Modeling Quasi-elastic Form Factors for Electron and Neutrino Scattering

H. Budd^{a}, A. Bodek^{a} and J. Arrington^{b}

aDepartment of Physics and Astronomy, University of Rochester, Rochester, New York 14618, USA

bArgonne National Laboratory, Argonne, Illinois 60439, USA

We calculate the total and differential quasielastic cross sections for neutrino and antineutrino scattering on nucleons using up to date fits to the nucleon elastic electromagnetic form factors G_{E}^{p}, G_{E}^{n}, G_{M}^{p}, G_{M}^{n}, and weak and pseudoscalar form factors. We find that using the updated non-zero value of G_{E}^{n} has a significant effect on both the total and differential neutrino and antineutrino quasielastic cross sections. Previous extractions of the weak axial form factor from neutrino scattering data are sensitive to the assumptions that were used for the vector form factors. We perform a re-analysis of previous neutrino data using updated form factors and obtain updated value of the axial vector mass. (Presented by Howard Budd at NuInt02, Dec. 2002, Irvine, CA, USA [1])

1. INTRODUCTION

Experimental evidence for oscillations among the three neutrino generations has been recently reported [2]. Since quasielastic (QE) scattering forms an important component of neutrino scattering at low energies, we have undertaken to investigate QE neutrino scattering using the latest information on nucleon form factors.

Recent experiments at SLAC and Jefferson Lab (JLab) have given precise measurements of the vector electromagnetic form factors for the proton and 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 for QE neutrino scattering.

2. EQUATIONS FOR QE SCATTERING

The hadronic current for QE neutrino scattering is given by [3]

< p(p_{2})|J_{λ}^{+}|n(p_{1}) >=

\overline{u}(p_{2}) [γ_{λ}F_{V}^{1} (q^{2}) + \frac{iσ_{λν}q^{ν}ξF_{V}^{2} (q^{2})}{2M}

+ γ_{λ}γ_{5}F_{A}(q^{2}) + \frac{q_{λ}γ_{5}F_{P} (q^{2})}{M}] u(p_{1}),

where q = k_{ν} − k_{μ}, ξ = (μ_{p} − 1) − μ_{n}, and M = (m_{p} + m_{n})/2. Here, μ_{p} and μ_{n} are the proton and neutron magnetic moments. We assume that there are no second class currents, so the scalar form factor F_{V}^{3} and the tensor form factor F_{A}^{3} need not be included. Using the above current, the cross section is

\frac{dσ^{ν, \overline{ν}}}{dq^{2}} = \frac{M^{2}G_{F}^{2} cos^{2}θ_{c}}{8πE_{ν}^{2}} ×

[A(q^{2}) ∓ \frac{(s − u)B(q^{2})}{M^{2}} + \frac{C(q^{2})(s − u)^{2}}{M^{4}} ] ,

where

A(q^{2}) = \frac{m^{2} − q^{2}}{4M^{2}} [(4 − \frac{q^{2}}{M^{2}} ) |F_{A}|^{2}

− (4 + \frac{q^{2}}{M^{2}} ) |F_{V}^{1} |^{2} − \frac{q^{2}}{M^{2}} |ξF_{V}^{2} |^{2} (1 + \frac{q^{2}}{4M^{2}} )

−\frac{4q^{2}ReF_{V}^{1∗}ξF_{V}^{2}}{M^{2}} ] ,

B(q^{2}) = −\frac{q^{2}}{M^{2}} ReF_{A}^{∗}(F_{V}^{1} + ξF_{V}^{2} ),

C(q^{2}) = \frac{1}{4} (|F_{A}|^{2} + |F_{V}^{1} |^{2} − \frac{q^{2}}{M^{2}} |\frac{ξF_{V}^{2}}{2} |^{2}) .

Although we have have not shown terms of order (m_{l}/M)^{2}, and terms including F_{P} (q^{2}) (which is multiplied by (m_{l}/M)^{2}), these terms are included in our calculations [3].) The form factors F_{V}^{1} (q^{2})

g_{A}  -1.267
G_{F}  1.1803×10^{−5} GeV^{−2}
cos θ_{c}  0.9740
μ_{p}  2.793 μ_{N}
μ_{n}  -1.913 μ_{N}
ξ  3.706 μ_{N}
M_{V}^{2}  0.71 GeV^{2}

Table 1 The most recent values of the parameters used in our calculations (Unless stated otherwise).

and ξF_{V}^{2} (q^{2}) are given by:

F_{V}^{1} (q^{2}) = \frac{G_{E}^{V}(q^{2}) − \frac{q^{2}}{4M^{2}} G_{M}^{V}(q^{2})}{1 − \frac{q^{2}}{4M^{2}}},

ξF_{V}^{2} (q^{2}) = \frac{G_{M}^{V}(q^{2}) − G_{E}^{V}(q^{2})}{1 − \frac{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}).

Previously, many neutrino experiment have assumed that the vector form factors are described by the dipole approximation.

G_{D}(q^{2}) = \frac{1}{(1 − \frac{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} = μ_{p}G_{D}(q^{2}), G_{M}^{n} = μ_{n}G_{D}(q^{2}).

We refer to the above combination of form factors as ‘Dipole Form Factors’. It is an approximation that is improved upon in this paper. We will refer to our updated form factors as ‘BBA-2003 Form Factors’ (Budd, Bodek, Arrington). Table 1 summarizes the most up to date values of the coupling constants and magnetic moments that we use in our calculations. Note that G_{E}^{p}, G_{M}^{p}, and G_{E}^{n} are Figure 1. Our fits to G_{E}^{p}/G_{D}, using cross section data only (solid), and with both the cross section and polarization transfer data (dashed). The diamonds are the from Rosenbluth extractions and the crosses are the Hall A polarization transfer data. Note that we fit to cross sections, rather than fitting directly to the extracted values of G_{E}^{p} shown here.

positive, while G_{M}^{n} and the axial form factor F_{A} are negative.

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 extracted from QE neutrino scattering. However, at low Q^{2}, this form factor can also be extracted from pion electroproduction data.

Previous neutrino experiments used g_{A}=−1.23, while the best current value is −1.267. The world average from neutrino experiments for M_{A} is 1.026 ± 0.020 GeV [4]. The value of M_{A} extracted from neutrino experiments depends on both the value of g_{A} and the values of the electromagnetic form factors which are assumed in the extraction process. Since we are updating these form factors, new values of M_{A} are extracted from previous neutrino data using the better known values for g_{A} and the vector form factors.

M_{A} can also be determined from pion electroproduction, which yields a world average value of 1.069 ± 0.016 GeV [4]. This value should be reduced by 0.055 GeV when compared to M_{A} as Figure 2. Our fits to G_{M}^{p}/μ_{p}G_{D}. The lines and symbols have the same meaning as Figure 1.

measured in neutrino data because of additional corrections [4]. Therefore, pion electroproduction experiments predict that M_{A} should be 1.014 ± 0.016 GeV in neutrino scattering.

In this communication, we show that the value of 1.026 ± 0.02 GeV [4] as measured from the average of all neutrino scattering should also be reduced by 0.025 GeV to account for incorrect vector form factors used in the past. This corrected value of 1.001 ± 0.020 GeV is in good agreement with the theoretically corrected value from pion electroproduction of 1.014 ± 0.016 GeV.

From PCAC, the pseudoscalar form factor F_{P} is predicted to be

F_{P} (q^{2}) = \frac{2M^{2}F_{A}(q^{2})}{M_{π}^{2} − q^{2}}.

In the expression for the cross section, F_{P} (q^{2}) is multiplied by (m_{l}/M)^{2}. Therefore, in muon neutrino interactions, this effect is very small except at very low energy, below 0.2 GeV. The effect is larger, about 5%, for tau neutrino interactions.

3. UPDATED FORM FACTORS

We have used an updated fit to the proton electromagnetic form factors. The fit is similar to the one described in Ref. [5], but using a slightly different fitting function (described below), and including additional data to constraint the fit at low Q^{2} values. Form factors can be determined Figure 3. Our fit to G_{M}^{n}/μ_{n}G_{D}. The lines and symbols have the same meaning as Figure 1.

from cross sections using the standard Rosenbluth separation technique [5], which is sensitive to radiative corrections, or from polarization measurements using the newer polarization transfer technique [6]. The polarization measurements do not directly measure the form factors, but measure the ratio G_{E}/G_{M}. Figures 1, 2, and 3 show the ratio of our fits divided by the dipole form, G_{D}.

Figure 4 shows our fits to μ_{p}G_{E}^{p}/G_{M}^{p}. The fit including only cross section data is roughly flat versus Q^{2} (Q^{2} = −q^{2}), while ratio decreases with Q^{2} in the combined fit to cross section and polarization transfer data. Although the polarization transfer measurement is believed to have smaller systematic error, especially at high Q^{2}, the origin of this disagreement is not known. If this disagreement comes from radiative corrections to the electron, in particular two-photon exchange terms, then the polarization transfer extraction will give the correct ratio, but the overall scale of G_{E}^{p} at low Q^{2} would be shifted down by ≈3%. Because the fit is constrained as Q^{2} → 0, there will not be an overall shift in G_{E}^{p} at low Q^{2}, but there will be some uncertainty in the low Q^{2} behavior. Current experiments at JLab aim to better understand the source of the disagreement by looking at the recoil proton in elastic electronproton scattering, thus minimizing the sensitivity to the dominant sources of uncertainty in previous Rosenbluth separations. However, since this discrepancy is most prominent at high Q^{2}, and the fit is constrained at low Q^{2}, it has only a rel-

data  a_{2}  a_{4}  a_{6}  a_{8}  a_{10}  a_{12}
G_{E}^{p} CS + Pol 3.253 1.422 0.08582 0.3318  -0.09371  0.01076
G_{M}^{p} CS + Pol 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
G_{E}^{p}  CS  3.226 1.508 -0.3773 0.6109  -0.1853  0.01596
G_{M}^{p}  CS  3.188 1.354 0.1511 -0.01135 0.0005330 -0.9005E-05

Table 2 The coefficients of the inverse polynomial fits for the G_{E}^{p}, G_{M}^{p}, and G_{M}^{n}. Fits using cross section data only, and using both cross section data and the Hall A polarization transfer data are shown separately. Note that these different polynomials replace G_{D} in the expression for G_{E}^{p}, G_{M}^{p}, and G_{M}^{n}. The first three rows of the table along with the fit of G_{M}^{p} Krutov et. al. [7] (see text) will be referred to as ‘BBA-2003 Form Factors’.

atively small effect on the neutrino QE scattering cross section.

To account for the fact that deviations from the dipole form are different for each of the different form factors, we fit electron scattering data for each of the form factors to an inverse polynomial

G_{E,M}^{N}(Q^{2}) = \frac{G_{E,M}^{N}(Q^{2} = 0)}{1 + a_{2}Q^{2} + a_{4}Q^{4} + a_{6}Q^{6} + ...}.

Table 2 shows the parameters of our fit. We have done fits using cross section data only (for the proton) and fits using both both cross section data and polarization transfer data from JLab Hall A. For G_{E}^{p}, the parameters in Table 2 are used for Q^{2} < 6 GeV^{2}. For Q^{2} > 6 GeV^{2}, the ratio of G_{E}^{p}/G_{M}^{p} is assumed to be constant:

G_{E}^{p}(Q^{2}) = G_{M}^{p}(Q^{2})\frac{G_{E}^{p}(6 GeV^{2})}{G_{M}^{p}(6 GeV^{2})}

Since the neutron has no charge, G_{E}^{n} must be zero at q^{2}=0, and previous neutrino experiments assumed G_{E}^{n}(q^{2})=0 for all q^{2} values. However, it is non-zero away from q^{2}=0, and its slope at q^{2}=0 is known precisely from neutron-electron scattering. At intermediate Q^{2}, recent polarization transfer data give precise values of G_{E}^{n}(q^{2}). Our analysis uses the parameterization of Krutov et. al. [7]:

G_{E}^{n}(Q^{2}) = −μ_{n} \frac{aτ}{1 + bτ} G_{D}(Q^{2}), τ = \frac{Q^{2}}{4M^{2}} ,

with a = 0.942 and b = 4.61. This parameterization is very similar to that of Galster et al. [8], as shown in Figure 5.

Figure 4. Ratio of G_{E}^{p} to G_{M}^{p} as extracted by Rosenbluth measurements and from polarization measurements. The lines and symbols have the same meaning as Figure 1.

Figure 5. Data and fits to G_{E}^{n}. The dashed line is the Galster et al. fit [8], and the solid line is the Krutovet al. fit [7].

Figure 6. Ratio versus energy of predicted neutrino (antineutrino) QE cross section using BBA-2003 Form Factors to the prediction using the dipole approximation with G_{E}^{n}=0.

The first three rows of Table 2, along with the fit of G_{E}^{n} of Krutov et. al. [7], will be referred to as ‘BBA-2003 Form Factors’. For BBA-2003 Form Factors, both the cross section and polarization data are used in the extraction of G_{E}^{p} and G_{M}^{p}.

4. CROSS SECTIONS AND FITS TO M_{A}

Figure 6 shows the ratio versus neutrino energy of the predicted neutrino (antineutrino) QE cross section using our BBA-2003 Form Factors to the prediction using the Dipole Form Factors. The same comparison was performed using the proton form factor extractions that included only the cross section data (i.e. excluding the polarization results). The results were nearly identical: the maximum difference between the cross sections is less then 0.3%. This is because the form factors extracted from the polarization transfer data and from the electron scattering cross section data are different only at high Q^{2}, while the neutrino cross sections are mostly sensitive to the form factors at low Q^{2}.

There is a large difference in the predicted neutrino (antineutrino) cross section between using the BBA-2003 Form Factors and using the Dipole Form Factor approximation. The difference is 3% at high energy and can become as much as Figure 7. Ratio versus energy of the predicted neutrino (antineutrino) QE cross section using G_{E}^{n} from Krutov [7] to the prediction using G_{E}^{n}=0. In both cases, the dipole approximation for the other form factors.

Figure 8. Ratio versus energy of predicted neutrino (antineutrino) QE cross section using our BBA-2003 Form Factor fits for G_{E}^{p}, G_{M}^{p}, and G_{N}^{n} to the prediction using the dipole approximation. In both cases, we use G_{E}^{n}=0.

6% at 1 GeV. As seen in Figure 7, which shows the difference between G_{E}^{n} = Krutov and G_{E}^{n} = 0, all the difference at high energy and most of the the difference at low energy is due to G_{E}^{n}. At the low energy region which is of interest for neutrino oscillation experiments, both a non-zero G_{E}^{n} and the deviations from the dipole form are important. This is also the case for the extraction of the axial form factor from neutrino data, since most of the neutrino differential cross section data are at low Q^{2}.

A 1% increase in either M_{A} or |g_{A}| increases the cross section about 1%. Replacing the old value of g_{A}=−1.23 with the more precise value of g_{A}=−1.267 increases the cross section by about 2.5%. Using the more recent value of M_{A} of 1.02 instead of the older value of 1.032 decreases the predicted cross section about 1%. F_{P} has almost no effect on the cross section except at very low E_{ν}. Therefore, even a very conservative error [4] on F_{P} of 50% has very little effect.

Previous neutrino measurements, mostly bubble chamber experiments, extracted M_{A} using the best known assumptions at the time. Changing these assumptions changes the extracted value of M_{A}. Hence, M_{A} needs to be updated using new form factors and up-to-date couplings. In this communication we will attempt to update the results from three previous deuterium bubble chamber experiments. These are Baker et al. [9], Barish et al. [10], Miller et al. [11], and Kitagaki et al. [12]. Barish et al. and Miller et al. are the same experiment, with the analysis of Miller et al. including the full data set, roughly three times the statistics included in the original analysis.

We start by calculating the shape of the Q^{2} distribution using the same form factors and couplings as used in the original extractions (including M_{A}). The flux is extracted from the flux figures shown in the original papers, which we parameterize using a spline fit. We extract the data and curves from their publications by picking points off the plots, and fitting the points on the curves to a spline fit. These experiments did not use a pure dipole approximation, but included a correction to the dipole form as parameterized by Olsson et al. [14]. They use g_{A}=−1.23, M_{V} =0.84 GeV (yielding M_{V}^{2} =0.7056 GeV^{2} in-Figure 9. Q^{2} distribution from Baker et al. [9]. The dotted curve is their calculation taken from their Q^{2} distribution histogram. The dashed curve is our calculation using their assumptions.

Figure 10. Q^{2} distribution from Barish et al. [10]. The dotted curve is their calculation taken from their Q^{2} distribution histogram. The dashed curve is our calculation using their assumptions.

Figure 11. Q^{2} distribution from Miller et al. [11]. The dotted curve is their calculation taken from their Q^{2} distribution histogram. The dashed curve is our calculation using their assumptions.

Figure 12. Q^{2} distribution from Kitagaki et al. [12]. The dotted curve is their calculation taken from their Q^{2} distribution histogram. The dashed curve is our calculation using their assumptions. stead of M_{V}^{2} =0.71 GeV^{2}), and a D_{2} Paulisuppression correction from Singh et al. [15]. Our calculations and their calculations are compared in Figures 9, 10, 11, and 12. In these figures, our curves are normalized such that they match the previous curves at one point in the low Q^{2} region, so that we can compare the Q^{2}-dependence of the spectra. The y-axis is weighted events/bin (corrected for efficiencies). We reproduce the shape (Q^{2}-dependence) of the calculations of Baker et al. [9], Barish et al. [10], and Kitagaki et al. [12], but not Miller et al. [11]. As Miller et al. gives an updated result of Barish et al., they should be using the same calculation. Therefore, we do not understand the origin of the curve shown in Miller et al.

Having reproduced the calculations under the same assumptions as in the original extractions, we perform our own extraction of M_{A}, using our calculation of the cross sections, but using the same input (g_{A} and form factors) as assumed in the original extractions. Due to inefficiencies in reconstruction for very low Q^{2} events, they do not use the first bin for fitting. Hence, we do not use the first bin for fitting or normalization either. We perform a binned maximum likelihood using a formula from the Particle Data Group [16]. The experiments performed an unbinned likelihood fit, which we cannot reproduce since we do not have the individual events. Table 3 gives the results of these fits. Using the same assumptions, we reproduce the fitted value of M_{A} from Baker et al., while disagreeing somewhat with the values from Barish et al., Miller et al., and Kitagaki et al. The difference may come from the fact that we are forced to use a binned likelihood fit, rather than being able to reproduce their unbinned fit.

Figures 13 and 14 show the difference between using their value of M_{A} and our value of M_{A} for Kitagaki et al. and Barish et al. The plot for Kitagaki et al. appears to show that our value of M_{A}=1.19 GeV is a better fit than their value of M_{A}. For Barish et al. its not clear which is a better fit. As previously stated, our Q^{2} distribution is slightly different than that of Miller et al. for the same M_{A}. Figure 15 shows their calculation for their best fit M_{A} versus our calculation using our best fit M_{A}. The two shapes agree very Figure 13. Q^{2} distribution from Kitagaki et al. [12]. The dash curve is our calculation using our fit value of M_{A}=1.19 GeV. The solid curve is our calculation using their fit value of M_{A}=1.05 GeV.

Figure 14. Q^{2} distribution from Barish et al. [10]. The dash curve is our calculation using our fit value of M_{A}=1.075 GeV. The solid curve is our calculation using their fit value of M_{A}=1.01 GeV.

Figure 15. Q^{2} distribution from Miller et al. [11]. The dotted curve is their calculation using their fit value of M_{A}=1.05 GeV. The dash curve is our calculation using our fit value of M_{A}=1.117 GeV.

well. Hence, we are able to reproduce the best fit shape of Miller et al. fit, but not with their value of M_{A}.

Finally, we extract M_{A} for each of these experiments using our calculations with the updated BBA-2003 Form Factors and g_{A} value. Comparing this to our extraction with their input parameters, we obtain change in M_{A} due to using updated values for g_{A} and the form factors. Because we sometimes obtain slightly different values of M_{A}, even with the same input parameters, we take the difference between our extraction with old and new form factors as the modification that should be applied to the previous extractions, which were able to do a more detailed comparison to their data. Table 3 gives the results of these fits, which indicate that we should shift the value of M_{A} determined from deuterium by −0.025 GeV from the value quoted by these experiments. We also show that a shift in M_{A} of −0.050 GeV is required in going from the Dipole Form Factors to BBA-2003 Form Factors, keeping g_{A} constant. If only cross section data were used for the form factor fits (Table 1, rows 4 and 5), the value of M_{A} would go up by 0.002 GeV, a

M_{A}  updated M_{A} updated M_{A}  ΔM_{A}  ΔM_{A}
(published) old params. new params. new–old BBA-2003–Dipole
Baker 1981 [9]  1.07 ± 0.06 1.079 ± 0.056 1.055 ± 0.055 −0.024  −0.049
Barish 1977 [10]  1.01 ± 0.09 1.075 ± 0.10 1.049 ± 0.099 −0.026  −0.046
Miller 1982 [11]  1.05 ± 0.05 1.117 ± 0.055 1.090 ± 0.055 −0.027  −0.046
Kitagaki 1983 [12] 1.05_{−0.16}^{+0.12}  1.194_{−0.11}^{+0.10}  1.175_{−0.11}^{+0.10}  −0.019  −0.050

Table 3 Published and updated extractions of M_{A} (GeV) from deuterium experiments. The first value of M_{A} is the values extracted in the original publications. For Barish and Miller, we give their ‘shape fit’ value, since this value most closely reflects how we can calculate their M_{A}. The second value of M_{A} is from the analysis presented here, using the same input parameters (form factors and g_{A}) as in the publications, while the third uses the updated parameters from tables 2 and 1. The last two columns show the change in M_{A} between the new and old input parameters, and the change when comparing the BBA-2003 and Dipole Form Factors (with g_{A} fixed).

Figure 16. A comparison of the Q^{2} distribution using 2 different sets of form factors. The data are from Baker et al. [9]. The dotted curve uses Dipole Form Factors with M_{A}=1.10 GeV. The dashed curve uses BBA-2003 Form Factors with M_{A}=1.05 GeV.

small effect.

Figure 16 shows the Q^{2} distribution for Dipole Form Factors and M_{A}=1.10 GeV with the distribution for BBA-2003 Form Factors and M_{A}=1.050 GeV. When we modify the electromagnetic form factors, the modification in M_{A} not only reproduces the original yield, it also reproduces the Q^{2} distribution. Because there is no modification of the Q^{2} dependence when strength is shifted between the electromagnetic and axial form factors, we conclude that the use of Dipole Form Factors will lead to an error in M_{A} of 0.050 GeV, independent of the details of the experiment.

Figures 17 and 18 show the QE cross section for ν and \overline{ν} using our most up to date assumptions. The normalization uncertainty in the data is approximately 10%. We have used BBA-2003 Form Factors, and have scaled down M_{A} from the old best fit of M_{A}=1.026 ± 0.021 GeV to M_{A}=1.00 GeV (which would have been obtained with the best vector form factors known today). The solid curve uses no nuclear correction, while the dotted curve [24] uses a NUANCE [26] calculation of a Smith and Moniz [25] based 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 binding energy and 220 MeV Fermi momentum. The updated form factors improve the agreement with neutrino QE cross section data and give a reasonable description of the cross sections from deuterium. However, the data from heavy targets, including all of the antineutrino data, are systematically below the calculation, even with the Fermi gas nuclear corrections. The only experiment on heavy nuclei which agrees with the calculation is SKAT [18] with neutrinos. The Fermi gas nuclear correction may not be sufficient. Tsushima et al. [27] Figure 17. The QE neutrino cross section along with data from various experiments. The calculation uses M_{A}=1.00 GeV, g_{A}=−1.267, M_{V}^{2} =0.71 GeV^{2} and BBA-2003 Form Factors. The solid curve uses no nuclear correction, while the dotted curve [24] uses a Fermi gas model for carbon with a 25 MeV binding energy and 220 Fermi momentum. The lower plot is identical to the upper plot with the E_{ν} axis limit changed to 2 GeV. The data shown are from FNAL 1983 [12], ANL 1977 [10], BNL 1981 [9], ANL 1973 [17], SKAT 1990 [18], GGM 1979 [19], LSND 2002 [20], Serpukov 1985 [21], and GGM 1977 [22].

Figure 18. The QE antineutrino cross section along with data from various experiments. The calculation uses M_{A}=1.00 GeV, g_{A}=−1.267, M_{V}^{2} =0.71 GeV^{2} and BBA-2003 Form Factors. The solid curve uses no nuclear correction, while the dotted curve [24] uses a Fermi gas model for carbon with a 25 MeV binding energy and 220 MeV Fermi momentum. The data shown are from SKAT 1990 [18], GGM 1979 [23], Serpukov 1985 [21], and GGM 1977 [22].

studied the effect of nuclear binding on the nucleon form factors. They stated that modifications in bound nucleon form factors reduce the cross section relative to free nucleon form factors by 8%. We plan to study the nuclear corrections, adopting models which have been used in precision electron scattering measurements from nuclei at SLAC and JLab. In addition, we will be updating the extraction of M_{A} from other experiments, using the updated versions of the input parameters and electromagnetic form factors.

Both the overall cross section and the Q^{2}- dependence depend on the form factors chosen. However, even for input form factors that yield identical Q^{2}-dependences, the cross section can differ significantly. Figure 19 compares the cross section for two sets of form factors with identical Q^{2} distribution (shown in Fig. 16). The figure shows that a cross section error as large as 7.5% can occur from using this combination of incorrect form factors, even though they match the ‘Q^{2} shape’ as well as the correct form factors Figure 19. Ratio of cross section versus energy using BBA-2003 Form Factors with M_{A}=1.00 GeV versus Dipole Form Factors with M_{A}=1.05 GeV.

Figure 20. Ratio of cross section versus energy using BBA-2003 Form Factors with M_{A}=1.00 GeV versus Dipole Form Factors with M_{A}=1.11 GeV.

do. The effects can be even larger than in this example, especially if the Q^{2}-dependence is not required to be identical. As shown by Y. Itow’s at NUINT02 talk on the results from K2K [1], the initial K2K analysis (on a water target) assumed Dipole Form Factors. They obtained a value of M_{A}=1.11 GeV and saw a low Q^{2} suppression with respect to the predicted distribution with Dipole Form Factors and M_{A} = 1.023. Figure 20 shows the ratio vs. energy of cross sections predicted using BBA-2003 Form Factors with M_{A}=1.00 GeV to the prediction with Dipole Form Factors with M_{A}=1.11 GeV. The figure shows differences as large as 12%. This will clearly be important for future neutrino experiments at Fermilab and elsewhere: Even if the form factors which are used in the model are adjusted to describe the Q^{2} distribution measured in a near detector neutrino experiment, one would be predicting an incorrect energy dependence of the QE cross section. Therefore, using the correct combination of form factors is important for determining the neutrino cross section and its energy dependence. An accurate measurement of QE scattering cross sections requires an accurate measurement of both the normalized cross section versus energy as well as the shape of the Q^{2} distribution.

5. CONCLUSION

We have made an updated extraction of nucleon electromagnetic form factors electron scattering, and have shown these have as much as a 6% effect for neutrino-nucleon QE scattering when compared with the standard dipole approximation. Inclusion of the new form factors yields to a reduction in the value of M_{A} extracted from neutrino scattering of 0.025 GeV. The agreement between the calculated neutrino and antineutrino free nucleon cross section and data is improved using the updated form factors, but is not spectacular, especially for data taken on heavy targets. We have shown that for a fixed Q^{2} dependence, the neutrino cross section and its energy dependence can be affected by more than 10% if the input shapes of the electromagnetic form factors used are not correct. Hence, a complete understanding of QE scattering scattering requires an accurate measurement of both the normalized cross section versus energy as well as the shape of the Q^{2} distribution. In addition, nuclear effects such as Pauli blocking and modification of nucleon form factors in bound nuclei need to be included. These effects are currently under investigation. For example, the simple Fermi gas model can be modified to include a high momentum tail [28] or more sophisticated spectral functions.

6. ACKNOWLEDGMENTS

This work is supported in part by the U. S. Department of Energy, Nuclear Physics Division, under contract W-31-109-ENG-38 (Argonne) and High Energy Physics Division under grant DE-FG02-91ER40685 (Rochester).

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