# Quark mass dependence of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$
resonances

F. Gil-Domínguez<EMAIL_ADDRESS>Departamento de Física Teórica and
IFIC, Centro Mixto Universidad de Valencia-CSIC, Parc Científic UV, C/
Catedrático José Beltrán, 2, 46980 Paterna, Spain  R. Molina
<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia-CSIC, Parc Científic UV, C/ Catedrático José Beltrán,
2, 46980 Paterna, Spain

###### Abstract

We determine the quark mass dependence - light and heavy - of the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ properties, such as, mass, coupling to
$D^{(*)}K$, scattering lengths and compositeness, from a global analysis of
$D^{(*)}K$ energy levels from LQCD. In particular, we analyze the HSC energy
levels for $DK$ scattering in $I=0$ for different boosts and two pion masses.
The formalism is based in the local hidden-gauge interaction of Weinberg-
Tomozawa type which respects both, chiral and heavy quark spin symmetries,
supplemented by a term that takes into account the $D^{(*)}K$ coupling to a
bare $c\bar{s}$ component. The isospin violating decay of the
$D_{s0}^{*}(2317)\to D_{s}^{+}\pi^{0}$ is also evaluated.

##  I Introduction

In order to describe the heavy meson spectrum, it is useful to think on the
restrictions arising from the heavy quark limit ($m_{Q}\to\infty$). As it is
well-known, the interaction becomes independent on the spin of the heavy quark
in this limit due to the inverse color-magnetic-moment dependence on the heavy
quark mass. Then, the heavy and light degrees of freedom are separately
conserved and heavy mesons form mass doublets whose components are related by
a flip of the heavy quark spin. Regarding the charmed-strange meson spectrum,
if one assumes that the mass of the charm quark is sufficiently heavy to
consider this limit, then, two degenerate states for $L=0$ emerge for the
ground state with spin-parity $J^{P}=0^{-},1^{-}$, being the total spin of the
light degrees of freedom $J^{P}_{l}=\frac{1}{2}^{-}$. These states correspond
naturally to the $D$ and $D^{*}$ mesons. On the other hand, for $l=1$, one has
two doublets, the first one for $J^{P}_{l}=\frac{1}{2}^{+}$,
$J^{P}=0^{+},1^{+}$, which in principle might be related to the observed
states $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ Aubert et al. (2003); Besson et
al. (2003), and the other for $J^{P}_{l}=\frac{3}{2}^{+}$,
$J^{P}=1^{+},2^{+}$, could be associated to the $D_{s1}(2536)$ and
$D_{s2}(2573)$. Still, the constituent quark model predicts the
$J^{P}_{l}=\frac{1}{2}^{+}$ doublet to be almost degenerate in mass and quite
broad decaying to $D^{(*)}K$ Godfrey and Isgur (1985); Godfrey and Kokoski
(1991); Di Pierro and Eichten (2001). Nevertheless, what is observed is a
couple of narrow states whose masses differ in about the $m_{D^{*}}-m_{D}$
splitting, being the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ very close to the
$DK$ and $D^{*}K$ thresholds respectively. Moreover, these two states have the
same binding energy relative to these thresholds, $\sim 45$ MeV. For these
reasons, molecular explanations have been proposed Barnes et al. (2003);
Kolomeitsev and Lutz (2004); Faessler et al. (2007a, b); Lutz and Soyeur
(2008); Gamermann et al. (2007); Gamermann and Oset (2007). From the
experimental point of view, several new exotic states in the charm and
strangeness sector have been observed in the last years. Remarkably, new
states close to $D^{*}K^{*}$ threshold, $T_{cs}(2900)$ and
$T_{c\bar{s}}(2900)$, have been recently observed, which were predicted
earlier in Molina et al. (2010). See also Molina and Oset (2023).
Consequently, the study of exotic states with charm and strangeness remains
interesting.

From lattice QCD simulations, reasonable descriptions of both, the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ have been obtained when $D^{(*)}K$
interpolators are included Bali et al. (2017); Mohler et al. (2013); Lang et
al. (2014); Liu et al. (2013); Cheung et al. (2021); Alexandrou et al. (2020),
while their masses are overestimated if omitted. In Martínez Torres et al.
(2015), the lattice energy levels of Mohler et al. (2013); Lang et al. (2014)
are reanalyzed using an auxiliary potential for $D^{(*)}K$, which comes from a
linear potential in the center-of-mass energy and a Castillejo-Dalitz-Dyson
(CDD) pole Castillejo et al. (1956). This results in a couple of bound states
with binding energy of $\sim 40$ MeV with respect to the $DK$ and $D^{*}K$
thresholds, with a probability of $70\%$ for the $DK$ and $D^{*}K$ components
in the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ states, respectively. This
outcome encounters consistency with predictions for $I=0$ $DK$ scattering from
previous analyses using Heavy Meson Chiral Perturbation Theory (HM$\chi$PT)
for different sets of lattice data on scattering lengths for $I=3/2$ $D\pi$,
$D_{s}\pi$, $D_{s}K$, $I=0,1$ $D\bar{K}$, with pion masses in the range of
$300-550$ Liu et al. (2013). In Liu et al. (2013), the quark mass dependence
of the scattering lengths are also extracted, and the isospin-breaking decay
width $\Gamma(D_{s0}^{*}(2317)\to D_{s}\pi)$ is predicted to be $133\pm 22$
KeV being consistent within the current experimental upper limit of $3.8$ MeV.
This decay seems to be very sensitive to the internal structure of the
$D_{s0}^{*}(2317)$. If the $D_{s0}^{*}(2317)$ were a $c\bar{s}$ state, in Han
et al. (2023), using the Bethe-Salpeter equation to describe a relativistic
bound state, a decay width of $\Gamma(D_{s0}^{*}(2317)\to
D_{s}\pi=7.83^{+1.97}_{-1.55}$ KeV has been predicted (see also Colangelo and
De Fazio (2003); Colangelo et al. (2005)). In contrast, the latest
calculations in the molecular model, give $120^{+18}_{-4}$ KeV Fu et al.
(2022) for this decay (see also Cleven et al. (2014)).

In Yao et al. (2015) the analysis of Liu et al. (2013) is extended to leading
one-loop order (next-to-next leading order or NNLO) in the charmed-meson
light-meson scattering amplitudes including the $N_{f}=2+1$ lattice data of
Liu et al. (2013) for pion masses in the range $300-620$ MeV and Mohler et al.
(2013) for $m_{\pi}=156$ MeV. However, with seven LECs to be fitted to the
lattice data that were available ($\sim 20$ data points), the result of the
LECs bear large uncertainties. In principle, the effect of the $c\bar{s}$
degree-of-freedom is encoded in the LECs beyond LO in the chiral expansion and
in the resummation required by unitarity. Nevertheless, if these kind of
configurations are close by the energy region under consideration, it could be
more accurate to consider this degree of freedom explicitly Albaladejo et al.
(2018); Yang et al. (2022). In Albaladejo et al. (2018), the lattice data of
Bali et al. (2017) are analyzed, and a reasonably good description can be
obtained coupling the LO $D^{(*)}K$ interaction from HM$\chi$PT with
$c\bar{s}$. The data set used supersedes Mohler et al. (2013); Lang et al.
(2014) since Bali et al. (2017) uses larger volumes in a finer lattice spacing
and one order of magnitude more gauge configurations. However, in the
$N_{f}=2$ simulation with dynamical fermions of Bali et al. (2017), effects
emerging from strange sea quarks are neglected. In Albaladejo et al. (2018),
$DK$ and $D_{s}\eta$ channels are included in the analysis. Notably, similar
results are obtained when NLO terms in the scattering amplitude are included,
and also, taking into account two different renormalization schemes. The
amount of the molecular component found is $70\pm 10\%$ (including both
channels, $54-63\%$ corresponding to $DK$). This result is in contradiction to
the one obtained by the constituent quark model (CQM) Vijande et al. (2005);
Valcarce et al. (2005); Segovia et al. (2013) which uses the ${}^{3}P_{0}$
transition operator Le Yaouanc et al. (1973) to couple the quark-antiquark and
meson-meson d.o.f’s. As a result, the dressed state had a $DK$ meson pair
probability of around $33\%$.

$DK$ femtoscopic correlation functions have been recently estimated Liu et al.
(2023); Albaladejo et al. (2023); Ikeno et al. (2023). These observables can
also be useful to extract the interaction and scattering parameters, being
sensitive to the nature of exotic hadrons Ikeno et al. (2023).

In Cleven et al. (2011) a prediction of the pion and kaon mass dependence of
the $D_{s0}^{*}(2317)$ is provided in the framework of NLO HM$\chi$PT. There,
it is argued that, the light ($u,d$) quark mass dependence of the
$D_{s0}^{*}(2317)$ mass should be different for a molecular state compared to
a $c\bar{s}$ state. The main reason is that this dependence comes through
meson-meson loops, which depend on the square of the coupling of the resonance
to the pair of mesons, being this coupling much larger for molecular states.
In addition, the loop function peaks right at the threshold, the molecular
state following the threshold, which is linear in the meson masses. While the
masses of the charmed mesons grow slightly with the squared of the pion mass
Gil-Domínguez and Molina (2023), the mass of the kaon grows linearly with it,
being the pion mass proportional to the square root of the light quark mass.
However, for a compact state, the light quark mass dependence is always an
analytic function in the light quark masses which is quadratic in the
Goldstone-Boson masses. In Cleven et al. (2011) the type of pion mass
dependence obtained for the $D_{s0}^{*}(2317)$ pole is quadratic, while being
linear with slope close to $1$ for the kaon mass dependence.

Nonetheless, the overall quark mass dependence of the $D_{s0}^{*}(2317)$ and
its properties have not been extracted from the available bulk of LQCD
simulations yet. In this work we provide a LQCD bare analysis of the quark
mass dependence of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$. Concretely, the
recent data of Cheung et al. (2021) that investigate $I=0$ $DK$ scattering for
two pions masses, $m_{\pi}=239$, $391$ MeV are analyzed for the first time,
comparing with previous lattice QCD analyses, and, finally, we perform a
global fit of the available LQCD data Lang et al. (2014); Mohler et al.
(2013); Martínez Torres et al. (2015); Bali et al. (2017); Cheung et al.
(2021), extracting accurately the quark mass dependence, light and heavy, of
the $D_{s0}^{*}(2317)$ parameters, and its degree of molecular state. The
meson-meson interaction is provided by the hidden gauge lagrangian, which at
leading order coincides with the one of HM$\chi$PT supplied by a CDD pole
term.

Compared with previous studies Liu et al. (2013); Martínez Torres et al.
(2015); Cleven et al. (2014); Yao et al. (2015); Albaladejo et al. (2018) this
work considers the analysis of the recent data Cheung et al. (2021) and of all
available LQCD data in $DK$, $D^{*}K$ scattering, extracting the overall quark
mass dependence of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ properties.

##  II Formalism

Figure 1: Feynman diagrams of the $DK\rightarrow DK$ process. Figure 2:
Feynman diagram of the $DK\rightarrow DK$ process with an interaction with a
bare $c\bar{s}$ state.

We consider that the $DK\to DK$ scattering is driven by vector meson exchange
and evaluate it using the lagrangian from the Hidden Gauge Formalism (HGF)
Bando et al. (1985, 1988); Nagahiro et al. (2009); Meissner (1988),

$\displaystyle{\cal
L}=ig\mathrm{Tr}\left([\partial_{\mu}\phi,\phi]V^{\mu}\right)$ (1)

with $g=m_{\rho}/2f_{\pi}$, $f_{\pi}$ the pion decay constant and $m_{\rho}$
the $\rho$ meson mass. The matrices $\phi$ and $V$ contain the 16-plet of
pseudoscalar and vector mesons and are given in the Appendix. This lagrangian
is consistent with both, chiral and heavy quark spin symmetries, for the light
vector-meson exchange, since there the heavy quark is acting as an spectator
and the interaction is broken to SU(3) Xiao et al. (2013). Considering only
light vector meson exchange, see Fig. 1, this interaction leads to,

$\displaystyle V_{DK}=-\frac{s-u}{2f_{\pi}^{2}},$ (2)

where $s$ and $u$ are the Mandelstam variables. Note that in the above
equation the pion decay constant is not taken as a constant parameter, but its
pion mass dependence is taken from Molina and Ruiz de Elvira (2020). Note also
that one arrives to this interaction through the LO HM$\chi$PT Liu et al.
(2013); Albaladejo et al. (2018). A term associated with the interaction of
$DK$ with a bare $c\bar{s}$ state of the $J^{P}_{l}=\frac{1}{2}^{+}$ Heavy
Quark Spin Symmetry (HQSS) doublet can be added as shown in Fig. 2. At LO in
the heavy quark expansion this gives Albaladejo et al. (2018),

$\displaystyle V_{\mathrm{ex}}=\frac{V_{c\bar{s}}^{2}}{s-m_{c\bar{s}}^{2}}\ ,$
(3)

with

$\displaystyle
V_{c\bar{s}}(s)=-\frac{c}{f_{\pi}}\sqrt{M_{D}m_{c\bar{s}}}\frac{s+m_{K}^{2}-M_{D}^{2}}{\sqrt{s}}\
,$ (4)

where $m_{c\bar{s}}$ is the mass of the bare $c\bar{s}$ component,
$m_{c\bar{s}}=m_{c\bar{s}}^{0^{+}}$, and $c$ is a dimensionless constant that
provides the strength of the coupling of this component to the $DK$ channel.
This constant was evaluated in Albaladejo et al. (2018) from a fit to the LQCD
energy levels of the $b\bar{s}$ $J^{P}=\left\\{0,1\right\\}^{+}$, obtaining
$c=0.74\pm 0.05$. In this work we will consider this coupling as a free
parameter, together with $m_{c\bar{s}}$.

Col. | $a$ | $L$ | $m_{\pi}$ | $m_{D_{s}}$ | $M_{avg}$
---|---|---|---|---|---
HSC | 0.12 | 3.8 | 239 | 1967 | 3011
$1.9-2.9$ | 391 | 1951 | 3006
RQCD | 0.071 | $1.7-4.5$ | 150 | 1978 | 3068
$3.4-4.5$ | 290 | 1977 | 3068
Prelovsek et al. | 0.0907 | 2.90 | 156 | 1809 | 2701
0.1239 | 1.98 | 266 | 1657 | 2429
Table 1: Comparison of the charm quark mass settings of the different
collaborations through their value of $m_{D_{s}}$. Note that the physical
value is $m_{D_{s}}=1968.35\pm 0.07$ MeV Zyla et al. (2020). $L$ and $a$ are
given in fm, and the masses in units of MeV. In the last column, $M_{avg}$
stands for $M_{avg}=1/4(m_{\eta_{c}}+3m_{J\Psi})$.

Note that $m_{c\bar{s}}$ can vary for every LQCD simulation with a different
setup. In fact, these simulations have different values for the $D_{s}$ meson
mass, as shown in Table 1 where HSC refers to the Hadron Spectrum
Collaboration Cheung et al. (2021), RQCD to the work of Bali et al. (2017) and
Prelovsek et al. to Lang et al. (2014); Mohler et al. (2013) 111Note that the
values are taken from the corresponding references dividing by $a$. Also, for
the data set ”Prelovsek et al.” we have taken the rest mass. This work the
kinetic mass was tuned to the physical mass, values of the charm meson masses
at rest and kinetic extracted from the dispersion relation of Eq. 27 are
given. Here we take the rest masses for consistency with the energy levels
extracted..

The potential $V(s)$ consistent with HQSS is then given by,

$\displaystyle V(s)=V_{DK}(s)+V_{\mathrm{ex}}(s)\ .$ (5)

Thus the interaction considered here, Eqs. (2) and (3), is derived by vector-
meson exchange in $DK$ interaction supplied by the interaction with a bare
state.

The interaction can also be extended to include the coupling of the $DK$ to
the $D_{s}\eta$ channel. However, in the LQCD analyses, $D_{s}\eta$
interpolators are generally neglected when energies are extracted Lang et al.
(2014); Mohler et al. (2013); Martínez Torres et al. (2015); Bali et al.
(2017); Cheung et al. (2021); Alexandrou et al. (2020), and thus, we omit here
this channel. Indeed, the $D_{s}\eta\to D_{s}\eta$ interaction gives zero at
LO using the hidden gauge lagrangian of Eq. (1), but not $DK\to D_{s}\eta$.
For the two-couple channel case, naming $1$ and $2$ to the $DK$ and
$D_{s}\eta$ channels, the term of Eq. (2) should be multiplied by a matrix of
coefficients,

$\displaystyle A=\left(\begin{array}[]{cc}1&\sqrt{2/3}\\\
\sqrt{2/3}&0\end{array}\right)$ (8)

while for $V_{\mathrm{ex}}$ one multiplies by

$\displaystyle B=\left(\begin{array}[]{cc}1&1/\sqrt{6}\\\
1/\sqrt{6}&1/6\end{array}\right)$ (11)

and replaces $V_{c\bar{s}}^{2}$ in Eq. (3) by
$V_{c\bar{s}}^{i}V_{c\bar{s}}^{j}$, with $i,j=1,2$ channels, changing also the
mass of the $K$ and $D$ mesons in Eq. (4) by the one of the $\eta$ and $D_{s}$
mesons respectively for channel $2$. The constant $c$ in Eq. (4) is assumed to
be the same in both channels, i.e., equal to the SU(3) limit. For the
$\\{D^{*}K,D^{*}_{s}\eta\\}$ system, the equations are the same as Eqs.
(3)-(11), but changing $D\to D^{*}$, and $m_{c\bar{s}}=m_{c\bar{s}}^{1^{+}}$

For the analysis of the LQCD energies, the Bethe Salpeter equation in the
finite volume is solved,

$\widetilde{T}^{-1}=V^{-1}-\widetilde{G}\ ,$ (12)

where $\widetilde{T}$ is the scattering matrix in the finite volume, $V$ is
given by Eq. (2), and $\widetilde{G}$ is the two-meson loop function in the
box,

$\displaystyle\widetilde{G}(P^{0},\vec{P}\ )=G^{DR}(P^{0},\vec{P}\
)+\mathrm{lim}_{q_{\mathrm{max}}\to\infty}\Delta
G(P_{0},\vec{P},q_{\mathrm{max}})\ ,$ (13)

where $P^{\mu}=(P^{0},\vec{P}\ )$ is the full four-momentum of the two meson
system. The Mandelstam variable $s$ is related to the momentum as
$s=P_{0}^{2}-\vec{P}^{2}=P_{0}^{2}$. The first term of Eq. (13) is the two-
meson loop function that can be evaluated in dimensional regularization as,

$\displaystyle G^{DR}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\left(\alpha+\mathrm{Log}\frac{m_{1}^{2}}{\mu^{2}}+\frac{m^{2}_{2}-m^{2}_{1}+s}{2s}\mathrm{Log}\frac{m^{2}_{2}}{m_{1}^{2}}\right.$
(14)
$\displaystyle\left.+\frac{p}{\sqrt{s}}\left[\mathrm{Log}\left(s-m_{2}^{2}+m_{1}^{2}+2p\sqrt{s}\right)\right.\right.$
$\displaystyle\left.-\mathrm{Log}\left(-s+m_{2}^{2}-m_{1}^{2}+2p\sqrt{s}\right)\right.$
$\displaystyle\left.+\mathrm{Log}\left(s+m_{2}^{2}-m_{1}^{2}+2p\sqrt{s}\right)\right.$
$\displaystyle\left.\left.-\mathrm{Log}\left(-s-m_{2}^{2}+m_{1}^{2}+2p\sqrt{s}\right)\right]\right)\
,$

$p$ being the on-shell momentum in the center-of-mass frame,
$p=\lambda^{1/2}(s,m_{1}^{2},m_{2}^{2})/2\sqrt{s}$. In particular, in the
center-of-mass frame (cm), where $\vec{P}=\vec{0}$, the second term in Eq.
(13), $\Delta G(P_{0},\vec{P},q_{\mathrm{max}})$, can be written as,

$\displaystyle\Delta
G=\frac{1}{L^{3}}\sum_{q_{i}}^{q_{\mathrm{max}}}I(\vec{q}_{i})-\int_{0}^{\mathrm{q_{\mathrm{max}}}}\frac{d^{3}q}{(2\pi)^{3}}I(\vec{q}\
)\ ,$ (15)

being $\vec{q}$ the momentum in the cm frame, which takes discrete values in
the finite box,
$\vec{q}_{i}=\frac{2\pi}{L}\vec{n}_{i},\,\vec{n}_{i}\in\mathbb{Z}^{3}$ and $L$
the spacial extent of the box. In the continuum limit, the loop function is

$\displaystyle
G=i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{1}{q^{2}-m_{1}^{2}+i\epsilon}\frac{1}{(P-q)^{2}-m_{2}^{2}+i\epsilon},$
(16)

where $m_{1}$ and $m_{2}$ refer to masses of mesons 1 and 2 in the loop. The
above formula can be evaluated, using a cutoff, $q_{\mathrm{max}}$,

$\displaystyle
G^{co}=\int_{0}^{q_{\mathrm{max}}}\frac{d^{3}q}{(2\pi)^{3}}I(\vec{q}\ )\ ,$
(17)

where $co$ refers to the evaluation of the G function with the cutoff method
and,

$\displaystyle I(\vec{q}\
)=\frac{\omega_{1}(q)+\omega_{2}(q)}{2\omega_{1}(q)\omega_{2}(q)\left[P^{2}_{0}-(\omega_{1}(q)+\omega_{2}(q))^{2}+i\epsilon\right]}\
,$ (18)

with $\omega_{i}=\sqrt{q^{2}+m_{i}^{2}}$, and $q=|\vec{q}\ |$.

For the moving frame and partial wave decomposition of the scattering
amplitude in the finite volume, we follow the procedure described in Doring et
al. (2012). The two-meson loop function in the finite volume, $\widetilde{G}$
in Eq. (13) projected in partial waves, is a matrix with non-diagonal elements
due to the partial weave mixing. Its elements are,

$\displaystyle\widetilde{G}_{lm,l^{\prime}m^{\prime}}=\frac{4\pi}{L^{3}}\sum_{n_{i}}^{q_{\mathrm{max}}}\frac{\sqrt{s}}{P_{0}}\left(\frac{q}{q^{\mathrm{on}}}\right)^{k}Y_{lm}^{*}(\hat{q})Y_{l^{\prime}m^{\prime}}(\hat{q})I(q)\
,$

with $k=0,1$ for $l+l^{\prime}=$ even, odd Doring et al. (2012). The relevant
equation to evaluate the energy levels in the finite volume, is,

$\displaystyle\mathrm{det}(\delta_{ll^{\prime}}\delta_{mm^{\prime}}-V_{l}\tilde{G}_{lm,l^{\prime}m^{\prime}})=0$
(20)

$V_{l}$ is the interaction projected in $l$-wave. Here, we include only $S$
and $P$ partial waves of the interaction given in Eq. (2). The irreducible
representations which appear for a given external momenta are discussed in
section II.E of Doring et al. (2012).

In the following section we show the results of the analysis of the LQCD
energy levels of HSC Cheung et al. (2021) for $m_{\pi}=239$ (HSC239) and $391$
MeV (HSC391). We will compare with previous analyses of LQCD data from other
collaborations Liu et al. (2013); Lang et al. (2014); Mohler et al. (2013);
Bali et al. (2017) done in Liu et al. (2013); Martínez Torres et al. (2015);
Albaladejo et al. (2018). A combined fit of the most recent LQCD energy levels
Lang et al. (2014); Mohler et al. (2013); Bali et al. (2017); Cheung et al.
(2021) will be performed, and the quark mass dependence of the
$D_{s0}^{*}(2317)$ extracted from this fit. Here, we use the dimensional
regularization formula of Eq. (14) to evaluate the two meson loop function in
the finite volume of Eqs. (13) and (15). If we consider only the meson-meson
interaction given by Eq. (2), the only free parameter is the subtraction
constant, $\alpha$ (we fix $\mu=1000$ MeV Oller and Meissner (2001)). However,
there are energy levels for two different pion masses in the case of Cheung et
al. (2021). The subtraction constant can be different for the energy levels of
the light and heavy pion mass. Thus, we assume a first order taylor expansion
of the subtraction constant in the pion mass,
$\alpha=\alpha_{1}+\alpha_{2}m_{\pi}^{2}$. The effect of $\alpha_{2}$ can be
understood as to reabsorb possibly relevant higher order contributions in the
potential of Eqs. (2) and (5) in the Bethe-Salpeter equation. Then, we end up
with two free parameters for every data set when it is analyzed separately. On
the other hand, we also compare the result using the interaction of Eq. (2)
with the one adding a bare component, Eqs. (3)-(5). In this case we have a
couple of two more free parameters, $c$, and $m_{c\bar{s}}$. The mass
$m_{c\bar{s}}$ is then tuned for every LQCD collaboration according to Table
1.

The scattering amplitude in the infinite volume is,

$\displaystyle T^{-1}(s)=V^{-1}(s)-G(s)\ .$ (21)

For the energies in the finite volume, $V^{-1}(E^{0})=\widetilde{G}(E^{0})$,
and Eq. (21) becomes equivalent to the Lüscher formalism Luscher (1986, 1991)
up to exponentially suppressed corrections Doring et al. (2011). Near to a
pole, $s_{0}$, the scattering amplitude behaves as,

$\displaystyle T\simeq\frac{g_{i}g_{j}}{s-s_{0}}$ (22)

We will study the quark mass dependence of the pole position and its
compositeness Weinberg (1965). For that, we evaluate the probability of
finding the $DK$ molecular component in the wave function Gamermann et al.
(2010); Hyodo et al. (2012); Guo and Oller (2016),

$P_{i}=-g^{2}_{i}\left(\frac{\partial G}{\partial s}\right)_{s=s_{0}}\ .$ (23)

where $i$ refers to the $DK$ channel. In general, one has the sum rule
Gamermann et al. (2010); Hyodo et al. (2012); Guo and Oller (2016),

$\displaystyle-\sum_{i}g_{i}^{2}\frac{\partial G_{i}}{\partial
s}-\sum_{i,j}g_{i}g_{j}G_{i}\frac{\partial V_{ij}}{\partial s}G_{j}=1\ ,$ (24)

where the first term is associated to the compositeness ($1-Z$), and the
second one to the elementariness ($Z$).

Finally, the scattering length and effective range can be evaluated using the
effective range approximation, which reads

$p\mathrm{cot}\,\delta=\frac{1}{a}+\frac{1}{2}r_{0}p^{2}\ .$ (25)

##  III Results

###  III.1 Energy levels and quark mass dependence

We analyze the energy levels of Cheung et al. (2021) for $DK$ scattering in
$I=0$, with the interaction given by Eqs. (2) and (5), by solving Eq. (20),
with $\tilde{G}_{l^{\prime}m^{\prime},lm}$ given by Eq. (II). As fitting
parameters, we have, $c$, $m_{c\bar{s}}$, $\alpha_{1}$ and $\alpha_{2}$ for
the subtraction constants. The $\chi^{2}$ is given by

$\displaystyle\chi^{2}=\Delta E^{T}C^{-1}\Delta E$ (26)

where $\Delta E_{i}=E_{i}-E^{0}_{i}$, with $E^{0}_{i}$ the lattice energy $i$,
and $C$ de covariance matrix. The covariance matrix is taken from the analysis
of Cheung et al. (2021).

We consider two situations, one where the bare $c\bar{s}$ component is not
included, i.e., $V_{\mathrm{ex}}=0$, and other with $V_{\mathrm{ex}}\neq 0$.
We also compare the results from the fits of energy levels of Cheung et al.
(2021) with previous LQCD analyses Albaladejo et al. (2018); Martínez Torres
et al. (2015), for the LQCD data of Bali et al. (2017); Mohler et al. (2013);
Lang et al. (2014), and with the quark mass dependence obtained for the
scattering lengths with previous works Liu et al. (2013).

First, we show the results for the LQCD energy levels, Fit I
($V_{\mathrm{ex}}=0$), and II ($V_{\mathrm{ex}}\neq 0$) in Fig. 3 for
$\vec{P}=\vec{0}$ and $\vec{P}=(0,0,1)2\pi/L$. See also Figs. 20 and 21 in the
Appendix, sec. VI.2, for other boosts. The energy levels are well described in
both models, as shown in these figures. These barely change between Fit I and
II.

Figure 3: Result from the fit for the dependence with the spatial extent (L)
of the energy levels of HSC for $P=(0,0,0)$ and $P=(0,0,1)2\pi/L$. Continuous
lines are for Fit I, and dashed line for Fit II. The result for other momenta
can be found in the Appendix, Sec. VI.2, Fig. 20 and Fig. 21.

The parameters obtained in both fits are shown in Table 2.

| $c$ | ${m}_{c\bar{s}}^{L}$ | ${m}_{c\bar{s}}^{H}$ | $\alpha_{1}$ | $\alpha_{2}[(m_{\pi}^{0})^{-2}]$
---|---|---|---|---|---
Fit I | - | - | - | $-1.86(1)$ | $-0.041(12)$
Fit II | $1.02(4)$ | $2508(61)$ | $2445(43)$ | $-1.34(6)$ | $0.000(2)$
Table 2: Values of the parameters obtained in the fit of the HSC data sets, in
both fits, I and II, as explained in the text. $\alpha_{2}$ is given in units
of $(m_{\pi}^{0})^{-2}$, with $m_{\pi}^{0}$ the physical pion mass. The
superscript $L$ means light pion mass and $H$ stands for the heavier in Table
1.

The magnitudes of the parameters $\alpha_{1}$, $\alpha_{2}$ are significantly
reduced when the bare component is included. Indeed, the pion mass dependence
of the subtraction constant is now reabsorbed in the new term of the
interaction given by the coupling of the $c\bar{s}$ bare component to $DK$, so
that, we obtain $\alpha_{2}=0$ in Fit II. The fact that the energy levels are
well described in both cases, means that the $\alpha_{1}$, $\alpha_{2}$
parameters are able to absorb the coupling to the genuine component in the
simple model of Fit I. We find only one pole in this sector. The pole position
and coupling of the $D_{s0}^{*}(2317)$ to $DK$ are evaluated by using Eqs.
(21) and (22). The results for the pole position, couplings, and compositeness
($1-Z$), are given in Tables 3 and 4, for the two different pion masses
studied and the physical pion mass.

$m_{\pi}$ | $m_{\eta_{c}}$ | $\sqrt{s_{0}}$ |  $g$ [GeV] | $1-Z$
---|---|---|---|---
$138$ | $2984$ | $2321(4)$ | $10.42(10)$ | $0.732(8)$
$239$ | $2986^{*}$ | $2350(12)$ | $10.34(12)$ | $0.724(13)$
$391$ | $2964^{*}$ | $2384(14)$ | $11.13(11)$ | $0.659(17)$
Table 3: Fit I: Values of the pole mass, the coupling of the state to $DK$, and its compositeness for the different values of the pion and the $\eta_{c}$ meson masses. The masses and pole position, $\sqrt{s_{0}}$, are given in units of MeV. 222The values of the $\eta_{c}$ mass for the light and heavy pion mass are obtained from $(a_{t}m_{\eta_{c}})^{2}=0.2412,0.2735$, respectively Cheung et al. (2016); Liu et al. (2012). $m_{\pi}$ | $m_{\eta_{c}}$ | $\sqrt{s_{0}}$ |  $g$ [GeV] | $1-Z$
---|---|---|---|---
$138$ | $2984$ | $2320(4)$ | $10.47(9)$ | $0.727(8)$
$239$ | $2986^{*}$ | $2349(12)$ | $10.38(11)$ | $0.720(12)$
$391$ | $2964^{*}$ | $2383(14)$ | $11.13(11)$ | $0.659(17)$
Table 4: Same as Table 3 but for Fit II.

The result obtained for the physical point is in very good agreement with the
experiment.

We can explore further the quark mass dependence of the $D_{s0}^{*}(2317)$
properties by taking as input the study of the low-lying charmed meson masses
of Gil-Domínguez and Molina (2023), where this dependence is studied for the
$D^{(*)}_{(s)}$ mesons from an analysis of LQCD data with HH$\chi$PT. The
results are shown in Fig. 4. In the top panel, the pole position and the
binding energy is shown. The analysis of Gil-Domínguez and Molina (2023) takes
into account the different charm quark mass of the HSC LQCD data sets, which
are also slightly different than the physical charm quark mass. The binding
energies obtained in Fits I and II are plotted in the middle panels, where we
also show the extrapolation to $m_{c}=m_{c}^{0}$ and $m_{s}=m_{s}^{0}$ 333The
superscript $0$ stands for physical value. as a continuous black line for Fit
I. The error band englobes the physical point.

The coupling of the state to the $DK$ component is shown in the lower panel of
Fig. 4. We see that the coupling has a value between $(10-11)\cdot 10^{3}$ MeV
in the range of pion masses studied here. With this, we can evaluate the
compositeness, shown in Fig. 5. One can conclude that in the range of pion
masses studied, this is almost constant around $0.7-0.8$, slightly decreasing
when the state is moving further from the threshold and higher in Fit II.
These values are consistent with previous works Liu et al. (2013); Martínez
Torres et al. (2015); Albaladejo et al. (2018); Yang et al. (2022).

Figure 4: Pole mass, binding energy and coupling of the state to the $DK$
component as a function of the pion mass for the two HSC data sets. Continuous
lines indicate Fit I, while the dashed lines are for Fit II. The black lines
are showing the extrapolation to $m_{c}=m_{c}^{0}$, and $m_{s}=m_{s}^{0}$.

Figure 5: Evolution of the compositeness, $1-Z$, with the pion mass, for the
two different HSC data sets as a result of Fits I and II.

Figure 6: Evolution of the $D_{s0}^{*}(2317)$ pole mass with the kaon mass,
fixing $m_{\pi}$ to its physical value.

The type of pion mass dependence obtained here for the $D_{s0}^{*}(2317)$ pole
is quadratic for very light pion masses, as in Ref. Cleven et al. (2011).
However, for larger pion masses it clearly becomes linear, see Fig. 4. Thus,
the prediction of this dependence here is milder than in Cleven et al. (2011).
We also plot the pole evolution with the mass of the kaon for
$m_{\pi}=m_{\pi}^{0}$, the physical value, in Fig. 6. We obtain a linear
dependence with slope near $1$. This is in agreement with Cleven et al.
(2011).

Next, we compare with the results of previous works. Looking at the energy
levels of Bali et al. (2017) which were analyzed in Albaladejo et al. (2018),
and Lang et al. (2014); Mohler et al. (2013), analyzed in Martínez Torres et
al. (2015), we can ask ourselves how good is the prediction of the energy
levels of Bali et al. (2017); Lang et al. (2014); Mohler et al. (2013) using
the parameters given in Table 2, where we have selected Fit I. The prediction
of the energy levels using $\alpha_{1},a_{2}$ for the pion masses of Bali et
al. (2017); Albaladejo et al. (2018) ($150$ and $290$ MeV), and ensemble (1)
of Lang et al. (2014); Mohler et al. (2013); Martínez Torres et al. (2015)
($266$ MeV), are shown in Fig. 7 and 8 with black continuous lines444In Lang
et al. (2014); Mohler et al. (2013) energy levels are also given for another
ensemble, called ensemble (2), with $m_{\pi}=156$ MeV. However, we do not
include this ensemble here because, even though we obtain a good agreement for
the energy levels in our fit, the mass of the $D_{s0}^{*}(2317)$ obtained
differs significantly from the one given in Lang et al. (2014); Mohler et al.
(2013).. The agreement is indeed quite good. We also conduct a fit of those,
being the result shown in the same figure (of Fit I type) with dashed lines.
The energy levels barely change for the RQCD Collaboration (first and second
panels are for $m_{\pi}=150$ and $290$ MeV, respectively), while for the
[Prelovsek et al.] data set, these change only slightly (third panel is for
$266$ MeV), being still inside the error band.

Figure 7: Result from the fit for the dependence with the spatial extent (L)

of the $DK$ energy levels obtained using the parameters $\alpha_{1},a_{2}$
(Fit I) in Table 2, for $m_{\pi}=150,290$, Bali et al. (2017) (first and
second panels from the top) and $m_{\pi}=266$ MeV Lang et al. (2014); Mohler
et al. (2013) (third panel), compared to the result from individual fits of
Bali et al. (2017) and Lang et al. (2014); Mohler et al. (2013).

Figure 8: Same as Fig. 7 but with $D^{*}K$ energy levels.

These results suggest that it may be possible to perform a global fit with the
energy levels of HSC Cheung et al. (2021), and from Bali et al. (2017); Mohler
et al. (2013); Lang et al. (2014). Note that in Bali et al. (2017); Mohler et
al. (2013); Lang et al. (2014) the $D_{s1}(2460)$ has also been simulated
through the $D^{*}K$ scattering. We also include here the energy levels
corresponding to the $D_{s1}(2460)$. Thus, we perform a combined fit of the
one-channel $DK$ and $D^{*}K$ energy levels. In Fig. 10 we show the result for
the energy levels at $P=0$, for HSC, RQCD, and [Prelovsek et al.] in one
channel. The two first energy levels for the different data sets are
reproduced well. Note that in Lang et al. (2014); Mohler et al. (2013) a non
relativistic dispersion relation based on the so-call Fermilab approach is
used:

$E_{D(D^{*})}(\vec{p})=M_{1}+\frac{\vec{p}^{2}}{2M_{2}}-\frac{(\vec{p}^{2})^{2}}{8M_{4}^{3}},$
(27)

where $M_{1}=1561$ MeV, $M_{2}=1763$ MeV and $M_{4}=1763$ MeV for the $D$
meson and $M_{1}=1693$ MeV, $M_{2}=2018$ MeV and $M_{4}=2110$ MeV for the
$D^{*}$ meson. Note that, in Eq. (27), the masses $M_{1}$, $M_{2}$ and $M_{4}$
do not correspond to the same masses obtained in lattice approaches with
relativistic charm quarks. However, they should become the same in the
continum limit. More discussion on this can be found in El-Khadra et al.
(1997). We have checked the result of Fit I using this dispersion relation, as
done in the reanalysis of Ref. Martínez Torres et al. (2015), and we have
found that, a result very close to the one with a relativistic dispersion
relation is obtained. Also, ensemble (1) included in the global fit has a
negligible effect in the final result of the global fit. Indeed, we obtain the
same central value and error band if we do not include this ensemble, as shown
in Fig. 9. However, as mentioned before, the two first energy levels are well
described, see Fig. 10.

Figure 9: Pion mass dependence of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$
binding energies and splitting. Black point correspond to the result of Lang
et al. (2014); Mohler et al. (2013), the results of our individual fits are
shown with a star (relativistic approach) and green square (non relativistic
dispersion relation).

We would like to make a consideration. In Table 3 we showed the result of the
$D_{s0}^{*}(2317)$ pole for the physical point for the analysis of the energy
levels of the HSC. One can see that the mass of the $D_{s0}^{*}(2317)$ is
compatible with the experimental point. However, at the physical point, there
is one more channel near the $DK$ threshold, the $D_{s}\eta$ channel. To
understand the role of this channel at the physical point, we predict, with
the result of the global fit, the effect of the $D_{s}\eta$ channel for the
HSC data. This is shown in Fig. 10 with dashed lines. The coupling to the
$D_{s}\eta$ channel tends to lower down the energy levels suggesting an
attractive interaction. This can be understood as one can eliminate the
$D_{s}\eta$ channel introducing an effective potential for the case of two
channels as, $V^{\prime}_{11}=V_{11}+V_{12}^{2}G_{2}$, with $2:D_{s}\eta$,
where $G_{2}$ is negative Aceti et al. (2014).

The set of parameters obtained in the global fit are collected in Tables 5 and
6. We obtain a reasonable value for the $D_{s1}-D_{s0}^{*}$ mass splitting by
letting the parameter $\alpha_{1}$ to sligthly vary for the $D_{s1}$ state,
that we call $\alpha_{1}^{1^{+}}$. As can be seen, these parameters barely
change respect to the ones in Table 2.

Figure 10: Fit I: Result from the global fit for the dependence with the
spatial extent (L)

of some energy levels of Cheung et al. (2021) (upper panel), Bali et al.
(2017) (middle), and Lang et al. (2014); Mohler et al. (2013) (lower panel).
In the upper panel, the effect of the $D_{s}\eta$ channel is shown in dashed
lines.

| $c$ | $\alpha_{1}^{0^{+}}$ | $\alpha_{1}^{1^{+}}$ | $\alpha_{2}[(m_{\pi}^{0})^{-2}]$
---|---|---|---|---
Fit I | - | $-1.87(1)$ | $-2.05(2)$ | $-0.040(2)$
Fit II | $1.05(1)$ | $-1.34(1)$ | $-1.44(1)$ | $-0.002(1)$
Table 5: Common parameters among all collaborations obtained in the combined LQCD anaylis (global fit), for Fits I and II. | ${m}_{c\bar{s}}^{0^{+}}$ | ${m}_{c\bar{s}}^{1^{+}}$
---|---|---
HSC239 | $2552(26)$ | -
HSC391 | $2476(25)$ | -
RQCD | $2475(25)$ | $2553(25)$
Prelovsek | $2277(23)$ | $2356(23)$
Table 6: Parameters of the ${m}_{c\bar{s}}$ CQM bare masses obtained in the
combined LQCD anaylis (global fit) for Fit II.

With these parameters and the input of the quark mass dependence of the
$D(D^{*})$ meson Gil-Domínguez and Molina (2023), we evaluate the dependence
of the $m_{D_{s0}^{*}}$, the binding energy and its coupling to the $DK$
channel in the one channel case. The results are given in Figs. 11 and 12. In
Fig. 11, the ratios, $m_{D_{s0}^{*}}/m_{\pi}$, $\Delta
E_{D_{s0}^{*}}/m_{\pi}$, as a function of $M_{avg}$, and
$m_{D_{s0}^{*}}/M_{avg}$, $\Delta E_{D_{s0}^{*}}/M_{avg}$, as a function of
$m_{\pi}$, are shown from top to bottom, respectively 555The dependence with
$M_{avg}$ is taken from the result of the analysis of Gil-Domínguez and Molina
(2023), where, the parameter depending on the heavy quark mass in the charm
meson masses, $m_{H}$, satisfies closely for all lattice data sets studied,
$m_{H}=0.635M_{avg}$.. The mass of the $D_{s0}^{*}(2317)$ increases with the
charm quark mass for all the collaborations. Remarkably, the binding energy
(with respect to the respective thresholds of the LQCD collaborations) becomes
larger for lower charm quark masses, suggesting that the interaction gets less
attractive when the charm quark mass increases, as depicted in the second
panels from the top. Note that, since the binding energy, $\Delta
E_{D_{s0}^{*}}=m_{K}+m_{D}-m_{D_{s0}^{*}}$, this means that the pole of the
$D_{{s_{0}}}^{*}$ is moving faster than the threshold when the charm quark
mass ($M_{avg}$) increases. For Fit I, the error band obtained for the
physical point is narrow and this effect is clear. However, as shown in Fig.
11, this error band is very broad in the case of Fit II666Only the error band
for the physical pion mass is shown for clarity.. This is because the number
of free parameters have increased. The dependence with the charm quark mass is
not unambiguously determined in this case. Therefore, more lattice data for
lower than the physical charm quark mass are needed to test this prediction.
Nevertheless, the behaviour with the light quark mass is opposite. For pion
masses larger than $200$ MeV, the binding energy clearly increases, becoming
the interaction more attractive.

Figure 11: Result of the global fits for the quark mass dependence of the pole
mass and binding energy for Fit I (left panels) and Fit II (right panels).

Figure 12: Result of the global fit (Fit I top panel, Fit II bottom panel) for
the coupling as a function of the pion mass.

Figure 13: Result of the global fit (Fit I top panel, Fit II bottom panel) for
the compositeness as a function of the pion mass (same legend than in Fig. 11
bottom).

In Fig. 13 and 14 we also show the pion mass dependence of the compositeness
and the scattering lengths, respectively, for both fits. The compositeness is
a bit lower for the Prelovsek data set, $1-Z\sim 0.6-0.7$, while it gets a
value $0.7-0.8$ for all the other collaborations. The scattering lengths are
in reasonably good agreement with previous works Martínez Torres et al.
(2015); Liu et al. (2013). However, the errors extracted here are smaller. The
scattering lengths obtained here are lower than in Liu et al. (2013). The
reason behind the improvement of the uncertainty is that, on one side, the
LQCD data analyzed here are much more precise than the ones for scattering
lengths of Liu et al. (2013). In particular, the data of Cheung et al. (2021)
present a great improvement in precision and also the covariance matrix of the
energy levels has been taken into account. On the other side, we have included
a larger amount of data than in Liu et al. (2013), where there are $20$
points, while we have $54$ data for the global fit.

Figure 14: Result of the global fit for the scattering lengths as a function
of the pion mass for Fit I (left panels) and Fit II (right panels), in
comparison with the result obtained in Liu et al. (2013) (dashed line with
blue band).

The effective range is depicted in Fig. 15 for both fits. We obtain that,
while the scattering length in Fits I and II are compatible, the effective
range is more negative for Fit I. These observables are sensitive to the type
of potential used.

Figure 15: Result of the global fits for the effective range $r_{0}$ as a
function of the pion mass for Fit I and II.

In Fig. 16 we show the value of bare mass as a function of the $D_{s}$ mass
for every ensemble. The overall result is compatible with a constant mass
difference: $m_{D_{s}}-m_{D_{s}^{0}}=m_{c\bar{s}}-m_{c\bar{s}^{0}}$, where
$m_{c\bar{s}^{0}}$ and $m_{D_{s}^{0}}$ are the values at the physical point.
This is in agreement with the assumption done in Albaladejo et al. (2018).

Finally, in Fig. 17 we plot the binding energies and mass splitting of the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ states as a function of the pion mass in
both fits for a physical charm quark mass. To plot Fig. 17 we have taken the
extrapolation of the bare mass (black star) to the physical point from Fig.
16. The binding energies of the $D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are
compatible with the experimental data for Fit II, while for the Fit I we
predict the $D_{s1}(2460)$ state with around $15$ MeV difference from the
physical value. They also increase with the pion mass for pion masses above
$200$ MeV, showing a quadratic behaviour. The 1S hyperfine splitting is
practically constant with the pion mass.

Finally, in the next section, we evaluate the isospin breaking decay rate
$\Gamma(D_{s0}^{*}(2317)\to D_{s}^{+}\pi^{0})$.

Figure 16: Values of the $m_{c\bar{s}}$ CQM bare masses from our fit as a
function of the $D_{s}$ meson mass.

Figure 17: Binding energy of the $D_{s0}^{*}(2317)$ (top), $D_{s1}(2460)$
(middle) and $1S$ hyperfine splitting (bottom) for physical charm and strange
quark masses.

###  III.2 Decay width of the $D_{s0}^{*}(2317)$ to $D_{s}^{+}\pi^{0}$

In this section we evaluate the decay width of the isospin violating decay
mode $D_{s0}^{*}(2317)\rightarrow D_{s}^{+}\pi^{0}$. This process can occur
via direct decay as depicted in Fig. 18,

Figure 18: Feynman diagrams of the $D_{s}^{*0}\rightarrow D_{s}^{+}\pi^{0}$
decay.

where one has $\bar{K}^{*}$ or $D^{*}$ exchange. Besides that, this decay can
go through $\pi^{0}-\eta$ mixing. The diagrams are the same than in Fig. 18
but instead of $\pi^{0}$ we have $\eta_{8}$ which mixes with the pion through
the mixing angle, that appears in the definition of mass eigenstates,

$\displaystyle\tilde{\pi}^{0}=\pi^{0}\cos\tilde{\epsilon}+\eta_{8}\sin\tilde{\epsilon}$
(28)
$\displaystyle\tilde{\eta}=-\pi^{0}\sin\tilde{\epsilon}+\eta_{8}\cos\tilde{\epsilon}$

with $\eta_{8}=\frac{2\sqrt{2}}{3}\eta-\frac{1}{3}\eta^{\prime}$, and is given
by,

$\displaystyle\tilde{\epsilon}=\frac{\sqrt{3}}{4}\frac{m_{d}-m_{u}}{m_{s}-\hat{m}}\
.$ (29)

In the above equation, $\hat{m}$ is the average mass of the $u$ and $d$
quarks. We proceed to evaluate the diagram of Fig. 19 where $X$ is the
$D_{s0}^{*}(2317)$, and $m_{1},m_{2}$ and $m_{3}$ are the masses of the three
pseudoscalars. The vertices are evaluated by means of the Lagrangian of Eq.
(1). The amplitude of the diagram of Fig. 19, is given by,

$\displaystyle t=\int\frac{d^{4}q}{(2\pi)^{4}}$ (30)
$\displaystyle\frac{g_{X}g^{2}c_{1}(p-p_{f}+q)(p+p_{f}-q)}{\left[(p-q)^{2}-m_{2}+i\epsilon\right]\left[(q-p+p_{f})^{2}-M+i\epsilon\right](q^{2}-m_{1}+i\epsilon)}$

where $g_{X}$ is the coupling of the $D_{s0}^{*}(2317)$ to $DK$, which can be
obtained straightforward. Since,

$\displaystyle\left|DK,I=0\right>$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}D^{+}K^{0}+\frac{1}{\sqrt{2}}D^{0}K^{+},$ (31)

one has, $g_{X}=\frac{g_{DK}^{(I=0)}}{\sqrt{2}}$.

$c_{1}$ | $m_{1}$ | $m_{2}$ | $M$ | $m_{1f}$ | $m_{2f}$
---|---|---|---|---|---
$-1/\sqrt{2}$ | ${D^{+}}$ | $K^{0}$ | ${\bar{K}^{0*}}$ | $D_{s}^{+}$ | ${\pi^{0}}$
$1/\sqrt{2}$ | ${D^{0}}$ | ${K^{+}}$ | ${K^{-*}}$ | ${D_{s}^{+}}$ | ${\pi^{0}}$
$-1/\sqrt{2}$ | ${D^{+}}$ | ${K^{0}}$ | ${\bar{D}^{+*}}$ | ${\pi^{0}}$ | ${D_{s}^{+}}$
$1/\sqrt{2}$ | ${D^{0}}$ | ${K^{+}}$ | ${D^{0*}}$ | ${\pi^{0}}$ | ${D_{s}^{+}}$
$\tilde{\epsilon}\sqrt{\frac{3}{2}}$ | ${D^{+}}$ | ${K^{0}}$ | ${\bar{K}^{0*}}$ | ${D_{s}^{+}}$ | ${\eta}$
$\tilde{\epsilon}\sqrt{\frac{3}{2}}$ | ${D^{0}}$ | ${K^{+}}$ | ${K^{-*}}$ | ${D_{s}^{+}}$ | ${\eta}$
$\tilde{\epsilon}\frac{1}{\sqrt{6}}$ | ${D^{+}}$ | ${K^{0}}$ | ${\bar{D}^{+*}}$ | ${\eta}$ | ${D_{s}^{+}}$
$\tilde{\epsilon}\frac{1}{\sqrt{6}}$ | ${D^{0}}$ | ${K^{+}}$ | ${D^{0*}}$ | ${\eta}$ | ${D_{s}^{+}}$
Table 7: Masses and coefficients for each diagram that contributes to the
$D_{s0}^{*}\rightarrow D_{s}^{+}\pi^{0}$ decay.

Figure 19: General Feynman diagram of the $D_{s0}^{*}\rightarrow
D_{s}^{+}\pi^{0}$ decay.

We assume that both couplings, $g_{X,D^{+}K^{0}}=g_{X,D^{0}K^{+}}=g_{X}$ are
equal, and take $g_{DK}$ from the analysis of LQCD data, where isospin
violation is not considered. We have also approximated
$\sum\epsilon_{\mu}\epsilon_{\nu}\simeq-g_{\mu\nu}$, the constant $c_{1}$ in
Eq. (30) is given in Table 7 for every diagram in Fig. 18. We integrate in the
$q^{0}$ variable applying the Residue Theorem. To make it easier, we decompose
the propagators in the positive and negative energy parts. For the two
diagramas in the top of Fig. 18, the propagator of $D$ meson ($m_{1}=m_{D}$)
is,

$\displaystyle\frac{1}{q_{0}^{2}-\vec{q}\,^{2}-m_{1}+i\epsilon}$
$\displaystyle=$
$\displaystyle\frac{1}{2\omega_{1}}\left(\frac{1}{q_{0}-\omega_{1}+i\epsilon}-\frac{1}{q_{0}+\omega_{1}-i\epsilon}\right)\
,$

where $\omega_{i}=\sqrt{\vec{q}\,^{2}_{i}+m_{i}^{2}}$. Since this is a heavy
particle, we can safely neglect the negative energy part of the propagator.
Finally, we obtain,

$\displaystyle t=$
$\displaystyle\frac{g_{X}g^{2}c_{1}}{32\pi^{2}}\int_{0}^{q_{max}}\frac{d^{3}q}{\omega_{1}\omega_{2}\omega_{M}}\left\\{\frac{m_{X}^{2}-(p_{f}^{0}-M_{X}+\omega_{2})^{2}+(\vec{p}_{f}-\vec{q})^{2}}{(M_{X}-\omega_{1}-\omega_{2}+i\epsilon)(p_{f}^{0}-\omega_{1}-\omega_{2})}+\frac{1}{M_{X}-p_{f}^{0}-\omega_{1}-\omega_{M}+i\epsilon)}\right.$
$\displaystyle\times\left.\left(\frac{M^{2}_{X}-(p_{f}^{0}-\omega_{1})^{2}+(\vec{p}_{f}-\vec{q})^{2}}{M_{X}-\omega_{1}-\omega_{2}+i\epsilon}-\frac{M_{X}^{2}-(2p_{f}-M_{X}+\omega_{M})^{2}+(\vec{p}_{f}-\vec{q})^{2}}{p_{f}^{0}+\omega_{M}+\omega_{2}}\right)\right\\}$

being $p_{f}$ the four-momentum of the pion, and
$\omega_{M}=\sqrt{\vec{q}\,^{2}+M^{2}}$. In the integral we have taken
$q_{max}=795$ MeV, which corresponds to the same value for $Re(G^{DR})$ in Eq.
14 at threshold for $\alpha=-1.91$ MeV, the result of the global fit. For the
diagrams in the bottom of Fig. 18, one makes the substitution in the above
formula of $p_{f}^{0}\rightarrow m_{X}-p_{f}^{0}$ and
$\vec{p}_{f}\rightarrow-\vec{p}_{f}$.

Finally, we can estimate the width of the $D_{s0}^{*}(2317)$ using the formula

$\displaystyle\Gamma_{X}=|\vec{p}_{f}|\frac{|t|^{2}}{8\pi m_{X}^{2}}.$ (34)

We obtain $\Gamma_{D_{s0}^{*}}=128\pm 40$ KeV, which supports the molecular
interpretation, being in agreement with previous calculations Liu et al.
(2013); Fu et al. (2022); Cleven et al. (2014). The error has been estimated
varying the couplings in the vertices involving $D^{*}$ mesons of the diagrams
in Fig. 18, from $g=4.20$ to $g=8.95$.

##  IV Conclusions

We have analyzed the available LQCD data on $D^{(*)}K$ scattering Lang et al.
(2014); Mohler et al. (2013); Bali et al. (2017); Cheung et al. (2021). In
particular, the HSC energy levels with different boost extracted recently
Cheung et al. (2021). The formalism is based on the local hidden gauge
lagrangian, which has been used in the past to predict several exotic states
Molina et al. (2010); Wu et al. (2010, 2011) observed in the past years Aaij
et al. (2015, 2019, 2020a, 2020b); tcs (2022). The LO term from the HGF is
supplemented by a term that accounts for the coupling of the $DK$ channel to a
bare $c\bar{s}$ component, respecting both, chiral and heavy quark spin
symmetries.

We have compared the results of the analysis of HSC data with previous LQCD
analysis, and with experiment. The results of the HSC data analysis done here
agree reasonably well with experiment, and with other previous LQCD studies on
$DK$, suggesting the possibility of a global fit. We have conducted this fit
and extracted for the first time, both, the quark mass dependence -light and
heavy- of the $D_{s0}^{*}(2317)$ resonance properties from the available LQCD
data. In this fit we also include LQCD data for $D^{*}K$ scattering from other
simulations. Our results suggest that the attractive $D^{(*)}K$ interaction
increases with the pion mass from $m_{\pi}\simeq 200$ MeV, but reduces with
the charm quark mass. Since most of the discretization error comes from the
charm quark, more data on $DK$ scattering at lower than the physical charm
quark mass would be needed to confirm this prediction. We obtain that the
$D_{s0}^{*}(2317)$ and $D_{s1}(2460)$ are predominantly molecular. Indeed, the
global fit of LQCD data suggest that the properties of these states are
sensitive to the light pion mass, reinforcing this statement. However, in
comparison with Cleven et al. (2011), the dependence obtained for the masses
of these resonances is only quadratic for very low pion masses, while it
becomes linear when the pion mass increases. We have also improved the
precision of the $DK$ scattering lengths extracted previously Lang et al.
(2014); Mohler et al. (2013); Bali et al. (2017); Albaladejo et al. (2018);
Martínez Torres et al. (2015); Liu et al. (2013), and studied its quark mass
dependence. Compared to Liu et al. (2013) we obtain smaller values for the
scattering lengths. Finally, we obtained a decay width of
$\Gamma_{D_{s0}^{*}\to D_{s}^{+}\pi^{0}}=128\pm 40$ KeV, in agreement with
previous works Liu et al. (2013); Fu et al. (2022); Cleven et al. (2014), and
also supporting the molecular interpretation. Future LQCD data analyses for
different pion masses than the ones included in our fit, could be a good test
for the study done here.

##  V Acknowledgments

R. Molina acknowledges support from the CIDEGENT program with Ref.
CIDEGENT/2019/015, the Spanish Ministerio de Economia y Competitividad and
European Union (NextGenerationEU/PRTR) by the grant with Ref. CNS2022-13614,
and from the spanish national grant PID2020-112777GB-I00. This project has
received funding from the European Union’s Horizon 2020 programme No. 824093
for the STRONG-2020 project.

##  VI Appendix

###  VI.1 Matrices $\phi$ and $V$ in Eq. (1)

$\displaystyle\phi=\left(\begin{array}[]{cccc}\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{3}}+\frac{\eta^{\prime}}{\sqrt{6}}&\pi^{+}&K^{+}&\bar{D}^{0}\\\
\pi^{-}&-\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta}{\sqrt{3}}+\frac{\eta^{\prime}}{\sqrt{6}}&K^{0}&D^{-}\\\
K^{-}&\bar{K}^{0}&-\frac{\eta}{\sqrt{3}}+\sqrt{\frac{2}{3}}\eta^{\prime}&D^{-}_{s}\\\
D^{0}&D^{+}&D_{s}^{+}&\eta_{c}\end{array}\right)\ ,$ (39) $\displaystyle
V_{\mu}=\left(\begin{array}[]{cccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}&\bar{D}^{*0}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}&D^{*-}\\\
K^{*-}&\bar{K}^{*0}&\phi&D^{*-}_{s}\\\
D^{0*}&D^{*+}&D^{*+}_{s}&J/\psi\end{array}\right)_{\mu}\ ,$ (44)

where we have considered ideal $\eta-\eta^{\prime}$ mixing Bramon et al.
(1992).

###  VI.2 Figures of HSC energy levels for different boosts

Figure 20: Result from the fit for the dependence with the spatial extent (L)

of HSC energy levels for $m_{\pi}=239$ MeV.

Figure 21: Result from the fit for the dependence with the spatial extent (L)

of HSC energy levels for $m_{\pi}=391$ MeV.

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