Time-reversed particle-vibration loops and nuclear Gamow-Teller response
The nuclear response theory for charge-exchange modes in the relativistic particle-vibration coupling approach is extended to include for the first time particle-vibration coupling effects in the ground state of the parent nucleus. In a parameter-free framework based on the effective meson-nucleon Lagrangian, we investigate the role of such complex ground-state correlations in the description of Gamow-Teller transitions in Zr in both (p,n) and (n,p) channels. We find that this new correlation mechanism is fully responsible for the appearance of the strength in the (n,p) branch. Comparison of our results to the available experimental data shows a very good agreement up to excitation energies beyond the giant resonance region when taking into account an admixture of the isovector spin monopole transitions to the data. The parent-daughter binding-energy differences are also greatly improved by the inclusion of the new correlations.
Charge-exchange transitions in nuclei, in particular Gamow-Teller (GT) modes, are important ingredients for the modeling of astrophysical environments Langanke and Martínez-Pinedo (2003). Transitions in the GT, or (p,n), branch, converting a neutron into a proton, determine in first approximation -decay which governs the rapid neutron-capture (”r-”) process. Alternatively, transitions in the GT, or (n,p), branch can occur in core-collapse supernovae via
electron capture on pf- or sdg-shell nuclei, in particular, those with a neutron number N greater than their proton number Z Cole et al. (2012).
In a simple independent-particle picture of such nuclei the transitions in the GT channel allowed by the selection rules are, however, typically blocked by the Pauli principle due to the neutron excess. The presence of correlations in the ground state of the parent nucleus, which can smear the occupancies of the single-particle shells, then constitute the only mechanism that can unlock transitions in this branch.
In methods based on the mean-field approximation doubly-magic nuclei constitute a great test case to study these effects. Indeed, in such systems no superfluid pairing correlations are present in the mean field, so that one can fully quantify
the effect of ground-state correlations (GSC) introduced beyond the mean field while avoiding possible interplays with pairing.
To study nuclear charge-exchange modes in nuclei in large single-particle spaces one can apply the linear response theory at different levels of approximations Ring and Schuck (2004). The proton-neutron Tamm-Dancoff Approximation (pn-TDA) describes such transitions as superpositions of interacting one-particle-one-hole (1p-1h) proton-neutron excitations of the mean-field ground state. The proton-neutron Random-Phase Approximation (pn-RPA) goes one step further and introduces additional one-hole-one-particle (1h-1p) transitions thus generating some correlations in the parent ground state. In the following we will refer to this type of GSC as GSC. While (pn-)RPA can generally reproduce the position of giant resonances (GRs) to a good accuracy, it is well known that this approximation typically leads to a rather poorly detailed description of the transition strength distributions and, in particular, is not able to describe the spreading width of the GRs. In order to correct for these deficiencies one should account for higher-order configurations of the nucleons in the description of nuclei. For instance, by introducing the coupling between single nucleons and collective nuclear vibrations, the particle-vibration coupling (PVC) scheme includes, in the leading approximation, configurations of the 1p-1h phonon type.
In the charge-exchange channel the PVC was recently implemented in consistent non-relativistic Niu et al. (2012, 2016, 2018) and relativistic Marketin et al. (2012); Litvinova et al. (2014); Robin and Litvinova (2016, 2018) frameworks. These studies, however, did not include the GSC induced by the PVC effects, which we will denote as GSC to differentiate them from GSC. In other words, configurations beyond pn-RPA were only introduced in the description of states of the daughter nucleus, but not in the ground state of the parent nucleus. It has been shown in Refs. Kamerdzhiev et al. (1994, 1997), that such GSC can in fact be consistently included. Their effect on neutral electric (E0, E2) and magnetic (M1) transitions were studied in a non-relativistic framework based on Landau-Migdal forces and they were found to be important for the description of the low-energy strength distributions.
In this Letter, we implement for the first time GSC in the description of charge-exchange transitions of non-superfluid nuclei in a relativistic framework. We apply the extended formalism to the description of GT and isovector spin-monopole modes in Zr. This nucleus is a particularly interesting case as it has been experimentally measured in both GT and GT channels up to energies beyond the giant resonance region Yako et al. (2005); Wakasa et al. (1997). Also, due to the weak pairing caused by the closed shell and sub-shell at and respectively, Zr can be considered in first approximation as a doubly-magic nucleus in which the effect of configuration mixing and ground-state correlations can be investigated carefully. We will find that GSC can induce new types of transitions from particle to particle states or from hole to hole states, that are crucial for the appearance of the strength in the GT branch.
Formalism. — We are interested in describing the response of a nucleus to a one-body charge-changing external field, i.e. converting a neutron (n) into a proton (p) () or vice versa (). The corresponding transition strength distribution is fully determined by the response function which describes the propagation of correlated proton-neutron pairs in the particle-hole channel:
where denotes the parent nuclear ground state, the states of the daughter nucleus, and the corresponding transition energies. In the following odd (resp. even) indices denote proton (resp. neutron) spherical single-particle states for the channel and vice versa for the channel. Letters will be used to denote single-particle states with unspecified isospin projection. We introduce a single-particle index where , if is a state above the Fermi level (”particle” state) and , if is a state below the Fermi level (”hole” state). The product will then be , if is a particle-hole or hole-particle pair, and will be , if is a particle-particle or hole-hole pair. The response function now has, in principle, four components , , and that denote
That is , ,
And similarly the one-body external field has two components and .
Since in the proton-neutron (relativistic) RPA (pn-(R)RPA) nuclear excitations are superpositions of 1p-1h or 1h-1p proton-neutron transitions on top of the parent ground state, , and cancel, so that and are the only components that remain. In order to account for higher-order configurations and have a more precise description of the strength, we go beyond the pn-RRPA by introducing the coupling between single nucleons and collective vibrations of the nucleus. Up to now we have done this in the resonant Time Blocking Approximation (TBA) Tselyaev (1989) by introducing PVC into the component via the kernel of the following Bethe-Salpeter equation Robin and Litvinova (2016, 2018):
In Eq. (LABEL:eq:BSE_pn) denotes the free particle-hole proton-neutron propagator calculated in the mean-field approximation, is the static meson-exchange interaction in the isovector channel (-meson, pion and Landau-Migdal term) Robin and Litvinova (2016) and the energy-dependent amplitude contains PVC effects induced by the ”forward-going” diagrams shown in Fig. 1.
Such diagrams introduce complex configurations of the 1p-1h phonon type in the state of the daughter nucleus, but do not modify the description of the parent ground state compared to pn-RRPA. The PVC can, however, generate ”backward-going” diagrams, as those shown in Fig. 2, which introduce complex configurations in the parent ground state. Such diagrams will modify further the function and also introduce non-zero , and components of the response.
The full derivation and expression of these additional terms for neutral (non isospin-flip) transitions in a non-relativistic framework can be found in e.g. Ref. Kamerdzhiev et al. (1997). Here we introduce them for the first time in the description of charge-exchange modes. All backward diagrams that are consistent with the TBA are included. In the following we refer to the present extended approach as proton-neutron Relativistic Time-Blocking Approximation (pn-RTBA) with GSC.
We apply the above formalism to the calculation of GT transitions which characterize the nuclear response to the spin-isospin flip operator , where is the relativistic spin operator and (resp. ) converts a neutron (resp. proton) into a proton (resp. neutron). We use the NL3* parametrization of the meson-exchange interaction Lalazissis et al. (2009) and follow the numerical scheme described in Ref. Robin and Litvinova (2018). The single-particle states participating in the PVC are limited to a window of MeV around the Fermi levels, allowing for convergence of the strength up to this energy.
The resulting transition strength distributions correspond to states in the daughter nuclei (Nb and Y). While the theoretical distributions are obtained with respect to the ground state of the parent nucleus, in order to compare to the data, it is usually necessary to relate the distribution to the ground state of the daughter nucleus. To this purpose we have calculated transitions of several angular momenta and parities ( to ) and identified the ground state of the daughter nucleus as the lowest peak. The energy of this state directly gives us the binding energy difference by which we need to shift our GT distribution. In Table 1 we show the corresponding binding energy differences calculated with PVC in the pn-RTBA, without (column 3) and with (column 4) the contribution of the new backward diagrams of the type shown in Fig. 2. These are compared to the experimental values Huang et al. (2017); Wang et al. (2017) shown in column 2. The new GSC induce extra binding of the parent ground state leading to a significant improvement of the results. Let us remind that we neglect superfluid pairing correlations which could explain the remaining discrepancy. We also note that the theoretical ground states of Nb and Y are found to have the same angular momentum and parity as the experimental ones, i.e. for Nb and for Y Audi et al. (2017), both with and without GSC.
|Huang et al. (2017); Wang et al. (2017)||no GSC||with GSC|
|BE( Zr) - BE(Nb)||6.893||2.760||5.600|
|BE(Zr) - BE(Y)||1.496||-1.090||1.555|
We show in Fig. 3 the GT strength distributions in Zr in both the GT (top panel) and GT (bottom panel) directions. They are obtained with a smearing parameter keV which provides a detailed description of the spectrum. Let us first examine the GT branch. The dashed and plain black curves show the results without PVC obtained at the pn-RTDA and pn-RRPA levels (i.e. without and with GSC), respectively. It is clear that the GSC are negligible for the present nucleus. The blue and red curves show the results when including PVC without and with the backward-going diagrams of the type shown in Fig. 2 (i.e. without and with GSC), respectively. PVC introduces fragmentation of the strength and the GSC induce further redistribution of the GT resonance (GTR) along with an upward shift of the low-energy peak by keV. Let us now turn to the GT channel. In the pn-RTDA limit, the possible proton neutron transitions respecting the GT selection rules are strongly hindered by the Pauli blocking as can be deduced from the single-particle spectrum of Fig. 4.
Then, the GSC can, in principle, unlock transitions from particle to hole states. However, the possible transitions appear only at excitation energies above 7 MeV and are very weak due to the small values of the corresponding matrix elements of the external field. The inclusion of PVC on top of pn-RRPA with only the forward-going diagrams of Fig. 1 (blue curve) induces almost no change. The final distribution, obtained when including the backward-going diagrams of Fig. 2, is depicted in red. Evidently, the GSC have a very strong effect in the GT channel. They induce fractional occupancies of the single-particle states of the parent nucleus and, thus, lead to new transitions from particle to particle state and from hole to hole state. In particular, the peak at MeV appears mainly due to the transitions from the proton- to the neutron- with a corresponding transition density of in absolute value, and from the proton- to the neutron- with a transition density of .
This is illustrated in Fig. 4.
In order to compare our results to the available data Yako et al. (2005); Wakasa et al. (1997), we have smeared the calculated GT and GT strength distributions with a parameter MeV and MeV, respectively, to match the experimental resolution. The resulting distributions are shown in Fig. 5. In the GT channel, the pn-RTBA with GSC (plain red curve) shows a very good agreement with the data up to MeV, except for a small shift of the low-energy state and a double-peak structure of the GTR which are expected to be corrected by the inclusion of superfluid pairing correlations. In the GT channel, the GSC induced by PVC are solely responsible for the appearance of the low-energy peak below MeV, which is observed experimentally, as well as for the presence of the higher-energy strength up to MeV. Above MeV the theoretical GT strength nevertheless largely underestimates the data. It is well known, however, that at large excitation energy contributions of other multipole modes can come into play. Among them the isovector spin monopole (IVSM) mode, or response to the operator , is expected to be the most important. The GT data points extracted from Refs. Yako et al. (2005); Wakasa et al. (1997) in fact also contain the contribution of such modes which could not be disentangled from the GT transitions due to the difficulty of such procedure. In order to have a meaningful comparison, we therefore follow the procedure adopted in Ref. Terasaki (2018), and plot with dashed lines the response to the hybrid operator where is a parameter that has the dimension of an inverse squared length, and that is adjusted to reproduce the magnitude of our theoretical low-energy GT strength. As seen from Fig. 5, the IVSM mode appears responsible for the strength above 25-30 MeV in the (p,n) branch. In the (n,p) channel it is found to be very important, even at low energy, above 5 MeV. This is in accordance with Ref. Miki et al. (2012). After adding the IVSM component we obtain a very good agreement of the overall strength distributions.
We note that, as was originally discussed in Ref. Tselyaev (2007) some of the new diagrams responsible for the GSC (those of fourth order in the PVC vertex) can induce a small violation of the Ikeda sum rule. Numerically we find a discrepancy of in the case of Zr when integrating the GT strength distributions up to 100 MeV and taking into account the contribution of the transitions to the Dirac sea Kurasawa et al. (2003). A procedure to correct for this deficiency by eliminating certain GSC diagrams has been proposed in Ref. Tselyaev (2007). Implementing such correction is beyond the scope of the present study and we leave it for a future work.
Finally, let us mention that calculations of GT modes in Zr using non-relativistic beyond-RPA methods including complex GSC have been performed in the past. In Ref. Rijsdijk et al. (1993) both GT and GT transitions were calculated with a perturbative dressed pn-RPA approach. That work also found some strength in the GT branch, but only for excitation energies above MeV. Ref. Drozdz et al. (1990) used second RPA and also studied the effect of perturbative GSC on GT in Ca. These were found to slightly enhance the strength distribution in the GTR region. Strong limitations of these methods on the configuration complexity, however, did not allow for a clear demonstration of the importance of complex GSC for the description of the GT transitions.
Summary and Outlook. — We have extended the proton-neutron relativistic time-blocking approximation in order to include consistently the GSC arising from the backward-going diagrams generated by the coupling between single nucleons and collective vibrations of the nucleus. We have implemented these new diagrams in a parameter-free framework based on the effective meson-nucleon Lagrangian. We studied the effect of these contributions on the description of GT and GT transitions in the nucleus Zr which constitutes a very clear test case. We found that the new correlations are decisive for the appearance of the GT strength and necessary to reproduce the low-energy transitions observed experimentally in this channel. This observation is generally verified in doubly-magic nuclei with NZ, for which the pure independent-particle model forbids GT transitions, and the GSC of pn-RPA do not allow for the appearance of such states either. Overall a very good agreement with experiment is found for Zr in both branches up to excitation energies of MeV when the contribution of the IVSM is taken into account. We also found that the new GSC are necessary to reproduce the binding-energy difference with the daughter odd-odd systems. An extension of this work to open-shell nuclei in the near future will allow us to perform a more systematic study of the GSC generated by PVC and their interplay with pairing correlations. Their joint effect on beta-decay and electron-capture rates will also be investigated.
Acknowledgments. — We thank V. I. Tselyaev and R. G. T. Zegers for enlightening discussion. This work was supported by the Institute for Nuclear Theory under US-DOE Grant DE-FG02-00ER41132, by JINA-CEE under US-NSF Grant PHY-1430152, and by US-NSF CAREER grant PHY-1654379.
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