Coherent and Incoherent Neutral Current Scattering for Supernova Detection

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 Ev < 100MeV . 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 → 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.


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 supernovanucleosynthesis 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 Advances in High Energy Physics 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 as 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 Z 0 bosons see Figure 1 b with a nucleus as 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 particlebound states usually decay by particle emission and, thus, they supply light particles that can cause further nuclear reactions.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 × 10 53 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 3.5 MeV, ν e has T 5 MeV, and all other flavors ν x have temperature T 8 MeV.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 40 Ar ν, ν 40 Ar * and 132 Xe ν, ν 132 Xe * 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 40 Ar and 132 Xe up to 40 MeV, in contrast to previous RPA calculations 42 concerning 40 Ar, 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.

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 3-8 MeV 41 .The FD energy distribution is given by where N 2 α is the normalization constant depending on the parameter α given by the relation 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 by 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.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:  The number of emitted neutrinos is where U ν 0.5 × 10 53 erg per neutrino flavor.Taking the temperature T to be 3.5, 5, and 8 MeV 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 /4πL 2 at a distance L 10 Kpc 3.1 × 10 22 cm is given in Table 3.
where G 1.1664 × 10 −5 GeV −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 form where ψ ν 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 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 n 1,2 .

3.4
Here τ 0 represents the nucleon isospin operator and θ W is the Weinberg angle sin 2 θ W 0.2325 .In 3.3 F A stands for the axial-vector form factor for which we employ the dipole ansatz given by 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 q 2 , the square of the momentum transfer, is written as 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 as 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 as where θ denotes the outgoing neutrino scattering angle and b ε i ε f /q 2 .In 3.9 the − sign corresponds to neutrino scattering and the sign to antineutrino one.

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 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 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 twobody 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 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 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 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 form where the quasi-particle pair creation A † αα ; JM and annihilation A αα ; JM operators are defined as 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 equation .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 particleparticle 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.

Calculated Cross Sections
In order to investigate neutrino scattering off the 40 Ar and 132 Xe 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 132 Xe we have adopted 40 Ca as inert core and the two oscillator 3 ω and 4 ω 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 40 Ar we have considered the major shells 0,1,2, and 3 ω 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-bosonexchange 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

5.1
in which A Z X 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 40 Ar and g p pair 1.08 and g n pair 0.89 for 132 Xe.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 d 2 σ 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 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: with Θ W being the weak mixing angle sin 2 Θ W ≈ 0.231 .F q 2 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 ν σ coh E ν dσ dΩ coh E ν dΩ.

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 132 Xe ν, ν 132 Xe and 40 Ar ν, ν 40 Ar, 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 ≤ 100 MeV.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. 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 40 Ar and 1 − for 132 Xe are the most pronounced channels.On the other hand, in high-energy region, the incoherent scattering for 40 Ar is dominated mostly by the 1 − transition while other transitions like 3 − and 2 start to contribute significantly.In the case of 132 Xe the channels 1 − , 3 − , 2 , and 4 are dominant.
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 .

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 T 0 , filled with noble gas such as Ar and Xe.In this case, the number of expected events in a year takes the form 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
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 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.
For a typical distance L 50 m we can take as neutrino flux for each neutrino flavor the value Φ ν, L 50 m 1.95 × 10 6 cm −2 s −1 .Summing over all the neutrino flavors we find the total cross sections σ tot 6.5 × 10 −39 cm 2 for A 40 and σ tot 7.0 × 10 −38 cm 2 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 at a distance L 10 Kpc 3.1 × 10 22 cm, the number of the observed events for each neutrino flavor is found to be In Table 7 the numbers of event rates are listed for two given radii R 6 and 9 m.As it seen, employing 132 Xe as a target nucleus one expects about 1761 events for a sphere of radius 6 m, while for 40 Ar 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 50 m from the source.The vessel is filled with gas under pressure P 10 Atm and temperature T 0 300 K.The total cross sections are obtained summing over all neutrino flavors.The chemical potential of the neutrino spectra is taken as α 0.

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 40 Ar and 132 Xe.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 132 Xe under pressure P 10 Atm and temperature T 0 300 K one expects about 1761 events for a sphere of radius 6 m.Employing 40 Ar 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.

Figure 1 :
Figure 1: Feynman-diagram of lowest order for: a the CC neutrino-nucleus reactions ν l 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.

Figure 3 :
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 .

Figure 4 :
Figure 4: Color on line .Coherent, incoherent, and total cross section as a function of the incoming neutrino energy E ν , in the NC reactions 132 Xe ν, ν 132 Xe a and 40 Ar ν, ν 40 Ar b .

Figure 6 :
Figure 6: Geometrical factor s R/L for a sphere of radius R whose center is at distance L from the source.

Table 1 :
The average supernova neutrino energies as a function of the parameters α and T .

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 10 58 .

Table 3 :
The time integrated neutrino flux, in units of 10 12 cm −2 , at a distance 10 kpc from the source.

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 as 4.7where 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 J αα ; ββ BCS A αα ; JM , H, A † ββ ; JM BCS , B J αα ; ββ − BCS A αα ; JM , H, A ββ ; JM BCS ,

Table 4 :
Coherent, incoherent, and total coherent plus incoherent neutrino nucleus cross section for40Ar and 132 Xe 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 for40Ar and 132 Xe targets.The supernova neutrino spectra described by a two parameter Fermi-Dirac distribution with α 0 and 3 for various temperatures T MeV .

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 T 0 300 K.The chemical potential of the neutrino spectra is taken as α 0.