# Radiative Transitions of Charmoniumlike Exotics in the Dynamical Diquark
Model

Justin M. Gens<EMAIL_ADDRESS>Jesse F. Giron<EMAIL_ADDRESS>Richard F. Lebed
<EMAIL_ADDRESS>Department of Physics, Arizona State University, Tempe,
AZ 85287, USA

(February, 2021)

###### Abstract

Using the dynamical diquark model, we calculate the electric-dipole radiative
decay widths to $X(3872)$ of the lightest negative-parity exotic candidates,
including the four $I=0$, $J^{PC}\\!=\\!1^{--}$ (“$Y$”) states. The
$O$(100–1000 keV) values obtained test the hypothesis of a common substructure
shared by all of these states. We also calculate the magnetic-dipole radiative
decay width for $Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$, and find it to be
rather smaller ($<$ 10 keV) than its predicted value in molecular models.

Exotic hadrons, radiative transitions, diquarks

##  I Introduction

The number of new heavy-quark exotic-hadron candidates, presumptive tetraquark
and pentaquark states, increases every year. In the past 18 years, over 40
candidates have been observed at multiple facilities and their hosted
experiments. However, no single theoretical picture to describe the structure
of these states has emerged as an undisputed favorite. Both the broad scope of
experimental results and competing theoretical interpretations have been
reviewed by many 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).

Among these competing physical approaches, the dynamical diquark picture
Brodsky  _et al._ (2014) was developed to provide a mechanism through which
diquark ($\delta^{\vphantom{1}}$)-antidiquark (${\bar{\delta}}$) states could
persist long enough to be identified as such experimentally. Diquarks are
formed through the attractive channels ${3}\otimes\bf{3}\\!\to\\!\bar{\bf{3}}$
[$\delta\\!\equiv\\!(Qq)_{\bar{\bf 3}}$] and $\bar{\bf 3}\otimes\bar{\bf
3}\\!\to\\!{\bf 3}$
[${\bar{\delta}}\\!\equiv\\!(\bar{Q}{\bar{q}}^{\prime})_{\bf 3}$] between
color-triplet quarks. In this physical picture, the heavy quark $Q$ must first
be created in closer spatial proximity to a light quark $q$ than to a light
antiquark $\bar{q}^{\prime}$ (and vice versa for $\bar{Q}$). This initial
configuration provides an opportunity for the formation of fairly compact
$\delta^{\vphantom{1}}$ and ${\bar{\delta}}$ quasiparticles, in distinction to
an initial state in which the strongly attractive
$\bf{3}\otimes\bar{\bf{3}}\\!\to\\!\bf{1}$ coupling immediately leads to
$(Q\bar{q}^{\prime})(\bar{Q}q)$ meson pairs. Second, the large energy release
of the production process (from a heavy-hadron decay or in a collider event)
drives apart the $\delta^{\vphantom{1}}$-${\bar{\delta}}$ pair before
immediate recombination into a meson pair can occur, creating an observable
resonance. A similar mechanism extends the picture to pentaquark formation
Lebed (2015), by means of using color-triplet “antitriquarks”
${\bar{\theta}}\\!\equiv\\![\bar{Q}_{\bar{\bf 3}}(q_{1}q_{2})_{\bar{\bf
3}}]^{\vphantom{(}}_{\bf 3}$.

This physical picture was subsequently developed into the dynamical diquark
model Lebed (2017): The separated $\delta^{\vphantom{1}}$-${\bar{\delta}}$
pair is connected by a color flux tube, whose quantized states are best
described in terms of the potentials computed using the Born-Oppenheimer (BO)
approximation. These are the same potentials as appear in QCD lattice gauge-
theory simulations that predict the spectrum of heavy-quarkonium hybrid mesons
Juge  _et al._ (1998, 1999, 2003); Mor ; Capitani  _et al._ (2019). The BO
potentials are introduced into coupled Schrödinger equations that are solved
numerically in order to produce predictions for the
$\delta^{\vphantom{1}}$-${\bar{\delta}}$ spectrum, as shown in Ref. Giron  _et
al._ (2019). As one of the primary results of that work, all the observed
exotic candidates are shown to be accommodated within the ground-state BO
potential $\Sigma^{+}_{g}$, with the specific multiplets in order of
increasing average mass being $1S$, $1P$, $2S$, $1D$, and $2P$. A full summary
of the BO potential notation is presented in Ref. Lebed (2017).

The mass spectrum and preferred decay modes (organized by eigenstates of
heavy-quark spin) of the 6 isosinglets and 6 isotriplets comprising the
${c\bar{c}q\bar{q}^{\prime}}$ positive-parity $\Sigma^{+}_{g}(1S)$ multiplet
(where $q,q^{\prime}\in\\{u,d\\}$) were studied in Ref. Giron  _et al._
(2020). This was the first work to differentiate $I\\!=\\!0$ and $I\\!=\\!1$
states in a diquark model. The specific model of Ref. Giron  _et al._ (2020)
naturally produces scenarios in which $X(3872)$ is the lightest
$\Sigma^{+}_{g}(1S)$ state, and moreover predicts that the lighter of the two
$I\\!=\\!1$, $J^{PC}\\!=\\!1^{+-}$ states in $\Sigma^{+}_{g}(1S)$
[$Z_{c}(3900)$] naturally decays almost exclusively to $J/\psi$ and the
heavier one [$Z_{c}(4020)$] to $h_{c}$, as is observed. The model of Ref.
Giron  _et al._ (2020) uses a 3-parameter Hamiltonian consisting of a common
multiplet mass, an internal diquark-spin coupling, and a long-distance
isospin- and spin-dependent coupling (analogous to $\pi$ exchange) between the
light quark $q$ in $\delta^{\vphantom{1}}$ and light antiquark
$\bar{q}^{\prime}$ in ${\bar{\delta}}$. Similar conclusions using QCD sum
rules have been obtained in Ref. Ghalenovi and Sorkhi (2020).

The dynamical diquark model was developed further through the corresponding
analysis Giron and Lebed (2020a) of the negative-parity
${c\bar{c}q\bar{q}^{\prime}}$ $\Sigma^{+}_{g}(1P)$ multiplet and its 28
constituent isomultiplets (14 isosinglets and 14 isotriplets), which includes
precisely four $Y$ ($I\\!=\\!0$, $J^{PC}\\!=\\!1^{--}$) states. In this case,
the simplest model has 5 parameters: the 3 listed above, plus spin-orbit and
tensor terms. An earlier diquark analysis using a similar Hamiltonian, but not
including isospin dependence, appears in Ref. Ali  _et al._ (2018).

The success of Ref. Giron  _et al._ (2019) in predicting the correct mass
splittings between the observed bands ($1S,1P,2S$) of exotic hadrons, and Ref.
Giron  _et al._ (2020) in effectively representing the fine structure within
the lowest multiplets [especially $\Sigma^{+}_{g}(1S)$] provides strong a
posteriori support for the applicability of the dynamical diquark model. In
particular, one may certainly question whether treating the exotics as quasi-
two-body states within a BO approximation, rather than including full 4- (or
5-) body interactions to represent the internal evolution of the
quasiparticles, is sensible. However, while such effects are undoubtedly
present at some level, the current experimental evidence appears to support
the presence of a scale separation that allows the quasiparticles to be
treated identifiable subunits within the hadrons. As an example, Ref. Giron
_et al._ (2019) showed in numerical simulations that the diquarks need not be
pointlike particles, but could have substantial spatial extent (characteristic
radii as large as $0.4$ fm) before the full hadron mass spectrum changes
significantly.

An analysis within this model of the 12 isomultiplets comprising the
${b\bar{b}q\bar{q}^{\prime}}$ $\Sigma_{g}^{+}(1S)$ multiplet and the 6 states
of the ${c\bar{c}}s\bar{s}$ $\Sigma_{g}^{+}(1S)$ multiplet appears in Ref.
Giron and Lebed (2020b). By using only experimental inputs for the states
$Z_{b}(10610)$ and $Z_{b}(10650)$, which includes their masses and relative
probability of decay into $h_{b}$ versus $\Upsilon$ states, the entire
${b\bar{b}q\bar{q}^{\prime}}$ mass spectrum is predicted. In particular, the
mass of the bottom analogue to $X(3872)$ is highly constrained
($\approx\\!10600$ MeV), and the lightest ${b\bar{b}q\bar{q}^{\prime}}$ state
($I\\!=\\!0$, $J^{PC}\\!=\\!0^{++}$) lies only a few MeV above the $B\bar{B}$
threshold. Furthermore, starting with the assumption that $X(3915)$ is the
lowest lying ${c\bar{c}}s\bar{s}$ state Lebed and Polosa (2016) and $Y(4140)$
is the sole $J^{PC}\\!=\\!1^{++}$ ${c\bar{c}}s\bar{s}$ state in
$\Sigma_{g}^{+}(1S)$, the remaining 4 masses in the multiplet are predicted.
Emerging naturally in the spectrum is $X(4350)$, a $J/\psi$-$\phi$ resonance
seen by Belle Shen  _et al._ (2010), while $Y(4626)$ and $X(4700)$ are found
to fit well within the $\Sigma_{g}^{+}(1P)$ and $\Sigma_{g}^{+}(2S)$
${c\bar{c}}s\bar{s}$ multiplets, respectively.

The dynamical diquark model has also recently been extended to the case in
which the light quarks $q$ are replaced with heavy quarks $Q$ to produce fully
heavy tetraquark states $Q_{1}\overline{Q}_{2}Q_{3}\overline{Q}_{4}$, where
$Q_{i}\\!=\\!c$ or $b$. Sparked by the recent LHCb report of at least one
di-$J/\psi$ resonance near $6900$ MeV Aaij  _et al._ (2020), Ref. Giron and
Lebed (2020c) determined the spectrum of $c\bar{c}c\bar{c}$ states in the
dynamical diquark model. In this system, the minimal model predicts each
$S$-wave multiplet to consist of 3 degenerate states
($J^{PC}\\!=\\!0^{++},\,1^{+-},\,2^{++}$) and 7 $P$-wave states. $X(6900)$ was
found to fit most naturally as a $\Sigma_{g}^{+}(2S)$ state, with other
structures in the measured di-$J/\psi$ spectrum appearing to match $C\\!=\\!+$
members of the $\Sigma^{+}_{g}(1P)$ multiplet.

In this paper we use the dynamical diquark model to predict radiative
transitions between exotic states. So far, very few theoretical papers have
investigated exotic-to-exotic transitions (and of these papers, only diquark
models have been considered Maiani  _et al._ (2014); Chen  _et al._ (2015)).
One of the distinctive features of the $P$-wave study in Ref. Giron and Lebed
(2020a) is the direct calculation of decay probabilities to eigenstates of
heavy-quark spin. Indeed, Ref. Giron and Lebed (2020a) uses the heavy quark-
spin content of states as the main criterion for associating observed
resonances with particular states in the $\Sigma^{+}_{g}(1P)$ multiplet, and
identifies using likelihood fits two particularly plausible assignments for
the states. Using the same decay probabilities, we calculate here the
transition amplitudes for
$\Sigma^{+}_{g}(1P)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$. We directly adapt the
well-known expression for electric dipole (E1) radiative transitions used to
great effect for conventional quarkonium. Since the E1 transition formula
depends sensitively upon the initial and final wave functions, a comparison
between our predictions and data provides an important test of the hypothesis
that the purported $\Sigma^{+}_{g}(1P)$ and $\Sigma^{+}_{g}(1S)$ states, such
as in $Y(4220)\\!\to\\!\gamma X(3872)$, truly share a common structure. The
corresponding magnetic dipole (M1) expression within this model is also
presented, in anticipation of the observation of relevant transitions such as
$\Sigma^{+}_{g}(2S)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$, or even between two
$\Sigma^{+}_{g}(1S)$ states such as $Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$.

This paper is organized as follows: In Sec. II we review the current
experimental data on transitions between ${c\bar{c}q\bar{q}^{\prime}}$ states.
Section III reprises the relevant phenomenological aspects of Ref. Giron and
Lebed (2020a). In Sec. IV we calculate the decay widths and decay
probabilities for exotic-to-exotic radiative transitions and focus upon two of
the more probable $P$-wave state assignments in Ref. Giron and Lebed (2020a).
We conclude in Sec. V.

##  II Experimental Review of Exotic-to-Exotic Transitions

Although the number of exotic-candidate discoveries continues to increase at a
remarkable pace, only a handful of exotic-to-exotic decays have been observed
to date, through radiative Ablikim  _et al._ (2014, 2019a) and pionic Ablikim
_et al._ (2013); Liu  _et al._ (2013); Ablikim  _et al._ (2020) transitions.

Considering first the radiative decays that form the topic of this work, thus
far only E1 transitions (as indicated by changing parity $\Delta P\\!=\\!-$)
have been observed in two states at BESIII, the $J^{PC}\\!=\\!1^{--}$
$Y(4260)$ Ablikim  _et al._ (2014) and $Y(4220)$ Ablikim  _et al._ (2019a),
both seen to decay to a photon and the $J^{PC}\\!=\\!1^{++}$ $X(3872)$.
Indeed, an increasing amount of evidence from BESIII (e.g., in Ref. Ablikim
_et al._ (2017a)) suggests that the well-known $Y(4260)$ is actually a
collection of resonances, of which $Y(4220)$ is just one component. Observed
exotic-to-conventional radiative transitions are also rather few in number,
due to the large decay widths of exotics that follows from the dominance of
their strong decay modes. To date, only $X(3872)\\!\to\\!\gamma J/\psi$ and
$\gamma\psi(2S)$, also both E1 transitions, have definitely been seen (e.g.,
in Ref. Aaij  _et al._ (2014a)). BESIII has also recently announced an
interesting negative result Ablikim  _et al._ (2021), an upper limit for
$Z_{c}(4020)^{0}\,(J^{PC}\\!=\\!1^{+-})\\!\to\\!\gamma X(3872)$. Indeed, to
date no M1 radiative decay ($\Delta P\\!=\\!+$) of any exotic candidate has
yet been seen at any experiment.

As for pionic transitions, both BESIII Ablikim  _et al._ (2013) and Belle Liu
_et al._ (2013) have observed (indeed, discovered) $Z_{c}(3900)^{\pm}$ through
$Y(4260)\\!\to\\!\pi^{+}\pi^{-}J/\psi$, and BESIII recently observed
$Z_{c}(3900)^{0}$ via $Y(4220)\\!\to\\!\pi^{0}\pi^{0}J/\psi$ Ablikim  _et al._
(2020). Assuming just a similarity of hadronic structure between various
exotic candidates, one may expect several more exotic-to-exotic pionic (or
other light-meson) transitions to be observed in the future. An essential
criterion for how such transitions may best be studied relies on the size of
the pion momentum $p_{\pi}$ in such processes; for example, in the decays
listed above, $p_{\pi}\\!\approx\\!300$ MeV. Processes with smaller $p_{\pi}$
values may be reliably studied using conventional chiral perturbation theory,
while studies of processes with larger $p_{\pi}$ values require modifications
to the perturbative calculation to improve their convergence. Since the
methods associated with radiative transitions (particularly E1 transitions)
present fewer computational ambiguities, we defer a study of exotic-to-exotic
pionic transitions for future work.

The expressions for E1 and M1 transition widths used below [Eqs. (14) and
(17), respectively] are almost identical to the forms derived in standard
quantum mechanics textbooks. As such, they are manifestly nonrelativistic, and
furthermore are developed using the photon long-wavelength approximation,
$\exp(i\bm{k}\cdot\bm{r})\\!\to\\!1$. Nevertheless, the expressions can also
be derived directly from the fundamental Lagrangian $j_{\mu}A^{\mu}$ couplings
of the electromagnetic current $j^{\mu}$ of charged quarks to the photon field
$A^{\mu}$ (see, e.g., Ref. Brambilla  _et al._ (2004)). In Sec. III we discuss
the effect of including certain corrections to the textbook expressions.

In the dynamical diquark model, all states in the multiplet
$\Sigma^{+}_{g}(1S)[(1P)]$ have $P\\!=\\!+[-]$ Lebed (2017). The current
observed properties of the $J^{PC}\\!=\\!1^{--}$ ($Y$) states identified with
the multiplet $\Sigma^{+}_{g}(1P)$, whose spectroscopy is analyzed extensively
in Ref. Giron and Lebed (2020a), are summarized in Table 1.

Table 1: $J^{PC}\\!=\\!1^{--}$ charmoniumlike exotic-meson candidates catalogued by the Particle Data Group (PDG) Zyla  _et al._ (2020), which are identified with specific states within the $\Sigma^{+}_{g}(1P)$ multiplet of the dynamical diquark model, as summarized by the cases presented in Ref. Giron and Lebed (2020a) and repeated in Sec. III. Both the particle name most commonly used in the literature and its label as given in the PDG are listed. Particle | PDG label | $I^{G}J^{PC}$ | Mass [MeV] | Width [MeV] | Production and decay
---|---|---|---|---|---
$Y(4220)$ | $\psi(4230)$ | $0^{-}1^{--}$ | $4218^{+5}_{-4}$ | $59^{+12}_{-10}$ |  $e^{+}e^{-}\to Y$; $Y\to\left\\{\begin{array}[]{l}\omega\chi_{c0}\\\ \eta J/\psi\\\ \pi^{+}\pi^{-}h_{c}\\\ \pi^{+}\pi^{-}\psi(2S)\\\ \pi^{+}D^{0}D^{*-}\\\ \pi^{0}Z_{c}^{0}(3900)\\\ \gamma X(3872)\\\ \end{array}\right.$
$Y(4260)$ | $\psi(4260)$ | $0^{-}1^{--}$ | $4230\pm 8$ | $55\pm 19$ |  $e^{+}e^{-}\to\gamma Y$ or $Y$; $Y\to\left\\{\begin{array}[]{l}\pi^{+}\pi^{-}J/\psi\\\ f_{0}(980)J/\psi\\\ \pi^{\mp}Z_{c}^{\pm}(3900)\\\ K^{+}K^{-}J/\psi\\\ \gamma X(3872)\end{array}\right.$
$Y(4360)$ | $\psi(4360)$ | $0^{-}1^{--}$ | $4368\pm 13$ | $96\pm 7$ |  $e^{+}e^{-}\to\gamma Y$ or $Y$; $Y\to\bigg{\\{}\begin{array}[]{l}\pi^{+}\pi^{-}\psi(2S)\\\ \pi^{0}\pi^{0}\psi(2S)\end{array}$
$Y(4390)$ | $\psi(4390)$ | $0^{-}1^{--}$ | $4392\pm 7$ | $140^{+16}_{-21}$ |  $e^{+}e^{-}\to Y$; $Y\to\bigg{\\{}\begin{array}[]{l}\eta J/\psi\\\ \pi^{+}\pi^{-}h_{c}\end{array}$
$Y(4660)$ | $\psi(4660)$ | $0^{-}1^{--}$ | $4643\pm 9$ | $72\pm 11$ | $e^{+}e^{-}\to\left\\{\begin{array}[]{l}\gamma Y;Y\to\pi^{+}\pi^{-}\psi(2S)\\\ \ \;Y;Y\to\Lambda^{+}_{c}\Lambda^{-}_{c}\end{array}\right.$

##  III Theoretical Review of $P$-Wave Exotic States

The full spectroscopy of diquark-antidiquark
($\delta^{\vphantom{1}}$-${\bar{\delta}}$) tetraquarks and diquark-
antitriquark ($\delta^{\vphantom{1}}$-${\bar{\theta}}$) pentaquarks connected
by a gluonic field of arbitrary excitation quantum numbers, and including
arbitrary orbital excitations between the
$\delta^{\vphantom{1}}$-${\bar{\delta}}$ or
$\delta^{\vphantom{1}}$-${\bar{\theta}}$ pair, is presented in Ref. Lebed
(2017). As discussed in that work, the gluonic-field excitations combined with
the quasiparticle sources
$\delta^{\vphantom{1}},{\bar{\delta}},{\bar{\theta}}$ produce states analogous
to ordinary quarkonium hybrids; therefore, these states may likewise be
classified according to the quantum numbers provided by BO-approximation
static gluonic-field potentials. The numerical studies of Ref. Giron  _et al._
(2019) show that the exotic analogues to hybrid quarkonium states lie above
the exotic states within the corresponding BO ground-state potential
$\Sigma^{+}_{g}$ by at least 1 GeV (just as for conventional quarkonium).
Since the entire range of observed hidden-charm exotic candidates [not
counting $c\bar{c}c\bar{c}$ candidates such as $X(6900)$] spans only about 800
MeV Lebed  _et al._ (2017), it is very likely that all known hidden-charm
exotic states occupy energy levels within the $\Sigma^{+}_{g}$ BO potential.
All known ${c\bar{c}q\bar{q}^{\prime}}$ candidates can be accommodated by the
lowest $\Sigma^{+}_{g}$ levels: $1S$, $1P$, $2S$, $1D$, and $2P$, in order of
increasing mass Giron  _et al._ (2019).

A detailed enumeration of the possible ${Q\bar{Q}}q\bar{q}^{\prime}$ states,
in which the light quarks $q$,$\bar{q}^{\prime}$ do not necessarily carry the
same flavor, is straightforward for the $S$ wave. Assuming zero relative
orbital angular momenta between the quarks, any two naming conventions for the
states differ only by the order in which the 4 quark spins are coupled. In the
diquark basis, defined by coupling in the order $(qQ)\\!+\\!(\bar{q}\bar{Q})$,
the 6 possible states are denoted by Maiani  _et al._ (2014):

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

where outer subscripts indicate total quark spin $S$. The same states may be
expressed in any other basis by using angular momentum recoupling coefficients
in the form of the relevant $9j$ symbol. For the purposes of this work, the
most useful alternate basis is that of definite heavy-quark (and light-quark)
spin, $({Q\bar{Q}})\\!+\\!({q\bar{q}})$:

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

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

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

Once light-quark flavor is included, one obtains 12 states: 6 each with
$I\\!=\\!0$ and $I\\!=\\!1$, and spin structures in the form of Eqs. (1) or
(6).111If strange quarks are included, one obtains 6 SU(3)flavor octets and 6
singlets. Using these states and the most minimal $3$-parameter Hamiltonian
[the $M_{0}$, $\kappa_{qQ}$, and $V_{0}$ terms of Eq. (7) below], Refs. Giron
_et al._ (2020); Giron and Lebed (2020b) calculate the masses of all $12$
$S$-wave states in the hidden-charm and hidden-bottom sectors using known
masses of $X(3872)$, $Z_{c}(3900)$, and $Z_{c}(4020)$ for the former; and the
known masses of $Z_{b}(10610)$, $Z_{b}(10650)$, and their relative $h_{b}$ to
$\Upsilon$ branching fractions for the latter. These results incorporate
isospin dependence (the $V_{0}$ term), a feature not explicitly integrated
into other diquark models.

Reference Giron and Lebed (2020a) extends this analysis by examining the
$P$-wave multiplet, whose mass spectrum is dictated by the most minimal
$5$-parameter Hamiltonian:

$\displaystyle H$ $\displaystyle=$ $\displaystyle M_{0}+2\kappa_{qQ}({\bf
s}_{q}\\!\cdot\\!{\bf s}_{Q}+{\bf s}_{\bar{q}}\\!\cdot\\!{\bf
s}_{\bar{Q}})+V_{LS}\,{\bf L}\cdot{\bf S}$ (7)
$\displaystyle+V_{0}\,{\bm{\tau}}_{q}\\!\cdot\\!{\bm{\tau}}_{\bar{q}}\;{\bm{\sigma}}_{q}\\!\cdot\\!{\bm{\sigma}}_{\bar{q}}+V_{T}\,{\bm{\tau}}_{q}\\!\cdot\\!{\bm{\tau}}_{\bar{q}}\;S_{12}^{({q\bar{q}})}\,,$

where $M_{0}$ is the common mass of the multiplet, $\kappa_{qQ}$ represents
the strength of the spin-spin coupling within each diquark, $V_{LS}$ is the
spin-orbit coupling strength, $V_{0}$ is the isospin-dependent
coupling,222$V_{0}$ in Eq. (7) is analogous to the axial coupling in $NN\pi$
interactions. $V_{T}$ represents the tensor coupling, and
$S_{12}^{({q\bar{q}})}$ is the tensor operator defined as

$S_{12}^{({q\bar{q}})}\equiv
3\,{\bm{\sigma}}_{q}\\!\cdot{\bm{r}}\,{\bm{\sigma}}_{\bar{q}}\\!\cdot{\bm{r}}/r^{2}-{\bm{\sigma}}_{q}\\!\cdot{\bm{\sigma}}_{\bar{q}}\,.$
(8)

The well-known tabulated expressions for matrix elements of
$S_{12}^{({q\bar{q}})}$ (e.g., in Ref. de Shalit and Talmi (1963)) directly
apply neither in the basis of $s_{q\bar{q}},s_{Q\bar{Q}}$ spins nor
$s_{\delta}^{\vphantom{1}},s_{\bar{\delta}}$ spins, but rather refer to the
basis of total light-quark angular momentum $J_{q\bar{q}}$:

${\bf J}_{q\bar{q}}\equiv{\bf L}_{q\bar{q}}+{\bf s}_{q\bar{q}}\,.$ (9)

Assuming that $\delta^{\vphantom{1}}$ and ${\bar{\delta}}$ have no internal
orbital excitation so that $L_{q\bar{q}}\\!=\\!L$, the matrix elements of
$S_{12}^{({q\bar{q}})}$ are most easily computed in the $J_{q\bar{q}}$ basis,
with results that are then related back to the $s_{q\bar{q}},s_{Q\bar{Q}}$
basis by means of recoupling using $6j$ symbols:

$\displaystyle\mathcal{M}_{J_{q\bar{q}}}\equiv\left<(L,s_{q\bar{q}}),J_{q\bar{q}},s_{Q\bar{Q}},J|L,(s_{{q\bar{q}}},s_{{Q\bar{Q}}}),S,J\right>$
$\displaystyle=$
$\displaystyle(-1)^{L+s_{q\bar{q}}+s_{Q\bar{Q}}+J}\sqrt{[J_{q\bar{q}}][S]}\left\\{\begin{array}[]{ccc}L&s_{q\bar{q}}&J_{q\bar{q}}\\\
s_{Q\bar{Q}}&J&S\end{array}\right\\}.$ (12)

Using this expression, $S_{12}^{({q\bar{q}})}$ matrix elements for all
relevant states are tabulated in Ref. Giron and Lebed (2020a).

The experimental status of the $P$-wave $J^{PC}\\!=\\!1^{--}$ exotic
candidates remains in flux, with BESIII providing the majority of the most
recent data. With reference to the information presented in Table 1, we have
already noted that the analysis of the BESIII Collaboration Ablikim  _et al._
(2017a) favors the interpretation of $Y(4260)$ as a superposition of states,
the lowest component of which is $Y(4220)$. They identify the higher component
with $Y(4360)$, although the previous mass measurements of this state given in
Table 1 are rather higher, and one of several scenarios considered in Ref.
Giron and Lebed (2020a) proposes that $Y(4360)$ and $Y(4390)$ are the same
state, while the higher-mass component in Ref. Ablikim  _et al._ (2017a) can
be interpreted as a distinct “$Y(4320)$”. Alternately, if the only lower
states are $Y(4220)$, $Y(4360)$, and $Y(4390)$, then $Y(4660)$ becomes the
fourth $I\\!=\\!0$, $1^{--}$ candidate state in $\Sigma^{+}_{g}(1P)$.

With the mass spectrum of these charmoniumlike states not yet entirely
settled, Ref. Giron and Lebed (2020a) also employs information on their
preferred charmonium decay modes as classified by heavy-quark spin: $\psi$
($s_{{Q\bar{Q}}}\\!=\\!1$) or $h_{c}$ ($s_{{Q\bar{Q}}}\\!=\\!0$). Assuming
heavy-quark spin symmetry as expressed by the conservation of $s_{{Q\bar{Q}}}$
in the decays, the heavy-quark spin content $P_{s_{{Q\bar{Q}}}}$ of each state
becomes an invaluable diagnostic in disentangling the $J^{PC}\\!=\\!1^{--}$
spectrum. For example, from Table 1 one sees that $Y(4220)$ decays to both
$\psi$ states and $h_{c}$, while if $Y(4360)$ and $Y(4390)$ are in fact one
state, the same can be said for them as well. Reference Giron and Lebed
(2020a) also introduces a parameter $\epsilon$ designed to enforce the
goodness-of-fit to a particular value $f$ of $P_{s_{{Q\bar{Q}}}}$, which in
the case of $s_{{Q\bar{Q}}}\\!=\\!0$ reads

$\Delta\chi^{2}=\left(\frac{\ln P_{s_{{Q\bar{Q}}}=0}-\ln
f}{\epsilon}\right)^{2}.$ (13)

In terms of the parameters $P_{s_{{Q\bar{Q}}}}$, $f$, and $\epsilon$, the 5
cases discussed in Ref. Giron and Lebed (2020a) designed to represent a
variety of interpretations of the current data are:

  1. 1.

$Y(4220)$, $Y(4260)$, $Y(4360)$, $Y(4390)$ masses are as given in the PDG
(Table 1). No constraint is placed upon $P_{s_{{Q\bar{Q}}}=0}^{Y(4220)}$ or
$P_{s_{{Q\bar{Q}}}=0}^{Y(4390)}$.

  2. 2.

$Y(4220)$, $Y(4260)$, $Y(4360)$, $Y(4390)$ masses are as given in the .
$P_{s_{{Q\bar{Q}}}=0}^{Y(4220)}$ is fit to $f\\!=\\!\frac{1}{3}$ with
$\epsilon\\!=\\!0.1$, and $P_{s_{{Q\bar{Q}}}=0}^{Y(4390)}$ is unconstrained.

  3. 3.

$Y(4220)$, $Y(4360)$, and $Y(4390)$ masses are as given in the PDG, while
$m_{Y(4260)}\\!=\\!4251\\!\pm\\!6$ MeV, which is the weighted average of the 3
PDG values not including the low BESIII value Ablikim  _et al._ (2017a).
$P_{s_{{Q\bar{Q}}}=0}^{Y(4220)}$ is fit to $f\\!=\\!\frac{1}{3}$ with
$\epsilon\\!=\\!0.2$, and $P_{s_{{Q\bar{Q}}}=0}^{Y(4390)}$ is fit to
$f\\!=\\!\frac{2}{3}$ with $\epsilon\\!=\\!0.05$.

  4. 4.

$Y(4360)$, $Y(4390)$, and $Y(4660)$ masses are as given in the PDG, but
$Y(4260)$ is assumed not to exist, and $m_{Y(4220)}\\!=\\!4220.1\\!\pm 2.9$
MeV is the weighted average of the PDG values combined with the newer BESIII
measurements Ablikim  _et al._ (2019b, c). $P_{s_{{Q\bar{Q}}}=0}$ values are
as given in Case 3.

  5. 5.

$m_{Y(4220)}$ is as given in Case 4; $m_{Y(4260)}$ is as given in Case 3;
$m_{\text{``$Y(4320)$''}}\\!=\\!4320\\!\pm\\!13$ MeV is the lower BESIII
$Y(4360)$ mass measurement from Ablikim  _et al._ (2017a);
$m_{Y(4390)}\\!=\\!4386\\!\pm\\!4$ MeV is the weighted average of the PDG
value and the upper BESIII $Y(4360)$ mass measurement from Ablikim  _et al._
(2017b). $P_{s_{{Q\bar{Q}}}=0}$ values are as given in Case 3.

We previously suggested the importance of heavy-quark spin-symmetry
($s_{Q\bar{Q}}$) conservation in the decays of exotics, particularly for
$Z_{c}(3900)$ and $Z_{c}(4020)$, but also for several other exotic candidates
that to date have only been observed to decay to charmonium states carrying
one specific value of $s_{Q\bar{Q}}$ (e.g., to $\psi$ or to $h_{c}$). We
assume that a state like $Y(4220)$ is able to decay to channels with either
value of $s_{Q\bar{Q}}$ due to the initial state being a mixture of
$s_{Q\bar{Q}}$ eigenstates, rather than to the value of $s_{Q\bar{Q}}$
changing in the decay process through a heavy-quark spin-symmetry violation.
In addition, in this analysis we take the well-known radiative transition
selection rules to apply to the light degrees of freedom, which carry the
total angular momentum $J_{q\bar{q}}$ defined in Eq. (9). As usual, the
operators defining E1 and M1 transitions transform as $J^{P}\\!=\\!1^{-}$ and
$J^{P}\\!=\\!1^{+}$, respectively.

Explicit expressions for radiative transitions between quarkonium states
(themselves transcribed from textbook atomic-physics formulae) appear in the
literature (e.g., Ref. Barnes  _et al._ (2005)), and may readily be adapted to
the present case. In particular, the quarkonium orbital angular momentum $L$
is replaced with $J_{q\bar{q}}$, and the heavy quark mass $m_{Q}$ is replaced
with the diquark mass $m_{\delta}^{\vphantom{1}}$. For E1 partial widths, one
has

$\Gamma_{\rm
E1}\left(n^{\,2s_{Q\bar{Q}}+1}\\!\left(J^{\vphantom{\prime}}_{{q\bar{q}}}\right)^{\vphantom{\dagger}}_{J}\rightarrow
n^{\prime\;2s_{Q\bar{Q}}^{\prime}+1}\\!\left(J^{\prime}_{{q\bar{q}}}\right)^{\vphantom{\dagger}}_{J^{\prime}}+\gamma\right)=\frac{4}{3}C_{fi}\,\delta_{s^{\vphantom{\prime}}_{Q\bar{Q}}s_{Q\bar{Q}}^{\prime}}\alpha\,Q_{\delta^{\vphantom{1}}}^{2}\left|\left\langle\psi_{f}\middle|r\middle|\psi_{i}\right\rangle\right|^{2}E_{\gamma}^{3}\frac{E_{f}^{({Q\bar{Q}}q\bar{q}^{\prime})}}{M_{i}^{({Q\bar{Q}}q\bar{q}^{\prime})}},$
(14)

where

$C_{fi}\equiv\mathrm{max}\left(J_{q\bar{q}}^{\vphantom{\prime}},J^{\prime}_{q\bar{q}}\right)\left(2J^{\prime}+1\right)\begin{Bmatrix}J^{\prime}_{q\bar{q}}&J^{\prime}&s_{Q\bar{Q}}\\\
J&J_{q\bar{q}}&1\end{Bmatrix}^{2}.$ (15)

The labels $i$ and $f$ refer to initial and final states, respectively. The
initial exotic state ${Q\bar{Q}}q\bar{q}^{\prime}$, of mass
$M_{i}^{({Q\bar{Q}}q\bar{q}^{\prime})}$, decays in its rest frame into a final
exotic state with the same flavor content and energy
$E_{f}^{({Q\bar{Q}}q\bar{q}^{\prime})}$, and a photon of energy $E_{\gamma}$.
$\alpha$ is the fine-structure constant. $\psi$ denotes radial wave functions
of the exotic hadrons, and $r$ is the spatial separation between the
$\delta^{\vphantom{1}}$-${\bar{\delta}}$ pair centers.
$Q_{\delta}^{\vphantom{1}}$ is the total electric charge (in units of proton
charge) to which the photon couples; in Ref. Chen  _et al._ (2015), the
diquarks are treated as pointlike, in which case one simply takes
$Q_{\delta}^{\vphantom{1}}\\!=\\!Q_{Q}\\!+\\!Q_{q}$. Alternately, one may
argue that the diquarks $\delta^{\vphantom{1}}$ are of sufficient spatial
extent that the photon couplings to the distinct quarks in
$\delta^{\vphantom{1}}$ should add through incoherent diagrams, in which case
one takes $Q_{\delta^{\vphantom{1}}}^{2}\\!=\\!Q_{Q}^{2}\\!+\\!Q_{q}^{2}$. In
our calculation we use the first option, but note in addition that a $Y$
state, being an isosinglet, contains an equal superposition of $u$ and $d$
quarks. We thus take

$Q_{\delta^{\vphantom{1}}}^{2}\to\frac{1}{2}\left[\left(Q_{c}+Q_{u}\right)^{2}+\left(Q_{c}+Q_{d}\right)^{2}\right]=\frac{17}{18}\,.$
(16)

Other schemes give rise to coefficients that differ from this value only at
$O(1)$. Corrections that arise from treating the distinct quarks within each
diquark as separated entities, for example through electromagnetic form
factors of the $\delta^{\vphantom{1}},{\bar{\delta}}$ composite
quasiparticles, would be incorporated in this model through the factor
$Q_{\delta^{\vphantom{1}}}^{2}$.

The corresponding expression for M1 partial widths, involving no change in
parity but a flip of the heavy-quark spin $s_{{Q\bar{Q}}}$ (hence breaking
heavy-quark spin symmetry), reads

$\Gamma_{\rm
M1}\left(n^{\,2s_{Q\bar{Q}}+1}\left(J^{\vphantom{\prime}}_{{q\bar{q}}}\right)^{\vphantom{\dagger}}_{J}\rightarrow
n^{\prime\;2s_{Q\bar{Q}}^{\prime}+1}\left(J^{\prime}_{{q\bar{q}}}\right)^{\vphantom{\prime}}_{J^{\prime}}+\gamma\right)=\frac{4}{3}\frac{2J^{\prime}+1}{2J_{q\bar{q}}+1}\delta_{J^{\vphantom{\prime}}_{q\bar{q}}J^{\prime}_{q\bar{q}}}\delta_{s^{\vphantom{\prime}}_{Q\bar{Q}},\,s_{Q\bar{Q}}^{\prime}\pm
1}\,Q_{\delta^{\vphantom{1}}}^{2}\frac{\alpha}{m_{\delta^{\vphantom{1}}}^{2}}\left|\left\langle\psi_{f}\middle|\psi_{i}\right\rangle\right|^{2}E_{\gamma}^{3}\frac{E_{f}^{({Q\bar{Q}}q\bar{q}^{\prime})}}{M_{i}^{({Q\bar{Q}}q\bar{q}^{\prime})}}.$
(17)

This expression is presented here for completeness, in light of the current
lack of experimental evidence for such transitions. However, in Sec. IV we use
it to calculate the expected radiative width for the yet-unobserved Ablikim
_et al._ (2021) transition $Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$.

As noted above, Eqs. (14) and (17) are almost identical to textbook
nonrelativistic results. The only exception in each case is the inclusion of a
factor $E_{f}/M_{i}$ to represent relativistic phase space associated with
recoil of the final-state hadron. In fact, Ref. Brambilla  _et al._ (2004)
discusses several distinct relativistic corrections that could be included in
a more complete study. Since this work represents the first attempt to
calculate the radiative widths for a spectrum of states whose experimental
interpretation remains ambiguous, we include only a minimal set of physical
effects in the analysis.

Lastly, corrections to the long-wavelength approximation discussed in Sec. III
that are derived by retaining the full photon plane-wave factor
$\exp(i\bm{k}\cdot\bm{r})$ have also been computed (e.g., Ref. Brambilla  _et
al._ (2004)). Explicitly, Eqs. (14) and (17) are modified through the
substitutions

$\langle\psi_{f}|r|\psi_{i}\rangle\to\left<\psi_{f}\\!\left|\frac{3}{k}\left[\frac{kr}{2}j_{0}\left(\frac{kr}{2}\right)-j_{1}\left(\frac{kr}{2}\right)\right]\right|\psi_{i}\right>\,,$
(18)

and

$\langle\psi_{f}|\psi_{i}\rangle\to\left<\psi_{f}\\!\left|\,j_{0}\left(\frac{kr}{2}\right)\right|\psi_{i}\right>\,,$
(19)

respectively,where $j_{0}$ and $j_{1}$ are spherical Bessel functions. The
corresponding series expansions of these functions read

$r-\frac{1}{20}k^{2}r^{3}+O(k^{4}r^{5})\,,$ (20)

and

$1-\frac{1}{24}k^{2}r^{2}+O(k^{4}r^{4})\,.$ (21)

Note especially the small subleading-term numerical coefficient in each case,
suggesting that the long-wavelength approximation holds relatively well even
for substantial values of $kr$. We examine specific examples in Sec. IV.

##  IV Analysis and Results

Possible assignments of observed $Y$ states to members of the
$\Sigma^{+}_{g}(1P)$ multiplet in this model are described by the 5 cases
discussed extensively in Ref. Giron and Lebed (2020a) and summarized in Sec.
III. Of these cases, all have excellent goodness-of-fit values $\chi^{2}_{\rm
min}/{\rm d.o.f.}$ except Case 3; however, we argue this case and Case 5 to be
the most phenomenologically relevant ones, since they enforce the important
physical constraint that both $Y(4220)$ and $Y(4390)$ are observed (see Table
1) to have substantial couplings to $h_{c}$ ($s_{{Q\bar{Q}}}\\!=\\!0$). Since
$\Sigma^{+}_{g}(1S)$ contains only one $I\\!=\\!0$, $J^{PC}\\!=\\!1^{--}$
state with $s_{{Q\bar{Q}}}\\!=\\!0$, the requirement of providing a
substantial component of this state to both of the well-separated $Y(4220)$
and $Y(4390)$ mass eigenstates is one of the primary obstacles to achieving a
good fit.

Case 5 relieves the tension of Case 3 by identifying, as discussed in Sec. II,
a new state “$Y(4320)$” from the data of Ref. Ablikim  _et al._ (2017a). In
addition, Cases 1, 2, 3, and 5 all predict the sole $I\\!=\\!1$,
$J^{PC}\\!=\\!0^{--}$ state in $\Sigma^{+}_{g}(1P)$ to lie in the range
4220–4235 MeV, which agrees well with the unconfirmed state $Z_{c}(4240)$
carrying these quantum numbers that is observed in the LHCb paper Aaij  _et
al._ (2014b) confirming the existence of $Z_{c}(4430)$.

Case 4 also satisfies the $Y(4220)/Y(4390)$ $s_{{Q\bar{Q}}}\\!=\\!0$
criterion, but additionally assigns the rather high-mass $Y(4660)$ to the
$\Sigma^{+}_{g}(1P)$ multiplet; the cost is a much higher prediction
($\approx\\!4440$ MeV) for the mass of the $\Sigma^{+}_{g}(1P)$ $I\\!=\\!1$,
$J^{PC}\\!=\\!0^{--}$ state, in conflict with the value of $m_{Z_{c}(4240)}$.

We therefore single out the fits of Cases 3 and 5 for the decomposition of $Y$
states with respect to the total light-quark angular momentum $J_{{q\bar{q}}}$
in Tables 2 and 3, respectively. For completeness, we also provide the
corresponding information for Cases 1, 2, and 4 in Table 4.

Table 2: Decomposition of $Y$ ($I\\!=\\!0$, $J^{PC}\\!=\\!1^{--}$) charmoniumlike exotic candidates into a basis of good light-quark spin $s_{{q\bar{q}}}$, heavy-quark spin $s_{{Q\bar{Q}}}$, and total light-quark angular momentum $J_{{q\bar{q}}}$, performed for the 4 experimentally observed candidate states as described in Case 3 above and in Ref. Giron and Lebed (2020a). A minus sign on the probability $(-|P|)$ means that the corresponding amplitude is $-|P|^{1/2}$, the same convention as is used for Clebsch-Gordan coefficients by the PDG Zyla  _et al._ (2020). Particle | $s_{{q\bar{q}}}$ | $s_{{c\bar{c}}}$ |  $J_{{q\bar{q}}}$ | Probability
---|---|---|---|---
$Y(4220)$ | $0$ | $0$ | $0$ | $+0.231$
| $1$ | $1$ | $0$ | $+0.012$
|  |  | $1$ | $-0.577$
|  |  | $2$ | $+0.181$
$Y(4260)$ | $0$ | $0$ | $0$ | $+0.061$
| $1$ | $1$ | $0$ | $+0.004$
|  |  | $1$ | $+0.352$
|  |  | $2$ | $+0.583$
$Y(4360)$ | $0$ | $0$ | $0$ | $+0.069$
| $1$ | $1$ | $0$ | $+0.835$
|  |  | $1$ | $+0.020$
|  |  | $2$ | $-0.075$
$Y(4390)$ | $0$ | $0$ | $0$ | $+0.638$
| $1$ | $1$ | $0$ | $-0.149$
|  |  | $1$ | $+0.051$
|  |  | $2$ | $-0.161$
Table 3: Decomposition of $Y$ ($I\\!=\\!0$, $J^{PC}\\!=\\!1^{--}$) charmoniumlike exotic candidates as in Table 2, except now performed for the 4 experimentally observed candidate states as described in Case 5 above and in Ref. Giron and Lebed (2020a). Particle | $s_{{q\bar{q}}}$ | $s_{{c\bar{c}}}$ |  $J_{{q\bar{q}}}$ | Probability
---|---|---|---|---
$Y(4220)$ | $0$ | $0$ | $0$ | $-0.264$
| $1$ | $1$ | $0$ | $-0.007$
|  |  | $1$ | $+0.543$
|  |  | $2$ | $-0.186$
$Y(4260)$ | $0$ | $0$ | $0$ | $+0.060$
| $1$ | $1$ | $0$ | $+0.036$
|  |  | $1$ | $+0.380$
|  |  | $2$ | $+0.523$
“$Y(4320)$” | $0$ | $0$ | $0$ | $+0.025$
| $1$ | $1$ | $0$ | $+0.870$
|  |  | $1$ | $+8\times 10^{-4}$
|  |  | $2$ | $-0.105$
$Y(4390)$ | $0$ | $0$ | $0$ | $-0.651$
| $1$ | $1$ | $0$ | $+0.086$
|  |  | $1$ | $-0.076$
|  |  | $2$ | $+0.187$
Table 4: Decomposition of $Y$ ($I\\!=\\!0$, $J^{PC}\\!=\\!1^{--}$) charmoniumlike exotic candidates as in Tables 2–3, except now performed for the 4 experimentally observed candidate states as described in Cases 1, 2, and 4 above and in Ref. Giron and Lebed (2020a). Case 1 | Case 2 | Case 4
---|---|---
Particle | $s_{{q\bar{q}}}$ | $s_{{c\bar{c}}}$ |  $J_{{q\bar{q}}}$ | Probability | Particle | $s_{{q\bar{q}}}$ | $s_{{c\bar{c}}}$ |  $J_{{q\bar{q}}}$ | Probability | Particle | $s_{{q\bar{q}}}$ | $s_{{c\bar{c}}}$ |  $J_{{q\bar{q}}}$ | Probability
$Y(4220)$ | $0$ | $0$ | $0$ | $+0.771$ | $Y(4220)$ | $0$ | $0$ | $0$ | $-0.336$ | $Y(4220)$ | $0$ | $0$ | $0$ | $-0.233$
| $1$ | $1$ | $0$ | $-0.019$ |  | $1$ | $1$ | $0$ | $+0.032$ |  | $1$ | $1$ | $0$ | $-0.048$
|  |  | $1$ | $-0.211$ |  |  |  | $1$ | $+0.631$ |  |  |  | $1$ | $+0.376$
|  |  | $2$ | $-4\times 10^{-7}$ |  |  |  | $2$ | $+8\times 10^{-4}$ |  |  |  | $2$ | $-0.343$
$Y(4260)$ | $0$ | $0$ | $0$ | $+0.212$ | $Y(4260)$ | $0$ | $0$ | $0$ | $+0.588$ | $Y(4360)$ | $0$ | $0$ | $0$ | $-0.119$
| $1$ | $1$ | $0$ | $+0.130$ |  | $1$ | $1$ | $0$ | $+0.056$ |  | $1$ | $1$ | $0$ | $+0.101$
|  |  | $1$ | $+0.597$ |  |  |  | $1$ | $+0.246$ |  |  |  | $1$ | $-0.473$
|  |  | $2$ | $+0.062$ |  |  |  | $2$ | $+0.109$ |  |  |  | $2$ | $-0.308$
$Y(4360)$ | $0$ | $0$ | $0$ | $+0.006$ | $Y(4360)$ | $0$ | $0$ | $0$ | $-0.046$ | $Y(4390)$ | $0$ | $0$ | $0$ | $+0.647$
| $1$ | $1$ | $0$ | $+0.252$ |  | $1$ | $1$ | $0$ | $-0.117$ |  | $1$ | $1$ | $0$ | $+0.003$
|  |  | $1$ | $-5\times 10^{-6}$ |  |  |  | $1$ | $-0.012$ |  |  |  | $1$ | $+0.009$
|  |  | $2$ | $-0.742$ |  |  |  | $2$ | $+0.824$ |  |  |  | $2$ | $-0.342$
$Y(4390)$ | $0$ | $0$ | $0$ | $-0.012$ | $Y(4390)$ | $0$ | $0$ | $0$ | $-0.029$ | $Y(4660)$ | $0$ | $0$ | $0$ | $-0.002$
| $1$ | $1$ | $0$ | $+0.599$ |  | $1$ | $1$ | $0$ | $+0.795$ |  | $1$ | $1$ | $0$ | $+0.848$
|  |  | $1$ | $-0.193$ |  |  |  | $1$ | $-0.111$ |  |  |  | $1$ | $+0.143$
|  |  | $2$ | $+0.196$ |  |  |  | $2$ | $+0.066$ |  |  |  | $2$ | $+0.007$

Using the mass eigenvalues for the $Y$ states in Table 1, the state
decompositions according to $J_{{q\bar{q}}}$ in Tables 2, 3, and 4, the
coefficient factors in Eq. (15), and the effective squared-charge
$Q_{\delta}^{2}$ from Eq. (16), one may calculate the E1 radiative partial
decay widths for $\Sigma^{+}_{g}(1P)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$
transitions from Eq. (14). The only nontrivial new input to the calculation is
that of the transition matrix element $\langle\psi_{f}|r|\psi_{i}\rangle$.
Using the numerical methods for solving Schrödinger equations developed in
Ref. Giron  _et al._ (2019), and particularly the fits performed in Ref. Giron
and Lebed (2020b) to obtain the fine structure of the $\Sigma^{+}_{g}(1S)$
multiplet, the optimal diquark mass is found to be

$m_{\delta^{\vphantom{1}}}=m_{{\bar{\delta}}}=1.933\pm 0.005\,{\rm GeV}\,,$
(22)

as one varies over the static gluonic-field potentials $\Sigma^{+}_{g}$
obtained in the lattice calculations of Refs. Juge  _et al._ (1998, 1999,
2003); Mor ; Capitani  _et al._ (2019). We then compute the relevant matrix
element to be

$\left<\psi_{f}(1S)\left|r\right|\psi_{i}(1P)\right>=0.402\pm 0.001\,{\rm
fm}\,.$ (23)

Note in particular that this numerical input appears in all
$\Sigma^{+}_{g}(1P)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$ transitions, not simply
those of $Y\\!\to\\!\gamma X(3872)$ that are compiled according to the 5 cases
in Table 5. Moreover, Table 5 and additional calculated width values presented
subsequently in this work exhibit only central values for $\Gamma$; the small
uncertainties in Eqs. (22) and (23) only refer to variation over different
lattice simulations, and do not take into account other much more significant
potential sources of uncertainty, such as effects due to finite diquark size.
Nevertheless, such effects were shown Giron and Lebed (2020b) to change
expectation values like $\langle r\rangle$ no more than 10%, a value that we
adopt as a benchmark uncertainty for all $\Gamma$ values computed here.

Also noteworthy is the magnitude of
$k\left<\psi_{f}(1S)\\!\left|r\right|\\!\psi_{i}(1P)\right>$ for each case,
which provides an indication of the reliability of the long-wavelength
approximation. Indeed, for $Y(4220)$, $k\\!=\\!334$ MeV, and using Eq. (23)
gives $kr\\!\to\\!0.680$, while the corresponding value for $Y(4660)$
($k\\!=\\!699$ MeV) is 1.424. However, the same simulations as in Eq. (23)
also produce

$\left<\psi_{f}(1S)\left|r^{3}\right|\psi_{i}(1P)\right>=0.135\pm 0.001\,{\rm
fm}^{3}\,,$ (24)

from which one computes the relative magnitude of the first correction term in
Eq. (20) to be only 0.033 for $Y(4220)$ and, surprisingly, only 0.300 for
$Y(4660)$.

Table 5: Radiative E1 partial widths (in keV) to $\gamma X(3872)$ calculated using Eq. (14), for the 5 cases of possible $Y$ state assignments defined above and in Ref. Giron and Lebed (2020a). For each case, note that two of the $Y$ states (indicated by dashes) are assumed either not to exist or not to belong to the $\Sigma^{+}_{g}(1P)$ multiplet. Case | $Y(4220)$ | $Y(4260)$ | “$Y(4320)$” | $Y(4360)$ | $Y(4390)$ | $Y(4660)$
---|---|---|---|---|---|---
$1$ | $30.4$ | $145.6$ | — | $721.3$ | $981.8$ | —
$2$ | $81.1$ | $80.6$ | — | $616.0$ | $1127.0$ | —
$3$ | $105.1$ | $211.2$ | — | $1016.1$ | $319.2$ | —
$4$ | $136.0$ | — | — | $432.1$ | $231.6$ | $3363.9$
$5$ | $102.4$ | $216.2$ | $807.6$ | — | $253.8$ | —

One observes from Table 5 that the widths $\Gamma_{Y(4220)\to\gamma X(3872)}$
and $\Gamma_{Y(4260)\to\gamma X(3872)}$ assume almost the same values in Cases
3 and 5 (102–105 keV and 211–216 keV, respectively). $\Gamma_{Y(4390)\to\gamma
X(3872)}$ also exhibits fairly modest variation, from 254–319 keV. Indeed,
some of the large radiative width values in Table 5, such as 3.4 MeV for
$Y(4660)\\!\to\\!\gamma X(3872)$ in Case 4, can serve as vital criteria for
eliminating possible assignments of $Y$ states to the $1P$ multiplet: Glancing
at the measured total $\Gamma_{Y(4660)}$ in Table 1, one sees that were
$Y(4660)$ truly a $1P$ state, then its large phase space for radiative decay
to $X(3872)$ [evident from the $E_{\gamma}^{3}$ factor of Eq. (14)] would
generate a radiative branching fraction of at least several percent.

The transition matrix element of Eq. (23) has already been noted to apply to
all $\Sigma^{+}_{g}(1P)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$ transitions. The
only observed hidden-charm tetraquark candidates with $P\\!=\\!-$ apart from
the $Y$ states are $Z_{c}(4240)$ and $Y(4626)$; the latter has thus far been
observed to decay only to various $D_{s}$ meson pairs Jia  _et al._ (2019,
2020), and therefore is very likely a $c\bar{c}s\bar{s}$ state Giron and Lebed
(2020b). As for $Z_{c}(4240)$, only its charged isobar has yet been observed,
but assuming the existence of a degenerate $Z_{c}(4240)^{0}$, one may input
its quantum numbers $s_{{Q\bar{Q}}}\\!=\\!1$, $J_{{q\bar{q}}}\\!=\\!1$,
$J\\!=\\!0$ Giron and Lebed (2020a) into Eq. (14) to obtain

$\Gamma\left[Z_{c}(4240)^{0}\to\gamma X(3872)\right]=503\ {\rm keV}\,.$ (25)

Lastly, we noted with Eq. (17) that M1 transitions occur only with a flip of
the heavy-quark spin. Such is the case for the
$\Sigma^{+}_{g}(1S)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$ transition
$Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$ ($s_{{Q\bar{Q}}}\\!=\\!0\\!\to
s_{{Q\bar{Q}}}\\!=\\!1$). Using Eq. (22), we calculate

$\Gamma\left[Z_{c}(4020)^{0}\to\gamma X(3872)\right]=7.91\ {\rm keV}\,,$ (26)

noting from Eq. (17) that the underlying matrix element
$\langle\psi_{f}|\psi_{i}\rangle\\!=\\!1$ since both states share the same
radial wave function. In comparison, the molecular model, in which $X(3872)$
and $Z_{c}(4020)$ are $D^{0}\bar{D}^{*0}\\!+\\!{\bar{D}}^{0}D^{*0}$ and
$D^{*}\bar{D}^{*}$ bound states, respectively, and the decay
$Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$ proceeds via $D^{*0}\\!\to\\!\gamma
D^{0}$, produces a rather larger radiative width: The calculation of Ref.
Voloshin (2019) predicts a branching fraction of about $5\\!\times\\!10^{-3}$,
which for $\Gamma_{Z_{c}(4020)}\\!=\\!13\\!\pm\\!5$ MeV Zyla  _et al._ (2020)
amounts to at least 40 keV.

In light of our investigations for
$\Sigma^{+}_{g}(1P)\\!\to\\!\Sigma^{+}_{g}(1S)$ E1 transitions, the long-
wavelength approximation for M1 transitions within the single multiplet
$\Sigma^{+}_{g}(1S)$ is undoubtedly satisfactory [for example, in
$Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$, $k$ is only 150 MeV]. Indeed, one
may press the approximation of Eq. (21) to consider a transition that is
forbidden in the long-wavelength limit of Eq. (17) due to the orthogonality of
wave functions, such as $\Sigma^{+}_{g}(2S)\\!\to\\!\gamma\Sigma^{+}_{g}(1S)$.
Assuming that $Z_{c}^{0}(4430)$ is the $2S$ partner to $Z_{c}^{0}(4020)$, then
$Z_{c}^{0}(4430)\\!\to\\!\gamma X(3872)$ has $k\\!\approx\\!565$ MeV, while we
compute

$\left<\psi_{f}(1S)\left|r^{2}\right|\psi_{i}(2S)\right>=0.152\pm 0.001\,{\rm
fm}^{2}\,,$ (27)

and the first nontrivial term of Eq. (21) evaluates to $-0.052$. Using this
correction in Eq. (17) leads to a radiative width
$\Gamma_{\gamma}\\!\simeq\\!1$ keV, to be compared with
$\Gamma_{Z_{c}^{+}(4430)}\\!\approx\\!180$ MeV Zyla  _et al._ (2020). The
observation a radiative transition with such a small branching fraction is not
impossible, but likely will not occur in the near future.

##  V Conclusions

In this paper we have calculated exotic-to-exotic hadronic radiative
transitions using the dynamical diquark model. The most phenomenologically
relevant final state is $X(3872)$, which is a member of the model’s hidden-
charm ground-state multiplet $\Sigma^{+}_{g}(1S)$. We use the results from a
recent study Giron and Lebed (2020a) of this model for the lowest $P$-wave
multiplet [$\Sigma^{+}_{g}(1P)$] of hidden-charm tetraquark states, in which
the $\Sigma^{+}_{g}(1P)$ states are identified with the observed $I\\!=\\!0$,
$J^{PC}\\!=\\!1^{--}$ ($Y$) states according to a variety of scenarios, based
upon both their mass spectra and preferred decay modes to eigenstates of
heavy-quark spin (e.g., $J/\psi$ vs. $\\!h_{c}$). We calculate E1 and M1
transition amplitudes for
$\Sigma_{g}^{+}(1P)\\!\to\\!\gamma\Sigma_{g}^{+}(1S)$ and
$\Sigma_{g}^{+}(1S)\\!\to\\!\gamma\Sigma_{g}^{+}(1S)$ processes, respectively,
and present corresponding values for the radiative decay widths of a number of
particular exclusive channels.

This analysis shows that if $Y(4220)$ and $X(3872)$ have a similar underlying
diquark structure, then one expects $\Gamma_{Y(4220)\to\gamma
X(3872)}\\!\approx\\!100$ keV. Moreover, similar values (albeit somewhat
larger due to increased $\gamma$ phase space) are expected for the heavier $Y$
states in $\Sigma^{+}_{g}(1P)$. The extreme possibility of $Y(4660)$ belonging
to the $1P$ multiplet would lead to a $\gamma X(3872)$ branching fraction of
several percent, and so the absence of such a remarkably large signal would
appear to relegate $Y(4660)$ instead to the $\Sigma^{+}_{g}(2P)$ multiplet.

Furthermore, we found that the observed but unconfirmed $Z_{c}(4240)$, a
candidate for the sole $I\\!=\\!1$, $J^{PC}=0^{--}$ state in
$\Sigma^{+}_{g}(1P)$, should have a substantial ($\approx\\!500$ keV)
radiative decay width to $X(3872)$ through its neutral isobar, and therefore
this decay is a good candidate for future experimental investigation. Indeed,
many of the $\Sigma^{+}_{g}(1P)$ states have not yet been observed, offering
multiple potential future tests of the model.

M1 transitions within a single multiplet, such as
$Z_{c}(4020)^{0}\\!\to\\!\gamma X(3872)$, produce much narrower widths
($<\\!10$ keV in this model), and can provide sensitive tests of substructure
(e.g., diquarks vs. meson molecules).

One may also study exotic-to-exotic radiative transitions in other heavy-quark
sectors (e.g., hidden-bottom or $c\bar{c}s\bar{s}$ exotics). Indeed, Eqs. (14)
and (17) are general for any tetraquark state in the diquark-antidiquark
configuration. For example, Ref. Giron and Lebed (2020b) calculates the mass
of $X_{b}$ [the hidden-bottom analogue to $X(3872)$] to lie in a rather narrow
range $m_{X_{b}}\in[10598,10607]$ MeV, only slightly below the observed
$Z_{b}(10610)^{0}$. The M1 transition $Z_{b}(10610)^{0}\\!\to\\!\gamma X_{b}$
is thus expected from Eq. (17) to produce a tiny [$O({\rm eV})$ or less]
radiative width, owing to not only the small phase space, but also the larger
($b$-containing) diquark mass. We conclude that even very coarse experimental
results in other sectors can be decisive in verifying or falsifying particular
models.

###### Acknowledgements.

This work was supported by the National Science Foundation (NSF) under Grant
No. PHY-1803912.

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