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\begin{document}

\title{Double Beta Decay in the Context of EMPM and STDA Methods}

\author{%
  Daria Denisova\thanks{denisova.d.i95@gmail.com} \and
  Petr Veselý \\
  \\
  Nuclear Physics Institute, Czech Academy of Sciences, \\
  250 68 Řež, Czech Republic
}

\date{} % leave empty if you don't want a date

\begin{document}
\maketitle



\begin{abstract}

In this work we study the $2\nu\beta\beta$ decay of $^{48}$Ca using the Second Tamm–Dancoff Approximation (STDA) and the Equation of Motion Phonon Method (EMPM). We demonstrate the numerical equivalence of the two approaches when up to two-phonon configurations are included in the EMPM. Our implementation yields consistent Fermi and Gamow–Teller NMEs, but also confirms that a strong quenching of the axial-vector coupling constant $g_A$ is required to reproduce the experimental half-life. We outline a strategy to incorporate more complex nuclear configurations to address this quenching issue and improve the predictive power of the methods for $0\nu\beta\beta$ NMEs.
\end{abstract}


\section{Introduction}

Double-beta decay links nuclear, particle, and atomic physics, while providing a potential probe of physics beyond the Standard Model. Two modes are distinguished: the two-neutrino process ($2\nu\beta\beta$), allowed for Dirac neutrinos, and the neutrinoless process ($0\nu\beta\beta$), which requires Majorana neutrinos. Observation of $0\nu\beta\beta$ would demonstrate lepton-number violation and yield information on the absolute neutrino mass scale, the mass hierarchy, and the origin of matter–antimatter asymmetry~\cite{Dolinski2019}.  

Nuclear matrix elements (NMEs) for $0\nu\beta\beta$ cannot be measured directly and must be calculated within nuclear structure models, which differ considerably~\cite{Engel2017,Simkovic2018}. In contrast, $2\nu\beta\beta$ decay has been observed in 11 isotopes, providing an important benchmark for testing such models~\cite{Barabash2020}. Since both modes involve the same initial and final nuclei and share common operators, precise $2\nu\beta\beta$ data can constrain parameters relevant to $0\nu\beta\beta$.  

%In earlier work we applied the Second Tamm–Dancoff Approximation (STDA) to $^{48}$Ca. Here we extend this study by employing both STDA and the Equation of Motion Phonon Method (EMPM), demonstrating their numerical equivalence. Our corrected formalism confirms that reproducing the experimental half-life requires strong quenching of the axial-vector coupling constant $g_A$, and suggests strategies to address this through more complex nuclear configurations.



\section{Theoretical Formalism}

We adopt the nuclear Hamiltonian in the form
\begin{equation}
\hat{H} = \sum_{i} e_i : a_i^\dagger a_i : +
\frac{1}{4} \sum_{ijkl} V^{NN}_{ijkl} : a_i^\dagger a_j^\dagger a_l a_k : ,
\end{equation}
where $a_i^\dagger$ and $a_i$ are the creation and annihilation operators of a nucleon,
respectively. The index $i=(n_i l_i j_i m_i \tau_i)$ denotes the set of single-particle quantum numbers:
principal quantum number $n$, orbital momentum $l$, total angular momentum $j$,
its projection $m$, and the isospin projection $\tau$ distinguishing protons and neutrons.
The symbols $:\ :$ denote normal ordering with respect to the unperturbed ground state
$|\Psi_0\rangle$, i.e.\ the Slater determinant where the lowest single-particle states are filled.
The single-particle energies $e_i$ are taken from a Coulomb-corrected Woods--Saxon
potential~\cite{Bohr1969}, while the interaction matrix elements $V^{NN}_{ijkl}$ are Brueckner
$G$-matrices based on the Argonne V18 nucleon--nucleon potential~\cite{Wiringa1995}.

\subsection*{Tamm--Dancoff Approximation (TDA)}

In the TDA method~\cite{Ring1980} the Hamiltonian is diagonalized in the space of all one
particle--hole excitations $a_p^\dagger a_h |\Psi_0\rangle$. The corresponding eigenvalue equation reads
\begin{equation}
\sum_{p'h'} \Big[ (e_p-e_h)\,\delta_{pp'}\delta_{hh'}
+ V^{NN}_{p'hh'p} \Big] c^{\mu}_{p'h'}
= E^\text{TDA}_\mu \, c^\mu_{ph},
\end{equation}
yielding excitation energies $E^\text{TDA}_\mu$ and amplitudes $c^\mu_{ph}$.
Here $p,p'$ ($h,h'$) denote particle (hole) states. Since particle--hole excitations may convert a neutron into a proton (and vice versa), the spectra of the $(A,Z)$ nucleus as well as $(A,Z\pm1)$ neighbors are obtained.
%As proton $\leftrightarrow$ neutron excitations are included, the method provides spectra not only of the parent nucleus $(A,Z)$ but also of neighboring $(A,Z\pm 1)$ isotopes.

\subsection*{Second Tamm--Dancoff Approximation (STDA)}

The STDA~\cite{Papakonstantinou2010} extends the configuration space by including two particle--two hole
states $a_{p_1}^\dagger a_{p_2}^\dagger a_{h_1} a_{h_2} |\Psi_0\rangle$. The eigenstates are
\begin{equation}
|\text{STDA};\mu\rangle =
\left( \sum_{ph} X^\mu_{ph} a_p^\dagger a_h
+ \sum_{p_1<p_2,\,h_1<h_2} X^\mu_{p_1 p_2 h_1 h_2}
a_{p_1}^\dagger a_{p_2}^\dagger a_{h_1} a_{h_2}
\right) |\Psi_0\rangle .
\end{equation}
These states describe not only the nucleus $(A,Z)$ but also its even--even
neighbors $(A,Z\pm 2)$, and odd--odd $(A,Z\pm 1)$ systems since particle--hole excitations may convert a neutron into a proton (and vice versa).

\subsection*{Equation of Motion Phonon Method (EMPM)}

The EMPM~\cite{DeGregorio2017} constructs recursively a basis of
$n$-phonon states $|\alpha_n\rangle$ generated from lower-order phonon states.
In the present work we restrict to $n=0,1,2$. The $n=0$ subspace corresponds
to the unperturbed ground state $|\Psi_0\rangle$, while $n=1$ spans the TDA
solutions,
the two-phonon ($n=2$) states represent correlated 2p–2h excitations.
When truncated to $n=0,1,2$, EMPM is numerically equivalent to STDA~\cite{Knapp2023},
but additionally allows center-of-mass spuriosity removal (e.g.\ via the SVD method~\cite{DeGregorio2021}).

\subsection*{Double Beta Decay Matrix Elements}

For the two-neutrino double beta decay ($2\nu\beta\beta$), the Fermi (F) and Gamow--Teller (GT) nuclear
matrix elements are defined as
\begin{equation}
M_{F/GT} = \sum_{\lambda}
\frac{\langle f \Vert \hat{O}_{F/GT} \Vert \lambda \rangle
\langle \lambda \Vert \hat{O}_{F/GT} \Vert \Psi_0 \rangle}
{E^\text{TDA}_\lambda - \tfrac{1}{2}\left( E_f^{\text{STDA/EMPM}} - E_i \right)} ,
\end{equation}
where $|f\rangle$ is the ground state of the daughter nucleus obtained in STDA or EMPM,
$E_i$ is the parent ground state energy, which is shifted to zero in our calculations, and $|\lambda\rangle$ are the TDA states
of the intermediate nucleus. The transition operators are $\hat{O}_F=\tau^+$ and
$\hat{O}_{GT}=\tau^+ \boldsymbol{\sigma}$, where $\tau^+$ changes a neutron to a proton.
For even--even nuclei, only intermediate $0^+$ (Fermi) and $1^+$ (Gamow--Teller) multipoles contribute.

Finally, the $2\nu\beta\beta$ half-life is given by
\begin{equation}
\left( T_{1/2}^{2\nu} \right)^{-1}
= g_A^4 m_e^2 \,
\left| M_{GT} - \frac{M_F}{g_A^2} \right|^2 \,
G^{2\nu},
\end{equation}
where $g_A$ is the axial-vector coupling constant, $m_e$ the electron mass,
and $G^{2\nu}$ the phase-space factor~\cite{Engel2017,Simkovic2018}.



% Reaction
%\begin{equation}
%^{48}\mathrm{Ca} \; \rightarrow \; ^{48}\mathrm{Ti} 
%+ 2e^- + 2\bar{\nu}_e .
%\end{equation}



\section{Results}
The $2\nu\beta\beta$ decay of $^{48}$Ca was studied with the TDA describing the
intermediate $^{48}$Sc, and the STDA or EMPM applied to the daughter $^{48}$Ti.
The calculations were performed in a single-particle model space up to
$N=6$ major oscillator shells, with Woods--Saxon single-particle energies and
a $G$-matrix interaction derived from the Argonne V18 potential~\cite{Wiringa1995}.

%The ground-state energy of the intermediate nucleus $^{48}$Sc relative to $^{48}$Ca was obtained as $E^{\text{TDA}}_{6^+_1}=-1.219$~MeV, about $0.43$~MeV below the experimental value corresponding to $Q_\beta=0.2795$~MeV. For the daughter nucleus $^{48}$Ti, both STDA and EMPM give essentially identical ground-state energies $E^{\text{STDA/EMPM}}_{0^+_1}=-4.425$~MeV, which is $0.86$~MeV higher than the value implied by the measured $Q_{\beta\beta}=4.268$~MeV.

The calculated Fermi and Gamow--Teller NMEs are summarized in
Table~\ref{tab:results}. The STDA and EMPM NMEs are numerically
equivalent within three digits, confirming the formal equivalence of the two
methods when two-phonon configurations are included.

\begin{table}[h!]
\centering
\caption{Nuclear Matrix Element (NME) results for $^{48}$Ca.}
\label{tab:results}
\vspace{0.5cm}
\begin{tabular}{lccc}
\hline
Method & $M_F$ (MeV$^{-1}$) & $M_{GT}$ (MeV$^{-1}$) & $ NME(tot)$ \\
\hline
STDA  & $-4.075 \times 10^{-2}$ & $0.1102$ & $0.1113$ \\
EMPM  & $-4.067 \times 10^{-2}$ & $0.1112$ & $0.1121$ \\
Expt. & ---                     & ---      & $0.0348$ \\
\hline
\end{tabular}
\end{table}

\begin{equation}
 NME(tot) = g_A^2 m_e \cdot \left| M_{GT} - \tfrac{M_F}{g_A^2}\right| 
\end{equation}
with $g_A = 1.27$ and $m_e = 0.511$ MeV.

Compared with our earlier STDA-only results~\cite{Vesely2025}, the new
implementation yields larger $|M_F|$ values, indicating stronger isospin-symmetry
breaking. While renormalization of the proton--neutron interaction
($g_{ph}$) had previously been suggested as a remedy, the present results show
that isospin restoration cannot be achieved this way. The discrepancy between
calculated and experimental NMEs instead points to the need for a substantial
quenching of $g_A$.%, in the range $q \approx 0.4$--$0.6$.



\section{Summary and Outlook}

We studied the $2\nu\beta\beta$ decay of $^{48}$Ca using the STDA and EMPM approaches for the daughter nucleus $^{48}$Ti, with the TDA describing the intermediate $^{48}$Sc, and found the STDA and EMPM to be numerically equivalent when two-phonon configurations are included in the EMPM~\cite{Knapp2023}.
  

Our calculations yield Fermi and Gamow–Teller
NMEs that are larger than values inferred from experiment. This implies a strong
quenching of the axial-vector coupling constant $g_A$. The large
Fermi contribution signals isospin-symmetry breaking that cannot be resolved by simple
renormalization of the proton–neutron interaction.  

Future work will extend the formalism by including 2p–2h or equivalently two-phonon correlations in the parent and
intermediate nuclei, applying center-of-mass corrections, and testing alternative NN
interactions. These steps are expected to clarify the origin of $g_A$ quenching and to
improve predictions of $0\nu\beta\beta$ NMEs in $^{48}$Ca and heavier systems.






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\end{thebibliography}


\end{document}
