# Nuclear charge-exchange excitations based on relativistic density-dependent
point-coupling model

D. Vale<EMAIL_ADDRESS>Gimnazija i strukovna škola Jurja Dobrile Pazin,
Šetalište Pazinske gimnazije 11, Pazin 52000, Croatia  Osnovna škola
Vodnjan/Scuola elementare Dignano, Žuka 6/Via delle ginestre 6,
Vodnjan/Dignano 52215, Croatia  Department of Physics, Faculty of Science,
University of Zagreb, Bijenička cesta 32, 10000 Zagreb, Croatia  Y. F. Niu
School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000,
China,
ELI-NP, Horia Hulubei National Institute for Physics and Nuclear Engineering,
30 Reactorului Street, RO-077125, Bucharest-Magurele, Romania  N. Paar
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, University of Zagreb,
Bijenička cesta 32, 10000 Zagreb, Croatia

###### Abstract

Spin-isospin transitions in nuclei away from the valley of stability are
essential for the description of astrophysically relevant weak interaction
processes. While they remain mainly beyond the reach of experiment,
theoretical modeling provides important insight into their properties. In
order to describe the spin-isospin response, the proton-neutron relativistic
quasiparticle random phase approximation (PN-RQRPA) is formulated using the
relativistic density-dependent point coupling interaction, and separable
pairing interaction in both the $T=1$ and $T=0$ pairing channels. By
implementing recently established DD-PCX interaction with improved isovector
properties relevant for the description of nuclei with neutron-to-proton
number asymmetry, the isobaric analog resonances (IAR) and Gamow-Teller
resonances (GTR) have been investigated. In contrast to other models that
usually underestimate the IAR excitation energies in Sn isotope chain, the
present model accurately reproduces the experimental data, while the GTR
properties depend on the isoscalar pairing interaction strength. This
framework provides not only an improved description of the spin-isospin
response in nuclei, but it also allows future large scale calculations of
charge-exchange excitations and weak interaction processes in stellar
environment.

###### pacs:

21.30.Fe, 21.60.Jz, 24.30.Cz, 25.40.Kv

##  I Introduction

Charge-exchange excitations in atomic nuclei correspond to a class of nuclear
transitions composed of the particle-hole configurations that contain the
exchange of the nucleon charge, described by the isospin projection lowering
(increasing) operator $\tau_{-}$ ($\tau_{+}$). The fundamental charge-exchange
excitation is the isobaric analog resonance (IAR) Auerbach  _et al._ (1972);
Auerbach (1983); Rumyantsev and Urin (1994); Pham  _et al._ (1995); Colò  _et
al._ (1998); Paar  _et al._ (2004); Fracasso and Colò (2005); Roca-Maza  _et
al._ (2012), with no changes in quantum numbers $\Delta J=\Delta L=\Delta
S=\text{0}$, thus the IAR corresponds to a collective excitation with
$J^{\pi}={\text{0}}^{+}$. The Gamow-Teller resonance (GTR) represents another
relevant charge-exchange mode, characterized by $J^{\pi}={\text{1}}^{+}$,
i.e., it corresponds to spin-flip excitations without changing the orbital
motion, $\Delta S=\text{1}$, $\Delta L=\text{0}$.

As it has been emphasized in Ref. Roca-Maza and Paar (2018), recent interest
in the GTR studies is motivated by its importance for understanding the spin
and spin-isospin dependence of modern effective interactions Bender  _et al._
(2002); Marketin  _et al._ (2012a); Litvinova  _et al._ (2014); Ekström  _et
al._ (2014); Zelevinsky  _et al._ (2017); Morita and Kanada-En’yo (2017); Nabi
_et al._ (2017); Ha and Cheoun (2017), nuclear beta decay Engel  _et al._
(1988); Borzov  _et al._ (1995); Engel  _et al._ (1999); Borzov (2003);
Madurga  _et al._ (2016); Marketin  _et al._ (2016); Moon  _et al._ (2017);
Wang  _et al._ (2016); Ravlić  _et al._ (2020a), beta delayed neutron emission
Caballero-Folch  _et al._ (2016), as well as double beta decay Faessler and
Simkovic (1998); Suhonen and Civitarese (1998); Šimkovic  _et al._ (2011);
Menéndez  _et al._ (2011); Vergados  _et al._ (2012); Menéndez  _et al._
(2014); Štefánik  _et al._ (2015); Navas-Nicolás and Sarriguren (2015); Delion
and Suhonen (2017). In addition, accurate description of GT± transitions,
including both in stable and exotic nuclei, is relevant for the description of
a variety of astrophysically relevant weak interaction processes Langanke and
Martinez-Pinedo (2000); Langanke and Martínez-Pinedo (2003); Janka  _et al._
(2007); Noji  _et al._ (2014); Paar  _et al._ (2015), electron capture in
presupernova stars Niu  _et al._ (2011, 2013); Ravlić  _et al._ (2020b),
r-process Arnould  _et al._ (2007); Mori  _et al._ (2016) and neutrino-nucleus
interaction of relevance for neutrino detectors and neutrino nucleosynthesis
in stellar environment Ejiri (2000); Suzuki and Sagawa (2003); Frekers  _et
al._ (2011); Cheoun  _et al._ (2010); Paar  _et al._ (2008, 2011); Karakoç
_et al._ (2014).

The properties of charge-exchange modes of excitation have extensively been
studied Osterfeld (1992); Fujita  _et al._ (2011). Following theoretical
prediction Ikeda  _et al._ (1963), in 1975 the GTR has been experimentally
confirmed in $(p,n)$ reactions Doering  _et al._ (1975). The GTR represents
one of the most extensively investigated collective excitation in nuclear
physics, both experimentally and theoretically (e.g., see Refs. Osterfeld
(1992); Halbleib and Sorensen (1967); Doering  _et al._ (1975); Towner and
Khanna (1979); Goodman  _et al._ (1980); Brown and Rho (1981); Bertsch and
Hamamoto (1982); Nakayama  _et al._ (1982); Van Giai and Sagawa (1981); Gaarde
(1983); Kuzmin and Soloviev (1984); Colò  _et al._ (1994); Hamamoto and Sagawa
(1993); Brown and Rykaczewski (1994); Suzuki and Otsuka (1997); De Conti  _et
al._ (1998); Caurier  _et al._ (1999); Langanke  _et al._ (2001); Bender  _et
al._ (2002); Algora  _et al._ (2003); Kalmykov  _et al._ (2006); Bai  _et al._
(2009); Lutostansky (2011); Sasano  _et al._ (2011); Fujita  _et al._ (2011);
Niu  _et al._ (2012); Ha and Cheoun (2013); Roca-Maza  _et al._ (2013);
Martini  _et al._ (2014); Ha and Cheoun (2016); Niu  _et al._ (2016); Liang
_et al._ (2018); Yasuda  _et al._ (2018)). More details about experimental
studies of spin-isospin excitations are also reviewed in Ref. Fujita  _et al._
(2011). Recent studies of the GTR in the framework based on relativistic
energy density functional include relativistic quasiparticle random phase
approximation (RQRPA) Paar  _et al._ (2004), the relativistic RPA based on the
relativistic Hartree-Fock (RHF) Liang  _et al._ (2008), relativistic QRPA
based on point coupling model with nonlinear interactions Niu  _et al._
(2013); Wang  _et al._ (2016), and relativistic QRPA formulated using the
relativistic Hatree–Fock–Bogoliubov (RHFB) model for the ground state Niu  _et
al._ (2017). In Ref. Finelli  _et al._ (2007) the nuclear density functional
framework, based on chiral dynamics and the symmetry breaking pattern of low-
energy QCD, has been used to formulate the proton–neutron QRPA to investigate
the role of chiral pion–nucleon dynamics in the description of charge-exchange
excitations. The GTR has also been studied by including couplings between
single nucleon and collective nuclear vibrations, e.g., particle-vibration
coupling (PVC) of the 1p-1h$\otimes$phonon type of coupling Niu  _et al._
(2012); Marketin  _et al._ (2012b); Niu  _et al._ (2016); Robin and Litvinova
(2019). The PVC allows to include important dynamical correlations missing in
the static self-consistent mean field models and it provides additional
fragmentation of the GTR strength when compared to the random phase
approximation studies based only on 1p-1h configurations Niu  _et al._ (2012).

At present, the knowledge about charge-exchange transitions in nuclei away
from the valley of stability is rather limited, and mainly beyond the reach of
experiment. Since these nuclei are especially important for their
astrophysical relevance in stellar evolution and nucleosynthesis, it is
crucial to develop microscopic theoretical approaches, that allow quantitative
and systematic analyses of the transition strength distributions of unstable
nuclei. In order to assess the overview into systematical model uncertainties
in modeling charge-exchange excitation phenomena, it is important to address
their properties from various approaches, by implementing different theory
frameworks and effective nuclear interactions.

In Ref. Paar  _et al._ (2004) charge-exchange excitations have been studied in
the framework based on the relativistic nuclear energy density functional
(EDF), within the approach that unifies the treatment of mean-field and
pairing correlations, relativistic quasiparticle random phase approximation
(RQRPA) formulated in the canonical single-nucleon basis of the relativistic
Hartree-Bogoliubov (RHB) model. In this implementation the relativistic EDF
with explicit density dependence of the meson-nucleon couplings is used, that
provides an improved description of asymmetric nuclear matter, neutron matter
and nuclei far from stability. The pairing correlations were described by the
pairing part of the finite range Gogny interaction Dechargé and Gogny (1980);
Berger  _et al._ (1991). However, the EDFs have usually been parameterized
with the experimental data on the ground state properties, supplemented with
the pseudo-observables on nuclear matter properties. In the case of density
dependent meson-exchange interactions, the neutron skin thickness in 208Pb has
been introduced as an additional constraint on the isovector channel of the
effective interaction. However, the results from measurements of the neutron-
skin thickness are usually model-dependent, and the pseudo-observables on
nuclear matter are often rather arbitrary. Recently, a novel EDF
parameterization has been established based on the relativistic point coupling
interaction, by using in the $\chi^{2}$ minimization the nuclear ground state
properties (binding energies, charge radii, pairing gaps) together with the
properties of collective excitations in nuclei, isoscalar giant monopole
resonance energy and dipole polarizability Yüksel  _et al._ (2019). In this
way an effective interaction DD-PCX has been established with improved
isovector properties, that is successful not only in the description of
nuclear ground state, but also of the excitation phenomena, incompressibility
of nuclear matter and the symmetry energy close to the saturation density
Yüksel  _et al._ (2019). The improved isovector channel for the DD-PCX
interaction is especially important not only for the symmetry energy of the
nuclear equation of state, but also for the description of ground-state and
excitation properties of $N\neq Z$ nuclei. Clearly, this is very important for
the implementation of the EDF based models to exotic nuclei, as well as for
applications in nuclear astrophysics.

In this work we establish the proton-neutron RQRPA in the canonical single-
nucleon basis of the RHB model based on density-dependent relativistic point
coupling interaction. Our study represents the first implementation of the
relativistic point coupling interaction with density dependent vertex
functions in formulating the RQRPA for the description of charge-exchange
excitations. In addition, the treatment of the pairing correlations is also
improved, by implementing the separable pairing force that allows accurate and
efficient calculations of the pairing properties Tian  _et al._ (2009). By
using recently established DD-PCX interaction with improved properties that
are essential for description of nuclei away from the valley of stability
Yüksel  _et al._ (2019), the proton-neutron RQRPA established in this work
will be employed in the investigation of the properties of collective charge-
exchange excitations, IAR and GTR.

Clearly, a study of both the IAR and GTR properties represents an important
benchmark test for novel theoretical approaches established not only for
description of charge-excitation modes, but also for modeling a variety of
astrophysically relevant processes in stellar environment. Therefore, in the
present work that introduces a microscopic approach to describe charge-
exchange excitations based on density-dependent relativistic point coupling
interaction, the novel theory framework will be employed in the analyses of
charge-exchange modes, the IAR and GTR, both for magic nuclei, as well as for
open-shell nuclei to probe the effect of the pairing correlations.

In Sec. II the formalism of the proton-neutron RQRPA based on the density
dependent point coupling interaction is introduced. In Sec. III the model is
employed in studies of charge-exchange modes of excitation, the IAR and the
GTR. The conclusions of this work are summarized in Section IV.

##  II Proton-neutron RQRPA based on relativistic point coupling interaction

In the previous implementation in the relativistic framework, the RQRPA has
been established on the ground of relativistic Hartree-Bogoliubov (RHB) model,
based on the effective Lagrangian with density-dependent meson-exchange
interaction terms Paar  _et al._ (2004). Therein the pairing correlations have
been described by the pairing part of the Gogny interaction Dechargé and Gogny
(1980); Berger  _et al._ (1991). In the present study, the RQRPA is
established using the relativistic point coupling interaction, while the
pairing correlations are described by the separable pairing force from Ref.
Tian  _et al._ (2009). Since the full RQRPA equations are rather complicated,
in the present study we solve the respective equations in the canonical basis,
where the Hartree-Bogoliubov wave functions can be expressed in the form of
the BCS-like wave functions. More details on the implementation of the
canonical basis in the RHB model and general formalism of the PN-RQRPA
equations in the canonical basis are given in Ref. Paar  _et al._ (2004). The
focus of this work is the implementation of the relativistic point coupling
interaction in deriving the PN-RQRPA equations. The nuclear ground state
properties are described in the RHB model for the point coupling interaction,
described in detail in Ref. Nikšić  _et al._ (2014).

Starting from the $0^{+}$ ground state of a spherical even-even nucleus,
transitions to $J^{\pi}$ excited state of the corresponding odd-odd daughter
nucleus are considered, using the charge-exchange operator ${\cal{O}}^{JM}$.
The general form of the PN-RQRPA equations read Paar  _et al._ (2004),

$\left(\begin{array}[c]{cc}A^{J}&B^{J}\\\
B^{{}^{\ast}J}&A^{{}^{\ast}J}\end{array}\right)\left(\begin{array}[c]{c}X^{\lambda
J}\\\ Y^{\lambda
J}\end{array}\right)=E_{\lambda}\left(\begin{array}[c]{cc}1&0\\\
0&-1\end{array}\right)\left(\begin{array}[c]{c}X^{\lambda J}\\\ Y^{\lambda
J}\end{array}\right)\;,$ (1)

where the $A$ and $B$ matrices are defined in the canonical basis,

$\displaystyle A_{pn,p^{\prime}n^{\prime}}^{J}$ $\displaystyle=$
$\displaystyle
H^{11}_{pp^{\prime}}\delta_{nn^{\prime}}+H^{11}_{nn^{\prime}}\delta_{pp^{\prime}}$
$\displaystyle+\left(u_{p}v_{n}u_{p^{\prime}}v_{n^{\prime}}+v_{p}u_{n}v_{p^{\prime}}u_{n^{\prime}}\right)V_{pn^{\prime}np^{\prime}}^{phJ}+\left(u_{p}u_{n}u_{p^{\prime}}u_{n^{\prime}}+v_{p}v_{n}v_{p^{\prime}}v_{n^{\prime}}\right)V_{pnp^{\prime}n^{\prime}}^{ppJ}$
$\displaystyle B_{pn,p^{\prime}n^{\prime}}^{J}$ $\displaystyle=$
$\displaystyle\left(u_{p}v_{n}v_{p^{\prime}}u_{n^{\prime}}+v_{p}u_{n}u_{p^{\prime}}v_{n^{\prime}}\right)V_{pn^{\prime}np^{\prime}}^{phJ}$
(2)
$\displaystyle-\left(u_{p}u_{n}v_{p^{\prime}}v_{n^{\prime}}+v_{p}v_{n}u_{p^{\prime}}u_{n^{\prime}}\right)V_{pnp^{\prime}n^{\prime}}^{ppJ}\;.$

The proton and neutron quasiparticle canonical states are denoted by $p$,
$p\prime$, and $n$, $n\prime$, respectively. $V^{ph}$ is the proton-neutron
particle-hole residual interaction, and $V^{pp}$ is the corresponding
particle-particle interaction, and $u$ and $v$ denote the occupation
amplitudes of the respective states. Since the canonical basis does not
diagonalize the Dirac single-nucleon mean-field Hamiltonian, the off-diagonal
matrix elements $H^{11}_{nn^{\prime}}$ and $H^{11}_{pp^{\prime}}$ are also
included in the $A$ matrix, as given in Ref. Paar  _et al._ (2004).
$E_{\lambda}$ denote the excitation energy, while $X^{\lambda J}$ and
$Y^{\lambda J}$ are the corresponding forward- and backward-going QRPA
amplitudes, respectively.

By solving the eigenvalue problem (1), the reduced transition strength can be
obtained between the ground state of the even-even $(N,Z)$ nucleus and the
excited state of the odd-odd $(N+1,Z-1)$ or $(N-1,Z+1)$ nucleus, using the
corresponding transition operators ${\cal{O}}^{JM}$ in both channels,

$\displaystyle B_{\lambda
J}^{-}=\left|\sum_{pn}<p||{\cal{O}}^{J}||n>\left(X_{pn}^{\lambda
J}u_{p}v_{n}+Y_{pn}^{\lambda J}v_{p}u_{n}\right)\right|^{2}\;,$ (3)
$\displaystyle B_{\lambda
J}^{+}=\left|\sum_{pn}(-1)^{j_{p}+j_{n}+J}<n||{\cal{O}}^{J}||p>\left(X_{np}^{\lambda
J}v_{p}u_{n}+Y_{np}^{\lambda J}u_{p}v_{n}\right)\right|^{2}\;.$ (4)

For the presentation purposes, the discrete strength distribution is folded by
the Lorentzian function of the width $\Gamma=1$ MeV,

$R(E)^{\pm}=\sum_{\lambda}B_{\lambda
J}^{\pm}\frac{1}{\pi}\frac{\Gamma/2}{(E-E_{\lambda_{\pm}})^{2}+(\Gamma/2)^{2}}\;.$
(5)

In the implementation of the relativistic point coupling interaction, the
spin-isospin dependent terms in the residual interaction of the PN-RQRPA are
induced by the isovector-vector and pseudovector terms. In comparison, for the
finite range meson-exchange interaction, these terms were obtained from the
$\rho$-and $\pi$-meson exchange, respectively Paar  _et al._ (2004). In the
present study, the PN-RQRPA residual interaction terms $V_{abcd}$ are derived
from the effective Lagrangian density for the point coupling interaction.

The isovector-vector part of PN-RQRPA contains only non-rearrangement terms of
the residual two-body interaction. For the respective spacelike components we
obtain,

$V_{abcd}^{(\text{TVs})}~{}=~{}-\int d^{3}r_{1}\int
d^{3}r_{2}\psi_{a}^{\dagger}(\vec{r}_{1})\left(\vec{\tau}\gamma_{0}\gamma_{i}\right)^{(1)}\psi_{c}(\vec{r}_{1})\alpha_{TV}(\rho)\delta(\vec{r}_{1}-\vec{r}_{2})\psi_{b}^{\dagger}(\vec{r}_{2})\left(\vec{\tau}\gamma_{0}\gamma^{i}\right)^{(2)}\psi_{d}(\vec{r}_{2}),$
(6)

and timelike components are given by

$V_{abcd}^{(\text{TVt})}~{}=~{}\int d^{3}r_{1}\int
d^{3}r_{2}\psi_{a}^{\dagger}(\vec{r}_{1})\vec{\tau}^{(1)}\psi_{c}(\vec{r}_{1})\alpha_{TV}(\rho)\delta(\vec{r}_{1}-\vec{r}_{2})\psi_{b}^{\dagger}(\vec{r}_{2})\vec{\tau}^{(2)}\psi_{d}(\vec{r}_{2}),$
(7)

where the coupling $\alpha_{TV}(\rho)$ is a function of baryon (vector)
density. For Dirac spinor given by,

$\psi_{am_{a}}(r)~{}=~{}\left(\begin{array}[]{clcr}f_{a}(r)\Omega_{\kappa_{a}m_{a}}(\Omega)\\\
\ ig_{a}(r)\Omega_{\bar{\kappa}_{a}m_{a}}(\Omega)\end{array}\right),$ (8)

where $a$ stands for all quantum numbers, with the exception of the projection
of total angular momentum $m_{a}$. Quantum number $\kappa$ is defined as
$\kappa=-(l+1)$ for $j=l+1/2$ and $\kappa=l$ for $j=l-1/2$, while
$\bar{l}=2j-l$ corresponds to the lower component orbital angular momentum
Greiner and Bromley (2000). The spacelike part of the matrix elements obtained
by the angular momentum coupling is given

$\displaystyle
V_{abcd}^{(\text{TVs})J}~{}=~{}\frac{2}{\left(2J+1\right)}\sum_{L}\int
drr^{2}\alpha_{TV}(\rho)\bigg{[}f_{a}(r)g_{c}(r)\langle(1/2\
l_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{c})j_{c}\rangle-$ $\displaystyle~{}g_{a}(r)f_{c}(r)\langle(1/2\
\bar{l}_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
l_{c})j_{c}\rangle\bigg{]}\bigg{[}f_{b}(r)g_{d}(r)\langle(1/2\
\bar{l}_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{b})j_{b}\rangle$
$\displaystyle~{}-g_{b}(r)f_{d}(r)\langle(1/2\
l_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{b})j_{b}\rangle\bigg{]},$ (9)

and the timelike part is

$\displaystyle V_{abcd}^{(TVt)J}~{}=$ $\displaystyle~{}\frac{2}{2J+1}\int
drr^{2}\alpha_{TV}(\rho)\bigg{[}f_{a}(r)f_{c}(r)+g_{a}(r)g_{c}(r)\bigg{]}\bigg{[}f_{b}(r)f_{d}(r)+g_{b}(r)g_{d}(r)\bigg{]}$
$\displaystyle~{}\times\langle(1/2\ l_{a})j_{a}||Y_{J}||(1/2\
l_{c})j_{c}\rangle\langle(1/2\ l_{d})j_{d}||Y_{J}||(1/2\ l_{b})j_{b}\rangle.$
(10)

When compared to standard R(Q)RPA matrix elements, in particular corresponding
direct term, there exists an additional factor of 2 in the numerator of Eqs.
(9) and (10) due to the difference in the isospin part of the matrix element
(see Appendix A). The isovector-pseudovector part of the point coupling
interaction is given by

$V_{PV}~{}=~{}-\alpha_{PV}\delta(\vec{r}_{1}-\vec{r}_{2})\left(\gamma_{0}\gamma_{5}\gamma_{\mu}\vec{\tau}\right)^{(1)}\left(\gamma_{0}\gamma_{5}\gamma^{\mu}\vec{\tau}\right)^{(2)},$
(11)

where $\alpha_{PV}$ denotes the strength parameter of the interaction. Since
$\alpha_{PV}$ remains a free parameter of the model that cannot be constrained
by the ground state properties, its value should be constrained by the
experimental data on charge-exchange excitations. For the corresponding
timelike part of the pseudovector matrix elements we obtain,

$\displaystyle V_{abcd}^{\text{PV(t)J}}~{}=$
$\displaystyle~{}\frac{2\alpha_{PV}}{2J+1}\int
drr^{2}\bigg{[}f_{a}(r)g_{c}(r)-g_{a}(r)f_{c}(r)\bigg{]}\bigg{[}f_{b}(r)g_{d}(r)-g_{b}(r)f_{d}(r)\bigg{]}$
$\displaystyle\times\langle(1/2\ l_{a})j_{a}||Y_{J}||(1/2\
\bar{l}_{c})j_{c}\rangle\langle(1/2\ \bar{l}_{d})j_{d}||Y_{J}||(1/2\
l_{c})j_{c}\rangle.$ (12)

We note that the timelike matrix elements are non-zero only in the case of
unnatural parity transitions, e.g., for Gamow-Teller transitions. The
spacelike part of the isovector-pseudovector matrix elements results,

$\displaystyle V_{abcd}^{PV(s)J}~{}=~{}\frac{2\alpha_{PV}}{2J+1}\sum_{L}\int
drr^{2}\bigg{[}f_{a}(r)f_{c}(r)\langle(1/2\
l_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{c})j_{c}\rangle$
$\displaystyle~{}+~{}g_{a}(r)g_{c}(r)\langle(1/2\
\bar{l}_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{c})j_{c}\rangle\bigg{]}\bigg{[}f_{b}(r)f_{d}(r)\langle(1/2\
l_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{b})j_{b}\rangle$
$\displaystyle~{}+~{}g_{b}(r)g_{d}(r)\langle(1/2\
\bar{l}_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{b})j_{b}\rangle\bigg{]}.$ (13)

In order to constrain the value of the pseudovector coupling $\alpha_{PV}$, we
follow the same procedure as used in the case of relativistic functionals with
meson-exchange Paar  _et al._ (2004), i.e., $\alpha_{PV}$ is adjusted to
reproduce the experimental value of excitation energy for Gamow-Teller
resonance in 208Pb, $E=19.2$ MeV Horen  _et al._ (1980); Akimune  _et al._
(1995a); Krasznahorkay  _et al._ (2001a). In this way, we obtain
$\alpha_{PV}=$ 0.734 for DD-PC1, and $\alpha_{PV}=$ 0.621 for DD-PCX point
coupling interaction, and use these values systematically in all further
investigations.

The PN-RQRPA also includes the pairing correlations. In most of the previous
applications of the RHB+RQRPA model, the pairing correlations have been
described by the paring part of the Gogny force D1S Dechargé and Gogny (1980);
Berger  _et al._ (1991). This interaction has already been used in the RHB
calculations of various ground state properties in nuclei Vretenar  _et al._
(2005). Since the calculations based on the finite range Gogny force require
considerable computational effort, in the present formulation of the PN-RQRPA
the separable form of the pairing interaction is used Tian  _et al._ (2009).
In Ref. Tian  _et al._ (2009) Y. Tian et al. introduced separable pairing
interaction in the gap equation of symmetric nuclear matter in ${}^{1}S_{0}$
channel:

$\Delta(k)~{}=~{}-\int_{0}^{\infty}\frac{k^{\prime
2}dk^{\prime}}{2\pi^{2}}\langle
k|V_{sep}^{{}^{1}S_{0}}|k^{\prime}\rangle\frac{\Delta(k^{\prime})}{2E(k^{\prime})},$
(14)

where

$\langle
k|V_{sep}^{{}^{1}S_{0}}|k^{\prime}\rangle~{}=~{}-G_{0}p(k)p(k^{\prime}),$ (15)

with Gaussian ansatz

$p(k)~{}=~{}e^{-a^{2}k^{2}}.$ (16)

The two parameters $a$ and $G_{0}$ were adjusted to density dependence of the
gap at the Fermi surface in nuclear matter, calculated with the Gogny force
Tian  _et al._ (2009). After transformation of the pairing force from momentum
to coordinate space one obtains,

$V(\vec{r}_{1},\vec{r}_{2},\vec{r}_{1}^{\prime},\vec{r}_{2}^{\prime})~{}=~{}-G_{0}\delta(\vec{R}_{1}-\vec{R}_{2})G(r)G(r^{\prime})\frac{1-\hat{P}^{\sigma}}{2}$
(17)

where $\vec{r}=1/\sqrt{2}(\vec{r}_{1}-\vec{r}_{2})$ and
$\vec{R}=1/\sqrt{2}(\vec{r}_{1}+\vec{r}_{2})$. $G(r)$ is the Fourier transform
of $p(k)$,

$G(r)~{}=~{}\frac{e^{-r^{2}/(2a^{2})}}{(4\pi a^{2})^{3/2}}.$ (18)

Thus, the pairing force has finite range, and due to the presence of the
factor $\delta(\vec{R}_{1}-\vec{R}_{2})$ it preserves the translational
invariance Tian  _et al._ (2009). Due to coordinate transformation from
laboratory to center of mass system and relative coordinates we need to use
Talmi-Moschinsky brackets,

$|n_{1}l_{1},n_{2}l_{2};\lambda\mu\rangle=\sum_{NLnl}M_{n_{1}l_{1}n_{2}l_{2}}^{NLnl}|NL,nl;\lambda\mu\rangle.$
(19)

The definition of $M^{NLnl}_{n_{1}l_{1}n_{2}l_{2}}$ is given in Refs. Buck and
Merchant (1996); Kamuntavičius  _et al._ (2001). If the transformation matrix
between two laboratory coordinates $\vec{r}_{1}$ and $\vec{r}_{2}$, and center
of mass $\vec{R}$ and relative coordinate $\vec{r}$ is given

$\left(\begin{array}[]{clcr}\vec{R}\\\
\vec{r}\end{array}\right)~{}=~{}\left(\begin{array}[]{clcr}\sqrt{\frac{d}{1+d}}&\sqrt{\frac{1}{1+d}}\\\
\sqrt{\frac{1}{1+d}}&-\sqrt{\frac{d}{1+d}}\end{array}\right)\left(\begin{array}[]{clcr}\vec{r}_{1}\\\
\vec{r}_{2}\end{array}\right),$ (20)

as a function of transformation parameter $d$ Kamuntavičius  _et al._ (2001),
then the most general definition of coefficient
$M^{NLnl}_{n_{1}l_{1}n_{2}l_{2}}$ is given by

$\displaystyle
M^{NLnl}_{n_{1}l_{1}n_{2}l_{2}}(d)~{}=~{}i^{-(l_{1}+l_{2}+L+l)}2^{-(l_{1}+l_{2}+L+l)/4}\sqrt{n_{1}!n_{2}!N!n![2(n_{1}+l_{1})+1]!![2(n_{2}+l_{2})+1]!!}$
$\displaystyle~{}~{}~{}\times\sqrt{[2(N+L)+1]!![2(n+l)+1]!!}\sum_{abcdl_{a}l_{b}l_{c}l_{d}}(-1)^{l_{a}+l_{b}+l_{c}}(-1)^{(l_{a}+l_{b}+l_{c}+l_{d})/2}d^{(2a+l_{a}+2d+l_{d})/2}$
$\displaystyle~{}~{}~{}\times\frac{[(2l_{a}+1)(2l_{b}+1)(2l_{c}+1)(2l_{d}+1)]}{a!b!c!d![2(a+l_{a})+1]!![2(b+l_{b})+1]!![(2(c+l_{c})+1]!![2(d+l_{d})+1]!!}$
$\displaystyle~{}~{}~{}\times(1+d)^{-(2a+l_{a}+2b+l_{b}+2c+l_{c}+2d+l_{d})/2}\langle
l_{a}0l_{c}0|L0\rangle\langle l_{b}0l_{d}0|l0\rangle\langle
l_{a}0l_{b}0|l_{1}0\rangle\langle l_{c}0l_{d}0|l_{2}0\rangle$
$\displaystyle~{}~{}~{}\times\left\\{\begin{array}[]{clcr}l_{a}&l_{b}&l_{1}\\\
l_{c}&l_{d}&l_{2}\\\ L&l&\Lambda\end{array}\right\\}.$ (24)

By employing the basis of spherical harmonic oscillator,

$\tilde{I}_{n}~{}=~{}\sqrt{4\pi}\int
R_{nl}(r)G(r)r^{2}dr~{}=~{}\frac{1}{2^{2/3}\pi^{3/4}b^{3/2}}\frac{(1-\alpha^{2})^{n}}{(1+\alpha^{2})^{n+3/2}}\frac{\sqrt{2n+1}}{2^{n}n!},$
(25)

the coupled matrix element for $T=1$ pairing is given by,

$\displaystyle V_{abcd}^{(pair)JM}~{}=$
$\displaystyle~{}-G_{0}\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\hat{j}_{d}(-1)^{l_{b}+l_{d}+j_{a}+j_{c}}\left\\{\begin{array}[]{clcr}l_{a}&j_{a}&1/2\\\
j_{b}&l_{b}&J\end{array}\right\\}\left\\{\begin{array}[]{clcr}l_{c}&j_{c}&1/2\\\
j_{d}&l_{d}&J\end{array}\right\\}$ (30)
$\displaystyle~{}\sum_{Nnn^{\prime}}\tilde{I}_{n}\tilde{I}_{n^{\prime}}M_{n_{a}l_{a}n_{b}l_{b}}^{NJn0}M_{n_{c}l_{c}n_{d}l_{d}}^{NJn^{\prime}0}.$
(31)

Furthermore, for Gamow-Teller transitions in open shell nuclei we need to
extend Eq. (31) to include both $T=0$ and $T=1$ channels. Therefore, we
introduce natural extension of the pairing:

$\displaystyle V_{abcd}^{(pair)JM}~{}=$
$\displaystyle~{}-G_{0}\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\hat{j}_{d}\sum_{LS}\sum_{T}\frac{1}{2}\left(1+(-1)^{S^{\prime}+T+1}\right)\tilde{f}(S,T)\hat{S}^{2}\hat{L}^{2}\left\\{\begin{array}[]{clcr}l_{b}&1/2&j_{b}\\\
l_{a}&1/2&j_{a}\\\ L&S&J\end{array}\right\\}$ (35)
$\displaystyle~{}\left\\{\begin{array}[]{clcr}l_{d}&1/2&j_{d}\\\
l_{c}&1/2&j_{c}\\\
L&S&J\end{array}\right\\}\sum_{nn^{\prime}}\tilde{I}_{n}\tilde{I}_{n}^{\prime}M_{n_{a}l_{a}n_{b}l_{b}}^{NLn0}M_{n_{c}l_{c}n_{d}l_{d}}^{NLn^{\prime}0}.$
(39)

This is non-vanishing only for $S=0$ and $T=1$ or $S=1$ and $T=0$ pairing.
Therefore, $\tilde{f}(S=0,T=1)~{}=~{}1$ case corresponds to the Eq. (31),
while the case $\tilde{f}(S=1,T=0)~{}=~{}V_{0pp}$. See Appendix B for detailed
derivation. We don’t know a priori the value of the isoscalar proton-neutron
pairing strength parameter $V_{0pp}$. It may be somewhat reduced or enhanced
compared to the case $T=1(S=0)$, which is only present at the RHB level, and
should be deduced from experimental data on excitations or charge-exchange
processes in open shell nuclei.

In comparison to the nuclear ground state based on the RHB, the PN-RQRPA
residual interaction includes an additional channel, described by the
pseudovector term and additional $T=0~{}(S=1)$ pairing term for Gamow-Teller
transitions in open shell nuclei, that are not present in the ground state
calculations. Apart from this, the PN-RQRPA introduced in this work is self-
consistent, i.e., the same interactions, both in the particle-hole and
particle-particle channels, are used in the RHB equation that determines the
canonical quasiparticle basis, and in the PN-RQRPA. In both channels, the same
strength parameters of the interactions are used in the RHB and RQRPA
calculations.

Similar to the previous implementations of the PN-RQRPA Paar  _et al._ (2004),
the two-quasiparticle configuration space includes states with both nucleons
in the discrete bound levels, states with one nucleon in the bound levels and
one nucleon in the continuum, and also states with both nucleons in the
continuum. The RQRPA configuration space also includes pair-configurations
formed from the fully or partially occupied states of positive energy and the
empty negative-energy states from the Dirac sea Paar  _et al._ (2004). As
pointed out in Ref. Paar  _et al._ (2004), the inclusion of configurations
built from occupied positive-energy states and empty negative-energy states is
essential for the consistency of the model, as well as to reproduce the model
independent sum rules.

Model calculations are based on 20 oscillator shells in the RHB model, as
given as standard in Ref. Nikšić  _et al._ (2014) to achieve the convergence
of the results. At the level of the PN-RQRPA calculations no truncations in
the maximal excitation energy for the $2qp$ configuration space are used. We
included a truncation of the $2qp$ configuration space by the condition on the
corresponding occupation factors Paar  _et al._ (2004),
$u_{qp1}v_{qp2}>0.001$, in order to exclude $2qp$ configurations with two
almost empty states. Further reducing of this limit value does not modify the
results.

##  III  Results

###  III.1 The isobaric analog resonance

The PN-R(Q)RPA based on point coupling interactions introduced in Sec. II is
first implemented in the case of $J^{\pi}=0^{+}$ charge-exchange transition,
isobaric analog resonance (IAR). It is induced by the Fermi isospin-flip
operator,

$\hat{T}_{\beta^{\pm}}^{F}=\sum_{i=1}^{A}\tau_{\pm}.$ (40)

Figure 1 shows the transition strength distributions for the IAR in closed
shell nuclei 48Ca, 90Zr and 208Pb, calculated with the PN-RRPA using two
density-dependent point coupling interactions, DD-PCX and DD-PC1, and density-
dependent meson-exchange effective interaction DD-ME2. As expected, for each
nucleus the response to the Fermi operator results in a pronounced single IAR
peak. The IAR peak energy and transition strength display rather moderate
model dependence. The most pronounced spread of the IAR excitation energies
for different interactions, about 1 MeV, is obtained for the heaviest system,
208Pb, while for 48Ca and 90Zr differences are smaller. The results of model
calculations are compared with the experimental data for IAR excitation
energies, denoted by arrows, obtained from $(p,n)$ scattering on 48Ca Anderson
_et al._ (1985a), 90Zr Bainum  _et al._ (1980a); Wakasa  _et al._ (1997a), and
208Pb Akimune  _et al._ (1995a). Good agreement of the PN-RRPA results with
the experimental data is obtained. In all three cases the calculated
transition strengths of the IAR fulfil the Fermi non-energy weighted sum rule,
consistent with Ref. Paar  _et al._ (2004).

Figure 1: The PN-RRPA isobaric analog resonance transition strength
distribution for 48Ca, 90Zr and 208Pb, calculated using DD-PCX, DD-PC1, and
DD-ME2 functionals.

Next we explore the evolution of the IAR within the Sn isotope chain, for
$A=$104$-$132\. In Fig. 3 the IAR transition strength distributions are shown
for representative cases, 112,116,122,130Sn. Model calculations are based on
the PN-RQRPA with DD-PCX interaction. The results without and with the $T=1$
proton-neutron pairing in the residual PN-RQRPA interaction are shown
separately, in comparison to the experimental data from a systematic study of
the (3He,t) charge-exchange reaction in stable Sn isotopes Pham  _et al._
(1995). As shown in Fig. 3, the full PN-RQRPA calculations result in a
pronounced single IAR peak, with the excitation energy that is in excellent
agreement with the experimental data for 112,116,122Sn Pham  _et al._ (1995).
Complete treatment of paring correlations both in the RHB and PN-RQRPA is
essential for description of the IAR Paar  _et al._ (2004). This is
illustrated in Fig. 3, where the strength functions are also shown without
including the $T=1$ proton-neutron residual pairing interaction, i.e., only
the the $ph$-channel of the RQRPA residual interaction is included. Without
the contributions of the $pp$-channel, pronounced fragmentation of the
transition strength is obtained for 112,116Sn, and the excitation energies are
overestimated. By including the attractive proton-neutron pairing interaction,
the transition strength becomes redistributed toward a single pronounced IAR
peak, that is consistent with the expectation of a narrow resonance peak from
the experimental study Pham  _et al._ (1995). More pronounced effect of the
residual pairing interaction is obtained for 112,116,122Sn, while for 130Sn
that is near the neutron closed shell the effect is rather small.

Figure 2: The isobaric analog resonance transition strength distribution for
112,116,122,130Sn, calculated with the PN-RQRPA using DD-PCX interaction.The
results without (dashed line) and with (solid line) proton-neutron pairing in
the residual PN-RQRPA interaction are shown separately, in comparison to the
experimental data from Ref. Pham  _et al._ (1995), denoted by arrows.

Figure 3 shows the evolution of the IAR excitation energy for the isotopic
chain 104-132Sn, with the proton-neutron pairing included in the PN-RQRPA. The
results are shown for the point coupling interactions DD-PCX and DD-PC1. For
comparison, the IAR excitation energies from the previous study based on DD-
ME1 interaction Paar  _et al._ (2004) and the experimental data from Ref. Pham
_et al._ (1995) are displayed. As one can observe in the figure, recently
established interaction DD-PCX reproduces the experimental data with high
accuracy, while the DD-PC1 and DD-ME1 interactions provide systematically
lower energies. We note that DD-PCX parameterization has been established
using additional constraints on nuclear collective transitions that resulted
with improved isovector properties, essential for the description of nuclear
ground state, excitation phenomena, and nuclear matter properties around the
saturation density Yüksel  _et al._ (2019). Clearly, the PN-RQRPA framework
based on DD-PCX interaction introduced in this work represents a considerable
progress in comparison to other approaches.

Figure 3: The PN-RQRPA isobaric analog resonance excitation energy for the
isotope chain 104-132Sn, with the proton-neutron pairing included.
Calculations are based on the point coupling interaction DD-PCX and DD-PC1.
The results from the previous study based on DD-ME1 interaction Paar  _et al._
(2004) and the experimental data from Ref. Pham  _et al._ (1995) are shown for
comparison.

###  III.2 The Gamow-Teller resonance

The Gamow-Teller transitions involve both the spin and isospin degrees of
freedom. In the charge-exchange excitation spectra, these transitions mainly
concentrate in a pronounced resonance peak - Gamow-Teller resonance (GTR),
representing a coherent superposition of $J^{\pi}=1^{+}$ proton-particle –
neutron-hole transitions of neutrons from orbitals with $j=l+\frac{1}{2}$ into
protons in orbitals with $j=l-\frac{1}{2}$. The GT transitions are excited by
the spin-isospin operator

$T_{\beta^{\pm}}^{GT}=\sum_{i=1}^{A}\bm{\Sigma}\tau_{\pm}\;.$ (41)

Figure 4 shows the GT- transition strength distribution for closed shell
nuclei 48Ca, 90Zr and 208Pb, calculated with the PN-RQRPA using DD-PCX and DD-
PC1 interactions. For comparison, the results of the meson exchange functional
with the DD-ME2 parameterization are also shown. The experimental values for
the main GT- peak are denoted with arrows for 48Ca Anderson  _et al._ (1985a),
90Zr Bainum  _et al._ (1980a); Wakasa  _et al._ (1997a), and 208Pb Horen  _et
al._ (1980); Akimune  _et al._ (1995a); Krasznahorkay  _et al._ (2001a). Since
the PN-RQRPA excitation energies are given with respect to the mother nucleus,
the experimental values given with respect to daughter nucleus are modified by
adding the mass difference between the daughter and mother isotopes as well as
the mass difference between neutron and proton which is missing when the GT is
built on the basis of p-h excitations Colò  _et al._ (1994); Yüksel  _et al._
(2020a). The transition strength distributions are dominated by the main
Gamow-Teller resonance peak, that is composed from direct spin-flip
transitions, ($j=l+\frac{1}{2}$ $\rightarrow$ $j=l-\frac{1}{2}$). In addition,
pronounced low-energy GT- strength is obtained, composed from the core-
polarization spin-flip ($j=l\pm\frac{1}{2}$ $\rightarrow$
$j=l\pm\frac{1}{2}$), and back spin-flip transitions ($j=l-\frac{1}{2}$
$\rightarrow$ $j=l+\frac{1}{2}$). As it is well known, quantitative
description of the low-energy GT- strength is essential in modelling beta
decay half-lives Marketin  _et al._ (2007, 2016). When using three different
effective interactions as shown in Fig. 4, the spread of values of the GT-
excitation energies within $\approx$ 1 MeV is obtained.

Table 1: Calculated and experimental GT- properties, strength contributions
for both GT- and GT+ channel, i.e. $\sum B(\text{GT}^{-})$ and $\sum
B(\text{GT}^{+})$, and total transition strengths $\sum
B(\text{GT}^{-})-B(\text{GT}^{+})$ for 48Ca, 90Zr and 208Pb, for DD-PCX and
DD-PC1 interactions. The total transition strengths are given in percentages
of the Ikeda sum rule value $3(N-Z)$ Ikeda  _et al._ (1963), and contributions
from the negative energy states of the Dirac sea are shown. The experimental
values for 48Ca

are from Ref. Anderson  _et al._ (1985a); Yako  _et al._ (2009), 90Zr from
Refs. Bainum  _et al._ (1980b); Wakasa  _et al._ (1997b); Raywood  _et al._
(1990); Condé  _et al._ (1992), and 208Pb from Ref. Akimune  _et al._ (1995b);
Krasznahorkay  _et al._ (2001b).

|  |  48Ca |  90Zr |  208Pb
---|---|---|---|---
Experiment |  $E_{1}$ ($E_{x}$) [MeV] | 8.3 ($\approx 10.5$ Anderson  _et al._ (1985a)) | 15.5 | 19.2
| FWHM1 | 1.5 | 3.8 | 4.1
|  $E_{2}$ [MeV] | 10.9 | - | -
| FWHM2 | 3.9 | - | -
| $\sum B(\text{GT}^{-}+\text{IVSM}^{-})$ |  $15.3\pm 2.2$ Yako  _et al._ (2009) |  $34.2\pm 1.6$ Wakasa  _et al._ (1997b) | -
| $\sum B(\text{GT}^{-})$ | - |  $28.0\pm 1.6$Wakasa  _et al._ (1997b) | -
| $\sum B(\text{GT}^{+}+\text{IVSM}^{+})$ |  $2.8\pm 0.3$ Yako  _et al._ (2009) | - | -
| $\sum B(\text{GT}^{+})$ |  $1.9\pm 0.5$ Yako  _et al._ (2009) |  |  $1.0\pm 0.3$Raywood  _et al._ (1990)
---
($1.7\pm 0.2$)Condé  _et al._ (1992)
-
| $3(N-Z)$ | 24 | 30 | 132
|  $\sum B(\text{GT}^{-})-B(\text{GT}^{+})$ [%] |  |  $(52\pm 9)$%
---
($\approx 70\%$ Anderson  _et al._ (1985a))
$(90\pm 5)\%$ Wakasa  _et al._ (1997b) |  $60-70\%$ Akimune  _et al._ (1995b)
DD-PCX |  $E_{x}$ [MeV] | 10.61 | 16.73 | 19.21
| $\sum B(\text{GT}^{-})$ | 27.34 | 38.99 | 147.33
| $\sum B(\text{GT}^{+})$ | 3.34 | 8.89 | 15.33
|  $\sum B(\text{GT}^{-})-B(\text{GT}^{+})$ [%] | 99.96% | 100.33% | 99.99%
| Dirac sea [%] | 6.48% | 7.90% | 8.23%
DD-PC1 |  $E_{x}$ [MeV] | 10.98 | 16.54 | 19.21
| $\sum B(\text{GT}^{-})$ | 27.09 | 37.21 | 145.92
| $\sum B(\text{GT}^{+})$ | 3.10 | 7.05 | 13.93
|  $\sum B(\text{GT}^{-})-B(\text{GT}^{+})$ [%] | 99.97% | 100.50% | 99.99%
| Dirac sea [%] | 5.93% | 7.46% | 7.48%

While the strength parameter of the pseudovector channel in the residual
PN-R(Q)RPA interactions is constrained by the GTR excitation energy in 208Pb,
reasonable agreement with experimental data is obtained for 48Ca and 90Zr
without additional adjustments of the effective interaction. As it has already
been discussed in previous studies, the Ikeda sum rule for GT transition
strength Ikeda  _et al._ (1963) is fully reproduced in a complete calculation
that includes both the configurations formed from occupied states in the Fermi
sea and empty negative-energy states in the Dirac sea Kurasawa  _et al._
(2003a, b); Ma  _et al._ (2004); Kurasawa and Suzuki (2004); Paar  _et al._
(2004). Table 1 shows the summary of the GTR properties for 48Ca, 90Zr and
208Pb for DD-PCX and DD-PC1 interactions: centroid excitation energies, the
total transition strength difference $\sum B(\text{GT}^{-})-B(\text{GT}^{+})$
in comparison to the Ikeda sum rule Ikeda  _et al._ (1963) (in $\%$),
contributions of the Dirac sea states to the sum rule (in $\%$), and the
respective experimental values Anderson  _et al._ (1985b); Bainum  _et al._
(1980b); Wakasa  _et al._ (1997b); Akimune  _et al._ (1995b); Yako  _et al._
(2009). While the calculations accurately exhaust the Ikeda sum rule values,
obtained total transition strengths in 48Ca and 90Zr for GT- channel are
somewhat larger than experimental ones for both point coupling interactions.
After subtraction of the estimated IVSM contribution (usually $\sim 10\%$) the
values of total GT- strengths are even lower Yako  _et al._ (2009). In the GT+
channel for 48Ca the difference between theoretical and experimental values is
reduced, i.e. $\sum B(\text{GT}^{+})=3.3$ ($3.0$) for DD-PCX (DD-PC1) and it
is very close to the experimental value of $\sum
B(\text{GT}^{+}+\text{IVSM}^{-})=2.8$ ($\sum B(\text{GT}^{+})=1.9$ without
IVSM). However, the experimental strengths in 48Ca may be significantly
underestimated for higher excitation energies above 15 MeV in the GT- spectrum
and above 8 MeV in the GT+ spectrum Yako  _et al._ (2009). The largest
discrepancy in the GT+ channel is observed for 90Zr, where calculated values
are at least few times greater than the highest estimates of experimental GT+
strengthsCondé  _et al._ (1992). Other nonrelativistic RPA approaches, such as
Extended RPA theories account also somewhat larger value of $\sum
B(\text{GT}^{-})$ (usually 10-20%) in lower part of 48Ca excitation spectrum
($E_{x}\lesssim 20$ MeV)Rijsdijk  _et al._ (1993). Both dressed and extended
RPA theories overestimate total experimental strengths in 90Zr for GT- channel
by $20-60\%$ for $E_{x}\lesssim 25$ MeV. However, in these calculations
significant percentage of the sum rule for both GT- and GT+ may be found for
excitation energies above $E_{x}\gtrsim$ 40 MeV Rijsdijk  _et al._ (1993),
which is not the case in our calculations. The nonrelativistic QRPA + PVC
calculations also overestimate the total experimental strengths in 48Ca (71%
of the RPA+PVC strength) and 208Pb (63% of the RPA+PVC strength) Niu  _et al._
(2014), still not providing the explanation for the missing experimental
strength. The contributions of the GT transitions to the empty Dirac sea in
doubly magic nuclei is from $6.5-8.2\%$ ($5.9-7.5\%$) of the total strength
for DD-PCX (DD-PC1), in agreement with the previous studies Paar  _et al._
(2004).

Figure 4: The GT- strength distribution for 48Ca, 90Zr and 208Pb, calculated
using the DD-PCX, DD-PC1, and DD-ME2 functionals. The experimental values of
the main GT- peak for 48Ca Anderson  _et al._ (1985a), 90Zr Wakasa  _et al._
(1997a) and 208Pb Akimune  _et al._ (1995a); Pham  _et al._ (1995); Horen  _et
al._ (1980) are denoted with arrows. Figure 5: The GT- strength distribution
for 112,116,122,130Sn, calculated with the PN-RQRPA using DD-PCX interaction,
for the range of values of the isoscalar proton-neutron pairing interaction
strength parameter, $V_{0}$=0-1.3. Experimental values for the main GT- peaks
for 112Sn, 116Sn and 122Sn are denoted with arrows Pham  _et al._ (1995).

In the following, the PN-RQRPA based on point coupling interaction DD-PCX is
employed in the study of GT- transitions in Sn istotope chain. For open shell
nuclei, in addition to the separable pairing interaction included in the RHB,
the PN-RQRPA residual interaction also includes the isoscalar proton-neutron
pairing as introduced in Sec. II. Since this pairing interaction channel is
not present within the RHB, its strength parameter $V_{0}$ can be constrained
by the experimental data, e.g., on GT- excitation energies or beta decay half-
lives. Rather than providing the optimal value of $V_{0}$, in the present
analysis we explore the pairing properties and sensitivity of the GT-
transitions by systematically varying $V_{0}$. In Fig.5, the GT- strength
distribution is shown for the isotopes 112,116,122,130Sn, calculated with the
PN-RQRPA using DD-PCX interaction. The isoscalar proton-neutron pairing
interaction strength parameter is varied within the range of $V_{0}$=0-1.3
MeV. At higher excitation energies, a pronounced GT resonance peak is
obtained, except for 122Sn, where the main peak is split. In all cases,
pronounced low-energy GT- strength is obtained, spreading over the energy
range of $\approx$ 5 MeV. While the direct spin-flip transitions ($\nu
j=l+\frac{1}{2}$ $\rightarrow$ $\pi j=l-\frac{1}{2}$) dominate the high-energy
region, the low-energy strength is dominated by core-polarization spin-flip
($\nu j=l\pm\frac{1}{2}$ $\rightarrow$ $\pi j=l\pm\frac{1}{2}$), and back
spin-flip ($\nu j=l-\frac{1}{2}$ $\rightarrow$ $\pi j=l+\frac{1}{2}$)
transitions. Considering the dependence of the GT- strength on the $T=0$
proton-neutron pairing, one can observe in Fig.5 that the main GTR peak
appears insensitive to this pairing interaction channel. In the low-energy
region, pronounced sensitivity of the GT- spectra on $V_{0}$ is obtained, i.e.
with increased $V_{0}$ the strength is shifted toward lower energies. This
result is in agreement with previous studies based on different effective
interactions both in the $ph$ and $pp$ channels Paar  _et al._ (2004).

Table 2: Calculated values of GTR properties for 112Sn, 116Sn, 122Sn and 130Sn with DD-PCX and DD-PC1 interactions and $T=0$ pairing strengths $V_{0}~{}=~{}0.7$ and $V_{0}~{}=~{}1.3$. The contributions of strengths in the GT- and GT+ channels are shown in separate columns. The total strength of GTR is also represented in percentages of the Ikeda sum rule value, $3(N-Z)$ Ikeda  _et al._ (1963). Contributions to total GTR strength from transitions to negative energy states of Dirac sea are also shown. |  | $3(N-Z)$ |  | Sum rule %
---|---|---|---|---
$\sum B(\text{GT}^{-})$ | $\sum B(\text{GT}^{+})$ | $\sum\left(B(\text{GT}^{-})-B(\text{GT}^{+})\right)$ | total | Dirac sea
DD-PCX |  112Sn | 36 | 47.22 | 11.68 | 35.53 | 98.69% | 7.36%
$(V_{0}=0.7)$ |  116Sn | 48 | 57.77 | 9.99 | 47.77 | 99.53% | 7.84%
|  122Sn | 66 | 75.26 | 9.34 | 65.91 | 99.87% | 7.87%
|  130Sn | 90 | 99.06 | 9.12 | 89.94 | 99.94% | 7.76%
DD-PC1 |  112Sn | 36 | 45.55 | 9.84 | 35.71 | 99.21% | 7.11%
$(V_{0}=0.7)$ |  116Sn | 48 | 57.03 | 9.26 | 47.77 | 99.53% | 7.85%
|  122Sn | 66 | 74.41 | 8.47 | 65.94 | 99.91% | 7.17%
|  130Sn | 90 | 98.23 | 8.26 | 89.96 | 99.96% | 7.01%
DD-PCX |  112Sn | 36 | 45.68 | 10.15 | 35.52 | 98.69% | 7.36%
$(V_{0}=1.3)$ |  116Sn | 48 | 56.79 | 7.59 | 49.20 | 99.53% | 7.85%
|  122Sn | 66 | 74.96 | 9.04 | 65.91 | 99.87% | 7.87%
|  130Sn | 90 | 99.01 | 9.06 | 89.94 | 99.94% | 7.76%
DD-PC1 |  112Sn | 36 | 44.67 | 8.95 | 35.71 | 99.21% | 7.12%
$(V_{0}=1.3)$ |  116Sn | 48 | 56.29 | 8.46 | 47.82 | 99.64% | 7.20%
|  122Sn | 66 | 74.26 | 8.32 | 65.94 | 99.91% | 7.18%
|  130Sn | 90 | 98.20 | 8.23 | 89.96 | 99.96% | 7.02%

In Tab. 2 the calculated GT strengths for the 112,116,122,130Sn isotopes are
shown for DD-PCX and DD-PC1 functionals, using $T~{}=~{}0$ strength parameter
set to $V_{0}~{}=~{}0.7$ and $V_{0}~{}=~{}1.3$. The Ikeda sum rule Ikeda  _et
al._ (1963) is reasonably well reproduced.

In Tab. 3 calculated values of moments $m_{0}(\text{GT}^{-})$ are shown for
the first three dominant GT- peaks, with the corresponding excitation energies
$m_{1}$/$m_{0}$ with respect to the IAS for 112Sn, 116Sn, 122Sn and 130Sn.
Calculations include the DD-PCX functional and $T=0$ pairing strength
$V_{0}~{}=~{}1.3$. We note that there are few methods one can use to constrain
the value of parameter $V_{0}$. One of them is based on comparison of
experimental and theoretical values of relative positions of the GT peaks with
respect to the IAS. As shown in Fig. 5, the number, position and intensity of
GT peaks strongly depend on the value of parameter $V_{0}$. In the GTR
spectrum of Sn isotopes one can observe two less intensier distinguishable
peaks for $V_{0}\lesssim 0.4$ in the lower part of spectrum (GT2 and (GT3),
besides one (GT1) or two dominant peaks (GT1a and GT1b), which is
characteristic for $A\gtrsim 122$. For values $V_{0}\gtrsim 0.4$ usually the
third peak (GT4) starts to show up, while for values $V_{0}\gtrsim 0.7$ we
have in general three smaller peaks in lower part of spectrum that may be
distinguished. The position of these peaks and respective transition strengths
are strongly influenced by $T~{}=~{}0$ pairing, i.e. with higher strength
$V_{0}$ their positions move towards lower energies. Furthermore, the strength
distribution is also modified, i.e. GT3 and GT4 peaks become more dominant
than the GT2 ones for values $V_{0}\gtrsim 1.3$. The influence of $T~{}=~{}0$
pairing becomes suppressed when approaching the magic neutron number N=82, as
observed in the GTR spectrum of 130Sn. Experimental values of GT positions and
cross sections obtained from Sn(3He,t)Sb reactions (see Ref. Pham  _et al._
(1995)) have some hierarchy, with descending values of cross sections
corresponding to each peak as the peak position is moved toward lower
energies, with minor exception of GT4 which intensity is similar to the GT3
one or somewhat larger. Therefore we may impose some constraints on the value
of $V_{0}$, which should be at least $\approx 0.7$ in order to reproduce the
GTR properties observed in experiments.

Table 3:  Calculated values of moments $m_{0}$ and corresponding energy $m_{1}/m_{0}$ with respect to the IAS for the first four dominant GT- peaks for 112Sn, 116Sn, 122Sn and 130Sn. The DD-PCX functional and $T=0$ pairing strengths $V_{0}~{}=~{}0.7$ and $V_{0}~{}=~{}1.3$ are used. The experimental results for the difference between GTR and IAS, i.e. $\Delta^{\text{exp}}~{}=~{}E_{\text{GTR}}^{\text{exp}}-E_{\text{IAS}}^{\text{exp}}$ and differential cross sections ${d\sigma}/{d\Omega}$ are taken from Ref. Pham  _et al._ (1995). †For 122Sn, the moments $m_{0}$ and $m_{1}/m_{0}$ contain contributions from two most dominant peaks.  |  | ——— $V_{0}~{}=~{}0.7$ ——— | ——— $V_{0}~{}=~{}1.3$ ———
---|---|---|---
| $\Delta^{\text{exp}}$ | ${d\sigma}/{d\Omega}$ | $m_{0}$ | $m_{1}/m_{0}$ | $\left(\frac{m_{1}}{m_{0}}-E_{\text{IAS}}^{\text{teo}}\right)$ | $m_{0}$ | $m_{1}/m_{0}$ | $\left(\frac{m_{1}}{m_{0}}-E_{\text{IAS}}^{\text{teo}}\right)$
|  | [MeV] | [mb/sr] |  | [MeV] | [MeV] |  | [MeV] | [MeV]
GT1 |  112Sn | 2.78 | 12.4 | 19.11 | 18.70 | 4.85 | 17.69 | 18.63 | 4.78
|  116Sn | 1.68 | 16.8 | 20.51 | 17.19 | 3.45 | 19.71 | 17.15 | 3.41
|  122Sn† | 1.01 | 21.9 | 31.68 | 16.71 | 3.14 | 29.62 | 15.82 | 2.25
|  130Sn | - | - | 50.03 | 14.22 | 0.83 | 48.55 | 14.12 | 0.73
GT2 |  112Sn | -2.08 | 4.9 | 6.30 | 13.27 | -0.58 | 3.69 | 13.05 | -0.80
|  116Sn | -3.32 | 6.2 | 10.02 | 11.85 | -1.89 | 6.54 | 11.61 | -2.13
|  122Sn | -4.59 | 9.2 | 14.26 | 9.94 | -3.63 | 11.27 | 9.78 | -3.79
|  130Sn | - | - | 15.75 | 7.53 | -5.86 | 16.07 | 7.40 | -5.99
GT3 |  112Sn | -3.67 | 1.5 | 5.17 | 11.31 | -2.54 | 2.54 | 11.06 | -2.79
|  116Sn | -5.18 | 2.1 | 3.04 | 10.64 | -3.10 | 8.71 | 9.52 | -4.22
|  122Sn | -7.87 | 6.3 | 10.27 | 8.33 | -5.24 | 12.81 | 7.87 | -5.70
|  130Sn | - | - | 9.74 | 5.87 | -7.52 | 10.90 | 5.80 | -7.59
GT4 |  112Sn | -4.83 | 2.0 | 3.14 | 10.71 | -3.14 | 8.94 | 9.01 | -4.84
|  116Sn | -6.52 | 2.5 | 2.05 | 8.12 | -5.62 | 6.65 | 7.33 | -6.41
|  122Sn | -9.79 | 1.2 | 3.32 | 5.86 | -7.71 | 5.64 | 5.59 | -7.98

In Fig. 6 the PN-RQRPA results for the GT- direct spin-flip transition
excitation energy centroid with the respect to the IAR energy are shown for
the chain of even-even isotopes 104-132Sn. The available experimental data
obtained from Sn(3He,t)Sb charge-exchange reactions are shown for comparison
Pham  _et al._ (1995). The PN-RQRPA calculations are performed for the range
of values of the isoscalar proton-neutron pairing interaction strength
parameter, $V_{0}$=0.4-2.5. The point coupling interaction DD-PCX is used and
the results based on the DD-ME2 interaction are also displayed Paar  _et al._
(2004). We noticed almost linear decrease of differences between the GTR and
IAS energies in Sn isotope chain as function of A or N (while keeping $Z=50$
fixed). Similar observations may be found in nonrelativistic calculations with
Skyrme parametrizations in Fracasso and Colò (2007). One can observe that the
proton-neutron pairing interaction systematically reduces the GT-IAR energy
splittings along the Sn isotope chain, resulting in good agreement for
$V_{0}\approx$2.5. We note that relatively higher values of $V_{0}$ are
required to reproduce the GT-IAR energy splittings. In Refs. Vretenar  _et
al._ (2003); Paar  _et al._ (2004), it has been emphasized that the energy
difference between the GTR and the IAS reflects the magnitude of the effective
spin-orbit potential. As one can see in Fig. 6, the GT-IAR energy splittings
reduce in neutron-rich Sn isotopes toward zero value, reflecting considerable
reduction of the spin-orbit potential and the corresponding increase of the
neutron skin thickness $r_{n}-r_{p}$ Vretenar  _et al._ (2003). Therefore, as
pointed out in Ref. Vretenar  _et al._ (2003), the energy difference $E_{\rm
GT}-E_{\rm IAS}$, obtained from the experiment, could be used to determine the
value of neutron skin thickness in a consistent framework that can
simultaneously describe the charge-exchange excitation properties and the
$r_{n}-r_{p}$ value, such as the RHB+PN-RQRPA approach.

In Ref. Yüksel  _et al._ (2020b) the GT- transition strength for 118Sn was
analyzed in more details, including the finite temperature effects, using the
Skyrme functional SkM∗. In the present study, for 118Sn without T=0 pairing
and DD-PCX interaction ($V_{0}$=0), we obtain pronounced low-lying GT- states
at energies E=7.32, 8.88, 9.64, 10.21, and 11.45 MeV with the corresponding
transition strengths B(GT-)=1.21, 3.25, 1.12, 1.14, and 14.30. The main GTR
peaks are obtained at E=16.46 and 19.83 MeV with B(GT-)=19.91 and 7.69,
respectively. The excitation energies are comparable with the respective
values obtained for the SkM∗ interaction in Ref. Yüksel  _et al._ (2020b),
E(GT-)= 6.2, 8.9, 10.5, 16.3 and 20.2 MeV Yüksel  _et al._ (2020b). Study in
Ref. Yüksel  _et al._ (2020b) also showed that relativistic calculations of
GT- transitions in 118Sn with density dependent meson-exchange interaction DD-
ME2 lead to the splitting between major (GT1a) and energetically higher
somewhat less intensive peak (GT1b), that exists only for $V_{0}\lesssim 200$
MeV in $T=0$ pairing channel. For $V_{0}\approx 200$ MeV the two peaks start
to merge, while for $V_{0}\gtrsim 240$ MeV they are completely replaced by one
somewhat stronger peak shifted $\approx 0.5$ MeV towards higher energies.
However, nonrelativistic calculations in Refs. Yüksel  _et al._ (2020b);
Fracasso and Colò (2007) show different behavior in GT- spectrum of 118Sn with
respect to $T=0$ pairing strength, i.e. the two dominant peaks never merge,
even for large values of overall strength of the $T=0$ pairing ($\approx 800$
MeV). The same feature is obtained in the present study, as shown in Fig. 5
for 122Sn. We note that relativistic calculation in Ref. Yüksel  _et al._
(2020b) used another pairing interaction in $T=0$ channel than the non-
relativistic one, i.e. two Gaussians with different relative strengths to
cover short and medium distances in coordinate space introduced separately for
the PN-RQRPA. This caused additional interplay between attractive and
repulsive terms which act differently for the short and medium ranges.
Therefore, behavior of the GTR spectrum obtained in our study is more similar
to the nonrelativistic calculations in Refs. Yüksel  _et al._ (2020b);
Fracasso and Colò (2007), where the zero range density dependent surface
pairing in both isospin channels is introduced separately in the PN-RQRPA.

Figure 6: The PN-RQRPA excitation energy for GT- direct spin-flip transitions
for 104-132Sn, for the range of values of the isoscalar proton-neutron pairing
interaction strength parameter $V_{0}$. Calculations are based on the density
dependent point coupling interaction DD-PCX and the results from the previous
study based on the DD-ME2 interaction Paar  _et al._ (2004) and the
experimental data from Ref. Pham  _et al._ (1995) are shown for comparison.

##  IV Conclusion

In this work we have formulated a consistent framework for description of
nuclear charge-exchange transitions based on the proton-neutron relativistic
quasiparticle random phase approximation in the canonical single-nucleon basis
of the relativistic Hartree-Bogoliubov model, using density-dependent
relativistic point coupling interactions. The implementation of recently
established DD-PCX interactionYüksel  _et al._ (2019), adjusted not only to
the nuclear ground state properties, but also with the symmetry energy of the
nuclear equation of state and the incompressibility of nuclear matter
constrained using collective excitation data, allows improved description of
ground-state and excitation properties of nuclei far from stability, that is
important for the studies of exotic nuclear structure and dynamics, as well as
for applications in nuclear astrophysics. The introduced formalism based on
point coupling interactions is also relevant from the practical point of view,
because it allows efficient systematic large-scale calculations of nuclear
properties and processes of relevance for the nucleosynthesis and stellar
evolution modeling. In the current formulation of the RHB+PN-RQRPA, the
pairing correlations are implemented by the separable pairing force Tian  _et
al._ (2009) that allows accurate and efficient calculations of the pairing
properties. The PN-RQRPA includes both the $T=1$ and $T=0$ pairing channels.
While the $T=1$ channel corresponds to the pairing interaction constrained at
the ground state level, $T=0$ proton-neutron pairing strength parameter can be
determined from the experimental data on charge-exchange transitions and beta
decay half lives. In order to validate the PN-R(Q)RPA framework introduced in
this work, spin and isospin excitations \- isobaric analog resonances and
Gamow-Teller transitions - have been investigated in several closed-shell
nuclei and Sn isotope chain. The results show very good agreement with the
experimental data, representing an improvement compared to previous studies
based on the relativistic nuclear energy density functionals. When compared to
other theoretical approaches that usually underestimate the IAR excitation
energies in Sn isotope chain, the present model using DD-PCX interaction
accurately reproduces the experimental data. Therefore, the framework
introduced in this work represents an important contribution for the future
studies of astrophysically relevant nuclear excitations and weak interaction
processes, in particular $\beta$-decay in neutron-rich nuclei, electron
capture in presupernova stage of massive stars, and neutrino-nucleus reactions
relevant for the synthesis of elements in the universe and stellar evolution.
Especially important is the extension of theory framework introduced in this
work to include both the finite temperature effects together with nuclear
pairing and apply it for the description of electron capture and beta decay at
finite temperature characteristic for stellar environment Yüksel  _et al._
(2017, 2020a). This work is currently in progress Ravlić  _et al._ (2020b, a).

## Acknowledgments

Stimulating discussions with Ante Ravlić are gratefully acknowledged. This
work is supported by the QuantiXLie Centre of Excellence, a project co
financed by the Croatian Government and European Union through the European
Regional Development Fund, the Competitiveness and Cohesion Operational
Programme (KK.01.1.1.01). Y. F. N. acknowledges support from the National
Natural Science Foundation of China under Grant No. 12075104

##  Appendix A Particle-hole matrix elements

For the isovector-pseudovector part of particle-hole channel in the PN-RQRPA
we use the contact type of interaction with constant coupling,

$V_{PV}=-\alpha_{PV}[\gamma_{5}\ \gamma^{\nu}\vec{\tau}]_{1}\
[\gamma_{5}\gamma_{\nu}\vec{\tau}]_{2}\ \delta(\vec{r}_{1}-\vec{r}_{2}).$ (42)

The uncoupled matrix element of isovector-pseudovector interaction is given by

$\langle a\ b|V|c\ d\rangle~{}=~{}-\alpha_{PV}\int
drr^{2}\sum_{L\lambda}\bigg{[}\int dr_{1}r_{1}^{2}\int
d\Omega_{1}\bar{\psi}_{a}({r}_{1},\Omega_{1})[\gamma_{5}\
\gamma^{\nu}\vec{\tau}]_{(1)}Y_{L\lambda}^{*}(\Omega_{1})\\\
~{}\delta(r-r_{1})\psi_{c}({r}_{1},\Omega_{1})\bigg{]}\bigg{[}\int
dr_{2}r_{2}^{2}\int d\Omega_{2}\bar{\psi}_{b}(r_{2},\Omega_{2})[\gamma_{5}\
\gamma_{\nu}\vec{\tau}]_{(2)}Y_{L\lambda}(\Omega_{2})\ \delta(r-r_{2})\
\psi_{d}({r}_{2},\Omega_{2})\bigg{]},$ (43)

where we used expansion of delta function in spherical coordinates Ring and
Schuck (2004),

$\displaystyle\delta(\vec{r}_{1}-\vec{r}_{2})~{}$
$\displaystyle=~{}\frac{\delta(r_{1}-r_{2})}{r_{1}r_{2}}\sum_{\lambda\mu}Y_{\lambda\mu}^{*}(\Omega_{1})Y_{\lambda\mu}(\Omega_{2})$
(44) $\displaystyle=~{}\int
drr^{2}\frac{\delta(r-r_{1})\delta(r-r_{2})}{r^{2}r_{1}r_{2}}\sum_{\lambda\mu}Y_{\lambda\mu}^{*}(\Omega_{1})Y_{\lambda\mu}(\Omega_{2}).$
(45)

Indices $a$, $b$, $c$ and $d$ refer to all quantum numbers involved, while the
Dirac’s conjugate is defined in standard way as

$\bar{\psi}_{a}=\psi_{a}^{+}\ \gamma_{0}.$ (46)

If we ignore for a moment the isospin part of the wave function (corresponding
quantum numbers isospin $t$ and its third projection $t_{0}$) one can rewrite
Eq. (43) in the following form:

$\langle am_{a}\ bm_{b}|V|cm_{c}\
dm_{d}\rangle^{\text{PV}}~{}=-\alpha_{PV}\int
drr^{2}\sum_{\lambda\mu}(-1)^{\mu}\sum_{\nu}\left[Q_{\lambda\mu}^{\nu}(r,\Omega_{1})\right]_{ac}\left[Q_{\lambda-\mu;\nu}(r,\Omega_{2})\right]_{bd}$
(47)

where the index $\nu$ refers to Dirac matrix $\gamma_{\nu}$ ($\gamma^{\nu}$)
in the vertex and we used (see Ref. Greiner and Bromley (2000) for the
sherical case of spinor):

$\displaystyle\left[Q_{\lambda\mu}^{\nu}(r,\Omega_{i})\right]_{am_{a}\
cm_{c}}~{}=$ $\displaystyle~{}\int
d\Omega_{i}\left(f_{n_{a}\kappa_{a}}(r)\Omega_{\kappa_{a}m_{a}}^{\dagger}(\Omega_{i})~{}~{}-ig_{n_{a}\kappa_{a}}(r)\Omega_{\bar{\kappa}_{a}m_{a}}^{\dagger}(\Omega_{i})\right)[\gamma_{0}\gamma_{5}\
\gamma^{\nu}]_{(i)}$ $\displaystyle~{}\times
Y_{\lambda\mu}(\Omega_{i})\left(\begin{array}[]{clcr}f_{n_{c}\kappa_{c}}(r)\Omega_{\kappa_{a}m_{c}}(\Omega_{i})\\\
ig_{n_{c}\kappa_{c}}(r)\Omega_{\bar{\kappa}_{c}m_{c}}(\Omega_{i})\end{array}\right),$
(50)

where the subscripts $1$ and $2$ refer to $\Omega_{\text{i =
1,2}}=\left(\theta_{i},\phi_{i}\right)$, while radial parts of bi-spinors
$f(r)$ and $g(r)$ are assumed to be real in our case. From Eq. (50) one can
easily obtain timelike

$\displaystyle\left[Q_{\lambda\mu}^{0}(r,\Omega_{i})\right]_{am_{a}\
cm_{c}}~{}=$ $\displaystyle~{}-\bigg{[}i\ f_{n_{a}\kappa_{a}}(r)\
g_{n_{c}\kappa_{c}}(r){\langle(1/2\ l_{a})j_{a}m_{a}|\
Y_{\lambda\mu}(\Omega_{i})|(1/2\bar{l_{c}})j_{c}m_{c}\rangle}$
$\displaystyle-i\ g_{n_{a}\kappa_{a}}(r)\
f_{n_{c}\kappa_{c}}(r){\langle(1/2\bar{l_{a}})j_{a}m_{a}|\
Y_{\lambda\mu}(\Omega_{i})|(1/2l_{c})j_{c}m_{c}\rangle}\bigg{]}$ (51)

and spacelike components ($k=1,2$ and $3$)

$\displaystyle\left[Q_{\lambda\mu}^{k}(r,\Omega_{i})\right]_{am_{a}\
cm_{c}}~{}=$
$\displaystyle~{}-\bigg{[}f_{n_{a}\kappa_{a}}(r)f_{n_{c}\kappa_{c}}(r)\langle(1/2l_{a})j_{a}m_{a}|\sigma_{k}Y_{\lambda\mu}(\Omega_{1})|(1/2l_{c})j_{c}m_{c}\rangle$
$\displaystyle~{}+~{}g_{n_{a}\kappa_{a}}(r)g_{n_{c}\kappa_{c}}(r)\langle(1/2\bar{l}_{a})j_{a}m_{a}|\sigma_{k}Y_{\lambda\mu}(\Omega_{1})|(1/2\bar{l}_{c})j_{c}m_{c}\rangle\bigg{]}.$
(52)

In order to couple particle and hole operator, i.e. states, into good total
angular momentum J (and projection M) in the matrix elements of the residual
particle-hole two body interaction, one needs to start from Suhonen (2007)

$\displaystyle|(ph^{-1})JM\rangle~{}$
$\displaystyle=~{}\left[c_{p}^{\dagger}h_{h}^{\dagger}\right]_{JM}|0\rangle$
(53)
$\displaystyle=~{}\sum_{m_{p}m_{h}}C_{j_{p}m_{p}j_{h}m_{h}}^{JM}c_{pm_{p}}^{\dagger}h_{hm_{h}}^{\dagger}|0\rangle$
(54)
$\displaystyle=\sum_{m_{p}m_{h}}(-1)^{j_{h}-m_{h}}C_{j_{p}m_{p}j_{h}-m_{h}}^{JM}a_{pm_{p}}^{\dagger}a_{h\
m_{h}}|0\rangle,$ (55)

from which $JJ$-coupled particle-hole matrix element follows directly

$\displaystyle\langle ac^{-1}|V|b^{-1}d\rangle^{JM}~{}=$
$\displaystyle~{}\sum_{m_{a}m_{c}}(-1)^{j_{c}-m_{c}}C_{j_{a}\ m_{a}\ jc\
-m_{c}}^{J\ M}\sum_{m_{b}m_{d}}(-1)^{j_{b}-m_{b}}C_{j_{d}\ m_{d}\ j_{b}\
-m_{b}}^{J\ M}$ $\displaystyle~{}~{}\langle am_{a}\ bm_{b}|V|cm_{c}\
dm_{d}\rangle.$ (56)

Using the Wigner-Eckart theorem for angular parts in Eqs. (51) and (52) after
some manipulations with Clebsch-Gordan coefficients one obtains the $ph$
matrix elements in $JJ$-coupled form. Therefore, after including isospin part
of the matrix elements, which gives a factor of 2, which we have ignored for
the moment, for the timelike part of pseudovector coupling we obtain:

$\displaystyle V_{abcd}^{(PVt)J}$
$\displaystyle=\frac{2\alpha_{PV}}{2J+1}\langle(1/2l_{a})j_{a}||Y_{J}(\Omega_{1})||(1/2\bar{l}_{c})j_{c}\rangle\langle(1/2{l}_{d})j_{d}||Y_{J}(\Omega_{2})||(1/2\bar{l}_{b})j_{b}\rangle$
$\displaystyle~{}~{}\int
drr^{2}\bigg{[}f_{a}(r)g_{c}(r)-g_{a}(r)f_{c}(r)\bigg{]}\bigg{[}f_{b}(r)g_{d}(r)-g_{b}(r)f_{d}(r)\bigg{]}$
(57)

and for spacelike part

$\displaystyle V_{abcd}^{PV(s)J}~{}=~{}\frac{2\alpha_{PV}}{2J+1}\sum_{L}\int
drr^{2}\bigg{[}f_{a}(r)f_{c}(r)\langle(1/2\
l_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{c})j_{c}\rangle$
$\displaystyle~{}+~{}g_{a}(r)g_{c}(r)\langle(1/2\
\bar{l}_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{c})j_{c}\rangle\bigg{]}\bigg{[}f_{b}(r)f_{d}(r)\langle(1/2\
l_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{b})j_{b}\rangle$
$\displaystyle~{}+~{}g_{b}(r)g_{d}(r)\langle(1/2\
\bar{l}_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{b})j_{b}\rangle\bigg{]},$ (58)

where the reducible angular part of matrix elements can be written in terms of
3jm symbols:

$\langle(1/2l_{a})j_{a}||Y_{J}||(1/2l_{c})j_{c}\rangle~{}=~{}\frac{1+(-1)^{l_{a}+l_{c}+J}}{2}\frac{\hat{j}_{a}\hat{j}_{c}\hat{J}}{\sqrt{4\pi}}(-1)^{j_{a}-1/2}\left(\begin{array}[]{clcr}j_{a}&J&j_{c}\\\
-1/2&0&1/2\end{array}\right),$ (59)

and

$\langle(1/2l_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2l_{c})j_{c}\rangle~{}=~{}\frac{1+(-1)^{l_{a}+l_{c}+J}}{2}\frac{\hat{j}_{a}\hat{j}_{c}\hat{J}\hat{L}}{\sqrt{4\pi}}(-1)^{l_{a}+L}\Bigg{[}(-1)^{l_{c}+j_{c}+1/2}\\\
~{}\left(\begin{array}[]{clcr}1&L&J\\\
0&0&0\end{array}\right)\left(\begin{array}[]{clcr}j_{a}&J&j_{c}\\\
-1/2&0&1/2\end{array}\right)-\sqrt{2}\left(\begin{array}[]{clcr}1&L&J\\\
-1&0&1\end{array}\right)\left(\begin{array}[]{clcr}j_{a}&J&j_{c}\\\
1/2&-1&1/2\end{array}\right)\Bigg{]}.$ (60)

Note that the timelike pseudovector matrix elements are non-zero only in the
case of unnatural parity transitions, like Gamow-Teller transition. The
isovector-vector part of two-body interaction is given by

$V_{V}=\alpha_{V}\left[\rho_{V}(r_{1})\right][\gamma^{\mu}\vec{\tau}]_{1}\
[\gamma_{\mu}\vec{\tau}]_{2}\ \delta(\vec{r}_{1}-\vec{r}_{2}).\\\ $ (61)

There are no additional rearrangement terms due to isospin restriction of the
PN-R(Q)RPA, i.e. changing the nucleon from neutron to proton or vice versa and
properties of the point coupling functional itself. Therefore, one should
start from Eq. (61) and follow the same procedure described before, from Eqs.
(43) to (58), replacing Eqs. (51) and (52) with

$\displaystyle\left[Q^{k}_{\lambda\mu}(r,\Omega_{i})\right]_{am_{a}\
cm_{c}}~{}=~{}\bigg{[}if_{n_{a}\kappa_{a}}(r)g_{n_{c}\kappa_{c}}(r)\langle(1/2\
l_{a})j_{a}m_{a}|\sigma_{k}Y_{\lambda\mu}(\Omega_{i})|(1/2\
\bar{l}_{c})j_{c}m_{c}\rangle$ $\displaystyle-
ig_{n_{a}\kappa_{a}}(r)f_{n_{c}\kappa_{c}}(r)\langle(1/2\
\bar{l}_{a})j_{a}m_{a}|\sigma_{k}Y_{\lambda\mu}(\Omega_{i})|(1/2l_{c})j_{c}m_{c}\rangle\bigg{]}$
(62)

and

$\displaystyle\left[Q^{0}_{\lambda\mu}(r,\Omega_{i})\right]_{am_{a}\
cm_{c}}~{}=$
$\displaystyle~{}\bigg{[}f_{n_{a}\kappa_{a}}(r)f_{n_{c}\kappa_{c}}(r)\langle(1/2\
l_{a})j_{a}m_{a}|Y_{\lambda\mu}(\Omega_{i})|(1/2\ {l}_{c})j_{c}m_{c}\rangle$
$\displaystyle~{}+~{}g_{n_{a}\kappa_{a}}(r)g_{n_{c}\kappa_{c}}(r)\langle(1/2\
\bar{l}_{a})j_{a}m_{a}|Y_{\lambda\mu}(\Omega_{i})|(1/2\
\bar{l}_{c})j_{c}m_{c}\rangle\bigg{]}$ (63)

in order to obtain $JJ$-coupled $ph$ matrix elements for this channel, i.e.
spacelike part

$\displaystyle V_{abcd}^{(TVs)J}~{}=~{}\frac{2}{2J+1}\sum_{L}\int
drr^{2}\alpha_{TV}\left[\rho_{V}(r)\right]\bigg{[}f_{a}(r)g_{c}(r)\langle(1/2\
l_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ \bar{l}_{c})j_{c}\rangle$
$\displaystyle~{}-g_{a}(r)f_{c}(r)\langle(1/2\
\bar{l}_{a})j_{a}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
l_{c})j_{c}\rangle\bigg{]}\bigg{[}f_{b}(r)g_{d}(r)\langle(1/2\
\bar{l}_{d})||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\ l_{b})j_{b}\rangle$
$\displaystyle~{}-g_{b}(r)f_{d}(r)\langle(1/2\
l_{d})j_{d}||\left[\sigma_{S}Y_{L}\right]_{J}||(1/2\
\bar{l}_{b})j_{b}\rangle\bigg{]}$ (64)

and timelike part

$\displaystyle V_{abcd}^{(TVt)J}~{}=$ $\displaystyle~{}\frac{2}{2J+1}\int
drr^{2}\alpha_{TV}\left[\rho_{V}(r)\right]\bigg{[}f_{a}(r)f_{c}(r)+g_{a}(r)g_{c}(r))\bigg{]}\bigg{[}f_{b}(r)f_{d}(r)+g_{b}(r)g_{d}(r)\bigg{]}$
$\displaystyle~{}\times\langle(1/2\ l_{a})j_{a}||Y_{J}||(1/2\
l_{c})j_{c}\rangle\langle(1/2\ l_{d})j_{d}||Y_{J}||(1/2\ l_{b})j_{b}\rangle.$
(65)

##  Appendix B Natural extension of separable pairing

In order to extend the standard $S~{}=~{}0$ ($T~{}=~{}1$) pairing, which was
used in the RHB and RQRPA, one needs to start from general expression for
particle-particle matrix element in the uncoupled form:

$V_{abcd}~{}=~{}\langle
n_{a}(1/2l_{a})j_{a}~{}n_{b}(1/2l_{b})j_{b}|V\left(1-\hat{P}^{r}\hat{P}^{\sigma}\hat{P}^{\tau}\right)|n_{c}(1/2l_{c})j_{c}~{}n_{d}(1/2l_{d})j_{d}\rangle$
(66)

where $\hat{P}^{r}$, $\hat{P}^{\sigma}$ and $\hat{P}^{\tau}$ represent the
exchange of relative spatial coordinate, spin and isospin, respectively. For
action of the spin exchange operator $\hat{P}^{\sigma}$ we simply have

$P^{\sigma}\left.|(1/2_{(1)}1/2_{(2)})SM_{S}\right\rangle~{}=~{}(-1)^{S+1}\left.|(1/2_{(1)}1/2_{(2)})SM_{S}\right\rangle,$
(67)

which affects just the order of coupling, i.e. the phase $(-1)^{S+1}$ of
symmetry transformation of the Clebsch-Gordan coefficient. Furthermore, one
may construct $P^{\sigma}$ mathematically in the following way:

$\hat{P}_{\sigma}~{}=~{}\frac{1+\vec{\sigma}_{1}\cdot\vec{\sigma}_{2}}{2},$
(68)

and isospin exchange operator in analogous way. However, the action of
$\hat{P}^{r}$ affects only relative spatial coordinates,

$\displaystyle\hat{P}^{r}|\left.NLM_{L}\right\rangle~{}=~{}|\left.NLM_{L}\right\rangle,$
(69)
$\displaystyle\hat{P}^{r}|\left.nlm_{l}\right\rangle~{}=~{}(-1)^{l}|\left.nlm_{l}\right\rangle,$
(70)

Therefore, as first step one needs to do the $LS$ recoupling in bra and ket
independently

$V_{abcd}^{\text{(pair)}J}~{}=~{}\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\hat{j}_{d}\sum_{\lambda
S}\sum_{\lambda^{\prime}S^{\prime}}\hat{\lambda}\hat{S}\hat{\lambda^{\prime}}\hat{S^{\prime}}\left\\{\begin{array}[]{clcr}s_{a}&l_{a}&j_{a}\\\
s_{b}&l_{b}&j_{b}\\\
S&\lambda&J\end{array}\right\\}\left\\{\begin{array}[]{clcr}s_{c}&l_{c}&j_{c}\\\
s_{d}&l_{d}&j_{d}\\\
S^{\prime}&\lambda^{\prime}&J\end{array}\right\\}\sum_{TM_{T}}\sum_{T^{\prime}M_{T^{\prime}}}C_{1/2-1/21/21/2}^{TM_{T}}\\\
C_{1/2-1/21/21/2}^{T^{\prime}M_{T^{\prime}}}\sum_{M_{S}M_{S}^{\prime}}C_{SM_{S}\lambda\mu}^{JM}C_{S^{\prime}M_{S}^{\prime}\lambda^{\prime}\mu^{\prime}}^{JM}\langle(1/2\
1/2)TM_{T}|\langle(s_{a}s_{b})SM_{S}(l_{a}l_{b})\lambda\mu|V(1-\hat{P}^{r}\hat{P}^{\sigma}\hat{P}^{\tau})\\\
|(s_{c}s_{d})S^{\prime}M_{S}^{\prime}(l_{c}l_{d})\lambda^{\prime}\mu^{\prime}\rangle|(1/2\
1/2)T^{\prime}M_{T^{\prime}}\rangle.$ (71)

In the second step one needs to substitute $V$ with generic separable form,
similar to Eq. (17) but now without any kind of projection,

$V(\vec{r}_{1},\vec{r}_{2},\vec{r}_{1}^{\prime},\vec{r}_{2}^{\prime})~{}=~{}-G_{0}\delta(\vec{R}_{1}-\vec{R}_{2})G(r)G(r^{\prime})$
(72)

into Eq. (71) and transform the laboratory coordinates $\vec{r}_{1}$ and
$\vec{r}_{2}$ ($\vec{r}_{1}^{\prime}$ and $\vec{r}_{2}^{\prime}$) in bra (ket)
into the center of mass $\vec{R}$ ($\vec{R}^{\prime}$) and relative
coordinates $\vec{r}$ ($\vec{r}^{\prime}$), i.e. the so-called Talmi-
Moschinsky transformation. Therefore we need to evaluate

$\displaystyle\sum_{TM_{T}}\sum_{T^{\prime}M_{T^{\prime}}}$ $\displaystyle
C_{1/2-1/21/21/2}^{TM_{T}}C_{1/2-1/21/21/2}^{T^{\prime}M_{T^{\prime}}}\langle(1/2\
1/2)TM_{T}|\langle(1/2\ 1/2)SM_{S}(l_{a}l_{b})\lambda\mu|$ $\displaystyle
V(1-\hat{P}^{r}\hat{P}^{\sigma}\hat{P}^{\tau})|(1/2\
1/2)S^{\prime}M_{S^{\prime}}(l_{c}l_{d})\lambda^{\prime}\mu^{\prime}\rangle|(1/2\
1/2)T^{\prime}M_{T^{\prime}}\rangle~{}=~{}$ $\displaystyle~{}=$
$\displaystyle-4\pi
G_{0}\delta_{SS^{\prime}}\delta_{M_{S}M_{S^{\prime}}}\delta_{\lambda
L}\delta_{\mu
m_{L}}\delta_{\lambda^{\prime}L}\delta_{\mu^{\prime}m_{L^{\prime}}}\sum_{NLnn^{\prime}}I_{n}I_{n^{\prime}}M_{n_{a}l_{a}n_{b}l_{b}}^{NLn0}M_{n_{c}l_{c}n_{d}l_{d}}^{NLn^{\prime}0}$
$\displaystyle\sum_{T}\frac{1}{2}\left(1+(-1)^{S^{\prime}+T+1}\right)$ (73)

Constraining ourself to the proton-neutron case only, note that the only
nonvanishing cases of the isospin coupling are $T=0$ and $M_{T}=0$ or $T=1$
and $M_{T}=0$, which both lead to factor $1/2$ in front of the round brackets
in Eq. (71). Therefore, substituting back Eq. (73) into Eq. (71) after some
mathematical manipulations we obtain

$\displaystyle V_{abcd}^{JM}~{}=$
$\displaystyle~{}-G_{0}\hat{j}_{a}\hat{j}_{b}\hat{j}_{c}\hat{j}_{d}\sum_{LS}\sum_{T}\frac{1}{2}\left(1+(-1)^{S^{\prime}+T+1}\right)\tilde{f}(S,T)\hat{S}^{2}\hat{L}^{2}\left\\{\begin{array}[]{clcr}l_{b}&1/2&j_{b}\\\
l_{a}&1/2&j_{a}\\\
L&S&J\end{array}\right\\}\left\\{\begin{array}[]{clcr}l_{d}&1/2&j_{d}\\\
l_{c}&1/2&j_{c}\\\ L&S&J\end{array}\right\\}$ (80)
$\displaystyle~{}~{}~{}\sum_{nn^{\prime}}\tilde{I}_{n}\tilde{I}_{n}^{\prime}M_{n_{a}l_{a}n_{b}l_{b}}^{NLn0}M_{n_{c}l_{c}n_{d}l_{d}}^{NLn^{\prime}0}.$
(81)

However, we also add multiplication with function $\tilde{f}(S,T)$ which
should take into account somewhat smaller or enhanced effect of the pairing in
$T~{}=~{}0$ case,

$\tilde{f}(S,T)~{}=~{}\begin{cases}1,~{}{\text{for}~{}S=0~{},~{}T=1}\\\
V_{0pp},~{}{\text{for}~{}S=1~{},~{}T=0}\\\ 0,~{}{\text{the rest}}\end{cases}$
(82)

while $\tilde{I}_{n}~{}=~{}\sqrt{4\pi}I_{n}$ is spatial integral of Gaussian
$G(r)$ derived analytically in the next section of Appendix.

##  Appendix C Analytical solution of radial part

Radial part of the eigenfunction of spherical 3D harmonic oscillator is given
by

$R_{nl}(r,b_{0})~{}=~{}b_{0}^{-3/2}R_{nl}(\xi^{2})~{}=~{}b_{0}^{-3/2}N_{nl}\xi^{l}L_{n}^{l+1/2}(\xi^{2})e^{-\xi^{2}/2},$
(83)

where $b_{0}$ represents oscillatory length, while $n$ corresponds to the
nodes number (in this notation we don’t take into account 0 and $\infty$).
Possible values are $n=0,1,2...$, and $\xi=r/b_{0}$. Normalization factor
$N_{nl}$ is given by

$N_{nl}~{}=~{}\left(\frac{2n!}{\Gamma(l+n+3/2)}\right)^{1/2}$ (84)

In the case of half-number arguments gamma function is given by

$\Gamma\left(\frac{1}{2}+n\right)~{}=~{}\frac{1\cdot 3\cdot
5\cdots(2n-1)}{2^{n}}\sqrt{\pi}~{}=~{}\frac{(2n-1)!!}{2^{n}}\sqrt{\pi},$ (85)

We are interested in the analytical solution of the following integral

$I_{n}~{}=~{}\int R_{nl}(r)G(r)r^{2}dr$ (86)

By inserting G(r) from Eq. (18) in Eq. (86) we obtain

$I_{n}~{}=~{}\frac{b_{0}^{3/2}N_{n0}}{(4\pi
a^{2})^{3/2}}\int_{0}^{\infty}\exp\left[-\frac{\xi^{2}}{2}\left(\frac{b_{0}^{2}}{a^{2}}+1\right)\right]L_{n}^{1/2}(\xi^{2})\xi^{2}d\xi.$
(87)

After substitution

$\alpha^{2}~{}=~{}\frac{a^{2}}{b_{0}^{2}},~{}~{}\xi^{2}=\eta\rightarrow\xi
d\xi=\frac{d\eta}{2},$ (88)

Eq. (87) can be rewritten as

$I_{n}~{}=~{}\frac{b_{0}^{-3/2}N_{n0}}{(4\pi\alpha^{2})^{3/2}}\frac{1}{2}\int_{0}^{\infty}\exp\left[-\eta+\frac{\eta}{2}\left(1-\frac{1}{\alpha^{2}}\right)\right]L_{n}^{(1/2)}(\eta)\sqrt{\eta}d\eta.$
(89)

Generating function for Laguerre polynomials with $\alpha~{}=~{}1/2$
($l~{}=~{}0$) is given by expression Abramowitz and Stegun (1965)

$\frac{1}{(1-z)^{3/2}}\exp\left(\frac{xz}{z-1}\right)~{}=~{}\sum_{n=0}^{\infty}L_{n}^{1/2}(x)z^{n}.$
(90)

By using substitution

$\frac{1}{2}\left(1-\frac{1}{\alpha^{2}}\right)~{}=~{}\frac{z}{z-1}~{}\rightarrow~{}z~{}=~{}\frac{1-{\alpha^{2}}}{1+{\alpha^{2}}}$
(91)

and Eq. (90), we can rewrite Eq. (89) in the following form,

$I_{n}~{}=~{}\frac{b_{0}^{-3/2}N_{n0}}{(4\pi\alpha^{2})^{3/2}}\frac{1}{2}(1-z)^{3/2}\sum_{m=0}^{\infty}z^{m}\int_{0}^{\infty}d\eta\eta^{1/2}e^{-\eta}L_{n}^{(1/2)}(\eta)L_{m}^{(1/2)}(\eta).$
(92)

The integral in Eq.(92) is just orthogonality condition for Laguerre’s
polynomials Abramowitz and Stegun (1965),

$\int_{0}^{\infty}x^{\alpha}e^{-\alpha}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx~{}=~{}\frac{\Gamma(n+\alpha+1)}{n!}\delta_{nm}.$
(93)

Therefore, Eq. (92) reduces to

$I_{n}~{}=~{}~{}\frac{1}{2}\frac{b_{0}^{-3/2}N_{n0}}{(4\pi\alpha^{2})^{3/2}}(1-z)^{3/2}\sum_{m=0}^{\infty}z^{m}\frac{\Gamma(m+3/2)}{m!}\delta_{nm}.$
(94)

Using Eq. (85), inserting normalization factor (Eq. (84)) and expressing
everything in terms of $\alpha$ and $n$, after some mathematical
manipulations, it can further be simplified to

$I_{n}~{}=~{}\frac{1}{2^{5/2}\pi^{5/4}b_{0}^{3/2}}\frac{(1-\alpha^{2})^{n}}{(1+\alpha^{2})^{n+3/2}}\frac{\sqrt{(2n+1)!}}{2^{n}n!}.$
(95)

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