Meson-exchange currents and quasielastic predictions for charged-current neutrino-12C scattering in the superscaling approach
G. D. Megias, T. W. Donnelly, O. Moreno, C. F. Williamson, J. A. Caballero, R. Gonzalez-Jimenez, A. De Pace, M. B. Barbaro, W. M. Alberico, M. Nardi, J. E. Amaro
MMeson-exchange currents and quasielastic predictions for charged-currentneutrino- C scattering in the superscaling approach
G. D. Megias ∗ , T. W. Donnelly, O. Moreno, C. F. Williamson, J. A. Caballero, R.Gonz´alez-Jim´enez, A. De Pace, M. B. Barbaro,
4, 3
W. M. Alberico,
4, 3
M. Nardi, and J. E. Amaro Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, 41080 Sevilla, SPAIN Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, ITALY Dipartimento di Fisica, Universit`a di Torino, Sezione di Torino, Via P. Giuria 1, 10125 Torino, ITALY Departamento de F´ısica At´omica, Molecular y Nuclear and Instituto Carlos I de F´ısica Te´orica y Computacional,Universidad de Granada, 18071 Granada, SPAIN
We evaluate and discuss the impact of meson-exchange currents (MEC) on charged-currentquasielastic (QE) neutrino cross sections. We consider the nuclear transverse response arising from2p-2h states excited by the action of electromagnetic, purely isovector meson-exchange currentsin a fully relativistic framework, based on the work by the Torino collaboration [1]. An accurateparametrization of this MEC response as a function of the momentum and energy transfers involvedis presented. Results of neutrino-nucleus cross sections using this MEC parametrization togetherwith a recent scaling approach for the 1p-1h contributions (SuSAv2) are compared with experimentaldata.
PACS numbers: 13.15.+g, 24.10.Jv, 25.30.Pt
I. INTRODUCTION
A correct interpretation of atmospheric andaccelerator-based neutrino oscillation experimentsstrongly relies on our understanding of neutrino-nucleusscattering at intermediate energies (from 0.5 to 10GeV) and in particular of the nuclear-structure effectsinvolved. One of the simplest descriptions of the nucleus,the relativistic Fermi gas (RFG) model, which is knownto be inadequate for inclusive electron scattering in theQE regime [2], also fails to reproduce recent measure-ments of QE neutrino and antineutrino scattering crosssections [3–8]. This supports the need for consideringmechanisms such as final-state interactions, nuclearcorrelations or MEC, in particular through their contri-bution to multinucleon knock-out around and beyondthe QE peak as suggested by explicit modeling [9–11].In particular, the recent muon neutrino charged-current quasielastic (CCQE) cross sections measured bythe MiniBooNE Collaboration [3, 4] show discrepancieswith a RFG description of the nuclear target. This sim-ple model, widely used in experimental analyses, under-estimates the total cross section, unless ad hoc assump-tions are made such as a larger mass parameter in thenucleon axial form factor ( M A = 1.35 GeV/c versus M A = 1.032 GeV/c ). Relativistic effects cannot be ne-glected for the kinematics of experiments such as Mini-BooNE, with neutrino energies as high as 3 GeV. Al-though the RFG model has the merit of accounting prop-erly for relativistic effects, it is too crude to account fordetailed nuclear dynamics, as is well known from com- ∗ Corresponding author: [email protected] parisons with QE electron scattering data [12]. Moresophisticated relativistic nuclear models have been ap-plied in recent years to neutrino reactions. In addition,phenomenological techniques have been proposed, suchas the superscaling approach (SuSA) [13] which assumesthe existence of universal scaling functions for the elec-tromagnetic and weak interactions. Analyses of inclusive( e, e (cid:48) ) data have shown that at energy transfers belowthe QE peak, superscaling is fulfilled rather well [14–16],which implies that the reduced cross section is largelyindependent of the momentum transfer (first-kind scal-ing) and of the nuclear target (second-kind scaling) whenexpressed as a function of the appropriate scaling vari-able. From these analyses a phenomenological scalingfunction was extracted from the longitudinal QE elec-tron scattering responses. It was subsequently used topredict neutrino-nucleus cross sections by multiplying itby the single-nucleon weak cross sections, assuming thatthe single universal scaling function was appropriate forall of the various responses involved, namely CC, CL, LL,T(VV), T(AA) and T (cid:48) (VA). In this work we will use arecently developed improved version of the superscalingmodel, called SuSAv2 [17], that incorporates relativisticmean field (RMF) effects [18–20] in the longitudinal andtransverse nuclear responses, as well as in the isovectorand isoscalar channels independently. Three referencescaling functions are provided to describe in a consis-tent way both electron- and (anti)neutrino-nucleus re-actions in the QE region: transverse ( ˜ f T ), longitudinalisovector ( ˜ f T =1 L ) and longitudinal isoscalar ( ˜ f T =0 L ). Thismodel also includes in a natural way an enhancement ofthe transverse response through RMF effects without re-sorting to inelastic processes or two-particle emission viaMEC.Strictly speaking only the longitudinal part of the re- a r X i v : . [ nu c l - t h ] A p r sponse appears to superscale; in the scaling region somedegree of scaling violation is found which can be at-tributed to the transverse part of the response. The as-sumption that the various types of response (CC, CL, LL,T(VV), T(AA) and T (cid:48) (VA)) scale the same way has beendenoted zeroth-kind scaling; the most recent SuSAv2 ap-proach builds in the degree of violation of zeroth-kindscaling demanded by the RMF results. Specifically, thelongitudinal contributions, apparently being essentiallyimpulsive at high energies, are usually used to determinethe basic nuclear physics of QE scattering, notably, in-cluding any correlations present in that sector, since theresults are obtained by fitting electron scattering data.Beyond the QE region it is natural to have scaling vio-lations, since the reaction mechanism there is not solelythe impulsive knockout of a nucleon, but may proceedvia meson production including baryon resonances suchas the ∆. It is known that the latter contributions aremuch more prominent in the transverse than in the lon-gitudinal responses [13, 21]. However, it is also knownthat even with only the 1p-1h contributions there are ex-pected to be violations of zeroth-kind scaling arising frompurely dynamical relativistic effects (see the discussionsof how the SuSAv2 approach is constructed).However, even below the meson production thresholdthere are scaling violations in the transverse response[16], one source of which could be the MEC contributions,again predominantly transverse. The MEC are two-bodycurrents that can excite both one-particle one-hole (1p-1h) and two-particle two-hole (2p-2h) states. Most stud-ies of electromagnetic ( e, e (cid:48) ) processes performed for low-to-intermediate momentum transfers with MEC in the1p-1h sector (see, e.g., [22–25]) have shown a small re-duction of the total response at the QE peak, mainlydue to diagrams involving the electroexcitation of the ∆resonance; they are roughly compensated by the positivecontributions of correlation diagrams, where the virtualphoton couples to a correlated pair of nucleons. In thepresent work we shall therefore neglect them and restrictour attention to 2p-2h final states, computed in a fullyrelativistic way. It has been found [9–11, 26, 27] thatthe MEC give a significant positive contribution to thecross section, which helps to account for the discrepancyobserved in ( e, e (cid:48) ) processes between theory and experi-ment in the “dip” region between the QE peak and ∆-resonance as well as for the discrepancies between somerecent neutrino CCQE measurements ( e.g., MiniBooNE,NOMAD, MINER ν A). In particular, in [28, 29] we useda parametrization of the results of [1] to evaluate the con-tribution of MEC to the vector transverse (anti)neutrinoresponse at MiniBooNE kinematics.The presence of nucleon-nucleon correlation interac-tions involving the one-nucleon current may lead to theexcitation of 2p-2h final states, and interference betweenthese processes and those involving MEC must also betaken into account. Results of calculations carried outwithin the Green’s Function Monte Carlo approach [30]suggest that these interference contributions may in fact be quite large. This is in agreement with our preliminarycalculation of the correlation current plus MEC effects inthe response functions within the scheme of the relativis-tic Fermi gas model [31]. These effects, also taken intoaccount in the RFG-based descriptions of 2p-2h providedby Nieves et al. [10] and Martini [9], are not included ex-plicitly in our RFG MEC model, that relies on a hybriddescription where the one-particle emission already con-tains contributions of nuclear ejections due to nuclearcorrelations — through the experimental scaling func-tion. Explicit calculations of the correlation-MEC inter-ference terms are still in progress and their contributionswill be presented in a forthcoming publication.This paper is organized as follows. In Sect. II webriefly describe the computation of the MEC consideredin this work and show for the first time the correspond-ing responses of C for several momentum transfers asa function of the QE scaling variable. We also show anew parametrization of these responses and compare itwith the one used in [11, 28, 29]. In Sect. III we applythe new MEC parametrization and the SuSAv2 modelto the computation of neutrino- C CCQE cross sectionsand compare the results with MiniBooNE, NOMAD andMINER ν A data. Finally, in Sect. IV we show the con-clusions of our analysis.
II. RESULTS FOR MEC RESPONSES
We consider in this work the purely isovector pion-exchange currents involving virtual ∆ resonances as wellas the seagull (contact) and pion-in-flight currents ob-tained in previous work [1, 32]. The evaluation was per-formed within the RFG model in which a fully Lorentzand transitionally invariant calculation of the MEC canbe developed. Deviations from the Fermi gas model 2p-2h responses produced by ingredients such as final-stateinteractions, finite nuclear effects or nuclear correlationsare expected to be moderate, which would result in smallcorrections in the impulsive cross section as the MECcontributions are also moderate.The previous statement is not insubstantial, and it re-quires further explanation. We expect the finite-size ef-fects to be moderate on the 2p-2h responses. This is inaccordance with the calculations performed by one of theauthors and presented in a series of papers, [see for in-stance [33, 34]]. These are the only calculations up todate concerning the inclusive 2p-2h transverse responsefunction at low-to-intermediate momentum transfers for C and Ca within the framework of the continuumshell model. The results were similar to those foundin nuclear matter by Van Orden and Donnelly [35], Al-berico, Ericson and Molinari [36], and Dekker, Brussaardand Tjon [37]. The non-relativistic 2p-2h response func-tion is a rather smooth function. Its general behavior isclearly dominated by the 2p-phase space and by the nu-cleon and pion electromagnetic form factors, whereas itis rather insensitive to details of the finite size nucleus.The previous works, together with [1, 38], are theonly calculations available for the 2p-2h electromagneticresponses for medium nuclei. The studies presented in [1,37] clearly showed that the relativistic effects, mainly inthe delta MEC, dominate the 2p-2h transverse response.It has been known for a long time that ground-statecorrelations deplete the occupation numbers of the holestates, the values of which drop from unity to ∼ .
8. Themain effect of such depletion is known to be a redistri-bution of the strength to higher energies. In the caseof the longitudinal response, dominated by the impulseapproximation, this is translated into a hardening of theresponse function with respect to an uncorrelated model,like the Fermi Gas or the semirelativistic shell model [39],with the appearance of a long tail at high energy. This isprecisely the shape of the scaling function we are using.Being a phenomenological observable, the scaling func-tion already contains all the physics embodied in the nu-clear structure details, including correlations, depletionsand final state interactions.In the case of the 2p-2h contributions, one expects thedepletion of the occupation numbers also to produce aredistribution of the strength to higher energies. Al-though this could modify the position of the peak in the2p-2h response function, the resulting redistribution isexpected to keep some resemblance with the behavior al-ready shown in the 1p-1h channel.As mentioned above, the kinematical regions containedunder the integral over the neutrino fluxes consideredhere extend to relativistic domains, so that a relativistictreatment of the process is required. As was discussed inthe previous work [1, 32], relativistic effects are importantto describe the nuclear transverse response function formomentum transfers above 500 MeV/c.All possible 2p-2h many-body diagrams containing twopionic lines and the virtual boson attached to the pion(pion-in-flight term), to the
N N π vertex (seagull or con-tact term) or involving the virtual ∆ resonance are takeninto account to compute the vector-vector transverseMEC response, R MECT, V V , of C [1]. These responses canbe given as a function of the energy transfer ω (cid:48) or of thethe scaling variable Ψ (cid:48) , related through:Ψ (cid:48) = 1 √ ξ F λ (cid:48) − τ (cid:48) (cid:113) (1 + λ (cid:48) ) τ (cid:48) + κ (cid:112) τ (cid:48) (1 + τ (cid:48) ) , (1)where ξ F is the dimensionless Fermi kinetic energy andthe following dimensionless transfer variables have beendefined: λ = ω/ m N , κ = q/ m N , τ = κ − λ . Primedvariables contain an energy transfer shift, ω (cid:48) = ω − E s ,which accounts (at least) for the binding energy of theejected nucleon, but is usually determined phenomeno-logically; for C we use E s = 20 MeV. The scaling vari-able considerably distorts the ω dependence, but it hasthe advantage of allowing us to easily locate the QE peakat Ψ (cid:48) = 0, from which the peaks of the MEC responsesare shifted. Over 100,000 terms are involved in the calcu-lation, with subsequent seven-dimensional integrations, which make it a highly non-trivial computational proce-dure. In order to include these results in the neutrinogenerators used in the analysis of neutrino experimentsa parametrization of the MEC responses is essential toreduce the computational burden of performing the cal-culation for a large number of kinematic conditions (mo-mentum and energy transfers).The MEC response functions for q ≥
400 MeV/c ex-hibit a peak that decreases with q together with a tailthat rises with Ψ (cid:48) and q . In order to parameterize thesefunctions we applied an expression with two terms, thefirst one mainly fitting the peak of the response and thesecond fitting the tail at larger Ψ (cid:48) : R MECT, V V (Ψ (cid:48) ) = 2 a e − (Ψ (cid:48)− a a e − (Ψ (cid:48)− a a + (cid:88) k =0 b k (Ψ (cid:48) ) k . (2)In this expression the parameters a i , b k are q -dependent,and they are used to fit the original R MECT, V V responsesshown in Fig. 1. We first fit each response for a given q to get the values of the a i , b k parameters for that spe-cific q -value, ensuring a smooth dependence on q for eachof them. The q -dependent values of the fitting param-eters are shown in Fig. 2. We then parametrize the q -dependence of the parameters themselves using a poly-nomial in q . The response in Eq. (2) then becomes explic-itly dependent on the momentum transfer, R MECT, V V (Ψ (cid:48) , q ),through the dependence in the parameters, a i ( q ), b k ( q ).For the fitting of the responses above q = 2000 MeV/c,which show almost no peak but a tail-like shape, we keeponly the second term in Eq. (2), namely a = 0; sincethese responses are very similar in the large- q region un-der consideration (up to 3500 MeV/c), we use the sameparametrization for all of them, namely b k ( q > b k ( q = 2000). In any case, as we can observe in Fig. 3,there are no significant MEC contributions for q > ω > q = 300 MeV/c we use again apolynomial to fit the results, R MECT, V V (Ψ (cid:48) , q < ) = (cid:88) k =0 c k ( q ) (Ψ (cid:48) ) k . (3)The results of the above parametrization of the MECresponses are presented as a function of the scaling vari-able Ψ (cid:48) in Fig. 1 where it is shown that it gives an excel-lent representation of the exact results in the full regionof q and Ψ (cid:48) explored.As already mentioned, in previous work [11, 28, 29] asimple parametrization of the exact MEC calculation wasused in order to evaluate the MiniBooNE (anti)neutrinocross sections. The present fit of the MEC responses im-proves the previous one in two respects: it uses data ina wider q range and includes the tail of the responsesat high Ψ (cid:48) or ω values. The previous parametrizationwas initially developed with electron scattering in mindand, since ( e, e (cid:48) ) data are rarely available when q → ω ,the high- ω region was ignored. Accordingly the oldparametrization missed the high energy tails arising inthe exact results and yielded lower peaks asymmetricallybroadened towards higher Ψ (cid:48) values. In contrast, forCCQE reactions one must integrate over a broad neu-trino spectrum and hence, potentially, the high- ω regionmay be relevant, and this motivated the re-evaluation ofthe MEC contributions. In Fig. 1, we also show the R MECT, V V results versus ω where it is noticed the negligiblecontribution below q <
300 MeV/c as well as the rele-vance of the tail in the response at q >
800 MeV/c. Onthe other hand, the tail of the MEC responses at high q ( q > ω (cid:38) f MECT, V V , defined analogously to the transverse scalingfunction coming from the transverse one-body response: f MECT,V V ( κ, λ ) = k F · R MECT,V V ( κ, λ ) G T ( κ, λ ) , (4)where the G T factor depends on the momentum and en-ergy transferred as well as on the isovector magnetic nu-cleon form factors and k F is the Fermi momentum of thenucleus. A detailed expression for G T , including higher-order relativistic corrections, can be found in [39] and hasbeen used in the calculation of f MECT, V V shown in Fig. 5.The remaining dependence on q of the scaling functionseen in Fig. 5 is consistent with the violation of first-kindscaling exhibited by the MEC [32]. The study of second-kind scaling violation, related to the dependence on thenuclear species, would require an in-depth study of theMEC contributions in other nuclei; some such studieswere presented in [32].For completeness, a comparison between our theoreti-cal predictions and electron scattering data [40] at kine-matics where MEC contributions are relevant, extendingfrom the non-relativistic to the highly-inelastic regime, isalso presented in Fig. 6. As shown, a model based solelyon impulsive response function is not able to reproducethe (e,e’) data. Contributions beyond the impulse ap-proximation such as 2p-2h MEC could provide part ofthe missing strength in the transverse channel. More-over, the addition of the impulsive inelastic contributionsis shown to be essential to analyze the (e,e’) data at highkinematics.In general the inelastic contributions can have a sig-nificant effect on the ( e, e (cid:48) ) cross section even in the QEregime, since the different domains can overlap. Thisagrees with the emerging pattern in Fig. 6 that suggeststhat the inclusion of inelastic processes — the contribu-tion of which clearly extends into the region dominated by quasielastic scattering—may lead to an enhancementof the theoretical results. The inelastic part of the crosssection is dominated by the delta peak (mainly trans-verse) that contributes to the transverse response func-tion. At low electron scattering angles the longitudinalresponse function dominates the cross section and theinelastic contribution is smaller. The opposite holds atlarge scattering angles, where the delta peak contributionis important. On the other hand, for increasing valuesof the transferred momentum the peaks correspondingto the Delta and QE domains become closer, and theiroverlap increases significantly. This general behaviour isclearly shown by our predictions compared with data.In those kinematical situations where inelastic processesare expected to be important, our results for the QE peakare clearly below the data. On the contrary, when theinelastic contributions are expected to be small, the QEtheoretical predictions get closer to data. It is importantto point out that the description presented in this workcorresponds to a semi-phenomenological model where thescaling function is fitted to the longitudinal ( e, e (cid:48) ) scat-tering data (and extended to the transverse response viathe RMF theory). Thus, it does not encode the inelas-ticities that dominate the transverse response.However, for completeness we also show in Fig. 6 someresults for the inelastic contributions. As observed, theinclusion of the inelastic processes does not necessarilyimply a “significant” enhancement of the cross section inthe region close to the QE peak. In fact, at the partic-ular kinematics considered in Fig. 6 the overlap betweenthe QE and inelastic regions is small and therefore theagreement with the data in the QE region is not spoiled.However, more detailed results are needed before moredefinitive conclusions can be reached. In this sense, anew analysis of the inelastic channel based on the use ofthe recent SuSAv2 and MEC models will be presented ina forthcoming paper [12]. III. EVALUATION OF NEUTRINO CROSSSECTIONS
In this section, we evaluate the CCQE double-differential and total cross sections of (anti)neutrino scat-tering off C using our latest SuSAv2 results and thenew 2p-2h MEC parametrization. We compare the re-sults with experimental data of MiniBooNE, NOMADand MINER ν A.As can be seen in Figs. 7 and 8, the inclusion of MECresults in an increase of the cross sections, yielding rea-sonable agreement with the MiniBooNE data for low an-gles, up to cos θ µ (cid:39) .
7. At larger scattering angles thedisagreement with the experiment becomes more signifi-cant, and the vector-vector transverse MEC do not seemto be sufficient to account for the discrepancy. The sameconclusion can be drawn by plotting the cross sectionversus the scattering angle (see Figs. 9 and 10) at fixedmuon momentum; the inclusion of MEC improves theagreement with the data at low scattering angles, butsome strength is missing at higher angles, especially forlow muon momenta, as observed in [41].The size of the MEC contribution to the cross sec-tion reported here — of the order of 10% — correspondsto the average value found within our particular RFGmodel. Our results show that processes involving MECare responsible for a sizable enhancement of the responsein the transverse channel. The extent to which this en-hancement affects the cross section, however, stronglydepends on the kinematics (see discussion in previoussection).We remark that axial-axial and axial-vector transverseMEC responses, R MECT, AA and R MECT (cid:48) , V A , are not consideredin this work and could partially explain the discrepancywith the data. Furthermore, additional nuclear correla-tions could contribute to the 2p-2h excitations as the onesinduced by MEC; however, since the longitudinal vec-tor contributions come directly from experimental dataand hence have all the correlations built in, such contri-butions would need to break zeroth-kind scaling whichhas not been demonstrated. Note that extended RFGor RMF models with 2p-2h, as well as 1p-1h, correla-tions are actually required to preserve gauge invariance,but their inclusion would call for consistent treatmentsto avoid double-counting.When comparing our theoretical results with the Mini-BooNE data one can observe a better agreement for an-tineutrinos than for neutrinos (see Fig. 11). This is dueto the fact that, in the neutrino case, the two missingMEC responses in our calculation are constructively com-bined, R MECT, AA + R MECT (cid:48) , V A , whereas they are destructivelycombined in the antineutrino case, R MECT, AA - R MECT (cid:48) , V A . Inother words, we expect a larger strength missing in ourcalculation in the neutrino case than in the antineutrinocase, whose origin possibly can be attributed to the miss-ing MEC pieces. Furthermore, one can see in the to-tal neutrino cross section (Fig. 11) that some strengthis missing at intermediate energies, 0.4-1.5 GeV, whichis the region where the VA QE component is peaked(Fig. 12); an extra contribution in this channel via 2p-2h MEC would thus improve the agreement with Mini-BooNE data. We can observe in Fig. 12 that below 1GeV the SuSAv2 VA response is higher than the VV oneand of the same order as the AA one. Other contributionsto the VA response, apart from the QE one (SuSAv2),can be estimated as follows( σ ν µ ) otherT (cid:48) , V A (cid:39) (cid:0) σ ν µ − σ ¯ ν µ (cid:1) exp − (cid:0) σ ν µ − σ ¯ ν µ (cid:1) SuSAv , (5)as long as one assumes no quenching of the axial currentwithin the nuclear medium with respect to the vector cur-rent, as is the case in the superscaling approach. If oneconsiders ( σ ν µ ) otherT (cid:48) , V A as mainly due to MEC, it is foundthat a VA MEC response as large as the computed VVMEC response would be needed to reproduce the data. InFig. 13 we show the experimental difference between neu-trino and antineutrino cross sections ( σ ν µ − σ ¯ ν µ ) exp from MiniBooNE, together with the corresponding theoreticalprediction from SuSAv2, which is approximately equalto 2 ( σ ν µ ) SuSAv T (cid:48) V A . The theoretical result from SuSAv2with VV MEC contributions is also shown in the figure,but is almost indistinguishable from the SuSAv2 resultdue to the VV character of the MEC used. Apart fromthe opposite sign in the
V A response, some minor differ-ences between neutrino and antineutrino cross sectionsarise from the different Coulomb distortions of the emit-ted lepton [13] and the final nuclei involved in the CCneutrino (Nitrogen) and antineutrino (Boron) scatteringprocesses.It can be seen that an extra contribution to the VAresponse from MEC would improve the agreement withthe data for the difference between neutrino and antineu-trino total cross sections of MiniBooNE, as was notedabove for just the neutrino case. In the same way, onecould deduce the suitability of extra AA and VA contri-butions via MEC in the double-differential MiniBooNEcross section by analyzing Figs. 14 and 15. At NOMADkinematics, Fig. 11, we observe a good agreement of theSuSAv2+MEC results, partly due to the negligible con-tribution of the VA response, whose MEC part is missingin our calculation, in such high-energy processes ( E ν be-tween 5 and 100 GeV). From Fig. 12 one sees that theVA interference becomes very small for E ν > θ µ → v T (cid:48) → f MECT,V V , to equal the axial-axial ( f MECT,AA )and vector-axial ( f MECT (cid:48) ,V A ) ones - as done for instance in[9] - a final result in agreement with MiniBooNE data isfound. On the contrary, the calculation slightly oversti-mates NOMAD data. However, such results cannot befully justified until a proper 2p-2h MEC calculation forthe axial-axial and vector-axial responses is completed.Moreover, one should take note of the different ways toanalyze the QE-like events in MiniBooNE and NOMAD,where in the latter [8] the combination of 1-track and2-track samples in the case of ν µ n → ν − p can help toreduce some uncertainties as well as some contributionsbeyond the Impulse Approximation, such as from MECor correlations that eject two nucleons. For complete-ness we also show in Fig. 16 recent results from the T2KCollaboration [43]. One should notice that, as they state,“there is consistency between the experiments within thecurrent statistical and systematic uncertainties.”Moreover, an analysis of the relevant kinematic regionsin the SuSAv2+MEC cross section is shown in Fig. 17,where it is observed that the main contribution to thetotal cross section comes from ω < q (cid:46) ω <
50 MeV and q <
250 MeV/c is not too significant for the cross section(less than 10%). This is in accordance with some previousworks [42, 44]. The same conclusion can be drawn byanalyzing the different kinematics in the total MEC crosssection (Fig. 3), where the low kinematic region ( ω < q <
250 MeV/c) is even less important ( < ν A kinematics, a good agreement arises forthe purely QE SuSAv2 model with the dσ/dQ QE datawithout additional assumptions, Fig. 18, as observed in[44] for other impulse-approximation based models. Anoverestimation of the data shows up at low Q QE whenadding 2p-2h MEC contributions. On the contrary, thiseffect is not observed in the same differential cross sec-tions of MiniBooNE, Fig. 19, which is an example ofthe discrepancies between the two experiments and theirdifferent ways to proceed in the data analysis. IV. CONCLUSIONS
We have obtained CCQE neutrino- C cross sec-tions using the SuSAv2 scaling procedure and a newparametrization of 2p-2h vector-vector transverse MEC.Both ingredients are based on relativistic models (RMF,RFG, RPWIA), as demanded by the kinematics ofpresent and future high-energy neutrino experiments,where traditional non-relativistic models are question-able. We do not include in this work axial-axial andvector-axial MEC contributions needed for the analysisof neutrino scattering processes, nor correlation diagrams— the calculation of the axial MEC contributions is cur-rently being considered using [26, 27].Any model aimed at providing a useful and reliabletool to be employed in the analysis of experimental stud-ies of neutrino oscillations needs their limits of applicabil-ity to be completely understood. This has been the casein our present study where the limits of the approachhave been stated clearly and discussed at length. Vari-ous models rely on different assumptions: non-relativisticexpansions, factorization approach, mean field, etc. , thatrestrict their reliability. However, in the absence of a“fully-unlimited” description of the reaction mechanism, the use of consistent, even limited, theoretical predic-tions to be contrasted with data allows one to get insightinto the physics underlying neutrino experiments. Hence,in spite of the limitations mentioned above, our presentmodel provides results that are in accordance with ( e, e (cid:48) )data in the region around the QE peak. This is of greatimportance, and it gives us confidence in the consistencyand validity of our calculations in order to analyze lepton-nucleus scattering.By comparing these results with the experimental dataof the MiniBooNE, NOMAD and MINER ν A collabora-tions we have shown that 2p-2h MEC play an impor-tant role in CCQE neutrino scattering and may help toresolve the controversy between theory and experiment.The main merit of the parametrization provided here isthat it translates a sophisticated and computationally de-manding microscopic calculation of MEC into a smoothparametrization which is dependent on the values of thetransfer variables of the process. The economy of thisMEC parametrization together with the one inherent ina scaling approach might be of interest to Monte Carloneutrino event simulations used in the analysis of exper-iments.
Acknowledgments
This work was partially supported by Spanish DGIFIS2011-28738-C02-01, Junta de Andaluc´ıa FQM-160,INFN, Spanish Consolider-Ingenio 2000 Program CPAN,U.S. Department of Energy under cooperative agree-ment de-sc0011090 (T.W.D), 7th European CommunityFramework Program Marie Curie IOF ELECTROWEAK(O.M.). G. D. M. acknowledges support from a fellowshipfrom the Junta de Andaluc´ıa (FQM-7632, Proyectos deExcelencia 2011) and financial help from VPPI-US (Uni-versidad de Sevilla). INFN under project MANYBODY(M.B.B. and A.D.P). DGI FIS2011-24149 and Junta deAndaluc´ıa FQM225 (J.E.A.). R.G.J. acknowledges finan-cial help from VPPI-US (Universidad de Sevilla). [1] A. D. Pace, M. Nardi, W. M. Alberico, T. W. Donnelly,and A. Molinari, Nucl.Phys. A , 303 (2003).[2] Note that, as is common in discussions of electron scat-tering, we define QE to mean the part of the cross sectionarising from nucleon knockout via one-body operators.This is to be distinguished from contributions that arisethrough the action of two-body operators such as theMEC effects discussed in the present work. The lattercan eject single nucleons or two nucleons (or in fact nonucleons at all, as in elastic scattering). In contrast, whatis referred to as “quasielastic” in the neutrino commu-nity really means the “no-pion cross section” and thatshould contain both one- and two-body current opera-tors, but should have no pions produced in the final state.Indeed, one common concern is how much model depen-dence occurs in defining this no-pion cross section, since corrections must be made for events where a pion is ac-tually produced, but is absorbed before being detectedand hence mistaken as a “quasielastic” event.[3] A. A. Aguilar-Arevalo et al. , Phys. Rev. D , 092005(2010), [MiniBooNE Collaboration].[4] A. A. Aguilar-Arevalo et al. , Phys. Rev. D , 032001(2013), [MiniBooNE Collaboration].[5] L. Fields et al. , Phys. Rev. Lett. , 022501 (2013),[MINER ν A Collaboration].[6] G. A. Fiorentini et al. , Phys. Rev. Lett. , 022502(2013), [MINER ν A Collaboration].[7] P. Adamson et al. , arXiv:1410.8613 [hep-ex] (2014).[8] V. Lyubushkin et al. , Eur. Phys. J. C , 355 (2009),[NOMAD Collaboration].[9] M. Martini, M. Ericson, G. Chanfray, , and J. Marteau,Phys. Rev. C , 065501 (2009). [10] J. Nieves, I. Ruiz-Simo, and M. J. Vicente-Vacas, Phys.Rev. C , 045501 (2011).[11] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, and J. M. Ud´ıas, Phys. Rev. D , 033004(2011).[12] In devising both the SuSA and SuSAv2 models compar-isons have been made with existing C QE inclusive elec-tron scattering data. The detailed results of those studieswill be presented in a future paper (in preparation).[13] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, A. Molinari, and I. Sick, Phys. Rev. C ,015501 (2005).[14] D. B. Day, J. S. McCarthy, T. W. Donnelly, and I. Sick,Annu. Rev. Nucl. Part. Sci. , 357 (1990).[15] T. W. Donnelly and I. Sick, Phys. Rev. Lett. , 3212(1999).[16] T. W. Donnelly and I. Sick, Phys. Rev. C , 065502(1999).[17] R. Gonz´alez-Jim´enez, G. D. Megias, M. B. Barbaro, J. A.Caballero, and T. W. Donnelly, Phys. Rev. C , 035501(2014).[18] J. A. Caballero, J. E. Amaro, M. B. Barbaro, T. W.Donnelly, C. Maieron, and J. M. Ud´ıas, Phys. Rev. Lett. , 252502 (2005).[19] J. A. Caballero, Phys. Rev. C , 15502 (2006).[20] J. A. Caballero, J. E. Amaro, M. B. Barbaro, T. W. Don-nelly, and J. M. Ud´ıas, Phys. Lett. B , 366 (2007).[21] C. Maieron, J. E. Amaro, M. B. Barbaro, J. A. Caballero,T. W. Donnelly, and C. F. Williamson, Phys. Rev. C ,035504 (2009).[22] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, , and A. Molinari, Phys. Rep. , 317 (2002).[23] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, , and A. Molinari, Nucl. Phys. A , 181(2003).[24] W. Alberico, T. W. Donnelly, and A. Molinari, Nucl.Phys. A , 541 (1998).[25] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, C. Maieron, and J. M. Udias, Phys. Rev. C , 014606 (2010).[26] I. Ruiz-Simo, C. Albertus, J. E. Amaro, M. B. Barbaro,J. A. Caballero, and T. W. Donnelly, Phys. Rev. D ,033012 (2014). [27] I. Ruiz-Simo, C. Albertus, J. E. Amaro, M. B. Barbaro,J. A. Caballero, and T. W. Donnelly, Phys. Rev. D ,053010 (2014).[28] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, and C. F. Williamson, Phys. Lett. B , 151(2011).[29] J. E. Amaro, M. B. Barbaro, J. A. Caballero, and T. W.Donnelly, Phys. Rev. Lett. , 152501 (2012).[30] O. Benhar, A. Lovato, and N. Rocco, arXiv:1312.1210[nucl-th] (2013).[31] J. E. Amaro, C. Maieron, M. B. Barbaro, J. A. Caballero,and T. W. Donnelly, Phys. Rev. C , 044601 (2010).[32] A. D. Pace, M. Nardi, W. M. Alberico, T. W. Donnelly,and A. Molinari, Nucl.Phys. A , 249 (2004).[33] J. E. Amaro, G. Co, E. M. V. Fasanelli, and A. M.Lallena, Phys. Lett. B , 249 (1992).[34] J. E. Amaro, G. Co, and A. M. Lallena, Nucl. Phys. A , 365 (1994).[35] J. W. V. Orden and T. W. Donnelly, Annals Phys. ,451 (1981).[36] W. M. Alberico, M. Ericson, and A. Molinari, AnnalsPhys. , 356 (1984).[37] M. J. Dekker, P. J. Brussaard, and J. A. Tjon, Phys.Rev. C , 2650 (1994).[38] A. Gil, J. Nieves, and E. Oset, Nucl. Phys. A , 543(1997).[39] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W.Donnelly, and C. Maieron, Phys. Rev. C , 065501(2005).[40] O. Benhar, D. Day, and I. Sick, Rev.Mod.Phys. , 189(2008).[41] M. V. Ivanov, A. N. Antonov, J. A. Caballero, G. D.Megias, et al. , Phys. Rev. C , 014607 (2014).[42] G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Ca-ballero, and T. W. Donnelly, Phys. Lett. B , 170(2013).[43] K. Abe et al. , arXiv:1411.6264 [hep-ex] (2014), [T2KCollaboration].[44] G. D. Megias, M. V. Ivanov, R. Gonz´alez-Jim´enez, J. A.Caballero, M. B. Barbaro, T. W. Donnelly, and J. M.Ud´ıas, Phys. Rev. D , 093002 (2014). -2 -1 0 1 2 3 4 5 6 700.0020.0040.0060.008 R T , VV M E C ( M e V - )
100 200 300400600800 -2 -1 0 1 2 3 4 5 6 700.00050.0010.00150.002 R T , VV M E C ( M e V - ) -2 0 2 4 6 8 Ψ ’0.00010.0010.01 R T , VV M E C ( M e V - ) ω (MeV)00.0010.0020.0030.0040.0050.0060.0070.008 R T M E C ( M e V - ) q=200-2000 MeV/c FIG. 1: (Color online) R MEC,paramT, V V (MeV − ) versus Ψ (cid:48) (firstthree panels) and versus ω (bottom panel), where for the lastone the curves are displayed from left to right in steps of q =200 MeV/c. The parameterized responses are shown asblack lines. Comparisons with Torino results (coloured thicklines) are also displayed. Note that the y-axis in the thirdpanel is shown as a logarithmic scale.
400 600 800 1000 1200 1400 1600 1800 2000-202468 a i ( q ) a a a a
400 600 800 1000 1200 1400 1600 1800 2000q (MeV/c)00.0010.0020.0030.004 a ( q ) , b k ( q ) ( M e V - ) a b b b FIG. 2: (Color online) Dependence on q of the fitting param-eters { a i , b k } . σ ν ( - c m ) MECMEC, q>100 MeV/cMEC, q>250 MeV/cMEC, q>500 MeV/cMEC, q>1000 MeV/cMEC, q>2000MeV/c ν (GeV)00.20.40.60.811.2 σ ν ( - c m ) MECMEC, ω >50 MeVMEC, ω >100 MeVMEC, ω >250 MeVMEC, ω >500 MeVMEC, ω >1000 MeVMEC, ω >2000MeV FIG. 3: (Color online) Total MEC neutrino cross section pertarget nucleon evaluated excluding all contributions comingfrom transferred momentum (upper panel) and energy (lowerpanel) below some selected values, as indicated in the figure. ν (GeV)00.20.40.60.811.2 σ ν ( - c m ) MEC old
MEC new
FIG. 4: (Color online) Comparison between the total MECcross section in the present and past parametrizations. -1 0 1 2 3 4 5 6 7 8 9 Ψ ’00.10.20.30.40.50.60.7 f T , VV M E C ( Ψ ’) FIG. 5: (Color online) Transverse 2p-2h MEC isovector scal-ing functions f MECT,V V versus the scaling variable Ψ (cid:48) from q =400MeV/c to 1500 MeV/c. d σ / d ω d Ω ( nb / M e V / s r) QEMECInelasticQE+MECTotal0 0.1 0.2 0.3 0.4 0.50102030 ε i =560MeV, θ e =36 o ,q QE =322.9MeV/c ε i =680MeV, θ e =36 o ,q QE =402.5MeV/c ω (GeV)0246 d σ / d ω d Ω ( nb / M e V / s r) ω (GeV)0123 ε i =961MeV, θ e =37.5 o ,q QE =585.8MeV/c ε i =3595MeV, θ e =16 o ,q QE =1043MeV/c FIG. 6: (Color online) Comparison of inclusive C( e, e (cid:48) ) crosssections and predictions of the QE(SuSAv2), MEC and In-elastic(SuSAv2) models at different set values of the positionof the QE peak ( q QE ), incident electron energy ( ε i ) and thescattering angle ( θ e ). Data taken from [40]. d σ / d c o s θ µ / d T µ ( - c m / G e V ) θ µ < 1.0 MiniBooNESuSAv2SuSAv2+MECMEC0.8 < cos θ µ < 0.9 θ µ < 0.8 µ (GeV)024681012141618 d σ / d c o s θ µ / d T µ ( - c m / G e V )) θ µ < 0.7 µ (GeV)024681012 θ µ < 0.3 µ (GeV)0246810 -0.2 < cos θ µ < -0.1 FIG. 7: (Color online) Flux-integrated double-differential cross section per target nucleon for the ν µ CCQE process on Cdisplayed versus the µ − kinetic energy T µ for various bins of cos θ µ obtained within the SuSAv2 and SuSAv2+MEC approaches.MEC results are also shown. The data are from Ref. [3]. d σ / d c o s θ µ / d T µ ( - c m / G e V ) θ µ < 1.0 MiniBooNESuSAv2SuSAv2+MECMEC0.8 < cos θ µ < 0.9 θ µ < 0.8 µ (GeV)01234 d σ / d T µ / d c o s θ µ ( - c m / G e V ) θ µ < 0.7 µ (GeV)00.511.52 θ µ < 0.3 µ (GeV)00.10.20.30.40.50.60.70.8 -0.2 < cos θ µ < -0.1 FIG. 8: (Color online) As for Fig. 7, but for ¯ ν µ scattering versus µ + kinetic energy T µ . The data are from Ref. [4]. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1024681012 d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ < 0.3 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10246810121416 MiniBooNESuSAv2SuSAv2+MECMEC 0.4 < T µ < 0.5 -0.2 0 0.2 0.4 0.6 0.8 102468101214161820 µ < 0.6 θ µ d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ < 0.8 θ µ µ < 1.0 θ µ µ < 1.3 FIG. 9: (Color online) Flux-integrated double-differential cross section per target nucleon for the ν µ CCQE process on Cdisplayed versus cos θ µ for various bins of T µ obtained within the SuSAv2 and SuSAv2+MEC approaches. MEC results arealso shown. The data are from Ref. [3]. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 101234 d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ < 0.3 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234567 MiniBooNESuSAv2SuSAv2+MECMECnew0.4 < T µ < 0.5 µ < 0.6 θ µ d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ < 0.8 θ µ µ < 1.0 θ µ µ < 1.3 FIG. 10: (Color online) As for Fig. 9, but for ¯ ν µ scattering versus cos θ µ . The data are from Ref. [4]. E ν (GeV) σ ν ( - c m ) MiniBooNENOMADSuSAv2MECSuSAv2+MEC E ν (GeV) σ ν ( - c m ) FIG. 11: (Color online) CCQE ν µ cross section per nu-cleon displayed versus neutrino energy E ν and evaluated usingthe SuSAv2 and the SuSAv2+MEC approaches (top panel).CCQE ¯ ν µ cross section is also shown (bottom panel). Resultsare compared with the MiniBooNE [3, 4] and NOMAD [8] ex-perimental data. Also presented for reference are the resultsfor the MEC contributions. ν (GeV)0246810121416 σ ν ( - c m ) SuSAv2SuSAv2+MECSuSAv2, T VV SuSAv2, T AA SuSAv2, T’ VA SuSAv2, TSuSAv2, LMEC
FIG. 12: (Color online) Separation into components of theCCQE ν µ cross section per nucleon displayed versus neu-trino energy E ν within the SuSAv2 approach. The MEC andSuSAv2+MEC curves are shown. The MiniBooNE [3] andNOMAD [8] data are also shown for reference. ν (GeV)0246810 σ ν - σ ν ( - c m ) SuSAv2 ( ν µ −ν µ )SuSAv2+MEC ( ν µ −ν µ )MiniBooNE ( ν µ −ν µ ) FIG. 13: (Color online) Experimental difference between neu-trino and antineutrino cross sections ( σ ν µ − σ ¯ ν µ ) from Mini-BooNE, together with the corresponding theoretical predic-tion from SuSAv2+MEC, whose difference with the SuSAv2prediction is negligible. d σ / d c o s θ µ / d T µ ( - c m / G e V ) MiniBooNESuSAv2SuSAv2+MECMECSuSAv2, LSuSAv2, TSuSAv2, T VV SuSAv2, T AA SuSAv2, T’ VA θ µ < 0.9 θ µ < 0.2 µ (GeV)0123456789 d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ (GeV)00.511.522.533.5 FIG. 14: (Color online) Separation into components of the MiniBooNE CCQE ν µ (top panel) and ¯ ν µ (botoom panel) double-differential cross section per nucleon displayed versus T µ for various bins of cos θ µ within the SuSAv2 approach. The MEC andSuSAv2+MEC curves are shown. The MiniBooNE [3, 4] data are also shown for reference. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1024681012 d σ / d c o s θ µ / d T µ ( - c m / G e V ) µ < 0.3 SuSAv2SuSAv2+MECMECSuSAv2, LSuSAv2, TSuSAv2, T VV SuSAv2, T AA SuSAv2, T’ VA µ < 0.8 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1cos θ µ d σ / d c o s θ µ / d T µ ( - c m / G e V ) θ µ FIG. 15: (Color online) Separation into components of the MiniBooNE CCQE ν µ (top panel) and ¯ ν µ (botoom panel) double-differential cross section per nucleon displayed versus cos θ µ for various bins of T µ within the SuSAv2 approach. The MEC andSuSAv2+MEC curves are shown. The MiniBooNE [3, 4] data are also shown for reference. E ν (GeV) σ ν ( - c m ) MiniBooNET2KNOMADSuSAv2MECSuSAv2+MEC