Charged-current neutrino-nucleus scattering off 95 , 97 Mo

Background: Reliable cross sections for the neutrino-nucleus scattering off relevant nuclei for supernova neutrinos are essential for various applications in neutrino physics and astrophysics (e.g., supernova mechanisms). Studies of the nuclear responses for the stable molybdenum isotopes are of great interest for the planned MOON (Mo Observatory of Neutrinos) experiment. Purpose: The purpose of the present work is, thus, to perform a detailed study of the charged-current nuclear responses to supernova neutrinos for the stable odd molybdenum isotopes. A special effort will be devoted to discuss in detail the structures of the most relevant ﬁnal states in the corresponding proton-odd nucleus. Method: The cross sections are computed by using the well-established framework for studies of semileptonic processes in nuclei developed by Donnelly and Walecka. The nuclear wave functions of the initial and the ﬁnal nuclear states are computed by using the microscopic quasiparticle-phonon model. The nuclear responses to supernova neutrinos are subsequently estimated by folding the cross sections with realistic energy proﬁles for the incoming neutrinos. Results: We present results for the cross sections of the charged-current neutrino and antineutrino scatterings off 95 Mo and 97 Mo. Nuclear responses to supernova neutrinos (both nonoscillating and oscillating ones) are also given. The inclusion of neutrino oscillations enhances signiﬁcantly the neutrino and antineutrino cross sections. Conclusions: We have found that the most important transitions are the Gamow-Teller–like ones which are mediated by the 1 + multipole. Furthermore, the three-quasiparticle degrees of freedom are essential in order to describe quantitatively the neutrino-nucleus scattering off odd open-shell nuclei.


I. INTRODUCTION
During the last decades a tremendous progress has been achieved concerning experimental and theoretical studies of the neutrino and its properties.Outstanding examples include the observations of neutrinos from the Sun and supernovae, the verification of the nonzero mass of the neutrino and of neutrino oscillations.For a review on advances in neutrino physics, see, e.g., Ref. [1].Recent experiments have also established that the third neutrino mixing angle, θ 13 , has a relatively large value of sin 2 θ 13 = 0.025 [2].
However, several open questions still remain to be answered such as the Dirac-or-Majorana nature of the neutrino, normalor-inverted neutrino-mass hierarchy and the absolute values of the neutrino masses.The studies of the neutrinoless double-beta (0νββ) decay of a selected set of even-even nuclei constitute a realistic way to determine the absolute mass spectrum of neutrinos [3,4].An observation of the 0νββ decay would also provide direct evidence that the neutrino is a Majorana particle.One of the proposed candidates for these studies is 100 Mo.Hence, the investigations of nuclear structure for this nucleus and close-by nuclei are of great interest.
From the astrophysics side, weak interactions in nuclei play a crucial role both for the supernova dynamics and the nucleosynthesis of heavy elements [5,6].The corecollapse supernovae constitute one of the proposed sites for the r-process [7].Neutrino-induced nucleosynthesis in the * e.ydrefors@gmail.com† jouni.suhonen@phys.jyu.fisupernova environment could also explain the abundances of some rare nuclei (see, e.g., Ref. [8]).It has been proposed [9] that the two nuclei 92 Nb and 98 Tc may have a neutrino-process origin.For the calculations of reaction rates of such processes reliable estimates of the neutrino-nucleus cross sections are needed.
A future detection of supernova neutrinos would be of great importance for both astrophysics and neutrino physics [10].From the astrophysics point of view predictions by current supernova models could be tested and the distance to the supernova could be determined.Moreover, neutrino oscillations in dense matter could be studied.
Neutrinos from a future galactic supernova can be detected by charged-current and/or neutral-current neutrino-nucleus scatterings off nuclei [11].Examples of current/planned possibilities for such measurements are the MOON (Mo Observatory of Neutrinos [12]), the HALO (Helium and Lead Observatory [13]), and the LAGUNA (Large Apparatus studying Grand Unification and Neutrino Astrophysics [14]).For the interpretation of the results from such measurements theoretical estimates of neutrino-nucleus cross sections are essential.
Computations of cross sections for the charged-current neutrino-nucleus scattering off medium-heavy odd nuclei are rarely found in the literature.The only published calculations are, to our knowledge, those of Ref. [15] for the neutrino reaction 127 I(ν e , e − ) 127 Xe, the ones of [16] for the charged-current scatterings off the stable even molybdenum isotopes, and the compilation of cross sections for r-process nuclei in Ref. [17].In this paper we extend this short list of calculations and compute the cross sections for the charged-current neutrino and antineutrino scatterings off the stable odd molybdenum isotopes, 95 Mo and 97 Mo, for energies of the impinging neutrino which are relevant for supernova neutrinos.Theoretical estimates of the nuclear responses are subsequently computed by folding the cross sections with realistic energy profiles of the incoming neutrinos.Recent studies (see, e.g., Ref. [18]) have shown that so-called collective neutrino oscillations could play a significant play a significant role for supernova neutrinos.The effects of such neutrino flavor conversions are taken into account by the neutrino spectra which are adopted in this work.Furthermore, we discuss in detail the nuclear structure of the most important final nuclear states.
The foundation for the present calculations is the socalled Donnelly-Walecka formalism which was introduced in the seminal paper [19].In Ref. [20] we developed an efficient method for the calculation of the required nuclear matrix elements which is based on the barycentric Lagrange interpolation [21].This method is adopted also in this work.The initial and final nuclear states are constructed by using the microscopic quasiparticle-phonon model (MQPM [22]).In the MQPM, the three-quasiparticle degrees of freedom are incorporated by using as the basic building blocks the BCS (Bardeen-Cooper-Schrieffer) quasiparticles and QRPA (quasiparticle random-phase approximation [23]) phonons.
This paper is outlined as follows.First, in Sec.II we summarize the theoretical framework for the calculations of the charged-current neutrino-nucleus scattering cross sections.We present shortly also the basic formalism of the MQPM.Then, in Sec.III we discuss our results.Finally, in Sec.IV we draw the conclusions.

A. Charged-current neutrino-nucleus scattering
In the present work the charged-current neutrino and antineutrino scatterings off a nucleus (A, Z) with mass number A and proton number Z are considered.We are interested in the neutrino reaction which proceed via the exchange of a W − boson and the antineutrino reaction where a W + boson is interchanged and l = e, μ, τ .We use in the present paper conventions which are similar to whose of Ref. [24].Hence, we define the four-momentum of the incoming neutrino as k κ = (E k , k) and that for the outgoing lepton as k κ = (E k , k ).The four momenta p κ and p κ of the initial and final nuclear states are defined correspondingly.Furthermore, the four-momentum transfer for the (anti)neutrino scattering is given by q For the low-energy neutrinos, such as, e.g., the supernova neutrinos, the transferred four-momentum is small compared to the mass of the exchanged particle, i.e., Q 2 = −q κ q κ M 2 W ± .The matrix element of the effective Hamiltonian for the neutrino reaction (1) can then be written in the form where J (+),κ (r) denotes the hadron current and the lepton matrix element l κ is defined as where j (−) κ (r) represents the lepton current.In Eq. (3) the coupling constant is G = G F cos θ C , where G F is the Fermi constant and θ C denotes the Cabibbo angle.For the antineutrino-induced reaction (2) the currents J (+),κ (r) and j (−)  κ (r) are to be replaced by their Hermitian conjugates, J (−),κ (r) and j (+) κ (r) respectively, in Eqs. ( 3) and (4).At the origin (r = 0) the one-nucleon matrix elements of the vector and axial-vector pieces of the current J κ can be written in the forms and where u(p, σ ) is the Dirac spinor and for a free particle with three-momentum p and spin projection σ , and ū(p, σ ) = u † (p, σ )γ 0 denotes the adjoint of u(p, σ ).Moreover, m N represents the nucleon mass and the nucleon form factors , and F CC P (Q 2 ) take into account the finite size of the nucleons.In the present computations we adopt the nucleon form factors of Ref. [16].The axial-vector form factor F CC A (Q 2 ) is, thus, of a momentum-dipole form with the quenched static value F CC A (0) = −1.0 for all multipoles J π .It is assumed that the final and initial nuclear states have well-defined angular momenta and parities.The doubledifferential cross section for the neutrino-nucleus scattering from an initial state i (with spin J i ) to a final state f (with spin J f ) is then given by where and In the expressions above the excitation energy E exc = E p − E p is computed with respect to the initial nuclear state and θ denotes the angle between the incoming and the outgoing leptons.In Eqs. ( 8) and ( 9) we have introduced the abbreviations and where m l denotes the mass of the outgoing lepton and the magnitude of the three-momentum transfer q is given by Furthermore, in Eq. ( 9) the minus sign is used for neutrinos and the plus sign for antineutrinos.The nuclear dependence of the cross sections is contained in the nuclear matrix elements of the multipole operators M J M (q), L J M (q), T el J M (q), and T mag J M (q).For the definitions of these operators, see, e.g., Ref. [25].The function F (±Z f , E k ) in Eq. ( 7) takes into account the distortion of the final lepton wave function by the Coulomb field of the final nucleus.The "+" sign is adopted for leptons and the "−" sign for antileptons.We adopt in this work the treatment of F introduced in Ref. [26].For more details, see, e.g., Refs.[16,27].

B. MQPM
In this work the wave functions of the initial and final nuclear states are computed by using the microscopic quasiparticle-phonon model (MQPM).Therefore, in this section we summarize the basic concepts of the MQPM.For a more comprehensive treatment we refer to [22].
The basic building blocks in the MQPM are the BCS quasiparticles and the QRPA (quasiparticle random-phase approximation) phonons.The quasiparticles are first defined via the Bogoliubov-Valatin transformation as where the index b includes the single-particle quantum numbers n b , l b and j b , and β = (b, m β ) where m β is the magnetic quantum number.In Eq. ( 14) c † β is the particle creation operator and cβ = (−1) j b +m β c −β , denotes the time-reversed particle annihilation operator where −β = (b, −m β ).Here the occupation amplitudes u b and v b are obtained by solving the BCS equations of motion (see, e.g., Ref. [28]).
The QRPA phonons are then formed by coupling twoquasiparticle configurations to good angular momentum J ω and parity π ω .Consequently, the QRPA creation operator for an excitation ω = (J ω , π ω , k ω ) is given by where σ bb = √ 1 + δ bb and the sum runs over all protonproton and neutron-neutron configurations, in the chosen valence space, so that none of them are counted twice.Here k ω enumerates the phonons with the same angular momentum J ω and parity π ω .
In the MQPM a state with angular momentum j and projection m in an odd-A nucleus is then created by using the operator † The MQPM amplitudes D k n and D k bω are solved from the MQPM equations of motion [22].While solving these equations special care has to be taken to handle the overcompleteness and nonorthogonality of the quasiparticle-phonon basis.
The MQPM states with one dominant amplitude D k n are called one-quasiparticle-like and the states with one, few or many important amplitudes D k bω are called three-quasiparticle-like ones.

III. RESULTS AND DISCUSSION
In this section we employ the formalism presented in Sec.II to compute the cross sections for the neutrino and antineutrino scatterings off 95 Mo and 97 Mo.In the present calculations we use for the initial nuclear states, i.e., the ground states of 95 Mo and 97 Mo, respectively, the nuclear wave functions computed in [20].The final nuclear states in 95 Tc, 97 Tc, 95 Nb, or 97 Nb are calculated similarly.Thus, the single-particle energies are computed from the Coulombcorrected Woods-Saxon potential with the parametrization of Ref. [29] and the Bonn one-boson-exchange potential [30] is used as the two-body interaction in both the BCS and the QRPA calculations.Furthermore, we adopt the large quasiparticle-phonon basis of [20] and the parameters of the BCS and the QRPA calculations are adjusted according to Ref. [31].
We subsequently compute the double-differential cross sections (7) for each final state, scattering angle θ and energy E k of the incoming (anti)neutrino.The total cross section, σ (E k ), as a function of the neutrino energy, is then calculated by summing up the contributions from all the final states and subsequently integrating (numerically) over the angle θ .
Table I shows the cross sections for the neutrino and antineutrino scatterings off 95 Mo and 97 Mo as functions of the energy of the impinging particle.It is seen in the table that both the neutrino and antineutrino cross sections increase significantly with the neutrino energy.The cross sections for the neutrinos are also substantially larger than the ones for antineutrinos.Furthermore, we can conclude that the neutrino cross sections increase with the neutron number and that for the antineutrinos the trend is the opposite.One important quantity from the experimental point of view is the averaged cross section, σ , which is obtained by folding the cross section with an appropriate energy profile for the incoming neutrinos.The energies of supernova neutrinos can reasonably well be described by a two-parameter Fermi-Dirac distribution where T ν represents the effective neutrino temperature and α ν is the pinching parameter.For a given value of α ν the temperature T ν can be computed from the average neutrino energy E ν by using the relation where Because of the large muon and tau rest masses only electron neutrinos (electron antineutrinos) from supernovae can be detected by charged-current neutrino-nucleus scattering.Neutrinos can, however, undergo flavor conversions due to interactions with the matter of the star.According to recent studies (see, e.g., Ref. [18]) collective neutrino oscillations caused by neutrino-neutrino interactions are also important.It is usually assumed that the neutrino-energy spectra of muon and tau neutrinos are the same.The energy profiles for electron neutrinos and electron antineutrinos which reach an Earth-bound detector can then be written in the forms and where p ( p) denotes the survival probability of electron neutrinos (electron antineutrinos).Here are the initial energy profiles of electron neutrinos (electron antineutrinos) and nonelectron neutrinos (nonelectron antineutrinos), respectively.Guided by Refs.[10,32] we use for the survival probability p(E k ) ( p(E k )) of electron neutrinos (electron antineutrinos) in the case of normal mass hierarchy the prescriptions with the split energy Ēs = 18.0 MeV [32].Similarly, for the inverted mass hierarchy we adopt the survival probabilities and for electron neutrinos and electron antineutrinos, respectively.
Here we adopt the value E s = 7 MeV [33].Equations ( 20) and ( 21) are valid for general energy profiles of the supernova   neutrinos.However, in the present calculations we adopt energy distributions of the Fermi-Dirac type (17).
In Table II we show the computed averaged cross sections for the charged-current neutrino and antineutrino scatterings off the odd stable Mo nuclei.In the table ν e (ν e ) denote nonoscillating neutrinos (antineutrinos).Moreover, ν NH ex (ν NH ex ) and ν IH ex (ν IH ex ) represent oscillating neutrinos (antineutrinos) in the cases of normal (NH) and inverted (IH) mass hierarchy, respectively.The results in the table correspond to the neutrino parameters of Table III.However, averaged cross sections for other neutrino profiles can be easily computed by using the cross sections which are tabulated in Table I.In the computations the corresponding neutrino temperatures T ν are computed from the average neutrino temperatures E ν of Table III by using Eq.(18).It is seen in Table II that the nuclear responses increase significantly when neutrino conversions are included in the calculations.Furthermore, the cross sections for the neutrino scattering are practically independent of the neutrino-mass hierarchy.Contrary to this, for the antineutrinos the cross sections are significantly larger for the normal mass hierarchy than for the inverted mass hierarchy.In Fig. 1 the contributions coming from the most important multipole channels of Eq. ( 7) to the averaged cross sections for the neutrino scattering off 95 Mo are shown.The results for the antineutrino scattering are similarly displayed in Fig. 2. In the figures we also show for each multipole J π the contributions coming from the vector and axial-vector parts of the nuclear current.We can conclude from Figs. 1 and 2 that for both the neutrino and antineutrino reactions the 1 + multipole transitions dominate the cross sections.Furthermore, the most important transitions are of the axial-vector type.Fermi-like transitions, mediated by the 0 + multipole, are also important for the neutrino-nucleus scattering.These general conclusions, i.e., that the neutrino cross sections are dominated by Gamow-Teller-like and Fermi-like transitions, are also in agreement with, e.g., the calculations performed for 56 Fe in Ref. [34].However, for the antineutrino reaction (2) the contribution from the 0 + multipole is largely suppressed and consequently 1 − and 2 − spin-dipole type of transitions become more important.
Moreover, in Fig. 3 (Fig. 4) we show the differential cross sections for the charged-current (anti)neutrino scattering leading to final states in 95 Tc ( 95 Nb).Similarly, the results for the neutrino and antineutrino scatterings off 97   displayed in Figs. 5 and 6.It is seen in the figures that the dominant states are 3/2 + , 5/2 + , and 7/2 + final states.This is in agreement with the fact that transitions mediated by the 1 + multipole are the most important ones as discussed above.However, the final nuclear states in the neutrino-induced reactions are spread over a large span of excitation energies.On the contrary, for the antineutrino scattering the most prominent final states have excitation energies around E exc ≈ 3.0 MeV.
For small energies of the incoming neutrino or antineutrino the 1 + transitions are mediated by Gamow-Teller-like operators j 0 (qr)σ τ ± .In the limit when the magnitude of the threemomentum transfer q approaches zero the aforementioned operators reduce to the usual Gamow-Teller operators σ τ ± .Consequently, in Fig. 7 are shown the strength functions for the Gamow-Teller β − type of transitions from the 5/2 + ground state of 95 Mo to 3/2 + , 5/2 + , and 7/2 + final states in 95 Tc.The results for the β + transitions to final states in 95 Nb are similarly presented in Fig. 8. Unfortunately, no experimental strength functions are available for comparison.As can be seen in the figures the total β − strength is much larger than the β + strength, which is in agreement with results for heavy even-even nuclei.It is well known that for the even-even nuclei the difference between the total Gamow-Teller β − and β + strengths satisfy the Ikeda sum rule [35], which for the reference nucleus 94 Mo would read 3(N − Z) = 30.In the present calculation for the odd-mass 95 Mo the corresponding difference in the total strengths reads 24.7 and is thus 0 0 .5 somewhat smaller than the Ikeda sum rule for the even-even reference nucleus.The difference stems from the fact that in the presently adopted formalism the phonons stem from the QRPA and not from the pnQRPA which automatically satisfies the Ikeda sum rule for the computed Gamow-Teller strengths [28].
An improvement in this respect can be achieved by taking the phonons from the pnQRPA and coupling them with the quasiparticles as was done in [36].However, the formalism of [36] lacks some important three-quasiparticle contributions that can contribute to collective excitations.An other source of missing the Ikeda sum rule in the present calculations is that for the sake of the CPU usage we have not employed the full set of QRPA phonons in the computations, but rather a truncated subset.This may reduce the β − strength at high excitation energies, beyond 15 MeV.As implied by Eq. (7) this missing strength at high energies would not influence notably the charged-current neutrino scattering due to the small momentum and kinetic energy of the final-state lepton.Furthermore, in Fig. 9 the Gamow-Teller strength functions are decomposed into the contributions coming from the possible 3/2 + , 5/2 + , and 7/2 + final states.As is seen in the figure the transitions to 5/2 + final states are the most important for the β − strength.Instead, for the β + mode the transitions to 7/2 + states are the most prominent.The differences between   the three groups of transitions are, however, quite small and the conclusions could be affected by uncertainties, e.g., in the nuclear structure.
More information about the prominent final states in Figs.3-6 can be obtained by studying the MQPM amplitudes D k n and D k bω of Eq. ( 16).Therefore, we tabulate in Table IV (Table V) the MQPM amplitudes for some of the prominent states for the charged-current (anti)neutrino scattering off 95 Mo, see also Figs. 3 and 4. The corresponding results for 97 Mo are shown in Tables VI and VII, respectively.The 5/2 + ground states of 95 Mo and 97 Mo can to a good approximation be considered as ν1d 5/2 one-quasiparticle states.As is seen in Tables IV and VI the neutrino scatterings to states in  IV has a rather large amplitude for the π 0g 7/2 one-quasiparticle configuration.But the nuclear matrix element for the l-forbidden ν1d 5/2 −→ π 0g 7/2 transition vanishes to lowest order in the three-momentum transfer q so that this component of the final state does not play a role in the transition strength.Furthermore, we can conclude from Tables V and VII that the most prominent single-particle  The reduced one-body transition densities corresponding to the β − and β + transitions between a proton-quasiparticlephonon state and a neutron-quasiparticle state are given by [22] ( and respectively.Moreover, it is well known (see, e.g., Ref. [28]) that the Gamow-Teller operators, σ τ ± , have nonzero matrix elements only between single-particle states a and b with the same n and l quantum numbers, i.e., one has the selection rules l = l a − l b = 0 and n = n a − n b = 0. Since the backward-going amplitudes Y ω f p f p are typically small we consider for the present discussion only the terms in Eqs.(26) and (27) which are proportional to Xω f n i n .Then the QRPA  15)] for the QRPA configurations given in Eqs. ( 28) and ( 29) for some of the phonons which are important for the neutrino scattering off 95 Mo, see Table IV.Moreover, in Table IX are tabulated the results for the QRPA phonons which are prominent in the antineutrino scattering (see also Table V).It is seen in the tables that most of the phonons have rather large amplitudes for one or more of the ν1d 5/2 ⊗ ν1d 5/2 , ν1d 5/2 ⊗ ν1d 3/2 , ν1d 5/2 ⊗ ν0g 9/2 , and ν1d 5/2 ⊗ ν0g 7/2 configurations.In particular, the importance of the ν1d 5/2 ⊗ ν0g 9/2 (ν1d 5/2 ⊗ ν0g 7/2 ) configuration for the neutrino (antineutrino) scattering should be noted.

IV. CONCLUSIONS
In this work we have computed the cross sections for the charged-current neutrino and antineutrino scatterings off the odd-mass A = 95, 97 molybdenum isotopes.In the calculations the initial and final nuclear states have been constructed by using the microscopic quasiparticle-phonon model.The nuclear responses to supernova neutrinos for the aforementioned nuclei have been calculated by folding the cross sections with a two-parameter Fermi-Dirac distribution.
According our calculations the nuclear responses to supernova neutrinos are not sensitive to whether the neutrino-mass hierarchy is normal or inverted.However, for antineutrinos the cross sections are significantly larger for the normal mass hierarchy than for the inverted hierarchy.

95
FIG.1.Contributions from the prominent multipole channels to the charged-current neutrino scattering off95 Mo.

95
FIG.2.Contributions from the prominent multipole channels to the charged-current antineutrino scattering off 95 Mo.

13 95
FIG.3.Differential cross sections for the charged-current neutrino scattering off95 Mo to final states in95 Tc.In the figure the states are labeled j π k where the number k enumerates states with the same angular momentum j and parity π in order of increasing excitation energy E exc .
FIG.5.Same as Fig.3but for the neutrino scattering off 97 Mo.

TABLE I .
Cross sections for the charged-current neutrino and antineutrino scatterings off95,97Mo in units of 10 −42 cm 2 as functions of the energy of the incoming neutrino.Exponents are given in parenthesis.

TABLE II .
Averaged cross sections for the charged-current neutrino and antineutrino scatterings off95Mo and 97 Mo in units of 10 −41 cm 2 computed with the neutrino parameters shown in TableIII.

TABLE III .
Adopted values of the average neutrino energy E ν and the pinching parameter α ν for the various neutrino flavors.The corresponding neutrino temperatures T ν are also given.

TABLE IV .
MQPM amplitudes X f for some important states in95Tc.

TABLE VI .
MQPM amplitudes X f for some important states in 97 Tc.

TABLE IX .
QRPA forward amplitudes for some states in94Mo which are important for the charged-current antineutrino scattering off95Mo.

TABLE VIII .
QRPA forward-going amplitudes for some states in94Mo which are important for the charged-current neutrino scattering off 95 Mo.