# Coherent and Incoherent Neutral Current Scattering for Supernova Detection

Authors: P. C. Divari
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

The total cross sections as well as the neutrino event rates are calculated in the neutral current neutrino scattering off40Ar and 132Xe isotopes at neutrino energies (Ev<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 (0gs+→0gs+) channel which is dominant with respect to incoherent (0gs+→Jf) 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.


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## Body

## 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 νl (antineutrino ν̅l) 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 as(1.1)νl+(A,Z)⟶l-+(A,Z+1)*,ν̅l+(A,Z)⟶l++(A,Z-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 heavyW± 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 as(1.2)ν+(A,Z)⟶ν′+(A,Z)*,ν̅+(A,Z)⟶ν̅′+(A,Z)*,
where ν (ν̅) 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.Feynman-diagram of lowest order for: (a) the CC neutrino-nucleus reactionsνl+(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.
(a)(b)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 (νe, νμ, ντ, ν̅e, ν̅μ, ν̅τ). 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], νe is emitted with temperature T≃3.5MeV, ν̅e has T≃5MeV, and all other flavors (νx) have temperature T≃8MeV. The temperatures differ from each other because ν̅e and νe 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 νx(x=μ,τ,μ̅,τ̅) 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 νe and ν̅e, which interact through charged current as well as neutral current. Since the number of neutrons is larger than the protons, νe loses energy much more than ν̅e and the average energy for ν̅e is more than νe.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 Ar40(ν,ν′)Ar*40 and Xe132(ν,ν′)Xe*132 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  Ar40 and Xe132 up to 40 MeV, in contrast to previous RPA calculations [42] concerning  Ar40, which seems to take only a few excited states known by experiment. Moreover, we have investigated the individual contributions coming from the coherent (0gs+→0gs+) and incoherent (0gs+→Jf) 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 range3–8MeV [41]. The FD energy distribution is given by(2.1)fi(Eν)=N2(α)T3Eν21+eEν/T-α,i=νe,ν̅e,orνx,
where N2(α) is the normalization constant depending on the parameter α given by the relation(2.2)Nk(α)=(∫0∞xk1+ex-αdx)-1
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 〈Eν〉 is given by(2.3)〈Eν〉=N2(α)N3(α)T.
Some characteristic values of 〈Eν〉 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 parametersα and T.
α〈Eν〉 (MeV)νeν̃eνxT=3.5 MeVT=5 MeVT=8 MeV011.0315.7625.210.7511.4516.3626.171.5012.0817.2627.612.0012.6218.0328.843.0013.9719.9631.944.0015.6322.3335.735.0017.5225.0240.04Figure 2
(Color on line). The normalized to unity Fermi-Dirac spectrum forα=0.(Color on line). Averaged neutrino energy as a function of the parameterα for various temperatures T (a). Averaged neutrino energy as a function of temperature for various parameters of chemical potential α (b).
(a)(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 neutrinosνx (x=μ,τ,μ̅,τ̅) 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 νe and ν̅e, which interact through charged current as well as neutral current. Since the number of neutrons is larger than the protons, νe loses energy much more than ν̅e and the average energy for ν̅e is more than νe. The numerical simulations give the following values of average energy for the different neutrino flavors:(2.4)〈Eν〉~{10-11MeVνe15–16MeVν̅e23–25MeVνx,x=μ,τ,μ̅,τ̅}.
Those average neutrino energies imply that for α=0(3) the values of temperature T are 3.5 MeV (2.75 MeV) for νe, 5 MeV(4 MeV) for ν̅e, and 8 MeV (6 MeV) for νx.The number of emitted neutrinos is(2.5)Nν=Uν〈Eν〉,
where Uν=0.5×1053 erg per neutrino flavor. Taking the temperature T to be 3.5, 5, and 8MeV for electron neutrinos (νe), electron antineutrinos (ν̅e), and all other flavors (νx) 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 Φν=Nν/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 parametersα and T in units of 1058.
αNν/1058νeν̅e∑xνxT=3.5 MeVT=5 MeVT=8 MeV00.280.200.500.750.270.190.481.500.260.180.462.000.250.170.443.000.220.160.394.000.200.140.355.000.180.120.32Table 3
The (time integrated) neutrino flux, in units of1012 cm−2, at a distance 10 kpc from the source.
αΦν/1012cm-2νeν̅e∑xνxT=3.5 MeVT=5 MeVT=8 MeV00.230.160.410.750.230.160.391.500.210.150.372.000.200.140.363.000.180.130.324.000.170.120.295.000.150.100.26


## 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 |Jπ〉=|0+〉 state.The corresponding standard model effective Hamiltonian in current-current interaction form is written as(3.1)H=G2jμ(x)Jμ(x),
where G=1.1664×10-5GeV-2 is the Fermi weak coupling constant. jμ and Jμ denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form(3.2)jμ=ψ̅νl(x)γμ(1-γ5)ψνl(x),
where ψνℓ 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 as(3.3)Jμ=Ψ̅N[F1γμ+F2iσμνqν2M+FAγμγ5]ΨN
(M stands for the nucleon mass and ΨN denote the nucleon spinors). Fi, 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 (Fip) and neutron (Fin) by the expressions [44](3.4)F1,2=(12-sin2θW)[F1,2p-F1,2n]τ0-sin2θW[F1,2p+F1,2n].
Here τ0 represents the nucleon isospin operator and θW is the Weinberg angle (sin2θW=0.2325). In (3.3) FA stands for the axial-vector form factor for which we employ the dipole ansatz given by(3.5)FA=-12gA(1-q2MA2)-2τ0,
where MA=1.05 GeV is the dipole mass and gA=1.258 is the static value (at q=0) of the axial form factor.In the convention we used in the present workq2, the square of the momentum transfer, is written as(3.6)q2=qμqμ=ω2-q2=(εi-εf)2-(pi-pf)2,
where ω=εi-εf is the excitation energy of the nucleus. εi denotes the energy of the incoming and εf that of the outgoing neutrino. pi, pf 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 as(3.7)(d2σi→fdΩdω)ν/ν̅=G2π|p⃗f|εf(2Ji+1)(∑J=0∞σCLJ+∑J=1∞σTJ).
The summations in (3.7) contain the contributions σCLJ, for the Coulomb ℳ̂J and longitudinal ℒ̂J, and σTJ, for the transverse electric 𝒯̂Jel and magnetic 𝒯̂Jmag multipole operators defined as in [47]. These operators include both polar-vector and axial-vector weak interaction components. The contributions σCLJ and σTJ are written as(3.8)σCLJ=(1+cos⁡θ)|〈Jf||M̂J(q)||Ji〉|2+(1+cos⁡θ-2bsin⁡2θ)|〈Jf||L̂J(q)||Ji〉|2+[ωq(1+cos⁡θ)]2Re〈Jf‖L̂J(q)‖Ji〉〈Jf‖M̂J(q)‖Ji〉*,(3.9)σTJ=(1-cos⁡θ+bsin⁡2θ)[|〈Jf‖T̂Jmag(q)‖Ji〉|2+|〈Jf‖T̂Jel(q)‖Ji〉|2]∓(εi+εf)q(1-cos⁡θ)2Re〈Jf‖T̂Jmag(q)‖Ji〉〈Jf‖T̂Jel(q)‖Ji〉*,
where θ denotes the outgoing neutrino scattering angle and b=εiεf/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) ϵα which runs over all values of quantum numbers α≡{nα,lα,jα,mα} and the second term which includes the two-body interaction V, that is(4.1)H=∑αϵαcα†cα+14∑αβγδV̅αβγδcα†cβ†cδcγ,
where the two-body term contains the antisymmetrised two-body interaction matrix element defined by V̅αβγδ=〈αβ|V|γδ〉-〈αβ|V|δγ〉. The operators cα† and cα 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 transformation(4.2)aα†=uαcα†-vαc̃αãα†=uαc̃α†+vαcα,
where c̃α†=c-α†(-1)jα+mα, ãα†=a-α†(-1)jα+mα and -α≡{nα,lα,jα,-mα}. The occupation amplitudes vα and uα 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(4.3)|BCS〉=∏α>0(uα-vαcα†c̃α†)|CORE〉,
where |CORE〉 represents the nuclear core (effective particle vacuum).After the transformation (4.2) the Hamiltonian can be written in its quasi-particle representation as(4.4)H=∑αEαaα†aα+Hqp,
where the first term gives the single quasi-particle energies Eα and the second one includes the different components of the residual interaction.In the present calculations we use a renormalization parametergpair 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 gpair. 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 form(4.5)Q̂†(JkπM)=∑α≤α′[Xαα′JkπA†(αα′;JM)-Yαα′JkπÃ(αα′;JM)],
where the quasi-particle pair creation A†(αα′;JM) and annihilation Ã(αα′;JM) operators are defined as(4.6)A†(αα′;JM)≡(1+δαα′)-1/2[aα†aα′†]JM,Ã(αα′;JM)≡(-1)J+MA(αα′;J-M),
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.TheX and Y forward and backward going amplitudes are determined from the QRPA matrix equation(4.7)(AB-B-A)(XJπYJπ)=ω(XJπYJπ),
where ω denotes the excitation energies of the nuclear state |Jπ〉. The QRPA matrices 𝒜 and ℬ, are deduced by the matrix elements of the double commutators of A† and A with the nuclear hamiltonian Ĥ defined as(4.8)AJ(αα′;ββ′)=〈BCS|[A(αα′;JM),Ĥ,A†(ββ′;JM)]|BCS〉,BJ(αα′;ββ′)=-〈BCS|[A(αα′;JM),Ĥ,Ã(ββ′;JM)]|BCS〉,
where 2[A,B,C]=[A,[B,C]]+[[A,B],C]. Finally the two-body matrix elements of each multipolarity Jπ, occurring in the QRPA matrices 𝒜 and ℬ, are multiplied by two phenomenological scaling constants, namely, the particle-hole strength gph and the particle-particle strength gpp. 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 the40Ar 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 h11/2 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 gpairp and gpairn 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](5.1)Δn(XZA)=-14[Sn(XZA+1)-2Sn(XZA)+Sn(XZA-1)],Δp(XZA)=-14[Sp(XZ+1A+1)-2Sp(XZA)+Sp(XZ-1A-1)]
in which ZAX stands for the doubly-even nucleus under consideration. The separation energies Sn/p are provided by [53]. The results of this procedure lead to the pairing-strength parameters gpairp = 1.05 and gpairn = 0.88 for 40Ar and gpairp = 1.08 and gpairn = 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 gph and separately the particle-particle channel of the interaction gpp 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 gph and gpp, 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 d2σi→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(Eν) 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 0gs+→0gs+ cross sections (elastic channel) as well as for incoherent cross sections 0gs+→Jf (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 Eν is given by [55](5.2)(dσdΩ)coh(Eν)=G216π2Eν2(1+cos⁡θ)QW2F2(q2),q2=2Eν2(1-cos⁡θ),
where θ denotes the scattering angle of the incident neutrino in the lab frame of the recoil nucleus, G is the Fermi constant, and QW is the weak charge of the nucleus with N neutrons and Z protons:(5.3)QW=N-(1-4sin2ΘW)Z
with ΘW being the weak mixing angle (sin2ΘW≈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 Eν(5.4)σcoh(Eν)=∫(dσdΩ)coh(Eν)dΩ.Figure4 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 Eν≤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 0gs+→0gs+ 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.(Color on line). Coherent, incoherent, and total cross section as a function of the incoming neutrino energyEν, in the NC reactions 132Xe(ν,ν′)132Xe (a) and 40Ar(ν,ν′)40Ar (b).
(a)(b)Figure5 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.(Color on line). Partial multipole distributions to the incoherent cross sections for40Ar and 132Xe, at the incoming neutrino energies Eν=20 and 100 MeV.
(a)(b)(c)(d)In order to obtain more information about supernova neutrinos, the total cross sectionσ(Eν) 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 0gs+→0gs+ 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 0gs+→0gs+.Table 4
Coherent, incoherent, and total (coherent plus incoherent) neutrino nucleus cross section for40Ar 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).
(T,α)(3.5,0)(5,0)(8,0)(2.75,3)(4,3)(6,3)σcoh(in10-39 cm2)40Ar0.330.631.400.300.611.26132Xe3.877.1714.543.637.0913.53σincoh(in10-42 cm2)40Ar0.080.281.560.060.210.95132Xe6.3736.23239.403.1923.10143.10σν=σcoh+σincoh(in10-39 cm2)40Ar0.320.631.400.300.611.26132Xe3.877.2014.783.647.1113.68Table 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).
σcoh(in10-39 cm2)(T,α)(3.5,0)(5,0)(8,0)(2.75,3)(4,3)(6,3)40Ar0.28 (0.33)0.56 (0.63)1.35 (1.40)0.26 (0.30)0.53 (0.61)1.16 (1.26)132Xe3.50 (3.87)6.86 (7.17)15.63 (14.54)3.23 (3.63)6.63 (7.09)13.87 (13.53)


### 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 volumeV 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 form(5.5)R=3.156×107t1yΦ(ν,L=50m)σtotPVkT0s(V,L),
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 by(5.6)s(R/L)=L2(4/3)πR32πL∫0R/Lx2dx∫0πdθsin⁡θ1+x2+2xcos⁡θ,x=rL.
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 factors(R/L) for a sphere of radius R whose center is at distance L from the source.For a typical distanceL=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 σtot=6.5×10-39cm2 for A=40 and  σtot=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)
(5.7)Φν=Nν4πL2
at a distance L=10 Kpc = 3.1×1022 cm, the number of the observed events for each neutrino flavor is found to be(5.8)Nev=ΦνσνPVkT0.
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 radiiR 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.
TargetR=1 mR=3 mR=6 mR=9 mR=12 m132Xe4.4×1031.2×1059.5×1053.2×1067.6×10640Ar4.1×1021.1×1048.8×1043.0×1057.0×105Table 7
The number of events rate for a spherical detector of various radiiR. 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.
NevR (m)νeν̅e∑xνxTotal40Ar6162212716595676430562132Xe619725513091761966686144185945


## 5.1. Calculated Cross Sections

In order to investigate neutrino scattering off the40Ar 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 h11/2 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 gpairp and gpairn 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](5.1)Δn(XZA)=-14[Sn(XZA+1)-2Sn(XZA)+Sn(XZA-1)],Δp(XZA)=-14[Sp(XZ+1A+1)-2Sp(XZA)+Sp(XZ-1A-1)]
in which ZAX stands for the doubly-even nucleus under consideration. The separation energies Sn/p are provided by [53]. The results of this procedure lead to the pairing-strength parameters gpairp = 1.05 and gpairn = 0.88 for 40Ar and gpairp = 1.08 and gpairn = 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 gph and separately the particle-particle channel of the interaction gpp 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 gph and gpp, 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 d2σi→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(Eν) 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 0gs+→0gs+ cross sections (elastic channel) as well as for incoherent cross sections 0gs+→Jf (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 Eν is given by [55](5.2)(dσdΩ)coh(Eν)=G216π2Eν2(1+cos⁡θ)QW2F2(q2),q2=2Eν2(1-cos⁡θ),
where θ denotes the scattering angle of the incident neutrino in the lab frame of the recoil nucleus, G is the Fermi constant, and QW is the weak charge of the nucleus with N neutrons and Z protons:(5.3)QW=N-(1-4sin2ΘW)Z
with ΘW being the weak mixing angle (sin2ΘW≈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 Eν(5.4)σcoh(Eν)=∫(dσdΩ)coh(Eν)dΩ.Figure4 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 Eν≤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 0gs+→0gs+ 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.(Color on line). Coherent, incoherent, and total cross section as a function of the incoming neutrino energyEν, in the NC reactions 132Xe(ν,ν′)132Xe (a) and 40Ar(ν,ν′)40Ar (b).
(a)(b)Figure5 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.(Color on line). Partial multipole distributions to the incoherent cross sections for40Ar and 132Xe, at the incoming neutrino energies Eν=20 and 100 MeV.
(a)(b)(c)(d)In order to obtain more information about supernova neutrinos, the total cross sectionσ(Eν) 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 0gs+→0gs+ 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 0gs+→0gs+.Table 4
Coherent, incoherent, and total (coherent plus incoherent) neutrino nucleus cross section for40Ar 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).
(T,α)(3.5,0)(5,0)(8,0)(2.75,3)(4,3)(6,3)σcoh(in10-39 cm2)40Ar0.330.631.400.300.611.26132Xe3.877.1714.543.637.0913.53σincoh(in10-42 cm2)40Ar0.080.281.560.060.210.95132Xe6.3736.23239.403.1923.10143.10σν=σcoh+σincoh(in10-39 cm2)40Ar0.320.631.400.300.611.26132Xe3.877.2014.783.647.1113.68Table 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).
σcoh(in10-39 cm2)(T,α)(3.5,0)(5,0)(8,0)(2.75,3)(4,3)(6,3)40Ar0.28 (0.33)0.56 (0.63)1.35 (1.40)0.26 (0.30)0.53 (0.61)1.16 (1.26)132Xe3.50 (3.87)6.86 (7.17)15.63 (14.54)3.23 (3.63)6.63 (7.09)13.87 (13.53)


## 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 volumeV 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 form(5.5)R=3.156×107t1yΦ(ν,L=50m)σtotPVkT0s(V,L),
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 by(5.6)s(R/L)=L2(4/3)πR32πL∫0R/Lx2dx∫0πdθsin⁡θ1+x2+2xcos⁡θ,x=rL.
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 factors(R/L) for a sphere of radius R whose center is at distance L from the source.For a typical distanceL=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 σtot=6.5×10-39cm2 for A=40 and  σtot=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)
(5.7)Φν=Nν4πL2
at a distance L=10 Kpc = 3.1×1022 cm, the number of the observed events for each neutrino flavor is found to be(5.8)Nev=ΦνσνPVkT0.
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 radiiR 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.
TargetR=1 mR=3 mR=6 mR=9 mR=12 m132Xe4.4×1031.2×1059.5×1053.2×1067.6×10640Ar4.1×1021.1×1048.8×1043.0×1057.0×105Table 7
The number of events rate for a spherical detector of various radiiR. 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.
NevR (m)νeν̅e∑xνxTotal40Ar6162212716595676430562132Xe619725513091761966686144185945


## 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 isotopes40Ar 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 (0gs+→0gs+) 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 gas132Xe 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.



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*Source: 379460-2012-05-29.xml*