Amaro, J. E.

Barbaro, Maria Benedetta

Caballero, J. A.

Donnelly, T. W.

English

Institutional Research Information System University of Turin

arXiv:1012.4265v1  [nucl-th]  20 Dec 2010Superscaling predictions for NC and CC quasi-elastic neutrino-nucleus scatteringJ.E. Amaroa, M.B. Barbarob, J.A. Caballeroc, T.W. Donnellyd,aDepartamento de F´ısica Ato´mica, Molecular y Nuclear, Universidad de Granada, Granada 18071, SpainbDipartimento di Fisica Teorica, Universita` di Torino and INFN,Sezione di Torino, Via P. Giuria 1, 10125 Torino, ItalycDepartamento de F´ısica Ato´mica, Molecular y Nuclear,Universidad de Sevilla, Apdo.1065, 41080 Sevilla, Spain anddCenter for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,Massachusetts Institute of Technology, Cambridge, MA 02139, USAQuasielastic double differential neutrino cross sections can be obtained in a phenomenologicalmodel based on the superscaling behavior of electron scattering data. In this talk the superscalingapproach (SuSA) is reviewed and its validity is tested in a relativistic shell model. Results includingmeson exchange currents for the kinematics of the MiniBoone experiment are presented.PACS numbers: 25.30.Pt,24.10.-i,25.30.FjAnalysis of inclusive (e, e′) data have demonstratedthat at energy transfers below the quasielastic (QE) peaksuperscaling is fulfilled rather well [1]—[3]. The gen-eral procedure consist on dividing the experimental (e, e′)cross section by an appropriate single-nucleon cross sec-tion to obtain the experimental scaling function f(ψ),which is then plotted as a function of the scaling vari-able ψ for several kinematics and for several nuclei. Ifthe results do not depend on the momentum transfer q,we say that scaling of the first kind occurs. If there isnot dependence on the nuclear species, one has scaling ofthe second kind. The simultaneous occurrence of scalingof both kinds is called superscaling. The Super-Scalingapproach (SuSA) is based on the assumed universality ofthe scaling function for electromagnetic and weak inter-actions [4].The superscaling property is exact in the relativistic12Cf L0.80.60.40.216Of L0.80.60.40.240Caψf L21.510.50-0.5-1-1.5-20.80.60.40.2q = 0.5 GeV/cf L0.80.60.40.2q = 0.7 GeV/cf L0.80.60.40.2q = 1 GeV/cf L0.80.60.40.2q = 1.3 GeV/cf L0.80.60.40.2q = 1.5 GeV/cψf L21.510.50-0.5-1-1.5-20.80.60.40.2FIG. 1: Scaling properties of the SR shell model. Left: scal-ing of the first kind. Curves for q = 0.5, 0.7, 1, 1.3, 1.5 GeVcollapse into one. Right: scaling of the second kind. Curvesfor 12C, 16O and 40Ca collapse into one.Fermi gas model (RFG) by construction, and it has beentested in more realistic models of the (e, e′) reaction [5]–[7]. A study of superscaling in the semirelativistic (SR)continuum shell model with Woods-Saxon (WS) meanpotential is summarized in Fig. 1. There we show thelongitudinal scaling function defined as fL = RL/GL,where RL is the longitudinal response function and GLis a single-nucleon factor. When fL(ψ) is plotted forvarious values of the momentum transfer and for severalclosed-shell nuclei, all the curves approximately collapseinto one. Small violations of scaling are seen at low valuesof ψ, coming from the low-energy potential resonances forq = 0.5 GeV/c, which disappear for higher q values.While the SR shell model superscales, it does not re-produce the experimental data of the phenomenologicalscaling function extracted from the longitudinal QE elec-tron scattering response. The WS potential used to de-scribe the final-state interaction (FSI) of the ejected pro-ton does not incorporate the appropriate reactions mech-anisms. A further improvement of the FSI consist in us-ing the Dirac-Equation based potential plus Darwin term(DEB+D). The DEB potential is obtained from the Diracequation for the upper component ψup(r) = K(r, E)φ(r),where the Darwin term K(r, E) is chosen in such a waythat the function φ(r) satisfies a Schro¨dinger-like equa-tion. The electromagnetic L and T scaling functionswithin this model are shown in Fig. 2. We use the samerelativistic Hartree potential as in the relativistic meanfield model of [5], and the scaling variable ψ′ includesa q-dependent energy shift Es(q). Although the scalingis not perfect, it is remarkable that our results give es-sentially the same scaling function for a wide range of qvalues, reproducing well the phenomenological data.In Fig. 2 (right) we compare DEB+D and Woods-Saxon predictions for the 12C(νµ, µ−) differential QEcross section for 1 GeV incident neutrinos and for sev-eral lepton scattering angles. The SuSA reconstructedcross section is also shown, using the theoretical scalingfunction extracted from (e, e′) results with the DEB+Dmodel (curves in Fig. 2, left). For angles above 15o theSuSA is applicable and its predictions can be considered2ψ′f L32.521.510.50-0.5-1-1.50.70.60.50.40.30.20.10ψ′f T32.521.510.50-0.5-1-1.50.70.60.50.40.30.20.10ψ′f L32.521.510.50-0.5-1-1.50.70.60.50.40.30.20.10ψ′f T32.521.510.50-0.5-1-1.50.70.60.50.40.30.20.10q [MeV/c]Es[MeV]160014001200100080060040080706050403020θ = 15oǫ = 1 GeV10009509008508006050403020100θ = 30oǫ = 1 GeV1000900800700600181614121086420θ = 45o100090080070060050040076543210θ = 60o90080070060050040030032.521.510.50θ = 75odσdΩ′dǫ′[10−15fm2/MeV]8007006005004003002001.61.20.80.40θ = 90o80070060050040030020010.80.60.40.20θ = 105o7006005004003002001000.70.60.50.40.30.20.10θ = 120o6005004003002001000.60.50.40.30.20.10θ = 135oǫ′ [MeV]6005004003002001000.40.30.20.10θ = 150oǫ′ [MeV]5004003002001000.30.20.10FIG. 2: Left: Scaling of 1st kind with DEB+D potential com-pared with experimental data. q = 0.5, 0.7 1.0, 1.3 and 1.5GEV/c. Right: Test of SuSA in the SR shell model for the12C(νµ, µ−) reaction with neutrino energy ǫ = 1 GeV. Dotted:Woods-Saxon potential. Solid: DEB+D potential. Dashed:SuSA reconstruction from the computed (e, e′) scaling func-tion.; N= 2008007006005004000.80.60.40.20; N= 6004003503002502001501005000.80.60.40.20; N= 200TN[MeV ℄d2=d=dEN[10 15fm2=sr=MeV℄8007006005004000.080.060.040.020; N= 600TN[MeV ℄4003503002502001501005000.40.30.20.10StrangegsA=  0:2s= 0:55No strange; p= 2008007006005004000.80.60.40.20; p= 60040030020010000.80.60.40.20; p= 200Tp[MeV ℄d2=dp=dEp[10 15fm2=sr=MeV℄8007006005004000.140.120.10.080.060.040.020; p= 600Tp[MeV ℄40030020010000.50.40.30.20.10FIG. 3: Neutral current proton knock-out cross section from12C. Left: RFG (solid lines), RFG with factorization (dottedlines), and SuSA (dashed lines). Right: study of nucleonstrangeness effect.H1 H2P ′1P ′2QK2(a)H1P ′1H2P ′2QK1(b)H1P ′1H2P ′2QK1 K2(c)H1 H2P ′1P ′2QK2(d)H1P ′1H2P ′2QK1(e)H1 H2P ′1P ′2QK2(f)H1P ′1H2P ′2QK1(g)FIG. 4: Diagrams contributing to the MECquite reasonable, with small error.The superscaling approach has been extended to in-clusive neutrino scattering via the weak neutral current[8]. In the (ν,N) reaction a nucleon is detected and thefinal neutrino must be integrated. We approximate theu-channel inclusive scattering cross sectiondσdΩNdpN≃ σ(u)sn F(u)(ψ(u)). (1)In Fig. 3 we see that the above factorization is a good30.8 < cos θ < 0.9Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)21.510.502520151050dataSuSA+MEC0.8 < cos θ < 0.9Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)21.510.5025201510500.7 < cos θ < 0.8Tµ (GeV)d2σ/dcosθdTµ(10−39cm2/GeV)21.510.5025201510500.7 < cos θ < 0.8Tµ (GeV)d2σ/dcosθdTµ(10−39cm2/GeV)21.510.5025201510500.6 < cos θ < 0.7Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)1.61.41.210.80.60.40.20201510500.6 < cos θ < 0.7Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)1.61.41.210.80.60.40.20201510500.5 < cos θ < 0.6Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)1.210.80.60.40.201816141210864200.5 < cos θ < 0.6Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)1.210.80.60.40.201816141210864200.4 < cos θ < 0.5Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)10.80.60.40.201816141210864200.4 < cos θ < 0.5Tµ (GeV)d2σ/dcosθ/dTµ(10−39cm2/GeV)10.80.60.40.201816141210864200.3 < cos θ < 0.4Tµ (GeV)d2σ/dcosθdTµ(10−39cm2/GeV)10.80.60.40.20141210864200.3 < cos θ < 0.4Tµ (GeV)d2σ/dcosθdTµ(10−39cm2/GeV)10.80.60.40.2014121086420FIG. 5: Computed QE (νµ, µ−) cross section compared toMiniboone data. Left: SuSA. Right: SuSA + MECapproximation in the RFG, and therefore one could ex-tend the scaling analysis used for CC reactions to NCscattering. In Fig. 3 we show examples of SuSA pre-dictions of NC cross sections using the phenomenological(e, e′) scaling function.Recently the effect of meson exchange currents (MEC)in (e, e′) for high momentum transfer has been investi-gated [9, 10], and the two-particle emission (2p-2h) di-agrams of Fig. 4 have been extended to the CC weakinteraction sector [11], in order to explore the role ofMEC in neutrino reactions. In Fig. 5 we show thatinclusion of 2p-2h contributions yields results for the QE(νµ, µ−) cross section that are comparable with the re-cent MiniBooNE collaboration data [12] for θ ≤ 50o, butlie below the data at larger angles where the predictedcross sections are smaller. The inclusion of the correla-tion diagrams which are required by gauge invariance [10]plus other relativistic effects might improve the agree-ment with the data.This work was partially supported by DGI (Spain):FIS2008-01143, FIS2008-04189, by the Junta de An-daluc´ıa, and part (TWD) by U.S. Department of Energyunder cooperative agreement DE-FC02-94ER40818.[1] D.B. Day, J.S. McCarthy, T.W. Donnelly, and I. Sick,Ann. Rev. Nucl. Part. Sci. 40, 357 (1990)[2] T.W. Donnelly and I. Sick, Phys. Rev. Lett. 82, 3212(1999).[3] T.W. Donnelly and I. Sick, Phys. Rev. C 60, 065502(1999).[4] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, A. Molinari, I. Sick, Phys. Rev. C 71, 015501(2005).[5] J.A. Caballero, J.E. Amaro, M.B. Barbaro, T.W. Don-nelly, C. Maieron, J.M. Udias, Phys. Rev. Lett. 95,252502 (2005).[6] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, and C. Maieron, Phys. Rev. C 71, 065501 (2005)[7] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, and J.M. Udias, Phys. Rev. C 75, 034613 (2007)[8] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, Phys. Rev. C 73, 035503 (2006)[9] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, C. Maieron, J.M. Udias, Phys. Rev. C 81, 014606(2010).[10] J.E. Amaro, C. Maieron, M.B. Barbaro, J.A. Caballero,T.W. Donnelly, Phys. Rev. C 82, 044601, (2010)[11] J.E. Amaro, M.B. Barbaro, J.A. Caballero, T.W. Don-nelly, C.F. Williamson, Phys. Lett. B, to be published.arXiv:1010.1708[12] A.A. Aguilar-Arevalo et al., [MiniBooNE Collaboration],Phys. Rev. D 81, 092005 (2010).

Superscaling predictions for NC and CC quasi-elastic neutrino-nucleus scattering

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Superscaling predictions for NC and CC quasi-elastic neutrino-nucleus scattering

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