#  $\mathcal{F}t$ values of the mirror $\beta$ transitions and the weak
magnetism induced current in allowed nuclear $\beta$ decay

N. Severijns<EMAIL_ADDRESS>KU Leuven University, Instituut
voor Kern- en Stralingsfysica, Celestijnenlaan 200D, B-3001 Leuven, Belgium
L. Hayen  KU Leuven University, Instituut voor Kern- en Stralingsfysica,
Celestijnenlaan 200D, B-3001 Leuven, Belgium  Department of Physics, North
Carolina State University, Raleigh, North Carolina 27695, USA  Triangle
Universities Nuclear Laboratory, Durham, North Carolina 27708, USA  V. De
Leebeeck  S. Vanlangendonck  KU Leuven University, Instituut voor Kern- en
Stralingsfysica, Celestijnenlaan 200D, B-3001 Leuven, Belgium  K. Bodek  D.
Rozpedzik  Institute of Physics, Jagiellonian University, 30348 Cracow, Poland
I.S. Towner  Cyclotron Institute, Texas A & M University, College Station,
Texas 77843, USA

###### Abstract

In recent years a number of correlation measurements in nuclear $\beta$ decay
have been performed reaching a precision of the order of 1% and below and it
is expected that even higher precision will be reached in the near future. At
these levels of precision higher-order corrections due to e.g. recoil terms
induced by the strong interaction and radiative corrections cannot necessarily
be neglected anymore when interpreting these results in terms of new physics
or extracting a value for the $V_{ud}$ quark-mixing matrix element. We provide
here an update of the $\mathcal{F}t$ values of the $T=1/2$ mirror $\beta$
decays as well as an overview of current experimental and theoretical
knowledge of the most important recoil term, weak magnetism, for both the
$T=1/2$ mirror $\beta$ transitions and a large set of $\beta$ decays in higher
isospin multiplets. The matrix elements determining weak magnetism were
calculated in the nuclear shell model and cross-checked against experimental
data, showing overall good agreement. Additionally, we show that further
insight can be obtained from properly deformed nuclear potentials, in
particular for mirror $T=1/2$ decays. The results provide new insights in the
size of weak magnetism, extending the available information to $\beta$
transitions of nuclei with masses up to $A=$ 75\. This provides important
guidance for the planning and interpretation of ongoing and new precise
correlation measurements in nuclear $\beta$ decay searching for new physics or
to extract the $V_{ud}$ quark-mixing matrix element in mirror $\beta$ decays.
This more detailed knowledge of weak magnetism can also be of interest for
further theoretical work related to the reactor neutrino problem.

blue

###### Contents

  1. I Introduction
    1. I.1 Searches for exotic scalar and tensor charged weak currents
    2. I.2 Determining the $V_{ud}$ quark-mixing matrix element from mirror beta transitions
    3. I.3 Reactor neutrino anomaly
    4. I.4 Motivation and outline of paper
  2. II ${\mathcal{F}}t$ values for mirror $\beta$ transitions
    1. II.1 Formalism
    2. II.2 Experimental data
      1. II.2.1 Data selection
      2. II.2.2 Adopted input values for the ${\mathcal{F}}t^{\mathrm{mirror}}$ values
    3. II.3 The ${\mathcal{F}}t^{\mathrm{mirror}}$ values
      1. II.3.1 The Gamow-Teller/Fermi mixing ratio $\rho$
      2. II.3.2 Note on the mirror transitions with $A>$61
  3. III Weak Magnetism
    1. III.1 Theoretical foundations
      1. III.1.1 Generalized Hamiltonian
      2. III.1.2 Conserved Vector Current
      3. III.1.3 Form factor evaluation
    2. III.2 Weak magnetism for the mirror $\beta$ transitions
      1. III.2.1 Magnetic moments of the $T=1/2$ mirror nuclei
      2. III.2.2 Experimental weak magnetism form factors $b$ and $b/Ac$
      3. III.2.3 Theoretical form factor comparison
      4. III.2.4 Impulse approximation: access to $M_{GT}$ and $M_{L}$ and comparison with shell-model calculations
      5. III.2.5 Discussion
    3. III.3 Isospin $T=$ 1, 3/2, and 2 multiplet decays
      1. III.3.1 Experimental values for Gamow-Teller and weak magnetism form factors
      2. III.3.2 Nuclear matrix elements $M_{GT}$ and $M_{L}$
    4. III.4 General discussion of the weak magnetism results
      1. III.4.1 Relation to nuclear structure
      2. III.4.2 Averages and their role for $\beta$ decay experiments and the reactor neutrino anomaly
      3. III.4.3 Higher-order form factors $d,f,g$, etc.
  4. IV Conclusion
  5. A Single particle evaluation of weak magnetism
    1. A.1 Overview
    2. A.2 Axially deformed potentials

##  I Introduction

In the past, so-called induced or recoil terms in nuclear $\beta$ decay have
received considerable attention, both theoretically and experimentally (e.g.
Holstein (1974a, 1976); Azuelos and Kitching (1975); Calaprice and Holstein
(1976); Calaprice  _et al._ (1977); Behrens  _et al._ (1978)). Their presence
originates from QCD effects in the weak interaction of a bound quark, and
folds in nuclear structure effects in heavier nuclei. Its study was motivated
mainly in relation to experimental tests of the Conserved Vector Current (CVC)
hypothesis Feynman and Gell-Mann (1958) (see e.g. Tribble and Garvey (1974);
Kaina  _et al._ (1977); Wu  _et al._ (1977); Elmbt  _et al._ (1987); Calaprice
and Alburger (1978); Camp (1990); Grenacs (1985); Sumikama  _et al._ (2011a))
and searches for second class currents Grenacs (1985) (e.g. Wilkinson (1970);
Holstein and Treiman (1971); Holstein (1974a); Holstein and Treiman (1976);
Tribble and Garvey (1975); Calaprice (1975); Calaprice and Holstein (1976);
Calaprice  _et al._ (1977); Fifield  _et al._ (1977); Behrens  _et al._
(1978); Dupuis-Rolin  _et al._ (1978); Tribble and May (1978); Tribble  _et
al._ (1981); Grenacs (1985); Rosa  _et al._ (1988); Wilkinson (2000);
Minamisono  _et al._ (1998, 2001); Smirnova and Volpe (2003); Sumikama  _et
al._ (2008); Minamisono  _et al._ (2011)). The most important of these induced
currents, the weak magnetism term, has recently received new attention for
several reasons. First, measurements of correlations between the spins and
momenta of the particles involved in nuclear $\beta$ decay, such as e.g.
$\beta-\nu$ correlation and $\beta$-asymmetry parameter measurements Jackson
_et al._ (1957a), have now reached the level of precision where recoil terms
cannot be neglected anymore (e.g. Pitcairn  _et al._ (2009); Wauters  _et al._
(2009, 2010); Soti  _et al._ (2014); Fenker  _et al._ (2018); Märkisch  _et
al._ (2019)). In such measurements one is searching either for new physics
beyond the standard model (SM), such as e.g. scalar and tensor weak currents
Severijns and Naviliat-Cuncic (2011); Naviliat-Cuncic and González-Alonso
(2013); Holstein (2014); Vos  _et al._ (2015); Gonzalez-Alonso  _et al._
(2019), or aiming at the determination of the $V_{ud}$ quark-mixing matrix
element in the $\beta$ decay of mirror isotopes Naviliat-Cuncic and Severijns
(2009a); Liénard et al. (2015); Brodeur  _et al._ (2016a); Beck  _et al._
(2020a). Secondly, weak magnetism is also important with respect to the so-
called reactor antineutrino anomaly Mention  _et al._ (2011); Mueller  _et
al._ (2011); Huber (2011); Hayes and Vogel (2016); Wang and Hayes (2017);
Hayes  _et al._ (2018); Hayen  _et al._ (2019a, b), i.e., the fact that the
ratio of the observed antineutrino event rate at nuclear reactors to the
predicted rate is lower than unity. Each of these respective topics is briefly
commented upon here below.

###  I.1 Searches for exotic scalar and tensor charged weak currents

In $\beta$ decay strong limits on the existence of charged weak scalar
currents have been obtained from the corrected $\mathcal{F}t$ values of the
superallowed $0^{+}\rightarrow 0^{+}$ pure Fermi $\beta$ transitions Hardy and
Towner (2015). Complimentary information as well as limits on tensor currents
are traditionally obtained from accurate measurements of correlations between
the spins and momenta of the particles involved in $\beta$ decay (reviews are
given in e.g. Severijns and Naviliat-Cuncic (2011); Naviliat-Cuncic and
González-Alonso (2013); Holstein (2014); Vos  _et al._ (2015); Gonzalez-Alonso
_et al._ (2019); Falkowski  _et al._ (2021)). The best-suited correlations are
the $\beta$-$\nu$ angular correlation and the $\beta$ asymmetry with respect
to the polarization axis of oriented parent nuclei Jackson  _et al._ (1957a);
Severijns  _et al._ (2006). In recent years the precision of such measurements
has improved significantly and several results with a precision of about 1% or
better have been reported already (see e.g. for the $\beta$-$\nu$ correlation
Adelberger  _et al._ (1999); Gorelov  _et al._ (2005); Vetter  _et al._
(2008); Fléchard  _et al._ (2011); Sternberg  _et al._ (2015); Beck  _et al._
(2020b), for the $\beta$-asymmetry parameter Wauters  _et al._ (2009, 2010);
Soti  _et al._ (2014); Fenker  _et al._ (2018); Brown  _et al._ (2018);
Märkisch  _et al._ (2019); Combs  _et al._ (2020), and for other correlations
Pitcairn  _et al._ (2009); Melconian  _et al._ (2007); Beck  _et al._ (2011);
Kozela  _et al._ (2012); Chupp  _et al._ (2012); Sun  _et al._ (2020); Saul
_et al._ (2020)). Many new experiments in search of exotic currents are
ongoing or planned. An overview can be found in Table 3 in Ref. Gonzalez-
Alonso  _et al._ (2019). Most of these are aiming at a precision down to the
few per-mille level. At such a precision, searches for physics beyond the
Standard Model in $\beta$ decay remain competitive to direct searches for new
bosons in collider experiments such as at the Large Hadron Collider (e.g.
Bhattacharya et al. (2012); Naviliat-Cuncic and González-Alonso (2013); V.
Cirigliano and Holstein (2013); V. Cirigliano and Graesser (2013); Falkowski
_et al._ (2021)). However, as experiments reach ever higher precision and dive
below the 1% level, additional theoretical corrections arise Hayen and Young
(2020). The most important of these is the so-called weak magnetism correction
which is nuclear structure dependent. Its importance and possible limitations
related to its exact knowledge were already demonstrated in the $\beta$-decay
correlation measurements with e.g. 37K Fenker  _et al._ (2018), 60Co Wauters
_et al._ (2010), 67Cu Soti  _et al._ (2014), 80Rb Pitcairn  _et al._ (2009)
and 114In Wauters  _et al._ (2009), and discussed recently in Hayen and Young
(2020); Vanlangendonck  _et al._ (2021).

###  I.2 Determining the $V_{ud}$ quark-mixing matrix element from mirror beta
transitions

Apart from searches for new physics, correlation measurements in nuclear
$\beta$ decay can also be used to determine the $V_{ud}$ quark-mixing matrix
element or perform an independent test of the CVC hypothesis, i.e. by checking
the constancy of the $V_{ud}$ values for the individual transitions, as was
shown for isospin $T$ = 1/2 mirror $\beta$ transitions Naviliat-Cuncic and
Severijns (2009a). Such measurements can also help to better understand the
isospin-symmetry breaking correction (noted as $\delta_{c}$) that is required
to obtain high-precision corrected ${\mathcal{F}}t$ values for superallowed
$\beta$ transitions (e.g. Towner and Hardy (2010); Hardy and Towner (2015)).
Recently, very precise results were reported for the $\beta$ asymmetry
parameter in de mirror beta decay of 37K Fenker  _et al._ (2018) and 19Ne
Combs  _et al._ (2020). In addition, a double-counting instance in the
extraction of $V_{ud}$ from the mirror $\beta$ transitions was corrected in
Ref. Hayen (2021). The current value for $V_{ud}$ from mirror $\beta$
transitions, i.e. $|V_{ud}|$ = 0.9739(10) Hayen (2021) (see also Falkowski
_et al._ (2021)), depends primarily on the availability of precise
experimental $\mathcal{F}t$ values Severijns  _et al._ (2008); Hayen and
Severijns (2019a) and existing experimental results, mainly for the
$\beta$-asymmetry parameter, $A$ Naviliat-Cuncic and Severijns (2009a);
Falkowski  _et al._ (2021).

In Refs. Naviliat-Cuncic and Severijns (2009b); Severijns and Naviliat-Cuncic
(2013); Hayen and Young (2020) prospects for a more precise determination of
$V_{ud}$ from these $T=1/2$ mirror $\beta$ transitions are given. The mirror
$\beta$ decays of 3H and 19Ne appear to be the most sensitive in case of
$\beta$-$\nu$ correlation measurements, with all other mirror transitions up
to 41Sc still having a quite good sensitivity Severijns and Naviliat-Cuncic
(2013); Hayen and Young (2020); Vanlangendonck  _et al._ (2021). Such
measurements have recently been performed with 19Ne and 35Ar Fabian et al.
(2014); Fabian  _et al._ (2018); Combs  _et al._ (2020) and new ones are
planned Liénard et al. ; Ron et al. ; Brodeur  _et al._ (2016b). For the
$\beta$-asymmetry parameter measurements in the mirror $\beta$ decays of 3H,
19Ne, 33Cl and 35Ar provide the best sensitivity Severijns and Naviliat-Cuncic
(2013); Combs  _et al._ (2020); Hayen and Young (2020); Vanlangendonck  _et
al._ (2021), with a measurement of the $\beta$-asymmetry parameter in the
decay of polarized 35Ar recently being attempted Velten (2014); Gins  _et al._
(2019); Gins (2019). Further improvement on the $\beta$ asymmetry parameter in
the mirror beta decay of 37K is anticipated as well Fenker  _et al._ (2018).
In order to obtain sufficiently precise values for $V_{ud}$ from such
measurements a relative precision of typically 0.5% or better is needed
Severijns and Naviliat-Cuncic (2013), again requiring that higher-order
effects are duly taken into account Hayen and Young (2020); Vanlangendonck
_et al._ (2021).

###  I.3 Reactor neutrino anomaly

As several modern reactor neutrino experiments have come online in the search
for neutrino oscillations, comparisons to theoretical predictions have
resulted in large discrepancies. Using two independent analysis methods, the
ratio of experimentally detected versus theoretically predicted antineutrinos
is found to be $R=0.943\pm 0.023$ Mention  _et al._ (2011). If genuine, this
could be related to the existence of a sterile neutrino with $\Delta
m^{2}\geq$ 1 eV2 Mention  _et al._ (2011); Kopp  _et al._ (2011). Over the
past few years, apart from other work, the treatment of forbidden transitions
Dwyer and Langford (2015); Hayes  _et al._ (2014); Hayen  _et al._ (2019a);
Petković  _et al._ (2019); Hayen  _et al._ (2019b); Estienne  _et al._ (2019)
and the evaluation of weak magnetism Hayes and Vogel (2016); Wang and Hayes
(2017) has undergone some scrutiny. Recently, it was suggested Hayen  _et al._
(2019b); Petković  _et al._ (2019) that a more proper treatment of forbidden
transitions and inclusion of more pandemonium-free data in calculating the
reactor antineutrino energy spectrum Estienne  _et al._ (2019) could possibly
explain the flux problem as well as the so-called spectral shoulder with
respect to the theoretical spectrum. As to weak magnetism, all calculations of
the expected antineutrino flux at reactors assume the weak magnetism
contribution to depend solely on the difference between the proton and neutron
magnetic moments and the $\beta$-particle energy. However, as will be shown
explicitly later in this document, its precise value depends strongly on the
isotope at hand. Using the same value for all transitions is then a
substantial oversimplification, with up to now poorly understood consequences.
As current experimental information on weak magnetism is limited to nuclei
with $A\leq 43$, a study including also heavier masses, and especially
measurements of weak magnetism for nuclei in the fission region, would be most
helpful. At present several precision $\beta$-spectrum shape measurements to
provide direct experimental information on weak magnetism are ongoing, e.g.
with the isotopes 6He Huyan  _et al._ (2016, 2018a, 2018b); Mukul  _et al._
(2018), 20F Naviliat-Cuncic (2021); Hughes  _et al._ (2018), and 114In De
Keukeleere and Severijns (2021); Vanlangendonck and Severijns (2021), the
latter effectively being in the fission isotope region. Here we present and
analyse the currently existing experimental data allowing to determine weak
magnetism for $\beta$ transitions up to mass $A$ = 75, and provide evidence
for a nuclear structure and mass dependence of the size of the weak magnetism
term.

###  I.4 Motivation and outline of paper

Based on the above formulated arguments, a more detailed knowledge of the weak
magnetism form factor in $\beta$ decay would clearly be of great value.
Previously, the weak magnetism term was evaluated for superallowed mixed
Fermi/Gamow-Teller (F/GT) mirror $\beta$ transitions between isospin $T=1/2$
doublets in mirror nuclei up to 43Ti Calaprice and Holstein (1976); Azuelos et
al. (1977), and for $T$ = 1 to $T$ = 0 Gamow-Teller decays in nuclei up to
mass $A$ = 32 Calaprice and Holstein (1976); Huber (2011).

In this paper, the currently available experimental data leading to the weak
magnetism form factor are analysed for both the isospin T$=1/2$ mixed
Fermi/Gamow-Teller (F/GT) $\beta$ transitions and Gamow-Teller transitions in
$T=1,3/2$ and $2$ multiplets, based on the CVC hypothesis. For the mirror
$\beta$ transitions, the input data for the corrected $\mathcal{F}t$ values
for these transitions are reviewed and updated $\mathcal{F}t$ values are
provided. Combining these with the nuclear magnetic moments of the mirror
transitions’ initial and final states allows determining values for the weak
magnetism form factor for these transitions up to 75Sr. For the Gamow-Teller
transitions, the data base of the Brookhaven Laboratory National Nuclear Data
Center database ENSDF  was scanned for $\beta$ transitions for which analog
(i.e. within the same isospin multiplet) gamma transitions are known and with
sufficient information being available for both transitions in order to
determine the weak magnetism form factor for the $\beta$ transition, again
using the CVC hypothesis.

In Sec. II we first review the currently available input data (the literature
cut-off was April 2021) leading to the corrected ${\mathcal{F}}t$ values for
the mirror $\beta$ transitions with $A=3-75$, thereby updating the previously
published $\mathcal{F}t$ values for the mirror $\beta$ transitions up to
$A=45$ Severijns  _et al._ (2008) and extending them further to $A=75$.

Sec. III focuses on the weak magnetism form factors for both the mirror
$\beta$ transitions between $T=1/2$ states, and the Gamow-Teller transitions
in higher-$T$ multiplets. In Sec. III.1, the general formalism with respect to
the induced form factors and the weak magnetism term is introduced. Then, in
Sec. III.2, the nuclear magnetic moments of the mirror $\beta$ transitions’
mother and daughter states are reviewed and the weak magnetism term is
extracted for the transitions up to $A=75$, using the CVC hypothesis.
Combining these with the $\mathcal{F}t$ values allows extracting both the
Gamow-Teller and orbital angular momentum matrix elements, $M_{GT}$ and
$M_{L}$, respectively. These are compared to shell model calculations in Sec.
III.2.3.

In Sec. III.3 the analysis of weak magnetism for Gamow-Teller $\beta$ decays
of states belonging to $T=1$, $T=3/2$, and $T=2$ isospin multiplets up to mass
$A=53$ is presented, again using the CVC hypothesis. The weak magnetism term
for the respective $\beta$ transitions is now obtained from the width of the
analog $E1$ gamma transition. Also for these transitions the $ft$ values are
calculated in order to extract the Gamow-Teller and orbital angular momentum
matrix elements, with these again being compared to shell model calculations
(for a subset of the transitions considered).

Finally, the nuclear structure and mass dependence of the weak magnetism term
and its relevance to the reactor anomaly in face of these new data, are
discussed.

##  II ${\mathcal{F}}t$ values for mirror $\beta$ transitions

In Ref. Severijns  _et al._ (2008) the corrected ${\mathcal{F}}t$ values for
mirror $\beta$ transitions in nuclei with mass $A$ up to 45 were presented for
the first time. Since then, many new experimental results for the half-life,
branching ratio and decay energies for these transitions, as well as for
mirror transitions in nuclei with higher masses have been reported. First, an
update of the relevant definitions and formalism will be given (Sec. II.1).
Thereafter, the updated data set is presented and evaluated, together with new
theoretical calculations of nuclear structure and radiative corrections (Sec.
II.2). Finally, the current values for the ${\mathcal{F}}t$ values for the
mirror $\beta$ transitions are reported (Sec. II.3).

###  II.1 Formalism

For a general $\beta$ transition, the uncorrected partial half-life can be
written as Severijns  _et al._ (2008)

$t=\frac{K}{G_{F}^{2}V_{ud}^{2}}\frac{1}{\xi f}$ (1)

where $G_{F}$ is the Fermi coupling constant, $V_{ud}$ is the
$\it{up}$-$\it{down}$ matrix element of the CKM matrix,

$\displaystyle K/(\hbar c)^{6}$ $\displaystyle=\frac{2\pi^{3}\ln
2\hbar}{(m_{e}c^{2})^{5}}$ (2) $\displaystyle=(8120.27648\pm 0.00026)\cdot
10^{-10}\text{GeV}^{-4}s\leavevmode\nobreak\ ,$ $\displaystyle\xi$
$\displaystyle=g_{V}^{2}M_{F}^{2}+g_{A}^{2}M_{GT}^{2}$ (3)

with $g_{V}$ and $g_{A}$ the strength of the vector and axial-vector
interactions (in units of $G_{F}$) as defined in the Hamiltonian of Jackson,
Treiman and Wyld Jackson  _et al._ (1957a). Here, $M_{F,GT}$ is the main Fermi
or Gamow-Teller matrix element, respectively. Further, $f$ is the statistical
rate function defined as the integral over the normalized spectrum shape

$\displaystyle f=\int_{1}^{W_{0}}$ $\displaystyle pW(W_{0}-W)^{2}F(Z,W)$
$\displaystyle\times C(Z,W)K(Z,W)dW\leavevmode\nobreak\ ,$ (4)

where $W$ is the total $\beta$ energy in units of its rest mass, i.e.
$W=E_{\mathrm{kin}}/m_{e}c^{2}+1$, and $W_{0}$ the spectral endpoint. Here
$F(Z,W)$ is the well-known Fermi function, $C(Z,W)$ the nuclear-structure
dependent shape factor, and $K(Z,W)$ all higher-order corrections Hayen  _et
al._ (2018).

The product $ft$ is then independent of the kinematics and produces the main
matrix elements. Its experimental value for a specific $\beta$ transition
depends on three quantities: ($i$) the total decay transition energy,
$Q_{EC}$, ($ii$) the half-life, $t_{1/2}$, of the decaying state and ($iii$)
the branching ratio, $BR$, of the particular transition of interest. The
$Q_{EC}$ value determines the statistical rate function, $f$. The half-life
and the branching ratio, together with the electron-capture fraction,
$P_{EC}$, determine the partial half-life, $t$, of the transition of interest,
the latter of which is defined as

$t=t_{1/2}\biggl{(}\frac{1+P_{EC}}{BR}\biggr{)}\leavevmode\nobreak\ .$ (5)

The superallowed mirror $\beta$ transitions between the analog $T=1/2$ states
situated at both sides of the $N=Z$ line are mixed transitions with both Fermi
and Gamow-Teller components. Despite this increased complexity, they harbour a
particular advantage. As the decay occurs within the same isospin multiplet -
and taking the isospin limit in QCD - the Fermi matrix element is determined
simply by using the Wigner-Eckart theorem. In the case of mirror $T=1/2$
transitions this is particularly simple and one finds $M_{F}^{0}=1$, with the
superscript denoting the isospin-conserving limit. In nuclei, however, the
presence of the Coulomb interaction between nucleons breaks the purported
isospin symmetry, albeit at only the few percent level. Small corrections must
therefore be introduced to the single particle matrix element

$M_{F}=M_{F}^{0}(1-\delta_{C})^{1/2},$ (6)

with $\delta_{C}$ a correction arising from the imperfect overlap of proton
and neutron radial wave functions due to their slightly different nuclear
potentials Towner and Hardy (2010). This must be provided by nuclear theory
and its estimate is of particular importance in, e.g., the $V_{ud}$ extraction
in superallowed $0^{+}\to 0^{+}$ Fermi decays and mirror decays, as discussed
below.

The weak axial-vector current, occurring in Gamow-Teller decays, is not
conserved and is renormalized through QCD effects. This prevents one, in
theory, from writing down an analogous isospin-invariant GT matrix element,
$M_{GT}^{0}$, although this has historically been commonplace Severijns  _et
al._ (2008); Naviliat-Cuncic and Severijns (2009a). With previous nuclear
many-body calculations, dominated by the nuclear shell model, this is at first
glance merely a notational matter as $M_{GT}$ could typically not be
calculated to high enough precision for the distinction to be relevant. With
the developing maturity of nuclear ab initio methods (see e.g. Pastore  _et
al._ (2018); Cirgiliano  _et al._ ), however, the distinction is much more
clearly defined and of interest as they enter the $A=10$-$19$ mass range and
beyond.

Besides strong interaction effects, additional electroweak corrections arise
both at the nucleon and nucleus level. Traditionally these have been separated
into a QCD-invariant, infrared-singular part (the outer correction,
$\delta_{R}^{\prime}$) and an energy-independent renormalization of the weak
coupling constants from the ultraviolet (the inner correction,
$\Delta_{R}^{V,A}$) Sirlin (1967). Both contribute roughly at the few percent
level. The latter is typically calculated for a single nucleon, i.e. in a loop
calculation the nucleon interacting with both weak gauge bosons is one and the
same. In a nucleus, however, this is not necessarily the case and additional
nuclear-structure dependent corrections arise, noted as $\delta_{NS}$ Towner
(1992). This once again requires nuclear theory input, and has garnered
significant attention following recent progress Seng  _et al._ (2018, 2019);
Czarnecki  _et al._ (2019).

In summary, Eq. (1) can be written as Severijns  _et al._ (2008); Towner and
Hardy (2010)

$\displaystyle 1/t$ $\displaystyle=\frac{G_{F}^{2}\leavevmode\nobreak\
V_{ud}^{2}}{K}\leavevmode\nobreak\ \left(1+\delta_{R}^{\prime}\right)$
$\displaystyle\times\biggl{(}f_{V}|M_{F}^{0}|^{2}\left(1+\delta_{NS}^{V}-\delta_{C}^{V}\right)g_{V}^{2}\left(1+\Delta_{R}^{V}\right)$
$\displaystyle+f_{A}|M_{GT}|^{2}\left(1+\delta_{NS}^{A}\right)g_{A}^{2}\left(1+\Delta_{R}^{A}\right)\biggr{)},$
(7)

where $f_{V}$ and $f_{A}$ are the statistical rate functions for,
respectively, the vector (Fermi) and axial-vector (Gamow-Teller) parts of the
transitions, and we write $M_{GT}$ as noted above. As denoted in Eq. (4),
these take into account higher-order nuclear corrections besides the main F/GT
matrix elements in the shape factor, $C(Z,W)$. Traditionally, these
corrections have been calculated in the elementary particle approach - i.e. a
form factor decomposition encoding the full nuclear response - and a matrix
element reduction using the impulse approximation (IA) which treats the
nucleus as a sum of independent nucleons in a mean-field potential. While the
results presented below are calculated in the Behrens-Bühring formalism
Behrens and Bühring (1982), we write the results in the Holstein formalism
Holstein (1974a) for convenience. For comparisons between them we refer the
reader to Refs. Behrens  _et al._ (1978); Hayen  _et al._ (2018); Hayen and
Young (2020).

We can absorb all constant corrections and define the Gamow-Teller/Fermi
mixing ratio as

$\displaystyle\rho\equiv\frac{c}{a}$
$\displaystyle\stackrel{{\scriptstyle\text{IA}}}{{=}}\frac{g_{A}\mathcal{F}_{GT}(0)}{g_{V}M_{F}^{0}}\left[\frac{1+\Delta_{R}^{A}-\Delta_{R}^{V}}{1+\delta_{NS}^{V}-\delta_{C}^{V}}\right]^{1/2}$
(8)
$\displaystyle=\frac{g_{A}M_{GT}^{0}}{g_{V}M_{F}^{0}}\left[\frac{(1+\delta_{NS}^{A}-\delta_{C}^{A})(1+\Delta_{R}^{A})}{(1+\delta_{NS}^{V}-\delta_{C}^{V})(1+\Delta_{R}^{V})}\right]^{1/2}\leavevmode\nobreak\
.$ (9)

with $a$ and $c$, respectively, the Fermi and Gamow-Teller form factors. The
first line contains a definition analogous to Ref. Hayen and Young (2020),
with $\mathcal{F}_{GT}(0)$ the form factor evaluated at zero momentum, while
the second has the more traditional notation. The benefit lies in the
generality of Eq. (8), meaning experimental analysis extracts the same
quantity, whether it comes from a $\beta$-asymmetry, a $\beta$-$\nu$
correlation or another correlation. One can define the “corrected” $ft$ value
for the mixed Gamow-Teller/Fermi $T=1/2$ mirror $\beta$ transitions to be
Severijns  _et al._ (2008); Towner and Hardy (2010)

$\displaystyle\mathcal{F}t^{\mathrm{mirror}}$ $\displaystyle\equiv
f_{V}t(1+\delta_{R}^{\prime})(1+\delta_{NS}^{V}-\delta_{C}^{V})$
$\displaystyle=\frac{2\mathcal{F}t^{0^{+}\rightarrow
0^{+}}}{|M_{0}^{F}|^{2}\left(1+\frac{f_{A}}{f_{V}}\rho^{2}\right)}\leavevmode\nobreak\
,$ (10)

where

$\displaystyle{\mathcal{F}}t^{0^{+}\rightarrow 0^{+}}$
$\displaystyle=\frac{K}{2G_{F}^{2}V_{ud}^{2}g_{V}^{2}(1+\Delta_{R}^{V})}$
$\displaystyle=3072.24(185)s\leavevmode\nobreak\ ,$ (11)

is the corrected ${\cal F}t$ value for the superallowed $0^{+}\rightarrow
0^{+}$ pure Fermi transitions Hardy and Towner (2020).

The Gamow-Teller/Fermi mixing ratio, $\rho$, can to good approximation be
written as Severijns  _et al._ (2008):

$\rho\cong\frac{g_{A}M_{GT}^{0}}{g_{V}M_{F}^{0}}\leavevmode\nobreak\ ,$ (12)

where we use the fact that the small radiative and isospin symmetry-breaking
corrections do not differ much for the vector and axial-vector parts of the
transition Severijns  _et al._ (2008). The radiative corrections
$\Delta_{R}^{V}$ and $\Delta_{R}^{A}$ were recently shown to indeed differ
only at the $10^{-4}$ level or even less Hayen (2021); Gorchtein and Seng
(2021). As the vector coupling constant $g_{V}=1$ and $|M_{F}^{0}|^{2}=1$ for
these $T=1/2$ mirror $\beta$ transitions, one has

$\rho\simeq g_{A}M_{GT}^{0}\leavevmode\nobreak\ .$ (13)

Before moving on, we comment on the evaluation of $f_{A}/f_{V}$ values in
relation to recent work Hayen and Severijns (2019b, c). It was found that a
double-counting instance occurs when combining results into Eq. (II.1) which
were typically calculated in different formalisms. Specifically, the
expressions used for $f_{A}$ and $f_{V}$ were obtained in the Behrens-Bühring
formalism, which was observed to contain parts of the Gamow-Teller-specific
inner radiative correction $\Delta_{R}^{A}$ (and missing cancellations from
the full $\mathcal{O}(\alpha)$ calculation) Hayen and Severijns (2019c); Hayen
(2021). Experimental results for $\rho$, on the other hand, were typically
obtained using (simplified) results in the Holstein formalism Holstein
(1974a), which do not contain this contribution and so return a fully
renormalized mixing ratio. When combining it into Eq. (II.1) then, the
electroweak renormalization was partially double-counted. Removing this issue
brings significantly improved internal consistency in the $V_{ud}$ mirror data
set and additional lowered uncertainty. We will come back to this point below
when reporting on $f_{A}/f_{V}$ for the data set presented here.

In the next section, a detailed survey of the data leading to the ${\cal F}t$
values for the $T=1/2$ mirror $\beta$ transitions is made. In addition to
bringing the previously published results for the mirror transitions up to 45V
Severijns  _et al._ (2008) up to date, we are now also incorporating existing
data on the heavier $T=1/2$ mirror $\beta$ transitions, leading to the ${\cal
F}t$ values for the transitions up to 75Sr.

###  II.2 Experimental data

####  II.2.1 Data selection

We considered all measurements published up to April 2021 as well as any
additional final results (available in preprint or as a PhD thesis) we were
aware of. Since the previous survey Severijns  _et al._ (2008) about 45 new
half-life measurements and four new branching ratios have been reported, for
almost 30 different isotopes. As in our previous survey Severijns  _et al._
(2008) the $Q_{EC}$ values have again been taken from the most recent Atomic
Mass Evaluation (AME2020) Huang  _et al._ (2021a, b); Wang  _et al._ (2021).
Since the Atomic Mass Evaluation 2003 (AME2003) Audi  _et al._ (2003), which
was used in Severijns  _et al._ (2008), the precision with which the $Q_{EC}$
value is known has been (significantly) improved for 11 out of the 19 mirror
$\beta$ transitions discussed in Severijns  _et al._ (2008).

In the tables and throughout this work, “error bars” and “uncertainties”
always refer to plus/minus one standard deviation (68% confidence level). In
analyzing the tabulated data we have followed the statistical procedures used
by the Particle Data Group (e.g. ref. Olive et al. (2014)) that were adopted
also for the surveys of the superallowed $0^{+}\rightarrow 0^{+}$ $\beta$
decays (e.g. Hardy and Towner (2005, 2015)) as well as for the previous survey
Severijns  _et al._ (2008). Thus, the one standard deviation error bars on
weighted averages were increased by a scaling factor

$S=\left[\chi^{2}/(N-1)\right]^{1/2}\leavevmode\nobreak\ ,$ (14)

with $\chi^{2}/(N-1)$ the reduced $\chi^{2}$. Further, data that were clearly
inconsistent and results with an uncertainty that is a factor 10 or more
larger than that of the most precise measurement for each quantity were
rejected. Also results that deviate 4 or more standard deviations from the
weighted average of mutually consistent results from other measurements (which
are in most cases more recent and also more precise), were not considered. For
the latter cases, the average values that would be obtained when the deviating
results are not omitted, are given as well. These all agree within error bars
with the finally adopted value, but have of course larger error bars and also
a larger scaling factor, $S$, than the adopted value.

All data that are now rejected on top of the ones that had already been
rejected in the initial survey Severijns  _et al._ (2008) are listed in Tables
3 and 4. As in e.g. Ref. Hardy and Towner (2005) an alphanumeric code
comprising the initial two letters of the first author’s name and the two last
digits of the publication date is used to link each datum appearing in the
tables to its original journal reference. The reference key linking these
alphanumeric codes with the actual reference numbers is given in Table 7.

Table LABEL:table:halflives_mirrors shows the collected half-lives of the
considered isotopes. A breakdown of the branching ratios is collected in Table
LABEL:table:branchings_mirrors. In comparison to the previous compilation
Severijns  _et al._ (2008) a more detailed analysis of the branching ratios
was now performed. It includes all known branchings to excited states in the
daughter isotopes of the mirror nuclei. Because of this, some branching ratios
for the mirror $\beta$ transitions have slightly shifted in central value or
some error bars have slightly increased. Of course, the more complete analysis
that was performed renders the newly adopted values more reliable.

Table 1: Half-lives, $t_{1/2}$, of the mirror nuclei from 3H to 39Ca, expressed in seconds unless specified differently under the name of the parent nucleus (days (d), minutes (min)). References to data listed in this table are given in Table 7. References to data that were not used are listed in Table 3 together with the reason for their rejection. The scale factor $S$ listed in the last column is defined in Eq. 14. Parent |  Measured half-lives, $t_{1/2}$ (s) | Average half-life | scale
---|---|---|---
Nucleus | 1 | 2 | 3 |  $t_{1/2}$ (s) | $S$
3H | $4419$ | $\pm$ |  $183$ | [No47] | $4551$ | $\pm$ |  $54$ | [Je50] | $4530$ | $\pm$ |  $27$ | [Jo51] |  |  |  |
(d) | $4479$ | $\pm$ |  $11$ | [Jo55] | $4596$ | $\pm$ |  $66$ | [Po58] | $4496$ | $\pm$ |  $16$ | [Me66] |  |  |  |
| $4474$ | $\pm$ |  $11$ | [Jo67] | $4501$ | $\pm$ |  $9$ | [Ru77] | $4498$ | $\pm$ |  $11$ | [Si87] |  |  |  |
| $4521$ | $\pm$ |  $11$ | [Ol87] | $4485$ | $\pm$ |  $12$ | [Ak88] | $4497$ | $\pm$ |  $11$ | [Bu91] |  |  |  |
| $4504$ | $\pm$ |  $9$ | [Un00] | $4500$ | $\pm$ |  $8$ | [Lu00] | $4479$ | $\pm$ |  $7$ | [Ak04] |  |  |  |
| $4497$ | $\pm$ |  $4$ | [Ma06] |  |  |  |  |  |  |  |  | $4497$ | $\pm$ |  $4$ d111We did not perform the analysis of the tritium half-lives ourselves, but rather used the value (and the references) from [Ma06]. An interesting effect is mentioned in [Ak04]; the half-life of molecular and atomic 3H would differ by about 9 days. Due to a lack of additional information on this effect we have not included it in the present compilation. All measurements, except for [Ak04], have been performed on molecular tritium. |
11C | $20.34$ | $\pm$ |  $0.04$ | [Ka64a] | $20.40$ | $\pm$ |  $0.04$ | [Aw69] | $20.38$ | $\pm$ |  $0.02$ | [Az75] |  |  |  |
(min) | $20.334$ | $\pm$ |  $0.024$ | [Wo02] | $20.3378$ | $\pm$ |  $0.0043$ | [Va18] |  |  |  |  | $20.3401$ | $\pm$ |  $0.0053$ min | 1.3
13N | $9.96$ | $\pm$ |  $0.03$ | [Ar58] | $9.965$ | $\pm$ |  $0.005$ | [Ja60] | $9.93$ | $\pm$ |  $0.05$ | [Ki60] |  |  |  |
(min) | $10.05$ | $\pm$ |  $0.05$ | [Bo65] | $9.96$ | $\pm$ |  $0.02$ | [Eb65] | $9.963$ | $\pm$ |  $0.009$ | [Ri68] |  |  |  |
| $9.965$ | $\pm$ |  $0.010$ | [Az77] |  |  |  |  |  |  |  |  | $9.9647$ | $\pm$ |  $0.0039$ min | 1
15O | $122.1$ | $\pm$ |  $0.1$ | [Ja60] | $122.23$ | $\pm$ |  $0.23$ | [Az77] | $122.308$ | $\pm$ |  $0.049$ | [Bu20] | $122.27$ | $\pm$ | $0.06$ | 1.3
17F | $64.50$ | $\pm$ |  $0.25$ | [Al72] | $64.31$ | $\pm$ |  $0.09$ | [Az77] | $64.347$ | $\pm$ |  $0.035$ | [Gr15a] |  |  |  |
| $64.402$ | $\pm$ |  $0.042$ | [Br16] |  |  |  |  |  |  |  |  | $64.366$ | $\pm$ |  $0.026$222If the results from [Wo69] and [Al77] (see Table 3) are included one has $t_{1/2}=64.411\pm 0.068$ s (S = 2.8). | 1
19Ne | $17.219$ | $\pm$ |  $0.017$ | [Az75] | $17.237$ | $\pm$ |  $0.014$ | [Pi85] | $17.262$ | $\pm$ |  $0.007$ | [Tr12] |  |  |  |
| $17.254$ | $\pm$ |  $0.005$ | [Uj13] | $17.2832$ | $\pm$ |  $0.0083$ | [Br14] | $17.2569$ | $\pm$ |  $0.0021$ | [Fo17] | $17.2573$ | $\pm$ |  $0.0034$333Note that $t_{1/2}=17.2561\pm 0.0025$ s (S = 1.4) when excluding [Br14] (which differs by 3.1$\sigma$ from the weighted average), while $t_{1/2}=17.2569\pm 0.0019$ s (S = 1) when considering only the three most precise results, i.e. [Tr12], [Uj13], and [Fo17]. | 1.9
21Na | $22.422$ | $\pm$ |  $0.010$ | [Gr15] | $22.4506$ | $\pm$ |  $0.0033$ | [Fi17] | $22.4615$ | $\pm$ |  $0.0042$ | [Sh18] | $22.4527$ | $\pm$ | $0.0067$ | 2.7
23Mg | $11.327$ | $\pm$ |  $0.014$ | [Az75] | $11.317$ | $\pm$ |  $0.011$ | [Az77] | $11.3027$ | $\pm$ |  $0.0033$ | [Ma17] | $11.3050$ | $\pm$ | $0.0044$ | 1.4
25Al | $7.23$ | $\pm$ |  $0.02$ | [Ju71] | $7.177$ | $\pm$ |  $0.023$ | [Ta73] | $7.174$ | $\pm$ |  $0.007$ | [Az75] |  |  |  |
| $7.1657$ | $\pm$ |  $0.0024$ | [Lo17] |  |  |  |  |  |  |  |  | $7.1674$ | $\pm$ | $0.0044$ | 1.9
27Si | $4.109$ | $\pm$ |  $0.004$ | [Az75] | $4.1117$ | $\pm$ |  $0.0020$ | [Ma17] |  |  |  |  | $4.1112$ | $\pm$ | $0.0018$ | 1
29P | $4.15$ | $\pm$ |  $0.03$ | [Sc70] | $4.083$ | $\pm$ |  $0.012$ | [Az75] | $4.084$ | $\pm$ |  $0.022$ | [Wi80] |  |  |  |
| $4.1055$ | $\pm$ |  $0.0044$ | [Lo20] |  |  |  |  |  |  |  |  | $4.1031$ | $\pm$ | $0.0058$ | 1.4
31S | $2.57$ | $\pm$ |  $0.01$ | [Ja60] | $2.543$ | $\pm$ |  $0.008$ | [Az77] | $2.562$ | $\pm$ |  $0.007$ | [Wi80] |  |  |  |
| $2.5534$ | $\pm$ |  $0.0018$ | [Ba12] |  |  |  |  |  |  |  |  | $2.5539$ | $\pm$ |  $0.0023$444If the result from [Al74] (see Table 3) is not omitted the weighted average is $t_{1/2}=2.5549\pm 0.0040$ s (S = 2.4). | 1.4
33Cl | $2.53$ | $\pm$ |  $0.02$ | [Mu58] | $2.51$ | $\pm$ |  $0.02$ | [Ja60] | $2.47$ | $\pm$ |  $0.02$ | [Sc70] |  |  |  |
| $2.513$ | $\pm$ |  $0.004$ | [Ta73] | $2.507$ | $\pm$ |  $0.008$ | [Az77] | $2.5038$ | $\pm$ |  $0.0022$ | [Gr15a] | $2.5059$ | $\pm$ | $0.0025$ | 1.3
35Ar | $1.79$ | $\pm$ |  $0.01$ | [Ja60] | $1.770$ | $\pm$ |  $0.006$ | [Wi69] | $1.774$ | $\pm$ |  $0.003$ | [Az77] |  |  |  |
| $1.7754$ | $\pm$ |  $0.0011$ | [Ia06] |  |  |  |  |  |  |  |  | $1.7752$ | $\pm$ | $0.0010$ | 1
37K | $1.223$ | $\pm$ |  $0.008$ | [Az77] | $1.23651$ | $\pm$ |  $0.00094$ | [Sh14] | $1.23635$ | $\pm$ |  $0.00088$ | [Ku17] | $1.23634$ | $\pm$ | $0.00076$ | 1.2
39Ca | $0.860$ | $\pm$ |  $0.005$ | [Mi58] | $0.873$ | $\pm$ |  $0.008$ | [Li60] | $0.8604$ | $\pm$ |  $0.0030$ | [Al73] |  |  |  |
| $0.8594$ | $\pm$ |  $0.0016$ | [Az77] | $0.8607$ | $\pm$ |  $0.0010$ | [Bl10] |  |  |  |  | $0.86046$ | $\pm$ |  $0.00080$555If the result from [Kl54] (see Table 3) is not omitted the weighted average is $t_{1/2}=0.861\pm 0.049$ s (S = 1.9). | 1
41Sc | $0.628$ | $\pm$ |  $0.014$ | [Ja60] | $0.596$ | $\pm$ |  $0.006$ | [Yo65] | $0.5963$ | $\pm$ |  $0.0017$ | [Al73] |  |  |  |
| $0.591$ | $\pm$ |  $0.005$ | [Ta73] |  |  |  |  |  |  |  |  | $0.5962$ | $\pm$ | $0.0022$ | 1.4
43Ti | $0.528$ | $\pm$ |  $0.003$ | [Ja60] | $0.56$ | $\pm$ |  $0.02$ | [Ja61] | $0.50$ | $\pm$ |  $0.02$ | [Pl62] |  |  |  |
| $0.49$ | $\pm$ |  $0.01$ | [Al67] | $0.54$ | $\pm$ |  $0.01$ | [Va69] | $0.509$ | $\pm$ |  $0.005$ | [Ho87] | $0.5223$ | $\pm$ | $0.0057$ | 2.4
45V | $0.539$ | $\pm$ |  $0.018$ | [Ho82] | $0.5472$ | $\pm$ |  $0.0053$ | [Ha87] |  |  |  |  | $0.5465$ | $\pm$ | $0.0051$ | 1
47Cr | $0.4600$ | $\pm$ |  $0.0015$ | [Ed77] | $0.4720$ | $\pm$ |  $0.0063$ | [Ha87] |  |  |  |  | $0.4606$ | $\pm$ |  $0.0027$666If the result from [Bu85] (see Table 3) is not omitted the weighted average is $t_{1/2}=0.4616\pm 0.0051$ s (S = 3.6). | 1.9
49Mn | $0.384$ | $\pm$ |  $0.017$ | [Ha80] | $0.3817$ | $\pm$ |  $0.0074$ | [Ha87] | $0.380$ | $\pm$ |  $0.030$ | [Ku17a] | $0.3820$ | $\pm$ | $0.0066$ | 1
51Fe | $0.310$ | $\pm$ |  $0.005$ | [Ay84] | $0.3050$ | $\pm$ |  $0.0043$ | [Ha87] | $0.298$ | $\pm$ |  $0.014$ | [Ho89] |  |  |  |
| $0.301$ | $\pm$ |  $0.004$ | [Su13] | $0.308$ | $\pm$ |  $0.005$ | [Sh15] | $0.288$ | $\pm$ |  $0.006$ | [Ku17a] | $0.3031$ | $\pm$ | $0.0029$ | 1.4
53Co | $0.240$ | $\pm$ |  $0.020$ | [Ho89] | $0.240$ | $\pm$ |  $0.009$ |  [Lo02]777See also [Bl02]. | $0.230$ | $\pm$ |  $0.017$ | [Su13] |  |  |  |
| $0.245$ | $\pm$ |  $0.003$ | [Ku17a] |  |  |  |  |  |  |  |  | $0.2440$ | $\pm$ | $0.0028$ | 1
55Ni | $0.189$ | $\pm$ |  $0.005$ | [Ho77] | $0.208$ | $\pm$ |  $0.005$ | [Ay84] | $0.2121$ | $\pm$ |  $0.0038$ | [Ha87] |  |  |  |
| $0.204$ | $\pm$ |  $0.003$ | [Re99] | $0.196$ | $\pm$ |  $0.005$ |  [Lo02]77footnotemark: 7 | $0.203$ | $\pm$ |  $0.002$ | [Ku17a] | $0.2032$ | $\pm$ | $0.0025$ | 1.8
57Cu | $0.1994$ | $\pm$ |  $0.0032$ | [Sh89] | $0.1963$ | $\pm$ |  $0.0007$ | [Se96] | $0.195$ | $\pm$ |  $0.004$ | [Ku17a] | $0.19640$ | $\pm$ | $0.00067$ | 1
59Zn | $0.1820$ | $\pm$ |  $0.0018$ | [Ar84] | $0.173$ | $\pm$ |  $0.014$ |  [Lo02]77footnotemark: 7 | $0.174$ | $\pm$ |  $0.002$ | [Ku17a] | $0.1784$ | $\pm$ | $0.0028$ | 2.1
61Ga | $0.150$ | $\pm$ |  $0.030$ | [Wi93] | $0.168$ | $\pm$ |  $0.003$ | [We02] | $0.148$ | $\pm$ |  $0.019$ |  [Lo02]77footnotemark: 7 |  |  |  |
| $0.162$ | $\pm$ |  $0.010$ | [Ro14] | $0.163$ | $\pm$ |  $0.005$ | [Ku17a] |  |  |  |  | $0.1660$ | $\pm$ | $0.0025$ | 1
63Ge | $0.095$ | ${}^{+}_{-}$ |  ${}^{0.023}_{0.020}$ |  [Wi93]888These asymmetric errors have been symmetrized for the analysis by using standard recommendations of the Particle Data Group. | $0.150$ | $\pm$ |  $0.009$ |  [Lo02]77footnotemark: 7 | $0.149$ | $\pm$ |  $0.004$ | [Ro14] |  |  |  |
| $0.156$ | $\pm$ |  $0.011$ | [Ku17a] |  |  |  |  |  |  |  |  | $0.1485$ | $\pm$ | $0.0051$ | 1.5
65As | $0.126$ | $\pm$ |  $0.016$ |  [Lo02]77footnotemark: 7 | $0.126$ | $\pm$ |  $0.007$ | [Ro14] |  |  |  |  | $0.1260$ | $\pm$ | $0.0064$ | 1
67Se | $0.107$ | $\pm$ |  $0.035$ | [Ba94] | $0.136$ | $\pm$ |  $0.012$ |  [Lo02]77footnotemark: 7 | $0.133$ | $\pm$ |  $0.004$ | [Ro14] | $0.1330$ | $\pm$ | $0.0038$ | 1
69Br |  $<100$ ns |  |  | [Bl95] |  $<24$ ns |  |  | [Pf96] |  $<24$ ns |  |  | [Ja99] |  $<24$ ns999Proton unbound. |  |  |
71Kr | $0.100$ | $\pm$ |  $0.003$ | [Oi97] | $0.0988$ | $\pm$ |  $0.0003$ | [Si19] |  |  |  |  | $0.0988$ | $\pm$ | $0.0003$ | 1
73Rb |  $<30$ ns |  |  | [Pf96] |  $<30$ ns |  |  | [Ja99] |  $<81$ ns |  |  | [Su17] |  $<30$ ns99footnotemark: 9 |  |  |
75Sr | $0.088$ | $\pm$ |  $0.003$ | [Hu03] | $0.0817$ | $\pm$ |  $0.0034$ | [Si19] |  |  |  |  | $0.0852$ | $\pm$ | $0.0031$ | 1.4
77Y | $0.057$ | ${}^{+}_{-}$ |  ${}^{0.022}_{0.012}$ |  [Ki01]88footnotemark: 8,101010Also reported in [Fa02]. |  |  |  |  |  |  |  |  | $0.065$ | $\pm$ | $0.017$ |
79Zr | $0.056$ | $\pm$ |  $0.030$ | [Bl99] |  |  |  |  |  |  |  |  | $0.056$ | $\pm$ | $0.030$ |
81Nb |  $<80$ ns |  |  | [Ja99] |  $<200$ ns |  |  | [Ki01] |  $<38$ ns |  |  | [Su13a] |  |  |  |
|  $<40$ ns |  |  | [Su17a] |  |  |  |  |  |  |  |  |  $<38$ ns99footnotemark: 9 |  |  |
83Mo | $0.006$ | ${}^{+}_{-}$ |  ${}^{0.030}_{0.003}$ |  [Ki01]88footnotemark: 8 |  |  |  |  |  |  |  |  | $0.028$ | $\pm$ | $0.019$ |
85Tc |  $<100$ ns |  |  | [Ja99] |  $<43$ ns |  |  | [Ja99] |  $<42$ ns |  |  | [Su13a] |  |  |  |
|  $<43$ ns |  |  | [Su17a] |  |  |  |  |  |  |  |  |  $<42$ ns99footnotemark: 9 |  |  |
89Rh |  $>500$ ns |  |  | [Ja99] |  $<120$ ns |  |  | [Ce16] |  |  |  |  |  $<120$ ns99footnotemark: 9 |  |  |
93Ag | $228$ | $\pm$ |  $16$ ns | [Ce16] |  |  |  |  |  |  |  |  | $228$ | $\pm$ |  $16$ ns |
97In |  $>3\leavevmode\nobreak\ \mu$s |  |  | [Ce16] |  |  |  |  |  |  |  |  |  $>3\leavevmode\nobreak\ \mu$s |  |  |
Table 2: Branching ratios, $B\\!R$, for the $T=1/2$ mirror $\beta$ transitions up to A = 75. References to data listed here are given in Table 7. References to rejected data are listed in Table 3. The scale factor, $S$, is defined in the caption of Table LABEL:table:halflives_mirrors. If branching ratios in literature were quoted relative to values that are to date known more accurately because of later measurements, the published values have been updated. When weighted averages of several input values are made the scale factor $S$ in Eq. 14 is also given. | Measured branching ratios to excited states (%) | Ground state
---|---|---
| 1 | 2 | 3 | 4 | 5 | branching ratio
|  |  |  |  |  |  |  |  |  |  |  $BR$ $(\%)$
3H |  |  |  |  |  |  |  |  |  |  |  $100$ [Ti87]
11C |  |  |  |  |  |  |  |  |  |  |  $100$ [Aj75]
13N |  |  |  |  |  |  |  |  |  |  |  $100$ [Aj70]
15O |  |  |  |  |  |  |  |  |  |  |  $100$ [Aj70]
17F |  |  |  |  |  |  |  |  |  |  |  $100$ [Aj82]
19Ne | 1.55 MeV |  | 0.11 MeV |  |  |  |  |  |  |  |
| $0.0021(3)$ | [Al76] | $0.012(2)$ | [Ad81] |  |  |  |  |  |  |
| $0.0023(3)$ | [Ad83] | $0.0099(7)$ | [Re19] |  |  |  |  |  |  |
| $0.0017(5)$ | [Re19] |  |  |  |  |  |  |  |  |
| 0.00212(20) | (S=1) | 0.0101(7) | (S=1) |  |  |  |  |  |  | 99.9878(7)
21Na | 0.35 MeV |  |  |  |  |  |  |  |  |  |
| $5.1(2)$ | [Al74] |  |  |  |  |  |  |  |  |
| $4.2(2)$ | [Az77] |  |  |  |  |  |  |  |  |
| $4.97(16)$ | [Wi80] |  |  |  |  |  |  |  |  |
| $4.74(4)$ | [Ia06] |  |  |  |  |  |  |  |  |
| 4.746(77) | (S=2.1) |  |  |  |  |  |  |  |  | 95.254(77)
23Mg | 2.39 MeV |  | 0.44 MeV |  |  |  |  |  |  |  |
| 0.0064(9) | [Ma74] | $9.1(5)$ | [Ta60] |  |  |  |  |  |  |
|  |  | $8.6(3)$ | [Go68a] |  |  |  |  |  |  |
|  |  | $9.1(4)$ | [Al74] |  |  |  |  |  |  |
|  |  | $8.1(4)$ | [Ma74] |  |  |  |  |  |  |
|  |  | $7.79(15)$ | [Az77] |  |  |  |  |  |  |
|  |  | $7.805(81)$ | [Ma17] |  |  |  |  |  |  |
| 0.0064(9) |  | 7.91(14) | (S=2.1) |  |  |  |  |  |  | 92.08(14)777A small correction because the 440 keV level is also fed in the decay of the 2390 keV level with a branching ratio of 0.0025(3)%, lowering the g.s. branching ratio by this amount, is negligible at the present level of precision.
25Al | 1.611 MeV |  | 0.975 MeV |  |  |  |  |  |  |  |
| $0.9(2)$ | [Ma69] | $0.036(17)$ | [Mo71] |  |  |  |  |  |  |
| $0.7(2)$ | [Ba70] | $0.044(7)$ | [Ma76] |  |  |  |  |  |  |
| $0.84(7)$ | [Ju71] | $0.040(14)$ | [Az77] |  |  |  |  |  |  |
| $0.794(35)$ | [Az77] |  |  |  |  |  |  |  |  |
| $0.776(28)$ | [Wi80] |  |  |  |  |  |  |  |  |
| 0.788(21) | (S=1) | 0.042(6) | (S=1) |  |  |  |  |  |  |  99.170(22) 888A branching ratio of 0.006(4)% to the level at 1.965 MeV is reported in [Az77] but is not retained in the ENSDF database.
27Si | 2.98 MeV |  | 2.73 MeV |  | 2.21 MeV |  | 1.01 MeV |  |  |  |
| $0.0249(40)$ | [De71] | $0.0155(17)$ | [Ma74] | $0.10(2)$ | [Go64] | $0.0024(8)$ | [De71] |  |  |
| $0.0259(24)$ | [Ma74] | $0.0142(10)$ | [Da85] | $0.18(5)$ | [Be71] | $0.0049(24)$ | [Ma74] |  |  |
| $0.0214(14)$ | [Da85] |  |  | $0.15(7)$ | [De71] | $0.0050(8)$ | [Da85] |  |  |
|  |  |  |  | $0.181(14)$ | [Ma74] |  |  |  |  |
|  |  |  |  | $0.148(7)$ | [Az77] |  |  |  |  |
|  |  |  |  | $0.164(28)$ | [Ma17] |  |  |  |  |
| 0.0227(14) | (S=1.2) | 0.0145(9) | (S=1) | 0.151(9) | (S=1.5) | 0.0038(9) | (S=1.7) |  |  | 99.808(9)
29P | 2.43 MeV |  | 1.27 MeV |  |  |  |  |  |  |  |
| $0.453(16)$ | [Wi80] | $0.8(2)$ | [Ro55] |  |  |  |  |  |  |
| $0.4(1)$ | [Lo02] | $1.2(2)$ | [Lo62] |  |  |  |  |  |  |
|  |  | $1.255(22)$ | [Wi80] |  |  |  |  |  |  |
| 0.455(14) | (S=1) | 1.249(35) | (S=1.6) |  |  |  |  |  |  | 98.296(38)
31S | 3.51 MeV |  | 3.13 MeV |  | 1.266 MeV |  |  |  |  |  |
| $0.011(3)$ | [De71] | $0.032(5)$ | [De71] | $1.1(1)$ | [Ta60] |  |  |  |  |
| $0.0121(10)$ | [Wi80] | $0.0326(16)$ | [Wi80] | $0.8(4)$ | [De71] |  |  |  |  |
|  |  |  |  | $1.25(6)$ | [Al74] |  |  |  |  |
|  |  |  |  | $0.98(20)$ | [Az77] |  |  |  |  |
|  |  |  |  | $1.097(33)$ | [Wi80] |  |  |  |  |
| 0.0120(9) | (S=1) | 0.0325(15) | (S=1) | 1.126(34) | (S=1.3) |  |  |  |  | 98.830(34)
33Cl | 4.75 MeV |  | 4.05 MeV |  | 2.87 MeV |  | 2.31 MeV |  | 1.97 MeV |  |
| 0.00041(21) | [Wi80] | 0.00047(15) | [Wi80] | 0.443(58) | [Wi80] | 0.0353(48) | [Wi80] | 0.460(60) | [Wi80] |
| 0.84 MeV |  |  |  |  |  |  |  |  |  |
| 0.479(64) | [Wi80] |  |  |  |  |  |  |  |  | 98.58(11)
35Ar | 3.97 MeV |  | 3.92 MeV |  | 3.00 MeV |  | 2.69 MeV |  | 1.76 MeV |  |
| $0.00744(87)$ | [Wi80] | $0.00790(52)$ | [Wi80] | $0.07(2)$ | [De71] | $0.19(3)$ | [De71] | $0.25(7)$ | [Wi69] |
| $0.00763(74)$ | [Ad84] | $0.0104(42)$ | [Ad84] | $0.11(3)$ | [Ge71] | $0.16(4)$ | [Ge71] | $0.234(13)$ | [Wi69] |
|  |  |  |  | $0.0901(33)$ | [Wi80] | $0.1606(65)$ | [Wi80] | $0.22(3)$ | [De71] |
|  |  |  |  | $0.0932(56)$ | [Ad84] | $0.171(11)$ | [Ad84] | $0.22(5)$ | [Ge71] |
|  |  |  |  | $0.084(6)$ | [Da85] | $0.160(9)$ | [Da85] | $0.272(10)$ | [Wi80] |
|  |  |  |  |  |  |  |  | $0.250(16)$ | [Ad84] |
|  |  |  |  |  |  |  |  | $0.260(11)$ | [Da85] |
| 0.00755(56) | (S=1) | 0.00794(52) | (S=1) | 0.0895(25) | (S=1) | 0.1630(47) | (S=1) | 0.2571(68) | (S=1.2) |
| 1.22 MeV |  |  |  |  |  |  |  |  |  |
| $1.04(24)$ | [Wi69] |  |  |  |  |  |  |  |  |
| $1.223(46)$ | [Wi69] |  |  |  |  |  |  |  |  |
| $1.22(20)$ | [Ge71] |  |  |  |  |  |  |  |  |
| $1.228(30)$ | [Wi80] |  |  |  |  |  |  |  |  |
| $1.23(7)$ | [Ad84] |  |  |  |  |  |  |  |  |
| 1.225(9) | (S=1) |  |  |  |  |  |  |  |  | 98.250(12)
37K | 3.94 MeV |  | 3.60 MeV |  | 3.17 MeV |  | 2.80 MeV |  | 2.49 MeV |  |
| 0.00116(13) | [Ha97] | 0.0224(12) | [Ha97] | 0.0027(2) | [Ha97] | $2.0(4)$ | [Ka64] | 0.0029(4) | [Ha97] |
|  |  |  |  |  |  | $2.22(21)$ | [Az77] |  |  |
|  |  |  |  |  |  | $2.07(11)$ | [Ha97] |  |  |
|  |  |  |  |  |  | $2.20(17)$ |  [Ku17]999See also [Se17] |  |  |
|  |  |  |  |  |  | 2.121(70) | (S=1) |  |  |
| 1.61 MeV |  | 1.41 MeV |  |  |  |  |  |  |  |
| 0.0025(20) | [Ha97] | 0.00422(75) | [Ha97] |  |  |  |  |  |  | 97.843(70)
39Ca | 0.35 MeV |  |  |  |  |  |  |  |  |  |
| $0.00226(58)$ | [Ad84] |  |  |  |  |  |  |  |  |
| $0.00250(27)$ | [Ha94] |  |  |  |  |  |  |  |  |
| 0.00246(24) | (S=1) |  |  |  |  |  |  |  |  |  99.9975(32)101010The error is obtained from the quadratic sum of the error bar on the BR to the state at 2.52 MeV and the upper limits for the BR to states at higher excitation energies reported in [Ha94].
41Sc | 2.96 MeV |  | 2.58 MeV |  |  |  |  |  |  |  |
| 0.0139(14) | [Wi80] | 0.0232(29) | [Wi80] |  |  |  |  |  |  |  99.9629(52)111111The error is obtained from the quadratic sum of the error bars on the BR to the states at 2.58 and 2.96 MeV and the upper limits for the BR to states at higher excitation energies reported in [Wi80].
43Ti | 3.63 MeV |  | 3.26 MeV |  | 2.76 MeV |  | 2.46 MeV |  | 2.34 MeV |  |
| 0.016(4) | [Ho87] | 0.011(3) | [Ho87] | 0.20(3) | [Ho87] | 0.92(13)) | [Ho87] | 0.39(6) | [Ho87] |
| 2.29 MeV |  | 1.96 MeV |  | 1.88 MeV |  | 1.41 MeV |  | 0.85 MeV |  |
| 4.7(7) | [Ho87] | 0.024(10) | [Ho87] | 0.20(4) | [Ho87] | 0.68(10) | [Ho87] | 2.7(8) | [Ho87] | 90.2(11)
45V | 0.040 MeV |  |  |  |  |  |  |  |  |  |
| 4.3(15)121212This value includes a correction for the internal conversion coefficient $\alpha$ = 0.223(10) for the 40 keV $\gamma$ ray to the ground state [Bu08]. | [Ho82] |  |  |  |  |  |  |  |  | 95.7(15)
47Cr | 0.087 MeV |  |  |  |  |  |  |  |  |  |
| 3.9(12)131313This value includes a correction for the internal conversion coefficient $\alpha$ = 0.041(4) for the 87 keV $\gamma$ ray to the ground state [Bu07]. | [Bu85] |  |  |  |  |  |  |  |  | 96.1(12)
49Mn | 2.50 MeV |  | 0.27 MeV |  |  |  |  |  |  |  |
| 2.3(9) | [Ho89] | $6.4(26)$ | [Ha80] |  |  |  |  |  |  |
|  |  | $5.8(26)$ | [Ho89] |  |  |  |  |  |  |
|  |  | 6.1(18) | (S=1) |  |  |  |  |  |  | 91.6(20)
51Fe | 3.56 MeV |  | 3.43 MeV |  | 2.91 MeV |  | 2.14 MeV |  | 1.82 MeV |  |
| 0.16(5) | [Ho89] | 0.20(6) | [Ho89] | 0.10(3) | [Ho89] | 0.24(7) | [Ho89] | 0.49(14) | [Ho89] |
| 0.237 MeV |  |  |  |  |  |  |  |  |  |
| 5.0(13) | [Ay84] |  |  |  |  |  |  |  |  | 93.8(13)
53Co | 1.33 MeV |  |  |  |  |  |  |  |  |  |
| 5.6(17) | [Ho89] |  |  |  |  |  |  |  |  | 94.4(17)
55Ni |  |  |  |  |  |  |  |  |  |  | 100.0(20)141414Based on the fact that the first excited state which may be populated by allowed $\beta$ decay is near 2.5 MeV, making 100% g.s. feeding very probable [Ju08]. A 2% error bar was assumed.
57Cu | 3.01 MeV |  | 2.44 MeV |  | 1.11 MeV |  | 0.77 MeV |  |  |  |
| 0.35(4) | [Se96] | 0.17(3) | [Se96] | 8.60(60) | [Se96] | 0.94(9) | [Se96] |  |  | 89.9(8)
59Zn | high-E |  | 0.914 MeV |  | 0.491 MeV |  |  |  |  |  |
| $0.09(2)$ | [Ar84] | $4.8(6)$ | [Ar84] | $1.1(2)$ | [Ar84] |  |  |  |  |
| $0.23(8)$ | [Ho81] | $5(2)$ | [Ho81] | $1.6(8)$ | [Ho81] |  |  |  |  |
| 0.098(33) | (S=1.2) | 4.82(57) | (S=1) | 1.13(19) | (S=1) |  |  |  |  | 93.95(60)
61Ga | 0.94 MeV |  | 0.76 MeV |  | 0.42 MeV |  | 0.12 MeV |  | 0.088 MeV |  |
| 0.68(18) | [Zu15] | 0.86(24) | [Zu15] | 1.3(4) | [Zu15] | 1.2(4) | [Zu15] | 1.9(6) | [Zu15] | 94.0(10)151515See also [We02].
67Se | high-E161616$\beta$ decay to proton-decaying level(s). |  |  |  |  |  |  |  |  |  |
| 0.5(1) | [Bl95] |  |  |  |  |  |  |  |  | 99.5(1)171717A level assumed to be at 0.35 MeV with a supposed 8% branching ratio reported in [Ba94] was not observed in (HI,xn$\gamma$) reactions [Ju05].
71Kr181818See [Ab11] for more details |  $\approx$ 4.25 MeV1616footnotemark: 16 |  | 0.407$?$ MeV191919Level uncertain (see [Ab10]) |  | 0.207 MeV |  |  |  |  |  |
|  $2.1(7)$202020The very different value of 5.2(6)% reported in [Bl95] might be due to the presence of an isomer at $\approx 10$ keV [Fi05, Ab11]. | [Oi97] |  |  | $15.8(14)$ | [Oi97] |  |  |  |  | 82.1(16)212121Value for [Oi97].
| $\approx 2$ |  [Fi05]222222No error bars given. Results in [Fi05] are based on a re-interpretation of the data in [Oi97]. | $\approx 15$ | [Fi05] | $\approx 15$ | [Fi05] |  |  |  |  |  to $\approx$ 68232323Value for [Fi05]. See also [Ab11] and Sec. II.3.2. Note that a minimum value of $68.1$% is required in order to avoid a negative value for $\rho^{2}$ when solving Eq. II.1.
| 2.1(7) |  | $\approx$ 15 |  | 15.8(14) |  |  |  |  |  |
75Sr |  5.50 MeV1616footnotemark: 16 |  | 0.144 MeV |  |  |  |  |  |  |  |
| $6.5(33)$ | [Bl95] |  $5.2(11)$242424This value includes a correction for the internal conversion coefficient $\alpha=0.15(10)$ for the 144 keV $\gamma$ ray to the ground state [Ne13]. | [Hu03] |  |  |  |  |  |  |
| $5.2(9)$ | [Hu03] |  |  |  |  |  |  |  |  |
| 5.3(9) | (S=1) | 5.2(11) |  |  |  |  |  |  |  | 89.5(14)
Table 3: References from which results for half-lives have been rejected because the error bars are a factor 10 or more larger than that of the most precise measurement (top part) or other reasons (lower part). The correlation between the alphabetical reference code used here and the actual reference numbers is listed in Table 7. |  $<$Error bar more than 10 times larger$>$
---|---
Parent nucleus |  Value [Reference]
11C (min) |  $20.35(8)$ [Sm41]; $20.0(1)$ [Di51]; $20.74(10)$ [Ku53]; $20.26(10)$ [Ba55]; $20.8(2)$ [Pr57]; $20.11(13)$ [Ar58]; $20.32(12)$ [Be75]
15O (s) |  $123.95(50)$ [Pe57]; $124.1(5)$ [Ki59]; $122.6(10)$ [Ne63]
19Ne (s) |  $17.43(6)$ [Ea62]; $17.36(6)$ [Go68]; $17.7(1)$ [Pe57]; $17.36(6)$ [Wi74]
21Na (s) |  $23.0(2)$ [Ar58]; $22.55(10)$ [Al74]; $22.48(4)$ [Az75, Az77]161616The value $t_{1/2}=22.47(3)$ s reported in [Az75] was reanalyzed with the updated value being reported in [Az77].
23Mg (s) |  $11.36(4)$ [Al74]; $11.26(8)$ [Az74]; $11.41(5)$ [Go68]; $12.1(1)$ [Mi58]
25Al (s) |  $7.24(3)$ [Mu58]
27Si (s) |  $4.21(3)$ [Gr71]; $4.14(3)$ [Mi58]; $4.16(3)$ [Su62]; $4.19(2)$ [Bl68]; $4.09(2)$ [Ba77]
31S (s) |  $2.80(5)$ [Cl58]; $2.66(3)$ [Ha52]; $2.40(7)$ [Hu54]; $2.61(5)$ [Li60]; $2.72(2)$ [Mi58]; $2.58(6)$ [Wa60]
37K (s) |  $1.25(4)$ [Ka64]; $1.23(2)$ [Sc58]
39Ca (s) |  $0.876(12)$ [Cl58]; $0.865^{+0.007}_{-0.017}$ [Ka68]
43Ti (s) |  $0.58(4)$ [Sc48]; $0.40(5)$ [Va63]
47Cr (s) |  $0.460(80)$ [Ku17a]
53Co (s) |  $0.267(109)$ [Ha87]
57Cu (s) |  $0.1994(32)$ [Sh84]; $0.183(17)$ [Lo02,Bl02]
59Zn (s) |  $0.213(34)$ [Ro14]
65As (s) |  $0.190(11)$ [Mo95]; $0.19^{+0.11}_{-0.07}$ [Wi93]
71Kr (s) |  $0.097(9)$ [Ew81]; $0.100(3)$ [Oi97]; $0.092(9)$ [Ro14]; $0.083(48)$ [Lo02,Bl02]
75Sr (s) |  $0.080^{+0.400}_{-0.040}$ [Ki01]171717Also reported in [Fa02]; $0.071^{+0.071}_{-0.024}$ [Bl95]
|  $<$Other reasons for exclusion$>$
Parent nucleus [Refs.] |  Explanation
17F [Al77], [Wo69] |  The quoted half-lives (resp. 64.80(9) s and 65.2(2) s) are significantly higher (by resp. 4.8 $\sigma$ and 4.2 $\sigma$) than the average of the other, more recent and also more precise results.
27Si[Ge76] |  The quoted half-life (i.e. 4.206(8) s) was measured with a spectrometer and is significantly higher (by 12 $\sigma$) than the average of the other, mutually consistent results, as is also the case for the other half-life (for 24Na) that was obtained with the same apparatus.
27Si[Go68] |  The quoted half-life (i.e. 4.17(1) s) is significantly higher (by 5.9$\sigma$) than the average of the other, mostly more recent and also more precise and mutually consistent results.
29P [Ja60], [Ta73] |  Both quoted half-lives (resp. 4.19(02) s and 4.149(5) s) are significantly higher (by resp. 4.3 $\sigma$ and 9.2 $\sigma$) than the weighted average of the other results.
31S [Al74] |  The quoted half-life (i.e. 2.605(12) s) is significantly higher (by 4.3 $\sigma$) than the average of the other, mostly more recent and also mutually consistent results.
39Ca [Kl54] |  The larger value reported (i.e. 0.90(1) s) is most probably due to a small contamination from 38K produced in the 40Ca($\gamma$,d)38K reaction with a threshold below the maximum energy used for the 40Ca($\gamma$,n)39Ca reaction.
47Cr [Bu85] |  The quoted half-life (i.e. 0.508(10) s) is significantly higher (by 4.7 $\sigma$) than the weighted average of the two other more precise, and mutually consistent results.
53Co [Ko73] |  The quoted half-life (i.e. 0.262(25) s) was obtained by a $\beta$-ray measurement possibly including the positrons from 53mCo [Ha87].
53Co, 55Ni, and 59Cu [Ru14] |  The quoted half-lives of 0.2460(18) s, 0.2018(19) s, and 0.1733(33) s, respectively, were later replaced by the new values of 0.245(3) s, 0.203(2) s, and 0.174(2) s respectively [Ku17a], following a re-analysis of the same data Rubio (2021).
63Ga [Sh93] |  The half-lives from [Wi93] and [Sh93] listed in Severijns  _et al._ (2008) were obtained from the same set of data.
67Se and 71Kr [Bl95] |  The quoted half-lives (i.e. $0.060^{+0.017}_{-0.011}$ s, resp. $0.064^{+0.008}_{-0.005}$ s) were obtained with very low statistics. They are a factor of respectively about 2 and 1.5 lower than the other results and also differ by more than 5 $\sigma$ from the weighted average of these.
Table 4: References from which results for branching ratios have been rejected. The correlation between the alphabetical reference code used here and the actual reference numbers is listed in Table 7. Parent nucleus [Reference(s)] |  Reason for exclusion
---|---
$<$Branching ratios$>$ |
19Ne [Sa93], 23Mg [Da85], |  Error bars are a factor 10 or more larger than that of the most precise measurement.
27Si [Go64], 33Cl [Ba70] |
21Na [Ac07], [Ac10] |  The result from the priv. comm. [Ac07] in Severijns  _et al._ (2008) has in the mean time been published as [Ac10] but it is not included here as the error bar is more than 10 times larger than the most precise measurement.
29P [Az77] |  The branching ratios for 29P in [Az77] have an error bar that is more than 10 times larger than the one of the most precise result (level at 1.27 MeV), or deviates by more than 12 $\sigma$ from the weighted average of the other results in the literature (level at 2.43 MeV).
|  The branching ratio for 29P to the level at 2.43 MeV in 29Si reported in
[Ro55] has an error bar that is 10 times larger than the one of the most
precise result.
35Ar [Az77], [Ha79] |  The branching ratios from [Az77] and [Ha79] were not considered as only three, resp. one, branch(es) were observed and the reported branching ratios differ significantly from the ones reported in other experiments.
37K [Ma76] |  The branching ratios from [Ma76] to the levels at 2.80 and 3.60 MeV were not considered as they are both about 4.2 $\sigma$ lower than the weighted average of other results in the literature.
57Cu [Sh84] |  The branching ratio result from [Sh84] was not considered as only the most intensive $\beta$ branch could be observed due to interference of the $\beta$ decays of 54gCo and 58Cu while, in addition, the lifetime for 57Cu reported in [Sh84] is significantly longer than more recent values in the literature.
61Ga [Oi99] |  The results reported in [Oi99] were not considered as this experiment suffered from strong 61Zn and 61Cu isobaric contaminations, resulting in a limited accuracy for the measurement of the 61Ga branching ratios [We02].
67Se [Ba84] |  The results reported in [Ba84] were not considered as no error bars are given.
71Kr [Fi05] |  The branching ratios to the different levels in the daughter isotope 71Br as shown in Fig. 10 of [Fi05] are only estimated values based on the $\gamma$ spectra shown in [Oi97].
Table 5: Adopted values for the half-lives, $t_{1/2}$, the branching ratios, $B\\!R$, and the decay transition energies, $Q_{EC}$ for the $T=1/2$ mirror $\beta$ transitions with $A\leq 83$ (i.e. $s$, $p$, $sd$, and $fp$ shells). Isotopes with half-lives in the range of $\mu s$ and below are not included here. The $Q_{EC}$ values are from the 2020 Atomic Mass Evaluation Huang  _et al._ (2021a, b); Wang  _et al._ (2021) unless otherwise indicated. Details on the calculation of the vector statistical rate factor, $f_{V}$, and the electron-capture fraction, $P_{EC}$, are given in the text. Parent | Half-life | Branching | $Q_{EC}$ | $f_{V}$ | $P_{EC}$ | Partial half-life
---|---|---|---|---|---|---
nucleus |  $t_{1/2}$(s) | $BR(\%)$ | (keV) |  | $(\%)$ | $t(s)$
3H | $38854(35)$ | $\times$ | $10^{4}$ |  | $100$ |  | $18.59202$ | $\pm$ |  $0.00006$181818From [My15] which is 17, respectively 24 times more precise than the two most precise previous results, i.e [Na06] resp. [Va93]. | $2.864604(26)$ | $\times$ | $10^{-6}$ | N/A | $(38854$ | $\pm$ | $35)\times 10^{4}$
11C | $1220.41$ | $\pm$ | $0.32$ |  | $100$ |  | $1981.689$ | $\pm$ | $0.061$ | $3.18289$ | $\pm$ | $0.00082$ | $0.232$ | $1223.24$ | $\pm$ | $0.32$
13N | $597.88$ | $\pm$ | $0.23$ |  | $100$ |  | $2220.47$ | $\pm$ | $0.27$ | $7.7143$ | $\pm$ | $0.0072$ | $0.196$ | $599.05$ | $\pm$ | $0.23$
15O | $122.27$ | $\pm$ | $0.06$ |  | $100$ |  | $2754.18$ | $\pm$ | $0.49$ | $35.496$ | $\pm$ | $0.043$ | $0.100$ | $122.392$ | $\pm$ | $0.060$
17F | $64.366$ | $\pm$ | $0.026$ |  | $100$ |  | $2760.47$ | $\pm$ | $0.25$ | $35.208$ | $\pm$ | $0.022$ | $0.146$ | $64.460$ | $\pm$ | $0.026$
19Ne | $17.2573$ | $\pm$ | $0.0034$ | $99.9878$ | $\pm$ | $0.0007$ | $3239.50$ | $\pm$ | $0.16$ | $98.650$ | $\pm$ | $0.031$ | $0.100$ | $17.2767$ | $\pm$ | $0.0034$
21Na | $22.4527$ | $\pm$ | $0.0067$ | $95.254$ | $\pm$ | $0.077$ | $3546.919$ | $\pm$ | $0.018$ | $170.7580$ | $\pm$ | $0.0054$ | $0.094$ | $23.594$ | $\pm$ | $0.020$
23Mg | $11.3050$ | $\pm$ | $0.0044$ | $92.08$ | $\pm$ | $0.14$ | $4056.179$ | $\pm$ | $0.032$ | $378.621$ | $\pm$ | $0.018$ | $0.073$ | $12.286$ | $\pm$ | $0.019$
25Al | $7.1674$ | $\pm$ | $0.0044$ | $99.170$ | $\pm$ | $0.022$ | $4276.808$ | $\pm$ | $0.045$ | $508.522$ | $\pm$ | $0.032$ | $0.079$ | $7.2331$ | $\pm$ | $0.0047$
27Si | $4.1112$ | $\pm$ | $0.0018$ | $99.808$ | $\pm$ | $0.009$ | $4812.358$ | $\pm$ | $0.096$ | $993.52$ | $\pm$ | $0.11$ | $0.065$ | $4.1218$ | $\pm$ | $0.0018$
29P | $4.1031$ | $\pm$ | $0.0058$ | $98.296$ | $\pm$ | $0.038$ | $4942.23$ | $\pm$ | $0.36$ | $1136.37$ | $\pm$ | $0.48$ | $0.075$ | $4.1774$ | $\pm$ | $0.0061$
31S | $2.5539$ | $\pm$ | $0.0023$ | $98.830$ | $\pm$ | $0.034$ | $5398.01$ | $\pm$ | $0.23$ | $1844.74$ | $\pm$ | $0.45$ | $0.068$ | $2.5859$ | $\pm$ | $0.0025$
33Cl | $2.5059$ | $\pm$ | $0.0025$ | $98.58$ | $\pm$ | $0.11$ | $5582.52$ | $\pm$ | $0.39$ | $2189.71$ | $\pm$ | $0.86$ | $0.075$ | $2.5439$ | $\pm$ | $0.0038$
35Ar | $1.7752$ | $\pm$ | $0.0010$ | $98.250$ | $\pm$ | $0.012$ | $5966.24$ | $\pm$ | $0.68$ | $3122.0$ | $\pm$ | $2.0$ | $0.072$ | $1.8081$ | $\pm$ | $0.0010$
37K | $1.23634$ | $\pm$ | $0.00076$ | $97.843$ | $\pm$ | $0.070$ | $6147.48$ | $\pm$ | $0.23$ | $3623.73$ | $\pm$ | $0.75$ | $0.079$ | $1.2646$ | $\pm$ | $0.0012$
39Ca | $0.86046$ | $\pm$ | $0.00080$ | $99.9975$ | $\pm$ | $0.0032$ | $6524.49$ | $\pm$ | $0.60$ | $4951.5$ | $\pm$ | $2.5$ | $0.077$ | $0.86114$ | $\pm$ | $0.00080$
41Sc | $0.5962$ | $\pm$ | $0.0022$ | $99.9629$ | $\pm$ | $0.0052$ | $6495.55$ | $\pm$ | $0.16$ | $4745.45$ | $\pm$ | $0.65$ | $0.094$ | $0.5970$ | $\pm$ | $0.0022$
43Ti | $0.5223$ | $\pm$ | $0.0057$ | $90.2$ | $\pm$ | $1.1$ | $6872.6$ | $\pm$ | $6.0$ | $6364$ | $\pm$ | $31$ | $0.092$ | $0.5796$ | $\pm$ | $0.0095$
45V | $0.5465$ | $\pm$ | $0.0051$ | $95.7$ | $\pm$ | $1.5$ | $7123.82$ | $\pm$ | $0.21$ | $7616.8$ | $\pm$ | $1.2$ | $0.096$ | $0.572$ | $\pm$ | $0.010$
47Cr | $0.4606$ | $\pm$ | $0.0027$ | $96.1$ | $\pm$ | $1.2$ | $7444.0$ | $\pm$ | $5.2$ | $9521$ | $\pm$ | $36$ | $0.097$ | $0.4798$ | $\pm$ | $0.0066$
49Mn | $0.3820$ | $\pm$ | $0.0066$ | $91.6$ | $\pm$ | $2.0$ | $7712.43$ | $\pm$ | $0.23$ | $11353.3$ | $\pm$ | $1.8$ | $0.101$ | $0.417$ | $\pm$ | $0.012$
51Fe | $0.3031$ | $\pm$ | $0.0029$ | $93.8$ | $\pm$ | $1.3$ | $8054.0$ | $\pm$ | $1.4$ | $14123$ | $\pm$ | $13$ | $0.101$ | $0.3235$ | $\pm$ | $0.0054$
53Co | $0.2440$ | $\pm$ | $0.0028$ | $94.4$ | $\pm$ | $1.7$ | $8288.11$ | $\pm$ | $0.44$ | $16222.0$ | $\pm$ | $4.6$ | $0.107$ | $0.2588$ | $\pm$ | $0.0055$
55Ni | $0.2032$ | $\pm$ | $0.0025$ | $100.0$ | $\pm$ | $2.0$ | $8694.04$ | $\pm$ | $0.58$ | $20642.3$ | $\pm$ | $7.4$ | $0.105$ | $0.2034$ | $\pm$ | $0.0048$
57Cu | $0.19640$ | $\pm$ | $0.00067$ | $89.9$ | $\pm$ | $0.8$ | $8774.95$ | $\pm$ | $0.44$ | $21374.2$ | $\pm$ | $5.7$ | $0.117$ | $0.2187$ | $\pm$ | $0.0021$
59Zn | $0.1784$ | $\pm$ | $0.0028$ | $93.95$ | $\pm$ | $0.60$ | $9142.78$ | $\pm$ | $0.60$ | $26205.3$ | $\pm$ | $9.2$ | $0.117$ | $0.1901$ | $\pm$ | $0.0032$
61Ga | $0.1660$ | $\pm$ | $0.0025$ | $94.0$ | $\pm$ | $1.0$ | $9214$ | $\pm$ | $38$ | $26914$ | $\pm$ | $597$ | $0.131$ | $0.1768$ | $\pm$ | $0.0033$
63Ge | $0.1485$ | $\pm$ | $0.0051$ |  | $-$ |  | $9626$ | $\pm$ | $37$ | $33452$ | $\pm$ | $688$ | $0.129$ |  | $-$ |
65As | $0.1260$ | $\pm$ | $0.0064$ |  | $-$ |  | $9541$ | $\pm$ | $85$ | $31458$ | $\pm$ | $1517$ | $0.151$ |  | $-$ |
67Se | $0.1330$ | $\pm$ | $0.0038$ |  $99.5$222Upper limit. See Sec. II.3.2. | $\pm$ | $0.1$ | $10007$ | $\pm$ | $67$ | $39923$ | $\pm$ | $1435$ | $0.146$ | $0.1339$ | $\pm$ | $0.0038$
|  |  |  |  $90.0$333Assumed lower limit. See Sec. II.3.2. | $\pm$ | $1.5$ |  |  |  |  |  |  |  | $0.1480$ | $\pm$ | $0.0049$
71Kr | $0.0988$ | $\pm$ | $0.0003$ |  $82.1$22footnotemark: 2 | $\pm$ | $1.6$ | $10175$ | $\pm$ | $129$ | $42371$ | $\pm$ | $2916$ | $0.177$ | $0.1206$ | $\pm$ | $0.0024$
|  |  |  |  $70.0$33footnotemark: 3 | $\pm$ | $1.5$ |  |  |  |  |  |  |  | $0.1414$ | $\pm$ | $0.0031$
75Sr | $0.0852$ | $\pm$ | $0.0031$ |  $89.5$22footnotemark: 2 | $\pm$ | $1.4$ | $10600$ | $\pm$ | $220$ | $51147$ | $\pm$ | $5841$ | $0.197$ | $0.0954$ | $\pm$ | $0.0038$
|  |  |  |  $80.0$33footnotemark: 3 | $\pm$ | $1.5$ |  |  |  |  |  |  |  | $0.1067$ | $\pm$ | $0.0044$
77Y | $0.065$ | $\pm$ | $0.017$ |  | $-$ |  | $11365$ | $\pm$ | $203$ | $71486$ | $\pm$ | $5734$ | $0.179$ |  | $-$ |
79Zr | $0.056$ | $\pm$ | $0.030$ |  | $-$ |  | $11033$ | $\pm$ | $310$ | $61468$ | $\pm$ | $10023$ | $0.217$ |  | $-$ |
83Mo | $0.028$ | $\pm$ | $0.019$ |  | $-$ |  | $11273$ | $\pm$ | $432$ | $67103$ | $\pm$ | $14649$ | $0.253$ |  | $-$ |

####  II.2.2 Adopted input values for the ${\mathcal{F}}t^{\mathrm{mirror}}$
values

The adopted values for the half-lives, $t_{1/2}$, the branching ratios, $BR$,
and the decay transtion energies, $Q_{EC}$, are listed in Table 5, together
with the deduced statistical rate function, $f_{V}$, the electron-capture
fraction, $P_{EC}$, and the resulting partial half-lives, $t$ (Eq. 5). Note
that the isotopes listed in Table LABEL:table:halflives_mirrors that are
unbound to proton decay are not further considered. The statistical rate
functions, $f_{V}$, listed in column 5 of Table 5 were calculated using the
same code as in the survey of the superallowed pure Fermi transitions Hardy
and Towner (2015). The basic methodology for the calculation is described in
the Appendix to Ref. Hardy and Towner (2005). They were evaluated with the
$Q_{EC}$ values and their uncertainties listed in column 4 of Table 5. The
$P_{EC}$ values, finally, were calculated as outlined in Ref. Hardy and Towner
(2005) using the tables of Bambynek et al. Bambynek  _et al._ (1977) and
Firestone Firestone (1996). No errors were assigned to these $P_{EC}$ values
as they are expected to be accurate to a few parts in 100 Bambynek  _et al._
(1977); Hardy and Towner (2005) such that they do not contribute perceptibly
to the overall uncertainties. All quantities listed in Table 5 finally allow
obtaining the partial half-life, $t$ (Eq. 5), for each $\beta$ transition
considered. These values are listed in the last column of Table 5.

###  II.3 The ${\mathcal{F}}t^{\mathrm{mirror}}$ values

Having surveyed the experimental data we can now turn to the determination of
the ${\cal F}t^{\mathrm{mirror}}$ values according to Eq. (II.1). To do this
we must, however, still deal with the different correction terms. The values
for the nucleus-dependent radiative correction $\delta_{R}^{\prime}$ have been
calculated using the $Q_{EC}$ values in Table 5 following the same procedures
as adopted for the superallowed $0^{+}\rightarrow 0^{+}$ $\beta$ decays Towner
and Hardy (2002) and are listed in column 4 of Table 6. As to the nuclear-
structure-dependent corrections $\delta_{C}^{V}-\delta_{NS}^{V}$ (column 5 in
Table 6), for the mirror $\beta$ decays from 3H to 45V the values previously
reported in Severijns  _et al._ (2008) were used. For the mirror decays from
47Cr to 75Sr estimates based on the systematic of the superallowed
$0^{+}\rightarrow 0^{+}$ $\beta$ decays Hardy and Towner (2020) and the lower-
mass mirror $\beta$ decays Severijns  _et al._ (2008) were used, with generous
error bars attached to them. In none of these cases the uncertainty on
$\delta_{C}^{V}-\delta_{NS}^{V}$ is the dominant contribution to the total
uncertainty on the ${\cal F}t^{\mathrm{mirror}}$ value.

The resulting ${\cal F}t^{\mathrm{mirror}}$ values are listed in column 6 of
Table 6. Since the previous survey Severijns  _et al._ (2008) the
${\mathcal{F}}t^{\mathrm{mirror}}$ values for all but three (i.e. 13N, 41Sc,
and 43Ti) of the 19 mirror $\beta$ transitions with $A$ = 3 to 45 have been
improved, with now a relative precision $\leq$ 0.2% for all transitions
(previously only 5) up to 39Ca, as can be seen in Fig. 1, and precisions
between 0.4% and 2.9% for the $fp$-shell mirror transitions up to 61Ga.
Figures 2 and 3 show the fractional uncertainties for each of the experimental
and theoretical input factors to the ${\cal F}t^{\mathrm{mirror}}$ values for
the $sp$-$sd$ shell transitions and the $fp$ shell transitions, respectively.
Note the difference in the vertical scales between both figures.

Figure 1: Progress in the relative uncertainties of the
${\mathcal{F}}t^{\mathrm{mirror}}$ values for the mirror $\beta$ transitions
up to $A=39$ from 2008 Severijns  _et al._ (2008) (dashed) till now (solid).
Figure 2: Fractional contribution of the experimental and theoretical input
factors to the final ${\mathcal{F}}t^{\mathrm{mirror}}$ values for the
transitions up to $A=39$. Figure 3: Fractional contribution of the
experimental and theoretical input factors to the final
${\mathcal{F}}t^{\mathrm{mirror}}$ values for the transitions in the mass
$A=41-61$ range.

####  II.3.1 The Gamow-Teller/Fermi mixing ratio $\rho$

From these $\mathcal{F}t^{\mathrm{mirror}}$ values the Gamow-Teller/Fermi
mixing ratio, $\rho$, can be calculated (Eq. II.1). Comparing this with values
obtained from e.g. beta-neutrino correlation or $\beta$ asymmetry parameter
measurements allows testing the Conserved Vector Current hypothesis Naviliat-
Cuncic and Severijns (2009a) and extracting a value for the $V_{ud}$ quark-
mixing matrix element Naviliat-Cuncic and Severijns (2009a); Fenker  _et al._
(2018); Karthein  _et al._ (2019).

Extracting $\rho$ from Eq. (II.1) requires the ratio of statistical rate
functions, $f_{A}/f_{V}$ to be known. The values for $f_{A}/f_{V}$ have been
calculated earlier for the mirror $\beta$ decays from 3H to 45V Severijns  _et
al._ (2008). However, it was recently pointed out Hayen (2021); Hayen and
Severijns (2019c) that a double-counting instance occurs when extracting the
mixing ratio $\rho$ from experimental data and then using it in combination
with Eq. II.1. The values for $f_{A}/f_{V}$ listed in Table 6 have therefore
been calculated according to Ref. Hayen (2021) resulting in values that differ
from the previous ones (e.g. Severijns  _et al._ (2008)) by up to about 2.5%
for the mirror $\beta$ transitions in the $sd$-shell. The effect of this shift
on the extraction of the mirror $\beta$ transition-independent
$\mathcal{F}t_{0}$ value (see Severijns  _et al._ (2008)) to test CVC and
extract a value for the $V_{ud}$ quark-mixing matrix element is discussed in
Hayen (2021). A shift of the central value towards the much more precise value
obtained from the superallowed pure Fermi-transitions is observed, bringing
the values obtained for both types of transitions in agreement with each
other.

To date only a limited number of mixing ratios for mirror $\beta$ transitions
have been determined experimentally Calaprice  _et al._ (1975); Garnett  _et
al._ (1988); Severijns  _et al._ (1989); Masson and Quin (1990); Converse  _et
al._ (1993); Melconian  _et al._ (2007); Vetter  _et al._ (2008); Fenker  _et
al._ (2018); Brown  _et al._ (2018); Märkisch  _et al._ (2019); Combs  _et
al._ (2020). With more measurements being planned and experimental precision
further improving (a non-exhaustive list of ongoing and planned measurements
with the precision that is aimed at is given in Table 3 of Gonzalez-Alonso
_et al._ (2019)) the value for the $V_{ud}$ quark-mixing matrix element
extracted from the mirror $\beta$ transitions (including the neutron) provides
a complementary approach to $V_{ud}$ compared to the superallowed pure Fermi
transitions Hardy and Towner (2020).

####  II.3.2 Note on the mirror transitions with $A>$61

For the mirror isotopes with $A>61$ no detailed decay spectroscopy has been
performed yet, as is indicated in Table 5 by a $"-"$ for several isotopes in
the column listing the branching ratios, $BR$. For three isotopes, i.e. 67Se,
71Kr, and 75Sr, some information is available which is, however, likely
incomplete as these isotopes are for now still difficult to produce copiously
at radioactive beam facilities. For the mirror $\beta$ decays of these three
isotopes we therefore list in Table 5 the upper limits on the branching
ratios, $BR$, based on the currently available spectroscopic data as well as
estimated values for the lower limits. Note that for 71Kr the lower limit of
$BR=68.1$% is the minimum value that is required in order to avoid a negative
value for $\rho^{2}$ when solving Eq. (II.1).

Table 6: Calculated quantities and corrections needed to obtain the ${\cal F}t^{\mbox{mirror}}$ values (Eq. (II.1)). Details are given in the text. Parent | $f_{V}t$ | $f_{A}/f_{V}$ | $\delta_{R}^{\prime}$ | $\delta^{V}_{C}-\delta^{V}_{NS}$ | ${\cal F}t^{\mathrm{mirror}}$ | $\delta({\cal F}t^{\mathrm{mirror}})$ | $\rho=c$
---|---|---|---|---|---|---|---
nucleus | $(s)$ |  | $(\%)$ | $(\%)$ | $(s)$ | $\%$ | (Eq. 13)
3H | $1113.0$ | $\pm$ | $1.0$ | $1.00027$ | $1.767(1)$ | $0.16(2)$ | $1130.9$ | $\pm$ | $1.0$ | 0.09 | $+2.1053(12)$
11C | $3893.4$ | $\pm$ | $1.4$ | $0.99923$ | $1.660(4)$ | $1.04(3)$ | $3916.9$ | $\pm$ | $1.9$ | 0.05 | $-0.75442(28)$
13N | $4621.3$ | $\pm$ | $4.7$ | $0.99802$ | $1.635(6)$ | $0.33(3)$ | $4681.3$ | $\pm$ | $4.9$ | 0.11 | $-0.55962(34)$
15O | $4344.3$ | $\pm$ | $5.7$ | $0.99637$ | $1.555(8)$ | $0.22(3)$ | $4402.3$ | $\pm$ | $5.9$ | 0.13 | $+0.63023(46)$
17F | $2269.5$ | $\pm$ | $1.7$ | $1.00196$ | $1.587(10)$ | $0.62(3)$ | $2291.2$ | $\pm$ | $1.9$ | 0.08 | $+1.29555(65)$
19Ne | $1704.31$ | $\pm$ | $0.63$ | $1.00110$ | $1.533(12)$ | $0.52(4)$ | $1721.5$ | $\pm$ | $1.0$ | 0.06 | $-1.60203(65)$
21Na | $4028.8$ | $\pm$ | $3.5$ | $1.00198$ | $1.513(14)$ | $0.41(3)$ | $4073.0$ | $\pm$ | $3.8$ | 0.09 | $+0.71245(39)$
23Mg | $4651.9$ | $\pm$ | $7.3$ | $0.99940$ | $1.476(17)$ | $0.40(3)$ | $4701.6$ | $\pm$ | $7.6$ | 0.16 | $-0.55413(47)$
25Al | $3678.2$ | $\pm$ | $2.4$ | $1.00193$ | $1.475(20)$ | $0.52(5)$ | $3713.0$ | $\pm$ | $3.2$ | 0.08 | $+0.80844(42)$
27Si | $4095.1$ | $\pm$ | $1.9$ | $1.00024$ | $1.443(23)$ | $0.42(4)$ | $4136.7$ | $\pm$ | $2.7$ | 0.07 | $-0.69659(30)$
29P | $4747.0$ | $\pm$ | $7.2$ | $1.00077$ | $1.453(26)$ | $1.07(6)$ | $4764.5$ | $\pm$ | $7.9$ | 0.17 | $+0.53798(47)$
31S | $4770.3$ | $\pm$ | $4.7$ | $0.99919$ | $1.430(29)$ | $0.79(4)$ | $4800.3$ | $\pm$ | $5.3$ | 0.11 | $-0.52939(33)$
33Cl | $5570.4$ | $\pm$ | $8.6$ | $0.98952$ | $1.435(32)$ | $0.93(6)$ | $5597.8$ | $\pm$ | $9.5$ | 0.17 | $-0.31416(28)$
35Ar | $5645.0$ | $\pm$ | $4.9$ | $0.99293$ | $1.421(35)$ | $0.53(5)$ | $5694.8$ | $\pm$ | $6.0$ | 0.11 | $+0.28199(17)$
37K | $4582.5$ | $\pm$ | $4.4$ | $0.99550$ | $1.431(39)$ | $0.79(6)$ | $4611.4$ | $\pm$ | $5.5$ | 0.12 | $-0.57789(39)$
39Ca | $4264.0$ | $\pm$ | $4.5$ | $0.99551$ | $1.422(43)$ | $0.95(8)$ | $4283.5$ | $\pm$ | $6.0$ | 0.14 | $+0.66061(50)$
41Sc | $2833$ | $\pm$ | $10$ | $1.00193$ | $1.454(47)$ | $0.86(7)$ | $2849$ | $\pm$ | $11$ | 0.38 | $+1.0743(21)$
43Ti | $3688$ | $\pm$ | $63$ | $0.99547$ | $1.444(50)$ | $0.63(11)$ | $3718$ | $\pm$ | $64$ | 1.7 | $-0.8097(69)$
45V | $4354$ | $\pm$ | $79$ | $1.00418$ | $1.438(53)$ | $0.93(12)$ | $4375$ | $\pm$ | $80$ | 1.8 | $+0.6346(58)$
47Cr | $4568$ | $\pm$ | $65$ | $1.00325$ | $1.439(58)$ | $0.8(2)$ | $4596$ | $\pm$ | $66$ | 1.4 | $-0.5794(42)$
49Mn | $4739$ | $\pm$ | $132$ | $0.99908$ | $1.438(61)$ | $0.8(2)$ | $4769$ | $\pm$ | $133$ | 2.8 | $+0.5373(75)$
51Fe | $4568$ | $\pm$ | $77$ | $0.99700$ | $1.442(66)$ | $0.8(2)$ | $4597$ | $\pm$ | $78$ | 1.7 | $-0.5811(49)$
53Co | $4197$ | $\pm$ | $90$ | $1.00385$ | $1.443(70)$ | $0.8(2)$ | $4224$ | $\pm$ | $91$ | 2.1 | $+0.6730(72)$
55Ni | $4199$ | $\pm$ | $99$ | $0.99650$ | $1.433(73)$ | $0.8(2)$ | $4225$ | $\pm$ | $100$ | 2.4 | $-0.6752(80)$
57Cu | $4675$ | $\pm$ | $45$ | $0.99118$ | $1.455(79)$ | $1.5(3)$ | $4672$ | $\pm$ | $47$ | 1.0 | $+0.5639(28)$
59Zn | $4982$ | $\pm$ | $84$ | $0.98563$ | $1.440(81)$ | $1.5(3)$ | $4978$ | $\pm$ | $86$ | 1.7 | $-0.4876(42)$
61Ga | $4759$ | $\pm$ | $137$ | $0.99331$ | $1.461(87)$ | $1.5(3)$ | $4756$ | $\pm$ | $138$ | 2.9 | $+0.5421(79)$
67Se111Values for upper limit of BR; see also Table 5 and Sec. II.3.2. | $5344$ | $\pm$ | $245$ | $1.01842$ | $1.461(99)$ | $1.7(3)$ | $5330$ | $\pm$ | $245$ | 4.6 | $-0.3873(89)$
67Se222Values for lower limit of BR; see also Table 5 and Sec. II.3.2. | $5908$ | $\pm$ | $289$ |  |  |  | $5893$ | $\pm$ | $288$ | 4.9 | $-0.2048(50)$
71Kr111The calculated moments for sgn$(\gamma_{\beta})=-1$ are preferred because of the agreement for 75Rb with the experimental value $\mu=+0.6544680(16)$ for the 3/2- g.s. state of the neighboring 77Rb [Du86, Th81], and because the 3/2- g.s. states of 79Sr [Bu90] and 81Rb [Th81] with, respectively, $\mu=-0.474(4)$ and $\mu=+2.0595(14)$, are farther away in mass and also situated above the N = 40 sub-shell closure. | $5108$ | $\pm$ | $366$ | $0.99758$ | $1.474(109)$ | $1.7(3)$ | $5095$ | $\pm$ | $365$ | 7.2 | $+0.454(16)$
71Kr22footnotemark: 2 | $5991$ | $\pm$ | $432$ |  |  |  | $5976$ | $\pm$ | $432$ | 7.2 | $+0.1682(61)$
75Sr111The calculated moments for sgn$(\gamma_{\beta})=-1$ are preferred because of the agreement for 75Rb with the experimental value $\mu=+0.6544680(16)$ for the 3/2- g.s. state of the neighboring 77Rb [Du86, Th81], and because the 3/2- g.s. states of 79Sr [Bu90] and 81Rb [Th81] with, respectively, $\mu=-0.474(4)$ and $\mu=+2.0595(14)$, are farther away in mass and also situated above the N = 40 sub-shell closure. | $4879$ | $\pm$ | $590$ | $0.95210$ | $1.484(118)$ | $1.7(3)$ | $4867$ | $\pm$ | $588$ | 12 | $+0.525(32)$
75Sr22footnotemark: 2 | $5458$ | $\pm$ | $662$ |  |  |  | $5445$ | $\pm$ | $661$ | 12 | $+0.367(22)$
Table 7: Reference key relating the reference codes used in Tables LABEL:table:halflives_mirrors and 5 to the actual reference numbers. Table | Reference | Table | Reference | Table | Reference | Table | Reference | Table | Reference | Table | Reference
---|---|---|---|---|---|---|---|---|---|---|---
code | No. | code | No. | code | No. | code | No. | code | No. | code | No.
Ab11 | Abusaleem and Singh (2011) | Ac07 | Achouri and Naviliat-Cuncic (2017) | Ac10 | Achouri  _et al._ (2010) | Ad81 | Adelberger et al. (1981) | Ad83 | Adelberger et al. (1983) | Ad84 | Adelberger et al. (1984)
Aj70 | Ajzenberg-Selove (1970) | Aj75 | Ajzenberg-Selove (1975) | Aj82 | Ajzenberg-Selove (1982) | Ak04 | Akulov et al. (2004) | Ak88 | Akulov et al. (1988) | Al67 | Aldridge et al. (1967)
Al72 | Alburger and Wilkinson (1972) | Al73 | Alburger and Wilkinson (1973) | Al74 | Alburger (1974) | Al76 | Alburger (1976) | Al77 | Alburger (1977) | Ar58 | Arnell et al. (1958)
Ar84 | Arai et al. (1984) | Aw69 | Awschalom et al. (1969) | Ay84 | Äystö et al. (1984) | Az74 | Azuelos et al. (1974) | Az75 | Azuelos and Kitching (1975) | Az77 | Azuelos et al. (1977)
Ba12 | Bacquias et al. (2012) | Ba55 | Bashkin et al. (1955) | Ba70 | Bardin et al. (1970) | Ba77 | Barker et al. (1977) | Ba94 | Baumann et al. (1994) | Be71 | Behrens et al. (1971)
Be75 | Behrens et al. (1975) | Bl02 | Blank (2002) | Bl10 | Blank et al. (2010) | Bl68 | Black and Mahieux (1968) | Bl95 | Blank  _et al._ (1995) | Bl99 | Blank (1999)
Bo65 | Bormann et al. (1965) | Br14 | Broussard et al. (2014) | Br16 | Brodeur  _et al._ (2016b) | Bu07 | Burrows (2007) | Bu08 | Burrows (2008) | Bu20 | Burdette  _et al._ (2020)
Bu85 | Burrows et al. (1985) | Bu91 | Budick et al. (1991) | Ce16 | Čeliković  _et al._ (2016) | Cl58 | Cline and Chagnon (1958) | Da85 | Daehnick and Rosa (1985) | De71 | Détraz et al. (1971)
Di51 | Dickson and Randle (1951) | Ea62 | Earwaker et al. (1962) | Eb65 | Ebrey and Gray (1965) | Ed77 | Edmiston et al. (1977) | Ew81 | Ewan et al. (1981) | Fa02 | Faestermann  _et al._ (2002)
Fi05 | Fischer  _et al._ (2005) | Fi17 | Finlay  _et al._ (2017) | Fo17 | Fontbonne  _et al._ (2017) | Ge71 | Geiger and Hooton (1971) | Ge76 | Genz  | Go64 | Gorodetzky et al.
Go68 | Goss et al. (1968) | Go68a | Gorodetzky et al. (1968) | Gr15 | Grinyer et al. (2015a) | Gr15a | Grinyer et al. (2015b) | Gr71 | Grober and Gruhle (1971) | Ha52 | Haslam et al. (1952)
Ha79 | Hagberg  _et al._ (1979) | Ha80 | Hardy et al. (1980) | Ha87 | Hama  _et al._ (1987) | Ha94 | Hagberg et al. (1994) | Ha97 | Hagberg et al. (1997) | Ho77 | Hornshoj et al. (1977)
Ho81 | Honkanen et al. (1981) | Ho82 | Hornshoj et al. (1982) | Ho87 | Honkanen et al. (1987) | Ho89 | Honkanen et al. (1989) | Hu03 | Huikari et al. (2003) | Hu54 | Hunt et al. (1954)
Ia06 | Iacob et al. (2006) | Ja60 | Janecke (1960) | Ja61 | Janecke (1961) | Ja99 | Janas  _et al._ (1999) | Je50 | Jenks et al. (1950) | Jo51 | Jones et al. (1951)
Jo55 | Jones et al. (1955) | Jo67 | Jones et al. (1967) | Ju05 | Junde  _et al._ (2005) | Ju08 | Junde (2008) | Ju71 | Jundt et al. (1971) | Ka14 | Kankainen et al. (2014)
Ka64 | Kavanagh and Goosman (1964) | Ka64a | Kavanagh et al. (1964) | Ka68 | Kadlecek (1968) | Ki01 | Kienle et al. (2001) | Ki59 | Kistner and Rustad (1959) | Ki60 | King et al. (1961)
Kl54 | Kline and Zaffarano (1954) | Ko73 | Kochan et al. (1973) | Ku53 | Kundu et al. (1953) | Ku17 | Kurtukian et al. (2017) | Ku17a | Kucuk, L.  _et al._ (2017) | Li60 | Lindenberger and Scheer (1960)
Lo02 | Lopez et al. (2002) | Lo17 | Long  _et al._ (2017) | Lo20 | Long  _et al._ (2020) | Lo62 | Lonsjo et al. (1962) | Lu00 | Lucas and Unterweger (2000) | Ma06 | MacMahon (2006)
Ma17 | Magron et al. (2017) | Ma69 | Makela et al. (1969) | Ma74 | Mann and Kavanagh (1974) | Ma76 | Mann et al. (1976) | Me66 | Merritt and Taylor  | Mi58 | Mihailovic and Povh (1958)
Mo71 | Moss et al. (1971) | Mo95 | Morrissey (1995) | Mu58 | Muller et al. (1958) | Ne13 | Negret and Singh (2013) | Ne63 | Nelson et al. (1963) | No47 | Novick (1947)
Oi97 | Oinonen et al. (1997) | Oi99 | Oinonen  _et al._ (1999) | Ol87 | Oliver et al. (1987) | Pe57 | Penning and Schmidt (1957) | Pf96 | Pfaff  _et al._ (1996) | Pi85 | Piilonen
Pl62 | Plendl et al. | Po58 | Povov et al. (1958) | Pr57 | Prokoshkin and Tiapkin (1957) | Re19 | Rebeiro  _et al._ (2019) | Re99 | Reusen et al. (1999) | Ri68 | Ritchie (1968)
Ro14 | Rogers et al. (2014) | Ro55 | Roderick  _et al._ (1955) | Ru14 | Rubio  _et al._ (2014) | Ru77 | Rudy and Jordan  | Sa93 | Saettler et al. (1993) | Sc48 | Schelberg et al. (1948)
Sc58 | F. (1958) | Sc70 | Scanlon and Crabtree (1970) | Se17 | Severijns and Blank (2017) | Se96 | Semon et al. (1996) | Sh14 | Shidling et al. (2014) | Sh15 | Shen  _et al._ (2015)
Sh18 | Shidling  _et al._ (2018) | Sh84 | Shinozuka  _et al._ (1984) | Sh89 | Shinozuka et al. | Sh93 | Sherrill et al. | Si19 | Sinclair  _et al._ (2019) | Si87 | Simpson (1987)
Sm41 | Smith and Cowie (1941) | Su13 | Su et al. (2013) | Su13a | Suzuki  _et al._ (2013) | Su17 | Suzuki  _et al._ (2017a) | Su17a | Suzuki  _et al._ (2017b) | Su62 | Sutton (1962)
Ta60 | Talbert and Stewart (1960) | Ta73 | Tanihata et al. (1973) | Ti87 | Tilleya et al. (1987) | Tr12 | Triambak et al. (2012) | Uj13 | Ujic et al. (2013) | Un00 | Unterweger and Lucas (2000)
Va18 | Valverde  _et al._ (2018) | Va63 | Vasilev and Shavtvalov (1963) | Va69 | Vasilev et al. (1969) | Wa60 | Wallace and J. A. Welsch (1960) | We02 | Weissmann et al. (2002) | Wi69 | Wick et al.
Wi74 | Wilkinson and Alburger (1974) | Wi80 | Wilson et al. (1980) | Wi93 | Winger et al. (1993) | Wo02 | Woods et al. (2002) | Wo69 | Wohlleben and Schuster (1969) | Yo65 | Wick et al. (1965)
Zu15 | Zuber and Singh (2015) |  |  |  |  |  |  |  |  |  |

##  III Weak Magnetism

The second part of this work focuses on the weak magnetism correction to the
$\beta$ spectrum shape (and corresponding $\mathcal{F}t$ values) and on
$\beta$-correlation measurements. Its importance has been hinted at throughout
the previous sections, and cannot be understated as experimental precision
reaches the sub-percent level. Its correct evaluation is an important factor
in the extraction of new physics as well as in the extraction of $V_{ud}$ from
precise $\beta$-correlation measurements, such that a thorough discussion of
its origin and properties is worthwhile as the field continues and grows.

We give here a fairly in-depth presentation of the theoretical origin and
state, followed by a discussion of experimental behaviour in mirror and other
isospin multiplet $\beta$ decays. In both cases we compare against results of
the nuclear shell model, assisted by simple single-particle results to aid in
its interpretation. Based on these results, we discuss the ability of nuclear
theory to correctly predict this quantity and the prospects in experimentally
unexplored territories.

###  III.1 Theoretical foundations

Throughout the years the effects of weak magnetism have been studied by
various groups of authors, using different formalisms (e.g. Holstein (1974a);
Calaprice and Holstein (1976); Behrens  _et al._ (1978); Behrens and Bühring
(1982)). Its precise evaluation comes with several subtleties and a number of
caveats. It is of interest then, to lay out the theoretical foundations
underlying the evaluation. To ease the interpretation of the reader, we will
present all final results in the formalism by Holstein Holstein (1974a),
although some intermediate results have been derived from other works.

We will first discuss the generalization of the Hamiltonian to include induced
currents and the electromagnetic interaction. We follow with the treatment of
the conserved vector current hypothesis and how it connects these two
components. Finally, we discuss the evaluation of the matrix elements that
appear when comparing to theoretical results.

####  III.1.1 Generalized Hamiltonian

#####  III.1.1.1 Generalized nuclear $\beta$ decay Hamiltonian

In the original Fermi approach, the $\beta$ decay Hamiltonian is constructed
as a simple current-current interaction, analogous to the electromagnetic
interaction

$H_{\beta}(x)=\frac{G_{F}\cos\theta_{C}}{\sqrt{2}}[H_{\mu}(x)L^{\mu}(x)+\text{h.c.}]\leavevmode\nobreak\
,$ (15)

where $G_{F}$ is the Fermi coupling constant obtained from muon decay,
$\theta_{C}$ is the Cabibbo angle and $H_{\mu}(x)$ and $L^{\mu}(x)$ are the
hadron and lepton currents, respectively. The Standard Model expressions for
the latter is defined as

$L^{\mu}=\bar{u}_{e}\gamma^{\mu}(1-\gamma^{5})v_{\nu}\leavevmode\nobreak\ ,$
(16)

with $u,v$ the lepton wave functions and $\gamma_{i}$ the Dirac $\gamma$
matrices, which couples only to left-handed particles. In the absence of any
other forces, all relevant particle states are simple plane waves. The nuclear
medium is, however, hardly a place devoid of additional forces. This is mainly
due to the strong interaction, which renormalizes the weak vertex. Further,
the presence of an electrostatic potential (i.e. QED) forces a slew of changes
compared to the simple plane wave picture. Since $\beta$ decay occurs only at
low momentum transfer, however, most of the SM intricacies are not
kinematically visible and instead serve to renormalize the coupling constants.
The renormalization of the weak vertex due to QCD effects results in two
changes. The first is a modification of the axial vector operator, $\langle
n|\bar{u}\gamma^{\mu}\gamma^{5}d|p\rangle$, which can, e.g., be calculated
using first principles on the lattice Chang  _et al._ (2018); Gupta  _et al._
(2018). The vector part, $\langle n|\bar{u}\gamma^{\mu}d|p\rangle$, is
protected through the Conserved Vector Current (CVC) hypothesis discussed in
the following section and remains unchanged, even to higher order Ademollo and
Gatto (1964). While in the Standard Model the simple $V$-$A$ behaviour is a
direct consequence of the electroweak interaction, the nuclear medium allows
for some modification. Additional operators which still adhere to the required
$V$-$A$ behaviour can be constructed through a combination with the nuclear
momentum transfer $q=p_{i}-p_{f}$. In preparation we write the hadronic
current of Eq. (15) more generally

$H(x)=\langle f|V_{\mu}(x)+A_{\mu}(x)|i\rangle\leavevmode\nobreak\ .$ (17)

The approach spearheaded by Holstein Holstein (1974a) and contemporaries
Behrens and Bühring (1982); Armstrong and Kim (1972) consists of treating the
nucleus as an elementary particle, and says nothing about its constituents or
internal turmoil. This allows one to encode all behaviour through form factors
that depend only on $q^{2}$ as mentioned above. For the simple neutron this
becomes

$\displaystyle V_{\mu}(x)$
$\displaystyle=[g_{V}\gamma_{\mu}-\frac{g_{M}-g_{V}}{2M}\sigma_{\mu\nu}q^{\nu}+i\frac{g_{S}}{2M}q_{\mu}]\tau^{+}\leavevmode\nobreak\
,$ (18) $\displaystyle A_{\mu}(x)$
$\displaystyle=[g_{A}\gamma_{5}\gamma_{\mu}-\frac{g_{II}}{2M}\sigma_{\mu\nu}\gamma_{5}q^{\nu}+\frac{g_{P}}{2M}\gamma_{5}q_{\mu}]\tau^{+}\leavevmode\nobreak\
,$ (19)

where all $g_{i}$ form factors are a dimensionless function of $q^{2}$, and
$\tau^{+}$ is the isospin ladder operator. The appearance of factors
$\mathcal{O}(q/M)$ traditionally lends them the name ”recoil” corrections. In
order of appearance, these are called the vector (Fermi), weak magnetism,
induced scalar, axial vector (Gamow-Teller), induced tensor and induced
pseudoscalar terms.

The presence of the electromagnetic field now has two additional consequences
for nuclear $\beta$ decay. The first entails a renormalization of the decay
vertex, as we have done before with the strong interaction in Eqs. (18)-(19).
Up to $\mathcal{O}(\alpha)$, with $\alpha$ the fine-structure constant, these
simply renormalize the leading order terms (i.e. $\Delta_{R}^{V,A}$ as in Eqs.
(II.1) and (8)) and do not introduce additional structure. The experimental
definition of the mixing ratio (and $g_{A}$ in the case of the neutron) simply
absorbs the difference of vector and axial vector inner radiative corrections,
in addition to the other small corrections in Eq. (8). The second consequence
concerns the operators in Eqs. (18)-(19), specifically those coupling to the
momentum $q_{\mu}$. In the traditional quantum mechanics calculation, turning
on the electromagnetic field requires the standard substitution
$\partial_{\mu}\to\partial_{\mu}-ieA_{\mu}$, where $A_{\mu}=(i\phi,\bm{A})$ is
the electromagnetic four vector. It is in part this effect which in the
traditional $\beta$ decay calculations Behrens and Bühring (1982) causes a
contribution to what is now understood to be part of the inner radiative
correction Hayen (2021). We will come back to this below.

We have, up to now, only discussed the generalization to a spin-$1/2$ system,
leading to Eqs. (18)-(19). For general allowed $\beta$ transitions where
$\Delta J=0,1$, several additional terms appear. Using the notation of
Holstein Holstein (1974b), the form factors are then written as $a$ (vector),
$b$ (weak magnetism), $c$ (axial vector), $d$ (induced tensor), $e$ (induced
scalar), $h$ (induced pseudoscalar), $f$, $g$, $j_{2}$ and $j_{3}$. Full
formulae can be found in several publications Holstein (1974a); Calaprice and
Holstein (1976). Once again, all form factors are a function of $q^{2}$.
Typically, however, only the dominant form factors are expanded as

$\displaystyle a(q^{2})$ $\displaystyle\approx
a_{1}+a_{2}q^{2}\leavevmode\nobreak\ ,$ (20a) $\displaystyle c(q^{2})$
$\displaystyle\approx c_{1}+c_{2}q^{2}\leavevmode\nobreak\ ,$ (20b)

while all others are approximated as their value for $q^{2}=0$. This is
possible because the momentum transfer in $\beta$ decay is sufficiently small
such that $qR\ll 1$, where $R$ is the nuclear radius.

#####  III.1.1.2 Nuclear electromagnetic interaction

The influence of the electromagnetic field on the weak vertex was summarized
in the previous section. As a charged particle, the nucleus also directly
couples to the electromagnetic field via

$\mathcal{L}(x)=ie\mathcal{J}_{\mu}A^{\mu}(x)\leavevmode\nobreak\ ,$ (21)

where $\mathcal{J_{\mu}}$ contains both nucleon, $J_{\mu}$, and electron,
$l_{\mu}$, parts. As before, we can treat the interaction of the latter in the
elementary particle approach and write

$\langle f|J_{\mu}(0)|i\rangle=i\langle
f|F_{1}\gamma_{\mu}+F_{2}\sigma_{\mu\nu}q^{\nu}|i\rangle\leavevmode\nobreak\
,$ (22)

where again all $F_{i}$ are a function of $q^{2}$. The first of these
represents the charge, while the second corresponds to the magnetic dipole
interaction. These can be split up into an isoscalar and isovector part,
writing $F_{i}=F_{i}^{(S)}+F_{i}^{(V)}\tau_{3}$. The absence of higher order
isospin operators in the electromagnetic interaction has obvious consequences
for selection rules between nuclear levels Weinberg (1958); Morpurgo (1959),
and allows one to write down the mass formula within isospin multiplets Blin-
Stoyle (1973)

$M(A,T,T_{3})=a(A,T)-b(A,T)T_{3}+c(A,T)T_{3}^{2}$ (23)

to only second order in $T_{3}$, where $A$ is the mass number, $T$ the isospin
and $T_{3}=\frac{1}{2}(Z-N)$. Here the parameters $a,b$ and $c$ can be found
in several text books Wilkinson (1969); Blin-Stoyle (1973) and in a recent
evaluation of the relevant experimental data MacCormick and Audi (2014), and
depend mainly on the different isomoments of the Coulomb displacement energy
and the difference in proton and neutron mass.

####  III.1.2 Conserved Vector Current

Several years before the Weinberg-Salam model of electroweak interactions was
proposed, a more intimate connection between electromagnetic and weak
interactions was put forward by Feynman and Gell-Mann Feynman and Gell-Mann
(1958) and Sudarshan and Marshak Sudarshan and Marshak (1958). Called the
Conserved Vector Current hypothesis, the name only covers the ‘weak’ part of
the conjecture, namely that

$\partial^{\mu}V_{\mu}=0\leavevmode\nobreak\ ,$ (24)

analogous to the classical continuity equation. Equation (24) strictly only
holds in the absence of electromagnetism, as the inclusion of the latter
requires the substitution $\partial^{\mu}\to\partial^{\mu}-ieA^{\mu}$ as
before. Application of the weak CVC principle forces $g_{V}(q^{2}=0)=1$ and
$g_{S}(q^{2}=0)=0$, and allows one to express the Fermi matrix element as
Behrens and Bühring (1982)

$\displaystyle\langle$ $\displaystyle
J_{f}M_{f}T_{f}T_{3f}|V_{0}(0)|J_{i}M_{i}T_{i}T_{3i}\rangle=$
$\displaystyle\sqrt{(T_{i}\pm T_{3i})(T_{i}\mp
T_{3i}+1)}\delta_{J_{i}J_{f}}\delta_{M_{i}M_{f}}\leavevmode\nobreak\ .$ (25)

Evaluating this for $0^{+}\to 0^{+}$ superallowed decays, one finds $\sqrt{2}$
and this is implicitly present in the factor $2$ in Eq. (II.1) Behrens and
Bühring (1982). The strong CVC principle states that the vector current from
$\beta$ decay forms an isospin triplet with the electromagnetic current Gell-
Mann (1958)

$V_{\mu}=\mp\left[\tau^{\pm},J_{\mu}^{\text{em}}\right]\leavevmode\nobreak\ ,$
(26)

where $\tau^{\pm}$ is the isospin ladder operator. Clearly the weak principle
directly follows from Eq. (26), and further allows one to relate weak matrix
elements to their electromagnetic analogues. This further implies that Eqs.
(18) and (22) can be directly compared. We consider here two cases.

The first deals with $\beta$ decays within the same isospin multiplet

$\langle TT_{3i}\pm 1|V_{\mu}^{\pm}|TT_{3i}\rangle\leavevmode\nobreak\ ,$ (27)

for which we then find191919The CVC hypothesis relates only the isovector part
of the electromagnetic interaction to the corresponding weak vector decay,
while the magnetic moment has both an isoscalar and isovector part. When
taking the difference within an isospin multiplet, the isoscalar component
obviously drops out.

$\displaystyle b(0)$ $\displaystyle=\pm
a(0)A\sqrt{\frac{J+1}{J}}(\mu_{f}-\mu_{i})\leavevmode\nobreak\ ,$ (28)

where the upper/lower sign is for $\beta^{-}$/$\beta^{+}$ decay, respectively,
$A$ is mass number, and $\mu_{i,f}$ are the nuclear dipole moments of
respectively the mother and daughter nuclear states. Evaluating this for the
neutron we find

$\frac{1}{\sqrt{3}}b(0)=g_{M}=\mu_{p}-\mu_{n}=4.706\leavevmode\nobreak\ ,$
(29)

where we used $a(0)=1$ for the neutron. The natural extension of neutron decay
from an isospin point of view are the mirror $\beta$ transitions, where
$T=1/2,T_{3}=\pm 1/2$ for initial and final states. When experimental nuclear
moment data is available, Eq. (28) allows for an unambiguous determination of
the weak magnetism contribution. For the higher multiplets, with $T=$1 and
higher, analog decays are typically energetically forbidden or connecting
unbound state due to the behaviour of Eq. (23), so that Eq. 28 cannot be used
anymore.

The second case is that of a decay where $\Delta T=1,\Delta T_{3}=\pm 1$. The
strong CVC conjecture then allows one to relate the weak magnetism form factor
to the isovector part of the decay width of the corresponding analog $M1$
$\gamma$ transition, $\Gamma_{M1}^{\text{iso}}$. In this case, one
finds202020Note that CVC only relates the square of $b(0)$, such that a sign
ambiguity remains.

$b^{2}(0)=\eta\frac{\Gamma_{M1}^{\text{iso}}6M^{2}}{\alpha
E_{\gamma}^{3}}\leavevmode\nobreak\ ,$ (30)

with $\alpha$ the fine structure constant and $\eta$ a constant. The factor
$\eta$ is unity in case the final state is the same for $\beta$ and $\gamma$
analog transitions, while it is $\eta=(2J_{i}+1)/(2J_{f}+1)$ when initial and
final states of the $\gamma$ transition (resp. $J_{i}$ and $J_{f}$) are
reversed relative to the $\beta$ decay, to compensate for the degeneracy in
the $\gamma$ transition phase space.

It is of interest to note that for $\beta$ transitions between members of a
common isotopic multiplet (like the mirror $\beta$ transitions) CVC requires
the form factors $e(q^{2})=f(q^{2})=0$, while the $d$ form factor can only be
non-zero if second-class currents would exist Holstein (1974a, b). In
addition, as many of the transitions that are considered here involve low-spin
states, the triangle inequality satisfied by Clesh-Gordan coefficients
requires the vanishing of one or more form factors, i.e. one has (with $J$ and
$J^{\prime}$ being the spin of the initial and final states of the $\beta$
transition, respectively):

$\displaystyle J=1,J^{\prime}=0:\leavevmode\nobreak\ \leavevmode\nobreak\
a=e=f=g=j_{2}=j_{3}=0$ $\displaystyle J=0,J^{\prime}=1:\leavevmode\nobreak\
\leavevmode\nobreak\ a=e=f=g=j_{2}=j_{3}=0$ $\displaystyle
J=J^{\prime}=1:\leavevmode\nobreak\ \leavevmode\nobreak\ j_{3}=0$
$\displaystyle J=1/2,J^{\prime}=1/2:\leavevmode\nobreak\ \leavevmode\nobreak\
f=g=j_{2}=j_{3}=0$ $\displaystyle J=1/2,J^{\prime}=3/2:\leavevmode\nobreak\
\leavevmode\nobreak\ a=e=j_{3}=0$ $\displaystyle
J=3/2,J^{\prime}=1/2:\leavevmode\nobreak\ \leavevmode\nobreak\
a=e=j_{3}=0\leavevmode\nobreak\ .$ (31)

Finally, the effect of the higher-order form factors $a_{2}$, $c_{2}$, and the
induced pseudoscalar form factor, $h$, that were not yet mentioned here, but
which appear in the $\beta$-spectrum shape (cf. Eq. (B7) in Ref. Holstein
(1974a)) and so determine the $ft$ values of a transition, are all reduced by
a factor $M$ (the average mass of the initial and final nuclei) with respect
to the $a$, $c$, $b$ and $d$ form factors. They can therefore be neglected at
the level of precision of the $ft$ values we are dealing with in this paper,
which is of the order of per mil to several percent.

####  III.1.3 Form factor evaluation

The weak magnetism correction shows up in a specific combination in the Gamow-
Teller $\beta$ spectrum as

$\delta_{\text{WM}}=\frac{b}{Ac_{1}}\leavevmode\nobreak\ ,$ (32)

where $c_{1}$ is the first component of the Gamow-Teller form factor as
defined in Eq. (20b). Considering we have generalized the nuclear current
using the elementary particle approach, and related the form factors to
electromagnetic observables for specific cases, one could consider the work to
be done and leave the determination of these form factors to experiment.
Often, however, application of CVC is impossible and one would like input from
nuclear models. We consider here the impulse approximation and discuss the
quenching of the axial vector coupling constant.

#####  III.1.3.1 Impulse Approximation

The easiest way through which we can accomplish such a feat is by using the
impulse approximation, where we consider the nuclear current a sum of
independent single particle nucleon currents. In doing this, one neglects
meson exchange, off-shell mass effects and other many-body effects. This is
typically a reasonable assumption for allowed transitions. Indeed,
discrepancies resulting from using the impulse approximation as compared to
the elementary particle treatment have been discussed in the light of CVC and
PCAC, and were found to be in good agreement for allowed transitions Armstrong
and Kim (1972); Hwang (1980).

The usual approach is performed by generalizing Eqs. (18)-(19) to a system of
$A$ nucleons, and performing a Foldy-Wouthuysen transformation to yield the
non-relativistic limit Foldy and Wouthuysen (1950). Comparing terms with that
of the elementary particle approach, one can write the corresponding reduction
of the form factors to proper nuclear matrix elements. A summary of the
relevant form factors is given in Table 8, with the required nuclear matrix
elements being listed in Table 9. For clarity, these are only given to first
order. The correction term of Eq. (32) then becomes

$\frac{b}{Ac_{1}}=\frac{1}{g_{A}}\left(g_{M}+g_{V}\frac{M_{L}}{M_{GT}}\right)\leavevmode\nobreak\
,$ (33)

such that we are left with the evaluation of two matrix elements. Note that
the value of $b/Ac$ measures the relative strength of $M_{L}$ and $M_{GT}$.
For a pure-spin transition, with $M_{L}\ll M_{GT}$ and $g_{A}=$ 1.00 (see Sec.
III.1.3.2), one obtains $b/Ac=$ 4.7. As $g_{V}=1$, values of $b/Ac>$ 5.7 thus
indicate a dominance of the orbital contribution. Finally, at the level of
precision we will be dealing with in the evaluation of $b/Ac$ later in this
paper, we can neglect the $q^{2}$ dependence of the $c$ form factor in Eq.
(20b) and so replace $c_{1}$ by $c$, i.e. the total Gamow-Teller form factor
as it is present in the $\mathcal{F}t^{\mathrm{mirror}}$ value (Eqs. (8 and
II.1)).

Table 8: Summary of the $a$, $b$, $c$, and $d$ form factors and their relation to the nuclear matrix elements defined in Table 9. Here, the impulse approximation is given to first order only, and the relativistic matrix elements are neglected, as is also done in Holstein (1974a). Further, $M=1/2(M_{i}+M_{f})$, with $M_{i}$ and $M_{f}$ being the masses of the initial and final states of the decay, is the nuclear mass, and $\Delta=M_{i}-M_{f}$ is the nuclear energy release. Important to note is that the induced tensor form factor, $d$, contains both a first-class and second-class contribution. The former can be shown to disappear for decays between analog states Holstein (1974a), while no second-class currents have been observed till now Grenacs (1985); Wilkinson (2000); Minamisono  _et al._ (2011). Form factor | Formula Imp. App. | Remark
---|---|---
a | $a\cong g_{V}M_{F}$ |  $g_{V}=1$ (CVC)
$b$ | $b\cong A(g_{M}M_{GT}+g_{V}M_{L})$ | $g_{M}=4.706$
$c$ | $c\cong g_{A}M_{GT}$ |  $g_{A}\rightarrow g_{A,eff}=1$ I.S. and Towner (1987)
$d$ | $d\cong A(g_{A}M_{\sigma L}\pm g_{II}M_{GT})$ |  $g_{II}\sim g_{T}\cong 0$212121This assumes the absence of second-class currents. Holstein (1974a)
$f$ | $f\cong(2/3)^{1/2}M\Delta g_{V}M_{Q}(\hbar c)^{2}$ | $M=1/2(M_{i}+M_{f})=\rm{nuclear\leavevmode\nobreak\ mass}$
$g$ | $g\cong(-4/3)M^{2}g_{V}M_{Q}(\hbar c)^{2}$ | $\Delta=M_{i}-M_{f}=\rm{nuclear\leavevmode\nobreak\ energy\leavevmode\nobreak\ release}$
$h$ | $h\cong(-2/\sqrt{(}10))M^{2}g_{A}M_{1y}(\hbar c)^{2}-A^{2}g_{P}M_{GT}$ | $g_{P}=-181.03$
$j_{k}$ | $j_{k}\cong(-2/3)M^{2}g_{A}M_{ky}(\hbar c)^{2}$ |
Table 9: Definitions of the reduced nuclear matrix elements relevant for the $a$, $b$, $c$, and $d$ form factors, from Ref. Calaprice  _et al._ (1977). Matrix element | Operator form
---|---
$M_{F}$ | $\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}\|\psi_{i}\rangle$
$M_{GT}$ | $\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}\overrightarrow{\sigma}_{i}\|\psi_{i}\rangle$
$M_{L}$ | $\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}\overrightarrow{l}_{i}\|\psi_{i}\rangle$
$M_{\sigma L}$ | $\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}i\overrightarrow{\sigma}_{i}\times\overrightarrow{l}_{i}\|\psi_{i}\rangle$
$M_{Q}$ | $(4\pi/5)^{1/2}\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}r^{2}_{i}Y_{2}(r_{i})\|\psi_{i}\rangle$
$M_{ky}$ | $(16\pi/5)^{1/2}\langle\psi_{f}\|\Sigma\tau^{\pm}_{i}\sigma^{2}_{i}C^{nn^{\prime}k}_{12k}Y^{n^{\prime}}_{2}(r_{i})\sigma_{in}\|\psi_{i}\rangle$

One can understand Eq. (30) more intuitively by writing down the M1 matrix
element in the long wavelength approximation, in which case it becomes
proportional to

$\langle
f|\sum_{i}^{A}\bm{l}_{i}\left(\frac{1+\tau_{3}^{(i)}}{2}\right)+\frac{\bm{\sigma}^{(i)}}{2}(\mu_{V}^{+}+\mu_{V}^{-}\tau_{3}^{(i)})|i\rangle\leavevmode\nobreak\
,$ (34)

where $\mu_{V}^{\pm}=(\mu_{p}\pm\mu_{n})$, with $\mu_{p}$ and $\mu_{n}$ the
proton and neutron magnetic moments. Using the experimental values for these,
it is clear that the isovector contribution is much more dominant than its
isoscalar counterpart, leading to several quasi-selection rules MacDonald
(1955); Gell-Mann and Telegdi (1953); Morpurgo (1958).

The use of the above equations depends on several approximations, with
increasing importance:

  1. 1.

Long wavelength: The expression for the $M1$ operator was derived for $kR\ll
1$, with $k$ the wave number of the emitted $\gamma$ ray. This is most
certainly valid for the low energies coming from nuclear transitions, and is
relevant only for energies upwards of 200 MeV.

  2. 2.

Nuclear currents: The orbital (convection) part of the nuclear current that
interacts with the vector potential is typically constructed using
$\bm{v}=\bm{p}/M$, whereas more precisely it should be
$\bm{v}=i[H,\bm{r}]/\hbar$ where $H$ is the full Hamiltonian, including spin-
orbit terms and exchange terms.

  3. 3.

Deformation: The comparison of weak magnetism and the $M1$ matrix elements
assumes the isobaric analog states are identical in all quantum numbers except
for $T_{3}$. When introducing deformation, however, this picture can quickly
turn around as often the analog state sits at high excitation energies and
differing deformation. Furthermore, states can be parts of different
rotational or vibrational bands, resulting in additional quenching of
transition amplitudes.

  4. 4.

Isospin purity: Typically we consider transitions for which $\Delta T=\pm 1$,
meaning the isovector part is automatically the only contributing factor.
This, however, assumes perfect isospin symmetry which is violated in practice
so that isoscalar contributions can sneak in.

Nevertheless, due to the smallness of the violation and the numerical weight
of the isovector moment, this is typically negligible Blin-Stoyle (1973).
However, in cases where the Gamow-Teller matrix element is strongly suppressed
the dominant contribution can arise from the angular momentum part in Eq. (34)
instead. The biggest culprit for isospin mixing is the isovector part of the
Coulomb interaction, resulting in an isospin impurity proportional to
$\sum_{j}[\langle f|V_{C}|j\rangle/(E_{f}-E_{j})]^{2}$ Blin-Stoyle (1973). Due
to a variety of mechanisms, the isospin purity remains rather low even for
higher $Z$ nuclei, though individual variation can change significantly Blin-
Stoyle (1973).

  5. 5.

Meson exchange: An important distinction between Eqs. (30) and (34) is the
treatment of meson exchange currents. The strong CVC principle (Eq. (26))
relates weak interaction and electromagnetic form factors, including all meson
exchange effects, induced currents, et cetera. While Eq. (30) is always
correct, the evaluation through Eq. (34) is not and requires a remedy. In
usual nuclear physics fashion this can be performed by introducing effective
charges to both proton and neutron De-Shalit and Feshbach (1990). Recently
much work was performed using ab initio methods Parzuchowski  _et al._ (2017).

It is clear that many subtleties can underlie the proper evaluation of weak
magnetism. Often, though, all of these approximations are sufficiently well-
behaved. In case of discrepancies, a breakdown of one or more of these
approximations can be investigated.

#####  III.1.3.2 Evaluation of $g_{A}$

In the neutron system, the value of $g_{A}/g_{V}$ is experimentally found to
be 1.2756(13) Particle Data Group, Zyla, P.A. et al. (2020). In more complex
systems, however, typically lower (quenched) values of $g_{A}$ are required to
reach agreement with experimental data when employing shell model or mean
field theories Wilkinson (1973a, b, 1974). The reason for this quenching lies
in a failure to take into account the required degrees of freedom inside the
nuclear medium meaning truncated valence spaces, three-body interactions and
nucleon-nucleon correlations. The quenching is then an attempt at a
parametrisation of theoretical deficiencies of commonly used theories. It
speaks for itself then that no quenching is unique, and is inherently coupled
to the details of the underlying theory with its effective interactions and
basis states. Intensive research was performed in the second half of the
twentieth century Ericson  _et al._ (1973); Delorme  _et al._ (1976); Towner
and Khanna (1979), and has recently become very actual again due to its
relevance in neutrinoless double $\beta$ decay searches Kostensalo  _et al._
(2017); Deppisch and Suhonen (2016); Suhonen (2017a, b). Some explanation is
required then, as we consider here both shell model and deformed single
particle results.

Two effects are at play here. The first is traditionally called core
polarization Fujita  _et al._ (1964); Koshigiri  _et al._ (1981). One usually
considers as starting point a single-particle Hamiltonian as $H_{0}=T+U$,
where $U$ is an average single particle potential. The true solutions of the
system, however, are solutions of the true Hamiltonian, $H=H_{0}+\mathcal{V}$,
where $\mathcal{V}=V-U$ is additional influence from the true nucleon-nucleon
potential. Further, solutions are constructed using single particle orbitals
generated from only a subspace of finite dimension. The correction required
using this approach is typically called core polarization, and represents the
lack of multi-particle correlations in the individual wave functions. This
includes residual interactions between valence particles, core deformation,
$\Delta(1236)$-isobar excitations and relativistic effects Towner and Khanna
(1979).

The second effect is that of meson exchange Barroso and Blin-Stoyle (1975);
Rho (1978); I.S. and Towner (1987). Typically, the proper nucleon-nucleon
interaction can be reduced in an effective manner to correspond to the
exchange of several mesons, principally the pion and heavier $\rho$ and
$\omega$ excitations. Combined with possible $\Delta$-isobars, this leads to
an impressive number of required calculations. A modern example can be found
in Ref. Siiskonen  _et al._ (2001).

The choice for a quenching factor then depends on how well these effects are
present in the models we employ Suhonen (2017b). In the case of the shell
model, one often uses $g_{A}=1.0$ (e.g. I.S. and Towner (1987); Siiskonen  _et
al._ (2001)) to counter model-space truncations and meson exchange effects.
The evaluation of the single particle results is slightly less
straightforward. For low mass nuclei and transitions close to closed shells,
meson exchange will be the dominating factor. Going to higher masses and more
exotic transitions, deformation effects will start to play a role. This can be
mitigated using a deformed potential (see Appendix). In mid-shell nuclei
proton-neutron residual interactions can become non-trivial and give dominant
contributions. It is clear and hardly unexpected that there is no easy
solution. As a way of making do, one often uses the $g_{A}$ quenching factors
extracted from early shell model calculations, i.e. those not containing
extensive core polarization and meson exchange corrections, i.e. $g_{A}=1.1$
for the $sd$ shell Brown and Wildenthal (1988), $g_{A}=1.0$ for the $pf$ shell
and onwards Martinez-Pinedo  _et al._ (1996) (see Suhonen (2017b) for a
comprehensive overview and discussion of the renormalization of $g_{A}$).

As here experimental results for the $b$ and $c$ form factors and the Gamow-
Teller and orbital current matrix elements, resp. $M_{GT}$ and $M_{L}$, will
be compared to shell model calculations that used $g_{A}=1$ (besides
$g_{V}=1$, $g_{M}=4.706$, $g_{P}=-181.03$, and $g_{S}=g_{T}=0$), we will also
use $g_{A}=1$ in dealing with the experimental quantities. Only for the the
decay of tritium ($A=$ 3), where there is very little core polarization, we
will use the free-nucleon value $g_{A}=1.27$.

###  III.2 Weak magnetism for the mirror $\beta$ transitions

For the superallowed $\beta$ transitions between $T=1/2$ isospin doublets in
mirror nuclei (${}^{A}_{Z+1}X_{N}\rightarrow^{A}_{Z}X^{\prime}_{N+1}$) the
vector form factor is well known, i.e. $a=g_{V}M_{F}=1$ (Eq. 25), while the
weak magnetism form factor, $b$, and the axial-vector form factor, $c$, can be
determined experimentally using the CVC hypothesis (Eq. 28) and the
experimental $ft$ value (Eq. II.1), respectively. Form factors $d$, $e$, $f$
and $j_{2}$ are set to zero as it was shown that the first-class contributions
to these form factors that arise in the impulse approximation vanish for
transitions between states of a common isospin multiplet Holstein (1974a).
Here we update the work of Ref. Calaprice and Holstein (1976), extending the
range of mirror transitions for which experimental data on the weak magnetism
term are available up to mass $A$ = 75.

####  III.2.1 Magnetic moments of the $T=1/2$ mirror nuclei

The experimental data required to calculate the weak magnetism form factor,
$b$, from experimental data, i.e. the nuclear magnetic moments of the mother
and the daughter isotopes (Eq. 28), are reviewed here.

Table LABEL:Magnetic-moments lists the available experimental values for the
magnetic moments of the mirror nuclei up to $A=$ 75, as published in the
recent survey of Ref. Stone (2014); Mertzimekis  _et al._ (2016) and updated
up to April 2021 via the database of the Brookhaven National Nuclear Data
Center ENSDF . The publications containing the respective input data are again
indicated with an alphanumeric code, with the reference key linking these
codes with the actual reference numbers given in Table 13.

The same data-selection procedure that was used for the data leading to the
$\mathcal{F}t$ values is used here for the analysis of the magnetic moments.
Thus, data with an uncertainty that is a factor 10 or more larger than that of
the most precise measurement, as well as values with other problems have again
been rejected. These are listed and commented in Table 11. The finally adopted
values for the magnetic moments that were used in the further analysis are
listed in column 4 of Table LABEL:Magnetic-moments. When two or more input
values are available and the error bars are of similar size, the weighted
average is used with the uncertainty being increased by a factor
$S=\sqrt{\chi^{2}/\nu}$ (Eq. (14)) if $\chi^{2}/\nu>1$. This scale factor is
then listed in column 5. When two, often very precise measurements, differ by
many standard deviations with no clear argument in favour of either one, the
unweighted average is used as the adopted value. This is then indicated in the
table.

For the heavier mirror nuclei, i.e. in the $fp$ shell, experimental magnetic
moment values are often not available. For these nuclei, the magnetic moments
were calculated from the strong linear correlations between the mirror pair
magnetic moments and their $ft$ values obtained in Ref. Perez  _et al._
(2008). For this, we used shell-model estimates for the $\it small$ quantities
$S_{e}$ and $J_{e}$, which represent the contributions from the type of
nucleon that is even in number to the $z$ components of the total spin $S$ and
total angular momentum $J$ of the mirror pair Perez  _et al._ (2008). Because
the quantities $S_{e}$ and $J_{e}$ are small Clement and Perez (1979), the
ground state gyromagnetic ratios, $\gamma_{p/n}=\mu_{p/n}/J$, for the odd-
proton and odd-neutron partners of the mirror nuclei could be related Perez
_et al._ (2008) to their superallowed $\beta$-transition strengths,
$\gamma_{\beta}$, defined as

$|\gamma_{\beta}|=\frac{1}{2}\sqrt{\left(\frac{6170}{\mathcal{F}t^{\mathrm{mirror}}}-1\right)\frac{1}{J(J+1)}}\leavevmode\nobreak\
,$ (35)

via

$(\gamma_{p}+\Delta\gamma_{p})=g_{p}+\frac{G_{p}-g_{p}}{R}\gamma_{\beta}\leavevmode\nobreak\
,$ (36)

and

$(\gamma_{n}+\Delta\gamma_{n})=g_{n}+\frac{G_{n}-g_{n}}{R}\gamma_{\beta}\leavevmode\nobreak\
.$ (37)

The sign of $\gamma_{\beta}$ is negative for isotopes with $j=l-1/2$ (with $j$
and $l$ being, respectively, the spin and orbital angular momentum of the
subshell of the odd nucleon), i.e. for the $p_{1/2}$ shell (the mirror nuclei
with $A=$ 13 and 15), the $d_{3/2}$ shell ($A=$ 33 to 39), and the $f_{5/2}$
shell ($A=$ 65 to 77). For all other mirror nuclei one has $j=l+1/2$ and a
positive sign for $\gamma_{\beta}$.

The quantities $\Delta\gamma_{p}$ and $\Delta\gamma_{n}$ are small quantities
representing the contributions to the total spin and total angular momentum of
the mirror pair generated by the even type of nucleon in these odd-even
nuclei. They were obtained from $0\hbar\omega$ shell model calculations based
on the Hamiltonians from Refs. Cohen and Kurath (1965); Brown and Richter
(2006); Honma  _et al._ (2002) and are listed in Perez  _et al._ (2008). As
one is dealing here with nucleons in nuclei, the free-nucleon values of the
$g$-factors, i.e. $g_{p}=1.0$, $g_{n}=0$, $G_{p}=5.586$, and $G_{n}=-3.826$,
as well as the ratio of the axial-vector to vector coupling constant
$R=|C_{A}/C_{V}|\simeq 1.27$, were replaced by effective values denoted by
$\widetilde{g}$, $\widetilde{G}$, and $\widetilde{R}$. The linear relations of
Eqs. (36) and (37) were used to determine values for $\widetilde{g}$, and
$(\widetilde{G}-\widetilde{g})/\widetilde{R}$ for protons and neutrons from
least-square fits to $(\gamma_{p,n}+\Delta_{\gamma_{p,n}})$ and
$\gamma_{\beta}$, thereby using the experimental magnetic dipole moments and
$\beta$-decay half-lives for the mirror nuclei with $A=11$ to 43, yielding
(’Fit (B)’ in Perez  _et al._ (2008)):

$\widetilde{g}_{p}=1.040\pm 0.020\leavevmode\nobreak\ ,$ (38)
$\widetilde{g}_{n}=-0.011\pm 0.016\leavevmode\nobreak\ ,$ (39)
$(\widetilde{G}_{p}-\widetilde{g}_{p})/\widetilde{R}=4.26\pm
0.07\leavevmode\nobreak\ ,$ (40)
$(\widetilde{G}_{n}-\widetilde{g}_{n})/\widetilde{R}=-3.70\pm
0.05\leavevmode\nobreak\ .$ (41)

Using the $\mathcal{F}t^{\mathrm{mirror}}$ values listed in Table 6 and Eqs.
(35) to (37) then yield the calculated magnetic moment values listed in column
7 of Table LABEL:Magnetic-moments for all mirror isotopes. For the mirror
isotopes in the $fp$ shell with $A=$ 57 to 75, no $\Delta\gamma_{p,n}$ values
were available. We therefore used the average values of these quantities for
the isotopes with $A=$ 45 to 57 in the $fp$ shell listed in Table V of Ref.
Perez  _et al._ (2008) with a one standard deviation error so as to cover the
full range of individual values listed there, i.e. $\Delta\gamma_{p}=+0.11(5)$
and $\Delta\gamma_{n}=-0.14(5)$. The errors due to the uncertainties on these
estimated values have been added in quadrature to the other contributions to
the error bars on the calculated magnetic moment values.

Good correspondence with the known experimental values is obtained as shows
from the fact that from the 43 cases for which experimental values are
available the difference between the experimental and calculated values,
listed in column 10, is less than 0.1 $\mu_{N}$ for 27 nuclei, between 0.1
$\mu_{N}$ and 0.2 $\mu_{N}$ for another 11 nuclei, and larger than 0.2
$\mu_{N}$ for 5 nuclei (i.e. 17F, 41Sc, 41Ca, 45Ti, and 59Cu). Reasonably good
confidence can thus also be given to the values calculated for those mirror
nuclei for which experimental magnetic moment values are not available yet.
For the heaviest isotopes, i.e. with mass $A=61$ to 75, this is shown in Table
LABEL:Magnetic-moments by comparing the calculated moment values with those of
neighbouring nuclei with the odd proton or neutron occupying the same shell
model orbital.

Table 10: Magnetic moments of the $T=1/2$ mirror nuclei. Column 2 lists the values available in the literature Stone (2014); Mertzimekis  _et al._ (2016); ENSDF , with adopted values in column 4. When an average was calculated the scale factor (Eq. 14) is listed in column 5. Columns 6 to 9 relate to the values calculated (column 8) from the linear relations between the mirror pair magnetic moments and their $ft$ values of Ref. Perez  _et al._ (2008). Columns 6 and 7 list the $\gamma_{\beta}$ values (calculated from the corrected $\mathcal{F}t^{\mathrm{mirror}}$ values in Table 6 using Eq. (35)), and the values for the quantities $\Delta\gamma_{p}$ and $\Delta\gamma_{n}$ to calculate the magnetic moments using Eqs. (36) and (37), respectively. Calculated values for isotopes for which no experimental data are available are shown in italics. The last column lists the difference between the calculated value and the adopted experimental magnetic moment value. Note that when the sign for a magnetic moment value in column 2 is given, the sign has been explicitly measured. The correlation between the alphabetical reference code used here and the actual reference numbers is listed in Table 13. Parent | Measured magnetic moment(s) | Adopted value | scale | $\gamma_{\beta}$ | $\Delta_{\gamma(p,n)}$ | Calculated | $|exp-calc|$
---|---|---|---|---|---|---|---
nucleus |  ($\mu_{N}$) |  ($\mu_{N}$) | $S$ |  |  |  value ($\mu_{N}$) |  ($\mu_{N}$)
11C | $-0.964(1)$ | [Wo70] | $-0.964(10)$ |  | $+0.19583(13)$ | $-0.1864$ | $-0.824(28)$ | $0.140$
11B | $+2.6886489(10)$ | [Mi39,Ep75] | $+2.6886489(10)$ |  | $+0.19583(13)$ | $+0.1804$ | $+2.541(36)$ | $0.148$
13N | $0.3222(4)$ | [Be64] | $-0.3222(4)$ |  | $-0.32558(71)$ | $+0.2646$ | $-0.306(15)$ | $0.016$
13C | $+0.7024118(14)$ | [Ro54] | $+0.7024118(14)$ |  | $-0.32558(71)$ | $-0.2586$ | $+0.726(11)$ | $0.024$
15O | $0.7189(8)$ | [Co63] | $+0.71950(12)$ | $1$ | $-0.36585(86)$ | $0.0000$ | $+0.671(12)$ | $0.048$
| $0.71951(12)$ | [Ta93] |  |  |  |  |  |
15N | $-0.28318884(5)$ | [Ba62] | $-0.28318884(5)$ |  | $-0.36585(86)$ | $0.0000$ | $-0.259(16)$ | $0.024$
17F | $+4.7223(12)$ | [Su66] | $+4.72136(29)$ | $1$ | $+0.21993(14)$ | $0.0000$ | $+4.942(63)$ | $0.221$
| $+4.7213(3)$ | [Mi93] |  |  |  |  |  |
17O | $-1.89379(9)$ | [Al51a] | $-1.89379(9)$ |  | $+0.21993(14)$ | $0.0000$ | $-2.062(49)$ | $0.168$