Comparison of optical potential for nucleons and $\Delta$ resonances

Precise modeling of neutrino interactions on nuclear targets is essential for neutrino oscillations experiments. The modeling of the energy of final state particles in quasielastic (QE) scattering and resonance production on bound nucleons requires knowledge of both the removal energy of the initial state bound nucleon as well as the Coulomb and nuclear optical potentials for final state leptons and hadrons. We extract the values of the nuclear optical potential for final state nucleons ($U_{opt}^{QE}$) from inclusive electron scattering data on nuclear targets ($\bf_{6}^{12}C$/$\bf_{8}^{16}O$, $\bf_{20}^{40}Ca$/$\bf_{18}^{40}Ar$, $\bf_{3}^{6}Li$, $\bf_{18}^{27}Al$, $\bf_{26}^{56}Fe$, $\bf_{82}^{208}Pb$/$\bf_{79}^{197}Au$) in the QE region and compare to theoretical calculations by Cooper et.al. We also extract for the first time values of the nuclear optical potential for a $\Delta(1232)$ resonance in the final state ($U^\Delta_{opt}$). We find that $U^\Delta_{opt}$ is more negative than $U_{opt}^{QE}$.


Introduction
Precise modeling of neutrino interactions on nuclear targets is essential for neutrino oscillations experiments. The modeling of the energy of final state particles in quasielastic (QE) scattering and resonance production on bound nucleons requires knowledge of both the removal energy of the initial state bound nucleon as well as the Coulomb and nuclear optical potentials for final state leptons and hadrons. In this communication we extract the values of the nuclear optical potential for final state nucleons (U QE opt ) from inclusive electron scattering data on nuclear targets in the QE region and compare to theoretical calculations by Cooper et.al. We also extract for the first time values of the nuclear optical potential for a ∆(1232) resonance in the final state (U ∆ opt ). First we summarize some of the results of our previous publication [1] on removal energies and the nuclear optical potential for final state nucleons extracted from inclusive quasielastic (QE) electron scattering on a variety of nuclei. The analysis was done within the framework of the impulse approximation.
The diagrams on the top two panels of Fig. 1 show the 1p1h process (one final state proton and one hole) for electron QE scattering from an off-shell bound proton (left) and neutron (right). The diagrams on the bottom two panels show antineutrino (ν) QE scattering from an off-shell bound proton producing a final state neutron (left), and neutrino (ν) scattering from an off shell bound neutron producing a final state proton (right). The electrons scatters from an off-shell nucleon of momentum p i =k bound in a nucleus of mass A. For electrons of incident energy E 0 and final state energy E , the energy transfer to the target is ν = E 0 − E . The square of the 4-momentum transfer (Q 2 ), and 3-momentum transfer (q 3 ) to a proton bound in the nucleus are: We include the effects of the interaction of initial and final state electrons with the Coulomb field of the nucleus by using published values of the average Coulomb energy at the interaction vertex V ef f extracted from a comparison of electron and positron inclusive QE differential cross sections [2]. For electron scattering from protons, the Coulomb energies at the interaction vertex for the final state proton (in QE scattering), final state ∆ + 1232 (in resonance production), and final state of mass W + (in inelastic scattering) are defined below.

Removal energy of initial state nucleons in a nucleus
In our analysis we use the impulse approximation. The nucleon is moving in the mean field (MF) of all the other nucleons in the nucleus. The on-shell recoil excited [A−1] * spectator nucleus has a momentum p (A−1) * = −k and a mean excitation energy E P,N x . The off-shell energy of the interacting nucleon is Here, M P = 0.938272 GeV is the mass of the proton, M N = 0.939565 is the mass of the proton, and S P,N the separation energy (obtained from mass differences of the initial and final state nuclei) needed to separate the nucleon from the nucleus. In Ref. [1] we extract the mean excitation energy E P,N x (or equivalently the removal energy P,N ) using exclusive ee P spectral function measurements. Some of the neutrino MC generators (e.g. current version of genie) do not include the effect of the excitation of the spectator nucleus, nor do they include the effects of the interaction of the final state nucleons and hadrons with the Coulomb [2] and nuclear optical potential of the nucleus.   . Extracted values of U QE opt for the final state nucleon in QE scattering (small black markers) for for 33 12 6 C and four 16 8 O inclusive electron scattering spectra. Also shown are theory prediction for U QE opt calculated by Artur. M. Ankowski [27,28] and Jose Manuel Udias [29] using the theoretical formalisms of Cooper 1993[30], and Cooper 2009 [31] The dashed grey lines are linear fits to the QE data. In addition, the larger markers are the values of U ∆ opt for the final state ∆(1232) (large markers) extracted from a subset of the data (15 12 6 C spectra) for which the measurements extend to higher invariant mass. Here, the solid grey lines are linear fits to the U ∆ opt values. The top and bottom panels show the measurements versus p 2 f 3 = (k + q3) 2 , and versus hadron kinetic energy T, respectively. the 3-momentum of the final state nucleon at the vertex. Alternatively, we also extract U QE opt (T P ) where T is the kinetic energy of the final state nucleon. In the analysis we make the assumption that U QE opt for the proton and neutron are the same.
The energy of the final state nucleon in QE electron scattering is given by the final expressions: and U QE opt (T ) from a comparison of the relativistic Fermi gas (RFG) model to measurements of inclusive QE e-A differential cross sections [3].
The data samples (see references [4]- [24]) include the following elements which are of interest to current neutrino experiments: 33 12 6 C spectra, five 16 8 O spectra, seven 29 40 20 Ca spectra, and two 40 18 Ar spectra. In addition, the data sample include four 6 3 Li spectra, 27 18 Alspectra, 30 56 26 Fe spectra, 23 208 82 Pb spectra and one 197 79 Au spectrum. Most of the QE differential cross sections are available on the QE electron scattering archive [3]. Figure 2, shows examples of three of the 33 fits to QE differential cross sections for 12 6 C. The solid black curve is the RFG fit with the best value of U QE opt for the final state nucleon. The blue dashed curve is a simple parabolic fit used to estimate the systematic error. The red dashed curve is the RFG model with U QE opt and V P ef f set to zero. In the extraction of the nuclear optical potential for final state nucleons in QE scattering we only fit to the data in the top 1/3 of the QE distribution and extract the best value of U QE opt (p 2 f 3 ) and U QE opt (T ). Here p f 3 is evaluated at the peak of the QE distribution. In the fit we let the normalization of the QE cross section float to agree with data. The difference between ν parabola peak and ν rf g peak is used as a systematic error in our extraction of U QE opt . The extracted values of U QE opt (p 2 f 3 ) versus p 2 f 3 from 33 Ca QE spectra and two 40 18 Ar QE spectra are shown in Fig. 6.
Similarly values extracted for five 6 3 Li QE spectra, and seven 27 18 AlQE spectra are shown in the top and bottom panels of Fig. 7. Values extracted for 30 56 26 Fe QE spectra and 23 208 82 Pb (including one 197 79 Au) QE spectra are shown in the top and bottom panel of Fig. 8.
We fit the extracted values of U QE opt (p 2 f 3 ) versus p 2 f 3 for p 2 f 3 > 0.1 GeV 2 to linear functions which are shown as as dashed grey lines in Figures 3-8. We also show linear fits to U QE opt as a function of final state kinetic energy T. The intercepts at p 2 f 3 = 0 and the slopes of the fits to U QE opt versus p 2 f 3 , and the intercepts and slopes of the fits to U QE opt as a function of T are given in Table 1.

Comparison of the values of U QE opt to theory
The solid red lines in Figures 3, 6 For electron scattering from a bound nucleon the optical potentials for QE electron scattering and ∆ resonance production, are given below.
where, M ∆ = 1.232 GeV is the mass of the ∆ resonance and |V ∆+ ef f | = |V P ef f | = Z−1 Z |V ef f |. in order to extract the nuclear optical potential for a ∆ resonance we need to model the cross section between the QE peak and the ∆ resonance. We use the effective spectral function [25] (which includes a 2p2h contribution) to model the region of the QE peak. In the calculation of the inelastic cross section for the production resonances and the continuum we use Jlab fits [26] to the structure functions for protons and neutrons in the resonance region and continuum. These structure functions were extracted from inclusive electron scattering cross sections on hydrogen and deuterium. The proton and neutron structure functions are combined with the relativistic Fermi gas (RFG) to model the resonance production from nuclei. Fig. 4. The top two panels show diagrams for electron scattering from an off-shell bound proton producing a ∆ + (left), and scattering from an off shell bound neutron producing a ∆ 0 (right). The bottom two panels show neutrino scattering from a bound neutron producing a ∆ + (left) and antineutrino scattering on a bound neutron producing a ∆ − (right).  5. Examples of fits for three out of 15 12 6 C ∆(1232) production differential cross sections. Here the QE peak is modeled with an effective spectral function (including 2p2h), and ∆ production is modeled by using RFG to smear fits to ∆ production structure functions on free nucleons. The solid black curve is the fit with the best value of U ∆ opt . The dashed red curve is the prediction with U ∆ opt = V ∆ ef f = 0.    Fig. 3 for 6 3 Li (top two panels) and 27 13 Al (bottom two panels).

Antineutrino Scattering on Neutron
We use a subset of the measured electron scattering cross sections on nuclei that includes measurements of both QE and resonance production. To extract values of the nuclear optical potential for a ∆ (1232) resonance in the final state (U ∆ opt ) we compare the data to predictions of the sum of QE and resonance production cross sections. In the fits the normalizations of the QE cross section, resonance cross sections and U ∆ opt are varied to fit the data. Examples of fits for three out of 15 ∆(1232) production differential cross sections on 12 6 C are shown in Fig. 5. The solid black curve is the fit with the best value of U ∆ opt . The dashed red curve is the same fit with U ∆ opt and |V ∆ ef f | set to zero. The extracted values of U ∆ opt versus p 2 f 3 from 15

Conclusion
We report on the extraction (from electron scattering data) of the nuclear optical potential for both nucleons and ∆(1232) resonances in the final state. This is the first measurement of the optical potential for the ∆(1232) resonance. The result indicate that: 2. We find that the optical potential for a ∆ resonance in the final state U ∆ opt is more negative than the optical potential for a final state nucleon U QE opt . There are no theory predictions available for U ∆ opt 3. Using the measurements of these four parameters P,N , U QE opt , U ∆ opt , and V ef f we can model the energy of electrons, nucleons and ∆(1232) resonance in the final state. For neutrino oscillations experiments these measurements can reduce the systematic uncertainty in the reconstruction of the neutrino energy (originating from uncertainties in the removal energy and nuclear optical potentials) from ± 20 MeV [32] to ± 5 MeV.