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\begin{document}
\title[Theoretical interpretations of NuTeV]{Interpretations of
the NuTeV  $\sin^2 \theta_W$ }
\author{ S Davidson   }
\address{ IPPP, University of Durham, Durham DH1 3LE,UK  }
\begin{abstract}
We summarize theoretical explanations of the
three $\sigma$ discrepancy between $\sin^2 \theta_W$ measured
by NuTeV and predicted by the Standard Model global
fit. Possible new physics explanations
({\it e.g.} an unmized $Z'$) are not compelling. 
The discrepancy would be reduced by a positive momentum
asymmetry  $s^-$ in the strange sea; present experimental
estimates of $s^-$ are unreliable or  incomplete.
 Upgrading the NuTeV analysis
to NLO would alleviate concerns that the discrepancy
is a QCD effect.
\end{abstract}
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%\pacs{00.00, 20.00, 42.10}
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%\submitto{\JPA}
\eads{\mailto{sacha.davidson@durham.ac.uk}}
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\section{Introduction}
The NuTeV collaboration studied 
$\nu_\mu$ Deep Inelastic Scattering ($\nu DIS$), and  measured 
$\sin^2 \theta_W$ on-shell, or $m_W^2/m_Z^2$, 
to be  $\sin^2 \theta_W  = 0.2276 \pm 0.0013({\rm stat})
\pm 0.0006({\rm syst})\pm 0.0006({\rm theo})$  ~\cite{NuTeV}.
 This is $\sim 3 \sigma$ from the
world average  $\sin^2 \theta_W  = 0.2226 \pm 0.0004$.
Is this  the long-awaited harbinger of
New Physics? 
 Neutrino $DIS$ is  a notoriously difficult environment in which to
do precision physics---is the discrepancy an overlooked
Standard Model (SM) effect?
Various explanations for this discrepancy have been put forward
\cite{Roy,sspna,Babu,nucl}.  In ref.
\cite{sspna}, we considered electroweak corrections, QCD effects,
new physics in loops and new physics at tree level.
\section{New Physics?}
It is difficult to saturate the NuTeV discrepancy with new physics in
loops; an $O(1 \%)$ effect is needed at NuTeV where $Q^2 \sim 20$ GeV$^2$,
but the new physics must not disrupt the part-per-mil
agreement between the SM and precision tests.  We found in
\cite{sspna} that oblique corrections, motivated
versions of the MSSM and modified $Z$ couplings 
\footnote{Some authors~\cite{Babu,Tacheuchi} have reconsidered models
where neutrinos mix with heavy singlets,
thereby reducing their couplings with the $Z$ and $W$ bosons
by a factor $1-\epsilon$ and $1-\epsilon/2$ respectively.
However, 
$\epsilon > 0$  reduces the NuTeV
anomaly  at the price of worsening the global fit \cite{sspna}. 
(Our equations differ from those of \cite{Babu} because
we place $\epsilon$ in different electroweak parameters.)}
cannot separately explain the whole 
NuTeV-LEP discrepancy. It has been observed in
\cite{Tacheuchi} that 
oblique corrections induced by new physics,  {\it and} 
 modified $Z$ couplings,
%together 
can fit all the data
\footnote{Bernstein, in these proceedings, has
a different interpretation of \cite{sspna} or \cite{Tacheuchi}} .
%, because the 
% new physics can cancel  the $\epsilon$-induced shift in the oblique
%parameters.
New tree-level physics offers more promising
explanations. A $\sim 1\%$ decrease with respect to the SM
of the coefficient of the operator $(\bar{\nu}_\mu \gamma^\alpha \nu_\mu)
(\bar{q_L}  \gamma_\alpha  q_L)$ is required, and could be provided
by a new $Z'$ boson, or by SU(2) triplet leptoquarks with judiciously
chosen unequal masses. A new $Z'$ must have
negligeable mixing with the $Z$ to satisfy
the oblique parameter and  precision bounds on the Z coupling: \cite{Z'}.
However, a $Z'$ coupled to $e.g.$ $B- 3 L_\mu$  would provide
the required four fermion operator at tree level.
(It would also induce the operator $(\bar{\nu}_\mu \gamma^\alpha \nu_\mu)
(\bar{q_R}  \gamma_\alpha  q_R)$; this is
acceptable  because the coefficient \footnote{ 
This coefficient has the wrong sign in the plots of
\cite{sspna}; the $Z'$ has vector couplings so makes
a negative contribution to both $g_L^2$ and $g_R^2$.
We thank Birgit Eberle for bringing this  to our
attention. }
this operator is measured less accurately by NuTeV. )
 The $Z'$  could have
$m_{Z'} > 600 $ GeV  for $g' \sim 1$, or  if the coupling
is small $g' \sim 10^{-3}$, it could have 2 GeV $< m_{Z'} < $ 10 GeV
consistently with all experimental constraints. A
$Z'$ with $m \simeq 3 $GeV  could  fit the  current
$g-2$ discrepancy \cite{g2}.
\section{Back to the Standard Model}
The NuTeV experiment measures the ratio of ``short''
(= muonless) to ``long'' (with a $\mu$) events for incident
$\nu_\mu$ and $\bar{\nu}_\mu$ beams.  From this
they extract the ratios $R^\nu$ and $R^{\bar{\nu}}$, where
%\begin{equation}
$R^{\nu} = %\frac
{ \sigma (\nu N \rightarrow \nu X)}/
{ \sigma (\nu N \rightarrow \mu X)}$ .
%\end{equation}
$R^\nu$ is more sensitive than  $R^{\bar{\nu}}$ to $\sin^2 \theta_W$,
 so $\sin^2 \theta_W$ is determined mainly from
$R^\nu$, after an effective ``charm mass'' is extracted
 from $R^{\bar{\nu}}$.
NuTeV uses leading order (LO) parton distribution
functions (pdfs), which are fit to their
data, they assume isospin symmetry ($u^p(x) = d^n(x)$),
and that $q(x) = \bar{q}(x)$ for second generation
quarks. 
%In \cite{sspna}, we studied the extraction of  $\sin^2 \theta_W$
%from the 
A theoretically cleaner ratio, where we studied  the effects
of isospin violation and $s \neq \bar{s}$ is the Paschos-Wolfenstein
ratio (related to $R^\nu$ and $R^{\bar{\nu}}$) :
\begin{eqnarray}
R_{PW} & \equiv & \frac{ \sigma (\nu N \rightarrow \nu X) - 
  \sigma (\bar{\nu} N \rightarrow \bar{\nu} X)}
{ \sigma (\nu N \rightarrow \mu X) - 
  \sigma (\bar{\nu} N \rightarrow \bar{\mu} X)} \nonumber \\
& =&   \frac{1}{2} - \sin^2 \theta_W + \left[ (1.3 + O(\alpha_s))(u^- - d^- - s^-)
\right] ~~~,
\label{RPW}
\end{eqnarray}
where the 1.3 is a simplification (see \cite{sspna}).
The square brackets contain the corrections
that arise if isospin is violated, or if
there is a momentum asymmetry in the strange sea:
$s^- \neq 0$, where $ s^- = \int dx x (s(x) - \bar{s}(x))$.
Most pdf fits assume $s^- = 0$.
This was not imposed in  
ref. \cite{BPZ} (BPZ), who   performed a NLO fit  to all
the cross section data available (this did not include
CCFR and NuTeV). They found that  $s^- \simeq .002$
was  a significantly better  fit ($\Delta \chi^2 = 25$ for 2 additional 
d.o.f.)  than $s^- = 0$.  
 Naively substituting
this into eqn. \ref{RPW}, one finds that
 $\sin^2 \theta_W |_{NuTeV} - \sin^2 \theta_W |_{LEP}$ decreases
to less than two $\sigma$. Realistically,
the effect of $s^-$ on  $\sin^2 \theta_W$  will be reduced by 
experimental cuts and sensitivities.
  NuTeV has published  a LO $ s \neq \bar{s}$ 
fit to their dimuon data \cite{Goncharov}, and found $s^-$
{\it negative}.  It is unclear whether the NuTeV dimuon data
is consistent with the CDHSW cross-section data, which
inconjunction with BCDMS,   drives
$s^-$ positive in the BPZ fit; a refit to all
the data  would be required to determine this. However, the
NuTeV analysis  \cite{Goncharov} had various peculiar features,
as outlined in the (post-publication) note added to \cite{sspna}.
After the appearance of \cite{sspna}, NuTeV
pointed out \cite{new} that according to their analysis \cite{Goncharov},
the asymmetry  had the wrong sign to reduce the  $\sin^2 \theta_W$
discrepancy: $s^- \sim -.0027 \pm .0013$.
 They have recently redone their $s \neq \bar{s}$
analysis at NLO\cite{ICHEP}, and   find a {\it positive}
asymmetry  $s^-  \sim .0003$. These determinations are affected by a
   theoretical uncertainty which is not included 
in the quoted error.
A more detailed discussion of the NuTeV $s^-$ extraction
can be found in the ``note added'' to \cite{sspna}.
$R_{PW}$ is theoretically attractive because the parton distributions
cancel out of the ratio at LO, and the NLO corrections are small.  However,
the $R^\nu$, $R^{\bar{\nu}}$ ratios measured  by NuTeV have some dependence
on the pdfs, which is exacerbated 
by  asymmetries between charged-current and neutral-current, 
or between $\nu$ and $\bar \nu$ events.
Such asymmetries could be induced by
experimental cuts and   by different
 $\nu$, $\bar{\nu}$ spectra.
It is therefore difficult
to estimate the size of the NLO corrections to the $\sin^2 \theta_W$
determination from NuTeV, particularily since NuTeV
fit their LO pdfs to their data, which could absorb
some of the NLO effects. A  NLO analysis of the
NuTeV experiment would be a welcome solution to these concerns.
\section*{References}
\begin{thebibliography}{999999999}
\bibitem{NuTeV}
G.~P.~Zeller {\it et al.}  [NuTeV Collaboration],
%``A precise determination of electroweak parameters in neutrino nucleon  scattering,''
Phys.\ Rev.\ Lett.\  {\bf 88} (2002) 091802; 
hep-ex/0110059.\\
%%CITATION = HEP-EX 0110059;%%
K. McFarland, seminar available at the internet address
www.pas.rochester.edu/\~{}ksmcf/NuTeV/seminar-only-fnaloct26.pdf
\bibitem{Roy}{E. Ma, D.P. Roy},[hep-ph/0111385]; 
E.~Ma and D.~P.~Roy,
%``Heavy triplet leptons and new gauge boson,''
[hep-ph/0206150].
%\cite{sspna}
\bibitem{sspna}
S.~Davidson, S.~Forte, P.~Gambino, N.~Rius and A.~Strumia,
%``Old and new physics interpretations of the NuTeV anomaly,''
JHEP {\bf 0202} (2002) 037
[hep-ph/0112302].
%%CITATION = HEP-PH 0112302;%%
%\cite{Babu:2002en}
\bibitem{Babu}
K.~S.~Babu and J.~C.~Pati,
%``Anomaly in neutrino nucleon scattering and vectorlike families,''
[hep-ph/0203029].
%%CITATION = HEP-PH 0203029;%%
\bibitem{nucl}
%\cite{Kovalenko:2002xe}
%\bibitem{Kovalenko:2002xe}
S.~Kovalenko, I.~Schmidt and J.~J.~Yang,
%``Nuclear effects on the extraction of sin**2 Theta(W),''
hep-ph/0207158.
%%CITATION = HEP-PH 0207158;%%
%\cite{Miller:2002xh}
%\bibitem{Miller:2002xh}
G.~A.~Miller and A.~W.~Thomas,
%``Comment on 'A precise determination of electroweak parameters in  neutrino nucleon scattering',''
 hep-ex/0204007.
%%CITATION = HEP-EX 0204007;%%
\bibitem{Z'} 
%\cite{Chivukula:2002ry}
%\bibitem{Chivukula:2002ry}
R.~S.~Chivukula and E.~H.~Simmons,
%``Electroweak limits on non-universal Z' bosons,''
Phys.\ Rev.\ D {\bf 66} (2002) 015006 ; hep-ph/0205064.
%%CITATION = HEP-PH 0205064;%%
%\cite{Erler:2002pr}
%\bibitem{Erler:2002pr}
J.~Erler, P.~Langacker and T.~j.~Li,
%``The Z - Z' mass hierarchy in a supersymmetric model with a secluded  U(1)'-breaking sector,''
Phys.\ Rev.\ D {\bf 66} (2002) 015002 ; hep-ph/0205001.
%%CITATION = HEP-PH 0205001;%%
\bibitem{g2}
%\cite{Bennett:2002jb}
%\bibitem{Bennett:2002jb}
G.~W.~Bennett  [Muon g-2 Collaboration],
%``Measurement of the positive muon anomalous magnetic moment to 0.7-ppm,''
[hep-ex/0208001].
%%CITATION = HEP-EX 0208001;%%
\bibitem{new}
G.~P.~Zeller {\it et al.}  [NuTeV Collaboration],
%``On the effect of asymmetric strange seas and isospin-violating 
%parton  distribution functions on sin**2(Theta(W)) measured in the NuTeV
%experiment,'' 
hep-ex/0203004.
%%CITATION = HEP-EX 0203004;%%
\bibitem{Goncharov}
The NuTeV collaboration, M.~Goncharov et al.{\PR}{D64} (2001),  {112006}.
\bibitem{BPZ}
{V.~Barone, C.~Pascaud, F.~Zomer}, {\it Eur.\ Phys.\ J.}
{\bf C12} (2000) {243}, hep-ph/9907512.
%\cite{Takeuchi}
\bibitem{Tacheuchi}
T.~Takeuchi,
%``The NuTeV Anomaly, Neutrino Mixing, and a Heavy Higgs,''
hep-ph/0209109; 
%%CITATION = HEP-PH 0209109;%%
W.~Loinaz, N.~Okamura, T.~Takeuchi and L.~C.~Wijewardhana,
%``The NuTeV anomaly, neutrino mixing, and a heavy Higgs,''
hep-ph/0210193.
%%CITATION = HEP-PH 0210193;%%
\bibitem{ICHEP}
NuTeV Collaboration talk at ICHEP 2002, Amsterdam.
\end{thebibliography}
\end{document}
