Coherent and Incoherent Neutral Current Scattering for Supernova Detection

Abstract

The total cross sections as well as the neutrino event rates are calculated in the neutral current neutrino scattering off ⁴⁰Ar and ¹³²Xe isotopes at neutrino energies (𝐸_𝑣<100 MeV). The individual contribution coming from coherent and incoherent channels is taking into account. An enhancement of the neutral current component is achieved via the coherent (0⁺_gs→0⁺_gs) channel which is dominant with respect to incoherent (0⁺_gs→𝐽_𝑓) one. The response of the above isotopes as a supernova neutrino detection has been considered, assuming a two parameter Fermi-Dirac distribution for the supernova neutrino energy spectra. The calculated total cross sections are tested on a gaseous spherical TPC detector dedicated for supernova neutrino detection.

1. Introduction

It is well known that neutrinos and their interactions with nuclei have attracted a great deal of attention, since they play a fundamental role in nuclear physics, cosmology, and in various astrophysical processes, especially in the dynamics of core-collapse supernova-nucleosynthesis [1–11]. Moreover, neutrinos proved to be interesting tools for testing weak interaction properties, by examining nuclear structure and for exploring the limits of the standard model [12]. In spite of the important role the neutrinos play in many phenomena in nature, numerous questions concerning their properties, oscillation characteristics, their role in star evolutions and in the dark matter of the universe, and so forth remain still unanswered. The main goal of experimental [13–17] and theoretical studies [18–27] is to shed light on the above open problems to which neutrinos are absolutely crucial.

Among the probes which involve neutrinos, the neutrino-nucleus interaction possess a prominent position [28–34]. Thus, the study of neutrino scattering with nuclei is a good way to detect or distinguish neutrinos of different flavor and explore the basic structure of the weak interactions. Also, specific neutrino-induced transitions between discrete nuclear states with good quantum numbers of spin, isospin, and parity allows us to study the structure of the weak hadronic currents. Furthermore, terrestrial experiments performed to detect astrophysical neutrinos, as well as neutrino-induced nucleosynthesis interpreted through several neutrino-nucleus interaction theories, constitute good sources of explanation for neutrino properties. There are four categories of neutrino-nucleus processes: the two types of charged-current (CC) reactions of neutrinos and antineutrinos and the two types of neutral-current (NC) ones. In the charged-current reactions a neutrino 𝜈_𝑙 (antineutrino 𝜈_𝑙) with 𝑙=𝑒,𝜇,𝜏 transforms one neutron (proton) of a nucleus to a proton (neutron), and a charged lepton 𝑙⁻ (anti-lepton 𝑙⁺) is emitted as𝜈_𝑙+(𝐴,𝑍)⟶𝑙⁻+(𝐴,𝑍+1)^∗,𝜈_𝑙+(𝐴,𝑍)⟶𝑙⁺+(𝐴,𝑍−1)^∗. (1.1)

These reactions are also called neutrino (antineutrino) capture, since they can be considered as the reverse processes of lepton capture. They are mediated by exchange of heavy 𝑊^± bosons according to the (lowest order) Feynman diagram shown in Figure 1(a). In neutral-current reactions (neutrino scattering) the neutrinos (antineutrinos) interact via the exchange of neutral 𝑍⁰ bosons (see Figure 1(b)) with a nucleus as𝜈+(𝐴,𝑍)⟶𝜈^′+(𝐴,𝑍)^∗,𝜈+(𝐴,𝑍)⟶𝜈^′+(𝐴,𝑍)^∗, (1.2) where 𝜈 (𝜈) denote neutrinos (antineutrinos) of any flavor. The neutrino-nucleus reactions leave the final nucleus mostly in an excited state lying below particle-emission thresholds (semi-inclusive processes) [26]. The transitions to energy levels higher than the particle-bound states usually decay by particle emission and, thus, they supply light particles that can cause further nuclear reactions.

(a)

(b)

Figure 1 

Feynman-diagram of lowest order for: (a) the CC neutrino-nucleus reactions 𝜈𝑙+(𝐴,𝑍)→𝑙−+(𝐴,𝑍+1)∗, and (b) the NC neutrino-nucleus processes 𝜈+(𝐴,𝑍)→𝜈′+(𝐴,𝑍)∗. The diagrams which correspond to the antineutrino reactions are similar.

When a massive star runs out of its nuclear fuel, it collapses under its own gravity [35–38]. As a consequence of this collapse, the density and temperature in its core increase and finally the outer shell of the star explodes, emitting a huge amount of energy. That procedure of violent energy emission in interstellar medium is called supernova (SN) explosion. Most part of this energy is carried in the space by neutrinos of all flavors (𝜈_𝑒, 𝜈_𝜇, 𝜈_𝜏, 𝜈_𝑒, 𝜈_𝜇, 𝜈_𝜏). Although the energy released by an SN explosion is shared equally between neutrinos of all flavors, their energy spectra differ due to the dependence of neutrinos flavor on their interaction with nuclei in the stellar gas. The change in gravitational binding energy between the initial stellar core and the final proton-neutron star is about 3×10⁵³ erg, 99% of which is carried off by all flavors of neutrinos and antineutrinos in about 10 s. The emission time is much longer than the light-crossing time of the protoneutron star because the neutrinos are trapped and then have to be diffused out, eventually escaping the star having energy distribution spectra which are approximated by the Fermi-Dirac (FD) energy distribution ones. In the canonical model [39–41], 𝜈_𝑒 is emitted with temperature 𝑇≃3.5MeV, 𝜈_𝑒 has 𝑇≃5MeV, and all other flavors (𝜈_𝑥) have temperature 𝑇≃8MeV. The temperatures differ from each other because 𝜈_𝑒 and 𝜈_𝑒 have charged-current opacities (in addition to the neutral-current opacities common to all flavors) and because the protoneutron star has more neutrons than protons. The neutrinos 𝜈_𝑥(𝑥=𝜇,𝜏,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. Since the number of neutrons is larger than the protons, 𝜈_𝑒 loses energy much more than 𝜈_𝑒 and the average energy for 𝜈_𝑒 is more than 𝜈_𝑒.

Precise theoretical estimates of neutrino-nucleus cross-sections, in low and intermediate neutrino energies, are extremely important in modern neutrino physics [28–34]. In the present work, we have performed realistic calculations for the differential and total cross sections of neutrino elastic (coherent) and inelastic (incoherent) scattering off ⁴⁰Ar(𝜈,𝜈′)⁴⁰Ar^∗ and ¹³²Xe(𝜈,𝜈′)¹³²Xe^∗ using the quasi-particle random phase approximation (QRPA). The response of noble gases Ar and Xe as a supernova neutrino detection is evaluated assuming a two-parameter FD distribution. Since neutrino energies from SN explosions are expected to be higher than those stemming from the solar neutrino, one needs to consider the contributions from higher multipole states. For this reason, we have considered all the QRPA excited states of ⁴⁰Ar and ¹³²Xe up to 40 MeV, in contrast to previous RPA calculations [42] concerning ⁴⁰Ar, which seems to take only a few excited states known by experiment. Moreover, we have investigated the individual contributions coming from the coherent (0⁺_gs→0⁺_gs) and incoherent (0⁺_gs→𝐽_𝑓) channels to total neutrino-nucleus cross sections. We found that the coherent channel is dominant versus the incoherent one.

2. The Primary Supernova Neutrino Flux

The neutrino spectrum of a core-collapse supernova is believed to be similar to an FD spectrum, with temperatures in the range 3-8MeV [41]. The FD energy distribution is given by𝑓_𝑖𝐸_𝜈=𝑁₂(𝛼)𝑇³𝐸²_𝜈1+𝑒^{𝐸𝜈/𝑇−𝛼},𝑖=𝜈_𝑒,𝜈_𝑒,or𝜈_𝑥, (2.1) where 𝑁₂(𝛼) is the normalization constant depending on the parameter 𝛼 given by the relation𝑁_𝑘(𝛼)=^∞₀𝑥^𝑘1+𝑒^𝑥−𝛼𝑑𝑥⁻¹ (2.2) for 𝑘=2. Characteristic of the FD energy distribution is that the peak shifts to higher neutrino energies and the width increases as the neutrino energy increases (Figure 2). According to [43], the average neutrino energy ⟨𝐸_𝜈⟩ is given by𝐸_𝜈=𝑁₂(𝛼)𝑁₃(𝛼)𝑇. (2.3) Some characteristic values of ⟨𝐸_𝜈⟩ are listed in Table 1. Figure 3(a) shows the averaged neutrino energy as a function of the parameter 𝛼 for various temperatures 𝑇. As it is seen the introduction of a chemical potential, 𝜇=𝛼𝑇, in the spectrum at fixed neutrino temperature increases the average neutrino energy. From Figure 3(b) it is also seen that at fixed neutrino temperature a nonvanishing chemical potential enhances the averaged neutrino energy.

Table 1 

The average supernova neutrino energies as a function of the parameters 𝛼 and 𝑇.

Figure 2 

(Color on line). The normalized to unity Fermi-Dirac spectrum for 𝛼=0.

(a)

(b)

Figure 3 

(Color on line). Averaged neutrino energy as a function of the parameter 𝛼 for various temperatures 𝑇 (a). Averaged neutrino energy as a function of temperature for various parameters of chemical potential 𝛼 (b).

The interaction of neutrinos with dense neutron rich matter in the core results in the different energy distributions for the various neutrino flavors. The neutrinos 𝜈_𝑥 (𝑥=𝜇,𝜏,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. Since the number of neutrons is larger than the protons, 𝜈_𝑒 loses energy much more than 𝜈_𝑒 and the average energy for 𝜈_𝑒 is more than 𝜈_𝑒. The numerical simulations give the following values of average energy for the different neutrino flavors:𝐸_𝜈∼10-11MeV𝜈_𝑒15−16MeV𝜈_𝑒23−25MeV𝜈_𝑥,𝑥=𝜇,𝜏,𝜇,𝜏. (2.4) Those average neutrino energies 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.

The number of emitted neutrinos is𝑁_𝜈=𝑈_𝜈𝐸_𝜈, (2.5) where 𝑈_𝜈=0.5×10⁵³ erg per neutrino flavor. Taking the temperature 𝑇 to be 3.5, 5, and 8MeV for electron neutrinos (𝜈_𝑒), electron antineutrinos (𝜈_𝑒), and all other flavors (𝜈_𝑥) respectively, and the parameter 𝛼 to be 0≤𝛼≤5, then the obtained results for the number of primary neutrinos emitted are shown in Table 2, while the (time averaged) neutrino flux Φ_𝜈=𝑁_𝜈/4𝜋𝐿² at a distance 𝐿=10 Kpc = 3.1×10²² cm is given in Table 3.

Table 2 

The number of primary neutrinos emitted in a typical supernova explosion as a function of the parameters 𝛼 and 𝑇 in units of 10⁵⁸.

Table 3 

The (time integrated) neutrino flux, in units of 10¹² cm⁻², at a distance 10 kpc from the source.

3. Brief Description of the Neutral-Current Neutrino-Nucleus Scattering Formalism

In the present work we consider neutral-current neutrino-nucleus interactions 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⁺⟩ state.

The corresponding standard model effective Hamiltonian in current-current interaction form is written as𝐺ℋ=√2𝑗_𝜇(𝐱)𝐽^𝜇(𝐱), (3.1) where 𝐺=1.1664×10⁻⁵GeV⁻² is the Fermi weak coupling constant. 𝑗_𝜇 and 𝐽^𝜇 denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form𝑗_𝜇=𝜓_𝜈ℓ(𝑥)𝛾_𝜇1−𝛾₅𝜓_𝜈ℓ(𝑥), (3.2) 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 processes of both vector and axial-vector components (neglecting the pseudo-scalar contributions) is written as𝐽_𝜇=Ψ_𝑁𝐹₁𝛾_𝜇+𝐹₂𝑖𝜎_𝜇𝜈𝑞^𝜈2𝑀+𝐹_𝐴𝛾_𝜇𝛾₅Ψ_𝑁 (3.3) (𝑀 stands for the nucleon mass and Ψ_𝑁 denote the nucleon spinors). 𝐹_𝑖, 𝑖=1,2, represent the weak nucleon form factors given in terms of the well-known charge and electromagnetic form factors (CVC theory) for proton (𝐹^𝑝_𝑖) and neutron (𝐹^𝑛_𝑖) by the expressions [44]𝐹_1,2=12−sin²𝜃_𝑊𝐹^𝑝_1,2−𝐹^𝑛_1,2𝜏₀−sin²𝜃_𝑊𝐹^𝑝_1,2+𝐹^𝑛_1,2. (3.4) Here 𝜏₀ represents the nucleon isospin operator and 𝜃_𝑊 is the Weinberg angle (sin²𝜃_𝑊=0.2325). In (3.3) 𝐹_𝐴 stands for the axial-vector form factor for which we employ the dipole ansatz given by𝐹_𝐴1=−2𝑔_𝐴1−𝑞²𝑀²_𝐴⁻²𝜏₀, (3.5) 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 work 𝑞², the square of the momentum transfer, is written as𝑞²=𝑞^𝜇𝑞_𝜇=𝜔²−𝐪²=𝜀_𝑖−𝜀_𝑓²−𝐩_𝑖−𝐩_𝑓², (3.6) where 𝜔=𝜀_𝑖−𝜀_𝑓 is the excitation energy of the nucleus. 𝜀_𝑖 denotes the energy of the incoming and 𝜀_𝑓 that of the outgoing neutrino. 𝐩_𝑖, 𝐩_𝑓 are the corresponding 3-momenta of the incoming and outgoing neutrino/antineutrino, respectively. In (3.4) we have not taken into account the strange quark contributions in the form factors. In the scattering reaction considered in this work only low-momentum transfers are involved and the contributions from strangeness can be neglected [45].

The neutral-current neutrino/antineutrino-nucleus differential cross section, after applying a multipole analysis of the weak hadronic current as in [46], is written asd²𝜎_𝑖→𝑓dΩd𝜔_𝜈/𝜈=𝐺²𝜋𝑝_𝑓⃗𝜀_𝑓2𝐽_𝑖+1^∞_𝐽=0𝜎^𝐽_𝐶𝐿+^∞_𝐽=1𝜎^𝐽_𝑇. (3.7) The summations in (3.7) 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 defined as in [47]. These operators include both polar-vector and axial-vector weak interaction components. The contributions 𝜎^𝐽_𝐶𝐿 and 𝜎^𝐽_𝑇 are written as𝜎^𝐽_𝐶𝐿=(1+cos𝜃)𝐽_𝑓ℳ_𝐽𝑞𝐽_𝑖²+1+cos𝜃−2𝑏sin²𝜃𝐽_𝑓ℒ_𝐽𝑞𝐽_𝑖²+𝜔𝑞(1+cos𝜃)2ℜ𝑒𝐽_𝑓ℒ_𝐽𝑞𝐽_𝑖𝐽_𝑓ℳ_𝐽𝑞𝐽_𝑖^∗,𝜎^𝐽_𝑇=1−cos𝜃+𝑏sin²𝜃𝐽_𝑓𝒯^mag_𝐽𝑞𝐽_𝑖²+𝐽_𝑓𝒯^el_𝐽𝑞𝐽_𝑖²∓𝜀_𝑖+𝜀_𝑓𝑞(1−cos𝜃)2ℜ𝑒𝐽_𝑓𝒯^mag_𝐽𝑞𝐽_𝑖𝐽_𝑓𝒯^el_𝐽𝑞𝐽_𝑖^∗, (3.8)(3.9) where 𝜃 denotes the outgoing neutrino scattering angle and 𝑏=𝜀_𝑖𝜀_𝑓/𝑞². In (3.9) the − sign corresponds to neutrino scattering and the + sign to antineutrino one.

4. 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 quasi-particle random phase approximation (QRPA) including two quasi-neutron and two quasi-proton excitations in the QRPA matrix [48] (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 is𝐻=_𝛼𝜖_𝛼𝑐^†_𝛼𝑐_𝛼+14_{𝛼𝛽𝛾𝛿}𝑉_{𝛼𝛽𝛾𝛿}𝑐^†_𝛼𝑐^†_𝛽𝑐_𝛿𝑐_𝛾, (4.1) where the two-body term contains the antisymmetrised 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 [49]. The simplest way to introduce these correlations in the wave function is to perform the Bogoliubov-Valatin transformation𝑎^†_𝛼=𝑢_𝛼𝑐^†_𝛼−𝑣_𝛼𝑐_𝛼̃𝑎^†_𝛼=𝑢_𝛼𝑐^†_𝛼̃+𝑣_𝛼𝑐_𝛼, (4.2) 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⟩, (4.3) where |CORE⟩ represents the nuclear core (effective particle vacuum).

After the transformation (4.2) the Hamiltonian can be written in its quasi-particle representation as𝐻=_𝛼𝐸_𝛼𝑎^†_𝛼𝑎_𝛼+𝐻_qp, (4.4) where the first term gives the single quasi-particle 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 when doing the BCS calculations. 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 quasi-particle energy to the phenomenological energy gap Δ obtained from the separation energies of the neighboring doubly-even nuclei for protons and neutrons separately.

In the next step 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 (QRPA phonon) has the form𝑄^†𝐽^𝜋_𝑘𝑀=_{𝛼≤𝛼′}𝑋^𝐽𝜋𝑘_𝛼𝛼′𝐴^†𝛼𝛼^′;𝐽𝑀−𝑌^𝐽𝜋𝑘_𝛼𝛼′𝐴𝛼𝛼^′;𝐽𝑀, (4.5) where the quasi-particle pair creation 𝐴^†(𝛼𝛼′;𝐽𝑀) and annihilation 𝐴(𝛼𝛼′;𝐽𝑀) operators are defined as𝐴^†𝛼𝛼^′;𝐽𝑀≡1+𝛿_𝛼𝛼′^−1/2𝑎^†_𝛼𝑎^†_{𝛼′𝐽𝑀},𝐴𝛼𝛼^′;𝐽𝑀≡(−1)^𝐽+𝑀𝐴𝛼𝛼^′;𝐽−𝑀, (4.6) where 𝛼 and 𝛼′ are either proton (𝑝) or neutron (𝑛) indices, 𝑀 labels the magnetic substates, and 𝑘 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 equation𝒜ℬ𝑋−ℬ−𝒜𝑌^{𝐽𝜋𝐽𝜋}𝑋=𝜔^𝐽𝜋𝑌^𝐽𝜋, (4.7) 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 as𝒜_𝐽𝛼𝛼^′;𝛽𝛽^′=BCS𝐴𝛼𝛼^′;𝐽𝑀,𝐻,𝐴^†𝛽𝛽^′,ℬ;𝐽𝑀BCS_𝐽𝛼𝛼^′;𝛽𝛽^′=−BCS𝐴𝛼𝛼^′;𝐽𝑀,𝐻,𝐴𝛽𝛽^′;𝐽𝑀BCS, (4.8) 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.

5. Results

5.1. Calculated Cross Sections

In order to investigate neutrino scattering off the 40Ar and 132Xe nuclei we followed the procedure of [30–34]. Specifically we have performed explicit state-by-state calculations for the nuclear transition matrix elements given by (3.8) and (3.9) in the framework of QRPA. The initial nucleus was assumed to be spherically symmetric having a 0⁺ ground state. In the case of 132Xe we have adopted 40Ca as inert core and the two oscillator 3ℏ𝜔 and 4ℏ𝜔 major shells, plus the intruder orbital ℎ_11/2 from the next higher oscillator major shell, as valence space for protons and neutrons. In the case of 40Ar we have considered the major shells 0,1,2, and 3ℏ𝜔 as the model space for both protons and neutrons. The corresponding single-particle energies (s.p.e) were produced by the Coulomb corrected Woods-Saxon potential using the parameters of Bohr and Mottelson [50].

The two-body interaction matrix elements were obtained from the Bonn one-boson-exchange potential applying G-matrix techniques [51]. 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 𝑔^𝑛_pair separately for protons and neutrons. The adjustment can be done by comparing the resulting lowest quasiparticle energy to reproduce the phenomenological pairing gap obtained by using the linear approximation [52]Δ_𝑛^𝐴_𝑍𝑋1=−4𝑆_𝑛𝑍^𝐴+1𝑋−2𝑆_𝑛^𝐴_𝑍𝑋+𝑆_𝑛𝑍^𝐴−1𝑋,Δ_𝑝^𝐴_𝑍𝑋1=−4𝑆_𝑝^𝐴+1_𝑍+1𝑋−2𝑆_𝑝^𝐴_𝑍𝑋+𝑆_𝑝^𝐴−1_𝑍−1𝑋 (5.1) in which ^𝐴_𝑍𝑋 stands for the doubly-even nucleus under consideration. The separation energies 𝑆_𝑛/𝑝 are provided by [53]. The results of this procedure lead to the pairing-strength parameters 𝑔^𝑝_pair = 1.05 and 𝑔^𝑛_pair = 0.88 for 40Ar and 𝑔^𝑝_pair = 1.08 and 𝑔^𝑛_pair = 0.89 for 132Xe. After settling the values of the pairing parameters, two other parameters are left to fix, the overall scale of the particle-hole interaction 𝑔_ph and separately the particle-particle channel of the interaction 𝑔_pp for each multipole up to 𝐽=8^±. The QRPA parameters are determined so that the low-lying energy spectrum fits to the experimental data [30–34]. An alternative fixing of the parameters 𝑔_ph and 𝑔_pp, especially for the charged-current neutrino-nucleus reactions, could be done on the giant dipole resonance of the studied nucleus. Using the formalism for the double differential cross section we have calculated d²𝜎_𝑖→𝑓/dΩd𝜔 (see (3.7)) for all QRPA states up to 40 MeV, in contrast to previous RPA calculations, which consider only a few states known by experiment [42]. The total cross section 𝜎(𝐸_𝜈) was obtained by integrating over the scattering angles 𝜃 and 𝜙 and subsequently summing over all discrete final sates. The results were obtained for coherent 0⁺_gs→0⁺_gs cross sections (elastic channel) as well as for incoherent cross sections 0⁺_gs→𝐽_𝑓 (inelastic channel).

The coherent neutrino-nucleus scattering (CNNS) is an important prediction of the Standard Model. It is worth mentioning that there is quite a wide literature describing CNNS mainly based on nuclear recoil signals [54]. The differential cross section versus neutrino energy 𝐸_𝜈 is given by [55]𝑑𝜎𝑑Ω_coh𝐸_𝜈=𝐺²16𝜋²𝐸²_𝜈(1+cos𝜃)𝑄²_𝑊𝐹²𝑞²,𝑞²=2𝐸²_𝜈(1−cos𝜃), (5.2) where 𝜃 denotes the scattering angle of the incident neutrino in the lab frame of the recoil nucleus, 𝐺 is the Fermi constant, and 𝑄_𝑊 is the weak charge of the nucleus with 𝑁 neutrons and 𝑍 protons:𝑄_𝑊=𝑁−1−4sin²Θ_𝑊𝑍 (5.3) with Θ_𝑊 being the weak mixing angle (sin²Θ_𝑊≈0.231). 𝐹(𝑞²) stands for the elastic form factor [56] that describes the distribution of weak charge within the nucleus. Integrating the differential cross section with respect to 𝑑Ω we obtain the CNNS cross section as a function of the neutrino energy 𝐸_𝜈𝜎_coh𝐸_𝜈=𝑑𝜎𝑑Ω_coh𝐸_𝜈𝑑Ω. (5.4)

Figure 4 shows the contributions of coherent and incoherent cross sections as a function of the incoming neutrino energies taken from the QRPA calculations. In Figures 4(a) and 4(b) we also present the total cross sections (coherent plus incoherent) for the reactions ¹³²Xe(𝜈,𝜈′)¹³²Xe and ⁴⁰Ar(𝜈,𝜈^′)⁴⁰Ar, respectively. As it is seen, the coherent cross sections are greater than incoherent ones by at least an order of magnitude in the relevant energy region and dominates the total cross section for all neutrino energies 𝐸_𝜈≤100MeV. These results are similar to the calculations performed by other nuclear systems [57]. In Figure 4 we also present the results for the coherent channel taken from (5.4). As it is seen, the coherent 0⁺_gs→0⁺_gs cross sections obtained from QRPA are in agreement with those taken from (5.4), especially for neutrinos with energies below 40 MeV. The theoretical uncertainty on the neutrino-nucleus scattering cross section comes from nuclear modelling in the form factor calculation.

(a)

(b)

Figure 4 

(Color on line). Coherent, incoherent, and total cross section as a function of the incoming neutrino energy 𝐸𝜈, in the NC reactions 132Xe(𝜈,𝜈′)132Xe (a) and 40Ar(𝜈,𝜈′)40Ar (b).

Figure 5 illustrates the corresponding distribution of the different multipolarities to the incoherent cross section for two impinging neutrino energies. As it is seen, in low-energy region, the transitions 0⁺ for 40Ar and 1⁻ for 132Xe are the most pronounced channels. On the other hand, in high-energy region, the incoherent scattering for 40Ar is dominated mostly by the 1⁻ transition while other transitions like 3⁻ and 2⁺ start to contribute significantly. In the case of 132Xe the channels 1⁻, 3⁻, 2⁺, and 4⁺ are dominant.

(a)

(b)

(c)

(d)

Figure 5 

(Color on line). Partial multipole distributions to the incoherent cross sections for 40Ar and 132Xe, at the incoming neutrino energies 𝐸𝜈=20 and 100 MeV.

In order to obtain more information about supernova neutrinos, the total cross section 𝜎(𝐸_𝜈) has to be folded with the FD neutrino energy distribution. The individual contributions into coherent, incoherent, and total (coherent plus incoherent) cross sections are given in Table 4. As it is seen from this table the coherent scattering clearly dominates the total cross sections. Finally in Table 5 we compare our results for the coherent 0⁺_gs→0⁺_gs cross sections folded with the FD spectra with those obtained from (5.4). As it is seen, the results obtained by means of the standard formula (5.4) are consistent with those taken by QRPA calculations. It is clear that the main contribution to the coherent channel comes from the transition 0⁺_gs→0⁺_gs.

Table 4 

Coherent, incoherent, and total (coherent plus incoherent) neutrino nucleus cross section for 40Ar and 132Xe targets.The supernova neutrino spectra described by a two-parameter Fermi-Dirac distribution with 𝛼=0 and 3 for various temperatures 𝑇 (MeV).

Table 5 

Coherent neutrino nucleus cross sections calculated by (5.4) and by the QRPA approach (numbers in parenthesis) for 40Ar and 132Xe targets. The supernova neutrino spectra described by a two parameter Fermi-Dirac distribution with 𝛼=0 and 3 for various temperatures 𝑇 (MeV).

5.2. Neutrino Detection with a TPC Detector

One of the most famous detectors for dedicated supernova detection is gaseous spherical TPC detector (Time Projection Chamber) [58]. TPC detector allows measurements of high multiplicity events (≃200) coming from relativistic nucleus-nucleus collisions. It has low threshold and high resolution. As it is known, a spherical TPC detector filled with either Xe or Ar has been proposed as a device able to detect low-energy neutrinos as those coming from a galactic supernova and, in particular, it will be able to observe coherent neutrino-nucleus scattering [59–64].

Taking into account our results concerning the total cross sections for Ar and Xe, it is a good opportunity to employ and test the spherical TPC gaseous detector of volume 𝑉 under pressure 𝑃 and temperature 𝑇₀, filled with noble gas such as Ar and Xe. In this case, the number of expected events in a year takes the formℛ=3.156×10⁷𝑡Φ()𝜎1𝑦𝜈,𝐿=50m_tot𝑃𝑉𝑘𝑇₀𝑠(𝑉,𝐿), (5.5) where the parameter 𝑠(𝑉,𝐿) is a geometrical factor needed when a large detector is close to the source [65]. It depends on the shape of the vessel and the distance 𝐿 of its geometric center from the source. In the case of sphere of radius 𝑅 with its center at a distance 𝐿 from the source, the function 𝑠(𝑉,𝑅) depends only on the ratio 𝑅/𝐿 and it is given by𝑠()=𝐿𝑅/𝐿²(4/3)𝜋𝑅³2𝜋𝐿₀^𝑅/𝐿𝑥²𝑑𝑥_𝜋0𝑑𝜃sin𝜃1+𝑥²𝑟+2𝑥cos𝜃,𝑥=𝐿. (5.6) Spherical coordinates (𝑟,𝜃,𝜙) are used to specify any point inside the sphere. The origin of coordinates was chosen at the center of the sphere with polar axis being the straight line from the source to the center. With the above choice the flux is independent of the angle 𝜙. A plot of the function 𝑠(𝑅/𝐿) is presented in Figure 6. The geometric factor 𝑠(𝑅/𝐿) is close to unity in the actual experimental setup where 𝐿≫𝑅.

Figure 6 

Geometrical factor 𝑠(𝑅/𝐿) for a sphere of radius 𝑅 whose center is at distance 𝐿 from the source.

For a typical distance 𝐿=50 mwe can take as neutrino flux for each neutrino flavor the value Φ(𝜈,𝐿=50m)=1.95×10⁶cm⁻²s⁻¹. Summing over all the neutrino flavors we find the total cross sections 𝜎_tot=6.5×10⁻³⁹cm² for 𝐴=40 and 𝜎_tot=7.0×10⁻³⁸ cm² for 𝐴=132. Finally the total number of events ℛ in a year is calculated using (5.5) and listed in Table 6. The parameters considered in our calculations are consistent with the experimental works of [59–66]. Moreover, for a primary supernova neutrino flux (time averaged) Φ_𝜈=𝑁_𝜈4𝜋𝐿² (5.7) at a distance 𝐿=10 Kpc = 3.1×10²² cm, the number of the observed events for each neutrino flavor is found to be𝑁_𝑒𝑣=Φ_𝜈𝜎_𝜈𝑃𝑉𝑘𝑇₀. (5.8) In Table 7 the numbers of event rates are listed for two given radii 𝑅=6 and 9 m. As it seen, employing 132Xe as a target nucleus one expects about 1761 events for a sphere of radius 6 m, while for 40Ar one expects about 562 events but with a vessel of larger radius (𝑅=9 m).

Table 6 

Number of events in a year for a spherical detector of various radii 𝑅 with its center at a distance 𝐿=50m from the source. The vessel is filled with gas under pressure 𝑃=10 Atm and temperature 𝑇₀=300K. The total cross sections are obtained summing over all neutrino flavors. The chemical potential of the neutrino spectra is taken as 𝛼=0.

Table 7 

The number of events rate for a spherical detector of various radii 𝑅. The neutrinos are emitted from a supernova at a distance 𝐿=10 Kpc. The spherical vessel of volume 𝑉 is filled with gas under pressure 𝑃=10 Atm and temperature 𝑇₀=300K. The chemical potential of the neutrino spectra is taken as 𝛼=0.

6. Conclusions

In this paper the coherent and incoherent contribution in neutrino-nucleus scattering due to neutral current has been examined considering as target materials the isotopes 40Ar and 132Xe. The differential as well as the total cross sections have been derived employing the quasi-particle random phase approximation. In order to obtain information appropriate for describing terrestrial detection of supernova neutrinos, the total cross sections (coherent+incoherent) were folded with a neutrino energy spectrum in the FD model. An enhancement of the neutral current component is achieved via the coherent channel (0⁺_gs→0⁺_gs) which is dominant with respect to incoherent one.

From the above results one can test a gaseous spherical TPC detector dedicated for SN neutrino detection. Filling the TPC detector with the noble gas 132Xe under pressure 𝑃=10 Atm and temperature 𝑇₀=300K one expects about 1761 events for a sphere of radius 6 m. Employing 40Ar one expects 562 events but with a vessel of larger radius (𝑅=9 m). This detector can also be tested with earth neutrino sources, which have a neutrino spectrum analogous to that of an SN. Neutral current detectors, which are not sensitive to neutrino oscillation effects, could provide a great deal of information about the primary supernova neutrino flux.

Acknowledgment

The author would like to thank Professor J. D. Vergados for useful discussions.