Using electron scattering superscaling to predict charge-changing neutrino cross sections in nuclei

PHYSICAL REVIEW C 71, 015501 (2005) Using electron scattering superscaling to predict charge-changing neutrino cross sections in nuclei J. E. Amaro,1M. B. Barbaro,2J. A. Caballero,3T. W. Donnelly,4A. Molinari,2and I. Sick5,∗ 1Departamento de F´ ısica Moderna, Universidad de Granada, E-18071 Granada, 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 5Departement f¨ ur Physik und Astronomie, Universit¨ at Basel, CH-4056 Basel, Switzerland (Received 1 October 2004; published 27 January 2005) Superscaling analyses of few-GeV inclusive electron scattering from nuclei are extended to include not only quasielastic processes, but also the region where excitation dominates. With reasonable assumptions about the basic nuclear scaling function extracted from data and information from other studies of the relative roles played by correlation and meson-exchange-current effects, it is shown that the residual strength in the resonance region can be accounted for through an extended scaling analysis. One observes scaling upon assuming that the elementary cross section by which one divides the residual to obtain a new scaling function is dominated by the N→transition and employing a new scaling variable suited to the resonance region. This yields a good representation of the electromagnetic response in both the quasielastic and regions. The scaling approach is then inverted and predictions are made for charge-changing neutrino reactions at energies of a few GeV, with focus placed on nuclei that are relevant to neutrino oscillation measurements. For this, a relativistic treatment of the required weak interaction vector and axial-vector currents for both quasielastic and -excitation processes is presented. DOI: 10.1103/PhysRevC.71.015501 PACS number(s): 25.30.Pt, 23.40.Bw, 24.10.Jv I. INTRODUCTION In recent studies of inclusive electron scattering at intermediate to high energies from nuclei we explored various aspects of scaling and superscaling [1–7]. The general procedure used in such analyses is to divide the experimental inclusive cross section by an appropriate single-nucleon cross section, having contributions from Zprotons and Nneutrons with their corresponding electromagnetic form factors, to obtain a reduced cross section. Here the word “appropriate” entails two things. First, the usual analysis in the region of the quasielastic (QE) peak assumes that the dominant process is elastic scattering from nucleons in the nuclear ground state followed by quasifree ejection of the nucleons from the nucleus, and hence the appropriate single-nucleon form factors are the elastic ones. Second, the nucleons in the nuclear ground state are moving (Fermi motion) and accordingly the single-nucleon cross section used must take this into account. Once one has the reduced cross section it can be plotted against one or more appropriately chosen variables; if the results do not depend on some of these variables and a universal behavior is found, one says that the results scale. Specifically, when the reduced cross section is plotted against a well-chosen scaling variable (see below) and no dependence on the momentum transfer qis observed, one says that one has scaling of the first kind. When no dependence occurs on the momentum scale that characterizes specific nuclei (essentially the Fermi momentum kFof a given nuclear species), one says that one has scaling of ∗To contact collaboration, send e-mail to: [email protected]. the second kind. If both types of scaling behavior are found, one says that superscaling occurs. At sufficiently high energies we have seen both types of scaling behavior. For specific nuclei one observes quite good first-kind scaling at excitation energies below the QE peak, namely, in the so-called scaling region. This is the familiar y-scaling behavior. At energies above the peak, where nucleon resonances (especially the ) are important, this type of scaling is broken for the total reduced cross section. On the other hand, from what data we have where longitudinaltransverse separations have been made, we know that these scaling violations apparently reside in the transverse response, but not in the longitudinal. The latter appears to superscale. In fact, this is not unexpected, since we know that there are contributions that do not scale arising from meson-exchange currents (MECs) plus the correlation effects1required by gauge invariance which must be considered together with the MEC [8–10] and from inelastic scattering from the nucleons [6]. As discussed below, it is important to observe that MEC and inelastic contributions are predominantly transverse in the kinematic regions of interest in the present work. Note that scaling of the second kind works very well in the scaling region and, even in the resonance region, it is only violated at roughly the 20% level. 1Note that such correlation effects are not the only ones. In particular, even in a factorized approach of the type presented here, there are both mean-field and short-range correlations in the initial state which are embodied in a nuclear spectral function and which lead to the scaling function deduced from data, as discussed later. 0556-2813/2005/71(1)/015501(17)/$23.00 015501-1 ©2005 The American Physical Society J. E. AMARO et al. PHYSICAL REVIEW C 71, 015501 (2005) Using these recent studies as a basis we now extend our analysis to encompass both the QE and regions. Our approach is the following: taking all of the high-quality data for two specific nuclei of relevance to our later discussions, namely for 12C and 16O, we proceed as outlined above and begin by performing a comprehensive scaling analysis in the QE region. Using our knowledge of the experimentally determined longitudinal superscaling function as the starting point, we work backward and predict the transverse response one would obtain strictly from the contributions that are present in that function. In other words, we reconstruct the part of the transverse cross section that has neither MEC effects nor inelasticity built into it. The net inclusive cross section so obtained is then, in effect, the QE contribution, except for corrections arising from MEC effects and their associated correlations. The next step is to subtract this from the data. What is left should now be dominated by the inelasticity; and, in particular, when not too far above the QE peak region, we expect the resonance to be the most important contribution. From other studies, we expect that a similar procedure can now be followed for the subtracted results. We again reduce the leftover cross section by dividing by the appropriate single-nucleon cross section, now for the N→transition, and display the result versus a new scaling variable in which the kinematics of resonance electro-production are respected. As discussed later, the results scale quite well, suggesting that this procedure has indeed identified the dominant contributions not only in the QE region, but also in the region. We check our analysis by assembling all of the pieces obtained via the scaling procedures to produce a total inclusive cross section which can be compared with data. Overall the answers are very encouraging, and only for specific kinematics do we see deviations as large as 10–20%. We believe that effects from MECs and their associated correlations (which are not incorporated using this approach) are probably responsible for these residuals. Having met with success in extending the scaling and superscaling analyses from the scaling region, through the QE peak region, and now into the region, we are in a position to take a step in a new direction. Since we have the scaling functions and can be sure that, upon being multiplied by the electromagnetic N→Nand N→single-nucleon cross sections, the total nuclear electromagnetic cross section is quite well reproduced, we can just as well multiply by the corresponding charge-changing weak interaction N→Nand N→single-nucleon cross sections to obtain predictions for neutrino reactions in nuclei at similar high-energy kinematics. Thus, the second motivation for the present investigation has been to work backward to predictions for these cross sections with the goal of providing high-quality results for use in ongoing experimental studies of neutrino oscillations at GeV energies [11–15]. These studies are presently being pursued in the MiniBooNE and K2K/T2K experiments [11]. Both of these and also the forthcoming MINOS, NOvA, and MINERvA [12–15] initiatives involve neutrino energies of several GeV where a fully relativistic treatment of the neutrino-nucleus scattering is mandatory, but hard to achieve. Targets of hydrocarbon or water are involved in the cases of MiniBooNE and K2K/T2K, and hence 12C and 16Oaretaken as the focus in the present work. For the others, iron and lead will also be considered, and in this regard we note that, to the degree that scaling of the second kind is reasonably well satisfied, one can focus on the nuclei where the most reliable electron scattering data are available and then subsequently obtain predictions for neutrino reactions not only for those nuclei, but also for a wide range of targets. Any reliable calculation for neutrino scattering should first be tested against electron scattering data. Here, instead of using a specific model to describe inclusive electron scattering at relatively high energies in the QE and regions, as stated above, we follow a different approach. Using as a basis the scaling behavior of the electron-nucleus cross section in both the QE and peaks, we extract the scaling functions directly from experimental data and use them to predict the neutrinonucleus cross section. This strategy is motivated by the fact that while relativistic modeling [16–25] of the nuclear dynamics in studies of highenergy inclusive lepton scattering is expected to be capable of getting the basic size and shape of the cross section, so far it has not been capable of accounting for important details of the response. Specifically, such modeling has been able to provide a reasonable representation of the eA inclusive cross section. Typically at high energies the cross sections obtained using wide classes of models, including those with relativistic meanfield dynamics and random-phase approximation (RPA)-type correlations included, are seen to be very similar to the results found using the relativistic Fermi gas (RFG) model, accordingly we will also compare the results obtained using the scaling approach with those obtained using the RFG model. At the peaks of the QE or responses, one finds that the two approaches differ by about 25% (with mean-field effects this discrepancy is reduced to perhaps 20%); however, as we shall see later, the phenomenological scaling approach requires a long tail, which is largely absent in most modeling. A possible reason for this disagreement is the absence of classes of short-range correlation effects in most of the relativistic modeling. It goes without saying that nonrelativistic modeling is completely inadequate at the energies of interest in the present work. At intermediate energies (below those considered in this study) it is, of course, important to include effects from final-state interactions and RPA correlations (see for instance recent work reported in [26,27]) as these can be significant. Since most semileptonic scattering processes at similar kinematics have much the same character, one should expect that failure at this level will also imply a similar level of disagreement for predictions made of neutrino reaction cross sections using the same types of modeling. Clearly, on the one hand, if all one wants is a rough estimate of neutrino reaction cross sections at similar energies, then existing relativistic modeling is probably adequate. On the other hand, if uncertainties of less than 25% are required [as, for example, when one wishes to see distortions in the energy distributions of the detected muons in (νµ,µ −) reactions with nuclei caused by neutrino oscillations], then one must use existing models with great caution. As we shall see below, the superscaling approach being followed in the present work appears to be capable of reducing the uncertainty to perhaps the 10% level, 015501-2 USING ELECTRON SCATTERING SUPERSCALING TO . . . PHYSICAL REVIEW C 71, 015501 (2005) at least when one limits the focus to the QE region and the region up to inelasticities where the contribution reaches its maximum. The paper is organized the following way. In Sec. II we begin with a brief discussion of kinematics, since we will be interested not only in electron-neutrino-induced reactions, where the lepton masses can safely be ignored, but also in muon-neutrino-induced reactions where the energies, although relatively high, are not high enough to safely ignore the muon mass. In Sec. III we present a summary of the formalism needed in studies of scaling and superscaling, both for the QE and regions. There we give the results of our analysis of inclusive high-energy inelastic electron scattering data for carbon and oxygen, using the procedures outlined above, and thereby validate the scaling functions we use in the rest of the paper. In Sec. IV we turn to the second theme of the paper and discuss charge-changing neutrino reactions with nuclei. We begin by presenting the basic formalism required in treating electroweak processes, followed by development of the single-nucleon responses in the QE region (Sec. IV A) and in the region (Sec. IV B), now of course with both vector and axial-vector N→Nand N→currents. Section IV C then contains a discussion of the formalism involved in obtaining the cross sections and response functions. Once these developments are in hand, we proceed in Sec. V with a presentation of our predictions for charge-changing neutrino reactions with nuclei. Finally, in Sec. VI we summarize our work and present our conclusions. II. LEPTON SCATTERING KINEMATICS In this section we begin with a brief discussion of the kinematics involved in studies of lepton scattering from nuclei including electron scattering and the subject of Sec. IV, charge-changing neutrino reactions. We start with a general scattering problem in which an incident beam of leptons with 4-momentum Kµ=(, k) scatters and a lepton with 4-momentum Kµ=(,k) emerges. In general, one has =m2+k2,(1) =m2+k2,(2) where mand mare the masses of the incident and outgoing leptons, respectively. Clearly, for electron scattering m= m=me(usually, but not always, this can be taken to be zero) and for electron-neutrino-induced charge-changing neutrino reactions m=mνe ∼ =0, whereas m=me(again, essentially zero). The difficult case is for muon-neutrino-induced chargechanging neutrino reactions where m=mνµ ∼ =0, whereas m=mµ; the last is clearly not negligible for the kinematics of interest in the present work. As usual, one has a 4-momentum transfer Qµ=(ω,q) with ω=−,(3) q=k−k,(4) the energy transfer and 3-momentum transfer, respectively. The momentum transfer is spacelike: −Q2=q2−ω2>0. For convenience we define an average leptonic mass as M≡1 2(m2+m2)⩾0,(5) and given an excitation from target rest mass Mito some final rest mass Mf⩾Mi(that is, the final hadronic rest frame total energy is W=Mf), we define a sort of excitation energy as ω0≡1 2MiM2 f−M2 i⩾0.(6) Then from energy-momentum conservation one has ω=ω0+|Q2| 2Mi .(7) Solving Eqs. (3) and (7) together produces expressions for the scattered lepton’s energy and 3-momentum. Defining 1≡M2 i+2Mi+m2+k2sin2θ, (8) 2≡Mi(−ω0)+M2,(9) where θis the lepton scattering angle (the angle between k and k), it can be shown that k=1 2 12 2(kcos θ)+(Mi+)4 2−m22 1,(10) =1 2 12 2(Mi+)+(kcos θ)4 2−m22 1,(11) where for the results to be real for all scattering angles the beam energy must be greater than min, where min =m+ω0+mω0+(m2−m2)/2 Mi−m.(12) Hence, for a given excitation energy ω0and for given beam energy and scattering angle, the quantities 1,2can be computed and through them the final lepton’s energy and 3-momentum are fixed. Clearly the 4-momentum transfer is then given as well. III. SCALING AND SUPERSCALING A. Scaling in the quasielastic peak region In this subsection we briefly review the structure of the nuclear responses in the region of the quasielastic peak (QEP). We begin with the basic relativistic Fermi gas model that has been used to motivate scaling and superscaling behavior in this region of kinematics [1–4]. Here a single parameter characterizes the dynamics, namely, the Fermi momentum kF. In the present work our goal is to use the electron scattering cross sections as input, perform a scaling analysis, and arrive at predictions for the charge-changing neutrino cross sections. Accordingly the focus is placed on kinematic regimes where the cross sections (induced by electrons or neutrinos) are substantial, and from past work it is known that under such circumstances it is a good approximation to work only to leading order in an expansion in ηF≡kF/mN.Also ξF≡√1+η2 F−1∼ =η2 F/2 is small. 015501-3 J. E. AMARO et al. PHYSICAL REVIEW C 71, 015501 (2005) The leading-order QE responses (denoted by subscript zero) may be written, in the non-Pauli-blocked domain, in the following form [1]: RQE L(κ, λ)0=0 κ2 τ[(1+τ)W2(τ)−W1(τ)]fRFG(ψ),(13) RQE T(κ, λ)0=0[2W1(τ)]×fRFG(ψ),(14) with 0≡NξF mNκη3 F∼ =N 2κkF ,(15) and W1,W 2the structure functions for elastic scattering. As usual the proton (N=Z) and neutron (N=N) contributions should be separately computed with the appropriate form factors and added together. The latter are linked to the Sachs form factors through the well-known relations (1+τ)W2(τ)−W1(τ)=G2 E(τ),(16) 2W1(τ)=2τG2 M(τ).(17) As usual, we employ dimensionless variables λ≡ω/2mN, κ≡q/2mN, and τ≡|Q2|/4m2 N=κ2−λ2. The RFG has the following universal form for the superscaling function: fRFG(ψ)=3 4(1 −ψ2)θ(1 −ψ2),(18) that is, when plotted against the scaling variable ψ, ψ≡1 √ξF λ−τ (1 +λ)τ+κ√τ(1 +τ),(19) a universal behavior is obtained with no dependence left either on momentum transfer (scaling of the first kind) or on nuclear species via kF(scaling of the second kind). In studies of electron scattering scaling, one usually includes a small energy shift by replacing ωby ω−Eshift in order to force the maximum of the QE response to occur for ψ=0 (see, for example [3–5,28]). This is equivalent to taking λ→λ=λ−λshift with λshift =Eshift/2mNand correspondingly τ→τ=κ2−λ2in Eq. (19). Often we shall use the pair of variables (κ, ψ) in place of the original pair (q,ω) to characterize the inclusive scattering responses—clearly they are functionally related by the above equations. Using the guidance provided by the RFG, the procedure to adopt in order to get the experimental scaling function FQE(κ, ψ) in the QE domain is then clear: Simply divide the experimental QE cross section by SQE ≡σMvLGQE L+vTGQE T,(20) where σMis the Mott cross section, vL,vTare the kinematic factors defined below in Eqs. (74) and (75), and the functions GQE L,T are [2–6] GQE L=κ 2τZG2 E,p +NG2 E,n+Oη2 F,(21) GQE T=τ κZG2 M,p +NG2 M,n+Oη2 F.(22) The factors involving κand τin Eqs. (21) and (22) arise partly from the Jacobian of the transformation from λto ψ[7] and partly from the explicit calculation leading to Eqs. (13) and (14). Finally, as in past discussions of scaling of the second kind, one multiplies FQE(κ, ψ)bykFto obtain the superscaling function fQE(κ, ψ). The nuclear response functions all have the general structure [R]QE =1 kF fQE(κ, ψ)N 2κ[R]s.n.,(23) where Nis the appropriate nucleon number. In particular, as stated above, one copy of this expression with proton form factors and N=Zshould be added to another with neutron form factors and N=Nfor electron scattering. Here [R]s.n. is the corresponding single-nucleon response. In previous work [2–4] we have shown that scaling works quite well for all nuclei and for energy loss ω<ω QEP, where ωQEP corresponds to the maximum of the quasielastic peak; the scaling function FQE(κ, ψ) is indeed largely independent of the momentum transfer as long as qis of the order of 2kFor larger. Deviations from scaling, which mainly occur at larger energy loss, are related to contributions beyond quasielastic scattering such as those from meson exchange currents and  excitation (see below). The scaling behavior becomes particularly clear if one studies the experimental response separated into its longitudinal (charge) and transverse (magnetic) pieces. The nonscaling contributions mentioned above mainly occur in the transverse response. Accordingly, one finds that the experimental longitudinal responses scale much better and to much larger energy loss. The approach taken in [3,4] therefore has been to use the experimental longitudinal responses to define the scaling function fQE(ψ). The total inclusive electron scattering response is then assumed to be composed of several contributions: (1) the entire longitudinal contribution appears to superscale and to be represented by the empirical scaling function fQE(ψ); (2) part of the transverse response arises from quasielastic knockout of nucleons from the nucleus and is also driven by the scaling function fQE(ψ); however, (3) the transverse response has additional contributions, at least from MEC effects with their associated correlations and from inelastic single-nucleon processes including the excitation of the . From our past work we know that typically the effects under item (3) break the scaling. The contributions from MEC effects together with their attendant correlations enter roughly at the 10% level [8–10,29–34] and, as we argue later, may be less important for charge-changing neutrino reactions than they are for electron scattering. Accordingly we shall ignore these nonscaling effects in the present work. In future work we hope to address this issue more directly with continued relativistic modeling of these contributions. Other effects can enter into the dynamics and invalidate the picture here (for example, at low q, the RPA correlation effects can modify the longitudinal and transverse responses in different ways because of the very different isospin character of these two channels); however, again, at the kinematics of interest in the present work these effects are thought to be relatively small. The largest nonscaling contribution to the transverse response is then believed to be the one arising from inelastic but impulsive processes, especially via the excitation of the for 015501-4 USING ELECTRON SCATTERING SUPERSCALING TO . . . PHYSICAL REVIEW C 71, 015501 (2005) FIG. 1. Averaged experimental fQE(ψ QE)versusψ QE in the quasielastic region together with a phenomenological parametrization of the data. The integral of the curve has been normalized to unity. the kinematics of interest in the current study; this provides the focus for the following subsection. In [3,4] the intercomparison of the scaling functions for various nuclei has been performed in terms of the functions fQE(κ, ψ) extracted from FQE(κ, ψ) as discussed above. Excellent scaling of the second kind, i.e., scaling functions fQE(κ, ψ) that closely match for different nuclei, was observed and, indeed, such second-kind scaling is actually significantly better realized than is scaling of the first kind. The combination of scaling of the first and second kind—superscaling—allows one to determine from the data a universal scaling function fQE(ψ). The scaling function (and quasielastic cross section) for individual nuclei can then be recalculated once the Fermi momentum of the nucleus is known. Reliable separations of data into their longitudinal and transverse contributions for A>4 are available only for a few nuclei [35]; all of these response functions have been used to extract the “universal” quasielastic response function fQE and to obtain a parametrization by a simple function. Figure 1 shows fQE(ψ QE) averaged over the nuclei employed, together with the corresponding fit.2Note that fQE(ψ QE) has a somewhat asymmetric shape and a tail that extends toward positive values of ψ QE. In contrast, the RFG [see Eq. (18)] is symmetric when plotted as in the figure, is limited strictly to the region −1⩽ψ QE ⩽+1, and has a maximum value of 3/4, whereas the empirical scaling function reaches only to about 0.6. One source for the difference could be typical mean-field dynamics in the initial and final nuclear states involved; however, for the kinematics of interest in the present work, both relativistic mean-field theory [17,20,21,23] and relativized shell-model studies [19] appear to provide only rather modest 2In the figure and henceforth, the QE scaling variable is denoted ψ QE to distinguish it from the scaling variable used in the region; see later discussion. differences from the RFG predictions. As stated above, nonrelativistic modeling is quite incapable for such kinematics (see, for instance, [19]). Another source for the differences seen between the RFG or mean-field descriptions and the empirically determined scaling function arises from high-momentum components in realistic wave functions that may be large enough to produce the results shown in the figure; although much more work, especially in a relativistic context, is required to put this on a solid footing. For the present we limit our approach to phenomenology and take the scaling function from experiment. It should be emphasized at this point that much of the inability of typical modeling to account for the inclusive response at these kinematics appears to stem directly from the inability of that modeling to account for the results in the figure. We shall see later that when the empirical scaling function is used, one obtains a good representation of measured (e, e) cross sections and, therefore, that one’s confidence in proceeding to predictions for neutrino reaction cross sections must be raised. Unseparated experimental quasielastic cross sections have been measured for several nuclei over a large range of momentum transfer (for a compilation see [4]). These data have been used to determine the values of kFfor the nuclei considered (in [5] a table with numerical values is given) and since the evolution of kFwith the nuclear mass number is slow, these values can easily be interpolated for any nucleus of interest. As was pointed out earlier, the description of the experimental scaling functions involves, besides the choice of the proper kFthat sets the overall momentum scale, the use of an energy shift Eshift that in an average way accounts for the nucleon removal energy. This small correction has been included in [3,4] in the analysis of the data, and [5] gives a table with the numerical values for various nuclei. We should add that in order to obtain fQE from the cross sections, we had to divide by the single-nucleon cross sections obtained from e-pand e-nscattering. We used the H¨ ohler parametrization 8.2 [36]. In the range of qof interest here, this parametrization agrees with more recent parametrizations fitted to a somewhat more extensive data set. B. Scaling in the region of the peak Following the framework of [5,37,38], let m∗be the mass of a generic nucleon excitation and µ∗≡m∗/mN; hence µ∗=1 for quasielastic scattering and µ∗=m/mN≡µ for electro-excitation in the region. Introducing β∗≡1 4(µ2 ∗−1),(24) ρ∗≡1+β∗/τ, (25) we generalize the dimensionless scaling variable of the quasielastic peak as ψ∗≡1 ξFκρ2 ∗+1/τ −λρ∗−11/2 ×+1λ⩾λ0 ∗, −1λ⩽λ0 ∗, (26) 015501-5 J. E. AMARO et al. PHYSICAL REVIEW C 71, 015501 (2005) which vanishes for λ=λ0 ∗=1 2µ2 ∗+4κ2−1,(27) or, in dimensionful variables, when ω=ω0 ∗=m2 ∗+q2−mN.(28) When µ∗=1, one recovers the QE answer in Eq. (19), where β∗=0, ρ∗=1 and at the peak ω0 QE =√m2 N+q2−mN.Asin the previous subsection where the QE scaling variable was discussed, here also we include the small energy shift Eshift by making the replacement ω→ω≡ω−Eshift with λ→λ and τ→τas before. Again these replacements are made in the above equations to yield a generic shifted scaling variable ψ ∗, and specifically for use in the region the shifted scaling variable ψ . When considering the N→transition structure functions, we change notation from the general quantities β∗,ρ∗, ψ∗, etc., to β,ρ,ψ, etc., and in addition introduce γ≡1 4(µ−1)2(29) and κ∗ =1 µτ+(τ+β)2,(30) which allow us to define ν 1≡(1+µ)2(τ+γ)(31) and ν 2≡ν 1 τ (µκ∗ )2.(32) Then the N→single-baryon responses will read w 1(τ)=ν 1G2 M,p(τ)+3G2 E,n(τ),(33) w 2(τ)=ν 2G2 M,p(τ)+3G2 E,n(τ)+4τ µ2  G2 C,(τ),(34) where the magnetic, electric, and Coulomb form factors (following [38]) are taken to be GM,p(τ)=2.97g(Q2),(35) GE,n(τ)=−0.03g(Q2),(36) GC,(τ)=−0.15GM,p(τ),(37) with g(Q2)≡GE,p(τ) √1+τ,(38) and GE,p(τ)=1 [1+4.97τ]2,(39) namely the dipole parametrization of the proton (elastic) electric form factor (see also Sec. IV C). As for the quasielastic region, in the domain we ignore terms of order η2 F. In this approximation (as above, denoted by the subscript 0) the RFG longitudinal and transverse N→ responses will read R L(κ, λ)0=1 20 κ2 τ1+τρ2 w 2(τ)−w 1(τ)fRFG(ψ), (40) R T(κ, λ)0=1 202w 1(τ)×fRFG(ψ),(41) where 0is given in Eq. (15). As usual, for electron scattering one should add the contribution obtained from Eqs. (40) and (41) computed with N=Zand the p→+structure functions to the one where Eqs. (40) and (41) are computed with N=Nand the n→0responses. Since these processes are purely isovector, clearly this is equivalent to using N=A with one set of the structure functions. Again, using the guidance provided by the RFG, this procedure is easily generalized to the experimental response in the region. Here, as long as density-dependent corrections (i.e., the corrections that go as η2 F) are ignored as they were above, one should divide the experimental inclusive electro-excitation cross section in the region by S≡σMvLG L+vTG T(42) to get the scaling function F(κ, λ). By comparing Eqs. (40) and (41) with Eqs. (13) and (14) one gets for the functions G L,T the expressions G L=κ 2τN1+τρ2 w 2(τ)−w 1(τ)+Oη2 F, (43) G T=1 κNw 1(τ)+Oη2 F.(44) Note that in Eq. (44) one has w 1(τ), whereas in Eq. (22) the factor τin W1(τ)=τG2 M(τ) has been taken out in front. As before, one should take the results for reactions with protons with the appropriate form factors and with N=Zand add them to the results for reactions with neutrons again with the appropriate form factors, but now with N=N. Finally, to get the superscaling function f(κ, λ) one multiplies F(κ, λ) by kF. With the formalism in hand, we now proceed in a manner that is analogous to our treatment of the data in the QE region; however, we are now focusing on the region. To isolate the contributions in the region, we subtract from the total experimental cross sections (with Coulomb distortion effects incorporated) the quasielastic cross section recalculated using the universal fQE(ψ QE) introduced above. That is, we remove the impulsive longitudinal and transverse contributions that arise from elastic eN scattering, leaving (at least) MEC effects with their associated correlations and impulsive contributions arising from inelastic eN scattering. As discussed earlier, the MEC effects will be ignored in the present work as they are believed to provide relatively small corrections, and thus this yields, at least for ψ <0, a response that is largely dominated by the . For energy losses beyond the maximum of the  peak, other resonances and, at the larger values of q, the tail of deep inelastic scattering contribute. As a consequence of the different qdependencies of the various contributions, it has not been possible using the present approach to further 015501-6 USING ELECTRON SCATTERING SUPERSCALING TO . . . PHYSICAL REVIEW C 71, 015501 (2005) FIG. 2. (Color) Averaged experimental values of f(ψ ) together with a phenomenological fit (whose validity is restricted to ψ <0). analyze this ψ >0 region. The region of validity of f(ψ ) therefore will be restricted to ψ <0. As shown by Eqs. (33) and (34), the determination of f(κ, λ) involves a division by a combination of the M, E, and Ccontributions with their appropriate qdependence. For the latter, we employ the parametrizations given in Eqs. (35)–(37) and used by Amaro et al. [38]. Obviously, the main contribution is due to the M1term, the C0and E2 contributions to the cross sections being minor. In Fig. 2 we show the resulting f(ψ ) extracted from the high-quality world data for inclusive electron scattering from 12C and 16O in the QE and regions. These data span energies extending from 300 MeV to 4 GeV and scattering angles from 12 to 145 degrees, depending on the beam energy. For this determination the larger-angle and higher-qdata are of particular importance. At small angles and lower q,the contribution is small and often not present in the available data due to limited coverage in energy loss. As for fQE, the experimental values of fhave been parametrized by a simple analytical function. We show in Fig. 2 the averaged experimental values together with this fit. As pointed out before, the validity of the fit is restricted to ωvalues below the peak, i.e., ψ <0. The data appear to scale reasonably well up to the peak of the , namely, the point where ψ ∼ =0; although clearly for still higher excitation energies, the scaling is broken by processes that are not well represented via dominance. There is also some excess at large negative ψ which breaks the scaling to some degree—this is thought to be due to contributions from MEC effects and their associated correlations [8–10,29–34], as discussed above. For reference we note that for electrons of 1 GeV scattering from 12C, the quasielastic peak (where ψ QE ∼ =0) occurs at ψ =−1.8, −1.2, and −1.0forθ=45, 90, and 135 degrees, respectively. From Fig. 1 we see that fQE peaks at roughly 0.6 and thus these nonscaling contributions typically occur at the 10–15% level in the total cross section. Below we discuss our expectations for the uncertainties incurred for our predictions FIG. 3. Experimental (e, e) cross section for 12C at an incident electron energy of 1.5 GeV and a scattering angle of 13.5 degrees, together with the calculated result obtained using fQE and f.The dashed curve is the QE contribution and the solid curve is the total including the . of charge-changing neutrino reactions when we ignore such effects (see Sec. V). In passing it is important to note that in the present study we have simply taken the residual scaling function ffrom experimental data. While similar to fQE it differs in detail: it is somewhat lower, is shifted slightly, and is more spread out over a wider range of scaling variable. This is perhaps not unexpected, since implicit in this approach is the fact that the brings with it its own width and shift. Only with a more microscopic model could one hope to be able to deconvolute these from the total response and see whether the underlying scaling function is indeed the basic fQE deduced above. Such an approach will be pursued in the future, although it only becomes practical when the MEC contributions are under control. For the present we limit the analysis to using two different functions fQE and f, both deduced from phenomenological fits to electron scattering data. With these ingredients, it is then possible to recalculate for every nucleus, incident electron energy, and scattering angle the inclusive cross section for ωbelow the maximum of the contribution. To demonstrate this, we show in Figs. 3–5 the experimental responses together with the calculated response obtained using the parametrized fQE and f. In particular, we have studied the accuracy of the predicted response using (e, e)for12C and 16O and for a variety of momentum transfers, since these are the most relevant nuclei for the MiniBooNE and K2K/T2K neutrino oscillation measurements discussed in the introduction. For the data sets that do cover the region, typical deviations are 10% or less. IV. CHARGE-CHANGING NEUTRINO REACTION FORMALISM Among several options available (see, e.g., [20,21,23,39, 40]) we choose to write the charge-changing neutrino cross 015501-7 J. E. AMARO et al. PHYSICAL REVIEW C 71, 015501 (2005) FIG. 4. As for Fig. 3, except for an electron energy of 1.3 GeV and a scattering angle of 37.5 degrees. section in the target laboratory frame in the form d2σ ddkχ≡σ0F2 χ,(45) where χ=+ for neutrino-induced reactions (for example, νl+n→l−+p, where l=e, µ, τ) and χ=− for antineutrino-induced reactions (for example, νl+p→l++ n). In Eq. (45) σ0≡(Gcos θc)2 2π2[kcos θ/2]2,(46) where G=1.16639 ×10−5GeV−2is the Fermi constant, θc is the Cabibbo angle (cos θc=0.9741), and the generalized scattering angle  θreads tan2 θ/2≡|Q2| v0 ,(47) FIG. 5. Experimental (e, e)crosssectionasinFigs.3and4, but now for 16O at an incident electron energy of 0.88 GeV and a scattering angle of 32 degrees. with v0≡(+)2−q2=4−|Q2|.(48) Henceforth we shall assume that m=mν=0, but will always keep mnonzero. One additional issue arises in computing the neutrino reaction cross sections having to do with the fact that the charged leptons in the final state are not plane waves but are influenced by the Coulomb potential provided by the nucleus. This implies that the 4-momentum of the scattered lepton, Kµ=(,k), is the local quantity in the sense that the 3-momentum kand energy =√m2+k2are determined using the sequence of steps outlined in Sec. II, culminating in Eqs. (10) and (11) for these variables. However, the asymptotic energy momentum is not the same as the local quantity. Following standard procedures (see, for instance, [21,41]) the Coulomb interaction can be incorporated, at least approximately, by shifting from (,k) to asymptotic energy-momentum ( ∞,k ∞) such that k ∞=D(k)k,(49)  ∞=m2+k2,(50) where D(k)=1−χ3Zα 2Rk,(51) and R∼ =1.2A1/3is the effective charge radius of the nucleus being studied. Thus, our procedure is to calculate the cross sections using the kinematics as discussed in Sec. II and then at the end present the results in terms of the asymptotic energies and momenta obtained in this approximate manner. The only remaining issue is that the local calculations must also be multiplied by the density-of-states factor [D(k)]−1to obtain the results we present in Sec. V. The nuclear-structure-dependent quantity F2 χmay be written as F2 χ=[ VCCRCC +2 VCLRCL + VLLRLL + VTRT] +χ[2 VTRT],(52) that is, as a generalized Rosenbluth decomposition having charge-charge (CC), charge-longitudinal (CL), longitudinallongitudinal (LL) and two types of transverse (T,T) responses. Next we expand these response functions into their vector and axial-vector contributions according to RCC =RVV CC +RAA CC,(53) RCL =RVV CL +RAA CL,(54) RLL =RVV LL +RAA LL ,(55) RT=RVV T+RAA T,(56) RT=RVA T.(57) The lepton kinematical factors are the following:  VCC =1−tan2 θ/2·δ2,(58)  VCL =ν+1 ρtan2 θ/2·δ2,(59) 015501-8 USING ELECTRON SCATTERING SUPERSCALING TO . . . PHYSICAL REVIEW C 71, 015501 (2005)  VLL =ν2+tan2 θ/21+2ν ρ+ρ·δ2·δ2,(60)  VT=1 2ρ+tan2 θ/2−1 ρtan2 θ/2 ×ν+1 2ρρ·δ2·δ2,(61)  VT=1 ρtan2 θ/2(1 −νρ·δ2).(62) Here mνhas been assumed to be zero, and the entire lepton mass dependence occurs via the dimensionless parameter δ≡m |Q2|.(63) Moreover, in the above we have defined ν≡ω q=λ κ,(64) ρ≡|Q2| q2=τ κ2=1−ν2,(65) ρ≡q +,(66) which turn out to be related as ρ=tan θ/2 ρ+tan2 θ/2 (67) and lie between zero and unity. In passing we note that, using the generalized scattering angle in Eq. (47), |Q2|=4sin2 θ/2,(68) v0=4cos2 θ/2.(69) Finally, for use later we define  VL≡ VCC −2ν VCL +ν2 VLL.(70) For comparisons with finite-mass corrections to electron scattering, see [42]. In the extreme relativistic limit (ERL), namely m→0, the kinematic factors are obtained by observing that in this situation  θbecomes θand all of the terms containing δcan be dropped. One then gets  VCC →vCC =1,(71)  VCL →vCL =ν, (72)  VLL →vLL =ν2,(73)  VL→vL=ρ2,(74)  VT→vT=1 2ρ+tan2θ/2,(75)  VT→vT=tan θ/2ρ+tan2θ/2,(76) where the last three kinematic factors coincide with those employed in electron scattering (spin observables, coincidence electron scattering, parity-violating electron scattering, etc.). In the case of the µ=0(C) and 3 (L) components of the vector current, which is assumed to be conserved, it is possible to collapse the contributions down to a single term. In particular, one has RVV CL =−νRVV CC ,(77) RVV LL =ν2RVV CC .(78) Since everything of the purely polar-vector type can be related to a single response, traditionally we call this the longitudinal contribution, defined by the equation RVV L≡RVV CC . Expressing the sum of the (µ, ν)=(0,0), (0,3), (3,0),and (3,3) contributions, one then ends up with the single term  VCCRVV CC +2 VCLRVV CL + VLLRVV LL = VLRVV L≡XVV L. (79) This collapse into a single expression does not occur for the AA terms. Indeed there one has  VCCRAA CC +2 VCLRAA CL + VLLRAA LL ≡XAA C/L.(80) To complete the analysis one should add the two contributions (µ, ν)=(1,1) and (2,2), which yield  VTRVV T+RAA T≡XT,(81) and, as well, consider the V/A interference term where (µ, ν)=(1,2) and (2,1), 2 VTRVA T≡XT.(82) The full response will then be [see Eq. (52)] F2 χ=XVV L+XAA C/L +XT+χXT.(83) A. Single-nucleon responses in the QE region The single-nucleon responses in the QE region all begin with the basic vector and axial-vector currents involving N→Nmatrix elements: jµ V=¯ u(P)F1γµ+i 2mN F2σµνQνu(P),(84) jµ A=¯ u(P)GAγµ+1 2mN GPQµγ5u(P),(85) with Qµ=Pµ−Pµ. Indeed, in the scaling analyses discussed in Sec. III A, the former was used together with the usual relationship between the Dirac/Pauli form factors and the Sachs form factors, GE=F1−τF2and GM=F1+F2, to obtain expressions such as those in Eqs. (16) and (17). The total current is then jµ=jµ V−jµ A. In fact, for the purely polar-vector contributions we have RVV L=1 ρG(1) E2(86) RVV T=2τG(1) M2,(87) which were used above. Here G(1) E,M are the nucleon’s EM isovector form factors. We reemphasize that, as written, these results contain effects from the motion of the nucleons in the nucleus up to first-order in ηFand only terms of order η2 Fand beyond have been neglected. 015501-9 J. E. AMARO et al. PHYSICAL REVIEW C 71, 015501 (2005) for the vector contributions, whereas they yield w1A=T2 31 4µ2  [4τ+(µ+1)2]16τ2−8τ(µ−1) +(µ−1)23µ2 +1CA2 3 +µ2 1+4τ−µ2 CA 4−2CV 52 +µCA 31+4τ+2µ−3µ2  ×1+4τ−µ2 CA 4−2CA 5,(A3) w2A=T2 34τ µ2 1+4τ+3µ2 CA2 3 +[4τ+(µ+1)2]µ2 CA2 4+CA2 5 4τ +µ1+4τ+4µ+3µ2 CA 4+2CA 5CA 3, (A4) u1A=−T2 31 16 τµ2  [4τ+(µ+1)2]CA 5−4τCA 6 ×CA 548τ2−µ2 −12+8τµ2 +1 +4τ4µCA 3+µCA 44τ+µ2 −1 −CA 616τ2+µ2 −12+8τµ2 +1,(A5) u2A=T2 31 4τµ2  [4τ+(µ+1)2]CA 5−4τCA 6 ×8τµCA 3+µCA 4+CA 54τ−µ2 +1, (A6) for the axial-vector contributions and w3=T2 31 2µ2 µ2CA 5−4τ−µ2 +1CA 4 ×1+4τ+4µ+3µ2 CV 3−µ ×4τ−µ2 +1CV 4−4τ+µ2 −1CV 5 −CA 321+8τ+16τ2+2µ2 −3µ4 CV 3 +µ1+4τ−4µ+3µ2 4τ−µ2 +1 ×CV 4−4τ+µ2 −1CV 5,(A7) for the interference pieces. Since the axial-vector current is not conserved, w1and w2are not sufficient to set up the AA hadronic tensor: hence, two extra functions u1Aand u2A, which vanish if CA 5=4τCA 6, come out from the traces. For the empirical functions entering above we take [22] CV 3(τ)=2.05 (1 +|Q2|/0.54GeV 2)2,(A8) CV 4(τ)=−CV 3 µ ,(A9) CV 5(τ)=0,(A10) CA 3(τ)=0,(A11) CA 4(τ)=−0.31−1.21|Q2| 2GeV 2+|Q2| ×1+|Q2| (1.28)2GeV 2−2 ,(A12) CA 5(τ)=1.21−1.21|Q2| 2GeV 2+|Q2| ×1+|Q2| (1.28)2GeV 2−2 ,(A13) CA 6(τ)=CA 5(τ)m2 N m2 π+|Q2|=CA 5 4τ+µ2 π ,(A14) which, inserted into Eqs. (A2)–(A7), lead to w1V=T2 31 4µ2  [4τ+(µ+1)2]2[4τ+(µ−1)2]CV2 3, (A15) w2V=T2 34τ µ2  [4 τ+(µ+1)2]CV2 3,(A16) for the vector contributions, w1A=T2 31 4[4τ+(µ+1)2] ×4τ−µ2 +1CA 4−2CA 52,(A17) w2A=T2 31 µ2  [4τ+(µ+1)2]4τµ2 CA2 4+CA2 5, (A18) u1A=−T2 31 16τµ2  [4τ+(µ+1)2]CA 5−4τCA 6 ×48τ2−µ2 −12+8τµ2 +1CA 5 +4τ4µ2 4τ+µ2 −1CA 4−16τ2+µ2 −12 +8τµ2 +1CA 6,(A19) u2A=T2 31 4τµ2  [4τ+(µ+1)2]CA 5−4τCA 6 ×8τµ2 CA 4+4τ−µ2 +1CA 5,(A20) for the axial-vector contributions; and w3=T2 3CV 3 µ [4τ+(µ+1)2] ×2CA 5−4τ−µ2 +1CA 4(A21) for the V/Ainterference. One finds that u1Aand u2A(which arise from PCAC) are negligible, whereas the other functions are all significant. The latter are seen to fall strongly with increasing q. 015501-16 USING ELECTRON SCATTERING SUPERSCALING TO . . . PHYSICAL REVIEW C 71, 015501 (2005) [1] W. M. Alberico, A. Molinari, T. W. Donnelly, E. L. Kronenberg, and J. W. Van Orden, Phys. Rev. C 38, 1801 (1988). 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