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\begin{document}
%\wideabs{

\draft

\title{A New Determination of $\alpha _{S}$ From Neutrino-Nucleon Scattering}

\author{W.~G.~Seligman,$^{2}$ C.~G.~Arroyo,$^{2}$ L.~de~Barbaro,$^{5}$ P.~de~Barbaro,%
$^{6}$ A.~O.~Bazarko,$^{2}$ R.~H.~Bernstein,$^{3}$ A.~Bodek,$^{6}$ T.~Bolton,%
$^{4}$ H.~Budd,$^{6}$ J.~Conrad,$^{2}$ D.~A.~Harris,$%
^{6} $ R.~A.~Johnson,$^{1}$ J.~H.~Kim,$^{2}$ B.~J.~King,$^{2}$ T.~Kinnel,$%
^{7}$ M.~J.~Lamm,$^{3}$ W.~C.~Lefmann,$^{2}$ W.~Marsh,$^{3}$ K.~S.~McFarland,%
$^{3}$ C.~McNulty,$^{2}$ S.~R.~Mishra,$^{2}$ D.~Naples,$^{4}$ P.~Z.~Quintas,$%
^{2}$ A.~Romosan,$^{2}$ W.~K.~Sakumoto,$^{6}$ H. Schellman,$^{5}$
F.~J.~Sciulli,$^{2}$ M.~H.~Shaevitz,$^{2}$ W.~H.~Smith,$^{7}$ P.~Spentzouris,%
$^{2}$ E.~G.~Stern,$^{2}$ M.~Vakili,$^{1}$ U.~K.~Yang,$^{6}$ and J.~Yu$^{3}$}

\address{$^{1}$ University of Cincinnati, Cincinnati, OH 45221 \\
$^{2}$ Columbia University, New York, NY 10027 \\
$^{3}$ Fermi National Accelerator Laboratory, Batavia, IL 60510 \\
$^{4}$ Kansas State University, Manhattan, KS 66506 \\
$^{5}$ Northwestern University, Evanston, IL 60208 \\
$^{6}$ University of Rochester, Rochester, NY 14627 \\
$^{7}$ University of Wisconsin, Madison, WI 53706 }

\date{28 January 1997}

\maketitle

\begin{abstract}
We present a new determination of the proton structure functions $F_{2}$ and 
$xF_{3}$ from the CCFR $\nu $-Fe deep inelastic scattering (DIS) experiment.
Comparisons to high-statistics charged-lepton scattering results for $F_{2}$
from the NMC, E665, SLAC, and BCDMS experiments, after correcting for
quark-charge and heavy-target effects, indicate good agreement for $x>0.1$
but some discrepancy at lower $x$. The $Q^{2}$ evolution of the structure
functions yields the quantum chromodynamics (QCD) scale parameter $\Lambda _{%
\overline{MS}}^{NLO,(4)}=337 \pm 28$(exp.)$\ MeV$. This corresponds to
a value of the strong coupling constant at the scale of mass of the $Z$-boson
of $\alpha _{S}(M_{Z}^{2})=0.119 \pm 0.002$(exp.)$\pm 0.004$(theory)
and is one of the most precise measurements of this quantity.

\end{abstract}

\pacs{PACS numbers: 13.15.+g, 12.38.Qk, 24.85.+p, 25.30.Pt}

%} % end wideabs

High-energy neutrinos are a unique probe for testing QCD and understanding
the parton properties of nucleon structure. Combinations of neutrino and
antineutrino scattering data are used to determine the $F_{2}$ and $xF_{3}$
structure functions which determine the valence, sea, and gluon parton
distributions in the nucleon \cite{GRVMRS,CTEQ}. The universalities of
parton distributions can also be studied by comparing neutrino and
charged-lepton scattering data. Past measurements have indicated that 
$F_{2}^{\nu }$ differs from $F_{2}^{e/\mu }$ by 10-20\% in the low-$x$ region.
These differences are larger than the quoted experimental errors of the
measurements and may indicate the need for modifications of the theoretical
modeling to include higher-order or new physics contributions. QCD predicts
the scaling violations ($Q^{2}$ dependence) of $F_{2}$ and $xF_{3}$ and,
experimentally, the observed scaling violations can be tested against those
predictions to determine $\alpha _{S}$ \cite{DGLAP} or the related QCD scale
parameter, $\Lambda _{QCD}$. The $\alpha _{S}$ determination from neutrino
scattering has a small theoretical uncertainty since the electroweak
radiative corrections, scale uncertainties, and next-to-leading order (NLO)
corrections are well understood. 

In this paper, we present an updated
analysis of the Columbia-Chicago-Fermilab-Rochester (CCFR) collaboration
neutrino scattering data with improved estimates of quark model parameters 
\cite{SAR} and systematic uncertainties. The $\alpha _{S}$ measurement from
this analysis is one of the most precise due to the high statistics of the
experiment, combined with small theoretical and experimental uncertainties.

The differential cross sections for the $\nu $-$N$ charged-current process 
$\nu _{\mu }\left( \overline{\nu}_{\mu }\right) +N\rightarrow \mu ^{-}\left(
\mu ^{+}\right) +X$, in terms of the Lorentz-invariant structure functions 
$F_{2} $, $2xF_{1}$, and $xF_{3}$ are: 
\begin{eqnarray}
\frac{d\sigma ^{\nu ,\overline{\nu }}}{dx\ dy} &=& \frac{G_{F}^{2}ME_{\nu }}{
\pi }\left[ \left( 1-y-\frac{Mxy}{2E_{\nu }}\right) F_{2}\left(
x,Q^{2}\right) \right. \nonumber \\
&&\left. +\frac{y^{2}}{2}2xF_{1}\left( x,Q^{2}\right) \pm y\left( 1-%
\frac{y}{2}\right) xF_{3}\left( x,Q^{2}\right) \right]  \label{eq:dxdy}
\end{eqnarray}
where $G_{F}$ is the weak Fermi coupling constant, $M$ is the nucleon mass, 
$E_{\nu }$ is the incident neutrino energy, $Q^{2}$ is the square of the
four-momentum transfer to the nucleon, the scaling variable 
$y=E_{HAD}/E_{\nu }$ is the fractional energy transferred to the hadronic
vertex with $E_{HAD}$ equal to the measured hadronic energy, and 
$x=Q^{2}/2ME_{\nu }y$, the Bjorken scaling variable, is the fractional
momentum carried by the struck quark. The structure function $2xF_{1}$ is
expressed in terms of $F_{2}$ by $2xF_{1}(x,Q^{2})=F_{2}(x,Q^{2})\times 
\frac{1+4M^{2}x^{2}}{1+R(x,Q^{2})}$, where $R=\frac{\sigma _{L}}{\sigma _{T}}
$ is the ratio of the cross-section of longitudinally to
transversely-polarized $W$-bosons. In the leading-order quark-parton model, 
$F_{2}$ is the singlet distribution $xq^{S}=x\sum \left( q+\overline{q}\right) $,
the sum of
the momentum densities of all interacting quark constituents, and $xF_{3}$
is the non-singlet distribution $xq^{NS}=x\sum \left( q-\overline{q}\right) =xu_{V}+xd_{V}$,
the valence quark momentum density; these relations are modified by
higher-order QCD corrections.

The neutrino DIS data were taken in two high-energy high-statistics runs,
FNAL E744 and E770, in the Fermilab Tevatron fixed-target quadrupole triplet
beam (QTB) line by the CCFR
collaboration. The detector, described in Refs.~\cite{hadcal,mucal}, consists
of a target calorimeter instrumented with both scintillators and drift
chambers for measuring the energy of the hadron shower $E_{HAD}$ and the
muon angle $\theta _{\mu }$, followed by a toroid spectrometer for measuring
the muon momentum $p_{\mu }$. There are 950,000 $\nu _{\mu }$ events and
170,000 $\overline{\nu }_{\mu }$ events in the data sample after
fiducial-volume cuts, geometric cuts, and kinematic cuts of $p_{\mu }$ $>15\
GeV$, $\theta _{\mu }<150\ mr$, $E_{HAD}$ $>10\ GeV$, and $30<E_{\nu }<360\
GeV$, to select regions of high efficiency and small systematic errors in
reconstruction.

In order to calculate the structure functions in Eq.~(\ref{eq:dxdy}) from the
number of observed $\nu _{\mu }$ and $\overline{\nu }_{\mu }$
events, it is necessary to determine the $\nu _{\mu }$ and $\overline{\nu }_{\mu }$
flux.
No direct measurement of the absolute flux was possible in the QTB. The
absolute normalization of the $\nu _{\mu }$ flux
was fixed to the constraint that the neutrino-nucleon
total cross-section equaled the world average of the isoscalar-corrected Fe
target experiments, $\sigma ^{\nu \,Fe}/E_{\nu }=(0.677\pm 0.014)\times
10^{-38}cm^{2}/GeV$ \cite{worldsig}. The relative flux determination, i.e.
the ratio of the flux between different energies and between $\nu _{\mu }$ and 
$\overline{\nu }_{\mu }$, was determined from the low- $E_{HAD}$ events using a technique
described in Ref.~\cite{WGSthesis}.

The cross-sections, multiplied by the flux, are compared to the observed
number of $\nu $-$N$ and $\overline{\nu }$ -$N$ events in an $x$ and $Q^{2}$ bin
to extract $F_{2}(x,Q^{2})$ and $xF_{3}(x,Q^{2})$. Structure functions
extracted from the CCFR data have been previously presented \cite{PZQprl}.
This paper presents a re-extraction of the structure functions that 
includes the following changes: the muon and hadron energy
calibrations are determined from a new analysis of test beam data collected
during the course of the experiment \cite{hadcal,mucal}; more complete
radiative corrections are applied \cite{radcorr}; the value of $R$ used in the
extraction comes from a global fit to the world's measurements \cite
{Rworld}. In addition, the estimates of the
experimental and theoretical systematic errors in the analysis are improved \cite
{WGSthesis}. The structure functions with statistical errors, along with the
QCD fits described below, are shown in Fig.~\ref{fig:SF}.

The structure function $F_{2}$ from $\nu $ DIS on iron can be compared to 
$F_{2}$ from $e$ and $\mu $ DIS on isoscalar targets. To make this
comparison, two corrections must be made to the charged-lepton data. For
deuterium data, a heavy nuclear target correction must be made to convert 
$F_{2}^{\ell D}$ to $F_{2}^{\ell \,Fe}$. This correction was made by
parameterizing the $F_{2}^{\ell N}/F_{2}^{\ell D}$ data from SLAC
and NMC \cite{heavytarget}. $F_{2}$ from electromagnetic interactions
couples to the constituent quarks with the square of the quark electric
charge. Thus a second correction is necessary: 
\begin{equation}
\frac{F_{2}^{\ell }}{F_{2}^{\nu }} = \frac{5}{18}\left( 1-\frac{3}{5}\frac{%
\left( s+\overline{s}-c-\overline{c}\right) }{\left( q+\overline{q}\right) }
\right)  \label{eq:five18ths}
\end{equation}

This formula is exact to all orders in QCD in the DIS renormalization
scheme, so for the purposes of this comparison the charged-lepton structure
functions were corrected according to Eq.~(\ref{eq:five18ths}) with quark
distributions given by CTEQ4D \cite{CTEQ}, which parameterizes the parton
densities in the DIS scheme. The corrected structure functions from NMC,
E665, SLAC, and BCDMS \cite{NMC,F2l} for selected x-bins are shown in
Fig.~\ref{fig:compare}. There is a 15\% discrepancy between the NMC
charged-lepton $F_{2}$ and the CCFR neutrino $F_{2}$ at $x=0.0125$. As the
value of $x$ increases, the discrepancy decreases, until there is agreement
between CCFR and the charged-lepton experiments above $x=0.1$.

The discrepancy between CCFR and NMC at low $x$ is outside the
experimental systematic errors
quoted by the groups and several suggestions for an explanation have been
put forward. One suggestion \cite{CTEQ1}, that the discrepancy can be
entirely explained by a large strange sea, is excluded by the CCFR dimuon
analysis which directly measures the strange sea \cite{AOB}. Other
suggestions are that the strange sea may have a different distribution than
the normally-assumed form \cite{ssbar}, or that the heavy nuclear target
correction may be different between neutrinos and charged leptons. More
experimental data will be necessary to resolve this issue.

According to perturbative QCD (PQCD), the $Q^{2}$ dependence of the quark
momentum densities is described by ``evolution equations'' \cite{DGLAP}.
The
evolution of the non-singlet distribution does not depend on assumptions
about the gluons, but the singlet distribution co-evolves with the gluon
distribution. This analysis takes
advantage of the ability of neutrino DIS to measure $xF_{3}$, and
simultaneously evolves the non-singlet, singlet, and gluon distributions
for a precision measurement of $\Lambda _{QCD}$.

Systematic uncertainties in the structure function
extraction were investigated, leading to correlated errors for each of
the data points in Fig.~\ref{fig:SF}. The largest sources of systematic
error in the determination of $\Lambda _{QCD}$ are the muon and hadron
absolute energy calibrations. The error in the energy calibration was
measured to be 1\% for $p_{\mu }$ \cite{mucal}, and 1\% for $E_{HAD}$ for
the E744 and E770 data separately \cite{hadcal}. Another major source of
systematic error is the error in the value of $\sigma ^{\overline{\nu }
}/\sigma ^{\nu }$, the ratio of the total $\overline{\nu }$ to $\nu $
cross-section. The value chosen was the world average of $\nu $-Fe DIS
experiments, including this one \cite{worldsig,WGSthesis}, $\sigma 
^{\overline{\nu }}/\sigma ^{\nu }=0.499\pm 0.007$. Other sources of
systematic error were investigated, including systematic
errors in the flux extraction and variations in the physics model used in
the Monte Carlo, but the effects of these other sources were small \cite{WGSthesis}.
To determine the
uncertainty for each source, the structure functions $F_{2}$ and $xF_{3}$
are extracted with the given systematic quantity changed by one error unit up and down,
where an ``error unit'' is the best estimate of the systematic error prior to the fit
described below.
The difference of these modified structure functions and the standard ones
gives the point-to-point correlated systematic errors in $F_{2}$ and $xF_{3}$
for each $(x,Q^{2})$ bin. Complete tables of errors can be found in
Ref.~\cite{WGSthesis}.

For the PQCD analysis of the structure functions, we performed a $\chi ^{2}$
fit which minimizes the difference between a theoretical prediction and the
measured values of $F_{2}$ and $xF_{3}$ in each $(x,Q^{2})$ bin. The
theoretical prediction is obtained using the Duke and Owens NLO QCD
evolution program \cite{DOprog,WGSthesis}. The prediction incorporates a
parameterization of the parton distributions for the singlet, non-singlet,
and gluon distributions at a reference value $Q_{0}^{2}=5\,GeV^{2}$ as shown
in Table \ref{tab:fitresults} and includes $\Lambda _{NLO}$ as a fit
parameter. The prediction is compared to the structure function data using a 
$\chi ^{2}$ that includes the statistical errors (including the $\Delta
F_{2}\Delta xF_{3}$ correlations) and the correlated systematic
uncertainties. The systematic errors are included by introducing a parameter 
$\delta \left( k\right) $ for each systematic uncertainty. This parameter
controls the amount of systematic deviation added to the structure function
and is also included in the $\chi ^{2}$ function (Eq.~(\ref{eq:chi2})). For
this procedure, we define the structure-function vector $\vec{F}$ = $\left( 
\begin{array}{cc}
F_{2} & xF_{3}
\end{array}
\right) ^{T}$ and the structure-function statistical error matrix $\widehat{V%
}=\left( \sigma _{ij}\right) $ for $i,j=\{F_{2},xF_{3}\}$. Then the $\chi
^{2}$ for a global fit is given by: 
\begin{equation}
\vec{F}^{diff}=\vec{F}^{data}-\vec{F}^{theory}+\sum\limits_{k}\delta \left(
k\right) \left( \vec{F}^{k}-\vec{F}^{data}\right)  \label{eq:fdiff}
\end{equation}
\begin{equation}
\chi ^{2}=\left( \vec{F}^{diff}\right) ^{T}\ \widehat{V}\ \left( \vec{F}%
^{diff}\right) +\sum\limits_{k}\delta \left( k\right) ^{2}  \label{eq:chi2}
\end{equation}
where $\vec{F}^{data}$ are the measured values as shown in Fig.~\ref{fig:SF}%
, $\vec{F}^{theory}$ are the predictions from the evolution program that
depend on fit parameters including 
$\Lambda _{\overline{MS}}$, and $\vec{F}^{k}$ are 
the structure functions measured with the $k$-th
systematic uncertainty changed by one standard error. 

The effects of target mass \cite{TMC} were included in the fit. Calculations of the
effects of higher-twist terms (HT) have recently become available \cite{DW} and 
are in agreement with the measurements of the $F_{2}$ HT \cite{VM}. However, the data
in Ref.~\cite{VM} were analyzed with a value of $\alpha _{S}$ smaller than our
present value which would increase the measurement of HT. An analysis of 
HT from preliminary CCFR $xF_{3}$ data \cite{Sidorov}
indicates that the calculation of Ref.~\cite{DW} yields HT that are too large. For this
analysis, the values of the $F_{2}$ and $xF_{3}$ HT corrections
were taken to be one-half the values from
Ref.~\cite{DW}, with a conservative systematic error given by repeating the fit with
no HT correction and with the full HT from Ref.~\cite{DW}.

Cuts of $Q^{2}>5\ GeV^{2}$ and the invariant
mass-squared of the hadronic system $W^{2}>10\ GeV^{2}$ were applied to the
data to include only the perturbative region, and an $x<0.7$ cut includes
the $x$-bins where the resolution corrections are insensitive to Fermi
motion. The $E_{\nu }<360\
GeV$ cut implies an effective limit of $Q^{2}< 125\ GeV^{2}$. The best QCD fits to the data are shown in Fig.~\ref{fig:SF}, and
the results of the fit are shown in Table \ref{tab:fitresults}. The $\delta \left( k\right) $
values from the fit are all zero within two standard deviations,
and have errors that range
from 0.12 to 0.98. The fact that these errors are all less than 1.0
indicates that the data coupled with the theory of QCD forms a more
restrictive constraint on the systematic error than the variations described above.

From this fit, we obtain a measured value of $\Lambda _{\overline{MS}}$ in
NLO QCD for 4 quark flavors of $337\pm 28$(exp.)$\pm 13$(HT)$\ MeV$, which yields $\alpha
_{S}(M_{Z}^{2})=0.119\pm 0.002$(exp.)$\pm 0.001$(HT)$\pm 0.004$(scale), 
where the
error due to the renormalization and factorization scales comes
from Ref.~\cite{VM}. 
The fit also yields a measurement of the
gluon distribution $xG(x,Q_{0}^{2}=5\ GeV^{2})=\left( 2.22\pm 0.34\right)
\times \left( 1-x\right) ^{4.65\pm 0.68}$ in the region $0.04<x<0.70$ which is
consistent with gluon distributions given in Refs.~\cite{GRVMRS,CTEQ}.
A fit to only the $xF_{3}$ data, which is not
coupled to the gluon distribution, gives $\Lambda _{\overline{MS}}
=381\pm 53$(exp.)$\pm 17$(HT)$\ MeV$, which is consistent with the result
of the combined $F_{2}$ and $xF_{3}$ fit but has larger errors because effectively only
half the data are used.
If the systematic uncertainties are not allowed to vary in the
$F_{2}$ and $xF_{3}$ fit and all effects of systematic 
uncertainties are added in quadrature, the value of 
$\Lambda _{\overline{MS}}$ is found to be 
$381\pm 23$(stat.)$\pm 58$(syst.)$\ MeV$.

This result is higher than our
previous measurement \cite{PZQprl}, $\alpha _{S}(M_{Z}^{2})=
0.111\pm 0.002$(stat.)$\pm 0.003$(syst.),
mainly due to effects of the new energy calibrations. It is also larger than the
muon DIS result by the SLAC/BCDMS collaboration \cite{VM}, $\alpha
_{S}(M_{Z}^{2})=0.113\pm 0.003$(exp.)$\pm 0.004$(theory); note that this
theoretical error and the CCFR theory error are from the same calculation.

In summary, a comparison of $F_{2}$ from $\nu $ DIS to that from
charged-lepton DIS shows good agreement above $x=0.1$
but a difference at smaller $x$ that grows
to 15\% at $x\approx 0.01$. We have presented a new, high-precision
measurement of $\Lambda _{\overline{MS}}=337\pm 28\ MeV$ from a
fit to the simultaneous $Q^{2}$ evolution of $F_{2}$ and $xF_{3}$. This
corresponds to a value of
$\alpha _{S}(M_{Z}^{2})=0.119\pm 0.002$(exp.)$\pm 0.004$(theory) and is the most
precise DIS measurement of this quantity.

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\begin{figure}
\centerline{\psfig{figure=wseligman-prl-fig1.eps,width=3.375in}}
\caption{The $F_{2}$ and $xF_{3}$ data (statistical errors) and the best QCD
fit (solid line). Cuts of $Q^{2}>5\ GeV^{2}$, $W^{2}>10\ GeV^{2}$, and $x<0.7
$ were applied for the NLO-QCD fit which include target mass corrections. The
dashed line extrapolates the QCD fit into the data regions excluded by the
cuts. Deviations of the data from the extrapolated fit are partly
due to non-perturbative effects.}
\label{fig:SF}
\end{figure}

\begin{figure}[tbp]
\centerline{\psfig{figure=wseligman-prl-fig2.eps,width=3.375in}}
\caption{$F_{2}$ from CCFR $\nu $-Fe DIS compared to $F_{2}$ from $e$D and 
$\mu $D DIS. Errors bars are the statistical and systematic errors added in
quadrature. The charged-lepton data have been corrected to an isoscalar Fe
target, and for quark-charge effects in the DIS
scheme which is valid to all orders (see text). The
NMC data plotted in the figure were extracted with the same $R$ as used in
the CCFR analysis \protect\cite{NMC}.}
\label{fig:compare}
\end{figure}

\begin{table}[tbp]
\caption{Results of the global systematic fit to the CCFR data. The parton
distributions at $Q_{0}^{2}=5\ GeV^{2}$ are parameterized by $%
xq^{NS}\left( x\right) =A_{NS}x^{\eta _{1}}\left( 1-x\right) ^{\eta _{2}}$, $%
xq^{S}\left( x\right) =xq^{NS}\left( x\right) +A_{S}(1-x)^{\eta _{S}},$ 
$xG\left( x\right) =A_{G}\left( 1-x\right) ^{\eta
_{G}}.$ $\delta \left( k\right) $ is the fractional shift for the best value
of systematic quantity $k$ as determined by the fit. Only the most important sources of
systematic error are shown. $\delta \left(
C_{HAD}^{744}\right) $ is the shift for the E744 hadron energy calibration,
$\delta \left( C_{HAD}^{770}\right) $ is the shift for the E770 hadron energy
calibration, $\delta \left( C_{\mu }\right) $ is the shift for the muon
energy calibration, and $\delta \left( \sigma ^{\overline{\nu }}/\sigma
^{\nu }\right) $ is the shift for the ratio of the total $\overline{\nu }$
to $\nu $ cross-section. The $\chi ^{2}$ of the fit is 158 for 164 degrees of freedom.}
\begin{center}
\begin{tabular}{l|r||l|r}                                 
Parameter                             & Fit Results        
  & Parameter                             & Fit Results        \\ \hline 
$\Lambda _{\overline{MS}}$            & $337\pm 28\ MeV$   
  & $A_{G}$                               & $2.22\pm 0.34$     \\ 
$\eta _{1}$                           & $0.805\pm 0.009$   
  & $\eta _{G}$                           & $4.65\pm 0.68$     \\ 
$\eta _{2}$                           & $3.94\pm 0.03$     
  & $\delta \left( C_{HAD}^{744}\right) $ & $0.95\pm 0.42$     \\ 
$A_{NS}$                              & $8.60\pm 0.18$     
  & $\delta \left( C_{HAD}^{770}\right) $ & $0.28\pm 0.27$     \\ 
$A_{S}$                               & $1.47\pm 0.04$     
  & $\delta \left( C_{\mu }\right) $      & $0.21\pm 0.18$     \\ 
$\eta _{S}$                           & $7.67\pm 0.13$     
  & $\delta \left( \sigma ^{\overline{\nu }}/\sigma ^{\nu }\right) $ 
                                      & $0.04\pm 0.50$ 
\end{tabular}
\end{center}
\label{tab:fitresults}
\end{table}

\end{document}
