Off-shell effects in the relativistic mean field model and their role in CC (anti)neutrino scattering at MiniBooNE kinematics
M.V. Ivanov, R. Gonzalez-Jimenez, J.A. Caballero, M.B. Barbaro, T.W. Donnelly, J.M. Udias
aa r X i v : . [ nu c l - t h ] O c t Off-shell effects in the relativistic mean field model andtheir role in CC (anti)neutrino scattering at MiniBooNEkinematics
M. V. Ivanov a,b, ∗ , R. Gonz´alez-Jim´enez c , J. A. Caballero c , M.B. Barbaro d ,T. W. Donnelly e , J.M. Ud´ıas a a Grupo de F´ısica Nuclear, Departamento de F´ısica At´omica, Molecular y Nuclear,Facultad de Ciencias F´ısicas, Universidad Complutense de Madrid,CEI Moncloa, Madrid E-28040, Spain b Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria c Departamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, 41080Sevilla, Spain d Dipartimento di Fisica, Universit`a di Torino and INFN, Sezione di Torino, Via P.Giuria 1, 10125 Torino, Italy e Center for Theoretical Physics, Laboratory for Nuclear Science and Department ofPhysics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abstract
The relativistic mean field (RMF) model is used to describe nucleons inthe nucleus and thereby to evaluate the effects of having dynamically off-shell spinors. Compared with free, on-shell nucleons as employed in someother models, within the RMF nucleons are described by relativistic spinorswith strongly enhanced lower components. In this work it is seen that forMiniBooNE kinematics, neutrino charged-current quasielastic cross sectionsshow some sensitivity to these off-shell effects, while for the antineutrino-nucleus case the total cross sections are seen to be essentially independentof the enhancement of the lower components. As was found to be the casewhen comparing the RMF results with the neutrino-nucleus data, the presentimpulse approximation predictions within the RMF also fall short of theMiniBooNE antineutrino-nucleus data.
Keywords: neutrino reactions, off-shell effects, nuclear effects
PACS: ∗ Corresponding author.
Preprint submitted to Nuclear Physics B August 31, 2018 . Introduction
An increased interest in neutrino interactions in the few GeV energyrange has emerged from the recent cross section measurements taken at dif-ferent laboratories. In particular, the MiniBooNE data on charged-currentquasielastic (CCQE) [1, 2] and neutral-current quasielastic (NCQE) [3] ν µ and ν µ scattering off C, with mean beam energy h E ν i = 788 and h E ν i =665 MeV, respectively, have stimulated important discussions about the roleplayed by both nuclear and nucleonic ingredients in the description of thereaction. To characterize CCQE (and similarly NCQE) neutrino scatteringfrom carbon, the MiniBooNE collaboration made use of the Relativistic FermiGas (RFG) model in Monte Carlo simulations. Indeed, the MiniBooNE crosssection is underestimated by the RFG unless the axial mass M A is signifi-cantly enlarged (1 .
35 GeV/c [1]) with respect to the world average value(1 .
03 GeV/c [4]) extracted from neutrino and antineutrino scattering dataoff the deuteron. However, previous data from the NOMAD collaboration [5]for higher beam energies (from 3 to 100 GeV) are in good agreement with thestandard value of the axial mass, and recent data from the MINER ν A collab-oration [6], corresponding to (anti)neutrino energies from 1 . M A = 1 .
35 GeV/c . Furthermore, as emergesfrom comparisons with electron scattering data, the RFG is too simplisticto account for the nuclear dynamics, and in particular it fails badly to re-produce the separated longitudinal and transverse response functions, whichis essential to make reliable predictions for neutrino scattering, where thebalance between the two channels is different from the electron scatteringcase. A larger axial mass within the RFG should simply be interpreted as acrude way effectively to incorporate nuclear effects.In addition to the RFG, other more sophisticated models based on theImpulse Approximation (IA) also underpredict the CCQE cross section mea-sured at MiniBooNE [7–11]. As well the phenomenological SuSA model de-scribed in [12], based on the superscaling function extracted from quasielasticelectron scattering cross sections [13], predicts neutrino cross sections whichare found to be lower than the MiniBooNE data [14–17].Different explanations have been proposed, based either on multi-nucleonknockout [8, 14–16, 18–21] or on particular treatments of final-state interac-tions through phenomenological optical potentials [22–24]. Although there is2eneral agreement that multi-nucleon effects produce a significant enhance-ment of the cross section, at a quantitative level the theoretical uncertaintiesrelated to the description of nuclear effects are rather large, as substantiallydifferent approximations are involved in each of the above-mentioned ap-proaches.The accurate interpretation of present experiments depends on the un-derstanding of all ingredients of the theory. Among them one would need toaddress the issue of nucleons being bound in the nuclei that form the nucleartargets and thus, necessarily, the cross sections have to be computed for off-shell nucleons. Indeed, within the IA the cross section is described as a sumof lepton-nucleon vertices, where the nucleons are bound and thus off-shell.The neutrino-nucleus reaction at intermediate energies, as is the case of theMiniBooNE experiment, would show sensitivity to off-shell effects [25]. Atintermediate or low energies, lepton-nucleon reactions are often describedwith models of the lepton-nucleon interaction [7, 8, 19–21] that incorporaterelativistic effects into the kinematics and in some cases also into the dynam-ics. In the SuSA approach with Meson Exchange Currents (MEC) of [14–16],while both the kinematics and the one- and two-body current operators arefully relativistic, only positive energy on-shell spinors are taken into account.Moreover, most non-relativistically inspired approaches implicitly assumeon-shell nucleons, that is, only positive energy spinors are involved in themodeling [26]. Thus they are less suitable for studying the influence of off-shell effects. Work has been done [27] in a relativistic context where off-shellness in the initial-state bound nucleons was the main focus, and ledto discussions of the break-down of factorization in ( e, e ′ p ) reactions. Inthis work, working at the mean field level using one-body effective operators( i.e., within the IA) we study the effects of off-shellness in both initial andfinal states within the context of the Relativistic Mean Field (RMF) model.The RMF has been successfully employed to describe electron-nucleus andneutrino-nucleus experiments [15, 28, 29] and the opportunity presented withthe availability of both neutrino and antineutrino data can shed light on off-shell effects, as we shall illustrate in this letter.It is worth noting that even if the RMF is a one-body model, that is, pro-cesses containing other particles in addition to the nucleon in the final state –including multi-nucleon knockout and pion production – are not explicitly in-corporated in this formalism, the one-body contribution from multi-nucleonknockout is to some extent incorporated into the model via the self-energyof the propagating nucleon. The RMF approach at mean field level includes3ll types of rescattering processes (elastic and inelastic) with the remainingnucleons. Here the redistribution of the strength and multi-nucleon knock-out are attributed to final-state interactions and not to explicit correlations.Notice that the RMF model provides the correct saturation properties fornuclear matter already at the mean field level [30] stemming from the com-bination of the strong scalar ( S ) and vector ( V ) potentials that incorporaterepulsive and attractive interactions. Further, within the RMF, initial andfinal nucleon wave functions are computed with the same mean field equa-tion and potentials, thus the current computed from these spinors fulfills thecontinuity equation.Within the RMF, the presence of strong S <
V > ψ down ( p ) = σ · p E + M N + S − V ψ up ( p ) . (1)That is, the nucleons are dynamically and strongly off-shell. This strong off-shellness is the main cause for the lack of exact factorization of the results,even at the IA level. Thus RMF is not factorized into a spectral functionand an elementary lepton-nucleus cross sections, as it is done at times in de-scribing these reactions [9, 10]. Factorization break-down is however not verystrong, as the results of the SuSA approach (that obviously is a factorizedscheme) do not depart much from the RMF predictions [12, 14–16, 33].In order to assess the influence of this strong off-shellness, also denotedin the past as spinor distortion [34], the fully relativistic results can be com-pared with the effective momentum approach (EMA) [26, 31, 34, 35]. WithinEMA, the spinors are put exactly on the mass shell, by enforcing the samerelationship between upper and lower components as for free spinors. EMAspinors lack the dynamical enhancement of the lower components due to thepresence of strong potentials. Lacking this spinor distortion, the EMA resultsshould lie closer to the so-called factorized approach [26, 31]. The comparisonbetween EMA and RMF is interesting because it allows one to estimate towhat extent dynamical off-shell effects may affect neutrino-nucleus observ-ables such as those involved in the analysis of MiniBooNE data — and thisin the context of a model that has been validated against inclusive electronscattering data in the quasielastic region at intermediate energies.4 . Results and discussion Before entering into a detailed study of results, it is helpful to keep inmind the following general properties. For CC inclusive neutrino-nucleusscattering, only the L , T and T ′ contributions to the cross section survive [29,36–43]: d σdε f d cos θ = d σ L dε f d cos θ + d σ T dε f d cos θ + h d σ T ′ dε f d cos θ (2)where h denotes the helicity of the incident lepton ( h = − h = +1 for antineutrinos), ε f and θ represent the energy and scatteringangle of the outgoing lepton. We describe the bound nucleon states as self-consistent Dirac-Hartree solutions, derived within a relativistic mean-fieldapproach using a Lagrangian containing σ , ω and ρ mesons [30, 44]. Theelectroweak current operators are the same as in recent work [12, 41, 45] andin a model-dependent way account for some aspects of off-shellness, namely“kinematical” off-shellness rather than our focus in the present work which is“dynamical” off-shellness stemming from the bound and continuum nucleon’sbeing off-shell with non-trivial lower components (see [46] for more discussionof kinematical off-shellness).Based on the use of the CC2 current (considered in this work) the L con-tribution is rather insensitive to off-shell effects, which can be traced backto the fact that within the RMF the matrix elements of the CC2 chargecurrent fulfill the continuity equation already at the one-body level. Actu-ally, within the RMF (and also under some other more general conditions,see [26, 27, 47]), the L contribution shows no sensitivity to dynamical off-shellness. However this is, at the kinematics of MiniBoone, a relatively smallcontribution for the neutrino case. The T and T ′ contributions are the dom-inant components of the cross section, and they exhibit a similar effect ofoff-shellness: off-shell effects tend to increase (in absolute magnitude) both T and T ′ contributions. Further the T contribution is the same for neutrinoand antineutrino, while the T ′ changes sign. As a consequence, while forneutrinos off-shell effects in the T and T ′ contributions (which add) are re-inforced and a net visible dependence of the off-shellness is seen in the totalcross section, for antineutrinos such effects are nearly perfectly canceled inthe cross section at MiniBooNE kinematics, since T and T ′ contributionstend to cancel. 5ith this guidance, it is easy to understand the results illustrated inFigs. 1–7. For instance in Fig. 1 we show the differential cross section pertarget nucleon for the (anti)neutrino CCQE process on C as a function ofmuon kinetic energy T µ . The incident (anti)neutrino energy is assumed tobe 1 GeV. In the figure results are given for the transverse ( T ), longitudinal( L ), and axial-transverse ( | T ′ | ) contributions using the EMA (dashed lines,black) approach and the RMF (solid lines, red) model, respectively.In Fig. 2 we present the flux-integrated double-differential cross sectionper target nucleon for the ν µ CCQE process on C. We display the crosssection, evaluated with the RMF model and EMA approach, versus the µ − kinetic energy T µ for two bins of cos θ µ (forward angles – top panel and back-ward angles – bottom panel of Fig. 2). Also, in Fig. 2 are shown the separatecontributions of longitudinal ( σ L ) and transverse ( σ T and σ T ′ ) componentscalculated within the RMF and EMA approaches. Here and in the followingfigures the results are compared to the MiniBooNE experimental data [1, 2].As shown, RMF and EMA results for the cross sections lie very close to-gether, so the effects linked to the enhancement of the lower componentsdue to the strong relativistic potentials are small. This conclusion also holdsfor antineutrino double-differential cross sections (Fig. 3) as well as for dif-ferential and total unfolded integrated neutrino/antineutrino cross sections(Figs. 6 and 7). Notice also the minor role played by the longitudinal com-ponent for angles in the range 0 ≦ θ µ ≦
45 degrees, this contribution beingalmost negligible for larger angles (as can be seen at backward angles – bot-tom panel of Fig. 2). On the contrary, as can be seen in Fig. 2, most of theneutrino CCQE cross section comes from the pure transverse contributiongiven by the sum σ T + σ T ′ . The T ′ contribution increases with the muonscattering angle θ µ : its contribution at forward angles (cos θ µ ∼
1) is close to L one, whereas at backward angles (cos θ µ ∼ −
1) it is almost equal to the T contribution.In Fig. 3 we present our predictions for the flux-averaged antineutrinoCCQE cross sections corresponding to the MiniBooNE experiment [2]. Here,in contrast with the neutrino case, the longitudinal contribution to the crosssection plays a significant role, increasing its strength as the muon scatteringangle θ µ goes up (as can be seen at backward angles – bottom panel of Fig. 3).This result can be understood from the destructive interference occurringbetween the two transverse responses, T and T ′ . Note that in the case ofantineutrinos the global transverse contribution to the cross section is giventhrough the difference σ T − σ T ′ . On the contrary, neutrino reactions involve6 constructive interference of both transverse responses. It is important topoint out that T and T ′ contributions are much larger than L ; however,they tend to cancel for antineutrinos, hence explaining the relatively moresignificant role played by the longitudinal component in this case.The difference between EMA and RMF results for T and T ′ is very sim-ilar. In the neutrino case we see in the total cross section mostly the samecomparison of EMA to RMF as for the separate T and T ′ responses, namelyabout a few percent difference. However, for antineutrinos the effect of RMFversus EMA in T and T ′ responses, due to the change of sign, is cancelledto a large extent, at the same level as the T and T ′ responses are cancelledout causing the L response to dominate the total cross section. Thus, forantineutrinos, for the cases where the total response is relatively small, thereis no effect or difference between EMA and RMF results (Figs. 3 and 5).In Fig. 4 (Fig. 5) we present the flux-integrated double-differential crosssection per target nucleon for the ν µ ( ν µ ) CCQE process on C for variousbins of cos θ µ . As discussed above, RMF slightly exceeds EMA results forneutrino scattering due to the sum of T and T ′ contributions which for allangles are bigger than EMA ones. For antineutrino CCQE process on C,results are almost identical within two approaches, only at large backwardangles there are small differences. As can be seen from Figs. 4 and 5, theo-retical predictions clearly underestimate the experimental cross sections forneutrinos and antineutrinos. This result is consistent with the additionalstrength, not included in our model, that may come from two-body currentsand multi-nucleon processes. While the RMF approach may account for someeffects linked to two-body contributions, there would certainly be additionalcontributions beyond the IA lacking in this model.In Fig. 6 (Fig. 7) results are presented for the MiniBooNE flux-averagedCCQE ν µ ( ν µ )- C differential cross section per nucleon as a function of themuon scattering angle (left-top panel, note that in order to compare withdata the integration is performed over the muon kinetic energies 0 . 3. Conclusions We have studied off-shell effects within a fully relativistic approach, theRelativistic Mean Field model, which displays strong off-shell, non-factorizingbehaviour. We can summarize our findings as follows:1) Most theoretical approaches to CCQE neutrino scattering are basedon factorization assumptions, or at least use on-shell, or almost on-shell,spinors to describe the nucleons. On the other hand the RMF model usesoff-shell spinors, with strongly enhanced lower components. In this workwe have studied the effect of this enhancement of the lower components for(anti)neutrino CCQE results. We have seen how these off-shell effects arevisible, although being relatively small, in the total cross section for neutrino-nucleus scattering while, for MiniBooNE kinematics they are negligible forthe antineutrino cross section. The effect of off-shell spinor distortion in theRMF cross-sections for neutrinos and antineutrinos can be compared withother ingredients considered in alternative approaches. For instance, in thecase of [14, 16], pionic MEC effects were studied and they were assumedto modify just the T response, which is enhanced, whereas the T ′ T − T ′ cancellation is less severe. On the contrary, Martini et al. [19] find a somewhatminor role of the 2p2h mechanisms for the antineutrino case. Finally, Nieves et al. [11] get similar relative multi-nucleon contributions for neutrinos andantineutrinos. Although the conclusions about off-shell effects leading tospinor distortion considered here are based on a specific model, one has torecall that it is always the case that off-shell effects can only be studied withina model. However, from what we see here one can be reasonably confidentthat for MiniBooNE kinematics, dynamical off-shellness leading to increasedlower components in the nucleon spinors would be a rather small effect forneutrino-nucleus scattering, and fully negligible for the antineutrino-nucleuscase. We have verified that the dynamical off-shell effects considered in thiswork affect the different contributions to the cross-section in a similar wayas found in this work, for higher (anti-)neutrino energies, upto 100 GeV.This is due to the fact that for higher projectile energies, the cross-section8s more and more forward peaked and then the momentum transfer is keptrelatively small, no matter how large is the incoming lepton energy. The onlything to keep in mind is that the almost complete insensitivity to off-shelleffects for the anti-neutrino case shown at MiniBooNE energies, dependson the cancellation of two contributions whose relative weight depends onthe kinematics. The cancellation of T and T ′ contributions breaks above2 GeV of incoming lepton energy. Actually, the T ′ cross section becomesnegligible at very high energies, so the sensitivity of the cross-sections to theenhancement of the lower components of the nucleon spinors would be similarfor neutrino and antineutrino for neutrino energies of several GeV and above.In this work we have shown that the dependence of the neutrino and anti-neutrino cross-section to spinor distortion ambiguity is relatively small. Thisambiguity would be hidden or remain unnoticed when using non-relativistic(in structure) models, but it should be kept in mind when trying to deriveneutrino properties from experiments.2) Neutrino MiniBooNE cross sections cannot be empirically fitted withinseveral IA approaches: RFG, realistic Spectral Function approach, Super-Scaling-Approximation, and RMF. An ad hoc enhancement of the axial mass,or what is the same, enhanced contribution from the axial term is needed forthese models to explain the data. We have shown that this remains the casefor antineutrino scattering. Acknowledgements This work was partially supported by Spanish DGI and FEDER funds(FIS2011-28738-C02-01, FPA2010-17142), by the Junta de Andalucia, by theSpanish Consolider-Ingenio 2000 program CPAN (CSD2007-00042), by theCampus of Excellence International (CEI) of Moncloa project (Madrid) andAndalucia Tech, by the Istituto Nazionale di Fisica Nucleare under ContractMB31, by the INFN-MICINN collaboration agreement (AIC-D-2011-0704),as well as by the Bulgarian National Science Fund under contracts No. DO-02-285 and DID-02/16-17.12.2009. M.V.I. is grateful for the warm hospitalitygiven by the UCM and for financial support during his stay there from theSiNuRSE action within the ENSAR european project. R.G.J. acknowledgessupport from the Ministerio de Educaci´on (Spain) and T.W.D acknowledgessupport from the US Department of Energy under cooperative agreementDE-FC02-94ER40818. 9 eferences [1] A. A. Aguilar-Arevalo et al. [MiniBooNE Collaboration], Phys. Rev. D81 (2010) 092005.[2] A. A. 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A 643 (1998) 189. 13 d s / d T m [ - c m / G e V ] T m [MeV] E n ( ‘ n ) = 1 GeV; M A = 1.03 GeVTotal, n Total, ‘ n |T’|TLRMFEMA Figure 1: (Color online) The differential cross section per target nucleon for the(anti)neutrino CCQE process on C as a function of muon kinetic energy T µ , as wellas transverse ( T ), longitudinal ( L ), and transverse-transverse ( | T ′ | ) contributions usingthe EMA approach (dashed, black online) and the RMF model (solid, red online). d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] 0.7 < cos q m < 0.8 n , M A = 1.03 GeV RMFEMARMF, LEMA, LRMF, TEMA, TRMF, T’EMA, T’ d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] -0.3 < cos q m < -0.2 RMFEMARMF, LEMA, LRMF, TEMA, TRMF, T’EMA, T’ Figure 2: (Color online) Flux-integrated double-differential cross section per target nucleonfor the ν µ CCQE process on C displayed versus the µ − kinetic energy T µ for two bins ofcos θ µ (forward angles – top panel and backward angles – bottom angles) obtained withinthe RMF model (solid thick line), EMA approach (solid thin line) and contribution oflongitudinal ( σ L , dash-dotted line) and transverse ( σ T – dotted line and σ T ′ – dash-dot-dot line) components within RMF and EMA models. The data are from [1]. d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] 0.7 < cos q m < 0.8 ‘ n , M A = 1.03 GeV RMFEMARMF, LEMA, LRMF, TEMA, TRMF, |T’|EMA, |T’| d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] -0.3 < cos q m < -0.2 RMFEMARMF, LEMA, LRMF, TEMA, TRMF, |T’|EMA, |T’| Figure 3: (Color online) As for Fig. 2, but for ν µ scattering versus µ + kinetic energy T µ : RMF model (solid thick line), EMA approach (solid thin line) and contribution oflongitudinal ( σ L , dash-dotted line) and transverse ( σ T – dotted line and σ T ′ – dash-dot-dot line) components within RMF and EMA models. The data are from [2]. d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] 0.8 < cos q m < 0.9 d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] 0.5 < cos q m < 0.6 d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] 0.2 < cos q m < 0.3 RMFEMAMiniBooNE 0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] -0.1 < cos q m < 0.0 0 1 2 3 4 5 6 7 8 9 0 100 200 300 400 500 d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] -0.4 < cos q m < -0.3 0 1 2 3 4 5 6 0 50 100 150 200 250 300 350 400 d s / d c o s q / d T m [ - c m / G e V ] T m [MeV] -0.7 < cos q m < -0.6 Figure 4: (Color online) Flux-integrated double-differential cross section per target nucleonfor the ν µ CCQE process on C displayed versus the µ − kinetic energy T µ for variousbins of cos θ µ obtained within the RMF model and EMA approach for M A = 1 .