We consider the charged-current quasielastic scattering of muon neutrinos on
an Oxygen 16 target, described within a relativistic shell model and, for
comparison, the relativistic Fermi gas. Final state interactions are described
in the distorted wave impulse approximation, using both a relativistic mean
field potential and a relativistic optical potential, with and without
imaginary part. We present results for inclusive cross sections at fixed
neutrino energies in the range $E_\nu =$ 200 MeV - 1 GeV, showing that final
state interaction effects can remain sizable even at large energies.Comment: 4 pages, 4 figures; poster session of the Third International
  Workshop on Neutrino-Nucleus Interactions in the Few GeV Region (NuInt04),
  Gran Sasso (Italy), March 17-21, 2004; to appear in the proceeding

Alberico

Bleve

C. Maieron

J.A. Caballero

J.M. Udías

M.C. Martínez

Maieron

Meucci

Udías

Zeller

English

arXiv

Crossref

Final state interaction effects in neutrino-nucleus quasielastic scattering

We consider the charged-current quasielastic scattering of muon neutrinos on an Oxygen 16 target, described within a relativistic shell model and, for comparison, the relativistic Fermi gas. Final state interactions are described in the distorted wave impulse approximation, using both a relativistic mean field potential and a relativistic optical potential, with and without imaginary part. We present results for inclusive cross sections at fixed neutrino energies in the range Eν = 200 MeV - 1 GeV, showing that final state interaction effects can remain sizable even at large energies

Maieron, C.

Martínez, M. C.

Caballero Carretero, Juan Antonio

Udías, J. M.

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:nucl-th/0407094v1  27 Jul 2004 Final state interaction effects in neutrino-nucleus quasiel tic scatteringC. Maierona, M.C. Martı́nezb, J.A. Caballerob, and J.M. Udı́ascaIstituto Nazionale di Fisica Nucleare, Sezione di Catania,Vi S. Sofia 64, I-95123 Catania, ItalybDepartamento de Fisica Atomica, Molecular y Nuclear, Universidad de Sevilla, E-41080 Sevilla, SpaincDepartamento de Fisica Atomica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid,SpainWe consider the charged-current quasielastic scattering of mu n neutrinos on an Oxygen 16 target, described within arelativistic shell model and, for comparison, the relativistic Fermi gas. Final state interactions are described in the distortedwave impulse approximation, using both a relativistic meanfield potential and a relativistic optical potential, with and withoutimaginary part. We present results for inclusive cross sections at fixed neutrino energies in the rangeEν = 200 MeV - 1 GeV,showing that final state interaction effects can remain sizable even at large energies.1. INTRODUCTION AND FORMALISMA precise and realistic description of neutrino-nucleus interaction in the intermediate energy regionis crucial for the interpretation of experimental resultsused to determine neutrino properties. At presentmost of the Monte Carlo codes [1] which have beendeveloped to simulate the response of the detectors inthese experiments are based on the Fermi gas model.Therefore they take into account the Fermi motion ofthe nucleons inside the nucleus and Pauli blocking ef-fects, but they neglect several other effects, which areknown to be important from electron scattering exper-iments [2].In order to study the relevance of some of theseeffects, in this contribution we compare relativisticFermi gas (RFG) calculations of neutrino-nucleusquasielastic scattering cross sections [3], with resultsobtained within a relativistic shell model (RSM), in-cluding, in particular, final state interactions (FSI) [4].More details about our calculations can be found inRef. [5]. FSI effects inν-nucleus scattering have alsobeen studied in Refs. [6,7].We consider the charged-current (CC) quasielasticscattering of muon neutrinos on16O at fixed neutrinoenergies of 200, 500 and 1000 MeV, describing thisprocess within the impulse approximation, schemati-cally represented in fig. 1.We thus assume that the incident neutrino interactswith only one nucleon, which is then emitted, whilethe remaining (A-1) nucleons in the target are spec-tators, that the nuclear current is the sum of singlenucleon currents and that the states of the target andresidual nuclei are described by independent particlemodel wave functions. We describe the ground stateof 16O as a closed shell configuration, the occupiedshells beings1/2, p3/2 and p1/2. For the removal ofa nucleon from a closed shell of angular momentumj, the cross section corresponding to the diagram infig. 1 has the following general form:d6σd3k′d3pN=∫δ4(q+ pA− pA−1− pN)G2F(2π)5(1)× (2 j +1)8EνEµ∑|ū(k′)γα (1+ γ5)u(k)Jα(q)|2d3pA−1,whereGF is the Fermi constant,∑ indicates the aver-age/sum over the initial/final spins,γ5 = −iγ0γ1γ2γ3and Dirac spinors are normalized according tou(k)†u(k) = 2k0. We then sum over the occupiedshells and integrate over the emitted nucleon and overthe direction of the outgoing muon in order to get theinclusive cross sectionsdσ/dTµ, and the integrated in-clusive cross sectionσ =∫(dσ/dTµ)dTµ, Tµ beingthe outgoing lepton kinetic energy.The main ingredient in eq. (1) is the single nucleon12Figure 1. Born approximation diagram for CCν–nucleus quasielastic scattering. The impulse approxi-mation is assumed in the hadronic vertex and the pos-sibility of final state interactions between the outgoingnucleon and the residual nuclear system is explicitlyshown.current matrix element,Jα(q) =√V∫d3reiq·r ψsN(pN,r)Γ̂α ψjmB (r) , (2)whereψ jmB (r) andψsN(pN, r) are the wave functionsfor the initial (bound) nucleon and for the emitted nu-cleon, respectively, and̂Γα is the single nucleon weakcharged-current operator. For the latter we assume thefree, on mass shell, nucleon expression:Γ̂α = |Vud|[FVγα +FMi2mNσαβqβ+ FAγαγ5−FPqαγ5] , (3)where FV,M are the CC single nucleon Pauli andDirac form factors,FA andFP are the axial and theinduced pseudoscalar form factor, respectively, and|Vud| is theud Cabibbo–Kobayashi–Maskawa matrixelement. ForFA we assume a dipole parameterizationwith cutoff massMA = 1.026 GeV.In eq. (2) the bound nucleon wave functionsψ jmB (r)are the self–consistent (Hartree) solutions of a Diracequation, derived, within a relativistic mean field ap-proach, from a Lagrangian containingσ, ω and ρmesons,which has been already successfully used in0501001502002503003500 20 40 60 80 100dσ/d Tµ (10-39 cm2 /GeV)200 MeVRSM-PWIARFG, eB=0RFG, eB=20 MeV0501001502002503003500 50 100 150 200 250 300 350 400dσ/d Tµ (10-39 cm2 /GeV)500 MeVRSM-PWIARFG, eB=0RFG, eB=20 MeV0501001502002503003500 100 200 300 400 500 600 700 800 900dσ/d Tµ (10-39 cm2 /GeV)Tµ (MeV)1 GeVRSM-PWIARFG, eB=0RFG, eB=20 MeVFigure 2. Differential cross section(dσ/dTµ) versusthe outgoing muon kinetic energy, for the quasielasticscattering of muon neutrinos on16O and incident neu-trino energy:Eν = 200 MeV (upper panel), 500 MeV(middle) and 1 GeV (lower panel).the study of(e,e′p) processes [4]. For the single-particle binding energies of the different shells we usethe corresponding experimental values, which deter-mine the threshold of the cross section for every shell.Concerning the emitted nucleon, as a starting pointwe describe it as a plane wave, thus neglecting its in-teraction with the residual nucleus (plane wave im-pulse approximation - PWIA). Then, to make our de-scription more realistic, we include the effects of FSIby using distorted waves (distorted wave impulse ap-proximation - DWIA). For the latter, we make the fol-lowing different choices.Complex ROP: following previous studies of ex-3 0 50 100 150 200 250 0  10  20  30  40  50  60  70  80  90dσ/d Tµ (10-39 cm2 /GeV)200 MeVRSM-PWIARMFreal ROPcomplex ROP 0 50 100 150 200 250 300 350 0  50  100  150  200  250  300  350  400dσ/d Tµ (10-39 cm2 /GeV)500 MeVRSM-PWIARMFreal ROPcomplex ROP 0 50 100 150 200 250 300 350 0  100  200  300  400  500  600  700  800  900dσ/d Tµ (10-39 cm2 /GeV)Tµ (MeV)1 GeVRSM-PWIARMFreal ROPcomplex ROPFigure 3. Same as fig. 2, but including FSI effects. Allcurves are calculated using the RSM model, in PWIA(solid) and describing FSI within the RMF (dashed),real ROP (dotted) and complex ROP (dot–dashed) ap-proaches.clusive electron scattering processes [4], we employdistorted waves which are obtained as solutions of aDirac equation containing a phenomenological rela-tivistic optical potential (ROP). The ROP has a realpart, which describes the rescattering of the ejectednucleon and an imaginary part, that accounts for theabsorption of it into unobserved channels.Real ROP:since, contrary to the(e,e′p) case, herewe are considering inclusive processes, where all fi-nal channels contribute, the presence of the imaginaryterm in the optical potential leads to an overestimationof FSI effects. For this reason we also consider thepotential obtained by setting the imaginary part of theROP to zero (a discussion on the use of real opticalpotentials has been presented in Ref. [7]).RMF: finally, we employ wave functions which areobtained as the solutions in the continuum of the sameDirac equation which is used to derive the bound nu-cleon wave functions. We refer to this approach asrelativistic mean field (RMF) and consider it appro-priate at low energy transfer.2. RESULTS AND CONCLUSIONSLet us now present the results we obtain forneutrino-nucleus quasielastic cross sections. As a firststep we neglect FSI effects and just compare RSM-PWIA results with the corresponding curves obtainedwithin the relativistic Fermi gas, for which we use aFermi momentumpF = 225 MeV and binding energyeB = 0 or 20 MeV. This comparison is presented infig. 2, where the differential cross sectionsdσ/dTµ isplotted as a function of the outgoing muon kinetic en-ergy. We observe that the differences between the twomodels are quite large at low neutrino energy, but theypractically disappear atEν = 1 GeV.The situation is different if we “turn on” FSI ef-fects, as illustrated in fig. 3, where the RSM-PWIAresults of fig. 2 are compared with the DWIA ap-proaches previously outlined.We see that FSI effects produce a reduction of thecross section, with respect to PWIA. The RMF andreal ROP curves are quite similar, showing a reductionof about 30÷40% forEν = 200 MeV and 20% for theother energy values. In the case of the complex ROPmodel, instead, the reduction of the cross section israther large, between 60% (Eν = 200 MeV) and 50%(at higher energy), due to the absorption introducedby the imaginary term.Let us note that the resonant structure appearing athigh Tµ is due to the use of real potentials (RMF andreal ROP) for describing the final nucleon state. Herewe only include single-particle excitations within amean field picture, while including residual interac-tions would make the width and number of resonancesto be considerably larger.Finally, the results of figures 2 and 3 are summa-rized in fig. 4, where the cross sectionσ, integratedover the muon energy, is plotted as a function ofthe incident neutrino energyEν. Again we see thatwithin the PWIA the discrepancy between different4 0 100 200 300 400 500 600 700 800 900 0  0.2  0.4  0.6  0.8  1σ (10-40 cm2 )Eν (GeV)RSM-PWIARMFreal ROPcomplex ROPRFG, eB=0RFG, eB=20 MeVFigure 4. Integrated cross sectionσ for the quasi-elastic scattering of muon neutrinos on16O as a func-tion of the incident neutrino energy. The curves arecalculated within the RFG model, while the pointscorrespond to RSM calculations without FSI (stars)and with FSI effects taken into account within theRMF (empty squares), real ROP (full squares) andcomplex ROP (circles) approaches.nuclear models is relatively small and decreases withincreasing neutrino energy, while FSI effects are stillpresent even at largeEν. The (more reliable) resultsfor the RMF and real ROP approaches show a not toolarge, but still appreciable reduction (∼ 15% atEν = 1GeV), while, again, the imaginary term in the com-plex ROP leads to a too large reduction (∼ 50%) ofthe cross section.In conclusion, within the Impulse Approximation,we observe that when FSI effects are neglected,the nuclear model dependence of the quasielasticν-nucleus cross section is large at low neutrino energybut becomes negligible atEν = 1 GeV. FSI effects arealso rather large atEν ≃ 200 MeV and decrease withincreasing neutrino energy, but, in the more realisticreal ROP approach, can still be as large as 15 % atEν= 1 GeV and thus must be carefully considered.As a final remark, we notice that if we considerNeutral Current quasielastic processes, where the out-going nucleon must be detected, then using the com-plex ROP description of FSI would be more appropri-ate. Therefore in this case rather large FSI effects areexpected to be present even at relatively high neutrinoenergy [3,8].REFERENCES1. G. P. Zeller, arXiv:hep-ex/0312061 and refer-ences therein.2. see for example the contributions by O. Benharand by C. Giusti to these proceedings.3. W. M. Albericoet al., Nucl. Phys. A 623, (1997)471; Phys. Lett. B 438 (1998) 9.4. J.M. Udı́aset al., Phys. Rev. C 48 (1993) 2731;Phys. Rev. C 51 (1995) 3246; Phys. Rev. C 53(1996) R1488; Phys. Rev. C 64 (2001) 024614.5. C. Maieron, M.C. Martı́nez, J.A. Caballero,J.M. Udı́as, Phys. Rev. C 68 (2003) 048501.6. C. Bleveet al., Astropart. Phys. 16 (2001) 145;G. Cò, C. Bleve, I. De Mitri and D. Martello,Nucl. Phys. Proc. Suppl. 112 (2002) 210.7. A. Meucci, C. Giusti and F. D. Pacati, Nucl. Phys.A 739 (2004) 277.8. A. Meucci, C. Giusti and F. D. Pacati,arXiv:nucl-th/0405004.

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Final state interaction effects in neutrino-nucleus quasielastic
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Final state interaction effects in neutrino-nucleus quasielastic scattering

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