#  Global calculation of two-neutrino double-$\beta$ decay
within the finite amplitude method in nuclear density functional theory

Nobuo Hinohara<EMAIL_ADDRESS>Center for Computational
Sciences, University of Tsukuba, Tsukuba, 305-8577, Japan  Faculty of Pure and
Applied Sciences, University of Tsukuba, Tsukuba, 305-8571, Japan  Jonathan
Engel<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of North Carolina, Chapel Hill, North Carolina 27516-3255, USA

###### Abstract

Two-neutrino double-beta ($2\nu\beta\beta$) decay has been used to constrain
the neutron-proton part of effective interactions, which in turn is used to
compute the nuclear matrix elements for neutrinoless double-beta decay, the
observation of which would have important consequences for fundamental
physics. We carefully examine $2\nu\beta\beta$ matrix elements within the
proton-neutron quasiparticle random-phase approximation with nuclear energy
density functionals. We work with functionals that are fit globally to single-
beta-decay half-lives and charge-exchange giant-resonance energies, but not to
$2\nu\beta\beta$ half-lives themselves, to evaluate the $2\nu\beta\beta$
nuclear matrix elements for all important nuclei, including those whose half-
lives have not yet been measured. Such a comprehensive evaluation in large
model spaces without configuration truncation requires an efficient
computational scheme; we employ a double contour integration within the finite
amplitude method. The results generally reproduce the nuclear matrix element
extracted from half-lives well, without the use of any of those half-lives in
the fitting procedure. We present predictions of the matrix elements in a
total of 27 nuclei with half-lives that are still unmeasured.

##  I Introduction

Experiments all over the world are attempting to observe neutrinoless double-
beta ($0\nu\beta\beta$) decay, which occurs only if neutrinos are Majorana
particles, at a rate that is related to neutrino masses. To learn anything
quantitative from an observed decay rate, one must know a nuclear matrix
element that cannot be measured independently and so must be computed [1, 2,
3, 4]. Such computations, which must handle the exchange of a virtual neutrino
among nucleons and mesons, are difficult, and so the matrix elements for
isotopes used in experiments are not known with high precision. A related
process, two-neutrino double-beta ($2\nu\beta\beta$) decay, has been observed,
however, and its rates are often a part of attempts to reduce the uncertainty
in $0\nu\beta\beta$ nuclear matrix elements.

The $2\nu\beta\beta$ nuclear matrix elements have been extracted from measured
half-lives in 11 nuclei at present [5]. To believe the results of many-body
computations of $0\nu\beta\beta$ decay, one would like to see similar
computations that reproduce these $2\nu\beta\beta$ matrix elements. Because
the closure approximation—replacing the energies of states in the decay’s
intermediate nucleus with an average—is accurate for $0\nu\beta\beta$, some
approaches rely on it and have a harder time with $2\nu\beta\beta$ decay, for
which the approximation is poor. Nuclear density functional theory (DFT) in
combination with the proton-neutron quasiparticle-random phase approximation
(pnQRPA) is not one of these approaches, however; $0\nu\beta\beta$ and
$2\nu\beta\beta$ matrix elements can be computed in similar ways. In many
applications of the pnQRPA, in fact, the $2\nu\beta\beta$ matrix element is
used to constrain the strength of the piece of the density functional
associated with isoscalar proton-neutron pairing, which suppresses both the
$0\nu\beta\beta$ and $2\nu\beta\beta$ matrix elements [6, 7].

The pnQRPA can be used in conjunction with a phenomenological Hamiltonian as
well as in nuclear DFT. The advantage of DFT is its large single-particle
model space and universality; a single energy-density functional (EDF) is
taken to describe all the isotopes in the nuclear chart. The authors of Ref.
[8] used a deformed-nucleus pnQRPA with a Skyrme EDF, computing
$2\nu\beta\beta$ and $0\nu\beta\beta$ matrix elements in several
experimentally important isotopes. They fit the strength of the isoscalar
pairing interaction, on which rates depend sensitively, and it obtains the
correct $2\nu\beta\beta$ matrix elements before computing the $0\nu\beta\beta$
matrix elements.

Nuclear EDFs are commonly optimized to reproduce a number of experimental
observables from a wide range of nuclei [9, 10, 11, 12, 13, 14]. The
optimization is easiest when the observables are ground-state expectation
values in even-even isotopes. The time-odd part of the EDF and the proton-
neutron pairing strength have no effect on even-even ground states, however,
and so cannot be fixed in the same way. Instead, they are usually optimized
globally, through the use of single-$\beta$-decay rates and Gamow-Teller and
spin-dipole giant-resonance energies in nuclei all over the table of isotopes
[15].

We would like to assess the ability of these globally determined EDFs to
reproduce 2$\nu\beta\beta$ matrix elements so that we can further optimize
them if necessary and then confidently apply them to $0\nu\beta\beta$ decay.
Standard pnQRPA calculations, however, require the construction of a QRPA
matrix Hamiltonian, which within large single-particle spaces can consume too
much computational time and memory. We can turn instead to the finite
amplitude method (FAM) within time-dependent DFT. The FAM, which is formally
equivalent to the QRPA [16, 17], computes the linear response induced by an
external field with a complex frequency. One-body induced fields and the
response of quasiparticle states are calculated by iteration, without the need
to compute the two-body QRPA matrix elements. The proton-neutron version of
the FAM (pnFAM) was developed and implemented in Ref. [18] in order to
calculate $\beta$-decay rates and Gamow-Teller strength distributions [19,
20]. Because of its efficiency, it was used in Ref. [15] in an attempt to
optimize the neutron-proton part of a particular nuclear EDF.

In this paper we show how to use the pnFAM to efficiently compute
$2\nu\beta\beta$ nuclear matrix elements. Our procedure, a preliminary version
of which was reported on in Ref. [21], employs a complex-plane integration
technique [22, 23] to perform the summation over intermediate states. We
compare our $2\nu\beta\beta$ nuclear matrix elements in 76Ge, 130Te, 136Xe,
and 150Nd to those obtained from matrix diagonalization with the same EDF in
Ref. [8]. Then we use the EDFs with time-odd terms fit in Ref. [15] to compute
the matrix elements for all 11 nuclei in which the $2\nu\beta\beta$ decay rate
has been measured, and for 27 nuclei in which it has not.

The rest of this paper is organized as follows: Section II briefly presents
the definition of the $2\nu\beta\beta$ matrix element and describes the
pnQRPA. Section III formulates our scheme for computing $2\nu\beta\beta$
nuclear matrix elements in the pnFAM. Section IV compares the pnFAM
$2\nu\beta\beta$ matrix elements with those obtained by matrix diagonalization
in the pnQRPA, and Sec. V assesses the performance of globally fit functionals
and offers predictions for unmeasured rates. Section VI is a conclusion.

##  II $2\nu\beta\beta$ matrix element and the QRPA

###  II.1 $2\nu\beta\beta$ matrix element

The nuclear matrix element governing the $2\nu\beta\beta$ decay of the nucleus
$(N,Z)$ to the ground state of the nucleus $(N-2,Z+2)$ contributes to the
half-life $T^{2\nu}_{1/2}$ as follows:

$[T^{2\nu}_{1/2}]^{-1}=G_{2\nu}(Q_{\beta\beta},Z)|M^{2\nu}|^{2}\,,$ (1)

where $G_{2\nu}$ is a phase space factor, and the 2$\nu\beta\beta$ matrix
element is a sum of Fermi and Gamow-Teller parts [1],

$\displaystyle M^{2\nu}$ $\displaystyle=M^{2\nu}_{\rm
GT}-\frac{g_{V}^{2}}{g_{A}^{2}}M^{2\nu}_{\rm F},$ (2) $\displaystyle
M^{2\nu}_{\rm F}$ $\displaystyle=\sum_{n}\frac{\displaystyle\langle
0^{+}_{f}|\sum_{a}\tau^{-}_{a}|n\rangle\langle
n|\sum_{b}\tau^{-}_{b}|0^{+}_{i}\rangle}{\displaystyle
E_{n}-\frac{M_{i}+M_{f}}{2}},$ (3) $\displaystyle M^{2\nu}_{\rm GT}$
$\displaystyle=\sum_{n}\frac{\displaystyle\langle
0^{+}_{f}|\sum_{a}\bm{\sigma}_{a}\tau^{-}_{a}|n\rangle\cdot\langle
n|\sum_{b}\bm{\sigma}_{b}\tau^{-}_{b}|0^{+}_{i}\rangle}{\displaystyle
E_{n}-\frac{M_{i}+M_{f}}{2}}.$ (4)

Here $\tau^{-}_{a}$ is the isospin-lowering operator for nucleon $a$,
$\bm{\sigma}_{a}$ is the corresponding spin operator, $M_{i}$ and $M_{f}$ are
the ground-state energies of the initial and final states of the decay, and
$|n\rangle$, with energy $E_{n}$ is one of a complete set of intermediate
states in the nucleus $(N-1,Z+1)$. The Fermi part of the $2\nu\beta\beta$
matrix element is very small because isospin is nearly conserved [24], and we
neglect it here.

###  II.2 The pnQRPA

The proton-neutron QRPA evaluates the transition matrix elements between the
initial or final state and the intermediate states that appear in the
numerator of Eq. (4), taking into account the effect of the proton-neutron
residual interaction beyond the mean-field approximation. In the pnQRPA, both
the initial and final states $|0_{i/f,{\rm QRPA}}^{+}\rangle$ are based on
Hartree-Fock-Bogoliubov (HFB) quasiparticle vacua, which incorporate axially-
symmetric deformation in our work. The intermediate states are related to the
initial or final state by a QRPA phonon operator

$\displaystyle|\lambda,K\rangle$ $\displaystyle=\hat{\cal
Q}^{\lambda{\dagger}}_{K}|0^{+}_{{\rm QRPA}}\rangle$ (5)
$\displaystyle\hat{\cal Q}^{\lambda{\dagger}}_{K}$
$\displaystyle=\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}pn\\\
j_{z,p}+j_{z,n}=K\end{subarray}}\\!\\!\\!\\!\\!\\!X_{pn,K}^{\lambda}\hat{a}^{{\dagger}}_{p}\hat{a}^{{\dagger}}_{n}-Y_{pn,K}^{\lambda}\hat{a}_{\bar{n}}\hat{a}_{\bar{p}}\,,$

where $\hat{a}_{\tau=n,p}$ is a neutron or proton quasiparticle operator,
defined so that $\hat{a}_{\tau}|0_{\rm HFB}^{+}\rangle=0$. Here, the indices
$p$ and $n$ label proton and neutron quasiparticles. $j_{z,\tau}$ and $K$ are
the projections along the symmetry axis of the quasiparticle and phonon
angular momentum, and the index $\bar{\tau}$ labels the time-reversal partner
of the state $\tau$ ($j_{z,\bar{\tau}}=-j_{z,\tau}$). From now on, for the
sake of simplicity we omit the restriction $j_{z,p}+j_{z,n}=K$ when summing
over the proton and neutron quasiparticle states.

The QRPA amplitudes $X_{pn,K}^{\lambda}$ and $Y_{pn,K}^{\lambda}$ are
solutions of the QRPA equations,

$\displaystyle\sum_{p^{\prime}n^{\prime}}\begin{pmatrix}A_{pn,p^{\prime}n^{\prime}}&B_{pn,p^{\prime}n^{\prime}}\\\
B^{\ast}_{pn,p^{\prime}n^{\prime}}&A^{\ast}_{pn,p^{\prime}n^{\prime}}\end{pmatrix}\begin{pmatrix}X^{\lambda}_{p^{\prime}n^{\prime},K}\\\
Y^{\lambda}_{p^{\prime}n^{\prime},K}\end{pmatrix}=\Omega^{\lambda}_{K}\begin{pmatrix}X^{\lambda}_{pn,K}\\\
-Y^{\lambda}_{pn,K}\end{pmatrix},$ (6)

where $\Omega^{\lambda}_{K}$ is an excitation energy, measured from the QRPA
ground state of the initial/final state. The $A$ and $B$ matrices contain
residual interactions, computed from the second functional derivative of the
EDF. The $2\nu\beta\beta$ matrix element can be calculated by combining the
pnQRPA transition matrix elements from the initial and final states of the
decay to the intermediate states. Because the procedure introduces two sets of
the intermediate states, an additional approximation for matching them is
necessary. We thus approximate the Gamow-Teller matrix element in Eq. (4) by

$\displaystyle M^{2\nu}_{\rm GT}$
$\displaystyle=\sum_{K=-1}^{1}(-1)^{K}\sum_{\begin{subarray}{c}\lambda_{i}>0\\\
\lambda_{f}>0\end{subarray}}\frac{\displaystyle\langle 0_{f,{\rm
QRPA}}^{+}|\hat{F}^{{\rm
GT}-}_{-K}|\lambda_{f},K\rangle\langle\lambda_{f},K|\lambda_{i},K\rangle\langle\lambda_{i},K|\hat{F}^{{\rm
GT}-}_{K}|0_{i,{\rm
QRPA}}^{+}\rangle}{\displaystyle\frac{\Omega^{\lambda_{i}}_{K}+\Omega^{\lambda_{f}}_{K}}{2}}.$
(7)

In the summation, the expression $\lambda>0$ denotes the states with
$\Omega_{K}^{\lambda}>0$.

The Gamow-Teller operator in the quasiparticle basis is

$\displaystyle\hat{F}^{{\rm GT}\pm}_{K}$
$\displaystyle=\sum_{a}(\sigma_{K})_{a}\tau^{\pm}_{a}$
$\displaystyle=\sum_{pn}\left[F_{20,K}^{{\rm
GT}\pm}(pn)\hat{a}^{\dagger}_{p}\hat{a}^{\dagger}_{n}+F_{02,K}^{{\rm
GT}\pm}(pn)\hat{a}_{\bar{n}}\hat{a}_{\bar{p}}\right]$
$\displaystyle\quad+(\hat{a}^{\dagger}\hat{a}{\rm-terms}),$ (8)

and its transition amplitudes in Eq. (7) are given by

$\displaystyle\langle\lambda_{i},K|\hat{F}^{{\rm GT-}}_{K}|0^{+}_{i,{\rm
QRPA}}\rangle=$ (9) $\displaystyle\sum_{pn}\left[F^{{\rm
GT-}}_{20,K}(pn)X^{\lambda_{i}\ast}_{pn,K}\right.$
$\displaystyle\left.+F^{{\rm
GT-}}_{02,K}(pn)Y^{\lambda_{i}\ast}_{pn,K}\right]$ $\displaystyle\langle
0^{+}_{f,{\rm QRPA}}|\hat{F}^{{\rm GT-}}_{-K}|\lambda_{f},K\rangle=$
$\displaystyle\sum_{pn}\left[F^{{\rm
GT-}}_{02,K}(pn)X^{\lambda_{f}}_{pn,K}\right.$ $\displaystyle\left.+F^{{\rm
GT-}}_{20,-K}(pn)Y^{\lambda_{f}}_{pn,-K}\right]\,.$

To compute the overlap of the two intermediate states
$\langle\lambda_{f},K|\lambda_{i},K\rangle$ we adapt expressions based on the
QRPA [25] and the quasiparticle Tamm-Dancoff approximation (QTDA) [8]. The
result is

$\displaystyle\langle\lambda_{f},K|\lambda_{i},K\rangle$
$\displaystyle=\sum_{pnp^{\prime}n^{\prime}}\left(X_{p^{\prime}n^{\prime},K}^{\lambda_{f}\ast}X_{pn,K}^{\lambda_{i}}-\alpha
Y_{p^{\prime}n^{\prime},K}^{\lambda_{f}\ast}Y_{pn,K}^{\lambda_{i}}\right)$
$\displaystyle\quad\times{\cal O}_{pp^{\prime}}(\alpha){\cal
O}_{nn^{\prime}}(\alpha)$
$\displaystyle=\sum_{pn}\left(\bar{X}_{pn,K}^{\lambda_{f}\ast}\bar{X}_{pn,K}^{\lambda_{i}}-\alpha\bar{Y}_{pn,K}^{\lambda_{f}\ast}\bar{Y}_{pn,K}^{\lambda_{i}}\right),$
(10)

where $\alpha$ is a parameter that is 0 for the QTDA overlap and 1 for the
QRPA overlap, and the ${\cal O}_{\tau\tau^{\prime}}(\alpha)$ are elements of
the matrix that connect the quasiparticles associated with the initial and
final states of the decay. Explicit expressions for these elements, together
with the derivation of Eq. (10), are in Appendix A. $\bar{X}$ and $\bar{Y}$
are defined by

$\displaystyle\bar{X}_{pn,K}^{\lambda_{i}}$
$\displaystyle=\sum_{p^{\prime}}{\cal
O}^{T}_{pp^{\prime}}(\alpha)X_{p^{\prime}n,K}^{\lambda_{i}},$ (11a)
$\displaystyle\bar{Y}_{pn,K}^{\lambda_{i}}$
$\displaystyle=\sum_{p^{\prime}}{\cal
O}^{T}_{pp^{\prime}}(\alpha)Y_{p^{\prime}n,K}^{\lambda_{i}},$ (11b)
$\displaystyle\bar{X}_{pn,K}^{\lambda_{f}}$
$\displaystyle=\sum_{n^{\prime}}X_{pn^{\prime},K}^{\lambda_{f}}{\cal
O}^{T}_{n^{\prime}n}(\alpha),$ (11c)
$\displaystyle\bar{Y}_{pn,K}^{\lambda_{f}}$
$\displaystyle=\sum_{n^{\prime}}Y_{pn^{\prime},K}^{\lambda_{f}}{\cal
O}^{T}_{n^{\prime}n}(\alpha).$ (11d)

##  III The FAM

###  III.1 pnFAM

The FAM is formally equivalent to the QRPA and enables us to compute DFT
response functions efficiently. A detailed formulation of the like-particle
FAM and the pnFAM in the presence of the pairing correlations appear,
respectively, in Refs. [17] and [18].

In the pnFAM, one applies a time-dependent external field of the form

$\displaystyle\hat{F}_{K}^{T}(t)=\eta(\hat{F}_{K}^{T}e^{i\omega
t}+\hat{F}^{T{\dagger}}_{K}e^{-i\omega t})\,,$ (12)

with $\hat{F}^{T}_{K}$ a one-body proton-neutron excitation operator and
$\omega$ a complex frequency. The excitation operator induces oscillations of
quasiparticle annihilation operators (e.g., for neutrons) of the form

$\displaystyle\delta\hat{a}_{n}(t)$
$\displaystyle=\eta\sum_{p}\hat{a}^{\dagger}_{p}\left[X_{pn}(\omega,\hat{F}^{T}_{K})e^{-i\omega
t}+Y^{\ast}_{pn}(\omega,\hat{F}^{T}_{K})e^{i\omega t}\right].$ (13)

Solving the time-dependent DFT equations results in the FAM amplitudes
$X_{pn}(\omega,\hat{F}^{T}_{K})$ and $Y_{pn}(\omega,\hat{F}^{T}_{K})$, which
are related to the QRPA amplitudes $X^{\lambda}_{pn,K}$ and
$Y^{\lambda}_{pn,K}$ through [22]

$\displaystyle X_{pn}(\omega,\hat{F}_{K}^{T})$
$\displaystyle=-\sum_{\lambda>0}\left\\{\frac{X_{pn,K}^{\lambda}\langle\lambda,K|\hat{F}_{K}^{T}|0^{+}\rangle}{\Omega^{\lambda}_{K}-\omega}\right.$
$\displaystyle\quad+\left.\frac{Y_{pn,K}^{\lambda\ast}\langle
0^{+}|\hat{F}_{K}^{T}|\lambda,-K\rangle}{\Omega^{\lambda}_{K}+\omega}\right\\},$
(14) $\displaystyle Y_{pn}(\omega,\hat{F}_{K}^{T})$
$\displaystyle=-\sum_{\lambda>0}\left\\{\frac{Y_{pn,K}^{\lambda}\langle\lambda,K|\hat{F}_{K}^{T}|0^{+}\rangle}{\Omega^{\lambda}_{K}-\omega}\right.$
$\displaystyle\quad+\left.\frac{X_{pn,K}^{\lambda\ast}\langle
0^{+}|\hat{F}_{K}^{T}|\lambda,-K\rangle}{\Omega^{\lambda}_{K}+\omega}\right\\}.$
(15)

###  III.2 $2\nu\beta\beta$ matrix elements in the pnFAM

To calculate the QRPA $2\nu\beta\beta$ nuclear matrix element in Eq. (7), we
separately solve the pnFAM computations in the initial and final nuclei,
distinguishing quantities from the two nuclei with the superscripts $(i)$ and
$(f)$. We then compute a quantity that is a combination of the two sets of
pnFAM amplitudes

$\displaystyle{\cal
T}(\alpha;\omega_{i},\hat{F}_{K_{i}}^{T_{i}};\omega_{f},\hat{F}_{K_{f}}^{T_{f}})$
$\displaystyle\equiv\sum_{pn}\left[\bar{Y}^{(f)}_{pn}(\omega_{f},\hat{F}_{K_{f}}^{T_{f}})\bar{X}^{(i)}_{pn}(\omega_{i},\hat{F}_{K_{i}}^{T_{i}})\right.$
$\displaystyle\quad\left.-\alpha\bar{X}^{(f)}_{pn}(\omega_{f},\hat{F}_{K_{f}}^{T_{f}})\bar{Y}^{(i)}_{pn}(\omega_{i},\hat{F}_{K_{i}}^{T_{i}})\right],$
(16)

Figure 1: Contours $C_{i}$ and $C_{f}$.

where $\bar{X}^{(i/f)}$ and $\bar{Y}^{(i/f)}$ are the amplitudes in Eqs. (14)
and (15), labeled in the same way as the QRPA amplitudes in Eq. (11). By
substituting Eqs. (14) and (15) into Eq. (16), we obtain an expression for
$\mathcal{T}$ in terms of the QRPA amplitudes:

$\displaystyle{\cal
T}(\alpha;\omega_{i},\hat{F}_{K_{i}}^{T_{i}};\omega_{f},\hat{F}_{K_{f}}^{T_{f}})$
$\displaystyle=\sum_{pn}\sum_{\begin{subarray}{c}\lambda_{i}>0\\\
\lambda_{f}>0\end{subarray}}\left[\frac{(\bar{X}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{X}^{\lambda_{i}}_{pn,K_{i}}-\alpha\bar{Y}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{Y}^{\lambda_{i}}_{pn,K_{i}})\langle
0_{f}^{+}|\hat{F}_{K_{f}}^{T_{f}}|\lambda_{f},-K_{f}\rangle\langle\lambda_{i},K_{i}|\hat{F}_{K_{i}}^{T_{i}}|0_{i}^{+}\rangle}{(\Omega^{\lambda_{f}}_{K_{f}}+\omega_{f})(\Omega^{\lambda_{i}}_{K_{i}}-\omega_{i})}\right.$
$\displaystyle\quad+\frac{(\bar{Y}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{X}^{\lambda_{i}}_{pn,K_{i}}-\alpha\bar{X}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{Y}^{\lambda_{i}}_{pn,K_{i}})\langle\lambda_{f},K_{f}|\hat{F}_{K_{f}}^{T_{f}}|0_{f}^{+}\rangle\langle\lambda_{i},K_{i}|\hat{F}_{K_{i}}^{T_{i}}|0_{i}^{+}\rangle}{(\Omega^{\lambda_{f}}_{K_{f}}-\omega_{f})(\Omega^{\lambda_{i}}_{K_{i}}-\omega_{i})}$
$\displaystyle\quad+\frac{(\bar{X}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{Y}^{\lambda_{i}}_{pn,K_{i}}-\alpha\bar{Y}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{X}^{\lambda_{i}}_{pn,K_{i}})\langle
0_{f}^{+}|\hat{F}_{K_{f}}^{T_{f}}|\lambda_{f},-K_{f}\rangle\langle
0_{i}^{+}|\hat{F}_{K_{i}}^{T_{i}}|\lambda_{i},-K_{i}\rangle}{(\Omega^{\lambda_{f}}_{K_{f}}+\omega_{f})(\Omega^{\lambda_{i}}_{K_{i}}+\omega_{i})}$
$\displaystyle\quad+\left.\frac{(\bar{Y}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{Y}^{\lambda_{i}}_{pn,K_{i}}-\alpha\bar{X}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{X}^{\lambda_{i}}_{pn,K_{i}})\langle\lambda_{f},K_{f}|\hat{F}_{K_{f}}^{T_{f}}|0_{f}^{+}\rangle\langle
0_{i}^{+}|\hat{F}_{K_{i}}^{T_{i}}|\lambda_{i},-K_{i}\rangle}{(\Omega^{\lambda_{f}}_{K_{f}}-\omega_{f})(\Omega^{\lambda_{i}}_{K_{i}}+\omega_{i})}\right].$
(17)

${\cal T}$ has first-order poles at
$\omega_{i}=\pm\Omega^{\lambda_{i}}_{K_{i}}$ and
$\omega_{f}=\pm\Omega^{\lambda_{f}}_{K_{f}}$. We choose a counterclockwise
contour $C_{i}$ ($C_{f}$) for $\omega_{i}$ ($\omega_{f}$) that includes
positive-energy (negative-energy) poles and excludes all the negative-energy
(positive-energy) poles, as in Fig. 1, to extract the residues from the first
term on the right side of Eq. (17).

Integrating $\mathcal{T}$ together with an arbitrary regular complex function
of $\omega_{i}$ and $\omega_{f}$ over those frequencies, we obtain an
expression that can be used for two-body matrix elements:

$\displaystyle{\cal
M}\bm{\bigl{(}}\alpha;\hat{F}_{K_{i}}^{T_{i}},\hat{F}_{K_{f}}^{T_{f}};f(\omega_{i},\omega_{f})\bm{\bigr{)}}$
$\displaystyle=-\frac{1}{4\pi^{2}}\oint_{C_{i}}d\omega_{i}\oint_{C_{f}}d\omega_{f}{\cal
T}(\alpha;\omega_{i},\hat{F}_{K_{i}}^{T_{i}};\omega_{f},\hat{F}_{K_{f}}^{T_{f}})f(\omega_{i},\omega_{f})$
$\displaystyle=\sum_{\begin{subarray}{c}\lambda_{i}>0\\\
\lambda_{f}>0\end{subarray}}\sum_{pn}\left(\bar{X}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{X}^{\lambda_{i}}_{pn,K_{i}}-\alpha\bar{Y}^{\lambda_{f}\ast}_{pn,K_{f}}\bar{Y}^{\lambda_{i}}_{pn,K_{i}}\right)$
$\displaystyle\quad\times
f(\Omega^{\lambda_{i}}_{K_{i}},-\Omega^{\lambda_{f}}_{-K_{f}})\langle
0_{f}^{+}|\hat{F}_{K_{f}}^{T_{f}}|\lambda_{f},-K_{f}\rangle\langle\lambda_{i},K_{i}|\hat{F}_{K_{i}}^{T_{i}}|0_{i}^{+}\rangle.$
(18)

The Fermi and Gamow-Teller $2\nu\beta\beta$ decay nuclear matrix elements are
then given by

$\displaystyle M^{2\nu}_{{\rm F}}$ $\displaystyle={\cal
M}\left(\alpha;\hat{F}^{{\rm F-}},\hat{F}^{{\rm
F-}};f=\frac{2}{\omega_{i}-\omega_{f}}\right),$ (19) $\displaystyle
M^{2\nu}_{{\rm GT}}$ $\displaystyle=\sum_{K=-1}^{1}(-1)^{K}{\cal
M}\left(\alpha;\hat{F}_{K}^{{\rm GT-}},\hat{F}_{-K}^{{\rm
GT-}};f=\frac{2}{\omega_{i}-\omega_{f}}\right),$ (20)

under the assumptions that $X^{\lambda_{f}}_{-K}=X^{\lambda_{f}}_{K}$,
$Y^{\lambda_{f}}_{-K}=Y^{\lambda_{f}}_{K}$, and
$\Omega^{\lambda_{f}}_{-K_{f}}=\Omega^{\lambda_{f}}_{K_{f}}$. Even when
starting from the final state we use the external operator
$\sigma_{-K}\tau^{-}$ that changes neutrons into protons to properly include
the backward amplitudes in Eqs. (14) and (15).

By setting $f=1$, $\alpha=1$, and taking the same HFB vacuum for the initial
and final states in Eq. (18), we can use that equation to compute the
unweighted summed strengths:

$\displaystyle{\cal M}^{i=f}(1;\hat{F}^{\rm F\mp},\hat{F}^{\rm F\pm};1)$
$\displaystyle=\sum_{\lambda>0}|\langle\lambda,0|\hat{F}^{\rm
F\mp}|0^{+}\rangle|^{2},$ (21) $\displaystyle{\cal M}^{i=f}(1;\hat{F}^{\rm
GT\mp}_{K},\hat{F}^{\rm GT\pm}_{-K};1)$ $\displaystyle=(-1)^{K}$
$\displaystyle\quad\times\sum_{\lambda>0}|\langle\lambda,K|\hat{F}^{{\rm
GT\mp}}_{K}|0^{+}\rangle|^{2}.$ (22)

Sum rules can be used to check the routines that compute matrix elements.

##  IV Results with SkM* and Comparison with Prior Work

Our calculation of $2\nu\beta\beta$ nuclear matrix elements uses an extension
of the pnFAM code developed in Ref. [18], which is in turn based on the
nuclear DFT solver hfbtho [26, 27, 28]. That last code uses the harmonic
oscillator basis in a cylindrical coordinate system and allows axial
deformation. In this section we provide details of our calculations with the
SkM* functional and compare our $2\nu\beta\beta$ matrix elements for 76Ge,
130Te, 136Xe, and 150Nd with those obtained in Ref. [8] by diagonalizing the
pnQRPA matrix.

###  IV.1 Parameter values

Table 1: Experimental values of $\tilde{\Delta}_{n}^{(3)}$ and $\tilde{\Delta}_{p}^{(3)}$ (in MeV) and the volume-pairing strengths $V_{n}$ and $V_{p}$ fit to those values (in MeV fm3). The averages of the strengths in the initial and final nuclei used in the pairing EDF. Experimental binding energies are taken from Ref. [29]. Data in parentheses are not used to fit the pairing strengths.  | $\tilde{\Delta}_{n}^{(3)}$ | $\tilde{\Delta}_{p}^{(3)}$ | $V_{n}$ | $V_{p}$
---|---|---|---|---
76Ge | 1.393 | 1.114 |  $-$182.70 |  $-$194.49
76Se | 1.551 | 1.392 |  $-$185.40 |  $-$202.22
average |  |  |  $-$184.05 |  $-$198.36
130Te | 1.114 | (0.801) |  $-$166.21 | N/A
130Xe | 1.170 | 1.016 |  $-$173.80 |  $-$194.00
average |  |  |  $-$170.01 |  $-$194.00
136Xe | (0.841) | 0.751 | N/A |  $-$148.66
136Ba | 0.960 | 1.005 |  $-$184.16 |  $-$172.54
average |  |  |  $-$184.16 |  $-$160.60
150Nd | 1.070 | 0.918 |  $-$181.64 |  $-$202.31
150Sm | 1.194 | 1.196 |  $-$184.84 |  $-$195.24
average |  |  |  $-$183.24 |  $-$198.78

To integrate in cylindrical coordinates, we use Gauss-Hermite quadrature with
$N_{\rm GH}=40$ points for the $z$ direction and Gauss-Laguerre quadrature
with $N_{\rm GL}=40$ points for the $r$ direction. To compute the direct
Coulomb mean field, we use the prescription described in Ref. [27] with length
parameter $L=50$ fm and $N_{\rm Leg}=80$ Gauss-Legendre points.

We include $N_{\rm sh}=20$ harmonic-oscillator major shells to describe the
HFB wave functions. This corresponds to 1771 single-particle states for
neutrons and protons (with axial and time-reversal symmetry taken into
account), and, in the pnFAM, to 257 686 $K=0$ two-quasiparticle states and 256
025 $K=\pm 1$ two-quasiparticle states. We include all such states, with no
additional model-space truncation, in the pnFAM calculations. The dimension of
the pnQRPA matrix corresponding to our pnFAM calculations is about 500 000 for
each $K$ quantum number.

We employ the same Skyrme SkM* functional [30] and volume-type pairing with
60-MeV energy cutoff (with $\hbar^{2}/2m=20.73$ MeV fm2 and the one-body
center-of-mass correction included in the kinetic term) as that in Ref. [8].
The HFB solver cited in that paper, however, works in a cylindrical box with
$r_{\rm max}=z_{\rm max}=20$ fm and a coordinate spacing of 0.7 fm, and is
thus different from ours.

Table 2: Properties of HFB ground states with the SkM* + volume pairing (with average pairing strengths) EDF. The table shows pairing gaps (in MeV), quadrupole deformation, and total HFB energies (in MeV) and compares the quadrupole deformation to the value in Ref. [8].  | $\Delta_{n}$ | $\Delta_{p}$ | $\beta$ | $E_{\rm HFB}$ |  $\beta$ (Ref [8])
---|---|---|---|---|---
76Ge | 1.609 | 1.473 |  $-$0.021 |  $-$661.804 |  $-$0.025
| 1.435 | 1.205 | 0.185 |  $-$662.274 | 0.184
| 1.612 | 1.475 | 0. |  $-$661.802 |
76Se | 1.589 | 1.648 | 0. |  $-$659.315 |  $-$0.018
| 1.508 | 1.257 |  $-$0.194 |  $-$659.594 |
130Te | 1.178 | 1.028 | 0. |  $-$1096.839 | 0.01
130Xe | 1.078 | 1.009 | 0.141 |  $-$1093.423 | 0.13
| 1.107 | 1.113 |  $-$0.124 |  $-$1093.152 |
| 1.359 | 1.351 | 0. |  $-$1092.393 |
136Xe | 0. | 0.878 | 0. |  $-$1143.253 | 0.004
136Ba | 1.025 | 0.931 |  $-$0.047 |  $-$1139.268 |  $-$0.021
| 0.928 | 0.735 | 0.094 |  $-$1139.538 |
| 1.057 | 0.985 | 0. |  $-$1139.231 |
150Nd | 1.129 | 0.764 | 0.292 |  $-$1235.794 | 0.27
| 1.375 | 1.358 |  $-$0.177 |  $-$1232.563 |
| 1.422 | 1.688 | 0. |  $-$1231.080 |
150Sm | 1.131 | 1.307 | 0.223 |  $-$1234.675 | 0.22
| 1.294 | 1.707 | 0. |  $-$1232.436 |
| 1.305 | 1.534 |  $-$0.137 |  $-$1233.068 |

We adjust the volume pairing strengths to reproduce experimental odd-even
staggering (OES) with the density-averaged pairing gap. To reduce fluctuations
[10], we take as the experimental data an average of the results of the three-
point formula evaluated at the two even-odd or odd-even systems:

$\displaystyle\tilde{\Delta}^{(3)}_{n}(N,Z)$
$\displaystyle=\frac{\Delta^{(3)}_{n}(N-1,Z)+\Delta^{(3)}_{n}(N+1,Z)}{2}$ (23)
$\displaystyle\tilde{\Delta}^{(3)}_{p}(N,Z)$
$\displaystyle=\frac{\Delta^{(3)}_{p}(N,Z-1)+\Delta^{(3)}_{p}(N,Z+1)}{2}\,,$

where $\Delta^{(3)}_{n/p}$ is the result of the three-point formula [31].
Table 1 lists the experimental values for this quantity and the neutron and
proton volume pairing strengths that best reproduce them. In order to use the
same EDF for both nuclei in the decay, we take the average of the pairing
strengths fit in the initial and final nuclei. We note that the experimental
$\Delta^{(3)}_{n/p}$ values do not provide useful information if the series of
isotopes used to calculate them includes closed-shell nuclei.
$\Delta^{(3)}_{p}$ in 130Te ($Z=50$ included) and $\Delta^{(3)}_{n}$ in 136Xe
and 136Ba ($N=82$ included) are such cases if the average of the results of
two odd-even mass formulas is used. We avoid using the pairing gap
$\tilde{\Delta}_{p}^{(3)}$ of 130Te to fit the proton pairing strength,
fitting the pairing strength instead to the proton gap in 130Xe. We do adopt
the neutron $\tilde{\Delta}^{(3)}$ of 136Ba, however, to determine the neutron
pairing strength because the strengths fit to $\Delta_{n}^{(3)}$ and
$\tilde{\Delta}_{n}^{(3)}$ are quite similar in that nucleus. The globally fit
EDFs described in Sec. V are free from these problems.

Table 2 shows the results of the DFT calculations for the initial and final
nuclei. The quadrupole deformations of the HFB states are quite close to those
in Ref. [8]. We choose the HFB solution in the top line for each nucleus in
which several HFB solutions coexist.

Table 3: $Q$ values for each double-beta decay in units of MeV. Experimental $Q$ values are obtained from atomic masses [29].  | This paper | SkM* (Ref. [8]) | Exp.
---|---|---|---
76Ge $\to$ 76Se | 4.05 | 4.84 | 2.04
130Te $\to$ 130Xe | 4.98 | 4.22 | 2.53
136Xe $\to$ 136Ba | 5.55 | 5.60 | 2.46
150Nd $\to$ 150Sm | 2.68 | 2.35 | 3.37

Table 3 shows $\beta\beta$ $Q$ values. Our calculation does not perfectly
reproduce the values in Ref. [8], which were obtained from the same SkM* EDF
but a different HFB code. We suspect that the differences are due to the
different representations of the oscillator basis states and treatments of
pairing.

Table 4: Neutron and proton parts of the HFB overlap $\langle 0^{+}_{f,{\rm HFB}}|0^{+}_{i,{\rm HFB}}\rangle$ between the initial and the final states compared with values from previous QRPA calculations.  | neutron | proton | total | Ref. [8] | Ref. [32] | Ref. [33]
---|---|---|---|---|---|---
76Ge | 0.907 | 0.886 | 0.803 |  | 0.81 | 0.72, 0.73
130Te | 0.329 | 0.403 | 0.133 |  |  | 0.73, 0.73
136Xe | 0.480 | 0.787 | 0.378 | 0.47 |  | 0.43, 0.39
150Nd | 0.679 | 0.589 | 0.400 |  | 0.52 | 0.51, 0.52

Table 4 lists the values of the HFB overlap included in the matrix ${\cal O}$.
Our values agree with those of previous calculations with similar nuclear
deformation. The overlap becomes small when the deformation of the initial and
final states are different. That situation arises in 130Te and 136Xe, where
the initial states are spherical while the final states are prolate and
oblate, respectively. The overlap also becomes small when the initial or final
state has no pairing gap. That is the case for neutrons in 136Xe.

###  IV.2 Contour

To use the expression in Eq. (18), we must choose the contours $C_{i}$ and
$C_{f}$. We take each to be centered on the real axis and circular, with the
circle specified by the two energies $\omega_{\rm L}$ and $\omega_{\rm R}$ at
they cross the real axis. The radius $r$ and the center of the contour
$\omega_{0}$ are then given by $r=(\omega_{\rm R}-\omega_{\rm L})/2$ and
$\omega_{0}=(\omega_{\rm L}+\omega_{\rm R})/2$, and every point on the contour
can be written in the form $\omega=\omega_{0}+re^{i\theta}$. We use
$\omega_{\rm L}=0.1$ MeV and $\omega_{\rm R}=120$ MeV for $C_{i}$ and
$\omega_{\rm L}=-120$ MeV and $\omega_{\rm R}=-0.1$ MeV for $C_{f}$ to include
all the unperturbed two-quasiparticle states within the quasiparticle-energy
cutoff.

Figure 2: Integrand in computation of summed Gamow-Teller strength and
$2\nu\beta\beta$ matrix element for 76Ge as a function of the angle $\theta$
for the contour $C_{i}$. The integrand is normalized to 1 at $\theta=\pi$.

Figure 2 shows the integrands for the summed GT strength and the
$2\nu\beta\beta$ matrix element in 76Ge as a function of the angle $\theta$
for the contour $C_{i}$. The main contribution to each comes from the peak at
$\theta=\pi$, the point at $\omega=\omega_{\rm L}$ where the two contours are
closest. While the integrand of the sum is distributed broadly along the whole
contour $C_{i}$, the energy denominator $2/(\omega_{i}-\omega_{f})$
concentrates the $2\nu\beta\beta$ contribution at $\theta=\pi$. To take this
contribution into account precisely and efficiently, we introduce a parameter
$\gamma$ to control the distribution of the discretized points near the
origin, i.e., we discretize the angle $\theta$ as follows:

$\displaystyle\theta_{k}$ $\displaystyle=(m+x_{k}^{\gamma})\pi,$ (24)
$\displaystyle x_{k}$
$\displaystyle=-1+2\frac{k-1}{n_{r}-1}\quad(k=1,2,\cdots,n_{r})\,,$ (25)

where $\gamma$ is an odd number, and $m$ is 1 for $C_{i}$ and $0$ for $C_{f}$.
The parameter $\theta_{k}$ runs from 0 to $2\pi$ for $C_{i}$ and $-\pi$ to
$\pi$ for $C_{f}$. We use $n_{r}=202$ and omit the contribution from
$(\omega_{i},\omega_{f})=(\omega_{\rm R},-\omega_{\rm L})$, because those
points are on the real axis and can be very close to the QRPA poles, although
their contribution to the nuclear matrix element should be small because of
the factor $2/(\omega_{i}-\omega_{f})$.

Table 5 shows the dependence of the summed strengths and the $2\nu\beta\beta$
matrix element in 76Ge on the parameter $\gamma$. The matrix element converges
by $\gamma=5$, which is the value we use.

###  IV.3 Summed Strengths

Table 6 shows the unweighted summed Fermi and Gamow-Teller strengths obtained
from the double contour integration for selected nuclei of interest to
experimentalists. Integration up to 120 MeV reproduces more than 99.9% of the
Ikeda sum rule in all these nuclei.

Table 5:  Dependence on the discretization parameter $\gamma$ in 76Ge of summed Fermi and Gamow-Teller strengths $m({\rm F\pm})=\sum_{\lambda>0}|\langle\lambda,0|\hat{F}^{\rm F\pm}|0^{+}\rangle|^{2}$ and $m({\rm GT\pm})=\sum_{K}(-1)^{K}\sum_{\lambda>0}|\langle\lambda,K|\hat{F}^{\rm GT\pm}_{K}|0^{+}\rangle|^{2}$ and of the dimensionless Gamow-Teller $2\nu\beta\beta$ nuclear matrix element. We use volume like-particle pairing and no isoscalar pairing.  $\gamma$ | 1 | 3 | 5 | 7
---|---|---|---|---
$m({\rm F-})$ | 12.0213 | 12.0209 | 12.0201 | 12.0189
$m({\rm F+})$ | 0.0252 | 0.0255 | 0.0260 | 0.0269
$m({\rm F-})-m({\rm F+})$ | 11.9961 | 11.9954 | 11.9940 | 11.9920
$m({\rm GT-})$ | 37.5860 | 37.5837 | 37.5811 | 37.5774
$m({\rm GT+})$ | 1.6065 | 1.6070 | 1.6085 | 1.6109
$m({\rm GT-})-m({\rm GT+})$ | 35.9795 | 35.9767 | 35.9726 | 35.9664
$M^{2\nu}m_{e}c^{2}$ | 0.1802 | 0.1574 | 0.1574 | 0.1575
Table 6: Summed Fermi and Gamow-Teller transitions, from double contour integration ($\omega_{\rm L}=0.1$ MeV and $\omega_{\rm R}=120$ MeV, $n_{r}=202$, and $\gamma=5$), as percentages of the corresponding sum rules.  | $\displaystyle\frac{m(\text{F}-)-m(\text{F}+)}{N-Z}$ | $\displaystyle\frac{m(\text{GT}-)-m(\text{GT}+)}{3(N-Z)}$
---|---|---
76Ge | 0.9995 | 0.9992
76Se | 0.9994 | 0.9992
130Te | 0.9996 | 0.9993
130Xe | 0.9996 | 0.9993
136Xe | 0.9998 | 0.9997
136Ba | 0.9996 | 0.9994
150Nd | 0.9996 | 0.9994
150Sm | 0.9996 | 0.9995

###  IV.4 $2\nu\beta\beta$ matrix element

We calculate the $2\nu\beta\beta$ matrix elements for 76Ge, 130Te, 136Xe, and
150Nd, setting the neutron-proton isovector pairing strength to the average of
the neutron and proton like-particle pairing strengths
[$V_{1}=(V_{n}+V_{p})/2$] and varying the isoscalar pairing strength $V_{0}$
from 0 to $-300$ MeV fm3. We use the QTDA ($\alpha=0$) to compute the overlap
among intermediate states. Figure 3 displays the dependence of the
$2\nu\beta\beta$ Gamow-Teller nuclear matrix elements on the isoscalar pairing
strength. Like the authors of that paper, we use two values of $g_{A}$: one
“unquenched” ($g_{A}=1.25$, though the currently accepted value is greater
than 1.27) and one quenched ($g_{A}=1.0$), and compare results for the EDF
SkM* with and without a modified proton-neutron piece [$C_{1}^{s}=100$ MeV
fm3, $C_{1}^{T}=C_{1}^{\nabla s}=0$, see Eq. (26)]. Our matrix elements agree
reasonably well with those of Ref. [8] in 130Te, 136Xe, and 150Nd, while they
are about twice as large in 76Ge.

Figure 3:  Dependence on the isoscalar pairing strength of the
2$\nu\beta\beta$ Gamow-Teller nuclear matrix element (in units of MeV-1) for
76Ge, 130Te, 136Xe, and 150Nd, with the SkM* + volume pairing EDF. Red curves
indicate the results with the time-odd functional derived from the SkM*
interaction and blue curves the results with the modified time-odd functional.
The thick solid and dotted curves correspond to $g_{A}=1.25$, and the dashed
and thin solid curves to the quenched value $g_{A}=1.0$.

##  V global edf

###  V.1 Performance of global EDFs

As Fig. 3 shows, the $2\nu\beta\beta$ and $0\nu\beta\beta$ nuclear matrix
elements are suppressed by isoscalar proton-neutron pairing correlations that
cannot be constrained from the ground-state properties of even-even nuclei.
The usual QRPA approach uses $2\nu\beta\beta$ decay rates to determine the
strength of isoscalar pairing separately in each decaying nucleus. The
philosophy of nuclear DFT, however, is that one EDF parameter set should, if
possible, describe all the $\beta\beta$-decaying nuclei in the nuclear chart.
In this section we assess the ability of globally-fit EDFs to describe
$2\nu\beta\beta$ decay, without using that observable at all in the fitting.
We use Skyrme-type EDFs, with the isovector time-odd and isoscalar pairing
parts globally fit to single-$\beta$ decay rates and to Gamow-Teller and spin-
dipole resonances. Reference [15], which deals with single-$\beta$ decay in
many isotopes, undertakes the global fitting and proposes ten parameter sets,
called 1A, 1B, 1C, 1D, 1E, 2, 3A, 3B, 4, and 5, each corresponding to a
different EDF. The time-even parts for all the sets except set 2 are taken
from the functional SkO′ [34]; that of parameter set 2 comes from the
functional SV-min [14] (though tensor-density terms are neglected, and the
neutron and proton have different masses). In both cases, center-of-mass
corrections to the mean field are neglected. The paper uses mixed volume-
surface like-particle isovector pairing terms, fit to reproduce odd-even
staggering in ten isotopes with $50\leq A\leq 250$; the strengths are
$V_{n}=-253.75$ MeV fm3, $V_{p}=-274.68$ MeV fm3 for SkO′ and $V_{n}=-244.06$
MeV fm3, $V_{p}=-257.90$ MeV fm3 for SV-min [35].

Table 7:  The neutron and proton pairing gaps (in MeV) and quadrupole deformation for the lowest-energy HFB solutions in the initial and final nuclei of the decay, computed with the SkO′ and SV-min EDFs, together with a mixed pairing EDF. Solutions with parentheses are not the lowest-energy ones, but we use them in addition when calculating matrix elements.  |  | SkO′ |  |  | SV-min |
---|---|---|---|---|---|---
| $\Delta_{n}$ | $\Delta_{p}$ | $\beta$ | $\Delta_{n}$ | $\Delta_{p}$ | $\beta$
48Ca | 0.771 | 0.000 | 0.000 | 0.793 | 0.000 | 0.000
48Ti | 1.270 | 1.386 | 0.000 | 1.275 | 1.309 | 0.000
76Ge | 1.063 | 1.189 | 0.136 | 1.123 | 1.094 | 0.131
76Se | 1.134 | 1.532 | 0.000 | 1.165 | 1.352 | 0.000
82Se | 0.619 | 1.106 | 0.152 | 0.689 | 1.124 | 0.134
82Kr | 1.014 | 1.353 | 0.112 | 1.041 | 1.230 | 0.101
96Zr | 1.153 | 1.133 |  $-$0.173 | 1.041 | 0.986 | 0.000
| (1.354 | 1.129 | 0.000) |  |  |
96Mo | 1.202 | 1.174 | 0.000 | 0.991 | 1.090 | 0.000
100Mo | 1.200 | 1.089 |  $-$0.192 | 1.299 | 1.078 | 0.000
| (1.123 | 1.246 | 0.214) |  |  |
100Ru | 0.994 | 1.092 | 0.186 | 1.189 | 1.137 | 0.000
116Cd | 1.430 | 0.854 | 0.000 | 1.463 | 0.492 | 0.120
116Sn | 1.406 | 0.000 | 0.000 | 1.553 | 0.000 | 0.000
128Te | 1.139 | 0.970 | 0.000 | 1.209 | 0.907 | 0.000
128Xe | 1.136 | 0.912 | 0.142 | 1.152 | 0.841 | 0.156
| (1.147 | 1.064 |  $-$0.112) | (1.179 | 0.986 |  $-$0.122)
130Te | 1.013 | 0.971 | 0.000 | 1.043 | 0.902 | 0.000
130Xe | 1.051 | 1.001 | 0.111 | 1.077 | 0.947 | 0.118
136Xe | 0.000 | 1.180 | 0.000 | 0.000 | 1.143 | 0.000
136Ba | 0.767 | 1.349 | 0.000 | 0.775 | 1.296 | 0.000
150Nd | 0.962 | 0.686 | 0.311 | 0.886 | 0.830 | 0.266
150Sm | 0.901 | 1.074 | 0.238 | 0.823 | 1.101 | 0.203
238U | 0.863 | 0.735 | 0.265 | 0.763 | 0.596 | 0.269
238Pu | 0.828 | 0.640 | 0.269 | 0.745 | 0.572 | 0.272

The isovector time-odd part of any Skyrme-type EDF is given by

$\displaystyle\chi^{\rm odd}_{1}(\bm{r})$
$\displaystyle=C_{1}^{s}[\rho_{0}]\bm{s}_{1}^{2}+C_{1}^{\Delta
s}\bm{s}_{1}\cdot\Delta\bm{s}_{1}+C_{1}^{j}\bm{j}_{1}^{2}$
$\displaystyle\quad+C_{1}^{T}\bm{s}_{1}\cdot\bm{T}_{1}+C_{1}^{\nabla
j}\bm{s}_{1}\cdot\bm{\nabla}\times\bm{j}_{1}$
$\displaystyle\quad+C_{1}^{F}\bm{s}_{1}\cdot\bm{F}_{1}+C_{1}^{\nabla
s}(\bm{\nabla}\cdot\bm{s}_{1})^{2}\,,$ (26)

where $\bm{s}_{1},\bm{j}_{1},\bm{T}_{1}$, and $\bm{F}_{1}$ are the isovector
spin, current, spin-kinetic, and tensor-kinetic densities, respectively. The
isoscalar pairing functional in all these parametrizations has the mixed
density dependence

$\displaystyle\tilde{\chi}_{0}(\bm{r})=\frac{V_{0}}{4}\left[1-\frac{1}{2}\frac{\rho_{0}(\bm{r})}{\rho_{c}}\right]|\tilde{\bm{s}}_{0}(\bm{r})|^{2}\,,$
(27)

where $\tilde{\bm{s}}_{0}$ is the isoscalar pair density, $\rho_{c}=0.16$
fm-3, and $\rho_{0}$ is the usual isoscalar density. In the parameter sets 1A,
1B, 1C, 1D, 1E, only $C_{1}^{s}$ (with no density dependence) and $V_{0}$ are
fit. In sets 3A and 3B $C_{1}^{T}$ and $C_{1}^{F}$ are fit as well. In the
parameter set 4, $C_{1}^{j},C_{1}^{\nabla j}$, and $C_{1}^{\nabla s}$ are
adjusted, while other parameters are the same as in set 3A. In set 5, $V_{0}$,
$C_{1}^{s}$, and $C_{1}^{j}$ are fit.

Table 8: The HFB overlap $\langle 0^{+}_{f,{\rm HFB}}|0^{+}_{{\rm HFB},i}\rangle$ between the initial and the final states of the double-beta decay, computed with SkO′ and SV-min EDFs. The numbers in parentheses denote powers of 10.  |  | SkO′ |  |  | SV-min |  |
---|---|---|---|---|---|---|---
| neutron | proton | total | neutron | proton | total |
48Ca | 0.764 | 0.513 | 0.392 | 0.776 | 0.512 | 0.398 |
76Ge | 0.577 | 0.559 | 0.323 | 0.586 | 0.587 | 0.344 |
82Se | 0.729 | 0.829 | 0.604 | 0.772 | 0.862 | 0.665 |
96Zr | 0.283 | 0.306 | 0.087 | 0.882 | 0.877 | 0.774 |
(sph.$\to$sph.) | 0.915 | 0.893 | 0.818 |  |  |  |
100Mo | $1.8(-3)$ | $1.4(-2)$ | $2.6(-5)$ | 0.914 | 0.905 | 0.828 |
(pro.$\to$pro.) | 0.864 | 0.875 | 0.755 |  |  |  |
116Cd | 0.932 | 0.521 | 0.485 | 0.507 | 0.293 | 0.148 |
128Te | 0.342 | 0.388 | 0.133 | 0.294 | 0.343 | 0.101 |
(obl.128Xe) | 0.440 | 0.533 | 0.235 | 0.403 | 0.487 | 0.197 |
130Te | 0.489 | 0.523 | 0.256 | 0.464 | 0.509 | 0.236 |
136Xe | 0.517 | 0.921 | 0.476 | 0.522 | 0.931 | 0.486 |
150Nd | 0.624 | 0.601 | 0.375 | 0.711 | 0.683 | 0.485 |
238U | 0.912 | 0.882 | 0.805 | 0.902 | 0.873 | 0.787 |

Table 7 lists the pairing gaps and quadrupole deformation of the HFB states
used to compute $2\nu\beta\beta$ nuclear matrix elements. Neutron pairing
collapses only in 136Xe and proton pairing collapses in 48Ca and 116Sn. SkO′
and SV-min cause different amounts of deformation. 96Zr, 100Mo, and 100Ru are
oblate, oblate, and prolate (respectively) with SkO′, while they are all
spherical with SV-min. 116Cd is spherical with SkO′, but is prolate with SV-
min.

Table 8 contains the overlaps of the initial and final HFB vacua. Significant
differences in deformation and pairing between the two HFB states lead to
small overlaps, and because the two EDFs can produce different levels of
deformation and pairing in any nucleus, the overlaps depend significantly on
the EDF. In 96Zr and 100Mo, the HFB overlaps with SkO′ are extremely small
because the initial state is oblate and the final state spherical or prolate.
In 116Cd, the HFB overlap with SV-min is smaller for a similar reason. The
QRPA may not be adequate when the overlaps, like those with SkO′ in 100Mo, are
very small. Our treatment omits both projection onto states with good angular
momentum, which involves the mixing of states with different orientations, and
the fluctuation in shape and pairing captured, e.g., by the generator
coordinate method [36, 37]. The effects of the physics we have neglected can
be significant when the matrix elements are small at the HFB or QRPA levels.

Figure 4: Comparison of dimensionless $2\nu\beta\beta$ nuclear matrix elements
obtained from global EDFs with experimental values. The matrix elements
computed with the lowest-energy HFB solutions are marked with crosses, while
those elements computed with the other HFB solutions are marked with
triangles. Orange symbols come from computations with the QTDA overlap and
blue symbols from computations with the QRPA overlap. The EDFs that give rise
to each particular point appear in Table 9 in Appendix B.  Figure 5:
Dimensionless $2\nu\beta\beta$ nuclear matrix elements calculated with the
global EDFs. The matrix elements from parameter set 4 are excluded.

In Fig. 4 we compare the Gamow-Teller $2\nu\beta\beta$ nuclear matrix element,
scaled by $g_{A}^{2}m_{e}c^{2}$ to be dimensionless, that results from
calculations with the ten different SkO′\- and SV-min-based EDFs discussed
just above. We also show the experimental matrix elements, extracted from the
half-lives in Ref. [5]. We use a quenched axial-vector coupling constant
$g_{A}=1.0$ to match the value from Ref. [15], which determines the EDF
parameters. Despite the differences among the EDFs in the pieces of the
functional that were fit and in the data chosen to fit them, the
$2\nu\beta\beta$ matrix elements that they produce are quite close to one
another in some of the heavier nuclei. This fact means that the parts of the
EDF that affect the $2\nu\beta\beta$ matrix element are determined almost
fully by the $\beta$ decay rates and giant resonance energies used in fitting
them. In some lighter isotopes such as 48Ca, 96Zr, and 100Mo, on the other
hand, the values of the nuclear matrix element, like those of the overlap,
depend significantly on the EDF. Although such matrix elements may provide an
additional constraint on the pnEDF, the disagreement also suggests, as we
noted earlier, that correlations that escape the QRPA are important [38, 39].
Table 9 in Appendix B contains more detail than Fig. 4, in particular the
values for each individual EDF of all the matrix elements.

In some nuclei, such as 76Ge, the EDFs all produce comparable values for the
matrix element, but those values are quite different from the experimental
one. The reason for the discrepancy, again, is the quite different degrees of
deformation in the initial and final nuclei, a difference that in reality is
probably made less significant by shape fluctuations. In other nuclei, 96Zr,
100Mo, and 128Te (and especially the first two), the values span a wide range.
The reason is that two local minima appear in the initial isotopes, and the
value of the $2\nu\beta\beta$ matrix element depends strongly on which minimum
is used. In these two cases, the HFB overlaps associated with the lowest
minima for SkO′ are very small (0.087 and 2.6$\times$10-5 in 96Zr and 100Mo),
and the $2\nu\beta\beta$ matrix elements from the lowest minimum are
consequently smaller than the experimental values, while the matrix elements
associated with the other HFB solutions are larger than or comparable to the
experimental values. Correlations that admix states near those other minima,
if they were taken into account, would probably increase the $2\nu\beta\beta$
matrix elements produced by the lowest minima. Such admixtures are beyond what
the QRPA includes, however, and seeing their effects would require an
approximation such as the generator-coordinate method.

Figure 4 also shows that overlaps computed with the QTDA prescription result
in larger matrix elements than those computed with the QRPA prescription.

###  V.2 Predictions

Using the same global EDFs as in the previous section, we compute the
$2\nu\beta\beta$ matrix elements for all the nuclei in which that decay might
conceivably be observed: 46Ca, 70Zn, 80Se, 86Kr, 94Zr, 98Mo, 104Ru, 110Pd,
114Cd, 122Sn, 124Sn, 134Xe, 142Ce, 146Nd, 148Nd, 154Sm, 160Gd, 170Er, 176Yb,
186W, 192Os, 198Pt, 204Hg, 226Ra, 232Th, 244Pu, and 248Cm. Figure 5 summarizes
the results, while Table 10 in Appendix B indicates the individual EDFs
responsible for each symbol in the figure. We emphasize that we are able to
make these predictions only because we use EDFs that are fit globally and
without considering $2\nu\beta\beta$ half-lives. In typical QRPA calculations,
by contrast, the strength of isoscalar pairing is adjusted in each nucleus
individually to reproduce the $2\nu\beta\beta$ half-life.

As we mentioned in the previous section, the QRPA nuclear matrix elements may
not be reliable if the deformations of the initial and final states of the
decay are different. The deformation parameters differ by more than 0.1 for
the following decays: 70Zn $\to$ 70Ge, 80Se $\to$ 80Kr, 134Xe $\to$ 134Ba, and
146Nd $\to$ 146Sm with SkO′ and 80Se $\to$ 80Kr, 114Cd $\to$ 114Sn, 122Sn
$\to$ 122Te, 134Xe $\to$ 134Ba, and 146Nd $\to$ 146Sm with SV-min. We also saw
earlier that the QRPA can go awry if shape mixing is important. A full
treatment of shape mixing requires something like the generator coordinate
method [40, 41], but we can get a good idea of when it will be significant by
examining potential energy curves. These turn out to be broad near the minimum
for the nuclei 46Ti, 70Ge, 94Zr (only SkO′), 98Mo, 104Ru, 110Pd, 114Cd, 122Te,
124Te, 134Ba, 142Ce, 198Pt, and 198Hg. Unfortunately, the generator coordinate
method, while it has been applied to $0\nu\beta\beta$ decay [37, 42, 43, 38,
44, 45, 46, 47, 48, 49, 50, 51, 52] is difficult to apply to $2\nu\beta\beta$
decay because the closure approximation is poor there and a complete set of
intermediate states is required.

With the parameter set 4, the pnFAM converges more slowly than with the other
parameter sets, and the resulting matrix elements are often quite different
from those produced by the other sets. Thus we exclude set 4 from the
distribution of the nuclear matrix elements shown in Fig. 5. We see better
agreement among the other EDFs in heavier isotopes as a general rule, and the
QTDA prescription for the overlap again leads to larger numbers than does QRPA
prescription.

##  VI Conclusions

We have presented a computationally efficient framework for calculating the
matrix elements for two-neutrino double-beta decay within nuclear density
functional theory. We employ the finite amplitude method to compute the QRPA
approximation to the matrix elements. Our approach allows large single-
particle model spaces and the use of a single nuclear EDF for all nuclei. It
also eliminates the need to truncate two-quasiparticle spaces.

We first used harmonic-oscillator-based HFB and FAM codes together with
familiar EDFs to compute the $2\nu\beta\beta$ matrix elements in a few
important nuclei, comparing the results with those obtained previously by
diagonalizing the QRPA matrix. Using EDFs that had been fit globally to
single-$\beta$ decay rates and giant-resonance energies, we then computed the
$2\nu\beta\beta$ matrix elements in all nuclei in which double-beta decay has
or could be observed. Agreement with the matrix elements extracted from
already measured half-lives is good in general, and we offered predictions for
those nuclei that have unmeasured half-lives.

Although we focus on $2\nu\beta\beta$ decay in this paper, we can also compute
double-electron capture matrix elements in the same way. The most interesting
extension of our work is to neutrinoless double-beta decay. The presence of a
neutrino propagator in that matrix element, however, will make that process
more challenging to treat than $2\nu\beta\beta$ decay.

## Acknowledgments

We are grateful to Mika T. Mustonen for the valuable discussions. This work is
supported by JSPS KAKENHI Grants No. 17H05194, No. 19KK0343, No. 20H05242, and
No. 20K03964, and by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under grant No. DE-FG02-97ER41019. This research was
conducted in part during the INT program INT-17-2a “Neutrinoless Double-Beta
Decay,” at the Institute for Nuclear Theory, University of Washington.
Numerical calculations were performed at the Oakforst-PACS Systems through the
Multidisciplinary Cooperative Research Program of the Center for Computational
Sciences, University of Tsukuba.

##  Appendix A Overlap

###  A.1 QRPA overlap

We follow the discussion in Ref. [25] to evaluate the overlap
$\langle\lambda_{f},K|\lambda_{i},K\rangle$ of two intermediate states. The
QRPA phonon operators that excite the initial HFB state are related to the
those that excite the final state by

$\displaystyle\hat{\cal
Q}^{\lambda_{i}{\dagger}}_{K}=\sum_{\lambda_{f}}\left(a_{\lambda_{i}\lambda_{f}}\hat{\cal
Q}^{\lambda_{f}{\dagger}}_{K}+b_{\lambda_{i}\lambda_{f}}\hat{\tilde{{\cal
Q}}}^{\lambda_{f}}_{K}\right)\,,$ (28)

where $\hat{\tilde{{\cal Q}}}_{K}^{\lambda}=\hat{{\cal Q}}_{-K}^{\lambda}$.
This relation is based on the fact that both operators span the complete set
of two-quasiparticle states with angular momentum projection $K$. The overlap
of the intermediate state can be written in terms of the phonon operators as

$\displaystyle\langle\lambda_{f},K|\lambda_{i},K\rangle$
$\displaystyle=\langle 0^{+}_{f,{\rm QRPA}}|\hat{\cal
Q}^{\lambda_{f}}_{K}\hat{\cal Q}^{\lambda_{i}{\dagger}}_{K}|0^{+}_{i,{\rm
QRPA}}\rangle$ $\displaystyle=\sum_{\lambda_{f}^{\prime}}\left(\langle
0^{+}_{f,{\rm QRPA}}|\hat{\cal Q}^{\lambda_{f}}_{K}\hat{\cal
Q}^{\lambda_{f}^{\prime}{\dagger}}_{K}|0^{+}_{i,{\rm QRPA}}\rangle
a_{\lambda_{i}\lambda_{f}^{\prime}}\right.$ $\displaystyle\quad+\left.\langle
0^{+}_{f,{\rm QRPA}}|\hat{\cal Q}^{\lambda_{f}}_{K}\hat{\cal
Q}^{\lambda_{f}^{\prime}}_{-K}|0^{+}_{i,{\rm QRPA}}\rangle
b_{\lambda_{i}\lambda_{f}^{\prime}}\right)$ $\displaystyle\approx
a_{\lambda_{i}\lambda_{f}}\langle 0^{+}_{f,{\rm HFB}}|0^{+}_{i,{\rm
HFB}}\rangle\,,$ (29)

where we neglect the term proportional to
$b_{\lambda_{i}\lambda_{f}^{\prime}}$, because it involves a two-phonon state,
and approximate the overlap between the two QRPA correlated ground states.

We have two sets of the quasiparticles, one defined for the initial HFB state
and the other for the final state:

$\displaystyle\hat{a}^{(i)}_{\mu}|0^{+}_{i,{\rm
HFB}}\rangle=0,\quad\hat{a}^{(f)}_{\mu}|0^{+}_{f,{\rm HFB}}\rangle=0,$ (30)

with $\mu$ a proton or neutron single-particle state with positive angular
momentum $j_{z}$ along the symmetry axis. We write the transformation between
the two sets of the quasiparticles in the form

$\displaystyle\hat{a}^{(i){\dagger}}_{\mu}$
$\displaystyle={\sum_{\nu\in\tau}}^{\prime}\left({\cal
R}_{\mu\nu}\hat{a}^{(f){\dagger}}_{\nu}+{\cal
S}_{\mu\bar{\nu}}\hat{a}^{(f)}_{\bar{\nu}}\right),$ (31)
$\displaystyle\hat{a}^{(i){\dagger}}_{\bar{\mu}}$
$\displaystyle={\sum_{\nu\in\tau}}^{\prime}\left({\cal
R}_{\bar{\mu}\bar{\nu}}\hat{a}^{(f){\dagger}}_{\bar{\nu}}+{\cal
S}_{\bar{\mu}\nu}\hat{a}^{(f)}_{\nu}\right),$ (32)

where ${\sum}^{\prime}$ means that the summation is only over states with
$j_{z}>0$, and the notation $\nu\in\tau$ means that index $\nu$ corresponds to
the same kind of particle (proton or neutron) as does the index $\mu$ on the
left side of the equation.

The relation

$\displaystyle{\cal R}^{T}{\cal R}^{\ast}+{\cal S}^{\dagger}{\cal S}$
$\displaystyle=I,$ (33) $\displaystyle{\cal R}^{T}{\cal S}^{\ast}+{\cal
S}^{\dagger}{\cal R}$ $\displaystyle=0$ (34)

follows from the unitarity of the transformation.

This transformation is defined in the full quasiparticle model space; any
quasiparticle cutoff thus breaks unitarity. Because the matrix composed of
${\cal R}$ and ${\cal S}$ is also unitary, the inverse transformation is given
by

$\displaystyle\hat{a}^{(f){\dagger}}_{\mu}$
$\displaystyle={\sum_{\nu\in\tau}}^{\prime}\left({\cal
S}_{\bar{\nu}\mu}\hat{a}^{(i)}_{\bar{\nu}}+{\cal
R}^{\ast}_{\nu\mu}\hat{a}^{(i){\dagger}}_{\nu}\right),$ (35)
$\displaystyle\hat{a}^{(f){\dagger}}_{\bar{\mu}}$
$\displaystyle={\sum_{\nu\in\tau}}^{\prime}\left({\cal
S}_{\nu\bar{\mu}}\hat{a}^{(i)}_{\nu}+{\cal
R}^{\ast}_{\bar{\nu}\bar{\mu}}\hat{a}^{(i){\dagger}}_{\bar{\nu}}\right).$ (36)

Using the Bogoliubov transformation

$\displaystyle\hat{a}^{(i/f){\dagger}}_{\mu}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i/f)}_{\bar{k}\mu}\hat{c}_{\bar{k}}+U^{(i/f)}_{k\mu}\hat{c}^{\dagger}_{k},$
(37) $\displaystyle\hat{a}^{(i/f){\dagger}}_{\bar{\mu}}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i/f)}_{k\bar{\mu}}\hat{c}_{k}+U^{(i/f)}_{\bar{k}\bar{\mu}}\hat{c}^{\dagger}_{\bar{k}},$
(38)

we can write the matrix elements of ${\cal R}$ and ${\cal S}$ in the form

$\displaystyle{\cal R}_{\mu\nu}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i)}_{\bar{k}\mu}V^{(f)\ast}_{\bar{k}\nu}+U^{(i)}_{k\mu}U^{(f)\ast}_{k\nu},$
(39) $\displaystyle{\cal R}_{\bar{\mu}\bar{\nu}}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i)}_{k\bar{\mu}}V^{(f)\ast}_{k\bar{\nu}}+U^{(i)}_{\bar{k}\bar{\mu}}U^{(f)\ast}_{\bar{k}\bar{\nu}},$
(40) $\displaystyle{\cal S}_{\mu\bar{\nu}}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i)}_{\bar{k}\mu}U^{(f)}_{\bar{k}\bar{\nu}}+U^{(i)}_{k\mu}V^{(f)}_{k\bar{\nu}},$
(41) $\displaystyle{\cal S}_{\bar{\mu}\nu}$
$\displaystyle={\sum_{k\in\tau}}^{\prime}V^{(i)}_{k\bar{\mu}}U^{(f)}_{k\nu}+U^{(i)}_{\bar{k}\bar{\mu}}V^{(f)}_{\bar{k}\nu}.$
(42)

Defining the proton-neutron two-quasiparticle creation and annihilation
operators

$\displaystyle\bm{A}^{(i){\dagger}}(pn,K)\equiv\hat{a}^{(i){\dagger}}_{p}\hat{a}^{(i){\dagger}}_{n},\quad\bm{A}^{(i){\dagger}}(\bar{p}\bar{n},K)\equiv\hat{a}^{(i){\dagger}}_{\bar{p}}\hat{a}^{(i){\dagger}}_{\bar{n}},$
(43)

we can relate the two-quasiparticle operators defined with respect to the
initial and final HFB states in the following way:

$\displaystyle\bm{A}^{(i){\dagger}}(pn,K)$
$\displaystyle={\sum_{p^{\prime}n^{\prime}}}^{\prime}\left[{\cal
R}_{pp^{\prime}}{\cal
R}_{nn^{\prime}}\bm{A}^{(f){\dagger}}(p^{\prime}n^{\prime},K)\right.$
$\displaystyle\quad\left.-{\cal S}_{p\bar{p}^{\prime}}{\cal
S}_{n\bar{n}^{\prime}}\bm{A}^{(f)}(\bar{p}^{\prime}\bar{n}^{\prime},K)\right]+(\hat{a}^{\dagger}\hat{a}{\rm-
terms}),$ (44) $\displaystyle\bm{A}^{(i)}(\bar{p}\bar{n},K)$
$\displaystyle={\sum_{p^{\prime}n^{\prime}}}^{\prime}\left[{\cal
R}_{\bar{p}\bar{p}^{\prime}}^{\ast}{\cal
R}_{\bar{n}\bar{n}^{\prime}}^{\ast}\bm{A}^{(f)}(\bar{p}^{\prime}\bar{n}^{\prime},K)\right.$
$\displaystyle\quad\left.-{\cal S}_{\bar{p}p^{\prime}}^{\ast}{\cal
S}_{\bar{n}n^{\prime}}^{\ast}\bm{A}^{(f){\dagger}}(p^{\prime}n^{\prime},K)\right]+(\hat{a}^{\dagger}\hat{a}{\rm-
terms}).$ (45)

The QRPA phonon operator is a combination of two-quasiparticle creation and
annihilation operators:

$\displaystyle\hat{\cal Q}^{\lambda_{i}{\dagger}}_{K}$
$\displaystyle={\sum_{pn}}^{\prime}\left[X^{\lambda_{i}}_{pn,K}\bm{A}^{(i){\dagger}}(pn,K)-Y^{\lambda_{i}}_{pn,K}\bm{A}^{(i)}(\bar{p}\bar{n},K)\right],$
(46) $\displaystyle\hat{\tilde{{\cal Q}}}^{\lambda_{i}}_{K}$
$\displaystyle={\sum_{pn}}^{\prime}\left[X^{\lambda_{i}}_{pn,K}\bm{A}^{(i)}(\bar{p}\bar{n},K)-Y^{\lambda_{i}}_{pn,K}\bm{A}^{(i){\dagger}}(pn,K)\right].$
(47)

Inverting this yields the relation

$\displaystyle\bm{A}^{(f){\dagger}}(pn,K)$
$\displaystyle=\sum_{\lambda_{f}}\left[X^{\lambda_{f}\ast}_{pn,K}\hat{\cal
Q}^{\lambda_{f}{\dagger}}_{K}+Y^{\lambda_{f}\ast}_{pn,K}\hat{\tilde{{\cal
Q}}}^{\lambda_{f}}_{K}\right],$ (48)
$\displaystyle\bm{A}^{(f)}(\bar{p}\bar{n},K)$
$\displaystyle=\sum_{\lambda_{f}}\left[X^{\lambda_{f}\ast}_{pn,K}\hat{\tilde{{\cal
Q}}}^{\lambda_{f}}_{K}+Y^{\lambda_{f}\ast}_{pn,K}\hat{\cal
Q}^{\lambda_{f}{\dagger}}_{K}\right],$ (49)

which leads to an expression for the $a$ matrix in Eq. (A.1):

$\displaystyle a_{\lambda_{i}\lambda_{f}}$
$\displaystyle={\sum_{pnp^{\prime}n^{\prime}}}^{\prime}\left[X^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}{\cal
R}_{pp^{\prime}}{\cal R}_{nn^{\prime}}X^{\lambda_{i}}_{pn,K}\right.$
$\displaystyle\quad\left.-Y^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}{\cal
R}_{\bar{p}\bar{p}^{\prime}}^{\ast}{\cal
R}_{\bar{n}\bar{n}^{\prime}}^{\ast}Y^{\lambda_{i}}_{pn,K}\right.$
$\displaystyle\quad\left.+X^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}{\cal
S}_{\bar{p}p^{\prime}}^{\ast}{\cal
S}_{\bar{n}n^{\prime}}^{\ast}Y^{\lambda_{i}}_{pn,K}\right.$
$\displaystyle\quad\left.-Y^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}{\cal
S}_{p\bar{p}^{\prime}}{\cal
S}_{n\bar{n}^{\prime}}X^{\lambda_{i}}_{pn,K}\right].$ (50)

In Ref. [25], the contribution from ${\cal S}$ in Eq. (50) is neglected.

The overlap between the HFB states is given by the Onishi formula, e.g., in
Eq. (E.49) of Ref. [40]:

$\displaystyle{\cal N}^{-1}$ $\displaystyle=\langle 0^{+}_{f,{\rm
HFB}}|0^{+}_{i,{\rm HFB}}\rangle=\langle 0^{+}_{i,{\rm HFB}}|0^{+}_{f,{\rm
HFB}}\rangle$ $\displaystyle=\left(\det{\cal
R}^{T}\right)^{\frac{1}{2}}=\left[\det\left(1+{\cal D}^{\dagger}{\cal
D}\right)\right]^{-\frac{1}{4}},$ (51)

where ${\cal D}$ is a skew-symmetric matrix that determines the relation
between the initial and final HFB states through

$\displaystyle|0^{+}_{i,{\rm HFB}}\rangle$ $\displaystyle={\cal
N}^{-1}\exp\left(\sum_{\tau}\sum_{\mu\nu\in\tau}{\cal
D}_{\mu\nu}\hat{a}^{(f){\dagger}}_{\mu}\hat{a}^{(f){\dagger}}_{\nu}\right)|0^{+}_{f,{\rm
HFB}}\rangle,$ (52)

and satisfies the relation

$\displaystyle{\cal D}={\cal S}^{\dagger}({\cal R}^{\dagger})^{-1}=-({\cal
R}^{-1}{\cal S})^{\ast}=-{\cal D}^{T}.$ (53)

Thus we end up with

$\displaystyle\langle\lambda_{f},K|\lambda_{i},K\rangle$
$\displaystyle=(\det{\cal
R})^{\frac{1}{2}}{\sum_{pnp^{\prime}n^{\prime}}}^{\prime}{\cal
R}_{pp^{\prime}}{\cal R}_{nn^{\prime}}$
$\displaystyle\quad\times\left(X^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}X^{\lambda_{i}}_{pn,K}-Y^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}Y^{\lambda_{i}}_{pn,K}\right).$
(54)

###  A.2 QTDA overlap

Reference [8] uses the QTDA to evaluate the overlap among intermediate states,
which are given by

$\displaystyle|\lambda_{i/f},K\rangle=\sum_{pn}X_{\mu\nu,K}^{\lambda_{i/f}}\hat{a}^{(i/f){\dagger}}_{p}\hat{a}^{(i/f){\dagger}}_{n}|0^{+}_{i/f,{\rm
HFB}}\rangle.$ (55)

From Eqs. (52) and (55), we find that

$\displaystyle\langle\lambda_{f},K|\lambda_{i},K\rangle$
$\displaystyle=(\det{\cal
R})^{\frac{1}{2}}{\sum_{pnp^{\prime}n^{\prime}}}^{\prime}X_{p^{\prime}n^{\prime},K}^{\lambda_{f}\ast}X_{pn,K}^{\lambda_{i}}$
$\displaystyle\quad\times\left({\cal
R}_{pp^{\prime}}+2\sum_{p^{\prime\prime}}{\cal S}_{pp^{\prime\prime}}{\cal
D}_{p^{\prime\prime}p^{\prime}}\right)$ $\displaystyle\quad\times\left({\cal
R}_{nn^{\prime}}+2\sum_{n^{\prime\prime}}{\cal S}_{nn^{\prime\prime}}{\cal
D}_{n^{\prime\prime}n^{\prime}}\right).$ (56)

The two QRPA and QTDA overlaps in Eqs. (A.1) and (A.2) can be written in the
same form as

$\displaystyle\langle\lambda_{f},K|\lambda_{i},K\rangle$
$\displaystyle={\sum_{pnp^{\prime}n^{\prime}}}^{\prime}\left(X^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}X^{\lambda_{i}}_{pn,K}-\alpha
Y^{\lambda_{f}\ast}_{p^{\prime}n^{\prime},K}Y^{\lambda_{i}}_{pn,K}\right)$
$\displaystyle\quad\times{\cal O}_{pp^{\prime}}(\alpha){\cal
O}_{nn^{\prime}}(\alpha),$ (57)

where ${\cal O}$ is a matrix that does not depend on the QRPA and includes the
HFB overlap and the transformation relating the initial and final
quasiparticle states:

$\displaystyle{\cal O}_{\rho\rho^{\prime}}(\alpha)=\left[\det{\cal
R}^{(\tau)}\right]^{\frac{1}{2}}\left[{\cal
R}_{\rho\rho^{\prime}}+2(1-\alpha){\sum_{\rho^{\prime\prime}\in\tau}}^{\prime}{\cal
S}_{\rho\rho^{\prime\prime}}{\cal
D}_{\rho^{\prime\prime}\rho^{\prime}}\right].$ (58)

Here $\rho,\rho^{\prime}$ are both proton or both neutron states, and ${\cal
R}^{(\tau)}$ is the neutron or proton part of the matrix ${\cal R}$. The QRPA
expression in Ref. [25] corresponds $\alpha=1$ and the QTDA expression in Ref.
[8] to $\alpha=0$.

##  Appendix B  Numerical Results for Matrix Elements in Individual Nuclei

Table 9 below provides details related to Fig. 4. Table 10 does the same for
Fig. 5.

Table 9: Dimensionless Gamow-Teller $2\nu\beta\beta$ nuclear matrix element $m_{e}c^{2}M_{\rm GT}^{2\nu}$ computed with the SkO′\- and SV-min-based EDFs, with globally fitted proton-neutron parts [15]. The value of the matrix element is compared with experimental values extracted from Ref. [5]. The overlap of the intermediate states is evaluated both with QTDA ($\alpha=0$, Ref. [8]) and QRPA ($\alpha=1$, Ref. [25]) prescriptions. The numbers in parentheses denote powers of 10.  | $\alpha$ | 1A | 1B | 1C | 1D | 1E | 2 | 3A | 3B | 4 | 5 | Exp.
---|---|---|---|---|---|---|---|---|---|---|---|---
48Ca | 0 | 0.0759 | 0.0734 | 0.0763 | 0.0722 | 0.0934 | 0.0698 | 0.0729 | 0.0879 | 0.0706 | 0.118 | 0.035$\pm$0.003
| 1 | 0.0399 | 0.0386 | 0.0401 | 0.0382 | 0.0489 | 0.0372 | 0.0386 | 0.0463 | 0.0385 | 0.0588 |
76Ge | 0 | 0.0496 | 0.0477 | 0.0496 | 0.0441 | 0.062 | 0.0469 | 0.0462 | 0.0588 | 0.0502 | 0.0426 | 0.106$\pm$ 0.004
| 1 | 0.0343 | 0.033 | 0.0344 | 0.0304 | 0.0431 | 0.0331 | 0.0319 | 0.0409 | 0.0355 | 0.0293 |
82Se | 0 | 0.0572 | 0.0547 | 0.0572 | 0.05 | 0.0736 | 0.0543 | 0.0528 | 0.0693 | 0.0567 | 0.061 | 0.085$\pm$ 0.001
| 1 | 0.0464 | 0.0444 | 0.0464 | 0.0404 | 0.0599 | 0.0463 | 0.0428 | 0.0564 | 0.0474 | 0.0485 |
96Zr | 0 | 0.0265 | 0.0257 | 0.0267 | 0.026 | 0.0321 | 0.123 | 0.0267 | 0.032 | 0.0296 | 0.0228 | 0.080$\pm$0.004
| 1 | 0.0133 | 0.0129 | 0.0134 | 0.013 | 0.0164 | 0.108 | 0.0134 | 0.0164 | 0.015 | 0.0113 |
(96Zr sph.) | 0 | 0.202 | 0.197 | 0.204 | 0.202 | 0.243 | 0.123 | 0.207 | 0.244 | 0.232 | 0.173 |
| 1 | 0.18 | 0.175 | 0.181 | 0.178 | 0.219 | 0.108 | 0.183 | 0.22 | 0.206 | 0.152 |
100Mo | 0 | $1.71(-5)$ | $1.67(-5)$ | $1.72(-5)$ | $1.69(-5)$ | $2.02(-5)$ | 0.234 | $1.73(-5)$ | $2.01(-5)$ | $1.7(-5)$ | $1.53(-5)$ | 0.151$\pm$ 0.005
| 1 | $2.67(-6)$ | $2.58(-6)$ | $2.68(-6)$ | $2.57(-6)$ | $3.25(-6)$ | 0.2 | $2.67(-6)$ | $3.24(-6)$ | $3.16(-6)$ | $2.32(-6)$ |
(100Mo pro.) | 0 | 0.166 | 0.161 | 0.167 | 0.162 | 0.201 | 0.234 | 0.166 | 0.199 | 0.179 | 0.147 |
| 1 | 0.142 | 0.138 | 0.143 | 0.137 | 0.175 | 0.2 | 0.142 | 0.173 | 0.16 | 0.125 |
116Cd | 0 | 0.11 | 0.107 | 0.111 | 0.108 | 0.131 | 0.0606 | 0.11 | 0.13 | 0.082 | 0.0997 | 0.108$\pm$0.003
| 1 | 0.0728 | 0.0708 | 0.0732 | 0.0707 | 0.0865 | 0.0132 | 0.0725 | 0.086 | 0.0541 | 0.0655 |
128Te | 0 | 0.0161 | 0.0153 | 0.0161 | 0.0137 | 0.0215 | 0.0124 | 0.0149 | 0.0207 | 0.00873 | 0.0131 | 0.043$\pm$0.003
| 1 | 0.00993 | 0.00944 | 0.00994 | 0.00848 | 0.0134 | 0.00695 | 0.00923 | 0.0129 | 0.00626 | 0.00808 |
(128Xe obl.) | 0 | 0.0306 | 0.0291 | 0.0306 | 0.0263 | 0.0407 | 0.0263 | 0.0285 | 0.0393 | 0.0155 | 0.0251 |
| 1 | 0.0195 | 0.0185 | 0.0195 | 0.0165 | 0.0264 | 0.0154 | 0.018 | 0.0253 | 0.0106 | 0.0159 |
130Te | 0 | 0.0227 | 0.0215 | 0.0227 | 0.0189 | 0.0308 | 0.0215 | 0.0208 | 0.0295 | 0.0149 | 0.0185 | 0.0293$\pm$0.0009
| 1 | 0.0168 | 0.0159 | 0.0168 | 0.0141 | 0.0229 | 0.0151 | 0.0154 | 0.0219 | 0.0118 | 0.0138 |
136Xe | 0 | 0.0222 | 0.0208 | 0.0221 | 0.0173 | 0.0318 | 0.0238 | 0.0194 | 0.0296 | 0.0184 | 0.0232 | 0.0181$\pm$ 0.0006
| 1 | 0.018 | 0.0169 | 0.018 | 0.0139 | 0.0261 | 0.0201 | 0.0156 | 0.0243 | 0.0136 | 0.0175 |
150Nd | 0 | 0.0413 | 0.0395 | 0.0414 | 0.0369 | 0.0541 | 0.0552 | 0.0399 | 0.0536 | 0.0511 | 0.0341 | 0.044 $\pm$ 0.005
| 1 | 0.0345 | 0.0329 | 0.0346 | 0.0308 | 0.0455 | 0.0463 | 0.0334 | 0.0451 | 0.0425 | 0.0284 |
238U | 0 | 0.0462 | 0.044 | 0.0462 | 0.039 | 0.0616 | 0.048 | 0.0434 | 0.0609 | 0.0717 | 0.0374 | 0.13${}^{+0.09}_{-0.07}$
| 1 | 0.0428 | 0.0407 | 0.0428 | 0.0359 | 0.0573 | 0.0431 | 0.04 | 0.0566 | 0.0597 | 0.0345 |
Table 10: Dimensionless Gamow-Teller $2\nu\beta\beta$ nuclear matrix element $m_{e}c^{2}M_{\rm GT}^{2\nu}$ for nuclei whose half lives have not been measured yet. The column and row labels are the same as in Table 9.  | $\alpha$ | 1A | 1B | 1C | 1D | 1E | 2 | 3A | 3B | 4 | 5
---|---|---|---|---|---|---|---|---|---|---|---
46Ca | 0 | 0.0868 | 0.0849 | 0.0874 | 0.0874 | 0.1 | 0.00611 | 0.0867 | 0.097 | 0.0831 | 0.117
| 1 | 0.0529 | 0.0518 | 0.0534 | 0.0532 | 0.0616 | 0.00332 | 0.0529 | 0.0595 | 0.0534 | 0.07
70Zn | 0 | 0.0848 | 0.0823 | 0.0854 | 0.0823 | 0.102 | 0.189 | 0.0831 | 0.0992 | 0.0982 | 0.0807
| 1 | 0.0263 | 0.0254 | 0.0265 | 0.0248 | 0.0326 | 0.138 | 0.0252 | 0.0312 | 0.0306 | 0.0246
80Se | 0 | 0.0673 | 0.0646 | 0.0674 | 0.0599 | 0.0849 | 0.0343 | 0.0626 | 0.0802 | 0.0657 | 0.0677
| 1 | 0.0505 | 0.0485 | 0.0505 | 0.0446 | 0.0638 | 0.0228 | 0.0468 | 0.0603 | 0.0507 | 0.0495
86Kr | 0 | 0.0308 | 0.0293 | 0.0308 | 0.0256 | 0.0411 | 0.0267 | 0.0274 | 0.0379 | 0.0276 | 0.0433
| 1 | 0.0228 | 0.0217 | 0.0227 | 0.0187 | 0.0303 | 0.0208 | 0.0201 | 0.028 | 0.0206 | 0.0294
94Zr | 0 | 0.166 | 0.162 | 0.168 | 0.169 | 0.195 | 0.0959 | 0.171 | 0.196 | 0.195 | 0.146
| 1 | 0.147 | 0.143 | 0.148 | 0.148 | 0.175 | 0.0841 | 0.151 | 0.175 | 0.172 | 0.127
98Mo | 0 | 0.186 | 0.181 | 0.188 | 0.186 | 0.223 | 0.171 | 0.189 | 0.222 | 0.208 | 0.169
| 1 | 0.149 | 0.144 | 0.15 | 0.146 | 0.181 | 0.148 | 0.149 | 0.18 | 0.168 | 0.133
104Ru | 0 | 0.18 | 0.175 | 0.181 | 0.177 | 0.217 | 0.198 | 0.181 | 0.216 |  $-$0.718 | 0.162
| 1 | 0.155 | 0.15 | 0.156 | 0.15 | 0.189 | 0.16 | 0.155 | 0.188 |  $-$0.149 | 0.137
110Pd | 0 | 0.179 | 0.173 | 0.18 | 0.175 | 0.215 | 0.187 | 0.18 | 0.214 | 0.279 | 0.16
| 1 | 0.127 | 0.123 | 0.128 | 0.123 | 0.154 | 0.111 | 0.127 | 0.153 | 0.134 | 0.113
114Cd | 0 | 0.126 | 0.123 | 0.127 | 0.126 | 0.149 | 0.0506 | 0.128 | 0.148 | 0.102 | 0.115
| 1 | 0.0771 | 0.0751 | 0.0776 | 0.0762 | 0.0911 | 0.00702 | 0.0775 | 0.0908 | 0.0623 | 0.0699
122Sn | 0 | 0.0279 | 0.0265 | 0.028 | 0.0239 | 0.0377 | 0.0193 | 0.026 | 0.0364 | 0.0645 | 0.0226
| 1 | 0.0171 | 0.0162 | 0.0172 | 0.0145 | 0.0234 | 0.00955 | 0.0159 | 0.0225 | 0.0285 | 0.0137
124Sn | 0 | 0.0488 | 0.0462 | 0.0489 | 0.041 | 0.0676 | 0.0263 | 0.0451 | 0.065 | 0.055 | 0.0391
| 1 | 0.0382 | 0.0361 | 0.0382 | 0.0319 | 0.053 | 0.0157 | 0.0351 | 0.0508 | 0.0386 | 0.0303
134Xe | 0 | 0.0203 | 0.0192 | 0.0202 | 0.017 | 0.0274 | 0.0218 | 0.0185 | 0.0261 | 0.0142 | 0.0176
| 1 | 0.015 | 0.0142 | 0.015 | 0.0126 | 0.0204 | 0.016 | 0.0138 | 0.0195 | 0.0114 | 0.0131
142Ce | 0 | 0.0289 | 0.0281 | 0.029 | 0.0277 | 0.0339 | 0.0168 | 0.0291 | 0.0343 | 0.0322 | 0.025
| 1 | 0.0224 | 0.0218 | 0.0225 | 0.0215 | 0.0264 | 0.013 | 0.0226 | 0.0266 | 0.026 | 0.0194
146Nd | 0 | 0.0117 | 0.0113 | 0.0117 | 0.0109 | 0.0145 | 0.0135 | 0.0116 | 0.0145 | 0.0145 | 0.00979
| 1 | 0.00512 | 0.00491 | 0.00514 | 0.0047 | 0.00655 | 0.00832 | 0.00505 | 0.00654 | 0.00663 | 0.00419
148Nd | 0 | 0.116 | 0.112 | 0.117 | 0.109 | 0.149 | 0.109 | 0.116 | 0.149 | 0.137 | 0.0979
| 1 | 0.0937 | 0.0898 | 0.0942 | 0.0869 | 0.121 | 0.0826 | 0.0928 | 0.121 | 0.112 | 0.0782
154Sm | 0 | 0.0725 | 0.0694 | 0.0728 | 0.0652 | 0.0944 | 0.0639 | 0.0701 | 0.0933 | 0.0958 | 0.0603
| 1 | 0.0658 | 0.0629 | 0.066 | 0.0589 | 0.0862 | 0.0571 | 0.0635 | 0.0852 | 0.083 | 0.0545
160Gd | 0 | 0.0847 | 0.081 | 0.085 | 0.0759 | 0.11 | 0.0807 | 0.0819 | 0.109 | 0.097 | 0.0704
| 1 | 0.0766 | 0.0732 | 0.0768 | 0.0682 | 0.1 | 0.0724 | 0.0737 | 0.0994 | 0.0914 | 0.0633
170Er | 0 | 0.0753 | 0.0722 | 0.0756 | 0.0677 | 0.0974 | 0.0844 | 0.0729 | 0.0965 |  $-$0.0389 | 0.0627
| 1 | 0.0651 | 0.0622 | 0.0652 | 0.0578 | 0.0847 | 0.0612 | 0.0626 | 0.0837 | 0.025 | 0.0538
176Yb | 0 | 0.0657 | 0.0627 | 0.0659 | 0.0585 | 0.087 | 0.0391 | 0.0635 | 0.0862 |  $-$0.0577 | 0.0542
| 1 | 0.0557 | 0.0531 | 0.0558 | 0.0493 | 0.074 | 0.026 | 0.0536 | 0.0733 | 0.0181 | 0.0457
186W | 0 | 0.0966 | 0.0923 | 0.0969 | 0.0853 | 0.127 | 0.0637 | 0.0923 | 0.125 |  $-$0.07 | 0.0799
| 1 | 0.0892 | 0.0851 | 0.0894 | 0.0781 | 0.118 | 0.0512 | 0.0848 | 0.116 | 0.00665 | 0.0733
192Os | 0 | 0.0672 | 0.0642 | 0.0673 | 0.0579 | 0.0881 | 0.0927 | 0.0631 | 0.0863 |  $-$0.0446 | 0.0553
| 1 | 0.0618 | 0.0589 | 0.0618 | 0.0529 | 0.0814 | 0.083 | 0.0578 | 0.0797 |  $-$0.0295 | 0.0506
198Pt | 0 | 0.0272 | 0.0262 | 0.0272 | 0.0238 | 0.0341 | 0.0478 | 0.0255 | 0.0333 | 0.396 | 0.0229
| 1 | 0.0167 | 0.016 | 0.0167 | 0.0144 | 0.0209 | 0.0279 | 0.0155 | 0.0204 |  $-$0.0265 | 0.014
204Hg | 0 | 0.0133 | 0.0127 | 0.0132 | 0.0107 | 0.0167 | 0.0195 | 0.0117 | 0.016 |  $-$0.023 | 0.0109
| 1 | 0.0108 | 0.0104 | 0.0107 | 0.0087 | 0.0136 | 0.0161 | 0.00954 | 0.013 | 0.0178 | 0.00885
226Ra | 0 | 0.0739 | 0.0703 | 0.074 | 0.064 | 0.0987 | 0.0706 | 0.0708 | 0.0986 | 0.343 | 0.06
| 1 | 0.068 | 0.0646 | 0.068 | 0.0585 | 0.0913 | 0.061 | 0.0649 | 0.091 | 0.172 | 0.0549
232Th | 0 | 0.0509 | 0.0485 | 0.0509 | 0.0434 | 0.0678 | 0.0531 | 0.0481 | 0.0672 | 0.171 | 0.0414
| 1 | 0.0465 | 0.0443 | 0.0465 | 0.0394 | 0.0623 | 0.0433 | 0.0438 | 0.0617 | 0.0894 | 0.0376
244Pu | 0 | 0.0431 | 0.0409 | 0.043 | 0.0359 | 0.0576 | 0.0454 | 0.0401 | 0.0568 | 0.0265 | 0.0347
| 1 | 0.0399 | 0.0379 | 0.0399 | 0.0331 | 0.0536 | 0.0404 | 0.037 | 0.0528 | 0.0384 | 0.0321
248Cm | 0 | 0.0415 | 0.0394 | 0.0414 | 0.0346 | 0.0552 | 0.0435 | 0.0387 | 0.0545 |  $-$0.00664 | 0.0335
| 1 | 0.0389 | 0.0369 | 0.0388 | 0.0322 | 0.052 | 0.0391 | 0.0361 | 0.0512 | 0.0184 | 0.0313

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Bo

# LabPipe: an extensible informatics platform to streamline management of
metabolomics data and metadata

Bo Zhao  Department of Respiratory Sciences, University of Leicester,
Leicester, UK  NIHR Leicester Biomedical Research Centre, Leicester, UK  Luke
Bryant  Department of Chemistry, University of Leicester, Leicester, UK
Michael Wilde  Department of Respiratory Sciences, University of Leicester,
Leicester, UK  Department of Chemistry, University of Leicester, Leicester, UK
Rebecca Cordell  Department of Chemistry, University of Leicester, Leicester,
UK  Dahlia Salman  Department of Respiratory Sciences, University of
Leicester, Leicester, UK  Dorota Ruszkiewicz  Department of Respiratory
Sciences, University of Leicester, Leicester, UK  Wadah Ibrahim  Department of
Respiratory Sciences, University of Leicester, Leicester, UK  Amisha Singapuri
NIHR Leicester Biomedical Research Centre, Leicester, UK  Tim Coats  NIHR
Leicester Biomedical Research Centre, Leicester, UK  Department of Chemistry,
University of Leicester, Leicester, UK  Department of Cardiovascular Sciences,
Cardiovascular Research Centre, University of Leicester, Leicester, UK  Erol
Gaillard  Department of Respiratory Sciences, University of Leicester,
Leicester, UK  Caroline Beardsmore  Department of Respiratory Sciences,
University of Leicester, Leicester, UK  Toru Suzuki  NIHR Leicester Biomedical
Research Centre, Leicester, UK  Department of Cardiovascular Sciences,
Cardiovascular Research Centre, University of Leicester, Leicester, UK  Leong
Ng  NIHR Leicester Biomedical Research Centre, Leicester, UK  Department of
Cardiovascular Sciences, Cardiovascular Research Centre, University of
Leicester, Leicester, UK  Neil Greening  NIHR Leicester Biomedical Research
Centre, Leicester, UK  Paul Thomas  Department of Respiratory Sciences,
University of Leicester, Leicester, UK  Paul S. Monks  Department of
Chemistry, University of Leicester, Leicester, UK  Christopher Brightling
Department of Respiratory Sciences, University of Leicester, Leicester, UK
NIHR Leicester Biomedical Research Centre, Leicester, UK  Salman Siddiqui
Department of Respiratory Sciences, University of Leicester, Leicester, UK
NIHR Leicester Biomedical Research Centre, Leicester, UK  Robert C. Free
Department of Respiratory Sciences, University of Leicester, Leicester, UK
NIHR Leicester Biomedical Research Centre, Leicester, UK

###### Abstract

Summary:
Data management in clinical metabolomics studies is often inadequate. To
improve this situation we created LabPipe to provide a guided, customisable
approach to study-specific sample collection. It is driven through a local
client which manages the process and pushes local data to a remote server
through an access controlled web API. The platform is able to support data
management for different sampling approaches across multiple sites / studies
and is now an essential study management component for supporting clinical
metabolomics locally at the EPSRC/MRC funded East Midlands Breathomics
Pathology Node.
Availability and Implementation:
LabPipe is freely available to download under a non-commercial open-source
license (NPOSL 3.0) along with documentation and installation instructions at
http://labpipe.org.

###### keywords:

Metabolomics | Data integration | Web services

rob.free and le.ac.uk

## Introduction

The nature of multi-disciplinary biomedical studies makes it difficult to
efficiently and effectively handle the collation and collection of data sets
from different sites and research groups. Effective data management is
particularly important in large clinical metabolomics studies, where equipment
is distributed across multiple sites with different operating protocols.
Moreover, the greater the number of groups and sites involved, the more
difficult it becomes to manage this data using manual approaches such as
removable storage and paper-based forms.

Software tools are available (1) which will manage some aspects of the data
collection process, however, these do not fully support key parts of data
management, which led to the emergence of LabPipe. This was created to
streamline data handling, by i) supporting semi-automated sample data
collection, linkage and transfer; ii) facilitating collection/data entry of
biomedical meta-data linked to the samples; and iii) enabling sample
collection notifications. This manuscript describes the LabPipe platform and
its successful deployment in a large-scale breathomics project, the East
Midlands Breathomics Pathology Node (EMBER) study, handling data from multiple
sites and instruments.

## Implementation

LabPipe was designed using a modular client/server approach (Figure
Implementation) with an extensible plugin architecture enabling new features
to be added when required.

Figure 1: Overview of the LabPipe technical architecture.

At the core of LabPipe is LPserver, a light-weight web service-based API. This
API provides role-based authorisation to prevent access from non-authorised
users. The back-end uses a NoSQL database which enables great flexibility when
storing metadata. It allows LabPipe to support different collection
configurations (which can be loaded using either the command line or API).
These configurations are fetched by each client upon set up and on each use
when networked. LPserver also provides web services allowing access to
metabolomics data/metadata that has been uploaded.

The LabPipe client tool (LPclient) was developed using the Electron framework,
so it can be deployed on multiple operating systems. LPclient is installed
locally on computers, providing an integrated front-end to support local
experimental data collection and to manage data on the server. LPclient
generates forms for different collection protocols using LPserver derived
configurations and form templates. Sample metadata can be entered at the
sample collection point, LPclient then links this metadata to data files.
Additionally, depending on the instrument and operating protocol, client form
processes can be customised to handle linkage of data in various ways. For
instance, LPclient can generate file IDs to use based on the metadata entered,
or identify new/changed files and automatically link these to the metadata.

## Results and Discussion

### EMBER

The LabPipe tool was developed initially to support the EMBER study (2), a
multi-disciplinary breathomics study involving staff from different
backgrounds. EMBER involved multiple clinic visits at which clinical staff
collected breathomics data and samples using analytical chemistry equipment.
Each visit generated multiple sets of metabolomics data which were effectively
managed using LabPipe. The tool also allowed pertinent metadata to be entered
at the time samples were taken.

Breath sampling instruments in the EMBER project included both online
technologies (in which samples were analysed in real-time on the instrument)
and offline technologies (in which samples were collected and later
transferred to be analysed in the lab). The approach to data management varied
by instrument, processing time and metabolomics software used. In EMBER,
LabPipe supported nine instruments (covering four approaches to breath
sampling) across two sites.

### Research and Collaboration

Prior to the deployment of LabPipe, data and metadata were handled manually in
the EMBER study. This was complex to handle for non-technical staff; required
paper-based records; and increased the likelihood of erroneous data re-entry
and data loss / poor quality data due to manual collation/linkage of data from
individual USB storage devices.

A close collaboration between informaticians, researchers and clinical staff
led to a tool which was accessible and easy to use; guiding non-technical
staff through the entire process of sample collection; uploading data to
LPserver; and providing data access as painlessly as possible. Development was
iterative and helped by qualitative surveys and informal team discussions to
assess pain-points in the software - resulting in a better user interface and
experience.

The introduction of LabPipe streamlined research by reducing manual data
handling and management and saving time and effort. Improved data handling
made it easier for researchers to carry out data analysis and link
metabolomics data with clinical data. Additionally, the extensible nature of
LabPipe meant the tool could evolve to meet the study’s needs by adding
extensions (e.g. sending researchers notification emails when samples were
collected).

Locally installed clients meant LabPipe could integrate with the file system
and handle file transfers automatically without requiring manual uploading of
data through a web portal. Despite, problems with intermittent wireless
network access, a built-in fall-back option meant LPclient would transfer
files to LPserver when a network connection became available.

### Other options

Before developing LabPipe, we investigated existing bioinformatics data
management tools using the BIBBOX resource as a reference (1). Most of the
documented tools specialised in particular aspects of data collection and
management. For instance, i2b2 (3), tranSMART (4) and LabKey (5) focused on
data warehousing; OpenSpecimen (6) tissue tracking; REDCap (7) and OpenClinica
(8) electronic Clinical Research Forms (eCRF) data collection; and SeedDMS (9)
document sharing and storing. While these are undoubtedly useful tools, none
of them supported our key need to create streamlined data flows with automated
transfer of data from local PCs. While there may be commercial proprietary
tools which provide some of the features of LabPipe, to the best of our
knowledge this side of data management is not handled by existing open/free
informatics solutions.

### Future work

Development of LabPipe has focused on metabolomics studies, but the platform
is generic enough that it could be used to handle data collection from other
lab-based approaches (e.g. flow cytometry, real time PCR etc.). Future work
will look at how LabPipe could be adapted to automatically transform/process
data into data standards such as ISA (10), for which integrated support for
ontologies would be required. To facilitate this, the form handling
capabilities are being improved to allow more complex setup such as multi-
faceted data entry, enabling eCRF data collection alongside the collection of
sample meta-data. Additionally, auditing capabilities are being enhanced so
that data access and entry is fully logged.

## Conclusion

LabPipe provides a user friendly, streamlined, yet fully configurable approach
to collecting data and metadata in metabolomics studies. It is now an
established component in the Leicester BRC and is supporting multi-
disciplinary metabolomics research in Leicester through its underpinning of
seven existing studies and similar upcoming research.

## Funding

This work was supported by the Medical Research Council (MRC), Engineering and
Physical Sciences Research Council (EPSRC) Stratified Medicine Grant for
Molecular Pathology Nodes (Grant No. MR/N005880/1), Midlands Asthma and
Allergy Research Association (MAARA) And British Lung Foundation (Grant No.
BLFPHD17-1). The work was carried out at the University Hospitals of Leicester
NHS Trust, University of Leicester and Loughborough University, supported by
the NIHR Leicester Biomedical Research Centre and the NIHR Leicester Clinical
Research Facility. The views expressed are those of the author(s) and not
necessarily those of the NHS, the NIHR or the Department of Health and Social
Care. The authors would like to acknowledge the invaluable efforts of the
research nurses responsible for the in-clinic sample collection as well as the
input from the wider East Midlands Breathomics Pathology Node consortium
(members list can be found at: https://ember.le.ac.uk/web).

###### Acknowledgements.

The authors would like to thank the EMBER nursing team for working with the
LabPipe team and providing feedback and comments.

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