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# Electron- versus neutrino-nucleus scattering

July 5, 2020
###### Abstract

We illustrate the connection between electron and neutrino scattering off nuclei and show how the former process can be used to constrain the description of the latter. After reviewing some of the nuclear models commonly used to study lepton-nucleus reactions, we describe in detail the SuSAv2 model and show how its predictions compare with the available electron- and neutrino-scattering data over the kinematical range going from the quasi-elastic peak to pion-production and highly inelastic scattering.

## I Introduction

Electron scattering represents the most powerful probe to study the internal structure of atomic nuclei, due to the weakness of the electromagnetic interaction as compared to the strong forces which govern the nuclear dynamics Kelly:1996hd; Frullani:1984nn; Donnelly:1985ry; Boffi:1996ikg; Waleckacvc. With this motivation, many electron-nucleus experiments were performed in the past, unraveling several interesting features of the many-body nuclear problem, although still leaving some open questions which would deserve further exploration. The field has seen renewed interest in the last decade due to its connection to neutrino-nucleus scattering and the importance of nuclear physics inputs for the interpretation of long baseline neutrino oscillation experiments White-paper; KatoriMartini. The main aim of these experiments, both past or ongoing (MiniBooNE, T2K, NOvA) and future (DUNE and T2HK), is to improve the precision on our knowledge of the Pontecorvo-Maki-Nakagawa-Sakata matrix, in particular the CP-violating phase which is related to the matter-antimatter asymmetry in the Universe. The achievement of this goal requires an extremely good control of the systematic uncertainties, which are mostly coming from the modelling of neutrino-nucleus interactions. Hence the accurate description of these reactions has become one of the top challenges for theoretical nuclear physics with electroweak probes White-paper.

In the energy regime relevant for the above mentioned experiments, going from about 100 MeV up to a few GeV, several mechanisms contribute to the nuclear response: from the excitation of nuclear collective states in the lowest energy part of the spectrum, up to the deep inelastic scattering at the highest energy transfers, encompassing the quasi-elastic region, corresponding to one-nucleon knockout, and the resonance region, corresponding to the excitation of nucleon resonances followed by their decay and subsequent production of pions and other mesons. Nuclear models used to analyze and interpret the data should be able to describe, as consistently as possible, all these processes with a few percent accuracy, a very challenging task which has triggered a lot of recent activity. Moreover, these models should be relativistic, as requested by the high energies involved in the experimental setup.

Due to the rareness of previous neutrino-nucleus experiments in this energy regime, nuclear models must be validated against other experiments. In particular, electron scattering is very closely related to neutrino scattering and therefore constitutes the ideal testing ground for nuclear models. In this paper we focus on the connection between these two processes, illustrating and inter-comparing some of the models used to describe them simultaneously.

The paper is organized as follows: in Sect. II we introduce the general formalism for lepton-nucleus scattering and we comment on the similarities and differences between the electron and neutrino cases. In Sect. III we present different models for the quasielastic (QE) reaction, going from the simplest Relativistic Fermi Gas to more realistic descriptions of the nucleus. Then we discuss relativistic models for the reactions occurring at higher energy transfer: two-particle-two-hole (2p2h) excitations (Sect. IV), pion production (Sect. V), higher inelastic and deep inelastic scattering (Sect. VI). In Sect. VII we concentrate on the SuSAv2 model, developed by our group for both electron and neutrino reactions, collecting in a coherent way the main results obtained in out past work and illustrating the possible implementation of the model in Monte Carlo generators. Finally, in Sect. VIII, we draw our conclusions and outline the future developments of our approach.

## Ii Connecting electron and neutrino scattering

In order to emphasize the similarities and differences between electron- and (anti)neutrino-nucleus scattering, here we present in parallel the general formalism for the two processes. Let us consider the scattering problem

 l+A→l′+X+B

where an incident lepton of mass and 4-momentum  111In this paper we shall use the units =1. scatters off a nuclear target of mass number (a nucleus or a single nucleon) and a lepton of mass and 4-momentum emerges, with and , together with a system and residual nucleus .

For electron scattering , whereas for neutrino scattering and for charged-current (CC) processes, where a boson is exchanged, or for neutral-current (NC) processes, associated to the exchange of a boson.

Let us first remind some basic definitions, commonly used in the nuclear physics language.

• The inclusive cross section, indicated by the notation , where only the outgoing lepton is detected, corresponds to the integration over all the final particles except the outgoing lepton and a sum over all open channels compatible with the kinematics.

• The semi-inclusive cross section, , corresponds to the situation in which the lepton and a particle (or a system of particles) are detected in coincidence in the final state. Note that can be any kinematically allowed system (a nucleon, a photon, a meson, a nucleus, etc.), but in general it is not the complete final state: depending on the kinematics, many different channels can be open, and the semi-inclusive cross section is the sum (or integral) over all the unobserved particles.

• The exclusive cross section corresponds to the simultaneous detection of all the final scattering products.

Exclusive processes are more challenging from both experimental and theoretical points of view: they are more difficult to measure and more sensitive to the details of the nuclear model description. On the other hand, different nuclear models can describe equally well inclusive data, while exclusive or semi-inclusive data constitute a more stringent test for the validity of a model and can better discriminate between different theories.

In this article we mainly focus on the easier case of inclusive reactions, where more data are available and nuclear models are more solid. It should be kept in mind, however, that the validity of nuclear models is better tested against semi-inclusive and exclusive data, and that a good description of the latter is desirable for models to be implemented in Monte Carlo generators used to analyze neutrino oscillation experiments.

The general formalism for electron and (anti)neutrino reactions is developed in great detail in Refs. Donnelly:1985ry; Raskin:1988kc and Moreno:2014kia, respectively. Here we outline some basic features which will be useful for further discussion.

The lepton-nucleus differential cross section is proportional to the contraction of the leptonic and hadronic tensors, defined as

 ημν(K,K′) = 2mm′¯¯¯¯¯¯¯¯∑ss′j∗μ(K′s′,Ks)jν(K′s′,Ks) (1) Wμν = ¯¯¯¯¯¯¯¯∑SS′Jμ∗S′SJνS′S, (2)

where the symbol represents average and sum over the appropriate initial and final leptonic or hadronic spin quantum numbers, which will depend on whether or not the initial lepton and hadron are polarized and on the possible measurement of the final polarizations. and are the leptonic and hadronic 4-current matrix elements, respectively.

### Leptonic tensor

In the case of unpolarized electron or (anti)neutrino scattering the leptonic current entering the tensor can be written as

 jμ(K′s′,Ks)=¯¯¯u(K′,s′)(aVγμ+aAγμγ5)u(K,s), (3)

where is the lepton Dirac spinor and, according to the Standard Model of electroweak interactions, while for unpolarized electron scattering (assuming only pure electromagnetic interaction 222Electrons can also interact weakly: in this case and .) and in the neutrino or antineutrino case. By inserting Eq. (3) into (1) and performing the traces one gets

 ημν(K,K′)=ηVVμν+ηAAμν+ηVAμν (4)

with

 ηVVμν = a2V[KμK′ν+K′μKν−gμν(K⋅K′−mm′)] (5) ηAAμν = a2A[KμK′ν+K′μKν−gμν(K⋅K′+mm′)] (6) ηVAμν = −2iaVaAεμναβKαK′β, (7)

where the upper labels correspond to the vector and axial components of the two leptonic currents.

The number of variables upon which the hadronic tensor , and hence the differential cross section, depend varies of course with the type of process: the inclusive cross section is a function of only 2 independent kinematic variables, e.g. the energy and scattering angle of the outgoing lepton; the semi-inclusive cross section, where the final lepton and a nucleon are detected in coincidence, depends on 5 variables, and so on. For semi-inclusive processes in which final particles are detected beyond the lepton, the kinematical variables are 2+3. Therefore the degree of complexity of nuclear response increases with the ”exclusiveness” of the process.

The number of components of the complex, hermitian , hadronic matrix contributing to the cross section also depends on the specific process. In the most general case, the contraction gives rise to 16 independent terms. These are usually organized into real linear combinations of the hadronic tensor components, the nuclear response functions

 RK,     K=CC,CL,LL,T,TT,TC,TL,T′,TC′,TL′,TT––––,TC––––,TL––––,CL′––––,TC′–––––,TL′––––, (8)

where the labels , , and correspond to a reference frame where the -axis () is parallel to the momentum transfer and refer to the charge (), longitudinal () and tranverse () projections of the two currents building up the response. A similar decomposition can be performed for the leptonic tensor, in such a way that the tensor contraction can be written as

 ημνWμν=v0∑KVKRK, (9)

where the coefficients are linear combinations of the leptonic tensor components. Their explicit expressions are given in Sect. III.

In the specific case of semi-inclusive neutrino scattering 6 of these coefficients vanish Moreno:2014kia, and therefore only 10 response functions contribute to the cross section:

 dσ(νl,l′N) ∝ VCCRCC+2VCLRCL+VLLRLL+VTRT+VTTRTT+VTCRTC+VTLRTL (10) + VT′RT′+VTC′RTC′+VTL′RTL′.

In the case of antineutrino the ”primed” terms, originating from the axial current, change sign.

In the case of semi-inclusive unpolarized electron scattering (pure electromagnetic interaction), due to absence of axial current and to the conservation of the vector current (CVC), the response functions reduce to 4:

 dσ(e,e′N)∝VLRL+VTRT+VTTRTT+VTLRTL. (11)

Obviously and are not the same in Eqs.(10,11) because the electromagnetic (e.m.) and weak currents are different. In particular, the e.m. currents are purely vector, while the weak ones contain an axial component. The vector e.m. and weak currents are simply related by an isospin rotation.

In the case of inclusive lepton scattering, as a consequence of the integration over the full phase space of the outgoing hadron, some of the responses do not contribute and one has:

 dσ(νl,l′) ∝ VCCRCC+2VCLRCL+VLLRLL+VTRT+VT′RT′ (12) dσ(e,e′) ∝ VLRL+VTRT. (13)

In the neutrino case, each response can be decomposed as

 RK=RVVK+RAAK+RVAK (14)

where the upper labels denote the vector or axial nature of the two hadronic currents entering the response in Eq. (2).

The above decomposition into response functions is valid in all kinematical regions (elastic, quasielastic, 2p2h excitations, inelastic), characterized by different nuclear currents. Further details for the response functions will be illustrated in Section III, where calculations of the quasielastic electromagnetic and weak cross sections will be presented in some specific nuclear models.

### What can we learn from electron scattering?

The formalism presented above shows that electron and neutrino scattering are closely related and validation against electron scattering data constrains models to be applied to neutrino scattering. It also shows that a model suitable to describe inclusive reactions is not necessarily a good model for exclusive processes. More nuclear responses are needed to describe the latter, and some details of the nuclear model can be washed out by the extra-integrations involved in the calculation of inclusive cross sections.

Comparison with inclusive electron scattering data is a necessary, but not sufficient, test for nuclear models, because of two main differences:

• The weak current carried by neutrinos has a vector and an axial component, while the electromagnetic current is purely vector. As a consequence, neutrinos can probe the axial nuclear response, not accessible via unpolarized electron scattering. This introduces uncertainties related to the knowledge of the axial and pseudoscalar form factors. In principle valuable information on the axial response could also be extracted from parity-violating (PV) electron scattering off nuclei Musolf:1992xm; Donnelly:1991qy; PRRaul, which could also provide important complementary information on nuclear correlations Alberico:1993ur; Barbaro:1993jg; Barbaro:1995ez; Barbaro:1995gp and on the radiative corrections entering in the isovector axial-vector sector Gonzalez-Jimenez:2015tla. However, few PV data on medium-heavy nuclei exist and are mostly limited to the elastic part of the spectrum.

• In typical electron scattering experiments the incident beam energy is known with good accuracy, hence the transferred energy, , and momentum, , can be precisely determined by measuring the outgoing lepton kinematics. In long baseline neutrino experiments the beam is not monochromatic: neutrinos are produced from meson decay with a more or less - depending on the experiment - broad distribution around an average value. This implies that each kinematic (momentum and scattering angle) of the outgoing lepton corresponds in general to a finite range of and , which can in turn correspond to various overlapping processes. An important example of this difference is represented by the mixing of 1p1h and 2p2h excitations: in the the former correspond to the quasielastic peak, while the latter are peaked in the ”dip” region between the QE and the resonance peaks. In data there is no way to disentangle the two channels and one has to rely on nuclear models to reconstruct the neutrino energy from lepton kinematics.

A very large set of high quality inclusive electron-nucleus scattering data has been collected in the past (see Ref. Benhar:2006wy for an exhaustive collection), covering a wide energy range (from 100 MeV to several GeV) and various nuclei (from up to ). Recently Dai:2018xhi; Dai:2018gch the cross section on argon and titanium has been measured at JLab, with the specific purpose of constraining models used in the analysis of neutrino-nucleus scattering. Some data are also available for the separated longitudinal and transverse responses. These in principle allow for a more stringent test of the models, since nuclear effects are different in the two channels.

### Inclusive electron-nucleus scattering and Superscaling

A powerful mean of extracting information on nuclear effects from data at different beam energies and on different nuclei is represented by the superscaling analysis proposed in Donnelly99a; Donnelly99b; Alberico:1988bv. This approach allows for the factorization of the cross section into a single-nucleon term times a function - the scaling function - which embodies the nuclear dynamics. Assuming that the latter is independent of the specific probe, scaling allows one to predict inclusive neutrino-nucleus cross sections using electron-nucleus data Amaro:2004bs.

Scaling phenomena occur in various different fields, including atomic, nuclear and hadronic physics, whenever a new scale is probed in a certain process. For example, -scaling occurs in lepton-nucleon scattering when the lepton, at high values of , is resolving the inner structure of the proton (or neutron), interacting with its constituents, the quarks. The typical manifestation of scaling is that a function of two variables (in this case the nucleon’s structure function becomes a function of only one variable (in this case the Bjorken variable ). In a similar way, in lepton-nucleus scattering, scaling occurs when the nuclear response, depending in general on the two variables , becomes function of only one scaling variable, indicating that the probe interacts with the nucleus’ constituents, the nucleons. Since these, unlike the quarks, are not point-like, the cross section has to be ”reduced”, i.e., divided by a single-nucleon function which takes into account the internal structure of the nucleon. In this context the scaling function is defined as (see Appendix A for details)

 F(q,ω)=d2σ/dΩedω˜σeN, (15)

the ratio between the double differential cross section and the half-off-shell single-nucleon cross section evaluated at the smallest possible values of the missing momentum and missing energy compatible with the kinematics and averaged over the ejected nucleon’s azimuthal angle. The corresponding scaling variable is a particular combination of and whose choice is not unique and may involve some approximation. The common choices for the scaling variables in the quasielastic domain are and , defined in detail in Appendix A. - (or equivalently -) scaling is satisfied when

 F(q,ω)⟶F(y). (16)

This occurs for high enough values of the momentum transfer , able to resolve the internal structure of the nucleus. Quantitavely this corresponds to the region 300-400 MeV/c, where collective effects become negligible. A further difference between - and -scaling resides in the fact that, while at high quarks become asymptotically free, nucleons are always interacting inside the nucleus, and the scaling function keeps track of these correlations, being different from that of a collection of free nucleons.

The analysis of experimental data performed in Refs. Day:1990mf; Donnelly99a; Donnelly99b shows that scaling is satisfied with good accuracy in the ”scaling region” , where corresponds to the maximum of the quasielastic peak. It is broken at higher energies loss, indicating that processes other than QE scattering, like pion production and excitation of two-particle two-hole (2p2h) states contribute to the cross section.

The above described -independence of the scaling function when expressed in terms of is also called scaling of first kind, in order to distinguish it from scaling of second kind, concerning the dependence of the function upon the specific nucleus. Scaling of second kind is formulated in terms of the scaling variable (see Appendix A) and consists in the fact that the function

 f(q,ω)=kFF(q,ω) (17)

only depends on

 f(q,ω)⟶f(ψ). (18)

In other words, the reduced cross section scales with the Fermi momentum as . This hypothesis is motivated by the RFG model (see Appendix C, where it is exactly fulfilled, and the analysis of data on different nuclei (with mass number A ranging from 4 to 197) proves that it is respected with very good accuracy in the scaling region.

Superscaling is the simultaneous occurrence of scaling of first and second kind and the function is called superscaling function.

The superscaling analysis can be extended to the separate longitudinal and transverse (relative to the transferred momentum ) responses. In this case two superscaling functions are introduced:

 fL = kFRL(q,ω)GL(q,ω), (19) fT = kFRT(q,ω)GT(q,ω), (20)

where and are the longitudinal and transverse response functions and the dividing factors , are defined in Appendix A.

The analysis of the separated and responses shows that scaling violations mainly reside in the transverse channel, whereas in the longitudinal one scaling works quite well in the full quasielastic region, i.e. also at . The reason for this difference is that the main contributions which violate scaling, namely the 2p2h and resonance excitations, are essentially transverse, with small longitudinal contamination of relativistic origin. From the analysis of the L/T separated data a phenomenological longitudinal superscaling function has been extracted Jourdan:1996aa. The agreement with this function represents a strong constrain for nuclear models used in neutrino-nucleus scattering simulations. This topic is examined in detail in subsequent sections. For example, in Fig. LABEL:fig:SuSA the comparison of the experimental superscaling function with the relativistic Fermi gas prediction is shown. The variable differs from by an energy shift related to the nucleon separation energy (see later). Note the striking difference between the RFG result and the experimental data: the RFG predicts a symmetric function around , with a maximum value of 0.75 (see Eq. (46)), while the data display a pronounced asymmetric tail at large (large ) and a maximum of 0.6. Figure 1: The experimental longitudinal scaling function defined in Eq.(19) displayed versus the scaling variable ψ′ and compared to the RFG prediction (red dashed curve). The solid blue curve is a fit of the data Jourdan:1996aa.

### Semi-inclusive electron-nucleus scattering

The study of reactions on nuclei provides essential information about nuclear structure and dynamics, not only on single-particle properties such as the spectral function and the nucleon’s momentum distribution, but also on more complex mechanisms as nucleon-nucleon correlations and meson-exchange currents  Frullani:1984nn; Boffi:1996ikg; Kelly:1996hd. These are essential inputs for the simulation of - scattering and therefore neutrino energy reconstruction.

The theoretical prediction of semi-inclusive reactions is a much harder task than modeling the inclusive process since the related observables are far more sensitive to the details of the dynamics. On the other hand the richer structure of the cross section (see Eqs. (10,11)) allows one to better discriminate among different models.

In Plane Wave Impulse Approximation (PWIA) the 6-th differential cross section can be written as Donnelly:2017aaa

 d6σdpNdΩNdk′edΩe=pNmNMA−1√(MA−1)2+p2σeN(q,ω;p,E)S(pm,Em), (21)

where , , , are the momenta and solid angles of the outgoing nucleon and electron, respectively, is the ground state mass of the recoiling system, is the missing momentum, is the missing energy, is the excitation energy of the residual nucleus having invariant mass , related to the missing energy and to the separation energy by . Finally, is the half-off-shell single-nucleon cross section DeForest:1983ahx and the nuclear spectral function, which yields the probability of removing a nucleon of momentum from the nuclear ground state leaving the residual system with excitation energy . The spectral function is related to the nucleon’s momentum distribution

 n(p)=∫∞0S(p,E)dp. (22)

The factorization (21) breaks if effects beyond the PWIA, as Final State Interactions (FSI) of the knocked-out nucleon with the residual nucleus and effects of two- or many-body currents, are taken into account. Moreover, even in the plane wave limit, the factorized expression (21) no longer holds if dynamical relativistic effects in the bound nucleons, arising from the lower components in the relativistic wave functions, are incorporated. As long as all these effects are properly taken into account in a Relativistic Distorted Wave Impulse Approximation (RDWIA) framework, which describes the distorsion of the ejected proton wave function, the experimental data for can provide reliable information on the nuclear spectral function.

Several experiments were performed in the past in various laboratories (Saclay, NIKHEF, MIT/Bates, Mainz, JLab) at different kinematic conditions and on a variety of nuclei, from to . New data on have recently been taken at JLab Benhar:2014nca and will provide a valuable input for the analyses of neutrino experiments. The comparison of existing data with theoretical models points to a systematic overestimation of the data from shell model based calculations. This discrepancy is interpreted as a probe of the limitations of the mean field approach and of the importance of NN correlations in the nuclear wave function. It is often quantified in terms of spectroscopic factors, , which measure the actual occupancy (different from 1) of each shell (being the set of quantum numbers characterizing a given orbital). The spectroscopic factors extracted from data are typically of the order of 0.5-0.7, depending on the orbital and on the nucleus. The knowledge of these factors, or better the understanding of their microscopic origin, is an important ingredient of the modelling of semi-inclusive reactions.

Summarizing, electron-nucleus scattering provides an essential input for the description of neutrino-nucleus reactions. The huge amount of high precision electron scattering data allows to discriminate between theoretical models much better than the comparison with neutrino scattering data: the latter have bigger errors which, most importantly, are largely due to nuclear model uncertainties. Therefore, it is mandatory that any model used in the analysis of neutrino oscillation experiment is first validated against electron scattering data. The comparison with inclusive data will be illustrated in the next sections for the models developed by our group. The next, more challenging, task, will be to develop reliable models for the description of semi-inclusive reactions, which require a more detailed knowledge of the nuclear structure and dynamics.

## Iii Models for QE in the Impulse Approximation

### iii.1 Relativistic Fermi Gas (RFG)

We follow closely the notation introduced in Ref. Alberico:1988bv. The quasielastic electroweak cross section is proportional to the hadronic tensor or response function for single-nucleon excitations transferring momentum and energy . The relativistic Fermi gas (RFG) gives the simplest approach to a fully relativistic nuclear system whose response functions can be analytically computed, because the uncorrelated, single-nucleon wave functions are free plane waves multiplied by Dirac spinors

 ψp,s(r)=1√Veip⋅rus(p) (23)

with single nucleon energy and the positive energy spinor, , with mass , is normalized to . In the ground state of the RFG all the momenta with (the Fermi momentum) are filled in the Lab system. In the impulse approximation, the electroweak current is approximated by a one-body operator, which can produce only one-particle one-hole (1p1h) excitations. Therefore the hadronic tensor is given in this model by

 Wμν(q,ω) = ∑p∑s,s′δ(E′−E−ω)m2NEE′Jμ∗s′s(p′,p)Jνs′s(p′,p)θ(kF−p)θ(p′−kF) (24)

where is the electroweak current matrix element while is the initial nucleon energy in the Fermi gas. The final momentum of the nucleon is and its energy is . Note that initial and final nucleons have spin component and , respectively.

In the thermodynamic limit, we substitute the sums by momentum integrations. Then the volume of the system is related to the Fermi momentum and proportional to the number of protons and/or neutrons participating in the process:

 Wμν(q,ω) = V(2π)3∫d3pδ(E′−E−ω)m2NEE′2wμνs.n.(p′,p)θ(kF−p)θ(p′−kF), (25)

where we have defined the single-nucleon tensor for the 1p1h excitation

 wμνs.n.(p′,p)=12∑ss′Jμ∗s′s(p′,p)Jνs′s(p′,p). (26)

In the case of electron scattering the electromagnetic current matrix element is given by

 Jμs′s(p′,p)=¯¯¯us′(p′)[F1(Q2)γμ+F2(Q2)iσμνQν2mN]us(p), (27)

where and are, respectively, the Dirac and Pauli electromagnetic form factors of proton or neutron.

In the case of neutrino or antineutrino CC scattering the weak current matrix element is the sum of vector and axial-vector terms , where the vector current is

 Vμs′s(p′,p)=¯¯¯us′(p′)[2FV1γμ+2FV2iσμνQν2mN]us(p) (28)

being the isovector form factors of the nucleon. The axial current is

 Aμs′s(p′,p)=¯¯¯us′(p′)[GAγμγ5+GPQμ2mNγ5]us(p), (29)

where is the nucleon axial-vector form factor and is the pseudo-scalar axial form factor. It is usual to assume the dipole parametrization of the axial form factor, with the axial mass GeV. From partial conservation of the axial current (PCAC) and pion-pole dominance, they are related by

 GP=4m2Nm2π+|Q2|GA, (30)

where .

The single nucleon tensor for the electroweak current is computed by performing the traces in Appendix B.

To obtain the quasielastic cross section for reaction we use the standard expansion in terms of response functions

 dσdΩ′dϵ′=σMott(vLRe.m.L+vTRe.m.T) (31)

where is the Mott cross section, and , with the scattering angle. The electromagnetic longitudinal and transverse response functions are the following components of the e.m. hadronic tensor in the scattering coordinate system with the -axis in the direction (longitudinal)

 Re.m.L(q,ω) = W00e.m. (32) Re.m.T(q,ω) = W11e.m.+W22e.m. (33)

Analogously the CCQE cross section for neutrino energy and final lepton energy , has been expanded in terms of five response functions. If the lepton scattering angle is , the double-differential cross section can be written as Amaro:2004bs; Amaro:2005dn

 d2σdEldcosθl = σ0{VCCRCC+2VCLRCL (34) +VLLRLL+VTRT±2VT′RT′},

where we have defined the cross section

 σ0=G2cos2θc4πk′ϵv0. (35)

In (35) the Fermi weak constant is , the Cabibbo angle is , is the final lepton momentum, and we have defined the factor . Note that the fifth response function is added (+) for neutrinos and subtracted () for antineutrinos. The lepton coefficients depend only on the lepton kinematics and they are defined by

 VCC = 1−δ2|Q2|v0 (36) VCL = ωq+δ2ρ′|Q2|v0 (37) VLL = ω2q2+(1+2ωqρ′+ρδ2)δ2|Q2|v0 (38) VT = |Q2|v0+ρ2−δ2ρ′(ωq+12ρρ′δ2)|Q2|v0 (39) VT′ = 1ρ′(1−ωρ′qδ2)|Q2|v0. (40)

Here we have defined the dimensionless factors , proportional to the charged lepton mass , , and .

The five nuclear response functions , (=Coulomb, =longitudinal, =transverse) in the same scattering coordinate system as for electrons are then given by the following components of the hadronic tensor:

 RCC = W00 (41) RCL = −12(W03+W30) (42) RLL = W33 (43) RT = W11+W22 (44) RT′ = −i2(W12−W21). (45)

It is convenient to introduce dimensionless variables measuring the energy and momentum in units of , namely , , , , and .

In Appendix C we calculate analytically the response functions of the RFG by integrating Eq. (25) for the different components of the single nucleon tensor computed in Appendix B. A good property of the RFG model is the factorization of the response function as a product of a “single nucleon” response multiplied by an universal function, called superscaling function:

 f(ψ)=34(1−ψ2)θ(1−ψ2). (46)

As shown in Appendix C, the scaling variable, , is related to the minimum kinetic energy allowed for the initial nucleon to absorb the momentum and energy transfer , given by

 ε0=Max{κ√1+1τ−λ,εF−2λ}, (47)

where is the Fermi energy in units of . The scaling variable is defined by

 ψ=√ε0−1εF−1sgn(λ−τ). (48)

Note that for (on the left side of the quasielastic peak). The meaning of is the following: it is the minimum kinetic energy of the initial nucleon divided by the kinetic Fermi energy.

Using this definition, the electromagnetic responses can be written as

 Re.m.K = GKf(ψ), (49) GK = Λ(ZUpK+NUnK), (50)

where () is the proton (neutron) number and

 Λ=ξFmNη3Fκ. (51)

Analogously, for the weak responses

 RK=NΛUKf(ψ),, (52)

where () for neutrino (antineutrino) scattering. This factorization of the scaling function inspires the superscaling models discussed in Sect. III.5 using a phenomenological scaling function instead of the universal superscaling function of the RFG.

The remaining quantities are the integrated single nucleon response functions. The electromagnetic ones, for protons or neutrons, are given by

 Up,nL = κ2τ[(Gp,nE)2+(Gp,nE)2+τ(Gp,nM)21+τΔ] (53) Up,nT = 2τ(Gp,nM)2+(Gp,nE)2+τ(Gp,nM)21+τΔ, (54)

where the quantity has been introduced

 Δ=τκ2[−(λ−τ)2τ+ξF[(1+λ)(1+ψ2)+ξF3(1+ψ2+ψ4)]]. (55)

In what follows we present the explicit expressions for the weak integrated single-nucleon responses.

The is the sum of vector and axial contributions. The vector part implements the conservation of the vector current (CVC). The axial part can be written as the sum of conserved (c.) plus non conserved (n.c.) parts. Then

 UCC = UVCC+(UACC)c.+(UACC)n.c.. (56)

For the vector CC response we have

 UVCC = κ2τ[(2GVE)2+(2GVE)2+τ(2GVM)21+τΔ] , (57)

where and are the isovector electric and magnetic nucleon form factors

 GVE = FV1−τFV2 (58) GVM = FV1+FV2. (59)

The axial-vector CC responses are

 (UACC)c. = κ2τG2AΔ (60) (UACC)n.c. = λ2τ(GA−τGP)2. (61)

Using current conservation for the conserved part of the integrated single-nucleon responses we can express them as

 UCL = −λκ[UVCC+(UACC)c.]+(UACL)n.c. (62) ULL = λ2κ2[UVCC+(UACC)c.]+(UALL)n.c. , (63)

where the n.c. parts are

 (UACL)n.c. = −λκτ(GA−τGP)2 (64) (UALL)n.c. = κ2τ(GA−τGP)2 . (65)

Finally, the transverse responses are given by

 UT = UVT+UAT (66) UVT = 2τ(2GVM)2+(2GVE)2+τ(2GVM)21+τΔ (67) UAT = G2A[2(1+τ)+Δ] (68) UT′ = 2GA(2GVM)√τ(1+τ)[1+˜Δ] (69)

with

 ˜Δ=1√τ(1+τ)[τκ(1+λ)−√τ(τ+1)+τκ12ξF(1−ψ2)]. (70)

### iii.2 Semi-relativistic shell models

Nuclear models based on the Fermi gas describe the single nucleon wave functions as plane waves, thus neglecting the finite size of the nucleus, assuming that the response per particle is the same as for infinite nuclear matter. The validity of this assumption has been investigated theoretically in the nuclear shell model (SM) for the kinematics of interest Amaro94; Amaro:1995kk; Amaro:2005sr; Amaro:2006if; Amaro:2006tf. It is obvious that the Fermi gas cannot describe very low energy transfer region of the nuclear response, dominated by excitation of discrete states and giant resonances. Besides, Pauli blocking, for small momentum transfer , forbids emission from a subset of occupied momentum states in the Fermi gas, even if there is enough energy transfer, by momentum conservation. In a real nucleus, however, the nucleons are not plane waves, and the ejected nucleons are plane waves only asymptotically, . This makes the RFG inappropriate to precise modeling of the nuclear response at low momentum transfer MeV/c Amaro94.

In ref. Amaro94 the finite size effects on the electromagnetic QE response functions were studied using a continuum shell model (CSM) in the non relativistic regime for MeV/c. In the CSM the single nucleon wave functions were obtained by solving the Schroedinger equation with a mean field potential of Woods-Saxon type, including central and spin orbit interactions, plus a Coulomb part for protons

 V(r)=−V01+e(r−R)/a+1m2π1rddr(−Vls1+e(r−R)/a)l⋅\boldmathσ+VCoul(r). (71)

This potential is solved numerically for negative and positive energies, and the parameters, are fitted to the experimental energies of the valence shells for protons and neutrons, obtained from the masses of the neighboring nuclei. The SM is the simplest model where the response of a confined quantum system of nucleons can be evaluated, and thus, it allows us to explore the importance of finite size against the infinite FG model. This study was made theoretically by comparing the separate longitudinal and transverse response functions. While the FG model reproduces the position and width of the separate SM responses, it cannot fully account for the detailed energy dependence. Specifically, the SM responses present tails in the low and high ends of the energy transfer, reminiscent of the momentum distribution of a finite system.

The finite size effects are often simulated by the local density approximation (LDA), which consists in replacing the Fermi momentum by a local value, , depending on the nuclear density . These local responses per unit volume are then integrated over the full nuclear volume to obtain the LDA responses. In Amaro94 it was shown that, while the LDA responses present in fact short tails closer to the SM, they fail to describe the SM responses around the center of the peak. As a result a bare Fermi gas with constant is more appropriate to describe the responses of a finite nucleus.

In Ref. Amaro:1995kk the shell model was extended to a semi-relativistic (SR) shell model by implementing relativistic kinematics and using a semi-relativistic expansion of the electromagnetic current. The relativistic electroweak current was expanded in powers of the initial nucleon momentum, while treating exactly the relativistic kinematics of the final ejected particle. The SR expansion of the current and relativistic kinematics was tested in the Fermi gas model by comparison of the RFG and SRFG response functions. The two models are in great accord for all values of the momentum transfer. This is seen in Fig. 2, where results for the semi relativistic shell model (SRSM) are compared to the RFG and the SRFG models for the reaction Amaro:2005dn. Two incident neutrino energies, and 1.5 GeV, and two scattering angles, , are shown. The cross section is plotted as a function of the final muon energy . Therefore a range of and values, which can be high or low depending on the kinematics, is being explored. The RFG and SRFG results are essentially equal, while the SRSM also gives similar results, with the exception of the characteristic finite size tails and a small shift of the maximum. Figure 2: Differential cross section of the reaction 12C(νμ,μ−) for incident neutrino energies ϵ=1 and 1.5 GeV and for two scattering angles: RFG with kF=215 MeV/c. SRFG and shell model from Amaro:2005dn.

The semi-relativistic shell model was extended in Amaro:2006if to improve the description of the final-state interaction of relativistic nucleons. The Dirac-equation-based potential plus the Darwin factor (DEB+D) was used instead of the non-relativistic Woods-Saxon potential of the SRSM. The DEB potential was taken from the relativistic mean field (RMF) model of ref. Caballero:2005sj by a Schrödinger reduction of the Dirac equation with scalar and vector potentials. The resulting DEB potential is local and energy-dependent. The solution of the Schrödinger equation, multiplied by the nonlocality Darwin factor, corresponds to the upper component of the Dirac spinor of the RMF. The Relativistic Mean Field model applied to electron and neutrino-nucleus scattering reactions is discussed in detail in next section. Here we simply summarize those aspects of the model that result of interest for the comparison with the semi-relativistic predictions discussed above. Figure 3: Electromagnetic scaling functions for three values of the momentum transfer computed with the SR model with DEB+D potential, SR model with Woods-Saxon potential (WS), and relativistic mean field with the CC2 current operator Amaro:2006if.

Results for the DEB+D potential in the semi-relativistic shell model are presented in Fig. 3, where we show the longitudinal and transverse scaling functions for electron scattering for three values of the momentum transfer , 0.7 and 1 GeV/c. We compare with the results of the RMF model with the CC2 current operator (see details in next section) and with the SR shell model with Woods-Saxon potential. The SR-DEB+D scaling functions are similar to the fully relativistic ones. They present the same amount of asymmetry with a long tail in the region, while the maximum of the scaling functions is shifted to the right with respect to the SRSM. Notice that the SR-DEB+D shell model contains the same physical ingredients in the FSI as the full RMF, except for the genuine off-shell properties of the CC2 current (see next section), not included in the SR current, which was obtained assuming on shell (relativistic) kinematics. This explains the differences between the results obtained with the two models, particularly in the transverse scaling function. As shown in next section, where a detailed discussion on the RMF results is presented, these differences are of the same order as the ones found in the RMF between the several prescriptions of the electromagnetic current.

### iii.3 Relativistic Mean Field (RMF)

The kinematics involved in electroweak reactions with very high energy and momentum values transferred in the processes implies the need of using a fully relativistic formalism. This means not only describing relativistically the kinematics but also the nuclear dynamics. This was the basic motivation in the past to introduce the Relativistic Mean Field (RMF) approach and to apply it to the description of electron and neutrino scattering reactions with nuclei. In this section we introduce the basic ideas embodied in the RMF approach as developed in detail in previous works Maieron03; Caballero05; Caballero:2006wi; Udias93; Udias01; Alberico97plus; Udias:1996iy; Udias:1993zs; Udias:1993xy, and show its capability to describe successfully observables for different type of reactions, namely, inclusive and semi-inclusive processes. The RMF model, as many other approaches applied to electroweak reactions, is used within the framework of the Impulse Approximation (IA). This means that the complex scattering process off a nucleus is simply described as an incoherent sum of single-nucleon scattering reactions. Although this is an oversimplified description of the real process, it accounts for the main effects that emerge in the case of the quasielastic (QE) kinematics. Obviously, ingredients beyond the IA, such as nucleon correlations, meson exchange currents (MEC), etc. should be taken into account for a proper description of the data in the whole range of kinematics. This topic is discussed at length in subsequent sections.

As already mentioned, the analysis of the world inclusive QE shows without ambiguity that scaling is fulfilled at high level. Thus any theoretical model aiming to explain electron scattering is constrained to satisfy scaling properties. Not only the theoretical scaling function should not depend on the momentum transfer , neither on the nuclear system, but also its particular shape, with a long tail extended to high values of the energy transfer (high positive values of the scaling variable), should emerge. Most of the models based on the IA satisfy scaling; this is the case of the RFG, the semirelativistic approach presented in the previous section, and diverse non relativistic approaches based on the use of the spectral function. However, most of them lead to symmetrical scaling functions that depart significantly from the data analysis, missing the tail at high . In this section we discuss the basic ingredients of the Relativistic Mean Field (RMF) approach. As it will be shown, this model, even being based on the IA, leads to an asymmetrical superscaling function in accordance with data. This proves the capability of the RMF model to describe inclusive electron scattering reactions and its extension to neutrino processes. Not only that, the model has been also applied to semi-inclusive reactions providing a good description of the reduced cross section (distorted momentum distribution), as well as the spectroscopic factors.

The main ingredient needed to evaluate the electromagnetic and weak tensor is the single-nucleon current matrix element,

 ⟨Jμ(Q)⟩=∫dreiq⋅r¯¯¯¯ψF(pF,r)^ΓμψjmB(r) (72)

with the corresponding single-nucleon current operator for electron or weak neutrino scattering, and () the wave function for the initial bound (emitted) nucleon. The RMF model incorporates a fully relativistic description of the scattering reaction based on the impulse approximation. We use the relativistic free nucleon expressions for the current operators corresponding to the usual options, denoted as CC1 and CC2,

 ^ΓμCC1 = (F1+F2)γμ−F22mN(¯¯¯¯P+PF)μ (73) ^ΓμCC2 = F1γμ+iF22mNσμνQν, (74)

where and are the Pauli and Dirac nucleon form factors, respectively, that depend only on . We have introduced the on-shell four-momentum with and the bound nucleon momentum. Notice that the two operators are equivalent for free on-shell nucleons, as occurs in the RFG model. However, in general one deals with off-shell bound and ejected nucleons, hence the and operators lead to different results. Moreover, the current is not strictly conserved and uncertainties linked to the election of the gauge also occur Caballero:2006wi; Caballero98a. In the case of neutrino scattering processes the current operator also includes axial and pseudoscalar terms,

 ^ΓμA=[GAγμ+GP2mNQμ]γ5 (75)

with and the axial-vector and pseudoscalar nucleon form factors, respectively. Figure 4: Relativistic attractive scalar and repulsive vector potentials corresponding to 12C and 16O. Figure 5: Real and imaginary terms of the relativistic optical potentials fitted to elastic nucleon-nucleus scattering. Results correspond to 16O. Top panels refer to the dependence with the final nucleon kinetic energy for the EDAIO potential. Bottom panels contain a comparison between the scalar and vector (real and imaginary terms) potentials for four different models. The kinetic energy of the ejected nucleon is fixed to TN=500 MeV (see text for details).

Concerning the nucleon wave functions, they are given as solutions of the Dirac equation in presence of phenomenological relativistic potentials with scalar and vector terms. We describe the bound nucleon states, , as self-consistent Dirac-Hartree solutions, derived within a Relativistic Mean Field (RMF) approach using a Lagrangian with local potentials fitted to saturation properties of nuclear matter, radii and nuclear masses. In Fig. 4 we present the potentials corresponding to the cases of C and O. As shown, large scalar (attractive) and vector (repulsive) potentials, that do not depend on the energy, are present. It is important to point out that the RMF does get saturation, even with no Fock terms neither explicit nucleon correlations included. This comes from the combination of the strong scalar and vector potentials that incorporate repulsive and attractive interactions, and it makes a big difference with most non-relativistic approaches where correlations are needed in order to get saturation. The use of strong local scalar and vector terms in the relativistic potentials leads to the presence of non-local terms when performing a non-relativistic reduction (the so-called Dirac equation based (DEB) approach. See ref. Udias:1993zs for details and the general discussion and results presented in the previous section for the SR model with DEB+D potential). Figure 6: (Left panel) Reduced cross sections versus missing momentum for the shells 3s1/2 and 2d5/2 of 208Pb. Small circles with error bars are data from Ref. Quint88. In the high missing momentum region the relativistic results obtained with the currents CC2 (solid line) and CC1 (long-dashed lines), as well as the nonrelativistic results (short-dashed lines) are compared with data from Ref.Bobeldijk:1994zz. Figure taken from Ref. Udias:1996iy. (Middle and right panels) Interference longitudinal-transverse response RTL and TL asymmetry ATL for proton knockout from 16O for the 1p1/2 (top panels) and 1p3/2 (bottom panels) orbits. Results correspond to a fully relativistic calculation using the Coulomb gauge and the current operator CC2 (solid line), a calculation performed by projecting the bound and scattered proton wave functions over positive-energy states (short-dashed line) and two nonrelativistic calculations with (long-dashed) and without (dotted) the spin-orbit correction term in the charge density operator.

Different options, that depend on the particular scattering process analyzed, have been considered to describe the ejected nucleon state. In what follows we describe briefly the model applied to semi-inclusive and inclusive and reactions. In the former FSI are described using phenomenological energy-dependent complex optical potentials fitted to elastic proton-nucleus scattering data. The presence of an imaginary term produces absorption, namely, flux lost into the unobserved non-exclusive channels. In Fig. 5 we present the real and imaginary (scalar and vector) terms corresponding to different models: EDAIO, EDAD1, EDAD2 and EDAD3 (see refs. Udias:1993zs; Udias93; Udias01; Udias:1996iy; Udias:1993xy for details), as well as their dependence with the energy of the outgoing nucleon. The RMF model with complex optical potentials yields very good agreement with data. Not only reasonable values for the spectroscopic factors are given Udias:1993zs; Udias:1993xy but also the shape of the reduced cross section fits the behavior of data even at high values of the missing momentum Udias:1996iy. It should be emphasized that this region (high -values) is very sensitive to theoretical models, and in particular, to the the strong enhancement of the lower components in the Dirac wave functions produced by the relativistic potentials mainly in the final scattering state. It is remarkable the agreement between the RMF predictions and data for the reduced cross sections (left panel in Fig. 6), and particularly, for the interference longitudinal-transverse response and asymmetry with the oscillations observed at moderate-to-high missing momentum values (middle and right panels in Fig. 6). This is a clear signal of the presence and crucial role played by dynamical effects of relativity affecting the lower components in lepton-nucleus scattering Udias:1999tm. Figure 7: Double differential cross section for (e,e′) scattering on 12C. Results correspond to the electron beam energy fixed to 1 GeV and scattering angle θ=45o. Top (bottom) panel refers to results obtained with the CC2 (CC1) current operators. Coulomb gauge has been considered. Figure 8: Longitudinal scaling function for 12C(e,e′) evaluated at different values of the momentum transfer. Results are shown for the three models discussed in the text: RPWIA (blue dot-dashed line), rROP (green dashed) and RMF (red solid). Also included for reference two fits (see Donnelly99a; Donnelly99b; Maieron:2001it for details). Figure 9: Same as Fig. 8 but restricted to the RMF model.

The extension of the RMF approach to inclusive processes such as and , requires to retain the contributions from the inelastic channels. A simple way of obtaining the inclusive strength is to use purely real potentials. We have considered in the past two approaches. The first uses the same phenomenological relativistic optical potentials considered in semi-inclusive reactions, but with their imaginary terms set to zero. This is known as rROP, and it clearly does not preserve orthogonalization as the bound and ejected nucleon states are evaluated using different potentials. A similar comment applies to the particular case in which FSI are turned off, that is, the Relativistic Plane Wave Impulse Approximation (RPWIA). Finally, a different approach consists of describing the scattered states as solutions of the Dirac equation with the same potential considered for the initial bound nucleon states, namely, the real energy-independent RMF potential. This model preserves orthogonality, verifies continuity equation and current conservation, and fulfils the dispersion relationship. In what follows we use RMF to name the previous model, that is, the use of the same RMF potential to describe both the initial bound and final scattered nucleon wave functions. In Fig. 7 we show the differential cross section for scattering on C at electron beam energy equal to 1 GeV. Results are presented for the two usual current operators, CC1 (bottom panel) and CC2 (top panel). In each case we compare the predictions provided by the RPWIA, rROP and RMF models. As noticed, the RMF-CC1 cross section is significantly larger than the result for CC2. This is a consequence of the extremely large effects introduced by the lower components of the nucleon wave functions, particularly the scattered one, when using the CC1 operator. This topic was studied in detail in Caballero:2006wi. A similar comment also applies to the rROP-CC1 versus rROP-CC2, although here the relative effects are weaker. Notice that the three models get the same value for the maximum in the cross section in the case of CC1, whereas a large discrepancy is observed for CC2 being the RMF prediction the lowest. Finally, only the RMF model leads to a significant tail extended to large values of the energy transfer, .

The RMF model has proved its capability to describe successfully inclusive electron scattering data in the QE domain. Not only superscaling emerges from the calculations, but also the specific shape of the scaling function with a long tail extended to high (large positive values of the scaling variable ) is present. However, in spite of its merits, the RMF also presents some drawbacks for increasing values of the momentum transfer . The strong energy-independent scalar and vector potentials involved in RMF lead to a very significant shift of the scaling functions to higher , and correspondingly, too severe a decrease in the maxima. In the text that follows we discuss some representative results for the scaling function and the separate and contributions (see section III.5 for explicit expressions) obtained with the different approaches discussed above, i.e., RPWIA, RMF and rROP. Figure 10: Global scaling function corresponding to (e,e′) evaluated with the RMF model. Results are presented for the two current operators: CC2 (left panels) and CC1 (right panels). Off-shell effects are shown in the top panels where results are presented for 12C and three gauges: Lorentz, Coulomb and Weyl. Bottom panels refer to the analysis of second kind scaling comparing the results obtained for carbon, oxygen and calcium. Figure 11: Double differential cross section for charged-current neutrino scattering on 12C. Results correspond to fixed values of the neutrino energy and muon scattering angle: εν=1 GeV and θμ=45o. Predictions of the RMF and RPWIA are compared with the results of a semirelativistic calculation (see text and Ref. Amaro:2005dn for details). Figure 12: Scaling function evaluated from (ν,μ−) on 12C. Results are shown for RPWIA (top panels), rROP (middle) and RMF (bottom). Scaling of the first kind is analyzed in the graphs on the left that compare results for different values of the energy beam. Scaling of the second kind is shown in the right panels that contain the scaling functions corresponding to 12C, 16O and 40Ca. Figure 13: (Color online) Longitudinal and transverse scaling functions for (e,e′) compared with f(ψ′) evaluated from (ν,μ−) and (¯¯¯ν,μ+). All results correspond to the RMF approach using the CC2 current operator. The kinematics selected corresponds to fixed values of the incident lepton energy, ε=1 GeV, and momentum transfer, q=0.7 GeV/c. The averaged experimental function extracted from longitudinal electron scattering data is also shown Maieron:2001it. Figure taken from Ref. Caballero:2007tz.

First in Fig. 8 we show the longitudinal scaling function for processes evaluated at several values of the momentum transfer ranging from low
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