Superscaling of non-quasielastic electron-nucleus scattering

PHYSICAL REVIEW C 80, 035504 (2009) Superscaling of non-quasielastic electron-nucleus scattering C. Maieron,1J. E. Amaro,2M. B. Barbaro,3J. A. Caballero,4T. W. Donnelly,5and C. F. Williamson5 1Dipartimento di Fisica, Universit` a del Salento and INFN, Sezione di Lecce, Via Arnesano, I-73100 Lecce, Italy 2Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain 3Dipartimento di Fisica Teorica, Universit` a di Torino and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy 4Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain 5Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 10 July 2009; published 17 September 2009) The present study focuses on the superscaling behavior of electron-nucleus cross sections in the region lying above the quasielastic peak, especially the region dominated by electroexcitation of the . Non-quasielastic cross sections are obtained from all available high-quality data for 12C by subtracting effective quasielastic cross sections based on the superscaling hypothesis. These residuals are then compared with results obtained within a scaling-based extension of the relativistic Fermi gas model, including an investigation of violations of scaling of the first kind in the region above the quasielastic peak. A way to potentially isolate effects related to meson-exchange currents by subtracting both impulsive quasielastic and impulsive inelastic contributions from the experimental cross sections is also presented. DOI: 10.1103/PhysRevC.80.035504 PACS number(s): 25.30.Fj, 24.10.Jv, 13.60.Hb I. INTRODUCTION In recent years, scaling [1,2] and superscaling [3,4] properties of electron-nucleus scattering have been studied in great detail. A first line of investigation has been focused on the behavior of experimental data and on the construction from them of suitable phenomenological models for lepton-nucleus scattering [3–8]. A second line, developed in parallel to the first, has instead been focused on more theoretical analyses; namely, the superscaling properties of cross sections obtained within specific nuclear models have been analyzed with the goals of testing the range of validity of the superscaling hypothesis and of finding and explaining possible scaling violations [9–19]. Lepton-nucleus scattering in the region of the resonance has been recently studied in Refs. [20,21], and an extension of the scaling formalism to neutral current neutrino processes has also been proposed [22–24]. The general procedure adopted in scaling analyses consists of dividing the experimental cross sections or separated response functions by an appropriate single-nucleon cross section, containing contributions from protons and neutrons, to obtain a reduced cross section which is then plotted as a function of an appropriate variable, itself a function of the energy and momentum transfer. If the result does not depend on the momentum transfer, we say that scaling of the first kind occurs. If, additionally, the reduced cross section has no dependence on the nuclear species, one has scaling of the second kind. The simultaneous occurrence of scaling of both kinds is called superscaling. The superscaling properties of electron-nucleus scattering data in the quasielastic (QE) region have been extensively studied in Refs. [3–5]: scaling of the first kind was found to be reasonably well respected at excitation energies below the QE peak, whereas scaling of second kind is excellent in the same region. At energies above the QE peak, both scaling of the first and, to a lesser extent, the second kind were shown to be violated because of the important contributions introduced by effects beyond the impulse approximation: inelastic scattering [5,6,25], correlations and meson-exchange currents (MEC) in both the 1p-1h and 2p-2h sectors [18,19,26–30], which mostly reside in the transverse channel. The variety and complexity of contributions that are present above the QE peak make it difficult to analyze inelastic data directly in terms of inelastic scaling variables and functions. Any analysis of this type requires some kind of theoretical assumption that allows one to focus on a specific kinematic region, having removed contributions from other processes (to the degree that one can). In Ref. [7], the scaling analysis of electron scattering data was extended to the resonance region. A non-quasielastic1(non-QE) cross section for the excitation region in which the plays a major role was obtained by subtracting QE-equivalent (see below) cross sections from the data and was found to scale reasonably well up to the peak. Phenomenologically determined QE and non-QE scaling functions were then used to obtain predictions for neutrino cross sections at similar kinematics [7,8]. This approach has been referred to as the superscaling analysis (SuSA). In this paper, one of our goals is to investigate superscaling, and its violations, in the region above the QE peak, starting from the idea presented in Ref. [7]. To this end, in Sec. II we begin by reviewing the basic formalism for scaling studies in the QE region; specifically, we summarize the essential features of the so-called SSM-QE model (to be defined in that section). We continue in that section by also considering 1In Ref. [7] this residual was called the “Delta” contribution, assuming the to be dominant. To avoid confusion with later discussions where dominance is assumed, in the present work we denote the entire residual after the quasielastic contribution is removed by “non-QE.” 0556-2813/2009/80(3)/035504(16) 035504-1 ©2009 The American Physical Society C. MAIERON et al. PHYSICAL REVIEW C 80, 035504 (2009) the region, reviewing and extending the SuSA approach of Ref. [7]. All available high-quality data for 12Care reconsidered and analyzed by applying a variety of kinematical cuts to illuminate the origins of the scaling violations that are observed. We then proceed to a deeper investigation of these scaling-violating contributions in the region between the QE and peaks. To do so, in Sec. III we present a model for inelastic electron-nucleus scattering within the impulse approximation based on the same superscaling ideas of Ref. [7], extending an earlier superscaling-based model for inelastic scattering [6]—this is the so-called SSM-inel approach and has a variant denoted SSM-—see that section for specific definitions. These models are used in Sec. III B to compute non-QE superscaling functions and to compare these with the experimental data and with the SuSA fit for several choices of kinematics. By subtracting theoretical inelastic cross sections from the experimental data, in Sec. IV we then use this model to isolate the non-impulsive components of the cross section and analyze their behavior in terms of 2p-2h MEC contributions obtained in previous studies. Finally, in Sec. Vwe summarize our study and draw our conclusions, including some remarks of relevance to studies of neutrino reactions with nuclei. II. FORMALISM AND PREVIOUS RESULTS A. Scaling in the QE region: The SSM-QE approach Here we present a summary of the relevant formalism for scaling studies in the QE region, focusing on the formulas and results that will be used in the rest of our study. We denote this the superscaling model for the QE response functions (SSM-QE). Our purpose is to illustrate how scaling ideas can be used to motivate the construction of superscaling-based models for electron-nucleon cross sections, in the spirit of Refs. [6,7], where more extensive discussions can be found. Within the relativistic Fermi gas (RFG) model, the only parameter characterizing the nuclear dynamics is the Fermi momentum kF. In the following we will retain only the lowest orders in an expansion in the parameter ηF=kF/mN,mN being the mass of the nucleon. Within this approximation, the RFG longitudinal (L) and transverse (T) quasielastic response functions, at momentum transfer qand energy transfer ω, can be written as RQE L,T (κ, λ)=1 kF fRFG(ψ)GQE L,T ,(1) where the scaling function is given by fRFG(ψ)=fL RFG(ψ)=fT RFG =3 4(1 −ψ2)θ(1 −ψ2),(2) and the scaling variable ψis ψ=1 √ξF λ−τ (1+λ)τ+κτ(1+τ) ,(3) with ξF≡1+η2 F−1. In these formulas, we have introduced the usual dimensionless variables: κ≡q/2mN,λ≡ω/2mN, and τ≡κ2−λ2. Retaining terms only up to order ηF,the functions GQE L,T are given by [5] GQE L=κ 2τZ(1 +τ)WQE 2,p −WQE 1,p  +N(1 +τ)WQE 2,n −WQE 1,n ,(4) GQE T=1 κZWQE 1,p +NWQE 1,n ,(5) WQE (1,2),p(n)being the single-proton (-neutron) electromagnetic structure functions, which are given in terms of electromagnetic form factors by WQE 1,p(n)=τG2 M,p(n)(τ),(6) WQE 2,p(n)=1 1+τG2 E,p(n)(τ)+τG2 M,p(n)(τ).(7) Given the response functions, the QE cross section is then obtained as dσ dd =σMvLRQE L+vTRQE T,(8) where is the outgoing electron energy and =(θ,φ)is the solid angle for the scattering. Here σMis the Mott cross section, and vL,T are the usual kinematic factors. The expressions above suggest that instead of using the RFG scaling function in Eq. (2), one may work backward to obtain an experimental scaling function by dividing the QE cross sections by the quantity SQE =σMvLGQE L+vTGQE T,(9) and then, for use in discussions of second-kind scaling, multiplying the result by kF: fQE(ψ,κ)=kF (dσ/dd)exp SQE .(10) Separate Land Tscaling functions can similarly be obtained as fQE L,T (ψ,κ)=kF Rexp L,T GQE L,T .(11) In our previous analyses of the world (e,e) data, we have found that for large enough momentum transfer (q>2kF), first-kind scaling works rather well for values of energy transfer ωbelow the QE peak value ωQE. For large values of ω,deviations are observed, coming from contributions beyond QE scattering, such as inelastic scattering and MEC effects. A separate analysis of the longitudinal and transverse channels shows that these deviations mainly occur in the transverse response, while the experimental longitudinal reduced cross sections scale much better and up to larger values of ω.This suggests that we can use the longitudinal QE experimental scaling function obtained in Refs. [3,4] to define a phenomenological scaling function. In particular, assuming that (i) indeed there is a universal superscaling function and (ii) it can be identified with the phenomenological function extracted from the analysis of the QE longitudinal response, we can now work backward and use this superscaling hypothesis to predict cross sections. To be more specific, we define the superscaling model for the 035504-2 SUPERSCALING OF NON-QUASIELASTIC ELECTRON- ... PHYSICAL REVIEW C 80, 035504 (2009) QE response functions (i.e., what we are calling the SSM-QE approach in this work). This consists in using Eq. (1), but with fSSM-QE(ψ)≡fQE L(ψ).(12) An important step has been taken here: only the longitudinal cross sections are employed in defining the phenomenological scaling function. This choice is based on the fact that the transverse cross sections can have significant non-QE or nonimpulsive contributions, for instance, the former from inelastic excitations of the nucleon (importantly the ) and the latter from 2p-2h MEC—see the discussions to follow in the present work. However, in lowest order, these are not very important in the longitudinal cross section, and thus it provides the only opportunity to isolate the impulsive contributions to the nuclear response. The phenomenological function fSSM-QE employed in the present approach is shown in Fig. 1(solid line), where it is compared with the RFG scaling function of Eq. (2) (dotted line). Also shown is the phenomenological non-QE scaling function fnon-QE SuSA to be defined below in the following subsection (dot-dashed line). Focusing on the phenomenological QE scaling function, one sees that it is significantly different from the RFG result: it is about 17% lower at the peak and is asymmetric, having a tail that extends to higher ω(in the positive ψdirection). In fact, subsequent to obtaining the phenomenological results shown in the figure [3,4], relativistic mean field theory (RMF) was employed to obtain theoretical scaling functions. This approach is especially relevant at high energies where relativistic effects are known to be important. These RMF studies yielded essentially the same longitudinal scaling function as the phenomenological model [11], and the required asymmetric shape of the scaling function was obtained theoretically. (We shall return below to comment on the RMF transverse scaling function.) Still later, a so-called semirelativistic approach was pursued [12], again yielding essentially the same results. More recently, a deceptively simple “BCS-inspired” model was developed [31], with the same outcome: a peak height that is significantly below the RFG SuSA-non-QE SSM-QE ψ f 3210-1-2-3 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 1. Phenomenological fits for the superscaling functions fSSM-QE and fnon-QE SuSA vs the appropriate scaling variable. The RFG superscaling function is also shown for comparison. RFG result and an asymmetric shape. Within the flexibility in each model and the experimental uncertainties, one can say that a single longitudinal QE scaling function has clearly emerged. In passing, we note that, as is usually done in studies of electron scattering in order to reproduce the correct position of the QE peak, in the present study we have introduced a small energy shift Eshift. Within the framework of the superscaling formalism outlined above, this amounts to considering a “shifted” scaling variable ψ, calculated according to Eq. (3), but with λ→λ=λ−Eshift/2mNand τ→τ=κ2−λ2. The values kF=228 MeV/c and Eshift =20 MeV have been used in all of the calculations for 12C presented here and in the following sections. Having found that the longitudinal QE scaling function is universal, whether treated phenomenologically or via models for 1p-1h knockout reactions, we now discuss the transverse QE response. In most approaches, one finds that once the single-nucleon cross section is removed in defining scaling functions as above, the longitudinal and transverse answers are basically the same, i.e., one has what has been called scaling of the zeroth kind with fT(ψ)=fL(ψ). However, in what is likely the best model employed so far, the RMF approach cited above, one finds that zeroth-kind scaling is mildly broken for momentum transfers in the 1 GeV region with fT(ψ)> fL(ψ). For instance, at q=500 MeV/c (1000 MeV/c)the transverse RMF scaling function is 13% (20%) larger at its peak than is the longitudinal one. On the other hand, from analyses of 1p-1h MEC contributions [26–30], one sees the opposite behavior; namely, the one-body (impulse approximation) and two-body MEC contributions to the 1p-1h response, which must occur coherently and hence can interfere, in fact do so destructively, and therefore a somewhat lower result is found for the total transverse scaling function. Neither of these effects is seen in the longitudinal response in leading order. Unfortunately, no single model exists in which one has adequate relativistic content (as in the RMF approach) and a consistent way to obtain the MEC contributions; indeed, the MEC studies cited above could not be attempted on the same footing as the one-body RMF computations and could only be undertaken using much simpler dynamics. Accordingly, we have no better option at present than to adopt some working procedure. Henceforth we shall assume that zeroth-kind scaling is obeyed and thus take fT(ψ)= fL(ψ) for the quasielastic response. One should remember, however, that this may not be completely true and that the QE transverse response could be either a bit larger or a bit smaller than the one obtained under this assumption. In Sec. IV, where the scaling-based cross sections are compared with data, we shall return to discuss these issues in somewhat more detail. B. The SuSA approach to scaling in the region We begin by summarizing the essentials of the SuSA approach taken in Ref. [7], where non-QE cross sections were obtained from experimental inclusive inelastic electronnucleus cross sections by subtracting QE cross sections given by the SSM-QE procedure described above. Namely, the 035504-3 C. MAIERON et al. PHYSICAL REVIEW C 80, 035504 (2009) following cross sections dσ ddnon-QE ≡dσ ddexp −dσ ddSSM-QE (13) were obtained as a first step. In the earlier work, it was assumed that dominance could be invoked. Namely, in analogy with the QE results of the previous section, a model in which only impulsive contributions proceeding via excitation of an onshell was employed. In that model, the leading-order RFG expressions for the electromagnetic response function can be written as [7,32] R L,T (κ, λ)=1 kF f(ψ)G L,T ,(14) with f(ψ)=fRFG(ψ) and ψ=1 √ξF λ−τρ (1+λρ)τ+κτ1+τρ2  ,(15) with ρ=1+µ2 −4τ 4τ,µ =m mN ,(16) and with G L=κ 4τA1+τρ2 +1w 2−w 1,(17) G T=1 2κAw 1.(18) In Eqs. (17) and (18), the single-hadron N→structure functions are2 w 1=1 2(µ+1)2(2τρ+1−µ) ×G2 M,p +3G2 E,n,(19) w 2=(µ+1)2(2τρ+1−µ) 1+τρ ×G2 M,p +3G2 E,n +4τ µ2  G2 C,,(20) where the magnetic, electric, and Coulomb form factors are taken to be GM,p =2.97g(τ),(21) GE,n =−0.03g(τ),(22) GC, =−0.15GM,p(τ),(23) with g(τ)=1 √1+τ 1 (1+4.97τ)2.(24) Starting from these expressions and assuming that the only non-QE contributions arise from this -dominance model, one 2Equations (19)and(20) should be taken with A=Zand the p→+structure functions and with A=Nand the n→0 structure functions, and then summed; but since these processes are purely isovector, we use A=N+Zwith one choice for the structure functions. can define a superscaling function in the region of the peak as follows: fnon-QE(ψ)≡kFdσ ddnon-QE S,(25) with S≡σMvLG L+vTG T.(26) We have performed an analysis similar to that presented in Ref. [7], and focusing on scaling of the first kind, we have considered all available high-quality data of inelastic electron scattering cross sections on 12C[33–42]. The functions fnon-QE we obtain are shown in Fig. 2. Note that, as above, we have introduced a small energy shift Eshift. In employing Eqs. (15) and (16), we do as in the QE case and replace λby λ, and τby τ. As before, for 12C the values kF=228 MeV/c and Eshift =20 MeV have been used in all of the calculations presented here and below. Overall we see a tendency for coalescence below and up to the peak for some, but not all, of the data. Specifically, for kinematics lying below the peak (ψ =0), these non-QE results scale reasonably well given the assumption of  dominance, showing scaling violations at the level of roughly 0.1 units of scaling function, versus the QE peak value of about 0.6, namely, scaling violations of approximately 15–20%. As discussed in more detail later, since we cannot have any inelasticity over much of this kinematic range (being below pion production threshold), we must suspect that effects such as from 2p-2h MEC contributions are playing a non-trivial role. Nevertheless, accepting this as a measure of the potential uncertainty in following the straightforward SuSA approach, in Ref. [7] an empirical fit to these results, fnon-QE SuSA , was obtained and then used to predict neutrino-nucleus cross sections in the region. It should be stressed that the assumption of  dominance is clearly only an approximation; in the following sections, we present a more microscopic approach in which the superscaling approach discussed above for the QE region is extended to the inelastic region (denoted the SSM-inel approach; see Sec. III) and where non-impulsive 2p-2h MEC effects are considered separately (see Sec. IV). Looking in more detail, let us first consider the two bottom panels of Fig. 2, which show all available high-quality data for 12C and the data with momentum transfers q>500 MeV/c. We observe that many (but not all) of the data indeed tend to collapse into a single function close to the peak. The spreading of the data is larger than what was observed in similar analyses of QE data, but a tendency to cluster (scale) is seen, at least for a subset of the data. To discuss scaling, and the breaking of it, one may consider three different regions. First, in the positive ψ region, the spreading of the data increases and the data themselves tend to diverge. This behavior is analogous to what happens for the QE case for large values of ψ, and it is due to the presence of contributions coming from higher resonances. Second, for ψ <−1,the data form a relatively uniform background showing no specific pattern. This is the range where effects from 2p-2h MEC are expected to play a significant role (see below). Finally, there is the region 035504-4 SUPERSCALING OF NON-QUASIELASTIC ELECTRON- ... PHYSICAL REVIEW C 80, 035504 (2009) q<0.5 fnon−QE 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 <q<1.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 <q<1.5 fnon−QE 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 <q<2.0 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 q>0.5 ψ ∆ fnon−QE 3210-1-2-3 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 all data ψ ∆ 3210-1-2-3 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 2. “Experimental” superscaling function fnon-QE for 12C, obtained by applying the QE-subtraction procedure described in the text to the available experimental data for 12C. The function is plotted vs the scaling variable ψ . Kinematical cuts on the values of the momentum transfer q(in GeV/c) are considered, as indicated in each panel. A phenomenological fit of the non-QE superscaling function fnon-QE SuSA is also shown for comparison by the solid line. −1<ψ  <0, where the spreading of the data is somehow less evident and where both types of scale-breaking effects can contribute. In a first attempt to disentangle these effects, in the top right-hand and two middle panels of the figure we apply a progression of cuts on the data, specifically taking those with 0.5GeV/c<q<q cut, where qcut goes from 1 to 2GeV/c. As the cut tightens, we expect to have fewer and fewer contributions from higher inelasticities. For completeness, and for a better understanding of the whole figure, in the upper left-hand panel we also report the data for low momentum transfer (q<0.5GeV/c). The results shown in the different panels seem to indicate that the presence of contributions from higher inelasticities correspond to values of fnon-QE,which lie above the average scaling function for −0.5<ψ  <0 and below it for −1<ψ  <−0.5 (for instance, compare the top and middle right-hand panels). In particular, we observe data sets that seem to cross the average function around ψ =−0.6. They correspond to JLab cross section data taken at an incident energy of 4.045 GeV and scattering angles between 23◦and 74◦, for which there is indeed a strong overlap of the and higher inelastic contributions. The observations above suggest that if we are interested in obtaining a phenomenological SuSA scaling function for the region alone, fnon-QE SuSA , these highly inelastic data sets should be excluded from the fit. Such a fit, similar to that obtained in Ref. [7], is indicated in Fig. 2by the solid line, and in Fig. 1it is compared with the phenomenological fit for the QE region and, for reference, with the RFG scaling function. We observe that fnon-QE SuSA differs significantly from fSSM-QE. This is expected, because, besides incorporating initial-state dynamics, the phenomenological non-QE scaling functions certainly contain additional effects, such as those due to the finite width of the resonance, as well as potential 2p-2h MEC contributions. However, it is interesting to investigate whether these differences can be explained only in terms of kinematics and of trivial effects, such as the finite width of the , or whether also differences in the nuclear dynamics at the QE and peaks can contribute to them. To address this issue we need to introduce some model for the cross sections in the region, and we will present this in the next section. Let us conclude this section by introducing the phenomenological SuSA model for the region [7] mentioned in the Introduction. Following the approach used in the previous section for the QE case, we can obtain the response functions 035504-5 C. MAIERON et al. PHYSICAL REVIEW C 80, 035504 (2009) for excitation from Eq. (14), by substituting the RFG expression for fwith the phenomenological fit obtained from the data, namely, RSuSA-, L,T (κ, λ)=1 kF fnon-QE SuSA (ψ)G L,T .(27) This model was tested in Ref. [7] for electron scattering over a range of kinematics, showing agreement with the data at the level of 10% or better. III. SSM-BASED MODELS FOR THE INELASTIC REGION In this section we develop a model for the response functions in the inelastic region lying above the QE peak, basing the approach on the assumption of universality of the superscaling function, i.e., using the same SSM approach employed for the QE region. This will allow us to address two issues. On the one hand, we will explore the origin of the difference between the phenomenological scaling functions obtained by fitting the data for the QE (fSSM-QE) and (fnon-QE SuSA ) regions, as discussed in the last section. We will start by assuming that this difference can be accounted for only by kinematics and finite width effects, and we will compare the scaling function obtained under this hypothesis with the experimental one. On the other hand, we will investigate further the role played by contributions from higher resonances in producing the scaling violations shown by the experimental fnon-QE in the region −1<ψ  <0. As the model we present here is based on the impulse approximation, it will not allow us to directly investigate MEC effects in the ψ <−1 region, but it will turn out to be useful later in Sec. IV, in presenting the experimental data in a different and more focused way. A. Formalism We follow closely the approach of Ref. [6], where a microscopic model based on the RFG and on superscaling was used to study highly inelastic electron-nucleus scattering. The RFG expressions for the inelastic nuclear response functions can be written as [6] Rinel L,T =1 kFµ2 µ1 dµXµXfRFG(ψX)Ginel L,T ,(28) where ψXis obtained from Eqs. (15) and (16) for a generic invariant mass WXof the final state reached by the nucleon, namely, by replacing µwith µX=WX/mN. The quantities Ginel L,T , neglecting terms of order η2 Fand higher as before, are given by Ginel L=mN κ 2τZ1+τρ2 X˜wp 2−˜wp 1 +N1+τρ2 X˜wn 2−˜wn 1,(29) Ginel T=mN 1 κZ˜wp 1+N˜wn 1,(30) where ˜w1,2are the inelastic single-nucleon structure functions, which depend on two variables, the four-momentum transfer Q2and the invariant mass WXor, equivalently, the singlenucleon Bjorken variable x=|Q2|/[W2 X−m2 N−Q2] (see also Ref. [6]). Note that the inelastic structure functions have dimension of E−1, at variance with the previous QE and  cases; for this reason, we indicate them as ˜w. The integration limits in Eq. (28)aregivenby µ1=1+µπ,(31) µ2=1+2λ−S, with µπ=mπ/mNand where S=ES/mNis the dimensionless version of the nucleon separation energy. The first limit is simply the threshold for pion production, while the second was derived in Ref. [6]. Following a procedure analogous to that illustrated for QE scattering, we can now generalize the RFG by making the substitution fRFG(ψX)→fSSM-QE(ψX)≡fSSM-non-QE(ψX)≡fSSM(ψX) (32) in Eq. (28). This modeling, which we will call SSM-inel in the following, is thus based on the assumption, suggested by the RFG, that there exists only a single universal scaling function and that the latter can be identified with the phenomenological fit obtained from the QE longitudinal data. Henceforth, for simplicity we denote the phenomenological (super-) universal scaling function to be used both for impulsive QE and inelastic contributions by fSSM. Important ingredients of the model are, of course, the single-nucleon structure functions. In our past work [6], which focused on the highly inelastic scattering region, we used the Bodek et al. [43] parametrizations of the proton and neutron structure functions that were available at the time. However, in recent years, new studies, both theoretical [44,45] and experimental [46–49], of the nucleon structure functions in the resonance region have been performed, indicating the need for more sophisticated parametrizations. As we are now studying this region, we have updated our calculations using more modern expressions for ˜wp,n 1,2. We thus use parametrizations recently obtained by Bosted and Christy [50], for both the proton [46] and neutron [47] structure functions. We note that all details regarding the nucleon resonances, such as finite widths, are automatically included in the parametrization and that we do not consider any possible medium modification of single-hadron properties. To better understand the role played by higher resonances, we also consider a variant of the full SSM-inel model. For this approach, denoted SSM-, we consider contributions coming only from N→excitations, which we describe in terms of form factors, and we take the finite width of the explicitly into account. Following Ref. [32], we start with R L,T =µ2 µ1 1 π (µX)/2mN (µX−µ)2+(µX)2/4m2 N ×R L,T (κ, λ, µX)dµX,(33) where R L,T (κ, λ, µX) are the RFG response functions of Eq. (14) calculated using a generic nucleon excitation invariant mass µX, and ψXis obtained from Eq. (15)forµ→µX. 035504-6 SUPERSCALING OF NON-QUASIELASTIC ELECTRON- ... PHYSICAL REVIEW C 80, 035504 (2009) Once again we then generalize the RFG model by substituting for the RFG scaling function in Eq. (14) the universal one, f(ψX)→fSSM(ψX). The integration limits in Eq. (33)are those of Eq. (31), and the µXdependence of the width is given by (µX)=0 µ µXp π pres π3 ,(34) with 0=120 MeV, p π=mN µXµ2 X−1−µ2 π2 4−µ2 π1 2 ,(35) and where pres πis obtained from Eq. (35) with µX=µ.We then compute inclusive cross sections using the response functioninEq.(33); and to obtain superscaling functions within this model, namely, fSSM-(ψ), as usual we divide the cross sections by S/kF, where Sis the factor given in Eq. (26). These superscaling functions may then be compared with the phenomenological SuSA one, fSSM-inel SuSA (ψ), discussed above [Eq. (25)]. B. Results In this section, we illustrate the results for the superscaling function obtained using the SSM-inel and SSM-models, together with the phenomenological SuSA fit. Before studying the behavior of the non-QE scaling function over the whole range of kinematics considered in Fig. 2, we will present a few selected examples of cross sections and scaling functions. The use of cross sections allows a direct comparison with “real” and more familiar data, and the selection of fixed kinematics can illustrate better the characteristics, and the limits, of the models. This comparison is shown in Fig. 3, where the left-hand panels show results for cross sections SuSA SSM-inel SSM-∆ (a1) dσ/dωdΩ 0.50.40.30.20.10 50 40 30 20 10 0 (a2) fnon−QE 3210-1-2-3 0.8 0.6 0.4 0.2 0 (b1) dσ/dωdΩ 0.60.50.40.30.20.10 3 2 1 0 (b2) fnon−QE 3210-1-2-3 0.8 0.6 0.4 0.2 0 (c1) dσ/dωdΩ 0.90.80.70.60.50.40.30.20.10 2 1 0 (c2) fnon−QE 3210-1-2-3 0.8 0.6 0.4 0.2 0 (d1) ω(GeV) dσ/dωdΩ 21.510.50 3 2 1 0 (d2) ψ ∆ fnon−QE 3210-1-2-3 0.8 0.6 0.4 0.2 0 FIG. 3. Cross sections in nb/(sr MeV) (left-hand panels; SSM-QE results also included) and non-QE superscaling functions (right-hand panels) for 12C, calculated within the SSM-inel and SSM-approaches, and compared with the phenomenological SuSA results. The kinematics selected here are summarized in Table I;the data are taken from Refs. [34–36]. 035504-7 C. MAIERON et al. PHYSICAL REVIEW C 80, 035504 (2009) TABLE I. 12C(e,e) kinematics considered. Case (MeV) θ(deg) qQE (MeV/c)q(MeV/c) a 620 36 366 460 b 680 60 606 600 c 1299 37.5 791 850 d 3595 16.02 1056 1189 and the right-hand panels show results for non-QE scaling functions. For illustration, we choose to consider kinematics covering a limited range of energy and momentum transfer, large enough so that the excitation is clearly present and small enough so that higher inelastic contributions do not overlap completely with the QE and peaks. Specifically, we select a lower limit case [panels (a1) and (a2)] corresponding to incident energy =620 MeV and scattering angle θ= 36◦, and an upper limit case [panels (d1) and (d2)] with =3595 MeV and θ=16◦. To explore the angle dependence of the cross sections and scaling functions, in the middle panels we show results for intermediate kinematics with two choices of scattering angle, =680 MeV, θ=60◦[panels (b1) and (b2)] and =1299 MeV, θ=37.5◦[panels (c1) and (c2)]. The kinematics are summarized in Table I, which contains as well the momentum transfers at the QE and peaks, qQE and q, respectively. In the figure, we compare the results obtained using our SSM-inel and SSM-models with those corresponding to the SuSA fit introduced at the end of Sec. II B. The SSM-model is certainly an overly simple one and, as can be seen from Fig. 3, the corresponding curves show the largest discrepancies with the data. However, the SSM-results are qualitatively interesting because, when compared with the SSM-inel results, they allow us to some extent to disentangle the effects related to contributions arising from higher resonances, which cannot be eliminated from the data. The cross sections plotted in the left-hand column of the figure include the QE contribution calculated within the SSM-QE modeling outlined in Sec. II A, which is the same for all models. Differences between the various curves in the QE region are therefore due to differences in the non-QE part of the cross sections obtained using the various models. By looking at the cross sections, we can clearly see that our inelastic model always underestimates the data in both QE and, especially, inelastic regions. More specifically, for small incident energy (upper panels) the QE peak is well reproduced. At the peak, both SSM models clearly underestimate the data, while SuSA obviously reproduces the peak reasonably, since it was fit to the data. All models are unable to reproduce the cross section completely in the region between the QE and peaks. Similar results hold for the -peak region at larger scattering angles [panel (b1)]. We notice that in this case the SSM and SuSA modeling underestimates the data even at the QE peak. This is related to the fact that at large scattering angles, the transverse contribution is dominant. Previous scaling studies [3,4] in fact showed that the transverse QE superscaling function extracted from the data differs from the longitudinal one and exhibits stronger scaling violations. We attribute these differences to contributions beyond the impulse approximation, such as 2p-2h MEC and correlations, which are not included in the models discussed in this section (see, however, Sec. IV). Moreover, at larger angles the overlap between the QE and peaks becomes more significant, which explains the difference between SuSA and SSM models at the QE peak. The same considerations can be extended to the case of higher incident energies [panels (c1) and (d1)]. We observe that in these cases the SSM-results decrease very rapidly at large energy transfer, as does the SuSA curve, because no higher inelastic contributions beyond the are included. The SSM-inel curve has an ωdependence similar to that of the data for large energy transfer, suggesting that the singlenucleon inelastic content has been correctly implemented in the model. However, the experimental cross sections are again underestimated even in the higher inelastic region. With these considerations about cross sections in mind, we can now examine the right-hand panels of the figure, which show the non-QE scaling functions. We can summarize our findings as follows. As already said, the SSM-inel model always underestimates the data. This difference, in both size and shape, is particularly relevant for small incident energies and, at all kinematics, for relatively large negative values of the scaling variable ψ . At very low energy (upper panels of the figure) or for ψ <−1 this is expected, because in these regions, effects stemming from correlations and 2p-2h MEC can play an important role [18,19] and they cannot be reproduced by models that assume impulsive, quasifree scattering on bound nucleons. In the region −1<ψ  <0,the theoretical SSM-inel curves still fall below the data, but their shape is similar to that displayed by the experimental scaling function. The discrepancies are larger below ψ =−0.5, where 2p-2h MEC may still contribute sizably, whereas when approaching the peak the theoretical curves lie closer to the data. The conclusion we draw from these observations is that the basic idea of the phenomenological superscaling-based model (SSM-inel) is probably correct and that it can account for most of the difference in shape between the experimental QE and non-QE scaling functions, but that the model presented here is still too simple and needs some improvements in order to be considered quantitatively reliable. In particular, as previously observed, the model assumes universality of the longitudinal and transverse QE scaling function, i.e., the so-called scaling of the zeroth kind, which has been shown to be violated by the QE data. While part of this violation can be ascribed to correlation and 2p-2h MEC effects, as discussed in the next section, a certain amount of it could be present even at the impulse approximation level, and, if so, should be incorporated in the model by using different scaling functions for the T and Lresponses. This would lead to a renormalization of the calculated cross sections and non-QE scaling functions, which may fill some of the discrepancy with the data at the  peak. Unfortunately, such an improvement of the model is not straightforward, although work is now in progress along this line. If we accept that at least some of the difference in normalization between the data and the calculated fnon-QE close to the 035504-8 SUPERSCALING OF NON-QUASIELASTIC ELECTRON- ... PHYSICAL REVIEW C 80, 035504 (2009) data fnon−QE 210-1-2-3 0.6 0.5 0.4 0.3 0.2 0.1 0 1300,11.95 1299,37.5 680,60 620,60 560,36 680,36 620,36 SSM-inel fnon−QE 210-1-2-3 0.6 0.5 0.4 0.3 0.2 0.1 0 SuSA 4045,15 3595,20 3595,16 2020,20 2020,15 1500,13.54 SSM-∆ ψ ∆ fnon−QE 210-1-2-3 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 4. (Color online) Experimental “QE-subtracted” data for fnon-QE for 12C for a variety of kinematics and corresponding results of the SSM-inel and SSM-models. The kinematics considered are labeled with (MeV) and θ(deg). The SuSA fit is shown by the solid (red) line. The values of the momentum transfer for the kinematics presented here fall approximately in the interval 0.5<q<1.5 GeV/c. peak can be accounted for by an improvement in the scaling functions used as ingredients in the model, then the SSM-inel results obtained so far can provide some useful additional insight into the behavior of the superscaling function. In Fig. 4, we plot the function fnon-QE for a relatively large set of kinematics (indicated in the key inside the figure) corresponding approximately to values of the momentum transfer in the range 500–1500 MeV/c. We show the experimental non-QE scaling functions, those obtained within the SSM-inel model, and those calculated with the SSM-model. We see that both the data and the SSM-inel scaling functions present the same type and degree of scaling violations in the region −1<ψ  <0, with the curves corresponding to the highest momentum transfer being the lowest ones for approximately ψ<−0.5 and then becoming the highest one for larger values of the scaling variable. In contrast, this behavior is practically absent in the SSM-results, suggesting that scaling violations in the region −1<ψ  <0 are essentially due to contributions from higher resonances. This observation has important consequences for SuSA modeling of neutrino cross sections [7], because it supports the validity of using the universal scaling function fSSM in predicting cross sections for kinematical conditions in which only contributions up to the excitation of the resonance are relevant. Still looking at Fig. 4, let us mention that both SSM models provide scaling violations at the peak which seem to be larger than those exhibited by the data. In our study, we have checked that this is due to kinematical effects, being related to the interplay between integration limits and the dependence of the variable ψ Xupon the invariant mass µX. The inclusion of some degree of scaling violation in the phenomenological scaling function used in the model may solve this problem. The differences in the behavior of the theoretical and experimental scaling functions at the peak of the may also be related to the different role of final-state interactions (FSIs) for QE scattering and excitation. In fact, previous studies in the QE region have shown that the phenomenological QE scaling function is affected by FSIs at the right of the QE peak where FSIs produce a tail, and partially at the peak, since a larger tail at positive ψresults in a smaller maximum value of the scaling function. While the tail of the phenomenological function contributes very little to the calculated non-QE scaling function at the left of the peak due to the limits of integration [see Eqs. (28), (31), and (33)], its maximum value may have some relevance at the peak. Some details concerning the limits of integration and the role of fSSM in determining the non-QE scaling function can be found in the Appendix. Finally, before proceeding in the next section to the analysis of the residual after impulsive contributions have been removed, and to complete the overview of our results for the function fnon-QE,weshowinFig.5the complete set of SSM-inel results for all kinematics for which data are available, with the same kinematical cuts used for the results presented in Fig. 2(see also Table I). Also, for comparison, in Fig. 6we show fQE [Eq. (10)] as a function of ψ QE. IV. RESIDUAL NON-IMPULSIVE CONTRIBUTIONS AND SYNTHESIS OF THE CROSS SECTION The superscaling-based model developed in the previous sections allows one to study the behavior of the superscaling function within the context of the impulse approximation and therefore to assess the size of any potential non-impulsive contributions. In particular, it is interesting to combine the two impulsive contributions denoted SSM-QE (Sec. II A) and SSM-inel (Sec. III A) and subtract this from the data to yield a residual: dσ ddres ≡dσ ddexp −dσ ddSSM-QE −dσ ddSSM-inel . (36) 035504-9 C. MAIERON et al. PHYSICAL REVIEW C 80, 035504 (2009) [42] See also http://faculty.virgina.edu/qes-archive. [43] A. Bodek and J. L. Ritchie, Phys. Rev. D 23, 1070 (1981); 24, 1400 (1981); A. Bodek et al.,ibid. 20, 1471 (1979). [44] O. Lalakulich and E. A. Paschos, Phys. Rev. D 71, 074003 (2005). [45] O. Benhar and D. Meloni, Nucl. Phys. A789, 379 (2007). [46] P. E. Bosted and M. E. Christy, Phys. Rev. C 77, 065206 (2008). [47] M. E. Christy and P. E. Bosted, arXiv:0712.3731 [hep-ph]. [48] P. E. Bosted et al. (CLAS Collaboration), Phys. Rev. C 78, 015202 (2008). [49] A. Psaker, W. Melnitchouk, M. E. Christy, and C. Keppel, Phys. Rev. C 78, 025206 (2008). [50] P. Bosted (private communication). [51] A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly, and A. Molinari (work in progress). 035504-16