# Theoretical analysis and predictions for the double electron capture of
124Xe

O. Niţescu  S. Ghinescu  V. A. Sevestrean  M. Horoi  F. Šimkovic  S. Stoica
<EMAIL_ADDRESS>
###### Abstract

We provide a complete theoretical description of the two-neutrino electron
capture in 124Xe, improving both the nuclear and the atomic structure
calculations. We improve the general formalism through the use of the Taylor
expansion method, leading to higher order terms in the decay rate of the
process. The nuclear part is treated with pn-QRPA and interacting shell model
(ISM) methods. The nuclear matrix elements (NMEs) are calculated with the pn-
QRPA method with spin restoration by fixing the input parameters so that the
experimental decay rate is reproduced, resulting in values significantly lower
than in previous calculations. The validity of the pn-QRPA NMEs is tested by
showing their values to be comparable with the ones for double-beta decay with
emission of two electrons of 128,130Te, which have similar pairing features.
Within the ISM, we reproduce the total experimental half-life within a factor
of two and predict the capture fraction to the KK channel of about 74%. We
also predict the capture fractions to other decay channels and show that for
the cumulative decay to the $\rm{KL_{1}}$-$\rm{KO_{1}}$ channels, a capture
fraction of about 24% could be observed experimentally. On the atomic side,
calculations are improved by accounting for the Pauli blocking of the decay of
innermost nucleon states and by considering all $s$-wave electrons available
for capture, expanding beyond the K and L1 orbitals considered in privious
studies. We also provide improved atomic relaxation energies of the final
atomic states of 124Te, which may be used as input for background modeling in
liquid Xenon experiments.

###### keywords:

double electron capture , nuclear structure , nuclear shell model , QRPA

††journal: Physics Letters B

[1]Faculty of Mathematics, Physics and Informatics, Comenius University in
Bratislava, 842 48 Bratislava, Slovakia [2]International Centre for Advanced
Training and Research in Physics, P.O. Box MG12, 077125 Magurele, Romania
[3]“Horia Hulubei” National Institute of Physics and Nuclear Engineering, 30
Reactorului, POB MG-6, RO-077125 Magurele, Romania

[4]Department of Physics, University of Bucharest, 405 Atomistilor, POB MG-11,
RO-077125 Magurele, Romania

[6]Department of Physics, Central Michigan University, Mount Pleasant, MI
48859, USA

[5]Institute of Experimental and Applied Physics, Czech Technical University
in Prague, 110 00 Prague, Czech Republic

##  1 Introduction

The study of double-beta decay (DBD) continues to be of great interest due to
its potential to yield novel, essential information about various subjects,
including the nuclear structure, the validity of some conservation laws, and
still-unknown properties of neutrinos. The experiments have so far largely
focused on measuring DBD with the emission of two electrons
($2\nu\beta^{-}\beta^{-}$), measured for a dozen nuclei, as a result of their
larger $Q$-values compared to other similar transitions and due to their
higher natural abundance [1, 2, 3, 4, 5, 6, 7]. This heightened attention to
DBD is largely attributed to the potential existence of the decay mode without
emission of neutrinos ($0\nu\beta^{-}\beta^{-}$), whose discovery would open a
rich avenue of investigation towards physics beyond the Standard Model (SM)
[8, 9, 10].

Besides these transitions, there are three other DBD modes less investigated
so far due to their lower transition probabilities, the double-positron
emitting ($2\nu/0\nu\beta^{+}\beta^{+}$) mode, the single electron-capture
with coincident positron emission ($2\nu/0\nu\textrm{EC}\beta^{+}$) mode and
the double electron capture ($2\nu/0\nu\textrm{ECEC}$) mode. However, in the
recent past, the ECEC mode has become attractive as well, since in addition to
being energetically favored compared to the other two, a resonant transition
can occur ($\textrm{R}0\nu\textrm{ECEC}$) if the initial and final states are
energetically degenerate, which may increase the transition probability by
several orders of magnitude [11]. Therefore, their sensitivity to probe
neutrinoless transitions can reach that of the $2\nu\beta^{-}\beta^{-}$
decays, so they are considered as an alternative venue to the discovery of
neutrinoless decay mode. Double-electron captures with the emission of two
neutrinos have been predicted for several isotopes but only measured in 78Kr,
124Xe, 130Ba and 132Ba [12]. Particularly, 124Xe is an interesting case since
it has the largest $Q$-value of all the isotopes that can undergo ECEC
transitions. It is currently investigated in large experiments together with
Dark Matter (DM) and DBD that use detectors with large-volume liquid-xenon
[13, 14, 15, 16, 17].

The experimental measurements also require a reliable theoretical support,
therefore there are several calculations of the ECEC decay rates and half-
lives of 124Xe [18, 19, 20], where the main goal focused on obtaining a more
reliable treatment of the nuclear part. In [18, 19], the half-lives were
calculated for the three possible channels, $\beta^{+}\beta^{+}$,
$\beta^{+}\rm{EC}$ and ECEC. For the double-capture transitions, only captures
of electrons from the K and L1 atomic shells were included in the calculation.
The phase space factors (PSFs) were calculated using the formalism from [21],
while the nuclear part was calculated with the pn-QRPA method (see [18, 19]
and Refs. therein). An important finding was that the calculated NME values
are close for all three channels, so the PSFs are decisive in establishing the
magnitude of the half-lives. In [20], the $2\nu\rm{ECEC}$ half-lives were
calculated using two different methods for the NME calculation, the effective
theory (ET) and large-scale Interacting Shell Model (ISM). However, the
electron capture processes involve important atomic effects, which were not
taken into account in the previous calculations and are relevant for an
accurate analysis of the experimental data.

In this work, we provide a complete description of the $2\nu\rm{ECEC}$ in
124Xe. We improve the general formalism through the use of the Taylor
expansion method [22, 23], where we include terms up to the fourth power in
the lepton energies in the derivation of the decay rate. Aside from increased
rigor, this method allows the definition of new NME ratios,
$\xi_{31}^{2\nu\textrm{ECEC}}$ and $\xi_{51}^{2\nu\textrm{ECEC}}$, whose
measurements could provide additional insight regarding the interplay between
low- and higher-lying states in the intermediate nucleus involved in the NME
calculations. Besides this general improvement, we re-examine the calculations
of both the atomic and nuclear parts. For the former, we employ the Dirac-
Hartree-Fock-Slater (DHFS) self-consistent framework to obtain the electron
wave functions. The atomic screening effect is accounted for more rigorously
and with better accuracy than in the previous works [21, 24, 25]. The diffuse
nuclear surface correction, realistic nuclear charge density, and electron
exchange-correlation effects are also taken into account within our atomic
structure calculations. We further improve the PSF calculation by accounting
for the Pauli blocking of the decay of innermost nucleon states and by
considering all $s$-wave electrons available for capture, expanding beyond the
K and L1 orbitals considered in prior studies. We also compute the atomic de-
excitation energies within the DHFS framework, which, along with the capture
fractions, can be used as input for background modeling in liquid xenon
experiments.

For the nuclear part we use the ISM [26, 27, 28, 29, 30, 31] and the pn-QRPA
with isospin restoration [32] methods. The 2$\nu$ECEC NMEs, as the
2$\nu\beta\beta$ NMEs, can only be calculated reliably if one includes the sum
on the excited $1^{+}$ states of the intermediate nucleus (see Eqs. (4)),
which is possible in the ISM and pn-QRPA approaches. Methods that cannot
calculate the full sum on the intermediate $1^{+}$ states rely on closure
approximation or single-state dominance approximation, which rarely provides
reliable results. In the ISM one can replace the direct sum on intermediate
states with a strength function approach, which converges much quicker to the
exact result. A full description of the strength function approach within the
ISM framework can be found in section 4 of Ref. [31]. This method is extended
in this work to higher powers in the denominators (see Eqs. (4)) necessary for
the Taylor expansion terms. The new ISM NMEs and the improved PSFs are used to
predict capture fractions for 2$\nu$ECEC channels that were not yet observed.
In the analysis we considered two often-used effective Hamiltonians, which
provide consistent nuclear structure results, thus reassuring of the
reliability of the calculated NMEs and of the newly predicted capture
fractions.

We also compute the NMEs with the pn-QRPA method with isospin restoration by
fixing the particle-particle strength parameter to reproduce the experimental
half-life. We found that our NME value is much smaller when compared with the
results of previous pn-QRPA calculations [18, 19].

We organize the manuscript as follows: Section 2 provides a description of the
improved $2\nu$ECEC capture formalism, including the most relevant equations.
Section 3 presents our results of the $2\nu$ECEC process of 124Xe, including
the refined PSFs, the atomic relaxation energies, the NMEs obtained with the
ISM and pn-QRPA, and the total and partial half-lives predictions. The
conclusions are summarized in Section 4.

##  2 Improved two-neutrino double electron capture formalism

We improve the $2\nu\textrm{ECEC}$ formalism through the use of a Taylor
expansion method. Following the same steps as for $2\nu\beta^{-}\beta^{-}$
decay [22, 23] and including terms up to the fourth power in the lepton
energies, we obtain the total inverse half-life of the $2\nu\textrm{ECEC}$
process in a similar form:

$\displaystyle\begin{aligned}
\left[T^{2\nu\rm{ECEC}}_{1/2}\right]^{-1}&=\left(g^{\rm
eff}_{A}\right)^{4}\left|M^{2\nu\rm{ECEC}}_{GT-1}\right|^{2}\left\\{G_{0}^{2\nu\rm{ECEC}}\right.\\\
&+\left.\xi^{2\nu\rm{ECEC}}_{31}G_{2}^{2\nu\rm{ECEC}}+\frac{1}{3}\left(\xi^{2\nu\rm{ECEC}}_{31}\right)^{2}G_{22}^{2\nu\rm{ECEC}}\right.\\\
&+\left.\left[\frac{1}{3}\left(\xi^{2\nu\rm{ECEC}}_{31}\right)^{2}+\xi^{2\nu\rm{ECEC}}_{51}\right]G_{4}^{2\nu\rm{ECEC}}\right\\},\end{aligned}$
(1)

The partial inverse half-life for $2\nu\textrm{xy}$ process, where the atomic
electrons are only captured from shells $\rm{x}$ and $\rm{y}$, can be written
as:

$\displaystyle\begin{aligned}
\left[T^{2\nu\rm{xy}}_{1/2}\right]^{-1}=&\left(2-\delta_{\rm{xy}}\right)\left(g^{\rm
eff}_{A}\right)^{4}\left|M^{2\nu\rm{ECEC}}_{GT-1}\right|^{2}\left\\{G_{0}^{2\nu\rm{xy}}+\right.\\\
&+\left.\xi^{2\nu\rm{ECEC}}_{31}G_{2}^{2\nu\rm{xy}}+\frac{1}{3}\left(\xi^{2\nu\rm{ECEC}}_{31}\right)^{2}G_{22}^{2\nu\rm{xy}}+\right.\\\
&+\left.\left[\frac{1}{3}\left(\xi^{2\nu\rm{ECEC}}_{31}\right)^{2}+\xi^{2\nu\rm{ECEC}}_{51}\right]G_{4}^{2\nu\rm{xy}}\right\\}.\end{aligned}$
(2)

Here, $g^{\rm eff}_{A}$ is an effective axial coupling constant and the
parameters:

$\xi^{2\nu\rm{ECEC}}_{31}=\frac{M^{2\nu\rm{ECEC}}_{GT-3}}{M^{2\nu\rm{ECEC}}_{GT-1}},\hskip
14.22636pt\xi^{2\nu\rm{ECEC}}_{51}=\frac{M^{2\nu\rm{ECEC}}_{GT-5}}{M^{2\nu\rm{ECEC}}_{GT-1}}$
(3)

are NME ratios arising from the Taylor expansion, with

$\displaystyle\begin{aligned}
M^{2\nu\rm{ECEC}}_{GT-1}&=\sum_{n}M^{2\nu}_{GT}(n)\frac{m_{e}}{E_{n}(1^{+})-(E_{i}+E_{f})/2},\\\
M^{2\nu\rm{ECEC}}_{GT-3}&=\sum_{n}M^{2\nu}_{GT}(n)\frac{4~{}m_{e}^{3}}{\left(E_{n}(1^{+})-(E_{i}+E_{f})/2\right)^{3}},\\\
M^{2\nu\rm{ECEC}}_{GT-5}&=\sum_{n}M^{2\nu}_{GT}(n)\frac{16~{}m_{e}^{5}}{\left(E_{n}(1^{+})-(E_{i}+E_{f})/2\right)^{5}}.\end{aligned}$
(4)

The summations are over all $1^{+}$ states of the intermediate nucleus and
$M^{2\nu}_{GT}(n)$ matrix elements depend on the $n^{\rm th}$ $1^{+}$
intermediate state, with energy $E_{n}(1^{+})$, as well as on the ground
states $|0^{+}_{i}\rangle$ and $|0^{+}_{f}\rangle$ of the initial and final
nuclei, with the energies $E_{i}$ and $E_{f}$:

$M^{2\nu}_{GT}(n)=\langle
0^{+}_{f}\|\sum_{m}\tau^{-}_{m}\sigma_{m}\|1^{+}_{n}\rangle\langle
1^{+}_{n}\|\sum_{m}\tau^{-}_{m}\sigma_{m}\|0^{+}_{i}\rangle,$ (5)

where $\tau^{-}_{m}$ is the isospin-lowering operator transforming a proton
into a neutron, and $\sigma_{m}$ is the nucleon spin operator.

The PSFs entering the inverse half-lives are given by

$\displaystyle\begin{aligned}
G_{N}^{2\nu\rm{ECEC}}&=\frac{m_{e}(G_{F}\left|V_{ud}\right|m_{e}^{2})^{4}}{2\pi^{3}\ln{(2)}}\frac{1}{m_{e}^{5}}\sum_{\rm{x,y}}\mathcal{B}_{\mathrm{x}}^{2}\mathcal{B}_{\mathrm{y}}^{2}\mathcal{I}_{N,\rm{xy}}\\\
&=\sum_{\rm{x,y}}G_{N}^{2\nu\rm{xy}},\end{aligned}$ (6)

where $G_{F}$ is the Fermi coupling constant, $V_{ud}$ the first element of
the Cabibbo-Kobayashi-Maskawa (CKM) matrix and the summations run over all
occupied atomic shells of the initial atom. The functions
$\mathcal{I}_{N,\rm{xy}}$ depend on the total energies of the electrons from
the initial atom’s orbitals $\rm{x}$ and $\rm{y}$, denoted as $e_{\textrm{x}}$
and $e_{\textrm{y}}$, respectively. Their expressions can be obtained from
Eqs. A4 and A5 of Ref. [23] by making the replacements, $E_{e_{1}}\rightarrow-
e_{\textrm{x}}$ and $E_{e_{2}}\rightarrow-e_{\textrm{y}}$. The localization
probability of an electron from shell $\rm{x}$ inside the nucleus, is given
by:

$\mathcal{B}^{2}_{\rm{x}}=\frac{1}{4\pi m_{e}^{3}}\left[\langle
g_{\rm{x}}\rangle^{2}+\langle f_{\rm{x}}\rangle^{2}\right],$ (7)

where

$\langle g_{\rm{x}}\rangle=\frac{\int
g_{\rm{x}}(r)\rho(r)r^{2}dr}{\int\rho(r)r^{2}dr}\hskip 28.45274pt\langle
f_{\rm{x}}\rangle=\frac{\int f_{\rm{x}}(r)\rho(r)r^{2}dr}{\int\rho(r)r^{2}dr}$
(8)

Here, $g_{\rm{x}}(r)$ and $f_{\rm{x}}(r)$ are, respectively, the large- and
small-component of the radial wave functions for the bound electron in shell
x, and the nuclear charge distribution normalized to $Z$ is given by:

$\rho(r)=\frac{1}{1+e^{\left(r-c_{\rm rms}\right)/a}}$ (9)

where the surface thickness $a=0.545$ fm was used and we found that
$c_{\rm{rms}}=5.569$ fm reproduces the experimental mean square radius,
$\sqrt{\langle r^{2}\rangle}=4.7661$ fm, for 124Xe [33]. The average of the
bound wave function, weighted with the nuclear charge distribution, accounts
for the Pauli blocking of the decay of innermost nucleon states while
preserving the intuitive picture of the electron being captured on the nuclear
surface.

##  3 Results and Discussions for the $2\nu\rm{ECEC}$ decay of
${}^{124}\rm{Xe}$

###  3.1 Atomic structure description, PSF calculation and atomic relaxation
energies

Accurate computation of the PSFs for the $2\nu\textrm{ECEC}$ process in 124Xe
and the determination of atomic relaxation energies for the final atom, 124Te,
requires precise atomic structure calculations for these atomic systems. For
this purpose, we employed the Dirac-Hartree-Fock-Slater (DHFS) self-consistent
framework. The nuclear, electronic and exchange components of the DHFS
potential and the convergence of the method were described in detail in [34,
35], based on the RADIAL subroutine package [36], that we also employ in this
paper. In a relativistic framework, the shells referred to as $\rm{x}$ or
$\rm{y}$, in previous sections, can be uniquely identified by the state
$n\kappa$, where $n$ represents the principal quantum number and $\kappa$
corresponds to the relativistic quantum number.

Usually, in the $2\nu$ECEC process, it is considered that the electrons are
captured from K shell and L1 subshell [21, 24, 25]. In particular for 124Xe
double electron capture process, even the electrons from $O$ shell have a
small but non-zero probability to be captured. In this paper, we assume that
the neutrinos are emitted with anti-parallel spins and that the electrons can
be captured from all occupied shells of the initial atom, so the summations in
Eq. 6 extend up to $O_{1}$ subshell. We have considered only the subshells
with $\kappa=-1$. Captures in which not all four leptons are in $s$-wave
states are strongly suppressed by additional terms in the NMEs [21]. This may
influence the interpretation of experimental results. For example, in [15],
the signal model uses captures from all combinations of subshells. In our
model, each subshell has only two electrons, hence there are two pairs of
electrons that can be captured for any given $\rm{x}\neq\rm{y}$ subshells.
This is accounted for in Eq. (6) by not requiring $\rm{x}<\rm{y}$.

The results of the PSFs, obtained using the Taylor expansion formalism for the
$2\nu\text{ECEC}$ process of 124Xe, are summarized in Table 1 according to Eq.
(6). The first line corresponds to the total PSFs, while the subsequent lines
display results for the first few most probable channels. As anticipated, with
the order of the Taylor expansion increasing, the values of the PSFs decrease.
This trend aligns with data for the case of $2\nu\beta^{-}\beta^{-}$ decay
[22, 23]. At first glance, one can compare the first entry in Table 1 with the
older PSF, e.g., $17200\times 10^{-24}$ yr-1 from Ref. [24], but the
comparison must be approached with care. Firstly, the result from Ref.[24]
must be combined with the closure NME, while our values are designed for the
Taylor expansion NMEs. Secondly, the older value includes only contributions
from K and L1 captures, whereas we take into account all possible captures of
124Xe. As a general result [37], we found that the DHFS screening decreases
the total PSF when compared with the Thomas-Fermi screening from [24, 25]. In
contrast, Pauli blocking effects and the inclusion of captures beyond K and L1
increase its value.

| $N=0$ | $N=2$ | $N=22$ | $N=4$
---|---|---|---|---
$G_{N}^{2\nu\rm{ECEC}}$ | $18332.0$ | $9802.2$ | $3056.1$ | $6116.3$
$G_{N}^{2\nu\rm{KK}}$ | $13605.3$ | $7230.0$ | $2241.2$ | $4482.4$
$G_{N}^{2\nu\rm{KL_{1}}}$ | $1741.9$ | $945.8$ | $298.9$ | $599.4$
$G_{N}^{2\nu\rm{KM_{1}}}$ | $355.4$ | $193.6$ | $61.3$ | $123.1$
$G_{N}^{2\nu\rm{KN_{1}}}$ | $78.6$ | $42.8$ | $13.6$ | $27.3$
$G_{N}^{2\nu\rm{KO_{1}}}$ | $11.8$ | $6.4$ | $2.0$ | $4.1$
$G_{N}^{2\nu\rm{L_{1}L_{1}}}$ | $222.9$ | $123.4$ | $39.9$ | $79.8$
$G_{N}^{2\nu\rm{L_{1}M_{1}}}$ | $45.5$ | $25.3$ | $8.2$ | $16.4$
Table 1:  The evaluated PSFs (Eq. 6) units of $10^{-24}\textrm{yr}^{-1}$ for
the total $2\nu\textrm{ECEC}$ process (first line) and the partial
$2\nu\textrm{xy}$ processes (the following lines) of 124Xe. We have used
$Q=2856.73$ keV [38].

After the capture takes place, the final atom is left in an excited state.
Experimentally, the detection of the de-excitation energies (in the form of
X-rays and Auger electrons) is used to identify 2$\nu$ECEC events, but it can
also interfere with other events from experiments where the decay of 124Xe is
an unavoidable source of background. The atomic relaxation energy subsequent
to the $2\nu\textrm{xy}$ process can be written in terms of the total electron
binding energies of 124Te in the ground state and in the excited state with
holes in x and y orbitals as

$R_{\rm xy}=B_{\rm g.s.}(^{124}\textrm{Te})-B_{\rm xy}(^{124}\textrm{Te}).$
(10)

The energies obtained from the DHFS framework are displayed in Table 6.
According to [35], our $R_{\mathrm{xy}}$ estimates are expected to have an
accuracy better than 1%. Considering the current experimental resolution of a
few keV [15], our estimation is a viable option for establishing the
background peak position in liquid xenon experiments [15, 39, 40].

###  3.2 The ISM evaluation of the NMEs

The nuclear matrix elements (NMEs), Eq. (4), have been calculated similarly as
in Ref. [20] but with few significant differences: (i) no single state
dominance (SSD) was assumed [31], (ii) two effective Hamiltonians were
employed, including GCN5082 [20] (name shortened to GCN in the tables below),
but also SVD [41], (iii) unique quenching factors ($q_{H}$) for the GT
$\tau^{-}\sigma$ operator, were employed for each effective Hamiltonian, which
describe well the two-neutrino double beta decay data of 136Xe for each case,
namely $q_{GCN}=0.4$ [20] and $q_{SVD}=0.7$ [29, 30], and (iv) up to four
nucleons were excited from the lower $g_{7/2}d_{5/2}$ orbitals into the higher
$jj55$-space orbitals.

As in Ref. [20] we calculate the NME in the $jj55$ model space consisting of
the $0g_{7/2}$, $1d_{5/2}$, $1d_{3/2}$, $2s_{1/2}$, and the $0h_{11/2}$
orbitals for both protons and neutrons. The shell model calculations in the
full model space for all nuclei involved in the transition are hindered by the
high dimensional basis needed, especially for 124Xe. As in Ref. [20], we rely
on truncations, by promoting nucleons from the lower $g_{7/2}d_{5/2}$ orbitals
into the higher $d_{3/2}s_{1/2}h_{11/2}$ orbitals. In this work we go beyond
the truncations used in Ref. [20], by allowing up to four nucleon excitations.
In addition, we go beyond the one state dominance approximation used in Ref.
[20] by performing a full summation on the intermediate $1^{+}$ states in Eq.
(4) [26] using a strength function approach described in detail in section 4
of Ref. [31]. The full summation reduces the NME for the GCN5082 Hamiltonian
by about 25%. In addition, as mentioned in the previous paragraph, we don’t
assume a range of unjustified quenching factors, but we use the same quenching
factors that describe reasonably well the NME for the two-neutrino double beta
decay of 136Xe calculated in the same model space and using the same effective
Hamiltonians.

Figure 1: The $M^{2\nu ECEC}_{GT-1}$ NME as a function of the number of
nucleons that were allowed to get excited from the lower $g_{7/2}d_{5/2}$
orbitals.

The results for the dominant matrix elements, $M_{GT-1}^{2\nu\rm{ECEC}}$, are
presented in Fig. 1, where g7d5-jump represents the number of nucleons that
were allowed to get excited (jump) from the lower $g_{7/2}d_{5/2}$ orbitals
into the higher $jj55$-space orbitals, maximum being 14. One can see that both
effective Hamiltonians provide similar results, which, although not yet
converged, do not significantly vary, especially for the SVD effective
Hamiltonian. We also calculated the other two nuclear matrix elements in Eq.
(4), $M_{GT-3}^{2\nu\rm{ECEC}}$ and $M_{GT-5}^{2\nu\rm{ECEC}}$, using similar
techniques by extending the approach presented in section 4 of Ref. [31]. In
these calculations the Gamow-Teller $\tau^{-}\sigma$ operator in Eq. (5) was
quenched by the $q_{H}$ factors given in the previous paragraph
($q_{SVD}=0.7,q_{GCN}=0.4$). The ISM point of view is that this quenching
effect is due to the renormalization of the $\tau^{-}\sigma$ operator in
reduced model spaces [42, 43, 44], while the axial coupling constant remains
that for free nucleons. Therefore in ISM the ”renormalized” $\tau^{-}\sigma$
operator in Eq. (5) is $q_{H}\tau^{-}\sigma$, and $g^{\textrm{eff}}_{A}$ in
Eqs. (1)-(2) is $g_{A}=1.276$ [45]. The NME results are summarized in Table 2,
where ”jump” is short for ”g7d5-jump”. These results are further used to
extract the $\xi_{31}^{2\nu\textrm{ECEC}}$ and $\xi_{51}^{2\nu\textrm{ECEC}}$
parameters, presented in Table 3, which are needed for calculating the Taylor
expansion corrections to the decay half-lives, Eqs. (1-2). One should
emphasize the $M_{GT-3}^{2\nu\rm{ECEC}}$ and $M_{GT-5}^{2\nu\rm{ECEC}}$ NMEs
exhibit similar behavior as that presented in Fig. 1 by the
$M_{GT-1}^{2\nu\rm{ECEC}}$ NMEs.

Model | NME type | jump=0 | jump=2 | jump=4
---|---|---|---|---
SVD | $M_{GT-1}^{2\nu\rm{ECEC}}$ | 0.0305 | 0.0329 | 0.0291
| $M_{GT-3}^{2\nu\rm{ECEC}}$ | 0.0066 | 0.0087 | 0.0064
| $M_{GT-5}^{2\nu\rm{ECEC}}$ | 0.0018 | 0.0026 | 0.0018
GCN | $M_{GT-1}^{2\nu\rm{ECEC}}$ | 0.0379 | 0.0354 | 0.0264
| $M_{GT-3}^{2\nu\rm{ECEC}}$ | 0.0108 | 0.0092 | 0.0057
| $M_{GT-5}^{2\nu\rm{ECEC}}$ | 0.0032 | 0.0027 | 0.0016
Table 2:  ISM results for the NMEs of Eq. (4) (see text for details). Model | $\xi_{i}^{2\nu\textrm{ECEC}}$ | jump=0 | jump=2 | jump=4
---|---|---|---|---
SVD | $\xi_{31}^{2\nu\textrm{ECEC}}$ | 0.216 | 0.264 | 0.220
| $\xi_{51}^{2\nu\textrm{ECEC}}$ | 0.059 | 0.079 | 0.062
GCN | $\xi_{31}^{2\nu\textrm{ECEC}}$ | 0.285 | 0.260 | 0.216
| $\xi_{51}^{2\nu\textrm{ECEC}}$ | 0.084 | 0.076 | 0.061
Table 3:  ISM results for the $\xi_{i}$ parameters of Eq. (3) (see text for
details).

###  3.3 The NME evaluation in pn-QRPA

The nuclear matrix elements $M^{2\nu\rm{ECEC}}_{GT-1}$,
$M^{2\nu\rm{ECEC}}_{GT-3}$ and $M^{2\nu\rm{ECEC}}_{GT-5}$ are calculated using
the pn-QRPA with isospin restoration [32]. They were obtained considering the
same large model space (23 subshells of N = 0–5 oscillator shells with the
addition of $i_{11/2}$ and $i_{13/2}$) and mean fields as in Ref. [32] for
double-beta of decay 128,130Te and 136Xe. Pairing and residual interactions
are derived from the same modern realistic nucleon-nucleon potentials, namely
from charge-dependent Bonn potential (CD-Bonn) and the Argonne V18 potential.
When solving the BCS pairing equations, the strengths of pairing interactions
are slightly renormalized so that experimental pairing gaps are correctly
reproduced [32]. The pn-QRPA equations contain three renormalization
adjustable parameters: $g_{ph}$ for particle-hole interaction, $g_{pp}^{T=1}$
and $g_{pp}^{T=0}$ for isovector and isoscalar parts of the particle-particle
interaction. While $g_{ph}=1.0$ is typically used [32], $g_{pp}^{T=1}$ is
fixed by the requirement that the $2\nu\beta^{-}\beta^{-}$ Fermi matrix
element vanishes, as it should. $g^{T=0}_{pp}$ is adjusted so that the half-
life of the $2\nu$ECEC of 124Xe is correctly reproduced for each considered
value of $g^{\rm eff}_{A}$. Unlike earlier QRPA calculations of
$2\nu\beta^{-}\beta^{-}$ NMEs, the BCS overlap of the initial and final vacua
is taken into account, leading to a reduction by factor 0.828 [46].

Figure 2: Dependence of the matrix elements $M^{2\nu\rm ECEC}_{GT-1}$, $M^{2\nu\rm ECEC}_{GT-3}$, and $M^{2\nu\rm{ECEC}}_{GT-5}$ on the effective value of axial-vector coupling constant $g_{A}^{\rm eff}$ for the case the calculated $2\nu$ECEC half-life is the same as its experimental value, i.e., $T^{2\nu\textrm{ECEC}}_{1/2}(^{124}\textrm{Xe})=1.1\times 10^{22}$ yrs [15]. Results were obtained within the QRPA with the restoration of the isospin by assuming the realistic Argonne V18 nucleon-nucleon potential. Table 4:  Nuclear matrix elements $M^{M}_{\rm GT-K}$ (M = $2\nu$ECEC and $2\nu\beta^{-}\beta^{-}$, K=1,3 and 5) for $2\nu$ECEC of 124Xe and $2\nu\beta^{-}\beta^{-}$-decay of 128,130Te calculated within pn-QRPA with isospin restoration. $\Delta_{p}$ and $\Delta_{n}$ denote experimental proton and neutron pairing gap, respectively. $g^{\rm eff}_{A}$ is the effective axial-vector coupling constant. $A$ and $B$ stand for the Argonne V18 and CD-Bonn nucleon-nucleon potential, respectively. The fixed value of the particle-particle strength parameter in the isovector channel $g^{T=1}_{pp}$ is 0.9626 (0.8884), 0.9672 (0.8934) and 0.9893 (0.9151) for $2\nu$ECEC of 124Xe, $2\nu\beta^{-}\beta^{-}$-decay of 128Te and 130Te, respectively, by exploiting the realistic Argonne V18 (CD-Bonn) nucleon-nucleon potential. $g^{T=0}_{pp}$ is the renormalization constant of particle-particle neutron-proton interaction in the isoscalar channel, which is fitted from the requirement that the calculated $2\nu$ECEC or $2\nu\beta^{-}\beta^{-}$ half-life is the same as its experimental value. $T^{2\nu\textrm{ECEC}}_{1/2}(^{124}\textrm{Xe})=1.1\times 10^{22}$ yrs [15], $T^{2\nu\beta^{-}\beta^{-}}_{1/2}(^{128}\textrm{Te})=2.3\times 10^{24}$ yrs [47] and $T^{2\nu\beta^{-}\beta^{-}}_{1/2}(^{130}\textrm{Te})=7.71\times 10^{20}$ yrs [48] are considered.  Nucl. tran. |  | $\Delta^{\rm Te}_{p}$ | $\Delta^{\rm Te}_{n}$ | $\Delta^{\rm Xe}_{p}$ | $\Delta^{\rm Xe}_{n}$ |  | $g^{\rm eff}_{A}$ | pot. | $g^{T=0}_{pp}$ | $M_{\rm GT-1}$ | $M_{\rm GT-3}$ | $M_{\rm GT-5}$ |  | $\xi_{31}$ | $\xi_{51}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
124Xe$\rightarrow^{124}$Te |  | 1.249 | 1.344 | 1.357 | 1.339 |  | 0.80 | A | 0.706 | 0.1044 | 0.018 | 0.0045 |  | 0.169 | 0.043
|  |  |  |  |  |  |  | B | 0.635 | 0.1050 | 0.019 | 0.0049 |  | 0.181 | 0.047
|  |  |  |  |  |  | 1.00 | A | 0.737 | 0.0662 | 0.016 | 0.0029 |  | 0.175 | 0.043
|  |  |  |  |  |  |  | B | 0.664 | 0.0677 | 0.014 | 0.0035 |  | 0.201 | 0.051
|  |  |  |  |  |  | 1.27 | A | 0.7542 | 0.0416 | 0.0074 | 0.0017 |  | 0.178 | 0.041
|  |  |  |  |  |  |  | B | 0.6814 | 0.0409 | 0.0094 | 0.0023 |  | 0.231 | 0.057
128Te$\rightarrow^{128}$Xe |  | 1.129 | 1.280 | 1.319 | 1.265 |  | 0.80 | A | 0.7432 | 0.0631 | 0.012 | 0.0038 |  | 0.196 | 0.060
|  |  |  |  |  |  |  | B | 0.6676 | 0.0631 | 0.014 | 0.0043 |  | 0.219 | 0.068
|  |  |  |  |  |  | 1.00 | A | 0.7645 | 0.0403 | 0.0081 | 0.0023 |  | 0.200 | 0.057
|  |  |  |  |  |  |  | B | 0.6886 | 0.0404 | 0.0098 | 0.0030 |  | 0.244 | 0.073
|  |  |  |  |  |  | 1.27 | A | 0.7773 | 0.0251 | 0.0051 | 0.0013 |  | 0.202 | 0.051
|  |  |  |  |  |  |  | B | 0.7013 | 0.0250 | 0.0070 | 0.0020 |  | 0.282 | 0.081
130Te$\rightarrow^{130}$Xe |  | 1.058 | 1.179 | 1.297 | 1.248 |  | 0.80 | A | 0.7490 | 0.0444 | 0.0082 | 0.0023 |  | 0.185 | 0.052
|  |  |  |  |  |  |  | B | 0.6726 | 0.0443 | 0.0093 | 0.0027 |  | 0.209 | 0.060
|  |  |  |  |  |  | 1.00 | A | 0.7680 | 0.0284 | 0.0054 | 0.0014 |  | 0.192 | 0.049
|  |  |  |  |  |  |  | B | 0.6915 | 0.0282 | 0.0067 | 0.0018 |  | 0.237 | 0.065
|  |  |  |  |  |  | 1.27 | A | 0.7797 | 0.0177 | 0.0035 | 0.0008 |  | 0.199 | 0.044
|  |  |  |  |  |  |  | B | 0.7031 | 0.0174 | 0.0049 | 0.0013 |  | 0.281 | 0.072

In Fig. 2, the calculated $2\nu$ECEC nuclear matrix elements for 124Xe within
the pn-QRPA with isospin restoration are presented as function of the
effective value of axial-vector coupling constant $g_{A}^{\rm eff}$ for fixed
value of $2\nu$ECEC half-life, which corresponds to its experimental value. We
see that $M^{2\nu\rm ECEC}_{GT-1}$ depend significantly on $g_{A}^{\rm eff}$
unlike $M^{2\nu\rm ECEC}_{GT-3}$ and $M^{2\nu\rm ECEC}_{GT-5}$. In Table 4,
the calculated values of $M^{2\nu\rm ECEC}_{GT-K}$ (K=1, 3 and 5) obtained
with the Argonne V18 and CD-Bonn potentials are displayed for unquenched and
modestly quenched value of $g_{A}^{\rm eff}$ ($g_{A}^{\rm eff}$ = 1.27, 1.00
and 0.80). They are compared with corresponding nuclear matrix elements for
$2\nu\beta^{-}\beta^{-}$ of 128Te and 130Te. Recall that all these transitions
connect the ground states of the tellurium and xenon isotopes. By glancing at
Table 4, we see that $2\nu$ECEC NMEs are only slightly larger, about up to a
factor of 2, when compared with $2\nu\beta^{-}\beta^{-}$ NMEs. It reflects the
fact that experimental paring gaps of all involved isotopes, which are also
shown in Table 4, are comparable. The ratios of NMEs $\xi_{13}$ and $\xi_{15}$
are presented as well. The maximal value $\xi_{13}$ evaluated for the
unquenched value of $g_{A}^{\rm eff}$ is 0.231, 0.282 and 0.281 in the case of
$2\nu$ECEC of 124Xe, $2\nu\beta^{-}\beta^{-}$ of 128Te and 130Te and CD-Bonn
potential. Recall that $\xi_{13}>0.26$ was excluded for the
$2\nu\beta^{-}\beta^{-}$-decay of 136Xe by the KamLAND-Zen Collaboration [49].
The measured value $\xi_{13}=0.45\pm 0.03(stat)\pm 0.05(syst)$ was achieved by
the CUPID-Mo experiment [50].

Before the half-life of the $2\nu$ECEC of 124Xe was measured, the nuclear
matrix element for this process was calculated within the pn-QRPA by choosing
a different way of adjusting the particle-particle interaction strength
parameter $g_{pp}$, in particular by using available experimental information
on single $\beta$-decays [18] or exploiting the formalism of statistical
approach, a novel possibility to reconcile QRPA results with experimental data
[51, 19]. Results of the second approach favor a strong quenching of axial-
vector coupling constant with $g^{\rm eff}_{A}\simeq 0.4-0.6$. The obtained
nuclear matrix elements in these studies are as follows: $M^{2\nu\rm
ECEC}_{GT}=0.10-0.20(g_{A}^{\rm eff}=1.25),0.34-0.71(g^{\rm eff}_{A}=1.00)$
[18] and $M^{2\nu\rm ECEC}_{GT}=0.296(g^{\rm eff}_{A}=0.60)$[19]. By comparing
them with NMEs values in Table 4 and with $M^{2\nu\rm ECEC}_{GT}$ = 0.191 and
0.186 (Argonne and CD-Bonn potentials) for $g^{\rm eff}_{A}$ =0.60 calculated
in the presented formalism, we see that they are too large leading to a
significant disagreement with measured half-life for $2\nu$ECEC of 124Xe.

###  3.4 Total and partial half-lives ISM calculations

Table 5 presents the total $2\nu\textrm{ECEC}$ half-life and the half-life for
the KK channel calculated with Eqs. (1-2), obtained using the PSF of Table 1,
the ISM NMEs of Table 2 and the ISM $\xi^{2\nu\textrm{ECEC}}_{i}$ parameters
in Table 3. Table 5 also includes the experimental values for comparison. The
overall conclusion is that the interacting shell model results seem to be
robust relative to changes of truncations and effective Hamiltonian and are in
good agreement with recently reported experimental data. One should also
remark that the Taylor expansion correction, as well as the newly more
accurate PSF, contributed to getting the calculated half-lives closer to the
experimental data.

Model | Channel | jump=0 | jump=2 | jump=4
---|---|---|---|---
SVD | Total | 1.94 | 1.61 | 2.12
| $KK$ | 2.61 | 2.17 | 2.86
GCN | Total | 1.20 | 1.40 | 2.58
| $KK$ | 1.62 | 1.88 | 3.48
Table 5:  The predicted $2\nu$ECEC half-lives for 124Xe (in units of $10^{22}$
yr) from Eqs. (1-2). To be compared with experimental data for the total half-
life, $(1.1\pm 0.2_{stat}\pm 0.1_{sys})\times 10^{22}$ yr and the inferred
data for the KK half-life, $(1.5\pm 0.3_{stat}\pm 0.1_{sys})\times 10^{22}$ yr
(see section III.F of Ref. [15]).

We note, for comparison, that various estimates of the half-life of 124Xe can
be found in the literature that employs various models for the NME
calculation, e.g., the pn-QRPA in Refs. [18, 19] and the ISM in Ref. [20]. For
example, Ref. [18] provides several predictions for the half-life of 124Xe for
the 2$\nu$ECEC process within the range of $(0.04$–$0.88)\times 10^{22}$ yr.
However, their NMEs are larger than the ones obtained here by a factor of
about 5, thus making them incompatible with the most accurate PSF shown in
Table I. In addition, the author refers to the expressions of [21] for the
calculation of PSFs, which employ the closure approximation. Similar
considerations apply to Ref. [19], which reports an estimate for the half-life
within the range of $(1.4$–$1.8)\times 10^{22}$ yr. Ref. [20] reports two
other ranges for the partial half-life of 124Xe, one obtained with ISM and one
with an effective theory (ET) extension to truncated ISM NME. The resulting
half-life expectations for the KK capture are $(1.3$–$18)\times 10^{22}$ y and
$(0.43$–$2.9)\times 10^{22}$ y, respectively. However, the PSF value adopted
from [24] also accounts for KL1 and L1L1 captures. Consequently, the reported
ranges are underestimated by about 20%. Moreover, the PSF value employed is
computed within the closure approximation while the NME is computed by summing
over all intermediate $1^{+}$ states in SM and by assuming the SSD hypothesis
in ET.

Decay Chanel |  $R_{\rm{xy}}$ (keV) | ISM CF (%)
---|---|---
$\rm{KK}$ | 64.62 | 74.13-74.15
$\rm{KL_{1}}$ | 37.05 | 18.76-18.83
$\rm{KM_{1}}$ | 32.98 | 3.83-3.84
$\rm{KN_{1}}$ | 32.11 | 0.83-0.85
$\rm{KO_{1}}$ | 31.93 | 0.13
$\rm{L_{1}L_{1}}$ | 10.04 | 1.22
$\rm{L_{1}M_{1}}$ | 6.01 | 0.49
Other | $<6$ | 0.52-0.55
Table 6: The atomic relaxation energies (Eq. 10) obtained within the DHFS
model (second column) and the capture fractions (CF) predicted by ISM (third
column). The captures with atomic relaxation energies below $6$ keV are
subsumed under the label ”other”. The ranges presented for the KK and KL1
channels correspond to the minimum and maximum values of the
$\xi_{31}^{2\nu\textrm{ECEC}}$ parameter predicted from ISM.

Given the success in describing the experimental data for the total $2\nu$ECEC
half-life of 124Xe, as well as the partial half-life for the KK channel, we
make predictions for other possible channels. Table 6 presents, in addition to
the relaxation energies calculated with Eq. 10, the predicted caption
fractions for different channels of the decay, including the measured total
and KK [15]. The ISM CF column shows the ISM predicted capture fraction (in %)
for different channels (the detailed results for different truncations and
effective Hamiltonians will be published elsewhere). We note that the
predicted CF for the KK channel (74.1%) differs from the value used in a
previous experimental search (72.4%) [15]. This difference stems from our
treatment considering only $s$-wave electrons, whereas in [15] all electrons
contribute to the process. In addition, the newly considered Taylor expansion
terms and the associated PSFs have a small contribution to the new caption
fractions. In addition, we predict that the next most probable channel after
KK is the $\rm{KL_{1}}$ with a CF of about 19%. Given the closeness of the
relaxation energies for the $\rm{KL_{1}}$-$\rm{KO_{1}}$ channels, we predict a
cumulative CF for those channels of about 24%, which is about one-third of the
$KK$ channel. This result suggests that the contribution from the
$\rm{KL_{1}}$-$\rm{KO_{1}}$ channels could be observed in future experiments.

It is important to note that the observables associated with the
$2\nu\rm{ECEC}$ process, such as the total and partial half-life, are
dependent on the $\xi_{31}^{2\nu\rm{ECEC}}$ and $\xi_{51}^{2\nu\rm{ECEC}}$
parameters in the Taylor expansion formalism. From an experimental
perspective, the ratio $\xi_{31}^{2\nu\rm{ECEC}}$ can be treated as a free
parameter when measuring these observables, and $\xi_{51}^{2\nu\rm{ECEC}}$ can
be fixed from a theoretical estimation as it has a much smaller influence.
Thus, an experimental constraint of $\xi_{31}^{2\nu\rm{ECEC}}$ can be obtained
from the measurement of the ratio between the half-lives of two channels, e.g.
$T^{2\nu\rm{KK}}_{1/2}/T^{2\nu\rm{KL}_{1}}_{1/2}$, similar to the method
proposed in Ref. [23] for the measurement of $\xi_{31}$ in
$2\nu\beta^{-}\beta^{-}$ decays from the angular correlation coefficient of
the emitted electrons. Although the half-lives ratios are rather weakly
dependent on $\xi_{31}^{2\nu\textrm{ECEC}}$, a future constraint on this
parameter might provide valuable insight and validation of different NMEs
calculations.

##  4 Conclusions

In conclusion, we studied the $2\nu$ECEC decay rates for the 124Xe isotope,
that was recently investigated experimentally [52, 15], improving the
calculation of both the phase space factors and the nuclear matrix elements.
The decay rate was obtained more rigorously using the Taylor expansion
approach, which leads to higher-order contributions.

In the calculation of the new PSFs, we made the following improvements: (i) we
used the DHFS self-consistent framework to incorporate effects such as atomic
screening, diffuse nuclear surface correction, realistic nuclear charge
density, and electron exchange-correlation effects, and (ii) we considered the
Pauli blocking of the decay of the innermost nucleon states. Moreover, we
consider captures of all $s$-wave electrons, expanding beyond the K and L1
orbitals considered in prior studies. Our findings indicate that a more
precise screening correction for bound states leads to a decrease in the decay
rate, while the Pauli blocking effect and the inclusion of new captures tend
to increase its value.

The nuclear part was treated with ISM and pn-QRPA methods. We extended the ISM
approach that calculates the sum on the $1^{+}$ states of the intermediate
nucleus to the new Taylor expansion terms of Eqs. (1-4). Additionally, we used
a larger shell model truncation in the $jj55$ single particle model space, and
we validated our results using two often-employed effective Hamiltonians in
this single particle model space. We calculated the new NMEs,
$M_{GT-3}^{2\nu\rm{ECEC}}$ and $M_{GT-5}^{2\nu\rm{ECEC}}$, by extending the
approach presented in section 4 of Ref. [31]. It is worth noting that the
reasonably good agreement between the ISM calculated NME and those that can be
extracted from experimental data reinforces confidence in the similar
prediction made for the $2\nu\beta^{-}\beta^{-}$ decay of 124Sn [30].

With the ISM approach we described with good accuracy, i.e., within a factor
of two, the existing experimental data for the total half-life. We also
predict the capture fraction of the KK channel to be 74.1%, which differs from
the value used in past experimental searches [15]. In addition, we made new
predictions for the capture fractions for a few more specific decay channels
(see Table 6) that may be accessible experimentally in the near future.
Specifically, we obtained a cumulative capture fraction of about 24%, which is
about one-third of that of the KK channel. This result suggests that the
contribution from the $\rm{KL_{1}}$-$\rm{KO_{1}}$ channels could be observed
at relaxation energies in the interval $37.05-31.93$ keV.

The pn-QRPA with a restoration of the isospin symmetry was used to calculate
nuclear matrix elements associated with the $2\nu$ECEC of 124Xe. The particle-
particle strength parameter $g^{T=0}_{pp}$ was fixed by the data on the
$2\nu$ECEC decay rate of 124Xe. We found that $M_{GT-1}^{2\nu\rm{ECEC}}$ is
much smaller when compared with results of previous pn-QRPA calculations [18,
19]. Our results show that values of $2\nu\rm{ECEC}$ NMEs for 124Xe are
comparable with those for $2\nu\beta^{-}\beta^{-}$ of 128,130Te. The
corresponding nuclear systems differ only in the number of neutrons and
exhibit comparable experimental pairing gaps, which constitute an important
input for the pn-QRPA calculations of nuclear matrix elements.

## Acknowledgements

O.N., S.G., V.A.S., S.S., and M.H. acknowledge support from project
PNRR-I8/C9-CF264, Contract No. 760100/23.05.2023 of the Romanian Ministry of
Research, Innovation and Digitization. M.H. acknowledges support from the US
Department of Energy grant DE-SC0022538 ”Nuclear Astrophysics and Fundamental
Symmetries”. F.Š. acknowledges support from the Slovak Research and
Development Agency under Contract No. APVV-22-0413 and by the Czech Science
Foundation (GAČR), project No. 24-10180S.

The authors wish to thank Dr. Andrei Neacşu for fruitful and constructive
discussions during all stages of the manuscript preparation.

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