Mean field and two-body nuclear effects in inclusive electron scattering on argon, carbon and titanium: the superscaling approach

TL;DR

This study validates the SuSAv2 model's predictions for inclusive electron scattering on various nuclei, emphasizing the importance of two-particle two-hole meson-exchange currents and scaling behaviors across different targets.

Contribution

It demonstrates the effectiveness of the SuSAv2 model in describing electron scattering data on multiple nuclei, highlighting the role of 2p2h responses and scaling properties.

Findings

Excellent agreement with JLab data across energy spectrum

2p2h response is crucial for accurate modeling

Distinct scaling behavior of 2p2h and quasielastic responses

Abstract

We compare the predictions of the SuSAv2 model including two-particle two-hole meson-exchange currents with the recent JLab data for inclusive electron scattering on three different targets (C, Ar and Ti). The agreement is very good over the full energy spectrum, with some discrepancy seen only in the deep inelastic region. The 2p2h response, peaked in the dip region between the quasielastic and $\Delta$-resonance peak, is essential to reproduce the data. We also analyze the $k_F$ (Fermi momentum) dependence of the data in terms of scaling of second kind, showing that the 2p2h response scales very differently from the quasielastic one, in full accord with what is predicted by the model. The results represent a valuable test of the applicability of the model to neutrino scattering processes on different nuclei.

Equations

$$f({\psi }^{\prime },q)\equiv k_F\frac{{\left[\frac{d\sigma }{d{\epsilon }^{\prime }d{\Omega }^{\prime }}\right]}_{(e,e^{\prime })}}{{\sigma }_M\left[V_L{G}_L(q,\omega )+V_T{G}_T(q,\omega )\right]},$$

$${\psi }^{\prime }\equiv \frac{1}{\sqrt{{\xi }_F}}\frac{{\lambda }^{\prime }-{\tau }^{\prime }}{\sqrt{(1+{\lambda }^{\prime }){\tau }^{\prime }+\kappa \sqrt{{\tau }^{\prime }(1+{\tau }^{\prime })}}}$$

$$G_L=\frac{\kappa {\tilde{G}}_E^2}{2\tau },G_T=\frac{\tau {\tilde{G}}_M^2}{\kappa }.$$

$${\tilde{G}}_E^2\equiv Z{G}_{Ep}^2+N{G}_{En}^2,{\tilde{G}}_M^2\equiv Z{G}_{Mp}^2+N{G}_{Mn}^2,$$

Full text

The precise description of electron scattering data on complex nuclei is not only interesting by itself, but also of crucial importance in connection with current and future neutrino oscillation experiments Acciarri et al. (2016); Abe et al. (2015, 2011); Ayres et al. (2007); Aguilar-Arevalo et al. (2009); Fleming (2012); Lu et al. (2018); Abe et al. (2018), which aim at measuring the leptonic CP violation phase, $\delta_{CP}$, assessing the neutrino mass hierarchy and improving the precision on the oscillation mixing angles. A few percent uncertainty in the description of neutrino-nucleus interactions occurring in the detectors of these experiments, typically composed of argon, water or mineral oil, is needed in order to meet the required precision in the experimental analyses (see Alvarez-Ruso et al. (2018); Katori and Martini (2017) for recent reviews of this subject). The importance of validating different nuclear models using high quality electron scattering data in the relevant energy domain has been often stressed Meucci et al. (2009); Gallmeister et al. (2016a); Pandey et al. (2015); Ivanov et al. (2016); Rocco et al. (2016); Lovato et al. (2016) and careful comparisons have been performed with the carbon and oxygen data collected in the QES archive Benhar et al. . The recent measurement of Ar$(e,e^{\prime})$X and Ti$(e,e^{\prime})$X cross sections performed at Jefferson Lab Dai et al. (2018a, b) offers a new opportunity to test nuclear models Ankowski and Sobczyk (2006); Butkevich (2012); Gallmeister et al. (2016b); Van Dessel et al. (2018) in a kinematic region and on nuclei that are specifically relevant for neutrino oscillation experiments.

In Amaro et al. (2005a) it was first suggested that the superscaling properties of inclusive $(e,e^{\prime})$ data represent a powerful tool to connect electron- and neutrino-scattering reactions. Detailed studies of scaling and superscaling for electron-nucleus cross sections have been presented in Donnelly and Sick (1999a, b); Maieron et al. (2002). The analysis of the $(e,e^{\prime})$ world data has shown the quality of the scaling behavior: scaling of the first kind (no dependence on the momentum transfer) is quite good at excitation energies below the QE peak, whereas scaling of second kind (no dependence on the nuclear species) works extremely well in the same region.

The Relativistic Mean Field (RMF) model is able to reproduce with good accuracy the superscaling properties of the data and gives a good microscopic description of quasi-elastic electron scattering data González-Jiménez et al. (2014). The RMF model describes the bound state nucleons as single-particle wave functions obtained by solving the Dirac equation with self-consistent relativistic mean-field potentials Serot and Walecka (1997); Horowitz and Serot (1981); Sharma et al. (1993). The outgoing nucleon is described as a continuum wave function, specifically, the solution of the Dirac equation with the same strong mean-field potentials used to describe the bound state. This implies that final-state interactions (FSI) are taken into account within a fully relativistic framework and the model is free from non-orthogonality effects. As shown in Meucci et al. (2009); Maieron et al. (2003); González-Jiménez et al. (2014), RMF predictions compare remarkably well with inclusive electron scattering data up to moderated $q$-values, i.e., up to values of the knock-out nucleon kinetic energy, $T_{N}$, of the order of 150 MeV. For larger values, i.e., high momentum transfer, the energy-independent scalar and vector potentials become unrealistically strong, leading to results that depart very significantly from data González-Jiménez et al. (2014). On the contrary, the relativistic plane wave impulse approximation (RPWIA), where FSI are turned off, yields better predictions. Notice that at these high nucleon momenta ($\geq 500$ MeV) the overlap between initial and final states is sufficiently small to prevent spurious contributions in the cross section.

The SuSAv2 model González-Jiménez et al. (2014); Megias et al. (2016a) takes advantage of these observations and, on the base of superscaling Donnelly and Sick (1999a, b), builds an effective model that incorporates both regimes, RMF and RPWIA, making use of a “blending” function. The model extended to the inelastic spectrum and including the important contribution of two-particle-two-hole (2p2h) excitations induced by Meson Exchange Currents (MEC) De Pace et al. (2003, 2004), provides amazingly good agreement with 12C$(e,e^{\prime})$ and 16O$(e,e^{\prime})$ data for very different kinematics and covering almost the whole range of energy transfer Megias et al. (2016a). Only at very low $q$ and $\omega$ values does one find that SuSAv2 fails, and this should be expected since this is the region where scaling breaks and collective nuclear effects are important. Likewise, the SuSAv2-MEC model has also demonstrated a good level of compromise with recent measurements of neutrino and antineutrino reactions on carbon and oxygen Megias et al. (2016b, 2018, 2014, ).

In this work we present and discuss the SuSAv2-MEC predictions with the new JLab data. We also perform an analysis of the nuclear dynamics and the scaling phenomenon for the different nuclear targets involved by means of the RMF and RPWIA approaches. Finally we study the superscaling behavior of the JLab data both in the quasi-elastic and in the 2p2h regimes to stress the validity of the SuSAv2-MEC model.

In Fig. 1 we present the predictions of the SuSAv2-MEC model compared with data for carbon (left panel), titanium (middle) and argon (right). The values used for the Fermi momentum, $k_{F}$, are 0.228 GeV/c for carbon and 0.241 GeV/c for titanium and argon. As also addressed in previous studies Megias et al. (2016a); Maieron et al. (2002), a shift in the transferred energy has also been included in all cases. Moreover, the extension of the SuSAv2 model to other nuclei, both symmetric and asymmetric, beyond 12C is based as well on RMF and RPWIA predictions González-Jiménez et al. (2014) but considering the neutron and proton form factors weighted by the corresponding neutron (N) and proton (Z) numbers. The differences between neutrino reactions on symmetric and asymmetric nuclear targets were carefully studied in Barbaro et al. (2018) within the Relativistic Fermi Gas (RFG) framework. Following this line, a dedicated analysis of the SuSAv2 predictions in asymmetric nuclei will be shortly presented Megias et al. (2019a). For the QE regime, we describe the electromagnetic nucleon form factors using the extended Gari-Krumpelmann (GKex) model Crawford et al. (2010). The sensitivity of the responses to other parametrizations has been discussed in Megias et al. (2013). With regard to the inelastic structure functions, we adopt the Bosted-Christy parametrization Bosted and Christy (2008); Christy and Bosted (2010) that provides a good description of the resonant structures in $(e,e^{\prime})$ cross sections covering a wide kinematic region. As shown in Megias (2017), the use of other choices such as the Bodek-Ritchie Bodek and Ritchie (1981) parametrization and models based on Parton Distribution Functions (PDF) leads to very large discrepancies with $(e,e^{\prime})$ data.

In each graph the separate contributions corresponding to the QE (red dashed line), the two-particle two-hole MEC (blue short-dashed) and inelastic (green dot-dashed) channels are shown. The total contribution is represented by the solid line. The excellent agreement between theory and data is excellent over most of the energy spectrum covering the QE, dip and a significant region in the inelastic domain. Only at the largest values of the transferred energy (smallest values for the ejected electron energy) do the theoretical predictions depart from data, these being larger.

The origin of this discrepancy is related to the specific procedure used to determine the RMF/RPWIA transition in the SuSAv2 model in the inelastic regime. Details on this analysis as well as a study on the sensitivity of the $(e,e^{\prime})$ cross sections to different choices of the parameters were given in Megias et al. (2016a). In Fig. 1 we use the model as presented in Megias et al. (2016a), even being aware that better agreement with the high inelastic data could be achieved by employing other options for the transition parameters. To illustrate this point, we show in the bottom panels of Fig. 1 the predictions of a modified model where the inelastic region is treated ignoring final-state interactions, namely keeping only the RPWIA component of the SuSAv2 model. It appears that the flat behavior of the cross section at low $E^{\prime}$ is better reproduced by the pure RPWIA inelastic contribution, indicating that FSI are somewhat overestimated at very high energy in the present SuSAv2-inelastic model. An updated version of the model, which takes into account also these new JLab data in order to improve description of the inelastic regime at very high kinematics, will be presented in a forthcoming publication Megias et al. (2019b), the focus of the present work being more on the QE and dip regions.

The level of accordance between data and theory in the dip region where similar contributions emerge from the three domains, viz. QE, 2p2h-MEC and inelastic, is also outstanding. This can be considered to be a crucial test for the validity of the model, and particularly, the description of the 2p2h-MEC contribution that reaches its maximum value in this region. Comparison between theory and data shows excellent agreement for the three nuclei, being even better in the case of the two heavier asymmetric systems, 48Ti and 40Ar. This is reassuring as this is the first time the SuSAv2-MEC model is applied to nuclei with isospin different from zero. It is important to point out that SuSAv2 is entirely based on results obtained with the Relativistic Mean Field (RMF) model and how well they fulfill scaling arguments. To make contact with results in Fig. 1, in what follows we discuss in detail RMF but applied for the first time to asymmetric nuclei. This is crucial to reinforce our confidence in the validity of the SuSAv2 approach extended to nonzero isospin nuclei.

In Fig. 2 we restrict the comparison to the QE regime. The solid line represents the RMF prediction, i.e., with FSI incorporated through the scalar and vector potentials in the final state, whereas the dashed line refers to the RPWIA result. As observed, the general behavior of the curves for the two asymmetric nuclei, 48Ti and 40Ar, is similar to the case of 12C. In the three cases, the RMF provides a cross section with a long tail extending to smaller values of the scattered electron energy (larger energy transferred). On the contrary, the RPWIA shows a much more symmetrical shape but with the maximum being higher and more in accordance with data (slightly overestimating them). The values of the energy and momentum transfer in the QE peak are sufficiently large ($q\approx 600$ MeV) to explain why in the SuSAv2 the RPWIA contribution becomes more important, being the main responsible for the QE response (see González-Jiménez et al. (2014); Megias et al. (2016a) for details).

In what follows we investigate the validity of scaling arguments within the RMF/RPWIA model, but including in the analysis cross sections corresponding to non-zero isospin nuclei, namely, titanium and argon. Note that this is the basis of the SuSAv2 model. To make the discussions clearer, we start by presenting the basic expressions needed to get the scaling function. For inclusive QE electron scattering processes, the superscaling function is evaluated by dividing the differential cross section by the appropriate single-nucleon $eN$ elastic cross section weighted by the corresponding proton and neutron numbers Donnelly and Sick (1999a, b); Maieron et al. (2002); Barbaro et al. (1998) involved in the process:

[TABLE]

where we have introduced the dimensionless scaling variable denoted as $\psi^{\prime}(q,\omega)$,

[TABLE]

with $\lambda^{\prime}\equiv(\omega-E_{shift})/2m_{N}$, $\kappa\equiv q/2m_{N}$, $\tau^{\prime}\equiv\kappa^{2}-\lambda^{\prime 2}$, and $\xi_{F}\equiv\sqrt{1+(k_{F}/m_{N})^{2}}-1$. The term $k_{F}$ is the Fermi momentum and the energy shift $E_{shift}$, taken from Maieron et al. (2002), has been introduced to force the maximum of the cross section to occur for $\psi^{\prime}=0$. As usual the notation $\psi$ refers to the scaling variable when $E_{shift}=0$. The single-nucleon functions $G_{L}$ and $G_{T}$ are given by111Here we retain only the lowest-order terms of $G_{L}$ and $G_{T}$ in powers of $\eta_{F}\equiv k_{F}/m_{N}$ (see Amaro et al. (2005a, b) for details)

[TABLE]

As usual one has

[TABLE]

involving the proton and neutron form factors weighted by the proton and neutron numbers $Z$ and $N$, respectively.

At sufficiently high energies the function $f$ extracted from the data depends only on the scaling variable $\psi$ but not on the transferred momentum $q$. Moreover, $f(\psi)$ also becomes independent of the momentum scale in the problem, that is, independent of $k_{F}$. In the context of the RMF and RWIA approaches, this is illustrated in Fig. 3 where we present $f(\psi)$ for 12C, 40Ca, 48Ti and 40Ar. The values of the Fermi momentum for carbon, titanium and argon are the ones mentioned before, while $k_{F}$ for Ca has been set to 241 MeV/c. Note the different shape in the scaling function when comparing RMF and RPWIA, the former with a long tail extended to high values of $\psi$ (large values of the transfer energy), whereas the latter presents a much more symmetrical shape with the maximum being significantly higher. In both models, the scaling function is very similar for all of the nuclei, particularly for the three heavier ones, Ca, Ti and Ar. Only the case of carbon departs a little bit in the tail (RMF) or in the maximum (RPWIA). This is partly connected with the very different bound nucleon shells involved in the various calculations. In spite of that, scaling of the second kind, i.e., independence on the nuclear system, is fulfilled to high precision in all of the cases, making it possible to define a general scaling function to be applicable also for asymmetric nuclei. This is the basis of the SuSAv2 model that here is applied for the first time to nuclei with different number of protons and neutrons providing results in excellent agreement with data (see Fig.1).

Next we perform an analysis of the superscaling behavior of the JLab data not only in the quasielastic region, but also in the “dip” region between the QE and $\Delta$ peaks. We recall that this domain is particularly relevant for the analyses of neutrino oscillation experiments, where, due to the broadly distributed neutrino flux, the 2p2h response is part of the so-called “QE-like” or “QE-0$\pi$” cross section and cannot be disentangled from the genuine QE response.

In the top left panel of Fig. 4 we present the superscaling function, as given in Eq. (1), but extracted directly from the experimental data. We show the whole energy spectrum for the three nuclei, C, Ti and Ar. As observed, scaling of second kind only works in the QE peak, then it breaks down because non-impulsive contributions (2p2h) and inelastic channels come into play. This was expected from superscaling Donnelly and Sick (1999a, b); Maieron et al. (2002) and was also noticed in Dai et al. (2018a, b). In the region dominated by 2p2h and inelastic contribution, the results for 12C are significantly below the ones for 48Ti and 40Ar which are more in accordance. In order to set how precise scaling of second kind occurs in the QE peak, in the top right panel of Fig. 4 we zoom in to the region at the left of the QE peak (up to $\psi^{\prime}\simeq 0.25$). Results prove without ambiguity that scaling of second kind is excellent, that is, the superscaling function in this region is universal, being valid up to high precision, for all nuclei. Notice that all data overlap, being aligned in an extremely narrow region. This is consistent with previous analyses.

The behavior of data with scaling is further investigated in the results shown in bottom panels of Fig. 4. Here we present the superscaling function $f(\psi^{\prime})$ but divided by $\eta_{F}^{3}$, where $\eta_{F}\equiv k_{F}/m_{N}$. The panel on the left presents the results for the whole energy spectrum, while the panel on the right is restricted to the dip region, namely, the region where the 2p2h-MEC contributions are more important. As shown, scaling is highly broken in the QE peak (results for carbon are significantly higher), however, results collapse into a single curve within the dip region. This is clearly observed in the right panel where it is shown that all data overlap within their error bars. This result confirms our previous study in Amaro et al. (2017), where we predicted that 2p2h response scales as $k_{F}^{3}$. It is important to point out that the minimum in the cross section shown in Fig. 4 (bottom right panel) corresponds to the maximum in the 2p2h contribution (see Fig. 1). Although contributions from the QE and inelastic domains also enter in the dip region, and this may at some level break the scaling behavior, results in bottom panels of Fig. 4 strongly reinforce our confidence in the validity of our 2p2h-MEC model, whose predictions are very successfully confirmed by the experimental data. It is also interesting to note that the same type of scaling, i.e, $f(\psi^{\prime})/\eta_{F}^{3}$, seems to work reasonbly well not only in the dip region but also in the resonance and DIS domains. Further studies on the origin of this behavior are underway.

To conclude, it is noteworthy that SuSAv2-MEC model has proved its capability to describe successfully not only electron but also charged-current neutrino scattering off isospin symmetrical, $N=Z$, nuclei. In the case of electron scattering, the model has been extended to the high inelastic domain providing a good description of data over the whole energy spectrum. In this work, the model is applied for the first time to non-zero isospin nuclei comparing its predictions with the recent data taken at JLab for inclusive electron scattering on carbon, titanium and argon. SuSAv2 makes use of a set of superscaling functions extracted from the predictions provided by the RMF/RPWIA approach. Thus, we have evaluated the RMF/RPWIA scaling functions corresponding to the asymmetric Ti and Ar nuclei, and have compared with the results for $N=Z$ systems, as carbon and calcium. The analysis performed shows unambiguously that scaling of second kind works to high precision which reinforces the validity of one of the main SuSAv2 assumptions. Not only are the scaling arguments strengthened, but also the model provides an excellent description of the recent JLab data over the whole energy spectrum. Some small discrepancy at very high energy transfer will be corrected in the next version of the model, which is currently under construction. The good agreement with these data gives us confidence in applying SuSAv2-MEC to neutrino scattering on asymmetric nuclei.

The scaling behavior of the QE and 2p2h responses has been investigated at depth by showing the superscaling function extracted directly from the JLab data. As in previous studies for symmetric nuclei, scaling of second kind works extremely well in the region at the left of the QE peak. On the contrary, the analysis of the dip region, where 2p2h get their largest contributions, proves that data obey a different scaling law, such that the ratio between the 2p2h and the QE responses scales as $k_{F}^{3}$. This result, predicted in Amaro et al. (2017), confirms and strengthens the validity of the 2p2h MEC model.

Finally, we emphasize the importance of scaling arguments applied also to non-zero isospin nuclei, and the successful description within the SuSAv2-MEC framework of the inclusive electron scattering data on C, Ti and Ar recently measured at JLab. This together with the capability of the SuSAv2-MEC approach to describe electron and neutrino interactions on different nuclei, translating sophisticated and demanding microscopic calculations into a straightforward formalism, makes this model a promising candidate to be employed in MonteCarlo event generators (NEUT Hayato (2009), GENIE Andreopoulos et al. (2010) and NuWro Golan et al. (2012)) used in neutrino oscillation analyses. Accordingly, collaborations with experimental groups at FermiLab (MINER$\nu$A, MicroBooNE) and J-PARC (T2K) are being carried out to implement the SuSAv2-MEC model for electron and neutrino reactions in event generators with rewarding preliminary results Megias et al. (2019c).

This work was partially supported by the Spanish Ministerio de Economia y Competitividad and ERDF (European Regional Development Fund) under contracts FIS2017-88410-P, by the Junta de Andalucia (grant No. FQM160), and part (T.W.D.) by the U.S. Department of Energy under cooperative agreement DE-FC02-94ER40818. M.B.B. and A.D.P. acknowledge support by the INFN under project Iniziativa Specifica MANYBODY and the University of Turin under Project BARM-RILO-17. R.G.J. was supported by Comunidad de Madrid and U.C.M. under the contract No. 2017-T2/TIC-5252. G.D.M. acknowledges support from the University of Seville and CEA-Irfu under a Junta de Andalucia fellowship (FQM7632, Proyectos de Excelencia 2011). We are in debt with J.M. Udías for providing us codes about the RMF model.