= } \nonumber \\ & \overline{u}(p_2)\left[ \gamma_{\lambda}F_V^1(q^2) +\frac{\D i\sigma_{\lambda\nu}q^{\nu}{\xi}F_V^2(q^2)}{\D 2M} %\right. % \nonumber \\& %\left. %~~~~~~~~~~~ +\gamma_{\lambda}\gamma_5F_A(q^2) +\frac{\D q_{\lambda}\gamma_5F_P(q^2)}{\D M} \right]u(p_1), \end{eqnarray*} where $q=k_{\nu}-k_{\mu}$, $\xi=(\mu_p-1)-\mu_n$, and $M=(m_p+m_n)/2$. Here, $\mu_p$ and $\mu_n$ are the proton and neutron magnetic moments. It is assumed that second-class currents are absent, and so the scalar form-factor $F_V^3$ and the tensor form-factor $F_A^3$ do not appear. The form factors $ F^1_V(q^2)$ and ${\xi}F^2_V(q^2)$ are given by: $$ F^1_V(q^2)= \frac{G_E^V(q^2)-\frac{\D q^2}{\D 4M^2}G_M^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}}, %$$ %$$ ~~~{\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-\frac{\D q^2}{\D 4M^2}}. $$ According to CVC, the form factors $ G_E^V(q^2)$ and $ G_M^V(q^2)$ are directly related to form factors determined via electron scattering $G_E^p(q^2)$, $G_E^n(q^2)$, $G_M^p(q^2)$, and $G_M^n(q^2)$: $$ G_E^V(q^2)=G_E^p(q^2)-G_E^n(q^2), %$$ %$$ ~~~G_M^V(q^2)=G_M^p(q^2)-G_M^n(q^2). $$ The axial form factor $F_A$ and the pseudoscalar form factor $F_P$ (related to $F_A$ by PCAC) are given by $$ F_A(q^2)=\frac{g_A}{\left(1-\frac{\D q^2}{\D M_A^2}\right)^2 }, %$$ ~~F_P(q^2)=\frac{2M^2F_A(q^2)}{M_{\pi}^2-q^2}. $$ In the quasi-elastic differential cross section, $F_P(q^2)$ is multiplied by $(m_l/M)^2$, consequently its contribution in muon neutrino interactions is very small except at very low energy, below 0.2~GeV. In general, the axial form factor $F_A(q^2)$ can only be extracted from quasi-elastic neutrino scattering. At low $Q^2$ however, the behavior of $F_A(q^2)$ can also be inferred from pion electroproduction data. Until recently, it has been common practice to assume that form factors are described by the dipole approximation. For example, the vector form factors are often described: $$ G_D(q^2)=\frac{1}{\left(1-\frac{\D q^2}{\D M_V^2}\right)^2 },~~M_V^2=0.71~GeV^2$$ $$ G_E^p=G_D(q^2),~~~G_E^n=0, %$$ %$$ ~~~G_M^p={\mu_p}G_D(q^2),~~~ G_M^n={\mu_n}G_D(q^2). $$ Note that $G_E^p$, $G_M^p$, and $G_E^n$ are positive, while $G_M^n$ and the axial form-factor $F_A$ are negative. \subsection{Axial Form Factor and Axial Mass} Electron scattering experiments continue to provide increasingly detailed determinations of the vector form factors. Neutrino scattering experiments however, are the only plausible route to comparable determinations of the axial form factors, the principal one being the axial form factor $F_A(Q^2)$. The fall-off of the form factor strength with increasing $Q^{2}$ is traditionally parameterized using an effective axial vector mass $M_A$. Its value is known to be $\approx$ 1.00 GeV to an accuracy of perhaps 5\%. This value is in agreement with with the theoretically corrected value from pion electroproduction~\cite{Bernard_01}, \mbox{1.014 $\pm$ 0.016 GeV}. Uncertainty in the value of $M_A$ contributes directly to uncertainty in the quasi-elastic cross section. Current values of other parameters which enter into calculations (as in BBA-2003) of $\sigma(E_{\nu})$ for quasi-elastic reactions, e.g. coupling constants and magnetic moments, are listed in Table~\ref{parameters}. %table 1 % \begin{table} \begin{center} \begin{tabular}{|l|c|} \noalign{\vspace{-8pt}} \hline $g_A$ & -1.267 \\ $G_F$ & 1.1803${\times}10^{-5}$ GeV$^{-2}$ \\ $\cos{\theta_c}$ & 0.9740 \\ $\mu_p$ & 2.793 $\mu_N$ \\ $\mu_n$ & -1.913 $\mu_N$ \\ $\xi$ & 3.706 $\mu_N$ \\ $M_V^2$ & 0.71 GeV$^2$\\ $M_A$ & 1.00 GeV\\ \hline \end{tabular} \end{center} \caption[Parameters of the BBA-2003 form-factors]{Values of parameters for the weak nucleon current used in the BBA-2003 calculation of quasi-elastic reaction cross sections.} \label{parameters} \end{table} The fractional contributions of $F_A$ and of $G_M^p$,$G_M^n$,$G_E^p$, and $G_E^n$ to the $Q^2$ distribution for quasi-elastic neutrino and anti-neutrino running with the NUMI beam configuration are shown in Figure~\ref{FFcontributions}. The contributions were determined by comparing the cross section calculated using BBA-2003 form-factors and the cross section with each of the form-factors set to zero. (Because of interference terms, the sum of the fractions does not have to add up to 100\%.) %figure4 \begin{figure}[htb] \centerline{\psfig{figure=show_gepgmp_ratio.eps,width=4.0in,height=3.in}} %\centerline{\missingFigure} \caption[Ratio of $G_E^p$ to $G_M^p$]{Ratio of $G_E^p$ to $G_M^p$ as extracted by Rosenbluth separation measurements (diamonds) and as obtained by polarization measurements(diamonds). There measurements are seen to be in disagreement at high $Q^2$; only a small uncertainty however is implied for the neutrino quasi-elastic total cross section. } \label{show_gepgmp} \end{figure} \subsection{Vector Form Factors; Discrepancy at High $Q^{2}$} Electron scattering experiments at SLAC and Jefferson Lab (JLab) have measured the vector electromagnetic form factors for the proton and neutron with high precision. The vector form factors can be determined from electron scattering cross sections using the standard Rosenbluth separation technique~\cite{JRA_03}, which is sensitive to radiative corrections, or from polarization measurements using the newer polarization transfer technique~\cite{halla}. The polarization measurements do not directly measure form factors, but measure the ratio $G_E$/$G_M$. These form factors can be related to the vector form factors for quasi-elastic neutrino scattering by the conserved vector current (CVC) hypothesis. Of course, more accurate form factors enable improved calculations for for quasi-elastic neutrino scattering. Recently however, discrepancies in electron scattering determinations of some vector form factors have appeared, the origins of which may be resolvable by study of quasi-elastic reactions in {\minerva}. Figure~\ref{show_gepgmp} shows the BBA-2003 fits to $\mu_p$ $G_E^p$/$G_M^p$. There appears a discrepancy between two different methods of measuring the ratio of electric and magnetic form factor for the proton. The fit including only cross section data (i.e. using Rosenbluth separation) is roughly flat versus $Q^2$ ($Q^2=-q^2$) and is consistent with form factor scaling. This is what is expected if the electric charge and magnetization distributions in the proton are the same. However, the new technique of polarization transfer yields a much lower ratio at high $Q^2$, and indicates a difference between the electric charge and magnetization distributions of the proton. The polarization transfer technique is believed to be more reliable and less sensitive to radiative effects from two-photon corrections. If the electric charge and magnetization distributions of the proton are indeed very different, a test of the high $Q^2$ behavior of the axial form factor can provide very useful additional input towards resolving differences among electron scattering measurements. An accurate mapping of axial behavior at high $Q^2$ can be done in MINER$\nu$A. Current experiments at JLab aim to better understand the source of the disagreement by looking at the recoil proton in elastic electron-proton scattering, thereby minimizing the sensitivity to the dominant sources of uncertainty in previous Rosenbluth separations. Fortunately, since this discrepancy is most prominent at high $Q^2$, it introduces a relatively small uncertainty to the $total$ neutrino quasi-elastic cross section. \subsection{Form Factor Deviations from Dipole Form} Electron scattering results show that dipole amplitudes provide only a first-order description of form factor trends at high $Q^{2}$. For example, Figure~\ref{show_gmp} shows the deviation of $G_M^p$ from dipole-type $Q^{2}$ fall-off. In general, the deviations are different for each of the form factors. %figure2 %\begin{figure}[p]%[htb] \begin{figure}[htb] \centerline{\psfig{figure=show_gmp_ratio.eps,width=4.0in,height=3.in}} %\centerline{\missingFigure} \caption[BBA-2003 fits to $G_M^p/{\mu}_{p}G_D$]{BBA-2003 fits to $G_M^p/{\mu}_{p}G_D$. The variation of the ratio from 1.0 indicates deviation from a pure dipole form; the deviation is quite pronounced for $Q^{2}$ above 1 $GeV^{2}$. } \label{show_gmp} \end{figure} %figure6 \begin{figure} \centerline{\psfig{figure=ratio_JhaKJhaJ_D0DD.eps,width=4.0in,height=3.in}} %\centerline{\missingFigure} \caption[Comparison between BBA-2003 cross-sections and dipole with $G_E^n=0$] {Ratio versus energy of the neutrino (anti-neutrino) quasi-elastic cross section using BBA-2003 form factors, to the expectation using the dipole approximation with $G_E^n$=0.} \label{ratio_JhaKJhaJ_D0DD} \end{figure} In the fits carried out by BBA-2003, the form factors are usually described using an inverse polynomial: $$ G_{E,M}^{N}(Q^2)=\frac{G_{E,M}^{N}(Q^2=0)}{1+a_2Q^2+a_4Q^4+a_6Q^6+...}. $$ The one exception is for $G_{E}^{N}$, for which a useful parameterization has been given by Krutov {\em et. al.}~\cite{Krutov_02}: $$G_E^n(Q^2) = -\mu_n\frac{a\tau}{1+b\tau}G_D(Q^2),~~~\tau=\frac{Q^2}{4M^2},$$ with $a=0.942$ and $b=4.61$. Figure~\ref{ratio_JhaKJhaJ_D0DD} shows the ratio of the predicted neutrino (anti-neutrino) quasi-elastic cross section on nucleons using the BBA-2003 vector form factors compared to the prediction using the dipole vector form factors (with $G_E^n$=0 and $M_A$ kept fixed). This plot shows that it is important to use most current form factor parameterizations. In {\minerva}, it will be possible to test for form factor structures which are more elaborate than simple dipole fall-off with increasing $Q^{2}$, the approximation which has been used by all neutrino experiments to date. \subsection{Nuclear Effects in Quasi-elastic Scattering} There are three important nuclear effects in quasi-elastic scattering from nuclear targets: Fermi motion, Pauli blocking, and corrections to the nucleon form factors due to distortion of the nucleon's size and its pion cloud in the nucleus. Figure~\ref{pauli} shows the nuclear suppression versus $Q^2$ from a NUANCE~\cite{Casper_02} calculation~\cite{Zeller_03} using the Smith and Moniz~\cite{Smith_72} Fermi gas model for carbon. This nuclear model includes Pauli blocking and Fermi motion but not final state interactions. The Fermi gas model was run with a 25~MeV nuclear potential binding energy $\epsilon$ and 220~MeV/c Fermi momentum $K_f$. Figure~\ref{monizPRL} from Moniz et. al. ~\cite{Smith_72} shows how the effective $k_f$ and nuclear potential binding energy $\epsilon$ (within a Fermi-gas model) for various nuclei is inferred from electron scattering data. The effective $k_f$ is extracted from the width of the electron scattered energy, and the nuclear potential binding energy $\epsilon$ is extracted from the shifted location of the quasi-elastic peak. %%figure8 \begin{figure} \centerline{\psfig{figure=NukeFermiGasC12EBind25.eps,width=2.91in,height=2.91in}} %\centerline{\missingFigure} \caption[Suppression of bound cross section in Fermi gas model]{Pauli blocking suppression for a Fermi gas model for carbon with binding energy and Fermi momentum cutoff set to 25~MeV and 220~MeV/c respectively. A suppression of this magnitude is expected for quasi-elastic reactions in MINER$\nu$A. } \label{pauli} \end{figure} \begin{figure}[htb] \centerline{\psfig{figure=monizPRL.eps,width=2.91in,height=4.3in}} %\centerline{\missingFigure} \caption[Fermi gas model parameters extracted from electron-scattering data] {Extraction of Fermi gas model parameters i.e. the effective $k_f$ and nuclear potential binding energy $\epsilon$, from 500 MeV electron scattering data (from Moniz.~\cite{Smith_72}. Distributions shown represent scattering from (a) carbon, (b) nickel, and (c)lead.} \label{monizPRL} \end{figure} The predicted distortion that nuclear binding exerts on the nucleon form factors in neutrino scattering is indicated in Figure~\ref{binding}. The effect is displayed as ratios of bound to free nucleon form factors for $F_1$, $F_2$, and $F_A$. %figure8 \begin{figure}[htb] \centerline{\psfig{figure=Tsushima_ff.eps,width=2.91in,height=2.91in}} %\centerline{\missingFigure} \caption[Ratio of bound to free nucleon form factors $F_1$, $F_2$, and $F_A$]{ The ratio of bound nucleon (in carbon) to free nucleon form-factors for $F_1$, $F_2$, and $F_A$ from ref ~\cite{Tsushima_03}. Binding effects on the form factors are expected to be very small at higher $Q^2$ (therefore, this model is not valid for $Q^2$ greater tha 1 $GeV^{2}$ } \label{binding} \end{figure} \begin{figure}[htb] \centerline{\psfig{figure=lowQ2.eps,width=4.in,height=2.91in}} \caption[Electron scattering data and predictions vs. recoil proton energy] {Comparison between $e-e'-P$ electron scattering data on carbon taken with incident electron energy of 2.2. GeV. Here both final-state electron and proton are both detected. Data and two theoretical models are compared as a function of the recoil proton momentum.} \label{fig:lowQ2.eps} \end{figure} Both the Pauli blocking and the nuclear modifications to bound nucleon form-factors reduce the quasi-elastic cross section relative to the cross section with free nucleons. However, it is possible that the low $Q^2$ deviations are not actual modifications of the nucleon form factors, but rather are effects of interaction with the pion cloud for $Q^2$ less than $1~\hbox{GeV}^2$. Note that data from JLab indicate that the binding effects on the form factors should be very small at higher $Q^2$. \subsection{Detection of Recoil Nucleons} In neutrino experiments, detection of the recoil nucleon is highly useful in distinguishing quasi-elastic from inelastic events. Consequently, knowledge of probabilities for outgoing protons to reinteract with the remaining target nucleons is highly desirable. Similarly, quasi-elastic scattering with nucleons in the high momenta region of the spectral functions needs to be understood. More sophisticated treatments than the simple Fermi Gas model are required. Conversely, inelastic events (such as in resonance production) may be misidentified as quasi-elastic events if the final state pion is absorbed in the nucleus. An optimal way to model these effects is to analyze samples of electron scattering data on nuclear targets (including the hadronic final states) and test the effects of the experimental cuts on the final-state nucleons. MINER$\nu$A can address the issues arising with proton intranuclear rescattering by investigating nuclear binding effects on neutrino scattering in carbon, and then comparing the data to observations obtained in similar kinematic regions as obtained by electron scattering at JLab. Indeed, MINER$\nu$A researchers of the Rochester group will be working with the CLAS collaboration to study hadronic final states in electron scattering on nuclear targets using existing JLab Hall~B CLAS data. This analysis will provide information on hadronic final states in quasi-elastic and inelastic resonance production in electron scattering, and will enable the testing of theoretical models to be used in both electron and neutrino experiments. In addition, collaborative work is underway with the Ghent nuclear physics group in Belgium~\cite{belgium}, to model both electron and neutrino quasi-elastic scattering on nuclei over the entire range of $Q^{2}$. This work will develop the theoretical tools needed to do a precise extraction of the axial form-factor of the nucleon using {\minerva} quasi-elastic data on carbon. It is envisaged that nearly identical analyses will be carried out on both neutrino and electron scattering data in the same range of $Q^{2}$. By way of illustration, Figure~\ref{fig:lowQ2.eps} shows electron scattering data in the quasi-elastic region (for carbon) in which the final-state electron and proton are both detected, compared to predictions of theoretical models, as a function of the recoil proton momentum. Extension of the models to neutrino scattering is currently under way. \subsection{Extraction of the Axial Vector Form Factor} Previous neutrino measurements, mostly bubble chamber experiments on deuterium, extracted $M_A$ using the best known assumptions at the time. Changing these assumptions changes the extracted value of $M_A$. Hence, a prerequisite fo new determinations of $M_A$ is that the most recent form factor parameterizations and couplings be used. %figure7 \begin{figure}[htb] \centerline{\psfig{figure=Baker_d0dd_110_JhaKJhaJ_105.eps,width=2.91in,height=2.5in}} %\centerline{\missingFigure} \caption[Comparison of $Q^2$ distributions for neutrinos with two sets of form-factors]{A comparison of the $Q^2$ distribution using two different sets of form-factors. The data are from Baker {\em et al.}~\cite{Baker_81}. The dotted curve uses dipole form factors with $G_E^n=0$ and $M_A=1.10~GeV$. The dashed curve uses the more accurate BBA-2003 form factors and require $M_A=1.05~GeV$. It is important to use the most current information on vector form factors from electron scattering experiments when extracting the axial form factor from neutrino data.} \label{Baker_d0dd_110_JhaKJhaJ_105} \end{figure} \begin{figure}%[htb] \centerline{\psfig{figure=FA_Minerva_ME_log.ps,width=4.in,height=2.5in}} %\centerline{\missingFigure} \caption[Extraction of $F_A$ in \minerva]{Estimation of values and errors (logarithmic scale) of $F_{A}$ from a sample of Monte-Carlo neutrino quasi-elastic events recorded in the {\minerva} active carbon target. Here, $F_{A}$ is assumed to be a pure dipole form factor with $M_{A}$ of 1 GeV. The reaction sample is the estimated yield from four years of NuMI running.} \label{log:ps} \end{figure} Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105} shows the $Q^2$ distribution from the Baker {\em et al.}~\cite{Baker_81} neutrino experiment compared to the prediction assuming dipole form factors with $G_E^n =0$ and $M_A =1.100~\hbox{GeV}$. Also shown are the prediction using BBA-2003 form factors and $M_A$=1.050~GeV. Utilization of more accurate electromagnetic form factors requires a different $M_A$ value in order to describe the same $Q^2$ distribution. Thus, with the same value of $g_A$, the use of dipole form factors (and $G_E^n =0$) instead of the BBA-2003 form factors may lead to an error in extracted value of $M_A$ of 0.050~GeV. \subsection{Measurement of the Axial Form Factor in {\minerva}} Current and future high-statistics neutrino experiments at low energies (e.g. MiniBooNE, J-PARC and {\minerva}) use an active nuclear target such as scintillator (mostly carbon). The maximum $Q^2$ values that can be reached with incident neutrino energies of 0.5, 1.0, 1.5 and 2 GeV are 0.5, 1.2, 2.1 and 2.9 $~\hbox{GeV}^2$ respectively. Since MiniBooNE and J-PARC energies are in the 0.7~GeV range, these experiments probe the low $Q^2< 1 ~\hbox{GeV}^2$ region where nuclear effects are large (see Figures~\ref{pauli} and ~\ref{binding}) and where the axial form factor is known rather well from neutrino data on deuterium (see Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105}). The low $Q^2$ ($Q^2< 1 ~\hbox{GeV}^2$) MiniBooNE and J-PARC experiments can begin to investigate the various nuclear and binding effects in carbon. At higher $Q^2$, as shown by the BBA-2003 fits, the dipole approximation can be in error by as much as a factor of two for the vector form factors when $Q^2> 2~\hbox{GeV}^2$. There is clearly no reason to assume this form will be valid for the axial form factors. As shown in Figure~\ref{Baker_d0dd_110_JhaKJhaJ_105} there is very little data for the axial form factor in the high $Q^2$ region (where nuclear effects are smaller). Both the low $Q^2$ ($Q^2< 1 ~\hbox{GeV}^2$) and high $Q^2$ ($Q^2> 2 ~\hbox{GeV}^2$) regions are accessible in higher energy experiments such as MINER$\nu$A which can span the 2-8~GeV energy neutrino range. A MINER$\nu$A determination of the axial form factor in the high $Q^2$ region is of keen interest to on-going investigations of the vector and axial structure of the neutron and proton. Figure~\ref{log:ps} shows the extracted values and errors of $F_{A}$ in bins of $Q^{2}$ from a sample of Monte-Carlo quasi-elastic interactions recorded in the {\minerva} active carbon target, from a four-year exposure in the NUMI beam. It can be seen that the high $Q^{2}$ regime which is inaccessible to MinibooNE and J-PARC, will be well-resolved in {\minerva}. Figure~\ref{csdip:ps} shows the these results as a ratio of $F_{A}$/$F_{A}$(Dipole). The lower plot assumes that this ratio is described by the ratio of $G_E^p$(Cross-Section)/$G_E^p$(dipole), which was the the accepted result for $G_E^p$ before the new polarization transfer measurement. The upper plot shows a ratio of 1.0 which is expected if the axial form factor is described exactly by the dipole form \begin{figure}%[htb] \centerline{\psfig{figure=GEP_rat_FA_err_CS_dip.ps,width=4.in,height=3.in}} \caption{ The extracted ratio and error of $F_{A}$/$F_{A}$(Dipole) from a sample of Monte-Carlo quasi-elastic interactions recorded in the {\minerva} active carbon target, from a four-year exposure in the NUMI beam. The lower plot assumes that this ratio is described by the ratio of $G_E^p$(Cross-Section)/$G_E^p$(dipole), which was the the accepted result for $G_E^p$ before the new polarization transfer measurement. 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