Neutrino Capture Reactions on Ar
Abstract
GamowTeller (GT) strength in Ar is studied by shellmodel calculations with monopolebased universal intearction, which has tensor components of +meson exchanges. Calculated GT strength is found to be consistent with the experimental data obtained by recent () reactions. Neutrino capture cross sections on Ar for solar neutrinos from B are found to be enhanced compared with previous calculations. The reaction cross sections for multipoles other than and are obtained by randomphase approximation (RPA). Their contributions become important for neutrino energies larger than 50 MeV.
pacs:
25.30.Pt, 21.60.Cs, 27.40.+zI Introduction
A liquid Argon detector has excellent potentialities to detect corecollapse supernova neutrinos. A liquid Argon TPC (time projection chamber), proposed by the ICARUS Collaborations ICA (), can provide threedimensional imaging of ionizing events. Recently, ArgoNeut Collaboration reported the first measurements of inclusive muon neutrino chargedcurrent differential cross sections on argon Arg (). Here, we are interested in chargedcurrent reactions on Ar induced by solar and supernova neutrinos. Direct measurements of the chargedcurrent reaction cross sections on Ar are accessible by using a liquid Argon TPC detector and a spallation neutron source for neutrinos Cav ().
Recently, GamowTeller (GT) strength distribution for Ar K was measured by using the () reaction up to an excitation energy of 8 MeV Bhat (). The measured GT distribution is found to be close to that obtained by decays of the mirror nucleus Ti up to an excitation energy of 6 MeV Bhat98 (). The GT strength obtained by a shellmodel calculation Orm () is smaller than the observed strength As the neutrino capture reaction cross section on Ar is sensitive to the magnitude and the distribution of the GT strength, it is a remarkable progress if we can improve shellmodel calculations to reproduce the experimental GT strength.
As shellmodel calculations of Ar involve both  and shells, crossshell correlations play important roles in the structure of the lowlying states in Ar. The choice of the crossshell interaction is important and it will affect the GT strength distribution considerably. Here, we adopt the monopolebaseduniversal interaction (VMU) VMU () for the  crossshell part. The VMU has tensor components of +meson exchanges. A proper inclusion of the tensor force is essential for a successful description of spindependent modes in nuclei. The use of VMU in the  cross shell part of the interaction proved to be successful in the description of GamowTeller transitions and magnetic moments in  shell nuclei Yuan ().
In the next section, we discuss GT strength in Ar obtained by shellmodel calculations and compare with experimental data. In Sect. III, cross sections of Ar (, ) K reaction are obtained by a hybrid model; shellmodel method is used for the GT and isobaric analog (IA) transitions while random phase approximation (RPA) is employed for other multipoles. Summary is given in Sect. IV.
Ii GamowTeller Strength in Ar with Monopolebased Universal Interaction
We study chargedcurrent neutrinoinduced reaction on Ar, Ar (, ) K, for neutrino energy below 100 MeV. We adopt a hybrid model in which the GamowTeller (GT) transitions and isobaricanalog (IA) transitions are evaluated by shellmodel calculations while contributions from other multipoles are treated by RPA.
We first discuss GT transitions in Ar. The sdpfm Utsuno () and GXPF1J Honma () interactions are used for shell and shell, respectively, in the shellmodel calculations. The monopolebaseduniversal interaction (VMU) VMU (), which has the tensor force of +meson exchanges, is adopted for the  crossshell part. The twobody spinorbit interaction due to meson exchanges is also added to the crossshell part of the interaction. This intercation will be referred as SDPFVMULS. Configurations within 2 excitations, , are taken with a quenching factor of = =0.775 Orm () for the axialvector coupling with =1.263. Singleparticle energies of shell for O core are tuned to reproduce those of GXPF1J in Ca. No further tunings are made. The shellmodel calculations are performed by using OXBASH OXB () and MSHELL64 Mizu () program codes. Calculated ground state of K is 4 consistent with the experiment. The energy of the first 1 state is also close to the experimental excitation energy. In case of WBT interaction WB (), for example, it is shifted to a higher energy region; excitation energy of the first 1 state is calculated to be 4.14 MeV.
Calculated cumulative sum of ,
(1) 
where =, up to the excitation energy of K () are shown in Fig. 1 as well as experimental values obtained by recent (, ) reaction Bhat (). The experimental GT strength from the () reaction is consistent with that obtained by decays from the mirror nucleus Ti Bhat98 () except that the relative strengths for the lowest two strong transitions are reversed.
The experimental cumulative sum of the GT strength is rather well described by SDPFVMULS while the strength in Ref. Orm () is smaller than the observed strength as shown in Fig. 1. The GT strength of the first strong peak is larger than that of the second strong peak in the present calculation, in agreement with the case of the GT strength obtained by the () reaction. The GT strength almost saturates around =18 MeV and the sum of the GT strength amounts to be 7.21, which is equal to 12. The sum of the strength exhausts the Ikeda sumrule value in the present configuration space.
Iii Neutrino Capture Reaction Cross Sections on Ar
Hamiltonian  GT  IA  GT+IA 

SDPFVMULS  11.95  2.10  14.05 
SDPFVMU  10.99  2.10  13.10 
Ref. Orm ()  7.70  3.80  11.50 
Neutrinonucleus reaction cross sections are evaluated by using the multipole expansion of the weak hadronic currents,
(2) 
for chargeexchange reactions (, ) and (), where and are vector and axialvector currents, respectively.
The reaction cross sections induced by or are given as follows Walecka (),
(3) 
where and are neutrino and lepton momenta, respectively, is the lepton energy, , , and .
cos with the Fermi coupling constant, is the Cabbibo angle, and the lepton is electron or positron for chargeexchange reactions. The Coulomb correction is taken into account by multiplying the Fermi function for chargeexchange reactions at low neutrino energires () while the correction by the effective momentum transfer approximation is used at higher energies. Lower value of the cross section obtained by the two methods is adopted at each Engel (). In Eq. (3), and are Coulomb, longitudinal, transverse electric and magnetic multipole operators for vector and axialvector currents
Calculated cross sections for Ar (, ) K induced by GT and IA transitions are shown in Fig. 2(a). Cross sections for Ar (, ) K folded over B neutrino spectrum Wint () are given in Table I. In Table I and Fig. 2, the cross sections are obtained with ICARUS condition, that is, electron energies are higher than 5 MeV. The cross section for the GT transition is enhanced for SDPFVMULS compared to that of Ref. Orm () about by 55. When the twobody spinorbit interaction is switched off in the  crossshell matrix elements, which is referred as SDPFVMU, the enhancement of the cross section is about 43. The spinorbit interaction is found to increase the cross section by about 9. Here, all the components of the GT transition, that is, the axial electric dipole, the magnetic dipole, the axial Coulomb and the axial longitudinal contributions are taken into acount while in Ref. Orm () only the axial electric dipole term is considered. As for the IA transition, both the Coulomb () and the longitudinal () terms are taken into account. The inclusion of the longitudinal term reduces the total contributions in the IA transition as
(4)  
This explains why the IA contribution in Ref. Orm (), where only the term is included, is smaller than the SDPFVMU case. The cross section for GT+IA is also found to be enhanced for SDPFVMULS (SDPFVMU) about by 22 (14) compared to ref. Orm ().
Cross sections for multipoles including up to =4 except for and are obtained by RPA with the use of SGII interaction SG (). The same quenching factor for as in the GT transitions is used. Calculated cross sections for GT, GT+IA, multipoles other than GT and IA, and the total cross section are shown in Fig. 2. The contributions from spindipole transitions and multipoles other than and are found to become important at 50 MeV.
Calculated total cross section obtained here is found to be rather close to that in Ref. Kolb () obtained by RPA calculations for all the multipoles. The cross section by the present hybrid model is enhanced about by 2040 compared to Ref. Kolb () at lower energies, =2040 MeV, where the GT contributions dominate. The calculated cross section in Ref. Chow () obtained by QRPA is smaller than our cross sections as well as in Ref. Kolb () about by three times. The GT distribution in Ref. Chow () is shifted toward higher energy region and quite a tiny strength is found below =8 MeV in contrast to the present case.
Iv Summary
In summary, we have carried out shellmodel calculations for GT and IA transitions in Ar with the use of VMU interaction. The GT strength obtained is found to reproduce the experimental values rather well. The GT strength obtained here is larger than that in Ref. Orm ().
Cross sections for the chargedcurrent induced reaction, Ar (, ) K, are evaluated by a hybrid model, where contributions from the GT and IA transtions are obtained by shellmodel calculations while other multipoles are treated by RPA. A better evaluation of the cross section for the GT contributions has been achieved in the present work. An enhancement of the cross section for solar neutrinos from B is found compared to previous calculations. The contributions from the GT and IA transitions dominate below 50 MeV while other multipoles become important at higher energies. It is strongy recommended that direct measurements of the cross section is carried out by using a liquid Ar TPC in near future. An accurate knowledge of the cross section is important for the advancement of the studies of supernova neutrinos and neutrino oscillations.
Here, we did not discuss neutralcurrent induced reactions on Ar as the present shellmodel configuration space is not large enough to evaluate the magnetic dipole (M1) transition Li () and neutralcurrent induced transitions and the expansion of the configuration space beyond is difficult due to slow convergence of wave functions Orm (). Study of neutralcurrent reactions with larger shellmodel configurations are left to future investigations.
This work has been supported in part by GrantsinAid for Scientific Research (C) 22540290 of the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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