Vector and Axial Form Factors Applied to Neutrino Quasielastic 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 quasielastic cross sections for neutrino 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 form factors. We show the extraction of F_{A}(q^{2}) for neutrino experiments. We show how well MINERνA, a new approved experiment at FNAL, can measure F_{A}(q^{2}). We show the that F_{A}(q^{2}) has a different contribution to the anti-neutrino cross section, and how the anti-neutrino data can be used to check F_{A}(q^{2}) extracted from neutrino scattering. (Presented by Howard Budd at NuInt04, Mar. 2004, Laboratori Nazionali del Gran Sasso - INFN - Assergi, Italy [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 and better determination of the axial form factor, F_{A}(q^{2}).

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.

The form factors F_{V}^{1} (q^{2}) 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}

Figure 1. 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 dashed curve [7] uses a Fermi gas model for carbon with a 25 MeV binding energy and 220 Fermi momentum. The dotted curve is the prediction for carbon including both Fermi gas Pauli blocking and the effect of nuclear binding on the nucleon form factors [10](bounded form factors). The data shown are from FNAL 1983 [11], ANL 1977 [12], BNL 1981 [13], ANL 1973 [14], SKAT 1990 [15], GGM 1979 [16], LSND 2002 [17], Serpukov 1985 [18], and GGM 1977 [19].

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 has been improved by us in a previous publication [4]. We use our updated form factors which we refer as ‘BBA-2003 Form Factors’ [4] [5] (Budd, Bodek, Arrington).

The axial form factor is given by

F_{A}(q^{2}) = \frac{g_{A}}{(1 − \frac{q^{2}}{M_{A}^{2}} )^{2}} .

We have used our updated value of M_{A}=1.00 ± 0.020 GeV [4] which is in good agreement with the theoretically corrected value from pion electroproduction of 1.014 ± 0.016 GeV [6]. For extraction of F_{A}(q^{2}) we use the value of M_{A} = 1.014, since it is independent of QE scattering measurements.

3. Comparison to Cross Section Data

Figures 1 shows the QE cross section for ν using BBA-2003 Form Factors and M_{A}=1.00 GeV The normalization uncertainty in the data is approximately 10%. The solid curve uses no nuclear correction, while the dotted curve [7] uses a NU-ANCE [8] calculation of a Smith and Moniz [9] 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 dotted curve is the prediction for carbon including both Fermi gas Pauli blocking and the effect of nuclear binding on the nucleon form factors as modeled by Tsushima et al. [10]. The ratio of bounded form factors to free form factors is set to 1 for Q^{2} > 2.0 GeV. The updated form factors improve the agreement with neutrino QE cross section data and give a reasonable description of the Figure 2. The percent change in the neutrino cross section for a 1% change in the form factors.

cross sections from deuterium. We plan to study the nuclear corrections, adopting models which have been used in precision electron scattering measurements from nuclei at SLAC and JLab.

4. Extraction of F_{A}(q^{2})

A substantial fraction of the cross section comes from the form factor F_{A}(q^{2}). Therefore, we can extract F_{A}(q^{2}) from the differential cross section. Figure 2 and 3 show the contribution of F_{A}(q^{2}) to dσ/dQ^{2}. Figure 2 shows the percent change in the neutrino cross section for a 1% change in the form factors. Figure 3 shows the fractional contribution of the form factor determined by setting the form factor to zero and by determining the fractional decrease in the differential cross section. Since some terms are products of different form factors, the sum of the curves do not have be 1.

To extract F_{A}, we write the equation for dσ/dq^{2}(q^{2}, E_{ν}) in terms of a quadratic function of F_{A}(q^{2}).

a(q^{2}, E_{ν})F_{A}(q^{2})^{2} + b(q^{2}, E_{ν})F_{A}(q^{2})

+ c(q^{2}, E_{ν}) − \frac{dσ}{dq^{2}} (q^{2}, E_{ν}) = 0

Figure 3. Fractional contribution of the form factor determined by setting the form factor to zero and by determining the fractional decrease in the differential cross section, 1 − (dσ/dQ^{2}(formfactor = 0))/(dσ/dQ^{2}).

For each q^{2} bin, we integrate the above equation over the q^{2} bin and the neutrino flux.

∫∫ dq^{2}dE_{ν}{a(q^{2}, E_{ν})F_{A}(q^{2})^{2} + b(q^{2}, E_{ν})F_{A}(q^{2})

+c(q^{2}, E_{ν}) − \frac{dσ}{dq^{2}} (q^{2}, E_{ν})\} = 0

The above equation can be written as a quadratic equation in F_{A} at the bin value q_{bin}^{2}.

αF_{A}(q_{bin}^{2})^{2} + βF_{A}(q_{bin}^{2}) + γ − Δ − N_{Bin}^{Data} = 0

The terms of this equation are given below:

α = ∫∫ dq^{2}dE_{ν}a(q^{2}, E_{ν})

β = ∫∫ dq^{2}dE_{ν}b(q^{2}, E_{ν})

γ = ∫∫ dq^{2}dE_{ν}c(q^{2}, E_{ν})

Figure 4. Extracted values of F_{A}(q^{2}) for the three deuterium bubble chamber experiments Baker et al. [13], Miller et al. [21], and Kitagaki et al. [11]. Also shown are the expected errors for MINERνA assuming a dipole form factor for F_{A}(q^{2}) with M_{A}=1.014.

To find q_{bin}^{2}, we assume a nominal F_{A}(q^{2}), written F_{A}^{N}(q^{2}). We determine q_{bin}^{2} from

αF_{A}^{N}(q_{bin}^{2})^{2} − ∫∫ dq^{2}dE_{ν}a(q^{2}, E_{ν})F_{A}^{N}(q^{2})^{2} = 0.

Δ is a bin center correction term which also uses F_{A}^{N}(q^{2}). Δ is determined by

Δ = βF_{A}^{N}(q_{bin}^{2}) − ∫∫ dq^{2}dE_{ν}b(q^{2}, E_{ν})F_{A}^{N}(q^{2}).

The number of events in the bin is given by N_{Bin}^{Data}. The number of events in the bin from theory is

N_{Bin}^{T hy} = ∫∫ dq^{2}dE_{ν} \frac{dσ}{dq^{2}} (q^{2}, E_{ν}).

The errors in the points are given by

\frac{\sqrt{N_{Bin}^{T hy}}}{2αF_{A}^{N}(q_{bin}^{2}) + β} .

Figure 5. Same as Figure 4 with a logarithmic scale.

Figure 4 and 5 show our extracted values of F_{A}(q^{2}) for the three deuterium bubble chamber experiments. For these plots the curve shown in the figures is a dipole with m_{A}=1.014, the value extracted from pion-electro production. The data and fluxes given in their papers are used in the extraction of F_{A}(q^{2}). These plots show the previous data is not sufficient to determine the form for F_{A}(q^{2}).

In addition, we have shown the expected values for MINERνA and its errors. We have plotted MINERνA assuming it is a dipole. We have assumed a 4 year run with 3 tons of fiducial volume and included the effects of inefficiencies and backgrounds. Resolution smearing and systematic errors are not included.

Figure 6 plots F_{A}(q^{2})/dipole to show how well MINERνA can measure F_{A}(q^{2}). G_{E}^{p}(q^{2}) from electron scattering experiments depends upon the measuring technique [5]. For MINERνA we show F_{A} under the assumption F_{A}(q^{2})/dipole = G_{E}^{p}(q^{2})/dipole from the cross section technique (Rosenbuth separation) and F_{A}(q^{2})/dipole = G_{E}^{p}(q^{2})/dipole from polarization transfer technique. The MINERνA errors are plotted assum-Figure 6. Extracted values of F_{A}(q^{2})/dipole for deuterium bubble chamber experiments Baker et al. [13], Miller et al. [21], and Kitagaki et al. [11]. For MINERνA the projected results are shown for two different assumptions: F_{A}/dipole=G_{E}^{p}/dipole from cross section and F_{A}/dipole=G_{E}^{p}/dipole from polarization. The MINERνA errors are for a 4 year run.

ing the plotted F_{A}(q^{2}) is the nominal F_{A}(q^{2}). We see that the measurement of F_{A}(q^{2}) from MINERνA can distinguish between these to the two possible forms. In addition, MINERνA can determine whether F_{A}(q^{2}) is a dipole or not.

5. Extraction of F_{A}(q^{2}) from anti-neutrinos

The determination of F_{A}(q^{2}) will have systematic errors from the flux, nuclear effects, QE identifications, background determination, etc. Antineutrino data can provide a check on F_{A}(q^{2}). Figure 7 and 8 show the contribution of F_{A}(q^{2}) to the cross section vs Q^{2} for anti-neutrinos. Figure 7 shows the percent change in the anti-neutrino cross section for a 1% change in the form factors. The plot shows that F_{A}(q^{2}) has a different contribution to the cross section for anti-neutrinos than neutrinos. At Q^{2} ∼ 3GeV^{2}, F_{A} is not con-Figure 7. The percent change in the anti-neutrino cross section for a 1% change in the form factors.

tributing to the cross section, and the cross section becomes independent of F_{A}(q^{2}). Hence, at higher Q^{2} the cross section can be predicted and compared to the data to determine errors to the neutrino extraction. Figure 8 shows the fractional contribution of the form factor determined by setting the form factor to zero and by determining the fractional decrease in the differential cross section. Note, since some terms are products of different form factors the sum of the curves do not have to sum to 1.

Figure 9 shows the errors on F_{A}/dipole for antineutrinos. The overall errors scale is arbitrary. As we expect, the errors on F_{A}(q^{2}) become large at Q^{2} around 3 GeV^{2} when the derivative of the cross section with respect to F_{A}(q^{2}) goes to 0.

6. Conclusions

We have used new form factors to show the cross sections for QE neutrino scattering. The cross sections give a reasonable description of the deuterium data, but the nuclear data is low. We have shown how to extract F_{A} and have shown show well MINERνA can measure F_{A}(q^{2}). For anti-neutrino data at high Q^{2}, F_{A} has a different Figure 8. 1−dσ/dQ^{2}(formfactor = 0)/dσ/dQ^{2}. The contribution of the form factors determined by setting the form factors = 0.

contribution, so anti-neutrinos provides a check for the extraction of F_{A} from neutrinos.

7. 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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