The quasielastic scattering of muon neutrinos on oxygen 16 is studied for
neutrino energies between 200 MeV and 1 GeV using a relativistic shell model.
Final state interactions are included within the distorted wave impulse
approximation, by means of a relativistic optical potential, with and without
imaginary part, and of a relativistic mean field potential. For comparison with
experimental data the inclusive charged--current quasielastic cross section for
$\nu_\mu$--$^{12}C$ scattering in the kinematical conditions of the LSND
experiment at Los Alamos is also presented and briefly discussed.Comment: 4 pages, 5 figures, two-column format. Accepted as brief report in
  Phys. Rev. 

B. D. Serot

C. Giusti

C. J. Horowitz

C. K. Jung

C. Maieron

E. Kolbe

J. A. Caballero

J. M. Udías

M. Albert

M. C. Martínez

P. Barreau

W. M. Alberico

English

arXiv

The  quasielastic  scattering  of  muon  neutrinos  on  oxygen  16  is  studied  for  neutrino  energies  between 200 MeV and 1 GeV using a relativistic shell model. Final state interactions are included within the distorted wave impulse approximation, by means of a relativistic optical potential, with and without imaginary part, and of  a  relativistic  mean  field  potential.  For  comparison  with  experimental  data  the  inclusive  charged-current quasielastic cross section for nm−12C scattering in the kinematical conditions of the LSND experiment at Los Alamos is also presented and briefly discussed

Maieron, C.

Martínez, M. C.

Caballero Carretero, Juan Antonio

Udías, J. M.

idUS. Depósito de Investigación Universidad de Sevilla

Nuclear model effects in charged-current neutrino-nucleus quasielastic scatteringC. Maieron, M. C. Martínez, and J. A. CaballeroDepartamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, E-41080 Sevilla, SpainJ. M. UdíasDepartamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain(Received 28 March 2003; published 8 October 2003)The quasielastic scattering of muon neutrinos on oxygen 16 is studied for neutrino energies between200 MeV and 1 GeV using a relativistic shell model. Final state interactions are included within the distortedwave impulse approximation, by means of a relativistic optical potential, with and without imaginary part, andof a relativistic mean field potential. For comparison with experimental data the inclusive charged-currentquasielastic cross section fornm−12C scattering in the kinematical conditions of the LSND experiment at LosAlamos is also presented and briefly discussed.DOI: 10.1103/PhysRevC.68.048501 PACS number(s): 25.30.Pt, 13.15.1g, 24.10.JvIn the past few years the observation of neutrino oscilla-tions at Super-Kamiokande[1] and the subsequent proposaland realization of new experiments, aimed at determiningneutrino properties with high accuracy[2], have renovatedthe interest towards neutrino scattering on complex nuclei. Infact, neutrino detectors usually contain carbon or oxygen nu-clei, and for a proper interpretation of the experimental re-sults the description of then-nucleus interaction must beaccurate[3].At intermediate neutrino energies, ranging from somehundreds MeV to a few GeV,n-nucleus quasielastic scatter-ing has been studied within several approaches[4]. Relativ-istic and nonrelativistic studies of random phase approxima-tion have shown nuclear structure effects to be relevant onlyat low momentum transfers, but indications have been foundthat for future and precise data analyses of, e.g., atmosphericneutrino measurements, more accurate theoretical estimatesmay be needed. Additionally, very recently attention hasbeen drawn towards final state interaction(FSI) effects,which, contrary to what is often assumed, may still be rel-evant even at the relatively high energyEn=1 GeV [5].In this contribution we study charged-current(CC)neutrino-nucleus quasielastic scattering within the frame-work of a relativistic shell model(RSM), already success-fully employed to study exclusive electron scattering[6] andneutral current neutrino scattering[7]. We compute inclusivenm-16O quasielastic cross sections for three values of the in-cident neutrino energy, namely, 200 MeV, 500 MeV, and1 GeV, which are representative of the kinematical rangewhere quasielastic scattering gives the main contribution tothe inclusiven-nucleus process.We describe the CC quasielastic scattering of neutrinos ona nuclear target within the impulse approximation(IA ), as-suming that the incident neutrino exchanges one vector bo-son with only one nucleon, which is then emitted, while theremainingsA−1d nucleons in the target are spectators. Thenuclear current is assumed to be the sum of single-nucleoncurrents, for which we employ the usual free nucleon expres-sion (see Ref.[7]) with the axial form factor parametrized asa dipole with cutoff massMA=1.026 GeV[8], and the statesof the target and residual nuclei to be adequately describedby an independent particle model wave function.To describe the bound nucleon states we use relativisticshell model wave functions, obtained as the self-consistent(Hartree) solutions of a Dirac equation, derived, within arelativistic mean field approach, from a Lagrangian contain-ing s, v, and r mesons[9]. As the single-particle bindingenergies determine the threshold of the cross section for ev-ery shell, in the numerical calculations, we have used theexperimental values corresponding to the binding energies ofthe different shells.For the outgoing nucleon the simplest choice is to useplane wave spinors, i.e., no interaction is considered betweenthe ejected nucleon and the residual nucleus[plane waveimpulse approximation(PWIA)]. For a more realistic de-scription, FSI effects should be taken into account. In ourformalism this is done by using distorted waves which are051015202530354040 60 80 100 120 140 160 180 200dσ/dΩdω (nb sr-1  MeV-1)ω (MeV)RMFreal ROPcomplex ROPRSM-PWIAFIG. 1. Differential cross sectionds/dVdv vs the energy trans-fer for inclusive quasielastic electron scattering on12C for a mo-mentum transferq.400 MeV/c. The solid curve corresponds to theRSM with FSI described by the RMF, while the long-dashed curvecorresponds to the real ROP and the short-dashed curve to the com-plex ROP. Finally, the dotted curve does not include FSI. Experi-mental data are from Ref.[11].PHYSICAL REVIEW C 68, 048501(2003)0556-2813/2003/68(4)/048501(4)/$20.00 ©2003 The American Physical Society68 048501-1given as solutions of a Dirac equation containing a phenom-enological relativistic optical potential(ROP), consisting of areal part, which describes the rescattering of the ejectednucleon and of an imaginary part that accounts for the ab-sorption of it into unobserved channels. In this work we usethe ROP corresponding to the energy-dependent-A-dependent-1(EDAD-1) single-nucleon parametrization pre-sented in Ref.[10]. The use of this phenomenological ROPleads to an excellent agreement between theoretical calcula-tions and data for exclusivese,e8Nd observables[6]; how-ever, some caution should be taken in extending the conclu-sions drawn from the analysis of exclusive reactions toinclusive ones. In the latter, unless a selection of the single-nucleon knockout contribution is experimentally feasible, allfinal channels are included and thus the imaginary term inthe optical potential leads to an overestimation of FSI effects.This is clearly illustrated in Fig. 1 where we compare ourtheoretical results with the experimental cross section corre-sponding to inclusive quasielastic electron scattering on12Cfor a momentum transfer equal to 400 MeV/c (similar resultsare obtained for 300 and 500 MeV/c). In our calculationbesides the complex ROP, we also consider the potential ob-tained by setting the imaginary part of the ROP to zero.Additionally, one may also use distorted waves which areobtained as the solutions in the continuum of the same Dirac 0 50 100 150 200 250 300 350 0  20  40  60  80  100dσ/d Tµ (10-39 cm2 /GeV)200 MeVRSM-PWIARFG, eB=0RFG, eB=20 MeV 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-PWIARFG, eB=0RFG, eB=20 MeV 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-PWIARFG, eB=0RFG, eB=20 MeVFIG. 2. Differential cross sectionds/dTm vs the outgoing muonkinetic energy, for the quasielastic scattering of muon neutrinos on16O and for three choices of the incident neutrino energy:En=200 MeV (upper panel), 500 MeV (middle panel), and 1 GeV(lower panel). The solid curves correspond to the RSM with no finalstate interaction, while the remaining curves are calculated withinthe RFG, with kF=225 MeV, and eB=0 (dot-dashed) and eB=20 MeV (dotted). 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 ROPFIG. 3. Same as Fig. 2, but including FSI effects. All curves arecalculated within the RSM model in PWIA(solid), and within theRMF (dashed), real ROP(dotted), and complex ROP(dot-dashed)approaches.BRIEF REPORTS PHYSICAL REVIEW C68, 048501(2003)048501-2equation used to describe the initial bound nucleon. We referto this approach, which should be adequate to describe FSI atmoderate energy transfer, as relativistic mean field(RMF).As shown in Fig. 1, the complex ROP results clearly under-estimate the data, while the reverse occurs for the PWIAcalculation. On the other hand, the RMF and purely realROP agree much better with experiment, particularly RMFfor small transfer energy and real ROP for higherv. Webelieve that the RMF and real ROP results indicate a reason-able “band” where FSI effects should lie.Let us now present our results for neutrino scattering on16O. To better illustrate our model, we start by neglecting FSIand comparing RSM-PWIA results with inclusive cross sec-tions obtained within the relativistic Fermi gas(RFG) [7].This is done in Fig. 2, which shows the differential crosssectionds/dTm as a function of the outgoing muon kineticenergy. With respect to the RFG curve calculated with nobinding energy we observe that the RSM cross section isreduced and shifted towards lowerTm values, in a way whichis similar to the effect of an average binding energy in theRFG. In addition, the RSM cross section has a differentshape, due to the different momentum distributions of thesingle-nucleon shells contributing to the process. Since thevarious shells have different binding energies, the corre-sponding contributions to the cross section go to zero at dif-ferent values ofTm and this gives rise to the structure ofsds/dTmd observed at largeTm. Figure 2 shows that nuclearmodel effects on the cross sections can be rather large at lowneutrino energy, but become less relevant asEn increases,practically disappearing atEn=1 GeV.On the other hand, the behavior of FSI effects is quitedifferent, as illustrated in Fig. 3. Here the results obtainedwith the RSM in PWIA are compared with the cases whereFSI are described within the RMF, the real ROP and thecomplex ROP approaches. The use of real potentials(RMFand real ROP) for describing the final nucleon states leads tothe resonant structure observed for relatively highTm (that is,small energy transferv). Note that in this work we onlyinclude single-particle excitations within a mean field pic-ture. Including residual interactions would make the widthand number of resonances to be considerably larger.We observe that FSI effects produce a reduction of thecross section, particularly important in the case of the com-plex ROP model due to the absorption introduced by theimaginary term: about 60% forEn=200 MeV and 50% forEn=500 MeV andEn=1 GeV in the region close to the maxi-mum. For the RMF and real ROP, the reduction, similar inboth cases, is about 30–40 % forEn=200 MeV and 20% forthe other energy values.Nuclear model effects on integrated cross sections arestudied in Fig. 4, where the cross sectionssEnd is plotted asa function of the incident neutrino energy. Here the contri-butions coming from the RMF and ROP resonances havebeen included in the calculation, in order to respect the com-pleteness of the set of final states predicted by the model.These contributions are important atEn=200 MeV, wherethey amount to about 10% of the integrated cross section,while at higher energies these effects are about 2%s500 MeVd and 1%s1 GeVd. Again we see that within thePWIA the discrepancy between different nuclear models isrelatively small and decreases with increasing neutrino en-ergy. On the contrary FSI effects remain sizable even at largeEn. As in the previous figure, the imaginary term in the ROPleads to a too large reductions,50%d of the integrated crosssection. The results for the RMF and real ROP models,hich are more reliable, show a smaller, but still sizable(,15% atEn=1 GeV) reduction.The results shown in Figs. 3 and 4 lead us to concludethat FSI effects should be carefully considered in neutrinoexperiments which use oxygen based detectors. Analogousconclusions can be drawn for the case of a12C target.For the purpose of comparison with the experiment, let usconsider the inclusive12Csnm,m−dX cross section measuredby the LSND Collaboration at Los Alamos, using a pion-decay-in-flightnm beam, with energies ranging from muonthreshold to 300 MeV, and a large liquid scintillator detector 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 MeVFIG. 4. Integrated cross sec-tion ssEnd for the quasielasticscattering of muon neutrinos on16O as a function of the incidentneutrino energy. The curves arecalculated within the RFG modelwith kF=225 MeV, and bindingenergy eB=0 (solid line) and eB=20 MeV (dashed). The pointscorrespond to RSM calculationswithout FSI (stars) and with FSIeffects taken into account withinthe RMF (empty squares), realROP (full squares), and complexROP(circles) approaches.BRIEF REPORTS PHYSICAL REVIEW C68, 048501(2003)048501-3[12]. This experiment has been compared to very differenttheoretical approaches[13], but an important discrepancystill remains. Although at these low energies processes dif-ferent from the quasielastic nucleon knockout affect the in-clusive cross section, we consider our FSI approach to beuseful to estimate integrated cross sections.In Fig. 5 we show the observed[12] and calculated dis-tribution of events, averaged over the 1994 Los Alamos neu-trino spectrum fsEnd within the energy range En=123.1–300 MeV. The shape and position of the maximumof the experimental distribution are approximately repro-duced by the three calculations, but the results that includeFSI with the RMF potential are clearly favored by the data.This is consistent with the fact that including FSI with themean field potential should be adequate at moderate kineticenergy of the ejected nucleon. However, the values we obtainfor the flux-averaged integrated cross section overestimatethe measured cross section by<50%. More precisely, in theRSM, we obtainksl=20.5 (PWIA), 16.8 (RMF), and 15.1(real ROP) 10−40 cm2. Additional corrections due to the out-going muon Coulomb distortion, evaluated within the effec-tive momentum approximation[14], further increase thesenumbers by 5–10 %. The corresponding final measured ex-perimental value iss10.6±0.3±1.8d310−40 cm2 [12].This work was partially supported by DGI(Spain) underContract Nos. BFM2002-03315, FPA2002-04181-C04-04,and BFM2000-0600 and by the Junta de Andalucía. C.M.acknowledges MEC(Spain) for a postdoctoral stay at Uni-versity of Sevilla(Contract No. SB2000-0427). M.C.M. ac-knowledges support from the Fundación Cámara(Universityof Sevilla).[1] Y. Fukudaet al., Super-Kamiokande Collaboration, Phys. Rev.Lett. 81, 1562(1998); C. K. Jung, C. McGrew, T. Kajita, andT. Mann, Annu. Rev. Nucl. Part. Sci.51, 451 (2001).[2] See, for example, A. De Santo, Int. J. Mod. Phys. A16, 4085(2001).[3] See Neutrino Nucleus Interactions in the Few Gev Region,Proceedings of the First International Workshop, Nuint01,Tsukuba, Japan, 2001, edited by J. G. Morfin, M. Sakuda, andY. Suzuki (North-Holland, Amsterdam, 2002).[4] T. K. Gaisser and J. S. O’Connell, Phys. Rev. D34, 822(1986); H. c. Kim, J. Piekarewicz, and C. J. Horowitz, Phys.Rev. C 51, 2739 (1995); H. c. Kim, S. Schramm, and C. J.Horowitz, ibid. 53, 2468(1996); 53, 3131(1996); S. K. Singhand E. Oset,ibid. 48, 1246 (1993); J. Engel, E. Kolbe, K.Langanke, and P. Vogel, Phys. Rev. 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Heyde, J. Ryckebusch, and S. Rombouts,Phys. Rev. C65, 025501(2002); C. Volpe et al., ibid. 62,015501(2000); E. Kolbe et al., ibid. 52, 3437 (1995); Nucl.Phys. A652, 91 (1999); Y. Umino, J. M. Udías, and P. J. Mul-ders, Phys. Rev. Lett.74, 4993 (1995); Y. Umino and J. M.Udías, Phys. Rev. C52, 3399(1995).[14] C. Giusti and F. D. Pacati, Nucl. Phys.A473, 717(1987), andreferences therein.05101520250 20 40 60 80 100 120dσ/dTµ (10-42 cm2 /MeV)Tµ (MeV)RMFreal ROPRSM-PWIAFIG. 5. Observed distribution of muon kinetic energiesTm com-pared with the flux-averaged predictions of our RSM, in PWIA(dotted line) and including FSI within the RMF(solid) and purelyreal ROP(dashed) frameworks. The theoretical distributions havebeen normalized to give the same integrated values as the experi-mental points, and have been folded in energy with a bin size of 5MeV, the same employed for the experimental data. Data are fromAlbert et al. [12].BRIEF REPORTS PHYSICAL REVIEW C68, 048501(2003)048501-4

Nuclear model effects in charged-current neutrino-nucleus quasielastic scattering

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Nuclear model effects in Charged Current neutrino--nucleus quasielastic
  scattering

Nuclear model effects in Charged Current neutrino--nucleus quasielastic scattering

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