Effective Spectral Function for Quasielastic Scattering on Nuclei from Deuterium to Lead
Abstract
Spectral functions do not fully describe quasielastic electron and neutrino scattering from nuclei because they only model the initial state. Final state interactions distort the shape of the differential cross section at the peak and increase the cross section at the tails of the distribution. We show that the kinematic distributions predicted by the superscaling formalism can be well described with a modified effective spectral function (ESF). By construction, models using ESF in combination with the transverse enhancement contribution correctly predict electron QE scattering data. Our values for the binding energy parameter are smaller than extracted within the Fermi gas model from pre 1971 data by Monizmoniz , probably because these early cross sections were not corrected for coulomb effects.
keywords:
Spectral Functions Quasielastic Electron Scattering Neutrino Scattering00 \journalnameNuclear Physics B Proceedings Supplement \runauthA. Bodek et al. \jidnuphbp \jnltitlelogoNuclear Physics B Proceedings Supplement
1 Introduction
Neutrino oscillation experiments make use of neutrino Monte Carlo (MC) event generators to model the cross sections and kinematic distributions of the leptonic and hadronic final state of neutrino interactions on nuclear targets. Because of the conservation of the vector current (CVC), the same models should be able to reliably predict the quasielaststic electron scattering cross section on nuclear targets. Unfortunately, none of the models that are currently implemented in neutrino MC generators are able to do it. Here we summarize an approach which guarantees agreement with QE electron scattering data by construction. A more detailed description is given in reference epicpaper
The left panel of Fig. 1 is the general diagram for QE lepton (election, muon or neutrino) scattering from a nucleon which is bound in a nucleus of mass . In this paper, we focus on charged current neutrino scattering. The scattering is from an offshell bound neutron of momentum . The onshell recoil (spectator) nucleus has a momentum . This process is referred to as the 1p1h process (one proton one hole). The * is used to indicate that the spectator nucleus is not in the ground state because it has one hole. The fourmomentum transfer to the nuclear target is defined as . Here is the energy transfer, and is the square of the fourmomentum transfer. For free nucleons the energy transfer is equal to where is the mass of the nucleon. At a fixed value of , QE scattering on nucleons bound in a nucleus yields a distribution in which peaks at . In this communication, the term ”normalized quasielastic distribution” refers to the normalized differential cross section where is the integral of over all values of (for a given value of ).
The right panel of Fig. 1 shows the same QE lepton scattering process, but now also including a final state interaction with another nucleon in the scattering process. This final state interaction modifies the scattering amplitude and therefore can change the kinematics of the final state lepton. In this paper, we refer to it as ”final state interaction of the first kind” (FSI). The final state nucleon can then undergo more interactions with other nucleons in the spectator nucleus. These interactions do not change the energy of the final state lepton. We refer to these final state interactions as ”final state interaction of the second kind”.
In general, neutrino event generators assume that the scattering occurs on independent nucleons which are bound in the nucleus. Generators such as GENIEgenie , NEUGENneugen , NEUTneut , NUANCEnuance NuWro nuwro and GiBUUgibuu account for nucleon binding effects by modeling the momentum distributions and removal energy of nucleons in nuclear targets. Functions that describe the momentum distributions and removal energy of nucleons from nuclei are referred to as spectral functions. Spectral functions describe the initial state.
Spectral functions can take the simple form of a momentum distribution and a fixed removal energy (e.g. Fermi gasmoniz ), or the more complicated form of a two dimensional (2D) distribution in momentum and removal energy (e.g. BenharFantoni spectral function bf ).
Fig. 2 shows the nucleon momentum distributions in a nucleus for some of the spectral functions that are currently being used. The solid green line is the nucleon momentum distribution for the Fermi gasmoniz which is currently implemented in all neutrino event generators. The solid black line is the projected momentum distribution of the BenharFantoni bf 2D spectral function as implemented in NuWro. The solid red line is the nucleon momentum distribution for the LocalThomasFermi gas (LTF).
It is known that theoretical calculations using spectral functions do not fully describe the shape of the quasielastic peak for electron scattering on nuclear targets . This is because the calculations only model the initial state (shown on the left panel of Fig. 1), and do not account for final state interactions of the first kind (shown on the right panel of Fig. 1) . Because FSI changes the amplitude of the scattering, it modifies the shape of . FSI reduces the cross section at the peak and increases the cross section at the tails of the distribution.
In contrast to the spectral function formalism, predictions using the superscaling formalismsuper1 ; super2 fully describe the longitudinal response function of quasielastic electron scattering data on nuclear targets. This is expected since the calculations use a superscaling function which is directly extracted from the longitudinal component of measured electron scattering QE differential cross sections.
In this communication we present the parameters for a new effective spectral function that reproduces the kinematics of the final state lepton predicted by superscaling. The momentum distribution for this ESF for is shown as the blue line in Fig. 2.
1.1 The superscaling functions for QE scattering
The superscaling variable includes a correction that accounts for the removal energy from the nucleus. This is achieved by replacing with , which forces the maximum of the QE response to occur at . QE scattering on all nuclei (except for the deuteron) is described using the same universal superscaling function. The only parameters which are specific to each nucleus are the Fermi broadening parameter and the energy shift parameter .
Fig. 3 shows two parametrizations of superscaling functions extracted from quasielastic electron scattering data on . Shown is the superscaling distribution extracted from a fit to electron scattering data used by Bosted and Mamyan super2 (solid black line labeled as 2012), and the superscaling function extracted from a recent updated fiteric to data from a large number of quasielastic electron scattering experiments on (dotted red line labeled as 2014). The top panel shows the superscaling functions on a a linear scale and the bottom panel shows the same superscaling functions on a logarithmic scale.
The superscaling function is extracted from the longitudinal QE cross section for GeV where there are no Pauli blocking effects. At very low values of , the QE differential cross sections predicted by the superscaling should be multiplied by a Pauli blocking factor which reduces the predicted cross sections at low . The Pauli suppression factor is givensuper2 by the function
(2) 
For , otherwise no Pauli suppression correction is made. Here is the absolute magnitude of the momentum transfer to the target nucleus,
(GeV)  (GeV)  

2  0.100  0.001 
3  0.115  0.001 
0.190  0.017  
0.228  0.0165  
0.230  0.023  
0.236  0.018  
0.241  0.028  
0.241  0.023  
0.245  0.018 
1.2 Comparison of models for quasielastic scattering
Fig. 4 shows predictions for the normalized QE differential cross sections for 10 GeV neutrinos on at =0.5 GeV for various spectral functions. Here is plotted versus . The prediction of the superscaling formalism for is shown as the solid black line. The solid green line is the prediction using the ”Global Fermi” gas moniz . The solid red line is the prediction using the Local Thomas Fermi gas (LTF) momentum distribution. The dotted purple line is the NuWro prediction using the full two dimensional BenharFantonibf spectral function. The predictions of all of these spectral functions for are in disagreement with the predictions of the superscaling formalism.
2 Effective Spectral Function for
2.1 Momentum Distribution
The probability distribution for a nucleon to have a momentum in the nucleus is defined as
For GeV, we parametrizebfit by the following function:
(3)  
For GeV we set = 0. Here, , is in GeV, N is a normalization factor to normalize the integral of the momentum distribution from =0 to =0.65 GeV to 1.0, and P() is in units of GeV.
2.2 Removal Energy
The kinematics for neutrino charged current quasielastic scattering from a offshell bound neutron with momentum and energy are given by:
(4)  
(5) 
For scattering from a single offshell nucleon, the term multiplying in Equations 4, 6, and 7 should be 1.0. However, we find that in order to make the spectral function predictions agree with superscaling at very low (e.g. ) we need to apply a dependent correction to reduce the removal energy, e.g. due to final state interaction (of the first kind) at low . This factor is given in equation 5. The value of the parameter =12.04 GeV was extracted from the fits at low values of . As mentioned earlier, is the momentum transfer to the neutron. We define the component of the initial neutron momentum which is parallel to as . The expression for depends on the process and is given by Equations 6 and 7 for the 1p1h, and 2p2h process, respectively.
We assume that the offshell energy () for a bound neutron with momentum can only take two possible valuesBodekRitchie . We refer to the first possibility as the 1p1h process (one proton, one hole in the final state). The second possibility is the 2p2h process(two protons and two holes in the final state).
In our effective spectral function model the 1p1h process occurs with probability , and the 2p2h process occurs with probability of . For simplicity, we assume that the probability is independent of the momentum of the bound nucleon.
2.2.1 The 1p1h process
The 1p1h process refers to scattering from an independent neutron in the nucleus resulting in a final state proton and a hole in the spectator nucleus. Fig. 1 illustrates the 1p1h process (for GeV), for the scattering from an offshell bound neutron of momentum in a nucleus of mass ABodekRitchie . In the 1p1h process, momentum is balanced by an onshell recoil nucleus which has momentum and an average binding energy parameter , where . The initial state offshell neutron has energy which is given by:
(6)  
The final state includes a proton and an nucleus in an excited state because the removal of the nucleon leaves hole in the energy levels of the nucleus.
2.2.2 The 2p2h process
In general, there are several processes which result in two (or more) nucleons and a spectator excited nucleus with two (or more) holes in final state:

Two nucleon correlations in initial state (quasi deuteron) which are often referred to as short range correlations (SRC).

Final state interaction (of the first kind) resulting in a larger energy transfer to the hadronic final state (as modeled by superscaling).

Enhancement of the transverse cross sections (”Transverse Enhancement”) from meson exchange currents (MEC) and isobar excitation.
In the effective spectral function approach the lepton energy spectrum for all three processes is modeled as originating from the two nucleon correlation process. This accounts for the additional energy shift resulting from the removal of two nucleons from the nucleus.
Fig. 5 illustrates the 2p2h process for scattering from an offshell bound neutron of momentum (for GeV). The momentum of the interacting nucleon in the initial state is balanced by a single onshell correlated recoil nucleon which has momentum . The spectator nucleus is left with two holes. The initial state offshell neutron has energy given by:
(7) 
where is given by eq. 5.
Benhar  ESF  ESF  

Fantoni  ESF  ESF  
Nucl.  
(MeV)  2Dspectral  12.5  0.13 
2Dspectral  0.808  0  
2Dspectral  0.192  1.00  
1.7  2.12  0.413475  
1.77  0.7366  1.75629  
1.5  12.94  8.29029  
0.8  10.62  3.621 x10  
2.823397  197.0  0.186987  
7.225905  9.94  6.24155  
0.00861524  4.36 x10  2.082 x10  
0.985  29.64  10.33 
Param.  

0.791  0.808  0.765  0.774  0.809  0.822  0.896  
2.14  2.12  1.82  1.73  1.67  1.79  1.52  
0.775  0.7366  0.610  0.621  0.615  0.597  0.585  
9.73  12.94  6.81  7.20  8.54  7.10  11.24  
7.57  10.62  6.08  6.73  8.62  6.26  13.33  
183.4  197.0  25.9  21.0  200.0  18.37  174.4  
5.53  9.94  0.59  0.59  6.25  0.505  5.29  
59.0x10  4.36 x10  221. x10  121.5 x10  269.0x10  141.0 x10  9.28x10  
18.94  29.64  4.507  4.065  40.1  3.645  37.96  
(MeV)  14.0  12.5  16.6  12.5  20.6  15.1  18.8 
super2  17.0  16.5  23.0  18.0  28.0  23.0  18.0 
super2  190  228  230  236  241  241  245 
2012  
Monizmoniz  17.0  25.0  32.0  32.0  28.0  36.0  44.0 
Monizmoniz  169  221  235  235  251  260  265 
(1971) 
In the effective spectral function approach, all effects of final state interaction (of the first kind) are absorbed in the initial state effective spectral function. The parameters of the effective spectral function are obtained by finding the parameters , , , , , , , , , , and for which the predictions of the effective spectral function best describe the predictions of the superscaling formalism for at values of 0.1, 0.3, 0.5 and 0.7 GeV.
Fig. 6 compares predictions for for as a function of at =0.5 GeV. The prediction of the effective spectral function is the dashed blue curve. The prediction of the superscaling model is the solid black curve. For =0.5 GeV the prediction of the effective spectral function is almost identical to the prediction of superscaling.
We find that the effective spectral function with only the 1p1h process provides a reasonable description of the prediction of superscaling. Including a contribution from the 2p2h process in the fit improves the agreement and results in a prediction which is almost identical to the prediction of superscaling.
The parameterizations of the effective spectral function for all nuclei from deuterium to lead are given in Tables 2 and 3. As shown in Table 3, the EFS values for the binding energy parameter are smaller than extracted within the Fermi gas model from pre 1971 electron scattering data by Monizmoniz . This may be because these early cross sections were not corrected for coulomb effects..
3 Conclusion
We present parameters for an effective spectral function that reproduce the prediction for from the formalism. We present parameters for a large number of nuclear targets from deuterium to lead.
Since most of the currently available neutrino MC event generators model neutrino scattering in terms of spectral functions, the effective spectral function can easily be implemented. For example, it has taken only a few days to implement the effective spectral function as an option in recent private versions of NEUT and GENIE. The predictions for QE scattering on nuclear targets using EFS with the inclusion of the the Trasverse Ehancementepicpaper ; TE contribution fully describe electron scattering data by construction.
References
 (1) A. Bodek, M. E. Christy, and B. Coopersmith, Xiv:1405.0583v3 (to be published in Eur. Phys. J. C, 2014)
 (2) C. Andreopoulos [GENIE Collaboration], Acta Phys. Polon. B 40, 2461(2009), C.Andreopoulos (GENIE), Nucl. Instrum. Meth.A614, 87,2010.
 (3) H. Gallagher, (NEUGEN) Nucl. Phys. Proc. Suppl. 112 (2002).
 (4) Y. Hayato (NEUT), Nucl Phys. Proc. Suppl.. 112, 171 (2002).
 (5) D. Casper (NUANCE) , Nucl. Phys. Proc. Suppl. 112, 161 (2002) (http://nuint.ps.uci.edu/nuance/).
 (6) J. Sobczyk (NuWro), PoS NUFACT 08, 141 (2008), C. Juszczak, Acta Phys. Polon. B40 (2009) 2507
 (7) T. Leitner, O. Buss, L. AlvarezRuso, U. Mosel (GiBUU), Phys. Rev. C79, 034601 (2009) (arXiv:0812.0587).
 (8) E. J. Moniz et al. Phys. Rev. Lett. 26, 445(1971); E. J. Moniz, Phys. Rev. 184, 1154 (1969); R. A. Smith and E. J. Moniz, Nucl. Phys. B43, 605 (1972).
 (9) O. Benhar and S. Fantoni and G. Lykasov, Eur Phys. J. A 7 (2000), 3, 415; O, Benhar et al., Phys. Rev. C55. 244 (1997).
 (10) J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Donnelly, A. Molinari, and I. Sick, Phys. Rev. C 71, 015501 (2005)
 (11) P. E. Bosted, V. Mamyan, arXiv:1203.2262.
 (12) M. E. Christy, private communication (2014).
 (13) A. Bodek, and J. L. Ritchie, Phys. Rev. D23, 1070 (1980). A. Bodek and J. L. Ritchie, Phys. Rev. D24, 1400 (1981).
 (14) S. Mishra, (NOMAD collaboration) private communication, fit parameters from E. Iacopini (1997).
 (15) A. Bodek, H. S. Budd and M. E. Christy, Eur. Phys. J. C 71, 1726 (2011).