\input epsf \documentstyle[12pt,psfig,epsfig]{article} \topmargin -0.5in % 1 inch at top of page \textheight 9.0in % 9.0 inches text ==> 1 inch bottom margin \oddsidemargin 0.25in % 1.25 inch left margin \textwidth 6.0in % 6.0 inches text % % release the floats \setcounter{topnumber}{4} \setcounter{bottomnumber}{4} \setcounter{totalnumber}{6} \renewcommand\topfraction{1.0} \renewcommand\bottomfraction{.5} \renewcommand\textfraction{0.05} \renewcommand\floatpagefraction{0.8} \setlength{\textfloatsep}{10pt plus 2pt minus 4pt} % \begin{document} \renewcommand{\thepage}{D/CCFR-NuTeV/NUMI-MINERvA--\arabic{page}} \input{my-macros.tex} %\input{my-macros-budd.tex} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\beas{\begin{eqnarray*}} \def\eeas{\end{eqnarray*}} \par \smallskip \noindent {\Large\bf Task D/CCFR-NuTeV/MINERvA} \\ \noindent(CCFR/NuTeV = FNAL Tevatron Experiments E744/E770/E815, NUMI-MINERvA = near detector neutrino detector at FNAL)\\ \\ \noindent{\bf A. Current Participants of the Rochester CCFR-NuTeV Group during 2003-2003}\vspace*{2mm}\\ Faculty: Prof. Arie Bodek, Prof. Kevin McFarland,\\ (Prof. Steve Manly, Collaborating Faculty on MINERvA) \\ Research Faculty: Dr. Pawel de Barbaro (CCFR/NuTeV), Dr. Howard Budd (CCFR/NuTeV/MINERvA), and\\ \hspace*{34mm}Dr. Willis K. Sakumoto (CCFR/NuTeV/MINERvA)\\ Graduate Students: Jesse Chovjka ((NUMI-MINERvA) REU summer Undergraduates: Frank Tompkins, Amee Slaughter (MINERvA) RET summer High School Teacher: Paul Conrow (MINERvA) Previous Participants: U. K. Yang, S. Avvakumov (PhD Students, CCFR/NuTeV)\\ \noindent{\bf Introduction}\\ \hspace*{0.196in} \begin{figure}[ht] \begin{center} \hspace{.75in} \epsfxsize=5.0in \epsfbox{labe.eps} \end{center} \caption{ The CCFR/NuTeV Detector } \label{fig:detector} \end{figure} The NuTeV experiment combined an upgraded CCFR detector (shown in Figure~\ref{fig:detector}), a new continuous hadron calibration beam, and a new SSQT neutrino beam. The new SSQT neutrino beam was sign-selected, thus allowing neutral current data to be taken separately with neutrinos and antineutrinos. The upgraded detector had new liquid scintillator and new phototubes. Upstream of the detector is a large volume filled with helium bags. It is a decay channel used in a search for neutral heavy leptons. The main physics goal of NuTeV is an improved measurement of the electroweak mixing angle by using separate neutrino and antineutrino beams. The Rochester effort on CCFR/NuTeV in 2002/2003 have focused on the electroweak mixing angle (led by Prof. McFarland), on structure functions (led by Bodek, and PhD. Thesis of Yang) and on limits on oscillatins (led by Bodek, and PhD thesis of Avvakumov).\vspace*{5mm} MINERvA (McFarland and Morfin co-sppokespersons), is a proposed experiment at the near detector hall at NUMI to investigate neutrino interactions at low enegies. It can run both on-axis and off-axis. The understading and modeling of neutrino interactions at the low energy range is of fundamental importance to both current neutrino oscillations experiments such as K2K and MINOS, and essential for future neutrino oscillations experiments such as JHF to SuperK and NUMI off-axis. The Rochester group is involved in both of these working groups, and MINERvA is expected to provide data which is useful to both efforts. IN 2002-2003 the Rochester effort related to MINERvA has focused on several areas. (1) McFarland is the co-spokesperson of the MINERvA collaboration. (2) Bodek and Yang phenomenological work on modeling inelastic electron and neutrino interactions at low energies. They have extracted effective PDFs which work at all Q2, through analysis of previous electron scattering data. Their method is is now used in the default Monte Carlo used by K2K, and will provide a framework for the analysis of the data in MINERvA. (23 Bodek and Budd (in collaboration with John Arrington from Argonne) have worked on improved modeling of quasielastic electron and neutrino reactions by doing a re-analysis of previous electron and neutrino data. Their work has improved on our knowledge of vector and axial vector form factors. (4) Bodek (with the help of Yang, Budd, McFarland and)Manly is now leading an a collaborative effort between the High Energy and Medium Energy communities focused on modeling the resonance region in both electron and neutrino reactions. (5) McFarland, Manly and Bodek, with the help of Professor Frank Wolfs, two REU undergraduate summer students and one RET High School Teacher have begun to do research and development on water based scintillators. Although the primary target at MINERvA is soid scintillator, the development active water neutrino target is of great interest to K2K and the JHF to SuperK program. \noindent{\bf CCFR/NuTeV/MINERvA progress in 2002-2003 and plans for future} \begin{enumerate} %\hspace*{0.196in} \item Structure Function Analysis \\ \hspace*{0.196in} The University of Rochester is working with the University of Pittsburgh on the NuTeV structure function analysis. This is a continuation of the work by Rochester Ph.D. student Un-ki Yang and Professor Arie Bodek. The Rochester group used CCFR data to perform the following structure function measurements: differential cross sections vs $x$ $y$ and $E_{\nu}$, $F_2$, $\Delta xF_3 = xF_3^{\nu}$-$xF_3^{\nub}$, and $R^\nu = {\sigma_L}/{\sigma_T}$. These were significant improvements to our knowledge of structure functions and parton distributions. (Note, additional work on structure functions by Yang and Bodek included analysis of electron, muon and neutrino scattering, and CDF W asymmetry data). \hspace*{0.196in} Dr. Un-Ki Yang received the Universities Research Association (URA) award for the best Fermilab thesis of 2001-2002. As an aside we tote that this is the second year in a row that a Rochester PhD student received this URA Award (a year earlier, the URA Prize was given to Rochester student Michael Fitch), and the first URA Tollestrup prize for best postdoctoral work at Fermilab was awarded to Rochester PhD. Juan Estrada). Yang's work on CCFR/NuTEV was also recognized by the Association of Korean Physicists in America, which presented a citation of Honorable Mention for the 2001 Outstanding Young Researcher Award to Dr. Un-Ki Yang for his truly outstanding scholarly and pioneering research in high energy physics. Yang was also awarded the Lobkowicz prize for best Ph.D. thesis in high energy physics in the department of physics and astronomy at the University of Rochester in 2001. \hspace*{0.196in} The sum of $\nu_\mu$ and $\nub_\mu$ differential cross sections for charged current interactions on isoscalar target is related to the structure functions as follows: $$ \left[\frac{d^2\sigma^{\nu }}{dxdy}+ \frac{d^2\sigma^{\overline \nu}}{dxdy} \right] \frac {(1-\epsilon)\pi}{y^2G_F^2ME_\nu} = 2xF_1 [ 1+\epsilon R ] - \frac {y(1-y/2)}{1+(1-y)^2} \Delta xF_3 . $$ Here $G_{F}$ is the weak Fermi coupling constant, $M$ is the nucleon mass, $E_{\nu}$ is the incident energy, the scaling variable $y=E_h/E_\nu$ is the fractional energy transferred to the hadronic vertex, $E_h$ is the final state hadronic energy, and $\epsilon\simeq2(1-y)/(1+(1-y)^2)$ is the polarization of virtual $W$ boson. The structure functions $2xF_1$, $R$, and $\Delta xF_3$ can be extracted from the measurements of $\frac{d^2\sigma^{\nu }}{dxdy}$ and $\frac{d^2\sigma^{\overline \nu}}{dxdy}$ at various values of $\epsilon$. The structure function $2xF_1$ is expressed in terms of $F_2$ by $2xF_1(x,Q^2)=F_2(x,Q^2)\times \frac{1+4M^2x^2/Q^2}{1+R(x,Q^2)}$, where $Q^2$ is the square of the four-momentum transfer to the nucleon, $x=Q^2/2ME_h$ (the Bjorken scaling variable) is the fractional momentum carried by the struck quark, and $R={\sigma_L}/{\sigma_T} $ is the ratio of the cross-section of longitudinally to transversely-polarized $W$-bosons. The term $\Delta xF_3 = xF_3^{\nu}$-$xF_3^{\nub}$ in leading order is $\simeq 4x(s-c)$. \hspace*{0.196in} Structure function measurements from NuTeV have several advantages over CCFR measurements. Three of these advantages are stated below: \begin{itemize} \item During the planning stage of NuTeV, Prof Arie Bodek conceived of the continuous hadron and muon calibration beam. In CCFR, the calibration beam entered the side of the building, and we had to move the detector and disrupt data taking to do calibrations. The hadron beam line was modified and redirected at the cener of the building. NuTeV ran the calibration beam concurent with neutrino data taking. The Rochester group analyzed the hadron energy calibration data which resulted in a smaller error of 0.43\% (versus 1.0\% in CCFR). Note that Rochester was also responsible for the hadron energy calibration for CCFR. \item Since the NuTeV beam is sign selected, even events for which the final state muon is not analyzed by the toroid spectrometer can be used in structure function analysis (because of the separate running in $\nu$ and $\overline{\nu}$ modes). These "range outs" extend the range to high $y$ and low $\epsilon$. \item Since the sign selected beam creates either $\nu$ or $\overline{\nu}$, the final state muons are always focused toward the center of the muon spectrometer. This results in higher acceptance for high $y$ events with low energy final state muons. \end{itemize} \begin{figure}[ht] \begin{center} \epsfxsize=4.5in \epsfbox{f2q2lowx.eps} \end{center} \caption{ NuTeV preliminary $F_2(x,Q^2)$. NLO curve is TR-VFS with MRTS99 (Thorne \& Roberts Phys.Lett B421,303(1998)) } \label{f2q2lowx} \end{figure} \hspace*{0.196in} the CCFR structure function results were published. Preliminary results of the NuTeV cross sections and structure functions, $F_2$, $xF_3$, and $R$ were presented in 2003. Figure~\ref{f2q2lowx} shows the NuTeV preliminary $F_2(x,Q^2)$. We see agreement with the previous CCFR $\nu$-Fe measurements. In addition, we see good agreement with the NLO QCD predictions. The preliminary NuTeV measurement of R and $xF_3$ is also in good agreement with the previous CCFR results. \hspace*{0.196in} The plan is to finalize the NuTeV cross section and structure functions in the latter part of 2003. The NuTeV "range-out" data, which measure the differential cross sections at high $y$, will be added to the sample. This will give a better $2xF_1$, $R$ fit. The same type of QCD analysis which CCFR performed will be done with the NuTeV structure functions. This includes the QCD fits to PDFs and extraction of $\alpha_s$. \item Precision Determination of $\sin^2 \theta_W$ \\ \hspace*{0.196in} Professor Kevin McFarland has been leading the CCFR and NuTeV analysis of $\sin^2 \theta_W$. \hspace*{0.196in} Neutrino-nucleon scattering is one of the most precise probes of the weak neutral current. The ratio of neutral current (NC) to charged current (CC) cross-sections for either $\nu$ or $\nub$ scattering from isoscalar targets of $u$ and $d$ quarks can be written as: %\cite{llewellyn} $$ R^{\nu(\nub)} \equiv \frac{\sigma(\nunub N\rightarrow\nunub X)} {\sigma(\nunub N\rightarrow\ell^{-(+)}X)} = (g_L^2+r^{(-1)}g_R^2), \mbox{ where } r \equiv \frac{\sigma({\overline \nu}N\rightarrow\ell^+X)} {\sigma(\nu N\rightarrow\ell^-X)} \sim \frac{1}{2}, $$ where $g_{L,R}^2$ are the average effective left and right-handed $\nu$-quark coupling. Charm production, which affects CC cross-sections, is a major theoretical uncertainty. Hence, NuTeV measures the Paschos-Wolfenstein variable $$ R^{-} \equiv \frac{\sigma(\nu_{\mu}N\rightarrow\nu_{\mu}X)- \sigma(\nub_{\mu}N\rightarrow\nub_{\mu}X)} {\sigma(\nu_{\mu}N\rightarrow\mu^-X)- \sigma(\nub_{\mu}N\rightarrow\mu^+X)} = \frac{\Rnu-r\Rnub}{1-r}=(g_L^2-g_R^2) = \frac{1}{2}-\sin^2 \theta_W. $$ Measuring R$^-$ required a new beam line to create separate $\nu$ and $\nub$ beams. This new NuTeV SSQT beam also reduces the uncertainty in the other major systematic error, including the $\nue$ contamination in the beam. In addition, the continuous calibration of the detector reduces the detector related systematic errors. \hspace*{0.196in} NuTeV measures the ratio of short to long events in the $\nu$ and $\nub$ beams to be: $$\Rmeasnu= 0.3916 \pm 0.0007~{\rm and}~\Rmeasnub= 0.4050 \pm 0.0016.$$ A detailed leading order (LO) Monte Carlo program converts \Rmeasnu and \Rmeasnub to $ \sin^2\theta_W$. This yields: $\sin^2\theta_W^{({\rms on-shell)}}=0.2277\pm0.0016$, assuming $M_{top}$=175 GeV and $M_{Higgs}$=150 GeV. This is in very good agreement with the world average of all previous neutrino experiments of $\sin^2 \theta_W=0.2277 \pm 0.0036$ (but the NuTeV errors are smaller). The leading terms in the one-loop electroweak radiative corrections %\cite{bardin} produce the small residual dependence of our result on M$_{top}$ and M$_{Higgs}$. The Standard Model fit to all other electroweak measurements excluding neutrino experiments gives $\sin^2 \theta_W=0.2227\pm0.00037$, approximately $3\sigma$ from the NuTeV result. \hspace*{0.196in} The NuTeV result has generated a great deal of interest. If new physics is the explanation, then the NuTeV result requires new physics at the tree level. This is difficult to accomplish within the various existing "natural" models for Physics beyond the Standard Model. Hence, much of the discussion on the NuTeV anomaly has been on Standard Model explanations. Our recent efforts have been devoted to the investigation of possible conventional explanations for the 3 $\sigma$ deviation from the Standard Model. These include the NLO and NNLO QCD corrections, PDF uncertainties, charm uncertainties, isospin breaking effects, and nuclear effects. Many of these effects are very small for $R^-$. However, NuTeV does not measure $R^-$ over the entire kinematic range. Note that $R^-$ is the ratio of cross sections, while NuTeV measures the ratio of events with exprimental cuts. The difference between $R^-$ and NuTeV's measurement includes the experimental cuts on ($E_{had}$), backgrounds, cross talk between NC and CC, differences between the NC and CC acceptance, charm production, etc. \hspace*{0.196in} The QCD corrections to DIS neutrino scattering for NLO and NNLO are known. The Paschos-Wolfenstein ratio can be written to include these higher order terms. This results in a NLO QCD correction to $R_{model}^-$=-0.00033 (which brings the value about 1/4$\sigma$ closer to the SM prediction). This correction includes the low y-cut from the 20 GeV $E_had$ cut, and an effective high y-cut from being unable to see very low energy muons in CC events. \hspace*{0.196in} It has been suggested that the NuTeV results might be explained by an asymmetric strange sea, i.e. if $\sav\neq\savbar$. However, this option has been ruled out in our recent publication, PRD 65:111103,2002 (hep-ex/0203004). Here we use the NuTeV opposite sign dimuon measurement to extract the quantity $\Sav-\Savbar = -0.0027 \pm 0.0013$. Instead of explaining the NuTeV results, our measurement of the strange sea asymmetry results in an increase in the NuTeV value of $\stw$, $\Delta\stw = +0.0020 \pm 0.0009$, which increases the discrepancy with respect to Standard Model expectation to $3.7\sigma$. The possibility of an asymmetric strange sea comes from fact that the CDHSW structure functions for neutrino and antineutrinos are inconsistent with each other at large x. The CDHSW structure functions deviate from the QCD prediction at high x, and an asymmetric strange sea is given as a possible explanation. However, we note that the CCFR structure functions do not show this effect and are consistent with QCD. Therefore there is no evidence for an asymmetric strange sea from the CCFR structure function data, and a limit has been placed on this asymmetry using the dimuon data. A preliminary analysis of the CCFR data for the $s$ and $\overline{s}$ asymmetry within an NLO cross-section model indicates that the momenta carried by the strange and antistrange seas are consistent with each other within experimental uncertainties. To best answer these and other questions, we are building a full NLO $\nu$ DIS event generator. However, ee do not believe this will be a big effect. This is a calculation to order $\alpha_s$ based on a 1978 paper of Altarelli, Ellis, Martinelli. As $R_L$ is included in our Monte Carlo program, we only need to include corrections to $xF_3$. The plan will be to do light quarks first and then put in NLO charm production. A NuTeV thesis topic (D. Mason, University of Oregon) is a NLO analysis of charm production, which includes an NLO study of the strange sea. This code will included in our Monte Carlo program for $\stw$. \begin{figure}[htp] \centerline{ \psfig{figure=osclimit_combo_color.eps,width=3. in,height=6.0in}} \caption{(a) Excluded region of $\sin^2 2\alpha$ and $\delta m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations from the NuTeV analysis at 90\% confidence is the area to the right of the dark, solid curve. (b) NuTeV limits for $\overline\nu_\mu \rightarrow \overline\nu_e$. (c) Combined NuTeV limits for $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$, assuming oscillation parameters for $\nu$ and $\nub$ are the same. } \label{fig:osc} \end{figure} \item Search for $\nu_\mu(\overline\nu_\mu)$ Oscillations \\ \hspace*{0.196in} S. Avvakumov, a Rochester PhD student on NuTeV (PhD 2002), extracted limits on $\nu_\mu \to \nu_e $ and $\overline{\nu}_\mu \to \overline{\nu}_e$ oscillations for his Ph.D. thesis. His thesis, done under the supervision of Professor Arie Bodek, was submitted in Jan 2002 . The results were published in PRL 89 011804,2002. (S.~Avvakumov {\it et al.}, A search for $\nu_\mu\to\nu_e$ and $\nubar_\mu\to\nubar_e$ oscillations at NuTeV (hep-ex/0203018)). \hspace*{0.196in} Since NuTeV had separate $\nu_\mu$ and $\overline{\nu}_\mu$ beams, we search for oscillations in both running modes, without the need to assume that the oscillations parameters for neutrinos and antineutrinos are the same. The oscillations are searched for using a statistical separation of $\nu_e N$ charged current interactions in the NuTeV detector at Fermilab. The $\nu_e$ interactions are identified by the difference in the longitudinal shower energy deposition pattern of $\nu_e N \rightarrow eX$ versus $\nu_\mu N \rightarrow \nu_\mu X$ interactions. Using this technique, the absolute flux of $\nu_e$'s at the detector is measured and is compared to the flux predicted by a detailed beam line simulation. Any excess could be interpreted as a signal of $\nu_\mu \rightarrow \nu_e$ oscillations. \hspace*{0.196in} At all $\Delta m^2$, the data are consistent with no observed $\nu_\mu \rightarrow \nu_e$ oscillations. The frequentist approach is used to set a 90\% confidence upper limit for each $\Delta m^2$. The limit in $\sin^2 2\alpha$ corresponds to a shift of 1.64 units in $\chi^2$ from the minimum. The 90\% confidence upper limit is shown in Fig. \ref{fig:osc}(a) for $\nu_\mu \rightarrow \nu_e$. Also shown are limits from BNL-E734 and BNL-E776. For $\sin^2 2\alpha = 1$, $\Delta m^2 > 2.4 $~${\rm eV^2}$ is excluded, and for $\Delta m^2 \gg 1000$~${\rm eV^2}$, $\sin^2 2\alpha > 1.6 \times 10^{-3}$. In the large $\Delta m^2$ %and small $\sin^2 2\alpha$ region, NuTeV provides improved limits for $\nu_\mu \rightarrow \nu_e$ oscillations. \hspace*{0.196in} Similarly, the limit for $\overline\nu_\mu \rightarrow \overline\nu_e$ is shown Fig. \ref{fig:osc}(b). Also shown are the LSND results and preliminary results from KARMEN. For the case of $\sin^2 2\alpha = 1$, $\Delta m^2 > 2.6 $~${\rm eV^2}$ is excluded, and for $\Delta m^2 \gg 1000$~${\rm eV^2}$, $\sin^2 2\alpha > 1.1 \times 10^{-3}$. In the $\nub_\mu$ mode, our results exclude the high $\Delta m^2$ end of $\overline\nu_\mu \rightarrow \overline\nu_e$ oscillations parameters favored by the LSND experiment, without the need to assume that the oscillation parameters for $\nu$ and $\nub$ are the same. These are the most stringent experimental limits %~\cite{karm} for $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$ oscillations in the large $\Delta m^2$ region. \hspace*{0.196in} If we assume that the oscillation parameters for $\nu$ and $\nub$ are the same, we can combine our $\nu$ and $\nub$ results and compare to the CCFR results with a mixed beam. The combined NuTeV results exclude $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$ oscillations with $\sin^2 2\alpha > 0.9 \times 10^{-3}$ for large $\Delta m^2 \gg 1000$~${\rm eV^2}$. For $\sin^2 2\alpha = 1$, $\Delta m^2 > 2.2 $~${\rm eV^2}$ is excluded. These are the most stringent experimental limits for $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$ oscillations in the large $\Delta m^2$ region. \end{enumerate} %------ \noindent{\bf Neutrino Oscillations and Electron and Neutrino Scattering at Low Energies} Arie Bodek, Howard Budd, Kevin McFarland (in collaboration with Prof. Steve Manly)\\ The recent discoveries of neutrino oscillations in atmospheric neutrinos~\cite{SK} and in neutrinos from the sun~\cite{SNO,KAMLAND} motivate the detailed studies of neutrino oscillations at future high intensity neutrino beams from accelerators. The two disparate mass scales observed in oscillations from these astrophysical sources, $\dmsq_{\rms atm}\approx \evsq{2\times10^{-3}}$ and $\dmsq_{\rms solar}\sim \evsq{10^{-4}}$, along with the stringent limits on $\nuebar$ disappearance at the atmospheric $L/E$ in the CHOOZ and Palo Verde reactor experiments experiments\cite{CHOOZ,PaloVerde}, have raised the possibility that there may be an observable CP-asymmetry in $\numu\to\nue$ transitions. This rare, sub-leading transition in the neutrino flavor sector is analogous to to searching for first and third generation mixing in the quark sector, which has led to a rich phenomenology of CP-violation, meson mixing and rare decays in the quark sector. These neutrino oscillation experiments are very challenging, because of the required $L/E$ of $400$~km/GeV. The experiments require megawatt proton sources, $1-3$~GeV neutrino beams and multi-kiloton detectors to make the observations. The measurements are further complicated by the low transition probability of $\numu\to\nue$ and the need to compare to $\numubar\to\nuebar$ at high precision. This requires a detailed knowledge of the neutrino interaction cross-sections both for the dominant signal processes and for background processes, such as $\nu N\to\nu N \pi^0$ where the $\pi^0$ is misidentified as an electron in a many kiloton sampling detector. \begin{figure} %[t] \begin{center} \epsfig{figure=Fig1Mex.eps,width=6.0in,height=6.0in} \caption{Electron and muon $F_2$ data (SLAC, BCDMS, NMC, H1 94) used to obtain the parameters of the Bodek-Yang modified GRV98 $\xi_w$ fit compared to the predictions of the unmodified GRV98 PDFs (LO, dashed line) and the modified GRV98 PDFs fits (LO+HT, solid line); [a] for $F_2$ proton, [b] for $F_2$ deuteron, and [c] for the H1 and NMC proton data at low $x$.} \label{fig:f1fit} \end{center} \end{figure} \noindent{\bf Modeling Electron and Neutrino Scattering at Low Energies in the Continuum Region} A. Bodek (in collaboration with U. K. Yang) The phenomenology of neutrino cross-sections is relatively simple when $E\nu\ll 1$~GeV or when $E\nu\gg$~few GeV since these regimes are dominated by (quasi)-elastic and deep inelastic processes, respectively. However in the $1$ to few GeV region, there are contributions to the cross-section from both of these processes as well as resonance-dominated hadroproduction. A successful phenomenological approach to modeling the resonance region in electron scattering is the use of quark-hadron duality to relate quark-model cross-sections to the cross-section over the discrete resonances~\cite{bodek-yang3} as shown in Fig.~\ref{fig:f1fit}, Fig.~\ref{fig:predict} and Fig.~\ref{fig:predictD}. Figure~\ref{fig:f1fit} shows a fit by Bodek and Yang to inelastic electron and muon scattering data with a modified scaling variable and GRV98 PDFs with additional corrections (based on consideration of the Adler and Gilman sum rules~\cite{Adler}. Figure~\ref{fig:predict} and Figure~\ref{fig:predictD} compare the predictions of the fit to data in the resonance region (which is not included in the fit, as well as other data %UPDATE such a photoproduction and high energy neutrino data). All predictions assume quark-model relations, and an empirical fit to R ($R_{1998}$). This approach requires the separation of the $F_{2}$, which has a simple representation in the quark model, and $R$ whose description requires a different prescription. We plan to build successful models of neutrino scattering using this same prescription with the addition of a quark-model representation of the axial vector component of the cross-section. Since neutrino data are measured in nuclear targets, even in the quark model case, the separated vector structure functions from electron scattering, $F_{2p}$, $F_{2n}$, $R_{p}$, $R_{n}$ for bound nucleons are needed in order to understand the axial structure function in neutrino scattering experiments. % \begin{figure} %[t] \begin{center} \epsfig{figure=Fig2Mex.eps,width=6.0in,height=6.0in} \caption{ Comparisons to proton and iron data not included in the Bodek-Yang GRV98 $\xi_w$ fit. (a) Comparison of SLAC and JLab (electron) $F_{2p}$ data in the resonance region (or fits to these data) and the predictions of the GRV98 PDFs with (LO+HT, solid) and without (LO, dashed) the Bodek-Yang modifications. (b) Comparison of photoproduction data on protons to predictions using Bodek-Yang modified GRV98 PDFs. (c) Comparison of representative CCFR $\nu_\mu$ and $\overline\nu_\mu$ charged-current differential cross sections on iron at 55 GeV and the predictions of the GRV98 PDFs with (LO+HT, solid) and without (LO, dashed) Bodek-Yang modifications. } \label{fig:predict} \end{center} \end{figure} \begin{figure} %[t] \begin{center} \epsfig{figure= Fig3Mex.ps,width=5.5in,height=3.2in} \caption{ Comparisons to data on deuterium which were not included in the Bodek-Yang GRV98 $\xi_w$ fit. (a) Comparison of SLAC and JLab (electron) $F_{2d}$ data in the resonance region and the predictions of the GRV98 PDFs with (LO+HT, solid) and without (LO, dashed) our modifications. (b) Comparison of photoproduction data on deuterium to predictions using Bodek-Yang modified GRV98 PDFs (including shadowing corrections). (c) The shadowing corrections that were applied to the PDFs for predicting the photoproduction cross section on deuterium.} \label{fig:predictD} \end{center} \end{figure} \noindent{\bf Modeling Electron and Neutrino Scattering at Low Energie in the Resonance Region} Work begun in collaboration with Cynthia Keppel of Hampton University and Jefferson Laboratory. In addition to our investigation of the inelastic continuum we plan to use data taken by the Rochester group at SLAC~\cite{e140} new precise data from Jefferson Lab~\cite{ejlab} to do a combined analysis of electron-nucleon and neutrino nucleon data in the resonance region within the Feynman quark-oscillator model as done years ago (with poor precision) by Rein and Seghal~\cite{rs}. The results of the updated Rein-Seghal type of analysis will be compared to an analysis which is based on duality. The full program of studies will use precise electron scattering data, in particular $\sigma_L$ and $\sigma_T$ %UPDATE (or equivalenty $F_{2}$ and $R$) % on Hydrogen and Deuterium~\cite{ejlab}. These studies will later be supllemented with data on nuclear targets targets (materials suited for future neutrino oscillation detectors -- water \cite{jhf}, hydrocarbons \cite{numi}, liquid argon -- and steel, where the most precise high energy neutrino cross-sections have been measured \cite{CCFR}) in the relevant kinematic regime. Later, as the new generation of high rate neutrino beams at Fermilab and J-PARC become available, the approach can be directly validated with comparisons to data from high rate neutrino cross-section experiments on the same targets \cite{eois}. % \noindent{\bf Modeling Quasi-elastic Form Factors for Electron and Neutrino Scattering} Arie Bodek, Howard Budd (in Collaboration with John Arrington of Argonne National Laboratory). This work is ongoing and has only been presented in conference proceedings late 2002 and 2003. It is being written up for publication. Since Quasielastic scattering forms an important component of neutrino scattering at low energies, we have undertaken to investigate QE neutrino scattering using the latest information. Recent experiments at SLAC and Jefferson Lab (JLAB) have given very precise measurements of the vector electro-magnetic form factors for the proton and the neutron. These form factors can be related to the form factors for QE neutrino scattering by conserved vector current hypothesis, CVC. These more recent form factors can be used to give better predictions of QE scattering. The hadronic current for QE neutrino scattering is given by $$
= \overline{u}(p_2)\left[ \gamma_{\lambda}F_V^1(q^2) +\frac{i\sigma_{\lambda\nu}q^{\nu}{\xi}F_V^2(q^2)}{2M} +\gamma_{\lambda}\gamma_5F_A(q^2) +\frac{q_{\lambda}\gamma_5F_P(q^2)}{M} \right]u(p_1) $$ We do not include second class currents, so the scaler form factor $F_V^3$ and the tensor form factor $F_A^3$ are not included. Using the above current, the cross section is $$ \frac{d\sigma^{\nu,~\overline{\nu}}}{dq^2} = \frac{M^2G_F^2cos^2\theta_c}{8{\pi}E^2_{\nu}} [A(q^2) \mp \frac{(s-u)B(q^2)}{M^2} + \frac{C(q^2)(s-u)^2}{M^4}], $$ where %$$ s-u = 4ME_{\nu} + q^2 - m_l^2$$ \begin{eqnarray*} %A(q^2) & = & A(q^2)= \frac{m^2-q^2}{4M^2}\left[ \left(4-\frac{q^2}{M^2}\right)|F_A|^2 -\left(4+\frac{q^2}{M^2}\right)|F_V^1|^2 -\frac{q^2}{M^2}|{\xi}F_V^2|^2\left(1+\frac{q^2}{4M^2}\right) -\frac{4q^2ReF_V^{1*}{\xi}F_V^2}{M^2} \right], %\right. \\ & & \left. % -\frac{m_l^2}{M^2}\left(|F_V^1+{\xi}F_V^2|^2 % \right) \end{eqnarray*} $$B(q^2) = -\frac{q^2}{M^2}ReF_A^*(F_V^1+{\xi}F_V^2), $$ $$ C = \frac{1}{4}\left(|F_A|^2 + |F_V^1|^2 - \frac{q^2}{M^2}\left|\frac{{\xi}F_V^2}{2}\right|^2\right). $$ We have not shown terms in $(m_l/M)^2$, and $F_P(q^2)$ is multiplied by $(m_l/M)^2$. (Note, $F_P(q^2)$ is included in the calculations.) The formulas for $ F^1_V(q^2)$ and $cF^2_V(q^2)$ are $$ F^1_V(q^2)= \frac{G_E^V(q^2)-\frac{q^2}{4M^2}G_M^V(q^2)}{1-\frac{q^2}{4M^2}},~~~ {\xi}F^2_V(q^2) =\frac{G_M^V(q^2)-G_E^V(q^2)}{1-q^2/4M^2} $$ We use the CVC to determine $ G_E^V(q^2)$ and $ G_M^V(q^2)$ from the electron scattering form factors $G_E^p(q^2)$, $G_E^n(q^2)$, $G_M^p(q^2)$, and $G_M^n(q^2)$. $$ G_E^V(q^2)=G_E^p(q^2)-G_E^n(q^2),~~~G_M^V(q^2)=G_M^p(q^2)-G_M^n(q^2) $$ Many of the neutrino experiment have assumed the form factors are the dipole approximation. $$ G_D(q^2)=\frac{1}{(1-q^2/M_V^2)^2},~~~M_V^2=0.71~GeV^2$$ $$ G_E^p=G_D(q^2),~~~G_E^n=0,~~~G_M^p={\mu_p}G_D(q^2),~~~ G_M^n={\mu_n}G_D(q^2) $$ The axial form factor is given by $$ F_A(q^2)=\frac{g_A}{(1-\frac{q^2}{M_A^2})^2 } $$ This form factor needs to be determined from QE neutrino scattering. Older experiments used $g_A=-1.23$, however the current value is -1.267. The world average from neutrino experiments is $M_A$ = 1.026 $\pm$ 0.02 GeV. The value of $M_A$ depends on the electro-magnetic form factors. Since we are updating these form factors, we need to determine a new value of $M_A$ using these latest form factors and $g_A$. $M_A$ can also be determined from pion electro-production, which gets 1.069 $\pm$ 0.016 GeV (it is expected that this determination is not as reliable as that from neutrino data because of theoretical corrections. These corrections bring the value into closer agreement with the value as measured in neutrino reactions). From PCAC, the pseudoscaler form factor $F_P$ is $$ F_P(q^2)=\frac{2M^2F_A(q^2)}{M_{\pi}^2-q^2}. $$ $F_P(q^2)$ is multiplied by $(m_l/M)^2$ so its effect is very small except at very low energy, \(< .2 GeV\). \begin{figure} \begin{center} \epsfxsize=3.01in \mbox{{\epsffile{show_gep_ratio.eps}}\epsfxsize=3.01in{\epsffile{show_gmp_ratio.eps}}}\\ \end{center} \caption{Our fits for $G_E^p/G_D$ and $G_M^p/{\mu}_{p}G_D$. The fits with and without the polarization measurements are shown. Polarization data is shown in cyan.} \label{show_gep} \end{figure} \begin{figure} \begin{center} \epsfxsize=4.01in \mbox{\epsffile[79 408 549 720]{show_gepgmp_ratio.eps}} \end{center} \caption{ Ratio of $G_E^p$ to $G_M^p$ as extracted by Rosenbluth measurements and from polarization measurements. } \label{gegm_proj} \end{figure} We have used almost all data from SLAC and JLAB to determine the form factors. Form factors can be determined by using older technique of Rosenbluth separation (cross section) or the newer technique of polarization transfer from JLAB. Figure~\ref{show_gep} shows the ratio of our fits divided by the dipole, $G_D$. The JLAB polarization measurement does not directly measure the form factors, but measures the ratio $G_E^p$/$G_M^p$. As we see from figure~\ref{gegm_proj}, $G_E^p$/$G_M^p$ is flat vs $Q^2$ ($Q^2=-q^2$) for the cross section measurement. However, $G_E^p$/$G_M^p$ decreases for the polarization measurement. Although the polarization measurement is believed to have smaller systematic error especially at high $Q^2$, the origin of this disagreement is not known. Experiments at JLAB hope to resolve this disagreement. We fit electron scattering data to an inverse polynomial $$ Poly^{-1}(Q^2)=\frac{1}{1+a_2Q^2+a_4Q^4+a_6Q^6+...}. $$ Table~\ref{JRA_HallA_coef} shows the results of our fit. These fits uses both cross section data and polarization transfer data from JLAB. In addition, we have fits which just use the cross section data. \begin{table} \begin{center} \begin{tabular}{|l|c|c|c|c|c|c|} \noalign{\vspace{-8pt}} \hline & $a_2$ & $a_4$ & $a_6$ & $a_8$ & $a_{10}$ & $a_{12}$ \\ \hline $G_E^p$ & 3.253 & 1.422 & 0.08582 & 0.3318 & -0.09371 & 0.01076 \\ $G_M^p$ & 3.104 & 1.428 & 0.1112 & -0.006981 & 0.0003705 & -0.7063E-05 \\ $G_M^n$ & 3.043 & 0.8548& 0.6806 & -0.1287 & 0.008912 & \\ \hline \end{tabular} \end{center} \caption{ The coefficients of the inverse polynomial for the $G_E^p$, $G_M^p$, and $G_M^n$. This fit uses both cross section data and polarization data from electron scattering. } \label{JRA_HallA_coef} \end{table} Previous experiments assumed $G_E^n(q^2)$ = 0. Since the neutron has no charge, $G_E^n(q^2)$ must be zero at $q^2$ = 0. However, it doesn't have to zero for $q^2 \neq$ 0. New JLAB polarization transfer data gives a precise non-zero value of $G_E^n(q^2)$. Our analysis uses $G_E^n(q^2)$ from Krutov et. al. (Hep-ph/0202183). $$G_E^n = -\mu_n\frac{a\tau}{1-b\tau}G_D(q^2),~~~\tau=\frac{Q^2}{4M^2}.$$ \begin{figure} \begin{center} \epsfxsize=3.01in \mbox{{\epsffile[7 5 550 400]{ratio_JKJJ_D0DD.eps}} \epsfxsize=3.01in{\epsffile[7 5 550 400]{ratio_JhaKJhaJ_D0DD.eps}}}\\ \end{center} \caption{ Ratio of cross section vs energy using the most updated form factors vs the dipole approximation. The left plot uses the Rosenbluth separation data and the polarization data. The right plot uses Rosenbluth separation data. } \label{ratio_JKJJ_D0DD} \end{figure} Figure~\ref{ratio_JKJJ_D0DD} shows the ratio of the QE cross section using the most updated form factors vs the dipole approximation. The most updated form factor are our fits to the form factors for $G_M^p$, $G_E^p$, and $G_M^n$ and Krutov $G_E^n$. The left plot uses the cross section data and the polarization data. The right plot uses Rosenbluth separation data. The cross section for neutrino QE scattering is independent of the polarization data. The differences between the polarization data and the cross section data are at high $Q^2$, while the form factors contribute to the cross section at low $Q^2$. There is a big effect in the cross section between using the latest form factors or not. The difference is 3\% at high energy and can become as much as 6\% at 1 GeV. Figure~\ref{ratio_JKJJ_D0DD} shows the difference between $G_E^n$ = Krutov vs $G_E^n$ = 0. We see all of the difference at high energy and most of the the difference at low energy is due to $G_E^n$. At low energy, which are the energies for neutrino oscillation experiments, the other form factors are important. \begin{figure} \begin{center} \epsfxsize=3.01in \mbox{{\epsffile[7 5 550 400]{ratio_DKDD_D0DD.eps}} \epsfxsize=3.01in{\epsffile[7 5 550 400]{ratio_Jha0JhaJ_D0DD.eps}}}\\ \end{center} \caption{ Ratio of cross section vs energy using different sets of form factors vs energy. The left plot looks at the difference between using $G_E^n$ = Kurtov vs $G_E^n$ = 0. The right plots looks at the difference between using our fits for $G_E^p$, $G_M^p$, and $G_N^n$ vs the dipole approximation. } \label{ratio_DKDD_D0DD} \end{figure} A 1\% increase in either $M_A$ or $|g_A|$ increases the cross section about 1\%. As the old value of $g_A$=-1.23. and more recent values of -1.267 increases the cross section by about 2.5\%. In addition the more recent value of $M_A$ of 1.02 vs the 1.032 causes the cross section to fall by about 1\%. $M_P$ has almost no effect on the cross section except at very low $E_{\nu}$. Previous neutrino experiment, mostly bubble chambers, extract $M_A$ using the best known assumptions at the time. Changing these assumptions changes $M_A$. Hence, we use published data to extract a corrections to $M_A$ using our form factors. They give their data in histograms of corrected events. Their flux is shown in figures, which we parameterize using a spline fit. We calculate the $Q^2$ distribution of their data and fit their data for $M_A$. We determine $M_A$ using their assumptions and our assumptions. Figure~\ref{Baker_ma107_nor99} shows a histograms of the $Q^2$ distribution for both Baker et. al. and Kitagaki et. al. Our curves agree very well with their curves. In addition our curves agree very well with Barish et al, but not quite so well with Miller et al. As Miller gives the final result of Barish, adding 3 times the data, they should be using the same code. Therefore, the discrepency between Miller and Barish is puzzling. \begin{figure} \begin{center} \epsfxsize=3.01in \mbox{{\epsffile[124 208 538 587]{Baker_81_ma107_nor100.eps}} \epsfxsize=3.01in{\epsffile[124 208 538 587]{Kit_83_105_flux_spline_area.eps}}}\\ \end{center} \caption{ $Q^2$ distribution from Baker et. al. and Kitagaki et. al. The red curve is our calculation using their assumptions. The blue curve is their calculation taken from their $Q^2$ distribution histogram.} \label{Baker_ma107_nor99} \end{figure} We fit for $M_A$ using their assumptions and our assumptions, and we determine the shift using our assumption. They calculate $M_A$ using unbinned maximum likelihood, which we can't do since we do not have the events. We use binned maximum likelihood. These experiment use a dipole correction from Ollson et. al. [PRD 17 2938 (1978)]. Table~\ref{MA_values} gives the result of the calculation. We also determine $M_A$ using the dipole. We agree with the value of $M_A$ of Baker, but disagree with the values of Barish, Miller, and Kitagaki. However, as previously stated our $Q^2$ distributions agree very well with Baker, Barish, and Kitagaki. Maybe their unbinned likelihood fit as opposed to our binned likelihood fit creates the difference. But then why did we get Baker's value correct? We do not have an explanation for the discrepency in $M_A$. The table indicates we should shift the value of $M_A$ determined from deuterium down by 0.025 GeV. \begin{table} \begin{center} \begin{tabular}{|l|c|c|c|c|c|c|} \noalign{\vspace{-8pt}} \hline & Their & Our Fit & Our & Our - Their & Our - Dipole \\ & Fit & Their Assum. & Assum. & $\delta M_A $& $\delta M_A$ \\ \hline Barish 77 & 1.01 $\pm$ 0.09 &1.087 $\pm$ 0.10 & 1.058 & -0.029 & -0.048 \\ Miller 82 & 1.05 $\pm$ 0.05 &1.118 $\pm$ 0.055& 1.091 & -0.027 & -0.046 \\ Kitagaki 83& 1.05$_{-0.16}^{+0.12}$& 1.139$\pm$ 0.10& 1.118 & -0.021 & -0.052 \\ Baker 81 & 1.07 $\pm$ 0.06 & 1.075 & 1.050 & -0.025 & -0.049 \\ \hline \end{tabular} \end{center} \caption{ Fit values and shifted values of $M_A$ (GeV) from deuterium experiments. Column 2 gives the fit values of $M_A$ from their papers. For Barish and Miller, we give their "shape fit" value, since this value most closely reflects how we can calculate their $M_A$. Column 3 gives our fit value of $M_A$ using their assumptions. Column 4 gives our fit value of $M_A$ with our assumptions. Column 5 gives $\delta M_A$ between our assumptions minus the experiments assumptions. Column 6 gives $\delta M_A$ between using our form factors and a dipole form factors. For this difference the value of $g_A$ is kept the constant.} \label{MA_values} \end{table} \begin{figure} \begin{center} \epsfxsize=5.01in \mbox{\epsffile[0 0 567 405]{elas_JhaKJhaJ.eps}} \end{center} \caption{ The QE cross section and $\nu$ and $\overline{\nu}$ along with data from various experiment. The calculation uses the latest form factors and $M_A$ = 1.00 and $G_A$ = -1.267} \label{elas_JhaKJhaJ} \end{figure} Figure~\ref{elas_JhaKJhaJ} shows the QE cross section for $\nu$ and $\overline{\nu}$ using our most up to date assumptions. We have used form factors from cross section and polarization measurements. We used $G_A$ = -1.267. We have scaled down $M_A$ from the old best fit of $M_A$=1.026 $\pm$ 0.021 to $M_A$=1.00 (which would have been obtained with the best vector form factors known today). Even with the most up to date assumptions on form factors the agreement between data and predicction is not spectacular. The data - flux errors are 10\%. The anti-neutrino data, which is on nuclear targets, is below the curves. This is most likely due to nuclear physics effects. Over the next year, we plan to study the nuclear corrections (using models which work in electron scattering~\cite{ejlab}). 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