# Spectroscopy of all charm tetraquark states

Rohit Tiwari  D P Rathaud<EMAIL_ADDRESS>A K Rai Department of
Physics, S. V. National Institute of Technology, Surat, Gujarat, India

###### Abstract

Abstract
The mass spectra of all-charm tetraquark states with the
[cc][$\bar{c}\bar{c}$] quark configuration are investigated. The coulomb plus
linear potential is used in conjunction with the relativistic mass correction
term $\mathcal{O}(\frac{1}{m})$. To determine the fitting parameters for all-
charm tetraquarks states [cc][$\bar{c}\bar{c}$], we first calculate the mass
spectra of charmonia [c$\bar{c}$] and its decay constants ($f^{2}_{P/V}$). We
estimated the masses of the tetraquark states in their ground and radially
excited states. For mass spectra of tetraquark states, we also included spin-
spin, spin-orbital, and tensor interactions. The mass spectra of charmonia
produced in this study are reasonably consistent with experimental and
theoretical predictions made by others, whilst the mass spectra of the
tetraquark states are consistent with previous theoretical predictions. We
propose that the X(6900) state, which has a mass range of 6.2 - 6.9 GeV and
was recently detected by LHCb, has the quantum numbers $0^{-+}$, $1^{-+}$,
$2^{-+}$ and belongs to the P-wave of the all-cham tetraquark state.

Keywords: Hadronic molecule, Exotic hadrons, Pentaquark, Potential model,
Hadron mass spectra
PACS No.: 12.39.Pn

###### keywords:

Non-relativistic model and Exotic hadrons and Tetraquarks and Mesons and
Potential model

##  1 Introduction

Exotic states are systems that do not fit within the quark-antiquark (meson)
or three quark combination (baryon) categories. Exotic states such as
tetraquarks, pentaquarks, and hexaquarks were proposed in 1964 [1], but the
first observation of exotic states was made in 2003 [2]. Due to improvements
in experimental facilities around the world, including as BaBar, Belle,
BESIII, CLEO, and LHCb, hadronic physics has seen a lot of progress in the
recent two decades [3, 4, 5].

Non-conventional states with quantum numbers and decay features other than
baryons and mesons were discovered as experimental equipment improved. Exotic
states are defined as states that defy the standard theory of hadrons, such as
X(3872) aka $\chi_{c1}$, X(3943), X(2900), Y(3943), Y(4260), and Z(3931) [6,
7, 8, 9]. The underlying structure of these exotic states necessitates a
thorough investigation to exclude out molecular tetraquark state [10] or
compact tetraquark as a viable explanation [11, 12, 13, 14, 15, 16].

In 2003, the Belle collaboration [6] presented $\chi_{c1}$(3872) as a narrow
charmonium-like exotic state formed only in the decay process
$B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}J/\psi$, with a mass extremely near
to the $M_{D^{0}}+M_{D^{*0}}$ threshold mass. Later, several research
collaborations in other decay modes proved the existence of $\chi_{c1}$(3872)
[17, 18, 19]. Many theoretical models and experimental analyses have been
produced since then in order to better understand the internal process of
tetraquark states [20, 21]. There are various well-established models that
explain the behaviours of tetraquark states, including lattice calculations
[22], QCD sum rules [23], coupled channel effects, and non-relativistic
effective field theories (e.g., see [20] and references therein). In Ref.
[24], author discusses the research of heavy tetraquarks in a relativistic
quark model. With diquark-antidiquark [qq-$\bar{q}\bar{q}$] pairs, the exact
nature of tetraquarks’ internal structure can be elucidated [25, 26]. A
diquark [qq] is a quark-quark pair bound, while an antidiquark
[$\bar{q}\bar{q}$] is an antiquark-antiquark pair bound. Later, the diquark-
antidiquark pair is thought to be in a non-singlet colour state, leading to a
color-singlet tetraquark state [27]. The mass spectroscopy of hidden charm
tetraquark states in a diquark-antidiquark model was investigated by the
author(s) in Ref. [28].

We studied the all-charm tetraquark state $T_{4c}$ (diquark-antidiquark
combination [cc][$\bar{c}\bar{c}$]) in this paper. In 1975, Iwasaki published
the first paper on all-charm tetraquarks[29]. Later, Chao discussed the
potential of all charm tetraquarks in the diquark-antidiquark model with
orbital excitation, and predicted their formation in the $e^{-}e^{+}$
annihilation [30]. The diquark-antidiquark model has proven to be an effective
tool for describing the tetraquarks found in Refs. [31, 32, 26, 33]. The LHCb
Collaboration [34] just announced the discovery of a full heavy charm
tetraquark, [cc][$\bar{c}\bar{c}$], in the J/$\psi$-pair spectrum with a mass
of roughly 6.9 GeV, however the quantum numbers ($J^{PC}$) have yet to be
confirmed. A tetraquark state $T_{4c}$ can decay into two c$\bar{c}$ states:
J/$\psi$ mesons or J/$\psi$ with a heavier charmonium state, which decays into
J/$\psi$ [34, 35]. The masses of all-charm tetraquarks are projected to be in
the range of 5.5-7 GeV in most theoretical and experimental predictions, which
is higher than the masses of predicted charmonium [c$\bar{c}$] states [36,
37]. In the J/$\psi$ productions at LHCb, CMS, and ATLAS collaboration, the
possible predictions of all-charm tetraquarks are highly promising [38, 39,
40]. Other possibilities include double c$\bar{c}$ production at Belle [41]
and differential production cross sections for J/$\psi$ pairings at LHCb [38]
around 6 to 8 GeV. The mass-spectra of four charm quark $T_{4c}$ system, which
has been classified in two body diquark-antidiquark [cc][$\bar{c}\bar{c}$]
system, have been obtained using a non-relativistic model with relativistic
correction in the current work.

The colour singlet tetraquark state $T_{4c}$ is considered to have diquark and
antidiquark constituents, which are selected to be in the colour antitriplet
and triplet representations, respectively. To research the dynamics of
tetraquark states, it is necessary to explore both the long and short range
behaviour of QCD, and Cornell-like potential has been particularly helpful in
this regard. A diquark-antidiquark pair will interact via a Cornell-like
potential and the relativistic mass correction $\mathcal{O}(\frac{1}{m})$. To
account for the splitting between the radial and orbital states, spin-
dependent factors (spin-spin, spin-orbit, and tensor) have been introduced as
perturbative corrections.
After a brief introduction in Sec. I, we will systematically discuss the
theoretical model using the diquark-antidiquark technique, i.e.
[cc][$\bar{c}\bar{c}$] in Sec. II. The results for mass spectra, as well as a
discussion of charmonia and S and P-wave tetraquark states, are reported in
Sec.III. Finally, we have concluded our work in Section IV.

##  2 Theoretical Model

In the diquark-antidiquark model, we present a non-relativistic [42, 43]
framework to analyse the spectroscopy of a hadronic bound state composed of
four charm quarks. The mass spectra of the all-charm tetraquark $T_{4c}$ were
estimated by solving the Schr$\ddot{o}$dinger equation numerically using code
initially created by W. Lucha et al., [44], which is based on the fourth order
Runge-Kutta (RK4) method [45, 46, 47]. The four-body system can be factorized
into two body system: the diquark and antidiquark, which are made up of two
quarks (antiquarks) that combine to form the colour antitriplet (triplet)
state. The two elements, diquark and antidiquark, are then studied, and their
interaction results in a colour singlet structure of tetraquarks. Obtaining
the mass-spectra of well-known charmonium [c$\bar{c}$] states has improved the
reliability and correctness of the current study, as it aids in the setting of
fitting parameters for obtaining mass-spectra of $T_{4c}$.

In heavy quark system the rest mass energy is comparatively higher than the
kinetic energy of the constituent quarks, hence the reasonable approximation
could be using of static potentials in a non-relativistic model [42]. We begin
to solve the two body problem with the time-independent Schr$\ddot{o}$dinger
equation;

$\left[\frac{1}{2\mu}\left(-\frac{d^{2}}{dr^{2}}\right)+V^{(0)}(r)\right]y(r)=Ey(r)$
(1)

It is more convenient to work in the center-of-mass frame (CM), which
incorporates two body problems in central potential [48]. Spherical harmonics
can separate the angular and radial terms of a wave function. The kinetic
energy of the system can be written as $\mu=\frac{m_{1}m_{2}}{m_{1}+m_{2}}$,
where $m_{1}$ and $m_{2}$ are the masses of charm(anticharm) quarks in
charmonium and diquarks, respectively, and the same input notation may be used
for diquark masses in $T_{4c}$. The Cornell-like potential model of zeroth-
order $V^{0}(r)$ has been applied in the spectroscopic research of the heavy-
quarkonium system. A coulomb term (($V^{(0)}_{C}$) is responsible for one
gluonic interaction between quarks and antiquarks, while a linear term
($V^{(0)}_{L}$) is responsible for quark confinement in the Cornell-like
potential model $V_{C+L}^{0}(r)$.

$V_{C+L}^{0}(r)=\frac{k_{s}\alpha_{s}}{r}+br$ (2)

where, $\alpha_{s}$ is known as the QCD running coupling constant, $k_{s}$ is
color factor, b is string tension. As we are dealing with the charm quark
which is considered in heavy-light mass limit as per data available in PDG
[49], we have incorporated the relativistic mass correction term $V^{1}(r)$
originally developed by Y. Coma et al. [50], in the central potential. The
final form of central potential is given by:

$V^{0}(r)=V_{C+L}^{0}(r)+V^{1}(r)\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)+\mathcal{O}\left(\frac{1}{m^{2}}\right)\\\
$ (3)

The non-perturbative form of relativistic mass correction term $V^{1}(r)$ is
not yet known but leading order perturbation theory yields,

$V^{1}(r)=-\frac{C_{F}C_{A}}{4}\frac{\alpha^{2}_{s}}{r^{2}}$ (4)

where $C_{F}=\frac{4}{3}$ and $C_{A}=3$ are the Casimir charges of the
fundamental and the adjoint representation respectively [50]. The relativistic
mass correction is found to be similar to the coulombic term of the static
potential when applied to the charmonium and to be one-fourth of the coulombic
term for bottomonium [51].

Along with the central interaction potential $V^{0}(r)$, we have also
incorporated spin-dependent interactions (spin-spin $V_{SS}$, spin-orbit
$V_{LS}$ and tensor $V_{T}$). These spin-dependent terms are included
perturbatively in the model.

###  2.1 Spin-dependent Terms

For better understanding about the splitting between orbital and radial
excitations for different combinations of quantum numbers of c$\bar{c}$ and
$T_{4c}$, it is necessary to incorporate the contributions from different
spin-dependent terms, i.e. spin-spin $V_{SS}$, spin-orbit $V_{LS}$ and tensor
$V_{T}$ which contributes significantly especially in excited states [53]. All
the three spin dependent terms are inspired with the Breit-Fermi Hamiltonian
for one-gluon exchange [55, 56], and yields;

$V_{SS}(r)=C_{SS}(r)S_{1}\cdot S_{2},$ (5) $V_{LS}(r)=C_{LS}(r)L\cdot S,$ (6)
$V_{T}(r)=C_{T}(r)S_{12}$ (7)

The matrix element of operator $\mathbf{S_{1}\cdot S_{2}}$ acts on wave
function and it generates a constant factor, still the ${V_{SS}}$ is a
function of r only and the expectation value of operator $\mathbf{\left\langle
S_{1}\cdot S_{2}\right\rangle}$ can be obtained by using quantum - mechanical
formula.

$\left\langle S_{1}\cdot
S_{2}\right\rangle=\left\langle\frac{1}{2}(S^{2}-S^{2}_{1}-S^{2}_{2})\right\rangle$
(8)

where, $S_{1}$ and $S_{2}$ are the spins of constituent quarks in case of
charmonium and diquarks in case of tetraquark respectively. $C_{SS}(r)$ may be
defined by;

$C_{SS}(r)=\frac{2}{3m^{2}}\nabla^{2}V_{V}(r)=-\frac{8k_{s}\alpha_{s}\pi}{3m^{2}}\delta^{3}(r),$
(9)

In heavy quarkonium spectroscopy a good agreement between theoretically
predicted states and experimental data available for c$\bar{c}$ can be
obtained by introducing a new parameter $\sigma$ Gaussian function in place of
the Dirac delta. So now $V_{SS}$ can be redefined as;

$V_{SS}(r)=-\frac{8\pi
k_{s}\alpha_{s}}{3m^{2}}(\frac{\sigma}{\sqrt{\pi}})^{3}\exp^{-\sigma^{2}r^{2}}S_{1}\cdot
S_{2},$ (10)

The expectation value of operator $\left\langle L\cdot S\right\rangle$ is
mainly dependent on the total angular momentum J which consists of $\bf
J=L+S$, and can be obtained by using the formula,

$\mathbf{\left\langle L\cdot
S\right\rangle=\left\langle\frac{1}{2}(J^{2}-L^{2}-S^{2})\right\rangle}$ (11)

where L is the total orbital angular momentum of quarks and diqaurks in the
case of charmonium and tetraquark respectively. $C_{LS}(r)$ can be calculated
by using relation given below;

$C_{LS}(r)=-\frac{3k_{s}\alpha_{s}\pi}{2m^{2}}\frac{1}{r^{2}}-\frac{b}{2m^{2}}\frac{1}{r}$
(12)

In the spin-orbit interaction the second term is known as Thomas precession
which is proportional to scalar term and it is assumed that confining
interaction arises from the Lorentz scalar structure. The contribution of
spin-tensor becomes very crucial in higher excited states which requires a bit
of algebra and can be calculated by;

$V_{T}(r)=C_{T}(r)\left(\frac{(S_{1}\cdot r)(S_{2}\cdot
r)}{r^{2}}-\frac{1}{3}(S_{1}\cdot S_{2})\right)$ (13)

where;

$C_{T}(r)=-\frac{12k_{s}\alpha_{s}\pi}{4m^{2}}\frac{1}{r^{3}}$ (14)

The results of $(S_{1}\cdot S_{2})$ can be obtained by solving the diagonal
matrix elements for the spin $\frac{1}{2}$ and spin 1 particles, for more one
can see in the following Refs.[56, 57]. The simplified formulation can be
expressed to solve the tensor interaction;

$\mathbf{S_{12}}=12\left(\mathbf{\frac{(S_{1}\cdot r)(S_{2}\cdot
r)}{r^{2}}-\frac{1}{3}(S_{1}\cdot S_{2}})\right)$ (15)

and which can be redefined as :

$\mathbf{S_{12}}=4\left[3\mathbf{(S_{1}\cdot\hat{r})(S_{2}\cdot\hat{r})-(S_{1}\cdot
S_{2})}\right]$ (16)

The results of $S_{12}$ term can be obtained with the help of Pauli matrices
and spherical harmonics with their corresponding eigenvalues. The following
results are valid for charmonium and diquarks are obtained as follows can be
found more on [57];

$\displaystyle\mathbf{\left\langle
S_{12}\right\rangle}_{\frac{1}{2}\otimes\frac{1}{2}\rightarrow S=1,~{}l\neq
0}=-\frac{2l}{2l+3},~{}for~{}J=l+1,$ (17)
$\displaystyle=-\frac{2(l+1)}{(2l-1)},~{}for~{}J=l-1,$ (18)
$\displaystyle=+2,~{}for~{}J=l$ (19)

For $l=0$ and $S=0$ the $\mathbf{\left\langle S_{12}\right\rangle}$ always
vanishes but it gives non-zero values for excited states in charmonium states,
equ.17 can be generalised as $\mathbf{\left\langle
S_{12}\right\rangle}=-\frac{2}{5},+2,-4$ for $J=2,1,0$ respectively. These
values are valid only in the case of spin-half particles specially used in
charmonium and diquarks, but in case of tetraquarks where spin-1 diquarks are
involved and it requires tedious algebra which we will not discuss here in
detail rather we use those results presented in Refs. [53, 57].
The tensor interaction of $T_{4c}$ will obtained by the same formula which is
used in case of charmonium except that the wavefunction obtained here will be
of spin 1 (anti)diquark.

$\displaystyle\mathbf{S_{{d}-{\bar{d}}}}=12\left(\mathbf{\frac{(S_{d}\cdot
r)(S_{\bar{d}}\cdot r)}{r^{2}}-\frac{1}{3}(S_{d}\cdot S_{\bar{d}})}\right)$
(20) $\displaystyle=\mathbf{S_{14}+S_{13}+S_{24}+S_{23}}$ (21)

where $S_{d}$ is the total spin of the diquark, $S_{\bar{d}}$ is the total
spin of the antidiquark. When solving the 2-body problem to get the mass of
the diquark, the interaction between the two quarks (whose indices are 1 and
2) inside the diquark is taken into account (and since we consider diquarks
only in S-wave state, only the spin-spin interaction is relevant; the spin-
orbit and tensor are identically zero, as we have already discussed). The
interaction between the two antiquarks (whose indices are 3 and 4) inside the
antidiquark is identical. Because the tetraquark radial wavefunction is
obtained by approximating the diquark and antidiquark as point-like
structures, it is reasonable to assume that the radial-dependence of the
tensor term is the same for these four [q-$\bar{q}$] interactions and can be
obtained using the radial wavefunction with Eq.(14). The expectation value of
the radial wavefunction between every q$\bar{q}$ pair is the same and may be
factorised since the tetraquark is viewed as a two-body system. The following
functional form for spin $\frac{1}{2}$ particles does not use any specific
relation or eigenvalues, instead relying on general angular momentum
elementary theory [58]. Within this approximation, generalization of tensor
operator can be consider a sum of four tensor interaction between four quark-
antiquark pair as illustrated in [57]. A thorough discussion on tensor
interaction can be found in Ref. [57]. All the spin-dependent (spin-spin,
spin-orbital and tensor) interactions have been treated perturbatively in the
model to observe the individual contributions.

###  2.2 Decay Constant

An important quantity is square modulus of wave function at the origin
$|\psi(0)|^{2}$, to calculate the decay widths, wavefunction or derivative of
wavefunction at the origin can be used [53]. In quarkonium models the
centrifugal term present in Schr$\ddot{o}$dinger equation creates a
“centrifugal barrier” and due to that wavefunction at the origin can be
calculated only for the $l=0$ i.e. S-states, and it vanishes in the excited
states for $l\neq 0$.

$|\psi(0)|^{2}=|Y_{0}^{0}(\theta,\phi)R_{nl}(0)|^{2}=\frac{|R_{nl}(0)|^{2}}{4\pi}$
(22)

The results of square modulus of radial wavefunction at the origin
$|R_{nl}(0)|^{2}$ can be obtained from numerical calculations. The values
obtained for pseudoscalar and vector charmonium states will be crucial
quantity to find out the decay constants which is again an important quantity
to understand the decay properties. In Ref.[54], author has examined the decay
constants of light-heavy mesons by relativistic treatment. We estimate the
decay constants in the non-relativistic limit by using the radial wave
function for pseudoscalar and vector charmonium states.

The formula is given by Van-Royen-Weisskopf [59, 60].

$f^{2}_{P/V}=\frac{3|R_{nsP/V}(0)|^{2}}{\pi M_{nsP/V}}$ (23)

where $|R_{nsP/V}(0)|^{2}$ and $M_{nsP/V}$ are wave-function at the origin and
mass of the pseudoscalar and vector states respectively.

##  3 Results and Discussion

###  3.1 Charmonium

To calculate the mass-spectra of diquarks and tetraquarks, first we estimate
the mass-spectra of charmonium states c$\bar{c}$, whose results are tabulated
in Table 1. The reliability and consistency of the model have been tested by
obtaining the mass-spectra and decay constant of the charmonium states which
is very close to the experimental data available in recent PDG [49]. The SU(3)
color symmetry allows only colorless quark combination $|q\bar{q}\rangle$ to
form any color singlet state [26, 61], as in our case c$\bar{c}$ is a meson
and exhibits $|q\bar{q}\rangle:\mathbf{3\otimes\bar{3}=1\oplus 8}$
representation which leads to carry a color factor $k_{s}=-\frac{4}{3}$ [57,
61]. The masses of the particular c$\bar{c}$ states can be calculated by
solving the Schr$\ddot{o}$dinger equation. Therefore :

$M_{(c\bar{c})}=2m_{c}+E_{c\bar{c}}+\langle V^{1}(r)\rangle$ (24)

The final mass $M_{f}$ obtained from above expression constitutes
contributions from spin-dependent terms (spin-spin, spin-orbital and spin-
tensor) along with the relativistic-mass correction term. The comparative
results like $M_{i}^{Exp}$ [49] (PDG) and study of all-charm tetraquark in a
non-relativistic model [26] are also tabulated in Table-1.
Parameters (adopted from PDG [49]) used in the present work are $m_{c}=1.4459$
GeV, $\alpha_{s}=0.5202$, b$=0.1463$ Ge$V^{2}$, and $\sigma=1.0831$ GeV. The
quantum numbers $J^{PC}$ of $q\bar{q}$ states i.e. parity and charge
conjugation can be calculated by; P = $(-1)^{L+1}$ and C = $(-1)^{L+S}$
respectively, where L is the orbital angular momentum and S being the total
spin of quarks.

Table 1: Mass-Spectra of charmonium states c$\bar{c}$ along with the
relativistic mass correction. (Units are in MeV)

$N^{2S+1}L_{J}$ | $\langle K.E.\rangle$ | $E^{(0)}$ | $\langle V^{(0)}_{C}\rangle$ | $\langle V^{(0)}_{L}\rangle$ | $\langle V^{(0)}_{SS}\rangle$ | $\langle V^{(1)}_{LS}\rangle$ | $\langle V^{(1)}_{T}\rangle$ | $V^{(1)}(r)$ | $M^{f}$ |  $M_{i}^{Exp}$ [49] | Meson
---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | 470.3 | 134.9 | -537.9 | 271.1 | -69.4 | 0 | 0 | -4.6 | 2983 | 2983.4$\pm$0.5 | $\eta_{c}(1S)$
$1^{3}S_{1}$ | 377.1 | 134.2 | -537.9 | 271.1 | 23.1 | 0 | 0 | -4.7 | 3075 | 3096.9$\pm$0.006 | J/$\psi(1S)$
$2^{1}S_{0}$ | 464.5 | 734.5 | -290.7 | 589.8 | -29.0 | 0 | 0 | -2.4 | 3623 | 3639.2$\pm$1.2 | $\eta_{c}(2S)$
$2^{3}S_{1}$ | 428.1 | 736.8 | -290.7 | 589.7 | 9.6 | 0 | 0 | -2.5 | 3664 | 3686.097$\pm$0.025 | J/$\psi(2S)$
$3^{1}S_{0}$ | 548.4 | 1148.2 | -220.7 | 840.7 | -20.1 | 0 | 0 | -1.7 | 4046 | - | -
$3^{3}S_{1}$ | 522.2 | 1148.9 | -220.7 | 840.7 | 6.7 | 0 | 0 | -1.6 | 4073 | 4039$\pm$1 | $\psi(4040)$
$4^{1}S_{0}$ | 635.7 | 1493.3 | -184.7 | 1058.2 | -15.9 | 0 | 0 | -1.2 | 4395 | - | -
$4^{3}S_{1}$ | 614.8 | 1493.7 | -184.7 | 1058.3 | 5.3 | 0 | 0 | -1.3 | 4417 | 4421$\pm$4 | $\psi(4415)$
$5^{1}S_{0}$ | 720.5 | 1800.1 | -161.9 | 1254.9 | -13.4 | 0 | 0 | -1.0 | 4704 | - | -
$5^{3}S_{1}$ | 702.9 | 1800.4 | -161.9 | 1254.9 | 4.4 | 0 | 0 | -1.0 | 4722 | - | -
$6^{1}S_{0}$ | 801.9 | 2081.5 | -145.7 | 1437.0 | -11.6 | 0 | 0 | -0.8 | 4987 | - | -
$6^{3}S_{1}$ | 786.6 | 2081.7 | -145.7 | 1437.0 | 3.8 | 0 | 0 | -0.8 | 5003 | - | -
$7^{1}S_{0}$ | 880.3 | 2344.3 | -133.5 | 1607.9 | -10.4 | 0 | 0 | -0.7 | 5251 | - | -
$7^{3}S_{1}$ | 866.5 | 2344.3 | -133.5 | 1607.9 | 3.4 | 0 | 0 | -0.7 | 5265 | - | -
$1^{3}P_{0}$ | 362.6 | 591.3 | -254.4 | 480.8 | 2.2 | -69.4 | -31.4 | -2.9 | 3410 | 3414.75$\pm$0.31 | $\chi_{c0}(1P)$
$1^{3}P_{1}$ | 362.5 | 591.2 | -254.4 | 480.8 | 2.2 | -34.7 | 15.7 | -3.1 | 3492 | 3510.66$\pm$0.07 | $\chi_{c1}(1P)$
$1^{1}P_{1}$ | 371.1 | 591.0 | -254.4 | 480.8 | -6.6 | 0 | 0 | -2.8 | 3502 | 3525.38$\pm$0.11 | $h_{c}(1P)$
$1^{3}P_{2}$ | 362.6 | 591.3 | -254.4 | 480.8 | 2.2 | 34.7 | -3.1 | -2.9 | 3543 | 3556.20$\pm$0.09 | $\chi_{c2}(1P)$
$2^{3}P_{0}$ | 463.3 | 1019.0 | -191.8 | 745.2 | 2.3 | -64.8 | -27.9 | -1.6 | 3846 | - | -
$2^{3}P_{1}$ | 464.5 | 1020.3 | -191.8 | 745.2 | 2.3 | -32.4 | 13.9 | -2.0 | 3922 | - | -
$2^{1}P_{1}$ | 473.6 | 1019.9 | -191.8 | 745.2 | -7.0 | 0 | 0 | -2.0 | 3930 | - | -
$2^{3}P_{2}$ | 464.2 | 1019.0 | -191.8 | 745.2 | 2.3 | 32.4 | -2.7 | -2.0 | 3969 | 3927.2$\pm$2.6 | $\chi_{c2}(2P)$
$3^{3}P_{0}$ | 561.2 | 1374.3 | -160.2 | 971.0 | 2.3 | -62.9 | -26.2 | -1.4 | 4205 | - | -
$3^{3}P_{1}$ | 561.8 | 1374.9 | -160.2 | 971.0 | 2.3 | -31.4 | 13.1 | -1.3 | 4276 | - | -
$3^{1}P_{1}$ | 571.1 | 1375.0 | -160.2 | 971.0 | -6.9 | 0 | 0 | -1.4 | 4286 | - | -
$3^{3}P_{2}$ | 561.9 | 1375.0 | -160.2 | 971.0 | 2.3 | 31.4 | -2.6 | -1.4 | 4324 | - | -
$4^{3}P_{0}$ | 653.6 | 1688.9 | -140.3 | 1173.4 | 2.2 | -61.9 | -25.2 | -1.1 | 4522 | - | -
$4^{3}P_{1}$ | 653.5 | 1688.9 | -140.3 | 1173.5 | 2.2 | -30.9 | 12.6 | -1.1 | 4590 | - | -
$4^{1}P_{1}$ | 662.5 | 1688.9 | -140.3 | 1173.5 | -6.7 | 0 | 0 | -1.1 | 4600 | - | -
$4^{3}P_{2}$ | 653.5 | 1688.9 | -140.3 | 1173.5 | 2.2 | 30.9 | -2.5 | -1.1 | 4637 | - | -
$5^{3}P_{0}$ | 739.9 | 1975.9 | -126.3 | 1359.9 | 2.1 | -61.2 | -24.5 | -0.9 | 4810 | - | -
$5^{3}P_{1}$ | 740.0 | 1975.9 | -126.3 | 1360.0 | 2.1 | -30.6 | 12.2 | -0.7 | 4877 | - | -
$5^{1}P_{1}$ | 748.8 | 1976.1 | -126.3 | 1360.0 | -6.4 | 0 | 0 | -0.9 | 4887 | - | -
$5^{3}P_{2}$ | 740.1 | 1976.9 | -126.3 | 1360.0 | 2.1 | 30.6 | -2.4 | -0.9 | 4924 | - | -
$6^{3}P_{0}$ | 822.2 | 2243.0 | -115.8 | 1534.5 | 2.0 | -60.7 | -24.0 | -0.7 | 5078 | - | -
$6^{3}P_{1}$ | 822.4 | 2243.1 | -115.8 | 1534.5 | 2.0 | -30.3 | 12.0 | -0.7 | 5144 | - | -
$6^{1}P_{1}$ | 823.5 | 2243.2 | -115.8 | 1534.5 | 2.0 | 0 | 0 | -0.8 | 5155 | - | -
$6^{3}P_{2}$ | 822.3 | 2243.1 | -115.8 | 1534.5 | 2.0 | 30.3 | -2.4 | -0.8 | 5191 | - | -
$1^{3}D_{1}$ | 409.3 | 873.7 | -179.7 | 643.9 | 0.2 | -9.7 | -4.0 | -2.3 | 3778 | 3773.13$\pm$0.35 | $\psi(3770)$
$1^{3}D_{2}$ | 409.2 | 873.6 | -179.7 | 643.9 | 0.2 | -3.2 | 4.0 | -2.3 | 3792 | - | -
$1^{1}D_{2}$ | 410.1 | 873.7 | -179.7 | 643.9 | -0.06 | 0 | 0 | -2.3 | 3791 | - | -
$1^{3}D_{3}$ | 409.2 | 873.7 | -179.7 | 643.9 | 0.2 | 6.5 | -1.1 | -2.3 | 3797 | - | -

Table 1: to be continued..

$2^{3}D_{1}$ | 512.2 | 1242.3 | -149.4 | 879.1 | 0.3 | -12.6 | -3.8 | -1.6 | 4144 | 4191$\pm$5 | $\psi(4160)$
---|---|---|---|---|---|---|---|---|---|---|---
$2^{3}D_{2}$ | 512.2 | 1242.3 | -149.4 | 879.1 | 0.3 | -4.2 | 3.8 | -1.5 | 4160 | - | -
$2^{1}D_{2}$ | 513.7 | 1242.4 | -149.4 | 879.1 | -0.1 | 0 | 0 | -1.6 | 4159 | - | -
$2^{3}D_{3}$ | 512.2 | 1242.3 | -149.4 | 879.1 | 0.3 | 8.4 | -0.1 | -1.6 | 4167 | - | -
$3^{3}D_{1}$ | 607.2 | 1565.4 | -130.5 | 1088.2 | 0.4 | -14.4 | -3.7 | -1.2 | 4465 | - | -
$3^{3}D_{2}$ | 607.6 | 1565.8 | -130.5 | 1088.2 | 0.4 | -4.8 | 3.7 | -1.2 | 4483 | - | -
$3^{1}D_{2}$ | 609.5 | 1565.8 | -130.5 | 1088.2 | -1.3 | 0 | 0 | -1.3 | 4482 | - | -
$3^{3}D_{3}$ | 607.7 | 1565.7 | -130.5 | 1088.2 | 0.4 | 9.6 | -1.0 | -1.3 | 4492 | - | -
$4^{3}D_{1}$ | 696.7 | 1859.5 | -117.3 | 1279.6 | 0.5 | -15.7 | -3.6 | -0.9 | 4758 | - | -
$4^{3}D_{2}$ | 697.0 | 1859.8 | -117.3 | 1279.7 | 0.5 | -5.2 | 3.6 | -1.0 | 4776 | - | -
$4^{1}D_{2}$ | 699.0 | 1859.7 | -117.3 | 1279.6 | -1.5 | 0 | 0 | -1.0 | 4776 | - | -
$4^{3}D_{3}$ | 696.9 | 1859.7 | -117.3 | 1279.7 | 0.5 | 10.4 | -1.0 | -1.0 | 4787 | - | -
$5^{3}D_{1}$ | 781.2 | 2132.4 | -107.5 | 1458.1 | 0.5 | -16.7 | -3.6 | -0.8 | 5030 | - | -
$5^{3}D_{2}$ | 781.4 | 2132.5 | -107.5 | 1458.1 | 0.5 | -5.5 | 3.6 | -0.8 | 5049 | - | -
$5^{1}D_{2}$ | 783.7 | 2132.5 | -107.5 | 1458.1 | -1.7 | 0 | 0 | -0.8 | 5048 | - | -
$5^{3}D_{3}$ | 781.4 | 2132.6 | -107.5 | 1458.1 | 0.5 | 11.1 | -1.0 | -0.8 | 5061 | - | -
$1^{3}F_{2}$ | 463.7 | 1105.8 | -143.5 | 785.6 | 0.001 | 6.4 | -1.4 | -1.9 | 4028 | - | -
$1^{3}F_{3}$ | 462.5 | 1104.7 | -143.5 | 785.6 | 0.001 | 1.6 | 1.8 | -2.0 | 4026 | - | -
$1^{1}F_{3}$ | 462.6 | 1104.7 | -143.5 | 785.6 | -0.005 | 0 | 0 | -1.5 | 4022 | - | -
$1^{3}F_{4}$ | 463.7 | 1105.2 | -143.5 | 785.6 | 0.001 | -4.8 | -0.6 | -2.0 | 4017 | - | -
$2^{3}F_{2}$ | 561.2 | 1437.9 | -124.9 | 1001.5 | 0.004 | 3.2 | -1.4 | -1.4 | 4357 | - | -
$2^{3}F_{3}$ | 561.7 | 1438.4 | -124.9 | 1001.6 | 0.004 | 0.8 | 1.7 | -1.4 | 4359 | - | -
$2^{1}F_{3}$ | 561.9 | 1438.4 | -124.9 | 1001.6 | -0.1 | 0 | 0 | -1.3 | 4356 | - | -
$2^{3}F_{4}$ | 561.8 | 1438.6 | -124.9 | 1001.6 | 0.004 | -2.4 | -0.5 | -1.4 | 4353 | - | -
$3^{3}F_{2}$ | 653.6 | 1739.8 | -112.1 | 1198.2 | 0.006 | 1.1 | -1.3 | -0.9 | 4657 | - | -
$3^{3}F_{3}$ | 654.1 | 1740.2 | -112.1 | 1198.2 | 0.006 | 0.2 | 1.7 | -1.1 | 4660 | - | -
$3^{1}F_{3}$ | 654.2 | 1740.1 | -112.1 | 1198.2 | -0.002 | 0 | 0 | -1.1 | 4657 | - | -
$3^{3}F_{4}$ | 653.9 | 1740.1 | -112.1 | 1198.2 | 0.006 | -0.8 | -0.5 | -1.1 | 4656 | - | -
$4^{3}F_{2}$ | 740.4 | 2018.7 | -102.5 | 1380.7 | 0.009 | -0.5 | -1.3 | -0.9 | 4934 | - | -
$4^{3}F_{3}$ | 740.6 | 2018.9 | -102.5 | 1380.7 | 0.009 | -0.1 | 1.7 | -0.9 | 4938 | - | -
$4^{1}F_{3}$ | 741.0 | 2018.9 | -102.5 | 1380.7 | -0.2 | 0 | 0 | -0.9 | 4936 | - | -
$4^{3}F_{4}$ | 740.6 | 2018.9 | -102.5 | 1380.7 | 0.009 | 0.3 | -0.5 | -0.9 | 4936 | - | -

The masses of charmonium S-wave states are tabulated in Table 1, they are in
good accord with experimental results. All three spin-dependent terms (spin-
spin, spin-orbital, and tensor) as well as the relativistic correction term
contribute to the final mass $M^{f}$. The stronger attractive strength of the
coulomb term $\langle V^{(0)}_{C}\rangle$ indicates that one gluon exchange
(OGE) dominates over other interactions, suppressing the masses of these
states. In all radially excited states, the spin-spin interaction is
attractive for pseudoscalar states and repulsive for vector states; however,
it weakens in higher orbitally excited states.

We find that at the lowest S-wave states, the spin-spin interaction $\langle
V^{(0)}_{SS}\rangle$ and the relativistic $\langle V^{1}(r)\rangle$ correction
contribute the most attractive strength, resulting in a mass extremely similar
to the empirically measured $\eta_{c}(1S)$ meson. In all radial and orbital
excited states, the relativistic-mass correction has attractive strength,
while its strength is insignificant in higher excited states. In S-wave
states, as well as higher orbital excited states when the total spin of the
quark and antiquark is zero, the spin-tensor and spin-orbital interactions
vanish, e.g. $1^{1}P_{1}$, $1^{1}D_{2}$, $1^{1}F_{3}$, and so forth. Our
model’s mass for the J/$\psi(1S)$ meson is 20 MeV lower than the experimental
mass [49]. Similarly, the masses of radially excited S-wave states
$\eta_{c}(2S)$, $\psi(2S)$, $\psi(4040)$ and $\psi(4415)$ accord well with the
experimental mass found in [49]. The masses of $\chi_{c0}(1P)$,
$\chi_{c1}(1P)$, $h_{c}(1P)$, and $\chi_{c2}(1P)$ in higher orbital excited
states, particularly in (1P) wave, differ by roughly 10 - 15 MeV from the
observed mass, which is owing to the different set of parameters we utilised
in our model. The mass spectra of two D-wave states obtained from our model,
$\psi(3770)$ and $\psi(4160)$, match the experimentally observed mesons. We
find that the uncertainty in the masses of c$\bar{c}$ states in total spin-1
states is higher (but only by a few MeV) than in total spin-0 quark states.
Table-2 shows the results of the radial wavefunction at the origin, as well as
the decay constants for pseudoscalar and vector charmonium states c$\bar{c}$.
Because of the relativistic correction term that has been included in the
central potential, we see that the wavefunction values for pseudoscalar and
vector states have a very minor differences. Table 2 also contains the results
of decay constants for c$\bar{c}$ states, which we compared to experimental
data and other prior publications. Our non-relativistic results for decay
constants of pseudoscalar and vector states are quite close to the findings
obtained using a complete relativistic approach. Some experimentally detected
S-wave mesons, such as the [62], have masses in the region of 3-4 GeV, and as
a result, they have the same quantum numbers $1^{--}$, causing a lot of
confusion. So, since decay properties are the instrument that allows us to
identify states, we’ve acquired the decay constants.

Table 2: Results of wavefunctions at the origin and the decay constants from
present work and others including experimental data [49], of pseudo scalar and
vector S-wave charmonium states c$\bar{c}$. (Units are in GeV)

$N^{2S+1}L_{J}$ | $|R_{nsP/V}(0)|^{2}$ | $f_{P/V}$ |  Exp.[49] | [51] | [63] | [64]
---|---|---|---|---|---|---
$1^{1}S_{0}$ | 1.053 | 0.355 |  0.335$\pm$0.075 | 0.501 | 0.471 | 0.404
$1^{3}S_{1}$ | 1.158 | 0.416 |  0.411$\pm$0.005 | 0.510 | 0.462 | 0.375
$2^{1}S_{0}$ | 0.823 | 0.178 | - | 0.301 | 0.344 | 0.331
$2^{3}S_{1}$ | 0.861 | 0.193 |  0.271$\pm$0.008 | 0.303 | 0.369 | 0.295
$3^{1}S_{0}$ | 0.754 | 0.134 | - | 0.264 | 0.332 | 0.291
$3^{3}S_{1}$ | 0.779 | 0.142 |  0.174$\pm$0.018 | 0.265 | 0.329 | 0.261
$4^{1}S_{0}$ | 0.718 | 0.112 | - | 0.245 | 0.312 | -
$4^{3}S_{1}$ | 0.738 | 0.117 | - | 0.240 | 0.310 | 0.240
$5^{1}S_{0}$ | 0.694 | 0.097 | - | 0.233 | - | -
$5^{3}S_{1}$ | 0.711 | 0.102 | - | 0.234 | 0.290 | -
$6^{1}S_{0}$ | 0.678 | 0.088 | - | 0.224 | - | -
$6^{3}S_{1}$ | 0.692 | 0.091 | - | 0.225 | - | -
$7^{1}S_{0}$ | 0.665 | 0.080 | - | - | - | -
$7^{3}S_{1}$ | 0.678 | 0.083 | - | - | - | -

###  3.2 Diquarks

A (anti)diquark is a bound state in which two (anti)charm quarks engage by a
single gluon exchange between quarks [65, 66]. By solving the
Schr$\ddot{o}dinger$ equation, we will be able to derive the mass spectra of
(anti)diquarks using the same formula as we did for charmonium.

When two quarks are combined in the fundamental (3) representation, we get
$\mathbf{3\otimes 3=\bar{3}\oplus 6}$, according to QCD colour symmetry.
Antiquarks are also combined in the $\mathbf{\bar{3}}$ representation,
resulting in antidiquark in the 3 reprsentation [48, 61].

In the antitriplet state, these QCD colour symmetry gives a colour factor
$k_{s}=-\frac{2}{3}$, which makes the short distance part of the interaction
$(\frac{1}{r})$ appealing [57]. We’ve opted to deal with an attractive
antitriplet colour state, thus the diquark’s colour wave function is
antisymmetric. To follow the Pauli exclusion principle, the diquarks’ spin
should be 1, resulting in an antisymmetric diquark [48] wavefunction. The
ground state ($1^{3}S_{1}$) diquark [cc] with no orbital or radial excitations
will be used to obtain the most compact diqaurk. (See [29, 67] and the
references therein for more information on diquarks.)

The masses of diquarks are calculated in the same way as the mass spectrum of
charmonium is calculated.

Table 3: The mass-spectra of (anti)diquarks [cc], generated from our model.
Parameters are $m_{c}=1.4459$ GeV, $\alpha_{s}=0.5202$, $b=0.07315$ $GeV^{2}$,
and $\sigma=1.0831$ GeV

$N^{2S+1}L_{J}$ | $\langle K.E.\rangle$ | $E^{(0)}$ | $\langle V^{(0)}_{C}\rangle$ | $\langle V^{(0)}_{L}\rangle$ | $\langle V^{(1)}_{SS}\rangle$ | $\langle V^{(1)}_{LS}\rangle$ | $\langle V^{(1)}_{T}\rangle$ | $\langle V^{(1)}(r)\rangle$ | $M_{f}$ | [26]
---|---|---|---|---|---|---|---|---|---|---
$1^{3}S_{1}$ | 181.6 | 200.9 | -181.9 | 196.2 | 5.0 | 0 | 0 | -2.7 | 3124 | 3133.4
$2^{3}S_{1}$ | 245.5 | 530.1 | -109.1 | 390.8 | 2.9 | 0 | 0 | -1.4 | 3451 | 3456.0
$1^{1}P_{1}$ | 206.7 | 430.2 | -95.0 | 319.5 | -0.9 | 0 | 0 | -1.8 | 3347 | 3353.3
$2^{1}P_{1}$ | 278.2 | 686.5 | -73.4 | 482.9 | -1.2 | 0 | 0 | -1.2 | 3603 | 3606.2

###  3.3 Tetraquark

A $T_{4c}$ is color singlet state and yields a color factor
$k_{s}=-\frac{4}{3}$, and which is a mixture of two body system diquark-
antidiquark in antitriplet and triplet color state respectively [26, 61].
These spin-1 diquark-antidiquark combined together to form color singlet
tetraquark [65] and that can be represented as;
$|QQ|^{3}\otimes|\bar{Q}\bar{Q}|^{\bar{3}}\rangle=\mathbf{1\oplus 8}$. We have
used the masses of diquark-antidiquark ($m_{cc}=m_{\bar{c}\bar{c}}$) in
$1^{3}S_{1}$ state as two inputs of four body system. The the mass-spectra of
all charm tetraquark (cc - $\bar{c}\bar{c}$) have been obtained with the same
formulation as we have obtained in the case of charmonium $c\bar{c}$ and
(anti)diquark.

$M_{(T_{4c})}=m_{cc}+m_{\bar{c}\bar{c}}+E_{[cc][\bar{c}\bar{c}]}+\langle
V^{1}(r)\rangle$ (25)

The Cornell potential, as well as the relativistic correction term $\langle
V^{1}(r)\rangle$, contribute to the mass of $T_{4c}$ produced from this
expression. All spin-dependent (spin-spin, spin-orbital, and spin-tensor)
factors’ contributions were estimated individually. For spin-1 diquarks and
spin 1 antidiquarks that pair to generate a colour singlet tetraquark with
spin $S_{T}$ = 0,1,2, all spin-dependent corrections will be calculated. By
coupling the total spin $S_{T}$ and total orbital angular momentum $L_{T}$, we
obtain the mass-spectrum of radial and orbital excitation, which gives rise to
the total angular momentum $J_{T}$. We’ll utilise notations for diquark spin
$S_{d}$ and antidiquark spin $S_{\bar{d}}$ to get the tetraquak’s total spin
$S_{T}$ and quantum number $J^{PC}$. The colour singlet state $S_{T}\otimes
L_{T}$ is created by coupling $S_{T}$ with the orbital angular momentum
$L_{T}$.

$|T_{4Q}\rangle=|S_{d},S_{\bar{d}},S_{T},L_{T}\rangle_{J_{T}}$ (26)

where $J_{T}$ is the total angular momentum of the tetraquark, which is
obtained by $S_{T}\otimes L_{T}$. To find out the $J^{PC}$ quantum numbers of
the tetraquark states, we will obtain the results of charge-conjugation and
parity from the given relations [48];

$C_{T}=(-1)^{L_{T}+S_{T}},P_{T}=(-1)^{L_{T}}.$ (27)

Parameters are $m_{cc}$=3.124 GeV, $\alpha_{s}$=0.5304, b=0.1480 $GeV^{2}$,
and $\sigma$= 1.048 GeV. In the case of charmonium (c$\bar{c}$) and diquark
(cc), there were only two potential total spin ($S_{T}$) combinations, however
in the case of tetraquark states $T_{4c}$, there are three potential spin
combinations because spin-1 diquark-antidiquark are used.

$\displaystyle|0^{++}\rangle_{T_{4c}}=|S_{cc}=1,S_{\bar{c}\bar{c}=1},S_{T}=0,L_{T}=0\rangle_{J_{T}=0};$
(28)
$\displaystyle|1^{+-}\rangle_{T_{4c}}=|S_{cc}=1,S_{\bar{c}\bar{c}=1},S_{T}=0,L_{T}=0\rangle_{J_{T}=1};$
(29)
$\displaystyle|2^{++}\rangle_{T_{4c}}=|S_{cc}=1,S_{\bar{c}\bar{c}=1},S_{T}=2,L_{T}=0\rangle_{J_{T}=2}$
(30)

Table 4: The mass-spectra of S and P-wave tetraquark $T_{4c}$, generated from
our model. $M_{th}$ [49] is threshold mass of two mesons. (Units are in MeV)

$N^{2S+1}L_{J}$ | $J^{PC}$ | $\langle K.E.\rangle$ | $E^{(0)}$ | $\langle V^{(0)}_{C}\rangle$ | $\langle V^{(0)}_{L}\rangle$ | $\langle V^{(1)}_{SS}\rangle$ | $\langle V^{(1)}_{LS}\rangle$ | $\langle V^{(1)}_{T}\rangle$ | $V^{(1)}(r)$ | $M_{f}$ |  $\mathbf{M_{th}}$ [49] | Threshold |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$1^{1}S_{0}$ | $0^{++}$ | 629.2 | -215.6 | -916.6 | 164.7 | -93.1 | 0 | 0 | -3.8 | 5939 | 5966.8 | $\eta_{c}(1S)\eta_{c}(1S)$ |
$1^{3}S_{1}$ | $1^{+-}$ | 583.3 | -215.6 | -916.6 | 164.7 | -46.5 | 0 | 0 | -3.7 | 5986 | 6080.3 | $\eta_{c}(1S)J/\psi(1S)$ |
$1^{5}S_{2}$ | $2^{++}$ | 490.4 | -214.8 | -916.6 | 164.7 | 46.5 | 0 | 0 | -3.7 | 6079 | 6193.8 | $J/\psi(1S)J/\psi(1S)$ |
$2^{1}S_{0}$ | $0^{++}$ | 434.9 | 417.4 | -409.7 | 414.8 | -22.7 | 0 | 0 | -2.0 | 6642 | - | - |
$2^{3}S_{1}$ | $1^{+-}$ | 423.6 | 417.4 | -409.7 | 414.8 | -11.3 | 0 | 0 | -2.0 | 6654 | - | - |
$2^{5}S_{2}$ | $2^{++}$ | 400.9 | 417.4 | -409.7 | 414.8 | 11.3 | 0 | 0 | -2.0 | 6676 | - | - |
$3^{1}S_{0}$ | $0^{++}$ | 471.7 | 776.8 | -297.6 | 617.1 | -14.3 | 0 | 0 | -1.4 | 7010 | - | - |
$3^{3}S_{1}$ | $1^{+-}$ | 464.5 | 776.8 | -297.6 | 617.1 | -7.1 | 0 | 0 | -1.4 | 7017 | - | - |
$3^{5}S_{2}$ | $2^{++}$ | 450.1 | 776.8 | -297.6 | 617.1 | 7.1 | 0 | 0 | -1.4 | 7032 | - | - |
$4^{1}S_{0}$ | $0^{++}$ | 528.8 | 1063.7 | -244.7 | 790.7 | -11.0 | 0 | 0 | -1.0 | 7300 | - | - |
$4^{3}S_{1}$ | $1^{+-}$ | 523.2 | 1063.7 | -244.7 | 790.7 | -5.5 | 0 | 0 | -1.0 | 7306 | - | - |
$4^{5}S_{2}$ | $2^{++}$ | 512.2 | 1063.7 | -244.7 | 790.7 | 5.5 | 0 | 0 | -1.0 | 7317 | - | - |
$5^{1}S_{0}$ | $0^{++}$ | 587.9 | 1312.9 | -212.4 | 946.7 | -9.2 | 0 | 0 | -0.8 | 7551 | - | - |
$5^{3}S_{1}$ | $1^{+-}$ | 583.4 | 1313.0 | -212.4 | 946.7 | -4.6 | 0 | 0 | -0.7 | 7556 | - | - |
$5^{5}S_{2}$ | $2^{++}$ | 574.1 | 1313.0 | -212.4 | 946.7 | 4.6 | 0 | 0 | -0.8 | 7565 | - | - |
$1^{1}P_{1}$ | $1^{--}$ | 363.9 | 320.3 | -366.7 | 337.5 | -14.4 | 0 | 0 | -2.6 | 6553 | - | - |
$1^{3}P_{0}$ | $0^{-+}$ | 356.7 | 320.2 | -366.7 | 337.5 | -7.2 | -56.9 | -43.1 | -2.6 | 6460 | 6398.1 | $\eta_{c}(1S)\chi_{c0}(1P)$ |
$1^{3}P_{1}$ | $1^{-+}$ | 356.6 | 320.3 | -366.7 | 337.5 | -7.2 | -28.4 | 21.5 | -2.7 | 6554 | 6494.1 | $\eta_{c}(1S)\chi_{c1}(1P)$ |
$1^{3}P_{2}$ | $2^{-+}$ | 356.6 | 320.2 | -366.7 | 337.5 | -7.2 | 28.4 | -2.1 | -2.4 | 6587 | 6539.6 | $\eta_{c}(1S)\chi_{c2}(1P)$ |
$1^{5}P_{1}$ | $1^{--}$ | 342.4 | 320.4 | -366.7 | 337.5 | 7.2 | -85.3 | -30.2 | -2.7 | 6459 | 6508.8 | $\eta_{c}(1S)h_{c1}(1P)$ |
$1^{5}P_{2}$ | $2^{--}$ | 342.2 | 320.2 | -366.7 | 337.5 | 7.2 | -28.4 | 30.2 | -2.5 | 6577 | 6607.6 | $J/\psi(1S)\chi_{c1}(1P)$ |
$1^{5}P_{3}$ | $3^{--}$ | 342.3 | 320.3 | -366.7 | 337.5 | 7.2 | 56.9 | -8.6 | -2.5 | 6623 | 6653.1 | $J/\psi(1S)\chi_{c2}(1P)$ |
$2^{1}P_{1}$ | $1^{--}$ | 414.7 | 688.7 | -263.4 | 548.6 | -11.2 | 0 | 0 | -1.6 | 6925 | - | - |
$2^{3}P_{0}$ | $0^{-+}$ | 410.0 | 689.6 | -263.4 | 548.6 | -5.6 | -46.2 | -34.5 | -1.7 | 6851 | - | - |
$2^{3}P_{1}$ | $1^{-+}$ | 410.0 | 689.6 | -263.4 | 548.6 | -5.6 | -23.1 | 17.2 | -1.6 | 6926 | - | - |
$2^{3}P_{2}$ | $2^{-+}$ | 410.0 | 689.6 | -263.4 | 548.7 | -5.6 | 23.1 | -3.4 | -1.7 | 6951 | - | - |
$2^{5}P_{1}$ | $1^{--}$ | 398.7 | 689.5 | -263.4 | 548.6 | -5.6 | -69.3 | -24.2 | -1.7 | 6849 | - | - |
$2^{5}P_{2}$ | $2^{--}$ | 398.7 | 689.5 | -263.4 | 548.6 | 5.6 | -23.1 | 24.2 | -1.5 | 6944 | - | - |
$2^{5}P_{3}$ | $3^{--}$ | 398.8 | 689.7 | -263.4 | 548.6 | 5.6 | 46.2 | -6.9 | -1.6 | 6982 | - | - |
$3^{1}P_{1}$ | $1^{--}$ | 479.8 | 982.2 | -215.5 | 727.8 | -9.3 | 0 | 0 | -1.1 | 7221 | - | - |
$3^{3}P_{0}$ | $0^{-+}$ | 475.2 | 982.7 | -215.5 | 727.7 | -4.6 | -41.9 | -31.0 | -1.2 | 7153 | - | - |
$3^{3}P_{1}$ | $1^{-+}$ | 475.1 | 982.6 | -215.5 | 727.7 | -4.6 | -20.9 | 15.5 | -1.2 | 7220 | - | - |
$3^{3}P_{2}$ | $2^{-+}$ | 475.1 | 982.6 | -215.5 | 727.8 | -4.6 | 20.9 | -3.1 | -1.0 | 7243 | - | - |
$3^{5}P_{1}$ | $1^{--}$ | 465.9 | 982.8 | -215.5 | 727.7 | 4.6 | -62.8 | -21.7 | -1.2 | 7150 | - | - |
$3^{5}P_{2}$ | $2^{--}$ | 465.7 | 982.6 | -215.5 | 727.8 | -4.6 | -20.9 | 21.7 | -1.1 | 7236 | - | - |
$3^{5}P_{3}$ | $3^{--}$ | 465.8 | 982.6 | -215.5 | 727.8 | 4.6 | 41.9 | -6.2 | -1.1 | 7271 | - | - |
$4^{1}P_{1}$ | $1^{--}$ | 544.5 | 1237.2 | -186.8 | 887.6 | -8.1 | 0 | 0 | -0.9 | 7477 | - | - |
$4^{3}P_{0}$ | $0^{-+}$ | 540.4 | 1237.1 | -186.8 | 887.6 | -4.0 | -39.7 | -29.1 | -0.8 | 7412 | - | - |
$4^{3}P_{1}$ | $1^{-+}$ | 540.5 | 1237.2 | -186.8 | 887.6 | -4.0 | -19.8 | 14.5 | -0.8 | 7475 | - | - |
$4^{3}P_{2}$ | $2^{-+}$ | 540.5 | 1237.2 | -186.8 | 887.6 | -4.0 | 19.8 | -2.9 | -0.9 | 7498 | - | - |
$4^{5}P_{1}$ | $1^{--}$ | 532.2 | 1237.2 | -186.8 | 887.6 | 4.0 | -59.6 | -20.4 | -0.9 | 7409 | - | - |
$4^{5}P_{2}$ | $2^{--}$ | 532.2 | 1237.1 | -186.8 | 887.6 | 4.0 | -19.8 | 20.4 | -0.9 | 7489 | - | - |
$4^{5}P_{3}$ | $3^{--}$ | 532.2 | 1237.1 | -186.8 | 887.6 | 4.0 | 39.7 | -5.8 | -0.8 | 7523 | - | - |

Table 4 summarises the results of all S and P-wave tetraquark $T_{4c}$ masses
found in the current work $M_{f}$, as well as comparisons. The coulombic
contact between two spin-1 (anti)diquarks predominate over other interactions,
according to our findings. This confirms the supremacy of the one-gluon
exchange (OGE) mechanism, which is responsible for the strong diquark-
antidiquark binding. The strength of $\langle V^{(0)}_{C}\rangle$ in S-wave
states is substantially stronger than in other states, suppressing mass below
threshold.

In higher radial and orbital excited states, the attractive strength of the
spin-spin interaction reduces, although the strength of the spin-spin
interaction is greater than in charmonia. This is a surprising conclusion, as
the spin-spin interaction contains a $\frac{1}{m^{2}_{cc}}$ term, implying
that the strength of this interaction should decrease rather than increase in
the case of the tetraquark $T_{4c}$. In contrast to charmonia, where charm and
anticharm interact, the colour interaction in tetraquark causes diquark and
antidiquark to interact substantially. The $T_{4c}$ states have been
principally described by the following channels $J/\psi$-$J/\psi$,
$\eta_{c}$-$\eta_{c}$, $J/\psi$-$\chi$, or the admixture of S and P-wave
charmoium like states [68]. The mass-spectra produced in this study also
demonstrate consistency with the above-mentioned states.

Author predicted the mass of S - wave [cc][$\bar{c}\bar{c}$] to be roughly 6.5
GeV in his QCD sum rule study [23]. In Ref. [69], the author suggests that the
probable quantum numbers of the [cc][$\bar{c}\bar{c}$] state in the di-
$\eta_{c}(1S)$ channel are $0^{++}$ and $2^{++}$, whereas in the same study,
the author proposes the possibility of a state with $1^{+-}$, mass roughly 6.5
GeV in the J/$\psi(1S)$ $\eta_{c}(1S)$ channel.

The masses of S-wave states such as $1^{1}S_{0}$, $1^{3}S_{1}$ and
$1^{5}S_{2}$ are in good agreement with $\eta_{c}(1S)-\eta_{c}(1S)$,
$J/\psi(1S)-\eta_{c}(1S)$ and $J/\psi(1S)-J/\psi(1S)$, respectively. The
compactness of these S-wave states suggests that a compact tetraquark, rather
than a loosely bound molecule, could exist. The LHCb has discovered a state
with a mass of nearly 6.9 GeV in the di-J/$\psi(1S)$ invariant mass spectrum
in the 6.2-6.8 GeV region. The masses found for P-wave $T_{4c}$ states in our
model are as follows: $\eta_{c}(1S)-\chi_{c0}(1P)$ \- 6460 MeV,
$\eta_{c}(1S)-\chi_{c1}(1P)$ \- 6554 MeV, $\eta_{c}(1S)-h_{c}(1P)$ \- 6459
MeV, $\eta_{c}(1S)-\chi_{c2}(1P)$ \- 6587 MeV, $J/\psi(1S)-\chi_{c1}(1P)$ \-
6577 MeV, $J/\psi(1S)-\chi_{c2}(1P)$ \- 6623 MeV. The masses of
[cc][$\bar{c}\bar{c}$] P-wave states are in the range mentioned in [37] and
[69]. We have not yet determined the masses of additional higher orbital
excited states because all stable tetraquark states have only been discovered
in S and P-wave modes. The internal structure of the P-wave
[cc][$\bar{c}\bar{c}$] state is difficult to comprehend due to its vast
spectrum; however, the decay properties and other experimental research will
eventually reveal its exact structure with its quantum numbers.

Table 5: Comparison of ground state all charm tetraquark masses (MeV)

$J^{PC}$ | Ours | [25] | [26] | [31] | [70] | [71] | [72] | [73] | [74]
---|---|---|---|---|---|---|---|---|---
$0^{++}$ | 5939 | 5960 | 5969 | 6695 | 5966 | 5990 | 6383 | 6518 | 6529
$1^{+-}$ | 5986 | 6009 | 6021 | 6528 | 6051 | 6050 | 6437 | 6500 | 6508
$2^{++}$ | 6079 | 6100 | 6115 | 6573 | 6223 | 6090 | 6437 | 6524 | 6565

##  4 Conclusion

We evaluated a model for all charm tetraquarks $T_{4c}$, which is thought to
be compact and consists of diquark [cc]-antidiquark [$\bar{c}\bar{c}$] pairs
in an antitriplet-triplet colour configuration. Using the Cornell inspired
potential and the relativistic correction, we determined the mass spectra of
tetraquarks. To observe the splitting between states with differing quantum
numbers, spin-dependent terms (spin-spin, spin-orbital, and tensor) have been
included. We started by formulating the model for charmonia [c$\bar{c}$] and
obtaining the most recent data for mesons, which we utilised to fit the
model’s parameters. We next calculated the masses of vector diquarks, which
are the tetraquark’s composite states. The values obtained from our model for
tetraquark masses are in good agreement with those obtained from other
publications listed in Table-5. The LHCb Collaboration[34] recently discovered
an all-charm tetraquark mass of roughly 6.9 GeV in the J/$\psi$ invariant mass
spectrum, suggesting that additional resonance structures in the 6.2-6.8 GeV
region are also feasible.

In Ref.[75], J.-Z Waang et al. suggests the possible channels of fully-heavy
charm tetraquark structures in the invariant mass spectrum of
J/$\psi(1S)-\psi$(3686), J/$\psi(1S)$\- $\psi$(3770),
$\psi$(3686)-$\psi$(3686). The masses of 1S and 2S states in this study are in
the range of 5.9 $\sim$ 6.6 GeV, implying that alternative structures for all-
charm tetraquarks could be found in the channels J/$\psi(1S)$-J/$\psi(1S)$,
$\eta_{c}(1S)$-$\eta_{c}(1S)$ and J/$\psi(1S)$-$\eta_{c}(1S)$. Similarly, the
masses of 1P and 2P states are in the range of 6.5 to 6.9 GeV, and this mass
range supports the good agreement between our results and experimentally
observed states. This mass range could be due to admixture of S-wave and
P-wave charmonium states like $\eta_{c}(1S)-\chi_{c0}(1P)$,
$\eta_{c}(1S)-\chi_{c1}(1P)$, $\eta_{c}(1S)-\chi_{c2}(1P)$,
$\eta_{c}(1S)-h_{c}(1P)$, $J/\psi(1S)-\chi_{c1}(1P)$,
$J/\psi(1S)-\chi_{c2}(1P)$. The X(6900) can be interpreted as a radially
excited state with quark content [cc][$\bar{c}\bar{c}$] in Ref.[76]. and an
orbitally excited 2P state with spin-parity $0^{++}$(3S) or $2^{++}$(3S). As
described in the previous section, discrete combinations of parity (P) and
charge conjugation (C) are suggested by the quantum numbers ($J^{PC}$)
predicted by various preceding publications and mentioned therein. We conclude
from the foregoing talks that X(6900), which has a mass range of 6.2-6.9 GeV,
may have the quantum numbers $0^{-+}$, $1^{-+}$, $2^{-+}$ and may belong to
the P-wave tetraquark state.

If a tetraquark state’s energy is lower than all possible two-meson
thresholds, it should be stable against the strong interaction. If the energy
of the tetraquark is above its threshold, strong interactions can break it
into two mesons. Strong decay into two mesons is forbidden if it falls below
the threshold, hence decay must take place via weak or electromagnetic
interactions. Our findings, along with those obtained in recent literature on
the $T_{4c}$ tetraquark, should motivate a comprehensive experimental search
for these states at the LHCb [38] and Belle II [41]. Our findings point to a
broad spectrum of $T_{4c}$ states in the charm sector, which will require
further investigation at different experimental facilities . Many other
tetraquark possibilities with light quarks (u,d,s) configurations, including
charm, will be scanned at experimental facilities like PANDA at FAIR in the
near future [77, 78, 79, 80, 81, 82, 83].

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\frac{\partial f_L(\vx; \vtheta)}{\partial \theta_l } &=  \mW_L\left(\prod_{k = 1}^{L-l + 1} \mSigma_{L-k}(\vx) \mW_{L-k} \right) \frac{\partial f_{l}(\vx; \vtheta)}{ \partial \mW_l }\\
&= \mW_L\left(\prod_{k = 1}^{L-l + 1} \mSigma_{L-k}(\vx) \mW_{L-k} \right) \left(\mSigma_l(\vx) \otimes f_{l-1}(\vx; \vtheta)\right)
\end{align*}

where the first equality arises from iterating (<ref>) and the second by applying (<ref>). Proceeding,

\begin{align*}
&\left \langle  \frac{\partial f_L(\vx_i)}{\partial \theta_l }, \frac{\partial f_L(\vx_j)}{\partial \theta_l } \right \rangle \\
&= \left( f_{l-1}(\vx_i)^T f_{l-1}(\vx_j) \right) \left( \left(\mSigma_l(\vx_i) \prod_{k = l+1}^{L-1} \mW_{k}^T \mSigma_{k}(\vx_i)  \right) \mW_L^T \right)^T \left(\left( \mSigma_l(\vx_j) \prod_{k = l+1}^{L-1} \mW_{k}^T \mSigma_{k}(\vx_j) \right) \mW_L^T \right) \\
&= \left( f_{l-1}(\vx_i)^T f_{l-1}(\vx_j) \right) \left( [\mB_{l}]_{i,:}^T [\mB_{l}]_{j,:} \right)
\end{align*}

as claimed.

§.§ Proof of Lemma <ref>

Let $\vz \in \R^d$ be a fixed vector and $\mW \in \R^{m \times d}$ a random matrix with mutually iid elements $[\mW]_{ij} \sim \mathcal{N}(0,1)$ for all $i \in [m]$ and $j \in [d]$. Consider the random vector $\vy \in \R^{m}$ defined as $\vy = \sigma(\mW \vz)$ where $\sigma$ denotes the ReLU function applied elementwise. For $\delta \in (0,1)$ if $m \gtrsim \delta^{-2}\log(1/ \epsilon)$ then

\[
\P\left((1 - \delta)\frac{m}{2}\| \vz \|^2 \leq \| \vy \|^2 \leq (1 + \delta)\frac{m}{2}\| \vz \|^2 \right) \geq 1 - \epsilon.
\]

For $i\in [m]$ define $Z_i = \tfrac{\vw_i^T\vz}{\| \vz \|}$, then $Z_i \sim \mathcal{N}(0,1)$ are mutually iid. Let $B_i = \mathbbm{1}(Z_i>0)$, note by symmetry $B_i \sim \text{Ber}(1/2)$, furthermore these random variables for $i \in [n]$ are also mutually iid with respect to one another. As $y_i = \| \vz \|B_i Z_i$ then

\begin{align*}
\| \vy \|_2^2
&= \| \vz \|^2 \sum_{i=1}^m B_i Z_i^2.
\end{align*}

For convenience let $\vy' = \vy / \| \vz\| $and define $\mathcal{S} = \{ i \in [n] \; : \; B_i = 1 \}$, then

\[
\|\vy' \|^2 = \sum_{i\in \mathcal{S}} Z_i^2 \sim \chi^2(|\mathcal{S}|).
\]

From <cit.> we have for any $t>0$

\begin{align*}
\P \left( | \left(\| \vy'\|^2 - | \mathcal{S}| \right) | \geq 2 \sqrt{ |\mathcal{S}| t}   \right) \leq 2\exp(-t).
\end{align*}

For $\delta_1 \in (0,1)$ let $t=\tfrac{|\mathcal{S}| \delta_1^2}{4}$, then

\[
\P \left( (1 - \delta_1)|\mathcal{S} | \leq \| \vy' \|^2 \leq (1 + \delta_1)|\mathcal{S} | \right) \geq 1 - 2 \exp \left( -\frac{|\mathcal{S}| \delta_1^2}{4}  \right).
\]

Observe $|\mathcal{S}| = \sum_{i=1}^{m} B_i \sim \text{Bin}(m, 1/2)$. With $\delta_2 \in (0,1)$ then applying Hoeffding's inequality we have

\begin{align*}
\P\left( (1 - \delta_2) \frac{m}{2} \leq \sum_{i=1}^{m} B_i \leq (1 + \delta_2) \frac{m}{2}  \right) &\geq 1 - 2 \exp \left( -\frac{\delta_2^2 m}{2}\right).
\end{align*}

Let $\omega$ denote the event that $(1 - \delta_2)\tfrac{m}{2} \leq |\mathcal{S} | \leq (1 + \delta_2)\tfrac{m}{2}$. If $m \geq \frac{16}{\delta_1^2\delta_2^2 (1 - \delta_2 )} \log(4 / \epsilon)$ then

\begin{align*}
&\P\left((1 - \delta_1)(1- \delta_2)\frac{m}{2} \leq \| \vy' \|^2 \leq (1 + \delta_1)(1 + \delta_2)\frac{m}{2} \right)\\ &\geq \P\left((1 - \delta_1)(1- \delta_2)\frac{m}{2} \leq \| \vy' \|^2 \leq (1 + \delta_1)(1 + \delta_2)\frac{m}{2}\; \mid \; \omega\right) \P(\omega)\\
&\geq  \P\left( (1 - \delta_1)|\mathcal{S} | \leq \| \vy'\|^2 \leq (1 + \delta_1)|\mathcal{S} | \; \mid \; \omega \;\right)\P(\omega)\\
& \geq \left( 1 - 2 \exp \left( -\frac{(1 - \delta_2) \delta_1^2m}{8}  \right) \right) \left(1 - 2 \exp \left( -\frac{\delta_2^2 m}{2}\right) \right)\\
& \geq \left( 1 - \frac{\epsilon}{2}\right)\left( 1 - \frac{\epsilon}{2}\right)\\
& \geq 1 -\epsilon.
\end{align*}

For some $\delta \in (0,1)$ let $\delta_2 = \delta_1 = \delta/3$, then if $m \geq 1944\delta^{-2}\log(4 / \epsilon)$ we have

\[
\P\left((1 - \delta)\frac{m}{2} \leq \| \vy' \|^2 \leq (1 + \delta)\frac{m}{2} \right) \geq 1 - \epsilon
\]

from which the result claimed follows.

For $k\in [l]$ let $\omega_k$ denote the event that the inequality

\[
\left(1 - \frac{1}{l} \right)^k \left( \prod_{h=1}^k \frac{d_h}{2}\right) \leq \| f_k(\vx) \|^2 \leq \left(1 + \frac{1}{l} \right)^k\left( \prod_{h=1}^k \frac{d_h}{2}\right)
\]

holds. We proceed by induction to establish that $\P(\omega_k) \geq ( 1 - \tfrac{\epsilon}{l})^k$ for all $k \in [l]$. For the base case note that $f_1(\vx) = \sigma(\mW_1 \vx)$ and $\| \vx \|^2 = 1$. Applying Lemma <ref> with $\delta = \frac{1}{l}$, if $d_1 \gtrsim l^2 \log(l/\epsilon)$ then $\P(\omega_1) \geq 1 - \tfrac{\epsilon}{l}$. Now suppose for $k \in [l-1]$ that $\P(\omega_k) \geq ( 1 - \tfrac{\epsilon}{l})^k$. Note

\begin{align*}
\P(\omega_{k+1}) \geq \P(\omega_{k+1} \mid \omega_k) \P(\omega_k) \geq \P(\omega_{k+1} \mid \omega_k) ( 1 - \tfrac{\epsilon}{l})^k
\end{align*}

Recall $f_{k+1}(\vx) = \sigma(\mW_1 f_{k}(\vx))$. Conditioned on $\omega_k$, then again applying Lemma <ref> with $\delta = \frac{1}{l}$ and as $d_{k+1} \gtrsim l^2 \log(l/\epsilon)$ we have

\[
\P(\omega_{k+1}\mid \omega_k) \geq 1 - \tfrac{\epsilon}{l}
\]

which completes the proof of the induction hypothesis. As $(1 - \epsilon/l)^l \geq 1 - \epsilon$ and $ e^{-1} \leq (1 - 1/l)^l \leq (1 + 1/l)^l \leq e$ then

\[
e^{-1} \left( \prod_{h=1}^l \frac{d_h}{2}\right) \leq \| f_l(\vx) \|^2 \leq e \left( \prod_{h=1}^l \frac{d_h}{2}\right)
\]

holds with probability at least $1-\epsilon$.

§.§ Proof of Lemma <ref>

Let $\vx \in \S^{d_0-1}$, $L \geq 2$ and assume $d_k \gtrsim L^2 \log \left(\frac{L}{\epsilon}\right)$ for all $k \in [L - 1]$. For any $l \in [L - 1]$ with probability at least $1 - \epsilon$ over the network parameters the following holds,

\[
\| \mS_{l}(\vx) \|_F^2 \asymp 2^{-L+l+1} \prod_{k = l}^{L-1} d_k.
\]

In what follows for convenience we define an empty product of scalars or matrices as the scalar one.
Let $K \in \{L - 1\}$, $l \in [K]$, and for some arbitrary $\vx \in \S^{d_0-1}$ define

\begin{align*}
\mS_{l,K} &= \mSigma_l(\vx) \prod_{k = l+1}^{K} \mW_k^T \bm{\Sigma}_k(\vx).
\end{align*}

Let $\omega_{l,K}$ denote the event

\begin{equation}\label{eq:frob-norm-K}
\frac{1}{2} \left(1 - \frac{1}{L}\right)^K \leq \|\mS_{l, K}\|_F^2\prod_{k = l}^K \frac{2}{d_l} \leq 2\left(1 + \frac{1}{L}\right)^K
\end{equation}

It suffices to lower bound the probability of the event $\omega_{l, L-1}$. Let $\mc{F}_{K}$ denote the $\sigma$-algebra generated by $\mW_1, \cdots, \mW_{K}$ and note that $\mS_{l,K} \in \mc{F}_{K}$. Let $\gamma_l$ denote the event that $f_l(\vx) \neq 0$, then

\begin{align*}
\P(\omega_{l, L-1}) &\geq \P(\omega_{l, L-1} \mid \omega_{l, L-2} ) \P(\omega_{l, L-2})\\
&\geq \P(\omega_{l, L-1} \mid \omega_{l, L-2} ) \P(\omega_{l, L-2} \mid \omega_{l, L-3})\P(\omega_{l, L-3})\\
& \geq \left(\prod_{h = l}^{L-2} \P(\omega_{l, h+1} \mid \omega_{l, h}) \right) \P(\omega_{l,l} \mid \gamma_l) \P(\gamma_l).
\end{align*}

Fixing $\epsilon \in (0,1)$, our goal is to show each term in this product is at least $(1 - \tfrac{\epsilon}{L})$: indeed, if this is true then

\begin{align*}
\P(\omega_{l, L-1}) \geq \left(1 - \frac{\epsilon}{L} \right)^{L-l} \geq 1 - \epsilon
\end{align*}

and our task is complete. To this end, first observe that as $d_k \gtrsim L^2 \log(L / \epsilon)$ for all $k \in [L-1]$, then $\P(\gamma_l) \geq 1 - \tfrac{\epsilon}{L}$ by Lemma <ref>. Proceeding to the term $\P(\omega_{l,l} \mid \gamma_l)$, recall $[\bm{\Sigma}_l(\vx)]_{jj}=\mathbbm{1}([\mW_l f_{l-1}(\vx)]_j>0)$. By symmetry the diagonal entries of $\bm{\Sigma}_l(\vx)$ are mutually iid Bernoulli random variables with parameter $\frac{1}{2}$. Therefore, using Hoeffding's inequality for all $t \geq 0$

\begin{align*}
\P\left(\left|\|\bm{\Sigma}_l(\vx)\|_F^2 - \frac{d_l}{2}\right| \geq t \; \middle|\; \gamma_l \right) \leq 2 \exp\left(-\frac{t^2}{d_l}\right).
\end{align*}

Let $t = d_l$, if $d_l \geq \log \frac{2L}{\epsilon}$ then with $K\geq 1$, $L\geq 2$ it follows that

\begin{align*}
\P(\omega_{l,l} \mid \gamma_l)
&= \P\left(\frac{1}{2}\left(1 - \frac{1}{L}\right)^K \leq \|\bm{\Sigma}_l(\vx)\|_F^2 \frac{2}{d_l} \leq 2\left(1 + \frac{1}{L}\right)^K \; \middle| \; \gamma_l \right)\\
&\geq \P\left(\frac{1}{2} \leq \|\bm{\Sigma}_l(\vx)\|_F^2 \frac{2}{d_l} \leq \frac{3}{2} \; \middle| \; \gamma_l \right)\\
&\geq 1 - \P\left(\left|\|\bm{\Sigma}_l(\vx)\|_F^2 - \frac{d_l}{2}\right| \geq \frac{d_l}{4} \; \middle|\; \gamma_l \right)\\
&\geq 1 - \frac{\epsilon}{L}.
\end{align*}

We now proceed to analyze $\P(\omega_{l, h+1} \mid \omega_{l, h})$ for $h \in [l, K-1]$. Note if $\omega_{l, h}$ is true then $\|\mS_{l, h} \|_F^2 > 0$. By definition this implies $\| \bm{\Sigma}_l(\vx) \|_F^2 > 0$, however, if $f_h(\vx) = 0$ then  $\| \bm{\Sigma}_l(\vx) \|_F^2 = 0$. Therefore $\omega_{l, h}$ being true implies $f_h(\vx) \neq 0$. For convenience in what follows we denote the $j$th column of $\mW_{h+1}$ as $\vw_j$. By definition

\[
\mS_{l, h+1} = \mS_{l, h}\mW_{h+1}^T \bm{\Sigma}_{h+1}(\vx),
\]

\begin{align*}
\E[\|\mS_{l, h+1}\|_F^2 \mid \mc{F}_{h} ] &= \E[\|\mS_{l, h} \mW_{h+1}^T \bm{\Sigma}_{h+1}(\vx)\|_F^2  \mid \mc{F}_{h} ]\\
&= \E\left[\sum_{j = 1}^{d_{h+1}} \left\|\mS_{l, h}\vw_j\right\|^2 \dot{\sigma}(\langle \vw_j, f_{h}(\vx) \rangle) \; \middle| \; \mc{F}_{h}\right].
\end{align*}

As highlighted already, if we condition on $\omega_{l, h}$ then $f_h(\vx)\neq 0$ and therefore the random variables $(\dot{\sigma}(\langle \vw_j, f_{h}(\vx) \rangle))_{j \in d_{h+1}}$ are mutually iid Bernoulli random variables with parameter $\frac{1}{2}$. Again by symmetry $\dot{\sigma}(\langle \vw_j, f_{h}(\vx) \rangle)$ is independent of $\| \mS_{l, h}\vw_j \|^2 $. Therefore conditioned on $\omega_{l,h}$

\begin{align*}
\sum_{j = 1}^{d_{h+1}}\E[\|\mS_{l, h}\vw_j\|^2 \mid \mc{F}_{d_{h+1}} ]\E[\dot{\sigma}(\langle \vw_j, f_{h}(\vx) \rangle) \mid \mc{F}_{h} ]
&= \frac{1}{2}\sum_{j = 1}^{d_{h+1}}\E[\|\mS_{l, h}\vw_j\|^2 \mid \mc{F}_{h} ]\\
&= \frac{1}{2}\sum_{j = 1}^{d_{h+1}}\|\mS_{l, h}\|_F^2\\
&= \frac{d_{h+1}}{2}\|\mS_{l, h}\|_F^2.
\end{align*}

Moreover, under the same conditioning

\begin{align*}
\left\|\left\|\mS_{l, h}\vw_j\right\|^2 \dot{\sigma}(\langle \vw_j, f_{h}(\vx) \rangle)  \right\|_{\psi_1} &\leq \| \|\mS_{l, h}\vw_j\|^2 \|_{\psi_1}\\
&= \| \|\mS_{l, h}\vw_j\| \|_{\psi_2}^2\\
&\lesssim \|\mS_{l, h}\|_F^2
\end{align*}

where the last line follows from Theorem 6.3.2 of [Vershynin, 2018]. As a result, conditioned on $\omega_{l,h}$ then using Bernstein's inequality <cit.> there exists an absolute constant $c$ such that for all $t \geq 0$

\begin{align*}
\P\left(\left|\|\mS_{l, h+1}\|_F^2 -\frac{d_{h+1}}{2}\|\mS_{l, h}\|_F^2\right| \geq  t \; \middle| \; \mc{F}_{h} \right) &\leq 2\exp\left(-c \min\left(\frac{t^2}{d_{h+1}\|\mS_{l, h}\|_F^4 }, \frac{t}{\|\mS_{l, h}\|_F^2} \right) \right).
\end{align*}

If $d_{h+1} \geq \frac{4L^2}{c} \log \frac{2L}{\epsilon}$ and $t = \frac{d_{h+1}\|\mS_{l, h}\|_F^2 }{2L }$ then conditioning on $\omega_{l,h}$ we obtain

\begin{align*}
\P\left(\left|\|\mS_{l, h+1}\|_F^2 -\frac{d_K}{2}\|\mS_{l, h}\|_F^2\right| \geq  \frac{d_{h+1} }{2L}\|\mS_{l, h}\|_F^2 \; \middle| \; \mc{F}_{h} \right) &\leq \frac{\epsilon}{L}.
\end{align*}

As a result, for any $h \in [l, K-1]$ we have $\P(\omega_{l, h+1} \mid \omega_{l,h }) \geq 1 - \tfrac{\epsilon}{L}$ from which the result claimed follows.

§.§ Proof of Lemma <ref>

Let $\vx \in \S^{d_0-1}$, $L \geq 3$ and assume $d_k \geq d_{k + 1}$ and $d_k \gtrsim  \sqrt{\log \frac{1}{\epsilon} }$ for all $k \in [L - 1]$. For any $l \in [L-1]$ with probability at least $1 - \epsilon$ over the network parameters the following holds,

\[
\left\| \mS_l(\vx) \right\|^2 \lesssim \prod_{k = l}^{L - 2}d_{k}.
\]

By Theorem 4.4.5 of [Vershynin, 2018], for any $k \in [L-1]$ and all $t \geq 0$

\[
\P\left(\|\mW_k\| \leq C(\sqrt{d_{k - 1}} + \sqrt{d_{k}} + t)  \right) \geq 1 - 2 e^{-t^2}.
\]

As $d_{k-1} \geq d_{k} \geq \sqrt{\log \frac{2L}{\epsilon} }$, then setting $t = \sqrt{\log \frac{2}{\epsilon} } $ yields

\begin{align*}
\P\left(\|\mW_k\| \leq 3C_1 \sqrt{d_{k - 1}} \right) &\geq \P\left(\|\mW_k\| \leq C(\sqrt{d_{k - 1}} + \sqrt{d_k} + t)\right)\\
&\geq 1 - \frac{\epsilon}{L}.
\end{align*}

Using a union bound it follows that

\[
\P\left(\|\mW_k\| \leq 3C_1 \max \{ \sqrt{d_{l -1 }},  \sqrt{d_{l }}\} \;\; \forall k \in [ L - 1] \right) \geq 1 - \epsilon.
\]

Note that $\| \mSigma_k(\vx) \| \leq 1 $ for all $ k \in [L-1]$, therefore conditional on the above event we have

\begin{align*}
\| \mS_{l}(\vx)\| &= \left\|\mSigma_{l}(\vx) \left(\prod_{k = l+1}^{L - 1}\mW_k^T \mSigma_k(\vx)\right) \right\| \\
&\leq \|\mSigma_{l}(\vx)\|\left(\prod_{k = l+1}^{L - 1}\|\mW_k\| \|\mSigma_k(\vx)\|\right)\\
&\leq \prod_{k = l+1}^{L - 1}\|\mW_k\|\\
&\lesssim \prod_{k = l}^{L - 2}\sqrt{d_{k}}.
\end{align*}

To conclude we square both sides.

§.§ Proof of Lemma <ref>

Let $\vx \in \S^{d_0-1}$ be arbitrary and recall $\mS_{l}(\vx) = \mSigma_l(\vx) \left( \prod_{k = l+1}^{L-1} \mW_k^T \mSigma_k(\vx) \right)$. Also recall that $\mW_L^T \in \R^{d_{L-1}}$ is distributed as $\mW_L^T \sim \mathcal{N}(\textbf{0}_{d_{L-1}}, \textit{I}_{d_{L_1}})$. Therefore by <cit.> for any $\mA \in \R^{d_2 \times d_{L-1}}$ and $t \geq 0$

\begin{align*}
\P( | \|\mA \mW_L^T \|_2 - \|\mA \|_F | \geq t) \leq 2 \exp \left( -\frac{Ct^2}{ \|\mA \|_2^2}\right)
\end{align*}

for some constant $C>0$. As a result, with $t = \tfrac{1}{2}\| \mA \|_F^2$ then for some constant $C>0$

\[
\P \left( \frac{1}{4} \| \mA \|_F^2 \leq \| \mA \mW_L^T \|_2^2 \leq \frac{3}{4} \| \mA \|_F^2 \right) \geq 1 - \exp\left( -C\frac{\| \mA \|_F^2}{\| \mA \|_2^2}\right).
\]

Therefore, in order to lower bound $\| \mS_{l}(\vx) \mW_L^T\|_2^2$ with high probability it suffices to condition on a suitable upper bound for $\| \mS_{L-1}(\vx) \|_2^2$ and a suitable lower bound for $\| \mS_{L-1}(\vx) \|_F^2$. Let $\omega$ denote the event that both

\[
\| \mS_l \|_F^2 \asymp 2^{L-l-1}\prod_{k=l}^{L-1} d_k
\]

\[
\| \mS_l(\vx) \|^2 \lesssim \prod_{k=l}^{L-2} d_{k}
\]

are true. Combining Lemmas <ref> and <ref> using a union bound, then as long as $L \geq 3$, $d_k \geq d_{k+1}$ and $d_k \gtrsim L^2\log \frac{nL}{\epsilon}$ for all $k \in [L - 1]$ then $\P(\omega) \geq 1 - \tfrac{\epsilon}{2}$. As a result and aslo as $d_{L-1} \gtrsim 2^L \log(2 / \epsilon)$ then

\begin{align*}
\P \left(  \| \mS_l(\vx)  \mW_L^T \|_2^2 \asymp 2^{L-l-1}\prod_{k=l}^{L-1} d_k  \right) & \geq \P \left(  \| \mS_l(\vx)  \mW_L^T \|_2^2 \asymp 2^{L-l-1}\prod_{k=l}^{L-1} d_k \; \mid \; \omega  \right) \P(\omega)\\
& \geq \P \left( \frac{1}{4} \| \mS_l(\vx)  \|_F^2 \leq \| \mS_l(\vx)  \mW_L^T \|_2^2 \leq \frac{3}{4} \| \mS_l(\vx) \|_F^2  \; \mid \; \omega \right)  \P(\omega) \\
&\geq 1 - \exp\left( -C 2^{-L} \frac{ \prod_{k=l}^{L-1} d_k}{\prod_{k=l}^{L-2} d_{k}} \right)\P(\omega)\\
& \geq 1 - \exp\left( -C 2^{-L} d_{L-1} \right)\P(\omega)\\
& \geq \left(1 - \frac{\epsilon}{2} \right)\left(1 - \frac{\epsilon}{2} \right)\\
& \geq 1 - \epsilon
\end{align*}

as claimed.

§.§ Proof of Theorem <ref>

Recall (<ref>),

\[
2^{L-1}\left( \prod_{l = 1}^{L - 1}\frac{1}{d_l}\right) \lambda_{\min}\left( \mF_{1} \mF_{1}^T \right) \min_{i \in [n]} \|[\mB_2]_{i,:} \|^2 \leq \lambda_{\min}(\mK) \leq  2^{L-1}\left( \prod_{l = 1}^{L - 1}\frac{1}{d_l}\right) \sum_{l=0}^{L-1} \| f_l(\vx_i) \|^2 \| [\mB_{l+1}]_{i,:} \|^2,
\]

where the upper bound holds for any $i \in [n]$. We start by analyzing the lower bound. Observe that $ \mF_1\mF_1^T= \sigma(\mW_1 \mX)^T \sigma(\mW_1 \mX)$ has the same distribution as $d_1 \mK_2$ in the shallow setting; see (<ref>). Let $\lambda_2$ be defined as in Lemma <ref>:

\[
\lambda_2 = d_0 \lambda_{\min} \left( \E_{\vu \sim U(\S^{d_0 -1})}\left[ \sigma(\vu^T\mX)^T\sigma(\vu^T\mX) \right]\right) = \lambda_{min}\left( \mK_{\sqrt{d_0} \sigma}^{\infty} \right).
\]

As the dataset $\vx_1, \vx_2, \cdots, \vx_n \in \S^{d_0 -1}$ is $\delta$-separated then by Lemma <ref>

\[
\lambda_2 \gtrsim  \left(1+ \frac{d_0\log(1/\delta) }{\log d_0}\right)^{-3}\delta^4.
\]

Furthermore, if $d_1 \gtrsim \tfrac{n}{\lambda_2} \log \left( \tfrac{n}{\lambda_2}\right) \log \left( \tfrac{n}{\epsilon}\right)$ then by Lemma <ref>

\[
\lambda_{\min}(\mF_1\mF_1^T) \gtrsim d_1 \lambda_2
\]

with probability at least least $1-\tfrac{\epsilon}{4}$ and as a result

\[
\lambda_{\min}(\mF_1\mF_1^T) \gtrsim d_1 \left(1 + \frac{\log(n / \epsilon)}{\log(d_0)} \right)^{-3} \delta^4
\]

with probability at least $1- \tfrac{\epsilon}{4}$. Furthermore, as $L \geq 3$, $d_l \geq d_{l+1}$ for all $l \in [L - 1]$ and $d_{L-1} \gtrsim 2^L \log \left (\frac{4nL}{\epsilon}\right)$ then

\[
\min_{i \in [n]} \|[\mB_2]_{i,:} \|^2 \gtrsim 2^{-L} \prod_{k = 2}^{L-1} d_k
\]

with probability at least $1 - \tfrac{\epsilon}{4}$. Via a union bound we conclude that the condition

\begin{align*}
2^{L-1}\left( \prod_{l = 1}^{L - 1}\frac{1}{d_l}\right) \lambda_{\min}(\mF_1\mF_1^T)  \min_{i \in [n]} \|[\mB_2]_{i,:} \|^2 & \gtrsim \left(1 + \frac{\log(n / \epsilon)}{\log(d_0)} \right)^{-3} \delta^4
\end{align*}

holds with probability at least $1 - \tfrac{\epsilon}{2}$. Fixing some $i\in [n]$, for the upper bound observe trivially by construction that

\begin{align*}
\| f_0(\vx_i) \|^2 \| [\mB_1]_{i,:} \|^2 = 1.
\end{align*}

By assumption $d_k \gtrsim L^2 \log(4L^2/\epsilon)$ for all $k \in [L-1]$. With $l \in [0, L-1]$ then by Lemma <ref>

\[
\| f_l(\vx_i) \|^2 \lesssim 2^{-l}\prod_{k=1}^{l} d_k
\]

holds with probability at least $1- \tfrac{\epsilon}{4L}$. Likewise by Lemma <ref> for $l \in [2, L]$,

\[
\| [\mB_{l}]_{i,:} \|^2 = \| \mS_l(\vx_i)\mW_L^T \| \lesssim 2^{-L+l+1} \prod_{k=l}^{L-1} d_k
\]

with probability at least $1- \tfrac{\epsilon}{4L}$. Combining these via a union bound then for any $l \in [0, L-1]$,

\[
\| f_l(\vx_i) \|^2 \| [\mB_{l}]_{i,:} \|^2 \lesssim 2^{-L+1} \prod_{k=1}^{L-1} d_k
\]

holds with probability at least $1- \tfrac{\epsilon}{2L}$. Again using a union bound now over the layers, it follows that

\begin{align}
2^{L-1}\left( \prod_{l = 1}^{L - 1}\frac{1}{d_l}\right) \sum_{l=0}^{L-1} \| f_l(\vx_i) \|^2 \| [\mB_{l+1}]_{i,:} \|^2 \lesssim L 2^{L-1}\left( \prod_{l = 1}^{L - 1}\frac{1}{d_l}\right)  2^{-L+1} \left( \prod_{l=1}^{L-1} d_l \right) = L
\end{align}

with probability at least $1 - \tfrac{\epsilon}{2}$. As a result, using a final union bound we conclude both the upper and lower bounds hold with probability at least $1-\epsilon$.