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\begin{document}
\subsubsection{Physics beyond the Standard Model}

\underline{2005.01.13: DRAFT by {\em V.~L.~Rykov and K.~Sudoh}}

RHIC-Spin potential for uncovering new physics beyond the Standard
Model (SM) has been explored in a number of last decade publications. Our
purpose in this section is to illustrate this new potentiality by
means of a few specific examples.

The non-SM modifications of parity-violating helicity asymmetry
%$A_{L}=\frac{\sigma^{+}-\sigma^{-}}{\sigma^{+}+\sigma^{-}}$ for
$A_{L}=(\sigma^{+}-\sigma^{-})/(\sigma^{+}+\sigma^{-})$ for
one-jet production in collisions of the longitudinally polarized
protons at unpolarized has been studied in Refs.~\cite{taxil:1,taxil:2}.
\label{page:beyond_sm_page1}
In the $A_{L}$ definition above,
$\sigma^{+}$ and $\sigma^{-}$ are for the cross sections\footnote{Or
  differential cross sections.} with the positive and negative
helicities of the initial protons, respectively. In the SM, inclusive
jet production is dominated by the pure QCD $gg$, $gq$, and $qq$
scattering which conserve a parity. 
However the existence of electroweak interactions through the $W^{\pm}$ 
and $Z$ gauge bosons gives a small contribution to $A_{L}$. 
Consequently, the $A_{L}$ is expected to be nonzero from
the QCD-electroweak interference (as shown in
Fig.~\ref{fig:beyond_sm_fig1}).
Additionally, a small peak near $E_{T}=M_{W,Z}/2$ is seen, which is 
the main signature of the purely electroweak contribution. 
The existence of new parity-violating
interactions could lead to large modifications of this SM
prediction~\cite{bourelly:1}.
%%%%%%%%%%%%%%%%%%%
\begin{figure}[tbh]
  \centerline{
    \includegraphics*[width=10.0cm,bb=243 396 495 657]{beyond_sm_fig1.eps} 
  }
  \caption[]{
    $A_{L}$, for one-jet inclusive production in $\vec{p}p$ collisions
    versus transverse energy, for $\sqrt{s}=$~500~GeV. The solid curve
    with error bars represents the SM expectations. The error bars
    show the sensitivity at RHIC for 800~pb$^{-1}$, for the STAR
    detector. The other solid curves, labeled by the product of
    $\mathcal{\epsilon\eta}$, correspond to the contact interaction at
    $\Lambda=$~2~TeV~\cite{taxil:1}. The dashed and dotted curves
    correspond to different leptophobic $Z^{\prime}$ models. 
    The calculations are at the leading order.
  }
\label{fig:beyond_sm_fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%

The modifications due to the presense of quark substructure have been 
analyzed in Ref.~\cite{taxil:1} in the framework of an effective Lagrangian 
approach.
Such effects are generally realized as quantum effects of new physics 
where new heavy particles are considered to be decoupled. 
The non-SM Lagrangian could be represented in terms of new quark-quark contact 
interactions\footnote{It is assumed here that only quarks are composite.} 
under the form:
\begin{equation}
\mathcal{L}_{qqqq} = \mathcal{\epsilon}\frac{g^{2}}{8\Lambda^2}
\overline{\Psi}\gamma_{\mu}(1-\mathcal{\eta}\gamma_{5})\Psi\cdot
\overline{\Psi}\gamma^{\mu}(1-\mathcal{\eta}\gamma_{5})\Psi\;\;,
\label{eq:beyond_sm_contact_ineraction}
\end{equation}
where $\Psi$ is a quark doublet, $g$ is a non-standard coupling, 
$\Lambda$ is a compositeness scale, and $\mathcal{\epsilon}=\pm 1$. 
If parity is maximally violated, $\eta =\pm 1$. 
Fig.~\ref{fig:beyond_sm_fig1} shows
how the SM prediction will be affected by such a new
interaction, assuming $\Lambda = 2$~TeV, which is close to the present
limit obtained for example by the D\O\, experiment at the
Tevatron~\cite{D0:1}. The statistical errors shown are for RHIC
luminosity of 800~pb$^{-1}$, and for the jets with rapidity $|y|<0.5$,
and include measuring $A_{L}$ using each beam, summing over the spin
states of the other beam. Due to the parity-violating signal's
sensitivity to new physics, RHIC is surprisingly sensitive to quark
substructure at the $\sim$2-TeV scale and is competitive with the
Tevatron, despite the different energy range of these
machines. Indeed, a parity-violating signal beyond the SM 
at RHIC would definitely indicate the presence of new
physics~\cite{bourelly:1}.

RHIC-Spin would also be sensitive to possible new neutral gauge
bosons~\cite{taxil:2,taxil:3}. A class of models, called leptophobic
$Z^{\prime}$, is poorly constrained up to now. Such models appear
naturally in several string-derived models~\cite{lykken:1}
(non-supersymmetric models may be also
constructed~\cite{agashe:1}). In addition, in the framework of
supersymmetric models with an additional Abelian $U(1)^{\prime}$ gauge, 
it has been shown~\cite{cvetic:1} that the
$Z^{\prime}$ boson could appear with a relatively low mass ($M_{Z}\leq
M_{Z^{\prime}}\leq 1$~TeV) and a mixing angle with the standard $Z$
close to zero. The effects of different representative models are also
shown in Fig.~\ref{fig:beyond_sm_fig1} (see Ref.~\cite{taxil:2} for
details). RHIC covers some regions of parameters space of the different
models that are unconstrained by present and forthcoming experiments,
%(e.g. Tevatron Run~II),
and RHIC would also uniquely obtain information on the chiral
structure of the new interaction. In Ref.~\cite{taxil:3}, it has been
suggested to extend this study to the collisions of polarized
neutrons, which could be performed with colliding at RHIC polarized
$^{3}He$ nuclei~\cite{courant:1}. The authors argue that, in case of
a discovery, a compilation of the information coming from both
polarized $\vec{p}\vec{p}$ and $\vec{n}\vec{n}$ collisions should
constrain the number of Higgs doublets and the presence or absence of
trilinear fermion mass terms in the underlying model of new physics.

%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
  \centerline{
    \includegraphics*[width=0.8\textwidth,bb=126 369 460
    689]{beyond_sm_fig2.eps}
  }
  \caption[]{
    The leading order $A_{L}$ predictions for sparticle production at
    RHIC (see Ref.~\cite{gehrmann:1} for details). Using the
    full-scale high-energy physics detector of $\sim 4\pi$ acceptance,
    similar to, for example, the one proposed in Ref.~\cite{harris:1},
    with the capability of measuring multi--jet events and missing
    transverse energy is assumed.
  }
\label{fig:beyond_sm_fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%
The study of the production cross sections for squarks and gluinos in
collisions of longitudinally polarized hadrons has been undertaken in
Ref.~\cite{gehrmann:1}. The resulting asymmetries are evaluated for
the polarized proton collider RHIC, as well as for hypothetical
polarized options of the Tevatron and the LHC. These asymmetries
turned out to be sizable over a wide range of supersymmetric particle
masses. Once supersymmetric particles are discovered in unpolarized
collisions, a measurement of the spin asymmetries would thus
potentially help to establish the properties of the newly discovered
particles and open a window to detailed sparticle spectroscopy at
future polarized colliders. Although non-observation of squark and
gluino signatures at the Tevatron thus turns into the stringent limits
on the squark and gluino masses in a frame of MSSM\footnote{Minimal
  Supersymmetric Standard Model.}~\cite{D0-CDF:1}:
$m_{\tilde{q}}>$~250~GeV, $m_{\tilde{g}}>$~195~GeV, these limits are
substantially weakened if more complicated supersymmetric models are
considered. RHIC energy up to $\sqrt{s}=$~500~GeV is not sufficient to
produce the MSSM sparticles; however they could be within its reach if
supersymmetry is realized in a more exotic scenario. Some results of
``scanning'' the space of squark and gluino mass parameters at RHIC
are shown in Fig.~\ref{fig:beyond_sm_fig2}. One can observe that, in
the low mass region, the asymmetry $A_{L}$ measurements at RHIC for
$\tilde{q}\overline{\tilde{q}}$ and $\tilde{q}\tilde{g}$ production
could be sensitive to gluino mass, although in
$\tilde{q}\overline{\tilde{q}}$ process, the gluino appears only as an
exchange particle. The authors of Ref.~\cite{gehrmann:1} conclude
that, assuming the design luminosities and beam polarization of 70\%,
the asymmetries are statistically measurable for sparticle masses up
to 75~GeV at RHIC, 350~GeV at the Tevatron and well above 1~TeV at
LHC, provided experimental uncertainties on them can be kept under
control.

The similar study for slepton production in polarized hadron collisions
has been recently presented in Ref.~\cite{bozzi:1}. However, this
channel might not be accessible at RHIC, because, even in the most
optimistic scenarios, the cross section is not expected to exceed
1~fb.

In the examples above, it is assumed that the polarized parton distribution 
functions (pol-PDFs) of initial longitudinally polarized hadrons would
be known at a sufficient accuracy for being able to detect $A_{L}$
deviations from the SM predictions\footnote{Presumably, the pol-PDFs 
  will be well measured as a part of the mainstream RHIC-Spin program
  discussed in the previous sections, as well as at the other
  facilities.}. Another venue (The best place? - J.~Soffer et
al.~\cite{kovalenko:1}) to look for a new physics beyond the SM, which
does not rely this much on the precise pol-PDF knowledge, is in the
observables that either vanish or are very suppressed in the SM. The
good representatives of such observables are transverse spin
asymmetries -- single or double -- for $W^{\pm}$ and $Z^{0}$
productions, since these are expected to be extremely small in the
SM~\cite{bourelly:2,boer:1,kovalenko:1}. Non-vanishing contributions
could arise here for example in the form of higher-twist terms, which
would be suppressed as powers of $M^{2}/M_{W,Z}^{2}$, where $M$ is a
hadronic mass scale and $M_{W,Z}$ is the $W^{\pm}$ or $Z^{0}$
mass. Other possible  contributions were demonstrated in
Ref.~\cite{boer:1} to be negligible as well. New physics effects, on
the contrary, might generate asymmetries at leading twist.

In Ref.~\cite{kovalenko:1}, the authors have argued that the existence
of {\em R}-parity violating MSSM interaction would generate the
single-spin azimuthal dependences\footnote{These are $A_{N}$ and
  $A_{T}$ asymmetries; see Refs.~\cite{kovalenko:1,ogawa:1} for
  details.} of the charged lepton production via $W^{\pm}$ in
collision of transversely polarized protons at unpolarized:
$p^{\uparrow}p\rightarrow W^{\pm}X\rightarrow e^{\pm}\nu X$ or
$\mu^{\pm}\nu X$ or $\tau^{\pm}\nu X$. The results
of~\cite{kovalenko:1} show that, in this particular extension of the
SM, the asymmetries are likely to be small and, at best,
could be just marginally detectable at RHIC. Nevertheless, this does
not exclude that other non-standard mechanisms produce larger
effects.

One more mechanism of generating non-zero $A_{N}$ and $A_{T}$
asymmetries in leptoproduction via $W^{\pm}$ and $Z^{0}$ decays is
due to anomalous electroweak dipole moments of
quarks~\cite{kovalenko:1,kane:1,ogawa:1}. 
Phenomenologically, 
the presence of anomalous dipole moments could be described as a
combination of tensor and (pseudo)scalar $q\overline{q}W$ and
$q\overline{q}Z$ couplings additional to the standard {\em V} and {\em A} 
couplings. 
The nonzero $A_{N}$ and $A_{T}$ arise from the interference of
these additional couplings with the SM's {\em V} and {\em A} couplings. 
The SM predictions for anomalous dipole moments of $u$ and $d$ quarks, 
which provide the main contribution to the $W^{\pm}$ and $Z^{0}$ production
at RHIC, are extremely small, and their effects are much below the
RHIC sensitivity. On the other hand, the current experimental limits
on anomalous dipole moments of quarks\footnote{And of $\tau$-lepton.}
are still far above the SM expectations. The most stringent
experimental constraints, applicable to {\em CP}-conserving components
of quark dipole moments, come from the analysis~\cite{escribano:1} of
electroweak data from high energy colliders. In this analysis, it has
been considered that theories beyond the SM, emerging at some
characteristic energy scale above $W/Z$ mass, 
%appear only in the couplings 
have effect 
at low energies $E\leq M_{W,Z}$, and can be introduced 
by taking account of an effective Lagrangian that extends the SM
Lagrangian $\mathcal{L}_{SM}$:
$\mathcal{L}_{eff}=\mathcal{L}_{SM}+\delta\mathcal{L}$. To preserve the
consistency of the low energy theory, it has been assumed that
the non-SM Lagrangian $\delta\mathcal{L}$ is 
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge invariant. 
The $W^{\pm}$ and $Z^{0}$ productions in $p^{\uparrow}p$
collisions at RHIC is expected to have a good sensitivity on
$\mathcal{L}_{SM}$--$\delta\mathcal{L}$ interference 
at the parton level 
due to strong correlations between the proton spin and
polarization of high-$x$ valence quarks, that participated in gauge
boson production~\cite{soffer:1}. As it has been estimated in
Ref.~\cite{ogawa:1}, the measurements at RHIC, carried out with
transversely polarized proton in the context of the physics discussed
in the previous sections, would improve the current experimental
limits~\cite{escribano:1} on electroweak dipole moments of $u$ and $d$
quarks by a factor of $\sim$5--10. But a non-zero result would be a
direct indication of a new physics beyond the SM.


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\end{document}