# Spectrum of Hidden-Charm, Open-Strange Exotics in the Dynamical Diquark
Model

Jesse F. Giron<EMAIL_ADDRESS>Richard F. Lebed<EMAIL_ADDRESS>Steven
R. Martinez<EMAIL_ADDRESS>Department of Physics, Arizona State University,
Tempe, AZ 85287, USA

(June, 2021)

###### Abstract

The observation by BESIII and LHCb of states with hidden charm and open
strangeness ($c\bar{c}q\bar{s}$) presents new opportunities for the
development of a global model of heavy-quark exotics. Here we extend the
dynamical diquark model to encompass such states, using the same values of
Hamiltonian parameters previously obtained from the nonstrange and hidden-
strange sectors. The large mass splitting between $Z_{cs}(4000)$ and
$Z_{cs}(4220)$ suggests substantial SU(3)flavor mixing between all
$J^{P}\\!=\\!1^{+}$ states, while their average mass compared to that of other
sectors offers a direct probe of flavor octet-singlet mixing among exotics. We
also explore the inclusion of $\eta$-like exchanges within the states, and
find their effects to be quite limited. In addition, using the same diquark-
mass parameters, we find $P_{c}(4312)$ and $P_{cs}(4459)$ to fit well as
corresponding nonstrange and open-strange pentaquarks.

Exotic hadrons, diquarks

##  I Introduction

The spectrum of known heavy-quark exotic hadrons continues to expand
frequently, with about 50 candidates observed to date. Almost all have a
valence light-flavor content consisting of only $q\\!\equiv\\!u$ and $d$
quarks, but very recently some states with open strangeness have been detected
in both the open-charm Aaij  _et al._ (2020a, b) ($\bar{c}d\bar{s}u$) and
hidden-charm sectors. In the latter, both a pentaquark Aaij  _et al._ (2021a)
($c\bar{c}uds$) state and tetraquark Ablikim  _et al._ (2021); Aaij  _et al._
(2021b) ($c\bar{c}u\bar{s}$) states have been observed.111Charge-conjugate
states are implied as included throughout.

Multiple reviews summarizing both experimental and theoretical advances in
this field have appeared in recent years Lebed  _et al._ (2017); Chen  _et
al._ (2016); Hosaka  _et al._ (2016); Esposito  _et al._ (2017); Guo  _et al._
(2018); Ali  _et al._ (2017); Olsen  _et al._ (2018); Karliner  _et al._
(2018); Yuan (2018); Liu  _et al._ (2019); Brambilla  _et al._ (2020). Several
competing theoretical frameworks (di-hadron molecular states, bound states of
diquarks, threshold enhancements, etc.) have been developed for years, but no
single scheme has yet emerged as a dominant paradigm to explain all the new
states, analogous to the way that quark-potential models successfully
elucidate the conventional $c\bar{c}$ and $b\bar{b}$ sectors Zyla  _et al._
(2020).

Since a number of the observed exotics decay to final states like
$J/\psi\,\phi$ or $D_{sJ}^{(*)}\bar{D}_{sJ}^{(*)}$, they possess a presumptive
$c\bar{c}s\bar{s}$ valence content. The advent of $\bar{c}c\bar{s}q$ states
thus introduces an intermediate case between $c\bar{c}q\bar{q}^{\prime}$ and
$c\bar{c}s\bar{s}$ cases, and therefore not only provides an opportunity to
examine whether a particular theoretical picture can successfully incorporate
data from all of these flavor sectors, but also examines the manifestation of
light-quark SU(3)flavor for the first time outside of conventional mesons and
baryons.

The new data in the hidden-charm, open-strange sector itself is quite
interesting. BESIII observes a structure Ablikim  _et al._ (2021) in the
$K^{+}$ recoil spectrum of
$e^{+}e^{-}\\!\to\\!K^{+}(D_{s}^{-}D^{*0}\\!+\\!D_{s}^{*-}D^{0})$ near the
2-charmed-meson thresholds (about 3975 and 3977 MeV, respectively). For this
$Z_{cs}(3985)$ state, they obtain:

$\displaystyle m_{Z_{cs}^{-}}$ $\displaystyle=$ $\displaystyle
3982.5^{+1.8}_{-2.6}\pm 2.1\ {\rm MeV}\,,$ $\displaystyle\Gamma_{Z_{cs}^{-}}$
$\displaystyle=$ $\displaystyle 12.8^{+5.3}_{-4.4}\pm 3.0/{\rm MeV}\,,$ (1)

and $J^{P}\\!=\\!1^{+}$. Meanwhile, LHCb reports 2 states Aaij  _et al._
(2021b) decaying to $J/\psi\,K^{+}$,

$\displaystyle m_{Z_{cs}^{+}(4000)}$ $\displaystyle=$ $\displaystyle 4003\pm
6^{+\ 4}_{-14}\ {\rm MeV}\,,$ $\displaystyle\Gamma_{Z_{cs}^{+}(4000)}$
$\displaystyle=$ $\displaystyle 131\pm 15\pm 26\ {\rm MeV}\,,$ $\displaystyle
m_{Z_{cs}^{+}(4220)}$ $\displaystyle=$ $\displaystyle 4216\pm 24^{+43}_{-30}\
{\rm MeV}\,,$ $\displaystyle\Gamma_{Z_{cs}^{+}(4220)}$ $\displaystyle=$
$\displaystyle 233\pm 52^{+97}_{-73}\ {\rm MeV}\,,$ (2)

with $Z_{cs}(4000)$ carrying $J^{P}\\!=\\!1^{+}$ and $Z_{cs}(4220)$ favored to
carry $J^{P}\\!=\\!1^{+}$. The masses of $Z_{cs}(3985)$ and $Z_{cs}(4000)$ are
compatible with them being the same state, but their measured widths are
wildly different. For the purposes of this paper, we assume that only a single
light $Z_{cs}$ state exists near 4.0 GeV, the discrepancy in width
measurements perhaps arising from effects caused by interactions with the
nearby charmed-meson thresholds.222This opinion is not universal. For example,
Ref. Maiani  _et al._ (2021a) treats $Z_{cs}(3985)$ and $Z_{cs}(4000)$ as
separate states belonging to distinct SU(3)flavor multiplets. LHCb also
observes a baryonic structure Aaij  _et al._ (2021a) $P_{cs}(4459)$ decaying
to $J/\psi\,\Lambda$:

$\displaystyle m_{P_{cs}(4459)}$ $\displaystyle=$ $\displaystyle 4458.8\pm
2.9^{+4.7}_{-1.1}\ {\rm MeV}\,,$ $\displaystyle\Gamma_{P_{cs}(4459)}$
$\displaystyle=$ $\displaystyle 17.3\pm 6.5^{+8.0}_{-5.7}\ {\rm MeV}\,,$ (3)

although its $J^{P}$ value is not yet determined.

Quite an extensive body of theoretical work on the hidden-charm, open-strange
hadrons exists. For example, in the meson sector whose study forms the bulk of
this work, several papers Lee  _et al._ (2009); Ebert  _et al._ (2008); Chen
_et al._ (2013); Dias  _et al._ (2013); Voloshin (2019); Ferretti and
Santopinto (2020) predate the experimental observations, while multiple
studies followed the announcement of the BESIII result but preceded the
appearance of the LHCb paper Wang  _et al._ (2021a); Yang  _et al._ (2021);
Meng  _et al._ (2020); Liu  _et al._ (2020); Wan and Qiao (2021); Chen and
Huang (2021); Du  _et al._ (2020); Cao  _et al._ (2021); Sun and Xiao (2020);
Wang  _et al._ (2020, 2021b); Wang (2021a); Azizi and Er (2021); Jin  _et al._
(2020); Süngü  _et al._ (2020); Ikeno  _et al._ (2021); Xu  _et al._ (2020);
Meng  _et al._ (2021a); Wang (2021b); Guo and Oller (2021); Albuquerque  _et
al._ (2021); Özdem and Azizi (2021); Yan  _et al._ (2021), and yet others
appeared subsequent to the LHCb results Maiani  _et al._ (2021a); Ge  _et al._
(2021); Chen  _et al._ (2021); Ortega  _et al._ (2021); Chen (2021); Maiani
_et al._ (2021b); Meng  _et al._ (2021b); Özdem and Karadeniz Yildirim (2021);
Shi  _et al._ (2021). As one may imagine, this body of work encompasses
multiple approaches, including molecular and diquark models, chiral-quark
models, and QCD sum rules, among others.

The present work uses the dynamical diquark model, initially introduced in
Ref. Brodsky  _et al._ (2014) as a theoretical picture to explain how
relatively compact color-triplet diquark quasiparticle pairs can form
spatially extended tetraquark states, and extended in Ref. Lebed (2015) to
describe pentaquarks as color-triplet diquark-triquark quasiparticle bound
states. The picture is developed in Ref. Lebed (2017) into a predictive model
by noting that the static interaction potential between the heavy color-
triplet quasiparticles is the same one as appearing in lattice simulations of
heavy quarkonium and its hybrids. The multiplet band structure for
$c\bar{c}q\bar{q}^{\prime}$ and $c\bar{c}qqq$ states is studied numerically in
Ref. Giron  _et al._ (2019); the fine structure of the ground-state ($S$-wave)
$c\bar{c}q\bar{q}^{\prime}$ multiplet is examined numerically in Ref. Giron
_et al._ (2020) and that of the $P$-wave multiplet appears in Ref. Giron and
Lebed (2020a). An analogous study of the $b\bar{b}q\bar{q}^{\prime}$ and
$c\bar{c}s\bar{s}$ systems is presented in Ref. Giron and Lebed (2020b), and
the $c\bar{c}c\bar{c}$ states are investigated in Ref. Giron and Lebed
(2020c). Radiative transitions between exotic states are computed in Ref. Gens
_et al._ (2021).

Our analysis of $c\bar{c}q\bar{s}$ exotics here directly interpolates between
the analysis of $c\bar{c}q\bar{q}^{\prime}$ in Ref. Giron  _et al._ (2020) and
$c\bar{c}s\bar{s}$ in Ref. Giron and Lebed (2020b), and uses the same
numerical inputs. However, we find that a careful treatment of the SU(3)flavor
structure introduces one new parameter, related to octet-singlet mixing. In
addition, we allow for the possibility of an $\eta$-like exchange between the
diquarks analogous to the $\pi$-like exchange already present in the original
model Giron  _et al._ (2020), but show that its effects are quite limited by
constraints from the phenomenology of the $c\bar{c}q\bar{q}^{\prime}$ sector.
We find that the known phenomenology of the open-strange sector does indeed
follow from that of the other sectors, despite superficially appearing quite
different. We also carry out an analogous exercise for nonstrange and open-
strange hidden-charm pentaquark states [$P_{cs}(4459)$ currently being the
only known example of the latter], and obtain remarkably satisfactory results.

This paper is organized as follows. In Sec. II we define the multiplets of
states in terms of eigenstates of both good diquark spin and good heavy-
quark/light-quark spins. Section III presents the Hamiltonian for the
$c\bar{c}q\bar{q}^{\prime}$ and $c\bar{c}s\bar{s}$ sectors, now including a
possible term from $\eta$-like exchanges, and computes all relevant matrix
elements. Section IV performs the same analysis for the $c\bar{c}q\bar{s}$
sector, and discusses possible mixing between multiplets whose nonstrange
members carry opposite $C$ parity. In Sec. V we discuss the effects of octet-
singlet mixing on the analysis and present numerical results, and in Sec. VI
we summarize and conclude.

##  II States of the Model

A cataloguing of the ${Q\bar{Q}}q\bar{q}^{\prime}$ or ${Q\bar{Q}}q\hskip
1.0ptq_{1}q_{2}$ states in the dynamical diquark model, where
$q,q^{\prime}\\!,q_{i}\\!\in\\!\\{u,d\\}$, first appears in Ref. Lebed (2017).
The same notation, with small modifications, is applied to $c\bar{c}s\bar{s}$
in Ref. Giron and Lebed (2020b) and to $c\bar{c}c\bar{c}$ in Ref. Giron and
Lebed (2020c). All confirmed exotic candidates to date have successfully been
accommodated within the lowest ($\Sigma^{+}_{g}$) Born-Oppenheimer potential
of the gluon field connecting the heavy diquark
[$\delta^{\vphantom{1}}\\!\equiv(Qq)$]-antidiquark
[${\bar{\delta}}\\!\equiv\\!(\bar{Q}\bar{q}^{\prime})$] or diquark-triquark
[${\bar{\theta}}\\!\equiv\\!(\bar{Q}(q_{1}q_{2}))$] quasiparticles. In all
cases, $\delta^{\vphantom{1}},{\bar{\delta}},{\bar{\theta}}$ are assumed to
transform as color triplets (or antitriplets) and each quasiparticle contains
no internal orbital angular momentum.

In the case of ${Q\bar{Q}}q\bar{q}^{\prime}$, the classification scheme then
begins with 6 possible core states in which the quasiparticle pair lie in a
relative $S$ wave. Indicating the total spin $s$ of a diquark
$\delta^{\vphantom{1}},{\bar{\delta}}$ by
$s_{\delta}^{\vphantom{1}},s_{\bar{\delta}}$ and using a subscript on the full
state to indicate its total spin, one obtains the spectrum

$\displaystyle J^{PC}=0^{++}:$ $\displaystyle
X_{0}\equiv\left|0_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{0}\,,\ \
X_{0}^{\prime}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{0}\,,$
$\displaystyle J^{PC}=1^{++}:$ $\displaystyle
X_{1}\equiv\frac{1}{\sqrt{2}}\left(\left|1_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{1}\\!+\left|0_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\right)\,,$
$\displaystyle J^{PC}=1^{+-}:$ $\displaystyle\,Z\
\equiv\frac{1}{\sqrt{2}}\left(\left|1_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{1}\\!-\left|0_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\right)\,,$
$\displaystyle\,Z^{\prime}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\,,$
$\displaystyle J^{PC}=2^{++}:$ $\displaystyle
X_{2}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{2}\,.$ (4)

Since 4 quark angular momenta are being combined, one may transform these
states into other convenient bases by means of $9j$ angular momentum
recoupling coefficients. In particular, in the basis of good total heavy-quark
(${Q\bar{Q}}$) and light-quark (${q\bar{q}^{\prime}}$) spin, the
transformation reads

$\displaystyle\left<(s_{q}\,s_{\bar{q}})s_{q\bar{q}^{\prime}},(s_{Q}\,s_{\bar{Q}})s_{Q\bar{Q}},S\,\right|\left.(s_{q}\,s_{Q})s_{\delta}^{\vphantom{1}},(s_{\bar{q}}\,s_{\bar{Q}})s_{\bar{\delta}},S\right>$
$\displaystyle=\left([s_{q\bar{q}^{\prime}}][s_{Q\bar{Q}}][s_{\delta}^{\vphantom{1}}][s_{\bar{\delta}}]\right)^{1/2}\begin{Bmatrix}s_{q}&s_{\bar{q}}&s_{q\bar{q}^{\prime}}\\\
s_{Q}&s_{\bar{Q}}&s_{Q\bar{Q}}\\\
s_{\delta}^{\vphantom{1}}&s_{\bar{\delta}}&S\end{Bmatrix}\,,\ \ $ (5)

with $[s]\\!\equiv\\!2s+1$ signifying the multiplicity of a spin-$s$ state.
Using Eqs. (4) and (5), one then obtains

$\displaystyle J^{PC}=0^{++}:$ $\displaystyle
X_{0}=\frac{1}{2}\left|0_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{0}+\frac{\sqrt{3}}{2}\left|1_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{0}\,,$
$\displaystyle
X_{0}^{\prime}=\frac{\sqrt{3}}{2}\left|0_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{0}-\frac{1}{2}\left|1_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{0}\,,$
$\displaystyle J^{PC}=1^{++}:$ $\displaystyle
X_{1}=\left|1_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{1}\,,$ $\displaystyle
J^{PC}=1^{+-}:$
$\displaystyle\,Z\;=\frac{1}{\sqrt{2}}\left(\left|1_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{1}\\!-\left|0_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{1}\right)\,,$
$\displaystyle\,Z^{\prime}=\frac{1}{\sqrt{2}}\left(\left|1_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{1}\\!+\left|0_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{1}\right)\,,$
$\displaystyle J^{PC}=2^{++}:$ $\displaystyle
X_{2}=\left|1_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{2}\,.$ (6)

In this work it is especially convenient to employ a basis of states carrying
a unique value of $s_{Q\bar{Q}}$ and of $s_{q\bar{q}^{\prime}}$. These states
are $X_{1}$ and $X_{2}$ [as seen in Eqs. (6)], and

$\displaystyle{\tilde{X}}_{0}$ $\displaystyle\equiv$
$\displaystyle\left|0_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{0}=+\frac{1}{2}X_{0}+\frac{\sqrt{3}}{2}X_{0}^{\prime}\,,$
$\displaystyle{\tilde{X}}_{0}^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left|1_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{0}=+\frac{\sqrt{3}}{2}X_{0}-\frac{1}{2}X_{0}^{\prime}\,,$
$\displaystyle{\tilde{Z}}$ $\displaystyle\equiv$
$\displaystyle\left|1_{q\bar{q}^{\prime}},0_{Q\bar{Q}}\right>_{1}=\frac{1}{\sqrt{2}}\left(Z^{\prime}\\!+Z\right)\,,$
$\displaystyle{\tilde{Z}}^{\prime}$ $\displaystyle\equiv$
$\displaystyle\left|0_{q\bar{q}^{\prime}},1_{Q\bar{Q}}\right>_{1}=\frac{1}{\sqrt{2}}\left(Z^{\prime}\\!-Z\right)\,.$
(7)

Including ($u,d$) light-quark flavor produces 12 states: 6 each with
$I\\!=\\!0$ and $I\\!=\\!1$, and spin structures in the form of Eqs. (4), (6),
or (7). The basis of Eqs. (7) in particular is ideal for discussing
SU(3)flavor multiplets: A state component like $1_{u\bar{s}}$ is a pure flavor
octet that transforms under spin and flavor analogously to $K^{*+}$ (although
in a diquark model it comprises a mixture of color-singlet and color-octet
components). The full SU(3)flavor structure of the multiplet
$\Sigma^{+}_{g}(1S)$ thus consists of 6 octets and 6 singlets. A study of the
possible mixing of states with the same $J^{P}$ between different SU(3)flavor
octets, or of octet-singlet mixing, form two principal theory innovations of
this work.

The ${Q\bar{Q}}q\bar{q}^{\prime}$ states in the multiplet $\Sigma^{+}_{g}(1S)$
are sufficient to accommodate all particles considered in this work. However,
we note that Ref. Lebed (2017) also provides a classification of orbitally
excited states (the multiplets $\Sigma^{+}_{g}(nP)$ appearing in Ref. Giron
and Lebed (2020a)), as well as states in excited-glue Born-Oppenheimer
potentials such as $\Pi_{u}^{+}$ (which are exotic analogues to hybrid
mesons), and pentaquark states ${Q\bar{Q}}q\hskip 1.0ptq_{1}q_{2}$.

##  III Review of ${c\bar{c}q\bar{q}^{\prime}}$ and ${c\bar{c}s\bar{s}}$
Sector

###  III.1 ${c\bar{c}q\bar{q}^{\prime}}$ Sector

For hidden heavy-flavor exotics containing only $u$ and/or $d$ light valence
quarks, we write the following Hamiltonian:

$\displaystyle H$ $\displaystyle=$ $\displaystyle M_{0}+\Delta
H_{{\kappa_{qQ}}}+\Delta H_{V_{0}}+\Delta H_{V_{8}}$ (8) $\displaystyle=$
$\displaystyle
M_{0}+2{\kappa_{qQ}}\left(\mathbf{s}_{q}\cdot\mathbf{s}_{Q}+\mathbf{s}_{\bar{q}}\cdot\mathbf{s}_{\bar{Q}}\right)+V_{0}\left(\bm{\tau}_{q}\\!\cdot\bm{\tau}_{\bar{q}}\right)\left(\bm{\sigma}_{q}\\!\cdot\bm{\sigma}_{\bar{q}}\right)$
$\displaystyle+V_{8}\left(\lambda^{8}_{a}\,\bm{\sigma}_{q}\right)\\!\cdot\\!\left(\lambda^{8}_{a}\,\bm{\sigma}_{\bar{q}}\right).$

Here, $M_{0}$ is the common $\Sigma^{+}_{g}(1S)$ multiplet mass, which depends
only upon the chosen diquark ($\delta^{\vphantom{1}},\,{\bar{\delta}}$) masses
and a central potential $V(r)$ computed numerically on the lattice from pure
glue configurations that connect $\bf{3}$ and $\bar{\bf{3}}$ sources, as
employed in Ref. Giron  _et al._ (2019). The second term represents the spin-
spin interaction within diquarks, assumed to couple only $q\leftrightarrow Q$
and $\bar{q}^{\prime}\leftrightarrow\bar{Q}$, and ${\kappa_{qQ}}$ indicates
the strength of this interaction. The prime (flavor) index on the light
antiquark has been suppressed throughout Eq. (8), since for the moment we
consider $q,q^{\prime}$ to be either a light-quark or strange-quark pair, so
that the same value of $\kappa_{qQ}$ appears for both spin-spin terms. An
isospin-spin-dependent interaction of strength $V_{0}$ between the light-quark
spins, which is modeled on the pion-nucleon coupling and was first introduced
in Ref. Giron  _et al._ (2020), comprises the third term. These 3 terms form
the full set included in the analysis of Ref. Giron  _et al._ (2020). The
final term is new to this work; it is modeled on an $\eta$-nucleon coupling
and evaluates in the relevant flavor sectors to:

$\displaystyle\Delta
M_{V_{8}}=\frac{1}{3}V_{8}\left[2s_{q\bar{q}^{\prime}}\\!\left(s_{q\bar{q}^{\prime}}+1\right)-3\right]\times\\!\begin{cases}1,\;q,q^{\prime}\in\\{u,d\\}\\\
4,\;q,q^{\prime}=s,s\end{cases}.$

To compute the mass expressions arising from Eq. (8), let us first abbreviate

$V_{-}\equiv V_{0}-\frac{1}{9}V_{8}\,,\ \ V_{+}\equiv
V_{0}+\frac{1}{3}V_{8}\,.$ (10)

Then the Hamiltonian matrix elements for the mixed $Q\bar{Q}q\bar{q}^{\prime}$
states of the $\Sigma^{+}_{g}(1S)$ multiplet, their components arranged in the
order $s_{{Q\bar{Q}}}\\!=\\!0,1$, read

$\displaystyle\tilde{M}_{0^{++}}^{I=0}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-{\kappa_{qQ}}\begin{pmatrix}0&\sqrt{3}\\\
\sqrt{3}&2\end{pmatrix}-3V_{-}\begin{pmatrix}-3&0\\\ 0&1\end{pmatrix}\\!,$
$\displaystyle\tilde{M}_{0^{++}}^{I=1}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-{\kappa_{qQ}}\begin{pmatrix}0&\sqrt{3}\\\
\sqrt{3}&2\end{pmatrix}+V_{+}\begin{pmatrix}-3&0\\\ 0&1\end{pmatrix},$
$\displaystyle\tilde{M}_{1^{+-}}^{I=0}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}+{\kappa_{qQ}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}-3V_{-}\begin{pmatrix}1&0\\\ 0&-3\end{pmatrix},$
$\displaystyle\tilde{M}_{1^{+-}}^{I=1}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}+{\kappa_{qQ}}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}+V_{+}\begin{pmatrix}1&0\\\ 0&-3\end{pmatrix}.$ (11)

Diagonalizing the expressions of Eq. (III.1) in order of increasing mass and
appending the corresponding (already diagonal) expressions for the remaining
$Q\bar{Q}q\bar{q}^{\prime}$ states of the $\Sigma^{+}_{g}(1S)$ multiplet, one
obtains

$\displaystyle M_{0^{++}}^{I=0}$ $\displaystyle=$
$\displaystyle\left(M_{0}-{\kappa_{qQ}}+3V_{-}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+2\tilde{V}_{1}^{{Q\bar{Q}q\bar{q}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},$ $\displaystyle M_{0^{++}}^{I=1}$ $\displaystyle=$
$\displaystyle\left(M_{0}-{\kappa_{qQ}}-V_{+}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+2\tilde{V}_{2}^{{Q\bar{Q}q\bar{q}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\,,$ $\displaystyle M_{1^{+-}}^{I=0}$ $\displaystyle=$
$\displaystyle\left(M_{0}+3V_{-}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\tilde{V_{3}}^{{Q\bar{Q}q\bar{q}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\,,$ $\displaystyle M_{1^{+-}}^{I=1}$ $\displaystyle=$
$\displaystyle\left(M_{0}-V_{+}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\tilde{V}_{4}^{{Q\bar{Q}q\bar{q}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\,,$ $\displaystyle M_{1^{++}}^{I=0}$ $\displaystyle=$
$\displaystyle M_{0}-{\kappa_{qQ}}-3V_{-}\,,$ $\displaystyle M_{1^{++}}^{I=1}$
$\displaystyle=$ $\displaystyle M_{0}-{\kappa_{qQ}}+V_{+}\,,$ $\displaystyle
M_{2^{++}}^{I=0}$ $\displaystyle=$ $\displaystyle
M_{0}+{\kappa_{qQ}}-3V_{-}\,,$ $\displaystyle M_{2^{++}}^{I=1}$
$\displaystyle=$ $\displaystyle M_{0}+{\kappa_{qQ}}+V_{+}\,,$ (12)

using the abbreviations

$\displaystyle\tilde{V}_{1}^{{Q\bar{Q}q\bar{q}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{qQ}}^{2}+3{\kappa_{qQ}}V_{-}+9V_{-}^{2}}\,,$
$\displaystyle\tilde{V}_{2}^{{Q\bar{Q}q\bar{q}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{qQ}}^{2}-{\kappa_{qQ}}V_{+}+V_{+}^{2}}\,,$
$\displaystyle\tilde{V}_{3}^{{Q\bar{Q}q\bar{q}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{qQ}}^{2}+36V_{-}^{2}}\,,$
$\displaystyle\tilde{V}_{4}^{{Q\bar{Q}q\bar{q}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{qQ}}^{2}+4V_{+}^{2}}\,.$ (13)

###  III.2 ${c\bar{c}s\bar{s}}$ Sector

The Hamiltonian relevant to the ${c\bar{c}s\bar{s}}$ sector is identical to
the one in Eq. (8), omitting the isospin-dependent $V_{0}$ term and performing
some $q\\!\to\\!s$ relabeling,

$\displaystyle H$ $\displaystyle=$ $\displaystyle M_{0}+\Delta
H_{{\kappa_{sQ}}}+\Delta H_{V_{0}}+\Delta H_{V_{8}}$ $\displaystyle=$
$\displaystyle
M_{0}+2{\kappa_{sQ}}\left(\mathbf{s}_{s}\cdot\mathbf{s}_{Q}+\mathbf{s}_{\bar{s}}\cdot\mathbf{s}_{\bar{Q}}\right)+V_{8}\left(\lambda^{8}_{a}\,\bm{\sigma}_{s}\right)\\!\cdot\\!\left(\lambda^{8}_{a}\,\bm{\sigma}_{\bar{s}}\right)\,.$

Equivalently, this expression generalizes the Hamiltonian used in the analysis
of Ref. Giron and Lebed (2020b) by the inclusion of the $V_{8}$ term. Note
that the value of $M_{0}\\!=\\!M_{0}^{{Q\bar{Q}s\bar{s}}}$ here differs from
$M_{0}\\!=\\!M_{0}^{{Q\bar{Q}q\bar{q}}}$ appearing in Eq. (8). The mass
eigenvalues for the 6 isosinglet $c\bar{c}s\bar{s}$ states evaluate to

$\displaystyle M_{0^{++}}$ $\displaystyle=$
$\displaystyle\left(M_{0}-{\kappa_{sQ}}-\frac{4}{3}V_{8}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\\!+2\tilde{V}_{1}^{{Q\bar{Q}s\bar{s}}}\\!\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\\!,$ $\displaystyle M_{1^{+-}}$ $\displaystyle=$
$\displaystyle\left(M_{0}-\frac{4}{3}V_{8}\right)\begin{pmatrix}1&0\\\
0&1\end{pmatrix}+\tilde{V}_{2}^{{Q\bar{Q}s\bar{s}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},$ $\displaystyle M_{1^{++}}$ $\displaystyle=$ $\displaystyle
M_{0}-{\kappa_{sQ}}+\frac{4}{3}V_{8}\,,$ $\displaystyle M_{2^{++}}$
$\displaystyle=$ $\displaystyle M_{0}+{\kappa_{sQ}}+\frac{4}{3}V_{8}\,,$ (15)

where

$\displaystyle\tilde{V}_{1}^{{Q\bar{Q}s\bar{s}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{sQ}}^{2}-\frac{4}{3}{\kappa_{sQ}}V_{8}+\frac{16}{9}V_{8}^{2}}\,,$
$\displaystyle\tilde{V}_{2}^{{Q\bar{Q}s\bar{s}}}$ $\displaystyle\equiv$
$\displaystyle\sqrt{{\kappa_{sQ}}^{2}+\frac{64}{9}V_{8}^{2}}\,.$ (16)

##  IV Hidden-Charm, Open-Strange Sector

In this sector, the Hamiltonian analogous to Eq. (8) becomes

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+\Delta
H_{{\kappa_{qQ}}}+\Delta H_{\kappa_{sQ}}+\Delta H_{V_{8}}$ (17)
$\displaystyle=$ $\displaystyle
H_{0}+2\left[{\kappa_{qQ}}\left(\mathbb{s}_{q}\cdot\bm{s}_{Q}\right)+\kappa_{sQ}\left(\mathbb{s}_{\bar{s}}\cdot\bm{s}_{\bar{Q}}\right)\right]$
$\displaystyle+V_{8}\left(\lambda^{8}_{a}\,\bm{\sigma}_{q}\right)\\!\cdot\\!\left(\lambda^{8}_{a}\,\bm{\sigma}_{\bar{s}}\right)\,.$

Without loss of generality, we have taken $q^{\prime}\\!\to\\!s$, with the
opposite choice $q\\!\to\\!s$ simply leading to the antiparticles of those
studied here. Then the spin couplings $\kappa_{sQ}$ and $\kappa_{qQ}$ are
numerically quite distinct, and we compute the mass contributions

$\displaystyle\Delta
M_{{\kappa_{qQ}}}=\frac{1}{2}{\kappa_{qQ}}\left[2s_{\delta}^{\vphantom{1}}\left(s_{\delta}^{\vphantom{1}}+1\right)-3\right]\,,$
(18) $\displaystyle\Delta
M_{\kappa_{\bar{s}\bar{Q}}}=\frac{1}{2}\kappa_{sQ}\left[2s_{\bar{\delta}}\left(s_{\bar{\delta}}+1\right)-3\right]\,,$
(19)

and

$\displaystyle\Delta
M_{V_{8}}=-\frac{2}{3}V_{8}\left[2s_{q\bar{s}}\left(s_{q\bar{s}}+1\right)-3\right]\,,$
(20)

with the final expression computed in the same manner as is performed to
obtain Eq. (III.1).

A notable feature of the open-strange exotics sector becomes apparent when
considering the full SU(3)flavor multiplet structure. $Q\bar{Q}q\bar{q}$
states with $I_{3}\\!=\\!0$ carry good $J^{PC}$ quantum numbers, and states
with different $J^{PC}$ values of course cannot mix with them. Inasmuch as
isospin is a nearly exact symmetry, one can extend $C$ parity to a full
isospin multiplet by defining the conserved $G$-parity quantum number (whose
eigenvalues for all hadrons are tabulated by the Particle Data Group (PDG)
Zyla  _et al._ (2020)). Specifically,

$G\equiv(-1)^{I}C\,,$ (21)

where the $C$-parity eigenvalue here is that of the $I_{3}\\!=\\!0$ member of
the isomultiplet. One could generalize the concept of $G$ parity to a full
SU(3)flavor multiplet, but since the corresponding flavor symmetry is broken,
mixing between the open-strange members of multiplets whose $I_{3}\\!=\\!0$,
$Y\\!=\\!0$ members have opposite $C$ parities can occur. Indeed, this
phenomenon is known among the conventional mesons: For example, the strange
partners to the lightest $1^{++}$ and $1^{+-}$ mesons are named $K_{1A}$ and
$K_{1B}$ respectively, and the observed $1^{+}$ strange-meson mass eigenstates
$K_{1}(1270)$ and $K_{1}(1400)$ are believed to be nearly equal admixtures of
$K_{1A}$ and $K_{1B}$ Zyla  _et al._ (2020).

In the exotics sector, the nonstrange $X_{1}$ ($1^{++}$) states cannot mix
with $\tilde{Z},\tilde{Z}^{\prime}$ ($1^{+-}$) due to $G$-parity conservation.
However, their open-strange $1^{+}$ partners can mix, leading to richer
phenomenological possibilities. To wit: The mass expressions obtained from Eq.
(17), prior to diagonalization, read

$\displaystyle\tilde{M}_{0^{+}}$ $\displaystyle=$ $\displaystyle
M_{0}\begin{pmatrix}1&0\\\ 0&1\\\
\end{pmatrix}-\frac{1}{2}\left({\kappa_{qQ}}+\kappa_{sQ}\right)\begin{pmatrix}0&\sqrt{3}\\\
\sqrt{3}&2\end{pmatrix}$ $\displaystyle-\frac{2}{3}V_{8}\begin{pmatrix}-3&0\\\
0&1\end{pmatrix}\,,$ $\displaystyle\tilde{M}_{1^{+}}$ $\displaystyle=$
$\displaystyle M_{0}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}+\frac{{\kappa_{qQ}}}{2}\begin{pmatrix}-1&-\sqrt{2}&+\sqrt{2}\\\
-\sqrt{2}&0&+1\\\ +\sqrt{2}&+1&0\end{pmatrix}\,,$
$\displaystyle+\frac{\kappa_{sQ}}{2}\begin{pmatrix}-1&+\sqrt{2}&-\sqrt{2}\\\
+\sqrt{2}&0&+1\\\
-\sqrt{2}&+1&0\end{pmatrix}-\frac{2}{3}V_{8}\begin{pmatrix}1&0&0\\\ 0&-3&0\\\
0&0&1\\\ \end{pmatrix}\,,$ $\displaystyle M_{2^{+}}$ $\displaystyle=$
$\displaystyle
M_{0}+\frac{1}{2}\left({\kappa_{qQ}}+\kappa_{sQ}\right)-\frac{2}{3}V_{8}\,.$
(22)

The elements of the matrices for $0^{+}$ are again arranged in order of
increasing heavy-quark spin. However, those for $1^{+}$ are arranged in the
order corresponding to increasing mass eigenvalues for their nonstrange
partners in the hidden-charm sector: $X_{1},{\tilde{Z}}^{\prime},{\tilde{Z}}$.

The mass eigenvalues for the $0^{+}$ sector read

$\displaystyle M_{0^{+}}$ $\displaystyle=$
$\displaystyle\left[M_{0}-\frac{1}{2}\left({\kappa_{qQ}}+\kappa_{sQ}\right)+\frac{2}{3}V_{8}\right]\begin{pmatrix}1&0\\\
0&1\end{pmatrix}$ (23)
$\displaystyle+2\tilde{V}_{1}^{{Q\bar{Q}q\bar{s}}}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix},$

where

$\tilde{V}_{1}^{{Q\bar{Q}q\bar{s}}}\equiv\sqrt{\left[\frac{1}{2}\left({\kappa_{qQ}}+\kappa_{sQ}\right)+\frac{1}{3}V_{8}\right]^{2}+\frac{1}{3}V_{8}^{2}}\,.$
(24)

The exact expressions for the $1^{+}$ eigenvalues are of course complicated
roots of a cubic equation, but anticipating that
$V_{8}\\!\ll\\!{\kappa_{qQ}}\\!\ll\\!\kappa_{sQ}$, one may perform a
perturbative expansion in $V_{8}$ to compute approximate values:

$\displaystyle M_{1^{+}}^{(1)}$ $\displaystyle=$ $\displaystyle
M_{0}^{{Q\bar{Q}q\bar{s}}}+\frac{1}{2}\left(-3\kappa_{sQ}+{\kappa_{qQ}}\right)+O(V_{8}^{2}/\kappa_{sQ})\,,$
$\displaystyle M_{1^{+}}^{(2)}$ $\displaystyle=$ $\displaystyle
M_{0}^{{Q\bar{Q}q\bar{s}}}+\frac{1}{2}\left(\kappa_{sQ}-3{\kappa_{qQ}}\right)+O(V_{8}^{2}/{\kappa_{qQ}})\,,$
$\displaystyle M_{1^{+}}^{(3)}$ $\displaystyle=$ $\displaystyle
M_{0}^{{Q\bar{Q}q\bar{s}}}+\frac{1}{2}\left(\kappa_{sQ}+{\kappa_{qQ}}\right)+\frac{2}{3}V_{8}+O(V_{8}^{2}/{\kappa_{qQ}})\,.$

##  V Analysis

###  V.1 Flavor SU(3) Multiplets and Mixing

The original analysis of $c\bar{c}s\bar{s}$ states in Ref. Lebed and Polosa
(2016), as well as its updated form in Ref. Giron and Lebed (2020b), takes the
$c\bar{c}s\bar{s}$ states to be completely unmixed with those in the
$c\bar{c}q\bar{q}^{\prime}$ sector, where $q,q^{\prime}\\!\in\\{u,d\\}$. If,
on the other hand, SU(3)flavor is exact, then the states should fill octets
and singlets of the flavor symmetry. Specifically, the flavor structure of
$c\bar{c}q\bar{q}^{\prime}$ states, now allowing
$q,q^{\prime}\\!\in\\{u,d,s\\}$, can be discussed using the same framework
that applies to conventional $q\bar{q}^{\prime}$ mesons. The $I\\!=\\!0$,
$I_{3}\\!=\\!0$, $Y\\!=\\!0$ unmixed octet and singlet combinations are, as
usual,

$\frac{1}{\sqrt{6}}\left(u\bar{u}+d\bar{d}-2s\bar{s}\right)\ {\rm and}\
\frac{1}{\sqrt{3}}\left(u\bar{u}+d\bar{d}+s\bar{s}\right)\,,$ (26)

respectively. In the lightest ($J^{PC}\\!=\\!0^{-+}$) meson multiplet, these
states correspond to $\eta$ and $\eta^{\prime}$, respectively, which remain
largely unmixed because the octet states are pseudo-Nambu-Goldstone bosons
whose masses vanish in the chiral limit, while the singlet has a nonzero mass
in this limit due to the anomalous breaking of the axial U(1) symmetry of
massless QCD.

Heavier meson multiplets, however, support much larger SU(3)flavor mixing
between the octet and singlet combinations. For example, the next-lightest
($1^{--}$) multiplet features the $\omega$ and $\phi$ as its $I\\!=\\!0$
states, which appear to be nearly ideally mixed into the flavor combinations
$\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$ and $s\bar{s}$, respectively. The
appearance of only the $J/\psi\,\phi$ decay mode for most of the purported
$c\bar{c}s\bar{s}$ candidates inspired the implicit adoption of an ideal-
mixing ansatz in Refs. Lebed and Polosa (2016); Giron and Lebed (2020b).

Moreover, the approach of treating $I\\!=\\!0$ states in the
$c\bar{c}q\bar{q}^{\prime}$ sector as containing no $s\bar{s}$ component,
which is implicit in Refs. Giron  _et al._ (2019, 2020); Giron and Lebed
(2020b), also introduces a hidden assumption of octet-singlet mixing into the
analysis. The fact that $\rho^{0}$ ($I\\!=\\!1$, pure octet) and $\omega$
($I\\!=\\!0$, ideally mixed) are nearly degenerate in mass, and likewise for
$Z_{c}(3900)^{0}$ ($I\\!=\\!1$, pure octet) and $X(3872)$ ($I\\!=\\!0$),
suggests that substantial octet-singlet flavor mixing is needed to understand
the spectrum of both $1^{--}$ conventional mesons and $\Sigma_{g}^{+}(1S)$
hidden-charm exotic mesons. Isospin symmetry then links the remaining
$Y\\!=\\!0$ states ($\rho^{\,\pm}$, $Z_{c}(3900)^{\pm}$). In the case of
exotics, the values of $M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ and
$M_{0}^{c\bar{c}s\bar{s}}$ extracted in previous work implicitly incorporate
ideal octet-singlet flavor mixing, with strangeness dependence entering only
through the differing diquark masses for $\delta^{\vphantom{1}}\\!=\\!(cq)$
and for $\delta^{\vphantom{1}}\\!=\\!(cs)$.

In contrast, the value of $M_{0}^{c\bar{c}q\bar{s}}$ for the open-strange
($K$-like) states refers to an unmixed SU(3)flavor octet. One should therefore
not be surprised that simply combining the values of $m_{(cq)}$ from
$M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ and $m_{(cs)}$ from
$M_{0}^{c\bar{c}s\bar{s}}$ with the lattice-computed glue potential $V(r)$
between heavy color-triplet sources generates a value for
$M_{0}^{c\bar{c}q\bar{s}}$ slightly different from the one that would be
obtained from starting with values of $M_{0}$ corresponding to unmixed
SU(3)-octet or -singlet $c\bar{c}q\bar{q}^{\prime}$ and $c\bar{c}s\bar{s}$
states. This effect should occur even in the absence of an explicit
SU(3)flavor-breaking difference between $m_{\delta^{\vphantom{1}}=(cq)}$ and
$m_{\delta^{\vphantom{1}}=(cs)}$.

We illustrate this mixing effect using a toy example well known from
elementary quantum mechanics: Ignore fine-structure effects and let the
unmixed mass parameters $M_{8}$ (pure octet) and $M_{1}$ (pure singlet) be
degenerate, $M\\!=\\!M_{8}\\!=\\!M_{1}$, in a 2-level system with an octet-
singlet mass-mixing parameter $\Delta$. Then the resulting mass eigenvalues
are $M\mp\Delta$, and the mixing angle of the system is maximal, $45^{\circ}$.
More generally, the lower mass eigenvalue is always smaller than the smaller
diagonal element, whether or not the unmixed octet and singlet mass parameters
are equal. In our case, the value of $M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ from
the previous analyses of Refs. Giron  _et al._ (2019, 2020); Lebed and Polosa
(2016); Giron and Lebed (2020b) is assumed to refer to ideally mixed states;
and since one expects the exotics observed thus far (which are used to extract
$M_{0}$ values) to be the lightest ones that exist, the derived
$M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ results represent the smaller mass
eigenvalues. Meanwhile, $M_{0}^{c\bar{c}s\bar{s}}$ is extracted from states
assumed to contain no light valence quarks, and therefore their component
diquarks are pure $(cs)$; thus, no mixing needs to be performed to extract the
parameter $m_{\delta^{\vphantom{1}}=(cs)}$. The result for
$M_{0}^{c\bar{c}q\bar{s}}$ naively obtained from using the (mixed)
$M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ and $M_{0}^{c\bar{c}s\bar{s}}$ values
should therefore be slightly lower than one determined entirely from the pure-
octet $c\bar{c}q\bar{s}$ sector. This expectation, in fact, is precisely what
occurs, as we see below.

###  V.2 $c\bar{c}q\bar{q}^{\prime}$ Sector and $V_{8}$

The analysis of the $c\bar{c}q\bar{q}^{\prime}$ sector here closely follows
that of Ref. Giron  _et al._ (2020), and especially Ref. Giron and Lebed
(2020b). The 3 primary inputs are the PDG averages Zyla  _et al._ (2020)

$\displaystyle m_{X(3872)}$ $\displaystyle=$ $\displaystyle 3871.69\pm 0.17\
{\rm MeV}\,,$ $\displaystyle m_{Z_{c}(3900)}$ $\displaystyle=$ $\displaystyle
3888.4\pm 2.5\ {\rm MeV}\,,$ $\displaystyle m_{Z_{c}(4020)}$ $\displaystyle=$
$\displaystyle 4024.1\pm 1.9\ {\rm MeV}\,,$ (27)

with only the value for $Z_{c}(3900)$ changing slightly since the previous
analyses. Since the Hamiltonian of Eq. (8) now has 4 parameters, the system is
underdetermined. However, one further constraint arises from noting the strong
charmonium decay preference Zyla  _et al._ (2020) of $Z_{c}(3900)$ to
$J/\psi$, and $Z_{c}(4020)$ to $h_{c}$, suggesting that these $Z_{c}$ states
are nearly pure $s_{c\bar{c}}\\!=\\!1$ and $s_{c\bar{c}}\\!=\\!0$ eigenstates,
respectively. Defining $P$ as the $s_{Q\bar{Q}}\\!=\\!1$ probability content
of the lower-mass $1^{+-}$, $I\\!=\\!1$ eigenstate of Eqs. (III.1) [i.e., the
square of the off-diagonal component of the unitary matrix diagonalizing
${\tilde{M}}^{I=1}_{1^{+-}}$ in Eqs. (III.1)], one obtains

$P=\frac{1}{2}\left[1+\frac{2\left(V_{0}+\frac{V_{8}}{3}\right)}{\sqrt{{\kappa_{qQ}}^{2}+4\left(V_{0}+\frac{V_{8}}{3}\right)^{2}}}\right]\,,$
(28)

which means that $V_{8}$ can be expressed as a function of $P$ (and the
parameters $V_{0}$ and ${\kappa_{qQ}}$). Using this constraint with the mass
expressions in Eqs. (III.1), the most convenient combinations of the 3 masses
in Eqs. (27) are

$\displaystyle\mu_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(m_{Z_{c}(4020)}+m_{Z_{c}(3900)}\right)=3956.3\pm
1.6\ {\rm MeV}$ $\displaystyle=$ $\displaystyle
M_{0}-\frac{1}{2}\left(P-\frac{1}{2}\right)\frac{{\kappa_{qc}}}{\sqrt{P(1-P)}}\,,$
$\displaystyle\mu_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(m_{Z_{c}(4020)}+m_{Z_{c}(3900)}\right)-m_{X(3872)}$
$\displaystyle=$ $\displaystyle 84.6\pm 1.6\ {\rm MeV}$ $\displaystyle=$
$\displaystyle{\kappa_{qc}}\left[1-\frac{\left(P-\frac{1}{2}\right)}{\sqrt{P(1-P)}}\right]+4V_{0}\,,$
$\displaystyle\mu_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(m_{Z_{c}(4020)}-m_{Z_{c}(3900)}\right)=67.9\pm
1.6\ {\rm MeV}$ (29) $\displaystyle=$
$\displaystyle\frac{{\kappa_{qc}}}{2\sqrt{P(1-P)}}\,.$

From Eqs. (29), one extracts

$\frac{4}{3}V_{8}=-\mu_{2}+2\mu_{3}\left[\left(P-\frac{1}{2}\right)+\sqrt{P(1-P)}\right]\,.$
(30)

The case $V_{8}\\!=\\!0$, which (in effect) is imposed in Ref. Giron and Lebed
(2020b), becomes

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[2+\frac{\mu_{2}}{\mu_{3}}\pm\sqrt{2-\left(\frac{\mu_{2}}{\mu_{3}}\right)^{2}}\,\right]\,,$
(31) $\displaystyle=$ $\displaystyle 0.979\pm 0.009\,,\ 0.644\pm 0.030\,.$

The only input used in Ref. Giron and Lebed (2020b) beyond those of Eqs. (27)
is the discrete choice of the larger $P$ value in Eq. (31) to recognize the
$Z_{c}$ charmonium decay preferences noted above.

In fact, Eq. (30) places a rather strong constraint upon $V_{8}$. While any
$P\\!\in\\![0,1]$ is in principle allowed, values of $P$ smaller (larger) than
the smaller (larger) root in Eq. (31) lead to negative values of $V_{8}$—and,
for sufficiently small values of $P$, values of $V_{8}$ that are also larger
in magnitude than $V_{0}$. Inasmuch as the accompanying operators in Eq. (8)
represent $\pi$-like and $\eta$-like exchanges, respectively, one expects the
analogy to the dynamics of true $\pi$ and $\eta$ exchanges between nucleons
(from, e.g., chiral perturbation theory) to hold. Under this assumption, the
$\eta$-like exchange should be attractive like the $\pi$-like exchange; hence
$V_{8}$, like $V_{0}$, should be positive. However, genuine $\eta$ exchange is
also weaker than $\pi$ exchange, both due to the $\eta$’s larger mass and
larger decay constant. We therefore take $V_{8}\\!>\\!0$ to be a natural
constraint of the model, which requires $P$ to lie between the roots given in
Eq. (31). According to Eq. (30), within this range $V_{8}$ reaches a maximum
at $P=\\!\frac{1}{2}\big{(}1+\frac{1}{\sqrt{2}}\big{)}\\!=\\!0.854$, at which

$V_{8}^{\rm max}=\frac{3}{4}\left(-\mu_{2}+\sqrt{2}\mu_{3}\right)=8.6\pm 2.3\
{\rm MeV}\,.$ (32)

We therefore expect the allowed range of $V_{8}$, as determined by the known
phenomenology of the $c\bar{c}q\bar{q}^{\prime}$ sector, to have a modest
effect compared to that provided by the other parameters in Eq. (8). Using
Eqs. (29), one obtains for the other Hamiltonian parameters:

$\displaystyle M_{0}$ $\displaystyle=$
$\displaystyle\mu_{1}+\left(P-\frac{1}{2}\right)\mu_{3}\,,$
$\displaystyle\kappa_{qc}$ $\displaystyle=$ $\displaystyle
2\mu_{3}\sqrt{P(1-P)}\,,$ $\displaystyle 4V_{0}$ $\displaystyle=$
$\displaystyle\mu_{2}+2\mu_{3}\left[\left(P-\frac{1}{2}\right)-\sqrt{P(1-P)}\right]\,.$
(33)

The values of $M_{0}$, ${\kappa_{qc}}$, and $V_{0}$ obtained for both
$V_{8}\\!=\\!0$ and for an optimized $V_{8}$ value obtained below from the
$c\bar{c}s\bar{s}$ spectrum [in Eqs. (V.3)] are presented in Table 1. The full
spectrum of masses for the $c\bar{c}q\bar{q}^{\prime}$ $\Sigma^{+}_{g}(1S)$
multiplet appears in Table 2.

Table 1: Hamiltonian parameters [Eq. (8)] of the dynamical diquark model obtained from fits to members of the $\Sigma^{+}_{g}(1S)$ multiplet in the $c\bar{c}q\bar{q}^{\prime}$ [$X(3872)$, $Z_{c}(3900)$, $Z_{c}(4020)$] and $c\bar{c}s\bar{s}$ [$X(3915)$, $X(4140)$, $X(4350)$] sectors. Also included is the $s_{c\bar{c}}\\!=\\!1$ content $P$ of the state $Z_{c}(3900)$, and the diquark mass $m_{\delta}$ derived from each case, as well as a value of $M_{0}^{c\bar{c}q\bar{s}}$ for the open-strange sector that supposes no SU(3)flavor octet-singlet mixing in the other flavor sectors. | $V_{8}=0$ | $\ V_{8}=3.9\pm 1.4\ {\rm MeV}$
---|---|---
$M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ | $3988.7\pm 1.5\ {\rm MeV}$ | $3987.7\pm 1.6\ {\rm MeV}$
$\kappa_{qc}$ | $19.6\pm 3.7\ {\rm MeV}$ | $25.6\pm 5.0\ {\rm MeV}$
$V_{0}$ | $32.5\pm 1.3\ {\rm MeV}$ | $30.4\pm 1.9\ {\rm MeV}$
$P$ | $0.979\pm 0.009$ | $0.963\pm 0.016$
$m_{\delta^{\vphantom{1}}(cq)}$ | $1927.0\pm 11.5\ {\rm MeV}$ | $1927.1\pm 11.0\ {\rm MeV}$
$M_{0}^{c\bar{c}s\bar{s}}$ | $\cdots$ | $4251.3\pm 2.8\ {\rm MeV}$
$\kappa_{sc}$ | $\cdots$ | $109.8\pm 1.1\ {\rm MeV}$
$m_{\delta^{\vphantom{1}}(cs)}$ | $\cdots$ | $2069.4\pm 10.9\ {\rm MeV}$
$M_{0}^{c\bar{c}q\bar{s}}$ | $\cdots$ | $4119.7\pm 1.7\ {\rm MeV}$

Using the value of $M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ from Table 1 with
$V_{8}\\!=\\!0$ and the lattice-simulated glue potentials $V(r)$ of Refs. Juge
_et al._ (2003); Mor  (JKM) and Capitani  _et al._ (2019) (CPRRW), we compute

$\displaystyle m_{\delta^{\vphantom{1}}(cq)}$ $\displaystyle=$ $\displaystyle
1938.5\pm 0.8\ {\rm MeV}\ {\rm(JKM)}\,,$ (34) $\displaystyle=$ $\displaystyle
1915.5\pm 0.8\ {\rm MeV}\ {\rm(CPRRW)}\,.$

A value spanning this spread is presented in Table 1. Again, these results for
$V_{8}\\!=\\!0$ differ from those in Ref. Giron and Lebed (2020b) only through
a small shift in the tabulated PDG value of $m_{Z_{c}(3900)}$ in Eq. (27).

###  V.3 $c\bar{c}s\bar{s}$ Sector: ${\kappa_{sc}}$ and $V_{8}$

The signature process used in Ref. Giron and Lebed (2020b) to identify
$c\bar{c}s\bar{s}$ exotics is the decay mode $J/\psi\,\phi$, although some
candidates are identifiable through $D_{s}$-type meson-pair decays.
Furthermore, the analysis of Ref. Giron and Lebed (2020b) argues that the
$J^{PC}\\!=\\!1^{++}$ $X(4274)$ is an excellent candidate for the conventional
charmonium state $\chi_{c1}(3P)$. The most unexpected addition to the
$c\bar{c}s\bar{s}$ spectrum is the peculiar state $X(3915)$, with likely
$0^{++}$ quantum numbers, as the candidate for the lightest $c\bar{c}s\bar{s}$
state in this model. In brief summary of the reasoning in Ref. Lebed and
Polosa (2016) and references therein, $X(3915)$ has no confirmed open-charm
decays, thus arguing against it being either the conventional charmonium state
$\chi_{c0}(2P)$ or $c\bar{c}q\bar{q}$. It has definitively been seen to couple
only to $\gamma\gamma$ and $J/\psi\,\omega$; with respect to the latter mode,
note that $X(3915)$ lies below the $J/\psi\,\phi$ threshold, so that
$\phi\\!\to\\!\omega$ mixing is proposed in Ref. Lebed and Polosa (2016) to be
responsible for the $J/\psi\,\omega$ decay mode. Furthermore, a recent lattice
calculation Prelovsek  _et al._ (2021) predicts the existence of a $0^{++}$
state in this mass region that has a strong coupling to $D_{s}\bar{D}_{s}$ but
a weak coupling to $D\bar{D}$. The mass used in this work is the PDG value
Zyla  _et al._ (2020):

$m_{X(3915)}=3921.7\pm 1.8\ {\rm MeV}\,.$ (35)

In the previous analysis Giron and Lebed (2020b), the $c\bar{c}s\bar{s}$
spectrum obtained for the multiplet is very simple. Referring to Eqs. (III.2),
the assumption that $\kappa_{sc}\\!\gg\\!V_{8}\\!>\\!0$ leads to the spectrum
(in increasing order of mass):

$\displaystyle M_{0^{++}}$ $\displaystyle=$ $\displaystyle
M_{0}-3\kappa_{sc}+O(V_{8}^{2}/\kappa_{sc})\,,$ $\displaystyle M_{1^{+-}}$
$\displaystyle=$ $\displaystyle
M_{0}-\kappa_{sc}-\frac{4}{3}V_{8}+O(V_{8}^{2}/\kappa_{sc})\,,$ $\displaystyle
M_{1^{++}}$ $\displaystyle=$ $\displaystyle
M_{0}-\kappa_{sc}+\frac{4}{3}V_{8}\,,$ $\displaystyle M^{\prime}_{0^{++}}$
$\displaystyle=$ $\displaystyle
M_{0}+\kappa_{sc}-\frac{8}{3}V_{8}+O(V_{8}^{2}/\kappa_{sc})\,,$ $\displaystyle
M^{\prime}_{1^{+-}}$ $\displaystyle=$ $\displaystyle
M_{0}+\kappa_{sc}-\frac{4}{3}V_{8}+O(V_{8}^{2}/\kappa_{sc})\,,$ $\displaystyle
M_{2^{++}}$ $\displaystyle=$ $\displaystyle
M_{0}+\kappa_{sc}+\frac{4}{3}V_{8}\,,$ (36)

which reduces to 3 degenerate sets in the case $V_{8}\\!=\\!0$, as listed in
Ref. Giron and Lebed (2020b). In particular, the lighter $0^{++}$ state
clearly lies far below the others, with the $1^{++}$ state (and the lighter
$1^{+-}$) being intermediate in mass, and the $2^{++}$ and heavier $0^{++}$
(and $1^{+-}$) states lying close together at a larger mass value. The overall
effect of Eq. (V.3) is to split the $\Sigma^{+}_{g}(1S)$ multiplet into 3
roughly equally spaced (by $2\kappa_{sc}$) clusters of $c\bar{c}s\bar{s}$
states.

The state $X(4140)$ is taken to be an unmistakable $c\bar{c}s\bar{s}$
candidate, the sole $1^{++}$ member of the multiplet $\Sigma^{+}_{g}(1S)$.
Therefore, the $c\bar{c}s\bar{s}$ spectrum should start with $X(3915)$ being
the distinct lightest member, a $1^{+-}$ state is predicted to appear with a
mass near $m_{X(4140)}$, and a trio of states ($0^{++}$, $1^{+-}$, $2^{++}$)
is predicted to appear at approximately
$m_{X(3915)}\\!+\\!2(m_{X(4140)}\\!-\\!m_{X(3915)})$. A complication arises,
however, with the latest LHCb measurement Aaij  _et al._ (2021b) of $X(4140)$:

$\displaystyle m_{X(4140)}$ $\displaystyle=$ $\displaystyle 4118\pm
11^{+19}_{-36}\ {\rm MeV},$ $\displaystyle\Gamma_{X(4140)}$ $\displaystyle=$
$\displaystyle 162\pm 21^{+24}_{-49}\ {\rm MeV}\,,$ (37)

which should be compared to the PDG average Zyla  _et al._ (2020),

$\displaystyle m_{X(4140)}$ $\displaystyle=$ $\displaystyle 4146.8\pm 2.4\
{\rm MeV},$ $\displaystyle\Gamma_{X(4140)}$ $\displaystyle=$ $\displaystyle
22^{+8}_{-7}\ {\rm MeV}\,,$ (38)

the mass differing by about $1.3\,\sigma$ (and the width differing radically).
LHCb observes $X(4140)$ with a $13\,\sigma$ total significance. On the other
hand, the previous LHCb observation of $X(4140)$ Aaij  _et al._ (2017) (at a
significance of $8.4\,\sigma$) forms part of the PDG averages of Eqs. (V.3),
and the mass value obtained in Ref. Aaij  _et al._ (2017) is much more in line
with the average mass value given in Eqs. (V.3):

$\displaystyle m_{X(4140)}$ $\displaystyle=$ $\displaystyle 4146.5\pm
4.5^{+4.6}_{-2.8}\ {\rm MeV}\,,$ $\displaystyle\Gamma_{X(4140)}$
$\displaystyle=$ $\displaystyle 83\pm 21^{+21}_{-14}\ {\rm MeV}\,.$ (39)

In fact, the data used in Ref. Aaij  _et al._ (2017) forms a small subset of
the LHCb data reported in Ref. Aaij  _et al._ (2021b). So how can a
measurement using much more data lead to a result with much larger
uncertainties? In large part, it arises from a new modeling of the $X(4140)$
lineshape, in which a naive Breit-Wigner profile is replaced with a Flatté
form Skwarnicki . To incorporate this new development, we reanalyze the PDG
mass average of Eqs. (V.3) by replacing the old LHCb mass measurement of Eqs.
(V.3) with the new one of Eqs. (V.3), producing the value to be used in our
analysis:

$m_{X(4140)}=4146.7\pm 2.7\ {\rm MeV}\,.$ (40)

The state $X(4350)$, although not yet confirmed at the same level of
confidence ($3.2\sigma$), is seen in $\gamma\gamma\\!\to\\!J/\psi\,\phi$ and
thus is an excellent $c\bar{c}s\bar{s}$ $0^{++}$ or $2^{++}$ candidate. Noting
that Zyla  _et al._ (2020)

$m_{X(4350)}=4351\pm 5\ {\rm MeV}\,,$ (41)

and using Eqs. (35) and (40), one finds

$m_{X(3915)}+2(m_{X(4140)}-m_{X(3915)})=4371.7\pm 5.7\ {\rm MeV}\,.$ (42)

One then sees (as in Refs. Lebed and Polosa (2016); Giron and Lebed (2020b))
that $X(4350)$ nearly satisifies the equal-spacing rule discussed above, which
confirms our previous result that $V_{8}$ is numerically small. In fact, at
linear order in $V_{8}$, Eqs. (V.3) give

$\displaystyle(M_{0^{++}}^{\prime}-M_{1^{++}})-(M_{1^{++}}-M_{0^{++}})$
$\displaystyle=$ $\displaystyle-\frac{16}{3}V_{8}\,,$
$\displaystyle(M_{2^{++}}-M_{1^{++}})-(M_{1^{++}}-M_{0^{++}})$
$\displaystyle=$ $\displaystyle-\frac{4}{3}V_{8}\,.$ (43)

Using $m_{X(4350)}$ from Eq. (41) for $M_{0^{++}}^{\prime}$ or $M_{2^{++}}$
gives

$\displaystyle{(m_{X(4350)}-m_{X(4140)})-(m_{X(4140)}-m_{X(3915)})}\hskip
20.00003pt$ $\displaystyle=-20.7\pm 7.6\ {\rm MeV}\,,\hskip 20.00003pt$ (44)

meaning that $V_{8}$ is small and positive, as anticipated in Eq. (32).
Returning to the full mass expressions of Eqs. (III.2), one uses Eqs. (35),
(40), and (41) to obtain

$\displaystyle M_{0}^{c\bar{c}s\bar{s}}$ $\displaystyle=$ $\displaystyle
4251.3\pm 2.8\ {\rm MeV}\,,$ $\displaystyle\kappa_{sc}$ $\displaystyle=$
$\displaystyle\ \,109.8\pm 1.1\ {\rm MeV}\,,$ $\displaystyle V_{8}$
$\displaystyle=$ $\displaystyle\ \ \ \ \,3.9\pm 1.4\ {\rm MeV}\,,$ (45)

assuming that $X(4350)$ is $0^{++}$, and

$\displaystyle M_{0}^{c\bar{c}s\bar{s}}$ $\displaystyle=$ $\displaystyle
4230.8\pm 7.0\ {\rm MeV}\,,$ $\displaystyle\kappa_{sc}$ $\displaystyle=$
$\displaystyle\ \,102.2\pm 2.8\ {\rm MeV}\,,$ $\displaystyle V_{8}$
$\displaystyle=$ $\displaystyle\ \ \;13.6\pm 4.3\ {\rm MeV}\,,$ (46)

assuming that $X(4350)$ is $2^{++}$. Obtaining these results requires the
resolution of a discrete ambiguity to impose the physical expectation
$\kappa_{sc}\\!>\\!0$, as discussed in Ref. Giron and Lebed (2020b). The
latter solution produces a slightly larger value of $V_{8}$ than allowed by
Eq. (32), but only by $1.0\,\sigma$, and therefore still viable. Nevertheless,
for purposes of illustration, we choose Eqs. (V.3) as the best-fit parameters
(also included in Table 1), and use them to compute the full spectrum of
masses for the $\Sigma^{+}_{g}(1S)$ $c\bar{c}s\bar{s}$ multiplet in Table 2.
In particular, using the value of $M_{0}^{c\bar{c}s\bar{s}}$ from Eqs. (V.3)
and the lattice-simulated glue potentials $V(r)$ of Refs. Juge  _et al._
(2003); Mor ; Capitani  _et al._ (2019), we compute

$\displaystyle m_{\delta^{\vphantom{1}}(cs)}$ $\displaystyle=$ $\displaystyle
2080.2\pm 1.5\ {\rm MeV}\ {\rm(JKM)}\,,$ (47) $\displaystyle=$ $\displaystyle
2058.5\pm 1.5\ {\rm MeV}\ {\rm(CPRRW)}\,.$

The $M_{0}^{c\bar{c}s\bar{s}}$ and $\kappa_{sc}$ values obtained in Eqs. (V.3)
and (47) differ rather little from those in Ref. Giron and Lebed (2020b), in
part because the previous work effectively takes $V_{8}\\!=\\!0$, and also
because the inputs of Eqs. (35) and (40) have changed little in the interim.

Feeding the value of nonzero $V_{8}$ back into the $c\bar{c}q\bar{q}^{\prime}$
expressions given by Eqs. (30) and (V.2), one obtains the $V_{8}\\!>\\!0$
values of $M_{0}^{c\bar{c}q\bar{q}^{\prime}}$, $\kappa_{qc}$, $V_{0}$, and $P$
given in Table 1. Using this value of $M_{0}^{c\bar{c}q\bar{q}^{\prime}}$ and
the lattice-simulated glue potentials $V(r)$ of Refs. Juge  _et al._ (2003);
Mor ; Capitani  _et al._ (2019), we compute

$\displaystyle m_{\delta^{\vphantom{1}}(cq)}$ $\displaystyle=$ $\displaystyle
1938.0\pm 0.9\ {\rm MeV}\ {\rm(JKM)}\,,$ (48) $\displaystyle=$ $\displaystyle
1916.2\pm 0.9\ {\rm MeV}\ {\rm(CPRRW)}\,.$

The averaged values for Eqs. (47) and (48) appear in Table 1.

###  V.4 $c\bar{c}q\bar{s}$ Sector

The results obtained from the $c\bar{c}q\bar{q}^{\prime}$ and
$c\bar{c}s\bar{s}$ $\Sigma^{+}_{g}(1S)$ multiplets in the previous two
subsections, with parameters collected in Table 1, are almost completely
sufficient to predict the entire $c\bar{c}q\bar{s}$ $\Sigma^{+}_{g}(1S)$
spectrum. However, as noted in Sec. V.1, the fact that the open-strange states
are pure SU(3)flavor octet, while $c\bar{c}q\bar{q}^{\prime}$ and
$c\bar{c}s\bar{s}$ are assumed to be ideally mixed octet-singlet combinations,
means that the value of $M_{0}^{c\bar{c}q\bar{s}}$ extracted using only inputs
from the other sectors is likely to be slightly too low to match observed
$Z_{cs}$ masses in Eqs. (I). On the other hand, the fine structure obtained in
this sector using values of $\kappa_{sc}$, $\kappa_{sq}$, and $V_{8}$ from
Table 1 should be predicted correctly.

Explicitly, using the $m_{\delta^{\vphantom{1}}(cq)}$ from Eqs. (48),
$m_{\delta^{\vphantom{1}}(cs)}$ from Eqs. (47), and the same lattice-
calculated potentials $V(r)$ as used previously, we compute

$\displaystyle M_{0}^{c\bar{c}q\bar{s}}$ $\displaystyle=$ $\displaystyle
4119.7\pm 1.7\ {\rm MeV}\ {\rm(JKM)}\,,$ (49) $\displaystyle=$ $\displaystyle
4119.7\pm 1.7\ {\rm MeV}\ {\rm(CPRRW)}\,,$

a remarkably stable result across simulations. Using this value along with the
other parameters in Table 1 in the $1^{+}$ expressions of Eqs. (IV), we
compute

$\displaystyle m_{Z_{cs}^{(1)}}$ $\displaystyle=$ $\displaystyle 3967.7\pm
3.4\ {\rm MeV}\,,$ $\displaystyle m_{Z_{cs}^{(2)}}$ $\displaystyle=$
$\displaystyle 4136.0\pm 7.1\ {\rm MeV}\,,$ $\displaystyle m_{Z_{cs}^{(3)}}$
$\displaystyle=$ $\displaystyle 4190.3\pm 3.1\ {\rm MeV}\,,$ (50)

which are lower than the measured values given in Eqs. (I). If, however, we
add an offset

$\Delta M_{0}^{c\bar{c}q\bar{s}}=35.3\pm 6.9\ {\rm MeV}\,,$ (51)

then the predictions of Eqs. (V.4) become

$\displaystyle m_{Z_{cs}^{(1)}}$ $\displaystyle=$ $\displaystyle 4003.0\pm
7.6\ {\rm MeV}\,,$ $\displaystyle m_{Z_{cs}^{(2)}}$ $\displaystyle=$
$\displaystyle 4171.3\pm 10.3\ {\rm MeV}\,,$ $\displaystyle m_{Z_{cs}^{(3)}}$
$\displaystyle=$ $\displaystyle 4225.6\pm 7.5\ {\rm MeV}\,,$ (52)

and so $m_{Z_{cs}^{(1)}}$ and $m_{Z_{cs}^{(3)}}$ beautifully match the
observed values in Eqs. (I).

Of course, this model predicts also a third open-strange $1^{+}$ state
$Z_{cs}^{(2)}$, which is not, as yet, reported by LHCb. In this regard, we
note that the reported mass uncertainty and width of $Z_{cs}(4220)$ in Eqs.
(I) are quite large, meaning that subsequent analysis might resolve the peak
as two states $Z_{cs}^{(2)}$ and $Z_{cs}^{(3)}$, as was found for the
pentaquark candidates $P_{c}(4440)$ and $P_{c}(4457)$ in Ref. Aaij  _et al._
(2019). Small hints of additional structure may be already visible in the LHCb
results above 4100 MeV (Fig. 3 of Aaij  _et al._ (2019), right inset). The
relative closeness of $Z_{cs}^{(2)}$ and $Z_{cs}^{(3)}$ in mass follows
directly in this model, as can be seen from Eqs. (IV), since (Table 1)
$\kappa_{qc}\\!\ll\\!\kappa_{sc}$. The large mass splitting
$m_{Z_{cs}^{(3)}}-m_{Z_{cs}^{(1)}}\\!\approx\\!2\kappa_{sc}\\!>\\!200$ MeV is
also explained naturally by the model; in this regard, note that such a large
mixing would not have occurred without the mixing of strange states between
$1^{++}$ and $1^{+-}$ multiplets, as discussed in Sec. IV.

One other criterion may be useful for disentangling the trio of $1^{+}$ states
$Z_{cs}^{(i)}$: The eigenvectors of Eqs. (IV) couple differently to the heavy-
and light-quark-spin eigenstates $X_{1}$ [Eqs. (6)], $\tilde{Z}^{\prime}$, and
$\tilde{Z}$ [defined in Eqs. (7)]. Explicitly, in the limit $V_{8}\\!=\\!0$,
the corresponding eigenvectors with respect to this basis are

$v^{(1),(2),(3)}=\frac{1}{2}\\!\left(\begin{array}[]{c}+\sqrt{2}\\\ -1\\\
+1\end{array}\\!\right),\ \frac{1}{2}\\!\left(\begin{array}[]{c}-\sqrt{2}\\\
-1\\\ +1\end{array}\\!\right),\
\frac{1}{\sqrt{2}}\\!\left(\begin{array}[]{c}0\\\ +1\\\
+1\end{array}\\!\right).$ (53)

Since only $\tilde{Z}$ has $s_{Q\bar{Q}}\\!=\\!0$, the third component of each
eigenvector indicates the relative strength of the coupling to $h_{c}$ (vs.
$J/\psi$) in decays that conserve heavy-quark spin. We note that
$Z_{cs}^{(2)}$ has the largest such coupling: 50% of its decays should be to
$s_{Q\bar{Q}}\\!=\\!0$ states. Therefore, a prediction of this model is not
only that $Z_{cs}(4220)$ resolves into two peaks, but also that the lower
state couples particularly strongly to $h_{c}$.

Using the values of $M_{0}^{c\bar{c}q\bar{s}}$, $\kappa_{sc}$, $\kappa_{qc}$,
and $V_{8}$ from Table 1 and the shifted multiplet average mass [using Eq.
(51)],

$\tilde{M}_{0}^{c\bar{c}q\bar{s}}\equiv M_{0}^{c\bar{c}q\bar{s}}+\Delta
M_{0}^{c\bar{c}q\bar{s}}=4155.0\pm 7.5\ {\rm MeV}\,,$ (54)

we tabulate mass values for all members of the $\Sigma^{+}_{g}(1S)$
$c\bar{c}q\bar{s}$ multiplet in Table 2.

Table 2: Predictions of hidden-charm plus light-quark tetraquark meson masses (in MeV) for all states in the lowest multiplet [$\Sigma^{+}_{g}(1S)$] of the dynamical diquark model, using the Hamiltonian parameters of Table 1 [and Eq. (54) for $c\bar{c}q\bar{s}$]. Observed masses (used to obtain the Hamiltonian parameters) are exhibited in boldface. Uncertainties are obtained by including those on all fit parameters. | $c\bar{c}q\bar{q^{\prime}}$ | $c\bar{c}s\bar{s}$ | $c\bar{c}q\bar{s}$
---|---|---|---
$J^{PC}$ | $I=0$ | $I=1$ |  |  | $J^{P}$ |  |
$0^{++}$ | $3841.9\pm 13.0$ | $4262.1\pm 17.1$ | $3872.1\pm 9.6$ | $3988.7\pm 2.3$ | $\mathbf{3921.7}\pm 4.3$ | $\mathbf{4350.9}\pm 4.7$ | $0^{+}$ | $3951.8\pm 7.7$ | $4228.0\pm 10.4$
$1^{+-}$ | $3896.0\pm 5.9$ | $4259.2\pm 17.1$ | $\mathbf{3887.2}\pm 6.1$ | $\mathbf{4024.4}\pm 3.0$ | $4135.8\pm 3.7$ | $4356.4\pm 3.4$ | $1^{+}$ | $\mathbf{4003.0}\pm 7.6$ | $4171.3\pm 10.3$
$1^{++}$ | $\mathbf{3872.2}\pm 7.8$ | $\cdots$ | $3993.8\pm 5.6$ | $\cdots$ | $\mathbf{4146.7}\pm 3.5$ | $\cdots$ |  | $\mathbf{4225.6}\pm 7.5$ | $\cdots$
$2^{++}$ | $3923.4\pm 7.8$ | $\cdots$ | $4045.0\pm 5.6$ | $\cdots$ | $4366.3\pm 3.5$ | $\cdots$ | $2^{+}$ | $4220.1\pm 7.5$ | $\cdots$

###  V.5 The $c\bar{c}uds$ Pentaquark

The numerical analysis of Ref. Giron  _et al._ (2019) builds upon the proposal
of pentaquarks as diquark-triquark bound states Lebed (2015), where the
triquark ${\bar{\theta}}$ is formed using a concatenation of the same color-
triplet-binding mechanism as appears within the diquarks:

${\bar{\theta}}\equiv\left[\bar{Q}(q_{1}q_{2})_{\bar{\bm{3}}}\right]_{\bm{3}}\,.$
(55)

The dynamical diquark model then uses the same lattice-calculated potentials
as before Juge  _et al._ (2003); Mor ; Capitani  _et al._ (2019) to connect
the diquark and triquark quasiparticles. In the original calculations of Ref.
Giron  _et al._ (2019), the pentaquark candidates $P_{c}(4312)$,
$P_{c}(4380)$, $P_{c}(4440)$, and $P_{c}(4457)$ observed at LHCb Aaij  _et
al._ (2015, 2019) are considered as $\delta^{\vphantom{1}}\\!=\\!(cu)$,
${\bar{\theta}}\\!=\\![\bar{c}(ud)]$ bound states using a coarse analysis:
LHCb identifies $P_{c}(4380)$ as having opposite parity to
$P_{c}(4440)/P_{c}(4457)$ Aaij  _et al._ (2015),333It is actually the
original, unresolved $P_{c}(4450)$ of Ref. Aaij  _et al._ (2015) that has
opposite parity to $P_{c}(4380)$, and an assumption of this work that the two
resolved components $P_{c}(4440)$ and $P_{c}(4457)$ Aaij  _et al._ (2019)
share this same parity eigenvalue. and such a small splitting between
multiplets of opposite parity in the model makes sense only if $P_{c}(4380)$
is a high-lying state in the $P\\!=\\!-$ multiplet444The $g$ quantum number is
lost in the asymmetric diquark-triquark case Lebed (2017). $\Sigma^{+}(1S)$
$\big{[}J^{P}\\!=\\!\big{(}\frac{1}{2},\frac{3}{2}\big{)}^{-}\big{]}$, while
the other states belong to the $P\\!=\\!+$ multiplet $\Sigma^{+}(1P)$
$\big{[}J^{P}\\!=\\!\big{(}\frac{1}{2},\frac{3}{2},\frac{5}{2}\big{)}^{+}\big{]}$.

Using $P_{c}(4312)$ to fix $M_{0}$ for the $\Sigma^{+}(1P)$ multiplet, where
Aaij  _et al._ (2019)

$m_{P_{c}(4312)}=4311.9\pm 6.8\ {\rm MeV}\,,$ (56)

and using $m_{\delta^{\vphantom{1}}=(cq)}$ obtained from the
$c\bar{c}q\bar{q}^{\prime}$ states, Ref. Giron  _et al._ (2019) computes a
value of $m_{{\bar{\theta}}}\\!\simeq\\!1.93$ GeV. Repeating the analysis here
using the new value of $m_{\delta^{\vphantom{1}}=(cq)}$ from Eq. (48), we find

$\displaystyle m_{{\bar{\theta}}=\bar{c}(q_{1}q_{2})}$ $\displaystyle=$
$\displaystyle 1884.6\pm 7.5\ {\rm MeV}\ {\rm(JKM)}\,,$ (57) $\displaystyle=$
$\displaystyle 1866.5\pm 7.5\ {\rm MeV}\ {\rm(CPRRW)}\,.$

As noted in Ref. Giron  _et al._ (2019), $m_{\delta^{\vphantom{1}}=(cq)}$ and
$m_{{\bar{\theta}}=[\bar{c}(q_{1}q_{2})]}$ are quite close in mass; indeed,
$m_{{\bar{\theta}}}$ is actually slightly smaller than
$m_{\delta^{\vphantom{1}}}$ in the new calculation. This peculiarity arises
from assigning the lowest observed $c\bar{c}q\bar{q}^{\prime}$ states to the
ground-state $\Sigma^{+}_{g}(1S)$ multiplet but assigning the lowest observed
pentaquark $P_{c}(4312)$ to the excited $\Sigma^{+}(1P)$ multiplet. Should the
opposite-parity $P_{c}(4380)$ disappear from future data, then $P_{c}(4312)$
and the other states would become suitable to belong to $\Sigma^{+}(1S)$, and
$m_{{\bar{\theta}}}$ (absorbing what was orbital excitation energy) would
become numerically larger. Even in the current circumstance, however, no
obvious physical requirement demands that
$m_{{\bar{\theta}}}\\!>\\!m_{\delta^{\vphantom{1}}}$. Indeed, one may argue
that ${\bar{\theta}}$ contains two significant sources of binding energy:
within the diquark $(q_{1}q_{2})$, and between this diquark and $\bar{c}$,
while $\delta^{\vphantom{1}}$ possesses only the first type of binding,
thereby allowing $m_{{\bar{\theta}}}\\!\lesssim\\!m_{\delta^{\vphantom{1}}}$.

Now suppose that the state $P_{cs}(4459)$ of Eqs. (I) is the open-strange
$\Sigma^{+}(1P)$ analogue to $P_{c}(4312)$, i.e., a
$\delta^{\vphantom{1}}$-${\bar{\theta}}$ state with
$\delta^{\vphantom{1}}\\!=\\!(cs)$, where $m_{\delta^{\vphantom{1}}=(cs)}$ is
given in Eqs. (47). Then we compute

$\displaystyle M_{0}^{c\bar{c}sq_{1}q_{2}}$ $\displaystyle=$ $\displaystyle
4441.8\pm 7.0\ {\rm MeV}\ {\rm(JKM)}\,,$ (58) $\displaystyle=$ $\displaystyle
4441.7\pm 7.0\ {\rm MeV}\ {\rm(CPRRW)}\,.$

This is a stunning result, being less than 2$\,\sigma$ lower than the value in
Eqs. (I). Indeed, no reason apart from convenience leads one to take the
$P_{c}(4312)$ [as opposed to, say, $P_{c}(4457)$] and $P_{cs}(4459)$ masses
equal to the $M_{0}$ values for their respective $\Sigma^{+}(1P)$ multiplets,
except that they are the lightest ones known. A complete analysis would
incorporate fine structure, as is done for the tetraquark sectors, but this
exercise has not yet been carried out in the pentaquark sectors of this model,
due to a lack of experimental clarity on $J^{P}$ quantum numbers for at least
some of the observed states. Nevertheless, the result of Eqs. (58) shows that
a single model can, in fact, accommodate exotics in all observed flavor
sectors.

The development of the dynamical diquark model to date has focused primarily
on spectroscopy, and to a lesser extent on identifying the dominant quarkonium
decay channels. Detailed quantitative calculations of strong decay widths,
particularly for open-heavy-flavor channels, have not yet been attempted,
because a precise description of couplings between diquark and hadron-hadron
configurations has not yet been developed. Qualitative statements to explain
the relative narrowness of exotic states have appeared since the initial
description of the picture in Refs. Brodsky  _et al._ (2014); Lebed (2015);
they originate from the significant spatial separation between the diquark or
triquark quasiparticles, which hinders the rearrangement of their component
quarks into color-singlet hadrons. Such effects may need to be quite potent in
the pentaquarks, since $P_{c}(4312)$, $P_{c}(4440)$, $P_{c}(4457)$,
$P_{cs}(4459)$ all have surprisingly small widths ($\lesssim\\!20$ MeV). In
addition, the proximity of these states to
$\overline{D}^{(*)}\Sigma_{c}^{(*)}$ or $\overline{D}^{(*)}\Xi_{c}$ thresholds
has been noted since the original experimental papers, and predicted earlier
in molecular models Wu  _et al._ (2010); Wang  _et al._ (2011); Yang  _et al._
(2012); Karliner and Rosner (2015). The next phase of the development of the
model will address the effect of mixing between diquark configurations and
hadron-hadron thresholds, thus providing critical insight into both the decay
properties of exotics and a connection to the successes of hadronic-molecule
pictures.

##  VI Conclusions

This work shows that the newly observed hidden-charm, open-strange exotic-
hadron candidates $Z_{cs}(4000)$, $Z_{cs}(4220)$, and $P_{cs}(4459)$ fit
naturally into the dynamical diquark model. Notably, the same lattice-
simulated potential $V(r)$ between two heavy, color-triplet sources in the
lowest Born-Oppenheimer configuration $\Sigma^{+}_{(g)}$ is seen to apply to
all cases studied here.

Among tetraquarks, the same Hamiltonian parameters, with numerical values
obtained from the $c\bar{c}q\bar{q}^{\prime}$ ($q$, $q^{\prime}$ being $u$ or
$d$) and $c\bar{c}s\bar{s}$ exotic $\Sigma^{+}_{g}(1S)$ multiplets,
successfully predict masses in the $c\bar{c}q\bar{s}$ sector. In particular,
the large $Z_{cs}(4220)$-$Z_{cs}(4000)$ mass splitting emerges naturally as
consequences of both the large $(cs)$ diquark internal spin-spin coupling
$\kappa_{sc}$ and the mixing of open-strange members of $J^{PC}\\!=\\!1^{++}$
and $1^{+-}$ multiplets, the latter an effect seen in conventional hadron
physics for strange mesons such as $K_{1A}$ and $K_{1B}$. The model also
predicts a third $1^{+}$ $Z_{cs}$ state lying not far below 4200 MeV.

The overall multiplet-average mass $M_{0}$ for $c\bar{c}q\bar{s}$ states also
receives a shift modification compared to those for
$c\bar{c}q\bar{q}^{\prime}$ and $c\bar{c}s\bar{s}$ states, since the former
are pure SU(3)flavor octet states, while states in the latter sets are assumed
in the numerical analysis to be ideally mixed octet-singlet combinations. The
size of this shift $\Delta M_{0}^{c\bar{c}q\bar{s}}$ is found to be
numerically not large, at most a few 10’s of MeV.

In addition, the model in all sectors has been expanded to allow not only a
$\pi$-like interaction operator between the diquarks (as in previous studies),
but an $\eta$-like interaction operator as well. The numerical size of the
coefficient $V_{8}$ of this operator is found to be much smaller than that
($V_{0}$) for $\pi$-like interactions, and has a fairly minimal effect on the
hadron spectra. Mass predictions for all states in the $\Sigma^{+}_{g}(1S)$
multiplet for each flavor content are presented.

Among pentaquarks, a crude calculation taking the nonstrange $P_{c}(4312)$ as
a base state for the positive-parity multiplet $\Sigma^{+}(1P)$ constructed of
a diquark-triquark pair $(cq)[\bar{c}(ud)]$, and replacing the $(cq)$ diquark
with a $(cs)$ diquark, produces a state very close in mass to that of
$P_{cs}(4459)$.

As new exotic hadrons continue to be uncovered—a rather safe expectation,
considering the rate of observational advances over the past few years—more
opportunities for sharpening our understanding of their mass spectrum and
transitions will emerge. Whether or not a diquark-based spectrum provides the
eventual global picture for these states, the dynamical diquark model supplies
a definite road map for the sort of spectrum to expect.

###### Acknowledgements.

This work was supported by the National Science Foundation (NSF) under Grant
No. PHY-1803912. We thank T. Skwarnicki for helpful insight into the LHCb
results.

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