# The Gallium Neutrino Absorption Cross Section and its Uncertainty

W. C. Haxton  Department of Physics, University of California, Berkeley, CA
94720, USA  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  E.
J. Rule  Department of Physics, University of California, Berkeley, CA 94720,
USA  S. R. Elliott  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
V. N. Gavrin  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow 117312, Russia  T. V. Ibragimova  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow 117312, Russia

###### Abstract

In the recent Baksan Experiment on Sterile Transitions (BEST), a suppressed
rate of neutrino absorption on a gallium target was observed, consistent with
earlier results from neutrino source calibrations of the SAGE and GALLEX/GNO
solar neutrino experiments. The BEST collaboration, utilizing a 3.4 MCi 51Cr
neutrino source, found observed-to-expected counting rates at two very short
baselines of $R=0.791\pm 0.05$ and $0.766\pm 0.05$, respectively. Among recent
neutrino experiments, BEST is notable for the simplicity of both its neutrino
spectrum, line neutrinos from an electron-capture source whose intensity can
be measured to a estimated precision of 0.23%, and its absorption cross
section, where the precisely known rate of electron capture to the gallium
ground state, 71Ge$(e^{-},\nu_{e})^{71}$Ga(g.s.), establishes a minimum value.
However, the absorption cross section uncertainty is a common systematic in
the BEST, SAGE, and GALLEX/GNO neutrino source experiments. Here we update
that cross section, considering a variety of electroweak corrections and the
role of transitions to excited states, to establish both a central value and
reasonable uncertainty, thereby enabling a more accurate assessment of the
statistical significance of the gallium anomalies. Results are given for 51Cr
and 37Ar sources. The revised neutrino capture rates are used in a re-
evaluation of the BEST and gallium anomalies.

##  I Introduction: The Ga neutrino anomaly

The possibility of additional, very weakly interacting “sterile” neutrinos,
beyond the three light neutrinos of the standard model, has been raised
frequently in the literature Abazajian2012 ; Gariazzo2015 ; Giunti2019 ;
Boser2020 ; Diaz2020 ; Seo2020 ; Dasgupta2021 ; Acero2022 . Theoretically,
sterile neutrinos arise naturally in extensions of the standard model that
account for nonzero neutrino masses. Experimentally, sterile neutrinos have
been discussed in connection with reactor neutrino anomalies, in SAGE and
GALLEX/GNO neutrino calibration experiments, and in reconciling oscillation
parameters derived from T2K and NOvA Acero2022 ; Seo2020 ; deGouvea2022 .

In the radiochemical SAGE and GALLEX/GNO solar neutrino experiments, a large
mass of Ga (30-50 tons) was exposed to the solar neutrino flux for a period of
about a month, during which neutrino capture occurs via
71Ga$(\nu_{e},e^{-})^{71}$Ge. The produced atoms of radioactive 71Ge,
$\tau_{1/2}=11.43~{}\pm$ 0.03 d, were then chemically extracted and counted as
they decay back to 71Ga via electron capture. These experiments established
capture rates that, in combination with those from the chlorine and Kamioka
experiments, indicated a pattern of solar neutrino fluxes that could not be
easily reconciled with solar models, helping to motivate a new generation of
solar neutrino detectors: Super-Kamiokande SK , the Sudbury Neutrino
Observatory SNO , and Borexino Borexino1 ; Borexino2 . This led to the
discovery of neutrino mass and oscillations and the detection of an energy-
dependent distortion of the solar neutrino flux, reflecting the interplay
between vacuum and matter-enhanced oscillations review1 ; review2 .

While both gallium experiments utilized tracers to demonstrate the reliability
of the chemical extraction, direct cross checks on their overall efficiencies
for neutrino detection were also performed. Intense 51Cr and 37Ar electron-
capture (EC) neutrino sources of known strength were placed at the center of
the Ga targets, and the additional production of 71Ge measured. Four such
calibrations Ansel1995 ; Ab1999 ; Hampel1998 ; Ab2006  were performed, which
when combined yield a ratio of the observed to expected counting rates of
$R=0.866\pm 0.054$. The discrepancy between this result and $R=1$ is known as
the gallium anomaly.

The gallium anomaly and other short-baseline neutrino discrepancies motivated
the recent Baksan Experiment on Sterile Transitions (BEST) BEST1 ; BEST2 .
BEST, employing an exceptionally intense 3.4MCi 51Cr neutrino source, measured
the rate of neutrino reactions at two distances by dividing the Ga target
reactor into inner and outer volumes. This opened up the possibility of
detecting an oscillation signal. While no distance dependence was seen, the
counting rates were again well below expectations, with $R=0.791\pm 0.05$ and
$0.766\pm 0.05$ for the inner (shorter baseline) and outer volumes,
respectively.

A critical issue in the analysis of BEST and earlier Ga neutrino source
experiments is the cross section for 71Ga$(\nu_{e},e^{-})^{71}$Ge, as this is
a common systematic in these measurements. The 71Ge states contributing to the
capture of 51Cr neutrinos (as well as 37Ar EC neutrinos) are shown in Fig. 1.
It is known that the Ga anomaly cannot be attributed entirely to uncertainties
in this cross section, due to the dominance of the strong ${3\over
2}^{-}\rightarrow{1\over 2}^{-}$ transition to the 71Ge ground state, as this
transition strength is precisely determined by the known EC rate of 71Ge. Even
if only this contribution is included, a discrepancy remains. In addition, two
allowed (GT) transitions to 71Ge excited states, the ${5\over 2}^{-}$ and
${3\over 2}^{-}$ levels at 175 and 500 keV, respectively, also contribute to
the total 51Cr neutrino absorption cross section. The contributions of these
transitions have generally been deduced from surrogate probes of Gamow-Teller
(GT) strength, forward-angle (p,n) or (3He,t) scattering — despite long-
established concerns about the reliability of these probes when applied to
specific weak transitions HH1 ; HH2 .

While one cannot attribute the Ga anomaly entirely to nuclear physics, the
central value and uncertainty of the cross section can influence the
statistical significance of the BEST result, its possible interpretation in
terms of new physics, and its consistency with other tests of neutrino
properties. The purpose of this letter is to 1) re-examine the relationship
between the 71Ge(g.s.)($e^{-},\nu_{e})^{71}$Ga(g.s.) electron capture rate and
the 71Ga(g.s.)$(\nu_{e},e^{-})^{71}$Ge(g.s.) cross section, in order to deduce
the best value and uncertainty of the latter; and 2) reconsider the excited-
state contributions in light of new data testing the proportionality between
(p,n) or (3He,t) cross sections and experimentally known weak rates. In 1), we
examine (or re-examine) several $\approx 1\%$ corrections that can impact the
proportionality between the $\mathrm{g.s.}\leftrightarrow\mathrm{g.s.}$
inverse reactions. In 2), our focus is on defining a reasonable uncertainty
for the excited state contribution, based on a critical examination of the
reliability of such surrogate interactions as probes of specific weak GT
transitions.

##  II The 71Ge electron capture rate

One would like to derive from the known electron capture rate for 71Ge the
strength of the ground-state GT transition of the inverse neutrino reaction
cross section. In addition to the half life Hampel1985 ,

$\tau_{1\over 2}[^{71}\mathrm{Ge}]=11.43\pm 0.03~{}\mathrm{d},$ (1)

relevant experimental information includes the $Q_{\mathrm{EC}}$ value for the
decay Frekers2016 , the difference in the atomic masses

$\displaystyle Q_{\mathrm{EC}}$ $\displaystyle=$ $\displaystyle
M[^{71}\mathrm{Ge}]-M[^{71}\mathrm{Ga}]$ (2) $\displaystyle=$ $\displaystyle
232.443\pm 0.093\mathrm{~{}keV},$

and the $P_{K}$, $P_{L}$, and $P_{M}$ electron-capture probabilities and
associated atomic binding energies Ab1999b ,

$\displaystyle P_{K}$ $\displaystyle=$ $\displaystyle
0.88,~{}~{}~{}~{}~{}~{}E_{\mathrm{bind}}=10.37~{}\mathrm{keV}$ $\displaystyle
P_{L}$ $\displaystyle=$ $\displaystyle
0.103,~{}~{}~{}~{}~{}E_{\mathrm{bind}}=1.2~{}\mathrm{keV}$ $\displaystyle
P_{M}$ $\displaystyle=$ $\displaystyle
0.017,~{}~{}~{}~{}~{}E_{\mathrm{bind}}=0.12~{}\mathrm{keV}.$ (3)

The 71Ge $\rightarrow$ 71Ga electron capture rate can then be written

$\omega={\mathrm{ln}[2]\over\tau_{1\over
2}}={G_{F}^{2}\cos^{2}{\theta_{C}}\over
2\pi}~{}|\phi_{1s}|_{\mathrm{avg}}^{2}~{}E_{\nu,1s}^{2}~{}{\textstyle\left[2(1+\epsilon_{o}^{1s})(1+{P_{L}+P_{M}\over
P_{K}})\right]}~{}~{}g_{A}^{2}~{}[2~{}\mathrm{B^{(\nu,e)}_{GT}}(\mathrm{gs})]~{}[1+g_{v,b}]_{\mathrm{EC}}~{}[1+\epsilon_{q}]$
(4)

The various terms appearing above are as follows:

  1. 1.

The neutrino energy $E_{\nu,1s}$ for electron K capture. Neglecting a very
small nuclear recoil correction, it is given by the energy constraint

$\displaystyle Q_{\mathrm{EC}}$ $\displaystyle=$ $\displaystyle
E_{\nu,1s}+10.37\mathrm{~{}keV}$ $\displaystyle\Rightarrow E_{\nu,1s}$
$\displaystyle=$ $\displaystyle 222.1\pm 0.1\mathrm{~{}keV}$

  2. 2.

The factor $\left[2(1+\epsilon_{o}^{1s})(1+{P_{L}+P_{M}\over P_{K}})\right]$
relates the total capture rate to the rate for capture of a single $1s$
electron, with the contributions of $L$ and $M$ capture included through use
of the experimentally known branching ratios. This procedure requires the
introduction of a rearrangement (or overlap-exchange) correction
$\epsilon_{0}^{1s}$ to account for the imperfect overlap of the state created
by annihilating a $1s$ electron in the 71Ge atomic ground state, with states
appropriate for the Coulomb field of 71Ga. That is, while the instantaneous
annihilation of the $1s$ electron in 71Ge will lead dominantly to a virtual
state that decays by emitting $K$-capture Auger electrons and X-rays, atomic
rearrangement generates small contributions from $L$ and $M$ capture. Similar
corrections would be needed for other channels. That is, the total rate would
be proportional to

$\displaystyle\sum_{i}|\phi_{i}|_{\mathrm{avg}}^{2}E_{\nu,i}^{2}(1+\epsilon_{o}^{i})$
$\displaystyle=$
$\displaystyle|\phi_{1s}|_{\mathrm{avg}}^{2}~{}E_{\nu,1s}^{2}~{}(1+\epsilon_{o}^{1s})\left[1+{\sum_{i\neq
1s}|\phi_{i}|_{\mathrm{avg}}^{2}E_{\nu,i}^{2}(1+\epsilon_{o}^{i})\over|\phi_{1s}|_{\mathrm{avg}}^{2}E_{\nu,1s}^{2}(1+\epsilon_{o}^{1s})}\right]$
(5) $\displaystyle=$
$\displaystyle|\phi_{1s}|_{\mathrm{avg}}^{2}~{}E_{\nu,1s}^{2}~{}(1+\epsilon_{o}^{1s}){\textstyle\left[1+{P_{L}+P_{M}\over
P_{K}}\right]}$

where $|\phi_{i}|_{\mathrm{avg}}^{2}$ is the $K$, $L$, or $M$ atomic density
at the nucleus, $E_{\nu,i}$ is the associated energy of the emitted neutrino,
and $\epsilon_{o}^{i}$ is the overlap and exchange correction needed in the
$i$th channel Benois1950 ; Odiot1956 ; Bahcall62 ; Vatai70 . Bahcall Bahcall62
noted that such corrections to theory were needed to reproduce precise
experimental $L/K$ capture ratios and estimated their sizes. As he has
emphasized, if the theoretical expression on the left-hand side of Eq. (5) is
used, the inclusion of the $\epsilon_{i}$ would have little net impact, as
these factors diminish $P_{K}$ but enhance $P_{L}$ and $P_{M}$. Here, however,
we make use of the experimentally measured probabilities $P_{K}$, $P_{L}$ and
$P_{M}$ to write the total rate in terms of the $1s$-capture rate, so
inclusion of overlap/exchange correction for the $1s$ channel is needed.

The values for $\epsilon_{o}^{1s}$ given by Bahcall Bahcall62 ; Bahcall63a ;
Bahcall63b ; Bahcall65  and by Vatai Vatai70  are -0.018 and -0.008,
respectively. Though the correction is small, there is a relatively large
fractional difference between the results. (In contrast, their corrections for
$L$ (0.083 and 0.088, respectively) and $M$ (0.247 and 0.188, respectively)
capture are in better agreement, with fractional differences of 6% and 31%,
respectively). We adopt as a nominal value the average and take twice the
standard deviation as the 95% confidence level (C.L.), yielding
$\epsilon_{o}^{1s}=-0.013\pm 0.014$ and

$(1+\epsilon_{o}^{1s})(1+{P_{L}+P_{M}\over P_{K}})=1.122\pm 0.016.$

  3. 3.

We adopt Particle Data Group values for the Fermi constant, $G_{F}/(\hbar
c)^{3}=1.1664\times 10^{-5}/$GeV2, and Cabibbo angle,
$\cos{\theta_{C}}=0.9743$, and the Perkeo II value Perkeo2019  for the axial
vector coupling $g_{A}=1.2764$. Uncertainties are neglected as they are below
the level of concern here.

  4. 4.

$|\phi_{1s}|^{2}_{\mathrm{avg}}$ is the 71Ge $1s$ atomic density at the
nucleus. The nuclear amplitudes for the EC transitions of interest involve
convolutions of the GT operator — the space-like component of the nuclear
axial current — with leptonic wave functions,

$\langle j_{f}|\int
d{\boldsymbol{r}}~{}\phi_{\nu_{e}}^{*}({\boldsymbol{r}})\phi_{1s}({\boldsymbol{r}})\sum_{i=1}^{A}{\boldsymbol{\sigma}}(i)\tau_{-}(i)~{}\delta({\boldsymbol{r}}-{\boldsymbol{r}}_{i})~{}|j_{i}\rangle$

where we abbreviate the nuclear ground states of 71Ge and 71Ga as
$|j_{i}\rangle$ and $|j_{f}\rangle$, respectively. As $q_{\nu}R_{N}<<1$, where
$R_{N}$ is the nuclear radius and $q_{\nu}$ is the magnitude of the neutrino’s
three-momentum, one can approximate the neutrino plane wave within the nucleus
by $\phi_{\nu_{e}}^{*}({\boldsymbol{r}})\sim 1$. (The leading correction to
this approximation will be evaluated below.) Similarly, given that the atomic
wave function varies slowly over the nuclear scale,
$\phi_{1s}({\boldsymbol{r}})$ can be removed from the integral and replaced by
an average value. Most commonly $|\phi_{1s}|_{\mathrm{avg}}$ is computed by
folding the electron probability density with the normalized 71Ge proton
charge distribution, then integrating over the nuclear volume.

In his 1997 work Bahcall97 , Bahcall used three relativistic, self-consistent
Hartree-Fock calculations that took into account the finite extent of the
nucleus, the Breit interaction, vacuum polarization, and self-energy
corrections, averaging the resulting wave functions over the nuclear volume to
obtain $|\phi_{i}|_{\mathrm{avg}}^{2}$ for $K$, $L$, and $M$ capture. The
calculations, performed by three independent groups, agreed at the $\pm 0.2\%$
level. We are not aware of any subsequent calculations that are as complete.
While Bahcall97  includes references to the atomic methods employed by the
three groups, the 71Ge results were provided as private communications and are
not described in separate publications. As the relationship between the
dimensionless numerical quantity given in Bahcall97  and the density
$|\phi_{1s}|_{\mathrm{avg}}$ may not be obvious to readers, we provide some of
the needed definitions here.

The dimensionless quantity evaluated in Bahcall97 , given by the quantity in
square brackets below, is related to the dimensionful quantities we define on
the left below by

$\sum_{i}|\phi_{i}|_{\mathrm{avg}}^{2}E_{\nu,i}^{2}={(m_{e}c^{2})^{5}\over(\hbar
c)^{3}}{1\over 4\pi}\left[\sum_{i}g_{i}^{2}\tilde{E}_{\nu,i}^{2}\right]$ (6)

where $\tilde{E}_{\nu,i}\equiv{E}_{\nu,i}/m_{e}c^{2}$. The factor of $1/4\pi$
appears because Bahcall evaluated the $s$-wave radial density, not the full
density. As the square-bracketed quantity depends on $Q_{\mathrm{EC}}$, a
small correction is needed because $Q_{\mathrm{EC}}=232.69$ keV was used in
Bahcall97 , while the current value is given by Eq. (2). Plugging in the
numerical values from Bahcall97  one finds

$\displaystyle\left[\sum_{i}g_{i}^{2}\tilde{E}_{\nu,i}^{2}\right]^{Q_{\mathrm{EC}}=232.443~{}\mathrm{keV}}$
(7) $\displaystyle\approx$ $\displaystyle
0.9978\left[\sum_{i}g_{i}^{2}\tilde{E}_{\nu,i}^{2}\right]^{Q_{\mathrm{EC}}=232.69~{}\mathrm{keV}}$
$\displaystyle=$ $\displaystyle 0.01440$

We again group terms so that we can use experimental EC ratios, finding from
Eqs. (6) and (7)

$\displaystyle{(m_{e}c^{2})^{5}\over(\hbar c)^{3}}~{}{0.01440\over 4\pi}$ (10)
$\displaystyle=$
$\displaystyle|\phi_{1s}|_{\mathrm{avg}}^{2}E_{\nu,1s}^{2}\left[1+{1+\epsilon_{o}^{1s}\over
1+\epsilon_{o}^{L}}{P_{L}\over P_{K}}+{1+\epsilon_{o}^{1s}\over
1+\epsilon_{o}^{M}}{P_{M}\over P_{K}}\right]$ $\displaystyle=$
$\displaystyle|\phi_{1s}|_{\mathrm{avg}}^{2}E_{\nu,1s}^{2}\left\\{\begin{array}[]{cc}1.121,&\mathrm{Bahcall}\\\
1.123,&\mathrm{Vatai}\end{array}\right.~{}~{}~{}$

where the overlap and exchange factors arise because, following Eq. (5),
canceling terms are implicitly included in the probabilities $P_{K}$, $P_{L}$,
and $P_{M}$. We find the the result depends only weakly on whether we take
these corrections from Bahcall62  or from Vatai70 . Evaluating this expression
yields

$\displaystyle~{}~{}~{}~{}(\hbar c)^{3}|\phi_{1s}|_{\mathrm{avg}}^{2}$
$\displaystyle=$ $\displaystyle(7.21\pm 0.03)\times
10^{-4}\mathrm{~{}MeV}^{3}$ (11) $\displaystyle=$ $\displaystyle R{(Z\alpha
m_{e}c^{2})^{3}\over\pi}\Big{|}_{Z=32}$

with $R=1.333$. The uncertainty is determined from the standard deviations of
the three atomic calculations reported in Bahcall97  and of the overlap and
the exchange corrections of Bahcall and Vatai. These are combined in
quadrature, then doubled to give the 95% C.L. range given in Eq. (11). This
procedure thus takes into account differences apparent from the spread among
competing calculations, but not those that could arise if the calculations
being compared employed common but flawed assumptions.

In the second line of Eq. (11) we express the result in terms of the
Schrödinger density for an electron bound to a point charge $Z$, evaluated at
the origin. The dimensionless proportionality factor $R$ is not too different
from 1.

  5. 5.

In this paper, all BGT values are given for the neutrino reaction direction,
71Ga$(\nu_{e},e^{-})^{71}$Ge. For the
$\mathrm{gs}~{}\rightarrow~{}\mathrm{gs}$ EC direction,

$\displaystyle\mathrm{B}_{\mathrm{GT}}^{\mathrm{EC}}(\mathrm{gs})$ (12)
$\displaystyle=$ $\displaystyle{1\over 2j_{i}+1}\left|\langle
j_{f}={\textstyle{3\over
2}}||\sum_{1=1}^{A}\boldsymbol{\sigma}(i)\tau_{-}(i)||j_{i}={\textstyle{1\over
2}}\rangle\right|^{2}$ $\displaystyle=$ $\displaystyle{2\over
2j_{i}+1}\left|\langle j_{f}={\textstyle{1\over
2}}||\sum_{1=1}^{A}\boldsymbol{\sigma}(i)\tau_{+}(i)||j_{i}={\textstyle{3\over
2}}\rangle\right|^{2}$ $\displaystyle\equiv$ $\displaystyle
2~{}\mathrm{B^{(\nu,e)}_{GT}}(\mathrm{gs})$

Thus, $\mathrm{B^{EC}_{GT}}(\mathrm{gs})$ is given as
2$\mathrm{B^{(\nu,e)}_{GT}}(\mathrm{gs})$ in Eq. (4).

  6. 6.

$[1+\epsilon_{q}]$ is the correction to $B_{\mathrm{GT}}$ arising from
contributions beyond the allowed approximation. As the momentum transfer $q$
to the nucleus is very low, one expects that this contribution will be
dominated by the term linear in the three-momentum $q$ arising from the
interference between the GT and weak-magnetism amplitudes. This contribution
is computed below and shown to be very small.

  7. 7.

The factor $[1+g_{v,b}]_{\mathrm{EC}}$ represents the impact of radiative
corrections on electron capture from the exchange of virtual photons and
bremsstrahlung. We also discuss this correction below, as radiative
corrections do appear to impact the relationship between electron capture and
the inverse process of neutrino capture.

Forbidden corrections: Because of the very low momentum transfer accompanying
electron capture, corrections to the ground-state transition probability are
expected to be very small and associated with the interference term between
the GT amplitude and weak magnetism. This interference generates a term linear
in the three-momentum transfer. We find a correction to the GT transition
probability of

$\epsilon_{q}={2E_{\nu,1s}\over 3m_{N}g_{A}}\left[\mu_{T=1}+{\langle{3\over
2}^{-}||\sum_{i=1}^{A}\boldsymbol{\ell}(i)\tau_{-}(i)||{1\over
2}^{-}\rangle\over\langle{3\over
2}^{-}||\sum_{i=1}^{A}\boldsymbol{\sigma}(i)\tau_{-}(i)||{1\over
2}^{-}\rangle}\right]$ (13)

where $m_{N}$ is the nucleon mass and $\mu_{T=1}\,\approx\,4.706$ is the
isovector magnetic moment. As the spin contribution to weak magnetism is
effectively determined by the EC capture rate, only the orbital contribution
must be taken from theory.

To estimate the orbital contribution as well as its uncertainty, we evaluated
the needed orbital matrix element in the shell model, retaining all Slater
determinants within the $2p_{3/2}1f_{5/2}2p_{1/2}1g_{9/2}$ model space, and
employing three well-known effective interactions designed for this space,
GCN2850 GCN2850 , jj44b jj44b , and JUN45 JUN45 . The dimension of the shell-
model space for 71Ge is about $1.5\times 10^{8}$. The diagonalizations were
performed with the Lanczos-algorithm code BIGSTICK Johnson:2013bna ; Calvin .
We found

${\textstyle\langle{3\over
2}^{-}}||\sum_{i=1}^{A}\boldsymbol{\ell}(i)\tau_{-}(i)||{\textstyle{1\over
2}^{-}\rangle}=\left\\{\begin{array}[]{ll}0.48&\mathrm{GCN2850}\\\
0.69&\mathrm{jj44b}\\\ 0.005&\mathrm{JUN45}\end{array}\right.$

so that the ratio that enters in Eq. (13) is

${{\textstyle\langle{3\over
2}^{-}}||\sum_{i=1}^{A}\boldsymbol{\ell}(i)\tau_{-}(i)||{\textstyle{1\over
2}^{-}\rangle}\over{\textstyle\langle{3\over
2}^{-}}||\sum_{i=1}^{A}\boldsymbol{\sigma}(i)\tau_{-}(i)||{\textstyle{1\over
2}^{-}\rangle}}_{\mathrm{exp}}=\left\\{\begin{array}[]{ll}-0.81&\mathrm{GCN2850}\\\
-1.18&\mathrm{jj44b}\\\ -0.01&\mathrm{JUN45}\end{array}\right.$

where the magnitude of the GT matrix element is taken from experiment, while
the relative sign is the shell-model prediction. From the average and standard
deviation of these theory results, we find

${{\textstyle\langle{3\over
2}^{-}}||\sum_{i=1}^{A}\boldsymbol{\ell}(i)\tau_{-}(i)||{\textstyle{1\over
2}^{-}\rangle}\over{\textstyle\langle{3\over
2}^{-}}||\sum_{i=1}^{A}\boldsymbol{\sigma}(i)\tau_{-}(i)||{\textstyle{1\over
2}^{-}\rangle}}_{\mathrm{exp}}=-0.7\pm 1.2~{}~{}(95\%~{}\mathrm{C.L.})$

where the assigned uncertainty is again twice the standard deviation. Because
the large isovector magnetic moment dominates Eq. (13), the estimate of the
forbidden corrections is relatively stable, despite substantial differences in
the shell model estimates of the orbital angular momentum matrix element. The
end result

$\epsilon_{q}=(4.9\pm 1.5)\times 10^{-4}~{}~{}~{}~{}(95\%~{}\mathrm{C.L.})$

shows that the weak magnetism correction is negligible. We have also evaluated
this correction using the full momentum dependence of the weak transition
amplitude, doing a standard multipole expansion, obtaining a result consistent
with the above to the precision shown.

Radiative corrections: Past work has either explicitly Bahcall97  or
implicitly assumed that radiative corrections would affect the electron
capture rate and the inverse neutrino capture cross section similarly, and
thus would be effectively included in calculations that extract an effective
GT matrix element from electron capture, then use that amplitude in computing
the inverse $(\nu_{e},e^{-})$ reaction. Sirlin Sirlin  has pointed out that
certain single-nucleon short-range contributions to radiative corrections are
universal. But other contributions, notably bremmstrahlung, affect electron
capture and neutrino reactions unequally, with the differences dependent on
the $Q$ value of the reaction Kurylov . Thus we will retain an unspecified
radiative correction $[1+g_{v,b}]_{\mathrm{EC}}$ in our expression for the EC
rate, then address the ratio of this correction to $[1+g_{v,b}]_{(\nu,e)}$ in
the next section.

Collecting all of the results from this section and utilizing the $11.43\pm
0.03$ d (1$\sigma$) half life of 71Ge we find

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle(7.019\pm 0.037)\times
10^{-7}/s~{}~{}~{}~{}~{}(95\%~{}\mathrm{C.L.})$ $\displaystyle=$
$\displaystyle(8.122\pm 0.122)\times
10^{-6}~{}\mathrm{B^{(\nu,e)}_{GT}}(\mathrm{gs})~{}[1+g_{v,b}]_{\mathrm{EC}}$

and therefore

$\displaystyle\mathrm{B}^{(\nu,e)}_{\mathrm{GT}}(\mathrm{gs})[1+g_{v,b}]_{\mathrm{EC}}\equiv\tilde{\mathrm{B}}^{(\nu,e)}_{\mathrm{GT}}(\mathrm{gs})$
$\displaystyle=0.0864\pm 0.0013~{}~{}~{}(95\%~{}\mathrm{\,C.L.})$ (14)

where the various uncertainties noted above have been combined in quadrature.
The ground-state transition probability is conservatively known to a precision
of about 1.5%: the primary motivation for the detailed discussion above was to
establish this uncertainty. Our recommended best value of 0.0864 is consistent
with most past estimates of this quantity, e.g., 0.087 HH1 , 0.0863 Bahcall97
, and 0.0864 Semenov2020  — though the agreement is a bit fortuitous, arising
because differences in rate components conspired to cancel.

##  III The 71Ga$(\nu_{e},e^{-})^{71}$Ge ground state cross section

The 71Ga$(\mathrm{gs})(\nu_{e},e^{-})^{71}$Ge$(\mathrm{gs})$ neutrino capture
cross section can be written in terms of
$\tilde{\mathrm{B}}^{(\nu,e)}_{\mathrm{GT}}(\mathrm{gs})$,

$\displaystyle\sigma_{\mathrm{gs}}={G_{F}^{2}\cos^{2}{\theta_{C}}\over\pi}~{}p_{e}E_{e}~{}{\cal
F}(Z_{f},E_{e})~{}g_{A}^{2}~{}\tilde{\mathrm{B}}^{(\nu,e)}_{\mathrm{GT}}(\mathrm{gs})~{}{[1+g_{v,b}]_{(\nu,e)}\over[1+g_{v,b}]_{\mathrm{EC}}}~{}[1+\epsilon_{q}]$
(15)

Nuclear recoil has been neglected as the target mass $M_{T}>>E_{e}$. We
evaluate the cross section for electron-capture neutrinos produced by 51Cr and
37Ar, for which the contributing lines are listed in Table 1.

Table 1: Neutrino source parameters and various correction factors that enter into the calculation of the cross section $\sigma_{\mathrm{gs}}(\nu_{e}+^{71}$Ga$\rightarrow e^{-}+^{71}$Ge$)$ for 51Cr and 37Ar neutrino sources. We report the energy of the incoming neutrino $E_{\nu}$, the corresponding neutrino branching ratio, and the energy of the final-state electron $E_{e}$. $\mathcal{F}(Z_{f},E_{e})$ is the Coulomb factor of the electron, obtained by combining the various correction factors $F_{0}$, $L_{0}$, $U$, and $S$ (see text). Finally, $\epsilon_{q}$ governs the strength of the forbidden corrections to the GT amplitude, and $[1+g_{v,b}]_{(\nu,e)}/[1+g_{v,b}]_{\mathrm{EC}}$ measures the difference in radiative corrections that enter into the calculation of the respective cross sections for neutrino capture and the inverse process of electron capture. Source |  $E_{\nu}$ (MeV) | Branching |  $E_{e}$ (MeV) | $F_{0}$ | $L_{0}$ | $U$ | S | $\mathcal{F}(Z_{f},E_{e})$ |  $1+\epsilon_{q}$ (95% C.L.) | $\frac{[1+g_{v,b}]_{(\nu,e)}}{[1+g_{v,b}]_{\mathrm{EC}}}$
---|---|---|---|---|---|---|---|---|---|---
51Cr | 0.7524 | 0.0140 | 1.031 | 2.791 | 1.0034 | 0.9986 | 0.9920 | 2.774 | $1.0034\pm 0.0010$ | 0.995
| 0.7518 | 0.0842 | 1.030 | 2.791 | 1.0034 | 0.9986 | 0.9920 | 2.775 | $1.0034\pm 0.0010$ | 0.995
| 0.7465 | 0.8025 | 1.025 | 2.795 | 1.0035 | 0.9986 | 0.9920 | 2.779 | $1.0034\pm 0.0010$ | 0.995
| 0.4323 | 0.0015 | 0.711 | 3.335 | 1.0053 | 0.9985 | 0.9876 | 3.306 | $1.0017\pm 0.0005$ | 0.997
| 0.4317 | 0.0092 | 0.710 | 3.338 | 1.0053 | 0.9985 | 0.9876 | 3.309 | $1.0017\pm 0.0005$ | 0.997
| 0.4264 | 0.0886 | 0.705 | 3.360 | 1.0053 | 0.9985 | 0.9874 | 3.330 | $1.0017\pm 0.0005$ | 0.997
37Ar | 0.8138 | 0.0111 | 1.092 | 2.750 | 1.0031 | 0.9986 | 0.9925 | 2.734 | $1.0037\pm 0.0011$ | 0.995
| 0.8135 | 0.0866 | 1.092 | 2.750 | 1.0031 | 0.9986 | 0.9925 | 2.734 | $1.0037\pm 0.0011$ | 0.995
| 0.8107 | 0.9023 | 1.089 | 2.752 | 1.0031 | 0.9986 | 0.9925 | 2.736 | $1.0037\pm 0.0011$ | 0.995

The neutrino energies $E_{\nu}$ and the corresponding branching ratios are
computed from the respective $Q$ values in 51Cr and 37Ar, 752.4 and 813.9 keV,
the $K$-shell binding energies, 5.99 and 3.21 keV, the $L$-shell binding
energies, 0.70 and 0.33 keV, the $M$-shell binding energies, 0.074 and 0.029
keV, the $K$/$L$/$M$ branching ratios of 0.891/0.094/0.016 and
0.902/0.0866/0.011, and the 9.93% branching ratio for 51Cr to decay to the
first excited $\textstyle{5\over 2}^{-}$ state in 51V at 321.1 keV.

We identify the various terms in Eq. (15):

  1. 1.

$E_{e}$ and $p_{e}$ are the energy and three-momentum magnitude of the
outgoing electron, respectively. For the neutrino reactions of interest off
71Ga

$E_{e}=E_{\nu}-Q_{\mathrm{EC}}+m_{e}-0.09\mathrm{~{}keV}$

where $Q_{\mathrm{EC}}$ in given in Eq. (2). We follow Bahcall Bahcall97  in
including a very small 0.09 keV correction for the energy lost to electronic
re-arrangement, as the electron cloud adjusts to the nuclear charge change.
For transitions to the ${\textstyle{5\over 2}^{-}}$(175 keV) and
${\textstyle{3\over 2}^{-}}$(500 keV) excited states in 71Ge, the nuclear
excitation energies would be added to $Q_{\mathrm{EC}}$.

  2. 2.

${\cal F}(Z_{f},E_{e})$ corrects the phase space for the Coulomb distortion of
the outgoing electron plane wave. Following Wilkinson , Hayen , and Behrens
this correction is decomposed as follows

$\displaystyle{\cal F}(Z_{f},E_{e})$ $\displaystyle=$ $\displaystyle
F_{0}(Z_{f},E_{e})~{}L_{0}(Z_{f},E_{e})~{}U(Z_{f},E_{e})~{}S(Z_{f},E_{e})$

with

$\displaystyle F_{0}(Z_{f},E_{e})$ $\displaystyle=$ $\displaystyle
4(2p_{e}R_{N})^{2(\gamma-1)}e^{\pi
y}{|\Gamma(\gamma+iy)|^{2}\over(\Gamma(1+2\gamma))^{2}}$ $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle\sqrt{1-(\alpha Z_{f})^{2}}$ $\displaystyle y$
$\displaystyle=$ $\displaystyle\alpha Z{E_{e}\over p_{e}}$

$F_{0}$ is taken from the solution of the Dirac equation for an electron of
momentum $p_{e}$ in a point Coulomb potential generated by a charge $Z_{f}$,
with $Z_{f}=32$ here. This correction is kept finite by its evaluation at the
nuclear surface, often taken to be $R_{N}\,\approx\,1.2A^{1/3}$ fm and
interpreted as the edge of a nucleus of uniform density. We fix the 71Ge
r.m.s. charge radius to 4.05 fm, the average of the charge radii for 70Ge and
72Ge, as measured in electron scattering Devries , then use the relationship
for a nucleus of uniform density to determine

$R_{N}=\sqrt{5\over 3}\sqrt{\langle
r^{2}\rangle}=5.23\mathrm{~{}fm}\,\approx\,1.263A^{1/3}~{}\mathrm{fm}\big{|}_{A=71}$

which we use in the evaluation. This initial estimate then must be corrected:

    1. (a)

$L_{0}$ accounts for most effects of the finite charge distribution. For a
nucleus with a uniform density and thus a sharp surface at $R_{N}$, the Dirac
solution can be continued to the origin by numerically integrating. We take
$L_{0}$ from the tables of Behrens and Janecke Behrens , who performed the
integration for $R_{N}=1.2A^{1/3}$ fm. We adjusted that result to account for
difference between this estimate of the r.m.s. charge radius and the
experimental value used here, using Eq. (16) of Hayen  (or Eq. (2) of
Wilkinson90 ).

    2. (b)

The factor $U(Z_{f},E_{e})$ represents the difference between the Coulomb
distortion computed for a uniform charge distribution and that resulting from
the use of a more realistic Fermi distribution with an equivalent r.m.s.
radius. We use the parameterization of Wilkinson Wilkinson , also recently
discussed in Hayen  (see Eqs. (29) and (30)).

    3. (c)

$S(Z_{f},E_{e})$ is a correction for atomic screening within the nuclear
volume, which we take from Rose Rose . A comparison of various prescriptions
for the atomic screening correction is presented in Hayen  (see Fig. 5 in this
reference), showing generally good agreement, except for very low
$E_{e}\lesssim 1.1m_{e}$.

Table 1 gives the Coulomb factors $\mathcal{F}(Z_{f},E_{e})$ and the
constituent corrections $F_{0}$, $L_{0}$, $U$, and $S$ for
71Ga$(\nu_{e},e^{-})^{71}$Ge for 51Cr and 37Ar neutrino sources.

  3. 3.

$[1+\epsilon_{q}]$ corrects for the emission of forbidden contributions, which
as we have noted almost entirely come from the interference between the GT
amplitude and weak magnetism. This correction is evaluated below.

  4. 4.

The ratio

${[1+g_{v,b}]_{(\nu,e)}\over[1+g_{v,b}]_{\mathrm{EC}}}$

accounts for the difference between the radiative corrections for neutrino
absorption and those for electron capture (contained in
$\mathrm{\tilde{B}_{GT}}(\mathrm{gs})$). This correction is also discussed
below.

Forbidden corrections: After integrating over electron angles, the correction
linear in weak magnetism takes on a form identical to Eq. (13)

$\displaystyle\epsilon_{q}$ $\displaystyle={2\over
3m_{N}g_{A}}\left(E_{\nu}+E_{e}-{m_{e}^{2}\over E_{e}}\right)$ (16)
$\displaystyle\times$ $\displaystyle\left[\mu_{T=1}+{\langle{3\over
2}^{-}||\sum_{i=1}^{A}\boldsymbol{\ell}(i)\tau_{-}(i)||{1\over
2}^{-}\rangle\over\langle{3\over
2}^{-}||\sum_{i=1}^{A}\boldsymbol{\sigma}(i)\tau_{-}(i)||{1\over
2}^{-}\rangle}_{\mathrm{exp}}\right]$

apart from the kinematic factor. The resulting forbidden correction
$\epsilon_{q}$ is shown in Table 1. The $2\sigma$ uncertainty reflects the
differences among the three shell model estimates of the orbital matrix
element, as discussed previously.

Radiative corrections: Although radiative corrections Sirlin  are significant
(few %Kurylov ) for low-energy neutrino reactions on nuclei, it is often
assumed Bahcall97  that these corrections are common to the inverse reactions
$(e^{-},\nu_{e})$ and $(\nu_{e},e^{-})$, and thus implicitly included when the
$(\nu_{e},e^{-})$ nuclear transition matrix elements are determined from known
electron-capture rates. Were this the case, Eq. (15) would simplify, with the
ratio of $(\nu,e)$ and EC radiative corrections replaced by 1. However, while
this universality assumption holds for charge-current reactions producing
electrons/positrons only in the final or only in the initial state, Kurylov et
al. Kurylov  have shown that it is not preserved in the comparison between
electron capture and $(\nu_{e},e^{-})$.

Kurylov et al. Kurylov  evaluated the one-nucleon $W\gamma$-loop and
bremsstrahlung contributions to the radiative corrections (Figs. 1a and 3 of
Kurylov ), finding that the bremsstrahlung contribution breaks the
universality due to its dependence on the $Q$ value. While the calculation
treats the electron in $(e^{-},\nu_{e})$ as a free state, the results
evaluated for $E_{e}\rightarrow m_{e}$ should approximate those needed for the
weakly bound electrons of interest here. (The radiative corrections describe
short-range loops and radiation associated with the strong Coulomb field near
the nucleus. The bound electron wave function varies over atomic scales, not
nuclear ones, providing justification for this assumption. In Czarnecki ,
similar issues are discussed in comparing muonium decay with free muon decay.)

The results shown in the last column of Table 1 were derived using Eqs. (4),
(5), (51), and (52) of Kurylov . The difference in the one-
nucleon/bremsstrahlung contributions to electron capture (implicitly absorbed
into $\tilde{\mathrm{B}}_{GT}(\mathrm{gs})$) and neutrino reactions yields a
correction to the neutrino absorption cross section of $\approx$ 0.5%.

In addition to the effects discussed above, there are nucleus-dependent
radiative corrections — contributions involving more than one nucleon (Fig. 1b
of Kurylov ) as well as the nuclear Green’s function corrections to terms
treated in leading order as one-nucleon contributions. Such corrections for
the axial current have not yet been estimated and thus are not included here.

Cross sections: Combining all of the results above yields

$\sigma_{\mathrm{gs}}=\left\\{\begin{array}[]{cc}(5.39\pm 0.08)\times
10^{-45}\mathrm{~{}cm}^{2}&~{}~{}~{}{}^{51}\mathrm{Cr}\\\ (6.45\pm 0.10)\times
10^{-45}\mathrm{~{}cm}^{2}&~{}~{}~{}{}^{37}\mathrm{Ar}\end{array}\right.$ (17)

at $95\%$ C.L.

##  IV Excited-state contributions

The uncertainty in the 71Ga$(\nu_{e},e^{-})$Ge cross section due to excited-
state contributions was noted by Kuzmin Kuzmin1966 , when he proposed 71Ga as
a solar neutrino detector in 1966. Neutrinos from 51Cr and 37Ar sources can
excite the 5/2- and 3/2- states at 175 and 500 keV, respectively. These same
states contribute to 7Be solar neutrino absorption: see Fig 1.

Figure 1: Level diagram for 71Ga$(\nu_{e},e^{-})^{71}$Ga showing the states
that contribute to the absorption of 51Cr and 37Ar EC neutrinos.

In his 1978 Ga cross section study, Bahcall Bahcall1978  used systematics to
constrain the excited-state contributions, identifying transitions in
neighboring nuclei that might be similar; that is, naïvely of a
$2p_{3/2}\leftrightarrow 1f_{5/2}$ character. Bahcall identified nine
$3/2^{-}\rightarrow 5/2^{-}$ transitions of known strength with log(ft) values
ranging from 5.9 to 7.5, and consequently assigned log(ft)$\gtrsim 6$ to the
transition to the 175 keV state in 71Ge. Similarly, he found eight
$3/2^{-}\rightarrow 3/2^{-}$ transitions with log(ft) values ranging from 5.0
to 5.8, assigning log(ft)$\gtrsim 5$ these to the transitions to the 500 and
710 keV states. Using these bounds, Bahcall argued that the excited-state
contribution to the 51Cr absorption cross section would be $\lesssim$ 14.6%.
But the unreliability of such arguments was pointed out in HH1 , where the
ground state transition 71Ga(3/2${}^{-})\rightarrow{}^{71}$Ge(1/2-) was
compared to the most closely analogous neighboring transition,
73Ga(3/2${}^{-})\rightarrow{}^{73}$Ge(1/2-, 67 keV). These two cases differ
only by the addition of a neutron pair that, in the Nilsson model, would not
be expected to affect the transition amplitude HH1 . Yet the BGT values for
these two cases differ by a factor of 50.

Alternatively, one might attempt a microscopic calculation of the strengths of
the excited-state transitions. Indeed, shell-model calculations of the BGT
values for exciting the 175 and 500 keV states were performed early on by
Baltz et al. Baltz1984  and by Mathews et al. Mathews1985 . But even today —
as we will describe later — this is a dubious undertaking due to the weakness
of these transitions. In the allowed approximation, the transition
probabilities are proportional to the BGT value

$\displaystyle\mathrm{B_{GT}}({j_{i}\alpha_{i}\rightarrow j_{f}\alpha_{f}})$
$\displaystyle=$ $\displaystyle{1\over 2j_{i}+1}|M_{\mathrm{GT}}|^{2}$
$\displaystyle=$ $\displaystyle{1\over 2j_{i}+1}|\langle
j_{f}\alpha_{f}||\hat{O}_{\mathrm{GT}}^{J=1}||j_{i}\alpha_{i}\rangle|^{2}$
$\displaystyle\hat{O}_{\mathrm{GT}}^{J=1}$ $\displaystyle\equiv$
$\displaystyle\sum_{j=1}^{A}{\boldsymbol{\sigma}}(j)\tau_{+}(j)$ (18)

where ${\boldsymbol{\sigma}}$ is the Pauli spin matrix, $\tau_{+}$ is the
isospin raising operator, $||$ denotes a matrix element reduced in angular
momentum, and $j_{i}\alpha_{i}$ and $j_{f}\alpha_{f}$ denote the quantum
numbers of the initial and final states, respectively, with the angular
momentum $j$ made explicit. From the known EC rate for 71Ge and from the lower
bounds Bahcall used for the transitions to the 175 and 500 keV levels, one
finds

$\displaystyle\mathrm{B_{GT}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}175\mathrm{~{}keV}\right)$
$\displaystyle\lesssim$ $\displaystyle 0.004$
$\displaystyle\mathrm{B_{GT}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}500\mathrm{~{}keV}\right)$
$\displaystyle\lesssim$ $\displaystyle 0.04$ (19)

values smaller than the ground-state BGT value of Eq. (14). As the total
$\mathrm{B}_{\mathrm{GT}}$ strength, summed over all final states, is given
approximately by the Ikeda sum rule $3(N-Z)=27$ Ikeda1964 , we see that the
transitions to the 5/2- and 3/2- states exhaust less than $\approx$ 0.01% and
$0.1\%$ of the sum-rule strength, respectively. Consequently, one expects
calculations to be sensitive to wave-function details, including the
interactions used, the adopted shell model spaces, etc. Indeed, early attempts
to estimate the needed excited-state contributions to the cross section, using
the shell model Baltz1984 ; Mathews1985 , schematic effective interactions,
and truncated model spaces, yielded results that varied by orders of
magnitude, depending on the specific simplifications adopted.

Today there exist several carefully tuned effective interactions designed for
the relevant model space, $2p_{3/2}1f_{5/2}2p_{1/2}1g_{9/2}$, and
diagonalizations using all of the $\gtrsim 10^{8}$ Slater determinants in this
space can be done. We in fact will employ this technology here, but not to
directly calculate the needed $\mathrm{B}_{\mathrm{GT}}$ strengths. Instead,
it will be used to help interpret the experimental calibrations described
below.

The possibility that excited-state GT strength could be measured through
surrogate reactions, (p,n) or (3He,t), generated significant interest in the
solar neutrino community. The approximate proportionality between medium-
energy forward-angle (p,n) cross sections and nuclear BGT profiles is well
established Taddeucci . This method was applied to the 71Ga transitions of
interest by Krofcheck et al. Krofcheck . From (p,n) measurements at 120 and
200 MeV they deduced

$\displaystyle\mathrm{B_{GT}^{(p,n)}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}175\mathrm{~{}keV}\right)$
$\displaystyle\lesssim$ $\displaystyle 0.005$
$\displaystyle\mathrm{B_{GT}^{(p,n)}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}500\mathrm{~{}keV}\right)$
$\displaystyle=$ $\displaystyle 0.011\pm 0.002$

results qualitatively consistent with Bahcall’s expectations based on
systematics. However, the use of this method in the case of weak transitions
can be problematic, as described in HH1 ; HH2 . Here we extend these previous
analyses with the goal of better quantifying the excited-state contributions
to the neutrino absorption cross section.

The same transitions were studied using (3He,t) at 420 MeV FrekersGa . This
method can achieve higher resolution, but has been applied less frequently to
the light nuclei we will later use to test the reliability of charge-exchange
mappings of Gamow-Teller strength. The results are

$\displaystyle\mathrm{B_{GT}^{(p,n)}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}175\mathrm{~{}keV}\right)$
$\displaystyle=$ $\displaystyle 0.0034\pm 0.0026$
$\displaystyle\mathrm{B_{GT}^{(p,n)}}\left({}^{71}\mathrm{Ga}~{}\mathrm{gs}\rightarrow{}^{71}\mathrm{Ge}~{}500\mathrm{~{}keV}\right)$
$\displaystyle=$ $\displaystyle 0.0176\pm 0.0014$

As the tension between Eqs. LABEL:eq:Krofcheck and LABEL:eq:Frekers for the
transition to the $\textstyle{3\over 2}^{-}$ state exceeds $3\sigma$, we will
treat the two data sets separately, rather than combining them.

The work in HH1 ; HH2  exploited the empirical observation Watson  that the
effective operator for forward-angle (p,n) scattering includes a subdominant
contribution from a tensor operator $\hat{O}^{J=1}_{\mathrm{T}}$,

$\displaystyle M^{\mathrm{(p,n)}}$ $\displaystyle\equiv$ $\displaystyle
M_{\mathrm{GT}}+\delta M_{\mathrm{T}},~{}~{}~{}~{}M_{\mathrm{T}}\equiv\langle
j_{f}\alpha_{f}||\hat{O}^{J=1}_{\mathrm{T}}||j_{i}\alpha_{i}\rangle$
$\displaystyle\hat{O}^{J=1}_{\mathrm{T}}$ $\displaystyle=$
$\displaystyle\sqrt{8\pi}\sum_{j=1}^{A}\left[Y_{2}(\Omega_{j})\otimes\boldsymbol{\sigma}(j)\right]_{J=1}\tau_{+}(j)$
(22)

where $\delta\,\approx\,0.1$, so that

$\displaystyle\mathrm{B_{GT}^{(p,n)}}={1\over 2j_{i}+1}|\langle
j_{f}\alpha_{f}||M^{\mathrm{(p,n)}}||j_{i}\alpha_{i}\rangle|^{2}~{}.$ (23)

The need for the tensor correction in forward-angle scattering, where the
momentum transfer is minimal and thus the interactions can occur at long
range, should not be a surprise: the central part of the one-pion-exchange
potential generates a target response proportional to $M_{\mathrm{GT}}$ while
the tensor part generates $M_{\mathrm{T}}$. In cases where $M_{\mathrm{GT}}$
is weak but $M_{\mathrm{T}}$ is strong, $M^{\mathrm{(p,n)}}$ will be an
unreliable probe of BGT strengths. An example where this would be the case is
an $\ell$-forbidden M1 transition, where the dominant amplitude links orbitals
with the quantum numbers $[n,\ell,j+\textstyle{1\over 2}]$ and
$[n-1,\ell+2,j=\ell+\textstyle{3\over 2}]$. Such transitions are often found
at low energy in nuclear spectra, as a consequence of an approximate
pseudospin symmetry Ginocchio . A candidate $\ell$-forbidden transition HH1
is 71Ga(3/2-) $\rightarrow^{71}$Ge(5/2-), which would be described in the
naïve shell model as $1f_{5/2}$ (n hole) $\rightarrow$ $2p_{3/2}$ (p
particle). State-of-the-art shell-model studies performed here and in another
recent study kostensalo  show that the transition density matrix does have a
dominant $\ell$-forbidden component.

The analysis in HH1  estimated $\delta$ by examining GT transitions in $1p$\-
and $2s1d$-shell nuclei, but did not evaluate the experimental and theoretical
errors in the determination, nor how they would propagate into an estimate of
$\delta$ and consequently the 71Ga excited-state cross section. Given the BEST
anomaly, it is now important to do so. The data examined in HH1  were sensibly
chosen, involving mirror transitions where $\beta$ decay and (p,n) transition
strengths were both available from experiment, including transitions near
closed shells where levels are well separated and thus their shell-model wave
functions less sensitive to small changes in effective interactions. However,
we have made some changes in the data set, reflecting new information that has
become available. We also assess theoretical uncertainties by employing
several available effective interactions in computing $M_{\mathrm{T}}$: to
relate $M^{\mathrm{(p,n)}}$ and $M_{\mathrm{GT}}$, $|M_{\mathrm{T}}|$ and the
sign of $M_{\mathrm{T}}/M_{\mathrm{GT}}$ must be computed.

The data we use in determining $\delta$ are given in Table 2 (compare to Table
1 of HH1 ) and consist primarily of isospin mirror transitions where both
(p,n) and $\beta$-decay strengths are known. Eight of these cases are taken
from the compilation of $\mathrm{B_{GT}^{(p,n)}}$ of Watson . To convert the
proportionality between (p,n) scattering and $\beta$ decay into an
equivalence, a normalization must be introduced. Often this is done by using,
for each target nucleus, a strong $\beta$ decay transition, which can still be
problematic if there are corrections due to $\hat{O}_{\mathrm{T}}^{J=1}$ that
affect normalizing transitions in differential ways. The study of Watson
instead computed normalizing cross sections in the distorted-wave impulse
approximation, employing a phenomenological interaction fitted to a large body
of data. This then avoids the issue of nucleus-by-nucleus normalization
systematics.

Table 2:  Weak transitions and their beta decay and (p,n) amplitudes deduced
from experiment. The sign of $M_{\mathrm{GT}}$ has been taken as positive.
$M_{\mathrm{T}}$ and its sign relative to $M_{\mathrm{GT}}$ have been taken
from theory, except for the case of the sign of 32S (see text).

Reaction***Transitions are between ground states unless otherwise specified.
log(ft)†††Taken from the ENSDF compilations. ($2J_{i}+1$)BGT
$\textstyle{\mathrm{B_{GT}^{\mathrm{(p,n)}}}\over\mathrm{B_{GT}}}$
$M_{\mathrm{GT}}$ $M^{\mathrm{(p,n)}}$ $M_{\mathrm{T}}$‡‡‡$2s1d$-shell
uncertainties (excluding 39K$\rightarrow^{39}$Ca) correspond to the $1\sigma$
spread of matrix elements computed from the USDA, USDB USD , and Brown-
Wildenthal Wildenthal  interactions. All other $M_{T}$ uncertainties are
assumed to be 10%. ${M_{\mathrm{T}}\over M^{\mathrm{(p,n)}}}$
13C$(\textstyle{1\over 2}^{-}$) $\leftrightarrow\,{}^{13}$N$(\textstyle{1\over
2}^{-}$) $3.6648(5)$ 0.404 $\pm$ 0.002 1.85$\pm$ 0.22 0.636$\pm$0.002
0.87$\pm$0.05 $2.8\pm 0.3$ 3.2
$\begin{array}[]{l}{}^{14}\mathrm{C}(\textstyle{0}^{+})\\\
{}^{14}\mathrm{O}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{14}\mathrm{N}(1^{+}\,3.95\,\mathrm{MeV})$
$3.131(17)$ 2.79 $\pm$ 0.11 0.97$\pm$0.12 1.67$\pm$0.03 1.65$\pm$0.10
$0.086\pm 0.009$ 0.052 15N$(\textstyle{1\over 2}^{-}$)
$\leftrightarrow\,^{15}$O$(\textstyle{1\over 2}^{-}$) $3.6377(8)$
0.509$\pm$0.003 2.04$\pm$0.24 0.713$\pm$0.002 1.02$\pm$0.06 $3.3\pm 0.3$ 3.2
17O$(\textstyle{5\over 2}^{+}$) $\leftrightarrow\,^{17}$F$(\textstyle{5\over
2}^{+}$) $3.3562(5)$ 6.280$\pm$0.011 0.91$\pm$0.11 2.506$\pm$0.002
2.39$\pm$0.14 $1.7\pm 0.2$ 0.69 18O$(\textstyle{0}^{+}$)
$\leftrightarrow\,^{18}$F$(\textstyle{1}^{+}$) $3.5700(19)$ 3.045$\pm$0.013
1.12$\pm$0.13 1.745$\pm$0.003 1.84$\pm$0.11 -0.04$\pm$0.03 -0.02
$\begin{array}[]{l}{}^{18}\mathrm{O}(\textstyle{0}^{+})\\\
{}^{18}\mathrm{Ne}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{18}\mathrm{F}(1^{+}\,1.70\,\mathrm{MeV})$
$4.470(15)$ 0.128$\pm$0.004 1.33$\pm$0.17 0.358$\pm$0.006 0.41$\pm$0.025
0.8$\pm$0.3 2 19F$(\textstyle{1\over 2}^{+}$)
$\leftrightarrow\,^{19}$Ne$(\textstyle{1\over 2}^{+}$) $3.2329(24)$
3.184$\pm$0.024 1.29$\pm 0.15$ 1.784$\pm$0.007 2.02$\pm$0.12 0.08$\pm$0.03
0.04 19F$(\textstyle{1\over 2}^{+}$)
$\leftrightarrow\,^{19}$Ne$(\textstyle{3\over 2}^{+}$ 1.55 MeV) $5.71(5)$
0.0294$\pm$0.0034 2.65$\pm 0.41$ 0.172$\pm$0.010 0.279$\pm$0.014 1.41$\pm$0.04
5.06 $\begin{array}[]{l}{}^{26}\mathrm{Mg}(\textstyle{0}^{+})\\\
{}^{26}\mathrm{Si}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{26}\mathrm{Al}(1^{+}\,1.06\,\mathrm{MeV})$
$3.550(11)$ 1.063$\pm$0.027 1.03$\pm$0.13 1.031$\pm$0.013 1.05$\pm$0.06
1.20$\pm$0.08 1.14 32S$(\textstyle{0}^{+}$)
$\leftrightarrow\,^{32}$Cl$(\textstyle{1}^{+}$) $6.74(18)$ 0.0021$\pm$0.0009
$6.9^{+7.3}_{-4.1}$ 0.046$\pm$0.010 0.116$\pm$0.043 0.99$\pm$0.05 8.6
39K$(\textstyle{3\over 2}^{+}$) $\leftrightarrow\,^{39}$Ca$(\textstyle{3\over
2}^{+}$) $3.6326(10)$ 1.060$\pm$0.008 1.41$\pm$0.17 1.030$\pm$0.004
1.22$\pm$0.07 $3.1\pm 0.3$ 2.5

Here we make two modifications in the compiled $\mathrm{B_{GT}^{(p,n)}}$
values. The first is a reduction by a factor of $(1.251/1.276)^{2}$ to account
for the current value of $g_{A}$. The second addresses the absence of
experimental errors on the compiled Watson  BGT values. In previous work a
value for $\delta$ was obtained by a simple fit, which weights all data points
equally and precludes a realistic estimate of the uncertainty for the derived
$\delta$. As discussed below, we now include several new transitions in our
analysis where uncertainties are available. For the transitions we retain from
the tabulation of Watson , we have estimated uncertainties using Anderson83 ,
which Watson  references for experimental details. The uncertainties tabulated
there include the efficiency determination ($\pm$8%), beam normalization
($\pm$5%), neutron attenuation ($\pm$5%), counting statistics ($\pm$3%), and
background subtractions ($\pm$5%). Combining these in quadrature yields an
estimated $\pm$12% uncertainty, which we adopt for all of the stronger
transitions in Table 2. (Ref. Anderson83  also includes a correction for
target water absorption, but that correction addresses an issue specific to
one target.) The uncertainty inherent in (p,n) mappings of BGT strength has
been frequently discussed, with most estimates in range of 10-20% Zegers . Our
choice of 12% is consistent with this range.

Two additional transitions used in HH1 ; HH2 ,
32S$(0^{+})\leftrightarrow\,^{32}$Cl($1^{+}$) and 39K(${3\over
2}^{+})\leftrightarrow\,^{39}$Cl(${1\over 2}^{+})$, are candidate
$\ell$-forbidden M1 transitions sensitive to the tensor amplitude. One expects
both to be dominated by the transition density $2s_{1/2}\leftrightarrow
1d_{3/2}$. The 39K $M^{\mathrm{(p,n)}}$ given in HH1  was extracted from raw
(p,n) cross sections as an order-of-magnitude estimate: there is no
experimentally extracted B${}_{\mathrm{GT}}^{\mathrm{(p,n)}}$ value. No
meaningful error can be assigned to this very weak transition, so it is not
included here. The 32S transition was reconsidered in Grewe , where a large
uncertainty was assigned, which we adopt for our analysis. The impact of this
transition on the current analysis is greatly diminished by the size of that
uncertainty.

We also include two $2s1d$-shell transitions not considered in earlier
analyses, one a recent result for
26Mg($0^{+})\rightarrow\,^{26}$Al$(1^{+},1.70\,$MeV), obtained from
$(^{3}$He,t). While this result was normalized to the 26Al ground state
$\beta$ decay rate, the unusually weak tensor contribution we predict for the
normalizing transition ($\lesssim$ 1%) allows us to accept this result. The
second is 19F(${1\over 2}^{+})\rightarrow\,^{19}$Ne$(\textstyle{3\over
2}^{+},1.55\,$MeV), a transition that is strongly $\ell$-forbidden, according
to the shell model, and thus potentially quite sensitive to
$\hat{O}_{\mathrm{T}}^{J=1}$.

The data displayed in Table 2 include $\beta$ decay log(ft) values taken from
the ENSDF data files ENSDF , from which we determine $M_{\mathrm{GT}}$ and its
uncertainty, and calculations of $M_{\mathrm{T}}$ — one needs both the
magnitude and sign of this quantity relative to $M_{\mathrm{GT}}$. In nine of
the tabulated cases the shell-model calculations we perform (see below) yield
a positive relative sign, and thus a positive $\delta$ to account for the
observed enhancement in $|M^{\mathrm{(p,n)}}|/|M_{\mathrm{GT}}|$. There are
two exceptions: The three shell model calculations we performed for
18O($0^{+})\rightarrow\,^{18}$F$(1^{+})$ all predict a negative
$M_{\mathrm{T}}$, but with a magnitude so small that it has no impact on our
study. For 19F($\textstyle{1\over
2}^{+})\rightarrow\,^{19}$Ne$(\textstyle{3\over 2}^{+},1.55\,$MeV), the shell
model calculations disagree on the sign, and all underestimate the already
quite suppressed $M_{\mathrm{GT}}$ derived from experiment. We have assumed
constructive interference as this is indicated by experiment and is consistent
with the calculation that best reproduces the known value of
$|M_{\mathrm{GT}}|$ (the USDA shell model result described below).

Our shell-model calculations of $M_{\mathrm{T}}$ were performed with the Cohen
and Kurath CK  interaction in the $1p$ shell and the Brown-Wildenthal
Wildenthal  and USDA/USDB USD  interactions in the $2s1d$ shell. The
availability of three $2s1d$ interactions that each do well in reproducing
$2s1d$ spectroscopy provides an opportunity to assess theory uncertainties.
The $2s1d$-shell values in the table are the means and standard deviations of
the three calculations. There is excellent consistency, typically at the level
of 10%. Even by eye, there is clearly a strong correlation between the cases
in Table 2 where $|M_{\mathrm{T}}|/|M_{\mathrm{GT}}|$ is large and those where
the experimental ratio
$|\mathrm{B}_{\mathrm{GT}}^{\mathrm{(p,n)}}|/|\mathrm{B}_{\mathrm{GT}}|$ is
significantly above 1. In the four cases where the
$|\mathrm{B}_{\mathrm{GT}}^{\mathrm{(p,n)}}|/|\mathrm{B}_{\mathrm{GT}}|$
exceeds 1.4, $|M_{\mathrm{T}}|/|M_{\mathrm{GT}}|$ ranges from 2.5 to 8.6.

We use these results to test whether the inclusion of the tensor operator in
Eq. (22) improves the agreement between $M^{\mathrm{(p,n)}}$ and
$M_{\mathrm{GT}}$. Evaluating the $\chi^{2}$ per degree of freedom with and
without $M_{\mathrm{T}}$ yields

$\displaystyle{1\over
11}\sum_{i=1}^{A}{(M_{\mathrm{GT}}(i)-M^{\mathrm{(p,n)}}(i))^{2}\over\sigma{(i)}^{2}}$
$\displaystyle\approx$ $\displaystyle 9.4$ $\displaystyle{1\over
10}\sum_{i=1}^{A}{(M_{\mathrm{GT}}(i)+\delta
M_{\mathrm{T}}(i)-M^{\mathrm{(p,n)}}(i))^{2}\over\sigma{(i)}^{2}}$
$\displaystyle\approx$ $\displaystyle 1.0$

where $\delta=0.076$, and $\sigma(i)$ is generated by combining uncertainties
for $M^{\mathrm{(p,n)}}$ and $M_{\mathrm{GT}}$ in quadrature. While a simple
proportionality between BGT strength and (p,n) scattering for individual
states is not supported by the data, the proportionality is restored with the
introduction of $M_{\mathrm{T}}$ to a level consistent with the statistical
fluctuations of the data. The variation around the minimum to achieve a unit
change in the total $\chi^{2}$ yields the estimate $\delta=0.076\pm 0.008$ at
$1\sigma$.

The analysis can also be done by examining each target separately: this
approach has advantages in understanding the relationship of the current work
to that of HH1 ; HH2 . For each target $i$ we determine a probability
distribution for $\delta(i)$ from the relation
$M_{\mathrm{GT}}(i)+\delta(i)M_{\mathrm{T}}(i)-M^{\mathrm{(p,n)}}(i)=0$,
treating the errors on $M_{\mathrm{GT}}(i)$, $M_{\mathrm{T}}(i)$, and
$M^{\mathrm{(p,n)}}(i)$ as Gaussian with the uncertainties listed in Table 2.
The convolution was done by discretizing the probability distributions in
bins, and independently by Monte Carlo, and the results cross-checked to
verify their numerical accuracy. The theory errors on $M_{\mathrm{T}}$ (see
Table 2) were computed from the standard deviations of the shell model results
in the cases where multiple effective interactions were explored; in the five
cases where only a single shell-model calculation was done, we assigned an
uncertainty of 10%, a value typical of the other cases. (In the simple
$\chi^{2}$ fit described previously, we used the best values for the
$M_{\mathrm{T}}$, neglecting the uncertainties.)

The resulting probability distributions for the $\delta(i)$, while not exactly
Gaussian, turn out to be nearly so in all cases. The equivalent Gaussian means
and standard deviations are given in Table 3. These results can then be
combined to form the overall uncertainty-weighted mean and standard deviation,
$\delta=0.074\pm 0.008~{}(1\sigma)$, a result nearly identical to that
obtained more simply from the $\chi^{2}$. This is our final result for
$\delta$.

In this fit, the result is dominated by four transitions, from 13C, 15N, 39K,
and 19F to the excited state, which are four of the five cases where
$M_{\mathrm{T}}$ exceeds $M_{\mathrm{GT}}$ by factors $\gtrsim$ 2.5. If only
these four transitions are retained, one obtains $\delta=0.078\pm 0.009$. The
other seven constraints have a minimal effect, shifting the mean by $\approx
5\%$ and improving the precision by only $\approx 6\%$. This reflects the fact
that in computing the weighted mean and uncertainty, the contributions of
these seven are diluted by their low weights, $w_{i}=1/\sigma_{i}^{2}$, for
the $\sigma_{i}$ of Table 3. Thus it is somewhat fortuitous that earlier work
HH1 ; HH2  in which central values were fit, thereby weighting each target
equally, gave results consistent with the range determined here. (The results
from HH1 ; HH2  are $\delta=0.096$ and 0.069 for the $1p$ and $2s1d$ shells,
respectively.)

These results and their relevance to the 71Ga($\nu_{e},e^{-})^{71}$Ge cross
section are apparent from Fig. 2. The agreement between the (p,n) amplitude
$|M^{\mathrm{(p,n)}}|$ and $|M_{\mathrm{GT}}|$, which is excellent for
transitions with strong BGT values, systematically deteriorates as BGT is
reduced. But this deterioration is corrected by the inclusion of
$M_{\mathrm{T}}$. The shaded region at small BGT is that relevant for the two
71Ga excited-state transitions: based on the trends apparent from the figure,
the interpretation of (p,n) data for these transitions would not be reliable
unless the effects of $M_{\mathrm{T}}$ are treated.

From Fig. 2 one sees that a fixed $\delta$ brings the (p,n) results into
accord with known Gamow-Teller strengths throughout the $1p$ and $2s1d$
shells. The absence of any evident dependence on mass number justifies the use
of the same $\delta$ in our 71Ga cross section work. It would be helpful to
verify this assumption by extending the results of Fig. 2 into the
$2p_{3/2}1f_{5/2}2p_{1/2}1g_{9/2}$ shell. Obstacles to doing this successfully
include the absence of an experimental compilation for heavier nuclei
analogous to that of Watson , fewer opportunities to exploit isospin mirror
transitions (which play a major role in the analysis presented here), and the
theory challenge of evaluating the tensor amplitudes in systems with higher
level densities and consequently more delicate level mixing.

Table 3: The Gaussian means $\bar{\delta}_{i}$ and standard deviations $\sigma_{i}$ for the distributions $\delta_{i}$ obtained from each of the reactions. The weighted combination of these results yields $\delta=0.074\pm 0.008~{}(1\sigma)$.  Reaction | $\bar{\delta}_{i}$ | $\sigma_{i}$
---|---|---
13C$(\textstyle{1\over 2}^{-}$) $\leftrightarrow\,{}^{13}$N$(\textstyle{1\over 2}^{-}$) | 0.082 | 0.020
$\begin{array}[]{l}{}^{14}\mathrm{C}(\textstyle{0}^{+})\\\ {}^{14}\mathrm{O}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{14}\mathrm{N}(1^{+}\,3.95\,\mathrm{MeV})$ | -0.24 | 1.21
15N$(\textstyle{1\over 2}^{-}$) $\leftrightarrow\,^{15}$O$(\textstyle{1\over 2}^{-}$) | 0.093 | 0.021
17O$(\textstyle{5\over 2}^{+}$) $\leftrightarrow\,^{17}$F$(\textstyle{5\over 2}^{+}$) | -0.070 | 0.085
18O$(\textstyle{0}^{+}$) $\leftrightarrow\,^{18}$F$(\textstyle{1}^{+}$) | -1.59 | 2.48
$\begin{array}[]{l}{}^{18}\mathrm{O}(\textstyle{0}^{+})\\\ {}^{18}\mathrm{Ne}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{18}\mathrm{F}(1^{+}\,1.70\,\mathrm{MeV})$ | 0.063 | 0.040
19F$(\textstyle{1\over 2}^{+}$) $\leftrightarrow\,^{19}$Ne$(\textstyle{1\over 2}^{+}$) | 2.69 | 1.80
19F$(\textstyle{1\over 2}^{+}$) $\leftrightarrow\,^{19}$Ne$(\textstyle{3\over 2}^{+}$ 1.55 MeV) | 0.076 | 0.012
$\begin{array}[]{l}{}^{26}\mathrm{Mg}(\textstyle{0}^{+})\\\ {}^{26}\mathrm{Si}(\textstyle{0}^{+})\end{array}\Big{\\}}\rightarrow{}^{26}\mathrm{Al}(1^{+}\,1.06\,\mathrm{MeV})$ | 0.015 | 0.051
32S$(\textstyle{0}^{+}$) $\leftrightarrow\,^{32}$Cl$(\textstyle{1}^{+}$) | 0.070 | 0.045
39K$(\textstyle{3\over 2}^{+}$) $\leftrightarrow\,^{39}$Ca$(\textstyle{3\over 2}^{+}$) | 0.061 | 0.023
Figure 2: In blue: correspondence between the (p,n) amplitude $|M^{\mathrm{(p,n)}}|$ and the beta decay amplitude $|M_{\mathrm{GT}}|$ is excellent when BGT is strong, but deteriorates for weaker BGT. In red: the agreement is restored with the inclusion of $|M_{\mathrm{T}}|$. The two excited states that contribute to the BEST cross section have weak transition strengths that would place them in the shaded region, where large tensor contributions would be anticipated. Table 4: Shell model results for the contributing 71Ga transitions for each of three interactions, from which we estimate best values and $1\sigma$ ranges. The predicted values and ranges for $M_{\mathrm{T}}$ are used in the extraction of $|M_{\mathrm{GT}}|$ for the two excited-state transitions from (p,n) results.  |  |  |  |  |
---|---|---|---|---|---
Transition | Interaction |  $M_{\mathrm{GT}}$ |  $M_{\mathrm{T}}$ |  $\overline{M}_{\mathrm{GT}}$ |  $\overline{M}_{\mathrm{T}}$
|  |  |  |  |
$\textstyle{3\over 2}^{-}\rightarrow\textstyle{1\over 2}^{-}$(gs) | JUN45 | $0.791$ | $-0.516$ |  |
| GCN2850 | $0.361$ | $-0.320$ | 0.48$\pm$0.27 | -0.37$\pm$0.13
| jj44b | $0.290$ | $-0.283$ |  |
$\textstyle{3\over 2}^{-}\rightarrow\textstyle{5\over 2}^{-}$(175 keV) | JUN45 | $0.145$ | $-0.764$ |  |
| GCN2850 | $0.159$ | $-0.410$ | 0.19$\pm$0.07 | -0.50$\pm$0.24
| jj44b | $0.264$ | $-0.311$ |  |
$\textstyle{3\over 2}^{-}\rightarrow\textstyle{3\over 2}^{-}$(500 keV) | JUN45 | $0.096$ | $0.062$ |  |
| GCN2850 | $0.196$ | $-0.178$ | 0.27$\pm$0.21 | -0.088$\pm$0.13
| jj44b | $0.505$ | $-0.148$ |  |

##  V Recommended excited-state BGT values

With $\delta=0.074\pm 0.008~{}(1\sigma)$ determined, we can now extract from
(p,n) measurements best values and estimated uncertainties for
$M_{\mathrm{GT}}$ for the two excited-state contributions to
71Ga$(\nu_{e},e^{-})^{71}$Ge, using

$M^{\mathrm{(p,n)}}=M_{\mathrm{GT}}+\delta M_{\mathrm{T}}.$

This requires us to compute the magnitude and relative sign of
$M_{\mathrm{T}}$. The theory task is more challenging than that of the
previous section because the effective interactions available for the relevant
shell-model space, $2p_{3/2}1f_{5/2}2p_{1/2}1g_{9/2}$, are known to be less
successful in their spectroscopic predictions. These nuclear physics
uncertainties should be reflected in the range of the predicted
$M_{\mathrm{T}}$.

As was done in the $2s1d$ shell, calculations were performed with three well-
tested interactions, GCN2850 GCN2850 , jj44b jj44b , and JUN45 JUN45 ,
including all $m$-scheme Slater determinants that can be formed in the valence
space ($\approx 10^{8}$ basis states). The results are shown in Table 4. There
is reasonable agreement among these calculations on magnitudes and signs, and
as before, we combine the results to obtain best values and $1\sigma$ ranges.
The combined results are denoted $\overline{M}_{\mathrm{GT}}$ and
$\overline{M}_{\mathrm{T}}$. The results for $\overline{M}_{\mathrm{T}}$ — its
magnitude and sign relative to $M_{\mathrm{GT}}$ — are needed in the analysis
below. An immediate cross check on the nuclear structure comes from the known
BGT value derived from the 71Ge electron capture rate,
$B_{\mathrm{GT}}=0.0864\pm 0.0013~{}(2\sigma)$, from which one finds
$M_{\mathrm{GT}}=0.588\pm 0.002~{}(1\sigma)$, in good agreement with the
shell-model result, $\overline{M}_{\mathrm{GT}}=0.48\pm 0.27~{}(1\sigma)$.
After we extract the ${M}_{\mathrm{GT}}$’s from the (p,n) results, we will be
able to make similar comparisons for the excited states. Note that the shell
model indicates largely destructive interference between the $M_{\mathrm{GT}}$
and $M_{\mathrm{T}}$ amplitudes for the three transitions of interest.

Analysis for Krofcheck et al.: The forward-angle (p,n) scattering results of
Krofcheck et al. Krofcheck  for exciting the 71Ge ground-state
($\textstyle{1\over 2}^{-}$), 175 keV ($\textstyle{5\over 2}^{-}$), and 500
keV ($\textstyle{3\over 2}^{-}$) levels yield
B${}_{\mathrm{GT}}^{\mathrm{(p,n)}}$ = 0.089$\pm$0.007, $<0.005$, and
0.011$\pm$0.002, respectively. These results were normalized to the analog
transition using an energy-dependent coefficient relating
B${}_{\mathrm{GT}}^{\mathrm{(p,n)}}$ to cross sections. As we are concerned
with states in a narrow energy band, we can avoid any issues with this choice
by forming the ratios

$\displaystyle{\mathrm{B_{GT}^{(p,n)}}(\textstyle{5\over
2}^{-})\over\mathrm{B_{GT}^{(p,n)}}(\mathrm{gs})}$ $\displaystyle\equiv$
$\displaystyle{|M_{\mathrm{GT}}(\textstyle{5\over
2}^{-})+\delta\overline{M}_{\mathrm{T}}(\textstyle{5\over
2}^{-})|^{2}\over|M_{\mathrm{GT}}(\mathrm{gs})+\delta\overline{M}_{\mathrm{T}}(\mathrm{gs})|^{2}}$
$\displaystyle<$ $\displaystyle 0.06~{}~{}~{}(68\%~{}\mathrm{C.L.})$
$\displaystyle{\mathrm{B_{GT}^{(p,n)}}(\textstyle{3\over
2}^{-})\over\mathrm{B_{GT}^{(p,n)}}(\mathrm{gs})}$ $\displaystyle\equiv$
$\displaystyle{|M_{\mathrm{GT}}(\textstyle{3\over
2}^{-})+\delta\overline{M}_{\mathrm{T}}(\textstyle{3\over
2}^{-})|^{2}\over|M_{\mathrm{GT}}(\mathrm{gs})+\delta\overline{M}_{\mathrm{T}}(\mathrm{gs})|^{2}}$
(26) $\displaystyle=$ $\displaystyle 0.124\pm
0.024~{}~{}~{}(68\%~{}\mathrm{C.L.})$

We first consider the transition to the $\textstyle{3\over 2}^{-}$ state.
Because the electron capture $M_{\mathrm{GT}}$ is so precisely known, we use
the central value in the analysis below. We insert the values for the two
tensor matrix elements $M_{\mathrm{T}}$ and $\delta$, including their
uncertainties. We take the (dominant) sign of $M_{\mathrm{T}}$ relative to
$M_{\mathrm{GT}}$ from theory. We assume all distributions are normal
(Gaussian), described by the specified standard deviations, then compute the
associated distribution for the needed BGT ratio, which we find is also well
represented by a normal distribution. This yields

$\displaystyle{\mathrm{B_{GT}}(\textstyle{3\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}$ $\displaystyle\equiv$
$\displaystyle{|M_{\mathrm{GT}}(\textstyle{3\over
2}^{-})|^{2}\over|M_{\mathrm{GT}}(\mathrm{gs})|^{2}}$ (27) $\displaystyle=$
$\displaystyle 0.121\pm 0.026~{}~{}~{}(68\%~{}\mathrm{C.L.})~{}.$

The tensor corrections have little impact, shifting the numerator and
denominator similarly, by 3.2% and 4.7% respectively, with these shifts
largely canceling when the ratio is formed. The central value obtained for
$|M_{\mathrm{GT}}(\textstyle{3\over 2}^{-})|\,\approx\,0.20$ is consistent
with the shell-model range in Table 4, 0.27 $\pm$ 0.21.

In our shell-model calculations the density matrices for the transition to the
175 keV $\textstyle{5\over 2}^{-}$ state are dominated by the the
$\ell$-forbidden amplitude $2p_{3/2}\rightarrow 1f_{5/2}$, which reaches
single-particle strength in the case of the JUN45 calculation. This is the
reason for the strength of $M_{\mathrm{T}}$ and the weakness of
$M_{\mathrm{GT}}$ in the shell-model studies of Table 4, and consequently
$\overline{M}_{\mathrm{T}}/\overline{M}_{\mathrm{GT}}\,\approx\,-2.6$. The
destructive interference allows for a larger $|M_{\mathrm{GT}}|$ than would be
the case if the tensor contributions to (p,n) scattering were ignored. As was
done for the 500 keV state, we take into account the uncertainties on the
various quantities by integrating over the probability distributions of each
input variable, taking the ranges of the $\overline{M}_{\mathrm{T}}$ from the
results of Table 4. In this calculation we interpret the experimental bound
given in the first of Eqs. (26) as a measurement of 0 with a one-sided normal
distribution described by $\sigma=0.06$. We find

$\displaystyle|M_{\mathrm{GT}}(\textstyle{5\over 2}^{-})|$
$\displaystyle\lesssim$
$\displaystyle\left\\{\begin{array}[]{ll}0.18&~{}68\%~{}\mathrm{C.L.}\\\\[1.9919pt]
0.24&~{}95\%~{}\mathrm{c.l}\end{array}\right.$ (30)
$\displaystyle\displaystyle{\mathrm{B_{GT}}(\textstyle{5\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}$ $\displaystyle\lesssim$
$\displaystyle\left\\{\begin{array}[]{ll}0.089&~{}68\%~{}\mathrm{C.L.}\\\\[1.9919pt]
0.160&~{}95\%~{}\mathrm{C.L.}\end{array}\right.$ (33)

Figure 3: Shaded region: numerically generated probability distribution for
the 51Cr cross section with excited-state contributions extracted from
forward-angle (p,n) cross sections of Krofcheck  (see text). Dashed line: the
analytic split-normal fit to these data.

Analysis for Frekers et al.: We repeat the analysis for the Frekers et al.
FrekersGa  data, as the use of the same effective operator for $(^{3}$He,t)
has support from both theory and experiment Frekers , while noting that a
separate derivation of $\delta$ based on data like those of Table 2 has not
been done for this reaction. We again form the ratios

$\displaystyle{\mathrm{B_{GT}^{(^{3}He,t)}}(\textstyle{5\over
2}^{-})\over\mathrm{B_{GT}^{(3He,t)}}(\mathrm{gs})}$ $\displaystyle\equiv$
$\displaystyle{|M_{\mathrm{GT}}(\textstyle{5\over
2}^{-})+\delta\overline{M}_{\mathrm{T}}(\textstyle{5\over
2}^{-})|^{2}\over|M_{\mathrm{GT}}(\mathrm{gs})+\delta\overline{M}_{\mathrm{T}}(\mathrm{gs})|^{2}}$
$\displaystyle=$ $\displaystyle 0.040\pm 0.031~{}~{}~{}(68\%~{}\mathrm{C.L.})$
$\displaystyle{\mathrm{B_{GT}^{(^{3}He,t)}}(\textstyle{3\over
2}^{-})\over\mathrm{B_{GT}^{(3He,t)}}(\mathrm{gs})}$ $\displaystyle\equiv$
$\displaystyle{|M_{\mathrm{GT}}(\textstyle{3\over
2}^{-})+\delta\overline{M}_{\mathrm{T}}(\textstyle{3\over
2}^{-})|^{2}\over|M_{\mathrm{GT}}(\mathrm{gs})+\delta\overline{M}_{\mathrm{T}}(\mathrm{gs})|^{2}}$
(34) $\displaystyle=$ $\displaystyle 0.207\pm
0.019~{}~{}~{}(68\%~{}\mathrm{C.L.})$

The calculation for the 500 keV $\textstyle{3\over 2}^{-}$ excited state state
proceeds as before, taking into account the values and uncertainties for the
two tensor matrix elements $M_{\mathrm{T}}$ and $\delta$, and taking the
relative signs of the $M_{\mathrm{T}}$ from theory. This yields

${\mathrm{B_{GT}}(\textstyle{3\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}=0.198\pm
0.024~{}~{}~{}(68\%~{}\mathrm{C.L.})~{}.$ (35)

Thus the effects of $M_{\mathrm{T}}$ are modest, for the reasons mentioned
above. The central value $M_{\mathrm{GT}}(\textstyle{3\over
2}^{-})\,\approx\,0.26$ is consistent with the theory range of Table 4,
$0.27\,\pm\,0.21$.

A similar analysis for the Frekers result for the transition to the 175 keV
state yields

$\displaystyle\left.\begin{array}[]{r}0.11\\\\[1.9919pt]
0.065\end{array}\right\\}$ $\displaystyle\lesssim$
$\displaystyle|M_{\mathrm{GT}}(\textstyle{5\over
2}^{-})|\lesssim\left\\{\begin{array}[]{ll}19&~{}68\%~{}\mathrm{C.L.}\\\\[1.9919pt]
0.23&~{}95\%~{}\mathrm{c.l}\end{array}\right.$ (40)
$\displaystyle\displaystyle{\mathrm{B_{GT}}(\textstyle{5\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}$ $\displaystyle=$ $\displaystyle
0.071\pm 0.036~{}~{}~{}(68\%~{}\mathrm{C.L.}).$ (41)

The central value for
$\left|M_{\mathrm{GT}}\left(\frac{5}{2}^{-}\right)\right|\,\approx\,0.15$ is
consistent with the shell-model range, 0.19$\pm$0.07.

##  VI Total Cross Sections

The total neutrino cross sections for 51Cr and 37Ar can be expressed in the
form HH1

$\sigma=\sigma_{\mathrm{gs}}\left[1+\xi(\textstyle{5\over
2}^{-})\,\displaystyle{\mathrm{B_{GT}}(\textstyle{5\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}+\xi(\textstyle{3\over
2}^{-})\,\displaystyle{\mathrm{B_{GT}}(\textstyle{3\over
2}^{-})\over\mathrm{B_{GT}}(\mathrm{gs})}\right]$ (42)

where the phase-space coefficients, computed from the results of Table 1, are

${}^{51}\mathrm{Cr}:~{}~{}\xi(\textstyle{5\over 2}^{-})$ $\displaystyle=$
$\displaystyle 0.669~{}~{}~{}\xi(\textstyle{3\over 2}^{-})=0.220$
${}^{37}\mathrm{Ar}:~{}~{}\xi(\textstyle{5\over 2}^{-})$ $\displaystyle=$
$\displaystyle 0.696~{}~{}~{}\xi(\textstyle{3\over 2}^{-})=0.264$ (43)

(For earlier calculations of these coefficients see HH1 ; Bahcall97 .)

Combining the excited-state BGT ratios derived from the data of Krofcheck et
al. Krofcheck , Eqs. (27) and (33), with the ground-state result of Eq. (17)
yields

$\displaystyle\sigma(^{51}\mathrm{Cr})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}5.71^{+0.27}_{-0.10}&68\%~{}\mathrm{C.L.}\\\\[5.69046pt]
5.71^{+0.51}_{-0.23}&95\%~{}\mathrm{C.L.}\end{array}\right\\}\times
10^{-45}\,\mathrm{cm}^{2}$ (46) $\displaystyle\sigma(^{37}\mathrm{Ar})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}6.88^{+0.34}_{-0.13}&68\%~{}\mathrm{C.L.}\\\\[5.69046pt]
6.88^{+0.63}_{-0.28}&95\%~{}\mathrm{C.L.}\end{array}\right\\}\times
10^{-45}\,\mathrm{cm}^{2}$ (50)

The probability distributions for the cross sections were computed numerically
by folding the ground-state and excited-state probabilities. The central
values correspond to the most probable cross section and the ranges contain
the 68% and 95% fractions of the most probable results. The excited-state
contributions increase the cross sections by $\approx$ 6.0% and 6.6% for 51Cr
and 37Ar, respectively.

Repeating this calculation using the excited-state BGT ratios derived from the
data of Frekers et al. FrekersGa , Eqs. (35) and (41), yields

$\displaystyle\sigma(^{51}\mathrm{Cr})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}5.85^{+0.19}_{-0.13}&68\%~{}\mathrm{C.L.}\\\\[5.69046pt]
5.85^{+0.36}_{-0.26}&95\%~{}\mathrm{C.L.}\end{array}\right\\}\times
10^{-45}\,\mathrm{cm}^{2}$ (53) $\displaystyle\sigma(^{37}\mathrm{Ar})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}7.01^{+0.22}_{-0.16}&68\%~{}\mathrm{C.L.}\\\\[5.69046pt]
7.01^{+0.43}_{-0.31}&95\%~{}\mathrm{C.L.}\end{array}\right\\}\times
10^{-45}\,\mathrm{cm}^{2}$ (57)

The excited-state contributions increase the cross sections by $\approx$ 8.6%
for both 51Cr and 37Ar sources. The results of Eqs. (50) and (57) agree at
$1\sigma$, but in our view should not be combined because they depend on input
strengths for the transition to the $\textstyle{3\over 2}^{-}$ state that
differ by significantly more.

The numerically evaluated cross section distributions can be accurately
described as split-normal probability distributions

$\displaystyle P(\sigma)\approx\sqrt{2\over\pi}$
$\displaystyle~{}\displaystyle{1\over\sigma_{1}+\sigma_{2}}~{}\bigg{[}\theta(\bar{\sigma}-\sigma)e^{-(\sigma-\bar{\sigma})^{2}/2\sigma_{1}^{2}}$
(58)
$\displaystyle+\theta(\sigma-\bar{\sigma})e^{-(\sigma-\bar{\sigma})^{2}/2\sigma_{2}^{2}}\bigg{]},$

where $\sigma$ is dimensionless, in units of $10^{-45}$ cm2. The best fits are
obtained by tuning the parameters given above to optimize the overall fit.
Figure 3, for the case of a 51Cr source and Krofcheck et al. Krofcheck
excited state contributions, illustrates the quality of the fit. These split-
normal distributions will enable users to adapt our results for any desired
confidence level. Table 5 gives the numerical values for the fit parameters.

Table 5: The split normal parameterization of our cross section results.  Source | Excited States | $\bar{\sigma}$ | $\bar{\sigma}_{1}$ | $\bar{\sigma}_{2}$
---|---|---|---|---
51Cr | Krofcheck  | 5.68 | 0.10 | 0.24
51Cr | Frekers  | 5.83 | 0.12 | 0.17
37Ar | Krofcheck  | 6.84 | 0.13 | 0.30
37Ar | Frekers  | 6.99 | 0.15 | 0.21

##  VII Impact on the BEST and Gallium Anomalies

The BEST 71Ge production rates for the outer and inner volumes, obtained from
the yields in the K and L peaks with a correction for the contribution of the
M peak BEST1 ; BEST2 , are

$\displaystyle R_{\mathrm{out}}$ $\displaystyle=$ $\displaystyle 55.6\pm
2.7\,\mathrm{(stat)}^{+1.6}_{-1.5}\,\mathrm{(syst)}~{}\mathrm{atoms/d}$
$\displaystyle=$ $\displaystyle 55.6\pm 3.1\mathrm{~{}atoms/d}$ $\displaystyle
R_{\mathrm{in}}$ $\displaystyle=$ $\displaystyle 54.9\pm
2.5\,\mathrm{(stat)}^{+1.6}_{-1.5}\,\mathrm{(syst)}~{}\mathrm{atoms/d}$ (59)
$\displaystyle=$ $\displaystyle 54.9^{+3.0}_{-2.9}\mathrm{~{}atoms/d}$

where the statistical and systematic errors have been combined in quadrature
to obtain a total error. From the neutrino source activity of 3.414$\pm$0.008
MCi and the cross section of Bahcall Bahcall97  used in BEST2 , one finds the
predicted production rates

$\displaystyle R_{\mathrm{out}}^{\mathrm{expected}}$ $\displaystyle=$
$\displaystyle 72.6^{+2.6}_{-2.1}\mathrm{~{}atoms/d}$ $\displaystyle
R_{\mathrm{in}}^{\mathrm{expected}}$ $\displaystyle=$ $\displaystyle
69.4^{+2.5}_{-2.0}\mathrm{~{}atoms/d}$ (60)

where again uncertainties have been combined in quadrature. The ratios of the
measured to predicted production rates are

$\displaystyle{R_{\mathrm{out}}\over R_{\mathrm{out}}^{\mathrm{expected}}}$
$\displaystyle=$ $\displaystyle{55.6\pm 3.1\over 72.6^{+2.6}_{-2.1}}=0.766\pm
0.05$ $\displaystyle{R_{\mathrm{in}}\over
R_{\mathrm{in}}^{\mathrm{expected}}}$ $\displaystyle=$
$\displaystyle~{}{54.9^{+3.0}_{-2.9}\over 69.4^{+2.5}_{-2.0}}~{}=0.791\pm
0.05$ (61)

showing discrepancies of 4.7 and 4.2 standard deviations, respectively. The
ratio of the outer and inner rates

$r={R_{\mathrm{out}}\over R_{\mathrm{in}}}={0.766\pm 0.05\over 0.791\pm
0.05}=0.97\pm 0.08$ (62)

is consistent with unity.

The Bahcall cross section used above employed the (p,n) data of Krofcheck et
al. Krofcheck  in estimating the excited-state contribution to the 71Ga cross
section. The analogous analysis presented here, updating the ground-state
contribution and correcting for the tensor contribution to the (p,n) results
of Krofcheck , yields

$\displaystyle R_{\mathrm{out}}^{\mathrm{expected}}$ $\displaystyle=$
$\displaystyle 71.4^{+3.4}_{-1.3}\mathrm{~{}atoms/d}$ $\displaystyle
R_{\mathrm{in}}^{\mathrm{expected}}$ $\displaystyle=$ $\displaystyle
68.2^{+3.2}_{-1.2}\mathrm{~{}atoms/d}$ (63)

The ratios of the measured to predicted production rates are

$\displaystyle{R_{\mathrm{out}}\over R_{\mathrm{out}}^{\mathrm{expected}}}$
$\displaystyle=$ $\displaystyle{55.6\pm 3.1\over
71.4^{+3.4}_{-1.3}}=0.799^{+0.046}_{-0.057}$
$\displaystyle{R_{\mathrm{in}}\over R_{\mathrm{in}}^{\mathrm{expected}}}$
$\displaystyle=$ $\displaystyle~{}{54.9^{+3.0}_{-2.9}\over
68.2^{+3.2}_{-1.2}}~{}=0.805^{+0.046}_{-0.057}$ (64)

The discrepancies are reduced slightly, to 4.3 and 3.8 standard deviations,
respectively.

We have also constrained the 71Ge excited-state contribution using the (3He,t)
data of Frekers et al. FrekersGa . This yields

$\displaystyle R_{\mathrm{out}}^{\mathrm{expected}}$ $\displaystyle=$
$\displaystyle 73.1^{+2.4}_{-1.7}\mathrm{~{}atoms/d}$ $\displaystyle
R_{\mathrm{in}}^{\mathrm{expected}}$ $\displaystyle=$ $\displaystyle
69.9^{+2.3}_{-1.6}\mathrm{~{}atoms/d}$ (65)

The ratios of the measured to predicted production rates are

$\displaystyle{R_{\mathrm{out}}\over R_{\mathrm{out}}^{\mathrm{expected}}}$
$\displaystyle=$ $\displaystyle{55.6\pm 3.1\over
73.1^{+2.4}_{-1.7}}=0.761^{+0.046}_{-0.049}$
$\displaystyle{R_{\mathrm{in}}\over R_{\mathrm{in}}^{\mathrm{expected}}}$
$\displaystyle=$ $\displaystyle~{}{54.9^{+3.0}_{-2.9}\over
69.9^{+2.3}_{-1.6}}~{}=0.786^{+0.046}_{-0.049}$ (66)

The discrepancies in this case are 5.0 and 4.5 standard deviations,
respectively.

The combined result from all six Ga calibration experiments BEST2  for the
ratio of the measured to expected rates are 0.82 $\pm$ 0.06 and 0.80 $\pm$
0.04, using the cross sections derived here from the excited-state data of
Krofcheck  and FrekersGa , respectively.

Although the cross section changes found here are modest, we have updated the
neutrino oscillation results of BEST1 ; BEST2 . Figure 4 gives the exclusion
contours corresponding to 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence
levels, using only the BEST inner and outer results. The cross sections used
are those derived here, with the excited-state contribution extracted from the
results of Krofcheck  (left panel) and FrekersGa  (right panel). The best-fit
points correspond to $\mathrm{sin}^{2}{2\theta}=0.41$ and
$\Delta\mathrm{m}^{2}$=6.1 eV2 and $\mathrm{sin}^{2}{2\theta}=0.45$ and
$\Delta\mathrm{m}^{2}$=6.5 eV2, respectively. However, the chi-square space is
quite shallow and flat, so solutions along a valley centered on the contours
of Fig. 4 (and Fig. 5) provide nearly equivalent fits.

Figure 5 gives the exclusion contours corresponding to 1$\sigma$, 2$\sigma$,
and 3$\sigma$ confidence levels, when the BEST inner and outer results are
combined with those of the two GALLEX and two SAGE calibrations. The best-fit
points correspond to $\mathrm{sin}^{2}{2\theta}=0.32$ and
$\Delta\mathrm{m}^{2}$=1.25 eV2 and $\mathrm{sin}^{2}{2\theta}=0.34$ and
$\Delta\mathrm{m}^{2}$=1.25 eV2, respectively, for the indicated cross
sections. The shift in the best-fit results from those of Fig. 4 reflect the
shallowness of the chi-square space.

Figure 4: The allowed regions for oscillations into a sterile state derived
from the BEST inner and outer results. Left: the neutrino absorption cross
section derived using the (p,n) data of Krofcheck  to constrain excited-state
contributions. Right: results using the (3He,t) data of FrekersGa . The best-
fit points are $\mathrm{sin}^{2}{2\theta}=0.41$ and $\Delta\mathrm{m}^{2}=6.1$
eV2 and $\mathrm{sin}^{2}{2\theta}=0.45$ and $\Delta\mathrm{m}^{2}=6.5$ eV2,
respectively (see text). Figure 5: As in Fig. 4, but combining the results of
the two GALLEX and two SAGE calibrations with those of BEST. The best-fit
points are $\mathrm{sin}^{2}{2\theta}=0.32$ and $\Delta\mathrm{m}^{2}=1.25$
eV2 (left) and $\mathrm{sin}^{2}{2\theta}=0.34$ and
$\Delta\mathrm{m}^{2}=1.25$ eV2 (right).

##  VIII Summary and Conclusions

The published BEST analysis employed an older 51Cr cross section from Bahcall
Bahcall97 , $\sigma=\left[5.81^{+0.21}_{-0.16}\right]\times
10^{-45}\,\mathrm{cm}^{2}\,(1\sigma)$. Due to the experiment’s surprising
result, there is good motivation for re-examining the neutrino capture cross
section on 71Ga, to determine both a more modern best value for the cross
section and its uncertainty. The latter is particularly important in judging
the significance of the BEST result. The cross section is dominated by the
transition to the ground state of 71Ge. Since the work of Bahcall97 , changes
impacting this cross section include a more accurate $Q$-value and updates in
the value of $g_{A}$ and other weak parameters. In addition, there are effects
such as weak magnetism and the lack of universality in radiative corrections
that have not previously been evaluated quantitatively. The first half of this
paper describes these and other corrections that typically each enter at the
0.5% level. We have evaluated these effects including their uncertainties,
finding that they combine to yield a ground-state cross section about 2.5%
smaller than that of Bahcall Bahcall97 .

However, the more serious potential uncertainty is that associated with the
transitions to the 175 keV $\textstyle{5\over 2}^{-}$ and 500 keV
$\textstyle{3\over 2}^{-}$ excited states in 71Ge. In Bahcall97  those cross
sections were taken from forward-angle (p,n) measurements by Krofcheck et al.
Krofcheck . As was stressed in HH1 ; HH2  and illustrated here in Fig. 2,
(p,n) scattering is not a reliable probe of weak BGT strengths due to the
presence of a subdominant spin-tensor interaction in the effective operator
for the scattering. While normally a correction, the tensor operator can
dominate when the competing GT amplitude is weak. Haxton and Hata stressed
that 71Ga is a problematic case, as the transition to the first excited state,
$\textstyle{3\over 2}^{-}\rightarrow\textstyle{5\over 2}^{-}$ (175 keV), is
naturally associated with the $\ell$-forbidden amplitude $2p_{3/2}\rightarrow
1f_{5/2}$. This observation was confirmed here in all three of the shell-model
calculations performed. For a pure $\ell$-forbidden transition, the GT
amplitude is zero, while the tensor amplitude would have approximately unit
strength HH1 .

The relationship between GT strength and (p,n) forward scattering would be
restored if one could quantitatively correct for the presence of the tensor
operator. This would be possible if one could 1) reliably determine the
coefficient $\delta$ of the tensor correction (its size and uncertainty) and
2) develop some means of determining the accuracy with which the accompanying
nuclear matrix element of the tensor operator can be determined. Despite
previous work Watson ; HH1 ; HH2  determining $\delta$ from experiment, the
simple fits performed and the rather uncritical selection of data left
questions about the certainty with which $\delta$ could be established.

In this paper we determine $\delta$ using only measurements where experimental
uncertainties have been assigned and focusing on transitions involving
relatively simple $1p$\- and $2s1d$-shell nuclei where structure differences
arising from the choice of shell-model effective interaction are small. Where
possible, nuclear structure differences were quantified by exploring several
effective interactions. An attractive aspect of this approach is that an
accurate evaluation of the tensor matrix element is most important in those
cases where it is strong — and these are cases where the shell model should do
well. We see from Table 2 that in the four $2s1d$-shell cases where the tensor
matrix element is $\approx$ 1, the variation among the calculations is
typically 10%. Consequently, in transitions where the GT amplitude is weak but
the tensor amplitude strong, one can use theory to estimate the latter (but
not the former) reliably, and thus subtract it — the 10% error enters in the
correction, not in end result. With a more quantitative relationship between
(p,n) measurements and GT amplitudes thus restored, even relatively weak
$M_{\mathrm{GT}}$ can then be extracted from the data.

In determining $\delta$, we found a very strong correlation between cases
where $(p,n)$ and weak transition strengths disagreed, and strong tensor
matrix elements producing a ratio of $|M_{\mathrm{T}}/M_{\mathrm{GT}}|$ well
above one. Our study also underscores the fact that the weakness of excited-
state contributions to the 71Ga cross section place them in a category of
transitions where important tensor corrections arise. After folding in
uncertainties from both experiment and theory, we found that $\delta=0.074\pm
0.008\,(1\sigma)$. As shown in Fig. 2, when the tensor correction is made with
this value of $\delta$, excellent agreement between (p,n) cross sections and
weak transition strengths is restored, even for weak GT transitions.

As this value of $\delta$ works well for a range of $2s1d$\- and $1p$-shell
nuclei, the use of the same $\delta$ in the $2p1f$ shell is reasonable. This
leaves the second issue mentioned above, the need to evaluate uncertainties
associated with theory estimates of the accompanying matrix element
$M_{\mathrm{T}}$. While the nuclear structure of 71Ga is more complex than
that of the $1p$\- and $2s1d$-shell nuclei used in our extraction of $\delta$,
the three large-basis, full-space $2p_{3/2}1f_{5/2}2p_{1/2}1g_{9/2}$ shell-
model calculations we performed were reasonably consistent in their
predictions of the magnitude of $M_{\mathrm{T}}$ and sign relative to
$M_{\mathrm{GT}}$. Though the spread in $M_{\mathrm{T}}$ is larger than found
in our $1p$\- and $2s1d$-shell calculations, this spread was incorporated into
a theory uncertainty that was then propagated through our analysis. With the
correction for $M_{\mathrm{T}}$ made, we then extracted the needed excited
state GT strengths from the (p,n) and $(^{3}$He,t) results of Krofcheck  and
Frekers , respectively.

The analysis shows that $M_{\mathrm{T}}$ and $M_{\mathrm{GT}}$ interfere
destructively in both excited-state transitions, which increases the
$|M_{\mathrm{GT}}|$ extracted from experiment. Consequently, the excited-state
contribution to the total 51Cr cross section is increased modestly, to
$\approx$ 6% and $\approx$ 9%, depending on whether the data from Krofcheck
or Frekers  is used. The result we obtained from the data of Krofcheck ,
$\left[5.71^{+0.27}_{-0.10}\right]\times 10^{-45}$ cm2 (1$\sigma$) can be
compared to the analogous result of Bahcall employed in the BEST analysis,
$\left[5.81^{+0.21}_{-0.16}\right]\times 10^{-45}\,\mathrm{cm}^{2}$
(1$\sigma$). The results are in agreement at $1\sigma$, reflecting in part
compensating changes in the present analysis due to a weaker ground-state and
stronger excited-state contributions.

Several objections that one might have raised to the use of an older cross
section — including a more naïve use of the (p,n) data, absence of radiative
and weak-magnetism corrections, and various changes in weak parameters and
Coulomb corrections — have been addressed and in combination have been found
to shift the recommended central value by only 2%. Most important, the
analysis presented here has propagated all identified errors — whether
experimental or theoretical — through to the end result. Thus the error bars
reflect all known uncertainties, to the precision currently possible. Finally,
taking into account uncertainties, the extracted values of $M_{\mathrm{GT}}$
are in agreement with the predictions of the shell model. While we would
certainly not advocate use of theory in estimating such weak BGT strengths,
nevertheless this consistency is of some comfort, as the shell model is
employed in the evaluation of the correction terms proportional to
$M_{\mathrm{T}}$.

Finally, we note that a 3% larger cross section is obtained if we base the
excited-state analysis on the (3He,t) data of Frekers . There would be some
value in repeating both the (p,n) and (3He,t) measurements: while the impact
on the cross section is modest, the difference in the cross sections for
exciting the $\textstyle{3\over 2}^{-}$ state exceeds expectations, given the
assigned error bars.

The lower cross section derived here from the (p,n) data of Krofcheck
slightly reduces the size of the BEST and Ga anomalies (by $\approx$ 2%), but
certainly does not remove them. We have demonstrated this by repeating the
sterile-neutrino oscillation analysis of BEST1 ; BEST2 , finding small shifts
in the confidence-level contours and best-fit values for
$\mathrm{sin}^{2}{2\theta}$ and $\Delta\mathrm{m}^{2}$. The very well measured
ground-state transition $\sigma_{gs}$ establishes a floor on the cross section
just 8% below the value used in the BEST and earlier Ga analyses. Even if this
minimum theoretical floor were to be used — the revised value found here is
$\left[5.39\pm 0.04\right]\times 10^{-45}\,\mathrm{cm}^{2}$ (1$\sigma$) — the
existing discrepancies would be reduced by about half, but not eliminated.
Furthermore, we stress that use of such an extreme minimum cross section would
not be consistent with the present analysis, as that value lies well beyond
the 95% C.L. lower bound on the total cross section derived here.

Acknowledgments: WCH and EJR thank Leendert Hayen and Ken McElvain for several
helpful discussions. This work was supported in part by the US Department of
Energy under grants DE-SC0004658, DE-SC0015376, and DE-AC02-05CH11231, by the
National Science Foundation under cooperative agreements 2020275 and 1630782,
and by the Heising-Simons Foundation under award 00F1C7.

## References

  * (1) K. N. Abazajian et al.,“Light sterile neutrinos: A white paper,” arXiv:1204.5379 (2012)
  * (2) S. Gariazzo, C. Giunti, M. Laveder, Y. F. Li, and E. M. Zavanin, J. Phys. G: Nucl. Part. Phys. 43, 033001 (2015)
  * (3) C. Giunti and T. Lasserre, Annu. Rev. Nucl. Part. Sci. 69, 163 (2019)
  * (4) S. Böser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft, and M. Wurm, Prog. Part. Nucl. Phys. 111, 103736 (2020)
  * (5) A. Diaz, C. Arguelles, G. Collin, J. Conrad, and M. Schaevitz, Phys. Rept. 884, 1 (2020)
  * (6) S.-H. Seo, “Review of sterile neutrino experiments,” talk presented at the 19th Lomonosov Conference, arXiv:2001.03349 (2020)
  * (7) B. Dasgupta and J. Kopp, Phys. Rep. 928, 1 (2021)
  * (8) M. A. Acero, C. A. Argüelles, N. Hostert, D. Kaira, G. Karagiorgi, et al., Snowmass white paper “Light Sterile Neutrino Searches and Related Phenomenology,” arXiv:2203.07323 (2022)
  * (9) A. de Gouvea, G. J. Sanchez, and K. J. Kelly, arXiv:2204.09130 (2022)
  * (10) K. Abe et al. (Super-Kamiokande Collaboration), Phys. Rev. D 94, 052010 (2016)
  * (11) B. Aharmim et al. (SNO Collaboration), Phys. Rev. C 88, 025501 (2013)
  * (12) M. Agostini et al. (Borexino Collaboration), Ap. J. 850, 21 (2017)
  * (13) S. Appel et al., Phys. Rev. Lett. 129, 252701 (2022)
  * (14) W. C. Haxton, R. G. H. Robertson, and A. M. Serenelli, Ann. Rev. Astron. Astrophys. 51, 21 (2013)
  * (15) G. D. Orebi Gann, K. Zuber, D. Bemmerer, and A. Serenelli, Ann. Rev. Nucl. Part./ Phys. 71, 491 (2021)
  * (16) GALLEX Collaboration, P. Anselmann, R. Frockenbrock, W. Hampel, G. Heusser, J. Kiko, et al., Phys. Lett. B 342, 440 (1995)
  * (17) J. N. Abdurashitov, V. N. Gavrin, S. V. Girin, V. V. Gorbachev, T. V. Ibragimova, A. V. Kalikhov, et al., Phys. Rev. C 59, 2246 (1999)
  * (18) W. Hampel, G. Heusser, J. Kiko, T. Kirsten, M. Laubenstein, et al., Phys. Lett. B 420, 114 (1998)
  * (19) J. N. Abdurashitov, V. N. Gavrin, S. V. Girin, V. V. Gorbachev, P. P. Gurkina, T. V. Ibragimova, et al., Phys. Rev. C 73, 045805 (2006)
  * (20) V. V. Barinov, B. T. Cleveland, S. N. Danshin, H. Ejiri, S. R. Elliott, D. Frekers, et al., Phys. Rev.Lett. 128, 232501 (2022)
  * (21) V. V. Barinov, S. N. Danshin, V. N. Gavrin, V. V. Gorbachev, D. S. Gorbunov, T. V. Ibragimova, et al., Phys. Rev. C 105, 065502 (2022)
  * (22) N. Hata and W. C. Haxton, Phys. Lett. B 353, 422 (1995)
  * (23) W. C. Haxton, Phys. Lett. B 431, 110 (1998)
  * (24) W. Hampel and L. P. Remsberg, Phys. Rev. C 31, 666 (1985)
  * (25) M. Allansary, D. Frekers, T. Eronen, L. Canete, J. Hakala, et al., Int. J. Mass Spectrom. 406, 1 (2016)
  * (26) J. N. Abdurashitov, V. N. Gavrin, S. V. Girin, V. V. Gorbachev, T. V. Ibragimova, A. V. Kalikhov, et al., Phys. Rev. C 60, 055801 (1999)
  * (27) P. Benois-Gueutal, Compt. Rend. 230, 624 (1950)
  * (28) S. Odiot and R. Daudel, J. Phys. Rad. 17, 60 (1956)
  * (29) J. N. Bahcall,. Phys. Rev. Lett. 9, 500 (1962)
  * (30) E. Vatai, Nucl. Phys. A156, 541 (1970)
  * (31) J. N. Bahcall,. Phys. Rev. 131, 1756 (1963)
  * (32) J. N. Bahcall,. Phys. Rev. 132, 362 (1963)
  * (33) J. N. Bahcall, Nucl. Phys. 71, 267 (1965)
  * (34) B. Märkisch, H. Mest, H. Saul, X. Wang, H. Abele, et al., Phys. Rev. Lett 122, 242501 (2019)
  * (35) J. N. Bahcall, Phys. Rev. C 56, 3391 (1997).
  * (36) A. Gniady, E. Caurier, and F. Nowacki, unpublished
  * (37) B. A. Brown, unpublished; B. Cheal, E. Mane, J. Billowes, M.L. Bissell, K. Blaum, B. A. Brown, et al., Phys. Rev. Lett. 104, 252502 (2010)
  * (38) M. Honma, T. Otsuka, T. Mizusaki, and M. Hjorth-Jensen, , Phys. Rev. C 80, 064323 (2009)
  * (39) C. W. Johnson, W. E. Ormand and P. G. Krastev, Comput. Phys. Commun. 184, 2761 (2013).
  * (40) C. W. Johnson, W. E. Ormand, K. S. McElvain, and H. Shan, arXiv:1801.08432.
  * (41) A. Sirlin, Rev. Mod. Phys. 50, 573 (1978)
  * (42) A. Kurylov, M. J. Ramsey-Musolf, and P. Vogel, Phys. Rev. C 67, 035502 (2003)
  * (43) S. V. Semenov, Phys. Atom. Nucl. ]bf 83, 1549 (2020)
  * (44) D. H. Wilkinson, Nucl. Instruments Methods Phys. Res. Sec. A 335, 182 (1993)
  * (45) L. Hayen, N. Severijns, K. Bodek, D. Rozpedzik, and X. Mougeot, Rev Mod. Phys. 90, 015008 (2018)
  * (46) H.Behrens and H. Jänecke, “Numerical Tables for Beta Decay and Electron Capture,” ed. K.-H. Hellwege (Springer-Verlag, Berlin, 1969)
  * (47) H. De Vries, C. W. De Jaeger, and C. De Vries, At. Data Nucl. Data Tables 36, 495 (1987)
  * (48) D. H. Wilkinson, Nucl. Instruments Methods Phys. Res. Sec. A 290, 509 (1990)
  * (49) M. E. Rose, Phys. Rev. 49, 727 (1936)
  * (50) A. Czarnecki, G. P. Lepage, and W. J. Marciano, Phys. Rev. D 61, 073001 (2000)
  * (51) V. A. Kuzmin, Sov Phys. JETP 22, 1051 (1966)
  * (52) J. N. Bahcall, Rev. Mod. Phys. 50, 881 (1978)
  * (53) A. J. Baltz, J. Weneser, B. A. Brown, and J. Rapaport, Phys. Rev. Lett. 53, 2078 (1984)
  * (54) G. J. Mathews, S. D. Bloom, G. M. Fuller, and J. N. Bahcall, Phys. Rev. C 32, 796 (1985)
  * (55) K. Ikeda, Prog. Theor. Phys. 31, 434 (1964); J. I. Fujita and K. Ikeda, Nucl. Phys. 67, 143 (1965)
  * (56) See, e.g., T.N. Taddeucci et al., Nucl. Phys. A 469, 125 (1987)
  * (57) D. Krofcheck et al., Phys. Rev. Lett. 55, 1051 (1985); D. Krofcheck, PhD thesis, Ohio State University (1987) (see Table D-7)
  * (58) D. Frekers et al., Phys. Lett. B 706, 134 (2011)
  * (59) J. W. Watson et al., Phys. Rev. Lett. 55, 1369 (1985)
  * (60) P. vonNeumann-Cosel and J. N. Ginocchio, Phys. Rev. C 62, 014308 (2000)
  * (61) J. Kontensalo, J. Suhonen, C. Giunti, and P. C. Srivastava, Phys. Lett. B 795, 542 (2019). The tabulated shell-model results included in this paper do not agree with the results of Table 4, with small differences arising for $|M_{\mathrm{GT}}|$ and substantial ones for $|M_{\mathrm{T}}|$.
  * (62) B. D. Anderson et al., Phys. Rev. C 27, 1387 (1983)
  * (63) R. G. T. Zegers, H. Akimune, S. M. Austin, D. Bazin, A. M. vandenBerg, G. P. A. Berg, et al., Phys. Rev. C 74, 024309 (2006)
  * (64) E. W. Grewe, C. Baumer, A. M. van den Berg, N. Blasi, B Davids, D. De Frenne, et al., Phys. Rev. C 69, 064325 (2004)
  * (65) See https://www.nndc.bnl.gov
  * (66) S. Cohen and D. Kurath, Nucl. Phys. 73, 1 (1965)
  * (67) B. A. Brown and H. Wildenthal, Annu. Rev. Nucl. Part. Sci. 38, 29 (1988).
  * (68) B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006).
  * (69) D. Frekers, P. Puppe, J. H. Thies, and H. Ejiri, Nucl. Phys. A 916, 219 (2013)