# Hunting for the prospective $T_{cc}$ family based on the diquark-antidiquark
configuration

Wen-Chao Dong1,2<EMAIL_ADDRESS>Zhi-Gang Wang1111Corresponding author
<EMAIL_ADDRESS>1Department of Physics, North China Electric Power
University, Baoding 071003, China
2School of Nuclear Science and Engineering, North China Electric Power
University, Beijing 102206, China

###### Abstract

Instigated by the first $T_{cc}$ observation at the LHCb Collaboration, the
spectroscopic properties of the entire isoscalar and isovector $T_{cc}$ family
are systematically unveiled by means of multiple sorts of relativized and
nonrelativistic diquark formalisms, encompassing the Godfrey-Isgur relativized
diquark model, the modified Godfrey-Isgur relativized diquark model
incorporating the color screening effects, the nonrelativistic diquark model
with the Gaussian type hyperfine potential, and the nonrelativistic diquark
model with the Yukawa type hyperfine potential. The theoretical outcomes of
various diquark-antidiquark scenarios are inclined to categorize the
$T_{cc}(3875)^{+}$ structure as the exemplary candidate of the $1S$-wave
isoscalar axial-vector double-charm tetraquark state. In light of the diquark-
antidiquark configuration, this work investigates the mixing angles of the
orbitally excited isovector $T_{cc}$ states and the magic mixing angles of the
ideal heavy-light tetraquarks for the first time. As the advancement of the
experimental detection capability, these phenomenological prognostications
will effectively boost the hunting for the prospective low-lying $T_{cc}$
states in the future.

##  I Introduction

Over the past two decades, a cornucopia of heavy flavored exotic hadrons
gradually emerged from diverse particle detection facilities conducted by
worldwide experimental collaborations Workman:2022ynf ; Olsen:2017bmm .
Nonetheless, it is rather arduous to cognize the authentic nature of these
novel hadron states. A well-known specimen is the hidden-charm
$\chi_{c1}(3872)$ state, whose composition is identified as the conventional
charmonium $c\bar{c}$, the hybrid charmonium $c\bar{c}g$, the charmoniumlike
tetraquark/molecule $c\bar{c}q\bar{q}$, the mixture of them, or other
configurations by discrepant arguments Olsen:2017bmm ; Hosaka:2016pey ;
Chen:2016qju ; Ali:2017jda ; Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ;
Karliner:2017qhf . Conspicuously, the advent of the $\chi_{c1}(3872)$ state
was a milestone in hadron physics, which incited multifarious phenomenological
explorations beyond the conventional quark model. Hitherto, there are all
sorts of theoretical propositions to construe the internal structure of exotic
hadron states Olsen:2017bmm ; Hosaka:2016pey ; Chen:2016qju ; Ali:2017jda ;
Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Karliner:2017qhf ;
Liu:2019zoy ; Brambilla:2019esw ; Chen:2022asf , comprising hybrid hadrons,
hadro-quarkonia, compact multiquarks, hadronic molecules, kinematical effects,
and so forth. Regrettably, none of them are capable of being predominant to
overwhelm the rest of the plausible perspectives on the so-called exotics,
which substantiates that both experimental and theoretical unremitting
exertions are still necessitous so as to disentangle the labyrinthine essence
of nonstandard heavy hadrons.

Nowadays, quite a large portion of heavy flavored exotic hadrons are the
states with hidden heavy flavors, e.g., the heavy quarkoniumlike
$T_{\psi}/T_{\Upsilon}$ states, the fully charmed tetraquark $T_{\psi\psi}$
states, and the hidden charmed pentaquark $P_{\psi}$ states Brambilla:2019esw
. By contrast, the open heavy flavored exotic hadrons established by various
experiments are fairly rare, merely touching upon the singly and doubly
charmed tetraquark $T_{cs}/T_{c\bar{s}}/T_{cc}$ states Chen:2022asf . Hence,
the quest for the ground and excited states of the double-charm tetraquark
$T_{cc}$ family is momentous to ascertain the properties of the exotic hadrons
with open heavy flavors. In 2021, a very narrow peaking structure, marked as
$T_{cc}(3875)^{+}$, was clearly detected in the $D^{0}D^{0}\pi^{+}$ invariant
mass distribution by the LHCb Collaboration LHCb:2021vvq ; LHCb:2021auc ,
which signified the first observation of the exotic states with
$cc\bar{u}\bar{d}$ quark component. In consideration of the absence of the
double-charm signal in the $D^{+}D^{0}\pi^{+}$ invariant mass distribution,
the $T_{cc}(3875)^{+}$ structure is presumably an isoscalar state. Manifestly,
the observed mass of the narrow $T_{cc}(3875)^{+}$ state is in close proximity
to the $D^{*+}D^{0}$ mass threshold. Thereupon, the spin-parity properties of
the $T_{cc}(3875)^{+}$ state are speculated as $J^{P}=1^{+}$ by postulating
that the relative angular momentum in the $D^{*}D$ pair is $S$-wave. The
experimental mass value of the $T_{cc}(3875)^{+}$ state is expressed as

$\displaystyle m_{T_{cc}(3875)^{+}}$ $\displaystyle=$ $\displaystyle
m_{D^{*+}}+m_{D^{0}}+\delta m,$ (1.1)

where $m_{D^{*+}}$, $m_{D^{0}}$, and $\delta m$ denote the observed mass of
the $D^{*+}$ meson, the observed mass of the $D^{0}$ meson, and the binding
energy of the $T_{cc}(3875)^{+}$ state with respect to the $D^{*+}D^{0}$ mass
threshold, respectively. For the sake of the determination of the binding
energy and the decay width of the $T_{cc}(3875)^{+}$ structure, the LHCb
Collaboration employed two sorts of Breit-Wigner (BW) parametrization schemes,
i.e., the generic BW parametrization and the unitarised BW parametrization. In
generic case LHCb:2021vvq , a relativistic $P$-wave two-body BW function with
a Blatt-Weisskopf form factor was adopted to attain the binding energy $\delta
m_{\text{BW}}$ and the decay width $\Gamma_{\text{BW}}$, i.e.,

$\displaystyle\delta m_{\text{BW}}$ $\displaystyle=$ $\displaystyle-273\pm
61\pm 5^{+11}_{-14}~{}\text{keV},$ (1.2) $\displaystyle\Gamma_{\text{BW}}$
$\displaystyle=$ $\displaystyle 410\pm 165\pm 43^{+18}_{-38}~{}\text{keV}.$
(1.3)

Moreover, the LHCb Collaboration extracted the position of the amplitude pole
of the narrow $T_{cc}(3875)^{+}$ peaking structure on the second Riemann
sheet, by making use of a unitarised three-body BW function and taking into
account a novel model of the detector mass resolution LHCb:2021auc . The
parameters $\delta m_{\text{pole}}$ and $\Gamma_{\text{pole}}$ of the
$T_{cc}(3875)^{+}$ pole were determined as

$\displaystyle\delta m_{\text{pole}}$ $\displaystyle=$ $\displaystyle-360\pm
40^{+4}_{-0}~{}\text{keV},$ (1.4) $\displaystyle\Gamma_{\text{pole}}$
$\displaystyle=$ $\displaystyle 48\pm 2^{+0}_{-14}~{}\text{keV}.$ (1.5)

It is evident to be conscious of that the decay width turned up in the
imaginary part of the $T_{cc}(3875)^{+}$ pole is much tinier than its
counterpart acquired by the generic BW parametrization, although the binding
energies obtained by two sorts of BW parametrization schemes are somewhat
near. Consequently, the visible information of the $T_{cc}(3875)^{+}$
structure is kind of scant compared to the famed $XYZ$ states, which entails
further experimental explorations and theoretical investigations of the exotic
$cc\bar{u}\bar{d}$ states.

Taking the perspectives of phenomenological theories, the delving of the
multiquark configuration is commenced by the seminal works of the quark model
Gell-Mann:1964ewy ; Zweig:1964ruk . In terms of the open-flavor tetraquark
states that carry two heavy quarks, a few pioneering surveys unraveled their
structural stabilities and existential feasibilities Ader:1981db ;
Ballot:1983iv ; Zouzou:1986qh ; Heller:1986bt ; Manohar:1992nd ; Lipkin:1986dw
. In 1986, the earliest theoretical inquiry with regard to $T_{cc}$
spectroscopy rendered an upper bound on the mass ambit of the lowest isoscalar
$cc\bar{u}\bar{d}$ state with $J^{P}=1^{+}$, by virtue of a model-independent
approach Lipkin:1986dw . Subsequently, the various properties of the
$cc\bar{u}\bar{d}$ states, e.g., mass spectra, decay properties, and
production mechanisms, were pervasively probed by Refs. Carlson:1987hh ;
Semay:1994ht ; Pepin:1996id ; Gelman:2002wf ; Vijande:2003ki ; Janc:2004qn ;
Ebert:2007rn ; Navarra:2007yw ; Vijande:2009kj ; Lee:2009rt ; Yang:2009zzp ;
Ohkoda:2012hv ; Li:2012ss ; Du:2012wp ; Feng:2013kea ; Ikeda:2013vwa ;
Luo:2017eub ; Mehen:2017nrh ; Cheung:2017tnt ; Karliner:2017qjm ;
Eichten:2017ffp ; Hyodo:2017hue ; Wang:2017uld ; Wang:2017dtg ; Yan:2018gik ;
Xing:2018bqt ; Zhu:2019iwm ; Liu:2019stu ; Maiani:2019lpu ; Park:2018wjk ;
Agaev:2019qqn ; Junnarkar:2018twb ; Yang:2019itm ; Tang:2019nwv ; Lu:2020rog ;
Ding:2020dio ; Deng:2018kly ; Tan:2020ldi ; Wallbott:2020jzh ; Jin:2021cxj ;
Feijoo:2021ppq ; Qin:2020zlg ; Cheng:2020wxa ; Noh:2021lqs ; Wang:2021yld ;
Zhang:2021yul ; Braaten:2020nwp ; Chen:2021vhg ; Huang:2021urd ; Meng:2020knc
; Fleming:2021wmk ; Meng:2021jnw ; Hu:2021gdg ; Dong:2021bvy ; Wang:2021ajy ;
Abreu:2022sra ; Xin:2021wcr ; Ling:2021bir ; Du:2021zzh ; Cheng:2022qcm ;
Kim:2022mpa ; Weng:2021hje ; Chen:2021tnn ; Ren:2021dsi ; Agaev:2022ast ;
Liu:2019yye ; Albuquerque:2022weq ; Zhao:2021cvg ; Guo:2021yws ; Gao:2020bvl ;
Abreu:2022lfy ; Agaev:2021vur ; Ke:2021rxd ; Dai:2021vgf ; Yan:2021wdl ;
Chen:2021cfl ; Deng:2021gnb ; Albaladejo:2021vln ; Chen:2022vpo ; Wang:2022clw
; Achasov:2022onn ; Braaten:2022elw ; Deng:2022cld ; Dai:2021wxi ;
Padmanath:2022cvl ; Liu:2023vrk ; Maiani:2022qze ; Meng:2023for ; Jia:2022qwr
; Peng:2023lfw ; Song:2023izj ; Ortega:2022efc ; Noh:2023zoq ;
Albuquerque:2023rrf ; He:2023ucd ; Jia:2023hvc ; Wang:2022jop ; Li:2023hpk ;
Dai:2023mxm ; Meng:2023jqk ; Lyu:2023xro ; Wu:2022gie ; Du:2023hlu ;
Asanuma:2023atv ; Guo:2023xyf ; Hua:2023zpa ; Sakai:2023syt ; Qiu:2023uno ;
Ma:2023int ; Wang:2024mop ; Kucab:2024nkv ; Hansen:2024ffk ; Zhai:2023ejo ;
Mutuk:2023oyz ; Chen:2023web ; Zhang:2024dth ; Wang:2024vjc ; Collins:2024sfi
; Meng:2023bmz ; Park:2024cic ; Li:2023wug ; Mutuk:2024vzv ; Abolnikov:2024key
; Sun:2024wxz ; Meng:2024yhu ; Whyte:2024ihh ; Sonnenschein:2024rzw ;
Lebed:2024zrp  with a myriad of theoretical prescriptions, including the MIT
bag model Carlson:1987hh ; Zhang:2021yul , the quark potential model
Semay:1994ht ; Pepin:1996id ; Gelman:2002wf ; Vijande:2003ki ; Janc:2004qn ;
Ebert:2007rn ; Vijande:2009kj ; Lee:2009rt ; Yang:2009zzp ; Luo:2017eub ;
Karliner:2017qjm ; Hyodo:2017hue ; Yan:2018gik ; Xing:2018bqt ; Zhu:2019iwm ;
Park:2018wjk ; Yang:2019itm ; Lu:2020rog ; Deng:2018kly ; Tan:2020ldi ;
Cheng:2020wxa ; Noh:2021lqs ; Meng:2020knc ; Kim:2022mpa ; Weng:2021hje ;
Chen:2021tnn ; Guo:2021yws ; Deng:2021gnb ; Wang:2022clw ; Deng:2022cld ;
Liu:2023vrk ; Meng:2023for ; Song:2023izj ; Ortega:2022efc ; Noh:2023zoq ;
He:2023ucd ; Meng:2023jqk ; Wu:2022gie ; Ma:2023int ; Mutuk:2023oyz ;
Wang:2024vjc ; Park:2024cic ; Li:2023wug ; Meng:2024yhu , the QCD sum rules
Navarra:2007yw ; Du:2012wp ; Wang:2017uld ; Wang:2017dtg ; Agaev:2019qqn ;
Tang:2019nwv ; Xin:2021wcr ; Agaev:2022ast ; Albuquerque:2022weq ; Gao:2020bvl
; Abreu:2022lfy ; Agaev:2021vur ; Albuquerque:2023rrf , the one-boson exchange
(OBE) model Ohkoda:2012hv ; Li:2012ss ; Liu:2019stu ; Wang:2021yld ;
Chen:2021vhg ; Dong:2021bvy ; Wang:2021ajy ; Cheng:2022qcm ; Ren:2021dsi ;
Peng:2023lfw ; Du:2023hlu ; Asanuma:2023atv ; Sakai:2023syt ; Qiu:2023uno ;
Zhang:2024dth ; Sun:2024wxz , the Bethe-Salpeter (BS) equation Feng:2013kea ;
Ding:2020dio ; Wallbott:2020jzh ; Zhao:2021cvg ; Ke:2021rxd , the lattice QCD
Ikeda:2013vwa ; Cheung:2017tnt ; Junnarkar:2018twb ; Chen:2022vpo ;
Padmanath:2022cvl ; Lyu:2023xro ; Collins:2024sfi ; Whyte:2024ihh , the heavy
quark symmetry Mehen:2017nrh ; Eichten:2017ffp ; Braaten:2020nwp , the
adiabatic (Born-Oppenheimer) approximation Maiani:2019lpu ; Maiani:2022qze ;
Mutuk:2024vzv ; Lebed:2024zrp , the coupled-channel approach Feijoo:2021ppq ;
Huang:2021urd ; Ling:2021bir ; Du:2021zzh ; Dai:2021vgf ; Albaladejo:2021vln ;
Achasov:2022onn ; Dai:2021wxi , the effective field theory (EFT) framework
Fleming:2021wmk ; Meng:2021jnw ; Abreu:2022sra ; Yan:2021wdl ; Chen:2021cfl ;
Braaten:2022elw ; Jia:2022qwr ; Jia:2023hvc ; Wang:2022jop ; Dai:2023mxm ;
Guo:2023xyf ; Hansen:2024ffk ; Zhai:2023ejo ; Meng:2023bmz ; Abolnikov:2024key
, the holographic QCD Liu:2019yye ; Sonnenschein:2024rzw , the chiral quark-
soliton model ($\chi$QSM) Kucab:2024nkv , the Regge trajectory relation
Chen:2023web , and so on. Lately, following the experimental establishment of
the $T_{cc}(3875)^{+}$ state LHCb:2021vvq ; LHCb:2021auc , sundry
phenomenological expositions for its intrinsic configuration were put forward,
containing the compact tetraquark (with the diquark-antidiquark configuration)
Jin:2021cxj ; Qin:2020zlg ; Zhang:2021yul ; Weng:2021hje ; Albuquerque:2022weq
; Guo:2021yws ; Gao:2020bvl ; Agaev:2021vur ; Liu:2023vrk ; Maiani:2022qze ;
Meng:2023for ; Noh:2023zoq ; Wu:2022gie ; Mutuk:2023oyz ; Wang:2024vjc ;
Park:2024cic ; Li:2023wug ; Sonnenschein:2024rzw , the hadronic molecule (with
the meson-meson configuration) Feijoo:2021ppq ; Chen:2021vhg ; Huang:2021urd ;
Fleming:2021wmk ; Meng:2021jnw ; Hu:2021gdg ; Dong:2021bvy ; Abreu:2022sra ;
Xin:2021wcr ; Ling:2021bir ; Du:2021zzh ; Cheng:2022qcm ; Chen:2021tnn ;
Ren:2021dsi ; Zhao:2021cvg ; Ke:2021rxd ; Chen:2021cfl ; Deng:2021gnb ;
Albaladejo:2021vln ; Chen:2022vpo ; Braaten:2022elw ; Deng:2022cld ;
Jia:2022qwr ; Ortega:2022efc ; Jia:2023hvc ; Wang:2022jop ; Li:2023hpk ;
Dai:2023mxm ; Meng:2023jqk ; Du:2023hlu ; Asanuma:2023atv ; Guo:2023xyf ;
Sakai:2023syt ; Qiu:2023uno ; Ma:2023int ; Wang:2024mop ; Zhai:2023ejo ;
Zhang:2024dth ; Meng:2023bmz ; Abolnikov:2024key ; Sun:2024wxz , the
tetraquark-molecule mixing (with an admixture of diquark-antidiquark and
meson-meson configurations) Yan:2021wdl ; Peng:2023lfw ; Song:2023izj ;
He:2023ucd ; Lebed:2024zrp , the Efimov state (with the meson-meson-sphaleron
configuration) Liu:2019yye , the virtual state (with the pole in the real axis
below threshold) Dai:2021wxi ; Padmanath:2022cvl ; Lyu:2023xro ;
Collins:2024sfi ; Whyte:2024ihh , etc. Ergo, the authentic nature of the
$T_{cc}(3875)^{+}$ state is still equivocal due to the dearth of the
phenomenological smoking gun.

As the generalised case of the $cc\bar{u}\bar{d}$ states, the double-heavy
tetraquarks $Q_{1}Q_{2}\bar{q}_{1}\bar{q}_{2}$ constituted by two heavy quarks
$Q$ and two light antiquarks $\bar{q}$ are the principal candidates of the
long-lived exotic states, which are relatively stable against the decays
engendered by strong interaction Karliner:2017qjm . Apparently, the doubly
heavy diquark $Q_{1}Q_{2}$ with antitriplet color will be capable of being
approximately deemed as a point-like heavy antiquark $\bar{Q}$ when the mass
of the heavy quark is large enough, implying the potential existence of such
doubly heavy tetraquark states Eichten:2017ffp . Recently, the QCD version of
the hydrogen bond was employed to treat the $cc$ pair as the heavy color
sources, having acquired the mass that conformed to the exotic
$T_{cc}(3875)^{+}$ structure Maiani:2022qze . In addition, the isoscalar
$T_{cc}$ tetraquark composed of the color-antitriplet diquark $cc$ and the
color-triplet antidiquark $\bar{u}\bar{d}$ was considered capable of
impeccably resuscitating the observed mass of the $T_{cc}(3875)^{+}$ state in
light of the heavy antiquark-diquark symmetry Wu:2022gie . Therefore, spurred
on by the first experimental discovery of the doubly charmed tetraquark
LHCb:2021vvq ; LHCb:2021auc , the theoretical disquisition on the ground and
excited $cc\bar{u}\bar{d}$ states with the diquark-antidiquark configuration
is undoubtedly of great phenomenological significance Lipkin:1986dw ;
Karliner:2017qjm ; Eichten:2017ffp ; Maiani:2022qze ; Wu:2022gie . In order to
shed light on the spectroscopic properties of the $T_{cc}$ family, this work
takes advantage of several diquark-antidiquark scenarios, involving the
Godfrey-Isgur (GI) relativized diquark model, the modified Godfrey-Isgur (MGI)
relativized diquark model (incorporating the color screening effects), and the
nonrelativistic (NR) diquark models. The compendium of this treatise is
encapsulated as follows. At the beginning, the experimental and theoretical
status quo of the double-charm tetraquark states is sententiously revisited in
Section I. Next, the corresponding diquark-antidiquark scenarios are
explicated in Section II. Whereafter, Section III exhibits the predicted
outcomes with regard to the low-lying $T_{cc}$ spectroscopy. Furthermore,
concerning the mass spectra and mixing angles of the $T_{cc}$ states, Section
IV discusses the connection between this work and other approaches.
Ultimately, Section V lays out a succinct summary of this work.

##  II Formalism

In this section, the Godfrey-Isgur (GI) relativized diquark model, the
modified Godfrey-Isgur (MGI) relativized diquark model with the color
screening effects, and the nonrelativistic (NR) diquark models whose hyperfine
interaction potentials are of the Gaussian or Yukawa forms are limned for the
sake of pinning down the spectroscopic properties of the low-lying
$cc\bar{u}\bar{d}$ states.

###  II.1 Godfrey-Isgur (GI) relativized diquark model

As one of the most eminent version among all sorts of quark models, the
Godfrey-Isgur relativized quark model (GI model) proposed by S. Godfrey and N.
Isgur is capable of successfully depicting the mass spectra of nearly all
types of mesons that consist of light or heavy (anti)quarks Godfrey:1985xj .
Fabulously, the GI model not only attained the universality of the parameters
in the one-gluon-exchange-plus-linear-confinement potential stimulated by QCD,
but also embraced pivotal relativistic effects. There was no doubt that the GI
model manifested that “all mesons—from the pion to the upsilon—can be
described in a unified framework”, as mentioned by Ref. Godfrey:1985xj . Up to
now, the spectroscopic properties of light mesons Godfrey:1985xj , singly
heavy mesons Godfrey:1985xj ; Godfrey:2016nwn ; Godfrey:2015dva , doubly heavy
mesons Godfrey:1985xj ; Godfrey:2004ya ; Barnes:2005pb ; Godfrey:2015dia ,
light baryons Capstick:1986ter , singly heavy baryons Capstick:1986ter ;
Lu:2016ctt , doubly heavy baryons Lu:2017meb , triply heavy baryons
Bedolla:2019zwg , light tetraquarks Lu:2019ira , open heavy tetraquarks
Lu:2016zhe , hidden heavy tetraquarks Lu:2016cwr , fully heavy tetraquarks
Dong:2022sef , and diquarks Lu:2016ctt ; Bedolla:2019zwg ; Lu:2019ira ;
Lu:2016zhe ; Lu:2016cwr ; Dong:2022sef ; Ferretti:2019zyh  have been amply
surveyed by the GI model. The particular introduction and pertinent details
with respect to the GI model are elaborated in Ref. Godfrey:1985xj . In terms
of mesons, the Hamiltonian of the GI model is decomposed into

$\displaystyle H_{\rm GI}$ $\displaystyle=$ $\displaystyle H_{\rm
GI}^{0}+V_{\rm GI}^{\rm si}+V_{\rm GI}^{\rm sd}$ (2.1) $\displaystyle=$
$\displaystyle H_{\rm GI}^{0}+V_{\rm GI}^{\rm conf}+V_{\rm GI}^{\rm
cont}+V_{\rm GI}^{\rm ten}+V_{\rm GI}^{\rm so},$

where $H_{\rm GI}^{0}$, $V_{\rm GI}^{\rm si}$, and $V_{\rm GI}^{\rm sd}$
denote the relativistic energy of entire (anti)quarks, the spin-independent
interaction potential (made up of the confinement potential $V_{\rm GI}^{\rm
conf}$), and the spin-dependent interaction potential (made up of the contact
potential $V_{\rm GI}^{\rm cont}$, the tensor potential $V_{\rm GI}^{\rm
ten}$, and the spin-orbit potential $V_{\rm GI}^{\rm so}$), respectively.
Concretely, these terms are

$\displaystyle H_{\rm GI}^{0}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}E_{i}(p),$ (2.2) $\displaystyle V_{\rm GI}^{\rm
conf}$ $\displaystyle=$ $\displaystyle\tilde{G}_{12}^{\rm
Coul}(p,r)+\tilde{S}_{12}(r),$ (2.3) $\displaystyle V_{\rm GI}^{\rm cont}$
$\displaystyle=$
$\displaystyle\frac{2}{3m_{1}m_{2}r^{2}}\frac{\partial}{\partial
r}\left[r^{2}\frac{\partial\tilde{G}_{12}^{\rm cont}(p,r)}{\partial
r}\right]\bm{S}_{1}\cdot\bm{S}_{2},$ (2.4) $\displaystyle V_{\rm GI}^{\rm
ten}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{1}m_{2}}\left(\frac{1}{r}-\frac{\partial}{\partial
r}\right)\frac{\partial\tilde{G}_{12}^{\rm ten}(p,r)}{\partial r}\mathbb{T},$
(2.5) $\displaystyle V_{\rm GI}^{\rm so}$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{1}^{2}r}\left[\frac{\partial\tilde{G}_{11}^{\rm
so(v)}(p,r)}{\partial r}-\frac{\partial\tilde{S}_{11}^{\rm
so(s)}(p,r)}{\partial r}\right]\bm{L}\cdot\bm{S}_{1}$ (2.6)
$\displaystyle+\frac{1}{2m_{2}^{2}r}\left[\frac{\partial\tilde{G}_{22}^{\rm
so(v)}(p,r)}{\partial r}-\frac{\partial\tilde{S}_{22}^{\rm
so(s)}(p,r)}{\partial r}\right]\bm{L}\cdot\bm{S}_{2}$
$\displaystyle+\frac{1}{m_{1}m_{2}r}\frac{\partial\tilde{G}_{12}^{\rm
so(v)}(p,r)}{\partial r}\bm{L}\cdot\bm{S},$

with

$\displaystyle E_{i}(p)=\left(p^{2}+m_{i}^{2}\right)^{\frac{1}{2}},$
$\displaystyle\tilde{G}_{ij}^{\rm Coul}(p,r)$ $\displaystyle=$
$\displaystyle\left[1+\frac{p^{2}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}}\tilde{G}_{ij}(r)\left[1+\frac{p^{2}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}},$
$\displaystyle\tilde{G}_{ij}^{\rm cont/ten/so(v)}(p,r)$ $\displaystyle=$
$\displaystyle\left[\frac{m_{i}m_{j}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}+\epsilon_{\rm
cont/ten/so(v)}}\tilde{G}_{ij}(r)\left[\frac{m_{i}m_{j}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}+\epsilon_{\rm
cont/ten/so(v)}},$ $\displaystyle\tilde{S}_{ij}^{\rm so(s)}(p,r)$
$\displaystyle=$
$\displaystyle\left[\frac{m_{i}m_{j}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}+\epsilon_{\rm
so(s)}}\tilde{S}_{ij}(r)\left[\frac{m_{i}m_{j}}{E_{i}(p)E_{j}(p)}\right]^{\frac{1}{2}+\epsilon_{\rm
so(s)}},$
$\displaystyle\mathbb{T}=\frac{\mathbb{S}_{12}}{12}=\frac{\left(\bm{S}_{1}\cdot\bm{r}\right)\left(\bm{S}_{2}\cdot\bm{r}\right)}{r^{2}}-\frac{1}{3}\bm{S}_{1}\cdot\bm{S}_{2}.$

Here, $E_{i}$, $m_{i}$, and $\mathbb{T}$ denote the relativistic energy of
each (anti)quark $i$, the mass of each (anti)quark $i$, and the operator of
the tensor coupling interaction (appraised by the identity from Landau and
Lifshitz Landau:1991wop ; Ali:2017wsf  or the Wigner-Eckart theorem
Varshalovich:1988krb ), respectively. Taking into account the momentum
dependence of the potentials, several sorts of factors are affixed to the
smeared Coulomb and linear confinement potentials ($\tilde{G}_{ij}$ and
$\tilde{S}_{ij}$), conducing to the momentum-dependent Coulomb, contact,
tensor, vector spin-orbit, and scalar spin-orbit potentials
($\tilde{G}_{ij}^{\rm Coul}$, $\tilde{G}_{ij}^{\rm cont}$,
$\tilde{G}_{ij}^{\rm ten}$, $\tilde{G}_{ij}^{\rm so(v)}$, and
$\tilde{S}_{ij}^{\rm so(s)}$) which contain universal parameters
($\epsilon_{\rm cont}$, $\epsilon_{\rm ten}$, $\epsilon_{\rm so(v)}$, and
$\epsilon_{\rm so(s)}$). As far as the smeared potential $\tilde{f}_{ij}(r)$
is concerned Godfrey:1985xj , the definition of the smearing procedure is

$\displaystyle\tilde{f}_{ij}(r)$ $\displaystyle\equiv$ $\displaystyle\int
d^{3}r^{\prime}\rho_{ij}({\bf r}-{\bf r}^{\prime})f(r^{\prime}),$ (2.7)

with

$\displaystyle\rho_{ij}({\bf r}-{\bf
r}^{\prime})=\frac{\sigma_{ij}^{3}}{\pi^{\frac{3}{2}}}e^{-\sigma_{ij}^{2}({\bf
r}-{\bf r}^{\prime})^{2}},$ (2.8)
$\displaystyle\sigma_{ij}=\sqrt{\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{i}m_{j}}{(m_{i}+m_{j})^{2}}\right)^{4}\right]+s^{2}\left(\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{2}}.$
(2.9)

Here, both $\sigma_{0}$ and $s$ are the universal parameters of the GI model
Godfrey:1985xj . Subtly, the Coulomb potential $G(r)$ with the
$\gamma^{\mu}\otimes\gamma_{\mu}$ short-range interaction and the linear
potential $S(r)$ with the $1\otimes 1$ long-range interaction are recast into
the smeared Coulomb potential $\tilde{G}_{ij}(r)$ and the smeared linear
potential $\tilde{S}_{ij}(r)$, respectively, by means of the employment of the
smearing function $\rho_{ij}({\bf r}-{\bf r}^{\prime})$. Conveniently, the
potentials $G(r)$ and $S(r)$ are expressed as

$\displaystyle G(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}(r)}{4r}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(2.10) $\displaystyle S(r)$ $\displaystyle=$
$\displaystyle-\frac{3}{16}(br+c)\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(2.11)

with

$\displaystyle\alpha_{s}(r)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{3}\alpha_{k}{\rm erf}(\gamma_{k}r),$ (2.12)
$\displaystyle{\rm erf}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi^{\frac{1}{2}}}\int_{0}^{x}e^{-t^{2}}dt.$ (2.13)

Here, all of the arisen parameters ($\alpha_{k=1,2,3}$, $\gamma_{k=1,2,3}$,
$b$, and $c$) are universal in the GI model Godfrey:1985xj . Besides,
$\bm{\lambda}_{i}$, $\alpha_{s}(r)$, and ${\rm erf}(x)$ denote the Gell-Mann
matrices acting on each (anti)quark $i$, the running coupling constant of QCD,
and the error function, respectively. Evidently, it is facile to appraise the
color operator $\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}$ by dint of the
eigenvalue of the quadratic Casimir operator Vijande:2009ac . Peculiarly, for
the case of tetraquarks constituted by diquark and antidiquark, the off-
diagonal part of the tensor potential $V_{\rm GI}^{\rm ten}$ can not only
cause ${}^{3}L_{J}$$\leftrightarrow$${}^{3}L^{\prime}_{J}$ and
${}^{5}L_{J}$$\leftrightarrow$${}^{5}L^{\prime}_{J}$ mixings but also cause
${}^{1}L_{J}$$\leftrightarrow$${}^{5}L_{J}$ mixing, quite dissimilar from the
counterpart of mesons that can only cause
${}^{3}L_{J}$$\leftrightarrow$${}^{3}L^{\prime}_{J}$ mixing. Moreover,
${}^{1}L_{J}$$\leftrightarrow$${}^{3}L_{J}$ and
${}^{3}L_{J}$$\leftrightarrow$${}^{5}L_{J}$ mixings are caused by the off-
diagonal part of the spin-orbit potential $V_{\rm GI}^{\rm so}$. Following
Refs. Godfrey:1985xj ; Godfrey:2016nwn ; Godfrey:2015dva ; Godfrey:2004ya ,
this work tackles ${}^{1}L_{J}$$\leftrightarrow$${}^{3}L_{J}$,
${}^{3}L_{J}$$\leftrightarrow$${}^{5}L_{J}$, and
${}^{1}L_{J}$$\leftrightarrow$${}^{5}L_{J}$ mixings perturbatively, omitting
${}^{3}L_{J}$$\leftrightarrow$${}^{3}L^{\prime}_{J}$ and
${}^{5}L_{J}$$\leftrightarrow$${}^{5}L^{\prime}_{J}$ mixings for convenience.

Currently, the perception of diquark has been pervasively employed to the
theoretical interpretations on the miscellaneous properties of baryons and
multiquarks Anselmino:1992vg ; Jaffe:2004ph ; Barabanov:2020jvn , for
instance, spectroscopy, production, magnetic moments, form factors, and decay
properties. On the basis of the color-triplet representation of the quark, the
color SU(3) representation of the diquark is antitriplet or sextet, i.e.,

$\displaystyle{\bf 3}_{q}\otimes{\bf 3}_{q}$ $\displaystyle=$
$\displaystyle\bar{\bf 3}_{qq}\oplus{\bf 6}_{qq},$ (2.14)
$\displaystyle\bar{\bf 3}_{\bar{q}}\otimes\bar{\bf 3}_{\bar{q}}$
$\displaystyle=$ $\displaystyle{\bf 3}_{\bar{q}\bar{q}}\oplus\bar{\bf
6}_{\bar{q}\bar{q}}.$ (2.15)

Remarkably, in regard to the color-(anti)sextet (anti)diquark, the matrix
element of the color operator $\bm{\lambda}_{i}\cdot\bm{\lambda}_{j}$ is
positive $4/3$, signifying that the repulsive interquark force deters the
forming of the color-(anti)sextet (anti)diquark in the diquark model
Lipkin:1986dw ; Karliner:2017qjm ; Eichten:2017ffp ; Bedolla:2019zwg ;
Lu:2019ira ; Lu:2016zhe ; Lu:2016cwr ; Dong:2022sef ; Ferretti:2019zyh . In
consequence, the diquark-antidiquark scenarios utilized by this work merely
treat the color-antitriplet diquark and the color-triplet antidiquark as the
genuinely effective (anti)diquark to decode the spectroscopy of the
$cc\bar{u}\bar{d}$ states. Overtly, the color-antitriplet diquark (color-
triplet antidiquark) is capable of being approximately deemed as the color-
antitriplet antiquark (color-triplet quark) owing to the equivalent color
between them. The GI relativized diquark model is effectuated in two steps.
Initially, the masses of the doubly charmed diquark, the isoscalar light
diquark, and the isovector light diquark are procured. Subsequently, the
spectroscopic properties of the low-lying $T_{cc}$ tetraquark states are
probed by reckoning the diquark (antidiquark) as the antiquark (quark). The
outcomes of the GI relativized diquark model are unveiled in Section III.

###  II.2 Modified Godfrey-Isgur (MGI) relativized diquark model with the
color screening effects

Albeit the long-range interaction between (anti)quarks can be delineated by
the Lorentz-scalar linear potential $S(r)$ Godfrey:1985xj , the vacuum
polarization effects of dynamical fermions may induce the fracture of the
color flux tube at large distances Born:1989iv , i.e., the color screening
effects (also known as the string breaking effects). A successful employment
of the nonrelativistic potential model with the color screening effects is the
spectroscopic investigation of heavy quarkonia Li:2009zu , via the replacement
of the linear potential $S(r)$ by the screened linear potential $S^{\rm
scr}(r)$ whose form is

$\displaystyle S^{\rm scr}(r)$ $\displaystyle=$
$\displaystyle-\frac{3}{16}\left(b\frac{1-e^{-\mu
r}}{\mu}+c\right)\bm{\lambda}_{1}\cdot\bm{\lambda}_{2}.$ (2.16)

Here, $\mu$ denotes the screening factor, flattening the linear potential with
the $1\otimes 1$ long-range interaction when the interquark distance $r$ is
large enough. In the tiny and large $r$ limits, the screened linear potential
$S^{\rm scr}(r)$ is approximated as

$\displaystyle S^{\rm scr}(r)\rightarrow\left\\{\begin{aligned}
S(r)=&-\frac{3}{16}(br+c)\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},&&r\rightarrow
0,\\\
c_{\mu}=&-\frac{3}{16}\left(\frac{b}{\mu}+c\right)\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},&&r\rightarrow\infty,\end{aligned}\right.$
(2.17)

which demonstrates that the screened linear potential $S^{\rm scr}(r)$ is
reverted to the linear potential $S(r)$ in the tiny $r$ limit, and reduced to
a particular constant $c_{\mu}$ that contains a certain saturation distance
$\mu^{-1}$ in the large $r$ limit Mezoir:2008vx .

The modified Godfrey-Isgur relativized quark model (MGI model) is elicited by
incorporating the color screening effects into the orthodox GI model
Song:2015nia . Specifically, the smeared linear potential $\tilde{S}_{ij}(r)$
in the GI model is substituted by the smeared screened linear potential
$\tilde{S}_{ij}^{\rm scr}(r)$ in the MGI model, acquired by embedding the
screened linear potential $S^{\rm scr}(r)$ in Eq. (2.7). The circumstantial
introduction and correlative details with respect to the MGI model are
unfolded in Ref. Song:2015nia . Thus far, the spectroscopic properties of
light mesons Pang:2017dlw , singly heavy mesons Song:2015nia ; Song:2015fha ,
doubly heavy mesons Wang:2018rjg , light tetraquarks Lu:2019ira , heavy
tetraquarks Lu:2016cwr ; Dong:2022sef , and diquarks Lu:2019ira ; Lu:2016cwr ;
Dong:2022sef  have been copiously explored by the MGI model. Mimicking the
effectuation of the GI relativized diquark model, the MGI relativized diquark
model with the color screening effects is carried out in two aforementioned
steps. The outcomes of the MGI relativized diquark model are laid out in
Section III.

###  II.3 Nonrelativistic (NR) diquark models

As is well known, a prestigious paragon of the nonrelativistic quark model (NR
model) is the Cornell potential model which superbly predicted the mass
spectra of heavy quarkonia Eichten:1979ms . Conventionally, the Hamiltonian of
the NR model is decomposed into

$\displaystyle H_{\rm NR}$ $\displaystyle=$ $\displaystyle H_{\rm
NR}^{0}+V_{\rm NR}^{\rm si}+V_{\rm NR}^{\rm sd}$ (2.18) $\displaystyle=$
$\displaystyle H_{\rm NR}^{0}+V_{\rm NR}^{\rm conf}+V_{\rm NR}^{\rm
cont}+V_{\rm NR}^{\rm ten}+V_{\rm NR}^{\rm so},$

where $H_{\rm NR}^{0}$, $V_{\rm NR}^{\rm si}$, and $V_{\rm NR}^{\rm sd}$
denote the nonrelativistic energy of entire (anti)quarks, the spin-independent
term (composed of the confinement potential $V_{\rm NR}^{\rm conf}$), and the
spin-dependent term (composed of the contact potential $V_{\rm NR}^{\rm
cont}$, the tensor potential $V_{\rm NR}^{\rm ten}$, and the spin-orbit
potential $V_{\rm NR}^{\rm so}$), respectively. The form of $H_{\rm NR}^{0}$
is expressed as

$\displaystyle H_{\rm NR}^{0}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\mathcal{E}_{i}(p),$ (2.19)

with

$\displaystyle\mathcal{E}_{i}(p)$ $\displaystyle=$ $\displaystyle
m_{i}+\frac{p^{2}}{2m_{i}}.$ (2.20)

Here, $\mathcal{E}_{i}$ is the nonrelativistic energy of each (anti)quark $i$.
As far as the forms of the spin-independent term $V_{\rm NR}^{\rm si}$ and the
spin-dependent term $V_{\rm NR}^{\rm sd}$ are concerned, there are two
scenarios which are capable of characterizing the spectroscopy of heavy-light
hadrons superiorly Kim:2020imk ; Kim:2021ywp . Based on the distinctive
functional forms of the hyperfine interaction potentials, these two scenarios
are elucidated as follows.

####  II.3.1 Scenario I: NR-G diquark model

Following Refs. Godfrey:1985xj ; Barnes:2005pb ; Kim:2020imk , the
nonrelativistic quark model with the Gaussian contact hyperfine interaction
(NR-G model) is employed in the first scenario. To be specific, the terms
$V_{\text{NR-G}}^{\rm conf}$, $V_{\text{NR-G}}^{\rm cont}$,
$V_{\text{NR-G}}^{\rm ten}$, and $V_{\text{NR-G}}^{\rm so}$ possess the forms
of

$\displaystyle V_{\text{NR-G}}^{\rm conf}$ $\displaystyle=$ $\displaystyle
G^{\rm Coul}(r)+S(r),$ (2.21) $\displaystyle V_{\text{NR-G}}^{\rm cont}$
$\displaystyle=$ $\displaystyle\frac{2}{3m_{1}m_{2}r^{2}}\frac{\rm d}{{\rm
d}r}\left[r^{2}\frac{{\rm d}G^{\rm Gauss}(r)}{{\rm
d}r}\right]\bm{S}_{1}\cdot\bm{S}_{2},$ (2.22) $\displaystyle
V_{\text{NR-G}}^{\rm ten}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{1}m_{2}}\left(\frac{1}{r}-\frac{\rm d}{{\rm
d}r}\right)\frac{{\rm d}G^{\rm Coul}(r)}{{\rm d}r}\mathbb{T},$ (2.23)
$\displaystyle V_{\text{NR-G}}^{\rm so}$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{1}^{2}r}\left[\frac{{\rm d}G^{\rm Coul}(r)}{{\rm
d}r}-\frac{{\rm d}S(r)}{{\rm d}r}\right]\bm{L}\cdot\bm{S}_{1}$ (2.24)
$\displaystyle+\frac{1}{2m_{2}^{2}r}\left[\frac{{\rm d}G^{\rm Coul}(r)}{{\rm
d}r}-\frac{{\rm d}S(r)}{{\rm d}r}\right]\bm{L}\cdot\bm{S}_{2}$
$\displaystyle+\frac{1}{m_{1}m_{2}r}\frac{{\rm d}G^{\rm Coul}(r)}{{\rm
d}r}\bm{L}\cdot\bm{S},$

with

$\displaystyle G^{\rm Coul}(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{c}}{4r}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(2.25) $\displaystyle G^{\rm Gauss}(r)$ $\displaystyle=$
$\displaystyle\frac{\mathfrak{\alpha}_{g}(r)}{4r}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(2.26) $\displaystyle\alpha_{g}(r)$ $\displaystyle=$
$\displaystyle\alpha_{c}{\rm erf}(\gamma_{c}r).$ (2.27)

Here, all of the corresponding parameters of the NR-G model ($\alpha_{c}$,
$\gamma_{c}$, $b$, and $c$) stem from the spectroscopic inquiries of
conventional heavy hadrons Barnes:2005pb ; Kim:2020imk . A salient feature of
the NR-G model is the emergence of a Gaussian function in the Laplace operator
of $G^{\rm Gauss}(r)$ which contains the error function ${\rm
erf}(\gamma_{c}r)$. Manifestly, the form of $\alpha_{g}(r)$ in $G^{\rm
Gauss}(r)$ is embodied as the approximation of $\alpha_{s}(r)$ in $G(r)$.
Additionally, the form of $G^{\rm Coul}(r)$ prevalently utilized in the
Cornell potential model Eichten:1979ms  will be capable of being looked upon
as the approximate form of $G(r)$, if the distance-dependent $\alpha_{s}(r)$
is simplified as the constant $\alpha_{c}$. By comparing Eqs. (2.21)-(2.24)
with Eqs. (2.3)-(2.6), it is facile to discern an exquisite similitude of the
forms of the interaction potentials between the NR-G model and the GI model
Godfrey:1985xj . Imitating Ref. Barnes:2005pb , the leading-order perturbation
theory is employed to regard the tensor potential $V_{\text{NR-G}}^{\rm ten}$
and the spin-orbit potential $V_{\text{NR-G}}^{\rm so}$ as mass shifts, which
touch upon the diagonal terms and the off-diagonal parts. The particular
introduction and pertinent details with respect to the NR-G model are
elaborated in Refs. Godfrey:1985xj ; Barnes:2005pb ; Kim:2020imk . In
consideration of the identical color SU(3) representation between the diquark
(antidiquark) and the antiquark (quark), the NR-G diquark model is performed
in the framework of the diquark-antidiquark configuration. Concerning the
spectroscopic properties of the doubly charmed tetraquark system, the
predicted outcomes of the NR-G diquark model are revealed in Section III.

####  II.3.2 Scenario II: NR-Y diquark model

The second scenario adopts the nonrelativistic quark model whose contact
hyperfine interaction is of the Yukawa form (NR-Y model) in light of Refs.
Kim:2020imk ; Kim:2021ywp ; Yoshida:2015tia . Further, the potentials
$V_{\text{NR-Y}}^{\rm conf}$, $V_{\text{NR-Y}}^{\rm cont}$,
$V_{\text{NR-Y}}^{\rm ten}$, and $V_{\text{NR-Y}}^{\rm so}$ are represented as

$\displaystyle V_{\text{NR-Y}}^{\rm conf}$ $\displaystyle=$
$\displaystyle\frac{m_{1}+m_{2}}{m_{1}m_{2}}G^{\rm Coul}(r)+S(r),$ (2.28)
$\displaystyle V_{\text{NR-Y}}^{\rm cont}$ $\displaystyle=$
$\displaystyle\frac{2}{3m_{1}m_{2}r^{2}}\frac{\rm d}{{\rm
d}r}\left[r^{2}\frac{{\rm d}G^{\rm Yukawa}(r)}{{\rm
d}r}\right]\bm{S}_{1}\cdot\bm{S}_{2},$ (2.29) $\displaystyle
V_{\text{NR-Y}}^{\rm ten}$ $\displaystyle=$
$\displaystyle\frac{(1-e^{-\gamma_{c}r})^{2}}{m_{1}m_{2}}\left(\frac{1}{r}-\frac{\rm
d}{{\rm d}r}\right)\frac{{\rm d}G^{\rm ten}(r)}{{\rm d}r}\mathbb{T},$ (2.30)
$\displaystyle V_{\text{NR-Y}}^{\rm so}$ $\displaystyle=$
$\displaystyle\frac{(1-e^{-\gamma_{c}r})^{2}}{2m_{1}^{2}r}\frac{{\rm d}G^{\rm
so}(r)}{{\rm d}r}\bm{L}\cdot\bm{S}_{1}$ (2.31)
$\displaystyle+\frac{(1-e^{-\gamma_{c}r})^{2}}{2m_{2}^{2}r}\frac{{\rm d}G^{\rm
so}(r)}{{\rm d}r}\bm{L}\cdot\bm{S}_{2}$
$\displaystyle+\frac{(1-e^{-\gamma_{c}r})^{2}}{m_{1}m_{2}r}\frac{{\rm d}G^{\rm
so}(r)}{{\rm d}r}\bm{L}\cdot\bm{S},$

with

$\displaystyle G^{\rm ten/so}(r)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\rm
ten/so}}{4r}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$ (2.32) $\displaystyle
G^{\rm Yukawa}(r)$ $\displaystyle=$
$\displaystyle\frac{\mathfrak{\alpha}_{y}(r)}{4r}\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(2.33) $\displaystyle\alpha_{y}(r)$ $\displaystyle=$
$\displaystyle-\alpha_{\rm cont}e^{-\gamma_{c}r}.$ (2.34)

It is notable that the form of $G^{\rm Yukawa}(r)$ which includes the
exponential function ${\rm exp}(-\gamma_{c}r)$ extraordinarily resembles the
canonical form of the Yukawa potential. In the NR-Y model, the total relevant
parameters ($\alpha_{\rm cont}$, $\alpha_{\rm ten}$, $\alpha_{\rm so}$,
$\alpha_{c}$, $\gamma_{c}$, $b$, and $c$) are determined by the mass spectra
of heavy-light hadrons Kim:2020imk ; Kim:2021ywp . In view of the latent
quark-mass dependence of the Coulombic parameter alluded by a lattice QCD
survey Kawanai:2011xb , the reciprocal of the reduced mass was ushered into
the Coulomb term of the NR-Y model Kim:2020imk ; Kim:2021ywp ; Yoshida:2015tia
, as exhibited in Eq. (2.28). The circumstantial introduction and correlative
details with respect to the NR-Y model are unfolded in Refs. Kim:2020imk ;
Kim:2021ywp ; Yoshida:2015tia . When it comes to the $T_{cc}$ tetraquark
comprised of the double-charm diquark and the light antidiquark, the advent of
five sorts of mixings (${}^{1}L_{J}$$\leftrightarrow$${}^{3}L_{J}$,
${}^{3}L_{J}$$\leftrightarrow$${}^{5}L_{J}$,
${}^{1}L_{J}$$\leftrightarrow$${}^{5}L_{J}$,
${}^{3}L_{J}$$\leftrightarrow$${}^{3}L^{\prime}_{J}$, and
${}^{5}L_{J}$$\leftrightarrow$${}^{5}L^{\prime}_{J}$) caused by the tensor and
spin-orbit potentials ($V_{\text{NR-Y}}^{\rm ten}$ and $V_{\text{NR-Y}}^{\rm
so}$) is inevitable. Following Ref. Kim:2021ywp ,
${}^{1}L_{J}$$\leftrightarrow$${}^{3}L_{J}$,
${}^{3}L_{J}$$\leftrightarrow$${}^{5}L_{J}$, and
${}^{1}L_{J}$$\leftrightarrow$${}^{5}L_{J}$ mixings are treated perturbatively
by the NR-Y diquark model, then remnant
${}^{3}L_{J}$$\leftrightarrow$${}^{3}L^{\prime}_{J}$ and
${}^{5}L_{J}$$\leftrightarrow$${}^{5}L^{\prime}_{J}$ mixings are left out for
convenience. As expounded in the GI relativized diquark model, the color-
antitriplet diquark $cc$ and the color-triplet antidiquark $\bar{u}\bar{d}$
are deemed as the genuinely effective (anti)diquark to unravel the
spectroscopy of the exotic $cc\bar{u}\bar{d}$ states in the NR-Y diquark
model. Subsequently, the particular results of the mass spectrum of the low-
lying $T_{cc}$ tetraquark family reaped by the NR-Y diquark model are
displayed in Section III.

##  III Results

Table 1: Parameters of the GI (MGI) relativized diquark model Godfrey:1985xj , NR-G diquark model Barnes:2005pb ; Kim:2020imk , and NR-Y diquark model Kim:2020imk ; Kim:2021ywp . GI (MGI) Godfrey:1985xj  | NR-G Barnes:2005pb ; Kim:2020imk  | NR-Y Kim:2020imk ; Kim:2021ywp
---|---|---
Parameter | Value | Parameter | Value | Parameter | Value | Parameter | Value
$m_{c}$ (GeV) | $1.628$ |  $m_{u,d}$ (GeV) | $0.220$ |  $m_{c}$ (GeV) | $1.4794$ |  $m_{c}$ (GeV) | $1.750$
$b$ (GeV2) | $0.18$ |  $c$ (GeV) | $-0.253$ |  $b$ (GeV2) | $0.1425$ |  $b$ (GeV2) | $0.165$
$\gamma_{1}$ (GeV) | $\sqrt{1/4}$ | $\alpha_{1}$ | $0.25$ |  $c$ (GeV) | $-0.191$ |  $c$ (GeV) | $-0.831$
$\gamma_{2}$ (GeV) | $\sqrt{10/4}$ | $\alpha_{2}$ | $0.15$ |  $\gamma_{c}$ (GeV) | $1.0946$ |  $\gamma_{c}$ (GeV) | $0.691$
$\gamma_{3}$ (GeV) | $\sqrt{1000/4}$ | $\alpha_{3}$ | $0.20$ | $\alpha_{c}$ | $0.5461$ |  $\alpha_{c}$ (GeV) | $0.045$
$\sigma_{0}$ (GeV) | $1.80$ | $s$ | $1.55$ | $\cdots$ | $\cdots$ | $\alpha_{\rm cont}$ | $0.9659$
$\epsilon_{\rm cont}$ | $-0.168$ | $\epsilon_{\rm so(v)}$ | $-0.035$ | $\cdots$ | $\cdots$ | $\alpha_{\rm ten}$ | $0.3741$
$\epsilon_{\rm ten}$ | $0.025$ | $\epsilon_{\rm so(s)}$ | $0.055$ | $\cdots$ | $\cdots$ | $\alpha_{\rm so}$ | $0.7482$

This section exhaustively reports the predicted outcomes on the spectroscopic
properties of the low-lying doubly charmed tetraquark family, in virtue of
four sorts of aforementioned diquark-antidiquark scenarios, i.e., GI
relativized diquark model, MGI relativized diquark model, NR-G diquark model,
and NR-Y diquark model.

###  III.1 Parameters

As enumerated in Table 1, the parameters of the GI (MGI) relativized diquark
model, NR-G diquark model, and NR-Y diquark model with respect to the mass
spectra of doubly charmed tetraquark states are designated to keep pace with
the counterparts employed by Ref. Godfrey:1985xj , Refs. Barnes:2005pb ;
Kim:2020imk , and Refs. Kim:2020imk ; Kim:2021ywp , respectively, in the cause
of retaining the model universality between conventional and exotic hadrons.

###  III.2 Diquarks

As the essential constituents of the doubly charmed tetraquark, doubly charmed
diquark $cc$ and light diquark $ud$ play a crucial role in deciphering
$T_{cc}$ spectroscopy. Constrained by the Pauli exclusion principle
Amsler:2018zkm , the spin quantum number of the ground state diquark $cc$ with
antitriplet color is endowed with $1$. Hence, the $S$-wave doubly charmed
diquark employed by this work is an axial-vector diquark due to the positive
internal parity Amsler:2018zkm . As delineated in Section II, the GI (MGI)
model procures the mass of the color-antitriplet diquark $cc$ via the
universal parameters Godfrey:1985xj . Analogously, the NR-G model reaps the
mass of the diquark $cc$ by dint of the parameters of the charmonium family
Barnes:2005pb . Taking into account the absence of the pertinent employment of
charmonium spectroscopy, the NR-Y model makes use of the mass relations of
heavy-light hadrons proposed by the heavy quark symmetry (HQS) to garner the
diquark $cc$ mass Eichten:2017ffp , i.e.,

$\displaystyle m_{\\{cc\\}[\bar{u}\bar{d}]}-m_{\\{cc\\}u}$ $\displaystyle=$
$\displaystyle m_{c[ud]}-m_{c\bar{u}},$ (3.1) $\displaystyle
m_{\\{cc\\}[\bar{u}\bar{d}]}-m_{c[ud]}$ $\displaystyle=$ $\displaystyle
m_{\\{cc\\}u}-m_{c\bar{u}},$ (3.2)

where the braces $\\{qq\\}$ and the brackets $[qq]$ denote the axial-vector
diquark and the scalar diquark Eichten:2017ffp , respectively. By averaging
the LHS and RHS of Eq. (3.2), the mass gap between doubly charmed diquark and
charm quark in the NR-Y model is acquired decently, i.e.,

$\displaystyle m_{\\{cc\\}}-m_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(m_{\\{cc\\}[\bar{u}\bar{d}]}-m_{c[ud]}+m_{\\{cc\\}u}-m_{c\bar{u}}),$
(3.3)

where the value of $m_{c}$ is rendered in the last column of Table 1.
Inserting the NR-Y model mass of charm quark and the experimental masses of
corresponding heavy-light hadrons into Eq. (3.3), the diquark $cc$ mass
adopted by the NR-Y model is determined legitimately. Alternatively, on the
basis of the heavy antiquark-diquark symmetry (HADS), the mass formula of the
doubly charmed diquark $cc$ in the NR-Y model is offered by Ref. Wu:2022gie ,
i.e.,

$\displaystyle m_{\\{cc\\}}$ $\displaystyle=$ $\displaystyle
2m_{c}+(A_{cc}\bm{S}_{1}\cdot\bm{S}_{2}+\frac{B_{cc}}{4})\bm{\lambda}_{1}\cdot\bm{\lambda}_{2},$
(3.4)

where the parameters $A_{cc}=-21.2$ MeV and $B_{cc}=217.7$ MeV are determined
from experimental data of the charmonium spectrum and the doubly charmed
baryon spectrum Wu:2022gie , respectively. After the substitution of the NR-Y
model mass of charm quark, the NR-Y model mass of the color-antitriplet
diquark $cc$ obtained by HADS is magically identical with the one previously
attained by HQS, which effectively illustrates the resemblance between HQS and
HADS. The particular introduction and pertinent details with respect to HQS
and HADS are elaborated in Refs. Eichten:2017ffp ; Wu:2022gie . Subsequently,
the masses of the ground state axial-vector doubly charmed diquark from these
four sorts of diquark-antidiquark scenarios are explicitly enumerated in Table
2. Concretely, the values of the diquark $cc$ masses fulfilled by the GI, MGI
($\mu=30$), MGI ($\mu=50$), MGI ($\mu=70$), NR-G, and NR-Y scenarios are 3329,
3320, 3314, 3309, 3152, and 3369 MeV, respectively.

Table 2: The masses of the doubly charmed and light diquarks from the GI, MGI, NR-G, and NR-Y scenarios (in unit of MeV). Scenario | $m_{\\{cc\\}}$ | $m_{[ud]}$ | $m_{\\{ud\\}}$
---|---|---|---
GI | $3329$ | $691$ | $840$
MGI ($\mu=30$) | $3320$ | $673$ | $814$
MGI ($\mu=50$) | $3314$ | $662$ | $796$
MGI ($\mu=70$) | $3309$ | $650$ | $778$
NR-G | $3152$ | $725$ | $1019$
NR-Y | $3369$ | $725$ | $973$

As far as the light diquark $ud$ is concerned, both scalar and axial-vector
cases are eligible owing to the symmetry of wave functions Amsler:2018zkm . In
accordance with the universal parameters stemmed from a variety of mesons with
discrepant flavors Godfrey:1985xj , the masses of scalar and axial-vector
light diquarks are obtained by the GI (MGI) model. On account of the
deficiency of adequate spectroscopic applications of light mesons, the NR
models take advantage of the light diquark masses deduced by unquenched
lattice QCD and chiral effective theory, whose circumstantial introduction and
correlative details are unfolded in Refs. Kim:2020imk ; Kim:2021ywp ;
Bi:2015ifa ; Harada:2019udr . Whereafter, the scalar and axial-vector light
diquark masses from aforementioned diquark-antidiquark scenarios are expressly
particularized in Table 2. Specifically, the values of scalar light diquark
masses carried out by the GI, MGI ($\mu=30$), MGI ($\mu=50$), MGI ($\mu=70$),
NR-G, and NR-Y scenarios are 691, 673, 662, 650, 725, and 725 MeV,
respectively. In the case of axial-vector light diquark, the masses performed
by the GI, MGI ($\mu=30$), MGI ($\mu=50$), MGI ($\mu=70$), NR-G, and NR-Y
scenarios are 840, 814, 796, 778, 1019, and 973 MeV, respectively.

###  III.3 Doubly charmed tetraquarks

Table 3: The mass spectrum of the $1S$-, $1P$-, $2S$-, and $1D$-wave doubly charmed tetraquark states procured by this work (in unit of MeV). State | Mass
---|---
$T_{cc}(n^{2S+1}L_{J})$ |  $I(J^{P})$ |  GI |  MGI ($\mu=30$) |  MGI ($\mu=50$) |  MGI ($\mu=70$) |  NR-G |  NR-Y
$T_{cc1}^{f}(1^{3}S_{1})$ |  $0(1^{+})$ |  $3948$ |  $3917$ |  $3897$ |  $3877$ |  $3884$ |  $3876$
$T_{cc0}^{a}(1^{1}S_{0})$ |  $1(0^{+})$ |  $3842$ |  $3809$ |  $3787$ |  $3766$ |  $3894$ |  $4029$
$T_{cc1}^{a}(1^{3}S_{1})$ |  $1(1^{+})$ |  $3960$ |  $3925$ |  $3902$ |  $3879$ |  $3991$ |  $4057$
$T_{cc2}^{a}(1^{5}S_{2})$ |  $1(2^{+})$ |  $4121$ |  $4083$ |  $4058$ |  $4032$ |  $4135$ |  $4105$
$T_{cc0}^{\eta}(1^{3}P_{0})$ |  $0(0^{-})$ |  $4349$ |  $4307$ |  $4279$ |  $4252$ |  $4289$ |  $4161$
$T_{cc1}^{\eta}(1^{3}P_{1})$ |  $0(1^{-})$ |  $4370$ |  $4327$ |  $4299$ |  $4271$ |  $4311$ |  $4174$
$T_{cc2}^{\eta}(1^{3}P_{2})$ |  $0(2^{-})$ |  $4408$ |  $4364$ |  $4335$ |  $4305$ |  $4356$ |  $4199$
$T_{cc0}^{\pi}(1^{3}P_{0})$ |  $1(0^{-})$ |  $4404$ |  $4357$ |  $4326$ |  $4295$ |  $4408$ |  $4310$
$T_{cc1}^{\pi}(1P_{1})$ |  $1(1^{-})$ |  $4477$ |  $4432$ |  $4400$ |  $4369$ |  $4534$ |  $4370$
$T_{cc1}^{\pi}(1P_{1}^{\prime})$ |  $1(1^{-})$ |  $4480$ |  $4427$ |  $4394$ |  $4361$ |  $4501$ |  $4331$
$T_{cc1}^{\pi}(1P_{1}^{\prime\prime})$ |  $1(1^{-})$ |  $4410$ |  $4362$ |  $4331$ |  $4300$ |  $4425$ |  $4308$
$T_{cc2}^{\pi}(1P_{2})$ |  $1(2^{-})$ |  $4501$ |  $4452$ |  $4419$ |  $4388$ |  $4554$ |  $4386$
$T_{cc2}^{\pi}(1P_{2}^{\prime})$ |  $1(2^{-})$ |  $4503$ |  $4453$ |  $4420$ |  $4385$ |  $4535$ |  $4354$
$T_{cc3}^{\pi}(1^{5}P_{3})$ |  $1(3^{-})$ |  $4529$ |  $4479$ |  $4446$ |  $4412$ |  $4577$ |  $4409$
$T_{cc1}^{f}(2^{3}S_{1})$ |  $0(1^{+})$ |  $4534$ |  $4478$ |  $4441$ |  $4405$ |  $4489$ |  $4388$
$T_{cc0}^{a}(2^{1}S_{0})$ |  $1(0^{+})$ |  $4554$ |  $4497$ |  $4460$ |  $4423$ |  $4609$ |  $4523$
$T_{cc1}^{a}(2^{3}S_{1})$ |  $1(1^{+})$ |  $4595$ |  $4537$ |  $4498$ |  $4459$ |  $4637$ |  $4539$
$T_{cc2}^{a}(2^{5}S_{2})$ |  $1(2^{+})$ |  $4674$ |  $4612$ |  $4570$ |  $4529$ |  $4694$ |  $4568$
$T_{cc1}^{f}(1^{3}D_{1})$ |  $0(1^{+})$ |  $4689$ |  $4631$ |  $4592$ |  $4554$ |  $4607$ |  $4428$
$T_{cc2}^{f}(1^{3}D_{2})$ |  $0(2^{+})$ |  $4701$ |  $4643$ |  $4603$ |  $4564$ |  $4616$ |  $4440$
$T_{cc3}^{f}(1^{3}D_{3})$ |  $0(3^{+})$ |  $4718$ |  $4659$ |  $4619$ |  $4579$ |  $4631$ |  $4457$
$T_{cc0}^{a}(1^{5}D_{0})$ |  $1(0^{+})$ |  $4776$ |  $4712$ |  $4670$ |  $4628$ |  $4790$ |  $4549$
$T_{cc1}^{a}(1D_{1})$ |  $1(1^{+})$ |  $4793$ |  $4729$ |  $4686$ |  $4643$ |  $4789$ |  $4575$
$T_{cc1}^{a}(1D_{1}^{\prime})$ |  $1(1^{+})$ |  $4783$ |  $4720$ |  $4678$ |  $4636$ |  $4803$ |  $4558$
$T_{cc2}^{a}(1D_{2})$ |  $1(2^{+})$ |  $4792$ |  $4730$ |  $4688$ |  $4646$ |  $4790$ |  $4616$
$T_{cc2}^{a}(1D_{2}^{\prime})$ |  $1(2^{+})$ |  $4822$ |  $4756$ |  $4712$ |  $4668$ |  $4828$ |  $4588$
$T_{cc2}^{a}(1D_{2}^{\prime\prime})$ |  $1(2^{+})$ |  $4799$ |  $4735$ |  $4693$ |  $4650$ |  $4801$ |  $4575$
$T_{cc3}^{a}(1D_{3})$ |  $1(3^{+})$ |  $4802$ |  $4739$ |  $4696$ |  $4654$ |  $4797$ |  $4628$
$T_{cc3}^{a}(1D_{3}^{\prime})$ |  $1(3^{+})$ |  $4821$ |  $4756$ |  $4713$ |  $4669$ |  $4820$ |  $4604$
$T_{cc4}^{a}(1^{5}D_{4})$ |  $1(4^{+})$ |  $4813$ |  $4750$ |  $4707$ |  $4664$ |  $4805$ |  $4644$
Table 4: The mass spectrum of the $2P$-, $3S$-, and $1F$-wave doubly charmed tetraquark states procured by this work (in unit of MeV). State | Mass
---|---
$T_{cc}(n^{2S+1}L_{J})$ |  $I(J^{P})$ |  GI |  MGI ($\mu=30$) |  MGI ($\mu=50$) |  MGI ($\mu=70$) |  NR-G |  NR-Y
$T_{cc0}^{\eta}(2^{3}P_{0})$ |  $0(0^{-})$ |  $4796$ |  $4725$ |  $4677$ |  $4629$ |  $4734$ |  $4605$
$T_{cc1}^{\eta}(2^{3}P_{1})$ |  $0(1^{-})$ |  $4810$ |  $4738$ |  $4689$ |  $4641$ |  $4755$ |  $4615$
$T_{cc2}^{\eta}(2^{3}P_{2})$ |  $0(2^{-})$ |  $4837$ |  $4762$ |  $4713$ |  $4663$ |  $4795$ |  $4634$
$T_{cc0}^{\pi}(2^{3}P_{0})$ |  $1(0^{-})$ |  $4872$ |  $4798$ |  $4748$ |  $4698$ |  $4833$ |  $4727$
$T_{cc1}^{\pi}(2P_{1})$ |  $1(1^{-})$ |  $4908$ |  $4832$ |  $4781$ |  $4730$ |  $4949$ |  $4774$
$T_{cc1}^{\pi}(2P_{1}^{\prime})$ |  $1(1^{-})$ |  $4916$ |  $4838$ |  $4786$ |  $4734$ |  $4913$ |  $4743$
$T_{cc1}^{\pi}(2P_{1}^{\prime\prime})$ |  $1(1^{-})$ |  $4881$ |  $4805$ |  $4755$ |  $4704$ |  $4851$ |  $4725$
$T_{cc2}^{\pi}(2P_{2})$ |  $1(2^{-})$ |  $4926$ |  $4849$ |  $4797$ |  $4745$ |  $4968$ |  $4786$
$T_{cc2}^{\pi}(2P_{2}^{\prime})$ |  $1(2^{-})$ |  $4934$ |  $4855$ |  $4802$ |  $4749$ |  $4947$ |  $4761$
$T_{cc3}^{\pi}(2^{5}P_{3})$ |  $1(3^{-})$ |  $4950$ |  $4871$ |  $4818$ |  $4765$ |  $4989$ |  $4805$
$T_{cc1}^{f}(3^{3}S_{1})$ |  $0(1^{+})$ |  $4956$ |  $4868$ |  $4809$ |  $4750$ |  $4913$ |  $4801$
$T_{cc0}^{a}(3^{1}S_{0})$ |  $1(0^{+})$ |  $4999$ |  $4912$ |  $4853$ |  $4795$ |  $5027$ |  $4911$
$T_{cc1}^{a}(3^{3}S_{1})$ |  $1(1^{+})$ |  $5026$ |  $4937$ |  $4877$ |  $4817$ |  $5046$ |  $4923$
$T_{cc2}^{a}(3^{5}S_{2})$ |  $1(2^{+})$ |  $5082$ |  $4988$ |  $4925$ |  $4862$ |  $5085$ |  $4946$
$T_{cc2}^{\eta}(1^{3}F_{2})$ |  $0(2^{-})$ |  $4958$ |  $4882$ |  $4831$ |  $4780$ |  $4852$ |  $4664$
$T_{cc3}^{\eta}(1^{3}F_{3})$ |  $0(3^{-})$ |  $4965$ |  $4888$ |  $4837$ |  $4785$ |  $4857$ |  $4674$
$T_{cc4}^{\eta}(1^{3}F_{4})$ |  $0(4^{-})$ |  $4973$ |  $4896$ |  $4845$ |  $4792$ |  $4864$ |  $4687$
$T_{cc1}^{\pi}(1^{5}F_{1})$ |  $1(1^{-})$ |  $5065$ |  $4984$ |  $4929$ |  $4874$ |  $5049$ |  $4777$
$T_{cc2}^{\pi}(1F_{2})$ |  $1(2^{-})$ |  $5056$ |  $4976$ |  $4922$ |  $4868$ |  $5029$ |  $4800$
$T_{cc2}^{\pi}(1F_{2}^{\prime})$ |  $1(2^{-})$ |  $5074$ |  $4992$ |  $4937$ |  $4881$ |  $5056$ |  $4785$
$T_{cc3}^{\pi}(1F_{3})$ |  $1(3^{-})$ |  $5043$ |  $4964$ |  $4912$ |  $4858$ |  $5003$ |  $4830$
$T_{cc3}^{\pi}(1F_{3}^{\prime})$ |  $1(3^{-})$ |  $5088$ |  $5005$ |  $4949$ |  $4893$ |  $5066$ |  $4809$
$T_{cc3}^{\pi}(1F_{3}^{\prime\prime})$ |  $1(3^{-})$ |  $5064$ |  $4983$ |  $4929$ |  $4874$ |  $5034$ |  $4798$
$T_{cc4}^{\pi}(1F_{4})$ |  $1(4^{-})$ |  $5047$ |  $4969$ |  $4916$ |  $4862$ |  $5005$ |  $4840$
$T_{cc4}^{\pi}(1F_{4}^{\prime})$ |  $1(4^{-})$ |  $5074$ |  $4993$ |  $4938$ |  $4882$ |  $5042$ |  $4822$
$T_{cc5}^{\pi}(1^{5}F_{5})$ |  $1(5^{-})$ |  $5053$ |  $4974$ |  $4921$ |  $4867$ |  $5009$ |  $4852$
Table 5: The mass spectrum of the $2D$-, $3P$-, and $4S$-wave doubly charmed tetraquark states procured by this work (in unit of MeV). State | Mass
---|---
$T_{cc}(n^{2S+1}L_{J})$ |  $I(J^{P})$ |  GI |  MGI ($\mu=30$) |  MGI ($\mu=50$) |  MGI ($\mu=70$) |  NR-G |  NR-Y
$T_{cc1}^{f}(2^{3}D_{1})$ |  $0(1^{+})$ |  $5058$ |  $4967$ |  $4906$ |  $4844$ |  $4989$ |  $4825$
$T_{cc2}^{f}(2^{3}D_{2})$ |  $0(2^{+})$ |  $5068$ |  $4976$ |  $4914$ |  $4852$ |  $4998$ |  $4836$
$T_{cc3}^{f}(2^{3}D_{3})$ |  $0(3^{+})$ |  $5081$ |  $4988$ |  $4926$ |  $4863$ |  $5013$ |  $4851$
$T_{cc0}^{a}(2^{5}D_{0})$ |  $1(0^{+})$ |  $5148$ |  $5053$ |  $4989$ |  $4925$ |  $5139$ |  $4922$
$T_{cc1}^{a}(2D_{1})$ |  $1(1^{+})$ |  $5160$ |  $5064$ |  $5000$ |  $4935$ |  $5151$ |  $4945$
$T_{cc1}^{a}(2D_{1}^{\prime})$ |  $1(1^{+})$ |  $5152$ |  $5057$ |  $4993$ |  $4929$ |  $5142$ |  $4929$
$T_{cc2}^{a}(2D_{2})$ |  $1(2^{+})$ |  $5156$ |  $5062$ |  $4999$ |  $4935$ |  $5149$ |  $4979$
$T_{cc2}^{a}(2D_{2}^{\prime})$ |  $1(2^{+})$ |  $5182$ |  $5084$ |  $5018$ |  $4952$ |  $5175$ |  $4955$
$T_{cc2}^{a}(2D_{2}^{\prime\prime})$ |  $1(2^{+})$ |  $5164$ |  $5068$ |  $5004$ |  $4939$ |  $5154$ |  $4944$
$T_{cc3}^{a}(2D_{3})$ |  $1(3^{+})$ |  $5165$ |  $5070$ |  $5006$ |  $4942$ |  $5156$ |  $4990$
$T_{cc3}^{a}(2D_{3}^{\prime})$ |  $1(3^{+})$ |  $5181$ |  $5084$ |  $5018$ |  $4953$ |  $5172$ |  $4970$
$T_{cc4}^{a}(2^{5}D_{4})$ |  $1(4^{+})$ |  $5175$ |  $5080$ |  $5015$ |  $4950$ |  $5165$ |  $5004$
$T_{cc0}^{\eta}(3^{3}P_{0})$ |  $0(0^{-})$ |  $5161$ |  $5055$ |  $4984$ |  $4912$ |  $5105$ |  $4986$
$T_{cc1}^{\eta}(3^{3}P_{1})$ |  $0(1^{-})$ |  $5172$ |  $5065$ |  $4993$ |  $4921$ |  $5124$ |  $4995$
$T_{cc2}^{\eta}(3^{3}P_{2})$ |  $0(2^{-})$ |  $5193$ |  $5084$ |  $5011$ |  $4937$ |  $5162$ |  $5012$
$T_{cc0}^{\pi}(3^{3}P_{0})$ |  $1(0^{-})$ |  $5241$ |  $5132$ |  $5060$ |  $4987$ |  $5182$ |  $5084$
$T_{cc1}^{\pi}(3P_{1})$ |  $1(1^{-})$ |  $5263$ |  $5154$ |  $5081$ |  $5008$ |  $5291$ |  $5123$
$T_{cc1}^{\pi}(3P_{1}^{\prime})$ |  $1(1^{-})$ |  $5272$ |  $5161$ |  $5087$ |  $5012$ |  $5255$ |  $5097$
$T_{cc1}^{\pi}(3P_{1}^{\prime\prime})$ |  $1(1^{-})$ |  $5249$ |  $5140$ |  $5066$ |  $4993$ |  $5196$ |  $5082$
$T_{cc2}^{\pi}(3P_{2})$ |  $1(2^{-})$ |  $5279$ |  $5169$ |  $5094$ |  $5020$ |  $5309$ |  $5134$
$T_{cc2}^{\pi}(3P_{2}^{\prime})$ |  $1(2^{-})$ |  $5287$ |  $5175$ |  $5099$ |  $5023$ |  $5287$ |  $5113$
$T_{cc3}^{\pi}(3^{5}P_{3})$ |  $1(3^{-})$ |  $5300$ |  $5188$ |  $5112$ |  $5036$ |  $5331$ |  $5150$
$T_{cc1}^{f}(4^{3}S_{1})$ |  $0(1^{+})$ |  $5305$ |  $5180$ |  $5096$ |  $5012$ |  $5270$ |  $5164$
$T_{cc0}^{a}(4^{1}S_{0})$ |  $1(0^{+})$ |  $5357$ |  $5234$ |  $5151$ |  $5069$ |  $5368$ |  $5250$
$T_{cc1}^{a}(4^{3}S_{1})$ |  $1(1^{+})$ |  $5377$ |  $5252$ |  $5169$ |  $5084$ |  $5382$ |  $5261$
$T_{cc2}^{a}(4^{5}S_{2})$ |  $1(2^{+})$ |  $5421$ |  $5292$ |  $5205$ |  $5118$ |  $5414$ |  $5280$

In the framework of the diquark-antidiquark configuration, the total angular
momentum $\bm{J}$ of the doubly charmed tetraquark $cc\bar{u}\bar{d}$ is
expressed as

$\displaystyle\bm{J}$ $\displaystyle=$
$\displaystyle\bm{J}_{cc}\otimes\bm{J}_{\bar{u}\bar{d}}\otimes\bm{L}_{\lambda},$
(3.5)

with

$\displaystyle\bm{J}_{cc}$ $\displaystyle=$
$\displaystyle\bm{L}_{cc}\otimes\bm{S}_{cc},$ (3.6)
$\displaystyle\bm{J}_{\bar{u}\bar{d}}$ $\displaystyle=$
$\displaystyle\bm{L}_{\bar{u}\bar{d}}\otimes\bm{S}_{\bar{u}\bar{d}},$ (3.7)
$\displaystyle\bm{S}_{cc}$ $\displaystyle=$
$\displaystyle\bm{S}_{c}\otimes\bm{S}_{c},$ (3.8)
$\displaystyle\bm{S}_{\bar{u}\bar{d}}$ $\displaystyle=$
$\displaystyle\bm{S}_{\bar{u}}\otimes\bm{S}_{\bar{d}}.$ (3.9)

Here, the total angular momentum $\bm{J}_{cc}$ of the doubly charmed diquark
$cc$ is acquired by coupling the relative orbital angular momentum
$\bm{L}_{cc}$ between two charm quarks and the spin quantum number
$\bm{S}_{cc}$ of the doubly charmed diquark $cc$. Likewise, the total angular
momentum $\bm{J}_{\bar{u}\bar{d}}$ of the light antidiquark $\bar{u}\bar{d}$
is procured by coupling the relative orbital angular momentum
$\bm{L}_{\bar{u}\bar{d}}$ between two light antiquarks and the spin quantum
number $\bm{S}_{\bar{u}\bar{d}}$ of the light antidiquark $\bar{u}\bar{d}$. In
addition, $\bm{L}_{\lambda}$ denotes the relative orbital angular momentum
between the doubly charmed diquark $cc$ and the light antidiquark
$\bar{u}\bar{d}$. For convenience, this work leaves out the relative orbital
excitations between two (anti)quarks within the (anti)diquark, i.e.,

$\displaystyle L_{cc}=L_{\bar{u}\bar{d}}$ $\displaystyle=$ $\displaystyle 0,$
(3.10) $\displaystyle J_{cc}=S_{cc}$ $\displaystyle=$ $\displaystyle 1,$
(3.11) $\displaystyle J_{\bar{u}\bar{d}}=S_{\bar{u}\bar{d}}$ $\displaystyle=$
$\displaystyle 0\ {\rm or}\ 1.$ (3.12)

In order to articulately discriminate multifarious low-lying states of the
doubly charmed tetraquark $cc\bar{u}\bar{d}$, the conventional mesonic
notation $n^{2S+1}L_{J}$ is utilized in this work, as Tables 3-5 lay out.
Accordingly, the doubly charmed tetraquark states with the total angular
momentum $\bm{J}$ are capable of being categorized by the principal quantum
number $n$, the orbital angular momentum $L$, and the spin quantum number
$\bm{S}$, i.e.,

$\displaystyle n$ $\displaystyle=$ $\displaystyle
n_{cc}+n_{\bar{u}\bar{d}}+n_{\lambda}+1,$ (3.13) $\displaystyle L$
$\displaystyle=$ $\displaystyle L_{cc}+L_{\bar{u}\bar{d}}+L_{\lambda},$ (3.14)
$\displaystyle\bm{S}$ $\displaystyle=$
$\displaystyle\bm{S}_{cc}\otimes\bm{S}_{\bar{u}\bar{d}}.$ (3.15)

Here, the principal quantum number $n$ of the doubly charmed tetraquark
$cc\bar{u}\bar{d}$ is made up of the radial quantum number $n_{cc}$ between
two charm quarks, the radial quantum number $n_{\bar{u}\bar{d}}$ between two
light antiquarks, and the radial quantum number $n_{\lambda}$ between the
diquark $cc$ and the antidiquark $\bar{u}\bar{d}$. For the sake of brevity,
this work omits the relative radial excitations between two (anti)quarks
within the (anti)diquark, leading to

$\displaystyle n$ $\displaystyle=$ $\displaystyle n_{\lambda}+1.$ (3.16)

Currently, there are several orthodox angular momentum coupling schemes,
including the $L-S$ and $j-j$ coupling schemes. On the basis of the $L-S$
coupling scheme, the total angular momentum $\bm{J}$ of the doubly charmed
tetraquark $cc\bar{u}\bar{d}$ is able to be written as Varshalovich:1988krb

$\displaystyle\left|\left[\left(\bm{L}_{cc}\otimes\bm{L}_{\bar{u}\bar{d}}\right)_{\bm{L}_{\rho}}\otimes\bm{L}_{\lambda}\right]_{\bm{L}_{t}}\otimes\left(\bm{S}_{cc}\otimes\bm{S}_{\bar{u}\bar{d}}\right)_{\bm{S}}\right\rangle_{\bm{J}}$
(3.23) $\displaystyle=$
$\displaystyle\sum_{J_{\rho}}\sum_{J_{cc}}\sum_{J_{\bar{u}\bar{d}}}(-1)^{L_{\lambda}+S+L_{t}+J_{\rho}}\sqrt{(2L_{t}+1)(2J_{\rho}+1)}$
$\displaystyle\times\sqrt{(2L_{\rho}+1)(2S+1)(2J_{cc}+1)(2J_{\bar{u}\bar{d}}+1)}$
$\displaystyle\times\left\\{\begin{array}[]{ccc}L_{\lambda}&L_{\rho}&L_{t}\\\
S&J&J_{\rho}\end{array}\right\\}\left\\{\begin{array}[]{ccc}L_{cc}&L_{\bar{u}\bar{d}}&L_{\rho}\\\
S_{cc}&S_{\bar{u}\bar{d}}&S\\\
J_{cc}&J_{\bar{u}\bar{d}}&J_{\rho}\end{array}\right\\}$
$\displaystyle\times\left|\left[\left(\bm{L}_{cc}\otimes\bm{S}_{cc}\right)_{\bm{J}_{cc}}\otimes\left(\bm{L}_{\bar{u}\bar{d}}\otimes\bm{S}_{\bar{u}\bar{d}}\right)_{\bm{J}_{\bar{u}\bar{d}}}\right]_{\bm{J}_{\rho}}\otimes\bm{L}_{\lambda}\right\rangle_{\bm{J}},\qquad\qquad$

with

$\displaystyle\bm{L}_{t}$ $\displaystyle=$
$\displaystyle\bm{L}_{\rho}\otimes\bm{L}_{\lambda},$ (3.24)
$\displaystyle\bm{L}_{\rho}$ $\displaystyle=$
$\displaystyle\bm{L}_{cc}\otimes\bm{L}_{\bar{u}\bar{d}},$ (3.25)
$\displaystyle\bm{J}_{\rho}$ $\displaystyle=$
$\displaystyle\bm{J}_{cc}\otimes\bm{J}_{\bar{u}\bar{d}}.$ (3.26)

Evidently, the $L-S$ coupling scheme is equivalent to the coupling scheme
employed by this work, since the internal orbital excitations inside the
(anti)diquark are left out, i.e.,

$\displaystyle\bm{J}$ $\displaystyle=$
$\displaystyle\bm{L}_{t}\otimes\bm{S}=\bm{L}_{\lambda}\otimes\bm{S}=\bm{L}_{\lambda}\otimes\bm{J}_{\rho}.$
(3.27)

Therefore, following the exemplary spectroscopic inquiries of heavy-light
hadrons Godfrey:1985xj ; Godfrey:2016nwn ; Godfrey:2015dva ; Godfrey:2004ya ;
Lu:2017meb ; Ali:2017wsf ; Song:2015nia ; Pang:2017dlw ; Song:2015fha ;
Kim:2021ywp , this work actually adopts the $L-S$ coupling scheme to unriddle
the exotic $cc\bar{u}\bar{d}$ states. Moreover, in light of the $j-j$ coupling
scheme, the total angular momentum $\bm{J}$ of the doubly charmed tetraquark
$cc\bar{u}\bar{d}$ is capable of being expressed as Varshalovich:1988krb

$\displaystyle\left|\left(\bm{L}_{cc}\otimes\bm{S}_{cc}\right)_{\bm{J}_{cc}}\otimes\left[\left(\bm{L}_{\bar{u}\bar{d}}\otimes\bm{S}_{\bar{u}\bar{d}}\right)_{\bm{J}_{\bar{u}\bar{d}}}\otimes\bm{L}_{\lambda}\right]_{\bm{J}_{l}}\right\rangle_{\bm{J}}$
(3.31) $\displaystyle=$
$\displaystyle\sum_{J_{\rho}}(-1)^{J_{cc}+J_{\bar{u}\bar{d}}+L_{\lambda}+J}\sqrt{(2J_{\rho}+1)(2J_{l}+1)}\left\\{\begin{array}[]{ccc}J_{cc}&J_{\bar{u}\bar{d}}&J_{\rho}\\\
L_{\lambda}&J&J_{l}\end{array}\right\\}$
$\displaystyle\times\left|\left[\left(\bm{L}_{cc}\otimes\bm{S}_{cc}\right)_{\bm{J}_{cc}}\otimes\left(\bm{L}_{\bar{u}\bar{d}}\otimes\bm{S}_{\bar{u}\bar{d}}\right)_{\bm{J}_{\bar{u}\bar{d}}}\right]_{\bm{J}_{\rho}}\otimes\bm{L}_{\lambda}\right\rangle_{\bm{J}},\qquad\qquad$

with

$\displaystyle\bm{J}_{l}$ $\displaystyle=$
$\displaystyle\bm{J}_{\bar{u}\bar{d}}\otimes\bm{L}_{\lambda}.$ (3.32)

In the doubly charmed tetraquark system, the isospin properties are determined
by the flavor wave function of the light antidiquark $\bar{u}\bar{d}$. On the
premise of the omission of the internal orbital excitations inside the
(anti)diquark, the isospin quantum number $I$ of the doubly charmed tetraquark
$cc\bar{u}\bar{d}$ is equal to the spin quantum number $S_{\bar{u}\bar{d}}$ of
the color-triplet ground state light antidiquark $\bar{u}\bar{d}$
Amsler:2018zkm , i.e.,

$\displaystyle I=S_{\bar{u}\bar{d}}$ $\displaystyle=$ $\displaystyle 0\ {\rm
or}\ 1.$ (3.33)

Additionally, in terms of the doubly charmed tetraquark $cc\bar{u}\bar{d}$
constituted by diquark $cc$ and antidiquark $\bar{u}\bar{d}$, the internal
parity $P$ is assessed as Amsler:2018zkm

$\displaystyle P$ $\displaystyle=$
$\displaystyle(-1)^{L_{\lambda}}P_{cc}P_{\bar{u}\bar{d}}=(-1)^{L},$ (3.34)

with

$\displaystyle P_{cc}=(-1)^{L_{cc}},$ $\displaystyle
P_{\bar{u}\bar{d}}=(-1)^{L_{\bar{u}\bar{d}}}.$ (3.35)

Here, $P_{cc}$ and $P_{\bar{u}\bar{d}}$ denote the internal parity of the
diquark $cc$ and the internal parity of the antidiquark $\bar{u}\bar{d}$,
respectively. Whereafter, in compliance with the definite $I(J^{P})$
characteristics of $cc\bar{u}\bar{d}$ states, the spectroscopic properties of
the low-lying $T_{cc}$ family established by the GI, MGI ($\mu=30$), MGI
($\mu=50$), MGI ($\mu=70$), NR-G, and NR-Y diquark-antidiquark scenarios are
explicitly exhibited in Tables 3-5.

##  IV Discussion

Table 6: A comparison of the ground state masses of the doubly charmed and light diquarks from this work (GI, MGI, and NR models) and other phenomenological approaches, categorized by the $I(J^{P})$ feature (in unit of MeV). {cc} | $0(1^{+})$ | {cc} | $0(1^{+})$ | {cc} | $0(1^{+})$ | {cc} | $0(1^{+})$
---|---|---|---|---|---|---|---
Song:2023izj  | $2865.5$ | NR-G | $3152$ | Wu:2022gie  | $3300.8$ | Yan:2018gik  | $3340$
Kucab:2024nkv  | $2912$ |  Feng:2013kea  I | $3170$ | MGI ($\mu=70$) | $3309$ | NR-Y | $3369$
Feng:2013kea  III | $3020$ | Weng:2021hje  | $3171.51$ | MGI ($\mu=50$) | $3314$ | Wallbott:2020jzh  | $3423\pm 8$
Lebed:2024zrp  | $3026.0$ | Guo:2021yws  | $3182.67\pm 30$ | MGI ($\mu=30$) | $3320$ | Mutuk:2024vzv  | $3510\pm 350$
Feng:2013kea  II | $3100$ | Ebert:2007rn  | $3226$ | GI Ferretti:2019zyh  | $3329$ |  Noh:2021lqs  I | $3609$
[ud] | $0(0^{+})$ | [ud] | $0(0^{+})$ | {ud} | $1(1^{+})$ | {ud} | $1(1^{+})$
Yan:2018gik  | $395$ | GI Ferretti:2019zyh  | $691$ | Yan:2018gik  | $395$ | MGI ($\mu=30$) | $814$
Song:2023izj  | $535$ | Ebert:2007rn  | $710$ | Guo:2021yws  | $723.86$ | Braaten:2020nwp  | $835.7\pm 11.1$
Braaten:2020nwp  | $627.4\pm 11.2$ | NR-G Bi:2015ifa  | $725$ | Weng:2021hje  | $724.85$ | GI Ferretti:2019zyh  | $840$
Feng:2013kea  III | $650$ | NR-Y Bi:2015ifa  | $725$ | Song:2023izj  | $745$ | Ebert:2007rn  | $909$
MGI ($\mu=70$) | $650$ |  Feng:2013kea  II | $750$ | MGI ($\mu=70$) | $778$ | NR-Y Kim:2021ywp  | $973$
MGI ($\mu=50$) | $662$ |  Feng:2013kea  I | $800$ | MGI ($\mu=50$) | $796$ | Wallbott:2020jzh  | $999\pm 60$
MGI ($\mu=30$) | $673$ | Wallbott:2020jzh  | $802\pm 77$ | Mutuk:2024vzv  | $810\pm 60$ | NR-G Kim:2021ywp  | $1019$
Table 7: The first part of the comparison of the $1S$-wave doubly charmed tetraquark masses from this work (GI, MGI, and NR models) and other phenomenological approaches, categorized by the $I(J^{P})$ feature (in unit of MeV). $1S$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $1S$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$
---|---|---|---|---|---|---|---|---|---
Pepin:1996id  II | $3580$ | $\cdots$ | $\cdots$ | $\cdots$ |  Gelman:2002wf  I | $3845$ | $\cdots$ | $\cdots$ | $\cdots$
Feng:2013kea  III | $3660$ | $\cdots$ | $\cdots$ | $\cdots$ |  Maiani:2022qze  IV | $3846$ | $3832$ | $3848$ | $3879$
He:2023ucd  II | $3696$ | $\cdots$ | $\cdots$ | $\cdots$ |  Deng:2018kly  III | $3847$ | $\cdots$ | $\cdots$ | $\cdots$
Yan:2018gik  | $3700$ | $4140$ | $4170$ | $4240$ |  Wang:2017uld  II | $3850\pm 90$ | $\cdots$ | $\cdots$ | $\cdots$
Yang:2009zzp  II | $3701.6$ | $4212.9$ | $4131.5$ | $4158.1$ |  Maiani:2022qze  II | $3851$ | $\cdots$ | $\cdots$ | $\cdots$
Chen:2021tnn  | $3704.8$ | $4086.6$ | $4133.3$ | $4159.4$ | Junnarkar:2018twb  | $3853\pm 11$ | $3761\pm 11$ | $\cdots$ | $\cdots$
Tan:2020ldi  | $3709.3$ | $3725.9$ | $3844.3$ | $3962.5$ | Meng:2020knc  | $3853$ | $\cdots$ | $\cdots$ | $\cdots$
Meng:2023jqk  III | $3724.2$ | $\cdots$ | $\cdots$ | $\cdots$ |  Vijande:2009kj  VI | $3856$ | $3862$ | $3914$ | $3991$
Maiani:2019lpu  II | $3725$ | $\cdots$ | $\cdots$ | $\cdots$ |  Park:2024cic  II | $3858.5$ | $\cdots$ | $\cdots$ | $\cdots$
Pepin:1996id  I | $3727$ | $\cdots$ | $\cdots$ | $\cdots$ |  Vijande:2009kj  III | $3860$ | $3911$ | $3975$ | $4031$
Deng:2018kly  I | $3731\pm 12$ | $3962\pm 8$ | $4017\pm 7$ | $4013\pm 7$ |  Xing:2018bqt  I | $3860$ | $\cdots$ | $\cdots$ | $\cdots$
Maiani:2019lpu  I | $3742$ | $\cdots$ | $\cdots$ | $\cdots$ | Zhu:2019iwm  | $\cdots$ | $3923\pm 59$ | $3957^{+63}_{-62}$ | $4026\pm 69$
Gao:2020bvl  | $3742^{+50}_{-40}$ | $\cdots$ | $\cdots$ | $\cdots$ |  Vijande:2009kj  VII | $3861$ | $3877$ | $3952$ | $\cdots$
Weng:2021hje  I | $3749.8$ | $3833.2$ | $3946.4$ | $4017.1$ |  Vijande:2009kj  V | $3861$ | $3905$ | $3972$ | $4025$
Deng:2018kly  IV | $3758$ | $\cdots$ | $\cdots$ | $\cdots$ |  Park:2024cic  I | $3861.5$ | $\cdots$ | $\cdots$ | $\cdots$
Ma:2023int  III | $3759$ | $\cdots$ | $\cdots$ | $\cdots$ | Meng:2024yhu  | $3863$ | $\cdots$ | $\cdots$ | $\cdots$
Guo:2021yws  | $3761.6$ | $3836.7$ | $3950.9$ | $4021.2$ |  Meng:2023for  I | $3863.1$ | $\cdots$ | $\cdots$ | $\cdots$
Vijande:2003ki  I | $3764$ | $4150$ | $4186$ | $4211$ |  Meng:2023jqk  I | $3864.5$ | $\cdots$ | $\cdots$ | $\cdots$
Li:2023wug  II | $3769.2$ | $3839.7$ | $3963.7$ | $4034.1$ | Agaev:2021vur  | $3868\pm 124$ | $\cdots$ | $\cdots$ | $\cdots$
Xing:2018bqt  II | $3770$ | $\cdots$ | $\cdots$ | $\cdots$ | Agaev:2019qqn  | $\cdots$ | $3845\pm 175$ | $\cdots$ | $\cdots$
Cheng:2020wxa  II | $3773.8$ | $3844.3$ | $3968.4$ | $4038.8$ |  Weng:2021hje  II | $3868.7$ | $3969.2$ | $4053.2$ | $4123.8$
Yang:2019itm  | $3778$ | $\cdots$ | $\cdots$ | $\cdots$ |  Meng:2023for  II | $3869.9$ | $\cdots$ | $\cdots$ | $\cdots$
Luo:2017eub  II | $3779$ | $3850$ | $3973$ | $4044$ |  Feng:2013kea  I | $3870$ | $\cdots$ | $\cdots$ | $\cdots$
Lee:2009rt  | $3796.5$ | $\cdots$ | $\cdots$ | $\cdots$ |  Sonnenschein:2024rzw  II | $3870$ | $\cdots$ | $3926\pm 74$ | $\cdots$
Feng:2013kea  II | $3800$ | $\cdots$ | $\cdots$ | $\cdots$ |  He:2023ucd  I | $3870.9$ | $\cdots$ | $\cdots$ | $\cdots$
Hyodo:2017hue  | $3805$ | $3934$ | $3966$ | $4030$ |  Maiani:2022qze  III | $3871$ | $3868$ | $3872$ | $3881$
Luo:2017eub  III | $3813$ | $\cdots$ | $\cdots$ | $\cdots$ |  Maiani:2022qze  I | $3872$ | $\cdots$ | $\cdots$ | $\cdots$
Deng:2018kly  II | $3817$ | $\cdots$ | $\cdots$ | $\cdots$ | Noh:2023zoq  | $3872$ | $\cdots$ | $\cdots$ | $\cdots$
Deng:2022cld  | $3820$ | $\cdots$ | $\cdots$ | $\cdots$ |  Noh:2021lqs  II | $3872.8$ | $\cdots$ | $\cdots$ | $\cdots$
Carlson:1987hh  | $3835$ | $3805$ | $3845$ | $3955$ | Wu:2022gie  | $3875.8\pm 7.6$ | $4035.4\pm 13.6$ | $4058.0\pm 9.5$ | $4103.2\pm 9.5$
Table 8: The second part of the comparison of the $1S$-wave doubly charmed tetraquark masses from this work (GI, MGI, and NR models) and other phenomenological approaches, categorized by the $I(J^{P})$ feature (in unit of MeV). $1S$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $1S$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$
---|---|---|---|---|---|---|---|---|---
Janc:2004qn  II | $3875.9$ | $\cdots$ | $\cdots$ | $\cdots$ |  Ma:2023int  II | $3916$ | $\cdots$ | $\cdots$ | $\cdots$
NR-Y | $3876$ | $4029$ | $4057$ | $4105$ | MGI ($\mu=30$) | $3917$ | $3809$ | $3925$ | $4083$
MGI ($\mu=70$) | $3877$ | $3766$ | $3879$ | $4032$ |  Noh:2021lqs  I | $3920$ | $\cdots$ | $\cdots$ | $\cdots$
Karliner:2017qjm  II | $3877.1\pm 12$ | $\cdots$ | $\cdots$ | $\cdots$ | Zhang:2021yul  | $3925$ | $4032$ | $4117$ | $4179$
Mehen:2017nrh  | $\cdots$ | $4043$ | $4064$ | $4107$ |  Vijande:2009kj  II | $3926$ | $4154$ | $4175$ | $4193$
Li:2023wug  I | $3878.2$ | $3948.8$ | $4072.8$ | $4143.2$ |  Vijande:2009kj  I | $3927$ | $4155$ | $4176$ | $4195$
Liu:2023vrk  | $3879.2$ | $3975.2$ | $4053.7$ | $4124.7$ |  Semay:1994ht  I | $3931$ | $\cdots$ | $\cdots$ | $\cdots$
Ma:2023int  I | $3880$ | $\cdots$ | $\cdots$ | $\cdots$ | Ebert:2007rn  | $3935$ | $4056$ | $4079$ | $4118$
Karliner:2017qjm  I | $3882.2\pm 12$ | $\cdots$ | $\cdots$ | $\cdots$ | Braaten:2020nwp  | $3947\pm 11$ | $4111\pm 11$ | $4133\pm 11$ | $4177\pm 11$
NR-G | $3884$ | $3894$ | $3991$ | $4135$ | GI | $3948$ | $3842$ | $3960$ | $4121$
Albuquerque:2022weq  III | $\cdots$ | $3878\pm 5$ | $\cdots$ | $\cdots$ | Kucab:2024nkv  | $3948$ | $4121$ | $4140$ | $4177$
Albuquerque:2022weq  I | $3885\pm 123$ | $3882\pm 129$ | $\cdots$ | $\cdots$ |  Cheng:2020wxa  I | $3961$ | $4087$ | $4122$ | $4194$
Albuquerque:2022weq  IV | $\cdots$ | $3883\pm 3$ | $\cdots$ | $\cdots$ | Kim:2022mpa  | $3961$ | $4132$ | $4151$ | $4185$
Albuquerque:2022weq  II | $3886\pm 4$ | $3885\pm 4$ | $\cdots$ | $\cdots$ |  Park:2018wjk  I | $3965$ | $4104$ | $4158$ | $\cdots$
Wang:2024vjc  | $3889$ | $4107$ | $4131$ | $4176$ |  Park:2018wjk  II | $3971$ | $4110$ | $4164$ | $\cdots$
Semay:1994ht  II | $3892$ | $\cdots$ | $\cdots$ | $\cdots$ |  Park:2018wjk  III | $3973.3$ | $\cdots$ | $\cdots$ | $\cdots$
Mutuk:2024vzv  | $\cdots$ | $3904$ | $3952$ | $4001$ | Eichten:2017ffp  | $3978$ | $4146$ | $4167$ | $4210$
Mutuk:2023oyz  | $3892$ | $4062$ | $4104$ | $4207$ |  Yang:2009zzp  I | $3996.3$ | $4137.5$ | $4211.7$ | $4253.5$
Meng:2023jqk  II | $3892.2$ | $\cdots$ | $\cdots$ | $\cdots$ | Song:2023izj  | $3997$ | $4163$ | $4185$ | $4229$
MGI ($\mu=50$) | $3897$ | $3787$ | $3902$ | $4058$ | Wang:2022clw  | $3998.90$ | $4069.69$ | $4092.34$ | $4134.59$
Semay:1994ht  IV | $3899$ | $\cdots$ | $\cdots$ | $\cdots$ |  Cheng:2020wxa  III | $3999.4$ | $4069.9$ | $4194.1$ | $4264.5$
Vijande:2009kj  IV | $3899$ | $\cdots$ | $\cdots$ | $\cdots$ | Navarra:2007yw  | $4000\pm 200$ | $\cdots$ | $\cdots$ | $\cdots$
Wallbott:2020jzh  | $3900\pm 80$ | $3800\pm 100$ | $4220\pm 440$ | $\cdots$ |  Li:2023wug  III | $4000.2$ | $4070.7$ | $4194.7$ | $4265.1$
Wang:2017uld  I | $3900\pm 90$ | $\cdots$ | $\cdots$ | $\cdots$ |  Noh:2021lqs  III | $4002.2$ | $\cdots$ | $\cdots$ | $\cdots$
Wang:2017dtg  | $\cdots$ | $3870\pm 90$ | $3900\pm 90$ | $3950\pm 90$ |  Luo:2017eub  I | $4007$ | $4078$ | $4201$ | $4271$
Park:2018wjk  IV | $3904.3$ | $\cdots$ | $\cdots$ | $\cdots$ | Lipkin:1986dw  | $4012$ | $\cdots$ | $\cdots$ | $\cdots$
Janc:2004qn  I | $3904.7$ | $\cdots$ | $\cdots$ | $\cdots$ |  Yang:2009zzp  III | $4037.3$ | $4235.4$ | $4175.5$ | $4201.2$
Gelman:2002wf  II | $3905$ | $\cdots$ | $\cdots$ | $\cdots$ |  Sonnenschein:2024rzw  I | $4050.5\pm 67.5$ | $\cdots$ | $4048\pm 67$ | $\cdots$
Semay:1994ht  V | $3915$ | $\cdots$ | $\cdots$ | $\cdots$ | Lu:2020rog  | $4053$ | $4241$ | $4268$ | $4318$
Semay:1994ht  III | $3916$ | $\cdots$ | $\cdots$ | $\cdots$ |  Vijande:2003ki  II | $4101$ | $4175$ | $4231$ | $4254$
Table 9: A comparison of the $2S$-, $3S$-, $4S$-, and $1D$-wave doubly charmed tetraquark masses from this work (GI, MGI, and NR models) and other phenomenological approaches, categorized by the $I(J^{P})$ feature (in unit of MeV). $2S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $\cdots$ | $2S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$
---|---|---|---|---|---|---|---|---|---|---|---|---
Meng:2024yhu  | $\cdots$ | $4028$ | $4717$ | $4667$ | $4775$ | $\cdots$ | MGI ($\mu=70$) | $\cdots$ | $4405$ | $4423$ | $4459$ | $4529$
Sonnenschein:2024rzw  II | $\cdots$ | $4271$ | $\cdots$ | $4318.5\pm 62.5$ | $\cdots$ | $\cdots$ | MGI ($\mu=50$) | $\cdots$ | $4441$ | $4460$ | $4498$ | $4570$
Chen:2021tnn  | $\cdots$ | $4304$ | $\cdots$ | $4639$ | $4687$ | $\cdots$ | MGI ($\mu=30$) | $\cdots$ | $4478$ | $4497$ | $4537$ | $4612$
Yang:2019itm  | $\cdots$ | $4310$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | NR-G | $\cdots$ | $4489$ | $4609$ | $4637$ | $4694$
Kim:2022mpa  | $\cdots$ | $4363$ | $4546$ | $4560$ | $4585$ | $\cdots$ | GI | $\cdots$ | $4534$ | $4554$ | $4595$ | $4674$
NR-Y | $\cdots$ | $4388$ | $4523$ | $4539$ | $4568$ | $\cdots$ |  Sonnenschein:2024rzw  I | $\cdots$ | $4554\pm 55$ | $\cdots$ | $4552\pm 55$ | $\cdots$
$3S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $\cdots$ | $3S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$
Chen:2023web  I | $\cdots$ | $4611$ | $\cdots$ | $4811$ | $\cdots$ | $\cdots$ |  Sonnenschein:2024rzw  I | $\cdots$ | $4867.5\pm 49.5$ | $\cdots$ | $4865.5\pm 49.5$ | $\cdots$
Sonnenschein:2024rzw  II | $\cdots$ | $4615$ | $\cdots$ | $4657\pm 55$ | $\cdots$ | $\cdots$ | MGI ($\mu=30$) | $\cdots$ | $4868$ | $4912$ | $4937$ | $4988$
Chen:2023web  II | $\cdots$ | $4615$ | $\cdots$ | $4809$ | $\cdots$ | $\cdots$ | NR-G | $\cdots$ | $4913$ | $5027$ | $5046$ | $5085$
MGI ($\mu=70$) | $\cdots$ | $4750$ | $4795$ | $4817$ | $4862$ | $\cdots$ | GI | $\cdots$ | $4956$ | $4999$ | $5026$ | $5082$
NR-Y | $\cdots$ | $4801$ | $4911$ | $4923$ | $4946$ | $\cdots$ | Meng:2024yhu  | $\cdots$ | $4986$ | $4958$ | $4958$ | $4956$
MGI ($\mu=50$) | $\cdots$ | $4809$ | $4853$ | $4877$ | $4925$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$4S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $\cdots$ | $4S$ | $\cdots$ | $0(1^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(2^{+})$
Chen:2023web  I | $\cdots$ | $4808$ | $\cdots$ | $5009$ | $\cdots$ | $\cdots$ |  Sonnenschein:2024rzw  I | $\cdots$ | $5152.5\pm 45.5$ | $\cdots$ | $5150.5\pm 45.5$ | $\cdots$
Chen:2023web  II | $\cdots$ | $4818$ | $\cdots$ | $5008$ | $\cdots$ | $\cdots$ | NR-Y | $\cdots$ | $5164$ | $5250$ | $5261$ | $5280$
Sonnenschein:2024rzw  II | $\cdots$ | $4922$ | $\cdots$ | $4960\pm 50$ | $\cdots$ | $\cdots$ | MGI ($\mu=30$) | $\cdots$ | $5180$ | $5234$ | $5252$ | $5292$
Meng:2024yhu  | $\cdots$ | $\cdots$ | $\cdots$ | $4985$ | $5027$ | $\cdots$ | NR-G | $\cdots$ | $5270$ | $5368$ | $5382$ | $5414$
MGI ($\mu=70$) | $\cdots$ | $5012$ | $5069$ | $5084$ | $5118$ | $\cdots$ | GI | $\cdots$ | $5305$ | $5357$ | $5377$ | $5421$
MGI ($\mu=50$) | $\cdots$ | $5096$ | $5151$ | $5169$ | $5205$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
$1D$ | $0(1^{+})$ | $0(2^{+})$ | $0(3^{+})$ | $1(0^{+})$ | $1(1^{+})$ | $1(1^{+})$ | $1(2^{+})$ | $1(2^{+})$ | $1(2^{+})$ | $1(3^{+})$ | $1(3^{+})$ | $1(4^{+})$
Sonnenschein:2024rzw  II | $\cdots$ | $\cdots$ | $4409$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4427\pm 60$ | $\cdots$ | $\cdots$
Chen:2023web  I | $\cdots$ | $\cdots$ | $4446$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4615$ | $\cdots$ | $\cdots$
Chen:2023web  II | $\cdots$ | $\cdots$ | $4447$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4613$ | $\cdots$ | $\cdots$
NR-Y | $4428$ | $4440$ | $4457$ | $4549$ | $4575$ | $4558$ | $4616$ | $4588$ | $4575$ | $4628$ | $4604$ | $4644$
Sonnenschein:2024rzw  I | $\cdots$ | $\cdots$ | $4554\pm 55$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4552\pm 55$ | $\cdots$ | $\cdots$
MGI ($\mu=70$) | $4554$ | $4564$ | $4579$ | $4628$ | $4643$ | $4636$ | $4646$ | $4668$ | $4650$ | $4654$ | $4669$ | $4664$
MGI ($\mu=50$) | $4592$ | $4603$ | $4619$ | $4670$ | $4686$ | $4678$ | $4688$ | $4712$ | $4693$ | $4696$ | $4713$ | $4707$
NR-G | $4607$ | $4616$ | $4631$ | $4790$ | $4789$ | $4803$ | $4790$ | $4828$ | $4801$ | $4797$ | $4820$ | $4805$
MGI ($\mu=30$) | $4631$ | $4643$ | $4659$ | $4712$ | $4729$ | $4720$ | $4730$ | $4756$ | $4735$ | $4739$ | $4756$ | $4750$
GI | $4689$ | $4701$ | $4718$ | $4776$ | $4793$ | $4783$ | $4792$ | $4822$ | $4799$ | $4802$ | $4821$ | $4813$
Table 10: A comparison of the $1P$\- and $1F$-wave doubly charmed tetraquark masses from this work (GI, MGI, and NR models) and other phenomenological approaches, categorized by the $I(J^{P})$ feature (in unit of MeV). $1P$ | $\cdots$ | $0(0^{-})$ | $0(1^{-})$ | $0(2^{-})$ | $1(0^{-})$ | $1(1^{-})$ | $1(1^{-})$ | $1(1^{-})$ | $1(2^{-})$ | $1(2^{-})$ | $1(3^{-})$ | $\cdots$
---|---|---|---|---|---|---|---|---|---|---|---|---
Yan:2018gik  | $\cdots$ | $\cdots$ | $3920$ | $\cdots$ | $\cdots$ | $4410$ | $4390$ | $4340$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Vijande:2009kj  IV | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4380$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4502$ | $\cdots$
Vijande:2009kj  VI | $\cdots$ | $\cdots$ | $\cdots$ | $3927$ | $\cdots$ | $4420$ | $\cdots$ | $\cdots$ | $3918$ | $\cdots$ | $4461$ | $\cdots$
Vijande:2009kj  V | $\cdots$ | $\cdots$ | $\cdots$ | $3996$ | $\cdots$ | $4426$ | $\cdots$ | $\cdots$ | $4004$ | $\cdots$ | $4461$ | $\cdots$
Semay:1994ht  II | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4464$ | $\cdots$
Semay:1994ht  III | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4477$ | $\cdots$
Semay:1994ht  IV | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4484$ | $\cdots$
Semay:1994ht  V | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4504$ | $\cdots$
Semay:1994ht  I | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4512$ | $\cdots$
Sonnenschein:2024rzw  II | $\cdots$ | $\cdots$ | $\cdots$ | $4155$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4175\pm 66$ | $\cdots$ | $\cdots$ | $\cdots$
Wang:2024vjc  | $\cdots$ | $4150$ | $4159$ | $4193$ | $4393$ | $4292$ | $4388$ | $\cdots$ | $4309$ | $4420$ | $4459$ | $\cdots$
NR-Y | $\cdots$ | $4161$ | $4174$ | $4199$ | $4310$ | $4370$ | $4331$ | $4308$ | $4386$ | $4354$ | $4409$ | $\cdots$
Kim:2022mpa  | $\cdots$ | $\cdots$ | $\cdots$ | $4253$ | $\cdots$ | $4423$ | $\cdots$ | $\cdots$ | $4430$ | $\cdots$ | $4442$ | $\cdots$
Song:2023izj  | $\cdots$ | $4253$ | $4268$ | $4298$ | $4429$ | $4446$ | $4447$ | $4487$ | $4484$ | $4499$ | $4517$ | $\cdots$
MGI ($\mu=70$) | $\cdots$ | $4252$ | $4271$ | $4305$ | $4295$ | $4369$ | $4361$ | $4300$ | $4388$ | $4385$ | $4412$ | $\cdots$
Sonnenschein:2024rzw  I | $\cdots$ | $\cdots$ | $\cdots$ | $4315\pm 60$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4313\pm 60$ | $\cdots$ | $\cdots$ | $\cdots$
MGI ($\mu=50$) | $\cdots$ | $4279$ | $4299$ | $4335$ | $4326$ | $4400$ | $4394$ | $4331$ | $4419$ | $4420$ | $4446$ | $\cdots$
NR-G | $\cdots$ | $4289$ | $4311$ | $4356$ | $4408$ | $4534$ | $4501$ | $4425$ | $4554$ | $4535$ | $4577$ | $\cdots$
MGI ($\mu=30$) | $\cdots$ | $4307$ | $4327$ | $4364$ | $4357$ | $4432$ | $4427$ | $4362$ | $4452$ | $4453$ | $4479$ | $\cdots$
GI | $\cdots$ | $4349$ | $4370$ | $4408$ | $4404$ | $4477$ | $4480$ | $4410$ | $4501$ | $4503$ | $4529$ | $\cdots$
$1F$ | $0(2^{-})$ | $0(3^{-})$ | $0(4^{-})$ | $1(1^{-})$ | $1(2^{-})$ | $1(2^{-})$ | $1(3^{-})$ | $1(3^{-})$ | $1(3^{-})$ | $1(4^{-})$ | $1(4^{-})$ | $1(5^{-})$
Chen:2023web  I | $\cdots$ | $\cdots$ | $4601$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4763$ | $\cdots$ | $\cdots$
Chen:2023web  II | $\cdots$ | $\cdots$ | $4604$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4761$ | $\cdots$ | $\cdots$
Sonnenschein:2024rzw  II | $\cdots$ | $\cdots$ | $4640$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4657\pm 55$ | $\cdots$ | $\cdots$
NR-Y | $4664$ | $4674$ | $4687$ | $4777$ | $4800$ | $4785$ | $4830$ | $4809$ | $4798$ | $4840$ | $4822$ | $4852$
Sonnenschein:2024rzw  I | $\cdots$ | $\cdots$ | $4774\pm 51$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | $4772\pm 51$ | $\cdots$ | $\cdots$
MGI ($\mu=70$) | $4780$ | $4785$ | $4792$ | $4874$ | $4868$ | $4881$ | $4858$ | $4893$ | $4874$ | $4862$ | $4882$ | $4867$
MGI ($\mu=50$) | $4831$ | $4837$ | $4845$ | $4929$ | $4922$ | $4937$ | $4912$ | $4949$ | $4929$ | $4916$ | $4938$ | $4921$
NR-G | $4852$ | $4857$ | $4864$ | $5049$ | $5029$ | $5056$ | $5003$ | $5066$ | $5034$ | $5005$ | $5042$ | $5009$
MGI ($\mu=30$) | $4882$ | $4888$ | $4896$ | $4984$ | $4976$ | $4992$ | $4964$ | $5005$ | $4983$ | $4969$ | $4993$ | $4974$
GI | $4958$ | $4965$ | $4973$ | $5065$ | $5056$ | $5074$ | $5043$ | $5088$ | $5064$ | $5047$ | $5074$ | $5053$

Concerning the ground state axial-vector doubly charmed diquark, the ground
state scalar and axial-vector light diquarks, and the low-lying doubly charmed
tetraquark states with the miscellaneous radial and orbital excitations, the
accessible comparisons on the spectroscopic disparities between this work (GI
relativized diquark model, MGI relativized diquark model, NR-G diquark model,
and NR-Y diquark model) and other phenomenological approaches are elaborately
portrayed in this section. What is more, the mixing angles of a variety of
isovector $T_{cc}$ states ($1P$-, $2P$-, $3P$-, $1D$-, $2D$-, and $1F$-wave)
and the so-called magic mixing angles of the orbitally excited double-heavy
tetraquarks in the ideal heavy quark limit are comprehensively dissected as
well.

###  IV.1 Comparison on the $cc$ and $ud$ diquarks

Admittedly, the masses of the doubly charmed and light diquarks are imperative
to educe the eventual outcome of the doubly charmed tetraquark spectroscopy in
the framework of the diquark-antidiquark picture. Currently, copious
theoretical approaches have investigated the mass spectra of the
$cc\bar{u}\bar{d}$ states based on the masses of the corresponding $cc$ and
$ud$ diquarks Ebert:2007rn ; Feng:2013kea ; Yan:2018gik ; Wallbott:2020jzh ;
Noh:2021lqs ; Braaten:2020nwp ; Weng:2021hje ; Guo:2021yws ; Song:2023izj ;
Wu:2022gie ; Kucab:2024nkv ; Mutuk:2024vzv ; Lebed:2024zrp . Nonetheless,
there is an overt diversity of the diquark masses performed to the $T_{cc}$
spectroscopy. For instance, a slightly spacious scope of the diquark $cc$
masses with the distribution between 2865.5 Song:2023izj  and 3609 MeV
Noh:2021lqs  is evidently displayed in Table 6. Conspicuously, the GI model
mass 3329 MeV procured by this work is in consonance with the existing result
offered by Ref. Ferretti:2019zyh . Moreover, the theoretical outcomes 3329 and
3369 MeV from the GI and NR-Y models are adjacent to the diquark $cc$ mass
3340 MeV drawn forth by Ref. Yan:2018gik . In addition, the theoretical masses
3309, 3314, and 3320 MeV of the diquark $cc$ acquired by the MGI model are
fairly close to the predicted outcome 3300.8 MeV reaped by Ref. Wu:2022gie .
Besides, the NR-G model mass 3152 MeV of the diquark $cc$ draws near to the
theoretical values 3170 and 3171.51 MeV employed by Refs. Feng:2013kea  and
Weng:2021hje , respectively.

When it comes to the scalar light diquark, the masses are spread across the
energy domain between 395 Yan:2018gik  and 802 MeV Wallbott:2020jzh . The
scalar diquark $ud$ masses 650, 662, and 673 MeV deduced by the MGI model are
in the vicinity of the theoretical value 650 MeV rendered by Ref. Feng:2013kea
. Particularly, the theoretical outcome 650 MeV attained by the MGI ($\mu=70$)
model is utterly in congruence with the mass of the scalar light diquark from
Ref. Feng:2013kea . Remarkably, the predicted value 710 MeV carried out by the
relativistic quark model in Ref. Ebert:2007rn  is rather proximate to the
scalar diquark $ud$ masses 691 and 725 MeV from the GI and NR models,
respectively. In terms of the axial-vector light diquark, the masses lie on
the extensive sector between 395 Yan:2018gik  and 1019 MeV Kim:2021ywp .
Thereinto, the MGI model masses 778, 796, and 814 MeV abut the axial-vector
diquark $ud$ mass 810 MeV utilized by Ref. Mutuk:2024vzv . Additionally, the
theoretical outcome 840 MeV of the GI model is quite close to the axial-vector
light diquark mass 835.7 MeV derived by Ref. Braaten:2020nwp . Apart from
that, the predicted mass 999 MeV of the axial-vector diquark $ud$ determined
by Ref. Wallbott:2020jzh  is sort of near to the pertinent results 973 and
1019 MeV adopted by the NR-Y and NR-G models, respectively. Manifestly, the
masses 691 and 840 MeV of the scalar and axial-vector light diquarks from the
GI model are in conformity with the counterparts garnered by Ref.
Ferretti:2019zyh .

###  IV.2 Comparison on the $cc\bar{u}\bar{d}$ tetraquarks

Regarding the mass spectra of the $T_{cc}$ family, this subsection is aimed at
the methodical discussion and bounteous comparison of the resemblance and
dissimilitude between this work (GI, MGI, and NR models) and other theoretical
recipes.

####  IV.2.1 $1S$-wave

There was no doubt that almost all state-of-the-art phenomenological
explorations of the doubly charmed tetraquark spectroscopy were focused on the
$1S$-wave isoscalar $T_{cc}$ state with the lowest spin-averaged mass. In
spite of this, a sizable discrepancy between the minimum theoretical value
3580 MeV Pepin:1996id  and the maximum theoretical value 4101 MeV
Vijande:2003ki  with regard to the ground state isoscalar $cc\bar{u}\bar{d}$
tetraquark mass was ferreted out by Tables 7-8, which indicated the tremendous
divarication of the existing $T_{cc}$ spectroscopic delving. The theoretical
outcomes of the $1S$-wave isoscalar $cc\bar{u}\bar{d}$ tetraquark mass
procured by this work (GI, MGI, NR-G, and NR-Y scenarios) are situated in the
energy interval between 3876 and 3948 MeV, dovetailing with the consequences
conducted by a wealth of phenomenological approaches which comprise the MIT
bag model Zhang:2021yul , the quark potential model Semay:1994ht ;
Gelman:2002wf ; Janc:2004qn ; Ebert:2007rn ; Vijande:2009kj ; Karliner:2017qjm
; Park:2018wjk ; Noh:2021lqs ; Weng:2021hje ; Liu:2023vrk ; Meng:2023for ;
Noh:2023zoq ; He:2023ucd ; Meng:2023jqk ; Wu:2022gie ; Ma:2023int ;
Mutuk:2023oyz ; Wang:2024vjc ; Li:2023wug , the QCD sum rules Wang:2017uld ;
Albuquerque:2022weq ; Agaev:2021vur , the Bethe-Salpeter (BS) equation
Feng:2013kea ; Wallbott:2020jzh , the heavy quark symmetry (HQS)
Braaten:2020nwp , the Born-Oppenheimer (BO) approximation Maiani:2022qze , the
holographic QCD Sonnenschein:2024rzw , and the chiral quark-soliton model
($\chi$QSM) Kucab:2024nkv . Visibly, the predicted results 3877, 3884, and
3876 MeV of the isoscalar $T_{cc}$ mass acquired by the MGI ($\mu=70$), NR-G,
and NR-Y scenarios accord with the observed mass of the $T_{cc}(3875)^{+}$
state. In the midst of other theoretical means, the quark potential model
Janc:2004qn ; Karliner:2017qjm ; Noh:2021lqs ; Liu:2023vrk ; Noh:2023zoq ;
He:2023ucd ; Wu:2022gie ; Ma:2023int ; Li:2023wug , the BS equation
Feng:2013kea , the BO approximation Maiani:2022qze , and the holographic QCD
Sonnenschein:2024rzw  garnered the outcomes in close contiguity to the
experimental value of the $T_{cc}(3875)^{+}$ mass, which inclined to the
tetraquark interpretation of the $T_{cc}(3875)^{+}$ state.

As the isospin siblings of the isoscalar $T_{cc}$ state, the $1S$-wave
isovector $T_{cc}$ states are indisputably the hot spots of the current
$cc\bar{u}\bar{d}$ tetraquark issues. As enumerated in Tables 7-8, this work
(GI, MGI, NR-G, and NR-Y scenarios) exhibits the prediction ambit of the
ground state isovector $T_{cc}$ masses between 3766 and 4135 MeV. As far as
the $1S$-wave scalar $cc\bar{u}\bar{d}$ state with $I=1$ is concerned, the
predicted masses from the NR models are distinctly higher than the
counterparts of the $1S$-wave axial-vector $cc\bar{u}\bar{d}$ state with
$I=0$, disparate with the akin circumstances from the GI (MGI) models.
Especially, the mass gap 153 MeV between the isovector scalar state and the
isoscalar axial-vector state elicited by the NR-Y model is greater than the
mass gap $-111$ MeV educed by the MGI ($\mu=70$) model, even though the masses
of the isoscalar $cc\bar{u}\bar{d}$ state from these two models are quite
parallel. On the one hand, the ground state $T_{cc}$ masses 3876, 4029, 4057,
and 4105 MeV reaped by the NR-Y model extremely resemble the masses 3876,
4035, 4058, and 4103 MeV rendered by the heavy antiquark-diquark symmetry
(HADS) Wu:2022gie . Likewise, the predictions of the NR-Y model and HADS are
corroborated by HQS Mehen:2017nrh . On the other hand, the case that the
isoscalar axial-vector $T_{cc}$ mass outweighs the isovector scalar $T_{cc}$
mass is espoused by the QCD sum rules Wang:2017uld ; Wang:2017dtg ;
Agaev:2019qqn ; Albuquerque:2022weq ; Agaev:2021vur , the BS equation
Wallbott:2020jzh , the lattice QCD Junnarkar:2018twb , and the BO
approximation Maiani:2022qze . Hence, the conundrum of the veritable mass gap
between the isovector scalar $cc\bar{u}\bar{d}$ state and the isoscalar axial-
vector $cc\bar{u}\bar{d}$ state necessitates the further quest of the
experiments. Furthermore, this work prognosticates that the isovector tensor
$cc\bar{u}\bar{d}$ tetraquark mass is in the energy interval between 4032 and
4135 MeV, not only conforming to the spectroscopic results predicted by Refs.
Ebert:2007rn ; Luo:2017eub ; Mehen:2017nrh ; Cheng:2020wxa ; Weng:2021hje ;
Wang:2022clw ; Liu:2023vrk ; Wu:2022gie ; Li:2023wug , but also delivering the
available hints to the latent experimental explorations.

####  IV.2.2 $2S$-, $3S$-, and $4S$-wave

Apparently, the $cc\bar{u}\bar{d}$ tetraquark states with radial excitations
are rarely probed by phenomenological theories Yang:2019itm ; Kim:2022mpa ;
Chen:2021tnn ; Chen:2023web ; Meng:2024yhu ; Sonnenschein:2024rzw . As
unveiled in Table 9, the minimum and maximum theoretical masses of the
$2S$-wave doubly charmed tetraquark states predicted by this work (GI, MGI,
NR-G, and NR-Y scenarios) are 4388 and 4694 MeV, respectively. Noteworthily,
the NR-Y model masses 4388, 4523, 4539, and 4568 MeV of the $2S$-wave
$cc\bar{u}\bar{d}$ tetraquarks jibe with the masses 4363, 4546, 4560, and 4585
MeV acquired by Ref. Kim:2022mpa . In addition, the predicted outcomes of the
$2S$-wave $T_{cc}$ masses procured by the GI model are close to the
counterparts from Ref. Sonnenschein:2024rzw . When it comes to the $3S$-wave
$cc\bar{u}\bar{d}$ tetraquarks, the predicted masses offered by this work are
located in the realm between 4750 and 5085 MeV, approaching to the
consequences garnered by Refs. Meng:2024yhu ; Sonnenschein:2024rzw .
Concerning the masses of the $4S$-wave doubly charmed tetraquark states, the
theoretical values achieved by this work are distributed in the gamut between
5012 and 5421 MeV, solely comporting with the outcomes reaped by Ref.
Sonnenschein:2024rzw . On the basis of the $1S$\- and $2S$-wave
$cc\bar{u}\bar{d}$ tetraquark masses predicted by Ref. Kim:2022mpa , the Regge
trajectory relation is exploited by Ref. Chen:2023web  to reveal the highly
excited $3S$\- and $4S$-wave $cc\bar{u}\bar{d}$ tetraquark masses which are
relatively lower than the results of this work.

####  IV.2.3 $1P$-, $1D$-, and $1F$-wave

So far, the spectroscopic properties of the orbitally excited doubly charmed
tetraquark states are merely surveyed by a minority of theoretical
prescriptions, including the quark potential model Semay:1994ht ;
Vijande:2009kj ; Yan:2018gik ; Kim:2022mpa ; Song:2023izj ; Wang:2024vjc , the
holographic QCD Sonnenschein:2024rzw , and the Regge trajectory relation
Chen:2023web . The masses of the $1P$-wave isoscalar and isovector doubly
charmed tetraquark states predicted by this work (GI, MGI, NR-G, and NR-Y
scenarios) lie on the range between 4161 and 4408 MeV and the extent between
4295 and 4577 MeV, respectively. As Table 10 demonstrates, the $1P$-wave
isoscalar $cc\bar{u}\bar{d}$ tetraquark masses predicted by Refs.
Vijande:2009kj ; Yan:2018gik  are somewhat lower than the theoretical outcomes
of this work. Howbeit, the predicted masses 4161, 4174, and 4199 MeV of the
$1P$-wave isoscalar $cc\bar{u}\bar{d}$ states from the NR-Y model are in
accordance with the masses 4150, 4159, and 4193 MeV earned by Ref.
Wang:2024vjc . What is more, the theoretical values 4253, 4268, and 4298 MeV
of the $1P$-wave isoscalar $T_{cc}$ masses derived by Ref. Song:2023izj  are
in good agreement with the outcomes 4252, 4271, and 4305 MeV acquired by the
MGI ($\mu=70$) model. Furthermore, the spectroscopic results of the $1P$-wave
isovector $cc\bar{u}\bar{d}$ states attained by the GI model possess a
similitude of the upshot deduced by Ref. Song:2023izj .

According to Tables 9-10, the holographic QCD Sonnenschein:2024rzw  and the
Regge trajectory relation Chen:2023web  are the few available theoretical
recipes with respect to the mass spectra of the $1D$\- and $1F$-wave $T_{cc}$
tetraquarks except this work. The masses of the $1D$-wave isoscalar and
isovector $cc\bar{u}\bar{d}$ states decoded by this work are situated in the
domain between 4428 and 4718 MeV and the realm between 4549 and 4828 MeV,
respectively. Apart from that, the $1F$-wave isoscalar and isovector $T_{cc}$
masses deciphered by this work are concentrated in the scope between 4664 and
4973 MeV and the interval between 4777 and 5088 MeV, respectively. In light of
the $1S$\- and $1P$-wave $T_{cc}$ masses reaped by the quark potential model
Kim:2022mpa , the Regge trajectory relation Chen:2023web  predicts the
$1D$-wave $cc\bar{u}\bar{d}$ tetraquark masses which adjoin the results of the
NR-Y model and the $1F$-wave $cc\bar{u}\bar{d}$ tetraquark masses which are
lower than the outcomes of this work. Owing to the sheer paucity of the
spectroscopic information about the $cc\bar{u}\bar{d}$ states with the joint
radial and orbital excitations, this work minutely investigates the mass
spectra of the $2P$-, $3P$-, and $2D$-wave $T_{cc}$ tetraquarks for the first
time. Thus it can be seen that the further phenomenological deliberation and
experimental probe are requisite to shed light on the spectroscopic properties
of the low-lying excited doubly charmed tetraquark states.

###  IV.3 Mixing angles of the $T_{cc}$ states

Table 11: The mixing angles of the $1P$-, $2P$-, $3P$-, $1D$-, $2D$-, and $1F$-wave isovector doubly charmed tetraquark states procured by this work. Mixing |  GI |  MGI ($\mu=30$) |  MGI ($\mu=50$) |  MGI ($\mu=70$) |  NR-G |  NR-Y |  HQL
---|---|---|---|---|---|---|---
$1^{1}P_{1}$$\leftrightarrow$$1^{3}P_{1}$ |  $-44.4^{\circ}$ |  $44.5^{\circ}$ |  $43.6^{\circ}$ |  $42.9^{\circ}$ |  $41.9^{\circ}$ |  $40.5^{\circ}$ |  $40.9^{\circ}$
$1^{3}P_{1}$$\leftrightarrow$$1^{5}P_{1}$ |  $9.4^{\circ}$ |  $-7.4^{\circ}$ |  $-7.0^{\circ}$ |  $-6.6^{\circ}$ |  $-7.7^{\circ}$ |  $12.4^{\circ}$ |  $40.9^{\circ}$
$1^{1}P_{1}$$\leftrightarrow$$1^{5}P_{1}$ |  $8.1^{\circ}$ |  $9.6^{\circ}$ |  $9.7^{\circ}$ |  $9.8^{\circ}$ |  $10.8^{\circ}$ |  $9.5^{\circ}$ |  $9.6^{\circ}$
$1^{3}P_{2}$$\leftrightarrow$$1^{5}P_{2}$ |  $7.4^{\circ}$ |  $-20.1^{\circ}$ |  $-41.2^{\circ}$ |  $39.7^{\circ}$ |  $28.3^{\circ}$ |  $28.7^{\circ}$ |  $30.0^{\circ}$
$2^{1}P_{1}$$\leftrightarrow$$2^{3}P_{1}$ |  $37.7^{\circ}$ |  $37.2^{\circ}$ |  $36.6^{\circ}$ |  $35.6^{\circ}$ |  $42.3^{\circ}$ |  $40.6^{\circ}$ |  $40.9^{\circ}$
$2^{3}P_{1}$$\leftrightarrow$$2^{5}P_{1}$ |  $-11.3^{\circ}$ |  $-10.6^{\circ}$ |  $-10.1^{\circ}$ |  $-9.4^{\circ}$ |  $-7.1^{\circ}$ |  $12.3^{\circ}$ |  $40.9^{\circ}$
$2^{1}P_{1}$$\leftrightarrow$$2^{5}P_{1}$ |  $11.5^{\circ}$ |  $11.4^{\circ}$ |  $11.3^{\circ}$ |  $11.3^{\circ}$ |  $11.1^{\circ}$ |  $9.6^{\circ}$ |  $9.6^{\circ}$
$2^{3}P_{2}$$\leftrightarrow$$2^{5}P_{2}$ |  $23.2^{\circ}$ |  $22.7^{\circ}$ |  $22.1^{\circ}$ |  $21.1^{\circ}$ |  $30.5^{\circ}$ |  $29.1^{\circ}$ |  $30.0^{\circ}$
$3^{1}P_{1}$$\leftrightarrow$$3^{3}P_{1}$ |  $38.1^{\circ}$ |  $37.4^{\circ}$ |  $36.8^{\circ}$ |  $35.9^{\circ}$ |  $42.3^{\circ}$ |  $40.7^{\circ}$ |  $40.9^{\circ}$
$3^{3}P_{1}$$\leftrightarrow$$3^{5}P_{1}$ |  $-14.1^{\circ}$ |  $-12.9^{\circ}$ |  $-12.1^{\circ}$ |  $-11.2^{\circ}$ |  $-6.1^{\circ}$ |  $12.3^{\circ}$ |  $40.9^{\circ}$
$3^{1}P_{1}$$\leftrightarrow$$3^{5}P_{1}$ |  $13.8^{\circ}$ |  $13.1^{\circ}$ |  $12.7^{\circ}$ |  $12.4^{\circ}$ |  $10.9^{\circ}$ |  $9.6^{\circ}$ |  $9.6^{\circ}$
$3^{3}P_{2}$$\leftrightarrow$$3^{5}P_{2}$ |  $21.8^{\circ}$ |  $21.2^{\circ}$ |  $20.7^{\circ}$ |  $19.7^{\circ}$ |  $31.1^{\circ}$ |  $29.4^{\circ}$ |  $30.0^{\circ}$
$1^{3}D_{1}$$\leftrightarrow$$1^{5}D_{1}$ |  $-35.8^{\circ}$ |  $-33.9^{\circ}$ |  $-32.4^{\circ}$ |  $-30.8^{\circ}$ |  $44.5^{\circ}$ |  $23.4^{\circ}$ |  $30.0^{\circ}$
$1^{1}D_{2}$$\leftrightarrow$$1^{3}D_{2}$ |  $43.7^{\circ}$ |  $44.0^{\circ}$ |  $44.3^{\circ}$ |  $44.7^{\circ}$ |  $45.3^{\circ}$ |  $43.8^{\circ}$ |  $45.0^{\circ}$
$1^{3}D_{2}$$\leftrightarrow$$1^{5}D_{2}$ |  $-25.6^{\circ}$ |  $-25.0^{\circ}$ |  $-24.7^{\circ}$ |  $-24.3^{\circ}$ |  $-27.4^{\circ}$ |  $38.5^{\circ}$ |  $52.2^{\circ}$
$1^{1}D_{2}$$\leftrightarrow$$1^{5}D_{2}$ |  $12.5^{\circ}$ |  $13.1^{\circ}$ |  $13.6^{\circ}$ |  $14.3^{\circ}$ |  $15.3^{\circ}$ |  $13.7^{\circ}$ |  $15.0^{\circ}$
$1^{3}D_{3}$$\leftrightarrow$$1^{5}D_{3}$ |  $34.3^{\circ}$ |  $34.7^{\circ}$ |  $35.0^{\circ}$ |  $35.3^{\circ}$ |  $35.9^{\circ}$ |  $32.4^{\circ}$ |  $35.3^{\circ}$
$2^{3}D_{1}$$\leftrightarrow$$2^{5}D_{1}$ |  $-42.7^{\circ}$ |  $-40.7^{\circ}$ |  $-39.2^{\circ}$ |  $-37.3^{\circ}$ |  $-43.0^{\circ}$ |  $23.8^{\circ}$ |  $30.0^{\circ}$
$2^{1}D_{2}$$\leftrightarrow$$2^{3}D_{2}$ |  $44.0^{\circ}$ |  $44.3^{\circ}$ |  $44.5^{\circ}$ |  $44.7^{\circ}$ |  $44.7^{\circ}$ |  $44.0^{\circ}$ |  $45.0^{\circ}$
$2^{3}D_{2}$$\leftrightarrow$$2^{5}D_{2}$ |  $-27.2^{\circ}$ |  $-26.6^{\circ}$ |  $-26.1^{\circ}$ |  $-25.6^{\circ}$ |  $-26.3^{\circ}$ |  $39.5^{\circ}$ |  $52.2^{\circ}$
$2^{1}D_{2}$$\leftrightarrow$$2^{5}D_{2}$ |  $13.3^{\circ}$ |  $13.7^{\circ}$ |  $14.0^{\circ}$ |  $14.4^{\circ}$ |  $14.1^{\circ}$ |  $13.9^{\circ}$ |  $15.0^{\circ}$
$2^{3}D_{3}$$\leftrightarrow$$2^{5}D_{3}$ |  $34.2^{\circ}$ |  $34.5^{\circ}$ |  $34.7^{\circ}$ |  $34.9^{\circ}$ |  $35.9^{\circ}$ |  $32.9^{\circ}$ |  $35.3^{\circ}$
$1^{3}F_{2}$$\leftrightarrow$$1^{5}F_{2}$ |  $38.0^{\circ}$ |  $38.4^{\circ}$ |  $38.8^{\circ}$ |  $39.1^{\circ}$ |  $37.1^{\circ}$ |  $29.6^{\circ}$ |  $35.3^{\circ}$
$1^{1}F_{3}$$\leftrightarrow$$1^{3}F_{3}$ |  $46.0^{\circ}$ |  $46.2^{\circ}$ |  $46.3^{\circ}$ |  $46.4^{\circ}$ |  $46.5^{\circ}$ |  $45.2^{\circ}$ |  $46.7^{\circ}$
$1^{3}F_{3}$$\leftrightarrow$$1^{5}F_{3}$ |  $-31.5^{\circ}$ |  $-31.1^{\circ}$ |  $-30.9^{\circ}$ |  $-30.7^{\circ}$ |  $-32.1^{\circ}$ |  $46.9^{\circ}$ |  $55.9^{\circ}$
$1^{1}F_{3}$$\leftrightarrow$$1^{5}F_{3}$ |  $16.3^{\circ}$ |  $16.6^{\circ}$ |  $16.8^{\circ}$ |  $17.0^{\circ}$ |  $17.1^{\circ}$ |  $15.5^{\circ}$ |  $17.4^{\circ}$
$1^{3}F_{4}$$\leftrightarrow$$1^{5}F_{4}$ |  $36.7^{\circ}$ |  $37.0^{\circ}$ |  $37.2^{\circ}$ |  $37.4^{\circ}$ |  $37.4^{\circ}$ |  $34.4^{\circ}$ |  $37.8^{\circ}$

Conventionally, the meson with the certain orbital angular momentum may exist
as the states that possess the identical total angular momentum and the
discrepant spin quantum number, e.g., the ${}^{1}P_{1}$ and ${}^{3}P_{1}$
states. For the case of the hidden-flavor quarkonium, the ${}^{1}P_{1}$ and
${}^{3}P_{1}$ states cannot mix since the equal mass of the quark and
antiquark engenders the diagonal spin-orbit interaction Barnes:2005pb ;
Godfrey:2015dia . On the contrary, the heavy-light meson constituted of a
quark and antiquark with unequal mass may manifest as an admixture of
${}^{1}P_{1}$ and ${}^{3}P_{1}$ states due to the advent of the off-diagonal
term in the spin-orbit interaction Godfrey:2016nwn ; Godfrey:2015dva . For
instance, the physical $P_{1}$ states are capable of being expressed as the
linear combinations of the unmixing ${}^{1}P_{1}$ and ${}^{3}P_{1}$ states
Godfrey:1985xj ; Godfrey:2016nwn ; Godfrey:2015dva ; Godfrey:2004ya , i.e.,

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle{{}^{1}}P_{1}\cos\theta_{nP_{1}}+{{}^{3}}P_{1}\sin\theta_{nP_{1}},$
$\displaystyle P_{1}^{\prime}$ $\displaystyle=$
$\displaystyle-{{}^{1}}P_{1}\sin\theta_{nP_{1}}+{{}^{3}}P_{1}\cos\theta_{nP_{1}}.$
(4.1)

Unlike a mixture of two states in the heavy-light meson, the isovector doubly
charmed tetraquark with the diquark-antidiquark configuration not only
involves the mixing between two states, but also touches upon the mixing among
three states, e.g., the ${}^{1}P_{1}$, ${}^{3}P_{1}$, and ${}^{5}P_{1}$
states. Concretely, the mixing of two states in the isovector doubly charmed
tetraquark holds the generic form of

$\displaystyle\left(\begin{array}[]{c}nL_{J}\\\ nL_{J}^{\prime}\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos\theta_{nL_{J}}&\sin\theta_{nL_{J}}\\\
-\sin\theta_{nL_{J}}&\cos\theta_{nL_{J}}\\\
\end{array}\right)\left(\begin{array}[]{c}n^{3}L_{J}\\\ n^{5}L_{J}\\\
\end{array}\right).$ (4.8)

Akin to the cause of mixing in the heavy-light meson, the mixture between
$n^{3}L_{J}$ and $n^{5}L_{J}$ states is solely elicited by the off-diagonal
term in the spin-orbit interaction. An intriguing fingerprint of the orbitally
excited isovector $T_{cc}$ states is the admixture among three states,
expressed as a $3\times 3$ unitary matrix by mimicking the prestigious
Cabibbo-Kobayashi-Maskawa (CKM) matrix Workman:2022ynf , i.e.,

$\displaystyle\left(\begin{array}[]{c}nL_{J}\\\ nL_{J}^{\prime}\\\
nL_{J}^{\prime\prime}\\\ \end{array}\right)$ $\displaystyle=$ $\displaystyle
V_{123}\left(\begin{array}[]{c}n^{1}L_{J}\\\ n^{3}L_{J}\\\ n^{5}L_{J}\\\
\end{array}\right),$ (4.15)

with

$\displaystyle V_{123}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}\\\ 0&1&0\\\
-s_{13}&0&c_{13}\\\
\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\ \end{array}\right)$ (4.25) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}&c_{12}c_{23}-s_{12}s_{23}s_{13}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}&-c_{12}s_{23}-s_{12}c_{23}s_{13}&c_{23}c_{13}\\\
\end{array}\right).$ (4.29)

Here, the $\cos\theta_{nL_{J}}$ and $\sin\theta_{nL_{J}}$ are abbreviated as
$c$ and $s$, respectively, for the sake of brevity. Apparently, any two of the
$n^{1}L_{J}$, $n^{3}L_{J}$, and $n^{5}L_{J}$ states are able to mix each other
when the total angular momentum is equal to the orbital angular momentum,
i.e., $J=L$. Thereinto, two sorts of mixings
(${}^{1}L_{J}$$\leftrightarrow$${}^{3}L_{J}$ and
${}^{3}L_{J}$$\leftrightarrow$${}^{5}L_{J}$) are educed from the off-diagonal
terms in the spin-orbit interaction. Moreover, the
${}^{1}L_{J}$$\leftrightarrow$${}^{5}L_{J}$ mixing is evoked from the off-
diagonal term in the tensor interaction, distinct from the conventional heavy-
light meson. In this work, the mixed states in Eq. (4.8) whose major
ingredients are the $n^{3}L_{J}$ and $n^{5}L_{J}$ states are assigned as the
$nL_{J}$ and $nL_{J}^{\prime}$ states, respectively. Analogously, the mixed
states in Eq. (4.15) whose major ingredients are the $n^{1}L_{J}$,
$n^{3}L_{J}$, and $n^{5}L_{J}$ states are assigned as the $nL_{J}$,
$nL_{J}^{\prime}$, and $nL_{J}^{\prime\prime}$ states, respectively. As
uncovered in Table 11, the sign of the mixing angle is contingent on two
facets: the sign of the off-diagonal matrix element and the order of the
$nL_{J}$, $nL_{J}^{\prime}$, and $nL_{J}^{\prime\prime}$ masses. Taking the
$1^{3}P_{2}$$\leftrightarrow$$1^{5}P_{2}$ mixing as an illustration, although
the signs of the off-diagonal matrix elements reaped by the GI and NR-G models
are inverse, the signs of the mixing angles exported by these two models are
fortuitously same inasmuch as the orders of the $1P_{2}$ and $1P_{2}^{\prime}$
masses stemming from the GI and NR-G models are also inverse.

In the ideal heavy quark limit (HQL), the total angular momentum $J$ of the
double-heavy tetraquark is limned by the spin quantum number $S_{QQ}$ of the
infinitely heavy diquark and the total angular momentum $J_{l}$ of the light
diquark degrees of freedom, on the basis of the $j-j$ coupling scheme.
According to the explicit relations between the $j-j$ and $L-S$ coupling
schemes with respect to the heavy-light hadrons rendered by the Appendix, the
magic mixing angles of the orbitally excited double-heavy tetraquarks are
systematically enumerated in Table 11. In the $m_{Q}\rightarrow\infty$ limit,
the $nL_{L-1}$, $nL_{L-1}^{\prime}$, $nL_{L+1}$, and $nL_{L+1}^{\prime}$
states garnered by Eq. (4.8) automatically degenerate into the
$nL_{L-1}(J_{l}=L)$, $nL_{L-1}(J_{l}=L-1)$, $nL_{L+1}(J_{l}=L+1)$, and
$nL_{L+1}(J_{l}=L)$ states, respectively. In a like manner, the $nL_{L}$,
$nL_{L}^{\prime}$, and $nL_{L}^{\prime\prime}$ states in the HQL derived by
Eq. (4.15) are prone to turn into the $nL_{L}(J_{l}=L+1)$, $nL_{L}(J_{l}=L)$,
and $nL_{L}(J_{l}=L-1)$ states, respectively. Conspicuously, there are two
fascinating equalities concerning the magic mixing angles of the doubly heavy
tetraquarks, i.e., the equality between the
${}^{1}P_{1}$$\leftrightarrow$${}^{3}P_{1}$ and
${}^{3}P_{1}$$\leftrightarrow$${}^{5}P_{1}$ mixing angles and the equality
between the ${}^{3}L_{L+1}$$\leftrightarrow$${}^{5}L_{L+1}$ and
${}^{3}(L+1)_{L}$$\leftrightarrow$${}^{5}(L+1)_{L}$ mixing angles. Apart from
that, the ${}^{3}L_{L-1}$$\leftrightarrow$${}^{5}L_{L-1}$ mixing is viable
only when the orbital excitation of the double-heavy tetraquark is higher than
$P$-wave ($L>1$), owing to the fact that the existence of the ${}^{5}P_{0}$
state is forbidden. One thing to point out here is that the definition of the
mixing angle is replete with ambiguities, e.g., the flip-flop sign of the
mixing angle induced by the charge conjugation. In order to refrain from the
latent perplexity, this work (GI, MGI, NR-G, NR-Y, and HQL scenarios) has
uniformly employed a phase convention corresponding to the order of coupling
$\vec{L}\times\vec{S}_{\bar{q}\bar{q}}\times\vec{S}_{QQ}$.

##  V Summary

Accompanied by the precision enhancement of the experimental detection, a zoo
of exotic hadrons is in the process of gradual formation. There is no doubt
that the phenomenological explorations dedicated to the spectroscopy of
multiquark states are supremely imperative to demystify the nature of these
novel hadrons. Conspicuously, the first $T_{cc}$ state observed by the LHCb
Collaboration offers a fantastic opportunity to the spectroscopists, thanks to
its narrow width and considerable significance conducing to the eminent signal
LHCb:2021vvq ; LHCb:2021auc . Accordingly, taking advantage of relativized and
nonrelativistic diquark-antidiquark scenarios, this work aspires to hunt for
the prospective double-charm tetraquark family. Specifically, these several
sorts of diquark formalisms muster the Godfrey-Isgur (GI) relativized diquark
model, the modified Godfrey-Isgur (MGI) relativized diquark model
incorporating the color screening effects, the nonrelativistic (NR-G) diquark
model with the Gaussian type hyperfine potential, and the nonrelativistic
(NR-Y) diquark model with the Yukawa type hyperfine potential.

To sum up, for the sake of shedding light on the low-lying isoscalar and
isovector members of the entire $T_{cc}$ family, this work methodically
investigates the spectroscopic properties of the doubly charmed tetraquarks
with the diverse orbital and radial excitations, comprising the $1S$-, $2S$-,
$3S$-, $4S$-, $1P$-, $2P$-, $3P$-, $1D$-, $2D$-, and $1F$-wave states.
Thereinto, with regard to the mass of the $1S$-wave isoscalar double-charm
tetraquark, the predicted values from the MGI and NR-Y diquark scenarios
powerfully corroborate the tetraquark assignment of the $T_{cc}(3875)^{+}$
state. Furthermore, in terms of the mass spectra of the $1S$-wave isovector
$T_{cc}$ tetraquark states, the theoretical outcomes performed by the NR-Y
diquark approach are highly compatible with the counterparts deduced by the
heavy antiquark-diquark symmetry (HADS) in Ref. Wu:2022gie  and the heavy
quark symmetry (HQS) in Ref. Mehen:2017nrh . In comparison with the existing
theoretical prescriptions, the spectroscopy of the $2P$-, $3P$-, and $2D$-wave
doubly charmed tetraquarks is deliberated by this work for the first time.
Moreover, this work carries out the first theoretical inquiry into the mixing
angles of the orbitally excited isovector $T_{cc}$ states and the magic mixing
angles of the heavy-light tetraquarks in the heavy quark limit. Ergo, the
spectroscopic predictions fulfilled by this work are not only capable of
facilitating the phenomenological construction of the complete double-charm
tetraquark family, but also capable of expediting the experimental quest for
the promising low-lying excited $T_{cc}$ states.

## ACKNOWLEDGMENTS

W. C. Dong would like to thank Duojie Jia for helpful discussions and valuable
comments. This work is supported by the National Natural Science Foundation of
China (Grant No. 12175068) and the Hebei Natural Science Foundation (Grant No.
A2024502002).

##  Appendix A Coupling relations

As far as the heavy-light hadrons ($Q\bar{q}$ mesons, $Qqq$ baryons, $QQq$
baryons, and $QQ\bar{q}\bar{q}$ tetraquarks) are concerned, the relations
between the $j-j$ and $L-S$ coupling schemes are unraveled in this appendix.
Additionally, the coupling relations of the singly heavy mesons and doubly
heavy baryons have been explored by Refs. Cahn:2003cw  and Matsuki:2021eck ,
respectively.

###  A.1 Singly heavy meson

For the case of the $Q\bar{q}$ meson, there is only one sort of coupling
relation corresponding to $J=L$, i.e.,

$\displaystyle\left(\begin{array}[]{cc}\left|J=L,J_{l}=L+1/2\right\rangle\\\
\left|J=L,J_{l}=L-1/2\right\rangle\end{array}\right)$ (A3)
$\displaystyle=\frac{1}{\sqrt{2L+1}}\left(\begin{array}[]{cc}\sqrt{L+1}&\sqrt{L}\\\
-\sqrt{L}&\sqrt{L+1}\end{array}\right)$ (A6)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L,S=0\right\rangle\\\
\left|J=L,S=1\right\rangle\end{array}\right).$ (A9)

Thereupon, the magic mixing angle of the ideal $Q\bar{q}$ meson is
$\theta=\arctan(\sqrt{L}/\sqrt{L+1})$ for $J=L$.

###  A.2 Singly heavy baryon

In regard to the $Qqq$ baryon, there are two sorts of coupling relations
corresponding to $J=L-1/2$ and $J=L+1/2$, i.e.,

$\displaystyle\left(\begin{array}[]{cc}\left|J=L-1/2,J_{l}=L\right\rangle\\\
\left|J=L-1/2,J_{l}=L-1\right\rangle\end{array}\right)$ (A12)
$\displaystyle=\frac{1}{\sqrt{3L}}\left(\begin{array}[]{cc}\sqrt{L+1}&\sqrt{2L-1}\\\
-\sqrt{2L-1}&\sqrt{L+1}\end{array}\right)$ (A15)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L-1/2,S=1/2\right\rangle\\\
\left|J=L-1/2,S=3/2\right\rangle\end{array}\right),$ (A18)
$\displaystyle\left(\begin{array}[]{cc}\left|J=L+1/2,J_{l}=L+1\right\rangle\\\
\left|J=L+1/2,J_{l}=L\right\rangle\end{array}\right)$ (A21)
$\displaystyle=\frac{1}{\sqrt{3(L+1)}}\left(\begin{array}[]{cc}\sqrt{2L+3}&\sqrt{L}\\\
-\sqrt{L}&\sqrt{2L+3}\end{array}\right)$ (A24)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L+1/2,S=1/2\right\rangle\\\
\left|J=L+1/2,S=3/2\right\rangle\end{array}\right).$ (A27)

Consequently, the magic mixing angles of the ideal $Qqq$ baryon are
$\theta=\arctan(\sqrt{2L-1}/\sqrt{L+1})$ for $J=L-1/2$ and
$\theta=\arctan(\sqrt{L}/\sqrt{2L+3})$ for $J=L+1/2$.

###  A.3 Doubly heavy baryon

In terms of the $QQq$ baryon, there are two sorts of coupling relations
corresponding to $J=L-1/2$ and $J=L+1/2$, i.e.,

$\displaystyle\left(\begin{array}[]{cc}\left|J=L-1/2,J_{l}=L+1/2\right\rangle\\\
\left|J=L-1/2,J_{l}=L-1/2\right\rangle\end{array}\right)$ (A30)
$\displaystyle=\frac{1}{\sqrt{3(2L+1)}}\left(\begin{array}[]{cc}2\sqrt{L+1}&\sqrt{2L-1}\\\
-\sqrt{2L-1}&2\sqrt{L+1}\end{array}\right)$ (A33)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L-1/2,S=1/2\right\rangle\\\
\left|J=L-1/2,S=3/2\right\rangle\end{array}\right),$ (A36)
$\displaystyle\left(\begin{array}[]{cc}\left|J=L+1/2,J_{l}=L+1/2\right\rangle\\\
\left|J=L+1/2,J_{l}=L-1/2\right\rangle\end{array}\right)$ (A39)
$\displaystyle=\frac{1}{\sqrt{3(2L+1)}}\left(\begin{array}[]{cc}\sqrt{2L+3}&2\sqrt{L}\\\
-2\sqrt{L}&\sqrt{2L+3}\end{array}\right)$ (A42)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L+1/2,S=1/2\right\rangle\\\
\left|J=L+1/2,S=3/2\right\rangle\end{array}\right).$ (A45)

Accordingly, the magic mixing angles of the ideal $QQq$ baryon are
$\theta=\arctan(\sqrt{2L-1}/(2\sqrt{L+1}))$ for $J=L-1/2$ and
$\theta=\arctan(2\sqrt{L}/\sqrt{2L+3})$ for $J=L+1/2$.

###  A.4 Doubly heavy tetraquark

With respect to the $QQ\bar{q}\bar{q}$ tetraquark, there are three sorts of
coupling relations corresponding to $J=L-1$, $J=L$, and $J=L+1$, i.e.,

$\displaystyle\left(\begin{array}[]{cc}\left|J=L-1,J_{l}=L\right\rangle\\\
\left|J=L-1,J_{l}=L-1\right\rangle\end{array}\right)$ (A48)
$\displaystyle=\frac{1}{\sqrt{2L}}\left(\begin{array}[]{cc}\sqrt{L+1}&\sqrt{L-1}\\\
-\sqrt{L-1}&\sqrt{L+1}\end{array}\right)$ (A51)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L-1,S=1\right\rangle\\\
\left|J=L-1,S=2\right\rangle\end{array}\right),$ (A54)
$\displaystyle\left(\begin{array}[]{ccc}\left|J=L,J_{l}=L+1\right\rangle\\\
\left|J=L,J_{l}=L\right\rangle\\\
\left|J=L,J_{l}=L-1\right\rangle\end{array}\right)$ (A58)
$\displaystyle=\left(\begin{array}[]{ccc}\displaystyle\sqrt{\frac{2L+3}{3(2L+1)}}&\displaystyle\sqrt{\frac{L(2L+3)}{2(L+1)(2L+1)}}&\displaystyle\sqrt{\frac{L(2L-1)}{6(L+1)(2L+1)}}\\\
\displaystyle-\sqrt{\frac{1}{3}}&\displaystyle\sqrt{\frac{1}{2L(L+1)}}&\displaystyle\sqrt{\frac{(2L-1)(2L+3)}{6L(L+1)}}\\\
\displaystyle\sqrt{\frac{2L-1}{3(2L+1)}}&\displaystyle-\sqrt{\frac{(L+1)(2L-1)}{2L(2L+1)}}&\displaystyle\sqrt{\frac{(L+1)(2L+3)}{6L(2L+1)}}\\\
\end{array}\right)$ (A62)
$\displaystyle\times\left(\begin{array}[]{ccc}\left|J=L,S=0\right\rangle\\\
\left|J=L,S=1\right\rangle\\\ \left|J=L,S=2\right\rangle\end{array}\right),$
(A66)
$\displaystyle\left(\begin{array}[]{cc}\left|J=L+1,J_{l}=L+1\right\rangle\\\
\left|J=L+1,J_{l}=L\right\rangle\end{array}\right)$ (A69)
$\displaystyle=\frac{1}{\sqrt{2(L+1)}}\left(\begin{array}[]{cc}\sqrt{L+2}&\sqrt{L}\\\
-\sqrt{L}&\sqrt{L+2}\end{array}\right)$ (A72)
$\displaystyle\times\left(\begin{array}[]{cc}\left|J=L+1,S=1\right\rangle\\\
\left|J=L+1,S=2\right\rangle\end{array}\right).$ (A75)

Hence, the magic mixing angles of the ideal $QQ\bar{q}\bar{q}$ tetraquark are
$\theta=\arctan(\sqrt{L-1}/\sqrt{L+1})$ for $J=L-1$ and
$\theta=\arctan(\sqrt{L}/\sqrt{L+2})$ for $J=L+1$. When it comes to $J=L$, the
magic mixing angles are enumerated in Table 11.

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