#  $DD^{\ast}$ chiral interactions with the local momentum-space
regularization up to the third order and the molecular nature of
${T_{cc}^{+}}$

Bo Wang  0000-0003-0985-2958<EMAIL_ADDRESS>School of Physical Science and
Technology, Hebei University, Baoding 071002, China  Key Laboratory of High-
precision Computation and Application of Quantum Field Theory of Hebei
Province, Baoding 071002, China

###### Abstract

We revisit the $DD^{\ast}$ interactions in chiral effective field theory up to
the third order for the first time. We deal with the pion-exchanged
interactions via local momentum-space regularization, in which we focus on
their long-range behavior through demanding their contributions vanish at the
origin in the coordinate space. The short-range contact interactions and
subleading pion-charmed meson couplings are estimated with the
phenomenological resonance saturation model. The subleading pion-charmed meson
couplings are much weaker than those in the pion-nucleon system, thus the
$DD^{\ast}$ binding mechanism is very different with that of the $NN$ system.
The binding solution only exists in the isoscalar channel. Our calculation
supports the molecular interpretation of ${T_{cc}^{+}}$.

##  I Introduction

The interactions between a pair of heavy-light hadrons can be fairly regarded
as the extension of the pattern of nuclear forces. The theoretical tools that
designed for the nucleon systems shall also be generalized to the heavy-light
systems via including the restriction of extra symmetries, such as the heavy
quark symmetry. Meanwhile, in recent years, the observations of many near-
threshold exotic states provide golden platforms to test and redevelop these
tools Chen  _et al._ (2016); Guo  _et al._ (2018); Liu  _et al._ (2019); Lebed
_et al._ (2017); Esposito  _et al._ (2017); Brambilla  _et al._ (2020); Chen
_et al._ (2021a, 2022a); Meng  _et al._ (2022), in which the successful
generalizations of the effective field theories (EFTs), e.g., the pionless and
pionful EFTs, is a epitome of the intimate connection between the nuclear
physics and the hadron physics Meng  _et al._ (2022).

Based on the instructive works of Weinberg Weinberg (1990, 1991), in the past
decades, the modern framework of nuclear forces was constructed upon the
chiral effective field theory ($\chi$EFT) Epelbaum  _et al._ (2009); Machleidt
and Entem (2011). In $\chi$EFT, the short-range part of the nuclear forces is
parameterized as the four-fermion contact interactions through integrating out
the heavy particle exchanging (e.g., the vector meson $\rho$ and $\omega$,
etc.), while the long- and intermediate-range parts are presented by the one-
pion exchange (OPE) and multi-pion exchange interactions, respectively Ordonez
_et al._ (1996); Kaiser  _et al._ (1997); Epelbaum  _et al._ (2000). The
latter can be derived from the chiral symmetry of QCD via a model-independent
way. The study of nucleon-nucleon ($NN$) interactions indicates that the
leading order (LO) two-pion exchange (TPE) potential is very weak and
insufficient to provide the appropriate attractive force at the intermediate
range, and which is in fact described by the subleading TPE potential with an
insertion of the subleading pion-nucleon vertices Epelbaum (2006); Epelbaum
_et al._ (2009); Machleidt and Entem (2011). It was found that the large
values of the low energy constants (LECs) in the subleading pion-nucleon
Lagrangians leads to the attractive source. The values of these LECs can be
quantitatively understood using the phenomenological resonance saturation
model (RSM) Bernard  _et al._ (1997). It was shown that these large value LECs
in the $\chi$EFT without explicit $\Delta$ resonance actually stem from the
‘high’ (note that $m_{\Delta}-m_{N}\approx 2m_{\pi}<m_{\rho}$, where
$m_{\rho}\sim 770$ MeV is usually regarded as the truly high energy scale in
chiral perturbation theory) energy scale $\Delta$ baryon as well as the pion-
pion correlation (or the $\sigma$ meson) Bernard  _et al._ (1997).

Epelbaum et al. noticed that the TPE loop diagrams calculated within the
dimensional regularization accompanying with the large value LECs in
subleading pion-nucleon vertices lead to unsatisfactory convergence of chiral
expansion and uncertain consequences in few-nucleon systems, e.g., the
unphysical deeply bound states in the low partial waves of isoscalar channel
Epelbaum  _et al._ (2002a). The expediency is to use the small value LECs, but
this is not compatible with the pion-nucleon scattering data Fettes  _et al._
(1998); Krebs  _et al._ (2007). In order to cure this problem, Epelbaum et al.
argued that one needs to suppress the high-momentum modes of the exchanged
pions, since they cannot be suitably handled in an EFT who only properly works
in the soft scales. They proposed to calculate the TPE loop diagrams using the
cutoff regularization and combining the spectral function representation
rather than the dimensional regularization Epelbaum  _et al._ (2002a); Reinert
_et al._ (2018); Epelbaum  _et al._ (2022). This is analogous to the means for
improving the convergence of chiral expansion in the SU(3) case Donoghue and
Holstein (1998); Donoghue  _et al._ (1999).

Obviously, one needs to consider the possible emergence of the above mentioned
problem when generalizing the $\chi$EFT to the heavy-light systems. The
application of $\chi$EFT in heavy-light systems for dealing with the hadronic
molecules has achieved much progress in recent years Meng  _et al._ (2022). In
Ref. Liu  _et al._ (2014), Liu et al. first calculated the $BB$ interactions
with considering the leading TPE contributions. Along this line, Xu et al.
studied the $DD^{\ast}$ interactions and used the RSM to determine the contact
LECs, in which they predicted a bound state in the isoscalar channel with
$J^{P}$ quantum numbers $1^{+}$ Xu  _et al._ (2019). Four years latter, the
LHCb Collaboration observed a state, the ${T_{cc}^{+}}$ in $D^{0}D^{0}\pi^{+}$
invariant mass spectrum Aaij  _et al._ (2022a, b). The ${T_{cc}^{+}}$ is below
the $D^{\ast+}D^{0}$ threshold about $300$ keV, thus it is the very good
candidate of $DD^{\ast}$ hadronic molecule. Similar to Ref. Xu  _et al._
(2019), Wang et al. studied the $B^{(\ast)}B^{(\ast)}$ interactions and
predicted the possible bound states in the isoscalar $BB^{\ast}$ and
$B^{\ast}B^{\ast}$ systems with $J^{P}=1^{+}$ Wang  _et al._ (2019a). The same
framework was also adopted to investigate the LHCb pentaquarks
$P_{\psi}^{N}(4312)$, $P_{\psi}^{N}(4440)$ and $P_{\psi}^{N}(4457)$ Meng  _et
al._ (2019); Wang  _et al._ (2019b) (throughout this paper, we use the new
naming scheme of the exotic states proposed by the LHCb Gershon (2022)), as
well as to predict the existence of molecular pentaquarks with strangeness in
$\Xi_{c}^{(\prime,\ast)}\bar{D}^{(\ast)}$ systems Wang  _et al._ (2020) (see
also the recent experimental measurements for the $P_{\psi s}^{\Lambda}$
states near the $\Xi_{c}\bar{D}^{\ast}$ Aaij  _et al._ (2021) and
$\Xi_{c}\bar{D}$ LHC (2022) thresholds), and the double-charm pentaquarks Chen
_et al._ (2021b). For a review of this topic, we refer to Ref. Meng  _et al._
(2022). In Ref. Chen  _et al._ (2022b), the study of $\Sigma_{c}\Sigma_{c}$
interactions turns out that there results in bad convergence and unnaturally
deep bound state in the lowest isospin channel if one calculates the leading
TPE diagrams with dimensional regularization. This demands us to properly
treat the TPE contributions for heavy-light systems as those in the $NN$ case.

In this work, we revisit the $DD^{\ast}$ interactions within $\chi$EFT, and
calculate the $DD^{\ast}$ interactions up to the third order [i.e. the next-
to-next-to-leading order (N2LO)] for the first time. We construct the
subleading $\pi D^{(\ast)}$ Lagrangians and determine the corresponding LECs
with the RSM. The TPE diagrams will be calculated with the cutoff
regularization, but we use the fully local momentum-space regularization
rather than the semi-local form as those in Ref. Reinert  _et al._ (2018). The
$DD^{\ast}$ interactions shall strongly correlate to the ${T_{cc}^{+}}$ inner
structures and its other properties. In contrast to the well-known $X(3872)$,
there is no coupling with the charmonia for ${T_{cc}^{+}}$. Thus it provides a
clean environment for investigating the interactions between the charmed
mesons. This is very similar to the $NN$ interactions.

The ${T_{cc}^{+}}$ state has been intensively studied from various aspects,
such as the decay behaviors Meng  _et al._ (2021); Agaev  _et al._ (2022);
Ling  _et al._ (2022); Yan and Valderrama (2022); Feijoo  _et al._ (2021); Ren
_et al._ (2022), the mass spectrum Dong  _et al._ (2021); Chen  _et al._
(2021c); Weng  _et al._ (2022); Xin and Wang (2022); Chen and Yang (2022);
Chen  _et al._ (2022c); Deng and Zhu (2022); Ke  _et al._ (2022); Padmanath
and Prelovsek (2022); Lin  _et al._ (2022); Kim  _et al._ (2022); Cheng  _et
al._ (2022), the productions Huang  _et al._ (2021); Jin  _et al._ (2021); Hu
_et al._ (2021); Abreu  _et al._ (2022); Braaten  _et al._ (2022), the
lineshapes Dai  _et al._ (2022); Fleming  _et al._ (2021); Du  _et al._
(2022), and the magnetic moment Azizi and Özdem (2021), etc.. In order to pin
down the inner configuration of ${T_{cc}^{+}}$, a systematic study of the
$DD^{\ast}$ interactions is very necessary.

This paper is organized as follows. The $DD^{\ast}$ effective potentials
within the local momentum-space regularization are shown in Sec. II. The pole
trajectory of $DD^{\ast}$ bound state and related discussions are given in
Sec. III. A short summary is given in Sec. IV. The estimations of LECs within
the RSM are listed in the Appendix A.

##  II Effective chiral potentials up to the third order

The effective potential of $DD^{\ast}$ can be extracted from their scattering
amplitude. In $\chi$EFT, the scattering amplitude of $DD^{\ast}$ is expanded
in powers of the ratio $\mathcal{Q}/\Lambda_{\rm{b}}$, where $\mathcal{Q}$
represents the soft scale, which could be the pion mass or the external
momenta of $D^{(\ast)}$, while $\Lambda_{\rm{b}}$ denotes the hard scale at
which the $\chi$EFT breaks down. The relative importance of the terms in the
expansion is weighed by the power $\nu$ of
$(\mathcal{Q}/\Lambda_{\rm{b}})^{\nu}$, this is known as the power counting
scheme. According to the naive dimensional analysis Weinberg (1990, 1991), the
power $\nu$ for a system with two matter fields (charmed mesons) is measured
as

$\displaystyle\nu=2L+\sum_{i}V_{i}\Delta_{i},\qquad\Delta_{i}=d_{i}+\frac{n_{i}}{2}-2,$
(1)

with $L$ the number of loops in a diagram, $V_{i}$ the number of vertices of
type-$i$. The $d_{i}$ is the number of derivatives (or the pion-mass
insertions), and $n_{i}$ is the number of charmed meson fields that involved
in the vertex-$i$.

The $DD^{\ast}$ interaction starts at $\nu=0$ (first order, the LO), and the
higher orders come as $\nu=2$ [second order, the next-to-leading order (NLO)],
$\nu=3$ (third order, the N2LO), etc.. At the given order, the number of the
corresponding irreducible diagrams are limited. In Fig. 1, we show the
pertinent Feynmann diagrams for the LO, NLO and N2LO interactions of the
$DD^{\ast}$ system. Then the effective potential of the $DD^{\ast}$ system can
be written as

$\displaystyle V_{\rm eff}=V_{\rm ct}+V_{1\pi}+V_{2\pi}+\dots,$ (2)

with

$\displaystyle V_{\rm ct}$ $\displaystyle=$ $\displaystyle V_{\rm
ct}^{(0)}+V_{\rm ct}^{(2)}+\dots,$ $\displaystyle V_{2\pi}$ $\displaystyle=$
$\displaystyle V_{2\pi}^{(2)}+V_{2\pi}^{(3)}+\dots,$ (3)

where $V_{\rm ct}$, $V_{1\pi}$ and $V_{2\pi}$ denote the contact, OPE and TPE
potentials, respectively. The numbers in the parentheses of the superscripts
represent the power $\nu$ [see Eq. (1)]. Each piece of the right hand side of
Eq. (2) can be further decomposed into the following form,

$\displaystyle
V_{i}=\left[V_{i,c}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}W_{i,c}\right]\mathcal{O}_{1}+\left[V_{i,t}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}W_{i,t}\right]\mathcal{O}_{2},$
(4)

where $i=\mathrm{ct},1\pi,2\pi$, and $\bm{\tau}_{1}\cdot\bm{\tau}_{2}$ denotes
the isospin-isospin interaction. The matrix element
$\langle\bm{\tau}_{1}\cdot\bm{\tau}_{2}\rangle=-3$ and $1$ for the isoscalar
and isovector channels, respectively. The operators $\mathcal{O}_{1}$ and
$\mathcal{O}_{2}$ are given as

$\displaystyle\mathcal{O}_{1}=\bm{\varepsilon}^{\prime\dagger}\cdot\bm{\varepsilon},\qquad\qquad\mathcal{O}_{2}=(\bm{q}\cdot\bm{\varepsilon}^{\prime\dagger})(\bm{q}\cdot\bm{\varepsilon}),$
(5)

where $\bm{q}=\bm{p}^{\prime}-\bm{p}$ ($\bm{p}$ and $\bm{p}^{\prime}$ denote
the initial and final state momenta in the center of mass system,
respectively) is the transferred momentum between $D$ and $D^{\ast}$,
$\bm{\varepsilon}$ and $\bm{\varepsilon}^{\prime\dagger}$ denote the
polarization vectors of the initial and final $D^{\ast}$ mesons, respectively.
In the heavy quark limit, we will not consider the $1/m$ (with $m$ the mass of
the charmed mesons) corrections of the charmed meson fields. Then only two
pertinent operators survive in the effective potentials of $DD^{\ast}$ (for
the $NN$ case, see Machleidt  _et al._ (1987)), i.e., the $\mathcal{O}_{1}$
and $\mathcal{O}_{2}$.

In the following subsections, we will derive the $V_{\rm ct}$, $V_{1\pi}$ and
$V_{2\pi}$, respectively.

Figure 1: The irreducible Feynmann diagrams for the $DD^{\ast}$ interactions
at the LO, NLO and N2LO. We use the single solid line, the double line (solid
plus dashed) and the dashed line to denote the $D$, $D^{\ast}$ and pion,
respectively. The small solid square [in (a1)] and large solid square [in
(b1)] denote the LO ($\Delta_{i}=0$) and NLO ($\Delta_{i}=2$) $DD^{\ast}$
contact vertices, respectively. The small solid dot [in (a2), (b2,…,11),
(c1,…,7)] and large solid dot [in (c1,…,7)] represent the LO ($\Delta_{i}=0$)
and NLO ($\Delta_{i}=1$) $\pi D^{(\ast)}$ vertices, respectively.

###  II.1 Short-range contact interactions

The contact potentials of $DD^{\ast}$ system at the order $\nu=0,2,4$ can be
respectively parameterized as

$\displaystyle V_{\mathrm{ct}}^{(0)}$ $\displaystyle=$
$\displaystyle\left(C_{1}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}C_{2}\right)\mathcal{O}_{1},$
(6) $\displaystyle V_{\mathrm{ct}}^{(2)}$ $\displaystyle=$
$\displaystyle(C_{3}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}C_{4})\bm{q}^{2}\mathcal{O}_{1}+\left(C_{5}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}C_{6}\right)\mathcal{O}_{2},$
(7) $\displaystyle V_{\mathrm{ct}}^{(4)}$ $\displaystyle=$
$\displaystyle(C_{7}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}C_{8})\bm{q}^{4}\mathcal{O}_{1}+(C_{9}+\bm{\tau}_{1}\cdot\bm{\tau}_{2}C_{10})\bm{q}^{2}\mathcal{O}_{2},$

where $C_{1,\dots,10}$ are the corresponding LECs. In the following
calculations, we will take the $V_{\mathrm{ct}}^{(4)}$ to test the convergence
of the expansion in different isospin channels. In Eqs. (7) and (II.1), we
ignore the pion-mass dependent terms for which are of irrelevance in our
studies. In calculations, the local form Gaussian regulator
$\exp{(-\bm{q}^{2}/\Lambda^{2})}$ is multiplied to Eqs. (6)-(II.1) to ensure
the convergence when they are inserted into the Lippmann-Schwinger equations
(LSEs).

In order to determine all the LECs in Eqs. (6)-(II.1), we resort to the
phenomenological RSM Ecker  _et al._ (1989); Epelbaum  _et al._ (2002b) (see
also the applications in heavy-light systems Xu  _et al._ (2019); Du  _et al._
(2016); Xu (2022); Peng  _et al._ (2022)). Within the RSM, we consider the
exchanging of the scalar, pseudoscalar, vector and axial-vector mesons [the
tensor exchanges (e.g., $a_{2},f_{2}$ mesons) are not considered, since their
contributions start at least at the fourth order Du  _et al._ (2016)]. The
derivation details are given in appendix A.1. Their numerical values are
listed in Table 1.

Table 1: The numerical values of the LECs [see Eqs. (6)-(II.1)] determined from the RSM. The $C_{1,2}$, $C_{3,\dots,6}$ and $C_{7,\dots,10}$ are in units of $\mathrm{GeV}^{-2}$, $\mathrm{GeV}^{-4}$ and $\mathrm{GeV}^{-6}$, respectively. The values for the isoscalar ($I=0$) and isovector ($I=1$) channels can be easily obtained via replacing the ${\rm sgn}$ with $(-1)^{I}$. $C_{1}$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$
---|---|---|---|---
$7.5$ | $10.9+0.3\mathrm{sgn}$ | $-26.0$ | $-18.8+23.1\mathrm{sgn}$ | $14.7$
$C_{6}$ | $C_{7}$ | $C_{8}$ | $C_{9}$ | $C_{10}$
$-23.1\mathrm{sgn}$ | $-5.2$ | $32.1-40.6\mathrm{sgn}$ | $-10.6$ | $40.6\mathrm{sgn}$

###  II.2 Long-range one-pion exchange interactions

The ${T_{cc}^{+}}$ was observed in the $D^{0}D^{0}\pi^{+}$ final state, and
its signal is absent in $D^{+}D^{0}\pi^{+}$ Aaij  _et al._ (2022a, b), which
implies that the ${T_{cc}^{+}}$ is an isoscalar state rather than the
isovector one. The flavor wave function of $DD^{\ast}$ in the isoscalar and
isovector channels read, respectively,

$\displaystyle|DD^{\ast},I=0,I_{3}=0\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[D^{0}D^{\ast+}-D^{\ast 0}D^{+}\right],$
(9) $\displaystyle|DD^{\ast},I=1,I_{3}=0\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[D^{0}D^{\ast+}+D^{\ast 0}D^{+}\right].$
(10)

We consider the explicit chiral dynamics for the light pion and relegate the
heavy $\eta$ ($m_{\eta}\simeq 4m_{\pi}$) contribution to the contact terms. In
the following, we show the complete LO ($\Delta_{i}=0$) chiral Lagrangian of
$\varphi D^{(\ast)}$ ($\varphi=\pi,\eta$) coupling Wise (1992); Manohar and
Wise (2000) for the latter convenience.

$\displaystyle\mathcal{L}_{\varphi\mathcal{H}}^{(0)}$ $\displaystyle=$
$\displaystyle
i\langle\mathcal{H}v\cdot\mathcal{D}\bar{\mathcal{H}}\rangle-\frac{1}{8}\delta_{b}\langle\mathcal{H}\sigma^{\mu\nu}\bar{\mathcal{H}}\sigma_{\mu\nu}\rangle$
(11)
$\displaystyle+g_{\varphi}\langle\mathcal{H}\not{u}\gamma_{5}\bar{\mathcal{H}}\rangle,$

where $v=(1,\bm{0})$ denotes the four-velocity of heavy mesons, and
$\mathcal{D}_{\mu}=\partial_{\mu}+\Gamma_{\mu}$, with
$\Gamma_{\mu}=[\xi^{\dagger},\partial_{\mu}\xi]/2$ the chiral connection.
$\delta_{b}=m_{D^{\ast}}-m_{D}\simeq 142$ MeV, and $g_{\varphi}=-0.59$. The
axial-vector current $u_{\mu}$ is defined as
$u_{\mu}=i\\{\xi^{\dagger},\partial_{\mu}\xi\\}/2$. Meanwhile, the
$\xi^{2}=U=\exp(i\varphi/f_{\varphi})$, and the matrix form of $\varphi$ reads

$\displaystyle\varphi=\left[\begin{array}[]{cc}\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}\pi^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta\end{array}\right].$ (14)

The $\mathcal{H}$ denotes the superfield of ($D,D^{\ast}$) doublet in the
heavy quark symmetry, which reads

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\frac{1+\not{v}}{2}\left(P_{\mu}^{\ast}\gamma^{\mu}+iP\gamma_{5}\right),\quad\bar{\mathcal{H}}=\gamma^{0}\mathcal{H}^{\dagger}\gamma^{0},$
(15)

with $P=(D^{0},D^{+})^{T}$ and $P^{\ast}=(D^{\ast 0},D^{\ast+})^{T}$.

With the OPE diagram in Fig. 1 ($\mathrm{a}_{2}$) and the LO chiral Lagrangian
in Eq. (11), one can easily get the OPE potential, which reads

$\displaystyle\mathcal{V}_{1\pi}(u_{\pi},\bm{q})$ $\displaystyle=$
$\displaystyle\mathrm{sgn}(\bm{\tau}_{1}\cdot\bm{\tau}_{2})\frac{g_{\varphi}^{2}}{4f_{\pi}^{2}}\frac{\mathcal{O}_{2}}{\bm{q}^{2}-u_{\pi}^{2}-i\epsilon},$
(16)

where $f_{\pi}=92.4$ MeV, and $u_{\pi}=\sqrt{\delta_{b}^{2}-m_{\pi}^{2}}$
(with $m_{\pi}\simeq 137$ MeV the pion mass). Eq. (16) contains two parts—the
principle-value and the imaginary parts. Its principle-value corresponds to an
oscillatory potential in the coordinate space, e.g., see Eq. (20), while the
imaginary part comes from the three-body ($DD\pi$) cut, it will contribute a
finite width to the bound state of $DD^{\ast}$. We then separate the operator
$\mathcal{O}_{2}$ into the ‘spin-spin’ part and the tensor part via the
equation

$\displaystyle\mathcal{O}_{2}=(\bm{q}\cdot\bm{\varepsilon}^{\prime\dagger})(\bm{q}\cdot\bm{\varepsilon})=\frac{1}{3}(\bm{\varepsilon}^{\prime\dagger}\cdot\bm{\varepsilon})\bm{q^{2}}+\bm{q}^{2}\mathcal{S}_{12},$
(17)

where
$\mathcal{S}_{12}=(\bm{\varepsilon}^{\prime\dagger}\cdot\hat{\bm{q}})(\bm{\varepsilon}\cdot\hat{\bm{q}})-\frac{1}{3}\bm{\varepsilon}^{\prime\dagger}\cdot\bm{\varepsilon}$,
with $\hat{\bm{q}}=\bm{q}/|\bm{q}|$. Then the principle-value part of Eq. (16)
can be transformed into

$\displaystyle\mathcal{V}_{1\pi}(u_{\pi},\bm{q})$ $\displaystyle=$
$\displaystyle\mathrm{sgn}(\bm{\tau}_{1}\cdot\bm{\tau}_{2})\frac{g_{\varphi}^{2}}{4f_{\pi}^{2}}\bigg{(}\frac{1}{3}\mathcal{O}_{1}+\frac{1}{3}\frac{u_{\pi}^{2}}{\bm{q}^{2}-u_{\pi}^{2}}\mathcal{O}_{1}$
(18)
$\displaystyle+\frac{\bm{q}^{2}}{\bm{q}^{2}-u_{\pi}^{2}}\mathcal{S}_{12}\bigg{)},$

in which the first term corresponds to a $\delta$-function in the coordinate
space after the Fourier transform. It is an artefact arising from the
idealized point-like $\pi D^{(\ast)}$ coupling. In reality, the OPE dominates
at the long-distance region, i.e., $r\gtrsim 2~{}\mathrm{fm}\simeq
1.5m_{\pi}^{-1}$ Ericson and Weise (1988). Therefore, it is better to subtract
the unphysical $\delta$-function part from the OPE potential. In Ref. Reinert
_et al._ (2018), Reinert et al. introduced a subtraction scheme for the $NN$
interaction with the following form

$\displaystyle V_{1\pi,\Lambda}(u_{\pi},\bm{q})$ $\displaystyle=$
$\displaystyle\mathrm{sgn}(\bm{\tau}_{1}\cdot\bm{\tau}_{2})\frac{g_{\varphi}^{2}}{4f_{\pi}^{2}}\Bigg{\\{}\frac{u_{\pi}^{2}}{3}\frac{1}{\bm{q}^{2}-u_{\pi}^{2}}\mathcal{O}_{1}$
(19)
$\displaystyle+\left[\frac{1}{3}+\mathcal{C}(u_{\pi},\Lambda)\right]\mathcal{O}_{1}+\frac{\bm{q}^{2}}{\bm{q}^{2}-u_{\pi}^{2}}\mathcal{S}_{12}\Bigg{\\}}$
$\displaystyle\times\exp\Big{(}-\frac{\bm{q}^{2}-u_{\pi}^{2}}{\Lambda^{2}}\Big{)},$

where an $u_{\pi}$-dependent term in the Gaussian regulator is introduced to
ensure the strength of OPE potential remains unchanged at the pion pole
Epelbaum  _et al._ (2022). The subtraction term $\mathcal{C}(u_{\pi},\Lambda)$
is determined by the requirement that the OPE potential vanishes at the
origin, i.e., when $r\to 0$. With the following relations of Fourier
transform,

$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\frac{1}{\bm{q}^{2}-u_{\pi}^{2}}=\frac{1}{4\pi
r}\cos(u_{\pi}r),$ (20)
$\displaystyle\mathcal{U}_{\Lambda}(u_{\pi},r)=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\frac{1}{\bm{q}^{2}-u_{\pi}^{2}}\exp\Big{(}-\frac{\bm{q}^{2}-u_{\pi}^{2}}{\Lambda^{2}}\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}=\frac{\cos(u_{\pi}r)}{8\pi
r}\left[\mathrm{erfc}\Big{(}\frac{u_{\pi}}{\Lambda}-\frac{\Lambda
r}{2}\Big{)}-\mathrm{erfc}\Big{(}\frac{u_{\pi}}{\Lambda}+\frac{\Lambda
r}{2}\Big{)}\right],$ (21)
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}\exp\Big{(}-\frac{\bm{q}^{2}-u_{\pi}^{2}}{\Lambda^{2}}\Big{)}$
$\displaystyle\qquad\qquad=\Big{(}\frac{\Lambda^{2}}{4\pi}\Big{)}^{3/2}\exp\Big{(}\frac{u_{\pi}^{2}}{\Lambda^{2}}-\frac{\Lambda^{2}r^{2}}{4}\Big{)},$
(22)

one easily obtains

$\displaystyle
V_{1\pi,\Lambda}(u_{\pi},r)=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\bm{q}\cdot\bm{r}}V_{1\pi,\Lambda}(u_{\pi},\bm{q})$
$\displaystyle\quad=-\mathrm{sgn}(\bm{\tau}_{1}\cdot\bm{\tau}_{2})\frac{g_{\varphi}^{2}}{4f_{\pi}^{2}}\Bigg{\\{}\mathcal{S}_{12}r\frac{\partial}{\partial
r}\left(\frac{1}{r}\frac{\partial}{\partial
r}\right)\mathcal{U}_{\Lambda}(u_{\pi},r)$
$\displaystyle\quad\quad-\mathcal{O}_{1}\bigg{[}\frac{u_{\pi}^{2}}{3}\mathcal{U}_{\Lambda}(u_{\pi},r)+\Big{(}\frac{1}{3}+\mathcal{C}(u_{\pi},\Lambda)\Big{)}\Big{(}\frac{\Lambda^{2}}{4\pi}\Big{)}^{3/2}$
$\displaystyle\quad\quad\times\exp\Big{(}\frac{u_{\pi}^{2}}{\Lambda^{2}}-\frac{\Lambda^{2}r^{2}}{4}\Big{)}\bigg{]}\Bigg{\\}}.$
(23)

With the constraints $V_{1\pi,\Lambda}(u_{\pi},r\to 0)=0$, we get

$\displaystyle\mathcal{C}(u_{\pi},\Lambda)=-\Big{(}\frac{1}{3}+\frac{2u_{\pi}^{2}}{3\Lambda^{2}}e^{-\frac{2u_{\pi}^{2}}{\Lambda^{2}}}\Big{)}.$
(24)

Note that, in Eq. (20) the $\mathrm{erfc}(x)$ represents the complementary
error function, i.e.,

$\displaystyle\mathrm{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}dte^{-t^{2}}.$
(25)

Figure 2: The central part (left panel) and tensor part (right panel) of the
subtracted OPE potential [see Eq. (II.2)] in isoscalar channel with the cutoff
$\Lambda=0.5$ GeV.

In Fig. 2, we show the behaviors of the central part [$\mathcal{O}_{1}$
related term in Eq. (II.2)] and tensor part [$\mathcal{S}_{12}$ related term
in Eq. (II.2)] of the $\delta$-function subtracted OPE potential for the $I=0$
case (the behaviors of the $I=1$ case are similar). One can see that both the
central and tensor potentials vanish for $r\to 0$, and the strength of the
central potential is much weaker than that of the tensor potential. Therefore,
though the central potential is attractive, it is too weak to form bound
state. However, if one does not subtract the $\delta$-function, then there
would result in very attractive central potential once using a large cutoff
when making the Fourier transform. This may also lead to the bound state, but
it is unreasonable. One also sees that the central potential can extend to
large distances since the effective mass $u_{\pi}$ in the pion propagator is
much smaller than the $m_{\pi}$, this is a very typical feature of the
$DD^{\ast}$ system.

###  II.3 Intermediate-range two-pion exchange interactions

We first show the LO ($\nu=2$) TPE contributions, which come from the diagrams
in Figs. 1 ($\text{b}_{2}$)-($\text{b}_{11}$). They can be obtained using the
Lagrangian (11) and calculating the loop integrals. We adopt the spectral
function representation for the TPE interactions. The long-range part of the
TPE interactions is determined by the non-analytic terms in momentum-space.
They have the following forms within the dimensional regularization,

$\displaystyle\mathcal{V}_{2\pi,c}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{4}}{512\pi\delta_{b}f_{\pi}^{4}}\left[6\bm{q}^{2}\mathrm{sgn}(4m_{\pi}^{2}-4\delta_{b}^{2}+\bm{q}^{2})A^{\prime}(q)-3(8m_{\pi}^{4}+10m_{\pi}^{2}\bm{q}^{2}+3\bm{q}^{4})A(q)-\frac{8\delta_{b}}{\pi}(5\delta_{b}^{2}+3\bm{q}^{2}\mathrm{sgn})L(q)\right],$
(26) $\displaystyle\mathcal{W}_{2\pi,c}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{4}}{256\pi\delta_{b}f_{\pi}^{4}}\bigg{\\{}\frac{1}{g_{\varphi}^{2}}(4m_{\pi}^{2}-4\delta_{b}^{2}+3\bm{q}^{2})\left[2\delta_{b}^{2}+g_{\varphi}^{2}(-2\delta_{b}^{2}+2m_{\pi}^{2}+\bm{q}^{2})\right]A^{\prime}(q)-2\mathrm{sgn}(4m_{\pi}^{2}+\bm{q}^{2})\bm{q}^{2}A(q)$
(27) $\displaystyle+\frac{2\delta_{b}}{3\pi
g_{\varphi}^{4}}\left[16\delta_{b}^{2}g_{\varphi}^{2}(5g_{\varphi}^{2}-3)+4(-5g_{\varphi}^{4}+4g_{\varphi}^{2}+1)m_{\pi}^{2}+(-23g_{\varphi}^{4}+10g_{\varphi}^{2}+1)\bm{q}^{2}\right]L(q)\bigg{\\}},$
$\displaystyle\mathcal{V}_{2\pi,t}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{4}}{512\pi\delta_{b}f_{\pi}^{4}}\left[\frac{3}{\bm{q}^{2}}(-8m_{\pi}^{4}-2m_{\pi}^{2}\bm{q}^{2}+\bm{q}^{4})A(q)-6\mathrm{sgn}(-4\delta_{b}^{2}+4m_{\pi}^{2}+\bm{q}^{2})A^{\prime}(q)+\frac{24\delta_{b}}{\pi}\mathrm{sgn}L(q)\right],$
(28) $\displaystyle\mathcal{W}_{2\pi,t}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{4}}{256\pi\delta_{b}f_{\pi}^{4}}\left\\{2\mathrm{sgn}(4m_{\pi}^{2}+\bm{q}^{2})A(q)-\frac{1}{g_{\varphi}^{2}\bm{q}^{2}}(4\delta_{b}^{2}-4m_{\pi}^{2}+\bm{q}^{2})\left[2\delta_{b}^{2}+g_{\varphi}^{2}(-2\delta_{b}^{2}+2m_{\pi}^{2}+\bm{q}^{2})\right]A^{\prime}(q)\right\\},$
(29)

where the three non-analytic functions $A(q)$, $A^{\prime}(q)$ and $L(q)$
respectively read

$\displaystyle A(q)$ $\displaystyle=$
$\displaystyle\frac{1}{2q}\arctan\frac{q}{2m_{\pi}},$ (30) $\displaystyle
A^{\prime}(q)$ $\displaystyle=$
$\displaystyle\frac{1}{2q}\arctan\frac{q}{2m^{\prime}},$ (31) $\displaystyle
L(q)$ $\displaystyle=$
$\displaystyle\frac{\varpi}{q}\ln\frac{q+\varpi}{2m_{\pi}},$ (32)

with $q=|\bm{q}|$, $m^{\prime}=[m_{\pi}^{2}-\delta_{b}^{2}-i\epsilon]^{1/2}$,
and $\varpi=[\bm{q}^{2}+4m_{\pi}^{2}-i\epsilon]^{1/2}$. The terms containing
the non-analytic functions
$\mathcal{F}(q,n,i)=\int_{-1/2}^{1/2}\left(y^{n}\zeta^{i}\arctan\frac{\delta_{b}}{\zeta}\right)dy$
(with $\zeta=q\sqrt{a^{2}-y^{2}}$, $a=\frac{\varpi^{\prime}}{2q}$, and
$\varpi^{\prime}=[\bm{q}^{2}+4(m_{\pi}^{2}-\delta_{b}^{2})-i\epsilon]^{1/2}$)
and their derivatives with respect to $\delta_{b}$ are ignored for simplicity
since we noticed that their contributions are much smaller than those in Eqs.
(26)-(29).

In order to obtain the subleading ($\nu=3$) TPE potential (see the diagrams in
the third column of Fig. 1), one needs an insertion of the subleading
($\Delta_{i}=1$) $\pi D^{(\ast)}$ Lagrangians. The Lagrangians read Meng  _et
al._ (2022)

$\displaystyle\mathcal{L}_{\varphi\mathcal{H}}^{(1)}$ $\displaystyle=$
$\displaystyle\tilde{c}_{1}\langle\mathcal{H}\bar{\mathcal{H}}\rangle\mathrm{Tr}(\chi_{+})+\tilde{c}_{2}\langle\mathcal{H}v\cdot
uv\cdot u\bar{\mathcal{H}}\rangle+\tilde{c}_{3}\langle\mathcal{H}u\cdot
u\bar{\mathcal{H}}\rangle$ (33)
$\displaystyle+i\tilde{c}_{4}\langle\mathcal{H}[u_{\mu},u_{\nu}]\sigma^{\mu\nu}\bar{\mathcal{H}}\rangle+\tilde{c}_{5}\langle\mathcal{H}\hat{\chi}_{+}\bar{\mathcal{H}}\rangle,$

where $\chi_{+}=\xi^{\dagger}\chi\xi^{\dagger}+\xi\chi^{\dagger}\xi$, with
$\chi=2B_{0}\mathrm{diag}(m_{u},m_{d})$, and
$\hat{\chi}_{+}=\chi_{+}-\frac{1}{2}\mathbf{\mathrm{Tr}}(\chi_{+})$. One can
see that the structure of $\mathcal{L}_{\varphi\mathcal{H}}^{(1)}$ is very
similar to the ones of $\pi N$ Lagrangians Bernard  _et al._ (1995).

Table 2: The numerical values of the LECs in Eq. (33) determined from the RSM (in units of GeV-1). $\tilde{c}_{1}$ | $\tilde{c}_{2}$ | $\tilde{c}_{3}$ | $\tilde{c}_{4}$ | $\tilde{c}_{5}$
---|---|---|---|---
$-0.21$ | $-0.83$ | $-0.55$ | $0.61$ | $0.26$

In literature, only the LECs in partial terms in Eq. (33) were determined for
certain problems (see Ref. Meng  _et al._ (2022)). Here, we again use the RSM
to estimate the $\tilde{c}_{i}$. One can consult appendix A.2 for details. The
numerical values of the LECs $\tilde{c}_{i}~{}(i=1,\dots,5)$ in Eq. (33) are
summarized in Table 2. From Table 2 one can see that the couplings of the
subleading $\pi D^{(\ast)}$ vertices are of natural size and are much smaller
than those of the $\pi N$ system Fettes  _et al._ (1998); Krebs  _et al._
(2007). In contrast to the $NN$ system, this makes the main contribution for
the binding forces of $DD^{\ast}$ come from the short-range contact
interactions.

The non-analytic terms of the subleading TPE potentials read

$\displaystyle\mathcal{V}_{2\pi,c}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{2}}{512\pi
f_{\pi}^{4}}\bigg{\\{}(4m_{\pi}^{2}+3\bm{q}^{2})\left[48\tilde{c}_{1}m_{\pi}^{2}-3\tilde{c}_{3}(2m_{\pi}^{2}+\bm{q}^{2})-8\tilde{c}_{5}\delta_{i}^{2}\right]A(q)+\big{\\{}3(-12\delta_{b}^{2}+12m_{\pi}^{2}+5\bm{q}^{2})$
(34)
$\displaystyle\times\left[16\tilde{c}_{1}m_{\pi}^{2}-2\tilde{c}_{2}\delta_{b}^{2}-\tilde{c}_{3}(2m_{\pi}^{2}+\bm{q}^{2})\right]-8\tilde{c}_{5}\delta_{i}^{2}(4m_{\pi}^{2}-4\delta_{b}^{2}+\bm{q}^{2})\big{\\}}A^{\prime}(q)$
$\displaystyle-\frac{8\delta_{b}}{\pi}\left[48\tilde{c}_{1}m_{\pi}^{2}+\tilde{c}_{2}(m_{\pi}^{2}-6\delta_{b}^{2}+\bm{q}^{2})+4\tilde{c}_{3}\delta_{b}^{2}-3\tilde{c}_{3}\bm{q}^{2}+4\tilde{c}_{5}\delta_{i}^{2}\right]L(q)\bigg{\\}},$
$\displaystyle\mathcal{W}_{2\pi,c}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{2}}{32\pi
f_{\pi}^{4}}\left\\{\tilde{c}_{4}\mathrm{sgn}(4m_{\pi}^{2}+\bm{q}^{2})\bm{q}^{2}A(q)-2\tilde{c}_{5}\delta_{i}^{2}(2m_{\pi}^{2}-2\delta_{b}^{2}+\bm{q}^{2})A^{\prime}(q)+\frac{6}{\pi}\tilde{c}_{5}\delta_{b}\delta_{i}^{2}L(q)\right\\},$
(35) $\displaystyle\mathcal{V}_{2\pi,t}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{2}}{512\pi
f_{\pi}^{4}\bm{q}^{2}}\bigg{\\{}(\bm{q}^{2}-4m_{\pi}^{2})\left[-48\tilde{c}_{1}m_{\pi}^{2}+3\tilde{c}_{3}(2m_{\pi}^{2}+\bm{q}^{2})+8\tilde{c}_{5}\delta_{i}^{2}\right]A(q)-(4\delta_{b}^{2}-4m_{\pi}^{2}+\bm{q}^{2})$
(36)
$\displaystyle\times\left[-48\tilde{c}_{1}m_{\pi}^{2}+6\tilde{c}_{2}\delta_{b}^{2}+3\tilde{c}_{3}(2m_{\pi}^{2}+\bm{q}^{2})+8\tilde{c}_{5}\delta_{i}^{2}\right]A^{\prime}(q)+\frac{4\delta_{b}}{\pi}\left[\tilde{c}_{2}\bm{q}^{2}-2m_{\pi}^{2}(\tilde{c}_{2}-6\tilde{c}_{3})\right]L(q)\bigg{\\}},$
$\displaystyle\mathcal{W}_{2\pi,t}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{2}}{32\pi
f_{\pi}^{4}}\left[-\tilde{c}_{4}\mathrm{sgn}(4m_{\pi}^{2}+\bm{q}^{2})A(q)\right],$
(37)

where $\delta_{i}^{2}=m_{K^{+}}^{2}-m_{K^{0}}^{2}$ denotes the $u,d$ quark
mass difference that stems from the $\hat{\chi}_{+}$ term in Eq. (33).

The LO and subleading TPE potentials can be obtained from the spectral
function representation associating with the local momentum-space
regularization Reinert  _et al._ (2018),

$\displaystyle V_{2\pi,i}^{(\nu)}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\exp\left(-\frac{\bm{q}^{2}}{2\Lambda^{2}}\right)\int_{0}^{\infty}d\mu\left[\mu\rho_{2\pi,i}^{(\nu)}(\mu)\right]\left[\frac{1}{\mu^{2}+\bm{q}^{2}}+\mathcal{C}_{1}(\mu,\Lambda)+\mathcal{C}_{2}(\mu,\Lambda)\bm{q}^{2}\right]\exp\left(-\frac{\mu^{2}}{2\Lambda^{2}}\right),$
(38) $\displaystyle W_{2\pi,i}^{(\nu)}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\exp\left(-\frac{\bm{q}^{2}}{2\Lambda^{2}}\right)\int_{0}^{\infty}d\mu\left[\mu\eta_{2\pi,i}^{(\nu)}(\mu)\right]\left[\frac{1}{\mu^{2}+\bm{q}^{2}}+\mathcal{C}_{1}(\mu,\Lambda)+\mathcal{C}_{2}(\mu,\Lambda)\bm{q}^{2}\right]\exp\left(-\frac{\mu^{2}}{2\Lambda^{2}}\right),$
(39)

where $i=c,t$ denotes the central and tensor parts, respectively, while the
$\nu$ is the chiral order defined in Eq. (1). The spectral functions
$\rho_{2\pi,i}^{(\nu)}(\mu)$ and $\eta_{2\pi,i}^{(\nu)}(\mu)$ respectively
read

$\displaystyle\rho_{2\pi,i}^{(\nu)}(\mu)$ $\displaystyle=$
$\displaystyle\Im\left[\mathcal{V}_{2\pi,i}^{(\nu)}(0^{+}-i\mu)\right],\qquad\qquad\eta_{2\pi,i}^{(\nu)}(\mu)=\Im\left[\mathcal{W}_{2\pi,i}^{(\nu)}(0^{+}-i\mu)\right].$
(40)

In order to get $\rho_{2\pi,i}^{(\nu)}(\mu)$ and $\eta_{2\pi,i}^{(\nu)}(\mu)$,
one also needs the following quantities Epelbaum  _et al._ (2002a),

$\displaystyle\Im A(0^{+}-i\mu)$ $\displaystyle=$
$\displaystyle\frac{\pi}{4\mu}\Theta(\mu-2m_{\pi}),$ (41) $\displaystyle\Im
A^{\prime}(0^{+}-i\mu)$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{\pi}{4\mu}\Theta(\mu-2m_{\pi})&m_{\pi}>\delta_{b}\\\
\frac{1}{2\mu}\arctan\frac{\mu}{2\sqrt{\delta_{b}^{2}-m_{\pi}^{2}}}&m_{\pi}<\delta_{b}\\\
\end{cases},$ (42) $\displaystyle\Im L(0^{+}-i\mu)$ $\displaystyle=$
$\displaystyle-\pi\frac{\sqrt{\mu^{2}-4m_{\pi}^{2}}}{2\mu}\Theta(\mu-2m_{\pi}),$
(43)

with $\Theta$ the Heaviside step function.

The subtraction terms $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are introduced
to minimize the mixture of the long- and short-range forces in TPE
interactions. They are determined by the following requirements Reinert  _et
al._ (2018)

$\displaystyle V_{2\pi,i}^{(\nu)}(r)\big{|}_{r\to
0}=\frac{d^{2}}{dr^{2}}V_{2\pi,i}^{(\nu)}(r)\big{|}_{r\to
0}=W_{2\pi,i}^{(\nu)}(r)\big{|}_{r\to
0}=\frac{d^{2}}{dr^{2}}W_{2\pi,i}^{(\nu)}(r)\big{|}_{r\to 0}=0,$ (44)

where $V_{2\pi,i}^{(\nu)}(r)$ and $W_{2\pi,i}^{(\nu)}(r)$ represent the
corresponding potentials in coordinate-space. They are obtained with the
following Fourier transform

$\displaystyle V_{2\pi,i}^{(\nu)}(r)=\frac{1}{2\pi^{2}}\int
dqq^{2}j_{0}(qr)V_{2\pi,i}^{(\nu)}(q),$ (45)

where the similar form holds for the $W_{2\pi,i}^{(\nu)}$, and $j_{0}(qr)$
represents the spherical Bessel function of the first kind. The expressions of
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are given as

$\displaystyle\mathcal{C}_{1}(\mu,\Lambda)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}\mu\exp\left(\frac{\mu^{2}}{2\Lambda^{2}}\right)\left(5\Lambda^{2}+\mu^{2}\right)\mathrm{erfc}\left(\frac{\mu}{\sqrt{2}\Lambda}\right)-2\Lambda\left(4\Lambda^{2}+\mu^{2}\right)}{4\Lambda^{5}},$
(46) $\displaystyle\mathcal{C}_{2}(\mu,\Lambda)$ $\displaystyle=$
$\displaystyle\frac{2\Lambda\left(2\Lambda^{2}+\mu^{2}\right)-\sqrt{2\pi}\mu\exp\left(\frac{\mu^{2}}{2\Lambda^{2}}\right)\left(3\Lambda^{2}+\mu^{2}\right)\mathrm{erfc}\left(\frac{\mu}{\sqrt{2}\Lambda}\right)}{12\Lambda^{7}}.$
(47)

##  III Numerical results and discussions

With the effective potentials, we solve the LSEs to analyze the pole
distributions in the physical Riemann sheet. The LSE in the partial wave basis
$|\ell sj\rangle$ reads

$\displaystyle T_{\ell^{\prime}\ell sj}(p^{\prime},p)$ $\displaystyle=$
$\displaystyle V_{\ell^{\prime}\ell
sj}(p^{\prime},p)+\sum_{\ell^{\prime\prime}}\int\frac{k^{2}dk}{(2\pi)^{3}}V_{\ell^{\prime}\ell^{\prime\prime}sj}(p^{\prime},k)$
(48)
$\displaystyle\times\frac{2\mu_{m}}{p^{2}-k^{2}+i\epsilon}T_{\ell^{\prime\prime}\ell
sj}(k,p),$

where $\mu_{m}$ denotes the reduced mass of $DD^{\ast}$ system, and
$V_{\ell^{\prime}\ell sj}(p^{\prime},p)=\langle\ell^{\prime}sj|V(\bm{q})|\ell
sj\rangle$ can be easily obtained with the approach in Ref. Golak  _et al._
(2010). The S-D wave coupling is considered in our calculations. Thus, the
$V_{\ell^{\prime}\ell sj}(p^{\prime},p)$ is given with the $2\times 2$ matrix
in the coupled-channel $|\ell sj\rangle$ basis.

We use the isospin average mass for $DD^{\ast}$ system in our calculations.
The threshold of $DD^{\ast}$ and the experimentally measured mass of
${T_{cc}^{+}}$ Aaij  _et al._ (2022a, b) are given as

$\displaystyle m_{\mathrm{th}}$ $\displaystyle=$ $\displaystyle
3875.8~{}\mathrm{MeV},$ $\displaystyle m_{\mathrm{exp}}$ $\displaystyle=$
$\displaystyle(m_{D^{\ast+}}+m_{D^{0}})-0.36=3874.7~{}\mathrm{MeV}.$ (49)

We first study the pole trajectory of $DD^{\ast}$ scattering T-matrix in the
isoscalar channel with the contact interactions being kept up to NLO
[including the Eqs. (6)-(7) in effective potentials], and the cutoff is in the
range $520-600$ MeV. We notice that the binding solution begins to appear when
$\Lambda\approx 600$ MeV in this case. The pole trajectory is very similar to
the case in which the contact interaction is kept up to N2LO [including the
Eqs. (6)-(II.1) in effective potentials, and the result in this case is shown
in Fig. 3]. In this case, the pole appears when $\Lambda\approx 520$ MeV, and
the pole mass approaches to the experimental value when $\Lambda\simeq 560$
MeV. From Fig. (3), one also sees that the binding becomes deeper with the
increasing of cutoff, while the half width is almost insensitive to the
cutoff. The width of ${T_{cc}^{+}}$ is around $80$ keV in our calculations,
which is about two times larger than the measured value $47.8\pm 1.9$ keV with
an unitarized Breit-Wigner profile in Ref. Aaij  _et al._ (2022a, b).

We also investigated the situation in isovector channel, but we did not find
binding solutions in this channel. This is consistent with the experimental
facts—there are no structures in the $D^{+}D^{0}\pi^{+}$ invariant mass
spectrum.

Figure 3: Pole trajectory of the $DD^{\ast}$ bound state in the physical
Riemann sheet with the change of cutoff $\Lambda\in[520,600]$ MeV. The arrow
denotes the direction that the cutoff becomes larger. The two dashed vertical
lines from left to right represent the experimentally measured mass of
$T_{cc}^{+}$ and the threshold of $DD^{\ast}$ in order.

##  IV Summary

We revisit the $DD^{\ast}$ interactions within the $\chi$EFT up to the third
order. The pion-exchanged interactions are carefully treated with the local
momentum-space regularization, in which their short-range components are
subtracted via demanding the pion-exchanged contributions vanish at the origin
in the coordinate space. This is consistent with the new developments of
nuclear forces in Ref. Reinert  _et al._ (2018).

The contact interactions and the subleading $\pi D^{(\ast)}$ vertices are
ascribed to the heavier meson exchanging, and consequently the LECs are
estimated with the resonance saturation model. We notice that the subleading
$\pi D^{(\ast)}$ couplings are much smaller that those in the $\pi N$ system,
which makes the binding force of $DD^{\ast}$ mainly come from the short-range
part. This is very different with that of the $NN$ system.

We investigate the pole trajectory of $DD^{\ast}$ scattering T-matrix in two
cases—keeping the contact interactions up to NLO and N2LO, respectively. The
pole trajectory are very similar in these two cases, which implies the low-
energy expansion works well in the isoscalar channel. The binding solution
only exists in the isoscalar channel, and this is consistent with the
experimental data. Our calculation favors the molecular explanation of
${T_{cc}^{+}}$.

## Acknowledgement

B.W is very grateful to Shi-Lin Zhu and Lu Meng for helpful discussions and
carefully reading the manuscript. This work is supported by the National
Natural Science Foundation of China under Grants No. 12105072, the Youth Funds
of Hebei Province (No. A2021201027) and the Start-up Funds for Young Talents
of Hebei University (No. 521100221021).

##  Appendix A Estimations of the LECs

###  A.1 The LECs in contact interactions

In what follows, we list the coupling Lagrangians of these resonances with the
($D,D^{\ast}$) doublet under the heavy quark symmetry, and estimate the LECs
in Eqs. (6)-(II.1).

  * •

Scalar exchange—$\sigma,a_{0},f_{0}$ mesons

The corresponding Lagrangians read

$\displaystyle\mathcal{L}_{\sigma\mathcal{H}}$ $\displaystyle=$ $\displaystyle
g_{\sigma}\langle\mathcal{H}\bar{\mathcal{H}}\rangle\sigma,$ (50)
$\displaystyle\mathcal{L}_{S\mathcal{H}}$ $\displaystyle=$ $\displaystyle
g_{s}\langle\mathcal{H}S\bar{\mathcal{H}}\rangle,$ (51)

where $g_{\sigma}$ and $g_{s}$ are the corresponding coupling constants. In
the SU(2) case, $g_{s}=\sqrt{2}g_{\sigma}$ in the large-$N_{c}$ limit Ecker
_et al._ (1989). The matrix form of $S$ is given as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{a_{0}^{0}}{\sqrt{2}}+\frac{f_{0}}{\sqrt{2}}&a_{0}^{+}\\\
a_{0}^{-}&-\frac{a_{0}^{0}}{\sqrt{2}}+\frac{f_{0}}{\sqrt{2}}\end{array}\right].$
(54)

Within the parity-doubling model Bardeen  _et al._ (2003), the $g_{\sigma}$
reads

$\displaystyle g_{\sigma}=-\frac{g_{\pi}}{2\sqrt{6}},\qquad
g_{\pi}=\frac{\Delta}{f_{\pi}},$ (55)

where $f_{\pi}=92.4$ MeV is the pion decay constant, and
$\Delta=m_{0^{+}}-m_{0^{-}}$ denotes the mass difference of the $J^{P}=0^{+}$
and $0^{-}$ charmed mesons. In most previous studies, such as the one-boson
exchange model, the $\Delta=m_{D_{s0}^{\ast}(2317)}-m_{D_{s}}\approx 350$ MeV
was usually used as the original work Bardeen  _et al._ (2003). In the SU(2)
case in this study, we chose $\Delta=m_{D_{0}^{\ast}(2300)}-m_{D}$. The nature
of the $D_{0}^{\ast}(2300)$ is still in controversy (one can consult the
recent review Meng  _et al._ (2022) for more details). The analyses in Moir
_et al._ (2016); Du  _et al._ (2021); Gayer  _et al._ (2021) shown that the
pole mass of $D_{0}^{\ast}(2300)$ is lower than the value in Review of
Particle Physics (RPP) Workman and Others (2022). Here, we adopt the value
$m_{D_{0}^{\ast}(2300)}=2196\pm 64$ MeV from the lattice calculation at pion
mass $m_{\pi}=239$ MeV Gayer  _et al._ (2021). Then we have $\Delta\simeq
330\pm 64$ MeV. Feeding this $\Delta$ into Eq. (55) one obtains

$g_{\sigma}=-0.73\pm 0.14.$ (56)

For the mass of $\sigma$ meson, we adopt the value that was determined in
Refs. Caprini  _et al._ (2006); Dai and Pennington (2014) with the model-
independent ways, which reads

$m_{\sigma}=441~{}\rm{MeV}.$ (57)

Meanwhile, for the masses of the $a_{0}$ and $f_{0}$ mesons, we ignore their
mass differences and use Workman and Others (2022)

$m_{a_{0}}=m_{f_{0}}\simeq 980~{}\rm{MeV}.$ (58)

  * •

Pseudoscalar exchange—$\eta$ meson

For the $\eta$ meson, its decay constant is $f_{\eta}=116$ MeV and the mass
$m_{\eta}=548$ MeV. The eta-exchange contribution to the contact interaction
is associated with the term in the second line of Eq. (11).

  * •

Vector exchange—$\rho,\omega$ mesons

For the vector exchange form, we use the Lagrangians from the local hidden-
gauge formalism Casalbuoni  _et al._ (1997), which read

$\displaystyle\mathcal{L}_{V\mathcal{H}}$ $\displaystyle=$ $\displaystyle
i\beta\langle\mathcal{H}v^{\mu}(\Gamma_{\mu}-\rho_{\mu})\bar{\mathcal{H}}\rangle+i\lambda\langle\mathcal{H}\sigma^{\mu\nu}F_{\mu\nu}\bar{\mathcal{H}}\rangle,$
(59)

where $\rho_{\mu}=i\frac{g_{v}}{\sqrt{2}}V_{\mu}$ and
$F_{\mu\nu}=\partial_{\mu}\rho_{\nu}-\partial_{\nu}\rho_{\mu}+[\rho_{\mu},\rho_{\nu}]$.
The matrix form of $V_{\mu}$ is given as

$\displaystyle
V_{\mu}=\left[\begin{array}[]{cc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}\end{array}\right]_{\mu}.$
(62)

The coupling constants $\beta$, $\lambda$, $g_{v}$ Casalbuoni  _et al._ (1992,
1997) and masses Workman and Others (2022) of the $\rho,\omega$ read

$\displaystyle\beta=0.9,\quad\lambda=-0.63~{}\mathrm{GeV}^{-1},\quad
g_{v}=5.8.$ $\displaystyle m_{\rho}=770~{}\mathrm{MeV},\quad
m_{\omega}=782~{}\mathrm{MeV}.$ (63)

  * •

Axial-vector exchange—$a_{1},f_{1}$ mesons

For involving the possible contribution of the heavier axial-vector mesons, we
construct the following effective Lagrangians,

$\displaystyle\mathcal{L}_{A\mathcal{H}}$ $\displaystyle=$ $\displaystyle
g_{a}\langle\mathcal{H}\gamma_{\mu}\gamma^{5}A^{\mu}\bar{\mathcal{H}}\rangle,$
(64)

in which we use the ideal mixing for the axial-vector quartet in the SU(2)
case,

$\displaystyle A^{\mu}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\frac{a_{1}^{0}}{\sqrt{2}}+\frac{f_{1}}{\sqrt{2}}&a_{1}^{+}\\\
a_{1}^{-}&-\frac{a_{1}^{0}}{\sqrt{2}}+\frac{f_{1}}{\sqrt{2}}\end{array}\right]^{\mu}.$
(67)

It is hard to determine the value of $g_{a}$ through a reliable way presently.
In Ref. Yan  _et al._ (2021), Yan et al. roughly estimated the $g_{a}$ via
introducing the field of $a_{1}$ in the axial-vector current $u_{\mu}$, and
they obtained the $g_{a}$ is about one order of magnitude large than the $g$
in Eq. (11). Here, we assume the coupling satisfies the naturalness, which
amounts to setting the order of $g_{a}$ to be unity. We naively use $g_{a}=1$
in this study. For the masses of $a_{1}$ and $f_{1}$, we use Workman and
Others (2022)

$\displaystyle m_{a_{1}}=1230~{}\mathrm{MeV},\quad
m_{f_{1}}=1282~{}\mathrm{MeV}.$ (68)

In order to obtain the LECs $C_{i}$ in Eqs. (6)-(II.1), one needs to sum up
the contributions from the scalar-, pseudoscalar-, vector- and axial-vector-
exchange interactions and use the expansion

$\displaystyle\frac{g_{i}^{2}\mathcal{O}_{j}}{\bm{q}^{2}+m_{i}^{2}}=\frac{g_{i}^{2}\mathcal{O}_{j}}{m_{i}^{2}}\left(1-\frac{\bm{q}^{2}}{m_{i}^{2}}+\frac{\bm{q}^{4}}{m_{i}^{4}}+\dots\right),$
(69)

where $i=\sigma,s,\varphi,v,a$ and $j=1,2$. The $m_{i}$ denotes either the
mass of the exchanged particle or the effective mass
$\sqrt{m_{i}^{2}-\delta_{b}^{2}}$. A matching with the terms in Eqs.
(6)-(II.1), one gets

$\displaystyle C_{1}$ $\displaystyle=$
$\displaystyle-\frac{g_{a}^{2}}{2u_{f_{1}}^{2}}-\frac{g_{s}^{2}}{2m_{f_{0}}^{2}}-\frac{g_{\sigma}^{2}}{m_{\sigma}^{2}}+\frac{\beta^{2}g_{v}^{2}}{4m_{\omega}^{2}},$
(70) $\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\mathrm{sgn}\frac{g_{a}^{2}}{2u_{a_{1}}^{2}}-\frac{g_{s}^{2}}{2m_{a_{0}}^{2}}+\frac{\beta^{2}g_{v}^{2}}{4m_{\rho}^{2}},$
(71) $\displaystyle C_{3}$ $\displaystyle=$
$\displaystyle\frac{g_{a}^{2}}{2u_{f_{1}}^{4}}+\frac{g_{s}^{2}}{2m_{f_{0}}^{4}}+\frac{g_{\sigma}^{2}}{m_{\sigma}^{4}}-\frac{\beta^{2}g_{v}^{2}}{4m_{\omega}^{4}}-\frac{g_{v}^{2}\lambda^{2}}{u_{\omega}^{2}},$
(72) $\displaystyle C_{4}$ $\displaystyle=$
$\displaystyle-\mathrm{sgn}\frac{g_{a}^{2}}{2u_{a_{1}}^{4}}+\frac{g_{s}^{2}}{2m_{a_{0}}^{4}}-\frac{\beta^{2}g_{v}^{2}}{4m_{\rho}^{4}}+\mathrm{sgn}\frac{g_{v}^{2}\lambda^{2}}{u_{\rho}^{2}},$
(73) $\displaystyle C_{5}$ $\displaystyle=$
$\displaystyle-\frac{g_{\varphi}^{2}}{12f_{\eta}^{2}u_{\eta}^{2}}-\frac{g_{a}^{2}}{2m_{f_{1}}^{2}u_{f_{1}}^{2}}+\frac{g_{v}^{2}\lambda^{2}}{u_{\omega}^{2}},$
(74) $\displaystyle C_{6}$ $\displaystyle=$
$\displaystyle\mathrm{sgn}\frac{g_{a}^{2}}{2m_{a_{1}}^{2}u_{a_{1}}^{2}}-\mathrm{sgn}\frac{g_{v}^{2}\lambda^{2}}{u_{\rho}^{2}},$
(75) $\displaystyle C_{7}$ $\displaystyle=$
$\displaystyle-\frac{g_{a}^{2}}{2u_{f_{1}}^{6}}-\frac{g_{s}^{2}}{2m_{f_{0}}^{6}}-\frac{g_{\sigma}^{2}}{m_{\sigma}^{6}}+\frac{\beta^{2}g_{v}^{2}}{4m_{\omega}^{6}}+\frac{g_{v}^{2}\lambda^{2}}{u_{\omega}^{4}},$
(76) $\displaystyle C_{8}$ $\displaystyle=$
$\displaystyle\mathrm{sgn}\frac{g_{a}^{2}}{2u_{a_{1}}^{6}}-\frac{g_{s}^{2}}{2m_{a_{0}}^{6}}+\frac{\beta^{2}g_{v}^{2}}{4m_{\rho}^{6}}-\mathrm{sgn}\frac{g_{v}^{2}\lambda^{2}}{u_{\rho}^{4}},$
(77) $\displaystyle C_{9}$ $\displaystyle=$
$\displaystyle\frac{g_{\varphi}^{2}}{12f_{\eta}^{2}u_{\eta}^{4}}+\frac{g_{a}^{2}}{2m_{f_{1}}^{2}u_{f_{1}}^{4}}-\frac{g_{v}^{2}\lambda^{2}}{u_{\omega}^{4}},$
(78) $\displaystyle C_{10}$ $\displaystyle=$
$\displaystyle\mathrm{sgn}\frac{g_{v}^{2}\lambda^{2}}{u_{\rho}^{4}}-\mathrm{sgn}\frac{g_{a}^{2}}{2m_{a_{1}}^{2}u_{a_{1}}^{4}},$
(79)

where ${\rm sgn}=(-1)^{I}$ (with $I$ the total isospin of $DD^{\ast}$), and
$u_{x}=\sqrt{m_{x}^{2}-\delta_{b}^{2}}$.

###  A.2 The LECs in subleading $\pi D^{(\ast)}$ Lagrangians

In the following, we estimate the LECs in Lagrangian (33) with the RSM.

  * •

$\tilde{c}_{1}$ and $\tilde{c}_{3}$—with the $\sigma$-exchange

One easily sees that the $\tilde{c}_{1}$ and $\tilde{c}_{3}$ related terms are
connected to the $\sigma$-exchange if we write out the Lagrangians of
$\sigma\pi\pi$ coupling,

$\displaystyle\mathcal{L}_{\pi\sigma}$ $\displaystyle=$ $\displaystyle
4\bar{c}_{d}\mathrm{Tr}(u\cdot
u)\sigma+\bar{c}_{m}\mathrm{Tr}(\chi_{+})\sigma.$ (80)

Combining the vertices in Eqs. (50) and (80) and integrating out the $\sigma$
field one obtains that

$\displaystyle\tilde{c}_{1}=\frac{\bar{c}_{m}g_{\sigma}}{m_{\sigma}^{2}},~{}~{}\tilde{c}_{3}=\frac{8\bar{c}_{d}g_{\sigma}}{m_{\sigma}^{2}}=8\frac{\bar{c}_{d}}{\bar{c}_{m}}\tilde{c}_{1},~{}~{}\bar{c}_{m}\bar{c}_{d}>0,$
(81)

in which $\bar{c}_{d,m}=\frac{1}{\sqrt{2}}c_{d,m}$ Bernard  _et al._ (1997),
and $|c_{d}|=26\pm 7$ MeV, $|c_{m}|=80\pm 21$ MeV Guo and Sanz-Cillero (2009).

  * •

$\tilde{c}_{2}$—with the $D_{0}^{\ast}(2300)/D_{1}(2430)^{0}$-exchange

$\displaystyle\mathcal{L}_{\pi\mathcal{H}\mathcal{S}}$ $\displaystyle=$
$\displaystyle
h\langle\mathcal{H}\gamma^{\mu}\gamma^{5}u_{\mu}\bar{\mathcal{S}}\rangle+\mathrm{H.c.},$
(82)

where
$\mathcal{S}=\frac{1+\not{v}}{2}\left[R^{*\mu}\gamma_{\mu}\gamma_{5}-R\right]$,
and $\bar{\mathcal{S}}=\gamma^{0}\mathcal{S}^{\dagger}\gamma^{0}$. The
$R^{*\mu}$ and $R$ denote the $P$-wave $1^{+}[D_{1}(2430)^{0}]$ and
$0^{+}[D_{0}^{\ast}(2300)]$ charmed meson fields, respectively. The coupling
constant $h$ can be extracted from the partial decay widthes of
$D_{0}^{\ast}(2300)\to D\pi$ or $D_{1}(2430)\to D^{\ast}\pi$. We use
$|h|=0.52$ Casalbuoni  _et al._ (1997) in our calculations. Considering both
the $t$\- and $u$-channel contributions one gets

$\displaystyle\tilde{c}_{2}$ $\displaystyle=$
$\displaystyle-\frac{h^{2}}{\Delta},$ (83)

where $\Delta=m_{D_{0}^{\ast}(2300)}-m_{D}\simeq 330$ MeV denotes the mass
difference of $D_{0}^{\ast}(2300)$ and $D$ mesons.

  * •

$\tilde{c}_{4}$—with the $\rho$-exchange

In addition to the $\rho\mathcal{H}$ coupling in the second term of Eq. (59),
we also need the $\rho\pi\pi$ Lagrangian, which reads Casalbuoni  _et al._
(1997)

$\displaystyle\mathcal{L}_{\pi\rho}$ $\displaystyle=$ $\displaystyle-
af_{\pi}^{2}\mathrm{Tr}[(\Gamma^{\mu}-\rho_{\mu})^{2}],\qquad a=2.$ (84)

Combining the vertices in Eqs. (59) and (84) and integrating out the $\rho$
field one gets

$\displaystyle\tilde{c}_{4}=-\frac{2\lambda
g_{V}^{2}f_{\pi}^{2}}{m_{\rho}^{2}}.$ (85)

  * •

$\tilde{c}_{5}$—with the mass splittings of the neutral and charged
$D^{(\ast)}$ mesons

The $\tilde{c}_{5}$-term is related to the isospin breaking considering the
$\hat{\chi}_{+}=2B_{0}\mathrm{diag}(m_{u}-m_{d},m_{d}-m_{u})$, with $m_{u,d}$
the masses of $u,d$ quarks. We first write out the relativistic Lagrangians of
$D$ and $D^{\ast}$, which read

$\displaystyle\mathcal{L}_{\mathcal{H}}^{\mathrm{rel}}$ $\displaystyle=$
$\displaystyle\mathcal{D}_{\mu}P\mathcal{D}^{\mu}P^{\dagger}-m^{2}_{0}PP^{\dagger}$
(86)
$\displaystyle-\mathcal{D}_{\mu}P^{\ast\nu}\mathcal{D}^{\mu}P_{\nu}^{\ast\dagger}+m^{2}_{0^{\ast}}P^{\ast\nu}P_{\nu}^{\ast\dagger},$

where $m_{0}$ and $m_{0^{\ast}}$ are the bare masses of $D$ and $D^{\ast}$,
respectively. Here, we ignore the electromagnetic interactions and assume the
mass splittings of the neutral and charged $D^{(\ast)}$ mesons come from the
mass difference of $u,d$ quarks. Then we have

$\displaystyle-m_{D^{0}}^{2}$ $\displaystyle=$ $\displaystyle-
m_{0}^{2}-4B_{0}\tilde{c}_{5}(m_{u}-m_{d})m_{D},$ $\displaystyle-
m_{D^{+}}^{2}$ $\displaystyle=$ $\displaystyle-
m_{0}^{2}-4B_{0}\tilde{c}_{5}(m_{d}-m_{u})m_{D},$ $\displaystyle-m_{D^{\ast
0}}^{2}$ $\displaystyle=$ $\displaystyle-
m_{0^{\ast}}^{2}-4B_{0}\tilde{c}_{5}(m_{u}-m_{d})m_{D^{\ast}},$
$\displaystyle-m_{D^{\ast+}}^{2}$ $\displaystyle=$ $\displaystyle-
m_{0^{\ast}}^{2}-4B_{0}\tilde{c}_{5}(m_{d}-m_{u})m_{D^{\ast}}.$ (87)

With these equations we finally get

$\displaystyle\tilde{c}_{5}=\frac{m_{D^{0}}^{2}-m_{D^{+}}^{2}+m_{D^{\ast
0}}^{2}-m_{D^{\ast+}}^{2}}{16\hat{m}_{D}(m_{K^{+}}^{2}-m_{K^{0}}^{2})},$ (88)

where $\hat{m}_{D}=\frac{m_{D}+m_{D^{\ast}}}{2}$, and we have used
$m_{K^{+}}^{2}=B_{0}(m_{u}+m_{s})$ and $m_{K^{0}}^{2}=B_{0}(m_{d}+m_{s})$.

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