# Tetraquark mass relations in quark and diquark models

Muhammad Naeem Anwar<EMAIL_ADDRESS>Timothy J. Burns
<EMAIL_ADDRESS>Department of Physics, Swansea University, Singleton
Park, Swansea, SA2 8PP, UK.

###### Abstract

We present new linear relations among the masses of S-wave tetraquarks with
either one flavour ($qq\bar{q}\bar{q}$) or two ($QQ\bar{q}\bar{q}$). Because
the relations are sensitive to the hidden-colour, spin, and spatial degrees of
freedom, comparison to experimental data can help to reveal the internal
structure of tetraquarks, and discriminate among different theoretical models.
Depending on the model, the relations are either exact, or valid in
perturbation theory, and a thorough comparison with existing literature
confirms their validity at the MeV level. Additionally, we explore the
connections among tetraquark models, and show how those with effective (quark
or diquark) masses are related to dynamical potential models. We also show how
the spectrum of diquark models is effectively a limiting case of (more
general) quark models, and in particular, that the diquark concept is most
relevant in the particular combination $QQ\bar{q}\bar{q}$, where $Q$ is much
heavier than $\bar{q}$.

##  I Introduction

The “November Revolution” provoked by the discovery of $J/\psi$ Augustin  _et
al._ (1974); Aubert  _et al._ (1974) is now sometimes known as the “first”
charm revolution, owing to a more recent sequence of discoveries which are
also deserving of revolutionary status. A characteristic feature of the
“second” charm revolution, which started at BaBar and Belle, and is still
ongoing at BESIII and the LHC experiments at CERN, is the discovery of states
which cannot apparently be understood as ordinary $q\bar{q}$ mesons or $qqq$
baryons; the experimental situation is reviewed in Refs. Lebed  _et al._
(2017); Brambilla  _et al._ (2020); Chen  _et al._ (2023). The new class of
hadrons, which includes states in the charm and bottom quark sector, poses a
significant challenge to our understanding of the strong interaction. Future
experiments, such as Belle II Altmannshofer  _et al._ (2019) and PANDA Lutz
_et al._ (2009), are designed to further explore these hadrons.

The initial flood of new states became known collectively as the “XYZ” states,
a name strongly suggestive of their mysterious characteristics. Some of the
states are by now so well-established that their nomenclature reflects the
standard conventions of the Particle Data Group (PDG) Workman  _et al._ (2022)
– so, for example, the state $X(3872)$ Choi  _et al._ (2003) which launched
the second charm revolution is now known as $\chi_{c1}(3872)$. Even so, the
underlying nature of many of these states is not well understood, and there is
considerable ongoing theoretical debate Burns (2015a); Burns and Swanson
(2016); Lebed  _et al._ (2017); Ali  _et al._ (2017); Lu  _et al._ (2017);
Brambilla  _et al._ (2020); Liu  _et al._ (2019a); Hanhart and Klempt (2019);
Burns and Swanson (2021a, b, 2022a); Dong  _et al._ (2021); Wu  _et al._
(2022); Burns and Swanson (2022b).

A key dividing line in these discussions is between molecular models and
“compact” multiquark models, which have characteristically different degrees
of freedom. In molecular models, the constituents are hadrons, whose
interactions can be modelled, for example, in terms of pion exchange, or as
effective field theory contact terms which are fit to data Tornqvist (1991,
1994); Swanson (2004a, b); Thomas and Close (2008); Fernandez-Carames  _et
al._ (2009); Guo  _et al._ (2018); Burns and Swanson (2019, 2021b); Anwar and
Lu (2021); Burns and Swanson (2022c). Such approaches are essentially an
extension into the heavy quark sector of ideas which are widely applied in
nuclear physics.

The focus of this paper is instead on compact multiquarks, which, by
comparison to molecular models, are more “exotic”, in the sense that there is
no effective description in terms of interacting hadrons. Instead, taking a
$QQ\bar{q}\bar{q}$ tetraquark as an example, the relevant degrees of freedom
are typically assumed to be four interacting quarks, or alternatively,
effective $QQ$ and $\bar{q}\bar{q}$ diquarks. One of our main motivations in
this paper is to distinguish between these two physical pictures, which we
refer to as quark models and diquark models, respectively. (We give citations
to the relevant literature in the main body of the paper, where the different
models are described in more detail.)

Models for compact multiquarks have parameters which are typically not well-
constrained, as they are usually fixed by comparison to the spectrum of
conventional mesons and baryons, which introduces a systematic uncertainty
which is difficult to quantify. Absolute predictions for the masses of states
are therefore not very reliable, and moreover, they cannot be used to
distinguish between quark and diquark models, whose predictions are similar
within (large) uncertainties. By contrast, predictions for relations among
masses (or mass splittings) are more general and, in some cases, are
completely independent of parameters. Such relations, which are the main focus
of this paper, obviously have more predictive power, and allow for more direct
tests of model assumptions.

For conventional hadrons, the Gell-Mann–Okubo formula Gell-Mann (1961, 1962);
Okubo (1962a, b) is a prototypical example of an empirically successful
relation among hadron masses. Additional relations among the masses of
conventional mesons and baryons have also been discovered and compared
favourably to experimental data, for example in Refs. Lipkin (1978); Cohen and
Lipkin (1980); Lipkin (1986a, 1989, 1990); Karliner and Lipkin (2007);
Karliner  _et al._ (2009). In this paper we uncover similar relations among
the masses of tetraquark states, and since they are based on similar symmetry
arguments, we expect them to be equally reliable.

An analogy with ordinary $Q\bar{Q}$ mesons is instructive. In that case,
absolute mass predictions (in quark potential models) have considerable
uncertainty, but a linear relation among the masses in the $P$-wave multiplet
is very reliable and is satisfied in experiments to less than an MeV Voloshin
(2008); Burns (2011a, b, 2014, 2015b), and also served as a benchmark for
exotic structures in that mass region Lebed and Swanson (2017); Anwar  _et
al._ (2019). The relations we find in this paper are conceptually very
similar. Note that in this paper we are concentrating on relations among the
masses of tetraquark states, as distinct from relations between their masses
and those of conventional mesons, which have also been discussed in the
literature Lipkin (1986b); Eichten and Quigg (2017); Anwar  _et al._ (2018a),
but remain to be confirmed experimentally.

Our main results in this paper are for tetraquarks with either two flavours in
the combination $QQ\bar{q}\bar{q}$, or just one flavour $qq\bar{q}\bar{q}$.
(Here $Q$ and $q$ are not necessarily heavy and light quarks, but rather, any
distinct quark flavours.) Our interest in these particular combinations is
partly due to recent experimental discoveries and lattice calculations. The
one-flavour case has been a particularly hot topic recently, owing to a
sequence of experimental observations showing apparent candidates for
$cc\bar{c}\bar{c}$ states in $J/\psi J/\psi$ decays Aaij  _et al._ (2020);
Hayrapetyan  _et al._ (2023); Aad  _et al._ (2023); many of the results of
this paper can be usefully applied to the phenomenology of these states Anwar
and Burns (2023). Similarly, there is a growing body of evidence from
experiment, lattice QCD and models, for the likely binding of
$bb\bar{q}\bar{q}^{\prime}$ and $cc\bar{q}\bar{q}^{\prime}$ tetraquarks, where
$\bar{q}$ and $\bar{q}^{\prime}$ are light flavours Lipkin (1986b); Zouzou
_et al._ (1986); Silvestre-Brac and Semay (1993a, b); Francis  _et al._
(2016); Bicudo  _et al._ (2017); Czarnecki  _et al._ (2018); Karliner and
Rosner (2017); Eichten and Quigg (2017); Mehen (2017); Francis  _et al._
(2019); Junnarkar  _et al._ (2019); Leskovec  _et al._ (2019); Lü  _et al._
(2020a); Aaij  _et al._ (2022). Note however that our results would only apply
where $\bar{q}$ and $\bar{q}^{\prime}$ are identical or, from isospin
symmetry, are an isovector $\bar{u}\bar{d}$ combination.

We begin (Section II) with some remarks on the main distinguishing feature of
quark and diquark models, namely the assumed colour wavefunctions. We then
discuss quark models (Section III), showing that the chromomagnetic quark
model (with effective quark masses) can be obtained in a symmetry limit from
the quark potential model. We then do a similar exercise for diquark models
(Section IV), and show how these are related to quark models in a truncated
basis of colour. Specialising to tetraquarks with one or two flavours (Section
V), we derive formulae for the masses of states in both quark and diquark
models, and show how the diquark model emerges as a limiting case of quark
models. Using the mass formulae, we identify linear relations among tetraquark
masses (Section VI), and show how these can discriminate among models. Finally
(Section VII) we summarise our results and suggest how they may be used to
inform comparisons with emerging experimental data.

##  II Colour, spin and flavour

The key distinction between quark and diquark models is the treatment of
colour. A pair of quarks can be coupled to colour $\bar{\mathbf{3}}$ or
$\mathbf{6}$, while a pair of antiquarks can be coupled to $\mathbf{3}$ or
$\bar{\mathbf{6}}$. To form an overall colour singlet, the possible
combinations are then $\bar{\mathbf{3}}\otimes\mathbf{3}$ or
$\mathbf{6}\otimes\bar{\mathbf{6}}$. A basic assumption of quark models is
that both possibilities should be considered, and in general, a quark model
state can be an admixture of the two. In diquark models, by construction, only
the $\bar{\mathbf{3}}\otimes\mathbf{3}$ configuration is included Lichtenberg
_et al._ (1996); Jaffe (2005).

As well as the treatment of colour, models are also distinguished according to
whether the constituents (quarks or diquarks) have effective masses, or
instead are dynamical objects whose contribution to the tetraquark mass is
obtained from the Schrödinger equation with some confining potential. We will
consider both of these approaches, and the relation between them.

In this paper we concentrate on tetraquarks with either two flavours (in the
combination $QQ\bar{q}\bar{q}$), or one ($qq\bar{q}\bar{q}$). Both systems are
subject to the same constraints, from the Pauli principle, on the allowed spin
and colour configurations. With reference to the $QQ\bar{q}\bar{q}$ system, an
S-wave $QQ$ pair can have (colour, spin) quantum numbers
($\bar{\mathbf{3}}$,1) or ($\mathbf{6}$,0), while an S-wave $\bar{q}\bar{q}$
pair can be ($\mathbf{3}$,1) or ($\bar{\mathbf{6}}$,0). Forming an overall
colour singlet, and combining the spins in S-wave to angular momentum $J$, the
allowed combinations (and their $J^{P(C)}$ quantum numbers) are

$\displaystyle\big{|}\varphi_{2}\big{>}$
$\displaystyle=\big{|}\\{(QQ)_{\bar{\mathbf{3}}}^{1}(\bar{q}\bar{q})_{\mathbf{3}}^{1}\\}^{2}\big{>}\quad[2^{+(+)}],$
(1) $\displaystyle\big{|}\varphi_{1}\big{>}$
$\displaystyle=\big{|}\\{(QQ)_{\bar{\mathbf{3}}}^{1}(\bar{q}\bar{q})_{\mathbf{3}}^{1}\\}^{1}\big{>}\quad[1^{+(-)}],$
(2) $\displaystyle\big{|}\varphi_{0}\big{>}$
$\displaystyle=\big{|}\\{(QQ)_{\bar{\mathbf{3}}}^{1}(\bar{q}\bar{q})_{\mathbf{3}}^{1}\\}^{0}\big{>}\quad[0^{+(+)}],$
(3) $\displaystyle\big{|}\varphi_{0}^{\prime}\big{>}$
$\displaystyle=|\\{(QQ)_{\mathbf{6}}^{0}(\bar{q}\bar{q})_{\bar{\mathbf{6}}}^{0}\\}^{0}\big{>}\quad[0^{+(+)}],$
(4)

where on the right-hand side, the subscripts are colour, and superscripts are
spin. The charge conjugation quantum number $C$ is relevant only for the one-
flavour case, corresponding to $Q=q$.

When counting the number of distinct quark flavours, we can treat $u$ and $d$
quarks as identical if they come in the isovector (symmetric) combination,
since they are subject to the same constraints from the Pauli principle
outlined above. So, for example, results we obtain for $(I,I_{3})=(1,\pm 1)$
states $QQ\bar{d}\bar{d}$ and $QQ\bar{u}\bar{u}$ apply equally to the
$(I,I_{3})=(1,0)$ partner $QQ\bar{u}\bar{d}$, but would not apply to an
$(I,I_{3})=(0,0)$ partner.

Diquark models are characterised by the inclusion of only colour triplet
combinations, meaning the spectrum has three states ($\varphi_{2}$,
$\varphi_{1}$ and $\varphi_{0}$). Quark models, by contrast, include both the
colour triplet and colour sextet combinations, so their are four states,
namely $\varphi_{2}$, $\varphi_{1}$, and two scalars, which are admixtures of
$\varphi_{0}$ and $\varphi_{0}^{\prime}$. Obviously an experimental
determination of the number of scalar states can distinguish diquark models
(one state) from quark models (two).

##  III Quark models

In the chromomagnetic quark model, also known as the colour-magnetic
interaction (CMI) model, the quark constituents have effective (rather than
dynamical) masses, and the splitting among the S-wave states is induced by
chromomagnetic interactions (one-gluon exchange) De Rujula  _et al._ (1975);
Jaffe (1977a, b); Mulders  _et al._ (1979); Aerts  _et al._ (1980); Hogaasen
_et al._ (2006); Buccella  _et al._ (2007); Cui  _et al._ (2007); Stancu
(2010); Hogaasen  _et al._ (2014); Karliner  _et al._ (2017); Wu  _et al._
(2018); Stancu (2016); Cheng  _et al._ (2020); Weng  _et al._ (2021). The
model has been widely applied to exotic hadron spectroscopy, as reviewed in
Ref. Liu  _et al._ (2019a). The Hamiltonian for S-wave states is

$H=\overline{M}-\sum_{i<j}C_{ij}~{}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}~{}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j},$
(5)

where the centre of mass

$\displaystyle\overline{M}=\sum_{i}m_{i}$ (6)

is the sum of quark masses, $\bm{\lambda}_{i}$ and $\bm{\sigma}_{i}$ are the
$SU(3)$ colour and $SU(2)$ spin (Pauli) matrices of quark $i$, and $C_{ij}$
are (positive) parameters which depend on quark flavours. The eigenstates of
$H$ are, in general, admixtures of $\bar{\mathbf{3}}\otimes\mathbf{3}$ and
$\mathbf{6}\otimes\bar{\mathbf{6}}$ colour configurations, with mixing induced
by the $\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}$ term.

The parameters $\overline{M}$ and $C_{ij}$ are typically fixed by applying the
same Hamiltonian to the spectrum of conventional mesons and/or baryons, and
fitting. An explicit assumption is that the same coefficients $C_{ij}$ control
the interactions between any pair of flavours $i$ and $j$, either as a quark-
quark ($q_{i}q_{j}$) or quark-antiquark ($q_{i}\bar{q}_{j}$) pair, and
regardless of whether these pairs are in a tetraquark, or in a conventional
meson or baryon. In some cases, it is further assumed that the coefficients
$C_{ij}$ scale inversely with quark masses,

$\displaystyle C_{ij}=\frac{c}{m_{i}m_{j}},$ (7)

for some constant $c$ (which will be identified later in this section).

Quark potential models Ader  _et al._ (1982); Weinstein and Isgur (1982,
1983); Zouzou  _et al._ (1986); Weinstein and Isgur (1990); Silvestre-Brac and
Semay (1993a, b); Brink and Stancu (1994, 1998); Lloyd and Vary (2004);
Vijande  _et al._ (2004); Barnea  _et al._ (2006); Vijande  _et al._ (2007a,
b, c); Vijande and Valcarce (2009a); Vijande  _et al._ (2009); Vijande and
Valcarce (2009b); Lu  _et al._ (2016); Anwar  _et al._ (2018a); Richard  _et
al._ (2017, 2018); Wang  _et al._ (2019); Hernández  _et al._ (2020); Liu  _et
al._ (2019b); Wang  _et al._ (2022); Zhang  _et al._ (2022) are a widely-used
(and somewhat more rigorous) alternative approach, in which the quark
constituents are dynamical, rather than having effective masses. A typical
quark model Hamiltonian

$\displaystyle H=T+V+U,$ (8)

has a potential with chromoelectric ($V$) and chromomagnetic ($U$)
contributions

$\displaystyle V$
$\displaystyle=-\sum_{i<j}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}~{}v(r_{ij}),$
(9) $\displaystyle U$
$\displaystyle=-\sum_{i<j}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}~{}\bm{\sigma}_{i}\cdot\bm{\sigma}_{j}~{}u(r_{ij}),$
(10)

whose radial parts are typically (but not necessarily) of the form

$\displaystyle v(r_{ij})$
$\displaystyle=\frac{3}{16}\left(b\,r_{ij}-\frac{4}{3}\frac{\alpha_{s}}{r_{ij}}+C_{0}\right),$
(11) $\displaystyle u(r_{ij})$
$\displaystyle=\frac{\pi}{6}\,\frac{\alpha_{s}}{m_{i}m_{j}}\,\delta^{3}(r_{ij})\,,$
(12)

where $b$ and $\alpha_{s}$ are the strengths of the confining force (string
tension) and color-Coulomb potential, respectively, and $C_{0}$ is mass
renormalization.

Comparing the two models, it is clear that $U$ in the potential model is
closely related to the interaction term in the chromomagnetic model (5). To
understand the relationship between the models, we treat $U$ as a perturbation
and consider the Hamiltonian

$\displaystyle\overline{H}=T+V\,,$ (13)

whose eigenstates are $\varphi_{0}$, $\varphi_{1}$, $\varphi_{2}$ and
$\varphi_{0}^{\prime}$ introduced in equations (1)-(4). (There is no term in
$\overline{H}$ which mixes $\varphi_{0}$ and $\varphi_{0}^{\prime}$, due to
the orthogonality of the spin wavefunctions.) Because $\overline{H}$ depends
on colour but not spin, there is degeneracy among the states $\varphi_{0}$,
$\varphi_{1}$, and $\varphi_{2}$, but not between these and
$\varphi_{0}^{\prime}$,

$\displaystyle\big{<}\varphi_{J}\big{|}\overline{H}\big{|}\varphi_{J}\big{>}\neq\big{<}\varphi_{0}^{\prime}\big{|}\overline{H}\big{|}\varphi_{0}^{\prime}\big{>}\,.$
(14)

We will point out that in order to make the connection with the chromomagnetic
model, an extra symmetry constraint is required, which restores this
degeneracy.

The matrix elements of $V$, using the colour matrix elements in Ref. Vijande
and Valcarce (2009b), are

$\displaystyle\big{<}\varphi_{J}\big{|}V\big{|}\varphi_{J}\big{>}$
$\displaystyle=\frac{8}{3}\big{<}v(r_{12})+v(r_{34})\big{>}+\frac{4}{3}\big{<}v(r_{13})+v(r_{14})+v(r_{23})+v(r_{24})\big{>}\,,$
(15)
$\displaystyle\big{<}\varphi_{0}^{\prime}\big{|}V\big{|}\varphi_{0}^{\prime}\big{>}$
$\displaystyle=-\frac{4}{3}\big{<}v(r_{12})+v(r_{34})\big{>}+\frac{10}{3}\big{<}v(r_{13})+v(r_{14})+v(r_{23})+v(r_{24})\big{>}$
(16)

where an integral over all spatial degrees of freedom is implied. From the
symmetries in $T$ and $V$, the ground state wavefunctions are symmetric under
the interchange of quarks ($1\leftrightarrow 2$), antiquarks
($3\leftrightarrow 4$), or both $(12\leftrightarrow 34)$, so the spatial
integral can be reduced to two independent terms

$\displaystyle\big{<}\varphi_{J}\big{|}V\big{|}\varphi_{J}\big{>}$
$\displaystyle=\frac{16}{3}\big{<}v(r_{12})\big{>}+\frac{16}{3}\big{<}v(r_{13})\big{>}\,,$
(17)
$\displaystyle\big{<}\varphi_{0}^{\prime}\big{|}V\big{|}\varphi_{0}^{\prime}\big{>}$
$\displaystyle=-\frac{8}{3}\big{<}v(r_{12})\big{>}+\frac{40}{3}\big{<}v(r_{13})\big{>}\,.$
(18)

Note however that the wavefunction does not have an additional symmetry under
the interchange of a quark and antiquark (such as $2\leftrightarrow 3$), so in
general, no further simplification is possible. This applies not only to
states with two flavours ($QQ\bar{q}\bar{q}$), but also states with one
flavour ($qq\bar{q}\bar{q}$): the Hamiltonian does not impose a symmetry under
$Q\leftrightarrow\bar{q}$ or $q\leftrightarrow\bar{q}$, so the wavefunction
does not have that symmetry. It turns out, however, that this additional
symmetry is often imposed on the wavefunction as an artefact of the
calculation. In particular this is often the case when the Gaussian Expansion
Method is applied to tetraquarks with one flavour ($qq\bar{q}\bar{q}$), as in
for example Refs. Liu  _et al._ (2019b); Zhang  _et al._ (2022). If we impose
this extra symmetry (under $2\leftrightarrow 3$), the spatial integral reduces
further, and the potentials for all states are identical

$\displaystyle\big{<}\varphi_{J}\big{|}V\big{|}\varphi_{J}\big{>}=\big{<}\varphi_{0}^{\prime}\big{|}V\big{|}\varphi_{0}^{\prime}\big{>}=\frac{32}{3}\big{<}v(r_{12})\big{>}\,,$
(19)

which further implies, as distinct from the general case (14), that the
eigenstates of $\overline{H}$ are degenerate. Identifying $\overline{M}$ in
equation (5) as the corresponding eigenvalue,

$\displaystyle\big{<}\varphi_{J}\big{|}\overline{H}\big{|}\varphi_{J}\big{>}=\big{<}\varphi_{0}^{\prime}\big{|}\overline{H}\big{|}\varphi_{0}^{\prime}\big{>}\equiv\overline{M}\,,$
(20)

we see that a perturbative treatment of the full quark model Hamiltonian

$\displaystyle H=\overline{H}+U\,,$ (21)

is equivalent to the chromomagnetic model (5), where the coefficients $C_{ij}$
are obtained from $u(r_{ij})$ by integrating over the spatial wavefunctions of
the eigenstates of $\overline{H}$,

$\displaystyle C_{ij}=\big{<}u(r_{ij})\big{>}\,.$ (22)

Note that with this interpretation, the centre of mass term $\overline{M}$ is
no longer just the sum of quark masses, but also absorbs the dynamical
contributions from the potential model, namely the kinetic energy and the
confining term. Also, the coefficients $C_{ij}$ depend not only on quark
masses, as in equation (7), but also depend on the spatial wavefunction of the
quarks.

In the symmetry limit we are working in, $\big{<}r_{ij}\big{>}$ is independent
of $i$ and $j$, hence so is $C_{ij}$. This validates the assumption, in the
chromomagnetic model, that the same $C_{ij}$ can be used for any pair of
flavours $i$ and $j$ in a tetraquark, both quark-quark ($q_{i}q_{j}$) and
quark-antiquark ($q_{i}\bar{q}_{j}$) pairs. However it does not establish that
one can use the same $C_{ij}$ in tetraquarks as in conventional mesons and
baryons: this remains an assumption of the model, which could however be
tested by evaluating equation (22) and the corresponding expression for mesons
and baryons.

With the specific form of $u(r_{ij})$ in equation (12), we reproduce the
inverse dependence of $C_{ij}$ on quark masses as in equation (7), with

$\displaystyle c=\frac{\pi}{6}\,\alpha_{s}\big{<}\delta^{3}(r_{ij})\big{>}\,.$
(23)

Note that, in the symmetry limit we are working in, $c$ is indeed constant, in
the sense of being independent of the flavours $i$ and $j$ within the
tetraquark.

##  IV Diquark models

A widespread implementation of the diquark model Maiani  _et al._ (2005a, b);
Drenska  _et al._ (2008, 2009); Ali  _et al._ (2010); Ali (2011); Ali and
Parkhomenko (2019) has a Hamiltonian which is very similar to that of the
chromomagnetic model,

$\displaystyle
H=\overline{M}+2\sum_{i<j}\kappa_{ij}~{}\mathbf{S}_{i}\cdot\mathbf{S}_{j}\,,$
(24)

where $\overline{M}$ is the sum of quark or diquark effective masses,
$\mathbf{S}_{i}=\bm{\sigma}_{i}/2$ is the spin of quark $i$, and $\kappa_{ij}$
are (positive) parameters which depend on quark flavours and which, unlike the
parameters $C_{ij}$ of the chromomagnetic model, are not assumed to be the
same for quark-quark ($q_{i}q_{j}$) and quark-antiquark ($q_{i}\bar{q}_{j}$)
combinations. Otherwise, the only distinction between the diquark model and
the chromomagnetic model is the use of a truncated colour basis
$\bar{\mathbf{3}}\otimes\mathbf{3}$. If we evaluate the chromomagnetic
Hamiltonian (5) in the same basis, the two models are equivalent provided
their couplings are related

$\displaystyle\kappa_{ij}=-2\big{<}\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}\big{>}C_{ij}$
(25)

namely

$\kappa_{ij}=\left\\{\begin{matrix}\frac{16}{3}C_{ij},&\textrm{ for
}q_{i}q_{j},\\\ \frac{8}{3}C_{ij},&\textrm{ for
}q_{i}\bar{q}_{j},\end{matrix}\right.$ (26)

where we have used the colour matrix elements in Ref. Vijande and Valcarce
(2009b). In this sense, the diquark model is identical to the chromomagnetic
quark model, but evaluated in a truncated colour basis. We will later use this
property to extract the spectrum of the diquark model as a limiting case of
the chromomagnetic quark model.

Referring to the Hamiltonian (24) as a diquark model is somewhat
counterintuitive, since the spin degrees of freedom are actually quarks (not
diquarks). The distinction turns out not to be important for the particular
flavour combinations of tetraquarks which are the focus of this paper. For
$QQ\bar{q}\bar{q}$ states there are three independent couplings

$\displaystyle\kappa_{QQ}$ $\displaystyle\equiv\kappa_{12}\,,$ (27)
$\displaystyle\kappa_{qq}$ $\displaystyle\equiv\kappa_{34}\,,$ (28)
$\displaystyle\kappa_{Q\bar{q}}$
$\displaystyle\equiv\kappa_{13}=\kappa_{14}=\kappa_{23}=\kappa_{24}\,,$ (29)

with the obvious simplification to two couplings ($\kappa_{qq}$ and
$\kappa_{q\bar{q}}$) in the special case $qq\bar{q}\bar{q}$. We consider the
(more general) $QQ\bar{q}\bar{q}$ case in detail. With the couplings above,
the Hamiltonian reduces to

$\displaystyle
H=\overline{M}+\frac{1}{2}\left(\kappa_{QQ}+\kappa_{qq}\right)+2\kappa_{Q\bar{q}}~{}\mathbf{S}_{12}\cdot\mathbf{S}_{34}\,,$
(30)

where here we have evaluated

$\displaystyle\big{<}\mathbf{S}_{1}\cdot\mathbf{S}_{2}\big{>}=\big{<}\mathbf{S}_{3}\cdot\mathbf{S}_{4}\big{>}=\frac{1}{4}\,,$
(31)

as appropriate to spin-1 diquarks. The key feature is that the spin-dependence
of the Hamiltonian is now expressed in terms of effective diquark spin
operators

$\displaystyle\mathbf{S}_{12}$
$\displaystyle=\mathbf{S}_{1}+\mathbf{S}_{2}\,,$ (32)
$\displaystyle\mathbf{S}_{34}$
$\displaystyle=\mathbf{S}_{3}+\mathbf{S}_{4}\,,$ (33)

corresponding to the total spin of the $QQ$ and $\bar{q}\bar{q}$ diquarks,
respectively. In this sense, the Hamiltonian defined at quark level can be
naturally interpreted in terms of diquark degrees of freedom. But this is a
peculiarity of the flavour combination $QQ\bar{q}\bar{q}$ (or
$qq\bar{q}\bar{q}$), which leads to equation (30). For other combinations of
flavours, such as $QQ\bar{Q}\bar{q}$ and $Qq\bar{Q}\bar{q}$, the same does not
apply, and in general an effective diquark description does not emerge in the
same way; this is because as well as effective diquark operators like (32) and
(33), there are other operators $\mathbf{S}_{1}-\mathbf{S}_{2}$ and/or
$\mathbf{S}_{3}-\mathbf{S}_{4}$ which mix “diquarks” with different spin.

Diquark potential models are more explicit in treating diquarks as effective
degrees of freedom. Here the mass spectrum of tetraquark states comes from the
Schrödinger equation, in which diquarks are massive (colour-triplet) objects
interacting through a confining potential Ebert  _et al._ (2006, 2007, 2008,
2009, 2011); Faustov  _et al._ (2020); Anwar  _et al._ (2018b); Bedolla  _et
al._ (2020); Anwar  _et al._ (2018a); Debastiani and Navarra (2019); Anwar
_et al._ (2018b); Lundhammar and Ohlsson (2020). The distinctions with the
previous model are that the mass spectrum comes from the Schrödinger equation
(rather than effective masses fit to data), and that the spin dependence is
expressed from the outset in terms of diquark (not quark) spin operators.

To clarify the relation between the different diquark models, we follow a
similar procedure to our previous discussion of quark models. In the
Hamiltonian,

$\displaystyle H=T+V+U\,,$ (34)

we isolate the spin-independent kinetic ($T$) and confining ($V$) terms, and a
spin-dependent term ($U$), which in all models is expressed (for S-wave
states) in terms of diquark spin operators,

$\displaystyle U=u(r)~{}\mathbf{S}_{12}\cdot\mathbf{S}_{34^{\prime}}\,,$ (35)

where here $r$ is the radial component of the vector joining the diquark and
antidiquark. There is considerable variation among the different approaches to
diquark potential models, for example, in how the effective diquark mass is
obtained, the assumed form of the confining potential $V$, and the precise
form of the radial component $u(r)$ of the spin-spin term. However these
differences are immaterial to the discussion.

As in the quark model case, we treat $U$ as a perturbation. If we identify the
eigenvalues of the spin-independent part

$\displaystyle\overline{H}=T+V$ (36)

with the spin-independent term in equation (30),

$\displaystyle\big{<}\overline{H}\big{>}\equiv\overline{M}+\frac{1}{2}\left(\kappa_{QQ}+\kappa_{qq}\right),$
(37)

we notice that a perturbative treatment of the full Hamiltonian (34) is
equivalent to the previous diquark model, with the couplings defined as
integrals over the eigenstates of $\overline{H}$,

$\displaystyle\kappa_{Q\bar{q}}\equiv\frac{1}{2}\big{<}u(r)\big{>}\,.$ (38)

##  V Mass formulae

At this stage we have established the underlying connections among four
different classes of models, distinguished according to the colour structure
(quarks versus diquarks), and the mass spectrum (effective masses versus
dynamical masses from the Schrödinger equation). Among the models with
effective masses, we showed that the quark model (5) and diquark model (24)
are equivalent, except that the latter is evaluated in a truncated colour
basis. We also showed that each of these models can be understood by applying
perturbation theory to a corresponding (quark or diquark) potential model,
which gives a dynamical interpretation for the effective masses, and implies
that the couplings are sensitive to the spatial wavefunctions.

Having established these connections among models, we will now obtain some
general results which apply to all four classes of model. As a framework for
our calculation, we will use the most general Hamiltonian (5). By evaluating
its spectrum in the full and truncated colour basis, respectively, we get
results which correspond to quark and diquark models.

When applying the Hamiltonian (5) to $QQ\bar{q}\bar{q}$ states, there are
three possible couplings,

$\displaystyle C_{QQ}$ $\displaystyle=C_{12}\,,$ (39) $\displaystyle C_{qq}$
$\displaystyle=C_{34}\,,$ (40) $\displaystyle C_{Q\bar{q}}$
$\displaystyle=C_{13}=C_{14}=C_{23}=C_{24}\,,$ (41)

whereas for $qq\bar{q}\bar{q}$ states there are only two,

$\displaystyle C_{qq}$ $\displaystyle=C_{12}=C_{34}\,,$ (42) $\displaystyle
C_{q\bar{q}}$ $\displaystyle=C_{13}=C_{14}=C_{23}=C_{24}\,.$ (43)

We will discuss the more general $QQ\bar{q}\bar{q}$ case, noting that
$qq\bar{q}\bar{q}$ is then a special case with $Q=q$.

We need the matrix elements of $H$ with respect to the basis states (1)-(4).
Using, for example, the colour and spin matrix elements of Ref. Vijande and
Valcarce (2009b), we find

$\displaystyle\big{<}\varphi_{J}\big{|}H\big{|}\varphi_{J}\big{>}$
$\displaystyle=\overline{M}+\frac{8}{3}\left(C_{QQ}+C_{qq}\right)+\frac{8}{3}C_{Q\bar{q}}\left[J(J+1)-4\right]\,,$
(44)
$\displaystyle\big{<}\varphi_{0}^{\prime}\big{|}H\big{|}\varphi_{0}^{\prime}\big{>}$
$\displaystyle=\overline{M}+4(C_{QQ}+C_{qq})\,,$ (45)
$\displaystyle\big{<}\varphi_{0}^{\prime}\big{|}H\big{|}\varphi_{0}\big{>}$
$\displaystyle=-8\sqrt{6}C_{Q\bar{q}}\,,$ (46)

which is consistent with the results of Refs. Buccella  _et al._ (2007); Wu
_et al._ (2018). Note that $C_{QQ}$ and $C_{qq}$ appear only in the
combination $C_{QQ}+C_{qq}$, and because of this, it turns out to be
convenient to introduce the dimensionless ratio

$\displaystyle R=\frac{2C_{Q\bar{q}}}{C_{QQ}+C_{qq}}$ (47)

which, in the case of $qq\bar{q}\bar{q}$ states, reduces to the simpler ratio
of $q\bar{q}$ and $qq$ couplings,

$\displaystyle R=\frac{C_{q\bar{q}}}{C_{qq}}\,.$ (48)

If the couplings are parameterised as in equation (7), then for two-flavour
states ($QQ\bar{q}\bar{q}$), $R$ depends only on the ratio of quark masses
$m_{q}$ and $m_{Q}$,

$\displaystyle R=\frac{2m_{q}/m_{Q}}{1+(m_{q}/m_{Q})^{2}}\,,$ (49)

and takes values in the range $0<R<1$, while for one-flavour states
($qq\bar{q}\bar{q}$) obviously $R=1$.

As discussed, the spectrum of the diquark model comes from truncating the
basis to include only the hidden colour-triplet states, namely $\varphi_{2}$,
$\varphi_{1}$ and $\varphi_{0}$, but not $\varphi_{0}^{\prime}$. Evaluating
equation (44), the masses of the tensor ($M_{2}$), axial ($M_{1}$) and scalar
($M_{0}$) are

$\displaystyle M_{2}$
$\displaystyle=\overline{M}+\frac{8}{3}\left(C_{QQ}+C_{qq}\right)(1+R)\,,$
(50) $\displaystyle M_{1}$
$\displaystyle=\overline{M}+\frac{8}{3}\left(C_{QQ}+C_{qq}\right)(1-R)\,,$
(51) $\displaystyle M_{0}$
$\displaystyle=\overline{M}+\frac{8}{3}\left(C_{QQ}+C_{qq}\right)(1-2R)\,.$
(52)

To get results for the quark model, we expand the basis to include
$\varphi_{0}^{\prime}$ which implies, as discussed previously, that the
spectrum includes two scalar states. The masses of the tensor ($M_{2}$) and
axial ($M_{1}$) are as above, but the masses of the scalars ($M_{0}$ and
$M_{0}^{\prime}$) are the eigenvalues of

$\displaystyle
H=\overline{M}+\left(C_{QQ}+C_{qq}\right)\begin{pmatrix}\frac{8}{3}(1-2R)&-4\sqrt{6}R\\\
-4\sqrt{6}R&4\end{pmatrix},$ (53)

namely

$\displaystyle M_{0}$
$\displaystyle=\overline{M}+\frac{2}{3}\left(C_{QQ}+C_{qq}\right)\left(5-4R-\Delta\right),$
(54) $\displaystyle M_{0}^{\prime}$
$\displaystyle=\overline{M}+\frac{2}{3}\left(C_{QQ}+C_{qq}\right)\left(5-4R+\Delta\right),$
(55)

where

$\displaystyle\Delta=\sqrt{232R^{2}+8R+1}\,,$ (56)

and we are adopting the convention that $M_{0}^{\prime}>M_{0}$. The
eigenstates $\psi_{0}$ and $\psi_{0}^{\prime}$ corresponding to masses $M_{0}$
and $M_{0}^{\prime}$ are orthogonally mixed

$\displaystyle\big{|}\psi_{0}\big{>}$
$\displaystyle=\cos\theta\big{|}\varphi_{0}\big{>}+\sin\theta\big{|}\varphi_{0}^{\prime}\big{>}\,,$
(57) $\displaystyle\big{|}\psi_{0}^{\prime}\big{>}$
$\displaystyle=-\sin\theta\big{|}\varphi_{0}\big{>}+\cos\theta\big{|}\varphi_{0}^{\prime}\big{>}\,,$
(58)

with mixing angle

$\displaystyle\theta=\tan^{-1}\left(\frac{\Delta-1-4R}{6\sqrt{6}R}\right).$
(59)

The mass formulae above imply unambiguous orderings for the masses of states,
regardless of parameters. For diquark models, the ordering is

$\displaystyle M_{0}<M_{1}<M_{2}\,,$ (60)

whereas in quark models it is

$\displaystyle M_{0}<M_{1}<M_{2}<M_{0}^{\prime}\,.$ (61)

This can help to assign quantum numbers to experimental candidates, as
discussed in Ref. Anwar and Burns (2023).

The results in this section are exact for (quark or diquark) models with
effective masses, whereas for potential models (whether quark or diquark),
they apply in the limit of perturbation theory. In the particular case of the
quark potential model, there is an additional caveat: recalling the discussion
at the end of Section III, the results derived above are valid only subject to
the additional assumption that the spatial wavefunction of the tetraquark is
totally symmetric under the interchange $Q\leftrightarrow\bar{q}$ (or
$q\leftrightarrow\bar{q}$, in the one flavour case). As discussed, in many
papers this assumption applies (even if implicitly), and we have found that in
such cases (for example Refs. Liu  _et al._ (2019b); Zhang  _et al._ (2022))
the results agree with all of our results above (for masses, mixing angles,
and mass orderings). In quark model studies which do not use that assumption,
there are some differences, which are immediately apparent in violations of
the mass ordering (61), for example in Ref. Wang  _et al._ (2019).

There is an intriguing connection between quark and diquark models in the
limit of small $R$. In this limit $\Delta\approx 1+4R$, meaning the scalar
masses are

$\displaystyle M_{0}$
$\displaystyle\approx\overline{M}+\frac{8}{3}\left(C_{QQ}+C_{qq}\right)(1-2R)\,,$
(62) $\displaystyle M_{0}^{\prime}$
$\displaystyle\approx\overline{M}+4\left(C_{QQ}+C_{qq}\right)\,,$ (63)

which is equivalent to a perturbative treatment of the Hamiltonian (53) to
first order in $R$. Note that the lighter scalar $M_{0}$ reproduces the result
of the diquark model, equation (52). In the same limit $\theta\approx 0$ so
the lighter scalar is purely $\varphi_{0}$, again coinciding with the diquark
model result. So apart from the existence of a heavier scalar state, we have
found that spectra of the quark and diquark models coincide in the limit of
small $R$, both in terms of masses and wavefunctions. With reference to the
definitions (47) and (48), small $R$ means that quark-antiquark interactions
are small compared to quark-quark interactions. That is precisely the limit in
which the diquark concept is physically reasonable.

From equation (49), the small $R$ limit applies to $QQ\bar{q}\bar{q}$
tetraquarks with $m_{Q}\gg m_{q}$. On this basis we suggest that the diquark
model is a sensible approximation to the quark model in such cases (modulo the
absent heavier scalar state). Otherwise, the spectra of the quark and diquark
models are rather different, and we explore this further in the next section.
The one-flavour case $qq\bar{q}\bar{q}$ actually deviates maximally from the
small $R$ limit, as it has $R=1$, which is the upper limit on $R$ from
equation (49).

Returning to the $QQ\bar{q}\bar{q}$ case, it is clear from above that as
$m_{Q}\to\infty$, the lighter scalar decouples from $\varphi_{0}^{\prime}$ and
becomes purely $\varphi_{0}$. This effect has been discussed previously for
$QQ\bar{q}\bar{q}$ states Heller and Tjon (1987); Vijande  _et al._ (2004). In
terms of colour, it is exactly what was observed also for the isoscalar
$QQ\bar{u}\bar{d}$ tetraquark Hernández  _et al._ (2020): as $m_{Q}\to\infty$,
the ground state decouples from $\mathbf{6}\otimes\bar{\mathbf{6}}$ and
becomes purely $\bar{\mathbf{3}}\otimes\mathbf{3}$. The comparison suggests
that the effect may be generic, noting that in spite of its apparent
similarity, the isoscalar $QQ\bar{u}\bar{d}$ system is very different to our
$QQ\bar{q}\bar{q}$ system, because of the isospin asymmetry, which implies
different spin-colour configurations for $\bar{u}\bar{d}$ compared to
$\bar{q}\bar{q}$.

##  VI Mass relations

The mass formulae in the previous section imply relations among the masses of
the states, and as far as we know these have not yet been identified in the
literature. For diquark models, the situation is very simple: equations (50),
(51) and (52) imply that the masses $M_{2}$, $M_{1}$, $M_{0}$ satisfy the
following linear relation,

$\displaystyle M_{1}=\frac{1}{3}\left(2M_{0}+M_{2}\right),$ (64)

independently of model parameters.

For quark models, the situation is only slightly more complicated. From
equations (50), (51), (54), and (55), it is clear that any mass splitting
among $M_{2}$, $M_{1}$, $M_{0}$ and $M_{0}^{\prime}$ is independent of
$\overline{M}$, while any ratio of such splittings is independent of
$C_{QQ}+C_{qq}$, leaving a function of $R$ only. By taking ratios of different
combinations of mass splittings, we get linear relations among the masses,
similar to equation (64) for the diquark model, but in this case involving
$R$. In this way, we can find a linear relation among any combination of three
masses out of the four ($M_{2}$, $M_{1}$, $M_{0}$ and $M_{0}^{\prime}$),
meaning a total of four relations. We concentrate on the following two:

$\displaystyle M_{1}$
$\displaystyle=M_{0}+\frac{\Delta-1}{\Delta-1+8R}(M_{2}-M_{0})\,,$ (65)
$\displaystyle M_{0}^{\prime}$
$\displaystyle=M_{0}+\frac{2\Delta}{\Delta-1+8R}(M_{2}-M_{0})\,,$ (66)

noting that the first of these is the closest analogue of the diquark model
result (64), in the sense of offering a formula for $M_{1}$ in terms of
$M_{0}$ and $M_{2}$. Indeed, it reduces to the diquark model result (64), in
the limit of small $R$ (taking $\Delta\approx 1+4R$). This reinforces our
previous observation that the quark model reduces (apart from the heavier
scalar) to the diquark model, in the limit of small $R$.

Ultimately the utility of these mass relations is that, given any two
experimental candidates, we may predict the mass of the other state (in
diquark models), or the other two states (in quark models). Since these
predictions are independent of (or only weakly dependent on) parameters, they
provide a very direct test of models, which can be checked against future
experimental data. In Ref. Anwar and Burns (2023) we apply this approach to a
putative multiplet of $cc\bar{c}\bar{c}$ states observed at LHCb, CMS and
ATLAS.

The relations also lead to a simple and very general understanding of the
pattern of masses which characterise quark and diquark models. In Figure 1 we
show the mass spectrum in arbitrary units, having fixed $M_{0}$ and $M_{2}$,
and using the relations (64), (65) and (66) to predict $M_{1}$ and
$M_{0}^{\prime}$. Note of course that in quark models, the pattern of masses
in the multiplet is sensitive to $R$, whereas in diquark models, it is not. As
a reminder, if the couplings $C_{ij}$ are parameterised as in equation (7),
then $qq\bar{q}\bar{q}$ states have $R=1$, while $QQ\bar{q}\bar{q}$ states
have $0<R<1$, with $R\approx 0$ for $m_{Q}\gg m_{q}$. Notice in Figure 1 that
as $R\to 0$, the scalar masses in the quark and diquark models become
degenerate, as anticipated above.

Comparison of the spectrum in Fig. 1 to experimental candidates can help to
reveal the underlying dynamics of a multiplet of tetraquark states. In
particular, apart from in the limit of small $R$, quark and diquark models can
be differentiated by the relative position of the axial ($M_{1}$) compared to
the scalar ($M_{0}$) and tensor ($M_{2}$). The difference is particularly
pronounced for larger $R$, including $R=1$ which applies to the single flavour
case ($qq\bar{q}\bar{q}$).

We have been assuming that the couplings satisfy equation (7), but empirically
this is not universally reliable; see for example the couplings fitted to
mesons and baryons in Refs. Buccella  _et al._ (2007); Cheng  _et al._ (2020).
Moreover, in quark potential models, it applies only in a symmetry limit which
is not strictly justified by the Hamiltonian (see Section III). If we no
longer assume equation (7), then of course $R$ is no longer constrained to
$0<R\leq 1$, and for this reason in Figure 1 we extend the plot to larger
values of $R$.

Without equation (7), for one-flavour states ($qq\bar{q}\bar{q}$) it is no
longer true that $R=1$ exactly. Using the more general definition of $R$ in
equation (48), we expect deviations from $R=1$ due to asymmetry in $qq$ and
$q\bar{q}$ spatial wavefunctions. In this context it is reassuring that the
spectrum in Fig. 1 is not very sensitive to the choice of $R$, for values near
$R=1$.

Figure 1: The mass spectrum (in arbitrary units) as a function of $R$, where
$M_{0}$ and $M_{2}$ are fixed, $M_{1}$ (diquark model) is given by equation
(64), while $M_{1}$ (quark model) and $M_{0}^{\prime}$ (quark model) are given
by equations (65) and (66), respectively. DPM Debastiani and Navarra (2019)DPM
Ebert  _et al._ (2007)DPM Lundhammar and Ohlsson (2020)DM Berezhnoy  _et al._
(2012)QPM Liu  _et al._ (2019b)QPM Zhang  _et al._ (2022)CQM Cheng  _et al._
(2020)CQM Wu  _et al._ (2018) Figure 2: The left and right panels show, for
the axial and heavy scalar, respectively, the difference between the mass
obtained from the relations (64), (65) and (66), and the quoted mass taken
from the literature, including examples of all four classes of model: the
chromomagnetic quark model (CQM), quark potential model (QPM), diquark model
(DM), and diquark potential model (DPM).

Although the relations (64), (65) and (66) have seemingly not been discussed
previously, they are actually apparent in the quoted mass predictions
throughout the literature, for each of the four classes of model we consider
in this paper. In the Table 1 (Appendix) we compile the masses $M_{0}$,
$M_{1}$, $M_{2}$ and $M_{0}^{\prime}$ quoted in many different model
calculations. To check the validity of the relations, for each model we have
taken $M_{0}$ and $M_{2}$ as inputs and, using the mass relations, we have
computed the masses of the axial $\widetilde{M}_{1}$ and (if appropriate) the
heavy scalar $\widetilde{M}_{0}^{\prime}$, which we can then compare to the
corresponding quoted values of $M_{1}$ and $M_{0}^{\prime}$. In the last two
columns in Table 1, and also in Figure 2, we show the differences
$\widetilde{M}_{1}-M_{1}$ and $\widetilde{M}_{0}^{\prime}-M_{0}^{\prime}$,
which in most cases are 1 MeV or less. This is a striking confirmation of the
validity of the relations, across all classes of model.

From previous discussions, we know that the mass relations are exact for
(quark and diquark) models based on effective masses (corresponding to refs Wu
_et al._ (2018); Cheng  _et al._ (2020); Berezhnoy  _et al._ (2012, 2011) in
Table 1 and Figure 2). In such cases, where $\widetilde{M}_{1}-M_{1}$ or
$\widetilde{M}_{0}^{\prime}-M_{0}^{\prime}$ deviate from zero, this can be
attributed to rounding errors, from having applied the relations using inputs
which are quoted to a particular number of significant figures.

In (quark and diquark) potential models, the relations apply strictly in the
limit of perturbation theory. Many of the masses in Table 1 and Figure 2 have
been computed in perturbation theory, so we would expect the mass relations to
be satisfied exactly, up to small rounding errors as mentioned previously.
Notably, the masses of Ref. Debastiani and Navarra (2019) are not computed in
perturbation theory, and the somewhat larger deviation from exact agreement in
this case can be understood for that reason.

In the specific case of quark (not diquark) potential models, we recall the
additional caveat mentioned previously: our mass formulae – hence the
resulting mass relations – apply only subject to the assumption that the
wavefunction is symmetric under the interchange $Q\leftrightarrow\bar{q}$ (or
$q\leftrightarrow\bar{q}$, in the one flavour case). The particular examples
shown in Table 1 and Figure 2 (Refs. Liu  _et al._ (2019b); Zhang  _et al._
(2022)) satisfy this requirement, because in their implementation of the
Gaussian Expansion Method, the symmetry is automatic for degenerate quarks
($cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$). This is to some extent an
artefact of the calculation, since the symmetry is not actually imposed by the
symmetries of the Hamiltonian (see Section III). In models where the symmetry
is not imposed (such as Refs. Wang  _et al._ (2019); Lü  _et al._ (2020b)) the
relations are not satisfied. Effectively this is because the chromoelectric
term in the Hamiltonian induces a splitting between the $\varphi_{0,1,2}$ and
$\varphi_{0}^{\prime}$ states which, before spin splitting, would otherwise be
degenerate. For precisely the same reason, the relations also do not apply to
the extended chromomagnetic model of Ref. Weng  _et al._ (2021) which, unlike
the ordinary chromomagnetic model, has a chromoelectric splitting.

##  VII Conclusions

One of the main obstacles to progress in understanding the nature of exotic
hadrons is that models are not very well constrained. This is because of the
intrinsic ambiguity in identifying the relevant degrees of freedom and their
interactions, and also because model parameters are only weakly constrained by
comparison with the spectra of conventional hadrons. Consequently, absolute
mass predictions for tetraquark states are subject to systematic uncertainties
which are large and difficult to quantify, so comparing these to experimental
candidates can hardly discriminate among models.

Our perspective is that it is considerably more useful to examine not absolute
mass predictions, but relations among masses. As well as being more reliable –
in the sense of depending only weakly on model parameters, or not at all –
such relations are considerably more direct, and therefore effective, as a way
of discriminating among competing models.

The most important distinguishing feature of models is whether they include
all colour configurations (quark models) or only a subset (diquark models).
For each class of model, we showed that the corresponding potential model is
equivalent (in perturbation theory) to a simpler model with effective (quark
or diquark) masses – though in the case of quark models, the equivalence
relies on an assumption of spatial symmetry which, though commonplace, is not
strictly justified.

We derived general formulae for the mass spectrum of S-wave $QQ\bar{q}\bar{q}$
and $qq\bar{q}\bar{q}$ states in quark and diquark models, and showed how the
two models coincide in an appropriate limit. From the formulae, we identified
several resulting linear relations which are independent of, or only weakly
dependent on, model parameters. The relations are exact for (quark or diquark)
models with effective masses, or valid in perturbation theory for (quark or
diquark) potential models. Although the relations have seemingly not been
discussed in the literature before, they are apparent in the quoted mass
predictions in all classes of models.

The relations reveal how quark and diquark models have a characteristically
different pattern of masses, which can be tested against future experimental
data. In particular, given any two experimental candidates, using the
relations one can predict the masses of the additional one or two states (in
diquark or quark models, respectively). In a forthcoming paper Anwar and Burns
(2023) we apply this concept (and some other results from the present work) to
the apparent $cc\bar{c}\bar{c}$ states observed at LHCb, CMS and ATLAS.

###### Acknowledgements.

This work is supported by The Royal Society through Newton International
Fellowship, and STFC grant ST/T000813/1.

APPENDIX

|  | $M_{0}~{}~{}$ | $M_{1}~{}~{}$ | $M_{2}~{}~{}$ | $M_{0}^{\prime}~{}~{}$ | $\widetilde{M}_{1}-M_{1}$ | $\widetilde{M}_{0}^{\prime}-M_{0}^{\prime}$
---|---|---|---|---|---|---|---
Chromomagnetic quark model Wu  _et al._ (2018) |  $bb\bar{b}\bar{b}$ | 20155.4 | 20211.6 | 20242.5 | 20275.5 | $0.0$ | $0.0$
|  $cc\bar{c}\bar{c}$ | 6796.6 | 6899.2 | 6955.7 | 7016 | $0.0$ | $-0.1$
|  $bb\bar{c}\bar{c}$ | 13496.5 | 13559.7 | 13594.9 | 13633.8 | $0.3$ | $-1.7$
Chromomagnetic quark model Cheng  _et al._ (2020) |  $bb\bar{u}\bar{u}$ | 10642 | 10676 | 10699 | 10738 | $0.6$ | $0.0$
|  $bb\bar{s}\bar{s}$ | 10858 | 10901 | 10926 | 10954 | $0.4$ | $0.8$
|  $cc\bar{u}\bar{u}$ | 4000 | 4124 | 4194 | 4277 | $-0.2$ | $-0.8$
|  $cc\bar{s}\bar{s}$ | 4227 | 4358 | 4430 | 4502 | $0.6$ | $0.1$
Quark potential model Zhang  _et al._ (2022) |  $bb\bar{b}\bar{b}$ | 19200 | 19216 | 19225 | 19235 | $0.1$ | $-0.5$
|  $cc\bar{c}\bar{c}$ | 6411 | 6453 | 6475 | 6500 | $-0.7$ | $-0.8$
Quark potential model Liu  _et al._ (2019b) |  $bb\bar{b}\bar{b}$ | 19306 | 19329 | 19341 | 19355 | $-0.4$ | $-0.8$
|  $cc\bar{c}\bar{c}$ | 6455 | 6500 | 6524 | 6550 | $-0.5$ | $0.1$
Diquark model Berezhnoy  _et al._ (2012, 2011) |  $bb\bar{b}\bar{b}$ | 18754 | 18808 | 18916 |  | $0.0$ |
|  $cc\bar{c}\bar{c}$ | 5966 | 6051 | 6223 |  | $0.7$ |
Diquark potential model Lundhammar and Ohlsson (2020) |  $bb\bar{b}\bar{b}$ | 18723 | 18738 | 18768 |  | $0.0$ |
|  $cc\bar{c}\bar{c}$ | 5960 | 6009 | 6100 |  | $-2.3$ |
Diquark potential model Ebert  _et al._ (2007) |  $bb\bar{u}\bar{u}$ | 10648 | 10657 | 10673 |  | $-0.7$ |
|  $bb\bar{s}\bar{s}$ | 10932 | 10939 | 10950 |  | $-1.0$ |
|  $cc\bar{u}\bar{u}$ | 4056 | 4079 | 4118 |  | $-2.3$ |
|  $cc\bar{s}\bar{s}$ | 4359 | 4375 | 4402 |  | $-1.7$ |
Diquark potential model Debastiani and Navarra (2019) |  $cc\bar{c}\bar{c}$ | 5969.4 | 6020.9 | 6115.4 |  | $-2.8$ |

Table 1: The masses $M_{0}$, $M_{1}$, $M_{2}$ and $M_{0}^{\prime}$ are taken
from literature calculations in different models, whereas $\widetilde{M}_{1}$
and $\widetilde{M}_{0}^{\prime}$ are computed from the mass relations (64),
(65) and (66), having taken $M_{0}$ and $M_{2}$ as inputs. In the last two
columns we show the differences $\widetilde{M}_{1}-M_{1}$ and
$\widetilde{M}_{0}^{\prime}-M_{0}^{\prime}$, as a measure of the accuracy of
the mass relations.

## References

  * Augustin  _et al._ (1974) J. E. Augustin _et al._ (SLAC-SP-017), Phys. Rev. Lett. 33, 1406 (1974).
  * Aubert  _et al._ (1974) J. J. Aubert _et al._ (E598), Phys. Rev. Lett. 33, 1404 (1974).
  * Lebed  _et al._ (2017) R. F. Lebed, R. E. Mitchell, and E. S. Swanson, Prog. Part. Nucl. Phys. 93, 143 (2017), arXiv:1610.04528 [hep-ph] .
  * Brambilla  _et al._ (2020) N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C.-P. Shen, C. E. Thomas, A. Vairo, and C.-Z. Yuan, Phys. Rept. 873, 1 (2020), arXiv:1907.07583 [hep-ex] .
  * Chen  _et al._ (2023) H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu, and S.-L. Zhu, Rept. Prog. Phys. 86, 026201 (2023), arXiv:2204.02649 [hep-ph] .
  * Altmannshofer  _et al._ (2019) W. Altmannshofer _et al._ (Belle-II), PTEP 2019, 123C01 (2019), [Erratum: PTEP 2020, 029201 (2020)], arXiv:1808.10567 [hep-ex] .
  * Lutz  _et al._ (2009) M. F. M. Lutz _et al._ (PANDA), (2009), arXiv:0903.3905 [hep-ex] .
  * Workman  _et al._ (2022) R. L. Workman _et al._ (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Choi  _et al._ (2003) S. K. Choi _et al._ (Belle), Phys. Rev. Lett. 91, 262001 (2003), arXiv:hep-ex/0309032 .
  * Burns (2015a) T. J. Burns, Eur. Phys. J. A51, 152 (2015a), arXiv:1509.02460 [hep-ph] .
  * Burns and Swanson (2016) T. J. Burns and E. S. Swanson, Phys. Lett. B760, 627 (2016), arXiv:1603.04366 [hep-ph] .
  * Ali  _et al._ (2017) A. Ali, J. S. Lange, and S. Stone, Prog. Part. Nucl. Phys. 97, 123 (2017), arXiv:1706.00610 [hep-ph] .
  * Lu  _et al._ (2017) Y. Lu, M. N. Anwar, and B.-S. Zou, Phys. Rev. D 96, 114022 (2017), arXiv:1705.00449 [hep-ph] .
  * Liu  _et al._ (2019a) Y.-R. Liu, H.-X. Chen, W. Chen, X. Liu, and S.-L. Zhu, Prog. Part. Nucl. Phys. 107, 237 (2019a), arXiv:1903.11976 [hep-ph] .
  * Hanhart and Klempt (2019) C. Hanhart and E. Klempt, Int. J. Mod. Phys. A 35, 2050019 (2019), arXiv:1906.11971 [hep-ph] .
  * Burns and Swanson (2021a) T. J. Burns and E. S. Swanson, Phys. Lett. B 813, 136057 (2021a), arXiv:2008.12838 [hep-ph] .
  * Burns and Swanson (2021b) T. Burns and E. Swanson, Phys. Rev. D 103, 014004 (2021b), arXiv:2009.05352 [hep-ph] .
  * Burns and Swanson (2022a) T. J. Burns and E. S. Swanson, Eur. Phys. J. A 58, 68 (2022a), arXiv:2112.11527 [hep-ph] .
  * Dong  _et al._ (2021) X.-K. Dong, F.-K. Guo, and B.-S. Zou, Commun. Theor. Phys. 73, 125201 (2021), arXiv:2108.02673 [hep-ph] .
  * Wu  _et al._ (2022) T.-W. Wu, Y.-W. Pan, M.-Z. Liu, and L.-S. Geng, (2022), arXiv:2208.00882 [hep-ph] .
  * Burns and Swanson (2022b) T. J. Burns and E. S. Swanson, Phys. Lett. B 838, 137715 (2022b), arXiv:2208.05106 [hep-ph] .
  * Tornqvist (1991) N. A. Tornqvist, Phys.Rev.Lett. 67, 556 (1991).
  * Tornqvist (1994) N. A. Tornqvist, Z. Phys. C61, 525 (1994), arXiv:hep-ph/9310247 .
  * Swanson (2004a) E. S. Swanson, Phys. Lett. B588, 189 (2004a), arXiv:hep-ph/0311229 .
  * Swanson (2004b) E. S. Swanson, Phys. Lett. B598, 197 (2004b), arXiv:hep-ph/0406080 .
  * Thomas and Close (2008) C. E. Thomas and F. E. Close, Phys. Rev. D78, 034007 (2008), arXiv:0805.3653 [hep-ph] .
  * Fernandez-Carames  _et al._ (2009) T. Fernandez-Carames, A. Valcarce, and J. Vijande, Phys. Rev. Lett. 103, 222001 (2009), arXiv:1001.4506 [hep-ph] .
  * Guo  _et al._ (2018) F.-K. Guo, C. Hanhart, U.-G. Meißner, Q. Wang, Q. Zhao, and B.-S. Zou, Rev. Mod. Phys. 90, 015004 (2018), [Erratum: Rev.Mod.Phys. 94, 029901 (2022)], arXiv:1705.00141 [hep-ph] .
  * Burns and Swanson (2019) T. J. Burns and E. S. Swanson, Phys. Rev. D100, 114033 (2019), arXiv:1908.03528 [hep-ph] .
  * Anwar and Lu (2021) M. N. Anwar and Y. Lu, Phys. Rev. D 104, 094006 (2021), arXiv:2109.02539 [hep-ph] .
  * Burns and Swanson (2022c) T. J. Burns and E. S. Swanson, Phys. Rev. D 106, 054029 (2022c), arXiv:2207.00511 [hep-ph] .
  * Gell-Mann (1961) M. Gell-Mann,  (1961), 10.2172/4008239.
  * Gell-Mann (1962) M. Gell-Mann, Phys. Rev. 125, 1067 (1962).
  * Okubo (1962a) S. Okubo, Prog. Theor. Phys. 27, 949 (1962a).
  * Okubo (1962b) S. Okubo, Prog. Theor. Phys. 28, 24 (1962b).
  * Lipkin (1978) H. J. Lipkin, Phys. Rev. Lett. 41, 1629 (1978).
  * Cohen and Lipkin (1980) I. Cohen and H. J. Lipkin, Phys. Lett. B93, 56 (1980).
  * Lipkin (1986a) H. J. Lipkin, Phys. Lett. B171, 293 (1986a).
  * Lipkin (1989) H. J. Lipkin, Phys. Lett. B233, 446 (1989).
  * Lipkin (1990) H. J. Lipkin, Nucl. Phys. A507, 205 (1990).
  * Karliner and Lipkin (2007) M. Karliner and H. J. Lipkin, Phys. Lett. B650, 185 (2007), arXiv:hep-ph/0608004 [hep-ph] .
  * Karliner  _et al._ (2009) M. Karliner, B. Keren-Zur, H. J. Lipkin, and J. L. Rosner, Annals Phys. 324, 2 (2009), arXiv:0804.1575 [hep-ph] .
  * Voloshin (2008) M. Voloshin, Prog.Part.Nucl.Phys. 61, 455 (2008), arXiv:0711.4556 [hep-ph] .
  * Burns (2011a) T. J. Burns, Phys.Rev. D84, 034021 (2011a), arXiv:1105.2533 [hep-ph] .
  * Burns (2011b) T. J. Burns, in  _14th International Conference on Hadron Spectroscopy_ (2011) arXiv:1108.5259 [hep-ph] .
  * Burns (2014) T. J. Burns, Phys.Rev. D90, 034009 (2014), arXiv:1403.7538 [hep-ph] .
  * Burns (2015b) T. J. Burns, Acta Phys. Polon. Supp. 8, 31 (2015b), arXiv:1411.2485 [hep-ph] .
  * Lebed and Swanson (2017) R. F. Lebed and E. S. Swanson, Phys. Rev. D96, 056015 (2017), arXiv:1705.03140 [hep-ph] .
  * Anwar  _et al._ (2019) M. N. Anwar, Y. Lu, and B.-S. Zou, Phys. Rev. D 99, 094005 (2019), arXiv:1806.01155 [hep-ph] .
  * Lipkin (1986b) H. J. Lipkin, Phys. Lett. B 172, 242 (1986b).
  * Eichten and Quigg (2017) E. J. Eichten and C. Quigg, Phys. Rev. Lett. 119, 202002 (2017), arXiv:1707.09575 [hep-ph] .
  * Anwar  _et al._ (2018a) M. N. Anwar, J. Ferretti, F.-K. Guo, E. Santopinto, and B.-S. Zou, Eur. Phys. J. C 78, 647 (2018a), arXiv:1710.02540 [hep-ph] .
  * Aaij  _et al._ (2020) R. Aaij _et al._ (LHCb), Sci. Bull. 65, 1983 (2020), arXiv:2006.16957 [hep-ex] .
  * Hayrapetyan  _et al._ (2023) A. Hayrapetyan _et al._ (CMS), (2023), arXiv:2306.07164 [hep-ex] .
  * Aad  _et al._ (2023) G. Aad _et al._ (ATLAS), (2023), arXiv:2304.08962 [hep-ex] .
  * Anwar and Burns (2023) M. N. Anwar and T. J. Burns, (2023), in preparation.
  * Zouzou  _et al._ (1986) S. Zouzou, B. Silvestre-Brac, C. Gignoux, and J. M. Richard, Z. Phys. C 30, 457 (1986).
  * Silvestre-Brac and Semay (1993a) B. Silvestre-Brac and C. Semay, Z. Phys. C 57, 273 (1993a).
  * Silvestre-Brac and Semay (1993b) B. Silvestre-Brac and C. Semay, Z. Phys. C 59, 457 (1993b).
  * Francis  _et al._ (2016) A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Phys. Rev. Lett. 118, 142001 (2016), arXiv:1607.05214 [hep-lat] .
  * Bicudo  _et al._ (2017) P. Bicudo, J. Scheunert, and M. Wagner, Phys. Rev. D 95, 034502 (2017), arXiv:1612.02758 [hep-lat] .
  * Czarnecki  _et al._ (2018) A. Czarnecki, B. Leng, and M. B. Voloshin, Phys. Lett. B 778, 233 (2018), arXiv:1708.04594 [hep-ph] .
  * Karliner and Rosner (2017) M. Karliner and J. L. Rosner, Phys. Rev. Lett. 119, 202001 (2017), arXiv:1707.07666 [hep-ph] .
  * Mehen (2017) T. Mehen, Phys. Rev. D 96, 094028 (2017), arXiv:1708.05020 [hep-ph] .
  * Francis  _et al._ (2019) A. Francis, R. J. Hudspith, R. Lewis, and K. Maltman, Phys. Rev. D 99, 054505 (2019), arXiv:1810.10550 [hep-lat] .
  * Junnarkar  _et al._ (2019) P. Junnarkar, N. Mathur, and M. Padmanath, Phys. Rev. D 99, 034507 (2019), arXiv:1810.12285 [hep-lat] .
  * Leskovec  _et al._ (2019) L. Leskovec, S. Meinel, M. Pflaumer, and M. Wagner, Phys. Rev. D 100, 014503 (2019), arXiv:1904.04197 [hep-lat] .
  * Lü  _et al._ (2020a) Q.-F. Lü, D.-Y. Chen, and Y.-B. Dong, Phys. Rev. D 102, 034012 (2020a), arXiv:2006.08087 [hep-ph] .
  * Aaij  _et al._ (2022) R. Aaij _et al._ (LHCb), Nature Phys. 18, 751 (2022), arXiv:2109.01038 [hep-ex] .
  * Lichtenberg  _et al._ (1996) D. Lichtenberg, R. Roncaglia, and E. Predazzi, (1996), arXiv:hep-ph/9611428 .
  * Jaffe (2005) R. L. Jaffe, Phys. Rept. 409, 1 (2005), arXiv:hep-ph/0409065 .
  * De Rujula  _et al._ (1975) A. De Rujula, H. Georgi, and S. L. Glashow, Phys. Rev. D12, 147 (1975).
  * Jaffe (1977a) R. L. Jaffe, Phys. Rev. D15, 267 (1977a).
  * Jaffe (1977b) R. L. Jaffe, Phys. Rev. D15, 281 (1977b).
  * Mulders  _et al._ (1979) P. J. Mulders, A. T. M. Aerts, and J. J. de Swart, Phys. Rev. D19, 2635 (1979).
  * Aerts  _et al._ (1980) A. T. M. Aerts, P. J. Mulders, and J. J. De Swart, Phys. Rev. D21, 1370 (1980).
  * Hogaasen  _et al._ (2006) H. Hogaasen, J. M. Richard, and P. Sorba, Phys. Rev. D73, 054013 (2006), arXiv:hep-ph/0511039 .
  * Buccella  _et al._ (2007) F. Buccella, H. Hogaasen, J.-M. Richard, and P. Sorba, Eur. Phys. J. C 49, 743 (2007), arXiv:hep-ph/0608001 .
  * Cui  _et al._ (2007) Y. Cui, X.-L. Chen, W.-Z. Deng, and S.-L. Zhu, HEPNP 31, 7 (2007), arXiv:hep-ph/0607226 [hep-ph] .
  * Stancu (2010) F. Stancu, J. Phys. G37, 075017 (2010), arXiv:0906.2485 [hep-ph] .
  * Hogaasen  _et al._ (2014) H. Hogaasen, E. Kou, J.-M. Richard, and P. Sorba, Phys.Lett. B732, 97 (2014), arXiv:1309.2049 [hep-ph] .
  * Karliner  _et al._ (2017) M. Karliner, S. Nussinov, and J. L. Rosner, Phys. Rev. D95, 034011 (2017), arXiv:1611.00348 [hep-ph] .
  * Wu  _et al._ (2018) J. Wu, Y.-R. Liu, K. Chen, X. Liu, and S.-L. Zhu, Phys. Rev. D 97, 094015 (2018), arXiv:1605.01134 [hep-ph] .
  * Stancu (2016) F. Stancu, J. Phys. G 43, 105001 (2016), arXiv:1603.03322 [hep-ph] .
  * Cheng  _et al._ (2020) J.-B. Cheng, S.-Y. Li, Y.-R. Liu, Z.-G. Si, and T. Yao, Chin. Phys. C 45, 043102 (2020), arXiv:2008.00737 [hep-ph] .
  * Weng  _et al._ (2021) X.-Z. Weng, X.-L. Chen, W.-Z. Deng, and S.-L. Zhu, Phys. Rev. D 103, 034001 (2021), arXiv:2010.05163 [hep-ph] .
  * Ader  _et al._ (1982) J. P. Ader, J. M. Richard, and P. Taxil, Phys. Rev. D 25, 2370 (1982).
  * Weinstein and Isgur (1982) J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659 (1982).
  * Weinstein and Isgur (1983) J. D. Weinstein and N. Isgur, Phys. Rev. D27, 588 (1983).
  * Weinstein and Isgur (1990) J. D. Weinstein and N. Isgur, Phys. Rev. D41, 2236 (1990).
  * Brink and Stancu (1994) D. M. Brink and F. Stancu, Phys. Rev. D49, 4665 (1994).
  * Brink and Stancu (1998) D. M. Brink and F. Stancu, Phys. Rev. D57, 6778 (1998).
  * Lloyd and Vary (2004) R. J. Lloyd and J. P. Vary, Phys. Rev. D 70, 014009 (2004), arXiv:hep-ph/0311179 .
  * Vijande  _et al._ (2004) J. Vijande, F. Fernandez, A. Valcarce, and B. Silvestre-Brac, Eur. Phys. J. A 19, 383 (2004), arXiv:hep-ph/0310007 .
  * Barnea  _et al._ (2006) N. Barnea, J. Vijande, and A. Valcarce, Phys. Rev. D 73, 054004 (2006), arXiv:hep-ph/0604010 .
  * Vijande  _et al._ (2007a) J. Vijande, N. Barnea, and A. Valcarce, Nucl. Phys. A790, 542 (2007a), arXiv:hep-ph/0610287 .
  * Vijande  _et al._ (2007b) J. Vijande, E. Weissman, A. Valcarce, and N. Barnea, Phys. Rev. D76, 094027 (2007b), arXiv:0710.2516 [hep-ph] .
  * Vijande  _et al._ (2007c) J. Vijande, E. Weissman, N. Barnea, and A. Valcarce, Phys. Rev. D76, 094022 (2007c), arXiv:0708.3285 [hep-ph] .
  * Vijande and Valcarce (2009a) J. Vijande and A. Valcarce, Phys. Rev. C80, 035204 (2009a), arXiv:0908.3254 [hep-ph] .
  * Vijande  _et al._ (2009) J. Vijande, A. Valcarce, and N. Barnea, Phys. Rev. D79, 074010 (2009), arXiv:0903.2949 [hep-ph] .
  * Vijande and Valcarce (2009b) J. Vijande and A. Valcarce, Symmetry 1, 155 (2009b), 0912.3605 .
  * Lu  _et al._ (2016) Y. Lu, M. N. Anwar, and B.-S. Zou, Phys. Rev. D 94, 034021 (2016), arXiv:1606.06927 [hep-ph] .
  * Richard  _et al._ (2017) J.-M. Richard, A. Valcarce, and J. Vijande, Phys. Rev. D95, 054019 (2017), arXiv:1703.00783 [hep-ph] .
  * Richard  _et al._ (2018) J.-M. Richard, A. Valcarce, and J. Vijande, Phys. Rev. C97, 035211 (2018), arXiv:1803.06155 [hep-ph] .
  * Wang  _et al._ (2019) G.-J. Wang, L. Meng, and S.-L. Zhu, Phys. Rev. D 100, 096013 (2019), arXiv:1907.05177 [hep-ph] .
  * Hernández  _et al._ (2020) E. Hernández, J. Vijande, A. Valcarce, and J.-M. Richard, Phys. Lett. B 800, 135073 (2020), arXiv:1910.13394 [hep-ph] .
  * Liu  _et al._ (2019b) M.-S. Liu, Q.-F. Lü, X.-H. Zhong, and Q. Zhao, (2019b), arXiv:1901.02564 [hep-ph] .
  * Wang  _et al._ (2022) G.-J. Wang, Q. Meng, and M. Oka, Phys. Rev. D 106, 096005 (2022), arXiv:2208.07292 [hep-ph] .
  * Zhang  _et al._ (2022) J. Zhang, J.-B. Wang, G. Li, C.-S. An, C.-R. Deng, and J.-J. Xie, Eur. Phys. J. C 82, 1126 (2022), arXiv:2209.13856 [hep-ph] .
  * Maiani  _et al._ (2005a) L. Maiani, F. Piccinini, A. D. Polosa, and V. Riquer, Phys. Rev. D71, 014028 (2005a), arXiv:hep-ph/0412098 .
  * Maiani  _et al._ (2005b) L. Maiani, V. Riquer, F. Piccinini, and A. D. Polosa, Phys. Rev. D72, 031502 (2005b), arXiv:hep-ph/0507062 .
  * Drenska  _et al._ (2008) N. V. Drenska, R. Faccini, and A. D. Polosa, Phys. Lett. B669, 160 (2008), arXiv:0807.0593 [hep-ph] .
  * Drenska  _et al._ (2009) N. V. Drenska, R. Faccini, and A. D. Polosa, Phys. Rev. D79, 077502 (2009), arXiv:0902.2803 [hep-ph] .
  * Ali  _et al._ (2010) A. Ali, C. Hambrock, I. Ahmed, and M. J. Aslam, Phys.Lett. B684, 28 (2010), arXiv:0911.2787 [hep-ph] .
  * Ali (2011) A. Ali, PoS BEAUTY2011, 002 (2011), 1108.2197 .
  * Ali and Parkhomenko (2019) A. Ali and A. Ya. Parkhomenko, Phys. Lett. B793, 365 (2019), arXiv:1904.00446 [hep-ph] .
  * Ebert  _et al._ (2006) D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B634, 214 (2006), arXiv:hep-ph/0512230 .
  * Ebert  _et al._ (2007) D. Ebert, R. N. Faustov, V. O. Galkin, and W. Lucha, Phys. Rev. D76, 114015 (2007), arXiv:0706.3853 [hep-ph] .
  * Ebert  _et al._ (2008) D. Ebert, R. N. Faustov, and V. O. Galkin, Eur. Phys. J. C58, 399 (2008), arXiv:0808.3912 [hep-ph] .
  * Ebert  _et al._ (2009) D. Ebert, R. N. Faustov, and V. O. Galkin, Mod. Phys. Lett. A24, 567 (2009), arXiv:0812.3477 [hep-ph] .
  * Ebert  _et al._ (2011) D. Ebert, R. N. Faustov, and V. O. Galkin, Phys. Lett. B696, 241 (2011), arXiv:1011.2677 [hep-ph] .
  * Faustov  _et al._ (2020) R. Faustov, V. Galkin, and E. Savchenko, Phys. Rev. D 102, 114030 (2020), arXiv:2009.13237 [hep-ph] .
  * Anwar  _et al._ (2018b) M. N. Anwar, J. Ferretti, and E. Santopinto, Phys. Rev. D 98, 094015 (2018b), arXiv:1805.06276 [hep-ph] .
  * Bedolla  _et al._ (2020) M. A. Bedolla, J. Ferretti, C. D. Roberts, and E. Santopinto, Eur. Phys. J. C 80, 1004 (2020), arXiv:1911.00960 [hep-ph] .
  * Debastiani and Navarra (2019) V. Debastiani and F. Navarra, Chin. Phys. C 43, 013105 (2019), arXiv:1706.07553 [hep-ph] .
  * Lundhammar and Ohlsson (2020) P. Lundhammar and T. Ohlsson, Phys. Rev. D 102, 054018 (2020), arXiv:2006.09393 [hep-ph] .
  * Heller and Tjon (1987) L. Heller and J. A. Tjon, Phys. Rev. D35, 969 (1987).
  * Berezhnoy  _et al._ (2012) A. Berezhnoy, A. Luchinsky, and A. Novoselov, Phys. Rev. D 86, 034004 (2012), arXiv:1111.1867 [hep-ph] .
  * Berezhnoy  _et al._ (2011) A. Berezhnoy, A. Likhoded, A. Luchinsky, and A. Novoselov, Phys.Rev. D84, 094023 (2011), arXiv:1101.5881 [hep-ph] .
  * Lü  _et al._ (2020b) Q.-F. Lü, D.-Y. Chen, and Y.-B. Dong, Eur. Phys. J. C 80, 871 (2020b), arXiv:2006.14445 [hep-ph] .