# Leverage Ratio: An empirical study of the European banking system

Jatin Dhingra  Department of Mathematics, Indian Institute of Technology
Guwahati, Guwahati-781039, India, e-mail<EMAIL_ADDRESS>Kartikeya Singh
Department of Mathematics, Indian Institute of Technology Guwahati,
Guwahati-781039, India, e-mail<EMAIL_ADDRESS>Siddhartha P.
Chakrabarty  Department of Mathematics, Indian Institute of Technology
Guwahati, Guwahati-781039, India, e-mail<EMAIL_ADDRESS>Phone:
+91-361-2582606

###### Abstract

This paper empirically analyzes a dataset published by the European Banking
Authority. Our main aim was to study how the Leverage Ratio is affected by
adverse financial scenarios. This was be followed by observing how Leverage
Ratio exposures are correlated to various other financial variables and how
various regression techniques can be used to explain the correlation.

Keywords: Leverage ratio; Financial variables; Correlation

##  1 Introduction

The 2008 financial crisis was exacerbated to a great extent by the acquisition
of huge leverage (by the large investment banks), while still maintaining a
robust risk-based capital requirements. Accordingly, in order to avoid the
recurrence of this trigger (of the crisis), a non-risk based regulatory
requirement in the form of the Leverage Ratio was incorporated, in addition to
the risk based regulatory requirements, in the Basel-III framework [1].
Restrictions imposed in terms of the Leverage Ratio was put in place with the
intent of inhibiting a built-up of over-leveraging within the banking setup.
Specifically, the Basel-III regulations defined the Leverage Ratio (expressed
in terms of percentage) as the ratio of “Capital Measure” to “Exposure
Measure”, with the regulatory requirement being that the Leverage Ratio cannot
exceed $3\%$. It may be noted that the numerator is the risk-based Tier 1
capital requirement, and the denominator is the sum-total of various defined
classes of the banks’ exposure (including off-balance-sheet items).

We begin with the question of whether the Leverage Ratio increases the
stability of banks. A theoretical and empirical analysis of stability of banks
in the context of Leverage Ratio is examined in [2]. A key observation from
the micro-economic model was that the imposition of the Leverage Ratio
requirement, while increasing the stability of a bank, can also act as an
incentive for banks to adopt more risk taking practices, and this was observed
to concur with the empirical analysis on a fairly large sample of banks in the
European Union (EU), for the period of 2005-2014. The model described in [3]
showed that in case of banks with more conservative (in terms of risk) lending
practices, the Leverage Ratio requirement may act as an inducement for such
banks to diversify into risky loan portfolios, while maintaining the Leverage
Ratio requirement. This in turn could contribute to the overall instability of
the banking sector. In conclusion, the authors recommended a much higher level
of Leverage Ratio requirement, as compared to the existing levels. Blum [4],
in his article, noted that limited supervisory authority (in terms of
identification and punitive action) incentivizes under-reporting of risks by
banks, and showed that Leverage Ratio would be more effective in eliciting
truthful reporting of risks by the banks. An empirical examination of the
impact of capital on the survival and market presence of banks, during a
financial crisis is studied in [5]. The two key observations were that in case
of smaller banks, capital helps in terms of survival, as well as market share
(in all market conditions), and that capital helps medium and large banks in
terms of survival and performance, even in a crisis situation.

Another aspect that merits examination is how the risk changes as a
consequence of enhancement of Leverage Ratio, with several articles in
literature asserting that risk taking practices can change as a result of
increase in capital requirements. An empirical study concluded that a positive
relation exists between changes in capital and risk [6]. Given this
simultaneous behavior, most banks increase their risk exposure, concurrently
with increase in capital requirement, and this holds even in case of banks
which are over-capitalized. A different conclusion, was, however arrived at in
a study for Swiss banks [7], wherein it was observed that while regulatory
enforcement compels banks to (naturally) increase their capital holdings, but
it does not result in concurrent increase in the level of risk. In case of a
study [8] on the impact of the FDIC Improvement Act. (FDICIA), which required
early intervention and prompt corrections, on capital requirements and credit
risk, the authors concluded that increase in the capital ratios did not result
in a concurrent increase in the risk. The relation between capital buffer and
adjustment of portfolio risk is examined in [9], with the conclusion being
that this relation is positive. Further, this risk adjustment of the portfolio
was observed to be contingent on the extent of capitalization.

In this paper, we empirically analyze the dataset of the EU wide stress test
conducted by European Banking Authority (EBA) in 2021, for 50 banks across 15
European Economic Area (EEA), to observe the dynamics of Leverage Ratio and
Leverage Ratio exposures. The remainder of this paper is organized as follows.
Section 2 presents a brief overview of the stress test dataset conducted by
EBA. In Section 3, we empirically analyze the results of EU wide stress test
to observe variations of Leverage Ratio for various banks across different
European countries, during the period of 2020-23. Section 4 empirically
analyzes the results of EU wide stress test, in order to observe co-relation
of Leverage Ratio exposures with other financial variables, namely, Risk
exposure for Credit, Market and Operational Risk, Profit or Loss for the year,
Tier 1 Capital and Net Interest Income. Finally Section 5 presents the
concluding remarks.

##  2 Dataset

The EBA conducted the stress test for 50 banks, pervading 15 EEA countries,
encompassing $70\%$ of the assets held by the European banks [10]. The primary
goal of the entire exercise was to test the resilience of the EU banking
system, under two scenarios, namely, the baseline and the adverse, over a time
window of three years, with a particular emphasis on taking into consideration
the impact of the pandemic. The stress test conducted, identified December
2020 as the starting point of the three year time window. The first of the two
cases considered in the test, namely the baseline scenario, emanating from the
projections of the central banks, envisaged that the GDP of the EU will
experience increases of $3.9\%$, $4.2\%$ and $2.3\%$ for 2021, 2022 and 2023
respectively. For the latter case, namely, the adverse scenario (whose design
was a collaborative exercise between the European Systemic Risk Board (ESRB)
and the European Central Bank (ECB)) was driven primarily by the anticipation
of a long-drawn pandemic, and assumed a real decrease in the GDP of $1.5\%$,
$-1.9\%$ and $–0.2\%$ for 2021, 2022 and 2023 respectively. The dataset
published by the EBA comprises of the values of several financial variables,
for the time window that is being considered, and was stored in two CSV files,
including several considered variables for each bank. In addition, a data
dictionary file is also available, and which can be used for the purpose of
deciphering the information included in the datasets.

##  3 Empirical Analysis: Leverage Ratio in Baseline and Adverse Scenario

In this Section, we empirically analyze the results of the 2021 EU wide stress
test, in order to observe the dynamics and variations of Leverage Ratio in
case of both the baseline as well as the adverse scenario. We graphically
observe the Leverage Ratio variations for various banks with each of the
several European countries, for the dataset considered, during the period of
2020 to 2023. Figure 1 shows the Leverage Ratio variations for several banks
across four different countries, namely, Germany, Finland, France and Italy.
It is clear from Figure 1 that the Leverage Ratios maintained by the banks in
an adverse scenario is significantly lower than the Leverage Ratio requirement
in a baseline scenario, indicating that banks are more susceptible to failure
in the case of an adverse scenario, as compared to the baseline scenario.
Further, the Leverage Ratio also falls below the BCBS recommendation of $3\%$,
in case of few banks, in the adverse scenario. This can be attributed to the
fact that the banks would hold lower capital in an adverse scenario.

Figure 1:  Variations of Leverage Ratio for both the scenarios

##  4 Empirical Analysis: Correlation of Leverage Ratio Exposures with other
Financial Variables

In this Section, we will empirically analyze the results of the 2021 EU wide
stress test, in order to observe how the Leverage Ratio exposures vary with
various other financial variables, namely, Risk exposure for Credit, Market
and Operational Risk, Profit or Loss for the year, Tier 1 Capital and Net
Interest Income. The financial variables correspond to Codes $213501$,
$213504$, $213505$, $213104$, $213110$ and $213601$, respectively, of the Item
Codes of the data dictionary file.

The Leverage ratio exposure of a bank is measured as the sum-total of four
kinds of exposures, namely, on-balance-sheet exposures, derivative exposures,
securities financing transactions (SFT) exposure and off-balance sheet items
[1]. This approach typically does not allow for consideration of collaterals,
guarantees or other credit-risk mitigation measures adopted, to achieve any
sort of reduction in the exposure measure. The on-balance sheet exposure
includes all assets on the balance sheet, except derivatives and SFT assets on
balance sheet, which, as already noted are considered separately. The former,
namely exposure due to derivatives leads to exposures from the underlying of
the derivative and exposure from Counterparty Credit Risk, while the latter,
namely, the SFTs are an inseparable component of the leverage exposure,
particularly securatized borrowing and lending through SFTs. Finally, the off-
balance sheet items, in case of Leverage ratio were adopted from its
definition incorporated in Basel II, and includes commitments, direct credit
substitutes, acceptances and letter of credit (both standby and credit).
Accordingly, in the following subsections, we present the results of the
correlation between Leverage Ratio exposures and various other financial
variables, as already identified in the preceding paragraph. The key metric
for the analysis, is the coefficient of determination, $R^{2}$, which measures
how much variability can be explained using the regression.

###  4.1 Risk Exposure for Credit Risk

Credit risk is defined as the likelihood of losses faced by a creditor, in the
event of failure to pay back the promised amount, by the debtor. These include
scenarios where the borrower fails to repay the loan amount to the bank or a
bond issuer fails to make coupon payments to the bond holder. Figure 2 shows
how the Leverage Ratio exposures vary with risk exposure amount for credit
risk. It is clear from the figure that both linear and polynomial regression
can be used to fit dataset as both result in a high value of $R^{2}$. It may
be noted that risk exposure amount for credit risk increases with an increase
in Leverage Ratio exposures.

Figure 2:  Correlation with risk exposure amount for credit risk

###  4.2 Risk Exposure for Market Risk

Market risk is the risk emanating from the factors that has repercussions
across all kinds of investments across the market. Figure 3 shows how the
Leverage Ratio exposures vary with risk exposure amount for market risk. It is
clear from the figure that linear regression produces a high value of $R^{2}$,
which implies that the variation between the variables can be explained by the
regression. So, it is better to use linear regression to fit the dataset.

Figure 3:  Correlation with risk exposure amount for market risk

###  4.3 Risk Exposure for Operational Risk

Finally, operational risk refers to the risk wherein the firm would loose
money, typically resulting from failure of internal processes. Figure 4 shows
how the Leverage Ratio exposures vary with risk exposure amount for
Operational risk. It is clear from the figure that both linear and polynomial
regression can be used to fit dataset as both produce a high value of $R^{2}$.

Figure 4: Correlation with risk exposure amount for operational risk

###  4.4 Profit or Loss for the Year

Profit and Loss statement is an accounting metric, reflecting the revenues, as
well as costs/expenses over a time window (yearly in our case). Naturally this
is indicative of the firm’s ability to enhance revenues and/or reducing
costs/expenses. Specifically, Profit and Loss account quantifies the overall
health of the firm in terms of reported “net-profit” earned or “net-loss”
incurred, as the case might be. Figure 5 shows how the Leverage Ratio
exposures vary with profit or loss for the year. In the baseline scenario,
profit or loss for the year increase with increase in Leverage Ratio exposure,
whereas we see inverse relation in case of the adverse scenario. The value of
$R^{2}$ is high for baseline scenario and small for adverse scenario, which
means regression technique can be used to fit dataset in only baseline
scenario.

Figure 5:  Correlation with risk exposure amount profit for profit or loss for
the year

###  4.5 Tier 1 Capital

Tier 1 Capital represents the pivotal regulatory capital reserve that a bank
must hold, and includes (among other things) sum-total of common shares and
stock surplus and retained earnings. It’s strong relevance in the context of
Leverage Ration stems from the fact that it appears in the numerator of its
definition itself. Figure 6 shows how the Leverage Ratio exposures vary with
Tier 1 Capital. In both scenarios, Tier 1 Capital increase with increase in
Leverage Ratio exposures. Both regression techniques can be used to fit the
dataset, as both have good values of $R^{2}$.

Figure 6:  Correlation with risk exposure amount for Tier 1 capital

###  4.6 Net Interest Income

Net Interest Income, as the name suggests is the difference between the
revenue from assets which generate interest and the payout on liabilities
emanating from interest driven assets. Examples of the former includes
outstanding personal and commercial loans, while the later primarily includes
interest paid on bonds and deposits. It may be noted that having a floating
interest rate, makes the bank’s Net Interest Income less susceptible to
interest rate movements, as compared to fixed interest rates. Figure 7 shows
how the Leverage Ratio exposures vary with Net Interest Income. In both the
scenarios, Net Interest Income increase with increase in Leverage Ratio
exposures. Polynomial regression should be used to fit the dataset, as it
produces large value of $R^{2}$ as compared to linear regression.

Figure 7:  Correlation with risk exposure amount for net interest income

###  4.7 $R^{2}$ Between Leverage Ratio Exposure and Other Variables

In this Subsection, we analyze how the Leverage Ratio exposures is correlated
to various other financial variables considered, using different regression
techniques, namely, linear, polynomial, decision tree, SVM and random forest
regression. Tables 4.7 and 4.7 represents the $R^{2}$ (coefficient of
determination) values in the baseline and the adverse scenario of the
regression between Leverage Ratio exposures and financial variables discussed
in previous sections, using different regression techniques. The value of
$R^{2}$ is a measure of the effectiveness of the regression technique, and
values close to $1$ reflects that the regression is able to predict the values
with high accuracy. The variable code in the Tables 4.7 and 4.7 correspond to
the variable codes as found in [10]. The values for all the variables have
been reported in the Appendix A. [ longtable =
—p0.14—p0.14—p0.14—p0.14—p0.14—p0.14—, table head=$R^{2}$ values in baseline
scenario.
Variable Code Linear Polynomial SVM Random Forest Decision Tree
, late after last line=
, ]r2_base2.csv  [ longtable = —p0.14—p0.14—p0.14—p0.14—p0.14—p0.14—, table
head=$R^{2}$ values in adverse scenario.
Variable Code Linear Polynomial SVM Random Forest Decision Tree
, late after last line=
, ]r2_adv2.csv

##  5 Conclusion

In this paper, we introduced the notion of Leverage Ratio as described in
Basel-III, and discussed its importance in the context of stability of the
banking system, as well as its repercussion on the risk-taking tendencies in
the banking sector. Then we empirically analyzed the EU-wide stress test
dataset to observe how Leverage Ratio varies in a baseline and an adverse
scenario. We observed that Leverage Ratio maintained by banks is significantly
lower in case of an adverse scenario, as compared to a normal scenario,
implying that bank stability increases with an increase in Leverage Ratio. In
the preceding section we studied the correlation of the Leverage Ratio with
several identified financial variables, and used various regression techniques
to find a correlation between Leverage Ratio and all other variables available
in the EU-wide stress test dataset.

##  Appendix A Tables

[ longtable = —p0.14—p0.14—p0.14—p0.14—p0.14—p0.14—, table head=$R^{2}$ values
in baseline scenario.
Variable Code Linear Polynomial SVM Random Forest Decision Tree
, late after last line=
, ]r2_base1.csv  [ longtable = —p0.14—p0.14—p0.14—p0.14—p0.14—p0.14—, table
head=$R^{2}$ values in adverse scenario.
Variable Code Linear Polynomial SVM Random Forest Decision Tree
, late after last line=
, ]r2_adv1.csv

## References

  * [1] Basel III leverage ratio framework and disclosure requirements, https://www.bis.org/publ/bcbs270.pdf, Accessed: 2022-06-20.
  * [2] J. Acosta-Smith, M. Grill, J. H. Lang, The leverage ratio, risk-taking and bank stability, Journal of Financial Stability (2020) 100833.
  * [3] I. Kiema, E. Jokivuolle, Does a leverage ratio requirement increase bank stability?, Journal of Banking & Finance 39 (2014) 240–254.
  * [4] J. M. Blum, Why ‘Basel II’may need a leverage ratio restriction, Journal of Banking & Finance 32 (8) (2008) 1699–1707.
  * [5] A. N. Berger, C. H. Bouwman, How does capital affect bank performance during financial crises?, Journal of Financial Economics 109 (1) (2013) 146–176.
  * [6] R. E. Shrieves, D. Dahl, The relationship between risk and capital in commercial banks, Journal of Banking & Finance 16 (2) (1992) 439–457.
  * [7] B. Rime, Capital requirements and bank behaviour: Empirical evidence for switzerland, Journal of Banking & Finance 25 (4) (2001) 789–805.
  * [8] R. Aggarwal, K. T. Jacques, The impact of fdicia and prompt corrective action on bank capital and risk: Estimates using a simultaneous equations model, Journal of Banking & Finance 25 (6) (2001) 1139–1160.
  * [9] T. Jokipii, A. Milne, Bank capital buffer and risk adjustment decisions, Journal of Financial Stability 7 (3) (2011) 165–178.
  * [10] European Banking Authority: EU-wide stress testing, https://www.eba.europa.eu/risk-analysis-and-data/eu-wide-stress-testing, Accessed: 2022-01-10.
  * [11] G. Dell’Ariccia, L. Laeven, R. Marquez, Real interest rates, leverage, and bank risk-taking, Journal of Economic Theory 149 (2014) 65–99.
  * [12] J. Hull, Risk management and financial institutions,+ Web Site, Vol. 733, John Wiley & Sons, 2012.


Molecular states of D^* D^* K̅^* nature

N. Ikeno
Department of Agricultural, Life and Environmental Sciences, Tottori University, Tottori 680-8551, Japan

M. Bayar
Department of Physics, Kocaeli University, 41380, Izmit, Turkey
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de
Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.
22085, 46071 Valencia, Spain

E. Oset
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de
Valencia-CSIC Institutos de Investigación de Paterna, Aptdo.
22085, 46071 Valencia, Spain

We study the interaction of two $ D^* $ and a $\bar{K}^{*}$
by using the Fixed Center Approximation to the Faddeev equations to search for bound states of the three body system. Since the $ D^*  D^* $ interaction is attractive and gives a bound state, and so is the case of the $D^* \bar{K}^{*}$ interaction, where the $J^{P}=0^{+}$ bound state is identified with the $X_0 (2900)$, the $ D^*  D^* \bar{K}^{*}$  system leads to manifestly exotic bound states with $ccs$ open quarks. We obtain bound states of isospin $I=1/2$, negative parity and total spin $J=0,1,2$. For $J=0$ we obtain one state, and for $J=1,2$ we obtain two states in each case. The binding energies range from $56$ MeV to $151$ MeV and the widths from $80$ MeV to $100$ MeV.

§ INTRODUCTION

While the study of general Few Body Systems has a long tradition, the study of Few Body Systems made of mesons has only a recent history. A review of such systems has been done in Ref. [1]. The study of the meson meson interaction with chiral Lagrangians [3] and the realization that this interaction, properly unitarized, led to meson–meson bound states that could be associated to known mesonic resonances [5, 7, 9, 11, 13] (see recent reviews in Refs. [15, 17, 19, 21]). The same formalism allowed one to address three body systems made with mesons, some of which could be associated to known mesonic states [1].

With a few exceptions [23], most of the states found in the past from the meson meson interaction correspond to states which are not manifestly exotic, in the sense that they could also be formed in principle from a conventional $q \bar q$. Yet, the recent experimental findings of the $X_0 (2900)$, $X_1 (2900)$ [25, 27] in the $D \bar K$ invariant mass, and the $T_{cc}$ state [29, 31] in the $D D \pi$ spectrum, revealed clear exotic mesonic structures, since one has $cs$ quarks in the first case and $cc$ quarks in the second one. These findings open the door to the formation of few body systems with several open quarks, having three of more quarks with flavors like $ccs$, $ccc$, $css$, etc., after eliminating the $q \bar q$ structures with no flavor as $u \bar u$, $d \bar d$, $s \bar s$, $c \bar c$, etc.
Since flavor is conserved in strong interactions, these systems can be relatively stable because they cannot decay into lighter systems with smaller number of mesons. In the present paper we report on calculations for the $D^* D^* \bar K^*$ system, which has open $ccs$ quarks.
The reason to choose this system is because we can establish connection with the experimental findings of Refs. [25, 27, 29, 31]. Indeed, the $X_0 (2900)$ was soon identified  as a likely $D^* \bar K^*$ molecular state [33, 35, 37, 39, 41, 43, 45, 47, 49, 51]. Other works favor compact tetraquarks [53, 54, 56],
yet, the relativized quark model of Ref. [58] disfavors the compact tetraquark structure and favors the molecular one.

A valuable information concerning the $D^* \bar K^*$ system comes from the work of Ref. [23], where 10 years prior to the observation of the $X_0 (2900)$, a $D^* \bar K^*$ molecule  had been predicted with the same quantum numbers as the $X_0 (2900)$ and a mass and width remarkably close to those of the observed ones.
Indeed, the mass predicted in [23] for the $D^* \bar K^*$ system with $ I=0 $, $J^{P}=0^{+}$ was $2848$ MeV and the width between $23-59$ MeV. Experimentally the  $X_0 (2900)$ was found with these quantum numbers, mass $M=2866\pm 7$ MeV, and width $ \Gamma =57.2 \pm 12.9  $ MeV. A fine tuning of the parameters to obtain the exact experimental values is done in [43].

The other part of the interaction is for the $D^* D^*$ subsystem. Here we rely upon the recent observation of the $T_{cc}$ state [29, 31]. Once again the $D^* D$ nature of the $ T_{cc} $ state has been advocated from the beginning [31] and supported by many theoretical papers [60, 17, 62, 64, 74, 76, 70, 78, 80, 82, 84, 86, 88, 66, 68, 90, 72].
A mapping of the $D^* D$ interaction to the $D^* D^*$ system has been done in Ref. [91] using an extrapolation of the local hidden gauge approach [93, 95, 97, 99] to the charm sector, which was already used in Ref. [23] to study the same $D^* D^*$ exotic system.
A similar approach using the boson exchange model is done in Ref. [101]. It was found in Ref. [23] that the $D^* D^*$ system was bound in $I=0$ and $J^P = 1^+$. In Ref. [91], the model was refined including decay channels and using the same cut off to regularize the loops that was used in Ref. [62] to describe the $T_{cc}$ state.

Three body systems of molecular nature containing two charmed quarks (or antiquarks) and a strange quark have been recently studied. In Table 1 of Ref. [1] one finds the $D D K$ molecule studied in Ref. [103, 105, 106], $D \bar{D}^* K$ studied in Ref. [107], and $D D^* K$ studied in Ref. [109]. More recently the $D \bar{D}^* K$ hexaquark state is also studied in [111] via QCD sum rules.
In these systems one has, however, $cc \bar s$ or $c \bar c \bar s$ quarks, but not the $ccs$ combination that we have in the system that we study, which makes it super exotic. These works are done using different technical approaches. However, for the purpose of justifying the Fixed Center Approximation (FCA) to the Faddeev equations that we follow, we find it interesting to mention two recent works on the study of the $D \bar D K$ system. In Ref. [112] it is studied using the Gaussian expansion method, minimizing the energy of the system. The same system is studied in Ref. [114] using the Fixed Center Approximation (FCA) and the results obtained are very similar. The coincidence of two very different methods in a similar system to the one we study gives us confidence in the FCA method that we use in the present approach to study the molecular $D^* D^* \bar K^*$ system. The fact that we use input tuned to the $T_{cc}$ state of $D^* D$ nature in Ref. [62] and the  $X_0 (2900)$ state of $D^* \bar K^*$ nature in Ref. [43], gives us further confidence, not only on the existence of the bound states that we find, but also on the values of the masses and widths that we predict.

§  FORMALISM

We will use the FCA to study the $ D^* D^* \bar{K}^{*} $ system. In this picture, there is a cluster of two bound particles and the third one collides with the components of this cluster without modifying its wave function. Certainly, if the third particle is lighter than the constituents of the cluster, the approximation is better, which suggests to consider the $ D^* D^* $ system as the cluster. This latter system has been shown to be bound in Refs. [116, 117, 91]. In Ref. [91] the local hidden gauge approach extrapolated to the charm sector has been used, and the loops are regularized with the cut off demanded to fit the binding of the $T_{cc}$ state as a $ D^* D$ molecule in Ref. [62]. A state with isospin $I=0$ and $J^{P}=1^{+}$ was found. The predicted binding should be realistic and we take the results of Ref. [91] as input for the $ D^* D^* $ cluster.

We follow the formalism of Ref. [118]. There, the amplitude for $\bar{K}^{*}$ collision with the $D^* D^*  $  cluster is given by the sum of the partition functions $T_{1}$, $T_{2}$, where $T_{1}$ sums all diagrams in Fig. <ref> where the $\bar{K}^{*}$  collides first with the particle $1$ of the cluster, while $T_{2}$ sums all diagrams where the $\bar{K}^{*}$ collides first with particle $2$ of the cluster.

Diagrams involved in the FCA for the collision of the  $\bar{K}^{*}$ with the cluster of $ D^* D^* $.

We have for the total amplitude $T$

\begin{eqnarray}
T \equiv  T_1 + T_2  \nonumber
\label{eq:TotalT}
\end{eqnarray}

and $T_{1}$, $T_{2}$ are coupled through

\begin{eqnarray}
T_1 &=&t_1 + t_1 G_0 T_2, \nonumber \\
T_2&=&t_2 + t_2 G_0 T_1, \nonumber
\label{eq:T1T2}
\end{eqnarray}

where $t_{1}$ is the scattering amplitude for $D^*(1) \bar{K}^{*} $, and $t_{2}$ the corresponding amplitude for $D^*(2) \bar{K}^{*} $  scattering, and $G_0$ is the $\bar{K}^{*} $ propagator folded with the cluster wave function.  However, we should take into account the isospin and spin of the $  D^*(i) \bar{K}^{*}$ amplitudes, where $i$ refers to any  $  D^* $ of the cluster.

§.§  ISOSPIN CONSIDERATIONS

With the isospin doublet $(D^{+}, - D^{0})$, the $I=0$ $D^* D^*$ state is given by

\begin{eqnarray}
\mid D^{*} D^{*}, ~I=0 \rangle =-\frac{1}{\sqrt{2}} ( D^{*+} D^{*0}- D^{*0} D^{*+}) \nonumber \\
\end{eqnarray}

and one should also bear in mind that this is accompanied by $ J=1 $ in each of the two components, which we will address below.

When evaluating $t_{1}$ of the scattering amplitude for $D^*(1) \bar{K}^{*}$, we have the following matrix element using the notation

$ \mid I_{3} (D^*(1)),I_{3} (D^*(2))\rangle \mid I_{3} (\bar{K}^{*})\rangle$, where we have chosen $\bar{K}^{*} $ to have the $ I_{3}=\frac{1}{2} $ component

\begin{eqnarray}
\frac{1}{\sqrt{2}} \frac{1}{\sqrt{2}} && \left(  \langle D^* D^*, \frac{1}{2}, -\frac{1}{2} \mid - \langle D^* D^*, -\frac{1}{2}, \frac{1}{2} \mid \right) \nonumber \\
&& \times \langle \bar{K}^{*}, \frac{1}{2} \mid  \mid t_1 \mid  \left(  \mid D^* D^*, \frac{1}{2}, -\frac{1}{2} \rangle - \mid D^* D^*, -\frac{1}{2}, \frac{1}{2} \rangle \right)  \mid \bar{K}^{*}, \frac{1}{2} \rangle.
\label{IsoXX}
\end{eqnarray}

To make connection with the $  D^* \bar{K}^{*}  $ isospin amplitudes of Refs. [23, 43], we combine the third component of $ D^*(1) $ with the one of the $ \bar{K}^{*}  $ to give states of $  D^* \bar{K}^{*}  $  isospin. Hence, we have with the notation
$ \mid  I (D^*(1) \bar{K}^{*}),~  I_{3} (D^*(1) \bar{K}^{*}) \rangle \mid  I_{3} (D^{*}(2))  \rangle $

\begin{eqnarray}
&&\dfrac{1}{2} \left[   \left( \langle 1,1 \mid \langle - \frac{1}{2}  \mid -\dfrac{1}{\sqrt{2}}  (\langle 1,0 \mid - \langle 0,0 \mid ) \langle \frac{1}{2}     \mid  \right)  \vert~t_1 ~ \vert \left(  \mid 1,1  \rangle  \mid - \frac{1}{2} \rangle - \dfrac{1}{\sqrt{2}} ( \mid 1 0 \rangle  -\mid 0 0\rangle)  \mid \frac{1}{2} \rangle \right) \right] \nonumber \\
&&= \dfrac{1}{2} \left( t^{I=1} + \dfrac{1}{2} t^{I=1} +  \dfrac{1}{2} t^{I=0} \right) =  \dfrac{3}{4} t^{I=1}+ \dfrac{1}{4} t^{I=0},
\label{IsospinTot}
\end{eqnarray}

where the final result stems since $t_1$ only affects  $  D^*(1) \bar{K}^{*}  $, considering that $ D^*(2)$ is a spectator and has to be the same in the bra and the ket of the matrix element. The amplitude $t_2$, for $D^*(2)\bar{K}^{*}$ scattering is obviously equal to $t_1$.

§.§  SPIN CONSIDERATIONS

We start with $ D^* D^* $ with $ J=1 $ and the $ \bar{K}^{*} $ has also $ J=1 $. Hence, we can have three total spins for $ D^* D^* \bar{K}^{*} $, $ J=0,1,2 $. Let us see each one of the cases.

§.§.§  TOTAL $J=0$

With the notation $ \mid  j_{3} (D^* D^* ), j_{3} (\bar{K}^{*}) \rangle $,  we have

\begin{equation}
\mid D^* D^* \bar{K}^{*}, J=0  \rangle =\dfrac{1}{\sqrt{3}} \left(  \mid 1,-1  \rangle - \mid 0,0  \rangle + \mid -1,1  \rangle \right)   \nonumber
\end{equation}

and now we write the state $ \mid  j (D^* D^* )=1 , j_{3} (D^* D^* ) \rangle $ in terms of the $  \mid j_{3} (D^*(1)) j_{3} (D^*(2))  \rangle$ and we have

\begin{eqnarray}
\dfrac{1}{\sqrt{3}} \left\lbrace \dfrac{1}{\sqrt{2}} (\mid 1,0  \rangle - \mid 0,1  \rangle) \mid -1  \rangle - \dfrac{1}{\sqrt{2}} (\mid 1,-1  \rangle - \mid -1,1  \rangle) \mid 0 \rangle + \dfrac{1}{\sqrt{2}} (\mid 0,-1  \rangle - \mid -1,0  \rangle) \mid 1 \rangle \right\rbrace  \nonumber
\end{eqnarray}

And now we write $ \mid j_{3} (D^*(1)) j_{3} (\bar{K}^{*}) \rangle$ in terms of $ \mid j(D^* \bar{K}^{*}) j_{3} (D^* \bar{K}^{*}) \rangle $,
and with the notation $ \mid  j(D^*(1) \bar{K}^{*}) , j_{3} (D^*(1) \bar{K}^{*}) \rangle  \mid j_{3} (D^*(2)) \rangle  $, we have

\begin{eqnarray}
&& \dfrac{1}{\sqrt{3}}  \dfrac{1}{\sqrt{2}} \left\lbrace  \left( \dfrac{1}{\sqrt{6}} \mid 2,0  \rangle  + \dfrac{1}{\sqrt{2}}  \mid 1,0  \rangle +  \dfrac{1}{\sqrt{3}} \mid 0,0  \rangle \right)  \mid 0  \rangle    - \left( \dfrac{1}{\sqrt{2}}  \mid 2,-1  \rangle  + \dfrac{1}{\sqrt{2}}  \mid 1,-1  \rangle \right) \mid 1 \rangle \right.  \nonumber \\
&& - \left( \dfrac{1}{\sqrt{2}}  \mid 2,1  \rangle  + \dfrac{1}{\sqrt{2}}  \mid 1,1  \rangle \right) \mid -1 \rangle + \dfrac{1}{\sqrt{2}} \left( \mid 2,-1  \rangle - \mid 1,-1  \rangle \right)   \mid 1  \rangle \nonumber \\
&& \left.   +\dfrac{1}{\sqrt{2}} \left( \mid 2,1  \rangle - \mid 1,1  \rangle \right)   \mid -1  \rangle - \left( \dfrac{1}{\sqrt{6}} \mid 2,0  \rangle - \dfrac{1}{\sqrt{2}}  \mid 1,0  \rangle +  \dfrac{1}{\sqrt{3}} \mid 0,0  \rangle \right) \mid 0  \rangle \right\rbrace \nonumber \\
&&= \dfrac{1}{\sqrt{3}}  \left\lbrace  \mid 1,0  \rangle \mid 0  \rangle -\mid 1,-1 \rangle \mid 1  \rangle - \mid 1,1  \rangle \mid -1  \rangle \right\rbrace   \nonumber
\end{eqnarray}

and the matrix element of $ t_1 $ sandwiched between this state, considering that the $ D^*(2) $ state is a spectator, becomes

\begin{eqnarray}
t_1 = \dfrac{1}{3}  \left( t^{j=1}+ t^{j=1} +t^{j=1} \right) = t^{j=1}. \nonumber
\end{eqnarray}

The result is expected since if we have $ \mid (D^*(1) \bar{K}^{*} ) D^*(2) \rangle $ with $j(D^*(2)) =1$, the spin of $  D^*(1)\bar{K}^{*}$ must necessarily  be $1$ to match total spin zero for the $ (D^*(1) \bar{K}^{*} ) D^*(2) $ system.

§.§.§  TOTAL $J=1$,  $ J_{3}=1 $

The result cannot depend on the third component and we choose it to be $ J_{3} =1 $ for simplicity.
Once again, with the $ \mid  j_{3} (D^* D^* ), j_{3} (\bar{K}^{*}) \rangle $ representation we have the state

\begin{equation}
\dfrac{1}{\sqrt{2}} \left\lbrace \mid 1,0  \rangle - \mid 0,1  \rangle \right\rbrace \nonumber
\end{equation}

which in the $ \mid  j_{3} (D^*(1)) j_{3} (D^*(2))  \rangle \mid  j_{3} (\bar{K}^{*})  \rangle $ representation reads

\begin{eqnarray}
\dfrac{1}{\sqrt{2}} \left\lbrace \left(  \dfrac{1}{\sqrt{2}} \mid 1,0  \rangle - \dfrac{1}{\sqrt{2}} \mid 0,1  \rangle \right)  \mid 0 \rangle - \left(  \dfrac{1}{\sqrt{2}} \mid 1,-1  \rangle - \dfrac{1}{\sqrt{2}} \mid -1,1  \rangle \right)  \mid 1 \rangle \right\rbrace  \nonumber
\end{eqnarray}

and written in terms of  $ \mid  j(D^* (1) \bar{K}^{*}) j_{3} (D^* (1) \bar{K}^{*})  \rangle  \mid  j_{3} (D^* (2) ) \rangle  $ is

\begin{eqnarray}
&&   \dfrac{1}{2} \left\lbrace  \left( \dfrac{1}{\sqrt{2}} \mid 2,1  \rangle  + \dfrac{1}{\sqrt{2}}  \mid 1,1  \rangle  \right)  \mid 0  \rangle    + \left( -\sqrt{\dfrac{2}{3}}  \mid 2,0  \rangle  + \dfrac{1}{\sqrt{3}}  \mid 0,0 \rangle \right) \mid 1 \rangle \right.  \nonumber \\
&&\left.  -  \mid 2,2  \rangle  \mid -1  \rangle + \left( \dfrac{1}{\sqrt{6}}  \mid 2,0  \rangle  - \dfrac{1}{\sqrt{2}}  \mid 1,0  \rangle + \dfrac{1}{\sqrt{3}}  \mid 0,0  \rangle \right) \mid 1 \rangle  \right\rbrace  \nonumber \\
&&=  \dfrac{1}{2} \left\lbrace  \left( \dfrac{1}{\sqrt{2}} \mid 2,1  \rangle  + \dfrac{1}{\sqrt{2}}  \mid 1,1  \rangle \right) \mid 0 \rangle - \dfrac{1}{\sqrt{6}} \mid 2,0  \rangle  \mid 1 \rangle +\dfrac{2}{\sqrt{3}} \mid 0,0  \rangle  \mid 1 \rangle \right.  \nonumber \\
&& \left. - \mid 2,2  \rangle  \mid -1 \rangle -\dfrac{1}{\sqrt{2}} \mid 1,0  \rangle  \mid 1 \rangle
\right\rbrace .  \nonumber
\end{eqnarray}

The $ t_1 $ matrix sandwiched between this state gives the combination

\begin{eqnarray}
t_1 = \dfrac{1}{4}  \left(\dfrac{5}{3}  t^{j=2}+ t^{j=1} + \dfrac{4}{3} t^{j=0} \right).  \nonumber
\end{eqnarray}

§.§.§  TOTAL $J=2$,  $ J_{3}=2 $

Once again the result does not depend on the third component, which we choose to be $ J_{3} =2 $.
In the $ \mid  j_{3} (D^* D^* ), j_{3} (\bar{K}^{*}) \rangle $  representation the state is

\begin{equation}
\mid 1,1 \rangle.  \nonumber
\end{equation}

Then with the $ \mid j_{3} (D^*(1)) j_{3} (D^*(2)) \rangle \mid j_{3} (\bar{K}^{*}) \rangle $ notation  we have

\begin{eqnarray}
\left(  \dfrac{1}{\sqrt{2}} \mid 1,0  \rangle - \dfrac{1}{\sqrt{2}} \mid 0,1  \rangle \right)  \mid 1 \rangle  \nonumber
\end{eqnarray}

which in the  $ \mid  j(D^* (1) \bar{K}^{*}) j_{3} (D^* (1) \bar{K}^{*}) \rangle \mid j_{3} (D^* (2) ) \rangle $ representation  reads

\begin{eqnarray}
\dfrac{1}{\sqrt{2}}  \mid 2,2  \rangle  \mid 0 \rangle -  \dfrac{1}{\sqrt{2}} \left(  \dfrac{1}{\sqrt{2}} \mid 2,1 \rangle - \dfrac{1}{\sqrt{2}} \mid 1,1  \rangle \right)  \mid 1 \rangle.  \nonumber
\end{eqnarray}

The $ t_1 $ matrix sandwiched between this state is then

\begin{eqnarray}
t_1 = \dfrac{3}{4}  t^{j=2} +  \dfrac{1}{4} t^{j=1}. \nonumber
\end{eqnarray}

§.§.§  COMBINED SPIN AND ISOSPIN AMPLITUDE

Combining the isospin and the spin decomposition of the amplitudes in subsections <ref> and <ref>, we find the final contributions

\begin{eqnarray}
t_1 &=& \dfrac{3}{4}  t^{I=1,~j=1} +  \dfrac{1}{4} t^{I=0,~j=1}, ~~~\text{for} ~~~J=0 \nonumber \\
t_1 &=& \dfrac{1}{16} \left\lbrace 5 t^{I=1,~j=2}+ 3 t^{I=1,~j=1}+ 4 t^{I=1,~j=0}+ \dfrac{5}{3} t^{I=0,~j=2}\right. \nonumber \\
&+& \left. t^{I=0,~j=1}+\dfrac{4}{3} t^{I=0,~j=0} \right\rbrace, ~~~\text{for} ~~~J=1\nonumber \\
t_1 &=& \dfrac{1}{16} \left\lbrace 9 t^{I=1,~j=2}+ 3 t^{I=1,~j=1}+ 3 t^{I=0,~j=2}+  t^{I=0,~j=1} \right\rbrace, ~~~\text{for} ~~~J=2 .
\label{eq:AmpJ012}
\end{eqnarray}

§.§  NORMALIZATION OF THE AMPLITUDES

By looking at the papers [119, 120] and the diagram of double scattering in Fig. <ref>, we find for the $ S $ matrix

\begin{eqnarray}
-i  (2\pi)^4  \delta^4 (p_{fin}-p_{in})   \frac{1}{ {\cal V}^2 } \nonumber \\
\frac{1}{ \sqrt{2\omega_{\bar{K}^{*}}} } \frac{1}{ \sqrt{2\omega_{\bar{K}^{*}}} }
\frac{1}{ \sqrt{2\omega_{D^*}} } \frac{1}{ \sqrt{2\omega_{D^*}} }
\frac{1}{ \sqrt{2\omega_{D^*}} } \frac{1}{ \sqrt{2\omega_{D^*}} } \nonumber \\
t_{1} t_{2} \int \frac{d^3 q}{(2\pi)^3} F(\vec{q}) \frac{1}{ q^{0^{2}} -\vec{q}~^2 - m^{2}_{{\bar{K}^{*}}}  + i \epsilon }   ,
\label{eq:S2a}
\end{eqnarray}

where $ \cal V $ is the volume of the normalization box,  $  p_{fin}$ and $p_{in} $ the final and initial momenta of the three particle system, $ q $ is the momentum of the exchanged $ \bar{K}^{*} $, $w_{i}=\sqrt{ m^{2}_{i}+\vec{q}_{i}^{~2}}$ and $  F(\vec{q}) $ the form factor of the cluster. If we look at the process from the macroscopic perspective of having $  (D^*(1) D^*(2))_{C} \bar{K}^{*}$, with $(D^*(1) D^*(2))_{C} $ meaning the $  D^* D^* $ cluster, the $S$ matrix reads

\begin{eqnarray}
-i  (2\pi)^4  \delta^4 (p_{fin}-p_{in})   \frac{1}{ {\cal V}^2 }
\frac{1}{ \sqrt{2\omega_{\bar{K}^{*}}} } \frac{1}{ \sqrt{2\omega_{\bar{K}^{*}}} }
\frac{1}{ \sqrt{2\omega_{C}} } \frac{1}{ \sqrt{2\omega_{C}} } T^{(2)}  ,
\label{eq:S2}
\end{eqnarray}

\begin{eqnarray}
T^{(2)}&=&2\omega_{C} ~t_{1} t_{2} ~\frac{1}{ 2\omega_{D^*} } \frac{1}{ 2\omega_{D^*} }
\int \frac{d^3 q}{(2\pi)^3} F(\vec{q}) \frac{1}{ q^{0^{2}} -\vec{q}~^2 - m^{2}_{{\bar{K}^{*}}}  + i \epsilon }  \nonumber \\
&=& \dfrac{2\omega_{C}}{2\omega_{D^*}} ~  \dfrac{2\omega_{C}}{2\omega_{D^*}} ~ \dfrac{1}{2\omega_{C}}
~t_{1} t_{2} \int \frac{d^3 q}{(2\pi)^3} F(\vec{q}) \frac{1}{ q^{0^{2}} -\vec{q}~^2 - m^{2}_{{\bar{K}^{*}}}  + i \epsilon }.
\label{eq:T2}
\end{eqnarray}

It is  thus convenient to write the partition functions suited to the macroscopic formalism as

\begin{equation}
\tilde{T_1} =\tilde{t_1} + \tilde{t_1} \tilde{G_0}  \tilde{T_2};~~~~   \tilde{T_2}=\tilde{t_2} + \tilde{t_2} \tilde{G_0} \tilde{T_1},
\label{eq:tildeT1T2}
\end{equation}

where approximating $ \omega_{D^*} = m_{D^*} $, $ \omega_{C} = m_{C} $

\begin{equation}
\tilde{t_1}=\frac{2m_C}{2m_{D^*}}~ t_1;~~~~\tilde{t_2}=\frac{2m_C}{2m_{D^*}}~ t_2, \nonumber
\label{eq:oran}
\end{equation}

\begin{equation}
\tilde{G_0}=\frac{1}{2 m_{C}}\int\frac{d^3q}{(2\pi)^3}F(\vec{q})\frac{1}{{q^0}^2-\vec{q}\,^2-m_{\bar{K}^{*}}^2+i\epsilon}.
\label{Eq:G0}
\end{equation}

The wave function of the cluster enters through $ \tilde{G_0} $ via the form factor $ F( \vec{q}) $. The function $ \tilde{G_0} $  corresponds to the propagator of $ K^* $ between two scatterings in Fig. <ref>, folded with the wave function of the cluster.

The form factor is $ F(q) = \int d^{3}r e^{i \vec{q}\cdot \vec{r} } \mid \Psi(\vec{r}) \mid^{2} $, but we find convenient to write it in terms of the wave function written in momentum space. For this purpose let us recall that the unitary approach that we use to obtain the $  D^* D^* $ bound states can be easily visualized as coming from the use of a separable potential of the type

\begin{equation}
V(\vec{q}, \vec{q}~')= V ~\theta(q_{max}-\mid \vec{q} \mid) ~ \theta(q_{max}-\mid \vec{q}~' \mid)
\label{Eq:Vvv}
\end{equation}

from where one can easily deduce the wave function in momentum space as [121]

\begin{equation}
\Psi(\vec{p})= g_{R}~\dfrac{\theta(q_{max}-\mid \vec{p} \mid)}{E-\omega_{1}(\vec{p})-\omega_{2}(\vec{p})}
\label{Eq:Psip}
\end{equation}

where $ g_{R} $ is the coupling of the state to the two components of the state and $ \omega_{i}= \sqrt{m^{2}_{i}+\vec{p}^{~2}}$. Then, using the Fourier transform of $ \Psi(\vec{q}) $ to go to $ \Psi(\vec{r}) $, one immediately finds the form factor $  F(\vec{q}) $, normalized as $  F(\vec{q}=0)=1 $, written in terms of $ \Psi(\vec{p}) $ as [118, 119],

\begin{eqnarray}
F(q) &=& \frac{1}{{\cal N}}
\int_{ \stackrel{p < q_{{ max} }}{|{\vec{p} - \vec{q}}| < q_{{max}}}}
d^3p ~
~ \frac{1}{m_{C}-\sqrt{m^{2}_{D^*}+ \vec{p}^{~2}} -\sqrt{m^{2}_{D^*}+ \vec{p}^{~2}}} \nonumber \\
\frac{1}{m_{C}-\sqrt{m^{2}_{D^*}+ (\vec{p}-\vec{q})^{2}} -\sqrt{m^{2}_{D^*}+ (\vec{p}-\vec{q})^{2}}},
\label{eq:formfactor}
\end{eqnarray}

with the normalization constant ${\cal N}$

\begin{eqnarray}
{{\cal N }}
\int_{p < q_{{max}}} d^3p \left( ~ \frac{1}{m_{C}-\sqrt{m^{2}_{D^*}+ \vec{p}^{~2}} -\sqrt{m^{2}_{D^*}+ \vec{p}^{~2}}} \right)^2  \nonumber
\label{eq:normformf}
\end{eqnarray}

With the potential of Eq. (<ref>) the $t$ matrix is also separable

\begin{equation}
T(\vec{q}, \vec{q}~')= \theta(q_{max}-\mid \vec{q} \mid) ~ \theta(q_{max}-\mid \vec{q}~' \mid) ~t
\label{Eq:tseperable}
\end{equation}

\begin{equation}
t=V+VG_{D^* D^*} t,
\label{Eq:BetheS}
\end{equation}

and $ G_{D^* D^*} $, the $ D^* D^* $ loop function, is given by

\begin{equation}
G_{D^* D^*}(M_{inv})= \int_{\mid q \mid < q_{{max}}} \frac{d^3q}{(2\pi)^3} \dfrac{\omega_{1}(q)+\omega_{2}(q)}{2\omega_{1}(q) \omega_{2}(q)} \dfrac{1}{M^{2}_{inv}-(\omega_{1}(q)+\omega_{2}(q))^{2}+i~\epsilon}.
\label{Eq:G0loopf}
\end{equation}

For the value of $ q_{max} $  we use $  q_{max}=420 $ MeV, which was used in Ref. [91]. This value is the one needed to get the $ T_{cc} $ state in Ref. [62]. With this representation of the wave function, $ F(\vec{q}) $ vanishes for $ |{\vec{q}}| > 2~q_{{max}} $.

Following Refs. [118, 119] we take $ q^{0} $ in the rest frame of the cluster, which in the present case is

\begin{equation}
q^0=\frac{s-m_{\bar{K}^{*}}^2-m_{C}^2}{2 m_{C}} \nonumber
\end{equation}

with $ s $ the square of the total rest energy of the $ (D^* D^* )_{C} \bar{K}^{*} $ system. We also need the argument of the  $ t_1 $ matrix of the  $ D^*  \bar{K}^{*}   $ subsystem. This is evaluated taking

\begin{eqnarray}
s^{'}&=&(p_{\bar{K}^{*}}+\dfrac{1}{2} P_{C})^{2} = m_{\bar{K}^{*}}^{2} + \dfrac{1}{4} m_{C}^{2} + q^0 m_{C}\nonumber \\
&=&\dfrac{1}{2} \left( s-m_{\bar{K}^{*}}^{2} - m_{C}^{2} \right) +\dfrac{1}{4} m_{C}^{2} + m_{\bar{K}^{*}}^{2}
\simeq  \dfrac{1}{2} \left( s-m_{\bar{K}^{*}}^{2} - m_{C}^{2} \right) + m_{D^*}^{2} + m_{\bar{K}^{*}}^{2},\nonumber
\end{eqnarray}

the last result corresponding to the formula used in Refs. [118, 119].

We  should note that, since $ t_1 = t_2 $, then  $ T_1 = T_2 $ and, thus, Eq. (<ref> ) leads to

\begin{equation}
\tilde{T_1} =\tilde{t_1} + \tilde{t_1} \tilde{G_0}  \tilde{T_1};~~~~   \tilde{T_1}=\dfrac{1}{\tilde{t_1}^{-1}-\tilde{G_0}  }; ~~ \tilde{T}= \tilde{T_1}+ \tilde{T_2}=2 \tilde{T_1}
\label{eq:tildeTtotal}
\end{equation}

and we shall plot $ \mid \tilde{T} \mid^{2} $ for real values of $s$, looking for the peaks, from where we deduce the mass and width of the states that we find.

§.§  ELEMENTARY AMPLITUDES

We need the $ D^* \bar{K}^{*} $ amplitudes for the different $ I,~j $ states, as shown in Eq. (<ref>). They are obtained in  [23] and  [43]. The $t$ matrices are obtained by means of the Bethe Salpeter equations, and the potentials $V^{I,~j}$ for all the cases that we have in Eq. (<ref>) are given in Tables XI, XII of Ref. [23].
To these potentials we add an imaginary part to account for the decay into $ D \bar{K} $ for $ I=0, ~ j( D^*  \bar{K}^{*} )=0,~2 $ and into   $   D^*  \bar{K}$ for $ I=0, ~ j( D^*  \bar{K}^{*} )=1,~2 $. For the decay into $ D^*  \bar{K} $ we find [43]

\begin{eqnarray}
i ~Im~V^{(D^{*} \bar{K} )}= -i~A_{j}~\dfrac{1}{8 \pi} (G^{'} g ~m_{D^*} )^{2} q^{5} \left( \dfrac{1}{(\omega_{K}(q)-p^{0}_{2})^{2}-\vec{q}^{2}-m_{\pi}^{2}} \right)^{2} \dfrac{1}{\sqrt{s}} F'^{~4}(q)
\label{eq:boxAnamolous}
\end{eqnarray}

where we have used  $q=\frac{\lambda^{1/2}(s',m^2_{D^*},m^2_{\bar{K}})}{2\sqrt{s'}}$, $p^{0}_{2}=\dfrac{s'+m_{\bar{K}^{*}}^{2}- m_{D^*}^{2}}{2\sqrt{s'}}$, $ F'(q)=e^{((w_K-p_2^0)^2-\vec{q}\,^2)/\Lambda^2}\ $ , $G'=\frac{3g'^{2}}{4\pi^2f};g'=-\frac{G_Vm_\rho}{\sqrt{2}f^2}$, $G_V\simeq 55$ MeV, $ f=93 $ MeV. As in [43] we take $ A_{j}=3/2 $ for $ j=1 $ and  $ A_{j}=9/10 $ for $ j=2 $.

The coupling $ G^{'} $ in Eq. (<ref>) is the one appearing in the anomalous Lagrangian for the vector-vector-pseudoscalar vertex

\begin{eqnarray}
{\cal{L}}_{VVP} &=& \dfrac{ G^{'}}{\sqrt{2}} \epsilon^{\mu\nu\alpha\beta} \,\langle \partial_\mu V_\nu \partial_\alpha V_\beta  P \rangle
\label{eq:Lagrangian_VVP}
\end{eqnarray}

where $V$, $ P $ are the $ q \bar{q} $ matrices written in terms of vectors and pseudoscalar respectively (see Appendix A). The expressions for $ G^{'} $ used above come from Ref. [122], and $ G_{V} $, $ f $ are couplings appearing in chiral Lagrangians. The function $ F'(q) $ is a form factor used in the box diagrams from where $Im V$ is obtained, which we take from Ref. [123].

The decay into  $ D \bar{K} $ is studied in [23] and a long formula is obtained for the box diagrams containing  $ D \bar{K} $ intermediate state.  The imaginary part alone is easier to evaluate and we sketch its derivation in Appendix A. We consider the imaginary part only for the $I=0$, $D^* \bar{K}^{*} $ states. The reason is that only the $I=0$ states of $ D^*  \bar{K}^{*}  $ are bound. The $  D^*  \bar{K}^{*} $ interaction in $I=1$ is repulsive. The amplitudes of $I=1$ are then small compared to those of $I=0$ which develop poles in the bound region, and then, neglecting a small imaginary part of the $I=1$ amplitudes has a negligible effect in the final results concerning the three body bound states.

The result for $I=0$ of the imaginary part for decay into $ D \bar{K} $ is given by

\begin{eqnarray}
i ~Im~V^{(D \bar{K} )}= -i~F_{j}~g^{4}~\dfrac{9}{15}~ \dfrac{1}{2 \pi} ~\dfrac{1}{\sqrt{s'}}~ q^{5} \left( \dfrac{1}{(\omega_{K}(q)-p^{0}_{2})^{2}-\vec{q}^{2}-m_{\pi}^{2}} \right)^{2}  F'^{~4}(q) F_{HQ}
\label{eq:box}
\end{eqnarray}

where $q=\frac{\lambda^{1/2}(s',m^2_{D},m^2_{\bar{K}})}{2\sqrt{s'}}$, $F_{HQ} = \left(  \dfrac{m_{D^*}}{m_{\bar{K}^{*}}} \right)^{2}$  and  $ F_{j}=5 $ for $ j=0 $ and  $ F_{j}=2$ for $ j=2 $.

Then the $ t_{1} $ matrix is obtained as

\begin{equation}
t_{1} =\dfrac{1}{V^{-1}-G_{D^*  \bar{K}^{*} }  },
\label{eq:t1tilde}
\end{equation}

and to get the same results as in [43], $ G_{D^*  \bar{K}^{*} } $ is regularized with dimensional regularization with the same input as in [43]. In the $ G_{D^*  \bar{K}^{*} } $  function we also perform a convolution to account for the width of the $ \bar{K}^{*} $ as done in [23, 43].

§  RESULTS

(Color in online) Modulus squared of the $ D^*  D^* \bar{K}^{*}$ scattering amplitude with total spin $J=0$. The dashed curve ignores the $\bar{K}^{*}$ width and the solid one accounts for it via a convolution of the $ G_{D^*  \bar{K}^{*} } $  function. The dotted vertical line indicates the $ D^*  D^* \bar{K}^{*}$ threshold.

In Fig. <ref> we show the results for $ \mid \tilde{T} \mid^{2} $ as a function of the energy of the   $ D^*  D^* \bar{K}^{*}$ system for $J=0$. We find a clear peak around $4845$ MeV, about $61$ MeV below the  $ D^*  D^* \bar{K}^{*}$ threshold.
To understand this binding we can recall that the  $ D^*  D^* $ state is bound by about $ 4-6 $ MeV, while the $D^* \bar{K}^{*}$ state, corresponding to the  $ X_{0} (2866) $ ($ X_{0} (2900) $ officially), is bound by about $30$ MeV.
The interaction of $\bar{K}^{*}$  with two  $ D^* $ would give rise to a binding about twice as big as the one of $ D^* \bar{K}^{*}$, and this accounts for the $60$ MeV binding of the $ D^*  D^* \bar{K}^{*}$ state. The width considering the convolution for the $\bar{K}^{*}$ width is about $80$ MeV,  while without the convolution it is about $77$ MeV. We see that the consideration of the $ \bar{K}^* $ width by means of a convolution of the $ G_{D^* \bar{K}^{*}} $ function reduces the strength of $ \mid \tilde{T} \mid^{2}$ and increases a bit the width, but barely changes the mass. As we have seen in Eq. (<ref>), in this case we have only $ j_{ D^* \bar{K}^{*}} =1 $ and the width comes from the imaginary part of $ t^{I=0,~j=1} $, which has its source in the decay to $ D^* \bar{K} $ (leaving apart the $ \bar{K}^* $ width effect).

In Fig. <ref> we show the same picture for total spin $J=1$. Interestingly, we see now two peaks, indicating two states. In this case it is easy to trace the origin of the peaks. As we can see in Eq. (<ref>), for $J=1$ we have now contributions from $j=0,1,2$. We can see in Table 5 of [43] that the $ 1^+ $ and  $ 0^+ $ states have about the same mass, $2861$ MeV and $2866$ MeV respectively, but the $2^+$ state is more bound, with a mass of $2775$ MeV. It is then clear that the first peak (higher energy), that we call state I, is due to $ t^{I=0,~j=0,1} $ while the second peak,  that we call state II, is due to $ t^{I=0,~j=2} $. The effect of the convolution due to the $K^{*}$ width is small, as in the case of $J=0$.

(Color in online) Modulus squared of the $ D^*  D^* \bar{K}^{*}$ scattering amplitude with total spin $J=1$. Same meaning of the lines as Fig. <ref>.

In Fig. <ref> we see a similar pattern for the case of total $J=2$. From Eq. (<ref>) one can also see that now one has $j=1,2$ contribution. The first peak (showing as a shoulder in the figure)  comes from the $ t^{I=0,~j=1} $ amplitude while the second peak is due to $ t^{I=0,~j=2} $ .

(Color in online) Modulus squared of the  $ D^*  D^* \bar{K}^{*}$ scattering amplitude with total spin $J=2$. Same meaning of the lines as Fig. <ref>.

Thus, in total we find 5 states which we summarize in Table <ref>. There we also show the main decay channel expected, based on the source of imaginary part of the $t_{1}$ amplitudes involved in the peak. The binding energies range from $56$ MeV to $158$ MeV and the widths
range from $80$ MeV to $100$ MeV.

Calculated  Mass ($M$), binding ($B$), width ($\Gamma$) (including the effects of the $ \bar{K}^{*}$ width) of the molecular  $ D^*  D^* \bar{K}^{*}$ states obtained. The binding energy $B$ is measured with respect to the threshold energy $ 2 m_{D^*}+ m_{K^*} $. We also show the main decay mode expected for each one of states.

$J$      $M[\mathrm{MeV}]$    $B[\mathrm{MeV}]$     $\Gamma[\mathrm{MeV}]$    Main decay mode

$0$    (State I )    $4845$     $61$    $80$     $D^* D^* \bar{K}$

$1$    (State I )    $4850$      $56$    $94$     $D^* D \bar{K}$, $D^* D^*\bar{K}$

$1$    (State II)    $4754$     $152$     $100$   $D^* D \bar{K}$, $D^* D^*\bar{K}$

$2$    (State I )    $4840$     $66$     $85$     $D^* D^*\bar{K}$

$2$    (State II )    $4755$     $151$     $100$     $D^* D \bar{K}$, $D^* D^*\bar{K}$

It is interesting to note that the strength of the peak of the two states for $J=1$ are similar, while for $J=2$ the strength of the second state is bigger than that of the first one. We can see the reason in Eq. (<ref>). As we mentioned, the reason that the state II is more bound than the state I, is due to the contribution of the $I=0,~~j=2$ amplitude of $  D^* \bar{K}^{*}$. This reflects the fact that the
$  D^* \bar{K}^{*}$  state of $I=0,~~j=2$ is more bound than those of $I=0,~~j=0,~ 1$. In Eq. (<ref>) we see that the weight of $t^{I=0,~~j=2}$ for $J=2$ is $3/16$ while for $J=1$ it is $5/48$, $1.8$ times smaller. On the other hand the weight of $t^{I=0,~~j=1}$ is the same for $J=1$ and $J=2$, and in the case of $J=1$ there is also contribution of $t^{I=0,~~j=0}$ which is absent in $J=2$.

We address here a different point. By looking at Table <ref> we can see that the states I for $J=0,1,2$, have similar energies, and so is the case for the two states II with  $J=1$ or $J=2$. It would be good to determine the spin experimentally, a task always difficult but addressed successfully in recent experimental analyses. Partial wave analysis in meson or photon nucleon experiments is performed by different groups and spin and parity of resonances is determined [124, 125, 126]. Concerning states in the charm sector, LHCb, BelleII and BESIII use also different methods of partial wave analysis to determine spin and parity of the states [127, 128, 129]. One method that also proved efficient for determining the spin of particles is the use of the moments of angular distributions, where cross sections (not the unknown amplitudes) are projected with spherical harmonics and useful relationships are obtained that help in the determination of the spin [130, 131, 90, 132, 133].

Another issue we would like to address here is the shape of the wave function that we obtained. We do that for the $J=0$ state. As we have discussed along the paper, we have a $ \bar{K}^{*}$ orbiting around the cluster of $D^* D^*$. The  $ \bar{K}^{*}$ will orbit around one $D^*$ and sometimes around the other $D^*$. In this picture we can have the $ \bar{K}^{*}$ distribution given by

\begin{equation}
\mid \Psi (r^{\prime}_{3})\mid^{2} =\int d^{3}r_{1} d^{3}r_{2} \left( \mid \phi (\vec{r}_{31})\mid^{2} +  \mid \phi (\vec{r}_{32})\mid^{2}  \right)  \mid  \phi^{\prime} (\vec{r}_{12})\mid^{2} \delta^{3} ( m_{D^*} \vec{r}_{1} + m_{D^*} \vec{r}_{2} + m_{\bar{K}^{*}} \vec{r}_{3})
\label{eq:distribution}
\end{equation}

where $ \vec{r}_{1} $,  $ \vec{r}_{2} $,  $ \vec{r}_{3} $ are the coordinates of the two  $D^*$ and $\bar{K}^{*}$ respectively and we refer $ \vec{r}^{~\prime}_{3} $ for the  $\bar{K}^{*}$ with respect to the center of mass of the three body system. The wave functions $ \phi $ stand for the $  D^* \bar{K}^{*}$  systems and $ \phi' $ for the  $D^* D^*$ one, and the arguments are given by  $ \vec{r}_{31}= \vec{r}_{3} - \vec{r}_{1}$,  $ \vec{r}_{32}= \vec{r}_{3} - \vec{r}_{2}$,  $ \vec{r}_{12}= \vec{r}_{1} - \vec{r}_{2}$. The wave functions are taken as Eq. (<ref>) in momentum space and by means of a Fourier transformation the wave functions  in coordinate space are obtained (See Appendix B).

(Color in online) The distribution of  $ r'^{2}_{3} \mid \Psi (r'_{3})\mid^{2} $ for the $ \bar{K}^{*}$  in the $ D^*  D^* \bar{K}^{*}$ system at rest. Arbitrary units in y-axis (we take $g_R=1$ in Eq. (<ref>)).

In Figure <ref> we show the distribution of the $ \bar{K}^{*}$  wave function by means of $ r'^{2}_{3} \mid \Psi (r'_{3})\mid^{2} $ with $ r'_{3}$
the $ \bar{K}^{*}$ coordinate with respect to the center of mass of the three body system. The  $ r'_{3}$ distribution is plotted without normalization to show the spatial distribution.

As we see, the $ r'_{3}$  distribution peaks around $0.7$ fm. One can evaluate the mean square radius from there and one obtains $ \langle r^{2} \rangle \simeq 1  $ fm  which is bigger than the mean square radius of the proton, $0.84$ fm, and smaller than the one of the deuteron, $2.1$ fm, which is lightly bound and governed by the range of the one pion exchange, while here the range of the interaction is shorter (vector exchange) and the molecule is much more bound.

§ CONCLUSIONS

We have conducted a search for possible bound states of the three body system $ D^*  D^* \bar{K}^{*}$. For this we have relied upon the FCA to the Faddeev equations which has proved reliable in the study of related systems compared to the other methods like the variational one using the Gaussian expansion method. For the study of this system we have chosen as the cluster the  $ D^*  D^* $, which appears bound in $I=0$ and $J^{P}=1^{+}$, and the $\bar{K}^{*}$  collides with it repeatedly. The FCA method has the virtue of allowing one to trace the structures obtained to the scattering amplitudes of $  D^* \bar{K}^{*}$. This latter system has bound states in $j=0,1,2$, with the $j=0,1$ states very close in energy and the $j=2$ state more bound. In the $ D^*  D^* \bar{K}^{*}$ system we can have total spin  $J=0,1,2$, and in each of these spins the $D^* \bar{K}^{*}$ amplitude appears with different combinations of isospin and spin. As a consequence, we obtain two peaks in the three-body partition functions which are tied to the $j=0,1$ and $j=2$  of $ D^* \bar{K}^{*}$, with the exception of $J=0$, where the $  D^* \bar{K}^{*}$system can only be in $j=1$, and here we only get one peak.

In total we obtained $5$ three-body states, one for  $J=0$, two for  $J=1$ and two for  $J=2$, and we evaluate the mass and width of these states. The states obtained, stemming from the cluster of  $ D^*  D^* $ with $I=0$ and $J^{P}=1^{+}$, have total isospin $I=1/2 $, negative parity and spin $J=0,1,2$. The widths range from $80$ MeV to $100$ MeV and the bindings range from $56$ MeV to $151$ MeV. We have also shown which is the main decay channel of each particular state to facilitate its experimental search. We hope the present work stimulates future searches for these super exotic states containing $ccs$ quarks. As mentioned in Ref. [134], since the flavor is conserved in the strong interactions, we can expect new states made of many mesons, relatively stable, which cannot decay to mesons with smaller meson number. This would make them similar to nuclei where the baryon number conservation is responsible for the elements of the periodic table, giving rise to a new periodic table of mesons. In coincidence with the point made in Ref. [134] we think that we are just at the beginning of this new era.

§ ACKNOWLEDGMENTS

The work of N. I. was partly supported by JSPS KAKENHI Grant Number JP19K14709.
This work is partly supported by the Spanish Ministerio de
Economía y Competitividad (MINECO) and European FEDER funds
under Contract No. PID2020-112777GB-I00, and by Generalitat Va-
lenciana under contract PROMETEO/2020/023. This project has re-
ceived funding from the European Union Horizon 2020 research
and innovation programme under the program H2020-INFRAIA-
2018-1, grant agreement No. 824093 of the STRONG-2020 project.

§ APPENDIX A

With the $D^*$, $\bar{K}^{*}$   doublets $(D^{*+}, - D^{*0})$ $(\bar{K}^{*0}, - K^{*-})$, the $I=0$ wave function is given by

\begin{eqnarray}
-\frac{1}{\sqrt{2}} ( D^{*+} K^{*-}- D^{*0} \bar{K}^{*0}) \nonumber
\end{eqnarray}

and the box diagrams that account for $  D \bar{K}  $ decay are given in Fig. <ref>

Box diagrams accounting for the decay of $D^{*}\bar{K}^{*}$ in $I=0$ into $  D \bar{K}  $.

The dynamics comes from the vector $ \rightarrow $ pseudoscalar-pseudoscalar vertex, $VPP$ given by the Lagrangian

\begin{eqnarray}
{\cal L}=-ig \langle [P,\partial_\mu P]V^\mu \rangle, ~~g=\dfrac{M_V}{2f} ~~(M_V=800 ~ \text{MeV}, ~~f=93 ~\text{MeV})\nonumber
\label{eq:lag}
\end{eqnarray}

where $P$, $V$ are $q \bar{q}$ matrices written in terms of pseudoscalar and vector mesons.

\begin{equation}\label{Psematrix}
P = \begin{pmatrix}
\frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{3}} \eta + \frac{1}{\sqrt{6}}\eta' & \pi^+ & K^+ & \bar{D}^0 \\
\pi^- & -\frac{1}{\sqrt{2}}\pi^0 + \frac{1}{\sqrt{3}} \eta + \frac{1}{\sqrt{6}}\eta' & K^0 & D^- \\
K^- & \bar{K}^0 & -\frac{1}{\sqrt{3}} \eta + \sqrt{\frac{2}{3}}\eta' & D_s^- \\
D^0  & D^+ & D_s^+ & \eta_c
\end{pmatrix},
\end{equation}

\begin{equation}\label{Vecmatrix}
V = \begin{pmatrix}
\frac{1}{\sqrt{2}}\rho^0 + \frac{1}{\sqrt{2}} \omega & \rho^+ & K^{* +} & \bar{D}^{* 0} \\
\rho^- & -\frac{1}{\sqrt{2}}\rho^0 + \frac{1}{\sqrt{2}} \omega  & K^{* 0} & D^{* -} \\
K^{* -} & \bar{K}^{* 0}  & \phi & D_s^{* -} \\
D^{* 0} & D^{* +} & D_s^{* +} & J/\psi
\end{pmatrix},
\end{equation}

To simplify the calculation we evaluate the diagrams at threshold of $D^* \bar{K}^{*}$. The external three momenta are zero and $\vec{q}$ is the running variable in the loop.  In this limit the $\epsilon^{0}$ component of the external vectors is zero and all the vertices are of the type of $\vec{\epsilon} \cdot \vec{q}$. Since one is concerned about the imaginary part, one realizes that the pions will be off shell because $D^* \rightarrow D \pi$ is possible but then $\bar{K}^* +\pi \rightarrow \bar{K}$ does not go.
The loop function is then given by

\begin{equation}
i\int\frac{d^4q}{(2\pi)^4} \left( \dfrac{1}{{q^0}^2-\vec{q}\,^2-m_{\pi}^2}\right)^{2}   \frac{1}{(p_{1}-q)^2-m_{D}^2+i\epsilon} \frac{1}{(p_{2}+q)^2-m_{K}^2+i\epsilon},\nonumber
\end{equation}

and we can perform the $q^0$ integration analytically using Cauchy's integration. Since we are concerned only about the imaginary part coming from the $  D \bar{K}  $ placed on shell, we consider only the poles of the $D$ and $\bar{K}  $ propagators and obtain

\begin{equation}
\int\frac{d^3q}{(2\pi)^3}  \left( \dfrac{1}{(w_{K}-{p_2}^{0})^2-\vec{q}\,^2-m_{\pi}^2}\right)^{2} \dfrac{1}{2 w_{K}} \dfrac{1}{2 w_{D}}
\dfrac{1}{{p_1}^{0}+{p_2}^{0}- w_{K} - w_{D} +i\epsilon}. \nonumber
\end{equation}

We have integrals of type

\begin{eqnarray}
\epsilon^{l} \epsilon^{m} \epsilon^{r} \epsilon^{s} \int\frac{d^3q}{(2\pi)^3}  f(\vec{q}\,^2)q^l q^m q^{r}q^{s} = \frac{1}{15}\int \frac{d^3q}{(2\pi)^3}f(\vec{q}\,^2)\vec{q}\,^4
(\delta_{lm}\delta_{rs}+\delta_{lr}\delta_{ms}+\delta_{ls}\delta_{mr})   \epsilon^{l} \epsilon^{m} \epsilon^{r} \epsilon^{s}, \nonumber
\end{eqnarray}

and considering the $VV$ spin projectors in $ j=0, 1, 2 $ [135]

\begin{eqnarray}
&&{\cal P}^{(0)}= \frac{1}{3}\eps^l \eps^l \eps^r \eps^r \nonumber\\
&&{\cal P}^{(1)}=\frac{1}{2}(\eps^l\eps^m \eps^l \eps^m-\eps^l\eps^m \eps^m \eps^l)\nonumber\\
&&{\cal P}^{(2)}= \frac{1}{2}(\eps^l\eps^m \eps^l \eps^m + \eps^l\eps^m \eps^m \eps^l )-\frac{1}{3} \eps^l \eps^l \epsilon^m \epsilon^m \nonumber
\end{eqnarray}

and the appropriate factors in the vertices, we obtain from the sum of all the diagrams

\begin{eqnarray}
i ~Im~V^{(D \bar{K} )}= -i~F_{j}~g^{4}~\dfrac{9}{15}~ \dfrac{1}{2 \pi} ~\dfrac{1}{\sqrt{s'}}~ q^{5} \left( \dfrac{1}{(\omega_{K}(q)-p^{0}_{2})^{2}-\vec{q}^{2}-m_{\pi}^{2}} \right)^{2}  F'^{~4}(q) F_{HQ},
\label{eq:box}
\end{eqnarray}

where we have also included a factor demanded by heavy quark symmetry $F_{HQ}$ to get the coupling $VPP$ in the case of heavy vectors [136] and used $q=\frac{\lambda^{1/2}(s',m^2_{D},m^2_{\bar{K}})}{2\sqrt{s'}}$, $F_{HQ} = \left(  \dfrac{m_{D^*}}{m_{\bar{K}^{*}}} \right)^{2}$, $p^{0}_{2}=\dfrac{s'+m_{\bar{K}^{*}}^{2}- m_{D^*}^{2}}{2\sqrt{s'}}$, $ F'(q)=e^{((w_K-p_2^0)^2-\vec{q}\,^2)/\Lambda^2}\ $.

Further details can be seen in the evaluation of the width of the analogous $ \bar{B}^{*} \bar{K}^{*} $ state in section D of Ref. [137].

§ APPENDIX B: WAVE FUNCTION OF THE  $ D^*  D^* \BAR{K}^{*}$ MOLECULE

We have the situation as in Fig. <ref>

Description of the coordinates of the $ D^*  D^* \bar{K}^{*}$ system.

The wave functions for the $\bar{K}^{*}$ relative to the center of mass of the $ D^*  D^* \bar{K}^{*}$ system is given by Eq. (<ref>). We need the wave functions $\phi (\vec{r}_{31})$,  $\phi (\vec{r}_{32})$, $ \phi^{\prime} (\vec{r}_{12}) $. The wave functions are given in momentum space by Eq. (<ref>) $ \langle \vec{q} \mid \phi \rangle  \equiv \phi (\vec{q})$. To obtain the wave function in coordinate space we write:

\begin{equation}
\phi (\vec{r}) \equiv \langle \vec{r} \mid \phi \rangle = \int \dfrac{d^{3}q}{(2 \pi )^{3/2}} e^{i \vec{q}\cdot \vec{r}}  \phi (\vec{q})
\label{eq:phi}
\end{equation}

\begin{equation}
e^{i \vec{q}\cdot \vec{r}} = 4 \pi \sum_{\ell} i^{\ell} j_{\ell}(qr) \sum_{m} Y^{*}_{\ell m} ( \hat{q}) Y_{\ell m} ( \hat{r})
\label{eq:eps}
\end{equation}

Performing the $ d\Omega(\hat{r}) $ integration we obtain

\begin{eqnarray}
\phi (\vec{r}) & =& \int \dfrac{{q^2}dq}{(2 \pi )^{3/2}} 4 \pi  \sqrt{4 \pi } j_{0}(qr) Y_{00} ( \hat{x}) \phi (\vec{q})\\ \nonumber
&=& \dfrac{2 \pi}{(2 \pi )^{3/2}} g_R ~ \frac{2}{r} \int^{q_{max}}_{0} q~dq \dfrac{sin(qr)}{E-\omega_{1}(q)-\omega_{2}(q)}
\label{eq:phir}
\end{eqnarray}

Using the $ \delta^{3} ( ~) $ function in Eq. (<ref>) we can eliminate $\vec{r}_{2}$ and have only the integral over $\vec{r}_{1}$, which requires two integrals, one over the modulus of $\vec{r}_{1}$ and the other one over the angle between $\vec{r}_{1}$ and $\vec{r}_{3}$. Altogether we need three integrals to evaluate $ \mid \Psi (r^{\prime}_{3})\mid^{2} $ of Eq. (<ref>).

As mentioned before, to obtain the binding of the $ T_{cc} $, the value $ q_{max}= 420$ MeV was used, and the same value is used to construct the wave function of the $ D^*  D^*$ cluster. On the other hand in Ref. [43] the loops were regularized with dimensional regularization. Since here we need the equivalent $ q_{max} $, we have seen which value is needed to obtain the same binding of the $D^* \bar{K}^{*}$ molecule, which is found for $ q_{max}= 1050$ MeV (see also Ref. [82]) that we use here to construct the $  D^* \bar{K}^*  $ wave function.

[1]

A. Martinez Torres, K. P. Khemchandani, L. Roca and E. Oset,

Few Body Syst. 61, 35 (2020).

[3]

J. Gasser and H. Leutwyler,

Annals Phys. 158, 142 (1984).

[5]

J. A. Oller and E. Oset,

Nucl. Phys. A 620, 438-456 (1997)
[erratum: Nucl. Phys. A 652, 407-409 (1999)].

[7]

J. A. Oller, E. Oset and J. R. Pelaez,

Phys. Rev. D 59, 074001 (1999)
[erratum: Phys. Rev. D 60, 099906 (1999); erratum: Phys. Rev. D 75, 099903 (2007)].

[9]

N. Kaiser,

Eur. Phys. J. A 3, 307-309 (1998).

[11]

M. P. Locher, V. E. Markushin and H. Q. Zheng,

Eur. Phys. J. C 4, 317-326 (1998).

[13]

J. Nieves and E. Ruiz Arriola,

Nucl. Phys. A 679, 57-117 (2000).

[15]

F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou,

Rev. Mod. Phys. 90, 015004 (2018)
[erratum: Rev. Mod. Phys. 94, 029901 (2022)].

[17]

X. K. Dong, F. K. Guo and B. S. Zou,

Commun. Theor. Phys. 73, 125201 (2021).

[19]

X. K. Dong, F. K. Guo and B. S. Zou,

Few Body Syst. 62, 61 (2021).

[21]

X. K. Dong, F. K. Guo and B. S. Zou,

Progr. Phys. 41, 65-93 (2021).

[23]

R. Molina, T. Branz and E. Oset,

Phys. Rev. D 82, 014010 (2010).

[25]

R. Aaij et al. [LHCb],

Phys. Rev. Lett. 125, 242001 (2020).

[27]

R. Aaij et al. [LHCb],

Phys. Rev. D 102, 112003 (2020).

[29]

R. Aaij et al. [LHCb],

Nature Phys. 18, 751-754 (2022).

[31]

R. Aaij et al. [LHCb],

Nature Commun. 13, 3351 (2022).

[33]

M. Z. Liu, J. J. Xie and L. S. Geng,

Phys. Rev. D 102, 091502 (2020).

[35]

X. G. He, W. Wang and R. Zhu,

Eur. Phys. J. C 80, 1026 (2020).

[37]

J. He and D. Y. Chen,

Chin. Phys. C 45, 063102 (2021).

[39]

Y. Huang, J. X. Lu, J. J. Xie and L. S. Geng,

Eur. Phys. J. C 80, 973 (2020).

[41]

M. W. Hu, X. Y. Lao, P. Ling and Q. Wang,

Chin. Phys. C 45, 021003 (2021).

[43]

R. Molina and E. Oset,

Phys. Lett. B 811, 135870 (2020).

[45]

S. S. Agaev, K. Azizi and H. Sundu,

J. Phys. G 48, 085012 (2021).

[47]

C. J. Xiao, D. Y. Chen, Y. B. Dong and G. W. Meng,

Phys. Rev. D 103, 034004 (2021).

[49]

B. Wang and S. L. Zhu,

Eur. Phys. J. C 82, 419 (2022).

[51]

H. X. Chen, W. Chen, R. R. Dong and N. Su,

Chin. Phys. Lett. 37, 101201 (2020).

[53]
Q. Xin, Z. G. Wang and X. S. Yang,

[arXiv:2207.09910 [hep-ph]].

[54]

M. Karliner and J. L. Rosner,

Phys. Rev. D 102, 094016 (2020).

[56]

G. J. Wang, L. Meng, L. Y. Xiao, M. Oka and S. L. Zhu,

Eur. Phys. J. C 81, 188 (2021).

[58]

Q. F. Lü, D. Y. Chen and Y. B. Dong,

Phys. Rev. D 102, no.7, 074021 (2020).

[60]

X. Z. Ling, M. Z. Liu, L. S. Geng, E. Wang and J. J. Xie,

Phys. Lett. B 826, 136897 (2022).

[62]

A. Feijoo, W. H. Liang and E. Oset,

Phys. Rev. D 104, 114015 (2021).

[64]

S. Fleming, R. Hodges and T. Mehen,

Phys. Rev. D 104, 116010 (2021).

[66]

H. Ren, F. Wu and R. Zhu,

Adv. High Energy Phys. 2022, 9103031 (2022).

[68]

K. Chen, R. Chen, L. Meng, B. Wang and S. L. Zhu,

Eur. Phys. J. C 82, 581 (2022).

[70]

M. Albaladejo,

Phys. Lett. B 829, 137052 (2022).

[72]

M. L. Du, V. Baru, X. K. Dong, A. Filin, F. K. Guo, C. Hanhart, A. Nefediev, J. Nieves and Q. Wang,

Phys. Rev. D 105, 014024 (2022).

[74]

N. Santowsky and C. S. Fischer,

Eur. Phys. J. C 82, 313 (2022).

[76]

C. Deng and S. L. Zhu,

Phys. Rev. D 105, 054015 (2022).

[78]

H. W. Ke, X. H. Liu and X. Q. Li,

Eur. Phys. J. C 82, 144 (2022).

[80]

S. S. Agaev, K. Azizi and H. Sundu,

JHEP 06, 057 (2022).

[82]

Y. Kamiya, T. Hyodo and A. Ohnishi,

Eur. Phys. J. A 58, 131 (2022).

[84]

L. Meng, B. Wang, G. J. Wang and S. L. Zhu,

arXiv:2204.08716 [hep-ph].

[86]

L. M. Abreu,

arXiv:2206.01166 [hep-ph].

[88]

S. Chen, C. Shi, Y. Chen, M. Gong, Z. Liu, W. Sun and R. Zhang,

arXiv:2206.06185 [hep-lat].

[90]
M. L. Du, M. Albaladejo, P. Fernández-Soler, F. K. Guo, C. Hanhart, U. G. Meißner, J. Nieves and D. L. Yao,

Phys. Rev. D 98, no.9, 094018 (2018).

[91]

L. R. Dai, R. Molina and E. Oset,

Phys. Rev. D 105, 016029 (2022).

[93]

M. Bando, T. Kugo and K. Yamawaki,

Phys. Rept. 164, 217-314 (1988).

[95]

M. Harada and K. Yamawaki,

Phys. Rept. 381, 1-233 (2003).

[97]

U. G. Meissner,

Phys. Rept. 161, 213 (1988).

[99]

H. Nagahiro, L. Roca, A. Hosaka and E. Oset,

Phys. Rev. D 79, 014015 (2009).

[101]

S. Q. Luo, T. W. Wu, M. Z. Liu, L. S. Geng and X. Liu,

Phys. Rev. D 105, 074033 (2022).

[103]

T. W. Wu, M. Z. Liu, L. S. Geng, E. Hiyama and M. P. Valderrama,

Phys. Rev. D 100, 034029 (2019).

[105]
A. Martinez Torres, K. P. Khemchandani and L. S. Geng,

Phys. Rev. D 99, no.7, 076017 (2019).

[106]
Y. Huang, M. Z. Liu, Y. W. Pan, L. S. Geng, A. Martínez Torres and K. P. Khemchandani,

Phys. Rev. D 101, no.1, 014022 (2020).

[107]

X. L. Ren, B. B. Malabarba, L. S. Geng, K. P. Khemchandani and A. Martínez Torres,

Phys. Lett. B 785, 112-117 (2018).

[109]

L. Ma, Q. Wang and U. G. Meißner,

Chin. Phys. C 43, 014102 (2019).

[111]
U. Ozdem,

arXiv:2207.14549 [hep-ph]

[112]

T. W. Wu, M. Z. Liu and L. S. Geng,

Phys. Rev. D 103, L031501 (2021).

[114]

X. Wei, Q. H. Shen and J. J. Xie,

arXiv:2205.12526 [hep-ph].

[116]
M. Z. Liu, T. W. Wu, M. Pavon Valderrama, J. J. Xie and L. S. Geng,

Phys. Rev. D 99, 094018 (2019).

[117]
S. Q. Luo, T. W. Wu, M. Z. Liu, L. S. Geng and X. Liu,

Phys. Rev. D 105, 074033 (2022).

[118]
L. Roca and E. Oset,

Phys. Rev. D 82, 054013 (2010).

[119]
J. Yamagata-Sekihara, L. Roca and E. Oset,

Phys. Rev. D 82, 094017 (2010)
[erratum: Phys. Rev. D 85, 119905 (2012)].

[120]
C. W. Xiao, M. Bayar and E. Oset,

Phys. Rev. D 86, 094019 (2012).

[121]
D. Gamermann, J. Nieves, E. Oset and E. Ruiz Arriola,

Phys. Rev. D 81, 014029 (2010).

[122]
A. Bramon, A. Grau and G. Pancheri,

Phys. Lett. B 283, 416-420 (1992).

[123]
F. S. Navarra, M. Nielsen and M. E. Bracco,

Phys. Rev. D 65, 037502 (2002).

[124]
A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma,

Eur. Phys. J. A 48, 15 (2012).

[125]
D. Ronchen, M. Doring, F. Huang, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, U. G. Meissner and K. Nakayama,

Eur. Phys. J. A 49, 44 (2013).

[126]
A. Švarc, M. Hadzimehmedovic, H. Osmanovic, J. Stahov, L. Tiator and R. L. Workman,

Phys. Rev. C 88, no.3, 035206 (2013).

[127]
R. Aaij et al. [LHCb],

Phys. Rev. D 92, no.1, 011102 (2015).

[128]
T. J. Moon et al. [Belle],

Phys. Rev. D 103, no.11, L111101 (2021).

[129]
M. Ablikim et al. [BESIII],

Phys. Rev. D 103, no.9, L091101 (2021).

[130]
R. Aaij et al. [LHCb],

Phys. Rev. D 94, no.7, 072001 (2016).

[131]
R. Aaij et al. [LHCb],

Phys. Rev. D 90, no.7, 072003 (2014).

[132]
V. Mathieu et al. [JPAC],

Phys. Rev. D 100, no.5, 054017 (2019).

[133]
M. Bayar and E. Oset,

Phys. Lett. B 833, 137364 (2022).

[134]
T. W. Wu, Y. W. Pan, M. Z. Liu and L. S. Geng,

arXiv:2208.00882 [hep-ph].

[135]
R. Molina, D. Nicmorus and E. Oset,

Phys. Rev. D 78, 114018 (2008).

[136]
W. H. Liang, C. W. Xiao and E. Oset,

Phys. Rev. D 89, 054023 (2014).

[137]
E. Oset and L. Roca,

Eur. Phys. J. C 82, no.10, 882 (2022).