We compare local Fermi gas and shell model in muon capture in nuclei in order
to estimate the effect of finite nuclear size in low energy weak reactions.Comment: 6 pages, 8 figures. To be published in the Proceedings of 20th Max
  Born Symposium, Wroclaw (Poland), December 7-10, 200

Amaro, J. E.

Maieron, C.

Nieves, J.

Valverde, M.

Acta Physica Polonica B

English

arXiv

Presented by M. Valverde at the XX Max Born Symposium “Nuclear Effects in Neutrino
Interactions”, Wrocław, Poland, December 7–10, 2005.We compare local Fermi gas and shell model in muon capture in nuclei in order to estimate the effect of finite nuclear size in low energy weak reactions.This work was supported by DGI and FEDER funds, contracts BFM2005-00810 and
BFM2003-00856 and by the Junta de Andalucía (FQM-225)

Amaro Soriano, José Enrique

Nieves Pamplona, Juan Miguel

Valverde Hermosilla, Manuel

Repositorio Institucional Universidad de Granada

Vol. 37 (2006) ACTA PHYSICA POLONICA B No 8LOCAL FERMI GAS IN INCLUSIVE MUON CAPTUREFROM NUCLEI∗ ∗∗J.E. Amaro, J. Nieves, M. ValverdeDepartamento de Física Atómica, Molecular y Nuclear, Universidad de Granada18071 Granada, SpainC. MaieronINFN, Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy(Received June 19, 2006)We compare local Fermi gas and shell model in muon capture in nucleiin order to estimate the effect of finite nuclear size in low energy weakreactions.PACS numbers: 23.40.Bw, 25.30.–c, 21.60.Cs1. IntroductionIn this paper we study the importance of nuclear finite size effects ininclusive muon capture reactions. The inclusive muon capture process innucleiµ− +AZ X → X + νµ , (1)is very similar to neutrino scattering off nucleiνµ +AZ X → X + µ− (2)and is experimentally more accessible so it serves as a benchmark for testingtheoretical models of the latter process.The motivation for this investigation comes from the results of a model(published in [1]), which describes rather well inclusive 12C(νµ, µ−)X and12C(νe, e−)X cross sections near threshold and inclusive muon capture by∗ Presented by M. Valverde at the XX Max Born Symposium “Nuclear Effects in Neu-trino Interactions”, Wrocław, Poland, December 7–10, 2005.∗∗ This work was supported by DGI and FEDER funds, contracts BFM2005-00810 andBFM2003-00856 and by the Junta de Andalucía (FQM-225).(2303)2304 J.E. Amaro et al.nuclei. This approach, which is an extension of the quasi–elastic (QE) inclu-sive electron scattering model of [2], is based on a Local Fermi Gas (LFG),where the simplicity of the model makes it possible to include a great varietyof effects into the reaction dynamics. The goal of this paper is to investigatewhether finite nucleus effects that are not addressed by a LFG can affectsignificantly the results of the model in [1].There already exist microscopic calculations of neutrino–nucleus reac-tions and muon capture that treat correctly the finite size of the system.However, there are some dynamical issues that are implemented in a differ-ent fashion in a LFG model so a direct comparison of these models is notpossible and we can not extract the effect of the inclusion of nuclear structuredetails. For this reason, we perform a comparison of the LFG model witha extreme shell model (SM), i.e. single particle states in a Woods–Saxon(WS) potential, where the finite size effects can be easily recognized.In the second section we will introduce the model of [1] as applied to in-clusive nuclear muon capture and give some numerical results to be comparedwith experiment. Then we will introduce the extreme SM and a simplifiedLFG model. In the fourth section we will show the comparison between thetwo models. We will finish with some conclusions.More details on this issue can be found in [3]. A more comprehensiveanalysis of the uncertainties in the model of [1] can be found in [4].2. Inclusive muon capture in nucleiThe evaluation of the decay width for inclusive muon capture in finitenuclei proceeds in two steps. In the first one we evaluate the spin averageddecay width for a muon at rest in a Fermi sea of protons and neutronsΓ̂ (ρp, ρn) with N 6= Z. In the second step, we use the LFG approximationto go to finite nuclei and evaluateΓ =∫d3r|φ1s(r)|2 Γ̂ (ρp(r), ρn(r)) , (3)where φ1s(~r ) is the muon wave function in the 1s state from where the cap-ture takes place. It has been obtained by solving the Schrödinger equationwith a Coulomb interaction taking into account the finite size of the nucleusand vacuum polarization. This approximation assumes a zero range of theinteraction, that becomes highly accurate as long as the ~q dependence of theinteraction is extremely weak for the µ-atom decay process.The spin averaged muon decay width is related to the imaginary partof the self-energy of a muon at rest in the medium. Further details andanalytical expressions can be found in the Appendix of [1].Local Fermi Gas in Inclusive Muon Capture From Nuclei 2305For kinematical reasons only the QE part of the W− self-energy con-tributes to the muon decay. Thus, both the muon decay in the medium(Eq. (1)) and the electroweak inclusive nuclear reactions (Eq. (2)) in theQE regime are sensitive to the same physical features. We can apply thesame nuclear physics corrections to the above model as in neutrino scatter-ing, that is Pauli blocking, RPA corrections and corrections to the energybalance, see [1]. The 1s muon binding energy, B1sµ > 0, is taken into accountby replacing mµ → m̂µ = mµ − B1sµ .In muon capture only very small nuclear excitation energies are explored,0–25MeV, so the kinematical regime of the muon capture process is the worstpossible for a LFG model of the nucleus. Nevertheless, the predictions ofthis model are in fairly good agreement with the experimental results as canbe seen in Table I.TABLE IExperimental and theoretical total muon capture widths for different nuclei. Wequote two different theoretical results: (i) Pauli+Q: obtained without includingRPA correlations, but taking into account the value of Q; (ii) RPA: the full cal-culation, including all nuclear effects. Experimental data (Exp) are a weightedaverage: Γ/σ2 =∑iΓi/σ2i, with 1/σ2 =∑i1/σ2iof the results cited in [5]. Fi-nally, in the last column we show the relative discrepancies existing between thetheoretical predictions given in the third column and the experimental data of thefourth column. δrelΓ =(ΓExp − ΓTh)/ΓExp)Pauli+Q [104 s−1] RPA [104 s−1] Exp [104 s−1] δrelΓ12C 5.42 3.21 3.78 ± 0.03 0.1516O 17.56 10.41 10.24 ± 0.06 −0.0218O 11.94 7.77 8.80 ± 0.15 0.1223Na 58.38 35.03 37.73 ± 0.14 0.0740Ca 465.5 257.9 252.5± 0.6 −0.0244Ca 318 189 179 ± 4 −0.0675As 1148 679 609±4 −0.11112Cd 1825 1078 1061±9 −0.02208Pb 1939 1310 1311±8 0.003. Comparison of non-correlated modelsIn order to simplify the calculations in the comparison between the twomodels we make a static approximation and expand the single nucleon weakcurrent Jµ in the nucleon momentum keeping terms up to order zero.In the SM we have to deal with an S matrix element of the kindSfi = −2πiδ (Ef − Ei − ω)G√2ℓµ〈f |J̃µ(−k′)|i〉 , (4)2306 J.E. Amaro et al.where the weak current is modified to include the muon wave functionJ̃µ(−k′) =∫d3r e−ik′·rJµ(r)φ1s(r) . (5)Now the states |i〉 and 〈f | are nuclear states in a shell model. This statesare single particle excitations of a N nucleon system in a WS potential andcan be not only in the continuum, like in a Fermi gas, but there can also bediscrete excitations so we have now two contributions to the decay width.We now have to get the nuclear wave functions as solutions of theSchrödinger equation for a WS potential where the parameters of the poten-tial are commonly fitted to the experimental energies of the valence shellsor the charge radius. In the present case of muon capture we fit the experi-mental Q-value for the decay reaction.For the LFG model we use simplified expression for the decay width,where we have not taken into account RPA correlations, but energy balanceand Pauli blocking effects are implemented. The only inputs remaining tobe fixed are the nuclear matter densities, that will be those provided by thewave functions of the WS potential.Up to now we have in both LFG and SM models the same physicalfeatures of Pauli blocking and correct energy balance, the only differencecoming from the more refined treatment of the nuclear wave function in theSM case. In this way we can compare both SM and LFG models.4. ResultsIn this section we present results for a set of closed-shell nuclei 12C, 16O,40Ca and 208Pb. Fixing the experimental Q-value only makes one conditionfor fixing the several parameters of the WS potential, so wherever possiblewe set the remaining parameters of the potential to values similar to thoseused in the literature. In our calculation we use different sets of parameters,denoted WS1, WS2 and WS3. In order to compare with the LFG, we use asinput the proton and neutron densities obtained in the corresponding shellmodel. The values of the different parameters sets can be seen in [3].In Table II we show results for the integrated inclusive widths for thefour nuclei. We can see that, in the case of WS1 and WS2, the LFG andWS results for 12C are quite similar, differing only in ∼ 2–3%. In the caseof WS3 the differences are larger, around 14%.This can be understood in terms of the values of the parametrizationWS3. For more attractive potentials the nucleus becomes more dense in theinterior. For this reason, the WS3 neutron density turns out to be the small-est one, while the proton density is around 3/2 the neutron one. Therefore,a proton near the Fermi surface can decay to a neutron above the neutronLocal Fermi Gas in Inclusive Muon Capture From Nuclei 2307TABLE IIIntegrated width in units of 105s−1 for the different nuclei and Woods–Saxon po-tentials, compared with the LFG results using the corresponding charge densities.The discrete contribution of the shell model is shown in the first column. Thecolumn labeled % gives the relative difference between WS and LFG results.discrete total LFG %12C WS1 0.3115 0.4406 0.4548 3.2WS2 0.3179 0.4289 0.4360 1.7WS3 0.2746 0.5510 0.4732 −14.116O WS1 1.124 1.267 1.346 6.2WS2 0.584 1.107 1.378 24.4WS3 1.143 1.316 1.373 4.340Ca WS1 27.72 34.87 34.81 −0.1WS2 26.34 31.70 33.07 4.3WS3 24.91 30.64 33.19 8.3208Pb WS1 128.5 191.0 187.27 −1.9WS2 159.6 243.4 213.64 −12.2Fermi surface with an energy decrement. This is an unrealistic situation,since precisely in this case the neutrons are less bound than protons in theSM, and therefore, lie at higher energies. Hence the LFG results are worsefor very different neutron and proton densities. Another argument to disre-gard this case is the well known property of closed (sub-)shell nuclei such as12C, for which the neutron and proton densities should be similar.In Fig. 1 we compare the SM results for the differential width to thecontinuum with the LFG distribution for the different WS parameters (leftpanels). The shapes of both distributions are completely different. Thepartial widths to the discrete states are shown in the right panels of Fig. 1.Considering these differences in shape between the LFG and the SM, it isa very notable result that the integrated widths (adding the discrete states)take similar values in both models as was shown in Table II.In the case of 16O the integrated widths computed in the LFG are alsovery close, ∼ 4–6%, to the SM results with the potentials WS1 and WS3(see Table II). The worst results are obtained for the WS2 parameterization;the corresponding width is 24% of the SM one. This can also been under-stood in terms of what was said for the case of 12C above, by looking at the16O densities shown in Fig. 2.The LFG results improve when the mass of the nucleus increases as in thecase of the nucleus 40Ca. In fact, from Table II we see that for this nucleusthe LFG integrated width is lower than 8% for all cases. This improvementwas expected because the Fermi gas description of the nucleus should workbetter for heavier nuclei.2308 J.E. Amaro et al.NP12C WS1ρ(~r)[fm−3]0.10.080.060.040.02WS2ρ(~r)[fm−3]0.10.080.060.040.02WS3r [fm]ρ(~r)[fm−3]5432100.10.080.060.040.02LDAWS112C2.521.510.5032.521.510.50LDAWS2dΓc/dǫ′[104s−1/MeV]0.30.250.20.150.10.050Γi→f[104s−1]32.521.510.50LDAWS3ǫ′ [MeV]10095908580757065600.350.30.250.20.150.10.050ǫ′ [MeV]32.521.510.50100908070Fig. 1. The left figure shows proton and neutron densities of 12C, for the several WSpotentials used in this work. The right figure shows the differential SM width of 12Cto the continuum (left panels) compared to the LFG, and partial widths contribu-tions to the discrete states (right panels), as a function of the neutrino energy.NP12C WS1ρ(~r)[fm−3]0.10.080.060.040.02WS2ρ(~r)[fm−3]0.10.080.060.040.02WS3r [fm]ρ(~r)[fm−3]5432100.10.080.060.040.02LFGWS116O10.90.80.70.60.50.40.30.20.1043.532.521.510.50LFGWS2dΓc/dǫ′[104s−1/MeV]0.90.80.70.60.50.40.30.20.10Γi→f[104s−1]3.532.521.510.50LFGWS3ǫ′ [MeV]100959085807570656010.90.80.70.60.50.40.30.20.10ǫ′ [MeV]43.532.521.510.50100908070Fig. 2. The same as Fig. 1 for 16O.Local Fermi Gas in Inclusive Muon Capture From Nuclei 2309NP40Ca WS1ρ(~r)[fm−3]0.10.080.060.040.02WS2ρ(~r)[fm−3]0.10.080.060.040.02WS3r [fm]ρ(~r)[fm−3]5432100.10.080.060.040.02LFGWS140Ca3530252015105080706050403020100LFGWS2dΓc/dǫ′[104s−1/MeV]20181614121086420Γi→f[104s−1]706050403020100LFGWS3ǫ′ [MeV]1101051009590858075702520151050ǫ′ [MeV]706050403020100100908070Fig. 3. The same as Fig. 1 for 40Ca.NP208Pb WS1ρ(~r)[fm−3]0.10.080.060.040.02WS2r [fm]ρ(~r)[fm−3]10864200.10.080.060.040.02LFGWS1208PbdΓc/dǫ′[×104s−1/MeV]500450400350300250200150100500LFGWS2ǫ′ [MeV]dΓc/dǫ′[104s−1/MeV]9590858075706560350300250200150100500Fig. 4. The same as Fig. 1 for 208Pb.For the closed-shell heavy nucleus 208Pb we present in Table II integratedwidths only for two sets of potential parameters, WS1 and WS2. In bothcases the LFG results are close to the SM ones.2310 J.E. Amaro et al.5. ConclusionsIn this paper we have estimated the magnitude of the finite nucleuseffects on inclusive muon capture, aiming at quantifying the uncertainty ofthe LFG results of [1].We have focused on a simple shell model without nuclear correlations,but that contains the relevant information about the finite nuclear structure,and we have compared it with the uncorrelated LFG using the same input.As expected, the neutrino spectrum is very different in the two models, inparticular the LFG cannot account for the resonances and discrete states.However, in the case of the lighter nuclei, 12C and 16O, the SM and LFGresults for the integrated width are close, within 3–6%, for WS parameterswith similar neutron and proton densities. For the medium and heavy nuclei,40Ca and 208Pb, the integrated widths are always very close, within 1–7%.The final neutrino spectra of the LFG become more similar to the SM,including the discrete part, for heavier nuclei.REFERENCES[1] J. Nieves, J.E. Amaro, M. Valverde, Phys. Rev. C70, 055503 (2004); ErratumC72, 019902 (2005).[2] A. Gil, J. Nieves, E. Oset, Nucl. Phys. A627, 543 (1997); Nucl. Phys. A627,598 (1997).[3] J.E. Amaro, C. Maieron, J. Nieves, M. Valverde, Eur. Phys. J. A24, 343(2005).[4] M. Valverde, J.E. Amaro, J. Nieves, Phys. Lett. B638, 325 (2006)[hep-ph/0604042].[5] T. Suzuki, D.F. Measday, J.P. Roalsvig, Phys. Rev. C35, 2212 (1987) andreferences therein.

Local Fermi Gas in inclusive muon capture from nuclei

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Local Fermi gas in inclusive muon capture from nuclei

Local Fermi gas in inclusive muon capture from nuclei

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