#  $T_{cc}$ in the Diabatic Diquark Model: Effects of $D^{*}D$ Isospin

Richard F. Lebed<EMAIL_ADDRESS>Steven R. Martinez<EMAIL_ADDRESS>Department of Physics, Arizona State University, Tempe, AZ 85287, USA

(June, 2024)

###### Abstract

$T_{cc}^{+}$ is an isoscalar 4-quark state with mass lying barely below the
$D^{*+}D^{0}$ threshold, and several times further below the $D^{*0}D^{+}$
threshold. It allows both di-meson molecular and elementary diquark-
antidiquark $(cc)(\bar{u}\bar{d})$ substructures. The diabatic generalization
of the adiabatic approximation within the Born-Oppenheimer formalism
rigorously incorporates the mixing of such elementary eigenstates with states
corresponding to two-particle thresholds. We examine the separate influence of
the two $D^{*}\\!D$ isospin channels and find that the influence of
$D^{*+}D^{0}$ is larger than that of $D^{*0}D^{+}$ but not overwhelmingly so,
and that $T_{cc}^{+}$ contains an $O(10\%)$ $(cc)(\bar{u}\bar{d})$ component.
We then explore the variation of these results if the isospin breaking between
the di-meson thresholds is varied, and also the sensitivity of our results to
variation of the mixing-potential parameters.

Exotic hadrons, diquarks

##  I Introduction

Even among the $>60$ heavy-quark exotics already observed, the state
$T_{cc}^{+}$ discovered by LHCb [1, 2] stands out as unique. First, it is the
only exotic state observed to date with open heavy flavor
($cc\bar{u}\bar{d}$). It lies only a few hundred keV below the $D^{*+}D^{0}$
threshold, and it has the smallest width of any hadron that decays strongly
through non-Okubo-Zweig-Iizuka (OZI)-suppressed modes (the only available
open-charm channels being $D^{+}\\!D^{0}\pi^{0}$ and its discovery mode
$D^{0}D^{0}\pi^{+}$). Its measured parameters are [3]:

$\displaystyle m_{T_{cc}^{+}}$ $\displaystyle=$ $\displaystyle\ 3874.83\pm
0.11\ {\rm MeV}\,,$ $\displaystyle m_{T_{cc}^{+}}-m_{D^{*+}}-m_{D^{0}}$
$\displaystyle=$ $\displaystyle\ \ \ \ -270\pm\ \hphantom{1}60\ {\rm keV}\,,$
$\displaystyle\Gamma_{T_{cc}^{+}}$ $\displaystyle=$ $\displaystyle\ \ \ \
\hphantom{+}410\pm\ 170\ {\rm keV}\,.$ (1)

In comparison, the famous $\chi_{c1}(3872)$ [or $X(3872)$] lies at almost the
same mass, extremely close to the $\bar{D}^{*0}\\!D^{0}$
($\\!=\\!\\!D^{*0}\bar{D}^{0}$) threshold, but has a larger width, chiefly due
to its hidden-charm content permitting charmonium decays [$J/\psi$,
$\chi_{c1}(1P)$, and $\psi(2S)$ modes being observed to date], as well as
open-charm decays [$D^{0}\bar{D}^{0}\pi^{0}$ and $\bar{D}^{*0}D^{0}$]:

$\displaystyle m_{\chi_{c1}(3872)}$ $\displaystyle=$ $\displaystyle 3871.65\pm
0.06\ {\rm MeV}\,,$ $\displaystyle
m_{\chi_{c1}(3872)}-m_{\bar{D}^{*0}}-m_{D^{0}}$ $\displaystyle=$
$\displaystyle\ \ \ \ -40\pm\ \ \ 90\ {\rm keV}\,,$
$\displaystyle\Gamma_{\chi_{c1}(3872)}$ $\displaystyle=$ $\displaystyle\ \ \
1190\pm\ \ 210\ {\rm keV}\,.$ (2)

In both cases, the isospin partner to the di-meson threshold nearest to the
resonance lies somewhat higher:111In calculating these values, we have used
the three best-determined independent mass splittings [3] among the $D^{(*)}$
mesons: $\left(m_{D^{\pm}}-m_{D^{0}}\right)=4.822(15)~{}{\rm MeV}$,
$\left(m_{D^{*+}}-m_{D^{0}}\right)=145.4258(17)~{}{\rm MeV}$, and
$\left(m_{D^{*0}}-m_{D^{0}}\right)=142.014(30)~{}{\rm MeV}$.

$\displaystyle\left(m_{D^{*0}}+m_{D^{+}}\right)-\left(m_{D^{*+}}+m_{D^{0}}\right)$
$\displaystyle=$ $\displaystyle 1.411\pm 0.034\ {\rm MeV}\,,$
$\displaystyle\left(m_{D^{*-}}+m_{D^{+}}\right)-\left(m_{\bar{D}^{*0}}+m_{D^{0}}\right)$
$\displaystyle=$ $\displaystyle 8.234\pm 0.034\ {\rm MeV}\,.$

The difference is significantly smaller for the $T_{cc}$ case than the
$\chi_{c1}(3872)$ case, suggesting that the structure of $T_{cc}$ is more
strongly influenced than $\chi_{c1}(3872)$ by isospin-dependent di-meson
threshold effects, while $\chi_{c1}(3872)$ has significant coupled-channel
charmonium decay effects (indicated by its larger decay width) that are
suppressed for $T_{cc}$, a point first clearly stated and explored in Ref.
[4].

Both $T^{+}_{cc}$ and $\chi_{c1}(3872)$ are isoscalar states; searches for
isospin (charge) partners (Refs. [2] and [5, 6], respectively) yield no
significant signals. However, $\chi_{c1}(3872)$ is observed to decay to
(indeed, was discovered in [7]) the channel $J/\psi\,\pi^{+}\pi^{-}$. The
$I=0$ $\chi_{c1}(3872)$ has $J^{PC}=1^{++}$ [8] and hence $G\equiv
C(-1)^{I}=+$, while the $I=0$ $J/\psi$ has $J^{PC}=1^{--}$ and hence $G=-$.
Then, by $C$-conservation in strong decays, the $\pi^{+}\pi^{-}$ pair ($G=+$)
must have $C=-$, and hence $I=1$; thus, $\chi_{c1}(3872)$ exhibits some
isospin-violating decay modes. For $T_{cc}^{+}$, $J^{P}=1^{+}$ is heavily
favored due both to the $S$-wave quantum number of the $D^{*+}D^{0}$ threshold
pair and to the absence of a signal in the $D^{+}\\!D^{0}$ channel (since two
$0^{-}$ mesons cannot form a $1^{+}$ in any partial wave).

While the di-meson pairs $D^{*0}\bar{D}^{0}$ and $D^{*+}D^{0}$ in their $S$
waves naturally provide the $J^{P}=1^{+}$, $I=0$ quantum numbers proven or
heavily favored for $\chi_{c1}(3872)$ and $T_{cc}^{+}$, respectively, these
composite quasi-molecular combinations do not represent the only natural
substructure capable of explaining the states. In addition, the diquark
combinations $(cq)_{\bar{\bf 3}}(\bar{c}\bar{q})^{\vphantom{X}}_{\bf 3}$ and
$(cc)_{\bar{\bf 3}}(\bar{u}\bar{d})^{\vphantom{X}}_{\bf 3}$ (color triplets
being the most attractive diquark combination) each naturally produce a
spectrum of states that includes $J^{P}=1^{+}$. These “elementary” diquark
combinations (in contrast to the composite di-meson components) provide a
natural alternative component for the full states.222And of course, the
conventional charmonium state $\chi_{c1}(2P)$ can contribute to
$\chi_{c1}(3872)$—an added complication for this state.

The complete spectrum of purely elementary diquark combinations
$(Qq)_{\bar{\bf 3}}(\bar{Q}\bar{q})_{\bf 3}$, where $Q$ denotes a heavy quark,
has been studied in the context of the dynamical diquark model [9, 10, 11, 12,
13, 14]. A key ingredient of this model is the presence of configurations in
which the diquark states do not always instantaneously reorganize into di-
meson $(Q\bar{q})(\bar{Q}q)$ combinations, which in this model is realized
through components of the 4-quark configuration for which the relative
momentum of the $(Qq)$ and ($\bar{Q}\bar{q}$) diquarks is larger than that
within either diquark. Each heavy quark $Q$,$\bar{Q}$ then serves to nucleate
a spatially localized diquark quasiparticle (due to its small Fermi momentum),
thus producing a configuration that can be distinguished from a di-meson pair.

In contrast, for $(Q_{1}Q_{2})(\bar{q}_{1}\bar{q}_{2})$ combinations like
those relevant to $T_{cc}$, the mechanism of nucleating the
$(\bar{q}_{1}\bar{q}_{2})$ diquark using a heavy quark is no longer available.
In this case, long-established phenomenology [15], as well as more recent
lattice simulations [16], assert that the $(\bar{u}\bar{d})_{\bf 3}$ diquark
in its “good” $S=0$, $L=0$, $I=0$ channel is the most tightly bound of all the
possibilities, while the $(cc)_{\bar{\bf 3}}$ diquark (antisymmetric in color)
is not only compact due to the small Fermi momentum of the two heavy quarks,
but in its ground state has $S=1$ and $L=0$ (i.e., is symmetric in spin and
space) in order to satisfy Fermi statistics. Assuming lastly that the ground
state of the $(cc)(\bar{u}\bar{d})$ combination also occurs in a relative
$S$-wave between the diquarks, then the state has the unique quantum numbers
$J^{P}=1^{+}$, $I=0$, exactly as appears to be true for $T_{cc}^{+}$.

Numerous calculations performing coupled-channel analyses of exotic hadrons
appear in the literature, including for $T_{cc}^{+}$. Analyses including both
molecular and elementary components for $T_{cc}^{+}$ include Refs. [4, 17, 18,
19, 20, 21, 22, 23]. In this work, we first consider $T_{cc}^{+}$ as an
elementary $(cc)_{\bar{\bf 3}}(\bar{u}\bar{d})_{\bf 3}$ state initially
interacting through a color-triplet static potential—i.e., using the Born-
Oppenheimer (BO) approximation—and then employ the rigorous diabatic formalism
[24] that generalizes the adiabatic formalism inherent in the BO
approximation, in order to include the effects of coupled di-meson channels on
the state. The diabatic approach was applied for the first time in hadronic
physics to treat exotic heavy-quark hadrons as mixtures of quarkonium with di-
meson states in Ref. [25], and later as mixtures of diquark-antidiquark states
with di-meson states in Ref. [26]. Very recently, the diabatic formalism was
generalized [27] to incorporate the combined effects of channels with distinct
di-meson quantum numbers (in particular, spin).

In the case of $T_{cc}^{+}$, the most important thresholds are of course the
two isospin channels $D^{*+}D^{0}$ and $D^{*0}D^{+}$, but a complete analysis
would also include their heavy-quark spin partner $D^{*+}D^{*0}$,
approximately 140 MeV higher. In this initial study, only the lower channels
are included, as our goal is to determine the separate effect of each of these
channels on the structure of the state. That is, we analyze the effect of not
only the nearest $D^{*+}D^{0}$ threshold, but also the effect of its
$D^{*0}D^{+}$ isospin partner about 1.4 MeV higher. As noted above,
$T_{cc}^{+}$ provides a cleaner laboratory than $\chi_{c1}(3872)$ for
examining the effects of isospin breaking, and we study its consequences both
for the observed value of isospin breaking, and also parametrically as this
number is varied.

This paper is organized as follows. In Sec. II we discuss the diquark
configuration relevant to $T_{cc}^{+}$ and other
$(Q_{1}Q_{2})(\bar{q}_{1}\bar{q}_{2})$ states in the BO approximation. Section
III presents a brief review of the diabatic formalism in its original form for
hadrons, while Sec. IV presents improvements to the formalism designed to
incorporate distinct spin and/or flavor di-hadron thresholds. In Sec. V we
analyze the $T_{cc}^{+}$ system in the diabatic formalism including both
$D^{*+}D^{0}$ and $D^{*0}D^{+}$ channels, and in Sec. VI we present our
conclusions and indicate directions for further improvements.

##  II Diquark Model for $T_{cc}^{+}$

As noted in the Introduction, the state $T_{cc}^{+}$—assuming the confirmation
of its $I=0$, $J^{P}=1^{+}$ quantum numbers—admits two natural substructures
to accommodate its valence-quark $cc\bar{u}\bar{d}$ content: As a $D^{*}D$
molecule (possibly with an admixture of $D^{*}D^{*}$), and as a bound state of
a compact color-$\bar{\bf 3}$ $(cc)$ diquark and a “good” (color 3, $I=0$,
$S=0$) $(\bar{u}\bar{d})$ antidiquark. While the extreme proximity of the
$D^{*+}D^{0}$ threshold [Eq. (1)] is quite suggestive of this di-meson
substructure, it does not rule out the possibility of a significant diquark
component. Moreover, even a small diquark component in the $T_{cc}^{+}$ state
does not indicate that the diquark structure is unimportant; as shown in the
diabatic formalism in Refs. [26, 28] to be discussed in Sec. III, a
$(cq)(\bar{c}\bar{q})$ diquark structure (where $q$ is a light quark) with
$I=0$ can easily serve as a “seed” for generating the famous $\chi_{c1}(3872)$
that lies so close to the $\bar{D}^{*0}\\!D^{0}$ threshold [Eq. (2)].

Indeed, the best corroborating evidence for the presence of a significant
$(cc)(\bar{u}\bar{d})$ component with a “good” diquark would be the absence of
a prominent state in a channel that relies upon a light diquark with “bad”
quantum numbers, such as $(\bar{u}\bar{d})$ with $S=1$ or $I=1$: i.e.,
$T_{cc}^{+}$ with $J^{P}=2^{+}$ or a $T_{cc}^{++}$ partner; or of a strange
analogue such as the charge-$2$ $(cc)(\bar{s}\bar{s})$. Since both
phenomenology [15] and recent lattice calculations [16] indicate a
“bad”-to-“good” light-diquark mass difference of at least 200 MeV, then no
additional $T_{cc}$ states much below 4100 MeV should arise under the
assumptions of this model.333For a differing view, see Ref. [29]. Additional
experimental evidence for this conclusion is provided by the Belle
Collaboration, which as yet has seen no evidence for $D_{s}^{+}D_{s}^{+}$ or
$D_{s}^{*+}D_{s}^{*+}$ resonances in $\Upsilon(1S)$ or $\Upsilon(2S)$ decays
[30].

Another way to visualize the exceptional nature of the state $T_{cc}^{+}$ is
by examining the full ground-state multiplet of tetraquark states in a
diquark-antidiquark ($\delta^{\vphantom{1}}$-${\bar{\delta}}$) picture. Self-
conjugate [$(Qq)(\bar{Q}\bar{q})$] states produce the spectrum [31, 10]:

$\displaystyle J^{PC}=0^{++}:$ $\displaystyle
X_{0}\equiv\left|0_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{0}\,,\ \
X_{0}^{\prime}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{0}\,,$
$\displaystyle J^{PC}=1^{++}:$ $\displaystyle
X_{1}\equiv\frac{1}{\sqrt{2}}\left(\left|1_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{1}\\!+\left|0_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\right)\,,$
$\displaystyle J^{PC}=1^{+-}:$ $\displaystyle\,Z\
\equiv\frac{1}{\sqrt{2}}\left(\left|1_{\delta}^{\vphantom{1}},0_{\bar{\delta}}\right>_{1}\\!-\left|0_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\right)\,,$
$\displaystyle\,Z^{\prime}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{1}\,,$
$\displaystyle J^{PC}=2^{++}:$ $\displaystyle
X_{2}\equiv\left|1_{\delta}^{\vphantom{1}},1_{\bar{\delta}}\right>_{2}\,.$ (4)

Here, the $\delta^{\vphantom{1}}({\bar{\delta}})$ spin is denoted by
$s^{\vphantom{\dagger}}_{\delta^{\vphantom{1}}}(s_{{\bar{\delta}}})$, and the
total state spin is designated by the outer subscript. In the present case,
$\delta^{\vphantom{1}}{\bar{\delta}}\\!=\\!(cc)(\bar{q}_{1}\bar{q}_{2})$
states are not self-conjugate, and therefore the $C$-parity quantum number is
lost. Furthermore, we have seen that $\delta^{\vphantom{1}}\\!=\\!(cc)$ in its
ground state gives $s_{\delta}^{\vphantom{1}}\\!=\\!1$; and since in addition
the “good” diquark ${\bar{\delta}}=(\bar{u}\bar{d})$ has
$s_{\bar{\delta}}\\!=\\!0$, then all states in Eqs. (4) disappear except for
the first components of $X_{1}$ and $Z$, which coalesce to leave a single
$J^{P}\\!=1^{+}$ state.

The Introduction also notes that the dynamical diquark model in its original
form is not applicable for doubly heavy open-flavor hadrons like $T_{cc}$,
because one of the two quasiparticle components (here, ${\bar{\delta}}$) lacks
a heavy quark, and hence the system has no typical configuration in which the
$\delta^{\vphantom{1}}$ and ${\bar{\delta}}$ components can be described as
spatially separated. In fact, the most natural diquark picture for
$T_{cc}^{+}$ much more closely resembles hadrocharmonium [32, 33], in which
the heavy quarks form a compact nucleus embedded within the light-quark cloud,
except that for $T_{cc}^{+}$ the “nucleus” is color-$\bar{\bf 3}$, and
${\bar{\delta}}$ is color-3. Directly using a 3-$\bar{\bf 3}$ interaction such
as the Cornell potential [34, 35],

$V(r)=-\frac{a}{r}+br\,,$ (5)

or a lattice simulation of the potential between two heavy colored sources
[36, 37, 38, 39] is highly questionable in the current circumstances. Instead,
here we model the state by supposing that the potential of Eq. (5) is valid
for pointlike sources, and then treat the diquark $(cc)$ as pointlike, but
regard the light diquark $(\bar{u}\bar{d})$ as a sphere with uniform density
for $0<r<R$, with $R$ given by its root-mean-square radius
$\left<r^{2}\right>_{(ud)}^{1/2}$ obtained from phenomenology or lattice
simulations. (Of course, one could choose to model ${\bar{\delta}}$ using any
other specific wave-function profile.) With this modification, $r$ indicates
the position of $(cc)$ with respect to the center of the $(\bar{u}\bar{d})$
wave function, and the full potential in this case is then straightforward to
compute:

$V(r)=\begin{cases}-\left(a-\frac{bR^{2}}{5}\right)\frac{1}{r}+br,&r\geq R,\\\
+\frac{a}{2R^{3}}\left(r^{2}-3R^{2}\right)\\\ \hskip
5.0pt-\frac{b}{20R^{3}}\left(r^{4}-10R^{2}r^{2}-15R^{4}\right),&r\leq
R.\end{cases}$ (6)

This expression, when restricted to $b\\!=\\!0$, is the textbook result
obtained from applying Gauss’ law to a uniform-density sphere in the $1/r$
potential of electrostatics or gravity. The analysis of the confining
($b\\!\neq\\!0$) part of the potential is slightly more complicated, since in
that case the potential at a given value of $r$ depends upon the effects of
sources both inside and outside the sphere of radius $r$. One finds,
interestingly, that the strength of the Coulomb term decreases outside the
sphere for any value of $R$.

The general form for the elementary $\delta^{\vphantom{1}}$-${\bar{\delta}}$
potential used in this work is the same as in Refs. [26, 28],

$V_{\delta^{\vphantom{1}}{\bar{\delta}}}(r)=-\frac{\alpha}{r}+\sigma
r+V_{0}+m_{\delta^{\vphantom{1}}}+m_{{\bar{\delta}}}\,,$ (7)

and uses lattice-determined values [40] of $\alpha,\sigma,$ and $V_{0}$:

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle 0.053\
\rm{GeV}\\!\cdot\rm{fm},$ $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle 1.097\ \rm{GeV\\!/fm},$ $\displaystyle V_{0}$ $\displaystyle=$
$\displaystyle-0.380\ \rm{GeV}.$ (8)

Of course, the parameters $a,b$ of the original Cornell potential Eq. (5) are
directly replaced in the current potential Eq. (7) with $\alpha,\sigma$,
respectively, and the specific value listed for $V_{0}$ is only taken as a
starting point for fits.

As for values of diquark masses and $\left<r^{2}\right>_{(ud)}^{1/2}$, one may
resort to results from phenomenology, QCD sum rules, or lattice calculations.
Values of $m_{(ud)}$ are broadly consistent; listing results from three papers
that provide uncertainties, we find $m_{(ud)}=640\pm 60$ MeV [41] (QCD sum
rules); $694\pm 22$ MeV [42] and $690\pm 47$ MeV [43] (lattice). For
definiteness, we use the last (most recent) of these determinations, which
employs an unquenched simulation. Values of $\left<r^{2}\right>_{(ud)}^{1/2}$
are less commonly presented; here we start with the estimate of 0.6 fm
presented in a rather recent lattice simulation [16]. Lastly, values of
$m_{(cc)}$ vary widely; here one finds results such as $3510\pm 350$ MeV [44]
(QCD sum rules); $3306.2$ MeV [45] and $3136\pm 10$ MeV [46] (phenomenology).
Due to this large spread, here we simply use the measured mass value of
$T_{cc}^{+}$ to fit for $m_{(cc)}$, and then study the results as the
parameters $m_{(ud)}$, $\left<r^{2}\right>_{(ud)}^{1/2}$, and $m_{(cc)}$ are
varied.

Lastly, note that the $(\bar{u}\bar{d})$ diquark itself contains no heavy
quarks, but its mass is nevertheless substantially larger than $\Lambda_{\rm
QCD}$ (and indeed is about the same as that of a pair of constituent light
quarks in a typical phenomenological quark model). For the purposes of these
calculations, we treat $(\bar{u}\bar{d})$ as a heavy—but not pointlike—source
in the BO approximation. In order for the BO formalism to be relevant here,
the light degrees of freedom (d.o.f.) associated with the potential $V(r)$
must be able to adjust quickly to changes in the configuration of the heavy
sources $(cc)$ and $(\bar{u}\bar{d})$. We take as evidence for this criterion
to be satisfied that typical values of the potential $V(r)$ for the state must
be small compared to $m_{(ud)}$, and we find explicitly that this requirement
is satisfied in our calculations.

##  III The Diabatic Formalism

The modeling of an exotic 4-quark hadron solely by a
$\delta^{\vphantom{1}}{\bar{\delta}}$ pair interacting through a potential
$V(r)$, such as in the context of the BO approximation, intrinsically neglects
the effects of coupling to di-meson thresholds. A coupled-channel formalism is
clearly required if one wishes to incorporate this important dynamical source.
While coupled-channel calculations are nothing new in the literature, they can
often seem rather ad hoc in their implementation. The BO approximation,
however, possesses a rigorous generalization called the diabatic formalism
that has become a standard, textbook approach in the context of atomic and
molecular physics [24]. It was first applied in the context of hadronic
physics relatively recently [25], in order to study the coupling of heavy
quarkonium to exotic hadrons with the same $J^{PC}$ quantum numbers. The first
introduction of $\delta^{\vphantom{1}}{\bar{\delta}}$ degrees of freedom
coupled to the di-meson thresholds followed in Ref. [26]. The diabatic
formalism was generalized to perform direct studies of the scattering
amplitudes in which the quarkoniumlike states appear in Ref. [47] for
$Q\bar{Q}$ states, and in Ref. [28] for $\delta^{\vphantom{1}}{\bar{\delta}}$
states. Calculations of mass shifts and strong decay widths induced by the
couplings to the di-meson thresholds were investigated for $Q\bar{Q}$ states
in Ref. [48] and for $\delta^{\vphantom{1}}{\bar{\delta}}$ states in Ref.
[49]. Since the diabatic approach is described in all of these papers, here we
present only a brief summary.

One begins with a Hamiltonian for a system of two heavy color sources
interacting through light fields:

$H=K_{\rm heavy}+H_{\rm light}=\frac{\mathbf{p}^{2}}{2\mu_{\rm heavy}}+H_{\rm
light}.$ (9)

Here, $H_{\rm light}$ contains both the light-field static energy and the
heavy-light interaction. Defining $\mathbf{r}$ as the separation vector for
the heavy-source pair (with corresponding eigenstates $|\mathbf{r}\rangle$)
and $|\xi_{i}(\mathbf{r})\rangle$ as the $i^{\rm th}$ eigenstate of $H_{\rm
light}$ with eigenvalue $E_{i}$, one may expand the solutions to the
corresponding Schrödinger equation as:

$|\psi\rangle=\sum_{i}\int
d\mathbf{r}\,\tilde{\psi}_{i}(\mathbf{r})\,|\mathbf{r}\rangle\>|\xi_{i}(\mathbf{r})\rangle.$
(10)

While this expansion already suggests the decoupling separation of the BO
approximation between heavy and light d.o.f., it still can be used in the
general case. The set $\\{|\xi_{i}(\mathbf{r})\rangle\\}$ forms a complete,
orthonormal basis for the light d.o.f. at any given $\mathbf{r}$, but
configuration mixing can be permitted at distinct values of $\mathbf{r}$:
$\langle\xi_{j}(\mathbf{r}^{\prime})|\xi_{i}(\mathbf{r^{\vphantom{\prime}}})\rangle\neq
0$ in general, even for $j\neq i$. The full BO approximation then consists of
two assumptions: $i$) the light d.o.f.’s in a given ($i^{\rm th}$) eigenstate
instantaneously (adiabatically) adapt to small changes
$\mathbf{r}^{\prime}\\!\\!\neq\\!\mathbf{r}$ in the heavy-source separation:
$\langle\xi_{i}(\mathbf{r}^{\prime})|\xi_{i}(\mathbf{r^{\vphantom{\prime}}})\rangle\approx
1$ (the adiabatic approximation); and $ii$) at comparable
$\mathbf{r},\mathbf{r}^{\prime}$ values, distinct light-field eigenstates do
not appreciably mix:
$\langle\xi_{j}(\mathbf{r}^{\prime})|\xi_{i}(\mathbf{r})\rangle\approx 0$ for
$j\neq i$, the single-channel approximation.

The rigorous generalization of the BO approximation to allow for the lifting
of these assumptions is called the diabatic formalism [24]. One introduces a
free parameter $\mathbf{r}_{0}$ and rewrites the expansion of the solution Eq.
(10) as:

$|\psi\rangle=\sum_{i}\int
d\mathbf{r}^{\prime}\tilde{\psi}_{i}(\mathbf{r}^{\prime}\\!,\mathbf{r}_{0})\>|\mathbf{r}^{\prime}\rangle\>|\xi_{i}(\mathbf{r}_{0})\rangle.$
(11)

Exploiting the completeness of the basis
$\\{|\xi_{i}(\mathbf{r}_{0})\rangle\\}$ for any specific value
$\mathbf{r}_{0}$, inserting the expansion Eq. (11) into the Schrödinger
equation for the Hamiltonian Eq. (9), and projecting onto
$\langle\xi_{j}(\mathbf{r}_{0})|$, one obtains:

$\sum_{i}\left[-\frac{\hbar^{2}}{2\mu_{i}}\delta^{\vphantom{1}}_{ji}\nabla^{2}+V_{ji}(\mathbf{r},\mathbf{r}_{0})-E^{\vphantom{\dagger}}_{i}\delta^{\vphantom{1}}_{ji}\right]\\!\tilde{\psi}_{i}(\mathbf{r},\mathbf{r}_{0})=0.$
(12)

The key development is the introduction of the diabatic potential matrix
$V_{ji}$, defined as:

$V_{ji}(\mathbf{r},\mathbf{r}_{0})\equiv\langle\xi_{j}(\mathbf{r}_{0})|H_{\rm
light}|\xi_{i}(\mathbf{r}_{0})\rangle.$ (13)

The parameter $\mathbf{r}_{0}$ may be chosen as any source separation that
gives a value of energy lying far from a potential-energy level crossing, in
which case the states $|\xi_{i}(\mathbf{r}_{0})\rangle$ are unambiguously
identified with pure, unmixed configurations identifiable with a single value
of $i$. Meanwhile, $H_{\rm light}$ still references the original source
separation $\mathbf{r}$.

Starting with an initial configuration of unmixed
$\delta^{\vphantom{1}}{\bar{\delta}}$ states and then introducing di-meson
states, the diagonal elements of $V$ represent the static light-field energy
$V_{\delta^{\vphantom{1}}{\bar{\delta}}}$ associated with a pure
$\delta^{\vphantom{1}}{\bar{\delta}}$ state ($V_{11}$), followed by the
potentials $V_{j+1,j+1}=V_{M_{1}\overline{M}_{2}}^{(j)}$, $j=1,2,\ldots,N$, of
the $N$ corresponding di-meson thresholds $(M_{1},\overline{M}_{2})^{(j)}$.
The explicit form of the potential matrix then reads:

$\text{V}=\begin{pmatrix}V_{\delta^{\vphantom{1}}{\bar{\delta}}}(\mathbf{r})&V_{\rm
mix}^{(1)}(\mathbf{r})&\cdots&V_{\rm
mix\vphantom{\overline{M}_{2}}}^{(N)}(\mathbf{r})\\\ V_{\rm
mix}^{(1)}(\mathbf{r})&V_{M_{1}\overline{M}_{2}}^{(1)}(\mathbf{r})&&\\\
\vdots&&\ddots\\\ V_{\rm
mix\vphantom{\overline{M}_{2}}}^{(N)}(\mathbf{r})&&&V_{M_{1}\overline{M}_{2}}^{(N)}(\mathbf{r})\\\
\end{pmatrix}.$ (14)

Note that we neglect direct mixing terms between any two di-meson
configurations by setting the suppressed elements to zero. Furthermore, for
simplicity we set each pure di-meson energy to equal the free energy of the
state:

$V_{M_{1}\overline{M}_{2}}^{(j)}(\mathbf{r})\to
T_{M_{1}\overline{M}_{2}}^{(j)}=M_{1}^{(j)}+M_{2}^{(j)}\,,$ (15)

although one could of course introduce explicit direct interactions between
different di-meson configurations $(j)$, or between the two mesons
$(M_{1}^{(j)},\overline{M}_{2})^{(j)}$ within any such configuration.

##  IV Distinct Diabatic Thresholds

Until very recently, all applications of the diabatic approach to heavy-quark
hadrons have assumed that each di-meson channel coupling to the elementary
($Q\bar{Q}$ or $\delta^{\vphantom{1}}{\bar{\delta}}$) state has the same
functional form and the same coupling. In the
$\delta^{\vphantom{1}}{\bar{\delta}}$ example,

$|V_{\rm
mix}^{(i)}(r)|=\frac{\Delta}{2}\exp\\!\left\\{-\frac{1}{2}\frac{\left[V^{\vphantom{\dagger}}_{\delta^{\vphantom{1}}{\bar{\delta}}}(r)-T_{M_{1}\overline{M}_{2}}^{(i)}\right]^{2}}{\Lambda^{2}}\right\\},$
(16)

specifically using the same value of $\Delta$ and $\Lambda$ for all channels.
Of course, this Ansatz falls short of elementary expectations even in the
heavy-quark limit, where channels composed of various hadron pairs (e.g.,
$D\bar{D}$ vs. $D\bar{D}^{*}$) must differ not only in the mass of the channel
(incorporated through $T_{M_{1}\overline{M}_{2}}^{(i)}$), but also in the spin
states of the component hadrons. The formalism for implementing this
improvement was developed in Ref. [27]; and while its original form refers to
$Q\bar{Q}$ states for which heavy-quark $CP$ is a good quantum number, it can
also be applied to $QQ$ states such as $T_{cc}^{+}$ for which only the $P$
eigenvalue of the heavy-quark pair is a good quantum number.

One of the essential ingredients for incorporating spin dependence into the
diabatic formalism is the specification of the overlap of the elementary state
with di-meson interpolating operators carrying the same quantum numbers. In
practice, these overlaps are obtained via a Fierz reordering of
$(\bar{Q}\Gamma_{1}q)(\bar{q}\,\Gamma_{2}Q)$ operators, where $\Gamma_{i}$ are
Dirac structures, into the form
$(\bar{Q}\Gamma^{\prime}_{1}Q)(\bar{q}\,\Gamma^{\prime}_{2}q)$. In the
original analysis of Ref. [27], the most interesting
$(\bar{Q}\Gamma_{1}q)(Q\Gamma_{2}\bar{q})$ operators are those that have the
same quantum numbers as pure $(\bar{Q}\Gamma^{\prime}_{1}Q)$ configurations,
and hence mix with heavy quarkonium. On the other hand,
$\delta^{\vphantom{1}}{\bar{\delta}}$ configurations, already possessing
$(Q\Gamma_{1}^{\prime\prime}q)(\bar{Q}\Gamma_{2}^{\prime\prime}\bar{q})$
structure, present additional opportunities for nonzero overlaps with
$(\bar{Q}\Gamma_{1}q)(\bar{q}\,\Gamma_{2}Q)$ operators [50].

In the present case of open heavy flavor, and specifically for the $1^{+}$
state $T_{cc}^{+}$, we require a Fierz reordering of the simplest $1^{+}$
current $J^{\mu}$ one can construct for $(cc)(\bar{u}\bar{d})$. The explicit
form of this operator, as well as its decomposition in terms of $1^{+}$
$(\bar{u}c)(\bar{d}c)$ operators, first appeared in Ref. [51], and is
presented explicitly in Appendix A. Significantly, although the current
$J^{\mu}$ is not a scalar, the derivation of its Fierz reordering directly
uses the original Fierz reordering theorem in its proof.444We thank Z.-G. Wang
for this insight. The relevant current and the subset of its couplings to
interpolating operators with the quantum numbers of a $(0^{-})(1^{-})$ di-
meson pair, as seen from Eq. (LABEL:eq:Fierz), reads:

$\displaystyle J^{\mu}(x)$ $\displaystyle=$
$\displaystyle\epsilon^{ijk}\epsilon^{imn}\bar{Q}^{c}_{j}\gamma^{\mu}Q_{k}\,\frac{1}{\sqrt{2}}\left(\bar{u}_{m}\,i\gamma^{5}d^{c}_{n}-\bar{d}_{m}\,i\gamma^{5}u^{c}_{n}\right)$
$\displaystyle\supset$
$\displaystyle-\frac{1}{\sqrt{2}}(\bar{u}i\gamma^{5}Q)(\bar{d}\gamma^{\mu}Q)+\frac{1}{\sqrt{2}}(\bar{d}i\gamma^{5}Q)(\bar{u}\gamma^{\mu}Q).$

Noting that the bilinears in the last line of this expression for $Q=c$
provide the simplest interpolating operators for meson pairs $D^{0}D^{*+}$ and
$D^{+}D^{*0}$, respectively, one may immediately construct the relevant
interaction matrix $G(r)$ of Ref. [27] [from which, as seen below, the
diabatic potential matrix $V(r)$ is derived], where the row/column indices
follow the ordering $(cc)(\bar{u}\bar{d})$, $D^{0}D^{*+}$, $D^{+}D^{*0}$:

$G(r)=\left(\begin{array}[]{ccc}V_{(cc)(\bar{u}\bar{d})}&-\frac{1}{\sqrt{2}}g(r)&+\frac{1}{\sqrt{2}}g(r)\\\
-\frac{1}{\sqrt{2}}g(r)&\Delta_{u}&0\\\
+\frac{1}{\sqrt{2}}g(r)&0&\Delta_{d}\end{array}\right)\,.$ (18)

The function $g(r)$ is defined as the universal string-breaking transition
amplitude between the $\delta^{\vphantom{1}}{\bar{\delta}}$ and di-meson
states. The latter states are assumed not to have direct transition couplings
amongst themselves, and indeed each di-meson state is assumed to behave as a
free-particle pair, apart from its $\delta^{\vphantom{1}}$-${\bar{\delta}}$
coupling $V_{\rm mix}$. Specifically, the constants $\Delta_{u,d}$ are defined
as the mass thresholds $\left(m_{D^{0}}+m_{D^{*+}}\right)$ and
$\left(m_{D^{+}}+m_{D^{*0}}\right)$, respectively, measured with respect to
their $I=0$ combination,

$\frac{1}{2}\left(m_{D^{0}}+m_{D^{+}}\right)+\frac{1}{2}\left(m_{D^{*0}}+m_{D^{*+}}\right)\,.$
(19)

which equals $3875.70\pm 0.05$ MeV. Therefore,

$\Delta_{u}\equiv\frac{1}{2}\left(m_{D^{0}}-m_{D^{+}}\right)-\frac{1}{2}\left(m_{D^{*0}}-m_{D^{*+}}\right)\equiv-\Delta_{d}\,,$
(20)

which, using the experimental values given in Footnote 1, is

$\Delta_{u}=-0.705\pm 0.017\ {\rm MeV}\,,$ (21)

precisely $-\frac{1}{2}$ times the 1st combination in Eq.
(LABEL:eq:TccIsospin). The combination $\Delta_{u}$ breaks both isospin
symmetry and heavy-quark spin symmetry. Thus, parametrically:

$\Delta_{u}\propto\Lambda_{\rm QCD}\,\frac{m_{d}-m_{u}}{m_{c}}\,.$ (22)

However, the interaction matrix $G(r)$ does not in general equal the full
diabatic potential matrix $V(r)$ appearing in the diabatic Schrödinger
equation Eq. (12). The motion of the heavy sources introduced by departures
from the static BO limit leads to the mixing of configurations with different
heavy-source quantum numbers, as we now discuss.

In the cases investigated in Ref. [27], the light degrees of freedom by
themselves are assumed to carry the trivial BO quantum numbers
$\Sigma^{+}_{g}$, which means a zero-spin projection along the axis of the
heavy sources ($\Sigma$), positive parity for reflections in a plane
containing this axis ($\epsilon=+$), and light-source $CP$ eigenvalue
$\eta=+$, which is denoted by “$g$”. In the present case of
$QQ\bar{u}\bar{d}$, its open-flavor quantum numbers mean that the state is not
a $C$ eigenstate, neither in total nor just in its heavy or just in its light
components. Nevertheless, the analysis of Ref. [27] remains valid upon
identifying $\eta$ eigenvalues for the heavy sources as those of $P$ alone,
rather than $CP$. In particular, the configuration for the heavy degrees of
freedom (here, the $QQ$ pair) is specified as $\lambda_{\eta}$, where
$\lambda$ is the projection of heavy-source spin (including sign) along a
chosen quantization axis $\hat{\bf z}$, which is not in general the same as
the axis $\hat{\bf r}$ defined by the heavy sources in the BO limit.

This mismatch between two natural axis choices—which one may denote as “space-
centered” versus “body-centered” axes, is the origin of the distinction
between the interaction matrix $G(r)$ and the diabatic potential matrix
$V(r)$. The relation between them is derived in Ref. [27] to be:

$\displaystyle
V^{\eta,J}_{i,i^{\prime},\ell,\ell^{\prime}}(r)=\sqrt{(2\ell+1)(2\ell^{\prime}+1)}$
(27) $\displaystyle\times$
$\displaystyle\sum_{\lambda}\left(\begin{array}[]{ccc}s_{i}&\ell&J\\\
\lambda&0&-\lambda\end{array}\right)\left(\begin{array}[]{ccc}s_{i^{\prime}}&\ell^{\prime}&J\\\
\lambda&0&-\lambda\end{array}\right)G^{\eta,\lambda}_{i,i^{\prime}}(r).$

Here, unprimed and primed variables refer, respectively, to initial (row) and
final (column) states labeled by $i^{(\prime)}$, and carry spin
$s_{i^{(\prime)}}$; the partial wave is labeled by orbital angular momentum
$\ell^{(\prime)}$, which combines with $s_{i^{(\prime)}}$ to give the total
angular momentum $J$ for the state; and large parentheses indicate Wigner $3j$
symbols.

Under the assumption that the heavy sources interact via a central potential,
the usual separation of variables in spherical coordinates also introduces the
centrifugal term in the effective potential for specific total $J^{P}$ quantum
numbers:

$V^{J^{P}}_{i,i^{\prime},\ell,\ell^{\prime}}(r)=V^{\eta,J}_{i,i^{\prime},\ell,\ell^{\prime}}(r)+\delta_{i,i^{\prime}}\delta_{\ell,\ell^{\prime}}\frac{\ell(\ell+1)}{2\mu
r^{2}},$ (28)

where $\mu$ is the reduced mass of the $(cc)(\bar{u}\bar{d})$ pair.

The specific case of the $J^{P}=1^{+}$ $T_{cc}^{+}$ is quite trivial in this
formalism. $J=1$, of course; and since we only consider $(cc)(\bar{u}\bar{d})$
and $DD^{*}$ components, then $i^{(\prime)}$ merely labels the
$(cc)(\bar{u}\bar{d})$ state and the two isospin states $D^{0}D^{*+}$,
$D^{+}D^{*0}$ in the order indicated by Eq. (18), so that
$s_{1}\\!=\\!s_{2}\\!=\\!s_{3}\\!=\\!1$; and the $(cc)$ diquark is pure spin-1
with no internal orbital excitation, so that $\lambda=+1,0,-1$ and $\eta=+$.
The $0^{-}0^{-}$ $D^{0}D^{+}$ state ($\simeq 140$ MeV lower than $T_{cc}^{+}$)
is forbidden from coupling to $1^{+}$ in any partial wave due to parity
conservation; and in this work we neglect the coupling to the combination
$D^{*0}D^{*+}$ ($\simeq 140$ MeV higher than $T_{cc}^{+}$), which allows
$s_{i^{(\prime)}}=0,1,2$ (as well as $\lambda=\pm 2$ for the di-meson
components), and thus would render Eq. (27) rather more complicated than for
the current case. In fact, while it is straightforward to evaluate the sum in
Eq. (27) term by term, in this case the calculation is especially simple
because all components have the same value of $s_{i}$, which implies that the
same interaction matrix $G(r)=G^{+,\lambda}(r)$ in Eq. (18) appears for each
of $\lambda=+1,0,-1$; thus, $G^{+,\lambda}_{i,i^{\prime}}(r)$ may be taken
outside the sum in Eq. (27). The remaining sum may be rewritten using a
completeness relation for $3j$ symbols [52]:

$\displaystyle\sum_{m_{1},m_{2}}\left(\begin{array}[]{ccc}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{array}\right)\left(\begin{array}[]{ccc}j_{1}&j_{2}&j_{3}^{\prime}\\\
m_{1}&m_{2}&m_{3}^{\prime}\end{array}\right)$ (33) $\displaystyle=$
$\displaystyle\frac{1}{2j_{3}+1}\,\delta_{j^{\vphantom{\prime}}_{3},j_{3}^{\prime}}\,\delta_{m^{\vphantom{\prime}}_{3},m_{3}^{\prime}}\,\delta(j_{1},j_{2},j_{3}),$
(34)

where the third $\delta$ factor is the triangle-rule constraint. Using the
invariance of $3j$ symbols under cyclic permutations of their columns [52] and
imposing the constraint $s_{i^{\prime}}=s_{i^{\vphantom{\prime}}}$ (unique to
this case), the sum in Eq. (27) becomes:

$\displaystyle\sum_{\lambda}\left(\begin{array}[]{ccc}J&s_{i}&\ell\\\
-\lambda&\lambda&0\end{array}\right)\left(\begin{array}[]{ccc}J&s_{i}&\ell^{\prime}\\\
-\lambda&\lambda&0\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2\ell+1}\,\delta_{\ell^{\vphantom{\prime}}\ell^{\prime}}\,\delta(J,s_{i},\ell),$
(39)

and, suppressing $i^{(\prime)}$ indices, Eq. (27) dramatically simplifies to:

$V^{+,1}_{\ell,\ell^{\prime}}(r)=G^{+,\lambda}(r)\,\delta_{\ell^{\vphantom{\prime}},\ell^{\prime}}\,\delta(1,1,\ell)=G(r)\,\delta_{\ell^{\vphantom{\prime}},\ell^{\prime}}\,\delta(1,1,\ell),$
(41)

so that the interaction matrix $G(r)$ in Eq. (18) equals the diabatic matrix
$V(r)$ in this particular case. Even though partial waves $\ell=0,1,2$ are
allowed, they combine in a very compact manner. The key ingredient in this
simplification is the absence of distinct couplings for different values of
$\lambda$, since only one di-meson spin combination (in two isospin channels)
occurs; the case of including $D^{*0}D^{*+}$ would be more involved.

##  V Analysis

As discussed in Sec. II, the new parameters introduced by the novel nature of
a diabatic dynamical-diquark $T^{+}_{cc}$ state are the
$V_{\delta^{\vphantom{1}}{\bar{\delta}}}$ parameter $R$ [Eq. (6)] indicating
the spatial extent of the “good” diquark ${\bar{\delta}}=(\bar{u}\bar{d})$ as
well as its mass $m_{\bar{\delta}}=m_{ud}$, and the doubly charmed diquark
mass $m_{\delta}=m_{cc}$. Additionally, one requires the mixing-potential [Eq.
(16)] parameters $\Lambda,\Delta$ introduced by the diabatic formalism [26,
25]. Following the practice of previous works, we set

$\Lambda=\rho\,\sigma,$ (42)

where the string-tension parameter $\sigma$ takes the same value as in Eq.
(II), and $\rho$ is a free parameter indicating the width of the mixing
potentials. We first assume that the previous $(\rho,\Delta)$ pair retrieved
in Ref. [28] from fitting to the PDG-averaged $\chi_{c1}(3872)$ mass [Eq.
(2)],

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle 0.165~{}\rm{fm},$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle 0.295~{}\rm{GeV},$ (43)

is appropriate for use in fitting $R$ and $m_{cc}$ to obtain the experimental
$T_{cc}^{+}$ mass. We later relax this assumption and fit to $(\rho,\Delta)$
as well.

Our first finding is that the modified potential Eq. (6), no longer being
singular as $r\to 0$, becomes too shallow to reproduce the experimental
$T^{+}_{cc}$ mass using the lattice-determined potential parameters of Eq.
(II) and any reasonable value for $m_{cc}$. However, one may note that the
eigenvalue $m_{T_{cc}^{+}}$ has an almost direct dependence on the combination
$m_{cc}+V_{0}$, since $m_{ud}\ll m_{cc}$. Thus, we introduce an additional
offset for the $R>0$ case, $\Delta_{V_{0}}=-0.150$ GeV:

$V(r)\to V(r)+\Delta_{V_{0}},$ (44)

so that, using $V_{0}$ from Eq. (II),

$V_{0}\to V_{0}+\Delta_{V_{0}}=-0.530~{}\rm{GeV}.$ (45)

With this modification, we find that the pair

$\displaystyle R$ $\displaystyle=$ $\displaystyle 0.4~{}\rm{fm},$
$\displaystyle m_{cc}$ $\displaystyle=$ $\displaystyle 3.0260~{}\rm{GeV},$
(46)

produces the experimental value as given in Eq. (1),

$m_{T_{cc}^{+}}=3874.83~{}\rm{MeV},$ (47)

and the full content of this eigenstate is:

$\displaystyle\delta^{\vphantom{1}}{\bar{\delta}}$ $\displaystyle:$
$\displaystyle\ \,9.84\%,$ $\displaystyle D^{*+}\\!D^{0}$ $\displaystyle:$
$\displaystyle 65.57\%,$ $\displaystyle D^{*0}D^{+}$ $\displaystyle:$
$\displaystyle 24.59\%.$ (48)

Since the entire range encompassing $m_{T_{cc}^{+}}$ and the two thresholds is
very narrow ($<\\!1.7~{}\rm{MeV}$), we see that even the relatively small mass
difference $-2\Delta_{u}$ between the $D^{*+}\\!D^{0}$ and $D^{*0}\\!D^{+}$
thresholds [Eq. (LABEL:eq:TccIsospin)] is enough to allow $D^{*+}\\!D^{0}$ to
dominate. This clear demonstration of isospin symmetry breaking can also be
directly explored by allowing the value of $\Delta_{u}$ to vary. In Fig. 1, we
show that adjusting $\Delta_{u}$ from $-0.705$ MeV to 0 MeV brings the
difference between $D^{0}D^{*+}$ and $D^{*0}D^{+}$ content to zero
quadratically in $\Delta_{u}$, the two meeting at around 44%. Notably, this
modification also relaxes the overall dominance of the di-meson thresholds on
the state content, allowing for the $\delta^{\vphantom{1}}{\bar{\delta}}$
content to rise to a maximum of about $11\%$, as seen in Fig. 2. Of course,
any change in $\Delta_{u}$ while holding other parameters fixed [particularly,
the isospin-averaged threshold of Eq. (19)] directly changes the eigenvalue
$m_{T_{cc}^{+}}$, as seen in Fig. 3.

Figure 1 also illustrates an interesting effect expected from the diabatic
formalism: The resonance mass eigenvalue (here, $m_{T_{cc}^{+}}$) prefers to
stay closer to the lower threshold, rather than reataining a special affinity
to the particular threshold ($D^{*+}\\!D^{0}$) to which it is closest at the
physical point. Moreover, the distance of the eigenvalue below the thresholds
is maximal (about 700 keV) when isospin breaking vanishes ($\Delta_{u}=0$) and
the thresholds coincide. Lastly, at larger values of isospin breaking
($|\Delta_{u}|>1.3$ MeV, where the $DD^{*}$ thresholds differ by more than 2.6
MeV), $m_{T_{cc}^{+}}$ rises above the lower threshold, and the $T_{cc}^{+}$
width would be expected to increase dramatically.

Although Figs. 1–3 are almost perfectly (anti)symmetric about $\Delta_{u}=0$,
one may note asymmetries about this point, especially in the quadratic fits in
Fig. 1, which have explicit discontinuities at $\Delta_{u}=0$. These jumps
originate from the fact that the physical point $\Delta_{u}=-0.705$ MeV is
special: i.e., the mass eigenvalue $m_{T_{cc}^{+}}$ (and thus the potential
parameters) is initially fitted at this value, and only then is $\Delta_{u}$
varied. While the quadratic fits of Fig. 1 are discontinuous at
$\Delta_{u}=0$, the content for each di-meson component is seen to vary
smoothly through this point. For a more quantitative description of this
asymmetry, we fit to the functional form $a\Delta^{2}_{u}+b\Delta_{u}+c$ and
for each di-meson component, providing the fit parameters $(a,b,c)$ (which
have units of MeV-(2,1,0), respectively):

$\displaystyle D^{*0}\\!D^{+}\;$ $\displaystyle(\Delta_{u}<0):$
$\displaystyle(-0.0623,-0.3651,0.4289),$ $\displaystyle D^{*0}\\!D^{+}\;$
$\displaystyle(\Delta_{u}>0):$ $\displaystyle(+0.0744,-0.3397,0.4412),$
$\displaystyle D^{0}D^{*+}\;$ $\displaystyle(\Delta_{u}<0):$
$\displaystyle(+0.0698,+0.3268,0.4404),$ $\displaystyle D^{0}D^{*+}\;$
$\displaystyle(\Delta_{u}>0):$ $\displaystyle(-0.0569,+0.3507,0.4282).$ (49)

As suggested above, one may use $m_{T^{+}_{cc}}$ as a fixed starting point,
and allow the diabatic parameters $\rho$ and $\Delta$ to be fit as well. In
doing so, $T^{+}_{cc}$ may be used as a laboratory to explore the
relationships of the relevant diabatic dynamical-diquark parameters. Since
fitting to the full parameter space of $(R,m_{cc},\Delta_{V_{0}},\rho,\Delta)$
using the single state $T^{+}_{cc}$ allows many solutions, we have performed
multiple experiments by holding some parameters constant and varying others.
Using the determination of $(R,m_{cc})$ from Eq. (46), we vary
$(\Delta_{V_{0}},\rho,\Delta)$. We find that a value of
$\Delta_{V_{0}}=-0.150~{}\rm{GeV}$ allows for the largest set of
$(\rho,\Delta)$ pairs that successfully fit the $m_{T^{+}_{cc}}$ (hence its
usage in the initial fit above), with a steep dropoff in the space of suitable
$(\rho,\Delta)$ pairs, whether one increases or decreases $\Delta_{V_{0}}$.
Within the $(\rho,\Delta)$ subspace for this $\Delta_{V_{0}}$ value, we
observe a quadratic relationship between $(\rho,\Delta)$ fit values, as shown
in Fig. 4. Clearly, the ability to freely vary such phenomenologically
critical parameters as $\rho,\Delta$ across such a wide range and still obtain
equivalent fits shows that a separate, precision determination of the
parameters from lattice QCD would have a significant impact on the
understanding of near-threshold states, as discussed in Refs. [26, 25].

Figure 1: Content of the normalized $T_{cc}^{+}$ eigenstate associated with
free di-meson configurations $D^{*+}\\!D^{0}$ and $D^{*0}D^{+}$ as a function
of the isospin-breaking parameter $\Delta_{u}$ [Eq. (20)] for potential
parameters given in Eqs. (V), (45), and (46). The isospin-averaged threshold
mass [Eq. (19)] is held fixed. Additionally, we fit quadratic curves to each
component separately for $\Delta_{u}<0$ [which contains the physical point,
represented by a dashed vertical line, from which agreement can be seen with
the results in Eq. (48)] and for $\Delta_{u}>0$. Figure 2: Content of the
normalized $T_{cc}^{+}$ eigenstate associated with the
$\delta^{\vphantom{1}}{\bar{\delta}}$ configuration as a function of the
isospin-breaking parameter $\Delta_{u}$ [Eq. (20)], using the same potential
parameters and isospin-averaged mass as in Fig. 1. The dashed vertical line
again represents the physical point, from which agreement can be seen with the
results in Eq. (48)]. Figure 3: The mass eigenvalue $m_{T^{+}_{cc}}$ as a
function of $\Delta_{u}$, using the same potential parameters and isospin-
averaged mass [Eq. (19)] as in Figs. 1, 2. The di-meson threshold energies are
included for reference. The dashed vertical line again represents the physical
point, from which agreement with the measured values in Eqs. (1) and
(LABEL:eq:TccIsospin) can be obtained. Figure 4: Pairs of diabatic-potential
parameters $\rho,\Delta$ [Eqs. (16), (42)] that give equivalent fits to the
experimental value of $m_{T_{cc}^{+}}$ [Eq. (1)], fixing
$\Delta_{V_{0}}=-0.150$ MeV [Eq. (45)]. Also presented is a quadratic fit to
the results.

##  VI Conclusions

We have performed the first study of the open-charm state $T_{cc}^{+}$ in the
diabatic generalization of the Born-Oppenheimer approximation. In particular,
the $T_{cc}^{+}$ substructure is dominated by its extreme proximity to the
$D^{*+}\\!D^{0}$ threshold, but it also lies not much further below the
$D^{*0}D^{+}$ threshold, and in addition permits a $(cc)(\bar{u}\bar{d})$
diquark-antidiquark ($\delta^{\vphantom{1}}{\bar{\delta}}$) component. As
expected [and in analogy to a similar situation for $\chi_{c1}(3872)$ with the
$D^{*0}\bar{D}^{0}+D^{0}\bar{D}^{*0}$ threshold], we find the mass eigenstate
to consist overwhelmingly ($>\\!90\%$) of di-meson components, but the
$D^{*0}D^{+}$ component is nevertheless more than 1/3 as large as the
$D^{*+}\\!D^{0}$ component that lies almost atop $m_{T_{cc}^{+}}$. Moreover,
the elementary $\delta^{\vphantom{1}}{\bar{\delta}}$ component persists in
both cases at a level of nearly $10\%$, providing a significant short-distance
component to the wave function of both $T_{cc}^{+}$ and $\chi_{c1}(3872)$.

The two states nevertheless differ in key respects. $\chi_{c1}(3872)$ has
several known decay channels to charmonium, which of course cannot occur for
the open-charm $T_{cc}^{+}$. But the two $D^{*}\\!D$ (or $D^{*}\\!\bar{D}$)
isospin-partner decay modes are much more closely spaced for $T_{cc}^{+}$ than
for $\chi_{c1}(3872)$, making $T_{cc}^{+}$ a superior laboratory for studying
isospin breaking in 4-quark states. We have used this feature to compute how
all 3 $T_{cc}^{+}$ components: $D^{*+}\\!D^{0}$, $D^{*0}D^{+}$, and
$\delta^{\vphantom{1}}{\bar{\delta}}$, change as the isospin-breaking mass
difference
$m^{\vphantom{\dagger}}_{D^{*+}\\!D^{0}}-m^{\vphantom{\dagger}}_{D^{*0}D^{+}}$
is adjusted. We find that the $\delta^{\vphantom{1}}{\bar{\delta}}$ component
is fairly stable (between $5$–$11\%$) over a large range of this difference,
with its maximum occurring close to the isospin-symmetric point. Since we have
fixed the measured value of $m_{T_{cc}^{+}}$ at the specific physical value of
isospin breaking, a small${}^{\vphantom{\dagger}}$ asymmetry corresponding to
exchanging the $D^{*+}\\!D^{0}$ and $D^{*0}D^{+}$ components arises.
Nevertheless, we conclude that the dominant parameter determining both the
variation of the $\delta^{\vphantom{1}}{\bar{\delta}}$ state content and the
eigenvalue $m_{T_{cc}^{+}}$ appears to be the magnitude of the isospin-
breaking mass difference.

We have noted above that the particular modeling of the $T_{cc}^{+}$ system in
this paper requires a very simple diabatic potential matrix, because we
neglected the more distant $D^{*+}\\!D^{*0}$ threshold. Incorporating this
threshold, which requires the use of a more involved spin-dependent formalism,
is one key direction of future research. Indeed, properly including spin
dependence in the diabatic potential matrix has not yet been performed in the
hidden-charm sector of this model, and constitutes its own set of projects
still to be carried out. We have also noted that, if the “good”
($\bar{u}\bar{d}$) diquark is in fact an essential component of $T_{cc}^{+}$,
then this state likely has no nearby multiplet partners, unlike the case of
the hidden-charm multiplet containing $\chi_{c1}(3872)$, $Z_{c}(3900)$,
$Z_{c}(4020)$, and presumably several other states. In several ways, the state
$T_{cc}^{+}$ provides an ideal laboratory in which to test different diabatic
hypotheses with a minimum of complications.

##  Appendix A Fierz Reordering of Isoscalar Diquark-Antidiquark Vector
Current

For an arbitrary fermion field $\psi$, define its charge-conjugate form in the
usual manner:

$\psi^{c}(x)\equiv C{\bar{\psi}}^{T}\,,$ (50)

where $C$ is the charge-conjugation Dirac matrix, which has the properties

$C\gamma^{\mu}C^{-1}=-\gamma^{\mu\,T}\,,\ \
C^{-1}=C^{T}=C^{\dagger}=-C=-C^{*}\,.$ (51)

From these relations also follows:

${\bar{\psi}}^{c}(x)=\psi^{T}\\!(x)\,C\,.$ (52)

The Hermitian interpolating operator for the “good” (spin-0, isoscalar, color-
triplet) light diquark ($\bar{u}\bar{d}$) is [15]:

$\epsilon_{ijk}\frac{1}{\sqrt{2}}\left(\bar{u}_{j}\,i\gamma^{5}d^{c}_{k}-\bar{d}_{j}\,i\gamma^{5}u^{c}_{k}\right),$
(53)

where $i,j,k$ are color indices.

Let $J^{\mu}(x)$ be the simplest current for creating a $J^{P}=1^{+}$ state
with flavor quantum numbers $QQ\bar{u}\bar{d}$, with the light quarks being in
the “good” diquark configuration of Eq. (53). Then it is a straightforward
(but lengthy) exercise to apply the conventional Fierz reordering to the
$Q\bar{q}$ and $q^{c}{\bar{Q}}^{c}$ field pairs to obtain:

$\displaystyle
J^{\mu}(x)=\epsilon^{ijk}\epsilon^{imn}\bar{Q}^{c}_{j}\gamma^{\mu}Q_{k}\,\frac{1}{\sqrt{2}}\left(\bar{u}_{m}\,i\gamma^{5}d^{c}_{n}-\bar{d}_{m}\,i\gamma^{5}u^{c}_{n}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}(\bar{u}i\gamma^{5}Q)(\bar{d}\gamma^{\mu}Q)+\frac{1}{\sqrt{2}}(\bar{d}i\gamma^{5}Q)(\bar{u}\gamma^{\mu}Q)$
$\displaystyle+\frac{i}{\sqrt{2}}(\bar{u}Q)(\bar{d}\gamma^{\mu}\gamma^{5}Q)-\frac{i}{\sqrt{2}}(\bar{d}Q)(\bar{u}\gamma^{\mu}\gamma^{5}Q)$
$\displaystyle-\frac{1}{\sqrt{2}}(\bar{u}\,\sigma^{\mu\nu}Q)(\bar{d}\gamma_{\nu}\gamma^{5}Q)+\frac{1}{\sqrt{2}}(\bar{d}\,\sigma^{\mu\nu}Q)(\bar{u}\gamma_{\nu}\gamma^{5}Q)$
$\displaystyle+\frac{1}{\sqrt{2}}(\bar{u}\,\sigma^{\mu\nu}\gamma^{5}Q)(\bar{d}\gamma_{\nu}Q)-\frac{1}{\sqrt{2}}(\bar{d}\,\sigma^{\mu\nu}\gamma^{5}Q)(\bar{u}\gamma_{\nu}Q)\,.$

The tetraquark state thus couples naturally not only to di-meson pairs with
quantum numbers $(0^{-})(1^{-})$, but also to $(0^{+})(1^{+})$,
$(2^{+})(1^{+})$, and $(2^{-})(1^{-})$. Moreover, the (pseudo-)tensor bilinear
operators in the final line of Eq. (LABEL:eq:Fierz) can also serve as
interpolating operators for vector particles, which is how the overlap with
$(1^{-})(1^{-})$ pairs (such as $D^{*}D^{*}$ for $T_{cc}^{+}$) occurs in this
formalism.

Analogous Fierz reorderings between diquark-antidiquark and di-meson scalar
operators, as would appear in Lagrangians, are presented in Appendix A of Ref.
[53].

###### Acknowledgements.

We thank R. Bruschini, E. Braaten, and Z.-G. Wang for insightful comments.
This work was supported by the National Science Foundation (NSF) under Grants
No. PHY-1803912 and PHY-2110278.

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