \beta_2 \beta_0}{8 \beta_1^2} - \frac{5}{4} \big) \big]~. \end{eqnarray} At large energy scales $\mu$, or equivalently at small distances, $\alpha_{\rm s}$ vanishes logarithmically; this behaviour of $\alpha_{\rm s}$ is called `asymptotic freedom'. In this report all calculations, equations and results refer to the `modified minimal subtraction scheme' ($\overline {\rm MS}$) \cite{msbar}. More detailed information about the basic QCD equations and for the treatment of heavy quark flavour thresholds can be found e.g. in \cite{pdg96,msbar,wernerb,qcd94}. \section{JET RATES IN $\rm e^+\rm e^-$ ANNIHILATION} Studies of hadron jets provide the most intuitive tests of the underlying parton (i.e. quark and gluon) structure of hadronic events. The most commonly used algorithm to reconstruct jets in $\rm e^+\rm e^-$ annihilation was introduced by the JADE collaboration \cite{jadejet2}: the scaled pair mass of any two resolvable jets $i$ and $j$ in a hadronic event, $y_{ij} = M_{ij}^2 / E_{\rm vis}^2$, is required to exceed a threshold value $y_{\rm cut}$, where $E_{\rm vis}$ is the total visible (measured) energy of the event. In a recursive process, the pair of particles or clusters of particles which has the smallest value of $y_{ij}$ is replaced by (or `recombined' into) a single jet $k$ with four-momentum $p_k = p_i + p_j$, as long as $y_{ij} < y_{\rm cut}$. The procedure is repeated until all $y_{ij}$ are larger than the jet resolution parameter $y_{\rm cut}$, and the remaining clusters of particles are called jets. \begin{figure}[htb] \epsfxsize7.5cm \epsffile{r3.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 1.} The ratio $r$ of 3-jet event rates, calculated from JETSET QCD shower model events before and after hadronisation, as a function of $E_{cm}$, for different jet algorithms. } \end{figure} \baselineskip=12.0pt \begin{figure}[htb] \epsfxsize7.5cm\epsffile{r3-q.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 2.} Energy dependence of three-jet event production rates $R_3$, using the JADE E0 jet scheme with $y_{\rm cut} = 0.08$. The measurements are compared with predictions of analytic ${\cal O}(\as^2)$ QCD calculations, with the hypothesis of an energy independent $\alpha_{\rm s}$ and with the abelian vector theory in ${\cal O}(\alpha_A^2)$. } \end{figure} \baselineskip=12.0pt Several methods of jet recombination and definitions of $M_{ij}$ exist, for which QCD predictions in complete ${\cal O}(\as^2)$ perturbation theory \cite{BKSS}, based on the matrix elements of Ellis, Ross and Terrano \cite{ert}, are available. In ${\cal O}(\as^2)$, the relative 2-, 3- and 4-jet production rates, $R_n = \sigma_{\rm n-jet} / \sigma_{\rm tot}$, where $\sigma_{\rm tot}$ is the total hadronic cross section and $\sigma_{\rm n-jet}$ are the cross sections for $n$-parton event production, are quadratic functions of the running coupling constant $\alpha_{\rm s} (\mu)$. In particular, \begin{equation} R_3(y_c,\mu) = A(y_c) \frac{\alpha_{\rm s}(\mu)}{2\pi} + B(y_c, x_{\mu}) \left(\frac{\alpha_{\rm s}(\mu)}{2\pi}\right)^2 , \end{equation} \noindent where $\mu = x_{\mu} E_{cm}$ is the renormalisation scale at which $\alpha_{\rm s}$ is evaluated, $x_{\mu}$ is the renormalisation scale factor and $y_c \equiv y_{\rm cut}$. The energy dependence of $R_3$ is only determined by the running $\alpha_{\rm s}$; the scale factor $x_\mu$ - although it's optimal value is not given by the theory - is not expected to change with energy. Jet production rates are therefore an ideal tool to test the energy dependence of $\alpha_{\rm s}$, without the need to actually determine $\alpha_{\rm s}$ itself. The first analysis in this sense was done by JADE \cite{jadejet2}. The original JADE (also called $E0$) scheme with $M_{ij}^2 = 2 E_i E_j (1-\cos{\theta_{ij}})$, where $E_i$ and $E_j$ are the energies of the particles and $\theta_{ij}$ is the angle between them, has the smallest hadronisation corrections with only a weak dependence on the centre of mass energy, $E_{cm}$. This is demonstrated in Figure~1, where the ratio $r$ = $R_3$(hadrons) / $R_3$(partons) predicted by the JETSET QCD shower model \cite{jetset} is plotted as a function of $E_{cm}$, for constant values of $y_{\rm cut}$ \cite{scotland}. For all jet algorithms, the quantity (1--$r$) shows an approximate 1/$E_{cm}$ behaviour at large $E_{cm}$, as expected for non-perturbative hadronisation effects. At smaller energies, usually for $\sqrt{y_{\rm cut}} E_{cm} < 7$ GeV, $r$ increases with decreasing $E_{cm}$ because of misassignments of jets, caused by hadronisation fluctuations and heavy quark decays. For the JADE E0 algorithm, $|1 - r|$ is small enough and the energy dependence of $r$ is sufficiently flat for $E_{cm}$ between 25 and 200 GeV to be approximated by a constant within a systematic uncertainty of $\pm 2\%$. This feature makes an important impact on the experimental evidence for asymptotic freedom. \begin{figure}[htb] \epsfxsize7.5cm\epsffile{r3-lnq.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 3.} The same data as shown in Fig.2, combined at similar energies, now as a function of 1/ln($E_{cm}$). } \end{figure} \baselineskip=12.0pt A compilation of the experimental results of $R_3$, analysed with the JADE (E0) jet finder at different $E_{cm}$ using $y_{\rm cut}$ = 0.08, is presented in Fig.~2 \cite{jadejet2,jetrates,o-133}. The data are compared with fit results of analytic ${\cal O}(\as^2)$ QCD calculations \cite{BKSS}, of the hypothesis of an energy $in$dependent coupling constant and of the abelian, QED-like vector theory in $O(\alpha_A^2)$, where $\alpha_A$ was adjusted such that the jet rates at $E_{cm}$ = 44 GeV are reproduced \cite{scotland}. For the QCD predictions and the hypothesis of $\alpha_{\rm s}$ = constant, the free parameters $\Lambda_{\overline{MS}}$ and a constant 3-jet rate $<R_3>$, respectively, were determined by minimising $\chi^2$ for the data from $E_{cm}$ = 29 to 133 GeV. The data points at $E_{cm}$ = 22 GeV were not included in the fit since hadronisation effects may already bias the measurements at this energy, see Fig. 1. For QCD, the fit results in $\Lambda_{\overline{MS}} = (253 \pm 12$~MeV), which corresponds to $\as(M_{\rm Z^0}) = 0.120 \pm 0.001$ (stat. error only), and $\chi^2 = 8.1$ for 13 degrees of freedom\footnote{ In order to account for the small, energy dependent hadronisation effects as predicted by the model calculations shown in Fig. 1, a relative systematic point-to-point uncertainty of $\pm2\%$ is included when calculating $\chi^2$.}, corresponding to a confidence level (CL) of 84\%. A linear fit through the data (not shown in Fig.~2) gives $\chi^2 = 12.7$ for 12 degrees of freedom (CL = 39\%). The hypothesis of $\alpha_{\rm s}$~= constant with $\chi^2 = 72$ (CL = $3.4 \times 10^{-10}$) and the abelian theory, the $\chi^2$ of which tends to infinity, are entirely ruled out by the data. The experimental evidence for asymptotic freedom is further demonstrated in Fig.~3, where the same experimental data, however combined at similar c.m. energies, are plotted as a function of $1 / \ln (E_{cm} )$. The dashed line is a fit to the leading order QCD prediction, namely $R_3 \propto \alpha_{\rm s} \propto 1 / \ln E_{cm}$. The corresponding prediction in ${\cal O}(\as^2)$ is also shown, indicating that higher order terms affect the energy dependence of $R_3$ only slightly. At infinite energies $( 1 / \ln (E_{cm} ) \rightarrow 0 )$, $R_3$ and $\alpha_{\rm s}$ are expected to vanish; an assumption which is in good agreement by the data. While the most recent data from LEP-1.5 ($E_{cm} \sim 133$~GeV) are statistically very limited, which will most likely not be improved in the future, it is expected that LEP-2 ($E_{cm} \sim 175$~GeV) will provide another data point with an absolute error of $\Delta(R_3) \sim 1 \%$. The significance of data from future high energy $\rm e^+\rm e^-$ linear colliders can be inferred from Figure~3: at $E_{cm} = 500$~GeV, the statistical error $\Delta(R_3)$ for 1000 hadronic events will be about 1\%. From the point of view of the previous experiments at PETRA and PEP and with the eyes of QCD, i.e. on a logarithmic energy scale, a linear collider at $E_{cm} = 500$~GeV is almost half-way to infinte energies! \section{RUNNING $\alpha_{\rm s}$ FROM e-p COLLISIONS} In deep inelastic electron-proton scattering (DIS), hadronic final states can be studied in a wide range of energy scales $Q^2$, the squared four-momentum transfer from the incident lepton. Both experiments at HERA, H1 and ZEUS, determined $\as (\q2 )$ from jet rates measured in the energy range of $10~{\rm GeV}^2 < Q^2 < 4000~{\rm GeV}^2$ \cite{h1jet,zeusjet}, using a modified JADE jet algorithm where the proton remnant is treated as a pseudoparticle which carries only longitudinal momentum (i.e. along the beam direction). The jet resolution parameter is given by $y_c = M_{ij}^2 / W^2$, where $W$ is the invariant mass of the hadronic system and $M_{ij}$ the invariant pair masses of all objects used including the pseudoparticle. This choice of algorithm ensures a jet classification which is similar to that used in $\rm e^+\rm e^-$ annihilation. The jet production rates $R_{N+1}$ are calculated from the number of events with $N$ resolvable jets inside the acceptance region, where \char'134 +1" denotes the pseudoparticle (or pseudojet) from the proton remnant. In DIS, jet rates by themselves cannot demonstrate the running of the coupling since theory predicts that the QCD coefficients of $\alpha_{\rm s}$ also depend, in contrast to the case of $\rm e^+\rm e^-$ annihilation (see Eq.~3), on $Q$ through the parton density functions. QCD predictions for (1+1) and (2+1)-jet events which are complete to ${\cal O}(\as^2)$ \cite{graudenz} are therefore used to extract $\alpha_{\rm s}$ for different bins of $Q^2$. \begin{figure}[tb] \epsfxsize7.5cm\epsffile{HERA-alphas.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 4.} Measurements of $\as (\q2 )$ from jet rates at HERA. The curves are the results of a QCD fit to the data (compilation from \cite{wwwh1}). } \end{figure} \baselineskip=12.0pt The results are compiled in Fig.~4 \cite{wwwh1}. Although the overall uncertainties, both statistical as well as systematic, are still rather large in these measurements, the general trend of a coupling which decreases with increasing $Q$ can clearly be seen. The lines in Fig.~4 indicate the results of a QCD fit through the measurements, which extrapolates to $\as(M_{\rm Z^0}) = 0.120 \pm 0.005 \pm 0.007$. In another study of the HERA data \cite{ball}, $\alpha_{\rm s}$ is determined from the proton structure function ${\rm F}^p_2 (x,Q^2)$ at small $x$ and $Q^2 < 100~{\rm GeV}^2$. $F^p_2$ is computed in next-to-leading order in $\alpha_{\rm s}$, including summations of all leading and subleading logarithms of $Q^2$ and $1/x$. In that study it is demonstrated that the structure function data of H1 and of ZEUS exhibit double logarithmic scaling in both $x$ and $Q^2$, which is regarded as direct evidence for the running $\alpha_{\rm s}$ \cite{ball}. A QCD fit to these data finally gives $\as(M_{\rm Z^0}) = 0.120 \pm 0.005 \pm 0.009$, where the first error is experimental and the second theoretical. This is in good agreement with the result from jets described above, and also with the world average of $\as(M_{\rm Z^0})$, see section~7. \section{JETS IN HADRON COLLISIONS} Similarly as in deep inelastic lepton-nucleon scattering, hadron colliders provide the opportunity to simultaneously probe QCD in a wide range of momentum transfers $Q$. In a recent study \cite{gielejets} based on the one-jet inclusive transverse energy ($E_T$) distribution measured at the Tevatron \cite{cdfjets}, values of $\alpha_{\rm s} (Q \equiv E_T)$ are determined over a wide range of energies, $E_T = 30\ {\rm to}\ 500$~GeV. \begin{figure}[tb] \epsfxsize7.5cm\epsffile{giele_alphas.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 5.} Values of $\alpha_{\rm s} (E_T)$ extracted from the one-jet inclusive jet cross sections from CDF as a function of the the jet transverse energy $E_T$, together with the QCD expectations based on NLO perturbation theory and the MRSA' particle density function (from \cite{gielejets}). } \end{figure} \baselineskip=12.0pt The results of this study are shown in Fig.~5. The values of $\alpha_{\rm s} (E_T)$ are seen to decrease with increasing $E_T$, in good agreement with the QCD expectations of a running coupling strength (shaded area). A simultaneous QCD fit to these data results in $\as(M_{\rm Z^0}) = 0.121 \pm 0.001\ {\rm (stat.)}\ \pm 0.008\ {\rm (syst.)}\ \pm 0.005\ {\rm (theor.)}$, which is in excellent agreement with other measurements, see section~7. \section{RUNNING $\alpha_{\rm s}$ FROM $\tau$ DECAYS} Measurements of the ratio of the hadronic and leptonic branching fractions of the $\tau$ lepton, $R_{\tau}$, have provided precise values of $\alpha_{\rm s}$ at the energy scale of the $\tau$-mass, $Q \equiv M_{\tau} = 1.777\ {\rm GeV}$, see e.g. \cite{qcd94,duflot} and references quoted therein. Recently, a new test of the energy dependence of $\alpha_{\rm s}$ was proposed \cite{neubert}, based on the $\tau$ decay rate into hadrons of invariant mass squared $s$ smaller than a threshold value $s_0$: \begin{eqnarray} R_{\tau} (s_0) &=& \frac{\Gamma (\tau \rightarrow \nu_{\tau} + {\rm hadrons};\ s < s_0)}{\Gamma (\tau \rightarrow \nu_{\tau} e \overline{\nu_e})} \nonumber \\ &=& \int^{s_0}_0 {\rm d} s \frac{{\rm d}R_{\tau} (s)}{{\rm d}s}. \end{eqnarray} The running coupling constant $\alpha_{\rm s} (s_0)$ is extracted from the inclusive hadronic spectrum ${\rm d}R_{\tau} (s) / {\rm d}s$ measured by ALEPH and CLEO \cite{duflot,cleotau}, in the low energy region $0.7\ {\rm GeV}^2 < s_0 < M_{\tau}^2$ where $\alpha_{\rm s}$ is expected to change by almost a factor of two. \begin{figure}[tb] \epsfxsize7.5cm\epsffile{neubert.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 6.} Values of $\alpha_{\rm s} (s_0)$ from the data on $R_{\tau} (s_0)$. The inner band represents experimental, the outer band the sum of experimental and theoretical uncertainties. The dashed line shows the running coupling constant in ${\cal O}(\as^3)$ QCD. (From \cite{neubert}) } \end{figure} \baselineskip=12.0pt Theoretical predictions for $R_{\tau} (s_0)$ include perturbative terms which are complete to ${\cal O}(\as^3)$ as well as estimates of nonperturbative contributions using the operator product expansion \cite{shifman,braaten}. Assuming global parton-hadron duality, $\alpha_{\rm s}$ can thus be determined from each measured value of $R_{\tau} (s_0)$. The results of the study of ref.\cite{neubert} are shown in Fig.~6. Since values of $\alpha_{\rm s}$ extracted from $R_{\tau} (s_0)$ are correlated with each other, the fit results are displayed as a band. The dashed curve shows the QCD expectation of the running coupling constant calculated in ${\cal O}(\as^3)$, normalised to the data at $s_0 = M^2_{\tau}$. The observed energy dependence is in excellent agreement with the QCD prediction of the running $\alpha_{\rm s}$ in ${\cal O}(\as^3)$. In addition, the data show a distinct preference for the 3-loop $\beta$-function (Eq.~ 2), compared to the leading order (1-loop) one \cite{neubert}. The overall value for $\alpha_{\rm s}$, including estimates of higher order perturbative uncertainties, results in $\alpha_{\rm s} (M_{\tau}) = 0.33 \pm 0.03$ or, equivalently, in $\as(M_{\rm Z^0}) = 0.119 \pm 0.004$. \begin{table*}[htb] \begin{center} \begin{tabular}{|l|c|c|l|l|c c|c|} \hline & & Q & & & \multicolumn{2}{c|} {$\Delta \as(M_{\rm Z^0}) $} & \\ Process & Ref. & [GeV] & $\alpha_s(Q)$ & $ \as(M_{\rm Z^0})$ & exp. & theor. & Theory \\ \hline \hline \normalsize & & & & & & & \\ DIS [$\nu$; Bj-SR] & \cite{bj-sr-ellis} & 1.58 & $0.375\ ^{+\ 0.062}_{-\ 0.081}$ & $0.122\ ^{+\ 0.005}_{-\ 0.009}$ & -- & -- & NNLO \\ DIS [$\nu$; GLS-SR] & \cite{gls-theory} & 1.73 & $0.32\pm 0.05$ & $0.115\pm 0.006$ & $ 0.005 $ & $ 0.003$ & NNLO \\ & & & & & & & \\ $\tau$-decays & \cite{narison-i1,neubert} & 1.78 & $0.330 \pm 0.030$ & $0.119 \pm 0.004$ & 0.001 & 0.004 & NNLO \\ & & & & & & & \\ DIS [$\nu$; ${\rm F_2\ and\ F_3}$] & \cite{ccfr} & 5.0 & $0.193\ ^{+\ 0.019\ }_{-\ 0.018\ }$ & $0.111\pm 0.006$ & $ 0.004 $ & $ 0.004$ & NLO \\ DIS [$\mu$; ${\rm F_2}$]& \cite{virchaux} & 7.1 & $0.180 \pm 0.014$ & $0.113 \pm 0.005$ & $ 0.003$ & $ 0.004$ & NLO \\ DIS [HERA; jets] & \cite{h1jet,zeusjet} & 10 - 60 & & $0.120 \pm 0.009$ & $ 0.005$ & $ 0.007$ & NLO \\ DIS [HERA; ${\rm F_2}$] & \cite{ball} & 2 - 10 & & $0.120 \pm 0.010$ & $ 0.005$ & $ 0.009$ & NLO \\ & & & & & & & \\ ${\rm Q\overline{Q}}$ states & \cite{davies} & 5.0 & $0.203 \pm 0.007$ & $0.115 \pm 0.002 $ & 0.000 & 0.002 & LGT \\ $J/\Psi + \Upsilon$ decays & \cite{kobel} & 10.0 & $0.167\ ^{+\ 0.015\ }_{-\ 0.011\ }$ & $0.113\ ^{+\ 0.007\ } _{-\ 0.005\ }$ & 0.001 & $^{+\ 0.007}_{-\ 0.005}$ & NLO \\ & & & & & & & \\ $\rm e^+\rm e^-$ [$\sigma_{\rm had}$] & \cite{haidt} & 34.0 & $0.146\ ^{+\ 0.031}_{-\ 0.026}$ & $0.124\ ^{+\ 0.021}_{-\ 0.019}$ & $^{+\ 0.021}_{-\ 0.019} $ & -- & NLO \\ $\rm e^+\rm e^-$ [ev. shapes] & \cite{budapest} & 35.0 & \ $0.14\pm 0.02$ & $0.119 \pm 0.014$ & -- & -- & NLO \\ $\rm e^+\rm e^-$ [ev. shapes] & \cite{topazas} & 58.0 & $0.132\pm 0.008$ & $0.123 \pm 0.007$ & 0.003 & 0.007 & resum. \\ & & & & & & & \\ $p\bar{p} \rightarrow {\rm b\bar{b}X}$ & \cite{ua1-bb} & 20.0 & $0.145\ ^{+\ 0.018\ }_{-\ 0.019\ }$ & $0.113 \pm 0.011$ & $^{+\ 0.007}_{-\ 0.006}$ & $^{+\ 0.008}_{-\ 0.009}$ & NLO \\ ${\rm p\bar{p},\ pp \rightarrow \gamma X}$ & \cite{a-ua6} & 24.2 & $0.137 \ ^{+\ 0.017}_{-\ 0.014}$ & $0.112\ ^{+\ 0.012\ }_{-\ 0.008\ }$ & 0.006 & $^{+\ 0.010}_{-\ 0.005}$ & NLO \\ ${\sigma (\rm p\bar{p} \rightarrow\ jets)}$ & \cite{gielejets} & 30 - 500 & & $0.121\pm 0.009$ & 0.001 & 0.009 & NLO \\ & & & & & & & \\ $\rm e^+\rm e^- \rightarrow \rm Z^0$: & & & & & & & \\ \ \ $\Gamma (\rm Z^0 \rightarrow {\rm had.})$ & \cite{lep-ewwg} & 91.2 & $0.126\pm 0.006$ & $0.126\pm 0.006$ & $ 0.005$ & $0.003$ & NNLO \\ \ \ had. event shapes & \cite{qcd94} & 91.2 & $0.119 \pm 0.006$ & $0.119 \pm 0.006$ &$ 0.001$ & $ 0.006$ & NLO\\ \ \ had. event shapes & \cite{qcd94} & 91.2 & $0.122 \pm 0.006$ & $0.122 \pm 0.006$ & $ 0.001$ & $ 0.006$ & resum. \\ & & & & & & & \\ $\rm e^+\rm e^-$ [ev. shapes] & Tab. 2 & 133.0 & $0.112\pm 0.009$ & $0.118 \pm 0.009$ & 0.003 & 0.009 & resum. \\ & & & & & & & \\ \hline \end{tabular} \end{center} \baselineskip=11.0pt {\small \noindent {\bf Table 1.} World summary of measurements of $\alpha_{\rm s}$. Abbreviations: DIS = deep inelastic scattering; GLS-SR = Gross-Llewellyn-Smith sum rules; Bj-SR = Bjorken sum rules; (N)NLO = (next-)next-to-leading order perturbation theory; LGT = lattice gauge theory; resum. = resummed next-to-leading order.} \end{table*} \section{WOLRD SUMMARY OF $\alpha_{\rm s}$} Having discussed the current tests of asymptotic freedom from measurements based on single observables like jet rates, jet-$E_T$-spectra and $\tau$-decays, the overall summary of all available $\alpha_{\rm s}$ determinations remains to be updated (see e.g. \cite{altarelli,sb-catani,qcd94,moriond95} for previous reviews). An update of the summary given in \cite{qcd94} is presented in Table~1. Graphical presentations of the running coupling $\alpha_{\rm s} (Q)$ and of the results extrapolated to $\as(M_{\rm Z^0})$, using the 3-loop expansion (Eq.~2) and treating heavy flavour thresholds according to ref. \cite{wernerb}, are given in Figs.~7 and~8, respectively. The most recent changes and additions are discussed in the following subsections; see \cite{qcd94} for comparison. \subsection{$\alpha_{\rm s}$ from Sum Rules} The results from the Gross-Llewellyn-Smith and the Bjorken sum rules \cite{bj-sr-ellis,gls-theory} have been retained; a preliminary update of the Bjorken sum rule result from the CCFR collaboration (see e.g. \cite{harris-m95}) exists but is not included here because the final publication is still missing. \subsection{$\alpha_{\rm s}$ from $\tau$ Decays} In the previous report \cite{qcd94}, $\alpha_{\rm s}$ from $\tau$ decays was obtained from a compilation of measurements of the ratio of the hadronic to the leptonic $\tau$ branching ratios, $R_\tau$. Meanwhile, several new measurements of this quantity became available (see \cite{duflot,opaltau} and references quoted therein). Instead of deriving an updated value of $\alpha_{\rm s}$ from $R_\tau$, the results from the study described in section~6, namely from the hadronic invariant mass distribution of $\tau$ decays, $R_{\tau} (s_0)$, is taken. This result is identical to the one derived from a recent evaluation of $\alpha_{\rm s}$ and its overall uncertainty from $R_\tau$ \cite{narison-i1}: $\alpha_{\rm s} (M_\tau) = 0.33 \pm 0.03$. \subsection{$\alpha_{\rm s}$ from Deep Inelastic Scattering} In addition to the earlier results from fixed target experiments \cite{ccfr,virchaux}, the new values of $\alpha_{\rm s}$ from HERA, discussed in Sect.~4, are included. \begin{figure}[tb] \epsfxsize7.5cm\epsffile{as-world.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 7.} A Summary of measurements of $\alpha_{\rm s}$, compared with QCD expectations for four different values of $\Lambda_{\overline{MS}}$ which are given for $N_f = 5$ quark flavours (relation between $\alpha_{\rm s}$ and $\Lambda_{\overline{MS}}$ in ${\cal O}(\as^3)$). } \end{figure} \baselineskip=12.0pt \begin{figure}[tb] \epsfxsize7.5cm\epsffile{as-mz.eps} \baselineskip=11.0pt {\small \noindent {\bf Figure 8.} A Summary of measurements of $\as(M_{\rm Z^0})$, as listed in Table~1. Filled symbols are derived using ${\cal O}(\as^3)$ QCD; open symbols are in ${\cal O}(\as^2)$ or based on lattice calculations.} \end{figure} \baselineskip=12.0pt \subsection{$\alpha_{\rm s}$ from Lattice QCD} The values of $\as(M_{\rm Z^0})$ from lattice QCD calculations, based on measurements of heavy quarkonia mass spectra, slowly but gradually increased during the past few years. These changes are mainly due to the availability of unquenched calculations (i.e. including dynamical light quark flavours) and to more refined procedures to convert the lattice coupling to the running coupling of perturbative QCD. A recent summary of these results gives $\as(M_{\rm Z^0}) = 0.115 \pm 0.002$ \cite{davies}, which is taken over for this review. There are, however, unpublished reports which result in $\as(M_{\rm Z^0}) = 0.118 \pm 0.002$, see e.g. \cite{lepage}. \subsection{$\alpha_{\rm s}$ from Hadron Collsions} The previous, preliminary determination of $\alpha_{\rm s}$ from a measurement of the $b \overline{b}$ cross section was updated and finally published in \cite{ua1-bb}. Results on $\alpha_{\rm s}$ from ${\rm p\bar{p} \rightarrow W\ jets}$ \cite{a-ua2,a-ua1} which were considered in previous compilations are no longer included since the QCD calculations on which they are based are not complete to next-to-leading order. A recent study of this process from D0 reports that calculations which are complete to NLO do not provide a reasonable fit of the data \cite{a-d0}. The result on $\alpha_{\rm s}$ from the one-jet inclusive $E_t$-distribution, as discussed in Section~5, is a new entry in Table~1. \subsection{$\alpha_{\rm s}$ from the $\rm Z^0$ Line Shape} The value of $\as(M_{\rm Z^0})$ derived from the hadronic width of the $\rm Z^0$ boson was continuously updated during the past years, according to the increasing data statistics of the four LEP experiments. Not all of these updates were published in journals, however they are documented as CERN preprints which are commonly available. In this review, the result which was documented last before this conference is taken, $\as(M_{\rm Z^0}) = 0.126 \pm 0.006$. \cite{lep-ewwg}. \subsection{$\alpha_{\rm s}$ from Event Shapes at LEP-1.5} Three of the LEP experiments have published determinations of $\alpha_{\rm s}$ from the data taken at $\rm e^+\rm e^-$ c.m. energies between 130 and 136~GeV \cite{l-133,a-133,o-133}. Each experiment collected about 5~$pb^{-1}$ of data, corresponding to only about 300 non-radiative hadronic events per experiment. Due to the large statistical uncertainty of each experiment, only the $combined$ value of $\alpha_{\rm s}$ from LEP-1.5 can provide a meaningful test of the running coupling. The results of $\alpha_{\rm s}$ from jet rates and from hadronic event shapes at LEP-1.5 are summarised in Table~2. There is good agreement between the experiments, within the statistical uncertainties. The average result is $\alpha_{\rm s} (133\ {\rm GeV}) = 0.112 \pm 0.009$ or, equivalently, $\as(M_{\rm Z^0}) = 0.118 \pm 0.009$, which is compatible, within the experimental errors, with the value which was directly obtained at the $\rm Z^0$ resonance, $\as(M_{\rm Z^0}) = 0.121 \pm 0.006$. \subsection{World Average of $\as(M_{\rm Z^0})$} Averaging the values of $\as(M_{\rm Z^0})$ from Table~1, either unweigthed or weigthed by the inverse square of their errors, gives \footnote{The result which is based on lattice QCD is not included when computing the weigthed average; see Section~7.4 for justification.} $\overline{\alpha_{\rm s}} (M_{\rm Z^0} ) = 0.118$ in both cases. This value has been remarkably stable during the past few years, see e.g. \cite{altarelli,sb-catani,qcd94,moriond95} for previous reviews. From Fig.~8 it can be seen that all results of $\alpha_{\rm s}$ are compatible with this world average, within the errors assigned to the measurements. The errors of most $\alpha_{\rm s}$ results are dominated by theoretical uncertainties, which are estimated using a variety of different methods and definitions. The significance of the quoted errors is largely unknown; they are neither gaussian nor are the correlations between different measurements known. A \char'134 correct" calculation of the overall uncertainty of $\overline{\alpha_{\rm s}} (M_{\rm Z^0} )$ is therefore not possible. Some methods were proposed to compute the overall error from the individual ones, either by rescaling the latter or by constructing an ad-hoc correlation matrix such that the overall $\chi^2$ deviation from the mean value is equal to the number of degrees of freedom (i.e. to $n-1$, where $n$ is the number of individual measurements) \cite{pdg96,schmelling}. If applied to the results listed in Table~1, these methods suggest that $\Delta \overline{\alpha_{\rm s}}(M_{\rm Z^0}) \sim 0.003 ... 0.005$. Since most of the errors listed in Table~1 are not gaussian but rather indicate probability distributions of rectangular shape (however still with unknown correlations between each other), a more pragmatic and conservative estimate of the overall uncertainty of $\overline{\alpha_{\rm s}}(M_{\rm Z^0})$ is therefore applied: counting the relative number of entries in Table~1 whose central values are within $\pm \Delta \overline{\alpha_{\rm s}}$ of $\overline{\alpha_{\rm s}} (M_{\rm Z^0} ) = 0.118$, one gets about 45\% for $\Delta \overline{\alpha_{\rm s}}= 0.003$, 60\% for 0.004, 75\% for 0.005, 90\% for 0.006, 95\% for 0.008 and 100\% for 0.008. A 90\% \char'134 confidence level" seems to be a reasonable and safe estimate for $\Delta \overline{\alpha_{\rm s}}$, such that the world average is quoted to be $$ \overline{\alpha_{\rm s}}(M_{\rm Z^0}) = 0.118 \pm 0.006\ ,$$ which corresponds, in ${\cal O}(\as^3)$ and for $N_f = 5$ or 4 flavours, to $$ \Lambda^{(5)}_{\overline{MS}} = 210^{+80}_{-65}\ {\rm MeV,\ or\ \ } \Lambda^{(4)}_{\overline{MS}} = 295^{+95}_{-80}\ {\rm MeV.} $$ The world average is indicated by the vertical line and the shaded area in Fig.~8. \begin{table*}[htb] \begin{center} \begin{tabular}{|l|c|c|c||c|c|} \hline Exp. & Ref. & $\alpha_{\rm s}$(133 GeV) & $\rightarrow \as(M_{\rm Z^0})$ [LEP-1.5]& $\as(M_{\rm Z^0})$ [LEP-I] & $\#\sigma$ \\ \hline ALEPH & \cite{a-133} & $0.119 \pm 0.005 \pm 0.007$ & $0.126 \pm 0.006 \pm 0.008$ & $0.120 \pm 0.002 \pm 0.007$ & 0.95 \\ L3 & \cite{l-133} & $0.107 \pm 0.005 \pm 0.006$ & $ 0.113 \pm 0.006 \pm 0.007$ & $ 0.125 \pm 0.003 \pm 0.008$ & 1.8 \\ OPAL & \cite{o-133} & $0.110 \pm 0.005 \pm 0.009$ & $ 0.116 \pm 0.006 \pm 0.010 $ & $0.120 \pm 0.002 \pm 0.006$& 0.63 \\ \hline Average & & $0.112 \pm 0.003 \pm 0.008$ & $ 0.118 \pm 0.003 \pm 0.009$ & $ 0.122 \pm 0.001 \pm 0.006$ & 0.95 \\ \hline \end{tabular} \end{center} \baselineskip=11.0pt {\small \noindent {\bf Table 2.} Summary of measurements of $\alpha_{\rm s}$ at LEP-1.5 ($<E_{cm} >$ = 133 GeV). The first errors are experimental, the second theoretical. The last two columns give the results of $\as(M_{\rm Z^0})$ previously obtained at the $\rm Z^0$ resonance (LEP-I) and the number of standard deviations between the LEP-I and LEP-1.5 results, respectively, taking only experimental errors into account.} \end{table*} \subsection{Systematic Differences in $\as(M_{\rm Z^0})$?} In previous reviews the observation was made that measurements which are obtained at energy scales of 5~GeV~$< Q <$~20~GeV are systematically low, corresponding to $\as(M_{\rm Z^0}) \approx 0.112$, while at $Q \approx M_\tau$ and $Q \ge 30$~GeV the results tend to be higher, $\as(M_{\rm Z^0}) \approx 0.120$ and 0.122, respectively. Speculations about the origin of these differences include the existence of a light, neutral, coloured object of spin 1/2 (e.g. a gluino), a possible dependence on the scattering process ($\rm e^+\rm e^-$ annihilation or deep inelastic scattering) or effects of quark masses which are not included in the current higher order QCD calculations \cite{qcd94}. In general, these systematic but hardly conclusive differences are still visible in this summary, see Table~1 and Figure~8. However, the most recent results of $\alpha_{\rm s}$ from deep inelastic scattering at HERA and from jet production in hadron collisions underline the tendency towards higher values of $\as(M_{\rm Z^0})$, in agreement with those from $\rm e^+\rm e^-$ annihilation, and the detailed studies of hadronic $\tau$-decays provide consistent results, too. Therefore the hyptheses of a process dependence of $\alpha_{\rm s}$ or of the existence of light gluinos are not very likely to explain the suspected differences. The most probable origin of those, if significant at all, could be the absence of heavy quark mass effects in the current QCD calculations, which would affect the results from data with energies close to the quark thresholds most. Next-to-leading order calculations including quark mass effects are currently being worked on. This, together with the ongoing efforts to determine $\alpha_{\rm s}$ from the yet increasing amount of data from various processes, has the potential to decrease systematic uncertainties and to resolve the cause of the differences which are still being observed. \section{Conclusion} Asymptotic freedom, which is $the$ key feature of the theory of strong interactions, has been successfully tested in various experimental studies. Perhaps the most intuitive and direct method, the study of the energy dependence of 3-jet event production rates in $\rm e^+\rm e^-$ annihilation, $R_3$, began to provide evidence for the running of $\alpha_{\rm s}$ already in 1988. These studies, carried out by various experiments in a large range of c.m. energies, are based on the JADE-E0 jet algorithm for which, at constant jet resolution, hadronisation corrections are small enough such that $R_3$ is directly proportional to $\alpha_{\rm s}$. With the availability of the LEP data the evidence developed into a proof of asymptotic freedom, demonstrating that $\alpha_{\rm s}$ decreases with increasing energy, as predicted by QCD. Further significant tests became available in the past two years: jet rates in ep-collisions at HERA and 1-jet inclusive transverse energy distributions in hadron collisions, measured by single experiments in large regions of the energy scale, provide the possibility to determine the energy dependence of $\alpha_{\rm s}$ while minimising systematic point-to-point uncertainties. A new analysis based on invariant mass distributions of hadronic $\tau$-decays demonstrates the running of $\alpha_{\rm s}$ in the energy range of $0.7~{\rm GeV}^2 < Q^2 < M^2_\tau$, where the coupling changes by almost a factor of two. In addition to these dedicated tests, the world summary of measurements of $\alpha_{\rm s}$, in the energy range of $M_\tau \le Q \le 133$~GeV, provides compelling evidence for asymptotic freedom. Within their assigned uncertainties, all measurements are compatible with the QCD expectation of a running $\alpha_{\rm s}$. Extrapolated to a common energy scale, using Equations 1~and~2 and treating quark flavour thresholds as described in \cite{wernerb,pdg96}, the measurements of $\alpha_{\rm s}$ average to $$\as(M_{\rm Z^0}) = 0.118 \pm 0.006\ .$$ The overall uncertainty of 0.006 corresponds to a simple estimate of a \char'134 90\% confidence level", derived from the scatter of the individual results. \bigskip \noindent {\bf Acknowledgements.} It is a pleasure to thank S. Narison for providing the possibility to present this review at this well organised and informative conference. I am grateful to E. Elsen, W. Giele, and M. Neubert for providing and authorising the use of Figures 4, 5 and 6, and to W. Bernreuther for many interesting discussions.

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\section*{Acknowledgement} We are grateful to D. Ebert, A.V. Efremov, E.A. Kuraev and L.N. Lipatov for fruitful discussions. HPP is grateful to the Deutsche Forschungsgemeinschaft for support under contract No. RO 905/11-1. The work of VNP and MKV was supported in part by the RFFI, grant No. 96-01-01223 and the Federal Minister for Research and Technology (BMFT) within Heisenberg-Landau Programme. Two of us (VNP and MKV) acknowledge the financial support provided by the Max-Planck-Gesellschaft and the hospitality of the MPG Arbeitsgruppe ''Theoretische Vielteilchenphysik'' at the University of Rostock, where part of this work has been done.\\ \vspace{5mm}\\ \noindent Note added: We wish to thank the referee for his comments and for pointing out Ref. \cite{narison} to us.

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\section{Introduction} Recent magnetotransport experiments on GaAs-AlGaAs heterostructures with a built-in 1D lateral electric modulation, in the $x$ direction, have focused on the internal structure of the Shubnikov-de Haas peaks of the resistivity $\rho_{xx}$.~\cite{WKP,MGT} This structure is determined by the density of states (DOS) of the 1D Landau bands and by their exchange-enhanced spin splitting. The interpretation of the experimental results is however difficult and still unclear. The electron-electron interaction has important effects. The electrostatic and the exchange components act oppositely: The Hartree screening reduces the energy dispersion of the Landau bands, {\em increasing} the DOS, while the negative exchange energy of the occupied states broadens the bands, {\em decreasing} the DOS.~\cite{MG} Our previous attempt to take into account such effects in a magnetotransport calculation suggested that the standard Hartree-Fock approximation (HFA) overestimates the exchange.~\cite{MGT} The resulting van Hove singularities (VHS) of the DOS become very sharp, and for modulation periods much larger than the magnetic length $l$ the competition between the long-range electrostatic screening and the exchange broadening may result in unrealistic short-range oscillations of the charge density. In the present paper we overcome such artefacts by using a screened HFA (SHFA), i.e.~we include the screening influence on the exchange interaction. We show that the enhanced spin splitting can coexist with the picture of compressible/incompressible strips. Our modulation model is a simple potential $V\!\cos Kx$, uniform on the $y$ axis. The material parameters are those for GaAs. \section{Screened Hartree-Fock Approximation} Our one-particle Hamiltonian has the form $H = H^0 + \Sigma^{ee} +\Sigma^{ei}$, in which $H^0$ describes an electron in the plane $\{{\bf r}=(x,y)\}$, in a perpendicular magnetic field $B$ and in the periodic potential, while $\Sigma^{ee}$ and $\Sigma^{ei}$ are the electron-electron and electron-impurity self-energies. We use the Hartree-Fock expression for $\Sigma^{ee}$, at a finite temperature $T$, and we include the polarisation loop in the interaction line of the exchange diagram by replacing the Coulomb potential $u(q)=2\pi/q$ by $u(q)/\varepsilon(q)$. Here $\varepsilon(q)$ is the {\it static} dielectric function of the 2D electron gas, which we consider quasi-homogeneous, characterised by $lV/a\ll \hbar\omega_c$, $a=2\pi/K$. We calculate $\varepsilon(q)$, for an arbitrary $q$, in the spirit of the random-phase approximation, using the Lindhard formula self-consistently with the eigenstates of the Hamiltonian $H$, within a numerical iterative scheme. The dominant screening corresponds to $ql\ll 1$ and is due to the intra-band transitions, which are determined by the DOS at the Fermi level, $D_F$. In our SHFA, charge-density instabilities of the homogeneous system ($V=0$) are no longer possible, in contrast to the standard HFA. In Fig.1 we show typical results in the limit of isolated wires, created by a long-period modulation. This picture combines the specific aspects of both Hartree~\cite{CSG,LG} and Hartree-Fock~\cite{MG,DGH} approximations. The former leads to strong screening, but only bare spin splitting (very small for GaAs), the latter to strong exchange enhancement, but poor screening. In the SHFA the spin splitting of the compressible edge states is accomplished by local fluctuations of the Landau bands with opposite spins, and of the spin density, but each band is individually pinned at the Fermi level. This result disagrees with the prediction of Dempsey et al.,~\cite{DGH} that when the lateral confinement decreases, a sharp transition from spin unpolarised to spin polarised edge states, with a steep energy dispersion at the Fermi level, should occur. We believe this prediction is related to the artefacts of the HFA. Note that, due to the exchange effects we can obtain stable incompressible strips in the bulk of the wires, even without impurity broadening. \section{Conductivities} We consider the electron-impurity interaction in a phenomenological self - consistent Born approximation, taking $\Sigma^{ei}$ as a c-number determined by a characteristic energy parameter $\Gamma=\gamma\sqrt{B{\rm [Tesla]}}$ [meV]. We calculate the conductivities using the Kubo formalism adapted to our system,~\cite{ZG} in which we define the velocity operators as ${\bf v}=i [H,{\bf r}]/\hbar$. The only contribution of the self-energy to the commutator is that of the (nonlocal) exchange interaction, which is in fact the current vertex correction required by the Ward identity.~\cite{G} The relation between the conductivities and the DOS is complicated. Both the diagonal conductivities $\sigma_{xx}$ and $\sigma_{yy}$ have inter-band-scattering components, proportional to $(\Gamma D_F)^2$ and sensitive to the VHS of the 1D Landau bands, when the impurity broadening $\Gamma$ is small. Due to the anisotropy of the system, only $\sigma_{yy}$ has an intra-band term, related to the dispersion of the Landau band, or, classically, related to the drift of the electronic orbits perpendicular to the modulated electric field. This band conductivity is, contrary to the one due to scattering, approximately proportional to $(\Gamma D_F)^{-2}$, thus vanishing near the VHS, but dominating in $\sigma_{yy}$ for $\Gamma\rightarrow 0$. For a comparison with the experiment we need to invert the conductivities into resistivities, $\rho_{xx,yy}= \sigma_{yy,xx}/(\sigma_{xx}\sigma_{yy}+\sigma_{xy}^2)$, and usually $\sigma_{xx}\sigma_{yy}\ll\sigma_{xy}^2$. For the results presented in Fig.2a we have chosen a sufficiently large impurity broadening, so that we have a small band conductivity and consequently $\sigma_{xx}\approx\sigma_{yy}$. In this case the spin splitting is not resolvable even without modulation. For a finite modulation amplitude the Shubnikov-de Haas oscillations of both longitudinal resistivities may have the double-peak structure of the DOS. Such a situation has been observed by Weiss et al.~\cite{WKP} for $\rho_{xx}$, in the second Landau band, $n=1$, while here we get it more clearly in the third band, $n=2$. At higher magnetic fields the stronger screening reduces the band width and the resolution of the VHS is rather poor. The spin-splitting is resolved in Fig.2b where both modulation amplitude and impurity broadening are small. For $n=2$ the spin splitted Landau bands still partly overlap. The VHS are now observable only in $\rho_{yy}$, and are covered by the band conductivity in $\rho_{xx}$. Recent measurements have shown a more complicated, double- and triple-peak profile in $\rho_{xx}$, which may be attributed to a combined effect of VHS and band conductivity.~\cite{MGT} But as we have shown, screening effects might be very strong. The screening can be reduced for a modulation with a shorter period, comparable to $l$. The steeper energy dispersion may allow the resolution of the VHS when the band conductivity is suppressed by disorder, as suggested in the experiments by Sfaxi et al.~\cite{SPL} However, a comparative measurement of both $\rho_{xx}$ and $\rho_{yy}$, in high magnetic fields, indicating the scattering and the band conductivity contributions is, to our knowledge, not available. In conclusion, our SHFA describes both screening and exchange-enhanced spin splitting of the Landau bands, interpolating in the expected manner between the contradictory results of the Hartree and the standard Hartree-Fock approximations. The theoretical resistivities are in qualitative agreement with the available experimental data. Calculational details will be published elsewhere. \section*{Acknowledgments} We are grateful to Gabriele Ernst, Behnam Farid, Marc Tornow and Dieter Weiss for stimulating discussions. \section*{References}

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\section{Introduction} This work is devoted to computer simulation of magnetization switching in single-domain ferromagnetic nanoparticles. For highly anisotropic systems, simple kinetic Ising models can qualitatively explain many experimental observations.\cite{MMM} However, to obtain more realistic results additional physical effects must be included in the model. An important aspect of real samples is heterogeneous droplet nucleation due to the presence of the system boundary and due to bulk disorder. The model used in our study is a square-lattice nearest-neighbor kinetic Ising ferromagnet with random updates using either Glauber or Metropolis single-spin-flip Monte Carlo dynamics. We first investigate systems with boundaries (but without disorder), then we consider disordered systems with periodic boundary conditions. We concentrate on two quantities: the lifetime and the switching field. To determine the metastable lifetime, we start the simulations with all spins $+1$ and impose a negative magnetic field. Then we measure (in Monte Carlo Steps per Spin, MCSS) the mean time the sample needs to reach zero magnetization. Having measured this ``lifetime'' $\tau$ for several values of the magnetic field, one can extract from the data the switching field $H_{\rm sw}$, which is defined as the field that produces a given lifetime. This quantity is often measured in experiments.\cite{Chang,Lederman,New} We performed Monte Carlo simulation studies of the switching dynamics of a kinetic Ising model defined on circular lattices (which we define as subsets of a square lattice, contained within a circle of given diameter $L$). The Hamiltonian is \begin{equation} {\cal H} = -J\sum_{<ij>} \sigma_i \sigma_j -H \sum_i \sigma_i -H_\Sigma \sum_{i \in \Omega} \sigma_i \ , \end{equation} where the first two terms represent the standard spin-spin interaction with positive $J$ and the coupling to the external field $H$, respectively. The last term, in which the summation runs only over the sites on the boundary of the lattice $\Omega$, is included to model the effect of the system boundary on nucleating droplets of the stable phase. For $H_\Sigma =0$ one has a free boundary, whereas positive (negative) $H_\Sigma$ mimic a boundary which effectively repels (attracts) the droplets. Figure~\ref{fig:cirb} shows the switching fields for a fixed waiting time $\tau$ versus the diameter $L$ of the circular lattice. For the smallest systems, $H_{\rm sw}$ is zero; for slightly larger systems, $H_{\rm sw}$ increases sharply with $L$. This is the coexistence region, in which the stable and metastable phases practically coexist.\cite{MMM,Rikvold} For larger systems, the behavior depends strongly on the affinity of the boundary for droplets of the equilibrium phase, which here is modeled by different values of $H_\Sigma$; and also on the waiting time $\tau$. In general, the increase in the coexistence region is followed by a decrease of the switching field in the single-droplet region (where switching is triggered by a single critical droplet\cite{MMM,Rikvold}). This results in a maximum located at the crossover between the coexistence and single-droplet regions.\cite{MMM} Similar switching-field peaks are observed in certain experiments on nanoscale ferromagnets.\cite{Chang} For larger systems, more than one droplet nucleates during the switching process, and the system is in the multidroplet region.\cite{MMM,Rikvold} There, the switching field becomes asymptotically independent of the system size. \begin{figure} \vspace*{2.7in} \special{psfile = cirb.ps angle = 0 hscale = 70 vscale = 70 hoffset = -80 voffset = -275} \caption{ The switching field $H_{\rm sw}$ vs.\ $L$ for different values of the boundary field $H_{\Sigma}$ for circular systems at $T=1.3J \approx 0.57 T_c$. The waiting time is $\tau=30000$ MCSS, the dynamics is Metropolis. Data for periodic systems (full line) are shown for comparison. Note that even lattices with $L=400$ are still in the single-droplet region. The decrease of the switching field would continue if we went to even larger systems. Eventually, for very large systems, it would converge to the same value in the multidroplet region, independent of $H_\Sigma$. } \label{fig:cirb} \end{figure} Just as in periodic systems,\cite{MMM} this behavior can be clearly observed in samples in which the boundaries repel droplets of the stable phase (positive $H_\Sigma$). For neutral boundaries ($H_\Sigma=0$) the maximum in the switching field can be much less pronounced or it can disappear completely, depending on the waiting time. For example, with $\tau = 1000$, the maximum of $H_{\rm sw}$ completely disappears for $H_\Sigma = 0$. In general, to observe the maximum, one needs a longer waiting time. Note that a long waiting time probably corresponds better to most real experimental situations. For theoretical considerations concerning magnetization switching in kinetic Ising systems without disorder, see Refs.\onlinecite{MMM} and \onlinecite{Richards}. To study effects of disorder, we simulated the above Ising model on periodic square lattices using Glauber dynamics. The first type of disorder we investigate is produced by defects generated by randomly deleting bonds of the lattice with concentration $c$. In order to understand how the disorder influences the switching, we measured various properties of the nucleating droplets of the stable phase during the simulation. All measured quantities were taken as mean values over all events in which a droplet of a given size appeared in the system. It turns out that with this type of disorder, the switching can be well described in terms of ``average'' nucleating droplets. Figure~\ref{fig:cdpr} shows the critical size of a droplet versus the defect concentration $c$. Droplets larger than the critical size are more likely to grow than to shrink, whereas smaller droplets do the opposite. As expected, the disorder leads to smaller critical droplets, which in turn results in a dramatic decrease of the metastable lifetime. \begin{figure} \vspace*{2.5in} \special{psfile = cdrprop.ps angle = 0 hscale = 70 vscale = 70 hoffset = -80 voffset = -280} \caption{ Size and defect content of a critical droplet as functions of the defect concentration. } \label{fig:cdpr} \end{figure} \begin{figure} \vspace*{2.6in} \special{psfile = hswi.ps angle = 0 hscale = 70 vscale = 70 hoffset = -80 voffset = -270} \caption{ Switching field as a function of system size for different defect concentrations, waiting time $\tau = 30000$ MCSS and Glauber dynamics at $T=1.3J$. } \label{fig:hswi} \end{figure} Another interesting property of the typical critical droplet is its defect content: the number of deleted bonds associated with the droplet. It can be deduced from Fig.~\ref{fig:cdpr}, as well as from direct measurements, that the effective concentration of defects within the droplets is considerably larger than the global concentration. This is caused by the fact that the droplets are preferentially nucleated in the vicinity of the defects. In Fig.~\ref{fig:hswi} we show the effect of the disorder on the switching field. Whereas the switching field is decreased by the disorder, its shape as a function of the system size remains approximately independent of $c$. Thus, in the intermediate-temperature region studied here, the switching dynamics of the kinetic Ising model on a bond-diluted lattice is essentially the same as in systems without disorder, although the metastable lifetimes and switching fields are considerably reduced by the disorder. The effect of the disorder can be approximately described in terms of an effective medium which affects the growing droplets of the stable phase. Details will be given in a future paper.\cite{Richards} \begin{figure} \vspace*{2.65in} \special{psfile = komb.ps angle = 0 hscale = 70 vscale = 70 hoffset = -80 voffset = -275} \caption{ The metastable lifetime $\tau$ as a function of disorder. The error bars show the width of the distributions of lifetimes for different realizations of the disorder. For the diluted-bond type of disorder (insert), this broadening is smaller than the symbol size. In both cases, $T=1.3J$, $H=-0.3J$ and $L=20$. \label{fig:tIItau} } \end{figure} Another type of disordered kinetic Ising model is defined by the Hamiltonian $ {\cal H} = -J\sum_{<ij>} a_i a_j \sigma_i \sigma_j - H \sum_i a_i \sigma_i $, where the amplitudes $a_i$ represent random local magnetic moments. Here, we take them uncorrelated with a ``box'' distribution of width $W$, centered around unity. The switching dynamics with this type of disorder differs from the one discussed above in several respects. First, the approach in terms of ``average'' nucleating droplets is not useful because the interesting quantities have broad distributions. Second, there is a lack of self-averaging, even in the intermediate-temperature region studied here. This is demonstrated in Fig.~\ref{fig:tIItau}, where we show how the mean lifetime is reduced by the disorder. At the same time, the relative width of the probability distribution of the mean individual-sample lifetimes increases. Whereas switching of a sample with a particular realization of the disorder is more deterministic than in a pure system, an ensemble of disordered systems exhibits a wide distribution of lifetimes and, consequently, of switching fields. The mechanism leading to this behavior is that the size of a critical droplet is different in different parts of the disordered lattice. In summary, we have studied magnetization switching in systems in which heterogeneous nucleation of the equilibrium phase may occur on the system boundary or be associated with bulk impurities. The presence of the system boundary strongly affects the magnetization switching in small systems since the critical fluctuations of the stable phase tend to nucleate in its vicinity. This considerably decreases the metastable lifetime as well as the switching field, compared to periodic systems. However, the basic features, such as the crossover from the coexistence region to a single-droplet region, and then to a multidroplet region with increasing $L$, can still be observed in systems with a boundary. The switching field as a function of the system size becomes less peaked, but a maximum, located near the crossover between the coexistence and the single-droplet regions, can still be observed if the waiting time is sufficiently long. We have studied two different types of disordered systems. Diluted-bond disorder can be a toy model for samples with impurities, whereas the model with random magnetic moments could represent films with fluctuating thickness. Our results show that whereas these types of bulk disorder result in shorter metastable lifetimes in general, the details of the magnetization switching can differ considerably for different types of disorder. Research supported by FSU-MARTECH, FSU-SCRI (DOE Contract No. DE-FC05-85ER25000), NSF Grants No. DMR-9315969, DMR-9520325, and INT-9512679, the Inoue Foundation, and NERSC supercomputer time.

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\section{Introduction} The Lagrangian BRST formalism of Batalin and Vilkovisky (BV) \cite{BV,HT,GPS} is presently considered the most powerful procedure for the quantisation of gauge theories. The application of this formalism to anomalous gauge theories was first discussed by Troost, van Nieuwenhuizen and Van Proeyen \cite{TPN}, that succeeded in using the Pauli Villars regularization in order to give a regularized meaning to the master equation at one loop order. The possibility of more general regularizations that enable the quantization of more complex gauge theories where the higher loop order terms play an important role is presently under study\cite{Paris,DPT}. For the case of irreducible gauge theories with closed algebra, it was shown that the BV procedure can be formulated in a superspace with one Grassmanian variable\cite{BD,AB}, where BRST transformations are realised as translations. The Chiral Schwinger Model (CSM) has been an important device to understand the quantisation of anomalous gauge theories. It was shown that the Wess Zumino term that restores the gauge invariance at the quantum level, realizing the mechanism proposed by Faddeev and Shatashvilli\cite{FS} can be generated by just solving the (regularized) master equation at one loop order\cite{BM} for this model, reproducing the results that have been found before by Harada and Tsutsui\cite{Harada} in a tricky way. The superspace formulation has the nice property of been, by construction, explicitly BRST invariant. The master equation is translated into the existence of a superfield structure associated to the quantum action in such a way that realizing the Wess Zumino mechanism is just equivalent to building up such a superfield without anomaly. We will see that the Chiral Schwinger Model illustrates this superspace formulation in an interesting way. This article is organized as follows: in section {\bf 2} we briefly review the superspace formulation for the BV formalism at one loop order. In section {\bf 3} we present our results for the CSM and section {\bf 4} is devoted to some concluding remarks. \section{Superspace Formulation for the BV formalism} The fermionic nature and the nillpotency of the BRST transformations makes it possible to build up a superspace representation where they are realized as translations in a Grassman variable\cite{SUP}. One adds to the original space-time variables one Grassmanian degree of freedom $\theta$ and associate to each original field $\phi (x)$ a superfield: \begin{equation} \label{SF} \Phi (x , \theta ) = \phi (x) + \theta \delta \phi (x) \end{equation} \noindent in such a way that: \begin{equation} \delta_{_{BRST}}\Phi ( x,\theta ) = {\partial \over \partial\theta }\,\Phi( x,\theta ) \end{equation} When one tries to apply this superspace realization to the case of the field antifield (FA) quantisation one faces a problem. The quantum master equation\cite{BV} \begin{equation} \label{Master} {1\over 2}(W,W) = i\hbar\Delta W \end{equation} \noindent involves the operator: \begin{equation} \label{Delta} \Delta \equiv {\delta_r \over\delta \phi^a }{ \delta_l \over \delta \phi^\ast_a }\; \end{equation} \noindent that represents the possibly non trivial behavior of the path integral measure.One then needs a superspace version for this operator and thus one should introduce functional derivatives with respect to superfields. However, superfields of the form (\ref{SF}) will in general be constrained, as the BRST transformations are in general not independent of the corresponding fields. For an unconstrained superfield: \begin{equation} \label{SFG} \Omega (x , \theta ) = A (x) + \theta B (x) \end{equation} \noindent where \begin{equation} \label{INDEP} {\delta B(x)\over \delta A(x^{\prime})} \,=\,0\,\,\,\,; \,\,\,\,\,\, {\delta A(x)\over \delta B(x^{\prime})}\,=\,0 \end{equation} \noindent we can define a functional derivative and relate it to the derivatives with respect to the component fields: \begin{equation} \label{COMP} {\delta \over \delta \Omega (x ,\theta )} = \,{\delta \over \delta B (x )} +\,\theta {\delta \over \delta A (x )} \end{equation} However, as discussed in \cite{AB}, functional differentiation and integration for constrained superfields is not in general well defined. This problem was circumvented in \cite{AB} by applying a procedure due to Alfaro and Damgaard\cite{COL}\footnote{See also \cite{DJ} for a review}, that we will call collective field approach to BV, where one gets the BV action by trivially enlarging the field content of the theory and then choosing an appropriate gauge fixing structure. Following this approach, as we will see, one is able to build up unconstrained superfields even when the original BRST transformations are not independent of the associated fields. The collective field approach to BV consists (in a very summarized way) in starting with a gauge field theory characterized by a classical action $S_0[\phi^i] $, introducing ghosts, antighosts and auxiliary fields associated to the original gauge invariances of $S_0$ in the usual way, getting an enlarged field set represented as $\phi^A$. These fields realize a BRST algebra: \begin{equation} \label{OGS} \delta_0 \phi^A = R^A \,[\phi ] \end{equation} Then we introduce a new set of fields called collective fields $\tilde \phi^A$ and replace everywhere $\phi^A$ by $\phi^A - \tilde \phi^A$. This way we double the field content of the theory and at the same time associate to each field a new trivial shift symmetry. In order to gauge fix these new symmetries we introduce new ghosts, antighosts and auxiliary fields, represented respectively as: $\pi^A$, $\phi^{\ast\,A}$ and $B^A$. We have a large freedom in choosing the BRST transformations for this enlarged set of fields. Following \cite{COL} we can define the enlarged BRST algebra as \begin{eqnarray} \label{ALG} \delta \phi^A &=& \pi^A\nonumber\\ \delta \tilde \phi^A &=& \pi^A - R^A [\phi -\tilde \phi ] \nonumber\\ \delta \pi^A &=& 0\nonumber\\ \delta \phi^{\ast\,A} &=& B^A\nonumber\\ \delta B^A &=& 0 \end{eqnarray} \noindent and the total action as \begin{equation} \label{A1} S = S_0 [\phi^i -\tilde \phi^i ] - \delta (\phi^{\ast\,A} \tilde \phi^A ) + \delta \psi [\phi^A] \end{equation} \noindent where $\psi [\phi^A]$ is a fermionic functional representing the gauge fixing of the original symmetries (\ref{OGS}). The BV gauge fixed classical action is obtained if one functionally integrates the vacuum functional associated with $S$ over $\pi^A$, $\tilde \phi^A$ and $B^A$. The interesting point is that in this collective field approach all the fields of the sets $\phi^A$ and $\phi^{\ast\,A}$ have BRST transformations that are independent quantities, unrelated to the associated field, as follows from (\ref{ALG}). Therefore, if we introduce superfields of the form (\ref{SF}), at least for this two sets, they will be unconstrained. The component decomposition (\ref{COMP}) for the functional derivatives then makes it easy to see that the operator: \begin{equation} \label{DS} \underline \Delta \equiv \int dx \int d\theta \int d\theta^\prime\,{\delta_r \over \delta \Phi^A (x,\theta ) }\,\, {\delta_l \over \delta \Phi^{\ast\,A} (x ,\theta^\prime \,)} \end{equation} \noindent with \begin{eqnarray} \label{EXTALG} \Phi^A (x,\theta ) &=& \phi^A (x) + \theta \pi^A (x) \nonumber\\ \tilde \Phi^A (x,\theta ) &=& \tilde \phi^A (x) + \theta ( \pi^A (x) - R^A [\,\phi -\tilde \phi \,]\, )\nonumber\\ \Phi^{\ast\,A} (x,\theta ) &=& \phi^{\ast\,A} (x) + \theta B^A (x) \nonumber\\ \end{eqnarray} \noindent represents the operator $\Delta$ in superspace. The naive action of this operator leads to an undefined singular result because of the two functional derivatives acting in the same space time point. It was shown in \cite{AB} that the Pauli Villars regularization developed in \cite{TPN} that gives a meaning to the one loop order master equation can be applied in the superspace formulation. We will see how to do it in next section. In the superspace formulation the quantum action, for non anomalous gauge theories, will have the component expansion \begin{equation} \label{SA} \underline W = W + \theta i\hbar \Delta W \end{equation} \noindent and the master equation will read: \begin{equation} \label{SMaster} {\partial \over \partial \theta} \underline W \,=\, i\hbar \underline \,\,\underline\Delta \underline W \end{equation} \noindent corresponding, order by order in $\hbar$, (we are considering quantum corrections only up to one loop order): \begin{equation} \label{SMO} {\partial \over \partial \theta} \underline S \,=\, 0 \,\,\,\,;\,\,\,\, {\partial \over \partial \theta} \underline M_1 \,=\, i \underline \Delta \,\, \underline S \end{equation} \vskip 1cm \section{The Chiral Schwinger Model} The classical action for the Chiral Schwinger Model (CSM) is: \begin{equation} S_0 = \int d^2x \Big[ -{1\over 4} F_{\mu\nu} F^{\mu\nu} + {1\over 2}\overline \psi {\slash\!\!\!\!D} (1-\gamma_5) \psi \Big] \end{equation} \noindent where $D_\mu = \partial_\mu + ie A_\mu$. The BRST transformations of the fields are: \begin{eqnarray} \label{TS} \delta A_\mu &=& \partial_\mu c\nonumber\\ \delta \psi &=& i\psi c\nonumber\\ \delta \overline\psi &=& - i \overline \psi c \nonumber\\ \delta c &=& 0 \end{eqnarray} \noindent where $c$ is the ghost field associated to the gauge invariance of $S_0$. We enlarge, as explained in the previous section, the field content of the theory, introducing the collective fields $\tilde A_\mu , \tilde \psi , \tilde {\overline \psi} , \tilde c $ and build up the associated superfields: \begin{eqnarray} \underline A_\mu (x,\theta) &=& A_\mu (x) + \theta \pi ^{\,[\,A_\mu\,]}_\mu (x)\nonumber\\ \underline {\tilde A}_\mu (x,\theta) &=& \tilde A_\mu (x) + \,\theta\, \Big[ \,\pi ^{\,[\,A_\mu\,]}_\mu (x) \,-\, \partial_\mu ( c(x) - \tilde c(x))\,\Big] \nonumber\\ \underline A_\mu^\ast (x,\theta) &=& A_\mu (x) + \theta B_\mu^{\,[\,A_\mu\,]}(x)\nonumber\\ & &\nonumber\\ \Psi ( x,\theta ) &=& \psi( x ) + \theta \pi^{\,[\,\psi\,]} (x)\nonumber\\ \tilde \Psi ( x,\theta ) &=& \tilde \psi (x) + \theta \,\big[\, \pi^{\,[\,\psi\,]} (x) - i (\psi (x) - \tilde \psi (x) ) ( c(x) - \tilde c (x) ) \Big] \nonumber\\ \Psi^\ast ( x,\theta ) &=& \psi^\ast (x) + \theta B^{\,[\psi\,]} (x) \nonumber\\ & &\nonumber\\ \overline \Psi ( x,\theta ) &=& \overline \psi ( x ) + \theta \pi^{\,[\,\overline \psi\,]} (x)\nonumber\\ \tilde {\overline \Psi} ( x,\theta ) &=& \tilde {\overline \psi} (x) + \theta \, \big[ \, \pi^{\,[\,\overline \psi\,]} (x) + i (\overline \psi (x) - \tilde {\overline \psi} (x) ) ( c(x) - \tilde c (x) ) \Big] \nonumber\\ \overline \Psi^\ast ( x,\theta ) &=& \overline \psi^\ast (x) + \theta B^{\,[\overline \psi\,]} (x) \nonumber\\ & &\nonumber\\ \eta ( x,\theta ) &=& c (x) + \theta \pi ^{\,[\,c\,]} (x) \nonumber\\ \tilde\eta ( x,\theta ) &=& \tilde c (x) + \theta \pi ^{\,[\,c\,]} (x)\nonumber\\ \eta^\ast ( x,\theta ) &=& c^\ast (x) + \theta B^{\,[\,c\,]} (x) \end{eqnarray} The total superfield action will be: \begin{equation} \underline S = \underline S_0 + \underline S_1 + \underline S_2 \end{equation} \noindent with the extended superspace version of the classical action: \begin{eqnarray} \label{S0} \underline S_0 &=& \int d^2x \Big( -{1\over 4} F_{\mu\nu}\, \big[ \underline A_\mu\,-\,\underline {\tilde A}_\mu\, \big] F^{\mu\nu} \big[\underline A_\mu\,- \underline {\tilde A}_\mu\,\big] \nonumber\\ &+& {1\over 2}( \overline \Psi - \tilde {\overline \Psi} ){\slash\!\!\!\!D}\, [\underline A_\mu\,- \underline {\tilde A}_\mu\,] (1-\gamma_5) ( \Psi - \tilde \Psi) \Big) \end{eqnarray} \noindent the gauge fixing of the shift symmetry: \begin{equation} \underline S_1 = - {\partial \over \partial\theta } \int d^2 x \Big[ {\underline A}^\ast_\mu \underline {\tilde A}^\mu + \Psi^{\ast} \,\tilde \Psi \,+\, \overline \Psi^\ast\, \tilde {\overline \Psi} +\, \eta^\ast\, \eta \,\Big] \end{equation} \noindent and the gauge fixing of the original symmetry of $S_0$: \begin{equation} \underline S_2 = {\partial \over \partial\theta } \int d^2x \,\Lambda\, \big[ {\underline A}_\mu , \Psi, \overline \Psi , \eta \big] \end{equation} \noindent with the collective field version of the classical action: We must now build up a superspace Pauli Villars (PV) action, that will regularize the action of the operator $\underline \Delta$ on the action. We only need PV partners for the fermionic fields as we can easily see from (\ref{TS}) that the gauge field and the ghost will not contribute to $\Delta S$ as their transformations are independent of the field itself. Following the prescriptions of \cite{AB} we associate with $\psi$ and $\overline\psi$ the PV fields $\chi$ and $\overline\chi$ and the corresponding collective tilde fields and introduce the action: \begin{eqnarray} \underline S_{PV} &=& {1\over 2} (\underline{\overline \chi} - \underline {\tilde {\overline \chi}} ) \,{\slash\!\!\!\!D}\, [\,{\underline A}_\mu - {\tilde {\underline A}}_\mu \,] \,(\underline \chi - \underline {\tilde \chi} )\nonumber\\ &-&{1\over 2} M (\underline {\overline \chi} - \underline {\tilde {\overline \chi}} ) (\underline \chi - \underline {\tilde \chi} ) - {\partial\over \partial \theta}\, \Big(\,\underline {\overline \chi}^{\ast\,} \underline {\tilde {\overline \chi}} + \,\underline \chi^{\ast\,} \underline {\tilde \chi} \Big) \end{eqnarray} \noindent that represents a copy of the fermionic part of action (\ref{S0}) but with a mass term that, after calculating the regularized $\delta S$, allows the removal of the PV fields by taking the limit $\,M\,\rightarrow \infty$. We define the BRST transformations of the PV fields to be similar to the ones from the corresponding fields given by (\ref{TS}). Thus the PV superfields will have the structure: \begin{eqnarray} \label{PVSUP} {\underline \chi} (x,\theta ) &=& \chi (x) + \theta \, \,\pi^{[\,\chi\,]\,} \nonumber\\ {\tilde {\underline \chi}} (x,\theta ) &=& \tilde \chi (x) + \theta \Big( \, \,\pi^{[\,\chi\,]\,} - i (\chi - \tilde\chi ) ( c - \tilde c ) \, \Big)\nonumber\\ {\underline \chi}^{\ast\,} (x,\theta ) &=& \chi^{\ast\,} (x) + \theta B^{\,[\,\chi\,]\,} \nonumber\\ \overline {\underline \chi} (x,\theta ) &=& \overline\chi (x) + \theta \, \,\pi^{[\,\overline\chi\,]\,} \nonumber\\ {\tilde {\overline {\underline \chi}}} (x,\theta ) &=& \tilde { \overline \chi} (x) + \theta \Big( \, \,\pi^{[\,\overline\chi\,]\,} + i (\overline \chi - \overline {\tilde\chi} ) ( c - \tilde c ) \, \Big)\nonumber\\ \overline {{\underline \chi}}^{\ast\,} (x,\theta ) &=& \overline\chi^{\ast\,} (x) + \theta B^{\,[\,\overline\chi\,]\,} \end{eqnarray} \noindent As usual, the PV fields are defined formally in such a way that their one loop contributions have a minus sign relative to the original fields\cite{TPN}. The action of the operator $\underline \Delta $ on the regularized total action, if we include the PV fields also in the operator (\ref{DS}), is then: \begin{equation} \label{DSR} \underline \Delta (\underline S + \underline S_{PV} ) \, = \, 0 \end{equation} \noindent The regularized form of $\Delta S$ when we use the PV regularization shows up as a violation of the zero order master equation associated to the presence of the mass term. In the present superspace formulation, this absence of BRST invariance of the total (regularized) classical action $S_{T} = S + S_{PV}$ is translated into the presence of a $\theta$ component in the corresponding superfield: \begin{equation} \underline S_{T} = \underline S + \underline S_{PV} = S_T + \theta \delta S_T \end{equation} The general form of $\delta S_T$ is \begin{equation} \label{1} \delta S_T = i M \, (\overline\chi - \tilde {\overline \chi} ) (\chi - \tilde\chi )\,( c - \tilde c ) \end{equation} \noindent Integration over the fields $\pi^{[\,\chi\,]\,A\,} \,,\, B^{\,[\,\chi\,]\,A}$ and $\tilde \chi^A$ removes the extended collective field structure, recovering the usual result as in \cite{TPN}, that corresponds in (\ref{1}) just to the absence of the collective tilde fields. The next step would be to integrate over the PV fields. We will not repeat this procedure here as it is exactly the same as in the component case, that is widely discussed in the literature\cite{GPS,TPN,DJ}. The result is: \begin{equation} (\Delta S)_{reg.} \,=\, {i\over 4\pi} \int d^2x ( c -\tilde c) \Big( (1-a) \partial_\mu (A^\mu - \tilde A^\mu ) - \epsilon^{\mu\nu}\partial_\mu (A_\nu -\tilde A_\nu ) \Big) \end{equation} Now going back to the one loop order master equation (\ref{SMO}) we have to look for a superfield $\overline M_1$ whose $\theta$ component is equal to $i (\Delta S)_{reg.}$. It is well known that for this model, that possess a genuine anomaly, a local solution for the master equation can not be found in the original space of fields and antifields. However the Wess Zumino mechanism of restoring gauge invariance can be realized in the field antifield formalism\cite{BM2,GP1} by introducing a field (and the corresponding antifield) that transforms as the gauge group elements. In the present superspace formulation we can introduce the superfields \begin{eqnarray} \Omega (x,\theta) &=& \omega (x) + \theta \pi^{\,[\,\omega\,]} (x)\nonumber\\ \tilde\Omega (x, \theta) &=& \tilde\omega (x) + \,\theta\, \Big[ \,\pi^{\,[\,\omega\,]} (x) \,-\, ( c - \tilde c)\,\Big] \nonumber\\ \Omega^\ast (x,\theta) &=& \omega^\ast + \theta B^{\,[\,\omega\,]}(x) \end{eqnarray} \noindent that realize in superspace the collective field version of the Wess Zumino field transformations. We include a gauge fixing term for the WZ field in the action defining: \begin{equation} \underline S^\prime = \underline S - {\partial\over\partial\theta} \, \int d^2 x \, \tilde\Omega \, \Omega^\ast \end{equation} From the transformation of $\Omega$ one easily realizes that $\Delta S= \Delta S^\prime$. Now we can write a superfield that satisfies $ \partial {\overline M}_1 / \partial \theta = i\,(\Delta S)_{reg.}$. \begin{eqnarray} {\overline M}_1 &=& {1\over 4\pi} \int d^2x \Big( {(a-1)\over 2} \partial_\mu (\Omega - \tilde\Omega) \partial^\mu (\Omega - \tilde\Omega) \nonumber\\ &-& \partial_\mu (\Omega - \tilde\Omega) \big( (a-1) (\underline A^\mu - \underline {\tilde A}^\mu) + \epsilon^{\mu\nu} ( \underline A_\nu - \underline {\tilde A}_\nu ) \big) \Big) \end{eqnarray} \noindent in components this superfield reads: $\underline M_1 = M_1 + \theta (i\Delta S)_{reg.}$ with: \begin{eqnarray} M_1 &=& {1\over 4\pi} \int d^2 x \Big( {(a-1)\over 2} \partial_\mu (\omega - \tilde\omega) \partial^\mu (\omega - \tilde\omega) \nonumber\\ &-& \partial_\mu (\omega - \tilde\omega) \big( (a-1) ( A^\mu - {\tilde A}^\mu) + \epsilon^{\mu\nu} ( A_\nu - {\tilde A}_\nu ) \big) \Big) \end{eqnarray} \noindent If we remove the collective fields, this corresponds just to the Wess Zumino term found in \cite{BM} in the non superspace approach. Therefore, the superfield $\underline W = \underline S^\prime + \underline M_1 $ satisfies the superspace version of the master equation (\ref{SMaster}), representing the superfield action, that includes, besides the quantum action, also the anomalous contribution from the path integral measure $\Delta S$. \section{Conclusion} We have shown to represent the quantum action of the Chiral Schwinger model in a BRST superspace. An interesting point of this formulation is that both the action and the $\Delta S$ term (that comes from the non trivial behaviour of the path integral measure) show up in the same superfield. The master equation corresponds thus just to a restriction on the structure of this object. We have also shown that the Wess Zumino mechanism can also be realised in this formulation, by adding a superfield that represents the gauge group elements. \vskip 1cm \section{Acknowledgements}This work was partially supported by CNPq, FINEP, FUJB and CAPES (Brazilian Research Agencies). \vfill\eject

proofpile-arXiv_065-568

\section{Introduction} Over the years it has been found that there exist many two-dimensional classical spin models, discrete and continuous alike, whose ground-state manifolds are macroscopically degenerate and, more interestingly, also exhibit critical behaviours, i.e., spin-spin correlation functions within the ground-state ensembles decay with distance as power laws. The classification of universality class for these models has always been a challenging problem\cite{Liebmann} An earlier example of this kind is the antiferromagnetic Ising model on the triangular lattice. The exact solution for this model by Stephenson\cite{Stephenson} showed that although this model remains paramagnetic at nonzero temperature, its ground state is critical. Later works by Bl\"ote {\sl et al} revealed yet another remarkable property of the ground-state ensemble of this model, namely, it permits a Solid-on-Solid (SOS) representation in which spin fluctuations are subsequently described by the fluctuating interface in the SOS model\cite{Blote}. Recent studies also demonstrated that this interfacial representation provides a valuable avenue for studying the ground-state ordering of quantum magnets\cite{quantum,Henley1} and the ground-state roughness of oriented elastic manifolds in random media\cite{Zeng}. Other recently studied models with critical ground states include three-state antiferromatic Potts model on the Kagom\'e lattice\cite{Huse,Chandra}, the $O(n)$ model on the honeycomb lattice\cite{Blote2,Kondev1}, the Four-Coloring model on the square lattice\cite{Kondev2,Kondev3}, and the square-lattice non-crossing dimer model and dimer-loop model\cite{Raghavan}. On the other hand, some very similar models with degenerate ground states exhibit long-range order, such as the constrained 4-state Potts antiferromagnet\cite{Burton}. In this article we study the ground-state properties of antiferromagnetic Ising model of general spin on a triangular lattice which also belongs to the class of models mentioned above. Recent numerical studies of this model include Monte Carlo simulations\cite{Nagai,Honda} and transfer matrix calculations\cite{Lipowski}. Here we revisit this model by performing Monte Carlo simulations. The motivation of the present work is two-fold: (1)unlike previous simulations, we utilize the interfacial representation directly in analyzing the simulation results, for example, we compute the stiffness constant of the fluctuating interface which, in turn, yields more accurate critical exponents of various operators; and (2) we also study the dynamical properties of this model for the first time making use of the interfacial representation. The body of the this paper is organized as follows. Section \ref{Model} describes the model Hamiltonian and maps it onto a spin-1 problem whose interfacial representation is then described. In Section \ref{height-rep}, we propose an effective continuum theory for the long-wavelength fluctuations of the interface. Here we also show how to relate scaling dimensions of various operators to the stiffness constant of the interface, and derive some other analytical results based on this ``height representation.'' This allows analytical understanding of the phase diagram (Sec.~\ref{PhaseDiag}). Details of Monte Carlo simulations and numerical results on both dynamical and static properties are presented in Section \ref{MC-results}, including a comparison of the new and old approaches to determining the exponents. As a conclusion, the paper is summarized and various possible extensions are outlined, in Section \ref{Conc-Disc}. \section{The Model} \label{Model} The antiferromagnetic Ising model of spin-$S$ on a triangular lattice can be described by the following Hamiltonian: \begin{equation} H = J \sum_ {{\bf r}} \sum_{{\bf e}} s({\bf r}) s({\bf r}+{\bf e}) \label{eq1} \end{equation} where the spin variable $s({\bf r})$ defined on lattice site ${\bf r}$ of the triangular lattice can take any value from a discrete set $[-S, -S+1, \cdot\cdot\cdot, S-1, S]$, and the sum over ${\bf e}$ runs over three nearest-neighbor vectors ${\bf e}_1$, ${\bf e}_2$ and ${\bf e}_3$ as shown in Fig.~\ref{fig1}. Here the coupling constant $J$ is positive describing the antiferromagnetic exchange interaction between two nearest-neighbor spins: $s({\bf r})$ and $s({\bf r}+{\bf e})$. One important reason for interest in this model is that the $S\to \infty $ limit\cite{FNSinfty} is the same as the Ising limit of the (classical or quantum) Heisenberg antiferromagnet on the triangular lattice with Ising-like anisotropic exchange. That model was shown to exhibit a continuous classical ground state degeneracy and unusual features of the selection by fluctuations of ground states\cite{Heisenberg}. The ground-state configurations of the above model given by Eq. (\ref{eq1}) consist of entirely of triangles on which one spin is $+S$, another is $-S$, and the third can be anything in $[-S,+S]$. Thus, if spin $s(\bf r)$ takes an intermediate value $-S<s({\bf r})<S$, this forces the six surrounding spins to alternate $+S$ and $-S$; exactly which intermediate value $s(\bf r)$ takes does not matter in determining whether a configuration is allowed. \subsection{Spin-1 mapping} Therefore, this allows us to reduce each state $\{s({\bf r})\}$ to a state $\{\sigma({\bf r})\}$ of a {\sl spin-1} model, by mapping $s({\bf r})=+S$ to $\sigma({\bf r})=+1$, $s({\bf r})=-S$ to $\sigma({\bf r})=-1$, and intermediate values $-S<s({\bf r})<+S$ to $\sigma({\bf r})=0$. In this {\sl spin-1} representation of the model, the rules for allowed configurations are exactly the same as for the $S=1$ model; however instead of being equal, the statistical weights have a factor $2S-1$ for each spin with $\sigma({\bf r})=0$. It should be noted that in the $S=1/2$ case, $s({\bf r})=\pm 1/2$ simply maps to $\sigma({\bf r})=\pm 1$. It can also be shown that the expectation of any polynomial in $\{ s({\bf r}) \}$, in the ground-state ensemble of the spin-$S$ model, can be written in terms of a similar expectation in the spin-1 model. Specifically, one must simply replace \begin{equation} s({\bf r})^m \to \cases {S^m \sigma({\bf r}), & m odd \cr S^m [ (1-C_m(S)) \sigma({\bf r})^2 + C_m(S)], & m even \cr} \end {equation} where (e.g.) $C_2(S) = {1\over 3} (1-S^{-1})$. Thus there is no loss of information in this mapping. Indeed, in some sense, the extra freedom to have $s({\bf r})$ vary from $-(S-1)$ to $S-1$ is trivial: once given that $s({\bf r})$ and $s({\bf r}')$ are intermediate spin values, there is no correlation between these values. So we henceforth restrict ourselves to the spin-1 mapped model whose partition function for its ground-state ensemble can be written as: \begin{equation} Z=\sum_{\{\sigma(\bf r)\}} (2S-1)^{n_s} \;\; , \label{eq2} \end{equation} where $n_s$ denotes the number of zero spins in a ground-state configuration ${\{\sigma(\bf r)\}}$. By varying the weight factor continuously in the spin-1 model, it would possible to give a precise meaning to {\it any} real value of $S$, and to simulate such an ensemble. However, in this article we perform Monte Carlo simulations for an ensemble in which $2S$ takes only integer values. The spin-1 representation could be further reduced to a spin-1/2 representation $\tilde \sigma({\bf r})$ as described in Refs.~\onlinecite{Lipowski2,Lipowski,Honda}. They let \begin{equation} \tilde \sigma ({\bf r}) \equiv \sigma({\bf r})+ k({\bf r}) \label{spinhalf} \end{equation} Here $k({\bf r})=0$ if $\sigma({\bf r})=\pm 1$ and if $\sigma({\bf r})=0$, $k({\bf r})=+1$ or $-1$ according to whether the surrounding spins are $(+1,-1,+1,-1,+1,-1)$ or the reverse. Note this mapping is not invertible. The spin-$1/2$ representation is less satisfactory in that is arbitrarily breaks the up-down symmetry of correlation functions, but it was desirable for the transfer-matrix calculations of Lipowski {\it et al}\cite{Lipowski} since it reduced the number of degrees of freedom. \subsection{Height mapping} We define a {\sl microscopic}, discrete-valued height function $z({\bf r})$ living on the vertex of the triangular lattice such that the step in $z({\bf r})$ between adjacent vertices is a function of the adjacent spins: \begin{equation} z({\bf r+e}) - z({\bf r}) = {1\over 2}+{3\over 2} \sigma({\bf r+e})\sigma({\bf r}) \;\; , \label{eq3} \end{equation} where $\sigma({\bf r})$ is the spin-1 operator and $\bf e$ can be any of the three nearest-neighbor vectors ${\bf e}_{1,2,3}$. It is easy to show that the total change in height function, when traversed along any smallest loop, i.e, an elementary triangle, is zero. Therefore, $z({\bf r})$ is well-defined everywhere for the ground-state configurations, but it is not well-defined in any excited state. This prescription generalizes that originally introduced by Bl\"ote et al for the case $S=1/2$\cite{Blote,Ogawa,FN1} (the prescriptions agree in that case). This type of height mapping differs from other sorts of mapping (e.g. dualities) in a crucial way: since the spin microstates of the spin-1 model are mapped essentially one-to-one to the height microstates, it is possible to perform Monte Carlo simulations and construct configurations $z({\bf r})$ after each time step. We have found that analysis of the $z({\bf r})$ correlations is much more efficient for extracting critical exponents than analysis of the spin correlations directly as was done in previous Monte Carlo simulations\cite{Nagai}. \section{Height Representation Theory} \label{height-rep} In this section we propose an effective continuum theory which describes the long-wavelength fluctuations of the interface. We also demonstrate how the critical exponents of various operators are determined by the stiffness constant of the interface. \subsection{Effective free energy} To describe the interface in the rough phase, we must define a smooth height field $h({\bf x})$ by coarse-graining the discrete field $z({\bf r})$. As a first stage, on every triangular plaquette formed by sites ${\bf r}_1, {\bf r}_2, {\bf r}_3$, define a new discrete height \begin{equation} h({\bf R}) \equiv {1\over 3} (z({\bf r}_1)+z({\bf r}_2)+z({\bf r}_3)) \;\; , \label{eq4} \end{equation} where ${\bf R}$ is the center of a triangle. The possible values of the $h({\bf R})$ are $\{ n/2 \}$, for any integer $n$. (For the case $S=1/2$, the only possible values are integers.) To each of these values corresponds a {\it unique} ground-state spin configuration of the spin-1 model on that triangle, i.e., \begin{equation} s({\bf r})=\Phi_s( h({\bf r+u})-h_0({\bf r}) ) \;\; , \label{eq5} \end{equation} where ${\bf u}$ is any vector from a site to the center of any adjoining triangle. The mapping is many-to-one: the function $\Phi_s(h)$ has period 6. Notice that the r.h.s. of Eq.(\ref{eq5}) turns out to be independent of $\bf u$, but the periodic dependence on $h$ is phase-shifted by a function $h_0({\bf r})$ which takes different values on each of the three $\sqrt3\times\sqrt3$ sublattices. Essentially, we have mapped the $T=0$ ensemble of the spin-1 problem into an equivalent interface problem. Note that, given a configuration of $\{ h({\bf R}) \}$, each $\sigma({\bf r})$ is specified (via Eq. (\ref{eq5})), once for each adjoining triangle. The requirement that these six values of $\sigma({\bf r})$ coincide translates to a somewhat complicated set of contraints between pairs $h({\bf R})$ and $h({\bf R'})$ on adjoining triangles; the difference $h({\bf R}) - h({\bf R'})$ may be 0, $\pm 1/2$, or $\pm 1$, but some of these are disallowed (depending on which $h()$ values are integer or half-odd-integer, and on the orientation of ${\bf R}-{\bf R'}$). The weight of each configuration is given, as in (\ref{eq2}), by by $(2S-1)^{n_s}$. Fig.~\ref{fig1} shows the $h({\bf R})$ mapping explicitly where the spins $\sigma({\bf r})$ take values from $\{+1,0,-1\}$. The twelve states are arranged in a circle because the pattern repeats when $h\to h\pm6$. There are certain special ``flat states'' in which $h({\bf R})$ is uniform on all triangles. Each of these is periodic with a $\sqrt3 \times \sqrt3$ unit cell -- in effect it is a repeat of one of the triangles in Fig.~\ref{fig1}. We shall name these states by writing the spins on the three sublattices,``$(+,+, -)$'' and ``$(+,-,0)$''; here ``$\pm$'' stands for $\sigma=\pm 1$. It should be noted that there are two non-equivalent species of flat state corresponding to integer, and half-integer valued $h({\bf R})$ respectively. They are non-equivalent in the sense that they are not related by {\sl lattice} symmetries. One of the species that is favored by the {\sl locking potential} (see Eq. (\ref{eq-Vlock}) below) is what is previously called ``ideal'' states\cite{Kondev2,Kondev3,Raghavan,Burton}. Thus we can imagine that all states can be described as domains of uniform $h({\bf R})$ separated by domain walls. Finally, by coarse-graining $h({\bf R})$ over distances large compared to the lattice constant, one obtains $h({\bf x})$ which enters the conjectured continuum formula for the free energy, which is entropic in origin\cite{Blote}, \begin{equation} F(\{ h({\bf x}) \} = \int d{\bf x} \left\lbrack {K\over2} |\nabla h({\bf x})|^2 + V(h({\bf x})) \right\rbrack \;\; , \label{eq6} \end{equation} where $K$ is the stiffness constant of the fluctuating interface. A lattice shift by one lattice constant leaves the free energy invariant, but induces global shifts in height space $h({\bf x}) \to h({\bf x})\pm 1$; hence the potential $V(\cdot)$ in (\ref{eq6}) must have period one. It is typically approximated as \begin{equation} V(h) \approx h_V \cos (2\pi h). \label{eq-Vlock} \end{equation} Such a periodic potential, usually referred as the {\sl locking term}\cite{Jose}, favors the heights to take their discrete values one of the two types of flat state, depending on the sign of $h_V$. For large $S$ we expect $h_V<0$, favoring the $(+,-,0)$ states, in view of the large entropy of flippable spins; it is not so sure which state is favored at smaller $S$, but this does not matter for the critical exponents (see Sec.~\ref{Scaling} and~\ref{Operators}, below. \subsection{Fluctuations and correlation functions} \label{Fluctuations} In the {\sl rough phase}, by definition, the locking term is irrelevent, and so the long-wavelength fluctations of height variable $h({\bf x})$ are governed by the Gaussian term of Eq. (\ref{eq6}): \begin{equation} F(\{ h({\bf x}) \} = \int d{\bf x} {K\over2} |\nabla h({\bf x})|^2 =\sum_{\bf q} {K\over2} {\bf q}^2 |h({\bf q})|^2 \;\; , \label{eq7} \end{equation} where we have performed the Fourier transform. Hence by equipartition theorem, \begin{equation} S_h({\bf q}) \equiv \langle |h({\bf q})|^2 \rangle = {1\over {K {\bf q}^2}} \;\; . \label{eq8} \end{equation} Similarly, we can also measure the {\sl height-height difference function} in the real space as: \begin{eqnarray} C_h({\bf R}) & \equiv & \frac{1}{2} \langle [h({\bf R})-h({\bf 0})]^2\rangle \nonumber \\ & = & \frac{1}{2\pi K} \ln(\pi R/a) + ... \;\; (R \gg 1) \;\; , \label{eq9} \end{eqnarray} where $a$ is the lattice spacing cutoff. \subsection{Scaling dimensions} \label{Scaling} Using Eq. (\ref{eq9}), we can compute the scaling dimension $x_O$ of any {\sl local} operator $O({\bf r})$, which is defined as in the correlation function, \begin{equation} \langle O^*({\bf r)} O({\bf 0}) \rangle \sim r^{-2x_O} \;\; . \label{eq10} \end{equation} By local operator, we mean that $O({\bf r})$ is a local function of spin operators in the vicinity of $\bf r$. Now, the same spin configuration is recovered when the height variable $h({\bf R})$ is increased by 6.\cite{ferroJ2} Thus any local operator $O({\bf r})$ is also a periodic function in the height space, and can consequently be expanded as a Fourier series: \begin{equation} O({\bf r}) = \sum_{G} O_{G} e^{i G h({\bf r})} \sim e^{i G_O h({\bf r})} \;\; , \label{eq11} \end{equation} where $G$ runs over height-space reciprocal-lattice vectors (i.e. multiples of $2\pi/6$). The last step of simplification in (\ref{eq11}) follows because the scaling dimension $x_O$ of the operator $O({\bf r})$ is determined by the leading relevant operator in the above expansion, i.e., $G_O$ is the smallest $G$ with nonzero coefficient in the sum. Inserting Eq. (\ref{eq11}) into Eq. (\ref{eq10}) and making use of Eq.~({\ref{eq9}), we obtain the following: \begin{eqnarray} \langle O^*({\bf r)} O({\bf 0}) \rangle & = & \langle e^{-i G_O h({\bf r})} e^{i G_O h({\bf 0})} \rangle \nonumber \\ & = & e^{-G_O^2 C_h({\bf r})} \sim r^{-\eta_O} \;\; . \label{eq12} \end{eqnarray} Therefore, the critical exponent $\eta_O$ associated with the operator $O({\bf r})$ is given by: \begin{equation} \eta_O \equiv 2 x_O = { 1 \over {2\pi K}} |G_O|^2 \;\; . \label{eq13} \end{equation} \subsection{Definition of operators} \label{Operators} In this paper, besides the usual spin operator $\sigma({\bf r})$, we also study the bond-energy operator $E({\bf r}+{\bf e}/2)$ for the reason that will become clear in the next section: \begin{equation} E({\bf r}+{\bf e}/2) = {1\over 2}+{3\over 2} \sigma({\bf r+e})\sigma({\bf r}) \;\; , \label{eq14} \end{equation} where ${\bf e}$ denotes one of the three nearest-neighbor vectors as before. As discussed already, the spin operator on a given site has a periodicity of $6$ in the height space, from which a simple inspection shows that the bond-energy operator is also periodic in the height space with a periodicity of $3$. Therefore, the reciprocal lattice vectors of the most relevant operator in the Fourier expansion in Eq. (\ref{eq11}) are \begin{equation} G_{\sigma} = {2\pi\over6} , \;\;\;\; G_{E} = {2\pi\over3} \;\; , \label{eq15} \end{equation} for spin and bond-energy operators respectively. If a magnetic field is implemented by adding a term $-H \sum _{\bf r} \sigma({\bf r}) $ to the Hamiltonian, then our dimensionless uniform ``magnetic field'' is defined by $H' \equiv H/T$. The exponents associated with $H'$ (and with the uniform magnetic susceptibility), are easily related to the correlation exponents of the uniform magnetization operator, \begin{equation} M({\bf R}) = {1\over 3} ( \sigma({\bf r}_1) + \sigma({\bf r}_2) +\sigma({\bf r}_3)) \;\; , \label{eq-M} \end{equation} where ${\bf R}$ is the center of a triangle formed by sites ${\bf r}_1, {\bf r}_2, {\bf r}_3$. A simple inspection of Fig.~\ref{fig1} shows that such an operator has a periodicity of $2$ in the height space, thus yielding: \begin{equation} G_{M} = {2\pi\over2} \;\; . \label{eq-GM} \end{equation} \subsection {Zone-corner singularities} \label{Zone-corner} Observe that the microscopic height variable $z({\bf r})$ in any flat state is not uniform but is rapidly modulated with the wave vector ${\bf Q}={4\pi\over3}(1,0)$. The amplitude of modulation itself is a periodic function of the {\sl coarse-grained} height field $h({\bf x})$ which in turn implies that the correlation function decays with distance as a power-law, and consequently that its structure factor has a power-law singularity at ${\bf Q}$. Such a zone-corner singularity is also directly connected to the singularity in the structure factor of the bond-energy operator. To see this, recall that there is a linear relation between the microscopic height variables and the bond-energy operator given by Eqs. (\ref{eq3}) and (\ref{eq14}), i.e., \begin{equation} E({\bf r}+{{\bf e}\over 2}) = z({\bf r}+{\bf e}) - z({\bf r}) \;\; . \label{eq-bond} \end{equation} Then it is interesting to note that the Fourier transform $E_{\bf e} ({\bf q})$ of bond-energy operator given above turns out to be \begin{eqnarray} E_{\bf e} ({\bf q}) & \equiv & N^{-1/2} \sum_{\bf r} e^{ i{\bf q}\cdot({\bf r}+ {{\bf e}\over2})} E({\bf r}+{{\bf e}\over2}) \nonumber \\ &=& -2i\sin ({1\over 2} {\bf q} \cdot {\bf e}) z({\bf q}) \;\; . \label{eq-Eq} \end{eqnarray} In other words, as a byproduct of measuring $\langle |z({\bf q})|^2\rangle$, we have at the same time measured the structure factor of, say, the bond-energy operator of the same orientation specified by the nearest-neighbor vector ${\bf e}$: \begin{equation} S_E({\bf q}) \sim \langle |E_{\bf e}({\bf q})|^2\rangle = 4 \sin^2 ({1\over 2} {\bf q} \cdot {\bf e}) \langle |z({\bf q})|^2\rangle \;\; . \label{eq-SE} \end{equation} We will utilize this relation in Sec.~\ref{Structure} to extract the exponent of bond-energy operator from the Monte Carlo simulations. \subsection {Exact solution for $S=1/2$} The $S=1/2$ triangular Ising antiferromagnet is exactly soluble, by the same techniques which solve the ferromagnetic two-dimensional Ising model, and was immediately recognized to have critical behavior as $T\to 0$. The spin and energy correlation functions were computed exactly by Stephenson; it transpires that $\eta_\sigma=1/2$ and $\eta_E=2$ exactly, implying through the arguments of Bl\"ote et al (see Sec.~\ref{Scaling} and~\ref{Operators}) that the effective stiffness in Eq.~(\ref{eq7}) is $K= \pi/9$ exactly. The exponents implied by the interface scenario\cite{Blote} -- in particular, the magnetic field exponent $\eta_M$ -- are fully confirmed by numerical transfer-matrix computations.\cite{Blote3} The Coulomb gas picture of Kondev {\em et al}\cite{Kondev4}, wherein the $S=1/2$ triangular Ising antiferromagnet is viewed as a fully-packed loop model\cite{Blote2} with fugacity 1, also predicts the exact exponents. \section {Phase Diagram} \label {PhaseDiag} In this section, we collect some consequences of the height representation for the phase diagram and the nature of the various phases within it.\cite{Chandra2}} \subsection{Kosterlitz-Thouless and locking transitions} \label{Locking} The locking potential $V(\cdot)$ in (\ref{eq6}) favors the flat states. In view of (\ref{eq-Vlock}), its leading reciprocal-lattice vector is $G_V=2\pi$, corresponding to a scaling index $ x_V = |G_V|^2 /{\pi K} = \pi/K $ for the corresponding conjugate field $h_V$. It is well known that if $ 2 - x_V >0$, then $h_V$ becomes relevant (under renormalization and the interface locks into one of the flat states.\cite{Jose} Since $K$ grows monotonically with $S$, such a locking transition occurs at a critical $S_L$ where $K_L =\pi/2=1.57079...$\cite{Blote,Lipowski}. In this ``smooth'' phase, any spin operator $O({\bf r})$ has long-range order, by arguments as in Sec.~\ref{Scaling}. \subsection{Fluctuations in smooth phase} \label{Smooth} One of our aims in this paper was to pinpoint the locking transition $S_L$, which demands that we have a criterion to distinguish these phases. We must supplement Eq.~(\ref{eq8}), which shows the expected qualitative behavior of height fluctuations $\langle |h({\bf q})|^2\rangle$ in the rough phase, with a parallel understanding of the smooth phase. In the smooth state, the symmetry (of height shifts) is broken and a fully equilibrated system has long-range order, such that $\langle h({\bf x}) \rangle$ is well defined and uniform throughout the system. Fluctuations around this height, then, have at most short-range (exponentially decaying) correlations. Thus we expect them to have a spatial ``white noise'' spectrum: \begin{equation} \langle |h({\bf q})|^2 \rangle \sim {\rm const} \label{eq-smooth} \end{equation} for small $\bf q$. A phase with ``hidden'' order was suggested by Lipowski and Horiguchi\cite{Lipowski,Lipowski2}. Numerical transfer-matrix calculations\cite{Lipowski} using the spin-1/2 representation indicated $0 < \eta_{\sigma} <1/9$ for $2S>6$, which is impossible if the spin correlations are derived from height fluctuations,\cite{Blote} as we reviewed in Sec.~\ref{height-rep}. An exotic phase to reconcile these facts was to postulate a phase in which the interface was smooth and $\langle \tilde\sigma({\bf r}) \rangle\neq 0$, yet for the real spins $\langle \sigma({\bf r}) \rangle = 0$ as suggested by spin correlation measurements. What does this imply for our height variable $h({\bf R})$, which has a one-to-one correspondence with the real spin configuration $\{ \sigma({\bf r}) \}$? If the interface is smooth, then the probability distribution of height values on a given plaquette, $P(h({\bf R}))$, is well defined. In order to ``hide'' the order, it is necessary that $P(h)$ correspond to zero expectations of the spins. Now, reversing $s({\bf r})$ on all three sites in the plaquette requires $h \to h\pm 3$, as seen from Fig.~\ref{fig1}. One can convince oneself that, to have ensemble average $\langle \sigma({\bf r})\rangle =0$, the distribution $P(h)$ must be at least as broad as ${1\over 2} \delta (h-h_1) + {1\over 2} \delta (h-h_2)$, with $h_1-h_2=\langle h \rangle \pm 3$, implying the bound \begin{equation} {Var}[h({\bf R})] \equiv \langle h({\bf R})^2 \rangle - \langle h({\bf R}) \rangle ^2 \ge (3/2)^2. \label{eq-dhbound} \end{equation} \subsection{Finite temperature behavior} \label{FiniteT} At $T>0$, plaquettes with non-minimal energy are present and they correspond to vortices in the function $h({\bf x})$. Thus, unfortunately, the height approach of analyzing simulations more or less breaks down. Nevertheless, one can still predict the $T>0$ phase diagram from knowledge of the $T=0$ stiffness constant derived from our simulations. The shape of this phase diagram has already been explained in Ref.~\cite{Lipowski}; here we note some additional interesting behaviors which can be predicted (following Ref.~\onlinecite{Blote}(b)) using the exponents associated with vortices. The other exponents in Kosterlitz-Thouless (KT) theory are associated with elementary defects (often called vortices). Indeed, it is easy to check (in this system) that the excess energy of a non-ground-state plaquette is directly proportional to its vortex charge (a Burgers vector in height-space), so the effect of nonzero temperature is simply to make the vortex fugacity nonzero. The vortex exponent is $\eta_v= 1/\eta_\sigma$, so as usual the vortex fugacity becomes relevant and defects unbind, destroying the critical state, at the KT transition defined by a spin exponent taking the critical value $\eta_{\sigma}=1/4$. If $\eta_{\sigma}>1/4$ at zero temperature, i.e. $K < K_{KT}\equiv 2\pi/9=0.69813...$, then defects unbind as soon as $T>0$. Thus a zero-temperature KT transition occurs at $S_{KT}$ defined by $K=K_{KT}$.\cite{Lipowski} Ref.~\onlinecite{Lipowski} did not, however, address the critical exponents of the correlation length $\xi(T)$ and the specific heat $C(T)$ as a function of temperature, which are also controlled by vortex exponents. Naively, if the energy cost creating one vortex is $E_c$, and if the minimum excitation is a vortex pair, then one would expect the low-temperature specific heat to behave as $C(T) \sim \exp (-2 E_c/T)$ and at $S=1/2$ this is indeed true\cite{Wannier}. However, the renormalization group\cite{Blote} shows the singular specific heat behaves as \begin{equation} f(T) \sim y(T)^{4/(4-\eta_v)} \end{equation} where $y(T) = \exp (-E_c/T)$ is the vortex fugacity; consequently when $\eta_v < 2$, the true behavior is \begin{equation} C(T) \sim \exp (-2 E_1/T) \end{equation} with $E_1 = 2 E_c /(4-\eta_v) < E_c$. (Physically, part of the excitation energy is cancelled by the large entropy due to the many places where the vortex pair could be placed.) This behavior has been observed in the 3-state Potts antiferromagnet on the Kagom\'e lattice\cite{Huse}, and should occur in the present system for all $S>1/2$. \subsection {Finite magnetic field} It is interesting to consider the effect of a nonzero magnetic field $H'$. It is known already that at $S=1/2$,\cite{Blote} such a field is an irrelevant perturbation, so that the system remains in a critical state, yet at sufficiently large $H$ it undergoes a locking into a smooth phase,\cite{Blote3} approximated by any of the three symmetry-equivalent flat states of type ``$(+,+,-)$'' with magnetization $S/3$ As also already noted\cite{Lipowski}, there is a critical value $S_{cH}$ defined by $\eta_\sigma(S_{cH})= 4/9$, beyond which $\eta_M = 9 \eta_\sigma < 4$ so that the system locks into long-range order as soon as $H'$ is turned on. Within this regime, there are still two subregimes with different behavior of $M(h)$ near $h=0$. For $2 < \eta_M < 4$, the initial slope is zero, i.e., the susceptibility is not divergent; when $\eta_M < 2$, as occurs for $S \ge 2$, there is a divergent susceptibility and correspondingly there should be a singularity at ${\bf q}=0$ in the spin structure factor $\langle |\sigma({\bf q})|^2 \rangle$. What do we expect in the locked phase at $S> S_{L}$? Here the difference between the two kinds of flat states becomes crucial. The $H'$ field favors the $(+,+,-)$ type of flat state, but entropy favors the $(+,-,0)$ type of flat state. Thus we expect a transition to the $(+,+,-)$ state only at a nonzero critical field $H'_c$. On reducing $H'$ through $H'_c$, a twofold symmetry breaking occurs, in which one of the $+$ sublattices becomes the $0$ (disordered) sublattice; hence, this transition should be in the Ising universality class. Presumably the line $H'_c(S)$ meets the $H'=0$ axis at $S=S_{L}$. There must also be line of locking transitions $S_{cH}(H')$, which terminates on the $H'=0$ axis at $S_{cH}$. For $S=1/2$, the effect of the magnetic field was confirmed numerically in Ref.~\onlinecite{Blote3}. \section{Monte Carlo Simulations and Results} \label{MC-results} In this section we describe the implementation details of Monte Carlo simulations performed for spin-1 model in which $2S$ takes only integer values from $1$ to $8$. We then present numerical results for the relaxation times of slow modes in the Monte Carlo dynamics. Two different methods of compute the critical exponents of the spin, bond-energy, and uniform-magnetization operators are described in different sub-sections: one in terms of the extrapolated stiffness constants of the interface and the other in terms of the singularities of the corrsponding structure factors. \subsection{Details of Monte Carlo Simulations} A spin is called {\sl flippable} if its six surrounding nearest-neighbor spins alternate between $+1$ and $-1$. Clearly, changing the value of this flippable spin results in another new spin configuration in the ground-state ensemble, provided that we start with a spin configuration in the ensemble. Moreover, such an update maintains the global tilt of the interface due to the {\sl local} nature of this update. This update will be used as our Monte Carlo update in this paper. Two slightly different cases arise for different values of $2S$: (1) for $2S=1$, the local update is precisely equivalent to a spin flip i.e., $\sigma({\bf r}) \rightarrow -\sigma({\bf r})$ due to the absence of zero spin; and (2) for all other values of $2S$, a random choice must be made in the local update: for example, $\sigma({\bf r})=0 \rightarrow \sigma({\bf r})=1$ or $-1$. (Recall $S$ denotes the spin magnitude of the original model.) Let $n_s$ and $n_f$ denote the number of zero-spins and flippable spins of configuration $\phi$. If an attempted single-spin update for $\phi$ results in a new configuration $\phi^{\prime}$ with $n_s^{\prime}$ and $n_f^{\prime}$, then the transition probability $W$ in accordance with the detailed balance principle is: \begin{equation} W= W_0 \cdot min \{ 1, {n_f\over{n_f^{\prime}}} \} \cdot min\{1, (2S-1)^{n_s^{\prime}-{n_s}} \} \;\; , \label{eq16} \end{equation} where $W_0$ denotes the {\sl bare} transition probability: $W_0={1\over n_f}$ for $2S=1$, and $W_0={1\over 2 n_f}$ for $2S \ge 2$ which reflects the random choice to be made in the local update as discussed above. With the transition probability given in Eq. (\ref{eq16}), it is straightforward to show that the detailed balance principle is satisfied, i.e., $P(\phi) W(\phi \rightarrow \phi^{\prime}) =P(\phi^{\prime}) W(\phi^{\prime} \rightarrow \phi)$, where $P(\phi)$ denotes the probability for configuration $\phi$ to occur and $P(\phi) \sim (2S-1)^{n_s}$ since each spin configuration in the original spin-$S$ model has equal probability to occur. Note also that $n_f/n_f' = 1 + O(1/N)$ for large $N$, so this rule is important only because of the finite system size. To implement in practice the transition probability given above, we randomly select a site out of a list of the $n_f$ flippable sites, and randomly update this spin to one of the two possible new spin values if $2S\ge 2$ or simply flip this spin if $2S=1$. The total numbers of zero spins $n_s^{\prime}$ and flippable spins $n_f^{\prime}$ in the resulting configuration are then computed. This update is subsequently accepted with a probability: $min \{ 1, {n_f/{n_f^{\prime}}} \}\cdot min \{ 1, (2S-1)^{n_s^{\prime}-{n_s}} \}$. A practical implementation of the transition probability given in Eq. (\ref{eq16}) is thus achieved. Throughout this paper, a unit time or one Monte Carlo Sweep (MSC) is defined such that there are $N_s$ attempts of updating within this unit of time (or one attempt per spin on average). Here $N_s$ denotes the total number of spins in the simulation cell. The simulation cell always contains $N_s=72 \times 72$ spins in this paper unless explicitly mentioned otherwise. Periodic boundary conditions are adopted. Since we always start with a flat state, the simulations are thus performed in the sector with a zero global tilt of the interface. \subsection{Dynamical scaling: the relaxation time $\tau_{\bf q}$ } \label{Dynamical} We now discuss the correlations between the configurations generated sequentially in the Monte Carlo simulations by studying the relaxation time of the slow modes in the model, namely, the Fourier modes $h_{\bf q}$ which play the role of an order parameter\cite{Henley1}. The linear-response dynamics of such a mode is usually formulated as a Langevin equation, \begin{equation} { {dh({\bf x},t)}\over{dt} } = -\Gamma { {\delta F(\{ h({\bf x}) \}) }\over{\delta h({\bf x})}} +\xi({\bf x}, t) \;\; , \label{eq17} \end{equation} where $\Gamma$ is the dissipation constant, and the static free energy functional $F(\{ h({\bf x}) \})$ is given by Eq. (\ref{eq6}). Here $\xi({\bf x}, t)$ is a stochastic noise generated in the Markov chain of Monte Carlo simulations. As it is expected that the correlation time of the slow mode under consideration is much longer than that of the noise, and since the update steps are local and independent, it is proper to model $\xi({\bf x}, t)$ as Gaussian noise, uncorrelated in space or time: \begin{equation} \langle \xi({\bf x}, t) \xi({\bf x}^{\prime}, t^{\prime}) \rangle =2 \Gamma \delta ({\bf x} - {\bf x}^{\prime}) \delta (t- t^{\prime}) \;\; , \label{eq18} \end{equation} in which the choice of $2\Gamma$ ensures that the steady-state of the interface under the Langevin equation (\ref{eq17}) agrees with its equilibrium state under the free energy (\ref{eq6}). This linear stochastic differential equation can be solved easily by performing Fourier transform. Eq. (\ref{eq17}) thus reduces to \begin{equation} { {dh({\bf q},t)}\over{dt} } = -\Gamma K |{\bf q}|^2 h({\bf q},t) +\xi({\bf q},t) \;\; , \label{eq19} \end{equation} which implies an exponentially decaying correlation function of $\langle h^{*}({\bf q},t) h({\bf q},0) \rangle \sim e^{-t/{\tau_{\bf q}}} $ with the relaxation time $\tau_{\bf q}$ given by \begin{equation} \tau_{\bf q} = {1\over{\Gamma K}} |{\bf q}|^{-2} \;\; . \label{eq20} \end{equation} Therefore, the dynamical scaling exponent for the Monte Carlo dynamics, defined by $\tau_{\bf q} \sim |{\bf q}|^{-z}$, is always $z=2$ in the rough phase. To check this prediction on the dynamical scaling exponent in practice where the above continuum theory is regularized on a lattice, we compute the following auto-correlation function $C({\bf q},t)$ of the {\sl microscopic} height variable $z({\bf q})$: \begin{equation} C({\bf q},t) = { {\langle z^*({\bf q},0) z({\bf q},t) \rangle -|\langle z({\bf q},0)\rangle|^2} \over {\langle z^*({\bf q},0) z({\bf q},0) \rangle -|\langle z({\bf q},0)\rangle|^2} } \;\; , \label{eq21} \end{equation} Here $\langle \rangle$ stands for the dynamical average, and the time $t$ is measured in unit of MCS. For each interger-valued $2S=1,2,...,8$, we perform $10^5$ MCS's with a flat initial configuration and compute the auto-correlation functions upto $t \le 50$ for modes that correspond to the five smallest $|{\bf q}|^2$ values. In Fig.~\ref{fig2}, we display the results so obtained for $2S=1$. Other cases of $2S$ are found to have very similar features. It is clear from Fig.~\ref{fig2} that $\log_{10} C({\bf q},t)$ can be fitted very well by $a - t/\tau_{\bf q}$ where $a$ and the relaxation time $\tau_{\bf q}$ are the fitting parameters. In other words, the relaxation is strictly exponential in all cases. Note that we used a cutoff $t=10$ in our fitting. The same fitting procedure is carried out for other cases of $2S$. The final results of the relaxation time $\tau_{\bf q}$ as a function of $|{\bf q}|^2$ for $2S=1, ..., 6$ are shown in Fig.~\ref{fig3}; and for $2S=6,7,8$ as an insert. The fact that $\tau_{\bf q}$ scales as $|{\bf q}|^2$ for $2S=1, ..., 5$ as indicated by the fitting in Fig.~\ref{fig3} thus shows that the ground-state ensembles for $2S=1, ..., 5$ are in the rough phase. On the other hand, it is indeed clear from the insert that for $2S=7$ and $8$, $\tau_{\bf q}$ curves downward as $|{\bf q}|^2 \rightarrow 0$ which is in sharp constrast to those of $2S=1, ..., 5$. From this, we conclude that ground-state ensembles for $2S=7$ and $8$ are in the flat phase. As for $2S=6$, it is not conclusive from the data available whether $\tau_{\bf q}$ scales as $|{\bf q}|^2$ or curves downward as $|{\bf q}|^2\rightarrow 0$. Nonetheless, the fact that the relaxation time of the slowest mode for $2S=6$ is longer than for any smaller {\it or larger} value of $S$, suggests that $2S=6$ is very close to the locking transition. Further support for this phase diagram is also obtained by explicit calculations of stiffness constants and critical exponents which is discussed in the next section. \subsection {Stiffness constants and critical exponents} \label{Stiffness} As implied by Sec.~\ref{Fluctuations} , the stiffness constant of the fluctuating interface can be directly measured by studying the long-wavelength fluctuations of the height variables, i.e., their structure factor as given by Eq. (\ref{eq8}). It should be noted that concerning the task of calculating the Fourier components $h({\bf q})$ in Eq. (\ref{eq8}), it can be replaced by the approximation in terms of the {\sl microscopic} height variables $z({\bf q})$ given by \begin{equation} h({\bf q}) \approx z({\bf q}) \equiv {w_0\over\sqrt{N_s}} \sum_{\bf r} e^{-i{\bf q}{\bf r}} z({\bf r}) \;\; , \label{eq22} \end{equation} where $\bf r$ labels a lattice site of the finite triangular lattice of total $N_s$ lattice sites used in the simulation. Here $w_0=\sqrt{3}/2$ is the {\sl weight} of a lattice site, i.e., the area of its Voronoi region, which is introduced so that the {\sl microscopic} height variable $z({\bf q})$ coincides with the {\sl coarse-grained} height variable $h({\bf q})$ in the long-wavelength limit (${\bf q} \rightarrow 0$). But unlike $h({\bf q})$, $z({\bf q})$ still contains features such as zone-corner singularities discussed in Sec.~\ref{Zone-corner} that are only manifested in miscroscopic height variables. Starting with a flat state, we perform $2\times 10^3$ MCS's as the equilibrium time; subsequent measurements of physical quantities are carried out at intervals of $20$ MCS's. This separation is a compromise between the correlation times of small $\bf q$ modes and of larger $\bf q$ modes, which are respectively much longer and somewhat shorter than 20 MCS -- see Fig.~\ref{fig2}. Each run consisted of $8 \times 10^5$ MCS, i.e. $4\times 10^4$ measurements were taken; these were subdivided into $20$ independent groups for the purpose of estimating statistical errors. The same procedure is used for all $2S=1,2,...,8$ reported in this paper. In Fig.~\ref{fig4}, we plot $\langle |z({\bf q})|^2\rangle ^{-1}$ vs. ${\bf q}^{2}$ for $2S=1$, including all ${\bf q}$ in the first Brillouin zone. From the plot, we observe that $\langle |z({\bf q})|^2\rangle ^{-1}$ is remarkably isotropic up to about ${\bf q}^{2} \sim 1.5$. This comes about because of the 6-fold rotational symmetry of the triangular lattice which ensures that {\sl anisotropy} occurs only in $q^6$ and higher order terms, assuming that the function is analytic. This is in constrast to other models defined on the square lattice where anisotropy already sets in at the order of $q^4$\cite{Raghavan,Kondev1}. The lower envelope of the data points in Fig.~\ref{fig4} corresponds to the line of $q_y=0$ in the $q$-vector space. Other cases of $2S$ are found to have very similar features as illustated in the insert of Fig.~\ref{fig4} where we plot the lower envelope for all $2S=1,2,...,8$. The structure factor of the height variables appears to diverge in the long-wavelength limit $|{\bf q}|^2 \rightarrow 0$ for all $S$ values, even for the largest $S$ values. (In the latter case, however, we believe one would see the plot asymptote to a constant value, in a sufficiently large system; see below.) Two other interesting features of the structure factor are also revealed in the insert in Fig.~\ref{fig4}: (1) for $2S\ge 2$, it appears to indicate yet another singularity at the zone corner ${\bf q} \rightarrow {\bf Q} \equiv {4\pi\over 3}(1,0)$ in the thermodynamic limit $N_s\rightarrow \infty$; and (2) for $2S=1$, it approaches a constant instead. As already discussed in Sec.~\ref{Zone-corner}, the appearance of zone-corner singularities is expected, the precise nature of such singularities, however, is discussed in the next section. In the remaining of this section, we analyze the zone-center singularity to check if height variables behave as required by Eq. (\ref{eq8}) for the rough phase and consequently extract the stiffness constants. To further study the nature of zone-center singularity in terms of how $\langle |z({\bf q})|^2\rangle$ scales as a function of ${\bf q}^{2}$ in the long-wavelength limit, we show the log-log plot of $\langle |z({\bf q})|^2\rangle^{-1}$ vs. ${\bf q}^{2}$ for $2S=1,...,8$ in Fig.~\ref{fig5}. Comparing the simulation results for different systems sizes of $L=36$, $48$ and and $72$, we notice that data are well converged down to accessible small ${\bf q}$ vectors -- except for the case of $2S=6$ and $7$, where the finite size effect is still discernible. This is, of course, consistent with the fact that $2S=6$ and $7$ are close to the locking transition where the correlation length diverges; it is interesting, however, to notice that their finite-size trends are different. In the case $2S=6$, the data plot for $L=72$ curves upwards less than that for $L=48$, while in the case $2S=7$, the $L=72$ data show {\em more} upwards curvature than the $L=48$ data. By fitting $\langle |z({\bf q})|^2\rangle^{-1}$ to a function $q^{2\alpha}$ with $\alpha$ being the fitting parameter, we obtain, using the data of system size $L=72$ and a cutoff ${\bf q}^2 \le 0.5$, the exponent $\alpha=0.990(1), 0.988(1), 0.986(2), 0.984(2), 0.974(2)$ and $0.935(1)$ respectively for $2S=1, 2, 3, 4, 5$, and $6$. Apart from the case of $2S=6$, these values agree with $\alpha=1$ as in the predicted ${\bf q}^{-2}$ power-law singularity of the structure factor in the rough phase, Eq. (\ref{eq8}). As for $2S=7$ and $8$, $\langle |z({\bf q})|^2\rangle^{-1}$ clearly deviates from a power-law scaling and instead curves upwards to level off, which indicates that models with $2S=7$ and $8$ are in the smooth phases where $\langle |z({\bf q})|^2\rangle$ remains finite as ${\bf q} \to 0$, as discussed in Sec.~\ref{Smooth}. This conclusion is in excellent agreement with that inferred from dynamical scaling analysis presented in Sec.~\ref{Dynamical}. It should be noted that in Fig.~\ref{fig5}, as a general procedure adopted throughout this paper in extracting numerical values of some physical quantities, we have averaged the data corresponding to the same magnitude of $|{\bf q}|^2$ to further reduce the effect due to statistical errors. The relative statistical error on each individual data point $\langle |z({\bf q})|^2\rangle$ of small ${\bf q}$, which is measured directly from the variance among the 20 groups, is found to range from $1\%$ to $3\%$. This is indeed consistent with the estimates of such relative errors from the relaxation times of the slowest modes of models with different values of $2S$ already given in Sec.~\ref{Dynamical}. It is perhaps also worth noting that another good check on the statistical errors on each data point is to compare the values of $\langle |z({\bf q})|^2\rangle$ for three ${\bf q}$ vectors which are related by $120^\circ$ rotations in reciprocal space, which ought to be equal by symmetry. For example, in the case of $2S=1$, the values of $\langle |z({\bf q})|^2\rangle$ for the three ${\bf q}$ vectors of the same smallest magnitude ${\bf q}^2=0.0101539$ of system size $L=72$ are, respectively, $285.551$, $280.528$, and $280.566$, from which one thus also obtain the relative error of about $1\%$. This observation therefore motivates the averaging procedure used in this paper. The stiffness constants can be subsequently determined by fitting ${\bf q}^{-2} \langle |z({\bf q})|^2\rangle^{-1}$ to the function $K + C_1 {\bf q}^2$ for the isotropic part of the data in which the stiffness constant $K$ and $C_1$ are the fitting parameters. The final fitting on the averaged data is shown in Fig.~\ref{fig6} where we used a cutoff ${\bf q}^2 \le 0.5$ in the fitting. We also tried other different cutoffs of ${\bf q}^2\le 0.1$ and ${\bf q}^2 \le 1.0$, and found as expected that the stiffness is not sensitive to the value of cutoff as long as it falls into the isotropic part of the data. For example, we obtain, in the case of $2S=1$, $K=0.3488\pm0.0022, 0.3490\pm0.0008$, and $0.3488\pm0.0006$ for cutoff ${\bf q}^2 \le 0.1, 0.5$, and $1.0$ respectively. Therefore, taking into account of the uncertainty introduced due to the cutoff, our final estimate for the stiffness constant is then $K=0.349\pm0.001$ which is in excellent agreement with the exact value $K_{\mbox{exact}}=0.349065...$. Similar procedure is carried out for other cases of $2S$ and the results are tabulated in Table I. In the same table, we also give the value for the critical exponents of spin, bond-energy and uniform magnetization operators which are obtained straightforwardly according to Eqs. (\ref{eq13}), (\ref{eq15}) and (\ref{eq-GM}). The agreement of our $\eta_\sigma^{(K)}$ values with the ``$\eta_\sigma$'' values from transfer-matrix eigenvalues (see Table~I of Ref.~\onlinecite{Lipowski}, is quite close and becomes better as $S$ grows (until $2S=6$.) As already discussed in Sec.~\ref{FiniteT}, a Kosterlitz-Thouless (KT) transition occurs at a critical value $S_{KT}$ where $\eta_{\sigma}=1/4$, such that for $S>S_{KT}$ algebraic correlations persist even at small finite temperatures. It is clear from our data that $S_{KT}>3/2$. As for $2S=6$, the value of ${\bf q}^{-2} \langle |z({\bf q})|^2\rangle^{-1} =1.75\pm 0.06$ at the smallest nonzero ${\bf q}^2=0.010153$ is already larger than $K_L=\pi/2=1.57079$. That is, even if the system may have a ``rough'' behavior at the length scales probed in the simulation, the stiffness constant is such that the locking potential is relevant and must dominates at sufficiently large length scales, as discussed in Sec.~\ref{Locking}. A similar observation has already been used to argue that the constrained Potts antiferromagnet is in a smooth phase\cite{Burton}. This fact together with the poor fitting using the formula suitable for the rough phase (see the top curve of Fig.~\ref{fig6}) leave us little choice but to conclude that the ground-state ensemble for $2S=6$ also falls into the smooth phase, or possibly is exactly at the locking transition. Just as the finite-size effect for $2S=6$ was severe both for the spin-spin correlations (measured via Monte Carlo\cite{Nagai,Honda}) and also in spin-operator eigenvalues (measured via tranfer-matrix,\cite{Lipowski}) we similarly find it is severe for height fluctuations. However, in view of the exponential relationship between the exponents and the stiffness constant, the latter measurements are much more decisive as to the true phase of the system. To sum up, based on the analysis on the nature of the singularity in the height structure factor at the long-wavelength limit and the numerical results on the stiffness constants, we thus conclude that the model exhibits three phases with a KT phase transition at ${3\over 2}<S_{KT}<2$ and a locking phase transition at ${5\over 2} < S_{L} \le 3$. \subsection{Structure factor and zone-corner singularity} \label{Structure} Another more traditional approach\cite{Nagai} in calculating the critical exponents of various operators is to compute the corresponding structure factors and analyze the power-law singularities at the appropriate ordering wave vectors. Namely, if the correlation function of an operator $O$ decays with distance as power-law (thus critical) \begin{equation} \langle O({\bf r}) O({\bf 0}) \rangle \sim { {e^{i{\bf Q}\cdot{\bf r}}}\over r^{\eta_O} } \;\; , \label{eq23} \end{equation} then its structure factor near the ordering vector ${\bf Q}$ shows a power-law singularity \begin{equation} S_O({\bf q=Q+k}) \sim {\bf k}^{2(x_O-1)} \;\; , \label{eq24} \end{equation} from which the critical exponent $\eta_O \equiv 2x_O$ can be numerically extracted. Here in this section, we adopt this approach to calculate the critical exponents of spin, bond-energy, and uniform-magnetization operators so as to compare with those obtained from the stiffness constant. As given by Eq. (\ref{eq-SE}), $S_E({\bf q=Q+k}) \sim \langle |z({\bf q=Q+k})|^2\rangle$. Here ${\bf Q}={4\pi\over3}(1,0)$ is the ordering vector of the bond-energy operator. Therefore the interesting feature of structure factor of height variables, namely, the appearance of zone-corner singularity as shown in Fig.~\ref{fig4}, is not only expected but also very useful in extracting the critical exponent $\eta_E$. Of course, such a zone-corner singularity can also be understood within the framework of interfacial representation, as in Sec.~\ref{height-rep}, particularly Subsec.~\ref{Zone-corner}. (Similar zone-corner singularities have been studied in Refs.~\onlinecite{Kondev2} and \onlinecite{Raghavan}.) Finally, according to the exact result $\eta_E=2$ ($x_E=1$) in the case of $2S=1$, i.e., $S_E({\bf q=Q+k}) \sim {\bf k}^{2(x_E-1)} \rightarrow const.$, the puzzling absence of the zone-corner singularity for $2S=1$ as shown in Fig.~\ref{fig4} is also resolved. In Fig.~\ref{fig7}, we plot $\log_{10}S_E({\bf q})$ vs. $\log_{10}|{\bf q-Q}|^2$ where we have averaged data points with the same magnitude of $|{\bf q-Q}|^2$. Fitting $S_E({\bf q})$ to the function $|{\bf q-Q}|^{2(x_E-1)}(C_1+C_2 |{\bf q-Q}|)$ where $x_E, C_1$ and $C_2$ are the fitting parameters, we obtain the critical exponents $\eta^{(S)}_E$ which are tabulated in Table I. In practice, we used two different cutoffs in the fitting: $|{\bf q-Q}|^2 \le 0.1$ and $\le 0.5$. The fitting for the latter is shown in Fig.~\ref{fig7}, and the final quoted errors take into account the uncertainty due to the cutoffs. Similarly, we also computed the structure factor for the spin operator $S_{\sigma}(\bf q)$ using Fast Fourier transform while computing the height-height correlation function within the same Monte Carlo simulations. Results are shown in Fig.~\ref{fig8} and the extracted exponents are also tabulated in Table I. Fitting precedure used is exactly the same as that for bond-energy except that we fit $S_{\sigma}({\bf q})$ to the function $C_1 |{\bf q-Q}|^{2(x_{\sigma}-1)}$ with $C_1$ and $x_{\sigma}$ being the fitting parameters. From Table I, we note that the critical exponents extracted in this way are in good agreement with those obtained from stiffness constant utilizing the interfacial representation, however, the latter yields much better statistical errors by an order of magnitude using the same Monte Carlo simulation data. This clearly demonstates the superiority of the interfacial representation in extracting critical exponents from numerical data. Similar points were made regarding other models, but based on much less extensive simulation data, in Refs.~\onlinecite{Kondev2} and \onlinecite{Raghavan}. Similar fits were attempted for $2S=6$, yielding $\eta^{(S)}_E (2S=6) = 0.53\pm0.41$ and $\eta^{(S)}_\sigma (2S=6)= 0.236\pm0.036$. While the statistical error on $\eta^{(S)}_E (2S=6)$ is too large to render the fitting meaningful, the increase in the value of $\eta^{(S)}_\sigma (2S=6)$ when compared with $\eta^{(S)}_\sigma (2S=5)$ is added evidence that $2S=6$ is {\sl not} in the rough phase; if it were still rough at that value of $S$, we would have expected a continuation of the decreasing trend of $\eta^{(S)}_\sigma$ with $S$. As for the cases of $2S=7$ and $8$, the structure factors of both the spin and bond-energy operators show {\it weaker} than power-law behavior as ${\bf q} \to {\bf Q}$, as in Figs.~\ref{fig7} and~\ref{fig8}, but they increase to a larger value (not seen in these logarithmic plots) right {\it at} $\bf Q$. This is indeed consistent with the $\delta$-function singularity. expected if these cases fall into the smooth phase with long-ranged order of the spin and bond-energy operators. Finally, we consider the uniform magnetization correlation exponent $\eta_M$. When $S>3/2$, it can be predicted (see $\eta^{(K)}_M$ in Table I) that $\eta_M< 2$, implying a divergent (ferromagnetic) susceptibility and a divergent structure factor $S_M({\bf q})$ as ${\bf q}\to 0$ Now, due to the linear relation (\ref{eq-M}) between $\{ M({\bf R}) \}$ and $\{ \sigma({\bf r}) \}$, we immediately obtain $S_M({\bf q}) \sim S_\sigma({\bf q})$ near ${\bf q}=0$, just as $S_E({\bf q}) \sim \langle |z({\bf q})|^2\rangle$ near ${\bf q}={\bf Q}$ (see Sec.~\ref{Zone-corner} and Eq.~(\ref{eq-SE})) Thus, a singularity at ${\bf q}=0$ is expected in the structure factor of spin operator which is plotted in Fig.~\ref{fig9}. From this figure, it appears that only for $2S=4$, $5$, and $6$ does $S_M({\bf q})$ show a power-law singularity indicated by a straight line in this log-log plot. This confirms the prediction based on the stiffness constant; however, the numerical values of $\eta_M$ extracted this way (see Table I) differ considerably from those calculated from the stiffness constant in the case of $2S=5$ and $6$. It is also apparent from Table I that $\eta_\sigma^{(S)}$ is systematically overestimated as compared with the more accurate value derived from height fluctuations. We suspect that a similar overestimation affected the values of $\eta_\sigma$ that were deduced from the finite-size scaling of the susceptibility of the staggered magnetization\cite{Nagai,Honda} (this obviously measures the same fluctuations seen in $S_\sigma({\bf q})$ near $\bf Q$.) Those data (also quoted in Ref.~\onlinecite{Lipowski}) have quoted errors about four times as large as ours for $\eta_\sigma^{(K)}$. Their exponent values are all noticeably larger than the accurate value ($\eta_\sigma^{(K)}$ or $\eta_\infty$ from Ref.~\onlinecite{Lipowski}) -- becoming {\it worse} as $S$ grows (for $2S=4,5$ the difference is twice their their quoted error.) Clearly the systematic contribution to their errors was underestimated. The transfer-matrix method\cite{Lipowski} ought to provide the effective exponent $\eta_\sigma$ for spin correlations on length scales comparable to the strip width, and hence is likewise expected to overestimate $\eta_\sigma$; indeed, every $\eta_\sigma$ value found in Ref.~\onlinecite{Lipowski} is slightly larger than our corresponding $\eta_\sigma^{(K)}$ value. \subsection{Smooth Phase} \label{MC-Smooth} Which type of flat state is actually selected in the smooth phase? Fig.~\ref{fig10} shows the measured expectation of $n_s$, the number of zero spins in the spin-1 representation, for $1 \leq 2 S \leq 8$. As $S$ grows, it is found that $\langle n_s\rangle$ approaches its maximum allowed value $N_s/3$ as in the $(+,-,0)$ state, rather than zero, as in the $(+,+,-)$ state. Thus, the flat states with half-integer valued $h({\bf R})$ in Fig.~\ref{fig1} are being selected in the smooth phase. Translating back to the spin-$S$ model, this means that spins on two sublattices of the triangular lattice take the extremal values, $+S$ and $-S$ respectively, while spins on the third sublattice remain disordered. It is perhaps more illuminating to study the distribution of height variables to probe the height fluctuations in the smooth phase. To this end, we also show, in Fig.~\ref{fig10}, the histogram of height variable $h({\bf R})$ in the cases of $2S=2$ and $2S=8$, which is measured for a {\sl typical} configuration generated in the Monte Carlo simulations.\cite {FN-fig10}. The broad distribution observed in the case of $2S=2$ ($S<S_L$) evolves to a narrowly peaked distribution in the case of $2S=8$ ($S>S_L$). (It decays as $\exp(-{\rm const}|h-\langle h \rangle|)$.) This supports the intuitive picture presented in Sec.~\ref{Smooth}. Furthermore, the center of this peaked distribution is half-integer valued. (Numerically, the mean is $\langle h\rangle =0.46$ for the distribution plotted in Fig.~\ref{fig10}.) In other words, the locking potential $V(h)$ favors the $(+,0,-)$ type of flat state, in which one sublattice is flippable, rather than the $(+,+,-)$ type of flat state. (See Fig.~\ref{fig1}). This kind of flat state was also expected analytically in the limit of large $S$ \cite{Horiguchi2,Horiguchi}. We have also computed ${\rm Var} (h)$ for each value of $S$, in two ways. First, ${\rm Var}(z)$ is just normalization factors times $\sum _{\bf q \neq 0} \langle |z({\bf q})|^2 \rangle$, which we accumulated throughout the Monte Carlo run, as described earlier in this section; then it can be shown that $Var(h) = Var(z) -{1\over 3} + {1 \over 2} \langle n_s \rangle$ exactly. For $N_s=72$ this gives ${\rm Var}(h) = 1.06$ and $0.20$ for $2S=2$ and $2S=8$, respectively, showing the contrast of the rough and smooth behavior. Secondly, we can compute ${\rm Var}(h)$ directly from the histogram (from one snapshot) seen in Fig.~\ref{fig10}; this gives respective values $1.1$ and $0.15$, in satisfactory agreement with the first method. The exotic ``hidden order'' phase\cite{Lipowski,Lipowski2} (see Sec.~\ref{Smooth}) can be ruled out on the basis of these data: according to Eq.~(\ref{eq-dhbound}) the variance of $h({\bf R})$ should be at least $(3/2)^2=2.25$ in the hidden-order phase, while our measurements indicate it is at most only $0.20$. Furthermore, for $2S=7$ and $8$, the structure factor $S_\sigma({\bf Q})$ at the zone-corner wave vector $\bf Q$ (not plotted) was much larger than at nearby $\bf q$; that direct suggests a $\delta$-function singularity in the thermodynamic limit, i.e., existence of long-ranged spin order in which $\langle s({\bf r})\rangle \ne 0$ on at least two of the sublattices. Additionally, the spin structure factor $S_\sigma({\bf q})$ near the zone-corner wave vector $\bf Q$ (Fig.~\ref{fig8}) showed a striking curvature in the ``smooth'' cases $2S=7$ and $8$, quite different from the behavior at smaller $S$. This makes it plausible that $S_\sigma({\bf q}) \to {\rm constant}$, so that spin fluctuations have short-range rather than power-law correlations for $S>S_L$. (It was not emphasized in Ref.~\onlinecite{Lipowski}, but power-law correlations are implied if one takes seriously their measured values $0< \eta_\sigma < 1/9$ for $2S=7,8$.) We propose, then, that actually $\eta_{\sigma} = \eta_{E} = \eta_{M} =0$ for $S> S_{c2}$, as in the simplest picture of the smooth phase, and that the observed nonzero values are simply finite-size effects due to the very slow crossover from rough to smooth behavior near a roughening transition (see Sec.~\ref{Disc-crossover}, below, for further discussion.) \section{Conclusion and Discussion} \label{Conc-Disc} To conclude, in this article, we have investigated the ground-state properties of the antiferromagnetic Ising model of general spin on the triangular lattice by performing Monte Carlo simulations. Utilizing the interfacial representation, we extrapolated the stiffness constants by studying the long-wavelength singularity in the height variables, which in turn lead to straightforward calculation of critical exponents of various operators within the framework of height representation. The results so obtained are further compared with those extracted from a more tranditional method, and demonstrate that the method in terms of height representation method is by far the preferable one for extracting the critical exponents. We also analyzed both the dynamical and static properties of the model in order to map out the phase diagram which consists of three phases with a Kosterlitz-Thouless phase transition at ${3\over 2}<S_{KT}<2$ and a locking phase transition at ${5\over 2} < S_{L} \le 3$. Even in the smooth state, analysis of the height fluctuations (as in ${\rm Var}(h)$ was helpful in resolving questions which are made difficult by the strong finite-size effects near the locking transition. \subsection{Rational exponents?} One of our initial motivations for this study was the possibility of finding rational exponents even for $S>1/2$. We believe the results in Table~I are the first which are accurate enough to rule out this possibility. Indeed, $\eta_\sigma(2S=4)\approx 3/16$ and $\eta_\sigma(2S=5)\approx4/27$, with differences similar to the error (0.001). But {\it any} random number differs from a rational number with denominator $<30$ by the same typical error. The exception is that $\eta_\sigma^{(K)}(2S=6)$ is quite close to $1/9$, but we have given other reasons to be suspicious of this value. \subsection {What is $S_{L}$?} \label{Disc-crossover} Another intriguing question was whether the critical values $2S_{KT}$ and $2 S_{L}$ are exactly integers. Previous data\cite{Lipowski} suggested that $S_L\equiv 3$ exactly, and had large enough errors that $S_{KT}=3/2$ could not be excluded. Since $\eta_\sigma(S_{KT})\equiv 1/4$ and $\eta_\sigma(S_L)\equiv 1/9$, this question was answered by the preceding subsection: we find that definitely $S_{KT}<3/2$. Furthermore, we suspect $S_{L} < 3$ as concluded in Sec.~\ref{Stiffness} since the effective stiffness at the length scale we access is more than enough to drive the system to the locked phase. The question of the value of $S_L$ suggests paying closer attention to the behavior of systems near the locking transition. It has been noted previously how the locked phase tends to behave qualitatively like the rough phase in a finite-size system, since the crossover is a very slow function of size.\cite{Blote3} This is consistent with the apparent power-law behaviors observed at $S>S_{L}$ in previous studies\cite{Nagai,Lipowski} and with the tendency of those studies to overestimate the exponents $\eta_{\sigma}$ and $\eta_{E}$ (as compared with our more accurate estimates.) This would suggest that, if extensive finite-size corrections were included in our analysis, they would reduce our estimate of $S_{L}$ a bit further, i.e. we would more definitely conclude that $2S=6$ is in the locked phase. Our analysis near the locking transition at $S_{L}$ suffers from our ignorance of the expected functional form of the critical behavior as a function of $S-S_{L}$. A study of the roughening transition\cite{Evertz} used the Kosterlitz-Thouless (KT) renormalization group to derive analytic approximations for the total height fluctuation (closely analogous to ${\rm Var}(h)$ in our problem), which made it possible to overcome very strong finite-size effects and fit the roughening temperature precisely. Use of KT finite-size corrections was also essential in extracting meaningful numbers from transfer-matrix calculations near the locking transition induced by a magnetic field in Ref.~\onlinecite{Blote3}. Thus, a similar adaptation of the KT renormalization group to give expressions for the behavior of $\langle | z({\bf q} |^2 \rangle $, as a function of (small) $|{\bf q}|$ and $S-S_{L}$, or the functional form of $K(S)$ near $S_{L}$, could make possible a more conclusive answer as to whether $S_{L}=3$ exactly. \subsection{Possible improved algorithms} Since the long-wavelength behavior in this model (in its rough phase) is purely Gaussian with $z=2$ (see Sec.~\ref{Dynamical}), the critical slowing down is particularly transparent. It seems feasible to take advantage of the existence of a height representation to develop an acceleration algorithm. For example, it might be possible to extend the cluster algorithms which are known for the $S=1/2$ triangular Ising antiferromagnet.\cite{cluster} These are well-defined at $T>0$, but their effectiveness seems to depend in a hidden fashion on the existence of the height representation when $T\to 0$. An intriguing alternative approach starts from the observation that at long wavelengths the system obeys Langevin dynamics (see Sec.~\ref{Dynamical} and Ref.~\onlinecite{Henley1}). Fourier acceleration\cite{Batrouni}, a nonlocal algorithm (efficient owing to use of the Fast Fourier Transform algorithm), is known to be effective in such cases. The key is to replace the uncorrelated noise function $\xi({\bf x},t)$ of Eq.~(\ref{eq18}) with a new correlated noise function having $\langle |\xi ({\bf q},t)|^2\rangle \sim 1/|{\bf q}|^2$. This might be implemented by first constructing a random function with such correlations, and then updating flippable spins with probabilities determined by that function, in such a fashion as to satisfy detailed balance. Additionally, it may be possible to analyze transfer-matrices using the height representation. Quite possibly this would yield an order-of-magnitude improvement in the accuracy of the numerical results, for the same size system, similar to the improvement in analysis of Monte Carlo data. The transfer matrix breaks up into sectors corresponding to the step made by $z({\bf r})$ upon following a loop transverse to the strip (across the periodic boundary conditions. Then the stiffness can be extracted directly from the ratio of the dominant eigenvalues of two such sectors; such an analysis is already standard for quasicrystal random tilings, for which the long-wavelength degree of freedom is also an effective interface \cite{quasicrystal}. \acknowledgements C.Z. gratefully acknowledges the support from NSF grant DMR-9419257 at Syracuse University. C.L.H. was supported by NSF grant DMR-9214943. \newpage \widetext \begin{table} \caption{Stiffness constant and critical exponents. Here $\eta_{\sigma}^{(K)}$, $\eta_{E}^{(K)}$ and $\eta_{M}^{(K)}$ are the estimates for the critical exponents of spin and bond-energy operators calculated from the stiffness constant $K$ as done in Sec. V(C), while $\eta_{\sigma}^{(S)}$, $\eta_{E}^{(S)}$, and $\eta_{M}^{(S)}$ stand for the same critical exponents, but extracted from the singularities of their respective structure factors in Sec. V(D). Estimated errors are also given in the parenthesis.} \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $2S$ & $K$ & $\eta_{\sigma}^{(K)}$ & $\eta_{E}^{(K)}$ & $\eta_{M}^{(K)}$ & $\eta_{\sigma}^{(S)}$ & $\eta_{E}^{(S)}$ & $\eta_{M}^{(S)}$ \\ \hline\hline 1 & 0.349(0.001) & 0.500(0.002) & 2.001(0.008) & 4.502(0.018) & 0.511(0.013) & 1.844(0.057) & \\ \hline 2 & 0.554(0.003) & 0.315(0.001) & 1.260(0.006) & 2.836(0.013) & 0.332(0.016) & 1.340(0.072) & \\ \hline 3 & 0.743(0.004) & 0.235(0.001) & 0.940(0.005) & 2.114(0.011) & 0.254(0.019) & 1.047(0.082) & \\ \hline 4 & 0.941(0.006) & 0.186(0.001) & 0.742(0.004) & 1.670(0.010) & 0.203(0.022) & 0.791(0.092) & 1.634(0.014)\\ \hline 5 & 1.188(0.008) & 0.147(0.001) & 0.588(0.004) & 1.322(0.009) & 0.180(0.026) & 0.504(0.115) & 1.560(0.015)\\ \hline 6 & 1.597(0.015) & 0.109(0.001) & 0.437(0.004) & 0.984(0.009) & 0.236(0.036) & 0.530(0.410) & 1.527(0.016)\\ \hline\hline \end{tabular} \end{table} \twocolumn

proofpile-arXiv_065-569

\section{Introduction} The very accurate measurement of the neutron spin-dependent structure function $g_1^n$ \cite{Hughes}, whose results have been presented recently, possess two remarkable properties. First, the values of neutron structure function at $Q^2=5GeV^2$ are rather large and negative in the region of moderately low $x$. Second, the data can be rather accurately fitted by the power function $x^{-0.8}$ and this power seems to be unexpectedly large. It is not obvious, where this large number, for which there is no indication in the proton data, is coming from, so it may affect the extrapolation to $x=0$ and cast some doubts on the validity of Bjorken sum rule. In the present paper we perform the isospin decomposition of the data for both proton and neutron. As a result, we conclude that the isovector contribution is well approximated by the power behaviour found earlier by an elaborate method based on NLO evolution \cite{BFR}. It may be interpreted either as the manifestation of $ln^2x$ terms \cite{rysns} or as the contribution of the Regge cut, produced by the $a_1$ meson and a BFKL pomeron which has a high intercept. At the same time, for the isoscalar part one has the signature for a rather singular singlet contribution, compatible with the similar behaviour predicted by QCD \cite{Ryskin}, although statistical errors are still rather large. This may imply a significant gluon polarization in the nucleon in this range of $x$, which could also clearly show up in double helicity asymmetries $A_{LL}$ at RHIC, if they turn out to be larger than estimated earlier\cite{RHIC}. \section{Bjorken sum rule validity} The main feature of the new neutron data \cite{Hughes} is large and negative $g_1^n \sim -g_1^p$, measured with a good accuracy up to small $x$, say $x \sim 0.01$. The SLAC proton data \cite{p} at $Q^2=3 GeV^2$ are positive and of roughly the same magnitude, but on the contrary, rather flat in this region. This could be understood, qualitatively, as a result of the interplay of a negative contribution at low $x$, responsible for the singular behaviour of the neutron, and a positive contribution at larger $x$. To check this assumption it is instructive to consider the isovector contribution. Since there is no clear evidence of scaling violations between $Q^2=3 GeV^2$ and $Q^2=5 GeV^2$ in any polarized deep inelastic scattering experiment, we may neglect, for the time being, the effects of QCD evolution, because we are only interested in the results, provided by the data at the present level of statistical accuracy. It is then possible, by combining the SLAC neutron data with the SLAC proton data, to determine the quantity $g_1^{p-n} \equiv g_1^p-g_1^n$, entering the Bjorken sum rule, with a higher accuracy, than for proton alone. This is because the neutron data are fortunately negative, so the difference is larger in magnitude than $g_1^p$, while the errors are practically the same for $g_1^{p-n}$ and $g_1^p$, given the better accuracy of the neutron data. Performing this analysis of the data, we see immediately a behaviour less sharp than that of neutron. Since phase space effects, as powers of $(1-x)$, are not so important in the region under consideration, we were looking for a simple power parametrization of the data, implied by Regge pole behaviour, namely \begin{equation} g_1^{p-n}(x)= g_1^{p-n}(x_0)\left({x\over {x_0}}\right)^{-a}, \end{equation} and we found that this works rather well, for $0.016 \leq x \leq 0.125$ with the following choice of parameters: \begin{equation} g_1^{p-n}(x)=0.147 x^{-0.45}, \end{equation} as shown in Fig.1. The power we obtain is significantly smaller than the expected contribution of the $a_1(1260)$ meson trajectory $(\sim 0.14)$. As a result, the contribution to the Bjorken integral from the region $0 \leq x \leq 0.125$ is large \begin{equation} \int_{0}^{0.125}dx g_1^{p-n}(x)=0.085. \end{equation} The region of higher $x$ corresponds to a neutron contribution much smaller than that of the proton, the latter also providing a large contribution to the integral, and we have \begin{equation} \int_{0.125}^{1}dx g_1^{p-n}(x)\approx \int_{0.125}^{1}dx g_1^{p}(x)=0.09. \end{equation} The total contribution to the Bjorken sum rule, \begin{equation} \int_0^{1}dx g_1^{p-n}(x)=0.175 \end{equation} appears to be in the fair agreement with the theoretical value. Note that when the final neutron data will be available, as well as more precise proton data, it will allow one a more serious analysis, taking also into account the effects of QCD evolution. At the present moment we would like to stress, that by combining the current neutron and proton data, one is led to good agreement with Bjorken sum rule. Let us stress that this power is compatible with the one obtained using next-to-leading order (NLO) fit \cite{BFR} to all previous data $(-0.56 \pm 0.21)$. It is consistent with the result of $ln^2x$ summation \cite{rysns}. We may also understand the observed power $\alpha=-0.45$ by considering the intercept of the Regge cut associated to the $a_1$ meson and a Pomeron, namely \begin{equation} \alpha=1-\alpha_P-\alpha_{a_1} \end{equation} provided we use the famous BFKL Pomeron \cite{BFKL} with the intercept $\alpha_P \sim 1.6$ and $\alpha_{a_1}=-0.14$. \section{Isoscalar channel and the singlet contribution}. Since we found, that the sharp neutron structure function is not seen in the difference between proton and neutron, it should be attributed to the isoscalar channel. Also, the partial cancellation, we suspect to be at the origin of a flat proton structure function, should be manifested in this channel as well. To check this, we calculated the quantity $g_1^{p+n} \equiv g_1^p+g_1^n$. It really shows a rather flat structure for $x \geq 0.035$. The relative errors are much larger in this case because $g_1^{p+n}$ is small due to the fact that $g_1^p$ and $g_1^n$ have opposite signs. Of course this fact also implies a small value of the deuteron structure function in this kinematic region, which is barely consistent with the existing data \cite{D}. This flat structure should be related to the interplay of the negative sharp contribution, showing itself in the fit $x^{-0.8}$ to the neutron data \cite{Hughes}, and a positive contribution with a smaller power, dominating at larger $x$. Since there is no counterpart for such a sharp behaviour in the conventional Regge analysis, we make a strong, but natural assumption. Namely, we suggest that it is manifested in the $SU(3)$-singlet channel. It is in this channel that a strong mixing between polarized quarks and gluons provides the anomalous gluon contribution to the first moment of $g_1$ \cite{EST}. Recent studies show that generalized anomalous gluon contribution appears also in all moments\cite{Muller,ST95}. At low $x$ the quark-gluon mixing provides a strong correction to the subleading behaviour \cite{Ryskin}, producing a power close to 1. One might expect that a similar effect is also present in the non-perturbative region where it can give rise to a strong $x$ dependence in gluon distribution at low $Q^2$, which is an initial condition in the approach just mentioned above. An example of such a non-perturbative contribution is given by instanton effects\cite{Dorohov} which,however, do not provide yet a reliable quantitative estimate. Moreover, the $SU(3)$- nonsinglet part receives the contributions from the $f_1(1285)$- and $\eta(547)$-mesons trajectories. The first one has an intercept close to that of $a_1$ occurring in the isovector channel, while the second one produces a smoother behaviour like $x^{0.3}$. For the first estimate we neglect the latter and find that the data are well described by the formula: \begin{equation} g_1^{p+n}(x)=0.145 x^{-0.45}-0.03 x^{-0.87}, \end{equation} as seen in Fig.2. This formula is suggesting that the isoscalar contribution is approximately equal to the isovector one, which is not so surprising, in order to have the neutron structure function, dominated by the most singular power only. This would mean, that in this region of $x$, say between 0.01 and 0.1, one has \begin{equation} \Delta u(x)-\Delta d(x) \sim \Delta d(x) - \Delta s(x), \end{equation} requiring a strong negative $s$-quark polarization. Apart from the difficulties of incorporating this result to current models of nucleon structure, it could also conflict with the Bjorken sum rule for the decay of strange baryons, implying that the integrals of both sides of this equation should be of opposite signs. Although the suggested above equality may be valid only in a limited region of $x$, and violations of $SU(3)$ symmetry may be possible, it is more likely, that the $\eta$ contribution makes the $x$ dependence of isovector and isoscalar combinations, different in the region under consideration, so we will have \begin{equation} g_1^{p+n}(x)=C_{ns}x^{-0.45}+C_\eta x^{0.3}-C_s x^{-a_s}. \end{equation} The numerical analysis shows, that the present inaccurate data, mainly for $x \leq 0.035$ are equally well described with a wide range of coefficients $C_f$ and $C_\eta$. Bearing in mind the problem with the second Bjorken sum rule mentioned above, one may suspect a negative value for $C_\eta$ in order to reduce the isoscalar integral at larger values of $x$. On the other hand, the parameters of the singlet contribution are rather stable, $C_s \sim 0.03, a_s \sim 1$, which is again related to the good accuracy of neutron data. Note that, if this neutron behaviour is really an isoscalar phenomenon, as suggested by our analysis, one should observe a decrease and sign change for the proton structure function not too far from $x \sim 0.005$ at low $Q^2$. This is the first major check of this result. However, negative proton structure function for much smaller values of $x$ and low $Q^2$ have been considered in the literature and a sign change may also come at large $Q^2$ from the effect of QCD evolution \cite{Blumlein}. Note that the obtained power is also compatible with the NLO fit \cite{BFR} result, but the accuracy of the neutron data would allow to reduce the error. It is, of course, too early to relate unambiguously the observed behaviour to the results of \cite{Ryskin}, because of the limited experimental accuracy and some theoretical problems. In particular, it is not absolutely clear, to what values of $x$ and $Q^2$ the results of \cite{Ryskin} should be applicable. Nethertheless, the relative closeness of the experimental and theoretical numbers may be a signal, that the low $x$ asymptotic behaviour is manifested rather early, especially in the neutron case, where it is not screened by a large nonsinglet contribution, like in the proton case. Note that clear evidence for negative $g_1^n$ no longer requires the negative gluon polarization, as was guessed in \cite{Ryskin}, relying on earlier data. Moreover, we found that the formula (3.24) of \cite {Ryskin} is not incompatible with the data, if some mean values (like suggested in \cite{Ryskin}) of parton distributions are taken as an input. However, this approach does not allow one to extract the $x$-dependent parton distribution, and we shall use now the continuity with the region of average $x$ in order to get an estimate. It is not clear, to which extent one should attribute a small $x$ singlet contribution to quarks or to gluons. However, according to \cite{Ryskin} the contribution of gluons is dominant, so we neglect the quark contribution in the present approach. Requiring the qualitative continuity in transition between low and average $x$, and applying the parton-like formula for the anomalous gluon contribution it seems natural to expect that the gluon distribution is behaving like \begin{equation} \Delta G(x) \sim x^{-0.87}. \end{equation} Note, that this formula is assuming the simple relation between singlet contribution to $g_1$ and $\Delta G$ at average $x$ \begin{equation} g_1^s(x)=-{\alpha_s\over {6 \pi}}\Delta G(x), \end{equation} which is, strictly speaking, is valid for the first moment only. More generally, one has \cite{Karl} \begin{equation} g_1^s(x)=-{\alpha_s\over {6 \pi}}\int_x^1 {\Delta G(z)\over z} E({x\over z})dz, \end{equation} where $E(y)$ is the coefficient function, describing the gluon-photon interaction. Recent studies, based on the non-local generalization of the axial anomaly\cite{Muller}, support the following choice \cite{ST95} \begin{equation} E(z)=2(1-z), \end{equation} leading, for the gluon distribution of the type $const \times x^{-a}$, to the relation \begin{equation} g_1^s(x)=-{2\over {a(a+1)}}{\alpha_s\over {6 \pi}}\Delta G(x). \end{equation} As a result, for $a=0.87$, $\Delta G(x)$ should be multiplied, for a given $g_1^s(x)$, by a factor $\sim 0.8$ and the integral of $\Delta G(x)$ in the range $0.01 \leq x \leq 0.1$ is about 1. The double helicity asymmetry $A_{LL}$ in prompt photon production in $pp$ collisions is directly related to $\Delta G(x)$ and for the center of mass energy $\sqrt{s}=500 GeV$, which will be reached at RHIC, one is probing precisely this kinematic region of $x$. So given such a strong gluon polarization, we anticipate a larger $A_{LL}$ than previously predicted \cite{RHIC}. Similar comments can be made for $A_{LL}$ in inclusive jet production. \section{Discussion and Conclusions} We have performed here an analysis based on the present level of the experimental accuracy, still not enough to see the effects of QCD evolution, but providing interesting information about isospin structure. For this reason we restrict ourself to the SLAC data, and neglect the effects of QCD evolution in our analysis. Note that the SMC data \cite{SMC} are unfortunately not accurate enough to be used for such a simple isospin decomposition. The results on $g_1^d$ are certainly not incompatible with our $g_1^{p+n}$ but in order to obtain $g_1^{p-n}$, it is necessary to extract $g_1^n$ from $g_1^d$ after subtracting $g_1^p$, which enhances substantially the statistical errors. However, the elaborate statistical analysis using NLO evolution lead to the very similar powers, although the error for the singlet case is still very large. The presented simple picture of the nucleon structure is based on the two observations. i) As suggested by the E154 Collaboration, the $g_1^n$ behaviour is well described by $\sim x^{-0.8}$. ii) From our simultaneous analysis of proton and neutron data, there is no indication of such a behaviour in $g_1^{p-n}$. Instead, it is well described by the $\sim x^{-0.45}$, which leads to a good saturation of Bjorken sum rule. Consequently, the existence of a strong negative isoscalar contribution is implied by these two facts. It seems rather well established, and leads to predict a negative $g_1^p$ for $x$ below $0.005$. Both the interpretation of the nonsinglet behaviour as a $ln^2x$ terms (or cut produced by the BFKL pomeron), as well as the relation of the sharp behaviour of the singlet contribution at low $x$, and even further, to a strong gluon polarization, can be considered more speculative. It would be an unusual coincidence, that two rather different aspects of small-$x$ physics manifest themselves in the same physical quantity. However, these assumptions seem to us possible, and they will be either supported or disproved by future more accurate data which will allow to elaborate a better analysis of the problem. We are indebted to C. Bourrely, A.V. Efremov, J. Ellis, S. Forte and E. Hughes for stimulating discussions and valuable comments. O.T. is grateful to Centre de Physique Th\'eorique for warm hospitality and to the Universit\'e de Provence for financial support. He was partially supported by Russian Foundation of Fundamental Investigation under Grant 96-02-17361. This investigation was supported in part by INTAS Grant 93-1180. \newpage \begin{figure}[ht] \hfill \begin{minipage}{6.5in} \label{fig1} \caption{Comparison for $g_1^{p-n}$ between the curve given by eq. (2) and the SLAC data refs.\protect\cite{Hughes,p}. For $g_1^p$ at $x=0.0165$ due to the absence of SLAC data we used the SMC data ref. \protect\cite{SMC}.} \end{minipage} \end{figure} \begin{figure}[ht] \hfill \begin{minipage}{6.5in} \label{fig2} \caption{Same as Fig.1 for $g_1^{p+n}$ with the curve given by eq. (7) } \end{minipage} \end{figure}

proofpile-arXiv_065-570

\section{Introduction} The Cal\'{a}n/Tololo Supernova Survey was begun in 1990 as a collaboration between astronomers at Cerro Tololo Inter-American Observatory (CTIO) and the Cerro Cal\'{a}n Observatory of the University of Chile with the principal goal of examining the Hubble diagram for type Ia supernovae (SNe Ia) out to redshifts of $\sim$0.1. During the course of the survey, which we completed in November 1993, a total of 32 SNe Ia were discovered and spectroscopically confirmed. Of these, useful follow-up CCD photometry was obtained for 27 events. In addition, as part of the same program, light curves were obtained of 2 SNe Ia discovered at other observatories. In this paper, we present the final reduced $BVRI$ light curves for these 29 SNe Ia, along with estimates of the maximum-light magnitudes in $BVI$ and the initial decline rate parameter $\Delta$m$_{15}$($B$) (\cite{phillips93}). Note that preliminary light curves for a few events have appeared in previous publications (\cite{hamuy93a}, hereafter referred to as Paper I; \cite{maza94}, hereafter referred to as Paper II; \cite{hamuy94}, hereafter referred to as Paper III). In two accompanying papers, we use these data to examine 1) the absolute luminosities of the sample (\cite{hamuy96a}, hereafter referred to as Paper V) and 2) the Hubble diagrams in $BVI$ and value of the Hubble constant (\cite{hamuy96b} hereafter referred to as Paper VI). \section{Observations} The search phase of the Cal\'{a}n/Tololo Supernova Survey consisted of photographic observations of 45 fields taken with the Curtis Schmidt Camera, with observations carried out approximately twice a month over the 1990-93 period. The details of these observations were described in considerable detail in Paper I, and therefore will not be repeated here. The follow-up phase consisted of two parts: 1) classification via optical spectroscopy, and 2) photometric monitoring via direct CCD imaging in the $BV$($RI$)$_{KC}$ system. Of the 50 SNe discovered in the course of the Survey for which classification spectra were obtained, 32 (64\%) were found to be type~Ia events. A complete listing of these SNe~Ia is found in Table~1 which gives: the SN and host galaxy names, the morphology and heliocentric redshift of the host galaxy; the line-of-sight extinction due to our own Galaxy (\cite{bs82}); the SN equatorial coordinates derived from an early-epoch CCD image using reference stars measured from the digitized sky survey plates available from the Space Telescope Science Institute; the offset of the SN from the host galaxy nucleus; the estimated photographic magnitude of the SN on the discovery plate; the name of the discoverer; and the UT discovery date. A V band CCD image of each SN is reproduced in Figure~1. Followup photometry was obtained for as many of these events as proved practical. Spectra of three of the SNe (1992O, 1992ai, and 1993af) showed that these had been caught several weeks or months past maximum light; hence, the decision was made not to concentrate on obtaining follow-up photometry of these events. For two of the more distant SNe, 1993M and 1993T, an insufficient number of observations were secured to provide adequate coverage of the light curves. Hence, the final number of Cal\'{a}n/Tololo SNe~Ia for which light curves were ultimately obtained was 27. In addition to these 27 SNe, we obtained $BVRI$ photometry of two other SNe~Ia, 1990O and 1992al, which were discovered at other observatories during the course of the survey. Information for these two events is included at the end of Table~1, and V band CCD images are shown in Figure~2. Thus, in this paper we present light curves for a total of 29 SNe~Ia. The follow-up photometry for all 29 SNe~Ia was obtained with CCD detectors on a total of 302 nights between 1990 July 4 and 1995 February 11 thanks to the extensive collaboration of many visiting astronomers and CTIO staff members. The vast majority (94\%) of these nights were at CTIO, with the remaining observations being carried out with the Las Campanas Observatory (LCO) 1.0-m telescope and four different telescopes at the European Southern Observatory (ESO). At CTIO, fully 90\% of the data were taken with the 0.9-m telescope, with the remainder coming from the 1.5-m and 4.0-m telescopes\footnote[1]{This project serves as an eloquent illustration of the capabilities of ``small'' telescopes equipped with state-of-the-art CCD detectors.}. A complete journal of the observations is given in Table~2, which contains the following information: the UT date, the telescope employed, the observatory, the identity of the CCD detector, and the observer(s). \section{Photometric Reductions} A detailed description of the procedures we followed to produce $BVRI$ magnitudes from the individual CCD images of each SN has been given in Paper III. The various steps are summarized as follows: \begin{enumerate} \item Several deep CCD images (in each color) were obtained of the SN field after the SN had faded from detection. These images were transformed geometrically to the same scale, and then coadded to produce a deep master image of the host galaxy. \item The master galaxy image was transformed and scaled to the flux scale of each individual SN image, and then subtracted. In order to save computing time, this galaxy subtraction was carried out over only a subset of the image centered on the SN (see Figure 2 of Paper III). \item Instrumental magnitudes of the SN and several field local standard stars were then measured from the galaxy-subtracted images via point spread function fitting. \item Finally, the instrumental magnitudes were transformed to the standard $BV$($RI$)$_{KC}$ system through the use of a photometric sequence set up in the same field surrounding the SN. (See Paper I for further details of the exact photometric transformations employed.) The photometric sequences for all 29 SNe are identified in the finder charts in Figures 1 and 2. Only three stars lie outside the observed fields and could not be identified in these charts, namely: c8 in the field of SN 1990Y which is located about 130 arcsec west from star c6; c10 in the field of SN 1990Y which is located about 160 arcsec west and 35 arcsec south from star c6; and c14 in the field of SN 1990af which is located about 10 arcsec south and 40 arcsec east from star c12. The magnitudes for the photometric sequences are listed in Table~3. In every case, these sequences were derived from observations made on several (typically 4-6) photometric nights. The uncertainties quoted correspond to the standard error of the mean. \end{enumerate} Table~4 lists the final reduced photometry for each SN. Please note that these magnitudes supersede all previously published values (papers I, II, and III) for the same SNe. The uncertainties quoted for each magnitude correspond to the sum in quadrature of the errors due to photon Poisson statistics and an {\it assumed} additional error of 0.03$^{m}$ in each individual observation. The latter uncertainty was included in order to account for errors involved in the transformation from our instrumental system to the standard system, and also due to the subtraction of the underlying host galaxy. \section{Maximum-Light Magnitudes \& Decline Rates} Figure~3 shows the $BVI$ light curves of the 29 SNe~Ia included in this study. Maximum-light magnitudes were derived for each SN in one of the following two methods: \begin{enumerate} \item {\it Direct Measurement.} For 11 SNe (slightly more than one third of the sample), photometry was obtained at or before maximum light allowing direct measurement of the maximum-light magnitudes in $B$ and $V$. However, for several of these objects (e.g., see the light curves of SN~1992ag in Figure 3), coverage of the $I$ light curve was insufficient to allow direct measurement of the maximum-light brightness in this band. In these cases, the best fitting template (see below) was used, often adjusting this to the first $I$ data point. The corresponding error in the peak magnitude was taken to be 0.03$^{m}$ in those cases where the coverage of the light curve started before maximum, and 0.05$^{m}$ when the observations started only one or two days before the peak. \item {\it Template Fitting.} For the majority of the SNe in our sample, the light curve observations did not begin until after maximum light. To estimate peak magnitudes for these events, we employed a template fitting procedure similar to that utilized in Paper III and \cite{hamuy95} (hereafter referred to as Paper IV). As detailed in a separate paper (\cite{hamuy96c}; hereafter referred to as Paper VIII), a family of six $BVI$ light curve templates, representing the range of observed decline rates of SNe~Ia, were produced from precise CCD photometry obtained at CTIO of seven well-observed events (1992bc, 1991T, 1992al, 1992A, 1992bo, 1993H, and 1991bg). These templates were fit to the observed photometry of each of the program SNe via a $\chi^2$-minimizing technique which solved simultaneously for the time of $B$ maximum and the peak magnitudes $B_{MAX}$, $V_{MAX}$, and $I_{MAX}$. (Note that in our previous papers, the $I$-band data was not included.) As detailed in Paper III, before performing these fits, the templates were first modified by the appropriate K terms (\cite{hamuy93b}) and were also stretched to account for time dilation. For about half of the SNe, one of the templates provided a much better fit (as judged by the value of the reduced $\chi^2$) than the others. An example is SN~1992ae (see Figure~3) whose $BVI$ light curves were found to be an excellent match to the SN~1992al templates. However, for many of the program SNe, the data were fit essentially equally well by two different templates. A good example of such an event is SN~1991ag (see Figure~3), for which the 1991T and 1992bc templates yielded similar values of the reduced $\chi^2$. Hence, we adopted the general rule that when the difference in the reduced $\chi^2$ of two template fits was $\leq$1.5, the peak magnitudes were obtained by averaging the results for the two templates. The corresponding errors were taken to be the greater of a) half of the difference between the peak magnitude estimates of the two templates, b) the 2$\sigma$ formal errors of the $\chi^2$ fits, or c) 0.05$^{m}$. When the difference in the reduced $\chi^2$ values was $>$1.5, the maximum-light magnitudes were taken from the single-best fitting template, with the adopted error being the larger of the 2$\sigma$ formal error of the $\chi^2$ fit or 0.05$^{m}$. Although these rules produced reasonable error estimates in most cases, we found that the errors derived for some SNe whose first light curve observations did not begin until $\sim$2 weeks after maximum were unrealistically low. Hence, in all cases where template fits indicated that the first photometry was not obtained until $\geq$10 days after $B$ maximum, we adopted the following error estimates: 0.2$^{m}$ in $B$, 0.15$^{m}$ in $V$, and 0.15$^{m}$ in $I$. \end{enumerate} For each of the 29 SNe in our sample, we also estimated the decline rate parameter $\Delta$m$_{15}$($B$) (\cite{phillips93}) which corresponds to the amount in magnitudes that the $B$ light curve decreases in brightness during the first 15 days after maximum . This parameter could be measured directly for the five best-observed SNe in the sample (1990af, 1992al, 1992bc, 1992bo, and 1993O). For the remaining events, $\Delta$m$_{15}$($B$) was estimated by fitting a parabola to the reduced $\chi^2$ values yielded by the six template fits (see Paper IV for further details of this procedure). Note that when the smallest value of the reduced $\chi^2$ corresponded to either of the two extremes of the range of $\Delta$m$_{15}$($B$) represented by our templates (SN~1992bc with $\Delta$m$_{15}$($B$) = 0.87 and SN~1991bg with $\Delta$m$_{15}$($B$) = 1.93), we set the inferred value of $\Delta$m$_{15}$($B$) to the same value as the template rather than attempting to extrapolate a value. Table~5 summarizes the resulting light curve parameters for all 29 SNe~Ia. Specifically, we give the epoch of $B$ maximum; the time with respect to $B_{MAX}$ of the first photometric observation in $B$, $V$, or $I$; the decline rate parameter $\Delta$m$_{15}$($B$); the apparent maximum-light magnitudes in $B$, $V$, and $I$; and the method employed to estimate the peak magnitudes where {\it Data} means that the values were measured directly from the photometry, {\it Single Fit} indicates that the best fitting template was used, and {\it Average} signifies that the results from the two best template fits were averaged. In Figure~4 we plot a histogram of the time with respect to $B_{MAX}$ of the first photometric observation in $B$, $V$, or $I$ for the 29 Cal\'{a}n/Tololo SNe Ia. In Table~6, we repeat the $\Delta$m$_{15}$($B$) values and give our final estimates of the peak magnitudes after correction for the extinction due to our own Galaxy (see Table~1) and the K~term. The uncertainties in the corrected magnitudes include errors in the observed magnitude (listed in Table~5), foreground reddening (0.06$^{m}$ in B, 0.045$^{m}$ in V, and 0.03$^{m}$ in I), as well as in the K~term (assumed to be $\pm$0.02$^{m}$). We also list the ``color'' of the SN, $B_{MAX} - V_{MAX}$. (Note that, strictly speaking, this is not a color since $B_{MAX}$ and $V_{MAX}$ occur at slightly different times.) The uncertainties in the color were estimated in the following manner: a) for the 11 SNe for which the photometry started before maximum light we adopted an error of 0.03$^{m}$ when the peak was very well observed (6 cases), or 0.05$^{m}$ otherwise (5 cases), b) when the coverage of the light curve started between days 1 and 10 (counted since B$_{MAX}$) the adopted error was the larger of half of the difference between the color estimates of the two templates or 0.05$^{m}$; if the single-best fitting template technique was used we adopted an error of 0.05$^{m}$, or c) when the observations started after day 10 (counted since B$_{MAX}$) the adopted error was the larger of half of the difference between the color estimates of the two templates or 0.10$^{m}$; if the single-best fitting template technique was used we adopted an error of 0.10$^{m}$. \acknowledgments This paper was possible thanks to grant 92/0312 from Fondo Nacional de Ciencias y Tecnolog\'{i}a (FONDECYT-Chile). MH acknowledges support provided for this work by the National Science Foundation through grant number GF-1002-96 from the Association of Universities for Research in Astronomy, Inc., under NSF Cooperative Agreement No. AST-8947990 and from Fundaci\'{o}n Andes under project C-12984. JM and MH acknowledge support by C\'{a}tedra Presidencial de Ciencias 1996-1997.

proofpile-arXiv_065-571

\section{Basics} Reverberation mapping is a technique in which variability data exhibiting light-travel-time effects are used to to derive information on the geometry and kinematics of a source. Reviews of the basic principles and methods of reverberation mapping as applied to AGNs have appeared in Peterson (1988; 1993) and in Netzer (1990). More detailed information is compiled in the proceedings of a workshop dedicated to the topic (Gondhalekar, Horne, \& Peterson 1994). The fundamental equation of reverberation mapping is the convolution equation relating the emission-line and continuum light curves, $L(t)$ and $C(t)$: $$ L(t)=\int \Psi(\tau) C(t-\tau) d\tau. $$ The kernel of the convolution is the transfer function, $\Psi(\tau)$, which contains the geometrical and physical information inherent to the reprocessing of energy that relates the two light curves. It is the ``Green's function'' of the system, or the hypothetical response of a BLR emission line to a very brief continuum burst. An analogous two-dimensional transfer function, $\Psi(\tau, v)$ relates every projected-velocity bin, $v$, in the line profile to the continuum light curve (see contribution by K. Horne). $\Psi$ provides a ``picture'' of the BLR in time-delay (or time-delay / projected-velocity) space. With some assumptions of symmetry, one can then guess the full six-dimensional geometry and kinematics of the BLR. In the next sections I will sketch my view of the current status of the field, including how well the basic assumptions of reverberation mapping seem to be faring, and what we have learned to date about the BLR. I will outline the directions I think echo mapping will take in the near future. \section{Problems} With the execution of increasingly ambitious reverberation-mapping campaigns (see this volume) and improved data, we have seen a basic prediction of the AGN photoionization model beautifully confirmed, namely, that emission-line light curves mimic the continuum behavior, but with a lag due to light-travel time effects. Among the most illuminating results are those from the joint {\it IUE}/optical campaign on NGC 5548 in 1989 (Clavel et al. 1991; Peterson et al. 1991; Dietrich et al. 1993; Maoz et al. 1993). The gross structure of the light curves for the strong lines, whose sum is shown and compared to the continuum in Figure 1, is just that expected. \begin{figure} \epsfxsize=330pt \epsfbox{maoz1.eps} \caption{UV (1350\AA) continuum light curve (empty circles, left vertical scale, in erg s$^{-1}$ cm$^{-2}$ \AA$^{-1}$) for NGC 5548, and total observed emission-line flux (filled circles, right vertical scale, in erg s$^{-1}$ cm$^{-2}$ ), during the 1989 {\it IUE}/optical monitoring campaign.} \end{figure} However, the NGC 5548 data also revealed various peculiarities, some of which point to inadequacies in the basic assumptions of reverberation mapping. Evidence for such effects has been seen in other objects as well. The three main complications which seem to be indicated by variability data are non-linear response of the emission lines, ``misrepresentation'' of the ionizing continuum by the observed optical or UV continuum, and significant structural evolution of the BLR on timescales of a year. \begin{figure} \epsfxsize=280pt \epsfbox{maoz2.eps} \caption{ Top panel: Jagged dashed line is the total emission line light curve of NGC 5548, same as in Fig. 1. Bottom panel: Transfer function obtained by maximum entropy inversion of the total light curve with the UV continuum light curve, excluding the third ``event''. The smooth solid line in the top panel is the reconstructed emission line light curve obtained by convolving the continuum light curve with the transfer function in the bottom panel. The third event is poorly reproduced for any such monotonically decreasing transfer function.} \end{figure} The nonlinear behavior is most clearly visible in the NGC 5548 1989 light curve of CIV $\lambda 1549$. The total energy in the third continuum ``event'' is much less than that of the previous two events and, if the line light curve were a linear convolution of the continuum light curve, would produce a correspondingly weak feature in the emission line light curve. Instead, the CIV flux rises to the same amplitude it had in response to the previous two events. In retrospect, evidence for nonlinearities in BLR response has been around for a long time. Wamsteker and Colina (1986) first pointed out the saturation in the CIV $\lambda 1549$ level with increasing continuum flux in Fairall-9. A more moderate manifestation of the same effect, whereby CIV responds less than Ly$\alpha$ to continuum changes, can be seen, e.g., in O'Brien et al. (1996) for the much smaller continuum variations of NGC 5548 in 1989. In all AGNs the fractional amplitude of the emission line variations is always considerably smaller than that of the continuum variations, once known constant components such as narrow lines and galaxy light have been accounted for. Clearly, some of the gas contributing to the broad line flux does not respond to the {\it observed} continuum variations. To a certain extent, non-linearities in the response of individual lines are predicted by conventional photoionization models (e.g. Goad, O'Brien \& Gondhalekar 1993). However, the details of the response of a particular line cannot be the entire story behind the problem, because as long as each BLR cloud remains optically thick to the ionizing radiation, the energy from each ionizing photon must come out as one line photon or another (or several). Therefore, the nonlinear aspects should disappear in the total emission line light curve.(This is not a new idea; see Blandford and McKee [1982], \S II.a.) Returning to Fig.1, the third-event problem is, indeed, considerably reduced in the total line light curve. However, the problem has not disappeared. Figure 2 (top panel) shows the same total line curve (now shown as a jagged dashed line). Superposed on it is a reconstructed light curve (smooth solid line) obtained with a maximum entropy inversion that was forced to produce a transfer function that is monotonically decreasing (bottom panel). Maximum entropy methods applied to the NGC 5548 data have generally produced separate aliasing peaks in the transfer function which ``conspire'' with previous continuum events in order to ameliorate the third-event problem. Figure 2 shows that if such aliasing is not allowed, the amplitude of the third event is still much too high, even in the total line flux. Possible explanations are either that there is a fraction of optically-thin gas in the BLR (Shields, Ferland, \& Peterson 1995; O'Brien et al. 1996); or the continuum behavior we see is not the ionizing continuum behavior that the BLR gas sees. This can come about if the continuum emission is not emitted isotropically, or if the ionizing continuum is not strictly linearly proportional to the observed continuum longward of the Lyman edge (as proposed, e.g., by Clavel \& Santos-Lleo 1990, to explain the CIV saturation in Fairall-9). There is already evidence for the latter possibility in NGC 5548, in the well-established hardening of the continuum as it rises (e.g. Maoz et al. 1993). Additional evidence for continuum ``misbehavior'' comes from the intensive 1993 multiwavelength campaign on NGC 4151 (Crenshaw et al. 1996; Kaspi et al. 1996a; Warwick et al. 1996; Edelson et al. 1996). Figure 9 of Kaspi et al. (1996a) shows a scaled version of the {\it IUE} 2700 \AA~ continuum light curve superposed on the 1275 \AA~ continuum light curve. The two light curves, just a factor of 2 apart in energy, are clearly not just linearly scaled versions of each other. Things might be just as bad or worse shortward of 912 \AA. Additional evidence for problems with the ``surrogate'' continuum is that the optical continuum variation amplitude is similar to that in a previous campaign on the same object (Maoz et al. 1991) but the Balmer-line variations are much smaller in the more recent campaign. If an imperfect correlation between the observed and ionizing continua is the source of the problems mentioned above, it could a pose a difficult hurdle, as the ionizing continuum itself cannot be directly observed. A third complication is the detection of a time-variable lag in several objects (Netzer \& Maoz 1990; Peterson et al. 1994; Wanders \& Horne 1994), suggesting a BLR that evolves on timescales of about one year. Possibly related is the fact that discrete velocity components are known to appear and disappear in AGN line profiles, in a manner unrelated to the continuum variations (Wanders and Peterson 1996). Wanders (this volume) has suggested that these components are fluctuations in the fraction of orbiting BLR clouds that are illuminated by an ionization cone, rather than real changes in the BLR geometry. In any case, the observed evolution in the BLR provides an exciting new dimension in reverberation mapping, but may mean that long, sparsely-sampled observations are not an alternative to intensive, season-long campaigns. Finally, there may have been some overinterpretation of the data. It has become common to take variability data and to attempt to invert the convolution equation directly to recover $\Psi$ (e.g., Maoz et al. 1991; Krolik et al. 1991; Horne, Welsh \& Peterson 1991; Peterson et al. 1994; Wanders \& Horne 1994). However, even the best currently available variability data are very noisy, and therefore may produce non-unique transfer functions when deconvolved. As an example, let us look at the H$\beta$ data for NGC 5548. Horne et al. (1991) and Peterson et al. (1994) have found that the maximum-entropy derived transfer function is peaked at 20 days, and has low-amplitude at zero lag. They interpreted this as meaning that there is little variable H$\beta$ line-emission coming from our line of sight to the nucleus, either because of the BLR geometry or due to optical-depth effects in the line. In Figure 3, I have taken the five-year-long optical continuum light curve of NGC 5548 (Peterson et al. 1994; Korista et al. 1995), linearly interpolated it to one-day intervals, and convolved it with three different transfer functions: a delta-function peaked at 20 days, a top-hat function that is positive from 0 to 40 days, and a triangular function peaked at 0 days and decreasing to zero at 60 days. In my talk, I argued that the differences between the light curves produced by these very different-shaped transfer functions are minute compared to the uncertainties in the line measurements themselves, and that these data cannot distinguish between these transfer functions, and in particular between transfer functions peaked at zero-lag and away from zero. After my talk, Keith Horne challenged me to add simulated measuring errors to the fake light curves, and send them to him, which I did. To my surprise, Keith's MEMECHO program recovered the transfer functions well enough for him to easily guess by a process of elimination the correct transfer function to assign to each light curve. On the other hand, Wanders and Peterson (1996) have recently concluded from analysis of a newly-reduced version of the same observations that the NGC 5548 H$\beta$ transfer function cannot yet be uniquely determined. \begin{figure} \epsfxsize=250pt \epsfbox{maoz8.eps} \caption{NGC 5548 5-year H$\beta$ light curve (empty circles). Some typical error bars are shown in upper right corner. The three solid lines are the convolution of the interpolated optical continuum light curves with three different-shaped transfer functions: a delta-function, a top-hat, and a triangle peaked at zero lag. Can the data distinguish between these models?} \end{figure} \section{What Have We Learned?} Despite the problems outlined above, we have learned quite a bit about the BLR from reverberation mapping. An important result from the 1989 NGC 5548 capaign, supported by the 1993 {\it IUE}+{\it HST} campaign of the same object (Korista et al. 1995), is that the BLR is stratified in ionization. Different emission lines respond to the continuum with different lags. The range in lags spans a factor of 5, perhaps more, and there is a trend for the most highly ionized species to have the smallest lags (i.e., be emitted at preferentially small radii). This result has outmoded the single-cloud photoionization models that were common for many years. It is now clear that BLR gas exists at a range of radii. Baldwin et al. (1995; see also Baldwin, this volume) have argued that such stratification is a natural consequence of a ``thick'' BLR, with different lines reaching their peak emission efficiencies at different radii. The question of BLR geometry and kinematics is still unresolved. I note that the data on the best-studied object, NGC 5548, have produced papers advocating a variety of models. Wanders et al. (1995) argue for a spherical BLR with randomly inclined circular orbits, illuminated by an ionizing bi-cone viewed approximately end-on. (This geometry is probably indistinguishable from one with random radial orbits, with the bi-cone approximately in the plane of the sky.) Done \& Krolik (1996) find the data consistent with a thick spherical geometry with Keplerian orbits. Chiang \& Murray (1996; see also Murray, this volume) and Rokaki (this volume) model the results with a Keplerian disk. The data cannot yet distinguish between these models. We probably {\it can} say that the BLR transfer function is resolved on timescales of days, and therefore has some structure. I hope that future experiments will see the details of this structure. We do know some things that the BLR is {\it not}. Models invoking pure radial flow (whether infall or outflow) as the dominant line broadening mechanism have been long discussed. A prediction of any such model is a lag between the blue and red wings of the profile, comparable to the lag of the total line flux behind the continuum. Every experiment that has tested this prediction with adequate temporal resolution has obtained a null result (e.g. Maoz et al. 1991, Wanders et al. 1995). It will be interesting to see if these null results hold up in higher luminosity objects; radial flows are utilized to explain the line shifts observed in quasars (see Espey, this volume). \section{The BLR at Higher Luminosities} Until recently, quasars have been largely neglected by echo mappers. This has been due to their faintness (and hence inaccesibility to {\it IUE} and to small telescopes), their frequent lack of narrow emission lines (still the most popular flux calibrator), and some prejudices about their (presumably long and non-paper/thesis-producing) variation and response timescales. The small amount of data that did exist on quasar emission-line variability (often only two or three epochs per object) produced controversial and sometimes contradicting interpretations (see Peterson 1993, for a review). Happily, reverberation results for higher-luminosity AGNs have begun to appear. Carone et al. (1996) measured a surprisingly large H$\beta$/continuum lag (100 days) in Mrk 509, a luminous Seyfert galaxy. For the past five years, a collaboration between Wise and Steward Observatories has been monitoring 28 bina-fide quasars from the PG sample (Maoz et al. 1994; Kaspi et al. 1996b; see Kaspi, this volume). This program has demonstrated that reverberation mapping indeed works in quasars, much as it does in Seyferts. The first clear lags that have been measured, in two quasars, are also of about 100 days. Does the BLR size scale with luminosity as $R_{BLR}\propto L^{1/2}$, as long predicted based on the overall great similarity of AGN spectra over many orders of magnitude in luminosity? Figure 3 in Kaspi's contribution (this volume) shows the BLR size as a function of luminosity. The two highest luminosity points are the two new quasar measurements. While more quasar BLR radii (which are upcoming from the Wise-Steward program) are desirable, the trend in the figure is certainly suggestive of the expected relation. \section{The Future} With the demise of {\it IUE} (after an incredibly long and fruitful service) a rethinking of strategy is required of echo mappers. To continue progress in understanding the BLR of Seyferts by this technique, a significant improvement is required in the sampling frequency (approximately daily sampling lasting about 6 months) and in the number of AGNs that are monitored. High spectral resolution and S/N may also start producing the long-sought 2-D transfer functions. Measurement errors should remain below the 2-3\% level, since we now know that relatively low-amplitude variations (tens of \%) are the norm in non-blazar AGNs. All these constraints point to the need for one or more dedicated AGN-monitoring ground-based telescopes. High efficiencies could be obtained with 2-3m class apertures, especially if a high degree of automation is implemented. Two telescopes can minimize to a large extent the gaps due to weather. Progress with quasars may be easier. Going down to $B=20$ mag, there are 15 quasars per square degree. Using multi-object spectrographs with wide fields, such as are available on many large telescopes, one can monitor tens of quasars simultaneously. The slower response time (further dilated by $1+z$ in high-$z$ objects) also means that the sampling constraint can be relaxed. With once-a-week observations on a 4m telescope over several years, the BLRs of hundreds of quasars, spanning both luminosity and look-back time, can be studied. \section{Conclusions} The work of the past decade has demonstrated that reverberation mapping works in both low and high luminosity AGNs, and has huge potential. While we cannot yet say that we have determined the structure or kinematics of any BLR, we are at the point where we can discern time-resolved transfer functions in the best-studied objects. Improved data will reveal the details of the transfer functions, and perhaps of the BLRs themselves. We have already learned about ionization stratification, and the inapplicability of some models, e.g. pure radial flow models for Seyferts. The steadily improving quality of the observations have also revealed complications, indicative of the approximations behind some of the basic assumptions: non-linear response, suggestive of the presence of some optically-thin gas, and BLR evolution, vs. the usually assumed stationarity. These are ``good'' complications, in that they point us to a more realistic picture of the BLR. A more worrisome complication is the possibility of a ``misbehaving'' observed continuum that does not completely represent the ionizing continuum seen by the BLR. Future, efficient, reverberation surveys, such as I have outlined, could show the degree to which such continuum misbehavior is a problem.

proofpile-arXiv_065-572

\section{Introduction} There has been a great deal of interest in the physics of heavy hadrons containing one heavy quark. The Heavy Quark Effective Theory (HQET) allows one to study the properties of the heavy hadrons in a systematic $1/m_Q$-expansion. The leading term of the expansion gives rise to the spin-flavour symmetry of Heavy Quark Symmetry (HQS). The corrections to the leading HQS results are determined by the small expansion parameter $\Lambda_{\rm QCD}/m_Q$, where $\Lambda_{\rm QCD}\approx 300\MeV$ is the scale of low energy physics (for a review of HQET see~\cite{HQET}, for a review of HQS and the sum rule approach for heavy mesons see~\cite{Neubert}). Among the well-known predictions of HQS are e.g.\ the relations between different heavy hadron transition form factors. Take for example, the $\Lambda_b\to\Lambda_c$ electro-weak transitions. The six form-factors describing this transition are reduced to one universal Isgur-Wise function in the HQS limit~\cite{Georgi,Mannel,Hussein}. Even then one still remains with many non-perturbative parameters characterizing the process and the heavy baryons participating in it. These concern the functional behaviour of the Isgur-Wise function itself, the masses and residues of the heavy baryons, and, at next-to-leading order in the heavy mass expansion, the average kinetic and chromomagnetic energy of the heavy quark in the heavy baryon. All these non-perturbative parameters can be determined by using non-perturbative methods as e.g.\ lattice calculations, QCD sum rule methods~\cite{Vain} or, in a less fundamental approach, by using potential models. In the present paper we study the correlator of two heavy baryon currents in the HQS limit when $m_Q\to\infty$. Using the QCD sum rule method we calculate the masses and residues of the heavy baryons associated with the heavy baryon currents. In its original form the QCD sum rule method was suggested by Shifman et al.~\cite{Vain} as a tool to investigate the properties of light meson systems. Later on the method was extended to the case of light baryons in~\cite{Ioffe,BeIo,Chung,Pivovarov}. The QCD sum rule approach has proven itself to be a very powerful non-perturbative QCD-based tool which takes into account the properties of the QCD vacuum. It allows one to obtain reliable estimates for hadron masses, their residues and their elastic as well as their transition form factors. In the heavy-light sector the first leading order analysis (leading both in $1/m_Q$ as well as in $\as$) of heavy meson properties within the QCD sum rule approach was performed in~\cite{Shuryak}. Later on the heavy meson sum rule calculation was extended to include next-to-leading order radiative corrections. The next-to-leading order corrections proved to be rather important~\cite{BrGr1,Braun,Neub1}. QCD sum rules for baryons with large but finite masses $m_Q$ were first studied in~\cite{Block,BaDo}. Later on the methods of HQET were incorporated in the sum rule analysis. The leading order QCD sum rules for heavy baryons were first considered in~\cite{Shuryak,GrYa,BaDo1}, again to leading order both in $1/m_Q$ as well as in $\as$. Finite mass corrections to these sum rules were discussed in~\cite{DaHu}. In order to improve on the accuracy of the existing QCD sum rule analysis of heavy baryons one needs to avail of the next-to-leading order radiative corrections to the sum rules. This forms the subject of the present paper. We calculate the QCD radiative corrections to the leading perturbative term in the Operator Product Expansion (OPE) and, from these, we derive next-to-leading order QCD sum rules for heavy baryons in the HQS limit. We then go on to analyze the sum rules and compute the values of the masses and the residues of the heavy baryons at next-to-leading order accuracy. \newpage The paper is organized as follows. In Sec.~2 we introduce heavy baryon currents as interpolating fields for the heavy ground state baryons. In Sec.~3 we construct the correlator of two heavy baryon currents by means of the OPE and define the spectral density. In Sec.~4 we present our results on the radiative corrections to the perturbative part of the spectral density and construct renormalization group invariant QCD sum rules by recapitulating some known results on the one- and two-loop anomalous dimensions of the currents. Sec.~5 contains the results of our numerical analysis. Sec.~6, finally, contains our summary and our conclusions. In Appendix A we provide a detailed collection of results on the calculation of the two- and three-loop contributions to the correlator of two heavy baryon currents. These results are quite general in that they are given for general space-time dimensions and for a general baryonic current structure. \section{Baryonic currents} The currents of the heavy baryon $\Lambda_Q$ and the heavy quark spin baryon doublet $\{\Sigma_Q,\Sigma_Q^*\}$ are associated with the spin-parity quantum numbers $j^P=0^+$ and $j^P=1^+$ for the light diquark system with antisymmetric and symmetric flavour structure, respectively. Adding the heavy quark to the light quark system, one obtains $j^P=\frac12^+$ for the $\Lambda_Q$ baryon and the pair of degenerate states $j^P=\frac12^+$ and $j^P=\frac32^+$ for the baryons $\Sigma_Q$ and $\Sigma_Q^*$. The general structure of the heavy baryon currents has the form (see e.g.~\cite{GrYa} and refs.\ therein) \begin{equation}\label{current} J=[q^{iT}C\Gamma\tau q^j]\Gamma'Q^k\epsilon_{ijk}. \end{equation} Here the index $T$ means transposition, $C$ is the charge conjugation matrix with the properties $C\gamma^T_\mu C^{-1}=-\gamma_\mu$ and $C\gamma^T_5C^{-1}=\gamma_5$, $i,j,k$ are colour indices and $\tau$ is a matrix in flavour space. The effective static field of the heavy quark is denoted by~$Q$. For each of the ground state baryons there are two independent interpolating currents $J_1$ and $J_2$ which both have the appropriate quantum numbers to interpolate to the respective ground state baryons. They are given by~\cite{Shuryak,GrYa} \begin{eqnarray}\label{currents} J_{\Lambda1}&=&[q^{iT}C\tau\gamma_5q^j]Q^k\varepsilon_{ijk},\qquad J_{\Lambda2}\ =\ [q^{iT}C\tau\gamma_5\gamma_0q^j]Q^k\varepsilon_{ijk}, \nonumber\\[7pt] J_{\Sigma1}&=&[q^{iT}C\tau\vec\gamma q^j] \cdot\vec\gamma\gamma_5Q^k\varepsilon_{ijk},\qquad J_{\Sigma2}\ =\ [q^{iT}C\tau\gamma_0\vec\gamma q^j] \cdot\vec\gamma\gamma_5Q^k\epsilon_{ijk},\nonumber\\ \vec J_{\Sigma^*1}&=&[q^{iT}C\tau\vec\gamma q^j]Q^k\varepsilon_{ijk} +\frac13\vec\gamma[q^{iT}C\tau\vec\gamma q^j] \cdot\vec\gamma Q^k\varepsilon_{ijk},\\ \vec J_{\Sigma^*2}&=&[q^{iT}C\tau\gamma_0\vec\gamma q^j]Q^k\varepsilon_{ijk} +\frac13\vec\gamma[q^{iT}C\gamma_0\vec\gamma q^j] \cdot\vec\gamma Q^k\varepsilon_{ijk},\nonumber \end{eqnarray} where $\vec J_{\Sigma^*1}$ and $\vec J_{\Sigma^*2}$ satisfy the spin-$3/2$ condition $\vec\gamma\vec J_{\Sigma^*i}=0$ ($i=1,2$). The flavour matrix $\tau$ is antisymmetric for $\Lambda_Q$ and symmetric for the heavy quark spin doublet $\{\Sigma_Q,\Sigma_Q^*\}$. The currents written down in Eq.~(\ref{currents}) are rest frame currents. The corresponding expressions in a general frame moving with velocity four-vector $v^\mu$ can be obtained by the substitutions $\gamma_0\rightarrow\slv$ and $\vec\gamma\rightarrow\gamma^\mu_\perp=\gamma^\mu-\slv v^\mu$. In the following analysis we shall be using both of these equivalent descriptions alternatively, i.e.\ we shall either use the static description with $v^\mu=(1,0,0,0)$ or a moving frame description with $v^\mu=(1,\vec{v})$ and $\vec{v}\neq 0$. For a general analysis it proves to be convenient to represent the general light-side Dirac structure of the currents in Eq.~(\ref{currents}) by an antisymmetrized product of $n$ Dirac matrices $\Gamma=\gamma^{[\mu_1}\cdots\gamma^{\mu_n]}$. When calculating the one- and two-loop vertex corrections one encounters $\gamma$-contractions of the form $\gamma_\alpha\Gamma\gamma^\alpha$ and $\gamma_0\Gamma\gamma_0$. The $\gamma_\alpha$-contraction leads to an $n$-dependence according to \begin{equation}\label{Dirac1} \gamma_\alpha\Gamma\gamma^\alpha=h\Gamma=(-1)^n(D-2n)\Gamma. \end{equation} The $\gamma_0$-contraction depends in addition on an additional parameter $s$ which takes the value ($s=+1$) and ($s=-1$) for an even or odd number of $\gamma_0$'s in $\Gamma$, respectively. The $\gamma_0$-contraction reads \begin{equation}\label{Dirac2} \gamma_0\Gamma\gamma_0=(-1)^ns\Gamma. \end{equation} In order to facilitate the use of Eqs.~(\ref{Dirac1}) and~(\ref{Dirac2}) we have compiled a table of the $(n,s)$-values relevant for the heavy baryon currents treated in this paper (see Table~1). \begin{table} \begin{tabular}{|l|c|c|l|} \hline $\Gamma$&$n$&$s$&particles\\\hline\hline $\gamma^{\rm AC}_5$&$0$&$+1$&$\Lambda_1$\\\hline $\gamma^{\rm AC}_5\gamma_0$&$1$&$-1$&$\Lambda_2$\\\hline $\vec\gamma$&$1$&$+1$&$\Sigma_1,\Sigma^*_1$\\\hline $\gamma_0\vec\gamma$&$2$&$-1$&$\Sigma_2,\Sigma^*_2$\\\hline\hline $\gamma^{\rm HV}_5$&$4$&$-1$&$\Lambda_1$\\\hline $\gamma^{\rm HV}_5\gamma_0$&$3$&$+1$&$\Lambda_2$\\\hline \end{tabular} \caption{Specific values of the parameter pair $(n,s)$ for particular cases of the light-side Dirac structure $\Gamma$. $\gamma^{\rm AC}_5$ refers to the naive $\gamma_5$-scheme with an anticommuting $\gamma_5$~\cite{Kreimer} and $\gamma_5^{\rm HV}$ to the $\gamma_5$-scheme due to Breitenlohner, Maison, 't~Hooft and Veltman~\cite{tHVt}.} \end{table} \section{Correlator of two baryonic currents} In this section we describe the steps needed for the evaluation of baryonic QCD sum rules. One starts with the correlator of two baryonic currents, \begin{equation}\label{correlator} \Pi(\omega=k\cdot v)=i\int d^4xe^{ikx}\langle 0|T\{J(x),\bar J(0)\}|0\rangle, \end{equation} where $k_\mu$ and $p_\mu$ are the residual and full momentum of the heavy quark and $v_\mu$ is the four-velocity using the momentum expansion $p_\mu=m_Qv_\mu+k_\mu$. As was mentioned before, there are two possible choices of interpolating currents for each of the heavy baryon states, given by $\Gamma_1$ and $\Gamma_2=\Gamma_1\slv$. Thus one may consider correlators of the same currents (diagonal correlators) or of different currents (non-diagonal correlators). In the general case, one may even consider correlators built from a linear combination $J=J_1+bJ_2$ of these currents with an arbitrary coefficient $b$. We mention that the choice $b=1$ corresponds to a constituent quark model current which has maximal overlap with the ground state baryons in the constituent quark model picture. In this paper we limit our attention to diagonal correlators only. The correlator in Eq.~(\ref{correlator}) depends only on the energy variable $\omega=k\cdot v$ because of the static nature of the heavy propagator. It can be factorized into a spinor dependent piece and a scalar correlator function $P(\omega)$ according to \begin{eqnarray} \Pi(\omega)=\Gamma'\frac{1+\slv}2\bar\Gamma' \frac14\spur(\Gamma\bar\Gamma)2\spur(\tau\tau^\dagger)P(\omega). \end{eqnarray} Following the standard QCD sum rule method~\cite{Vain}, the correlator is calculated in the region $-\omega\approx 1-2\GeV$, including perturbative and non-perturbative contributions, where the non-perturbative contributions can in general be quite large. The non-perturbative effects are taken into account by employing an Operator Product Expansion (OPE) for the time-ordered product of currents in Eq.~(\ref{correlator}). One then has \begin{equation} T\{J(x),\bar J(0)\}=\sum_d C_d(x^2)O_d, \end{equation} where the operators $O_d$ are local operators with a given dimension $d$, $O_0=\hat 1$, $O_3=\langle\bar qq\rangle$, $O_4=\langle GG\rangle$, \dots\ , and the expansion coefficients $C_d(x^2)$ are the corresponding coefficient functions or Wilson coefficients of the OPE. A straightforward dimensional analysis shows that the OPE of the diagonal correlator contains only even-dimensional terms. We take into account the perturbative term for $d=0$, the gluon condensate term for $d=4$ and a condensate term with four quark fields for $d=6$. The four-quark operator will be factorized into a product of two two-quark operators, $\langle\bar q(0)q(x)\rangle^2$~\cite{Vain}. Accordingly the Fourier transform of the scalar correlator function $P(\omega)$ reads \begin{eqnarray}\label{OPE} P(t)=P_{\rm OPE}(t)=i\theta(t)N_c!\Big(\frac1{\pi^4t^6} +\frac{c\as\langle GG\rangle}{32N_c(N_c-1)\pi^3t^2} -\frac1{4N_c^2}\langle\bar q(0)q(t)\rangle^2\Big), \end{eqnarray} where $c=1$ for $\Lambda_Q$, $c=-1/3$ for $\{\Sigma_Q,\Sigma_Q^*\}$ and $N_c$ is the number of colours. For the non-local quark condensate $\langle\bar q(0)q(t)\rangle$ one may use the OPE about $\langle\bar qq\rangle:=\langle\bar q(0)q(0)\rangle$, namely \begin{eqnarray} \langle\bar q(0)q(t)\rangle=\langle\bar qq\rangle\Big(1 +\frac1{16}m^2_0t^2+\pi\as\langle GG\rangle\frac{t^4}{96N_c}+\ldots\ \Big), \end{eqnarray} where the parameter $m_0$ is defined in Eq.~(\ref{condensates}). Alternatively one may use the Gaussian ansatz~\cite{Rady} \begin{eqnarray} \langle\bar q(0)q(t)\rangle =\langle\bar qq\rangle\exp(\frac1{16}m^2_0t^2). \end{eqnarray} When expanding the Gaussian ansatz one sees that the two forms agree up to the term linear in $t^2$. Thus the two representations of the non-local quark condensate are quite similar to one another for small values of $t$. In our sum rule analysis we shall make use of the Gaussian ansatz because it provides for better stability of the sum rules. For the condensates we use the numerical values \begin{eqnarray} \langle\bar qq\rangle&=&-(0.23\GeV)^3,\nonumber\\ \as\langle GG\rangle&=&0.04\GeV^4,\label{condensates}\\ g_S\langle\bar q\sigma_{\mu\nu}G^{\mu\nu}q\rangle &=&m_0^2\langle\bar qq\rangle \quad\mbox{with\ }m^2_0=0.8\GeV^2.\nonumber \end{eqnarray} With these condensate values one sees that the OPE in Eq.~(\ref{OPE}) with Euclidian time $\tau=it$ converges nicely for $1/\tau>0.3\GeV$. In this region one may thus safely truncate the OPE series after the second term. At $1/\tau=0.3\GeV$ the contribution of the first term is two times larger than the last quark condensate term. Its contribution grows rapidly with $1/\tau$. When $1/\tau$ is further increased we see that the correlator becomes dominated by the perturbative contribution. For example, at $1/\tau=0.6\GeV$ the perturbative term is two orders of magnitude larger than the contribution of the condensate terms. Note, however, that at $1/\tau=0.4\GeV$ the contribution of the ground state to the correlator is ten times smaller than the contribution of the excited states and the continuum. This would imply that if the theoretical and phenomenological continuum contributions differ by about $10\%$ (and are not equal to each other as assumed here), this difference would induce a $100\%$ change in the contribution of the ground state. Thus the sum rules can only be trusted at values $1/\tau<0.4\GeV$ (see also the discussions of the numerical results of the sum rules). In the next section we will show that the perturbative corrections become even more important at small Euclidian distances in comparison to the non-perturbative condensate contributions. As a next step one determines the spectral density using the coordinate space representation $P(t)$ of the current correlator. The simplest way to proceed is as follows. The scalar correlator function $P_{\rm OPE}(\omega)$ satisfies a dispersion relation \begin{eqnarray}\label{dispersion} P_{\rm OPE}(\omega)=P(\omega)=\int_0^\infty \frac{\rho(\omega')d\omega'}{\omega'-\omega-i0}+P'(\omega), \end{eqnarray} where $\rho(\omega)={\sl Im}(P(\omega))/\pi$ is the spectral density and $P'(\omega)$ is a polynomial in $\omega$, which takes into account possible subtractions in the dispersion representation. The Fourier transform of the polynomial $P'(\omega)$ consists of the $\delta$-function $\delta(t)$ and derivatives $\delta^{(n)}(t)$ of the $\delta$-function. A comparison with Eq.~(\ref{OPE}) shows that one does not in fact need any subtractions. We therefore set $P'(\omega)=0$. Taking the Fourier transform of Eq.~(\ref{dispersion}) according to \begin{equation} P(t)=\int\frac{d\omega}{2\pi}e^{-i\omega t}P(\omega) \end{equation} we obtain \begin{eqnarray}\label{dispersion1} P(t)=i\int^{\infty}_0\rho(\omega)e^{-i\omega t}d\omega. \end{eqnarray} Then we analytically continue $P(t)$ from $t>0$ to imaginary times by introducing the Euclidian time $\tau=it$. After this transformation, Eq.~(\ref{dispersion1}) becomes the well known Laplace transformation. One may thus use an inverse Laplace transformation in order to obtain an Euclidean time representation of the spectral density, \begin{eqnarray}\label{dispersion2} \rho(\omega) =\frac1{2\pi}\int^{c+i\infty}_{c-i\infty}P(-i\tau)e^{\omega\tau}d\tau, \end{eqnarray} where $c$ is to be chosen as a real constant to the right of all singularities of $P(t)$. It is then easy to check that the form $P(t)=\theta(t)/t^n$ gives the spectral density $\rho(\omega)=i^{n+1}\theta(\omega)\omega^{n-1}/(n-1)\!$, whereas $P(t)=\theta(t)t^n$ results in $\rho(\omega)=-(-i)^{n-1}\delta^{(n)}(\omega)$ for $n>0$. Following the argumentation in~\cite{BeIo} we do not include forms of the second kind into the spectral density $\rho$. So the leading order perturbative contribution and the next-to-leading order contribution of the gluon condensate to the spectral density are given by \begin{eqnarray}\label{leadingSD} \rho(\omega)&=&\rho_0(\omega)+\rho_4(\omega),\qquad\mbox{where}\\\nonumber \rho_0(\omega)&=&\frac{\omega^5}{20\pi^4}\quad\mbox{and}\quad \rho_4(\omega)=c\frac{\as\langle GG\rangle}{32\pi^3}\omega. \end{eqnarray} \section{Radiative correction to the perturbative term} Next we consider radiative corrections to the leading order spectral density in Eq.~(\ref{leadingSD}). There are altogether four different three-loop graphs that contribute to the correlator of two baryonic currents, which are shown in Fig.~1. Contrary to the experience in the two-loop case, the most convenient way to calculate the three-loop contributions is to evaluate them in momentum space. The fact that all graphs in Fig.~1 have two-point two-loop subgraphs greatly simplifies the calculational task. One can first evaluate the respective subgraphs such that one remains with a one-loop integration. The subgraph two-loop integration can be performed by using the algebraic methods described in~\cite{BrGr}. It is important to note that the results of the two-loop integration can be expressed in terms of a polynomial function of the external momentum that flows into the subgraph. Hence, the remaining integration is a one-loop type integration, where the power of one of the propagators has become a non-integer number due to the use of dimensional integration. The upshot of this is that all steps of the three-loop integration can be reduced to purely algebraic manipulations. We present the results of calculating the two-loop and three-loop contributions to the correlator in the form \begin{equation}\label{results} P(\omega)=\lambda_0C_0B_0+\lambda_1\sum_{i=1}^4C_iB_i, \end{equation} where we have used the abbreviations $\lambda_0=(-2\omega/\mu)^{(2D-3)}$, $\lambda_1=\gs^2/(4\pi)^D(-2\omega/\mu)^{(3D-7)}$, and where $D=4-2\epsilon$ is the space-time dimension. Concerning the colour structure we have defined the colour factors $C_i$ ($i=0,\ldots,4$) according to the labelling of the graphs in Fig.~1. Their values are given by $C_0=N_c!$, $C_1=C_2=-N_c!C_B$ and $C_3=C_4=N_c!C_F$, where $C_F=(N_c^2-1)/2N_c$ and $C_B=(N_c+1)/2N_c$. Values for the scalar coefficients $B_i$ defined in Eq.~(\ref{results}) are listed in Appendix~A. Putting everything together, the two-loop and three-loop scalar correlation factor $P(\omega)$ defined in Eq.~(\ref{results}) is given by \begin{eqnarray}\label{results1} P(\omega)\!\!&=&\!\!-\frac{32\omega^5}{(4\pi)^4}\Bigg[ \left(\frac{-2\omega}{\mu}\right)^{-4\epsilon} \frac1{40}\Bigg(\frac1{\epsilon}+\frac{107}{15}\Bigg)\\&& +\frac{\as}{4\pi}\left(\frac{-2\omega}{\mu}\right)^{-6\epsilon} \Bigg(\frac{n^2-4n+6}{45\epsilon^2} +\frac{40\zeta(2)+61n^2-234n+396}{225\epsilon} +\frac{(n-2)s}{90}\nonumber\\&& +\frac{5(195n^2-780n+1946)\zeta(2)-2200\zeta(3) +4907n^2-18408n+34352}{2250}\Bigg)\Bigg].\nonumber \end{eqnarray} The scalar correlation function $P(\omega)$ is renormalized by the square of the renormalization factor $Z_J$ of the baryonic current derived in~\cite{GrYa}. Accordingly one has \begin{equation} P(\omega)=Z^2_JP^{\rm ren}(\omega)\qquad\hbox{\rm with\ } Z_J=1+\frac{\as C_B}{4\pi\epsilon}(n^2-4n+6). \end{equation} The multiplication of $P(\omega)$ in Eq.~(\ref{results1}) with $Z^2_J$ results in the cancellation of the second power in $1/\epsilon$. The remaining $1/\epsilon$-singularity is purely real and hence does not contribute to the spectral density. Since the renormalized spectral density $\rho^{\rm ren}(\omega)={\sl Im}(P^{\rm ren}(\omega))/\pi$ has to be finite, this provides a check on our calculation. The spectral density can be read off from Eq.~(\ref{results1}) and is given by \begin{eqnarray}\label{ro} \rho^{\rm ren}(\omega,\mu)&=&\rho_0(\omega)\Bigg[ 1+\frac{\as}{4\pi}r(\omega/\mu)\Bigg],\quad\mbox{where}\\ \rho_0(\omega)&=&\frac{\omega^5}{20\pi^4}\quad\mbox{and}\quad r(\omega/\mu)\ =\ r_1\ln\left(\frac\mu{2\omega}\right)+r_2 \quad\mbox{with}\nonumber\\ r_1&:=&\frac83(n^2-4n+6)\quad\mbox{and}\quad r_2\ :=\ \frac8{45}(60\zeta(2)+38n^2-137n+273).\nonumber \end{eqnarray} The coefficient $r_1$ of the logarithmic term in Eq.~(\ref{ro}) coincides with twice the one-loop anomalous dimension given in Eq.~(\ref{oneloopAD}), as expected. The reason is that the evolution of $\rho(\omega,\mu)$ is controlled by the renormalization group equation and that the anomalous dimension of $\langle J\bar J\rangle$ and $\rho(\omega,\mu)$ coincide. The $\as$-correction can be seen to depend on the properties of the light-side Dirac matrix $\Gamma$ in the heavy baryon current, as specified in Table~1. As an explicit result we list representations of the $r(\omega/\mu)$-functions of the four baryon currents in the naively anticommuting $\gamma_5$-scheme (AC). They read \begin{eqnarray} r_{\Lambda 1}(\omega/\mu)&=&16\ln\left(\frac\mu{2\omega}\right) +\underbrace{\frac{8(20\zeta(2)+91)}{15}}_{\approx 66.04},\nonumber\\ \nonumber\\ r_{\Lambda 2,\Sigma 1}(\omega/\mu)&=&8\ln\left(\frac\mu{2\omega}\right) +\underbrace{\frac{16(10\zeta(2)+29)}{15}}_{\approx 48.48},\label{ro1}\\ \nonumber\\ r_{\Sigma 2}(\omega/\mu)&=&\frac83\ln\left(\frac\mu{2\omega}\right) +\underbrace{\frac{8(60\zeta(2)+151)}{45}}_{\approx 44.40}. \nonumber \end{eqnarray} The results for the two different baryon currents $\Lambda_1$ and $\Lambda_2$ in the 't~Hooft-Veltman $\gamma_5$-scheme (HV) differ from those presented above. It is well known that currents in different $\gamma_5$-schemes are connected by a finite renormalization factor $Z$ such that \begin{equation}\label{finiteRen} J_{AC}=ZJ_{HV}. \end{equation} These finite factors also appear in the calculation of two-loop anomalous dimensions of baryonic currents~\cite{GKY}. From the results of~\cite{GKY} one has \begin{equation}Z_{\Lambda 1}=1-\frac{4\as}{3\pi}\quad\mbox{and}\quad Z_{\Lambda 2}=1-\frac{2\as}{3\pi}. \end{equation} Using these finite renormalization factors one may convert the results in the naively anticommuting $\gamma_5$-scheme given in Eq.~(\ref{ro1}) to the corresponding results in the 't~Hooft-Veltman scheme. Least the reader worry that we do not list the corresponding $\Sigma$-type conversion factors we remind him that the $\gamma_5$ in the $\Sigma$-type currents act on the heavy quark line and thus there are no $\gamma_5$-ambiguities. Nevertheless, the 't~Hooft-Velman $\gamma_5$-scheme needs some counter terms to satisfy some kind of Ward identities. To avoid this complification, we will henceforth concentrate on the naively anticommuting $\gamma_5$-scheme, where such counterterms are not necessary at all. We only mention that the finite renormalization in Eq.~(\ref{finiteRen}) will bring the results of the two $\Gamma_5$-schemes in line. In order to allow for a quick appraisal of the importance of the perturbative corrections we have exhibited the numerical values of the second terms in Eq.~(\ref{ro1}). For $\as$ we use the running coupling constant, which we normalized to the value of $\as(m_Z)=0.118$ at the mass of the $Z$-boson for $N_f=5$ active flavours. By doing so one has $\as(\mu)=0.333$ at $\mu=1\GeV$ for $N_f=3$ active flavours. Using this value for $\as(\mu=1\GeV)$, the above results show that the perturbative $\as$-corrections to the spectral density amount to about $100\%$. This highlights the importance of perturbative QCD radiative corrections in QCD sum rule applications. The same observation was made in the heavy meson sector \cite{BrGr1,Braun,Neub1}. As in the heavy meson sector on remains with several unsettled questions: \begin{enumerate} \item Are there any special reasons for such big QCD ``corrections''? \item Can we trust the QCD sum rule predictions and the $\as$-expansion\\ when the $\as$-corrections are so big? \item How big are the $\as^2$-corrections? Is it possible to estimate them? \end{enumerate} These questions should be clarified in the near future. \subsection{Residues and QCD sum rules} To proceed with the usual QCD sum rules analysis, we evaluate the scalar correlator function $P(\omega)$ using the theoretical result $P_{\rm OPE}(t)$ given in Eq.~(\ref{OPE}) and equate this to the dispersion integral over the contributions of hadron states. These consist of the lowest lying ground state with bound state energy $\bar\Lambda$ plus the excited states and the continuum. To leading order in $1/m_Q$ the bound state energy of the ground state is defined by \begin{equation} m_{\rm baryon}=m_Q+\bar\Lambda, \end{equation} where $m_Q$ is the pole mass of the heavy quark. Note that the leading order sum rules do not depend on $m_Q$ at all since the heavy mass dependence has been eliminated by employing the heavy mass expansion. We assume that the continuum is given by the OPE expression above a certain threshold energy $E_C$~\cite{Vain}. For the hadron-side (h.s.) contribution to the spectral density we thus write \begin{equation} \rho_{\rm h.s.}(\omega)=\rho_{\rm g.s.}(\omega)+\rho_{\rm cont}(\omega), \end{equation} where the contribution of the lowest-lying ground state (g.s.) baryon is contained in $\rho_{\rm g.s.}$ and is given by \begin{eqnarray} \rho_{\rm g.s.}(\omega)=\frac12F^2\delta(\omega-\bar\Lambda). \end{eqnarray} In this expression $F$ is the absolute value of one of the residues $F_i$ ($i=\Lambda,\Sigma,\Sigma^*$) of the baryonic currents defined by \begin{equation}\label{residue} \langle 0|J|\Lambda_Q\rangle=F_\Lambda u,\qquad \langle 0|J|\Sigma_Q\rangle=F_\Sigma u\quad\mbox{and}\quad \langle 0|J_\nu|\Sigma^*_Q\rangle=\frac1{\sqrt 3}F_{\Sigma^*}u_{\nu}, \end{equation} where $u$ and $u_{\mu}$ are the usual spin-$1/2$ and spin-$3/2$ spinors. Note that $F_{\Sigma^*}$ coincides with $F_\Sigma$ in the lowest order of the heavy quark mass expansion that we are working in. As is usual we assume hadron-parton duality for the contribution of excited states and continuum contributions and take $\rho_{\rm cont}(\omega)=\theta(\omega-E_C)\rho(\omega)$, where $\rho$ is the result of the OPE calculations given in Eqs.~(\ref{OPE}) and~(\ref{leadingSD}). With these assumptions we arrive at the sum rule \begin{equation} P_{\rm OPE}(\omega)=\frac{\frac12F^2}{\bar\Lambda-\omega-i0} +\int_{E_C}^\infty\frac{\rho(\omega')d\omega'}{\omega'-\omega-i0} \end{equation} or \begin{equation}\label{presumrule} \frac{\frac12F^2}{\bar\Lambda-\omega-i0}=\int_0^{E_C} \frac{\rho(\omega')d\omega'}{\omega'-\omega-i0} +P_{\rm p.c.}(\omega), \end{equation} where the power counting part $P_{\rm p.c.}(\omega)$ is defined as the Fourier transform of that part of the correlator function $P(t)$ which contains non-negative powers $(t^2)^n$ ($n\geq 0$). Finally we apply the Borel transformation \begin{equation} \hat B_T=\lim\frac{\omega^n}{\Gamma(n)}\left(-\frac{d}{d\omega}\right)^n \qquad n,-\omega\to\infty\quad(T=-\omega/n\hbox{\rm\ fixed}) \end{equation} to the sum rule in Eq.~(\ref{presumrule}). Using $\hat B_T(1/(\omega-\omega'))=\exp(-\omega'/T)/T$ we obtain the Borel sum rule \begin{equation}\label{sumrule} \frac12F^2(\mu)e^{-\bar\Lambda/T} =\int_0^{E_C}\rho(\omega',\mu)e^{-\omega'/T}d\omega' +\hat BP_{\rm p.c.}(T)=:K(E_C,T,\mu), \end{equation} where we reintroduced the $\mu$-dependence of the spectral density, which causes a $\mu$-dependence for the residue. The Borel-transformed $\hat BP_{\rm p.c.}(T)$ can be obtained directly from $P_{\rm p.c.}(t)$ by the substitution $t\rightarrow-i/T$ (see the discussion in~\cite{GrYa}). Note that the bound state energy $\bar\Lambda$ can be obtained from the sum rule in Eq.~(\ref{sumrule}) by taking the logarithmic derivative with respect to the inverse Borel parameter according to \begin{equation} \bar\Lambda=-\frac{d\ln(K(E_C,T,\mu))}{dT^{-1}}. \end{equation} Returning to the sum rule in Eq.~(\ref{sumrule}), one has \begin{eqnarray} \frac12F^2(\mu)e^{-\bar\Lambda/T}&=&\frac{N!}{\pi^4}\Bigg[T^6\left( f_5(x_C)+\frac{\as}{4\pi}\left(\left(\ln\Big(\frac\mu{2T}\Big) f_5(x_C)-f_5^l(x_C)\right)r_1+r_2\right)\right)\nonumber\\&&\qquad\qquad +cE_G^4T^2f_1(x_C)+E_Q^6\exp\left(-\frac{2E_0^2}{T^2}\right)\Bigg] \label{sumrulework} \end{eqnarray} with the polynomials $r_1$ and $r_2$ presented in Eq.~(\ref{ro}) and the functions \begin{eqnarray} f_n(x)&:=&\int_0^x\frac{x'^n}{n!}e^{-x'}dx' \ =\ 1-e^{-x}\sum_{m=0}^n\frac{x^m}{m!},\nonumber\\ f_n^l(x)&:=&\int_0^x\frac{x'^n}{n!}\ln x'e^{-x'}dx'. \end{eqnarray} In order to simplify the notation we have introduced the abbreviations \begin{equation} x_C:=\frac{E_C}T,\quad E_0:=\frac{m_0}4,\quad (E_Q)^3:=-\frac{\pi^2}{2N}\langle\bar qq\rangle\quad\mbox{and}\quad (E_G)^4:=\frac{\pi\as\langle GG\rangle}{32N(N-1)}.\qquad \end{equation} The numerical analysis of the Borel sum rule is the subject of the section 5. \subsection{Anomalous dimensions} The one-loop renormalization of the effective heavy baryon currents was considered in~\cite{GrYa}, the two-loop case was studied in~\cite{GKY}. In general they differ from those in conventional QCD. The one-loop anomalous dimension of baryonic currents, namely the first coefficient in the expansion $\gamma=\sum_k(\as/4\pi)^k\gamma_k$, only depends on $n$ and is given by~\cite{GrYa,GKY} \begin{equation}\label{oneloopAD} \gamma_1=-\frac43((n-2)^2+2). \end{equation} The general $(n,s)$-dependent formula for the two-loop anomalous dimension case is rather lengthy and can be found in~\cite{GKY}. As an illustration we list explicit values for the two-loop anomalous dimensions as calculated in the MS-scheme using the naive $\gamma_5$-scheme. One has (with explicit values given for $N_f=3$) \begin{eqnarray} \gamma_{\Lambda1}&=&-8\left(\frac{\as}{4\pi}\right) +\underbrace{\frac19(16\zeta(2)+40N_f-796)}_{\approx-72.19} \left(\frac{\as}{4\pi}\right)^2,\label{andimlam1}\\ \gamma_{\Lambda2}&=&-4\left(\frac{\as}{4\pi}\right) +\underbrace{\frac19(16\zeta(2)+20N_f-322)}_{\approx-26.19} \left(\frac{\as}{4\pi}\right)^2,\label{andimlam2}\\ \gamma_{\Sigma1}&=&-4\left(\frac{\as}{4\pi}\right) +\underbrace{\frac19(16\zeta(2)+20N_f-290)}_{\approx-22.63} \left(\frac{\as}{4\pi}\right)^2,\\ \gamma_{\Sigma2}&=&-\frac83\left(\frac{\as}{4\pi}\right) +\underbrace{\frac1{27}(48\zeta(2)+8N_f+324)}_{\approx 15.81} \left(\frac{\as}{4\pi}\right)^2. \end{eqnarray} \subsection{Renormalization group invariant sum rules} It is clear that the currents $J(\mu)$ depend on the renormalization scale $\mu$. This dependence can be expressed by the renormalization group equation \begin{equation} (\mu\frac{d}{d\mu}+\gamma)J(\mu)=0,\qquad\gamma:=\frac{d\ln Z_J}{d\ln\mu}, \end{equation} arising from the scale independence of the bare current $J_0=Z_J(\mu)J(\mu)$, where $\gamma$ is the anomalous dimension of the current discussed in the preceding subsection. To construct a renormalization group invariant quantity $J_{\rm inv}$, the renormalized current $J(\mu)$ is multiplied by some Wilson coefficient $C(\as(\mu))$, $J_{\rm inv}=J(\mu)C(\as(\mu))$, which is subject to the ``dual'' renormalization group equation (see also~\cite{Neub1,BaDo}) \begin{equation}\label{dualRGE} (\mu\frac{d}{d\mu}-\gamma)C(\as(\mu))=0\quad\Rightarrow\quad (\as\beta(\as)\frac\partial{\partial\as}-\gamma(\as))C(\as)=0, \end{equation} where $\beta:=d\ln\as/d\ln\mu=\sum_k(\as/4\pi)^k\beta_k$ is the beta function of QCD with \begin{equation}\label{beta} \beta_1=-2(11-\frac23N_f)\quad\mbox{and}\quad \beta_2=-4(51-\frac{19}3N_f). \end{equation} The formal solution of Eq.~(\ref{dualRGE}) is given by \begin{equation} C(\as(\mu))=\exp\left(\int^{\as(\mu)}\frac{d\alpha}\alpha \frac{\gamma(\alpha)}{\beta(\alpha)}\right). \end{equation} Finally, the perturbative expansion of the beta function and the anomalous dimension up to second order in $\as$ gives \begin{equation}\label{running} C(\as(\mu))=\as(\mu)^{\gamma_1/\beta_1}\left(1+\frac{\as(\mu)}{4\pi} \frac{\gamma_1}{\beta_1}\left(\frac{\gamma_2}{\gamma_1} -\frac{\beta_2}{\beta_1}\right)\right). \end{equation} The first factor in Eq.~(\ref{running}) is the result of resumming the leading logarithmic terms $(\as\ln\mu)^n$, where the result is valid only in the logarithmic approximation. As Eq.~(\ref{running}) shows, one needs to know also the two-loop anomalous dimension of the baryon current in order to obtain the evolution at next-to-leading log accuracy, e.g.\ in the order $\as(\as\ln\mu)^n$. The usage of the invariance property of $J_{\rm inv}$ also provides a connection between currents at different renormalization scales, \begin{eqnarray} &&J(\mu_2)C(\as(\mu_2))=J(\mu_1)C(\as(\mu_1)) \quad\Rightarrow\nonumber\\[7pt] J(\mu_2)&=&J(\mu_1)C(\as(\mu_1))C(\as(\mu_2))^{-1} =:J(\mu_1)U(\mu_1,\mu_2),\\[7pt] U(\mu_1,\mu_2)&=&\exp\left(\int^{\as(\mu_1)}_{\as(\mu_2)} \frac{d\alpha}{\alpha}\frac{\gamma(\alpha)}{\beta(\alpha)}\right) \nonumber\\&=&\left(\frac{\as(\mu_1)}{\as(\mu_2)} \right)^{\gamma_1/\beta_1}\left(1+\frac{\as(\mu_1)-\as(\mu_2)} {4\pi}\frac{\gamma_1}{\beta_1}\left(\frac{\gamma_2}{\gamma_1} -\frac{\beta_2}{\beta_1}\right)\right), \end{eqnarray} where $U(\mu_1,\mu_2)$ is perturbatively evaluated up to next-to-leading order in $\as$ (see also the discussion in~\cite{Neubert,Bardeen,JiMu}). As being evident from Eq.~(\ref{residue}), also the residues are functions of the renormalization scale parameter $\mu$, the functional form for this dependence is the same as for the currents. So one can construct the renormalization group invariant $F_{\rm inv}=F(\mu)C(\as(\mu))$ by means of the same Wilson coefficient. A renormalization group invariant sum rule can then be constructed by considering the expression \begin{equation}\label{Invsumrules} \frac12F_{\rm inv}^2\exp(-\bar\Lambda/T)=K(E_C,T,\mu)C(\as(\mu))^2 =:K_{\rm inv}(E_C,T). \end{equation} The theoretical part of the sum rule $K(E_C,T,\mu)$ depends on the renormalization scale $\mu$ through the QCD perturbative corrections which contain the logarithmic factor $\ln(\mu)$. On the other hand, the left hand side of Eq.~(\ref{Invsumrules}) is independent of the renormalization scale $\mu$ by construction, and thus the right hand side must also be renormalization scale independent. It is easy to check this to first order in $\as$ by introducing a second scale $\mu'$ and writing \begin{equation} \frac{\as(\mu)}{\as(\mu')}=1-\frac{\as(\mu)}{8\pi}\beta_1 \ln(\frac{\mu'^2}{\mu^2}). \end{equation} Remembering that $\rho(\omega,\mu)$ in Eq.~(\ref{ro}) appears as an integrand of $K(E_C,T,\mu)$, one obtains cancellations (in first order of $\as$) of the logarithmic factors $\ln(\mu)$ in $\rho(\omega,\mu)C(\as(\mu))^2$ and thereby in $K(E_C,T,\mu)C(\as(\mu))^2$. The cancellation occurs because of $r_1=-2\gamma_1$. In this paper we will make no usage of the renormalization group invariant sum rule in Eq.~(\ref{Invsumrules}). Instead of this we analyse the sum rule~(\ref{sumrule}) at some fixed point $\mu'=1\GeV$ in order to estimate the bound state energy $\bar\Lambda$ and the residue $F(\mu')$. The value of the residue $F(\mu)$ at other scales can then be obtained by using the evolution function $U(\mu',\mu)$, while the $\mu$-independent function $F_{\rm inv}$ can be immediately obtained by multiplying with $C(\as(\mu))$. \section{Numerical results} Let us discuss the sum rule analysis in some detail. We start by discussing the sum rules without radiative corrections and execute the analysis in consecutive steps. As Eq.~(\ref{sumrulework}) shows, the analysis of the non-radiatively corrected sum rules does not depend on which of the two different current cases are being discussed. First, we analyse the dependence of the bound state energy $\bar\Lambda$ on the threshold parameter $E_C$ and the Borel parameter $T$ in a large window of parameter space. The aim is to try and find regions of stability in $T$ and $E_C$. By looking at the three-dimensional plots for $\bar\Lambda$ as functions of $T$ and $E_C$ we found a stability of the sum rules only in the case of the exponential ansatz for the non-local operator $\langle\bar q(0)q(x)\rangle$. Keeping in mind the rather narrow window of $0.3\GeV<T<0.4\GeV$ mentioned in the connection with Eq.~(\ref{OPE}) for the consecutive replacement $\tau\rightarrow it\rightarrow 1/T$, one ends up with a more reasonable discussion of the stability. We mention that the range of acceptable values for $T$ is extended down to $T>0.2\GeV$ when radiative corrections are included, which enlarge the perturbative contributions. This, however, does not bring in a new region of stability. Returning to the analysis of the non-radiatively corrected sum rules for the $\Lambda_Q$-baryon, we find areas of stability around $E_C=1.2\GeV$ in the window $0.3\GeV<T<0.4\GeV$. The range of confidence for $E_C$ is $1.0\GeV<E_C<1.4\GeV$. Therefore in Fig.~2(a) we show plots for five values of $E_C$ around $E_C^{\rm best}=1.2\GeV$, namely for $E_C=E_C^{\rm best}$, $E_C=E_C^{\rm best}\pm 0.1\GeV$ and $E_C^{\rm best}\pm 0.2\GeV$. From these curves we then can read off values for $E_C$ and $\bar\Lambda$ with good sum rule stability, namely \begin{equation}\label{LambdaEnergies} \bar\Lambda(\Lambda)=0.78\pm 0.05\GeV\qquad\hbox{\rm in the range}\qquad E_C(\Lambda)=1.2\pm 0.1\GeV, \end{equation} where the quoted errors present rough error estimates taken from Fig.~2(a) according to the interval for $E_C$ in Eq.~(\ref{LambdaEnergies}). Next we estimate the value of the residue. The sum rules depends on the three parameters $\bar\Lambda$, $E_C$ and $T$. In Fig.~2(b) we plot $|F_\Lambda|$ for a fixed bound state energy $\bar\Lambda(\Lambda)=0.78\GeV$ in the indicated window for the Borel parameter $T$. The five different curves again correspond to the above five different values of $E_C$. Sum rule stability is found at \begin{equation} |F_\Lambda|=0.023\pm 0.002\GeV^3, \end{equation} where the errors again represent rough error estimates taken from Fig.~2(b). Next we take into account the $\as$-correction to the spectral density. As is evident from Eq.~(\ref{ro}), the sum rule analysis now depends on which of the two types of baryonic currents are used. The results for the bound state energy for both cases are displayed in Fig.~2(c). Using the same analysis as for Fig.~2(a) we obtain \begin{equation} \bar\Lambda(\Lambda)=0.78\pm 0.05\GeV\qquad\hbox{\rm in the range}\qquad E_C(\Lambda)=1.1\pm 0.1\GeV. \end{equation} Here we note the nice technical effect that the $\as$-corrected sum rule is more stable at the lower value $E_C=1.1\GeV$ for the continuum threshold but predicts the same bound state energy~$\bar\Lambda$. So there occurs some ``stabilization'' in $\bar\Lambda$. Using the central value $\bar\Lambda(\Lambda)=0.78\GeV$ one can then obtain values for the residue looking at Fig.~2(d), which give rise to the value \begin{equation} |F_\Lambda|=0.028\pm 0.002\GeV^3. \end{equation} Doing the same analysis for the $\Sigma$ baryon, at leading order in $\as$ we obtain \begin{eqnarray} \bar\Lambda(\Sigma)&=&0.90\pm 0.05\GeV,\qquad E_C(\Sigma)=1.3\pm 0.1\GeV \qquad\mbox{and}\nonumber\\|F_\Sigma|&=&0.026\pm 0.002\GeV^3 \end{eqnarray} and including the $\as$ radiative corrections we have \begin{eqnarray} \bar\Lambda(\Sigma)&=&0.95\pm 0.05\GeV,\qquad E_C(\Sigma)=1.3\pm 0.1\GeV \qquad\mbox{and}\nonumber\\ |F_\Sigma|&=&0.039\pm 0.003\GeV^3 \end{eqnarray} The results are displayed graphically in Fig.~3(a,b) and in Fig.~3(c,d), respectively. Our predictions for the bound state energy $\bar\Lambda$ combined with the experimental charm and bottom baryon masses may be taken to calculate the charm and bottom quark pole masses $m_Q$. Taking into account the experimental results as given by the Particle Data Group~\cite{PDG}, namely $m(\Lambda_c)=2284.9\pm 0.6\MeV$, $m(\Sigma^+_c)=2453.5\pm 0.9\MeV$ and $m(\Lambda_b)=5641\pm 50\MeV$, we obtain the pole masses $m_c\approx 1500\MeV$ and $m_b\approx 4860\MeV$ for the heavy quarks. The experimental difference of $m(\Lambda_c)-m(\Sigma_c)\approx 167\MeV$ \cite{PDG} is quite near to our prediction $m(\Lambda)-m(\Sigma)\approx 170\MeV$. Here we present only central values. As was discussed above, the accuracy of our predictions is connected with the internal accuracy of the QCD sum rules method (mainly because of the dependence on the energy of continuum) and is probably not better than 20\%. All the results are summarized in Table~2 where we compare our results with the leading order results obtained in~\cite{GrYa,DaHu,Cola}. This concludes our analysis. \begin{table} \begin{tabular}{|r||c|c|c||c|c|}\hline &\cite{GrYa}&\cite{DaHu}&\cite{Cola}&L.O.&N.L.O.\\\hline\hline $E_C(\Lambda)$&$1.20$&$1.20\pm0.15$&$1.2\pm0.1$&$1.2\pm0.1$&$1.1\pm0.1$\\ $E_C(\Sigma)$&$1.46$&$1.30\pm0.15$&$1.4\pm0.1$&$1.3\pm0.1$&$1.3\pm0.1$\\ \hline\noalign{\smallskip} $\bar\Lambda (\Lambda )$&$0.78$&$0.79\pm0.05$&$0.9\pm0.1$ &$0.78\pm0.05$&$0.78\pm0.05$\\ $\bar\Lambda( \Sigma)$&$0.99$&$0.96\pm0.05$& &$0.90\pm0.05$&$0.95\pm0.05$\\ $\bar\Lambda(\Sigma)-\bar\Lambda(\Lambda)$&$0.21$&$0.17$&&$0.12$&$0.17$\\ \hline\hline $|F_\Lambda|$&$2.3\pm0.5$&$1.7\pm0.6$&$2.5\pm0.5$&$2.3\pm0.1$&$2.8\pm0.2$\\ $|F_\Sigma|$&$3.5\pm0.6$&$4.1\pm0.6$&$4.0\pm0.5$&$2.6\pm0.2$&$3.9\pm0.3$\\ \hline \end{tabular} \caption{Sum rule results on non-perturbative and sum rule parameters of heavy ground state baryons. The continuum threshold parameter $E_C$, the bound state energy $\bar\Lambda$ and the difference between the two bound state energies are given in $\GeV$, whereas the residues are listed in units of $10^{-2}\GeV^3$. The value of the Borel parameter is $T=0.35\GeV$.} \end{table} \section{Conclusions} We have considered the Operator Product Expansion of the correlator of two static heavy baryon currents at small Euclidian distances and determined the $\as$ radiative corrections to the first Wilson coefficient in the expansion. Based on this expansion we formulated and analyzed heavy baryon sum rules for the $\Lambda$-type and $\Sigma$-type heavy baryons using two different types of interpolating fields for the baryons in each case. We have discussed in some detail the scale independence of the $\as$ sum rules which requires the consideration of the anomalous dimensions of the heavy baryon currents at the two-loop level. Similar to the case of heavy mesons the QCD radiative correction to the first term in the OPE is quite large and amounts to a $100\%$ change in the perturbative contribution. The radiative correction to the perturbative term increase the calculated sum rule values for the baryon masses by about $10\%$ and the residues by about $20-50\%$ relative to the corresponding lowest order values. The sum rule results do not depend very much on which of the two possible interpolating fields is used in each case. The sum rule analysis is, however, quite sensitive to changes in the assumed threshold energy of the continuum. This sensitivity is the main source of uncertainty in our results and is partly due to the use of diagonal correlators. QCD sum rules based on the diagonal correlators feature a leading order spectral density which grows rapidly as $\rho(\omega)\approx\omega^5$. This rapid growth introduces a strong dependence of the sum rule results on the assumed energy of continuum. Second, the QCD radiative correction to the leading order spectral density is about $100\%$ at the renormalization scale $\mu=1\GeV$. We may try to make the coefficient at $\as$ in these corrections to be moderate considering the very low renormalization scale $\mu=10\MeV$. We have not considered non-diagonal sum rules which come in when one considers correlators between two different currents with the same quantum numbers. These non-diagonal sum rules bring in some new features such as a more ``normal'' behaviour of the spectral density $\rho(\omega)\approx\langle\bar qq\rangle\omega^2$ and thus probably more moderate QCD corrections to this spectral density. On the other hand, the leading term for non-diagonal sum rules is proportional to the quark condensate, whose value $\langle\bar qq\rangle=(-0.23\pm0.02\GeV)^3$ is known only with an accuracy of 10\%, which gives an additional uncertainty in the result for non-diagonal sum rules. The analysis of the non-diagonal sum rules will form the subject of a subsequent paper. \vspace{.5truecm} \noindent{\large \bf Acknowledgments:}\smallskip\\ This work was partially supported by the BMBF, FRG, under contract 06MZ566, and by the Human Capital and Mobility program under contract CHRX-CT94-0579. We would like to thank A.~Grozin and B.~Tausk for valuable discussions. \newpage \section*{Appendix A: Diagrammatic contributions} \setcounter{equation}{0} \defA\arabic{equation}{A\arabic{equation}} In this Appendix we collect together results on the calculation of the two-loop and three-loop contributions to the correlator of two heavy baryon currents. We start with the two-loop contribution depicted in Fig.~1 ($i=0$) where one has \begin{eqnarray} B_0&=&\frac{(D-2)E_2}{16(2D-7)(2D-5)(2D-3)E_1}\tilde b_0 \spur(\bar\Gamma\slv\Gamma\slv)\qquad\mbox{with}\nonumber\\ \tilde b_0&=&\frac{E_1}{(D-4)(D-3)}. \end{eqnarray} We have introduced the abbreviation $E_n=\Gamma(1-\epsilon)^n\Gamma(1+n\epsilon)$ (with natural numbers $n=1,2,3,\ldots\ $) which is also used in the subsequent presentation of the three-loop results. For the three-loop contributions $i=1,2$ and $4$ depicted in Fig.~1 one has \begin{eqnarray} B_i&=&\frac{2(D-2)(2D-7)E_3}{9(3D-11)(3D-10)(3D-8)(3D-7)E_2} \tilde b_i\spur(\bar\Gamma\slv\Gamma\slv)\nonumber\\ \noalign{\noindent\mbox{with}\bigskip} \tilde b_1&=&\frac{2(D-2)E_1^2}{(D-4)^3(D-3)^2} -\frac{(D-2)(3D-10)E_2}{(D-4)^3(D-3)^2(2D-7)},\\ \tilde b_3&=&\frac{(D-2)E_2}{2(D-4)^2(D-3)(2D-7)},\nonumber\\ \tilde b_4&=&\frac{-(D-2)E_2}{(D-4)^2(D-3)^2(2D-7)}.\nonumber \end{eqnarray} The contribution of diagram ($2$) in Fig.~1 is the most involved one. In order to be able to write the results in a compact form we introduce the abbreviations \begin{eqnarray} Q_1&=&\Gamma(1-\epsilon)^2\Gamma(1+\epsilon)/\Gamma(1-2\epsilon) \quad\mbox{and}\nonumber\\ Q_2&=&\Gamma(1-\epsilon)^3\Gamma(1+2\epsilon)/\Gamma(1-3\epsilon). \end{eqnarray} In terms of the basic structure terms \begin{equation} \tilde\Gamma_0=\spur(\bar\Gamma\slv\Gamma\slv),\qquad \tilde\Gamma_1=\spur(\bar\Gamma\gamma_\mu\Gamma\gamma^\mu) \quad\mbox{and}\quad\tilde\Gamma_2=\spur(\bar\Gamma\gamma_\mu \gamma_\nu\slv\Gamma\slv\gamma^\nu\gamma^\mu),\nonumber \end{equation} one obtains \begin{eqnarray} B_2&=&\frac{E_3}{36(D-3)(3D-11)(3D-7)Q_2} \sum_{j=0}^2\tilde b_{2,j}\tilde\Gamma_j\qquad\mbox{with}\nonumber\\ \tilde b_{2,0}&=&\frac{12(D-2)^2Q_1^2}{(D-4)^3(D-3)^2(D-1)} -\frac{24D(D-2)^2Q_2}{(D-4)^3(D-1)(3D-10)(3D-8)},\nonumber\\ \tilde b_{2,1}&=&\frac{(D^2-7D+16)Q_1^2}{(D-4)^2(D-3)^2(D-1)} -\frac{4(D^2-4D+8)Q_2}{(D-4)^2(D-1)(3D-10)(3D-8)},\\ \tilde b_{2,2}&=&\frac{3Q_1^2}{(D-4)^2(D-3)(D-1)} -\frac{4Q_2}{(D-4)(D-1)(3D-10)(3D-8)}.\nonumber \end{eqnarray} \newpage

proofpile-arXiv_065-573

\section{Introduction} In two previous articles (Gebhard, Bott, Scheidler, Thomas, and Koch I and II 1996) we have studied the optical absorption of non-interacting and strongly interacting tight-binding electrons in Peierls-distorted chains. These two different cases are idealizations of real almost ideal one-dimensional semiconductors (Farges 1994). While polymers like polyacetylen (Heeger, Kivelson, Schrieffer, and Su 1988); (Baeriswyl, Campbell, and Mazumdar 1992); (Schott and Nechtschein 1994) are commonly believed to be Peierls insulators some charge-transfer salts (Alc\'{a}cer 1994); (Brau and Farges 1994) should be understood as Mott-Hubbard insulators. The extended Hubbard model for interacting electrons on a distorted chain at half-filling is considered appropriate for the latter class of materials (Mazumdar and Dixit 1986); (Fritsch and Ducasse 1991); (Mila 1995). Unfortunately, the study of the optical, i.e., finite frequency properties of correlated electron systems poses a very difficult many-body problem that cannot be solved analytically without further approximations on the Hamiltonian or the calculations (Kohn 1964); (Maldague 1977); (Lyo and Galinar 1977); (Lyo 1978); (Galinar 1979); (Campbell, Gammel, and Loh 1989); (Mahan 1990); (Shastry and Sutherland 1990); (Stafford, Millis, and Shastry 1991); (Fye, Martins, Scalapino, Wagner, and Hanke 1992); (Stafford and Millis 1993). In our second article (Gebhard {\em et al.} II 1996) we focused on the optical absorption in the dimerized extended Harris-Lange model which is equivalent to the Hubbard model to lowest order in the strong-coupling expansion. We could exactly map the charge degrees of freedom onto two parallel (Hubbard-)bands for free spinless Fermions of band-width~$W$. The eigenstates of the Harris-Lange model are highly spin-degenerate which allowed us to exactly calculate the optical absorption. The results apply to the ``hot-spin'' regime of the Hubbard model when the temperature is large compared to the spin exchange energy~$J\sim W^2/(U-V)$. Real experiments are carried out at low temperatures for which the system is in an unique ground state with antiferromagnetic correlations. Unfortunately, this problem cannot be solved analytically. In this article we will employ the analogy to an ordinary semiconductor (electrons and holes in a phonon bath) to design a ``no-recoil approximation'' for the ``chargeons'' in a ``spinon bath''. It will allow us to determine the coherent absorption features of the Hubbard model at large~$U/W$. The paper is organized as follows. In section~\ref{Hamilts} we derive our approximate Hamiltonian from the Hubbard model. The charge dynamics will be governed by the Harris-Lange model while the ground state is determined by the Heisenberg model. Basic results of our articles (Gebhard {\em et al.} I and II 1996) for the optical absorption and the current operator are recalled in section~\ref{optabssec}. In section~\ref{norecoilsec} we introduce the ``no-recoil approximation'' which allows us to obtain informations on the coherent part of the absorption spectrum at low temperatures. The corresponding results for the Peierls-distorted extended Harris-Lange model are presented in section~\ref{optabsHubbard}. Summary and conclusions close our presentation. \section{Model Hamiltonians} \label{Hamilts} \subsection{Charge degrees of freedom at strong coupling: the Harris-Lange model} As shown in (Gebhard {\em et al.} II 1996) the Peierls-distorted extended Hubbard model (Hubbard 1963); (E\ss ler and Korepin 1994) can be mapped onto the Harris-Lange model in the limit of strong correlations. In standard notation of second quantization the latter model reads \begin{mathletters} \begin{eqnarray} \hat{H}_{\rm HL}^{\rm dim, \, ext} &=& \hat{T}_{\rm LHB}(\delta) + \hat{T}_{\rm UHB}(\delta) + U\hat{D} + V \hat{V}\\[6pt] \hat{T}_{\rm LHB} &=& (-t) \sum_{l,\sigma} (1+(-1)^l\delta) \left(1-\hat{n}_{l,-\sigma}\right) \left( \hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}} + \hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}} \right) \left( 1-\hat{n}_{l+1,-\sigma}\right) \\[6pt] \hat{T}_{\rm UHB} &=& (-t) \sum_{l,\sigma} (1+(-1)^l\delta) \hat{n}_{l,-\sigma} \left( \hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}} + \hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}} \right) \hat{n}_{l+1,-\sigma} \\[6pt] \hat{D}&=& \sum_l \hat{n}_{l,\uparrow}\hat{n}_{l,\downarrow}\\[6pt] \hat{V} &=& \sum_l (\hat{n}_l-1)(\hat{n}_{l+1}-1) \end{eqnarray} \end{mathletters}% where $\hat{n}_{l,\sigma}=\hat{c}_{\l,\sigma}^+\hat{c}_{\l,\sigma}^{\phantom{+}}$ is the local density of $\sigma$-electrons and $\hat{n}_{l}=\hat{n}_{l,\uparrow}+\hat{n}_{l,\downarrow}$ is the total local electron density. $U\hat{D}$ is the Hubbard interaction between electrons on the same site, $V\hat{V}$ is the nearest-neighbor interaction between charged objects like double occupancies and holes, and $\hat{T}_{\rm LHB}$ ($\hat{T}_{\rm UHB}$) describes the motion of holes (double occupancies) in the lower (upper) Hubbard band. The Harris-Lange model is equivalent to the Hubbard model to order~$t(t/U)^{-1}$, $t(t/U)^{0}$, and $t(V/U)^{0}$. Due to the Peierls distortion the electron transfer between two lattice sites is modulated by~$\pm t\delta$. We are only interested in the half-filled case where the number of electrons~$N$ equals the even number of lattice sites~$L$. \subsection{Band structure interpretation} For~$V=0$ and in the absence of a lattice distortion the exact eigenenergies of the Harris-Lange model are the same as those of independent spinless Fermions moving in two parallel bands separated by~$U$. For the case of linear optical absorption we may thus work with the effective band structure Hamiltonian \begin{equation} \hat{H}_{\rm HL}^{\rm band} = \sum_{|k|\leq \pi/a}\left[ (U+\epsilon(k)) \hat{n}_{k}^{u} + \epsilon(k) \hat{n}_{k}^{l} \right] \label{effHL} \end{equation} with $\epsilon(k)=-2ta \cos(ka)$, and $\hat{n}_{k}^{u}=\hat{u}_{k}^+ \hat{u}_{k}^{\phantom{+}}$, $\hat{n}_{k}^{l}=\hat{l}_{k}^+ \hat{l}_{k}^{\phantom{+}}$ for our fermionic quasi-particles (chargeons) in the upper and lower Hubbard band. Their band width is~$W=4t$, the lattice spacing is~$a$, and the $k$-values of the first Brillouin zone are spaced by~$\delta k=2\pi/(La)$. The lattice distortion results in a Peierls splitting of the upper and lower Hubbard band. In this case the effective Hamiltonian for linear optical absorption becomes (Gebhard {\em et al.} II 1996) \begin{equation} \hat{H}_{\rm HL}^{\rm dim,\, band} = \sum_{|k|\leq \pi/(2a),\tau=\pm 1}\biggl[ (U+\tau E(k)) \hat{n}_{k,\tau}^{u} + \tau E(k) \hat{n}_{k,\tau}^{l} \biggr] \; . \label{bandhldim} \end{equation} with the dispersion relation \begin{mathletters} \begin{eqnarray} E(k)&=&\sqrt{\epsilon(k)^2+\Delta(k)^2} \\[3pt] \epsilon(k) &=& -2t \cos(ka) \\[3pt] \Delta(k)&=& 2t\delta \sin (ka) \end{eqnarray} \end{mathletters}% where $\hat{n}_{k,\tau}^{u}$ ($\hat{n}_{k,\tau}^{l}$) are the number operators for the quasi-particles for the upper ($\tau=+$) and lower ($\tau=-$) Peierls subband in the upper~($u$) and lower~($l$) Hubbard band. \subsection{Spin degrees of freedom at strong coupling: the Heisenberg model} \label{spinhamilT} At half-filling the ground state of the Harris-Lange model is~$2^L$-fold degenerate. We are ultimately interested in the optical properties of the (Peierls-distorted extended) Hubbard model at strong correlations for which the ground state for large~$U/W$ is unique. Consequently, we have to go to the next order in the expansion of the Hubbard model in~$W/U$ to lift the above degeneracy. The Harris-Lange model is unsatisfactory in yet another aspect: at half filling a finite lattice distortion cannot be sustained within the model because there is no gain in electronic energy. The electronic part of the ground state energy is zero, irrespective of the Peierls parameter~$\delta$. A consistent treatment of the expansion in~$W/U$ will also produce corrections to the Harris-Lange model (Harris and Lange 1967); (van Dongen 1994) which would render the problem intractable. Thus we argue that the degeneracy of the ground state will inevitable be lifted by a residual spin-spin interaction which may have its origin in the itinerant exchange in the Hubbard model (next term in the expansion in $W/U$) or in their direct exchange which is not taken into account in the extended Hubbard model where only the direct Coulomb terms were kept (``Zero Differential Overlap Approximation'' (Kivelson, Su, Schrieffer, and Heeger 1987); (Wu, Sun, and Nasu 1987); (Baeriswyl, Horsch, and Maki 1988); (Gammel and Campbell 1988); (Kivelson, Su, Schrieffer, and Heeger 1988); (Campbell, Gammel, and Loh 1988); (Painelli and Girlando 1989); (Campbell, Gammel, and Loh 1990)). The additional ``Hubbard-$W$-terms'' might very well be important in strongly-correlated narrow-band materials. Thus we {\em assume\/} that the {\em charge\/} dynamics is still governed by the Harris-Lange model. In particular, we assume that the chargeons do not scatter from the spinons. For the {\em spin\/} dynamics we choose the (dimerized, anisotropic) Heisenberg Hamiltonian with an anisotropy into the $z$-direction \begin{equation} \hat{H}_{\rm Heis} = \sum_{l} \left(1+(-1)^l\delta_S\right) \left[ J_{\perp}\left(\hat{S}_l^x \hat{S}_{l+1}^x + \hat{S}_l^y \hat{S}_{l+1}^y \right) + J_{z} \hat{S}_l^z \hat{S}_{l+1}^z - \frac{1}{4} \right] \label{dimHeisHamilt} \end{equation} where~$\hat{\rm\bf S}_l$ is the vector operator for the spin at site~$l$, $\hat{S}_l^+=\hat{S}_l^x+i\hat{S}_l^y= \hat{c}_{l,\uparrow}^+\hat{c}_{l,\downarrow}^{\phantom{+}}$, $\hat{S}_l^-=\hat{S}_l^x-i\hat{S}_l^y= \hat{c}_{l,\downarrow}^+\hat{c}_{l,\uparrow}^{\phantom{+}}$, and $\hat{S}_l^z=(\hat{n}_{l,\uparrow}-\hat{n}_{l,\downarrow})/2$. $J_{\perp}>0$ and $J_z>0$ are two generally different antiferromagnetic coupling constants, and $0\leq \delta_S\leq 1$ determines the degree of spin dimerization. It is well-known that a one-dimensional spin-system that is coupled to the lattice degrees of freedom shows the spin-Peierls effect, $\delta_S >0$. If the itinerant exchange was responsible for the antiferromagnetic coupling we would have~$J\equiv J_{\perp}=J_z=(1+\delta^2) 2t^2/U$ and $\delta_S=2\delta/(1+\delta^2)$. In general, however, we do not assume a simple relation between~$J_{\perp}$, $J_z$ and~$t$, or~$\delta_S$ and~$\delta$. We state, however, that $J_{\perp}$, $J_z$ are supposed to be {small\/} energy scales compared to the Coulomb energies~$U$,~$V$. Its influence on the exact optical excitation energies will thus be neglected in this work. This implies, for example, that we cannot distinguish between singlet and triplet excitons. \section{Optical absorption and effective current operator} \label{optabssec} \subsection{Optical conductivity and optical absorption} The dielectric function~$\widetilde{\epsilon}(\omega)$ and the coefficient for the linear optical absorption~$\widetilde{\alpha}(\omega)$ are given by (Haug and Koch 1990) \begin{mathletters} \begin{eqnarray} \widetilde{\epsilon}(\omega) &=& 1 +\frac{4\pi i \sigma(\omega)}{\omega} \label{epssigma}\\[6pt] \widetilde{\alpha}(\omega) &=& \frac{4\pi {\rm Re}\{\sigma(\omega)\}}{n_b c} \end{eqnarray} \end{mathletters}% where ${\rm Re}\{\ldots\}$ denotes the real part and $n_b$ is the background refractive index. It is supposed to be frequency independent near a resonance. Hence, the real part of the optical conductivity directly gives the absorption spectrum of the system. The standard result (Maldague 1977); (Mahan 1990) for the real part of the optical conductivity in terms of the current-current correlation function~$\chi(\omega)$ is \begin{eqnarray} {\rm Re}\{ \sigma(\omega) \} &=&\frac{{\rm Im}\{\chi(\omega)\}}{\omega} \\[6pt] \chi(\omega) & =& \frac{{\cal N}_{\perp}}{La} i \int_0^{\infty} dt e^{i\omega t} \langle \left[\hat{\jmath}(t),\hat{\jmath}\right]_- \rangle \end{eqnarray} where~${\cal N}_{\perp}$ is the number of chains per unit area perpendicular to the chain direction. The current-current correlation function can be spectrally decomposed in terms of exact eigenstates of the system as \begin{equation} \chi(\omega) = \frac{{\cal N}_{\perp}}{La} \sum_n |\langle 0 | \hat{\jmath} | n\rangle|^2 \left[ \frac{1}{\omega +(E_n-E_0) +i\gamma} - \frac{1}{\omega -(E_n-E_0) +i\gamma} \right] \; . \label{decomp} \end{equation} Here, $|0\rangle$ is the exact ground state (energy $E_0$), $|n\rangle$ are exact excited states (energy $E_n$), and $\left|\langle 0 | \hat{\jmath} | n\rangle\right|^2$ are the oscillator strengths for optical transitions between them. Although $\gamma =0^+$ is infinitesimal we may introduce $\gamma>0$ as a phenomenological broadening of the resonances at $\omega = \pm(E_n-E_0)$. The spectral decomposition of the real part of the optical conductivity reads \begin{equation} {\rm Re}\{ \sigma(\omega) \} = \frac{{\cal N}_{\perp} \pi}{La \omega} \sum_n \left| \langle 0 | \hat{\jmath} | n\rangle\right|^2 \left[ \delta\left( \omega -(E_n-E_0)\right) -\delta\left( \omega +(E_n-E_0)\right) \right] \label{speccomp} \end{equation} which is positive for all~$\omega$. In the following we will always plot the dimensionless reduced optical conductivity \begin{equation} \sigma_{\rm red}(\omega >0) = \frac{\omega {\rm Re}\{\sigma(\omega>0)\} }% {{\cal N}_{\perp}a e^2 W } \; . \label{sigmared} \end{equation} Furthermore we replace the energy conservation~$\delta(x)$ by the smeared function \begin{equation} \widetilde{\delta}(x) = \frac{\gamma}{\pi(x^2+\gamma^2)} \end{equation} to include effects of phonons and experimental resolution. \subsection{Effective current operator} In (Gebhard {\em et al.} II 1996) we showed that the current operator for an excitation from the filled lower Hubbard band to the empty upper Hubbard band can be written as \begin{eqnarray} \hat{\jmath}_{{\rm inter},+}^{\rm H} &=& -(itea) \sum_{l,\sigma}\left(1+(-1)^l \delta\right) \left(1+(-1)^l \eta\right) \nonumber \\[3pt] && \phantom{-(itea)\sum} \left[ \hat{n}_{l+1,-\sigma} \hat{c}_{l+1,\sigma}^+ \hat{c}_{l,\sigma}^{\phantom{+}} \left( 1- \hat{n}_{l,-\sigma} \right) - \hat{n}_{l,-\sigma} \hat{c}_{l,\sigma}^+ \hat{c}_{l+1,\sigma}^{\phantom{+}} \left( 1- \hat{n}_{l+1,-\sigma} \right) \right] \end{eqnarray} where $\eta=-|R_{l+1}-R_l-a|/a<0$ is the relative change of lattice distances due to the Peierls distortion (Gebhard {\em et al.} I 1996). $\hat{\jmath}_{\rm inter,+}^{\rm H}$ creates a neighboring pair of a double occupancy and a hole. As a central result of (Gebhard {\em et al.} II 1996) we found the band structure representation of the current operator as \begin{mathletters} \begin{equation} \hat{\jmath}_{\rm inter,+}^{\rm band}= \sum_{|k|,|q|\leq \pi/a} iea\epsilon(k) \hat{u}_{k+q/2}^+\hat{l}_{k-q/2}^{\phantom{+}} \hat{x}_{q}^{\phantom{+}} \; . \label{jhlbandpicture} \end{equation} for the translational invariant case, and \begin{eqnarray} \hat{\jmath}_{{\rm inter},+}^{\rm band,\, dim}&=& \sum_{|q|,|k|\leq \pi/(2a)}\biggl\{ iea \epsilon(k) \Bigl[ \hat{u}_{k+q/2}^+ \hat{l}_{k-q/2}^{\phantom{+}} - \hat{u}_{k+q/2+\pi/a}^+ \hat{l}_{k-q/2+\pi/a}^{\phantom{+}} \Bigr] \hat{x}_{q}^{\phantom{+}} \label{jinterband}\\[9pt] && \phantom{\sum_{|q|,|k|\leq \pi/(2a)}\biggl\{ } +ea \frac{\Delta(k)}{\delta} \Bigl[ \hat{u}_{k+q/2+\pi/a}^+ \hat{l}_{k-q/2}^{\phantom{+}} - \hat{u}_{k+q/2}^+ \hat{l}_{k-q/2+\pi/a}^{\phantom{+}} \Bigr] \hat{x}_{q+\pi/a}^{\phantom{+}} \biggr\} \nonumber \end{eqnarray} \end{mathletters}% for a Peierls-distorted lattice. The diagonalized form of the interband current operator is given in (Gebhard {\em et al.} II 1996). Since the excitation energy only depends on the charge configuration the complicated spin problem could be hidden in the operators~$\hat{x}_q$ which are defined only in terms of their product, {\arraycolsep=0pt\begin{eqnarray} \hat{x}_q^{+}(\delta,\eta)\hat{x}_{q'}^{\phantom{+}}(\delta,\eta) &=& \sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}} \frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} \bigl(1+\eta\delta +(-1)^l(\delta+\eta)\bigr) \bigl(1+\eta\delta +(-1)^r(\delta+\eta)\bigr) \nonumber \\[6pt] && \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}} \frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} } \langle 0 | S_1^{\prime},\ldots S_{l-1}^{\prime}, \left( \uparrow_{l}\downarrow_{l+1}-\downarrow_{l}\uparrow_{l+1}\right), S_{l}^{\prime},\ldots S_{L-2}^{\prime} \label{xqxqprime} \rangle \\[6pt] && \phantom{\sum_{S_1^{\prime},\ldots S_{L-2}^{\prime}} \frac{1}{L^2} \sum_{l,r} e^{i(ql-q'r)} } \langle S_{L-2}^{\prime},\ldots S_{r}^{\prime}, \left( \downarrow_{r+1} \uparrow_{r}-\uparrow_{r+1}\downarrow_{r}\right), S_{r-1}^{\prime},\ldots S_{1}^{\prime} | 0 \rangle \; . \nonumber \end{eqnarray} In practice, $q'=q$ or $q'=q+\pi/a$.} In (Gebhard {\em et al.} II 1996) we studied the ``hot-spin'' case where all $2^L$~spin configurations were equally possible ground states. Now we will set up the ``no-recoil'' approximation for the case of the (unique) ground state of the Heisenberg model, equation~(\ref{dimHeisHamilt}). \section{No-recoil approximation} \label{norecoilsec} \subsection{Basic ideas} Although the eigenenergies and even the eigenstates of the Hubbard model at strong coupling nicely display charge-spin separation (Ogata and Shiba 1990); (Parola and Sorella 1990) the two dynamical degrees of freedom are coupled again by the current operator that creates an optical excitation. The oscillator strength for such an excitation, however, factorizes into a charge and a spin part. For the charge part our band structure picture applies. The spin part is determined by a ground state correlation function for nearest-neighbor singlets at all lattice distances, see below. Hence, the calculation of the optical absorption spectrum remains a difficult many-particle problem even for a formally charge-spin separated system. If we want to make further progress we have to make assumptions on the behavior of the spin system. In~(Gebhard {\em et al.} II 1996) we treated the case of a degenerate spin background (``hot-spin case'') where all spin configurations were equivalent. Now we are interested in the more realistic case where the optical excitation starts from the unique ground state of the Heisenberg model, equation~(\ref{dimHeisHamilt}). Our band structure interpretation suggests an analogy between our strongly correlated Mott-Hubbard system and the situation in an intrinsic semiconductor with a direct gap. The Hubbard bands for the charges correspond to conduction and valence bands, while the spin degrees of freedom play the role of phonons in a semiconductor or metal. In both cases the energy scales are well separated, the spinon (phonon) energy being much smaller than the chargeon (electron) energy. The spin spectrum is gaped in the presence of dimerization ($\delta_S\neq 0$) such that the spin excitations correspond to optical rather than acoustical phonons in that case. This appealing analogy is also the basic idea behind the tJ-model approach to high-Tc superconductivity (Zhang and Rice 1988); (Dagotto 1994). In electron-phonon systems one may often {\em ignore\/ } phonon emission or absorption during an excitation at low temperatures. This is an approximation even at zero temperatures since phonons can always be emitted even if phonon absorption is impossible. Effectively there is a finite probability (``Debye-Waller factor'') that there is no momentum transferred to the phonons during an excitation by photons. This constitutes the ``no-recoil approximation''. It has many applications in solid state science (e.g., X-ray scattering, M\"{o}\ss bauer-effect), and it is also very successfully applied for the explanation of optical spectra of semiconductors (Haug and Koch 1990). Thus we are confident that a (modified) ``no-recoil approximation'' will also be valid for the strongly correlated Hubbard model where chargeons and spinons are energetically well separated. \subsection{Formulation of the approximation} To formulate the approximation for our case we will scrutinize equation~(\ref{xqxqprime}) for the spin sector. Recall that the charge sector has already been mapped onto a standard band structure picture. We apply {\arraycolsep=0pt\begin{eqnarray} | S_1^{\prime},\ldots S_{l-1}^{\prime}, \left( \uparrow_{l},\downarrow_{l+1}-\downarrow_{l},\uparrow_{l+1}\right), S_{l}^{\prime},\ldots S_{L-2}^{\prime} \rangle &=& \\[6pt] && \hspace*{-6pt} \hat{{\cal T}}_S^{(l-2)}\hat{{\cal T}}_{S'}^{-(l-2)} | S_1^{\prime} \left( \uparrow_{2},\downarrow_{3}-\downarrow_{2},\uparrow_{3}\right), S_{2}^{\prime},\ldots S_{L-2}^{\prime} \rangle \nonumber \end{eqnarray} for both expectation values in equation~(\ref{xqxqprime}) where~$\hat{\cal T}_{S}$ shifts all spins by one site and $\hat{\cal T}_{S'}$ shifts all spins but those at sites~$2$ and~$3$.} Furthermore, we use the identity \begin{equation} 2 \sum_{S_2,S_3} \left(\frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} \right) | S_2, S_3\rangle\langle S_3, S_2 | \left(\frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} \right) = | \left( \uparrow_2, \downarrow_3 - \uparrow_3, \downarrow_2\right) \rangle \langle \left( \downarrow_3, \uparrow_2 - \downarrow_2, \uparrow_3\right)| \; . \end{equation} Now that we have eliminated the restrictions on the intermediate spin sum we can carry it out and are left with a complicated ground state expectation value, \begin{mathletters} \begin{eqnarray} \hat{x}_q^{+}\hat{x}_{q'}^{\phantom{+}} &=& 2 \langle 0 | \hat{Z}_{2,3}^{+}(q) \left(\frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} \right) \hat{Z}_{2,3}^{\phantom{+}}(q') | 0 \rangle \\[6pt] \hat{Z}_{2,3}^{\phantom{+}}(q) &=& \frac{1}{L} \sum_l e^{-iqla} \hat{{\cal T}}_S^{(l-2)}\hat{{\cal T}}_{S'}^{-(l-2)} \left( 1+ (-1)^l\delta\right)\left( 1+ (-1)^l\eta\right) \; . \end{eqnarray} \end{mathletters}% Thus far the expressions are exact. Note that $\hat{x}_q^+\hat{x}_{q'}^{\phantom{+}}$ is almost --but not quite-- the double Fourier transform of the correlation function of nearest-neighbor singlets at site~$l$ and~$r$. To obtain further insight into the problem we first analyze two special cases which can exactly be solved before we formulate the general approximation. \subsubsection{N\'{e}el state} The N\'{e}el state is the ground state of the anisotropic Heisenberg model, equation~(\ref{dimHeisHamilt}), for $J_{\perp}=0$ (Ising model), \begin{equation} | \hbox{AF}\rangle = \prod_{l=1}^{L/2} \hat{c}_{2l,\uparrow}^+\hat{c}_{2l+1,\downarrow}^+|\hbox{vacuum}\rangle \; . \end{equation} It displays a perfect double-periodic structure, i.~e., if we shift all spins by two lattice sites we recover $|\hbox{AF}\rangle$. Hence, \begin{equation} \hat{{\cal T}}_S^{(l-2)}\hat{{\cal T}}_{S'}^{-(l-2)} |\hbox{AF}\rangle =\frac{1}{2} \left[\left(1+(-1)^l\right) + \left( 1 - (-1)^l\right) \hat{{\cal T}}_S^{\phantom{1}} \hat{{\cal T}}_{S'}^{-1}\right] |\hbox{AF}\rangle \; . \label{AFshift1} \end{equation} The wave function renormalization factors become \begin{mathletters} \label{AFshift2} \begin{eqnarray} \hat{Z}_{2,3}^{\phantom{+}}(q) |\hbox{AF}\rangle &=& \frac{\delta_{q,0}+\delta_{q,\pi/a}}{2}|\hbox{AF}(q)\rangle \\[6pt] |\hbox{AF}(q)\rangle &=& \left( (1+\delta)(1+\eta) + e^{iqa} (1-\delta)(1-\eta) \hat{{\cal T}}_S^{\phantom{1}}\hat{{\cal T}}_{S'}^{-1} \right) |\hbox{AF}\rangle \; . \end{eqnarray} \end{mathletters}% Thus we may write \begin{equation} \left. \hat{x}_q^+\hat{x}_{q'}^{\phantom{+}} \right|_{\rm AF} = \frac{1}{2} \left(\delta_{q,0}+\delta_{q,\pi/a} \right) \left(\delta_{q',0}+\delta_{q',\pi/a} \right) \langle \hbox{AF}(q) | \frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} | \hbox{AF}(q') \rangle \; . \label{neuneuneuAF} \label{AFshift3} \end{equation} For later use we define \begin{mathletters} \label{theZs} \begin{eqnarray} Z_0^{\rm AF} &=& \langle \hbox{AF}(0) | \frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} | \hbox{AF}(0) \rangle \\[6pt] Z_{\pi}^{\rm AF} &=& \langle \hbox{AF}(\pi/a) | \frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} | \hbox{AF}(\pi/a) \rangle \\[6pt] Z_M^{\rm AF} &=& \left[ \langle \hbox{AF}(0) | \frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} | \hbox{AF}(\pi/a) \rangle + {\rm h.c.}\right] \end{eqnarray} \end{mathletters}% such that $\left. \hat{x}_0^+\hat{x}_{0}^{\phantom{+}} \right|_{\rm AF} =Z_0^{\hbox{\scriptsize AF}}/2$, $\left. \hat{x}_{\pi/a}^+\hat{x}_{\pi/a}^{\phantom{+}} \right|_{\rm AF} =Z_{\pi}^{\hbox{\scriptsize AF}}/2$, and $\left. \left(\hat{x}_{\pi/a}^+\hat{x}_{0}^{\phantom{+}} + \hat{x}_{0}^+\hat{x}_{\pi/a}^{\phantom{+}} \right) \right|_{\rm AF} =Z_M^{\hbox{\scriptsize AF}}/2$. Explicitly, \begin{equation} \begin{array}{@{}rcl@{}} Z_0^{\hbox{\scriptsize AF}} &=& 2(\delta+\eta)^2\\ Z_{\pi}^{\hbox{\scriptsize AF}} &=& 2(1+\delta\eta)^2 \\ Z_M^{\hbox{\scriptsize AF}} &=& 4(\delta+\eta)(1+\delta\eta) \end{array} \; . \end{equation} The N\'{e}el state dominantly provides the momentum $\Delta q_S=\pi/a$ to the charge system during an optical excitation since $Z_{\pi}^{\hbox{\scriptsize AF}} > Z_M^{\hbox{\scriptsize AF}}> Z_0^{\hbox{\scriptsize AF}}$ for generic values for $\delta$, $\eta$. We note that for $\delta=\eta=0$ only $Z_{\pi}^{\hbox{\scriptsize AF}}$ contributes. This implies that for a N\'{e}el state one of the two Hubbard bands can be thought as being shifted by $\pi/a$ which results in vertical transitions between {\em antiparallel\/} bands as for non-interacting electrons in a Peierls-distorted lattice (Gebhard {\em et al.} I 1996). \subsubsection{Dimer state} The ground state of the fully dimerized Heisenberg model, equation~(\ref{dimHeisHamilt}) for $J_{\perp}=J_z\equiv J$ and $\delta_S=1$, is the dimer state \begin{equation} | \hbox{DIM}\rangle = \prod_{l=1}^{L/2} \sqrt{\frac{1}{2}} \left( \hat{c}_{2l,\uparrow}^+\hat{c}_{2l+1,\downarrow}^+ - \hat{c}_{2l,\downarrow}^+\hat{c}_{2l+1,\uparrow}^+\right)|\hbox{vacuum} \rangle \; . \end{equation} Again, the dimer state is invariant under a shift of all spins by two lattice sites. Hence, equations~(\ref{AFshift1}), (\ref{AFshift2}), (\ref{AFshift3}), and (\ref{theZs}) analogously hold for the dimer state. In particular, \begin{mathletters} \begin{eqnarray} \left(\frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3}\right) |\hbox{DIM}\rangle&= & |\hbox{DIM}\rangle \\[3pt] \left(\frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} \right) \hat{{\cal T}}_S^{\phantom{1}}\hat{{\cal T}}_{S'}^{-1} |\hbox{DIM}\rangle&= & -\frac{1}{2} |\hbox{DIM}\rangle \end{eqnarray}\end{mathletters}% holds such that the $Z$-factors for the dimer state become \begin{equation} \begin{array}{@{}rcl@{}} Z_0^{\hbox{\scriptsize DIM}} &=& \left[1+\delta\eta +3(\delta+\eta)\right]^2/4\\ Z_{\pi}^{\hbox{\scriptsize DIM}} &=&\left[3(1+\delta\eta) +\delta+\eta\right]^2/4\\ Z_M^{\hbox{\scriptsize DIM}} &=& \left[1+\delta\eta +3(\delta+\eta)\right] \left[3(1+\delta\eta) +\delta+\eta\right]/2 \end{array} \; . \end{equation} We see again that momentum transfer by $\Delta q_S=\pi/a$ dominates in the dimer state but that there is zero momentum transfer even for $\delta=\eta=0$, in contrast to the N\'{e}el state. \subsubsection{General case} The above examples are limiting cases for the ground state of the general Heisenberg model in equation~(\ref{dimHeisHamilt}). The correlation function for nearest-neighbor spin singlets is long-ranged ordered such that the spin system only allows for momentum transfers~$\Delta q_S=0, \pi/a$. It has recently been shown by (Talstra, Strong, and Anderson 1995) that this order persists in the standard Heisenberg model ($J_{\perp}=J_z=J$, $\delta_S=0$) as ``hidden'' long-range order although their is neither dimer nor antiferromagnetic order in the ground state. These observations make us confident that the following ``no-recoil'' approximation will give the coherent features of the optical absorption spectrum for all values of $J_{\perp}/J_z>0$ and $\delta_S$ in the dimerized Heisenberg model. \begin{equation} \hat{{\cal T}}_S^{l}\hat{{\cal T}}_{S'}^{-l} |0\rangle = \left[ w_{\rm DW} \left(\frac{1+(-1)^l}{2}\right) + \overline{w_{\rm DW}} \left(\frac{1-(-1)^l}{2}\right) \hat{{\cal T}}_S^{\phantom{1}}\hat{{\cal T}}_{S'}^{-1} \right]|0\rangle \quad + {\rm rest} \; . \label{norecoildim} \end{equation} Due to the ``hidden'' long-range order of spin singlets in the ground state we expect that we need two different ``Debye-Waller factors'' $w_{\rm DW}$, $\overline{w_{\rm DW}}$ as in the cases of the N\'{e}el and dimer state where they both were unity. In general, the square of their absolute value is smaller than one as can be checked from sum rules. It is clear that more elegant approximations than this can be designed, e.g., an~$l$-dependent factor~$w_{\rm DW}(l)$ could be introduced to mimic a finite correlation length for finite temperatures as in (Gebhard {\em et al.} II 1996). We will not follow the latter ideas here. The wave function renormalization factors become \begin{mathletters} \begin{eqnarray} \hat{Z}_{2,3}^{\phantom{+}}(q) |0\rangle &=& \frac{\delta_{q,0}+\delta_{q,\pi/a}}{2}|\Psi(q)\rangle \quad + {\rm rest} \\[6pt] |\Psi(q)\rangle &=& \left( w_{\rm DW}(1+\delta)(1+\eta) + e^{iqa} \overline{w_{\rm DW}}(1-\delta)(1-\eta) \hat{{\cal T}}_S^{\phantom{1}}\hat{{\cal T}}_{S'}^{-1} \right) |0\rangle \; . \end{eqnarray} \end{mathletters}% Thus we may write \begin{equation} \left. \hat{x}_q^+\hat{x}_{q'}^{\phantom{+}} \right|_{\rm coh} = \frac{1}{2} \left(\delta_{q,0}+\delta_{q,\pi/a} \right) \left(\delta_{q',0}+\delta_{q',\pi/a} \right) \langle \Psi(q) | \frac{1}{4}-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3} | \Psi(q') \rangle \; . \label{neuneuneu} \end{equation} In the presence of a Peierls distortion there is no recoil from the spin system since the reciprocal lattice vector is given by~$Q=\pi/a$. Even if the lattice distortion was absent, however, the translational symmetry is broken by the ``hidden'' long-range order in the nearest-neighbor singlet correlation function. The coherent part of the spin contribution can then be expressed with the help of $\left. \hat{x}_0^+\hat{x}_{0}^{\phantom{+}} \right|_{\rm coh} =Z_0/2$, $\left. \hat{x}_{\pi/a}^+\hat{x}_{\pi/a}^{\phantom{+}} \right|_{\rm coh} =Z_{\pi}/2$, and $\left. \left(\hat{x}_{\pi/a}^+\hat{x}_{0}^{\phantom{+}} + \hat{x}_{0}^+\hat{x}_{\pi/a}^{\phantom{+}} \right) \right|_{\rm coh} =Z_M/2$. \section{Optical conductivity in the Hubbard model at strong coupling} \label{sechubb} \label{optabsHubbard} In~(Gebhard {\em et al.} II 1996) we calculated the optical absorption for the case of an incoherent spin background. In the present case we only have to replace the expressions for the operator products~$\hat{x}_q^+\hat{x}_{q'}^{\phantom{+}}$ by those for the ``no-recoil approximation'', equations~(\ref{norecoildim}) and~(\ref{neuneuneu}). \subsection{Band case: $V=0$} Since the ground state implicitly breaks the translational symmetry of our system even for $\delta=\eta=0$ we directly approach the case of non-vanishing $\delta$, $\eta$. For the dimerized case we have to diagonalize the current operator, equation~(\ref{jinterband}), in terms of the quasi-particle operators for the respective Peierls bands. The result for the optical conductivity can be written as (Gebhard {\em et al.} II 1996) \begin{mathletters} \label{monstersigma} \begin{equation} {\rm Re}\{\sigma(\omega >0, \delta,\eta) \} = \frac{\pi {\cal N}_{\perp}}{L a \omega} \sum_{\tau,\tau'=\pm 1} \sum_{|q|,|k| \leq \pi/(2a)} \left| \lambda_{\tau,\tau'} (k,q)\right|^2 \delta(\omega - E_{\tau,\tau'}(k,q)) \end{equation} with the absorption energies between the respective Peierls subbands \begin{equation} E_{\tau,\tau'}(k,q) = U +\tau' E(k+q/2)-\tau E(k-q/2) \; . \label{Etautauprime} \end{equation} \end{mathletters}% The transition matrix elements are given by $\lambda_{+,+}(k,q)=-\lambda_{-,-}(k,q)$, $\lambda_{-,+}(k,q)= \lambda_{+,-}(k,q)$, and \begin{mathletters} \label{thelambdas} \begin{eqnarray} \lambda_{+,+}(k,q) &=& iea \left[ \epsilon(k) (\alpha_{+}^{\phantom{*}}\alpha_{-}^* -\beta_{+}^{\phantom{*}}\beta_{-}^*) \hat{x}_q^+ +\frac{\Delta(k)}{\delta} (\alpha_{+}^{\phantom{*}}\beta_{-}^* +\beta_{+}^{\phantom{*}}\alpha_{-}^*) \hat{x}_{q+\pi/a}^+ \right] \\[6pt] \lambda_{+,-}(k,q) &=& iea \left[ - \epsilon(k) (\alpha_{+}^{\phantom{*}}\beta_{-}^* +\beta_{+}^{\phantom{*}}\alpha_{-}^*) \hat{x}_q^+ +\frac{\Delta(k)}{\delta} (\alpha_{+}^{\phantom{*}}\alpha_{-}^* -\beta_{+}^{\phantom{*}}\beta_{-}^*) \hat{x}_{q+\pi/a}^+ \right] \; . \end{eqnarray} \end{mathletters}% Here we used the short-hand notation $\alpha_{\pm}=\alpha_{k\pm q/2}$ etc.\ for the mixing amplitudes in the standard Bogoliubov transformation (Gebhard {\em et al.} I and II 1996). For the coherent part of the optical absorption we only need their values at~$q=0$. In this case, $\alpha_k^2-\beta_k^2=-\epsilon(k)/E(k)$ and $2 \alpha_k\beta_k =-\Delta(k)/E(k)$. We thus find $\left. |\lambda_{+,\pm }(k,q)|^2\right|_{\rm coh} = \delta_{q,0} |\lambda_{+,\pm}(k)|^2$ with \begin{mathletters} \begin{eqnarray} |\lambda_{+,+}(k)|^2 &=& \frac{(ea)^2}{2} \left[ Z_0 \frac{\epsilon(k)^4}{E(k)^2} + \delta^2 Z_{\pi} \frac{\Delta(k)^4}{\delta^4 E(k)^2} + \delta Z_M \left(\frac{\epsilon(k)\Delta(k)}{\delta E(k)}\right)^2\right] \\[6pt] |\lambda_{+,-}(k)|^2 &=& \frac{(ea)^2}{2} \left( \frac{\epsilon(k)\Delta(k)}{\delta E(k)} \right)^2 \left[ \delta^2 Z_0+Z_{\pi}-\delta Z_M\right] \; . \end{eqnarray} \end{mathletters}% Each of the two quantities above can be expressed in terms of an expectation value of the projection operator $(1/4-\hat{\rm\bf S}_2\hat{\rm\bf S}_{3})$. Hence we can be sure that the result is positive or zero. For vanishing nearest-neighbor interaction we may directly use equation~(\ref{monstersigma}) to arrive at \begin{eqnarray} {\rm Re}\{\sigma_{\rm coh}(\omega >0, \delta,\eta) \} & = & \frac{\pi {\cal N}_{\perp}}{a \omega} \biggl[ \frac{2}{L} \sum_{|k| \leq \pi/(2a)} \delta(\omega-U) \left| \lambda_{+,+} (k)\right|^2 \\[6pt] && + \frac{1}{L} \sum_{|k| \leq \pi/(2a)} \left| \lambda_{+,-} (k)\right|^2 \Bigl( \delta(\omega - U- 2E(k)) +\delta(\omega - U+ 2E(k)) \Bigr)\biggr] \; . \nonumber \end{eqnarray} For later use we define the abbreviations \begin{mathletters} \label{handx} \begin{eqnarray} H_1&=& \frac{16}{W^2 L}\sum_{|k|\leq \pi/(2a)} \frac{\epsilon(k)^4}{E(k)^2} = 1-\delta^2 H_3\\[6pt] H_2&=& \frac{16}{W^2 L}\sum_{|k|\leq \pi/(2a)} \frac{\Delta(k)^4}{\delta^4 E(k)^2} = \frac{1}{\delta^2}\left( 1 -H_3\right)\\[6pt] H_3&=& \frac{16}{W^2 L}\sum_{|k|\leq \pi/(2a)} \left( \frac{\epsilon(k)\Delta(k)}{\delta E(k)}\right)^2 = \left(\frac{1}{1+|\delta|}\right)^2 \\[6pt] X(\omega) &=& -\frac{1}{\pi}\frac{16}{W^2 L}\sum_{|k|\leq \pi/(2a)} \left( \frac{\epsilon(k)\Delta(k)}{\delta E(k)}\right)^2 \frac{\omega-U}{(\omega-U)^2-4E(k)^2} \; . \end{eqnarray} \end{mathletters}% For vanishing nearest-neighbor interaction we only need the imaginary part of the function~$X(\omega)$ which can easily be calculated. The real part of the coherent optical conductivity finally becomes \begin{eqnarray} {\rm Re}\{\sigma_{\rm coh}(\omega >0, \delta,\eta) \} &=& \frac{\pi {\cal N}_{\perp}(Wea)^2}{ 16 a\omega} \biggl[ \left(Z_0 H_1+\delta^2 Z_{\pi}H_2+\delta Z_MH_3\right) \delta(\omega-U) \label{whynotnumbered} \\[6pt] && + \frac{2(\delta^2 Z_0+Z_{\pi}-\delta Z_M) \sqrt{\left[(\omega-U)^2-(W\delta)^2\right]\left[W^2-(\omega-U)^2\right]} }{ \pi W^2(1-\delta^2)^2|\omega-U|} \nonumber \end{eqnarray} in the regions~$\omega=U$ and $W\delta\leq |\omega-U|\leq W$. Thus the generic coherent features are (i)~a $\delta$-peak at $\omega=U$ which results from vertical transitions between the parallel Hubbard bands ($\Delta q_C=\Delta q_S=0$), and (ii)~a band-to-band transition feature for $W\delta\leq |\omega-U|\leq W$ which results from vertical transitions between two antiparallel bands ($\Delta q_C=\Delta q_S=\pi/a$). To check the sum rule we integrate over the reduced coherent optical conductivity and obtain \begin{eqnarray} \int_0^{\infty} \frac{d\omega}{W} \sigma_{\rm red, coh}(\omega,\delta,\eta) &=& \frac{\pi}{16}\left(Z_0+Z_{\pi}\right) \\[3pt] &=& \frac{\pi}{4}\left[ |w_{\rm DW}|^2 (1+\delta)^2 (1+\eta)^2 C_S^{\rm even} + |\overline{w}_{\rm DW}|^2 (1-\delta)^2 (1-\eta)^2 C_S^{\rm odd} \right]\nonumber \end{eqnarray} where $C_S^{\rm even,odd}$ is nearest-neighbor spin-correlation function on even/odd sites, see equation~(\ref{CSevenCSodd}) of the appendix. The comparison with the sum rule, equation~(\ref{sumrulesigmared}), shows that~$|w_{\rm DW}|^2\leq 1$ and $|\overline{w_{\rm DW}}|^2 \leq 1$ have to hold which indeed allows to interpret them as Debye-Waller factors. For $\delta=1$, $\eta=0$ ($L/2$ independent two-site systems) we have $Z_0=Z_{\pi}=Z_M/2=8 |w_{\rm DW}|^2 C_S^{\rm even}$. The contribution of the band vanishes again, and we obtain a single peak at $\omega=U$ with the coherent fraction $|w_{\rm DW}|^2=1$ of the total oscillator strength. The case of a vanishing Peierls distortion ($\delta=\eta=0$) is particularly interesting. Recall that in the case of an incoherent spin background (``hot-spin'' case) (Gebhard {\em et al.} II) we observed a logarithmic divergence at $\omega=U$. Now we find from equation~(\ref{whynotnumbered}) that \begin{equation} {\rm Re}\{\sigma_{\rm red, coh}(\omega >0) \} = \frac{1}{16}\left[W\pi Z_0 \delta(\omega-U) + 2Z_{\pi}\sqrt{1-\left( \frac{\omega-U}{W}\right)^2}\, \right] \end{equation} in which the proper $Z$-values for $\delta=\eta=0$ have to be inserted. In general, the absorption consists of a $\delta$-peak at $\omega=U$ and a semielliptic contribution which come from vertical transitions between the parallel ($\Delta q_C=\Delta q_S=0$) and antiparallel ($\Delta q_S=\Delta q_C=\pi/a$) bands. For the N\'{e}el state $Z_0^{\hbox{\scriptsize AF}}(\delta=\eta=0)=0$, and $Z_{\pi}^{\hbox{\scriptsize AF}}(\delta=\eta=0)=2$, and the result of (Lyo and Galinar 1977); (Lyo 1978) is recovered. It is {\em only\/} the N\'{e}el state that suppresses the physics of the parallel Hubbard bands for $\delta=\eta=0$. If we had only considered this state as reference state, the physics of the parallel Hubbard bands would have been missed. The dimer state with $Z_0^{\hbox{\scriptsize DIM}}(\delta=\eta=0)=1/4$, and $Z_{\pi}^{\hbox{\scriptsize DIM}}(\delta=\eta=0)=9/4$ does show the generic features. The exact results for both states are depicted in figure~\ref{heisfig00}. For nonzero lattice dimerization ($\delta\neq 0$, $\eta\neq 0$) even the N\'{e}el state reproduces the generic situation. The coherent peak at $\omega=U$ is not at all a consequence of the lattice distortion but a consequence of the parallel Hubbard bands. This has been overlooked in previous analytical investigations (Lyo and Galinar 1977); (Lyo 1978); (Galinar 1979). Numerical calculations (Campbell, Gammel, and Loh 1989) for $\delta=\eta=0$ essentially give the results from the N\'{e}el ground state since this state dominates for small system sizes. Small traces of a peak at $\omega=U$ may have been washed out by the adopted smoothing procedure. For non-zero lattice distortion, however, the $\delta$-resonance at $\omega=U$ becomes clearly visible even in the numerical simulations. The exact results for the optical absorption of the N\'{e}el and dimer state in the presence of a lattice distortion are shown in figure~\ref{heisfig10}. For realistic states the $Z$-factors will not be too different such that the figure should reproduce the generic situation for the Peierls-distorted Hubbard model at strong correlations and low temperatures. This observation further supports our ``no-recoil approximation''. Figure~\ref{heisfig10} has to be compared to figure~4 of (Gebhard {\em et al.} II 1996). It is seen that the linear absorption at the threshold~$\omega=U-W$ now shows a square-root behavior. This is also the case in the ``hot-spin'' case for large enough~$\delta$ when the Peierls gap has not been smeared out. It is again seen that the significant features of the absorption spectrum have not changed much when we go from the Harris-Lange model to the strongly correlated Hubbard model. \subsection{Exciton case: $V\neq 0$} In the presence of a nearest-neighbor interaction we have to solve an integral equation when the lattice is distorted. Note that only the absolute position of the resonances will be determined by the Coulomb parameter~$U/t$ while their relative position depends on $V/t$. For illustrative purposes we tune $V/t$ independently of~$U/t$ and put aside the question of the stability of the ground state against the formation of a charge density wave. The calculations are lengthy and will not be redone here, see appendix~B of (Gebhard {\em et al.} II 1996) for details. The result can be expressed in terms of two operator-valued functions~$G_{1,2}(q)$, \begin{mathletters} \label{CapitalGred} \begin{eqnarray} G_1(q)&=& itea \left[ \hat{x}_q^+ F_1(q) - \hat{x}_{q+\pi/a}^+ F_3(q)\right]\\[6pt] G_2(q)&=& itea \left[ -\hat{x}_q^+ F_3(q) + \hat{x}_{q+\pi/a}^+ F_2(q)\right]\; , \end{eqnarray}\end{mathletters}% and three $q$-dependent functions $F_{1,2,3}(q)$ as {\arraycolsep=0pt\begin{eqnarray} {\rm Re}\{\sigma(\omega >0, V, \delta,\eta) \} &=& {\rm Re}\{\sigma(\omega >0,\delta,\eta) \} + \frac{2V{\cal N}_{\perp}}{a\omega} \label{mostimportant} \\[6pt] &&{\rm Im}\Biggl\{ \sum_{|q| \leq \pi/(2a) } \frac{1}{(1+VF_1)(1+VF_2)-(VF_3)^2} \biggl[G_1^{\phantom{+}}G_1^+ + G_2^{\phantom{+}}G_2^+ \nonumber \\[6pt] && \phantom{ {\rm Im}\biggl\{ \sum_{|q| \leq \pi/(2a) } } +V \left( G_1^{\phantom{+}}G_1^+F_2 + G_2^{\phantom{+}}G_2^+F_1 + (G_1^{\phantom{+}}G_2^+ + G_2^{\phantom{+}}G_1^+)F_3 \right)\biggr]\Biggr\} \; . \nonumber\end{eqnarray} In the ``no-recoil approximation'' we only need the $q=0$ value of the functions $F_{1,2,3}$. We have} \begin{mathletters} \begin{eqnarray} F_1(q=0)&=& \frac{H_1}{\omega-U} - \pi\delta^2 X(\omega) \\[3pt] F_2(q=0)&=& \frac{\delta^2 H_2}{\omega-U} - \pi X(\omega) \\[3pt] F_3(q=0)&=& -\delta \left[ \frac{H_3}{\omega-U} + \pi X(\omega)\right] \; , \end{eqnarray} \end{mathletters}% see eqs.~(\ref{handx}). The definitions~(\ref{theZs}) allow us to reduce equation~(\ref{mostimportant}) to \begin{eqnarray} {\rm Re}\{\sigma_{\rm coh}(\omega >0, V, \delta,\eta) \} &=& {\rm Re}\{\sigma_{\rm coh}(\omega >0,\delta,\eta) \} \nonumber \\[6pt] && + \frac{V{\cal N}_{\perp}(Wea)^2}{16 a\omega} {\rm Im}\Biggl\{ \frac{1}{(1+VF_1)(1+VF_2)-(VF_3)^2} \label{thefinalresultHubb} \\[6pt] && \phantom{+} \biggl[ Z_0(F_1^2+F_3^2)+Z_{\pi}(F_2^2+F_3^2)-Z_M(F_1+F_2)F_3 \nonumber \\[3pt] && \phantom{+\biggl[ } +V\left(F_1F_2-F_3^2\right)\left(Z_0F_1+Z_{\pi}F_2-Z_MF_3\right)\biggr] \Biggr\}\; . \nonumber \end{eqnarray} Note that the {\em positions\/} of the excitons are determined by the zeros of the denominator which is a function of $\omega$, $V$, and $\delta$ but is independent of the $Z$-factors and $\eta$. The oscillator strength of an exciton, however, strongly depends on the $Z$-factors. For the translational invariant case, $\delta=\eta=0$, equation~(\ref{thefinalresultHubb}) can be simplified to ($F_3=0$, $F_1=1/(\omega-U)$, $F_2=-\pi X(\omega)$) \begin{equation} {\rm Re}\{\sigma(\omega >0,V) \} = - \frac{(Wea)^2{\cal N}_{\perp}}{16 a\omega} {\rm Im}\left\{ \frac{Z_0F_1}{1+VF_1} + \frac{Z_{\pi}F_2}{1+VF_2} \right\} \; . \end{equation} {}From the first part we see that the peak at $\omega=U$ for $V=0$ becomes a $q=0$-exciton at frequency $\omega_0=U-V$. For the N\'{e}el state this exciton is absent while it is present {\em in all generic cases}, e.~g., in the dimer state. The second part describes the formation of the $q=\pi/a$-exciton from the semielliptic ``band absorption'' part for $V=0$. The formation of this exciton leads to a redshift in the ``band absorption'' part as already noticed by (Galinar 1979). The exact result for $\delta=\eta=0$ for the dimer state is displayed in figure~\ref{heisfig01} for various values of $V/t$. For $V>W/2$ the $q=\pi/a$-exciton becomes a true bound state at $\omega_{\pi/a}=U-V-W^2/(4V)$ below the band absorption edge at $\omega_{\rm edge}=U-W$. For large enough~$V/t$ the fact that~$Z_{\pi} > Z_0$ implies that the $q=\pi/a$-exciton resonance is stronger than the one from the $q=0$-exciton. Since they are only separated by an energy difference $\delta\omega=W^2/(4V)$ thermal and disorder broadening could merge the two exciton lines into a single asymmetric absorption line. The general case of the optical absorption of the dimer state in the presence of a lattice distortion is shown in figure~\ref{heisfig11}. The N\'{e}el state results in a very similar curve such that we are confident that it reproduces the generic features for the coherent absorption in the Hubbard model at strong correlations. At $V=0$ we had the resonance at $\omega=U$ and a Peierls-split ``band absorption'' in the range $W\delta \leq |\omega-U|\leq W$. For moderate~$V/t$ the two bands evolve into two peaks such that three lines are visible. The peak at energy $\omega_{\pi/a}=U-V-W^2/(4V)$ corresponds to the $q=\pi/a$-exciton while the dominant peak is the~$q=0$-exciton at~$\omega_0=U-V$. The peak near $\omega=U$ corresponds to the {\em anti-bound\/} exciton which quickly looses oscillator strength as $V/t$ increases. For large~$V/t$ the $q=0$- and $q=\pi/a$-excitons dominate the absorption spectrum with a preference of the $q=\pi/a$-exciton since $Z_{\pi}>Z_0$. In addition, a weak structure remains near $\omega=U$ which does not play a role for the optical absorption but may become visible in electroabsorption, i.e., optical absorption in the presence of a static electrical field. The dimer state underestimates the factors~$Z_{\pi}$, $Z_M$ as compared to $Z_0$. For the N\'{e}el state the strength of the $q=0$-exciton is much weaker for realistic values for the lattice distortion. The result for the N\'{e}el state is shown in figure~\ref{heisfigextra}. It is now seen that the $q=\pi/a$-exciton strongly dominates the $q=0$-exciton. This is to be expected since the components which imply a momentum transfer $q=0$ to the spin system ($Z_0$, $Z_M$) are much weaker now. Even a large value of~$\delta$ does not drastically change this situation. It should be clear that the strengths of all exciton peaks above the exciton at~$\omega_{\pi/a}=U-V-W^2/(4V)$ are a direct measure for the dimerization degree {\em both\/} of the charge ($\delta,\eta \neq 0$) {\em and\/} the spin ($\delta_S\neq 0$) system. \section{Summary and conclusions} In this paper we have further studied the optical absorption for strongly-correlated electrons in half-filled Peierls-distorted chains. In (Gebhard {\em et al.} II 1996) we exactly solved the case of the Harris-Lange model with its incoherent spin background (``hot-spin case''). For the more realistic situation of low temperatures, i.~e., a unique ground state, exact solutions are impossible for the generic cases. To make further progress we assumed that the spin dynamics is irrelevant: (i)~we neglected corrections of the order~$J$ to the optical excitation energies and thus a singlet/triplet-splitting of the excitations, and (ii)~we assumed that no spin excitations are created during an optical excitation (``no-recoil approximation''). Then we could analytically investigate the problem even in the presence of a lattice dimerization~$\delta$ and a nearest-neighbor interaction~$V$ between the electrons. In the ``no-recoil approximation'' for the strongly correlated Hubbard model all the features already seen in the ``hot-spin case'' become more prominent since now only transitions with $\Delta q_S=0$ and $\Delta q_S=\pi/a$ are allowed. This corresponds to vertical transitions between parallel Hubbard bands ($\Delta q_S=\Delta q_C=0$) and antiparallel bands ($\Delta q_S=\Delta q_C=\pi/a$). The relative strength of the two processes can be measured by three parameters ($Z_0$, $Z_{\pi}$, $Z_M$) which depend on both the spin structure of the ground state and the lattice distortion. In the generic case the parallel Hubbard bands result in a single peak at $\omega=U$. The $\Delta q_S=\Delta q_C=\pi/a$ part supplements the optical absorption spectrum by two absorption bands for $W\delta \leq |\omega-U| \leq W$. An additional nearest-neighbor interaction results in a tightly bound exciton at $\omega_0=U-V$ and another exciton at $\omega_{\pi/a}= U-V - W^2/(4V)$. Simpson's exciton band has seized to exist since the exciton's center of mass momentum has to be $\Delta q_C=0$ or $\Delta q_C=\pi/a$. Again, the excitons draw the oscillator strength from the ``band absorption'' part. Although the location of the exciton resonances is independent of the spin configuration of the ground state their relative amplitude is a very sensitive measure of the lattice and spin dimerization. We thus come to the important conclusion that the linear {\em optical\/} absorption in strongly correlated electron systems might serve as a subtle probe for the {\em magnetic\/} structure of the ground state. In particular, optical absorption allows to measure the ``hidden'' long-range order of nearest-neighbor spin singlets (Talstra, Strong, and Anderson 1995). One might think that it is fairly simple to experimentally distinguish between a Peierls and a Mott-Hubbard insulator on the basis of their optical absorption. At low temperatures we expect a Peierls insulator to show a broad band-to-band transition while a Mott-Hubbard insulator with appreciable values for the Coulomb interaction ($U \gg W$, $V >W/2$) should display sharp exciton lines in the optical absorption spectrum. Unfortunately, a direct comparison to experiment is difficult for two reasons. First, disorder effects can inhomogeneously broaden single lines. Hence, an experimentally observed band can very well be a sign of disorder rather than an argument against a Mott-Hubbard insulator. Secondly, when a residual electron-electron interaction is included in a Peierls insulator (Abe, Yu, and Su 1992); (Abe, Schreiber, Su, and Yu 1992) one can equally well obtain excitons which draw the oscillator strength form the band transitions. Hence, single exciton lines are not a clear-cut indication against the presence of a Peierls insulator either. It is clear that one needs to compare model calculations and experimental observations on many more physical quantities than just the linear optical absorption before definite conclusions can be reached. \section*{Acknowledgments} We thank H.~B\"{a}\ss ler, A.~Horv\'{a}th, M.~Lindberg, S.~Mazumdar, M.~Schott, and G.~Weiser for useful discussions. The project was supported in part by the Sonderforschungsbereich~383 ``Unordnung in Festk\"{o}rpern auf mesoskopischen Skalen'' of the Deutsche Forschungsgemeinschaft. \newpage \begin{appendix} \section*{Sum rule} \setcounter{section}{1} We briefly account for the sum rule. We have \begin{equation} \int_{0}^{\infty} d\omega\ {\rm Im}\{\chi(\omega)\} = \pi \frac{{\cal N}_{\perp}}{La} \sum_n \left| \langle 0 | \hat{\jmath}^2|n\rangle\right|^2 = \pi \frac{{\cal N}_{\perp}}{La} \langle 0 | \hat{\jmath}^2|0\rangle \; . \end{equation} It is a standard exercise to show that \begin{equation} \langle 0 | \hat{\jmath}^2 | 0 \rangle = (2tea)^2 \sum_l \left(1+(-1)^l\delta\right)^2\left(1+(-1)^l\eta\right)^2 \langle 0 | \left( \frac{1}{4} -\hat{\rm\bf S}_l\hat{\rm\bf S}_{l+1}\right) | 0 \rangle \; . \end{equation} We define the positive quantities \begin{equation} C_S^{\rm even, odd}= \frac{1}{L} \sum_l \frac{1\pm (-1)^l}{2} \langle 0 | \left( \frac{1}{4} -\hat{\rm\bf S}_l\hat{\rm\bf S}_{l+1}\right) | 0 \rangle \label{CSevenCSodd} \end{equation} such that $C_S=C_S^{\rm odd}+C_S^{\rm odd}$ is the value of the nearest-neighbor spin-spin correlation function. Then we may write \begin{equation} \int_{0}^{\infty} d\omega\ {\rm Im}\{\chi(\omega)\} = \pi {\cal N}_{\perp}a (2te)^2 \left[ \left(1+\delta\right)^2\left(1+\eta\right)^2 C_S^{\rm even} + \left(1-\delta\right)^2\left(1-\eta\right)^2 C_S^{\rm odd} \right] \; . \end{equation} The area under the curves for~$\sigma_{\rm red}(\omega)$, equation~(\ref{sigmared}), is thus given by \begin{equation} \int_0^{\infty} \frac{d\omega}{W} \sigma_{\rm red}(\omega) = \frac{\pi}{4} \left[ \left(1+\delta\right)^2\left(1+\eta\right)^2 C_S^{\rm even} + \left(1-\delta\right)^2\left(1-\eta\right)^2 C_S^{\rm odd} \right] \label{sumrulesigmared} \; . \end{equation} \end{appendix} \newpage \begin{center} {\bf REFERENCES} \end{center} \begin{itemize} \item S.~Abe, M.~Schreiber, W.~P.~Su, and J.~Yu, Phys.~Rev.~B~{\bf 45}, 9432 (1992). \item S.~Abe, J.~Yu, and W.~P.~Su, Phys.~Rev.~B~{\bf 45}, 8264 (1992). \item L.~Alc\'{a}cer, in: {\sl Organic Conductors}, ed. by J.-P.\ Farges, (Marcel Dekker, New York (1994)). \item D.~Baeriswyl, P.~Horsch, and K.~Maki, Phys.~Rev.~Lett.~{\bf 60} (C), 70 (1988). \item D.~Baeriswyl, D.~K.~Campbell, and S.~Mazumdar, in: {\sl Conjugated Conducting Polymers}, ed.~by H.~Kiess, (Springer Series in Solid State Sciences~{\bf 102}, Springer, Berlin (1992)). \item A.~Brau and J.-P.~Farges, in: {\sl Organic Conductors}, ed. by J.-P.\ Farges, (Marcel Dekker, New York (1994)). \item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh, Phys.~Rev.~B~{\bf 38}, 12043 (1988). \item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh, Int.~J.~Mod.~Phys.~B~{\bf 3}, 2131 (1989). \item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh, in: {\sl Interacting Electrons in Reduced Dimensions}, ed.~by D.~Baeriswyl and D.~K.~Campbell, (NATO ASI Series~B~{\bf 213}, Plenum Press, New York (1989)), p.~171. \item D.~K.~Campbell, J.~T.~Gammel, and E.~Y.~Loh, Phys.~Rev.~B~{\bf 42}, 475 (1990). \item E.~Dagotto, Rev.~Mod.~Phys.~{\bf 66}, 763 (1994). \item P.~G.~J.~van Dongen, Phys.~Rev.~B~{\bf 49}, 7904 (1994). \item P.~G.~J.~van Dongen, Phys.~Rev.~B~{\bf 50}, 14016 (1994). \item F.~H.~L.~E\ss ler and V.~E.~Korepin (ed.), {\sl Exactly Solvable Models of Strongly Correlated Electrons}, (World Scientific, Singapore (1994)). \item J.-P.\ Farges (ed.), {\sl Organic Conductors}, (Marcel Dekker, New York (1994)). \item J.~L.~Fave, in: {\sl Electronic Properties of Polymers}, ed.~by H.~Kuzmany, M.~Mehring, and S.~Roth, (Springer Series in Solid State Sciences~{\bf 107}, Springer, Berlin (1992)). \item A.~Fritsch and L.~Ducasse, J.~Physique~I~{\bf 1}, 855 (1991). \item R.~M.~Fye, M.~J.~Martins, D.~J.~Scalapino, J.~Wagner, and W.~Hanke, Phys.~Rev.~B~{\bf 45}, 7311 (1992). \item J.-P.~Galinar, J.~Phys.~C~{\bf 12}, L335 (1979). \item J.~T.~Gammel and D.~K.~Campbell, Phys.~Rev.~Lett.~{\bf 60} (C), 71 (1988). \item F.~Gebhard, K. 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A broadening of~$\gamma=0.01W$ has been included.} \label{heisfig00} \end{figure} \typeout{figures} \begin{figure}[th] \caption{Reduced optical conductivity, $\sigma_{\rm red}(\omega >0, \delta,\eta)$, for the dimer and the N\'{e}el state for $U=2W$ in the presence of a lattice distortion, $\delta=0.2$, $\eta=-0.06$. A broadening of~$\gamma=0.01W$ has been included.} \label{heisfig10} \end{figure} \begin{figure}[th] \caption{Reduced optical conductivity, $\sigma_{\rm red}(\omega >0, V)$, for the dimer state for $U=2W$ in the presence of a nearest-neighbor interaction, $V=0,W/2,W$. A broadening of~$\gamma=0.01W$ has been included.} \label{heisfig01} \end{figure} \begin{figure}[th] \caption{Reduced optical conductivity, $\sigma_{\rm red}(\omega >0, V, \delta,\eta)$, for the dimer state for $U=2W$ in the presence of a lattice distortion, $\delta=0.2$, $\eta=-0.06$, and a nearest-neighbor interaction, $V=0, W/2, W$. A broadening of~$\gamma=0.01W$ has been included.} \label{heisfig11} \end{figure} \begin{figure}[th] \caption{Reduced optical conductivity, $\sigma_{\rm red}(\omega >0, V, \delta,\eta)$, for the N\'{e}el state for $U=2W$ in the presence of a lattice distortion, $\delta=0.2$, $\eta=-0.06$, and a nearest-neighbor interaction, $V=0, W/2, W$. A broadening of~$\gamma=0.01W$ has been included.} \label{heisfigextra} \end{figure} \end{document}

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\section{Improved Actions} In lattice QCD, one of the fundamental challenges is to minimize the errors caused by discretizing space-time. This is accomplished through a combination of advances in computer technology, and advances in the formulation of methods to solve the problem computationally, including the development of improved numerical algorithms. We show an example of how an improved gauge field action can be used to suppress artificial lattice contributions to physical measurements. We consider gluon actions that are constructed in a gauge invariant fashion, from a combination of Casimir invariants of closed Wilson loops. In principle, a lattice action of this type can consist of an arbitrary sum of Wilson loops, but a truncation to a small set of localized loops is necessary due to computational expense. We study actions constructed from $(1\times1)$ and $(1\times2)$ loops: \setlength{\tabcolsep}{0.00pc} \vspace{0.25cm} \begin{tabular}{@{\hspace{-0.5pc}}lcl@{\hspace{0.0pc}}} \multicolumn{2}{@{\hspace{-0.5pc}}l@{\hspace{0.0pc}}}{$S= K^{1\times1} $}&$ \displaystyle\sum {\rm Re}\,{\rm Tr}\, W^{(1 \times 1)} $ \cr $ + $&$ K^{1\times2} $&$ \displaystyle\sum {\rm Re}\,{\rm Tr}\, W^{(1 \times 2)} $ \cr $ + $&$ K_6^{1 \times 1} $&$ \displaystyle\sum {\rm Re}\,[ {3 \over 2} ({\rm Tr}\, W^{(1 \times 1)} )^2 - {1 \over 2} {\rm Tr}\, W^{(1 \times 1)} ] $ \cr $ + $&$ K_8^{1 \times 1} $&$ \displaystyle\sum [ {9 \over 8} {\rm Tr}\, W^{(1 \times 1)} |^2 - {1 \over 8} ]$ \cr \end{tabular} \vspace{0.25cm} where the actions and coefficients, in order of increasing improvement (approximation to the renormalization trajectory) are given by: \vspace{0.5cm} \setlength{\tabcolsep}{0.60pc} \begin{tabular}{@{\hspace{-0.5pc}}lcccc@{\hspace{0.0pc}}} Action &$ K^{1\times1} $&$ K^{1\times2} $&$K_6^{1 \times 1}$&$ K_8^{1 \times 1} $\cr WA &$ 1 $&$ 0 $&$ 0 $&$ 0 $ \cr SW &$ 5/3$&$-1/12$&$ 0 $&$ 0 $ \cr RGT &$ k $&$ -0.04k $&$ -0.12k $&$ -0.12k $ . \cr \end{tabular} \vspace{0.5cm} To compare these three actions, we generate four ensembles: \vspace{0.5cm} \setlength{\tabcolsep}{0.45pc} \begin{tabular}{@{\hspace{-0.5pc}}lrrrr@{\hspace{0.0pc}}} Action & \multicolumn{1}{c}{Size} &$\beta$& \multicolumn{1}{r}{$\approx\beta_{Wil}$}&$\ \ N$ \cr Wilson &$ (16^3\times40) $ &$\ \ 6.0 $&$ 6.0\,\,\, $&$ 35 $\cr SW &$ (16^3\times32) $ &$\ \ 4.2 $&$ 5.8\,\,\, $&$ 36 $\cr SW &$ (16^3\times32) $ &$\ \ 4.43$&$ 6.0\,\,\, $&$ 40 $\cr RGT &$ (18^3\times36) $ &$k=10.58$&$ 6.0\,\,\, $&$ 28 $\cr \end{tabular} \vspace{0.5cm} The two SW ensembles allow us to study the effect of increasing $\beta$, we have used estimates of corresponding Wilson action $\beta$ from the deconfining phase transition temperature calculation by Cella {\it et al.\/}\cite{Cella}. Since we have used a modest number of configurations in each case, we focus on the qualitative comparison between Wilson and improved actions. Further calculations with a larger number of lattices would be needed for quantitative studies, for example, to determine the consistency and scaling of $\chi_t/m_\rho$. \section{Topology: Comparing Actions} Lattice topology provides a test case for comparing various gauge field actions. There are several prescriptions for measuring topological charge $$ Q = {1\over32\pi^2} \int d^4x F(x) \tilde{F}(x) $$ on the lattice, and each prescription is subject to a different set of lattice cutoffs and renormalizations which affect the measurement of the topological susceptibility $\chi_t=\left\langle Q^2\right\rangle/V$. In the plaquette method the topological density $F(x)\tilde{F}(x)$ is constructed from a product of lattice $(1\times1)$ Wilson loops. This method in general gives noninteger values of the topological charge, and is affected by large multiplicative and additive lattice renormalizations\cite{mixing}. The geometric method\cite{Luscher} does guarantee an integer topological charge (except for ``exceptional'' configurations) but is not guaranteed to obey physical scaling in the continuum limit, and is in fact known to violate scaling for the Wilson action\cite{Gockeler89}. Low-action dislocations which can be suppressed by improving the action\cite{Gockeler89} contaminate the geometric $\chi_t$. In the cooling prescription, ultraviolet fluctuations in the fields are removed by locally minimizing the action in successive sweeps, isolating instanton-like configurations. After cooling, a single instanton configuration spanning several lattice spacings has a computed charge of nearly one using either the geometric or plaquette formula; we therefore apply the plaquette formula to the cooled configurations to obtain a value for $Q$. Lattice artifacts are very different among these methods, and we can in general get different results for plaquette ($Q_p$), the geometric ($Q_g$), and the cooling ($Q_c$) topological charges computed on the same original configuration. For improved actions, we expect lattice artifacts such as dislocations to be suppressed, therefore we test this prediction by comparing the different topological charge methods {\it with each other\/}. The cooling prescription actually encompasses a family of cooling algorithms. Typically one cools by selecting a link $U$ to minimize some action $S_c$, and since cooling is merely used as a tool to isolate instantons, there is no reason to tie $S_c$ to the Monte Carlo gauge action $S$. The cooling algorithms $S_c$ we consider here are linear combinations of Wilson loops with coefficients $c_{(1\times1)}$ and $c_{(1\times2)}$, and since action is minimized only the ratio $r_{12} = c_{(1\times2)}/c_{(1\times1)}$ is significant. The cooling algorithm with $r_{12}=-0.05$ removes the leading scaling violation from the classical instanton action, and we also include cooling algorithms with $r_{12}=0$ and $r_{12}=-0.093$, which has been derived from a linear weak coupling approximation to the RGT action, for comparison. For the case $r_{12}=0$, the lack of barrier to a decrease in the instanton size causes the instanton to disappear by implosion during the cooling process, and for $r_{12}=-0.093$ a large instanton expands until halted by the boundary. We cool for $200$ sweeps for all three algorithms, and the comparison between these three in an indication of the systematic effect of picking some particular means of cooling. We note that with $200$ sweeps of Wilson cooling most of the topological charges are retained, since the large instantons haven't had enough time to implode. In general, we do not see any effect from the selection of the cooling algorithm, except perhaps in one ensemble. \begin{figure}[t] \epsfxsize=7.0cm \epsfbox{wagcs1.eps} \caption{Comparison of geometric and cooled topological charges for Wilson action lattices at $a \approx 0.1{\rm fm}$. Each point represents one configuration, with the cooled charge as the absicca and the geometric charge on the uncooled lattice as the ordinate. The least squares linear fit is shown. Due to close overlaps there appear to be fewer that $35$ points.} \label{fig:qgcwa} \end{figure} \begin{figure}[t] \epsfxsize=7.0cm \epsfbox{gagcs1.eps} \caption{Comparison of geometric and cooled topological charges for RGT action lattices at $a \approx 0.1{\rm fm}$.} \label{fig:qgcra} \end{figure} \section{Results} As described above, we compute $Q_p$, $Q_g$, and $Q_c$ on all of our lattices. We show two scatter plots (Figures \ref{fig:qgcwa}, \ref{fig:qgcra}) highlighting the discrepancy between $Q_g$ and $Q_c$. The best fit line is constructed through the points on a scatter plot. The slope of this line is an estimate of the ratio of multiplicative renormalizations, and should be close to $1$ since both the geometric and cooling methods give integer charges. The correlation $$z_{gc} = \frac{\left\langle\left( Q_g - \bar{Q}_g\right) \left( Q_c - \bar{Q}_c\right)\right\rangle } {\sqrt{\left\langle\left(Q_g-\bar{Q}_g\right)^2\right\rangle \left\langle\left(Q_c-\bar{Q}_c\right)^2\right\rangle} } $$ between $Q_g$ and $Q_c$ is a measure of random additive artifacts seen by one method but not the other, such as dislocations which disappear during cooling therefore contributing only to $Q_g$ but not to $Q_c$. The scatter plots show a strong correlation between $Q_g$ and $Q_c$ for the RGT action, and a far weaker correlation for the WA, suggesting that the effect of lattice artifacts on topological charge is far less pronounced for the improved action. \begin{figure}[t] \epsfxsize=7.2cm \epsfbox{corrs.eps} \vspace{-0.5cm} \caption{Correlation $z_{gc}$ between geometric and cooled topological charges, for WA, SW, and RGT ensembles. Results for three different cooling methods are shown.} \label{fig:zgc} \end{figure} \begin{figure}[t] \epsfxsize=7.2cm \epsfbox{corrp.eps} \vspace{-0.5cm} \caption{Statistical correlation $z_{pc}$ between plaquette and cooled topological charges, for same ensembles, three cooling methods.} \label{fig:zpc} \end{figure} \begin{table}[b] \caption{Correlations, SW Cooling Linear Fits} \setlength{\tabcolsep}{0.39pc} \begin{tabular}{@{\hspace{0.2pc}}lllll@{\hspace{0.0pc}}} Corr. & \multicolumn{1}{c}{WA} & \multicolumn{1}{c}{SW, $4.2$} & \multicolumn{1}{c}{SW, $4.43$} & \multicolumn{1}{c}{RGT} \cr $z_{gc} $&$ 0.28(13) $&$ 0.77(8) $&$ 0.80(6) $&$ 0.88(5) $\cr $z_{pc} $&$ 0.04(16) $&$ 0.29(16) $&$ 0.31(13) $&$ 0.32(16)$\cr \end{tabular} \label{tab:corr} \end{table} \vspace{0.25cm} In Fig.~\ref{fig:zgc} we show a comparison between the WA, SW, and RGT actions of the correlation between $Q_g$ and $Q_c$. Fig.~\ref{fig:zpc} similarly shows the correlation between $Q_p$ and $Q_c$, computed in the same manner as $z_{gc}$, and numerical values are in Table \ref{tab:corr}. The SW action serves as an intermediate point between the other two actions, since the RGT action represents a better estimate of the renormalization trajectory than the WA and SW actions. It is unclear whether the spread in $z_{gc}$ at $\beta=4.43$ is due to any systematic effect of the cooling algorithm. It is possible that for better improved actions, where the exponential suppression of dislocations is greater than for the S-W action, increasing beta will have a more profound effect than we have seen here for the S-W action. For the plaquette method, we have shown previously\cite{lat94} that the multiplicative $Z_P$ becomes less severe as the action improves, and the increased correlation $z_{pc}$ suggests that the additive renormalization also decreases. In addition, improving the action is far more effective than increasing $\beta$ for suppressing lattice artifacts. Future calculations can include a more comprehensive study of other improved actions. Other methods for $\chi_t$, including the fermionic method\cite{SmVink}, and an indirect measurement by calculating the $\eta'$ mass\cite{Tsuk94,lat95}, should also be tested. Having established a correlation between cooled and uncooled topology and located individual instantons, we are now prepared to investigate Shuryak's picture of a dilute instanton gas, and the influence of instantons on hadronic physics by working directly on uncooled lattices. \noindent{\bf Acknowledgement.} The calculations were performed at the Ohio (OSC) and National Energy Research (NERSC) supercomputer centers, and the Advanced Computing Laboratory.

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\section*{Figure Captions} \begin{description} \item{\bf Fig.\ 1}\ \ \ The average transverse momentum of kaons in the reaction plane for Au+Au reactions at $P_{beam}/A=$ 4 GeV/c (upper window) and 12 GeV/C (lower window) at impact parameters less than 4 fm. The open (filled) circles are the results obtained with (without) the kaon potential. \item{\bf Fig.\ 2}\ \ \ Inclusive kaon transverse momentum distributions around midrapidity (left window) and projectile rapidity (right window) for the reaction of Au+Au at $P_{beam}/A=$ 4 GeV/c and impact parameters less than 4 fm. The open (filled) circles are the results obtained with (without) the kaon potential. \item{\bf Fig.\ 3}\ \ \ The azimuthal angle distribution of kaons with respect to the reaction plane in the Au+Au reactions described in Fig.\ 1. \end{description} \end{document}

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\section{The Robertson-Walker spacetime and observational parameters} The metric of the Robertson-Walker spacetime is of the form $$g=dt^2-R^2(t)\left\{\frac{dr^2}{1-kr^2}+r^2d\theta^2+ r^2\sin^2\theta\,d\varphi^2\right\},\quad k=-1,0,1. \eqno{(1.1)}$$ In models of the universe related to this spacetime, matter is described by density, $\rho(t)$, and pressure, $p(t)$. The models differ from one another by equations for $R$, $\rho$, and $p$. Observational parameters are Hubble constant $$H_0=\frac{\dot R_0}{R_0}, \eqno{(1.2)}$$ deceleration parameter $$q_0=-\frac{\ddot R_0R_0}{\dot R_0^2}, \eqno{(1.3)}$$ and density parameter $$\Omega_0=\left(\frac{\rho}{\rho_c}\right)_0=\frac{\rho_0} {3H_0^2/8\pi\kappa},\qquad \rho_c=\frac{3H^2}{8\pi\kappa}, \eqno{(1.4)}$$ where dot denotes $d/dt$, $\kappa$ is the gravitational constant, and subscript 0 indicates present-day values. \section{The Friedmann universe and the problem of the missing dark matter} The Friedmann universe is a model in which the complete Einstein equation is used [5,3]. The latter amounts to two equations: $$2\ddot RR+\dot R^2+k=-8\pi\kappa pR^2, \eqno{(2.1)}$$ $$\dot R^2+k=\frac{8\pi\kappa}{3}\rho R^2. \eqno{(2.2)}$$ From eqs.(2.1),(2.2) it follows $$\dot\rho R^3+3(\rho+p)R^2\dot R=0, \eqno{(2.3)}$$ which is equivalent to $$dE=-pdV,\quad E=\rho V,\quad V\sim R^3. \eqno{(2.4)}$$ For present-day values, eqs.(2.1),(2.2) reduce to $$p_0=-\frac{1}{8\pi\kappa}\left[\frac{k}{R_0^2}+H_0^2 (1-2q_0)\right], \eqno{(2.5)}$$ $$\rho_0=\frac{3}{8\pi\kappa}\left(\frac{k}{R_0^2}+ H_0^2\right), \eqno{(2.6)}$$ or $$\Omega_0=1+\frac{k}{R_0^2H_0^2}. \eqno{(2.7)}$$ For the closed universe $(k=1)$, it follows from eq.(2.2) for the maximal value of the radius $$R_{\rm max}=\sqrt{\frac{3}{8\pi\kappa\rho}}, \eqno{(2.8)}$$ so that the infimum of the set of $R_{\rm max}{}'$s is equal to zero: $$\inf\{R_{\rm max}:R_{\rm max}<\infty\}=0. \eqno{(2.9)}$$ As $$p_0\ll\frac{1}{3}\rho_0, \eqno{(2.10)}$$ it follows from eqs.(2.5),(2.6) $$\frac{k}{R_0^2}=(2q_0-1)H_0^2. \eqno{(2.11)}$$ Eqs.(2.7),(2.11) give $$\Omega_0=2q_0. \eqno{(2.12)}$$ It is the equation (2.12) that is the primary source of the problem of the missing dark matter: According to observational data $$\Omega_0<2q_0, \eqno{(2.13)}$$ or $$\rho_0<\rho_{0\rm Friedmann}, \eqno{(2.14)}$$ where $$\rho_{0\rm Friedmann}=\frac{3H_0^2q_0}{4\pi\kappa} \eqno{(2.15)}$$ is a value of $\rho_0$ given in the Friedmann universe. The value $$\Delta\rho_0=\rho_{0\rm Friedmann}-\rho_0 \eqno{(2.16)}$$ is the missing part of the density. As long as $$p\ll\frac{1}{3}\rho \eqno{(2.17)}$$ holds, the equality $$\rho R^3=\rho_0R_0^3 \eqno{(2.18)}$$ is fulfilled. Then the equation of motion for $R$ $$\left(\frac{\dot R}{R_0}\right)^2=H_0^2\left[1-2q_0+2q_0 \left(\frac{R_0}{R}\right)\right] \eqno{(2.19)}$$ takes place [5]. \section{The cosmic-length universe and the lifting of the problem} In cosmodynamics [3], the dynamical equation (2.1) holds, whereas the constraint (2.2) is not valid. The equation for matter is (2.4). Thus in place of eqs.(2.1),(2.2), we have equations $$2\ddot RR+\dot R^2+k=-8\pi\kappa pR^2, \eqno{(3.1)}$$ $$\frac{d(\rho R^3)}{dR}=-3pR^2 \eqno{(3.2)}$$ ((3.2) is equivalent to (2.4)). It is pertinent to note that $p$, $\dot\rho$, and, as a consequence, $\ddot R$ undergo sudden changes at quantum jumps of the state of matter. As $$2\ddot RR+\dot R^2=\frac{d}{dR}(R\dot R^2), \eqno{(3.3)}$$ we obtain from eqs.(3.1),(3.2) $$\frac{d}{dR}\left( R\dot R^2+kR-\frac{8\pi\kappa}{3}\rho R^3\right)=0, \eqno{(3.4)}$$ whence $$R\dot R^2+kR-\frac{8\pi\kappa}{3}\rho R^3=L=const. \eqno{(3.5)}$$ The length $L$---an integral of motion---we call cosmic length. In accordance with this, the model considered is called the cosmic-length universe. It is seen from eqs.(2.2),(3.5) that the Friedmann universe corresponds to a particular value of the cosmic length, $$L_{\rm Friedmann}=0. \eqno{(3.6)}$$ In this sense, the Friedmann universe is the zero-length universe. The results for present-day values are as follows. From eq.(3.1) we obtain $$p_0=-\frac{1}{8\pi\kappa}\left[\frac{k}{R_0^2}+ H_0^2(1-2q_0)\right], \eqno{(3.7)}$$ i.e., eq.(2.5). From eq.(3.5) it follows $$\rho_0=\frac{3}{8\pi\kappa}\left(\frac{k}{R_0^2}+H_0^2- \frac{L}{R_0^3}\right), \eqno{(3.8)}$$ or $$\Omega_0=1+\frac{k-L/R_0}{R_0^2H_0^2} \eqno{(3.9)}$$ in place of eqs.(2.6),(2.7) respectively. As the inequality (2.10) holds, we obtain from eqs.(3.7),(3.8) $$\frac{k}{R_0^2}=(2q_0-1)H_0^2, \eqno{(3.10)}$$ i.e., eq.(2.11). Eqs.(3.9),(3.10) give $$\Omega_0=2q_0-\frac{L/R_0}{R_0^2H_0^2} \eqno{(3.11)}$$ in place of eq.(2.12). Eqs.(3.11),(3.9) lift the problem of the missing matter for the universe as a whole. In the cosmic-length universe this problem does not exist. In indeterministic quantum gravity---a theory being developed in this series of papers---the universe is closed, i.e., $$k=1, \eqno{(3.12)}$$ so that by eq.(3.10) $$q_0>\frac{1}{2} \eqno{(3.13)}$$ in agreement with observational data. We proceed to consider the cosmic-length universe. From eqs. (3.5),(2.18),(3.12) we obtain $$R_{\rm max}=\frac{8\pi\kappa}{3}\rho_0R_0^3+L \eqno{(3.14)}$$ and, using eqs.(3.11),(1.4), $$R_{\rm max}=2q_0H_0^2R_0^3, \eqno{(3.15)}$$ which is independent of $L$. From eqs.(3.11),(3.15) it follows $$L=\frac{2q_0-\Omega_0}{2q_0}R_{\rm max}. \eqno{(3.16)}$$ We obtain from eqs.(3.5),(2.18),(3.10),(3.11) $$\left(\frac{\dot R}{R_0}\right)^2=H_0^2\left[1-2q_0+ 2q_0\left(\frac{R_0}{R}\right)\right], \eqno{(3.17)}$$ i.e., eq.(2.19). It follows from eqs.(3.10),(3.11),(3.16),(3.12) $$R_0=\frac{1}{\sqrt{2q_0-1}H_0}, \eqno{(3.18)}$$ $$L=\frac{2q_0-\Omega_0}{2q_0-1}R_0, \eqno{(3.19)}$$ $$R_{\rm max}=\frac{2q_0}{2q_0-1}R_0. \eqno{(3.20)}$$ In the Friedmann universe, eqs.(3.18),(3.20) are fulfilled, whereas eq.(3.19) reduces to the identity 0=0. \section{The cosmic length as the infimum of the set of maximal radii} We have from eqs.(3.18),(3.19) for the cosmic length $$L\approx R_0\approx\frac{1}{H_0}. \eqno{(4.1)}$$ It follows from eq.(3.5) with $k=1$ $$R_{\rm max}=L+\frac{8\pi\kappa}{3}(\rho R^3)_{R=R_{\rm max}}. \eqno{(4.2)}$$ Thus, in the cosmic-length universe, the infimum of the set of $R_{\rm max}{}'$s is equal to $L$: $$L=\inf\{R_{\rm max}:R_{\rm max}<\infty\}. \eqno{(4.3)}$$ \section{What is small and what is great?} The equations of the cosmic-length universe involve two constants: the gravitational constant $$\kappa=t_P^2, \eqno{(5.1)}$$ where $t_P$ is Planck time, and the cosmic length $L$. Now that we have two universal constants---Planck length $l_P$ and the cosmic length $L$---we may answer the question: What is small and what is great? It is these constants that are measures of the smallness and greatness respectively. And the greatness dwarfs the smallness: $$L\approx 10^{26}\:{\rm m}>>>1.6\cdot10^{-35}\:{\rm m}=l_P. \eqno{(5.2)}$$ The universe is great in the sense of this inequality.

proofpile-arXiv_065-577

\section{Introduction} \vspace*{-0.5pt} \noindent In three dimensional spacetime, besides the Maxwell term, the parity-violating Chern-Simons term can be the kinetic part for the gauge field.\cite{siegel,deser} The Chern-Simons Lagrangian for an abelian gauge field $A_\mu$ is given as \begin{equation} {\cal L}_{CS} =\frac{\kappa}{2} \epsilon^{\mu\nu\rho}A_\mu\partial_\nu A_\rho, \label{cs1} \end{equation} and the corresponding term for a nonabelian gauge field $A^a_\mu$ is \begin{equation} {\cal L}_{CS}' = \frac{\kappa}{2} \epsilon^{\mu\nu\rho}(A_\mu^a\partial_\nu A_\rho^a + \frac{1}{3}f^{abc}A^a_\mu A^b_\nu A^c_\rho ), \label{cs2} \end{equation} where the coefficients $f^{abc}$ are the structure constants of the gauge group. The Lagrangians (\ref{cs1}) and (\ref{cs2}) are invariant under infinitesimal gauge transformations which vanish at spatial infinity. For quantum amplitude $\exp( i\int d^3x {\cal L}'_{CS})$ to be invariant under large gauge transformations, the coefficient $\kappa$ of Eq. (\ref{cs2}) should be quantized.\cite{deser} The abelian theory of a complex scalar field $\phi$ coupled to $A_\mu$ with the Chern-Simons kinetic term is defined by the Lagrangian \begin{equation} {\cal L}_1 = {\cal L}_{CS} + D_\mu\phi^\dagger D^\mu \phi - U(\phi), \label{acsh} \end{equation} where $D_\mu \phi = (\partial_\mu - iA_\mu)\phi$. We will consider the case of the pure Chern-Simons kinetic term as the Maxwell term, if present additionally, will affect only the short distance physics. The Gauss law of the theory (\ref{acsh}) is \begin{equation} \kappa F_{12} = J_0, \label{gauss} \end{equation} where $J_\mu = i(D_\mu \phi^\dagger \phi - \phi^\dagger D_\mu \phi)$. Thus total magnetic flux $\Psi = \int d^2x\, F_{12}$ is related to total charge $Q=\int d^2x \, J_0$ by $\kappa \Psi = Q$. The basic excitations of the system are either charge neutral particles or charge-flux composites. The conserved angular momentum for the Lagrangian (\ref{acsh}) is \begin{equation} J = - \int d^2x \, \epsilon^{ij} x^i \left( D_0\phi^*D_j \phi + D_j\phi^*D_0\phi \right), \end{equation} whose density is gauge-invariant and localized. Under the CTP symmetry the sign of the angular momentum does not change, and so particles and antiparticles carry the same spin. In the symmetric phase particles carry nonzero spin $1/(4\pi\kappa)$. In the broken phase there are elementary neutral scalar and vector bosons, and also charged magnetic flux vortices whose spin is $-\pi\kappa$. Note the sign difference of anyon spins in the symmetric and broken phases. In two dimensional space particles of fractional spin, anyons, are possibilities.\cite{anyon} In the symmetric phase of the Chern-Simons-Higgs systems, anyons are represented by charge-flux composites. The fractional statistics can be understood by considering the orbital angular momentum of a pair of anyons or anyon-antianyon interacting under a central force. In the system of two identical anyons of spin $s$, the allowed orbital angular momentum is $L=2l+2s $ with an integer $l$ and so the total angular momentum is $J=L+2s = 2l+4s$. For the system of anyon-antianyon, the allowed orbital angular momentum is $L= 2l-2s$ so that the total angular momentum is $J=L+2s= 2l$. This makes possible to creat pairs of anyon and antianyon by vacuum fluctuations. The fractional statistics arise when we gauge away the flux carried by charged bosons. For the detail of anyon physics, the readers can consult many review articles.\cite{lerda} There are already a few reviews on the Chern-Simons Higgs systems.\cite{khare,dunne3} This talk is a brief survey of the relativistic Chern-Simons systems, focusing on work done by my collaborators and myself. In Sec.~2, I summerize the salient features of the self-dual abelian Chern-Simons-Higgs model. In Sec.~3, the self-dual models with nonabelian gauge symmetry are discussed. In Sec.~4, the self-duality is generalized to the sigma and $CP(N)$ models. In Sec.~5, some general ideas, like supersymmetry, the correction to the Chern-Simons coefficient, and moduli space approximation of low energy dynamics of vortices, are discussed. Here I summerize some questions whose answer seems not known. Section~6 contains concluding remarks. \section{ Self-dual Abelian Chern-Simons Higgs Systems} \noindent One of the first self-dual models found is the self-dual Maxwell-Higgs system.\cite{bogo} The relativistic self-dual Chern-Simons-Higgs system has been found later on.\cite{hong} The self-dual Lagrangian is given by Eq.({\ref{acsh}) with a specific potential \begin{eqnarray} U(\phi) = \frac{1}{\kappa^2}|\phi|^2(|\phi|^2-v^2)^2. \end{eqnarray} The theory is renormalizable and the only dimensionful parameter is $v^2$. When $v^2$ vanishes, there is a classical scaling symmetry which may be broken quantum mechanically. With the help of the Gauss law (\ref{gauss}), the energy of the model can be expressed as \begin{eqnarray} E = \int d^2 x \left\{ |D_0\phi \pm \frac{i}{\kappa}(|\phi|^2-v^2)\phi|^2 + |D_1\phi \pm i D_2\phi |^2 \right\} \pm v^2 \Psi, \end{eqnarray} where there is no boundary contribution as we consider only finite energy configurations. The Bogomolny bound on the energy is then \begin{equation} E \ge \pm \frac{v^2}{\kappa} Q. \label{bogom} \end{equation} This bound is saturated by configurations satisfying the self-dual equations, \begin{eqnarray} D_0\phi \pm \frac{i}{\kappa}(|\phi|^2-v^2)\phi = 0, \,\,\,\,\,\,\,\, D_1 \phi \pm i D_2\phi =0. \end{eqnarray} The above equations imply that $\partial_0 |\phi|=0$ and so the field configuration can be static in time in a given gauge choice. Combined with the Gauss law (\ref{gauss}), the self-dual equations can be put to \begin{equation} \partial_i^2 \ln |\phi|^2 -4|\phi|^2(|\phi|^2-v^2) = 4\pi \sum_a \delta^2(x^i-q^i_a), \end{equation} where $q^i_a$'s are the positions of vortices. The potential has two degenerate minima; the symmetric phase where $<\phi>=0$ and the broken phase where $<\phi>=v$. As mentioned before, there are elementary excitations in both phases and self-dual anyonic vortices in the broken phase. In the symmetric phase there are also self-dual anyonic nontopological solitons of unquantized magnetic flux.\cite{jack1} In the broken phase the self-dual configurations are determined unquely by vortex positions.\cite{wang} While the energy is degenerate, the angular momentum is a complicated function of vortex positions.\cite{kim1} The statistics of anyons in the symmetric phase is decided by the Aharonov-Bohm phase. The statistics of anyonic vortices in the broken phase have the contributions from both the Aharonov-Bohm phase and a phase originated from the Magnus force. These two phases can be combined into a single dual phase in the dual formalism where vortices appear as charged elementary particles, explaining the fore-mentioned sign difference of anyon spins.\cite{kim1,mark} The nonrelativistic limit of this self-dual model has been found and studied.\cite{jack2} The self-dual systems with the both Maxwell and Chern-Simons kinetic terms have been also found.\cite{clee1} This self-dual model interpolates smoothly between the Maxwell-Higgs and Chern-Simons-Higgs systems. A further generalization of these self-dual models by including unform background charge has been found.\cite{klee2} The whole structure of this model is quite rich. Some phase appears to be infinitely degenerate, some self-dual solitons have a negative rest mass even though their kinetic mass is positive, in some phase there is a roton mode among elementary excitations, some phase appears to be inhomogeneous, implying spontaneous breaking of translation symmetry, etc. The definition of the angular momentum becomes delicate as in the Maxwell-Higgs case with the background charge.\cite{klee1} \section{Self-dual Nonabelian Chern-Simons-Higgs Systems} \noindent The previous abelian self-dual model can be generalized to the self-dual systems with nonabelian gauge group.\cite{klee3,dunne1} The crucial point is to require that there exists at least a global $U(1)$ symmetry. For simplicity we consider the theory of a complex scalar field $\phi$ in a given irreducible representation of the gauge group. If the representation is real, we need two real scalar fields to make the matter field complex. The generators of the symmetry group in this representation are hermitian matrices $T^a$. The conserved global $U(1)$ symmetry is generated by a global phase rotation, $\phi \rightarrow e^{i\alpha}\phi$. The self-dual Lagrangian is then \begin{eqnarray} {\cal L}_2 = {\cal L}_{CS}' + |D_\mu \phi|^2 -\frac{1}{\kappa^2} |T^a\phi \phi^\dagger T^a \phi -v^2 \phi|^2, \end{eqnarray} where $D_\mu \phi = (\partial_\mu -iT^aA^a_\mu)\phi $. The energy bound is given by Eq.(\ref{bogom}) with the global charge $Q=i(D_0\phi^+ \phi -\phi^\dagger D_0\phi)$. Similar self-dual models with the pure Yang-Mills kinetic term are possible. However, there seem no interesting solitons here. These models may be regarded as a bosonic part of theories with an extended supersymmetry. Interesting vacuum and soliton structures show up when the matter field is in adjoint representation. The vacuum expectation value of the potential satisfies the algebraic equation \begin{equation} [ [ \phi,\phi^\dagger],\phi]=v^2\phi, \label{vacuum} \end{equation} where $\phi =\phi^a T^a$. This equation is the $SU(2)$ Lie algebra with identification $J_3 = [\phi,\phi^\dagger] /v^2$ and $J_+ = \phi/v$. This allows the detail analysis of vacuum and mass spectrum.\cite{kao1,dunne2} The solitonic structure in the $SU(3)$ case has been studied in detail.\cite{kao1} The nonrelativistic limit of this theory represents a theory of anyons with nonabelian statistical phase. There are extensive work and review of self-dual solitons in this limit.\cite{dunne4,dunne3} The dynamics of vortices in the broken phase may involve the nonabelian generalization of the Magnus force. \section{Sigma and $CP(N)$ Models} \noindent The sigma model has been studied extensively, where self-duality has also been explored.\cite{zak} The self-dual field configurations are topological lumps which are characterized by the second homotopy of the field as a mapping from two dimensional space to the internal field space. By gauging a part of the global symmetry of the sigma model, one can have another self-duality. Especially a new self-dual sigma model with the Maxwell term has been found recently.\cite{schroers} This has been further generalized to the models with the Chern-Simons term.\cite{ghosh,kimm1} These models have been generalized further to the $CP(N)$ models where the matter field is a complex vector field $z=(z_1,...,z_{N+1})$ of unit length. The nontrivial generalization is achieved by gauging the part of the global $SU(N+1)$ symmetry such that there exists at least one conserved global $U(1)$ group which commutes with the gauge group.\cite{kimm2} With the gauge symmetry generators $T^a$ and the global symmetry generator $R$, the covariant derivative is \begin{equation} \nabla_\mu z = (\partial_\mu - iT^aA^a)z - z (\bar{z} (\partial_\mu - iT^aA^a)z ), \end{equation} and the Lagrangian is \begin{eqnarray} {\cal L}_{CP(N)} = {\cal L}_{CS}' + |\nabla_\mu z|^2 - \frac{1}{\kappa^2} \biggl| \bigl(T^a z - z(\bar{z}T^az) \bigr) \bar{z}T^a z - v \bigl(Rz - z(\bar{z}Rz)\bigr)\biggr|^2. \end{eqnarray} The conserved topological current is then $K^\mu = -i\epsilon^{\mu\nu\rho} \partial_\nu (\bar{z}(\partial_\rho -iA_\rho) z)$. The global current for $R$ is $J^\mu = i(\nabla^\mu \bar{z} (Rz - z(\bar{z}Rz)) - h.c.)$. The Bogomolny bound is given by $E\ge |T|$ where $T = \int d^2x (K^0+ vJ^0/\kappa)$. In certain limits the self-dual Chern-Simons $CP(N)$ models approach all known self-dual Chern-Simons Higgs models, implying that the $CP(N)$ models have all the complicated vacuum and soliton structures as the Higgs cases, and more. Especially with the matter in adjoint representation, the vacuum condition is identical to Eq.(\ref{vacuum}) for an appropriate range of $v$. The structure of the self-dual configurations is not yet fully explored. \section{General Ideas and Questions} \noindent The self-dual Chern-Simons-Higgs systems are renormalizable while the self-dual $CP(N)$ models are not. We can use the perturbative approach to calculate the quantum effect in the Higgs case, which has not been fully explored even at one-loop. When one introduces uniform background charge, the theory is still renormalizable as far as the dimensional counting is concerned. As the Lorentz symmetry is lost in this case, there may be some surprises. In the abelian case uniform external charge can be introduced and is neutralized by the Higgs charge in the broken phase. One can ask whether there exists a similar configuration in nonabelian self-dual systems. I think that it does because one can imagine a configuration where the global $U(1)$ charge is distributed unformly, exactly like Q-matters.\cite{coleman1} There is also a curious homogeneous configuration whose properties are not fully explored. The vector potential here is uniformly rotating; $(A_1+iA_2) = c e^{iwt}$. The energy density is homogeneous and constant parameters $c,w$ are determined by the field equations.\cite{ezawa} We do not know whether such configuration is classically or quantum mechanically stable. This case may be somewhat analogous to the uniform current case in the Maxwell-Higgs case where the current decays by nucleating vortex loops.\cite{kao2} \newpage \subsection{Supersymmetry} \noindent For every self-dual model we expect that there exits an underlying supersymmetry.\cite{witten1} The $N=2$ supersymmetric models behind the self-dual Chern-Simons-Higgs system have been found.\cite{clee2} The central charge of the $N=2$ supersymmetry gives the Bogomolny bound on the Hamiltonian. In the broken phase, the supermultiplet of massive neutral vector bosons can have the spin structure $\pm (1,1/2,0,-1/2)$ at most as there is only one degree of freedom associated with vector bosons. Thus the maximal supersymmetry allowed in this models is $N=3$.\cite{kao3} When there is uniform background charge, the supersymmetry is not obvious at all as the mass spectrum of a given supermultiplet does not show the degeneracy.\cite{klee2,klee1} In the $N=2$ or $N=3$ supersymmetric cases the Bogomolny bound is expected to be exact because the charged sector saturating the energy bound has a reduced representation and the supersymmetry is not supposed to be broken. The $N=2$ supersymmetric theories needs the infinite renormalization.\cite{leblanc,avdeev} On the other hand the $N=3$ supersymmetric theories seem to be finite at least one loop.\cite{kao4,kao5} It would be interesting to find out whether the $N=3$ models are finite in all orders. When the parameter $v$ vanishes, the classical field theory has the scaling symmetry, which may be broken quantum mechanically by the Coleman-Weinberg mechanism.\cite{coleman2} Indeed recently such mechanism is shown to work here by calculating two-loop diagrams.\cite{hosotani} The scale symmetry may be preserved quantum mechanically for the $N=2,3$ supersymmetric models. If this is the case, these supersymmetric theories have a quantum superconformal symmetry. Recently there has been a considerable progress in understanding of the low energy nature of the self-dual Yang-Mills Higgs systems with the $N=4$ supersymmetry in three dimensions.\cite{seiberg} Similar to these systems, in Chern-Simons-Higgs systems magnetic monopole instantons exist.\cite{klee4} It would be interesting if one can make similar exact statements for the $N=2,3$ supersymmetric Chern-Simons-Higgs systems. Following an argument similar to that for getting $N=3$ for the maximally supersymmetric Chern-Simons-Higgs systems, one can see the maximal supergravity theory with massive gravitons should be $N=7$. It would be interesting to see whether this theory, if constructed, is finite. \subsection{Chern-Simons Coefficient in the Broken Phase} In the abelian Chern-Simons theories the Coleman-Hill theorem states that the Chern-Simons coefficient does not get corrected except by the fermion contribution at one loop when the gauge symmetry is not spontaneously broken and there is no massless charged particle.\cite{coleman3} The vacuum polarization by the fermion loop renormalizes the bare Chern-Simons coefficient at the scale larger than the fermion Compton length. When the gauge symmetry is partially broken, the correction to the coefficient for the unbroken gauge group is shown to be still quantized\cite{chen}. This theorem has been extended to the broken phase, where the `total' Chern-Simons term is argued to be a sum of the `pure' renormalized Chern-Simons term plus an `effective' term involving the scalar field which looks like the Chern-Simons coefficient.\cite{khare2} One-loop correction to the pure coefficient is quantized. That to the effective coefficient is not quantized in general. In the self-dual abelian Chern-Simons-Higgs system, the correction to the effective term is however quantized.\cite{kao4} This may be true even with nonabelian gauge symmetry with the pure Chern-Simons kinetic term. It would also be interesting to find out whether there is a quantum correction to the vortex spin and if it does, whether it is related to the correction to the coefficient. Suppose that many family of bosons become massive fermions by a Chern-Simons interaction and they are coupled to another gauge field. The natural question is whether these composite fermions induce the Chern-Simons term to another gauge field. If it does, the composite fermions can be treated as fundamental fermions. \subsection{Low Energy Dynamics of Vortices} \noindent In the broken phase of the theory considered in Sec.2, the self-dual configurations for $n$ vortices, with gauge equivalent configurations identified, form a finite dimensional moduli space. The natural coordinates for this moduli space are the vortex positions $q^i_a,a=1,,,n$. One expects the low energy dynamics of these vortices can be described as dynamics on the moduli space.\cite{manton} There is no potential energy for vortices as the energy is degenerate. However there exists a term linear in velocities as the total angular momentum depends on vortex positions.\cite{kim1} This linear term leads to the statistical interaction between vortices and is originated from the sum of the naive gauge interaction and the Magnus force.\cite{kim1} The most general nonrelativistic Lagrangian for the moduli coordinates is then \begin{equation} L = \frac{1}{2} T_{ab}^{ij}(q^k_c) \, \dot{q}_a^i \dot{q}_b^j + H^i_a(q^k_c) q^i_a. \end{equation} One may interpret $T_{ab}^{ij}$ as the metric and $H^i_a$ as a linear connection or a vector potential. The connection $H^i_a$ has been obtained explicitly in terms of the self-dual configurations.\cite{kim1} However no satisfactory answer for $T^{ij}_{ab}$ has been found in spite of several attempts.\cite{kim1,dziarmaga} This situation contrasts to the self-dual Maxwell-Higgs case.\cite{samols} When there is a uniform background charge, moduli space approximation becomes more interesting. Again $H^i_a$ is known but $T_{ab}^{ij}$ is not.\cite{klee1,klee2} If moduli space approximation is reasonable, a single vortex moves a circular motion on this background due to the Magnus force. In some range of the parameter space, the rest mass of vortices becomes negative. (The total energy of a pair of vortex and antivortex is positive and so the system is stable.) I do not have any clue for the method to calculate the kinetic mass in this case. On the other hand the energy difference of the Landau levels after quantization can be larger than the rest mass of some elementary neutral quanta in the broken phase. This contradicts the spirit of moduli space approximation where we expect only zero modes to be excited at low energy. This makes me to wonder whether moduli space approximation is good at all here. \section{Concluding Remarks} \noindent The relativistic self-dual Chern-Simons systems come with many flavors. Their vacuum and soliton structures are rich and diverse. They have been a playing ground for testing and sharpening our understanding of quantum field theory of anyons and solitons. I have discussed many ideas and questions related the Chern-Simons systems. There are also many interesting topics I have not discussed at all: the potential force between vortices away from self-duality, the finite temperature correction to the Chern-Simons coefficient in the symmetric phase, the theories on compact Riemann surfaces, quantum Hall effects and boundary states, semi-local solitons, supergravity models behind the self-dual models, etc. I believe that there are still more surprises and insights to be discovered in this field. \nonumsection{Acknowledgements} I thank the organizers, Daniel Cangemi and Gerald Dunne, and many participants of the Low Dimensional Field Theory Workshop at Telluride (August 1996) for warm hospitality and relaxed atmosphere. This work is supported in part by the NSF Presidential Young Investigator program. \nonumsection{References}

proofpile-arXiv_065-578

\section{Introduction} \label{sec1} In the last few years strong evidence was given that for certain types of glass formers {\it at least} the universal predictions of the so-called mode-coupling theory (MCT) are correct in that it was shown that, e.g., there exists a critical temperature $T_c$, that the factorization property in the $\beta$-relaxation regime holds, or that two distinct diverging time scales can be observed. An introduction to the theory can be found in some recent review articles~\cite{bibles,schilling92,cummins94} and in Ref.~\cite{yip95} the reader will find a compendium of many investigations performed to test the validity of the theory. The outcome of most of these tests is that, at least for fragile glassformers, MCT is a valid theory, although recent calculations have shown that the theory might even be applicable to relatively strong glassformers, such as glycerol~\cite{franosch96}. Apart from some noticeable exceptions, discussed below, most of the tests done to check the validity of the theory investigated only whether the {\it universal} predictions of the theory are correct. The reason for this is the fact that the {\it nonuniversal} predictions of the theory, e.g. the value of $T_c$ or the details of the wave vector dependence of the nonergodicity parameters, can be tested only for those systems for which the temperature dependence of the structure factor (or of the partial structure factors in the case of a multi component system) is known with a fairly high accuracy. Since in most cases these structure factors were not available with the required accuracy, only the universal predictions of the theory could be tested. The drawback with these types of tests is that the various (nonuniversal) parameters occurring in the theory, such as the exponent parameter $\lambda$, the critical temperature $T_c$, or the nonergodicity parameter $f_c(q)$, had to be considered as fit parameters of the theory, thus making the tests less stringent. There are two exceptions to this. The first one is a system of colloidal particles whose glass transition was studied extensively in light scattering experiments by Pusey, van Megen and Underwood~\cite{colloids}. The structure of such systems is believed to be modelled well by a system of hard spheres. For the structure factor of the latter very reliable analytical expressions are available~\cite{fluid_books} and thus were used by G\"otze and Sj\"ogren to demonstrate that in the $\beta$-relaxation regime the dynamics of the colloidal particles could be described very well with MCT~\cite{goetze91}. The second system is a model of soft spheres for which Barrat and Latz showed~\cite{barrat90} that MCT gives a fair quantitative description of quantities like the nonergodicity parameter and the critical coupling constant, which were determined by means of computer simulations~\cite{bernu87,roux89}. Thus these two examples show that MCT is able to give not only a {\it qualitative} correct description of the dynamics of supercooled liquids, but that, {\it at least} in some cases, it also gives a {\it quantitative} correct description. In a recent computer simulation we studied the dynamics of a mixture of Lennard-Jones particles~\cite{kob94,kob95a,kob95b,kob95c,kob95d}. It was shown that at low temperatures the dynamics of this system could be described very well by MCT. However, in these papers only the universal properties of the theory were tested since we used the data from the simulation to fit the occurring parameters of the theory. The goal of the present paper is now to compare the results of the simulation with the predictions of the theory without using any fit parameter at all. The only input to the theory will be the partial structure factors that were obtained from the simulation. In this way it will be possible to make a more stringent test of the theory than it was done in Refs.~\cite{kob94,kob95a,kob95b,kob95c,kob95d} and therefore to test whether also for this system the theory is correct not only in a qualitative way but also in a quantitative way. The rest of the paper is organized as follows: In Sec.~\ref{sec2} we collect the MCT equations needed to compute the various quantities we investigate. In Sec.~\ref{sec3} we give the details of the model and of the numerical calculations. In Sec.~\ref{sec4} we present our results and summarize and discuss them in Sec.~\ref{sec5}. \section{Mode-Coupling Theory} \label{sec2} In this section we summarize the equations that are necessary to describe the dynamics of the system within the framework of MCT. Most of these equations can also be found in Refs.~\cite{barrat90,gotze85,gotze87,bosse87,fuchs93,fuchs_phd}. We consider a two-component system of classical particles with particle concentrations $x_i$ and masses $m_i$, $i=1,2$. In the following the dynamics of the system will be described by means of the partial intermediate scattering functions \begin{equation} F_{ij}(q,t)=\langle \delta\rho_i(q,0)\delta\rho_j^{\star}(q,t)\rangle\quad , \label{eq0} \end{equation} where $\delta\rho_i(q,t)$ is the density fluctuation for wave vector $q$ at time $t$ of species $i$. For a binary system it is useful to collect these functions in a $2\times 2$ matrix ${\bf F}(q,t)$ with $[{\bf F}(q,t)]_{ij}=F_{ij}(q,t)$. The equation of motion of ${\bf F}$ is given by \begin{equation} \ddot{{\bf F}}(q,t)+{\mbox{\boldmath $\Omega $}}^2(q){\bf F}(q,t)+ \int_0^td\tau {\bf M}(q,t-\tau) \dot{{\bf F}}(q,\tau) = 0 \quad , \label{eq1} \end{equation} where the frequency matrix ${\mbox{\boldmath $\Omega $}}^2$ is given by \begin{equation} \left[{\mbox{\boldmath $\Omega $}}^2(q)\right]_{ij}=q^2k_B T (x_i/m_i)\sum_{k}\delta_{ik} \left[{\bf S}^{-1}(q)\right]_{kj}\quad. \label{eq2} \end{equation} Here ${\bf S}(q)$ stands for the $2\times 2$ matrix consisting of the partial structure factors $S_{ij}(q)$. Within the mode-coupling approximation the memory term ${\bf M}$ is given at long times by \begin{equation} M_{ij}({\bf q},t)=\frac{k_B T}{2n m_i x_j}\int\frac{d {\bf k}}{(2\pi)^3} \sum_{\alpha\beta}\sum_{\alpha'\beta'}V_{i\alpha\beta}({\bf q},{\bf k}) V_{j\alpha'\beta'}({\bf q},{\bf q-k}) F_{\alpha\alpha'}({\bf k},t) F_{\beta\beta'}({\bf q-k},t)\quad , \label{eq3} \end{equation} where $n$ is the particle density and the vertex $V_{i\alpha\beta}({\bf q},{\bf k})$ is given by \begin{equation} V_{i\alpha\beta}({\bf q},{\bf k})=\frac{{\bf q}\cdot {\bf k}}{q}\delta_{i\beta} c_{i\alpha}({\bf k})+ \frac{{\bf q}\cdot ({\bf q}-{\bf k})}{q} \delta_{i\alpha} c_{i\beta} ({\bf q}-{\bf k}) \label{eq4} \end{equation} and the matrix of the direct correlation function is defined by \begin{equation} c_{ij}({\bf q})=\frac{\delta_{ij}}{x_i}- \left[{\bf S}^{-1}({\bf q})\right]_{ij} \quad . \label{eq5} \end{equation} Making use of the isotropy of the system the expression for $M_{ij}({\bf q},t)$ can be reduced to a two dimensional integral: \begin{eqnarray} M_{ij}(q,t) & = & \frac{k_B T}{32n x_j m_i\pi^2 q^3}\int_0^{\infty} dk k \int_{|q-k|}^{q+k} dp p \sum_{\alpha\beta}\sum_{\alpha' \beta'} F_{\alpha\alpha'}(q,t) F_{\beta\beta'}(p,t) \cdot \nonumber \\ & & \left\{ \left(q^2+k^2-p^2\right) \delta_{i\beta}c_{i\alpha}(k)+ \left(q^2+p^2-k^2\right)\delta_{i\alpha}c_{i\beta}(p)\right\} \cdot \nonumber \\ & & \left\{ \left(q^2+k^2-p^2\right)\delta_{j\beta'}c_{j\alpha'}(k)+ \left(q^2+p^2-k^2\right)\delta_{j\alpha'}c_{j\beta'}(p)\right\} \quad . \label{eq6} \end{eqnarray} The memory function for the incoherent intermediate scattering function $F_i^{(s)}$ is given by: \begin{eqnarray} M_{i}^{(s)}({\bf q},t) & = & \int \frac{d{\bf k}}{(2\pi)^3} \frac{1}{n} \left(\frac{{\bf q}\cdot {\bf k}}{q}\right) (cF)_i ({\bf k},t) F_{i}^{(s)}({\bf q}-{\bf k},t) \nonumber \\ & = & \frac{1}{16\pi^2n q^3}\int_0^{\infty} dk \int_{|q-k|}^{q+k} dp p \left\{q^2+k^2-p^2\right\}^2 (cF)_i(k,t) F_{i}^{(s)}(p,t)\quad , \label{eq7} \end{eqnarray} with \begin{equation} (cF)_i(k,t)=(c_{ii}(q))^2 F_{ii}(q,t)+2c_{ii}(q)c_{ij}(q)F_{ij}(q,t) +(c_{ij}(q))^2F_{jj}(q,t)\quad j\neq i \quad . \label{eq8} \end{equation} The matrix of the nonergodicity parameters ${\bf f}({\bf q})$ for the coherent intermediate scattering function is given by the solution of Eq.~(\ref{eq1}) at long times, i.e. $f_{ij}({\bf q})=lim_{t\to \infty} F_{ij}({\bf q},t)$. It can be shown that ${\bf f}({\bf q})$ can be computed via the following iterative procedure~\cite{fuchs_phd}: \begin{equation} {\bf f}^{(l+1)}(q) = \frac{ {\bf S}(q) \cdot {\bf N}[{\bf f}^{(l)},{\bf f}^{(l)}](q) \cdot {\bf S} (q) + q^{-2}|{\bf S}(q)| |{\bf N}[{\bf f}^{(l)},{\bf f}^{(l)}](q)| {\bf S}(q) }{ q^2+Tr({\bf S}(q) \cdot {\bf N}[{\bf f}^{(l)},{\bf f}^{(l)}](q)) + q^{-2}| {\bf S}(q)| | {\bf N}[{\bf f}^{(l)},{\bf f}^{(l)}](q)|} \quad, \label{eq9} \end{equation} where the matrix ${\bf N}(q)$ is given by \begin{equation} N_{ij}(q)=\frac{m_i}{x_i k_B T} M_{ij}(q) \quad. \label{eq10} \end{equation} For temperatures above the critical temperature $T_c$ this iteration converges to the trivial solution ${\bf f}(q)=0$ whereas for $T<T_c$ it converges to a nontrivial solution ${\bf f}(q)>0$. The incoherent nonergodicity parameter $f_i^{(s)}$ can be computed from the following iterative procedure: \begin{equation} q^2 \frac{f_i^{(s,l+1)}(q)}{1-f_i^{(s,l+1)}(q)} = M_i^{(s)}[{\bf f}, f_i^{(s,l)}](q) \quad . \label{eq11} \end{equation} In order to determine the critical point it is useful to consider the so-called stability matrix ${\bf C}$ which is defined by its action on a vector $\delta {\bf f}(q)=(\delta {\bf f}_{11}(q), \delta {\bf f}_{12}(q), \delta {\bf f}_{22}(q))$: \begin{equation} ({\bf C}\cdot \delta {\bf f})(q)=\frac{1}{q^2} ({\bf S}(q)-{\bf f}(q)) \cdot \left[ {\bf M}[{\bf f},\delta {\bf f}](q) + {\bf M}[\delta {\bf f},{\bf f}](q) \right] \cdot ({\bf S}(q)-{\bf f}(q)) \quad . \label{eq12} \end{equation} We define $E_0$ to be the largest eigenvalue of this matrix, ${\bf e}(k) = (e_{11}(k),e_{12}(k),e_{22}(k))$ as the corresponding right and $\hat{\bf e}(k)=(\hat{e}_{11}(k),\hat{e}_{12}(k),\hat{e}_{22}(k))$ as the corresponding left eigenvector of this matrix. The normalization of these eigenvectors is given by~\cite{fuchs_priv}: \begin{eqnarray} \int_0^{\infty} dk \sum_{n=11,12,22} \hat{e}_{n}(k) e_n(k) & = & 1 \label{eq13a} \\ \int_0^{\infty} dk \sum_{n=11,12,22} \hat{e}_n(k)\left[{\bf e}(k)\cdot \left[ {\bf S}(k)-{\bf f}(k) \right]^{-1}\cdot {\bf e}(k)\right]_n & = & 1 \quad . \label{eq13b} \end{eqnarray} The critical amplitudes ${\bf h}(q)=(h_{11}(q),h_{12}(q),h_{22}(q))$ describe the dynamics of the system in the $\beta$-relaxation regime, i.e. \begin{equation} F_{ij}(q,t)=f_{c,ij}(q)+h_{ij}(q)g(t)\quad , \label{eq14} \end{equation} where $g(t)$ is a function which is independent of $q$, and whose form depends on temperature and the so-called exponent parameter $\lambda$, and $f_{c,ij}$ are the nonergodicity parameters at the critical temperature. This critical amplitude is given by the value of the right eigenvector at the critical temperature $T_c$, i.e.: \begin{equation} h_{ij}(q)=e_{c,ij}(q)\quad . \label{eq15} \end{equation} The value of the mentioned exponent parameter $\lambda$ is given by: \begin{equation} \lambda = \int_0^{\infty} dq \sum_{n=11,12,22} \hat{e}_{c,n}(q) \left(\frac{1}{q^2}({\bf S}(q)-{\bf f}_c(q))\cdot {\bf M}[{\bf e}_c, {\bf e}_c]\cdot ({\bf S}(q)-{\bf f}_c(q))\right)_{n} \label{eq16} \end{equation} The procedure to compute the nonergodicity parameters is now the following: Given the partial structure factors for a temperature $T$ which corresponds to the glass state one computes from Eqs.~(\ref{eq5}) and (\ref{eq6}) the memory kernel and iterates Eq.~(\ref{eq9}) until ${\bf f}^{(l)}(q)$ has converged. Then the stability matrix ${\bf C}(q)$ and its largest eigenvalue $E_0$ are computed. It can be shown that in the vicinity of the critical temperature $T_c$ the relation \begin{equation} (1-E_0)^2= A (T_c-T) +O((T_c-T)^2) \label{eq17} \end{equation} holds, which can thus be used for a precise determination of $T_c$. Having determined $T_c$ we can compute the right and left eigenvalue of ${\bf C}$ at $T_c$ and thus obtain the critical amplitudes $h_{ij}(q)$ and the exponent parameter $\lambda$ (Eqs.~(\ref{eq15}) and (\ref{eq16})). Using the nonergodicity parameters of the coherent intermediate scattering function we can use Eqs.~(\ref{eq7}) and (\ref{eq11}) to finally compute the nonergodicity parameter for the incoherent intermediate scattering function. \section{Model and Computational Details} \label{sec3} In this section we introduce the system we investigate and give some of the details of our numerical calculations. More details on these calculations can be found in Ref.~\cite{nauroth95}. The model we are studying is a binary mixture of classical Lennard-Jones particles all of them having mass $m$. The interaction between two particles of type $i$ and $j$ ($i,j \in \{A,B\})$ is given by $V_{ij}(r)=4\epsilon_{ij}\left[ (\sigma_{ij}/r)^{12} - (\sigma_{ij}/r)^6\right]$. The parameters $\epsilon_{\alpha\beta}$ and $\sigma_{\alpha\beta}$ are given by: $\epsilon_{AA}=1.0$, $\epsilon_{AB}=1.5$, $\epsilon_{BB}=0.5$, $\sigma_{AA}=1.0$, $\sigma_{AB}=0.8$ and $\sigma_{BB}=0.88$. The composition of the mixture is such that $x_A=0.8$ and $x_B=0.2$. In the following we will measure length scales in units of $\sigma_{AA}$ and energy in units of $\epsilon_{AA}$ and set $k_B=1$. In a recent simulation the dynamics of this system was investigated by means of a molecular dynamics computer simulation~\cite{kob94,kob95a,kob95b,kob95c,kob95d}. This simulation used 800 particles of type $A$ and 200 particles of type $B$. The size of the cubic box was held fixed at $L=9.4$. In order to lower the computational burden the Lennard-Jones potential was truncated and shifted at a distance of $2.5\sigma_{\alpha\beta}$. More details on that simulation can be found in the original papers. In that work also the partial structure factors $S_{ij}(q)$ were calculated. This was done by computing the space Fourier transform of the radial distribution function $g_{ij}(r)$. Because of the finite size of the system this Fourier transform gave rise to unphysical oscillations in the structure factors at small values of $q$. Since these structure factors are the (only) input in the mode-coupling equations, such unphysical oscillations would possibly modify the outcome of the MCT calculations. Therefore we repeated the simulations and computed the partial structure factor directly from the positions of the particles by means of Eq.~(\ref{eq0}) and thus avoided the above mentioned Fourier transform. In order to filter out high frequency noise in $q$ the so determined structure factors were smoothed with a spline under tension. Because of the finite size of the box, wave vectors with modulus less than $2\pi/L$ are not accessible. Therefore we extrapolated the determined partial structure factors to $q=0$. These new simulations were done only for a few selected values of the temperature, all of them in the vicinity of the critical temperature, i.e. at $T=1.0$, $T=0.8$ and $T=0.6$. For a precise determination of the critical temperature we also needed the structure factors at intermediate values of the temperature. Therefore we used the structure factors at the three mentioned temperatures and a quadratic interpolation scheme to compute the structure factors for intermediate temperatures. Since in this temperature region the structure factors show only a weak variation with temperature such an interpolation scheme should be fairly reliable. More details on the so obtained structure factors are given in the next section. The integral equations presented in the previous section were solved iteratively in the way described in that section. The occurring integrals were computed using a high order Simpson scheme. (Note that it is necessary to use an integration scheme that accesses only points that are spaced in an equidistant way since the integrals involve convolutions. Therefore more efficient integration schemes like Gaussian quadrature cannot be used.) The upper limit of the integrals was set to $q_{co}=40$, which is sufficiently large to allow the direct correlation function to be negligible small for $q>q_{co}$. In order to perform the integration we used 300 grid points in the interval $[0,q_{co}]$. A few calculations with a larger number of points showed that this number is sufficiently large to neglect the dependence of the final results on the used discretisation scheme. Close to the glass transition the convergence of the iteration scheme given by Eq.~(\ref{eq9}) was quite slow, since the maximum eigenvalue was very close to unity ($1-E_0\approx 3\cdot 10^{-3}$, which corresponds to $T-T_c \approx 1\cdot 10^{-5}$), and only after 1000-2000 iterations reliable results could be obtained. Thus such an iteration took about 24 hrs on a medium level workstation. Note that in order to get results that are accurate to within one percent it is indeed necessary to determine $T_c$ that precisely, since, e.g., quantities like the nonergodicity parameters show a square root dependence on $(T-T_c)$. \section{Results} \label{sec4} In this section we present the obtained results. In the first part we investigate whether MCT, as presented in Sec~\ref{sec2}, is able to predict correctly various quantities that are relevant in the dynamics of the supercooled liquid. In the second part we test whether two possible modifications of MCT lead to an even better agreement between theory and the simulation. The partial structure factors $S_{ij}(q)$, crucial input for the theory, were computed as described in Sec.~\ref{sec3}. The resulting structure factors are shown in Fig.~\ref{fig1} for $T=1.0$, $T=0.8$ and $T=0.6$. (Note that, although in our computation we used $S_{ij}(q)$ for all values of $q$ up to $q_{co}=40$ we show only the range $0\leq q \leq 20$, since outside this interval the structure factors are almost constant.) From this figures we see that in that range of temperature the dependence of the structure factors on temperature is very smooth, thus justifying the interpolation procedure described in Sec.~\ref{sec3} to obtain the structure factors at intermediate values of $T$. Using the procedure described in Sec.~\ref{sec2}, we determined the critical temperature $T_c$ to be around 0.922. This value has to be compared with the result that was obtained from the molecular dynamics computer simulation, which was $T=0.435\pm0.003$~\cite{kob94,kob95a,kob95b,kob95c,kob95d}. Thus we find that, for the system considered here, MCT overestimates the critical temperature by about a factor of two. Since the idealized theory neglects certain types of relaxation processes, which are usually called hopping processes, it can be expected that the critical temperature predicted by the theory is too high. Nevertheless, {\it at first sight}, the factor of two seems to be surprisingly large when compared with the results of similar comparisons between the prediction of MCT for the value of the critical coupling parameter and the results of experiments or computer simulations. E.g. it was found in light scattering experiments on colloidal particles, a system which is considered to be described well by a hard sphere model, that these systems undergo a glass transition at a packing density $\phi_c$ which is between 0.56 and 0.58~\cite{colloids}. This value compares well with the critical packing density of $0.52\pm0.01$ of MCT for a system of hard spheres~\cite{bgs}. Thus in this case the discrepancy between the experiments and the theory is about 10\%. In the case of a binary system of soft spheres, i.e. a pair interaction which is proportional to $r^{-12}$, it was found by means of computer simulations that the glass transition occurs at a value of the effective coupling constant $\Gamma$ of 1.46~\cite{bernu87,roux89}. It can be shown that $\Gamma$ is the only relevant parameter for the thermodynamic state of such a system and that \begin{equation} \Gamma \propto nT^{-1/4}, \label{eq18} \end{equation} where $n$ is the particle density. Using the Roger-Young integral equations to compute the structure factors for this system, Barrat and Latz computed the critical value of the coupling constant within the framework of MCT and found it to be 1.32~\cite{barrat90}. Using expression~(\ref{eq18}) we thus find that the discrepancy is about 10\% in the density, comparable to the above mentioned discrepancy for the hard sphere system, but about 50\% in the critical temperature. Expression~(\ref{eq18}) is valid only for a soft sphere system. However, since at low temperatures it is mainly the repulsive core of the particles that is important for the structure of the liquid, it can be expected that for a Lennard-Jones system, which has the same type of hard core as the soft sphere system, expression~(\ref{eq18}) is a reasonable approximation. This expectation is corroborated by the calculations of Bengtzelius in which the critical temperature $T_c$ was determined at different densities for the case of a one component Lennard-Jones system (the structure factor was computed from the so-called optimized random-phase approximation)~\cite{bengtzelius86}. In that work it was shown that a change of 10\% in density gives rise to a change of a factor of two in $T_c$, in qualitative agreement with the results for the soft sphere system. Thus we conclude that a discrepancy of a factor of two in the critical temperature correspond to a discrepancy between theory and simulation of only 20\% in the coupling constant, which is comparable to the discrepancy found in the above mentioned hard sphere system and the soft sphere system. The next quantity for which we compare the prediction of the theory with the result of the computer simulation is the exponent parameter $\lambda$. In the simulation this parameter was determined by fitting the functional form provided by MCT for the $\beta$-correlator~\cite{bibles} to the corresponding master curves found in the $\beta$-relaxation regime~\cite{kob95b}. Depending on the type of correlator investigated, the value of $\lambda$ was found to vary between 0.75 and 0.83 with the most likely value of $\lambda=0.78\pm 0.02$. Our MCT calculations showed that for this system the theory predicts a value of 0.708, which compares quite well with the one found previously in the simulation. The discrepancy in $\lambda$ between theory and computer simulation is in our case smaller than the one found for the soft sphere system, for which the theory predicted $\lambda$=0.73~\cite{barrat90} and for which $\lambda\approx 0.61$ was found in the simulation. This latter value has, however, probably a relatively large error bar, since it was determined from the critical exponent of the diffusion constant close to $\Gamma_c$. This critical exponent was determined to be around 2.0~\cite{bernu87,roux89} and, if it is assumed that the connection predicted by MCT between this critical exponent and the exponent parameter $\lambda$ holds~\cite{bibles}, one obtains the quoted value of $\lambda$. However, there are two reasons why the so determined value of $\lambda$ might be slightly wrong: First the value of the critical exponent is not known very precisely~\cite{bernu87,roux89} and secondly it might well be that the above mentioned connection between the critical exponent for the diffusion constant and the exponent parameter holds only very close to the critical temperature (or coupling parameter). E.g., in the simulation of the binary Lennard-Jones system it has been found that the critical exponent for the diffusion constant and the one for the $\alpha$-relaxation times differ by about 20\%~\cite{kob95b}, although MCT predicts these exponents to be the same. Thus this shows that either the connection proposed by MCT is not always valid or that, in order to see the correct critical behavior, one has to be much closer to the critical temperature than it is currently possible with computer simulations (because very close to the critical point the time scales of the $\alpha$-relaxation exceeds the time scale accessible to such simulations). Furthermore it can be that in the vicinity of $T_c$ the so-called hopping processes become important for the soft sphere system and thus give rise to an {\it effective} exponent which is different from the one predicted by the theory. Thus because this connection between the critical exponent of the diffusion constant and the exponent parameter $\lambda$ is not beyond any doubt, the value of $\lambda$ in the soft sphere model is not known very precisely. Finally we mention that in the comparisons between the prediction of MCT and the results of the experiments on colloidal systems the latter were always assumed to be hard spheres and thus the exponent parameter was not a fit parameter~\cite{goetze91}. We now turn our attention to the wave-vector dependence of the nonergodicity parameter. This quantity was determined in the simulation~\cite{kob95b} and the results are shown for the coherent intermediate scattering functions for the $AA$, the $AB$ and the $BB$ correlation (bold dashed lines), as well as for the incoherent intermediate scattering function for the $A$ and the $B$ particles in Fig.~\ref{fig2} (thin dashed lines). For small values and very large values of $q$ it was not possible to determine $f_c$ from the simulation because of problems with finite size effects and statistics. Also included in the figure are the predictions of MCT (solid lines). First we consider Fig.~\ref{fig2}a which shows $f_c(q)$ for the $AA$ correlation and the $A$ particles. The first observation is that the theoretical curves match the ones of the simulations qualitatively well for all values of $q$ in that the location of the various extrema in the $f_c(q)$ for the coherent scattering function are reproduced correctly. In addition also the fact that for large values of $q$ the nonergodicity parameter for the coherent scattering function oscillates around the one for the incoherent one is reproduced correctly by the theory. For wave-vectors in the vicinity of the first peak of the structure factor, also the {\it quantitative} accordance between theory and simulation is very good. This agreement is less good for wave-vectors larger than the second peak in the structure factor. This can be due to two reasons: The first one is that the nonergodicity parameters as determined from the simulation are affected by systematic errors of unknown magnitude~\cite{kob95b}. From the way these quantities were measured it can be expected that these errors increase with increasing wave-vector which might be the reason for the increasing discrepancy between simulation and theory. The second possible reason is that for large values of $q$ MCT is no longer reliable. This can be understood as follows: The large wave-vectors correspond to distances which are relatively small compared to the diameter of the particles. Now one should remember that in the derivation of the MCT equations a factorization ansatz was made. This ansatz is reasonable for distances on the order of the diameter of a particle but is likely to be bad for much smaller distances. Thus it is expected that the vertices $V_{i\alpha\beta}$ of Eq.~(\ref{eq4}) are not quite correct for large values of $q$ or $k$. Therefore it is not surprising that the accordance between the results of the simulations and the predictions of MCT is not as good for large values of $q$ as it is for wave-vectors in the vicinity of the peak of the structure factor. Furthermore we comment on two smaller features in the curves. First we see that the curve for the nonergodicity parameter for the coherent scattering functions, as computed from the simulation, shows to the left and to the right of the large peaks (at $q\approx 7$ and $q\approx 12$) a small peak. These small peaks are a finite size effect which is due to the method we computed the intermediate scattering function~\cite{kob95b}. Thus the fact that these small peaks are not present at all in the curve as computed from MCT should not be viewed as a failure of the theory to reproduce this feature. Secondly we see that the MCT curve for the coherent intermediate scattering function shows some small peaks for wave-vectors smaller than 2. This behavior is likely to be due to numerical instabilities in the computation of this curve and therefore has no physical relevance. The wave-vector dependence of the nonergodicity parameter for the $AB$ correlation is shown in Fig.~\ref{fig2}b. In order to make clear where the measured points actually are, we show them as open squares and the connecting dashed line should be considered just as a guide to the eye. We recognize that this $q$-dependence is very different from the one found for the $AA$ correlation. We see that for values of $q$ near $q=7$ and near $q=10$ there is a gap in the data of the simulation. The reason for this is that in the vicinity of these wave-vectors the partial structure factor of the $AB$ correlation changes sign which in turn leads to a singularity and hence to numerical difficulties in determining the corresponding intermediate scattering function. Also included in the figure is the prediction of MCT for this nonergodicity parameter. We see that for intermediate values of $q$ the theoretical curve describes the data very well and we see that MCT correctly predicts the presence of the just mentioned singular behavior of the nonergodicity parameter. For larger values of $q$ the agreement is only qualitatively correct and the probable reason for this has been given above. In Fig.~\ref{fig2}c we show the wave-vector dependence of the nonergodicity parameter for the coherent and incoherent intermediate scattering function for the $B$ particles. From this figure we see that for the coherent part this dependence is very different from the one for the $A$ particles in that it resembles much more the $q$-dependence of the incoherent part. This can be qualitatively understood by remembering that the number of $B$ particles is smaller by a factor of 5 than the number of $A$ particles. Thus since the $B$ particles are relatively dilute the coherent intermediate scattering function behaves very similar to the single particle correlation function, i.e. the incoherent intermediate scattering function. As we can see from the figure, MCT is able to reproduce the wave-vector dependence also of these two nonergodicity parameters very well. Although the theoretical curves lie below the ones from the computer simulation for all values of $q$, the agreement is nevertheless on the order of about 5\% for intermediate values of $q$. From figure~\ref{fig2} we recognize that in all cases the nonergodicity parameters as determined from the simulations is a bit larger than the ones predicted by MCT. This is in qualitative agreement with our observation that the critical temperature as found in the simulation is quite a bit lower than the one predicted by the theory, which, in a hand wavy fashion, can be understood as follows: The nonergodicity parameter is some sort of measure for how much a particle can move in the cage formed by its surrounding particles. Since it can be expected that this movement is smaller the lower the temperature is, it follows that the nonergodicity parameter increases with decreasing temperature. Thus we see that if MCT would have predicted a critical temperature which is lower than the one it predicts now, we would expect that the theoretical nonergodicity parameters would be larger than the ones the theory predicts now. Thus we have evidence that the too high critical temperature and the nonergodicity parameters that are too small are related phenomena. We will come back to this point below. The fact that the predicted nonergodicity parameters are smaller than the ones determined from the simulation animated us to compare the former with the amplitude of the $\alpha$-relaxation. It should be remembered that in the simulation the nonergodicity parameter was determined from the height of the plateau in the time correlation function. In these simulations it was shown that the height of this plateau is {\it not} equivalent to the amplitude of the Kohlrausch-Williams-Watts (KWW) function which describes the relaxation on time scales beyond the $\beta$-relaxation time scale, i.e. $\phi(t)= A\exp(-(t/\tau)^{\beta})$. Since we have found that this KWW-amplitude $A$ is always a bit smaller than the nonergodicity parameter~\cite{kob95b,kob95d}, it is interesting to compare the $q$-dependence of this measured amplitude with the $q$-dependence of the nonergodicity parameter as predicted by MCT. This is done in Fig.~\ref{fig3}. We see that for all correlation functions the agreement between these two quantities is very good. To our surprise we find that this agreement is always better than the one between the nonergodicity parameter of the simulation and the one of MCT, which was shown in Fig.~\ref{fig2}. (An exception is the $AB$ correlation in the range $5\leq q \leq 7$, where the experimental point for $A$ are now clearly below the MCT curve.) At the moment it is not clear to us whether this surprising accordance between the KWW-amplitude and the nonergodicity parameter of MCT is just a coincidence or whether there is some underlying reason for it. One possibility might be that the corrections to the asymptotic scaling laws of MCT are larger for the $\beta$-relaxation regime than for the $\alpha$-relaxation regime. Thus it would be interesting to compute the full time dependence of the correlation functions within the framework of MCT and to compare the so obtained results with the results from the simulations. In addition it would be helpful to make similar comparisons with other systems in order to see whether the just described phenomenon holds for other systems as well. The last quantity we investigate is the wave-vector dependence of the critical amplitudes $h(q)$. These amplitudes are used to describe the time dependence of a correlation function in the $\beta$-relaxation regime, see Eq.~(\ref{eq14}). In the computer simulation it was found that in the $\beta$-relaxation regime the various intermediate scattering functions are indeed of the form of Eq.~(\ref{eq14})~\cite{kob95a} in that it was demonstrated that the left hand side of \begin{equation} \frac{\Phi(r,t)-\Phi(r,t')}{\Phi(r',t)-\Phi(r',t')}=\frac{H(r)}{H(r')} \quad, \label{eq20} \end{equation} which is obtained from the space Fourier transform of Eq.~(\ref{eq14}), holds if the times $t$ and $t'$ are on the time scale of the $\beta$-relaxation. Here $r'$ can be chosen arbitrarily. From that calculation it was possible to estimate an upper and lower bound for $H(r)$, and these bounds are shown in Fig.~\ref{fig4} for the three coherent intermediate scattering functions (thin solid lines). The value of $r'$ was 1.095, 0.9 and 1.73 for the $AA$, the $AB$ and the $BB$ correlation. Also included in the figures are the prediction of MCT for these quantities (bold solid lines). From these figures we recognize that the agreement between the results of the computer simulation and the prediction of the theory is, in the case of the $AA$ and the $AB$ correlation, qualitatively as well as quantitatively very good in that also small details in the curves, such as the small dip in the peak at around $1.8$, are reproduced correctly. The agreement between simulation and theory is not that good for the case of the $BB$ correlation in that the amplitude of the various peaks is not predicted correctly. However, the position of these peaks is in accordance with the theory and thus MCT is correct for this correlation function at least qualitatively. We also notice that in all three cases the agreement between theory and simulation is not very good at small distances. This is not surprising, since we have already explained above why MCT is not very accurate for large wave-vectors, i.e. small distances. In order to gain some insight into the nature of the various peaks in $H(r)$ we have included in the figures also the corresponding radial distribution functions $g_{ij}(r)$ at $T=0.466$, the lowest temperature considered in the computer simulation~\cite{kob95a} (dashed curves). We see that, for values of $r$ larger than the first nearest neighbor peak, the different maxima and minima in $H(r)$ occur at the same location at which the corresponding $g_{ij}(r)$ shows its extrema. This means that in $q$-space $h_{ij}(q)$ shows extrema at the same values of $q$ as the corresponding partial structure factor. Since the latter shows a similar $q$-dependence as the corresponding coherent nonergodicity parameter, see Fig.~\ref{fig2}, the nonergodicity parameter and $h_{ij}(q)$ will show extrema at the same values of $q$. A similar observation was made in the case of the soft sphere system~\cite{barrat90} and thus it can be conjectured that this is a general rule. In the remaining of this section we present the results we obtained by considering two small modification of MCT. These modifications were done in an attempt to improve the agreement between the results of the computer simulation and the prediction of the theory. The basic idea of the first modification is as follows. In the first part of this section we have shown that MCT is able to give a surprisingly good description of the $q$-dependence of the nonergodicity parameters and the critical amplitudes. The most severe disagreement seems to be that the theory overestimates the critical temperature by quite a bit. Furthermore we have seen that the theory underestimates the nonergodicity parameter and that this disagreement is most pronounced at large values of $q$. In the discussion of this effect we argued that one of the reason for its occurrence might be that a factorization ansatz, which is used in the derivation of the MCT equations, breaks down for small distances and that therefore the MCT equations are not accurate for these values of $q$. Therefore one could argue that it is better to leave out from the calculation of the memory kernel in Eq.~(\ref{eq3}) that part of the wave-vector integration altogether, i.e. to restrict the integration to wave-vectors with modulus less than a certain limit $q_{co}$. This approximation is equivalent to the assumption that the structure factor is constant for $q\geq q_{co}$. Thus the value of $q_{co}$ can be used as a fit parameter in order to match the critical temperature as predicted by MCT with the one determined from the simulation. The hope is that this fix of the critical temperature will lead to theoretical nonergodicity parameters that are in better agreement with the ones of the simulation (of course at the cost that for wave-vectors larger than $q_{co}$ the theory does not give any nonergodicity parameter at all). Thus we proceeded as follows. Using the partial structure factors that we determined at $T=0.466$, $T=0.475$ and $T=0.5$ we made an extrapolation to determine the partial structure factors at $T=0.435$, the value of the critical temperature as determined from the computer simulation. Since at these low temperatures the temperature dependence of the structure factors is only weak and very regular, such an extrapolation is not problematic. Equipped with the structure factors at the correct critical temperature we determined $q_{co}$ such that the critical temperature as determined from MCT is exactly at $T=0.435$, i.e. the critical temperature from the simulation. The value of $q_{co}$ we obtained was around 11.7, i.e. a bit to the right of the first minimum in the partial structure factor for the $AA$ correlation. This value shows on the one hand that it is mainly the first peak in the structure factor that is relevant to give a {\it qualitative} correct description of the transition and on the other hand that for a {\it quantitative} correct calculation of the transition temperature it is necessary to take into account the structure also for larger values of $q$. With this value of $q_{co}$ we computed the $q$-dependence of the nonergodicity parameters and the result for the $AA$ correlation are shown in Fig.~\ref{fig5} (solid line). Also included is the curve from the simulation (bold dashed line) and, as a reference, the curve when $q_{co}$ is 40, i.e. the value of $q_{co}$ used to compute the results of the first part of this section (thin dashed line). From this figure we recognize that the curve for the new value of $q_{co}$ is now significantly below the curve from the simulation and that the discrepancy between the (modified) theory and simulation is now quite a bit larger than it was with the original theory. Thus this shows two things: First that the contribution to the memory kernel that come from values of $q$ larger than $q_{co}=11.7$ are important in order to get quantitatively correct results, despite the above discussed fact that the integrand is not quite appropriate for such large values of $q$, and secondly that it is not that easy to improve the theory qualitatively by introducing a fit parameter in order to fix certain shortcomings of the theory (as e.g. the not so satisfactory prediction of the critical temperature). The second modification of the theory we did was to ignore the fact that we have to compare the results for the $q$-dependence of the nonergodicity parameter as obtained from the simulation with the prediction of MCT for the nonergodicity parameter {\it at the critical temperature $T_c$}. Since we have seen that MCT underestimates the nonergodicity parameters and we know that for temperatures $T<T_c$ the theory predicts that the nonergodicity parameters increase, we tried to correct this discrepancy by computing $f_A^{(s)}(q)$ at a temperature $T_{eff}$ below $T_c$ and to determine $T_{eff}$ by requiring that this nonergodicity parameter, which we will call $f_{eff}^{(s)}$, fits the corresponding quantity from the simulation well. The reason for choosing this type of nonergodicity parameter, instead of e.g. the one for the $AA$ correlation, is that in the simulation it can be determined with the best accuracy. This fit gave a value for $T_{eff}$ around 0.91, thus quite close to the original critical temperature $T_c=0.922$. The resulting $q$-dependence of $f_{eff}^{(s)}$ is shown in Fig.~\ref{fig6}a together with $f_c^{(s)}$ from the simulation. We see that for small and intermediate values of $q$ the agreement between $f_{eff}^{(s)}$ and $f_c^{(s)}$ is very good. Only for large values of $q$ significant, but not large, discrepancies occur. We also computed the $q$ dependence of the other nonergodicity parameters for the {\it same temperature $T_{eff}$} and the results are shown in Fig.~\ref{fig6}. From Fig.~\ref{fig6}a we see that for the coherent nonergodicity parameter for the $AA$ correlation the agreement between $f_c(q)$ and $f_{eff}(q)$ has improved significantly compared to the agreement when the original MCT function is used (see Fig.~\ref{fig2}a) and that for $q$ values in the vicinity of the first peak and the first maximum the agreement is perfect to within the noise of the simulation data. Also for the $AB$ correlation function (Fig.~\ref{fig6}b) the agreement between simulation and theory has improved considerably compared to the original MCT and the same conclusion holds for the coherent and incoherent nonergodicity parameters for the $B$ particles (Fig.~\ref{fig6}c). Thus we conclude that introducing one fit parameter, namely the temperature $T_{eff}$ at which the nonergodicity parameters are evaluated within the framework of MCT leads to a significant improvement of the agreement between theory and simulation. \section{Summary and Discussion} \label{sec5} We have presented the results of a numerical calculation in which the mode-coupling equations were solved for a binary Lennard-Jones mixture. The goal of these calculations was to test whether the agreement between the predictions of MCT for the dynamics, which was investigated by means of a computer simulation~\cite{kob94,kob95a,kob95b,kob95c,kob95d}, holds only for the universal predictions of the theory or also for the nonuniversal ones. Using the partial structure factors, as determined from a computer simulation, as input, we computed within the framework of MCT the critical temperature, the exponent parameter, the $q$-dependence of the various nonergodicity parameters and the various critical amplitudes. Although the critical temperature as predicted by MCT is a factor of two larger than the one determined from the computer simulation, we argue that this discrepancy is significantly smaller when expressed through the effective coupling constants, and is then comparable with the discrepancies found for this quantity for systems like hard spheres~\cite{colloids} or soft spheres~\cite{barrat90}. The exponent parameter as predicted by the theory is in fair agreement with the one determined from the simulation. MCT makes a very good quantitative prediction for the wave-vector dependence of the nonergodicity parameter for values of $q$ in the vicinity of the first maximum and the first minimum. For large values of $q$ the agreement is still fair and we can rationalize the increasing discrepancy between theory and simulation by arguing that some of the approximations used to derive the mode-coupling equations no longer hold in this limit. In addition we also showed that the theory is also able to make a quantitatively correct prediction of the various critical amplitudes $H(r)$. We also compared the $q$-dependence of the Kohlrausch-Williams-Watts amplitude, as determined from the simulation, with the $q$-dependence of the nonergodicity parameter as predicted by MCT and found that the two quantities match surprisingly well. So far it is not clear why this is the case and how general this observation is. Therefore it would be very valuable to make similar comparisons for different types of systems. In an attempt to improve the agreement between the measured and theoretical nonergodicity parameters we introduced an upper cut-off $q_{co}$ in the integral of the memory kernel and used $q_{co}$ to match the critical temperature $T_c$ between MCT and simulation. We found that the introduction of this fit parameter leads to a worsening of the agreement between the measured and theoretical nonergodicity parameter which shows that the contributions to the memory kernel from large values of $q$ are important for a quantitative correct description of the nonergodicity parameter. In a second ``modification'' of the theory we used temperature as a fit parameter and determined a temperature $T_{eff}<T_c$ such that the resulting incoherent nonergodicity parameter for the $A$ particles fits the simulation data well. We found that at this temperature also all the other nonergodicity parameters fit the data from the simulation well, in some cases even very well. Thus it seems that there exists a temperature $T_{eff}$ for which MCT is able to predict very well the $q$-dependence of the various nonergodicity parameters as measured in the simulation {\it at $T_c$}. This shows that the intrinsic structure of the mode-coupling equations are clearly able to correctly describe such quantities and that it is perhaps only through the omission of certain contributions to the memory-kernel that there is no perfect quantitative agreement between the prediction of the theory and the results of the simulation. To summarize we can say that our calculations have shown that MCT is able to give a correct {\it quantitative} description of the dynamics of a simple liquid if one restricts oneself to quantities like the critical temperature, the exponent parameter, the nonergodicity parameter or the critical amplitudes. Whether MCT is also able to give a correct description of the {\it full} time dependence of the various correlation functions, as it is the case for the $\beta$-relaxation regime in colloidal systems~\cite{goetze91}, remains to be tested and is subject of ongoing work. Acknowledgements: We thank M. Fuchs for extensive help and enlightening discussions during this work, W. G\"otze for valuable discussions and useful comments on the manuscript, K. Binder for helpful comments on the manuscript and J.L. Barrat for providing us with some programs which allowed to check our programs. This work was supported by SFB 262/D1 of the Deutsche Forschungsgemeinschaft.

proofpile-arXiv_065-579

\section{Introduction} In previous microlensing models the rates and optical depth per star are usually estimated (Kiraga and Paczy\'nski, 1994 Han and Gould, 1995, Mollerach and Roulet , 1996). Below some limiting magnitude these stars can not be resolved individually and form a dense stellar background on all the surface of the images. I will call these star "background stars". These background stars can be lensed, and this issue has already been investigated in the case of the LMC (Bouquet, 1993), and for the Galaxy (Nemiroff, 1994). However the stellar fields of the current Microlensing experiments MACHO (Alcock {\it et al.} 1996, Alcock {\it et al.} 1993), OGLE (Udalski{ \it et al.} 1994), DUO (Alard 1995a, Alard {\it et al.} 1995b) , EROS (Aubourg {\it et al.} 1993.) are extremly crowded, so tha inside the resolution radius of a monitored stars, there are plenty of these unresolved background stars. Thus if one of these background stars is lensed, we will see a blended event. The amplification, duration and impact parameter will be biased by the blending. We may now wonder if their contribution to the total number of events observed is significant, compared to those expected from the resolved stars. As the blending of a background star with a resolved star will be high we expect a large reduction of the effective Einsten Radius (Di Stefano and Esin, 1995). Consequently the observed number of events will be a competition between the decrease of the Einsten radius, and the increasing number of stars at larger magnitudes. Also the effective duration of the event will be much shorter than it is for to events on resolved stars, and consequently it may produce a tail of short events. These short events could be confused with low mass lens events, and especially lensing by brown dwarf, so that a detailed study of these events is particulary important. Another problem will be to recognise these blended events. The blending of light produces a modified light curve which might be distinguished from the unblended light curve. However to make an unambiguous separation a good time sampling is required, and a very accurate photometry is required. In the case of these short events, we expect that it will be difficult to achieve a sampling dense enough. \section{The Bulge luminosity function} The study of lensing of unresolved stars is very sensitive to the shape of the adopted luminosity function. In previous works (Kiraga and Pazcy\'nski, Roulet and Mollerach, Han and Gould), the number of stars at a given luminosity $L$ is expressed as: $dN=L^{-\alpha}$, with usually $\alpha \simeq 2$. A more realistic luminosity function can be derived from the Holtzman (Holtzman {\it et al.} 1993) observations of a field in Baade Window with the HST. The final Holtzman {\it et al.} luminosity function can be approximated by three continuous straight segments with different slopes. The first change in the slope occur at $V \simeq 20$ . With the extinction and distance to the Galactic Center adopted by Holtzman {\it et al.} , it correponds to a value of $M_V=4.9$. . The Holtzman {\it et al.} study agrees well with Terndrup (Terndrup 1989) investigations of the Galactic Bulge which found a value of $M_V=4.07$ for the Bulge turn-off at $b=-8$. In the Holtzman luminosity function the turn-off is situated around $M_V\simeq 4.0$ also. The second change in the slope occur around $V=21.5$ and seeems to continue until the Holtzman {\it et al.} limit situated at $V \simeq 22.5$. A comparison with the Luminosity function of the globular clusters reviewed by Mould (Mould 1996) indicates that the shape of their luminosity function is almost the same in this range of magnitude. The Mould diagram allows also to a better determination of the last segment of the luminosity function. This straight segment seems to hold until $M_V=11$ where a possible turnover in the luminosity function is observed. However $M_V=11$ is about 7 magnitudes fainter than the limiting magnitude of the current microlensing surveys, and such faint unresolved stars do not contribute to the optical depth. The duration of the events on stars blended with a star 7 magnitudes brighter would be so short that it is not observable. Consequently the exact behavior of the luminosity function in this region is not important for our study, and we will make the reasonable choice to cut the luminosity function at $M_V=11$. Looking now at the bright side of the luminosity function, we find that at magnitudes brighter than the turnoff , the slope is very steep indicating that the number of stars drops rapidly. In this region the Bulge stars are essentially sub giants. The density rises again in the clump giant region, however these stars do not makes more than 5 percent of the stars in the current surveys. Consequently they have not a very significant influence on the total luminosity function. These stars should be treated separately (Alard 1996), and in the present study, we will ignore them. This will put the bright end of the luminosity function at $M_V \simeq 3$ in the Holtzman {\it et al.} diagram. The adopted value for the distance to the Galactic center in this study is 8.5 Kpc. The final adopted luminosity function is shown in figure 1. \begin{figure} \centerline{\psfig{angle=0,figure=lfa.ps,width=8cm}} \caption{ The adopted Luminosity function for the present study.} \end{figure} \section{Estimation of the lensing rates and optical depth} It is clear that, in an investigation of the lensing of unresolved stars, consistency requires that we calculate the rates and optical depth also with the same luminosity function. This involves some slight changes from previous analysis. Therefore I will first investigate the case of resolved stars considering a more general form for the luminosity function. I first calculate the change in the optical depth. \subsection{Optical depth} The Numbers of source stars at distance $D_s$, and with absolute luminosity L can be written as: $$ dN(D_s,L)= k \ lf(L)\ n(D_s) \ D_s^2 \ dL \ dD_s $$ Where k is a constant. \\ Suppose that the experiment is able to find a total of $N_{tot}$ stars up to a limit in apparent luminosity $l_0$. This leads to the following expressions: \begin{equation} \hspace{1cm} \frac{dN(D_s,L)}{N_{tot}} = \frac{1}{I_d} \times lf(L)\ n(D_s) \ D_s^2 \ dL \ dD_s \end{equation} with: $$ I_d = {\int_{D_s} \int_{L=L_{min}}^{L=L_{max}} lf(L)\ n(D_s) \ D_s^2 \ dL \ dD_s} $$ and: $L_{min} = l_0 \times D_0^2/D_s^2$, where $l_0$ is the apparent limiting luminosity. \\ $L_{max} = l_{sat} \times D_s^2/D_0^2$. Where $l_{sat}$ is defined as the saturation limit of the detector. It can be expressed as: $l_{sat}=d \times l_0$, where $d$ is the detector dynamic range, a reasonable value is about 6 magnitudes in crowded fields, either for CCD or photographic plates. The optical depth associated with these stars can be calculated as (Paczy\'nski 1991, Kiraga and Paczy\'nski 1994, hereafter refered as KP): \begin{equation} \hspace{1.5cm} d\tau_0(D_s,L)=\frac{dN(D_s,L)}{N_{tot}} \times f(D_s) \end{equation} with: $$ f(D_s)=\frac{4 \pi G}{c^2} \int_0^{D_s} \rho_d(D_d) \times \frac {D_d(D_s-D_d)}{D_s} \ dD_d $$ and $l=L\times \frac{D_0^2}{D_s^2}$ \\ To get the total optical depth we have now to integrate eq (1) over the source distances $D_s$ and the absolute luminosity $L$ of these stars. \\ $$ \tau_0 = \frac{1}{I_k} \int_{D_s} \ N_{L0}(D_s) \ f(D_s) \ n(D_s) \ D_s^{2-2 \beta} \ dD_s $$ with: $$ N_{L0}(D_s)= \frac{I_k}{I_d} \times D_s^{2 \beta} \int_{L=L_{min}}^{L=L_{max}} \ lf(L) \ dL $$ and: $$ I_k = {\int_{D_s} \ n(D_s) \ D_s^{ \ 2-2 \beta} \ dL \ dD_s} $$ Here the factor $D_s^{2 \beta}$ is put in this expression to allow direct comparison with the standard optical depth formulae (Kiraga and Paczy\'nski 1994): $$ \tau_k= \frac{1}{I_k}\int_{D_s} \ f(D_s) \ n(D_s) \ D_s^{ \ 2-2 \beta} \ dD_s $$ We see that the Kiraga and Paczy\'nski expression is simply equivalant to taking $N_{L0}(D_s)=1$ at all distances. A slight improvement is to take $N_{L0}(D_s)=0$ beyond a given distance (Roulet and Mollerach, 1995). The plot in Figure 2 allows a direct comparison between $N_{L0}(D_s)$, and the approximation $N_{L0}(D_s)=1$. Figure 2 illustrates the calculations both for DUO and for the OGLE experiment, which is about one magnitude deeper. \begin{figure} \centerline{\psfig{angle=0,figure=N_L0.ps,width=8cm}} \caption{Plot of the $N_{L0}(D_s)$ function as a function of distance. For the DUO experiment a limiting magnitude of $M_V=3.5$ at the distance of the Galactic Center was adopted, OGLE is assumed to be one magnitude deeper. The dotted line is for DUO, and the dashed line for OGLE.} \end{figure} Let's see now the changes in the total optical depth introduced by the $N_{L0}(D_s)$ function. The optical depths are computed using the COBE bar model for the Bulge (Dwek {\it et al.}, ..), and the Bahcall model for the Disk (Bahcall and Soneira, 1984) The ratio of the modified optical depth to the standard one is shown in table 1. The change is already important for OGLE, and is more significant again for DUO. It is easy to understand that the lower optical depth of the new formulae is essentially due to the fact that due to the limiting magnitude, the experiments are more sensitive to sources in the near end of the bar, for which the density of lenses on the line of sight is smaller. \begin{table} \caption{Optical depth ratio for DUO and OGLE} \label{} \begin{tabular}{llll} \hline\noalign{\smallskip} Optical depth ratio & Bulge lenses & Disk lenses & Total \\ \hline\noalign{\smallskip} OGLE & 0.85 & 0.89 & 0.86 \\ DUO & 0.75 & 0.8 & 0.77 \\ \hline\noalign{\smallskip} \noalign{\smallskip} \end{tabular} \end{table} \subsection{Event rates.} The Formulae given for the lensing rates by Kiraga and Paczy\'nski is: $$ \Gamma_k = \frac{1}{I_k} \int_{D_s} \ f_{\gamma}(D_s) \ n(D_s) \ D_s^{ \ 2-2 \beta} \ dD_s $$ Where: $$ f_{\gamma}(D_s) = \frac{4 \ G^{1/2}}{c M^{1/2}} \int_0^{D_s} \int_{V_i} \rho_d(D_d) \ V(D_s,D_d) $$ $$ \times {[\frac {D_d(D_s-D_d)}{D_s}]}^{1/2} \ dD_d \ dV_i $$ $V(D_s,D_d)$ is the relative velocity between lens and source, and the $V_i$ are the four components of this velocity. \\ Now, it is straightforward to introduce the $N_{L0}(D_s)$ function in this expression, exactly as was done for the optical depth. \\ \\ This leads to: $$ \Gamma_0 = \frac{1}{I_k} \int_{D_s} \ N_{L0}(D_s) \ f_{\gamma}(D_s) \ n(D_s) \ D_s^{2-2 \beta} \ dD_s $$ The result of the calculation of the $\Gamma_0$ and $\Gamma_k$ rates is shown in figure 3 for DUO, and in figure 4 for OGLE. For all the figures the velocity dispersion given by Han and Gould (Han and Gould 1995) was adopted. The mass function is a Salpeter mass function with a lower cut off at $0.08 M_{\sun}$ and an upper cut off at $1 M_{\sun}$. \begin{figure} \centerline{\psfig{angle=0,figure=duo_rates0.ps,width=8cm}} \caption{Comparison of the total rates of events for Bulge sources using KP formulae (dashed line), and the formulae described in this text (solid line). The total rate is estimated for the 1994 season of the DUO experiment using the DUO efficiencies (Alard and Guibert 1996). The two bumps are due to the particular shape of the DUO efficiencies.} \end{figure} \begin{figure} \centerline{\psfig{angle=0,figure=ogle_rates0.ps,width=8cm}} \caption{Comparison of the total rates of events for Bulge sources using KP formulae (dashed line), and the formulae described in this text (solid line). The total rate is estimated for $10^6$ stars in Baade window using the OGLE efficiency (Udalsky {\it et al.} 1994)} \end{figure} \section{Modelling the lensing of blended unresolved stars} \subsection{Basic Principles} The Einsten radius is defined as the distance for which the amplification of a microlensing event is a factor of 1.34. In the case of an unresolved background star of luminosity $l_0$, blended with a resolved star of magnitude $L_0$, the background star amplification required to make a total amplification in the combined image of 1.34 is: $$ A_b = 0.34 \times f_b+1.34 $$ Hereafter, I will refer to $f_b$ as the blending factor: $$ f_b=L_0/l_0 $$ This gives the following expression for the blended Einsten radius: $$ R_b(f_b)= R_e \times re_b(f_b) $$ with: $$ re_b(f_b)= \left [2(-1+A_b/\sqrt{(A_b^2-1.0)} \right ]^{1/2} $$ \subsection{Optical Depth due to the Unresolved stars.} Let us consider unresolved background stars with absolute luminosity L at a distance $D_s$, whose number is dN($D_s$,L), and let us suppose also that they are blended with stars of apparent luminosity $l_b$. The optical depth per resolved star associated with these stars can be calculated using equation (2) by replacing the Einsten radius by the blended Einsten radius. $$ d\tau(D_s,L)=\frac{dN(D_s,L)}{N_{tot}} \times r_b(l_b,l) \times f(D_s) \times R $$ The expression for $d\tau$ contains the factor $R$, which takes into account that only those unresolved objects which are lensed close to a resolved star will be seen. The Value of $R$ will be close to the ratio of the area of the seeing disk, to the mean space occupied by a star on the image.\\ with: \\ $r_b(l_b,l)=[re_b(f_b)]^2$, and $f_b = l_b/l$. \\ \\ We have now to take into account that the luminosity $l_b$ of the resolved star is not constant, but comes from the luminosity function. It was shown by Zhao (Zhao 1995) using the OGLE data that the number of resolved stars in Baade'swindow can be well described by a power law, with an exponent close to $-2$. To be precise we have a fraction of stars near luminosity $l_b$: \begin{equation} \hspace{2.5cm} dF(l_b)= \beta_2 \ l_b^{-\alpha_2} \ l_0^{\beta_2}\ dl_b \end{equation} with: $\alpha_2 \simeq 2$ and $\beta_2=\alpha_2-1$ \\ consequently, $$ d\tau(D_s,L,l_b)=d\tau(D_s,L) \times dF(l_b) $$ To get the total optical depth associated with the unresolved stars, we have now to integrate over the luminosity range of these stars, from the absolute luminosity ($L_{min}$) to the faint end of their luminosity function ($L_{end}$). In addition, we have also to integrate over all the possible distances for the sources, and all apparent luminosities for the blends.\\ \\ This leads to: $$ \tau= \int_{L=L_{end}}^{L=L_{min}}\int_{D_s} \int_{l_b=l_0}^{l_b=l_{sat}} \frac{dN(D_s,L)}{N_{tot}} $$ $$ \times dF(l_b) \ R \ r_b(l_b,l) \ f(D_s) \ dD_s \ dL \ dl_b $$ \\ If now we uses eqs, (1), (2), (3), we can express $\tau$ as: \\ \\ $$ \tau= \frac{R}{I_d} \times \beta_2 \ l_0^{\beta_2} \int_{D_s} \int_{L=L_{end}}^{L=L_{min}} \int_{l_b=l_0}^{l_b=l_{sat}} \ l_b^{-\alpha_2} \ r_b(l_b,l) $$ $$ \times \ lf(L) \ f(D_s) \ n(D_s) \ D_s^2 \ dL \ dl_b \ dD_s $$ This expression can be integrated over the variables $L$ and $l_b$, consquently: $$ \tau= \frac{1}{I_k} \int_{D_s} \ N_L(D_s) \ f(D_s) \ n(D_s) \ D_s^{2-2\beta} \ dD_s $$ with: $$ N_L(D_s)= \frac{I_k}{I_d} \ R \ \beta_2 \ l_0^{\beta_2} \ D_s^{2 \beta}\int_{L=L_{end}}^{L=L_{min}} \int_{l_b=l_0}^{l_b=l_{sat}} \ l_b^{-\alpha_2} \ r_b(l_b,l) $$ $$ \times lf(L) \ dL \ dl_b $$ Note that the dependance in distance is hidden in the variable $l=L \times D_0^2/D_s^2$, and also in the boundary $L_{min}$. \\ \\ Figure 5 illustrates the new $N_L(D_s)$ function for the unresolved stars in the case of the DUO and OGLE experiments. Note that these functiond reach their maxima at larger distances than the previous $N_{L0}(D_s)$ functions, it means that a large contribution will come from unresolved sources on the far side of the bar. \begin{figure} \centerline{\psfig{angle=0,figure=N_L.ps,width=8cm}} \caption{Plot of the $N_L(D_s)/R$ function as a function of distance for unresolved sources. The limiting magnitude defined for figure 1 were kept for this illustration. The dotted line is for DUO, and the dashed line for OGLE.} \end{figure} It is now possible to quantify the contribution of the unresolved sources to the total optical depth. In table 2 the ratio of the optical depth of unresolved sources to the the optical depth of resolved sources is indicated for DUO and OGLE. The calculation of this ratio require an estimation of the R constant. A crude estimate of R is the ratio of the surface covered by the resolution radius to the mean area occupied by a star. For the DUO experiment the resolution radius is close to 3 pixels, and the mean area occupied by a star is about 60 pixels (Alard and Guibert 1996). This gives $R \simeq 0.5$. For OGLE the resolution radius is probably close to the seeing value (this is due to smaller pixels). Consequently with a mean seeing of $1.25 \arcsec$ in Las Campanas, and a pixel of $0.44 \arcsec$ we can estimate the resolution radius as 2.7 pixels. The OGLE experiment follows $1.3 \ 10^6$ stars on 14 images of $2048 \times 2048$ pixels each. This gives again a value for R, of $R \simeq 0.5$. \begin{table} \caption{Ratio of the optical depth of unresolved sources to the the optical depth of resolved sources . A value of $R=0.5$ is adopted for the calculations (see text for discussions).} \begin{tabular}{llll} \hline\noalign{\smallskip} $ \tau/\tau_0$ & Bulge lenses & Disk lenses & Total \\ \hline\noalign{\smallskip} OGLE & 0.58 & 0.55 & 0.57 \\ DUO & 1.93 & 1.81 & 1.89\\ \hline\noalign{\smallskip} \noalign{\smallskip} \end{tabular} \end{table} \subsection{The rates of microlensing events from unresolved stars.} The rates per resolved star for unresolved events can be calculated in the same way as for the Optical Depth. \\ We have: \begin{equation} \Gamma = \frac{1}{I_k} \int_{D_s} \ N_{L\gamma}(D_s) \ f_{\gamma}(D_s) \ n(D_s) \ D_s^{2-2 \beta} \ dD_s \end{equation} with: $$ N_{L\gamma}(D_s)= \frac{I_k}{I_d} \ R \ \beta_2 \ l_0^{\beta_2} \ D_s^{2 \beta} \int_{L=L_{end}}^{L=L_{min}} \int_{l_b=l_0}^{l_b=l_{sat}} \ l_b^{-\alpha_2} \ re_b(l_b,l) $$ $$ \times lf(L) \ dL \ dl_b $$ It is clear that the change of the Einsten radius by the factor $re_b(l_b,l)$ changes also the duration by the same factor. Consequently in the differential rates calculation, each duration $t_E=r_E/V$ will have to be scaled by the factor $re_b(l_b,l)$. However, we see in equation 4 that for a given Einsten radius there is a distribution of the scaling factors $re_b(l_b,l)$ hidden in the double integral $N_{L\gamma}(D_s)$. This distribution can be calculated for each distance $D_s$, and for the realisation of the calculations, the scaling factor of the Einsten radius will be choosen by a Monte-Carlo method using this distribution. Some examples of such distributions for different distances are shown in figure 6, in the case of the DUO experiment. \\ \\ \begin{figure} \centerline{\psfig{angle=-90,figure=table.ps,width=8cm}} \caption{Examples of duration scaling factor distribution functions, for different distances. The solid line is for a distance of 8 Kpc, the long dashed line is for 15 Kpc, and the short dashed line is for 6 Kpc.} \end{figure} The result of the calculation of the $\Gamma_0$ and $\Gamma_k$ rates is shown in figure 7 for DUO, and in figure 8 for OGLE. It is evident that the DUO experiment is dominated by unresolved stars. This result is in very good agreement with the data (Alard and Guibert 1996). Even for OGLE which has the deepest photometry, the bias is still very significant. It is rather straightforward to relate these events to the excess in the rates and optical depth observed towards the Bulge. This excess was not been explained, even using a bar in the Galactic model. We find here a natural explaination. \begin{figure} \centerline{\psfig{angle=0,figure=duo_rates.ps,width=8cm}} \caption{Comparison of the the contribution to Bulge microlensing event rates for resolved stars (short dashed line), and unresolved stars (solid line). The total rate is represented by a long dashed line. The rates are estimated for the 1994 season of the DUO experiment using the DUO efficiencies (Alard and Guibert 1996). Note the dominant contribution of the unresolved stars.} \end{figure} \begin{figure} \centerline{\psfig{angle=0,figure=ogle_rates.ps,width=8cm}} \caption{Comparison of the contribution to Bulge microlensing event rates for resolved stars (short dashed line), and unresolved stars (solid line). The total rate is represented by a long dashed line. The rates are estimated for $10^6$ stars in Baade window using the OGLE efficiency (Udalsky {\it et al.} 1994).} \end{figure} \section{Can we identify the blending events using the light curves ?} The light curve of a blended event is affected by the additional light coming from the companion, so that the light curve is modified. This blended light curve has more extended wings than an unblended event,so that it might be possible to differentiate it from the unblended one using a chi-square test. \\ Another important issue is the possibilty that the amplified star and its companion may have different colors. For instance if the companion is more blue than the amplified star, we expect more blending in the blue, and thus a larger reduction of the amplitude in this color than in the red. \subsection{Color variations} Let us now investigate the problem of color variation during the event in more detail. In the current microlensening experiments the sources are concentated in the range $18<V<20$. An examination of Terndrup (Terndrup, 1988) color magnitude diagram in Baade'swindow indicates\ that the mean $V-I$ color is about 1.3 to 1.4. For fainter stars, where expect to find the unresolved sources, the Holtzman {\it et al.} color magnitude diagram shows that the color changes again very slightly. The mean color a close to 1.4 at $V=22$. We see that in the apparent magnitude range of interest we expect a diferential color variation of about 0.1 magnitude at most between the unresolved source and the blending companion. we have to reach $V=23$ to expect more significant color variation between sources and blend (about 0.2 magnitudes). Unfortunately the photographic technique does not perform very well in the red band, thus an accurate investigation of the color changes during the event is not possible. It means that the DUO survey will not be able to detect the slight color changes expected. The OGLE survey has a good coverage only in the I band, the sampling in V is very sparse, and does not allow color analysis during the event. In the MACHO case, photometric data are available in two large bandpasses, which is transformed to a $V-R$ color index. The analysis of the LMC events by MACHO shows that they can identify color variations due to blending for some of the events. However there is a small residual noise on the color variations of all the events of about 0.02 to 0.03 magnitudes, associated with the photometric errors. At a 3 $\sigma$ confidence level we may require a color variation of about $0.075$ magnitude to firmly established the chromaticity of the event. Converted to $V-I$ it requires a color changes of about 0.15 magnitudes at least to recognise a blended event with a good confidence level. \\ We consider now with a such accuracy how many of the Bulge microlensing events which involves unresolved sourves could be identified as such. \\ \\ Let us make the following simple model: \\ \\ And we assume that the difference of color can be described by a gaussian distribution, shifted by a systematic value. Thus the number density $n(C)$ of stars with color a difference in color C will be expressed as: \begin{equation} n(C)=\exp \left ( \frac{(C-{\rm shift_c})^2}{-2 \ \sigma_c^2} \right ) \end{equation} The Holtzman {\it et al.} color magnitude diagram indicates that for the gaussian we can assume $\sigma_c \simeq 0.07$. In the range $V=20$ to $V=22$, a conservative value for the shift is ${\rm shift_c} \simeq 0.15$. The color shift is then given by: $$ \Delta_C=2.5 \log(A_i/A_v) $$ i and v stand where the subscripts for the two photometric colors. with: $$ A_i=\frac{1+a f_b}{1+f_b} $$ and: $$ A_v=\frac{1+a f_b c}{1+f_b c} $$ \\ a and c are defined by the following expressions: \\ \\ $$ a = \frac{u^2+2}{u \times \sqrt{u^2+4}} \hspace{0.75cm} c=10^{ \ C/2.5}. $$ The number of events with color blending signature identified will be simply the number of events with a color shift $\Delta_C>0.15$. It can be easily calculated if we assume that the unblended impact parameter $u$ has a uniform distribution. In this case it is sufficient to integrate the number of events with $\Delta_C>0.15$ over the $u$ distribution with a given color difference $C$. The calculation is completed by suming over the color difference distribution defined in eq (5).The maximum value of the impact parameter is set by the blending factor $fb$ using the formulae: \begin{equation} u_{max}=\left [2(-1+A_b/\sqrt{(A_b^2-1.0)} \right ]^{1/2} \end{equation} with: $$ A_b = 0.34 \times f_b+1.34 $$ The calculation of the percentage of events with color shifts is performed for several values of the blending factor $f_b$. An illustration of these calculations is given in Figure 9, where the color shift is expressed as a function of the impact parameter for a few $f_b$ values, and color diffrence of 0.2 magnitude. Looking at this diagram we can already guess that the number of events with detectable color shitfts will be small. The final values of the events with color shifts identified is given in table 3. The values are extremly small. A shift of 0.15 mags in the color difference distribution, even for high values of the blending factor. The values for a shift of 0.3 magnitudes are also given, to illustrate the case of the very faint unresolved stars. However these stars will be extremly blended, and consequently will give events will very short duration which will be almost all removed by the low efficiency of the experiments in this range. Consequently the color shift method will give poor results in the case of the Galactic Bulge. In the case of the Magellanic Clouds, the color changes rapidly with the magnitude close to the limit of the microlensing experiments, which explains why so many events with color shifts are found by MACHO (Alcock, {\it et al.}, 1996)towards the LMC. \begin{figure} \centerline{\psfig{angle=0,figure=colshift.ps,width=8cm}} \caption{The distribution of the color shift for blending parameters of 2,5,10,50, and a color difference of 0.2. The curve with $f_b=50$ is upper, and the values 10,5,2, are below in a decreasing order.} \end{figure} \begin{table} \caption{The fraction of events with detectable color shifts.} \begin{tabular}{lllll} \hline\noalign{\smallskip} Blending factor &2 & 5 & 10 & 50 \\ \hline\noalign{\smallskip} ${\rm shift_c}=0.15$ & 0.0025 & 0.0219 & 0.0364 & 0.0485 \\ ${\rm shift_c}=0.3$ & 0.1677 & 0.3079 & 0.3559 & 0.3906 \\ \hline\noalign{\smallskip} \noalign{\smallskip} \end{tabular} \end{table} \subsection{The shape of the light curves.} I will investigate another possibility to identify a blended microlensing event based on the shape of the light curve in this section. The blending of light modifies the shape of the light curve, and consequently we may use a statistical test to see if a light curves differs significantly from an unblended one. However we have to realise that the likely difference due to blending is rather small, and we may expect that with the current available photometry it will be difficult to identify a blended light curve at a significant level of confidence. It is possible to adress this problem in more general terms using Monte-Carlo simulation of microlensing events. Assuming a noise distribution, it is easy to simulate microlensing events with different blending factors. These blended events can be analysed by fitting an unblended curve to the simulated data set. We expect a systematic difference in the chi-square of the fir due to the different shape of the blended light curve, especially in the wings. However the problem is to know how significant is this difference compared to the normal chi-square variations for unblended events due to noise. This problem can be easily addressed if we are able to build the chi-square distribution for a series of blended light curves. Such chi-square distributions are illustrated in Fig. 10, the line shows the limit within which 95 percent of the chi-square distribution for unblended events is contained. Beyond this limit we have a 95 percent confidence level that the chi-square indicates a light curve which is systematically different from the unblended model. \begin{figure} \centerline{\psfig{angle=0,figure=hist_chi.ps,width=8cm}} \caption{An example of simulated Chi-square distributions for unblended events (full line), and for blended events (dashed line). The vertical line indicates the 95 percent limit for the unblended distribution (see text for explanations). For this example the events are simulated with a duration of 40 days, and an impact parameter of 0.05. The blending factor has a value of 4.} \end{figure} It is now sufficient to sum the fraction of the blended distribution beyond this limit to get the fraction of blended events $R_b$ which can be identified with a good confidence level. The ability to recognise a blended event will of course depend on the amplitude of the event, for a given blending factor $fb$ and a given duration $t_0$. The amplitude is related to the impact parameter $u$. Consequently to get the expected fraction $R_b$ , at $fb$ and $t_0$ it is sufficient to sum on the impact parameter $u$, in the range 0 to $u_{max}$ ( $u_{max}$ is defined in eq 6). The result of the corresponding calculation is illustrated in Fig 11, as a function of the blending factor, for a series of durations. The blended events are well identified only for small impact parameters. Beyond $u_{max}/10$, the efficiency drops significantly, which makes the total efficiency quite low. \begin{figure} \centerline{\psfig{angle=0,figure=mc.ps,width=8cm}} \caption{The fraction of blended events identified as a function of the blending factor for different durations. The durations are respectively 100,63,25,15,10,6,4,2.5 days (from top to bottom).} \end{figure} These tabulated expressions of $R_b$ as a function of $fb$ and $t_0$ are now introduced directly in the calculation of the unresolved event rates. Then new rates corrected for the number of blended events which could be recognised are computed. Fig 12 shows a comparison between this rate and the uncorrected rate in a case resembling to the OGLE experiment. Typical errors distributions are extrpolated from the DUO experiment (Alard and Guibert 1996), and are divided by a scaling factor of 2 to take into account the better quality of the OGLE photometry. The resulting errors distribution has a sigma about 0.07 magnitude for most of the points, and reaches 0.1 or slightly more in 10 percent of the cases; it seems to be an acceptable description of the OGLE photometry on stars close to the limiting magnitude, which represents most of the sample. We see that the difference is small, and considering that OGLE has the best photometric accuracy, it is certainly worse again for the others experiments. \\ \\ The conclusion is again, as for the color test, that given the mean quality of the photometry available in the microlensing experiments only, a slight percentage of blended events should be recognised on the basis of the light curve shape. Only the the events with good photometry and small impact parameter should be identified as blended. However for some of the monitored stars the OGLE experiment is able to perform photometry much better again than for the majority of the stars. This gives an interesting opportunity to look for a blending signature in the light curve. \begin{figure} \centerline{\psfig{angle=0,figure=ogle_rates_mc.ps,width=8cm}} \caption{The event rate statiscally corrected for the fraction of blended which might be identified on the basis of their light curve shape. The corrected rate is represented by a full line, and the initial rate by a dashed line.} \end{figure} \section{OGLE 5: lensing of an unresolved star} The photometry achieved on the OGLE 5 event is of an interesting quality, the errors bars are as good as a few percent on many points. However this event is not well fitted with the standard unblended model (see figure 13). The discrepancy is especially large in the wings of the event, so that it is natural to try to fit this event with blending. Fig. 14 shows the dramatic improvement of the chi-square when a blended light curve is fitted. The best fit is obtained for a blending factor of 2.45 (Fig 15.). Thus makes the lensed source 1.34 magnitude fainter, and places it below the OGLE detection limit. Consequently it is very likely that OGLE 5 is an example of lensing on an unresolved star. The better Chi-Square per degree of freedom is different from unity, but we have to recall that an event on an unresolved star is seen only by blending on a resolved star. For the photometry, the position of the resolved star is used, but it might be significantly different from the position of the true magnified star. The achieved errors in a PSF fitting routine are rather sensitive to the quality of the positioning, thus we expect somewhat larger errors in the case of an unresolved event. The small discrepancy in the Chi-Square per degree of freedom is thus perfectly consistent with the scenario of an unresolved event. \begin{figure} \centerline{\psfig{angle=-90,figure=b0.ps,width=8cm}} \caption{The fit of an unblended light curve to the data, note the large discrepancy in the wings.} \end{figure} \begin{figure} \centerline{\psfig{angle=0,figure=chi.ps,width=8cm}} \caption{The Chi-Square per degree of freedom obtained for a fit to the data for OGLE 5 with different blending factors.} \end{figure} \begin{figure} \centerline{\psfig{angle=-90,figure=b2.45.ps,width=8cm}} \caption{The best fit with blending, note the large change in the estimated duration in comparison with the unblended fit.} \end{figure} \section{Discussion} In this paper I show that the rates of microlensing events and optical depth to microlensing due to the unresolved stars can be modelled. However I emphasize that the exact rates and optical depth are closely related to the number of unresolved stars which will contribute per resolved star. This number is well constrained in crowded fields, essentially because the surface occupied by a star on the image is set by the crowding limit. On the other hand, this crowding limit is closely related to the resolution radius, thus the surface occupied by a star is just a function of this resolution radius. A value of 0.5 was found for $R$ (the ratio of the surface covered by the resolution radius to the mean surface occupied by a star), both for DUO and OGLE. This simply means that probably there is a linear relation between the resolution radius and the mean radius per star. \\ However in the crowded fields, if a new star appear at more than a resolution radius, it does not mean necessarily that it will be resolved. There are so many stars that it might well be confused with another one. This issue should be clarified later using Monte-Carlo simulation of crowded fields. This effect will probably lead to a slight increase of $R$. Another problem is that if the amplified star is situated at some distance from the resolved star the photometry which assumes fixed positions (except for DUO) is certainly affected. This possibilty was already discussed in the case of OGLE 5 and will cause a slight drop in the efficiency, which will increase as the unresolved star will be more distant form the resolved one. This issue could be also clarified later using Monte-Carlo simulation. But globally, this effect will more or less cancell out with the confusion effect exposed just before. Thus to conclude, the value $R=0.5$ might be an acceptable approximation. \begin{figure} \centerline{\psfig{angle=0,figure=rates_bias.ps,width=8cm}} \caption{ Comparison of the total microlensing rates (unresolved+resolved stars) with the rates for resolved stars for different cut-offs in the mass function in case of the OGLE experiment. The cut-offs are: 0.02 $M\sun$ (long and short dashed line), 0.04 $M\sun$ (long dashed line), 0.06 $M\sun$ (dotted line), and 0.08 $M\sun$ (short dashed line). The scaling factors are respectively: 1.09, 1.24, 1.18, 1.28 (see text for explainations).} \end{figure} \section{Conclusion.} The modeling of the Unresolved star microlensing rates demonstrates that they induce an important bias for the short events. I show that given the current photometric errors, most of these events would be hard to distinguish from the unblended events. Thus this is an important and annoying bias. If this bias is not taken into account, it will certainly influence the shape of the lens mass function estimated from the data. For instance, if a Salpeter mass function is used, we expect that the lower cut-off will be shifted towards the brown dwarfs. This idea is illustrated in Fig. 16 where the total OGLE rates (resolved+unresolved) are compared to the rates for Salpeter mass functions with different lower cut-offs (I recall that initially I used a Salpeter mass function with a lower cut-off of $0.08 M\sun$ to compute the OGLE and DUO rates). A better agreement with the total rates is found if the different trial models are slightly scaled. I calculated this scaling factor, so that the trial models give the same total rates in the range 0 to 80 days as the total rate (see Fig 16.). The case of the DUO experiment will be treated in another paper (Alard and Guibert 1996). While the unresolved star bias has to be taken into account, as demonstrated in the previous section, the modelling includes some uncertainties. However for the data set already assembled by the microlensing experiments, it is certainly an acceptable solution to the unresolved star bias. For the future, a very interesting possibilty is offered by the PLANET (Sackett, 1995) collaboration, who will provide a dense and accurate sampling of the events. It will allow a much better control of the bias caused by the unresolved, using both color shift but importantly the shape of the light curve, as demonstrated for OGLE 5. \\ To conclude, I will also emphasize that the contribution of the unresolved stars drops if the magnitude limit is increased. This is well illustrated by the comparison OGLE vs. DUO. Consequently a first solution is to try to reach stars as faint as possible to minimise the bias. It is certainly a possible orientation for the OGLEII experiment, which will achieve improved resolution better again with a new telescope. On the other hand for MACHO and EROSII experiments, which use bigger pixels, and cover larger fields, an interesting solution is certainly to monitor the bright clump giants. These stars are much brighter (about 3 magnitudes) than most of the bulge stars, and their density per image is also much lower. Thus, the unresolved stars per giant will be low and they will be so blended that the effective Einsten radius will be very small. This idea leads to the conclusion that the unresolved star bias will be very small on clump giants. Some other advantages of the clump giants include the good photometry expected, and also the possibility to have a sample with a very high completeness (Gould 1995). \begin{acknowledgements} It is a great pleasure to thank B. Paczy\'nski, Gerry Gilmore, Olivier Bienaym\'e and Gary Mamon for interesting discussions. \end{acknowledgements}

proofpile-arXiv_065-580

\section{Introduction} The phenomenology of nuclear photoabsorption is governed by two characteristic features. First, all nuclei with mass numbers $A$ ranging from 10 to more than 200 obey the same fundamental curve $\sigma (\omega )/A$ for the total photoabsorption cross section devided by $A$ as a function of the photon energy $\omega $. Second, the $\Delta $-isobar excitation of the nucleon is responsible for the main properties of this curve in the energy region between 200 and 400 MeV. Models, which focus on the behaviour of the $\Delta $-isobar in a nuclear environment, namely the $\Delta $-hole calculations \cite{bb,cc}, have proven to be highly successful in explaining the experimental findings for pion scattering processes \cite{aa}. Indeed, on grounds of the $\Delta $-hole formalism a wide variety of pion-nucleus reactions can be described within one consistent framework \cite{dd}. In the case of photonuclear reactions, however, serious descrepancies remain, which partially have been accounted for by including non-resonant background terms \cite{koch,ee}. Nevertheless, such a procedure, in particular for nuclear photoabsorption, either leads to contradictions with previous $\Delta $-hole results or lacks the complete agreement with experimental data \cite{koch}. The question arises, whether theoretical ingredients other than in-medium $\Delta $-hole propagations can lead to a similar degree of accuracy. Therefore it is natural to address the situation from a different point of view asking to what extend the absorption process can be described, when only a very simple $\Delta $-nucleon interaction is used and all additional effects are accounted for in a purely diagrammatic approach. When combined with a simple form of nucleon momentum distribution inside the nucleus, namely a Fermi gas model, this leads to analytical expressions, in which different corrections to this lowest-order approximation can be studied. This is the aim of the present article. In our formalism we follow closely Wakamatsu and Matsumoto \cite{waka}. However, we do not introduce a phenomenological potential to account for the deviation of the nucleon wave functions from plane waves, but study the influence of such corrections in a perturbative way, similar to our previous work \cite{hm}. In addition, our focus is on the energy-dependence of the total photoabsorption, rather than on the differential cross section as a function of the momentum of the outgoing proton. A characteristic feature of Wakamatsu's and Matsumoto's approach is the technically equal treatment of the ($\gamma $,pn) and the ($\gamma $,pp) knock-out process, which allowed them to obtain a natural explanation for the supression of the two-proton knock-out. This characteristic property is also present in our calculation, where it is related to a vanishing trace in spin space. In the last years the ratio ($\gamma $,pp)/($\gamma $,pn) has been investigated in depth within different formalisms. In an extension of Wakamatsu's and Matsumoto's work by Boato and Giannini \cite{ff}, finite-size effects have been calculated and, more recently, a combination of pion exchange and shell-model wave functions was used to investigate this quantity \cite{ryck1}. Currently, two complementary approaches for the description of nuclear knock-out reactions exist. Carrasco and Oset \cite{oset1} used a diagram-oriented many-body expansion in a Fermi gas. The evaluation of self-energy diagrams leads to an accuracy for medium effects high enough to study knock-out reactions in great detail. With their primary goal being thus different from ours, their formalism does not yield isolated expressions for the resonant and non-resonant parts of the mechanisms of nuclear photoabsorption A different approach is used by the Gent group \cite{ryck2,ryck4,ryck5}, where the main emphasis lies in the construction of realistic shell-model wave functions, rather than on a microscopic description fully based on the evaluation of Feynman diagrams. As the nuclear photoabsorption is almost insensitive to structural differences between nuclei, the quality of their approach becomes obvious in the investigation of differential cross sections for nucleon knock-out, rather than of photoabsorption. Nuclear photoabsorption provides an interesting tool to study the interplay between one-nucleon and two-nucleon contributions. We obtained analytical expressions for these contributions, as well as for their resonant and non-resonant parts. This set of results can be used as a starting point to include (and test) further nuclear or nucleonic effects. In Section 2 the basic notations are listed, as well as the interaction terms and the most important model assumptions and approximations, which are present in this calculation. The main results and their most prominent properties, e.g. the effects of relativistic corrections and nuclear structure, are discussed in Section 3, where also the following possible extension of such an approach is considered: As the angular dependence of the two-nucleon process is not very strong (cf. \cite{angle}), the different mechanisms contributing to the photoabsorption curve can also be used to understand qualitative features of experimental data for nucleon knock-out processes as a function of the photon energy. In Section 4 some concluding remarks are made, with an emphasis on the applicability of the partial cross sections, whose analytical forms are given in the Appendix. \section{Formalism and Notation} The starting point of our investigation is the static Hamiltonian for the pion-nucleon interaction, \begin{equation} H_{\pi NN}=-{f \over m}\;\vec \sigma \cdot \vec \nabla \;\underline \tau \cdot \underline \phi \label{s1eq1} \end{equation} together with a minimal coupling to the photon field. In eq.\,(\ref{s1eq1}) $m$ is the pion mass. For all coupling constants we use the same notation and values as given in \cite{ew}, in particular $f^2/(4\pi )$=0.08. In eq.\,(\ref{s1eq1}) underlined symbols denote vectors in (cartesian) isospin space, while an arrow indicates a vector in coordinate space. In the static limit the interactions with the $\Delta $-isobar excitation of the nucleon are determined by the following Hamiltonians (see e.g. \cite{ew}): \begin{equation} H_{\gamma N\Delta }=-{{ef_{\gamma N\Delta }} \over m}\,\vec S^+\cdot \left( {\vec \nabla \times \vec A} \right)\,\,T_z^+ \label{s1eq2} \end{equation} and \begin{equation} H_{\pi N\Delta }=-{{f_\Delta } \over m}\,\vec S^+\cdot \vec \nabla \underline T^+\underline \phi \,\, , \label{s1eq3} \end{equation} with the hermitian conjugate to be added in both cases. Here $\vec S$ and $\underline T$ are the 1/2-to-3/2 transition operators in spin space and isospin space, respectively; $e$ is the proton charge, $e^2=1/137$. For the coupling constants we have $f_\Delta $=2 and $f_{\gamma N\Delta }$=0.35. In all cases, where high momentum transfers occur at the pion-nucleon vertices we regularize the vertex functions by introducing dipole form factors \begin{equation} g_{\pi}(q)={{\Lambda ^2-m^2} \over {\Lambda ^2- q^2}} \label{s1eq4} \end{equation} as was also done e.g. in \cite{waka,Riska}. The value for the cut-off parameter $\Lambda $ has been taken to be 800 MeV. This value gives the best agreement of our predictions with the experimental data. In addition, a similar value has been used in \cite{sasa,waka}. The general expression for the total cross section $\sigma_1(\omega)$ of the photoabsorption with one nucleon outside the Fermi sphere and one pion in the final state (one-nucleon process) is of the form \begin{eqnarray} & &\sigma _1(\omega )=\int {{{V\,d\vec p} \over {(2\pi )^3}}}\,\int {{{d\vec q} \over {2\varepsilon _q(2\pi )^3}}}\,{{4\pi }\over {2\omega }}\;\left| {T_1} \right|^2\times \nonumber \\ & &2\pi \,\delta \left( {\omega +{{p^2} \over {2M}}-\varepsilon _q-{{(\vec k+\vec p-\vec q)^2} \over {2M}}} \right)\;n(\vec p)\;\left[ {1-n(\vec k+\vec p-\vec q)}\right]\; . \label{s1eq5} \end{eqnarray} For the total cross section $\sigma_2(\omega)$ of the process with two free nucleons in the final state it is given by \begin{eqnarray} & &\sigma _2(\omega )=\int {{{V\,d\vec p_1\,V\,d\vec p_2} \over {(2\pi )^6}}}\,\int {{{d\vec p_3\,d\vec p_4\,} \over {(2\pi )^6}}}\,{{4\pi } \over {2\omega }}\;\left| {T_2} \right|^2\,\times \nonumber \\ & &\delta \left( {\omega +{{p_1^2+p_2^2-p_3^2-p_4^2} \over {2M}}} \right)\,(2\pi )^4\,\delta (\vec p_1+\vec p_2+\vec k-\vec p_3-\vec p_4) \times \nonumber \\ & &n(\vec p_1)n(\vec p_2)\left[ {1-n(\vec p_3)\;} \right]\left[ {1-n(\vec p_4)\;} \right] \label{s1eq6}\; . \end{eqnarray} The notation for the external momenta is shown in Fig.\,(\ref{figA}). The function $n(\vec p)=\theta (p_F-\left| {\vec p}\right|)$ is the occupation number, $\theta(x)$ is the step function. Furthermore, $V$ is the nuclear volume, $p_F$ is the Fermi momentum, $M$ is the mass of the proton, $V$ is the nuclear volume, $V=3\pi ^2A/(2p_F^3)$, and $\varepsilon _q=\sqrt {\left| {\vec q} \right|^2+m^2}$ is the energy of the outgoing pion. Both amplitudes $T_1$ and $T_2$ consist of non-resonant and resonant parts, $T_i=T_i^{(NR)}+T_i^{(R)}$. Diagrammatically this decomposition is shown in Fig.\,(\ref{figB}). The second (crossed) contribution to the resonant part is small due to the big energy denominator and will be skipped in the following. In the energy $\delta $-function in eq.\,(\ref{s1eq5}) and eq.\,(\ref{s1eq6}) we will usually not consider the smallest term connected with the kinetic energy of the incoming nucleon, as it is smaller than $p_F^2/2M\approx 37\,$MeV. Indeed, as we expect its contribution to introduce only a small modification of the actual $p$-dependence in the integrand in (\ref{s1eq5}), we substitute it by its average value $\left\langle {p^2} \right\rangle /2M=(3/5)p_F^2/2M$, which results in an overall shift of the absorption cross section. For the one-nucleon process we find first-order relativistic corrections to be essential for a successful treatment of the absorption process. In the case of the resonant contribution, such corrections are accounted for by making the following substitutions in the vertices \cite{waka}: \begin{eqnarray} &\vec q&\buildrel {} \over \longrightarrow \;\vec q-{{\varepsilon _{\vec q}} \over {M_\Delta }}(\vec p+\vec k) , \label{s1eq7} \\ &\vec k&\buildrel {} \over \longrightarrow \;\vec k\left( {1+{\Delta \over M}}\right)-{\Delta \over M}\,\vec p \, , \label{s1eq8} \end{eqnarray} where $M_\Delta $ is the $\Delta$-isobar mass and $\Delta=M_\Delta -M$. For the non-resonant part, the corrections give amplitude $T_1$ of the following form: \begin{eqnarray} T_1^{(NR)}&=&{{\sqrt 2ef} \over m}\,\left\{ {\matrix{{}\cr {}\cr }} \right.i\,\vec \sigma \cdot \vec \varepsilon \,\left[ {1+{{\varepsilon _q} \over {2M}}} \right]-{{2i(\vec \sigma \cdot (\vec q-\vec k))(\vec \varepsilon \cdot \vec q)}\over {(\vec q-\vec k)^2+m^2-(\varepsilon _q-\omega )^2}} \nonumber \\ &-&{{2i(\vec \sigma \cdot \vec q)(\vec \varepsilon \cdot (\vec k+\vec p-\vec q))}\over {\,\left[ {(\vec k+\vec p-\vec q)^2-2M\omega -(\vec p-\vec q)^2}\right]}}\left. {\matrix{{}\cr {}\cr }} \right\} , \label{s1eq9} \end{eqnarray} which corresponds to the production of a $\pi ^-$. In comparison with \cite{waka} we neglected those terms in (\ref{s1eq9}), which modify the result by less than 2 per cent. It should be noted that no free parameters are present in our approach. A certain model dependence exists, however, in the selection of diagrams. We neglect all those diagrams, which are suppressed by some mechanism. In Fig.\,(\ref{figAA}) two examples for suppression mechanisms are given. For Fig.\,(\ref{figAA}a) the contribution is small because in the case of infinite nuclear matter due to momentum conservation the four-momentum of the photon should be equal to that of the outgoing pion, which is impossible. For Fig.\,(\ref{figAA}b) let us consider the case, where the upper two nucleons (incoming and outgoing) are identified. Furthermore, let us select a $\Delta $-isobar as an intermediate state. Then, one has a vanishing trace in spin space: $$ \mbox{Sp}\left[ (\vec S^+\cdot (\vec k\times \vec \varepsilon ))\;(\vec S\cdot \vec q)\right]\,=\, 0 $$ at $\vec q\,=\vec k$. Therefore, the contribution in this case vanishes in the non-relativistic limit considered here. As a result of applying such methods to the various diagrams, we have obtained that only those displayed in Fig.\,(\ref{figA}) should be taken into acount. \section{Results} Explicit evaluation of the diagrams shown in Fig.\,(\ref{figA}) leads to amplitudes for the one- and two-nucleon contribution to nuclear photoabsorption. Squaring these amplitudes and summing over spin and isospin states of the nucleons by using trace methods as previously \cite{hm} one finds the expressions $\left| {T_1} \right|^2$ and $\left| {T_2} \right|^2$, which enter into eqs.\,(\ref{s1eq5}) and (\ref{s1eq6}). It has turned out to be convenient to investigate the resonant and non-resonant parts of each of these contributions separately. This can be done by neglecting the interference terms, which are highly suppressed (cf. Fig.\,(\ref{figH})). Shown here exemplary for the resonant parts, one obtains the following expressions, which serve as a starting point for the integrations with respect to nucleon momenta: \begin{eqnarray} & &\sigma _1^{(R)}(\omega )={{4e^2} \over {27\pi \omega }}\,\left( {{{f_\Delta f_{\gamma N\Delta }} \over {m^2}}} \right)^2\int {{{V\,d\vec p} \over {(2\pi )^3}}}\,\int {d\vec q}\;\times \nonumber \\ & &\delta \left( {\omega ^2+{\omega \over M}\left[ {(\vec p-\vec k)^2-(\vec p-\vec q)^2} \right]-q^2-m^2} \right)\times \nonumber \\ & &n(\vec p-\vec k)\;\left[ {1-n(\vec p-\vec q)} \right]\,{{3\left[ {(\vec Q\times \vec K)\cdot \vec \varepsilon } \right]^2+Q^2(\vec K\times \vec \varepsilon )^2} \over {\left( {\omega -\Delta - p^2/2M} \right)^2+ \Gamma^2 / 4}} , \label{s2eq1} \end{eqnarray} where \begin{displaymath} \vec Q=\vec q-{\omega \over {M_\Delta }}\,\vec p\;,\quad \vec K=\vec k\left( {1+{\Delta\over M}} \right)-{\Delta \over M}\,\vec p \end{displaymath} and \begin{eqnarray} & &\sigma _2^{(R)}(\omega )={{16e^2} \over {81\pi }}\,\left( {{{f_\Delta f_{\gamma N\Delta }}\over {m^3}}} \right)^2\,{\omega \over {\left( {\omega -\Delta } \right)^2+{{\Gamma ^2}/4}}}\times \nonumber \\ & &\int {{{V\,d\vec p_1\,V\,d\vec p_2} \over {(2\pi )^6}}}\,\int {d\vec p_3\,d\vec p_4\,}n(\vec p_1)n(\vec p_2)\left[ {1-n(\vec p_3)\;} \right]\left[ {1-n(\vec p_4)\;} \right]\times \nonumber \\ & &\delta \left( {\omega -{{p_3^2+p_4^2} \over {2M}}} \right)\,\delta (\vec p_1+\vec p_2+\vec k-\vec p_3-\vec p_4)\times \nonumber \\ & &\left\{ {{{2a^2\left[ {a^2+3(\vec \varepsilon \cdot \vec a)^2} \right]} \over {(a^2+m^2)^2}}\,g_\pi ^4(a)+{{2\omega ^2(\vec \varepsilon \cdot \vec a)(\vec \varepsilon \cdot \vec b)} \over {(a^2+m^2)(b^2+m^2)}}\,g_\pi ^2(a)g_\pi ^2(b)} \right\} \label{s2eq2} \end{eqnarray} with \begin{displaymath} \vec a=\vec p_4-\vec p_2\;,\quad \vec b=\vec p_1-\vec p_3 . \end{displaymath} The analytical expressions for the resulting partial cross sections are given in the Appendix. For the total absorption cross section $\sigma (\omega )$ a comparison with experimental data is shown in Fig.\,(\ref{figC}). The result of this model calculation compares favorably with the data. The one-nucleon and two-nucleon parts of the cross section are equally important at energies around 250 MeV. As can be seen in this plot, significant features of the data, e.g. the position of the peak, are only obtained due to the interplay between the two mechanisms. As mentioned before, each of the contributions to the full curve in Fig.\,(\ref{figC}) has a resonant and a non-resonant part. In Fig.\,(\ref{figD}) and (\ref{figE}) this decomposition is shown for the one-nucleon and the two-nucleon mechanism, respectively. Here it is clearly seen that the non-resonant parts give an important contribution at lower energies. In the two-nucleon case the non-resonant part decreases with energy, while in the one-nucleon process it remains almost constant. It is interesting to see, in what way the one-nucleon partial cross sections are affected by the use of only the static limit of the interaction. Neglecting the first-order relativistic corrections in the current one has \begin{eqnarray} \tilde \sigma ^{(R)}_1(\omega )={{Ae^2(f_\Delta f_{\gamma N\Delta })^2} \over {27m^2\left[ {\left( {\omega -\Delta } \right)^2+{{\Gamma ^2} \over 4}} \right]}}\,\int\limits_{q_{min}}^{q_{max}} {q\,dq}\,\left( {1+S_0(q)} \right)\times & & \nonumber \\ \left\{ {\left[ {\omega ^2-m^2-{{\omega q^2} \over M}} \right]+{3 \over 2} \left( {q^2-\left[ {q^2\left( {1+{\omega \over M}} \right)+m^2} \right]^2{1 \over {4\omega ^2}}} \right)} \right\} & & \label{s2eq3} \end{eqnarray} and \begin{eqnarray} & &\tilde \sigma ^{(NR)}_1(\omega )={{Ae^2f^2} \over {m^2\omega ^2}}\,\int\limits_{q_{min}}^{q_{max}} {q\,dq}\,\left( {1+S_0(q)} \right)\left[ {\matrix{{}\cr {}\cr }} \right.1-{{2m^2q^2} \over {(q^2+m^2)^2}}\times \nonumber \\ & &\left\{ {1-{1 \over {4q^2\omega ^2}}\left[ {q^2\left( {1+{\omega \over M}} \right)+m^2} \right]^2} \right\}\left. {\matrix{{}\cr {}\cr }} \right] , \label{s2eq4} \end{eqnarray} where the integration limits in both cases are given by \begin{equation} q_{\min ,\max }=\left( {1+{\omega \over M}} \right)\,\left\{ {\omega \mp \sqrt {\omega ^2-m^2\left( {1+{\omega \over M}} \right)\,}} \right\} \label{s2eq5} \end{equation} and the integrand contains the function \begin{equation} S_0(q)=-\left( {1-{q \over {2p_F}}} \right)^2\left( {1+{q \over {4p_F}}} \right)\,\theta (2p_F-q) . \label{s2eq6} \end{equation} In Fig.\,(\ref{figF}) the corresponding cross sections are compared with those resulting from eq.\,(\ref{s2eq1}) and its non-resonant counterpart. The relativistic corrections lead mainly to a shift of the one-nucleon curve. This effect is essential for obtaining a good agreement with the experimental data. Although no relativistic corrections in the current are included, note that in (\ref{s2eq3}) and (\ref{s2eq4}) terms of that order have been kept in the kinematical contributions to the integrand. As the two-nucleon mechanism gives a comparatively small contribution at higher energies, we neglect relativistic corrections in $\sigma _2(\omega )$. This has also been done in \cite{waka}. A more difficult problem is the influence, which nucleon correlations inside the nucleus can have on the nuclear photoabsorption process. We investigate this aspect for the non-relativistic forms (\ref{s2eq3}) and (\ref{s2eq4}) of the one-nucleon case. The main reason for doing so is the fact that due to the square of the amplitudes we can express part of the integrand via the standard lowest-order central correlation function of a Fermi gas (see e.g. \cite{Molinari}). The object, which is obtained by diagrammatically squaring the one-nucleon contribution of Fig.\,(\ref{figA}a), can be coupled to an additional nucleon. In the incoherent case of the diagrammatical square also a further pion exchange can be allowed. The modifications of the correlation function, which occur due to such effects, have been investigated analytically in \cite{hm}, where a corrected central correlator $S_C(q)$ has been constructed. The function $S_C$ is given in Fig.\,(\ref{figJ}). By making the substitution $ S_0(q)\buildrel {} \over \longrightarrow S_C(q) $ these further correlations can be incorporated effectively. In Fig.\,(\ref{figG}) the one-nucleon contribution resulting from $S_C$ is compared with the original form, in which $S_0$ has been used. It can be seen that such medium effects modify the result by about 15 per cent. Naturally, the use of this substitution technique can only be used to obtain an estimate for such a medium-induced modification. A full examination should involve the inclusion of the additional two-and three-nucleon diagrams in eq.\,(\ref{s1eq5}). The limit of our approach is certainly reached, when a comparison with differential cross sections is attempted. An extreme case is the comparison with $^4$He, for which a recent measurement of both, the one-nucleon and the two-nucleon channel exists \cite{wich}. We compare the calculated average cross section with the data for the differential cross section in c.m. frame, as we expect the angular dependence not to be strong in that frame. The corresponding plot is shown in Fig.\,(\ref{figI}). Although no full agreement is obtained, it is interesting to note that the general features of the two cross sections are well reproduced, such as the peak positions and the relative size of the two processes. By these means it is possible to unambiguously identify the physical mechanisms behind the data points. A similar degree of agreement is obtained for other data \cite{homma2}. \section{Conclusion} In the present paper we have developed a diagrammatical description of the nuclear photoabsorption process. The main result of our investigation is that the total photoabsorption cross section can be fully understood in terms of a simple physical picture, where point-like nucleons and $\Delta $-isobars interacting via pion exchange are the relevant degrees of freedom. Due to the diagram-oriented formalism and the Fermi gas model as an approximate description of the nucleons in momentum space, we could obtain analytical expressions for all the relevant contributions to the photoabsorption curve. In this way a flexible and efficient description has been obtained, which can be used as a starting point for the investigation of additional effects. Especially in the low-energy part of our calculated curve, the agreement with experiment comes about as a non-trivial interplay between the one-nucleon and two-nucleon contributions. It is worth noting that, as long as a comparatively low cut-off parameter in the vertex form factor is used, there seems to be no need for an explicit diagrammatical inclusion of the $\rho $-meson as an additional mechanism of the nucleon-nucleon interaction. We found relativistic corrections in the case of the one-nucleon process to be crucial for obtaining a good agreement. The aspect of additional nucleon correlations, which can be accounted for as a deviation of the nucleon wave functions from plane waves, deserves some further attention in future investigations. We could estimate the overall effect to be of the order of 15 per cent. \section*{Acknowledgement} We are most grateful to A.I. L'vov for useful comments and discussions. One of us (M.T.H.) wishes to thank the Budker Institute, Novosibirsk, for the kind hospitality accorded him during his stay, when part of this work was done. \section*{Appendix: Analytical expressions for absorption cross sections} Here we present the explicit expressions for the four contributions to the photoabsorption curve, which have been used to obtain the figures shown in Section 3. As was mentioned earlier, the interference terms between the resonant and the non-resonant contributions are small (cf. Fig.\,(\ref{figH})). Therefore, we can write the absorption cross section as a sum of four parts, $\sigma (\omega )=\sigma _1^{(NR)}(\omega )+ \sigma _1^{(R)}(\omega )+\sigma _2^{(NR)}(\omega )+\sigma _2^{(R)}(\omega )$. First, we deal with the one-nucleon case. In the case of the non-resonant contribution the absorption cross section can be represented in the following form: \begin{eqnarray} &\displaystyle \sigma _1^{(NR)}(\omega )= \frac{3Ae^2}{4m^2\omega ^2p_F^3}\, \int\limits_{\omega_{-}}^{\omega_{+}} dq\int\limits_{l(q)}^ {q+p_F} dp\,g_\pi^2(q)\,p\,[p_F^2-(q-p)^2]\times \nonumber\\ &\displaystyle G(p,q,\omega)\,\theta(2q\omega-m^2-q^2-\frac{\omega}{M}p^2) , \label{app1} \end{eqnarray} where \begin{equation} G(p,q,\omega )\!=\!\left(1+\!\frac{\omega}{2M}\right)^2\!-\! \frac{2}{D}\left(\frac{m^2}{D}+\frac{\omega}{2M}\right) \left[q^2-{1 \over {4\omega ^2}}\left( D+{{\omega}\over M}p^2\right)^2 \right] \label{app4} \end{equation} and $l(q)=\mbox{Max}[p_F,|q-p_F|]$ , $D=q^2+m^2$ , $\omega_{\pm}=\omega \pm\sqrt{\omega ^2-m^2}$. In eq.\,(\ref{app1}) the form factor $g_\pi ^2(q)$ is the same as in (\ref{s1eq4}). Note that the integration with respect to the variable $p$ can easily be performed, but the result is too lengthy to be given here explicitely. The resonant part of the one-nucleon process has the following form: \begin{eqnarray} &\displaystyle \sigma _1^{(R)}(\omega )= \frac{4Ae^2}{9\omega p_F^3}\left(\frac{f_{\gamma N\Delta }} {m^2}\right)^2\left(1+{\Delta\over M}\right)^2 \left(1+{\omega\over M}\right)^2 \int\limits_{-1}^{+1} {dy} \int\limits_0^b dp\, p^2F\times \nonumber \\ &\displaystyle \frac{\theta \left(p_F^2-p^2\left(1+\omega/M \right)^2 -\omega ^2+2\omega py\left(1+\omega/M \right)\right)} {\left[\omega -\Delta -p^2(1+\omega/M)^2/2M_{\Delta}\right]^2+ \Gamma^2/4}\; H(p,y,\omega) \label{app6} \end{eqnarray} where $b=(\omega+p_F)/(1+\omega/M)$ and the integrand is given by \begin{eqnarray} &\displaystyle H(p,y,\omega)= 2A_1\, [\theta(F-p-p_F)+\theta(F)\,\theta(p-p_F-F)\,]+ \nonumber\\ &\displaystyle [(x+1)A_1-{1 \over 8}(x-x^3)A_2]\theta(F-|p-p_F|)\,\theta(p+p_F-F) \, , \end{eqnarray} with \begin{eqnarray} &\displaystyle F=\left[\omega^2\left(1-\frac{2py}{M}\right) -\frac{m^2}{1+\omega/M}\right]^{1/2}\; , \; x=\frac{p^2+F^2-p_F^2}{2pF}\, , \nonumber \\ &\displaystyle A_1=\omega^2-2\omega ay+a^2\; ,\; A_2=\omega^2(1-3y^2)+4\omega ay-2a^2\; , \nonumber \\ &\displaystyle a=\frac{p\Delta}{M}\left(\frac{1+\omega/M}{1+\Delta/M}\right)\, . \end{eqnarray} The mass difference $\Delta $ between the proton and the $\Delta $-excitation is $\Delta=292$ MeV, the width $\Gamma $ of the $\Delta$-isobar has been taken to be 115 MeV. Again, the integration with respect to $y$ in eq.\,(\ref{app6}) can be performed analytically, but due to its length the result is not presented here. For the partial cross sections of the two-nucleon case, we obtained the following result: \begin{eqnarray} &\displaystyle \sigma _2^{(NR)}(\omega )+\sigma _2^{(R)}(\omega )=\\ &\displaystyle \Biggl\{ \theta (\omega -\varepsilon )\,\theta (5\varepsilon -\omega) \,\Biggl[\,\int\limits_{L_1(Q)}^{L_2(Q)} \Phi(\beta_1,\, Q)+ \int\limits_{L_2(Q)}^2 \Phi(\beta_2,\, Q)\Biggr]+\nonumber \\ &\displaystyle \theta(\omega -5\varepsilon )\,\theta (9\varepsilon - \omega )\,\int\limits_{L_1(Q)}^{L_3(Q)}\Phi(\beta_1,\, Q) \Biggr\}g_\pi ^4(p)\left[G^{(NR)}(p,\omega )+G^{(R)}(p,\omega )\right]dp , \nonumber \label{app10} \end{eqnarray} where $Q=\sqrt {M\omega }$ , $\varepsilon =p_F^2/M$ , $$ \beta_1=\arccos \frac{p+p_F}{\sqrt{2}Q} \; , \beta_2=\arccos \frac{p_F}{\sqrt {2}Q}\; . $$ The elementary function $\Phi(\beta,\, Q)$ is \begin{eqnarray} &\displaystyle \Phi(\beta,\,Q)= \Biggl\{ \frac{1}{2}(p^2-p_F^2)(2Q^2+p^2-p_F^2)\sin ^2x\,+Q^4 (\sin ^4x-\frac{2}{3}\sin ^6x)\, + \nonumber \\ &\displaystyle p^2Q^2(x-\frac{1}{4}\sin 4x\,) -\sqrt 8pQ\, [\frac{1}{3}(2Q^2+p^2-p_F^2)\sin ^3x- \frac{2}{5}Q^2\sin ^5x\,]\, + \nonumber \\ &\displaystyle \sqrt 8pQ\,[\frac{1}{3}(2Q^2+p^2-p_F^2)\cos ^3x -\frac{2}{5}Q^2\cos ^5x\,]\Biggr\}\, \Biggl.\Biggr|_{x=\beta}^{x=\pi /2 -\beta}\, . \label{app13} \end{eqnarray} The functions in the integrand of eq.\,(\ref{app10}), which characterize the resonant and non-resonant part, are given by \begin{equation} G^{R}(p,\omega )=\frac{8e^2Af_{\gamma N\Delta }^2M^2} {27\pi^2m^6p_F^3}\,\frac{\omega ^2}{(\omega -\Delta)^2+\Gamma ^2/4} \cdot\frac{p^4}{(p^2+m^2)^2}\, \label{app14} \end{equation} and \begin{equation} G^{NR}(p,\omega )=\frac{2e^2AM^2}{\pi ^2m^4p_F^3} \,\left[1+\frac{m^4}{(p^2+m^2)^2}\right] \frac{p^2}{(p^2+m^2)^2} \label{app15} \end{equation} respectively. In eq.\,(\ref{app10}) the integration limits are \begin{displaymath} L_1(Q)=Q-p_F\;,\quad L_2(Q)=\sqrt {2Q^2-p_F^2}-p_F \end{displaymath} and \begin{displaymath} L_3(Q)=\sqrt {Q^2-p_F^2} . \end{displaymath} \newpage

proofpile-arXiv_065-581

\section{Introduction} \noindent The quantum field theory in curved space-time has been a matter of great interest in recent years because of its applications to cosmology and astrophysics. The evidence of existence of strong gravitational fields in our Universe led to the study of the quantum effects of material fields in external classical gravitational field. After the appearance of Parker's paper on scalar fields \cite{Par1} and spin-$\frac{1}{2}$ fields \cite{Par2}, several authors have studied this subject. Although the Universe seems homogenous and isotropic at present, there are no observational data guarantying the isotropy in the era prior to the recombination. In fact, there are theoretical arguments that sustain the existence of an anisotropic phase that approaches an isotropic one \cite{Mis}. Interest in studying Klein-Gordon and Dirac equations in anisotropic models has increased since Hu and Parker \cite{Hu0} have shown that the creation of scalar particles in anisotropic backgrounds can dissipate the anisotropy as the Universe expands. \noindent A Bianchi type-I (B-I) Universe, being the straightforward generalization of the flat Robertson-Walker (RW) Universe, is one of the simplest models of an anisotropic Universe that describes a homogenous and spatially flat Universe. Unlike the RW Universe which has the same scale factor for each of the three spatial directions, a B-I Universe has a different scale factor in each direction, thereby introducing an anisotropy to the system. It moreover has the agreeable property that near the singularity it behaves like a Kasner Universe even in the presence of matter and consequently falls within the general analysis of the singularity given by Belinskii et al \cite{Bel}. And in a Universe filled with matter for $p\,=\,\gamma\,\varepsilon, \quad \gamma < 1$, it has been shown that any initial anisotropy in a B-I Universe quickly dies away and a B-I Universe eventually evolve into a RW Universe \cite{Jac}. Since the present-day Universe is surprisingly isotropic, this feature of the B-I Universe makes it a prime candidate for studying the possible effects of an anisotropy in the early Universe on present-day observations. In light of the importance of mentioned above, several authors have studied linear spinor field equations \cite{Chim}, \cite{Cas} and the behavior of gravitational waves (GW's) \cite{Hu}, \cite{Mied}, \cite{Cho} in B-I Universe. Nonlinear spinor field (NLSF) in external cosmological gravitation field was first studied by G. N. Shikin in 1991 \cite{Shik}. This study was extended by us for more general case where we consider nonlinear term as an arbitrary function of all possible invariants generated from spinor bilinear forms. In that paper we also studied the possibility of elimination of initial singularity specially for Kasner Universe \cite{Ryb1}. In a recent paper \cite{Ryb} we studied the behavior of self-consistent NLSF in B-I Universe that was followed by the papers \cite{Alv}, \cite{AlvIz} where we studied the self-consistent system of interacting spinor and scalar fields. The purpose of the paper is to extend our study for more general NLSF in presence of perfect fluid. In the section 2 we derive fundamental equations corresponding to the Lagrangian for the self-consistent system of spinor and gravitational fields in presence of perfect fluid and seek their general solutions. In section 3 we give a detail analysis of the solutions obtained for different kinds of nonlinearity. In section 4 we study the role of perfect fluid and in section 5 we sum up the results obtained. \vskip 5mm \section{Fundamental equations and general solutions} \setcounter{equation}{0} \noindent The Lagrangian for the self-consistent system of spinor and gravitation fields in presence of perfect fluid is \begin{equation} L=\frac{R}{2\kappa}+\frac{i}{2} \biggl[ \bar \psi \gamma^{\mu} \nabla_{\mu} \psi- \nabla_{\mu} \bar \psi \gamma^{\mu} \psi \biggr] - m\bar \psi \psi + L_N +L_m, \end{equation} with $R$ being the scalar curvature and $\kappa$ being the Einstein's gravitational constant. The nonlinear term $L_N$ describes the self-interaction of spinor field and can be presented as some arbitrary functions of invariants generated from the real bilinear forms of spinor field having the form: $$S\,=\, \bar \psi \psi, \quad P\,=\,i \bar \psi \gamma^5 \psi, \quad v^\mu\,=\,(\bar \psi \gamma^\mu \psi), \quad A^\mu\,=\,(\bar \psi \gamma^5 \gamma^\mu \psi), \quad T^{\mu\nu}\,=\,(\bar \psi \sigma^{\mu\nu} \psi),$$ where $\sigma^{\mu\nu}\,=\,(i/2)[\gamma^\mu\gamma^\nu\,-\, \gamma^\nu\gamma^\mu]$. Invariants, corresponding to the bilnear forms, look $$ I = S^2, \quad J = P^2, \quad I_v = v_\mu\,v^\mu\,=\,(\bar \psi \gamma^\mu \psi)\,g_{\mu\nu} (\bar \psi \gamma^\nu \psi),$$ $$I_A = A_\mu\,A^\mu\,=\,(\bar \psi \gamma^5 \gamma^\mu \psi)\,g_{\mu\nu} (\bar \psi \gamma^5 \gamma^\nu \psi), \quad I_T = T_{\mu\nu}\,T^{\mu\nu}\,=\,(\bar \psi \sigma^{\mu\nu} \psi)\, g_{\mu\alpha}g_{\nu\beta}(\bar \psi \sigma^{\alpha\beta} \psi).$$ According to the Pauli-Fierz theorem \cite{Ber} among the five invariants only $I$ and $J$ are independent as all other can be expressed by them: $I_v = - I_A = I + J$ and $I_T = I - J.$ Therefore we choose the nonlinear term $L_N = F(I, J)$, thus claiming that it describes the nonlinearity in the most general of its form. $L_m$ is the Lagrangian of perfect fluid.\\[2mm] \noindent We choose B-I space-time metric in the form \begin{equation} ds^2\,=\,dt^2 - \gamma_{ij}(t)\,dx^i\,dx^j. \end{equation} As it admits no rotational matter, the spatial metric $\gamma_{ij}(t)$ can be put into diagonal form. Now we can rewrite the B-I space-time metric in the form \cite{Zel}: \begin{equation} ds^2 = dt^2 - a^{2}(t)\,dx^{2} - b^{2}(t)\,dy^{2} - c^{2}(t)\,dz^2, \end{equation} where the velocity of light is taken to be unity. Einstein equations for $a(t), b(t)$ and $c(t)$ corresponding to the metric (2.3) and Lagrangian (2.1) read~\cite{Zel}: \begin{eqnarray} \frac{\ddot a}{a} +\frac{\dot a}{a} \biggl(\frac{\dot b}{b}+\frac{\dot c}{c}\biggr)= -\kappa \biggl(T_{1}^{1}- \frac{1}{2}T\biggr), \\ \frac{\ddot b}{b} +\frac{\dot b}{b} \biggl(\frac{\dot a}{a}+\frac{\dot c}{c}\biggr)= -\kappa \biggl(T_{2}^{2}- \frac{1}{2}T\biggr), \\ \frac{\ddot c}{c} +\frac{\dot c}{c} \biggl(\frac{\dot a}{a}+\frac{\dot b}{b}\biggr)= -\kappa \biggl(T_{3}^{3}- \frac{1}{2}T\biggr), \\ \frac{\ddot a}{a} +\frac{\ddot b}{b} +\frac{\ddot c}{c}= -\kappa \biggl(T_{0}^{0}- \frac{1}{2}T\biggr), \end{eqnarray} where points denote differentiation with respect to t, and $T=T_{\mu}^{\mu}.$ \noindent NLSF equations and components of energy-momentum tensor for the spinor field and perfect fluid corresponding to (2.1) are \begin{eqnarray} i\gamma^\mu \nabla_\mu \psi -m\psi + F_{I} 2 S \psi + F_{J} 2P i \gamma^5 \psi\,&=&\,0, \nonumber \\ i \nabla_\mu \bar \psi \gamma^\mu + m \bar\psi - F_{I} 2 S \bar \psi - F_{J} 2P i \bar \psi \gamma^5 \,&=&\,0, \end{eqnarray} where $F_{I}:= \partial F/\partial I$ and $F_{J}:= \partial F/\partial J.$ \begin{equation} T_{\mu}^{\rho}=\frac{i}{4} g^{\rho\nu} \biggl(\bar \psi \gamma_\mu \nabla_\nu \psi + \bar \psi \gamma_\nu \nabla_\mu \psi - \nabla_\mu \bar \psi \gamma_\nu \psi - \nabla_\nu \bar \psi \gamma_\mu \psi \biggr) \,- \delta_{\mu}^{\rho}L_{sp}+ T_{\mu\,(m)}^{\rho}, \end{equation} while $L_{sp}$ on account of spinor field equations takes the form: $$ L_{sp}\,=\,-\biggl[\frac{1}{2}\biggl(\bar \psi \frac{\partial L_N}{\partial \bar \psi}+ \frac{\partial L_N}{\partial \psi} \psi \biggr)-L_N\biggr]\,= -\bigl[2 I\,F_{I}\, + 2 J\, F_{J} - L_N\bigr].$$ Here $T_{\mu\,(m)}^{\rho}$ is the energy-momentum tensor of perfect fluid. For a Universe filled with perfect fluid, in the concomitant system of reference $(u^0=1, \, u^i=0, i=1,2,3)$ we have \begin{equation} T_{\mu (m)}^{\nu}\,=\, (p + \varepsilon) u_\mu u^\nu - \delta_{\mu}^{\nu} p \,=\,(\varepsilon,\,- p,\,- p,\,- p), \end{equation} where energy $\varepsilon$ is related to the pressure $p$ by the equation of state $p\,=\,\gamma\,\varepsilon$, the general solution has been derived by Jacobs \cite{Jac}. $\gamma$ varies between the interval $0\,\le\, \gamma\,\le\,1$, whereas $\gamma\,=\,0$ describes the dust Universe, $\gamma\,=\,\frac{1}{3}$ presents radiation Universe, $\frac{1}{3}\,<\,\gamma\,<\,1$ ascribes hard Universe and $\gamma\,=\,1$ corresponds to the stiff matter. In (2.8) and (2.9) $\nabla_\mu$ denotes the covariant derivative of spinor, having the form \cite{Zelnor}: \begin{equation} \nabla_\mu \psi=\frac{\partial \psi}{\partial x^\mu} -\Gamma_\mu \psi, \end{equation} where $\Gamma_\mu(x)$ are spinor affine connection matrices. $\gamma^\mu(x)$ matrices are defined for the metric (2.3) as follows. Using the equalities \cite{Brill}, \cite{Wein} $$ g_{\mu \nu} (x)= e_{\mu}^{a}(x) e_{\nu}^{b}(x) \eta_{ab}, \qquad \gamma_\mu(x)\,=\,e_{\mu}^{a}(x)\bar\gamma^a,$$ where $\eta_{ab}= \mbox{diag}(1,-1,-1,-1)$, $\bar \gamma_\alpha$ are the Dirac matrices of Minkowski space and $e_{\mu}^{a}(x)$ are the set of tetradic 4-vectors, we obtain the Dirac matrices $\gamma^\mu(x)$ of curved space-time $$ \gamma^0=\bar \gamma^0,\quad \gamma^1 =\bar \gamma^1 /a(t),\quad \gamma^2= \bar \gamma^2 /b(t),\quad \gamma^3 = \bar \gamma^3 /c(t), $$ $$ \gamma_0=\bar \gamma_0,\quad \gamma_1 =\bar \gamma_1 a(t),\quad \gamma_2= \bar \gamma_2 b(t),\quad \gamma_3 = \bar \gamma_3 c(t). $$ $\Gamma_\mu(x)$ matrices are defined by the equality $$\Gamma_\mu (x)= \frac{1}{4}g_{\rho\sigma}(x)\biggl(\partial_\mu e_{\delta}^{b}e_{b}^{\rho} - \Gamma_{\mu\delta}^{\rho}\biggr)\gamma^\sigma\gamma^\delta, $$ which gives \begin{equation} \Gamma_0=0, \quad \Gamma_1=\frac{1}{2}\dot a(t) \bar \gamma^1 \bar \gamma^0, \quad \Gamma_2=\frac{1}{2}\dot b(t) \bar \gamma^2 \bar \gamma^0, \quad \Gamma_3=\frac{1}{2}\dot c(t) \bar \gamma^3 \bar \gamma^0.\end{equation} Flat space-time matrices we choose in the form, given in \cite{Bog}: \begin{eqnarray} \bar \gamma^0&=&\left(\begin{array}{cccc}1&0&0&0\\0&1&0&0\\ 0&0&-1&0\\0&0&0&-1\end{array}\right), \quad \bar \gamma^1\,=\,\left(\begin{array}{cccc}0&0&0&1\\0&0&1&0\\ 0&-1&0&0\\-1&0&0&0\end{array}\right), \nonumber\\ \bar \gamma^2&=&\left(\begin{array}{cccc}0&0&0&-i\\0&0&i&0\\ 0&i&0&0\\-i&0&0&0\end{array}\right), \quad \bar \gamma^3\,=\,\left(\begin{array}{cccc}0&0&1&0\\0&0&0&-1\\ -1&0&0&0\\0&1&0&0\end{array}\right). \nonumber \end{eqnarray} Defining $\gamma^5$ as follows \begin{eqnarray} \gamma^5&=&-\frac{i}{4} E_{\mu\nu\sigma\rho}\gamma^\mu\gamma^\nu \gamma^\sigma\gamma^\rho, \quad E_{\mu\nu\sigma\rho}= \sqrt{-g} \varepsilon_{\mu\nu\sigma\rho}, \quad \varepsilon_{0123}=1,\nonumber \\ \gamma^5&=&-i\sqrt{-g} \gamma^0 \gamma^1 \gamma^2 \gamma^3 \,=\,-i\bar \gamma^0\bar \gamma^1\bar \gamma^2\bar \gamma^3 = \bar \gamma^5, \nonumber \end{eqnarray} we obtain \begin{eqnarray} \bar \gamma^5&=&\left(\begin{array}{cccc}0&0&-1&0\\0&0&0&-1\\ -1&0&0&0\\0&-1&0&0\end{array}\right).\nonumber \end{eqnarray} We study the space-independent solutions to NLSF equation (2.8). In this case the first equation of the system (2.8) together with (2.11) and (2.12) is \begin{equation} i\bar \gamma^0 \biggl(\frac{\partial}{\partial t} +\frac{\dot \tau}{2 \tau} \biggr) \psi -m \psi +{\cal D} \psi + i {\cal G} \gamma^5\psi=0, \quad \tau(t)=a(t)b(t)c(t), \end{equation} where we denote ${\cal D} := \, 2 S\, F_{I}$ and ${\cal G}:=\, 2 P\,F_{J}.$ For the components $\psi_\rho= V_\rho(t)$, where $\rho=1,2,3,4,$ from (2.13) one deduces the following system of equations: \begin{eqnarray} {\dot V}_1 +\frac{\dot \tau}{2 \tau} V_1 +i(m- {\cal D}) V_1 - {\cal G}V_3 &=& 0, \nonumber\\ {\dot V}_2 +\frac{\dot \tau}{2 \tau} V_2 +i(m- {\cal D}) V_2 - {\cal G}V_4 &=& 0, \nonumber\\ {\dot V}_3 +\frac{\dot \tau}{2 \tau} V_3 -i(m- {\cal D}) V_3 + {\cal G}V_1 &=& 0, \nonumber \\ {\dot V}_4 +\frac{\dot \tau}{2 \tau} V_4 -i(m- {\cal D}) V_4 + {\cal G}V_2 &=& 0. \end{eqnarray} \noindent Let us now define the equations for \begin{eqnarray} P\,=\,i(V_1 V_{3}^{*}-V_{1}^{*}V_3 +V_2V_{4}^{*}-V_{2}^{*}V_4), \nonumber \\ R\,=\,(V_1 V_{3}^{*}+V_{1}^{*}V_3 +V_2V_{4}^{*}+V_{2}^{*}V_4), \nonumber \\ S\,=\,(V_{1}^{*} V_{1}+V_{2}^{*}V_2 -V_{3}^{*}V_{3}-V_{4}^{*}V_4). \end{eqnarray} After a little manipulation one finds \begin{eqnarray} \frac{d S_0}{d t} -2 {\cal G}\, R_0\,=\,0, \nonumber\\ \frac{d R_0}{d t}+2 (m- {\cal D})\, P_0 + 2 {\cal G} S_0\,=\,0, \nonumber\\ \frac{d P_0}{d t}-2 (m- {\cal D})\, R_0\,=\,0, \end{eqnarray} where $S_0 = \tau S, \quad P_0 = \tau P, \quad R_0 = \tau R$. From this system we obtain \begin{eqnarray} S_0 {\dot S}_0 + R_0 {\dot R}_0 +P_0 {\dot P}_0\,=\,0, \nonumber \end{eqnarray} that gives \begin{equation} S^2 + R^2 + P^2 \,=\, C^2/ \tau^2, \qquad C^2 = \mbox{const.} \end{equation} \noindent Let us go back to the system of equations (2.14). It can be written as follows if one defines $W_\alpha\,=\,\sqrt{\tau}\,V_\alpha$: \begin{eqnarray} {\dot W}_1 +i\Phi W_1 - {\cal G}W_3 &=& 0, \quad {\dot W}_2 +i\Phi W_2 - {\cal G}W_4 = 0, \nonumber\\ {\dot W}_3 -i\Phi W_3 + {\cal G}W_1 &=& 0, \quad {\dot W}_4 -i\Phi W_4 + {\cal G}W_2 = 0, \end{eqnarray} where $\Phi\,=\,m- {\cal D}$. Defining $U(\sigma) = W (t)$, where $\sigma = \int\,{\cal G} dt$, we rewrite the foregoing system as: \begin{eqnarray} U_{1}^{\prime} + i (\Phi/{\cal G}) U_{1} - U_{3} &=& 0, \qquad U_{2}^{\prime} + i (\Phi/{\cal G}) U_{2} - U_{4} = 0, \nonumber\\ U_{3}^{\prime} - i (\Phi/{\cal G}) U_{3} + U_{1} &=& 0, \qquad U_{4}^{\prime} - i (\Phi/{\cal G}) U_{4} + U_{2} = 0, \end{eqnarray} where prime ($^\prime$) denotes differentiation with respect to $\sigma$. \noindent Let us now solve the Einstein equations. To do it we first write the expressions for the components of the energy-momentum tensor explicitly. Using the property of flat space-time Dirac matrices and the explicit form of covariant derivative $\nabla_\mu$ one can easily find \begin{equation} T_{0}^{0}= m\,S \,-\,F(I,\,J) + \varepsilon, \quad T_{1}^{1}=T_{2}^{2}=T_{3}^{3}= 2 I\,F_{I}\,+\,2 J\,F_{J} - F(I,\,J) -\,p. \end{equation} Summation of Einstein equations (2.4), (2.5) and (2.6) leads to the equation \begin{equation} \frac{\ddot \tau}{\tau}=-\kappa(T_{1}^{1}+T_{2}^{2}+T_{3}^{3}-\frac{3}{2}T)= \frac{3\kappa}{2}\,\bigl(mS +2 I\,F_{I}\,+\,2 J\,F_{J} \,-\,2\, F(I,\,J) \,+\varepsilon - \,p\bigr). \end{equation} In case if the right hand side of (2.21) be the function of $\tau(t)\,=\,a(t)b(t)c(t)$, this equation takes the form \begin{equation} \ddot \tau+\Phi(\tau)=0. \end{equation} As is known this equation possesses exact solutions for arbitrary function $\Phi(\tau)$. Giving the explicit form of $L_N \,=\,F(I,\,J)$, from (2.21) one can find concrete function $\tau(t)=abc$. Once the value of $\tau$ is obtained, one can get expressions for components $V_\alpha(t), \quad \alpha =1,2,3,4.$ Let us express $a, b, c$ through $\tau$. For this we notice that subtraction of Einstein equations (2.4) - (2.5) leads to the equation \begin{equation} \frac{\ddot a}{a}-\frac{\ddot b}{b}+\frac{\dot a \dot c}{ac}- \frac{\dot b \dot c}{bc}= \frac{d}{dt}\biggl(\frac{\dot a}{a}- \frac{\dot b}{b}\biggr)+\biggl(\frac{\dot a}{a}- \frac{\dot b}{b} \biggr) \biggl (\frac{\dot a}{a}+\frac{\dot b}{b}+ \frac{\dot c}{c}\biggr)= 0. \end{equation} Equation (2.23) possesses the solution \begin{equation} \frac{a}{b}= D_1 \mbox{exp} \biggl(X_1 \int \frac{dt}{\tau}\biggr), \quad D_1=\mbox{const.}, \quad X_1= \mbox{const.} \end{equation} Subtracting equations (2.4) - (2.6) and (2.5) - (2.6) one finds the equations similar to (2.23), having solutions \begin{equation} \frac{a}{c}= D_2 \mbox{exp} \biggl(X_2 \int \frac{dt}{\tau}\biggr), \quad \frac{b}{c}= D_3 \mbox{exp} \biggl(X_3 \int \frac{dt}{\tau}\biggr), \end{equation} where $D_2, D_3, X_2, X_3 $ are integration constants. There is a functional dependence between the constants $D_1,\, D_2,\, D_3,\, X_1,\, X_2,\, X_3 $: $$ D_2=D_1\, D_3, \qquad X_2= X_1\,+\,X_3.$$ Using the equations (2.24) and (2.25), we rewrite $a(t), b(t), c(t)$ in the explicit form: \begin{eqnarray} a(t) &=& (D_{1}^{2}D_{3})^{\frac{1}{3}}\tau^{\frac{1}{3}}\mbox{exp}\biggl[\frac{2X_1 +X_3}{3} \int\,\frac{dt}{\tau (t)} \biggr], \nonumber \\ b(t) &=& (D_{1}^{-1}D_{3})^{\frac{1}{3}}\tau^{\frac{1}{3}}\mbox{exp}\biggl[-\frac{X_1 -X_3}{3} \int\,\frac{dt}{\tau (t)} \biggr], \nonumber \\ c(t) &=& (D_{1}D_{3}^{2})^{-\frac{1}{3}}\tau^{\frac{1}{3}}\mbox{exp}\biggl[-\frac{X_1 +2X_3}{3} \int\,\frac{dt}{\tau (t)} \biggr]. \end{eqnarray} Thus the previous system of Einstein equations is completely integrated. In this process of integration only first three of the complete system of Einstein equations have been used. General solutions to these three second order equations have been obtained. The solutions contain six arbitrary constants: $D_1, D_3, X_1, X_3 $ and two others, that were obtained while solving equation (2.22). Equation (2.7) is the consequence of first three of Einstein equations. To verify the correctness of obtained solutions, it is necessary to put $a, b, c$ into (2.7). It should lead either to identity or to some additional constraint between the constants. Putting $a, b, c$ from (2.26) into (2.7) one can get the following equality: \begin{equation} \frac{1}{3 \tau}\biggl[3 \ddot \tau - 2\frac{\dot \tau^2}{\tau}+ \frac{2}{3 \tau}\biggl(X_{1}^{2}+X_1 X_3 +X_{3}^{2}\biggr)\biggr] = -\kappa \biggl(T_{0}^{0}-\frac{1}{2}T\biggr), \end{equation} that guaranties the correctness of the solutions obtained. In fact we can rewrite (2.21) and (2.27) as \begin{equation} \frac{\ddot \tau}{\tau} \, = \, \frac{3 \kappa}{2} \,(T_{0}^{0} + T_{1}^{1}), \end{equation} and \begin{equation} \frac{\ddot \tau}{\tau} - \frac{2}{3}\,\frac{\dot{\tau}^2}{\tau^2} + \frac{2}{9 \tau^2} {\cal X} \, = \, - \frac{\kappa}{2}\, (T_{0}^{0} - 3 T_{1}^{1}), \end{equation} where ${\cal X} := X_{1}^{2} + X_1 X_3 + X_{3}^{2}$. Combining (2.28) and (2.29) together one gets the solution for $\tau$ in quadrature: \begin{equation} \int\,\frac{d\tau}{\sqrt{3\kappa \tau^2 T_{0}^{0} + {\cal X}/3}}\,=\,t. \end{equation} Let us note that in our further study we exploit the equations (2.21) to obtain $\tau$ and (2.27) to estimate integration constants. \noindent It should be emphasized that we are dealing with cosmological problem and our main goal is to investigate the initial and the asymptotic behavior of the field functions and the metric ones. As one sees, all these functions are in some functional dependence with $\tau$: $\psi \sim 1/\sqrt{\tau}$ and $a_i \sim \tau^{1/3} e^{\pm \int dt/\tau}$. Therefore in our further investigation we mainly look for $\tau$, though in some particular cases we write down field and metric functions explicitly. \vskip 5mm \section{Analysis of the solutions obtained for some special choice of nonlinearity} \setcounter{equation}{0} \noindent Let us now study the system for some special choice of $L_N$. First we analyze the system only for the NLSF which will be followed by the study when the Universe is filled with perfect fluid. But first of all we study the linear case. The reason to get the solution to the self-consistent system of equations for the linear spinor and gravitational fields is the necessity of comparing this solution with that for the system of equations for the nonlinear spinor and gravitational fields that permits to clarify the role of nonlinear spinor terms in the evolution of the cosmological model in question. Using the equation (2.21) one gets \begin{equation} \tau(t)\,=\,(1/2)\,Mt^2\,+\,y_1t\,+\,y_0 \end{equation} where $M=\frac{3}{2}\kappa mC_0,$\,\,$C_0 = C_{1}^{2}+C_{2}^{2}-C_{3}^{2} -C_{4}^{2}$ and $y_1, y_0$ are the constants. In this case we get explicit expressions for the components of spinor field functions and metric functions: \begin{equation} V_r(t)=(C_r/\sqrt{\tau})\,e^{-imt},\quad r = 1,2; \quad V_l(t)=(C_l/\sqrt{\tau})\,e^{imt}, \quad l = 3,4. \end{equation} \begin{eqnarray} a(t) &=& (D_{1}^{2}D_{3})^{\frac{1}{3}}(\frac{1}{2}Mt^2+y_1t+y_0)^{\frac{1}{3}} Z^{2(2X_1+X_3)/3B}, \nonumber \\ b(t) &=& (D_{1}^{-1}D_{3})^{\frac{1}{3}}(\frac{1}{2}Mt^2+y_1t+y_0)^{\frac{1}{3}} Z^{-2(X_1-X_3)/3B}, \nonumber \\ c(t) &=& (D_{1}D_{3}^{2})^{-\frac{1}{3}}(\frac{1}{2}Mt^2+y_1t+y_0)^{\frac{1}{3}} Z^{-2(X_1+2X_3)/3B}, \end{eqnarray} where $Z = \frac{(t-t_1)}{(t-t_2)},\,\, B = M(t_1 -t_2),$ and $t_{1,2} = -y_1/M \pm \sqrt{(y_1/M)^2 - 2y_0/M}$\, are the roots of the quadratic equation \, $Mt^2+2y_1t+2y_0 = 0.$ Substituting $\tau(t)$ into (2.27), one gets \begin{equation} y_{1}^{2}- 2My_0\,=\,(X_{1}^{2}+X_1X_3+X_{3}^{2})/3\,=\,{\cal X}/3 \,>\,0. \end{equation} This means that the quadratic polynomial in (3.1) possesses real roots, i.e. $\tau(t)$ in (3.1) turns into zero at $t=t_{1,2}$ and the solution obtained is the singular one. Let us now study the solutions (3.1) - (3.3) at $t \to \infty$. In this case we have $$\tau(t) \approx \frac{3}{4}\kappa mC_0 t^2, \qquad a(t) \approx b(t) \approx c(t) \approx t^{2/3}, $$ that leads to the conclusion about the asymptotical isotropization of the expansion process for the initially anisotropic B-I space. Thus the solution to the self-consistent system of equations for the linear spinor and gravitational fields is the singular one at the initial time. In the initial state of evolution of the field system the expansion process of space is anisotropic, but at $t \to \infty$ there happens isotropization of the expansion process. \noindent Once the solutions to the linear spinor field equations and corresponding to them metric functions are obtained, let us now study the nonlinear case. \vskip 3mm \noindent {\bf I.} Let us consider the case when $L_N\,=\,F(I)$. It is clear that in this case ${\cal G}\,=\,0$. From (2.16) we find \begin{equation} S = C_0/\tau, \quad C_0= \mbox{const.} \end{equation} As in the considered case $L_N\,=\,F$ depends only on $S$, from (3.5) it follows that $F(I)$ and $F_I(I)$ are functions of $\tau= abc$. Taking this fact into account, integration of the system of equations (2.14) leads to the expressions \begin{eqnarray} V_{r}(t) = (C_r/\sqrt{\tau})\,e^{-i\Omega}, \quad r=1,2, \quad V_{l}(t) = (C_l/\sqrt{\tau})\,e^{i\Omega}, \quad l=3,4. \end{eqnarray} where $ C_r$ and $C_l$ are integration constants. Putting (3.6) into (2.15) one gets \begin{equation} S = (C_{1}^{2}+C_{2}^{2}-C_{3}^{2}-C_{4}^{2})/\tau. \end{equation} Comparison of (3.5) with (3.7) gives $C_0 = C_{1}^{2}+C_{2}^{2}-C_{3}^{2}-C_{4}^{2}.$ \noindent Let us consider the concrete type of NLSF equation with $F(I) =\lambda I^{(n/2)}=\lambda S^n$ where $\lambda$ is the coupling constant, $n>1$. In this case for $\tau$ one gets: \begin{equation} \ddot \tau = (3/2)\kappa C_0 \bigl[m+ \lambda (n-2) C_{0}^{n-1}/\tau^{n-1}\bigr]. \end{equation} The first integral of the foregoing equation takes form: \begin{equation} \dot \tau^2 = 3\kappa C_0 \bigl[m\tau - \lambda \, C_{0}^{n-1}/\tau^{n-2} + g^2\bigr], \end{equation} where from (2.27) one determines $g^2\,=\, {\cal X}/9\kappa\,C_0$. The sign $C_0$ is determined by the positivity of the energy-density $T_{0}^{0}$ of linear spinor field: \begin{equation} T_{0}^{0} = m C_0/\tau > 0. \end{equation} It is obvious from (3.10) that $C_0 >0.$ Now one can write the solution to the equation (3.9) in quadratures: \begin{equation} \int \frac{\tau^{(n-2)/2}d\tau}{\sqrt{m \tau^{n-1} +g^2 \tau^{n-2}-\lambda C_{0}^{n-1}}}= \sqrt{3\kappa C_0}\,t \end{equation} The constant of integration in (3.11) has been taken zero, as it only gives the shift of the initial time. Let us study the properties of solution to equation (3.8) for $n>2$. From (3.11) one gets \begin{equation} \tau(t)\mid_{t \to \infty} \approx (3/4) \kappa mC_0t^2, \end{equation} which coincides with the asymptotic solution to the equation (3.3). It leads to the conclusion about isotropization of the expansion process of the B-I space. It should be remarked that the isotropization takes place if and only if the spinor field equation contains the massive term [cf. the parameter m in (3.12)]. If m=0 the isotropization does not take place. In this case from (3.11) we get \begin{equation} \tau(t)\mid_{t \to \infty} \approx \sqrt{3\kappa C_0 g^2}\,t. \end{equation} Substituting (3.13) into (2.26) one comes to the conclusion that the functions $a(t), b(t)$ and $c(t)$ are different. Let us consider the properties of solutions to equation (3.8) when $t \to 0.$ For $\lambda<0$ from (3.11) we get \begin{equation} \tau(t)= \bigl[(3/4) n^2 \kappa |\lambda| C_{0}^{n}\bigr]^{1/n}t^{2/n} \to 0, \end{equation} i.e. solutions are singular. For $\lambda>0,$ from (3.11) it follows that $\tau=0$ cannot be reached for any value of $t$ as in this case the denominator of the integrand in (3.11) becomes imaginary. It means that for $\lambda>0$ there exist regular solutions to the previous system of equations \cite{Ryb}. The absence of the initial singularity in the considered cosmological solution appears to be consistent with the violation for $\lambda>0$, of the dominant energy condition in the Hawking-Penrose theorem \cite{Zel}. \noindent Let us consider the Heisenberg-Ivanenko equation when in (3.8) n=2 \cite{Ivan}. In this case the equation for $\tau(t)$ does not contain the nonlinear term and its solution coincides with that of the linear equation (3.3). With such $n$ chosen the metric functions $a, b, c$ are given by the equality (3.2), and the spinor field functions are written as follows: \begin{equation} V_r = (C_r/\sqrt{\tau})\,e^{-imt}Z^{4i\lambda C_0/B}, \quad V_l = (C_l/\sqrt{\tau})\,e^{imt}Z^{-4i\lambda C_0/B} \end{equation} As in the linear case, the obtained solution is singular at initial time and asymptotically isotropic as $t \to \infty$. \noindent We now study the properties of solutions to equation (3.8) for $1<n<2.$ In this case it is convenient to present the solution (3.11) in the form: \begin{equation} \int \frac{d \tau}{\sqrt{m\tau -\lambda \tau^{2-n} C_{0}^{n-1}+g^2}}=\sqrt{3\kappa C_0}\,t \end{equation} As $t \to \infty$, from (3.16) we get the equality (3.12), leading to the isotropization of the expansion process. If $m=0$ and $\lambda>0,$ \quad $\tau(t)$ lies on the interval $$0 \le \tau(t) \le \bigl(g^2/\lambda C_{0}^{n-1}\bigr)^{1/(2-n)}.$$ If m=0 and $\lambda<0,$ the relation (3.16) at $t \to \infty$ leads to the equality: \begin{equation} \tau(t) \approx \bigl[(3/4)n^2 \kappa |\lambda| C_{0}^{n} \bigr]^{1/n}t^{2/n}. \end{equation} Substituting (3.17) into (2.26) and taking into account that at $t \to \infty$ $$ \int \frac{dt}{\tau} \approx \frac{n(3\kappa |\lambda| n^2C_{0}^{n})^{1/n}}{(n-2)2^{2/n}}t^{-2/n+1} \to 0 $$ due to $-2/n+1 < 0,$ we obtain \begin{equation} a(t) \sim b(t) \sim c(t) \sim [\tau(t)]^{1/3} \sim t^{2/3n} \to \infty. \end{equation} It means that the solution obtained tends to the isotropic one. In this case the isotropization is provided not by the massive parameter, but by the degree $n$ in the term $L_N = \lambda S^n.$ (3.16) implies \begin{equation} \tau(t)\mid_{t \to 0} \approx \sqrt{3\kappa C_0 g^2}\,t \to 0, \end{equation} which means the solution obtained is initially singular. Thus, for $1<n<2$ there exist only singular solutions at initial time. At $t \to \infty$ the isotropization of the expansion process of B-I space takes place both for $m\not= 0$ and for $m=0.$ \noindent Let us finally study the properties of the solution to the equation (3.8) for $0<n<1.$ In this case we use the solution in the form (3.16). As now $2-n>1,$ then with the increasing of $\tau(t)$ in the denominator of the integrand in (3.16) the second term $\lambda \tau^{2-n} C_{0}^{n-1}$ increases faster than the first one. Therefore the solution describing the space expansion can be possible only for $\lambda<0.$ In this case at $t\to \infty$, for $m=0$ as well as for $m\not= 0,$ one can get the asymptotic representation (3.17) of the solution. This solution, as for the choice $1<n<2,$ provides asymptotically isotropic expansion of the B-I space. For $t \to 0$ in this case we shall get only singular solution of the form (3.19). \vskip 3mm \noindent {\bf II.} We study the system when $L_N\,=\,F(J)$, which means in the case considered ${\cal D}\,=\,0$. Let us note that, in the unified nonlinear spinor theory of Heisenberg the massive term remains absent, as according to Heisenberg, the particle mass should be obtained as a result of quantization of spinor prematter \cite{Hei}. In nonlinear generalization of classical field equations, the massive term does not possess the significance that it possesses in linear one, as by no means it defines total energy (or mass) of nonlinear field system. Thus without losing the generality we can consider massless spinor field putting $m\,=\,0$ that leads to $\Phi\,=\,0.$ This assumption metamorphoses (2.16) to get \begin{equation} P(t)\,=\,D_0/\tau, \,\, D_0=\,\mbox{const.} \end{equation} The system of equations (2.19) in this case reads \begin{eqnarray} U_{1}^{\prime} - U_{3} &=& 0, \qquad U_{2}^{\prime} - U_{4} = 0, \nonumber\\ U_{3}^{\prime} + U_{1} &=& 0, \qquad U_{4}^{\prime} + U_{2} = 0. \end{eqnarray} Differentiating the first equation of system (3.21) and taking into account the third one we get \begin{equation} U_{1}^{\prime \prime} +U_{1} =\,0, \end{equation} which leads to the solution \begin{equation} U_1 = D_1 e^{i \sigma} + iD_3 e^{-i \sigma},\quad U_3 = i D_1 e^{i \sigma} + D_3 e^{-i \sigma}. \end{equation} Analogically for $U_2$ and $U_4$ one gets \begin{equation} U_2 = D_2 e^{i \sigma} + iD_4 e^{-i \sigma},\quad U_4 = i D_2 e^{i \sigma} + D_4 e^{-i \sigma}, \end{equation} where $D_i$ are the constants of integration. Finally, we can write \begin{eqnarray} V_1&=&(1/\sqrt{\tau}) (D_1 e^{i \sigma} + iD_3 e^{-i\sigma}), \quad V_2 = (1/\sqrt{\tau}) (D_2 e^{i \sigma} + iD_4 e^{-i\sigma}), \nonumber \\ V_3&=&(1/\sqrt{\tau}) (iD_1 e^{i \sigma} + D_3 e^{-i \sigma}), \quad V_4 = (1/\sqrt{\tau}) (iD_2 e^{i \sigma} + D_4 e^{-i\sigma}). \end{eqnarray} Putting (3.25) into the expressions (2.15) one finds \begin{equation} P=2\,(D_{1}^{2} + D_{2}^{2} - D_{3}^{2} -D_{4}^{2})/\tau. \end{equation} Comparison of (3.20) with (3.26) gives $D_0=2\,(D_{1}^{2} + D_{2}^{2} - D_{3}^{2} -D_{4}^{2}).$ Let us now estimate $\tau$ using the equation \begin{equation} \ddot{\tau}/\tau\,=\,3 \kappa \,\lambda (n - 1) P^{2n}, \end{equation} where we chose $L_N\,=\,\lambda P^{2n}$. Putting the value of $P$ into (3.20) and integrating one gets \begin{equation} \dot{\tau}^2 \, = \,- 3\kappa\,\lambda D_{0}^{2n} \tau^{2 - 2n} + y^2, \end{equation} where $y^2$ is the integration constant and can be defined from (2.27): $y^2\,=\,{\cal X}/3 > 0$. The solution to the equation (3.28) in quadrature reads \begin{equation} \int\,\frac{d\tau}{\sqrt{- 3 \kappa\lambda D_{0}^{2n}\tau^{2 - 2n} + y^2}} \, = \, t. \end{equation} Let us now analyze the solution obtained here. As one can see the case $n = 1$ is the linear one. In case of $\lambda < 0$ for $n > 1$ i.e. $2 - 2n < 0$, we get $$ \tau(t)\mid_{t \to 0} \approx [(\sqrt{3 \kappa|\lambda|} D_{0}^{n}n) t]^{1/n},$$ and $$ \tau\mid_{t \to \infty} \approx \sqrt{3\kappa y^2} \,t, $$ it means that for the term $L_N$ considered with $\lambda < 0$ and $n > 1$ the solution is initially singular and the space-time is anisotropic at $t \to \infty.$ Let us now study it for $n < 1$. In this case we obtain $$ \tau\mid_{t \to 0} \approx \sqrt{3\kappa y^2}\, t, $$ and $$ \tau\mid_{t \to \infty} \approx [(\sqrt{3\kappa|\lambda|} D_{0}^{n} n) t]^{1/n}.$$ The solution is initially singular as in previous case, but as far as $ 1/n > 1$, it provides asymptotically isotropic expansion of B-I space-time. \vskip 3mm \noindent {\bf III.} In this case we study $L_N\,=\,F(I,\,J)$. Choosing \begin{equation} L_N\,=\,F(K_{\pm}), \quad K_{+} = I + J = I_v = -I_A, \quad K_{-} = I - J = I_T, \end{equation} in case of massless NLSF we find $$ {\cal D}\,=\,2 S F_{K_{\pm}}, \quad {\cal G}\,=\, \pm 2 P F_{K_{\pm}}, \quad F_{K_{\pm}} = dF/dK_{\pm}. $$ Putting them into (2.16) we find \begin{equation} S_{0}^{2} \pm P_{0}^{2} = D_{\pm}. \end{equation} Choosing $F = \lambda K_{\pm}^{n}$ from (2.21) we get \begin{equation} \ddot{\tau}\,=\,3 \kappa \lambda (n - 1)\,D_{\pm}^{n}\,\tau^{1-2n}, \end{equation} with the solution \begin{equation} \int\,\frac{\tau^{n-1} d \tau}{\sqrt{g^2\tau^{2n - 2} - 3\kappa\lambda D_{\pm}^{n}}} \,=\, t, \end{equation} where $g^2\,=\,{\cal X}/3.$ Let us study the case with $\lambda < 0$. For $n < 1$ from (3.33) one gets \begin{equation} \tau (t)\mid_{t \to 0} \approx g t \to 0, \end{equation} i.e. the solutions are initially singular, and \begin{equation} \tau (t)\mid_{t \to \infty} \approx [\sqrt{(3\kappa |\lambda| D_{\pm}^{n})}t]^{1/n}, \end{equation} which means that the anisotropy disappears as the Universe expands. In case of $n > 1$ we get $$ \tau (t)\mid_{t \to 0} \approx t^{1/n} \to 0, $$ and $$ \tau (t)\mid_{t \to \infty} \approx gt, $$ i.e. the solutions are initially singular and the metric functions $a(t), b(t), c(t)$ are different at $ t \to \infty$, i.e. the isotropization process remains absent. For $\lambda > 0$ we get the solutions those are initially regular, but it violates the dominant energy condition in Hawking-Penrose theorem \cite{Zel}. Note that one comes to the analogical conclusion choosing $L_N\,=\,\lambda S^{2n}P^{2n}.$ \vskip 5mm \section{Analysis of the results obtained when the B-I Universe is filled with perfect fluid} \setcounter{equation}{0} \noindent Let us now analyze the system filled with perfect fluid. Let us recall that the energy-momentum tensor of perfect fluid is \begin{equation} T_{\mu (m)}^{\nu}\,=\, (p + \varepsilon) u_\mu u^\nu - \delta_{\mu}^{\nu} p\,=\,(\varepsilon, - p, - p, - p). \end{equation} As we saw earlier the introduction of perfect fluid does not change the field equations, thus leaving the solutions to the NLSF equations externally unchanged. Changes in the solutions performed by perfect fluid carried out through Einstein equations, namely through $\tau$. So, let us first see how the quantities $\varepsilon$ and $p$ connected with $\tau$. In doing this we use the well-known equality $T_{\mu;\nu}^{\nu}\,=\,0$, that leads to \begin{equation} \frac{d}{dt}(\tau \varepsilon) + {\dot \tau} p\,=\,0, \end{equation} with the solution \begin{equation} \mbox{ln} \tau\,=\,-\int\,\frac{d \varepsilon}{(\varepsilon + p)}. \end{equation} Recalling the equation of state $p\,=\,\xi \varepsilon,\,\, 0 \le \xi \le 1$ finally we get \begin{equation} T_{0 (m)}^{0}\,=\,\varepsilon\,=\,\frac{\varepsilon_0}{\tau^{1+\xi}}, \,\, T_{1 (m)}^{1}\,=\,T_{2 (m)}^{2}\,=\,T_{3 (m)}^{3}\,=\,- p\,=\,- \frac{\varepsilon_0 \xi}{\tau^{1+\xi}}, \end{equation} where $\varepsilon_0$ is the integration constant. Putting them into (2.21) we get \begin{equation} \frac{\ddot \tau}{\tau}\,=\,\frac{3 \kappa}{2}\frac{(\xi - 1) \varepsilon_0}{\tau^{(\xi + 1)}} \end{equation} which shows that for stiff matter $(\xi = 1)$ the contribution of fluid to the solution is missing. Let us now study the system with nonlinearity type {\bf I}. In this case we get \begin{equation} \int\,\frac{d \tau}{\sqrt{m C_0 \tau - \lambda C_{0}^{n}/\tau^{(n-2)} + \varepsilon_0 \tau^{(1 - \xi)} + g^2}}\,=\,\pm \sqrt{3 \kappa} t. \end{equation} As one can see in case of dust $(\xi = 0)$ the fluid term can be combined with the massive one, whereas in case of stiff matter $(\xi = 1)$ it mixes up with the constant. Analyzing the equation (4.6) one comes to the conclusion that the presence of perfect fluid does not influence the result obtained earlier for the nonlinear term type {\bf I}. One comes to the same conclusion analyzing the system with perfect fluid for the other types of nonlinear terms considered here. At least both at $t \to 0$ and at $t \to \infty$ the key role is played by the other terms rather than the term presenting fluid. \vskip 5mm \section{Conclusions} \noindent Exact solutions to the NLSF equations have been obtained for the nonlinear terms being arbitrary functions of the invariant $I = S^2$ and $J = P^2$, where $S=\bar \psi \psi$ and $P= i \bar \psi \gamma^5 \psi$ are the real bilinear forms of spinor field, for B-I space-time. Equations with power nonlinearity in spinor field Lagrangian $L_N = \lambda S^n$, where $\lambda$ is the coupling constant, have been thoroughly studied. In this case it is shown that equations mentioned possess solutions both regular and singular at the initial moment of time for $n>2$ . Singularity remains absent for the case of field system with broken dominant energy condition. It is also shown that if in the NLSF equation the massive parameter $m \ne 0$ and $n\ge 2$ then at $t \to \infty$ isotropization of B-I space-time expansion takes place, while for $m=0$ the expansion is anisotropic. Properties of solutions to the spinor field equation for $1<n<2$ and $0<n<1$ we also studied. It was found that in these cases there does not exist solution that is initially regular. At $t \to \infty$ the isotropization process of B-I space-time takes place both for $m \ne 0$ and for $m = 0$. In case of nonlinear term $L_N = \lambda P^{2n}$, we found the solutions those are initially singular and the isotropization process of B-I space-time depends on the choice of $n$. For $L_N = \lambda (I \pm J)^{n}$ we obtained the solutions those may be initially singular or regular depends on the sign of coupling constant $\lambda$, but isotropization process depends on the value of power $n$. It is also shown that the results remain unchanged even in the case when the B-I space-time is filled with perfect fluid. \noindent

proofpile-arXiv_065-582

\section{Introduction.} Vortices in the layered high-T$_c$ materials have remarkably strong thermal fluctuations, which have been extensively studied\cite{review}. At sufficiently low temperatures, vortex lines are also expected to be subject to \underline{quantum} fluctuations. Quantum effects should manifest themselves in the zero-point motion of vortex lines. If these are large enough, the flux lattice can melt even at temperature T=0. Indeed, many experiments suggest that vortex lattice melting, both in high-T$_c$ materials\cite{andrade,schilling,fukuzumi} and in low-T$_c$ films and multilayers\cite{attanasio}, is strongly influenced by quantum fluctuations. Several authors have already considered the possibility of quantum melting in the high-T$_c$ superconductors. Blatter and Ivlev\cite{blatter1} have examined the influence of quantum fluctuations at finite temperatures. They estimated the shift in the melting curve using a Lindemann criterion, assuming overdamped dynamics combined with a Matsubara formalism. Chudnovsky\cite{chudnovsky} has studied a hypothetical two-dimensional (2D) quantum vortex liquid state at temperature $T = 0$. Onogi and Doniach\cite{onogi} computed the $T=0$ melting field for a 2D superconductor using quantum Monte Carlo (QMC) techniques without dissipative quantum tunneling. By taking into account a fictitious magnetic field arising from the Magnus force on the vortex pancakes\cite{ao}, they also found strong numerical evidence for fractional quantum Hall (FQHE) states in the vortex liquid. Such FQHE states had been predicted by several authors\cite{choi,stern}, but principally in the context of Josephson junction arrays. In this paper, we describe two simple, quasi-analytical models for estimating the conditions for quantum melting of a 2D vortex lattice at T = 0, in which the fictitious magnetic field is explicitly included. The same model might also apply to one layer in a 3D stack of uncoupled layers of high-T$_c$ material. The first estimate is a simple Lindemann criterion. The second involves a simple comparison of internal energies in the crystalline and liquid phases. \section{Fictitious Magnetic Field and Lindemann Melting Criterion.} In our model, the vortex pancakes experience two types of forces: those due to other pancakes, and the Magnus force arising from the density of Cooper pairs. We neglect dissipative forces from the ``viscous'' normal electron background, as may be acceptable in the so-called superclean limit\cite{blatter95}. Of the two remaining forces, the Magnus force usually dominates (see below). The melting Lindemann melting criterion proves \underline{independent} of the vortex mass. By contrast, in the opposite limit where the intervortex forces dominate, the melting field depends sensitively on the vortex mass\cite{onogi}. The so-called Magnus force\cite{ao} acting on a single two-dimensional (``pancake'') vortex, in its rest frame, is \begin{equation} {\bf F}_p = q_v h{\bf v} \times {\bf \hat{z}}n_p \equiv \frac{2e}{c}{\bf v} \times {\bf \hat{z}}B_{eff}. \end{equation} Here $q_v = \pm 1$ is the effective charge of the pancake vortex, $h$ is Planck's constant, ${\bf v}$ is the pancake velocity, n$_p$ is the effective areal number density of Cooper pairs (discussed further below), $B_{eff} = \Phi_0n_p$ is the fictitious field, $\Phi_0 = hc/2e$ is the flux quantum, and the film is assumed perpendicular to the $z$-axis. We now wish to show that the intervortex force is typically small compared to the Magnus force. For simplicity, consider a superconducting film of thickness $d$ and London penetration depth $\lambda$. The direct interaction potential between two pancakes separated by $r$ is \begin{equation} \Pi(r) = 2\epsilon_0 d \K \left({r\over\lambda_{\perp}}\right), \end{equation} where $\lambda_{\perp} = \lambda^2/d$ is the transverse penetration depth, $\epsilon_0=\Phi_{0}^2/(16\pi^2\lambda^2)$ gives the energy scale of the interaction per unit length, and $K_0(x)$ is the modified Bessel function of zeroth order. To estimate the effects of the vortex-vortex interaction, we assume that the vortices are ordered into a triangular lattice, and calculate the change in potential energy per vortex, $\Delta U_{harm}$, due to harmonic vibrations about this lattice. After some algebra, this extra energy is found to take the form $\Delta U_{harm} =\sum_{\bf l} {\chi(l)\over 4}\langle|{\bf u}_0 - {\bf u}_{\bf l}|^2\rangle$. Here ${\bf l}$ is a lattice vector of the triangular lattice, ${\bf u}_{\bf l}$ is the displacement of the ${\bf l}^{th}$ vortex from equilibrium, and $\chi(l)=\frac{\epsilon_0 d}{\lp^2}\left[{\lp\over l}\K^{'} \left({l\over \lp}\right)+\K^{''}\left({l\over\lp}\right)\right]$, where $\ell = |{\bf \ell}|$, and the primes denote differentiation. We estimate this energy as follows. First, since the vortex-vortex interaction is assumed small, we neglect $\langle {\bf u_0}\cdot {\bf u}_{\bf l}\rangle$. Secondly, in the weak-screening regime where the nearest neighbor intervortex distance $a_0<<\lp$, the summation may reasonably be replaced by an integral. With these approximations, and using several identities for derivatives of Bessel functions, we finally obtain \begin{equation} \Delta U_{harm} \approx (\epsilon_0 d)\times\pi n_v \langle|{\bf u}_0|^2\rangle, \end{equation} where $n_v = 2/(\sqrt{3}a_0^2)$ is the areal vortex density. Similarly, for a pancake of mass $m_v$ moving in a fictitious field $B_{eff}$, the zero-point energy per pancake $\Delta U_{mag}$ for a pancake in the lowest Landau level is \begin{equation} \Delta U_{mag} = \frac{1}{2}\hbar\omega_c^{eff}, \end{equation} where $\omega_c^{eff} = 2eB_{eff}/(m_vc)$. To show that the zero point motion is usually dominated by $B_{eff}$. we will demonstrate that $\hbar\omega_c \ll \hbar \omega_c^{eff}$, where $\omega_c$ is the frequency for zero-point motion of the harmonic lattice in the absence of B$_{eff}$. Now $\omega_c = \sqrt{k/m_v}$, where $k$ is the effective spring constant of the harmonic lattice. It follows from eq.\ (3) that $k = 2\epsilon_0d\pi n_v$. To compare $\omega_c$ and $\omega_c^{eff}$, we use the London estimate for the penetration depth $\lambda^2(T)=(m_p c^2)/(4\pi q^2 n_p^{3D})=(m_p c^2 d)/(4 \pi q^2 n_p)$, where $n_p^{3D}$ is the pair density per unit volume, $m_p$ is the pair mass, and $q$ the pair charge. Then a little algebra reveals that $\omega_c \ll \omega_c^{eff}$ provided that \begin{equation} \frac{m_v}{m_p} \ll \frac{2n_p}{n_v}, \end{equation} where $m_p$ is the Cooper pair mass . As will be shown below, $n_v /n_p \approx 0.1$ at the melting point. Then inequality (9) is satisfied so long as $m_v/m_p \ll 20$. Now in BiSr$_2$Ca$_2$Cu$_2$O$_{8+x}$, the mass of a single pancake vortex, assuming a thickness $d \approx 10 \AA$ (appropriate for a single layer of high-T$_c$ material) has been estimated as one electron mass\cite{onogi}. Thus, in this regime, the inequality is satisfied and $\Delta U_{harm} \ll \Delta U_{mag}$ as required. Hence, in calculating melting behavior for vortices of this mass, we apparently need consider only $\Delta U_{mag}$. Our results based on considering only $\Delta U_{mag}$ do indeed give $n_v/n_p \approx 0.1$, thereby confirming the self-consistency of our approach. We now obtain a simple Lindemann melting criterion, assuming that the dominant contribution to zero-point vortex motion arises from $B_{eff}$. Although $\omega_c^{eff}$ clearly depends on $m_v$, the corresponding zero-point displacement does not. We calculate this displacement assuming the symmetric gauge for the fictitious vector potential, ${\bf A}_{eff} = \frac{1}{2}{\bf B}_{eff} \times {\bf r}$. Then in the lowest Landau level, one finds \begin{equation} \langle |{\bf u}_0|^2\rangle \equiv \langle (u_x^2 + u_y^2) \rangle = \frac{\Phi_0}{\pi B_{eff}}= {1\over{\pi n_p}}, \end{equation} \underline{independent} of vortex mass. According to the Lindemann criterion, melting occurs the zero-point amplitude is a certain fraction, say $\alpha_L$, of $a_0$. In most conventional materials, $\alpha_L \approx 0.1-0.2$. Since $a_0 = (2\Phi_0/\sqrt{3}B)^{1/2}$, the Lindemann criterion becomes \begin{equation} \frac{n_v}{n_p} = \frac{2\pi}{\sqrt{3}}\alpha_L^2 \approx 0.07, \end{equation} using the estimate $\alpha_L^2 \approx 0.02$. Thus, the Lindemann picture predicts quantum melting at T = 0 at a vortex density of around 7\% of the effective density of Cooper pairs per layer. \section{Laughlin Liquid Versus Wigner Crystal.} Next, we describe an alternative way of estimating the melting temperature in a 2D lattice. We treat the pancake vortices as bosons, moving in the effective magnetic field $B_{eff}$. To describe the bosons, we use a Wigner crystal (WC) wave function in the solid phase, and a properly symmetrized Laughlin wave function in the liquid state. The melting point is determined by the condition that the energies $E_{WC}$ and $E_{LL}$ of the solid and liquid states should be equal. A related approach has been used to discuss melting of the 2D electron lattice in a magnetic field\cite{maki,lam} The WC wave function may be written \begin{equation} \Psi_{WC}=A\ {\cal S}\left(\prod_{\bf l}\psi({\bf r_l}-{\bf l})\right). \end{equation} Here $\psi({\bf r})$ denotes the zero-momentum single-particle wave-function of the lowest Landau level, ${\cal S}$ is the symmetrization operator, and $A$ is a normalization constant. We wish to calculate the averaged vortex-vortex interaction energy in this state, i.\ e.\ $E_{WC}/(2\epsilon_0dS) =\langle\Psi|\sum_{\bf l_1}\sum_{\bf l_2\neq l_1}\K\Bigl({{| {\bf r_{l_1}}-{\bf r_{l_2}}|}\over\lp}\Bigr)|\Psi\rangle/2S$, where $S$ is the sample surface area. We simplify the calculation by several approximations. First, since $a_0 ^2 \gg \langle |{\bf u}_0|^2\rangle$, the wave function symmetrization is quantitatively unimportant for calculating $E_{WC}$. Indeed, for large argument, the single-particle wave function $\psi({\bf r})$ decays exponentially, and the overlap integral between $\psi({\bf r}-{\bf l}_1)$ and $\psi({\bf r}-{\bf l}_2)$ is almost zero, unless ${\bf l}_1={\bf l}_2$. In view of this degree of localization, $E_{WC}$ can be expanded in powers of the small ratio $\langle |{\bf u}_0|^2\rangle/\lp^2$, keeping only the first two terms. The result is $E_{WC}/(2\epsilon_0 dS)=(n_v/2)\sum_{\bf l \neq 0} \K\Bigl({l\over\lp}\Bigr) +n_v{\Delta U_{harm}\over{2\epsilon_0 d}}$, where $\Delta U_{harm}$ is given by eq.\ (3). The fluctuations $\langle |{\bf u}_0|^2\rangle$ appearing in eq.\ (3) are, as noted previously, the sum of two parts: one due to $B_{eff}$ and the other to the intervortex potential. Of these, the former is usually much larger, as noted above, and has already been evaluated in eq.\ (6). We substitute this value into eq.\ (3) and hence into the expression for $E_{WC}$. In the limit $a_0 \ll \lambda_{\perp}$, one can evaluate this sum numerically. The result is very well fitted numerically by the form $\sum_{\bf l \neq 0}\K\Bigl({l\over\lp}\Bigr) \approx n_v\int{d^2{\bf r} \K\Bigl( {r\over\lp}\Bigr)}-0.500{\rm ln}\Bigl(\lp^2 n_v\Bigr) -1.437$. Collecting all these results, we finally obtain \begin{equation} {E_{WC}\over{2\epsilon_0 dS}}={{n_v}^2\over 2}\int{d^2{\bf r}\K\Bigl( {r\over\lp}\Bigr)}- 0.25\ n_v\ {\rm ln}\Bigl(\lp^2 n_v\Bigr) - \end{equation} $$ -0.719\ n_v+{n_v^2\over 2 n_p}. $$ For the liquid phase, the wave function symmetry matters since the pancakes are delocalized. We use as a trial wave function an (unnormalized) wave function of the Laughlin form\cite{laughlin}: \begin{equation} \Psi_{LL,m}= \prod_{j < k}(z_j-z_k)^m\exp(-\frac{1}{4}\sum_{\ell}|z_{\ell}|^2). \end{equation} Here $z_j = x_j+iy_j$ is the position coordinate of the j$^{th}$ pancake, and all lengths are expressed in units of the ``magnetic length'' $\ell_0 \equiv (\Phi_0/(2\pi B_{eff}))^{1/2}$. Since the vortex pancakes are bosons, $m$ must be an even integer. In the Laughlin theory of the fractional quantum Hall effect, $1/m$ is the filling fraction of the first Landau level. Laughlin's prescription for obtaining the minimizing value of $m$ is readily translated to the present problem, in which the role of charges and magnetic field are reversed. The generalized prescription is that the minimizing $m$ occurs when the number density n$_p$ of vortices of the fictitious magnetic field equals $m$ times the number density $n_v$ of fictitious charges, i.\ e. $m = n_p/n_v$. We next calculate the internal energy of the Laughlin liquid at various values of $m$. With a change of scale, the vortex-vortex interaction energy of the liquid becomes $E_{LL}/(2\epsilon_0 dS)=(n_v/2\pi)\int{d^2{\bf x}\K\Bigl( {x\over\lp\sqrt{\pi n_v}}\Bigr)g(x)}$, where $g(x)$ is the dimensionless density-density correlation function for the Laughlin liquid (normalized to unity at large $x$), and ${\bf x}$ is a dimensionless coordinate defined by ${\bf x}={\bf r}\sqrt{\pi n_v}$. Since $g(x)$ differs significantly from unity mainly in the region $x<1$, it is convenient to decompose the interaction energy as follows: \begin{equation} {E_{LL}\over{2\epsilon_0 dS}}={n_v^2\over 2}\int{d^2{\bf r}\K\Bigl( {r\over\lp}\Bigr)}+ \end{equation} $$ {n_v\over 2\pi}\int{d^2{\bf x} \K\Bigl( {x\over\lp\sqrt{\pi n_v}}\Bigr)\Bigl(g(x)-1\Bigr)}\approx $$ $$ {n_v^2\over 2}\int{d^2{\bf r}\K\Bigl({r\over\lp}\Bigr)}- $$ $$ -n_v\int_0^{\infty}{xdx\Biggl({\rm ln}\Bigl({x\over 2\lp\sqrt{\pi n_v}}\Bigr)+ \gamma\Biggr)\Bigl(g(x)-1\Bigr)}, $$ where $\gamma \approx 0.577...$ is Euler's constant and we have used the small-$x$ approximation for K$_0$(x). As noted by Laughlin, the correlation function g(r) for the Laughlin liquid state is just that of the 2D one-component classical plasma (OCP), in which the particles interact logarithmically. The last term on the right is, to within a factor, just the internal energy of the OCP. We can therefore use standard numerical results for the OCP, as obtained by Monte Carlo methods by Caillol {\it et al}\cite{levesque}. Using the analytical fit of these authors to their own numerical results for the integral $\int_0^{\infty}{xdx{\rm ln}\/x(g(x)-1)}$, we find \begin{equation} -\int_0^{\infty}{xdx\Biggl({\rm ln}\Bigl({x\over 2\lp\sqrt{\pi n_v}}\Bigr)+ \gamma\Biggr)\Bigl(g(x)-1\Bigr)}= \end{equation} $$ -\int_0^{\infty}{xdx{\rm ln}\/x\Bigl(g(x)-1\Bigr)- {1\over4}{\rm ln}\Bigl(4\pi\lp^2 n_v\Bigr)+{\gamma\over2}}\approx $$ $$ -0.3755+0.4400\Bigl({n_v\over2n_p}\Bigr)^{0.74}-{1\over4}{\rm ln} \Bigl(\lp^2 n_v\Bigr)-{1\over4}{\rm ln}(4\pi)+{\gamma\over2}. $$ Hence, the energy of the Laughlin liquid can be written as \begin{equation} {E_{LL}\over{2\epsilon_0 dS}}={{n_v}^2\over 2}\int{d^2{\bf r}\K\Bigl( {r\over\lp}\Bigr)}-{1\over 4}n_v\ {\rm ln}\Bigl(\lp^2 n_v\Bigr) - \end{equation} $$ 0.720\ n_v+0.4400\ n_v\Bigl({n_v\over 2 n_p}\Bigr)^{0.74} $$ Finally, the zero-temperature melting transition is defined by the equation $E_{WC}=E_{LL}$, or ${n_v\over2 n_p}\approx0.440\Bigl({n_v\over2 n_p}\Bigr)^{0.74}$, or equivalently \begin{equation} {n_v\over n_p}\approx 0.09. \end{equation} We see that this result agrees remarkably well with the Lindemann criterion. \section{Discussion.} We now evaluate these predictions for two materials of interest, using a simplified approximation for $n_p$. As noted by Ao and Thouless, $n_p$ is not simply the areal density of Cooper pairs per unit area, but that of \underline{superconducting} Cooper pairs - that is, those not pinned by lattice disorder. Since it is not obvious how to evaluate this quantity, we simply use the London equation to estimate $n_p$ at zero field. To get $n_p(B)$, we use the Ginzburg-Landau approximation $\lambda(B,0)=\lambda(0,0)/[1-B/B_{c2}]^{1/2}$, where $B_{c2}$ is the $T=0$ upper critical field, and $\lambda$(B, T) is the magnetic field and temperature dependent penetration depth. The melting condition, from either the Lindemann criterion or from equating solid and liquid energies, is $n_v/n_p = \beta$, where $\beta \approx 0.1$. Substituting the above expressions into this melting condition, we obtain \begin{equation} \frac{B_m}{B_{c2}}= \frac{B_0}{B_0+B_{c2}}, \end{equation} where $B_0 = \beta m_pc^2d\Phi_0/[4\pi\lambda^2(0,0)q^2]$. First, we apply this result to an amorphous MoGe film, an extensively studied 2D extreme Type-II superconductor. An amorphous Mo$_{0.43}$Ge$_{0.57}$ film of thickness $30 \AA$ has $\lambda(0,0) \approx 8000\AA$ and $B_{c2} \approx 10^4$ G\cite{ephron}. Taking $B \approx H$ (a good approximation in the extreme Type-II limit), and using $\beta = 0.1$, we find $B_0 \approx 7 \times 10^4$ G, and therefore $B_m/B_{c2} \approx 0.8-0.9$. This is consistent with the observations of Ephron {\it et al}\cite{ephron}, who find a superconducting-insulating transition at around 10 kG, quite close to the estimated $B_{c2}$. The transition in \cite{ephron} is undoubtedly \underline{not} uncomplicated quantum melting, since it occurs in highly disordered samples. Indeed, it is undoubtedly better described as a continuous phase transition from a vortex glass to a Cooper pair glass\cite{fisher}. Nonetheless, it is gratifying that our predicted field, estimated for a \underline{clean} sample, falls rather close to the observed transition. Of at least equal interest is possible quantum melting in high-T$_c$ superconductors. Since our model is strictly 2D, we consider only a single layer of a high-T$_c$ material. The result may conceivably be extrapolated to the most anisotropic CuO$_2$-based high-T$_c$ materials, such as BiSr$_2$Ca$_2$Cu$_2$O$_{8+x}$. Assuming $d = 10 \AA$ and $\lambda(0,0) = 1400 \AA$, we obtain $B_0 \approx 1.5\times 10^6$ G. Estimating $B_{c2} = 3 \times 10^6$ G, we find $B_m \approx 10^6$ G. Since $T_c$ is smaller and $\lambda(0,0)$ is larger in an underdoped sample, however, we may expect $B_m$ also to decrease in such materials. Finally, we comment on the connection between our results and the calculations of \cite{onogi}. While these authors find FQHE-like commensuration effects in the flux liquid state, their observed melting scales with m$_v$ as if there were no influence of $B_{eff}$ on $B_m$. Our simplified analytical calculations suggest that $B_{eff}$ may dominate the melting behavior for sufficiently light pancake masses ($m_v \ll 40m_e$). Presumably, this influence of $B_{eff}$ would show up in QMC studies at sufficiently low values of m$_v$. To conclude, we have calculated the quantum melting criterion for a 2D vortex lattice at $T = 0$, by comparing the internal energies of the vortex solid and vortex fluid states in a hypothetical superclean limit. We find that, at sufficiently low vortex masses, melting behavior seems to be dominated by a fictitious magnetic field acting on the vortices and produced by the Cooper pair density. The calculated melting field is close to the superconducting-insulating transition observed in certain thin films of amorphous $MoGe$, and may be within reach of pulsed magnetic fields in some underdoped CuO$_2$-based high-T$_c$ materials. \section{Acknowledgments.} One of us (DS) gratefully acknowledges many valuable conversations with Professor S.\ Doniach, as well as the warm hospitality of the Department of Applied Physics at Stanford University, where this calculation was initiated. This work was supported by NSF Grant DMR94-02131 and by the Department of Energy through the Midwest Superconductivity Consortium at Purdue University, through Grant DE-FG90-02ER-45427.

proofpile-arXiv_065-583

\section{Introduction} The usual path chosen to reduce the independent parameters of a theory is the introduction of a symmetry. Grand Unified Theories (GUTs) are representative examples of such attempts. A natural gradual extension of the GUT idea, which preserves their successes and enhances the predictions, may be to attempt to relate the gauge and Yukawa couplings, or in other words, to achieve Gauge-Yukawa Unification (GYU). In recent papers, we have proposed an alternative way to achieve unification of couplings, which is based on the principles of reduction of couplings and finiteness \footnote{Appropriate references may be found in ref. \cite{kmoz1}.}. These principles, which are formulated in perturbation theory, are not explicit symmetry principles, although they might imply symmetries. The former principle is based on the existence of renormalization group (RG) invariant relations among couplings, which do not necessarily result from a symmetry, but nevertheless preserve perturbative renormalizability. Similarly, the latter one is based on the fact that it is possible to find RG invariant relations among couplings that keep finiteness in perturbation theory. We have found that various supersymmetric GYU models predict mass values for the top and bottom quarks, $M_t$ and $M_b$, which are consistent with the experimental data, and that under certain circumstances the different models can be distinguished from each other if $M_t$ and $M_b$ can be more accurately measured \cite{kmoz2}. The most arbitrary part of a phenomenologically viable supersymmetric model is the breaking of supersymmetry. It is widely believed that the breaking of supersymmetry is soft whatever its origin is. If the model is coupled to supergravity, for instance, one can compute in principle the soft supersymmetry--breaking (SSB) terms. In fact, this is an attractive way to reduce the arbitrariness of the SSB terms, where the gravitino mass $m_{2/3}$ defines the scale of the supersymmetry--breaking \cite{nilles1}. In this letter, we would like to extend our unification idea to include the SSB sector. That is, we want to find RG invariant relations among the SSB parameters that are consistent with perturbative renormalizability \footnote{A similar but different idea has been recently proposed in refs. \cite{jack1,kazakov1}.}. To be definite, we will consider the minimal SUSY $SU(5)$ model with the GYU in the third generation \cite{mondragon1}. We will find that, if one requires the breaking of the electroweak symmetry to occur in the desired manner, the SSB sector of the model can be completely fixed by the gaugino mass parameter $M$. It will turn out that the asymptotic freedom in the SSB sector of the Gauge-Yukawa unified model can be achieved only through the reduction of the SSB parameters. We will then calculate within a certain approximation the SSB parameters of the minimal supersymmetric standard model (MSSM), which will turn out to be consistent with the experimental data. More details of our results will be published elsewhere. \section{Formalism} The reduction of couplings was originally formulated for massless theories on the basis of the Callan-Symanzik equation \cite{zim1}. The extension to theories with massive parameters is not straightforward if one wants to keep the generality and the rigor on the same level as for the massless case; one has to fulfill a set of requirements coming from the renormalization group equations, the Callan-Symanzik equations, etc. along with the normalization conditions imposed on irreducible Green's functions \cite{piguet1}. There has been some progress in this direction \cite{zim2}. Here, to simplify the situation, we would like to assume that a mass-independent renormalization scheme has been employed so that all the RG functions have only trivial dependencies of dimensional parameters. To be general, we consider a renormalizable theory which contain a set of $(N+1)$ dimension-zero couplings, $\{\hat{g}_0,\hat{g}_1,\dots,\hat{g}_N\}$, a set of $L$ parameters with dimension one, $\{\hat{h}_1,\dots,\hat{h}_L\}$, and a set of $M$ parameters with dimension two, $\{\hat{m}_{1}^{2},\dots,\hat{m}_{M}^{2}\}$. The renormalized irreducible vertex function satisfies the RG equation \begin{eqnarray} 0 &=& {\cal D}\Gamma [~{\bf \Phi}'s;\hat{g}_0,\hat{g}_1,\dots,\hat{g}_N;\hat{h}_1,\dots,\hat{h}_L; \hat{m}^{2}_{1},\dots,\hat{m}^{2}_{M};\mu~]~,\\ {\cal D} &=& \mu\frac{\partial}{\partial \mu}+ ~\sum_{i=0}^{N}\,\beta_i\, \frac{\partial}{\partial \hat{g}_i}+ \sum_{a=1}^{L}\,\gamma_{a}^{h}\, \frac{\partial}{\partial \hat{h}_a}+ \sum_{\alpha=1}^{M}\,\gamma^{m^2}_{\alpha}\frac{\partial} {\partial \hat{m}_{\alpha}^{2}}+ ~\sum_{J}\,\Phi_I \gamma^{\phi I}_{~~~J} \frac{\delta}{\delta \Phi_J}~.\nonumber \end{eqnarray} Since we assume a mass-independent renormalization scheme, the $\gamma$'s have the form \begin{eqnarray} \gamma_{a}^{h} &=& \sum_{b=1}^{L}\, \gamma_{a}^{h,b}(g_0,\dots,g_N)\hat{h}_b~,\nonumber\\ \gamma_{\alpha}^{m^2} &=& \sum_{\beta=1}^{M}\,\gamma_{\alpha}^{m^2,\beta}(g_0,\dots,g_N) \hat{m}_{\beta}^{2}+ \sum_{a,b=1}^{L}\,\gamma_{\alpha}^{m^2,a b} (g_0,\dots,g_N)\hat{h}_a \hat{h}_b~, \end{eqnarray} where $\gamma_{a}^{h,b}, \gamma_{\alpha}^{m^2,\beta}$ and $\gamma_{a}^{m^2,a b}$ are power series of the dimension-zero couplings $g$'s in perturbation theory. As in the massless case, we then look for conditions under which the reduction of parameters, \begin{eqnarray} \hat{g}_i &=&\hat{g}_i(g)~,~(i=1,\dots,N)~,\\ ~\hat{h}_a &= &\sum_{b=1}^{P}\, f_{a}^{b}(g) h_b~,~(a=P+1,\dots,L)~,\\ ~\hat{m}_{\alpha}^{2} &= &\sum_{\beta=1}^{Q}\, e_{\alpha}^{\beta}(g) m_{\beta}^{2}+ \sum_{a,b=1}^{P}\,k_{\alpha}^{a b}(g) h_a h_b~,~(\alpha=Q+1,\dots,M)~, \end{eqnarray} is consistent with the RG equation (1), where we assume that $g\equiv g_0$, $h_a \equiv \hat{h}_a~~(1 \leq a \leq P)$ and $m_{\alpha}^{2} \equiv \hat{m}_{\alpha}^{2}~~(1 \leq \alpha \leq Q)$ are independent parameters of the reduced theory. We find that the following set of equations has to be satisfied: \begin{eqnarray} \beta_g\,\frac{\partial \hat{g}_{i}}{\partial g} & =& \beta_i ~,~(i=1,\dots,N)~,\\ \beta_g\,\frac{\partial \hat{h}_{a}}{\partial g}+\sum_{b=1}^{P}\gamma_{b}^{h} \frac{\partial \hat{h}_{a}}{\partial h_b} &=&\gamma_{a}^{h}~,~(a=P+1,\dots,L)~,\\ \beta_g\,\frac{\partial \hat{m}^{2}_{\alpha}}{\partial g} +\sum_{a=1}^{P}\gamma_{a}^{h} \frac{\partial \hat{m}^{2}_{\alpha}}{\partial h_a}+ \sum_{\beta=1}^{Q}\gamma_{\beta}^{m^2} \frac{\partial \hat{m}^{2}_{\alpha}}{\partial m^{2}_{\beta}} &=&\gamma_{\alpha}^{m^2}~,~(\alpha=Q+1,\dots,M)~. \end{eqnarray} Using eq. (2) for $\gamma$'s, one finds that eqs. (6)--(8) reduce to \begin{eqnarray} & &\beta_g\,\frac{d f_{a}^{b}}{d g}+ \sum_{c=1}^{P}\, f_{a}^{c} [\,\gamma_{c}^{h,b}+\sum_{d=P+1}^{L}\, \gamma_{c}^{h,d}f_{d}^{ b}\,] -\gamma_{a}^{h,b}-\sum_{d=P+1}^{L}\, \gamma_{a}^{h,d}f_{d}^{ b}~=0~,\\ & &~(a=P+1,\dots,L; b=1,\dots,P)~,\nonumber\\ & &\beta_g\,\frac{d e_{\alpha}^{\beta}}{d g}+ \sum_{\gamma=1}^{Q}\, e_{\alpha}^{\gamma} [\,\gamma_{\gamma}^{m^2,\beta}+\sum_{\delta=Q+1}^{M}\, \gamma_{\gamma}^{m^2,\delta}e_{\delta}^{\beta}\,] -\gamma_{\alpha}^{m^2,\beta}-\sum_{\delta=Q+1}^{M}\, \gamma_{\alpha}^{m^2,\delta}e_{\delta}^{\beta}~=0~,\\ & &~(\alpha=Q+1,\dots,M; \beta=1,\dots,Q)~,\nonumber\\ & &\beta_g\,\frac{d k_{\alpha}^{a b}}{d g}+2\sum_{c=1}^{P}\, (\,\gamma_{c}^{h,a}+\sum_{d=P+1}^{L}\, \gamma_{c}^{h,d}f_{d}^{a}\,)k_{\alpha}^{c b}+ \sum_{\beta=1}^{Q}\, e_{\alpha}^{\beta} [\,\gamma_{\beta}^{m^2,a b}+\sum_{c,d=P+1}^{L}\, \gamma_{\beta}^{m^2,c d}f_{c}^{a} f_{d}^{b}\nonumber\\ & &+2\sum_{c=P+1}^{L}\,\gamma_{\beta}^{m^2,c b}f_{c}^{a}+ \sum_{\delta=Q+1}^{M}\,\gamma_{\beta}^{m^2,\delta} k_{\delta}^{a b} \,]- [\,\gamma_{\alpha}^{m^2,a b}+\sum_{c,d=P+1}^{L}\, \gamma_{\alpha}^{m^2,c d}f_{c}^{a} f_{d}^{b}\nonumber\\ & &+ 2\sum_{c=P+1}^{L}\,\gamma_{\alpha}^{m^2,c b}f_{c}^{a} +\sum_{\delta=Q+1}^{M}\,\gamma_{\alpha}^{m^2,\delta} k_{\delta}^{a b} \,]~=0~,\\ & &(\alpha=Q+1,\dots,M; a,b=1,\dots,P)~.\nonumber \end{eqnarray} If these equations are satisfied, the irreducible vertex function of the reduced theory \begin{eqnarray} & &\Gamma_R [~{\bf \Phi}'s; g; h_1,\dots,h_P; m^{2}_{1}, \dots,\hat{m}^{2}_{Q};\mu~]~\nonumber\\ &\equiv& \Gamma [~{\bf \Phi}'s; g,\hat{g}_1(g),\dots,\hat{g}_N (g); h_1,\dots,h_P, \hat{h}_{P+1}(g,h),\dots,\hat{h}_L(g,h);\nonumber\\ & & m^{2}_{1},\dots,\hat{m}^{2}_{Q},\hat{m}^{2}_{Q+1}(g,h,m^2), \dots,\hat{m}^{2}_{M}(g,h,m^2);\mu~] \end{eqnarray} has the same renormalization group flow as the original one. The requirement for the reduced theory to be perturbative renormalizable means that the functions $\hat{g}_i $, $f_{a}^{b} $, $e_{\alpha}^{\beta}$ and $k_{\alpha}^{a b}$, defined in eq. (3)--(5), should have a power series expansion in the primary coupling $g$: \begin{eqnarray} \hat{g}_{i} &=& g\,\sum_{n=0}^{\infty} \rho_{i}^{(n)} g^{n}~,~ f_{a}^{b}= g\sum_{n=0}^{\infty} \eta_{a}^{b~(n)} g^{n}~,\nonumber\\~ e_{\alpha}^{\beta} &= &\sum_{n=0}^{\infty} \xi_{\alpha}^{\beta~(n)} g^{n}~,~ k_{\alpha}^{a b }= \sum_{n=0}^{\infty} \chi_{\alpha}^{a b~(n)} g^{n}~, \end{eqnarray} To obtain the expansion coefficients, we insert the power series ansatz above into eqs. (6), (9)--(11) and require that the equations are satisfied at each order in $g$. Note that the existence of a unique power series solution is a non-trivial matter: It depends on the theory as well as on the choice of the set of independent parameters. In a concrete model we will consider below, we will discuss this issue more in detail. \section{Application to the minimal SUSY $SU(5)$ GUT} {\bf 3.1} {\em The model and its RG functions} \newline The three generations of quarks and leptons are accommodated by three chiral superfields in $\Psi^{I}({\bf 10})$ and $\Phi^{I}(\overline{\bf 5})$, where $I$ runs over the three generations. A $\Sigma({\bf 24})$ is used to break $SU(5)$ down to $SU(3)_{\rm C} \times SU(2)_{\rm L} \times U(1)_{\rm Y}$, and $H({\bf 5})$ and $\overline{H}({\overline{\bf 5}})$ to describe the two Higgs superfields appropriate for electroweak symmetry breaking \cite{sakai}. The superpotential of the model is \cite{sakai} \footnote{We suppress the hat on the couplings from now on, which was used in the previous section to distinguish the independent parameters from the dependent ones.} \begin{eqnarray} W &=& \frac{g_{t}}{4}\, \epsilon^{\alpha\beta\gamma\delta\tau}\, \Psi^{(3)}_{\alpha\beta}\Psi^{(3)}_{\gamma\delta}H_{\tau}+ \sqrt{2}g_b\,\Phi^{(3) \alpha} \Psi^{(3)}_{\alpha\beta}\overline{H}^{\beta}+ \frac{g_{\lambda}}{3}\,\Sigma_{\alpha}^{\beta} \Sigma_{\beta}^{\gamma}\Sigma_{\gamma}^{\alpha}+ g_{f}\,\overline{H}^{\alpha}\Sigma_{\alpha}^{\beta} H_{\beta}\nonumber\\ & &+ \frac{\mu_{\Sigma}}{2}\, \Sigma_{\alpha}^{\gamma}\Sigma_{\gamma}^{\alpha}+ +\mu_{H}\,\overline{H}^{\alpha} H_{\alpha}~, \end{eqnarray} where $\alpha,\beta,\ldots$ are the $SU(5)$ indices, and we have suppressed the Yukawa couplings of the first two generations. The Lagrangian containing the SSB terms is \begin{eqnarray} -{\cal L}_{\rm soft} &=& m_{H_u}^{2}{\hat H}^{* \alpha}{\hat H}_{\alpha} +m_{H_d}^{2} \hat{\overline {H}}^{*}_{\alpha}\hat{\overline {H}}^{\alpha} +m_{\Sigma}^{2}{\hat \Sigma}^{\dag~\alpha}_{\beta} {\hat \Sigma}_{\alpha}^{\beta} +\sum_{I=1,2,3}\,[\, m_{\Phi^I}^{2}{\hat \Phi}^{* ~(I)}_{\alpha}{\hat \Phi}^{(I)\alpha} \nonumber\\ & &+\,m_{\Psi^I}^{2}{\hat \Psi}^{\dag~(I)\alpha\beta} {\hat \Psi}^{(I)}_{\beta\alpha}\,] +\{ \, \frac{1}{2}M\lambda \lambda+ B_H\hat{\overline {H}}^{\alpha}{\hat H}_{\alpha} +B_{\Sigma}{\hat \Sigma}^{\alpha}_{\beta} {\hat \Sigma}_{\alpha}^{\beta} +h_{f}\,\hat{\overline{H}}^{\alpha} {\hat \Sigma}_{\alpha}^{\beta} {\hat H}_{\beta}\nonumber\\ & &+\frac{h_{\lambda}}{3}\,{\hat \Sigma}_{\alpha}^{\beta} {\hat \Sigma}_{\beta}^{\gamma}{\hat \Sigma}_{\gamma}^{\alpha}+ \frac{h_{t}}{4}\, \epsilon^{\alpha\beta\gamma\delta\tau}\, {\hat \Psi}^{(3)}_{\alpha\beta} {\hat \Psi}^{(3)}_{\gamma\delta}{\hat H}_{\tau}+ \sqrt{2}h_{b}\,{\hat \Phi}^{(3) \alpha} {\hat \Psi^{(3)}}_{\alpha\beta}\hat{\overline{H}}^{\beta} +\mbox{h.c.}\, \}~, \end{eqnarray} where a hat is used to denote the scalar component of each chiral superfield. The RG functions of this model may be found in refs. \cite{mondragon1,polonsky1,kazakov1}, and we employ the usual normalization of the RG functions, $d {\rm A}/d \ln \mu ~=~ [\beta^{(1)}(A) ~~\mbox{or}~~ \gamma^{(1)}(A)]/16 \pi^2+\dots$, where $\dots$ are higher orders, and $\mu$ is the renormalization scale: \begin{eqnarray} \beta^{(1)}(g) &=& -3 g^3~,~ \beta^{(1)}(g_t) = [\,-\frac{96}{5}\,g^2+9\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+ 4\,g_{b}^{2}\,]\,g_{t}~,\nonumber\\ \beta^{(1)}(g_b) &=& [\,-\frac{84}{5}\,g^2+3\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+ 10\,g_{b}^{2}\,]\,g_{b}~,\nonumber\\ \beta^{(1)}(g_\lambda) &=& [\,-30\,g^2+\frac{63}{5}\,g_{\lambda}^2+3\,g_{f}^{2} \,]\,g_{\lambda}~,\nonumber\\ \beta^{(1)}(g_f) &=& [\,-\frac{98}{5}\,g^2+3\,g_{t}^{2} +4\,g_{b}^{2} +\frac{53}{5}\,g_{f}^{2}+\frac{21}{5}\,g_{\lambda}^{2} \,]\,g_{f}~,~ \gamma^{(1)}(M) = -6g^2 \,M~,\nonumber\\ \gamma^{(1)}(\mu_{\Sigma}) &=& [\, -20g^2 +2g_{f}^{2} +\frac{42}{5} g_{\lambda}^{2}\,]\,\mu_{\Sigma} ~,~ \gamma^{(1)}(\mu_H) = [\, -\frac{48}{5}g^2 +\frac{48}{5}g_{f}^{2} +4 g_{b}^{2}+3g_{t}^{2}\, ]\,\mu_H ~,\nonumber\\ \gamma^{(1)}(B_H) &=& [\, -\frac{48}{5}g^2 +\frac{48}{5}g_{f}^{2} +4 g_{b}^{2}+3g_{t}^{2} \,]\, B_H \nonumber\\ & &+ [\,\frac{96}{5}g^2 M+\frac{96}{5}h_{f}g_{f} +8 g_b h_{b}+6 g_t h_{t}]\, \mu_H ~, \nonumber\\ \gamma^{(1)}(B_{\Sigma}) &=& [\, -20 g^2 +2g_{f}^{2} +\frac{42}{5} g_{\lambda}^{2} \,]\, B_{\Sigma}+ [\,40 g^2 M+4 h_{f}g_{f} +\frac{84}{5} g_{\lambda} h_{\lambda}]\, \mu_{\Sigma} ~, \nonumber\\ \gamma^{(1)}(h_t) &=& [\,-\frac{96}{5}\,g^2+9\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+ 4\,g_{b}^{2}\,]\,h_t\nonumber\\ & &+[\, \frac{192}{5} M g^2+ 18 h_t g_t+8 h_b g_b +\frac{48}{5}h_f g_f\,] \, g_t~,\nonumber\\ \gamma^{(1)}(h_b) &=& [\,-\frac{84}{5}\,g^2+3\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+ 10\,g_{b}^{2}\,]\, h_b\nonumber\\ & &+[\, \frac{168}{5} M g^2+ 6 h_t g_t+20 h_b g_b +\frac{48}{5}h_f g_f\,]\, g_b~~,\\ \gamma^{(1)}(h_{\lambda}) &=& [\,-30\,g^2+\frac{63}{5}\,g_{\lambda}^2+3\,g_{f}^{2} \,]\,h_{\lambda} +[\, 60 M g^2+ \frac{126}{5}h_{\lambda} g_{\lambda} +6h_f g_f\,] \,g_{\lambda}~,\nonumber\\ \gamma^{(1)}(h_f) &=& [\,-\frac{98}{5}\,g^2+3\,g_{t}^{2} +4\,g_{b}^{2} +\frac{53}{5}\,g_{f}^{2}+\frac{21}{5}\,g_{\lambda}^{2} \,]\, h_f\nonumber\\ & &~+[\, \frac{196}{5} M g^2+6 h_t g_t+8 h_b g_b+ \frac{42}{5}h_{\lambda} g_{\lambda} +\frac{106}{5}h_f g_f\,] \, g_{f} ~,\nonumber\\ \gamma^{(1)}(m_{H_d}^{2}) &=& -\frac{96}{5} g^2 M^{2}+\frac{48}{5}g_{f}^{2}(m_{H_u}^{2}+ m_{H_d}^{2}+m_{\Sigma}^{2}) \nonumber\\ & &+ 8 g_{b}^{2}( m_{H_d}^{2}+m_{\Psi^3}^{2}+m_{\Phi^3}^{2})+ \frac{48}{5} h_{f}^{2}+8 h_{b}^{2}~, \nonumber\\ \gamma^{(1)}(m_{H_u}^{2}) &=& -\frac{96}{5} g^2 M^{2}+\frac{48}{5}g_{f}^{2}(m_{H_u}^{2}+ m_{H_d}^{2}+m_{\Sigma^3}^{2})+ 6 g_{t}^{2}( m_{H_u}^{2}+2m_{\Psi^3}^{2})+ \frac{48}{5} h_{f}^{2}+6 h_{t}^{2}~, \nonumber\\ \gamma^{(1)}(m_{\Sigma}^{2}) &=& -40 g^2 M^{2}+2g_{f}^{2}(m_{H_u}^{2}+ m_{H_d}^{2}+m_{\Sigma}^{2}) + \frac{126}{5} g_{\lambda}^{2}m_{\Sigma}^{2}+ 2 h_{f}^{2}+\frac{42}{5} h_{\lambda}^{2}~,\nonumber\\ \gamma^{(1)}(m_{\Phi^3}^{2}) &=& -\frac{96}{5} g^2 M^{2}+ 8 g_{b}^{2}( m_{H_d}^{2}+m_{\Psi^3}^{2}+m_{\Phi^3}^{2})+ 8 h_{b}^{2} ~,\nonumber\\ \gamma^{(1)}(m_{\Psi^3}^{2}) &=& -\frac{144}{5} g^2 M^{2}+6g_{t}^{2}( m_{H_u}^{2}+2m_{\Psi^3}^{2})+ 4 g_{b}^{2}( m_{H_d}^{2}+m_{\Psi^3}^{2}+m_{\Phi^3}^{2})+ 6h_{t}^{2}+4 h_{b}^{2}~,\nonumber\\ \gamma^{(1)}(m_{\Phi^{1,2}}^{2}) &=& -\frac{96}{5} g^2 M^{2} ~,~ \gamma^{(1)}(m_{\Psi^{1,2}}^{2}) = -\frac{144}{5} g^2 M^{2} ~,\nonumber \end{eqnarray} where $g$ stands for the gauge coupling. \vspace{0.3cm} \noindent {\bf 3.2} {\em The reduction solution} \newline We require that the reduced theory should contain the minimal number of the SSB parameters that are consistent with perturbative renormalizability. We will find that the set of the perturbatively unified SSB parameters significantly differ from the so-called universal SSB parameters. Without loss of generality, one can assume that the gauge coupling $g$ is the primary coupling. Note that the reduction solutions in the dimension-zero sector is independent of the dimensionfull sector (under the assumption of a mass independent renormalization scheme). It has been found \cite{mondragon1} that there exist two asymptotically free (AF) solutions that make a Gauge-Yukawa Unification possible in the present model: \begin{eqnarray} a & :& g_t=\sqrt{\frac{2533}{2605}} g +0(g^3)~,~ g_b=\sqrt{\frac{1491}{2605}} g +0(g^3)~,~ g_{\lambda}=0~,~ g_f=\sqrt{\frac{560}{521}} g +0(g^3)~,\nonumber\\ b & :& g_t=\sqrt{\frac{89}{65}} g +0(g^3)~,~ g_b=\sqrt{\frac{63}{65}} g +0(g^3)~,~ g_{\lambda}=0~,~g_f=0~, \end{eqnarray} where the higher order terms denote uniquely computable power series in $g$. It has been also found that the two solutions in (17) describe the boundaries of an asymptotically free RG-invariant surface in the space of the couplings, on which $g_{\lambda}$ and $g_f$ can be different from zero. This observation has enabled us to obtain a partial reduction of couplings for which the $g_{\lambda}$ and $g_f$ can be treated as (non-vanishing) independent parameters without loosing AF. Later we have found \cite{kmoz2} that the region on the AF surface consistent with the proton decay constraint has to be very close to the solution $a$. Therefore, we assume in the following discussion that we are exactly at the boundary defined by the solution $a$ \footnote{How to go away slightly from this boundary will be discussed elsewhere. Note that $ g_{\lambda}=0 $ is inconsistent, but $g_{\lambda} < \sim 0.005$ has to be fulfilled to satisfy the proton decay constraint \cite{kmoz2}. We expect that the inclusion of a small $g_{\lambda} $ will not affect the prediction of the perturbative unification of the SSB parameters.}. In the dimensionful sector, we seek the reduction of the parameters in the form (4) and (5). First, one can realize that the supersymmetric mass parameters, $\mu_{\Sigma}$ and $\mu_H$, and the gaugino mass parameter $M$ cannot be reduced; that is, there is no solution in the desired form. Therefore, they should be treated as independent parameters. We find the following lowest order reduction solution: \begin{eqnarray} B_H &=& \frac{1029}{521}\,\mu_H M~,~ B_{\Sigma}=-\frac{3100}{521}\,\mu_{\Sigma} M~, \end{eqnarray} \begin{eqnarray} h_t &=&-g_t\,M~,~h_b =-g_b\,M~, ~h_f =-g_f\,M~,~h_{\lambda}=0~,\nonumber\\ m_{H_u}^{2} &=&-\frac{569}{521} M^{2}~,~ m_{H_d}^{2} =-\frac{460}{521} M^{2}~, ~m_{\Sigma}^{2} = \frac{1550}{521} M^{2}~,\nonumber\\ m_{\Phi^3}^{2} & = &\frac{436}{521} M^{2}~,~ m_{\Phi^{1,2}}^{2} =\frac{8}{5} M^{2}~,~ m_{\Psi^3}^{2} =\frac{545}{521} M^{2}~,~ m_{\Psi^{1,2}}^{2} =\frac{12}{5} M^{2}~. \end{eqnarray} So, the gaugino mass parameter $M$ plays a similar role as the gravitino mass $m_{2/3}$ in supergravity coupled GUTs and characterizes the scale of the supersymmetry--breaking. In addition to the $\mu_{\Sigma}$, $\mu_H$ and $M$, it is possible to include also $B_H$ and $B_{\Sigma}$ as independent parameters without changing the one-loop reduction solution (19). \vspace{0.3cm} \noindent {\bf 3.3} {\em Uniqueness of the reduction}\newline We next address the question of whether the lowest-order solution given in (18) and (19) can be uniquely extended to a power series solution in higher orders. In ref. \cite{mondragon1}, the uniqueness in the dimension-zero sector is proved, and so we assume here that the reduction in this sector has been performed. Let us begin with the case of $h_a~~ (a=t,b,f)$. We prove the uniqueness by induction; we assume that the reduction is unique to $O(g^{n-1})$ and show that the expansion coefficients in the next order can be uniquely calculated. We then insert the ansatz \begin{eqnarray} h_a &=&-g_a M+\dots+g g^n \eta_{a}^{(n)} M~,~ a=t,b,f~, \end{eqnarray} along with the solution $a$ in the dimension-zero sector (17), into the reduction equation (9) using eq. (13). Then collecting terms of $O(g^{n+3})$, one obtains $\sum_{c=t,b,f}\,L_{a c}(n) \eta_{c}^{(n)} = \cdots$, where $\cdots$ in the r.h. side is known by assumption. One finds that the determinant, \begin{eqnarray} \det L(n) &=& \frac{38423832921}{6786025} + \frac{21646499373}{6786025}n + \frac{1423971}{2605}n^2 + 27 n^3~, \end{eqnarray} for integer $n > 0$ never vanishes, implying that the expansion coefficients $\eta_{a}^{(n)}$ can be uniquely calculated. Since the one-loop reduction (19) is unique, the $\eta$'s exist uniquely to any finite order. The uniqueness in the dimension-two sector proceeds similarly. Note that the uniqueness of the expansion coefficients for $B_H$, $B_{\Sigma}$, $m_{\Phi^{1,2}}^{2}$ and $m_{\Psi^{1,2}}^{2}$ can be easily shown, because their one-loop anomalous dimensions are such that there exists no mixing among the coefficients (see eq. (16)). In the case of $m_{\alpha}^{2}~ (\alpha=H_d,H_u,\Sigma,\Phi^3,\Psi^3)$, we have to do a similar investigation as for the $h$'s. So we start with $ m_{\alpha}^{2} = \xi_{\alpha}^{(0)} M^{2}+\dots +g^n \xi_{\alpha}^{(n)} M^{2}$, where \footnote{As for the case of $h_a$'s, we have assumed that the $\gamma (m^{2})$'s are independent of the supersymmetric mass parameters $\mu_H$ and $\mu_{\Sigma}$.} the lowest order coefficients $\xi_{\alpha}^{(0)}$ can be read off from (19), and we assume that the lower order terms denoted by $\dots$ are known. After some algebraic calculations, one finds that the $\xi_{i}^{(n)}$ also can be uniquely calculated to any finite order \footnote{The approach of unifying the SSB parameters of ref. \cite{jack1} is based on a condition on the anomalous dimensions (the $P=Q/3$ condition). This condition is more restrictive than simply requiring the complete reduction of parameters, because the number of the anomalous dimensions usually exceeds that of parameters. It has turned out to be very difficult to satisfy the $P=Q/3$ condition in higher orders in non-finite theories \cite{jack2}.}. \vspace{0.3cm} \noindent {\bf 3.4} {\em Asymptotic freedom (AF) and the stability of the reduction solution} \newline If a reduction solution is unstable, the aysmptotic freedom requirement and the requirement on a power series reduction solution are equivalent in general. In what follows, we show that the reduction solution (19) is an unstable asymptotically free solution and exhibits the Pendleton--Ross infrared fixed point \cite{pendleton1}. That is, the AF requirement forces all the $h_a$'s and $m_{\alpha}^{2}$'s to be reduced according to the reduction solution (19). On contrary, $B_H$ and $B_{\Sigma}$ behave asymptotically free, and their reduction solution (18) will turn out to be stable. To see these, we first derive the asymptotic behavior of the independent parameters, $\mu_{\Sigma}$, $\mu_{H}$ and $M$: \begin{eqnarray} \mu_{\Sigma} & \sim & g^{3100/1653}~,~ \mu_{H} \sim g^{-1029/521}~,~ M \sim g^{2}~~\mbox{as}~~g \to 0~, \end{eqnarray} where we have used eq. (17) and $ d/d \ln \mu = (-3 g^3 +O(g^5))d/d g$. So, the $ \mu_{H} $ does not vanish asymptotically. Note, however, that thanks to the AF in the Gauge-Yukawa sector the asymptotic behavior given in (22) becomes exact in the ultraviolet limit. Moreover, in a mass independent renormalization scheme (which we are assuming throughout), the supersymmetric mass parameters $\mu_H$ and $\mu_{\Sigma}$ do not enter in the anomalous dimensions for $h$'s and $m^{2}$'s \cite{gates1} so that the investigation below is not affected by the bad asymptotic behavior of $\mu_{H} $. To proceed, we introduce $\tilde{h}_a \equiv h_a/M$ and $ \tilde{m}_{\alpha}^{2} \equiv m_{\alpha}^{2}/M^{2}$, and consider a solution near the reduction solution (19): $\tilde{h}_a (g) =-g_a +\Delta^{h}_{a}(g)~,~a=t,b,f$. Then we derive from eq. (7) the linearized equations \begin{eqnarray} \frac{d \Delta^{h}_{a}(g)}{d g}&=& \sum_{c=t,b,f} Y_{a c}\Delta^{h}_{c}(g)/g~. \end{eqnarray} The asymptotic behavior of the system is dictated by the eigenvalues of the matrix $Y$, and one finds that the three basis vectors ${\bf v}^{h}_{i}(g)$ behave like \begin{eqnarray} {\bf v}^{h}_{i} &\sim & g^{\lambda_i}~,~ \lambda_i=-11.64\dots, -4.98\dots, -3.61\dots~, \end{eqnarray} as $g \to 0$, where the $\lambda_i$'s are the eigenvalues of $Y$, implying that the reduction solution for $h_a$'s is ultraviolet unstable. One also sees that AF requires the $h_a$'s to be reduced because $M \sim g^2$ as $g \to 0$. The $m^{2}$-sector can be discussed similarly. Assuming that $\tilde{m}_{\alpha}^{2} (g) = \xi_{(0)}^{5} +\Delta^{m^2}_{\alpha}(g)~,~ \alpha=H_d,H_u,\Sigma,\Phi^{1,2,3},\Psi^{1,2,3}$, and that the $h_a$'s are reduced, we find that the eigenvalues of the matrix $Z$ which enters in the linearized equations, $ d \Delta^{m^2}_{\alpha}(g)/d g = \sum_{\beta=H_d,H_u,\Sigma,\Phi^3,\Psi^3} Z_{\alpha\beta}\Delta^{m^2}_{\beta}(g)/g$, are given by $(\,-14.64\dots, -7.98\dots, -6.61\dots,-4,-4,-4,-4\,)$. Therefore, the reduction solution for $m_{\alpha}^{2}$'s is also ultraviolet unstable, and one, moreover, sees that the AF of $m_{\alpha}^{2}$'s is ensured only by the reduction (19) because $M^{2} \sim g^4~~\mbox{as}~~g \to 0$. As for $B_H$ and $B_{\Sigma}$, we find that as $g \to 0$, \begin{eqnarray} B_H &\simeq &\frac{1029}{521}\mu_H M+c_H\,g^{1.97\dots}~,~ B_{\Sigma} \simeq -\frac{3100}{521}\mu_{\Sigma} M+c_{\Sigma} \,g^{0.64\dots}~ \end{eqnarray} near the reduction solution, where $c$'s are integration constants. Therefore, the $B$'s are asymptotically free ($\mu_H M \sim g^{0.024\dots}~, ~\mu_{\Sigma} M \sim g^{3.8\dots}$), and so the reduction solution for the $B$'s are asymptotically stable. This is good news, because, as we will see later, the reduction solution (19) including (18) is not consistent with the radiative breaking of the electroweak symmetry at low energy. To make the radiative breaking possible, we have to treat $B_H$ as an independent parameter. But, as we have just seen, this can be done without loosing AF of the model. The solution (19) exhibits the one-loop infrared fixed point, which therefore could be used for the infrared-fixed-point approach \cite{lanross}. This approach is based on the assumption that infrared fixed points found in first order in perturbation theory persist in higher orders and that the ratio of the compactification scale $\Lambda_{\rm C}$ (or the Planck scale $M_{\rm P}$) to $M_{\rm GUT}$ is large enough for various parameters to come very close to their infrared values when running from $\Lambda_{\rm C}$ down to $M_{\rm GUT}$. Therefore, this approach may yield similar results to ours, because the reduction solution in one-loop order (19) is the infrared fixed point. Here we would like to see how fast the desired infrared fixed point can be approached in our concrete model. To this end, we assume that $h_a$, $a=t,b,f$ and $m_{\alpha}^{2}$, $\alpha= H_d,H_u,\Sigma,\Phi^{1,2,3}, \Psi^{1,2,3} $ vanish at $\Lambda_{\rm C}$, while we treat $M$ as independent. The one-loop evolution of $ m_{\Phi^{1,2}}^{2} $ and $ m_{\Psi^{1,2}}^{2} $ can be discussed analytically: \begin{eqnarray} \frac{m_{\Phi^{1,2}}^{2}}{M^{2}} &=& \frac{8}{5}+c_{\Phi^{1,2}} g^{-4}~,~ \frac{m_{\Psi^{1,2}}^{2}}{M^{2}} = \frac{12}{5}+c_{\Psi^{1,2}} g^{-4}~, \end{eqnarray} where $c$'s are integration constants. Imposing the above mentioned boundary condition at $\Lambda_{\rm C}$, one finds at $M_{\rm GUT}$ \begin{eqnarray} \frac{m_{\Phi^{1,2}}^{2}}{M^{2}} &\simeq& 0.25, 0.35, 0.52 ~,~\frac{m_{\Psi^{1,2}}^{2}}{M^{2}}\simeq 0.37, 0.53, 0.79~~\mbox{for}~~\frac{\Lambda_{\rm C}}{M_{\rm GUT}} = 10^2, 10^3, 10^5 ~, \end{eqnarray} respectively, where we have used $\alpha=g^2/4\pi=0.04$ at $M_{\rm GUT}$. Unfortunately, we see that the infrared fixed point, $1.6$ and $2.4$, is quite far from the approached points. We have checked numerically that this also holds for the other SSB parameters. \vspace{0.3cm} \noindent {\bf 3.5} {\em Prediction}\newline Since the $SU(5)$ symmetry is spontaneously broken at $M_{\rm GUT}$, the reduction relations (17)-(19) exhibit a boundary condition on the gauge and Yukawa couplings and also on the SSB parameters at this energy scale \footnote{Here we examine the evolution of these parameters according to their renormalization group equations in two-loop order for the gauge and Yukawa couplings and in one-loop order for the SSB parameters.}. To make our unification idea and its consequence transparent, we shall make an oversimplifying assumption that below $M_{\rm GUT}$ their evolution is governed by the MSSM and that there exists a unique threshold $M_{\rm SUSY}$, which we identify with $M$, for all superpartners of the MSSM, so that below $M_{\rm SUSY}$ the standard model (SM) is the correct effective theory. We recall that it is most convenient to fix $\tan\beta$ through the matching condition on the Yukawa couplings at $M_{\rm SUSY}$ in the Gauge-Yukawa Unification scenario \cite{mondragon1,kmoz2}. That is, the Higgs sector is partly fixed by the dimension-zero sector. This is the reason why the complete reduction in the dimensionfull sector, defined by (18) and (19), is inconsistent with the radiative breaking of the electroweak symmetry, as we will see below. Since we are not stressing the accuracy of the approximation, we assume that the potential of the MSSM at $\mu=M$ takes the tree-level form. The minimization of the potential yields two conditions at $M_{\rm SUSY}$ \cite{ibanez1}, \begin{eqnarray} 0 &=& m_{H_d}^{2}-m_{H_u}^{2}+ M_{Z}^{2} \frac{1-\tan^2\beta}{1+\tan^2\beta}+ B_H\frac{\tan^2\beta-1}{\tan\beta}~,\\ 0 &=& 2\mu_{H}^{2}+m_{H_d}^{2}+m_{H_u}^{2}+ B_H\frac{\tan^2\beta+1}{\tan\beta}~, \end{eqnarray} where $\tan\beta=v_2/v_1 ~,~M_Z= (1/2)\sqrt{(3 g_{1}^{2}/5 +g_{2}^{2})( v_{1}^{2}+v_{2}^{2})}~, ~v_{1,2}=(1/\sqrt{2})<\hat{H}_{d,u}>$. Using the unification condition given by (18) and (19) under the assumption that $M_Z $ and $\tan\beta $ at $M_{\rm SUSY}$ are given, these two conditions could fix the $M$ and $\mu_H$ at $M_{\rm GUT}$. Unfortunately, this is not the case. We have numerically checked that the unification condition given by (17)-(19) does not satisfy eqs. (28) and (29). Therefore, we have to treat one of $m_{H_u}$, $m_{H_d}$ and $B_H$ as an independent parameter to make the radiative breaking at $M_{\rm SUSY}$ possible. From the discussion of sect. 3.4 it is clear that the most natural choice is $B_H$, because this is the unique possibility to keep AF. In addition, the lowest order unification condition (19) remains the same; otherwise it would be modified. We use \begin{eqnarray} \alpha_1(M_Z) &=&0.0169~,~ \alpha_2(M_Z) =0.0337~,~ \alpha_{\tau}(M_Z) =8.005\times 10^{-6}~ \end{eqnarray} as input parameters and fix $M_{\rm SUSY}=M$ at $500$ GeV. Then the prediction from the Gauge-Yukawa Unification (17) is: \begin{eqnarray} M_{t} &\simeq& 1.8 \times 10^2 ~\mbox{GeV}~,~ M_{b} \simeq 5.4 ~\mbox{GeV}~,~\alpha_3 (M_Z)\simeq 0.12~,\nonumber\\ M_{\rm GUT} &\simeq& 1.7\times 10^{16}~\mbox{GeV}~, \alpha_{\rm GUT} \simeq 0.040~, ~\tan\beta (M_{\rm SUSY})\simeq 48~, \end{eqnarray} where $M_{t}$ and $M_b$ are the physical top and bottom quark masses. These values suffer from corrections coming from different sources such as threshold effects, which are partly taken into account and estimated in ref. \cite{kmoz2}. In table 1, we show the prediction of the SSB parameters. \begin{center} \begin{tabular}{|c||c|c||c|} \hline $M_1$ (TeV) & 0.22 & $m_{L_3}^{2}$ (TeV$ ^2$) & 0.30 \\ \hline $M_2$ (TeV) & 0.42 & $m_{\tau}^{2}$ (TeV$ ^2$) & 0.23 \\ \hline $M_3 $ (TeV) & 1.2 & $m_{Q_3}^{2}$ (TeV$ ^2$) & 1.1 \\ \hline $h_t $ (TeV) & -0.89 & $m_{b}^{2}$ (TeV$ ^2$) & 0.95 \\ \hline $h_b $ (TeV) & -0.88 & $m_{t}^{2}$ (TeV$ ^2$) & 0.93 \\ \hline $h_{\tau} $ (TeV) & -0.12 & $m_{L_1}^{2}=m_{L_2}^{2}$ (TeV$ ^2$) & 0.52 \\ \hline $B_H~(\mbox{TeV}^2)$ & -0.0027 & $m_{e}^{2}=m_{\mu}^{2}$ (TeV$ ^2$) & 0.64 \\ \hline $\mu_H$ (TeV) & $\pm$ 0.94 & $m_{Q_1}^{2}=m_{Q_2}^{2}$ (TeV$ ^2$) & 1.9 \\ \hline $m_{H_d}^{2}$ (TeV$ ^2$) & -0.76 & $m_{d}^{2}=m_{s}^{2}$ (TeV$ ^2$) & 1.6 \\ \hline $m_{H_u}^{2}$ (TeV$ ^2$) & -0.90 & $m_{u}^{2}=m_{c}^{2}$ (TeV$ ^2$) & 1.8 \\ \hline \end{tabular} \vspace{0.2cm} {\bf Table 1}: Prediction of the SSB parameters. \end{center} \noindent For the SSB parameters above we have used the notation of ref. \cite{martin1}. Using these parameters, one can then compute the superpartner spectrum. We have checked that it is consistent with the experimental data. The LSP, for instance, is found to be a neutralino of $\sim 220$ GeV with a dominant component of the photino \footnote{The present example, however, does not satisfy the naturalness constraints \cite{dimopoulos1}.}. Details of our calculations and results will be presented elsewhere. \vspace{0.3cm} \noindent We thank B. Ananthanarayan, M. Olechowski, R. Oehme, K. Sibold, and W. Zimmermann for useful discussions and suggestions. \newpage

proofpile-arXiv_065-584

\section{ } Polarized deep inelastic scattering (DIS) experiments have proved invaluable for providing information on the spin-dependent parton densities of the nucleon \cite{emc,smc,slac} through measurement of the polarized structure functions such as $g^p_1(x,Q^2)$. Data from such experiments have traditionally been used in phenomenological analyses which attempt to constrain the polarized parton densities and evolve them in $Q^2$ through the use of the spin-dependent Altarelli-Parisi evolution equations. Up until fairly recently, all such analyses had to be performed in leading order (LO) QCD since the spin-dependent two loop splitting functions were unknown. Since the two-loop splitting functions have been calculated \cite{mertig,vogelsang}, recent analyses have been performed in next-to-leading order (NLO) QCD \cite{grsv,gs,forte} and parametrizations of the polarized parton densities have been provided which can be used in calculations of other polarized processes. In these analyses extensive use is made of theoretical constraints such as the Bjorken sum rule in order to extract the parton densities from the polarized structure function measurements. Despite this the data does not provide tight constraints on sizes, and even more so the shapes of either the polarized quark $(\Delta q)$ or gluon $(\Delta G)$ densities. This is true in particular for $\Delta G$. Each group provides more than one different parametrization of the parton densities, each set fitting the structure function data, but they can have very different individual parton densities reflecting the limited constraints on these quantities provided by DIS experiments. It is thus very important, if our understanding of the spin structure of the nucleon is to be improved, that accurate measurements of these distributions be obtained from other processes. Since the cross section for prompt photon production is dominated by the subprocess $qg\rightarrow\gamma q$ in hadronic collisions already in leading order, it has proved very useful for providing information on the unpolarized gluon densities, $g(x,Q^2)$, of hadrons at fixed target facilities. It has therefore been suggested that it may prove equally useful in pinning down the polarized gluon densities \cite{bergerqiu}. In this context it has been examined quite extensively. The NLO corrections were calculated in \cite{contogouris} and \cite{gorvogel}, where numerical estimates were also presented and it was established that the LO results were perturbatively stable. In both those analyses only LO polarized parton densities were available and the fragmentation contributions were not included. More recently, prompt photon production with polarized beam and target was examined in ref.\cite{gorvogel1,gorvogel2} using NLO polarized parton densities and including the fragmentation contributions. Although single inclusive prompt photon production will definitely be very important for constraining the size of $\Delta G$, information on the detailed $x$-shape of the distribution may not be as easily extracted. This is because the calculation of the inclusive cross section involves one convolution over the momentum fractions, $x$, of the initial partons. The practical effect of this is that a measurement of the kinematic variables of the photon is not sufficient to determine the value of $x$. If, on the other hand, one or more of the jets produced in the reaction is also tagged, no convolution is involved in the calculation and the cross section is directly proportional to the parton densities. Thus the double longitudinal asymmetry $A_{LL}$ is directly proportional to the ratio $\Delta G/g$ \cite{nowak} in kinematic regions where other subprocesses such as $q\bar{q}$ scattering can be neglected. Schematically the cross section is given in terms of the polarized and unpolarized hard subprocess cross sections $(\Delta)\hat{\sigma}_{ij}$ and parton densities $(\Delta) f^a_i(x_a,M^2)$ (the presence of the $\Delta$ before a quantity means the polarized version of the quantity) by \begin{equation} (\Delta)\sigma=\sum_{i,j}(\Delta)f^a_i(x_a,M^2)*(\Delta)f^b_j(x_b,M^2)* \left[ (\Delta)\hat{\sigma}^{dir}_{ij}+\sum_{c} (\Delta)\hat{\sigma}^{frag}_{ij\rightarrow c}*D^{\gamma}_c(z,M_F^2)\right]. \end{equation} The $*$ indicates either a product or convolution and explicit dependence of the hard subprocess cross sections on the kinematic variables of the observed products and the renormalization and factorization scales have been omitted. The sum over $i,j$ extends over all initial partons and $D^\gamma_c(z,M_F^2)$ is the unpolarized fragmentation function for a parton, $c$, into a photon at scale $M_F^2$. Thus, just as in the single photon case, there are two classes of contributions to the cross section labelled the $direct$ and $fragmentation$ contributions. The polarized and unpolarized parton densities are defined by \begin{equation} \Delta f_i(x,M^2)=f_i^+(x,M^2)-f^-_i(x,M^2) \end{equation} and \begin{equation} f_i(x,M^2)=f_i^+(x,M^2)+f^-_i(x,M^2), \end{equation} where the $f^+_i$ and $f^-_i$ represent the distribution of partons of type $i$ with positive and negative helicities respectively, with respect to that of the parent hadron. The hard subprocess cross sections are defined by \begin{equation} \Delta\hat{\sigma}_{ij}=\frac{1}{2}\left( \sigma^{++}-\sigma^{+-}\right) \end{equation} and \begin{equation} \hat{\sigma}_{ij}=\frac{1}{2}\left( \sigma^{++}+\sigma^{+-}\right). \end{equation} One of the main quantities studied in polarized experiments is the longitudinal asymmetry $A_{LL}$. This quantity gives a measure of the sensitivity of the process to polarization effects. The double spin asymmetry studied in this paper is defined by \begin{equation} A_{LL}=\frac{\Delta\sigma}{\sigma}, \end{equation} the ratio of the cross section for longitudinal polarization of the incoming hadrons to the corresponding unpolarized cross section. In LO the direct processes contributing to the cross section are $qg\rightarrow \gamma q$ and $q\bar{q}\rightarrow \gamma g$. In a LO jet calculation, the simple approximation `parton = jet' is used irrespective of the jet definition used in the experiments, hence either the final state gluon or quark will form the jet. It is only in NLO that more than one parton may make up a jet, and one finds a dependence of the cross section on the jet definition and parameters. In addition there are the fragmentation processes: \begin{eqnarray} qg &\rightarrow& q g \nonumber \\ qq &\rightarrow& q q \nonumber \\ qq' &\rightarrow& q q' \nonumber \\ q\bar{q} &\rightarrow& q \bar{q} \nonumber \\ qg &\rightarrow& q g \nonumber \\ q\bar{q} &\rightarrow& g g \nonumber \\ gg &\rightarrow& g g \nonumber \\ gg &\rightarrow& q \bar{q} \end{eqnarray} where one of the final state partons fragments to produce the photon, i.e., $q (g)\rightarrow \gamma + X$. The cross section of interest here is the triple differential cross section \begin{displaymath} \frac{d^3\Delta\sigma^{\gamma J}}{dp_T^\gamma d\eta^\gamma d\eta^J}, \end{displaymath} where $\eta^\gamma$ and $\eta^J$ are the pseudorapidities of the photon and jet respectively and $p_T^\gamma$ is the transverse momentum of the photon. The direct contribution to the cross section is given in LO by \begin{equation} \frac{d^3\Delta\sigma^{\gamma J}}{dp_T^\gamma d\eta^\gamma d\eta^J }= 2\pi p_T^\gamma\sum_{i,j}x_a\Delta f^a_i(x_i,M^2) x_b \Delta f^b_j(x_j,M^2)\frac{d\Delta\hat{\sigma}_{ij}}{d\hat{t}}. \end{equation} The corresponding fragmentation contributions will involve a convolution over the fragmentation variable $z$. $\hat{s}=x_a x_b S$ where $\sqrt{S}$ is the cms energy in the hadron-hadron system and as usual, $\hat{t}=(p_1-p_{\gamma})^2$ where $p_1$ and $p_{\gamma}$ are the momenta of one of the initial state partons and the observed final state trigger photon respectively. The Bjorken variables, $x_{a,b}$, are given in terms of the kinematic variables of the photon and jet by \begin{eqnarray} x_a&=&\frac{p^\gamma_T}{\sqrt{S}}\left( e^{\eta^\gamma}+e^{\eta^J}\right)\nonumber \\ x_b&=&\frac{p^\gamma_T}{\sqrt{S}}\left(e^{-\eta^\gamma}+e^{-\eta^J}\right). \end{eqnarray} In leading order $p^\gamma_T=p_T^J$ and thus a measurement of $p_T^\gamma$, $\eta^\gamma$ and $\eta^J$ means that $x_{a,b}$ can be determined. In other words, a measurement of $d^3\Delta\sigma^{\gamma J}/dp_T^\gamma d\eta^\gamma d\eta^J $ corresponds to a measurement of $d^2\Delta\sigma^{\gamma J} /d x_a d x_b$. In NLO due to the presence of a third parton in the process this simple formula is no longer applicable. In that case, in order to determine $x_{a,b}$ unambiguously, one would need to measure the kinematic variables of all three final state partons. The values of $x_{a,b}$ are only approximately determined by the transverse momenta and pseudorapidities of the photon and jet in this case, due to the need to implement a jet definition. If photon isolation requirements are also implemented, then the determination of $x_{a,b}$ from the photon and jet kinematic variables would be even less precise. Such a measurement would still nevertheless yield some very useful information on the $x$-dependence of the polarized parton densities as will be shown. In NLO, $O(\alpha\alpha_s^2)$, there are virtual corrections to the LO non-fragmentation processes, as well as the further three-body processes: \begin{mathletters}\label{eq:1} \begin{eqnarray} q &+g \rightarrow q + g + \gamma\label{eq:11}\\ g &+ g \rightarrow q +\bar{q} + \gamma\label{eq:12}\\ q &+ \bar{q} \rightarrow g +g + \gamma\label{eq:13}\\ q &+ q \rightarrow q + q + \gamma\label{eq:14}\\ \bar{q} &+ q \rightarrow \bar{q} + q + \gamma\label{eq:15}\\ q &+ \bar{q} \rightarrow q' + \bar{q}' + \gamma\label{eq:16}\\ q &+ q' \rightarrow q' + q + \gamma\label{eq:17} \end{eqnarray} \end{mathletters} The virtual corrections as well as the three-body matrix elements were calculated in ref.\cite{gorvogel} and I use these matrix elements in this calculation. In principle the the fragmentation processes of eq.(1.7) should now be calculated to $O(\alpha_s^3)$ and convoluted with the NLO photon fragmentation functions whose leading behavior is $O(\alpha/\alpha_s)$, but the hard subprocess matrix elements are not yet available in the polarized case. Hence unless otherwise stated, in both the polarized and unpolarized cases, I include the leading order contributions to these processes only. Numerically the fragmentation processes are not significant except at extremely low $p_T$ due to the low cms energy of the fixed target experiment as I shall show later, but for a theoretically consistent calculation they should nevertheless be included as they help to reduce uncertainties from scale dependence. For this calculation I use the Monte Carlo method first used in \cite{ohnemus} for the unpolarized case of the photon plus jet calculation performed here, as well as for calculations of some other important processes, to deal with the phase space integrals. In this letter I shall not give any details of the calculation except to say that the calculation was performed in the $\overline{MS}$ scheme using the t'Hooft-Veltman (HVBM) scheme \cite{hvbm} to treat $\gamma_5$. I refer to ref.\cite{gordon} for the details of the calculation. At NLO, since there are two-to-three scattering processes included in the calculation it is possible that two partons may be too close together to be resolved as two separate jets. In this case a jet definition is required. In this calculation I use the cone definition proposed at Snowmass \cite{snowmass}, which defines a jet as hadronic energy deposited in a cone of radius $R_J=\sqrt{(\Delta\eta)^2+(\Delta\phi)^2}$. At the partonic level, if one parton forms the jet then the kinematic variable of the parton is set equal to that of the jet, $p_J=p_i$, $\eta_J=\eta_i$ and $\phi_J=\phi_i$, and the jet is centered on the parton. If another parton falls inside the jet cone then the jet variables are the weighted averages of those of the two partons: \begin{eqnarray} p_J&=&p_1+p_2 \nonumber \\ \eta_J&=&\frac{1}{p_J}\left(p_1\eta_1+p_2\eta_2\right)\nonumber \\ \phi_J&=&\frac{1}{p_J}\left(p_1\phi_1+p_2\phi_2\right). \end{eqnarray} All results are displayed for $\vec{p}\vec{p}$ collisions at the center-of-mass energy $\sqrt{s}=40$ GeV appropriate for the proposed HERA-$\vec{\rm N}$ experiment at DESY \cite{nowak}. For the unpolarized cross section the CTEQ3M parton densities are used throughout, and the value of $\Lambda_{\overline{MS}}$ corresponding to this distribution is also used. Use of other unpolarized parton densities at the $x$-values probed here do not yield significantly different results. For the polarized case the GRSV \cite{grsv} and GS \cite{gs} distributions are used with the corresponding values for $\Lambda_{\overline{MS}}$. The authors of ref.\cite{grsv} and \cite{gs} have proposed various parametrizations of the polarized parton densities differing mainly in the choice of input for the polarized gluon density $\Delta G$. In the case of the GRSV distributions we use the `valence' set which corresponds to a fit of the available DIS data (referred to by the authors as the `fitted' $\Delta G$ scenario), the large gluon fit which assumes that $\Delta G(x,Q_0^2)=g(x,Q_0^2)$ at input (the '$\Delta G=g$' scenario) and the small gluon fit which uses $\Delta G(x,Q_0^2)=0$ at the input scale (the '$\Delta G=0$' scenario), which in this case starts at the very low value of $Q_0^2=0.34$ GeV$^2$. The latter two distributions are intended to represent extreme choices for $\Delta G$. These parametrizations give gluon densities which differ in their absolute sizes as well as in their $x$-shape. The GS parametrizations provide three fits to the data; GS A, GS B and GS C. It has been shown that the GS A and GS B distributions do not differ very much from the $\Delta G=g$ and fitted $\Delta G$ sets of GRSV respectively, whereas the the GS C set is widely different from any of the others. I shall present distributions using the three GRSV sets discussed above, along with the GS C set for comparison. For the fragmentation functions I use the LO asymptotic parametrization of ref.\cite{owens}. As will be shown, the choice of fragmentation functions makes very little difference to the predictions, since these processes account for only a small fraction of the cross section. The renormalization, factorization, and fragmentation scales are set to a common value $\mu = p_T^{\gamma}$ unless otherwise stated. Dependence on $\mu$ is examined in one of the figures below. Since there are two `particles' in the final state, the jet and the photon, both of whose transverse momenta are large, an alternative choice might be $\mu = p_T^J$ or some function of $p_T^{\gamma}$ and $p_T^J$. The results of the calculations show that the magnitudes of $p_T^{\gamma}$ and $p_T^J$ tend to be comparable and that dependence of the asymmetries on $\mu$ is slight, although the individual cross sections may vary significantly with $\mu$. Therefore, choices of $\mu$ different from $\mu = p_T^{\gamma}$ should not produce significantly different predictions for the asymmetries . The two loop expression for $\alpha_s(Q^2)$ is used throughout, with the number of flavors fixed at $N_f=4$, although the contribution from charm was verified to be negligible at the energies considered. The value of the jet cone size is fixed at $R_J=0.5$ unless otherwise stated. Fig.1a shows the triple differential cross section as a function of $p^\gamma_T$ of the photon for the various parametrizations. The unpolarized cross section is shown for comparison. The curves were obtained by averaging over $\Delta \eta=1$ and $\Delta p_T^\gamma=1$ GeV and the restriction $p_T^J\geq p_T^\gamma$ was imposed. The upper dotted curve is for the GS A parametrization, verifying that it is very similar in both shape and size to the $\Delta G= g$ parametrization of GRSV. All the remaining parametrizations give distributions which are distinctly different in both their shapes and sizes. If one compares these distributions with the corresponding curves for single inclusive prompt photon production \cite{gorvogel1} one finds, apart from the expected fall in the absolute size of the cross section, that the corresponding distributions differ also in their shapes. This is most obvious for the GS C parametrization which has the most distinctly different shape. Fig.1b also shows significantly larger asymmetries than the inclusive photon case for equivalent $p_T^\gamma$'s. Most of these differences can be traced to the fact that for the inclusive photon case, a given value of $p_T^\gamma$ corresponds to a much less sharply defined value of $x$ than in the photon plus jet case. In Fig.1a,b the range $4\leq p_T^\gamma\leq 10$ GeV corresponds to $0.2 \leq x_{a,b} \leq 0.5$. It was also verified that for all parametrizations, the $qg$ scattering process accounts for more than $90\%$ of the cross sections. Also plotted in fig.1b are the projected statistical errors in the asymmetries, $\delta A_{LL}$, as estimated in ref.\cite{nowak} from the formula \begin{equation} \delta A_{LL}=0.17/\sqrt{\sigma(pb)}. \end{equation} In \cite{nowak} $\delta A_{LL}$ has been calculated by assuming an integrated luminosity of $240$ pb$^{-1}$ and beam and target polarizations of $P_B=0.6$ and $P_T=0.8$, and including a trigger and reconstruction efficiency of $50\%$ with no acceptance correction. Similar errors for $\delta A_{LL}$ were estimated in \cite{gorvogel1} by integrating the unpolarized cross section over bins of sizes $\Delta \eta=1$ and $\Delta p_T=1$ GeV. We use the same proceedure here. The results indicate that for $p_T\leq 7-8$ GeV the asymmetries will be distinguishable. Fig.1c and Fig.1d are similar to Figs.1a,b but in this case both the jet and photon are restricted to the forward rapidity regions. $\eta^\gamma$ and $\eta^J$ are centered at $+1$, with $\Delta \eta=0.2$. At $p_T^\gamma=5$ GeV this corresponds to probing $x_{a,b}$ in bins centered at the points $x_{a,b}=0.07$ and $x_{b,a}=0.55$. The asymmetries are smaller than in fig.1b but again there are clear distinctions in sizes and shapes for the different parametrizations. In Fig.2 we look at distributions in $\eta^\gamma$ at $p_T^\gamma=5$ GeV and allow $p_T^J$ to vary between $5$ and $20$ GeV. In Fig.2a the jet is restricted to be in the central region, $-0.5\leq \eta^J \leq 0.5$, and in Fig.2c it is restricted to the forward region, $0.5\leq \eta^J\leq 1.5$. The asymmetry plots of Fig.2b shows important differences between the various parametrizations but that of Fig.2d is very striking. The asymmetries are rather flat, although showing differences in shape for the various parametrizations in the positive rapidity regions, but are very large and increase very sharply in the negative rapidity region, almost approaching $1$ at the edge of phase space for the $\Delta G=g$ scenario. An examination of Fig.2c shows that this effect originates with the difference in shapes between the polarized and the unpolarized rapidity distributions. It was argued in ref.\cite{berger} that positive rapidity correlations at collider energies are an inherent property of the hard subprocess matrix elements. Thus if one restricts the jet to be at positive rapidity then the rapidity distribution of the photon would be expected to peak in the positive rapidity region. This was verified recently in ref.\cite{bailey} where prompt photon plus charm quark correlations were studied at NLO. Clearly in Fig.2c this expectation is confirmed for the unpolarized cross section, but in contrast the polarized cross sections show {\it negative} correlations. Since the propagators in the hard subprocess matrix elements are the same this can only be due to the differences between the polarized and unpolarized parton densities. This fact was verified by using artificial polarized parton densities of exactly the same shape as the unpolarized ones along with the polarized matrix elements. In that case positive rapidity correlations between the photon and the jet were obtained. In ref.\cite{berger} it is suggested that it is the behavior at small-$x_a, x_b$ of the product $x_a x_b f^a(x_a,Q^2) f^b(x_b,Q^2)$ along with the structure of the hard subprocess matrix elements which generate the positive rapidity correlations between two final state particles. It is well established that the small-$x$ behavior of the ratio $\Delta G/g$ is $\sim x$ as $x\rightarrow 0$. Hence there is an additional power of $x$ in the polarized parton distributions at small-$x$. This also applies in the set for which $\Delta G=g$ at the input scale. At the scale we are considering ($Q^2\sim 25$ GeV$^2$) the evolution equations determine the shape of the parton densities regardless of the shape at the input scale. Therefore given the fact that the behavior of the ratio $\Delta G/g$ at small-$x$ is generated by the polarized Altarelli-Parisi equations themselves \cite{bergerqiu} and that the GRSV input scale is so low ($Q^2_0=0.34$ GeV$^2$), it is understandable that even the $\Delta G=g$ scenario will also show the same small-$x$ behavior characteristic of the polarized parton densities. It is easy to show from eq.(1.9) \cite{berger} that when only two final state particles are present, as is the case in LO \begin{equation} x_a x_b=\frac{2 (p_T^\gamma)^2}{S}(1+\cosh(\eta^\gamma-\eta^J)). \end{equation} The effect of this additional factor is to suppress the polarized rapidity distributions at the point $\eta^\gamma=\eta^J$ and produce two symmetrical peaks on both sides of this point. There are many other factors such as, for example, available phase phase which also affect the rapidity correlations and may tend to obscure the effect of the small-$x$ behavior, but from Fig.2d the effect is a significant rise in the predicted asymmetries at negative rapidities. This strong sensitivity to polarization effects in this region should serve as a very good test of the underlying QCD mechanism as well as a check on assumptions about the small-$x$ behavior of the polarized parton densities. In order to test the sensitivity of the predictions to the choice of factorization and renormalization scales, Fig.3a shows predictions for the asymmetry of a sample of the the $p_T^\gamma$ distributions of Fig.1a for three different scales. All hard scales are kept equal and varied simultaneously. Although, as in the case of single prompt photon production, varying the scales over the full range shown can change the individual cross sections by up to $50\%$, the predicted asymmetries are relatively stable. In Fig.3b the sensitivity of the predicted asymmetries to the cone size of the jet $R_J$ and to the inclusion of the fragmentation contributions is tested. The solid lines are again the predictions of Fig.1b. Setting $D_{c/\gamma}(z,M_F^2)=0$ clearly has very little effect on the asymmetries, meaning that the predictions are likely to be very similar if NLO corrections to these contributions were included. This could be expected since at this cms energy, fragmentation is less the $10\%$ of the cross section except at the very lowest $p_T$ values. The dotted line in Fig.3b shows the effect of setting $R_J=1$. The effect on the cross sections is generally less than $10\%$ and clearly it has no effect at all on the asymmetries. Since in this calculation a jet definition is involved in NLO, the calculation of a so-called K-factor to estimate the size of the NLO corrections is not meaningful, as it would depend on the value of $R_J$ chosen. Thus in Fig.3c, a comparison is made between predictions for the cross section using the LO and NLO matrix elements but both using the NLO structure functions. The aim is to examine whether the NLO matrix elements and jet definition have any effect on the shapes of the distributions. As stated earlier, in NLO, measuring the kinematic variables of the photon and jet does not serve to uniquely determine the values of $x_a$ and $x_b$. Any significant differences between the values of these variables probed in LO and NLO could possible show up as a difference in the shapes of the distributions. Figs.3a,b clearly show that there are no significant shape differences between the LO and NLO distributions, and therefore one can assume that estimating the values of $x$ probed for a given kinematical configuration of the photon and jet by the LO formula may not lead to unacceptably large errors. In conclusion, I have examined the possibility that both the size and $x$-shape of the polarized gluon distribution of the proton, $\Delta G$, may be measured at HERA-$\vec{\rm N}$ via a measurement of the photon plus jet cross section. Control over the kinematic variables of both the photon and jet allows a much better determination of the $x$-value probed when compared to inclusive prompt photon production. A comparison of the predictions obtained using different polarized parton densities show that a clear distinction between both the sizes and shapes should be possible. Assuming that the `fitted $\Delta G$' scenario is the most plausible distribution, then a typical value for the asymmetry, $A_{LL}$ is $10\%$, but given the uncertainty in $\Delta G$ the asymmetry could be as small as $2\%$ or as large as $40\%$. The expected small-$x$ behavior of the polarized distributions lead to predictions of negative correlations between the rapidities of the photon and jet. The effect is to produce very large asymmetries, even approaching the maximum value of $A_{LL}=1$ in certain kinematic regions, which should make them more easily measurable in the experiments. \section{Acknowledgments} I am indebted to Ed Berger for many helpful discussions and for reading the manuscript, and to Werner Vogelsang for some helpful comments. The work at Argonne National Laboratory was supported by the US Department of Energy, Division of High Energy Physics, Contract number W-31-109-ENG-38. \pagebreak

proofpile-arXiv_065-585

\section*{INTRODUCTION} Parton distributions contain a wealth of information concerning the non-perturbative structure of hadrons. While most attention has so far been paid to the distributions of the nucleon, through the Drell-Yan process one can also get information on the parton distributions of the hyperons. We shall be concerned with the predictions of the distibutions for the $\Sigma^{\pm}$ hyperons in various models, including: (1) a quark and meson model, (2) an SU(3) model and (3) a model involving the coupling of octets of mesons and baryons (without regard to mass). In particular, a comparison of $\Sigma^{\pm}$ and proton structure functions can be used to differentiate between these models. We show that measurements of the Drell-Yan process for $\Sigma^{\pm}$ scattering on protons and neutrons (in deuteron targets) allow one to extract information on the parton distributions of the $\Sigma^{\pm}$ and that the expectations are quantitatively quite different for (1), (2) and (3), above. In particular, we find that measurements of the $\bar{u} /\bar{d}$ ratio in the $\Sigma^{+}$ are very sensitive to the model chosen, and that a quark-diquark model of the hyperon predicts large SU(3) violations in the valence quark distributions. One of the surprises in the structure of the proton is that the sea appears to have a flavor asymmetry, an excess of $\bar{d}$ compared to $\bar{u}$ \cite{NMC91,NMC94,NA51}. Although the experimental results could imply some violation of isospin, this appears to be less likely, and we interpret them as an $SU(2)_{Q}$ flavor asymmetry in the sea \cite{Forte93}. Thus, the $\bar{d}$ excess in the proton is expected to be reflected in an excess of $\bar{u}$ in the neutron; isospin symmetry would be broken if this were not the case. The evidence for flavor asymmetry in the proton sea is based on analyses of deep inelastic muon scattering \cite{NMC91,NMC94} and Drell-Yan processes \cite{NA51}. One explanation that has been offered is that the excess of $\bar{d}$ over $\bar{u}$ is due to the Pauli exclusion principle \cite{Feynman77,Signal88,Signal89}. A more likely explanation, in our view, is that offered by Thomas and colleagues \cite{Thomas83,Ericson84,Thomas87,Melnitchouk91,Signal91}, Henley and Miller \cite{Henley90}, and others \cite{Eichten92,Kumano91a,Kumano91b,Hwang91,Szczurek93,Szczurek94,Szczurek96,% Holtmann96}, namely that the presence of a pion cloud surrounding a proton favors $\bar{d}$ over $\bar{u}$ because of the excess positive charge of the meson cloud. It is interesting to apply these arguments to the strange baryons and to compare them with SU(3). Here we focus on the charged $\Sigma^{+}$ ($\Sigma^{-}$), composed of $uus$ ($dds$) valence quarks. Thus the main difference from the $p(n)$ case is the replacement of a valence $d(u)$ quark by an $s$ quark. In the following, the quark distribution $q(u,d,$ or $s)$ without subscripts refers to the proton, and with a $\Sigma$ subscript refers to the $\Sigma^{+}$. The $x$-dependence of these distributions is implied, but not shown explicitly. With neglect of mass effects or under SU(3), \begin{equation} \bar{r}\equiv \frac{\bar{u}}{\bar{d}} =\frac{\bar{u}_{\Sigma}} {\bar{s}_{\Sigma}}=0.51\pm0.04\pm0.05 , \end{equation} where the experimental ratio \cite{NA51} is that obtained for the proton at $x\approx 0.18$. The ratio $\kappa \equiv 2 \bar{s}/(\bar{d}+\bar{u})$ is a measure of the strange quark content of the nucleon. It has been determined experimentally in neutrino-induced charm production \cite{CDHS,CCFR,E733} to be in the range $0.373^{+0.048}_{-0.041}\pm{0.018}\leq \kappa \leq 0.57 \pm 0.09$. The CTEQ \cite{CTEQ} determination of parton distributions from global QCD analyses of experimental data uses the value $\kappa = 0.5$, from which \begin{equation} \bar{r}_{s}\equiv \frac{\bar{s}}{\bar{u}+\bar{d}}=\frac{\kappa}{2}=0.25, \end{equation} so with $\bar{d}\approx 2\bar{u}$ from Eq. 1, \begin{equation} \frac{1}{\bar{r}_{\Sigma}}\equiv \frac{\bar{d}_{\Sigma}}{\bar{u}_{\Sigma}}=\frac{\bar{s}}{\bar{u}}\approx 0.75 . \end{equation} We also find \begin{equation} \frac{\bar{d}_{\Sigma}}{\bar{s}_{\Sigma}}=\frac{\bar{s}}{\bar{d}}\approx 0.38 \end{equation} from the same analysis. For the model of quarks surrounded by light pseudoscalar mesons, the $\Sigma^{+}$ has an excess of $\bar{d}$ over $\bar{u}$ (and the opposite for the $\Sigma^{-}$) in contradistinction to the SU(3) prediction. If we neglect higher masses, then the $\Sigma^{+}(uus)$ will have components $\Lambda^{0}(uds)\pi^{+}(u\bar{d})$, $\Sigma^{0}(uds)\pi^{+}(u\bar{d})$, $\Sigma^{+}(uus)\pi^{0}(\frac{1}{\sqrt{2}}[d\bar{d}-u\bar{u}])$, or $p(uud)\bar{K^{0}}(\bar{d}s)$; similarly a $\Sigma^{-}(dds)$ can be $\Lambda^{0}(uds)\pi^{-}(d\bar{u})$, $\Sigma^{0}(uds)\pi^{-}(d\bar{u})$, $\Sigma^{-}(dds)\pi^{0}(\frac{1}{\sqrt{2}}[d\bar{d}-u\bar{u}])$, or $n(udd)K^{-}(\bar{u}s)$. Thus there is a clear enhancement of $\bar{d}$ for $\Sigma^{+}$ and $\bar{u}$ for $\Sigma^{-}$ and we expect $ {\bar{r}_{\Sigma}} \leq$ 0.5. We also consider an SU(3) model in which a baryon is composed of octets of baryons and mesons. We use the SU(3) isoscalar factors and representation matrices given by the Particle Data Group\cite{PDG}. For \mbox{\boldmath $8_{1}\rightarrow 8 \otimes 8$} \begin{equation} N \rightarrow \frac{g_{1}}{\sqrt{20}}\left[3N\pi-N\eta-3\Sigma K-\Lambda K\right], \end{equation} and for \mbox{\boldmath $8_{2}\rightarrow 8 \otimes 8$} \begin{equation} N \rightarrow \frac{g_{2}}{\sqrt{12}}\left[\sqrt{3}N\pi+\sqrt{3}N\eta+ \sqrt{3}\Sigma K-\sqrt{3}\Lambda K\right]. \end{equation} The standard $D$ and $F$ couplings are related to $g_{1}$ and $g_{2}$ by $D=\frac{\sqrt{30}}{40}g_{1}$ and $F=\frac{\sqrt{6}}{24}g_{2}$, so \begin{eqnarray} p& \rightarrow &\sqrt{8}D\left[p\left\{(1+\alpha)\pi^{0}+ \sqrt{3}\left(\alpha-\frac{1}{3}\right)\eta\right\} -\sqrt{2}(1+\alpha)n\pi^{+}+\sqrt{2}(\alpha-1)\Sigma^{+}K^{0}\right.% \nonumber \\ & &\left.+(1-\alpha)\Sigma^{0}K^{+}-\sqrt{3}\left(\alpha+\frac{1}{3}\right)% \Lambda K^{+}\right], \end{eqnarray} \begin{eqnarray} p& \rightarrow & \sqrt{8}D\left[p\left\{(1+\alpha)\frac{u\bar{u}-d\bar{d}} {\sqrt{2}}+\sqrt{3}\left(\alpha-\frac{1}{3}\right) \frac{u\bar{u}+d\bar{d}-2s\bar{s}}{\sqrt{6}}\right\}-\sqrt{2} (1+\alpha)nu\bar{d}\right.\nonumber \\ & & \left.+ \sqrt{2}(\alpha-1)\Sigma^{+}d\bar{s}+(1-\alpha)\Sigma^{0}u\bar{s}- \sqrt{3}\left(\alpha+\frac{1}{3}\right)\Lambda u\bar{s}\right], \end{eqnarray} with $\alpha \equiv F/D$, and which leads to relative probabilities, averaged over $x$, \begin{eqnarray} \bar{u} & \approx & \frac{2}{9}\left[9\alpha^{2}+6\alpha+1\right],\\ \bar{d} & \approx & \frac{2}{9}\left[9\alpha^{2}+18\alpha+13\right],\\ \bar{s} & \approx & \frac{4}{9}\left[18\alpha^{2}-12\alpha+8\right]. \end{eqnarray} Then \begin{equation} \bar{r}\equiv \frac{\bar{u}}{\bar{d}}=\frac{\bar{u}_{\Sigma}} {\bar{s}_{\Sigma}}= \frac{1+6F/D+9(F/D)^2}{13+18F/D+9(F/D)^2} \end{equation} and \begin{equation} \bar{r}_{s}\equiv \frac{\bar{s}}{\bar{u}+\bar{d}}=\frac{\bar{d}_{\Sigma}} {\bar{u}_{\Sigma}+\bar{s}_{\Sigma}}=\frac{8-12F/D+18(F/D)^2}{7+12F/D+9(F/D)^2}. \end{equation} With $\alpha = 0.6$, consistent with a recent analysis\cite{Ratcliffe}, we obtain for the proton \begin{equation} \bar{r}=.29,\: \bar{r}_{s}=.42 \end{equation} which differ significantly from the experimental result ($\bar{r}=.51$) and parameter ($\bar{r}_{s}= \kappa /2 = .25$). We also show in Table I the prediction of this model, $\bar{r}_{\Sigma}=0.54$, which, like the meson cloud model, disagrees with the SU(3) expectation of 4/3. Deviations from SU(3) predictions are also expected for the valence quark distributions in $\Sigma^{+}$ ($\Sigma^{-}$). On the basis of SU(3) symmetry we expect \begin{equation} r_{\Sigma} \equiv \frac {s_{\Sigma}}{u_{\Sigma}} \approx \frac {d}{u} \approx 0.57 (1-x). \label{SU3} \end{equation} The functional form is taken from a fit by CDHS \cite{CDHSV}, and agrees with the latest parton distribution analysis of CTEQ \cite{CTEQ} within $20\%$, which is adequate for our calculations. We find that for $x \geq 0.2$, quark models predict valence quark flavor asymmetries in the $\Sigma^{+}$ that are greater than the SU(3) result, e.g. by a factor of 3.4 at $x=0.7$. Our approach to estimating valence quark distributions in the $\Sigma^{+}$ is based on a quark-diquark model initiated at Adelaide \cite{Signal89,Schreiber90} which has led to the study of charge symmetry violation in the nucleon \cite{CSV}. It was found that the dominant contribution to the structure function $q(x)$ in the valence region comes from a state in which the two spectator quarks are in their ground states. The effective mass of this diquark state will deviate from 3/4 of the nucleon mass (in the MIT bag model, 2/3 in the constituent quark model) because of the hyperfine interaction. The mass difference between the two spin states of the diquark leads to spin and flavor dependence of $q(x)$ \cite{Close88}. Let $x_{q}$ represent the most probable momentum fraction carried by the quark $q$, and $x_{qq}$ represent the most probable momentum fraction carried by the diquark. Then the peak in $q(x)$ can be estimated from \begin{equation} x_{q}+x_{qq}=1,\: x_{q}=1-x_{qq}\approx 1-\frac{m_{qq}}{m_{B}}, \end{equation} in which $m_{qq}$ and $m_{B}$ are the diquark and baryon masses, respectively. For the nucleon, the $N-\Delta$ splitting leads to $m_{qq}=650$ MeV in the spin singlet state, and $m_{qq}=850$ MeV in the spin triplet. Then in the proton, $d(x)$ peaks at $x_{d} \approx 0.10$, whereas $u(x)$ peaks at $x_{u} \approx 0.31$ -- at the scale appropriate to the model. After QCD evolution, these estimates are in reasonable agreement with recent parton distribution analyses \cite{CTEQ}. These same arguments may be applied to the $\Sigma^{+}$. In this case the diquark $uu$ must be in a spin triplet, so from the $\Lambda-\Sigma$ splitting, $m_{uu} \approx 850$ MeV, and $s_{\Sigma}(x)$ peaks at $x_{s} \approx 0.28$. This is close to the value found for $u(x)$ in the proton, so we set $s_{\Sigma}(x)\approx u(x)/2$ (the factor of 2 comes from normalization). To estimate the $u_{\Sigma}$ distribution we note that the $su$ diquark mass is increased by $\approx 180$ MeV because of $m_{s}$, and with the hyperfine splitting, $m_{su} \approx 900$ MeV in the singlet, leading to a peak $x_{u}(S=0) \approx 0.24$ -- i.e., a ``harder'' distribution, like that of $u(x)$ in the proton -- and $m_{su} \approx 1050$ MeV in the triplet, leading to $x_{u}(S=1) \approx 0.10$ -- a ``softer'' distribution like $d(x)$ in the proton. Since the singlet and triplet diquark states are equally probable, we approximate $u_{\Sigma}(x) \approx d(x)+u(x)/2$. Then \begin{equation} r_{\Sigma} \equiv \frac{s_{\Sigma}}{u_{\Sigma}}\approx \frac {u}{2(d+u/2)} \approx \frac{1}{1+2d/u} =\frac{1}{1+1.14(1-x)}. \label{QM} \end{equation} In the valence quark region, this ratio is considerably in excess of that predicted by SU(3), as can be seen in Fig. 1, in which we plot the ratio $R$ \begin{equation} R=\frac{(r_{\Sigma})_{th}}{(r_{\Sigma})_{SU(3)}} \approx \frac{1}{[1+1.14(1-x)]0.57(1-x)} \end{equation} There are a number of ways that these arguments and models can be tested for $\Sigma$ hyperons. The most practical to us appears to be in terms of the Drell-Yan cross sections for $\Sigma^{\pm}p$ and $\Sigma^{\pm}n$ (i.e. $d$) -- e.g., in the inclusive reactions $\Sigma^{\pm}p\rightarrow l^{+} l^{-} X$, where $l^{\pm}$ are muons or electrons and $X$ is unmeasured. Beams of $\Sigma^{\pm}$ appear to be adequate for this purpose, but $\pi$ contamination will lead to problems which need to be overcome. We first consider the determination of sea quark flavor asymmetry for $\Sigma^{\pm}$. We find that extracting the ratio $\bar{r}_{\Sigma}(x)\equiv \bar{u}_{\Sigma}(x)/\bar{d}_{\Sigma}(x)$ for the $\Sigma^{+}$ depends on the known ratios $r(x)\equiv u(x)/d(x)$ and $\bar{r}(x)\equiv \bar{u}/\bar{d}$ in the proton. The former is well-determined, and the recent determination of the latter has been discussed above. Ratios involving $\bar{s}$ in the $\Sigma^{\pm}$ cannot be tested easily because they involve second order annihilations ($\bar{s}_{\Sigma}$ on $s$), terms which we neglect because the present accuracy of Drell-Yan measurements is insufficient to be sensitive to them. Drell-Yan cross-sections are proportional to the products $q(x)\bar{q}(x^{\prime})$, weighted by the product of the quark charges, and summed over contributions from beam and target. We neglect sea-quark - sea-quark collisions, which would contribute below the likely level of accuracy of the experiment. We assume isospin reflection (charge) symmetry: $u(x)=d_{n}(x)$, $\bar{u}(x)=\bar{d}_{n}(x)$, $u_{\Sigma^{+}}(x)=d_{\Sigma^{-}}(x)$, $\bar{u}_{\Sigma^{+}}(x)=\bar{d}_{\Sigma^{-}}(x)$, and $s_{\Sigma^{+}}(x) =s_{\Sigma^{-}}(x)$. In the following equations, $q(x)$ represents valence quarks and $\bar{q}(x)$ represents sea quarks. The valence quark normalizations are: $\int u(x)\,dx = 2$ and $\int d(x)\,dx = 1$. Consider the Drell-Yan process for $\Sigma N$. Let $\sigma(\Sigma N)$ represent the cross-section for inclusive dilepton production \begin{equation} \sigma(\Sigma N) \equiv s\frac{d^{2}\sigma (\Sigma N \rightarrow l^{+}l^{-}X)}{d\sqrt{\tau}dy}=\frac{8\pi\alpha^2}{9\sqrt{\tau}} K(x_{\Sigma},x_{N})\sum_{i}e_{i}^{2}\{q_{i}(x_{\Sigma})\bar{q}_{i}(x_{N}) +[\Sigma \leftrightarrow N] \label{ddcross} \end{equation} with $M$ the mass of the dilepton pair and $\sqrt{\tau}=M/\sqrt{s}$. The factor $K(x_{\Sigma},x_{N})$ accounts for higher-order QCD corrections. If the c.m. rapidity $y \approx 0$, then $x_{\Sigma}\approx x_{N}\approx x$, and \begin{equation} \sigma (\Sigma^{+}p)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}K(x) \{ \frac{4}{9}[u(x)\bar{u}_{\Sigma}(x)+u_{\Sigma}(x)\bar{u}(x)]+ \frac{1}{9}[d(x)\bar{d}_{\Sigma}(x)+s_{\Sigma}(x)\bar{s}(x)]\}. \label{cross1} \end{equation} Then by charge symmetry \begin{equation} \sigma (\Sigma^{-}n)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}K(x) \{ \frac{1}{9}[u(x)\bar{u}_{\Sigma}(x)+u_{\Sigma}(x)\bar{u}(x)+ s_{\Sigma}(x)\bar{s}(x)]+\frac{4}{9}d(x)\bar{d}_{\Sigma}(x)\}. \end{equation} We also find \begin{equation} \sigma (\Sigma^{+}n)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}K(x) \{\frac{4}{9}[d(x)\bar{u}_{\Sigma}(x)+u_{\Sigma}(x)\bar{d}(x)]+ \frac{1}{9}[u(x)\bar{d}_{\Sigma}(x)+s_{\Sigma}(x)\bar{s}(x)]\}, \end{equation} and again by charge symmetry \begin{equation} \sigma (\Sigma^{-}p)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}K(x) \{\frac{1}{9}[d(x)\bar{u}_{\Sigma}(x)+u_{\Sigma}(x)\bar{d}(x)+ s_{\Sigma}(x)\bar{s}(x)]+\frac{4}{9}u(x)\bar{d}_{\Sigma}(x)\}. \label{cross4} \end{equation} As we note below, if $K(x)$ is known, and all four cross sections are measured, $\bar{u}_{\Sigma}$, $\bar{d}_{\Sigma}$, ${u}_{\Sigma}$, and ${s}_{\Sigma}$ can be determined. The uncertainties in $K(x)$ can be factored out by taking ratios of cross sections; two independent ratios can be constructed. We first define a ratio $R^{\prime}(x)$ determined from the Drell-Yan cross-sections so as to eliminate all unknowns except for $\bar{r}_{\Sigma}(x)$ \begin{equation} R^{\prime}(x)\equiv \frac {[\sigma (\Sigma^{+}p)-\sigma (\Sigma^{-}n)]+\bar{r}(x)[\sigma (\Sigma^{-}p)-\sigma (\Sigma^{+}n]} {[\sigma (\Sigma^{+}p)-\sigma (\Sigma^{+}n)]+4[\sigma (\Sigma^{-}p)-\sigma (\Sigma^{-}n)]}, \end{equation} and use Eq. \ref{cross1} - Eq. \ref{cross4} to write $R^{\prime}(x)$ in terms of the ratios $\bar{r}_{\Sigma}(x)$, $r(x)$ and $\bar{r}(x)$: \begin{equation} R'(x)=\frac{\bar{r}_{\Sigma}(x)[r(x)-\bar{r}(x)]-[1-\bar{r}(x)r(x)]} {5[r(x)-1]}. \label{R'} \end{equation} Thus for $r(x)\approx 2$ and $\bar{r}(x)\approx 0.5$, $R{^\prime}(x)\approx 0.3 \,\bar{r}_{\Sigma}(x)$. If $K(x)$ is known, $\bar{d}_{\Sigma}(x)$ can be determined directly from the cross sections: \begin{equation} \bar{d}_{\Sigma}(x)=\frac{27\sqrt{\tau}}{40\pi\alpha^2 K(x)}\frac {[\sigma (\Sigma^{+}p)-\sigma (\Sigma^{+}n)]+4[\sigma (\Sigma^{-}p)- \sigma (\Sigma^{-}n)]}{[u(x)-d(x)]}, \end{equation} and $s_{\Sigma}(x)$ can be determined from the cross sections and $\bar{s}(x)$: \begin{equation} s_{\Sigma}(x)=\frac{27\sqrt{\tau}}{8\pi\alpha^2 K(x)}\frac {[\sigma (\Sigma^{+}n)-4\sigma (\Sigma^{-}p)]-r(x)[\sigma (\Sigma^{+}p)-4\sigma (\Sigma^{-}n)]}{\bar{s}(x)[r(x)-1]}. \end{equation} (Recall that, because of the higher mass of the strange quark, we expect $s_{\Sigma}(x)$ to peak at a larger $x$ than $d(x)$ --c.f., Eq. 11.) Quark models with a meson cloud predict the sea quark distributions $\bar{q}(x)$; they also predict that the difference $D\equiv x [\bar{d}(x)-\bar{u}(x)]$ peaks at $x\approx 0.1$ \cite{Eichten92,Kumano91a,Kumano91b,Hwang91,Szczurek93,Szczurek94,Szczurek96}. On the basis of meson cloud models\footnote{We are undertaking a calculation of the $\Sigma^{\pm}$ sea quark distributions.}, the distributions of sea quarks in the $\Sigma^{\pm}$ may differ somewhat from those in the nucleon due to the presence of kaons; this may shift the maximum of $D$ to somewhat smaller values of $x$. Nevertheless, the region $0\leq x \leq 0.2$ should be a good one in which to determine $\bar{r}_{\Sigma}$. We believe that the measurement of $R^{\prime}$ should be possible to within $\approx 20\%$ and this is sufficient to establish the preponderance of $\bar{d}$ over $\bar{u}$ in the $\Sigma^{+}$, as predicted by the octet and meson cloud models. From Eq. \ref{R'}, an error, $e$, in the measurement of $R^{\prime}$ leads to an error of approximately $3e$ in $\bar{r}_{\Sigma}$. If $\bar{r}_{\Sigma}$ were found to be $\leq 0.5$, together with the known value of $\bar{r}\approx 0.51$, this measurement would help to reinforce the necessity to include pseudoscalar mesons in quark models of baryons. To measure valence quark asymmetries we consider the Drell-Yan process for $\Sigma^+$ and $\Sigma^-$ on isoscalar targets -- with cross sections $\sigma_+$ and $\sigma_-$, respectively. We fix $x_{\Sigma}$ to be above 0.3 so that valence quarks in the hyperons dominate. Then from Eq. \ref{ddcross}- Eq. \ref{cross4}, with $x\equiv x_{\Sigma}$ and $x'\equiv x_{N}$, \begin{equation} \sigma_{+}\equiv \sigma (\Sigma^{+}A)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}\frac{A}{2} K(x,x') \{ \frac{4}{9}u_{\Sigma}(x)[\bar{u}(x')+\bar{d}(x')]+\frac{2}{9} [s_{\Sigma}(x)\bar{s}(x')]\}, \end{equation} and \begin{equation} \sigma_{-}\equiv \sigma (\Sigma^{-}A)\approx \frac{8\pi\alpha^2}{9\sqrt{\tau}}\frac{A}{2} K(x,x') \{ \frac{1}{9}u_{\Sigma}(x)[\bar{u}(x')+\bar{d}(x')] +\frac{2}{9}[s_{\Sigma}(x)\bar{s}(x')]\}. \end{equation} We approximate $K(x,x')$ by an average $\bar{K}$, and integrate over $x^{\prime}$ in the nucleons, so that \begin{equation} \int dx^{\prime}\,\sigma_{+}(x^{\prime},x)= \frac{8\pi\alpha^2}{9\sqrt{\tau}}\frac{A}{2} \bar{K} \{ \frac{4}{9}u_{\Sigma}(x)[\bar{u}+\bar{d}]+ \frac{2}{9}[s_{\Sigma}(x)\bar{s}]\}, \end{equation} and similarly for $\sigma_{-}$, with $\bar{q} = \int dx^{\prime}\,\bar{q}(x^{\prime})$. Then \begin{equation} R_{v}(x)\equiv \frac{\int dx^{\prime}\,\sigma_{-}(x^{\prime},x)} {\int dx^{\prime}\,\sigma_{+}(x^{\prime},x)} =\frac{u_{\Sigma}(x)(\bar{d}+\bar{u})+2s_{\Sigma}(x)\bar{s}} {4u_{\Sigma}(x)(\bar{d}+\bar{u})+2s_{\Sigma}(x)\bar{s}} =\frac{1+\kappa r_{\Sigma}}{4+\kappa r_{\Sigma}}. \label{exp} \end{equation} We again use the CTEQ \cite{CTEQ} value, $\kappa=0.5$, and evaluate $R_{v}$ for both the SU(3) prediction for $r_{\Sigma}$ (Eq. \ref{SU3}) and for our quark model (Eq. \ref{QM}). In Fig. 2 we plot the ratio $D$ \begin{equation} D(x) \equiv \frac{R_{v}{\rm (quark \: model)}}{R_{v}{\rm (SU(3))}}. \end{equation} We note that the predicted asymmetry exceeds the SU(3) prediction by about $10\%$ at $x=0.5$, increasing to $20\%$ at $x=0.75$. An accuracy of $\approx 5\%$ should be possible for these integrated cross sections. Thus these measurements will test the SU(3) violations in the valence quark distributions of $\Sigma^{\pm}$ predicted by quark models. In summary, there are substantial differences expected between the valence and sea parton distributions associated with several models of hyperon structure. We have seen that Drell-Yan experiments based on existing hyperon beams should be capable of testing these ideas. The substantial violations of SU(3) flavor symmetry in the valence distributions are probably the easiest to test as they require only an isoscalar target and a semi-integrated cross-section. However, the enormous interest in the underlying cause of the flavor asymmetry of the proton sea should also make the tests of sea quark distributions an important priority as well. This work has been supported in part by the U.S. Department of Energy, Contract \# DOE/ER/4027-6-N96, by the National Institute for Nuclear Theory and by the Australian Research Council. We wish to thank Joel Moss, Jen-chieh Peng and other participants in the program INT-96-1, ``Quark and Gluon Structure of Nucleons and Nuclei'' for helpful discussions, and Jen-chieh Peng for a constructive critique of this manuscript. XJ and AWT also thank the Institute for Nuclear Theory for its hospitality during the time that part of this work was done.

proofpile-arXiv_065-586

\section{Introduction} Most studies of supersymmetric phenomenology have focused on models in which supersymmetry is broken in a hidden sector at a scale of order $M_{\rm SUSY} \simeq 10^{10}$ GeV. In these models, supersymmetry breaking is communicated to the visible sector by gravitational interactions. Such models give rise to superpartner masses in the 100 -- 1000 GeV range, and to a gravitino with approximately the same mass. The couplings of the gravitino to matter are negligible, suppressed by $(E/M_{\rm SUSY})^2 \simeq 10^{-16}$. The now-standard ``minimal supergravity'' model is of this type, with the familiar unification-scale boundary conditions on $M_0,$ $M_{1/2},$ and $A_0$. Recently, there has been a resurgence of interest in models where supersymmetry is broken at a scale of order $M_{\rm SUSY}\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 10^5$ GeV \cite{DN} -- \cite{Riotto}. In these models, supersymmetry breaking is transmitted to the superpartners of the quarks, leptons, and gauge bosons (and to the Higgs bosons themselves) via the usual SU(3) $\times$ SU(2) $\times$ U(1) gauge interactions. Because gauge interactions are flavor diagonal, these models naturally suppress the flavor-changing neutral currents associated with the soft squark and slepton masses. Models with gauge-mediated symmetry breaking also have superpartner masses in the 100 -- 1000 GeV range. However, because of the low scale of supersymmetry breaking, the gravitino is essentially massless; its couplings to the superpartners are suppressed by $(E/M_{\rm SUSY})^2 \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 10^{-6}$. Typically, superparticles decay by cascading down to the next-to-lightest supersymmetric particle (NLSP), which in turn decays to its partner and the gravitino. Such decays give rise to characteristic experimental signatures, e.g. final states containing two photons and missing energy \cite{pheno,Martin et al}. In this paper we take a detailed look at the low-energy spectrum of the simplest models with gauge-mediated supersymmetry breaking. We begin by describing the models, elaborating on the issues associated with electroweak symmetry breaking. We use two-loop renormalization group equations and the full one-loop threshold corrections to determine the parameter space which gives rise to consistent radiative electroweak symmetry breaking. We find the Higgs boson and superpartner masses to the same level of accuracy. We comment on how our results are affected by higher-order messenger-sector corrections. We compare the spectra of the simplest gauge-mediated models with those from the minimal supergravity-inspired model. Finally, we identify the next-to-lightest supersymmetric particle and compute its lifetime and branching ratios across the allowed parameter space. \section{Electroweak symmetry breaking} In models of gauge-mediated supersymmetry breaking, the SU(3) $\times$ SU(2) $\times$ U(1) gauge interactions of ``messenger" fields communicate supersymmetry breaking from a hidden sector to the fields of the visible world. In the simplest models \cite{DN}, the messenger sector contains a set of vector-like fields, $M_i$ and $\overline{M}_i$, coupled to a standard-model singlet, $S$, through the superpotential interaction \begin{equation} W_{\rm messenger} = \lambda_i S M_i \overline{M}_i. \end{equation} The lowest and $F$-components of the singlet superfield $S$ acquire vevs through their interactions with the hidden fields. This breaks supersymmetry and $R$-symmetry. To maintain the near unification of the standard-model gauge couplings, we will take the fields $M_i$ and $\overline{M}_i$ to transform in complete SU(5) representations. In the same spirit, we will also require that the gauge couplings remain perturbative up to the unification scale. This implies that we can consider at most four $5 + \overline{5}$ pairs, or one $10 + \overline{10}$ and one $5 + \overline{5}$ pair. In what follows we will take the fields $M_i$ and $\overline{M}_i$ to lie in $(n_5,n_{10})$ $5 + \overline{5}$ and $10 + \overline{10}$ SU(5) representations. We will assume that these fields couple to the singlet $S$ through a single Yukawa coupling $\lambda$ at the unification scale. (See ref.~\cite{Faraggi,Martin} for a discussion of variations.) Below that scale, the SU(5) representations split apart, and the Yukawa couplings evolve according to their own renormalization group equations \cite{Carone}. We will ignore this splitting in most of what follows; we will remark on it briefly in sect.~3. The lowest and $F$ components of the superfield $S$ acquire vevs through interactions with the sector which dynamically breaks supersymmetry. These vevs induce masses and mixings for the messenger fields. The messenger fermions gain mass $M=\lambda \langle S \rangle$, while the messenger scalars obtain the mass matrix \begin{equation} \left(\begin{array}{cc} M^2 & \lambda \langle F_S\rangle\\ \lambda \langle F_S\rangle & M^2 \end{array}\right) . \end{equation} {}From this point on, we omit the brackets which denote the vevs. Following the philosophy of dynamical models, we will assume that the standard-model $\mu$-term and all soft supersymmetry-breaking terms arise dynamically. For the case at hand, the messenger fields transmit the supersymmetry breaking to the visible sector through loop diagrams which contain insertions of the $S$ superfield. Such diagrams induce weak-scale masses for the gauginos and scalars of the minimal supersymmetric standard model. However, they cannot give sizable values for the soft supersymmetry-breaking $A$ parameters. In what follows, we shall set all $A$-terms to zero at the messenger scale. The messenger fields induce the following gaugino \begin{equation} \tilde M_i(M) = (n_5+3n_{10}) g\left({\Lambda\over M}\right) {\alpha_i(M)\over4\pi} \Lambda \label{bc1} \end{equation} and scalar \begin{equation} \tilde m^2(M) = 2 (n_5+3n_{10}) f\left({\Lambda\over M}\right) \sum_{i=1}^3 k_i C_i \biggl({\alpha_i(M)\over4\pi}\biggr)^2 \Lambda^2 \label{bc2} \end{equation} masses at the scale $M$, where $\Lambda=F_S/S$ and $k_i=1,1,3/5$ for SU(3), SU(2), and U(1), respectively. The $C_i$ are zero for gauge singlets, and 4/3, 3/4, and $Y^2$ for the fundamental representations of SU(3), SU(2), and U(1). Here $Y=Q-I_3$ denotes the usual hypercharge and we use the grand unification normalization for $\alpha_1$. Because a pair of $10 + \overline{10}$ fields contributes to the soft masses as if $n_5=3$, we will set $n_{10}=0$ and only consider changes in $n_5$. The messenger-scale threshold functions \cite{Martin,Pomarol} \begin{equation} g(x)={1+x\over x^2}\log(1+x) + (x\rightarrow-x)\ , \end{equation} \begin{equation} f(x)={1+x\over x^2}\biggl[\log(1+x) -2{\rm Li}_2 \left({x\over1+x}\right)+\ {1\over2}{\rm Li}_2 \left({2x\over1+x}\right)\biggr] + (x\rightarrow-x)\ , \end{equation} have the property that $g(x), f(x)\rightarrow1$ as $x\rightarrow0$, or $\Lambda \ll M.$ In this limit, the expressions (\ref{bc1}) and (\ref{bc2}) take the characteristic simple forms \cite{DN} that are often associated with gauge-mediated models. The region $\Lambda \rightarrow M$ corresponds to $x \rightarrow 1$, where $g(1) \simeq 1.4$ and $f(1) \simeq 0.7$. We must exclude the limit $M=\Lambda$ because it gives rise to a massless messenger scalar. For the purposes of this paper, we restrict ourselves to the region $x<0.97$, or $M/\Lambda>1.03$. This corresponds to an upper limit on the fine tuning of the messenger masses; it is obtained by requiring that the average scalar mass-squared be less than 30 times the light scalar mass-squared. Equations (\ref{bc1}) and (\ref{bc2}) serve as boundary conditions for the renormalization group equations at the messenger scale, $M$. They give rise to rather generic predictions for the soft supersymmetry-breaking gaugino and scalar masses. In contrast, the boundary conditions for $B(M)$ and $\mu(M)$ are more model-dependent.\footnote{The parameter $\mu$ is the standard supersymmetric Higgsino mass; $B$ is the dimension-two soft mass that is often denoted $B\mu$ in supergravity-inspired models.} New interactions are necessary to induce these terms because they violate a Peccei-Quinn symmetry in the effective action \cite{mu-term,DNS}. The new interactions can give rise to additional contributions to the scalar masses beyond those in (\ref{bc2}). In particular, they can give additional contributions to the Higgs masses, $m_{H_1}^2$ and $m_{H_2}^2$. In this paper, we will not commit ourselves to a particular model for $B(M)$ and $\mu(M)$. Instead, we will take a more phenomenological approach and treat them as free parameters. We will, however, assume that the soft Higgs masses $m_{H_1}^2$ and $m_{H_2}^2$ are given by eq.~(\ref{bc2}). We will also require that electroweak symmetry be radiatively broken. Our approach is as follows. We start at the messenger scale, $M$, and fix the boundary conditions (\ref{bc1}) and (\ref{bc2}). We use the two-loop renormalization group equations to run the soft masses down to the squark mass scale, $M_{\tilde q}\simeq \sqrt{n_5} \Lambda/90$, where we impose electroweak symmetry breaking and calculate the supersymmetric mass spectrum. At the squark scale, we consistently include all one-loop weak-scale threshold corrections. These corrections play an important role in determining $B(M_{\tilde q})$ and $\mu(M_{\tilde q})$. We then run $B$, $\mu$ and the gauge and Yukawa couplings back to the messenger scale. We repeat the procedure until we find a self-consistent solution for $B(M)$ and $\mu(M)$ in terms of the $Z$-boson pole mass, $M_Z$, and the ratio of Higgs vacuum expectation values, $\tan\beta$. (We take the top- and bottom-quark pole masses to be $m_t = 175$ GeV, $m_b = 4.9$ GeV, and the {\footnotesize{$\overline{{\rm MS}}$}} value of the strong coupling to be $\alpha_s(M_Z) = 0.118.$) We note that there are messenger-scale threshold corrections which we do not take into account, even though they are formally of the same order as the weak-scale threshold corrections. We choose to ignore these corrections because they are model dependent. We will, however, estimate their importance by determining the sensitivity of our results to small changes in the messenger-scale boundary conditions. With these assumptions, the parameter space of this minimal model is described by $\tan\beta$, the scale $\Lambda$, the messenger scale $M$, along with $n_5$, the effective number of $5+\overline{5}$ messenger fields, and the sign of $\mu$. We will first set $n_5=1$ and $M=2\Lambda$. This leaves a two-dimensional parameter space, $(\Lambda,\tan\beta)$, for each sign of $\mu$. \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig1.ps} \begin{center} \parbox{5.5in}{ \caption[]{\small Contours of $\mu(M)$ and $B(M)/\mu(M)$ in the $\Lambda, \tan\beta$ plane, with $n_5=1$ and $M/\Lambda=2$. The contours are labeled in GeV. \label{f.Bmu}}} \end{center} \end{figure} In Fig.~\ref{f.Bmu} we show our results for $B(M)/\mu(M)$ and $\mu(M)$ in the $(\Lambda, \tan\beta)$ plane. We see that consistent electroweak symmetry breaking requires $|\mu(M)|$ to be in the range from 150 GeV for small $\Lambda$ to over 1 TeV for large $\Lambda$. We also find that $B(M)/\mu(M)$ ranges from near zero, at the largest values of $\tan\beta$, to $ \raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 500$ GeV at small $\tan\beta$. We do not consider $\Lambda>300$ TeV because of fine tuning considerations. Various constraints exclude the border regions in Fig.~\ref{f.Bmu} (and the remaining contour plots). The region of small $\tan\beta$ is excluded because the top Yukawa coupling diverges below the unification scale. This is not necessarily fatal, but since we wish to preserve perturbative grand unification, we require $\lambda_t(M_{\rm GUT}) < 3.5$. The region of very large $\tan\beta$ is excluded because electroweak symmetry is not broken. Typically, we find $m_A^2<0$ in the large $\tan\beta$ region. The region at small $\Lambda$ is excluded because of the usual experimental bounds on the masses of supersymmetric particles, most notably $m_{\tilde\chi^+} > 65$ GeV. (We use the bounds listed in Ref.~\cite{PBMZ}.) These bounds hold when the LSP is the lightest neutralino and is stable. They also hold in gauge-mediated models when the NLSP is $\tilde\chi_1^0$ and it decays outside the detector. In sect.~4 we delineate the regions of the parameter space where this applies. If the NLSP decays inside the detector, the collider bounds are stronger because the decay modes (e.g. $\tilde\chi_1^0\tilde\chi_1^0\rightarrow \gamma\gamma +$ missing energy) have much smaller backgrounds \cite{pheno}. \begin{figure}[t] \epsfysize=2.5in \epsffile[-140 215 0 540]{fig2.ps} \begin{center} \parbox{5.5in}{ \caption[]{The parameters $\mu(M)$ and $B(M)/\mu(M)$ as a function of the scale, $Q$, where the renormalization group evolution is stopped and electroweak symmetry breaking is imposed. The dotted (dot-dashed) lines include the one-loop (two-loop) evolution equations, but no weak-scale thresholds. The dashed (solid) lines include the one-loop (two-loop) evolution equations, plus the full set of one-loop weak-scale thresholds. The dotted vertical lines on each plot denote the squark scales $M_{\tilde q}/2$, $M_{\tilde q}$ and $2M_{\tilde q}$. \label{f.scale}}} \end{center} \end{figure} In Fig.~\ref{f.scale} we plot the scale dependence of $B(M)/\mu(M)$ and $\mu(M)$ to illustrate the importance of the weak-scale threshold corrections. We see that our results, which consistently incorporate the weak-scale thresholds, have significantly less scale dependence than they would if the corrections were ignored. Furthermore, we see that the threshold corrections are especially important at large $\tan\beta$, where $B(M)$ is small. The largest threshold corrections arise from squark loops, so the threshold corrections are smallest in the vicinity of the squark scale, $M_{\tilde q} = \sqrt{n_5} \Lambda/90$. Indeed, in Fig.~\ref{f.scale} we see that the tree-level result is nearly equal to the full result at the scale $M_{\tilde q}/2$. If we vary the scale over a reasonable range, e.g. $M_{\tilde q}/2$ to $2M_{\tilde q}$, we find that the weak-scale threshold corrections (between $5$\% and $15$\%) are generally larger than the corrections from the two-loop evolution (about $2\%$), as might be expected because of the relatively small amount of running. The region $B(M) \simeq 0$ is of considerable phenomenological interest \cite{DNS,BKW}. For instance, models where $B(M) \simeq 0$ give rise to large $\tan\beta$ without fine tuning \cite{DNS}. Furthermore, since the $A$-terms are also small at the scale $M$, such models have a naturally small neutron electric dipole moment \cite{DNS}. Indeed, only the small ${\cal O}(\alpha M_2/\pi)$ starting values of $A(M)$ and $B(M)/\mu(M)$ can contribute to the neutron EDM. The pieces of $A$ and $B$ which are generated by the renormalization group, $A_{RG}$ and $B_{RG}$, do not contribute because they are proportional to $M_2$ and $\mu M_2$, so the invariant CP-violating phases \cite{CP} arg($A_{RG}^*M_2$) and arg($B_{RG}^*\mu M_2$) vanish. The region of $B(M) \simeq 0$ is associated with the region of large $\tan\beta$. At tree level, we have \begin{equation} B = {m_{H_1}^2 - m_{H_2}^2 \over \tan\beta-\cot\beta} - {M_Z^2\over\tan\beta+\cot\beta} , \end{equation} which implies that $B$ decreases as $\tan\beta$ increases. From Fig.~\ref{f.Bmu} we find, for $n_5=1$ and $M=2\Lambda$, that $B(M)$ is small (but non-zero) along the excluded region at large $\tan\beta$. (Our results should be contrasted with those of ref.~\cite{BKW}, which do not include one-loop thresholds. The authors of ref.~\cite{BKW} find $B(M)=0$ for $\tan\beta \simeq 20$ and $\mu < 0$.) Until now we have restricted our attention to the simple case where $M/\Lambda =2.$ In principle, $M/\Lambda$ can be much larger. A large hierarchy $M\gg\Lambda$ can arise from a small Yukawa coupling or from loop factors \cite{Hotta}. An upper bound on this splitting can be obtained from the cosmological constraint that the gravitino relic density not overclose the universe. In the usual cosmological scenario, this translates into an upper bound on the gravitino mass of about 1 keV \cite{grav bound}. (Note, however, that larger masses can be accommodated if the gravitino relic density is suppressed by a brief inflationary epoch.) If we assume that the singlet $F$-term is of the same order of magnitude as the largest $F$-term in the theory, we have \begin{equation} m_{\tilde G} \simeq {F_S \over M_P} = \lambda^{-1} \left({M\over\Lambda}\right){\Lambda^2\over M_P}, \end{equation} Setting $\Lambda=30$ TeV and $\lambda=1$, we find that $m_{\tilde G} \simeq 1$ keV if $M/\Lambda\simeq10^4$. Hence we will not consider values of $M/\Lambda$ larger than $10^4$. Note that for $\lambda < 1$ the upper bound on $M/\Lambda$ is correspondingly reduced. \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig3.ps} \begin{center} \parbox{5.5in}{ \caption[]{The same as Fig.~\ref{f.Bmu}, with $M/\Lambda = 10^4$ and $n_5 = 1$. The magnitude of $\mu(M)$ is increased relative to that of Fig.~1. \label{f.MoL}}} \end{center} \end{figure} As we increase $M$ for a fixed $\Lambda$, there is more renormalization group evolution, so there are larger splittings between the soft Higgs masses. These, in turn, give somewhat larger values of $\mu(M)$. We show this in Figs.~\ref{f.MoL}(a-b), where we plot contours of $\mu(M)$ in the $(\Lambda, \tan\beta)$ plane, for $n_5=1$ and $M/\Lambda =10^4$. We see that the values of $\mu(M)$ are slightly larger than those in Fig.~\ref{f.Bmu}. The situation for $B(M)$ is more subtle because the region $B(M) \simeq 0$ is sensitive to additive radiative corrections. In Figs.~\ref{f.MoL}(c-d) we show our results for $M/\Lambda=10^4$. We find that $B(M)=0$ occurs for $\mu<0$ with $\tan\beta \simeq 20$ to 40, depending on $\Lambda$. For smaller $M/\Lambda$, the $B(M)=0$ contour in Fig.~\ref{f.MoL}(d) moves to the right, to larger values of $\tan\beta$. \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig4.ps} \begin{center} \parbox{5.5in}{ \caption[]{Figures (a) and (b) show contours of the light CP-even Higgs-boson mass, $m_h$, in GeV, while Figs.~(c) and (d) show the CP-odd Higgs-boson mass, $m_A$, in GeV. We set $n_5 = 1$, and the solid (dashed) lines correspond to $M/\Lambda = 2 (10^4)$. \label{f.higgs}}} \end{center} \end{figure} In Fig.~\ref{f.higgs} we plot the pole mass of the lightest Higgs boson, $m_h$, and of the CP-odd Higgs boson, $m_A$, in the $(\Lambda, \tan\beta)$ plane, for $M/\Lambda = 2$ and $10^4$. We see that $m_h \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 130$ GeV (see, however, Ref.~\cite{Riotto}), and that $m_A$ is nearly always larger than 200 GeV. For such large values of $m_A$, all three heavy Higgs bosons are nearly degenerate in mass. \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig5.ps} \begin{center} \parbox{5.5in}{ \caption[]{The same as Fig.~\ref{f.Bmu} for varying $n_5$. The dashed, solid and dot-dashed lines correspond to $n_5 = 2, 3, 4$, respectively. The values of $\mu(M)$ and $B(M)/\mu(M)$ scale like $\sqrt n_5$ for a fixed value of $\Lambda$, except for $B(M) \simeq 0$, where radiative corrections are important. \label{f.n5}}} \end{center} \end{figure} The above results depend sensitively on $n_5$, the number of $5 + \overline{5}$ pairs. In Fig.~\ref{f.n5} we show, for $M=2\Lambda$, contour plots with $n_5 = 2, 3$ and $4$. The change in the parameter $\mu(M)$ is dominated by the change in the scalar masses. Indeed, we find $\mu(M)/\Lambda \propto \sqrt{n_5}$, as expected from eq.~(\ref{bc2}). The ratio $B(M)/\mu(M)$ obeys the same scaling at small $\tan\beta$. At large $\tan \beta$, however, the change in $B(M)$ is more complicated. This is the region where $B(M)$ is small, so its value depends sensitively on radiative corrections. For $\mu > 0$, there is a line in parameter space where $B(M) = 0$ at large $\Lambda$. For $\mu < 0$, however, $B(M)$ is never zero. We find $B(M)/\mu(M) > 10$ GeV for $n_5 = 4$. The results presented here have varying degrees of sensitivity to the input parameters. We illustrate this sensitivity by computing the changes in $B(M)/\mu(M)$ and $\mu(M)$. We first vary the quark masses, $m_b=4.9 \pm 0.5$ GeV and $m_t=175 \pm 5$ GeV, and find a shift $\Delta|B/\mu| \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 10$ GeV and $|\Delta\mu/\mu| \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 7\%$ in the region $B(M) \simeq 0$. To estimate the sensitivity to messenger-scale threshold corrections, we randomly vary the soft masses at the messenger scale by $5\%$. In the region $B(M) \simeq 0$, we find shifts in $B(M)/\mu(M)$ of up to $10$ GeV, and changes in $\mu$ of order $5$\%. \section{The spectrum} In this section we find the masses of the supersymmetric particles that arise in the gauge-mediated scheme. We examine how the spectrum depends on the parameter space, and comment on the sensitivity to low- and high-energy threshold corrections. As above, we follow a self-consistent procedure. We start with the masses (\ref{bc1}) and (\ref{bc2}) at the messenger scale, $M$. We then use the two-loop renormalization group equations to run these masses to the squark scale, $M_{\tilde q}=\sqrt{n_5} \Lambda/90$. At that scale we apply the one-loop threshold corrections and impose electroweak symmetry breaking. We then calculate the superpartner masses, and run the gauge couplings back to the messenger scale. We iterate this procedure to determine the consistent one-loop superpartner pole masses. The weak-scale threshold corrections for all the superpartner masses are contained in ref.~\cite{PBMZ}. \begin{figure}[t] \epsfysize=2.4in \epsffile[-15 320 125 550]{fig6.ps} \begin{center} \parbox{5.5in}{ \caption[]{The ratio of various superpartner masses to $m_{\tilde\chi_1^0}$, versus $m_{\tilde\chi_1^0}$, for $M=2\Lambda$ and $n_5=1$. The different lines for each particle correspond to the cases of small/large $\tan\beta$ and positive/negative $\mu$. Except for $m_A$ and $m_{\tau_1}$, the large $\tan\beta$ curves are essentially independent of the sign of $\mu$. \label{f.spec}}} \end{center} \end{figure} Figure \ref{f.spec} shows the spectrum in the canonical case where the messenger scale $M$ is equal to $2 \Lambda$. We plot the various masses against the mass of the lightest neutralino, $\tilde\chi_1^0$. The first point to note is that, for $m_{\tilde\chi_1^0} \raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 100$ GeV, most of the curves are rather flat. This simply reflects the fact that most of the masses, including $m_{\tilde\chi_1^0}$, scale with $\Lambda$. The only exceptions are the light Higgs, $h$, whose mass is determined by radiative corrections (the correction to its mass-squared grows like $\ln\Lambda$), and the light tau slepton, whose mass is significantly affected by left-right mixing. \begin{figure}[t] \epsfysize=1.8in \epsffile[-10 378 0 548]{fig7.ps} \begin{center} \parbox{5.5in}{ \caption[]{(a) The light tau slepton mass versus $\tan\beta$, for $n_5=1$, $\Lambda=100$ TeV, $M=2\Lambda$, and $\mu<0$. The one-loop result (solid) and tree-level results for $Q=M_{\tilde q}/2,$ $M_{\tilde q},$ and $2M_{\tilde q}$ are shown. The tree-level curve for $Q=M_{\tilde q}/2$ is closest to the full result. (b) The scale dependence of the tau slepton mass for the same parameters, with $\tan\beta=50$. The full result (solid) is contrasted with the tree-level result (dot-dashed). The corresponding curves obtained using one-loop renormalization group evolution are shown (dashed, dotted). \label{f.stau}}} \end{center} \end{figure} Because of the large mixing in the tau slepton mass matrix, the light tau slepton mass is especially sensitive to radiative corrections. The one-loop self-energy correction is typically less than ${\cal O} (3\%)$ (see Ref.~\cite{PBMZ}). However, for large $\tan\beta$, the large threshold corrections to $\mu$ (see Fig.~\ref{f.scale}) induce a significant shift in the light tau slepton mass. In Fig.~\ref{f.stau}(a) we plot the light tau slepton mass versus $\tan\beta$. We show the one-loop and tree-level results, for various choices of the renormalization scale, $Q$. At large $\tan\beta$, the corrections can be of order 30\%. In Fig.~\ref{f.stau}(b) we show the scale dependence of the one-loop and tree-level tau slepton masses, as well as the effect of the two-loop renormalization group evolution. The weak-scale threshold corrections significantly reduce the scale dependence of the pole mass. As expected, the two-loop renormalization group effects are small because of the small amount of running. The tau slepton mass matrix should be contrasted with those of the top and bottom squarks. These matrices can also have substantial mixing, but the mixing does not significantly change the mass eigenvalues. This implies that the masses of the light top and bottom squarks cannot be much less than the masses of the other squarks. Indeed, we find that the mass of the light top squark is typically 10 to 20\% less than that of the other squarks, although it can be as much as 35\% less at the smallest values of $\tan\beta$. Similarly, the light bottom squark mass is usually 0 to 10\% less than that of the other squarks, but it can be as much as 25\% less at the largest values of $\tan\beta$. \begin{figure}[t] \epsfysize=2in \epsffile[-180 240 -40 490]{fig8.ps} \begin{center} \parbox{5.5in}{ \caption[]{The ratio $m_{\tilde\chi_1^0}/\Lambda$ versus $\Lambda$, for the same choice of parameters as in Fig.~\ref{f.spec}. \label{f.mchi0}}} \end{center} \end{figure} In Fig.~\ref{f.spec} we see that for $m_{\tilde\chi_1^0} \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 100$ GeV, the curves leave their linear trajectories, especially for $\tan\beta =2$ with $\mu$ negative. This is because for small $m_{\tilde\chi_1^0}$, the mixing in the neutralino mass matrix become increasingly important, so the $\tilde\chi_1^0$ mass no longer scales with $\Lambda$. This is illustrated in Fig.~\ref{f.mchi0}, where we show the $\tilde\chi_1^0$ mass as a function of $\Lambda$. The previous spectra were all computed with $M = 2 \Lambda$. If $M$ is increased with respect to $\Lambda$, the masses change in two ways. First, the initial conditions are different because the masses at the scale $M$ depend on $\alpha_i(M)$. Second, more running is needed to reach the squark scale, $M_{\tilde q}$. \begin{figure}[t] \epsfysize=2.4in \epsffile[-20 320 120 550]{fig9.ps} \begin{center} \parbox{5.5in}{ \caption[]{Various sparticle masses versus the ratio $M/\Lambda$, for $\Lambda=50$ TeV, $\tan\beta=2$ and $\mu>0$. The solid (dashed) lines correspond to $n_5=1$ ($n_5=3$). \label{f.spec.MoL}}} \end{center} \end{figure} These effects almost cancel for the gaugino masses because the gaugino masses obey the same one-loop renormalization group equations as the gauge couplings. They do, however, change the scalar masses. For squarks, the two effects work in opposite directions. It turns out that for $n_5=1$ the change in the boundary conditions is more important, so the squark masses are reduced. For $n_5=3$ the two effects largely cancel. For sleptons, the effects work in the same direction, and give a slight increase in the slepton masses. The shifts in the scalar masses are illustrated in Fig.~\ref{f.spec.MoL}. The small changes indicate that it is safe to ignore any factor of two or three change in the messenger quarks scale with respect to the messenger lepton scale, such as might be induced by renormalization group evolution of the messenger Yukawa coupling $\lambda$. Figure \ref{f.spec.MoL} also illustrates the importance of the threshold functions $f(x)$ and $g(x)$. In the region $1.03 \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} M/\Lambda \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 2$ the threshold functions lead to significant increases in the gaugino masses, and to small decreases in the scalar masses, relative to the case when $M/\Lambda \raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 2$. The threshold corrections are so important that the light tau slepton becomes the NLSP for $M$ near $\Lambda$. In Fig.~\ref{f.spec.MoL} we also show the effect of increasing $n_5$ from 1 to 3. The gaugino masses scale like $n_5$, while the scalar masses go like $\sqrt n_5$. Because of this fact, for the case of a $10+ \overline{10}$ pair of messenger fields ($n_5=3$), the light tau slepton is the NLSP over most of the parameter space. We discuss this in more detail in the next section. Note that the squark masses are light enough that they will be produced in the next generation of colliders over most of the parameter space. As above, we can estimate the sensitivity of the supersymmetric spectrum to messenger-scale thresholds by varying the soft masses at the messenger scale by $5\%$. We find that most of the pole masses vary by about $5$\%, except for the light Higgs, which varies by $\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 1$\%, and the top and bottom squarks, which change by up to $ 10$\%. The light tau slepton mass varies by up to $15$\%, except at a few exceptional points, where the variation can be as large as 70\%. This large sensitivity in the tau slepton mass occurs at large $\tan\beta$, in the region where it is potentially the NLSP. We will conclude this section by contrasting the predictions of the gauge-mediated models with the predictions of the minimal supergravity model. At the unification scale, the inputs to the supergravity model are a universal scalar mass, $M_0$, a common gaugino mass, $M_{1/2}$, and a trilinear scalar coupling $A_0$. As in the gauge-mediated model, radiative breaking of electroweak symmetry breaking is imposed. \begin{figure}[t] \epsfysize=2in \epsffile[-30 370 110 570]{fig10.ps} \begin{center} \parbox{5.5in}{ \caption[]{(a) The ratio of the (first or second generation) squark mass to the gluino mass in the supergravity-inspired (dots) and gauge-mediated models (bands). The four bands in the gauge-mediated case correspond to $n_5=1,2,3,4$. The largest ratio (uppermost band) is for $n_5 = 1$. (b) The ratio of the left-handed to the right-handed selectron mass in the two models. The scatter plot shows the supergravity model prediction. The bands mark, in the gauge-mediated model with $n_5=4$, the regions where $m_{\tilde e_R}<M_Z$ (grey) and $m_{\tilde e_R}>M_Z$ (hatched). \label{f.compare}}} \end{center} \end{figure} In general, the two models predict qualitatively different spectra. The supergravity model includes separate scales for the gauginos and the scalars, so the scalars can be much heavier than the gauginos. The minimal gauge-mediated model has only one scale, so the ratios of the gaugino to the scalar masses are more or less fixed. We illustrate this in Fig.~\ref{f.compare}(a), where we show the ratio of the (first or second generation) squark mass to the gluino mass in the two models.\footnote{The supergravity scatter plots were generated as in Ref.~\cite{PBMZ}.} In the supergravity case, this ratio varies from about 0.9 to 5. For gauge-mediated models, the ratio depends on $n_5$. For $n_5=1$, the squarks are 20 to 40\% heavier than the gluinos, while for $n_5=4$ the squarks are about 5 to 10\% lighter. (These numbers and the bands on the figure ignore the region $M \lesssim2 \Lambda$, where the ratio $m_{\tilde q}/m_{\tilde g}$ falls by 20\%.) Note that gauge-mediated models with $n_5=4$ predict the same ratio of squark to gluino mass as the supergravity models with $M_0 \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} M_{1/2}$. In the region where the two models predict the same gluino and squark masses, they do not typically predict the same slepton masses. We illustrate this in Fig.~\ref{f.compare}(b), where we show a supergravity scatter plot of the ratio of the left-handed to the right-handed selectron mass, versus $M_0/M_{1/2}$. On the scatter plot we superimpose the prediction of the gauge-mediated model, for $n_5=4$, divided up into regions of small and large right-handed selectron mass ($m_{\tilde e_R}<M_Z$ and $m_{\tilde e_R}>M_Z$). In the small $m_{\tilde e_R}$ region, the $D$-term contributions to the slepton masses are important. The $D$-term contributions can make the selectron ratios coincide with those from the supergravity model in the region $M_0 \ll M_{1/2}$ if $\tan\beta$ is less than 3 in supergravity model and larger than 5 in the gauge-mediated model. \begin{table}[t] \begin{center} \begin{tabular}{ccc|cccc} \multicolumn{3}{c|}{$\Delta m / m < 2 \%$} & \multicolumn{4}{ c }{$\Delta m / m > 2 \%$} \\ \hline & GMSB & SUGRA & & GMSB & SUGRA & \% diff. \\ \hline $m_{\tilde g}$ & 361 & 366 & $m_h$ & 104 & 77 & 26 \\ $m_{\tilde\chi_1^+}$ & 83 & 82 & $m_{\tilde\chi_2^+}$ & 212 & 303 & $-43$ \\ $m_{\tilde\chi_2^0}$ & 87 & 87 & $m_{\tilde \chi_1^0}$ & 49 & 45 & 9 \\ $m_{\tilde e_L}$ & 107 & 106 & $m_{\tilde\chi_3^0}$ & 175 & 276 & $-58$ \\ $m_{\tilde e_R}$ & 64 & 64 & $m_{\tilde\chi_4^0}$ & 210 & 307 & $-46$ \\ $m_{\tilde u_L}$ & 336 & 337 & $m_{\tilde\nu_e}$ & 72 & 85 & $-19$ \\ $m_{\tilde u_R}$ & 328 & 329 & $m_{\tilde\nu_\tau}$ & 71 & 85 & $- 19$ \\ $m_{\tilde d_L}$ & 345 & 343 & $m_{\tilde\tau_L}$ & 115 & 105 & 8 \\ $m_{\tilde d_R}$ & 330 & 329 & $m_{\tilde\tau_R}$ & 46 & 63 & $-37$ \\ $m_{\tilde t_L}$ & 382 & 382 & $m_{\tilde t_R}$ & 281 & 235 & 16 \\ $m_{\tilde b_L}$ & 309 & 304 & $m_{\tilde b_R}$ & 330 & 322 & 3 \\ && & $m_A$ & 170 & 318 & $-87$ \\ \end{tabular} \parbox{5.5in}{ \caption[]{Comparison of the spectra of a gauge-mediated (GMSB) and supergravity-inspired (SUGRA) model. The masses are in units of GeV.}} \end{center} \end{table} If we go even further, we find that the two models can predict the same gluino, squark, charged slepton and light chargino masses. We give an example in Table~1. This table lists the spectra for the supergravity model parameters $\tan\beta=2.1$, $M_0=5$ GeV, $M_{1/2}=145$ GeV, $A_0=686$ GeV, and $\mu<0$, and for the gauge-mediated model parameters $\tan\beta=14$, $\Lambda=11$ TeV, $M/\Lambda=3716$, $n_5=4$ and $\mu<0$. The overlap can occur only for small selectron and $\tilde\chi_1^0$ masses, where the slepton $D$-terms and the gaugino/Higgsino mixing come into play. Table 1 also illustrates that the models differ in their predictions for other observables. If we scan over the parameter spaces of the two models and match the gluino masses to 8\%, and the light chargino and (first and second generation) squark and selectron masses to 2\%, we find that the two models predict different values of $\tan\beta$, as well as significantly different masses for the neutralinos, the sneutrinos, the heavy chargino and the light Higgs boson. They also predict different values for the Higgs parameters $\mu$ and $m_A$, and, typically, different masses for the third-generation squarks and sleptons. \begin{figure}[t] \epsfysize=2in \epsffile[0 358 140 548]{fig11.ps} \begin{center} \parbox{5.5in}{ \caption[]{Scatter plots of observables in the regions of parameter space where the gauge-mediated and supergravity models predict the same gluino, squark, slepton and light chargino masses (see text). Figure (a) shows a scatter plot of $m_h$ versus $\tan\beta$ in the gauge-mediated (GMSB) and supergravity models (SUGRA). Figure (b) shows $m_{\tilde\chi_2^+}$ versus $m_A$. \label{f.comp}}} \end{center} \end{figure} We illustrate these differences in Fig.~\ref{f.comp}, where we show scatter plots of the values of $\tan\beta, m_h, m_A,$ and $m_{\tilde\chi_2^+}$ in the two models. We see that the models are clearly distinguished. Note that since both models have essentially one scale, there is a strong correlation between the heavy Higgs and Higgsino masses. (The two scales are $\Lambda$ and $M_{1/2}$, since $M_0\simeq0$.) We conclude that it is possible, but not probable, that measurements of the gluino, squark, slepton and light chargino masses may not be enough to discriminate the gauge-mediated from the supergravity model. However, additional measurements would easily rule out one (or both) of the models. \section{Properties of the NLSP} In the preceding section, we saw that the NLSP is either the light tau slepton, $\tilde\tau_1$, or the lightest neutralino, $\tilde\chi_1^0$. Over most of the parameter space, the number of $5+\overline{5}$ pairs determines the NLSP. For $n_5 = 1$, the $\tilde\chi_1^0$ tends to be the NLSP. For larger $n_5$, the balance tips towards the $\tilde\tau_1$ because as $n_5$ increases, the scalar masses increase less than their gaugino counterparts. In Fig.~\ref{f.Xtau} we plot contours of $m_{\tilde\chi_1^0} = m_{\tilde\tau_1}$ in the $(m_{\tilde\chi_1^0},\tan\beta)$ plane, for each value of $n_5$ between 1 and 4. Each subfigure contains four contours, two for each sign of $\mu$, and two for $M/\Lambda=2$ and $10^4$. We see that for $n_5=1$, the neutralino is the NLSP, except for small regions at large $\tan\beta \raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 25$ and small $m_{\tilde\chi_1^0}$. For $n_5=2$ and $M/\Lambda = 10^4$, $\chi_1^0$ is the NLSP for $\tan\beta \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 20-30$. For $n_5=2$ and $M/\Lambda = 2$, the light tau slepton is the NLSP over most of the parameter space, as it is for $n_5=3$ or 4 and arbitrary $M/\Lambda$. For $n_5 > 2$, the neutralino is the NLSP at very small $m_{\tilde\chi_1^0} \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 100$ GeV. These results are subject to significant uncertainties from the unknown messenger-scale thresholds. In particular, the line where $m_{\tilde\chi_1^0} = m_{\tilde\tau_1}$ is sensitive to potentially large one-loop corrections to the mass of the light tau slepton. We find that 5\% variations in the messenger-scale boundary conditions give rise to a $m_{\tilde\chi_1^0} - m_{\tilde\tau_1}$ mass difference of up to 30 GeV in the region where the difference is less than 100 GeV. The collider phenomenology in gauge-mediated models depends crucially on the nature of the NLSP, especially when it decays inside the detector. The lifetime of the NLSP depends on its mass, $m$, and on the ratio $M/\Lambda$, as follows, \begin{equation} \tau \propto {F_S^2\over m^5} \propto {1\over m}\left({M\over\lambda\Lambda}\right)^2 . \label{ctau} \end{equation} For the case at hand, where the NLSP is either the $B$-ino or the $\tilde\tau_1$, we can put in the appropriate factors of $\pi$ and $\alpha_1$ and write the lifetime in units of meters, \begin{equation} c\tau \simeq \left({100~{\rm GeV}\over m}\right) \left({M\over\lambda\Lambda}\right)^2 \times 10^{-5}~\rm{meters} . \label{length} \end{equation} For $\beta\gamma = \sqrt{E^2/m^2 - 1} \simeq 1$, this is the approximate decay length. We see that for $\beta\gamma \simeq 1$, $\lambda \simeq 1$ and $M/\Lambda \raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}} 100$, the NLSP will decay inside the detector. However, for sufficiently small $\lambda$ it will decay outside the detector. In Fig.~\ref{f.Xlife} we illustrate the neutralino lifetime in different regions of parameter space. In Fig.~\ref{f.Xlife}(a) we plot the lifetime versus the neutralino mass, for $M/\Lambda=2$ and 100, with $n_5=1$, $\lambda=1$ and $\tan\beta=2$. In Fig.~\ref{f.Xlife}(b) we show a scatter plot of the neutralino lifetime versus $M/\Lambda$. The lifetime increases by eight orders of magnitude as we increase $M/\Lambda$ from 1 to $10^4$. In this and the next scatter plot, we vary the parameters $n_5$ from 1 to 4, $\Lambda$ from 16 TeV$/n_5$ to 300 TeV$/\sqrt{n_5}$, $M/\Lambda$ from 1.03 to $10^4$, and $\tan\beta$ from 1.2 to 70. The latter three variables are sampled on logarithmic measures, subject to the phenomenological constraints discussed in sect. 2. If the tau slepton is the NLSP, it will decay to a $\tau$ lepton and a gravitino with a branching fraction that is essentially 100\%. The lightest neutralino has many more decay modes because it has Higgsino and gaugino components. The $\tilde\chi_1^0$ can decay to either a Higgs or a gauge boson, plus a gravitino, $\tilde G$. In Fig.~\ref{f.ki} we show the Higgsino and photino components of the $\tilde\chi_1^0$. In Figs.~\ref{f.ki}(a) and \ref{f.ki}(b) we set $n_5=1$ and see that the NLSP is approximately 60 to 90\% (40\% to 70\%) photino for $\mu > 0$ $(\mu < 0)$. The Higgsino component is quite small. The Higgsino component can be larger if $n_5$ is larger, because $|\mu|$ is reduced relative to the gaugino masses. In Fig.~\ref{f.ki}(c) we see that for $n_5=4$ and $\mu>0$, the Higgsino component of the NLSP can be large. In fact, it can be as large as 50\%. The $\tilde \chi_i^0$ partial widths can be readily computed. We express them in units of \begin{equation} {\cal A} = {m_{\tilde\chi^0_i}^5\over96\pi M_{\tilde G}^2M_P^2} = {m_{\tilde\chi^0_i}^5\over32\pi F_S^2} . \end{equation} We first write the well-known decay rate to a photon and the gravitino, \begin{equation} {\cal A}^{-1}\Gamma(\tilde\chi_i^0\rightarrow\tilde G\gamma) = 2 \kappa_{i\gamma}. \label{phot} \end{equation} For the remaining decay modes, we give the three body decay formula to account for the decay to standard model fermions via real or virtual boson exchange. If we sum over final-state standard-model fermions, we find \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig12.ps} \begin{center} \parbox{5.5in}{ \caption[]{Contours marking the boundary where the mass of the light tau slepton is equal to the mass of the lightest neutralino, for $n_5 = 1, 2, 3$ and $4$. The solid (dashed) line corresponds to $\mu > 0$, $M/\Lambda = 2$ $(10^4)$. The dot-dashed (dotted) line corresponds to $\mu < 0$, $M/\Lambda = 2$ $(10^4)$. The excluded region is plotted for $\mu > 0$ and $M/\Lambda = 2$. The label $\tilde\chi_1^0$ ($\tilde\tau_1$) marks the region where the neutralino (tau slepton) is the NLSP. \label{f.Xtau}}} \end{center} \end{figure} \begin{eqnarray} {\cal A}^{-1} \Gamma(\tilde\chi_i^0\rightarrow\tilde Gf^+f^-) &=& 2 \kappa_{iZ_T} I_1(Z) + \kappa_{iZ_L} I_0(Z) \\[2mm] &+&80s_Wc_W\biggl[{3-8s_W^2 \over63-120s_W^2+160s_W^4}\biggr]\kappa_{iZ\gamma} \left(I_1(Z)-I_0(Z)\right)\nonumber\\[2mm] &+& \kappa_{i\gamma}{2\alpha\over3\pi}\sum_f N_c^f e_f^2 \left(-{25\over12} -\ln {s^f_{\rm min}\over m_{\tilde\chi_i^0}^2}\right)\nonumber\\ &+& \sum_{\varphi=h,H,A} {\rm BR}(\varphi\rightarrow f^+f^-) \kappa_{i\varphi} I_1(\varphi)\nonumber\\ &-& 2 \Biggl[{\rm BR}(h\rightarrow f^+f^-){\rm BR}(H\rightarrow f^+f^-) \Biggr]^{1\over2} \kappa_{iHh} I_{10}(h,H) ,\nonumber \end{eqnarray} where the sum $\sum_f$ is over all fermions with mass less than $m_{\tilde\chi_i^0}/2$, and $N_c^f$ is the number of colors: 3 for quarks and 1 for leptons. In the virtual photon contribution, $s^f_{\rm min}$ is a detector dependent cut-off on the fermion pair invariant mass. Also, we define \begin{eqnarray} \kappa_{i\gamma} &=& |N_{i1}c_W+N_{i2}s_W|^2 ,\\ \kappa_{iZ_T} &=& |N_{i2}c_W-N_{i1}s_W|^2 ,\nonumber\\ \kappa_{iZ\gamma} &=& c_Ws_W\left(|N_{i2}|^2-|N_{i1}|^2\right) + (c_W^2-s_W^2) {\cal R}e \left(N_{i1}^*N_{i2}\right) ,\nonumber\\ \kappa_{iZ_L} &=& |N_{i3}c_\beta-N_{i4}s_\beta|^2 ,\nonumber\\ \kappa_{iA} &=& |N_{i4}c_\beta+N_{i3}s_\beta|^2 ,\nonumber\\ \kappa_{iH} &=& |N_{i3}c_\alpha+N_{i4}s_\alpha|^2 ,\nonumber\\ \kappa_{ih} &=& |N_{i4}c_\alpha-N_{i3}s_\alpha|^2 ,\nonumber\\ \kappa_{iHh} &=& s_\alpha c_\alpha\left(|N_{i4}|^2-|N_{i3}|^2\right)+ (c_\alpha^2-s_\alpha^2) {\cal R}e \left(N_{i3}^*N_{i4}\right) .\nonumber \end{eqnarray} In writing the $H$-$h$ interference contribution, we have neglected all Yukawa couplings except those of the the bottom and tau. The $\gamma$-$Z$ ($H$-$h$) interference terms go to zero in the limit $m_{\tilde\chi_i^0}\gg M_Z$ ($m_{\tilde\chi_i^0}\gg m_H$). Note that if $m_{\tilde\chi_i^0} > M_\varphi$ ($\varphi = h,H,A$), the decay rate ${\cal A}^{-1} \Gamma(\tilde\chi_i^0\rightarrow\tilde G\varphi) = \kappa_{i \varphi} I_1(\varphi)$. In these expressions, the integrals $I_n$ are given by \begin{equation} I_n(\varphi) = {\epsilon_\varphi\over\pi}\int_0^1 dx {(1-x)^4(x/R_\varphi)^n\over(x-R_\varphi)^2+\epsilon_\varphi^2} , \end{equation} with $R_\varphi=M_\varphi^2/m_{\tilde\chi_i^0}^2$ and $\epsilon_\varphi=\Gamma_\varphi M_\varphi/m_{\tilde\chi_i^0}^2$. These integrals reduce to $(1-R_\varphi)^4$ for small $\epsilon_\varphi$ (i.e. in the narrow width approximation). The integral $I_{10}(h,H)$ is \begin{equation} I_{10}(h,H) = {1\over\pi} \sqrt{{\epsilon_h\epsilon_H\over R_h R_H}}\int_0^1 dx {x (1-x)^4 \biggl[\left(x-R_H\right)\left(x-R_h\right) + \epsilon_h\epsilon_H\biggr]\over \left((x-R_h)^2+\epsilon_h^2\right) \left((x-R_H)^2+\epsilon_H^2\right)} . \end{equation} \begin{figure}[t] \epsfysize=2in \epsffile[-10 390 130 580]{fig13.ps} \begin{center} \parbox{5.5in}{ \caption[]{(a) The lifetime of the NLSP, for $\tan\beta=2$ and $n_5 =1$, with $\mu>0$ (solid line) or $\mu<0$ (dot-dashed). For fixed $m_{\chi_1^0}$, the lifetime scales like $(M/\Lambda)^2$. (b) A scatter plot of the neutralino lifetime, restricted to cases where the neutralino is the NLSP. \label{f.Xlife}}} \end{center} \end{figure} Our results complete the formulae for the two-body decay rates of the $\chi_i^0$ given in ref.~\cite{Martin et al}. Our results include the contribution of the virtual photon and the contributions from $Z$-$\gamma$ and $H$-$h$ interference. Note that the formula for $\Gamma( \tilde \chi_i^0 \rightarrow \tilde G +$ Higgs boson) contains the function $I_1$, while the formula for $\Gamma( \tilde \chi_i^0 \rightarrow \tilde G +$ longitudinal Z boson) contains the function $I_0$. At first sight one might think that this violates the electroweak equivalence theorem, since the decay rate to the longitudinal $Z$ is not equal to the decay rate to the Goldstone boson. However, the equivalence principle does indeed hold in the applicable regime, since the two functions approach each other for $m_{\tilde\chi_i^0}\gg M_Z$. \begin{figure}[t] \epsfysize=2.5in \epsffile[-180 180 -40 545]{fig14.ps} \begin{center} \parbox{5.5in}{ \caption[]{Figures (a) and (b) [(c) and (d)] show the photino component of the lightest neutralino, $\kappa_{1\gamma}$ [the Higgsino component, $\kappa_{1h}$], in per cent, for $M/\Lambda = 2$ and $n_5 = 1$ [$n_5 = 4$]. \label{f.ki}}} \end{center} \end{figure} In Fig.~\ref{f.BR}(a) we show the branching fractions of the $\tilde\chi_1^0$ versus $m_{\tilde\chi_1^0}$, for $\tan\beta=2$, $n_5=1,$ and $M=2\Lambda$. (We plot the magnitude of the $Z$-$\gamma$ interference contribution, $|\Gamma_{Z\gamma}| /\Gamma_{\rm tot}$, and we take into account 5 quark flavors.) The decay to the photon dominates, as expected, since the $\tilde\chi_1^0$ has a large photino component. The branching fraction to fermions via an off-shell photon varies from about 3 to 4\% for $s^f_{\rm min}=1$ GeV. In the parameter space associated with the minimal models, the branching ratio to fermion pairs via heavy Higgs boson exchange is negligible. For the parameters corresponding to Fig.~\ref{f.BR}, the heavy Higgs exchange branching ratio is of order $10^{-12}$. The branching ratio associated with $H$-$h$ interference and the branching fraction associated with virtual $h$-boson exchange are about an order of magnitude larger. Because of the kinematical suppression, the branching fraction to the $Z$-boson rises slowly above threshold, to about 15\% (20\%) for $m_{\tilde\chi_1^0} = 2M_Z$ $(3M_Z)$. We illustrate this in Fig.~\ref{f.BR}(b), where we show a scatter plot of the branching fraction of the lightest neutralino to a gravitino and a fermion pair. We see that the branching ratio depends only on the neutralino mass. This would not be true if the part of parameter space in which the $\tilde\chi_1^0$ has a large Higgsino component ($n_5=4, \mu>0, $small $\Lambda$, see Fig.~\ref{f.ki}(c)) led to a sizable branching ratio to the Higgs boson. However, in this region of parameter space either the $\tilde\tau_1$ is the NLSP or the $\tilde\chi_1^0$ mass is below the $h$-boson threshold. Hence, the associated partial width is negligible. As discussed above, the mechanism that generates $B(M)$ and $\mu(M)$ might result in extra terms to the scalar masses beyond those in eq.~(\ref{bc2}). In this case, the NLSP can decay predominately into a Higgs boson. To study this possibility, we take $\mu$ and $B$ to be independent low-energy parameters, and focus our attention on the case of small $\mu$, so the lightest neutralino is predominantly Higgsino. In Fig.~\ref{f.Xhiggs} we plot the lifetime and branching fractions of the $\tilde\chi_1^0$ as a function of $m_{\tilde\chi_1^0}$, for various choices of parameters. Note that in the region $m_{\tilde\chi_1^0}\lesssim250$ GeV the lightest neutralino is largely Higgsino, and $m_{\tilde\chi_1^0} \simeq |\mu|$. The lifetime and branching fractions of the NLSP in the Higgsino region are shown in Fig.~\ref{f.Xhiggs}. In Fig.~\ref{f.Xhiggs}(a) we see that the lifetime varies by 6 or 7 orders of magnitude as $m_{\tilde\chi^0_1}$ varies over less than one order of magnitude. This can be contrasted with the case where the neutralino is dominantly $B$-ino, where the lifetime varies by less than one order of magnitude (see eq.~(\ref{ctau}) and Fig.~\ref{f.Xlife}). The branching fractions of the lightest neutralino are shown in Figs.~\ref{f.Xhiggs}(b-d). \begin{figure}[t] \epsfysize=2in \epsffile[-20 410 120 600]{fig15.ps} \begin{center} \parbox{5.5in}{ \caption[]{(a) The branching ratios of the $\tilde\chi^0_1$ in the case $M/\Lambda = 2$, $n_5 = 1$ and $\mu > 0$. The solid line near 1 is the branching ratio to a photon and the gravitino, and the dashed line is the branching ratio to fermions via a virtual photon. (b) A scatter plot of the neutralino branching fraction to fermion pairs, in the cases where the neutralino is the NLSP. \label{f.BR}}} \end{center} \end{figure} These results can be understood as follows. In the small $\mu$ region the NLSP is primarily Higgsino, so it prefers to decay to a Higgs boson. Below $h$ and $Z$ threshold, however, it is forced to decay through its suppressed photino component. This accounts for the long lifetime and the large photino branching fraction for small $m_{\tilde\chi^0_1}$. As the mass increases, the $h$ and $Z$ channels open, so the lifetime decreases and the branching fractions are determined by $\kappa_{1 Z_L}$ and $\kappa_{1h}$. For small $\tan\beta$, the decay to $h$ ($Z$) dominates for $\mu$ positive (negative). For large $\tan\beta$, $\kappa_{1 Z_L} \simeq \kappa_{1h}$ for each sign of $\mu$. In this case the branching fraction to $Z$ is larger because the Higgs is heavier, so $I_0(Z) > I_1(h)$. Finally, as $m_{\tilde\chi_1^0}$ increases still further, the decay to photons again dominates, because in this region, the NLSP is predominately photino. For $m_{\tilde\chi_1^0} \simeq 300$ GeV, the decay length approaches $10^{-5}$ meters. The neutralino lifetime in the Higgsino region can be readily scaled for other values of $n_5$ and $M/\Lambda$. As before, for fixed $m_{\tilde\chi_1^0}$, $c\tau$ scales like $(M/\Lambda)^2$. For fixed $M/\Lambda$ and fixed gaugino masses, it scales like $1/n_5^4$ because $\tau \propto F^2_S/\mu^5 \propto \Lambda^4/\mu^5$. These results should be contrasted with those of the minimal models, where we found that, above $h$ threshold, the branching ratio to the $h$ varies from $10^{-8}$ to $10^{-4}$. Therefore collider events with four $b$-jets and missing energy would imply a non-minimal Higgs sector in these models. \begin{figure}[t] \epsfysize=2.5in \epsffile[-140 220 0 555]{fig16.ps} \begin{center} \parbox{5.5in}{ \caption[]{The (a) lifetime and (b-d) branching fractions of the NLSP, for the case where the $\tilde\chi^0_1$ is primarily Higgsino, with $n_5 = 1$, $\lambda=1$ and $\Lambda = 200$ TeV. The $m_{\tilde\chi_1^0}$ values are obtained by varying $\mu$ from 50 to 350 GeV. The solid (dashed) lines are for $\tan\beta=2$, with $\mu$ positive (negative). The dot-dashed lines correspond to $\tan\beta=40$, $\mu>0$. (The curves for $\tan\beta=40$, $\mu<0$ are almost identical.) \label{f.Xhiggs}}} \end{center} \end{figure} \section{Conclusions} In this paper we examined the detailed low-energy spectrum of gauge-mediated supersymmetry breaking models. We used two-loop renormalization group equations for the gauge and Yukawa couplings, and for the soft supersymmetry breaking parameters. We imposed consistent one-loop radiative electroweak symmetry breaking under the assumption that the mechanism that generates the $\mu$ and $B$ parameters does not induce any extra contributions to the scalar masses. We examined the phenomenology in the case of an arbitrary number of $5+\overline{5}$ messenger fields, and in the case that the messenger scale $M$ is greater than $\Lambda$. We began by examining electroweak symmetry breaking. We initially considered the minimal case, $n_5=1$ and $M=2 \Lambda$. In the parameter space we considered, with $\Lambda < 300$ TeV, we found that $|\mu(M)|$ varies from 150 GeV to over 1 TeV. We also found that $|B(M)/\mu(M)|$ varies from near (but not equal to) zero to about 500 GeV. We found the phenomenologically interesting region $B(M) \simeq 0$ to be compatible with electroweak symmetry breaking in the region of large $\tan\beta$, for either sign of $\mu$. For larger $M/\Lambda$, we found $B(M) \simeq 0 $ requires $\mu < 0$. We examined the spectrum and illustrated how it depends on $n_5$ and $M/\Lambda$. We found that the spectrum is not qualitatively affected by an increase in $M/\Lambda$. For $n_5>1$, we found that the light tau slepton is the NLSP over most of the parameter space. It decays to a tau lepton and a gravitino. By varying the boundary conditions of the soft parameters at the messenger scale by $5$\%, we determined the sensitivity of the supersymmetric spectrum to higher-order messenger-sector corrections. We found that in the region $B(M) \simeq 0$, $B(M)$ varies by $10$ GeV. Furthermore, the spectrum varies by $5\%$, except for the light Higgs mass, which is essentially unchanged, and the third-generation squark and slepton masses. In the most extreme case, we found a $70\%$ variation in the light tau slepton mass, which implies a substantial uncertainty in the identification of the NLSP. We compared the predictions for the spectra in the gauge-mediated and supergravity-inspired models. In general, the two models predict qualitatively different spectra. We found that it is possible for the two models to give the same gluino, light chargino, and (first and second generation) squark and slepton masses. Such a match is only possible for very light masses, where $D$-terms and gaugino/Higgsino mixing is important. In this region, the models can be distinguished by their predictions for other observables, such as $\tan\beta$, as well as the sneutrino, neutralino, heavy chargino, and Higgs boson masses. For $n_5=1$, we found that the $\tilde\chi_1^0$ is usually the NLSP. The $\tilde\chi_1^0$ decays to the gravitino and either an (on- or off-shell) gauge or Higgs boson. We derived the neutralino decay rate and branching fractions. The lightest neutralino tends to be gaugino-like, and decays to a photon and a gravitino. However, in the region $m_{\tilde\chi_1^0} \raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}} 100$ GeV, the branching fraction to the $Z$ can be larger than 20\%. Leptons are easier to track than photons, so this mode has the potential to permit a precise neutralino lifetime measurement. We found that for $n_5=4$ the Higgsino component of the $\tilde\chi_1^0$ can be as large as 50\%. Nevertheless, over the entire parameter space the branching fraction to the light Higgs boson is less than $10^{-4}$. We also examined the branching fraction of the neutralino in the Higgsino region. In this case the branching fraction to the on-shell $h$-boson can reach over 80\% well above threshold. Since this Higgsino region can only occur for models with non-minimal Higgs sectors, the observation of 4 $b$-jets plus missing energy would be an important step towards understanding the origin of the $\mu$ term. Models with gauge-mediated supersymmetry breaking offer the appealing possibility that the origin of supersymmetry breaking is experimentally accessible. Once supersymmetry is discovered, detailed study of the supersymmetric particles, along the lines suggested here, might well prove to be the first step towards uncovering the mechanism of supersymmetry breaking. \section*{Acknowledgements} One of us (D.M.P.) thanks S. Thomas and J. Wells for useful conversations.

proofpile-arXiv_065-587

\section{INTRODUCTION} It was shown by ${\rm M\phi ller}^{1)}$ that a tetrad description of gravitational field allows a more satisfactory treatment of the energy-momentum complex than general relativity (GR). The Lagrangian formulation of the theory was given by Pellegrini and ${\rm Plebanski.}^{2)}$ Hayashi and ${\rm Nakano}^{3)}$ independently formulated the tetrad theory of gravitation as a gauge theory of the spacetime translation group. In the earlier attempts, admissible Lagrangians were limited by the assumption that the field equation has the Schwarzschild solution. ${\rm M\phi ller}^{4)}$ later suggested to abandon this assumption and to look for a wider class of Lagrangians. ${\rm Meyer}^{5)}$ formulated the tetrad theory as a special case of Poincar$\acute{e}$ gauge ${\rm theory.}^{6),7)}$ ${\rm S\acute{a} ez}^{8)}$ generalized the theory into a scalar-tetrad theory. Hayashi and ${\rm Shirafuji}^{9)}$ studied the geometrical and observational basis of the tetrad theory of gravitation. Geometrically the tetrad fields are identified with the parallel vector fields defined by the underlying absolute parallelism. Incidentally they gave the name, new general relativity (NGR), to the theory of gravitation based on absolute parallelism, since ${\rm Einstein}^{10)}$ was the first to introduce the notion of absolute parallelism in physics. They assumed the Lagrangian with three unknown parameters, denoted by $a_1$, $a_2$ and $a_3$. In order to reproduce the correct Newtonian limit, the first two parameters should satisfy a condition called the Newtonian approximation condition, which allows us to express these two parameters in terms of an unknown dimensionless parameter $\epsilon$. An exact, vacuum solution of the gravitational field equation was found for ${\it static}$, spherically symmetric case, where the parallel vector fields take a diagonal form. This solution describes the spacetime around a mass point located at the origin, and will be referred to as the "exact spherically symmetric solution" hereafter. Then comparison with solar-system experiments showed that the value of $\epsilon$ should be very small. By contrast only an upper bound has been estimated for the remaining parameter ${\it a}_3.^{9),11)}$ The singularity problem of the exact solution has been ${\rm studied.}^{12),13)}$ If $(a_1+a_2)$=0, then the theory reduces to the one studied by Hayashi and ${\rm Nakano}^{3)}$ and ${\rm M\phi ller.}^{4)}$ The equation of motion for a test particle was discussed based on the response equation of the energy-momentum tensor of matter ${\rm fields.}^{9)}$ In particular, when the intrinsic spin of the fundamental particles constituting the test particle can be ignored, the response equation reduces to the covariant conservation law of GR, and hence the world line of the test particle is a geodesics of the metric defined by the parallel vector fields. Accordingly, for spinless test particles the passive gravitational mass is equal to the inertial mass. In GR the equality between passive gravitational mass and inertial mass is predicted to be valid also for massive bodies like planets which contains appreciable fraction (about $5\times10^{-10}$ for the Earth) of gravitational self-energy. This equality means that gravity pulls on gravitational binding energy of a massive body just as it does on other forms of mass-energy, and is referred to as the strong equivalence ${\rm principle.}^{14)}$ ${\rm Nordtvedt}^{15)}$ developed the parametrized post-Newtonian (PPN) formalism and calculated the diviation from the unity of the ratio of gravitational mass to inertial mass in terms of the PPN parameters. Analysis of the lunar laser ranging (LLR) over past 24 years has confirmed that the Earth and the Moon accelerate equally to the sun within fractional difference less than $5\times10^{-13}$, which yields the Nordtvedt parameter $\eta=-0.0010 \pm 0.0010$.$^{16),17),18)}$ This parameter $\eta$ ($\equiv 0$ in GR) can be expressed as $\eta=4\beta-\gamma-3$, in fully conservative theory which possesses a full complement of the post-Newtonian conservation laws: energy, momentum, angular momentum and center-of-mass motion.$^{14)}$ Here $\beta$ and $\gamma$ are the so-called Eddington-Robertson parameters.$^{19),20)}$ Thus, if NGR is a fully conservative theory in the above sense, then the Nordtvedt parameter $\eta$ vanishes, since the Eddington-Robertson parameters are expressed as $\beta=1-\epsilon/2$ and $\gamma=1-2\epsilon$ in terms of the dimensionless parameter $\epsilon$ mentioned above. It can be shown that energy, momentum and angular momentum are conserved in NGR. We must be careful not to conclude that $\eta=0$, however, since NGR is not a metric theory: Accordingly we are now trying to develop its PPN formalism. It is the purpose of this paper to study a different aspect of the equivalence principle, namely the problem of whether or not the active gravitational mass (or simply the gravitational mass) of an isolated system is equal to its inertial mass, i.e. the total energy divided by the square of the velocity of light.\footnote{Since we use the unit $c=1$, we need not draw a distinction between the inertial mass and the total energy} As is well known, this problem is settled affirmatively in ${\rm GR.}^{21)}$ In the case of $(a_1+a_2)=0$ Mikhail et ${\rm al.}^{22)}$ calculated the energy of two spherically symmetric solutions, and found that the energy in one of the two solutions does not coincide with the gravitational mass. Shirafuji et ${\rm al.}^{23)}$ extended the calculation to all the stationary asymptotically flat solutions with spherical symmetry, dividing them into two classes: The one in which the components, $({b^a}_{0})$ and $({b^{(0)}}_{\alpha})$, of the parallel vector fields $({b^k}_{\mu})$ tend to zero faster than $1/\sqrt{r}$ for large $r$ and the other in which those components go to zero like $1/\sqrt{r}$ .\footnote{In this paper Latin indices $(i,j,k,...)$ represent the vector number, which runs from $(0)$ to $(3)$, while Greek indices $(\mu,\nu,\rho, ...)$ represent the world-vector components running from 0 to 3. The spatial part of Latin indices are denoted by $(a,b,c,...)$, while that of Greek indices by $(\alpha, \beta,\gamma,...)$.} It was found that the equality of the gravitational and inertial masses holds true only in the first class. In this paper we study the problem for stationary, spherically symmetric systems without making any assumption on the parameters, $a_1$, $a_2$ and $a_3$ other than the Newtonian approximation condition. We organize this paper as follows. In {\S 2} we give a brief review of the NGR. In \S 3 we derive the superpotential of the total energy-momentum complex, and apply it to the exact spherically symmetric solution. In \S 4 we study stationary spherically symmetric solutions at far distances in the linear approximation. In \S 5 we calculate the energy in the special case of $(a_1-4a_3/9)=0$, taking all the leading terms into account beyond the linear approximation. The final section is devoted to conclusion and discussion. \newsection{A brief review of the NGR} We assume spacetime to admit absolute parallelism, i.e. to have a quadruplet of linearly independent parallel vector fields $({b^k}_{\mu})$ satisfying \begin{equation {D_\nu} {b^k}_\mu = {b^k}_{\mu,\nu}- {\Gamma^\lambda}_{\mu \nu}{b^k}_\lambda=0 \end{equation} with ${b^k}_{\mu,\nu}= {\partial}_\nu {b^k}_{\mu}$. Solving (1), we get the nonsymmetric connection \begin{equation {\Gamma^\lambda}_{\mu \nu} ={b_k}^\lambda {b^k}_{\mu,\nu}, \end{equation} which defines the torsion tensor as \begin{equation {T^\lambda}_{\mu \nu} = {\Gamma^\lambda}_{\mu \nu}-{\Gamma^\lambda}_{\nu \mu} ={b_k}^\lambda ({b^k}_{\mu,\nu}-{b^k}_{\nu,\mu}). \end{equation} The curvature tensor defined by ${\Gamma^\lambda}_{\mu \nu}$ is identically vanishing, however. The metric tensor is given by the parallel vector fields as \begin{equation g_{\mu \nu}= b_{k \mu} {b^k}_ \nu, \end{equation} where we raise or lower Latin indices by the Minkowski metric $\eta_{i j}$=$\eta^{i j}$ =diag(-1,+1,+1,+1). Assuming the invariance under \\ a) the group of general coordinate transformations,\\ b) the group of global Lorentz transformations, and \\ c) the parity operation,\\ we write the gravitational Lagrangian density in the form \footnote{ Throughout this paper we use the relativistic units, $c=G=1$. The Einstein constant ${\kappa}$ is then equal to $8 \pi $. We will denote the symmetric part by the parenthesis (\ ) and the antisymmetric part by the square bracket [\ ].} \begin{equation {\cal L}_G= {\sqrt{-g}\over \kappa} \left[a_1(t^{\mu \nu \lambda} t_{\mu \nu \lambda})+a_2(v^{\mu} v_{\mu})+a_3(a^{\mu}a_{\mu}) \right], \end{equation} where $a_1$, $a_2$ and $a_3$ are dimensionless parameters of the theory, \footnote{ The dimensionless parameters $\kappa a_i$ of Ref.9) are here denoted by $a_i$ for convenience.} and \begin{eqnarray t_{\mu \nu \lambda} &\!\!\! = &\!\!\! {1 \over 2} \left(T_{\mu \nu \lambda} +T_{\nu \mu \lambda} \right) +{1 \over 6} \left(g_{\lambda \mu} v_\nu +g_{\lambda \nu} v_{\mu} \right)-{1 \over 3} g_{\mu \nu} v_\lambda, \\ v_{\mu} &\!\!\! = &\!\!\! {T^\lambda}_{\lambda \mu}, \\ a_{\mu} &\!\!\! = &\!\!\! {1 \over 6}{\epsilon}_{\mu \nu \rho \sigma} T^{\nu \rho \sigma} \end{eqnarray} with ${\epsilon}_{\mu \nu \rho \sigma}$ being the completely antisymmetric tensor normalized as ${\epsilon}_{0123}=\sqrt{-g}$. By applying variational principle to the above Lagrangian, we get the field equation: \begin{equation I^{\mu \nu}= {\kappa}T^{\mu \nu} \end{equation} with \begin{equation I^{\mu \nu}=2{\kappa}[{D}_\lambda F^{\mu \nu \lambda}+ v_\lambda F^{\mu \nu \lambda}+H^{\mu \nu} -{1 \over 2} g^{\mu \nu}L_G], \end{equation} where \begin{eqnarray F^{\mu \nu \lambda} &\!\!\! = &\!\!\! {1 \over 2} b^{k \mu} {\partial L_G \over \partial {b^k}_{\nu,\lambda}} \nonumber \\[.5mm] &\!\!\! = &\!\!\! {1 \over \kappa} \left[ a_1 \left(t^{\mu \nu \lambda} -t^{\mu \lambda \nu} \right)+a_2 \left(g^{\mu \nu} v^\lambda -g^{\mu \lambda} v^\nu \right) -{a_3 \over 3} \epsilon^{\mu \nu \lambda \rho} a_\rho \right] =-F^{\mu \lambda \nu},\\ H^{\mu \nu} &\!\!\! = &\!\!\! T^{\rho \sigma \mu} {F_{\rho \sigma}}^\nu - {1 \over 2} T^{\nu \rho \sigma} {F^\mu}_{\rho \sigma}=H^{\nu \mu},\\ {L_G} &\!\!\! = &\!\!\! {{\cal L}_G \over \sqrt{-g}},\\ T^{\mu \nu} &\!\!\! = &\!\!\! {1 \over \sqrt{-g}} {\delta {\cal L}_M \over \delta {b^k}_\nu} b^{k \mu}. \end{eqnarray} Here ${\cal L}_M$ denotes the Lagrangian density of material fields and $T^{\mu \nu}$ is the material energy-momentum tensor which is nonsymmetric in general. In order to reproduce the correct Newtonian limit, we require the parameters $a_1$ and $a_2$ to satisfy the condition \begin{equation a_1+4a_2+9a_1a_2=0 \end{equation} called the Newtonian approximation ${\rm condition,}^{9)}$ which can be solved to give \begin{equation a_1=-{1 \over 3(1-\epsilon)}, \quad a_2={1 \over 3(1-4\epsilon)} \end{equation} with $\epsilon$ being a dimensionless parameter. The comparison with solar-system experiments shows that $\epsilon$ should be given by$^{\rm 9)}$ \begin{equation \epsilon=-0.004 \pm0.004, \end{equation} which we assume throughout this paper. It is well known that the conservation law in GR is given by \begin{equation {{T_{GR}}^{\mu \nu}}_{;\nu}= 0, \end{equation} where ${T_{GR}}^{\mu \nu}$ is the symmetric material energy-momentum tensor of GR and the semicolon denotes covariant derivative with respect to the Christoffel symbol. This law does not follow from (9), however. Instead, we can derive the response equation \begin{equation {T^{\mu \nu}}_{;\nu}=K^{\nu \lambda \mu}T_{[\nu \lambda]}, \end{equation} where $K^{\nu \lambda \mu}$ is the contortion tensor given by \begin{equation K^{\nu \lambda \mu}={1 \over 2} \left( T^{\nu \lambda \mu}-T^{\lambda \nu \mu} -T^{\mu \nu \lambda} \right)=-K^{\lambda \nu \mu}. \end{equation} The antisymmetric part $T^{[\mu \nu]}$ is due to the contribution from the intrinsic spin of fundamental spin-$1/2$ particles. For macroscopic test particles for which the effects due to intrinsic spin can be ignored, their energy-momentum tensor can be supposed to be symmetric and satisfy (18). The equation of motion for such macroscopic test particles is then the geodesic equation of the metric. In ${\it static}$, spherically symmetric spacetime the parallel vector fields take a diagonal form, and the field equation (9) can be exactly solved in vacuum to give \begin{equation \left( {b^k}_\mu \right) = \left( \matrix{ {\displaystyle \left( 1-\displaystyle{m \over pr} \right)^{p/2} \over \left( 1+\displaystyle{m \over qr} \right)^{q/2}} & 0 \vspace{3mm} \cr 0 & \displaystyle \left( 1-\displaystyle{m \over pr} \right)^{(2-p)/2} \displaystyle \left( 1+\displaystyle{m \over qr} \right)^{(2+q)/2} {\delta^a}_\alpha \cr}\right), \end{equation} where $m$ is a constant of integration, and the constants $p$ and $q$ are given by \begin{equation p={2 \over (1-5\epsilon)}[(1-5\epsilon+4\epsilon^2)^{1 \over 2}-2\epsilon], \quad q={2 \over (1-5\epsilon)}[(1-5\epsilon+4\epsilon^2)^{1 \over 2}+2\epsilon]. \end{equation} In spherical polar coordinates (21) gives the line-element \begin{equation ds^2=-{\displaystyle \left( 1-{m \over pr} \right)^p \over \displaystyle \left( 1+{m \over qr} \right)^q} dt^2+\left( 1-{m \over pr} \right)^{2-p} \left( 1+{m \over qr} \right)^{2+q} \left [dr^2+r^2(d\theta^2 +{\rm sin^2} \theta d\phi^2) \right]. \end{equation} >From the asymptotic behavior of the component $g_{0 0}$ of the metric tensor, the constant ${\it m}$ can be identified with the gravitational mass of the central gravitating system. It is clear that if the parameter $\epsilon$ vanishes, the line-element (23) coincides with the Schwarzschild solution written in the isotropic coordinates. \newsection{Superpotential of the NGR and calculation of the energy} The following identity can be derived from the invariance of ${\cal L}_G$ under general coordinate ${\rm transformations:}^{1),24)}$ \begin{equation -{\delta {\cal L}_G \over \delta{{b^k}_{\nu}}}{{b^k}_{\mu}} -{\partial {\cal L}_G \over \partial{{b^k}_{\lambda,\nu}}}{{b^k}_{\lambda,\mu}} +{\delta^\nu}_\mu{\cal L}_G-{\partial_{\lambda}} \left({\partial {\cal L}_G \over \partial{{b^k}_{\nu,\lambda}}}{{b^k}_{\mu}} \right) \equiv 0. \end{equation} When $({b^k}_{\mu})$ satisfies the gravitational field equation (9), this implies \begin{equation \sqrt{-g} \left( {T_\mu}^\nu+{t_\mu}^\nu \right)=\partial_{\lambda} \left( 2\sqrt{-g} {F_\mu}^{\nu \lambda} \right), \end{equation} where ${t_\mu}^\nu$ is the canonical energy-momentum complex of the gravitational field \begin{equation \sqrt{-g}{t_\mu}^\nu = -{\partial {\cal L}_G \over \partial{{b^k}_{\lambda,\nu}}} {{b^k}_{\lambda,\mu}}+{\delta_\mu}^\nu {\cal L}_G . \end{equation} The total energy-momentum complex is then defined by \begin{equation {\cal M_\mu}^\nu = \sqrt{-g}({T_\mu}^\nu+{t_\mu}^\nu)= {{\cal U_\mu}^{\nu \lambda},_{\lambda}} \end{equation} with ${\cal U_\mu}^{\nu \lambda}$ being the superpotential \footnote{The Lagrangian used by ${\rm M\phi ller}^{4)}$ is different from that used by Hayashi and ${\rm Shirafuji}^{9)}$ by a factor (-2). Accordingly, the definition (28) is different from that of M$\phi$ller by a factor (-2).} \begin{equation {\cal U_\mu}^{\nu \lambda} = 2 \sqrt{-g}{F_\mu}^{\nu \lambda}. \end{equation} Since the tensor ${F_\mu}^{\nu \lambda}$ is antisymmetric with respect to $\nu$ and $\lambda$, the ${\cal M_\mu}^\nu$ of (27) satisfies ordinary conservation law, \begin{equation \partial_{\nu}{\cal M_\mu}^\nu=0. \end{equation} The total energy is now given by \begin{equation E=-\int {{\cal M}_0}^0 d^3 x= - \lim_{r \rightarrow \infty}\int_{r=constant} {{\cal U}_0}^{0 \alpha} n_\alpha dS, \end{equation} where $n_\alpha$ is the outward unit 3-vector normal to the surface element $dS$. Let us calculate the superpotential by writing the Lagrangian (5) in the form \begin{equation {\cal L}_G={\sqrt{-g} \over \kappa} \left[ a_1 L^{(1)}+a_2L^{(2)}+a_3 L^{(3)} \right], \end{equation} where $L^{(1)}= t^{\lambda \mu \nu}t_{\lambda \mu \nu}$, $L^{(2)}= v^\mu v_\mu$ and $L^{(3)}= a^\mu a_\mu$. Writing (11) in the form \begin{equation {F_\mu}^{\nu \lambda}={1 \over \kappa} \left[ a_1 {{F^{(1)}}_\mu}^{\nu \lambda} +a_2 {{F^{(2)}}_\mu}^{\nu \lambda}+a_3 {{F^{(3)}}_\mu}^{\nu \lambda} \right], \end{equation} we get \begin{eqnarray {{F^{(1)}}_\mu}^{\nu \lambda} &\!\!\! = &\!\!\! {1 \over 2} \left[ 2{T_\mu}^{\nu \lambda}+{T^{\lambda \nu}}_{\mu}- {T^{\nu \lambda}}_{\mu}-({\delta_\mu}^{\nu}v^\lambda- {\delta_\mu}^{\lambda}v^\nu) \right], \\ {{F^{(2)}}_\mu}^{\nu \lambda} &\!\!\! = &\!\!\! \left( {\delta_\mu}^{\nu}v^\lambda-{\delta_\mu}^{\lambda}v^\nu \right), \\ {{F^{(3)}}_\mu}^{\nu \lambda} &\!\!\! = &\!\!\! -{1 \over 9} \left[ {T_\mu}^{\nu \lambda}-{T^{\lambda \nu}}_{\mu}+ {T^{\nu \lambda}}_{\mu}\right], \end{eqnarray} where ${{F^{(1)}}_\mu}^{\nu \lambda}$, ${{F^{(2)}}_\mu}^{\nu \lambda}$ and ${{F^{(3)}}_\mu}^{\nu \lambda}$ correspond to $L^{(1)}$, $L^{(2)}$ and $L^{(3)}$, respectively. So with the help of (28) the superpotential of the NGR can be written as \begin{eqnarray {{\cal U}_\mu}^{\nu \lambda} &\!\!\! = &\!\!\! {2 \sqrt{-g} \over \kappa} \Biggl[ \left( a_1-{a_3 \over 9} \right) {T_\mu}^{\nu \lambda} +\left( {a_1 \over 2}+{a_3 \over 9} \right) \left( {T^{\lambda \nu}}_{\mu}-{T^{\nu \lambda}}_\mu \right) \nonumber \\[.5mm] &\!\!\! &\!\!\! \qquad \qquad \qquad -\left( {a_1 \over 2} - a_2 \right) \left( {\delta_\mu}^\nu v^\lambda-{\delta_\mu}^\lambda v^\nu \right) \Biggr]. \end{eqnarray} As an example, let us apply (36) to the exact solution (21). The appropriate components ${{\cal U}_0}^{0 \alpha}$ are given by \begin{eqnarray {{\cal U}_0}^{0 \alpha} &\!\!\!= &\!\!\!-{m \over 2 \kappa pq}{n^\alpha \over r^2} \Biggl[ (p-q-8)(a_1-2a_2){m \over r}+4(p-q) ( a_1-2 a_2) \nonumber \\[.5mm] &\!\!\! &\!\!\! \qquad \qquad \qquad +4pq(2 a_1-a_2)+3(p-q){a_1 m \over r} \Biggr] \end{eqnarray} with $p$ and $q$ given by (22). Using (37) in (30), we get \begin{eqnarray E &\!\!\!= &\!\!\! {8\pi m \over \kappa pq} \left[ (p-q)(a_1 -2 a_2)+pq(2 a_1 -a_2) \right] \nonumber \\[.5mm]% &\!\!\! = &\!\!\! m \left[ (2 a_1 - a_2)-2\epsilon(a_1 -2 a_2) \right]=m, \end{eqnarray} where the relation $(p-q)/pq=-2\epsilon$, which follows from (22), is used in the second equality and (16) is employed in the last one. This means that the total energy of the exact solution is just the same as the gravitational mass of the central gravitating system. \newsection{The spherically symmetric parallel vector fields and its energy in the linear approximation} The solution (21) is the only exact spherically symmetric solution in vacuum that we know at present, and it is not clear to us whether there exist any other spherically symmetric solutions in vacuum. In view of this let us consider a wider class of spherically symmetric solutions at far distances from the source. Consider an isolated, gravitating system with spherical symmetry, and restrict attention to the weak field far from the source. The most general form of the parallel vector fields is given in the Cartesian coordinates ${\rm by}^{25)}$ \begin{equation \left({b^k}_\mu \right)= \left( \matrix{ C(r) & G(r) n^\alpha \vspace{3mm} \cr H(r) n^a & {\delta^a}_\alpha D(r) + E(r) n^a n^\alpha + F(r) \epsilon_{a \alpha \beta} n^\beta \cr } \right), \end{equation} where the two real functions ${\it C(r)}$ and ${\it D(r)}$ are supposed to approach 1 at infinity, while the remaining four real functions, ${\it E(r)}$, ${\it F(r)}$, ${\it G(r)}$ and ${\it H(r)}$, must tend to zero there. Here we define the radial unit vector $n^a$ and $n^\alpha$ by \begin{equation n^\alpha={x^\alpha \over r}={\delta^\alpha}_a n^a, \end{equation} without making distinction between upper and lower indices. Using the freedom to redefine the radial coordinate ${\it r}$, we can eliminate the function ${\it E(r)}$ from the components $({b^a}_{\alpha})$. Accordingly we can put ${\it E(r)}=0$ without loss of generality. The metric tensor $g_{\mu \nu}$ is then written as \begin{eqnarray g_{0 0} &\!\!\!=&\!\!\! -(C^2-H^2), \nonumber \\[.5mm] g_{0 \alpha} &\!\!\!=&\!\!\! \{ -CG +DH \}n_\alpha, \nonumber \\[.5mm] g_{\alpha \beta} &\!\!\!=&\!\!\! (D^2+F^2) \delta_{\alpha \beta}-(F^2+G^2) n_\alpha n_\beta. \end{eqnarray} According to (30), the total energy of an isolated system can be calculated if the superpotential is known up to order $O(1/r^2)$. It is then enough to know the parallel vector fields up to order $O(1/r)$. So let us restrict our attention to the weak field far from the source, and analyze the field equation in vacuum up to order $O(1/r^3)$. This has been performed in G${\rm R,}^{21)}$ showing quite generally that the gravitational mass is equal to the total energy for any stationary isolated system. In the NGR, however, the analysis of the field equation in vacuum has not yet been done for the general case of $(a_1+a_2) \neq 0$ even in the weak field approximation. This is due to the fact that the gravitational field equation of the NGR is more complicated than the Einstein equation of GR. Let us suppose that the leading term of the five unknown functions is given by some power of $1/r$, and that ${(b^k}_\mu)$ can be represented as \footnote{ The constant ${\it b}$ should not be confused with det$({b^k}_\mu)$, which we denote by $\sqrt{-g}$.} \begin{equation \left({b^k}_\mu \right)= \left( \matrix{ (1+\displaystyle{b \over r^s}) & \displaystyle{j \over r^t} n^\alpha \vspace{3mm} \cr \displaystyle{h \over r^u} n^a & (1+\displaystyle{d \over r^v}) {\delta_\alpha}^a +\displaystyle{f \over r^w}\epsilon_{a \alpha \beta} n^\beta \cr} \right). \end{equation} Here the powers, ${\it s}$, ${\it t}$, ${\it u}$, ${\it v}$ and ${\it w}$, are positive unknown constants at the beginning of the calculation, and their value will be determined by the field equation of NGR. Here the constant coefficients, ${\it b}$, ${\it j}$, ${\it h}$, ${\it d}$ and ${\it f}$, are also unknown, but they can be assumed to be nonvanishing without loss of generality since the powers of $r$ are left unknown. Use (42) in (41) gives the asymptotic behavior of the metric tensor which involves linear and quadratic terms of the unknown constants, ${\it b}$, ${\it j}$, ${\it h}$, ${\it d}$ and ${\it f}$: The linear terms are dominant if the powers satisfy the following inequality \begin{equation s<2u, \qquad v<2t, \qquad v<2w, \end{equation} which we call the condition of the linear approximation. Now apply the vacuum field equation (9) to (42), assuming the condition (43). Keeping only the leading terms, which are shown to be linear in the five unknown constants due to (43), we get the nonvanishing components of $I_{\mu \nu}$; \begin{eqnarray I_{0 0} &\!\!\!=&\!\!\! {-2 \over r^2} \left \{ \displaystyle{s(s-1)(a_1+a_2)b \over r^s}- \displaystyle {v(v-1)(a_1-2 a_2)d \over r^v} \right \},\\ I_{\alpha 0} &\!\!\!=&\!\!\! 2(u+1)(u-2)\displaystyle {(a_1+a_2)h \over r^{u+2}}n_\alpha, \end{eqnarray} \begin{eqnarray I_{\alpha \beta} &\!\!\!=&\!\!\! {n_\alpha n_\beta \over r^2} \left \{ \displaystyle{s(s+2)(a_1-2 a_2)b \over r^s}- \displaystyle{v(v+2)(a_1+4 a_2)d \over r^v} \right \} \nonumber \\[.5mm] &\!\!\! &\!\!\! -{\delta_{\alpha \beta} \over r^2} \left \{ \displaystyle{s^2 (a_1-2 a_2)b \over r^s}- \displaystyle{v^2 (a_1+4 a_2)d \over r^v} \right \}\nonumber \\[.5mm] &\!\!\! &\!\!\! +(w+1)(w-2)\displaystyle{(a_1-\displaystyle{4 \over 9} a_3)f \over r^{w+2}} \epsilon_{\alpha \beta \gamma} n^\gamma. \end{eqnarray} Here it is important to notice that $I_{\alpha 0}$ and $I_{[\alpha \beta]}$ are dominated by the linear term irrespectively of the condition of the linear approximation. Also using (42) in (36), we get the leading terms for the appropriate components of the superpotential; \begin{equation {{\cal U}_0}^{0 \alpha} =- {2n^\alpha \over \kappa r} \left[{s(a_1+a_2)b \over r^s}- {v(a_1-2 a_2)d \over r^v} \right]. \end{equation} By virtue of (44) and (46) we can show that the equations, $I_{0 0}=0$ and $I_{(\alpha \beta)}=0$, give the following results: 1) if $s<v$ then $b=0$, because $(a_1-2a_2)\neq 0$, 2) if $s>v$ then $d=0$, because $(a_1+4a_2)\neq 0$, and 3) if $s=v\neq 1$ then $b=d=0$, because \begin{equation {\rm det} \left( \matrix{ (a_1+a_2) & -(a_1-2a_2) \vspace{3mm} \cr (a_1-2a_2) & -(a_1+4a_2) \cr} \right) \neq 0. \end{equation} All the above three cases contradict our assumption that $b\neq 0$ and $d\neq 0$. Therefore, we find that the powers $s$ and $v$ should be given by \begin{equation s=v=1. \end{equation} Then the equation $I_{0 0}=0$ is automatically satisfied and the remaining one, $I_{(\alpha \beta)}=0$, gives \begin{equation d=(2\epsilon-1)b. \end{equation} We notice that the asymptotic behavior of the diagonal exact solution (21) satisfy (49) and (50) with $b=-m$. Also this result of (49) is physically very satisfactory in view of the superpotential (47), because we can then get a finite value of the energy. Next the $(\alpha 0)$-component of the field equation in vacuum, $I_{\alpha 0}=0$, implies that \begin{equation (u-2)(a_1+a_2)=0. \end{equation} When $(a_1+a_2)\neq0$, this gives \begin{equation u=2, \end{equation} compatibly with the condition of the linear approximation (43). Using (49) and (51) in (41), we see that the component $g_{00}$ of the metric tensor behaves asymptotically like \begin{equation g_{0 0}=-\left( 1+\displaystyle{2b \over r} \right), \end{equation} which indicates that the constant $c$ is related to the gravitational mass $m$ of the isolated system by \begin{equation b=-m, \end{equation} since as shown in \S 2 the world line of a spinless test particle is a geodesics of the metric. So (50) can now be written as \begin{equation d=(1-2\epsilon)m. \end{equation} In the special case of $(a_1+a_2)=0$, on the other hand, the power $u$ is not constrained by the field equation $I_{\alpha 0}=0$. However, the exact form of all the spherically symmetric solutions was found in this special case, and the energy of those solutions was ${\rm calculated,}^{23)}$ as will be explained at the end of this section. Finally from the skew part, $I_{[\alpha \beta]}=0$, we get \begin{equation (w-2) \left(a_1-{4 a_3 \over 9} \right)=0. \end{equation} When $(a_1-4 a_3/9) \neq 0$, it follows from the condition (56) that \begin{equation w=2, \end{equation} satisfying the condition of the linear approximation (43). In the special case of \\ $( a_1-4 a_3/9)=0$, on the other hand, the power $w$ is not restricted by the field equation. We shall discuss this case separately in the next section. Collecting the above arguments together, we see that when the parameters satisfy $( a_1+ a_2) ( a_1-4a_3/9) \neq 0$, the field equation $I_{\mu \nu}=0$ can be solved in the linear approximation to give the following asymptotic form of $({b^k}_\mu)$: \begin{eqnarray \left({{b^k}_\mu} \right) &\!\!\! = &\!\!\! \left( \matrix{ (1-\displaystyle{m \over r}) & j \displaystyle{n_\alpha \over r^t} \vspace{3mm} \cr h \displaystyle{n^a \over r^2} & (1+\displaystyle{m (1-2\epsilon) \over r}){\delta^a}_{\alpha}+ {f \over r^2} \epsilon_{\alpha \beta \gamma} n^\gamma \cr}\right) \nonumber \\[.5mm] &\!\!\!=&\!\!\! \left( {{ b^k}_{\mu}{\rm (exact)}} \right)+ \left( \matrix{ 0 & j \displaystyle{n_\alpha \over r^t} \vspace{3mm} \cr 0 & 0 \cr }\right)+O \left({1 \over r^2} \right), \end{eqnarray} where $\left( {{ b^k}_{\mu}{(\rm exact)}} \right)$ denotes the exact solution (21) in the asymptotic form. Substituting (58) into (36) gives \begin{equation {{\cal U}_0}^{0 \alpha} = {2m n^\alpha \over \kappa r^2} \left[(a_1+a_2)+(1-2 \epsilon) (a_1-2 a_2) \right]= -{2m n^\alpha \over \kappa r^2}, \end{equation} where we have used (16) in the last equation. From (30) we then get \begin{equation E=m. \end{equation} Thus, the gravitational mass is equal to the total energy for an isolated spherically symmetric system. For completeness let us recapitulate the known results$^{\rm 23)}$ about the exact, spherically symmetric solutions and their total energy. Referring to the general expression for the parallel vector fields given by (39), the solutions are divided into two classes: (1) the solution with $F(r)=0$ and (2) the solution with $F(r) \neq 0$. The general solution of the class (1) involves an arbitrary function of $r$, and therefore there exist solutions whose components $({b^a}_0)$ asymptotically behave like $1/r^u$ for any positive value of $u$. The calculated energy was shown to coincide with the gravitational mass only when the components $({b^a}_0)$ go to zero faster than $1/\sqrt{r}$. When $({b^a}_0) \sim h n^\alpha/\sqrt{r}$ for large $r$, the quadratic terms of $h$ must be taken into account in the superpotential, and the energy does not coincide with the gravitational mass, being given by \begin{equation E=m+{h^2 \over 2}. \end{equation} If $({b^a}_0)$ behave like $1/r^u$ with $0<u<1/2$, the calculated energy is infinite, implying that the solutions with such an asymptotic behavior is physically unacceptable. Next let us turn to the general solution of the class (2), which was shown to involve an arbitrary constant parameter, and to have the asymptotic behavior of (42) with $h=0$ and $w=2$: The calculated energy was shown to coincide with the gravitational mass. \newsection{The energy in the special case of $(a_1-4 a_3/9)=0$} Now let us turn to the special case of $(a_1 -4 a_3/9)=0$, in which we must go beyond the linear approximation discussed in the preceding section since the field equation does not impose any restriction on the power $w$. We notice that when the parameters, $a_1$, $a_2$ and $a_3$ satisfy the conditions, $(a_1+a_2)=0$ and $(a_1-4a_3/9)=0$, together with the Newtonian approximation condition (15), the gravitational Lagrangian (5) reduces to the Einstein-Hilbert Lagrangian of GR written in terms of the tetrad field, thus leading to inconsistency of the field equation (9): For example, the $I_{\mu \nu}$ is symmetric, while $T_{\mu \nu}$ of spinor fields is nonsymmetric. Accordingly we assume $(a_1+a_2) \neq 0$ in this special case. Now consider the parallel vector fields $({b^k}_\mu)$ which asymptotically behave like (42) with $u=2$. Since the power $w$ cannot be fixed by the field equation, we shall discuss the three cases, $w>1/2$, $w=1/2$ and $w<1/2$, separately.\\ \noindent (i) When $w>1/2$, it can be shown by the same argument as in the previous section that the powers $s$ and $v$ satisfy either $s=v=1$, or $s>1$ and $v>1$. If $s=v=1$, the linear approximation studied in the preceding section is still valid, and the parallel vector fields behave asymptotically like \begin{equation \left({{b^k}_\mu} \right)= \left( {{ b^k}_{\mu}(exact)} \right)+ \left( \matrix{ 0 & j \displaystyle{n_\alpha \over r^t} \vspace{3mm} \cr 0 & \displaystyle{f \over r^w} \epsilon_{a \alpha \beta}n^\beta \cr }\right), \end{equation} and the energy coincides with the gravitational mass. If it happens that $s>1$ and $v>1$, on the other hand, the superpotential dies out at infinity faster than $1/r^2$, and hence the calculated energy will be vanishing. The gravitational mass is also vanishing since $g_{0 0}$ tends to 1 for large $r$ faster than $1/r$. Therefore, such a solution, if it exists, is devoid of physical meaning, although we cannot exclude its existence by the present approximate treatment.\\ \noindent (ii) When $w=1/2$, the field equation $I_{\alpha \beta}=0$ implies that the powers $s$ and $v$ must satisfy one of the three possible alternatives: (a) $s=v=1$, (b) $s=1$ and $v>1$ or (c) $s>1$ and $v=1$. Let us start with the case (a) of $s=v=1$. The left-hand side of (9) is given by \begin{eqnarray I_{0 0}&\!\!\! =&\!\!\! I_{0 \alpha}= I_{\alpha 0}= 0, \nonumber \\[.5mm] I_{\alpha \beta} &\!\!\!=&\!\!\! -\left[ b(a_1-2a_2) - d(a_1+4a_2)+2f^2(a_1+a_2) \right] \left( \displaystyle{{\delta_{\alpha \beta} -3n_\alpha n_\beta} \over r^3} \right) \end{eqnarray} up to order $O(1/r^3)$, where we have used $(a_1-4a_3/9)=0$, and the constant $c$ satisfies (54) since $s=1$ and $u=2$. The vacuum field equation then provides the condition \begin{equation d={1 \over a_1+4 a_2} \left[ b(a_1-2 a_2) +2f^2(a_1+a_2) \right] = (1-2\epsilon)m+2\epsilon f^2. \end{equation} Similarly the superpotential is expressed by \begin{eqnarray {{\cal U}_0}^{0 \alpha} &\!\!\!=&\!\!\! -{2 n^\alpha \over \kappa r^2} \left[ (a_1+a_2)b- (a_1-2 a_2)d+{1 \over 2}(a_1-2 a_2)f^2 \right]\nonumber \\[.5mm] &\!\!\!=&\!\!\! -{2m n^\alpha \over \kappa r^2} \left[ 1-\displaystyle{1-2\epsilon \over 1-\epsilon}\displaystyle{f^2 \over 2m} \right], \end{eqnarray} where we have used (16) as well as (64). With the help of (30) we then have \begin{equation E=m-{1-2\epsilon \over 2(1-\epsilon)}f^2 . \end{equation} In the case (b) of $s=1$ and $v>1$, the terms proportional to the constant $d$ do not contribute to the leading term of $I_{\alpha \beta}$ and ${{\cal U}_0}^{0 \alpha}$. Accordingly, the calculated energy is written as \begin{equation E={m \over 4\epsilon(1-\epsilon)}, \end{equation} which goes to infinity as $\epsilon$ tends to zero (i.e., $(a_1+a_2)\rightarrow 0$ ). This means that when the parameter $\epsilon$ is varied, the solution of the case (b), if it exists, will be singular at $\epsilon=0$. \noindent The last case (c) of $s>1$ and $v=1$ can be examined in the similar manner. Since $s>1$, the gravitational mass of the central body must be vanishing, and furthermore the energy takes a value \begin{equation E=-{1-2\epsilon \over 2(1-\epsilon)}f^2, \end{equation} which is negative due to the experimental value of $\epsilon$ given by (17). Thus, the case (c) should be discarded. Therefore, when the power $w$ is equal to $1/2$, the energy of the spherically symmetric body differs from the gravitational mass as measured from far distances. It must be noticed, however, that we have not yet actually found a solution with such asymptotic behavior with $w=1/2$. The above result merely implies that the existence of such a solution is not excluded by the field equation which considers only up to order $O(1/r^3)$: Detailed analysis based on exact solutions is desirable. \noindent (iii) When $w<1/2$, quadratic terms of $f$ contribute to the superpotential with the asymptotic behavior like $1/r^{2w}$, and hence the energy integral will diverge, giving an infinite value. Thus, the solution with $w<1/2$, if it exists, must be discarded. \newsection{Conclusion and discussion} We studied the problem whether or not the equality of the active gravitational mass and the inertial mass (i.e., the total energy) is ensured by the gravitational field equation in NGR, restricting ourselves to isolated spherically symmetric systems. Based on the identity following from the general coordinate invariance, we defined the gravitational energy-momentum complex and derived the explicit expression for the superpotential which allows us to calculate the total energy of isolated systems. We first applied it to the exact spherically symmetric solution of diagonal matrix form, and found that the equality of the active gravitational mass and the total energy is satisfied. Since we do not know whether the spherically symmetric solution is unique, we then discussed the asymptotic method to calculate the total energy. Namely, we started from a quite general expression for the parallel vector fields with spherical symmetry, and solved the vacuum field equation at far distances from the source. The main results can be summarized as follows:\\ (a) In the generic case of $(a_1+a_2) (a_1-4a_3/9)\neq0$, the linear approximation can be applied to solve the field equation. All the components of the parallel vector fields other than $({b^{(0)}}_\alpha)$ are asymptotically determined by the field equation, and coincide with those of the exact solution up to order $O(1/r)$. The components $({b^{(0)}}_\alpha)$ do not contribute to the total energy, however, and the calculated value of the energy agrees with the gravitational mass.\\ (b) In the special case of $(a_1+a_2)=0$ the asymptotic behavior of the components $({b^a}_0)$ is not fixed at all by the field equation. We know, however, that for any asymptotic behavior of $({b^a}_0)$, there exist exact solutions with such asymptotic behavior.$^{23)}$ The total energy is finite only when $({b^a}_0)\sim 1/r^u$ with $u\geq1/2$, and furthermore if $u=1/2$, the quadratic terms of $({b^a}_0)$ contribute to the total energy, violating the equality of the total energy and the gravitational mass.\\ (c) In the special case of $(a_1-4a_3/9)=0$, the gravitational field equation does not impose any condition on the $\epsilon_{a \alpha \beta} n^\beta$-term of $({b^a}_\alpha)$. We showed that if this term tends to zero at infinity like $1/\sqrt{r}$, the equality between the gravitational mass and the total energy is violated. It is yet to be studied, however, whether exact solutions with such asymptotic behavior really exist. Thus, in the special cases of (b) and (c) above, there are anomalous solutions which violate the equality of the energy and the gravitational mass. The characteristic feature of these solutions is that specific components of the parallel vector fields tend to zero as $1/\sqrt{r}$ for large $r$. The physical meaning of these anomalous solutions is not yet clear, however. Is it reasonable to rule out such anomalous solutions by demanding that all the components of the parallel vector fields should tend to flat-space value faster than $1/\sqrt{r}$? As we noticed, if the parameters satisfy both the conditions, $(a_1+a_2)=0$ and $(a_1-4a_3/9)=0$, then the gravitational Lagrangian in the NGR coincides with the Einstein-Hilbert Lagrangian in GR expressed in terms of the tetrad field. The gravitational field equation of the NGR then becomes inconsistent: In fact, the L.H.S. of (9) will be symmetric but the R.H.S. is nonsymmetric. Therefore, we assumed in this paper that the combinations of the parameters, $(a_1+a_2)$ and $(a_1-4a_3/9)$, do not vanish at the same time. The special case of $(a_1+a_2)=0$ has been studied in rather detail.$^{3),4),9),22),23)}$ Another case of $(a_1-4a_3/9)=0$ deserves more attention in this respect. We assumed the spherical symmetry to obtain the solution at large distances. In GR, however, it is well known that such assumption of spherical symmetry can be removed. In fact, Einstein equation ensures that the metric tensor at far distances is the same as the Schwarzschild metric up to order $O(1/r)$ for any isolated stationary system. Is it possible in NGR to investigate the vacuum solution at far distances without assuming spherical symmetry? We have ignored the parity-violating term, $v_\mu$$a^\mu$, throughout this paper. The possibility that this term plays an important role was suggested by M\"uller-Hoissen and ${\rm Nitsch}^{26)}$ in the case of $(a_1+a_2)=0$. It seems interesting to take into account this parity-violating term also in the case of $(a_1-4 a_3/9)=0$. We shall address ourselves to these problems in future work. \newpage \bigskip \bigskip \centerline{\Large{\bf Acknowledgements}} The authors would like to thank Prof.\ K. Hayashi for a careful reading of the manuscript. Also one of the authors (G.N.) would like to thank Japanese Government for supporting him with a Monbusho Scholarship and also wishes to express his deep gratitude to all the members of Physics Department at Saitama University, especially to Prof.\ T. Saso, Prof. Y. Tanii and Dr.\ S. Yamaguchi. \bigskip \newpage \centerline{\Large{\bf References}} \bigskip \begin{enumerate} \item[{1)}] C. M$\phi$ller, {\it Mat.\ Fys.\ Medd.\ Dan.\ Vid.\ Selsk.\ } {\bf 1} (1961), 10.\ \item[{2)}] C. Pellegrini and J. Plebanski, {\it Mat.\ Fys.\ Skr.\ Dan.\ Vid.\ Selsk.\ } {\bf 2} (1963), 4.\ \item[{3)}] K. Hayashi and T. Nakano, {\it Prog.\ Theor.\ Phys.\ } {\bf 38} (1967), 491.\ \item[{4)}] C. M$\phi$ller, {\it Mat.\ Fys.\ Medd.\ Dan.\ Vid.\ Selsk.\ } {\bf 39} (1978), 13.\ \item[{5)}] H. Meyer, {\it Gen.\ Rev.\ Grav.\ } {\bf 14} (1982), 531.\ \item[{6)}] F.W. Hehl, J. Nitsch, and P. Von der Heyde, in {\it General Relativity and Gravitation}, A. held, ed. (Plenum Press, New York, 1980), Vol.\ {\bf 1}, pp.\ 329-355.\ \item[{7)}] K. Hayashi and T. Shirafuji, {\it Prog.\ Theor.\ Phys.\ } {\bf 64} (1980), 866, 883, 1435, 2222; {\bf 65} (1980), 525.\ \item[{8)}] D. S\'aez, {\it Phys.\ Rev.\ } {\bf D27} (1983), 2839.\ \item[{9)}] K. Hayashi and T. Shirafuji, {\it Phys.\ Rev.\ } {\bf D19} (1979), 3524; {\bf D24} (1981), 3312. \ \item[{10)}] A. Einstein, {\it Sitzungsber.\ Preuss.\ Akad.\ Wiss.\ } (1928), 217. \item[{11)}] S. Miyamoto and T. Nakano, {\it Prog.\ Theor.\ Phys.\ } {\bf 45} (1971), 295.\ \item[{12)}] T. Kawai and N. Toma, {\it Prog.\ Theor.\ Phys.\ } {\bf 83} (1990), 1.\ \item[{13)}] K. Hayashi and T. Shirafuji, {\it Prog.\ Theor.\ Phys.\ } {\bf 84} (1990), 36.\ \item[{14)}] C.M. Will, {"\it Theory and Experiment in Gravitational Physics" } (Cambridge U.P., Cambridge, 1993).\ \item[{15)}] K. Nordtvedt, {\it Phy.\ Rev.\ } {\bf 169} (1968), 1014, 1017; {\bf 170} (1968), 1186.\ \item[{16)}] J.O. Dickey et al., {\it Science } {\bf 265} (1994), 482.\ \item[{17)}] J.G. Williams, X.X. Newhall and J.O. Dickey, {\it Phy.\ Rev.\ } {\bf D53} (1996), 6730.\ \item[{18)}] K. Nordtvedt, {\it Icarus } {\bf 114} (1995), 51.\ \item[{19)}] A.S. Eddington, {\it "The Mathematical Theory of Relativity"} (Cambridge U.P., Cambridge, 1957), p.\ 105.\ \item[{20)}] H.P. Robertson, in {\it"Space Age Astronomy"}, A.J. Deutsch and W.E. Klemperer, ed.\ (Academic Press, New York, 1962), p.\ 228.\ \item[{21)}] C.W. Misner, K.S. Thorne and J.A. Wheeler, {\it Gravitation} (Freeman, San Francisco, 1973), pp.\ 448-457.\ \item[{22)}] F.I. Mikhail, M.I. Wanas, A. Hindawi and E.I. Lashin, {\it Int.\ J.\ Theor.\ Phys.\ } {\bf 32} (1993), 1627.\ \item[{23)}] T. Shirafuji, G.G.L. Nashed and K. Hayashi, {\it Prog.\ Theor.\ Phys.\ } {\bf 95} (1996), 665.\ \item[{24)}] C. M$\phi$ller, {\it Ann.\ Phy.\ } {\bf 4}, (1958) 347.\ \item[{25)}] H.P. Robertson, {\it Ann.\ Math.\ (Princeton)} {\bf 33} (1932), 496.\ \item[{26)}] F. M\"uller-Hoissen and J. Nitsch, {\it Phys.\ Rev.\ } {\bf D28}, (1983) 718.\ \end{enumerate} \end{document}

proofpile-arXiv_065-588

\section{Introduction} Precision measurements of the total and as well as partial Z decay rates have provided one of the the most important and, from the theoretical viewpoint, clean determination of the strong coupling constant $\alpha_s$ with a present value of $\alpha_s= 0.1202 \pm 0.0033$ \cite{Blondel}. Theoretical ingredients were the knowledge of QCD corrections to order $\alpha_s^3$ in the limit of massless quarks plus charm and bottom quarks effects (see, e.g. \cite{review} and references therein). These mass corrections which indeed are relevant at the present level of accuracy have been calculated up to the order $\alpha_s^3 m_q^2/s$ for the vector and $\alpha_s^2 m_q^2/s$ for the axial current induced decay rate. In this short note the prediction is extended to include $\alpha_s^3 m_q^2/s$ terms for the (non-singlet part) of the axial current induced rate. At the same time results are obtained for the non-diagonal current correlator with two different masses -- a case of relevance e.g. for the W decay rate into charmed and bottom quarks. The same formulae can also be applied to a subclass of corrections which enter single top production in the Drell-Yan like reaction $q \overline{q}\to t \overline{b}$ far above threshold. The calculation is based on an approach introduced in Refs.~\cite{ChetKuhn90,CheKueKwi92}. Knowledge of the polarization function to order $\alpha_s^2$, the appropriate anomalous dimensions at order $\alpha_s^3$, combined with the renormalization group equation allows one to predict the corresponding logarithmic terms of order $\alpha_s^3$ and hence the constant terms of the imaginary part. The first of these ingredients has been available since some time \cite{GorKatLarSur90,Chetyrkin93,Karl94,levan94} while the anomalous dimension can been obtained from Ref.~\cite{gssq} in a straightforward way. In this short note only the theoretical framework and the analytical results are presented -- numerical studies will presented elsewhere. \section{Renormalization Group Analysis} In analogy to the vector case, we take as a starting point the generic vector/axial quark current correlator $\Pi^{V/A}_{\mu\nu}$ which is defined by \begin{equation} \ba{ll} \Pi^{V/A}_{\mu\nu}(q,m_u,m_d,m{},\mu,\alpha_s) & = \ds i \int dx e^{iqx} \langle T[\, j^{V/A}_{\mu}(x) (j^{V/A}_\nu)^{\dagger}(0)\, ] \rangle \\ & = \ds g_{\mu\nu} \Pi^{(1)}_{V/A}(Q^2) + q_{\mu}q_{\nu} \Pi^{(2)}_{V/A}(Q^2)) {}. \ea \label{correlator} \end{equation} with $Q^2=-q^2$, $m^2_q = \sum_f m_f^2$ and $j^{V/A}_{\mu} = \bar{q}\gamma_{\mu}(\gamma_5) q'$. Here $q$ and $q'$ are just two (generically different) quarks with masses $m_u$ and $m_d$ respectively. Note that the vector and axial correlators are related through \begin{equation} \Pi^{A}_{\mu\nu}(q,m_u,m_d,m{},\mu,\alpha_s) = \Pi^{V}_{\mu\nu}(q,m_u,-m_d,m{},\mu,\alpha_s) \label{VAidentity} \end{equation} The polarization function $\Pi^{(1)}_{V/A}$ and the spectral density $R^{V/A}(s)$ which in turn governs the $Z$ and $W$ decays rate obey the following dispersion relation \begin{equation} \displaystyle \Pi_{A}^{(1)} (Q^2) = \frac{-1}{12\pi^2} \int_{(m_u+m_d)^2}^{\infty}ds \frac{s R^{V/A}(s,m_u,m_d,m{},\mu,\alpha_s)}{s+Q^2} \ \ \ \;\;{\rm mod \;sub.} \label{dispersion.rel} \end{equation} Whereas $R^{V/A}$ as a physical quantity is invariant under renormalization group transformations, the function $ \langle T[\, j^{V/A}_{\mu}(x) (j^{V/A}_\nu)^{\dagger}(0)\, ] \rangle $ contains some non-integrable singularities in the vicinity of the point $x=0$. These cannot be removed by standard quark mass and coupling constant renormalizations, but must be subtracted independently. As a result the relevant renormalization group equation assumes the form \cite{review} \begin{equation} \label{rgea} \mu^2\frac{d}{d\mu^2} \Pi^{V/A}_{\mu\nu} = (q_{\mu}q_{\nu}-g_{\mu\nu}q^2) \gamma^\pm_q(\alpha_s) \frac{1}{16\pi^2} + (m_u \mp m_d)^2 g_{\mu\nu} \gamma^\pm_m (\alpha_s) \frac{1}{16\pi^2}, \end{equation} where \begin{equation} \label{rgdef} \mu^2\frac{d}{d\mu^2} = \mu^2\frac{\partial}{\partial\mu^2} + \pi\beta(\alpha_s) \frac{\partial}{\partial \alpha_s} +\gamma_m(\alpha_s) \sum_f \bar{m_f} \frac{\partial}{\partial \bar{m_f}} {}. \end{equation} Here and below the upper and lower signs give the results for vector and axial vector correlators respectively. {}From the identity (\ref{VAidentity}) we infer that both anomalous dimensions $\gamma^\pm_q$ and $\gamma^\pm_m$, being not dependent on any masses, also do not depend on the sign. In what follows we will denote \[ \gamma^\pm_q = \gamma^{VV}_q \ \ \ \mbox{and} \ \ \ \gamma^\pm_m = \gamma^{VV}_m \ {}. \] The $\beta$-function and the quark mass anomalous dimension $\gamma_m$ are defined in the usual way \begin{equation} \mu^2\frac{d}{d\mu^2} \left( \frac{\alpha_s(\mu)}{\pi} \right) = \alpha_s \beta(\alpha_s) \equiv -\sum_{i\geq0}\beta_i\left(\frac{\alpha_s}{\pi}\right)^{i+1}, \end{equation} \begin{equation} \mu^2\frac{d}{d\mu^2} \bar{m}(\mu) = \bar{m}(\mu)\gamma_m(\alpha_s) \equiv -\bar{m}\sum_{i\geq0}\gamma_m^i\left(\frac{\alpha_s}{\pi}\right)^{i+1}. \end{equation} Their expansion coefficients up to order ${\cal O}(\alpha_s^3)$ are well known \cite{beta,Larin:betaQCD,gamma,Larin:massQCD} and read ($n_f$ is the number of quark flavours) \begin{equation} \ba{c}\displaystyle \beta_0=\left(11-\frac{2}{3}n_f\right)/4, \ \ \beta_1=\left(102-\frac{38}{3}f\right)/16, \\ \displaystyle \beta_2=\left(\frac{2857}{2}-\frac{5033}{18}n_f+ \frac{325}{54}n_f^2\right)/64, \ea \label{beta3} \end{equation} \begin{equation} \ba{c}\displaystyle \gamma^0_m=1, \ \ \ \gamma^1_m=\left(\frac{202}{3}-\frac{20}{9}f\right)/16, \\ \displaystyle \gamma^2_m=\left(1249 - \left[\frac{2216}{27}+\frac{160}{3}\zeta(3)\right] n_f-\frac{140}{81}f^2\right)/64. \ea \label{anom.mass3} \end{equation} Another useful and closely related object is the correlator of the (pseudo)scalar quark currents \begin{equation} \label{SP} \Pi^{{\rm S/P}}(Q^2,m_u,m_d,m{},\mu,\alpha_s) = \int e^{iqx}\langle0|\; T\; [\,j^{{\rm S/P}}(x) (j^{{\rm S/P})\dagger}\,](0) \;|0\rangle {}\, . \end{equation} Scalar and pseudoscalar current correlators are also related in a simple manner: \begin{equation} \Pi^{{\rm S}}(Q^2,m_u,m_d,m{},\mu) = \Pi^{{\rm P}}(Q^2,m_u,-m_d,m{},\mu) {}. \label{SPidentity} \end{equation} For vanishing quark masses scalar and pseudoscalar correlators are, therefore, identical: $\Pi^{{\rm S}}= \Pi^{{\rm P}}$ and meet the following RG equation \begin{equation} \label{rg:sc} \left( \mu^2\frac{d}{d\mu^2} + 2 \gamma_m(\alpha_s) \right) \Pi^{{\rm S/P}} = Q^2 \gamma^{{\rm SS}}_q(\alpha_s) \frac{1}{16\pi^2} {}. \end{equation} The (axial) vector and (pseudo)scalar correlators are connected through a Ward identity \cite{david75} \begin{equation} q_\mu q_\nu \Pi^{{\rm V/A}}_{\mu\nu} = (m_u \mp m_d)^2 \Pi^{\rm S/P} + (m_u \mp m_d) ( \langle \overline{\psi}_{{\rm q}} \psi_{{\rm q}} \rangle \mp \langle \overline{\psi}_{{\rm q'}} \psi_{{\rm q'}} \rangle ) {}\, , \label{axial-ward} \end{equation} where the vacuum expectation values on the r.h.s. are understood within the framework of perturbation theory and the minimal subtractions. Equation~(\ref{axial-ward}) leads to the following relation between the corresponding anomalous dimensions \cite{CheKueKwi92}: \begin{equation} \gamma_{{m}}^{\rm VV} \equiv - \gamma_{{q}}^{{\rm SS}} {}\, . \label{VV-SS} \end{equation} This relation was used in Ref.~\cite{CheKueKwi92} in order to find the anomalous dimension $\gamma_{{m}}^{\rm AA}$ at the $\alpha_s^2$ order starting from the results of Ref.~\cite{GorKatLarSur90}. In what follows we will be interested in quadratic mass corrections to the polarization operator $\Pi^{(1)}_{A}$ which is convenient to represent in the form (${\bf m} = \{ m_u, m_d, m{} \}$): \begin{equation} \Pi^{(1)}_{V/A}( Q^2,{\bf m},\mu,\alpha_s) = \frac{3}{16\pi^2}\Pi^{(1)}_{V/A,0}(\frac{\mu^2}{Q^2}, \alpha_s) + \frac{3}{16\pi^2}\Pi^{(1)}_{V/A,2}(\frac{\mu^2}{Q^2},{\bf m},\alpha_s) + {\cal O}({\bf m^4}) {}. \label{mass-exp} \end{equation} Here the first term on the rhs corresponds to the massless limit while the second term stands for quadratic mass corrections. Note that $\Pi^{(1)}_{V/A,2}$ is a second order polynomial in quark masses: a logarithmic dependence on quark masses may appear starting from $m^4$ terms only\footnote{Provided of course that one uses a mass independent renormalization scheme like the $\overline{\mbox{MS}}$-scheme employed in this work.}. {}From the RG equation (\ref{rgea}) we arrive at the following equation for $\Pi^{(1)}_{V/A,2}$: \begin{equation} \mu^2\frac{d}{d\mu^2} \Pi^{(1)}_{V/A,2} = \frac{1}{3}(m_u \mp m_d)^2 \gamma^{VV}_m(\alpha_s) \label{rgPi1} {} \end{equation} or, equivalently, ($L_q = \ln\frac{\mu^2}{Q^2}$) \begin{equation} \frac{\partial }{\partial L_q} \Pi^{(1)}_{V/A,2} = \frac{1}{3}(m_u \mp m_d) \gamma^{VV}_m -\left( \beta \alpha_s \frac{\partial }{\partial \alpha_s} + 2\gamma_m \right) \Pi^{(1)}_{V/A,2} \label{rgPi2} {}. \end{equation} The last relation explicitly demonstrates that $R^{V/A}_2$ --- the absorptive part of $\Pi^{(1)}_{V/A,2}$ --- depends in order $\alpha_s^n$ on the very function $\Pi^{(1)}_{V/A,2}$ which is multiplied by at least one factor of $\alpha_s$. This means that one needs to know $\Pi^{(1)}_{V/A,2}$ up to order $\alpha_s^{n-1}$ only to unambiguously reconstruct all $Q$-dependent terms in $\Pi^{(1)}_{V/A,2}$ to $\alpha_s^n$, provided, of course, the beta function and anomalous dimensions $\gamma_m$ and $\gamma^{VV}_m$ are known to $\alpha_s^n$. This observation was made first in \cite{ChetKuhn90} where it was used to find the absorptive part $R^{V}_2$ in order $\alpha_s^3$ for the case of the diagonal vector current (that is for the case of $m_u = m_d$). In the present paper we will use the results of a recent calculation of $\gamma^{SS}_q$ \cite{gssq} to order $\alpha_s^3$ to determine the absorptive part $R^{V/A}_2$ to the same order in the general case of non-diagonal currents. \section{Calculation and results} The result for the function $\Pi^{(2)}_{V/A,2}$ in the general non-diagonal case to order $\alpha_s^2$ was first published in Ref.~\cite{Chetyrkin93}. On the other hand, the Ward identity (\ref{axial-ward}) expresses the combination $\Pi^{(1)}_{V/A,2}/Q^2 - \Pi^{(2)}_{V/A,2} $ in terms of the {\em massless} polarization operator $\Pi^S$ known from Refs.~\cite{GorKatLarSur90,Karl94}. A sum of these two functions leads us to the following result for $\Pi^{(1)}_{V,2}$ \begin{eqnarray} \lefteqn{\Pi^{(1)}_{V,2} = {}\frac{m_{-}^2}{Q^2} \left[ 2 +2 {\,\rm ln }\frac{\mu^2}{Q^2} \right] {+}\frac{m_{+}^2}{Q^2} \left[ -2\right ]} \nonumber\\ &{+}&\frac{m_{-}^2}{Q^2} \frac{\alpha_s}{\pi} \left[ \frac{107}{6} -8 \,\zeta(3) +\frac{22}{3} {\,\rm ln }\frac{\mu^2}{Q^2} +2 {\,\rm ln }^2\frac{\mu^2}{Q^2} \right] {+}\frac{m_{+}^2}{Q^2} \frac{\alpha_s}{\pi} \left[ -\frac{16}{3} -4 {\,\rm ln }\frac{\mu^2}{Q^2} \right] \nonumber\\ &{+}&\frac{m_{-}^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{3241}{18} -129 \,\zeta(3) -\frac{1}{2} \,\zeta(4) +55 \,\zeta(5) -\frac{857}{108} \,n_f +\frac{32}{9} \,\zeta(3) \,n_f \right. \nonumber \\ &{}& \left. \phantom{+\frac{m_{-}^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2} +\frac{8221}{72} {\,\rm ln }\frac{\mu^2}{Q^2} -39 \,\zeta(3) {\,\rm ln }\frac{\mu^2}{Q^2} -\frac{151}{36} \,n_f {\,\rm ln }\frac{\mu^2}{Q^2} +\frac{4}{3} \,\zeta(3) \,n_f {\,\rm ln }\frac{\mu^2}{Q^2} \right. \nonumber \\ &{}& \left. \phantom{+\frac{m_{-}^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2} +\frac{155}{6} {\,\rm ln }^2\frac{\mu^2}{Q^2} -\frac{8}{9} \,n_f {\,\rm ln }^2\frac{\mu^2}{Q^2} +\frac{19}{6} {\,\rm ln }^3\frac{\mu^2}{Q^2} -\frac{1}{9} \,n_f {\,\rm ln }^3\frac{\mu^2}{Q^2} \right] \nonumber\\ &{+}&\frac{m_{+}^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{19691}{216} -\frac{124}{27} \,\zeta(3) +\frac{1045}{27} \,\zeta(5) +\frac{95}{36} \,n_f -\frac{253}{6} {\,\rm ln }\frac{\mu^2}{Q^2} \right. \nonumber \\ &{}& \left. \phantom{+\frac{m_{+}^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2} +\frac{13}{9} \,n_f {\,\rm ln }\frac{\mu^2}{Q^2} -\frac{19}{2} {\,\rm ln }^2\frac{\mu^2}{Q^2} +\frac{1}{3} \,n_f {\,\rm ln }^2\frac{\mu^2}{Q^2} \right] \nonumber\\ &{+}&\frac{m^2}{Q^2}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{128}{9} -\frac{32}{3} \,\zeta(3) \right] {}. \label{Pi1V2} \end{eqnarray} Here $m_- = m_u - m_d$ and $m_+ = m_u + m_d$, $Q^2 = -q^2$, all masses as well as QCD coupling constant $\alpha_s$ are understood to be taken at a generic value of the t' Hooft~mass $\mu$. All correlators are renormalized within $\overline{\mbox{MS}}$-scheme. We have also checked (\ref{Pi1V2}) by a direct calculation with the help of the program MINCER \cite{mincer2} written for the symbolic manipulation system FORM \cite{Ver91}. In a particular case of $m_u = m_d$ Eq.~(\ref{Pi1V2}) is in agreement with Refs.~\cite{GorKatLar86,levan94}. Now, as was shown in \cite{ChetKuhn90} the anomalous dimension $\gamma^{AA}_m \equiv = -\gamma^{SS}_q$, and, thus, from the results of \cite{gssq} we have: \begin{eqnarray} \lefteqn{\gamma^{VV}_m = -\gamma^{SS}_q = 6\left\{ 1 {+} \frac{5}{3}\frac{\alpha_s}{\pi} \right. {+}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{455}{72} -\frac{1}{2} \,\zeta(3) -\frac{1}{3} \,n_f \right]} \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{157697}{5184} -\frac{1645}{216} \,\zeta(3) +\frac{15}{8} \,\zeta(4) +\frac{65}{12} \,\zeta(5) -\frac{14131}{7776} \,n_f \right. \nonumber \\ &{}& \left. \left. \phantom{+\left(\frac{\alpha_s}{\pi}\right)^3} -\frac{13}{9} \,\zeta(3) \,n_f -\frac{11}{12} \,\zeta(4) \,n_f -\frac{1625}{11664} \, n_f^2 +\frac{1}{9} \,\zeta(3) \, n_f^2 \right] \right\} {}. \label{gVVm} \end{eqnarray} At last, integrating eq.~(\ref{rgPi2}) we find the spectral density $R^{V}_2$ in general case to order $\alpha_s^3$: \begin{equation} R^V_2 = 3\left\{ \frac{m_+^2}{s} r^V_{2,+} + \frac{m_-^2}{s}r^V_{2,-} + \frac{m^2}{s}r^V_{2,0} \right\} {}, \label{R2V} \end{equation} where the functions $r^V$ are \begin{eqnarray} \lefteqn{r_{2,+}^{V} = 3 \frac{\alpha_s}{\pi} {+} \left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{253}{8} -\frac{13}{12} \,n_f +\frac{57}{4} {\,\rm ln }\frac{\mu^2}{s} -\frac{1}{2} \,n_f {\,\rm ln }\frac{\mu^2}{s} \right]} \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{1261}{2} -\frac{285}{16} \pi^2 +\frac{155}{6} \,\zeta(3) -\frac{5225}{24} \,\zeta(5) -\frac{2471}{54} \,n_f +\frac{17}{12} \pi^2 \,n_f \label{r2+ } \right. \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{197}{54} \,\zeta(3) \,n_f +\frac{1045}{108} \,\zeta(5) \,n_f +\frac{125}{216} \, n_f^2 -\frac{1}{36} \pi^2 \, n_f^2 +\frac{4505}{16} {\,\rm ln }\frac{\mu^2}{s} \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{175}{8} \,n_f {\,\rm ln }\frac{\mu^2}{s} +\frac{13}{36} \, n_f^2 {\,\rm ln }\frac{\mu^2}{s} +\frac{855}{16} {\,\rm ln }^2\frac{\mu^2}{s} -\frac{17}{4} \,n_f {\,\rm ln }^2\frac{\mu^2}{s} +\frac{1}{12} \, n_f^2 {\,\rm ln }^2\frac{\mu^2}{s} \right] {}, \nonumber \end{eqnarray} \begin{eqnarray} \lefteqn{r_{2,-}^{V} = -\frac{3}{2} {+} \frac{\alpha_s}{\pi} \left[ -\frac{11}{2} -3 {\,\rm ln }\frac{\mu^2}{s} \right] } \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{8221}{96} +\frac{19}{8} \pi^2 +\frac{117}{4} \,\zeta(3) +\frac{151}{48} \,n_f -\frac{1}{12} \pi^2 \,n_f \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^2} - \,\zeta(3) \,n_f -\frac{155}{4} {\,\rm ln }\frac{\mu^2}{s} +\frac{4}{3} \,n_f {\,\rm ln }\frac{\mu^2}{s} -\frac{57}{8} {\,\rm ln }^2\frac{\mu^2}{s} +\frac{1}{4} \,n_f {\,\rm ln }^2\frac{\mu^2}{s} \right] \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^3 \left[ -\frac{4544045}{3456} +\frac{335}{6} \pi^2 +\frac{118915}{144} \,\zeta(3) -\frac{635}{2} \,\zeta(5) +\frac{71621}{648} \,n_f \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{209}{48} \pi^2 \,n_f -54 \,\zeta(3) \,n_f +\frac{5}{4} \,\zeta(4) \,n_f +\frac{55}{4} \,\zeta(5) \,n_f -\frac{13171}{7776} \, n_f^2 \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{2}{27} \pi^2 \, n_f^2 +\frac{13}{18} \,\zeta(3) \, n_f^2 -\frac{4693}{6} {\,\rm ln }\frac{\mu^2}{s} +\frac{285}{16} \pi^2 {\,\rm ln }\frac{\mu^2}{s} +\frac{1755}{8} \,\zeta(3) {\,\rm ln }\frac{\mu^2}{s} \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{8909}{144} \,n_f {\,\rm ln }\frac{\mu^2}{s} -\frac{17}{12} \pi^2 \,n_f {\,\rm ln }\frac{\mu^2}{s} -\frac{59}{4} \,\zeta(3) \,n_f {\,\rm ln }\frac{\mu^2}{s} -\frac{209}{216} \, n_f^2 {\,\rm ln }\frac{\mu^2}{s} \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{1}{36} \pi^2 \, n_f^2 {\,\rm ln }\frac{\mu^2}{s} +\frac{1}{3} \,\zeta(3) \, n_f^2 {\,\rm ln }\frac{\mu^2}{s} -\frac{335}{2} {\,\rm ln }^2\frac{\mu^2}{s} +\frac{209}{16} \,n_f {\,\rm ln }^2\frac{\mu^2}{s} \right. \nonumber \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{2}{9} \, n_f^2 {\,\rm ln }^2\frac{\mu^2}{s} -\frac{285}{16} {\,\rm ln }^3\frac{\mu^2}{s} +\frac{17}{12} \,n_f {\,\rm ln }^3\frac{\mu^2}{s} -\frac{1}{36} \, n_f^2 {\,\rm ln }^3\frac{\mu^2}{s} \right] {}, \nonumber\\ \label{r2-} \end{eqnarray} \begin{equation} r_{2,0}^{V} = \left(\frac{\alpha_s}{\pi}\right)^3 \left[ -80 +60 \,\zeta(3) +\frac{32}{9} \,n_f -\frac{8}{3} \,\zeta(3) \,n_f \right] {}. \label{r20} \end{equation} The expressions for the hadronic decay rates of the intermediate bosons read: \begin{eqnarray} \Gamma(Z \to \mbox{hadrons} ) = \Gamma^Z_0 \left[ \right. &{}& \sum_f ((g^f_V)^2 + (g^f_A)^2)\left(R_0(s) + R_2^V(s,0,0,\sqrt{m^2_b+m_c^2}\right) \nonumber \\ &+&\sum_{f=b,c} (g^f_V)^2 R_2^V(s,m_f,m_f,0) \nonumber \\ &+& \left. \sum_{f=b,c}(g^f_A)^2 R_2^V(s,m_f,m_f,0) \right] \label{Zff} {}, \end{eqnarray} \begin{eqnarray} \Gamma(W \to \mbox{hadrons}) = \Gamma^W_0 \left[ \right. &{}& 2 \left( R_0(s) +R_2^{V}(s,0,0,\sqrt{m^2_b+m_c^2}) ) \right) \label{Wff} \\ &+& \frac{1}{2}\sum_{\displaystyle {i=u,c}\atop{\displaystyle j=d,s,b}} |V_{i,j}|^2( R_2^V(s,m_{i},m_{j},0) + R_2^A(s,m_{i},m_{j},0) ) \left. \right] \nonumber {}, \end{eqnarray} with $\Gamma_0 = \frac{G_F M_{Z/W}^3}{6\pi \sqrt{2}}$, $g^{{f}}_V = I^{{f}}_3 - 2Q_{{f}}\sin^2 \theta_{{\rm w}}, \ \ g^{{f}}_A = I^{{f}}_3$ and $V_{ij}$ being the CKM matrix. Here $R_0(s)$ is the (non-singlet part) of the ratio $R(s)$ in massless QCD; it was computed to $\alpha_s^3$ in \cite{GorKatLar91,SurSam91} and confirmed in \cite{gluino}; it reads: \begin{equation}\label{apxc2} \begin{array}{rl}\displaystyle R_0(s) = & \displaystyle 3 \Bigg\{ 1+\frac{\alpha_s}{\pi} \\ & \displaystyle +\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{365}{24}-11\zeta(3) +n_f\left( -\frac{11}{12}+\frac{2}{3}\zeta(3) \right) +\left( -\frac{11}{4}+\frac{1}{6}n_f \right)\ln\,\frac{s}{\mu^2} \right] \\ & \displaystyle +\left(\frac{\alpha_s}{\pi}\right)^3 \Bigg[ \frac{87029}{288}-\frac{121}{48}\pi^2 -\frac{1103}{4}\zeta(3) +\frac{275}{6}\zeta(5) \\ & \displaystyle \hphantom{\left(\frac{}{}\right)^2} +n_f\left( -\frac{7847}{216}+\frac{11}{36}\pi^2 +\frac{262}{9}\zeta(3)-\frac{25}{9}\zeta(5) \right) +n_f^2\left( \frac{151}{162}-\frac{1}{108}\pi^2 -\frac{19}{27}\zeta(3) \right) \\ & \displaystyle \hphantom{\left(\frac{}{}\right)^2} +\left( -\frac{4321}{48}+\frac{121}{2}\zeta(3) +n_f\left[ \frac{785}{72}-\frac{22}{3}\zeta(3) \right] +n_f^2\left[ -\frac{11}{36}+\frac{2}{9}\zeta(3) \right] \right)\ln\,\frac{s}{\mu^2}W \\ & \displaystyle \hphantom{\left(\frac{}{}\right)^2} +\left( \frac{121}{16}-\frac{11}{12}n_f +\frac{1}{36}n_f^2 \right)\ln^2\frac{s}{\mu^2} \Bigg] \Bigg\} {}. \end{array} \end{equation} In deriving Eqs.~(\ref{Zff}) and (\ref{Wff}) we have assumed that (i) the top quark is completely decoupled (the power suppressed corrections to this approximation start from the order $\frac{s}{m_t^2}\alpha_s^2$ and have been studied in Refs.~\cite{Kniehl90,me93,Soper94}); (ii) all other quarks except for the charmed and bottom ones are massless. Note that for the case of {\em diagonal} currents there exist also so-called singlet contributions to $R(s)$. We will ignore these contributions in what follows as they are absent for the case of non-diagonal currents relevant for the $W$-decay (a detailed discussion of the $Z$-decay rate including singlet contributions can be found in \cite{review}). Taking into account the peculiar structure of the general result (\ref{R2V}), the last formula can be written in a simpler form, viz. \begin{eqnarray} \Gamma(W \to \mbox{hadrons}) = \Gamma^W_0 \left[ \right. &{}& 2 \left( R_0(s) +R_2^V(s,0,0,\sqrt{m^2_b+m_c^2}) ) \right) \nonumber \\ &+& R_2^V(s,m_{eff},0,0) \left. \right] \label{Wff2} {}. \end{eqnarray} Here \[ m_{eff}^2 = \sum_{\displaystyle {i=u,c}\atop{\displaystyle j=d,s,b}} |V_{i,j}|^2 (m_i^2 + m_j^2) \] and we have taken into account the fact that \[ R^V(s,m_i,m_j,0) + R^A(s,m_i,m_j,0) = 2 R^V(s,\sqrt{m^2_i+m^2_j},0,0) = 2 R^A(s,\sqrt{m^2_i+m^2_j},0,0) \] As a direct consequence of Eqs.~(2,\ref{R2V}) we obtain the following expressions for particular functions entering into (\ref{Zff},\ref{Wff}) \begin{eqnarray} R_2^{V}(s,m,m,0) &=& \frac{4m^2}{s} 3 r^V_{2,+}, \label{part1} \\ R_2^{A}(s,m,m,0) &=& \frac{4m^2}{s} 3 r^V_{2,-}, \\ \label{part2} R_2^{V}(s,m,0,0) &=& R_2^{A}(s,m,0,0)= \frac{m^2}{s} 3 ( r^V_{2,+}+ r^V_{2,-} ), \\ R_2^{V}(s,0,0,m) &=& R_2^{A}(s,0,0,m) = \frac{m^2}{s}3r_{2,0} {}. \label{part3} \end{eqnarray} At last, with $n_f=5$ and $\mu^2= s$ the above formulas are simplified to \begin{eqnarray} R_2^{V}(s,m,m,0) = &{}&\frac{m^2}{s}3\left\{ 12\frac{\alpha_s}{\pi} {+}\frac{629}{6} \left(\frac{\alpha_s}{\pi}\right)^2 \right. \label{R2Vmm0nf5} \\ &{+}& \left. \left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{89893}{54} -\frac{1645}{36} \pi^2 +\frac{820}{27} \,\zeta(3) -\frac{36575}{54} \,\zeta(5) \right] \right\} \nonumber \end{eqnarray} \begin{eqnarray} &{}&\!\!\!\!\!\!\!\!\!R_2^{A}(s,m,m,0) = {}\frac{m^2}{s} 3\left\{ -6 -22 \frac{\alpha_s}{\pi} +\left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{2237}{8} +\frac{47}{6} \pi^2 +97 \,\zeta(3) \right] \right. \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left. \left[ -\frac{25024465}{7776} +\frac{15515}{108} \pi^2 +\frac{27545}{12} \,\zeta(3) +25 \,\zeta(4) -995 \,\zeta(5) \right] \right\} \label{R2Amm0nf5} \end{eqnarray} \begin{eqnarray} &{}&R_2^{V}(s,m,0,0)= \frac{m^2}{s} 3\left\{ -\frac{3}{2} -\frac{5}{2} \frac{\alpha_s}{\pi} {+}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{4195}{96} +\frac{47}{24} \pi^2 +\frac{97}{4} \,\zeta(3) \right] \right. \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left. \left[ -\frac{12079873}{31104} +\frac{2645}{108} \pi^2 +\frac{251185}{432} \,\zeta(3) +\frac{25}{4} \,\zeta(4) -\frac{90305}{216} \,\zeta(5) \right] \right\} \label{ } \end{eqnarray} or, in the numerical form, \begin{equation} R_2^{V}(s,m,m,0) = \frac{m^2}{s} 3\left\{ 12 \frac{\alpha_s}{\pi} {+}104.833\left(\frac{\alpha_s}{\pi}\right)^2 {+}547.879\left(\frac{\alpha_s}{\pi}\right)^3 \right\} \label{R2Vmm0NN} {}, \end{equation} \begin{equation} R_2^{A}(s,m,m,0) = \frac{m_{}^2}{s} 3\left\{ -6 -22 \frac{\alpha_s}{\pi} -85.7136\left(\frac{\alpha_s}{\pi}\right)^2 -45.7886 \left(\frac{\alpha_s}{\pi}\right)^3 \right\} {}, \label{R2Amm0NN} \end{equation} \begin{equation} R_2^{V}(s,m,0,0) \equiv R_2^{A}(s,m,0,0) = \frac{m_{}^2}{s} 3\left\{ -1.5 -2.5 \frac{\alpha_s}{\pi} + 4.7799\left(\frac{\alpha_s}{\pi}\right)^2 + 125.523\left(\frac{\alpha_s}{\pi}\right)^3 \right\} {}. \label{R2Vm00NN} \end{equation} \section{ Acknowledgments} \noindent This work was supported by BMFT under Contract 057KA92P(0) and INTAS under Contract INTAS-93-0744. \sloppy \raggedright \def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3} \def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf #1} (#2) #3} \defProc. LHC Workshop, CERN 90-10{Proc. LHC Workshop, CERN 90-10} \def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3} \def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3} \def\partial#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3} \def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3} \def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3} \def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3} \def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3} \def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3} \def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3} \def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3} \def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3} \def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3} \def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3} \def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3} \def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3} \def\mpl#1#2#3{{\it Mod. Phys. Lett.~}{\bf A #1} (#2) #3}

proofpile-arXiv_065-589

\section{Introduction} In perturbation theory, one considers evolution of soft gluon density in $x$, $Q^2$ or both. In the double log approximation of DGLAP formalism, terms of the form $\alpha \ln (x) \ln (Q^2)$ are summed. This is done by considering the so-called ladder diagrams where both longitudenal and transverse momenta are ordered. For very small values of $x$ it may be more appropirate to sum terms of the form $\alpha \ln (x)$. This is done in BFKL formalism where one has ordering in longitudenal momenta while assuming that everything is happening at more or less the same transverse area. It is important to realize that both DGLAP and BFKL are evolution equations and as such require initial parton distributions as input. Regardless of whether BFKL or DGLAP is more appropirate for description of soft gluons, they both have one outstanding feature in common; they both predict a sharp rise of soft gluon density at very small values of $x$ and thus would eventually lead to violation of unitary bound on physical cross sections. However, the above picture of harder partons spliting into softer ones with no further interaction among them is perhaps a bit too naive. Physically we know that at very small values of $x$, the parton density will be high and they will spatially overlap. The effects of parton recombination, screening etc., will therefore be important. This will eventually lead to saturation of soft gluon density at small $x$. It may be more natural to consider soft gluons not as quasi-free partons but rather as classical fields with large amplitudes. After summing the effects of high density into the clasical fields one can do perturbation theory in the background of these large classical fields. \section{McLerran-Venugopalan Model} In the Mclerran-Venugopalan model of small $x$ gluons, one considers a large nucleus traveling with very high velocities and thus highly Lorentz contacted. It therefore has a high valence quark surface density. This approach can also be used for hadrons at sufficiently small values of $x$ where the parton density is large. The color charge density $\rho$ is the only dimentionfull parameter in the problem. Therefore the strong coupling constant $\alpha _{s}$, as a function of $\rho$, will be small for large charge densities so that one can use weak coupling methods. Since the coupling is weak, the valence quarks do not lose an appreciable fraction of their momenta and stay on straight line trajectories and are therefore static sources of color charge. This is the no-recoil approximation. Also, due to the high density of glue, one can treat them as classical charges. This corresponds to taking a higher dimentional representation of the color algebra. Treating the valence charges as classical leads to computing expectation values with a Gaussian distribution of the form \begin{equation} \int [d\rho]\,\, exp\big{\{}{{-1}\over {2\mu ^2}} \int d^2 x_t \rho ^2 (x_t)\big{\}} \label{eq:GD} \end{equation} where $\mu^2 $ is the average color charge squared per unit area. Alternatively, one can solve the Yang-Mills equations in the presence of this static color charge. Once a solution is found, one can then compute the distribution function with the above Gaussian weight. It was shown in~\cite{MV} that this leads to a distribution function of the Weizs\"{a}cker-Williams form for soft gluons. Quantum corrections to the classical results were also computed and similar to standard perturbation theory, it was found that there are potentially large logs of ratio of longitudenal momenta~\cite{QC}. This led to a longitudenal structure for the color charge density $\rho$. Then the problem to consider was to solve the classical equation of motion in the presence of this rapidity dependant charge density. Working in light-cone gauge $A^+=0$ and using light cone notation $x^{\pm} \sim (t\pm z)$ and $y \sim \ln (x^-)$, we have \begin{equation} D_i {{d}\over {dy}} A_i = g^2 \rho (y, x_t ). \end{equation} One can write a formal solution to this equation in terms of $\rho$ and then compute the corelation function $G^{aa}_{ii}\sim < A^{a}_{i} A^{a}_{i} >_{\rho} $ where now both $\rho $ and $ \mu ^2$ depend on $y$. The result is \begin{equation} G^{aa}_{ii}= {{4(N^2_c - 1)} \over {N_c x^2_t}} \big[ 1- (x^2_t \Lambda ^2_{QCD})^{{{g^4 N_c }\over {8\pi}} \chi (y, Q^2) x^2_t } \big] \end{equation} where \begin{equation} \chi(y,Q^2)= \int_{y}^{\infty} dy^{\prime} \mu^2 (y^{\prime} ,Q^2) \nonumber \end{equation} is the average color charge squared per unit area at rapidity $y$ and resolution scale $Q^2$. \section{RG Equation for $\chi$} Next we need to determine $\chi(y, Q^2)$. We start with the valence charge density and then include hard gluons into the valence charge step by step. In other words, we integrate out higher x gluons perturbatively. This generates an effective Lagrangian and renormalizes the charge density. Iterating this procedure one can derive a RG equation for $\chi$. Diagramatically this can be represented as \hskip 2cm \epsfxsize=8.0cm \epsffile{dpf96fig1.eps} \label{fig:fig1} In the first diagram, the propagator of the hard field is in the background of classical fields. This propagator was computed in~\cite{PR}. In the second diagram the hard field propagates in the background of soft modes. This is where Sudakov and non-Sudakov form factors are expected to show up. There are also diagrams where the hard and soft modes are mixed. As a first approxiamation, we will use the following ladder diagram on the right hand side of our RG equation where the vertices are eikonalized. It is know that $\ln (x)$ terms come from eikonal vertices. \hskip 4cm \epsfxsize=8.0cm \epsffile{dpf96fig2.eps} \label{fig:fig2} Solid and dashed lines are hard and soft gluons respectively. With this approximation, the RG equation becomes \begin{equation} {{d^2 \chi }\over {dy dQ^2 }} \sim G^{aa}_{ii}(y,Q^2). \end{equation} It should be emphasized that the right hand side is still a non-linear functional of $\chi(y,Q^2) $. \section{Solution to RG Equation} In the high $Q^2$ region ($Q^2 \gg \alpha ^2 \chi $), it can be shown that our RG equation reduces to the standard DGLAP evolution equation. Assuming $\alpha $ to be independent of $Q^2$, the approximate solution is \begin{equation} \chi \sim exp \big( 2 \sqrt{{{N_c \alpha_s}\over {\pi}} y \ln Q^2 }\big). \nonumber \end{equation} In the intermediate $Q^2$ region ($Q^2 \sim \alpha^2 \chi $), the effects of the background fields as well as the soft fields are expected to be important quantitatively. However, here we are interested in the qualitative behavior of our RGE. Assuming that, as in BFKL, the transverse phase space is a constant, we get the following BFKL like behavior \begin{equation} \chi \sim exp \big(\# {{N_c \alpha_s }\over {\pi}}y\big). \nonumber \end{equation} Finally in the low $Q^2$ region ($Q^2 \ll \alpha^2 \chi $), our RG equation saturates and the solution is of the form \begin{equation} \chi = \chi_\circ + \kappa (y_\circ - y) Q^2 \nonumber \end{equation} where $\kappa $ is a slowly varying function of $Q^2$. This leads to saturation of the gluon distribution function at small $x$. As a result, cross sections computed with this distribution function would satisfy unitarity bounds. To illustrate this saturation, let us consider the behavior of $\chi(y,Q^2)$ at some fixed $Q^2$ as $x$ decreases. In the high transverse momentum region ($k^2_t \gg \alpha^2 \chi$), the distribution function will grow as some power of $x$. As $x$ gets smaller, $\alpha^2 \chi$ will eventually become larger than $Q^2$ and we will be in the saturation region (the region between $\Lambda^2_{QCD}$ and $\alpha^2 \chi$) where the distribution function is a slowly varying function of $k_t^2$. This is shown below. \hskip 1cm \epsfxsize=8.0cm \epsffile{dpf96fig3.eps} \label{fig:fig3} Notice that this saturation is due to shrinking of the transverse phase space as indicated above. \section{Acknowledgments} The work presented here has been done in collaboration with Alex Kovner, Larry McLerran and Heribert Weigert. For a full derivation of the results presented here see~\cite{JJM}.

proofpile-arXiv_065-590

\section{Introduction} In this letter we explore issues related to the interpretation of recent putative observations of deuterium in seemingly chemically unevolved hydrogen clouds along the line of sight to QSOs. These observations presently do not provide a consistent value for the deuterium abundance, D/H, in high redshift Lyman limit sytems. Measurements in several clouds suggest a \lq\lq high\rq\rq\ value, D/H$\sim 2\times{10}^{-4}$ (Songalia {\it et al.} 1994; Carswell {\it et al.} 1994; Rugers \& Hogan 1996a; Rugers \& Hogan 1996b; Carswell {\it et al.} 1996; Wampler {\it et al.} 1996); while determinations in two systems yield a \lq\lq low\rq\rq\ value, D/H$\sim 2\times{10}^{-5}$ (Tytler, Fan, \& Burles 1996; Burles \& Tytler 1996a). It is widely accepted that at least some of these observational inferences of D/H reflect the primordial value of this quantity at the conclusion of the big bang nucleosynthesis (hereafter; BBN) epoch. This belief is founded on the absence of viable alternative sites/mechanisms which could produce significant amounts of deuterium without overproducing other light elements such as $^6$Li, $^7$Li, and $^3$He (Epstein, Lattimer, \& Schramm 1976; Sigl {\it et al.} 1995). It is also widely noted that the low metallicities inferred for hydrogen clouds at high redshift generally imply only negligible amounts of deuterium depletion by stars. It is important to resolve which (if any) of the various inferred D/H values represent the cosmic average primordial abundance ({\it cf.} Cardall \& Fuller 1996; Hata {\it et al.} 1996). Here we use the term \lq\lq average\rq\rq\ since, in principle, there could exist intrinsic, primordial, super-horizon scale inhomogeneity at the BBN epoch ({\it e.g.}, isocurvature fluctuations {\it cf.} Jedamzik \& Fuller 1995). Such intrinsic inhomogeneity could give rise to the apparent discordance in observed D/H, but only if the cosmic average of this quantity is D/H$\sim {10}^{-4}$ (Jedamzik \& Fuller 1995). A real discordance in D/H is, however, not well established by the data. If the apparent discordance {\it is} established by future observations, and it does not arise from intrinsic inhomogeneity, then it must result from processes operating after the BBN epoch. It may be that the apparent discordance is simply a result of some subset of the data being wrong because, for example, hydrogen \lq\lq interlopers\rq\rq\ are mistaken for isotope-shifted Lyman-$\alpha$ lines (Steigman 1994). An erroneous (high) D/H would result if a low column density Lyman-$\alpha$ forest line by chance happened to reside at the position in velocity space where the deuterium isotope-shifted Lyman-$\alpha$ line is expected. From the observed frequency of Lyman-$\alpha$ forest lines in quasar spectra (Hu {\it et al.} 1995), one can estimate the {\it a priori} probability for any one Lyman-limit system (hereafter; LLS) to have such an interloper. This probability is given by, \begin{equation} P\approx 9\times 10^{-3}\biggl({({\rm D}/{\rm H})_p\over 10^{-4}}\biggr)^{-0.46} \biggl({N_{\rm HI}\over 3\times 10^{17} {\rm cm^{-2}}}\biggr)^{-0.46} \biggl({1+z\over 4}\biggr)\biggl({R_{\rm v}\over 10\ {\rm km\ s^{-1}}}\biggr) \Bigl(1+\xi(\Delta v)\Bigr)\ , \end{equation} and is seen to depend on the primordial $({\rm D}/{\rm H})_p$ ratio, the column density $N_{\rm HI}$ and redshift $z$ of the LLS, and the observational velocity resolution $R_{\rm v}$. The quantity $1+\xi(\Delta v)$ accounts for the possibility that Lyman-forest clouds may be \lq\lq clustered\rq\rq\ in velocity space around LLSs. A similar quantity, the clustering of forest clouds around each other, has been observationally estimated to be approximately $\xi(\Delta v)\sim 1$ for absorber velocity separations $\Delta {\rm v}\mathrel{{}^<_\sim} 100$km s$^{-1}$ (Chernomordik 1995; Meiksin \& Bouchet 1995). In practice, there is a strong observational bias to claim deuterium detections in only those clouds which show the smallest Doppler broadening of absorption lines. The expected narrow width of the deuterium line, as well as the relative widths of the deuterium and hydrogen lines, may then be used to argue against the interloper possibility on statistical grounds (Rugers \& Hogan 1996a; Burles \& Tytler 1996b). Even should this issue be resolved, there are a plethora of usually hidden and implicit assumptions and decisions which must be made in any assessment of the observational data to extract a {\it primordial} D/H. These assumptions revolve around issues of chemical evolution and formation histories of LLSs which show deuterium. Any such assumptions may be worrisome, given that even such basic aspects of LLSs as morphology, environment, and their masses are poorly understood. LLSs are clouds or sheets of highly ionized gas with temperatures around a few times $10^4$ K and with approximate neutral column densities $N_{\rm HI}\approx 3\times 10^{17}$cm$^{-2}$. It is commonly assumed that the bulk of the gas in LLSs is ionized by the diffuse UV background at high redshift. Nevertheless, local sources for the ionizing radiation such as young blue stars (York {\it et al.} 1990; Gruenwald \& Viegas 1993) or hot galactic halo gas (Viegas \& Fria\c{c}a 1995) have also been proposed. It is even difficult to eliminate entirely the possibility that a particular LLS is the result of looking through the gas of one, or a few, planetary nebulae. It is instructive to estimate typical parameters of a LLS such as total baryon mass, spatial dimension, and total hydrogen density. Under the assumption of heating/cooling equilibrium and/or ionization equilibrium of the cloud with the background ionizing radiation, and further assuming spherical geometry for the cloud with a line-of-sight passing close to the center of the cloud, one finds: the total baryon mass, \begin{equation} M_b\approx 4\times 10^6 M_{\odot} \biggl({U\over 10^{-3}}\biggr)^{5.2} \biggl({J_0\over 10^{-21}{\rm ergs\ cm^{-2}s^{-1}Hz^{-1}}}\biggr)^{-2} \biggl({N_{\rm HI}\over 3\times 10^{17}{\rm cm^{-2}}}\biggr)^3\ ; \end{equation} the radius, \begin{equation} R\approx 2\ {\rm kpc} \biggl({U\over 10^{-3}}\biggr)^{2.07} \biggl({J_0\over 10^{-21}{\rm ergs\ cm^{-2}s^{-1}Hz^{-1}}}\biggr)^{-1} \biggl({N_{\rm HI}\over 3\times 10^{17}{\rm cm^{-2}}}\biggr)\ ; \end{equation} and the total hydrogen density for the cloud, \begin{equation} n_H\approx 5\times 10^{-3}{1\over {\rm cm^3}} \biggl({U\over 10^{-3}}\biggr)^{-1} \biggl({J_0\over 10^{-21}{\rm ergs\ cm^{-2}s^{-1}Hz^{-1}}}\biggr)\ . \end{equation} In these expressions $U$ is the ionization parameter, i.e. the ratio of the density of ionizing photons (with energies $E_{\gamma}>13.6$ eV) to the total hydrogen number density, and $J_0$ is the specific intensity of ionizing photons at $E_{\gamma}=13.6$ eV. The ionization parameter is inferred from either the relative abundances of ionization states of \lq\lq metals\rq\rq\ or the inferred temperature of the cloud (Donahue \& Shull 1991). Typical uncertainties in $U$ are about one order of magnitude implying a five order of magnitude uncertainty in the mass scale of a spherical LLS. Similarly, even under the assumption that the diffuse UV-background is the source of the ionization of the cloud, there is considerable uncertainty in $J_0$, translating into uncertainty in the basic cloud parameters. We conclude that not only is it difficult to determine the masses of the objects in which D/H ratios are observationally inferred, but it is also uncertain how to translate these D/H ratios into a cosmic average. In principle, it is difficult to rule out very small masses for LLSs. Such small clouds could have been subject to significant local deuterium destruction or production. Numerical simulations (Cen {\it et al.} 1994; Katz {\it et al.} 1996) suggest that there are two broad classes of hydrogen absorption systems with hydrogen column densities sufficiently high $(\mathrel{{}^>_\sim} 3\times 10^{17}{\rm cm^{-2}})$ to be considered Lyman limit absorbers: (1) \lq\lq field\rq\rq\ clouds which are distinct and isolated from (proto) galactic systems; and (2) the tenuous outer regions of an otherwise massive (proto) galactic disk or halo. In the first case of isolated field clouds, the geometries are not well determined and they could be compact spherical systems or extended sheets. One may imagine that the formation and chemical evolution histories of these two classes of Lyman limit absorbers are different. The question of whether different chemical evolution histories in clouds could give rise to inhomogeneity in the observed $D/H$ ratios requires resolution. Chemical evolution calculations are characterized by specifications of an initial mass function (IMF) and a star formation rate. We follow the notation of \markcite{MC96} Malaney \& Chaboyer (1996) and take the star formation rate $\Psi$ (in Gyr$^{-1}$) and the IMF $\phi (m)$ in $M_{\odot}^{-2}$, so that $\Psi /\Omega_g$ represents a typical inverse time scale for consumption of baryons into stars and $m\phi (m) dm$ is the fraction of mass going into stars within the stellar mass range $m$ and $m+dm$. Here $\Omega_g$ is the fractional contribution of cold gas in damped Lyman-$\alpha$ systems to the critical density and takes values of $\Omega_g\sim 0.003$ at redshift $z\approx 3-4$ (Lanzetta, Wolfe, \& Turnshek 1995; Storrie-Lombardi {\it et al.} 1995). In this notation the evolution of cold gas with time can be written as, \begin{equation} {d\Omega_g(t)\over dt}=-\Psi (t) +\int_{m_l(t)}^{m_{up}}\bigl(m-m_r)\Psi \bigl(t(z)-\tau (m)\bigr)\phi (m)dm\ , \end{equation} whereas the evolution of the deuterium mass fraction $X_D$ with time is given by, \begin{equation} {dX_D(t)\over dt}=-{X_D(t)\over \Omega_g(t)}\int_{m_l(t)}^{m_{up}}\bigl(m-m_r) \Psi\bigl(t(z)-\tau (m)\bigr)\phi (m)dm\ . \end{equation} Here $m_{up}$ represents the mass of the largest stars formed, $m_r$ is the remnant mass of a star of mass $m$, and $m_l(t)$ is the lowest stellar mass which could have returned its gas to the interstellar medium within the age of the universe $t(z)$ (i.e. the lifetime $\tau$ of a star with mass $m_l(t)$ has to satisfy $\tau (m_l)=t(z)$). We may approximate the evolution of the deuterium mass fraction if we assume a constant star formation rate (and IMF), neglect remnant masses, and approximate $\Omega_g$ and $m_l(t)$ as constant. This yields $X_D(t)=X_D(0){\rm exp}(-t/\tau_D)$, with $\tau_D$ the typical time scale for deuterium destruction, \begin{equation} {1\over\tau_D}={\Psi\over\Omega_g}\int_{m_l\bigl(t(z)\bigr)}^{m_{up}}m\phi (m)dm\ , \end{equation} such that $\Psi /\Omega_g$ is the characteristic time scale for incorporation of baryons into stars and the integral is the fraction of stellar material which has been returned to the ISM by redshift z. It has become possible recently to derive constraints on the average star formation rate and IMF from observations of damped Lyman-$\alpha$ systems (Timmes, Lauroesch, \& Truran 1995; Malaney \& Chaboyer 1996). In order to be consistent with the observed decline in $\Omega_g(z)$ with decreasing redshift, Malaney \& Chaboyer (1996) argue that typical average star formation rates are $\Psi\approx 10^{-2.5}$Gyr$^{-1}$ for $3\mathrel{{}^<_\sim} z\mathrel{{}^<_\sim} 4$. Star formation rates in this range would imply a characteristic time scale for incorporation of baryons into stars of only $\sim 1$Gyr. Discounting the possibility of outflow, and assuming IMF's close to standard (Salpeter), the predicted metal enrichment by Malaney \& Chaboyer (1996) is also in rough agreement with the observed metallicities in damped systems (Lu, Sargent, \& Barlow 1996). However, average deuterium destruction factors ${\rm exp}(-t/\tau_D)$ are predicted to be small ($\sim 1-5\%$) in the redshift range $3\mathrel{{}^<_\sim} z\mathrel{{}^<_\sim} 4$, mainly because it is thought that only a small fraction of stellar material (0.1-0.2) has been returned to the ISM. The question arises as to how one could change the IMF and/or star formation rate in evolving LLSs in order to \lq\lq achieve\rq\rq\ significant deuterium destruction. This may be done locally in stochastic chemical evolution scenarios or globally by using non-standard chemical evolution scenarios which incorporate, for example, a peaked IMF and/or mass/metal outflows. In an example taken from {\it galactic} chemical evolution, it has been shown recently that destruction of deuterium by a factor of 10 between epochs at high redshift and the time of solar system fromation may be possible in models which employ an early metal-rich galactic wind (Scully {\it et al.} 1996). Nevertheless, stringent limits can be placed on the maximum possible deuterium destruction in individual LLSs at high redshift by stars with masses below $M\mathrel{{}^<_\sim} 40M_{\odot}$, provided the abundances of certain key isotopes are determined confidently. Stars have to be massive enough so that their main-sequence lifetimes are shorter than the age of the universe at redshift $z\sim 3-4$. This implies that only stars with masses $M\mathrel{{}^>_\sim} 2M_{\odot}$ could have contributed to a possible deuterium depletion in the interstellar medium. Note that this lower mass cutoff is fairly insensitive to the adopted cosmology, the value of the Hubble parameter, and the precise redshift of the LLS. Stars in the mass range $2M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 4M_{\odot}$$2M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 4M_{\odot}$, are generally believed to be significant ${}^{12}$C producers. The $^{12}$C is transported to the surface of the star during dredge-ups, when the base of the convective zone reaches shells which are highly carbon enriched, and subsequently returned to the ISM in planetary nebulae ejecta. The ejecta of AGB stars with $2M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 4M_{\odot}$ have typical ${}^{12}$C/H ratios which are between 0.1 and 10 times the solar ratio, depending on stellar mass, metallicity, and the details of the dredge-up processes (Iben \& Truran 1978; Renzini \& Violi 1981; Forestini \& Charbonnel 1996). Most models predict ${}^{12}$C/H ratios a few times solar. More massive AGB stars, $4M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 8M_{\odot}$, may in fact be net destroyers of ${}^{12}$C/H (cf. Forestini \& Charbonnel 1996). The ejecta of massive stars $M\mathrel{{}^>_\sim} 8M_{\odot}$, which undergo Type II supernova explosions, are generally expected to be enriched in ${}^{12}$C, but also heavier isotopes such as ${}^{28}$Si and ${}^{56}$Fe with typical mass fractions of one to a few times the corresponding solar mass fraction (Woosley \& Weaver 1995). Here production factors become less certain for massive stars $M\mathrel{{}^>_\sim} 30-40M_{\odot}$, in particular for the heavier isotopes. The observational determination of carbon and silicon abundances in LLSs ({\it e.g.}, [C/H]=-2.2 and -3.0 for the two clouds in the system at $z=3.572$ determined by Tytler {\it et al.} 1996 from the observations of the carbon ionization states CII, CIII, and CIV) can be used to constrain stellar deuterium depletion. Adopting moderate carbon production of one times solar over the stellar mass range $2M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 4M_{\odot}$ and $8M_{\odot}\mathrel{{}^<_\sim} M\mathrel{{}^<_\sim} 40M_{\odot}$, and using $[C/H]=-2$ for the LLS, one can infer that not more than $\sim 1\%$ of the gas in the LLS could have been cycled through stars in the above given mass range. This implies that deuterium depletion by most stars with $M\mathrel{{}^<_\sim} 40M_{\odot}$ cannot exceed about $1\%$. Note that this constraint can {\it not} be circumvented by metal-rich winds (outflow), because the {\it same} stars which deplete deuterium also produce ${}^{12}$C abundantly. Moreover, low observed ${}^{12}$C abundances significantly reduces the possibility that a given LLS results from a line-of-sight passing through one or a few deuterium-depleted planetary nebulae. If one imposes the constraint that significant deuterium depletion by stars must have occurred, there are only a few, seemingly highly unlikely, possibilities. Chemical evolution could have proceeded via a sharply peaked IMF at $M\approx 6M_{\odot}$. Observational consequences of such a scenario may include the significant enrichment of the LLS in other isotopes, such as ${}^{14}$N. As a second possibility, a large fraction of material may have been cycled through an early generation of supermassive stars $M\mathrel{{}^>_\sim} 1000M_{\odot}$ which eject a substantial fraction of their initial mass in deuterium-depleted radiation-driven winds (Fuller, Woosley, \& Weaver 1986) enriched only in $^4$He. Perhaps direct inference of black hole remnants is the only way to establish the viability of such a scenario. It may be possible that the carbon abundance in a LLS is underestimated, since either the dominant carbon ionization state is CI or carbon is depleted on grains. Whereas one can place observational constraints on the CI abundance (Burles 1996), the existence of dust in LLSs is not easily observationally constrained. However, it seems unlikely that significant amounts of dust in LLSs could survive evaporation by the ambient ionizing radiation field at high redshift. Lastly, it may be that carbon production, and particularly the dredge-up processes in AGB stars, are not well understood for low-metallicity stars. Deuterium may also be produced or destroyed by nuclear photo-disintegration in the presence of a $\gamma$-ray source: ${}^4$He($\gamma$,pn)${}^2$H; ${}^4$He($\gamma$,${}^2$H)${}^2$H; or ${}^2$H($\gamma$,n)p. For most $\gamma$-ray sources, production of $^2$H dominates over destruction because the number density of ${}^4$He targets is much larger than that of ${}^2$H targets. In fact, ${}^4$He photo-disintegration has been proposed as an efficient non-BBN source for deuterium (Gnedin \& Ostriker 1992), even though it has been subsequently shown that this would yield anomalously large ${}^3$He/${}^2$H$\sim 10$ ratios in conflict with the presolar abundance ratio ${}^3$He/${}^2$H$\sim 1$ (Sigl {\it et al.} 1995). In any case, in the absence of direct ${}^3$He abundance determinations, one may posit that a LLS is enhanced (or depleted) in deuterium since it had once been close to a powerful $\gamma$-ray source. Assume, for example, the existence of a population of $\gamma$-ray bursters at redshift $z_b\mathrel{{}^<_\sim} 1000$ each of which radiates a flux with spectrum hard enough to produce $\gamma$-ray energies slightly above the ${}^4$He($\gamma$,${}^2$H)${}^2$H threshold, $E_{th}\approx 23\,{\rm MeV}$. In order for these $\gamma$-ray bursters not to overproduce the diffuse x/$\gamma$-ray background at the present epoch, the comoving $\gamma$-ray burster density has to be smaller than, \begin{equation} N_{\gamma}^c\mathrel{{}^<_\sim} {1\over (10{\rm Mpc})^3}{1\over (1+z_b)} \biggl({j_{\gamma}(z=0,{E_{th}/(1+z_b)})\over 10^{-5}{\rm MeV^{-1}cm^{-2}s^{-1}sr^{-1}}}\biggr) \biggl({E\over 10^{60}{\rm ergs}}\biggr)^{-1}\ , \end{equation} where $j_{\gamma}$ is the specific x/$\gamma$-ray intensity at the present epoch determined at the energy $E_{th}/(1+z_b)$ and $10^{-5}{\rm MeV^{-1}cm^{-2}s^{-1}sr^{-1}}$ is the approximate present specific intensity at $E_{\gamma}\approx 20\,{\rm MeV}$ (Fichtel {\it et al.} 1977). Here $E$ is the total energy in $\gamma$-rays above threshold in a single burst. An adopted approximate comoving distance between $\gamma$-ray bursters of $r_c\sim 10\,{\rm Mpc}$ should be compared to the maximum distance by which an individual LLS could have been separated from a $\gamma$-ray burst in order to still have had significant deuterium production by ${}^4$He photo-disintegration. This comoving distance is, \begin{equation} r_p^c\mathrel{{}^<_\sim} 10^{-2}{\rm kpc}\ (1+z_b)\biggl({E\over 10^{60}{\rm ergs}}\biggr)^{1\over 2} \biggl({({}^4{\rm He}/{}^2{\rm H})_p\over 2.8\times 10^3} \biggr)^{1\over 2}\ , \end{equation} where (${}^4$He/${}^2$H)$_p$ is a primordial number ratio. These distances indicate that significant deuterium production, as well as destruction, by ${}^4$He/${}^2$H photo-disintegration should be regarded as an improbable process. Spatially varying (D/H) ratios at high redshift, if they exist, may have their origin in the intermediate mass scale primordial inhomogeneity of the baryon-to-photon ratio. Jedamzik \& Fuller 1995 pointed out that such primordial isocurvature fluctuations may yield order unity (D/H) fluctuations on galactic mass scales ($M\simeq 10^{10}-10^{12}M_{\odot}$) and fluctuations in (D/H) by a factor $\sim 10$ on the post-recombination Jeans mass scale ($M_J\simeq 10^5-10^6 M_{\odot}$). Nevertheless, such scenarios of BBN can only agree with observationally inferred primordial abundance constraints if a variety of criteria are met, such as the efficient collapse of high-density regions, the presence of a cutoff for isocurvature fluctuations on mass scales $M\mathrel{{}^<_\sim} M_J$ (cf. Jedamzik \& Fuller 1995; Gnedin, Ostriker, \& Rees 1995; Kurki-Suonio, Jedamzik, \& Mathews 1996), and the moderate to significant ${}^7$Li-depletion in low-metallicity PopII stars. Note that contrary to recent claims (Copi, Olive, \& Schramm 1996) models which predict intrinsic fluctuations in (D/H) on the LLS-scale are {\it not} generally ruled out by the isotropy of the CMBR. Future observations of (D/H) ratios in different LLSs may constitute the first test for the presence or absence of baryon-to-photon fluctuations on intermediate mass scales. In conclusion, it is difficult to envision a {\it compelling} model for differential D/H destruction/production in LLSs that could explain the apparent observationally-inferred discordance. The logical leading candidate for such a model is anomalous/stochastic chemical evolution involving a finely tuned star formation rate or IMF. However, we have argued that most of such models may be ruled out by $^{12}$C overproduction. In any case, future observations of additional LLSs showing deuterium may give insight into the resolution of this problem: either (1) mis- identification or -analysis of deuterium lines in LLSs; (2) super-horizon scale primordial inhomogeneity at the BBN epoch; or (3) very finely tuned IMF and star formation rates ({\it i.e.} quite different from those inferred from galactic chemical evolution considerations) in some LLSs. With the advent of the Sloan Digital Sky Survey one may expect a substantial increase in the number of known bright quasars in the near future. It has been estimated that this may yield of the order $\sim 100$ LLSs suitable for the determination of (D/H) ratios (Hogan 1996). With the help of this data one may gain important new insights into chemical/stellar evolution and the galaxy formation problem. \acknowledgments We wish to thank B. Balick, S. Burles, C. Cardall, C. Hogan. J. Prochaska, D. Tytler, S. Viegas, and A. Wolfe for useful discussions. We also acknowledge the hospitality of the Institute for Nuclear Theory at the University of Washington where a substantial part of this research has been performed. This work was supported by NSF Grant PHY-9503384 and NASA Theory Grant NAG5-3062 at UCSD, and under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48 and DoE Nuclear Theory grant SF-ENG-48.

proofpile-arXiv_065-591

\section{Introduction} The environment at the centers of galaxy cluster cooling flows is exotic and extreme compared to that found near the average galaxy in the Universe. The intracluster medium (ICM) thermal pressure is a factor of 10-100 times higher than that near other cluster galaxies. The cooling flow may deposit $\ge$100 M$_{\odot}$/yr onto the central galaxy in such clusters. Thus, this gaseous environment may be a particularly good one to trigger radio source production, confine the extended radio plasma, and shape the extended radio emission if asymmetries (i.e., pressure gradients) exist in the flow. With such a potentially strong relationship between the environment and the radio source, radio emission associated with central dominant galaxies may be a useful, generally unappreciated probe of cooling flows. In $\S$2 of this paper, we begin by presenting a brief overview of cluster radio sources so that radio galaxies in cooling flows can be placed into a broader relational context. Next, in $\S$3, the radio continuum properties of supergiant cDs which lie at the centers of clusters are described. In $\S$4, we discuss the apparent strong relationship between radio structure and the cooling flow environment by comparing VLA and new {\it ROSAT} HRI images. Then, in $\S$5, numerical simulations of supersonic radio jets in a cooling flow atmosphere are presented in an effort to gain insight into the radio/X-ray relationships. In $\S$6, we briefly discuss clusters without cooling flows and present new hydro/N-body simulations which show how cluster-cluster mergers can destroy cooling flows. Finally, in $\S$7, we summarize our conclusions. \section{Overview of Cluster Radio Sources} Much of what follows in this section has been taken from a series of 7 papers which have now been published on the results of the VLA 20-cm survey of $z\le0.09$ Abell clusters (see e.g., Ledlow \& Owen 1995a,b,1996; Owen \& Ledlow 1997, and references therein). Cluster radio source morphologies are predominately Fanaroff \& Riley (1974) class I (i.e., low power, edge darkened). Examples include narrow-angle tailed (NAT) U-shaped sources such as NGC 1265 in Perseus, wide-angle tailed (WAT) sources such as 3C 465 in A2634, and twin-jet sources such as 3C31 (see Owen \& Ledlow 1997 for many examples). In general, cluster radio source structures are very complex and probably reflect the local ``weather conditions" within the ICM (Burns 1996). It should be noted that Fanaroff-Riley II sources (i.e., high power, edge-brightened, classical doubles) are rare in nearby rich clusters, but do occur $\approx$7\% of the time in a statistical sample of radio galaxies in Abell clusters (Ledlow \& Owen 1996). Another intriguing, and poorly understood, property of cluster radio sources is the fact that the radio luminosity function (RLF) is virtually identical for sources inside and outside of rich clusters (Fanti 1984; Ledlow \& Owen 1996). This is very surprising since the higher thermal gas pressure inside rich clusters is expected to better confine extended radio sources (and reduce adiabatic expansion losses) and enhance source B-fields, thus creating more luminous radio sources than in radio galaxies outside rich clusters. However, this picture is clearly too simple. The probability of a cluster galaxy being a radio source was found to be independent of the global cluster galaxy density (Ledlow \& Owen 1995a). This, too, seems contrary to the conventional wisdom where (for, at least, the more powerful radio galaxies) the richness of the galaxy neighborhood was thought to be important in triggering radio emission in galaxies (e.g., Heckman et al.~1986). In an effort to gain some understanding of the complex relationship between extended radio sources and their gaseous environs, we have compared the radio structures with X-ray images of the ICM from {\it{Einstein}} and {\it{ROSAT}} for samples of Abell clusters. The initial results are intriguing. We found that larger radio sources in Abell clusters often coincide with clumps of X-ray emission (Burns et al.~1994; Burns 1996) or appear in clusters with significant substructure. For example, for a sample of 17 clusters with 26 NATs, 88\% have significant X-ray substructure whereas only 23\% of a comparable sample of radio-quiet Abell clusters have substructure (Bliton et al. 1997). We also investigated the X-ray emission around a complete sample of 25 $z<0.05$ radio galaxies not cataloged to be in rich clusters; we found that 90\% have X-ray emission similar to the X-ray clumps around radio galaxies in Abell clusters and their optical environments are similar to those in poor clusters (Owen et al.~1996). From these new data, it appears that the local, not the global, cluster environment is most important in influencing the radio structure. Similarities in morphologies and RLFs between radio galaxies inside and outside of rich clusters may be caused, at least in part, by similarities in local environments. It appears that radio galaxies prefer to exist within clusters with complex structures/potentials, which may be evolving as a result of recent cluster/subcluster merging. \section{Radio Continuum Properties of cD Galaxies} Supergiant (D \& cD) galaxies are commonly found at the centers of clusters, especially those clusters with cooling flows. These galaxies are exceptional at virtually every wavelength band, including the radio (Ball, Burns, \& Loken 1993). Whereas a typical elliptical in a cluster has only a 14\% probability of being radio-loud (Ledlow \& Owen 1996), a cD galaxy in a non-cooling flow cluster has a slightly higher 20\% likelihood of being a radio source. This increases to 60-70\% for a cD in a cooling flow cluster (Burns 1990; Ledlow et al.~1997). Thus, the large mass and optical brightness of the cD enhances its probability of being a radio source over that of an average elliptical, but its presence in a cooling flow greatly increases the chances of being radio-loud. Furthermore, the average radio luminosity ($\sim$10$^{42}$ ergs/sec) suggests that the radio plasma energy can be a significant fraction of the thermal energy in the cooling flow if the radio luminosity is only $\sim$1\% of the plasma kinetic energy; thus, the radio sources may be dynamically important influences on the cooling flow as discussed further in the next section. For a sample of cDs in Abell clusters with {\it{Einstein}} or {\it{ROSAT}} all-sky survey (RASS) images, we do not find any correlation between $L_X$ and the $P_{6cm}$ (6-cm radio power). However, there may be a weak relationship between \.M and $P_{6cm}$, such that low \.M clusters tend to have low power radio sources (Burns 1990). There are, though, prominent exceptions such as clusters with WATs which have powerful, extended radio sources yet are rarely ever found in a cooling flow. A wide variety of radio morphologies are associated with cD galaxies. Many cDs have typical jet/lobe radio structures (e.g., sources in A1795, A2029, A2199), others have compact ($<$10 kpc) morphologies (e.g., A496), and those not in cooling flows sometimes have wide-angle tails (e.g., A2634). Interestingly, there is a class of radio morphology that appears to exist only in clusters with cooling flows, which have been dubbed ``amorphous" (Burns 1990). These sources are typically 100-400 kpc in diameter, have steep radio spectra, strong cores, and diffuse, quasi-spherical structure with little signs of collimated emission such as jets or lobes. Examples include 3C 84/NGC 1275 in Perseus (Burns et al.~1992) and 3C317 in A2052 (Zhao et al. 1993). It appears that the cooling flows in such clusters have either disrupted the radio jets or prevented them from forming (Soker \& Sarazin 1988; Loken et al.~1993). Most of the radio sources associated with cDs in cooling flow clusters lie within the optical envelopes of the radio galaxies ($\le$50 kpc). Virtually all these radio sources have exceptionally steep radio spectra ($\alpha \approx 1-2$, where $S\propto {\nu}^{-\alpha}$). This suggests that the sources are confined by the high thermal gas pressure at the cores of the cooling flows and allowed to spectrally age. However, cluster radio sources are often found to have equipartition pressures that are lower than the ICM thermal pressure (e.g., Taylor et al.~1990; Feretti et al.~1992; Burns et al.~1995). This may suggest that entrainment of ICM thermal gas by the radio sources is an important component of the radio plasma which is not included in the equipartition calculation. \section{Relationship Between Radio Structure \& Cooling Flow Environment} Since the radio sources identified with central galaxies in cluster cooling flows are generally small, high resolution X-ray observations are needed if one is to investigate the relationship between the radio and the ICM/ISM plasmas. {\it ROSAT} HRI imaging with $\approx$5$\arcsec$ resolution is now becoming available for a number of cooling flow clusters which allows for detailed comparison with the radio structures. Although the numbers of observations are still small, some possible trends are beginning to emerge. We will present our initial impressions here of these trends by dividing these clusters into 5 categories. {\it Radio Sources Blowing Bubbles.} 3C 84/NGC 1275 in Perseus is a beautiful example of radio plasma in the core of the central galaxy apparently evacuating X-ray cavities. The HRI images presented by B\"ohringer et al.~(1993) show an anticorrelation between the inner radio structure and the X-ray emission within a radius of $\approx$1$\arcmin$ of the AGN core. The X-ray emission associated with the inner cooling flow is highly asymmetric and appears to have been strongly influenced by the presence of the radio source. The radio source has a relatively high total radio power of $P_{6cm}=10^{25}$ W/Hz. {\it X-ray Enhancements on the Edges of Radio Sources.} We have recently analyzed the HRI images of 2 nearby cooling flow clusters, A133 and A2626, which contain steep-spectrum radio sources. These are only moderately powerful radio sources at 6-cm ($P_{6cm}=4\times10^{22}$ W/Hz for A133 and $P_{6cm}=5\times10^{22}$ W/Hz for A2626). After subtracting out circular model-fits to the X-ray isophotes, we find X-ray excesses which nicely correlate with the steep-spectrum extended radio emission. The example of A133 is shown in Fig. 1. We find that inverse Compton scattering of microwave background photons by the relativistic electrons in the radio plasma predicts X-ray emission which is $\sim$100 times less than what is observed. So, it appears that the X-ray excess is more likely to be hot gas (possibly ICM/ISM gas compressed by the bow shock of the radio source, see below); we estimate the X-ray ``clouds'' to have masses of $10^5-10^6$ M$_{\odot}$. \begin{figure}[tbh] \epsfxsize=4.2in \epsfbox{fig1.ps} \caption{Grey-scale image of the radio source in Abell 133 overlaid with contours representing the residual X-ray emission after subtraction of a circularly-symmetric model of the cluster emission. Contours to the south of the radio core show a negative hole. Note the strong spatial correlation between the positive excess X-ray emission to the north of the radio core ($\approx$10$\sigma$) and the radio-emitting plasma.} \end{figure} {\it Bubbles and X-ray Enhancements.} Cygnus A is an example of a cooling flow cluster which shows both X-ray enhancements near the outer hot spots of the radio lobes as well as X-ray holes within the lobes closer to the nucleus (Carilli, Perley, \& Harris 1994). Clarke \& Harris (1996) used hydro simulations to explain these observations -- the enhancements are compressed gas near the source bow shock and the X-ray holes are produced by gas which has been evacuated by the radio plasma propagating through the cooling flow. Cygnus A is, of course, one of the most powerful classical double sources, $P_{6cm}=2\times10^{27}$ W/Hz. {\it X-ray Extensions Between the Radio Tails/Lobes.} We have also begun to see a few interesting examples of correlations between asymmetries in the inner cooling flows and bent radio structures. A2597 (Sarazin et al.~1995; $P_{6cm}=5\times10^{24}$ W/Hz) and A2199/3C 338 (Owen \& Eilek 1997; $P_{6cm}=8\times10^{23}$ W/Hz) show excess X-ray emission between the radio tails or lobes of these small linear size sources. Both the X-ray asymmetries and the bent radio sources suggest that there are important pressure gradients within the inner cooling flow regions of these cD galaxies. {\it Alignment of Radio Source with Dust \& Emission-line Gas.} Finally, a recent, spectacular example of an unanticipated correlation between inner dust lanes, outer emission-line gas, and extended radio features has been revealed for A1795 using new HST imaging (Pinkney et al.~1996; McNamara et al.~1996). Clearly, these phenomena are all related in a yet-to-be-determined manner, making the inner cooling flow environment far more complex than previously believed. In summary, the new HRI and HST images for cooling flows with central radio sources clearly show them to be highly nonspherical. Both the radio luminosities and now the X-ray/radio correlations suggest that the radio plasma has an important impact on the central structure of the cooling flow. The highest radio power sources (e.g., Cygnus A \& 3C 84) are capable of blowing bubbles or evacuating cavities in the inner cooling flow region, whereas lower radio power sources can produce X-ray excesses which coincide with the outer portions of the radio structures. \section{Numerical Simulations of Radio Jets in a Cluster Cooling Flow} In an effort to understand the complex relationship between extended radio sources and the cooling flow environment, Loken et al.~(1993) performed the first numerical simulations of supersonic radio jets propagating through a realistic cooling flow atmosphere. These 2-D hydrodynamics simulations launched 4 jets of differing Mach numbers (3, 6, 12, and 50 as measured with respect to the jet internal sound speed) into a steady-state cooling flow. One example of the end state of a 1.3 kpc long, M=12 jet is shown in Fig. 2. \begin{figure}[h] \plotone{fig2.ps} \caption{Grey-scale image of gas density at a late stage in the evolution of a perturbed, 2D, Mach 12 jet in a cooling flow atmosphere. The jet has disrupted and is forming a large, low-density (dark) turbulent lobe. The region shown is 200 jet radii long. See Loken et al.~(1993) for more details. } \end{figure} For a laminar jet, one can show using simple ram pressure balance that the morphology and length of a radio jet should be a strong function of jet and cooling inflow Mach numbers (Loken et al.~1993). Clearly, lower Mach number jets (lower thrust) should not travel as far into a high inflow velocity cooling flow as would a high Mach number jet in a low velocity cooling flow. However, this naive calculation does not consider the fact that fluid shear or Kelvin-Helmholtz instabilities play a dominant role as the jet fluid attempts to propagate through the inflowing cooling cluster gas. The jets become unstable, ultimately disrupt, and stagnate as shown in Fig. 2. The disruption length of a jet is once again a strong function of jet and cooling flow Mach numbers. Thus, a M=3 jet stagnates close to the radio core, whereas a high Mach number jet can ``blow through" the central cooling flow with little effect. This may be why low luminosity radio sources in central cooling flow galaxies are generally small (low thrust jets), whereas a powerful source such as Cygnus A is able to grow to a diameter of $\approx$200 kpc. For the M=12 jet simulation shown in Fig. 2, we have computed the projected X-ray brightness distribution in the {\it ROSAT} band produced by the hot ISM/ICM surrounding the jet. We find that the X-ray brightness around the jet is $\approx$10-20\% higher than would be expected from the cooling flow alone. It appears that the jet bow shock has compressed and heated the gas in a shell around the radio source. Since this jet is within the inner $\approx$2 kpc of the cooling flow center where the gas temperature is only $\approx$10$^6$ K, the bow shock heating produces a significant enhancement in X-ray emissivity in the {\it ROSAT} band. Thus, X-ray excesses on the edges of radio components such as that seen in Fig. 1 may be understood as shocked, compressed gas produced as the radio plasma propagates through the cores of cooling flows. \section{Cluster-Cluster Mergers \& the Death of Cooling Flows} Although many (possibly most) galaxy clusters contain cooling flows, there are also a number of clusters whose temperature and X-ray surface brightness profiles suggest a more isothermal gas at the cluster core. Two good examples of such classes of non-cooling flow clusters are those which contain wide-angle tailed radio galaxies (e.g., G\'omez et al.~1997) and radio halos (Burns et al.~1995). Both of these classes of clusters show strong evidence for substructure in X-ray images (e.g., Briel et al.~1991 for the radio halo cluster A2256) and in cluster velocity distributions (e.g., Pinkney et al.~1993 for the WAT cluster A2634). These data have been used to model these clusters as candidates for cluster-cluster mergers (e.g., Roettiger et al.~1995; 1997). Do such mergers also destroy previously existing cooling flows? To address this question, we performed 2-D simulations of a merger between two idealized spherical clusters with a mass ratio of 4:1 using our hydro/N-body code (see Roettiger et al.~1997 for details). In the larger cluster with $10^{15}$ M$_{\odot}$, $T$=11.7 keV, and $r_{core}$=250 kpc, a steady-state cooling flow with mass dropout was initially evolved in 1-D and then placed within the 2-D grid. The second cluster was initially isothermal with $T$=6.6 keV and $r_{core}$=157 kpc. The two clusters were allowed to fall together under the influence of gravity. In this simulation, cooling via bremsstrahlung and line radiation was activated using the cooling function described in Westbury \& Henriksen (1992). Three epochs during the cluster-cluster collision are displayed in Fig. 3. The net effect of the collision is strong shock heating of the cooling flow core which causes it to expand, the temperature increases in the core from 1.5 keV to 15 keV, and the cooling time increases to $>20$ Gyrs in the final epoch. Thus, for the particular parameters in this simulation, the cooling flow is effectively destroyed and will not become re-established within a Hubble time. \begin{figure}[bt] \plotone{fig3.ps} \caption{Contours of gas density at 3 stages during the evolution of a cluster merger. The subcluster falls from the right towards the 4-times more massive, cooling flow cluster (first panel). Shortly after the cores overlap, there are elongations in the gas distribution both perpendicular and parallel to the merger axis (middle panel). 3 Gyrs after the merger (final panel), the main cluster is relatively spherical but there is no evidence for a cooling flow (note less compact core than in first panel). Axes are labelled in Mpc; times are relative to core-crossing. } \end{figure} It is probably premature to generalize this result to the life history of most clusters. This is a ``hard", head-on collision (high momenta) between two massive clusters which may be relatively rare in the Universe. Although cosmological large-scale structure simulations do find that clusters grow hierarchically through mergers and accretion of gas and dark matter along filaments (e.g., Bryan et al.~1994; Loken et al.~1997), this process may generally be softer and not destroy a cooling flow in all cases. Such a picture must be true to explain both the preponderance of substructure and cooling flows in nearby clusters. We are in the process of performing additional simulations with higher mass ratios to explore this issue further. \section{Summary} There appears to be a very strong correlation between radio emission and the presence of cooling flows around central galaxies in clusters. Nearly three-quarters of cD galaxies at the centers of cooling flows are radio-loud whereas only 14\% of average cluster ellipticals have radio emission in surveys of Abell clusters. Radio sources associated with dominant galaxies in cooling flows tend to be small ($<50$ kpc in diameter) and have steep radio spectra ($\alpha$=1-2). A class of radio source, termed amorphous, which have little collimated emission (i.e., jets or lobes) and instead have a core-halo (steep spectrum halo) morphology are found to exist only in cooling flows. New {\it ROSAT} HRI images reveal strong interactions between radio plasma and the centers of cooling flows. ``Bubbles" or evacuated cavities are seen in the X-ray emission in cooling flows around the most powerful radio sources such as 3C 84 in the Perseus cluster and near the classical double Cygnus A. In other cases such as A133 and A2626, enhancements of X-ray emission are seen on the edges of the extended radio emission, possibly produced by compression and heating of the ISM/ICM by the radio source bow shock as suggested by numerical simulations. Finally, in yet other cases such as A2199 and A2597, we observe alignments between bent radio jets/tails and extended central X-ray emission. In virtually every cooling flow cluster with a central radio source, the X-ray emission is observed to be highly asymmetric. Such properties need to be considered in modeling cooling flows (see e.g., Garasi et al.~1997). New hydro/N-body simulations suggest that cooling flows can be destroyed by low mass ratio cluster-cluster collisions. Such collisions are consistent with X-ray and optical observations of clusters which contain WAT radio galaxies or cluster halos -- neither of which possess cooling flows. Further simulations will be needed to reconcile the observed abundance of both cooling flow clusters and substructure (i.e., merging) within clusters. \acknowledgments We thank our colleagues who have contributed to this work including Wolfgang Voges, Anatoly Klypin, Kurt Roettiger, Neal Miller, Rusty Ball, and Jason Pinkney. This research was supported by grants from the U.S. National Science Foundation (AST-9317596) and NASA (NAGW-3152). NRAO is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation.

proofpile-arXiv_065-592

\section{Introduction} {\Large 1. Introduction} Path integrations over the group manifolds or over the homogeneous spaces are frequenly used for solving the path integrals of the quantum mechanical potentials \cite{kn:gros},\cite{kn:com},\cite{kn:per}.For example, P\"{o}schl-Teller, Hulthen and Wood-Saxon potentials can be related to the group SU(2) \cite{kn:dur}.Path integrals over the SU(1,1) manifold is studied for solving the modified P\"{o}schl-Teller potential \cite{kn:bom}. $V=-\cosh^{-2} \omega x $ i.e. the symmetric Rosen-Morse potential is the special case of the potentials already mentioned in the above paragraph, which can also be solved by transforming its Green function into the Green function of the particle motion over the SO(3) manifold \cite{kn:ih}. When the sign of the above potential is changed, that is when we consider the well known potential barrier $V=\cosh^{-2} \omega x$, which is related to the single soliton solution of KdV equation, special attention is required. It is not the special case of the Rosen-Morse or the modified P\"{o}schl- Teller potentials anymore. It is not related by coordinate transformation to them either.When one writes the e-value equation for the Laplace-Beltrami (LB) operator in the space of matrix elements of the unitary irreducile representations of SO(1,2) realized in the compact basis one arrives at the Schr\"{o}dinger equation for the potential $V=\sinh^{-2} \omega x$ from which by the substitution $\omega x=\omega x'+i\frac{\pi}{2}$ we come to the potential $V=-\cosh^{-2} \omega x'$. To obtain the positive sign before the latter potential one should diagonolize the LB operator in the space of matrix elements of the unitary irreducible representation constructed in the mixed basis i.e. between compact and non-compact basis. Such a necessity requries the derivation of new addition theorem for these matrix elements to get the path integral solution. In fact the construction of the unitary ireducible representations in the mixed basis and the harmonic analysis on the double-sheeted hyperboloid in the hyperbolic coordinate system are the basic ingredients of the present note. The approach we adopt is of general nature which can be used to obtain the path integral solutions in any homogenous space in any parametrization.It leads to the path integral solution for the new class of potentials (see sec. 2). The wave functions of these potentials correspond to the matrix elements of the unitary irreducible representations in the basis defined by the choice of the group decomposition. In Section 2 we briefly rewiew the several possible decompositions of SO(1,2) relevant to the coordinates employed in the coset spaces which are double-sheeted and single- sheeted hyperboloids and the cone. In Section 3 we formulate the LB operator for the coset space SO(1,2)/SO(2). Following the derivation of SO(1,2) matrix elements in the mixed basis, we diagonolize LB operator, and then arrive at the Schr\"{o}dinger equation of the potential barrier $V=\cosh^{-2} \omega x$. Normalized wave functions and spectrum are given. In Section 4 we present the path integral formulation over the homogeneous space SO(1,2)/SO(2). Starting from the short time interval Kernel and making use of the newly derived addition theorems we expand the short time interval Kernel in terms of the group matrix elements. In Section V,we study the path integration for the potential barrier $V=\cosh^{-2} \omega x$.Transmittion and Reflection coefficients are given. Formulas for the barrier moving with a constant speed $g_0$, i.e. for $V=\cosh^{-2}\omega (x-g_0t)$, which is more relavent to the solitonic potential are also presented. \vfill \eject {\Large 2.Decompositions of the group SO(1,2) and the related quantum systems} To express the group SO(1,2) in the decomposed forms the following one parameter subgroups can be employed: \begin{eqnarray} a= \left( \begin{array}{ccc} \cosh \alpha & 0 & \sinh \alpha \\ 0 & 1 & 0 \\ \sinh\alpha & 0 & \cosh\alpha \end{array} \right) h= \left( \begin{array}{ccc} \cosh \beta & \sinh \beta & 0 \\ \sinh \beta & \cosh \beta & 0 \\ 0 & 0 & 1 \end{array} \right) \nonumber \\ k= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos \psi & -\sin \psi \\ 0 & \sin \psi & \cos \psi \end{array} \right) n= \left( \begin{array}{ccc} 1+\frac{t^2}{2} & t & \frac{t^2}{2} \\ t & 1 & t \\ -\frac{t^2}{2} & -t & 1-\frac{t^2}{2} \end{array} \right) \end{eqnarray} where \begin {equation} \alpha,\beta \in (-\infty,\infty), \ \ \ \ \ \psi \in (0,2\pi), \ \ \ \ \ t\in(-\infty ,\infty ) \end {equation} G=SO(1,2) leaves the form $(x,x)=x_0^2-x_1^2-x_2^2$ invariant. There are three possibilities: \ (I) $\underline{Double - Sheeted \ hyperboloid \ M \ : \ (x,x) \ > \ 0}$ \cite{kn:vil},\cite{kn:v}. The vector $\dot{\xi}=(1,0,0)\in M$ is the stationary point of the compact subgroup k as $\dot{\xi}k=\dot{\xi}$. Thus the decompositions of the group SO(1,2) related to the double-sheeted hyperboloid M is \begin{equation} g=kb,\ \ g\in SO(1,2), \ \ k\in SO(2) \end{equation} The choice of the boost b defines the coordinate systems on M: (i) b=ak or the Cartan decomposition of the group g=kak' defines the spherical coordinate parametrization of M. This choice is convenient for studying the quantum mechanical potential $\frac{1}{\sinh^2\alpha}$. (ii) b=ah or the non-compact Cartan decomposition of the group g=kah defines the hyperbolic coordinates on M.It is suitable to the quantum mechanical system with potential $\frac{1}{\cosh^2\alpha}$. This case is the subject of the present work. (iii) b=an or the Iwasawa decomposition of the group g=kan defines the parabolic coordinates on M which leads to the quantum mechanical system with potential $V=\exp(\alpha)$. \ (II) $\underline{Single - Sheeted \ hyperboloid \ \overline{M} \ : \ (x,x) \ < \ 0}$ \cite{kn:vil},\cite{kn:v} The vector $\dot{z}=(0,0,1)\in \overline{M}$ is the stationary point of the non-compact subgroup h as $\dot{z}h=\dot{z}$.The decompositions of the group SO(1,2) related to the single-sheeted hyperboloid $\overline{M}$ has the form \begin{equation} g=hb,\ \ g\in SO(1,2), \ \ h\in SO(1,1) \end{equation} with the possible choices of the boost b are given as the following: (i) b=ak or the non-compact Cartan decomposition of the group g=hak defines the spherical coordinate parametrization of $\overline{M}$ which produces the potential $-\frac{1}{\cosh^2\alpha}$. (ii) b=$(aI^{\varepsilon} h,kI^{\varepsilon} h)$ or decomposition of the group g=$(h aI^\varepsilon h,h kI^\varepsilon h)$ \cite{kn:ver} defines the hyperbolic coordinates on $\overline{M}$. Here I is the metric tensor given by \begin {equation} I=diag(1,-1,-1) \end {equation} and $\varepsilon $=0,1. This decomposition is suitable to the quantum mechanical system with potentials $-\frac{1}{\sinh^2\alpha}$ and $-\frac{1}{\sin^2\phi}$. (iii) $b=aI^\varepsilon n$ leads to the non-compact Iwasawa decomposition of the group $g=kaI^\varepsilon n $ and defines the parabolic coordinates on $\overline{M}$.The related quantum mecanical system is $V=-\exp(\alpha)$. \ (III) $\underline{Cone \ M_0 \ : \ (x,x) \ = \ 0}$ The vector $\dot{y}=(1,0,1)\in M_0$ is the stationary point of the nilpotent subgroup n as $\dot{y}n=\dot{y}$.The decompositions of the group SO(1,2) related to the cone $M_0$ has the form $g=nb$ with b having the following forms : (i) b=ak is the Iwasawa decomposition which defines spherical coordinates on $M_0$. (ii) b=a$I^\varepsilon$h is the non-compact Iwasawa decomposition which defines hyperbolic coordinates on $M_0$. (iii) b=a$n^T$ is the Gauss decomposition which defines spherical coordinates on $M_0$ It is impossible to relate quantum systems with the cone because the metric tensor of $M_0$ is degenerate. This space is used for the construction of the irreducible representations \cite{kn:gel}. To construct the irreducible representations in the mixed basis we simultaneously have to use the realizations given by (i) and (ii) (see Appendix A). \vfill \eject {\Large 3.The Double-Sheeted Hyperboloid in the Hyperbolic Coordinates} We decompose the group G=SO(1,2) as \begin{equation} g=hak \end{equation} Starting from the stationary point $\dot{\xi}=(1,0,0)$ we cover all the homogeneous space M by the act of the group elements as $x=\dot{\xi}g$. Using (1) we get the parametrization of M \begin {equation} \xi=\dot{\xi}g=\dot{\xi}ah =(\cosh \alpha \cosh \beta ,\cosh\alpha\sinh\beta ,\sinh\alpha ) \end {equation} The metric tensor and the Laplace-Beltrami operator \cite{kn:hel} of M are \begin {equation} g_{M}=diag(-1,-\cosh^{2}\alpha),\ \ \ \ \ detg_{M}=\cosh^{2}\alpha \end {equation} and \begin {equation} \bigtriangleup=-\partial_{\alpha}^{2}-\tanh\alpha\ \partial_{\alpha}- \cosh^{-2}\alpha\ \partial_{\beta}^{2} \end {equation} We will write down the e-value equation for the above LB operator in the space of SO(1,2) matrix elements which are evaluated between the compact and non-compact basis. The matrix elements of the unitary principle series of the group SO(1,2) in the mixed basis [see Appendix A] given by \begin {equation} d_{\mu k}^{\sigma}(g)=\langle \mu \mid T^{\sigma}(g)\mid k \rangle \end {equation} are the e-functions of the invariant differentiale operator $\bigtriangleup$ \begin {equation} \bigtriangleup d_{\mu k}^{\sigma}(g)=-\sigma(1+\sigma)d_{\mu k}^{\sigma}(g) \end{equation} Here $\sigma$ is the weight of the representation \begin {equation} \sigma=-1/2+i\rho, \ \ \ \ \ \rho\in(0,\infty), \end {equation} and $\mid k \rangle\ and \mid \mu\rangle$ are the compact and non-compact basis corresponding to the the degrees of freedom $\phi$ and $\beta$ respectfully. Since we are dealing with the coset space M=G/K with K=SO(2), we do not need the full set of the matrix elements (10), instead we employ \begin{equation} d_{\mu 0_k}^{\sigma}(ha)=\langle\mu\mid T^\sigma (ha)\mid 0_k\rangle \end {equation} Writing $d_{\mu 0_k}^\sigma (ha)$ as \begin{equation} d_{\mu 0}^\sigma (ha)=\exp (i\mu\beta )d_{\mu 0}^{\sigma}(a) =\exp (i\mu\beta ) (\det g_{M})^{-1/4} \Psi_{\mu}^{\sigma}(\alpha ) \end{equation} the e-value equation (12) becomes \begin {equation} (-\partial_{\alpha}^{2}+(\mu^{2}+1/4)\cosh^{-2}\alpha+1/4) \Psi_{\mu}^{\sigma}(\alpha)= -\sigma(\sigma+1) \Psi_{\mu}^{\sigma}(\alpha) \end {equation} which is equivalent to the Schr\"{o}dinger equation for the potential barrier $V=\cosh^{-2}\alpha$ with an extra constant energy shift of $\frac{1}{4}$. Note that if we would diagonolize the operator (9) in the space of matrix elements written between purely compact basis, the sign of potential would be negative. The wave functions of the Sch\"{o}dinger equation (15) are given in terms of the Legendre functions \cite{kn:ar} by \begin {equation} \Psi_{\mu}^{\sigma}(\alpha)= \frac{\cosh^{1/2}(\pi\rho)\cosh^{-1/2}(\alpha)}{\cosh(\pi\rho) -isinh(\pi\mu)} (P_{\sigma}^{i\mu}(i\sinh\alpha)+P_{\sigma}^{i\mu}(-i\sinh\alpha)) \end {equation} which are normalized as [see Appendix B] \begin {equation} \int_{-\infty}^{\infty}d\alpha \Psi_{\mu}^{\sigma} \overline{\Psi_{\mu'}^{\sigma'}}= \frac{\delta(\mu-\mu')\delta(\sigma-\sigma')}{\rho\tanh\pi\rho}. \end {equation} \vfill \eject {\Large 4. Path Integration over the Coset Space M=G/K} The probability amplitude for the particle of ``moment of inertia" m, to travel in the space M from the point $\xi_{a}$ to $\xi_{b}$ in the time interval T is expressed by the path integral which in the time graded formulation is given by \begin{equation} K(\xi_{a},\xi_{b};T)=\lim_{n\rightarrow \infty } \int\prod_{j=1}^{n} d\xi_{j} \prod_{j=1}^{n+1}K(\xi_{j-1},\xi_{j};\varepsilon) \end{equation} with $T=(n+1)\varepsilon$\ \ and \ \ $d\xi_{j}=(detg_{M})^{1/2}d\alpha_{j}d\beta_{j}$. The Kernel connecting the points $\xi_{j-1}$ and $\xi_{j}$ separated by the small time interval $t_{j}-t_{j-1}=\varepsilon$\ \ is \cite{kn:dur} \begin{equation} K_{j}=K(\xi_{j-1},\xi_{j};\varepsilon)=(\frac{m} {2i\pi\varepsilon})^{3/2}\exp(iS_{j}) \end{equation} where $S_j$ is the short time interval action \begin{equation} S_j =\frac{m}{2\varepsilon} \delta^2_{j-1,j} \end{equation} The invariant distance between the points is \begin {equation} \delta^{2}_{j,j-1}= (\xi_{j}-\xi_{j-1},\xi_{j}-\xi_{j-1})=2-2\cosh\theta_{j-1,j} \end {equation} with \begin{equation} \cosh\theta_{j-1,j}=\cosh\alpha_{j-1}\cosh\alpha_{j}\cosh(\beta_{j-1}-\beta_{j})- \sinh\alpha_{j-1}\sinh\alpha_{j} \end{equation} The short time interval Kernel (19) can be expanded in terms of the Legendre functions $P_{\sigma}(\cosh\theta)$ as ( with $\sigma=-1/2+i\rho$ ) \begin {equation} K_{j-1,j}= \int_{0}^{\infty}d\rho\rho\tanh(\pi\rho)C_{\sigma} P_{\sigma}(\cosh\theta_{j-1,j}) \end {equation} From the orthogonality condition \begin {equation} \int_{1}^{\infty}dzP_{\sigma}(z)\overline{P_{\sigma'}(z)}=\frac{1} {\rho\tanh\pi\rho}\delta(\sigma-\sigma') \end {equation} the coefficients $C_\sigma$ is obtained : \begin{equation} C_{\sigma}=-\frac{m}{\surd\pi\varepsilon}\exp(-\frac{i\pi}{\varepsilon}) K_{i\rho}(-\frac{m}{\varepsilon}) \end{equation} Here $K_{i\rho}$ is the MacDonald function. In $\varepsilon\rightarrow 0$ limit,\ \ by using the asymptotic form of the MacDonald function \cite{kn:jun} we can write the short time interval Kernel (19) as \begin{equation} K_{j}\simeq\int_{0}^{\infty}d\rho\rho\tanh\pi\rho \exp(\frac{i\varepsilon}{2m}\sigma(\sigma+1)) P_{\sigma}(\cosh\theta_{j}) \end{equation} By the help of the addition theorem [see Appendix C] for the complete set of functions on the homogeneous space M we get \begin{equation} K_j=\int_{-\infty}^\infty d\mu\int_0^\infty d\rho\rho\tanh\pi\rho \exp(\frac{i\varepsilon}{2m}\sigma(\sigma+1)) d_{\mu 0}^\sigma(\xi_{j-1}) \overline{d_{\mu 0}^\sigma (\xi_j)} \end{equation} We first insert the above form of the short time interval Kernel into (18); then by making use of the orthogonality condition (17) we can execute the $\prod_{j-1}^n d\xi$ integrals: \begin{eqnarray} K(\xi_a ,\xi_b;T)=\int_{-\infty}^\infty d\mu\int_0^\infty d\rho\rho\tanh\pi\rho \nonumber \\ \exp(-\frac{i(\rho^2+1/4)}{2m}T) d_{\mu 0}^\sigma(\xi_a)\overline{d_{\mu 0}^\sigma(\xi_b)} \end{eqnarray} By using the addition theorem the Kernel (28) can be written in the form \begin{equation} K(\xi_a ,\xi_b;T)=\int_0^\infty d\rho\rho\tanh\pi\rho\exp(-i\frac{\rho^2+1/4}{2m}T) P_\sigma(\cosh\theta_{ab}) \end{equation} where $\cosh\theta_{ab}$ depends on the coordinates of the points a and b through the relation defined by (22). The Fourier transform of (29) can be calculated to obtain the energy dependent Green function $G(\xi_a ,\xi_b ;E)$ \begin{equation} G(\xi_a ,\xi_b ;E)=\int_0^\infty\exp(iET) K(\xi_a ,\xi_b;T) =2mQ_{-1/2-i\surd(2mE-\frac{1}{4})}(\cosh\theta_{ab}) \end{equation} where Q is the Legendre function of the second kind. In deriving (30) we used the connection between the Legendre functions of the first and second kind \cite{kn:grad} \begin{equation} \int_0^\infty dx\frac{x\tanh\pi x}{a^2+x^2}P_{-1/2+ix}(z)=Q_{a-1/2}(z) \end{equation} \vfill \eject {\Large 5.Path Integral for the Potential Barrier $V=V_0\cosh^{-2}(\omega x)$} Phase space path integral for the particle of mass m moving under the influence of the potential barrier $V=V_0\cosh^{-2}(\omega x)$ is \begin{equation} K(x_a ,x_b;T)=\int Dx Dp_x \exp(i\int_0^T dt(px-p_x^2/2m-V_0\cosh^{-2}\omega x)) \end{equation} which is in the time graded formulation equal to \begin{eqnarray} K(x_a ,x_b;T)=\lim_{n\rightarrow\infty}\int\prod_{j=1}^ndx_j \prod_{j=1}^{n+1}d\frac{p_{xj}}{2\pi} \prod_{j=1}^{n+1} \nonumber \\ \exp[i(p_{xj}(x_j-x_{j-1})-\varepsilon p_{xj}^2/2m- \varepsilon V_0\cosh^{-2}\omega x_j)] \end{eqnarray} The phase space formulation (32) easily enable us to establish the connection between our quantum mechanical problem and the path integration over the coset space M=SO(1,2)/K. In fact when we consider the Hamiltonian $H_M$ for the particle motion over the coset space M=G/K (recall the Sch\"{o}dinger equation (15)): \begin{equation} H_M -\frac{1}{4} \Rightarrow\frac{1}{2m}(p_\alpha^2+p_\beta^2\cosh^{-2}\alpha) -\frac{1}{4}(\frac{\omega^2}{2m}) \end{equation} we observe that the Hamiltonian in the action of (32) resembles the above Hamiltonian with the momentum $p_\beta$ fixed to the value $ p_\beta=\surd(2mV_0) $. In writing (34) we introduced the corrections due to parameters $\omega$ and (2m) which were equal to 1 in sections 3 and 4. Thus we can convert the path integral (32) into the path integral for the motion in the space M=G/K. We first rescale x by $ \omega x=\alpha $ ;\ \ \ $ p_x=\omega p_\alpha $ and arrive at: \begin{equation} K(x_a ,x_b;T)=\omega\int D\alpha Dp_\alpha\exp[i\int_0^{\omega^2 T}dt (p_\alpha\alpha -p_\alpha^2/2m-V_0\cosh^{-2}\alpha)] \end{equation} We then rewrite the potential term in the above path integral by introducing an auxilary dynamics by extending the phase space with the identity \begin{eqnarray} \exp[-i(\int_0^{\omega^2 T} dtV_0\cosh^{-2}\alpha)]= \int d\beta_b\exp(-\surd(2mV_0/\omega^2)(\beta_b-\beta_a)) \nonumber \\ \lim_{n\rightarrow\infty}\int\prod_{j=1}^nd\beta_j \prod_{j=1}^{n+1} d\frac{p_{\beta j}}{2\pi} \prod_{j=1}^{n+1} \exp(i(p_{\beta j}(\beta_j-\beta_{j-1})- \frac{\omega^2\varepsilon p_{\beta_j}^2}{2m\cosh^2\alpha}) \end{eqnarray} which can be proven by direct calculation. Note that the phase space formulation is rather essential for the above identity which establishes the connection between our quantum mechanical problem and the partical mation over M. The identity of (36) enables us to reexpress (35) as \begin{equation} K(x_a ,x_b;T)=\omega\int d\beta_b\exp(-\surd(2mV_0/\omega^2)(\beta_b-\beta_a)) \exp(i\frac{\omega^2}{8m}T) K_M(\xi_a ,\xi_b;T) \end{equation} where $ K_M $ is the Kernel for the motion over the manifold M=G/K which is studied in the previous chapter. The factor $\exp(i\frac{\omega^2}{8m}T)$ in the above equation reflects (see eqs. (15) and (34)) the 1/4 energy difference between the potential barrier $\cosh^{-2}omega x$ and the particle motion over the coset space M. We then insert the expression (28) into (37), use (14) for the matrix elements $ d_{\mu0}^\sigma $ and arrive at \begin{eqnarray} K(x_a ,x_b;T)=2\pi\omega(\cosh\omega x_a\cosh\omega x_b)^{-1/2} \nonumber \\ \int d\rho\rho\tanh\pi\rho exp(-i\frac{\rho^2\omega^2}{2m}T) \psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega x_a) \overline{\psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega x_b)} \end{eqnarray} which displays the wave functions.The asymptotic form of the wave functions are \begin{equation} \lim_{x\rightarrow\infty} \psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega x) \simeq \frac{\Gamma (1/2-i\rho)} {\Gamma (1/2-i(\rho + \frac{(2mV_0)^{1/2} }{\omega}))} \exp(-i\rho\omega x) \end{equation} \begin{equation} \lim_{x\rightarrow -\infty} \psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega x) \simeq \frac{\Gamma (1/2-i\rho)} {\Gamma (1/2-i(\rho + \frac{(2mV_0)^{1/2} } {\omega}))} (T\exp(-i\rho\omega x)+R\exp(i\rho\omega x)) \end{equation} in which the transition and the reflection coefficients are identified as: \begin{equation} T=\frac {\Gamma (1/2+i(\rho + \frac{(2mV_0)^{1/2}}{\omega} )) \Gamma (1/2+i(\rho - \frac{(2mV_0)^{1/2}}{\omega}))} {\Gamma (i\rho)\Gamma(1+i\rho)} \end{equation} \begin{equation} R=\frac {\Gamma (1/2+i(\rho + \frac{(2mV_0)^{1/2}}{\omega} ) ) \Gamma (1/2+i(\rho - \frac{(2mV_0)^{1/2}}{\omega}) ) \Gamma (-i\rho)} {\Gamma (1/2+i \frac{(2mV_0)^{1/2}}{\omega} ) \Gamma (1/2-i\frac{(2mV_0)^{1/2}}{\omega} ) \Gamma (i\rho)} \end{equation} If one considers the same potential barrier in motion with a constant speed \begin {equation} V(x,t)=\frac{V_0}{\cosh^2 \omega (x-g_0t)} \ \ ; \ \ \ \ g_0=const. \end{equation} the Kernel becomes \cite{kn:hak}: \begin{equation} K_{g_0}(x_a,x_b;T)=\exp(-i\frac{m}{2}g_0^2T)\exp(-img_0(x_b-x_a)) K(x_a-g_0t_a,x_b-g_0t_b:T) \end{equation} Here the form of K is given by (38).From the above formula the wave functions are recognized as \begin{equation} \psi_{g_0}(x,t)=\exp(-i\frac{m}{2}g_0^2t)\exp(-img_0x) \psi^\sigma_{\frac{(2mV_0)^{1/2}}{\omega}}(\omega (x-g_0t)) \end{equation} where $\psi(\omega (x-g_0t))$ is obtained from the static one given in (16) and (38) by simply replacing $x$ by $x-g_0t$. For finite values of time variable $t$ the transition and the reflection coefficients remain in the static forms of (41) and (42). Inspecting the limiting forms of T and R we obtains \begin{equation} \mid\frac{T}{R}\mid\rightarrow\rightarrow\infty \ as \ \rho\rightarrow\infty, \ and \ \mid\frac{T}{R}\mid\rightarrow0 \ as \ \rho\rightarrow 0 \end{equation} We see that the low energy waves are mostly reflected, while the high energy waves are more easily transmitted through the barrier. Inspecting (47) and the asymptotic forms as $x\rightarrow\infty$ we observe that the barrier motion contributes the following constant additional term to the energy \begin{equation} \bigtriangleup E=\frac{mg_{0}^2}{2}-\rho\omega g_0 \end{equation} The first term of the above extra energy is of the kinetic energy type (for $g_0$ has the dimension of velocity). It is also interesting that the barrier motion introduces the extra ondulation of Doppler nature trough $\exp(-img_0x)$ term in the wave function. \vfill \eject \begin {center} \huge {Appendix} \end {center} {\Large A. The Unitary Irreducible Representations of the Group SO(1,2) in the Mixed Basis} \renewcommand{\theequation}{A.\arabic{equation}} \setcounter{equation}{0} We will construct the irreducible representations of the pseudo-orthogonal group G=SO(1,2) in the space of the infinitely differentiable homogeneous functions $F(y)$ with the homogenity degree $\sigma$ on the cone $Y:\ \ [y,y]=0 $. \begin {equation} T^\sigma (g)F(y)=F(yg),\ \ \ \ \ \ g\in G ,\ y\in Y , \end {equation} \begin {equation} F(ay)=a^\sigma F(y) ,\ \ \ \ \ \ a\in R ,\ \sigma\in C \end {equation} In order to construct the matrix elements of the representation in the mixed basis we have to define the cone in two coordinate systems corresponding to these basis. Use the Iwasawa decompositions for the group SO(1,2) \cite{kn:vil}: \begin{equation} g=n(t)a(\theta )k(\phi ) ,\ \ \ \ \ g\in G \end{equation} \begin{equation} g=n(t)I^\varepsilon a(\gamma )h(\beta ) ,\ \ \ \ \ g\in G \end{equation} Here the element n of the nilpotent subgroup and other matrices $a,h,k$ are the ones defined in (2). The stationary point of the nilpotent subgroup is $\dot{y}=(1,0,1)$ \begin {equation} \dot{y}n(t)=\dot{y}, \end {equation} The coset space Y=G/N is equivalent to the cone Y given by $y=\dot{y}g$ which can be defined in two realizations as: \begin{equation} y=\exp(\gamma_k )s_k ,\ \ \ s_k=\dot{y}k(\phi )=(1,\sin\phi ,\cos\phi) \end{equation} \begin{equation} y=\exp(\gamma_h )s_h ,\ \ \ s_h=\dot{y}h(\beta )= (\cosh\beta ,\sinh\beta ,(-1)^\varepsilon ) \end{equation} The connection between the above realizations is: \begin{equation} \exp(\gamma_h )s_h=\exp(\gamma_k )s_k \end{equation} or \begin{equation} \cosh\beta\exp(\gamma_h )=\exp(\gamma_k ),\ \ \ \cos\phi =\frac{(-1)^\varepsilon} {\cosh\beta},\ \ \ \sin\phi=\tanh\beta \end{equation} Using (A.2) we get: \begin{equation} F(y)=\exp(\gamma_h\sigma)F(s_h)=\exp(\gamma_k\sigma)F(s_k) \end{equation} We know that the principal series of the irreducible representation in the compact basis (the group decomposition is $g=kak$ ) is unitary with respect to the scalar product \begin{equation} (F_1,F_2)=\frac{1}{2\pi}\int_{0}^{2\pi} d\phi F(s_k)\overline{F(s_k)} \end{equation} if $\sigma=-1/2+i\rho ,\rho\in(0, \infty) $ [11]. Using the relation (A.10) we can introduce the scalar product in the space of representation in the mixed basis: \begin{equation} \langle F_1,F_2\rangle=\frac{1}{2\pi}\int_0^{2\pi} d\phi\overline{\exp( (\gamma_h-\gamma_k)\sigma ))}\overline{F(s_h)}F(s_k) \end{equation} The invariant bilinear Hermitian form (A.12) can be written as \begin{equation} \langle F_1,F_2\rangle=\frac{1}{2\pi}\int_{-\infty}^{\infty} d\beta \cosh\beta^\sigma \overline{F(s_h)}F(s_k) \end{equation} From (A.1) we get the representation formulas corresponding to the above mentioned realizations: \begin {equation} T^\sigma (g)F(s_k)=\exp((\gamma^g_k-\gamma_k)\sigma )F(s_{k^g}) ,\ \ \ \ \ g\in G \end {equation} \begin {equation} T^\sigma (g)F(s_h)=\exp((\gamma^g_h-\gamma_h)\sigma )F(s_{h^g}) ,\ \ \ \ \ g\in G \end {equation} Here $s_{k^g}$ \ \ and $s_{h^g}$ are defined as \begin {eqnarray} \exp(\gamma^g_k ) s_{k^g}=\exp(\gamma_k ) s_kg \nonumber \\ \exp(\gamma^g_h ) s_{h^g}=\exp(\gamma_h ) s_hg \end {eqnarray} We see that the natural representations for the maximal compact k=SO(2) and noncompact h=SO(1,1) subgroups corresponding to the realizations of the representations (A.14) and (A.15) are \begin {equation} T^\sigma (k(\phi_0))F(s_{k(\phi)})=F(s_{k(\phi +\phi_0)}) \end {equation} and \begin {equation} T^\sigma (h(\beta_0)F(s_{h(\beta)})=F(s_{h(\beta +\beta_0)}) \end {equation} By the help of the expansion formulas \begin {eqnarray} F(s_k)=\sum_{n=-\infty}^\infty C_n\exp(in\phi) \nonumber \\ F(s_h)=\int_{-\infty}^\infty d\mu C_\mu\exp(i\mu\beta) \end {eqnarray} we can rewrite (A.17)and (A.18) as \begin {eqnarray} T(k(\phi_0))\exp(in\phi)=\exp(in\phi_0)\exp(in\phi) \nonumber \\ T(h(\beta_0)\exp(i\mu\beta)=\exp(i\mu\beta_0)\exp(i\mu\beta) \end {eqnarray} which coincide with the unitary irreducible representations of the subgroups SO(2) and SO(1,1). Now we are ready to construct the unitary irreducible representation for the group SO(1,2) in the mixed basis. Let us introduce the function D(g) \begin{equation} D(g)=\langle F_1\mid T^\sigma (g)\mid F_2\rangle \end{equation} in terms of the Hermitian bilinear form given by (A.13). Using the expansion formulas (A.19) we obtain \begin{equation} D(g)=\sum_{n=-\infty}^{\infty}\int_{-\infty}^\infty d\mu C_n \overline {C_\mu} d_{\mu n}^\sigma (g) \end{equation} where $d_{\mu n}^\sigma (g)$ are the matrix elements of the unitary irreducible representation \begin{equation} d_{\mu n}^\sigma (g)=\langle\mu\mid T^\sigma (g)\mid n\rangle,\ \ \ \ g=hak\in G \end{equation} By the help of the group property $T^\sigma (hak)=T(h)T^\sigma (a)T(k)$ and the expressions in (A.20) we obtain \begin{equation} d_{\mu n}^\sigma (g)=\exp(-i\mu\beta )\langle\mu\mid T^\sigma (a) \mid n\rangle\exp(in\phi) \end{equation} The integral representation for the matrix elements corresponding to the subgroup $a(\alpha )$ is (in the case n=0) \begin {equation} d_{\mu 0_k}^{\sigma}(a)=\sum_{\varepsilon=0}^{1} \int_{-\infty}^\infty d\beta \exp(-i\mu\beta ) (\cosh\beta\cosh\alpha+(-1)^\varepsilon\sinh\alpha )^ \sigma \end {equation} Evaluting this integral we get: \begin {equation} d_{\mu 0_k}^{\sigma}(a(\alpha))=\frac{\cosh^{1/2}(\pi\rho)} {\cosh(\pi\rho)-isinh(\pi\mu)} (P_{\sigma}^{i\mu}(i\sinh\alpha)+P_{\sigma}^{i\mu}(-i\sinh\alpha)) \end {equation} {\Large B. The Orthogonality Condition} \renewcommand{\theequation}{B.\arabic{equation}} \setcounter{equation}{0} Consider the expression \begin{equation} B^{\sigma\sigma '}_{\mu\mu '}=\int_G dg d^\sigma_{\mu 0_k}(g)\overline {d^{\sigma '}_{\mu '0_k}(g)} \end{equation} with g=hak. We first change the variables in the above integral \begin{equation} hak=k'a'k'' \end{equation} which is equivalent to passing from $g=hak$ to the Cartan decomposition $g'=k'a'k''$ \cite{kn:vil}. Using the completness condition for the matrix elements of the maximal compact subgroup K=SO(2): \begin{equation} \sum_{n=-\infty}^{\infty}\mid n\rangle \langle n\mid=1 \end{equation} and the equality \begin{eqnarray} d^\sigma_{\mu 0_k}(g)=\langle\mu\mid T^\sigma (g)\mid 0_k\rangle= \sum_{n=-\infty}^\infty\langle\mu\mid n\rangle \langle n\mid T^\sigma (g)\mid 0_k\rangle= \nonumber \\ =\sum_{n=-\infty}^\infty\langle\mu\mid n\rangle d^\sigma_{n0_k}(g) \end{eqnarray} we get \begin{eqnarray} B^{\sigma\sigma '}_{\mu\mu '}= \sum_{n,n'=-\infty}^{\infty} \langle \mu \mid n \rangle \langle \mu ' \mid n ' \rangle \int_{G'} dg' d^\sigma_{n0_k}(g')\overline{d^{\sigma '}_{n' 0_k} (g')} \nonumber \\ =\sum_{n,n'=-\infty}^{\infty}\langle\mu\mid n\rangle \overline{\langle\mu '\mid n'\rangle} \frac{\delta(\rho-\rho ')\delta_{n n'}}{\rho\tanh\pi\rho}= \frac{\delta(\rho-\rho ')\delta (\mu -\mu ')}{\rho\tanh\pi\rho} \end{eqnarray} In (B.5) we used the orthogonality condition of the matrix elements in the Cartan basis \cite{kn:vil}, \cite{kn:ver}. together with the orthogonality condition for the matrix elements of the maximal noncompact subgroup SO(1,1) \begin{equation} \langle\mu\mid\mu '\rangle=\delta(\mu-\mu ') \end{equation} It is obviously equivalent to \begin{equation} \int_G dg d^\sigma_{\mu 0_k}(g)\overline{d^{\sigma '}_{\mu '0_k}(g)}= \frac{\delta(\rho-\rho ')\delta (\mu -\mu ')}{\rho\tanh\pi\rho} \end{equation} where the invariant measure is $dg=(det_m g)^{1/2}d\alpha d\beta d\phi$ Taking into account (14) we get the orthogonality condition for the wave functions: \begin {equation} \int_{-\infty}^{\infty}d\alpha \Psi_{\mu}^{\sigma} \overline{\Psi_{\mu'}^{\sigma'}}= \frac{\delta(\mu-\mu')\delta(\sigma-\sigma')}{\rho\tanh\pi\rho} \end {equation} {\Large C. The Addition Teorem and the Completeness Condition} \renewcommand{\theequation}{C.\arabic{equation}} \setcounter{equation}{0} Note that the Legendre functions appear as the zonal spherical functions of the representation of the group SO(1,2) if the elements of the group G has the Cartan decomposition $g=kak'$: \begin {equation} P^\sigma (\cosh\theta )=d^\sigma_{0_k,0_k}(a(\theta )) \end {equation} Suppose that $g_1$ and $g_2$ are the elements of the group G which have the following decompositions \begin {equation} g_j=h_ja_jk_j,\ \ j=1,2 \ \ and\ \ g_{12}=h_{12}a_{12}k_{12}=g_1^{-1}g_2 \end{equation} Write the element $g_{12}$ in the Cartan decomposition: \begin{equation} g_{12}= k'_{12}a'_{12}k_{12} \end{equation} Using (C.1),(C.2) and (C.3) we obtain: \begin {eqnarray} P^\sigma (\cosh\theta_{12})=d^\sigma_{0_k,0_k}(a(\theta_{12}))= d^\sigma_{0_k,0_k}(k'_{12}a(\theta_{12})k_{12})= \nonumber \\ =d^\sigma_{0_k,0_k}(g^{-1}_1g_2)=\int_{-\infty}^\infty d\mu d^\sigma_{0_k,\mu}(g^{-1}_1) d^\sigma_{\mu ,0_k}(g_2) \end {eqnarray} Making use of the property of the matrix elements \begin {equation} d^{-\sigma-1}_{0_k,\mu}(g^{-1})= \overline{d^\sigma_{\mu ,0_k}(g)} \end {equation} and the equivalence of the representations $T^\sigma$ and $T^{-\sigma-1}$ \cite{kn:ver} we get the final result: \begin {equation} P^\sigma (\cosh\theta_{12})= \int_{-\infty}^\infty d\mu \overline{d^\sigma_{\mu,0_k}(g_1)} d^\sigma_{\mu ,0_k}(g_2) \end {equation} Here $\cosh\theta_{12}$ is defined from the algebraic equation (C.3) and is given by \begin{equation} \cosh\theta_{1,2}=\cosh\alpha_{1}\cosh\alpha_{2} \cosh(\beta_{1}-\beta_{2})- \sinh\alpha_{1}\sinh\alpha_{2} \end{equation} which coincide with (22). The completness condition for the matrix elements on the homogeneous space $g\in M$ are given by \begin{equation} \int_0^\infty d\rho\rho\tanh\pi\rho\int_{-\infty}^\infty d\mu d^\sigma_{\mu 0_k}(g)\overline{d^\sigma_{\mu 0_k}(g')}=\delta (g-g') \end{equation} To prove the above relation one considers the connection between the invariant differential operator ( Laplace-Beltrami operator ) on the manifold M with the Schr\"{o}dinger equation.Since the Schr\"{o}dinger equation has only the continuous specrum, the spectrum of the invariant differential operator should also be continuous. Therefore the discrete unitary series of the irreducible representation does not make contribution and can be ignored. From the physical point of view the complementary series can also be ignored. \vfill \eject \begin {thebibliography}{99} \bibitem{kn:gros} C. Grosche, {\em Path Integrals, Hyperbolic Spaces and Selberg Trace Formuae} World Scientific, Singapore,1996. \bibitem{kn:com} R. Comporesi, {\em Harmonic Analysis and Propagators on the Homogeneous Spaces} (Elsevier Science, North-Holland, 1990). \bibitem{kn:per} M.A. Olshanetsky and A.M. Perelomov, {\em Phys.Rep. $\underline{94}$,No 6,313-404},(1983). \bibitem{kn:dur} I.H.Duru, {\em Phys.Rev., $\underline{D30}$,2121 (1984); and Phys.Lett.$\underline{A119}$,163(1986)}. \bibitem{kn:bom} M.Bohm and G.Junker, {\em J.Math.Phys.$\underline{28}$,1978},(1987); and C.Grosche, {\em J.Math.Phys. $\underline{32}$,1984},(1991). For similar problem see also C.Grosche,{\em Fortschr.Phys.$\underline{38}$},(1990); and A.Frank.B.Wolf, {\em J.Math.Phys. $\underline{26}$},(1985); and H.Kleinert, I.Mustapic {\em J.Math.Phys. $\underline{33}$},(1992); \bibitem{kn:ih} I.H.Duru, {\em Path Integral Representation of the Symmetric Rosen-Morse Potential}, ICTP Trieste report 95-021, IC/83/178 \bibitem{kn:vil} N.Ya.Vilenkin and A.O.Klimyk, {\em Representation of Lie Groups and Special Functions}, vol.3 (Kluwer Akademy,1992). \bibitem{kn:v} Dane C. and Verdiev Yi.A., {\em J.Math.Phys.$\underline{37}$},(1996). \bibitem{kn:ver} Yi.A.Verdiev, {\em Harmonic Analysis on Homogeneous Spaces of SO(1,2)}, (Hadronic Press, Massachusetts, 1988). \bibitem{kn:gel} I.M.Gelfand and M.I.Graev, {\em Geometry of Homogeneous Spaces, Representations of Groupsin Homogeneous Spaces and Related Questions}, I.Amer.Math.Soc.Transl.,Ser2$\underline{37}$,(1964). \bibitem{kn:hel} S.Helgason, {\em Group and Geometric Analysis,Integral Geometry,Invariant Differential Operators and Spherical Functions} (Academy Press, New York, 1984). \bibitem{kn:ar}H.Bateman and A.Erdelyi, {\em Higher Trancendental Functions, vol.1},(1953). \bibitem{kn:jun} G.Junker, in M.C.Gurtzwiller,A.Inomata,J.R.Klauder and L.Streit eds.{\em Path Integral From mev to Mev},World Scientific,Singapure (1989). \bibitem{kn:grad}I.S.Gradstein and I.M.Rhyzik, {\em Tables of Integrals Series and Products}, (Academy Press, New-York, 1969). \bibitem{kn:hak} I.H.Duru, {\em J. Phys. A: Math. Gen. $\underline{22}$, 4827} (1989). \end{thebibliography} \end{document}

proofpile-arXiv_065-593

\section{INTRODUCTION AND PROBLEM STATEMENT} Coarsening underlies various natural non-equilibrium processes, {\it e.\ g.}, phase separation in binary alloys, grain growth, and growth of soap bubbles\cite{langer}. A common feature of coarsening phenomena is the scale-invariant morphology that arises in the late stage\cite{langer,bray}. Such a behavior is a signature of dynamical scaling. If dynamical scaling holds, the average domain size, $\ell(t)$, typically exhibits algebraic growth, $\ell(t)\sim t^{1/z}$. It has recently been appreciated that knowledge of the dynamical exponent $z$ does {\it not} provide a comprehensive description of the coarsening dynamics. In particular, the exponent $\lambda$ which describes the dependence of the autocorrelation function $A(t)\equiv \langle s(x,0)s(x,t)\rangle$, where $s(x,t)$ is the order parameter at position $x$ and time $t$, on the average domain size, $A(t)\sim\ell(t)^{-\lambda}$\cite{bray,fisher,desai}, and the exponent $\theta$ which characterizes (in magnetic language) the fraction of spins which have never flipped, $P_0(t)\sim t^{-\theta}$\cite{dbg,kbr,bdg}, were found to be independent of the dynamical exponent $z$. The latter quantity, $P_0(t)$, naturally suggests the generalization to $P_n(t)$, the fraction of spins which have flipped exactly $n$ times up to time $t$\cite{elp} as a detailed and fundamental characterization of the temporal history of spin flips. In this study, we investigate a related aspect of this temporal history by focusing on the time $\tau$ when the last spin flip occurs (Fig.~1). More generally, we may introduce $P_n(\tau,t)$ as the probability that a given spin flips $n$ times up to time $t$ {\it and\/} that the last spin flip occurs at time $\tau$. Here we investigate $P_+(\tau,t)$ which focuses on the last spin flip and does not specify the total number of flips, $P_+(\tau,t)=\sum_{n\geq 1}P_n(\tau,t)$. If we view a spin as being ``reborn'' each time it flips, then $P_+(\tau,t)$ gives the density of spins of ``age'' $t-\tau$. There is also a finite fraction of spins which never flipped yet; these spins should be treated as spins of age $t$. The total age distribution density of the spins is therefore \begin{equation} P(\tau,t)=P_0(t)\delta(\tau)+P_+(\tau,t). \label{total} \end{equation} The density $P(\tau,t)$ should satisfy the normalization condition $\int_0^t d\tau P(\tau,t)=1$, while the average age of the system is defined via \begin{eqnarray} T &=&\int_0^t d\tau (t-\tau)P(\tau,t)\nonumber\\ &=&tP_0(t)+\int_0^t d\tau (t-\tau)P_+(\tau,t). \label{age} \end{eqnarray} \begin{figure} \narrowtext \epsfxsize=\hsize \epsfbox{f1.eps} \vskip 0.2in \caption{ Graphical definition of $P(\tau,t)$ for one-dimensional coarsening processes. At the point marked by the dashed line, the spin last flips, or equivalently, is visited by a domain wall, at time $\tau$. The specific examples shown are: (a) the infinite-state Potts model (in which the domain walls undergo diffusive single-species coalescence) and (b), the deterministic coarsening of a 3-state system with cyclic interactions (in which the domain walls undergo ballistic single-species annihilation). \label{fig1}} \end{figure} The age distribution $P(\tau,t)$ will be of primarily importance in systems with history-dependent dynamics, such as glassy systems\cite{angell}, and in systems with infinite memory where actual aging takes place. Generally, when a two-time correlation function ${\cal C}(\tau,t)=\langle s(x,\tau)s(x,t)\rangle$ becomes a function of a single variable $\tau/t$, instead of being a function of $t-\tau$ (as in an equilibrium system), this is interpreted as a signature of aging\cite{bou1,bou2,bou3}. According to this definition, aging is a characteristic of coarsening processes and the scaling dependence $P(\tau,t)\simeq t^{-1}f(\tau/t)$ has been found in a number of pertinent examples\cite{kolja,slava}. The age distribution will also play a fundamental role when the dynamics of a system is explicitly time dependent. A potentially interesting situation is that of the ``adaptive'' voter model. The conventional voter model\cite{lig} is a two state lattice system in which a voter (site) randomly chooses one of its nearest neighbors and assumes the state of this neighbor. In the adaptive extension of this model the probability that a given voter changes its opinion depends on the local environment (as in the usual voter model) {\it and\/} on the time interval since this particular voter last changed its opinion. This might be viewed as a model to describe the increasing conservatism of people when they are not stimulated by contact with those of differing opinions. This adaptive voter model exhibits rather unexpected coarsening dynamics which is ultimately driven by the underlying age distribution \cite{adaptive}. In particular, we find coarsening for all spatial dimensions, while the conventional voter model coarsens only for spatial dimension $d\leq 2$. In the following two sections, we consider the age distribution for two specific one-dimensional coarsening processes for which exact results can be obtained. In Sec.\ II, we first treat a deterministic 3-state model of coarsening in which the dynamics of the domain walls is simply that of two-velocity ballistic annihilation. Because of this equivalence, it is possible to obtain the exact expression for $P(\tau,t)$ by simple means. In Sec.\ III, we investigate the age distribution in two stochastic coarsening models. The first is the infinite-state Potts model in which the domain wall dynamics is simply diffusion-limited coalescence process, which may be represented as $A+A\to A$. We find that the scaling form of the age distribution is identical to that found in the deterministic coarsening process. We also consider the age distribution for the Ising model with zero-temperature Glauber dynamics in which the domain wall dynamics coincides with single-species diffusion-limited annihilation process, which may be represented as $A+A\to 0$. In this case, the age distribution has a bimodal ``smiling'' form as a function of $\tau$, a result which can be understood intuitively. We then discuss the age distribution for the dynamical Ising model in dimension $d\geq 2$ and give an exact expression for the distribution in the mean-field limit. Sec.\ IV gives a brief summary and outlook. \section{AGING IN A DETERMINISTIC MODEL OF COARSENING} We first examine the age distribution in a deterministic coarsening model which describes phase ordering dynamics in a cyclic one-dimensional system with three equilibrium states, $A$, $B$, and $C$. The dynamics is cyclic so that the $B$ phase invades the $A$ phase, $C$ invades $B$, and $A$ invades $C$. Corresponding to this dynamics, interfaces between dissimilar domains move toward the subordinate domain with a fixed velocity. A domain which is besieged by two dominant domains shrinks and eventually disappears, leading to the merging of the neighboring domains. The interfaces therefore undergo ballistic motion with annihilation occurring whenever two interfaces meet. These rules are precisely those of the ballistically-driven single-species annihilation reaction. The simplicity and rich phenomenology of this reaction has stimulated extensive fundamental work\cite{els,brl,krl,Piasecki,Droz}, as well as related applications to growth processes\cite{Krug,sekimoto,ben,gold}, and the dynamics of interacting populations\cite{bramson,fisch,lpe}. We start by describing the behavior\cite{els,Piasecki} of the ballistic annihilation model for the domain walls. In this model, the density of right-moving and left-moving walls is equal, with velocities which can be taken to be $\pm 1$ without loss of generality. From the exact solution\cite{els}, the probability $S(t)$ for an arbitrary interface to survive up to time $t$ is \begin{equation} S(t)=e^{-2t}[I_0(2t)+I_1(2t)]. \label{st} \end{equation} Here $I_j$ denotes the modified Bessel function of order $j$, the initial spatial distribution of interfaces is assumed to be Poissonian (no correlations), with the initial densities of $\pm$ interfaces taken to be equal 1/2. To obtain the age distribution for the coarsening process induced by this domain wall dynamics, first consider $P_0(t)$, the fraction of space that has not been crossed by any interface in the time interval $(0,t)$. One can interpret $P_0(t)$ as the probability that a stationary ``target'' particle, which is placed at the origin, for example, is not hit by any moving domain wall. It is convenient to consider an auxiliary one-sided problem with interfaces distributed only to the right of the origin. For this case, the survival probability of the stationary particle, $S_0(t)$, is\cite{krl} \begin{equation} S_0(t)=e^{-t}[I_0(t)+I_1(t)]. \label{s0t} \end{equation} Indeed, the relative velocity between a stationary particle and its reaction partner is a factor of two smaller than the relative velocity between two moving reaction partners. Hence, $S_0(t)=S(t/2)$, and Eq.~(\ref{s0t}) follows from Eq.~(\ref{st}). Clearly, the survival probability $P_0(t)$ in the original two-sided problem is \begin{equation} P_0(t)=S_0(t)^2. \label{p0t} \end{equation} The continuous part of the age distribution $P_+(\tau,t)$ can also be expressed in terms of the survival probabilities $S(t)$ and $S_0(t)$. We first note that for the origin to be crossed by an interface during the time interval $(\tau,\tau+d\tau)$, a left moving interface should be initially located in the spatial interval $\tau<x<\tau+d\tau$, or a right moving interface should be located in the spatial interval $-\tau-d\tau<x<-\tau$. Each of these events occurs with probability $d\tau/2$ for an initial interface density of unity. Suppose that the origin is crossed by a left moving interface (Fig.~2). Then this interface will ultimately be annihilated with some right moving interface at some future time $t_1$, which satisfies $t_1>\tau$. If $t_1>(t+\tau)/2$, then the origin cannot be crossed by a right moving interface during the time interval $(\tau,t)$. The contribution of these type of configurations to $P_+(\tau,t)$ is \begin{equation} S\left({t+\tau\over 2}\right)S_0(t-\tau). \label{conf1} \end{equation} The first factor is just the probability that the left moving interface survives up to time $(t+\tau)/2$. The latter factor in Eq.~(\ref{conf1}) is the probability that the initial location of the left moving interface has not been crossed by any other left moving interface during the time interval $(0,t-\tau)$ which, in turn, ensures that the origin remains uncrossed from the right during the time interval $(\tau,t)$. \begin{figure} \narrowtext \epsfxsize=\hsize \epsfbox{f2.eps} \vskip 0.2in \caption{ Illustration of a typical configuration which contributes to $P_+(\tau,t)$ in the deterministic coarsening process generated by domain walls which undergo ballistic single-species annihilation. The left-moving domain wall trajectory which crosses the origin at time $\tau$ is shown as a heavy line. This domain wall is annihilated at a time $t_1>\tau $ such that any right-moving trajectory cannot reach the origin before time $t=2t_1-\tau$. \label{fig2}} \end{figure} Consider now the complementary situation when the left-moving interface which crosses the origin in the time interval $(\tau,\tau+d\tau)$ survives to time $t_1$ with $\tau<t_1<(t+\tau)/2$. In this case, additional right-moving interfaces can cross the origin before time $t$. The contribution of such configurations to $P_+(\tau,t)$ is \begin{equation} S_0(t-\tau)\int_{\tau}^{t+\tau\over 2} S_0(t-2t_1+\tau)[-\dot S(t_1)]dt_1. \label{conf2} \end{equation} Here $S_0(t-\tau)$ again ensures that the origin remains uncrossed from the right during the time interval $(\tau,t)$. Similarly, $S_0(t-2t_1+\tau)$ guarantees that the origin remains uncrossed from the left. Finally, $-\dot S(t_1)dt_1$ is the probability that the left moving interface is annihilated in the time interval $(t_1,t_1+dt_1)$. Combining these contributions, gives the final exact expression for the age distribution density \begin{eqnarray} P(\tau,t)&=&S_0(t)^2\delta(\tau) +S\left({t+\tau\over 2}\right)S_0(t-\tau)\nonumber\\ &-&S_0(t-\tau)\int_{\tau}^{t+\tau\over 2} S_0(t-2t_1+\tau)\dot S(t_1)dt_1. \label{main} \end{eqnarray} The singular part of the age distribution, $S_0(t)^2\delta(\tau)$, corresponds to the fraction of space that has not been traversed by any interface; in the long-time limit, this fraction decays as $t^{-1}$. To determine the asymptotic behavior of the continuous part of the age distribution, we substitute into Eq.~(\ref{main}) the asymptotic expressions, $S(t)\sim 1/\sqrt{\pi t}$ and $S_0(t)\sim \sqrt{2/\pi t}$, which are found by using the asymptotic relations for the modified Bessel functions, $I_j(z)\to e^z/\sqrt{2\pi z}$ as $z\to \infty$ and $j$ fixed\cite{bender}. The contribution of the third term of Eq.~(\ref{main}) turns out to be asymptotically negligible, while the second term leads to the scaling form, \begin{equation} P_+(\tau,t)\simeq t^{-1}f(\xi), \label{scal} \end{equation} in the scaling limit \begin{equation} t\to\infty, \quad \tau\to\infty, \quad \xi=\tau/t, \label{scalv} \end{equation} with the scaling function given by \begin{equation} f(\xi)={2\over\pi}\,{1\over \sqrt{1-\xi^2}}. \label{scalf} \end{equation} A prominent feature of the age distribution is that $\tau$ scales as $t$. That is, the average age, \begin{equation} T=\langle t-\tau\rangle\simeq t\int_0^1 d\xi\,(1-\xi)f(\xi)\simeq \left(1-{2\over\pi}\right)t, \label{time} \end{equation} grows linearly with the observation time $t$. \section{AGING IN STOCHASTIC MODELS OF COARSENING} The ballistic annihilation model is perhaps the simplest one-dimensional coarsening process with {\it deterministic} dynamics. We now consider simple examples of one-dimensional coarsening processes with {\it stochastic} dynamics. Consider first the $q$-state Potts model for $q=\infty$, with zero temperature Glauber dynamics and with the initial condition where each spin is in a different state. The dynamics proceeds as follows: during the time interval $dt$ a given spin assumes the state of one of its nearest neighbor with overall probability $dt/2$. In one dimension, the interfaces between domains of identical spins therefore diffuse and coalesce whenever two domains meet. The domain wall dynamics is thus identical to the diffusion-limited coalescence reaction, which may be represented as $A+A\rightarrow A$. Because of this equivalence between the Potts model and the coalescence reaction, the age distribution can be calculated exactly. Since interfaces coalesce upon colliding, only the interfaces which are the nearest neighbors of a particular site are important in determining its age distribution. In constructing the age distribution, first note that the spin will not change its color up to time $t$ if neither of the two neighboring interfaces reaches the spin. The probability $P_0(t)$ is thus equivalent to the square of the probability $Q(t,1)$ that a random walker on a lattice starting at position $x_0=1$ will not reach the origin up to time $t$. The probability $Q(t,1)$ is readily computable\cite{glauber} and gives the fraction of ``persistent'' spins: \begin{equation} P_0(t)=\left(e^{-t}[I_0(t)+I_1(t)]\right)^2. \label{p0potts} \end{equation} To compute the contribution to the age distribution from configurations where an interface has previously reached the spin (which we may take to be at the origin), let us assume that this spin takes on a new color from its left neighbor at time $\tau$. This spin is now the right extremity of a domain of same color spins (see Fig.~3). \begin{figure} \narrowtext \epsfxsize=\hsize \epsfbox{f3.eps} \vskip 0.2in \caption{ Illustration of one process which enters in the computation of $P_+(\tau,t)$ for the infinite-state Potts model. Shown is the spin configuration at times $\tau$ and $\tau+d\tau$ just as one spin changes its state. For the state this spin to remain unchanged until time $t$, both the domain wall a distance 1 to the right and the domain wall a distance $n$ to the left must not reach the position of the newly-flipped spin. \label{fig3}} \end{figure} Let the size of this domain be $n$. The position of the interface which defines the left edge of this domain is distributed according to the domain size distribution $F(n-1,\tau)$. The spin at the origin will then not change its color up to time $t$ if the two surrounding interfaces do not cross the origin. The continuous part of the age distribution can thus be written as \begin{equation} P_+(\tau,t)=\sum_{n=2}^{\infty}F(n-1,\tau)Q(t-\tau,n)Q(t-\tau,1). \label{p+potts} \end{equation} The last factor is just the probability that the domain which is one lattice spacing to the right of the spin at the origin does not reach the origin between time $\tau$ and time $t$, while the first two factors given the corresponding probability for the left-neighboring domain which is a distance $n$ from the origin. Each of the factors in this equation are well known. The domain size distribution is given by $F(n-1,\tau)= E(n-1,\tau)- 2E(n,\tau)+ E(n+1,\tau)$, where $E(k,t)$ is the probability to find at least $k$ successive spins of the same color at time $t$\cite{ben-avraham}. For a discrete lattice system, this latter distribution satisfies a lattice diffusion equation, with boundary condition $E(0,t)=1$ and initial condition $E(k,0)=\delta_{k,0}$, corresponding to the initial condition where each spin is different. The expression for $E(k,t)$ is\cite{glauber} \begin{equation} E(k,t)=1-e^{-2t}\left[I_0(2t)+2\sum_{j=1}^{k-1}I_j(2t)+I_k(2t)\right] \label{ekt} \end{equation} and thus \begin{equation} F(n-1,\tau)={e^{-2\tau}\over\tau}\,nI_n(2\tau). \label{fkt} \end{equation} In a similar vein, the probability $Q(t,k)$ that a random walker which starts at $x=k$ does not hit the origin during the time interval $(0,t)$ is\cite{glauber} \begin{equation} Q(t,k)=e^{-t}\left[I_0(t)+2\sum_{j=1}^{k-1}I_j(t)+I_k(t)\right]. \label{qkt} \end{equation} So we finally obtain \begin{eqnarray} P_+(\tau,t)={e^{-2t}\over\tau}\,\left[I_0(t-\tau)+I_1(t-\tau)\right] \,\sum_{n=1}^\infty nI_n(\tau)\nonumber\\ \times\left[I_0(t-\tau)+2\sum_{k=1}^{n-1}I_k(t-\tau)+I_{n}(t-\tau)\right]. \label{ptpottssol} \end{eqnarray} In the scaling limit (\ref{scalv}), the dominant contribution to the sum in Eq.~(\ref{ptpottssol}) is provided by terms with $n\propto\sqrt{t}$. In this region we use the asymptotic form of the Bessel functions $I_n(t)\simeq \exp(t-n^2/2t)/\sqrt{2\pi t}$. A lengthy but elementary computation then yields \begin{equation} P_+(\tau,t)\simeq {2\over \pi \sqrt{t^2-\tau^2}} \label{scalptpotts} \end{equation} which is exactly of the same form as Eqs.~(\ref{scalv})--(\ref{scalf}). At first sight, it may seem surprising to find the same scaling function, as well as the same expression for $P_0(t)$, as in the ballistic annihilation problem. Indeed, Eq.~(\ref{s0t}) can be computed from a mapping of the initial distribution of the interfaces onto a random walk process. $S_0(t)$ can then be computed in the same way as the probability $Q(t,1)$ shown above. Whenever we can determine a property of the infinite-state Potts model via the behavior of two independent random walks, we should recover the same results as in the ballistic annihilation problem. Nevertheless, some properties of these two systems are very different. For example, the domain size distribution in ballistic annihilation exhibits a non-trivial behavior which is characterized by an infinite number of singularities \cite{Piasecki,laupaul}. Let us now consider the age distribution of spins in the 2-state Potts model with zero temperature spin-flip dynamics, {\it i.\ e.}, the kinetic Ising-Glauber model\cite{glauber}. Since the solution for $P_0(t)$ in the Ising-Glauber model is difficult\cite{der}, one can anticipate that calculation of $P_+(\tau,t)$ is also subtle. We therefore study this problem numerically and give heuristic arguments to explain the limiting behaviors of the age distribution $P_+(\tau,t)$. Our numerical results, which are based on simulations of the equivalent $A+A\to 0$ reaction process, confirm that the scaling ansatz (\ref{scal})--(\ref{scalv}) still applies (Fig.~4). \begin{figure} \narrowtext \epsfxsize=\hsize \epsfbox{f4.eps} \caption{Simulation data for the age distribution in the one-dimensional Ising-Glauber model. Shown is the scaling function $f(\xi)$ versus $\xi$ for $t=(1.5)^{12}$ $(+)$ and $t=(1.5)^{17}$ $(\circ)$, with the latter data averaged (smoothed) over 5 consecutive points. The solid line is the guess $f_{\rm guess}(\xi) ={\cal B}\, \xi^{-5/8}(1-\xi)^{-1/2}$, with ${\cal B}= 0.259349\ldots$ as explained in the text. \label{fig4}} \end{figure} The singular behavior of the scaling part of the age distribution function $f(\xi)$ in the limits $\xi\downarrow 0$ and $\xi\uparrow 1$ can be accounted for by matching to the known behaviors in these limits. When $\tau={\cal O}(1)$, $P_+(\tau,t)\sim P_0(t)\sim t^{-3/8}$\cite{der}. Matching this with Eq.~(\ref{scal}) at $\xi=\tau/t={\cal O}(t^{-1})$ implies the $f(\xi)\sim \xi^{-5/8}$ as $\xi\downarrow 0$. This asymptotic behavior agrees well with our simulations. In the opposite limit of $\tau\to t$, the corresponding limiting form of the age distribution is determined by domain walls which have crossed the origin at time $\tau$ close to $t$ --- this happens with probability $t^{-1/2}$, since the number of domain walls decreases with time as $t^{-1/2}$\cite{glauber}. The diffusing domain wall should then not cross the origin again in the following time interval $(\tau,t)$ --- this happens with probability $(t-\tau)^{-1/2}$\cite{feller}. Thus, $P(\tau,t)\sim t^{-1/2}(t-\tau)^{-1/2}$, which implies that $f(\xi)\sim (1-\xi)^{-1/2}$ as $\xi\uparrow 1$, in agreement with our numerical results. Indeed, the product of these two asymptotic forms, $f_{\rm guess}(\xi)={\cal B}\, \xi^{-5/8}(1-\xi)^{-1/2}$ provides a reasonable fit to the data over most of the range of $\xi$. If one uses this guess over the entire range of $\xi$, then the normalization condition $\int d\xi f_{\rm guess}(\xi)=1$, requires the numerical prefactor to be ${\cal B}={\Gamma(7/8)\over \Gamma(3/8)\Gamma(1/2)} = 0.259349\ldots$. For the general $q$-state Potts model with zero-temperature Glauber dynamics, we may also expect that the age distribution scales, with the limiting behaviors of the scaling function given by \begin{equation} f(\xi)\sim \cases{\xi^{\theta(q)-1} & $\xi\downarrow 0$,\cr &\cr (1-\xi)^{-1/2} & $\xi\uparrow 1$. \cr} \label{limits} \end{equation} The persistence exponent $\theta(q)$, found analytically in Ref.~\cite{der}, increases from 3/8 to 1 as $q$ increases from 2 to $\infty$. Thus the ``smiling'' form of the age distribution in the Ising case (Fig.~4) gradually transforms into the half-smiling form of the infinite-state model (see Eq.~(\ref{scalptpotts})). In more than one dimension, aging of spins in the kinetic Ising model is expected to depend on the temperature. If an initially disordered system is quenched to a final temperature $T_f>0$, the average age is expected to be finite for all $d>1$. This follows because for non-conserved dynamics, even spins embedded within a large region of aligned spins will flip at a finite rate for all positive temperatures. On the other hand, for a quench to zero temperature, we anticipate that the average age will grow with time, since spin flips can occur only at interfaces, and these eventually disappear. To test this expectation, we performed numerical simulations of the two-dimensional kinetic Ising-Glauber model on the square lattice and found that the average age of the spins grows linearly in time and that scaling still applies. Moreover, the age distribution function has the same qualitative ``smiling'' form of the one-dimensional system (Fig.~4). In the small-age limit, $t-\tau\ll t$, the numerical data suggests a behavior of the age distribution which is consistent with $P(\tau,t)\sim t^{-1/2}(t-\tau)^{-1/2}$. To understand this result, which is identical to that of the one-dimensional counterpart, first note that the density of domain walls decays as $t^{-1/2}$. This arises because for non-conserved dynamics, the average domain size grows as $t^{1/2}$\cite{bray} and domains appear to be compact. Consequently, the domain wall density is expected to be the reciprocal of the average domain size. The perimeter of a domain has typically a vicinal shape, with the kinks and antikinks which define terraces undergoing diffusive motion (this diffusion does not cost energy and is therefore allowed at zero temperature). This diffusional motion is one-dimensional in character and thus a step (either kink or antikink) which has crossed a bond at time $\tau$ will not cross it again in the following time interval $(\tau,t)$ with probability $(t-\tau)^{-1/2}$. The age distribution is then given by the product of step density and the above no return probability, which gives $P(\tau,t)\sim t^{-1/2}(t-\tau)^{-1/2}$. In fact, the evolution of interfaces is much more involved process -- kinks and antikinks annihilate upon colliding, spin-flips at the corner give birth to a pair of steps (horizontal and vertical) -- but in the small-age limit these additional complexities should not qualitatively affect the age distribution. In the large-age limit, $\tau\ll t$, the scaled age distribution is expected to behave as $f(\xi)\sim \xi^{\theta-1}$, similarly to one dimension. Indeed, we confirmed numerically such power-law behavior and found that $\theta\approx 0.21$ provides the best fit to our data. This is consistent with previous simulations of the two-dimensional Ising-Glauber model for which the fraction of persistent spins, $P_0(t)$, was found to decay as $t^{-0.22}$\cite{dbg,stauf}. To determine the form of the age distribution for the kinetic Ising-Glauber model in higher dimensions, we apply a mean-field approach. It is simple to solve for $P(\tau,t)$ in the mean-field limit ({\it e.\ g.}, for the Ising model on a complete graph) since the dynamics in the zero-temperature case is simple: Spins from the majority phase do not change their state, while spins from the minority phase change their state with a constant rate which we may set equal to one. Suppose that the system starts from an initial condition where the fraction of $+$ and $-$ spins is equal to $p$ and $q=1-p$, respectively (with $p\geq q$ without loss of generality). Clearly, the fraction of spins which never change their state until time $t$ is equal to $p+qe^{-t}$. The probability that a minority spin changes its state in the time interval $(\tau,\tau+d\tau)$ is equal to $e^{-\tau}d\tau$. Thus, \begin{equation} P(\tau,t)=\left(p+qe^{-t}\right)\delta(\tau)+qe^{-\tau}. \label{mfising} \end{equation} This result violates the scaling form of Eq.~(\ref{scal}) but still implies that the average age (see Eq.~(\ref{age})) increases linearly in time: \begin{eqnarray} T &=&\left(p+qe^{-t}\right)t+q\left(t-1+e^{-t}\right)\nonumber\\ &=& t-q\left(1-e^{-t}-te^{-t}\right). \label{mfage} \end{eqnarray} \section{SUMMARY AND OUTLOOK} The age distribution in one-dimensional coarsening processes has been investigated by analytical and numerical techniques. These approaches indicate that the average age grows linearly with the observation time of the system. Exact results for two prototypical coarsening processes, the deterministic ballistic annihilation and the stochastic infinite-state Potts model with zero temperature Glauber dynamics have been obtained. For the general $q$-state Potts model with zero temperature Glauber dynamics, asymptotic behaviors have been established. Various results for the aging of spins in the Ising-Glauber model in general dimension have been obtained. The interesting situation, for non-conserved spin-flip dynamics, is that of zero temperature where domain walls ultimately disappear so that the system undergoes aging. In particular, numerical results in two dimensions were found to be qualitatively similar to corresponding one-dimensional results. We anticipate that the bimodal ``smiling'' form of the age distribution will arise for all spatial dimension $d<4$. When $d\geq 4$, however, the age distribution is expected to exhibit features similar to the easily-derived mean-field solution (see Eq.~(\ref{mfising})). In particular, the fraction of spins which never flip should saturate at a finite value even in the symmetric case of $p=q=1/2$. This has apparently been observed \cite{stauf}, although it is hard to definitively settle this issue by numerical means, especially in the marginal case of $d=4$. It is worth noting that for the models discussed in this work, the only possibilities found are systems where the average age saturates to a finite value or where the average age increases linearly in time. The saturation of the age in first class of systems arises because a steady state is reached. On the other hand, for systems which coarsen is is perhaps worth investigating whether there are examples where the average age grows slower than linear in time. Numerical evidence shows that the average age in the two-dimensional voter model is growing slower than linearly and perhaps logarithmically in time. This intriguing possibility merits further consideration. For the coarsening processes examined in this work, the dynamics determines the age distribution. It may be instructive to study models with feedback, in which the aging process influences the coarsening dynamics\cite{adaptive}. The adaptive voter model is one such example. Another possibly intriguing extension would be to consider coarsening processes with conservative dynamics. \vskip 0.16in The research reported here was supported in part by the Swiss National Foundation, by the ARO (grant DAAH04-93-G-0021), and by the NSF (grant DMR-9219845). We also thank J. F. Mendes for helpful discussions.

proofpile-arXiv_065-594

\section{Introduction} The main motivation behind this work lay in developing approximate analytical tools for dealing with the long range physics of Quantum Chromodynamics. The large-$N_c$ approximation is an attractive candidate for this project. Initiated by 't Hooft, Ref. \cite{Hoo1}, it was immediately applied to solve large-$N_c $ 1+1 dimensional QCD, Refs. \cite{Hoo2}, \cite{Col}, \cite{Cal}, \cite{Ein} and led to extremely fruitful phenomenological insights and models, Refs. \cite{Col}, \cite{Wit}, \cite{Adk}. The factorization property of correlators of gauge invariant operators suggested the idea of the master field -- the gauge field configuration which dominates the path integral in the large-$N_c$ limit, cf. Refs. \cite{Col}, \cite{Wit}, \cite{Mak}. Knowledge of the master field should allow to calculate any gauge invariant observable as if it were a classical non-fluctuating object. A concrete example of a master field was provided by the exact solution of a model which described the dynamics (classical as well as quantum mechanical) of a single hermitian $N\times N$ matrix, Ref. \cite{Bre1}. It was found that in the large-$N$ limit the Boltzmann integral (in the classical case) and the ground state properties (in the quantum case) of this model were completely determined by an ensemble of matrices with a frozen distribution of eigenvalues given by a solution of a certain integral equation of the mean field type. The ``angular'' orientations of the matrices in this ensemble, i.e. the unitary matrices $W$ in the decomposition $M = WmW^{+}$ with $m= {\rm diag}(m_1, m_2, . . . , m_N)$ were completely free, i.e. distributed with equal probability according to the unbiased $U(N)$ group theoretical Haar measure. This work and, in its sequel, Ref. \cite{Itz} gave rise to an intense interest in solutions of large-$N$ matrix models and made contact with the study of such models in other fields of physics, most notably nuclear physics, cf. Ref. \cite{Meh} and, more recently, two-dimensional quantum gravity, cf. Ref.~\cite{Bre2} and quantum chaos, cf. Ref. \cite{Alt}. The general interest in large-$N$ matrix models has been behind a steady progress in understanding solutions of various versions and generalizations of such models as reported e.g. in Refs. \cite{Amb}, \cite{Bre3}, \cite{Bre4}, \cite{Kaz}, \cite{Kazm}. A new direction in this activity opened up when the physics community became acquainted with recent advances in the mathematical literature, associated with the work of Voiculescu, Ref. \cite{Voi}, on non-commutative probability theory, Ref. \cite{Sin}. The statistical mechanics (classical or quantum mechanical) of matrix (non-commuting) variables or (gauge) fields is an example of probability theory of non-commutative variables. A concept which turned out to be especially useful for the study of large-$N$ matrix and gauge theories, cf. Refs. \cite{Gop} and \cite{Dou}, was the introduction in Ref. \cite{Voi} of the so-called {\em free random variables}. The simplest examples of such variables occur in independent large-$N$ matrix models, i.e. in the classical statistical mechanics of several $N\times N, \;\; N\to\infty $ hermitian matrices $M_i,\, i=1, \ldots ,D$, with the probability distribution (the Boltzmann factor) given as $const \cdot \exp(-N^2 \sum_i V_i (M_i))$ with $V_i (M_i ) = (1/N) \mbox{Tr} \, P_i (M_i )$ and $P_i $ polynomial functions. Typical ``observables'' in such a model are correlators \begin{equation} \frac{1}{N}\langle \mbox{Tr} \, (M_{i_1} \ldots M_{i_k})\rangle \equiv \int \, \frac{1}{N} \, \mbox{Tr} \, (M_{i_1} \ldots M_{i_k}) \exp\{\sum_{i=1}^D [F_i - N^2 V_i(M_i)]\}\prod_{k=1}^{D} D\mu[M_k] \; , \end{equation} with the integration measure \begin{equation} D\mu[M_k] \equiv \prod_{\gamma=1}^N d(M_k)_{\gamma\gamma} \prod_{\gamma > \nu =1}^N d \mbox{Re} (M_k)_{\gamma\nu} \;d \mbox{Im} ( M_k)_{\gamma\nu} \; , \label{intmes} \end{equation} and $F_i = -\ln \int D\mu[M_i] \exp[-N^2 V_i(M_i)] \; . $ Were the $M_i $ regular commuting variables, one would call them independent since $\langle M_{i_1} \ldots M_{i_k}\rangle = \langle M_{i_1}\rangle \ldots \langle M_{i_k}\rangle $ for any selection $i_1 ,\ldots , i_k$. For the matrix model this is obviously not the case. The probability distribution depends only upon the eigenvalues $m_i$ of the $M_i$. The expectation of products of the $M_i$ on the other hand involves also the ``angular'' variables $W_i$ in the decomposition $M_i = W_i m_i W_i^{+}$. These variables however are ``free'', meaning that they are weighted with the unbiased Haar $U(N)$ measure. The integrals over the $W_i$ can therefore be evaluated in a universal manner independent of what the potentials $V_i$ are. The remaining integrals over the $m_i$ can then be evaluated in the limit $N\to\infty$ by a saddle point approximation as in Ref. \cite{Bre1}. The remarkable fact is that this procedure can be formulated in terms of very general rules relating any correlator of independent matrices to a linear combination of products of various individual moments of these matrices, cf. \cite{Voi}. These rules will be reviewed in Section \ref{tracfr} below. Any non-commutative variables distributed in such a manner that they satisfy these rules are termed ``free random variables''. Refs. \cite{Gop} and \cite{Dou} provide a review of the techniques which were developed to deal with such things as sums and products of free random variables, to define a certain operator algebra in a suitably defined Fock space, and further applications, cf. also \cite{Zee}, \cite{Are}, \cite{Nsp}. It has been possible to prove that a master field for any matrix model can be constructed on the basis of the operator algebra developed for free models, Ref. \cite{Gop}, \cite{Dou}. However, so far no dynamical principle has been advanced in order to carry out this construction explicitely for interacting matrix and field theories. This paper presents an attempt to make progress in this direction by using a variational method based on the inequality $\langle e^{-S} \rangle \ge e^{-\langle S\rangle } $. This approach is fairly standard in statistical physics. The new element will be to use the free matrix models as the trial variational space. The details of the development are outlined in Section \ref{secone} below. Given a certain set of variables in terms of which the action is formulated, one can give a complete solution of the variational problem in the space of free matrix models by defining what will be called the free partner of the exact action of the model. Subsequently one observes however that the available variational freedom is even wider since one can first transform to a new set of variables, $M_i\to M_i({B_j})$. In the new $B_i$ it is again possible to define the free partner of the action and give the general variational solution. Thus the ``art part'' of the proposed variational method is the choice of the optimal $B_i$ or in other words of the optimal parameters of the transformation $M_i\to M_i({B_j})$. Unfortunately it is hard to give estimates of the accuracy of this variational approximation. As usual, one must gain experience from examples and comparison with exact solutions. In Section \ref{secexa}, a number of such examples are considered starting with classical interacting matrices and then going to quantum models of first one and then two interacting quantum matrices. Only linear transformations $M_i=\sum_j c_{ij}B_j $ are considered and the variational results using free random variables are compared with exact Monte Carlo calculations and, where available, with analytic results. In almost all models considered, e.g. in the models of two interacting matrices with the action $$ S= \frac{1}{N} \mbox{Tr} \, (M_1^4 + M_2^4 + g M_1^2 M_2^2) $$ and various values of $g$, of many interacting matrices with $$ S=\frac{1}{D-1} \sum_{i\ne j}^D \frac{1}{N} \mbox{Tr} \, M_i^2 M_j^2 \;\;\; {\rm and} \;\;\; S = \frac{1}{D-1} \sum_{i\ne j}^D \frac{1}{N} \mbox{Tr} \, M_i^4 M_j^4 $$ and in the quantum models of one and two interacting matrices, impressively good agreement is found with exact results. Some models are also presented for which the method (with the additional limitation to the linear $M\to B$ transformations) failed completely. Presumably nonlinear transformations could improve the agreement but the authors have not investigated this. \section{Variational Method for Matrix Models} \label{secone} In this paper, matrix models described by partition functions of the type \begin{equation} Z\equiv e^{-F} = \int \exp (-N^2 S[M_1,. . .,M_D]) \prod_{k=1}^D D\mu[M_k] \label{matint} \end{equation} are considered, with $M_k$, $k = 1,\ldots ,D$, representing hermitian $N \times N$ dimensional matrices, in the limit $N\rightarrow \infty $. A factor $N^2 $ has been pulled out in the exponent and in return, all matrix traces appearing in actions $S$ will be accompanied by a factor $1/N$. Note that the term ``action'' is used very loosely here to denote essentially any weight appearing in the exponent of the Boltzmann factor in a partition function. Only actions $S[M_1,\ldots ,M_D]$ which are invariant under the global unitary ``rotations'' $M_i\to U M_i U^{+}$ will be considered. One can further distinguish between ``classical'' and ``quantum'' matrix models. By classical one means the models for which the above integral is just the ordinary integral with the integration measure given by (\ref{intmes}) whereas in quantum models this is a path integral with $M_i$ depending on (imaginary) time $\tau$, $M_i(\tau)$, and the measure \begin{equation} {\cal N}\prod_{0<\tau<\beta} D\mu[M_k(\tau)] = {\cal N}\prod_{0<\tau<\beta}\prod_{\gamma=1}^N d(M_k(\tau))_{\gamma\gamma} \prod_{\gamma > \nu =1}^N d \mbox{Re} (M_k(\tau))_{\gamma\nu} \;d \mbox{Im} (M_k(\tau))_{\gamma \nu} \; . \end{equation} Here $\beta$ is the inverse temperature and $\cal N$ is a normalization factor which in principle depends on $\Delta \tau$, but will not be important in the following. In both the classical and quantum cases one deals with integrals over many matrix variables and the real distinction will lie in the form of the action. In the classical case the action (the classical potential) is a {\em real function} usually represented as a sum of monomials in powers of the $M_i$, \begin{equation} S_{\rm cl} \equiv V(M_1, \ldots ,M_D) = \frac{1}{N} \, \sum_{i_1,i_2 ,\ldots ,i_D} \sum_{m_1,m_2 ,\ldots ,m_D} C_{m_1 m_2 \ldots m_D}^{i_1,i_2,\ldots ,i_D} \mbox{Tr} \, \left(M_{i_1}^{m_1}M_{i_2}^{m_2} \ldots M_{i_D}^{m_D}\right)\; , \label{clact} \end{equation} with real coefficients $C_{m_1 m_2 \ldots m_D}^{i_1,i_2, \ldots ,i_D}$. In the quantum case the action will be a sum of kinetic and potential terms \begin{equation} S_Q = \frac{1}{2\Delta \tau}\sum_{0<\tau<\beta}\;\;\sum_{i=1}^D \frac{1}{N} \mbox{Tr} \, \left[M_i(\tau +\Delta \tau) - M_i(\tau)\right]^2 + \sum_{0<\tau<\beta} \Delta\tau V(M_1(\tau), \ldots , M_D(\tau)) \; , \end{equation} with $V$ as in (\ref{clact}). \subsection{The Variational Principle} As was reviewed in the Introduction, only a limited number of matrix models are amenable to analytical solution so that various methods of approximation are called for. Here, the powers of the variational method, combined with the large-$N$ saddle point evaluation, will be explored. As is common with statistical integrals, the variational approximation will be developed on the basis of the inequality, cf. Ref. \cite{Fey}, \begin{equation} \Big< e^{- x}\Big> \ge e^{-\left<x\right>} \; , \label{ineq} \end{equation} which expresses the convexity property of the exponential function -- the average value of the exponential of a random variable $x$ is greater than or equal to the exponential of the average provided $x$ is real and the weights used in the averaging are positive. If one can find an action $S_0[M]$ which is simple enough to allow the calculation of integrals like $$ \int \prod_k D\mu[M_k ]e^{-N^2 S_0[M]}\;\;\;\; {\rm and }\;\;\;\; \int \prod_k D\mu[M_k] K(M_1, \ldots , M_D)e^{-N^2 S_0[M]} \; $$ for some simple functions $K$ then one can approximate the integral (\ref{matint}) as follows. One can represent \begin{equation} e^{-F} = \int \prod_k D\mu[M_k]e^{-N^2 S[M]} = \int \prod_k D\mu[M_k] e^{-N^2 (S[M] - S_0[M])} e^{-N^2 S_0[M]} = e^{-F_0} \Big< e^{-N^2 (S - S_0)} \Big>_{S_0 } \; . \end{equation} where $\langle O\rangle_{S}$ means $\int O\exp(F - N^2 S) D\mu(M)$. Using the inequality (\ref{ineq}) one obtains \begin{equation} e^{-F} = e^{-F_0}\Big< e^{-N^2 (S - S_0)} \Big>_{S_0} \ge e^{-[F_0 + N^2 \big<S - S_0 \big>_{S_0} ] } \;, \end{equation} or \begin{equation} F\le F_0 + N^2 \big<S - S_0 \big>_{S_0} \; . \label{minp} \end{equation} If $S_0$ depends on adjustable parameters, one can find their optimal values by minimizing the right hand side of the above inequality. The resulting $S_0$ can then be adopted as an approximation for $S$ with which one should calculate the relevant correlators. As always with variational methods, it is hard to assess the accuracy of the approximation. Here, it will be tested in different classes of matrix models, using the so-called free matrix models (cf. below) as the variational space, by comparing (where available) with exact analytical results or with Monte Carlo simulations. \subsection{Free Matrix Models} \label{tracfr} Freeness is a concept recently developed in the mathematical literature associated with noncommuting random variables \cite{Voi}. While freeness is abstractly defined without reference to a specific realization of the random variables, in the context discussed here, namely in relation to large-$N$ hermitian multi-matrix models, it essentially means that the weight $\exp(F-N^2 S)$ according to which the matrices are distributed contains no bias with respect to the relative $SU(N\rightarrow \infty )$ ``orientations'' of the matrices. In terms of the decomposition $M_k = W_k m_k W_k^{+}$ with diagonal $m_k$ and unitary $W_k$ this means that the action $S$ in the free models depends only on the $m_k$. Consequently all the angular integrals in any given correlation function are weighted only with the group theoretical $U(N)$ Haar measure and give universal results independent of what the action is. The remaining integrals over the $m_k$ in the large-$N$ limit are controlled by the saddle point of the effective action which includes the contribution from the Vandermonde determinants for each $k$, cf. examples below. As a consequence of these properties all correlation functions for free multi--matrix models are systematically accessible in terms of the {\em moments of individual matrices} through the recently developed techniques, as will be elaborated upon below, cf. also Refs. \cite{Voi}, \cite{Gop}, \cite{Dou}. A family of noncommuting random variables $M_i$ is called free if and only if \begin{equation} \langle f_1 (M_{i_1 } ) f_2 (M_{i_2 } ) \ldots f_n (M_{i_n } ) \rangle =0 \ \mbox{whenever} \ \langle f_j (M_{i_j } ) \rangle = 0 \ \mbox{for all} \ j \end{equation} where $i_j \neq i_{j+1} $ (and $i_{n+1} $ is identified with $i_1 $ ). Note that the bracket notation introduced here following \cite{Voi} means taking the expectation value of the trace of the expression inside, divided by $N$. In order to evaluate a general correlation function $\langle p_1 (M_{i_1 } ) p_2 (M_{i_2 } ) \ldots p_n (M_{i_n } ) \rangle $, one can use \begin{equation} \langle (p_1 (M_{i_1 } ) - \langle p_1 (M_{i_1 } ) \rangle ) \; (p_2 (M_{i_2 } ) - \langle p_2 (M_{i_2 } ) \rangle ) \; \ldots \; (p_n (M_{i_n } ) - \langle p_n (M_{i_n } ) \rangle ) \rangle = 0 \label{algo} \end{equation} and multiply out the terms in the expectation value. This recursively determines correlation functions of a certain order in terms of lower-order correlation functions until one arrives at an expression containing only {\em the moments of the individual matrices}. For two free matrices $M_1 $ and $M_2 $ this algorithm yields, e.g., \begin{eqnarray} \langle p_1 (M_1 ) p_2 (M_2 ) \rangle &=& \langle p_1 (M_1 ) \rangle \, \langle p_2 (M_2 ) \rangle \\ \langle M_1 M_2 M_1 M_2 \rangle &=& \langle M_1^2 \rangle \, \langle M_2 \rangle^{2} + \langle M_2^2 \rangle \, \langle M_1 \rangle^{2} - \langle M_1 \rangle^{2} \, \langle M_2 \rangle^{2} \label{m1m2e} \end{eqnarray} Building on this information, it has been possible, e.g., to determine, \cite{Voi}, how the eigenvalue distributions of matrices which are free with respect to one another are additively and multiplicatively convoluted. Freeness in matrix models is a slightly more general concept than what is usually meant by the term ``independent matrix model'', in which the potential governing the distribution of the matrices is a sum of terms for each individual matrix. Independent matrix models are certainly free; however, so is e.g. a two-matrix model with potential \begin{equation} V(M_1 ,M_2 ) = \frac{1}{N} \mbox{Tr} \, M_1^2 + \frac{1}{N} \mbox{Tr} \, M_2^2 + \frac{1}{N^2 } \mbox{Tr} \, M_1^2 \, \mbox{Tr} \, M_2^2 \label{freex} \; . \end{equation} More generally, the potential of a free matrix model can be any function of traces over functions of the individual matrices\footnote{To the authors' knowledge, this type of model was first considered in \cite{Wadia}.}. Such more general free matrix models are almost as easy to solve as the independent ones. This is best illustrated by an example. Consider the model described by (\ref{freex}). After the transformation of variables $M_i = W_i m_i W_i^{+}$, with diagonal $m_i$ and unitary $W_i$, the potential $V(M_1, M_2)$ depends only on the $m_i$. Moreover, the transformation introduces the Vandermonde Jacobians which, when exponentiated, modify the potential into \begin{eqnarray} N^2 V_{eff} &=& \sum_{i=1,2} \;\; \sum_{\alpha\ne\beta}^N\;\ln|m_{i,\alpha } - m_{i,\beta }| + N^2 V(m_1,m_2) = \nonumber \hspace{5.0cm} \\ &=& N^2\left[\sum_{i=1,2}\left(\int_{-\infty}^{\infty} \rho_i(\mu)\;\ln|\mu-\mu'|\;\rho_i(\mu')\; d\mu \;d\mu' +\int_{-\infty}^{\infty}\rho_i(\mu)\mu^2\right) + \prod_{i=1,2}\int_{-\infty}^{\infty}\rho_i(\mu)\mu^2 d\mu\right] \; . \nonumber \end{eqnarray} In the large-$N$ limit, the distributions $\rho_i(\mu)$ of the eigenvalues are ``frozen'' at a saddle point, so that the moments of the matrices have definite values \begin{equation} \frac{1}{N} \mbox{Tr} \, M_1^2 = x_1 \ \ \ \ \ \ \ \ \frac{1}{N} \mbox{Tr} \, M_2^2 = x_2 \label{freeea} \end{equation} This means that the saddle point eigenvalue distributions $\rho_i(\mu)$ can be solved for as if they were controlled by the potentials \begin{equation} V(M_1 ) = (x_2 +1) \frac{1}{N} \mbox{Tr} \, M_1^2 \ \ \ \ \ \ \ \ V(M_2 ) = (x_1 +1) \frac{1}{N} \mbox{Tr} \, M_2^2 \; . \label{freeeb} \end{equation} For such quadratic potentials, one obtains the well-known semicircular distributions, cf. Ref. \cite{Meh},\cite{Bre1} \begin{equation} \rho_1 (\mu ) = \frac{x_2 +1}{\pi } \sqrt{\frac{2}{x_2 +1} -\mu^{2} } \ \ \ \ \ \ \ \ \rho_2 (\mu ) = \frac{x_1 +1}{\pi } \sqrt{\frac{2}{x_1 +1} -\mu^{2} }\; . \end{equation} The constants $x_1 $ and $x_2 $ can now be determined by the self-consistency conditions \begin{equation} \left. \begin{array}{c} x_1 = \langle M_1^2 \rangle = \int d\mu \, \mu^{2} \rho_1(\mu ) \\ x_2 = \langle M_2^2 \rangle = \int d\mu \, \mu^{2} \rho_2 (\mu ) \end{array} \right\} \Rightarrow x_1 = x_2 = \frac{\sqrt{3} -1}{2}\; . \end{equation} Including such self-consistency conditions on selected moments of the matrices is the only additional step needed compared with solving independent matrix models. Knowledge of the eigenvalue distributions $\rho_i (\mu)$ is sufficient to calculate any correlation function since the latter can be reduced using Eq. (\ref{algo}) to a sum of products of individual moments, i.e. to the integrals $\langle M_i^k \rangle = \int\mu^k\rho_i(\mu) d\mu $. It was argued e.g. in Refs. \cite{Gop},\cite{Dou} that a free behaviour of the matrix variables representing physical degrees of freedom may capture correctly the main features of the dynamics in the large-$N$ limit of nonabelian gauge theories. This hope and the wealth of available analytical techniques presented above render the space of free matrix models a good candidate for variational calculations. The main idea of the present approach is to choose the optimal action $S_0$ in the variational principle (\ref{minp}) from among all actions which leave the matrices in terms of which $S$ is given, free with respect to one another. \subsection{General Variational Solution in the Space of Free Matrix Models} In making a variational approximation based on free matrix models one must first decide which combinations of the original matrix variables $M_i$ will be assumed to be free, i.e. in which variables to formulate the exact action $S$. One should realize that e.g. free $M_1 $, $M_2 $ do not imply free $M_1 +M_2 $, $M_1 -M_2 $ nor vice versa (more about this below). The second step is to find the best trial action $S_0$ which is free in the variables one has settled for. In this subsection, the solution of the second step is addressed. In other words, consider $S_0 $ to be {\em free in terms of the original} $M_i$. Under this limitation, it is possible to give a {\em general recipe} for finding the trial $S_0$ which minimizes the right hand side of (\ref{minp}). For this it is useful to introduce the following algorithm. Given a trace of any function of matrices $M_i$, $P=(1/N) \mbox{Tr} \,p(M_1, \ldots , M_D)$, write down its expectation value in terms of the moments of the individual matrices under the assumption that the matrices are free with respect to one another. This is easily accomplished for any multinomial expression using Eq. (\ref{algo}). Now rewrite the resulting expression with $\langle M_i^j \rangle$ replaced by Tr$\, M_i^j /N$. This defines what will be called the ``free partner'' $P_{f}$ of $P$ (the ``liberated'' $P$, so to speak). For instance, the free partner of $P=(1/N) \mbox{Tr} \,(M_1M_2M_1M_2)$ is $$ P_{f} =\frac{1}{N^3 } \mbox{Tr} \, M_1^2 \, (\mbox{Tr} \, M_2)^2 +\frac{1}{N^3 } (\mbox{Tr} \, M_1)^2 \, \mbox{Tr} \, M_2^2 - \frac{1}{N^4 } (\mbox{Tr} \, M_1)^2 \, (\mbox{Tr} \, M_2)^2 \; , $$ cf. Eq. (\ref{m1m2e}). Evidently, the original function and its free partner have the same expectation values when evaluated in any free matrix model for the $M_i $ (note that the expectation values of the products of traces appearing in the free partner factorize at large $N$). In general, the two expectation values will of course differ. Furthermore, if the free partner $S_{f}$ of any action $S$ is used as a trial action, it will generate a free matrix model; hence the terminology. With the help of the above definitions, one can give the general solution of the minimization principle (\ref{minp}) in the space of matrix models which are free in terms of the variables $M_i $. The crucial observation is that, when $S_0 $ is to be chosen from among all free actions, the original action $S$ and its free partner $S_{f} $ lead to the same minimization problem, i.e. identical right hand sides of Eq. (\ref{minp}), since $\langle S\rangle_{S_0 } =\langle S_{f} \rangle_{S_0 } $ by construction for any free $S_0$, as mentioned above. However, the minimization problem for $S_{f} $ is trivial to solve: Since $S_{f} $ itself describes a free matrix model, the best approximation to $S_{f} $ in the space of free actions is $S_0 =S_{f} $ itself. Since the approximation to $S$ is governed by the same minimization problem, $S_0 =S_{f} $ is at the same time the best free approximation to $S$. This result provides a general solution of the minimization problem in the space of matrix models which are free {\em in terms of the original variables} $M_i $. Note that the solution $S_f $ of the minimization problem defines a type of mean field approximation to the exact action $S$. Interaction terms in $S$ which are sensitive to angular correlations between the different matrices are replaced by terms in which each matrix couples only to selected moments of the other matrices, i.e. to some mean properties independent of the angular orientations. This is entirely analogous to the standard development of the Hartree or Hartree-Fock mean field theories from a variational principle using the space of (properly symmetrized or antisymmetrized) product states. There, in second-quantized language, the combinations in which the mean field enters the single-particle Hamiltonian are determined by Wick's theorem; here, in complete analogy, the combinations in which the moments enter the free partner are determined by the free random variable axioms as encoded in Eq. (\ref{algo}). In fact, (\ref{algo}) is nothing but Wick's theorem generalized from the usual bosonic and fermionic cases to the case of objects obeying the Cuntz algebra (this is the algebra obeyed by the Fock space operator representation of free variables, cf. Ref. \cite{Gop}). \subsection{Variable Transformations} The idea of the present variational approach was stated as choosing the optimal action $S_0 $ from among all matrix model actions which leave the matrices in terms of which $S$ is given, free with respect to one another. In this there exists an additional freedom of choice which will now be exploited, namely the choice of matrix variables in terms of which $S$ is given, i.e. in terms of which one develops the mean field approximation by constructing the free partner $S_f $. Up to now, only a fixed set of variables was considered -- the original $M_i $. However, if one rewrites the action $S$ in terms of other variables, \begin{equation} S(\{ M_i \} ) = S( \{ M_i ( \{ B_j \} ) \} ) = \tilde{S} ( \{ B_j \} ) \end{equation} and again looks for the optimal free action, now in terms of the new variables $B_j $, approximating $\tilde{S} $, one may find a better approximation than the one found using the variables $M_i $. It should be emphasized that free $B_j $ in general do not imply free $M_i $ nor vice versa. Therefore, the accessible variational space using free matrix models is in fact much larger than was apparent in the last section. By allowing different sets of variables in which to formulate the problem to be approximated, one can even include into the variation models in which the original matrices $M_i $ are not free with respect to one another. To formalize this idea, it is necessary to reexamine the derivation of the variational principle (\ref{minp}). Carrying out a variable transformation $M_i \rightarrow M_i ( \{ B_j \} )$, one has \begin{equation} e^{-F} = \int \prod_j D\mu[B_j] e^{-N^2 \tilde{S} [B_j] + \ln J [B_j ] } \ge \exp \left[ -( F_0 + N^2 \langle \tilde{S} - (\ln J)/N^2 -S_0 \rangle_{S_0 } ) \right] \end{equation} i.e. \begin{equation} F \le F_0 + N^2 \langle \tilde{S} - (\ln J)/N^2 - S_0 \rangle_{S_0 } \label{linv} \end{equation} with the Jacobian (which must be non-vanishing to have a legitimate change of variables) \begin{equation} J=\left|\frac{D\mu [M_i ] }{D\mu [B_j ] }\right| \end{equation} Note that this is the determinant of a $DN^2 \times DN^2 $ matrix, where $D$ denotes the number of matrix variables. Now, one can use the theorem of the previous section with the immediate result that the optimal free choice of $S_0 $ in the new variables is the free partner of $\tilde{S} - (\ln J)/N^2 $. Thus, while for any given set of variables, the optimal $S_0 $ is known, one may now try to further improve the approximation (i.e. diminish the right hand side of (\ref{linv})) by trying different sets of variables leading to different $\tilde{S} $ and $J$. In general, this is technically difficult, since $\ln J $ is not a multinomial expression, for which the algorithm of finding the free partner is easiest to carry out. Therefore, in this paper the treatment will be specialized to linear transformations, \begin{equation} M_i = \sum_{j} c_{ij} B_j \end{equation} Then $J=(\det (c_{ij} ) )^{N^2 } $ is independent of the matrix variables $B_j $. This greatly simplifies the treatment of (\ref{linv}); the Jacobian does not play any role in the determination of the optimal free action $S_0 $ for given new variables $B_j $ and thus the best free approximation $S_0 $ to the given $\tilde{S} $ is the free partner $\tilde{S}_f $ of $\tilde{S} $. Using furthermore that $\langle \tilde{S} -\tilde{S}_f \rangle_{\tilde{S}_f } =0 $, one arrives at the residual minimization problem \begin{equation} F \le \tilde{F}_f - \ln J \label{linvs} \end{equation} where $\tilde{F}_f$ is the free energy associated with $\tilde{S}_f $. It now remains to try different choices of $c_{ij} $, leading to different $\tilde{S} $, $\tilde{S}_f $, and finally $\tilde{F}_f$, such as to minimize the right hand side of (\ref{linvs}). \subsection{Calculation of the Free Energy} Before actually carrying out the minimization procedure, it is convenient to clarify a technical point concerning the actual evaluation of $\tilde{F}_f $ given $\tilde{S}_f$. The best way to achieve this seems to be the following. By subtracting a constant $F_{ref} $, independent of $c_{ij} $, from both sides of (\ref{linvs}), one arrives at the equivalent problem of minimizing $\tilde{F}_f - \ln J -F_{ref} $. For $F_{ref} $, one can choose a reference free energy, preferably of an independent matrix model with the same number of variables as $S$ and whose potential $V_{ref} $ is a polynomial of the same degree as $S$. This specification is not necessary, but convenient from the point of view of keeping the expressions one deals with in practice as regular as possible. In many applications, $V_{ref}(B_i ) = (1/N) \mbox{Tr} \, \sum_{i} B_i^2 $ is adequate. Then one can write \begin{eqnarray} \tilde{F}_f - F_{ref} &=& - \ln \int \prod_j D\mu[B_j] e^{-N^2 \tilde{S}_f[B] } + \ln \int \prod_j D\mu[B_j] e^{-N^2 V_{ref} [B] } \nonumber\\ &=& \left. - \ln \int \prod_j D\mu[B_j] e^{-N^2 (\alpha \tilde{S}_f[B] + (1-\alpha ) V_{ref} [B] )} \right|^{\alpha =1 }_{\alpha =0 } \nonumber\\ &=& -\int_{0}^{1} d\alpha \, \frac{\partial}{\partial \alpha}\ln \int\prod_j D\mu[B_j] e^{-N^2 \tilde{S}^{\prime}_f [B,\alpha] } \\ &=& N^2 \int_{0}^{1} d\alpha \, \langle \tilde{S}_f [B] - V_{ref} [B] \rangle_{\tilde{S}^{\prime}_f } \label{linmod} \end{eqnarray} where \begin{equation} \tilde{S}_f^{\prime} [B,\alpha] = \alpha \tilde{S}_f [B] + (1-\alpha ) V_{ref} [B ] \label{dpridef} \end{equation} In practice, the matrix model described by $\tilde{S}^{\prime}_f $ is only insignificantly harder to solve than the one described by $\tilde{S}_f $; on the other hand, expectation values like the ones contained in (\ref{linmod}) are very easy to evaluate and thus constitute a convenient way of evaluating the right hand side of the variational principle (\ref{linvs}). Thus, in applications, the best formulation of the variational problem seems to be to demand minimization of \begin{equation} \int_{0}^{1} d\alpha \, \langle \tilde{S}_f [B] - V_{ref} [B] \rangle_{\tilde{S}^{\prime}_f } -(\ln J)/N^2 \label{modprin} \end{equation} as a function of the transformation matrix $c_{ij} $, where, to recapitulate, $\tilde{S}^{\prime}_f $ is given by (\ref{dpridef}) and $\tilde{S}_f $ is the free partner of $\tilde{S} $, which in turn arises from performing linear transformations on the original matrix variables $M_i $, i.e. $S(M_i ) = S(c_{ij} B_j ) \equiv \tilde{S} (B_j ) $. Furthermore, $J=(\det (c_{ij} ) )^{N^2 } $, and $V_{ref}[B]$ is a conveniently chosen reference potential independent of $c_{ij} $ generating an independent matrix model. As a last remark, if one has succeeded in minimizing (\ref{modprin}) by varying the transformation $c_{ij} $, one will usually be interested in transforming back to the original variables; e.g., given the eigenvalue distributions of the matrices $B_j $, one may wish to extract the eigenvalue distributions of the matrices $M_i $. Here, the additive convolution techniques for free variables developed in \cite{Voi} find fruitful application, as will become clear in the examples to be treated further below. \section{Examples} \label{secexa} \subsection{Models Involving Two Classical Matrices} The simplest type of interacting matrix model involves just two classical matrices; this provides a first testing ground for the variational approach developed in the previous sections. Without any deeper motivation, the actions $$ S_1 = \frac{1}{N} \mbox{Tr} \, (M_1^4 +M_2^4 + gM_1^2 M_2^2 ) $$ and $$ S_2 = \frac{1}{N} \mbox{Tr} \, (M_1^2 + M_2^2 + gM_1 M_2 M_1 M_2 ) $$ will be considered. The variational approximation will be compared mainly with Monte Carlo experiments, although there also already exist exact analytical solutions to some simple models of this type, see e.g. \cite{Che} in the case of $S_2 $. The models considered in this section are invariant under the transformation $M_i \rightarrow -M_i $, and throughout the treatment it will be assumed for simplicity that, at the large-$N$ saddle point, $\langle M_i \rangle =0$. Of course, it constitutes no problem in principle to work without this assumption and to verify it from the solution; this would just unnecessarily complicate the notation. In more complicated models, it may become an interesting issue whether such a reflection symmetry can be broken spontaneously. Consider now the model described by $S_1 $. This model will be treated here in some detail to exhibit the new techniques in practice. Later examples will receive a more cursory treatment. Allowing for a general linear transformation of variables, \begin{equation} M_i = \sum_{j} c_{ij} B_j \label{glt} \end{equation} and inserting into $S_1 $ to arrive at $\tilde{S}_{1} (B_j )$, one obtains for the free partner of the latter \begin{equation} \tilde{S}_{1f} = b_1 \frac{1}{N} \mbox{Tr} \, B_1^4 + b_2 \frac{1}{N} \mbox{Tr} \,B_2^4 + b_{12} \frac{1}{N^2 } \mbox{Tr} \, B_1^2 \mbox{Tr} \, B_2^2 \end{equation} with the notations \begin{eqnarray} b_1 &=& c_{11}^4 + c_{21}^4 + gc_{11}^2 c_{21}^2 \nonumber \\ b_2 &=& c_{12}^4 + c_{22}^4 + gc_{12}^2 c_{22}^2 \\ b_{12} &=& 4(c_{11}^2 c_{12}^2 + c_{21}^2 c_{22}^2 ) + g(c_{11}^2 c_{22}^2 + c_{12}^2 c_{21}^2 + 2 c_{11} c_{12} c_{21} c_{22} ) \; .\nonumber \end{eqnarray} Here the reflection symmetry $\langle B_j \rangle =0$ has been assumed to carry over from $\langle M_i \rangle =0$. Choosing furthermore the reference action \begin{equation} V_{ref} = \frac{1}{N} \mbox{Tr} \, (B_1^4 + B_2^4 ) \end{equation} one has (cf. Eq. (\ref{dpridef})) \begin{equation} \tilde{S}^{\prime}_{1f} = (\alpha b_1 +1-\alpha ) \frac{1}{N} \mbox{Tr} \, B_1^4 + (\alpha b_2 +1-\alpha ) \frac{1}{N} \mbox{Tr} \, B_2^4 + \alpha b_{12} \frac{1}{N^2 } \mbox{Tr} \, B_1^2 \, \mbox{Tr} \, B_2^2 \end{equation} Treating the last term in analogy to (\ref{freex}) in conjunction with (\ref{freeea}) and (\ref{freeeb}), one has to solve one-matrix problems for $B_1 $ and $B_2 $ with the moments $\langle B_i^2 \rangle $ to be determined self-consistently. The explicit solution of the one-matrix model with arbitrary symmetric quartic potential is listed in Appendix \ref{appa}; the expressions for the second moments yield equations for the $\langle B_i^2 \rangle $ which can be solved numerically for given $\alpha $ and $c_{ij} $. Then $\langle \tilde{S}_{1f} -V_{ref}\rangle $ can be evaluated, again using the expressions for the moments given in Appendix \ref{appa}. Finally, evaluating the integral over $\alpha $ leads to the free energy, cf. Eq. (\ref{modprin}), with the additional Jacobian piece $(\ln J)/N^2 = \ln (c_{11} c_{22} - c_{12} c_{21} )$. Minimizing the free energy numerically as a function of the $c_{ij} $ yields the following result, cf. also Fig. \ref{fig0}: For sufficiently small $|g|$, the original variables $M_1 $ and $M_2 $ are the optimal ones for a free approximation, whereas for large $g$, it becomes favorable to switch variables by substituting $M_1 = B_1 + B_2 $ and $M_2 = B_1 - B_2 $. This change in behavior is of course simply induced by the competition between the different terms in the action. For sufficiently low $g$, the quartic part, which is already independent in the original variables $M_1 $, $M_2 $, dominates. For larger $g$, it becomes favorable to accomodate the interaction term as much as possible. The competition is essentially linear in the sense that one optimizes the treatment of either one or the other part of the action and the switch between the optimal choices of variables $M_1 $, $M_2 $ or $B_1 = (M_1 + M_2 )/2$, $B_2 = (M_1 - M_2 )/2$ happens suddenly at $g_{\rm cr} = 2$; the optimal choice does not change continuously with $g$. The authors have considered the possibility that this transition in the variational approximation may signal a large-$N$ phase transition also in the exact solution of the model, visible e.g. as a discontinuity of the derivative of the exact free energy as a function of the coupling constant $g$. However, the Monte Carlo results do not show such a transition -- the expectation of $\mbox{Tr} \, M_1^2 M_2^2$ displays a smooth behaviour near $g_{\rm cr} $. Thus, the transition in the variational calculation seems to merely reflect a crossover between two physically different regimes rather than a genuine large-$N$ phase transition. A higher order phase transition can however not be ruled out on the basis of the data taken by the authors. Having ascertained the optimal choice of variables, one can proceed to give the corresponding approximations to the eigenvalue distributions. For $g=\pm 1$, the original variables $M_1 $, $M_2 $ are best; the free partner of $S_1 $ is \begin{equation} S_{1f} = \frac{1}{N} \mbox{Tr} \, (M_1^4 + M_2^4 ) \pm \frac{1}{N^2 } \mbox{Tr} \, M_1^2 \mbox{Tr} \, M_2^2 \end{equation} Using the formulae of Appendix \ref{appa} to solve these models, one obtains (assuming the eigenvalue distributions of $M_1 $ and $M_2 $ to be identical) the consistency conditions \begin{equation} \langle M_1^2 \rangle = \langle M_2^2 \rangle = x_M = \frac{1}{108} \left[ \mp x_M (x_M^2 +18) + (x_M^2 +12)^{3/2} \right] \end{equation} solved by $x_M =0.334$ for $g=1$ and $x_M =0.476$ for $g=-1$ (the relevant solution can be picked out by using positivity of $x_M $, etc.). The corresponding eigenvalue distributions are given by (cf. Appendix \ref{appa}) \begin{equation} \rho_{M} (\lambda ) = \frac{1}{\pi } (2\lambda^{2} +x_M +m^2 ) \sqrt{m^2 -\lambda^{2} } \ \ \ \ \mbox{with} \ \ \ \ m^2 = \frac{1}{3} (\sqrt{x_M^2 +12} -x_M ) \end{equation} These distributions are plotted in Figs.~\ref{fig1} and \ref{fig2}, and compared with Monte Carlo results for $10\times 10 $ matrices. In the case of $g=1$, the dependence of the Monte Carlo results on the size of the matrices is exhibited in Fig. \ref{fig3}. In this, as in all other cases tested, $N=10$ already seems to embody the large-$N$ asymptotic bulk behavior rather well, up to the characteristic oscillations in the eigenvalue density induced by the eigenvalue repulsion. In this respect, no deviation was observed from the well-known finite-$N$ phenomena observed in one-matrix models. For $g=4$, on the other hand, one considers the best free approximation after the variable substitution $M_1 = B_1 + B_2 $ and $M_2 = B_1 - B_2 $, i.e. \begin{equation} \tilde{S}_{1f} = \frac{6}{N} \mbox{Tr} \, (B_1^4 + B_2^4 ) + \frac{8}{N^2 } \mbox{Tr} \, B_1^2 \mbox{Tr} \, B_2^2 \label{s1fjk} \end{equation} Here, the consistency condition becomes \begin{equation} \langle B_1^2 \rangle = \langle B_2^2 \rangle = x_B = \frac{1}{486} \left[ -4x_B (16 x_B^2 +27) + (16 x_B^2 +18)^{3/2} \right] \label{s1fly} \end{equation} solved by $x_B =0.131$. Eqs. (\ref{s1fjk}) and (\ref{s1fly}) imply that the $B_i $ are both governed by the quartic potential $(1/N) \mbox{Tr} \, (6B_i^4 + 8x_B B_i^2 )$, and the corresponding eigenvalue distribution is (cf. Appendix \ref{appa}) \begin{equation} \rho_{B} (\lambda ) = \frac{1}{\pi } (12\lambda^{2} +8x_B +6m^2 ) \sqrt{m^2 - \lambda^{2} } \ \ \ \ \mbox{with} \ \ \ \ m^2 = \frac{1}{18} (\sqrt{64x_B^2 + 72} - 8x_B ) \end{equation} In order to obtain the eigenvalue distributions of the original variables $M_1 $ and $M_2 $ from this, one must use the free additive convolution techniques derived in \cite{Voi}. In Appendix \ref{appb}, the convolution of two identical distributions governed by an arbitrary symmetric quartic potential is discussed. The resulting eigenvalue distribution $\rho_{M_1 } = \rho_{M_2 } $ is plotted in Fig.~\ref{fig4} along with the corresponding Monte Carlo results for $N=10$. Also, the comparison with the free approximation in the original variables $M_1 $, $M_2 $ is plotted. Evidently, allowing the choice of free variables to vary leads to a vastly improved approximation in this case. In exactly the same vein as the case $g=4$ one can treat the extreme case (corresponding, up to a rescaling, to $g\rightarrow \infty $) \begin{equation} S_1^{red} = \frac{1}{N} \mbox{Tr} \, M_1^2 M_2^2 \; . \label{s1redef} \end{equation} Here, in the absence of any piece in the action which is free in the original variables and which even in the case of quite strong coupling $g=4$ (see above) kept the behavior of the model reasonably regular, the disagreement between the variational calculation and the exact result is rather catastrophic, cf. Fig.~\ref{fig5}. This is not hard to understand; in the case of the action $S_1^{red} $, the variables $M_1 $, $M_2 $ can use configurations such as \begin{equation} M_1 = \left( \begin{array}{cc} m_1 & 0 \\ 0 & 0 \end{array} \right) \ \ \ \ \ \ \ \ \ \ \ \ \ M_2 = \left( \begin{array}{cc} 0 & 0 \\ 0 & m_2 \end{array} \right) \label{redco} \end{equation} where $m_1 $, $m_2 $ are roughly $N/2 \times N/2 $ matrices, to preserve a low value of $S_1^{red} $ while realizing very large eigenvalues (together with a concentration of very small ones). This is indeed what is seen in Fig.~\ref{fig5}. Dominance of configurations such as (\ref{redco}) implies a strong angular correlation between the matrices; if one rotates the above matrices with respect to one another, e.g. such that the eigenvalues are arbitrarily reordered, $S_1^{red} $ will in general take a very large value. Such angular correlations are completely lost in the free partner $S_{1f}^{red} = (1/N^2 ) \mbox{Tr} \, M_1^2 \mbox{Tr} \, M_2^2 $ and evidently even the best choice of linearly transformed variables $M_1 = B_1 + B_2 $ and $M_2 = B_1 -B_2 $, though incorporating some angular correlations between $M_1 $ and $M_2 $, cannot do much to alleviate this problem. It is however possible that a nonlinear change of variables exists for which the difficulty will be satisfactorily removed. The authors did not explore this direction. The dismal failure of the variational approximation in this case thus does not come as a complete surprise. Rather, it is gratifying to see how well the variational approximation still works in the previous, not quite as pathological, example described by $S_1 $ with the strong coupling of $g=4 $. One striking feature of the Monte Carlo eigenvalue distribution in the case of $S_{1}^{red} $ is the absence of the characteristic finite-$N$ oscillations observed in all other cases. This can be understood as follows: Usually, the potential confines the eigenvalues to a compact domain and, due to the eigenvalue repulsion, they tend to be equidistant. This (fluctuating) lattice has a favored equilibrium position, leading to the oscillations in the eigenvalue density. However, when the action allows the eigenvalues to spread over the entire real axis, the correlations induced by the eigenvalue repulsion become insignificant and the oscillations in the eigenvalue density disappear\footnote{The authors are indebted to U.-J. Wiese for this remark.}. Apart from plotting the eigenvalue distributions, which essentially contain the information about all the moments of the individual matrices, one can test mixed correlators for freeness properties. In particular, if the matrices $M_1 $ and $M_2 $ are free with respect to each other, then there are e.g. the following relations between correlators, following from the axioms of freeness: \begin{eqnarray} C_1 \equiv \langle M_1^2 M_2^2 \rangle &=& \langle M_1^2 \rangle \langle M_2^2 \rangle \equiv C_{1f} \nonumber \\ C_2 \equiv \langle M_1^4 M_2^4 \rangle &=& \langle M_1^4 \rangle \langle M_2^4 \rangle \equiv C_{2f} \label{mixmf} \\ C_3 \equiv \langle M_1^2 M_2^2 M_1^2 M_2^2 \rangle &=& \langle M_1^4 \rangle \langle M_2^2 \rangle^{2} + \langle M_1^2 \rangle^{2} \langle M_2^4 \rangle - \langle M_1^2 \rangle^{2} \langle M_2^2 \rangle^{2} \equiv C_{3f} \nonumber \end{eqnarray} In the Monte Carlo calculations, both sides of these equations were sampled in order to give a measure for the deviation from freeness (i.e. strength of angular correlations) contained in the exact models. The results are tabulated in Table~1 for different values of $g$ in $S_1 $. Note that for $S_1^{red} $, the moments of the exact distribution diverge due to the evidently too slow fall-off of the eigenvalue distribution for large eigenvalues, making such a comparison impossible. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_i^2 \rangle $ & $\langle M_i^4 \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline -1 & .483 & .377 & .254 & .233 & .171 & .142 & .150 & .121 \\ \hline 1 & .336 & .197 & .106 & .113 & .0339 & .0387 & .0272 & .0317 \\ \hline 4 & .267 & .133 & .0585 & .0713 & .0116 & .0177 & .0085 & .0139 \\ \hline \end{tabular} \vspace{0.3cm} Table 1 : Mixed correlators, Eqs. (\ref{mixmf}), for the model $S_1$ compared to the predictions for free $M_i$ \end{center} One observes how, for growing $g$, the interaction term introduces stronger angular correlations, evident in the stronger deviations from free predictions. Now, for sufficiently large $g$, e.g. $g=4$, the variational principle asserts that a better approximation is obtained with a new set of mutually free variables $B_i $. Using this new set of free variables, of course also the predictions for mixed correlators, derived in (\ref{mixmf}) for free $M_i $, change. This shall be illustrated here for the correlator $\langle M_1^2 M_2^2 \rangle $. Inserting $M_1 = B_1 + B_2 $ and $M_2 = B_1 - B_2 $, using the axioms of freeness on $B_1 $ and $B_2 $ together with $\langle B_i \rangle =0$, and then substituting back $B_1 = (M_1 + M_2 )/2$ and $B_2 = (M_1 - M_2 )/2$, one obtains \begin{eqnarray} \langle M_1^2 M_2^2 \rangle &=& \langle B_1^4 + B_2^4 \rangle = \frac{1}{8} \langle M_1^4 + M_2^4 +4 M_1^2 M_2^2 + 2 M_1 M_2 M_1 M_2 \rangle \\ \langle M_1 M_2 M_1 M_2 \rangle &=& \langle B_1^4 + B_2^4 \rangle -4 \langle B_1^2 \rangle \langle B_2^2 \rangle = \langle M_1^2 M_2^2 \rangle - \frac{1}{4} (\langle M_1^2 + M_2^2 \rangle^{2} -4 \langle M_1 M_2 \rangle^{2} ) \\ \langle M_1 M_2 \rangle &=& \langle B_1^2 \rangle - \langle B_2^2 \rangle =0 \end{eqnarray} where in the last line, identical distributions for the $B_i $ were assumed. Putting these relations together yields $\langle M_1^2 M_2^2 \rangle $ in terms of the individual moments, \begin{equation} \langle M_1^2 M_2^2 \rangle = \frac{1}{2} \langle M_1^4 + M_2^4 \rangle - \frac{1}{4} \langle M_1^2 + M_2^2 \rangle^{2} \label{mixbf} \end{equation} In the case $g=4$, the right hand side, using the individual moments from the Monte Carlo experiment, takes the value .0617, which indeed is closer to the exact value for $\langle M_1^2 M_2^2 \rangle \equiv C_1 $ than the prediction using free variables $M_i $ quoted in Table 1. Consider now the model described by $$ S_2 = \frac{1}{N} \mbox{Tr} \, (M_1^2 + M_2^2 + gM_1 M_2 M_1 M_2 ) \; . $$ After performing a general linear transformation as in (\ref{glt}), one arrives at the free partner \begin{eqnarray} \tilde{S}_{2f} = (c_{11}^{2} + c_{21}^{2} ) \frac{1}{N} \mbox{Tr} \, B_1^2 + (c_{12}^{2} + c_{22}^{2} ) \frac{1}{N} \mbox{Tr} \, B_2^2 + gc_{11}^{2} c_{21}^{2} \frac{1}{N} \mbox{Tr} \, B_1^4 + \nonumber \\ + gc_{12}^{2} c_{22}^{2} \frac{1}{N} \mbox{Tr} \, B_2^4 + 4gc_{11} c_{12} c_{21} c_{22} \frac{1}{N^2 } \mbox{Tr} \, B_1^2 \mbox{Tr} \, B_2^2 \; . \label{ts2f} \end{eqnarray} Evidently, from the point of view of the variational approximation, this model is slightly simpler than the one considered above. Since the noninteracting part of $S_2 $ is invariant under linear transformations of the variables and subsequent construction of the free partner, up to trivial variable rescalings (note also that, as before, $\langle B_i \rangle =0 $ was used in deriving (\ref{ts2f})), the choice of variables will always be such as to best accomodate the interacting term, for any coupling $g$. Carrying out the calculation of the free energy in complete analogy to the calculation for $S_1 $, this turns out to be the choice $M_1 = B_1 + B_2 $, $M_2 = B_1 - B_2 $, corresponding to the free partner \begin{equation} \tilde{S}_{2f} = \frac{1}{N} \mbox{Tr} \, ( gB_1^4 + gB_2^4 + 2B_1^2 + 2B_2^2 ) - \frac{4g}{N^2 } \mbox{Tr} \, B_1^2 \mbox{Tr} \, B_2^2 \label{ts2fopt} \end{equation} The corresponding consistency condition is \begin{equation} \langle B_1^2 \rangle = \langle B_2^2 \rangle = x_B = \frac{1}{108g^2 } \left[ -(2-4gx_B )^3 -18g(2-4gx_B )+((2-4gx_B )^2 +12g)^{3/2} \right] \label{ts2foptc} \end{equation} From the resulting eigenvalue distributions of the matrices $B_i $ one again obtains the eigenvalue distributions of the original variables $M_i $ by additive convolution. The result for $g=2/5$, leading to $x_B = .2522 $ in (\ref{ts2foptc}), is displayed in Fig.~\ref{fig6}, compared with the Monte Carlo result for $N=40$. Also, various correlators are tabulated in Table 2. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_i^2 \rangle $ & $\langle M_i^4 \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline 2/5 & .53 & .56 & .30 & .28 & .37 & .32 & .28 & .24 \\ \hline \end{tabular} \vspace{0.3cm} Table 2 : Mixed correlators, Eqs. (\ref{mixmf}), for the model $S_2$, compared to the predictions for free $M_i$ \end{center} The exact eigenvalue distribution turns out to be surprisingly similar to the semicircle of radius $\sqrt{2} $ obtained at $g=0$. This is well reproduced by the variational calculation despite a quite strong dependence on $g$ of the individual coefficients in $\tilde{S}_{2f} $. The effects of the $g$-dependence of the quadratic and quartic coefficients nearly cancel in the complete potential, leading to a quite stable eigenvalue distribution. Turning to the mixed correlators in Table 2, one does not observe a strong deviation from free behavior of the $M_i $. In this case, there is almost no room for improvement by assuming free $B_i $ instead of $M_i $, leading to equation (\ref{mixbf}) for $\langle M_1^2 M_2^2 \rangle $. Indeed, one here obtains for the right hand side of (\ref{mixbf}) the value $.28$, the same as $C_{1f} $ in Table 2. The reader may wonder why only the relatively small value $g=2/5$ was displayed in the comparison above. In fact, the model described by $S_2 $ becomes unstable for too large values of $|g|$. Note that the product of two hermitian matrices $M_1 M_2 $ is in general not hermitian; therefore, the interaction term, proportional to $(M_1 M_2 )^2 $, is not bounded from below even for positive $g$. Strictly speaking, the model is unstable for all $g$, but for sufficiently small $g$, there is effectively a barrier posed by the independent part of the action preventing the system from spilling over to the region of large eigenvalues where the interaction will allow eigenvalues to grow without bound. A glimpse of this instability (for negative $g$, namely setting in at $g=-4/9$) was already given in \cite{Che}\footnote{Presumably, the model described by $S_1 $ will display a similar instability when the coupling $g$ becomes too negative. This was not investigated further by the authors.}. Here, instead the instability at positive $g$ was investigated in slightly more detail. According to Monte Carlo simulations at $N=10$, $20$, and $40$, the critical coupling lies in the interval $g \in [0.4,0.45]$. On the other hand, the variational approximation in the best basis, described by the action (\ref{ts2fopt}), is stable up to $g=2$. Beyond this point, there is no solution to the consistency condition (\ref{ts2foptc}). The authors also checked that there is no two-cut solution to (\ref{ts2fopt}) above $g=2$ (note that (\ref{ts2foptc}) is valid only for one-cut solutions). Thus, the variational approximation does qualitatively capture the phase structure of the model described by $S_2 $, albeit with a badly overestimated critical coupling. In this respect, the choice of variables $B_1 = (M_1 + M_2 )/2$ and $B_2 = (M_1 - M_2 )/2$ represents a drastic improvement over the original variables $M_i $. In terms of the latter, the free counterpart of $S_2 $ is simply $S_{2f} = (1/N) \mbox{Tr} \, (M_1^2 + M_2^2 )$, which entirely misses the unstable phase of the model (in fact, all the dependence on the coupling $g$). \subsection{Classical Constant Matrices in Many Dimensions} Consider a model of $D$ classical matrices with an action of the type \begin{equation} S_D = \frac{1}{D-1} \sum_{i\neq j}^{D} S (M_i , M_j ) \label{ldtyp} \end{equation} Note the scaling of this action with $D$, which is necessary to retain a meaningful balance between it and the eigenvalue repulsion originating in the Haar measure of the $M_i $. Before proceeding, a comment is in order regarding the special form of (\ref{ldtyp}) and the interpretation of $D$ as the number of space-time dimensions. In (\ref{ldtyp}), each matrix degree of freedom interacts with all others in the same way. If one wishes to attach physical meaning to the matrices as degrees of freedom living in space-time, this occurs most naturally when all the degrees of freedom are attached to the same space-time point. Such a model may become relevant if one has managed to decouple the different space-time points of a $D$-dimensional large-$N$ field theory, e.g. as the result of an Eguchi-Kawai reduction (for a review, see Ref. \cite{Egu}). Now, in view of the scaling of the individual terms in (\ref{ldtyp}), one might hope that the detailed angular correlations between pairs of matrices become unimportant at large $D$ and a free approximation becomes exact. This argument can be made rigorous e.g. in the Kazakov-Migdal model, where the leading large-$D$ behavior is described by a free matrix model \cite{Mak1}. The proof however depends on the specific link variable structure of the Kazakov-Migdal model. In order to test this idea more generally, the models described by \begin{equation} S_{D2} = \frac{1}{D-1} \sum_{i\neq j}^{D} \frac{1}{N} \mbox{Tr} \, M_i^2 M_j^2 \ \ \ \ \ \ \ \mbox{and} \ \ \ \ \ \ \ S_{D4} = \frac{1}{D-1} \sum_{i\neq j}^{D} \frac{1}{N} \mbox{Tr} \, M_i^4 M_j^4 \label{ladmo} \end{equation} were investigated in the present framework for different $D$. The reader is reminded that in the case of $S_{D2} $ for $D=2$, the free variational approximation failed miserably (cf. previous section). In analyzing the models (\ref{ladmo}) in the free variational approximation, the authors did not treat different choices of free variables; for simplicity, only the best free approximation in the original variables $M_i $ was considered. Then the best free approximation is easily derived: The free counterparts of (\ref{ladmo}) are \begin{equation} S_{D2f} = \frac{1}{D-1} \sum_{i\neq j}^{D} \frac{1}{N^2 } \mbox{Tr} \, M_i^2 \mbox{Tr} \, M_j^2 \ \ \ \ \ \ \ \mbox{and} \ \ \ \ \ \ \ S_{D4f} = \frac{1}{D-1} \sum_{i\neq j}^{D} \frac{1}{N^2 } \mbox{Tr} \, M_i^4 \mbox{Tr} \, M_j^4 \end{equation} and thus each variable $M_i $ is described by the action \begin{equation} \bar{S}_{D2} (M_i ) = 2 x_{D2} \frac{1}{N} \mbox{Tr} \, M_i^2 \ \ \ \ \ \ \ \mbox{or} \ \ \ \ \ \ \ \bar{S}_{D4} (M_i ) = 2 x_{D4} \frac{1}{N} \mbox{Tr} \, M_i^4 \end{equation} independent of $D$, with $x_{D2} = \langle M_i^2 \rangle $ and $x_{D4} = \langle M_i^4 \rangle $ to be determined self-consistently. One obtains $x_{D2} = 1/2 $ and $x_{D4} = 1/\sqrt{8} $ and therefore the eigenvalue distributions \begin{equation} \rho_{2} (\lambda ) = \frac{1}{\pi } \sqrt{2-\lambda^{2} } \ \ \ \ \ \ \ \mbox{and} \ \ \ \ \ \ \ \rho_{4} (\lambda ) = \frac{1}{\pi } \left( \sqrt{2} \lambda^{2} + \sqrt{\frac{\sqrt{8} }{3} } \right) \sqrt{\sqrt{\frac{2\sqrt{8} }{3} } -\lambda^{2} } \end{equation} Comparing this with the results of Monte Carlo experiments, cf. Figs. \ref{fig7} and \ref{fig8}, the correspondence is quite satisfying already at $D=10$ (the simulations were carried out for $N=10$, some checks for $N=20$ did not reveal any significant differences). Also, comparing mixed correlators of two matrices (without loss of generality, $M_1 $ and $M_2 $) with free predictions (cf. equations (\ref{mixmf})) in the Monte Carlo experiments, one obtains good agreement for $D=10$, cf. Table 3. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} & $\langle M_i^2 \rangle $ & $\langle M_i^4 \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline $ S_{D2}, D=3 $ & .62 & 1.00 & .25 & .38 & .33 & 1.00 & .18 & .62 \\ \hline $ S_{D2}, D=10 $ & .51 & .53 & .25 & .26 & .25 & .28 & .18 & .21 \\ \hline \hline $ S_{D4}, D=3 $ & .49 & .46 & .20 & .25 & .12 & .21 & .09 & .16 \\ \hline $ S_{D4}, D=10 $ & .46 & .37 & .21 & .22 & .13 & .13 & .11 & .11 \\ \hline \end{tabular} \vspace{0.3cm} Table 3 : Mixed correlators, Eq. (\ref{mixmf}), for the models (\ref{ladmo}), and predictions for free $M_i $ \end{center} As mentioned above, this agreement does not come unexpected. However, while in the cases considered here the Monte Carlo solution seemed to converge towards the free approximation at large $D$, it seems doubtful that this should be true in general. On the one hand, a solution with angular correlations, say with angles restricted to a certain sector as compared with some fixed reference matrix, costs a weight $\exp (-\alpha N^2 D )$ in the partition sum (there are $O(N^2 D )$ angles, and the coefficient $\alpha $ depends on the details of the angular distributions). On the other hand, the interaction itself is also scaled to be of order $O(N^2 D )$. Therefore, there should be a genuine competition between the two terms, the former favoring freeness and the latter preventing it. \subsection{Quantum Mechanics of One Matrix} For a wide class of potentials, the quantum mechanical ground state of a single large-$N$ matrix can be solved for exactly \cite{Bre1}. This is achieved by working directly with continuous (Euclidean) time and mapping the problem to a noninteracting fermion problem. If one discretizes time as in the original definition of the path integral, the problem can be interpreted in terms of many interacting classical matrices. In this form, the quantum mechanics of one matrix is hard to solve exactly; on the other hand, it becomes amenable to the variational method dealt with here. Interest in such a discretized form has arisen in recent applications to quantum gravity, \cite{GroK}. For definiteness, the model described by \begin{equation} S_{Q1} = \frac{1}{N} \mbox{Tr} \, \left( \sum_{i=0}^{L-1} (M_{i+1} -M_i )^2 + gM_i^4 \right) \label{sq1def} \end{equation} will be considered here, where $L$ is the number of time slices, and $M_0 $ is identified with $M_L $, i.e. periodic boundary conditions are posited. Generically, there is a competition between two different interaction terms: The nearest-neighbor coupling and the potential term, which acts instantaneously. In order to best accomodate the former in a free variational approximation, one will want to change matrix variables to a Fourier basis, which decouples the kinetic part of the action. On the other hand, the potential term will prefer the original variables. Thus, in the present treatment, not a general linear change of variables will be considered, but only the aforementioned two discrete choices; the variational minimization principle will decide which choice is preferred. The Fourier basis is defined as \begin{equation} M_i = \sum_{j} U_{ij} B_j \label{foutra} \end{equation} with ($L$ is for simplicity taken to be even) \begin{equation} U = \sqrt{\frac{2}{L} } \left( \begin{array}{cccccccc} 0 & \cdots & 0 & 1/\sqrt{2} & 1 & \cdots & 1 & 1/\sqrt{2} \\ \sin k_{L/2 -1} & \cdots & \sin k_{1} & 1/\sqrt{2} & \cos k_{1} & \cdots & \cos k_{L/2 -1} & -1/\sqrt{2} \\ \sin 2k_{L/2 -1} & \cdots & \sin 2k_{1} & 1/\sqrt{2} & \cos 2k_{1} & \cdots & \cos 2k_{L/2 -1} & 1/\sqrt{2} \\ \sin 3k_{L/2 -1} & \cdots & \sin 3k_{1} & 1/\sqrt{2} & \cos 3k_{1} & \cdots & \cos 3k_{L/2 -1} & -1/\sqrt{2} \\ \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ \sin (L-1)k_{L/2 -1}& \cdots & \sin (L-1)k_{1} & 1/\sqrt{2} & \cos (L-1)k_{1} & \cdots & \cos (L-1)k_{L/2 -1} & -1/\sqrt{2} \end{array} \right) \label{umat} \end{equation} Here, the wavevectors are $k_j = 2\pi j/L$; note that $U$ is orthogonal and therefore the Jacobian term in the free energy (\ref{modprin}) gives no contribution for the Fourier choice of variables. For convenience in notation, the index $j$ labeling the variables $B_j $ in (\ref{foutra}), i.e. the columns in (\ref{umat}), will be taken to run from $-L/2 +1 $ to $L/2$. In the Fourier basis, the kinetic part of the action (\ref{sq1def}) is exactly diagonalized and coincides with its free partner \begin{equation} \tilde{S}_{Q1}^{kin} (B_i ) = \frac{2}{N} \mbox{Tr} \, \sum_{j=-L/2 +1}^{L/2} (1-\cos k_j ) B_j^2 =\tilde{S}_{Q1f}^{kin} (B_i ) \end{equation} On the other hand, the potential part in the new variables is \begin{equation} \tilde{S}_{Q1}^{pot} (B_i ) = \frac{g}{N} \mbox{Tr} \, \sum_{i} \sum_{jklm} U_{ij} B_j U_{ik} B_k U_{il} B_l U_{im} B_m \end{equation} Assuming as before $\langle B_i \rangle =0$, the only terms which will give a contribution to the free partner are ones in which the four indices $j,k,l,m$ all take the same value or pairwise two different values. Thus, one has the free partner \begin{equation} \tilde{S}_{Q1f}^{pot} (B_i ) = g\left[ 4\sum_{j<k} \frac{1}{N^2 } \mbox{Tr} \, B_j^2 \mbox{Tr} \, B_k^2 \left( \sum_{i} U_{ij}^2 U_{ik}^2 \right) +\sum_{j} \frac{1}{N} \mbox{Tr} \, B_j^4 \left( \sum_{i} U_{ij}^4 \right) \right] \end{equation} By explicit calculation, $\sum_{i} U_{ij}^4 $ is of order $O(1/L)$ and therefore the quartic term can be dropped for a large number of time slices $L$. On the other hand, again by explicit calculation, \begin{equation} \sum_{i=0}^{L-1} U_{ij}^2 U_{ik}^2 = \frac{1}{L} \ \ \ \ \ \ \ \ \mbox{for all} \ j,k \end{equation} and therefore one finally has for the free partner of the action (\ref{sq1def}) in the Fourier variables $B_i $, \begin{equation} \tilde{S}_{Q1f} (B_i ) = 2 \sum_{i=-L/2 +1}^{L/2} (1-\cos k_i ) \frac{1}{N} \mbox{Tr} \, B_i^2 +\frac{4g}{L} \sum_{i<j=-L/2 +1}^{L/2} \frac{1}{N^2 } \mbox{Tr} \, B_i^2 \mbox{Tr} \, B_j^2 \end{equation} On the other hand, the free partner of (\ref{sq1def}) in the original variables is \begin{equation} S_{Q1f} (M_i ) = \frac{1}{N} \mbox{Tr} \, \sum_{i=0}^{L-1} (2M_i^2 + gM_i^4 ) \label{sforig} \end{equation} again assuming $\langle M_i \rangle =0$. Here, the potential term is treated exactly, whereas the nearest-neighbor coupling has been completely truncated. The free models are now easily solved. For the purpose of calculating the free energies, the reference action $V_{ref} (C_i ) = (1/N) \mbox{Tr} \, \sum_{i} C_i^2 $ will be used; then one considers (cf. Eq. (\ref{dpridef})) \begin{equation} S^{\prime }_{Q1f} (M_i ) = \frac{1}{N} \mbox{Tr} \, \sum_{i} \left[ (1+\alpha ) M_i^2 + \alpha g M_i^4 \right] \label{sfcorig} \end{equation} for the original variables and \begin{equation} \tilde{S}^{\prime }_{Q1f} (B_i ) = \sum_{i} (1+\alpha -2\alpha \cos k_i ) \frac{1}{N} \mbox{Tr} \, B_i^2 + \frac{4\alpha g}{L} \sum_{i<j} \frac{1}{N^2 } \mbox{Tr} \, B_i^2 \mbox{Tr} \, B_j^2 \end{equation} for the Fourier variables. In the latter, introducing the abbreviation \begin{equation} x_B =\frac{2}{L} \left\langle \sum_{i} B_i^2 \right\rangle \end{equation} one has semicircular distributions for the matrix variables $B_i $ of radius squared \begin{equation} r_i^2 = \frac{2}{1+\alpha -2\alpha \cos k_i + \alpha gx_B } \label{rsbi} \end{equation} implying second moments $\langle B_i^2 \rangle =r_i^2 /4 $, which self-consistently determines $x_B $: \begin{eqnarray} x_B = \frac{2}{L} \left\langle \sum_{i} B_i^2 \right\rangle &=& \frac{1}{L} \sum_{i} \frac{1}{1+\alpha -2\alpha \cos (2\pi i/L) +\alpha gx_B } \nonumber \\ & \stackrel{L\rightarrow \infty }{\longrightarrow } & \int_{-1/2}^{1/2} dk \, \frac{1}{1+\alpha +\alpha gx_B -2\alpha \cos 2\pi k} \\ &=& \frac{1}{\sqrt{(\alpha +\alpha gx_B +1)^2 - 4\alpha^{2} } } \nonumber \end{eqnarray} or, \begin{equation} x_B^2 (\alpha +\alpha gx_B +1)^2 -4\alpha^{2} x_B^2 -1 =0 \label{xq1eq} \end{equation} The free energy is then \begin{eqnarray} F_B &=& \int_{0}^{1} d\alpha \, \sum_{i} \frac{1}{2} \frac{2-2\cos (2\pi i/L) + gx_B -1}{1+\alpha -2\alpha \cos (2\pi i/L) +\alpha gx_B } \nonumber \\ & \stackrel{L\rightarrow \infty }{\longrightarrow } & L \int_{0}^{1} d\alpha \, \int_{-1/2}^{1/2} dk \, \frac{1}{2} \frac{2-2\cos 2\pi k + gx_B -1}{1+\alpha -2\alpha \cos 2\pi k +\alpha gx_B } \\ &=& L \int_{0}^{1} d\alpha \, \frac{1}{2\alpha } \left[ 1-\frac{1}{\sqrt{(\alpha +\alpha gx_B +1)^2 -4\alpha^{2} } } \right] \nonumber \end{eqnarray} which must be calculated numerically (remember that at every $\alpha $, $x_B $ is determined by the equation (\ref{xq1eq})). On the other hand, in the decoupled basis, one has the quartic action (\ref{sfcorig}), identical and independent for all the $M_i $. Using Appendix \ref{appa}, the resulting moments are \begin{eqnarray} \langle M^2 \rangle &=& \frac{1}{108\alpha^{2} g^2 } \left[ -(1+\alpha )^3 - 18\alpha g (1+\alpha ) +((1+\alpha )^2 + 12\alpha g)^{3/2} \right] \\ \langle M^4 \rangle &=& \frac{1}{216\alpha^{3} g^3 } \left[ (1+\alpha )^4 + 18\alpha g (1+\alpha )^2 + 54\alpha^{2} g^2 -(1+\alpha ) ((1+\alpha )^2 + 12\alpha g)^{3/2} \right] \end{eqnarray} and, inserting in the free energy, \begin{eqnarray} F_M &=& \int_{0}^{1} d\alpha \, 2L \langle M^2 \rangle + Lg \langle M^4 \rangle - L \langle M^2 \rangle \nonumber \\ &=& \frac{L}{108 g^2 } \left[ -4-36g-\frac{81g^2 }{2} + (4+30g)\sqrt{1+3g} + 54g^2 \ln (1+\sqrt{1+3g} ) \right] \end{eqnarray} Numerically, $F_M = F_B $ at $g=1.2306$. For smaller $g$, the Fourier basis provides the better approximation, for larger $g$ the original choice of variables does. It has been argued in \cite{GroK} that matrix models such as the one considered here exhibit a Kosterlitz-Thouless transition at some value of the lattice spacing separating the two regimes where the matrix chain behaves more like a collection of independent sites and the one where it supports spin wave type of modes. The variational approach presented here, while giving no details of the transition region, is capable of capturing these two different possible regimes. It remains to give the eigenvalue distributions in the two regimes. In the original variables, described by the action $S_{Q1f} (M_i )$, cf. equation (\ref{sforig}), one immediately has, using Appendix \ref{appa}, \begin{equation} \rho_{M} (\lambda ) = \frac{1}{\pi } (2g\lambda^{2} + 2 + m^2 g) \sqrt{m^2 -\lambda^{2} } \ \ \ \ \ \ \mbox{with} \ \ m^2 = \frac{2}{3g} (\sqrt{1+3g} -1) \end{equation} for all the variables $M_i $. In the Fourier case, one must convolute the semicircular distributions of the $B_i $ to obtain the distributions of the $M_i $. It is well known \cite{Voi} that the semicircular distributions are closed under additive convolution and that the square radii add up to the square radius of the resulting semicircle. I.e., \begin{equation} r_{M_i }^{2} = \sum_{j} U_{ij}^{2} r_{B_j }^{2} \end{equation} Inserting (\ref{rsbi}) for $\alpha =1$ and (\ref{umat}), one obtains identical radii for all the $M_i $, \begin{eqnarray} r_{M_i }^{2} &=& \frac{1}{L} \sum_{j=-L/2 +1}^{L/2} \frac{2}{2-2\cos (2\pi j/L) + gx_B } \nonumber \\ & \stackrel{L\rightarrow \infty }{\longrightarrow } & \int_{-1/2}^{1/2} dk \, \frac{2}{2-2\cos 2\pi k +gx_B } \\ &=& \frac{2}{\sqrt{4gx_B + g^2 x_B^2 } } \nonumber \\ &=& 2x_B \nonumber \end{eqnarray} where in the last line it has been used that $x_B =(2/L) \langle \sum_{i} B_i^2 \rangle $ solves (cf. (\ref{xq1eq}) for $\alpha =1$) \begin{equation} 4gx_B^3 + g^2 x_B^4 =1 \label{q1bx} \end{equation} The corresponding semicircular distribution for the $M_i $ is then \begin{equation} \rho_{M} (\lambda ) = \frac{1}{x_B \pi } \sqrt{2x_B -\lambda^{2} } \label{q1brho} \end{equation} The variational approximation derived above should now be compared to Monte Carlo experiments and the exact results known about this model. In Figs. \ref{fig9}-\ref{fig11} Monte Carlo results for $L=10$ and $N=10 $ are exhibited together with variational results at different couplings $g$. Checks with $N=20$ and $L=20$ revealed no significant differences. One indeed observes, as already borne out by the calculation of the free energies, that the Fourier basis is preferable in a weak potential and the original basis is favored by a strong potential. This is corroborated by the Monte Carlo evaluation of correlators between matrices at neighboring time slices, cf. Table 4. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_i^2 \rangle $ & $\langle M_i^4 \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline .1 & .55 & .57 & .41 & .31 & .59 & .33 & .52 & .26 \\ \hline 1 & .24 & .11 & .066 & .059 & .015 & .012 & .014 & .0093 \\ \hline 10 & .098 & .017 & .0099 & .0097 & .00031 & .00029 & .00025 & .00024 \\ \hline \end{tabular} \vspace{0.3cm} Table 4 : Mixed correlators, Eqs. (\ref{mixmf}), between matrices $M_i $ at neighboring time slices and predictions for free $M_i $ \end{center} At large coupling, neighboring matrices are to a very good approximation free, meaning the original decoupled basis gives a good description. By contrast, at low $g$, there are significant angular correlations induced by the nearest-neighbor coupling. These angular correlations are washed out by the fluctuations when one considers matrices far apart on the temporal lattice, cf. Table 5. There is virtually no deviation from free behavior between distant matrices even at low values of $g$. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_i^2 \rangle $ & $\langle M_i^4 \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline .1 & .55 & .57 & .31 & .31 & .35 & .33 & .28 & .26 \\ \hline 1 & .24 & .11 & .059 & .059 & .012 & .012 & .0093 & .0093 \\ \hline 10 & .098 & .017 & .0097 & .0097 & .00029 & .00029 & .00024 & .00024 \\ \hline \end{tabular} \vspace{0.3cm} Table 5 : Mixed correlators, Eqs. (\ref{mixmf}), between matrices $M_i $ separated by half of the total length of the temporal lattice, and predictions for free $M_i $ \end{center} In order to make contact with physical (still Euclidean) time, one should write the discretized action of the quantum one-matrix model as \begin{equation} S_{Q1}^{\prime } = \frac{1}{N} \mbox{Tr} \, \left( \sum_{i=0}^{L-1} \frac{(M_{i+1}^{\prime } -M_i^{\prime } )^2 }{2} \frac{L}{T} + \tilde{g} \frac{T}{L} (M_i^{\prime } )^4 \right) \end{equation} where $T$ is now the length of the circle in time direction. Rescaling $M_i^{\prime } = M_i \sqrt{2T/L} $, one regains the form (\ref{sq1def}) of the action, with the identification $g=4\tilde{g} T^3 /L^3 $. Therefore, $g$ behaves like the third power of the lattice spacing. In the continuum limit, the ground state distribution of the eigenvalues of the quantum mechanical one-matrix model has been found analytically in \cite{Bre1}. This can now be compared to the variational solution. According to the correspondence established above, the limit of vanishing lattice spacing corresponds to vanishing $g$; therefore the Fourier basis is the preferable one. The eigenvalue distribution for the rescaled variables $M_i^{\prime } $ is (cf. equation (\ref{q1brho})) \begin{equation} \rho_{M^{\prime } } (\lambda ) = \sqrt{\frac{L}{2T} } \rho_{M} \left( \sqrt{\frac{L}{2T} } \lambda \right) = \sqrt{\frac{L}{2T} } \frac{1}{x_B \pi } \sqrt{2x_B -\frac{L}{2T} \lambda^{2} } \end{equation} and $x_B $ is determined by (cf. equation (\ref{q1bx})) \begin{equation} 16\tilde{g} \frac{T^3 }{L^3 } x_B^3 + 16\tilde{g}^{2} \frac{T^6 }{L^6 } x_B^4 =1 \end{equation} which for large $L$ is solved by \begin{equation} x_B =\frac{L}{T} (16\tilde{g} )^{-1/3} + O(1) \end{equation} Therefore \begin{equation} \rho_{M^{\prime } } (\lambda ) = \frac{2}{\pi } \left( \frac{\tilde{g} }{4} \right)^{1/3} \sqrt{ \left( \frac{4}{\tilde{g} } \right)^{1/3} -\lambda^{2} } \end{equation} Note that there is no $T$-dependence left in the continuum limit. By contrast, the exact solution is \cite{Bre1} \begin{equation} \rho_{M^{\prime } }^{exact} (\lambda ) = \frac{1}{\pi } \sqrt{2\epsilon -2\tilde{g} \lambda^{4} } \ \ \ \ \ \ \ \mbox{with} \ \ \ \ \epsilon^{3} = 81\pi^{6} \tilde{g} \left( \frac{1}{\Gamma (1/4) } \right)^{8} \end{equation} determined by the normalization condition. The two solutions are compared for $\tilde{g} =1$ in Fig. \ref{fig12}. \subsection{Quantum Mechanics of Two Matrices} While the case of one quantum mechanical matrix discussed in the previous section is amenable to exact treatment \cite{Bre1} due to its special form, an analogous solution is not possible once more than one quantum mechanical matrix is involved. On the other hand, the approximate variational approach, using discretized time, is easily generalized to the case of more than one matrix. Here, a commutator-type interaction will be considered, as it is especially interesting from the point of view of applications to Yang-Mills theories: \begin{equation} S_{Q2} = \frac{1}{N} \mbox{Tr} \, \left( \sum_{i=0}^{L-1} (M_{1,i+1} -M_{1,i} )^2 + (M_{2,i+1} -M_{2,i} )^2 + g(i [ M_{1,i},M_{2,i} ] )^2 \right) \label{sq2def} \end{equation} where the first index labels the two different matrix variables and the second one the different time slices. Again, only the original choice of variables will be compared to a Fourier basis (cf. (\ref{umat})) using the variational criterion. Note that the action $S_{Q2} $ is invariant under simultaneous shifts of all the matrices by a multiple of the unit matrix. This trivial freedom should be removed in order to obtain stable Monte Carlo results; otherwise, the system just performs a random walk in the trace of the matrices. Thus, the $M_{n,i} $ here are constrained to be traceless, i.e. the condition $\langle M_{n,i} \rangle =0$, assumed to be realized dynamically in prior examples, is enforced by hand. The free partner to (\ref{sq2def}) in the original basis is \begin{equation} S_{Q2f} (M_{n,i} ) = \sum_{i=0}^{L-1} \frac{1}{N} \mbox{Tr} \, (2M_{1,i}^2 + 2M_{2,i}^2 ) + \frac{2g}{N^2 } \mbox{Tr} \, M_{1,i}^2 \mbox{Tr} \, M_{2,i}^2 \label{sq2fdef} \end{equation} (using $\langle M_{1,i} \rangle = \langle M_{2,i} \rangle =0$). On the other hand, in the Fourier basis \begin{equation} M_{n,i} = \sum_{j} U_{ij} B_{n,j} \end{equation} with $U$ as in (\ref{umat}), the kinetic part becomes \begin{equation} \tilde{S}_{Q2}^{kin} (B_{n,i} ) = \tilde{S}_{Q2f}^{kin} (B_{n,i} ) = \frac{2}{N} \mbox{Tr} \, \sum_{j=-L/2 +1}^{L/2} (1-\cos k_j ) (B_{1,j}^2 + B_{2,j}^2 ) \end{equation} The potential part on the other hand becomes \begin{equation} -\frac{g}{N} \mbox{Tr} \, \sum_{i} [ M_{1,i},M_{2,i} ]^2 = \frac{2g}{N} \mbox{Tr} \, \sum_{i} \sum_{jklm} U_{ij} U_{ik} U_{il} U_{im} (B_{1,j} B_{1,k} B_{2,l} B_{2,m} - B_{1,j} B_{2,k} B_{1,l} B_{2,m} ) \end{equation} Using again the assumption $\langle B_{n,i} \rangle =0$, the second term in the round brackets on the right hand side never gives a contribution to the free partner, whereas the first term only gives a contribution if $j=k$ and $l=m$. Thus one has \begin{equation} \tilde{S}_{Q2f}^{pot} (B_{n,i} ) = \frac{2g}{N^2 } \sum_{j,k} \mbox{Tr} \, B_{1,j}^2 \mbox{Tr} \, B_{2,k}^2 \sum_{i} U_{ij} U_{ij} U_{ik} U_{ik} =\frac{2g}{N^2 L} \sum_{j,k} \mbox{Tr} \, B_{1,j}^2 \mbox{Tr} \, B_{2,k}^2 \end{equation} Now one can again easily solve the free models. For the purpose of calculating the free energies, the reference action \begin{equation} V_{ref} (C_{n,i} ) = \frac{1}{N} \mbox{Tr} \, \sum_{i} (C_{1,i}^2 + C_{2,i}^2 ) \end{equation} is convenient; then one has to solve the models \begin{equation} S^{\prime }_{Q2f} (M_{n,i} ) = \sum_{i=0}^{L-1} \frac{1+\alpha }{N} \mbox{Tr} \, (M_{1,i}^2 + M_{2,i}^2 ) + \frac{2\alpha g}{N^2 } \mbox{Tr} \, M_{1,i}^2 \mbox{Tr} \, M_{2,i}^2 \label{q2fcor} \end{equation} for the original variables $M_{n,i} $ and \begin{equation} \tilde{S}^{\prime }_{Q2f} (B_{n,i} ) = \sum_{j} (1+\alpha -2\alpha \cos k_j ) \frac{1}{N} \mbox{Tr} \, (B_{1,j}^2 + B_{2,j}^2 ) +\frac{2g\alpha}{N^2 L} \sum_{j,k} \mbox{Tr} \, B_{1,j}^2 \mbox{Tr} \, B_{2,k}^2 \label{q2fcfo} \end{equation} for the Fourier variables $B_{n,i} $. Starting with (\ref{q2fcor}), abbreviating $\langle M_{n,i}^{2} \rangle =x_M $, one obtains identical semicircular distributions for all matrix variables with radius squared \begin{equation} r^2 = \frac{2}{\alpha +2\alpha gx_M +1} \end{equation} whereupon the consistency condition determining $x_M $ becomes \begin{equation} x_M =\frac{r^2 }{4} = \frac{1}{2} \frac{1}{\alpha +2\alpha gx_M +1} \end{equation} solved by \begin{equation} x_M =\frac{1}{4\alpha g} (\sqrt{(1+\alpha )^2 +4g\alpha } -(1+\alpha )) \end{equation} The free energy then is given by \begin{equation} F_M = L \int_{0}^{1} d\alpha \, ( 4x_M + 2gx_M^2 - 2x_M ) \end{equation} On the other hand, in the Fourier case (\ref{q2fcfo}), abbreviating \begin{equation} \left\langle \frac{2}{L} \sum_{j} B_{1,j}^2 \right\rangle = \left\langle \frac{2}{L} \sum_{j} B_{2,j}^2 \right\rangle = x_B \end{equation} one notices that the variables $B_{1,j} $ and $B_{2,j} $ are controlled by exactly the same potential as the variables $B_j $ in the case of the quantum mechanical one-matrix model discussed in the previous section. In particular, one has the consistency condition \begin{equation} x_B^2 (\alpha +\alpha gx_B +1)^2 -4\alpha^{2} x_B^2 -1 =0 \end{equation} and the second moments \begin{equation} \langle B_{1,j}^2 \rangle = \langle B_{2,j}^2 \rangle = \frac{1}{2} \frac{1}{1+\alpha -2\alpha \cos k_j + \alpha gx_B } \end{equation} The free energy now is \begin{equation} F_B = L \int_{0}^{1} d\alpha \, \left[ 2x_B - \frac{1}{L} \sum_{j=-L/2 +1}^{L/2} \frac{2\cos (2\pi j/L)}{1+\alpha -2\alpha \cos (2\pi j/L) +\alpha gx_B } +\frac{gx_B^2 }{2} -x_B \right] \end{equation} Numerical evaluation of the two free energies yields $F_B \leq F_M $ for all $g$, i.e. the Fourier basis is always preferable over the original basis. While this is to be expected at small $g$, it is not necessarily an indication that the Fourier basis represents a particularly good approximation for large $g$ as well: Rather, working in the original basis still truncates the interaction term rather badly at every time slice (cf. equation (\ref{sq2fdef})). This is different from the one-matrix case, where assuming the original variables to be free merely implied decoupling variables at different time slices, which certainly should become exact for large coupling $g$. In the two-matrix case, by contrast, there is still an additional truncation at every time slice which prevents convergence to the exact problem at large $g$. In light of this, it is not so surprising that the Fourier basis is preferable for all $g$. Nevertheless, in Figs. \ref{fig13}-\ref{fig15}, this basis appears to provide quite a satisfactory description even for $g=10$; the agreement with Monte Carlo results ($N=10$ and $L=10$ were used in the latter) is not much worse at $g=10$ than at $g=1/10$. Also in the correlators, cf. Tables 6 and 7, one observes that the angular correlations involving the same matrix variable at different time slices are stronger than the angular correlations between the two different matrix variables at a fixed time, up to quite high $g$. At $g=10$, these two correlations seem to be roughly equally strong. This again corroborates the result that using the Fourier basis, which treats the nearest-neighbor coupling exactly, provides a good approximation for a large range of $g$. \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_{n,i}^{2} \rangle $ & $\langle M_{n,i}^{4} \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline .1 & .86 & 1.5 & 1.2 & .75 & 5.6 & 2.3 & 5.1 & 1.7 \\ \hline 1 & .38 & .29 & .19 & .14 & .16 & .084 & .13 & .063 \\ \hline 10 & .18 & .068 & .038 & .033 & .0064 & .0047 & .005 & .0034 \\ \hline \end{tabular} \vspace{0.3cm} Table 6 : Mixed correlators, Eqs. (\ref{mixmf}), between matrices $M_{n,i} $ for one fixed $n$ at neighboring time slices $i$ and predictions for free $M_{n,i} $ \end{center} \begin{center} \begin{tabular}{|c||c|c||c|c||c|c||c|c|} \hline \rule[-1.5ex]{0ex}{4.5ex} $g$ & $\langle M_{n,i}^{2} \rangle $ & $\langle M_{n,i}^{4} \rangle $ & $C_1 $ & $C_{1f} $ & $C_2 $ & $C_{2f} $ & $C_3 $ & $C_{3f} $ \\ \hline \hline .1 & .86 & 1.5 & .66 & .75 & 1.7 & 2.3 & 1.3 & 1.7 \\ \hline 1 & .38 & .29 & .12 & .14 & .059 & .084 & .043 & .063 \\ \hline 10 & .18 & .068 & .027 & .033 & .0027 & .0047 & .002 & .0034 \\ \hline \end{tabular} \vspace{0.3cm} Table 7 : Mixed correlators between the matrices $M_{1,i} $ and $M_{2,i} $ at a fixed time slice $i$ and predictions for free $M_{1,i} $, $M_{2,i} $. \end{center} Ultimately, though, one would expect a local treatment, which assumes different time slices to be free with respect to one another, to provide better agreement at high $g$ if it manages to well approximate the commutator interaction. This is precisely what is not achieved if the original variables are assumed to be free. On the contrary, a nontrivial mixing of the variables $M_{1,i} $ and $M_{2,i} $ at every time slice is necessary such as to better accomodate the commutator-type interaction in (\ref{sq2def}). Note that linear combinations of $M_{1,i} $ and $M_{2,i} $ will not effect any improvement, since the commutator is invariant under such linear transformations (up to trivial rescalings of the variables). It is disappointing that precisely this phenomenologically very interesting type of interaction term seems particularly resistent to free approximation combined with linear transformations. Here, it seems that using (technically more complicated) nonlinear variable transformations must be contemplated in order to capture the essential angular correlations. Note also that the commutator term in the action on its own does not confine the eigenvalue distributions of the involved variables to a compact support, even if one enforces tracelessness; the eigenvalues can become arbitrarily large if the matrices are located in the regions of configuration space where they commute. This pathology however seems to disappear (according to Monte Carlo experiments carried out by the authors) when more than two matrices are involved (i.e. a higher-dimensional Yang-Mills type of action). Presumably there, the regions of configuration space where all the matrices commute are too small compared with the whole space such that entropy suppresses configurations with arbitrarily large eigenvalues. \section{Summary} In this work, a new variational approach to interacting large-$N$ multi-matrix models was developed. The partition function was approximated using the variational principle $F\le F_0 + \langle S - S_0 \rangle_{S_0 }$ with $S_0 $ initially taken from the space of all matrix models which are free in the set of variables the original interacting problem is given in. It turned out to be possible to give a general solution to this variational problem for a fixed set of matrix variables in terms of the concept of the ``free partner'' introduced by the authors. The free partner defines a type of mean field approximation to the original action and is constructed using the axioms of freeness in a fashion analogous to the construction of more conventional mean field theories for fermionic or bosonic particles using Wick's theorem. The freeness axioms constitute the analog of Wick's theorem for objects obeying the Cuntz algebra, which is the algebra obeyed by the Fock space operator representation of free random variables \cite{Gop}. However, the variational approach presented here goes beyond simply acting as a device to derive a mean field theory. Above, the variational space was characterized as being the space of all matrix models which are free in the set of variables the original interacting problem is given in. This implies that one can considerably enlarge the variational space by first allowing for a change of variables in the original problem and only then varying over all matrix models which are free in the variables one has settled for. In this way one includes into the variation models which are not free in the original variables. Every set of variables defines a different mean field theory given by the corresponding free partner. The variational principle not only allows to derive these mean field theories, but further allows to decide which out of a set of mean field theories provides the best approximation to the exact problem. A number of examples were considered in order to assess the accuracy of the variational method; the variational results were compared with Monte Carlo simulations as well as analytical results, where available. To begin with, classical two-matrix models described by actions with at most quartic terms were investigated allowing for a general linear transformation of matrix variables. Impressive agreement was observed, except in the case of the action $S_{1}^{red} $, Eq. (\ref{s1redef}). The authors did not attempt to improve the approximation by allowing for nonlinear variable transformations, which may relieve the discrepancy. Classical models with a large number of matrices were considered in which all matrices pairwise interact in the same way. Such models e.g. become relevant when one has managed to decouple the different space-time points of a large-$N$ matrix field theory in a large number of dimensions $D$, e.g. in the framework of an Eguchi-Kawai reduction. Among the models considered was one which reduces to $S_{1}^{red} $ for $D=2$. It turned out that the quality of the variational approximation using only the original set of variables improves as $D$ rises. The variational approximation was also considered for the problems of a single and two coupled matrix chains simulating path integrals for one and two quantum matrices. Two discrete choices of free variables, namely the original variables and a Fourier basis, were considered. In the single chain case, as expected, for small (large) lattice spacing, the Fourier (original) basis provided the better approximation. In the continuum limit of the one-matrix case, good agreement was achieved with the available analytical solution. For two matrix chains coupled via a commutator-type interaction, the Fourier basis provided a better approximation for all couplings; it also compares quite favorably with the exact Monte Carlo results for a large range of coupling constants. However, for very large coupling, one would expect a local basis, constructed such as to capture the angular correlations introduced by the commutator interaction, to ultimately be more appropriate. Here, again it seemed that technically more involved nonlinear transformations must be contemplated in order to achieve further progress.

proofpile-arXiv_065-595

\section{Topological Objects in QCD} \indent The appearance of QCD-monopole was pointed out by 't Hooft using abelian gauge fixing \cite{thooft}. Lattice QCD calculations \cite{{kron},{pol}} show that this topological object plays an essential role on color confinement through its condensation, which is characterized by the appearance of the large monopole clusters and can be interpreted as a Kosterlitz-Thouless-type phase transition \cite{ezawa}. In QCD, there is also another non-trivial topological object, a instanton. Instantons and QCD-monopoles are thought to be hardly related to each other since these topological objects appear from different non-trivial homotopy groups. Recently, however, the existence of a relation between the two objects has been shown by analytic studies and lattice QCD simulations \cite{{pol},{suga},{mar}}. In our previous analytical works, we conjectured that the existence of instantons promotes the clustering of monopole trajectories as a signal of monopole condensation \cite{suga}. First, we examine an evidence for our conjecture by measuring the monopole-loop length and the number of instantons simultaneously in the lattice QCD simulation. For this discussion, we take the SU(2) gauge theory. \section{Lattice Study for Monopole Trajectory and Instantons} \indent We study the correlation between the total monopole-loop length $L_{\rm total}$ and the integral of the absolute value of the topological density $I_{Q}$, which corresponds to the total number of instantons and anti-instantons, by use of the Monte Carlo simulation in the lattice gauge theory of the SU(2) Wilson action. We can measure $L_{\rm total}$ and $I_{Q}$ using the following procedure. In the maximally abelian (MA) gauge, the abelian gauge fixing is done by maximizing $R \equiv \sum_{s, \mu} {\rm Tr}\{ U_{\mu}(s) \tau^{3} U_{\mu}^{-1}(s) \tau^{3}\}$. The SU(2) link variable $U_{\mu}(s)$ is then factorized into the abelian link variable; $u_{\mu}(s)=\exp \{ i\tau_{3}\theta(s) \}$ and off-diagonal part; $M_{\mu}(s)\equiv\exp \{ i\tau_{1} C^{1}_{\mu}(s)+i\tau_{2}C^{2}_{\mu}(s) \}$ as $U_{\mu}(s)=M_{\mu}(s)u_{\mu}(s)$. The Dirac string is extracted from the abelian field strength $\theta_{\mu \nu}\equiv \partial_{\mu}\theta_{\nu}-\partial_{\nu}\theta_{\mu}$ by decomposition as $\theta_{\mu \nu}={\bar \theta}_{\mu \nu}+2\pi M_{\mu \nu}$ with $-\pi \leq{\bar \theta}_{\mu \nu}< \pi$ and $M_{\mu \nu}\in {\bf Z}$. Here, ${\bar \theta}_{\mu \nu}$ and $M_{\mu \nu}$ correspond to the regular part and the Dirac string part, respectively \cite{{kron},{suga}}. The monopole current is derived from the Dirac string part as \begin{equation} k_{\mu}(s)={1 \over 2} \varepsilon_{\mu \nu \rho \sigma}\partial_{\nu}M_{\rho \sigma} (s+\hat \mu)\;\;\;, \end{equation} which forms a closed loop, since the monopole current is topologically conserved. The total monopole-loop length is measured as $L_{\rm total}=\sum_{\mu, s}\mid k_{\mu}(s)\mid$. In order to examine the total number of topological pseudoparticles, the integral of the absolute value of the topological density is defined as \begin{equation} I_{Q}={1 \over 32\pi^{2}}\sum_{s} \varepsilon_{\mu \nu \rho \sigma} \mid{\rm Tr} \{ U_{\mu \nu}(s)U_{\rho \sigma}(s) \}\mid \end{equation} where $U_{\mu \nu}(s)$ is the plaquette variable. This value is measured with the Cabibbo-Marinari cooling method in the same way as the topological charge. We make lattice calculations for various $\beta=2.2\sim 2.35$ on the $16^{3}\times 4$ lattice, where the deconfinement transition occurs at $\beta_{c}\simeq 2.3$. As shown in Fig.1, we find the almost linear correlation between $I_{Q}$ after 3 cooling sweeps and $L_{\rm total}$. Hence, the monopole-loop length would be largely enhanced in the dense instanton system. \section{Clustering of Monopoles in the Multi-Instanton System} \indent We study the multi-instanton system by measuring the monopole clustering in the abelian gauge in order to understand the role of instantons on confinement \cite{fukushima}. The field configuration for single instanton with the size $\rho$ and the center $z_{\mu}$ in the singular gauge is \begin{equation} A^{I}_{\mu}(x;z_{k},\rho_{k},O_{k})= {{i\rho^{2}\tau_{a}O^{ai}{\bar \eta}^{i}_{\mu \nu}(x-z)_{\nu}} \over {(x-z)^{2}[(x-z)^{2}+\rho^{2}]}}\;\;, \end{equation} where $O^{ai}$ is the color orientation matrix and ${\bar \eta}^{i}_{\mu \nu}$ the 't Hooft symbol. For {\it anti}-instantons $A^{I}_{\mu}(x;z_{k},\rho_{k},O_{k})$, one has to replace the ${\bar \eta}^{i}_{\mu \nu}$ symbol by $\eta^{i}_{\mu \nu}$. The multi-instanton configurations are assumed as the sum of instanton ($I$) and anti-instanton ($\bar I$) solutions, \begin{equation} A_{\mu}(x)=\sum_{k}A^{I}_{\mu}(x;z_{k},\rho_{k},O_{k}) +\sum_{k}A^{\bar I}_{\mu}(x;z_{k},\rho_{k},O_{k})\;\;\;. \label{suminst} \end{equation} We generate ensembles of $N$ pseudoparticles with random orientations and random positions. The size of pseudoparticles are taken according to the size distribution $f(\rho)$. Here, we adopt the following instanton size distribution \cite{fukushima} as \begin{equation} f(\rho)={1 \over {{\left({\rho \over \rho_{_{\rm IR}}}\right)^{\nu}} +{\left({\rho_{_{\rm UV}} \over \rho}\right)^{b-5}} } } \label{dis} \end{equation} with $b = {11 \over 3}N_{c}$. Here, $\rho_{_{\rm IR}}$ and $\rho_{_{\rm UV}}$ are certain parameters such that the distribution (\ref{dis}) is normalized to unity, while the maximum of the distribution is fixed to a give value $\rho_{0}$, which is the most probable size of the pseudoparticles in the ensemble. It is noted that $f(\rho)$ is reduced to the one obtained in the dilute instanton case; $f(\rho)\sim \rho^{b - 5}$ in the limit of $\rho \rightarrow 0$. For large size instantons, $f(\rho)$ falls off with negative power of $\nu$, since large size instantons are suppressed by the instanton interaction. Here, we adopt $\nu=3$, which is proposed by Diakonov and Petrov \cite{fukushima}. We then introduce a lattice and express the gauge field in terms of the unitary matrices $U_{\mu}=\exp(iaA_{\mu})$ living on the links. Then, we can do exactly the same procedure as done in lattice simulations in order to extract the monopole trajectories in the MA gauge. We take a $16^{4}$ lattice with the lattice spacing of $a=0.15 {\rm fm}$ and the most probable instanton size, $\rho_{0}=0.4 {\rm fm}$. The volume is thus fixed and equal to $V=(2.4{\rm fm})^{4}$. In this calculation, we take equal numbers of instantons and anti-instantons; $N_{I}=N_{\bar I}=N/2$. In Fig.2, we show the histograms of monopole loop length for two typical cases with the total pseudoparticle number $N=20$ and 60, which correspond to the density $(N/V)^{1 \over 4}=174$ and $228 {\rm MeV}$, respectively. At low instanton density, only relatively short monopole loops are found. At high density, there appears one very long monopole loop in {\it each} gauge configuration. \section{Summary} \indent From above results, instantons would be the source of large size monopole clustering, which indicates occurring of monopole condensation in the similar argument as the Kosterlitz-Thouless-type phase transition \cite{ezawa}. Thus, instantons seem to play a relevant role on color confinement. \section*{References}

proofpile-arXiv_065-596

\section{#1}} \renewcommand{\theequation}{\thesection.\arabic{equation}} \newcommand{\Big[}{\Big[} \newcommand{\Big]}{\Big]} \newcommand{\Big\{}{\Big\{} \newcommand{\Big\}}{\Big\}} \newcommand{\Big[\hspace{-3pt}\Big[}{\Big[\hspace{-3pt}\Big[} \newcommand{\Big]\hspace{-3pt}\Big]}{\Big]\hspace{-3pt}\Big]} \newcommand{{\Bbb Z}}{{\Bbb Z}} \newcommand{{\Bbb C}}{{\Bbb C}} \newcommand{{\ZZ'}}{{{\Bbb Z}'}} \newcommand{\delta}{\delta} \newcommand{\varepsilon}{\varepsilon} \newcommand{{\frak a}}{{\frak a}} \newcommand{{\frak b}}{{\frak b}} \newcommand{\frac{1}{2}}{\frac{1}{2}} \newcommand{\smbox}[1]{\ \mbox{#1}\ } \newcommand{\medbox}[1]{\quad\mbox{#1}\quad} \newcommand{\bigbox}[1]{\qquad\mbox{#1}\qquad} \newcommand{{\cal A}}{{\cal A}} \newcommand{{\cal E}}{{\cal E}} \newcommand{{\cal U}}{{\cal U}} \newcommand{{\tilde a}}{{\tilde a}} \newcommand{{\tilde A}}{{\tilde A}} \newcommand{{\tilde K}}{{\tilde K}} \newcommand{{\widehat A(M-1,N-1)}}{{\widehat A(M-1,N-1)}} \newcommand{{\cU_q(\widehat A(M-1,N-1))}}{{{\cal U}_q(\widehat A(M-1,N-1))}} \newcommand{{\cU_q(A(M-1,N-1))}}{{{\cal U}_q(A(M-1,N-1))}} \newtheorem{theorem}{Theorem} \begin{document} \newpage \pagestyle{empty} \setcounter{page}{0} \vfill \begin{center} {\Large {\bf ANYONIC REALIZATION OF THE \vspace{5mm} QUANTUM AFFINE LIE SUPERALGEBRAS \vspace{5mm} ${\cU_q(\widehat A(M-1,N-1))}$}}\\[1cm] \vspace{7mm} {\large L. Frappat$^{a}$ \footnote{On leave of absence from Laboratoire de Physique Th\'eorique ENSLAPP.}, A. Sciarrino$^{b}$, S. Sciuto$^{c}$, P. Sorba$^{d}$ } \vspace{7mm} {\em $^{a}$ Centre de Recherches Math\'ematiques} \\ {\em Universit\'e de Montr\'eal, Canada} \vspace{5mm} {\em $^{b}$ Dipartimento di Scienze Fisiche, Universit\`a di Napoli ``Federico II''} \\ {\em and I.N.F.N., Sezione di Napoli, Italy} \vspace{5mm} {\em $^{c}$ Dipartimento di Fisica Teorica, Universit\`a di Torino} \\ {\em and I.N.F.N., Sezione di Torino, Italy} \vspace{5mm} {\em $^{d}$ Laboratoire de Physique Th\'eorique ENSLAPP} \\ {\em Annecy-le-Vieux et Lyon, France} \end{center} \vfill \begin{abstract} We give a realization of the quantum affine Lie superalgebras ${\cU_q(\widehat A(M-1,N-1))}$ in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter $q$ being related to the statistical parameter $\nu$ of the anyons by $q = e^{i\pi\nu}$. The construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of the quantum superalgebras ${\cU_q(A(M-1,N-1))}$ is obtained. \end{abstract} \vfill \vfill \rightline{CRM-2383} \rightline{DSF-T-35/96} \rightline{DFTT-39/96} \rightline{ENSLAPP-AL-603/96} \rightline{q-alg/9609033} \rightline{September 1996} \newpage \pagestyle{plain} \indent \sect{Introduction} \label{sectintro} Superalgebras, which are the mathematical framework for describing symmetry betwen bosons and fermions have by now found several interesting applications in physics, even if the {\em fundamental} supersymmetry between elementary constituents of the matter has not yet been supported by experimental evidence. A further enlarged concept of symmetry represented by quantum algebras has shown up in a large number of areas in physics.The fusion of these new enlarged symmetry structures is a natural step and it leads to the so-called $q$-superalgebras. Moreover, the connection between quantum algebras and generalized statistics has been pointed out in several contexts. {\em Anyons} are typical objects of generalized statistics whose importance in two-dimensional physics is relevant. They have been used to construct Schwinger-Jordan like realization of the deformed classical finite Lie algebras \cite{LS93,FMS94} and of the deformed affine Lie algebras of the unitary and symplectic series \cite{F3S96,Conf96}. So it is natural to ask which kind of oscillators are necessary to build realizations of $q$-superalgebras. We will show that the deformation of the affine unitary superalgebras (and therefore also of the finite unitary superalgebras) can be realized by means of anyons and of a new type of generalized statistics objects which satisfy braiding relations and which will be called {\em bosonic anyons} for reasons which will be clear from their definitions, see Sec. \ref{sect3}. Let us emphasize that the construction we propose may be interesting in the study of systems of correlated electrons. In fact the so-called $t-J$ model \cite{ZR88}, which has been suggested as an appropriate starting point for the theory of the high-temperature superconductivity, is supersymmetric for particular values of the coupling constants and of the chemical potential, the Hamiltonian commuting with a $su(1|2)$. Moreover in ref. \cite{LS93b} it has been shown that the $t-J$ model at the supersymmetric point can be written in terms of anyons, which gives a new realization of supersymmetry. Although in this model no deformation appears, it is conceivable that anyons can be used to describe further deformed generalizations of this or of similar models, like the Hubbard model \cite{Mon94}. The article is organized in the following way: in Sec. \ref{sect1} we briefly recall the structure of ${\widehat A(M-1,N-1)}$ in the Cartan-Weyl and in the Serre-Chevalley bases and then we write its deformation in the distinguished Serre-Chevalley basis; in Sec. \ref{sect2} the fermionic-bosonic oscillators realization of ${\widehat A(M-1,N-1)}$ is presented and finally in Sec. \ref{sect3} the anyonic realization of ${\cU_q(\widehat A(M-1,N-1))}$ is given in terms of one-dimensional anyons and bosonic anyons; in Sec. \ref{sect4} the generalization of the construction to two-dimensional anyons is discussed and a few conclusions are presented. \sect{Presentation of the superalgebra ${\widehat A(M-1,N-1)}$} \label{sect1} We will recall in this section the presentation of the affine Lie superalgebra ${\widehat A(M-1,N-1)}$, where $M,N \ge 1$, both in the Cartan--Weyl basis and in the Serre--Chevalley basis. We set $R=M+N-1$ and we exclude the case $R=1$ (obtained when $M=N=1$). \subsection{Cartan--Weyl presentation of ${\widehat A(M-1,N-1)}$} In the Cartan--Weyl basis, the generators of the affine Lie superalgebra ${\widehat A(M-1,N-1)}$ are denoted by $h_a^m$ (Cartan generators) and $e_{\frak a}^m$ (root generators) where $a=1,\dots,R$ and $m \in {\Bbb Z}$. The even root system of ${\widehat A(M-1,N-1)}$ is given by $\Delta_0 = \{ \pm(\varepsilon_i - \varepsilon_j), \pm(\delta_k - \delta_l) \}$ and the odd root system by $\Delta_1 = \{ \pm(\varepsilon_i - \delta_k) \}$ where $1 \le i < j \le M$ and $1 \le k < l \le N$, the $\varepsilon_i$ and $\delta_k$ spanning the dual of the Cartan subalgebra of $gl(M|N)$. To each root generator $e_{\frak a}^m$ one assigns a ${\Bbb Z}_2$-grading defined by $\deg(e_{\frak a}^m)=0$ if ${\frak a} \in \Delta_0$ and $\deg(e_{\frak a}^m)=1$ if ${\frak a} \in \Delta_1$. The generators satisfy for $M \ne N$ the following commutation relations: \subequations \begin{eqnarray} && \Big[ h_a^m,h_b^n \Big] = \gamma \,m \,\delta_{m+n,0} \,K(h_a,h_b) \label{eq1a} \\ && \Big[ h_a^m,e_{\frak a}^n \Big] = {\frak a}_a ~ e_{\frak a}^{m+n} \label{eq1b} \\ && \Big[\hspace{-3pt}\Big[ e_{\frak a}^m,e_{\frak b}^n \Big]\hspace{-3pt}\Big] = \left\{ \begin{array}{ll} \varepsilon({\frak a},{\frak b}) ~ e_{{\frak a}+{\frak b}}^{m+n} & \quad \smbox{if} {\frak a} + {\frak b} \smbox{is a root} \\ {\frak a}^a h_a^{m+n} + \gamma \,m \,\delta_{m+n,0} \,K(e_{\frak a},e_{-{\frak a}}) & \quad \smbox{if} {\frak b} = -{\frak a} \\ 0 & \quad \smbox{otherwise} \end{array} \right. \label{eq1c} \\ && \Big[ h_a^m,\gamma \Big] = \Big[ e_{\frak a}^m,\gamma \Big] = 0 \label{eq1d} \end{eqnarray} \endsubequations where $\varepsilon({\frak a},{\frak b}) = \pm 1$ is the usual 2-cocycle, $K$ is the (non-degenerate) Killing form on the horizontal superalgebra $A(M-1,N-1)$ and $\gamma$ is the central charge. $[\hspace{-2pt}[ ~,~ ]\hspace{-2pt}]$ denotes the super-commutator: $[\hspace{-2pt}[ e_{\frak a}^m,e_{\frak b}^n ]\hspace{-2pt}] = e_{\frak a}^m \, e_{\frak b}^n - (-1)^{\deg(e_{\frak a}^m).\deg(e_{\frak b}^n)} e_{\frak b}^n \, e_{\frak a}^m$. Note that by virtue of Eqs. (\ref{eq1a}-d) the value of the central charge of $\hat A_{M-1}$ is opposite to that of $\hat A_{N-1}$. \\ In the case $M=N$, although the Killing form is zero, it is possible to define a non-degenerate bilinear form $K$ on $A(N-1,N-1)$ such that Eqs. (\ref{eq1a}-d) still hold. \subsection{Serre--Chevalley presentation of ${\widehat A(M-1,N-1)}$} In the Serre--Chevalley basis, the algebra is described in terms of simple root and Cartan generators, the only data being the entries of the Cartan matrix $(a_{\alpha\beta})$ of the algebra. Let us denote the generators in the Serre--Chevalley basis by $h_\alpha$ and $e_\alpha^\pm$ where $\alpha=0,1,\dots,R$. If $\tau$ is a subset of $\{0,1,\dots,R\}$, the ${\Bbb Z}_2$-gradation of the superalgebra is defined by setting $\deg(e_\alpha^\pm) = 0$ if $\alpha \notin \tau$ and $\deg(e_\alpha^\pm) = 1$ if $\alpha \in \tau$. The superalgebra is described by the (super)commutation relations \subequations \begin{eqnarray} && \Big[ h_\alpha,h_\beta \Big] = 0 \label{eq2a} \\ && \Big[ h_\alpha,e^\pm_\beta \Big] = \pm a_{\alpha\beta} e^\pm_\beta \label{eq2b} \\ && \Big[\hspace{-3pt}\Big[ e^+_\alpha,e^-_\beta \Big]\hspace{-3pt}\Big] = e^+_\alpha e^-_\beta - (-1)^{\deg(e^+_\alpha)\deg(e^-_\beta)} ~ e^-_\beta e^+_\alpha = h_\alpha ~ \delta_{\alpha\beta} \label{eq2c} \\ && \Big\{ e_\alpha^\pm,e_\alpha^\pm \Big\} = 0 \medbox{if} a_{\alpha\alpha} = 0 \label{eq2d} \end{eqnarray} \endsubequations and by the following relations: \begin{itemize} \item Serre relations for all $\alpha \ne \beta$ \begin{equation} (\mbox{ad}\, e_\alpha^\pm)^{1-{\tilde a}_{\alpha\beta}} ~ e_\beta^\pm = 0 \label{eq3} \end{equation} \item supplementary relations for $\alpha$ such that $a_{\alpha\alpha}=0$ \begin{equation} \Big[\hspace{-3pt}\Big[ (\mbox{ad}\,e_{\alpha-1}^\pm) \,e_\alpha^\pm, (\mbox{ad}\,e_{\alpha+1}^\pm) \,e_\alpha^\pm \Big]\hspace{-3pt}\Big] = 0 \label{eq4} \end{equation} \end{itemize} where the matrix ${\tilde A} = ({\tilde a}_{\alpha\beta})$ is deduced {from} the Cartan matrix $A = (a_{\alpha\beta})$ of ${\widehat A(M-1,N-1)}$ by replacing all its positive off-diagonal entries by $-1$. Here ad denotes the adjoint action: $(\mbox{ad}\,X) ~ Y = XY - (-1)^{\deg X.\deg Y} ~ YX$. \medskip One has to emphasize that for superalgebras, the description given by the Serre relations (\ref{eq3}) leads in general to a bigger superalgebra than the superalgebra under consideration. It is necessary to write supplementary relations involving more than two generators, that for $A(M-1,N-1)$ take the form (\ref{eq4}), in order to quotient the bigger superalgebra and recover the original one, see \cite{Sch93} for more details. As one can imagine, these supplementary conditions appear when one deals with isotropic fermionic simple roots (that is $a_{\alpha\alpha} = 0$). Note that these supplementary relations are unnecessary when $M=1$ or $N=1$. \medskip In the following, we will only use the Serre--Chevalley description of the affine Lie superalgebra in the {\em distinguished} basis, such that the number of odd simple roots is the smallest one. In the case of ${\widehat A(M-1,N-1)}$, the distinguished basis is defined by taking $\tau = \{0,M\}$. The corresponding Dynkin diagram is the following, with the labels identifying the corresponding simple roots: \begin{center} \begin{picture}(180,80) \thicklines \multiput(0,20)(42,0){5}{\circle{14}} \put(84,65){\circle{14}} \put(7,20){\dashbox{3}(28,0)} \put(49,20){\line(1,0){28}} \put(91,20){\line(1,0){28}} \put(5,25){\line(2,1){72}} \put(163,25){\line(-2,1){72}} \put(133,20){\dashbox{3}(28,0)} \put(79,15){\line(1,1){10}}\put(79,25){\line(1,-1){10}} \put(79,60){\line(1,1){10}}\put(79,70){\line(1,-1){10}} \put(0,0){\makebox(0.4,0.6){\scriptsize{$1$}}} \put(42,0){\makebox(0.4,0.6){\scriptsize{$M-1$}}} \put(84,0){\makebox(0.4,0.6){\scriptsize{$M$}}} \put(126,0){\makebox(0.4,0.6){\scriptsize{$M+1$}}} \put(168,0){\makebox(0.4,0.6){\scriptsize{$M+N-1$}}} \put(84,50){\makebox(0.4,0.6){\scriptsize{$0$}}} \end{picture} \end{center} associated to the Cartan matrix \begin{equation} \left(\begin{array}{rrrrrrrrrrrr} 0 & 1 & 0 & \cdots & \cdots &&&& \cdots & \cdots & 0 & -1 \\ -1 & 2 & -1 & 0 &&&&&&&& 0 \\ 0 & -1 & \ddots & \ddots & \ddots &&&&&&& \vdots \\ \vdots & 0 & \ddots && \ddots & 0 &&&&&& \vdots \\ \vdots && \ddots & \ddots & \ddots & -1 & \ddots &&&&& \\ &&& 0 & -1 & 2 & -1 & \ddots &&&& \\ &&&& \ddots & -1 & 0 & 1 & \ddots &&& \\ &&&&& \ddots & -1 & 2 & -1 & 0 && \vdots \\ \vdots &&&&&& \ddots & -1 & \ddots & \ddots & \ddots & \vdots \\ \vdots &&&&&&& 0 & \ddots && \ddots & 0 \\ 0 &&&&&&&& \ddots & \ddots & \ddots & -1 \\ -1 & 0 & \cdots & \cdots && \cdots & \cdots & 0 & \cdots & 0 & -1 & 2 \\ \end{array}\right) \label{eq5} \end{equation} The correspondence between the distinguished Serre--Chevalley and the Cartan--Weyl bases is the following ($i=1,\dots,M-1$ and $k=1,\dots,N-1$): \begin{equation} \begin{array}{lll} \bigg. h_i = h_i^0 \qquad & e_i^+ = e_{\varepsilon_i-\varepsilon_{i+1}}^0 \qquad & e_i^- = e_{\varepsilon_{i+1}-\varepsilon_i}^0 \\ \bigg. h_M = h_M^0 \qquad & e_M^+ = e_{\varepsilon_M-\delta_1}^0 \qquad & e_M^- = e_{\delta_1-\varepsilon_M}^0 \\ \bigg. h_{M+k} = h_{M+k}^0 \qquad & e_{M+k}^+ = e_{\delta_k-\delta_{k+1}}^0 \qquad & e_{M+k}^- = e_{\delta_{k+1}-\delta_k}^0 \\ \bigg. h_0 = -\gamma + \sum_{i=1}^{M} h_i^0 - \sum_{k=1}^{N-1} h_{M+k}^0 \qquad & e_0^+ = e_{\delta_N-\varepsilon_1}^1 \qquad & e_0^- = e_{\varepsilon_1-\delta_N}^{-1} \end{array} \label{eq6} \end{equation} Notice that in the Serre--Chevalley picture the central charge $\gamma$ \ is uniquely defined by the following equation: \begin{equation} h_0 = -\gamma + \sum_{i=1}^{M} h_i - \sum_{k=1}^{N-1} h_{M+k} \label{eq6bis} \end{equation} \subsection{Serre--Chevalley presentation of ${\cU_q(\widehat A(M-1,N-1))}$} We consider now the universal quantum affine Lie superalgebra ${\cU_q(\widehat A(M-1,N-1))}$. The Serre--Chevalley description in the quantum case is very similar. The defining relations take the form \subequations \begin{eqnarray} && \Big[ h_\alpha,h_\beta \Big] = 0 \label{eq7a} \\ && \Big[ h_\alpha,e_\beta^\pm \Big] = \pm a_{\alpha\beta} e_\beta^\pm \label{eq7b} \\ && \Big[\hspace{-3pt}\Big[ e_\alpha^+,e_\beta^- \Big]\hspace{-3pt}\Big] = e^+_\alpha e^-_\beta - (-1)^{\deg(e^+_\alpha)\deg(e^-_\beta)} ~ e^-_\beta e^+_\alpha = \frac{q_\alpha^{h_\alpha} - q_\alpha^{-h_\alpha}}{q_\alpha - q_\alpha^{-1}} ~ \delta_{\alpha\beta} \label{eq7c} \\ && \Big\{ e_\alpha^\pm,e_\alpha^\pm \Big\} = 0 \medbox{if} a_{\alpha\alpha} = 0 \label{eq7d} \end{eqnarray} \endsubequations where $q_{\alpha} = q^{d_\alpha}$ and the numbers $d_\alpha$ symmetrize the Cartan matrix $\bar a_{\alpha\beta}$ of $A(M-1,N-1)$: $d_\alpha \bar a_{\alpha\beta} = d_\beta \bar a_{\beta\alpha}$ ($\alpha,\beta \ne 0$) and $d_0 = 1$ (in the distinguished basis, $q_\alpha = q$ for $\alpha=0,\dots,M$ and $q_\alpha = q^{-1}$ for $\alpha=M+1,\dots,M+N-1$). \\ In terms of the generators ${\cal E}_\alpha^\pm = e_\alpha^\pm ~ q_\alpha^{-h_\alpha/2}$ the usual Serre relations are for all $\alpha \ne \beta$ \begin{equation} (\mbox{ad}_q\,{\cal E}_\alpha^\pm)^{1-{\tilde a}_{\alpha\beta}} ~ {\cal E}_\beta^\pm = 0 \label{eq8} \end{equation} while the supplementary relations read now for $\alpha$ such that $a_{\alpha\alpha}=0$ (the definition of the quantum adjoint action $\mbox{ad}_q$ is given below Eq. \ref{eq12}) \cite{FLV91,Sch93,Yam94} \begin{equation} \Big[\hspace{-3pt}\Big[ (\mbox{ad}_q\,{\cal E}_{\alpha-1}^\pm) \,{\cal E}_\alpha^\pm, (\mbox{ad}_q\,{\cal E}_{\alpha+1}^\pm) \,{\cal E}_\alpha^\pm \Big]\hspace{-3pt}\Big] = 0 \label{eq10} \end{equation} or in terms of the generators $e_\alpha^\pm$ \begin{equation} \Big[\hspace{-3pt}\Big[ \Big[ e_{\alpha-1}^\pm , e_\alpha^\pm \Big]_q , \Big[ e_\alpha^\pm , e_{\alpha+1}^\pm \Big]_q \Big]\hspace{-3pt}\Big] = 0 \label{eq9} \end{equation} the $q$-commutator being defined as usual by $\Big[ X,Y \Big]_q = XY - q YX$. \medskip The universal quantum affine Lie superalgebra ${\cal U} \equiv {\cU_q(\widehat A(M-1,N-1))}$ is endowed with a Hopf algebra structure, with coproduct $\Delta: {\cal U} \rightarrow {\cal U} \otimes {\cal U}$, counit $\varepsilon: {\cal U} \rightarrow {\Bbb C}$ and antipode $S: {\cal U} \rightarrow {\cal U}$ such that ($\alpha=0,1,\dots,R$) \subequations \begin{eqnarray} && \Delta(h_\alpha) = 1 \otimes h_\alpha + h_\alpha \otimes 1 \medbox{and} \Delta(e_\alpha^\pm) = e_\alpha^\pm \otimes q_\alpha^{h_\alpha/2} + q_\alpha^{-h_\alpha/2} \otimes e_\alpha^\pm \label{eq11a} \\ && \varepsilon(h_\alpha) = \varepsilon(e_\alpha^\pm) = 0 \medbox{and} \varepsilon(1) = 1 \label{eq11b} \\ && S(h_\alpha) = -h_\alpha \medbox{and} S(e_\alpha^\pm) = -q_\alpha^{\pm a_{\alpha\alpha}/2} e_\alpha^\pm \label{eq11c} \end{eqnarray} \endsubequations The quantum adjoint action $\mbox{ad}_q$ can be explicitly written in terms of the coproduct and the antipode as \begin{equation} (\mbox{ad}_q\,X) ~ Y = (-1)^{\deg X_{(2)}.\deg Y} ~ X_{(1)}\,Y\,S(X_{(2)}) \label{eq12} \end{equation} using the Sweedler notation for the coproduct: $\Delta(X) = X_{(1)} \otimes X_{(2)}$ (summation is understood). \sect{Oscillator realization of the affine Lie superalgebra ${\widehat A(M-1,N-1)}$} \label{sect2} Let us recall now the oscillator realization of ${\widehat A(M-1,N-1)}$ in terms of creation and annihilation operators. We consider an infinite number of fermionic oscillators $c_i(r), c_i^\dagger(r)$ with $i=1,\dots,M$ and $r \in {\Bbb Z}+1/2 = {\ZZ'}$, which satisfy the anticommutation relations \begin{equation} \Big\{ c_i(r),c_j(s) \Big\} = \Big\{ c_i^\dagger(r), c_j^\dagger(s) \Big\} = 0 \bigbox{and} \Big\{ c_i(r),c_j^\dagger(s) \Big\} = \delta_{ij} \delta_{rs} \label{eq20} \end{equation} and an infinite number of bosonic oscillators $d_k(r), d_k^\dagger(r)$ with $k=1,\dots,N$ and $r \in {\ZZ'}$, which satisfy the commutation relations \begin{equation} \Big[ d_k(r),d_l(s) \Big] = \Big[ d_k^\dagger(r), d_l^\dagger(s) \Big] = 0 \bigbox{and} \Big[ d_k(r),d_l^\dagger(s) \Big] = \delta_{kl} \delta_{rs} \label{eq21} \end{equation} the two sets $c_i(r), c_i^\dagger(r)$ and $d_k(r), d_k^\dagger(r)$ commuting each other: \begin{equation} \Big[ c_i(r),d_k(s) \Big] = \Big[ c_i(r),d_k^\dagger(s) \Big] = \Big[ c_i^\dagger(r),d_k(s) \Big] = \Big[ c_i^\dagger(r),d_k^\dagger(s) \Big] = 0 \label{eq30} \end{equation} The fermionic and bosonic number operators are defined as usual by $n_i(r) = c_i^\dagger(r)c_i(r)$ and $n'_k(r) = d_k^\dagger(r)d_k(r)$. \\ These oscillators are equipped with a normal ordered product such that \begin{equation} :c_i^\dagger(r) c_j(s): = \left\{ \begin{array}{ll} c_i^\dagger(r) c_j(s) & \smbox{if} s > 0 \\ - c_j(s) c_i^\dagger(r) & \smbox{if} s < 0 \end{array} \right. \label{eq22} \end{equation} and \begin{equation} :d_k^\dagger(r) d_l(s): = \left\{ \begin{array}{ll} d_k^\dagger(r) d_l(s) & \smbox{if} s > 0 \\ d_l(s) d_k^\dagger(r) & \smbox{if} s < 0 \end{array} \right. \label{eq23} \end{equation} Therefore \begin{equation} :n_i(r): = \left\{ \begin{array}{ll} n_i(r) & \smbox{if} r > 0 \\ n_i(r) - 1 & \smbox{if} r < 0 \end{array} \right. \label{eq24} \end{equation} and \begin{equation} :n'_k(r): = \left\{ \begin{array}{ll} n'_k(r) & \smbox{if} r > 0 \\ n'_k(r) + 1 & \smbox{if} r < 0 \\ \end{array} \right. \label{eq25} \end{equation} Then an oscillator realization of the generators of ${\widehat A(M-1,N-1)}$ in the Cartan--Weyl basis with $\gamma =1$ is given by \subequations \begin{eqnarray} && \hspace{-5mm} h_i^m = \sum_{r \in {\ZZ'}} \left(:c_i^\dagger(r) c_i(r+m): - :c_{i+1}^\dagger(r) c_{i+1}(r+m): \right) ~~~ (i=1,\dots,M-1) \,, \label{eq26a} \\ && \hspace{-5mm} h_M^m = \sum_{r \in {\ZZ'}} \left(:c_M^\dagger(r) c_M(r+m): + :d_1^\dagger(r) d_1(r+m): \right) \,, \label{eq26b} \\ && \hspace{-5mm} h_{M+k}^m = \sum_{r \in {\ZZ'}} \left(:d_k^\dagger(r) d_k(r+m): \- :d_{k+1}^\dagger(r) d_{k+1}(r+m): \right) ~~~ (k=1,\dots,N-1) \,, \label{eq26c} \\ && \hspace{-5mm} e_{\varepsilon_i-\varepsilon_j}^m = \sum_{r \in {\ZZ'}} c_i^\dagger(r) c_j(r+m) \label{eq26d} \\ && \hspace{-5mm} e_{\delta_k-\delta_l}^m = \sum_{r \in {\ZZ'}} d_k^\dagger(r) d_l(r+m) \label{eq26e} \\ && \hspace{-5mm} e_{\varepsilon_i-\delta_k}^m = \sum_{r \in {\ZZ'}} c_i^\dagger(r) d_k(r+m) \label{eq26f} \\ && \hspace{-5mm} e_{\delta_k-\varepsilon_i}^m = \sum_{r \in {\ZZ'}} d_k^\dagger(r) c_i(r+m) \label{eq26g} \end{eqnarray} \endsubequations A fermionic oscillator realization of the simple generators of ${\widehat A(M-1,N-1)}$ in the distinguished Serre-Chevalley basis is given by $(\alpha=0,1,\dots,R)$ \begin{eqnarray} h_\alpha = \sum_{r \in {\ZZ'}} h_\alpha(r) \bigbox{and} e_\alpha^\pm = \sum_{r \in {\ZZ'}} e_\alpha^\pm (r) \label{eq27} \end{eqnarray} where ($i=1,\dots,M-1$ and $k=1,\dots,N-1$) \subequations \begin{eqnarray} && h_i(r) = n_i(r) - n_{i+1}(r) = ~ :n_i(r): - :n_{i+1}(r): \label{eq28a} \\ && h_M(r) = n_M(r) + n'_1(r) = ~ :n_M(r): + :n'_1(r): \label{eq28b} \\ && h_{M+k}(r) = n'_k(r) - n'_{k+1}(r) = ~ :n'_k(r): - :n'_{k+1}(r): \label{eq28c} \\ && h_0(r) = n'_N(r) + n_1(r+1) = ~ :n'_N(r): + :n_1(r+1): - ~ \delta_{r+1/2,0} \label{eq28d} \\ && e_i^+(r) = c_i^\dagger(r) c_{i+1}(r) \,, \hspace{18mm} e_i^-(r) = c_{i+1}^\dagger(r) c_i(r) \label{eq28e} \\ && e_M^+(r) = c_M^\dagger(r) d_1(r) \,, \hspace{18.5mm} e_M^-(r) = d_1^\dagger(r) c_M(r) \label{eq28f} \\ && e_{M+k}^+(r) = d_k^\dagger(r) d_{k+1}(r) \,, \hspace{12mm} e_{M+k}^-(r) = d_{k+1}^\dagger(r) d_k(r) \label{eq28g} \\ && e_0^+(r) = d_N^\dagger(r) c_1(r+1) \,, \hspace{14mm} e_0^-(r) = c_1^\dagger(r+1) d_N(r) \label{eq28h} \end{eqnarray} \endsubequations Inserting Eq. (\ref{eq28d}) into Eq. (\ref{eq27}) and taking into account that the sum over $r$ can be splitted into a sum of two convergent series only after normal ordering, one can check that \begin{equation} h_0 = -1 + \sum_{r \in {\ZZ'}} :n'_N(r): + \sum_{r \in {\ZZ'}} :n_1(r): ~ = -1 + \sum_{i=1}^{M-1} h_i^0 + h_M^0 - \sum_{k=1}^{N-1} h_{M+k}^0 \,, \label{eq29} \end{equation} that is the central charge is $\gamma = 1$. \\ Note that the value of the central charge is related to the definition of the normal ordered product. A different definition like ($i = 1,\dots,M$ and $k=1,\dots,N$) \begin{equation} :n_i(r): = n_i(r) \smbox{and} :n'_k(r): = n'_k(r) \smbox{for any} r \in {\ZZ'} \label{eq31} \end{equation} would lead to $\gamma = 0$. \sect{Anyonic realization of ${\cU_q(\widehat A(M-1,N-1))}$} \label{sect3} In order to obtain an anyonic realization of ${\cU_q(\widehat A(M-1,N-1))}$, we will replace the fermionic and bosonic oscillators by suitable anyons in the expressions of the simple generators of ${\cU_q(\widehat A(M-1,N-1))}$ in the distinguished Serre--Chevalley basis. Since we have to deal with fermionic and bosonic generators, we have to introduce two different types of anyons. \medskip Let us first define fermionic anyons on a one-dimensional lattice ${\ZZ'}$ \cite{LS93,FLS96}: \begin{equation} a_i(r) = K_i(r) c_i(r) \bigbox{and} {\tilde a}_i(r) = {\tilde K}_i(r) c_i(r) \label{eq40} \end{equation} and similar expressions for the conjugated operators $a_i^\dagger(r)$ and ${\tilde a}_i^\dagger(r)$, where the disorder factors $K_i(r)$ and ${\tilde K}_i(r)$ are expressed as \subequations \begin{eqnarray} && K_i(r) = q^{-\frac{1}{2}\sum_{t\in{\ZZ'}} \varepsilon(t-r) :n_i(t):} \label{eq41a} \\ && {\tilde K}_i(r) = q^{\frac{1}{2}\sum_{t\in{\ZZ'}} \varepsilon(t-r) :n_i(t):} \label{eq41b} \end{eqnarray} \endsubequations The function $\varepsilon(t) = |t|/t$ if $t \ne 0$ and $\varepsilon(0) = 0$ is the sign function. \\ It is easy to prove that the $a$-type anyons satisfy the following braiding relations for $r>s$: \begin{eqnarray} && a_i(r) a_i(s) + q^{-1} a_i(s) a_i(r) = 0 \nonumber \\ && a_i^\dagger(r) a_i^\dagger(s) + q^{-1} a_i^\dagger(s) a_i^\dagger(r) = 0 \nonumber \\ && a^\dagger_i(r) a_i(s) + q ~ a_i(s) a^\dagger_i(r) = 0 \nonumber \\ && a_i(r) a_i^\dagger(s) + q ~ a_i^\dagger(s) a_i(r) = 0 \label{eq42} \end{eqnarray} and \begin{eqnarray} && a_i(r) a_i^\dagger(r) + a_i^\dagger(r) a_i(r) = 1 \nonumber\\ && a_i(r)^2 = a_i^\dagger(r)^2 = 0 \label{eq43} \end{eqnarray} The braiding relations between the ${\tilde a}$-type anyons are obtained from eqs. (\ref{eq42}) and (\ref{eq43}) by replacing $q \leftrightarrow q^{-1}$. \\ Finally, the braiding relations between $a$-type and ${\tilde a}$-type anyons are given by \begin{eqnarray} && \Big\{ {{\tilde a}}_i(r), a_i(s) \Big\} = \Big\{ {{\tilde a}}^\dagger_i(r),a^\dagger_i(s) \Big\} = 0 \medbox{for all} r,s \in {\ZZ'} \label{eq44} \\ && \Big\{ {{\tilde a}}^\dagger_i(r), a_i(s) \Big\} = \Big\{ {{\tilde a}}_i(r),a^\dagger_i(s) \Big\} = 0 \medbox{for all} r \ne s \in {\ZZ'} \label{eq45} \end{eqnarray} and \begin{eqnarray} && \Big\{ {{\tilde a}}_i(r), a^\dagger_i(r) \Big\} = q^{ ~ \sum_{t \in {\ZZ'}} \varepsilon(t-r) :n_i(t): } \nonumber \\ && \Big\{ {{\tilde a}}^\dagger_i(r), a_i(r) \Big\} = q^{ - \sum_{t \in {\ZZ'}} \varepsilon(t-r) :n_i(t): } \label{eq46} \end{eqnarray} Moreover, the following identity holds: \begin{equation} a^\dagger_i(r) a_i(r) = {{\tilde a}}^\dagger_i(r) {{\tilde a}}_i(r) = n_i(r) \label{eq47} \end{equation} the normal ordering between $a$-type and ${\tilde a}$-type anyons being defined as in eq. (\ref{eq22}). \medskip Now we will define anyonic-like operators based on $q$-deformed bosons. Let us recall that $q$-deformed bosons can be constructed from ordinary ones by the following procedure \cite{Son90}: \subequations \begin{eqnarray} && n'_k(r) = d_k^\dagger(r) d_k(r) \label{eq48a} \\ && b_k(r) = d_k(r) ~ \sqrt{\frac{[n'_k(r)]_q}{n'_k(r)}} = \sqrt{\frac{[n'_k(r)+1]_q}{n'_k(r)+1}} ~ d_k(r) \label{eq48b} \\ && b_k^\dagger(r) = \sqrt{\frac{[n'_k(r)]_q}{n'_k(r)}} ~ d_k^\dagger(r) = d_k^\dagger(r) ~ \sqrt{\frac{[n'_k(r)+1]_q}{n'_k(r)+1}} \label{eq48c} \end{eqnarray} \endsubequations The $q$-deformed bosons $b_k(r),b_k^\dagger(r)$ satisfy the following $q$-commutation relations: \subequations \begin{eqnarray} && b_k(r) b_l^\dagger(s) - q^{\delta_{kl}\delta_{rs}} b_l^\dagger(s) b_k(r) = q^{-n'_k(r)} \delta_{kl} \delta_{rs} \label{eq49a} \\ && b_k(r) b_l^\dagger(s) - q^{-\delta_{kl}\delta_{rs}} b_l^\dagger(s) b_k(r) = q^{n'_k(r)} \delta_{kl} \delta_{rs} \label{eq49b} \\ && b_k(r) b_l(s) - b_l(s) b_k(r) = b_k^\dagger(r) b_l^\dagger(s) - b_l^\dagger(s) b_k^\dagger(r) = 0 \label{eq49c} \\ && \Big[ n'_k(r),b_l(s) \Big] = -b_k(r) \delta_{kl}\delta_{rs} \label{eq49d} \\ && \Big[ n'_k(r),b_l^\dagger(s) \Big] = b_k^\dagger(r) \delta_{kl}\delta_{rs} \label{eq49e} \end{eqnarray} \endsubequations {from} which it follows that \begin{equation} b_k^\dagger(r) b_k(r) = [n'_k(r)]_q \bigbox{and} b_k(r) b_k^\dagger(r) = [n'_k(r)+1]_q \label{eq50} \end{equation} Now, let us define anyonic-like operators as follows: \begin{equation} A_k(r) = K'_k(r) b_k(r) \bigbox{and} {\tilde A}_k(r) = {\tilde K}'_k(r) b_k(r) \label{eq51} \end{equation} and similar expressions for the conjugated operators $A_k^\dagger(r)$ and ${\tilde A}_k^\dagger(r)$, where the disorder factors are given by \subequations \begin{eqnarray} && K'_k(r) = q^{\frac{1}{2}\sum_{t\in{\ZZ'}} \varepsilon(t-r) :n'_k(t):} \label{eq52a} \\ && {\tilde K}'_k(r) = q^{-\frac{1}{2}\sum_{t\in{\ZZ'}} \varepsilon(t-r) :n'_k(t):} \label{eq52b} \end{eqnarray} \endsubequations It can be proved that the operators $A_k(r),A_k^\dagger(r)$ satisfy the following braiding relations for $r>s$: \begin{eqnarray} && A_k(r) A_k(s) - q A_k(s) A_k(r) = 0 \nonumber \\ && A_k^\dagger(r) A_k^\dagger(s) - q A_k^\dagger(s) A_k^\dagger(r) = 0 \nonumber \\ && A^\dagger_i(r) A_k(s) - q^{-1} ~ A_k(s) A^\dagger_i(r) = 0 \nonumber \\ && A_k(r) A_k^\dagger(s) - q^{-1} ~ A_k^\dagger(s) A_k(r) = 0 \label{eq53} \end{eqnarray} and \begin{eqnarray} && A_k(r) A_k^\dagger(r) - q A_k^\dagger(r) A_k(r) = q^{-n'_k(r)} \nonumber \\ && A_k(r) A_k^\dagger(r) - q^{-1} A_k^\dagger(r) A_k(r) = q^{n'_k(r)} \label{eq54} \end{eqnarray} Therefore, the operators $A_k(r),A_k^\dagger(r)$ satisfy the $q$-commutation relations of the $q$-deformed bosonic oscillator at the same point, while they satisfy braiding relations when taken at different points. \\ The braiding relations between the ${\tilde A}$-type anyons are obtained from eq. (\ref{eq53}) by replacing $q \leftrightarrow q^{-1}$. Note that for these anyonic $q$-deformed bosons defined by the above relations, we do not have any physical interpretations, on the contrary of the $a$-type anyons, see refs. \cite{LS93,FLS96}. Is is worth to point out that the above introduced {\em bosonic anyons} differ from the ones introduced in ref. \cite{LMR96} by the non trivial fact that our anyons are defined on a lattice while in ref. \cite{LMR96} are defined in the continuum and by the local braiding relation (\ref{eq54}). Replacing in Eq. (\ref{eq51}) the $q$-boson by a standard boson, we find in the lattice the bosonic anyons of ref. \cite{LMR96}. We will come back on the difference between the two approaches in the next section. \medskip Now we can build an anyonic realization of ${\cU_q(\widehat A(M-1,N-1))}$ by "anyonizing" the oscillator realization eqs. (\ref{eq28a}-\ref{eq28h}), that is replacing the fermionic oscillators $c_i$ by the anyonic oscillators $a_i$ and ${\tilde a}_i$ and the bosonic oscillators $b_i$ by the operators $A_i$ and ${\tilde A}_i$. More precisely, one has: \begin{theorem} An anyonic realization of the simple generators of the quantum affine Lie superalgebra ${\cU_q(\widehat A(M-1,N-1))}$ with central charge $\gamma = 1$ is given by (with $\alpha=0,1,\dots,R$) \begin{equation} H_\alpha = \sum_{r \in {\ZZ'}} H_\alpha(r) \bigbox{and} E_\alpha^\pm = \sum_{r \in {\ZZ'}} E_\alpha^\pm (r) \label{eq55} \end{equation} where ($i=1,\dots,M-1$ and $k=1,\dots,N-1$) \subequations \begin{eqnarray} && H_i(r) = n_i(r) - n_{i+1}(r) = :n_i(r): - :n_{i+1}(r): \label{eq56a} \\ && H_M(r) = n_M(r) + n'_1(r) = :n_M(r): + :n'_1(r): \label{eq56b} \\ && H_{M+k}(r) = n'_k(r) - n'_{k+1}(r) = :n'_k(r): - :n'_{k+1}(r): \label{eq56c} \\ && H_0(r) = n'_N(r) + n_1(r+1) = :n'_N(r): + :n_1(r+1): - \delta_{r+1/2,0} \label{eq56d} \\ && E_i^+(r) = a_i^\dagger(r) a_{i+1}(r) \,, \hspace{20mm} E_i^-(r) = {\tilde a}_{i+1}^\dagger(r) {\tilde a}_i(r) \label{eq56e} \\ && E_M^+(r) = a_M^\dagger(r) A_1(r) \,, \hspace{19mm} E_M^-(r) = {\tilde A}_1^\dagger(r) {\tilde a}_M(r) \label{eq56f} \\ && E_{M+k}^+(r) = A_k^\dagger(r) A_{k+1}(r) \,, \hspace{12mm} E_{M+k}^-(r) = {\tilde A}_{k+1}^\dagger(r) {\tilde A}_k(r) \label{eq56g} \\ && E_0^+(r) = A_N^\dagger(r) a_1(r+1) \,, \hspace{14mm} E_0^-(r) = {\tilde a}_1^\dagger(r+1) {\tilde A}_N(r) \label{eq56h} \end{eqnarray} \endsubequations \end{theorem} \noindent {\bf Proof} We must check that the realization eqs. (\ref{eq55}) and (\ref{eq56a}-\ref{eq56h}) indeed satisfy the quantum affine Lie superalgebra ${\cU_q(\widehat A(M-1,N-1))}$ in the distinguished Serre--Chevalley basis (\ref{eq7a}-\ref{eq7d}) together with the quantum Serre relations (\ref{eq8}) and (\ref{eq9}). The proof follows the lines of the algebraic case \cite{F3S96}: the equations (\ref{eq7a}-\ref{eq9}) which define a generic deformed affine superalgebra ${\cal U}_q(\widehat{\cal A})$ reduce to ${\cal U}_q({\cal A})$ when the affine dot is removed and to another finite dimensional superalgebra ${\cal U}_q({\cal A}')$ if the affine dot is kept and one or more other suitable dots are removed. The relations defining ${\cal U}_q(\widehat{\cal A})$ coincide with the union of those defining ${\cal U}_q({\cal A})$ and ${\cal U}_q({\cal A}')$: therefore, it will be enough to check that the equations defining ${\cal U}_q({\cal A})$ and ${\cal U}_q({\cal A}')$ are satisfied. Consider the non-extended Dynkin diagram of $A(M-1,N-1)$ to which the set of generators $\{ H_\alpha, E^\pm_\alpha \}$ (with $\alpha \ne 0$) corresponds. Inserting Eqs. (\ref{eq40}), (\ref{eq41a}-b), (\ref{eq50}), (\ref{eq51}), the expressions (\ref{eq56e}-g) become \begin{equation} E^\pm_\alpha(r) = \hat e^\pm_\alpha(r) ~ q_\alpha^{\frac{1}{2} \sum_{t \in {\ZZ'}} \varepsilon(t-r) \, :h_\alpha(t):} \label{eq57} \end{equation} where the generators $\hat e^\pm_\alpha(r)$ are obtained from the generators $e^\pm_\alpha(r)$ in Eqs. (\ref{eq28e}-g) replacing the bosonic oscillators $d_k$ by the $q$-deformed bosons $b_k$. The generators $\hat e^\pm_\alpha(r)$ coincide locally, i.e. for fixed $r$, with the generators of ${\cU_q(A(M-1,N-1))}$ of refs. \cite{FLV91,FSV91} as the $q$-deformed fermions $\psi_i$ of \cite{FSV91} are equivalent to the usual fermionic oscillators $c_i$. It follows that the generators $\{ H_\alpha, E^\pm_\alpha \}$ of Eq. (\ref{eq55}) are a representation of ${\cU_q(A(M-1,N-1))}$ as they are obtained with the correct coproduct, see Eqs. (\ref{eq11a}) and (\ref{eq57}), by the representation in terms of $\{ h_\alpha, \hat e^\pm_\alpha \}$. Let us remark that due to the equivalence betwen $q$-fermions and standard fermions the realization of finite $q$-superalgebras of ref. \cite{FSV91} are realizations of deformed algebras only for the subalgebra realized in terms of $q$-bosons while the subalgebra realized in terms of $q$-fermions is left undeformed. On the contrary the here presented anyonic realization is completely deformed in any sector. Finally let us remark that the difference $q \rightarrow q^{-1}$ in the disorder factor of the $A$-type anyons (in the site $r$) with respect to the $a$-type anyons (in the same site) -- see the $q_\alpha$-factor in Eq. (\ref{eq57}) -- is essential for the consistency of the $q$-superalgebra structure. We consider then the {\em extended} Dynkin diagram of $A(M-1,N-1)$ and we delete a dot which is not the affine dot. For example, cutting the dot number 2, we obtain the following Dynkin diagram: \begin{center} \begin{picture}(280,20) \thicklines \multiput(0,0)(42,0){7}{\circle{14}} \put(0,15){\makebox(0.4,0.6){1}} \put(42,15){\makebox(0.4,0.6){0}} \put(84,15){\makebox(0.4,0.6){$M$+$N$-1}} \put(126,15){\makebox(0.4,0.6){$M$+1}} \put(168,15){\makebox(0.4,0.6){$M$}} \put(210,15){\makebox(0.4,0.6){$M$-1}} \put(252,15){\makebox(0.4,0.6){3}} \put(37,-5){\line(1,1){10}}\put(37,5){\line(1,-1){10}} \put(163,-5){\line(1,1){10}}\put(163,5){\line(1,-1){10}} \put(7,0){\line(1,0){28}} \put(49,0){\line(1,0){28}} \put(91,0){\dashbox{3}(28,0)} \put(133,0){\line(1,0){28}} \put(175,0){\line(1,0){28}} \put(217,0){\dashbox{3}(28,0)} \end{picture} \end{center} which corresponds to the Lie superalgebra $A(M-1,N-1)$ in a {\em non-distinguished} basis. \\ For a fixed $r \in {\ZZ'}$, it is possible to show that the set $\{ h_j(r), \hat e^\pm_j(r) , h_1(r+1), \hat e^\pm_1(r+1) \}$ ($j = 0,3,\dots,M+N-1$) is a representation of ${\cU_q(A(M-1,N-1))}$ in the non-distinguished basis specified by the above Dynkin diagram. We emphasize that in this case we have to satisfy two more supplementary Serre relations than in the distinguished basis. Of course for particular values of $M$ and $N$ one or both relations can be absent. Note that deleting the $M$-th dot, we reobtain the superalgebra ${\cU_q(A(M-1,N-1))}$ in the distinguished basis again. Then it follows that the generators $\{ H_\alpha, E^\pm_\alpha \}$ with $\alpha \ne 2$ are a representation of ${\cU_q(A(M-1,N-1))}$ as they are obtained by the generators of a representation of the finite $q$-superalgebra with the correct coproduct. This completes the proof. \sect{General representations and conclusions} \label{sect4} In the previous section, we have built a representation of the deformed affine Lie superalgebras ${\cU_q(\widehat A(M-1,N-1))}$ by means of anyons defined on an infinite linear chain; as the corresponding fermionic representation, it has central charge $\gamma = 1$. Representations with vanishing central charge could be built in the same way by using alternative normal ordering prescriptions Eq. (\ref{eq31}). Representations with $\gamma=0$ and $\gamma=1$ can be combined together to get representations with arbitrary positive integer central charges (we do not discuss here the problem of the irreducibility of these representations). Associating a representation to any horizontal line of a two-dimensional square lattice, infinite in one direction (say the horizontal one), and taking $K$ copies of representations in one-dimensional lattice with central charge equal to 1, one can get representations with the value of the central charge equal to $K$. Note that by combining one representation with central charge equal to $K$ with a finite number of representations (in one-dimensional lattice) with vanishing value of the central charge one obtains an inequivalent representation with the same value of the central charge. The extensions to two-dimensional lattice infinite in both directions can also be done, but it requires some care in the definition in order to avoid convergence problems. \medskip We have shown in ref. \cite{F3S96} that the use of $a$-anyons on a two-dimensional lattice naturally gives the correct coproduct with the correct powers of the deformation of the representations of a $q$-algebra defined in a fixed site of the lattice. For completeness we recall that each site of the two-dimensional lattice is labelled by a vector $\vec{x} = (x_1,x_2)$, the first component $x_1 \in {\ZZ'}$ being the coordinate of a site on the line $x_2 \in {\Bbb Z}$. The angle $\Theta(\vec{x},\vec{y})$ which enters in the definition of two-dimensional $a$-anyons through the disorder factor, see e.g. ref. \cite{FLS96}, \begin{equation} K(\vec{x}) = \exp\left(i\nu{\sum\limits_{\vec{y}\ne \vec{x}} \Theta(\vec{x},\vec{y}) ~ n(\vec{y})} \right) \label{eq61} \end{equation} may be chosen in such a way that \begin{equation} \Theta(\vec{x},\vec{y}) = \left\{ \begin{array}{ll} +\pi/2 & \qquad \smbox{if} x_2 > y_2 \cr -\pi/2 & \qquad \smbox{if} x_2 < y_2 \end{array} \right. \label{eq60} \end{equation} while if $\vec{x}$ and $\vec{y}$ lie on the same horizontal line, that is $x_2 = y_2$, the definitions of Sec. \ref{sect3} hold. Two-dimensional anyons still satisfy the braiding and anticommutations relations expressed in the general form in Eqs. (\ref{eq42})-(\ref{eq46}). Analogous relations hold for the $A$-anyons. Let us replace in the equations of Sec. \ref{sect3} the one-dimensional anyons by two-dimensional ones and sum over the sites of the two-dimensional lattice. This sum has to be read as a sum over the infinite line $x_1$ and a sum over the finite number of lines labelled by $x_2$. Then the generators are given by a sum, with the correct coproduct, of the generators of a ${\cU_q(\widehat A(M-1,N-1))}$ representations defined in a line. Therefore they define a ${\cU_q(\widehat A(M-1,N-1))}$ representation with value of the central charge given by the sum of the values (0 or 1, see discussion in Sec. \ref{sect3}) of the central charges associated to each line of the two-dimensional lattice. \medskip In the previous sections we have discussed the case of $\vert q \vert = 1$. The case of $q$ real can also be discussed and we refer to \cite{LS93} for the definition of anyons for generic $q$. \medskip One can naturally ask if the realization in terms of $a$-anyons and $A$-anyons here presented can be used to realize the deformation of other finite or affine superalgebras. It seems that this procedure can be extended to the other basic finite superalgebras, i.e. the series $B(0, N)$, $B(M,N)$, $C(N+1)$, $D(M,N)$, while it is not clear its extension to the exceptional or strange finite superalgebras or to the affine case. Finally we want briefly to comment on the difference between our approach and the approach of \cite{LMR96}, even if here we present the realization of a $q$-superalgebra and in \cite{LMR96} a realization of a $q$-algebra is presented. The approach of \cite{LMR96} is made on the continuum and the authors do not use, as already remarked, $q$-bosons, as in the present paper, but standard bosons before ``anyonization''. However one has to stress that their approach is based on a Fock space realization which guarantees the consistency of the commutation relations. It is worth noticing that on the Fock space fundamental representation of the deformed algebra is indistinguishable from the fundamental representation of the undeformed algebra. It follows that a sum with the correct product of the fundamental representation gives a representation of the deformed algebra. On the contrary as we fulfill the commutation relations in abstract way, we are not allowed to consider only the fundamental representation of the deformed algebra realized by bosons and in order to achieve the consistency of the representation we are lead to use $q$-bosons. \section*{Acknowledgements} Work supported by the European Commission TMR programme ERBFMRX-CT96-0045. A.S. thanks the Laboratoire de Physique Th\'eorique ENSLAPP for kind hospitality during the period in which this paper was finished. \newpage

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\section{INTRODUCTION} The field of Nuclear Astrophysics has to provide nuclear reaction rates suited for a wide range of astrophysical applications. Therefore, there is not only need for rates involving all possible (stable and unstable) nuclei across the nuclear chart but also for temperatures ranging from $0<T_9<10$ . For the majority of reactions, the statistical model (Hauser-Feshbach, SM) will be suited for the calculation and prediction of the rates. However, at low level densities the SM is not applicable anymore and single resonances and direct capture contributions have to be taken into account. For the user of such rates it would be convenient to have a simple means to determine the validity of the SM approach, showing which rates need special attention and probably further experimental investigation. In this work, we provide a ``map'' for the applicability of the SM depending on the interacting nuclei and the temperature. \section{THE NUCLEAR LEVEL DENSITY} Considerable effort has been put into the improvement of the input for the SM calculations (e.g.~\cite{cowan}). However, the nuclear level density still has shown the largest uncertainties among the properties entering the SM. For calculating the level densities in the given context one does not only have to find reliable methods, but also computationally feasible ones. In dealing with thousands of nuclei one has to resort to simple models in order to minimize computer time. Such a simple model is the back-shifted Fermi-gas description~\cite{cowan,bethe}, recently improved by introducing an energy-dependent level density parameter $a$~\cite{igna,reis,ilji}. More sophisticated Monte Carlo shell model calculations~\cite{karli} have shown excellent agreement with this phenomenological approach and justified the application of the Fermi-gas description at and above the neutron separation energy. Assuming equally distributed even and odd parities, one obtains the following form: \begin{equation} \label{levden} \rho(U,J,\pi)={1 \over 2} f(U,J) \rho(U)\quad, \end{equation} with \begin{eqnarray} \rho(U)={1 \over \sqrt{2\pi} \sigma}{\sqrt{\pi} \over 12a^{1/4}}{\exp(2\sqrt{aU}) \over U^{5/4}}\ ,\qquad f(U,J)={2J+1 \over 2\sigma^2} \exp\left({-J(J+1) \over 2\sigma^2}\right) \\ \sigma^2={\Theta_{\mathrm{rigid}} \over \hbar^2} \sqrt{U \over a}\ ,\qquad \Theta_{\mathrm{rigid}}={2 \over 5}m_{\mathrm{u}}AR^2\ ,\qquad U=E-\delta\quad. \nonumber \end{eqnarray} An improved approach has to consider the energy dependence of the microscopic effects which are known to vanish at high excitation energies~\cite{ilji}, i.e.\ the thermal damping of microscopic effects. The level density parameter $a$ is then described by~\cite{igna} \begin{equation} \label{endepa} a(U,Z,N)=\tilde{a}(A)\left[1+C(Z,N){f(U) \over U}\right]\quad, \end{equation} where \begin{equation} \tilde{a}(A)=\alpha A+\beta A^{2/3} \end{equation} and \begin{equation} f(U)=1-\exp(-\gamma U)\quad. \end{equation} The shape of the function $f(U)$ was found by approximation of numerical microscopic calculations based on the shell model. Thus, one is left with three open parameters, namely $\alpha$, $\beta$, and $\gamma$. The values of these parameters are determined by a fit to experimental s-wave neutron resonance spacing at the neutron separation energy~\cite{ilji}. The values $\alpha=0.1336$, $\beta=-0.06712$, $\gamma=0.04862$ result in a highly improved fit with an averaged global deviation of 1.5~\cite{tomturin,tom}, when taking the microscopic corrections $C(Z,N)$ from the latest FRDM mass formula~\cite{moeller} and consistently computing the backshift $\delta(Z,N)$=1/2\{$\Delta_{\mathrm{n}}(Z,N)+\Delta_{\mathrm{p}}(Z,N)$\} with the neutron and proton pairing gaps $\Delta_{\mathrm{n,p}}$ from the same source. \section{APPLICABILITY OF THE STATISTICAL MODEL} Having found a suitable method to calculate level densities one can apply it to determine the range of validity of the SM. It is often colloquially termed that the SM is only applicable for intermediate and heavy nuclei. However, the only necessary condition for its application is a large number of resonances at the appropriate bombarding energies, so that the cross section can be described by an average over resonances. \begin{figure} \psfig{file=nfig.ps,height=21.9cm,rheight=20.8cm} \caption{\label{nfig}Temperatures (in T$_9$) for which the statistical model can be used. Plotted is the compound nucleus of the neutron-induced reaction.} \end{figure} The nuclear reaction rate per particle pair at a given stellar temperature $T$ is determined by folding the reaction cross section with the Maxwell-Boltzmann (MB) velocity distribution of the projectiles~\cite{fowler} \begin{equation} \left< \sigma v \right>=\left( \frac{8}{\pi \mu} \right) ^{1/2} \frac{1}{\left( kT \right) ^{3/2}} \int_0^{\infty} \sigma(E) E \exp \left( -\frac{E}{kT} \right) dE \quad. \end{equation} An effective energy window is then found around the peak of the integrand at $E_0$. For charged particles this is the so-called Gamow peak at $E_0=E_{\mathrm{G}}^{1/3}(kT/2)^{2/3}$ (with the Gamow energy $E_{\mathrm{G}}$). For s-wave neutrons the effective peak coincides with the peak of the MB distribution at $E_0=kT$ (close to the neutron separation energy), for higher partial waves the energy window is shifted to slightly higher energies (similarily to the Gamow peak) due to the centrifugal barrier~\cite{wagoner}. The effective energy window for a given nucleus and temperature has to contain sufficiently many resonances in order to make it possible to solve the integral with the assumption of an average level density instead of calculating the exact sum over the individual levels. Numerical test calculations~\cite{tom} have shown that an average number of 5--10 contributing resonances is sufficient. Choosing a lower limit for the number of resonances, determining the width and location of the effective energy window at a given temperature and using the above level density description to calculate the number of resonances in this window, we derive a lower limit for the temperature at which the SM can still be used. Those temperature limits are shown in Fig.~\ref{nfig} for neutron-induced reactions. It should be noted that the derived temperatures will not change considerably even if changing the required level number within a factor of two, because of the exponential dependence of the level density on the excitation energy. \section{SUMMARY} Making use of an improved level density description we presented a method to determine the applicability of the SM, also providing clues on which reactions might be of special in\-ter\-est for experimental investigations. In principle, the method can be used for any pro\-jec\-ti\-les and also with different incident energy distributions (e.g.\ for experimental beams).

proofpile-arXiv_065-598

\section{Introduction} It is a common problem in heavy ion reactions between 25 MeV/N and 200 GeV/N that the single particle spectra do not allow to ascertain the underlying reaction mechanism. To mention only two examples: At low energies despite of many years of efforts the fragmentation of nuclei into many intermediate mass fragments remains still a process whose origin is heavily debated. At high energies it turned out to be very difficult to rule out a hadronic scenario which may produce the same spectra as those proposed as a signal for the creation of a quark gluon plasma. In such a situation a search for experimental information beyond the single particle spectra is obvious. Most valuable would be an information on the spatial structure of the reaction. It would allow to calculate key quantities like densities or energy densities. This information is, however, hard to obtain. The only promising method proposed up to now is based on the interferometry of identical particles. The interference of the amplitudes of two indistinguishable processes gives rise to a correlation function which in principle allows to extract the radius of the emitting source. This approach has been very successfully applied by Hanbury-Brown and Twiss \cite{hbt} in astronomy to determine the angular radius of stars by measuring the spatial correlations between two photons. Later Goldhaber\cite{gol} and Kopylov and Podgoretsky \cite{Podgor72} advanced its application in particle physics by showing that measurable momentum space correlations may contain information on the size of the emitting source. In the ideal case of a large, randomly emitting source of known shape this method is indeed very powerful and the experimental results can be directly related to the source radius of the emitting object. In particle and heavy ion physics the situation is, however, much more difficult. There we encounter quite a number of problems. The size of the emitting sources is of the order of the radius of a nucleus and therefore not small as compared to the size of the wave function of the emitted particles. This renders some approximations impossible. The signal may be distorted by final state interactions between the emitted particles or by to the long range Coulomb force of the source. The emission time point cannot be defined unambiguously. The decay of resonances into identical particles or correlations between the momenta of the emitted particles and the coordinates of the emission point may pretend a wrong size of the source. For a discussion of these problems we refer to ref. \cite{zaj} ,\cite{gyu}. Recently is has been discussed that the HBT correlation function for an expanding source, as encountered frequently in heavy ion reactions, yields a much more difficult relation between the space time structure of the emitting source and the correlation function as that for a static source\cite{hei}. Due to these problems the measured correlation function in heavy ion collisions cannot be directly related to the parameters of the emitting source even if its form were known. In this situation there are two possibilities. Either one {\it assumes} the form of the source and uses the measured correlation function to fix the source parameters. Unfortunately this procedure makes these parameters model dependent. Hence they cannot be used for more than a comparison between different experiments and yield little information on the actual source properties. Or one tries to describe the reaction in its entity. This turns out to be a quite complicated procedure but became very popular recently. In this approach one follows the time evolution of the system with help of one of the standard transport models. Unfortunately none of them propagates (anti)symmetrized wave functions but at most a direct product wave function. Since the HBT effect is based on the (anti)symmetrization of the wave function of identical particles the transport model themselves cannot predict the correlation function. Rather one assumes that each particle "freezes out" at some time point. The freeze out time is different for each particle. At high energies it is assumed that the freeze out time is that time at which the particle encounters its last collision with another particle of the system. At low energies, where potential interactions are important as well, the freeze out time cannot be unambiguously defined. The freeze out times as well as the particle momenta and positions form then the input for the subsequent calculation of the Hanbury-Brown and Twiss (HBT) correlation function \cite{led} which is then compared with experiment. Agreement is usually interpreted as a sign that the underlying transport model gives a realistic space time evolution of the different particles. Since these transport codes provide not only the momentum space coordinates of the particles but also that of the coordinate space they can then be used to calculate the time evolution of key quantities like the energy density or the density. The weak point in this procedure is the transition between the transport model and the subsequent program which calculates the correlation function. Does the transport model provide the correct time evolution of those quantities which are essential for the calculation of the correlation function? It is the purpose of this article to show that this is hardly the case. In order to understand the reason one has to understand in detail the derivation of the different transport models from the fundamental quantal equations as well as the derivation of the equation which is employed to determine the correlation function. We will perform this investigation for the three types of present day simulation programs: The Quantum Molecular Dynamics approach (QMD)\cite{aichelin91}, BUU type models like Boltzmann \"Uhling Uhlenbeck (BUU) \cite{ai85a}-\cite{cas90}, Vlasov \"Uhling Uhlenbeck (VUU) \cite{st86} or Landau Vlasov (LV) \cite{greg87} and cascade models. The later class includes also the high energy simulation programs like VENUS \cite{wer}, RQMD \cite{rqmd} and ARC \cite{arc}. This problem is independent of the relativistic or nonrelativistic nature of the approach. It also does neither depend on the time between the emissions of the two identical particles nor on the presence of resonances. It is also independent of a possible final state interaction between the particles which is therefore omitted. The common demand on all transport programs is that an emission time point can be defined. Essential is the information the programs provide at that time point. This information is quite different for the three types of programs mentioned above and hence the systematic errors are specific to each of the different transport models. In two cases (QMD and BUU) this procedure implies systematic errors which question the usefulness of the approach for its original purpose: The discrimination between different reaction mechanisms which yield the same single particle spectra. For the high energy simulation programs the correlation function is completely artificial. For clarity we limit our formalism to the simplest form possible by assuming that we are dealing with two bosons which are simultaneously emitted and can be treated nonrelativistically. For this simple case the formalism is very transparent. More realistic but also more complicated scenarios \cite{hei} may add additional problems but do not overcome the problems discussed here. \section{The correlation function} We start with the derivation of the correlation function which relates the freeze out points with the measurable two body correlation function. We assume that a source, which is considered as classical, emits simultaneously two identical bosons with momenta $\vec p_1$ and $\vec p_2$. The differential two body probability W reads then as follows \cite{Pratt84} ($\hbar,c =1$): \begin{equation} \label{1} {d^2W \over d\vec p_1 d\vec p_2} = |T_S(\vec p_1, \vec p_2, \alpha ,\beta)|^2 \end{equation} \noindent $\alpha $ and $\beta $ characterize the emitting source. The (anti)symmetrized production amplitude $T_S$ is given by \begin{equation} \label{2} T_S(\vec p_1,\vec p_2,\alpha ,\beta) = {1\over\sqrt{2}} (T (\vec p_1, \vec p_2,\alpha ,\beta) \pm T(\vec p_2, \vec p_1,\alpha ,\beta)). \end{equation} \noindent where $T(\vec p_1, \vec p_2,\alpha ,\beta)$ is the Fourier transform of the wave function \begin{equation} \label{3} T(\vec p_1, \vec p_2,\alpha ,\beta) = \int{d^3x_1d^3x_2 \over (2\pi)^3} e^{-i(\vec p_1\vec x_1+\vec p_2\vec x_2)}<\vec x_1,\vec x_2| \psi(\alpha ,\beta)>. \end{equation} Introducing the Wigner density of the two body density matrix $\rho_2 = |\psi_2><\psi_2|$ \begin{equation} \label{4} D(\vec x_1,\vec p_1,\vec x_2,\vec p_2) = {1\over (2\pi)^6}\int \prod_{i=1,2}d^3y_i e^{i\vec p_i\vec y_i} <\vec x_1-\vec y_1/2,\vec x_2-\vec y_2/2|\psi_2><\psi_2|\vec x_1+\vec y_1/2, \vec x_2+\vec y_2/2> \end{equation} we can express the probability as a function of the two particle Wigner density \cite{gyu} \begin{equation} \label{1} {d^2W \over d\vec p_1 d\vec p_2} = \int d^3x_1d^3x_2[ D(\vec x_1,\vec p_1,\vec x_2,\vec p_2)\pm D(\vec x_1,{\vec p_1+\vec p_2\over 2}, \vec x_2,{\vec p_1+\vec p_2\over 2})cos(\vec p_1-\vec p_2)(\vec x_1-\vec x_2)]. \end{equation} Hence for the calculation of this probability the transport theories have to provide the two body Wigner density. Unfortunately most of them do not permit to calculate this quantity. Therefore one has introduced an approximation, called smoothness assumption (SA) : \begin{equation} D(\vec x_1,{\vec p_1+\vec p_2\over 2},\vec x_2,{\vec p_1+\vec p_2\over 2}) \approx D(\vec x_1,\vec p_1,\vec x_2,\vec p_2). \end{equation} We will discuss the limits of its validity which turns out to be crucial in the course of the article. Employing the smoothness assumption the two body probability reads as \begin{equation} {d^2W^{SA} \over d\vec p_1 d\vec p_2} = \int d^3x_1d^3x_2 D(\vec x_1,\vec p_1,\vec x_2,\vec p_2) (1 \pm cos(\vec p_1-\vec p_2)(\vec x_1-\vec x_2)). \end{equation} This is the standard expression for the two particle probability employed in numerous publications to relate the measured cross section with the radius of the emitting source. Depending on the available information, for actual calculations one may have to employ further approximations. For BUU type equations, which propagate the one particle Wigner density only, one assumes that the correlations between particles are negligible \begin{equation} D(\vec x_1,\vec p_1,\vec x_2,\vec p_2)\approx D(\vec x_1,\vec p_1)\cdot D(\vec x_2,\vec p_2) \end{equation} whereas for classical cascade calculations one assumes that the quantal two body Wigner density can be replaced by the classical 2 body phase space density $F_{cl}$ \begin{equation} D(\vec x_1,\vec p_1,\vec x_2,\vec p_2)\approx F_{cl}(\vec x_1,\vec p_1,\vec x_2,\vec p_2). \end{equation} For a static source without any correlation between the emission point and the momentum of the emitted particle the correlation function, the quantity one compares with experiment, is independent of the center of center of mass motion of the emitted pair and is given by \begin{equation} C(\vec p) ={\int {d^2 W \over d\vec p_1 d\vec p_2} d^3P \over \int { dW \over d\vec p_1 }{dW \over d\vec p_2}d^3P} \end{equation} where $\vec P = {(\vec p_1+\vec p_2)\over 2}$ is the center of mass momentum, $\vec p = \vec p_1-\vec p_2$ is the relative momentum and $ {dW \over d\vec p_1 }$ is the one particle momentum distribution \begin{equation} {dW \over d\vec p_1 }= \int d^3x_1 D(\vec x_1, \vec p_1). \end{equation} In the general case, where correlations are present, C depends on the center of mass motion as well. As we will see the correlation function $C(\vec p)$ contains the desired information about the spatial properties of the emitting source. \section{Consequences of the smoothness assumption for cascade calculations} One of the classes of models employed to extract source radii by comparing experimental results with model predictions are the so called cascade models. These are classical n-body models which solve the Hamilton equations of a n-body system and are presently the only models available to simulate heavy ion reactions at CERN and Brookhaven energies. They include VENUS \cite{wer}, RQMD \cite{rqmd} (in its usually employed cascade version) and ARC \cite{arc} as well as now less frequently employed programs for heavy ion reactions at an energy of around 1 GeV/N \cite{cug}. In these models the particles do not interact via potentials but suffer two body collisions if they come sufficiently close in coordinate space. In between the collisions the particles move on straight lines. In the computer programs they are treated as classical particles with a sharp momentum and a sharp position. One may ask how classical models can be employed to calculate a correlation function which is solely based on the interference of amplitudes and hence a genuine quantal effect. For an understanding we have to make a detour. In order to employ eq. 5 we have to construct the Wigner density out of the classical two body phase space density. This is of course not unique but the approach \begin{eqnarray} F_{cl} &=& \delta(\x -\vec x_{\alpha})\delta(\vec p_1 -\vec p_{\beta}) \delta({\vec{x}}_2 -\vec x_{\beta})\delta({\vec{p}}_2 -\vec p_{\beta}) \nonumber \\ &=& \lim_{C\to \infty,D\to 0} {C^3D^3\over \pi^6} e^{-(\vec P -\vec K(t))^2C/4-(\vec X -\vec R(t))^24/D} e^{-(\vec p -\vec k(t))^2C-(\vec x -\vec r(t))^2/D} \nonumber \\ &\equiv& D_{cl}(\vec x_1,\vec p_1,\vec x_2, \vec p_2) \end{eqnarray} serves our purpose. The expression in the last line will be considered as Wigner density.Here we have used the definitions \begin{eqnarray} \vec P = \vec p_1+\vec p_2&;\vec K = \vec p_\alpha+\vec p_\beta\nonumber \\ \vec X = {\vec x_1+\vec x_2 \over 2}&; \vec R = {\vec x_\alpha+\vec x_\beta \over 2}\nonumber \\ \vec p = {\vec p_1-\vec p_2 \over 2}&; \vec k = {\vec p_\alpha-\vec p_\beta \over 2}\nonumber \\ \vec x = \vec x_1-\vec x_2&; \vec r = \vec x_\alpha-\vec x_\beta. \end{eqnarray} The Wigner density $ D_{cl}(\vec x_1,{\vec p_1+\vec p_2\over 2}, \vec x_2,{\vec p_1+\vec p_2\over 2})$ is obtained by replacing $\vec p_1$ and $\vec p_2$ by ${\vec p_1+\vec p_2\over 2}$. Please note that this Wigner density does not respect the uncertainty relation. Inserting this expression in eqs. 5 and 7 and performing the limit procedure we obtain for the two particle correlator without smoothness assumption \begin{eqnarray} {d^2W \over d\vec p_1 d\vec p_2}d^3P &=& \int d^3x_1d^3x_2d^3P[ D_{cl}(\vec x_1,\vec p_1,\vec x_2,\vec p_2)\pm D_{cl}(\vec x_1,{\vec p_1+\vec p_2\over 2}, \vec x_2,{\vec p_1+\vec p_2\over 2})cos(\vec p_1-\vec p_2)(\vec x_1-\vec x_2)] \nonumber \\ &=& \delta(\vec p- \vec k(t)) \pm \delta(\vec k(t))cos(2 \vec p \vec r). \end{eqnarray} This expression differs from \begin{equation} \int { dW \over d\vec p_1 }{dW \over d\vec p_2}d^3P =\delta(\vec p- \vec k(t)) \end{equation} only for the case that the relative momentum of the emitted classical particles is zero what in practical terms never happens. For all other cases we find \begin{equation} C(\vec p) = 1. \end{equation} Applying the smoothness assumption we find, however, \begin{equation} \int{d^2W^{SA} \over d\vec p_1 d\vec p_2} d^3P = \delta(\vec p-\vec k(t))(1+cos 2{\vec{p}}\vec r ) \end{equation} and hence \begin{equation} C^{SA}(\vec p) = 1+cos 2{\vec{p}}\vec r. \end{equation} Thus we observe that here the smoothness assumption creates correlations out of nothing. One faces the somewhat surprising result that the correlation function and hence the extracted radii are artificial and are only due to the differences between the approximate and the exact formula for the correlation function. Applying the correct formula the cascade calculations do not yield any correlation function, as the exact result shows. The truth of this observation can even easily be verified without any calculation. If two particles with a sharp momentum are emitted from two localized sources one can measure the momentum sufficiently precise in order to identify the source from which each particle has been emitted provided the both momenta are not identical. Thus there are no alternative processes, hence no interference of their amplitudes and there is, as a consequence, no HBT correlation function. {\it This has unfortunately the consequence that there is presently no microscopic model which may be used for the interpretation of the correlation data measured with ultrarelativistic heavy ion beams at CERN and AGS}. \section{Quantum Molecular Dynamics (QMD)} The discussion of models which provide sufficient information to construct a correlation function we begin with the QMD approach because it is the only one which allows to calculate the 2 body Wigner density. Hence one can calculate ${d^2W \over d\vec p_1 d\vec p_2}$ (eq.5) without any approximation. One can furthermore introduce the smoothness assumption and can calculate then ${d^2W^{SA} \over d\vec p_1 d\vec p_2}$ applying eq. 7. This may serve as a test for the validity of this approximation in the situation of a heavy ion reaction and hence for the judgement of the predictive power of the correlation function calculated in the framework of the other models. The QMD model is a n body theory which simulates heavy ion reactions between 30 MeV/N and 2 GeV/N on an event by event basis. Each nucleon is represented by a coherent state of the form \begin{equation} \phi_\alpha (\vec p_1,t) = \left({\frac{L }{2\pi}}\right)^{3/4}\, e^{-(\vec p_1 - \vec p_\alpha(t))^2L/4} \,e^{-i\vec p_1 \vec x_\alpha(t)}\, e^{+i p_\alpha^2(t)t/2m} \end{equation} Thus the wave function has two time dependent parameters $x_\alpha, p_\alpha$% , L is fixed. As we will see this wave function serves as a test wave function for a variational principle. Hence it is an {\it input} of the calculation and not the result of the solution of the Schr\"odinger equation. It relies heavily on intuition; other test wave functions may yield a different time evolution of the system. The total n body wave function is assumed to be the direct product of n coherent states \begin{equation} \phi = \phi _\alpha (\x, \vec x_\alpha, \vec p_\alpha, t) \phi _\beta (\vec x_2,\vec x_\beta,\vec p_\beta, t)\cdots, \end{equation} thus antisymmetrization is neglected. The initial values of the parameters are chosen in a way that the ensemble of $A_T$ + $A_P$ nucleons gives a proper density distribution as well as a proper momentum distribution of the projectile and target nuclei. The time evolution of the system is calculated by means of a generalized variational principle: We assume that $\vec p_\alpha$ and $\vec x_\alpha$contain the essential time dependence of the n-body wave function. The Lagrange function ${\cal L}$ can then be written as a functional of these parameters where H is the n - body Hamiltonian . \begin{equation} {\cal L} = \left(\phi \left\vert i\hbar ({\frac{\partial }{\partial t}} +{\frac{d\vec p_\alpha }{dt}} {\frac{d}{d\vec p_\alpha}}+{\frac{d\vec x_\alpha }{dt}} {\frac{d}{d\vec x_\alpha}}) - H \right\vert \phi\right). \end{equation} The time evolution of the parameters is obtained by the requirement that the action \begin{equation} S = \int\limits ^{t_2} _{t_1} {\cal L} [\phi, \phi^\ast] dt \end{equation} is stationary under the allowed variation of the wave function. For the wave function of eq. 20 the Lagrange function is given up to a constant by \begin{equation} {\cal L} = \sum_\alpha(\vec p_{\alpha} \dot{\vec x_{\alpha}} - {\vec p_{\alpha} \dot{\vec p_{\alpha}}t \over m} - {p_\alpha^2 \over 2m} - {1\over 2}\sum_{\beta}V(\vec x_\alpha,\vec x_\beta)). \end{equation} $V( \vec x_\alpha, \vec x_\beta)$ is the expectation value of the (density dependent) 2 body potential. The variation of the Lagrange function gives Euler Lagrange equations for each of the 6 parameters \begin{equation} \dot{\vec p_{\alpha}} = - \vec \nabla_{\vec x_a} \sum_\beta V( \vec x_\alpha, \vec x_\beta) \end{equation} and \begin{equation} \dot{\vec x_{\alpha}} = \vec p_\alpha/m. \end{equation} With these two equations one has reduced the problem of solving a n - body Schr\"odinger equation to that of solving 6 n ordinary differential equations. In reality V is a parametrization of the real part of the Br\"uckner G- matrix. The imaginary part is approximated by the measured cross section. For details we refer to ref.\cite{aichelin91}. Hence in QMD the centroids of the Gaussians in momentum and coordinate space are the only quantities which change in time. The form of the wave function around the centroids is fixed. This is a consequence of the ansatz ( eq. 19). From the test wave function eq.(19) we calculate the Wigner density of a pair of particles \begin{equation} D_{QMD}(\vec x_1 ,\vec p_1,\vec x_2,\vec p_2) = {1\over \pi^6} e^{-(\vec P -\vec K(t))^2L/4-(\vec X -\vec R(t))^24/L} e^{-(\vec p -\vec k(t))^2L-(\vec x -\vec r(t))^2/L}. \end{equation} Inserting this Wigner density in eq. 5 one obtains after integration over the pair's center of mass momentum \begin{equation} \int{d^2W \over d\vec p_1 d\vec p_2} d^3P = ({L\over \pi})^{3/2} (e^{-(\vec p -\vec k(t))^2L} \pm e^{- p^2L - k(t)^2L} \cos{2\vec p \vec r}) \end{equation} where t is the (assumed common) freeze out time. This is the probability to find two particles with a relative momentum $\vec p$, which have been emitted from two classical sources at a relative distance of $\vec r$ and a relative momentum of $\vec k$. \section{One Body Transport Theories} In order to derive the equation for the time evolution of the one--body Wigner density of a particle moving in a selfconsistent potential $V({\vec{x}})$ we start from that for the one body density matrix $\rho_1=|\psi_1><\psi_1|$ \begin{equation} \dot \rho_1 = -i [H,\rho_1]. \end{equation} Applying to this equation the Wigner transformation for an operator O \begin{equation} O_W ({\vec{x}},{\vec{p}}) = {1\over (2\pi)^3} \int d^3 y e^{i{\vec{p}}\vec y} <{\vec{x}} - {\vec y\over 2}|O|{\vec{x}} + {\vec y\over 2}> \end{equation} one obtains the differential equation \begin{equation} ({\partial \over \partial t} + {{\vec{p}}_1 \over m}\vec \nabla_{\vec x_1}) D(\x,\vec p_1,t) =\int d^3p_1'K_1({\vec{p}}_1-\vec{p}\,'_1,{\vec{x}}_1)D({\vec{x}}_1,\vec{p}\,'_1,t) \end{equation} D being the Wigner density of the one body density operator and $K_1$ is defined as \begin{equation} K_1({\vec{p}}_1-\vec{p}\,'_1,{\vec{x}})= {1\over i\hbar} \int {d^3y \over (2\pi\hbar)^3}e^{-i({\vec{p}}_1-\vec{p}\,'_1)\vec y/\hbar} (V({\vec{x}}+\vec y/2)- V({\vec{x}}-\vec y/2)). \end{equation} We have restored $\hbar$ here for reasons which will soon become obvious. One can expand the integration kernel around $x_1$ \begin{equation} K_1({\vec{p}}_1-\vec{p}\,'_1,{\vec{x}}_1)= {2\over \hbar} \sin({\hbar\vec \nabla_{\vec x_1}\vec \nabla_{\vec p_1}\over 2}) V({\vec{x}}_1)\delta({\vec{p}}_1-\vec{p}\,'_1). \end{equation} We see that $ K_1 $ can be viewed as a series with the expansion coefficient $\hbar\vec \nabla_{\vec x_1}\vec \nabla_{\vec p_1}$. Hence in the limit that the expansion can be terminated after the first term the Schr\"odinger equation in its Wigner representation is equivalent to the classical Vlasov equation :\par \begin{equation} ({\partial \over \partial t} + {{\vec{p}}_1 \over m}\vec \nabla_{\vec x_1}) D(\x,\vec p_1,t) = (\vec \nabla_{\vec x_1} V({\vec{x}}_1))\vec \nabla_{\vec p_1} D(\x,\vec p_1,t) \end{equation} The Vlasov equation describes the time evolution of the phase space density of particles which move on classical orbits specified by the Hamilton equations ${\partial \x \over \partial t} = {\vec p_1 \over m}$ and $ {\partial \vec p_1 \over \partial t} = - \vec \nabla_{\vec x_1} V $. As in QMD V presents the real part of the Br\"uckner G- matrix and the imaginary part is added as a cross section. \par There are two approaches to solve the above equation. Either one solves the differential equation directly or one creates a swarm of test particles which are subject to the Hamilton equations and fulfil the initial condition $D(\x,\vec p_1,t_0)$. One propagates this swarm with help of the Hamilton equations until a time t and then constructs the Wigner density $D(\x,\vec p_1,t)$ by coarse graining. The latter solution method is called test particle method and is employed in the BUU, VUU and LV approaches. When calculating the observables, i.e. the expectation values of operators, the transition from the first to the second method corresponds to the replacement of an analytical integration by a Monte Carlo procedure. Using the swarm of test particles the analytical solution \begin{equation} \langle O(t) \rangle = \int D(\x,\vec p_1,t) O(\x,\vec p_1) \, d^3x_1\, d^3 p_1 \end{equation} is replaced by the corresponding Monte Carlo type integral \begin{equation} \langle O(t) \rangle = \frac{1}{N} \sum_{i=1}^{N} O(\vec r_i(t),\vec k_i(t)) \end{equation} where the $\vec r_i(t)$ and $\vec k_i(t)$ are the phase space coordinates of the N test particles propagated with the Hamilton equations. As said, they are distributed like $D(\x,\vec p_1,t)$. According to the theory of the Monte Carlo integration both integration procedures yield the same result in the limit of an infinite number of test particles. In practice one has to verify that the results do not depend on this number. Usually 100 test particles per physical nucleon in the system are considered as sufficient. It is very important to realize that these test particles have nothing to do with physical nucleons. They serve only as a representation of the one body Wigner density $D(\x,\vec p_1,t)$. All observables which require more than its knowledge are beyond the scope of applicability of these theories. Hence the possibility to extract source radii and hence correlation functions from the one body theories requires: \begin{itemize} \item The smoothness assumption is valid \item $D(\x,\vec p_1,{\vec{x}}_2,{\vec{p}}_2,t) \approx D(\x,\vec p_1,t)D({\vec{x}}_2,{\vec{p}}_2,t)$ \end{itemize} They are a consequence of the impossibility to create two body Wigner densities or Wigner densities of two body observables like $\vec p_1+\vec p_2$ from the swarm of test particles defined as above. If both approximation were valid the correlation function is given by \begin{eqnarray} {d^2W \over d\vec p_1 d\vec p_2} &= \int d^3x_1d^3x_2 D(\vec x_1,\vec p_1)D(\vec x_2,\vec p_2) (1 \pm cos(\vec p_1-\vec p_2)(\vec x_1-\vec x_2)) \nonumber \\ & = {1 \over N(N-1)}\sum_{i\neq j} (1\pm cos(\vec p_i(t_0)-\vec p_j(t_0))(\vec x_i(t_0)-\vec x_j (t_0))) \end{eqnarray} $t_0$ is the (assumed common) freeze out time. The second approximation, the absence of two body correlations, is hard to control. The importance of many particle correlations for the fragment formation has been discussed in \cite{go} but its relevance for the proton or pion emission has not yet been investigated. \section{Results for a given source distribution} To interpret the different results given above it is useful to apply them to a situation where the source is known. To keep the things simple we assume a completely chaotic source without any correlation between coordinate and momentum space: \begin{equation} S(\vec k_1,\vec r_1) = ({B\over A\pi^2})^{3/2}e^{-k_1^2B/2 -r_1^22/A}. \end{equation} We start out from the Wigner density (eq.26) for a pair of particles as given in the QMD simulation. Averaging over the Gaussian source distribution we obtain for the correlation function as in eq.5 \begin{equation} C(\vec p) ={\int {d^2d W \over d\vec p_1 d\vec p_2} d^3PS(\vec k,\vec r)d^3kd^3r \over \int { dW \over d\vec p_1 }{dW \over d\vec p_2}d^3PS(\vec k,\vec r)d^3kd^3r} = 1 \pm e^{-p^2(L+A-{LB\over L+B})} \end{equation} where $S(\vec k, \vec r) = \int S(\vec k_1,\vec r_1)\cdot S(\vec k_2,\vec r_2) d^3K d^3R$. If we apply the smoothness assumption (eq.6) we obtain \begin{equation} C^{SA}(\vec p) = {\int {d^2d W^{SA} \over d\vec p_1 d\vec p_2} d^3P S(\vec k,\vec r) d^3kd^3r \over \int { dW \over d\vec p_1 }{dW \over d\vec p_2}d^3PS(\vec k,\vec r) d^3kd^3r} = 1 \pm e^{- p^2 (L+A)}. \end{equation} If we assume as in BUU, VUU or LV that the one body Wigner density is not given by Gaussians but as a sum over test particles (TP) each represented by a delta function in coordinate and momentum space \begin{equation} D^{TP}(\vec x_1,\vec p_1,t) = {1 \over N}\sum_{\alpha =1}^N \delta (\vec x_1 - \vec x_\alpha(t)) \delta (\vec p_1 - \vec p_\alpha(t)) \end{equation} where the $p_\alpha $'s and $x_\alpha$ 's are distributed according to our source function we obtain as a correlation function \begin{equation} C^{TP}(\vec p) = 1 \pm e^{- p^2 A}. \end{equation} Defining the square of the source radii as $ {\int C(\vec p) d^3p \over \int C(\vec p) p^2d^3p}$ and comparing eqs. 38,39,41 we observe that for the same measured correlation function $C(\vec p)$ we obtain different source radii depending on the simulation programs and the approximations used. From a mathematical point of view the difference between $C^{TP}(\vec p)$ and $C^{SA}(\vec p)$ is easy to understand. Because the wave function used for the calculation of $C^{SA}(\vec p)$ has a width of L, the true distribution of the source is the convolution of the distribution of the centers given by $S(\vec r, \vec k)$ with the distribution of the one particle density around the centers. For $C^{TP}(\vec p)$ one assumes that the true source distribution is given by $S(\vec r, \vec k)$. In order to make both quantities comparable, both mean square radii have to be the same and hence one has to replace in $C^{TP}(\vec p)$ A by A'= L+A. However, being purely mathematical, this argument has an essential drawback. We have started out from the approximation (eq.1) that the source can be treated classically, and hence that the distance between two sources is large as compared to L \cite{Podgor72}. Hence, either the difference between A' and A is small and can be neglected. Then our approximation is valid. Or this difference is not negligible. Then our classical source approximation breaks down. That the wave function plays indeed a nontrivial role can be seen if one compares $C^{SA}(\vec p)$ and $C(\vec p)$. In both cases the same single particle wave function has been employed. The result for the correlation function is, however, different. Only if $L << A$ the difference between both is negligible. Hence our quantitative result confirms the well known qualitative argument that the smoothness assumption is only valid if the source can be assumed to be classical, i.e. if the width of the wave function is small as compared to the size of the emitting system. Opposite, if L is of the same order as A as in all presently employed simulation models, the difference becomes important as we will see below and hence the smoothness assumption will break down. Hence we are confronted with the fact that present day simulation programs use a value of L which neither justifies the classical treatment of the source nor confirms the validity of the smoothness assumption. Nevertheless it seems that the community has agreed upon a pragmatic point of view in pretending that at least the classical treatment of the source is acceptable in modeling heavy ion collisions although a proof has not be given yet. Hence it may be useful to see whether under this assumption a quantitative prediction is possible. This includes the answer to two questions: To what precision we desire to measure the density and is the systematic error of the correlation function sufficiently small to obtain the desired precision. The study of the space time correlation is born out of the demand to measure the size of the system at the moment when the particles are emitted. If we study nucleons of the fireball, the density has to be in between twice and half normal nuclear matter density, because if the expanding fireball passes the latter density, there are no interactions anymore and hence the emission of particles defined as the time point of the last collision has ceased. For nucleons emitted from the spectator matter which remains at normal nuclear matter density one would like to know the source size. At lower energies the interest is to study whether the emitted nucleons come from a compound nucleus or whether the they are emitted from a subsystem called hot spot. Also here the density varies little around normal nuclear matter density. Whereas in the first case an uncertainty of the density determination of about 20\% may be tolerable, the latter two require a precision of the determination if the source radius by about 3\% (and hence of the mass number of about 10\%) if one would like to avoid that the uncertainty is already as large as the possible variation of the size of the system under investigation. In order to see whether this precision can be obtained we have to calculate the values for A, L and B for the cases of interest. If we assume that the rms radius of the source corresponds to the size of a nucleus at normal nuclear matter density $R_0 = 1.2 A_M^{1/3}$ where $A_M$ is the atomic number of the nucleus we obtain \begin{equation} A = A_M^{2/3} [fm^2] \end{equation} i.e. A = 21.5 $fm^2$ for $A_M = 100$ and A = 34 $fm^2$ for $A_M = 200$. Because our source emits particles according to a Maxwell Boltzmann distribution we can relate the slope in momentum space with the temperature of the emitting source and find that \begin{equation} B = { 40 \over T [MeV]} [fm^2] \end{equation} In the standard versions of QMD resp. IQMD the parameter L has the value 4.33 and 8.66 $fm^2$, respectively. Hence first of all we observe that L is not at all negligible as compared to A. However, as mentioned above, accepting a classical treatment of the source we can correct for this. It remains to be seen whether the smoothness assumption can be justified. Comparing the mean square radii obtained with and without the smoothness assumption \begin{eqnarray} F={R\over R^{SA}} &=& {(L+A-{LB\over L+B})\over (L+A)} \nonumber \\ &=& 1 - {LB\over (L+B)(L+A)} \end{eqnarray} we find that the smoothness assumption pretends a larger radius of the system. The value of F ranges between .87 for small systems at low temperature (5 MeV) and .98 for large systems at high temperature (80 MeV). Hence for particles emitted from a compound nucleus or from the spectator matter the error in the determination of the mass number due to the smoothness approximation is of the order of 20\% even if the source is completely chaotic and of known form and the classical approximation of the source remains valid. For fireball nucleons the smoothness assumption produces an error of about 4\% on the density. Of course if we were sure that we have a source of a given temperature we could also correct for the temperature, however such a source is not encountered in heavy ion physics where the excitation energy and hence the temperature changes in the course of time. \section{Realistic Simulations} We have seen that for the most favourable condition (chaotic source of known form without any momentum space coordinate space correlation) the smoothness assumption enlarges the apparent source size by about 20\%. If one applies now the simulation programs to real experiments one has to inspect the consequences of two facts: 1) Nature most probably does not keep the rms radius of a nuclear wave function constant during a heavy ion reactions, QMD does. For observables which do not depend on the width of the wave function explicitly this may be of minor importance, the influence on observables which depend explicitly on the width, like the correlation function, is hard to judge since no calculations are available for a more sophisticated treatment of the reaction as done in QMD. If the width of the wave function has changed in the course of the reaction the difference between $C^{TP}(\vec p)$ and $C^{SA}(\vec p)$ cannot be corrected anymore by use of the known initial density distribution. In the QMD calculations the width of the wave function L serves two purposes. First it is used to have the proper one body density distribution when one initialize the nuclei. This is, however, a very weak condition because with much larger widths than that actually employed one can obtain the same one particle density distribution. Second, it appears in the time evolution equations but only in form of the expectation value of the potential. Thus what counts for the time evolution is the convolution of the potential range and the width of the wave functions. Hence one can obtain the same expectation value of the potential for a smaller width and a larger potential range. Hence there is no need for an exact determination of the width L in the QMD calculation or, vice versa, the success of these calculations cannot be used to determine L. 2) The source is as simulation programs show not at all chaotic and shows strong correlations between momenta and positions. Momentum space coordinate space correlations decrease the source size extracted from the correlation function as compared to the geometrical size of the source. This can be easily understood if one goes to the extreme. If the momentum is a monotonic function of the position, two particles with a small relative momentum have to come from the places very close in coordinates space. Thus the correlation function measures only that region in coordinate space from where these particles can come. Hence the stronger the momentum space coordinate space correlations are the smaller is the source size measured by the correlation function. As a consequence the value of A becomes smaller and the importance of the width of the wave function increases. Thus the stronger these correlations are the larger becomes the difference between $R^{SA}$ and R (eq. 44). Thus for realistic calculations the situation becomes worse as compared to a static source. For a given size of the system correlations make A smaller and hence increase the importance of the width of the wave function if one compares eqs. 39 and 41 . They also do not allow to corrected for the smoothness assumption (eqs. 38 and 39) because the temperature is not anymore a global variable. The calculation of the systematic error of the value of the radius determined by eqs. 39 or 41 requires more than the present models can predict, however the above arguments show that it will be larger than that for a static source. \section{conclusion} We have discussed the possibility to extract source radii by comparing the experimental results with the prediction of simulation programs. There is no doubt that the experimental results indeed show momentum space correlations caused by the bosonic or fermionic nature of the observed hadrons. These correlations carry information about the space time structure of the reaction. The goal to relate the observed correlation functions in momentum space with physical parameters in coordinate space like source radii, densities or energy densities can presently only achieved by use of transport theories. None of these transport models takes the bosonic or fermionic nature of the hadrons into account. Whereas this may be no essential drawback for many observables it makes it impossible to calculate the correlation function in a straight forward manner. Every model requires the introduction of the (anti)symmetrization of the wave function in an ad hoc fashion in order to predict a correlation function. We have found that for all presently existing models, which can be subdivided into three classes, this introduction poses problems. Cascade models, in which classical particles are propagated, do not allow the calculation of a correlation function. The quoted values of source radii are totally artificial being a consequence of the employed approximation and not of physical origin. QMD, LV, BUU and VUU models allow the calculation of a correlation function. We find, however, that the basic approximation of the whole approach, namely that the source can be considered as classical, is not fulfilled. Even if it were fulfilled, the systematic error of the extracted density introduced by the smoothness approximation is for the most favourable case of a chaotic source of known form up to 20\%. For realistic cases where space momentum space correlations are present and where we do not know the form of the source we have shown that the error will increase. This questions the possibility that in nuclear physics the HBT method allows a determination of the density to a precision which allows to discriminate between different proposed interaction mechanisms. Of course this raises the question how to proceed. As we have seen we are plagued with systematic errors of the order L/A. There is first of all the open question whether the wave function of a emitted nucleon is smaller than L. Mean field calculation yield a much broader wave function and consequently the approximation of a classical source cannot be justified any more. Short range correlations, however, may distort this wavefunction. Hence it may be justified to address the question if there is a possibility to construct dynamical theories which can provide a prediction for the correlation function? Either one can try to decrease L or to avoid the systematic errors. The first suggestion implies a localization of the particles with a precision of about 1fm. This will be hardly possible. Not only because in a nuclear environment the root mean square radius of the wave function of the nucleon is considerably larger than the radius of a free nucleon but also because it implies an uncertainty of 200 MeV/c for the momentum of the nucleons which poses several severe technical problems for transport theories: \begin{itemize} \item How to propagate particles in semiclassical theories whose velocity uncertainty is about 0.2c is unknown. \item The sequence of collisions becomes undetermined \item The applied scattering cross sections have to be modified because the scattering partners are asymptotically not in a plane wave state. \end{itemize} The second suggestion implies the construction of transport theories which propagates at least two particle wave functions and not parameters of the wave function. Presently such an approach is not available. Before a solution to these problems has been found the Hanbury Brown Twiss effect is a very nice quantal effect. Its application in nuclear physics to study the space time structure remains, however, premature. Interesting discussions with Drs. Ardouin, Erazmus, Gyulassy, Heinz, Lednicky and Werner are gratefully acknowledged. Furthermore I would like to thank Dr. Heinz for a careful reading of the manuscript.

proofpile-arXiv_065-599