---
abstract: 'The quark condensate in nuclear matter contains a term of order $\rho m_\pi$, arising from the contribution of low-momentum virtual pions to the $\pi N$ sigma commutator.  Standard treatments of QCD sum rules for a nucleon in matter generate a similar term in the nucleon effective mass, although this is inconsistent with chiral perturbation theory.  We show how an improved treatment of pionic contributions on the phenomenological side of the sum rules can cancel out this unwanted piece.  Our results also show that factorisation ansatz for the four-quark condensate cannot be valid in matter.' 
address: |
    Theoretical Physics Group, Department of Physics and Astronomy\
    University of Manchester, Manchester, M13 9PL, UK\
author:
- 'Michael C. Birse and Boris Krippa[^1]'
title: |
    Chiral symmetry and QCD sum rules\
    for the nucleon mass in matter
---

The presence of dense nuclear matter could significantly influence the QCD vacuum, for example by partially restoring the chiral symmetry of QCD.  There is much interest at present in elucidating the effects this might have on the properties of nucleons and other hadrons in matter[@csrevs].  In this context, the method of QCD sum rules[@svz] provides a promising approach to connecting hadron properties with underlying QCD matrix elements.  Applications of this to the energy of a nucleon in matter[@nmsr] have shown that changes to the condensates in matter can give rise to large scalar and vector self-energies that are similar to those in relativistic nuclear phenomenology[@dirph]. 

A major ingredient in QCD sum rules for the nucleon mass is the scalar quark condensate.  To first order in the density the change in this condensate in matter is given by the model-independent result[@dl; @cfg] $${\langle\Psi|\overline qq|\Psi\rangle\over\langle 0|\overline qq|0\rangle}
=1-{\sigma_{\pi{\scriptscriptstyle N}}\over f_\pi^2 m_\pi^2}\rho,
\label{ratio}$$ where where $|\Psi\rangle$ denotes our nuclear matter ground-state and $\sigma_{\pi{\scriptscriptstyle N}}$ is the pion-nucleon sigma commutator.  The leading nonanalytic dependence of the sigma commutator on the current quark mass $\overline m$ is given by a term of order $\overline m^{3/2} \propto
m_\pi^3$[@gl].  This arises from the long-ranged part of the pion cloud of the nucleon. 

The leading correction term in the chiral expansion of the density-dependent part of the quark condensate in matter is thus of order $\rho m_\pi$. However chiral counting rules[@wein] tell us that no term of order $m_\pi$ appears in the $NN$ interaction[@orvk] and hence that no term of this order should be present in the self-energy of a nucleon in matter.  As one of us has pointed out recently[@birqq], the nucleon mass in matter thus cannot simply be a function of the quark condensate alone.  Other long-ranged effects must contribute to ensure that no term of order $\rho m_\pi$ appears.  This presents a potential problem for QCD sum rules for a nucleon in matter, since the operator-product expansion (OPE) involves matrix elements of local operators like $\overline qq$ which cannot distinguish between short- and long-ranged contributions.  The cancellation of the parts of these condensates that are of order $\rho m_\pi$ must therefore come from the phenomenological side of the sum rules, and in particular from states involving low-momentum pions.  Similar cancellations have been found previously in sum rules for the mass of a nucleon at finite temperature[@el; @koi] and in vacuum[@lccg]. 

Sum rules are obtained by matching two expressions for a correlator of interpolating fields with the quantum numbers of the particle of interest.  The first of these is an OPE of the correlator at large (space-like) momentum which introduces information about the nonperturbative features of the QCD ground state (either the vacuum or matter) through the various condensates that appear in it.  The second takes the form a dispersion relation based on a spectral representation of the correlator, for which one makes a simple phenomenological model.  This representation contains a pole corresponding to the hadron state whose properties one is interested in, as well as a continuum of higher-energy states with the same quantum numbers.  Standard sum rule analyses assume that the continuum below some threshold energy can be neglected.  They include only a high-energy continuum, which is modelled by the QCD continuum from the OPE.  Current applications of the method to a nucleon in matter thus give rise to an effective mass containing a term of order $\rho
m_\pi$, produced by the corresponding term in the quark condensate.  We show here how, with a better treatment of the continuum, this term can be cancelled by contributions of the same order from states with low-momentum pions. 

The leading nonanalytic term in the sigma commutator, which we denote by $\sigma_{\pi{\scriptscriptstyle N}}^{(3)}$, has the form[@gl] $$\sigma_{\pi{\scriptscriptstyle N}}^{(3)}=-{9\over 16\pi}\left({g_{\pi 
{\scriptscriptstyle NN}}\over 2 M_{\scriptscriptstyle N}}\right)^2m_\pi^3.$$ It is produced by the lowest-momentum virtual pions in the cloud surrounding the nucleon.  Since these are very-low momentum pions, their interactions with a second nucleon are suppressed by two powers of $m_\pi$ relative to their contributions to the scalar quark density ($\sigma_{\pi{\scriptscriptstyle
N}}^{(3)}/m_\pi^2$) and so the leading nonanalytic terms in the $NN$ interaction are of order $m_\pi^3$[@wein; @orvk].  At mean-field level, these terms in the interaction give rise to leading nonanalytic terms in the self-energy of a nucleon in matter of order $\rho m_\pi^3$ and not $\rho
m_\pi$.

One can relate $\sigma_{\pi{\scriptscriptstyle N}}^{(3)}$ to the leading nonanalytic contribution of the pion mass term to the nucleon mass (cf.[@bmcg]) $${\textstyle{1\over 2}}m_\pi^2\langle N(\hbox{\bf p})|
:\hbox{\boldmath$\pi$}(0)^2:|N(\hbox{\bf p}')\rangle\simeq
(2\pi)^{-3}\sigma_{\pi{\scriptscriptstyle N}}^{(3)}+\cdots,
\label{signonan}$$ where [$\pi$]{}([**r**]{}) is the pion field.[^2] There should, in principle, also be a term of order $m_\pi^2$ in this matrix element.  This has been “renormalised away" by lumping it with other terms of the same order, all of which contribute to the quark condensate at order $m_\pi^0$. We can therefore to relate the ratio in (\[ratio\]) to its value in the chiral limit $\overline m=0$ by $${\langle\Psi|\overline qq|\Psi\rangle\over\langle 0|\overline qq|0\rangle}
\simeq\left.{\langle\Psi|\overline qq|\Psi\rangle\over\langle 0|\overline qq
|0\rangle}\right|_{\overline m=0}(1-\xi),
\label{qqlc}$$ where we have defined $$\xi={\rho\sigma_{\pi{\scriptscriptstyle N}}^{(3)}\over f_\pi^2 m_\pi^2}.$$

Sum rules for a nucleon in matter are obtained from the correlator of two interpolating nucleon fields[@iof] $$\Pi(p)=i\!\int d^4x e^{ip\cdot x}\langle \Psi|T\{\eta(x)\,\overline{\eta}(0)\}
|\Psi\rangle. 
\label{correl}$$ Since we are interested in the leading chiral corrections to this, we need to examine the piece of this correlator that is of order $\rho m_\pi$. These corrections arise from pieces of the correlator where the interpolating fields couple directly to a low-momentum pion from the cloud of one of the nucleons forming the matter.  A convenient way to obtain these terms is to use partial conservation of the axial current (PCAC)[@dgh] in the way suggested by Eletsky[@el] (see also[@dei; @koi]).  The main difference is that in the present case we are dealing with virtual pions and so all possible time-orderings need to be considered. 

To the order we are interested, the piece of the correlator where the “probe" nucleon created by the interpolating fields interacts with a pion that is emitted and then absorbed by the matter can be written in the form $$\sum_{a,b}\int {d^3\hbox{\bf k}\over 2\omega_k}
{d^3\hbox{\bf k}'\over 2\omega_{k'}}\langle\Psi|a^{a\dagger}(\hbox{\bf k})
a^b(\hbox{\bf k}')|\Psi\rangle
\,i\!\int d^4x e^{ip\cdot x}\langle \Psi\,\pi^a(\hbox{\bf k})|T\{\eta(x)\,
\overline{\eta}(0)\}|\Psi\,\pi^b(\hbox{\bf k}')\rangle,
\label{piinout}$$ similar to that needed in the case of thermal correction.  In addition, we need the other time-orderings where two pions are either emitted or absorbed by the matter.  These are, respectively,[^3] $${1\over 2}\sum_{a,b}\int {d^3\hbox{\bf k}\over 2\omega_k}
{d^3\hbox{\bf k}'\over 2\omega_{k'}}\langle\Psi|a^a(\hbox{\bf k})
a^b(\hbox{\bf k}')|\Psi\rangle
\,i\!\int d^4x e^{ip\cdot x}\langle \Psi|T\{\eta(x)\,\overline{\eta}(0)\}
|\Psi\,\pi^a(\hbox{\bf k})\pi^b(\hbox{\bf k}')\rangle,
\label{twopiin}$$ and $${1\over 2}\sum_{a,b}\int {d^3\hbox{\bf k}\over 2\omega_k}
{d^3\hbox{\bf k}'\over 2\omega_{k'}}\langle\Psi|a^{a\dagger}(\hbox{\bf k})
a^{b\dagger}(\hbox{\bf k}')
|\Psi\rangle
\,i\!\int d^4x e^{ip\cdot x}\langle \Psi\,\pi^a(\hbox{\bf k})\pi^b(\hbox{\bf 
k}')|
T\{\eta(x)\,\overline{\eta}(0)\}|\Psi\rangle
\label{twopiout}$$ where $\omega_k=\sqrt{\hbox{\bf k}^2+m_\pi^2}$.

Note we are ignoring pieces of the correlator where the pions transfer nonzero momentum to the probe nucleon and that momentum is transfered back by another interaction while the nucleon is propagating through the matter.  Such terms give rise to two-loop diagrams, with two integrals over pion momenta, and so are of higher-order in the chiral expansion.  This has allowed us to write the matrix elements in the factorised forms (\[piinout\],\[twopiin\],\[twopiout\]), with the ground state of nuclear matter $|\Psi\rangle$ appearing in in all of them. 

For the leading chiral correction we need only the contributions of the lowest-momentum pions in the cloud, and hence we may approximate the matrix elements of the interpolating fields in the expressions above by their values in the soft-pion limit[@el].  In this limit we can use the soft-pion theorem of PCAC[@dgh] to rewrite (\[piinout\]) in terms of $$i\int d^4x e^{ip\cdot x}\langle \Psi\,\pi^a(\hbox{\bf k})|T\{\eta(x)\,
\overline{\eta}(0)\}|\Psi\,\pi^b(\hbox{\bf k}')\rangle
\simeq {-i\over f_\pi^2}\int d^4x e^{ip\cdot x}\langle \Psi|\left[Q_5^a,
\left[Q_5^b, T\{\eta(x)\,\overline{\eta}(0)\}\right]\right]|\Psi\rangle,
\label{soft}$$ while the matrix elements in both (\[twopiin\]) and (\[twopiout\]) also reduce to the same expression.  If we use Ioffe’s choice of interpolating field for the nucleon[@iof], this transforms under axial isospin rotations as[@el] $$\left[Q_5^a, \eta(x)\right]=-\gamma_5{\tau^a\over 2}\eta(x). 
\label{fchtr}$$ The matrix element of the double commutator in (\[soft\]) can then be reduced to $$i\int d^4x e^{ip\cdot x}\langle \Psi\,\pi^a(\hbox{\bf k})|T\{\eta(x)\,
\overline{\eta}(0)\}|\Psi\,\pi^b(\hbox{\bf k}')\rangle
\simeq-{1\over f_\pi^2}\delta^{ab}
{1\over 2}\Bigl(\Pi(p)+\gamma_5\Pi(p)\gamma_5\Bigr). 
\label{soft2}$$

The full expression for the leading correction to the correlator can be obtained by adding up (\[piinout\],\[twopiin\],\[twopiout\]), using the soft-pion approximation (\[soft2\]).  Making use of the expansion of the pion field, $$\pi^b(\hbox{\bf r})={1\over(2\pi)^{3/2}}\int {d^3\hbox{\bf k}
\over 2\omega_k}\left(a^b(\hbox{\bf k})\exp(i\hbox{\bf k}\cdot\hbox{\bf r})
+a^{b\dagger}(\hbox{\bf k})\exp(-i\hbox{\bf k}\cdot\hbox{\bf r})\right),$$ the sum of these contributions can be conveniently expressed in the form $$(2\pi)^3\langle\Psi|:{\textstyle{1\over 2}}\hbox{\boldmath$\pi$}
(0)^2:|\Psi\rangle\left(-{1\over f_\pi^2}\right)
{1\over 2}\Bigl(\Pi(p)+\gamma_5\Pi(p)\gamma_5\Bigr).$$ Comparing this with the expression for the leading nonanalytic term in the sigma commutator (\[signonan\]) above, we see that to first order in the density the pionic matrix element is $$\begin{aligned}
(2\pi)^3\langle\Psi|:{\textstyle{1\over 2}}\hbox{\boldmath$\pi$}
(0)^2:|\Psi\rangle&\simeq&\rho(2\pi)^3\langle N|:{\textstyle{1\over 2}}
\hbox{\boldmath$\pi$}(0)^2:|N \rangle\nonumber\\
&\simeq&{\rho\sigma_{\pi{\scriptscriptstyle N}}^{(3)}\over m_\pi^2}.\end{aligned}$$ Hence we may write the chiral expansion of the correlator (\[correl\]) to the order of interest, ${\cal O}(\rho m_\pi)$, as $$\Pi(p)\simeq\overcirc\Pi(p)-{\xi\over 2}\Bigl(\overcirc\Pi(p)+\gamma_5 
\overcirc\Pi(p)\gamma_5\Bigr),
\label{correxp}$$ where $\overcirc\Pi(p)$ denotes the correlator in the chiral limit. 

The nucleon correlator in matter (\[correl\]) can be decomposed into three terms with diferent Dirac matrix structures[@nmsr], $$\Pi(p)=\Pi^{(m)}(p)+\Pi^{(p)}(p)p\llap/+\Pi^{(u)}(p)u\llap/,$$ where $u^\mu$ is a unit four-vector specifying the rest-frame of the matter.  From this we can see that only the piece $\Pi^{(m)}(p)$, corresponding to the Dirac unit matrix, contributes to the ${\cal O}(\rho m_\pi)$ correction in (\[correxp\]).  Hence to this order, the three pieces of the correlator are $$\begin{aligned}
\Pi^{(m)}(p)&\simeq&(1-\xi)\overcirc \Pi^{(m)}(p),\nonumber\\
\Pi^{(p)}(p)&\simeq&\overcirc \Pi^{(p)}(p),\label{cpexp}\\
\Pi^{(u)}(p)&\simeq&\overcirc \Pi^{(u)}(p).\nonumber\end{aligned}$$

The sum rules are obtained by equating the OPE and phenomenogical representations of the correlator.  To demonstrate the cancellation of the leading chiral corrections to the nucleon energy, we need to examine the chiral expansion of the pole and continuum pieces separately, as well as that of the OPE.  Consider first the OPE of the correlator.  From (\[cpexp\]) we see that the leading correction to each condensate in the OPE of $\Pi^{(m)}(p)$ takes the form of a factor of $1-\xi$, as for the quark condensate in (\[qqlc\]).  This also follows from the fact that all these condensates are matrix elements of operators that transform like $\overline
qq$ under chiral isospin rotations.  In contrast the OPE’s of $\Pi^{(p,u)}(p)$ have no corrections to this order, reflecting the fact that they involve chirally invariant operators, such as $G_{\mu\nu}G^{\mu\nu}$ or $q^\dagger q$ respectively[@nmsr].  This common form for the leading corrections to all the terms with a given Dirac structure was also noted in[@lccg] for nucleon sum rules in vacuum. 

By inserting a complete set of states with nucleon quantum numbers in the correlator $\overcirc \Pi(p)$ we can break (\[correxp\]) up into chiral-limit nucleon-pole and continuum pieces: $$\Pi(p)\simeq\left(1-{\xi\over 2}\right)\overcirc\Pi_{pole}(p)
-{\xi\over 2}\gamma_5 \overcirc\Pi_{pole}(p)\gamma_5
+\left(1-{\xi\over 2}\right)\overcirc\Pi_{cont}(p)
-{\xi\over 2}\gamma_5 \overcirc\Pi_{cont}(p)\gamma_5. 
\label{corrpcexp}$$ To interpret the various pieces of this expression in terms of the pole and continuum parts of the correlator away from the chiral limit, we need to look again at the double commutator in (\[soft\]).  Expanding this out gives four terms $$\begin{aligned}
\langle \Psi|\Bigl[Q_5^a,\Bigl[Q_5^b, T\{\eta(x)&&
\,\overline{\eta}(0)\}\Bigr]\Bigr]|\Psi)\rangle\label{dcomm}\\
=&&\langle \Psi|T\Bigl\{\left[Q_5^a,\left[Q_5^b,\eta(x)\right]\right]
\overline{\eta}(0)
+\eta(x)\Bigl[Q_5^a,\Bigl[Q_5^b,\overline{\eta}(0)\Bigr]\Bigr]
\nonumber\\
&&\qquad\quad+\Bigl[Q_5^a,\eta(x)\Bigr]\Bigl[Q_5^b,\overline{\eta}(0)
\Bigr]+\Bigl[Q_5^b,\eta(x)\Bigr]\Bigl[Q_5^a,\overline{\eta}(0)\Bigr]
\Bigr\}|\Psi\rangle.\nonumber\end{aligned}$$

The first two terms contain double commutators of axial charges with the same interpolating nucleon field.  Applying the commutator (\[fchtr\]) twice shows that these give rise to the correction term without factors of $\gamma_5$ in (\[correxp\]).  If we insert a complete set of states into these terms expressed in the form (\[dcomm\]) then we can see that they involve matrix elements such as $$\langle \Psi|\eta(x)|\Psi N(\hbox{\bf p})\rangle\langle\Psi N(\hbox{\bf p})|
\Bigl[Q_5^a,\Bigl[Q_5^b,\overline{\eta}(0)\Bigr]\Bigr]|\Psi\rangle,$$ which comes from the soft-pion limit of $$\langle \Psi|\eta(x)|\Psi N(\hbox{\bf p})\rangle\langle\Psi N(\hbox{\bf p})|
\overline{\eta}(0)|\Psi\,\pi^a(\hbox{\bf k})\pi^b(\hbox{\bf k}')\rangle.$$ where $N(\hbox{\bf p})$ in these states denotes the probe nucleon created by the interpolating fields.  These terms are thus the soft-pion limits of the pieces of (\[piinout\],\[twopiin\],\[twopiout\]) where both pions couple to the same interpolating field.  They contribute to the residue of the nucleon pole in (\[correl\]) but do not alter its position, and hence they are corrections in matter to the strength with which the interpolating field couples to the nucleon.  If we apply the same procedure that we used to get from (\[piinout\],\[twopiin\],\[twopiout\]) to (\[correxp\]) to the pole terms alone, we obtain $$\Pi_{pole}(p)\simeq\left(1-{\xi\over 2}\right)\overcirc\Pi(p)_{pole}.$$ Note that these terms provide the correction to the pole term in (\[correxp\]) without factors of $\gamma_5$.

The second chiral-limit pole term in (\[correxp\]), with $\gamma_5$’s, is in fact part of the continuum of the correlator away from the chiral limit.  Like the similar term in the expression for the vacuum correlator[@lccg], this arises from the last two terms in (\[dcomm\]), which are the soft-pion limits of terms such as $$\langle \Psi|\eta(x)|\Psi N(\hbox{\bf p})\pi^b(\hbox{\bf k}')\rangle
\langle\Psi N(\hbox{\bf p})\pi^a(\hbox{\bf k})|\overline{\eta}(0)
|\Psi\rangle,$$ where one pion couples to each interpolating field.  Their form shows that they arise from the lowest-energy part of the $\pi N$ continuum where the pion scatters from one of the nucleons in the matter. 

Hence to the order of interest, the phenomenological representation of the correlator can be written $$\Pi(p)\simeq\Pi_{pole}(p)-{\xi\over 2}\gamma_5\Pi_{pole}(p)\gamma_5
+\left(1-{\xi\over 2}\right)\overcirc\Pi_{cont}(p)
-{\xi\over 2}\gamma_5 \overcirc\Pi_{cont}(p)\gamma_5,
\label{corrphen}$$ where the last three terms form the continuum piece.  For a nucleon in the presence of mean scalar and vector potentials, the pole term of the correlator in matter has the form[@nmsr] $$\Pi_{pole}(p)=-\lambda^{*2}{p\llap/+M^*+V\gamma_0\over 2E(\hbox{\bf p})
[p^0-E(\hbox{\bf p})]},
\label{pole}$$ where we have chosen to work in the rest frame of the matter.  In (\[pole\]) $M^*$ is the nucleon mass modified by the scalar potential, $V$ is the vector potential and $$E(\hbox{\bf p})=\sqrt{\hbox{\bf p}^2+M^{*2}}+V.$$ Here $\lambda^*$ denotes the strength with which the interpolating field couples to the nucleon in matter. 

The sum rules are obtained by equating the pieces of (\[corrphen\]) with the corresponding ones of the OPE and then integrating over $p$ with some convenient weighting function, such as the Borel transform[@svz; @nmsr].  For the nucleon in matter, this gives three sum rules corresponding to the three Dirac structures: $$\begin{aligned}
-\left(1-{\xi\over2}\right)\lambda^{*2}M^*\int d^4p\,{w(p)\over 
2E(\hbox{\bf p})[p^0-E(\hbox{\bf p})]}&\simeq&(1-\xi)\int d^4p\, w(p)\left[
\overcirc\Pi^{(m)}_{OPE}(p)-\overcirc\Pi^{(m)}_{cont}(p)\right],\nonumber\\
-\left(1+{\xi\over2}\right)\lambda^{*2}\int d^4p\,{w(p)\over 2E(\hbox{\bf p})
[p^0-E(\hbox{\bf p})]}&\simeq&\int d^4p\, w(p)\left[\overcirc\Pi^{(p)}_{OPE}(p)
-\overcirc\Pi^{(p)}_{cont}(p)\right],\label{sumrules}\\
-\left(1+{\xi\over2}\right)\lambda^{*2}V\int d^4p\,{w(p)\over 2E(\hbox{\bf p})
[p^0-E(\hbox{\bf p})]}&\simeq&\int d^4p\, w(p)\left[\overcirc\Pi^{(u)}_{OPE}(p)
-\overcirc\Pi^{(u)}_{cont}(p)\right].\nonumber\end{aligned}$$ By taking ratios of these sum rules one can see that to order $\rho m_\pi$ there are no chiral corrections to either $M^*$ or $V$. The self-energy of a nucleon in matter thus contains no term of this order, in agreement with the chiral counting rules[@wein; @birqq]. 

We note in passing that the form of the cancellation between the pieces of this order in the condensates on the OPE side of the sum rules with pieces from the phenomenological side is the same as that for the leading chiral corrections to the sum rule in vacuum[@lccg] and the order $T^2$ pieces at finite temperature[@el; @koi].  This reflects the fact that all of these contributions are produced by low-momentum pions, whose interactions with the probe nucleon are subject to the same constraints due to chiral symmetry. 

The leading nonanalytic term in the sigma commutator $\sigma_{\pi{\scriptscriptstyle N}}^{(3)}$ is about $-25$ MeV, while the full value is $\sigma_{\pi{\scriptscriptstyle N}} =45\pm 7$ MeV[@gls].  The effects discussed here are thus significant on the scale of the change in the quark condensate in matter given by (\[ratio\]).  In particular, the factor of $1-\xi$ in the unit-matrix piece of the OPE, as in (\[cpexp\]), is thus $\sim 1.2$. If one keeps the terms of this order on the phenomenological side that cancel this out, as in (\[sumrules\]), then one would expect to see a significant additional decrease in the nucleon mass, on top of that already found in[@nmsr].  However one should remember that the size of the decrease in the mass is also sensitive to assumptions about other condensates, in particular the four-quark condensate[@nmsr]. 

Finally, our results show that the factorisation ansatz often used for the four-quark condensate cannot be valid in matter.  That condensate, usually denoted $\langle\Psi|(\overline qq)^2|\Psi\rangle$, appears in the OPE of $\Pi^{(p)}(p)$ and so is a matrix element of a chirally invariant operator.  Replacing it by the factorised expression $\langle\Psi|\overline qq|\Psi\rangle^2$, which is not chirally invariant, introduces a term of order $\rho m_\pi$. Such a term is inconsistent with the chiral expansion (\[cpexp\]) of the correlator and cannot be cancelled by pionic contributions from the phenomenological side of the correlator in the way that we have seen happens for the ${\cal O}(\rho m_\pi)$ term in the two-quark condensate. 

Acknowledgements {#acknowledgements .unnumbered}
================

We are grateful to J. A. McGovern for a critical reading of the manuscript.  This work was supported by the EPSRC and PPARC. 

[999]{} C. Adami and G. E. Brown, Phys.  Reports [**224**]{} (1993) 1;\
M. C. Birse, J. Phys.  G: Nucl.  Part.  Phys.  [**20**]{} (1994) 1537;\
G. E. Brown and M. Rho, hep-ph/9504250, to be published in Phys.  Reports.  M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl.  Phys.  [**B147**]{} (1979) 385, 448, 519.  T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys.  Rev.  Lett.  [**67**]{} (1991) 961;\
T. D. Cohen, D. K. Griegel and R. J. Furnstahl, Phys.  Rev. [**C46**]{} (1992) 1507;\
X. Jin, T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys.  Rev. [**C47**]{} (1993) 2882;\
X. Jin, M. Nielsen, T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys.  Rev. [**C49**]{} (1995) 364.  B. D. Serot and J. D. Walecka, Adv.  Nucl.  Phys.  [**16**]{} (1986) 1;\
S. J. Wallace, Annu.  Rev. Nucl.  Part.  Sci.  [**37**]{} (1987) 267.  E. G. Drukarev and E. M. Levin, Nucl.  Phys.  [**A511**]{} (1990) 679; [**A516**]{} (1990) 715(E).  T. D. Cohen, R. J. Furnstahl and D. K. Griegel, Phys.  Rev. [**C45**]{} (1992) 1881.  J. Gasser, Ann.  Phys.  [**136**]{} (1981) 62;\
J. Gasser and H. Leutwyler, Phys.  Reports [**87**]{} (1982) 77.  S. Weinberg, Phys.  Lett.  [**251**]{} (1990) 288; Nucl.  Phys.  [**B363**]{} (1991) 1; Phys.  Lett.  [**B295**]{} (1992) 114.  C. Ordoñez, L. Ray and U. van Kolck, Phys.  Rev. Lett.  [**72**]{} (1994) 1982; hep-ph/9511380.  M. C. Birse, University of Manchester preprint MC/TH 96/09, hep-ph/9602266.  V. L. Eletsky, Phys.  Lett.  [**B245**]{} (1990) 229.  Y. Koike, Phys.  Rev. [**D48**]{} (1993) 2313.  S. H. Lee, S. Choe, T. D. Cohen and D. K. Griegel, Phys.  Lett.  [**B348**]{} (1995) 263.  M. C. Birse and J. A. McGovern, Phys.  Lett.  [**B292**]{} (1992) 242;\
I. Jameson, A. W. Thomas and G. Chanfray, J. Phys.  G: Nucl.  Part.  Phys.  [**18**]{} (1992) L159.  B. L. Ioffe, Nucl.  Phys.  [**B188**]{} (1981) 317; [**B191**]{} 591(E).  J. F. Donoghue, E. Golowich and B. R. Holstein, [*Dynamics of the standard model*]{} (Cambridge University Press, Cambridge, 1992).  M. Dey, V. L. Eletsky and B. L. Ioffe, Phys.  Lett.  [**B252**]{} (1990) 620.  J. Gasser, H. Leutwyler and M. E. Sainio, Phys.  Lett.  [**B253**]{} (1991) 252, 260. 

[^1]: Permanent address: Institute for Nuclear Research of the Russian Academy of Sciences, Moscow Region 117312, Russia. 

[^2]: The nucleon states here are normalised so that $\langle N(\hbox{\bf p})|N(\hbox{\bf p}')\rangle
    =(E_p/M)\delta(\hbox{\bf p}-\hbox{\bf p}')$. Only the leading term in powers of $m_\pi$ and momenta has been written out in (\[signonan\]). 

[^3]: In these expressions we have used a covariant normalisation for the pions such that $\langle
    \pi^a(\hbox{\bf k})|\pi^b(\hbox{\bf k}')\rangle=2\omega_k\delta^{ab}
    \delta(\hbox{\bf k}-\hbox{\bf k}')$.
