Cross Sections of Charged Current Neutrino Scattering off ¹³²Xe for the Supernova Detection

Abstract

The total cross sections as well as the neutrino event rates are calculated in the charged current neutrino and antineutrino scattering off ¹³²Xe isotope at neutrino energies 𝐸_𝑣<100 MeV. Transitions to excited nuclear states are calculated in the framework of quasiparticle random-phase approximation. The contributions from different multipoles are shown for various neutrino energies. Flux-averaged cross sections are obtained by convolving the cross sections with a two-parameter Fermi-Dirac distribution. The flux-averaged cross sections are also calculated using terrestrial neutrino sources based on conventional sources (muon decay at rest) or on low-energy beta-beams.

1. Introduction

The detection of neutrinos and their properties is one of the top priorities of modern nuclear and particle physics as well as astrophysics. Among the probes which involve neutrinos, the neutrino-nucleus reactions possess a prominent position. Detailed predictions of neutrino-nucleus cross sections (NNCS) are crucial to detect or distinguish neutrinos of different flavor and explore the basic structure of the weak interactions [1–14]. Mured cross sections for neutrino-nucleus scattering at neutrino energies which are relevant for supernova neutrinos are available in only a few cases, that is, for ⁵⁶Fe [15], ¹²C [15, 16], and the deuteron [17]. The use of microscopic nuclear structure models is therefore essential, for a quantitative description of neutrino-nucleus reactions. These include the nuclear shell model [18, 19], the random-phase approximation (RPA), relativistic RPA [20, 21], continuum RPA (CRPA) [22], quasiparticle RPA (QRPA) [23–26], projected quasiparticle RPA (PQRPA) [27], hybrid models of CRPA, the shell model [28, 29], and the Fermi gas model [30]. The shell model provides a very accurate description of ground-state wave functions. The description of high-lying excitations, however, necessitates the use of large-model spaces, and this often leads to computational difficulties, making the approach applicable essentially only to light- and medium-mass nuclei. Therefore, for, systematic studies of weak interaction rates for relevant heavy nuclei of mass number around 𝐴=128−132, microscopic calculations must be performed using models based on the RPA [23, 25].

The signature of supernova neutrino interaction taking place in various detectors is the observation of electrons, positrons, photons, and other particles which are produced through the charged and neutral current interactions. Two processes that contribute to the total event rates in the detectors are the charged current (CC) reactions 𝜈_𝑒+_𝑍𝑋_𝑁→_𝑍+1𝑋^∗_𝑁−1+𝑒⁻,𝜈_𝑒+_𝑍𝑋_𝑁→_𝑍−1𝑋^∗_𝑁+1+𝑒⁺ (1) and the neutral current (NC) reactions 𝜈_𝑥𝜈_𝑥+_𝑍𝑋_𝑁→𝜈_𝑥𝜈_𝑥+_𝑍𝑋^∗_𝑁,𝑥=𝑒,𝜇,𝜏. (2) The neutrinos 𝜈_𝑥 (or antineutrinos 𝜈_𝑥) with 𝑥=𝜇,𝜏 do not have sufficient energy to produce corresponding leptons in charged current reactions and interact only through neutral current interactions and therefore have a higher-average energy than 𝜈_𝑒 and 𝜈_𝑒, which interact through charged current as well as neutral current interactions. Numerical simulations give the following values of average energy for the different neutrino flavors, that is, ⟨𝐸_𝜈𝑒⟩∼10-11 MeV, ⟨𝐸_𝜈𝑒⟩∼15-16 MeV, and ⟨𝐸_𝜈𝑥⟩∼23−25 MeV, and are consistent with the supernova neutrino spectrum given by a Fermi-Dirac distribution [31, 32]: 𝜙𝐸_𝜈=𝑁₂(𝛼)𝑇³𝐸²_𝜈1+exp𝐸_𝜈,/𝑇−𝛼 (3) where 𝑇 is the neutrino temperature, 𝛼 is a degeneracy parameter taken to be either 0 or 3. 𝑁₂(𝛼) denotes the normalization factor depending on 𝛼 given from 𝑁_𝑘(𝛼)=^∞₀𝑥^𝑘1+𝑒^𝑥−𝛼𝑑𝑥⁻¹, (4) for 𝑘=2. Following [33], the average neutrino energy ⟨𝐸_𝜈⟩ can be written in terms of the functions of (4) as 𝐸_𝜈=𝑁₂(𝛼)𝑁₃(𝛼)𝑇. (5) Most calculations of neutrino-nucleus cross sections have been taken the value 𝛼=0. However, in astrophysical applications, it might be important to perform studies of reaction rates for different values of 𝛼 depending on the simulation performed and on the specific supernova phase considered [29, 34]. In our study, the value 𝛼=3 has also been used. The average energy values for the various neutrino species imply that for 𝛼=0(3), the values of temperature 𝑇 are 3.5 MeV (2.75 MeV) for 𝜈_𝑒, 5 MeV (4 MeV) for 𝜈_𝑒, and 8 MeV (6 MeV) for 𝜈_𝑥 (𝑥=𝜇,𝜏,𝜇,𝜏). The recent theoretical studies predict a smaller value of temperature for 𝜈_𝑥 which is closer to 𝜈_𝑒 [34–37].

Systematic neutrino-nucleus interaction measurements could be an ideal tool to explore the weak nuclear response. At present, new experiments on various nuclei are being proposed with a new facility using muon decay at rest [38]. Another possibility could be offered by beta-beams. This is a new method to produce pure and well-known electron neutrino beams, exploiting the beta-decay of boosted radioactive ions [39]. The idea of establishing a low-energy beta-beam facility has been first proposed in [40] and discussed in nuclear structure studies, core-collapse supernova physics, and the study of fundamental interactions [40–49].

A detector whose active target consists of the noble liquid Xenon can offer unique detection capabilities in the field of neutrino physics [47, 50] as well as the ability to detect very low-energy signals in the context of dark matter searches [51, 52]. The new concept of a spherical TPC detector, filled with high-pressure Xenon, has also been proposed as a device able to detect low-energy neutrinos as those coming from a galactic supernova. In particular, a TPC detector can be used to observe coherent NC as well as CC neutrino-nucleus scattering [37, 53–58].

In this paper, we present microscopic calculations of the CC 𝜈_𝑒𝜈_𝑒+¹³²Xe→¹³²Cs^∗132I^∗+𝑒⁻𝑒⁺ (6) reaction cross sections. The corresponding-reduced matrix elements in the low- and intermediate-neutrino energy range have been calculated in the framework of quasiparticle random-phase approximation (QRPA). We present the total neutrino-nucleus cross sections as well as the contribution of the various multipoles and discuss how their importance evolves, as a function of neutrino energy. A comparison between the CC cross sections and those involved by the coherent NC ones [37, 55] is also presented. Finally, we give the flux-averaged cross sections associated to the Fermi-Dirac distribution as well as to distributions based on terrestrial neutrino sources such as the low-energy beta-beams or to conventional sources (muon decay at rest).

2. The Formalism for Neutrino-Nucleus Cross Sections Calculations

Let us consider a neutral or charged current neutrino-nucleus interaction in which a low- or intermediate-energy neutrino (or antineutrino) is scattered inelastically from a nucleus (𝐴,𝑍). The initial nucleus is assumed to be spherically symmetric having ground state a |𝐽^𝜋⟩=|0⁺_gs⟩ state.

The corresponding standard model effective Hamiltonian of the current-current interaction can be written as ℋ=𝐺𝑎^CC,NC√2𝑗_𝜇(𝐱)𝐽^𝜇(𝐱), (7) where 𝐺=1.1664×10⁻⁵GeV⁻² is the Fermi weak coupling constant, 𝑎^CC=cos𝜃_𝑐 for charged current reaction, and 𝑎^NC=1 for neutral current reaction. 𝑗_𝜇 and 𝐽^𝜇 denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form 𝑗_𝜇=𝜓_𝜈ℓ(𝑥)𝛾_𝜇1−𝛾₅𝜓_𝜈ℓ(𝑥), (8) where 𝜓_𝜈ℓ are the neutrino/antineutrino spinors.

From a nuclear physics point of view, only the hadronic current is important. The structure for neutral current (NC) and charged current (CC) processes of both vector and axial-vector components (neglecting the pseudoscalar contributions) is written as 𝐽^CC,NC_𝜇=𝜓_𝑁𝐹^CC,NC₁𝑞²𝛾_𝜇+𝐹^CC,NC₂𝑞²𝑖𝜎_𝜇𝜈𝑞^𝜈2𝑀+𝐹^CC,NC_𝐴𝑞²𝛾_𝜇𝛾₅𝜓_𝑁, (9) where 𝑀 stands for the nucleon mass, and 𝜓_𝑁 denote the nucleon spinors. The form factors 𝐹^CC_1,2(𝑞²) and 𝐹^CC_𝐴(𝑞²) are defined as 𝐹^CC_1,2𝑞²=𝐹^p_1,2𝑞²−𝐹ⁿ_1,2𝑞²,𝐹^CC_𝐴𝑞²=𝐹_𝐴𝑞² (10) and the neutral current form factors 𝐹^NC_1,2(𝑞²) and 𝐹^NC_𝐴(𝑞²) as 𝐹^NC_1,2𝑞²=12−sin²𝜃_𝑊𝐹^p_1,2𝑞²−𝐹ⁿ_1,2𝑞²𝜏₀−sin²𝜃_𝑊𝐹^p_1,2𝑞²+𝐹ⁿ_1,2𝑞²,𝐹^NC_𝐴𝑞²=12𝐹_𝐴𝑞²𝜏₀. (11) Here, 𝜏₀ represents the nucleon isospin operator, and 𝜃_𝑊 is the Weinberg angle (sin²𝜃_𝑊=0.2325). The detailed expressions of nucleonic form factors 𝐹^p,n_1,2(𝑞²) are given in [59]. The axial-vector form factor 𝐹_𝐴(𝑞²) is given by [60] 𝐹_𝐴=−𝑔_𝐴𝑞1−²𝑀²_𝐴⁻², (12) where 𝑀_𝐴=1.05 GeV is the dipole mass, and 𝑔_𝐴=1.258 is the static value (at 𝑞=0) of the axial form factor.

In the convention we used in the present paper, 𝑞², the square of the momentum transfer, is written as 𝑞²=𝑞^𝜇𝑞_𝜇=𝜔²−𝐪²=𝜀_𝑖−𝜀_𝑓²−𝐩_𝑖−𝐩_𝑓², (13) where 𝜔=𝜀_𝑖−𝜀_𝑓 is the excitation energy of the nucleus. 𝜀_𝑖 denotes the energy of the incoming neutrino and 𝜀_𝑓 denotes the energy of the outgoing lepton. 𝐩_𝑖, 𝐩_𝑓 are the corresponding 3-momenta. In (11), we have not taken into account the strange quark contributions in the form factors. In the scattering reaction considered in our paper, only low-momentum transfers are involved, and the contributions from strangeness can be neglected [61].

The neutrino/antineutrino-nucleus differential cross section, after applying a multipole analysis of the weak hadronic current, is written as 𝜎𝜀_𝑖=2𝐺²𝑎^CC,NC22𝐽_𝑖+1_𝑓^→𝑝_𝑓𝜀_𝑓×¹₋₁𝑑(cos𝜃)𝐹𝜀_𝑓,𝑍_𝑓×^∞_𝐽=0𝜎^𝐽_CL+^∞_𝐽=1𝜎^𝐽_𝑇, (14) where 𝜃 denotes the lepton scattering angle. The summations in (14) contain the contributions 𝜎^𝐽_CL, for the Coulomb ℳ_𝐽 and longitudinal ℒ_𝐽, and 𝜎^𝐽_𝑇, for the transverse electric 𝒯^el_𝐽 and magnetic 𝒯^mag_𝐽 multipole operators [62]. These operators include both polar-vector and axial-vector weak interaction components. The contributions 𝜎^𝐽_CL and 𝜎^𝐽_𝑇 are written as 𝜎^𝐽_CL=(1+𝑎cos𝜃)𝐽_𝑓ℳ_𝐽𝑞𝐽_𝑖²+1+𝑎cos𝜃−2𝑏sin²𝜃𝐽_𝑓ℒ_𝐽𝑞𝐽_𝑖²+𝜔𝑞()𝐽1+𝑎cos𝜃+𝑐×2ℜ𝑒_𝑓ℒ_𝐽𝑞𝐽_𝑖𝐽_𝑓ℳ_𝐽𝑞𝐽_𝑖^∗,𝜎^𝐽_𝑇=1−𝑎cos𝜃+𝑏sin²𝜃×𝐽_𝑓𝒯^mag_𝐽𝑞𝐽_𝑖²+𝐽_𝑓𝒯^el_𝐽𝑞𝐽_𝑖²∓𝜀_𝑖+𝜀_𝑓𝑞()𝐽1−𝑎cos𝜃−𝑐×2ℜ𝑒_𝑓𝒯^mag_𝐽𝑞𝐽_𝑖𝐽_𝑓𝒯^el_𝐽𝑞𝐽_𝑖^∗, (15)(16) where 𝑏=𝜀_𝑖𝜀_𝑓/𝑞², 𝑎=(1−(𝑚_𝑓𝑐²/𝜀_𝑓)²)^1/2, and 𝑐=(𝑚_𝑓𝑐²)²/𝑞𝜀_𝑓. In (16), the (−) sign corresponds to neutrino scattering and the (+) sign to antineutrino. The absolute value of the three momentum transfers is given by 𝑞=^→𝑞=𝜔²+2𝜀_𝑓𝜀_𝑖(1−𝑎cos𝜃)−𝑚_𝑓𝑐²². (17) For charged current reactions, the cross-sectional equation (14) must be corrected for the distortion of the outgoing lepton wave function by the Coulomb field of the daughter nucleus. The cross section can either be multiplied by the Fermi function 𝐹(𝜀_𝑓,𝑍_𝑓) obtained from the numerical solution of the Dirac equation for an extended nuclear charge distribution [29, 63], or, at higher energies, the effect of the Coulomb field can be described by the effective momentum approximation (EMA) [63–65]. In this approximation, the lepton momentum 𝑝_𝑓 and energy 𝜀_𝑓 are modified as 𝑝^eff_𝑓=1𝑐𝜀^eff_𝑓²−𝑚_𝑓𝑐²²𝜀^eff_𝑓=𝜀_𝑓−𝑉^eff_𝐶, (18) where 𝑉^eff_𝐶 is the effective Coulomb potential. In a recent study using exact Dirac wave functions, it has been shown that an accurate approximation for the effective electron momenta is obtained by using the mean value of the Coulomb potential, 𝑉^eff_𝐶=4𝑉_𝐶(0)/5, where 𝑉_𝐶(0)=−3𝑍_𝑓𝛼/(2𝑅) corresponds to the electrostatic potential evaluated at the center of the nucleus [66, 67]. 𝑍_𝑓 is the charge of the daughter nucleus, and 𝑅 is its radius assuming spherical charge distribution. 𝛼 denotes the fine structure constant. In calculations with EMA, the original lepton momentum 𝑝_𝑓 and energy 𝜀_𝑓 appearing in the expression for the cross section are replaced by the above effective quantities.

3. Energies and Wave Functions

For neutral-current-neutrino-nucleus-induced reactions, the ground state and the excited states of the even-even nucleus are created using the quasiparticle random-phase approximation (QRPA) including two quasineutron and two quasiproton excitations in the QRPA matrix [68] (hereafter denoted by pp-nn QRPA). We start by writing the A-fermion Hamiltonian 𝐻, in the occupation-number representation, as a sum of two terms. One is the sum of the single-particle energies (spe) 𝜖_𝛼 which runs over all values of quantum numbers 𝛼≡{𝑛_𝛼,𝑙_𝛼,𝑗_𝛼,𝑚_𝛼} and the second term which includes the two-body interaction 𝑉, that is, 𝐻=_𝛼𝜖_𝛼𝑐^†_𝛼𝑐_𝛼+14_{𝛼𝛽𝛾𝛿}𝑉_{𝛼𝛽𝛾𝛿}𝑐^†_𝛼𝑐^†_𝛽𝑐_𝛿𝑐_𝛾, (19) where the two-body term contains the antisymmetric two-body interaction matrix element defined by 𝑉_{𝛼𝛽𝛾𝛿}=⟨𝛼𝛽|𝑉|𝛾𝛿⟩−⟨𝛼𝛽|𝑉|𝛿𝛾⟩. The operators 𝑐^†_𝛼 and 𝑐_𝛼 stand for the usual creation and destruction operators of nucleons in the state 𝛼.

For spherical nuclei with partially filled shells, the most important effect of the two-body force is to produce pairing correlations. The pairing interaction is taken into account by using the BCS theory [69]. The simplest way to introduce these correlations in the wave function is to perform the Bogoliubov-Valatin transformation: 𝑎^†_𝛼=𝑢_𝛼𝑐^†_𝛼−𝑣_𝛼𝑐_𝛼̃,𝑎^†_𝛼=𝑢_𝛼𝑐^†_𝛼̃+𝑣_𝛼𝑐_𝛼, (20) where 𝑐^†_𝛼̃=𝑐^†_−𝛼(−1)^{𝑗𝛼+𝑚𝛼}, 𝑎^†_𝛼=𝑎^†_−𝛼(−1)^{𝑗𝛼+𝑚𝛼}, and −𝛼≡{𝑛_𝛼,𝑙_𝛼,𝑗_𝛼,−𝑚_𝛼}. The occupation amplitudes 𝑣_𝛼 and 𝑢_𝛼 are determined via variational procedure for minimizing the energy of the BCS ground state for protons and neutrons, separately. In the BCS approach, the ground state of an even-even nucleus is described as a superconducting medium, where all the nucleons have formed pairs that effectively act as bosons. The BCS ground state is defined as |BCS⟩=_𝛼>0𝑢_𝛼−𝑣_𝛼𝑐^†_𝛼𝑐^†_𝛼̃|CORE⟩, (21) where |CORE⟩ represents the nuclear core (effective particle vacuum).

After the transformation (20), the Hamiltonian can be written in its quasiparticle representation as 𝐻=_𝛼𝐸_𝛼𝑎^†_𝛼𝑎_𝛼+𝐻_qp, (22) where the first term gives the single-quasiparticle energies 𝐸_𝛼, and the second one includes the different components of the residual interaction.

In the present calculations, we use a renormalization parameter 𝑔_pair which can be adjusted solving the BCS equations. The monopole matrix elements ⟨𝛼𝛼;𝐽=0|𝑉|𝛽𝛽;𝐽=0⟩ of the two-body interaction are multiplied by a factor 𝑔_pair. The adjustment can be done by comparing the resulting lowest-quasiparticle energy to the phenomenological energy gap Δ obtained from the separation energies of the neighboring doubly even nuclei for protons and neutrons, separately.

The excited states of the even-even reference nucleus are constructed by use of the QRPA. In the QRPA, the creation operator for an excited state |𝜔;𝐽^𝜋𝑀⟩ has the form 𝑄^†𝐽^𝜋_𝜔𝑀=_{𝛼≤𝛼′}𝑋^𝐽𝜋𝜔_𝛼𝛼′𝐴^†𝛼𝛼^′;𝐽𝑀−𝑌^𝐽𝜋𝜔_𝛼𝛼′𝐴𝛼𝛼^′;𝐽𝑀, (23) where the quasiparticle pair creation 𝐴^†(𝛼𝛼^′;𝐽𝑀) and annihilation 𝐴(𝛼𝛼^′;𝐽𝑀) operators are defined as 𝐴^†𝛼𝛼^′;𝐽𝑀≡1+𝛿_𝛼𝛼′^−1/2𝑎^†_𝛼𝑎^†_{𝛼′𝐽𝑀},𝐴𝛼𝛼^′;𝐽𝑀≡(−1)^𝐽+𝑀𝐴𝛼𝛼^′;𝐽−𝑀, (24) where 𝛼 and 𝛼^′ are either proton (p) or neutron (n) indices, 𝑀 labels the magnetic substates, and 𝜔 numbers the states for particular angular momentum 𝐽 and parity 𝜋.

The 𝑋 and 𝑌 forward- and backward-going amplitudes are determined from the QRPA matrix equation 𝒜ℬ𝑋−ℬ−𝒜𝑌^{𝐽𝜋𝐽𝜋}𝑋=𝜔^𝐽𝜋𝑌^𝐽𝜋, (25) where 𝜔 denotes the excitation energies of the nuclear state |𝐽^𝜋⟩. The QRPA matrices, 𝒜 and ℬ, are deduced by the matrix elements of the double commutators of 𝐴^† and 𝐴 with the nuclear Hamiltonian 𝐻 defined as 𝒜_{𝛼𝛼′,𝛽𝛽′}=BCS𝐴𝛼𝛼^′;𝐽𝑀,𝐻,𝐴^†𝛽𝛽^′,ℬ;𝐽𝑀BCS_{𝛼𝛼′,𝛽𝛽′}=−BCS𝐴𝛼𝛼^′;𝐽𝑀,𝐻,𝐴𝛽𝛽^′;𝐽𝑀BCS, (26) where 2[𝐴,𝐵,𝐶]=[𝐴,[𝐵,𝐶]]+[[𝐴,𝐵],𝐶]. Finally, the two-body matrix elements of each multipolarity 𝐽^𝜋, occurring in the QRPA matrices 𝒜 and ℬ, are multiplied by two phenomenological scaling constants, namely, the particle-hole strength 𝑔_ph and the particle-particle strength 𝑔_pp. These parameter values are determined by comparing the resulting lowest-phonon energy with the corresponding lowest-collective vibrational excitation of the doubly even nucleus and by reproducing some giant resonances which play crucial role.

For charged current neutrino-nucleus reactions, the excited states |𝜔;𝐽^𝜋𝑀⟩ of the odd-odd nucleus are generated adopting the proton-neutron QRPA(pnQRPA). The QRPA in its proton-neutron form contains phonons made out of proton-neutron pairs as follows: 𝑄^†𝐽^𝜋_𝜔𝑀=_pn𝑋^𝐽𝜋𝜔_pn𝐴^†pn;𝐽𝑀−𝑌^𝐽𝜋𝜔_pn𝐴pn;𝐽𝑀. (27) The matrices 𝒜 and ℬ defined in the canonical basis are 𝒜_pn,p′n′=𝛿_pn,p′n′𝐸_p+𝐸_n+𝑔_pp𝑢_p𝑢_n𝑢_p′𝑢_n′+𝑣_p𝑣_n𝑣_p′𝑣_n′𝑉^pp_pn,p′n′+𝑔_ph𝑢_p𝑣_n𝑢_p′𝑣_n′+𝑣_p𝑢_n𝑣_p′𝑢_n′𝑉^ph_pn,p′n′,ℬ_pn,p′n′=−𝑔_pp𝑢_p𝑢_n𝑣_p′𝑣_n′+𝑣_p𝑣_n𝑢_p′𝑢_n′𝑉^pp_pn,p′n′+𝑔_ph𝑢_p𝑣_n𝑣_p′𝑢_n′+𝑣_p𝑢_n𝑢_p′𝑣_n′𝑉^ph_pn,p′n′, (28) where 𝐸_p and 𝐸_n are the two-quasiparticle excitation energies, and 𝑉^ph_pn,p′n′ and 𝑉^pp_pn,p′n′ are the p-h and p-p matrix elements of the residual nucleon-nucleon interaction 𝑉, respectively. For charged current reactions, the matrix elements of any transition operator 𝒪_𝜆 between the ground state |0⁺_gs⟩ and the excited |𝜔;𝐽^𝜋𝑀⟩ can be factored as follows: 0⁺_gs𝒪_𝜆𝜔;𝐽^𝜋𝑀=_pnp𝒪_𝜆n𝑋^𝐽𝜋𝜔_pn𝑢_p𝑣_n+𝑌^𝐽𝜋𝜔_pn𝑣_p𝑢_n, (29) where ⟨p‖𝒪_𝜆‖n⟩ are the reduced matrix elements calculated independently for a given single-particle basis [70, 71].

4. Results and Discussion

Transition matrix elements of the type entering in (15) and (16) can be calculated in the framework of pnQRPA. The initial nucleus ¹³²Xe was assumed to be spherically symmetric having a 0⁺ ground state. Two-oscillator (3ℏ𝜔 and 4ℏ𝜔) major shells, plus the intruder orbital ℎ_11/2 from the next higher-oscillator major shell, were used for both protons and neutrons as the valence space of the studied nuclei. The corresponding single-particle energies (SPE) were produced by the Coulomb corrected Woods-Saxon potential using the parameters of Bohr and Mottelson [72].

The two-body interaction matrix elements were obtained from the Bonn one-boson-exchange potential applying G-matrix techniques [73]. The strong pairing interaction between the nucleons can be adjusted by solving the BCS equations. The monopole matrix elements of the two-body interaction are scaled by the pairing-strength parameters 𝑔^p_pair and 𝑔ⁿ_pair, separately, for protons and neutrons. The adjustment can be done by comparing the resulting lowest-quasiparticle energy to reproduce the phenomenological pairing gap [74]. The results of this procedure lead to the pairing-strength parameters 𝑔^p_pair=0.98 and 𝑔ⁿ_pair=1.3. The particle-particle matrix elements as well as the particle-hole ones are renormalized by means of the parameters 𝑔_pp and 𝑔_ph, respectively. These parameters were adjusted for each multipole state separately in order to reproduce few of the experimental known energies of the low-lying states in the ¹³²Cs and ¹³²I nucleus, respectively. The obtained values for the corresponding parameters lie in the range 0.6≤𝑔_pp≤1.2 and 0.5≤𝑔_ph≤1.0. Especially, for the 1⁻ multipolarity, the values 𝑔_pp=1.0 and 𝑔_ph=0.5 were used, while for the 1⁺ multipolarity, the values 𝑔_pp=1 and 𝑔_ph=1 were used. Moreover, for ¹³²I, the values 𝑔_pp=1 and 𝑔_ph=1 were used with the exception of 4⁺ multipolarity for which 𝑔_pp=0.3 and 𝑔_ph=1.2 were used. All the states up to 𝐽=5^± have been included.

In Figure 1, we present the numerical results of the total scattering cross section 𝜎(𝐸_𝜈) (14) as a function of the incoming neutrino energy 𝐸_𝜈 for the reactions ¹³²Xe(𝜈_𝑒,𝑒⁻)¹³²Cs and ¹³²Xe(𝜈_𝑒,𝑒⁺)¹³²I, respectively. The 𝑄 values of the reactions are 2.12 MeV and 3.58 MeV, respectively. Here, we have considered a hybrid prescription already used in previous calculations [19, 75, 76], where the Fermi function for the Coulomb correction is used below the energy region on which both approaches predict the same values, while EMA is adopted above this energy region.

(a)

(b)

Figure 1 

(Color on line). Total cross section as a function of the incoming neutrino energy 𝐸𝜈, in the CC reactions 132Xe(𝜈𝑒,𝑒−)132Cs (a) and 132Xe(𝜈𝑒,𝑒+)132I (b).

The contribution of the different multipoles to the total cross section for the impinging neutrino energies 𝐸_𝜈𝑒=20,60, and 80MeV is shown in Figure 2. When 𝐸_𝜈𝑒=20MeV, the total cross section 𝜎_𝜈𝑒 is mainly ascribed to the Gamow-Teller (1⁺) and the Fermi (0⁺) transitions. Other transitions contribute only a few percent to the total cross section. As the neutrino energy increases, the multipole states 𝐽^𝜋=1⁻,2⁻, and 2⁺ become important as well. Finally, beyond 80 MeV, all states contribute, and the cross section is being spread over many multipoles.

(a)

(b)

(c)

Figure 2 

(Color on line). Partial multipole distributions to the total cross sections for 132Xe(𝜈𝑒,𝑒−)132Cs, at the incoming neutrino energies 𝐸𝜈=20, 60, and 80 MeV.

Figure 3 shows the cross sections of coherent neutral and charged current processes as a function of neutrino energy. As it is seen, the coherent neutral current (𝜈−NC) process [55] presents cross sections which are an order of magnitude greater than the electron neutrino charged current cross sections (𝜈_𝑒−CC). Both of them are even bigger than those from electron antineutrino charge current cross section (𝜈_𝑒−CC) events. At 𝐸_𝜈=80 MeV, the difference between 𝜈_𝑒−CC and 𝜈_𝑒−CC turns out to be a factor of 5. This can be understood in terms of the energy threshold and nuclear effects of the reactions. Since the 𝑄 value for the 𝜈_𝑒 reactions is 1.46 MeV greater than 𝜈_𝑒 one, it decreases the incident neutrino energy as 𝐸_𝜈→𝐸_𝜈−𝑄 and therefore reduces the 𝜈_𝑒 cross section for a given energy. In Table 1, the total (anti-) neutrino cross sections are listed in units of 10⁻⁴² cm².

Table 1 

Total cross sections for the indicated neutrino-nucleus charged current reactions as a function of incoming neutrino energy. The cross sections are given in units of 10⁻⁴² cm², and exponents are given in parentheses.

Figure 3 

(Color on line). Comparison of cross sections for relevant neutrino reactions on 132Xe.

The flux-averaged total cross sections ⟨𝜎⟩ can be calculated by folding the cross sections shown in Figure 3 with the Fermi-Dirac spectrum given by (3) as follows: ⟨𝜎⟩=^∞₀𝜎𝐸_𝜈𝜙𝐸_𝜈𝑑𝐸_𝜈. (30) Table 2 shows the flux-averaged total cross sections for different values of temperature 𝑇. The chemical potential parameters 𝛼=0 and 𝛼=3 have been used in order to describe the supernova spectrum [32]. The corresponding average neutrino energy ⟨𝐸_𝜈⟩ has been calculated by means of (4) and (5). As it is seen, the calculated flux-averaged cross section increases as the average neutrino energy increases. The introduction of a chemical potential in the spectrum at fixed neutrino temperature increases the average neutrino energy. In Figure 4, a contour plot is used to display lines of constant flux-averaged cross sections (in units of 10⁻³⁹ cm²) of ¹³²Xe(𝜈_𝑒,𝑒⁻)¹³²Cs^∗ reaction, as a function of 𝑇 and 𝛼. At lower temperatures, the flux-averaged cross sections depend only very weakly on 𝛼. However, already above 𝑇=2 MeV, the flux-averaged cross sections increase much faster for higher values of the chemical potential 𝛼.

Table 2 

Flux-averaged cross sections (10⁻⁴⁰ cm²) obtained by convoluting the cross sections of Figure 3 with (3). Different temperatures 𝑇 (MeV) and 𝛼 values are considered. The average neutrino energy ⟨𝐸_𝜈⟩ is given in MeV.

Figure 4 

(Color online). Contour plot of supernova neutrino-nucleus flux-averaged cross sections (in units of 10−39 cm2) for the 132Xe(𝜈𝑒,𝑒−)132Cs∗ reaction as functions of the temperature 𝑇 and chemical potential 𝛼, that determine the Fermi-Dirac neutrino spectrum. The solid lines in the plot correspond to constant values of flux-averaged cross section. The lighter shading area denotes the increase of flux-averaged cross sections.

In Table 3, we present the number of expected events for supernova explosion occurring at a distance of 10 kpc from earth, releasing an energy of 3×10⁵³ ergs. These event rate calculations have been done for 3 kT Xenon detector corresponding to various values of temperature 𝑇 with 𝛼=0 and 3. Using 𝛼=3, we find the total event rate of 485 which corresponds to a decrease of 32% as compared to the 𝛼=0 supernova neutrino spectrum.

Table 3 

Expected event rates for a 3 kT xenon detector for a supernova at 10 kpc corresponding to ⟨𝐸_𝜈𝑒⟩=11 MeV and ⟨𝐸_𝜈𝑒⟩=16 MeV.

In the literature, there are suggestions to look for charged current neutrino-nucleus scattering at several neutrino sources. We propose to look for this reaction with a terrestrial neutrino sources with spectra similar to those of SN neutrinos, using a near detector whose active target consists of a noble liquid gas such as ¹³²Xe. In this paper, we examine two possibilities: (i) the low-energy neutrino spectra corresponding to conventional neutrino sources, that is, muon decay at rest (DAR) given by the well-known Michel spectrum from muons decaying at rest and (ii) the low-energy beta-beams with a boosted parameter 𝛾.

Several experiments to be done at low-energy beta-beam have been proposed. Throughout our calculations, we have assumed that the boosted ions are storage in a ring similar to that used in [77]. Its total length is 𝐿=450 m with straight section length 150 m, while the detector is located 10 m away from the straight section. The radius of the cylindrical detector is 2.13 m with thickness 5 m. As seen in Figure 5, the DAR spectrum has quite similar shape to the low-energy beta-beam spectrum with 𝛾=10. Note that, in principle, since the cross sections approximately grow as the neutrino energy square, the flux-averaged cross sections can show differences due to the high energy part of the neutrino spectrum.

Figure 5 

Normalized neutrino spectra stemming from the decay of 18Ne ions boosted at 𝛾=6 (dot-dashed line), 𝛾=10 (dotted line) and 𝛾=14 (dashed line). The full line presents the Michel spectrum for neutrinos from muon decay-at-rest.

Table 4 presents the contribution of the different states to the flux-averaged cross section. One can see that the results for 𝛾=10 are similar to the DAR case. The neutrino-Xenon cross section is dominated by the 0⁺,1⁺, and 1⁻ multipoles. When the ion boost parameter 𝛾 increases, the relative contribution of the 1⁺ decreases in favor of all other multipoles except 0⁺ which seems to be almost constant. For 𝛾=14, the contribution of all states becomes important in agreement with previously published results [76]. Table 5 presents the results for the antineutrino scattering, where the antineutrino fluxes are produced by the decay of boosted ⁶He ions. As it can be seen, the contribution of both 0⁺ and 1⁺ transitions to the flux-averaged cross sections is lying between 86% for boosted ions at 𝛾=6 and 72% for 𝛾=14.

Table 4 

Fraction (in %) of the flux-averaged cross section associated to states of a given multipolarity with respect to the total flux-averaged cross section, that is, ⟨𝜎⟩_𝐽^𝜋/⟨𝜎⟩_tot. The first column tells if the results correspond to low-energy beta-beams or to a conventional source (DAR is for the decay at rest of muons). The neutrino fluxes are those produced by boosted ¹⁸Ne ions. The Lorentz ion boost parameter 𝛾 takes the values 6, 10, and 14. The last column gives the total flux-averaged cross sections (10⁻⁴⁰ cm²).

Table 5 

Same as Table 4, but for the antineutrino-nucleus cross sections and the antineutrino fluxes produced by boosted ⁶He ions.

5. Conclusions

Detailed microscopic calculations of charged current and neutral current neutrino-nucleus reaction rates are of crucial importance for models of neutrino oscillations, detection of supernova neutrinos, and studies of the 𝑟-process nucleosynthesis. In this paper we have calculated charged-current-neutrino-induced reactions on ¹³²Xe by including multipole transitions up to 𝐽=5^±. Excited states up to a few tens of MeV are taken into account. The ground state of ¹³²Xe is described with the BCS model, and the neutrino-induced transitions to excited nuclear states are computed in the quasiparticle random-phase approximation.

In addition to the total neutrino-nucleus cross sections, we have also analyzed the evolution of the contributions of different multipole excitations as a function of neutrino energy. It has been shown that except at relatively low-neutrino energies 𝐸_𝜈≤30 MeV for which the reactions are dominated by the transitions to 0⁺ and 1⁺ states, at higher energies, the inclusion of spin-dipole transitions, as well as excitations of higher multipolarities, is essential for a quantitative description of neutrino-nucleus cross sections. It is found that the 𝜈_𝑒 cross section on ¹³²Xe is about 5 times greater than the 𝜈_𝑒 one. This difference is anticipated because of (i) the different 𝑄 values of the corresponding reactions, (ii) the fact that there are less excited states that one can populate in the 𝜈_𝑒 channel with respect to the 𝜈_𝑒 one and (iii) the different sign (minus for neutrino plus for antineutrino) of the interference term of magnetic and electric transitions introduced in (16).

Finally, we have given the contribution of the different states to the flux-averaged cross section considering low energy neutrino beams.

These are either based on conventional sources (muon decay at rest) or on low-energy beta-beams. We found that the Gamow-Teller (1⁺) and the Fermi (0⁺) transitions are the main components. When the Lorentz ion boost parameter 𝛾 increases, the relative contribution of 1⁺ decreases in favor of all other multipole states except 0⁺ which seems to be almost constant, while the contribution of other states like 1⁻, 2⁻, 2⁺, 3⁻, and 3⁺ become important as well.