Semirelativistic description of quasielastic neutrino reactions and superscaling in a continuum shell model

PHYSICAL REVIEW C 71, 065501 (2005) Semirelativistic description of quasielastic neutrino reactions and superscaling in a continuum shell model J. E. Amaro,1M. B. Barbaro,2J. A. Caballero,3T. W. Donnelly,4and C. Maieron5 1Departamento de F´ ısica Moderna, Universidad de Granada, Granada E-18071, Spain 2Dipartimento di Fisica Teorica, Universit` a di Torino and INFN, Sezione di Torino Via P. Giuria 1, I-10125 Torino, Italy 3Departamento de F´ ısica At´ omica, Molecular y Nuclear Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain 4Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 5INFN, Sezione di Catania, Via Santa Sofia 64, I-95123 Catania, Italy (Received 23 March 2005; published 9 June 2005) The so-called semirelativistic expansion of the weakly charged current in powers of the initial nucleon momentum is performed to describe charge-changing, quasielastic neutrino reactions (νµ,µ −) at intermediate energies. The quality of the expansion is tested by comparing it with the relativistic Fermi gas model using several choices of kinematics of interest for ongoing neutrino oscillation experiments. The new current is then implemented in a continuum shell model together with relativistic kinematics to investigate the scaling properties of (e, e)and(νµ,µ −) cross sections. DOI: 10.1103/PhysRevC.71.065501 PACS number(s): 25.30.Pt, 25.30.Fj, 24.10.Jv, 21.60.Cs I. INTRODUCTION The importance of neutrino-induced reactions in nuclei has been stressed in connection with the neutrino oscillation experiments performed by the KARMEN and LSND Collaborations [1–8]. In experiments of this type, relatively low νeor νµenergies are involved (at most a few hundreds of MeV), and so the nuclear excitations involved can be described by standard nonrelativistic models of the reaction including the relevant machinery random-phase-approximation (RPA) correlations, large-basis shell models, -hole excitations, final-state interactions, etc. for this energy regime, where in particular giant resonances may play an important role [9–16]. However, when passing to the ongoing and next generation of neutrino experiment, MiniBooNE, K2K/T2K, MINOS, NOνA, and MINERνA [17–22], the neutrino beam energies increase to the GeV level, and typically large energies and momenta are transferred to the nucleus. For these kinematics, relativity is important and nonrelativistic models of the reaction such as those listed above are bound to fail unless the relevant relativistic ingredients are included. First of all, the use of relativistic kinematics is required and must be implemented in the model. From quasielastic (QE) electron scattering studies we know that a good approximation to the correct kinematics consists in the substitution λ−→ λ(1 +λ),(1) with λ=ω/2mN, where ωis the energy transfer and mNthe nucleon mass. This substitution can easily be performed in all places in the calculation—except in the nucleon form factors, where the correct value of momentum transfer Qµ=(ω,q) must be used. Second, a good approximation to the current matrix elements is required. The current is traditionally obtained from the fully relativistic one by some expansion procedure, usually through the Foldy-Wouthuysen (FW) transformation, which is valid for small momenta compared to the nucleon mass. Within this procedure, relativistic effects were studied for neutrino energies up to 300 MeV in [23], showing that a good agreement is obtained between fully relativistic and non relativistic Fermi gas calculations, when terms up to order (q/mN)3are included in the latter. Similar results were obtained at larger energies, for the scattering of atmospheric neutrinos from oxygen [24], where, however, it must be noted that because of the steep decrease of the atmospheric neutrino spectrum at large energies (see, for example, [25]), the most important contributions to the process correspond to relatively small energy transfers. Obviously, the Foldy-Wouthuysen expansion is not applicable to the high values of the momentum transfer qwhich are of interest to the new experiments, i.e., for values around 1 GeV/c. In this case, a different expansion procedure must be performed in which qand ωcan be arbitrarily large. Thus, the first goal of this paper is to develop an approximation to the nuclear charged current (CC) that accounts for specific relativistic effects relevant to intermediate-energy quasielastic neutrino reactions. The CC is obtained here through an expansion that only requires the momentum of the initial nucleon to be small, while it treats exactly the dependence on (ω,q). Moreover, the resulting semirelativistic (SR) current is simple enough to be easily implemented in already existing nonrelativistic models of (νl,l−) and (νl,l+) reactions, where l=eor µ. It is presented here as an extension of the electromagnetic and weak-neutral current expansion originally derived in [26] and that in recent years has been widely tested and applied in several Collaborations to describe a wide variety of inclusive and exclusive electron scattering observables for intermediate energies and excitations in the vicinity of the QE peak [27–37]. Extensions of the SR expansion to meson-exchange currents have also been developed [38,39], and a detailed description of their application to two-body currents can be found in a recent review article [40]. 0556-2813/2005/71(6)/065501(15)/$23.00 065501-1 ©2005 The American Physical Society AMARO, BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 71, 065501 (2005) In this paper, we apply the SR model to neutrino-induced reactions, for the first time. Apart from the different isospin dependence, the SR expansion coincides essentially with that of the weak neutral current performed in [26]. The only difference is that in the present reaction, we have to include the time component of the axial-vector current. That component was not considered in [26] because it does not contribute significantly to parity-violating electron scattering. We check the quality of the SR expansion in the context of the relativistic Fermi gas (RFG) model. This is a very convenient model for our purposes, since it is fully relativistic and simple enough to be solved exactly. Also this model is capable of getting the basic size and shape of the QE (e, e) cross section. Thus we will show that starting from the nonrelativistic Fermi gas, performing the replacement in Eq. (1) in the kinematics, and implementing the new SR charged current, we reproduce basically the RFG results for (νµ,µ −) reactions. This “relativizing” procedure can easily be extrapolated to more sophisticated finite nuclei models of the reaction. The result will at least reproduce appropriately the allowed kinematical region and the relevant relativistic content of the current operator. Of course, features related to the relativistic aspects of the dynamics cannot be accounted for by our procedure. In context we note that studies of relativistic nuclear dynamics in charged and neutral-current neutrino-nucleus QE scattering have already been presented in some previous work [41–45]. In these studies, a basic focus was to analyze the effects introduced by various descriptions of the final nucleon relativistic states upon the integrated cross sections. We illustrate the relativizing procedure by applying it to the continuum shell model (CSM), i.e., nucleons in a mean field taken here to be a Woods-Saxon potential. We use the same (real) potential for initial and final states in order to maintain orthogonality between nuclear states. In this way, we extend the SR shell model of [26] to neutrino reactions at the QE peak. Since the use of relativistic kinematics is potentially equivalent to solving a Klein-Gordon equation, the present SR continuum shell model includes some aspects of a relativistic mean field. With such a model, we are in a position to fulfill the second goal of this paper, which is to investigate superscaling properties of both (e, e) and (νµ,µ −) inclusive cross sections at the QE peak for intermediate energies, namely, the degree to which one finds that the reduced cross sections are independent of the momentum transfer (scaling of the first kind) or the nuclear species (scaling of the second kind) or both (superscaling). Exhaustive analyses of the (e, e) world data and explorations of various aspects of their scaling properties have been performed in [46–52]. In particular, in recent work [53], the approach has been extended to the peak, allowing one to construct a semiempirical model based on scaling which is very successful in describing the experimental (e, e) cross sectionuptothepeak for high energies. This model allowed us in [53] to generate predictions for the (νµ,µ −) cross section, under the reasonable hypothesis that it presents, for high energies, scaling properties similar to those of the (e, e)cross section. This hypothesis is true by construction in the RFG and can be also demonstrated at least for the conventional plane-wave impulse approximation (PWIA) in the high q limit.1However, when distortion of the ejected nucleon is present, a general proof of scaling cannot be provided, while for lower energies, it is clear that the axial-vector and vector matrix elements are renormalized differently in the nuclear medium because of RPA correlations, and this can obviously modify the scaling properties of the neutrino cross section. For higher energies, one expects that nuclear effects such as from RPA correlations are less important than at low energies. In this work, we use the SR shell model to investigate the degree of violation of the scaling hypothesis for the QE peak region within the distorted-wave impulse approximation (DWIA), where the distortion of the ejected nucleon is described with a real potential. We perform this study in two steps. First we focus on the (e, e) cross section and study the scaling properties of the separate response functions. Once the superscaling has been verified for electromagnetic processes, we are able to reconstruct the (νµ,µ −) cross section from the (e, e) one using the scaling hypothesis and compare it with the one computed directly using the SR shell model. Thus, at least within the context of the SR shell model discussed in the present work, following this procedure we shall be able to check the consistency of our approach and quantify the degree to which scale breaking effects are expected to enter. As we shall see below, there appears to be very little impact from this source of scale breaking on the scaling approach used in [53] to predict neutrino-induced cross sections. The paper is organized in the following way. In Sec. II we begin with a brief review of the general formalism for neutrino scattering and present the expansion of the CC. We particularize the formalism for the shell model and introduce the general multipole expansion of the responses, with some details on the derivation of the Coulomb multipoles of the axial-vector current placed in Appendix A. We also provide the expressions for the factorized PWIA in Appendix B. In Sec. III, we present results for the (νµ,µ −) reaction for several choices of kinematics and several nuclei of interest. We first perform an analysis of the quality of the relativizing procedure by comparison with the RFG for the relevant kinematics (the RFG model is further discussed in Appendix C.) The quality of the various components of the SR current can be checked separately by examining the individual response functions that contribute to the process. We then focus on the SR shell model and perform the scaling analysis of the longitudinal and transverse electromagnetic responses as functions of the momentum transfer for various nuclei. We apply the scaling hypothesis to reconstruct the (νµ,µ −) cross section starting from the electromagnetic scaling function and compare, our findings with the shell model results. Finally, in Sec. IV we present our conclusions. II. FORMALISM In this section, we briefly present the basic formalism for neutrino-induced reactions. Some of the previous approaches 1However, this is broken to some degree by the relativistic PWIA, where factorization no longer obtains. 065501-2 SEMIRELATIVISTIC DESCRIPTION OF . . . PHYSICAL REVIEW C 71, 065501 (2005) to the general formalism for these reactions can be found in [41,54–57]. A. Charge-changing neutrino cross section Here we focus on the particular case of (νµ,µ −), while the cases of antineutrinos or other lepton species can be easily obtained with obvious changes. The four momenta of the incident neutrino and detected muon are kµ=(, k) and kµ=(,k), respectively. The four-momentum transfer is Qµ=kµ−kµ=(ω,q). We use a coordinate system with the zaxis pointing along qand the xaxis along the transverse component of the incident momentum, i.e., k⊥=k−1 q2(k· q)q. We follow the formalism of [53], where the cross section is written as dσ dd=σ0F2 +,(2) with σ0=G2cos2θc 2π2kcos2˜ θ 2.(3) Here G=1.166 ×10−11 MeV−2is the Fermi constant, θcis the Cabibbo angle, cos θc=0.975, and the angle ˜ θis defined as tan2˜ θ 2=|Q2| (+)2−q2,(4) with Q2=ω2−q2<0. The nuclear structure information is contained in F2 +, defined by F2 += VCCRCC +2 VCLRCL + VLLRLL + VTRT+2 VTRT, (5) where the kinematical factors  VKcoming from the leptonic tensor are defined by  VCC =1−δ2tan2˜ θ 2,(6)  VCL =ω q+δ2 ρtan2˜ θ 2,(7)  VLL =ω2 q2+1+2ω qρ+ρδ2δ2tan2˜ θ 2,(8)  VT=tan2˜ θ 2+ρ 2−δ2 ρω q+1 2ρρδ2tan2˜ θ 2,(9)  VT=1 ρ1−ωρ qδ2tan2˜ θ 2.(10) In Eqs. (6)–(10), following [53], we have defined δ=m |Q2| ,(11) ρ=|Q2| q2,(12) ρ=q +.(13) Note that the only dependence on the muon mass mis contained in the δcoefficient. Finally, the weak response functions are given by RCC =W00,(14) RCL =−1 2(W03 +W30),(15) RLL =W33,(16) RT=W11 +W22,(17) RT=−i 2(W12 −W21),(18) in terms of the inclusive hadronic tensor [54] Wµν(q,ω)= fi δ(Ef−Ei−ω)f|Jµ(Q)|i∗f|Jν(Q)|i. (19) In Eq. (19), Jµ(Q) is the hadronic CC current operator, to be specified below, and a sum over final states and an average over initial spin is assumed. B. Semirelativistic charge-changing current We begin with the basic relativistic charged weak current of the nucleon, jµ=jµ V−jµ A. In this work, we employ only the standard model of electroweak interactions at tree level and thus, for example, do not include radiative corrections or contributions from second-class currents (see [54]). We use the conventions of [58]. The vector and axial-vector currents are given by jµ V(p,p)=u(p)2FV 1γµ+iFV 2 mN σµνQνu(p),(20) jµ A(p,p)=u(p)GAγµ+GP Qµ 2mNγ5u(p),(21) where for the isovector nucleon form factors FV 1,2=(Fp 1,2− Fn 1,2)/2 we use the Galster parametrization [59], and u(p)is the free Dirac spinor of the nucleon. The axial-vector and pseudoscalar form factors are parametrized as GA=gA 1−Q2/M2 A ,(22) GP=4m2 N m2 π−Q2GA,(23) with gA=1.26,M A=1032 MeV. The SR approximation to this current is then obtained by inserting the appropriate free spinors u(p),u(p), and γmatrices, and performing an expansion in powers of η≡p/mNto first order. The procedure was developed in [26,38,40] for the electromagnetic and transverse neutral, vector, and axial-vector currents and is based on the fact that inside the nucleus |η|is a good expansion parameter, the characteristic dimensionless nuclear scale being ηF=kF/mN≃1/4. We exploit QE kinematics, while further reasonable simplifications are eventually needed in order to 065501-3 AMARO, BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 71, 065501 (2005) arrive at simple expressions that are easily implementable in traditional nonrelativistic calculations. For the vector current, we use the following SR approximation: J0 V=ξ0+iξ 0(κ×η)·σ,(24) J⊥ V=ξ1η⊥+iξ 1σ×κ,(25) where ξ0=κ √τ2GV E,ξ  0=2GV M−GV E √1+τ,(26) ξ 1=2GV M √τ κ,ξ 1=2GV E √τ κ,(27) and use has been made of the dimensionless variables κ= q/2mNand τ=κ2−λ2. From vector current conservation, the longitudinal component is given by J3 V=λ κJ0 V. Note that in Eqs. (24) and (25), the various terms making up the current are similar to the ones that can be found in traditional nonrelativistic expansions commonly used for the charged current (see, for instance, [16]), except for the κ- and τ-dependent factors, ξi,ξ i, that provide the required relativistic behavior. In J0 V we include the first-order O(η) contribution. This spin-orbit term is proportional to the operator (κ×η)·σand is of some importance for high qvalues. On the other hand, the transverse component J⊥ Vis the sum of the usual magnetization (σ×κ) piece plus a first-order term, the convection term which is proportional to η⊥, that gives in general a very small2 contribution to the cross section for high q[26,28]. In the case of the axial-vector sector, only the transverse component of the weak neutral current was expanded in [26]. We use the following version of the corresponding SR current J⊥ A=ζ 1σ⊥,ζ  1=√1+τGA.(28) Note that we have neglected the terms of order ηsince, as we shall show in the next section (and also demonstrated in [26] in the context of parity-violating electron scattering), they are small and add unnecessary complications to the shell model calculation. They can be safely neglected for our purposes. Remarkably, the factor √1+τin Eq. (28) already accounts for relativistic effects in this current (see [26] for the full expansion of this current to first order in η). We are left with the 0 and zcomponents of the axial-vector current. Their SR expressions are presented here for the first time. Using a notation reminiscent of that used in Appendix A of [26], we write them to first order in ηas J0 A=ζ 0κ·σ+ζ 0η⊥·σ,(29) Jz A=ζ 3κ·σ+ζ 3η⊥·σ,(30) 2Note that η=p/mNis essentially the velocity vof the initial struck nucleon in units of c. where ζ 0=1 √τ λ κG A,ζ  0=κ √τGA−λ2 κ2+κ√τ(τ+1)G A, (31) ζ 3=1 √τG A,ζ  3=λ √τGA−κ κ+√τ(τ+1)G A, (32) and we have introduced the following combination of axialvector and pseudoscalar form factors G A=GA−τGP.(33) For these components of the axial-vector current, Eqs. (29) and (30), we have performed the expansion to first order in η. The first-order axial-convective term is proportional to σ·η⊥, and, as for the spin-orbit and convection terms, only the perpendicular velocity η⊥appears. As we shall show below, for the kinematics of interest in this work, the G Aform factor which drives the zeroth-order terms turns out to be small because of cancellations in Eq. (33) between GAand τGP. In such cases, the O(η) term is dominant in these current components. C. The continuum shell model In this work, we restrict our attention to the closed-shell3 nuclei 12C, 16O, and 40Ca, which we describe in a continuum shell model (CSM). Thus the present model does not include nuclear correlations. Such effects are important for low energy, in particular in the axial responses, and have been estimated, for instance, in the RPA approach of Ref. [15], but are expected to be smaller at the GeV energies considered in this paper. The initial state |iappearing in the hadronic tensor, Eq. (19), is described as a Slater determinant representing the uncorrelated nuclear core with all shells occupied. Since we are working in the impulse approximation, the final states are particle-hole excitations coupled to total angular momentum J, namely, |f=|(ph−1)J. The single hole |h=|hlhjh and particle |p=|plpjpwave functions are obtained by solving the Schr¨ odinger equation with a Woods-Saxon potential V(r)=−V0f(r, R0,a 0)+Vls m2 πr df (r, R0,a 0) dr l·σ+VC(r), (34) where f(r, R, a)=1 1+e(r−R)/a ,(35) and VC(r) is the Coulomb potential of a charged sphere of charge Z−1 and radius R0(it is equal to zero for neutrons). The parameters of the potential are fitted to the experimental energies of the valence shells and are given in Table I. 3By shell we mean a subshell with quantum numbers (nlj). 065501-4 SEMIRELATIVISTIC DESCRIPTION OF . . . PHYSICAL REVIEW C 71, 065501 (2005) TABLE I. Woods-Saxon potential parameters for protons pand neutrons n. Units are MeV for Vi, and fm for a0and r0. The reduced radius parameter r0is defined by R0=r0A1/3. Vp 0Vp LS Vn 0Vn LS r0a0 12C 62.0 3.20 60.00 3.15 1.25 0.57 16O 52.5 7.00 52.50 6.54 1.27 0.53 40Ca 57.5 11.11 55.00 8.50 1.20 0.53 In the shell model, the energy transfer is computed as the difference between the (nonrelativistic) single-particle energies of particle and hole ω=p−h. The relativistic kinematics are taken into account by the substitution p→p(1 +p/2mN) (36) as the eigenvalue of the Schr¨ odinger equation for the particle; cf., Eq. (1). Since the nuclear states have good angular momentum, it is convenient for the shell model calculation to perform a multipole expansion of the components of the current operator in terms of the usual Coulomb (C), longitudinal (L), transverse electric (E), and transverse magnetic (M) operators [26,54,60], defined by ˆ CJ0(q)=d3rjJ(qr)YJ0(ˆ r)J0(r),(37) ˆ LJ0(q)=i qd3r∇[jJ(qr)YJ0(ˆ r)]·J(r),(38) ˆ EJm(q)=1 qd3r∇×[jJ(qr)YJJm(ˆ r)] ·J(r),(39) ˆ MJm(q)=d3rjJ(qr)YJJm(ˆ r)·J(r),(40) where jJ(qr) is a spherical Bessel function and YJJm(ˆ r)isa vector spherical harmonic. The nuclear response functions are then written as RCC =4π αCV α2+CA α2,(41) RCL =2π αCV∗ αLV α+CV αLV∗ α+CA∗ αLA α+CA αLA∗ α, (42) RLL =4π αLV α2+LA α2,(43) RT=4π αEV α2+MV α2+EA α2+MA α2,(44) RT=2π αEV∗ αMA α+EV αMA∗ α+EA∗ αMV α+EA αMV∗ α, (45) where in the above sums over α, the particle energy p= h+ωis modified according to Eq. (36), and we use the index αto label the quantum numbers α=(h, lp,j p,J). Moreover, the C,L, E, and Mmultipoles of the vector and axial-vector currents are defined by the reduced matrix elements of the corresponding operators CV α+iCA α=fˆ CJ(q)i,(46) LV α+iLA α=fˆ LJ(q)i,(47) EV α+iEA α=fˆ EJ(q)i,(48) −iMV α−MA α=fˆ MJ(q)i.(49) These reduced matrix elements are given in [26] for the vector and axial-vector transverse operators and in [60] for the leading-order longitudinal axial-vector component. In the present paper, we add the first-order convective term of the axial-vector current η⊥·σappearing in Eqs. (29) and (30). The CA αand LA αmultipoles of this new term are presented in Appendix A. The sums over αin Eqs. (41)–(45) are infinite and have to be truncated in the calculation once convergence is reached. In our approach, the number of multipoles is determined by the maximum, of the total angular momentum Jmax.Wefix this quantity by computing the response functions setting the potential in the final state to zero and comparing with the factorized PWIA (see Appendix B for details). The number Jmax increases with qand with the number of nucleons A.In the next section up to 41 multipoles have to be summed for the case where q=1.5GeV/c. III. RESULTS A. Test of the SR approach The quality of the SR expansion of the charged current is illustrated in Figs. 1–3. There we show the separate vector, axial-vector, and total response functions of 12Cfor three values of the momentum transfer, q=0.5, 1, and 1.5 GeV/c. The Fermi momentum is chosen to be kF= 220 MeV/c. We show the separate response functions instead of cross sections because in these functions we can appreciate better the contribution of the various current components and the behavior of the SR approximation as a function of the momentum and energy transfers. This can be seen in the figures by comparing the solid lines, representing the RFG result (exact relativistic answer) with the dashed lines, corresponding to the semirelativistic Fermi gas (SRFG). This last model is obtained by implementing the SR expansion of the current in a nonrelativistic Fermi gas, although with relativistic kinematics. The accord of the two models is almost perfect for the three values of qconsidered. In Fig. 1, we only show RV CC and RV T, since RV CL and RV LL are related to RV CC by current conservation. In Fig. 2, we give instead the three response functions, since the axial-vector current is not conserved. Finally, in Fig. 3, we display the sums of Figs. 1 and 2 and also the interference response RT. In Figs. 1–3 we also plot (as dotted lines) the CSM responses, computed using the relativizing procedure. For comparison, we show the PWIA results obtained by setting the potential in the final state to zero (or, equivalently, by integration of the factorized exclusive responses, as shown in 065501-5 AMARO, BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 71, 065501 (2005) q=0.5 GeV/c RV CC [GeV−1] 15 10 5 q= 1 GeV/c 1.2 1 0.8 0.6 0.4 0.2 q=1.5 GeV/c 0.2 0.15 0.1 0.05 ω[MeV] RV T[GeV−1] 4003002001000 40 30 20 10 ω[MeV] 800600400200 12 10 8 6 4 2 ω[MeV] 12001000800600 3 2 1 FIG. 1. Vector response functions of 12C for three values of the momentum transfer. Solid lines: RFG with kF=220 MeV/c. Dashed lines: SRFG. Dotted: CSM. Dot-dashed: PWIA. Appendix B). The CSM response functions are quantitatively similar in magnitude to the RFG and are centered approximately around the same value of ω. The major discrepancy between the two models is found for the axial-vector RA CC,RA CL, and RA LL response functions, where the CSM responses are slightly larger in magnitude than the RFG ones. These responses are in general small compared with the corresponding vector responses. This is a consequence of the suppression of the axial-vector current in the longitudinal channel becasue of the small value of the form factor G A, defined in Eq. (33), for these kinematics. In fact, from the definition in Eq. (23) of the pseudoscalar form factor, one has G A=1−Q2 Q2−m2 πGA.(50) At the intermediate energies of interest, |Q2|m2 π, and hence the factor inside the parenthesis in Eq. (50) is also small, of order O(m2 π/Q2). In this situation the first-order, axial-vector convective term of the current, which is proportional to η⊥·σ q=0.5 GeV/c RA CC [GeV−1] 1 0.8 0.6 0.4 0.2 q= 1 GeV/c 0.15 0.1 0.05 q=1.5 GeV/c 0.03 0.02 0.01 RA CL [GeV−1] 0 -0.1 -0.2 -0.3 0 -0.02 -0.04 -0.06 0 -0.005 -0.01 -0.015 RA LL [GeV−1] 0.3 0.2 0.1 0.03 0.02 0.01 0.012 0.008 0.004 ω[MeV] RA T[GeV−1] 4003002001000 70 50 30 10 ω[MeV] 800600400200 8 6 4 2 ω[MeV] 12001000800600 2 1.5 1 0.5 FIG. 2. Same as Fig. 1, but for the axial-vector responses. 065501-6 SEMIRELATIVISTIC DESCRIPTION OF . . . PHYSICAL REVIEW C 71, 065501 (2005) q=0.5 GeV/c RCC [GeV−1] 14 10 6 2 q= 1 GeV/c 1.4 1 0.6 0.2 q=1.5 GeV/c 0.2 0.15 0.1 0.05 RCL [GeV−1] 0 -1 -2 -3 -4 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 -0.04 -0.08 -0.12 RLL [GeV−1] 2 1.5 1 0.5 0.3 0.2 0.1 0.08 0.06 0.04 0.02 RT[GeV−1] 100 80 60 40 20 20 15 10 5 5 4 3 2 1 ω[MeV] RT[GeV−1] 4003002001000 50 40 30 20 10 ω[MeV] 800600400200 10 8 6 4 2 ω[MeV] 12001000800600 2.5 2 1.5 1 0.5 FIG. 3. Same as Fig. 1, but for the total (vector plus axial-vector) responses. [see Eqs. (29) and (30)], is dominant over the zeroth-order contribution, and the corresponding response functions are in general small compared with the vector ones. An example is shown in Fig. 4, where we display the separate contributions of the zeroth- and first-order terms to the axial-vector response RA CC of 12Cforq=500 MeV/cin the CSM. For higher values of q the zeroth-order contribution is much smaller than the others. From inspection of the responses RCC,R T, and RT,we observe that the PWIA results are clearly shifted to the right of the CSM by roughly the averaged depth of the potential ∼35 MeV. This shift is present also in the separate vector and axial-vector responses. In the case of the CL and LL,theshiftis larger because of the energy factors used to compute the Lcomponent of the current. The origin of the shift in PWIA is related to the different treatment of the nuclear Hamiltonian in the initial and final states [27]: while the energy of the initial bound neutron is the sum of kinetic plus potential, h=th+vh,the exiting proton is a plane wave and has only kinetic energy p=tp. Accordingly, for fixed ωthere is an imbalance between potential energies in the initial and final states, yielding ω=tp−th−vh. However, in the CSM we use the same potential for the initial and final state, which now has p=tp+vp. Hence ω=tp−th+vp−vh. Thus, in the CSM the potential energies of particle and hole partially cancel q=0.5 GeV/c ω[MeV] RA CC [GeV−1] 4003002001000 1 0.8 0.6 0.4 0.2 FIG. 4. Axial-vector response RA CC of 12C in the CSM. Solid: contribution of the zeroth-order term in the SR expansion. Dashed: first-order contribution. 065501-7 AMARO, BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 71, 065501 (2005) θ=45 o =1GeV dσ dΩd[10−15fm2/MeV] 1000900800700600500 6 5 4 3 2 1 θ=45 o =1.5GeV 12001000800 3 2 1 0 θ= 135o [MeV] dσ dΩd[10−15fm2/MeV] 600500400300200 0.5 0.4 0.3 0.2 0.1 0 θ= 135o [MeV] 600500400300200 0.15 0.1 0.05 FIG. 5. Differential cross section of the reaction 12C(νµ,µ −) for incident neutrino energies =1 and 1.5 GeV and for two scattering angles. Solid: RFG with kF=215 MeV/c. Dashed: SRFG. Dotted: CSM. out; this explains why the position of the peak is close to the RFG, where only kinetic energies enter. Of course, the cancellation is not perfect, and a slight shift to the right of the RFG is also observed in the CSM. This small shift of the CSM is the behavior expected from previous theoretical studies [61]. Another test of the SR approximation is illustrated by the results shown in Fig. 5 for the differential cross section 12C(νµ,µ −). Therein we show examples for two incident neutrino energies, =1 and 1.5 GeV, and for two scattering angles θ=45o, and 135o, as a function of the exiting muon energy ; so in this case, we are testing different ranges of q and ω, which can be high or low depending on the kinematics. We see again that the RFG and SRFG predictions are almost equal in all cases, while the CSM also gives similar results, with the exception of the characteristic tails and slight shift, which now is to the left of the RFG because ωdecreases with . Summarizing this subsection, using the Fermi gas as “testing arena” for the approximated CC current, we have found that the accord between the results obtained using the SR expansion and the exact relativistic result is almost perfect for the intermediate and high values of qand ωconsidered. The CSM relativized using our procedure gives rise to cross sections that are within the allowed kinematical region and have a magnitude similar to those of the RFG. This behavior of the CSM is similar to what was found for (e, e) reactions in [26]. We should underscore the fact that the CSM is used here just to illustrate how one can use our relativizing procedure in a more complex model than the Fermi gas and to show the results expected from it. In particular, especially when discussing lower energies where nonrelativistic approaches are valid, more elaborate models exist that describe the (e, e) experimental data rather well. For example, it is well known from other work that final-state interactions can significantly modify the bare CSM or RPA responses, mainly through medium renormalization of the particle-hole excitations. This mechanism can be approximately taken into account by using an effective nucleon mass that produces a shift and by a folding of the bare responses, producing an asymmetric broadening which improves the agreement with the experimental data [62–64]. 12C fL 0.8 0.6 0.4 0.2 16O fL 0.8 0.6 0.4 0.2 40Ca ψ fL 21.510.50-0.5-1-1.5-2 0.8 0.6 0.4 0.2 FIG. 6. Scaling of the first kind in the CSM with scaling functions obtained from the longitudinal electromagnetic responses in the CSM. In each panel we include q=0.5, 0.7, 1, 1.3, and 1.5 GeV/c. 065501-8 SEMIRELATIVISTIC DESCRIPTION OF . . . PHYSICAL REVIEW C 71, 065501 (2005) q=0.5 GeV/c fL 0. 8 0. 6 0. 4 0. 2 q=0.7 GeV/c fL 0. 8 0. 6 0. 4 0. 2 q= 1 GeV/c fL 0. 8 0. 6 0. 4 0. 2 q=1.3 GeV/c fL 0. 8 0. 6 0. 4 0. 2 q=1.5 GeV/c ψ fL 21. 510. 50-0.5-1-1.5-2 0. 8 0. 6 0. 4 0. 2 FIG. 7. Scaling of the second kind including in each panel the nuclei 12C, 16O, and 40Ca. B. Scaling In this subsection we work within the CSM. We first focus on the inclusive electron scattering reaction and investigate the scaling properties of the electromagnetic responses for various choices of kinematics for 12C, 16O, and 40Ca. The (e, e) cross section is given by dσ dd=σMott (vLRL+vTRT).(51) The same SR expansion for the vector current, Eqs. (24) and (25), is employed here for the electromagnetic sector. We compute the CSM scaling functions as fL=RL GL ,(52) fT=RT GT ,(53) q=0.5 GeV/c 16O 0.8 0.6 0.4 0.2 q=0.5 GeV/c 40Ca q=0.7 GeV/c 0.8 0.6 0.4 0.2 q=0.7 GeV/c q= 1 GeV/c fL,fT 0.8 0.6 0.4 0.2 q= 1 GeV/c q=1.3 GeV/c 0.8 0.6 0.4 0.2 q=1.3 GeV/c q=1.5 GeV/c ψ 21.510.50-0.5-1-1.5-2 0.8 0.6 0.4 0.2 q=1.5 GeV/c ψ 21.510.50-0.5-1-1.5-2 FIG. 8. Scaling function fLobtained from the longitudinal electromagnetic response (solid lines), compared with the scaling function fTobtained from the transverse response (dashed lines), for several values of q. with GK=0ZUp K+NUn KK=L, T, (54) where 0is given in Eq. (C2) in Appendix C, and the electromagnetic single-nucleon functions for protons Up Kand neutrons Un Kare defined similarly to the ones given in Appendix C for the vector current UV CC and UV T. The scaling behavior can be studied by plotting these functions against the scaling variable [46] ψ=1 √ξF λ−τ (1 +λ)τ+κ√τ(1 +τ)(55) for different kinematics and for different nuclei. Here ξF= 1+k2 F/m2 N−1. A study of the behavior of fL, computed for various values of the momentum transfer, is summarized in Fig. 6. We plot together fLfor q=0.5, 0.7, 1, 1.3, and 1.5 GeV/c. All the curves approximately collapse into one. Small violations of the scaling are seen at low ψcoming from the low-energy 065501-9