General study of superscaling in quasielastic ( e , e )and( ν, μ ) reactions using the relativistic impulse approximation

PHYSICAL REVIEW C 74, 015502 (2006) General study of superscaling in quasielastic (e,e)and(ν, µ) reactions using the relativistic impulse approximation J. A. Caballero Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, E-41080 Sevilla, Spain (Received 28 March 2006; published 21 July 2006) The phenomenon of superscaling for quasielastic lepton induced reactions at energies of a few GeV is investigated within the framework of the relativistic impulse approximation. A global analysis of quasielastic inclusive electron and charged-current neutrino scattering reactions on nuclei is presented. Scaling and superscaling properties are shown to emerge from both types of processes. The crucial role played by final state interactions is evaluated by using different approaches. The asymmetric shape presented by the experimental scaling function, with a long tail in the region of positive values of the scaling variable, is reproduced when the interaction in the final state between the knockout nucleon and the residual nucleus is described within the relativistic mean field approach. The impact of gauge ambiguities and off-shell effects in the scaling function is also analyzed. DOI: 10.1103/PhysRevC.74.015502 PACS number(s): 25.30.Pt, 23.40.Bw, 24.10.Jv, 25.30.Fj I. INTRODUCTION Scaling is a very general phenomenon [1] occurring in various areas of physics that deal with probes weakly interacting with many-body systems in which a single constituent in the target system absorbs the energy and momentum transfer. The validity of the concepts of scaling [2] and superscaling [3] applied to inclusive quasielastic (QE) electron scattering at intermediate to high energies has been investigated in depth [3–5]. From an exhaustive analysis of the (e, e) world data, one concludes that the scaling behavior is highly fulfilled [4]. One may distinguish between scaling of the first kind, which corresponds to reduced cross sections being independent of the momentum transfer q, and scaling of the second kind, namely, no dependence on the nuclear species. The data analysis shows that scaling of the first kind is reasonably well respected at excitation energies below the QE peak, usually called the scaling region, whereas scaling of the second kind is excellent in the same region. The simultaneous occurrence of both kinds of scaling is named superscaling. At energies above the QE peak, where nucleon resonances are important, both types of scaling, first and, to a lesser extent, second, are broken. Scaling violations are shown to reside mostly in the transverse response, but not in the longitudinal which appears to superscale [5]. These results are in accordance with the important contributions expected in the transverse channel due to effects beyond the impulse approximation: inelastic scattering [6,7], correlations, and meson exchange currents (MEC) in both the 1p-1h and 2p-2h sectors [8–12]. The scaling analysis of QE (e, e) data was extended into the region [13], leading to the extraction of two different scaling functions which embody the nuclear dynamics in the two regions. The scaling approach has been exploited to predict inclusive charged-current (CC) neutrino-nucleus cross sections. This strategy is based on the assumption that a universal scaling function exists, which is valid for both electron and neutrino scattering reactions, provided that the corresponding kinematics are similar. This hypothesis of a universal scaling function, which is true by construction in the relativistic Fermi gas (RFG) model [14], was further investigated when final state interactions (FSI) were included. The analysis performed in [15] within the context of a nonrelativistic mean field calculation, but incorporating important aspects of relativity at the level of the current operators, called the semirelativistic (SR) approach, proved that scaling properties were highly fulfilled for the electromagnetic responses at intermediate to high energies. Moreover, the scaling function extracted from these coincides with that from the (ν,µ) reaction. However, it presents a symmetric shape which is not supported by the analysis of (e, e) data. This is not unexpected because additional dynamical effects, which are beyond the nonrelativistic mean field picture considered in [15], are needed in order to reproduce the asymmetry extracted from the experiment. An investigation of the QE scaling properties of CC neutrino-nucleus scattering within the context of the relativistic impulse approximation (RIA) has been presented in [16]. Although resorting only to one-body excitations, the RIA has been shown to provide the required asymmetry of the scaling function when strong relativistic potentials are included in the model. This makes an important difference with previous nonrelativistic, or SR, calculations based on the impulse approximation [15,17,18]. The superscaling function evaluated from QE (ν,µ) calculations was compared with the (e, e) phenomenological one, showing the capability of RIA models to yield the required properties of data. In this paper we extend the investigation on scaling by performing a global analysis of (e, e) and (ν,µ) reactions within the RIA framework. We follow the general procedure of scaling and superscaling studies [5,13,15,16]. First, we calculate inclusive cross sections within a specific model and then obtain scaling functions by dividing them by the relevant single-nucleon cross sections weighted by the corresponding proton and neutron numbers [5,19]. The scaling function so 0556-2813/2006/74(1)/015502(12) 015502-1 ©2006 The American Physical Society J. A. CABALLERO PHYSICAL REVIEW C 74, 015502 (2006) obtained is plotted against the scaling variable ψ(q,ω), and its scaling properties analyzed, i.e., we explore its dependence on the transfer momentum (first-kind scaling) and on the specific target nucleus (second-kind scaling). A detailed study of the off-shell effects and gauge ambiguities in the scaling function is also presented. The analysis performed and its comparison with data give us important clues as to the validity of the different theoretical descriptions considered. Furthermore, the consistency between (e, e) and (ν,µ) calculations, which reflects the universality property of the superscaling function, is also clearly illustrated. Finally, we show that the RIA approach, in spite of its simplicity, gives rise to the required shape presented by the experimental scaling function. Scaling and superscaling ideas have been carried a step further to include neutral-current (NC) neutrino-nucleus scattering processes in [20]. Here, NC differential cross sections were obtained by making use of the phenomenological (e, e) scaling function, i.e., the universality property was assumed. It will be very interesting to investigate scaling for NC reactions within the RIA framework. This will allow us to prove if various RIA-FSI models superscale, and moreover, if the universal asymmetric scaling function also emerges from the RIA calculations on NC processes. The paper is organized as follows. In Sec. II, we present a brief summary of the basic formalism involved in describing inclusive QE electron-nucleus and CC neutrino-nucleus scattering processes. We restrict ourselves to the plane-wave Born approximation (PWBA) and assume the RIA. The analysis of scaling and superscaling with the general expressions implied is discussed in Sec. III. Here we also show the experimental scaling function together with a phenomenological fit and a comparison with the simple RFG result. In Sec. IV, we discuss the results starting with a global analysis of QE (e, e) reactions where off-shell effects in the differential cross sections are investigated at depth. We continue with the study of scaling properties, showing that our theoretical results do scale even when strong relativistic potentials are present. A separate analysis of the different channel contributions is also presented. Comparison with data shows that our model calculations do agree with experiment for specific descriptions of FSI. To conclude, we compare the scaling functions evaluated from (e, e) calculations with those obtained from (ν, µ) reactions [16]. Results show that they almost coincide, hence the model is consistent with the fulfillment of the universality property. Finally, in Sec. Vwe present our conclusions. II. INCLUSIVE QUASIELASTIC LEPTON SCATTERING FORMALISM: THE RELATIVISTIC IMPULSE APPROXIMATION This work deals with lepton induced reactions at energies of a few GeV and QE kinematics. In particular, we focus on inclusive electron scattering and CC neutrino (antineutrino) scattering on nuclei and assume the Born approximation (BA). The leptonic variables (in the laboratory system) involved in the processes are Kµ=(ε, k), the 4-momentum of the incident lepton (eor νµ) beam, and Kµ=(ε,k), the 4-momentum of the scattered lepton (eor µ). The process is mediated by the exchange of a virtual photon (electron scattering) or a charged vector boson (CC neutrino scattering) with 4-momentum Qµ=(K−K)µ=(ω,q). The general formalism for (e, e) and (ν, µ) reactions has been presented in previous works [4,8,13,15,21]. Here we simply summarize those basic aspects needed for later discussion. Assuming PWBA, i.e., one virtual particle exchanged and leptons described as free particles, the QE differential cross section can be expressed in terms of separate nuclear response functions. In the case of (e, e) reactions, we may write dσ dεd(e,e)=σM[vLRL(q,ω)+vTRT(q,ω)],(1) where is the scattered electron solid angle and the term σMrepresents the Mott cross section. Analogously, for CC neutrino scattering reactions the differential cross section can be written in the form [13,15] dσ dεdχ=σ0[ˆvCC ˆ RCC +2ˆvCL ˆ RCL +ˆvLL ˆ RLL +ˆvTˆ RT+2χˆvTˆ RT],(2) with (ε, ) the muon kinematic variables. The symbol χ specifies neutrino-induced reactions (χ=+) or antineutrinoinduced reactions (χ=−), and the term σ0depends on the Fermi constant and the Cabibbo angle (see [13] for its explicit expression). The kinematic factors vKand ˆvKcome solely from the electromagnetic and weak leptonic tensors, respectively, and their explicit expressions can be found in [4,8,13]. The electromagnetic RKand weak ˆ RKresponse functions contain the whole dependence on the nuclear vertex coupling and are expressed by taking the appropriate components of the nuclear tensor [4,8,13]. This involves the matrix elements of the virtual photon or charged boson interaction with the nuclear electromagnetic or weak current. The inclusive hadronic electromagnetic tensor reads Wµν(q,ω)= i  f δ(Ef−Ei−ω)f|ˆ Jµ em(Q)|i∗ ×f|ˆ Jν em(Q)|i,(3) where |idescribes the initial target state and |frepresents a specific many-body final nuclear state. The term ˆ Jµ em(Q) refers to the nuclear electromagnetic many-body current operator. A similar expression to (3) should be written for the weak tensor ˆ Wµν in terms of the nuclear weak many-body current operator ˆ Jµ w(Q). The electromagnetic tensor given in (3)is an exceedingly complicated object which includes all possible final states that can be connected with the initial ground state through the action of the many-body current operator. In this paper, we restrict ourselves to the QE kinematic regime and we adopt the relativistic impulse approximation. Within the RIA, the many-body nuclear current operator is simply given as a sum of single-nucleon current operators that only couple the target ground state to scattering states lying in the one-body knockout space. The RIA approach has been extensively applied in investigations of exclusive electron scattering reactions [22–25]. Further details on the model for 015502-2 GENERAL STUDY OF SUPERSCALING IN . . . PHYSICAL REVIEW C 74, 015502 (2006) neutrino-nucleus scattering reactions have been presented in [21,26–28]. Within the RIA framework, the main ingredient needed to evaluate the electromagnetic and weak tensor is the single-nucleon current matrix element, ˆ Jµ(Q)=dreiq·rψF(pF,r)ˆ µψjm B(r).(4) Here ψjm B(r) and ψF(pF,r) are the wave functions for the initial (bound) nucleon and for the emitted nucleon, respectively, and ˆ µis the corresponding single-nucleon current operator for electron ( ˆ µ em) or weak CC neutrino ( ˆ µ w) scattering. We describe the bound nucleon states as self-consistent Dirac-Hartree solutions, derived within a relativistic mean field (RMF) approach using a Lagrangian containing σ, ω, and ρmesons [29,30]. The outgoing nucleon state is described as a relativistic scattering wave function. Different options have been considered: first, the relativistic plane-wave impulse approximation (RPWIA), namely, the description of the knockout nucleons by means of plane-wave spinors; second, the effects due to FSI between the ejected nucleon and the residual nucleus. In our model, FSI effects are described by using Dirac equation solutions in the presence of relativistic potentials. This constitutes the relativistic distorted-wave impulse approximation (RDWIA) [22]. The use of energy-dependent complex relativistic optical potentials fitted to elastic proton scattering data has proven to be successful in describing exclusive (e, ep) scattering reactions under QE kinematics [22–25,31]. In this case of exclusive reactions, the optical potentials are built to reproduce the contribution from the elastic channel. For inclusive processes such as (e, e) and (ν, µ), the contribution from the inelastic channels should be retained. Ignoring them would lead to an underestimation of the inclusive cross section [27,32,33]. Multiple nucleon knockout effects have been treated in detail within the context of the Green function method [34–37]. A simple way of obtaining the inclusive strength within the RIA is to use purely real potentials. We consider two choices for the real part. The first uses the phenomenological relativistic optical potential from the energy-dependent, A-independent parametrizations (EDAIC, EDAIO, EDAICa) derived by Clark et al. [38], but with their imaginary parts set to zero. The second approach consists of describing the outgoing nucleon by means of distorted waves obtained with the same relativistic mean field used to describe the initial bound nucleon states. We refer to these two FSI descriptions as real relativistic optical potential (rROP) and RMF, respectively. Dispersion relation and Green function techniques [34–37] lead to results which are close to those obtained in the impulse approximation with either the rROP [36,37] or the mean field [34]. Concerning the current operator, we use the relativistic free nucleon expressions [13,39,40]. For electromagnetic (e, e) processes, the three usual options, denoted as CC1, CC2, and CC3, are considered: ˆ µ CC1p(n) em =Fp(n) 1+Fp(n) 2γµ−Fp(n) 2 2mN (P+PF)µ,(5) ˆ µ CC2p(n) em =Fp(n) 1γµ+iFp(n) 2 2mN σµνQν,(6) ˆ µ CC3p(n) em =Fp(n) 1 2mN Pµ+iFp(n) 1+Fp(n) 2 2mN σµνQν,(7) where Fp(n) 1and Fp(n) 2are the Pauli and Dirac proton (neutron) form factors, respectively, that depend only on Q2, and the on-shell 4-momentum Pµ=(E,p) with E=p2+m2 N, and pthe bound nucleon momentum has been introduced. Note that the three operators are equivalent for free on-shell nucleons (they are connected by the Gordon transformation). However, the RIA deals in general with off-shell bound and ejected nucleons. Hence the three operators lead to different results. Moreover, the current is not strictly conserved and uncertainties dealing with the election of gauge also occur [41–43]. The relativistic charged weak current of the nucleon is given as ˆ µ w=ˆ µ V−ˆ µ A, where the vector and axial-vector current operators read ˆ µ V=FV 1γµ+iFV 2 2mN σµνQν,(8) ˆ µ A=GAγµ+GP 2mN Qµγ5,(9) with FV 1,2the isovector nucleon form factors given in terms of the electromagnetic ones as FV 1,2=Fp 1,2−Fn 1,2. The axialvector and pseudoscalar form factors are parametrized as GA=gA 1−Q2/M2 A ,(10) GP=4m2 N m2 π−Q2GA,(11) with gA=1.26 and MA=1032 MeV (see [44,45]). III. SCALING AND SUPERSCALING AT THE QUASIELASTIC PEAK Detailed studies of scaling and superscaling for electronnucleus cross sections have been presented in [3–5]. The analysis of the (e, e) world data has shown the quality of the scaling behavior: scaling of the first kind (no dependence on momentum transfer) is quite good at excitation energies below the QE peak, whereas scaling of the second kind (no dependence on nuclear species) works extremely well in the same region. In this paper, our aim is to investigate the QE scaling properties of electron-nucleus and CC neutrinonucleus scattering within the context of the RIA. Assuming various RIA models, we prove that they do superscale, and we compare the associated scaling functions with the (e, e) phenomenological one. In what follows, we present the basic expressions needed to get the scaling functions. Several choices have been proposed 015502-3 J. A. CABALLERO PHYSICAL REVIEW C 74, 015502 (2006) in the literature for the appropriate scaling variable. Here, following the analysis of the RFG model, we adopt the dimensionless variable denoted as ψ(q,ω), ψ≡1 √ξF λ−τ (1 +λ)τ+κ√τ(1 +τ) ,(12) where λ≡(ω−Eshift)/2mN,κ ≡q/2mN,τ≡κ2−λ2, and ξF≡1+(kF/mN)2−1. The term kFis the Fermi momentum, and the energy shift Eshift, taken from [5], has been introduced to force the maximum of the cross section to occur for ψ=0. As usual, the notation ψrefers to the scaling variable when Eshift =0. For inclusive QE electron scattering processes, the superscaling function is evaluated by dividing the differential cross section (1) by the appropriate single-nucleon eN elastic cross section weighted by the corresponding proton and neutron numbers [4,5,19] involved in the process. We may write f(ψ,q)≡kF dσ dεd(e,e) σM[vLGL(q,ω)+vTGT(q,ω)].(13) The scaling behavior can also be analyzed by taking into account the separate electromagnetic longitudinal Land transverse Tcontributions. Thus the following scaling functions are introduced: fL(ψ,q)≡kF RL(q,ω) GL(q,ω),(14) fT(ψ,q)≡kF RT(q,ω) GT(q,ω).(15) The single-nucleon functions GLand GTare given by GL=(κ2/τ)˜ G2 E+˜ W2 2κ1+ξF(1 +ψ2)/2,(16) GT=2τ˜ G2 M+˜ W2 2κ1+ξF(1 +ψ2)/2,(17) where the function reads =ξF(1 −ψ2)√τ(1 +τ) κ+1 3ξF(1 −ψ2)τ κ2.(18) As usual, one has ˜ G2 E≡ZG2 Ep +NG2 En,˜ G2 M≡ZG2 Mp +NG2 Mn, (19) ˜ W2=1 1+τ˜ G2 E+τ˜ G2 M, involving the proton and neutron form factors weighted by the proton and neutron numbers Zand N, respectively. At sufficiently high energies, the function fdepends only on the scaling variable ψbut not on the transferred momentum q. Moreover, f(ψ) becomes also independent of the momentum scale in the problem, that is, independent of kF. The scaling behavior has been clearly demonstrated from the analysis of the QE (e, e) world data [3,4]. The investigation of the separate contribution of the longitudinal and transverse response functions has shown that scaling violations occur mainly exp fit RFG 12C ψ f(ψ) 21.510.50-0.5-1-1.5 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 1. RFG superscaling function compared to data and a parametrization of the results. in the Tchannel because of significant effects introduced by MEC, correlations, and inelastic scattering. From the global analysis presented in [3,4], a universal experimental superscaling function fexp(ψ) has been defined. In Fig. 1, we present fexp(ψ) averaged over the nuclei employed in the analysis, together with the corresponding fit. As noted, the experimental scaling function presents an asymmetric shape with a tail that extends toward positive values of the scaling variable ψ. This is in contrast with the RFG superscaling function given by fRFG(ψ)=(3/4)(1 −ψ2)θ(1 −ψ2), that is symmetric, limited strictly to the region |ψ|⩽1, and with a maximum value of 3/4. The general analysis of superscaling for CC neutrinonucleus scattering has been presented in previous works [13,15,16]. Here, the superscaling function is obtained by dividing the differential cross section evaluated within the RIA (2) by the corresponding weak single-nucleon cross section as given explicitly in Eqs. (15), (45), (52), and (86)–(94) of [13]. Details and specific expressions are also given in Appendix C of [15]. This theoretical scaling function calculated from CC neutrino-nucleus cross sections can be compared directly with the one corresponding to (e, e) scattering calculations as well as with fexp(ψ) shown above. This allows a check on the universality assumption of f(ψ) and on the capabilities of different RIA models to yield the required properties of the experimental scaling function. By analogy to (e, e), and in addition to the scaling function obtained from the CC neutrino-nucleus cross section, one may also construct the separate contributions given by the longitudinal L, transverse T, and axial-vector transverse Tresponses, fL,T,T (ψ). Here, the Lweak response includes the contribution from the three terms in (2), namely, ˆvCC ˆ RCC +2ˆvCL ˆ RCL +ˆvLL ˆ RLL.Inthe next section, we present a detailed study of scaling and superscaling properties for (e, e) and (ν, µ) reactions within the RIA scheme. IV. RESULTS In this work, we consider the PWBA; i.e., Coulomb distortion of the leptons is neglected. Checks made for light-to015502-4 GENERAL STUDY OF SUPERSCALING IN . . . PHYSICAL REVIEW C 74, 015502 (2006) medium nuclei within the effective momentum approximation [21,27] show that these effects are within a few percent for the high-energy lepton kinematics considered in this work. In this sense, our general conclusions about scaling are not modified by them. Obviously, the analysis of heavier nuclear systems requires a careful description of Coulomb distortion effects in the leptons (electrons and muons) involved in both processes (e, e) and (ν, µ). A. Inclusive QE (e,e) reactions 1. Differential cross sections In what follows, we present predictions for QE (e, e) reactions on 12C within the framework of the RIA. Results correspond to fixed values of the incident electron beam energy ε=1 GeV and scattering angle θe=45◦. The Lorentz gauge has been selected. In the next section, we present results for the scaling function concerning gauge effects. Figure 2shows the differential cross section evaluated for the two current operators, CC1 and CC2, and FSI included through the rROP and RMF potentials. As observed, the CC1 choice leads to a significantly larger cross section, particularly within the RMF approach. Concerning FSI, the use of the RMF potential gives rise to a clear asymmetry in the cross section with a pronounced tail extending toward higher values of the transfer energy ω. For reference we also include in Fig. 2the curve corresponding to RPWIA, i.e., no FSI. In such a case, the effects introduced by the current operator choice are minor. The difference observed between both FSI descriptions at high ωvalues is linked to the behavior of the two relativistic potentials: whereas the RMF contains strong energy-independent scalar and vector potentials, the energy dependence of the rROP makes its scalar and vector terms to be importantly reduced for high nucleon kinetic energies (high transfer energy). Hence, rROP results get close to the RPWIA ones for large ωvalues. The asymmetry in the differential cross section is proved to be an effect entirely linked to the FSI description. Only in the presence of relativistic optical potentials with strong scalar and vector terms (RMF approach) is a significant shift RPWIA rROP-CC2 rROP-CC1 RMF-CC2 RMF-CC1 ω(MeV ) dσ/dΩedε [nb/(MeV sr)] 7006005004003002001000 2.5 2 1.5 1 0.5 0 FIG. 2. Differential cross section for QE (e, e)on12C. FSI are described within the RMF and the rROP models. Results correspond to CC1 and CC2 current operators. For reference, the RPWIA result is presented. Incident electron energy ε=1 GeV, scattering angle θe=45◦. T(CC2) L(CC2) T(CC1) L(CC1) ω(MeV) dσ/d dΩe[nb/(MeV sr)] 600500400300200100 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 T (CC2) L (CC2) T (CC1) L (CC1) ω(MeV) 600500400300200100 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 FIG. 3. Longitudinal Land transverse Tcontributions to the cross section. Left panel corresponds to RMF description of FSI; right panel to rROP. Results are presented for CC1 and CC2 prescriptions. of strength to higher values of ωshown to occur. Details on the specific mechanism that produces the asymmetric tail in the cross section within the RMF-FSI approach are given in [46]. The importance of the current operator choice for the two FSI models is further investigated in Fig. 3, which presents a separate analysis of the longitudinal and transverse contributions to the cross section. As observed, the pure longitudinal response is almost identical with both current operators. In the case in which FSI are neglected, this result has been also found previously [41], and it was proven to be as due to the validity of the Gordon transformation for the longitudinal contributions. For the case of the transverse channel, the CC1 contribution is much larger than that of CC2 (likewise for CC3). Notice that the magnitude of this discrepancy depends on the specific FSI description, being larger for RMF, whereas in the plane-wave limit (RPWIA) both currents lead to very similar results. The ambiguity introduced by the current choice for inclusive (e, e) reactions has already been signaled in some previous works [47,48]. The large effects introduced by the current operator within the RMF (likewise for rROP) approach can also be analyzed by directly comparing theoretical and experimental cross sections. This may allow us to determine which particular choice is more appropriate. In Fig. 4, we compare data [49–54] with the results corresponding to the RIA-RMF approach with CC1 and CC2 at very different kinematics. As a general rule, we conclude that CC1 tends to overpredict data whereas the reverse applies to CC2. This outcome clearly favors the CC2 option in that the effects beyond the QE peak (and MEC effects) may also play a significant role in the analysis of the data even at the maximum of the peak. However, data are not conclusive yet because the and MEC contributions have not been evaluated. 2. Analysis of scaling behavior The important effects already shown in the differential cross sections are also visible in the scaling function. In the following, we rely on the investigation of the superscaling properties and present results for the scaling function f(ψ) (13), as well as the separate longitudinal and transverse contributions, i.e., fL(ψ) and fT(ψ). A comparison between the three scaling functions is presented in Fig. 5, where for simplicity only the RMF-FSI model has been considered. 015502-5 J. A. CABALLERO PHYSICAL REVIEW C 74, 015502 (2006) εe= 500 MeV, θe=60° dσ dΩe[nb (MeV sr)] 300250200150100500 9 8 7 6 5 4 3 2 1 0 εe= 1650 MeV, θe=135° 300250200150100500 400 350 300 250 200 150 100 50 0 εe= 620 MeV, θe=60° dσ dΩe[nb (MeV sr)] 4003002001000 4 3.5 3 2.5 2 1.5 1 0.5 0 εe= 2020 MeV, θe=15° 4003002001000 90 80 70 60 50 40 30 20 10 εe= 730 MeV, θe=371° ω[MeV] dσ dΩe[nb (MeV sr)] 300250200150100500 30 25 20 15 10 5 0 εe= 2020 MeV, θe=20° ω[MeV] 4003002001000 14 12 10 8 6 4 2 0 FIG. 4. Differential cross section for (e, e) reactions on 12C. Theoretical predictions correspond to the RMF-FSI approach with CC1 (solid line) and CC2 (dashed line). Different kinematics (see labels) have been considered. Similar conclusions are drawn for the rROP approach. In addition to the usual CC1 and CC2 prescriptions, we also include the results for the CC3 choice. As observed, the difference between f(ψ) and the contributions fL(ψ) and fT(ψ) is very large for CC1, being much smaller for CC2 and almost negligible for CC3. This means that zero-kind scaling is fully broken for CC1 and RMF, whereas only a mild (negligible) violation is observed for CC2 (CC3). This behavior of the superscaling function is in accordance with the very diverse contributions given in the Tchannel by the different current operators. In particular, the CC1 current and RMF description of FSI lead to an important increase in the Tchannel strength compared with the single-nucleon contribution. On the contrary, the longitudinal function fLis found to be basically the same for the three choices of the operator (likewise for rROP). Scaling of the first kind is explored in Fig. 6, where we present f(ψ) for three different values of the incident electron energy, ε=1,1.5, and 2 GeV. Results are shown for the two different descriptions of the FSI: RMF and rROP. In each case, we make predictions for the two usual current prescriptions, CC1 and CC2. As observed, the scaling function for the rROP model shows a very mild dependence on the fT(ψ) fL(ψ) f(ψ) CC1 32.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 fT(ψ) fL(ψ) f(ψ) CC2 32.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 fT(ψ) fL(ψ) f(ψ) CC3 ψ 32.521.510.50-0.5-1-1.5 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 5. Analysis of zero-kind scaling. Global scaling function f(ψ) compared with separate Land Tcontributions. All results correspond to RMF description of FSI and current operators: CC1, CC2, and CC3. transfer momentum in both positive and negative ψregions, i.e., first-kind scaling is well satisfied. In the case of the RMF model, a slight shift occurs in the “scaling region” ψ<0, whereas for ψpositive, the model breaks scaling 2GeV 1.5 GeV 1GeV RMF CC1 f(ψ) 32.521.510.50-0.5-1-1.5 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2GeV 1.5 GeV 1GeV RMF CC2 32.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2GeV 1.5 GeV 1GeV rROP CC1 ψ f(ψ) 32.521.510.50-0.5-1-1.5 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2GeV 1.5 GeV 1GeV rROP CC2 ψ 32.521.510.50-0.5-1-1.5 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 6. Analysis of first-kind scaling. Scaling functions for three values of the incident electron energy. Kinematics and target as in previous figures. Results correspond to RMF and rROP models and the two choices of current operators: CC1 and CC2. 015502-6 GENERAL STUDY OF SUPERSCALING IN . . . PHYSICAL REVIEW C 74, 015502 (2006) 2GeV 1.5 GeV 1GeV fL(ψ) 32.521.510.50-0.5-1-1.5 0.6 0.5 0.4 0.3 0.2 0.1 0 2GeV 1.5 GeV 1GeV ψ fT(ψ) 32.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 7. Same as Fig. 6, but for the separate Land Tcontributions. Only the RMF approach is considered. In the case of the Tresponse, the larger functions correspond to the CC1 operator and the smaller ones to CC2. In the Lchannel, almost no distinction is seen between the operators. at roughly 30%. This violation is not in conflict with (e, e) data that indeed leave room for some violation of the first-kind scaling in this region, due partly to production and partly to other contributions, such as MEC and correlations. A separate analysis of the Land Tchannels in the scaling function (see Fig. 7) shows that the breakdown of scaling within the RMF description of FSI is similar for both channels independently of which current choice is considered. In the case of rROP (and RPWIA) calculations, scaling of the first kind is excellent for all current operators and in both channels. Scaling of the second-kind, i.e., independence of the specific nuclear system, is analyzed in Fig. 8. Here we present the results corresponding to CC1 and CC2 operators and the two descriptions of FSI: RMF and rROP. In each case, we compare the superscaling function evaluated for three different nuclei: 12C, 16O, and 40Ca. The values of the Fermi momentum considered [5] correspond to kF=216 MeV/c (16O), kF= 228 MeV/c (12C), and kF=241 MeV/c (40Ca). As shown, the effects introduced by changing the nucleus are very small for the CC2 current choice and the two FSI descriptions. This is in complete accordance with data which show that second-kind scaling is excellent. On the contrary, the CC1 operator leads to a significant breakdown of scaling behavior which affects both FSI descriptions. This violation of the second-kind scaling comes totally from the transverse response. This is clearly illustrated in Fig. 9, where we present results for the separate functions fL(ψ) and fT(ψ) corresponding to the RMF model. Similar results are obtained for the rROP approach. From these theoretical results and the exhaustive analysis of the (e, e) world data, which proves the excellent quality of second-kind scaling behavior, one may question the validity of the CC1 operator used in describing QE (e, e) processes. 40Ca 16O 12C RMF CC1 f(ψ) 32.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 40Ca 16O 12C RMF CC2 32.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 40Ca 16O 12C rROP CC1 ψ f(ψ) 32.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 40Ca 16O 12C rROP CC2 ψ 32.521.510.50-0.5-1-1.5 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 8. Analysis of second-kind scaling. Scaling functions for 12C, 16O, and 40Ca. Kinematics as in previous figures. Results for the RMF and rROP models are presented for CC1 and CC2 prescriptions. 3. Comparison with experiment A comparison between the theoretical superscaling functions and the averaged QE phenomenological function obtained from the analysis of (e, e) data is presented in Fig. 10. We have selected the CC2 operator and show results for the three different descriptions of the continuum final state, namely, the RPWIA, rROP, and RMF. Lorentz gauge has been assumed. The symmetric character of the RPWIA and rROP curves differs clearly from the experimental analysis. On the contrary, the RMF approach displays an asymmetric shape with a long tail extended to positive values of the 40Ca 16O 12C fL(ψ) 32.521.510.50-0.5-1-1.5 0.6 0.5 0.4 0.3 0.2 0.1 0 40Ca 16O 12C ψ fT(ψ) 32.521.510.50-0.5-1-1.5 1.2 1 0.8 0.6 0.4 0.2 0 FIG. 9. Separate Land Tcontributions to second-kind scaling analysis. Results correspond only to the RMF model. 015502-7 J. A. CABALLERO PHYSICAL REVIEW C 74, 015502 (2006) exp fit RPWIA rROP RMF 12C ψ f(ψ) 21.510.50-0.5-1-1.5 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 10. Scaling function for 12C(e, e) evaluated with the RPWIA, rROP, and RMF approaches compared to experimental function together with a phenomenological parametrization. scaling variable ψwhich follows closely the behavior of the phenomenological function. As already mentioned in previous work [16], this asymmetry of the RMF description constitutes a basic difference from other models presented in the literature [55–57] where the long tail in f(ψ) is largely absent. The asymmetry in data has usually been ascribed to ingredients beyond the mean field, such as short-range correlations, induced nonlocalities, and two-body currents. Within a nonrelativistic aproach, such ingredients are needed in order to get the asymmetry [17,18,58]. However, here we show that a large amount of the asymmetry is indeed obtained within the framework of the RIA and a RMF description of the final continuum nucleon states. This is in accordance with some previous works [59–62] where a comparison between Dirac-Brueckner-Hartree-Fock (DBHF) and Dirac-Hartree calculations indicates that effects from correlations and Fock terms in the DBHF calculation can be accounted for by the simple Dirac-Hartree approach fitted to saturation properties of nuclear matter. This is at variance with the nonrelativistic mean field case. In this respect, note that the Dirac equation in the presence of scalar and vector local potentials can be reduced to a nonrelativistic Schr¨ odinger-like equation with energydependent and nonlocal terms [23]. Results in Fig. 10 show that the asymmetry in the scaling function can be produced via local, energy-independent relativistic potentials within the impulse approximation, and moreover, such asymmetry is very close to the experiment. This outcome does not contradict the additional role that may be played by correlations and exchange currents not accounted for within the relativistic mean field calculation. However, the small magnitude of the local central and spin-orbit potentials involved in the nonrelativistic approach cannot yield a significant asymmetry. Only a strong nonlocality of the potentials (effective values of the mass and energy) may give rise to an asymmetric differential cross section [46]. To complete the analysis, in Fig. 11 (left panels) we select the RMF description of FSI and compare data and the fit curve with the theoretical results for the three choices of the current operator: CC1, CC2, and CC3. In each case, we also present a separate analysis of the Land Tchannels involved in the process. As already shown in previous results, the longitudinal contribution fL(ψ) does not depend on the exp fit fT(ψ) fL(ψ) f(ψ) CC1 12C 2.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 exp fit Weyl Coulomb Lorentz CC1 12C 2.521.510.50-0.5-1-1.5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 exp fit fT(ψ) fL(ψ) f(ψ) CC2 12C 2.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 exp fit Weyl Coulomb Lorentz CC2 12C 2.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 exp fit fT(ψ) fL(ψ) f(ψ) CC3 12C ψ 2.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 exp fit Weyl Coulomb Lorentz CC3 12C ψ 2.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 11. Scaling functions evaluated for different currents and gauges compared to experiment. Left panels show f(ψ) as well as the contributions fL(ψ)andfT(ψ) for the three current choices and the gauge fixed to Lorentz. Right panels show the results for the longitudinal scaling function fL(ψ) for the three currents and the three gauges: Lorentz, Coulomb, and Weyl. current operator choice and agrees nicely with experiment. On the contrary, the strength of the transverse response with the CC1 current leads to a function fT(ψ) that is roughly twice the data. Discrepancies between fL(ψ) and fT(ψ) are mild (negligible) for CC2 (CC3) leading to a global scaling function f(ψ) which is in accord in both cases with the experiment. All the curves presented for the longitudinal contribution fL(ψ) satisfy the Coulomb sum rule, i.e., they integrate to unity. Up to now we have only discussed calculations corresponding to the Lorentz gauge. Results for the Coulomb gauge are similar for all FSI descriptions, whereas the Weyl gauge leads in most of the cases to very important discrepancies. This is clearly illustrated in the right panels of Fig. 11 where we show the longitudinal scaling function fL(ψ) evaluated for the three gauges and the three current operator choices. The RMF approach for the final state has been assumed. The discussion of results follows in general similar trends for the rROP model. Note that the gauge only affects the longitudinal response. As observed in Fig. 11, Lorentz and Coulomb gauges lead to very close results for all currents, particularly for CC2, where there is no distinction between the two curves. In the case of the Weyl gauge, an important difference emerges between the CC2 operator and the other two, CC1 and CC3. In the former, the scaling function fL(ψ)isalmostthesamefor 015502-8 GENERAL STUDY OF SUPERSCALING IN . . . PHYSICAL REVIEW C 74, 015502 (2006) the three gauges. This is in accordance with the fact that the continuity equation is highly fulfilled within the RMF model and the CC2 current operator. By contrast, the CC1 and CC3 operators within the Weyl gauge give rise to longitudinal scaling functions which depart significantly from data being much larger (smaller) for CC1 (CC3). In both cases the Coulomb sum rule is clearly violated. These results reinforce our confidence in the adequacy of descriptions of inclusive (e, e) reactions when based on the RMF-FSI approach and the CC2 current operator. Only in this case, the results do not depend on the specific gauge selected;1moreover, they are also shown not to be modified by the dynamic enhancement of the lower components [46]. B. Inclusive QE charged-current (ν, µ) reactions The analysis of scaling and superscaling for CC neutrinonucleus reactions has been presented in [13,16]. It is important to point out that any reliable calculation of neutrino-nucleus cross sections must first be tested against electron scattering data. Hence, two different approaches can be pursued. First, using the scaling behavior of (e, e) cross sections and the universality property of the superscaling function, we can make predictions for inclusive (ν,µ) reactions by taking the empirical electron scattering scaling function fexp(ψ). This strategy, applied not only to the QE regime but also to the kinematic region, was analyzed at depth in [13]. The second approach, considered in [16], consists of evaluating explicitly f(ψ)for(ν, µ) reactions within a specific model, namely, RDWIA. The scaling function obtained in this way can be compared directly with the model predictions given for (e, e) processes. This allows us to check not only the scaling behavior of the calculations, but also the consistency of the universality assumption of f(ψ) and the capability of the model to reproduce the experimental data. In [16], we presented a detailed investigation of CC neutrino-nucleus scattering reactions within the RIA framework. We proved that superscaling is verified to high accuracy by the model calculations even in the presence of strong relativistic potentials. Importantly, the results obtained when FSI were described by means of the RMF potential presented the right asymmetry compared with data. This is fully consistent with the discussion outlined in the previous section concerning the study of inclusive (e, e) reactions. This consistency is clearly illustrated in Fig. 12, which compares the scaling functions evaluated from (e, e) and (ν, µ) reactions. We only consider the RMF-FSI case as this is the only model which is in accordance with data. However, the consistency between electron and neutrino scattering calculations applies also to rROP and RPWIA approaches. In the left panel of Fig. 12, the CC2 prescription has been assumed for (e, e) and the separate contributions of both channels, Land T,arealso shown. For completeness, the case of the CC3 operator is 1Although not shown, the Weyl gauge leads to different results when the rROP model is assumed independently of the current operator selected. (νµ,µ) fT fL (e, e ) CC2 RMF ψ f(ψ) 32.521.510.50-0.5-1-1.5 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 fit (νµ,µ) (e, e ) CC3 RMF ψ 32.521.510.50-0.5-1-1.5 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 12. Consistency of scaling functions evaluated for (e, e)and (ν, µ) reactions for CC2 and CC3 current choices. Incident electron (neutrino) energy is fixed to 1 GeV and the scattering angle to 45◦. Separate Land Tcontributions are also presented. considered in the right panel, and the fit curve to experiment is also drawn for reference. From these results, it is clear that the universality assumption of the scaling function is highly fulfilled within the present RIA model. This supports the use of the experimental (e, e) scaling function in order to predict reliable neutrino-nucleus scattering cross sections [13]. Although not presented for simplicity, a separate analysis of the various channels, L, T , and T, contributing to (ν,µ) reactions shows that fT(ψ)=fT(ψ)=f(ψ), i.e., scaling of zero kind is verified. The longitudinal contribution leads to a scaling function fL(ψ) which departs significantly from f(ψ). However, one should be cautious because the Lcontribution to inclusive (ν,µ) cross sections is almost negligible compared with the transverse, T,T, ones. To conclude with the analysis of results, we present again in Fig. 13 the scaling function predictions for both (e, e) and (ν,µ) reactions with the RMF description in the final state, and compare them with data and the fit curve. Here we use a logarithmic scale in f(ψ) in order to enlarge the discrepancies between theory and experiment in the scaling region, i.e., negative ψvalues. In this region, our model predictions for electron and neutrino processes are in full agreement and tend to underpredict the data. This is not unexpected because the model is entirely based on one-body phase space. Ingredients beyond the impulse approximation, i.e., multinucleon knockout, either induced by exchange currents, correlations, or rescattering effects, are surely needed to get more strength in the scaling function which will be then closer to the experiment. This is in fact the case of the coherent density fluctuation model (CDFM) for correlations presented in [55–57]. This model is an extension of the RFG applied to finite nuclei, and its prediction is also shown for comparison in Fig. 13. The CDFM result, compared with the RMF calculations, presents more strength in the negative ψregion, being closer to data. However, CDFM is manifestly symmetrical around the QE peak.2As discussed in previous sections, the asymmetry of the scaling function comes mainly from the inclusion of FSI in the reaction mechanism. This 2The author is aware of a new development of the CDFM model in which asymmetry is incorporated in an effective way [63]. 015502-9