Inclusive electron scattering within the SuSAv2 meson-exchange current approach

Inclusive electron scattering within the SuSAv2 meson-exchange current approach G. D. Megias,1,* J. E. Amaro,2M. B. Barbaro,3J. A. Caballero,1and T. W. Donnelly4 1Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla, Spain 2Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain 3Dipartimento di Fisica,Università di Torino and INFN, Sezione di Torino, Via P.Giuria1, 10125 Torino, Italy 4Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 28 March 2016; published 25 July 2016) We present our recent progress on the relativistic modeling of electron-nucleus reactions and compare our predictions with inclusive 12C(e,e0) experimental data in a wide kinematical region. The model, originally based on the superscaling phenomenon shown by electron-nucleus scattering data, has recently been improved through the inclusion of relativistic mean field theory effects that take into account the enhancement of the quasielastic transverse scaling function compared with its longitudinal counterpart. In this work, we extend the model to include the complete inelastic spectrum—resonant, nonresonant and deep inelastic scattering. We also discuss the impact of meson-exchange currents through the analysis of two-particle two-hole contributions to electromagnetic response functions evaluated within the framework of the relativistic Fermi gas, considering for the first time not only the transverse but also the longitudinal channel. The results show quite good agreement with data over the whole range of energy transfer, including the dip region between the quasielastic peak and the Δresonance. DOI: 10.1103/PhysRevD.94.013012 I. INTRODUCTION One of the challenging goals of current neutrino oscillation experiments is a proper and precise description of neutrinonucleus scattering at intermediate energies (from a few hundred MeV to a few GeV). Particular emphasis is placed on the evaluation of effects linked to the nuclear structure involved in the analysis of experiments. In recent years, several models, originally developed to study electronnucleus scattering, have been further extended to the description of neutrino-nucleus cross sections [1–8]. These models are required to provide a precise enough description of electron scattering data before they can be applied to neutrino reactions. In some cases, such as the simple and commonly used relativistic Fermi gas model (RFG), they fail to reproduce both inclusive electron scattering in the quasielastic (QE) regime as well as recent measurements of QE neutrino and antineutrino scattering cross sections. This is connected with the approaches assumed by the specific nuclear models and, more importantly, with the simplified description of the reaction mechanism that, in most of the cases, is based on the impulse approximation (IA) with additional nonrelativistic reductions. Hence, a proper evaluation of the effects introduced by final-state interactions (FSI) and mechanisms beyond the IA, such as nuclear correlations and two-particle two-hole excitations, are needed. In this context, a consistent and complete description of the electron scattering cross section that includes not only the QE regime but also regions at higher energy transfer (nucleon resonances, inelastic spectrum) is essential for the analysis of current neutrino oscillation experiments. This provides a critical baseline for the validation of theoretical neutrino-nucleus interaction models. In recent years, the scaling [9] and superscaling properties [10,11] of electron-nucleus interactions have been analyzed in detail and used to construct a semi-phenomenological model for lepton-nucleus scattering [1]. This model, denoted as the superscaling approach (SuSA) [10–12], assumes the existence of universal scaling functions for electromagnetic and weak interactions. The general procedure adopted in this analysis consists of dividing the (e,e0) experimental cross section by an appropriate single-nucleon one to obtain a reduced cross section. When this is plotted as a function of the “scaling” variable (ψ), itself a function of the energy (ω) and momentum transfer (q), some particular properties emerge. Specifically, analyses of inclusive (e,e0) data have shown that at energy transfers below the QE peak, the reduced cross section is largely independent of the momentum transfer, which is called scaling of the first kind, and of the nuclear target, which is defined as scaling of the second kind. This simultaneous occurrence of scaling of both kinds is denoted as superscaling. At higher energies, above the QE peak, both kinds of scaling are shown to be violated as a consequence of the contributions introduced by effects *Corresponding author. [email protected] PHYSICAL REVIEW D 94, 013012 (2016) 2470-0010=2016=94(1)=013012(13) 013012-1 © 2016 American Physical Society beyond the impulse approximation (IA), such as mesonexchange currents (MEC) and inelastic scattering. An extension of the scaling formalism, originally introduced to describe the QE domain, to the region of the Δresonance and the complete inelastic spectrum—resonant, nonresonant and deep inelastic scattering (DIS)—has also been proposed in [13–15]. Recently, we have developed an improved version of the superscaling model, called SuSAv2 [16], that incorporates relativistic mean field (RMF) effects [17–19] in the longitudinal and transverse nuclear responses, as well as in the isovector and isoscalar channels independently. Note that the RMF model leads to a natural enhancement of the transverse response through RMF effects without resorting to inelastic processes or two-particle emission via MEC. The RMF works properly at low to intermediate values of the momentum transfer, q. However, because of the strong energyindependent scalar and vector potentials involved, the RMF does less well at higher values of q, wherethe relativistic plane wave impulse approximation (RPWIA) gives better predictions. Hence, both regimes are incorporated in SuSAv2 by making use of a reasonable “blending”function [16]. While the original SuSAv2 was based exclusively on the IA, and used to describe the QE domain, in this work the model is extended to the inelastic spectrum. Following previous studies on the inelastic RFG modeling [13],we achieve this goal by employing phenomenological fits to the single-nucleon inelastic structure functions. Ingredients beyond the IA, namely, 2p-2h MEC effects, havebeenshowntoplayanimportantroleinthe“dip” region between the QE and the Δpeaks. In this work, the SuSAv2 model also incorporates contributions in both longitudinal and transverse reaction channels arising from 2p-2h states excited by the action of electromagnetic, purely isovector meson-exchange currents within a fully relativistic framework (see [20–23] for details). Therefore, the new “SuSAv2-MEC”predictions can be compared with data for very different kinematical situations, covering the entire energy spectrum. The accordance between theory and data gives us confidence in the extension of the model and its validity when applied to recent neutrino oscillation experiments where all the different kinematical regions may contribute and, in particular, effects linked to 2p-2h MEC have been claimed to be essential in order to reproduce the neutrino-nucleus scattering cross sections [3,23,24]. This paper is organized as follows. In Sec. II, we briefly introduce the formalism for QE and inelastic leptonnucleus reactions and describe how the MEC have been computed. In Sec. III, we compare our predictions with inclusive (e,e0) experimental data in a wide kinematical region. The analysis is presented for the cross sections paying a special attention to the relevance of the RMF and RPWIA effects at different kinematics. Finally, in Sec. IV, we show the conclusions of our analysis, including some remarks related to studies of neutrino reactions with nuclei. II. GENERAL FORMALISM: THE MODEL A. SuSAv2 in the QE region Following the Rosenbluth prescription [25], the double differential (e,e0) inclusive cross section (differential with respect to the electron scattering angle Ωeand the transferred energy ω) is given as the sum of two response functions corresponding to the longitudinal, RL, and transverse, RT, channels (Land Trefer to the direction of the transferred momentum, q), d2σ dΩedω¼σMottðvLRLþvTRTÞ;ð1Þ where σMott is the Mott cross section and the vs are kinematical factors that involve leptonic variables (see [9] for explicit expressions). Assuming charge symmetry, these two channels can be decomposed as a sum of the isoscalar (T¼0) and isovector (T¼1) contributions. In terms of the scaling functions the nuclear responses are RL;Tðq; ωÞ¼ 1 kF ½fT¼1 L;T ðψ0ÞGT¼1 L;T ðq; ωÞ þfT¼0 L;T ðψ0ÞGT¼0 L;T ðq; ωÞ;ð2Þ where kFis the Fermi momentum and the fs are the scaling functions, that only depend on the scaling variable ψ0. This scaling variable depends on q,ωand on the energy shift, Eshift, needed in order to have the corresponding scaling function peak located at Ψ0¼0, as described in [16]. The functions GT¼0;1 L;T are defined as the isoscalar and isovector responses of a moving nucleon and include relativistic corrections arising from the presence of the medium. Their explicit expressions, not reported here for the sake of brevity, can be found in [9,16]. In Fig. 1, we present the scaling functions of relevance for electron-nucleus reactions, based on results from [16]. -1 0123 ψ’ 0 0.2 0.4 0.6 0.8 fL T=1 fL T=0 fT T=1 fL RPWIA fT RPWIA FIG. 1. Reference scaling functions in the SuSAv2 model. G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016) 013012-2 Some basic conclusions emerge from the analysis of the scaling functions in the RMF and RPWIA models. First, the two models differ in the treatment of the final state. Whereas the RPWIA describes the outgoing nucleon as a relativistic plane wave, the RMF takes into account FSI between the outgoing nucleon and the residual nucleus using the same mean field as considered for the bound nucleon. This leads to a violation of the so-called zerothkind scaling; that is, the RMF transverse and longitudinal scaling functions differ from each other, the former being larger by an amount of the order of 20%. This is directly linked to the distortion introduced by FSI in the lower components of the outgoing nucleon Dirac wave functions. Secondly, it is also noteworthy that the tail exhibited by the scaling function at large values of ωis significantly higher and more extended in the transverse channel. On the contrary, the results obtained within the RPWIA show that the two types of scaling functions are roughly the same, having a shape that is much more symmetric, i.e., lacking the long tail extending to large values of ω. In spite of the merits of the RMF description, a particular drawback of the RMF concerns its dependence upon the momentum transfer q: indeed, the RMF peak position keeps growing with q, calling into question the validity of the model at very high q. In fact, the large kinetic energy of the outgoing nucleon at very high qshould make the FSI effects negligible. Thus, it would be desirable that the RMF scaling functions approach the RPWIA ones for increasing momentum transfer [16]. This was a basic motivation in the development of a new superscaling approach as a combination of RMF and RPWIA scaling functions where the first dominates at low to intermediate qand the latter at high q. This implies that the scaling functions in Eq. (2) should be replaced by linear combinations of RMF-based (~ fL;T) and RPWIA (~ fRPWIA L;T ) scaling functions: FT¼0;1 L≡cos2χðqÞ~ fT¼0;1 Lþsin2χðqÞ~ fRPWIA L FT≡cos2χðqÞ~ fTþsin2χðqÞ~ fRPWIA T;ð3Þ where χðqÞis a q-dependent angle given by χðqÞ≡π 2ð1−½1þeððq−q0Þ ω0Þ −1 Þ;ð4Þ and the transition between RMF and RPWIA behaviors occurs at intermediate qvalues (q0) in a region of width ω0, which is fixed at 200 MeV. Notice that the separation into isoscalar (T¼0) and isovector (T¼1) contributions is only taken into account for the RMF longitudinal function as in the transverse component the isoscalar contribution is negligible. In contrast, for the RPWIA longitudinal and transverse scaling functions, the isovector and isoscalar contributions collapse into a single curve. The electromagnetic response functions are now defined as RLðq; ωÞ¼ 1 kF ½FT¼1 Lðψ0ÞGT¼1 Lðq; ωÞ þFT¼0 Lðψ0ÞGT¼0 Lðq; ωÞ ð5Þ RTðq; ωÞ¼ 1 kF FT¼1 Tðψ0Þ½GT¼1 Tðq; ωÞþGT¼0 Tðq; ωÞ:ð6Þ Thus, the transition between the two models depends on the particular kinematics involved, namely, on the momentum transfer q. Accordingly, the transition parameter, q0,is expected to increase with qin such a way that the RMF contribution will be dominant at low kinematics whereas the RPWIA one starts to be relevant at higher energies. Therefore we introduce a dependence of the parameter q0 on the momentum transfer qthat determines the relative RMF and RPWIA contributions at different kinematics. The particular procedure to determine the q0behavior with qis in accordance to the best fit to a large amount of (e,e0) experimental data, covering from low to high q values (q:239–3432 MeV=c). The method applied is based on a reduced-χ2analysis of the data sets. This analysis is performed in conjunction with the inelastic one, taking into account the 2p-2h MEC contributions as well, and will be detailed in Sec. II D. B. Inelastic electron-nucleus scattering in the superscaling approach The general formalism describing inclusive inelastic electron-nucleus scattering in the superscaling approach has been presented in previous work [13]. Here we consider a more sophisticated description of the lepton-nucleus reactions via RMF and RPWIA ingredients (SuSAv2 model). The hadronic tensor for inelastic processes can be written in the form [13] Wμν inelðq;ωÞ¼ 3N 4πk3 FZF dhmN ¯ Eh wμν inelðH; Q; ωþ¯ EhÞ;ð7Þ with kFthe Fermi momentum and Hand ¯ Eh¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2þm2 N p the 4-momentum and energy of the on-shell nucleon in the nucleus attached to the virtual photon. The inelastic longitudinal and transverse responses functions, given by specific components of the hadronic tensor, RL inel ¼W00 inel and RT inel ¼W11 inel þW22 inel, can be expressed as RL;T inel ðq;ωÞ¼ N η3 FκξFZ1þ2λ−εS μthresh dμXμXFL;Tðψ0 XÞUL;T;ð8Þ where we have introduced the dimensionless variables κ¼q=2mN,ξF¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þðkF=mNÞ2 p−1and εS¼ES=mN with mNthe nucleon mass and ESthe separation energy (see [13] for details). The parameter μXis the dimensionless invariant mass and μthresh refers to the pion-production threshold. The terms FL;T are the inelastic scaling INCLUSIVE ELECTRON SCATTERING WITHIN THE …PHYSICAL REVIEW D 94, 013012 (2016) 013012-3 functions which exhibit the same structure as in Eq. (3), but using the inelastic scaling variable ψ0 X. Finally, the functions UL;T, first introduced in [13], depend on the single-nucleon inelastic structure functions w1;2which are described in our case by using empirical fits of the inelastic electron-proton and electron-deuteron cross sections [26,27]. As already commented on for the QE case, the determination of the RMF=RPWIA transition parameter (q0) in the inelastic regime also depends on the particular kinematics involved and it will be discussed in detail in Sec. II D. C. Electromagnetic 2p-2h MEC contributions The evaluation of the 2p-2h pionic MEC contributions is performed within the RFG model in which a fully Lorentz covariant calculation of the MEC can be performed (see [20,21,23]). Although MEC clearly dominate in the transverse channel, our present study also includes, for the first time, MEC contributions in the longitudinal sector. In Fig. 2, we present the separate 2p-2h MEC responses in the two channels. As shown, the transverse sector clearly dominates up to q∼1800 MeV=c, while the Land T contributions are of the same order for larger values of the momentum transfer. However, note that the kinematics where the MEC give the largest contribution to the cross section corresponds to q≲1000–1500 MeV=c, as stated in [23]. As discussed in previous work [20–22,28,29], relativity is an essential ingredient in the analysis of 2p-2h processes at momentum transfers above 500 MeV=c. At these q values, the static approximation used for the Δpropagator in the nonrelativistic calculations of 2p−2htransverse response function [30] fails to explain the “dip”region. A fully relativistic calculation of the 2p-2h MEC response functions in the RFG model requires one to compute the spin-isospin traces of all the many-body MEC diagrams. This involves the analytical calculation of more than 100,000 terms some of which involve subsequent numerical seven-dimensional integrations. This makes the computation highly nontrivial. In order to reduce the computational time as well as to ease the implementation of the results in Monte Carlo generators used in the analysis of neutrino experiments, where a wide range of kinematic conditions—momentum and energy transfers—are involved, we make use of a parametrization of the MEC responses. The parametrization form employed for the transverse electromagnetic response was analyzed in [23]. In the present work, we follow a similar procedure to get a description for the longitudinal one. As shown in Fig. 3the 2p-2h MEC longitudinal response function is reproduced with a high accuracy, a result very similar to the situation already presented in the transverse channel (see [23]). Notice that no approximations are involved in the present calculation. The MEC parametrization considered here takes care of the complete relativistic calculation, making it suitable to be applied at very high values of the momentum and energy transfers. D. Determination of q0parameters The procedure to determine the RMF=RPWIA transition in the SuSAv2 model in both QE and inelastic regimes is based on the analysis of the (e,e0) data in a wide kinematical region. The transition parameter, q0[see Eq. (4)], must exhibit a dependence on the particular kinematics involved in such a way that at higher energies, which imply higher momentum transfers, the RPWIA contribution is more relevant than the RMF one, whereas the opposite occurs at lower energies. With these assumptions, we perform a χ2analysis of the electron-nucleus experimental data which is first focused on the QE region 0500 1000 1500 2000 ω (MeV) 0 0.002 0.004 0.006 0.008 RL,T (MeV-1) q: 200-2000 MeV/c (steps: 200 MeV/c) L T FIG. 2. Comparison between 2p-2h MEC RLand RTresponse functions versus ω. The curves are displayed from left to right in steps of q¼200 MeV=c. 0500 1000 1500 2000 ω (MeV) 0 0.001 0.002 RL (MeV-1) q: 200-2000 MeV/c (steps: 200 MeV/c) FIG. 3. Comparison between the longitudinal 2p-2h MEC response functions (dashed lines) and the parameterized ones (thick solid lines) versus ω. The curves are displayed from left to right in step of q¼200 MeV=c. G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016) 013012-4 (qQE 0) and after that extended to the inelastic domain (qinel 0). In the whole analysis we take into account the SuSAv2 model for both QE and inelastic regimes as well as the 2p-2h MEC calculations. After analyzing the experimental data set, we get the qQE 0and qinel 0parameters as functions of q. Figure 4illustrates the behavior of both parameters, qQE 0(top and middle panels) and qinel 0(bottom). The data points and their error bands represent the values of the parameters that best fit the data at different kinematics (within a ∼10% in the χ2minimum). As shown, qQE 0 increases moderately with qat low to intermediate values whereas the slope goes up significantly at higher kinematics (q≳700 MeV=c). This suggests the following parametrization, qQE 0ðqÞ¼AþBq; q < q1 CþDq; q > q1 ;ð9Þ with q1¼700 MeV=c, A¼377.629 MeV=c, B¼0.407, C¼−5.322 MeV=c and D¼0.968. Imposing continuity of the above function, we are left with three free parameters, A,B,C, in the fit. A similar parametrization is found for qinel 0ðqÞ, but in this case only one linear function is used for the whole region of qexplored, qinel 0ðqÞ¼A0þB0q; ð10Þ with A0¼494.439 MeV=c and B0¼0.706. Finally, it is also worth mentioning that an even better agreement with the (e,e0) data could be achieved by employing a nonlinear fit of the q0parameters as well as including a dependence on the incident energy (Ei) or the scattering angle (θe) in the transition parameters (q0,ω0); however, the simpler assumptions made in this work are felt to be adequate for our purposes. III. RESULTS In this section, we present our results for 12Cðe; e0Þcross sections. In the following, we adopt the Bosted and Christy parametrization for the single-nucleon inelastic structure functions [26,27] which describes DIS, resonant and nonresonant regions. For the QE regime, we employ the electromagnetic form factors of the extended GariKrumpelmann (GKex) model [31–33]. The sensitivity of the QE results to the different parametrizations has been discussed in [34]. Additionally, for the Fermi momentum we employ the values obtained in [12], namely kF¼ 228 MeV=c for 12C. A. Differential cross sections In this section, we present the double differential inclusive 12Cðe; e0Þcross section versus the energy transferred to the nucleus (ω), confronting our predictions with the available experimental data [35,36]. Results are shown in Figs. 5,6and 7: in each panel we show the three separate contributions to the inclusive cross section, namely, QE, 2p-2h MEC and inelastic. The comparisons are carried out for a very wide range of kinematics from low-intermediate energies to the highly-inelastic regime. Each panel 400 800 1200 1600 2000 2400 2800 3200 q (MeV/c) 400 800 1200 1600 2000 2400 2800 3200 q0,QE (MeV/c) 400 600 800 1000 q (MeV/c) 400 500 600 700 800 900 1000 q0,QE (MeV/c) 1000 2000 3000 q (MeV/c) 1000 2000 3000 q0,inel (MeV/c) FIG. 4. Parametrization of q0;QE in terms of q(top and middle panels). Data points represent the q0value that best fits each case. Parametrization of q0;inel in terms of q(bottom panel). INCLUSIVE ELECTRON SCATTERING WITHIN THE …PHYSICAL REVIEW D 94, 013012 (2016) 013012-5 00.05 0.1 0.15 0 1e+05 2e+05 00.05 0.1 0.15 0.23 0.23 0.24 0.24 0.25 E=400 MeV, θ=36o, qQE=239 MeV/c 00.05 0.1 0 10000 20000 30000 40000 50000 0.24 0.25 0.26 0.27 0.28 0.29 E=280 MeV, θ=60o, qQE=263.739 MeV/c 0 0.1 0.2 0.3 0 25000 50000 75000 1e+05 0.28 0.3 0.32 0.34 E=480 MeV, θ=36o, qQE=286.4 MeV/c 00.05 0.1 0.15 0 10000 20000 30000 0.27 0.28 0.29 0.3 0.31 0.32 0.33 E=320 MeV, θ=60o, qQE=299.4 MeV/c 0 0.1 0.2 0.3 0.4 0 2e+05 4e+05 6e+05 8e+05 0.3 0.35 0.4 0.45 0.5 E=1500 MeV, θ=11.95o, qQE=311 MeV/c 0 0.1 0.2 0.3 0.4 0 10000 20000 30000 40000 50000 60000 70000 0.32 0.34 0.36 0.38 0.4 0.42 0.44 E=560 MeV, θ=36o, qQE=332.9 MeV/c 00.05 0.1 0.15 0.2 0 5000 10000 15000 20000 25000 0.31 0.32 0.33 0.34 0.35 0.36 E=361 MeV, θ=60o, qQE=335.7 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 E=1650 MeV, θ=11.95o, qQE=343 MeV/c 0 0.1 0.2 0.3 0.4 0 1e+05 2e+05 3e+05 4e+05 0.3 0.35 0.4 0.45 0.5 0.55 E=1500 MeV, θ=13.5o, qQE=352 MeV/c 00.05 0.1 0.15 0 2000 4000 6000 8000 0.3 0.32 0.35 0.37 E=280 MeV, θ=90o, qQE=352 MeV/c 0 0.1 0.2 0.3 0.4 0 10000 20000 30000 40000 50000 0.37 0.4 0.42 0.45 0.47 E=620 MeV, θ=36o, qQE=367 MeV/c 00.05 0.1 0.15 0.2 0.25 0 5000 10000 15000 0.34 0.36 0.38 0.4 E=401 MeV, θ=60o, qQE=370.8 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 1e+05 2e+05 3e+05 ω (GeV) 0.4 0.5 0.6 E=1650 MeV, θ=13.5o, qQE=388 MeV/c 0 0.1 0.2 0.3 0.4 0.5 ω (GeV) 0 10000 20000 30000 40000 0.35 0.4 0.45 0.5 0.55 E=680 MeV, θ=36o, qQE=402.5 MeV/c 00.05 0.1 0.15 0.2 0.25 0 5000 10000 15000 00.05 0.1 0.15 0.2 0.25 ω (GeV) 0.38 0.39 0.4 0.41 0.42 0.43 0.44 E=440 MeV, θ=60o, qQE=404.7 MeV/c FIG. 5. Comparison of inclusive 12Cðe; e0Þcross sections and predictions of the QE-SuSAv2 model (long-dashed red line), 2p-2h MEC model (dot-dashed brown line) and inelastic-SuSAv2 model (long dot-dashed orange line). The sum of the three contributions is represented with a solid blue line. The qdependence with ωis also shown (short-dashed black line). The yaxis on the left represents d2σ=dΩ=dωin nb=GeV=sr, whereas the one on the right represents the qvalue in GeV=c. G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016) 013012-6 0 0.1 0.2 0.3 0 2000 4000 6000 8000 0.4 0.42 0.44 0.46 0.48 E=480 MeV, θ=60o, qQE=439.4 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5000 10000 15000 20000 25000 0.4 0.45 0.5 0.55 0.6 0.65 E=730 MeV, θ=37.1o, qQE=443 MeV/c 0 0.1 0.2 0.3 0 2000 4000 6000 8000 0.43 0.44 0.45 0.46 0.47 0.48 0.49 E=500 MeV, θ=60o, qQE=456.6 MeV/c 0 0.1 0.2 0.3 0.4 0 1000 2000 3000 4000 5000 6000 7000 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 E=519 MeV, θ=60o, qQE=472.9 MeV/c 00.05 0.1 0.15 0.2 0.25 0 500 1000 1500 2000 2500 3000 0.4 0.42 0.45 0.47 0.5 0.52 0.55 E=400 MeV, θ=90o, qQE=489 MeV/c 00.05 0.1 0.15 0.2 0.25 0 400 800 1200 1600 0.4 0.45 0.5 0.55 E=320 MeV, θ=145o, qQE=495 MeV/c 0 0.1 0.2 0.3 0.4 0 1000 2000 3000 4000 5000 0.48 0.5 0.52 0.54 0.56 E=560 MeV, θ=60o, qQE=508 MeV/c 0.1 0.2 0.3 0.4 0 10000 20000 30000 40000 50000 60000 70000 0.05 0.15 0.25 0.35 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 E=2020 MeV, θ=15o, qQE=528.1 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 50000 60000 70000 0.525 0.55 0.575 0.6 0.625 0.65 0.675 E=1930 MeV, θ=16o, qQE=536.3 MeV/c 0.05 0.1 0.15 0.2 0.25 0 400 800 1200 0.4 0.45 0.5 0.55 0.6 0.65 E=360 MeV, θ=145o, qQE=547 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 1000 2000 3000 4000 0.52 0.54 0.56 0.58 0.6 E=620 MeV, θ=60o, qQE=559.1 MeV/c 0 0.1 0.2 0.3 0.4 0 500 1000 1500 2000 0.45 0.5 0.55 0.6 0.65 E=479 MeV, θ=90o, qQE=576 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0.6 0 1000 2000 3000 4000 5000 6000 7000 ω (GeV) 0.6 0.65 0.7 0.75 E=961 MeV, θ=37.5o, qQE=585.8 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0.6 0 10000 20000 30000 40000 ω (GeV) 0.6 0.65 0.7 0.75 0.8 E=2130 MeV, θ=16o, qQE=593.9 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 10000 20000 30000 40000 ω (GeV) 0.6 0.65 0.7 0.75 E=1930 MeV, θ=18o, qQE=601 MeV/c FIG. 6. As for Fig. 5, but now for kinematics corresponding to higher qQE values. INCLUSIVE ELECTRON SCATTERING WITHIN THE …PHYSICAL REVIEW D 94, 013012 (2016) 013012-7 0 0.1 0.2 0.3 0.4 0.5 0 500 1000 1500 2000 2500 0.58 0.6 0.62 0.64 0.66 0.68 E=680 MeV, θ=60o, qQE=610 MeV/c 0 0.1 0.2 0.3 0.4 0.5 0 5000 10000 15000 20000 25000 0.65 0.7 0.75 0.8 E=2130 MeV, θ=18o, qQE=640 MeV/c 0.1 0.15 0.2 0.25 0.3 0.35 0 100 200 300 400 500 600 700 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 E=440 MeV, θ=145o, qQE=650 MeV/c 0 0.2 0.4 0.6 0.8 0 5000 10000 15000 20000 25000 30000 0.7 0.8 0.9 E=2500 MeV, θ=15o, qQE=658.6 MeV/c 0.2 0.4 0.6 0 1000 2000 3000 4000 0.65 0.7 0.75 0.8 E=1108 MeV, θ=37.5o, qQE=674.6 MeV/c 0.1 0.2 0.3 0.4 0 2000 4000 6000 8000 10000 12000 14000 16000 0.7 0.72 0.74 0.76 E=2020 MeV, θ=20o, qQE=700.2 MeV/c 0.2 0.4 0.6 0.8 0 500 1000 1500 2000 0.8 0.85 0.9 0.95 E=1299 MeV, θ=37.5o, qQE=792 MeV/c 0.2 0.3 0.4 0.5 0 100 200 300 400 500 0.65 0.7 0.75 0.8 0.85 E=560 MeV, θ=145o, qQE=795 MeV/c 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 250 500 750 1000 0.9 0.95 1 1.05 1.1 E=1501 MeV, θ=37.5o, qQE=917 MeV/c 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 1000 2000 3000 1 1.25 1.5 1.75 E=3595 MeV, θ=16o, qQE=1044.3 MeV/c 00.20.4 0.6 0.8 1 0 500 1000 1500 2000 2500 1 1.1 1.2 1.3 1.4 1.5 E=4045 MeV, θ=15o, qQE=1113.8 MeV/c 0.2 0.4 0.6 0.8 0 100 200 300 400 500 600 1.2 1.25 1.3 1.35 1.4 1.45 E=3595 MeV, θ=20o, qQE=1316 MeV/c 0.4 0.8 1 0 50 100 150 0.6 ω (GeV) 1.5 1.55 1.6 1.65 1.7 1.75 1.8 E=3595 MeV, θ=25o, qQE=1640 MeV/c 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ω (GeV) 0 10 20 30 40 50 2 2.1 2.2 2.3 2.4 2.5 E=4045 MeV, θ=30o, qQE=2247 MeV/c 2.4 2.6 2.8 3 ω (GeV) 0 1 2 3 4 5 3.35 3.4 3.45 3.5 3.55 3.6 E=4045 MeV, θ=55o, qQE=3432.2 MeV/c FIG. 7. As for Fig. 5, but now for kinematics corresponding to the highest qQE values considered. G. D. MEGIAS et al. PHYSICAL REVIEW D 94, 013012 (2016) 013012-8 corresponds to fixed values of the incident electron energy (Ei) and the scattering angle (θe): Ei∶ 280–4045 MeV and θe∶12°–145°. To make it easier to discuss the results to follow, the ordering of the panels has been done according to the corresponding value for the momentum transfer at the quasielastic peak, denoted as qQE. This gives us the value of qwhere the maximum in the QE peak appears. However, it is important to point out that as ωvaries, qalso varies. This is important in order to estimate the value of the RMF=RPWIA transition parameter q0in both regimes, QE and inelastic. Hence, we also include in each panel a curve that shows how the momentum transfer changes with ω. Results illustrate that, at very forward angles, the value of qincreases with the energy transfer, whereas this trend tends to reverse at backward angles. Thus, for electrons scattered backwards, the q values corresponding to the inelastic process are smaller than those ascribed to the QE regime. However, notice that in this situation the cross section is clearly dominated by the QE peak. On the contrary, at very forward kinematics, the inelastic process takes place at larger values of q. Thus, the two regimes, QE and inelastic, overlap strongly, the inelastic processes being the main ones responsible for the large cross sections observed at increasing values of ω. Finally, for intermediate scattering angles the behavior of q exhibits a region where it decreases (QE-dominated process), whereas for higher ω(inelastic regime) the behavior of qreverses and starts to go up. In these situations the QE peak, although significantly overlapped with the inelastic contributions, is clealy visible even for very high electron energies. The systematic analysis presented in Figs. 5,6and 7 demonstrates that the present SuSAv2-MEC model provides a very successful description of the whole set of (e,e0) data, validating the reliability of our predictions. The positions, widths and heights of the QE peak are nicely reproduced by the model taking into account not only the QE domain but also the contributions given by the 2p-2h MEC terms (around ∼10%–15%). Only at very particular kinematics, i.e.,θe¼145° and E¼320 (360) MeV (Fig. 6) and 440 MeV (Fig. 7), does the model clearly underpredict data at the QE peak as also observed in [37]. However, notice that the dip region is successfully reproduced by the theory. Moreover, the remaining kinematics corresponding to very backward angles, E¼560 MeV, θe¼145° (Fig. 7), is well described by the model with a very high tail ascribed to the inelastic processes. Another kinematical situation whose discussion can be of interest concerns the scattering angle θe¼37.5°. Four cases are shown, one in Fig. 6and three in Fig. 7. As noted, the model does very well for the lower values of qQE starting to depart from data as qQE goes up. Note that this is the case at qQE ¼792 MeV=c and, particularly, at qQE ¼917 MeV=c where the theoretical predictions overestimate data by 5% and 10%, respectively, at the QE peak as well as in the dip region where the QE and inelastic contributions overlap and 2p-2h MEC are sizeable. This overestimation of cross section occurs only for the set of data of [38], while a good agreement is observed at similar scattering angles, but for lower momentum transfers, namely, qQE ¼402.5MeV=c (Fig. 5) and qQE ¼443 MeV=c (Fig. 6), which correspond to different experimental setups. Some comments concerning the “dip”region between the QE and the Δpeaks are also in order. This is the region where the QE and the inelastic contributions overlap the most and where FSI effects that modify in a significant way the tail of the QE curve at large ωvalues can introduce an important impact. Moreover, the role of the 2p-2h MEC effects is essential because its maximum contribution occurs in this region. Thus, only a realistic calculation of these ingredients beyond the IA can describe successfully the behavior of the cross section. To conclude, the accordance between theory and data in the inelastic regime, where a wide variety of effects are taken into account, also gives us great confidence in the reliability of our calculations. Note the excellent agreement in some situations even being aware of the limitations and particular difficulties in order to obtain phenomenological fits of the inelastic structure functions, and the poorer quality of some experimental data sets at these kinematics. B. Sensitivity of the model It is important to point out the novelties introduced in this work compared with some previous (preliminary) studies. With regards to the results shown in [14], that were based only on the superscaling function extracted from the analysis of the longitudinal (e,e0) data and assuming the transverse function to be equal (scaling of the zeroth kind), in the present paper the enhancement in the transverse channel introduced by the RMF model is incorporated. Moreover, the role of FSI is carefully examined by making use of the evolution of the scaling funtion from the RMF responses to the RPWIA ones as the momentum transfer goes up. This explains why the present analysis provides a much more accurate description of the data. Notice that the new SuSAv2 makes both the QE and the inelastic results higher. This outcome also can be observed in [16], where the study was restricted to the QE region and a fixed value of q0that can be appropriate for the specific kinematics considered was used. On the contrary, here the aim is to provide a model capable of reproducing (e,e0) cross sections for a very wide selection of kinematics and including in each case the whole energy spectrum. This is consistent with the qdependence shown by q0in both regimes, QE and inelastic. We have also tested the sensitivity of our results to different choices in the values of ω0,q0and Eshift for two representative kinematical situations (see Fig. 8). With regards to ω0, a variation of INCLUSIVE ELECTRON SCATTERING WITHIN THE …PHYSICAL REVIEW D 94, 013012 (2016) 013012-9