Advances in High Energy Physics

Volume 2012 (2012), Article ID 379460, 18 pages

doi:10.1155/2012/379460

Research Article

Coherent and Incoherent Neutral Current Scattering for Supernova Detection

P. C. Divari

Department of Physical Sciences and Applications, Hellenic Army Academy, Vari 16673, Attica, Greece

Received 28 December 2011; Revised 15 March 2012; Accepted 30 March 2012

Academic Editor: Guey-Lin Lin

Copyright © 2012 P. C. Divari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ( 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 () channel which is dominant with respect to incoherent () 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 (antineutrino ) with transforms one neutron (proton) of a nucleus to a proton (neutron), and a charged lepton (anti-lepton ) is emitted as

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 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 bosons (see Figure 1(b)) with a nucleus as

where () 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 , and (b) the NC neutrino-nucleus processes . 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 (, , , , , ). 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  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], is emitted with temperature MeV, has MeV, and all other flavors () have temperature MeV. The temperatures differ from each other because and 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 ) 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 and , which interact through charged current as well as neutral current. Since the number of neutrons is larger than the protons, loses energy much more than and the average energy for is more than .

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 and 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 and up to 40 MeV, in contrast to previous RPA calculations [42] concerning , which seems to take only a few excited states known by experiment. Moreover, we have investigated the individual contributions coming from the coherent () and incoherent () 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 MeV [41]. The FD energy distribution is given by

where is the normalization constant depending on the parameter given by the relation

for . 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 is given by

Some characteristic values of are listed in Table 1. Figure 3(a) shows the averaged neutrino energy as a function of the parameter for various temperatures . As it is seen the introduction of a chemical potential, , 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 and .

(MeV)

 MeV
 MeV
 MeV

0
11.03
15.76
25.21

0.75
11.45
16.36
26.17

1.50
12.08
17.26
27.61

2.00
12.62
18.03
28.84

3.00
13.97
19.96
31.94

4.00
15.63
22.33
35.73

5.00
17.52
25.02
40.04

Figure 2: (Color on line). The normalized to unity Fermi-Dirac spectrum for .

(a)

(b)

Figure 3: (Color on line). Averaged neutrino energy as a function of the parameter for various temperatures (a). Averaged neutrino energy as a function of temperature for various parameters of chemical potential (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 () 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 and , which interact through charged current as well as neutral current. Since the number of neutrons is larger than the protons, loses energy much more than and the average energy for is more than . The numerical simulations give the following values of average energy for the different neutrino flavors:

Those average neutrino energies imply that for the values of temperature are 3.5 MeV (2.75 MeV) for , 5 MeV(4 MeV) for , and 8 MeV (6 MeV) for .

The number of emitted neutrinos is

where  erg per neutrino flavor. Taking the temperature to be 3.5, 5, and MeV for electron neutrinos (), electron antineutrinos (), and all other flavors () respectively, and the parameter to be , then the obtained results for the number of primary neutrinos emitted are shown in Table 2, while the (time averaged) neutrino flux at a distance  Kpc =  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 and in units of .

 MeV
 MeV
 MeV

0
0.28
0.20
0.50

0.75
0.27
0.19
0.48

1.50
0.26
0.18
0.46

2.00
0.25
0.17
0.44

3.00
0.22
0.16
0.39

4.00
0.20
0.14
0.35

5.00
0.18
0.12
0.32

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

 MeV
 MeV
 MeV

0
0.23
0.16
0.41

0.75
0.23
0.16
0.39

1.50
0.21
0.15
0.37

2.00
0.20
0.14
0.36

3.00
0.18
0.13
0.32

4.00
0.17
0.12
0.29

5.00
0.15
0.10
0.26

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 . The initial nucleus is assumed to be spherically symmetric having ground state a state.

The corresponding standard model effective Hamiltonian in current-current interaction form is written as

where is the Fermi weak coupling constant. and denote the leptonic and hadronic currents, respectively. According to V-A theory, the leptonic current takes the form

where 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

( stands for the nucleon mass and denote the nucleon spinors). , , represent the weak nucleon form factors given in terms of the well-known charge and electromagnetic form factors (CVC theory) for proton () and neutron () by the expressions [44]

Here represents the nucleon isospin operator and is the Weinberg angle (. In (3.3) stands for the axial-vector form factor for which we employ the dipole ansatz given by

where GeV is the dipole mass and is the static value (at ) of the axial form factor.

In the convention we used in the present work , the square of the momentum transfer, is written as

where is the excitation energy of the nucleus. denotes the energy of the incoming and that of the outgoing neutrino. , 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

The summations in (3.7) contain the contributions , for the Coulomb and longitudinal , and , for the transverse electric and magnetic multipole operators defined as in [47]. These operators include both polar-vector and axial-vector weak interaction components. The contributions and are written as

where denotes the outgoing neutrino scattering angle and . 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) which runs over all values of quantum numbers and the second term which includes the two-body interaction , that is

where the two-body term contains the antisymmetrised two-body interaction matrix element defined by . The operators and stand for the usual creation and destruction operators of nucleons in the state .

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

where , and . The occupation amplitudes and 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 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 and the second one includes the different components of the residual interaction.

In the present calculations we use a renormalization parameter which can be adjusted when doing the BCS calculations. The monopole matrix elements of the two-body interaction are multiplied by a factor . The adjustment can be done by comparing the resulting lowest quasi-particle energy to the phenomenological energy gap 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 and annihilation operators are defined as

where and are either proton () or neutron () indices, labels the magnetic substates, and numbers the states for particular angular momentum and parity .

The and forward and backward going amplitudes are determined from the QRPA matrix equation

where denotes the excitation energies of the nuclear state . The QRPA matrices and , are deduced by the matrix elements of the double commutators of and with the nuclear hamiltonian defined as

where . Finally the two-body matrix elements of each multipolarity , occurring in the QRPA matrices and , are multiplied by two phenomenological scaling constants, namely, the particle-hole strength and the particle-particle strength . 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 Ar and 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 ground state. In the case of Xe we have adopted Ca as inert core and the two oscillator and major shells, plus the intruder orbital from the next higher oscillator major shell, as valence space for protons and neutrons. In the case of Ar we have considered the major shells 0,1,2, and 3 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 and 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]

in which stands for the doubly-even nucleus under consideration. The separation energies are provided by [53]. The results of this procedure lead to the pairing-strength parameters = 1.05 and = 0.88 for Ar and = 1.08 and = 0.89 for Xe. After settling the values of the pairing parameters, two other parameters are left to fix, the overall scale of the particle-hole interaction and separately the particle-particle channel of the interaction for each multipole up to . 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 and , 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 (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 was obtained by integrating over the scattering angles and and subsequently summing over all discrete final sates. The results were obtained for coherent cross sections (elastic channel) as well as for incoherent cross sections (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 is given by [55]

where denotes the scattering angle of the incident neutrino in the lab frame of the recoil nucleus, is the Fermi constant, and is the weak charge of the nucleus with neutrons and protons:

with being the weak mixing angle (). 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 we obtain the CNNS cross section as a function of the neutrino energy

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 and , 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 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 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 , in the NC reactions (a) and (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 for Ar and for Xe are the most pronounced channels. On the other hand, in high-energy region, the incoherent scattering for Ar is dominated mostly by the transition while other transitions like and start to contribute significantly. In the case of Xe the channels , , , and are dominant.

(a)

(b)

(c)

(d)

Figure 5: (Color on line). Partial multipole distributions to the incoherent cross sections for Ar and Xe, at the incoming neutrino energies and 100 MeV.

In order to obtain more information about supernova neutrinos, the total cross section 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 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 .

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

(3.5,0)
(5,0)
(8,0)
(2.75,3)
(4,3)
(6,3)

 cm²

Ar
0.33
0.63
1.40
0.30
0.61
1.26

Xe
3.87
7.17
14.54
3.63
7.09
13.53

 cm²)

Ar
0.08
0.28
1.56
0.06
0.21
0.95

Xe
6.37
36.23
239.40
3.19
23.10
143.10

 cm²)

Ar
0.32
0.63
1.40
0.30
0.61
1.26

Xe
3.87
7.20
14.78
3.64
7.11
13.68

Table 5: Coherent neutrino nucleus cross sections calculated by (5.4) and by the QRPA approach (numbers in parenthesis) for Ar and Xe targets. The supernova neutrino spectra described by a two parameter Fermi-Dirac distribution with and 3 for various temperatures (MeV).

 cm²

(3.5,0)
(5,0)
(8,0)
(2.75,3)
(4,3)
(6,3)

Ar
0.28 (0.33)
0.56 (0.63)
1.35 (1.40)
0.26 (0.30)
0.53 (0.61)
1.16 (1.26)

Xe
3.50 (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 volume under pressure and temperature , 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 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 of its geometric center from the source. In the case of sphere of radius with its center at a distance from the source, the function depends only on the ratio and it is given by

Spherical coordinates 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 . A plot of the function is presented in Figure 6. The geometric factor is close to unity in the actual experimental setup where .

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

For a typical distance  mwe can take as neutrino flux for each neutrino flavor the value . Summing over all the neutrino flavors we find the total cross sections for and  cm² for . Finally the total number of events 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  Kpc =  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 and 9 m. As it seen, employing Xe as a target nucleus one expects about 1761 events for a sphere of radius 6 m, while for Ar one expects about 562 events but with a vessel of larger radius ( m).

Table 6: Number of events in a year for a spherical detector of various radii with its center at a distance m from the source. The vessel is filled with gas under pressure  Atm and temperature K. The total cross sections are obtained summing over all neutrino flavors. The chemical potential of the neutrino spectra is taken as .

Target
 m
 m
 m
 m
 m

Xe

Ar

Table 7: The number of events rate for a spherical detector of various radii . The neutrinos are emitted from a supernova at a distance  Kpc. The spherical vessel of volume is filled with gas under pressure  Atm and temperature K. The chemical potential of the neutrino spectra is taken as .

(m)

Total

6
16
22
127
165

9
56
76
430
562

6
197
255
1309
1761

9
666
861
4418
5945

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 Ar and 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 () 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 Xe under pressure  Atm and temperature K one expects about 1761 events for a sphere of radius 6 m. Employing Ar one expects 562 events but with a vessel of larger radius ( 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.

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