# Quantized mass-energy effects in an Unruh-DeWitt detector

Carolyn E. Wood  Magdalena Zych  Australian Research Council Centre for
Engineered Quantum Systems, School of Mathematics and Physics, The University
of Queensland, St Lucia, QLD 4072, Australia

###### Abstract

A simple but powerful particle detector model consists of a two-level system
coupled to a field, where the detected particles are the field excitations.
This is known as the Unruh-DeWitt detector. Research using this model has
often focused on either a completely classical description of the external
degrees of freedom of the detector, or a full field-theoretic treatment, where
the detector itself is described as a field. Recently there has been much
interest in quantum aspects of the detector’s center of mass—either described
as moving in superposition along semiclassical trajectories, or dynamically
evolving under a nonrelativistic Hamiltonian. However, the processes of
interest—the absorption or emission of field particles—necessarily change the
detector’s rest mass by the amount of energy of the absorbed or emitted field
quanta. Neither of the above models can capture such effects. Here we
incorporate the quantization of the detector’s mass-energy into the Unruh-
DeWitt model. We show that internal energy changes due to emission or
absorption are relevant even in the lowest energy limit. Specifically,
corrections to transition rates due to the detector’s mass changing cannot be
ignored unless the entire center of mass dynamics is also ignored. Our results
imply that one cannot have a consistent model of the Unruh-DeWitt detector as
a massive particle without including the mass-energy equivalence.

##  I Introduction

The mass-energy equivalence of special relativity implies a universal coupling
between the internal degrees of freedom (DoFs) of a composite particle, which
describe its total internal energy, and the particle’s center of mass (CoM)
[1]. The resulting simple model of a particle with quantized mass-energy
opened new avenues for exploring phenomena at the quantum-and-gravity
interface, including gravitational decoherence and time dilation [2, 3, 4, 5].
The coupling leads to relativistic effects even at low CoM energies, thus also
opening new avenues for laboratory experiments [6, 7]. The coupling also has
consequences for the free propagation of composite particles [8] and their
boosts [9], and ultimately leads to the requirement of a class of states which
describe semiclassical propagation of composite particles—which are different
to the usually expected pure Gaussian state of the particle’s CoM [10].

Importantly, the universality of the coupling also leads to signatures of the
mass-energy equivalence emerging where it would not have been expected. For
example, failure to incorporate the mass-energy equivalence in the spontaneous
emission process leads to an “anomalous friction” effect [11, 12]. Mass-energy
equivalence resolves this problem by allowing the particle’s momentum to
change during emission without changing its velocity.

The above naturally implies that the mass of a particle interacting with a
field must be altered by the outgoing or incoming energy in any emission or
absorption process. This in turn implies that the count made by a physical
detector, of particles in some field state (when the detector is modeled as a
physical system with its own dynamics, including the CoM), may also be
affected if the mass-energy equivalence is not incorporated. The simplest
model of such a detector, the Unruh-DeWitt (UDW) detector, is used widely to
probe quantum fields in curved spacetime and to explore applications to
relativistic quantum information.

Traditionally, the UDW detector model consists of a pointlike particle with
some internal degrees of freedom interacting with a quantum field along a
classically prescribed worldline [13, 14]. However, it was originally
formulated through quantum field theory, beginning with the seminal
explorations of the Unruh effect and Hawking radiation [15, 16, 17]. See ref.
[18] for a review of applications of the Unruh effect, including particle
detectors. The field-theoretic treatment continues to be explored to gain
insights into vacuum entanglement, e.g. [19], and recently in the context of
quantum reference frames (associated with the detector) [20]. Nonetheless,
many recent works introduce additional features to both the detector and the
field as needed to apply the model to more diverse or more realistic
situations. These range from explorations of light-matter interactions [21]
and the Casimir-Polder effect [22] to axion detection [23] and neutrino
oscillations [24].

Additionally, a formalism has been developed to describe an UDW detector
following a superposition of different classical paths [25, 26, 27, 28]. This
has further allowed the exploration of quantum causal structures through
superpositions of detector-field interaction times [29], as well as through
describing the dynamics of the UDW detector in genuine quantum spacetimes [30,
31].

Recently there has been interest in a yet different regime, motivated by
operational considerations: An UDW detector described as a first-quantized
particle with internal states coupled to a field, with the addition of
external dynamics—i.e. quantized CoM degrees of freedom. This has been
explored, for example, to study the effects of the CoM motion [32] and
coherent spreading of the CoM [33], and to study the recoil of the detector’s
CoM due to interactions with field quanta [34]. An understanding of the
dynamics of the UDW detector’s CoM is also relevant for models of atom-light
interactions. However, thus far the models either considered a nonrelativistic
CoM of the detector, thus not including the mass-energy equivalence (a fact
also noted in Ref. [35]), or looked at formal expressions for transition
amplitudes for a relativistic CoM.

In this work, we introduce relativistic corrections to the UDW detector model
which account for the mass-energy equivalence—whereby the mass is different
for each of the UDW detector’s ground and excited states. As the detector
changes mass only at the moment of excitation (or emission) one might expect
no differences, compared to earlier models, for transitions between initial
internal eigenstates. Additionally, as the resulting effects are relativistic
corrections one might further expect that there is a sufficiently low energy
regime where the mass-energy equivalence effects can be neglected. We show
that neither of these intuitions hold. Incorporating quantized mass-energy
into the UDW model is necessary for a consistent description as it leads to
corrections to the detector’s emission and absorption rates that are on the
same order as the effects of even the nonrelativistic CoM dynamics.

The article is structured as follows: We present our generalization of the UDW
detector dynamics and derive the emission and absorption amplitudes for our
model in Sec. II. We then find the emission and absorption rates in Secs.
III.1 and III.2, and discuss the relevance of the relativistic corrections. We
also discuss an interesting further subtlety related to the interpretation of
the nonrelativistic notion of mass in light of the role of mass in the
relativistic model. We compare the rates obtained in this work to those
arising from a nonrelativistic CoM model [33] in Sec. III.3. Finally, we
discuss implications of our results in Sec. IV. Throughout this work we set
$\hbar=1$ but keep the speed of light $c$ in the expressions for ease of
discussing relativistic vs nonrelativistic effects.

##  II Unruh-DeWitt detector with external dynamics and mass-energy
equivalence

The UDW detector is a model of a pointlike particle coupling linearly through
the monopole operator $\vec{\mu}$ to a scalar field along the detector’s
predefined, classical worldline $\vec{x}(\tau)$. The associated Schrödinger
picture interaction Hamiltonian has the form

$\hat{H}_{\mathrm{int}}=\lambda\hat{\mu}\hat{\phi}(\vec{x}),$ (1)

where $\lambda$ is the coupling strength, $\hat{\mu}$ the monopole moment
operator, and $\hat{\phi}(\vec{x})$ the scalar field evaluated at the point
$\vec{x}$ which represents the classical position of the detector.

In the case of a detector with two internal energy states, the monopole
operator takes the form

$\hat{\mu}=\ket{e}\bra{g}+\ket{g}\bra{e},$ (2)

where $\ket{g}$ and $\ket{e}$ are respectively the ground and excited energy
states.

In the majority of recent works, the detector’s free Hamiltonian included just
the energy associated with its internal state [18]. Here we also include the
CoM energy required to accurately describe the dynamics of all relevant DoFs
of the detector for its interaction with the field.

The energy of a pointlike first-quantized particle with internal DoFs and in
flat spacetime (Minkowski metric) reads
$\sqrt{\hat{M}^{2}c^{4}+\hat{\vec{p}}^{2}c^{2}}$ [1, 2, 36, 37, 38, 4, 10]
where $\hat{M}c^{2}$ is the total rest mass-energy of the UDW detector. This
form is required for detector models which aim to include both quantum and
relativistic effects, see also Ref. [20] for the corresponding second-
quantized model. Here we are interested in the limit of low CoM energy—where
first-quantization is accurately describing the system and where the
detector’s CoM is moving slowly—and investigate the impact of the mass-energy
equivalence in this regime.

The CoM energy of the composite particle in such a low-energy limit reduces to

$\hat{M}c^{2}+\frac{\hat{\vec{p}}^{2}}{2\hat{M}}.$ (3)

Note, a fully nonrelativistic model of the detector would be governed by the
Galilei symmetry group, however with the inclusion of internal dynamics
through $\hat{M}$, the symmetry group of the resulting model in Eq. (3) is the
central extension of the Galilei group [39, 40]. Since this inclusion of the
internal dynamics comes from relativistic mass-energy equivalence, we will
hereafter refer to our model as the ‘semirelativistic’ model, to distinguish
it from a fully nonrelativistic model where the mass features only as a fixed
parameter.

We restrict the model to just two internal states corresponding to the
detector’s ground $\ket{g}$ and excited $\ket{e}$ states, and introduce the
following split

$\begin{split}\hat{M}&=m_{g}\ket{g}\bra{g}+m_{e}\ket{e}\bra{e}\\\
&=m_{g}\ket{g}\bra{g}+\left(m_{g}+\frac{E}{c^{2}}\right)\ket{e}\bra{e}.\end{split}$
(4)

Thus, $m_{g},m_{e}$ are the masses associated with the ground and the excited
state of the detector, respectively, and $E$ is the energy gap between them.
Note that the first term in Eq. (3) gives the usual internal Hamiltonian of
the UDW detector, while the second term is the kinetic energy with mass-energy
incorporated.

Note that the fully nonrelativistic limit of the detector’s dynamics is
obtained when $E/m_{g}c^{2}\rightarrow 0$. This is in addition to the low-
energy limit for the CoM already present in Eq. (3), which requires
$\vec{p}^{2}\ll(m_{g(e)}c)^{2}$. The full nonrelativistic limit thus results
in the kinetic term in Eq. (3) becoming
$\hat{p}^{2}/2\hat{M}\rightarrow\hat{p}^{2}/2m_{g}\otimes\hat{\mathbb{I}}_{\mathrm{int}}$
where $\hat{\mathbb{I}}_{\mathrm{int}}$ is the identity operator on the
internal states and where the first factor acts only on the CoM. Importantly,
the rest energy term in Eq. (4) remains unchanged in this limit. The full
nonrelativistic limit of Eq. (3), dropping dynamically irrelevant term
$m_{g}c^{2}\hat{\mathbb{I}}$ arising from $\hat{M}c^{2}$, reads

$E\ket{e}\bra{e}+\frac{\vec{p}^{2}}{2m_{g}}.$ (5)

As expected in nonrelativistic physics, the above is simply a sum of the
internal energy (acting only on the internal DoF) and the kinetic energy
(acting only on the CoM) which features one mass parameter $m_{g}$ for all
internal states. See refs [40, 41] for detailed discussions of the
nonrelativistic limit of dynamics for composite particles, which relate to the
aforementioned underlying group structure.

The full free Hamiltonian in our model consists of the energy of the detector,
Eq. (3), and the energy of the massless scalar field $\int
d^{3}k~{}c|\vec{k}|~{}\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}}$, with
$\hat{a}_{\vec{k}},\hat{a}_{\vec{k}}^{\dagger}$ the creation and annihilation
operators of the field quanta with three-momentum $\vec{k}$, satisfying the
canonical commutation relation
$[\hat{a}_{\vec{k}},\hat{a}_{\vec{k}^{\prime}}^{\dagger}]=\delta^{(3)}(\vec{k}-\vec{k}^{\prime})$.
Dropping the term proportional to the identity from Eq. (4), the total free
Hamiltonian for our model reads

$\hat{H}_{0}=E\ket{e}\bra{e}+\frac{\hat{\vec{p}}^{2}}{2\hat{M}}+\int
d^{3}k~{}c|\vec{k}|~{}\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}}.$ (6)

The detector couples to the field at the CoM position, described by the
position operator $\hat{\vec{x}}$. Equation (1) is modified to include this
using the spectral expansion of the position operator

$\hat{H}_{\mathrm{int}}=\lambda\int
d^{3}x\ket{\vec{x}}\bra{\vec{x}}\otimes\hat{\mu}\otimes\hat{\phi}(\vec{x}),$
(7)

with CoM position eigenvalues $\vec{x}$, and eigenstates $\ket{\vec{x}}$, and
with the field operator defined as

$\hat{\phi}(\vec{x})=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sqrt{\frac{c^{2}}{2|\vec{k}|}}\left[e^{i\vec{k}\vec{x}}\hat{a}_{\vec{k}}+e^{-i\vec{k}\vec{x}}\hat{a}^{\dagger}_{\vec{k}}\right],$
(8)

where normalization of the field has been chosen such that $\lambda$ remains
dimensionless.

To find the transition amplitude for the system, we work in the interaction
picture and perform a perturbative expansion. The interaction picture form of
the interaction Hamiltonian is obtained from
$e^{iH_{0}t}\ket{\vec{x}}\bra{\vec{x}}\hat{\mu}\hat{\Phi}(\vec{x})e^{-iH_{0}t}$.
Note that due to the mass-energy equivalence, in the kinetic term of (3) there
is coupling between the CoM and internal DoFs of the detector and so the time
evolution of $\hat{\mu}$ and $\ket{\vec{x}}\bra{\vec{x}}$ has to be computed
jointly. In summary, Eqs (6) and (7) result in the following interaction-
picture Hamiltonian

$\hat{H}_{\mathrm{int}}(t)=\lambda\int\hat{\phi}(\vec{x},t)\Bigg{[}e^{it\left[\frac{\hat{\vec{p}}^{2}}{2m_{e}}+E\right]}\ket{e}\bra{g}\\\
\ket{\vec{x}(t)}\bra{\vec{x}(t)}e^{-it\frac{\hat{\vec{p}}^{2}}{2m_{g}}}+\mathrm{H.c.}\Bigg{]}d^{3}x,$
(9)

where the time-dependent position eigenstates can be expressed as

$\ket{\vec{x}(t)}\ket{i}=\int\frac{d^{3}p}{(2\pi)^{3/2}}e^{-i\vec{p}\vec{x}+it\frac{\vec{p}^{2}}{2}\left(\frac{1}{m_{g}}\ket{g}\bra{g}+\frac{1}{m_{e}}\ket{e}\bra{e}\right)}\ket{\vec{p}}\ket{i}.$
(10)

where $\ket{i}$ is an arbitrary state of the internal DoF.

Note that in the fully nonrelativistic limit of the CoM we recover the model
of the UDW detector with mass as a nondynamical parameter and fixed internal
energy gap $E$ which is used in ref. [33]. The question of whether the ground
state mass-energy $m_{g}$ or the excited state’s $m_{e}$ becomes the mass
parameter of the nonrelativistic model is discussed in Sec. III.3.

Transition amplitudes between any pair of mutually orthogonal states, to first
order perturbation in the coupling strength $\lambda$, are given by

$\mathcal{A}=-i\bra{\Psi_{f}}\int^{t_{f}}_{t_{i}}dt~{}\hat{H}_{\mathrm{int}}(t)\ket{\Psi_{i}}$
(11)

with $\ket{\Psi_{i}}$ and $\ket{\Psi_{f}}$ the initial and final states of the
system, respectively.

We first look at the emission process and thus consider the following states

$\ket{\Psi_{i}}=\ket{\psi_{0}}\otimes\ket{e}\otimes\ket{0},~{}~{}\ket{\Psi_{f}}=\ket{\vec{p}^{\prime}}\otimes\ket{g}\otimes\hat{a}^{\dagger}_{\vec{k}^{\prime}}\ket{0},$
(12)

describing the detector beginning in the excited state, with an arbitrary CoM
state denoted by $\ket{\psi_{0}}$, and where the field is initially in the
Minkowski vacuum state. The final state is then taken as the ground state of
the detector with some CoM momentum $\vec{p}^{\prime}$, and a one-particle
excitation in the field. We will later sum over the final states of the
detector and the field to derive the total emission rate, discussed in Sec.
III.1.

The states in Eq. (12), incorporated into Eq. (11), result in the following
emission amplitude:

$\mathcal{A}^{\mathrm{em}}_{\mathrm{semirel.}}=-\frac{i\lambda}{(2\pi)^{9/2}}\sqrt{\frac{c^{2}}{2|\vec{k}|}}\psi_{0}(\vec{p}^{\prime}+\vec{k})\\\
\times\int^{t_{f}}_{t_{i}}dt~{}e^{it(\frac{\vec{p}^{\prime
2}}{2m_{g}}-\frac{(\vec{p}^{\prime}+\vec{k})^{2}}{2m_{e}}-E+c|\vec{k}|)}.$
(13)

Both masses $m_{g}$ and $m_{e}$ are present in the time integral and in
general affect the amplitude. This integral (with bounds extended to infinity)
expresses the energy conservation in the process in which the internal state
and thus also the mass-energy of the detector changes. This clarifies why the
amplitude in general depends on the masses of both internal states. This also
indicates that in general the mass-energy equivalence will have consequences
for observable effects, which we quantitatively assess in the next section.
Note that the difference between the masses disappears from Eq. (13), for
arbitrary final states of the field, only when the initial momentum of the
detector is strictly zero. That is, when
$\psi_{0}(\vec{p})\propto\delta^{(3)}(\vec{p})$, in which case the amplitude
depends only on the ground state mass $m_{g}$.

We now look at the absorption amplitude, where the detector begins in the
ground state. In the present case, as the absorption or emission of a field
particle affects the initial and final mass of the detector, the two processes
exhibit differences that are not accessible in an UDW detector with a
preassigned trajectory nor in a model with fully nonrelativistic CoM. The
initial and final states of all DoFs, in analogy to the emission case, read:

$\ket{\Psi_{i}}=\ket{\psi_{0}}\otimes\ket{g}\otimes\hat{a}^{\dagger}_{\vec{k}^{\prime}}\ket{0},~{}~{}\ket{\Psi_{f}}=\ket{\vec{p}^{\prime}}\otimes\ket{e}\otimes\ket{0}.$
(14)

Note that for comparison between the two processes we take the same initial
state $\ket{\psi_{0}}$ for the CoM as in Eq. (12). The resulting amplitude for
the absorption process is

$\mathcal{A}^{\mathrm{abs}}_{\mathrm{semirel.}}=-\frac{i\lambda}{(2\pi)^{9/2}}\sqrt{\frac{c^{2}}{2|\vec{k}|}}~{}e^{-it_{f}\left(\frac{\vec{p}^{\prime
2}}{2m_{e}}+E\right)}~{}\psi_{0}(\vec{p}^{\prime}-\vec{k})\\\
\times\int^{t_{f}}_{t_{i}}dt~{}e^{it\left(\frac{\vec{p}^{\prime
2}}{2m_{e}}-\frac{(\vec{p}^{\prime}-\vec{k})^{2}}{2m_{g}}+E-c|\vec{k}|\right)}.$
(15)

Compared to the emission case, the signs of $E$ and $k$, as well as the masses
$m_{g},m_{e}$ have swapped places, as expected. We thus again see both masses
in the time integral, contributing to conservation of energy (for
$t_{i},t_{f}$ taken to infinity). As in the emission case, only one of the
masses plays a role in the final amplitude for the initial CoM state being
$\psi_{0}(\vec{p})\propto\delta^{(3)}(\vec{p})$, however, for the absorption
process this is the excited state mass $m_{e}$. In general, the two masses
contribute differently to the two amplitudes (true also for other CoM states),
which has consequences for the interpretation of the nonrelativistic mass
parameter of the detector, further discussed in the next two sections. The
rate of absorption corresponding to Eq. (15) is explored in Section III.2.

##  III Mass-energy effect on detector emission and absorption rates

###  III.1 Emission rate

The total emission probability corresponding to the amplitude Eq. (13) is
obtained by summing over the final momenta of the CoM and of the field. The
corresponding transition rate is then obtained by taking its time derivative,
and then taking the limits of the time integral to infinity. The resulting
transition rate for the emission process is

$\mathcal{R}^{\mathrm{em}}_{\mathrm{semirel.}}=\frac{\lambda^{2}c^{2}m_{g}}{4\pi}\int
d^{3}p~{}\left|\psi_{0}(\vec{p})\right|^{2}\mathcal{T}^{\mathrm{em}}(p),$ (16)

where $p\equiv|\vec{p}|$ and

$\begin{split}\mathcal{T}^{\mathrm{em}}_{\mathrm{semirel.}}(p)&:=2-\frac{1}{p}\sqrt{\frac{p^{2}m_{g}}{m_{e}}+2pm_{g}c+m_{g}^{2}c^{2}+2Em_{g}}\\\
&\qquad+\frac{1}{p}\sqrt{\frac{p^{2}m_{g}}{m_{e}}-2pm_{g}c+m_{g}^{2}c^{2}+2Em_{g}}.\end{split}$
(17)

We refer to $\mathcal{T}(p)$ as a “template function” following the
terminology used in Ref. [33]. In the nonrelativistic limit for the detector’s
CoM, discussed in the previous section, we have $m_{g}=m_{e}$, and Eq. (17)
reduces to the expression found in that paper. In the infinite mass limit
(while keeping $E$ finite), for any normalized initial CoM state $\psi_{0}$,
Eqs (16), (17) reduce to the “classical” case, i.e., a detector on a fixed
classical trajectory:

$\mathcal{R}^{\mathrm{em}}_{\mathrm{class.}}=\frac{\lambda^{2}}{2\pi}E.$ (18)

Because our Hamiltonian is valid for low CoM energies, we expand around small
CoM momentum, specifically, about $p/m_{g}c=0$. Note that this is also the
regime where we expect the least impact of the mass-energy equivalence on the
final result, as discussed below Eqs (13), (15). The expansion results in a
template function

$\mathcal{T}^{\mathrm{em}}_{\mathrm{semirel.}}(p)=2-\frac{2}{\sqrt{1+\frac{2E}{m_{g}c^{2}}}}+\frac{p^{2}(c^{2}(m_{g}-m_{e})+2E)}{c^{4}m_{g}^{2}m_{e}\left(1+\frac{2E}{m_{g}c^{2}}\right)^{\frac{5}{2}}}.$
(19)

We now take for the CoM state a three-dimensional Gaussian wave packet
centered at $\vec{p}=0$, with position-space wave function

$\psi_{0}(\vec{x})=\left(\frac{2}{\pi
L^{2}}\right)^{3/4}e^{-\frac{|\vec{x}-\vec{x}_{0}|^{2}}{L^{2}}},$ (20)

where $L$ is the initial (spatial) spread of the wave packet, centered in
space at $\vec{x}=\vec{x}_{0}$. The resulting emission rate reads

$\mathcal{R}^{em}_{\mathrm{semirel.}}=\frac{\lambda^{2}c^{2}m_{g}}{2\pi}\\\
\Bigg{[}1-\frac{1}{\sqrt{1+\frac{2E}{m_{g}c^{2}}}}+\frac{3~{}(c^{2}(m_{g}-m_{e})+2E)}{2L^{2}c^{4}m_{g}^{2}m_{e}\left(1+\frac{2E}{m_{g}c^{2}}\right)^{\frac{5}{2}}}\Bigg{]}.$
(21)

We note that the above holds provided that $L>1/m_{g}c$, i.e., the
localization of the wave packet, is larger than the Compton wavelength, as
localization below this scale can lead to unphysical results (due to the
breakdown of 1st-quantization). Equivalently, the expression holds when the
initial momentum spread is small, $L_{p}:=1/L<m_{g}c$, so that the initial
state is consistent with the low-energy limit of the detector’s CoM dynamics.

For $m_{g}=m_{e}\equiv M$ in Eq. (21), we again recover the result of ref.
[33] for the nonrelativistic CoM:

$\mathcal{R}^{\mathrm{em}}_{\mathrm{nonrel.}}=\frac{\lambda^{2}c^{2}M}{2\pi}\\\
\Bigg{[}1-\frac{1}{\sqrt{1+\frac{2E}{Mc^{2}}}}+\frac{3E}{L^{2}c^{4}M^{3}\left(1+\frac{2E}{Mc^{2}}\right)^{\frac{5}{2}}}\Bigg{]},$
(22)

Note that this limit cannot be achieved by simply setting $E$ to zero in the
final rate, Eq. (21), since $E$ plays two roles in any relativistic model of
an emitter/absorber— it is the “gap” between masses associated with different
internal states in the free Hamiltonian, and the energy gap between internal
states in the interaction Hamiltonian. In prior studies, the UDW detector had
no mass gap, but a finite internal energy gap, and so to recover such models
it is more convenient to keep the general notation for the masses
$m_{g},m_{e}$ (despite the fact that their difference is simply $E/c^{2}$).

From the physics perspective, in these prior models interaction with the field
would only alter the linear momentum of the CoM of the detector but not its
mass. As discussed above, by incorporating the mass-energy equivalence, known
in atomic physics as the Röntgen term [35] or the mass-defect [42, 43, 44],
our model can take into account the energy change of the CoM arising also due
to the change in the mass of the detector.

Importantly, this mass change cannot be neglected even if one is looking only
at transitions between internal states, as we do here. Indeed, to lowest order
in $E/m_{g}c^{2}$, our Eq. (21) reads
$\mathcal{R}^{\mathrm{em}}_{\mathrm{semirel.}}\sim
E(1+\frac{3}{2}\frac{1}{(Lm_{g}c)^{2}})$ while the nonrelativistic rate Eq.
(22) reads $\mathcal{R}^{\mathrm{em}}_{\mathrm{nonrel.}}\sim
E(1+3\frac{1}{(Lm_{g}c)^{2}})$. This is the same for both choices of $M=m_{g}$
and $M=m_{e}=m_{g}+E/c^{2}$ in Eq. (22). Thus, the mass-energy effects cannot
be ignored even at lowest order; they are only irrelevant in the “classical”
limit where the dynamics of the CoM is entirely ignored, Eq. (18).

We further compare the full emission rates Eqs (21) and (22) numerically in
Sec. III.3.

###  III.2 Absorption rate

Following the same steps as for the emission process, the absorption rate
arising from Eq. (15) is

$\mathcal{R}^{\mathrm{abs}}_{\mathrm{semirel.}}=\frac{\lambda^{2}c^{2}m_{e}}{2\pi}\int
d^{3}p\left|\psi_{0}(\vec{p})\right|^{2}\mathcal{T}^{\mathrm{abs}}(p),$ (23)

with template function

$\mathcal{T}^{\mathrm{abs}}_{\mathrm{semirel.}}(p):=\frac{1}{p}\sqrt{\frac{p^{2}m_{e}}{m_{g}}+2pm_{e}c+m_{e}^{2}c^{2}-2m_{e}E}\\\
-\frac{1}{p}\sqrt{\frac{p^{2}m_{e}}{m_{g}}-2pm_{e}c+m_{e}^{2}c^{2}-2m_{e}E}.$
(24)

The expression for the rate formally requires us to introduce a cutoff $K$ in
the momentum integral (i.e. $p<K$) and to restrict the energy gap to
$E/m_{g}c^{2}<(\frac{K}{m_{g}c}-1)^{2}$. We treat the width $L_{p}$ of the
Gaussian state as an effective cutoff $K$ henceforth. As anticipated, in the
absorption case above the masses $m_{g}$ and $m_{e}$ have swapped compared to
the emission case, Eq. (17), and the sign of the energy (in $2m_{e}E$) has
changed.

Taking again the expansion about $p/m_{g}c=0$, and using the Eq. (20)
Gaussian, we find the absorption rate

$\mathcal{R}^{\mathrm{abs}}_{\mathrm{semirel.}}=\frac{\lambda^{2}c^{2}m_{e}}{\pi}\\\
\Bigg{[}\frac{1}{\sqrt{1-\frac{2E}{m_{e}c^{2}}}}+\frac{3\left(c^{2}(m_{g}-m_{e})+2E\right)}{2L^{2}c^{4}m_{e}^{2}m_{g}\left(1-\frac{2E}{m_{e}c^{2}}\right)^{\frac{5}{2}}}\Bigg{]}.$
(25)

As in the emission case we can take $m_{g}=m_{e}\equiv M$ in Eq. (25), to
obtain the result for a nonrelativistic CoM:

$\mathcal{R}^{\mathrm{abs}}_{\mathrm{nonrel.}}=\frac{\lambda^{2}c^{2}M}{\pi}\\\
\Bigg{[}\frac{1}{\sqrt{1-\frac{2E}{Mc^{2}}}}+\frac{3E}{L^{2}c^{4}M^{3}\left(1-\frac{2E}{Mc^{2}}\right)^{\frac{5}{2}}}\Bigg{]}.$
(26)

We now look again at the rates to lowest order in mass-energy corrections
$E/m_{g}c^{2}$. Our result reads
$\mathcal{R}^{\mathrm{abs}}_{\mathrm{semirel.}}\sim
1+2E(1+\frac{3}{4}\frac{1}{(Lm_{g}c)^{2}})$, while the nonrelativistic rate is
$\mathcal{R}^{\mathrm{abs}}_{\mathrm{nonrel.}}\sim
1+2E(1+\frac{3}{2}\frac{1}{(Lm_{g}c)^{2}})$ for $M=m_{e}$ and reads
$\mathcal{R}^{\mathrm{abs}}_{\mathrm{non-rel.}}\sim
1+E(1+3\frac{1}{(Lm_{g}c)^{2}})$ for $M=m_{g}$. We again see that mass-energy
corrections cannot be ignored. However, we also see that with the choice of
$M=m_{e}$ the nonrelativistic expression better approximates the relativistic
result—a conclusion which also holds for full rates, which we will further
discuss based on the figures in the next section.

Finally, we note that for the absorption process, the classical limit is
undefined in the sense that the above rates diverge in the limit of large
mass. We thus will not be comparing the absorption rates against such a
classical limit of a detector.

###  III.3 Comparing to prior models

In previous models that looked at a first-quantized UDW detector, or in a
related model of a dipole interacting with an optical field, the CoM of the
detector/dipole was either pre-fixed (nondynamical), or dynamical but
nonrelativistic, having a fixed mass parameter $M$, e.g. [32]. In this work,
we incorporate the relativistic correction to the CoM which allows the mass of
the detector to change upon the exchange of energy with the field (where in an
emission process, the mass is decreasing, while in an absorption process it is
increasing). Therefore, in order to compare our results with nonrelativistic
models, we must interpret how the mass-energies $m_{g},m_{e}$ relate to the
nonrelativistic mass parameter $M$.

One choice is to equate $M$ with the ground state mass $m_{g}$. This choice
naturally arises when looking at a nonrelativistic limit for the free
Hamiltonian discussed in Sec. II. It is worthwhile to note that this itself
arises from taking the split of $\hat{M}$ as in Eq. (4), and so defining
internal energy such that the internal ground state $\ket{g}$ has (internal)
energy zero and the energy of the excited state $\ket{e}$ is non-negative.

Another choice is to equate the nonrelativistic mass $M$ with the mass-energy
which allows the relativistic and nonrelativistic dynamics of the full
interacting system to align more closely (in the limit of small difference
between $m_{e},m_{g}$) . Surprisingly, this approach dictates different
choices for $M$ for the emission and for the absorption processes. It is only
in the full nonrelativistic limit, where $m_{e}\equiv m_{g}$, that the
ambiguity is resolved, as there is strictly only one mass parameter. In order
to understand which choice for the value of the non-relativistic mass $M$ best
approximates the relativistic corrections—which albeit small are nonetheless
finite—we explore both choices and discuss them further in this section.

Before presenting plots as illustrations of our analytical results, we
reiterate that the form of the absorption rates Eqs (25), (26) requires the
parameters to be constrained so that $2E<m_{e}c^{2}$ and $2E<Mc^{2}$,
respectively. As a result, we have chosen to present both the absorption and
the emission rates’ dependence on $E$ in this regime for consistency, even
though the latter does not technically require this restriction.

####  III.3.1 Emission rates compared

In Fig. 1 the emission rate for the classical (infinite mass) case, Eq. (18),
is compared directly against our result (21) (the “semirelativistic” model)
and the rate for the nonrelativistic CoM model (22), for each of the choices
of $M\equiv m_{g}$ (top plot) and $M\equiv m_{e}$ (bottom plot). All
quantities are presented in units of the energy gap $E$ (setting $c=1$), with
the rates scaled by $(\lambda^{2}/2\pi)^{-1}$, resulting in a value of 1 for
the classical rate.

The convergence of the other two rates to the classical case previously
mentioned in Sec. III.1 is evident here. Our model is exactly the same in the
top and bottom plots, while the curve for the non-relativistic model changes
depending on whether its mass $M$ is identified with $m_{g}$ or $m_{e}$. The
form of the non-relativistic curves thus differs from the top to the bottom
plot because of the different mass used in the non-relativistic emission rate,
Eq. (22). In the bottom plot this is $M=m_{e}=m_{g}+E$, while in the top plot
$M=m_{g}$. This will continue to underpin the behaviour of the non-
relativistic model in subsequent figures below.

Figure 1: Emission rates as a function of mass for the UDW detector: With
classical (fixed) CoM—the green, dashed-dotted line; with dynamical CoM
including relativistic mass-energy corrections (our semirelativistic
model)—the solid blue line; and with nonrelativistic CoM—dashed yellow line.
Top: for the nonrelativistic mass set to $M=m_{g}$. Bottom: for $M=m_{e}$. The
classical UDW case can be seen as the infinite-mass limit of both models (ours
and the nonrelativistic one) and for both choices of the mass, as discussed in
the main text. All quantities are in units of the energy gap (setting $c=1$),
thus we plot starting from $m_{g}/E=1$ and we use $L\cdot E=1$ which
guarantees that $L>1/m_{g}$. The rates are additionally scaled by
$(\lambda^{2}/2\pi)^{-1}$ setting the classical rate to $1$. Finally, both
axes are on a log scale. Figure 2: Emission rates
$\mathcal{R}^{\mathrm{em}}_{\mathrm{semirel.}}$ for our model of the UDW
detector with relativistic corrections from the mass-energy equivalence (solid
lines) and for the model with nonrelativistic CoM (dashed lines). Left column:
emission rate dependence on the initial CoM spread in momentum, $L_{p}$, for
the two choices of the nonrelativistic model’s mass $M=m_{g}(m_{e})$ top
(bottom) panels. Right column: emission rate dependence on the energy gap $E$
between ground and excited states, top/bottom panels as in the left column.
All quantities are in units set by $m_{g}$, that is: rates in units of the
Compton frequency $1/m_{g}c^{2}$, $L_{p}$ in units of $m_{g}c$, and $E$ in
units of $m_{g}c^{2}$. The vertical dotted line at $L_{p}=1\cdot(m_{g}c)$
marks the limit of the applicability of the first-quantized model.

We now further explore the behavior of the emission rate of our model compared
to that of the nonrelativistic CoM model for both choices of the
nonrelativistic model’s mass. We plot Eq. (21) and Eq. (22) for these two
choices in Figure 2—the $M\equiv m_{g}$ case on the top row, and the $M\equiv
m_{e}$ case on the bottom row. The plots in the left column show the emission
rate dependence on the initial spread in the CoM momentum $L_{p}=1/L$; the
plots in the right column show the rate dependence on the energy gap $E$. We
stress that in all these plots, as in Fig. 1, our model is fixed while the
nonrelativistic model’s $M$ is different between the top and the bottom rows.

First we find that in general the inclusion of mass-energy equivalence
suppresses the emission rate compared to the nonrelativistic CoM model. For
increasing initial spread in momentum, $L_{p}$, the rates arising from the two
models diverge as expected. That is, increasing $L_{p}$ means the higher CoM
momenta are contributing significantly to the initial state and so
relativistic corrections due to ${p}^{2}/m_{g}\neq{p}^{2}/m_{e}$ become more
relevant. The difference is greater for the $M\equiv m_{e}$ case where it also
increases with the energy gap.

Similar conclusions could be drawn for emission rate dependence on the energy
gap $E$ based on the regime plotted. However, note that when looking at the
limit of large energy $E\gg m_{g}c^{2}$ (or $E\gg Mc^{2}$), our emission rate
converges to $\sim m_{g}$ while the nonrelativistic rate converges to $\sim
M$. Hence the two converge to each other for $M=m_{g}$. This can be understood
by looking at the template function Eq (17) which approaches a constant value
of 2 for large $E$, independently of whether $m_{g}\neq m_{e}$ or
$m_{g}=m_{e}\equiv M$ as in the nonrelativistic case. Physically, this is the
limit where the CoM dynamics are essentially completely suppressed by the rest
energy. The first term in Eq. (3) then dominates over the second term, and the
dynamics of the two models become effectively the same.

As we have mentioned earlier in this section, the choice of $M=m_{g}$ or
$M=m_{e}$ for any finite $E$ results in different emission rates predicted by
the nonrelativistic model. Considering that the “correct” choice here is the
one which arises in the nonrelativistic limit of small $L_{p}$, this favors
the choice of $M=m_{g}$. This is visible in all the emission rate plots in
Fig. 2, most notably from the left-hand column (where the rates are plotted as
a function of $L_{p}$ for these two choices of $M$).

Figure 3: Absorption rates $\mathcal{R}^{\mathrm{abs}}_{\mathrm{semirel.}}$
for our model of the UDW detector with relativistic corrections from the mass-
energy equivalence (solid lines) and for the model with nonrelativistic CoM
(dashed lines). Left column: absorption rate dependence on the initial
momentum spread $L_{p}$. Right column: absorption rate dependence on the
energy gap $E$. Top (bottom) panels in both columns correspond to the two
choices for the nonrelativistic model’s mass $M=m_{g}(m_{e})$. Units of all
quantities set by $m_{g}$: Rates are in the units of the Compton frequency
$1/m_{g}c^{2}$, $L_{p}$ is in units of $m_{g}c$, and $E$ is in units of
$m_{g}c^{2}$. The vertical dotted line at $L_{p}=1$ marks the limit of the
applicability of the first-quantized model; the vertical lines in the right-
column mark asymptotes of the absorption rates which are at $2E=m_{e}c^{2}$
(equivalent to $E=m_{g}c^{2}$) for our model and at $2E=Mc^{2}$ for the
nonrelativistic model.

####  III.3.2 Absorption rates compared

We now turn to the absorption rate. As mentioned in the preceding section,
absorption rates for both models considered here diverge in the large-mass
limit. We thus move on to comparing the absorption rate of our model to that
of the UDW detector with nonrelativistic CoM and we again do this for both
choices for the nonrelativistic mass. Figure 3 thus plots the absorption rate
for our model, Eq. (25), and the equivalent for the UDW detector with
nonrelativistic CoM, Eq. (26). As for the emission rate, the left-hand column
shows the rates as a function of the initial CoM spread $L_{p}$ and the right-
hand column the rates as a function of the energy gap. For the top row plots,
the mass in the nonrelativistic model is set $M=m_{g}$ and in the bottom row
it is set $M=m_{e}$. The asymptotes in the right column plots are at
$2E=Mc^{2}$ for the nonrelativistic model and at $2E=m_{e}c^{2}$
(equivalently, at $E=m_{g}c^{2}$, due to $m_{e}=m_{g}+E/c^{2}$) for our model.
This is explicitly visible from the Eqs (25), (26).

Note that in this case, when looking at the limit of small momenta, small
$L_{p}$, the two rates converge if we identify the nonrelativistic mass $M$
with the excited state mass $m_{e}$. This is the opposite of the emission
case. For this choice (and thus for the bottom-row plots) we again see that
the inclusion of relativistic corrections overall suppresses the rate compared
to the nonrelativistic model.

##  IV Discussion

We derived transition amplitudes and rates for an Unruh-DeWitt detector with
dynamical internal mass-energy, thus incorporating a relativistic correction
to the CoM, modeled in other contexts by the Röntgen term or a mass defect.
Our goal has been to extend a nonrelativistic model of the detector’s CoM in a
general manner, agnostic as to the details of the implementation of the
detector. We have demonstrated that at least to lowest order perturbation in
the interaction strength, the transition rates can be derived analytically.

The main result of this work is the necessity of including the relativistic
corrections in the UDW model if any CoM dynamics is to be included at all. We
carried out our derivations in the low-energy regime where relativistic
effects have the least impact, yet, we found that mass-energy corrections are
on the same order as the effects of the inclusion of the CoM dynamics itself.
As a result there is no consistent model of an UDW detector with fully
nonrelativistic CoM—the mass-energy corrections are necessary.

Furthermore, there is the ambiguity of choosing the value of the mass in the
nonrelativistic limit, again relevant as the mass difference is also on the
order of $E/mc^{2}$. We have argued that this choice should be made by
defining the nonrelativistic limit as the one stemming from the interacting
relativistic model in its low energy regime (here low CoM momentum) whilst
keeping a small but finite energy gap. We have found that this approach
essentially identifies the nonrelativistic mass with the final mass-energy in
the given process. That is, $M$ is identified with the ground-state mass
$m_{g}$ in the emission case and the excited-state mass $m_{e}$ in the
absorption case.

One particular advantage of working in the low-energy regime is the potential
transfer of the theory into the laboratory [6, 7, 45]. Our result, and indeed
already the existing literature, suggest that relativistic mass-energy
corrections will again be necessary for consistent treatment of the detector
recoil during the absorption or emission processes. As the UDW detector’s
recoil has been identified as a potential signature of the Unruh effect that
could be experimentally observed [34], it presents a natural context in which
to implement our model and approach. Apart from formal consistency of the
results, the inclusion of the mass-energy equivalence also brings such a
detector model closer to a composite particle, such as an atom, and thus also
closer to the experimental realm.

From a theory perspective, inclusion of mass-energy equivalence in the UDW
detector’s dynamics results in a novel probe of joint gravitational and
quantum phenomena—e.g. the Hawking radiation surrounding a black hole. We
expect this direction to extend a recent model of an UDW detector in a
quantum-controlled superposition of classical trajectories [25, 26]. The
question of thermalization in our model is of particular interest, as such a
system, with CoM and internal states coupled, would constitute a genuinely
different model of a thermometer.

There are also direct applications to relativistic quantum information, where
particle detectors have long been a tool of choice, and entanglement
“harvesting” with UDW detectors is a current area of particular interest.
Already the impact of the inclusion of nonrelativistic CoM dynamics on the
harvested entanglement, compared to classical models, has been shown [46]. It
would be interesting to see whether the addition of the relativistic mass-
energy effects in this context again leads to relevant corrections. Beyond the
intuition arising from the present work, this may in particular be expected
since the considered relativistic correction leads to internal states
interacting, and so entangling, with the CoM. This suggests some interplay
with the entanglement that these states can develop with the field.

###### Acknowledgements.

We thank A. Kempf and N. Stritzelberger for early discussions, and also V.
Baccetti, J. Foo, E. Gale, L. Henderson, and N. Menicucci for further helpful
discussions and advice. M.Z. acknowledges support through Australian Research
Council (ARC) Future Fellowship grant No. FT210100675 and the ARC Centre of
Excellence for Engineered Quantum Systems (EQUS), grant No. CE170100009. The
authors acknowledge the traditional owners of the land on which the University
of Queensland is situated.

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# Radiative decays of the neutral $Z_{c}(3900)$ and $Z_{c}(4020)$

Xiao-Yun Wang1, Gang Li1 111<EMAIL_ADDRESS>Chun-Sheng An2
<EMAIL_ADDRESS>Ju-Jun Xie3,4,5 1College of Physics and Engineering, Qufu
Normal University, Qufu 273165, People’s Republic of China  2School of
Physical Science and Technology, Southwest University, Chongqing 400715, China
3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  4School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 101408, China  5Lanzhou Center for Theoretical
Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou
University, Lanzhou, Gansu 730000, China

###### Abstract

We study the radiative decays
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$
($J=0,1,2$), with the assumption that the $Z_{c}(3900)$ and $Z_{c}(4020)$
couple strongly to $D\bar{D}^{*}+c.c$ and $D^{*}{\bar{D}}^{*}$ channel,
respectively. By considering the contributions of intermediate charmed mesons
triangle loops within an effective Lagrangian approach, it is shown that the
calculated partial widths of $Z_{c}(3900)\to\gamma\chi_{cJ}$ are about a few
hundreds keVs, while the obtained partial widths
$Z_{c}(4020)\to\gamma\chi_{cJ}$ are about tens of keVs. The predicted partial
widths of $Z_{c}(3900)\to\gamma\chi_{c0,1}^{\prime}$ are less than 1 keV,
which mainly due to the very small phase space. For
$Z_{c}(4020)\to\gamma\chi_{c0,2}^{\prime}$, the calculated partial widths are
usually smaller than 1 keV. For the $Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$
process, the obtained partial widths can reach up to the order of 10 keV.
Furthermore, the dependence of these ratios between different decay modes on
the masses of $Z_{c}(3900)$ or $Z_{c}(4020)$ are also investigated, which may
be a good quantity for the experiments. It is hoped that these calculations
here could be tested by future experiments.

##  I Introduction

The discovery of $X(3872)$ in 2003 opened the gate to the abundance of the
$XYZ$ structures in the heavy quarkonium region Belle:2003nnu . Many of them
cannot be accommodated in the conventional quark model as $Q\bar{Q}$ and thus
turn out to be excellent candidates for exotic state. A large amount of
experimental and theoretical studies are devoted to those $XYZ$ states
Chen:2016qju ; Chen:2016spr ; Esposito:2016noz ; Guo:2017jvc ; Olsen:2017bmm ;
Liu:2019zoy ; Brambilla:2019esw ; Guo:2019twa . Among these states, the
charged charmoniumlike states $Z_{c}(3900)$ BESIII:2013ris ; Belle:2013yex
and $Z_{c}(4020)$ BESIII:2013ouc  have attracted special attention due to
their four-quark nature. In 2013, the BESIII Collaboration first observed a
new charged state $Z_{c}(3900)$ in the $\pi^{\pm}J/\psi$ invariant mass
spectra of the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ reaction BESIII:2013ris ,
and it was confirmed by the Belle Collaboration in the same process
Belle:2013yex . Later, the charged $Z_{c}(3900)$ is also observed in the
invariant mass spectrum of $D\bar{D}^{*}$ in the open charm process
$e^{+}e^{-}\to\pi^{\pm}(D\bar{D}^{*})^{\mp}$ BESIII:2013qmu . In addition, the
first evidence of the neutral $Z_{c}(3900)$ decaying into $J/\psi\pi^{0}$, was
reported in Ref. Xiao:2013iha  using the CLEO-c data. While the BESIII
Collaboration reported the observation of neutral $Z_{c}(3900)$ in the
$e^{+}e^{-}\to\pi^{0}\pi^{0}J/\psi$ BESIII:2015cld  and
$e^{+}e^{-}\to\pi^{0}(D{\bar{D}}^{*})^{0}$ BESIII:2015ntl , respectively.
Through the partial wave analysis of process
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$, the quantum numbers of $Z_{c}(3900)$ are
determined as $I^{G}(J^{PC})=1^{+}(1^{+-})$ BESIII:2017bua ; Workman:2022ynf .

The BESIII Collaboration found charged $Z_{c}(4020)$ state in the invariant
mass spectrum of $\pi^{\pm}h_{c}$ in the process of
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ BESIII:2013ouc . The charged $Z_{c}(4020)$
states were confirmed in the $D^{*}\bar{D}^{*}$ invariant mass spectrum of the
$e^{+}e^{-}\to\pi^{\pm}(D^{*}\bar{D}^{*})^{\mp}$ reaction BESIII:2013mhi .
While its neutral partner was reported in the $\pi^{0}h_{c}$ invariant mass
spectrum of the $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ reaction BESIII:2014gnk
and in the $(D^{*}\bar{D}^{*})^{0}$ invariant mass spectrum of the
$e^{+}e^{-}\to(D^{*}\bar{D}^{*})^{0}\pi^{0}$ reaction BESIII:2015tix ,
respectively.

After the observation of the $Z_{c}(3900)$ and $Z_{c}(4020)$ states, many
theoretical discussions have been carried out to explore their inner
structures Wang:2013cya ; Aceti:2014uea ; Guo:2013sya ; Cui:2013yva ;
Zhang:2013aoa ; Chen:2013omd ; Soleymaninia:2013cxa ; Braaten:2013boa ;
Faccini:2013lda ; Wang:2013llv ; Qiao:2013dda ; Wang:2022clw ; Chen:2013coa ;
Swanson:2014tra ; Szczepaniak:2015eza ; Wilbring:2013cha ; Dong:2013iqa ;
Dong:2013kta ; Li:2014pfa ; Gutsche:2014zda ; Esposito:2014hsa . Since the
measured masses of the $Z_{c}(3900)$ and $Z_{c}(4020)$ lie slightly above the
$D{\bar{D}}^{*}$ and $D^{*}{\bar{D}}^{*}$ mass thresholds, it may indicate
that these two states are good candidates of the hadronic molecule
Wang:2013cya ; Aceti:2014uea ; Guo:2013sya ; Cui:2013yva ; Zhang:2013aoa ;
Chen:2013omd . Besides the molecule explanation, these two states are also
identified as tetraquark states Soleymaninia:2013cxa ; Braaten:2013boa ;
Faccini:2013lda ; Wang:2013llv ; Qiao:2013dda ; Wang:2022clw , or kinematical
threshold effects Chen:2013coa ; Swanson:2014tra ; Szczepaniak:2015eza .
Recently, the production and decay behaviors of $Z_{c}(3900)$ and
$Z_{c}(4020)$ states were extensively investigated in $D{\bar{D}}^{*}$ and
$D^{*}{\bar{D}}^{*}$ hadronic molecule scenario in Refs. Wilbring:2013cha ;
Dong:2013iqa ; Dong:2013kta ; Li:2014pfa ; Gutsche:2014zda ; Esposito:2014hsa
, where the theoretical calculations are consistent with the corresponding
experimental measurements.

It is known that the intermediate meson loop (IML) transition is regarded as
an important nonperturbative transition mechanism which has a long history
Lipkin:1986bi ; Lipkin:1988tg ; Moxhay:1988ri  and recently it is widely used
to study the production and decays of exotic states Liu:2013vfa ; Liu:2020orv
; Guo:2013zbw ; Li:2013yla ; Chen:2013bha ; Liu:2016xly ; Li:2013xia ;
Chen:2015igx ; Voloshin:2019ivc ; He:2013nwa ; Wu:2016ypc ; Li:2012as . The
radiative transitions between neutral $Z_{c}(3900)/Z_{c}(4020)$ and the
charmonia states are particular modes compared to the charged
$Z_{c}(3900)/Z_{c}(4020)$. The quark and antiquark in the different components
of neutral $Z_{c}(3900)/Z_{c}(4020)$ can annihilate into a photon, and the
rest charm and anticharm quarks form a charmonium in the final state. In Ref.
Chen:2015igx , the radiative decays of the $Z_{c}(3900)\to\gamma\chi_{c0,1}$
was studied in hadronic molecule picture using an effective Lagrangian
approach. In Ref. Voloshin:2019ivc , the radiative transitions from the
neutral exotic $Z_{c}(4020)$ resonance to $X(3872)$ was investigated.

Along this line, in this work, we estimate the radiative decays
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$
($J=0,1,2$) with the non-relativistic effective field theory (NREFT) under
assumption that the $Z_{c}(3900)$ and $Z_{c}(4020)$ couple strongly to
$D\bar{D}^{*}+c.c$ and $D^{*}{\bar{D}}^{*}$ channel in $S$-wave, respectively.
Based on this picture, in the present work, the quantum numbers of the neutral
$Z_{c}(4020)$ state are taken to be $I^{G}(J^{PC})=1^{+}(1^{+-})$, which are
consistent with those previous theoretical results, for instance, in Refs.
Wang:2013cya ; Chen:2015igx ; Voloshin:2019ivc ; He:2013nwa ; Aceti:2014uea .

The rest of this paper is organized as follows. In Sec. II, we present the
theoretical framework used in this work. In Sec. III, numerical results are
presented, and a brief summary is given in Sec. IV.

##  II Theoretical Framework

###  II.1 Triangle diagrams

We study the radiative decays of $Z_{c}(3900)$ and $Z_{c}(4020)$ states within
the effective Lagrangian approach. Based on the strong couplings of
$Z_{c}(3900)$ and $Z_{c}(4020)$ to $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$,
respectively, their radiative decays can be proceed via those triangle
diagrams as shown in Fig. 1, where there are three charmed mesons in the
triangle loop. To be specific, we denote the one connecting the initial
$Z_{c}(3900)/Z_{c}(4020)$ and the photon as $M1$, the one connecting the
$Z_{c}(3900)/Z_{c}(4020)$ and the final $\chi_{cJ}^{(\prime)}$ as $M2$ and the
exchanged meson between $\gamma$ and $\chi_{cJ}$ as $M3$. For example, in Fig.
1 (a), $M1$, $M2$ and $M3$ are the $D$, ${\bar{D}}^{*}$ and $D^{*}$,
respectively. In addition, all the loops contributing to each decay are listed
in Table 1.

Figure 1: The hadron-level diagrams of the radiative decay processes $Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{c0}^{(\prime)}$. The charge conjugated diagrams are not shown but included in the calculations. Table 1:  All the possible loops contributing to each transition. The mesons are listed as $[M1,M2,M3]$. $Z_{c}(3900)\to\gamma\chi_{c0}^{(\prime)}$ | [$D$, ${\bar{D}^{*}}$, $D^{*}$], [$D^{*}$, ${\bar{D}}$, $D$]
---|---
$Z_{c}(3900)\to\gamma\chi_{c1}^{(\prime)}$ | [$D^{*}$, $\bar{D}$, $D^{*}$]
$Z_{c}(3900)\to\gamma\chi_{c2}^{(\prime)}$ | [$D$, ${\bar{D}}^{*}$, $D^{*}$]
$Z_{c}(4020)\to\gamma\chi_{c0}^{(\prime)}$ | [$D^{*}$, ${\bar{D}^{*}}$, $D^{*}$]
$Z_{c}(4020)\to\gamma\chi_{c1}^{(\prime)}$ | [$D^{*}$, $\bar{D}^{*}$, $D$]
$Z_{c}(4020)\to\gamma\chi_{c2}^{(\prime)}$ | [$D^{*}$, $\bar{D}^{*}$, $D^{*}$]

###  II.2 Effective interaction Lagrangians

In order to calculate the triangle loops shown in Fig. 1, we firstly need the
effective couplings of the $Z_{c}(3900)$ and $Z_{c}(4020)$ states to
$D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ channels, respectively. The effective
Lagrangians describing the couplings of $Z_{c}(3900)/Z_{c}(4020)$ to
$D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ channels via $S$-wave can be easily
written as Cleven:2013sq ,

$\displaystyle{\cal L}_{Z_{c}^{(\prime)}}$ $\displaystyle=$ $\displaystyle
z^{\prime}\epsilon^{ijk}{\bar{V}}_{a}^{\dagger i}Z_{c}^{\prime
j}V_{a}^{\dagger k}+z[{\bar{V}}_{a}^{\dagger i}Z_{c}^{i}P_{a}^{\dagger}$ (1)
$\displaystyle-{\bar{P}}_{a}^{\dagger}Z_{c}^{i}V_{a}^{\dagger
i}]+{\mbox{H.C.}}$

where $Z_{c}$ and $Z^{\prime}_{c}$ stand for $Z_{c}(3900)$ and $Z_{c}(4020)$,
respectively. While $V_{a}$ and $P_{a}$ are the vector and pseudoscalar
charmed mesons, respectively, i.e.
$P_{a}(V_{a})=(D^{(*)0},D^{(*)+},D_{s}^{(*)+})$. In the two-component notation
of Ref. Hu:2005gf , the charmed mesons are represented by
$H_{a}=\vec{V_{a}}\cdot\vec{\sigma}+P_{a}$, with $\vec{\sigma}$ being the
Pauli matrices, and $a$ is the light flavor index. Besides, $z$ and
$z^{\prime}$ are the effective couplings. With the above effective
Lagrangians, we can obtain

$\displaystyle\Gamma_{Z_{c}(3900)\to D^{0}\bar{D}^{*0}}$ $\displaystyle=$
$\displaystyle\frac{|z|^{2}}{4\pi}\frac{|\vec{q}_{D}|}{M_{Z_{c}(3900)}}M_{D}M_{D^{*}},$
(2) $\displaystyle\Gamma_{Z_{c}(4020)\to D^{*0}\bar{D}^{*0}}$ $\displaystyle=$
$\displaystyle\frac{|z^{\prime}|^{2}}{4\pi}\frac{|\vec{q}_{D^{*}}|}{M_{Z_{c}^{\prime}(4020)}}M_{D^{*}}^{2},$
(3)

with $\vec{q}_{D}$ and $\vec{q}_{D^{*}}$ being the three-momenta of $D^{0}$
and $D^{*0}$ meson in the rest frame of $Z_{c}(3900)$ and $Z_{c}(4020)$,
respectively. Here, we have assumed that the total widths of $Z_{c}(3900)$ and
$Z_{c}(4020)$ are saturated by the decays $Z_{c}(3900)\to
D^{0}{\bar{D}}^{*0}+{\bar{D}}^{0}D^{*0}$ and $Z_{c}(4020)\to
D^{*0}{\bar{D}}^{*0}$. With these values for $\Gamma_{Z_{c}(3900)\to
D^{0}{\bar{D}}^{*0}+{\bar{D}}^{0}D^{*0}}=35\pm 19$ MeV BESIII:2015ntl  and
$\Gamma_{Z_{c}(4020)\to D^{*0}\bar{D}^{*0}}=23\pm 6$ MeV BESIII:2015tix , one
can get the relevant coupling constants as follows,

$\displaystyle|z|$ $\displaystyle=$ $\displaystyle(1.67\pm 0.45)~{}{\rm
GeV}^{-1/2}\,,$ (4) $\displaystyle|z^{\prime}|$ $\displaystyle=$
$\displaystyle(0.43\pm 0.06)~{}{\rm GeV}^{-1/2}\,.$ (5)

where the errors are obtained with the uncertainties of the partial width of
$Z_{c}(3900)\to D^{0}{\bar{D}}^{*0}+{\bar{D}}^{0}D^{*0}$ and $Z_{c}(4020)\to
D^{*0}\bar{D}^{*0}$ decays BESIII:2015ntl ; BESIII:2015tix . Note that to get
these above coupling constants, we have used $M_{Z_{c}(3900)}=3885.7$ MeV
BESIII:2015ntl , $M_{Z_{c}(4020)}=4025.5$ MeV BESIII:2015tix , $M_{D}=1864.83$
MeV and $M_{D^{*}}=2006.85$ MeV as quoted in the PDG Zyla:2020zbs .

On the other hand, the leading order Lagrangian for the coupling of the P-wave
charmonium fields to the charmed and anticharmed mesons can be constructed
considering parity, charge conjugation conjugation, and spin symmetry
Casalbuoni:1996pg , which can be written as

$\displaystyle{\cal L}_{\chi}$ $\displaystyle=$ $\displaystyle
i\frac{g_{1}}{2}Tr[\chi^{\dagger i}H_{a}\sigma^{i}\bar{H}_{a}]+H.c.\,,$ (6)

where ${\bar{H}}_{a}=-\vec{\bar{V}}_{a}\cdot\vec{\sigma}+\bar{P}_{a}$ is the
anti-charmed mesons fields. $g_{1}$ is the coupling constant of the ground
P-wave charmonia to the charmed and anticharmed mesons. The Lagrangians for
the coupling of the radial excited charmonia to the charmed and anticharmed
mesons have the same form as Eq. (6) with the coupling constants changed to
those for the excited states $g_{1}^{\prime}$. Then, the Lagrangian of
$\chi_{cJ}$ reads

$\displaystyle{\cal L}_{\chi}$ $\displaystyle=$ $\displaystyle
ig_{1}\chi_{c2}^{\dagger
ij}(V_{a}^{i}{\bar{V}}_{a}^{j}+V_{a}^{j}{\bar{V}}_{a}^{i})+\sqrt{2}g_{1}\chi_{c1}^{\dagger
i}(V_{a}^{i}{\bar{P}}_{a}+P_{a}{\bar{V}}_{a}^{i})$ (7)
$\displaystyle+\frac{i}{\sqrt{3}}g_{1}\chi_{c0}^{\dagger}(\vec{V}_{a}\cdot\vec{\bar{V}}_{a}+3P_{a}{\bar{P}}_{a})\,,$

where the trace and symmetry properties $\chi_{c2}^{ij}\delta^{ij}=0$ and
$\chi_{c2}^{ij}\epsilon^{ijk}=0$ have been used in the derivations. In this
work we take $g_{1}=4$ ${\rm GeV}^{-1/2}$ which is from an estimation using
the vector meson dominance in Ref. Colangelo:2003sa . With
$M_{\chi_{c0}^{\prime}}=3.869$ GeV and $\Gamma_{\chi_{c0}^{\prime}\to
D\bar{D}}=22.3$ MeV calculated in LP model Deng:2016stx ; Gui:2018rvv , we get
$g_{1}^{\prime}\simeq 1.28$ ${\rm GeV}^{-1/2}$.

Finally, we also need the effective Lagrangians for these vertexes of charmed
mesons and photon, which are Amundson:1992yp ; Hu:2005gf ; Guo:2013zbw

$\displaystyle{\cal L}_{\gamma}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{e\beta}{2}Tr[H_{a}^{\dagger}H_{b}\vec{\sigma}\cdot\vec{B}Q_{ab}]+\frac{eQ^{\prime}}{2m_{Q}}Tr[H_{a}^{\dagger}\vec{\sigma}\cdot\vec{B}H_{a}]\,,$
(8)

where $Q={\rm diag}\\{2/3,-1/3,-1/3\\}$ is the light quark charge matrix, and
$Q^{\prime}$ is the heavy quark electric charge (in units of $e$). $\beta$ is
and effective coupling constant and, in this work, we take $\beta=3.0$ GeV-1
that is determined in the nonrelativistic constituent quark model and is
adopted in the study of radiative $D^{*}$ decays Amundson:1992yp . In Eq. (8),
the first term is the magnetic moment coupling of the light quarks, while the
second one is the magnetic moment coupling of the heavy quark and hence is
suppressed by $1/m_{Q}$.

Then, follow the works of Ref. Cleven:2013sq , where the decays of
$Z_{b}(10610)/Z_{b}(10650)\to\gamma\chi_{bJ}(J=1,2,3)$ were investigated
within the same framework as here. The transition amplitudes for
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$ are
similar to those given in Ref. Cleven:2013sq , where one just needs to change
bottom quark to charm quark sector. Thus the decay amplitudes of these
triangle diagrams shown in Fig. 1 can be easily obtained. We present the
explicit transition amplitudes for
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$ in the
Appendix A.

Finally, the partial decay width of
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$ are
given by

$\displaystyle\Gamma(Z_{c}(3900)\to\gamma\chi_{cJ}^{(\prime)})$
$\displaystyle=$ $\displaystyle\frac{E_{\gamma}|{\cal
M}_{Z_{c}(3900)\to\gamma\chi_{cJ}^{(\prime)}}|^{2}}{24\pi
M_{Z_{c}(3900)}^{2}}\,,$ (9)
$\displaystyle\Gamma(Z_{c}(4020)\to\gamma\chi_{cJ}^{(\prime)})$
$\displaystyle=$ $\displaystyle\frac{E_{\gamma}^{\prime}|{\cal
M}_{Z_{c}(4020)\to\gamma\chi_{cJ}^{(\prime)}}|^{2}}{24\pi
M_{Z_{c}(4020)}^{2}}\,,$ (10)

where $E_{\gamma}$ and $E^{\prime}_{\gamma}$ are the photon energies in the
$Z_{c}(3900)$ and $Z_{c}(4020)$ rest frame, respectively.

##  III Numerical results and discussions

The $X(3927)$ was observed in the $\gamma\gamma\to D\bar{D}$ process by Belle
Uehara:2005qd  and Babar Aubert:2010ab  Collaborations, and has been a good
candidate for $\chi_{c2}^{\prime}$ state Li:2009zu . In this work, we take
$M_{\chi_{c2}^{\prime}}=3.927$ GeV as measured by the experiments Zyla:2020zbs
. In the observed spectrum of the charmonia, the only candidate of the
$\chi_{c1}^{\prime}$ with well-established quantum numbers is $X(3872)$, which
was discovered by the Belle Collaboration Belle:2003nnu . However, since the
proximity to the $D{\bar{D}}^{*}$ threshold, the interpretation of the
$X(3872)$ as a molecular state or virtual state is very intriguing. The
measured mass of $X(3872)$ is much lighter mass than potential quark model
predictions Deng:2016stx ; Li:2009zu ; Barnes:2005pb . Thus we will not
identify directly the $X(3872)$ as the $\chi_{c1}^{\prime}$, and the mass of
the $\chi_{c1}^{\prime}$ will also be allowed to vary. To be specific, a range
from $3.83$ to $3.93$ GeV for the mass of $\chi^{\prime}_{c1}$ and $X(3872)$
will be chosen that covers the predicted values from quark models Deng:2016stx
; Li:2009zu ; Barnes:2005pb .

The $X(3860)$ observed in the process $e^{+}e^{-}\to J/\psi D\bar{D}$ by the
Belle collaboration Chilikin:2017evr  serves as a good candidate for the
$\chi_{c0}^{\prime}$ state. The measured mass and width fit the expectation of
the $\chi_{c0}^{\prime}$ state predicted in the potential models Deng:2016stx
. Thus, we will take the mass of the $\chi_{c0}^{\prime}$ in a range from
$3800$ to $3900$ MeV which covers the predicted values of the mass of
$\chi_{c0}^{\prime}$ from these quark models Deng:2016stx ; Li:2009zu ;
Barnes:2005pb  and the experimental measurements Chilikin:2017evr . In Table.
2, we list the obtained charmonium masses in previous works. One can see that
these values from different models are consistent with each other.

Table 2: The final charmonium masses (in unit of GeV) are taken from the PDG Zyla:2020zbs , the calculated with screened potential (SP) in Ref. Li:2009zu , linear potential (LP) in Ref. Barnes:2005pb , and the results with SP and LP in Ref. Deng:2016stx . Name | $J^{PC}$ | Exp. Zyla:2020zbs  | Barnes:2005pb  | Li:2009zu  | LP Deng:2016stx  | SP Deng:2016stx
---|---|---|---|---|---|---
$\chi_{c0}$ | $0^{++}$ | $3.415$ | $3.424$ | $3.433$ | $3.415$ | $3.415$
$\chi_{c1}$ | $1^{++}$ | $3.511$ | $3.505$ | $3.510$ | $3.516$ | $3.521$
$\chi_{c2}$ | $2^{++}$ | $3.556$ | $3.556$ | $3.554$ | $3.552$ | $3.553$
$\chi_{c0}^{\prime}$ | $0^{++}$ | $3.862?$ | $3.852$ | $3.842$ | $3.869$ | $3.848$
$\chi_{c1}^{\prime}$ | $1^{++}$ | $...$ | $3.925$ | $3.901$ | $3.937$ | $3.914$
$\chi_{c2}^{\prime}$ | $2^{++}$ | $3.927$ | $3.972$ | $3.937$ | $3.967$ | $3.937$

In Table 3, we listed the calculated partial widths of
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}$, which are obtained with
$M_{Z_{c}(3900)}=3885.7$ MeV, $M_{Z_{c}(4020)}=4025.5$ MeV,
$M_{\chi_{c0}}=3414.7$ MeV, $M_{\chi_{c1}}=3510.7$ MeV, and
$M_{\chi_{c2}}=3556.2$ MeV. It is found that these partial decay widths of
$Z_{c}(3900)\to\gamma\chi_{cJ}$ are about a few hundreds of keVs, while the
obtained partial decay widths $Z_{c}(4020)\to\gamma\chi_{cJ}$ are about tens
of keVs.

Table 3: The decay widths of $Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}$ in the unit of keV. Decay mode | $\gamma\chi_{c0}$ | $\gamma\chi_{c1}$ | $\gamma\chi_{c2}$
---|---|---|---
$Z_{c}(3900)$ | 499.5 | 608.6 | 593.4
$Z_{c}(4020)$ | 33.4 | 99.4 | 108.9

Next we study the mass effects of $\chi^{(\prime)}_{cJ}$ on these partial
decay widths. In Fig. 2, we show the predicted partial widths of
$Z_{c}(3900)\to\gamma\chi_{c0}^{\prime}$ (solid line) and
$Z_{c}(4020)\to\gamma\chi_{c0}^{\prime}$ (dashed line) as a function of the
mass of $\chi_{c0}^{\prime}$. The results show that the partial widths of
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{c0}^{\prime}$ are less than 1 keV. For
the decay of $Z_{c}(3900)\to\gamma\chi_{c0}^{\prime}$ in Fig. 1, there are two
subdiagrams: $[D{\bar{D}}^{*}]D^{*}$ and $[D^{*}{\bar{D}}]D$. The threshold of
the $D{\bar{D}}$ is approached at the lower end. Consequently, the curve for
the width shows an increasing tendency at the lower end. While for
$Z_{c}(4020)\to\gamma\chi_{c0}^{\prime}$ in Fig. 1, only
$[D^{*}{\bar{D}}^{*}]D^{*}$ subdiagram contributes. Due to the threshold of
$D^{*}{\bar{D}}^{*}$ is far away the range of $M_{\chi_{c0}^{\prime}}$ that we
have chosen, the results show an monotonous behavior. Besides, with
$M_{\chi_{c0}^{\prime}}=3.862$ GeV obtained in Ref. Chilikin:2017evr , the
predicted partial widths are

$\displaystyle\Gamma(Z_{c}(3900)\to\gamma\chi_{c0}^{\prime})$ $\displaystyle=$
$\displaystyle 0.03\,{\rm keV}\,,$
$\displaystyle\Gamma(Z_{c}(4020)\to\gamma\chi_{c0}^{\prime})$ $\displaystyle=$
$\displaystyle 0.33\,{\rm keV}\,.$ (11)

Figure 2: Dependence of the decay widths of the
$Z_{c}(3900)\to\gamma\chi_{c0}^{\prime}$ (solid line) and
$Z_{c}(4020)\to\gamma\chi_{c0}^{\prime}$ (dashed line) on the mass of
$\chi_{c0}^{\prime}$.

In Fig. 3, we show the numerical results for the partial widths of
$Z_{c}(3900)\to\gamma\chi_{c1}^{\prime}$ (solid line) and
$Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ (dashed line) as a function of the
mass of $\chi_{c1}^{\prime}$. The predicted partial widths of
$Z_{c}(3900)\to\gamma\chi_{c1}^{\prime}$ are only a few KeVs, while the
partial widths of $Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ can reach up to
$12$ KeV. For the $Z_{c}(3900)\to\gamma\chi_{c1}^{\prime}$ process, the
$[D^{*}{\bar{D}}]D^{*}$ subdiagram contributes, and it has very small phase
space. While for the $Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ process, the
$[D^{*}{\bar{D}}^{*}]D$ subdiagram contributes. From Fig. 3 one can see that
there are two obvious cusps near the thresholds of $D^{*0}{\bar{D}}^{0}$ and
$D^{*+}D^{-}$, respectively. These cusps appear due to the triangle
singularities near the thresholds of neutral and charged $D{\bar{D}}^{*}$
mesons. And if we take $M_{\chi_{c1}^{\prime}}=3.872$ GeV as the mass of
$X(3872)$, the predicted partial widths are

$\displaystyle\Gamma(Z_{c}(3900)\to\gamma\chi_{c1}^{\prime})$ $\displaystyle=$
$\displaystyle 0.11{\rm keV}\,,$
$\displaystyle\Gamma(Z_{c}(4020)\to\gamma\chi_{c1}^{\prime})$ $\displaystyle=$
$\displaystyle 8.76{\rm keV}\,.$ (12)

Figure 3:  Dependence of the decay widths of the
$Z_{c}(3900)\to\gamma\chi_{c1}^{\prime}$ (solid line) and
$Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ (dashed line) on the mass of the
$\chi_{c1}^{\prime}$.

In Fig. 4, we present the dependence of the decay widths of the
$Z_{c}(4020)\to\gamma\chi_{c2}^{\prime}$ on the mass of the
$\chi_{c2}^{\prime}$. The predicted partial widths of
$Z_{c}(4020)\to\gamma\chi_{c2}^{\prime}$ are less than $1$ keV. The
$[D^{*}{\bar{D}}^{*}]D^{*}$ subdiagram contributes this process. From Fig. 4,
it is shown that the partial width monotonically decrease with the mass of
$\chi_{c2}^{\prime}$. On the other hand, if we take
$M_{\chi_{c2}^{\prime}}=3.927$ GeV as the mass of the $X(3927)$ state, the
predicted partial width is

$\displaystyle\Gamma(Z_{c}(4020)\to\gamma\chi_{c2}^{\prime})$ $\displaystyle=$
$\displaystyle 0.62{\rm keV}\,.$ (13)

Figure 4: Dependence of the decay widths of the
$Z_{c}(4020)\to\gamma\chi_{c2}^{\prime}$ on the mass of the
$\chi_{c2}^{\prime}$.

From Eqs. (9) and (10) we know that these partial decay widths are
proportional to the product of the coupling constants
$|z^{(\prime)}g_{1}^{(\prime)}|^{2}$, which will be cancelled in the ratio
between different partial decay widths. Therefore, the ratios among these
partial decay widths are interesting, and we define

$\displaystyle R_{1}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z_{c}(3900)\to\gamma\chi_{c1}^{\prime})}{\Gamma(Z_{c}(3900)\to\gamma\chi_{c0}^{\prime})}\,,$
(14) $\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z_{c}(4020)\to\gamma\chi_{c1}^{\prime})}{\Gamma(Z_{c}(4020)\to\gamma\chi_{c0}^{\prime})}\,,$
(15) $\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(Z_{c}(4020)\to\gamma\chi_{c2}^{\prime})}{\Gamma(Z_{c}(4020)\to\gamma\chi_{c0}^{\prime})}\,.$
(16)

The numerical results of ratio $R_{1}$ in terms of the mass of $Z_{c}(3900)$
are shown in Fig. 5. It is seen that there is no cusp structure because the
mass of $Z_{c}(3900)$ chosen here is above the mass threshold of
$D{\bar{D}}^{*}$. In addition, the ratio is larger than one in the considered
mass range of $Z_{c}(3900)$.

Figure 5: Numerical results of $R_{1}$ as a function of the mass of
$Z_{c}(3900)$.

In Fig. 6, we present the theoretical results of the ratios $r_{1}$ (solid
curve) and $r_{2}$ (dashed curve) as a function of the mass of $Z_{c}(4020)$.
The ratio $r_{1}$ can reach up to about one thousand, which shows that the
$Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ is dominant and could be easier
measured by experiments. For the ratio $r_{1}$, there is a double-cusp
structure, which correspond to the thresholds of the neutral and charged
$D^{*}{\bar{D}}^{*}$ mesons. On the other hand, the ratio $r_{2}$ are much
less dependent on the mass of $Z_{c}(4020)$ and its value is about one.

Figure 6: Numerical results for $r_{1}$ (solid line) and $r_{2}$ (dashed line)
as a function of the mass of $Z_{c}(4020)$.

##  IV SUMMARY

In this work, we have investigated the radiative decays
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}(\gamma\chi_{cJ}^{\prime})$
($J=0,1,2$), with the assumption that the $Z_{c}(3900)$ and $Z_{c}(4020)$
couple strongly to $D\bar{D}^{*}+c.c$ and $D^{*}{\bar{D}}^{*}$ channel,
respectively. Including the contributions of intermediate charmed mesons
triangle loops, these partial decay widths of $Z_{c}(3900)\to\gamma\chi_{cJ}$
and $Z_{c}(4020)\to\gamma\chi_{cJ}$ are calculated within an effective
Lagrangian approach. With the masses quoted in the PDG Zyla:2020zbs , the
obtained partial decay widths of $Z_{c}(3900)\to\gamma\chi_{cJ}$ are about of
few hundreds keVs, while the obtained partial widths of
$Z_{c}(4020)\to\gamma\chi_{cJ}$ are about tens of keVs. In addition, for the
$Z_{c}(3900)\to\gamma\chi_{c0,1}^{\prime}$ and
$Z_{c}(4020)\to\gamma\chi_{c0,2}^{\prime}$ decays, the partial decay width are
less than 1 keV, which mainly due to the very small phase space. For the
$Z_{c}(4020)\to\gamma\chi_{c1}^{\prime}$ process, the partial width can reach
up to about 10 keV. Moreover, the dependence of these ratios between different
decay modes on the masses of $Z_{c}(3900)$ or $Z_{c}(4020)$ are also
investigated, which may be a good quantity for the experiments. These
calculations here could be tested by future experiments.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China,
under Grants Nos. 12075133, 11835015, 11975165, 12075288, 11735003, and
11961141012. It is also partly supported by Taishan Scholar Project of
Shandong Province (Grant No. tsqn202103062), the Higher Educational Youth
Innovation Science and Technology Program Shandong Province (Grant No.
2020KJJ004), the Youth Innovation Promotion Association CAS, and the Chongqing
Natural Science Foundation under Project No. cstc2021jcyj-msxmX0078.

##  Appendix A The transition amplitudes

Here we give the amplitudes for the transitions
$Z_{c}(3900)/Z_{c}(4020)\to\gamma\chi_{cJ}$. $\epsilon_{1}$, $\epsilon_{2}$
and $\epsilon_{3}$ are the polarization vector of initial state, final photon
and final charmonium state, respectively.

(i) $Z_{c}(3900)\to\gamma\chi_{c0}$

$\displaystyle\mathcal{M}_{[D,\bar{D}^{*},D^{*}]}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}ezg_{1}\epsilon_{ijk}\epsilon^{i}_{1}\epsilon^{j}_{2}q^{k}\left(\beta
Q+\frac{Q^{\prime}}{m_{c}}\right)$ $\displaystyle\times
I(q,D,\bar{D}^{*},D^{*})\,,$ $\displaystyle\mathcal{M}_{[D^{*},\bar{D},D]}$
$\displaystyle=$
$\displaystyle-\frac{3}{\sqrt{6}}ezg_{1}\epsilon_{ijk}\epsilon^{i}_{1}\epsilon^{j}_{2}q^{k}\left(\beta
Q+\frac{Q^{\prime}}{m_{c}}\right)$ (17) $\displaystyle\times
I(q,D^{*},\bar{D},D)\,.$

(ii) $Z_{c}(3900)\to\gamma\chi_{c1}$

$\displaystyle\mathcal{M}_{[D^{*},\bar{D},D^{*}]}$ $\displaystyle=$
$\displaystyle-ezg_{1}\left[\epsilon_{1}\cdot
q\epsilon_{2}\cdot\epsilon_{3}-q\cdot\epsilon_{3}\epsilon_{1}\cdot\epsilon_{2}\right]\times$
(18) $\displaystyle\left(\beta
Q-\frac{Q^{\prime}}{m_{c}}\right)I(q,D^{*},\bar{D},D^{*})\,.$

(iii) $Z_{c}(3900)\to\gamma\chi_{c2}$

$\displaystyle\mathcal{M}_{[D,\bar{D}^{*},D^{*}]}$ $\displaystyle=$
$\displaystyle-\sqrt{2}ezg_{1}\epsilon_{ijk}\epsilon^{il}_{3}\epsilon^{j}_{2}q^{k}\epsilon_{1l}\left(\beta
Q+\frac{Q^{\prime}}{m_{c}}\right)$ (19) $\displaystyle\times
I(q,D,\bar{D}^{*},D^{*}).$

(iv) $Z_{c}(4020)\to\gamma\chi_{c0}$

$\displaystyle\mathcal{M}_{[D^{*},\bar{D}^{*},D^{*}]}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{6}}iez^{\prime}g_{1}\epsilon_{ijk}\epsilon^{i}_{1}\left(q^{k}\epsilon_{2}^{j}-q^{j}\epsilon_{2}^{k}\right)\times$
(20) $\displaystyle\left(\beta
Q-\frac{Q^{\prime}}{m_{c}}\right)I(q,D^{*},\bar{D}^{*},D^{*})\,.$

(v) $Z_{c}(4020)\to\gamma\chi_{c1}$

$\displaystyle\mathcal{M}_{[D^{*},\bar{D}^{*},D]}$ $\displaystyle=$
$\displaystyle
iez^{\prime}g_{1}\left[\epsilon_{1}\cdot\epsilon_{2}q\cdot\epsilon_{3}-\epsilon_{2}\cdot\epsilon_{3}q\cdot\epsilon_{1}\right]$
(21) $\displaystyle\times(\beta
Q+\frac{Q^{\prime}}{m_{c}})I(q,D^{*},\bar{D}^{*},D).$

(vi) $Z_{c}(4020)\to\gamma\chi_{c2}$

$\displaystyle\mathcal{M}_{[D^{*},\bar{D}^{*},D^{*}]}$ $\displaystyle=$
$\displaystyle\sqrt{2}iez^{\prime}g_{1}\epsilon_{ijk}\epsilon^{i}_{1}\epsilon^{jl}_{3}\left[q_{l}\epsilon^{k}_{2}-q^{k}\epsilon_{2l}\right]\times$
(22) $\displaystyle\left(\beta
Q-\frac{Q^{\prime}}{m_{c}}\right)I(q,D^{*},\bar{D}^{*},D^{*})\,.$

In the above amplitudes, the basic three-point loop function $I(q)$ as follows
Guo:2010ak :

$\displaystyle I(q)$ $\displaystyle=$ $\displaystyle
i\int\frac{d^{d}l}{(2\pi)^{d}}\frac{1}{(l^{2}-m_{1}^{2}+i\epsilon)[(P-l)^{2}-m_{2}^{2}+i\epsilon]}\frac{1}{[(l-q)^{2}-m_{3}^{2}]+i\epsilon}$
(23) $\displaystyle=$ $\displaystyle\frac{\mu_{12}\mu_{23}}{16\pi
m_{1}m_{2}m_{3}}\frac{1}{\sqrt{a}}\left[{\rm
tan}^{-1}\left(\frac{c^{\prime}-c}{2\sqrt{ac}}\right)+{\rm
tan}^{-1}\left(\frac{2a+c^{\prime}-c}{2\sqrt{a(c^{\prime}-a)}}\right)\right]\,,$

where the $\mu_{ij}=m_{i}m_{j}/{(m_{i}+m_{j})}$ are the reduced masses,
$b_{12}=m_{1}+m_{2}-M$, $b_{23}=m_{2}+m_{3}+q^{0}-M$, and the $M$ represents
the mass of initial particle.
$a=\left(\mu_{23}/{m_{3}}\right)^{2}\vec{q}^{2}$, $c=2\mu_{12}b_{12}$,
$c^{\prime}=2\mu_{23}b_{23}+\mu_{23}\vec{q}^{2}/{m_{3}}$. $m_{1}$, $m_{2}$ and
$m_{3}$ represent the masses of up, down and right charmed mesons in the
triangle loop, respectively.

It is worth to mention that considering the non-relativistic normalization of
the charmonium and charmed meson fields, a factor
$\sqrt{M_{i}M_{f}}m_{1}m_{2}m_{3}$ should be multiplied in each amplitude.
Therefore, the $I(q,M1,M2,M3)=\sqrt{M_{i}M_{f}}m_{1}m_{2}m_{3}I(q)$, where
$M_{i}$ and $M_{f}$ represent the masses of initial and final particle,
respectively.

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