# The effects of charmonium on the properties of the $1^{++}$ hidden charm
poles in effective field theory

E. Cincioglu<EMAIL_ADDRESS>Department of Physics, Middle East
Technical University, Ankara, Turkey  A. Ozpineci  Department of Physics,
Middle East Technical University, Ankara, Turkey  D. Yildirim Yilmaz
Department of Physics, Faculty of Sciences, Ankara University, Ankara, Turkey
Department of Physics, Faculty of Sciences and Arts, Amasya University,
Amasya, Turkey<EMAIL_ADDRESS>
###### Abstract

In this study, the properties of the $J^{PC}=1^{++}$ hidden charm poles are
analyzed under the variation of the bare 2P charmonium mass within the
effective field theory proposed in Ref. Cincioglu  _et al._ (2016). The main
focus of the current work is on the pole trajectory of the $\chi_{c1}(2P)$
charmonium dressed by the $D\bar{D}^{*}$ meson loops. It is shown that the
trajectories of the pole change radically for values of the bare charmonium
mass above a certain value and also depending on how close the pole is to the
threshold.

##  I Introduction

In the previous decades, many states were found experimentally in the mass
range of heavy hadrons that can not be easily explained within a quark model
and a standard quarkonium picture Olsen (2015); Tanabashi  _et al._ (2018). To
understand the structure of these states is one of the important aims of
hadron physics. For most of these states, only their masses are experimentally
known, and information about their quantum numbers and decays modes are very
scarce Olsen  _et al._ (2018); Lebed  _et al._ (2017). Especially the content
of $X(3872)$ among all the exotics has been the subject of various studies
since its discovery. The mass of $X(3872)$, which is very close to the
$D^{0}D^{*0}$ threshold Choi  _et al._ (2003); Acosta  _et al._ (2004); Abazov
_et al._ (2004); Aubert  _et al._ (2005); Chatrchyan  _et al._ (2013); Aaij
_et al._ (2013, 2020a, 2020b), and the isospin-breaking decay
$X(3872)\rightarrow J/\psi\rho$, make it an ideal candidate for a
$D\bar{D}^{*}$ hadronic molecule Chen  _et al._ (2016); Guo  _et al._ (2018).
Another important measurement that might give insight into the structure of
X(3872) is its electromagnetic decay into $\psi(2S)$ and $J/\psi$. The ratio
of the branching ratios of these decays is measured as Aubert  _et al._
(2009); Aaij  _et al._ (2014):

$R_{\psi\gamma}=\frac{B_{r}(X\to\psi(2S)\gamma)}{B_{r}(X\to
J/\psi\gamma)}=2.46\pm 0.64\pm 0.29\,.$ (1)

This implies that, since the phase space for the decay
$X\rightarrow\psi(2S)\gamma$ is much smaller than the phase space for the
$X\rightarrow J/\psi\gamma$ decay, the amplitude for
$X\rightarrow\psi(2S)\gamma$ is much larger. This is naturally expected in the
quark model since in the quark model $X\rightarrow\psi(2S)\gamma$ is a $\Delta
L=1$ transition. But, in the quark model, the predicted mass of the
$1^{++}(J^{PC})$ charm- anticharm state, called $\chi_{c1}(2P)$, is around
$3.95$ GeV Godfrey and Isgur (1985); Ebert  _et al._ (2011), about 70 MeV
higher than the observed state. On the other hand, in Ref. Guo  _et al._
(2015), triangular $DD^{*}\bar{D}^{*}$ and simple $D\bar{D}^{*}$ loop
contributions to the radiative amplitude were computed. It was concluded that
the observed ratio allows the $X(3872)$ to be a hadronic molecule with the
dominant component $D\bar{D}^{*}$. In Ref. Cincioglu and Ozpineci (2019), the
effects of short-range contributions to the radiative decays of the $X(3872)$
were analyzed. It was demonstrated that the possible constructive or
destructive interferences between the meson-loop and the short-distance
contact term are important to determine whether the charmonium content of the
$X(3872)$ is nontrivial.

In Cincioglu  _et al._ (2016), an effective theory using heavy quark spin
symmetry (HQSS) is presented in which $X(3872)$ is described as a
superposition of molecular components and a compact core, taken to be 2P
charmonium state throughout this study111 There are plenty of studies that
considered X(3872) as a mixture in the literature Dong  _et al._ (2011);
Takizawa and Takeuchi (2013); Chen  _et al._ (2015). . An important source of
uncertainty in the model is bare masses of the compact components. In
Cincioglu  _et al._ (2016), a bare mass was taken to be the charmonium mass
predicted by potential quark models. This work aims to draw attention to the
effect of the bare charmonium mass, which affects the trajectories on the
predictions of the model, and, as shown, how it alters the molecular weight of
the observation. It is noted that the bare mass is UV regulator dependent and
it is a free parameter in the presented scheme. Although the bare mass is not
physically observable, theoretically it is still relevant, because its value
can be obtained from schemes that ignore the coupling of charmonium states to
the mesons ($d=0$). A major problem is to set the UV regulator to match the
quark model and the EFT approaches. Thus, all calculations have been performed
with two different UV cutoffs, spanning a physically motivated range of
values. The expectation is that the cutoff dependence will be absorbed into
the low energy constants (LECs) and thus predictions for observables could
become at most mildly regulator dependent.

This paper is organized as follows: In section II, the main points of the
model proposed in Cincioglu  _et al._ (2016) is presented, followed by the our
results and discussion.

##  II Formalism

Due to the presence of heavy quarks in $X(3872)$, it can be described within a
HQSS framework. To describe the molecular component of $X(3872)$, interactions
of $D$ and $D^{*}$ are needed. In the HQSS, these mesons group in a HQSS
doublet, which can be written as:

$H^{(Q)}_{a}=\frac{1+\not{v}}{2}(P^{*(Q)}_{a\mu}\gamma^{\mu}-P^{(Q)}_{a}\gamma_{5})\,.$
(2)

Due to the very low momentum exchange between the mesons in the molecule,
contact interactions are sufficient to describe the D mesons’ interactions in
X(3872) Nieves and Valderrama (2012). Except for contact interaction, other
interactions like one pion exchange and coupled channel are sub-leading order
Nieves and Valderrama (2012). Therefore their contribution can be ignored
safely. Since the interaction among the heavy hadrons forming a molecule is
nonperturbative, the potential should be iterated by solving Lippmann-
Schwinger equation. Lippmann-Schwinger equation shows ill-defined ultraviolet
behavior resulting from a contact interaction. In consequence, it requires
regularization. As a regulator function, a Gaussian function
$f_{\Lambda}(\vec{p})$ is employed

$\langle\vec{p}^{\prime};D^{(*)}\bar{D}^{(*)}|V_{\Lambda}|\vec{p};D^{(*)}\bar{D}^{(*)}\rangle=C_{0X}f_{\Lambda}(\vec{p}^{\prime})f_{\Lambda}(\vec{p})$
(3)
$\langle\vec{p};D^{(*)}\bar{D}^{(*)}|V_{c\bar{c};\Lambda}|\Psi_{c\bar{c}}(2P)\rangle=df_{\Lambda}(\vec{p})$
(4)

where $d$ and $C_{0X}$ are low energy constants of related interactions, and
in this paper we take $\Lambda$ cutoff values a $0.5-1.0$ GeV.

At leading order the interaction of four heavy mesons with contact interaction
potentials can be described as below Nieves and Valderrama (2012)

$\begin{array}[]{l l l l
l}\mathcal{L}_{4H}&=&D_{0a}Tr\left[\bar{H}^{(Q)a}H^{(Q)}_{a}\gamma_{\mu}\right]Tr\left[H^{(\bar{Q})b}\bar{H}^{(\bar{Q})}_{b}\gamma^{\mu}\right]\\\
&+&D_{0b}Tr\left[\bar{H}^{(Q)a}H^{(Q)}_{a}\gamma_{\mu}\gamma_{5}\right]Tr\left[H^{(\bar{Q})b}\bar{H}^{(\bar{Q})}_{b}\gamma^{\mu}\gamma_{5}\right]\\\
&+&E_{0a}Tr\left[\bar{H}^{(Q)a}\vec{\tau}^{b}_{a}H^{(Q)}_{b}\gamma_{\mu}\right]Tr\left[H^{(\bar{Q})r}\vec{\tau}^{s}_{r}\bar{H}^{(\bar{Q})}_{s}\gamma^{\mu}\right]\\\
&+&E_{0b}Tr\left[\bar{H}^{(Q)a}\vec{\tau}^{b}_{a}H^{(Q)}_{b}\gamma_{\mu}\gamma_{5}\right]Tr\left[H^{(\bar{Q})r}\vec{\tau}^{s}_{r}\bar{H}^{(\bar{Q})}_{s}\gamma^{\mu}\gamma_{5}\right]\,,\end{array}$
(5)

where $D_{0i}$ and $E_{0i}$ are LECs. To include the compact core component,
it is necessary to identify the HQSS multiplet that can be used to describe
it. For this purpose, the $P$-wave quarkonium multiplet, which can be written
as Casalbuoni  _et al._ (1993):

$J^{\mu}=\frac{1+\not{v}}{2}\left(\chi_{2}^{\mu\alpha}\gamma_{\alpha}+\frac{i}{\sqrt{2}}\epsilon^{\mu\alpha\beta\gamma}\chi_{1\gamma}v_{\alpha}\gamma_{\beta}+\frac{1}{\sqrt{3}}\chi_{0}(\gamma^{\mu}-v^{\mu})+h^{\mu}\gamma_{5}\right)\frac{1-\not{v}}{2}~{}.$
(6)

HQSS restricts the possible contact interactions between the $J^{\mu}$
multiplet, and the $D$ meson multiplets. The only interaction term between two
$D$ mesons and $J^{\mu}$ that does not contain any derivative interactions at
the leading order, and the only consistent Lagrangian with HQSS is:

$\mathcal{L}_{HHQ\bar{Q}}=\frac{d}{2}Tr[H^{a(\bar{Q})}\bar{J}_{\mu}H_{a}^{(Q)}\gamma^{\mu}]+\frac{d}{2}Tr[\bar{H}^{a(Q)}J_{\mu}\bar{H}_{a}^{(\bar{Q})}\gamma^{\mu}]\,,$
(7)

where the parameter $d$ is an unknown LEC that causes molecular and compact
components to mix (For more details, see e.g. Cincioglu  _et al._ (2016);
Hanhart  _et al._ (2014); Colangelo  _et al._ (2004)).

For bound states, to study the weights of the molecular and compact components
in $X(3872)$, a method put forward by Weinberg Weinberg (1963, 1965) can be
used. The method is crucial to examine interaction couplings and the
probabilistic interpretations of the components. For small binding energies
(s-wave), the approach is model-independent. With the help of the sum rule
Garcia-Recio  _et al._ (2015); Weinberg (1965)

$-1=\sum_{ij}g_{i}g_{j}\left(\delta_{ij}\left[\frac{\partial
G_{i}^{II}(E)}{\partial E}\right]_{E=E_{R}}+\left[G_{i}^{II}(E)\frac{\partial
V_{ij}(E)}{\partial E}G_{j}^{II}(E)\right]_{E=E_{R}}\right)\,,$ (8)

the probabilistic interpretation of the compositeness condition can be made;
moreover, Eq. (8) is valid for both bound states and resonance statesGarcia-
Recio  _et al._ (2015). For resonance (bound) states, $G$ should be taken as
$G^{II}$ ($G^{I}$). Each term in Eq.(8) can be identified differently222 The
imaginary parts of $\tilde{X}_{i}$ and $\tilde{Z}$ must cancel each other..

$X_{i}=Re\tilde{X}_{i}=Re\left(-g_{i}^{2}\left[\frac{\partial
G_{i}^{II}(E)}{\partial E}\right]_{E=E_{R}}\right)$ (9)
$Z=Re\tilde{Z}_{i}=Re\left(-\sum_{ij}\left[g_{i}G_{i}^{II}(E)\frac{\partial
V_{ij}(E)}{\partial E}G_{j}^{II}(E)g_{j}\right]_{E=E_{R}}\right)$ (10)

With the definitions in Eqs.(9) and (10), we obtain the compositeness and
elementariness, respectively. $\tilde{X}_{i}$ quantifies the probability of
finding a two-body component in the wave function of a hadron, and
$\tilde{Z}_{i}$ is related to other components and thus is understood as the
elementariness. Hence Z close to 1 signifies that its compact component
dominates the bound state.

On the other hand, in the case of a resonance, probabilistic interpretations
are not entirely accurate because of $\tilde{X}_{i}$’s negative imaginary
values. But in Ref. Guo and Oller (2016); Aceti  _et al._ (2014), it was
claimed that the absolute value of $\tilde{X}_{i}$ can be used as a measure of
the weight of the i-th channel333In Ref. Guo and Oller (2016) a probabilistic
interpretation of the compositeness relation at the pole of a resonance with
only positive coefficients thanks to a suitable transformation of the S matrix
has been derived. Absolute value of $\tilde{Z}_{i}$ gives the weight of
finding a specific component in the wave function of a hadron, but it is only
valid when $Re(E_{R})>M_{i,th}$, with $E_{R}$ resonance energy and $M_{i,th}$
the corresponding threshold of the channel $i$..

$T$-matrix, which can be obtained as a solution of an LSE equation, develops
poles on the complex energy plane. If $T$-matrix is close to the pole, its
elements are approximately

$T_{ij}\approx\frac{g_{i}g_{j}}{E-E_{R}}\,,$ (11)

where $g_{i}$ is the coupling of the state to the i-th channel.

When considering a specific $(1^{++})$ state, $T$-matrix, which gives dynamics
of the system, can be formed as Cincioglu  _et al._ (2016):

$T(E)=\frac{\Sigma_{c\bar{c}}}{1-G^{0}_{c\bar{c}}\Sigma_{c\bar{c}}}\left(\begin{array}[]{cc}f^{2}_{\Lambda}(E)[\frac{1}{d^{2}G_{QM}^{2}}-\frac{1-G^{0}_{c\bar{c}}\Sigma_{c\bar{c}}}{G_{QM}\Sigma_{c\bar{c}}}]&f_{\Lambda}(E)\frac{1}{dG_{QM}}\\\
f_{\Lambda}(E)\frac{1}{dG_{QM}}&1\end{array}\right)\,,$ (12)

where $\Sigma_{c\bar{c}}$, $G^{0}_{c\bar{c}}$, $f_{\Lambda}$444 All
calculations have been performed with an UV cutoff $\Lambda=0.5-1$ GeV.  , and
$G_{QM}$ 555 QM stands for non-relativistic quantum mechanics. are the
charmonium self-energy induced by the meson loops, the non-relativistic bare
charmonium propagator, the Gaussian regulator, and the diagonal meson loop
function, respectively. In Eq. (12), while the first channel is molecular
type, the second is charmonium Baru  _et al._ (2010). Besides, poles of the
transition matrix are given by zeros of the inverse of the dressed propagator
Cincioglu  _et al._ (2016)

$1-G^{0}_{c\bar{c}}(E_{R})\Sigma_{c\bar{c}}(E_{R})=0\,,$ (13)

where $\Sigma_{c\bar{c}}$ is the quarkonium self-energy

$\Sigma_{c\bar{c}}(E)=\left[V_{c\bar{c}}^{\rm QM}\right]^{t}G_{\rm
QM}(E)\Gamma_{c\bar{c}}(E)\,,$ (14)

with the dressed vertex function, $\Gamma_{c\bar{c}}$, reads

$\Gamma_{c\bar{c}}(E)=\left(1-V^{\rm QM}G_{\rm
QM}(E)\right)^{-1}V_{c\bar{c}}^{\rm QM}\,.$ (15)

where $V^{QM}$ and $V_{c\bar{c}}^{QM}$ are the molecular contact potential and
the $\chi_{c_{1}}(2P)-D\bar{D}^{(*)}$ transition amplitudes, respectively (see
Eq. (30) and Eq. (34) of Ref. Cincioglu  _et al._ (2016)). Finally, for an
arbitrary $E$, the mesonic loop function is given by Albaladejo  _et al._
(2013)

$\displaystyle G_{\rm QM}(E)$
$\displaystyle=\int\frac{\text{d}^{3}\vec{q}}{(2\pi)^{3}}\frac{e^{-2\vec{q}^{\,2}/\Lambda^{2}}}{E-M_{1}-M_{2}-\vec{q}^{\,\,2}/2\mu+i0^{+}}$
$\displaystyle=-\frac{\mu\Lambda}{(2\pi)^{3/2}}+\frac{\mu
k}{\pi^{3/2}}\phi\left(\sqrt{2}k/\Lambda\right)-i\frac{\mu
k}{2\pi}e^{-2k^{2}/\Lambda^{2}}~{},$ (16)

with $\mu^{-1}=M_{1}^{-1}+M_{2}^{-1}$, $k^{2}=2\mu(E-M_{1}-M_{2})$ and
$\phi(x)$ the Dawson integral given by:

$\phi(x)=e^{-x^{2}}\int_{0}^{x}e^{y^{2}}\text{d}y~{}.$ (17)

Poles on the complex E-energy plane represent the observable states. The mass
and width of a pole can be obtained from the pole position on the complex
energy plane. On the complex energy E-plane, poles can be located on the
different Riemann sheets. Indeed, $G_{\rm QM}(E)$ has two Riemann sheets. In
the first Riemann sheet (FRS), $0\leqslant{\rm Arg}(E-M_{1}-M_{2})<2\pi$,
there is a discontinuity $G_{\rm QM}^{I}(E+i\epsilon)-G_{\rm
QM}^{I}(E-i\epsilon)=2i\,{\rm Im}G_{\rm QM}^{I}(E+i\epsilon)$ for
$E>(M_{1}+M_{2})$. For those poles located on the FRS, on the real axis, and
below the threshold are named bound states. In the second Riemann sheet (SRS),
$2\pi\leqslant{\rm Arg}(E-M_{1}-M_{2})<4\pi$, one can find $G_{\rm
QM}^{II}(E-i\epsilon)=G_{\rm QM}^{I}(E+i\epsilon)$, for real energies and
above threshold. Poles located below the real axis, and above the threshold on
the SRS are called resonances666 The more detailed information about poles can
be found in Refs. Guo  _et al._ (2018); Hanhart  _et al._ (2014). There is no
restriction for the location of the poles on the second Riemann (unphysical)
sheets. Hermitian analyticity requires that if there is a pole at a complex
value of s (resonance), there must be another pole at its complex conjugate
value, $s^{*}$ (anti-resonance). In this study, the properties of the
conjugate pole are not given since this pole corresponds to the same
resonance..

When it is a narrow resonance in SRS, the pole with a negative imaginary part
(the pole located in the lower half-plane) is closer to the physical Riemann
sheet than the pole with a positive imaginary part; thus, it influences the
observables more strongly in the vicinity of the resonance region Baru  _et
al._ (2010). Moreover, when the poles’ real part reaches the threshold with
increasing d, both resonance and anti-resonance poles are equally essential.
Those nearby poles only significantly influence resonance behavior in the
experiment region that could be extracted from the experiment data in a
phenomenological study. However, when they have large imaginary parts, they
lose their width interpretation.

The position of the pole might give further insight into the structure of the
state. It appears that if the bound state is mostly compact, there are two
near-threshold poles, one is on the first Riemann sheet, and the other is on
the second Riemann sheet. Furthermore, if it is a predominantly molecular
state, there is a single near-threshold pole on the first Riemann sheet Baru
_et al._ (2010); Morgan (1992).

We look carefully at the trajectories of 2P charmonium poles located in nearby
threshold zone in scattering amplitudes. There are qualitative differences
between the pole trajectories of resonances that couple to the related
continuum channel with changing bare 2P charmonium masses.

###  II.1 General remarks

In the model of Ref. Cincioglu  _et al._ (2016), two poles in the $1^{++}$
sector are expected. One is located in the FRS as a bound state. This state is
identified as $X(3872)$ bound state in the FRS, and its mass is fixed at
$3871.69$ MeV to determine the LEC $C_{0X}$ defined as

$C_{0X}=C_{0A}+C_{0B},\qquad
C_{0\phi}=D_{0\phi}+3E_{0\phi},\qquad\text{for}\qquad\phi=a,\,b.$ (18)

The other one is identified as a dressed $\chi_{c1}(2P)$. The dependence of
the second pole position on the bare 2P charmonium mass results from the non-
relativistic bare propagator:

$G^{0}_{c\bar{c}}(E)=\frac{1}{E-\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}}\,,$
(19)

where $\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}$ is the mass of the 2P
bare charmonium state. As mentioned above, it is dressed by the $D\bar{D}^{*}$
meson loops and gives rise to the physical mass of the charmonium states, as
$d$ increases.

On the other hand, there exist some uncertainties in the presented model of
Ref. Cincioglu  _et al._ (2016). One of the most considerable uncertainties is
the mass of the bare charmonium state. As can be seen in Table 1, most recent
constituent quark models give the mass of the $\chi_{c1}(2P)$ in quite a broad
range777 Furthermore, if the compact component is identified as a tetraquark,
its mass is completely unknown.. In Ref. Cincioglu  _et al._ (2016), the bare
$\chi_{c1}(2P)$ mass is taken 3906 MeV from Ref. Ebert  _et al._ (2011) due to
the closest prediction to the experimental mass of the $\chi_{c2}(2P)$ state.
Another uncertainty is a sizable error in the predicted masses that can reach
up to $10\%$. Besides, different models in Table 1 predict masses for the two-
meson threshold and the bare $\chi_{c1}(2P)$ mass differences ranging from
around 35 MeV to 82 MeV 888To see effects of the a lower mass of the
threshold(for $3865$ MeV see Tables 2 and 10), we searched the hypothetical
values below the threshold. Contrary to the values higher than the threshold,
as $C_{0X}$ decreases, $d$ increases, the pole moves towards the threshold,
gaining small width.. Also, it might be interesting to see the impact of the
bare charmonium mass, which is $10$ MeV below the $3906$ MeV mass value, on
the properties of the $1^{++}$ hidden charm poles (charmonium content in
$X(3872)$, dressed charmonium $\chi_{c1}(2P)$, $DD^{*}-\chi_{c1}(2P)$
coupling, etc.). For $m_{c\bar{c}}^{0}=3906$ MeV, the state is just $\sim 35$
MeV above the $DD^{*}$ threshold, and dressed $\chi_{c1}(2P)$ pole becomes
below threshold in the SRS with relatively small $X(3872)$ charmonium content.
However, as the larger bare $\chi_{c1}(2P)$ mass is taken, it is seen that on
the SRS, a larger charmonium content is needed to move the $\chi_{c1}(2P)$
state below the $DD^{*}$ threshold.

Ref. |  $m_{\mathcal{X}_{c1}(2P)}$ [MeV] |  $\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}-m_{DD^{*}}$ [MeV]
---|---|---
Ebert  _et al._ (2011) | 3906 | 35
Sng and Jumasahatov (2019) | 3924 | 53
Barnes  _et al._ (2005) | 3925 | 54
Ebert  _et al._ (2003) | 3929 | 58
Gui  _et al._ (2018); Deng  _et al._ (2017) | 3937 | 66
Segovia  _et al._ (2013); Ortega  _et al._ (2013) | 3947 | 76
Godfrey and Isgur (1985) | 3953 | 82
Table 1: The $\chi_{c1}(2P)$ masses in the literature. Constituent quark model
Segovia  _et al._ (2013); Ortega  _et al._ (2013), Regge trajectory Ebert  _et
al._ (2011), relativistic quark model Ebert  _et al._ (2003), non-relativistic
potential model Barnes  _et al._ (2005); Godfrey and Isgur (1985), non-
relativistic quark model - ${}^{3}P_{0}$ model Gui  _et al._ (2018); Deng  _et
al._ (2017), and QCD sum rule Sng and Jumasahatov (2019) were used to obtain
these mass values.

##  III Results and Discussion

In Tables 3–9 and Table I of Ref. Cincioglu  _et al._ (2016)999 Table I
includes properties of $\chi_{c1}(2P)$ states, taking the bare charmonium mass
as 3906 MeV. , the properties of the poles found in the $1^{++}$ hidden charm
sector are studied as a function of the LEC $d$, which controls the admixtures
of charmonium and $DD^{*}$ molecule. The location of the dressed
$\chi_{c1}(2P)$ pole depends on the mixing parameter $d$ and the bare
charmonium mass. Within the scheme presented here and in Ref. Cincioglu  _et
al._ (2016) of bare charmonium mass is a free parameter and it is not an
observable, as mentioned above, which gets dressed by the $D^{(*)}D^{(*)}$
meson loops and gives rise to the physical mass of the charmonium states.
Couplings of the $\chi_{c1}(2P)$ state to $D^{(*)}$ mesons causes the bare
mass to be renormalised. Since, in the effective theory, the difference
between the bare and the physical charmonium masses is a finite
renormalization. This shift depends on the UV regulator since the bare mass
itself depends on the renormalization scheme. To show the cutoff dependency of
the properties of the poles found in the $1^{++}$ hidden charm sector, all
calculation have been carried out with UV cutoff values $\Lambda=0.5\\--1$
GeV. The results obtained for both cutoffs with values of d are qualitatively
similar, though some quantitative differences appear, as can be seen in Tables
3–9 and Tables 11–16 of the appendix. Moreover, it is observed that the
behavior of the physical $\chi_{c1}(2P)$ state alters around
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}\approx 3908.5$ MeV.

For $\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3896,\,3906$ and $3908.5$
MeV values, the trajectories of the corresponding dressed $\chi_{c1}(2P)$ pole
are depicted in Fig. 1. When $d$ is $0$, the pole in the SRS is on the real
axis. With the increasing value of $d$, the pole gradually gets away from the
real axis by gaining width. As d continues to increase, the pole moves below
the threshold. At some point, it reaches the real axis again. When the pole
reaches the real axis with $\tilde{X}_{X(3872)}\sim 0.39$ for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3896$ MeV,
$\tilde{X}_{X(3872)}\sim 0.43$ for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3906$ MeV and
$\tilde{X}_{X(3872)}\sim 0.47$ for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3908.5$ MeV. As can be seen at
Tables 3–4, for $m_{c\bar{c}}^{0}=3896$ MeV charmonium content
$|\tilde{X}_{\chi_{c1}}|$ is about 35$\%$, and for $m_{c\bar{c}}^{0}=3908.5$
MeV charmonium content $|\tilde{X}_{\chi_{c1}}|$ is about 83$\%$. With the
conjugate pairs coinciding in the real axis, two poles appear in the SRS below
threshold located at $m_{R}-0i$. As one pole moves along the real axis toward
the threshold, a second pole departs from the threshold and leaves the real
axis forming another conjugate pair, with $d$ increasing. These new-formed
poles are either far below the threshold or above the threshold but deep in
the complex plane. It means that, in that region, the width interpretation is
lost. Since it will not produce any observable effects, its behavior is not
illustrated in Fig. 1, and also its properties are not given in Tables 3 and
4. Additionally, values smaller than $3908.5$ MeV bare charmonium mass have
similar trajectories.

In the case of $\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}\gtrsim 3908.5$
MeV, the behaviors of the pole trajectories, depicted in Fig. 2, are different
from Fig. 1. However, with increasing $d$ values, the poles stay in the SRS
above or below the threshold. But they do not reach the real axis until d’s
high values. For larger bare mass values than $3908.5$ MeV, until around
$3930$ MeV bare charmonium mass, the pole trajectories cross the threshold.
For those who cross the threshold, while the bare charmonium mass is getting
bigger, their molecular weight of the $X(3872)$ state is getting smaller.
Besides this, for some specific values of $d$ where $d>d^{\rm{crit}}$, there
exist a double pole in the SRS. One of the poles approaches the threshold,
with $\Sigma_{c\bar{c}}^{\prime}$ decreasing and moves along the real axis in
the SRS becomes quite close to the threshold, where SRS and FRS are connected,
it might have visible effects in scattering observables as the line shape will
be determined by both the pole and this virtual state. The other goes away
from the real axis with gaining width but eventually reaches the real axis far
from the threshold. These trajectories of the poles (illustrated with crosses)
as $d$ increases are either below the threshold or above the threshold but
located in much above the real axis. As mentioned above, since these poles
will not have any observable consequences, their details are not also included
in Tables 5–9 in the appendix. In the $d\gg d^{\rm{crit}}$ limit, $X(3872)$
appears to be a 2P charmonium state where the molecular weight in $X(3872)$,
$X_{X(3872)}\sim 0$, mirror in the FRS of the pole found in the SRS. Also, the
proximity of the bare mass to the threshold is essential within the model.

As mentioned in the introduction, the radiative decays of $X(3872)$ were
analyzed in Ref. Cincioglu and Ozpineci (2019) within the effective theory of
Ref. Cincioglu  _et al._ (2016) to constrain the charmonium content in
$X(3872)$. In the case of the destructive interferences between the meson
loops and the counter-term modeled by a charm quark loop, a strong restriction
on the charmonium admixture was found. In this work, the contribution from
short range interaction to the ratio $R{\psi\gamma}$ depends on
$\tilde{Z}_{X(3872)}$101010In the presented study, it is defined as
$\tilde{Z}_{X(3872)}=1-\tilde{X}_{X(3872)}$, which is the weight of finding
the charmonium component $\chi_{c1}(2P)$ in the physical wave function of
$X(3872)$, and position of the $\chi_{c1}(2P)$ pole, as can be seen from Eq.
(4) of that reference. It was claimed that the behavior of the predictions of
the ratio of radiative decays is different when $\tilde{Z}_{X(3872)}\lesssim
0.55$ and when $\tilde{Z}_{X(3872)}\gtrsim 0.55$111111As can be seen from Fig.
2 of that reference there is a bump around $\tilde{Z}_{X(3872)}\sim$ 0.55..
Indeed, in the vicinity of $C_{0X}$=0 ($\tilde{Z}_{X(3872)}\sim 0.55$), which
controls the four-meson contact interaction, the dressed $\chi_{c1}(2P)$ pole
becomes below threshold with its mass decreasing rapidly and quite wide. As
$\tilde{Z}_{X(3872)}$ increases, the mass and width of the charmonium state
decrease, up to $\tilde{Z}_{X(3872)}\sim 0.57$ ($\chi_{c1}(2P)$ pole reaches
the real axis with that weight), while $C_{0X}$ increases and takes large
positive values which creates a strong repulsive force between the $D$ and
$D^{*}$ mesons. Thus, the contribution of the molecular component in the
$X(3872)$ is suppressed. However, this mentioned behavior of radiative
branching ratio is valid only for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3906$ MeV. For larger
(smaller) values of bare charmonium, smaller (larger) $\chi_{c1}(2P)$ contents
are required to reach the real axis of the pole. For instance, the
$\chi_{c1}(2P)$ pole appears on the real axis below threshold when
$\tilde{Z}_{X(3872)}\sim$ 0.61 for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3896$ MeV and when
$\tilde{Z}_{X(3872)}\sim$ 0.39 for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3937$ MeV, as can be seen in
Tables 3–9 of the appendix. Due to these non-trivial effects, it is difficult
to put a firm restriction on the charmonium admixture in $X(3872)$.

Figure 1:  Pole trajectories of the $\chi_{c1}(2P)$, located in the SRS, for
the different bare charmonium masses for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3896,\,3906$ and 3908.5 MeV.
The $D\bar{D}^{*}$ threshold is shown as a vertical black dashed line. The
lines and the crosses are obtained with the help of Tables 3, 4 and Table I of
Ref. Cincioglu  _et al._ (2016). While the crosses show the trajectories which
come close to the threshold after colliding the real axis, and the solid
circles show the trajectories which move away from the threshold. Note that
the properties of the poles that depart from the threshold are not given in
Tables 3, 4.

Figure 2:  Pole trajectories of the $\chi_{c1}(2P)$, located in the SRS, for
the different bare charmonium masses for
$\stackrel{{\scriptstyle\circ}}{{m}}_{c\bar{c}}=3910,\,3925,\,3937,\,3947$ and
3953 MeV. The $D\bar{D}^{*}$ threshold is shown as a vertical black dashed
line. To present a general picture, those located in the much above the real
axis do not have any observable effects and are illustrated with crosses. The
circles show other pole trajectories which come close to the threshold from
deep. The lines and the circles are obtained with the help of the values in
Tables 5–9. Note that the properties of the poles that depart from the
threshold are not given in Tables 5–9.

### Acknowledgement

This research has been supported by TUBITAK (The Scientific and Technological
Research Council of Turkey) under the grant no F117090.

##  IV Appendix

In this appendix, for the values whose pole trajectories are shown with the
Figs. 1 and 2, their tables are shown below. The calculations have been
carried out with an UV cutoff $\Lambda=0.5-1$ GeV. In the tables, $d^{crit}$
indicates the point that $C_{0X}$ is zero. After $d^{crit}$ values, $C_{0X}$
becomes positive, which means that the interaction becomes repulsive. While
details of the Fig. 1 values are given by Tables 3 and 4, values in the Fig. 2
are given by Tables 5–9. Moreover, the properties of the $1^{++}$ hidden charm
poles are compiled as a function of the mixing parameter $d$, when an UV
cutoff is used to regularized the molecular interactions (see Tables 11–16).
Note that $X(3872)$ is assumed as a bound state in the FRS; therefore, the
pole position of $X(3872)$ is fixed at 3871.69 MeV in the FRS. Finally, the
position of the $\chi_{1c}(2P)$ is located in the SRS.

Numerical results for an UV cutoff $\Lambda=1$ GeV:

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.01 | -0.792 | 0.891 | 0.978 | (3865.0, 0.1) | 0.02-0.05i | 0.002 | 1.00
0.05 | -0.862 | 0.723 | 0.645 | (3866.1, 2.0) | 0.11-0.20i | 0.055 | 1.05-0.01i
0.10 | -1.084 | 0.503 | 0.312 | (3868.7, 3.17) | 0.25-0.20i | 0.122 | 1.11+0.06i
0.20 | -1.968 | 0.288 | 0.102 | (3870.8, 1.38) | 0.21-0.07i | 0.071 | 1.03+0.06i
0.40 | -5.508 | 0.150 | 0.028 | (3871.46, 0.39) | 0.11-0.03i | 0.022 | 1.01+0.02i
1.00 | -30.285 | 0.061 | 0.005 | (3871.65, 0.06) | 0.05-0.01i | 0.004 | 1.00
3.00 | -266.251 | 0.020 | 0.000 | (3871.69, 0.007) | 0.015-0.003i | 0.00 | 1.00
10.00 | -2950.37 | 0.006 | 0.000 | (3871.69, 0.000) | 0.005-0.001i | 0.00 | 1.00

Table 2: For $m^{0}_{c\bar{c}}=$ 3865 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$.

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3896.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.768 | 0.88 | 0.96 | (3896.7, 2.2) | 0.01-0.19i | 0.03 | 0.98+0.02i
0.10 | -0.707 | 0.83 | 0.86 | (3898.6, 9.6) | 0.00-0.36i | 0.10 | 0.93+0.07i
0.20 | -0.464 | 0.70 | 0.60 | (3900.2, 50.0) | 0.16+0.63i | 0.35 | 0.80+0.29i
0.30 | -0.058 | 0.57 | 0.40 | (3821.0, 123.1) | 0.82+1.02i |  $>$1 | 0.64+1.79i
0.305 | -0.034 | 0.56 | 0.39 | (3797.6, 94.2) | 1.10+1.25i |  $>$1 | 0.62+3.02i
0.307 | -0.024 | 0.56 | 0.39 | (3784.8, 60.5) | 1.48+1.60i |  $>$1 | 0.60+5.33i
0.3075 | -0.021 | 0.56 | 0.39 | (3781.1, 43.9) | 1.79+1.88i |  $>$1 | 0.59+7.59i
0.3078 | -0.019 | 0.56 | 0.39 | (3778.8, 28.1) | 2.28+2.35i |  $>$1 | 0.59+12.12i
0.30798 | -0.019 | 0.56 | 0.39 | (3777.4, 7.9) | 4.35+4.39i |  $>$1 | 0.59+43.38i
0.309 | -0.014 | 0.56 | 0.39 | (3734.2, 0.0) | 0.00+2.25i |  $>$1 | -4.84
0.31 | -0.008 | 0.56 | 0.38 | (3810.7, 0.0) | 1.73 | $|\tilde{X}_{\chi_{c1}}|<1$ | 4.44
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.55 | 0.38 | (3818.8, 0.0) | 1.45 | $|\tilde{X}_{\chi_{c1}}|<1$ | 3.45
0.35 | 0.206 | 0.52 | 0.33 | (3853.3, 0.0) | 0.66 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.57
0.40 | 0.510 | 0.47 | 0.27 | (3861.6, 0.0) | 0.47 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.33
0.50 | 1.241 | 0.40 | 0.19 | (3866.8, 0.0) | 0.33 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.18
1.00 | 7.328 | 0.21 | 0.06 | (3870.8, 0.0) | 0.14 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.04
1.50 | 17.475 | 0.15 | 0.03 | (3871.3, 0.0) | 0.10 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.02
2.00 | 31.680 | 0.11 | 0.02 | (3871.5, 0.0) | 0.07 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01
3.00 | 72.265 | 0.07 | 0.01 | (3871.6, 0.0) | 0.05 | 0.00 | 1.00

Table 3: For $m^{0}_{c\bar{c}}=$ 3896 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.311708$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3908.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.775 | 0.89 | 0.98 | (3909.6, 1.8) | 0.01+0.15i | 0.01 | 0.99+0.01i
0.10 | -0.735 | 0.87 | 0.93 | (3910.6, 7.6) | 0.03+0.30i | 0.06 | 0.97+0.05i
0.20 | -0.574 | 0.79 | 0.77 | (3915.0, 35.8) | 0.14+0.54i | 0.21 | 0.89+0.18i
0.30 | -0.306 | 0.70 | 0.60 | (3910.4, 104.9) | 0.35+0.71i | 0.49 | 0.79+0.44i
0.35 | -0.132 | 0.65 | 0.53 | (3886.1, 172.1) | 0.55+0.80i | 0.83 | 0.73+0.78i
0.38 | -0.014 | 0.63 | 0.49 | (3827.9, 230.5) | 0.80+0.97 |  $>$1 | 0.60+1.55i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.63 | 0.48 | (3811.2, 236.5) | 0.87+1.04i |  $>$1 | 0.54+1.84i
0.39 | 0.027 | 0.62 | 0.47 | (3756.2, 233.0) | 1.13+1.34i |  $>$1 | 0.16+3.31i
0.391 | 0.031 | 0.62 | 0.47 | (3739.8, 225.7) | 1.24+1.47i |  $>$1 | -0.05+4.02i
0.393 | 0.039 | 0.62 | 0.47 | (3672.0, 162.6) | 1.98+2.58i |  $>$1 | -3.05+12.07i
0.3932 | 0.040 | 0.62 | 0.47 | (3652.2, 131.8) | 2.49+3.39i |  $>$1 | -6.48+20.28i
0.39334 | 0.041 | 0.62 | 0.47 | (3616.6, 49.3) | 6.50+9.19i |  $>$1 | -54.59+148.92i
0.393345 | 0.041 | 0.62 | 0.47 | (3611.2, 27.1) | 10.84+14.09 |  $>$1 | -104.57+383.37i
0.393346 | 0.041 | 0.62 | 0.47 | (3609.5, 16.3) | 15.72+18.77 |  $>$1 | -136.33+742.74i
0.395 | 0.048 | 0.62 | 0.47 | (3741.6, 0.0) | 2.14 | $|\tilde{X}_{\chi_{c1}}|<1$ | 6.25
0.398 | 0.061 | 0.61 | 0.46 | (3774.3, 0.0) | 1.58 | $|\tilde{X}_{\chi_{c1}}|<1$ | 3.85
0.40 | 0.069 | 0.61 | 0.46 | (3785.7, 0.0) | 1.43 | $|\tilde{X}_{\chi_{c1}}|<1$ | 3.31
1.00 | 4.572 | 0.31 | 0.12 | (3869.4, 0.0) | 0.22 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.09
2.00 | 20.654 | 0.16 | 0.03 | (3871.2, 0.0) | 0.11 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.02
3.00 | 47.458 | 0.11 | 0.02 | (3871.5, 0.0) | 0.07 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01

Table 4:  For $m^{0}_{c\bar{c}}=$ 3908.5 MeV bare charmonium mass, dressed
mass value of the $\chi_{1c}(2P)$ and its other properties as a function of
$d$ ($d^{crit}=0.383565$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3910.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.776 | 0.89 | 0.98 | (3910.6, 1.8) | 0.01+0.15i | 0.01 | 0.99+0.01i
0.10 | -0.737 | 0.87 | 0.94 | (3912.1, 7.5) | 0.04+0.29i | 0.05 | 0.97+0.04i
0.20 | -0.583 | 0.80 | 0.79 | (3916.6, 34.8) | 0.14+0.53i | 0.20 | 0.89+0.17i
0.30 | -0.325 | 0.71 | 0.62 | (3913.7, 100.7) | 0.34+0.70i | 0.46 | 0.80+0.42i
0.35 | -0.158 | 0.67 | 0.55 | (3895.0, 164.3) | 0.51+0.78i | 0.74 | 0.75+0.70i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.63 | 0.49 | (3818.6, 249.8) | 0.85+1.01i |  $>$1 | 0.54+1.72i
0.40 | 0.035 | 0.63 | 0.48 | (3163.8, 0.0) | 0.86 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.22
0.402 | 0.044 | 0.62 | 0.48 | (3514.7, 0.0) | 3.30 | $|\tilde{X}_{\chi_{c1}}|<1$ | 15.71
0.40205 | 0.044 | 0.62 | 0.48 | (3529.6, 0.0) | 3.61 | $|\tilde{X}_{\chi_{c1}}|<1$ | 18.40
0.4021 | 0.044 | 0.62 | 0.48 | (3545.8, 0.0) | 3.97 | $|\tilde{X}_{\chi_{c1}}|<1$ | 21.75
0.403 | 0.048 | 0.62 | 0.48 | (3694.4, 0.0) | 2.82 | $|\tilde{X}_{\chi_{c1}}|<1$ | 0.52
0.403 | 0.050 | 0.62 | 0.48 | (3714.3, 0.0) | 2.42 | $|\tilde{X}_{\chi_{c1}}|<1$ | 0.52
0.45 | 0.254 | 0.59 | 0.42 | (3838.0, 0.0) | 0.78 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.73
0.50 | 0.499 | 0.55 | 0.37 | (3852.3, 0.0) | 0.59 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.46
1.00 | 4.362 | 0.32 | 0.13 | (3869.2, 0.0) | 0.23 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.10
2.00 | 19.815 | 0.17 | 0.36 | (3871.1, 0.0) | 0.11 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.03
3.00 | 45.569 | 0.11 | 0.02 | (3871.5, 0.0) | 0.07 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01

Table 5:  For $m^{0}_{c\bar{c}}=$ 3910 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.391302$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3925.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.780 | 0.90 | 0.99 | (3925.5, 1.5) | 0.03+0.12i | 0.01 | 1.00+00i
0.10 | -0.752 | 0.89 | 0.97 | (3926.8, 6.2) | 0.06+0.24i | 0.03 | 0.98+0.03i
0.20 | -0.641 | 0.84 | 0.88 | (3931.5, 27.5) | 0.15+0.45i | 0.13 | 0.94+0.11i
0.30 | -0.456 | 0.79 | 0.76 | (3935.9, 73.3) | 0.29+0.62i | 0.29 | 0.87+0.26i
0.40 | -0.196 | 0.72 | 0.64 | (3927.9, 172.3) | 0.50+0.74i | 0.59 | 0.79+0.56i
0.45 | -0.039 | 0.69 | 0.59 | (3898.6, 280.6) | 0.69+0.84i |  $>$1 | 0.67+0.96i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.68 | 0.57 | (3882.1, 325.8) | 0.76+0.90i |  $>$1 | 0.58+1.18i
0.48 | 0.064 | 0.67 | 0.56 | (3373.4, 0.0) | 0.99 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.44
0.50 | 0.137 | 0.66 | 0.54 | (3749.4, 0.0) | 1.20 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.66
0.60 | 0.544 | 0.60 | 0.44 | (3842.3, 0.0) | 0.65 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.53
1.00 | 2.913 | 0.43 | 0.22 | (3866.2, 0.0) | 0.33 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.18
2.00 | 14.017 | 0.23 | 0.07 | (3870.6, 0.0) | 0.16 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.05
4.00 | 58.44 | 0.12 | 0.02 | (3871.4, 0.0) | 0.08 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01

Table 6:  For $m^{0}_{c\bar{c}}=$ 3925 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.461594$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3937.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.781 | 0.90 | 0.99 | (3937.42, 1.36) | 0.03+0.11i | 0.01 | 1.00+0.01i
0.10 | -0.758 | 0.89 | 0.98 | (3938.6, 5.54) | 0.07+0.21i | 0.02 | 0.99+0.02i
0.20 | -0.668 | 0.86 | 0.92 | (3943.2, 23.9) | 0.16+0.40i | 0.09 | 0.96+0.09
0.30 | -0.517 | 0.82 | 0.83 | (3949.1, 61.5) | 0.28+0.56i | 0.22 | 0.91+0.20i
0.40 | -0.305 | 0.77 | 0.73 | (3951.7, 134.3) | 0.44+0.68i | 0.42 | 0.85+0.39i
0.50 | -0.033 | 0.72 | 0.63 | (3930.6, 311.5) | 0.70+0.82i | 0.94 | 0.67+0.88i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.71 | 0.62 | (3923.5, 352.8) | 0.74+0.86i |  $>$1 | 0.59+1.01i
0.53 | 0.060 | 0.70 | 0.61 | (3081.0, 0.0) | 0.47 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.39
0.54 | 0.092 | 0.70 | 0.60 | (3457.4, 0.0) | 0.94 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.24
0.60 | 0.299 | 0.67 | 0.55 | (3795.6, 0.0) | 0.87 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.86
1.00 | 2.233 | 0.49 | 0.30 | (3862.4, 0.0) | 0.40 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.24
2.00 | 11.297 | 0.28 | 0.10 | (3870.0, 0.0) | 0.19 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.07
4.00 | 47.554 | 0.15 | 0.03 | (3871.3, 0.0) | 0.09 | $|\tilde{X}_{chi_{c1}}|<1$ | 1.02

Table 7:  For $m^{0}_{c\bar{c}}$= 3937 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.510903$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3947.0, 0.0) | 0.00 | 0\. 00 | 1.00
0.05 | -0.782 | 0.90 | 1.00 | (3947.4, 1.3) | 0.04 + 0.10i | 0.01 | 0.99 + 0.01i
0.10 | -0.762 | 0.89 | 0.98 | (3948.6, 5.1) | 0.08 + 0.19i | 0.02 | 0.99 + 0.02i
0.20 | -0.683 | 0.87 | 0.93 | (3953.0, 21.8) | 0.16 + 0.37i | 0.08 | 0.97 + 0.07i
0.30 | -0.553 | 0.84 | 0.86 | (3959.3, 54.7) | 0.27 + 0.52i | 0.18 | 0.93 + 0.17i
0.40 | -0.369 | 0.80 | 0.78 | (3965.2, 115.0) | 0.42 + 0.63i | 0.34 | 0.88 + 0.31i
0.50 | -0.134 | 0.75 | 0.70 | (3963.3, 237.0) | 0.61 + 0.73i | 0.64 | 0.78 + 0.60i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.73 | 0.66 | (3953.8, 365.3) | 0.74 + 0.83i | 0.99 | 0.61 + 0.91i
0.60 | 0.155 | 0.71 | 0.61 | (3628.0, 0.0) | 1.00 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.24
1.00 | 1.832 | 0.54 | 0.37 | (3858.0, 0.0) | 0.46 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.30
2.00 | 9.692 | 0.32 | 0.13 | (3869.4, 0.0) | 0.22 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.09
4.00 | 41.135 | 0.17 | 0.03 | (3871.2, 0.0) | 0.11 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.03
6.00 | 93.538 | 0.11 | 0.02 | (3871.5, 0.0) | 0.07 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01

Table 8:  For $m^{0}_{c\bar{c}}=$ 3947 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.548624$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -0.789 | 1.00 | 1.00 | (3953.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -0.783 | 0.90 | 1.00 | (3953.4, 1.2) | 0.04+0.09i | 0.01 | 1.00+0.00i
0.10 | -0.764 | 0.89 | 0.99 | (3954.5, 4.9) | 0.08+0.18i | 0.02 | 0.99+0.02i
0.20 | -0.692 | 0.88 | 0.94 | (3958.8, 20.7) | 0.17+0.35i | 0.07 | 0.97+0.07i
0.30 | -0.570 | 0.85 | 0.88 | (3965.3, 51.4) | 0.27+0.50i | 0.16 | 0.94+0.15i
0.40 | -0.400 | 0.81 | 0.81 | (3972.2, 106.2) | 0.41+0.61i | 0.30 | 0.89+0.28i
0.50 | -0.182 | 0.77 | 0.73 | (3975.2, 210.5) | 0.58+0.70i | 0.54 | 0.81+0.51i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.74 | 0.67 | (3970.6, 370.1) | 0.74+0.82i | 0.95 | 0.61+0.87i
0.60 | 0.085 | 0.73 | 0.65 | (3185.6, 0.0) | 0.50 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.40
0.65 | 0.237 | 0.70 | 0.61 | (3716.5, 0.0) | 0.93 | $|\tilde{X}_{\chi_{c1}}|<1$ | 2.00
1.00 | 1.638 | 0.57 | 0.40 | (3854.6, 0.0) | 0.50 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.33
2.00 | 8.919 | 0.34 | 0.14 | (3868.9, 0.0) | 0.24 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.10
4.00 | 38.041 | 0.18 | 0.04 | (3871.1, 0.0) | 0.12 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.03
6.00 | 86.578 | 0.12 | 0.02 | (3871.4, 0.0) | 0.08 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01

Table 9:  For $m^{0}_{c\bar{c}}=$ 3953 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=$0.57006 fm1/2).

Numerical results for an UV cutoff $\Lambda=0.5$ GeV:

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.01 | -1.941 | 1.052 | 0.995 | (3865.0, 0.21) | -0.02i | 0.001 | 1.00
0.05 | -2.011 | 0.997 | 0.894 | (3865.3, 0.11) | 0.04-0.02i | 0.013 | 1.01-0.01i
0.10 | -2.232 | 0.868 | 0.677 | (3866.1, 1.65) | 0.05-0.20i | 0.048 | 1.05-0.01i
0.20 | -3.118 | 0.619 | 0.344 | (3868.5, 2.80) | 0.17-0.25i | 0.110 | 1.11+0.03i
0.40 | -6.657 | 0.359 | 0.116 | (3870.7, 1.34) | 0.18-0.12i | 0.070 | 1.04+0.06i
1.00 | -31.434 | 0.151 | 0.021 | (3871.5, 0.25) | 0.08-0.04i | 0.015 | 1.01+0.01i
3.00 | -267.4 | 0.051 | 0.002 | (3871.67, 0.03) | 0.03-0.01i | 0.002 | 1.00
10.00 | -2951.5 | 0.015 | 0.000 | (3871.69, 0.00) | 0.01 | 0.000 | 1.00

Table 10: For $m^{0}_{c\bar{c}}=$ 3865 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$.

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3896.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.917 | 1.05 | 0.99 | (3896.2, 0.4) | 0.02-0.18i | 0.006 | 0.99+0.04i
0.10 | -1.856 | 1.03 | 0.96 | (3896.8, 1.8) | 0.04-0.16i | 0.025 | 0.98+0.02i
0.20 | -1.613 | 0.98 | 0.87 | (3899.2, 7.95) | 0.10+0.30i | 0.09 | 0.93+0.07i
0.30 | -1.207 | 0.91 | 0.75 | (3903.1, 21.1) | 0.19+0.4i | 0.21 | 0.86+0.16i
0.40 | -0.639 | 0.83 | 0.63 | (3908.2, 48.3) | 0.33+0.47i | 0.40 | 0.78+0.33i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.77 | 0.53 | (3917.1, 105.9) | 0.48+0.55i | 0.80 | 0.53+0.65i
0.50 | 0.091 | 0.76 | 0.52 | (3920.2, 118.4) | 0.49+0.57i | 0.89 | 0.43+0.69i
0.70 | 2.039 | 0.63 | 0.36 | (3856.6, 0.0) | 0.35 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.28
1.00 | 6.179 | 0.49 | 0.21 | (3866.9, 0.0) | 0.23 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.13
2.00 | 30.53 | 0.27 | 0.06 | (3870.7, 0.0) | 0.113 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.03

Table 11: For $m^{0}_{c\bar{c}}=$ 3896 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.488589$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3910.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.925 | 1.05 | 0.99 | (3910.1, 0.3) | 0.03-0.05i | 0.003 | 0.99+0.02i
0.10 | -1.886 | 1.04 | 0.98 | (3910.6, 1.4) | 0.06-0.11i | 0.013 | 0.99+0.01i
0.20 | -1.731 | 1.024 | 0.94 | (3912.6, 5.8) | 0.13+0.22i | 0.05 | 0.97+0.04i
0.30 | -1.474 | 0.99 | 0.88 | (3916.0, 14.2) | 0.21+0.30i | 0.11 | 0.93+0.10i
0.40 | -1.113 | 0.94 | 0.81 | (3921.0, 28.4) | 0.29+0.37i | 0.21 | 0.88+0.18i
0.50 | -0.650 | 0.90 | 0.73 | (3928.5, 52.3) | 0.39+0.42i | 0.36 | 0.81+0.31i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.85 | 0.64 | (3946.1, 100.5) | 0.51+0.48i | 0.67 | 0.57+0.52i
0.70 | 0.586 | 0.80 | 0.58 | (3979.1, 139.1) | 0.52+0.54i | 0.89 | 0.23+0.46i
1.00 | 3.213 | 0.67 | 0.40 | (3855.4, 0.0) | 0.32 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.24
2.00 | 18.665 | 0.40 | 0.14 | (3869.2, 0.0) | 0.174 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.07
3.00 | 44.419 | 0.28 | 0.07 | (3870.6, 0.0) | 0.117 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.03

Table 12: For $m^{0}_{c\bar{c}}=$ 3910 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.613349$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3925.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.928 | 1.05 | 0.99 | (3925.1, 0.2) | 0.03-0.04i | 0.002 | 0.99+0.02i
0.10 | -1.900 | 1.05 | 0.99 | (3925.5, 1.1) | 0.07-0.08i | 0.008 | 0.99+0.07i
0.20 | -1.789 | 1.04 | 0.97 | (3927.3, 4.5) | 0.13+0.16i | 0.03 | 0.98+0.03i
0.30 | -1.604 | 1.02 | 0.93 | (3930.2, 10.7) | 0.21+0.23i | 0.07 | 0.96+0.06i
0.40 | -1.345 | 0.99 | 0.89 | (3934.7, 20.4) | 0.28+0.28i | 0.14 | 0.93+0.12i
0.50 | -1.012 | 0.96 | 0.84 | (3941.0, 34.9) | 0.36+0.33i | 0.23 | 0.88+0.20i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.89 | 0.72 | (3970.9, 91.9) | 0.53+0.42i | 0.59 | 0.61+0.44i
1.00 | 1.763 | 0.79 | 0.57 | (3822.8, 0.0) | 0.33 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.26
2.00 | 12.868 | 0.52 | 0.25 | (3866.4, 0.0) | 0.23 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.12
4.00 | 57.286 | 0.29 | 0.07 | (3870.6, 0.0) | 0.12 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.04

Table 13: For $m^{0}_{c\bar{c}}=$ 3925 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.723529$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3937.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.930 | 1.05 | 0.99 | (3937.1, 0.2) | 0.03-0.03i | 0.001 | 0.99+0.00i
0.10 | -1.907 | 1.05 | 0.99 | (3937.5, 0.9) | 0.07-0.06i | 0.006 | 0.99+0.00i
0.20 | -1.817 | 1.04 | 0.98 | (3939.1, 3.8) | 0.13+0.12i | 0.02 | 0.98+0.02i
0.30 | -1.665 | 1.03 | 0.95 | (3941.8, 9.0) | 0.20+0.18i | 0.06 | 0.97+0.05i
0.40 | -1.454 | 1.01 | 0.92 | (3945.9, 16.7) | 0.27+0.23i | 0.11 | 0.94+0.01i
0.50 | -1.182 | 0.99 | 0.89 | (3951.5, 27.7) | 0.34+0.27i | 0.17 | 0.91+0.15i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.91 | 0.75 | (3988.2, 84.8) | 0.53+0.37i | 0.54 | 0.62+0.39i
1.00 | 1.083 | 0.86 | 0.66 | (3762.3, 0.0) | 0.23 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.16
1.50 | 4.860 | 0.72 | 0.47 | (3852.0, 0.0) | 0.31 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.21
2.00 | 10.1478 | 0.61 | 0.33 | (3869.0, 0.0) | 0.33 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.48
4.00 | 46.404 | 0.35 | 0.11 | (3869.9, 0.0) | 0.14 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.05

Table 14: For $m^{0}_{c\bar{c}}=$ 3937 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.800832$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3947.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.931 | 1.05 | 0.99 | (3947.1, 0.2) | 0.03-0.02i | 0.001 | 0.99+0.00i
0.10 | -1.911 | 1.05 | 0.99 | (3947.5, 0.8) | 0.06-0.05i | 0.005 | 0.99+0.00i
0.20 | -1.833 | 1.04 | 0.98 | (3949.0, 3.4) | 0.13+0.10i | 0.02 | 0.98+0.02i
0.30 | -1.702 | 1.03 | 0.96 | (3951.8, 7.9) | 0.20+0.15i | 0.05 | 0.97+0.04i
0.40 | -1.518 | 1.02 | 0.94 | (3955.3, 14.5) | 0.26+0.19i | 0.10 | 0.95+0.08i
0.50 | -1.282 | 1.00 | 0.91 | (3960.6, 23.6) | 0.33+0.23i | 0.14 | 0.93+0.12i
0.70 | -0.653 | 0.968 | 0.84 | (3977.5, 50.9) | 0.45+0.28i | 0.31 | 0.82+0.25i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.93 | 0.78 | (4001.6, 79.0) | 0.53+0.33i | 0.50 | 0.64+0.36i
1.00 | 0.682 | 0.90 | 0.72 | (3660.8, 0.0) | 0.12 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.05
2.00 | 8.543 | 0.66 | 0.39 | (3859.3, 0.0) | 0.28 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.18
4.00 | 41.13 | 0.16 | 0.03 | (3871.1, 0.0) | 0.11 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.02

Table 15: For $m^{0}_{c\bar{c}}=$ 3947 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.859959$ fm1/2).

d [fm1/2] | C0X [$fm^{2}$] |  $g_{D\bar{D}^{*}}^{X(3872)}$[GeV-1/2] | $\tilde{X}_{X(3872)}$ | ($m_{\chi_{c1}},\Gamma_{\chi_{c1}}$)[MeV] |  $g_{D\bar{D}^{*}}^{\chi_{c1}}$[GeV-1/2] | $|\tilde{X}_{\chi_{c1}}|$ | $\tilde{Z}_{\chi_{c1}}$
---|---|---|---|---|---|---|---
0.00 | -1.938 | 1.00 | 1.00 | (3953.0, 0.0) | 0.00 | 0.00 | 1.00
0.05 | -1.931 | 1.05 | 0.99 | (3953.1, 0.2) | 0.03-0.02i | 0.001 | 0.99+0.00i
0.10 | -1.913 | 1.05 | 0.99 | (3953.5, 0.8) | 0.06-0.04i | 0.005 | 0.99+0.00i
0.20 | -1.840 | 1.04 | 0.98 | (3954.9, 3.2) | 0.13+0.10i | 0.02 | 0.99+0.01i
0.30 | -1.719 | 1.04 | 0.97 | (3957.4, 7.3) | 0.19+0.13i | 0.04 | 0.98+0.04i
0.40 | -1.549 | 1.03 | 0.95 | (3961.1, 13.4) | 0.26+0.17i | 0.08 | 0.96+0.07i
0.50 | -1.331 | 1.01 | 0.92 | (3966.1, 21.7) | 0.32+0.20i | 0.13 | 0.93+0.11i
0.70 | -0.748 | 0.979 | 0.86 | (3981.9, 45.8) | 0.44+0.26i | 0.28 | 0.84+0.23i
$\emph{d}^{\emph{crit}}$ | 0.000 | 0.94 | 0.79 | (4009.2, 75.7) | 0.53+0.31i | 0.48 | 0.65+0.34i
1.00 | 0.489 | 0.91 | 0.75 | (3550.4, 0.0) | 0.06 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.01
2.00 | 7.769 | 0.69 | 0.43 | (3856.6, 0.0) | 0.29 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.19
4.00 | 36.89 | 0.42 | 0.16 | (3868.9, 0.0) | 0.17 | $|\tilde{X}_{\chi_{c1}}|<1$ | 1.08

Table 16: For $m^{0}_{c\bar{c}}=$ 3953 MeV bare charmonium mass, dressed mass
value of the $\chi_{1c}(2P)$ and its other properties as a function of $d$
($d^{crit}=0.893559$ fm1/2).

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