RPA puzzle in C(12) weak decay processes

We explain the origin of the difficulties that appear in a straightforward application of the QRPA in C(12), and we demonstrate that it is imperative to use the projected QRPA (PQRPA). Satisfactory results, not only for the weak processes among the ground states of the triad B(12),C(12),N(12), but also for the inclusive ones are obtained. We sketch as well a new formalism for the neutrino-nucleus interaction that furnishes very simple final formulae for the muon capture rate and neutrino induced cross sections.

New types of nuclear weak processes have been measured in recent years.They are based on neutrino and antineutrino interactions with complex nuclei and, rather than being used to study the corresponding cross sections, they are mainly aimed to inquire on possible exotic properties of neutrino themselves, such as neutrino oscillations and the associate neutrino massiveness, which are not contained in the Standard Model (SM) of elementary particles.
So, in neutrino oscillation experiments with liquid scintillators, the charge-exchange reactions 12 C(ν e , e − ) 12 N and 12 C(ν µ , µ − ) 12 N, both exclusive (to the 1 + ground state) and inclusive (to all final states), are just tools.As such, and to be useful, the corresponding cross sections σ exc e,µ and σ inc e,µ must be accurately accounted for by nuclear structure models.¿From the recent works [1,2,3,4] we have learned, however, that neither RPA nor QRPA are able to explain the weak processes (β-decays, µ-capture, and neutrino induced reactions) among the ground states of the triad { 12 B, 12 C, 12 N }.In fact, in the RPA a rescaling factor of the order of 4 is needed to bring the calculations and data in agreement [1], and a subsequent ad hoc inclusion of partial occupancy of the p 1/2 subshell reduces this factor to less than 2 [2,3].But, when the RPA is supplemented with the pairing correlations in a self-consistent way, i.e., in the framework of a full QRPA [4], the exclusive cross sections turn out to be again overestimated by a factor of ∼ = 4.Moreover, Volpe et al [4] have called attention to "difficulties in choosing the ground state of 12 N because the lowest state is not the most collective one" when the QRPA is used.
In the present paper we explain the origin of the difficulties that appear in a straightforward application of the BCS approximation in a light nucleus such as 12 C, and we demonstrate that the problem is circumvented by the employment of the particle number projected BCS (PBCS).We show simultaneously that the proton-neutron QRPA is not a recommended approach, and that the aforementioned RPA puzzle is solved within the projected QRPA (PQRPA) for the charge-exchange excitations [5].The later approach furnishes satisfactory results not only for the weak processes among the ground states of the triad { 12 B, 12 C, 12 N }, but also for the inclusive weak processes.For numerical evaluation of the weak decay observables we have found it suitable to develop a new theoretical framework, which is similar to that build up by Barbero et al. [6] for the neutrinoless double beta decay.The motivation for that and the complete formulation will be exposed elsewhere.Here we just explain the notation and exhibit the final formulae.
The weak Hamiltonian is expressed in the form [6,7,8] where is the hadronic current operator, and is the plane waves approximation for the matrix element of the leptonic current; G = (3.04545±0.00006)×10 −12 is the Fermi coupling constant (in natural units) [9], is the momentum transfer (P i and P f are momenta of the initial and final nucleon (nucleus)), M is the nucleon mass, m ℓ is the mass of the charged lepton, and g V , g A , g M and g P are, respectively, the vector, axial-vector, weak-magnetism and pseudoscalar effective dimensionless coupling constants.Their numerical values are [7,8,9]: The above estimates for g M and g P come from the (well tested) conserved vector current (CVC) hypothesis, and from the partially conserved axial vector current (PCAC) hypothesis, respectively.In the numerical calculation we will use an effective axial-vector coupling g A = 1 [10,11,12].The finite nuclear size (FNS) effect is incorporated via the dipole form factor with a cutoff Λ = 850 MeV, i.e., as [13]: To use (1) with the non-relativistic nuclear wave functions, the Foldy-Wouthuysen transformation has to be performed on the hadronic current (2) .When the velocity dependent terms are neglected1 , this yields [16]: where the following short notation has been introduced: For the neutrino-nucleus reaction k = p − q, with p ≡ {E ℓ , p} and q ≡ {E ν , q}, and the corresponding cross section from the initial state |J i to the final state |J f reads where F (Z + 1, E ℓ ) is the Fermi function, θ ≡ q • p, and with ψ i (r) ≡ r|J i M i and ψ f (r) ≡ r|J f M f being the nuclear wave functions.The transition amplitude can be cast in the form: where are the nuclear matrix elements, and are the effective coupling constants, which contain the lepton traces with and the momentum transfer k is along the z axis ( k ≡ ẑ ≡ ǫ 0 ).In going from the results for the neutrino-nucleus reaction cross section to that for the muon capture rate one should keep in mind that: i) the roles of p and q are interchanged within the matrix element of the leptonic current, which brings in a minus sign in the last term of L ±1 , ii) the momentum transfer turns out to be k = q − p, and therefore the signs on the right hand sides of (15) have to be changed, and iii) the threshold values (p → 0 : k → q, k 0 → E ν − m ℓ ) must be used for the lepton traces.All this yields: Finally, one should remember that instead of summing over the initial lepton spins s ℓ , as done in (10), one has now to average on the same quantum number.The resulting transition amplitude T Λ (J f ) is again of the form (11) but the effective charges are here: with For the capture rate one gets [17] Λ where φ 1S is the muonic bound state wave function evaluated at the origin.Note that the neutrino energy is fixed by the energy of the final state through the relation: is the binding energy of the muon in the 1S orbit.Lastly, we mention that the B-values for the GT beta transitions are defined and related to the ft-values as [9]: To start the discussion on the difficulties found by Volpe et al. [4], it should be remembered that, the pn-QRPA yields the same energy spectra for the four (Z ±1, N ∓1) and (Z ±1, N ±1) nuclei, when the BCS equations are solved in the parent (Z, N) nucleus under the constraint This is a physically sound zero order approximation when the nuclei in question are far from the closed shells and possess a significant neutron excess.Yet, as we show below, the use of the QRPA in N = Z nuclei is not free from care.
Let us define the quasiparticle energies relative to the Fermi levels: where E j k and λ k are the BCS quasiparticle energies and chemical potentials, respectively.In the particle-hole limit the energies ) correspond to the single-particle (-hole) excitations for the levels above (below) the Fermi surface [18], and to the 2p1h (1p2h) seniorityone excitations for levels below (above) the Fermi surface.In nuclei with large neutron excess jn are in general quite different, but in N = Z nuclei the proton and neutron spectra are almost equal, except for the Coulomb energy displacement.As a consequence the unperturbed QRPA energies: are almost degenerate with E j ′ p jn i.e., E jpj ′ n ∼ = E j ′ p jn , for all four odd-odd (Z ± 1, N ∓ 1) and (Z ± 1, N ± 1) nuclei.Moreover, in the case of 12 C, both the proton and the neutron Fermi energies are placed almost in the middle between the 1p 3/2 and 1p 1/2 shells.This causes an additional degeneracy, namely for 12 N, 12 B, 14 N and 10 B. But we know that the physically sound energy sequences are: as can be easily seen from the scrutiny of the particle-hole limits for the seniority-two pn-states, which are indicated parenthetically in (25).The RPA correlations are unable to remedy the situation and the degeneracy (24) among four lowest E pn is the cause for the problems found in Ref. [4] regarding the ground state of 12 N.
Well aware of all these difficulties, Cha [19], in his study of the Gamow-Teller (GT) resonances, has solved the BCS equations in the daughter nuclei under the constraints k=n(p) which gives way to the energy orderings (25).Thus, the problem risen by Volpe et al [4] can, in principle, be solved by using the Cha's recipe.However, the price to pay is that a different QRPA equation has to be worked out for each nucleus separately, i.e., one for the (Z + 1, N − 1) nucleus and the other for the (Z − 1, N + 1) nucleus, being in each case significant only the positive energy frequencies.This means that we have to abandon the nice properties of the particle-hole charge-exchange RPA, where: (1) only one RPA equation is solved for the (Z ± 1, N ∓ 1) nuclei, and (2) both the positive and negative solutions are physically meaningful, with the β + spectrum viewed as an extension of the β − spectrum to negative energies [20,21,22].Note also that, in order to fulfill the GT sum rule, Cha has evaluated the transition probabilities with the usual pairing factors u's and v's, obtained from (21).None of the undesirable features of the Cha's method appear within the charge-exchange PQRPA.This approach has been presented in detail in Ref. [5], and we just mention here that the PBCS quasiparticle energies read: where K = N, Z and the quantities R K and I K are defined in [5].
Table 1: BCS and PBCS results for neutrons.E exp j stand for the experimental energies used in the fitting procedure, and e j are the resulting s.p.e.The underlined quasiparticle energies correspond to single-hole excitations (for j = 1s 1/2 , 1p 3/2 ) and to single-particle excitations (for j = 1p 1/2 , 1d 5/2 , 2s 1/2 , 1d 3/2 , 1f 7/2 , 2p 3/2 ).The non-underlined energies are mostly two hole-one particle and two particle-one hole excitations.The fitted values of the pairing strengths v pair s in units of MeV-fm 3 are also displayed.The numerical calculations were performed within the nl = (1s, 1p, 1d, 2s, 1f, 2p) configuration space, and for the residual interaction we adopted the delta force
The BCS and PBCS results for neutrons are displayed in Table 1.The underlined quasiparticle energies correspond to single-hole and single-particle excitations, while the non-underlined ones are basically 2h1p and 2p1h excitations.Note that, while the first ones are fairly similar within the BCS and PBCS approaches, the last ones are quite different.(The resulting s.p.e. are also quite similar.)Analogous results are obtained for protons, with the same value of v pair s .The unperturbed energies E jpj ′ n of lowest four pn quasiparticle states within the BCS and PBCS approximations are shown in Table 2.For comparison, the results obtained with the Cha's method are also displayed in the same table.It is easy to see that, while the standard BCS approximation exhibits the degeneracy (24), the Cha's recipe and the PBCS approach produce the energy sequences (25), being the energy separations between the 1p1h, 2p2h and 3p3h-like states significantly larger in the later case.This does not take us by surprise since the role of the number projection is precisely that of restoring the correct number of particles and holes.
Table 2: Unperturbed energies E jpj ′ n (in units of MeV) of lowest four proton-neutron quasiparticle states in the neighborhood of 12 C, within the approximations: (a) BCS equations are solved in 12 C with the condition ( 21), (b) BCS equations are solved in daughter nuclei, employing (24) as suggested by Cha [19], and (c) number projected BCS (PBCS) equations are employed.The underlined energies are equal for all three cases, because they are adjusted to the experimental data via the IGP procedure [23].After having established truthful unperturbed PQRPA energies we proceed with full calculations, where the values of v s and v t within the particle-particle (pp) and particle-hole (ph) channels are needed.In similar calculations of double beta decaying nuclei [6,24], which possess significant neutron excess, the following procedure has been pursued: (i) v ph s and v ph t were taken from the study of energetics of the GT resonances done by Nakayama et al. [25] (see also ref. [11]), and (ii) the pp strengths were fixed on the basis of the isospin and SU(4) symmetries as: v pp s ≡ v pair s , and Different to what happens in the N > Z nuclei, such a parametrization is not suitable for the N = Z nuclei, and the best agreement with data is obtained when the pp channel is totally switched off.Thus we will exhibit here only the results for v pp s = v pp t = 0, and the next three sets of ph parameters: .92 MeV-fm 3 , and v ph t = v ph s /0.6 = 39.86 MeV-fm 3 .That is, the singlet ph strength is taken to be the same as v pair s obtained from the gap equation, while the triplet ph depth is estimated from the relation used by Goswami and Pal [26] in the RPA calculation of 12 C.
Calculation III: v ph s = v ph t = 45 MeV-fm 3 .This parametrization gives fairly good results for the energies of the J π = 0 + 1 and 1 + 1 states in 12 B and 12 N.
Table 5: Results for the energies (in units of MeV) and the partial muon capture rates (in units of 10 3 s −1 ) the bound excited states in 12 B.In the upper part of the table are shown the previous theoretical calculations based on the RPA [1,41] (where only the results for Λ are reported) and the on shell model [42].We wish to restate the ingredients that play a part in the agreement between the data and calculations for the ground state processes within the triad { 12 B, 12 C, 12 N }.They are: (a) the pairing short range correlations, which are added to improve the description of the 12 C ground state, (b) the RPA-type correlations, which are repulsive in the particle-hole channel, and (c) the effective axial-vector coupling constant g A = 1, which in principle simulates the removal of the spin strength due to the coupling to the ∆ resonance [10,11,12,42].For instance, these effects reduce the bare single-particle value B(GT ) = (16/9)g 2 A by factors 1.7, 1.8 − 2.2 and 1.6, respectively.It is worthy of note that the PBCS by itself reproduces better the data than the majority of previous RPA and QRPA calculations [1,2,3,4].We have considered all orbitals from 1s 1/2 up to 1f 5/2 , but the valence p-shell correlations (both pairing and RPA like) are definitely the most important ones for the quenching of the 1 + 1 ↔ 0 + 1 transition rates.Yet, as discussed by Vogel et al. [3,45], the effect of these correlations on the dipole and quadrupole operators is very tiny.
In addition to the total µ capture rates in Table 3, we show the individual rates to the individual bound states of 12 B in Table 5.They represent another test for our calculation and have been derived from the intensities of the observed de-excitation γ rays following the µ capture [43,44].The agreement between the experiment and the present PQRPA estimate for the energies of the J π f = 2 + 1 , 2 − 1 and 1 − 1 states is only moderate, but that for the capture rates is as good or even better than in a recent shell model (SM) study [42].
In summary, we have shown that to account for the weak decay observables around 12 C in the framework of the RPA, besides including the BCS correlations, it is imperative to perform the particle number projection, and this is the way out of the RPA puzzle in 12 C. More, as far as we are acquainted with, such an important effect of the projected linear response theory for charge-exchange excitations has never before been observed, indicating that it could be more relevant in light N = Z nuclei than in heavy nuclei with large neutron excess [5].Thus, it could be interesting enough to investigate the consequences of the PQRPA in other light N = Z and N ∼ = Z nuclei.On the other hand, the fact that we have been forced to switch off completely the residual interaction in the particle-particle channel could indicate that some relevant piece of physics is still lacking in our approach.In this sense it would be very illuminating to redo the PQRPA calculations with more realistic forces than the one used here.