#### To the Annual Report of the 02/23/99

#### WSU Nuclear Science Group V. L. Rykov

#### Search for CP-violation and new physics beyond SM in polarized proton collisions at RHIC

The measurable asymmetries due to physics beyond SM, including CP-
and/or T-violation, in charged and neutral current leptoproduction at
polarized colliders are discussed.

#### Introduction

Just in months ahead, the first high luminosity collisions of two
polarized proton beams are expected to occur at RHIC in BNL at
$\sqrt{s}$ up to 500 GeV, bringing a new quality to the collider
physics. In the collisions of polarized particles, the presence of two
axial vectors of initial polarizations, fully controlled by
experimenters, may dramatically increase the number of available for
test correlations between participating vectors, generating asymmetries,
which could relatively easily be measured. In frame of Standard Model
(SM), many of these asymmetries are either strongly suppressed or
strictly prohibited. Therefore, if some of them were found nonzero, this
could be a strong indication of new physics beyond SM.

In this paper we will show a few examples of measurable asymmetries for
hadronic leptoproduction via vector $W^{\pm}$- and $Z^{0}$-boson
exchange:

> $q\overline{q} \rightarrow W^{\pm} \rightarrow l\overline{l}$ and
> $q\overline{q} \rightarrow Z^{0} \rightarrow l\overline{l}$ (1)

which are probably the cleanest processes to search for new physics
beyond SM at RHIC energies. If certain criteria met, it might be
difficult to explain the observed nonzero correlations in theories
without CP- and/or T-violation at the energy scale of hundreds of GeV,
just above the *W*- and *Z*-mass range. At this energy scale, the
CP/T-violating mechanism behind these correlations could virtually be
anywhere and everywhere \[1,2\].

1.  **Phenomenological model and asymmetries**

In the lowest order, processes (1) are represented by the usual
*s-*channel annihilation graphs. The CP- and/or T-violation may occur in
either or both of two vertices. In this report, for the sake of
simplicity, the presence of CP-odd phases has been assumed in the quark
sector only, and the usual (*V-A*)-interactions were kept for lepton
coupling to gauge bosons. To carry out some particular calculations, the
phenomenological lagrangian \[2\] has been used. For the charged current
$q_{u}q_{d}W$-coupling, this lagrangian looks as follows:

$L_{c} = \frac{g}{2\sqrt{2}}\left\{ W_{\mu}^{-}{\overline{q}}_{d}\gamma^{\mu}(h_{V}^{+} + h_{V}^{-}\gamma_{5})q_{u} + \frac{1}{\Lambda}\lbrack\partial_{\nu}W_{\mu}^{-}{\overline{q}}_{d}\sigma^{\text{μν}}(h_{T}^{+} + h_{T}^{-}\gamma_{5})q_{u}\rbrack + h\text{.}c\text{.} \right\}$
(2)

where *g* is a coupling constant, presumably on the order of the
electroweak one, and Λ is the energy scale of the "full strength"
tensor[^1] interactions. The notations $q_{u}$ and $q_{d}$ are for the
"upper" (*u* and *c*)[^2] and "lower" (*d*, *s*, and *b*) quarks,
respectively. The CP-symmetry of model (2) is broken if any or all
formfactors $h_{V,T}^{\pm}$ are complex. In the neutral current
lagrangian, $L_{n}$, of type (2) for $q\overline{q}Z$-interactions[^3],
three formfactors, $h_{V}^{\pm}$ and $h_{T}^{+}$, must be real. The
CP-violation may occur only due to tensor coupling with purely imaginary
$h_{T}^{-} \neq 0$.

In model (2), the tree-level cross section for the polarized quark and
antiquark annihilation (1) in their center-of-mass system of reference
is:

> $\frac{d\sigma}{d\Omega} = \frac{g^{4}s\left\{ \left| M_{\text{fi}}^{\text{VV}} \right|^{2} + \frac{\sqrt{s}}{\Lambda}\left| M_{\text{fi}}^{\text{VT}} \right|^{2} + \frac{s}{\Lambda^{2}}\left| M_{\text{fi}}^{\text{TT}} \right|^{2} \right\}}{\text{64}\left\{ (s - M_{W,Z}^{2})^{2} + \frac{\Gamma_{W,Z}^{2}}{2}(s + M_{W,Z}^{2}) + \frac{\Gamma_{W,Z}^{4}}{\text{16}} \right\}}$
> (3)

where $M_{W,Z}$ and $\Gamma_{W,Z}$ are for the *W*- or *Z*-peak mass and
width, respectively. With two transversely polarized quarks, the
double-spin CP- and T-odd asymmetries could be generated if CP-symmetry
is broken in either vector or tensor coupling or both:

$\begin{array}{r}
\left| M_{\text{fi}}^{\text{VV}} \right|^{2} \simeq \frac{\left| h_{V}^{+} + h_{V}^{-} \right|^{2}}{4}\left\{ 1 + {\overrightarrow{n}}_{p} \cdot {\overrightarrow{n}}_{k} \right\}^{2} + \frac{\left| h_{V}^{+} - h_{V}^{-} \right|^{2}}{4}\left\{ 1 - {\overrightarrow{n}}_{p} \cdot {\overrightarrow{n}}_{k} \right\}^{2} + \\
 + \left\{ \left| h_{V}^{+} \right|^{2} - \left| h_{V}^{-} \right|^{2} \right\} \times \left\{ ({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{n}}_{k})({\overrightarrow{\zeta}}_{\overline{q}}^{} \cdot {\overrightarrow{n}}_{k}) - \frac{1}{2}({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{\zeta}}_{\overline{q}}^{})\lbrack 1 - ({\overrightarrow{n}}_{p} \cdot {\overrightarrow{n}}_{k})^{2}\rbrack \right\} \pm \\
 \pm \text{Im}\left( h_{V}^{+}h_{V}^{-} \right) \times \left\{ {\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{q}^{} \times {\overrightarrow{n}}_{p}\rbrack({\overrightarrow{\zeta}}_{\overline{q}}^{} \cdot {\overrightarrow{n}}_{k}) + {\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{\overline{q}}^{} \times {\overrightarrow{n}}_{p}\rbrack({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{n}}_{k}) \right\},
\end{array}$ (4)

$\begin{array}{r}
\left| M_{\text{fi}}^{\text{TT}} \right|^{2} \simeq \frac{1}{2}\left\{ 1 - ({\overrightarrow{n}}_{p} \cdot {\overrightarrow{n}}_{k})^{2} \right\} \times \left\{ \left| h_{T}^{+} \right|^{2} + \left| h_{T}^{-} \right|^{2} - (\left| h_{T}^{+} \right|^{2} - \left| h_{T}^{-} \right|^{2})({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{\zeta}}_{\overline{q}}^{}) + \middle| \middle| \middle| \middle| + 2\text{Im}\left( h_{T}^{+}h_{T}^{-} \right)({\overrightarrow{n}}_{p} \cdot \lbrack{\overrightarrow{\zeta}}_{q}^{} \times {\overrightarrow{\zeta}}_{\overline{q}}^{}\rbrack) \right\}\text{.}
\end{array}$ (5)

In formulae (4)-(5), ${\overrightarrow{\zeta}}_{q}^{}$ and
${\overrightarrow{\zeta}}_{\overline{q}}^{}$ are transverse
polarizations of initial quark and antiquark;
${\overrightarrow{n}}_{p} = \overrightarrow{p}/\left| \overrightarrow{p} \right|$
and
${\overrightarrow{n}}_{k} = \overrightarrow{k}/\left| \overrightarrow{k} \right|$
where $\overrightarrow{p} \equiv {\overrightarrow{p}}_{q}$ is the
momentum of the initial quark and
$\overrightarrow{k} \equiv {\overrightarrow{k}}_{e^{-}}$ or
${\overrightarrow{k}}_{\nu}$ is the momentum of the final lepton. The
upper and lower signs before the T-odd term in Eq. (4) correspond to
$W^{+}$ and $W^{-}$ productions, respectively. Only the main
contributions to the cross-section are kept in Eqs. (4)-(5), neglecting
all others, suppressed by powers of $m_{q,l}/\sqrt{s}$.

In the ultrarelativistic limit of $m_{q,l}/\sqrt{s}\text{<<}1$, the term
$\left| M_{\text{fi}}^{\text{VT}} \right|^{2}$ does not contribute into
the $q\overline{q}$-annihilation cross sections of unpolarized or just
longitudinally polarized quarks. However, in the case of transverse
polarizations, "vector-tensor" interference could generate a number of
T-even and T-odd single-spin asymmetries, as well as double-spin
asymmetries with one quark polarized transversely and the other one
polarized longitudinally:

$\begin{array}{r}
\left| M_{\text{fi}}^{\text{VT}} \right|^{2} \simeq ({\overrightarrow{n}}_{p} \cdot {\overrightarrow{n}}_{k}) \times \left\{ \text{Re}\lbrack(h_{V}^{+}h_{T}^{+} \mp h_{V}^{-}h_{T}^{-}) + \lambda_{\overline{q}}( \mp h_{V}^{+}h_{T}^{-} + h_{V}^{-}h_{T}^{+})\rbrack({\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{q}^{} \times {\overrightarrow{n}}_{p}\rbrack) - \middle| \middle| \middle| \middle| - \text{Im}\lbrack(h_{V}^{+}h_{T}^{-} \mp h_{V}^{-}h_{T}^{+}) + \lambda_{\overline{q}}( \mp h_{V}^{+}h_{T}^{+} + h_{V}^{-}h_{T}^{-})\rbrack({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{n}}_{k}) + \middle| \middle| + \text{Re}\lbrack(h_{V}^{+}h_{T}^{+} \pm h_{V}^{-}h_{T}^{-}) - \lambda_{q}( \pm h_{V}^{+}h_{T}^{-} + h_{V}^{-}h_{T}^{+})\rbrack({\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{\overline{q}}^{} \times {\overrightarrow{n}}_{p}\rbrack) + \middle| \middle| + \text{Im}\lbrack(h_{V}^{+}h_{T}^{-} \pm h_{V}^{-}h_{T}^{+}) - \lambda_{q}( \pm h_{V}^{+}h_{T}^{+} + h_{V}^{-}h_{T}^{-})\rbrack({\overrightarrow{\zeta}}_{\overline{q}}^{} \cdot {\overrightarrow{n}}_{k}) \right\} + \\
 + \text{Re}\lbrack( \mp h_{V}^{+}h_{T}^{+} + h_{V}^{-}h_{T}^{+}) + \lambda_{\overline{q}}(h_{V}^{+}h_{T}^{+} \mp h_{V}^{-}h_{T}^{-})\rbrack({\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{q}^{} \times {\overrightarrow{n}}_{p}\rbrack) - \\
 - \text{Im}\lbrack( \mp h_{V}^{+}h_{T}^{+} + h_{V}^{-}h_{T}^{-}) + \lambda_{\overline{q}}(h_{V}^{+}h_{T}^{-} \mp h_{V}^{-}h_{T}^{+})\rbrack({\overrightarrow{\zeta}}_{q}^{} \cdot {\overrightarrow{n}}_{k}) + \\
 + \text{Re}\lbrack( \pm h_{V}^{+}h_{T}^{-} + h_{V}^{-}h_{T}^{+}) - \lambda_{q}(h_{V}^{+}h_{T}^{-} \pm h_{V}^{-}h_{T}^{-})\rbrack({\overrightarrow{n}}_{k} \cdot \lbrack{\overrightarrow{\zeta}}_{\overline{q}}^{} \times {\overrightarrow{n}}_{p}\rbrack) + \\
 + \text{Im}\lbrack( \pm h_{V}^{+}h_{T}^{-} + h_{V}^{-}h_{T}^{-}) - \lambda_{q}(h_{V}^{+}h_{T}^{-} \pm h_{V}^{-}h_{T}^{+})\rbrack({\overrightarrow{\zeta}}_{\overline{q}}^{} \cdot {\overrightarrow{n}}_{k})
\end{array}$ (6)

where
$\lambda_{q} = ({\overrightarrow{\zeta}}_{q} \cdot {\overrightarrow{p}}_{q})/\left| {\overrightarrow{p}}_{q} \right|$
and
$\lambda_{\overline{q}} = ({\overrightarrow{\zeta}}_{\overline{q}} \cdot {\overrightarrow{p}}_{\overline{q}})/\left| {\overrightarrow{p}}_{\overline{q}} \right|$
are quark and antiquark helicities, respectively; the upper and lower
signs follow the convention of Eq. (4).

2.  **Discussion**

The double-spin T-odd correlations (4) of the vector coupling could be
detected as a transverse "quadrupole" azimuthal anisotropy of lepton
production. In two CP-conjugate processes of $W^{\pm}$-production, these
"true" CP-odd asymmetries are of the opposite signs. This makes them
distinguishable from "spurious" CP-even ones because the latter do not
change signs after CP-conjugation. Therefore, if this anisotropy is
detected in the neutral current annihilation, it could never be "true",
but only a "spurious\'" one due to some CP-even initial and/or final
state interactions.

CP-violation in "tensor" coupling makes the cross section dependent on
the sign of product
$({\overrightarrow{n}}_{p} \cdot \lbrack{\overrightarrow{\zeta}}_{q} \times {\overrightarrow{\zeta}}_{\overline{q}}\rbrack)$
with the measurable ${\Delta\sigma}_{} = \sigma_{}^{R} - \sigma_{}^{L}$
where cross sections $\sigma_{}^{R,L}$ are for the "right"- and
"left\'"-handed mutual orientations of vectors
${\overrightarrow{\zeta}}_{q}$,
${\overrightarrow{\zeta}}_{\overline{q}}$, and
${\overrightarrow{n}}_{p}$, respectively. The "true" CP-odd
${\Delta\sigma}_{} \propto \text{Im}\left( h_{T}^{+}h_{T}^{-} \right)$
should be of the same sign in $W^{+}$-and $W^{-}$-productions, but
"spurious" ones change signs after CP-conjugation. In the neutral
current annihilation, neither CP-even theory can generate a nonzero
${\Delta\sigma}_{}$. Therefore, a detection of a nonzero neutral-current
${\Delta\sigma}_{}$ in quark (or lepton) and [its]{.underline} antiquark
(antilepton) annihilation would be an unambiguous evidence of
CP-violation.

The single-spin and double-spin asymmetries of type (6) are strongly
suppressed at the tree-level of Standard Model. On the other hand, for
the equal strength of unusual interactions at the quark level, the
single-spin asymmetries are less diluted, compared to double-spin ones,
by not so strong correlations of quark's, but particularly antiquark's
polarizations to the proton's one. Therefore, single-spin asymmetries
could be the most sensitive to a new physics beyond SM. Like in the case
of nonzero double-spin asymmetries (4)-(5), to determine, whether
CP-violation takes place, the relative signs of detected asymmetries in
CP-conjugated processes have to be compared. It should be underlined
that CP-violation may show up not only in nonzero T-odd correlations,
but also in purely T-even ones. An example is fomula (6) where CP-odd
imaginary parts are before the T-even scalar products while before the
T-odd products of three vectors are the real, CP-even parts of
participating amplitudes. This, to some extent surprising result, is
apparently the feature of the particular interaction model with the
lagrangian (2), although it is probably not so difficult to invent
models where CP-violation generates T-odd single-spin asymmetries as
well and vice versa.

To detect, but particularly to distinguish CP-odd and CP-even
asymmetries, the directions of motion of the incident quark and
antiquark have to be known. In polarized $p\overline{p}$- (or
$e^{+}e^{-}$, or $\mu^{+}\mu^{-}$, or ...)-collisions, this won't be a
problem. This is not the case at *pp-*colliders. As a result, for
example, T-odd double-spin asymmetries (4)-(5) and some single-spin
asymmetries integrated over the full phase space will obviously be zero.
However, if a valence quark participated in the CP-odd process of *W-*
or *Z-*production, the produced boson would likely be moving in the
direction of the incident "hard\'" valence quark rather then in the
direction of a "soft" sea antiquark. As a result, the nonzero T-odd
asymmetries might separately be detectable in the forward and backward
hemispheres. In the most favorable case of CP-violating *udW-*coupling,
the "true" CP-odd correlations in $W^{\pm}$-production could even be
distinguishable from the "spurious\'" ones.

3.  **Conclusion**

An expected RHIC sensitivity to the asymmetries above at *pp-*level
would be \~$\text{10}^{- 2} - \text{10}^{- 3}$. It is yet to be studied,
how this sensitivity should be transferred back to the quark level, as
well as the limitations to CP-odd amplitudes, arising from already
accomplished experiments.

This work was supported in part by the U.S. Department of Energy Grant
DE-FG0292ER40713. At various stages, it has been reported to a number of
workshops and seminars in BNL and also in the Indiana University, to the
plenary session of the STAR Collaboration Meeting at BNL in July 1998.
It will be published in the Proceedings of the 13^th^ International
Symposium on High-Energy Spin Physics, which had been held in Protvino,
Russia, in September 1998.

#### References

1.  See for examples S.Weinberg, *Phys. Rev. Lett.* **37**, 657 (1976);
    A.J.Davies, Xiao-Gang He, *Phys. Rev.* **D43**, 225 (1991); G.Eilam,
    J.L.Hewett, A.Soni, *Phys. Rev. Lett* **67**, 1979 (1991);
    G.Belanger, C.Q.Geng, *Phys. Rev.* **D44**, 2789 (1991); A.DeRujula,
    M.B.Gavela, O.Pene, F.J.Vegas, *Nucl. Phys.* **B357**, 311 (1991);
    R.Garisto, G.Kane, *Phys. Rev.* **D44**, 2038 (1991) and *TRI-PP-1*
    (1993); C.J.-C.Im, G.L.Kane, P.J.Malde, *Phys. Lett.* **B317**, 454
    (1993); E.A.Paschos*, D0-TH 96/01, FERMILAB-Conf-96/045-T* (1996);
    *Future ElectroWeak Physics at the Fermilab Tevatron: Report of the
    tev-2000 Study Group*, Eds. D.Amidei and R.Brock,
    *FERMILAB-Pub-96/082* and references therein.

2.  G.L.Kane, G.A.Ladinsky and C.-P.Yuan, *Phys. Rev.* **D45**, 124
    (1992).

[^1]: The shortcuts "tensor" and "vector" are used for the couplings
    with and without derivatives $\partial_{\nu}$, respectively.

[^2]: *t*-quark is virtually not reachable at RHIC.

[^3]: ... of a truly neutral $q\overline{q}$-pair, consisting of a
    fermion *q* and [its]{.underline} antiparticle.