Time-reversed particle-vibration loops and nuclear Gamow-Teller response

TL;DR

This paper extends nuclear response theory to include particle-vibration coupling effects in the ground state, significantly improving the description of Gamow-Teller transitions and matching experimental data.

Contribution

It introduces for the first time particle-vibration coupling effects in the ground state within a relativistic framework for charge-exchange modes.

Findings

Particle-vibration coupling explains the (n,p) strength in Gamow-Teller transitions.

Results show excellent agreement with experimental data up to high excitation energies.

Parent-daughter binding-energy differences are significantly improved.

Abstract

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 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 $^{90}$Zr in both (p,n) and (n,p) channels. The particle-vibration coupling effects are calculated without introducing new parameters. 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 a phenomenological admixture of the isovector spin monopole transitions. The parent-daughter binding-energy differences are also greatly improved by the inclusion of the new correlations.

Figures

Tables

Table 1. Table 1 : BE differences in MeV.

	EXP	pn-RRPA	pn-RTBA	pn-RTBA
	Huang et al. (2017); Wang et al. (2017)		without	with
			GSCPVC	GSCPVC
BE( 90Zr) - BE(90Nb)	6.893	6.160	2.760	5.430
BE(90Zr) - BE(90Y)	1.496	2.200	-1.090	1.880

Equations

$$S_{\pm }(E)$$

$$R_{1423}^{({\sigma }_{12},{\sigma }_{34})}\equiv R_{1423}^{{\sigma }_1{\sigma }_4{\sigma }_2{\sigma }_3}.$$

Full text

Introduction. Charge-exchange transitions in nuclei, in particular Gamow-Teller (GT) modes, are important probes of the nuclear forces and key 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 $\beta^{-}$-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 orbits, then constitutes the only mechanism that can unlock transitions in this branch. In methods based on the mean-field approximation doubly-magic nuclei represent a test case to study these effects. Indeed, in such systems no superfluid pairing correlations are present in the mean field, and one can fully quantify the effect of ground-state correlations (GSC) introduced beyond the mean field while avoiding possible interplays with pairing.

To study charge-exchange modes 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 on top 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 GSCRPA. While (pn-)RPA reproduces the position of giant resonances (GRs) to a good accuracy, it is well known that it typically generates a poorly detailed description of the transition strength distributions and, in particular, is not able to describe the spreading width of the GRs. To correct for these deficiencies one should account for higher-order configurations of the nucleons. 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 $\otimes$ 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 GSCPVC to differentiate them from GSCRPA. In other words, configurations beyond pn-RPA were introduced in the description of states of the daughter nucleus, but not in the ground state of the parent system. It was shown in Refs. Kamerdzhiev et al. (1994, 1997), that such GSCPVC can in fact be consistently included. Their effect on electromagnetic transitions was studied in a non-relativistic framework based on Landau-Migdal forces and they were found important for the description of the low-energy strength distributions.

In this Letter, we implement for the first time GSCPVC in the description of charge-exchange transitions of non-superfluid nuclei in a relativistic consistent framework. We apply the extended formalism to the calculation of GT and isovector spin-monopole (IVSM) modes in 90Zr. This nucleus is particularly interesting as it has been experimentally measured in both GT+ and GT- channels up to energies beyond the GR region Yako et al. (2005); Wakasa et al. (1997). Due to the weak pairing caused by the closed shell at $Z=50$ and sub-shell at $N=40$, 90Zr can be well approximated as doubly-magic nucleus where the effect of configuration mixing and GSC can be investigated carefully. We will find that GSCPVC 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 GT+ strength.

Formalism. We want to describe the response of a nucleus to a one-body charge-changing external field, i.e. converting a neutron (n) into a proton (p) ($\hat{F}_{-}=\sum_{pn}F_{pn}a^{\dagger}_{p}a_{n}$) or vice versa ($\hat{F}_{+}=(\hat{F}_{-})^{\dagger}$). The corresponding transition strength distribution $S_{\pm}$ is fully determined by the response function $R$ describing the propagation of correlated proton-neutron pairs in the particle-hole channel:

[TABLE]

where $\ket{\Psi_{0}}$ denotes the parent nuclear ground state, $\ket{\Psi_{N}}$ the states of the daughter nucleus, and $\Omega_{N}=E_{N}-E_{0}$ the corresponding transition energies. In the following odd (resp. even) indices denote proton (resp. neutron) spherical single-particle states for the $(p,n)$ channel and vice versa for the $(n,p)$ channel. Letters $i,j$ will be used to denote states with unspecified isospin projection. We introduce an index $\sigma_{i}$ where $\sigma_{i}=+$, if $i$ is a state above the Fermi level (”particle” state) and $\sigma_{i}=-$, if $i$ is below the Fermi level (”hole” state). The product $\sigma_{ij}\equiv\sigma_{i}\sigma_{j}$ will then be $-$ if $(i,j)$ is a particle-hole or hole-particle pair, and will be $+$ if $(i,j)$ is a particle-particle or hole-hole pair. The response function now has, in principle, four components $R^{(--)}$, $R^{(-+)}$, $R^{(+-)}$ and $R^{(++)}$ that denote

[TABLE]

That is $R^{(--)}_{1423}=\{R_{1423}^{+--+},R_{1423}^{-++-},R_{1423}^{++--},R_{1423}^{--++}\}$, $R^{(++)}_{1423}=\{R_{1423}^{++++},R_{1423}^{----},R_{1423}^{+-+-},R_{1423}^{-+-+}\}$, $R^{(+-)}_{1423}=\{R_{1423}^{+-++},R_{1423}^{+++-},R_{1423}^{---+},R_{1423}^{-+--}\}$ and $R^{(-+)}_{1423}=\{R_{1423}^{++-+},R_{1423}^{-+++},R_{1423}^{+---},R_{1423}^{--+-}\}$. Similarly the external field has two components $F^{(-)}_{12}=\{F^{+-}_{12},F^{-+}_{12}\}$ and $F^{(+)}_{12}=\{F^{++}_{12},F^{--}_{12}\}$.

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, $R^{(+-)}$, $R^{(-+)}$ and $R^{(++)}$ cancel, so that $R^{(--)}$ and $F^{(-)}$ are the only components that remain. In order to account for higher-order configurations and describe nuclear excitations more precisely, 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 $R^{(--)}$ component via the Bethe-Salpeter equation (BSE) Robin and Litvinova (2016, 2018) shown in Fig. 1.

Such diagrams introduce complex configurations of the 1p-1h $\otimes$ phonon type in the states of the daughter nucleus, but do not modify the description of the parent ground state compared to pn-RRPA. The PVC, however, generates ”backward-going” diagrams, as those shown in Fig. 2, which introduce complex configurations in the parent ground state.

Such diagrams modify further the $R^{(--)}$ function of Fig. 1 and also introduce non-zero $R^{(-+)}$, $R^{(+-)}$ and $R^{(++)}$ components of the response. The derivation of these additional terms for non-isospin-flip transitions in a non-relativistic framework are available 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 consistent with the TBA are included, and the full mathematical expressions are in the supplemental material. In the following we refer to the present extended approach as proton-neutron Relativistic Time-Blocking Approximation (pn-RTBA) with GSCPVC.

Results. We apply the above formalism to the calculation of GT transitions which characterize the nuclear response to the spin-isospin-flip operator $F_{GT_{\pm}}=\sum_{i}\boldsymbol{\Sigma}^{(i)}\tau_{\pm}^{(i)}$, where $\boldsymbol{\Sigma}$ is the relativistic spin operator and $\tau_{-}$ (resp. $\tau_{+}$) 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 pion, which does not contribute to the mean field, is included in the response with free-space coupling constant ($\frac{f_{\pi}^{2}}{4\pi}=0.08$) and associated zero-range parameter $g^{\prime}=0.6$ Paar et al. (2004).The spectrum of RRPA phonons coupled to nucleons include natural-parity neutral phonons from $0^{+}$ to $6^{+}$ and all-parity charge-exchange phonons from $0^{\pm}$ to $7^{\pm}$ with excitation energy up to 20 MeV. The single-particle states participating in the PVC are limited to a window of $50$ MeV around the Fermi levels, allowing for convergence of the strength up to this energy.

The resulting strength distributions correspond to states in the daughter nuclei (90Nb and 90Y). 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 them to the ground state of the daughter system. To this end we calculated transitions of several multipolarities ($J^{\pi}=0^{\pm}$ to $9^{\pm}$) and identified the daughter ground state as the lowest peak. The energy of this state directly gives us the binding energy (BE) difference by which we should shift our GT distribution. Table 1 shows the corresponding BE differences calculated in the pn-RRPA, pn-RTBA, without and with 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).

As expected pn-RRPA gives satisfactory BE differences due to the fitting of NL3* force. The inclusion of PVC without GSC worsens these values due to the lack of consistency in the description of parent and daughter nuclei. The new GSCPVC induce extra binding of the parent system significantly improving the results that are brought back close to the experimental values. We also find the theoretical daughter ground states with the same angular momentum and parity as the experimental ones: $J^{\pi}=8^{+}$ for 90Nb and $J^{\pi}=2^{-}$ for 90Y Audi et al. (2017), both with and without GSCPVC. pn-RRPA predicts $J^{\pi}=5^{+}$ for the 90Nb ground state and $J^{\pi}=2^{-}$ for 90Y.

We show in Fig. 3 the GT strength distributions in 90Zr in the GT- (top panel) and GT+ (bottom panel) directions. To obtain a detailed spectrum we used a smearing parameter $\Delta=200$ keV (see Eq. (1)). Let us first examine the GT- branch. The dotted and dashed black curves show the results without PVC obtained at the pn-RTDA and pn-RRPA levels (i.e. without and with GSCRPA), respectively. Clearly the GSCRPA are negligible in 90Zr. 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 GSCPVC), respectively. PVC introduces fragmentation of the strength and the GSCPVC induce further redistribution of the GT resonance (GTR) along with an upward shift of the low-energy peak by $\sim 500$ keV. Let us now turn to the GT+ channel. In the pn-RTDA limit, the possible proton $\rightarrow$ 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 GSCRPA 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 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, including the backward-going diagrams as in Fig. 2, is depicted in red. Evidently, the GSCPVC have a very strong effect in the GT+ channel. They induce fractional occupancies of the single-particle states of the parent nucleus, leading to new transitions from particle to particle state and from hole to hole state. In particular, the peak around $4.5$ MeV appears mainly due to transitions from the proton-$1g_{9/2}$ to the neutron-$1g_{7/2}$ and from the proton-$2p_{3/2}$ to the neutron-$2p_{1/2}$, with corresponding absolute transition densities of $0.347$ and $0.182$, respectively.

This is illustrated in Fig. 4.

For a comparison to the available data Yako et al. (2005); Wakasa et al. (1997), in Fig. 5 we have smeared the calculated GT+ and GT- distributions with a parameter $\Delta=2$ and $1$ MeV, respectively, to match the experimental resolutions. In the GT- channel, the pn-RTBA with GSCPVC (plain red curve) shows a good agreement with the data up to $\sim 25$ MeV, besides a small shift of the low-energy state. In the GT+ channel, the GSC induced by PVC are solely responsible for the appearance of the low-energy peak below $6$ MeV, which is observed experimentally, as well as for higher-energy strength up to $\sim 50$ MeV. Above $\sim 6$ MeV the theoretical GT+ strength alone 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 IVSM mode, or response to the operator $F_{IVSM^{\pm}}=\sum_{i}r_{(i)}^{2}\boldsymbol{\Sigma}^{(i)}\tau_{\pm}^{(i)}$, 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 of Ref. Terasaki (2018), and plot the response to the hybrid operator $F^{hyb}_{\pm}=\sum_{i}(1+\alpha r_{(i)}^{2})\boldsymbol{\Sigma}^{(i)}\tau_{\pm}^{(i)}$ where $\alpha$ is a parameter that we adjust to reproduce the magnitude of our theoretical low-energy GT strength. We find $\alpha=9.1\times 10^{-3}$ and $7.5\times 10^{-3}$ fm*-2* for the GT+ and GT- branch respectively. 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 clearly very important, even at low energy, above 5 MeV. This observation is in accordance with Ref. Miki et al. (2012). After adding the IVSM component we obtain a very good agreement of the overall strength distribution.

We note that, as originally discussed in Ref. Tselyaev (2007), some of the new diagrams responsible for the GSCPVC (those of fourth order in the PVC vertex) can induce a small violation of the Ikeda sum rule. Numerically we find a discrepancy of $1.23\%$ in the case of 90Zr when using a cut-off energy of 50 MeV on the nucleon states entering the PVC mechanism, and accounting for the contribution of the transitions to the Dirac sea Kurasawa et al. (2003). Based on a convergence study conducted in smaller single-particle bases, we expect the sum rule violation to be stable when increasing the PVC cut-off energy. A procedure to correct for the violation by eliminating certain GSCPVC diagrams was proposed in Ref. Tselyaev (2007). Implementing it is beyond the scope of the present study and we leave it for a future work.

Finally, calculations of GT modes using non-relativistic beyond-RPA methods including complex GSC were performed in the past. Ref. Rijsdijk et al. (1993) used a perturbative dressed pn-RPA approach and found some strength in the GT+ branch of 90Zr, but only for energies above $\sim 5$ MeV. Ref. Drozdz et al. (1990) included perturbative GSC within second RPA which produced a slight enhancement of the GTR region in 48Ca. 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 GT transitions.

Summary and Outlook. We extended the pn-RTBA to consistently include the GSC arising from the coupling between single nucleons and collective nuclear vibrations. We implemented the corresponding new diagrams in a framework based on the effective meson-nucleon Lagrangian and studied their effect on the description of GT*-* and GT*+* transitions in the nucleus 90Zr which constitutes a very clear test case. 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 N$>$Z, where 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 90Zr in both branches up to excitation energies of $\sim 50$ MeV when the contribution of the IVSM is accounted for. The new GSC were also necessary to reproduce the binding-energy differences with the daughter odd-odd systems. An extension of this work to open-shell nuclei in the future will allow more systematic studies 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. Ultimately these developments can strongly impact astrophysical simulations which up to now use inputs from quasiparticle RPA. In the context of searches beyond the Standard Model, such as the neutrinoless double-beta decay for which an accurate description of GSC in both branches is of the utmost importance, this work could improve the predictions of RPA-based approaches and reduce the large discrepancy that exists between the many theoretical methods Engel and Menéndez (2017). GSCPVC, being one of the most difficult aspects of the quantum many-body problem, also provides a refined modeling in other areas of mesoscopic physics, such as condensed matter and quantum chemistry.

Acknowledgments. We thank V. I. Tselyaev and R. G. T. Zegers for enlightening discussions. 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.