We present a continuum random phase approximation approach to study electron-
and neutrino-nucleus scattering cross sections, in the kinematic region where
quasielastic scattering is the dominant process. We show the validity of the
formalism by confronting inclusive ($e,e'$) cross sections with the available
data. We calculate flux-folded cross sections for charged-current quasielastic
antineutrino scattering off $^{12}$C and compare them with the MiniBooNE
cross-section measurements. We pay special emphasis to the contribution of
low-energy nuclear excitations in the signal of accelerator-based
neutrino-oscillation experiments.Comment: 5 pages, 5 figures. Contribution to the proceedings of the 16th
  International Workshop on Neutrino Factories and Future Neutrino Beam
  Facilities (NUFACT-2014

Jachowicz, N.

Martini, M.

Pandey, V.

Ryckebusch, J.

Van Cuyck, T.

English

arXiv

We present a continuum random phase approximation approach to study electron- and neutrino-nucleus scattering cross sections, in the kinematic region where quasielastic scattering is the dominant process. We show the validity of the formalism by confronting inclusive ($e,e'$) cross sections with the available data. We calculate flux-folded cross sections for charged-current quasielastic antineutrino scattering off $^{12}$C and compare them with the MiniBooNE cross-section measurements. We pay special emphasis to the contribution of low-energy nuclear excitations in the signal of accelerator-based neutrino-oscillation experiments

Pandey, Vishvas

Jachowicz, Natalie

Van Cuyck, Tom

Ryckebusch, Jan

Martini, Marco

Ghent University Academic Bibliography

PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleusscattering in a continuum random phaseapproximation approachV. Pandey∗Ghent UniversityE-mail: vishvas.pandey@ugent.beN. JachowiczGhent UniversityE-mail: Natalie.Jachowicz@UGent.beT. Van CuyckGhent UniversityE-mail: Tom.VanCuyck@UGent.beJ. RyckebuschGhent UniversityE-mail: Jan.Ryckebusch@UGent.beM. MartiniGhent UniversityE-mail: Marco.Martini@UGent.beWe present a continuum random phase approximation approach to study electron- and neutrino-nucleus scattering cross sections, in the kinematic region where quasielastic scattering is the dom-inant process. We show the validity of the formalism by confronting inclusive (e,e′) cross sectionswith the available data. We calculate flux-folded cross sections for charged-current quasielasticantineutrino scattering off 12C and compare them with the MiniBooNE cross-section measure-ments. We pay special emphasis to the contribution of low-energy nuclear excitations in thesignal of accelerator-based neutrino-oscillation experiments.16th International Workshop on Neutrino Factories and Future Neutrino Beam Facilities25 -30 August, 2014University of Glasgow, United Kingdom∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandey1. IntroductionRecent years have seen an enormous enhancement in the understanding of neutrino-oscillationparameters in accelerator-based experiments. These experiments are however confronted with anumber of problems. These are related to the large systematic uncertainties associated with theneutrino-nucleus signal in the detector. Major issues arise from the fact that the neutrino energy-flux in experiments is distributed over a wide range of energies from very low to a few GeV. Hencea number of nuclear effects over a broad kinematical range (from low-energy nuclear excitations tomultinucleon emission) simultaneously come into play. The simulation codes used in the analysisof the experimental results are predominantly based on relativistic Fermi gas (RFG) models. RFGcan describe the quasielastic (QE) cross section sufficiently accurate for medium momentum (q ≈500 MeV/c) transfer reactions, but its description becomes poor for low momentum (q . 300MeV/c) transfer processes, where nuclear effects are prominent. For the broad neutrino energy-flux used in the experiments, more realistic models are required.In this work, we present a self-consistent continuum random phase approximation (CRPA)approach to calculate QE electron and neutrino-scattering cross-sections off the nucleus. This for-malism was used to describe exclusive photo-induced and electron-induced QE scattering [1, 2],inclusive neutrino scattering at supernova energies [3, 4, 5, 6, 7, 8] and charged-current quasielastic(CCQE) antineutrino scattering at intermediate energies [9]. We will briefly describe the essenceof our model, for an updated version of the formalism we refer the reader to Ref. [10]. The mainupdate in Ref. [10] from Ref. [9], are the inclusion of relativistic corrections and a suppression ofthe RPA quenching at high Q2. We start with a mean-field (MF) description of the nucleus wherewe solve the Hartree-Fock (HF) equations with a Skyrme (SkE2) two-body interaction [2, 11] toobtain the MF potential. We obtain the continuum wave functions by integrating the positive energySchrödinger equation with appropriate boundary conditions, hence taking into account final-stateinteractions in this manner. Long-range correlations are implemented by means of a CRPA ap-proach based on a Green’s function formalism. The polarization propagator is approximated byiteration of its first-order contribution. In this way, the formalism takes into account one-particleone-hole excitations out of the correlated nuclear ground state. Within the RPA an excited nu-clear state is represented as the coherent superposition of the particle-hole (ph−1) and hole-particle(hp−1) excitations out of a correlated ground state|ΨCRPA〉= ∑C′[XC,C′ |p′h′−1〉− YC,C′ |h′p′−1〉], (1.1)where C denotes all quantum numbers identifying an accessible channel. The RPA polarizationpropagator can be written asΠ(RPA)(x1,x2;Ex) = Π(0)(x1,x2;Ex)+1h¯∫dxdx′Π0(x1,x;Ex) ˜V (x,x′)Π(RPA)(x′,x2;Ex),(1.2)where Ex is the excitation energy of the nucleus and x is a short-hand notation for the combinationof spatial, spin and isospin coordinates. The Π(0) corresponds to the MF contribution and ˜V is theantisymmetrized nucleon-nucleon interaction.2PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandey0 50 (nb/MeV sr)Ωdω/dσ2 d0100200o = 60θE = 161 MeV, (a)20 40 60 800500o = 36θE = 200 MeV, (b)200 40000.10.2o = 145θE = 560 MeV, (c) (MeV) ω200 400 600 (nb/MeV sr)Ωdω/dσ2 d024o = 37.5θE = 1108 MeV, (d) (MeV)ω0 200 400 60001020o = 18θE = 2130 MeV, (e) (MeV)ω500 100002o = 16θE = 3595 MeV, (f)Figure 1: The double-differential cross-sections for 12C(e,e′) plotted as a function of excitation energyω . The incident electron energy E and lepton scattering angle θ are listed on top of each panel. Solidlines are CRPA cross sections and dashed-lines are HF cross sections. Experimental data is taken fromRefs. [14, 15, 16, 17].We used the modified effective momentum approximation (MEMA) [12], in order to takeinto account the influence of the nuclear Coulomb field on the ejected lepton. In order to preventthe SkE2 force from becoming unrealistically strong at high virtuality Q2, we introduce a dipolehadronic form factor at the nucleon-nucleon interaction vertices [10]. Further, we have imple-mented relativistic kinematic corrections [13] in an effective manner.We first test the reliability of the formalism by confronting (e,e′) scattering cross sectionswith the data of Refs. [14, 15, 16, 17]. Thereby, we present updated results of flux-folded charged-current quasielastic (CCQE) antineutrino scattering off 12C and compare them with the MiniBooNEmeasurements [18]. Further, we discuss the contribution of neutrino-induced low-energy nuclearexcitations in the signal of the accelerator-based neutrino oscillation experiments.2. Cross section resultsWe start this section by showing some examples of electron-scattering results. In Fig. 1, weshow our prediction of QE 12C(e,e′) scattering cross-sections and compare them with the measure-ments of the Refs. [14, 15, 16, 17]. Our predictions successfully describe the data over the broadkinematical range considered here. The formalism successfully describes low-energy excitations3PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandeyµθcos-1 0 1)-1MeV2cm-42 (10〉 µθdcosµ/dTσ2 d〈 024 < 300 MeVµ200 < T(a)µθcos0.5 1 0510 < 600 MeVµ500 < T(b)µθcos0.6 0.8 1 0510 < 900 MeVµ800 < T(c)(MeV)µT0 500 1000 1500)-1MeV2cm-42 (10〉 µθdcosµ/dTσ2 d〈 0510 < 0.9µθ0.8 < cos(d)(MeV)µT0 500 1000 024 < 0.6 µθ0.5 < cos(e)(MeV)µT0 200 400 600 012  < 0.3µθ0.2 < cos(f)Figure 2: (Color online) MiniBooNE flux-folded double-differential cross section per target proton for12C( ¯νµ ,µ+)X plotted as a function of the direction of the scattering lepton cosθµ for different values of itskinetic energy Tµ values (top), as a function Tµ for different ranges of cosθµ (bottom). Solid lines are CRPAand dashed lines are HF calculations. MiniBooNE data [18] are filled squares, error bars represent the shapeuncertainties and error boxes represent the 17.2% normalization uncertainty.(panel (a) and (b)) below the QE peak. The forward scattering cross sections, even for higher in-coming electron energies, are dominated by the QE contribution. However, the data include crosssection contributions beyond the QE channel, like ∆-excitations and other inelastic channels. Ourcalculations are intended to predict only the QE behavior. A detailed comparison of (e,e′) crosssection on 12C, 16O and 40Ca is performed in Ref. [10]. An overall successful description of QE(e,e′) cross section data and especially low-energy excitations, validates the reliability of our for-malism.We show the double-differential cross section for 12C( ¯νµ ,µ+)X , folded with the MiniBooNEantineutrino flux [18], in Fig. 2. The top panels show the cross section in Tµ bins and the bottompanels show the cross section in cosθµ bins. The cross section is integrated over the correspondingbin width. We adopt an axial mass value of MA = 1.03 GeV, in the dipole axial form factor. HF andCRPA cross sections are compared with the MiniBooNE measurements of Ref. [18]. MiniBooNEdata is presented with both shape and normalization uncertainties. Overall, CRPA and HF calcu-lations successfully reproduce the gross features of the measured cross section. The predictionstend to underestimate the data. It has been suggested in Refs. [19, 20, 21, 22, 23] that the inclusionof multinucleon contributions, which are not included in our calculations, are essential for a morecomplete reproduction of the data.The flux-folded differential cross section as a function of cos θµ , is shown in Fig. 3. For com-parison with data, we integrate MiniBooNE data over Tµ . It is interesting to note that in the veryforward direction the CRPA results are larger than the HF ones. This is due to the collective giantresonance contributions which are absent in the HF approximation but appear in CRPA results that4PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandey µθ cos0 0.5 1)2cm-42 (10〉 µθ/dcosσd〈 0510 HFCRPA MiniBooNEµνFigure 3: MiniBooNE flux-folded cross section per target proton for 12C( ¯νµ ,µ+)X as a function of cosθµ .The MiniBooNE data [18] are integrated over Tµ . (GeV)µν E0.5 1 1.5 2/nucleon)2cm-39 (10σ 0510HFCRPA MiniBooNEµνFigure 4: Flux unfolded total cross section per target proton for 12C( ¯νµ ,µ+)X as a function of E ¯νµ , com-pared with MiniBooNE data [18].include long-range correlations. In Fig. 4, we present total cross sections per target proton as a func-tion of neutrino energy and compare them with the experimental data. Unlike double-differentialcross sections, this quantity is model dependent. The theoretical calculations are function of a trueantineutrino energy while the experimental data are function of reconstructed antineutrino energy.Up to E¯ν = 0.4 GeV, the HF results essentially coincide with the CRPA ones. This is due to a com-pensation between a reduction in the QE region and an enhancement in the giant resonance part ofthe CRPA results. For E¯ν & 0.4 GeV, the CRPA results are slightly smaller than the HF ones.In order to illustrate the impact of the low-energy nuclear excitations, in Fig. 5, we show thedouble-differential cross-section for fixed neutrino energies and fixed scattering angles. As it ap-pears in panel (a), 150 MeV energy neutrinos induce low-lying nuclear excitations at all scattering5PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandey (MeV)µT0 10 20)-1MeV2cm-42(10 µθdcosµ/dTσ2 d 010 = 0.99 µθcos          = 0.65          = 0.35          = 0.15 = 150 MeVνE(a) (MeV)µT500 6000500  = 0.99 µθcos          = 0.95          = 0.90          = 0.80 = 800 MeVνE(b)Figure 5: Neutrino induced low-energy nuclear excitations in double differential cross section for12C(νµ ,µ−) plotted as a function Tµ , for different cosθµ values.angles. For neutrino energies of 800 MeV, which is near the mean energy of the MiniBooNE [24]and T2K [25] fluxes, in panel (b), the forward scatterings still show sizable low-energy excitations.This feature can have a non-negligible contribution to the neutrino signals in these experiments, butcan not be accounted for within the RFG-based simulation codes. As already mentioned, one canobserve in Fig. 3, that at very forward scatterings, cosθµ ≈ 1, the CRPA cross section generatesmore strength (emerging from the low-lying excitation) than the HF.3. ConclusionsWe have presented a continuum random phase approximation approach for quasielastic electron-and neutrino-nucleus scattering. We validated the reliability of our formalism, in the quasielasticregion, by comparing (e,e′) cross section with the available data. An interesting feature of ourCRPA formalism is the successful prediction of low-energy nuclear excitations. We calculatedflux-folded 12C( ¯νµ ,µ+)X cross sections and compared them with the MiniBooNE antineutrinocross-section measurements. CRPA predictions are successful in describing the gross features ofthe cross section but seem to underestimate slightly the measured cross section. We illustratedhow low-energy nuclear excitations can possibly account for non-negligible contributions to theneutrino signal in accelerator-based neutrino-oscillation experiments.AcknowledgmentsThis research was funded by the Interuniversity Attraction Poles Programme initiated by theBelgian Science Policy Office, the Research Foundation Flanders (FWO-Flanders) and by the Eras-mus Mundus External Cooperations Window’s Eurindia Project.References[1] J. Ryckebusch, M. Waroquier, K. Heyde, J. Moreau, and D. Ryckbosch, Nucl. Phys. A476, 237(1988).6PoS(NUFACT2014)055Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approachV. Pandey[2] J. Ryckebusch, K. Heyde, D. Van Neck, and M. Waroquier, Nucl. Phys. A503, 694 (1989).[3] N. Jachowicz, K. Heyde, J. Ryckebusch, and S. Rombouts, Phys. Rev. C59, 3246 (1999).[4] N. Jachowicz, K. Heyde, J. Ryckebusch, and S. Rombouts, Phys. Rev. C65, 025501 (2002).[5] N. Jachowicz, K. Vantournhout, J. Ryckebusch, and K. Heyde, Phys. Rev. Lett. 93, 082501 (2004).[6] N. Jachowicz, G. C. McLaughlin, Phys. Rev. Lett. 96, 172301 (2006).[7] N. Jachowicz, C. Praet, and J. Ryckebusch, Acta Physica Polonica B, 40(9), 2559-2564 (2009).[8] N. Jachowicz, and V. Pandey, proceedings NuInt12, to be published.[9] V. Pandey, N. Jachowicz, J. Ryckebusch, T. Van Cuyck, and W. Cosyn, Phys. Rev. C89, 024601(2014).[10] V. Pandey, N. Jachowicz, T. Van Cuyck, J. Ryckebusch, and M. Martini, arXiv:1412.4624 [nucl-th].[11] M. Waroquier, J. Ryckebusch, J. Moreau, K. Heyde, N. Blasi, S.Y. van de Werf, and G. Wenes,Phys. Rep. 148, 249 (1987).[12] J. Engel, Phys. Rev. C57, 2004 (1998).[13] S. Jeschonnek, T. W. Donnelly, Phys. Rev. C57, 2438 (1998).[14] P. Barreau et al., Nucl. Phys. A402, 515 (1983).[15] R. M. Sealock et al., Phys. Rev. Lett.62, 1350 (1989).[16] D. S. Bagdasaryan et al., YERPHI-1077-40-88 (1988).[17] D. B. Day et al., Phys. Rev. C 48, 1849 (1993).[18] A. A. 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Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approach

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Quasielastic electron- and neutrino-nucleus scattering in a continuum random phase approximation approach

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