# Isospin-symmetry implications for nuclear two-body distributions and short-
range correlations

Ronen Weiss${}^{\,{\rm a}}$, Alessandro Lovato ${}^{\,{\rm b,c,d}}$, R. B.
Wiringa ${}^{\,{\rm b}}$ ${}^{\,{\rm a}}$Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA
${}^{\,{\rm b}}$Physics Division, Argonne National Laboratory, Argonne,
Illinois 60439, USA
${}^{\,{\rm c}}$Computational Science Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA
${}^{\,{\rm d}}$INFN-TIFPA Trento Institute of Fundamental Physics and
Applications, Via Sommarive, 14, 38123 Trento, Italy

###### Abstract

We study the implications of isospin symmetry in nuclear systems on two-body
distributions and derive relations between different densities or momentum
distributions of a given nucleus or of different nuclei in the same isospin
multiplet. A connection between neutrinoless double beta decay transition of
mirror nuclei and two-body densities is also obtained. The relations are
numerically verified by means of quantum Monte Carlo calculations. We discuss
the relevance of these results to the analysis of nuclear short-range
correlations and also show that isospin symmetry allows us to extract
information about the spectator nucleons when a short-range correlated pair is
formed.

††preprint: LA-UR-22-23223

##  I Introduction

Although the proton has a positive charge, and the neutron is neutral, they
are very similar particles, with nearly degenerate masses. Back in 1932,
Heisenberg argued that the proton and neutron can be regarded as different
isospin states of the same underlying nucleon particle Heisenberg (1932).
Specifically, the protons and neutrons are considered to form an isospin
doublet, with total isospin $T=1/2$, and isospin projection $t_{z}=1/2$ and
$t_{z}=-1/2$ for the proton and the neutron, respectively.

The proton-proton ($pp$), neutron-neutron ($nn$), and proton-neutron ($pn$)
interactions in a given total isospin channel are approximately equal, as the
nuclear Hamiltonian commutes with the total isospin operator, up to small
corrections due to the Coulomb force and other small isospin-breaking terms.
Therefore, nuclear eigenstates have well-defined total isospin $T$ and isospin
projection $T_{z}$. Consequences of this approximate isospin symmetry can be
seen, for example, by comparing the energy levels of different nuclei with the
same total number of nucleons. In addition, when modeling the interactions
between two nucleons, we are mainly concerned with only two isospin states
instead of dealing with four charge states.

In this work, we carry out a detailed analysis of the implications of isospin
symmetry on two-body distribution functions. The latter encode nuclear
correlation effects and are key quantities to model the interaction of atomic
nuclei with electroweak probes. Two-body momentum distributions provide
important input for the description of correlation effects in electron-
scattering experiments Weiss  _et al._ (2019); Andreoli  _et al._ (2022). In
coordinate space, information regarding short-range correlations (SRCs) can be
extracted from two-body densities Cruz-Torres  _et al._ (2021). As another
example, neutrinoless double beta ($0\nu\beta\beta$) decay matrix elements can
be directly calculated using appropriate two-body transition densities
Cirigliano  _et al._ (2022); Agostini  _et al._ (2022); Yao  _et al._ (2022);
Šimkovic  _et al._ (2008, 2018); Cirigliano  _et al._ (2019); Weiss  _et al._
(2022). Finally, two-body current contributions in electroweak transitions can
be expressed in terms of transition densities that exhibit a universal
behavior at short distances King  _et al._ (2020).

In this work, we derive analytic relations between different diagonal and off-
diagonal two-body distribution functions, including two-body densities,
momentum distributions, and $0\nu\beta\beta$ decay transition densities, of a
given nucleus or of different nuclei in the same isospin multiplet. These
relations are completely general, as they do not depend on the specific
nuclear Hamiltonian of choice, provided that isospin-breaking terms are small.
Our relations enable sanity checks in ab-initio many-body calculations of
diagonal and off-diagonal two-body density distributions. They are also
relevant for experimental investigations of nuclear SRCs, i.e. nucleon pairs
found in close proximity inside the nucleus.

This paper is organized as follows. In Section II we present a general
expansion of nuclear wave functions assuming isospin symmetry to be exact. In
Section III, we derive analytic relations between different diagonal and off-
diagonal two-body distribution functions. Their implications to the study of
nuclear SRCs are discussed in Section IV. Finally, in Section V we summarize
and discuss the importance of our findings.

##  II wave function expansion

We consider a nuclear wave function $\Psi$ of $A$ nucleons with $T$ and
$T_{z}$ isospin quantum numbers, neglecting the effect of isospin-breaking
terms. To analyze two-body quantities, it is convenient to expand it using a
complete set of orthonormal anti-symmetric two-body functions of the form

$\varphi_{m}^{t,t_{z}}(\boldsymbol{r}_{ij})\equiv\varphi_{m}^{t}(\boldsymbol{r}_{ij})\eta_{t,t_{z}},$
(1)

where $t$ is the isospin of the pair, $t_{z}$ its projection, $m$ denotes all
the remaining quantum numbers, and
$\boldsymbol{r}_{ij}=\boldsymbol{r}_{i}-\boldsymbol{r}_{j}$ is the relative
coordinate. The radial and spin part is given by
$\varphi_{m}^{t}(\boldsymbol{r})$, while the isospin part is given by
$\eta_{t,t_{z}}$. The latter is obtained from the coupling of two isospin-half
nucleons and is normalized to 1, i.e.
$\eta_{t,t_{z}}^{\dagger}\eta_{t,t_{z}}=1$.

Expanding $\Psi$ using this basis we obtain

$\displaystyle\Psi(\boldsymbol{r}_{1},...,\boldsymbol{r}_{A})$
$\displaystyle=\sum_{m}\varphi_{m}^{1}(\boldsymbol{r}_{12})\sum_{t_{z}=-1}^{1}\eta_{1,t_{z}}S_{m}^{1,t_{z}}(\boldsymbol{R}_{12},\boldsymbol{r}_{3},...,\boldsymbol{r}_{A})$
$\displaystyle+\sum_{m}\varphi_{m}^{0}(\boldsymbol{r}_{12})\eta_{0,0}S_{m}^{0,0}(\boldsymbol{R}_{12},\boldsymbol{r}_{3},...,\boldsymbol{r}_{A}),$
(2)

where $\boldsymbol{r}_{i}$ are single-nucleon coordinates,
$\boldsymbol{R}_{ij}=(\boldsymbol{r}_{i}+\boldsymbol{r}_{j})/2$ is the center-
of-mass (CM) coordinate of the pair, and $S_{m}^{t,t_{z}}$ serve as the
coefficients in this expansion. Note that the superscripts $t,t_{z}$ and
subscript $m$ in $S_{m}^{t,t_{z}}$ are just indices and do not represent its
quantum numbers. We stress that this is an exact expansion of $\Psi$ as the
functions $\varphi_{m}^{t,t_{z}}(\boldsymbol{r}_{ij})$ form a complete basis.
In particular, the anti-symmetry of $\Psi$ for the exchange of any two
nucleons is respected in this expansion.

Since $\varphi_{m}^{t}$ are orthogonal, we conclude that the functions

$\sum_{t_{z}=-1}^{1}\eta_{1,t_{z}}S_{m}^{1,t_{z}}(\boldsymbol{R}_{12},\boldsymbol{r}_{3},...,\boldsymbol{r}_{A})$
(3)

and

$\eta_{0,0}S_{m}^{0,0}(\boldsymbol{R}_{12},\boldsymbol{r}_{3},...,\boldsymbol{r}_{A})$
(4)

must each separately have the same $T,T_{z}$ quantum numbers as $\Psi$, for
any $m$ (see Appendix A for more details). In the second expression,
$\eta_{0,0}$ is an isospin-zero function. Therefore, $S_{m}^{0,0}$ must be an
isospin eigenstate with $T,T_{z}$ quantum numbers and we will thus use the
notation $S_{m}^{0,0}(T,T_{z})$. Regarding the first expression, because it
has $T,T_{z}$ quantum numbers, it can be written as

$\displaystyle\sum_{t_{z}=-1}^{1}\eta_{1,t_{z}}S_{m}^{1,t_{z}}=\sum_{T^{A-2}}\left[\tilde{S}_{m}(T^{A-2})\otimes\eta_{1}\right]^{T,T_{z}}$
$\displaystyle=\sum_{t_{z}}\eta_{1,t_{z}}\sum_{T^{A-2}}\langle
T^{A-2}(T_{z}-t_{z})1t_{z}|TT_{z}\rangle\tilde{S}_{m}(T^{A-2},T_{z}-t_{z}).$
(5)

The sum over $T^{A-2}$ accounts for the different possible values of the $A-2$
nucleon isospin, such that coupled to $t=1$ it can produce the total isospin
$T$. $\tilde{S}_{m}(T^{A-2},T_{z}^{A-2})$ is a set of functions with well-
defined isospin quantum numbers of the $A-2$ nucleons. Their dependence on
$\boldsymbol{R}_{12},\boldsymbol{r}_{3},...,\boldsymbol{r}_{A}$ is suppressed
here. $\langle T^{A-2}T_{z}^{A-2}tt_{z}|TT_{z}\rangle$ are Clebsch-Gordan (CG)
coefficients. The last equation allows us to express $S_{m}^{1,t_{z}}$ using
the $\tilde{S}_{m}(T^{A-2},T_{z}^{A-2})$ functions

$S_{m}^{1,t_{z}}=\sum_{T^{A-2}}\langle
T^{A-2}(T_{z}-t_{z})1t_{z}|TT_{z}\rangle\tilde{S}_{m}(T^{A-2},T_{z}-t_{z}).$
(6)

This relation, with the given set of $\tilde{S}_{m}(T^{A-2},T_{z}^{A-2})$
functions, is applicable for all nuclei in the isospin multiplet of $\Psi$,
i.e. for any value of $T_{z}$. This is the starting point for the relations
derived in this paper.

##  III two-body distributions

We start by deriving relations involving the probability of finding two
nucleons with a relative distance $r$ and isospin quantum numbers $t,t_{z}$,
i.e. the two-body density $\rho_{t,t_{z}}(r)$ defined as

$\rho_{t,t_{z}}(r)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\langle\Psi|\delta(r-r_{12})\hat{\cal{P}}_{12}^{t,t_{z}}|\Psi\rangle.$
(7)

$\hat{\cal{P}}_{12}^{t,t_{z}}$ is a $t,t_{z}$ projection for the combined
isospin of particles $1$ and $2$. $\rho_{t,t_{z}}(r)$ obeys $\int dr4\pi
r^{2}\rho_{t,t_{z}}(r)=N_{t,t_{z}}$, where $N_{t,t_{z}}$ is the total number
of $t,t_{z}$ pairs in the nucleus.

In general, we derive here two kinds of relations. The first kind connects
different $\rho_{t,t_{z}}(r)$ densities of a given nucleus, while the second
kind connects different $\rho_{t,t_{z}}(r)$ densities of different nuclei that
are part of the same isospin multiplet.

###  III.1 Relations for a single nucleus

Starting with the first kind of relations, we focus on the $t=1$ densities.
Using Eqs. (II) and (6) and the orthogonality of $\tilde{S}_{m}$ with respect
to $T^{A-2}$, we obtain

$\displaystyle\rho_{1,t_{z}}$ $\displaystyle(r)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\sum_{T^{A-2}}|\langle T^{A-2}(T_{z}-t_{z})1t_{z}|TT_{z}\rangle|^{2}$
$\displaystyle\times\sum_{m,m^{\prime}}\langle\varphi_{m}^{1}|\delta(r-r_{12})|\varphi_{m^{\prime}}^{1}\rangle\langle\tilde{S}_{m}(T^{A-2})|\tilde{S}_{m^{\prime}}(T^{A-2})\rangle.$
(8)

Notice that the $\langle\tilde{S}_{m}|\tilde{S}_{m^{\prime}}\rangle$ matrix
element is independent of the $T_{z}^{A-2}$ component, due to the Wigner-
Eckart theorem, and, therefore, it is suppressed above. As a result, the only
dependence on $t_{z}$ appears in the CG coefficients. This observation allows
us to derive relations between different $t_{z}$ densities for certain values
of $T$.

Starting with $T=0$ nuclei, the only possible contribution comes from
$T^{A-2}=1$. In addition, ${|\langle 1-t_{z}1t_{z}|00\rangle|^{2}=1/3}$ for
all $t_{z}=\pm 1,0$. Therefore, the $t_{z}$ dependence disappears and we
obtain the expected relation for $T=0$ nuclei

$\rho_{1,1}(r)=\rho_{1,0}(r)=\rho_{1,-1}(r)\;\;\;\;\;\;\;(T=0\;\textrm{nuclei}),$
(9)

i.e. the $pp$, $nn$, and $t=1$ $pn$ densities are equal. The $pp$ and $nn$
densities, but not necessarily the $t=1$ $pn$ density $\rho_{1,0}$, are
expected to be identical for any state with $T_{z}=0$ (and any integer value
of $T$). This is simply obtained using this formulation based on the CG
identity

$|\langle T^{A-2}\;-t_{z}\;1\;t_{z}|T\;0\rangle|^{2}=|\langle
T^{A-2}\;t_{z}\;1\;-t_{z}|T\;0\rangle|^{2},$ (10)

relating the $t_{z}=1$ and $t_{z}=-1$ densities.

We move now to $T=1/2$ nuclei. In this case, both $T^{A-2}=1/2$ and
$T^{A-2}=3/2$ contribute. Looking at Eq. (III.1), the three
$\rho_{1,t_{z}}(r)$ densities, for $t_{z}=\pm 1,0$, are expressed using only
two quantities of the form

$\sum_{m,m^{\prime}}\langle\varphi_{m}^{1}|\delta(r-r_{12})|\varphi_{m^{\prime}}^{1}\rangle\langle\tilde{S}_{m}(T^{A-2})|\tilde{S}_{m^{\prime}}(T^{A-2})\rangle,$
(11)

for $T^{A-2}=1/2,3/2$. Therefore, there must be a linear relation between
these three densities. A simple calculation, for both $T_{z}=\pm 1/2$, leads
to the result (see Appendix B for more details):

$2\rho_{1,0}(r)=\rho_{1,1}(r)+\rho_{1,-1}(r)\;\;\;\;\;\;\;(T=\frac{1}{2}\;\textrm{nuclei}),$
(12)

i.e. the $t=1$ $pn$ density is the average of the $pp$ and $nn$ densities for
$T=1/2$ nuclei. The same relation holds also if the densities are projected to
specific spin $s=0$ or $s=1$ of the pair. This is a non-trivial result of
isospin symmetry. We can of course also integrate over the densities and
obtain expressions for the total number of $t=1$ and $t=0$ pairs. They are
consistent with the more general formulas for the number of $t=1$ and $t=0$
pairs derived in Ref. Forest  _et al._ (1996) for any value of $T$, see Eqs.
(6.1) and (6.2).

For $T\geq 1$ there are three possible values of $T^{A-2}$. There are also
three $\rho_{1,t_{z}}(r)$ densities. Therefore, generally, no internal
relation between the $\rho_{1,t_{z}}(r)$ densities can be obtained based on
isospin symmetry. This is also a non-trivial statement regarding the
implications of isospin symmetry: the three $\rho_{1,t_{z}}(r)$ densities of a
given nucleus are related to each other only for $T=0$ and $T=1/2$ nuclei (or
$T_{z}=0$ nuclei, for which the $pp$ and $nn$ densities are equal
$\rho_{pp}=\rho_{nn}$).

We numerically test the above relations against variational Monte Carlo (VMC)
calculations of light nuclei. The VMC method Carlson  _et al._ (2015)
approximates the solution of the nuclear Schrödinger equation with a trial
wave function of the form

$|\Psi_{T}\rangle=\Big{(}1+\sum_{i<j<k}F_{ijk}\Big{)}\Big{(}\mathcal{S}\prod_{i<j}F_{ij}\Big{)}|\Psi_{J}\rangle\,$
(13)

where $F_{ij}$ and $F_{ijk}$ are two- and three-body correlation operators,
respectively. The symbol $\mathcal{S}$ indicates a symmetrized product over
nucleon pairs since, in general, the spin-isospin dependent $F_{ij}$ do not
commute. The strong $\alpha$-cluster structure of light nuclei is explicitly
accounted for by the antisymmetric Jastrow wave function $|\Psi_{J}\rangle$
that is constructed from a sum over independent-particle terms, each having
four nucleons in an $\alpha$-like core and the remaining $(A-4)$ nucleons in
p-shell orbitals Pieper  _et al._ (2002). The optimal set of variational
parameters defining $F_{ij}$, $F_{ijk}$, and $|\Psi_{J}\rangle$ is found by
minimizing the expectation value of the energy

$E_{T}\equiv\frac{\langle\Psi_{T}|H|\Psi_{T}\rangle}{\langle\Psi_{T}|\Psi_{T}\rangle}\geq
E_{0},$ (14)

where $E_{0}$ is the true ground-state energy of the system. Evaluating the
above expectation value requires carrying out a multi-dimensional integration
over the $3A$ spatial coordinates of the nucleons. This is done in a
stochastic fashion using the Metropolis-Hastings algorithm Metropolis  _et
al._ (1953); Hastings (1970).

Calculations for 7Li $(T=1/2)$ are presented in Fig. 1. The $T=1/2$ relation,
Eq. (12), is satisfied to a very good accuracy for both $s=0$ and $s=1$.
Notice that the calculations are performed using the Argonne v18 (AV18) two-
body interaction Wiringa  _et al._ (1995), including the full electromagnetic
interaction, and the Urbana X (UX) three-body interaction, which is a
truncation of the Illinois-7 model Pieper  _et al._ (2001). These models
include small isospin-breaking terms, but explicit isospin-breaking operators
are not included in the $F_{ij}$ or $F_{ijk}$.

Figure 1:  $t=1$ VMC two-body densities for 7Li using the AV18+UX potential,
with spin $s=0$ (top) and $s=1$ (bottom).

###  III.2 Relations for nuclei in the same isospin multiplet

We now discuss a second kind of relations, which connect densities of
different nuclei in the same isospin multiplet.

We can start with the $t=0$ case, i.e. isospin-zero $pn$ pairs. Based on Eqs.
(II) and (7) we obtain an expression for the $t=0$ two-body density for a
nucleus with $T,T_{z}$ quantum numbers

$\displaystyle\rho_{0,0}(r)$ $\displaystyle=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}$
$\displaystyle\times\sum_{m,m^{\prime}}\langle\varphi_{m}^{0}|\delta(r-r_{12})|\varphi_{m^{\prime}}^{0}\rangle\langle
S_{m}^{0,0}(T,T_{z})|S_{m^{\prime}}^{0,0}(T,T_{z})\rangle.$ (15)

The isospin quantum numbers of $S_{m}^{0,0}$, as identified above, are written
here explicitly. Based on the Wigner-Eckart theorem the matrix element
$\langle S_{m}^{0,0}(T,T_{z})|S_{m^{\prime}}^{0,0}(T,T_{z})\rangle$ does not
depend on $T_{z}$. Therefore, $\rho_{0,0}$ does not depend on $T_{z}$ and we
conclude that the $t=0$ two-body density is identical for all nuclei in the
same isospin multiplet, assuming isospin symmetry. This remains true if the
densities are projected to a specific spin $s$.

As for the previous relations, we can verify the above relation through a
comparison with VMC calculations. Results for the 6He-6Li*-6Be
$(J^{\pi}=0^{+},T=1)$ multiplet, where 6Li* is an excited state, are displayed
in Fig. 2. We can see that indeed the $t=0$ densities are equal for all nuclei
in this multiplet to a good accuracy, with some small deviation for $s=0$,
possibly due to the statistics of the calculations or small isospin-breaking
effects. Notice that this $s=0$ density is significantly smaller than the
$s=1$ density for $t=0$ pairs.

Figure 2:  $t=0$ VMC two-body densities for 6Be, 6Li*, and 6He using the
AV18+UX potential, with spin $s=0$ (top) and $s=1$ (bottom).

Relations for the $t=1$ densities can also be derived. The general expression
for these densities is given in Eq. (III.1), where there are at most three
relevant values of $T^{A-2}$. Therefore, the three $t=1$ densities in a given
nucleus, corresponding to three $t_{z}$ values, depend on no more than three
quantities of the form given in Eq. (11). Considering a specific $(T,T_{z})$
nucleus, the three (or less) quantities of Eq. (11) can be expressed using the
three $\rho_{1,t_{z}}(r)$ densities. Notice that these quantities are
identical for all nuclei in the same multiplet and the only $T_{z}$ dependence
appears in the CG coefficients in Eq. (III.1). Therefore, the
$\rho_{1,t_{z}}(r)$ densities of the remaining nuclei in the same multiplet
can be directly expressed using the same densities of the $(T,T_{z})$ nucleus,
i.e. knowing the three $\rho_{1,t_{z}}(r)$ densities of one nucleus allows us
to obtain the three $\rho_{1,t_{z}}(r)$ densities of any nucleus in the same
isospin multiplet.

Such relations exist for any value of $T$. We provide here explicit relations
for a few cases but other cases can be easily derived. We use here the
notation $\rho_{t,t_{z}}(T,T_{z})$ for the $t,t_{z}$ density in a $(T,T_{z})$
nucleus, suppressing the $r$ dependence. Following the above arguments, for
$T=1/2$ nuclei we obtain the following relations:

$\displaystyle\rho_{1,0}\left(\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle=\rho_{1,0}\left(\frac{1}{2},-\frac{1}{2}\right),$ (16)
$\displaystyle\rho_{1,\pm 1}\left(\frac{1}{2},\frac{1}{2}\right)$
$\displaystyle=\rho_{1,\mp 1}\left(\frac{1}{2},-\frac{1}{2}\right),$ (17)

i.e. the $pn$ $t=1$ density is identical for the $T_{z}=\pm 1/2$ nuclei while
the $pp$ density in one nucleus is identical to the $nn$ density in the other.
These relations are very much expected as these two nuclei are mirror nuclei.
Using this formalism and CG identities, it can be shown that they hold for any
pair of mirror nuclei, i.e.

$\displaystyle\rho_{1,0}\left(T,T_{z}\right)$
$\displaystyle=\rho_{1,0}\left(T,-T_{z}\right),$ (18)
$\displaystyle\rho_{1,\pm 1}\left(T,T_{z}\right)$ $\displaystyle=\rho_{1,\mp
1}\left(T,-T_{z}\right).$ (19)

Following the same ideas, much less trivial relations are obtained for the
$T=1$ multiplet:

$\displaystyle\rho_{1,1}(1,0)=\rho_{1,-1}(1,0)=\rho_{1,0}(1,1)$ (20)
$\displaystyle\rho_{1,0}(1,0)=\rho_{1,1}(1,1)-\rho_{1,0}(1,1)+\rho_{1,-1}(1,1).$
(21)

In this case, the $t=1$ densities of the $(T=1,T_{z}=0)$ nucleus are expressed
using the $t=1$ densities of the $(T=1,T_{z}=1)$ nucleus. Notice that the
former is most likely an excited state, while the latter can be a ground
state. This allows us to access information on excited states using ground-
state quantities. Specifically, the $pp$ and $nn$ densities of the
$(T=1,T_{z}=0)$ nucleus are equal to the $pn$ $t=1$ density of the
$(T=1,T_{z}=1)$ nucleus. The $pn$ $t=1$ density of the $(T=1,T_{z}=0)$ nucleus
is equal to a specific linear combination of the $pp$, $nn$ and $pn$ $t=1$
densities of the $(T=1,T_{z}=1)$ nucleus. Notice that the $(T=1,T_{z}=-1)$
nucleus is a mirror of the $(T=1,T_{z}=1)$ nucleus and its densities can be
obtained using Eqs. (18) and (19). All these relations hold also if the
densities are projected to a specific spin $s$.

For the $T=3/2$ multiplet we can similarly express the excited-state
$T_{z}=1/2$ densities using the $T_{z}=3/2$ densities:

$\displaystyle\rho_{1,1}\left(\frac{3}{2},\frac{1}{2}\right)=\frac{1}{3}\rho_{1,1}\left(\frac{3}{2},\frac{3}{2}\right)+\frac{2}{3}\rho_{1,0}\left(\frac{3}{2},\frac{3}{2}\right)$
(22) $\displaystyle\rho_{1,0}\left(\frac{3}{2},\frac{1}{2}\right)$
$\displaystyle=\frac{2}{3}\rho_{1,1}\left(\frac{3}{2},\frac{3}{2}\right)-\frac{1}{3}\rho_{1,0}\left(\frac{3}{2},\frac{3}{2}\right)$
$\displaystyle+\frac{2}{3}\rho_{1,-1}\left(\frac{3}{2},\frac{3}{2}\right)$
(23)
$\displaystyle\rho_{1,-1}\left(\frac{3}{2},\frac{1}{2}\right)=\frac{2}{3}\rho_{1,0}\left(\frac{3}{2},\frac{3}{2}\right)+\frac{1}{3}\rho_{1,-1}\left(\frac{3}{2},\frac{3}{2}\right).$
(24)

Also here the relations for mirror nuclei allow us to obtain the $T_{z}=-1/2$
and $T_{z}=-3/2$ densities, and again all these relations hold also if the
densities are projected to a specific spin $s$.

We can provide a comparison to VMC calculations also for these relations.
Results for the 6He-6Li*-6Be $(J^{\pi}=0^{+},T=1)$ multiplet are presented in
Figs. 3 and 4, investigating the $T=1$ relations given in Eqs. (20) and (21),
respectively. We can see that the relations are satisfied to a good accuracy,
with some small deviation for $s=1$, possibly due to isospin-breaking effects
or the statistics of the calculations. Notice that this $s=1$ density is
significantly smaller than the $s=0$ density for $t=1$ pairs.

Results for the 7He-7Li*-7Be*-7B $(J^{\pi}=3/2^{-},T=3/2)$ multiplet are shown
in Fig. 5. Once again we observe that VMC calculations obey the relations
given in Eqs. (22), (III.2) and (24) for spin $s=0$. The $s=1$ calculations
(not shown here) are also in good agreement with some small deviations,
similar to the $T=1$ case.

Figure 3:  $t=1$ VMC two-body densities involved in Eq. (20) for 6Be and 6Li*
using the AV18+UX potential, with spin $s=0$ (top) and $s=1$ (bottom).

Figure 4:  $t=1$ VMC two-body densities involved in Eq. (21) for 6Be and 6Li*
using the AV18+UX potential, with spin $s=0$ (top) and $s=1$ (bottom). The
legend of the bottom panel is identical to the top panel just with $s=1$.

Figure 5:  $t=1$ VMC two-body densities involved in Eqs. (22) (top), (III.2)
(middle) and (24) (bottom) for 7B and 7Be* using the AV18+UX potential, with
spin $s=0$.

Looking on the above $T=1/2$, $T=1$ and $T=3/2$ relations we can see that the
sum over three $\rho_{1,t_{z}}$ densities is the same for all nuclei in the
multiplet, i.e. the sum $\sum_{t_{z}}\rho_{1,t_{z}}(T,T_{z})$ is independent
of $T_{z}$. Based on Eq. (III.1) and the orthogonality relation of the CG
coefficients we can conclude that this is a general property valid for all
values of $T$.

We note that all the relations discussed in Secs. III.1 and III.2 for the two-
body densities hold equivalently for two-body momentum distributions,
describing the probability of finding two nucleons with a relative momentum
$k$ and isospin quantum number $t,t_{z}$.

###  III.3 $0\nu\beta\beta$ decay

Isospin-symmetry also has implications regarding the transition densities of
$0\nu\beta\beta$ decay between mirror nuclei. $0\nu\beta\beta$ decay is a
beyond-Standard-Model (BSM) process in which two neutrons decay into two
protons and two electrons without any accompanying anti-neutrinos. If
measured, it will have significant implications to our understanding of the
origin of the neutrino mass and of the matter dominance in the universe (see
Refs. Cirigliano  _et al._ (2022); Agostini  _et al._ (2022); Yao  _et al._
(2022) for recent reviews). Experimentally, there is a potential to observe
such a decay only in certain nuclei, all of them involve an isospin change of
$\Delta T=2$ between the initial and final nucleus. Nevertheless, theoretical
studies of $0\nu\beta\beta$ involve also decay between mirror nuclei ($\Delta
T=0$), e.g. for the purpose of testing different models Pastore  _et al._
(2018); Wang  _et al._ (2019); Yao  _et al._ (2021); Novario  _et al._ (2021).

Extracting quantitative BSM information from $0\nu\beta\beta$ decay
measurement requires the calculations of nuclear matrix elements (NMEs). These
NMEs can be calculated by integrating over transition densities (multiplied by
the appropriate neutrino potential and other factors), that can be separated
into Fermi (F), Gamow-Teller (GT), and tensor components. In this work, we
focus on the first two transitions, which are defined as (see e.g. Šimkovic
_et al._ (2008, 2018); Cirigliano  _et al._ (2019); Weiss  _et al._ (2022))

$\rho_{F}(r)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\langle\Psi_{f}|\delta(r-r_{12})\tau_{1}^{+}\tau_{2}^{+}|\Psi_{i}\rangle$
(25) $\rho_{GT}(r)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\langle\Psi_{f}|\delta(r-r_{12})\boldsymbol{\sigma}_{1}\cdot\boldsymbol{\sigma}_{2}\tau_{1}^{+}\tau_{2}^{+}|\Psi_{i}\rangle.$
(26)

Here, $\Psi_{i}$ and $\Psi_{f}$ are the initial and final nuclei, and
$\boldsymbol{\sigma}_{a}$ and $\boldsymbol{\tau}_{a}$ are the nucleon spin and
isospin operators of nucleon $a$, respectively.

We consider here a decay between two ground-state mirror nuclei. Therefore,
these are necessarily $T=1$ nuclei, where the initial and final states have
isospin projection $T_{z}(i)=-1$ and $T_{z}(f)=1$, respectively. The expansion
given in Eq. (II) and the relation given in Eq. (6) can be now utilized. For
the Fermi transition we obtain

$\displaystyle\rho_{F}(r)$ $\displaystyle=\frac{A(A-1)}{8\pi
r^{2}}\sum_{T^{A-2}}\langle T^{A-2}011|11\rangle^{*}\langle
T^{A-2}01-1|1-1\rangle$
$\displaystyle\times\sum_{m,m^{\prime}}\langle\varphi_{m}^{1}|\delta(r-r_{12})|\varphi_{m^{\prime}}^{1}\rangle\langle\tilde{S}_{m}(T^{A-2})|\tilde{S}_{m^{\prime}}(T^{A-2})\rangle.$
(27)

Notice that only $nn$ pairs in the initial state and $pp$ pairs in the final
state contribute. There are three values of $T^{A-2}$ that contribute here
$(0,1,2)$. In addition, the expression on the second line of Eq. (III.3) is
identical to the expression appearing in the two-body densities, Eq. (III.1).
As discussed above for the two-body densities, these expressions for the three
values of $T^{A-2}$ are directly related to the three densities
$\rho_{1,t_{z}}$ (of the initial state, for example). Therefore, we obtain a
relation between the Fermi $0\nu\beta\beta$ transition density of mirror
nuclei and the two-body densities

$\rho_{F}(r)=\rho_{1,-1}(r)+\rho_{1,1}(r)-2\rho_{1,0}(r),$ (28)

where on the right-hand side we have here the densities of the initial-state
nucleus. A similar relation for the GT transition can also be derived,
involving spin-projected two-body densities $\rho_{t,t_{z}}^{s}$ to account
for the effect of the $\sigma_{a}\cdot\sigma_{b}$ operator. We obtain

$\displaystyle\rho_{GT}(r)=$
$\displaystyle-3\left[\rho_{1,-1}^{s=0}(r)+\rho_{1,1}^{s=0}(r)-2\rho_{1,0}^{s=0}(r)\right]$
$\displaystyle+\left[\rho_{1,-1}^{s=1}(r)+\rho_{1,1}^{s=1}(r)-2\rho_{1,0}^{s=1}(r)\right].$
(29)

We can compare these relations to VMC calculations. Results of the
10Be$\rightarrow^{10}$C $\Delta T=0$ transition are presented in Fig. 6. A
remarkably good agreement is observed.

Figure 6:  (top) $\rho_{1,t_{z}}$ two-body densities for 10Be and the
10Be$\rightarrow^{10}$C $0\nu\beta\beta$ Fermi transition density using the
VMC method and the AV18+UX potential. The dashed line labeled as ”F relation”
shows the linear combination of $\rho_{1,t_{z}}$ for 10Be given in Eq. (28).
(bottom) The same, but for the spin-projected $\rho_{1,t_{z}}^{s}$ two-body
densities and the GT transition. The label ”GT relation” refers to the linear
combination of $\rho_{1,t_{z}}^{s}$ given in Eq. (III.3).

##  IV short-range correlations

In this section, we consider the implications of isospin symmetry on nuclear
SRCs. The study of nuclear SRCs is focused on describing the properties of
nucleons that are found close to each other inside the nucleus, and their
impact on different nuclear quantities. Currently, there is a comprehensive
understanding of the nature of two-nucleon SRCs, i.e. pairs of nucleons at
close proximity or with large relative momentum and small CM momentum. Such
pairs are predominantly back-to-back $pn$ pairs, due to the effect of the
tensor force, and are the origin for most of high-momentum nucleons in the
nucleus. This picture is corroborated by both experimental measurements,
mainly high-energy quasi-elastic electron- and proton-scattering measurements,
and theoretical studies, including ab-initio calculations of two-body
densities and momentum distributions — see, e.g., Refs. Ciofi degli Atti
(2015); Hen  _et al._ (2017); Arrington  _et al._ (2022); Wiringa  _et al._
(2014) and references therein.

The generalized contact formalism (GCF) is an effective theory for describing
SRCs. It is based on the asymptotic factorization of the nuclear wave
function, when two particles are close to each other, to a part describing the
correlated pair and another one describing the spectator $A-2$ nucleons Weiss
_et al._ (2015a):

$\Psi\xrightarrow[r_{ij}\rightarrow
0]{}\sum_{\alpha}\varphi^{\alpha}(\boldsymbol{r}_{ij})A^{\alpha}(\boldsymbol{R}_{ij},\\{\boldsymbol{r}_{k}\\}_{k\neq
i,j}).$ (30)

In the above equation, $\alpha$ denotes the quantum numbers of the pair, i.e.
parity $\pi_{\alpha}$, spin $s_{\alpha}$, total angular momentum $j_{\alpha}$
and projection $j_{\alpha z}$, and total isospin $t_{\alpha}$ and projection
$t_{\alpha z}$. The universal function $\varphi^{\alpha}(\boldsymbol{r}_{ij})$
is Hamiltonian dependent but nucleus independent. It describes the pair
dynamics and is defined as the solution of the zero-energy two-body
Schrödinger equation. This factorization of the wave function was tested
against ab-initio calculations Weiss  _et al._ (2015a, 2018); Cruz-Torres  _et
al._ (2021), and allows us to quantitatively study the properties of SRCs and
their impact on different quantities such as momentum distributions and two-
body densities Weiss  _et al._ (2015a, 2018); Cruz-Torres  _et al._ (2021),
electron-scattering cross sections Weiss  _et al._ (2019); Schmidt  _et al._
(2020); Pybus  _et al._ (2020); Duer  _et al._ (2019); Weiss  _et al._ (2021);
Korover  _et al._ (2021); Patsyuk  _et al._ (2021), $0\nu\beta\beta$ matrix
elements Weiss  _et al._ (2022), the Coulomb sum rule Weiss  _et al._ (2017)
and photo-absorption cross sections Weiss  _et al._ (2015b, 2016). The GCF is
also used in the planning of new experiments, e.g. involving photon-induced
reactions Hen  _et al._ (2020) and for measurements in the future Electron Ion
Collider Hauenstein  _et al._ (2022).

The nuclear contacts for a nucleus with $A$ nucleons are defined as Weiss  _et
al._ (2015a)

$C^{\alpha\beta}=\frac{A(A-1)}{2}\langle A^{\alpha}|A^{\beta}\rangle.$ (31)

The factor $A(A-1)/2$ appears in place of the number of proton-proton,
neutron-proton or neutron-neutron pairs present in most previous publications
because here the wave function is anti-symmetric under permutation of any two
nucleons. The diagonal contacts $C^{\alpha\alpha}$ are proportional to the
number of correlated pairs in the nucleus with quantum numbers $\alpha$.

Among all possible channels, accounted for by the sum over $\alpha$, it was
identified that the two channels that include an s-wave component are the most
significant Weiss  _et al._ (2018); Cruz-Torres  _et al._ (2021). These
channels are the so-called isospin-zero deuteron channel, defined by positive
parity, $s_{\alpha}=1$, $j_{\alpha}=1$, $t_{\alpha}=0$, and the isospin-one
channel defined by positive parity, $s_{\alpha}=0$, $j_{\alpha}=0$,
$t_{\alpha}=1$. Focusing on the $t=1$ contribution, the factorized wave
function can be written as

$\Psi\xrightarrow[r_{ij}\rightarrow
0]{}\varphi^{t=1}(\boldsymbol{r}_{ij})\sum_{t_{z}=-1}^{1}\eta_{1,t_{z}}A^{t=1}_{t_{z}}(\boldsymbol{R}_{ij},\\{\boldsymbol{r}_{k}\\}_{k\neq
i,j}).$ (32)

The index $t=1$ refers here specifically to the above-mentioned leading
isospin-one channel. The two-body function is independent of $t_{z}$ assuming
isospin symmetry, and the isospin part $\eta_{1,t_{z}}$ is written explicitly.
The corresponding contact terms are given by

$C^{t=1}_{t_{z}}=\frac{A(A-1)}{2}\langle
A^{t=1}_{t_{z}}|A^{t=1}_{t_{z}}\rangle.$ (33)

Therefore, the $t=1$ two-body density obey the following asymptotic expression

$\rho_{t=1,t_{z}}(r)\xrightarrow[r_{ij}\rightarrow
0]{}C^{t=1}_{t_{z}}\frac{1}{4\pi
r^{2}}\langle\varphi^{t=1}|\delta(r-r_{12})|\varphi^{t=1}\rangle.$ (34)

Based on this expression, some of the results of the previous section have
direct implication on the contact values. For example, Eqs. (9) and (12),
which connect the different $t=1$ densities for $T=0$ and $T=1/2$ nuclei,
respectively, imply similar relations for the nuclear contacts

$C^{t=1}_{t_{z}=1}=C^{t=1}_{t_{z}=0}=C^{t=1}_{t_{z}=-1}\;\;\;\;\;\;\;(T=0\;\textrm{nuclei}),$
(35)
$2C^{t=1}_{t_{z}=0}=C^{t=1}_{t_{z}=1}+C^{t=1}_{t_{z}=-1}\;\;\;\;\;\;\;(T=\frac{1}{2}\;\textrm{nuclei}).$
(36)

The $T=0$ relation was realized and utilized before Weiss  _et al._ (2018),
but the $T=1/2$ relation was not, and can be helpful in extracting more
accurate contact values. Knowing the contact values is crucial for analyzing
the effect of SRCs on different observables. This relation can be important
for extracting information from inclusive and exclusive electron-scattering
data, where disentangling the contribution of different channels can be
difficult. In the inclusive case, the combined contribution from all pairs is
measured. In exclusive measurements, where the knocked-out pair is detected,
it is possible in principle to measure the $pp$, $nn$ and $pn$ contributions,
but the above relation can allow disentangling the isospin-one and isospin-
zero contributions for $T=1/2$ nuclei. Additional relations between nuclear
contacts arise similarly from the relations connecting the densities of
different nuclei in the same isospin multiplet, Eqs. (18)-(24) (and their
generalization to $T>3/2$ multiplets). These relations allow us to extract the
$t=1$ nuclear contacts of all nuclei in an isospin multiplet if they are known
for one nucleus in this multiplet. The $t=0$ contact is the same for all
nuclei in the multiplet. Therefore, nuclear contact values of excited states
can be obtained using nuclear contact values of ground-state nuclei in the
same multiplet.

$0\nu\beta\beta$ transitions were recently analyzed using the GCF, introducing
new contact parameters $C(f,i)$, describing the transition from the initial
($i$) to final ($f$) nucleus Weiss  _et al._ (2022). The relations between the
two-body densities and the $0\nu\beta\beta$ transition densities, given in
Eqs. (28) and (III.3), lead to a direct relation between the $C_{t_{z}}^{1}$
contacts and the $C(f,i)$ $0\nu\beta\beta$ contacts for $\Delta T=0$
transitions

$C(f,i)=C^{t=1}_{t_{z}=-1}(i)+C^{t=1}_{t_{z}=1}(i)-2C^{t=1}_{t_{z}=0}(i),$
(37)

where on the right-hand side we have the contacts of the initial nucleus.
Since $C_{t_{z}}^{1}$ contact values can be extracted from electron-scattering
experiments Schmidt  _et al._ (2020), this relation provides a connection
between such electron-scattering reactions and $\Delta T=0$ $0\nu\beta\beta$
transitions.

On top of these direct consequences of the general two-body density relations,
isospin symmetry has an additional implication for the description of SRCs: it
allows us to extract information regarding the isospin structure of the
spectator $A-2$ nucleon system. It can be seen most simply if we consider a
$pn$ $t=0$ SRC pair. In this case, the $A-2$ nucleon subsystem must have the
same $T,T_{z}$ quantum numbers of the full wave function $\Psi$. Considering a
$t=1$ SRC pair, the analysis becomes non trivial. For $T=0$ nuclei, the $A-2$
nucleon part must have $T^{A-2}=1$, but for $T=1/2$ nuclei there are two
possible values $T^{A-2}=1/2,3/2$ and for $T\geq 1$ there are three possible
values.

Similar to the analysis in Sec. II, based on isospin symmetry we can write

$A^{t=1}_{t_{z}}=\sum_{T^{A-2}}\langle
T^{A-2}(T_{z}-t_{z})1t_{z}|TT_{z}\rangle\tilde{A}^{t=1}(T^{A-2},T_{z}-t_{z}),$
(38)

for a nucleus with $T,T_{z}$ quantum numbers.
$\tilde{A}^{t=1}(T^{A-2},T_{z}-t_{z})$ are functions with well-defined isospin
quantum number of the $A-2$ nucleon system. Then, using Eq. (33), the $t=1$
contacts are given by,

$\displaystyle C_{t_{z}}^{t=1}$
$\displaystyle=\frac{A(A-1)}{2}\sum_{T^{A-2}}|\langle
T^{A-2}(T_{z}-t_{z})1t_{z}|TT_{z}\rangle|^{2}$
$\displaystyle\times\langle\tilde{A}^{t=1}(T^{A-2})|\tilde{A}^{t=1}(T^{A-2})\rangle.$
(39)

It has to be noted that the $\langle\tilde{A}^{t=1}|\tilde{A}^{t=1}\rangle$
matrix element is independent of the $T_{z}^{A-2}$ component, due to the
Wigner-Eckart theorem. We see that the three $t=1$ contacts of a given nucleus
depend on the same elements
$\langle\tilde{A}^{t=1}(T^{A-2})|\tilde{A}^{t=1}(T^{A-2})\rangle$. There are
no more than three such elements, as discussed above. Therefore, the relation
can be reversed and we can express
$\langle\tilde{A}(T^{A-2})|\tilde{A}(T^{A-2})\rangle$ using the three $t=1$
contacts.

As an example, for $T=1/2$, $T_{z}=-1/2$ nucleus we get

$\displaystyle\langle\tilde{A}({1}/{2})|\tilde{A}({1}/{2})\rangle=\frac{2}{A(A-1)}\left[\frac{3}{2}C^{t=1}_{t_{z}=-1}-\frac{1}{2}C^{t=1}_{t_{z}=1}\right]$
(40)
$\displaystyle\langle\tilde{A}(3/2)|\tilde{A}(3/2)\rangle=\frac{2}{A(A-1)}2C^{t=1}_{t_{z}=1}.$
(41)

Now, looking for example on a $t=1$, $t_{z}=-1$ ($nn$) pair with small
relative distance $r$ (inside a $T=1/2$, $T_{z}=-1/2$ nucleus), we can ask
what is the relative probability of finding the $A-2$ nucleon system in a
state with $T^{A-2}=1/2$ or $T^{A-2}=3/2$. Based on Eq. (IV) we obtain

$\displaystyle\frac{P_{nn}(T^{A-2}=1/2)}{P_{nn}(T^{A-2}=3/2)}$
$\displaystyle=\frac{|\langle\frac{1}{2}\frac{1}{2}1-1|\frac{1}{2}-\frac{1}{2}\rangle|^{2}\langle\tilde{A}(1/2)|\tilde{A}(1/2)\rangle}{|\langle\frac{3}{2}\frac{1}{2}1-1|\frac{1}{2}-\frac{1}{2}\rangle|^{2}\langle\tilde{A}(3/2)|\tilde{A}(3/2)\rangle}$
$\displaystyle=\frac{3C^{t=1}_{t_{z}=-1}-C^{t=1}_{t_{z}=1}}{C^{t=1}_{t_{z}=1}}.$
(42)

Therefore, we can extract the relative contributions of $T^{A-2}=1/2$ and
$T^{A-2}=3/2$ using only the $pp$ and $nn$ contact values. Since the two-body
densities at short distances are proportional to the contacts, we can also
express it directly using the two-body densities

$\frac{P_{nn}(1/2)}{P_{nn}(3/2)}=\frac{3\rho_{1,-1}^{s=0}(r\rightarrow
0)-\rho_{1,1}^{s=0}(r\rightarrow 0)}{\rho_{1,1}^{s=0}(r\rightarrow 0)}.$ (43)

We used here the spin-zero densities because this is the spin of the pair in
the leading $t=1$ channel, although the spin-one $t=1$ contribution is
negligible at short distances and using the total $\rho_{t,t_{z}}(r)$
densities leads to very similar results. The last ratio is presented in Fig. 7
for the 7Li nucleus based on VMC calculations. We can deduce that when a $nn$
pair is close together in 7Li, the $A-2$ spectators have approximately 4.6
larger probability to be in a $T=1/2$ state than a $T=3/2$ state.

Figure 7:  Ratio of VMC spin-zero two-body densities of 7Li ground state
($T=1/2,T_{z}=-1/2$) using the AV18+UX potential, following Eq. (43). The
horizontal dashed line shows the value of this ratio at short distances.

This example was specifically for a $nn$ pair in a $T=1/2,T_{z}=-1/2$ nucleus,
but similar relations can be derived for any nucleus and any $t=1$ pair. For
$T\geq 1$ it will involve the three $t=1$ contacts and three $T^{A-2}$
components. The contact values are needed to extract the relative
probabilities of the $A-2$ nucleon isospin components. We used here VMC
calculations for this purpose, but contact values can also be extracted, for
example, from exclusive electron-scattering data where the knocked-out pair is
detected Schmidt  _et al._ (2020). Therefore, it can be applicable for heavy
nuclei beyond the reach of ab-initio calculations. This also means that we can
extract information about the $A-2$ nucleon system by only measuring the
knocked-out pair and not the $A-2$ nucleons.

This information about the $T^{A-2}$ probabilities is relevant for modeling
the $A-2$ nucleon part of nuclei. It can be used, for example, to estimate the
maximum probability that the $A-2$ nucleon subsystem is found in its ground
state. It should also be useful for models of the spectral function, used to
calculate lepton scattering cross sections, where the average excitation
energy of the $A-2$ spectators is required Weiss  _et al._ (2019); Andreoli
_et al._ (2022). This might also be relevant for the analysis of inverse-
kinematics scattering experiments, where the $A-2$ nucleon system is detected
Patsyuk  _et al._ (2021). For example, if the probability of the A-2 nucleon
system to be in different energy levels can be measured in such experiments
using $\gamma$-ray detection, our predictions regarding the $T^{A-2}$
probabilities can be tested. Such studies will allow a further experimental
examination of the short-distance wave function factorization to a single
leading $t=1$ channel, which is required to obtain our predictions.

##  V Summary

In this work, we have derived general relations concerning two-body
distributions based on isospin symmetry. These relations would be exact if
isospin symmetry was exact in nuclear systems. They were successfully tested
against ab-initio VMC calculations with nuclear potential models that include
realistic (and small) isospin-breaking terms. Some of these relations connect
isospin-one $pp$, $nn$, and $pn$ two-body densities (or momentum
distributions) in a given $T=0$ or $T=1/2$ nucleus. We also concluded that no
such relations exist for $T\geq 1$ nuclei (aside from $T_{z}=0$ nuclei, where
the $pp$ and $nn$ densities are equal). Another set of connections shows that,
for any isospin multiplet, one can obtain the two-body densities
$\rho_{t,t_{z}}(r)$ (or the equivalent two-body momentum distributions) of any
nucleus in the multiplet if they are known for a single nucleus in this
multiplet. We have explicitly derived the relations for mirror nuclei and
$T=1$ and $T=3/2$ multiplets. The relations for $T>3/2$ multiplets can be
similarly obtained. A connection between $0\nu\beta\beta$ decay densities of
mirror nuclei and the two-body densities $\rho_{t,t_{z}}(r)$ was also
revealed.

These results can be useful for guiding different models or estimations that
rely on two-body densities, which are needed for the calculation of two-body
observables. They can also be utilized for benchmarking numerical codes and
reducing the cost of calculations by calculating only independent quantities
that are not related via isospin symmetry (assuming isospin breaking effects
are negligible). In addition, isospin-breaking effects can be studied by
looking for violations of these relations. As discussed before, they can also
be used to obtain information regarding excited states from ground-state
nuclei in the same multiplet.

Finally, we have focused on the implications of isospin symmetry on the study
of nuclear SRCs. The relations discussed above lead to equivalent connections
between different nuclear contacts, that measure the number of SRC pairs in
nuclei. These connections will allow a more accurate interpretation of ab-
initio calculations and experimental data sensitive to SRCs, and specifically
a more accurate extraction of contact values for $T=1/2$ nuclei. This might be
relevant for some of the upcoming experiments in the field, see e.g. Ref.
Arrington  _et al._ (2006). The $0\nu\beta\beta$ relations result in a
connection between the relevant contact values, which reveals a connection
between electron-scattering reactions and $\Delta T=0$ $0\nu\beta\beta$
transitions. Additionally, we have shown that isospin-symmetry arguments allow
us to obtain information regarding the isospin structure of the $A-2$ nucleon
subsystem when an SRC pair is formed. The relative probability of the
different isospin components of that subsystem can be calculated using only
the number of SRC pairs in each channel. As mentioned above, this can be
relevant for spectral-function models and inverse-kinematic experiments.

##  VI Acknowledgement

We would like to thank E. Piasetzky, J. Kahlbow and N. Barnea for their
comments and suggestions. The present research is supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, under
contract DE-AC02-06CH11357, the NUCLEI SciDAC program (A.L., R.B.W.), and by
the Laboratory Directed Research and Development program of Los Alamos
National Laboratory under project number 20210763PRD1 (R.W.). The work of A.L.
is also supported by the DOE Early Career Research Program award. Quantum
Monte Carlo calculations were performed on the parallel computers of the
Laboratory Computing Resource Center, Argonne National Laboratory, the
computers of the Argonne Leadership Computing Facility via the INCITE grant
“Ab-initio nuclear structure and nuclear reactions”, and the 2020/2021 ALCC
grant “Chiral Nuclear Interactions from Nuclei to Nucleonic Matter”.

##  Appendix A Isospin eigenstates

We explain here in detail our claim that the functions given in Eqs. (3) and
(4) are isospin eigenstates with the same $T,T_{z}$ quantum numbers as the
full wave function $\Psi$. $\Psi$ obeys

$\hat{T}^{2}|\Psi\rangle=T(T+1)|\Psi\rangle$ (44)

and

$\hat{T}_{z}|\Psi\rangle=T_{z}|\Psi\rangle.$ (45)

Starting with $\hat{T}_{z}$, we can multiply the last equation by
$\varphi_{m}^{t\dagger}(\boldsymbol{r}_{12})$ and integrate over
$\boldsymbol{r}_{12}$. $\varphi_{m}^{t\dagger}(\boldsymbol{r}_{12})$ is
isospin independent [because the isospin part was explicitly separated, see
Eq. (1)] so it can be exchanged with the operator $\hat{T}_{z}$ and we get

$\hat{T}_{z}\int
d^{3}r_{12}\varphi_{m}^{t\dagger}(\boldsymbol{r}_{12})\Psi(\boldsymbol{r}_{1},...,\boldsymbol{r}_{A})=T_{z}\int
d^{3}r_{12}\varphi_{m}^{t\dagger}(\boldsymbol{r}_{12})\Psi.$ (46)

The basis states $\varphi_{m}^{t,t_{z}}$ are orthonormal. This directly means
that for a given $t$ (and different values of $m$), the functions
$\varphi_{m}^{t}$ are orthonormal as well because
$\eta_{t,t_{z}}^{\dagger}\eta_{t,t_{z}}=1$. Now consider two functions with
different isospin, i.e. $\varphi_{m}^{t=0}$ and
$\varphi_{m^{\prime}}^{t^{\prime}=1}$. Different isospin values dictate that
either the spin or the parity of these two functions must also be different,
due to the Pauli exclusion principle. Therefore, they are orthogonal as well.
As a result, using the expansion of Eq. (II), we obtain

$\hat{T}_{z}\sum_{t_{z}}\eta_{t,t_{z}}S_{m}^{t,t_{z}}=T_{z}\sum_{t_{z}}\eta_{t,t_{z}}S_{m}^{t,t_{z}}.$
(47)

An equivalent result can be similarly obtained for the operator $\hat{T}^{2}$,
and so we conclude that the functions of the form
$\sum_{t_{z}}\eta_{t,t_{z}}S_{m}^{t,t_{z}}$ have the same $T,T_{z}$ quantum
numbers as $\Psi$. These are exactly the functions appearing in Eqs. (3) and
(4).

##  Appendix B $T=1/2$ relation

We provide here more details on the derivation of Eq. (12). Based on Eq.
(III.1) we obtain the following expressions for the
$\rho_{1,t_{z}}(T=1/2,T_{z}=1/2)$ densities:

$\rho_{1,1}\left(\frac{1}{2},\frac{1}{2}\right)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\left[\frac{2}{3}D\left(\frac{1}{2}\right)+\frac{1}{6}D\left(\frac{3}{2}\right)\right]$
(48)
$\rho_{1,0}\left(\frac{1}{2},\frac{1}{2}\right)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\left[\frac{1}{3}D\left(\frac{1}{2}\right)+\frac{1}{3}D\left(\frac{3}{2}\right)\right]$
(49)
$\rho_{1,-1}\left(\frac{1}{2},\frac{1}{2}\right)=\frac{A(A-1)}{2}\frac{1}{4\pi
r^{2}}\left[\frac{1}{2}D\left(\frac{3}{2}\right)\right],$ (50)

where we denote by $D(T^{A-2})$ the expression given in Eq. (11). We can now
directly see that indeed

$2\rho_{1,0}\left(\frac{1}{2},\frac{1}{2}\right)=\rho_{1,1}\left(\frac{1}{2},\frac{1}{2}\right)+\rho_{1,-1}\left(\frac{1}{2},\frac{1}{2}\right),$
(51)

as argued in Eq. (12). The same relation is obtained for $T_{z}=-1/2$.

## References

  * Heisenberg (1932) W. Heisenberg, Z. Phys. 77, 1 (1932).
  * Weiss  _et al._ (2019) R. Weiss, I. Korover, E. Piasetzky, O. Hen, and N. Barnea, Phys. Lett. B791, 242 (2019), arXiv:1806.10217 [nucl-th] .
  * Andreoli  _et al._ (2022) L. Andreoli, J. Carlson, A. Lovato, S. Pastore, N. Rocco, and R. B. Wiringa, Phys. Rev. C 105, 014002 (2022), arXiv:2108.10824 [nucl-th] .
  * Cruz-Torres  _et al._ (2021) R. Cruz-Torres, D. Lonardoni, R. Weiss, M. Piarulli, N. Barnea, D. W. Higinbotham, E. Piasetzky, A. Schmidt, L. B. Weinstein, R. B. Wiringa, and O. Hen, Nature Phys. 17, 306 (2021), arXiv:1907.03658 [nucl-th] .
  * Cirigliano  _et al._ (2022) V. Cirigliano, Z. Davoudi, W. Dekens, J. de Vries, J. Engel, X. Feng, J. Gehrlein, M. L. Graesser, L. Gráf, H. Hergert, L. Jin, E. Mereghetti, A. Nicholson, S. Pastore, M. J. Ramsey-Musolf, R. Ruiz, M. Spinrath, U. van Kolck, and A. Walker-Loud, “Neutrinoless double-beta decay: A roadmap for matching theory to experiment,”  (2022).
  * Agostini  _et al._ (2022) M. Agostini, G. Benato, J. A. Detwiler, J. Menéndez, and F. Vissani, “Toward the discovery of matter creation with neutrinoless double-beta decay,”  (2022).
  * Yao  _et al._ (2022) J. Yao, J. Meng, Y. Niu, and P. Ring, Progress in Particle and Nuclear Physics 126, 103965 (2022).
  * Šimkovic  _et al._ (2008) F. Šimkovic, A. Faessler, V. Rodin, P. Vogel, and J. Engel, Phys. Rev. C 77, 045503 (2008).
  * Šimkovic  _et al._ (2018) F. Šimkovic, A. Smetana, and P. Vogel, Phys. Rev. C 98, 064325 (2018).
  * Cirigliano  _et al._ (2019) V. Cirigliano, W. Dekens, J. De Vries, M. L. Graesser, E. Mereghetti, S. Pastore, M. Piarulli, U. Van Kolck, and R. B. Wiringa, Phys. Rev. C 100, 055504 (2019), arXiv:1907.11254 [nucl-th] .
  * Weiss  _et al._ (2022) R. Weiss, P. Soriano, A. Lovato, J. Menendez, and R. B. Wiringa, Phys. Rev. C 106, 065501 (2022).
  * 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), arXiv:2004.05263 [nucl-th] .
  * Forest  _et al._ (1996) J. L. Forest, V. R. Pandharipande, S. C. Pieper, R. B. Wiringa, R. Schiavilla, and A. Arriaga, Phys. Rev. C 54, 646 (1996).
  * Carlson  _et al._ (2015) J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015), arXiv:1412.3081 [nucl-th] .
  * Pieper  _et al._ (2002) S. C. Pieper, K. Varga, and R. B. Wiringa, Phys. Rev. C 66, 044310 (2002), arXiv:nucl-th/0206061 .
  * Metropolis  _et al._ (1953) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, The Journal of Chemical Physics 21, 1087 (1953), https://doi.org/10.1063/1.1699114 .
  * Hastings (1970) W. K. Hastings, Biometrika 57, 97 (1970), https://academic.oup.com/biomet/article-pdf/57/1/97/23940249/57-1-97.pdf .
  * Wiringa  _et al._ (1995) R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995), arXiv:nucl-th/9408016 .
  * Pieper  _et al._ (2001) S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, and J. Carlson, Phys. Rev. C 64, 014001 (2001).
  * Pastore  _et al._ (2018) S. Pastore, J. Carlson, V. Cirigliano, W. Dekens, E. Mereghetti, and R. Wiringa, Phys. Rev. C 97, 014606 (2018), arXiv:1710.05026 [nucl-th] .
  * Wang  _et al._ (2019) X. B. Wang, A. C. Hayes, J. Carlson, G. X. Dong, E. Mereghetti, S. Pastore, and R. B. Wiringa, Phys. Lett. B 798, 134974 (2019), arXiv:1906.06662 [nucl-th] .
  * Yao  _et al._ (2021) J. M. Yao, A. Belley, R. Wirth, T. Miyagi, C. G. Payne, S. R. Stroberg, H. Hergert, and J. D. Holt, Phys. Rev. C 103, 014315 (2021), arXiv:2010.08609 [nucl-th] .
  * Novario  _et al._ (2021) S. Novario, P. Gysbers, J. Engel, G. Hagen, G. R. Jansen, T. D. Morris, P. Navrátil, T. Papenbrock, and S. Quaglioni, Phys. Rev. Lett. 126, 182502 (2021), arXiv:2008.09696 [nucl-th] .
  * Ciofi degli Atti (2015) C. Ciofi degli Atti, Phys. Rept. 590, 1 (2015).
  * Hen  _et al._ (2017) O. Hen, G. A. Miller, E. Piasetzky, and L. B. Weinstein, Rev. Mod. Phys. 89, 045002 (2017).
  * Arrington  _et al._ (2022) J. Arrington, N. Fomin, and A. Schmidt, Annual Review of Nuclear and Particle Science 72, 307 (2022), https://doi.org/10.1146/annurev-nucl-102020-022253 .
  * Wiringa  _et al._ (2014) R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, Phys. Rev. C 89, 024305 (2014), arXiv:1309.3794 [nucl-th] .
  * Weiss  _et al._ (2015a) R. Weiss, B. Bazak, and N. Barnea, Phys. Rev. C92, 054311 (2015a), arXiv:1503.07047 [nucl-th] .
  * Weiss  _et al._ (2018) R. Weiss, R. Cruz-Torres, N. Barnea, E. Piasetzky, and O. Hen, Phys. Lett. B 780, 211 (2018).
  * Schmidt  _et al._ (2020) A. Schmidt _et al._ (CLAS), Nature 578, 540 (2020), arXiv:2004.11221 [nucl-ex] .
  * Pybus  _et al._ (2020) J. Pybus, I. Korover, R. Weiss, A. Schmidt, N. Barnea, D. Higinbotham, E. Piasetzky, M. Strikman, L. Weinstein, and O. Hen, Phys. Lett. B 805, 135429 (2020), arXiv:2003.02318 [nucl-th] .
  * Duer  _et al._ (2019) M. Duer _et al._ (CLAS Collaboration), Phys. Rev. Lett. 122, 172502 (2019), arXiv:1810.05343 [nucl-ex] .
  * Weiss  _et al._ (2021) R. Weiss, A. W. Denniston, J. R. Pybus, O. Hen, E. Piasetzky, A. Schmidt, L. B. Weinstein, and N. Barnea, Phys. Rev. C 103, L031301 (2021).
  * Korover  _et al._ (2021) I. Korover _et al._ (CLAS), Phys. Lett. B 820, 136523 (2021), arXiv:2004.07304 [nucl-ex] .
  * Patsyuk  _et al._ (2021) M. Patsyuk _et al._ , Nature Phys. 17, 693 (2021), arXiv:2102.02626 [nucl-ex] .
  * Weiss  _et al._ (2017) R. Weiss, E. Pazy, and N. Barnea, Few Body Syst. 58, 9 (2017), arXiv:1612.01059 [nucl-th] .
  * Weiss  _et al._ (2015b) R. Weiss, B. Bazak, and N. Barnea, Phys. Rev. Lett. 114, 012501 (2015b), arXiv:1405.2734 [nucl-th] .
  * Weiss  _et al._ (2016) R. Weiss, B. Bazak, and N. Barnea, Eur. Phys. J. A 52, 92 (2016), arXiv:1511.04722 [nucl-th] .
  * Hen  _et al._ (2020) O. Hen _et al._ (GlueX), (2020), arXiv:2009.09617 [nucl-ex] .
  * Hauenstein  _et al._ (2022) F. Hauenstein _et al._ , Phys. Rev. C 105, 034001 (2022), arXiv:2109.09509 [physics.ins-det] .
  * Arrington  _et al._ (2006) J. Arrington, D. Day, N. Fomin, and P. Solvignon-Slifer, “E12-06-105: Inclusive scattering from nuclei at $x>1$ in the quasielastic and deeply inelastic regimes,” (2006).