Electron-neutrino scattering off nuclei from two different theoretical perspectives

We analyze charged-current electron-neutrino cross sections on Carbon. We consider two different theoretical approaches, on one hand the Continuum Random Phase Approximation (CRPA) which allows a description of giant resonances and quasielastic excitations, on the other hand the RPA-based calculations which are able to describe multinucleon emission and coherent and incoherent pion production as well as quasielastic excitations. We compare the two approaches in the genuine quasielastic channel, and find a satisfactory agreement between them at large energies while at low energies the collective giant resonances show up only in the CRPA approach. We also compare electron-neutrino cross sections with the corresponding muon-neutrino ones in order to investigate the impact of the different charged-lepton masses. Finally, restricting to the RPA-based approach we compare the sum of quasielastic, multinucleon emission, coherent and incoherent one-pion production cross sections (folded with the electron-neutrino T2K flux) with the charged-current inclusive electron-neutrino differential cross sections on Carbon measured by T2K. We find a good agreement with the data. The multinucleon component is needed in order to reproduce the T2K electron-neutrino inclusive cross sections.

The wealth of experimental and theoretical results on muon-neutrino cross sections contrasts with the few published results on electron-neutrino cross sections. After the inclusive ν e CC total cross sections measured by the Gargamelle bubble chamber in 1978 [36], the first measurement of inclusive ν e CC differential cross sections on Carbon was performed by T2K [37]. Recently the measurement performed by MINERvA of quasielastic and quasielasticlike differential cross sections on Carbon also appeared [38]. A precise knowledge of ν µ and ν e cross sections is important in connection to the ν µ → ν e oscillation experiments which aim at the determination of the neutrino mass hierarchy and the search for CP violation in the lepton sector. A theoretical comparison of the ν µ and ν e cross sections was performed by Day and McFarland [39] who analyzed the influence of the final lepton-mass difference on the cross sections as a function of the neutrino energy and of Q 2 . Here we study these differences focusing on the ν µ and ν e differential cross sections. In a first part we consider the electron-neutrino cross sections on Carbon using two different theoretical models. The first one is the one of Martini et al. [20] which is based on nuclear response functions, treated in the random phase approximation (RPA) on top of a local relativistic Fermi gas (LRFG) calculation. It includes the quasielastic cross section, multinucleon emission and coherent and incoherent single pion production. The second model is the one of Jachowicz et al. [40] which is based on the continuum random phase approximation (CRPA) on top of Hartree-Fock (HF) calculations. It was originally developed to study electroweak reactions in the giant resonance region and then extended by Pandey et al. [41,42] to the quasielastic regime. The common channel where the two approaches can be compared is hence the quasielastic one. After a description of the two theoretical models, we confront their results in the quasielastic channel, first for fixed kinematics. We also illustrate in both models the differences between ν µ and ν e cross sections. The difference between the two approaches manifests itself for small values of the neutrino energy (E ν ≃ 200 MeV). However when the cross sections are folded with the T2K and the MiniBooNE ν e fluxes which are peaked at E ν ≃ 500 MeV it turns out that the differences are washed out. This justifies the use of the RPA approach of Martini et al., which incorporates, beyond the quasielastic, one pion production and multinucleon emission to compare with the inclusive ν e CC differential cross sections on Carbon recently measured by T2K [37]. We postpone the comparison with the very recent MINERvA results [38] to a future paper.

II. THEORETICAL MODELS
We summarize here the basic ingredients of the two models. Both approaches calculate the polarization propagator Π in the random phase approximation (RPA) which allows the inclusion of long-range nucleon-nucleon correlations. This amounts to solving integral equations which have the generic form where Π 0 is the bare polarization propagator and V denotes the effective particle-hole interaction. However, the bare polarization propagator and the residual interaction differ in the two approaches.
For Martini et al. [20] the bare polarization propagator is evaluated in momentum space.
In a finite system it is non-diagonal and writes Π 0 (ω, q, q ′ ). In order to account for the finite-size effects, it is evaluated in a semi-classical approximation [43,44] where it can be cast in the form This semiclassical calculation is a Wigner-Kirwood expansion in restricted to first order.
Its validity has been explored by Stroth et al. [45] and by Chanfray and Schuck [43] with the conclusion that the semiclassical description agrees with a quantum mechanical calculation but for small momenta (q < 200 MeV/c for light nuclei). The non-diagonality in the momenta of Π 0 (ω, q, q ′ ) shows that this description goes beyond the infinite nuclear matter one. In order to obtain the quantity in the integrand of Eq. (2), a local density approximation is used which relates it to the relativistic Fermi gas polarization propagator according to The local Fermi momentum k F (r) is related to the experimental nuclear density through : k F (r) = (3/2 π 2 ρ(r)) 1/3 . The density profile of 12 C is taken from the Sum-of-Gaussians nuclear charge density distribution parameters according to Ref. [46]. As for the collective RPA effects, the microscopic approach first introduced by Delorme et al. [47] is used. The method is described in detail in Ref. [20]. In the bubble RPA chain for the spin-isospin channel the possibility of ∆-hole excitation is taken into account.
In the approach of Jachowicz et al. [40], the starting point is the continuum Hartree-Fock model which evaluates the bound and the continuum single-nucleon wave functions through the solution of the Schrödinger equation with a mean field potential. The bare polarization propagator, in this case the HF one, is then calculated in coordinate space. According to Ref. [42], the maximum multipole number to calculate the multipole operators is set to J = 16 which turns out to be sufficient to reach the convergence of the cross sections.
For the Landau-Migdal parameter f ′ the value f ′ = 0.6 is taken. As for the spin-isospin parameters g ′ the following values are used: g ′ N N = 0.7, g ′ N ∆ = g ′ ∆∆ = 0.5. In the Jachowicz et al. model, the zero-range Skyrme effective interaction, in its parametrization called SkE2 [48,49], is used. The same interaction which enters in the mean field calculation is employed to generate the continuum RPA (CRPA) solution. In this way, the calculation becomes self-consistent with respect to the interaction. In Ref. [42] this residual Skyrme-type interaction is multiplied by a dipole form factor which controls the influence of the residual interaction at high Q 2 values [50]. At variance with respect to Ref. [42], here the CRPA results are not folded with a Lorentz function which would spread the strength of giant-resonances without modification of the position of the quasielastic peak.
Concerning the RPA differences, an important point should be mentioned. The possibility of ∆-hole excitation in the RPA chain is included explicitly in the case of Martini et al. This is reflected in the appearance in certain kinematical regions of a sizable quenching in the RPA results, due to the mixing of nucleon-hole states with ∆-hole ones, the Ericson-Ericson-Lorentz-Lorenz (EELL) effect [51]. This quenching has been introduced and established in pion scattering [51]. It has been discussed also in relation with electron [52] and neutrino [20,25,53] scattering.

III. COMPARISON BETWEEN THEORETICAL CALCULATIONS
A. LRFG+RPA vs HF+CRPA In this subsection, we compare the theoretical results in the one nucleon-one hole sector obtained in the two different approaches. We consider the ν e -12 C double differential cross sections for different values of the neutrino energy and lepton scattering angle. These cross sections are purely theoretical quantities since the experimental ones depend on the neutrino fluxes and hence are specific for each experiment. In Fig.1 we display the results of the two approaches by switching on and off the residual particle-hole interaction. We keep the same notations as in the previous papers of the groups. Namely we call "bare-LRFG" the results of Martini et al. when the particle-hole interaction is switched off (these are relativistic Fermi gas results in the local density approximation) and "RPA" the results obtained by switching   tail is more important. As discussed in Ref. [45] this is a consequence of the non-locality of the mean field which has a similar effect as the residual transverse repulsive ph interaction: it quenches and hardens the responses.
Turning to RPA effects, the important difference is the large RPA quenching in the As it clearly appears, at low neutrino energy and forward scattering angles, the cross section, with its giant resonance structures is dominated by the 1 − , 2 − and 2 + multipole contributions. On the contrary at larger energies and scattering angles the contribution of the various multipoles is more evenly distributed, as discussed in Ref. [40]. One can appreciate this point by noticing that the first three major multipole contributions are of the same order of magnitude and are not dominant, hence one needs to include all multipole contributions up to J = 16 with natural and unnatural parity.
Turning to the flux folded cross sections we consider the T2K [54] and MiniBooNE [55] ν e normalized fluxes, which are shown in Fig. 4. We discuss single-differential quasielastic cross sections, dσ dpe and dσ d cos θe , their theoretical evaluation is displayed in Fig. 5. One observes that the giant resonance effects are no longer apparent in the CRPA and that in general the differences between HF and CRPA calculations are largely washed out by the flux folding.  After the discussion of the differences between the two approaches for the ν e case, we turn to a comparison between the charged current ν e and ν µ cross sections. We perform this comparison for different final state channels. For the sake of illustration, we present in Fig. 6 the double differential cross sections in different channels (genuine quasielastic, multinucleon excitations and one pion production) for fixed values of the scattering angle (60 degrees) and neutrino energy (E ν =500 MeV) as a function of the lepton kinetic energy T l , a measurable quantity. The role of the different charged lepton masses appears not only in the trivial relative shift between the ν e and ν µ CC cross sections, according to the identities     but also in the strength and in the shape of the cross sections, particularly in the pion production channel. In order to eliminate trivial mass shift effects we plot in the following figures (Figs. 7, 9 and 11) the differential cross sections as a function of the energy transfer We start with the CRPA case which allows a simultaneous treatment of giant resonances and quasielastic excitations. In Fig. 7, we display the double-differential cross sections for different values of incoming neutrino energy and lepton scattering angle, both for ν e and ν µ .
In most cases the ν e and ν µ results are quite similar, sometimes practically indistinguishable.
However, in some cases interesting differences appear. The first one is a consequence of the stringent limit on the maximum transferred energy ω max = E ν − m lepton which has smaller values in the muon case. This threshold effect can be observed in Fig. 7 for E ν =150 MeV and for E ν =200 MeV in the case of 60 degrees. Other differences can be appreciated by observing the evolution with the scattering angle of the cross sections at small neutrino energies such as E ν =150 MeV or E ν =200 MeV. For small scattering angles such as 5 degrees, ν µ cross other. This angular behavior weakly survives at E ν =500 MeV while for E ν =750 MeV the ν e and ν µ cross sections practically coincide for all the scattering angles. The differences discussed above are also related to the differences in the momentum transfer between the ν e and ν µ scattering. For completeness we show in Fig. 8 the values of the momentum transfer spanned in the 16 panels of Fig. 7. As expected, the major differences between ν e and ν µ appear at small neutrino energies where threshold effects are more evident. Furthermore at small scattering angle the momentum transfer is always larger for ν µ than for ν e while at 60 degrees this is not always the case.
It is also interesting to illustrate the behavior of ν e and ν µ cross sections by separating their contributions, as shown in Fig.9 for incoming neutrino energies of E ν =200 MeV and E ν =750 MeV. According to the notation of Ref. [41], the global contribution related to the Coulomb and longitudinal multipole excitation operators (containing vector and axial com-ponents) is labeled as CL. In the language of Refs. [20,21] it represents the sum of isovector and isospin spin-longitudinal response contributions. The sum of transverse contributions, including the vector-axial interference term, is labeled as T. These are the terms containing the isospin spin-transverse response in the language of Refs. [20,21]. As one can observe in Fig.9, for E ν =200 MeV and θ=5 degrees (and in general for very forward scattering) the neutrino cross section is dominated by the CL contribution while for larger angles, such as 60 degrees, the transverse contribution T is dominant. At larger energies the transverse part dominates everywhere except for very small scattering angles. At E ν =200 MeV and θ = 5 degrees the dominant CL contribution to the cross sections, as well as the smaller T one, are larger for ν µ than for ν e , hence the larger ν µ cross sections for this case. The relative weight of CL and T contributions is the result of a subtle interplay between lepton kinematic factors and response functions. The competition for dominance of the cross section between both, is very sensitive to energy and momentum transfer. The surprising dominance of ν µ over ν e cross sections for small scattering angles is related to this and dictated by the nontrivial dependence of momentum transfer on lepton mass and scattering angle for forward scattering, as illustrated in Fig. 8.
The non-trivial behaviour of the ν e cross sections with respect to the ν µ ones is also illustrated in Fig. 10 where the ratio of the single differential cross section dσν e d cos θ / dσν µ d cos θ is shown for E ν =200 MeV and E ν =750 MeV. In the low energy case of E ν =200 MeV where the semiclassical description breaks down, we restrict ourself to the CRPA approach. In this case the ratio deviates very appreciably from 1 while at larger neutrino energies, such as E ν =750 MeV, it gets closer to 1 in the CRPA as well as in the RPA case with or without the EELL effect. In Fig. 10 this quantity dσν e d cos θ / dσν µ d cos θ at E ν =750 MeV is given also for two other channels, the pion production and multinucleon excitations in the RPA approach of Martini et al. where they are available. This dσν e d cos θ / dσν µ d cos θ ratio, always larger than 1, is characterized by a smooth decreasing behavior. For the pion emission channel (via ∆ excitation) this ratio is larger than the one for the np-nh and 1p-1h excitations.
Concerning the pion production and multinucleon excitations, we display for completeness in Fig. 11 the ν e and ν µ results obtained for these channels (as well as for the QE one) in the RPA approach of Martini et al. for the double differential cross sections at incoming neutrino energies of E ν = 500 MeV and E ν = 750 MeV and scattering angles of 30 and 60 degrees. One observes the clear energy separation between the three channels, the highest  the scattering angle increases. This behavior with the scattering angle appears also in the previous Fig. 10.

IV. COMPARISON WITH THE T2K ν e INCLUSIVE CROSS SECTIONS
The T2K collaboration published the first results for ν e charged-current inclusive differential cross sections on Carbon [37]. In this section we compare these experimental results with our predictions, restricting to the Martini et al. [20] RPA approach which describes all final channels. We compute the ν e T2K flux integrated differential cross sections dσ dpe and dσ d cos θe in the different excitation channels, namely quasielastic, multinucleon excitations (npnh) and one-pion (coherent and incoherent) production. Ericson [31] in connection with the T2K inclusive ν µ double differential cross sections [9].
This agreement with both ν µ and ν e CC inclusive T2K flux folded differential cross sections is not systematically obtained in other approaches. For instance the SuSAv2 model by Ivanov et al. [56] reproduces well the CC inclusive T2K flux folded ν µ double differential cross section but underestimates the CC inclusive T2K flux folded ν e single differential cross section. A comparison with these quantities has also been performed by Meucci and Giusti using the Relativistic Green's function model which turned to underestimate the ν µ and ν e CC inclusive T2K data [57]. The experimental T2K points are taken from Ref. [37].

V. SUMMARY AND CONCLUSIONS
In conclusion, our study has dealt with several facets of the neutrino interaction with We have compared the two approaches for the one nucleon -one hole excitations finding a reasonable agreement between them in the quasielastic peak region, with a trend for the RPA approach to lead to lower cross sections than the CRPA presumably due to the mixing with ∆ excitations. Other general trends are related to the more important high transferredenergy tail in the CRPA results and to a relative shift of the cross sections of ω ≃ 18 MeV, reflecting the presence of the nucleon separation energy in the CRPA calculations. The most striking difference is the appearance of giant resonance peaks in the CRPA results.
The comparison of the two approaches has been performed for fixed values of the incoming neutrino energy as well as for the ν e T2K and MiniBooNE flux-folded cross sections.
We have also compared the ν e cross sections with the corresponding ν µ ones for fixed values of the neutrino energy in order to investigate the impact of different charged lepton masses. We have found some non trivial behaviour, in particular for the 1p-1h excitations at low neutrino energies, such as an inversion with the scattering angle of the relative strength of ν e and ν µ cross sections. Due to the different kinematical limits, the ν e cross sections are in general expected to be larger than the ν µ ones, however for forward scattering angles this hierarchy is opposite. In the precision era of neutrino oscillation physics the ν e cross sections should be known with the same accuracy as the ν µ ones. Trying to deduce the ν e cross sections from the experimental ν µ ones can be considered only as a first approximation in the study of the ν e interactions.
Concerning the comparison with experiment, we have considered the inclusive ν e T2K flux-folded single-differential cross sections on Carbon where the inclusion of np-nh and pion production effects beyond the genuine quasielastic is mandatory. For this reason we have compared the data with the only RPA-based approach of Martini et al. which includes all these channels. We have found a good agreement with the data. This success obtained with a new flux, the ν e T2K one, complements those already reached with the three different fluxes, such as the MiniBooNE ν µ andν µ , and T2K ν µ ones.