We present a detailed study of charged-current quasielastic neutrino-nucleus
scattering and of the influence of correlations on one- and two-nucleon
knockout processes. The quasielastic neutrino-nucleus scattering cross
sections, including the influence of long-range correlations, are evaluated
within a continuum random phase approximation approach. The short-range
correlation formalism is implemented in the impulse approximation by shifting
the complexity induced by the correlations from the wave functions to the
operators. The model is validated by confronting $(e,e^\prime)$ cross-section
predictions with electron scattering data in the kinematic region where the
quasielastic channel is expected to dominate. Further, the
$^{12}$C$(\nu,\mu^-)$ experiments are studied. Double differential cross
sections relevant for neutrino-oscillation $^{12}$C$(\nu,\mu^-)$ cross
sections, accounting for long- and short-range correlations in the one-particle
emission channel and short-range correlations in the two-particle two-hole
channel, are presented for kinematics relevant for recent neutrino-nucleus
scattering measurements.Comment: 10 pages, 4 figures. Contribution to the proceedings of the 17th
  International Workshop on Neutrino Factories and Future Neutrino Beam
  Facilities (NUFACT-2015

González-Jiménez, Raul

Jachowicz, Natalie

Martini, Marco

Pandey, Vishvas

Ryckebusch, Jan

Van Cuyck, Tom

Van Dessel, Nils

English

arXiv

We present a detailed study of charged-current quasielastic neutrino-nucleus scattering and of the influence of correlations on one- and two-nucleon knockout processes. The quasielastic neutrino-nucleus scattering cross sections, including the influence of long-range correlations, are evaluated within a continuum random phase approximation approach. The short-range correlation formalism is implemented in the impulse approximation by shifting the complexity induced by the correlations from the wave functions to the operators. The model is validated by confronting (e,e′ ) cross-section predictions with electron scattering data in the kinematic region where the quasielastic channel is expected to dominate. Further, the 12C(ν,μ−) experiments are studied. Double differential cross sections relevant for neutrino-oscillation 12C(ν,μ−) cross sections, accounting for long- and short-range correlations in the one-particle emission channel and short-range correlations in the two-particle two-hole channel, are presented for kinematics relevant for recent neutrino-nucleus scattering measurements

González Jiménez, Raúl

Ghent University Academic Bibliography

NuFact15 - Rio de Janeiro, Brazil - August 2015Correlations in neutrino-nucleus scattering∗T. Van Cuyck,† V. Pandey,‡ N. Jachowicz,§ R. Gonza´lez-Jime´nez,M. Martini, J. Ryckebusch, and N. Van DesselDepartment of Physics and Astronomy,Ghent University,Proeftuinstraat 86,B-9000 Gent, Belgium(Dated: November 13, 2018)1arXiv:1606.08636v1  [nucl-th]  28 Jun 2016NuFact15 - Rio de Janeiro, Brazil - August 2015 2AbstractWe present a detailed study of charged-current quasielastic neutrino-nucleus scattering and of theinfluence of correlations on one- and two-nucleon knockout processes. The quasielastic neutrino-nucleus scattering cross sections, including the influence of long-range correlations, are evaluatedwithin a continuum random phase approximation approach. The short-range correlation formalismis implemented in the impulse approximation by shifting the complexity induced by the correlationsfrom the wave functions to the operators. The model is validated by confronting (e, e′) cross-sectionpredictions with electron scattering data in the kinematic region where the quasielastic channelis expected to dominate. Further, the 12C(νµ, µ−) cross sections relevant for neutrino-oscillationexperiments are studied. Double differential 12C(νµ, µ−) cross sections, accounting for long- andshort-range correlations in the one-particle emission channel and short-range correlations in thetwo-particle two-hole channel, are presented for kinematics relevant for recent neutrino-nucleusscattering measurements.INTRODUCTIONOne of the major issues in accelerator-based neutrino-oscillation experiments is the needfor accurate predictions of neutrino-nucleus scattering cross sections at intermediate (0.01- 2 GeV) energies. A model where the W boson interacts with a single nucleon, whichsubsequently leaves the residual nucleus unhindered, does not accurately describe recent ex-perimental measurements of neutrino and antineutrino cross sections. A major complicationstems from the fact that typical neutrino-nucleus measurements do not uniquely determinethe nuclear final state, as only the energy-momentum of the final muon are measured. Inorder to explain the discrepancy between theory and experiment, one needs a model thatincludes nuclear correlations, meson-exchange currents and final-state interactions. In thiswork, we focus on the influence of nuclear correlations on inclusive quasielastic (QE) crosssections. First we will discuss long-range correlations in a continuum random phase approx-imation (CRPA) approach, and secondly, the influence of short-range correlations (SRCs).The model described below was used successfully in the description of exclusive electron-scattering processes [1, 2], low-energy and supernova neutrino processes [3–5], and extendedto the description of inclusive quasielastic electroweak scattering cross sections at interme-diate energies in [6, 7].NuFact15 - Rio de Janeiro, Brazil - August 2015 3QUASIELASTIC NEUTRINO-NUCLEUS SCATTERING CROSS SECTIONIn this section, we briefly describe the approach for the calculation of the nuclear responsefor inclusive electron- and neutrino-nucleus scattering in the QE region. Considering electronscattering off a nucleus, the double differential A(e, e′) cross section is written asdσdEe′dΩe′=(α cos(θe′/2)2Ee sin2(θe′/2))2 [veLWCC + veTWT], (1)with α the fine-structure constant and θe′ the scattering angle of the electron. For CCneutrino-nucleus A(νµ, µ−) interactions, the cross section is expressed asdσdEµdΩµ=(GF cos(θc)Eµ2pi)2ζ[vCCWCC + vCLWCL + vLLWLL + vTWT − vT ′WT ′], (2)with GF the Fermi constant, θc the Cabibbo angle and the kinematic factor ζζ =√1− m2µE2µ. (3)The functions v contain the leptonic information and the W are nuclear response functions,they are defined as products of transition matrix elements JλJλ = 〈Ψf|Ĵnuclλ |Ψi〉, (4)with |Ψf〉 and |Ψi〉 the final and initial nuclear state and Ĵnuclλ the spherical components ofthe nuclear current. The expressions for the v and W can be found in Ref. [7].HARTREE-FOCK MEAN FIELD MODELA key element in the model presented here is the non-relativistic impulse approximation.The Hartree-Fock (HF) single-particle bound-states and the continuum wave functions areobtained by solving the Schro¨dinger equation using an effective Skyrme interaction. TheSkE2 Skyrme parameterization is based on a fit to ground-state and low-lying excited stateproperties of spherical nuclei [1, 2]. The fact that the outgoing nucleon’s wave function isgenerated in a (real) nuclear potential partially includes final-state interactions, in a naturalway. The influence of the spreading width of the particle states is taken into accountby a folding procedure [7]. The impact of the Coulomb potential of the nucleus on theoutgoing lepton is implemented using a modified effective momentum approach (MEMA)[8]. As the description of the nuclear dynamics is non-relativistic, relativistic corrections areimplemented based on the effective scheme proposed in [9].NuFact15 - Rio de Janeiro, Brazil - August 2015 4LONG-RANGE CORRELATIONSLong-range correlations are introduced using a continuum random phase approximationapproach. The CRPA is based on a Green’s function formalism, where the CRPA propagatoris obtained by the iteration to all orders of the first-order contribution to the particle-holeGreen’s functionΠ(RPA)(x1, x2;Ex) = Π(0)(x1, x2;Ex)+1h¯∫dxdx′Π(0)(x1, x;Ex)V˜ (x, x′)Π(RPA)(x′, x2;Ex), (5)with V˜ the antisymmetrized Skyrme residual interaction. The same Skyrme SkE2 param-eterization that is used to generate the HF single-particle wave functions is used as ph-interaction in the RPA calculation, assuring consistency of the formalism with regards tothe nucleon interaction that is used. The Q2 running of the residual interaction is controledby a dipole form factor at the nucleon vertices [7]. The CRPA wave-functions |ΨRPAC 〉 andtransition densities are then related to the unperturbed wave-functions |ph−1〉 through|ΨRPAC 〉 =∑C′[XC,C′ |p′h′−1〉 − YC,C′ |h′p′−1〉], (6)withXC,C′(E, εp′) = δC,C′ δ(E − εp′h′) + P∫dx1∫dx2 V˜ (x1, x2)ψh′(x1)ψ†p′(x1, εp′)E − εp′h′〈Ψ0∣∣∣ψˆ†(x2)ψˆ(x2)∣∣∣ΨC(E)〉 , (7)andYC,C′(E, εp′) =∫dx1∫dx2 V˜ (x1, x2)ψ†h′(x1)ψp′(x1, εp′)E + εp′h′〈Ψ0∣∣∣ψˆ†(x2)ψˆ(x2)∣∣∣ΨC(E)〉 , (8)with C denoting all quantum numbers representing an accessible channel. These equationsreflect the fact that RPA wave functions are a superposition of ph- and hp-excitations outof a correlated ground state.In FIG. 1, the HF and CRPA predictions are compared with double-differential electron-scattering data for three different target nuclei. In view of the fact that our descriptionNuFact15 - Rio de Janeiro, Brazil - August 2015 500.511.520 100 200 300 40000.511.520 100 200 300 400 500 600010203040500 100 200 300 400 500 600051015202530350 100 200 300 400 5000246810121416180 100 200 300 400 50005101520250 100 200 300 400 50012C, Ee = 480 MeV, θe′ = 90◦12C, Ee = 1299 MeV, θe′ = 37.5◦12C, Ee = 2130 MeV, θe′ = 16◦HFCRPAdσ/dEe′ dΩe′ (10−33cm2/MeV)ω (MeV)16O, Ee = 880 MeV, θe′ = 32◦ω (MeV)16O, Ee = 1080 MeV, θe′ = 32◦ω (MeV)40Ca, Ee = 739 MeV, θe′ = 45.5◦FIG. 1: Double differential (e, e′) cross section with 12C, 16O and 40Ca as target nuclei. Data arefrom Refs. [10–14].only considers the QE channel, while the measurements include contributions such as ∆excitations and two-particle knockout, our numerical calculations provide a fair agreementwith the data in the kinematic range presented here. The difference between the HF andCRPA results are sizable for Q2 ≤ 0.25 (GeV/c)2, see Ref. [7]. For the results presentedhere, which account for higher Q2 values, the HF and CRPA cross sections are comparable.SHORT-RANGE CORRELATIONSTo account for SRCs in neutrino-nucleus scattering, we rely on a model developed forexclusive as well as semi-exclusive electron-nucleus scattering cross sections [15–18]. Thiswork is a first step in the extension of this model towards the weak sector.The correlated wave functions |Ψ〉 are constructed by applying a many-body correlationoperator Ĝ to the uncorrelated wave functions |Φ〉|Ψ〉 = 1√N Ĝ|Φ〉, (9)with N = 〈Φ|Ĝ†Ĝ|Φ〉 the normalization constant. In the construction of the correlationoperator, one is guided by the features of the one-boson exchange nucleon-nucleon force.In this work, only the central and tensor part of the correlation operator are considered,NuFact15 - Rio de Janeiro, Brazil - August 2015 6spin-isospin correlations will be included in future work,Ĝ = Ŝ(A∏i<j[1 + l̂(i, j)]), (10)withl̂(i, j) = −ĝ(i, j) + t̂(i, j) (11)= −gc(rij) + ftτ (rij)Ŝij (~τi · ~τj) , (12)where Ŝ is the symmetrization operator, Ŝij the tensor operator3r2ij(~σi · ~rij) (~σj · ~rij) −(~σi · ~σj), gc(rij) the central correlation function and ftτ (rij) the tensor correlation function.In the calculations presented in this work we used the central correlation function parame-terization by Gearhaert and Dickhoff [19] and the tensor correlation function by Pieper etal. [20]. Ref. [21] provides arguments and evidence to support the fact that these correlationfunctions can be considered realistic.When calculating transition matrix elements between correlated states |Ψ〉, one can shiftthe effect of the correlations towards the transition operators and calculate matrix elementsbetween uncorrelated states |Φ〉 with an effective transition operator. In the IA, the many-body nuclear current operator Ĵnuclλ can be written as a sum of one-body currents Ĵ[1]λ (i).To account for SRCs, the current in Eq. 4 is replaced with an effective current〈Ψf|Ĵnuclλ |Ψi〉 =1N 〈Φf|Ĝ†Ĵnuclλ Ĝ|Φi〉 =1N 〈Φf|Ĵeffλ |Φi〉, (13)withĴeffλ =(A∏j<k[1 + l̂(j, k)])† A∑i=1Ĵ[1]λ (i)(A∏l<m[1 + l̂(l,m)]). (14)Relying on the short-range behavior of the correlations, the effective current is approximatedasĴeffλ ≈A∑i=1Ĵ[1]λ (i) +A∑i<jĴ[1],inλ (i, j) +[A∑i<jĴ[1],inλ (i, j)]†, (15)where the first term is the nuclear current in the IA, and the second term is a two-bodycurrent which is the product of a one-body current and a correlation operatorĴ[1],inλ (i, j) =[Ĵ[1]λ (i) + Ĵ[1]λ (j)]l̂(i, j). (16)NuFact15 - Rio de Janeiro, Brazil - August 2015 7dσ/dEµdΩµdEbdΩbdΩa(10−45cm2/MeV2)90180270360θa (◦)090180270360θb (◦)00.511.522.533.544.5FIG. 2: Exclusive 12C(νµ, µ−NaNb) cross section at Eνµ = 750 MeV, Eµ = 550 MeV, θµ = 15◦ andTp = 50 MeV with outgoing nucleons in the lepton scattering plane.This model is used to study the effect of SRCs on the quasielastic double differential neutrino-nucleus scattering cross section. Due to the two-body structure of the effective operator,the SRCs influence the 1p1h as well as the 2p2h channel.FIG. 2 shows the result of an exclusive cross section calculation. A striking feature of thedisplayed cross section is the dominance of back-to-back nucleon knockout, reminiscent ofthe ’hammer events’ seen by the ArgoNeuT collaboration [22]. This feature is independentof the interacting lepton or the type of two-body interaction [16, 23].The contribution of the 2p2h channel to the double differential cross section involves anintegration over the phase space of the undetected nucleons as outlined in Refs. [15, 23]dσdEµdΩµ(νµ, µ−) =∫dTbdΩbdΩadσdEµdΩµdTbdΩbdΩa(νµ, µ−NaNb). (17)In FIG. 3, double differential CRPA 12C(e, e′) calculations are compared with the modelincluding SRCs in the 1p1h and 2p2h channels. The CRPA suppression in the QE-regionis visible as well as the increase of the cross section in the dip-region due to the two-particle knockout of short-range correlated pairs. FIG. 4 compares the influence of long-and short-range correlations, accounting for one- and two-particle knockout, on the mean-field 12C(νµ, µ−) cross section, for three kinematics relevant for accelerator-based neutrino-oscillation experiments. Both models result in a decrease of the cross section at the QE-peak.NuFact15 - Rio de Janeiro, Brazil - August 2015 8010203040500 50 100 150 200 250 30000.511.522.533.50 100 200 300 400 5000510152025300 100 200 300 400 500dσ/dEe′ dΩe′ (10−33cm2/MeV)ω (MeV)12C, Ee = 680 MeV, θe′ = 36◦BarreauCRPASRCω (MeV)12C, Ee = 680 MeV, θe′ = 60◦Barreauω (MeV)12C, Ee = 730 MeV, θe′ = 37.1◦O’ConnellFIG. 3: Double differential 12C(e, e′) cross section for three kinematics. Data are from Refs. [12, 24].0501001502002503003504004505000 50 100 150 200 2500204060801001201401600 100 200 300 400 50005101520253035400 100 200 300 400 500 600dσ/dEµdΩµ(10−42cm2/MeV)ω (MeV)12C, Eνµ = 750 MeV, θµ = 15◦HFCRPASRCω (MeV)12C, Eνµ = 750 MeV, θµ = 30◦ω (MeV)12C, Eνµ = 750 MeV, θµ = 60◦FIG. 4: Double differential 12C(νµ, µ−) cross section for three kinematics.SUMMARYWe have presented a discussion of long- and short-range correlations in quasielasticcharged-current neutrino-nucleus scattering. We confronted our numerical results withdouble-differential inclusive (e, e′) electron scattering data and calculated double differential(νµ, µ−) neutrino-nucleus scattering cross sections at energies relevant for recent measure-ments. A fair agreement with electron-scattering data was reached in the region wherethe quasielastic channel is expected to dominate. Furthermore, the framework allows forthe prediction of exclusive cross sections, which might provide deeper insight in neutrinoexperiments detecting the nuclear final state.This work was supported by the Interuniversity Attraction Poles Programme P7/12 ini-tiated by the Belgian Science Policy Office and the Research Foundation Flanders (FWO-Flanders). The computational resources (Stevin Supercomputer Infrastructure) and servicesused in this work were provided by Ghent University, the Hercules Foundation and the Flem-ish Government.NuFact15 - Rio de Janeiro, Brazil - August 2015 9∗ Presented at NuFact15, 10-15 Aug 2015, Rio de Janeiro, Brazil [C15-08-10.2]† Electronic address: Tom.VanCuyck@UGent.be; Speaker‡ Electronic address: Vishvas.Pandey@UGent.be§ Electronic address: Natalie.Jachowicz@UGent.be[1] J. Ryckebusch, M. Waroquier, K. Heyde, J. Moreau, and D. Ryckbosch, Nucl.Phys. A476,237 (1988).[2] J. Ryckebusch, K. Heyde, D. Van Neck, and M. Waroquier, Nucl.Phys. 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Correlations in neutrino-nucleus scattering

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Correlations in neutrino-nucleus scattering

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