# Determining $g_{A}/g_{V}$ with High Resolution Spectral Measurements Using
an LiInSe2 Bolometer

A.F. Leder<EMAIL_ADDRESS>Massachusetts Institute of Technology, 77
Massachusetts Ave. Cambridge, MA 02139, USA  Department of Nuclear
Engineering, University of California - Berkeley, 2521 Hearst Ave, Berkeley,
CA 94709, USA  D. Mayer<EMAIL_ADDRESS>Massachusetts Institute of Technology,
77 Massachusetts Ave. Cambridge, MA 02139, USA  J. L. Ouellet  Massachusetts
Institute of Technology, 77 Massachusetts Ave. Cambridge, MA 02139, USA  F. A.
Danevich  Institute for Nuclear Research of NASU, Kyiv 03028, Ukraine  L.
Dumoulin  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  A.
Giuliani  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  J.
Kostensalo  Natural Resources Institute Finland, Yliopistokatu 6B, FI-80100
Joensuu, Finland  J. Kotila  Department of Physics, University of Jyväskylä,
P.O. Box 35, FI-40014 Jyväskylä, Finland  Finnish Institute for Educational
Research, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
Center for Theoretical Physics, Sloane Physics Laboratory Yale University, New
Haven, Connecticut 06520-8120, USA  P. de Marcillac  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay, France  C. Nones  Commissariat á l’Énergie
Atomique (CEA)- Saclay, 91191 Gif-sur-Yvette, France  V. Novati  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  E. Olivieri  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  D. Poda  Université
Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  J. Suhonen  Department
of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland
V.I. Tretyak  Institute for Nuclear Research of NASU, Kyiv 03028, Ukraine  L.
Winslow<EMAIL_ADDRESS>Massachusetts Institute of Technology, 77
Massachusetts Ave. Cambridge, MA 02139, USA  A. Zolotarova  Commissariat á
l’Énergie Atomique (CEA)- Saclay, 91191 Gif-sur-Yvette, France

###### Abstract

Neutrinoless Double-Beta decay (0$\nu\beta\beta$) processes sample a wide
range of intermediate forbidden nuclear transitions, which may be impacted by
quenching of the axial vector coupling constant ($g_{A}/g_{V}$), the
uncertainty of which plays a pivotal role in determining the sensitivity reach
of 0$\nu\beta\beta$ experiments. In this Letter, we present measurements
performed on a high-resolution LiInSe2 bolometer in a “source=detector”
configuration to measure the spectral shape of the 4-fold forbidden
$\beta$-decay of 115In. The value of $g_{A}/g_{V}$ is determined by comparing
the spectral shape of theoretical predictions to the experimental $\beta$
spectrum taking into account various simulated background components as well
as a variety of detector effects. We find evidence of quenching of
$g_{A}/g_{V}$ at $>5\sigma$ with a model-dependent quenching factor of
$0.655\pm 0.002$ as compared to the free-nucleon value for the Interacting
Shell Model. We also measured the 115In half-life to be [$5.18\pm
0.06(\text{stat.})^{+0.005}_{-0.015}(\text{sys.})]\times{10}^{14}$ yr within
the Interacting Shell Model framework. This work demonstrates the power of the
bolometeric technique to perform precision nuclear physics single-$\beta$
decay measurements, which can help reduce the uncertainties in the calculation
of $0\nu\beta\beta$ nuclear matrix elements.

††preprint: APS/123-QED

CUORE

    Cryogenic Underground Observatory for Rare Events
0$\nu\beta\beta$

    Neutrinoless Double-Beta decay
SM

    Standard Model
NTD

    Neutron Transmutation Doped
FWHM

    full-width half-max
MC

    Monte Carlo
MCMC

    Markov-Chain Monte Carlo
ROI

    region of interest
NME

    Nuclear Matrix Element
OT

    Optimum Trigger
LD

    light detector
IBM

    Interacting Boson Model
ISM

    Interacting Shell Model
MQPM

    Microscopic Quasiparticle-Phonon Model
ML

    maximum likelihood
BCR

    Bayesian Credibility Region
MAP

    maximum a posteriori

##  I Introduction

From the first observation of single $\beta$-decay Chadwick (1914) that led W.
Pauli to propose the neutrino Pauli (1930) and the subsequent efforts to
develop a theory of $\beta$-decay by E. Fermi Fermi (1934) to C.S. Wu’s
ground-breaking work to determine the vector and axial vector form of the weak
interaction Wu  _et al._ (1957), the study of $\beta$-decay has been used to
elucidate the hidden world of nuclear and particle physics. Modern efforts
continue this legacy, using nuclear $\beta$-decay to investigate the
properties of neutrino mass including its absolute scale through endpoint
measurements Aker  _et al._ (2019, 2021, 2022), and possible Majorana origin
through searches for Neutrinoless Double-Beta decay ($\nu\beta\beta$) Gando
_et al._ (2016); Alvis  _et al._ (2019); Anton  _et al._ (2019); Agostini  _et
al._ (2020); Adams  _et al._ (2020, 2022); Armengaud  _et al._ (2021);
Azzolini  _et al._ (2019a); Alenkov  _et al._ (2019).

In recent years, cryogenic bolometers have established themselves as a
powerful technology in rare event searches for $\nu\beta\beta$ Adams  _et al._
(2020, 2022); Armengaud  _et al._ (2021); Azzolini  _et al._ (2019a); Agostini
_et al._ (2020); Alvis  _et al._ (2019); Alenkov  _et al._ (2019), direct Dark
Matter detection Abdelhameed  _et al._ (2019); Alkhatib  _et al._ (2021a);
Agnese  _et al._ (2019), and more Augier  _et al._ (2021); Alkhatib  _et al._
(2021b); Pyle  _et al._ (2015); Poda (2021); De Marcillac  _et al._ (2003).
Such detectors operate at milli-kelvin temperatures and measure energy
deposition events by converting phonons into a temperature increase within a
sensitive thermistor. Bolometers benefit from excellent energy resolution,
high electron containment efficiencies, low energy trigger thresholds, and
strong particle-ID capabilities when equipped with a dual heat/light or
heat/ionization readout Poda (2021); Huang  _et al._ (2021); Azzolini  _et
al._ (2019a). Additionally, the ability to operate nearly any crystalline
material as a bolometer provides practical means to study a very wide range of
long-lived nuclear processes for which sufficient quantities of isotope may be
procured and grown into crystalline form.

As pointed out by Barea  _et al._ (2013), theoretical calculations of the
nuclear physics contributions to the $\nu\beta\beta$ half-life have often
assumed an axial-to-vector coupling ratio equal to that of the free neutron,
$g_{A}/g_{V}$ = 1.276 Mund  _et al._ (2013); Markisch  _et al._ (2019), though
it is common to use a quenched value to obtain agreement with observed
single-$\beta$ transition rates Kumar  _et al._ (2016); Chou  _et al._ (1993);
Brown (1988); Martínez-Pinedo  _et al._ (1996). The exact impact on
$\nu\beta\beta$ will depend on the underlying physics of axial quenching
Suhonen (2017); recently Ref. Gysbers  _et al._ (2019) provided evidence that
the inclusion of two-nucleon currents and additional correlations may provide
an explanation within light ($A\leq 14$) nuclei and certain super-allowed
heavy nuclei $\beta$-decay transitions. Axial quenching creates a significant
uncertainty in the interpretation of any $\nu\beta\beta$ search when
converting isotope-specific half-lives back to the underlying physics of
interest Dolinski  _et al._ (2019), on top of the existing spread in the value
of calculated Nuclear Matrix Elements (NMEs) for $\nu\beta\beta$ isotopes
Engel and Menéndez (2017).

As was proposed in Haaranen  _et al._ (2016), the _shape_ of highly-forbidden
$\beta$-decay spectra can be very sensitive to $g_{A}/g_{V}$, and studying
such decays of nuclei with mass around $A\sim 100$ could shed light on axial
quenching in a similar nuclear environment as those found in $\nu\beta\beta$
decays. This analysis technique could also have applications in explaining
reactor flux anomalies through examination of 1st-order forbidden
$\beta$-decay transitions Hayen  _et al._ (2019). This spectral shape
technique was used for the first time in Alessandrello  _et al._ (1994), where
the experimental data from a CdWO4 scintillation detector Belli  _et al._
(2007) were compared to theoretical spectra in order to extract a value for
$g_{A}$ in the range of 0.90–0.93. More recently, COBRA has applied this
spectral shape approach to their data of CdZnTe detectors in order to obtain a
range for $g_{A}$ between 0.92 and 0.96 depending on the theoretical models
used Bodenstein-Dresler  _et al._ (2020). In this Letter, we make a precision
$\beta$-decay spectral shape measurement using a high-resolution
“source=detector” bolometer. In particular, we study the 4-fold forbidden
$\beta$-decay of ${}^{115}{\rm In}\rightarrow{}^{115}{\rm Sn}$ with
$Q_{\beta}=497.489$ keV Wang  _et al._ (2021) with the most recent previously
measured half-life of ($4.41\pm 0.25)\times 10^{14}$ years Pfeiffer  _et al._
(1979). This decay occurs in a mass range relevant to $\nu\beta\beta$ isotopes
of interest and provides a benchmark to test whether many-body nuclear
calculations are capable of simultaneously explaining the $\beta$-decay
spectral shape and rate. Recently, interest has been growing to measure this
particular ${}^{115}{\rm In}$ decay mode by examining an In2O3 bolometer in
order to provide a measurement of $g_{A}/g_{V}$ Celi  _et al._ (2022). Here we
use a LiInSe2 crystal with a natural abundance of ${}^{115}{\rm In}$ (95.72 %
Meija  _et al._ (2016)), to evaluate $g_{A}/g_{V}$ for leading nuclear models,
and make the most precise measurement of the ${}^{115}{\rm In}$ half-life to
date.

##  II Methods

Figure 1: (Left) Photo of the LiInSe2 bolometer with an NTD thermistor
attached to the crystal. (Right) The combined detector setup in a tower
configuration with two pairs of bolometers stacked in two stages. The light
detector is placed above each ”stage” of the tower for maximum photon
absorption

The LiInSe2 crystal was grown by RMD Inc. 11144 Hunt Street, Watertown, MA
02472, email<EMAIL_ADDRESS>website: rmdinc.com using the vertical Bridgman
process Isaenko  _et al._ (2005); Tower  _et al._ (2020). The crystal was
enriched in 6Li to 95% for potential use as a neutron detector Bell  _et al._
(2015); Tupitsyn  _et al._ (2012), however, that analysis is beyond the scope
of this work and does not affect the $\beta$-decay analysis. The LiInSe2
crystal was instrumented with a Neutron Transmutation Doped (NTD) thermistor
Haller  _et al._ (1984), and installed inside a cryostat at IJCLab (ex. CSNSM)
in Orsay, France Danevich  _et al._ (2018), see Fig. 1. The LiInSe2
scintillation signal was monitored by a separate Neganov-Trofimov-Luke Ge
light detector (LD) Novati  _et al._ (2019), which allowed us to perform
particle identification and pile-up rejection. 42.2 g$\cdot$days of data was
collected over two weeks, with the pertinent measured/derived experimental
parameters summarized in Table 1.

Table 1: Experimental parameters of the LiInSe2 crystal during the October-November 2017 data runs. Detector Parameter | LiInSe2 Crystal
---|---
Crystal Dimensions |  $1.3\times 1.6\times 0.7$ cm
Total Crystal Mass | 10.3 grams
Effective ${}^{115}{\rm In}$ Mass | 4.1 grams
Noise Level | 1.1 keV (1$\sigma$)
Avg. Energy Resolution | 2.4 keV (1$\sigma$)
100 % Trigger Threshold | 20.0 keV
Analysis Threshold | 160 keV
Containment Eff. | 96.6% @ 497 keV
Data Selection Cut Eff. | 47.6(2)% ($160-500$ keV)
Livetime Fraction | 52.54(8)%
Total Exposure | 39.7 g$\cdot$days

The data was processed using the Apollo/Diana software developed by the CUORE
Alduino  _et al._ (2016)/CUPID-0 Azzolini  _et al._ (2018)/CUORICINO Andreotti
_et al._ (2011) collaborations. Events are triggered with the Optimum Trigger
(OT) Di Domizio  _et al._ (2011) and processed following a procedure similar
to Alduino  _et al._ (2016); Adams  _et al._ (2020). The trigger threshold was
determined by injecting a series of low energy pulses through the attached
Joule heater Andreotti  _et al._ (2012), achieving $\sim 100$% trigger
efficiency above 20 keV. The LiInSe2 detector is calibrated with a set of
dedicated runs with a 133Ba source using the four most prominent $\gamma$
peaks in the energy range 250–400 keV.

Figure 2: LiInSe2 detector events with $3\sigma$ cut bands, analysis and
trigger thresholds superimposed. The corresponding rise times were collected
in 10 keV energy bins running between 20-450 keV and then each bin of rise
times were individually fit to a Gaussian. The cut band was then defined by
interpolating between the individual 3$\sigma$ profiles cuts as a function of
energy. Outside of the 20–450 keV energy range, the cut values were kept
constant due to large uncertainties in the profile fit parameters as a result
of non-Gaussian parameter distributions or low statistics at the low/high
energy ranges respectively.

The internal ${}^{115}{\rm In}$ decay on its own results in an expected event
rate of $\approx 1.2$ Hz in the 10.3 gram LiInSe2 detector, which means that
internal event pile-up is expected to be a significant background. The
recovery time after an event is $\sim$200 ms, and the event window around each
event includes 100 ms before the trigger and 500 ms after. Together, these
lead to a significant paralyzable deadtime. The faster response time of the LD
allows us to efficiently tag and remove these pile-up events that might
otherwise slip through the LiInSe2 data quality cuts (see Fig. 2), along with
tagging $\alpha$ events through particle-identification via event-by-event
light-yield cuts.

In order to filter out spurious events from 115In $\beta^{-}$ events, a series
of loose pulse shape cuts were employed to filter out electrical glitch and
badly reconstructed events. Then a rise time pulse quality cut (see Fig. 2)
was defined by a 3$\sigma$ cut band determined by fitting the resulting pulse
shape variable profiles across each energy bin. We also employ a coincidence
cut that enforces a single-event criterion. We require that an event is
included in the final spectrum if it appears on both the LiInSe2 and the LD
detectors within 20 ms and no other events are recorded on the LiInSe2
detector within a broader 600 ms window. Over the region of 160–500 keV, we
find a cut efficiency of ($47.6\pm 0.2$)%, dominated by the LD single-event
criterion. The 160 keV threshold was selected as the lowest energy where
multiple event pile up was well handled by the autoconvolution background
component. The resulting events that pass all the above cuts are then compiled
into the input LiInSe2 spectrum as shown in Fig. 3.

Figure 3: Spectral fit to the collected LiInSe2 spectrum over the region
160–1520 keV. Component normalizations and the ${}^{115}{\rm In}$ spectral
shape correspond to the best-fit values for the Interacting Shell Model (ISM)
exhibiting a $\chi^{2}$ value of 160 with 101 degrees of freedom. Fits to the
Microscopic Quasi-particle-Phonon Model (MQPM) and Interacting Boson Model
(IBM) result in similar reconstructions. The bottom panel displays Data/Fit
ratios for the reconstruction, along with 1$\sigma$ (purple), 2$\sigma$ (red)
and 3$\sigma$ (yellow) fit credibility regions. The spectrum is binned by 5
keV up until 530 keV and by 30 keV above 530 keV in order to maintain
reasonable statistics per bin above the ${}^{115}{\rm In}$ endpoint.

To extract $g_{A}/g_{V}$ from the measured LiInSe2 spectrum, we follow a
procedure similar to Adams  _et al._ (2021); Azzolini  _et al._ (2019b);
Armengaud  _et al._ (2020); Alduino  _et al._ (2017) and decompose it into
various components: a model-dependent signal component from the $\beta$-decay
of ${}^{115}{\rm In}$ which will depend on $g_{A}/g_{V}$, an untagged pile-up
component, and other radioactive background contributions. The fit is
implemented using the Bayesian Analysis Toolkit package Caldwell  _et al._
(2009), which implements a Markov Chain Monte-Carlo (MCMC) to sample the full
joint posterior. We perform this decomposition on the spectrum in Fig. 3,
which has a binning of 5/30 keV below/above 530 keV respectively up until the
analysis cut-off at 1520 keV. This binning scheme allows for the fitting of as
many broad spectral/peak features as possible present in the experimental data
while maintaining the highest possible statistics per bin in the region beyond
530 keV. Despite the low trigger threshold of the LiInSe2 crystal, we
implement an analysis threshold of 160 keV to avoid low-energy pile-up events
which are difficult to separate in time and can distort the spectrum.

To implement the MCMC, we define our binned likelihood as:

$\mathcal{L}=\prod_{i}\text{Pois}\left(k_{i};\sum_{j}a_{j}\lambda_{ij}\right),$
(1)

enumerating bins by $i$ and fitted components by $j$. Here, $k_{i}$ is the
number of observed counts within a given bin, $\lambda_{ij}$ is the normalized
density of the $j^{\text{th}}$ component within the $i^{\text{th}}$ bin, and
$a_{j}$ are the fitted normalizations for the different components. The
densities $\lambda$ corresponding to ${}^{115}{\rm In}$ are
$g_{A}/g_{V}$-dependent.

A numerical calculations for the structure of ${}^{115}{\rm In}$ are performed
using ISM Iwata  _et al._ (2016); Menéndez  _et al._ (2009); Horoi and Neacsu
(2016), IBM Barea  _et al._ (2015) and MQPM Toivanen and Suhonen (1998). The
resulting $\beta$-decay spectrum is generated as a function of energy for each
of these structural models taking $g_{A}/g_{V}$ as an input. We generate a
library of 200 discrete $\beta$-decay spectra for $g_{A}/g_{V}$ uniformly
spaced across the range $0.6<g_{A}/g_{V}<1.3$ and then perform an
interpolation for the spectral shape for $g_{A}/g_{V}$ values not in our
library. Each ${}^{115}{\rm In}$ spectrum is then convolved with an energy-
dependent detector response function to account for energy losses as well as
shifts in the spectral shape from $\beta$-particles that escape the absorber.
This is calculated through a Geant4 simulation Allison  _et al._ (2016) that
only simulates the LiInSe2 crystal and the copper plate it rests on. These
simulations find that 96.6% of electrons at the $\beta$-decay endpoint will be
fully contained within the detector, which represents the minimum containment
efficiency over the ${}^{115}{\rm In}$ spectrum. Background component spectra
are obtained by simulating in Geant4 various possible radiogenic
contaminations including from daughter nuclei on various detector/cryostat
components, for example the copper cryostat cold plates and lead shielding
surrounding the detector. In total, we simulated the $\gamma$/$\beta$ spectra
stemming from 238U/232Th decay chains as well as 60Co, and 40K decays present
uniformly throughout the LiInSe2 detector, to simulate possible contamination
of the various cyrostat components particularly the copper plate near the
detector, and from external environmental sources. In addition, we simulated a
separate background contribution coming solely from possible surface
contaminations of the LiInSe2 crystal. All these input spectra components were
then convolved with the detectors’ measured energy resolution and binned into
the same binning scheme as the data before their use as a potential component
of the MCMC fit. The inclusion of a pile-up component (the autoconvolution of
the ${}^{115}{\rm In}$ $\beta$-spectrum) was designed to account for the
inability to separate events which occur too closely in time and could then be
mis-reconstructed as a single higher energy event.

The final MCMC fit only included the four most-dominant background components:
1/2) internal crystal contamination stemming from the 238U decay chains and
60Co decays, 3) 232Th decay chain events on the copper plate underneath the
LiInSe2 crystal, and 4) 232Th decay chain events from external sources mostly
in the form of $\gamma$s. Contamination from $\alpha$ surface backgrounds can
be ignored, thanks to the strong pulse shape and coincidence cuts that were
applied to the collected data, resulting in predominantly bulk $\gamma$
backgrounds. All other simulated background components were found to have only
a negligible effect on the final fit parameters. This results in a
satisfactory description of background features in the collected spectrum
without introducing degeneracies in the fit from additional components which
may not be differentiated with available data. We perform a separate fit for
each nuclear model tested, and apply uniform priors to the normalizations of
each fitted component within the regions of $g_{A}/g_{V}$ discussed below.

##  III Discussion

For all three nuclear models examined, the likelihood function within the fit
is bi-modal with respect to $g_{A}/g_{V}$, exhibiting a local minimum both at
low-$g_{A}/g_{V}$ values below 0.95, and at high-$g_{A}/g_{V}$ values above
1.05. Fits arising from the high-$g_{A}/g_{V}$ minimum result in a poor match
to the observed spectral shape, with decreases in log-likelihood as compared
to the low-$g_{A}/g_{V}$ minimum of at least 65 (IBM), 90 (MQPM) and 118
(ISM). Despite resulting in an overall worse fit, the high-$g_{A}/g_{V}$ fit
minima are still sufficiently favored that without a restricted prior, the
MCMC chain will take an unreasonably long time to achieve convergence. In
order to ensure a good convergence of the MCMC chain about the global minimum
while avoiding numerical instabilities, we restrict ourselves to a uniform
prior on $g_{A}/g_{V}$ $\in[0.6,1.0]$.

We extract the best-fit values from the maximum a posteriori point (which we
will refer to as the “best-fit” values), along with Bayesian Credibility
Regions (BCRs) for parameters of interest pertaining to the ${}^{115}{\rm In}$
decay rate and value of $g_{A}/g_{V}$. We marginalize over all background
component normalizations as nuisance parameters; all three fits result in
compatible contributions from each of the included background components. The
best-fit values for $g_{A}/g_{V}$ along with the central $1\sigma$ BCRs
arising from the fits are summarized in Table 2. Unsurprisingly, the various
nuclear calculations prefer different values of $g_{A}/g_{V}$, however all
models strongly reject the free-nucleon value of $g_{A}/g_{V}$ $=1.276$ at
$>5\sigma$ as determined by the $\Delta\log\mathcal{L}$ between the best-fit
values and the free-nucleon value, assuming Wilk’s theorem Wilks (1938).

Table 2: Fit results for each of the three nuclear models considered. For the parameters of interest of $g_{A}/g_{V}$ and $T_{1/2}$ for ${}^{115}{\rm In}$, we quote the best fit value with uncertainty given by the width of the central 68% Bayesian credibility interval, along with the reduced-$\chi^{2}$ value for the best-fit reconstruction. Model | $g_{A}/g_{V}$ |  $T_{1/2}$ ($10^{14}$ yr) | Reduced $\chi^{2}$
---|---|---|---
ISM | $0.830\pm 0.002$ | $5.177\pm 0.060$ | 1.58
IBM | $0.845\pm 0.006$ | $5.031\pm 0.065$ | 1.50
MQPM | $0.936\pm 0.003$ | $5.222\pm 0.061$ | 1.60
Pfeiffer et al. Pfeiffer  _et al._ (1979) |  | $4.41\pm 0.25$ |
Watt and Glover Watt and Glover (1962) |  | $5.1\pm 0.4$ |
Beard and Kelly Beard and Kelly (1961) |  | $6.9\pm 1.5$ |

Additionally, using the normalization of the ${}^{115}{\rm In}$ component, we
can extract the value of the half-life $T_{1/2}(^{115}\mathrm{In})=[5.18\pm
0.06(\text{stat.})^{+0.005}_{-0.015}(\text{sys.})]\times{10}^{14}$ years. Here
we quote the best-fit value arising from the ISM model fit, with statistical
uncertainty determined by the width of the $1\sigma$ central BCR with
negligible contributions from uncertainties in the cut and live-time
efficiencies which are propagated on top of the fitted ${}^{115}{\rm In}$
normalization. We choose to quote the spread in half-life with respect to the
IBM and MQPM best-fit values (shown in Table 2) as a systematic uncertainty.
This is slower by 3$\sigma$ with respect to the measurement within Pfeiffer
_et al._ (1979), but falls within 2$\sigma$ of the older, less precise
measurements Watt and Glover (1962); Beard and Kelly (1961). Figure 4b)
displays the joint 2-dimensional Bayesian credibility regions for
$g_{A}/g_{V}$ and $T_{1/2}$ for each fitted nuclear model, along with the
best-fit points.

Each of the nuclear models calculations discussed in this letter are able to
simultaneously calculate the $T_{1/2}$ as a function of $g_{A}/g_{V}$ values
Haaranen  _et al._ (2017) as shown by the dash-doted lines in Figure 4. In our
analysis, our best fit values for the half-life given by each model
overestimates the half-lives by factors of 1.2 (IBM), 2.2 (MQPM), and 2.0
(ISM) compared to Pfeiffer  _et al._ (1979), and simultaneously does not fall
upon one of the theory curves. This suggests that quenching-dependent
calculations that we used are not yet able to simultaneously match the
spectral shape and decay rate in ${}^{115}{\rm In}$. It is worth noting that
the half-life in Pfeiffer  _et al._ (1979) is similarly incapable of
simultaneously matching the spectral shape and decay rate.

Previous work with COBRA 113Cd data has shown that the tension between the
independently measured half-life and the quenched $g_{A}/g_{V}$ values
extracted from the spectral shape analysis can be relaxed via the introduction
of a small relativistic nuclear matrix element correction that affects the
spectral shape due to the enforcement of the conserved vector current
assumption Kostensalo  _et al._ (2021). Additionally due to the closeness of
our results with the measurements presented in Watt and Glover (1962); Beard
and Kelly (1961), we do not present any conclusion regarding the accuracy of
any single nuclear model presented here. This letter seeks to showcase the
ability of this technique to simultaneously provide two additional
experimental cross checks to any nuclear calculation model, namely spectral
shape and half-life, on any provided nuclear model able to address highly
forbidden nuclear $\beta$-decays.

Figure 4: Top: Half-lives verses $g_{A}/g_{V}$ theory curves (dash-dot lines),
assuming a conserved vector current Feynman and Gell-Mann (1958), for
${}^{115}{\rm In}$ as well as the best fit half-lives and $g_{A}/g_{V}$ values
(markers) resulting from the spectral-shape fits for the IBM (red), ISM
(yellow), and MQPM (cyan) models considered in this Letter. Bottom: Inset of
above, focused around the experimentally determined half-life values for
${}^{115}{\rm In}$. Contours about the best fit values represent the joint
two-dimensional Bayesian credibility regions produced from the fit posteriors
and only include statistical uncertainties. The previous half-life measurement
from Pfeiffer  _et al._ (1979) is shown in gray with $1\sigma$ uncertainty
(other measurements omitted for clarity).

##  IV Conclusion

From these data, it is clear that the value of $g_{A}/g_{V}$ that governs this
highly forbidden decay process is quenched by approximately 0.65–0.75 compared
to the decay of the free neutron. Interestingly, for each of the three nuclear
models examined there is strong disagreement between the measured half-life
from Pfeiffer  _et al._ (1979) and the predicted half-life value for the
favored value of $g_{A}/g_{V}$ calculated from spectral shape analysis. This
tension could point to possible issues with regards to the many-body
approaches and Hamiltonians used in the various calculation frameworks. At the
same time, our better agreement with the older measurements of Watt and Glover
(1962); Beard and Kelly (1961) may point to additional systematics effects
that could play a vital role in the determination of any half-life
measurement/calculation.

This measurement shows the utility of cryogenic bolometers for precision
studies across multiple energy bins to test various spectral shapes that stem
from rare/forbidden nuclear processes. Further developments in cryogenic
detectors which exhibit faster timing resolution, such as those using TESs for
heat and/or light readout, would provide better separation of low-energy pile-
up events and could offer even better energy resolutions than the NTDs used in
this experiment Armatol  _et al._ (2021); Huang  _et al._ (2021). Coupled with
further improvements in the theory calculations of the nuclear matrix elements
Kumar  _et al._ (2020, 2021), this would allow for future studies of
${}^{115}{\rm In}$ and other candidate isotopes for such as 113Cd Kostensalo
_et al._ (2021) (for an expanded list see Ejiri  _et al._ (2019)) further
increasing the sensitivity to $g_{A}/g_{V}$ and opening the door to reducing
this source of uncertainty on the nuclear matrix elements utilized by
$\nu\beta\beta$ experiments in their current and projected sensitivity limits.

###### Acknowledgements.

The dilution refrigerator used for the tests and installed at IJCLab (Orsay,
France) was donated by the Dipartimento di Scienza e Alta Tecnologia of the
Insubria University (Como, Italy). This work makes use of the Diana data
analysis and Apollo data acquisition software which has been developed by the
CUORICINO, CUORE, LUCIFER, CUPID-Mo and CUPID-0 collaborations. A.F.L. also
acknowledges the support of the California Alliance Fellowship. J.K. would
like to acknowledge funding from Academy of Finland Grant Nos. 314733 and
345869. This work has also been partially supported by the Academy of Finland
under the Academy Project No. 318043. F.A.D. and V.I.T. were supported in part
by the National Research Foundation of Ukraine Grant No. 2020.02/0011 and
would like to acknowledge the heroic efforts of the Armed Forces of Ukraine.

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  *[$\nu\beta\beta$]: Neutrinoless Double-Beta decay
  *[NTD]: Neutron Transmutation Doped
  *[LD]: light detector
  *[OT]: Optimum Trigger
  *[ISM]: Interacting Shell Model
  *[MQPM]: Microscopic Quasiparticle-Phonon Model
  *[IBM]: Interacting Boson Model