Neutrino-nucleus scattering in the SuSA model

Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 https://doi.org/10.1140/epjs/s11734-021-00289-5 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Neutrino-nucleus scattering in the SuSA model J. E. Amaro1,M.B.Barbaro 2,3,a, J. A. Caballero4,5,T.W.Donnelly 6,R.Gonz´alez-Jim´enez7, G. D. Megias4,8, and I. Ruiz Simo1 1Departamento de F´ısica At´omica, Molecular y Nuclear, and Instituto de F´ısica Te´orica y Computacional Carlos I, Universidad de Granada, 18071 Granada, Spain 2Dipartimento di Fisica, Universit`a di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Turin, Italy 3Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France 4Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, 41080 Seville, Spain 5Instituto de F´ısica Te´orica y Computacional Carlos I, 18071 Granada, Spain 6Laboratory for Nuclear Science, Department of Physics, Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 7Grupo de F´ısica Nuclear, Departamento de Estructura de la Materia, F´ısica T´ermica y Electr´onica, Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid and IPARCOS, 28040 Madrid, Spain 8Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Japan Received 5 June 2021 / Accepted 19 September 2021 / Published online 25 October 2021 ©The Author(s) 2021 Abstract The super-scaling approach (SuSA) model, based on the analogies between electron and neutrino interactions with nuclei, is reviewed and its application to the description of neutrino-nucleus scattering is presented. The contribution of both one- and two-body relativistic currents is considered. The model is validated with the (e, e) data, including also inelastic contributions for the inclusive reaction. A discussion of semi-inclusive reactions and their implications for charge-changing reactions of neutrinos is also presented. A selection of results for the inclusive neutrino reactions with change of charge is presented where theoretical predictions are compared with cross-section measurements from the main ongoing neutrino oscillation experiments. 1 Introduction A huge experimental programme on accelerator-based neutrino oscillation experiments has been developed in recent years with the scope of improving our knowledge of neutrino properties and of potentially answering one of the fundamental open questions of modern physics, that is the origin of the matter/antimatter asymmetry in the universe, by measuring the weak CP- violating phase [1]. The success of this program partly relies on the control of systematic errors, largely due to uncertainties in the description of the neutrino interactions with the detector, typically made of mediumweight nuclei like carbon, oxygen or argon. With this motivation, intense theoretical activity has been carried out in parallel to provide an accurate description of neutrino-nucleus reactions in the GeV region, relevant for the experiments, for different processes ranging from quasielastic up to deep inelastic scattering, encompassing the excitation of nucleon resonances and the emission of two or more nucleons (see Ref. [2]for a comprehensive review of recent progresses and open challenges in the field). ae-mail: [email protected] (corresponding author) The scattering of energetic neutrinos with atomic nuclei has been studied in depth from a theoretical point of view for more than a decade. There is abundant bibliography and review articles have been written on the subject [3–10]. However, the problem is non-trivial and there are still differences between the different existing theoretical models. The reasons are many and profound. The main reason is that the atomic nucleus is a system of many strongly interacting particles that undergo the Pauli exclusion principle. This implies that the probability of interaction with a nucleus is not the sum of the probabilities of interaction with its components (nucleons). The nucleons are not free particles but they are correlated with each other and the interaction in the final state of the hadrons that emerge from the reaction (nucleons, pions, ...) must also be accounted for. Furthermore, the energies of the neutrinos in the accelerators are of the order of the GeV and this implies large transfers of energy and momentum to the nucleus, which requires a relativistic treatment of the nuclear problem. In this regard, the models that treat relativity in an appropriate way have been the most used (relativistic fermi gas, relativistic mean field). When it comes to the interaction of neutrinos with many-particle systems, there is the added difficulty of 123 4322 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 multinucleon emission, either due to the interaction with two correlated nucleons, or due to the action of meson exchange currents (MEC). The main emission channel of two nucleons (2p2h) can represent more than 20% of the quasi-elastic cross section depending on the kinematics. Its main contribution comes from the excitation of a Δthat within the medium can decay, emitting two nucleons. The excitation of the Δis also the largest contribution to the pion emission channel, where the interaction in the final state of the pions must also be taken into account. An alternative to microscopic nuclear models are the phenomenological scaling models applied to the quasielastic response functions. The phenomenon of scaling is sometimes found to occur in different fields, including solid state, atomic, molecular, nuclear and hadronic physics, when an interacting probe scatters from composite many-body systems. Under some circumstances, it is found that the response of the complex system no longer depends on two independent variables, but only on a particular combination of those, called the scaling variable. This phenomenon is very well known in high energy physics where the inelastic nucleon response functions are shown to depend only on the Bjorken variable x[11]. A similar phenomenon is observed in lepton-nucleus scattering, where the nuclear response functions are found to scale with a single scaling variable, denoted as y. This indicates that the probe (lepton) interacts with the nucleus’ constituents, in this case with the nucleons in the nucleus. The phenomenon of y-scaling emerges from the analysis of quasielastic (QE) (e, e) reactions and has been studied in detail in Refs. [12–14]. The scaling function is defined as the QE (e, e) differential cross section divided by a single-nucleon cross section averaged over the Fermi gas. For high enough values of the momentum transfer, q, the scaling function does only depend on a single variable, y, given as a particular combination of the energy (ω) and momentum (q) transferred in the process. The scaling variable yis (up to a sign) the minimum value of the missing momentum allowed by kinematics [12–14]. In this article, we review our work on the SuSA (super-scaling approach) model, originally developed in Ref. [15] and subsequently refined and improved to the updated version SuSAv2 [16,17] (see also Refs. [18,19] for other recent reviews). The original SuSA, as will be explained in more detail in Sect. 2.2, is a phenomenological model which, while retaining the relativistic aspect of the relativistic fermi gas (RFG), provides by construction—unlike the RFG—a good description of inclusive electron scattering data (e, e) in the quasielastic region. In the model initial and final state interaction effects, absent in the RFG, are directly extracted from (e, e) data. The updated version, SuSAv2, implements inputs from the relativistic mean field model (RMF), which provides a microscopic interpretation of the basic features of SuSA and also includes a more detailed description of the different spin and isospin channels. Moreover, the model has been extended to include two-body currents, able to excite two-particle-two-hole (2p2h) states; these, as will be shown in Sect. 4,play an important role in the analysis of neutrino-nucleus experimental data. The scheme of this review is the following. In Sect. 2 we focus on the inclusive reaction, presenting the formalism, describing the model and testing it with electron scattering data. In Sect. 3we discuss the semiinclusive reaction an its relation to the inclusive one. In Sect. 4we show some selected results and discuss the comparison with experimental data. Finally, in Sect. 5 we draw our conclusion and outline the future developments of our work. 2 Inclusive neutrino-nucleus scattering 2.1 General formalism In this review, we mainly focus on the charged current (CC) inclusive process νl(νl)+A−→ l−(l+)+X, (1) where a neutrino (antineutrino) of a given flavour l having four-momentum Kμ=(ε, k) hits a nucleus A and a charged lepton l−(l+) is detected in the final state with four-momentum Kμ=(ε,k) and scattering angle θ, while the residual system Xis unobserved. The four-momentum transferred from the probe to the nucleus is Qμ=(ω,q), with ω=ε−εand q=k−k. We shall assume the neutrino to be massless, while the outgoing lepton has finite mass ml. The corresponding cross section is obtained by contracting the leptonic and hadronic tensors lμν(q,ω) =2KμK ν+K μKν−K·Kgμν +iχμνρσKρKσ, (2) Wμν(q,ω) = n <A|Jμ†(q,ω)|n><n|Jν(q,ω)|A> δ(ω+EA−En),(3) where χ=+1(−1) in the neutrino (antineutrino) case, |A>is the initial nuclear ground state and |n>are all the intermediate hadronic states that can be reached through the weak current operator Jμ. The double differential cross section with respect to the momentum and scattering angle of the outgoing lepton can be expressed in terms of five response functions [15] (for the semi-inclusive reaction (νl,l−N), which will be briefly discussed in Sect. 3, the response functions are ten) 123 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 4323 d2σ dεdcos θ=(GFcos θc)2 4πv0 k ε(VCCRCC +2VCLRCL +VLLRLL +VTRT+2χVTRT),(4) where GFthe Fermi weak constant, θcthe Cabibbo angle and v0=(ε+ε)2−q2. The indices C,L,Trefer to the Coulomb, longitudinal and transverse components of the leptonic and hadronic currents with respect to q. The leptonic coefficients VKare related to the components of the tensor (2) and are given by VCC =1−δ2|Q2| v0 ,V CL =ω q+δ2 ρ|Q2| v0 , VLL =ω2 q2+δ2|Q2| v01+ 2ω qρ+ρδ2, VT=|Q2| v0 +ρ 2−δ2|Q2| v0ω qρ+ρδ2 2, VT=|Q2| v01 ρ−ωδ2 q,(5) where the variables δ2=m2 l |Q2|,ρ=|Q2| q2and ρ=q ε+ε have been introduced. The response functions RCC =W00,R CL =−1 2W03 +W30, RLL =W33,R T=W11 +W22, RT=−i 2W12 −W21,(6) embody the nuclear dynamics and are specific components of the hadronic tensor Wμν, depending only upon ωand q. The above decomposition into response functions is valid for all reaction channels—elastic, quasielastic, inelastic—each of which characterized by a different current operator Jμin the hadronic tensor (3). In the case of quasielastic scattering, corresponding to the interaction of the probe with a single nucleon, the weak current operator is Jμ(q,ω)=F1(Q2)γμ+iF2(Q2) 2mN σμνQν−GA(Q2)γμγ5 −GP(Q2) mN Qμγ5,(7) where F1,F2,GAand GPare the Pauli, Dirac, axial and pseudoscalar weak form factors, respectively. In neutrino oscillation experiments the incident beam is not monochromatic. As a consequence, before comparing with experimental data, the cross section (4) must be folded with the normalized neutrino flux φ(ε) (dε φ(ε) = 1): d2σ dεdcos θ=dε φ(ε)d2σ dεdcos θ.(8) 2.2 Scaling and super-scaling: the SuSAv2 model In the QE domain the basic mechanism in (e, e) reactions on nuclei corresponds to elastic scattering from individual nucleons in the nuclear medium. This implies that the inclusive (e, e) cross section is mainly constructed from the exclusive (e, eN) process, including the contribution of all nucleons in the target and integrating over all (unobserved) ejected nucleon variables (see also the following section). This approach constitutes the basis of the impulse approximation (IA). Thus, the double differential (e, e) inclusive cross section is given as the sum of two response functions corresponding to the longitudinal, L, and transverse, T, channels, d2σ dΩedω =σMott [vLRL(q,ω)+vTRT(q,ω)] ,(9) where σMott is the Mott cross section and the vL,T are kinematical factors that involve leptonic variables (see Refs. [18,20] for explicit expressions). In terms of the scaling functions the nuclear responses are RL,T (q,ω)= 1 kFfL,T (q,ω)GL,T (q,ω),(10) where kFis the Fermi momentum and GL,T are defined as the responses of a moving nucleon and include relativistic corrections arising from the presence of the medium. Their explicit expressions can be found in Refs. [16,18,21]. The terms fL,T are the scaling functions that show a mild dependence upon the momentum transfer q(first-kind scaling) and very weak dependence on the nuclear system considered (second-kind scaling). The occurrence of both types of scaling is denoted as superscaling. Hence to the extent that at some kinematics the above ffunctions are the same for all nuclei and do only depend on a single variable, denoted as the scaling variable ψ(q,ω), but not separately on q,onesays that superscaling occurs. The superscaling variable ψis given in terms of qand ωas ψ=1 √ξF λ−τ (1 + λ)τ+κτ(1 + τ) ,(11) where the dimensionless Fermi kinetic energy ξF=TF mN, energy transfer λ=ω 2mN, momentum transfer κ=q 2mN and squared four-momentum transfer τ=κ2−λ2have been introduced. Here the y-variable introduced above is given approximately by y∼ =kFψ, (12) as discussed in Ref. [14]. A phenomenological energy shift Eshift, fitted for each nucleus to the experimental position of the quasielastic peak [22], is included in the definition of the scaling variable ψ≡ψ(q,ω −Eshift). 123 4324 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 -1 0 1 2 3 ψ’ 0 0.1 0.2 0.3 0.4 0.5 0.6 fL (RMF model) q=550 MeV/c q=650 MeV/c q=800 MeV/c q=1100 MeV/c q=1400 MeV/c 500 600 700 800 q (MeV/c) 1 1.5 2 2.5 3 3.5 χ2/dof Fig. 1 Left panel: Phenomenological SuSA scaling function versus ψin the QE region together with (e, e) longitudinal scaling data from Ref. [26]. The RFG scaling function is also shown as reference. Right panel: RMF longitudinal scaling functions for (e, e) at different qvalues compared with (e, e) longitudinal scaling data from Ref. [26]. In the inner smaller plot a reduced-χ2analyses shows a minimum at q= 650 MeV/c. Figure adapted from Ref. [16] Scaling and superscaling properties of electronnucleus interactions have been analyzed in detail in a series of previous works [12–14,22–25]. The importance of this phenomenon to test the validity of any nuclear model aiming to describe electron scattering reactions has been clearly proven. The model, denoted as the super-scaling approach (SuSA), is entirely based on the phenomenology, making use of a unique, universal, scaling function extracted from the analysis of the longitudinal electron scattering data (see Fig. 1). Notice that the behavior and properties of the experimental superscaling function constitute a strong constraint for any theoretical model describing QE electron scattering. Not only should the superscaling behavior be fulfilled, but also the specific shape of the longitudinal scaling function, fexp L, must be reproduced. The SuSA model assumes the longitudinal phenomenological scaling function to be valid also in describing the transverse channel, i.e., fL=fT. In recent years, we have explored in detail the extension of SuSA to neutrino-nucleus scattering. Due to the complexity of the weak process, with an increased number of nuclear responses compared with the pure electromagnetic one, and the particular role played by the axial term in the weak current, we have developed an improved version of the superscaling model, called SuSAv2 [16]. Contrary to the original SuSA [15,21,27] where a universal scaling function based on electron scattering data is used, the new SuSAv2 model incorporates relativistic mean field (RMF) effects [24,25]in the longitudinal and transverse nuclear responses, as well as in the isovector and isoscalar channels that is of great importance to describe charged-current (CC) neutrino reactions that are purely isovector [28]. The origin of the SuSAv2 approach is based on the capability of the RMF to describe properly the scaling behavior of the electron scattering data. As shown in previous works [24,25], RMF is one of the few microscopic models capable of reproducing the asymmetric shape of the phenomenological scaling function with a long tail extended to high values of the transfer energy (large values of ψ). Moreover, RMF produces an enhancement in the transverse scaling function, a genuine dynamical relativistic effect linked to the lower components in the wave functions, that is supported by the analysis of data. The RMF framework to finite nuclei has proven to successfully reproduce the scaling behavior shown by data at low to intermediate qvalues (see Fig. 1). However, the model clearly fails at higher momentum transfers where final state interactions (FSI) are expected to be weaker. This is due to the RMF strong energy-independent scalar and vector potentials used in the final state that lead to too much asymmetry in the scaling functions and shift the QE peak to very high transfer energies, in clear disagreement with data. To remedy this shortfall of the RMF model, the SuSAv2 incorporates both the RMF scaling functions at low-to-intermediate qvalues and the relativistic plane wave impulse approximation (RPWIA) ones at higher qusing a q-dependent blending function that smoothly connects the two regimes (see Refs. [17,29] for details). A similar solution to this drawback of the RMF model has been taken in the recent energydependent RMF (ED-RMF) approach [30,31] where RMF potentials are multiplied by a blending function inspired by the SuSAv2 one that scales them down as the kinetic energy of the scattered nucleon increases, also preventing non-orthogonality issues. This model predicts both lepton and nucleon kinematics, showing a similar agreement on electron and neutrino data with SuSAv2. In summary, the SuSAv2 model and, for extension, the ED-RMF one reproduce the experimental longitudinal scaling data, gives rise to an enhancement in the electromagnetic transverse channel, i.e., f(e,e) T>f (e,e) L, takes into account the differences in the 123 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 4325 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 500 1000 1500 2000 2500 B-C B-R GRV98 E=680 MeV, θ=60o, qQE=610 MeV/c 0,2 0,4 0,6 ω (GeV) 0 1000 2000 3000 4000 E=1108 MeV, θ=37.5o, qQE=674.6 MeV/c Fig. 2 Comparison of inclusive 12C(e, e) double differential cross sections and predictions for the inelastic regime of the Bosted–Christy parametrization [33,34] (dot-dashed lines), Bodek–Ritchie parametrization [35,36] (solid lines) and GRV98 PDFs [37,38] (dashed lines) at different kinematics (incident electron beam and scattering angle) in terms of the energy transferred to the nucleus (ω). Experimental data taken from [39,40]. The y-axis represents d2σ/dΩ/dω in nb/GeV/sr. The value of qat the QE peak (qQE) is shown as reference. Figure adapted from [41] isoscalar/isovector scaling functions, of crucial interest for neutrino scattering processes, and finally avoids the problems of the RMF model in the region of high momentum transfer, where FSI effects are negligible. One of the basic merits of SuSAv2 is the translation of sophisticated and demanding microscopic calculations into relatively straightforward parametrizations and, hence, easing its implementation in the Monte- Carlo simulations employed in the analysis of neutrino oscillation experiments. To conclude, it is important to point out that SuSAv2 is not restricted to the QE kinematics domain. On the contrary, it can be extended to the inelastic region by generalizing the superscaling variable (11) to the excitation of any inelastic state having invariant mass W. One can thus define [32] ψW=1 √ξF λ−τρW (1 + λρW)τ+κτ(1 + τρ2 W) , ρW=1+ 1 4τW2/m2 N−1,(13) and evaluate the inelastic hadronic tensor as the integral Wμν inel =1 m2 NWmax Wmin dWWf(ψW)wμν inel,(14) between the limits Wmin,W max imposed by the kinematics, where f(ψW) is the SuSAv2 superscaling function previously described. This procedure is based on the assumption that the nuclear effects, encoded in the function f, are the same in the QE and inelastic regions and requires the knowledge of the elementary tensor wμν inel across the full inelastic spectrum. The SuSAv2 inelastic model has been applied to the electron scattering case using phenomenological fits of the single-nucleon inelastic structure functions w1 and w2extracted from e-p and e-d data [33,34]. In Fig. 2, a comparison of the SuSAv2-inelastic model with 12C(e, e) data using different inelastic structure functions shows a preference for the Bosted–Christy parametrization [33,34]. In general, the SuSAv2 model (QE+inelastic) provides a very good description of data for very different kinematics once meson exchange currents (MEC) are also incorporated, denoted as SuSAv2- MEC (see Sect. 2.4,[17,42,43]). The inclusion of the full inelastic spectrum in the SuSAv2 model for weak interactions is still in progress. An alternative approach to the study of the resonant pion production was taken in Refs. [15,44,45], where a scaling function to be used in the Δresonance region, different from the quasielastic one, was extracted from electron scattering data and multiplied by the appropriate weak N→Δtransition form factors to get the neutrino-nucleus cross section in this region. This method provides a phenomenological description valid at transferred energies below the Δpeak, while at higher ωit fails due to the opening of other inelastic channels. Some results corresponding to this method, referred to as SuSA-Δapproach, will be shown in Sect. 4. 2.3 Relativistic model for CC MEC and 2p2h responses Multinucleon knockout processes give a non-negligible contribution to the inclusive neutrino cross section for the intermediate energies involved in the experiments [17,46–49]. In Ref. [50] we developed a model of twoparticle two-hole excitations of the RFG induced by 123 4326 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 Fig. 3 Feynman diagrams for the electroweak MEC (a) (b) (c) (d) (e) (f) (g) (h) weak meson-exchange currents for inclusive CC neutrino scattering. The model is fully relativistic and includes the diagrams of Fig. 3, involving one-pion exchange and Δexcitation, taken from the pion production model of [51]. The 2p2h matrix element of MEC depends on the momenta, spin and isospin coordinates (1,2;1,2) ≡ (p 1s 1t 1,p 2s 2t 2;h1s1t1,h2s2t2) of the two holes, h1,h2, and the two particles, p 1,p 2.Itisthesumoffourcontributions jμ(1,2;1,2) ≡jμ(p 1s 1t 1,p 2s 2t 2;h1s1t1,h2s2t2) =jμ sea +jμ π+jμ pole +jμ Δ,(15) corresponding in Fig. 3to the seagull (diagrams a,b), pion in flight (c), pion-pole (d,e) and Δ(1232) excitation (f,g,h,i). Their explicit expressions are given in Ref. [50]. The inclusive hadronic tensor in the 2p-2h channel is computed by integration over all the 2p-2h excitations of the RFG. Momentum conservation enforces p 2= h1+h2+q−p 1. Hence Wμν 2p2h =V (2π)9d3p 1d3h1d3h2 m4 N E1E2E 1E 2 wμν(p 1,p 2;h1,h2)Θ(p 1,h 1)Θ(p 2,h 2) δ(E 1+E 2−E1−E2−ω),(16) where Θ(p,h)≡θ(p−kF)θ(kF−h). The tensor inside the integral is wμν(p 1,p 2;h1,h2) ≡1 4 s1s2s 1s 2 t1t2t 1t 2 jμ(1,2;1,2)∗ Ajν(1,2;1,2)A, (17) where jμ(1,2,1,2)Ais the antisymetrized MEC matrix element jμ(1,2,1,2)A≡jμ(1,2,1,2) −jμ(1,2,2,1) . (18) The factor 1/4inEq.(17) accounts for the antisymmetry of the two-body wave function in isospin formalism, to avoid double counting in the number of final 2p-2h states. Due to azimuthal symmetry around the z-axis—in the qdirection—we fix the azimuthal angle of particle 1’ φ 1= 0, and multiply by a factor 2π. The energy deltafunction enables integrating over p 1. Then Eq. (16)is reduced to a seven dimensions integral that is computed numerically [52,53]. The Dirac matrix elements of the currents are also computed numerically. The 2p-2h inclusive cross section requires one to compute the five weak response functions, RCC,CL,LL,T,T , 123 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 4327 for (νμ,μ −). All of these responses were computed and analyzed in Ref. [50]. The five response functions have been parametrized in the kinematic range 100 <q< 2000 [17]; the parametrization is convenient because in neutrino scattering there is an additional integration over the incident neutrino flux. This parametrization has been implemented in the Monte Carlo event generator GENIE [43,54]. Using the parametrization of Ref. [17] does not allow one to modify the internal parameters of the MEC model. This is why we have developed approximations to the 2p2h responses to speed up the calculation. Specifically, in the so-called frozen nucleon approximation we set the momenta of the two holes h1=h2=0, allowing us to integrate over the initial states analytically [55]. In the modified convolution approximation (MCA) [56] we write the 2p2h responses as a convolution of two 1p-1h responses multiplied by the elementary hadronic tensor wμν (p 1,p 2;h1,h2) evaluated for convenient averaged values of h1and h2. The resulting MCA responses are a good approximation to the exact result with 4D integration only over the momentum and energy communicated to one of the nucleons. 2.4 Validation of the SuSAv2 model versus electron scattering data The model introduced in the previous sections allows one to describe inclusive lepton-nucleus scattering in the kinematic region including the quasielastic reaction, the excitation of 2p2h states and the inelastic spectrum. Validation against the large amount of existing highprecision electron scattering data is a necessary test to be performed before using nuclear models in the analysis of neutrino oscillation experiments. Such tests have been successfully carried out for the SuSAv2 model in a very wide range of kinematics from low/intermediate energies up to the highly inelastic regime. Only at very low energy and momentum transfers (q<300 MeV/c and ω<50 MeV) does the model fail to reproduce the data, as do all models based on the impulse approximation, which is clearly not appropriate in the low-energy regime. SuSAv2 predictions have been extensively compared with the available (e, e) data on carbon and oxygen in Refs. [29,42] and with the more recent JLab data on argon and titanium in Ref. [57]. Here we just show a few representative examples, choosing the kinematics of particular interest for ongoing neutrino experiments. In Fig. 4(top left panel), the double differential 12C(e, e) cross section is shown as a function of the energy transfer ω, for an incident electron beam of energy Eand scattering angle θ, corresponding to the momentum transfer qQE at the quasielastic peak. The separate QE, 2p2h and inelastic contributions are shown. The agreement with the data is excellent across the full spectrum. Note that 2p2h excitations are essential to describe the “dip” region between the QE and Δpeaks. An important feature of the SuSAv2 model, based on the RMF theory, is its capability to reproduce not only the (e, e) cross sections but also the separated longitudinal and transverse responses, as shown in the top right panel of Fig. 4, where RL(solid) and RT (dashed) are plotted versus ωfor q= 570 MeV/c, together with the separate QE, 2p2h and inelastic contributions. Note that while the longitudinal response is almost purely quasielastic, with a small 2p2h contributions arising from relativistic corrections, the transverse response receives contributions from all the three processes, which sizeably overlap at these kinematics. In Fig. 4(bottom panels), the same kind of comparison is shown for the oxygen (left) and argon (right) nuclei, showing that the superscaling approach allows for a consistent description of different nuclei. As already mentioned, electron scattering is an important test for the description of the nuclear response in a wide kinematical region. Such a detailed comparison with data is not possible for neutrino-nucleus scattering because the energy and momentum transfers are not experimentally known and because the large experimental errors do not allow for a clear discrimination between different models. The excellent agreement of the SuSAv2 predictions with electron scattering data proves that the model can be safely used to describe also neutrino-nucleus scattering, where the leptonic tensor is different but the nuclear dynamics, contained in the hadronic tensor, is the same as in charged-lepton scattering. Before showing the comparison with neutrino data, in the next section we address the study of semi-inclusive electron scattering. 3 Semi-inclusive electron-nucleus scattering A pivotal difference between electron and neutrino scattering experiments is that the neutrino energy is known only as a broad distribution, while the energy in electron beams is typically defined with high precision. Hence, in electron scattering experiments by, for instance, detecting one nucleon in coincidence with the scattered electron, the hadronic final state is completely determined if one restricts one’s attention to excitation energies of the residual system below the two-nucleon emission threshold. On the contrary, in the neutrino case, even in scenarios in which one or a few hadrons are detected in coincidence with the scattered lepton, there is no way of fully determining the hadronic final state because the beam energy and consequently the excitation energy of the residual system is unknown. This means that one needs to ‘integrate’ for all possible final states compatible with the given kinematics. Formally, this integration is equivalent to the integration over the neutrino flux (or flux folding procedure) introduced in Sect. 2.1. The semi-inclusive process for CCQE reactions was recently discussed in Refs. [60,61]. To obtain a somewhat deeper understanding of how the inclusive cross sections and scaling emerge, in this section we briefly discuss semi-inclusive (coincidence) 123 4328 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 0 0.1 0.2 0.3 0.4 ω (GeV) 0 1000 2000 3000 4000 5000 d2σ/dΩdω (nb/GeV/sr) QE 2p-2h MEC Inelastic Total E=560 MeV, θ=60o, qQE=508 MeV/c 050 100 150 200 250 300 350 ω (MeV) 0 0.005 0.01 0.015 0.02 0.025 R (MeV-1) q=570 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0.6 ω (GeV) 0 5000 10000 15000 20000 25000 30000 d2σ/dΩ/dω (nb/GeV/sr) QE 2p-2h MEC Inelastic Total 16O, E=737MeV, θ=37.1o 1.6 1.2 1.4 1.8 2 2.2 E’(GeV) 0 20 40 60 80 100 120 140 d2σ/dΩ/dω (nb/GeV/sr) QE 2p-2h MEC Inelastic Total 40Ar, E=2.222 GeV, θ=15.541o Fig. 4 Top left panel: The 12C(e, e) cross sections from Refs. [39,40] compared with SuSAv2-MEC predictions. The separate QE, 2p-2h and inelastic contributions are also shown. Figure adapted from Ref. [29]. Top right panel: 12C(e, e) longitudinal (solid) and transverse (dashed) responses at q= 570 MeV/c. QE, 2p2h and inelastic contributions are shown, respectively, as green, blue and orange lines. The total response is shown by the black lines. Data from Ref. [26]. Figure adapted from Refs. [18,41]. Bottom panels: The (e, e) cross section (left) for 16OfromRef. [58] and for 40Ar (right) from Ref. [59] compared with the SuSAv2-MEC model. The separate QE, 2p-2h and inelastic contributions are also displayed. Figures adapted from Refs. [42]and[57], respectively electron scattering, focusing on the reaction (e, eN) in which the scattering electron and a nucleon are assumed to be detected in the final state. Then, in addition to the electron kinematical variables introduced above, we have an outgoing nucleon with 4-momentum Pμ N=(EN,pN) involving 3-momentum pNand polar and azimuthal angles θNand φN, respectively, together with energy EN=p2 N+m2 N. No other particles are assumed to be detected, although, depending on the specific kinematics, they must be present (see below). The magnitude of the nucleon’s 3-momentum is given by pN=|pN|. Apart from the detected nucleon, the final state contains an undetected hadronic system having missing 4-momentum (EB,pB), namely, a total energy of EBand a missing 3-momentum pB≡pm.In the following we shall assume that the detected nucleon is a proton. One then has pm=q−pN.(19) The undetected hadronic system has invariant mass MB(M0 Bat threshold with MB≥M0 B) and total energy EB=TB+MB=(MB)2+pm2,(20) which defines the kinetic energy of the unobserved finalstate system, TB.FromEq.(20) one has EB=ε−ε−TN+(M0 A−mN),(21) where M0 Ais the target ground-state mass and TN= EN−mNis the kinetic energy of the detected nucleon. This leads to the following expression for the so-called missing-energy, Em=ε−ε−TN−TB.(22) 123 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 4329 Defining the excitation energy of the residual system E=EB−E0 B≃MB−M0 B,(23) where the (typically very small) recoil kinetic energy difference has been neglected, we have Em=E+Es,(24) where Es=M0 B+mN−M0 Ais the separation energy. The magnitude of the missing-momentum pmis given by pm=k2+k2+p2 N−2kkcos θl−2kpNcos θN +2kpN(cos θlcos θN+sinθlsin θNcos φN)1 2. (25) Depending on the value of the missing-energy, the residual system may be the daughter nucleus in its ground state (this defines the threshold for the reaction to become possible); or it may be in a discrete excited state (lying below the threshold where a second nucleon can be ejected), and, although they de-excite by γ-decay, that process is slow on the nuclear timescale and thus these states may be treated effectively as stationary states. Then, at a well-defined threshold a second nucleon must be emitted (this is not optional: there are no nuclear states involving one nucleon and a residual bound nucleus above this point). As Emcontinues to increase more and more particles enter in the final state in addition to the one special nucleon that is assumed to be detected. At even larger missing energy (roughly 140 MeV) pion production becomes possible (still with the lepton and one nucleon assumed to be detected). From Eqs. (22–25) it is clear that for fixed values of the observable parameters the values of Emand pmare determined for each value of ε. In Fig. 5trajectories are shown for selected kinematics, namely, ε= 3800 MeV, θl=7 ◦,TN= 140 MeV, φN= 180◦, representing a “typical” situation. What is varied here is the polar angle for the detected proton, θN. As stated above, as one goes along a given trajectory the electron energy ε that determines where on the trajectory one finds oneself must vary. The lower boundary defines the threshold for the semi-inclusive reaction to occur. In effect, each event where an electron and a proton are detected in coincidence corresponds to a specific trajectory and point on that trajectory. All realistic representations of the missing-energy and missing-momentum distribution of the nucleons in the target nuclei, the so-called spectral function, share essentially the same ingredients, that is, a mean field part that describes the shells, with their widths somehow adjusted to the available experimental information, and a background contribution corresponding to the high missing-energy and high missing-momentum part of the spectrum, that is associated to short-range correlations. In this work, we will consider the spectral - - - - - - - - - 0 100 200 300 400 500 600 700 800 pm (MeV) 0 50 100 150 200 250 300 Em (MeV) θN=40 θN=50 θN=60 θN=70 θN=80 θN=90 θN=100 θN=110 θN=120 Fig. 5 The Em−pmtrajectories are shown for selected “typical” kinematics: ε= 3800 MeV, θl=7 ◦,TN= 140 MeV, φN= 180◦for the reaction 16O(e, ep). Each line corresponds to a different value of the proton scattering angle θN(in degrees). Here, we plot the Rome spectral function as a background to allow one to easily identify the different regions of the spectral function that are crossed by the trajectories function of Refs. [62,63], labeled for brevity as “Rome spectral function”. Though other models are available in the literature, e.g., Refs. [64,65], all of them have been constrained by electron scattering experiments, and hence they will provide similar results. When going along a given trajectory in Fig. 5, one sees a striking pattern in the behavior one should expect. The strength is extremely localized. The largest concentration occurs where the p-shells are located (at around Em=20 MeV), with less strength where the broad s-shell is located (at around Em= 50 MeV); at still larger values of Em(and pm) the spectral function does have some strength, although it is spread over a wide region in the Em−pmplane and is too small to be seen in this representation. Furthermore, we note that pion production cannot occur until one reaches Em∼mπand that it is not appreciable until Em∼mΔ−mN∼300 MeV. We notice that in the case of the CCQE neutrino interaction, studied in Ref. [61], one gets a Em−pm trajectory plot that is almost identical to the one in Fig. 5. The reason is that the muon mass does not play a strong role in these trajectories, especially for the kinematics shown here for which εis much larger than the muon mass. In passing we note that, while other choices of two variables to replace Emand pmcan of course be made, the generic behavior seen here strongly suggests that the present choice is a good one and that other choices may not reflect the highly localized nature of the nuclear response. In Fig. 6, we represent the sixfold differential cross section for fixed electron and proton kinematics as a function of the missing-energy. The model used was PWIA together with the Rome spectral function [62, 63], as described, e.g., in Refs. [60,66]. Equation (22) 123 4336 Eur. Phys. J. Spec. Top. (2021) 230:4321–4338 5 Conclusions In this paper, we have summarized the basic ingredients that go into scaling analyses of inclusive electron scattering and charge-changing neutrino reactions with nuclei. Importantly, good agreement in the former case is viewed as a pre-requisite to being able to predict the latter—clearly if one fails to account for inclusive electron scattering then it is unreasonable to expect that inclusive CCνreactions will be adequately modeled. In particular, we have emphasized the use of scaling of the first kind (independence of qat high energies) and of scaling of the second kind (independence of nuclear species). When both kinds of scaling are invoked we refer to it as super-scaling and accordingly our present focus has been placed on the so-called super-scaling approach (SuSA) in its original form and in extensions of that original form. Such extensions have been introduced to account for the modest level of scaling violations seen in direct comparisons with inclusive electron scattering data. In addition we have introduced the way 2p-2h MEC effects have been incorporated, showing that, while they are usually corrections to the cross sections being represented via the scaling functions obtained either phenomenologically or through use of specific models, they typically are required to get a successful picture of the response. We have briefly summarized these basic ideas and then proceeded to show examples of the excellent agreement found with inclusive electron scattering cross sections measured for several light to medium-weight nuclei. To place the discussions in context, we have also provided a section outlining how semi-inclusive scattering and inclusive scattering are related, and specifically how the former probes particular regions of missingenergy and -momentum (characterized by so-called trajectories), while the latter involves integrations over the Em−pmplane of the semi-inclusive response. On the one hand, inclusive scattering, being a total hadronic cross section, is less dependent on the underlying details of nuclear structure. In fact, given that the kinematics of the single-nucleon knockout are handled relativistically and that there are sum rules that determine the integral of the inclusive response, then a single parameter can be used to fix the width of the response. For this reason many models yield the rough behaviour of the inclusive cross section, even the relativistic Fermi gas model. On the other hand, the semi-inclusive cross section depends critically on how the strength is distributed as a function of Emand pm. Accordingly, models that fare reasonably well for inclusive scattering cannot be relied upon to properly represent the underlying nuclear structure needed for semi-inclusive reactions. Finally, having discussed the foundations of superscaling for electron scattering we show some selected results for CCνreactions. The agreements with existing data are found to be excellent, giving us confidence that at least for inclusive neutrino reactions the problem appears to be well in hand. Note, however, from the statements made in the previous paragraph, that this is not to be taken as proof that semi-inclusive CCνmodeling should be expected to be robust. Indeed, different models yield quite different results for cross sections in which both a charged lepton and a nucleon are detected in coincidence (a trend in modern experiments), even when the corresponding inclusive cross sections do not differ significantly. The modeling of semi-inclusive reactions together with the analysis of nuclear-medium effects for different targets are part of our present projects. These have become essential ingredients for current neutrino experiments, being the origin of major experimental uncertainties on neutrino energy reconstruction and oscillation analyses. These aspects will be also of paramount relevance for the next generation of oscillation experiments, HyperKamiokande [104] and DUNE [105], which will enlarge the capabilities of current neutrino facilities to produce high-precision measurements and gain enough sensitivity to improve the determination of neutrino oscillation parameters, mass hierarchy and CP- violating phase and where energy reconstruction and nuclear-medium uncertainties could be an important limiting factor. Acknowledgements This work was supported in part by the Project BARM-RILO-20 of University of Turin and from INFN, National Project NUCSYS (M.B.B.); by the Madrid Government and Complutense University under, project PR65/19-22430 (R.G.-J.); by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades and ERDF (European Regional Development Fund) under contracts FIS2017- 88410-P, and by the Junta de Andalucia (grants No. FQM160 and SOMM17/6105/UGR) (J.A.C.); by the University of Tokyo ICRR’s Inter-University Research Program FY2020&FY2021 and by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 839481 (G.D.M.); by the Spanish Ministry of Science through grant FIS2017- 85053-C2-1-P, and by Junta de Andalucia (grant No. FQM- 225) (J.E.A. and I.R.-S); by the Office of Nuclear Physics of the US Department of Energy under Grant Contract DEFG02-94ER40818 (T.W.D.). Funding Open access funding provided by Universit`adegli Studi di Torino within the CRUI-CARE Agreement. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. 123 Eur. Phys. J. Spec. Top. 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