# Impact of clustering on the ${}^{8}\mathrm{Li}$ $\beta$ decay and recoil
form factors

G. H. Sargsyan  Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA 70803, USA  K. D. Launey  Department of Physics
and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA  M. T.
Burkey  Department of Physics, University of Chicago, Chicago, Illinois 60637,
USA  Physics Division, Argonne National Laboratory, Argonne, Illinois 60439,
USA  Lawrence Livermore National Laboratory, Livermore, California 94550, USA
A. T. Gallant  Lawrence Livermore National Laboratory, Livermore, California
94550, USA  N. D. Scielzo  Lawrence Livermore National Laboratory, Livermore,
California 94550, USA  G. Savard  Physics Division, Argonne National
Laboratory, Argonne, Illinois 60439, USA  A. Mercenne  Department of Physics
and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA  Center
for Theoretical Physics, Sloane Physics Laboratory, Yale University, New
Haven, Connecticut 06520, USA  T. Dytrych  Department of Physics and
Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA  Nuclear
Physics Institute of the Czech Academy of Sciences, 250 68 Řež, Czech Republic
D. Langr  Department of Computer Systems, Faculty of Information Technology,
Czech Technical University in Prague, Prague 16000, Czech Republic  L.
Varriano  Department of Physics, University of Chicago, Chicago, Illinois
60637, USA  Physics Division, Argonne National Laboratory, Argonne, Illinois
60439, USA  B. Longfellow  Lawrence Livermore National Laboratory, Livermore,
California 94550, USA  T. Y. Hirsh  Soreq Nuclear Research Center, Yavne
81800, Israel  J. P. Draayer  Department of Physics and Astronomy, Louisiana
State University, Baton Rouge, LA 70803, USA

###### Abstract

We place unprecedented constraints on recoil corrections in the $\beta$ decay
of 8Li, by identifying a strong correlation between them and the 8Li ground
state quadrupole moment in large-scale ab initio calculations. The results are
essential for improving the sensitivity of high-precision experiments that
probe the weak interaction theory and test physics beyond the Standard Model
(BSM). In addition, our calculations predict a $2^{+}$ state of the
$\alpha+\alpha$ system that is energetically accessible to $\beta$ decay but
has not been observed in the experimental 8Be energy spectrum, and has an
important effect on the recoil corrections and $\beta$ decay for the $A=8$
systems. This state and an associated $0^{+}$ state are notoriously difficult
to model due to their cluster structure and collective correlations, but
become feasible for calculations in the ab initio symmetry-adapted no-core
shell-model framework.

_Introduction.–_ The left-handed vector minus axial-vector (V$-$A) structure
of the weak interaction was postulated in late 1950’s and early 1960’s Feynman
and Gell-Mann (1958); Sudarshan and Marshak (1958) guided in large part by a
series of $\beta$-decay experiments Wu  _et al._ (1957); Herrmannsfeldt  _et
al._ (1957); Johnson  _et al._ (1963), and later was incorporated in the
Standard Model of particle physics. However, in its most general form, the
weak interaction can also have scalar, tensor, and pseudoscalar terms as well
as right-handed currents.

Today, $\beta$-decay experiments continue to pursue increasingly sensitive
searches for additional contributions to the weak interaction. Various
experiments Johnson  _et al._ (1963); Fléchard  _et al._ (2011); Li  _et al._
(2013); Sternberg  _et al._ (2015) have constrained the tensor part of the
interaction, although the limits are less stringent compared to the other non-
standard-model terms Severijns  _et al._ (2006); Wauters  _et al._ (2014).
While these experiments have achieved remarkable precision, further
improvements require confronting the systematic uncertainties that stem from
higher-order corrections (referred to as recoil-order terms) in nuclear
$\beta$ decay. These terms are inherently small compared to the allowed
$\beta$ decay terms; however, current experiments have reached a precision
where even subtle distortions matter. Measurements of recoil-order terms are
also interesting in their own right as they can test additional symmetries of
the Standard Model, such as the existence of second-class currents Grenacs
(1985); De Braeckeleer  _et al._ (1995); Sumikama  _et al._ (2008) and the
accuracy of the conserved vector current (CVC) hypothesis Holstein (1974);
Tribble and Garvey (1975); De Braeckeleer  _et al._ (1995); Sumikama  _et al._
(2011).

Figure 1: Decay scheme for $\beta$ decay of ${}^{8}\mathrm{Li}$
$\mathrm{g.s.}$ to the broad low-lying $2^{+}$ state in ${}^{8}\mathrm{Be}$.
Energies, in MeV, are relative to ${}^{8}\mathrm{Be}$ $\mathrm{g.s.}$ A small
$\beta$-decay branching is observed to the doublet 2+ states due to their
resonant nature.

The $\beta$ decay of 8Li to ${}^{8}\mathrm{Be}$, which subsequently breaks up
into two $\alpha$ particles, has long been recognized as an excellent testing
ground to search for new physics Barnes  _et al._ (1958); Holstein (1974);
Grenacs (1985) due to the high decay energy and the ease of detecting the
$\beta$ and two $\alpha$ particles. Recently, by taking advantage of ion-
trapping techniques, high-precision measurements of
$\beta$-$\bar{\nu}$-$\alpha$ correlations Scielzo  _et al._ (2012); Li  _et
al._ (2013); Sternberg  _et al._ (2015) have been performed that set the most
stringent limit on a tensor contribution to date Burkey  _et al._ (2021).
However, in this type of experiment, one of the largest uncertainties comes
from the several different recoil-order corrections that contribute to the
decay. A number of other experiments have taken advantage of the presence of
certain recoil-order terms in the $\beta$ decay of ${}^{8}\mathrm{Li}$ (Fig.
1) to perform CVC tests by studying $\beta$-$\alpha$ angular correlations
Tribble and Garvey (1975); McKeown  _et al._ (1980) and $\beta$-spin alignment
correlations Sumikama  _et al._ (2008, 2011). In addition, these terms have
been deduced from $\gamma$ decays of the doublet 2+ states near 17 MeV in the
${}^{8}\mathrm{Be}$ spectrum, which contain the isobaric analogue of the
${}^{8}\mathrm{Li}$ ground state ($\mathrm{g.s.}$) De Braeckeleer  _et al._
(1995). Due to their small size, and the fact that there are several terms
that contribute to decay observables, most of the experimentally extracted
recoil-order terms have large uncertainties.

In this Letter, we report the first ab initio calculations of recoil-order
terms in the $\beta$ decay of ${}^{8}\mathrm{Li}$. These calculations achieve
highly reduced uncertainties compared to the experimentally extracted values
of Sumikama  _et al._ (2011). They help decrease the systematic uncertainties
on the tensor-current estimates in the weak interaction reported in Burkey
_et al._ (2021), and are of interest to experimental tests of the CVC
hypothesis De Braeckeleer  _et al._ (1995). We also provide evidence that the
$\beta$-transition strength of the ${}^{8}\mathrm{Li}$ decay is affected by a
disputed low-lying 2+ state (sometimes referred to as an “intruder” state)
below 16 MeV in the ${}^{8}\mathrm{Be}$ spectrum. Our calculations in
unprecedentedly large model spaces support the existence of low-lying states
with a large overlap with the $\alpha+\alpha$ $s$\- and $d$-waves. Indeed, a
very broad 2+ state along with a lower 0+ were initially proposed by Barker
from the R-matrix analysis of $\alpha+\alpha$ scattering and the $\beta$
decays of ${}^{8}\mathrm{Li}$ and 8B Barker  _et al._ (1968); Barker (1969,
1989). Even though such states have not been directly observed experimentally,
some earlier theoretical studies have predicted them in the low-lying spectrum
of ${}^{8}\mathrm{Be}$ Caurier  _et al._ (2001); Maris (2013); Rodkin and
Tchuvil’sky (2020). Furthermore, there has been a recent experimental
indication in favor of intruder states below 16 MeV Munch  _et al._ (2018).

SA-NCSM framework.– For this work, we employ the ab initio symmetry-adapted
no-core shell model (SA-NCSM) Launey  _et al._ (2016); Dytrych  _et al._
(2020); Launey  _et al._ (2021). The use of chiral effective-field-theory
interactions Bedaque and van Kolck (2002); Epelbaum  _et al._ (2002); Entem
and Machleidt (2003); Epelbaum (2006) enables nuclear calculations informed by
elementary particle physics, while the symmetry-adapted (SA) basis allows us
to achieve ultra-large model spaces imperative for the description of
challenging features in the ${}^{8}\mathrm{Be}$ states, such as clustering and
collectivity. It uses a harmonic oscillator (HO) basis with frequency
$\hbar\Omega$ and a model space with an $N_{\mathrm{max}}$ cutoff, which is
the maximum total HO excitation quanta above the lowest HO configuration for a
given nucleus. These parameters are related to infrared (IR) and ultraviolet
(UV) cutoffs Wendt  _et al._ (2015), which can be understood as the effective
size of the model space in which the nucleus resides, and its grid resolution,
respectively. The calculations become independent of $\hbar\Omega$ at
$N_{\mathrm{max}}$ $\rightarrow\infty$, providing a parameter-free ab initio
prediction. The SA-NCSM results exactly reproduce those of the NCSM Navrátil
_et al._ (2000, 2000) for the same nuclear interaction. However, by utilizing
the emergent symplectic $\mathrm{Sp}(3,\mathbb{R})$-symmetry in nuclei Dytrych
_et al._ (2020), the SA-NCSM can expand the model space by a physically
relevant subspace, which is only a fraction of the complete NCSM space,
thereby including localized-$\alpha$ degrees of freedom within the interaction
effective range Dreyfuss  _et al._ (2020).

Figure 2: (a) Ab initio ${}^{8}\mathrm{Li}$ g.s. quadrupole moment $Q(2^{+})$
compared to experiment Borremans  _et al._ (2005) (denoted as “Expt.”).
Calculations use the NNLOopt NN for different model-space sizes and
resolutions (open symbols), along with the infinite-size extrapolated value
(dashed line) and the corresponding many-body uncertainty (shaded area). (b)
Calculated $j_{2}/A^{2}c_{0}$ and $j_{3}/A^{2}c_{0}$ (squares and triangles,
respectively) and their predicted values (upper and lower horizontal lines,
respectively) for the ${}^{8}\mathrm{Li}\;\beta$ decay to $2_{1}^{+}$ in
${}^{8}\mathrm{Be}$ vs. the calculated ${}^{8}\mathrm{Li}$ $Q(2^{+}$). The
first (second) set of the uncertainties in Eq. (2) is shown as the line
thickness (gray bands). Calculations use the NNLOopt, NNLOsat and ${\rm
N}^{3}{\rm LO}$ chiral potentials, and the JISP16 NN, in $N_{\mathrm{max}}$
=6-12 model spaces. Inset: subset of calculated $j_{2}/A^{2}c_{0}$ vs.
$Q(2^{+}$) for selected IR (in fm)/UV (in MeV) many-body cutoffs across all
interactions, and for two interactions across several cutoffs.

We adopt various chiral potentials without renormalization in nuclear medium:
N3LO-EM Entem and Machleidt (2003), NNLOopt Ekström  _et al._ (2013), as well
as NNLOsat Ekström  _et al._ (2015) with the three-nucleon (3N) forces,
hierarchically smaller than their nucleon-nucleon (NN) forces, added as
averages Launey  _et al._ (2021). For comparison, we present results with the
soft JISP16 phase-equivalent NN interaction Shirokov  _et al._ (2010). We use
$\hbar\Omega$ =15-25 MeV for N3LO-EM, NNLOopt, and JISP16, and $\hbar\Omega$
=16 and 22 MeV for NNLOsat, unless otherwise stated. The NNLOopt is used
without 3N forces, which have been shown to contribute minimally to the 3- and
4-nucleon binding energy Ekström  _et al._ (2013). Furthermore, the NNLOopt NN
potential has been found to reproduce various observables, including the 4He
electric dipole polarizability Baker  _et al._ (2020); the challenging
analyzing power for elastic proton scattering on 4He, 12C, and 16O Burrows
_et al._ (2019); along with B(E2) transition strengths for 21Mg and 21F
Ruotsalainen  _et al._ (2019) in the SA-NCSM without effective charges.

For the purposes of this study, the quadrupole moment of the
${}^{8}\mathrm{Li}$ $\mathrm{g.s.}$, $Q(2_{\mathrm{g.s.}}^{+})$, for which SA-
NCSM calculations with the NNLOopt NN are extrapolated to an infinite model-
space size, is shown to reproduce the experimental value within the many-body
model uncertainties (Fig. 2a). The result is in close agreement with the
extrapolated value of McCracken  _et al._ (2021) that uses renormalized NN+3N
chiral potentials. The model uncertainties are based on variations in the
model-space size and resolution, and extrapolations use the Shanks method
Dytrych  _et al._ (2020).

_Recoil-order corrections.–_ The recoil-order form factors are generally
neglected in $\beta$-decay theory since they are of the order of $q/m_{N}$ or
higher, where $q$ is the momentum transfer (typically several MeV/c) and
$m_{N}$ is the nucleon mass Holstein (1974). Thus, for most $\beta$ decays,
the recoil effects are typically less than a percent of the dominant Fermi and
Gamow-Teller (GT) contributions (for an example see Glick-Magid  _et al._
(2021)). However, for measurements of sufficiently high precision, these terms
must be included in the analysis especially when the leading contributions are
suppressed or the recoil-order terms are unusually large. These recoil-order
form factors include, the second forbidden axial vectors ($j_{2}$ and
$j_{3}$), induced tensor ($d$), and weak magnetism ($b$), and along with the
GT ($c_{0}$) are given in the impulse approximation (IA) as:

$\displaystyle c_{0}(q^{2})$ $\displaystyle=$
$\displaystyle(-)^{(J^{\prime}-J)}\frac{g_{A}(q^{2})}{\sqrt{2J+1}}\left\langle
J^{\prime}\right||\sum_{i=1}^{A}\tau^{\pm}_{i}\sigma_{i}|\left|J\right\rangle=(-)^{(J^{\prime}-J)}\frac{g_{A}(q^{2})}{\sqrt{2J+1}}M_{GT},$
$\displaystyle j_{K}(q^{2})$ $\displaystyle=$
$\displaystyle-(-)^{(J^{\prime}-J)}\frac{2}{3}\frac{g_{A}(q^{2})}{\sqrt{2J+1}}\frac{(Am_{N}c^{2})^{2}}{(\hbar
c)^{2}}\left\langle
J^{\prime}\right||\sum_{i=1}^{A}\tau^{\pm}_{i}[Q_{i}\times\sigma_{i}]^{K}|\left|J\right\rangle,{\rm
with\,}K=2,3,$ $\displaystyle d(q^{2})$ $\displaystyle=$
$\displaystyle(-)^{(J^{\prime}-J)}A\frac{g_{A}(q^{2})}{\sqrt{2J+1}}\left\langle
J^{\prime}\right||\sum_{i=1}^{A}\tau^{\pm}_{i}\sqrt{2}[L_{i}\times\sigma_{i}]^{1}|\left|J\right\rangle,$
$\displaystyle b(q^{2})$ $\displaystyle=$ $\displaystyle
A\frac{(-)^{(J^{\prime}-J)}}{\sqrt{2J+1}}\Big{[}g_{M}(q^{2})\left\langle
J^{\prime}\right||\sum_{i=1}^{A}\tau^{\pm}_{i}\sigma_{i}|\left|J\right\rangle+g_{V}(q^{2})\left\langle
J^{\prime}\right||\sum_{i=1}^{A}\tau^{\pm}_{i}L_{i}|\left|J\right\rangle\Big{]},$
(1)

where $g_{V}(0)=1$, $g_{A}(0)\approx 1.27$ and $g_{M}(0)\approx 4.70$ are the
vector, axial and weak magnetism coupling constants, $A$ is the mass number
and $J(J^{\prime})$ is the total angular momentum of the initial (final)
nucleus. The $\tau_{i}$/2, $\sigma_{i}/2$,
$Q_{i}=\sqrt{16\pi/5}r_{i}^{2}Y_{2\mu}(\hat{r}_{i})$, and $L_{i}$ are the
isospin, intrinsic spin, quadrupole moment and angular momentum operators,
respectively, of the $i^{\mathrm{th}}$ particle. $M_{GT}$ is the conventional
GT matrix element. The matrix elements in Eq. (1) are computed translationally
invariant in the SA-NCSM. These recoil-order form factors, usually reported as
the ratios $j_{2,3}/A^{2}c_{0}$, $d/Ac_{0}$, and $b/Ac_{0}$, enter into the
expression of the $\beta$-decay rate for nuclei undergoing delayed
$\alpha$-particle emission Holstein (1974); Sternberg  _et al._ (2015); Burkey
(2019); Burkey  _et al._ (2021).

Remarkably, we identify a strong correlation between $j_{2,3}/A^{2}c_{0}$ and
the ${}^{8}\mathrm{Li}$ $\mathrm{g.s.}$ quadrupole moment based on
calculations across several interactions, $N_{\mathrm{max}}$ and $\hbar\Omega$
parameters (Fig. 2b, using $N_{\mathrm{max}}$ = 6 to 12 for NNLOopt, to 10 for
N3LO-EM and JISP16, and to 8 for NNLOsat). As can be seen in the Fig. 2b
inset, the linear dependence is observed regardless of any errors that may
arise from the many-body truncation and from the higher-order effects (e.g.,
Filin  _et al._ (2021); Maris  _et al._ (2021)) associated with various
interactions. An identical spread is found for $j_{3}/A^{2}c_{0}$ due to the
strong correlation between $j_{2}$ and $j_{3}$ (see Supplemental Material sup
). A linear regression along with the combination of the correlation to
$Q(2_{\rm g.s.}^{+})$ and its experimental value of 3.14(2) $e\,\rm{fm}^{2}$
Borremans  _et al._ (2005) lead to reduced uncertainties on our predictions:

$\frac{j_{2}}{A^{2}c_{0}}=-966\pm 13\pm
33,\quad\frac{j_{3}}{A^{2}c_{0}}=-1546\pm 19\pm 40,$ (2)

Here, the first set of uncertainties uses the quadrupole moment experimental
uncertainties given the linear regression slope, and the second set arises
from the regression uncertainty using Student’s $t$-distribution and a $99\%$
confidence level. This correlation is important, as we can reduce the problem
of calculating a matrix element that depends on cluster physics in
${}^{8}\mathrm{Be}$ to a bound state observable in 8Li.

Most significantly, with the values in Eq. (2), the uncertainty from the
recoil-order corrections on the tensor current contribution to the weak
interaction presented in Burkey (2019) is reduced by over $50\%$ Burkey  _et
al._ (2021). The recoil-order terms, including the $b$ and $d$ terms, for the
lowest four SA-NCSM $2^{+}$ states, are summarized in Table 1. The $d/Ac_{0}$
prediction for $2_{1}^{+}$ is based on a correlation similar to the one for
$j_{2,3}/A^{2}c_{0}$ (see Supplemental Material sup ). These predictions can
be used in future experiments to constrain BSM tensor currents, while these
$b$ weak magnetism predictions are of interest to experiments that test the
CVC hypothesis and $d$ is of importance to determining the existence of
second-class currents De Braeckeleer  _et al._ (1995).

Table 1: The recoil-order terms from SA-NCSM. Results for the 2${}^{+}_{1}$ $j_{2,3}/A^{2}c_{0}$ and $d/Ac_{0}$ are based on the correlation to $Q(2_{\rm g.s.}^{+})$; all other calculations use NNLOopt and have error bars from variations in $\hbar\Omega$ by 5 MeV and in model-space sizes up to $N_{\mathrm{max}}$ =16 (12) for $j_{2,3}/A^{2}c_{0}$ ($d/Ac_{0}$ and $b/Ac_{0}$).  | $j_{2}/A^{2}c_{0}$ | $j_{3}/A^{2}c_{0}$ | $d/Ac_{0}$ | $b/Ac_{0}$
---|---|---|---|---
$2_{1}^{+}$ | $-966\pm 36$ | $-1546\pm 44$ | $10.0\pm 1.0$ | $6.0\pm 0.4$
$2^{+}_{2}\mathrm{(new)}$ | $-10\pm 10$ | $-80\pm 30$ | $-0.5\pm 0.5$ | $3.7\pm 0.4$
$2^{+}_{3}\rm{(doublet\,1)}$ | $12\pm 5$ | $-60\pm 15$ | $0.3\pm 0.2$ | $3.8\pm 0.2$
$2^{+}_{4}\rm{(doublet\,2)}$ | $11\pm 3$ | $-65\pm 11$ | $0.2\pm 0.2$ | $3.8\pm 0.2$

New final state for $\beta$ decays to ${}^{8}\mathrm{Be}$.– The experimentally
deduced values presented in Sumikama  _et al._ (2011),
$j_{2}/A^{2}c_{0}=-490\pm 70$, $j_{3}/A^{2}c_{0}=-980\pm 280$,
$d/Ac_{0}=5.5\pm 1.7$ and $b/Ac_{0}=7.5\pm 0.2$, are comparable but different
from our predicted values. These experimental results were obtained through a
global fit to $\beta$-spin alignment Sumikama  _et al._ (2011) and
$\beta$-$\alpha$ angular correlation data McKeown  _et al._ (1980) from
${}^{8}\mathrm{Li}$ and 8B $\beta$ decays. Due to the small size of higher-
order effects and relatively large statistical uncertainties, the
$j_{2,3}/A^{2}c_{0}$ and $d/Ac_{0}$ were assumed in Sumikama  _et al._ (2011)
to be independent of the ${}^{8}\mathrm{Be}$ excitation energy. Thus, the
results were averaged over the entire $\beta$ decay spectrum. In contrast, the
SA-NCSM wavefunctions are for individual states, hence, the predictions in Eq.
(2) are for the lowest 2+ state only, which is the dominant transition for the
${}^{8}\mathrm{Li}$ $\beta$ decay. The SA-NCSM reveals large differences
between the recoil-order terms to the lowest 2${}_{1}^{+}$ and higher-lying
states, the most notable being for the $j_{K}/A^{2}c_{0}$ terms where the
values differ by almost two orders of magnitude (see Table 1). Hence, the
angular-correlation experiment in Burkey  _et al._ (2021) minimizes the
sensitivity to the higher-lying states by restricting their analysis to decays
centered on the broad $2_{1}^{+}$ state.

Figure 3: (a) Calculated ${}^{8}\mathrm{Be}$ low-lying $0^{+}$ state energies
illustrated for the NNLOopt chiral potential ($\hbar\Omega$ =15 MeV) vs. the
model-space size, together with the extrapolation values (dotted lines) and
uncertainties (bands). Extrapolations use complete model spaces up to
$N_{\mathrm{max}}$ =12 and do not include the $N_{\mathrm{max}}$ =14 and 16 SA
selected model spaces shown with uncertainties determined by the selection.
The measured $0_{\rm g.s.}^{+}$ energy is $-56.5$ MeV Huang  _et al._ (2021).
Inset: $\alpha+\alpha$ $s$-wave for $0^{+}_{\rm g.s.}$ (blue solid) and
$0^{+}_{2}$ (orange dashed). (b) Ab initio low-lying states from
extrapolations for ${}^{8}\mathrm{Be}$, compared to experiment (Expt.). The
extrapolation uncertainties (error bars) for the $0^{+}_{2}$ and $2^{+}_{2}$
states (dashed levels) are based on variations in the model-space size and
selection. For NNLOopt, $\alpha$ width estimates (shaded areas) for the lowest
two $0^{+}$ and $2^{+}$ states are shown with uncertainties (lighter shades)
determined from the energy extrapolation uncertainties; the small $0_{\rm
g.s.}^{+}$ width (not shown) is estimated to be 5.7 eV, compared to 5.57 eV
Tilley  _et al._ (2004).

Importantly, the SA-NCSM indicates the existence of another $2^{+}$ state
below 16 MeV – accessible to the ${}^{8}\mathrm{Li}$ or 8B $\beta$ decays
through allowed transitions – and a corresponding lower $0^{+}$ state that
largely overlaps with the $\alpha$+$\alpha$ system (Fig. 3, see also
Supplemental Material sup ). In the SA-NCSM, these states quickly decrease in
energy as the model space increases (Fig. 3a) regardless of the realistic
interaction used, similar to the Hoyle-state rotational band in 12C Dreyfuss
_et al._ (2013). The extrapolations are performed using the three-parameter
exponential formula from Maris  _et al._ (2009). Notably, $0_{3}^{+}$
converges to $20.1\pm 1.5$ MeV and has a structure similar to the doublet 2+
states and isospin $T=1$. This state is not seen in the currently-available
experimental spectrum and it is likely to be the isobaric analogue of the low-
lying 0+ state in ${}^{8}\mathrm{Li}$ predicted by recent ab initio
calculations Navrátil  _et al._ (2010); McCracken  _et al._ (2021).

The calculated low-lying states in ${}^{8}\mathrm{Be}$ are in good agreement
with experiment (Fig. 3b).The NNLOsat results include the average 3N
contribution determined for a given isospin (for ${}^{8}\mathrm{Be}$, the
contribution to the binding energy in $N_{\mathrm{max}}$ =12 is 1.51 MeV,
resulting in a total extrapolated binding energy of 56.8 MeV). The
extrapolations determine the energies of $0_{2}^{+}$ and $2_{2}^{+}$ between 5
and 15 MeV above the $\mathrm{g.s.}$, corroborating earlier estimates Barker
_et al._ (1968); Barker (1969); Caurier  _et al._ (2001).

For NNLOopt NN and the case of the fastest energy convergence of the
$0^{+}_{2}$ and $2^{+}_{2}$ states ($\hbar\Omega$ =15 MeV), we estimate
$\alpha$ widths (Fig. 3b), by projecting the $N_{\mathrm{max}}$ =16 SA-NCSM
wavefunctions onto $\alpha+\alpha$ cluster states and considering the exact
continuum Coulomb wavefunctions outside of the interaction effective range,
following the procedure of Dreyfuss  _et al._ (2020). For this, the
${}^{8}\mathrm{Be}$ and ${}^{4}\mathrm{He}$ states are expressed in the
$\mathrm{Sp}(3,\mathbb{R})$ basis, associated with intrinsic shapes Dytrych
_et al._ (2020). For ${}^{8}\mathrm{Be}$, we consider three dominant prolate
shapes with contributions of 75%, 4%, and 3% to $0_{\rm g.s.}^{+}$ (totaling
82%), and 46%, 15%, and 11% to $0_{2}^{+}$ (totaling 72%), and similarly for
the $2^{+}$ states (see Supplemental Material sup ). These shapes extend to 18
HO shells and start at the most deformed configurations among those in the
valence shell: 2$\hbar\Omega$ and 4$\hbar\Omega$ excitations. Except for the
$0_{\rm g.s.}^{+}$ width that uses the experimental threshold of $-92$ keV
relative to the ${}^{8}\mathrm{Be}$ g.s., all the widths use the
$\alpha$+$\alpha$ threshold estimated at $-104$ keV from the SA-NCSM
extrapolations of the 4He and 8Be binding energies with NNLOopt. These widths
are in good agreement with experimentally deduced values Tilley  _et al._
(2004) and earlier theoretical studies Kravvaris and Volya (2017); Kravvaris
_et al._ (2020); Elhatisari  _et al._ (2015).

Intruder 0+ and 2+ states in the low-lying spectrum of ${}^{8}\mathrm{Be}$
were proposed in the late 1960’s by Barker from concurrent R-matrix fits to
scattering, reaction, and decay data associated with the ${}^{8}\mathrm{Be}$
nucleus Barker  _et al._ (1968); Barker (1969). The inclusion of an intruder
2+ state below 16 MeV in the R-matrix fits of $\beta$ decays in Warburton
(1986) results in a decrease of the extracted $M_{\rm GT}$ for a decay to
$2_{1}^{+}$ by almost 1.5 times, which yields a closer agreement with the SA-
NCSM $M_{\rm GT}$ (see Table 2; the calculated $M_{\rm GT}$ are not used in
the experimental analysis of Burkey  _et al._ (2021)). Note that, depending on
the interaction, two-body axial currents may significantly affect $M_{\rm GT}$
King  _et al._ (2020), however, here we are interested only in the IA part.
Due to the large uncertainty on the $2_{2}^{+}$ state in the calculations, we
provide only the lower limits on the $\mathrm{log}(ft)$ based on the
convergence pattern. The energies from Barker’s R-matrix fits for the intruder
0+ and 2+ states are $\sim 6$ MeV and 9 MeV, respectively, with $\alpha$
widths $>7$ MeV. These excitation energies agree with the SA-NCSM extrapolated
results given the error bars (Fig. 3b), as well as with the predicted widths.
The strong excitation-energy dependence of the recoil-order terms due to the
presence of $2_{2}^{+}$ has a small effect on the weak tensor currents
constraints in the low excitation-energy range (see systematic uncertainty in
Table I in Burkey  _et al._ (2021)), but is imperative to consider in analyses
over the entire $\beta$ decay spectrum.

Table 2: Ab initio $M_{\rm GT}$, $c_{0}$ and log$(ft)$ in IA, compared to the experimentally deduced values. Ref. Warburton (1986) includes evaluations both with an intruder $2^{+}$ state (denoted by *) around 8 MeV similar to Ref. Barker (1989), and without it.  | $2_{1}^{+}$ | $2^{+}_{2}$
---|---|---
| $|M_{\rm GT}|$ | $|c_{0}|$ | log$(ft)$ | log$(ft)$
NNLOopt | $0.16(1)$ | $0.09(1)$ | $5.90$ | $\quad>5.06$
$\mathrm{NNLO}_{\mathrm{sat}}$ | $0.21(3)$ | $0.12(2)$ | $5.64$ | $\quad>5.05$
$\mathrm{JISP16}$ | $0.23(4)$ | $0.13(2)$ | $5.54$ | $\quad>4.28$
Expt., Ref. Barker (1989) | $0.190$ | $0.108$ | $5.72$ | $\quad 5.27$
Expt., Ref. Warburton (1986)∗ | $0.204$ | $0.116$ | $5.66$ | $\quad 5.2$
Expt., Ref. Warburton (1986) | $0.284$ | $0.163$ | $5.37$ | $\quad-$

Summary.– The ab initio SA-NCSM has determined the size of the recoil-order
form factors in the $\beta$ decay of 8Li. It has shown that states of the
$\alpha+\alpha$ system not included in the evaluated 8Be energy spectrum have
an important effect on all $j_{2,3}/A^{2}c_{0}$, $b/Ac_{0}$ and $d/Ac_{0}$
terms, and can explain the $M_{\rm GT}$ discrepancy in the $A=8$ systems. The
outcomes reduce – by over $50\%$ – the uncertainty on these recoil-order
corrections, and help improve the sensitivity of high-precision $\beta$-decay
experiments that probe the V$-$A structure of the weak interaction Burkey  _et
al._ (2021). Furthermore, our predicted $b/Ac_{0}$ and $d/Ac_{0}$ values are
important for other investigations of the Standard Model symmetries, such as
the CVC hypothesis and the existence of second-class currents.

_Acknowledgments.-_ We thank Scott Marley and Konstantinos Kravvaris for
useful discussions. This work was supported in part by the U.S. National
Science Foundation (PHY-1913728), SURA, Czech Science Foundation (22-14497S),
and the Czech Ministry of Education, Youth and Sports under Grant No.
CZ.02.1.01/0.0/0.0/16_019/0000765. It benefited from high performance
computational resources provided by LSU (www.hpc.lsu.edu), the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy
Office of Science User Facility operated under Contract No. DE-AC02-05CH11231,
as well as the Frontera computing project at the Texas Advanced Computing
Center, made possible by National Science Foundation award OAC-1818253. L.
Varriano was supported by a National Science Foundation Graduate Research
Fellowship under Grant No. DGE-1746045. This work was performed under the
auspices of the U.S. Department of Energy by Lawrence Livermore National
Laboratory under Contract DE-AC52-07NA27344.

##  I Supplemental Material

###  I.1 Correlation of the induced tensor term $d/Ac_{0}$ with $Q(2_{\rm
g.s.}^{+})$ of ${}^{8}\mathrm{Li}$

For completeness, we present results for the correlation of the induced tensor
term $d/Ac_{0}$, defined in Eq. (1), with the quadrupole moment of the
${}^{8}\mathrm{Li}$ $\mathrm{g.s.}$ $Q(2_{\rm g.s.}^{+})$. We note that all of
the reduced matrix elements in Eq. (1) are with respect to the total angular
momentum but not isospin. The additional factor,
$(-)^{(J^{\prime}-J)}/\sqrt{2J+1}$, compared to Ref. Holstein (1974) arises
from the difference in the convention of Wigner-Eckart theorem. The convention
we use can be found in Ref. Suhonen (2007).

Similar to the $j_{K}/A^{2}c_{0}$ terms, the SA-NCSM calculations for
$d/Ac_{0}$ with NNLOopt for the lowest $2^{+}$ state in ${}^{8}\mathrm{Be}$
exhibit a strong correlation with the ${}^{8}\mathrm{Li}$ $\mathrm{g.s.}$
quadrupole moment calculations. The $d/Ac_{0}$ value in Table 1 is the
prediction using a linear regression and the experimental value for the
${}^{8}\mathrm{Li}$ $Q(2_{\rm g.s.}^{+})$ (Fig. 4). The calculations are
performed in model spaces with up to $N_{\mathrm{max}}$ =12 and for
$\hbar\Omega$ =15, 20 and 25 MeV. The quoted value of $10.0\pm 1.0$ has a
comparatively large regression uncertainty due to the comparatively small
number of calculations with a single chiral potential. This uncertainty is
calculated using the slope uncertainty multiplied by the $Q(2_{\rm g.s.}^{+})$
experimental value. Further improvement of the $d/Ac_{0}$ uncertainty is
underway: with the present study reducing the largest uncertainties hitherto,
namely, for the $j_{2}/A^{2}c_{0}$ and $j_{3}/A^{2}c_{0}$ terms, the
$d/Ac_{0}$ uncertainty becomes now important for further investigations of
tensor currents and other physics beyond the Standard Model Burkey  _et al._
(2021),

Figure 4: Calculated $d/Ac_{0}$ for the ${}^{8}\mathrm{Li}$ $\beta$ decay to
the lowest $2^{+}$ state in ${}^{8}\mathrm{Be}$ vs. the calculated
${}^{8}\mathrm{Li}$ ground state quadrupole moment Q(2+). Calculations use the
NNLOopt chiral potential in model spaces $N_{\mathrm{max}}$ =6 to 12. The
horizontal line corresponds to the $d/Ac_{0}$ value predicted from the linear
regression using the experimental ${}^{8}\mathrm{Li}$ quadrupole moment
Borremans  _et al._ (2005). The blue band indicates the uncertainty from
experiment. The total uncertainty, which also includes the linear regression
slope uncertainty, is shown by the gray band.

###  I.2 Correlation of $j_{2}/A^{2}c_{0}$ with $j_{3}/A^{2}c_{0}$

Since both $j_{2}/A^{2}c_{0}$ and $j_{3}/A^{2}c_{0}$ for the
${}^{8}\mathrm{Be}$ lowest $2^{+}$ state are strongly correlated with the
${}^{8}\mathrm{Li}$ g.s. quadrupole moment (Fig. 2), it is expected that there
is a strong correlation between $j_{2}$ and $j_{3}$. Indeed, the calculated
$j_{2}/A^{2}c_{0}$ vs. $j_{3}/A^{2}c_{0}$ for all four interactions under
consideration follow an almost perfect linear trendline (Fig. 5). It is
important to note that the experimentally deduced values from Ref. Sumikama
_et al._ (2011), which have been determined over the whole range of the
$\beta$ energy spectrum, also agree with the calculated correlation.

Figure 5: Calculated $j_{3}/A^{2}c_{0}$ vs. $j_{2}/A^{2}c_{0}$ values with
NNLOopt, NNLOsat, N3LO-EM and JISP16 interactions for the same model spaces as
in Fig. 2b, along with the values from Sumikama _et al._ Sumikama  _et al._
(2011) (orange cross). The linear fit (dotted green line) is added to guide
the eye.

###  I.3 Overlaps of ${}^{8}\mathrm{Be}$ wavefunctions with the
$\alpha+\alpha$ and ${}^{7}\mathrm{Li}+$p systems

To study the cluster structure of the lowest $2_{1}^{+}$ and $2_{2}^{+}$
states, we calculate the overlaps of the many-body ${}^{8}\mathrm{Be}$
wavefunctions with the $\alpha+\alpha$ and ${}^{7}\mathrm{Li}+$p systems, and
present the interior $\alpha+\alpha$ cluster wave function (Fig. 6). The
experimental threshold of these systems with respect to the
${}^{8}\mathrm{Be}$ ground state are $-0.092$ MeV for $\alpha+\alpha$ and
$17.255$ MeV for ${}^{7}\mathrm{Li}+$p, thus the lowest two $2^{+}$ states are
below the proton separation energy. For the calculations of $\alpha+\alpha$
cluster wavefunctions (Fig. 6a), we follow the procedure in Ref. Dreyfuss  _et
al._ (2020) using the ${}^{8}\mathrm{Be}$ $2^{+}$ states and
${}^{4}\mathrm{He}$ ground state calculated in $\mathrm{Sp}(3,\mathbb{R})$
basis. To compose a $J=2$ cluster state with two $\alpha$’s in their ground
state and channel spin $s=0$, we consider the relative orbital momentum $l=2$.
Both $2^{+}$ states have larger peaks at longer distances suggesting alpha
clustering, whereas the $2_{2}^{+}$ state is more spatially expanded.
Calculations of the single-nucleon overlaps
$\left\langle{}^{8}\mathrm{Be}|^{7}\mathrm{Li}+{\rm p}\right\rangle$ follow
the formalism of Ref. Navrátil (2004) (Fig. 6b). Here, the $p$-wave is
described by the ${}^{7}\mathrm{Li}$ ground state, which couples with the
proton spin to a channel spin $s=1$. We find a large overlap between the
${}^{7}\mathrm{Li}+$p and the lowest $2^{+}$ state, and a comparatively small
but non-negligible overlap with the $2_{2}^{+}$ state.

Figure 6: Interior cluster wavefunctions for the lowest two
${}^{8}\mathrm{Be}$ $2^{+}$ states with (a) $l=2,\,s=0$
$\left\langle{}^{8}\mathrm{Be}|\alpha+\alpha\right\rangle$ in
$N_{\mathrm{max}}$ =16 and (b) $l=1,\,s=1$
$\left\langle{}^{8}\mathrm{Be}|^{7}\mathrm{Li}+p\right\rangle$ in
$N_{\mathrm{max}}$ =10 (SA-NCSM calculations use NNLOopt and $\hbar\Omega$ =
15 MeV).

###  I.4 ${}^{8}\mathrm{Be}$ wavefunctions in the symplectic basis

In SA-NCSM the many-body wavefunctions can be decomposed in terms of
collective $\mathrm{Sp}(3,\mathbb{R})$ basis states. As shown in Ref. Dytrych
_et al._ (2020), each irreducible representation (irrep) of the symplectic
$\mathrm{Sp}(3,\mathbb{R})$ group describes an equilibrium shape within a
nucleus that vibrates and rotates. These irreps are labeled by the
$\mathrm{SU}(3)$ quantum numbers $(\lambda\,\mu)$ of the equilibrium shape
with an average deformation given by the familiar shape parameters, the
deformation $\beta$ and triaxiality $\gamma$ Castaños  _et al._ (1988);
Mustonen  _et al._ (2018). Typically, the nuclei are comprised of only a few
shapes, often a single shape (a single symplectic irrep) as for the
${}^{8}\mathrm{Be}$ ground state and the corresponding lowest $2^{+}$ state
(Fig. 7a). However, for the $0_{2}^{+}$ and $2_{2}^{+}$ states, the
wavefunction consists of a mixture of three dominant prolate shapes (Fig. 7b).
Thus, one needs calculations in ultra large model spaces and consideration of
all the major shapes to obtain an accurate description of these states.

Figure 7: Symplectic $\mathrm{Sp}(3,\mathbb{R})$ irreps that compose the
lowest $0_{\rm g.s.}^{+}$ and $2_{1}^{+}$ states (a), and the second
$0_{2}^{+}$ and $2_{2}^{+}$ states of ${}^{8}\mathrm{Be}$; each irrep is
specified by its equilibrium shape, labeled by $\beta$ and the corresponding
SU(3) labels $(\lambda\,\mu)$ together with total intrinsic spin $S$. $N$
denotes the total HO excitations. Insets: the same irreps but without the
terms larger than 5%.

## References

  * Feynman and Gell-Mann (1958) R. P. Feynman and M. Gell-Mann, Phys. Rev. 109, 193 (1958).
  * Sudarshan and Marshak (1958) E. C. Sudarshan and R. Marshak, Phys. Rev. 109, 1860 (1958).
  * Wu  _et al._ (1957) C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 105, 1413 (1957).
  * Herrmannsfeldt  _et al._ (1957) W. B. Herrmannsfeldt, D. R. Maxson, P. Stähelin, and J. S. Allen, Phys. Rev. 107, 641 (1957).
  * Johnson  _et al._ (1963) C. Johnson, F. Pleasonton, and T. Carlson, Phys. Rev. 132, 1149 (1963).
  * Fléchard  _et al._ (2011) X. Fléchard, P. Velten, E. Liénard, A. Méry, D. Rodríguez, G. Ban, D. Durand, F. Mauger, O. Naviliat-Cuncic, and J. Thomas, J. Phys. G: Nucl. Part. Phys. 38, 055101 (2011).
  * Li  _et al._ (2013) G. Li, R. Segel, N. D. Scielzo, P. F. Bertone, F. Buchinger, S. Caldwell, A. Chaudhuri, J. A. Clark, J. E. Crawford, C. M. Deibel, J. Fallis, S. Gulick, G. Gwinner, D. Lascar, A. F. Levand, M. Pedretti, G. Savard, K. S. Sharma, M. G. Sternberg, T. Sun, J. Van Schelt, R. M. Yee, and B. J. Zabransky, Phys. Rev. Lett. 110, 092502 (2013).
  * Sternberg  _et al._ (2015) M. Sternberg, R. Segel, N. Scielzo, G. Savard, J. Clark, P. Bertone, F. Buchinger, M. Burkey, S. Caldwell, A. Chaudhuri, _et al._ , Phys. Rev. Lett. 115, 182501 (2015).
  * Severijns  _et al._ (2006) N. Severijns, M. Beck, and O. Naviliat-Cuncic, Rev. Mod. Phys. 78, 991 (2006).
  * Wauters  _et al._ (2014) F. Wauters, A. García, and R. Hong, Phys. Rev. C 89, 025501 (2014).
  * Grenacs (1985) L. Grenacs, Annu. Rev. Nucl. Part. Sci. 35, 455 (1985).
  * De Braeckeleer  _et al._ (1995) L. De Braeckeleer, E. G. Adelberger, J. H. Gundlach, M. Kaplan, D. Markoff, A. M. Nathan, W. Schieff, K. A. Snover, D. W. Storm, K. B. Swartz, D. Wright, and B. A. Brown, Phys. Rev. C 51, 2778 (1995).
  * Sumikama  _et al._ (2008) T. Sumikama, K. Matsuta, T. Nagatomo, M. Ogura, T. Iwakoshi, Y. Nakashima, H. Fujiwara, M. Fukuda, M. Mihara, K. Minamisono, _et al._ , Phys. Lett. B 664, 235 (2008).
  * Holstein (1974) B. R. Holstein, Rev. Mod. Phys. 46, 789 (1974).
  * Tribble and Garvey (1975) R. E. Tribble and G. T. Garvey, Phys. Rev. C 12, 967 (1975).
  * Sumikama  _et al._ (2011) T. Sumikama, K. Matsuta, T. Nagatomo, M. Ogura, T. Iwakoshi, Y. Nakashima, H. Fujiwara, M. Fukuda, M. Mihara, K. Minamisono, T. Yamaguchi, and T. Minamisono, Phys. Rev. C 83, 065501 (2011).
  * Barnes  _et al._ (1958) C. A. Barnes, W. A. Fowler, H. B. Greenstein, C. C. Lauritsen, and M. E. Nordberg, Phys. Rev. Lett. 1, 328 (1958).
  * Scielzo  _et al._ (2012) N. Scielzo, G. Li, M. Sternberg, G. Savard, P. Bertone, F. Buchinger, S. Caldwell, J. Clark, J. Crawford, C. Deibel, _et al._ , Nucl. Instrum. Methods Phys. Res. A: Accel. Spectrom. Detect. Assoc. Equip. 681, 94 (2012).
  * Burkey  _et al._ (2021) M. Burkey, G. Savard, A. Gallant, N. Scielzo, J. Clark, T. Hirsh, L. Varriano, G. Sargsyan, K. Launey, M. Brodeur, D. Burdette, _et al._ , Submitted, Phys. Rev. Lett. (2021).
  * McKeown  _et al._ (1980) R. D. McKeown, G. T. Garvey, and C. A. Gagliardi, Phys. Rev. C 22, 738 (1980).
  * Barker  _et al._ (1968) F. Barker, H. Hay, and P. Treacy, Aust. J. Phys. 21, 239 (1968).
  * Barker (1969) F. Barker, Aust. J. Phys. 22, 293 (1969).
  * Barker (1989) F. Barker, Aust. J. Phys. 42, 25 (1989).
  * Caurier  _et al._ (2001) E. Caurier, P. Navrátil, W. E. Ormand, and J. P. Vary, Phys. Rev. C 64, 051301(R) (2001).
  * Maris (2013) P. Maris, in  _J. Phys. Conf. Ser._ , Vol. 445 (IOP Publishing, 2013) p. 012035.
  * Rodkin and Tchuvil’sky (2020) D. Rodkin and Y. M. Tchuvil’sky, Chin. Phys. C 44, 124105 (2020).
  * Munch  _et al._ (2018) M. Munch, O. Sølund Kirsebom, J. A. Swartz, K. Riisager, and H. O. U. Fynbo, Phys. Lett. B 782, 779 (2018).
  * Launey  _et al._ (2016) K. D. Launey, T. Dytrych, and J. P. Draayer, Prog. Part. Nucl. Phys. 89, 101 (2016).
  * Dytrych  _et al._ (2020) T. Dytrych, K. D. Launey, J. P. Draayer, D. J. Rowe, J. L. Wood, G. Rosensteel, C. Bahri, D. Langr, and R. B. Baker, Phys. Rev. Lett. 124, 042501 (2020).
  * Launey  _et al._ (2021) K. D. Launey, A. Mercenne, and T. Dytrych, Annu. Rev. Nucl. Part. Sci. 71 (2021).
  * Bedaque and van Kolck (2002) P. F. Bedaque and U. van Kolck, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002).
  * Epelbaum  _et al._ (2002) E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. Meißner, and H. Witala, Phys. Rev. C 66, 064001 (2002).
  * Entem and Machleidt (2003) D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001(R) (2003).
  * Epelbaum (2006) E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006).
  * Wendt  _et al._ (2015) K. A. Wendt, C. Forssén, T. Papenbrock, and D. Sääf, Phys. Rev. C 91, 061301(R) (2015).
  * Navrátil  _et al._ (2000) P. Navrátil, J. P. Vary, and B. R. Barrett, Phys. Rev. Lett. 84, 5728 (2000).
  * Navrátil  _et al._ (2000) P. Navrátil, J. Vary, and B. Barrett, Phys. Rev. C 62, 054311 (2000).
  * Dreyfuss  _et al._ (2020) A. C. Dreyfuss, K. D. Launey, J. E. Escher, G. H. Sargsyan, R. B. Baker, T. Dytrych, and J. P. Draayer, Phys. Rev. C 102, 044608 (2020).
  * Borremans  _et al._ (2005) D. Borremans, D. L. Balabanski, K. Blaum, W. Geithner, S. Gheysen, P. Himpe, M. Kowalska, J. Lassen, P. Lievens, S. Mallion, R. Neugart, G. Neyens, N. Vermeulen, and D. Yordanov, Phys. Rev. C 72, 044309 (2005).
  * Ekström  _et al._ (2013) A. Ekström, G. Baardsen, C. Forssén, G. Hagen, M. Hjorth-Jensen, G. R. Jansen, R. Machleidt, W. Nazarewicz, _et al._ , Phys. Rev. Lett. 110, 192502 (2013).
  * Ekström  _et al._ (2015) A. Ekström, G. R. Jansen, K. A. Wendt, G. Hagen, T. Papenbrock, B. D. Carlsson, C. Forssén, M. Hjorth-Jensen, P. Navrátil, and W. Nazarewicz, Phys. Rev. C 91, 051301 (2015).
  * Shirokov  _et al._ (2010) A. Shirokov, V. Kulikov, P. Maris, A. Mazur, E. Mazur, and J. Vary, in  _EPJ Web of Conferences_ , Vol. 3 (EDP Sciences, 2010) p. 05015.
  * Baker  _et al._ (2020) R. B. Baker, K. D. Launey, S. Bacca, N. N. Dinur, and T. Dytrych, Phys. Rev. C 102, 014320 (2020).
  * Burrows  _et al._ (2019) M. Burrows, C. Elster, S. P. Weppner, K. D. Launey, P. Maris, A. Nogga, and G. Popa, Phys. Rev. C 99, 044603 (2019).
  * Ruotsalainen  _et al._ (2019) P. Ruotsalainen, J. Henderson, G. Hackman, G. H. Sargsyan, K. D. Launey, A. Saxena, P. C. Srivastava, S. R. Stroberg, T. Grahn, J. Pakarinen, G. C. Ball, R. Julin, P. T. Greenlees, J. Smallcombe, C. Andreoiu, N. Bernier, M. Bowry, M. Buckner, R. Caballero-Folch, A. Chester, S. Cruz, L. J. Evitts, R. Frederick, A. B. Garnsworthy, M. Holl, A. Kurkjian, D. Kisliuk, K. G. Leach, E. McGee, J. Measures, D. Mücher, J. Park, F. Sarazin, J. K. Smith, D. Southall, K. Starosta, C. E. Svensson, K. Whitmore, M. Williams, and C. Y. Wu, Phys. Rev. C 99, 051301(R) (2019).
  * McCracken  _et al._ (2021) C. McCracken, P. Navrátil, A. McCoy, S. Quaglioni, and G. Hupin, Phys. Rev. C 103, 035801 (2021).
  * Glick-Magid  _et al._ (2021) A. Glick-Magid, C. Forssén, D. Gazda, D. Gazit, P. Gysbers, and P. Navrátil, arXiv preprint arXiv:2107.10212 (2021).
  * Burkey (2019) M. T. Burkey,  _Searching for tensor currents in the weak interaction using lithium-8 $\beta$ decay_, Ph.D. thesis, Chicago U. (2019).
  * Filin  _et al._ (2021) A. A. Filin, D. Möller, V. Baru, E. Epelbaum, H. Krebs, and P. Reinert, Phys. Rev. C 103, 024313 (2021).
  * Maris  _et al._ (2021) P. Maris, E. Epelbaum, R. J. Furnstahl, J. Golak, K. Hebeler, T. Hüther, H. Kamada, H. Krebs, U.-G. Meißner, J. A. Melendez, A. Nogga, P. Reinert, R. Roth, R. Skibiński, V. Soloviov, K. Topolnicki, J. P. Vary, Y. Volkotrub, H. Witała, and T. Wolfgruber (LENPIC Collaboration), Phys. Rev. C 103, 054001 (2021).
  * (51) See Supplemental Material at [url], which includes Refs. [52–55], for additional correlation plots, intercluster wavefunctions used in calculations of widths, as well as the decomposition of the ground and intruder states in the SA basis .
  * Suhonen (2007) J. Suhonen,  _From nucleons to nucleus: concepts of microscopic nuclear theory_ (Springer Science & Business Media, 2007).
  * Navrátil (2004) P. Navrátil, Phys. Rev. C 70, 054324 (2004).
  * Castaños  _et al._ (1988) O. Castaños, J. P. Draayer, and Y. Leschber, Z. Phys. A 329, 33 (1988).
  * Mustonen  _et al._ (2018) M. T. Mustonen, C. N. Gilbreth, Y. Alhassid, and G. F. Bertsch, Phys. Rev. C 98, 034317 (2018).
  * Huang  _et al._ (2021) W. Huang, M. Wang, F. Kondev, G. Audi, and S. Naimi, Chin. Phys. C 45, 030002 (2021).
  * Tilley  _et al._ (2004) D. Tilley, J. Kelley, J. Godwin, D. Millener, J. Purcell, C. Sheu, and H. Weller, Nucl. Phys. A 745, 155 (2004).
  * Dreyfuss  _et al._ (2013) A. C. Dreyfuss, K. D. Launey, T. Dytrych, J. P. Draayer, and C. Bahri, Phys. Lett. B 727, 511 (2013).
  * Maris  _et al._ (2009) P. Maris, J. P. Vary, and A. M. Shirokov, Phys. Rev. C 79, 014308 (2009).
  * Navrátil  _et al._ (2010) P. Navrátil, R. Roth, and S. Quaglioni, Phys. Rev. C 82, 034609 (2010).
  * Kravvaris and Volya (2017) K. Kravvaris and A. Volya, Phys. Rev. Lett. 119, 062501 (2017).
  * Kravvaris  _et al._ (2020) K. Kravvaris, S. Quaglioni, G. Hupin, and P. Navratil, arXiv preprint arXiv:2012.00228 (2020).
  * Elhatisari  _et al._ (2015) S. Elhatisari, D. Lee, G. Rupak, E. Epelbaum, _et al._ , Nature 528, 111 (2015).
  * Warburton (1986) E. K. Warburton, Phys. Rev. C 33, 303 (1986).
  * King  _et al._ (2020) G. B. King, L. Andreoli, S. Pastore, M. Piarulli, R. Schiavilla, R. B. Wiringa, J. Carlson, and S. Gandolfi, Phys. Rev. C 102, 025501 (2020).