# Properties of Low-Lying Charmonium States in a Phenomenological Approach

Praveen P D’Souza 1,2 Antony Prakash Monteiro 1<EMAIL_ADDRESS>1P.
G. Department of Physics, St Philomena college Darbe, Puttur 574 202, India
K. B. Vijaya Kumar 2 2Department of Physics, Mangalore University,
Mangalagangothri P.O., Mangalore - 574199, INDIA

###### Abstract

We investigate the spectrum and decay rates of low lying charmonium states
within the framework of the non relativistic quark model by employing a
Coulomb like potential from the perturbative one gluon exchange and the linear
confining potential along with the potential derived from instanton vacuum to
account for the hyperfine mass splitting of charmonium states in variational
approach. We predict radiative E1, M1, two-photon, leptonic and two-gluon
decay rates of low lying charmonium states. An overall agreement is obtained
with the experimental masses and decay widths.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED††preprint: APS/123-QED

##  I INTRODUCTION

Charmonia are bound states of a charm and an anticharm quark $(c\bar{c})$, and
represent an important testing ground for the properties of the strong
interaction. There has been a great progress in the observation of the
charmonium states from the past few years. The discovery of the first
charmonium state $J/\psi$Aubert  _et al._ (1974); Augustin  _et al._ (1974)
has revolutionized the field of hadron spectroscopy. This led to a clear
understanding of the prevailing theory of particle physics. Several quarkonium
states have been observed after the discovery of the charmonium state $J/\psi$
at BNL and SLAC(Olive  _et al._ , 2014). The first observation of a singlet
ground state of charmonium $\eta_{c}$ was done by Mark II and Crystal Ball
experiments in the radiative decays of $J/\psi$ and $\psi^{\prime}$(Olive  _et
al._ , 2014). The discoveries of conventional states $h_{c}(1P)$, $h_{c}(2P)$,
$\chi_{c}(1P)$, $\chi_{c}(2P)$, $\eta_{c}(1S)$ and the observation of the
exotic states like X(3872), X(3915), Y(4260), Z(3930) at Belle, BaBar, LHC,
BESIII,CLEO, etc have created a renewed interest in quarkonium physicsOlive
_et al._ (2014). These new observations have given a deeper understanding of
the charmonium physics and have unraveled many mysteriesDeng  _et al._ (2017).
Charmonium system is a powerful tool for the study of forces between quarks in
QCD in non-perturbative regime. Studies of charmonia production can improve
our understanding of heavy quark production and the formation of bound states.

The exploration and understanding of the substructure of hadrons, presented in
terms of quarks and gluons by quantum chromodynamics(QCD), has led to a
considerable progress in the study of charmonium states. Though there have
been numerous calculations of charmonium spectra from first principles such as
Lattice QCDDavies  _et al._ (1995); Dudek and Rrapaj (2008) and NRQCD which
provide rigorous theoretical implications for the experimental
observations,quark model calculations provide more intuitive insights and
supply important phenomenological guidance towards their study. The quark
antiquark potential cannot be obtained from the first principles of QCD.
Therefore, one has to use potential models to explain the observed hadronic
properties. The QCD inspired potential models have been playing an important
role in investigating heavy quarkonium. Most of the quark potential models
Eichten  _et al._ (1976, 1978, 1980); Godfrey and Isgur (1985); Stanley and
Robson (1980); Ebert  _et al._ (2003); Gershtein  _et al._ (1995a); Fulcher
(1991, 1999, 1994); Gupta  _et al._ (1994); Gupta and Johnson (1996) have
common ingredients in the non relativistic quark models(NRQM). Recently it is
shown that the NRQM with instanton induced interaction(III) explains
${}^{1}S_{0}$ and ${}^{3}S_{1}$ the nucleon- nucleon potentialVanamali and
Kumar (2016).

The NRQM formalism provides a systematic treatment of the perturbative and non
perturbative components of QCD at hadronic scale Bhaduri  _et al._ (1981);
Burakovsky and Goldman (1997). These QCD inspired potential models use a short
range part motivated by perturbative QCD (Coulomb like or one gluon exchange
potential) Bhaduri  _et al._ (1981); Gershtein  _et al._ (1988); Chen and
Kuang (1992); Gershtein  _et al._ (1995b); Baldicchi and Prosperi (2000);
Kwong and Rosner (1991); Vijaya Kumar  _et al._ (2013); Monteiro and Vijaya
Kumar (2010a, b, c) and a phenomenological long range part accounting for
confinement (i.e. linear, logarithmic or quadratic potential)Monteiro  _et
al._ (2017); Bhat  _et al._ (2017); Vijaya Kumar  _et al._ (1997); Monteiro
_et al._ (2011); Bhaghyesh  _et al._ (2012, 2011a); Bhaghyesh and Vijaya Kumar
(2013, 2011). Our Model uses the NRQM formalism for the study of low lying
charmonium states using a Hamiltonian which has the heavy quark potential from
the instanton vacuum depending on r, the inter quark distance. The heavy quark
potential derived from the instanton ensemble rises linearly as the relative
distance between the quark and antiquark increases, then it gets saturated. As
the quark and the antiquark distance increases the central potential turns out
to be Coulomb like potential. Therefore to study the mass spectra of the
quarkonia we have added Coulombic type potential to central instanton
potential. This can be understood as a non perturbative contribution to the
perturbative potential from instanton vacuum at large inter quark distances
greater than the instanton size. At large distance the instanton vacuum cannot
explain quark confinement, hence in our phenomenological potential model we
have included a linear confinement potential. Taking into consideration the
above factors, we have developed a nonrelativistic potential model to obtain a
reliable estimate of the masses of the low lying charmonium states and their
decay widths.

The paper is organized in 4 sections. In sec. II we briefly review the
theoretical background for the non relativistic model. In sec. III we present
the results and discussions. In sec. IV we draw up conclusions.

##  II Theoretical Background

###  II.1 The Model

In a potential model approach the entire dynamics of quarks in a meson is
governed by a Hamiltonian which is composed of a kinetic energy term $K$ and a
potential energy term V, that is,

$H=K+V.$ (1)

The kinetic energy K is given by,

$K=M+\frac{p^{2}}{2\mu}$ (2)

Here p is the relative momentum,
$\mu=\frac{m_{Q}m_{\bar{Q}}}{m_{Q}+m_{\bar{Q}}}$ is the reduced mass of the
$Q\bar{Q}$ system, where $m_{Q}$ and $m_{\bar{Q}}$ are the masses of the
individual quark and anti quark respectively and M is the total mass of quark
and antiquarkBhaghyesh and Vijyaya Kumar (2012).

The potential energy V is the sum of the heavy-quark potential
$V_{Q\bar{Q}}(\vec{r})$,confining potential $V_{conf}(\vec{r})$ and Coulomb
potential $V_{coul}(\vec{r})$,that is

$V(\vec{r})=V_{Q\bar{Q}}(\vec{r})+V_{coul}(\vec{r})+V_{conf}(\vec{r})$ (3)

The heavy-quark potential $V_{Q\bar{Q}}(\vec{r})$ is,

$V_{Q\bar{Q}}(\vec{r})=V_{C}(\vec{r})+V_{SD}(\vec{r}).$ (4)

Here $V_{C}(\vec{r})$ and $V_{SD}(\vec{r})$ are central and spin dependent
potentials due to instanton vacuum respectivelyYakhshiev  _et al._ (2017).
$V_{C}(\vec{r})$ is given by the following expression

$V_{C}(\vec{r}){\simeq}\frac{4\pi\bar{\rho}^{3}}{\bar{R^{4}}N_{c}}\Bigg{(}1.345\frac{r^{2}}{\bar{\rho}^{2}}-0.501\frac{r^{4}}{\bar{\rho}^{4}}\Bigg{)}$
(5)

Here, $\rho=\frac{1}{3}$ fm the average size of the instanton, $\bar{R}=1$ fm
the average separation between instantons and number of colors $N_{C}$ is 3.

The spin- spin interaction $V_{SS}(\vec{r})$, the spin-orbit coupling term
$V_{LS}(\vec{r})$ and the tensor part $V_{T}(\vec{r})$ contribute to the spin
dependent potential;

$\begin{split}V_{SS}(\vec{r})=\frac{1}{{3m_{Q}}^{2}}\nabla^{2}V_{C}(\vec{r});~{}V_{LS}(\vec{r})=\frac{1}{{2m_{Q}}^{2}}\frac{1}{r}\frac{dV_{C}(\vec{r})}{dr};~{}\\\
V_{T}(\vec{r})=\frac{1}{{3m_{Q}}^{2}}\Bigg{(}\frac{1}{r}\frac{dV_{C}(\vec{r})}{dr}-\frac{d^{2}V_{C}(\vec{r})}{dr^{2}}\Bigg{)}.\end{split}$
(6)

The coulomb-like (perturbative) one gluon exchange part of the potential is
given by

$V_{coul}(\vec{r})=\frac{-4\alpha_{s}}{3r}$ (7)

with the strong coupling constant $\alpha_{s}$ and inter quark distance r.

The confinement term represents the non perturbative effect of QCD which
includes the spin-independent linear confinement termBhaghyesh and Vijyaya
Kumar (2012)

$V_{conf}(\vec{r})=-\Bigg{[}\frac{3}{4}V_{0}+\frac{3}{4}cr\Bigg{]}F_{1}{\cdot}F_{2}$
(8)

where c and $V_{0}$ are constants. F is related to the Gell-Mann matrix,
$F_{1}=\frac{\lambda_{1}}{2}$ and $F_{2}=\frac{\lambda_{2}^{*}}{2}$ and
$F_{1}{\cdot}F_{2}=\frac{-4}{3}$ for the mesons.

###  II.2 Radiative Transitions

The study of radiative transitions can help in understanding the theory of
strong interaction in the nonperturbative regime of QCD. The non-relativistic
treatment adopted for the study of charmonium systems allows us to apply the
usual multi-pole expansion in electrodynamics to compute the transition
between the quarkonia states with the emission of a photon. The lowest order
of multi-pole expansion tends to dominate the transition. The resulting
transitions are the magnetic dipole $M_{1}$ and electric dipole $E_{1}$
transitions. The electric and magnetic dipole operators are decomposed by a
multi-pole expansion into components with definite spherical tensor ranks.

In the phenomenological potential model the expression used for the radiative
decay width does not directly influence the structure of the potential model
Hamiltonian. Thus the transition probabilities are influenced by the chosen
Hamiltonian only through the chosen wave function. In a $M_{1}$ transition
only the spin of the quarkonium state is changed, while the parity and the
orbital angular momentum remain unchanged. The expression for the decay width
of a spin flip $M_{1}$ transition between heavy quarkonium states depend on
the radial matrix. The different radial matrix elements can be obtained from
the corresponding spin-flip magnetic moment operator. The $M_{1}$ decay widths
of charmonium are calculated using an assumption that the effective confining
interaction is purely scalar. In electric dipole $E_{1}$ transition, the
parity of the states changes while spin remains unchanged. The one gluon
exchange contribution survives in the $E_{1}$ transition, whereas the
contribution from the scalar confining interaction term vanishes.

In calculating the radiative decay widths, we have assumed that in the non
relativistic limit, the dipole radial matrix elements are independent of J,
i.e all states within the same angular momentum multiplet have the same wave
function Deng  _et al._ (2016a). Radiative transitions could play an important
role in the discovery and identification of charmonium states. They are
sensitive to the internal structure of states, in particular to
${}^{3}L_{L}-^{1}L_{L}$ mixing for states with J = L. For our study of low
lying charmonium states we have taken $E_{1}$ and $M_{1}$ radiative
transitions since the other order transitions contribute a little to the
radiative decays. The partial width for an $E_{1}$ radiative transition is
given by,

$\Gamma(i\rightarrow f+\gamma)=\frac{4\alpha
e_{c}^{2}}{3}(2J_{f}+1)S_{if}^{E}k^{3}_{0}\left|{\cal E}_{if}\right|^{2}$ (9)

where $k_{0}=m_{i}-m_{f}$ is the energy of the emitted photon, $\alpha$ is the
fine structure constant. $e_{c}=2/3$ is the charge of the c quark in units of
$|e|$, $m_{i}$ and $m_{f}$ are the masses of initial and final mesons,
$S_{if}^{E}=\rm{max}(L_{i},L_{f})\left\\{\begin{array}[]{ccc}J_{i}&1&J_{f}\\\
L_{f}&S&L_{i}\end{array}\right\\}^{2}$ is the statistical factor, $J_{i}$ and
$J_{f}$ are the total angular momenta of initial and final mesons, $L_{i}$ and
$L_{f}$ are the orbital angular momenta of initial and final mesons and $S$ is
the spin of the initial meson. The radial overlap integral which has the
dimension of length is,

${\cal
E}_{if}=\frac{3}{k_{0}}\int^{\infty}_{0}r^{3}R_{nl}(r)R^{\prime}_{nl}(r)dr\left[\frac{k_{0}r}{2}j_{0}\left(\frac{k_{0}r}{2}\right)-j_{1}\left(\frac{k_{0}r}{2}\right)\right]$
(10)

with $R_{nl}(r)$ and $R^{\prime}_{nl}(r)$ as the normalized radial wave
functions for the corresponding states and $j_{0}$ and $j_{1}$ are spherical
Bessel functions.

The $M_{1}$ transitions between S-wave $c\bar{c}$ states are given in the non
relativistic approximation by Kwong and Rosner (1988); Novikov  _et al._
(1978); Brambilla  _et al._ (2004); Caswell and Lepage (1986); Bodwin  _et
al._ (1995); Brambilla  _et al._ (1999); Thacker and Lepage (1991),

$\Gamma(i\rightarrow f+\gamma)=\frac{4\alpha
e_{c}^{2}}{3m_{c}^{2}}\frac{2J_{f}+1}{2L_{i}+1}\delta_{L_{i}L_{f}}\delta_{S_{i}S_{f}}k^{3}_{0}|M_{if}(r)|^{2}$
(11)

Here $M_{if}$ is the radial overlap integral which has the dimension of
length,

$M_{if}=\int^{\infty}_{0}4\pi r^{3}R_{nl}(r)j_{0}(kr/2)R^{\prime}_{nl}(r)dr.$
(12)

In the overlap integral for unit operator between the coordinate wave
functions of the initial and the final meson states, $j_{0}(kr/2)$ is the
spherical Bessel function, $m_{c}$ is the mass of charm quark, $J_{f}$ is the
total angular momentum of final meson state, $L_{i}$ is the orbital angular
momentum of the initial meson state.

###  II.3 Annihilation Decays

The annihilation decays of charmonium states into gluons and light quarks make
significant contributions to the total decay width of the states. The
annihilation decays allow us to determine wave function at the origin. The
annihilation decays of some $c\bar{c}$ states into photons can be used as a
tool for the production and identification of the resonances.

####  II.3.1 Two Photon Decays

Two-photon branching fraction for the charmonium provides a probe for the
strong coupling constant at the charmonium scale via the two-photon decay
width. This can be utilized as a sensitive test for the corrections for the
non-relativistic approximation in the potential models or in the effective
field theories such as non relativistic QCD (NRQCD). The two-photon decays of
P-wave charmonia are helpful for better understanding the nature of inter-
quark forces and decay mechanisms.

The $q\bar{q}$ quark pair in charge conjugation even states with $J\neq 1$ can
annihilate into two photons. The expressions for the decay rates of
$n~{}^{1}S_{0}$, $n~{}^{3}P_{0}$ and $n~{}^{3}P_{2}$ states into two photons
with the first order QCD radiative corrections are given by Kwong  _et al._
(1988).

$\displaystyle\Gamma(n~{}^{1}S_{0}\to\gamma\gamma)=\frac{3e^{4}_{c}\alpha^{2}}{m^{2}_{c}}|R_{nS}(0)|^{2}\left(1-\frac{3.4\alpha_{s}}{\pi}\right)$
(13)
$\displaystyle\Gamma(n~{}^{3}P_{0}\to\gamma\gamma)=\frac{27e^{4}_{c}\alpha^{2}}{m^{4}_{c}}|R^{\prime}_{nP}(0)|^{2}\left(1+\frac{0.2\alpha_{s}}{\pi}\right)$
(14)
$\displaystyle\Gamma(n~{}^{3}P_{2}\to\gamma\gamma)=\frac{36e^{4}_{c}\alpha^{2}}{5m^{4}_{c}}|R^{\prime}_{nP}(0)|^{2}\left(1-\frac{16\alpha_{s}}{\pi}\right)$
(15)

The two photon decay widths of P wave charmonium states depend on the
derivative of the radial wave function at the origin.

####  II.3.2 Two Gluon Decays

The even states in charge conjugation of quarkonium with $J\neq 1$ can
annihilate into two gluons, much in the same way as they decay into two
photons. The charmonium states ${}^{1}S_{0}$, ${}^{3}P_{0}$, ${}^{3}P_{2}$ and
${}^{1}D_{2}$ can decay into two gluons, which account for a substantial
portion of the hadronic decays for states below $c\bar{c}$ threshold. The two
gluon decay widths are given by Royen and Weisskopf (1967); Kwong  _et al._
(1988).

$\Gamma(n~{}^{1}S_{0}\to
2g)=\frac{2\alpha^{2}_{s}}{3m^{2}_{c}}|R_{nS}(0)|^{2}\left(1+\frac{4.8\alpha_{s}}{\pi}\right)$
(16) $\Gamma(n~{}^{3}P_{0}\to
2g)=\frac{6\alpha^{2}_{s}}{m^{4}_{c}}|R^{\prime}_{nP}(0)|^{2}\left(1+\frac{9.5\alpha_{s}}{\pi}\right)$
(17) $\Gamma(n~{}^{3}P_{2}\to
2g)=\frac{8\alpha^{2}_{s}}{5m^{4}_{c}}|R^{\prime}_{nP}(0)|^{2}\left(1-\frac{2.2\alpha_{s}}{\pi}\right)$
(18) $\Gamma(n~{}^{1}D_{2}\to 2g)=\frac{2\alpha^{2}_{s}}{3\pi
m^{6}_{c}}|R^{\prime\prime}_{nD}(0)|^{2}$ (19)

It is natural that in the non-relativistic potential model of charmonium, the
ratio of the two-photon and two-gluon widths of the charmonium decays does not
depend on the wave function and slowly grows with increase of the charmonium
mass because of the proportionality to $\frac{1}{\alpha_{s}^{2}}$. According
to QCD, the decay of charmonium is due to the annihilation of $c\bar{c}$ pair.
The mass of $c\bar{c}$ pair is large and the annihilations of $c\bar{c}$ into
gluons are perturbative, so the two-gluon decay mode is dominant in the
charmonium. The two gluon decay widths are sensitive to the behavior of the
$q\bar{q}$ wave function and its derivatives near the origin.

####  II.3.3 Leptonic Decays

The vector mesons decay leptonically through interaction with the
electromagnetic current. The leptonic decay width is proportional to the
average value of the squared charge and the squared wave function at the
origin. Which gives the probability that the quark and antiquark will interact
with the photon at the origin of their relative coordinates and the mass of
the vector mesons. The quark-antiquark assignments for the vector mesons, as
well as the fractional values for the quark charges, may be experimented from
the values of their leptonic decay widths.

The decay of vector meson into charged leptons proceeds through the virtual
photon ($q\bar{q}\to l^{+}l^{-}$ where $l=e^{-},\mu^{-},\tau^{-}$). The
${}^{3}S_{1}$ and ${}^{3}D_{1}$ states have quantum numbers of a virtual
photon, $J^{PC}=1^{--}$ and can annihilate into lepton pairs through one
photon.

The leptonic decay width of the vector meson (${}^{3}S_{1}$ charmonium)
including the first order radiative QCD correction is given by Royen and
Weisskopf (1967); Kwong  _et al._ (1988)

$\Gamma(n~{}^{3}S_{1}\to
e^{+}e^{-})=\frac{4\alpha^{2}e^{2}_{c}|R_{nS}(0)|^{2}}{M^{2}_{nS}}\left(1-\frac{16\alpha_{s}}{3\pi}\right)$
(20)

where $\alpha\approx\frac{1}{137}$ is the fine structure constant, $M_{nS}$ is
the mass of the decaying charmonium state and $e_{c}=2/3$ is the charge of the
charm quark in units of the electron charge. For D wave charmonium states the
leptonic decay width with leading order QCD correction is given by

$\Gamma(n~{}^{3}D_{1}\to
e^{+}e^{-})=\frac{25\alpha^{2}e^{2}_{c}|R^{\prime\prime}_{nD}(0)|^{2}}{2m^{4}_{c}M^{2}_{nD}}\left(1-\frac{16\alpha_{s}}{3\pi}\right)$
(21)

where $M_{nD}$ is the mass of the decaying charmonium state. The leptonic
partial widths are an exploration of the compactness of the quarkonium system
and provide important information supplementary to level spacings. The quark-
antiquark assignments for the vector mesons, as well as the fractional values
for the quark charges, may be experimented from the values of their leptonic
decay widths.

##  III Results and Discussions

In our work, we have used the three-dimensional harmonic oscillator wave
function which has been extensively used in atomic and nuclear physics is used
as the trial wave function for obtaining the $Q\bar{Q}$ mass spectrum.

$\psi_{nlm}(r,\theta,\phi)=N\left(\frac{r}{b}\right)^{l}L_{n}^{l+1/2}(\frac{r}{b})exp\left(-\frac{r^{2}}{2b^{2}}\right)Y_{lm}(\theta,\phi)$
(22)

where $|N|$ is the normalizing constant given by

$|N|^{2}=\frac{2\alpha^{3}n!}{\sqrt{\pi}}\frac{2^{(2(n+l)+1)}}{(2n+2l+1)!}(n+l)!$
(23)

and $L_{n}^{l+1/2}(x)$ are the associated Laguerre polynomials, The harmonic
oscillator wave function allows the separation of motion of the center of mass
and has been used to study the spectra of baryons and mesons Faiman and Hendry
(1969); Feynman  _et al._ (1971). If the basic states are harmonic oscillator
wave functions it is rather easy to evaluate the matrix elements of a few body
systems such as mesons or baryons. In the harmonic oscillator wave function b
is treated as a variational parameter, which is determined for each state by
minimizing the expectation value of the Hamiltonian. The obtained b value is
used in the harmonic oscillator wave function to find the mass spectrum
Bhaghyesh  _et al._ (2011b). We have two important parameters characterizing
the dilute instanton liquid; the average size of the instanton
$\rho=\frac{1}{3}$ fm, the values of $\rho$ is less effective in the spin-
dependent parts of the potential. The average separation between instantons is
$\bar{R}=1$ fm Dyakonov and Petrov (1984); Shuryak (1982). The strength of
each part of the potential becomes stronger when smaller value of $\bar{R}$ is
employed. The instanton density is given as $N/V\simeq(200MeV)^{4}$ and number
of colors $N_{C}$ is 3. Other parameters in our potential model are, the
coupling constant $\alpha_{s}$, the charm quark mass $m_{c}$, the confinement
strength c and a constant $V_{0}$. The confinement strength $c$ is fixed by
the stability condition for variation of mass of the vector meson against the
size parameter $b$. The mass of the charm quark $m_{c}$ and constant $V_{0}$
were fixed so as to reproduce the ground state masses. We start with a set of
reasonable values of $m_{c}$ and $V_{0}$. We used the following set of
parameters in our work.

$\begin{split}m_{c}=1475~{}{\rm MeV};~{}~{}~{}~{}~{}\alpha_{s}=0.3\\\
~{}~{}c=260{\rm MeV~{}fm^{-1}};~{}~{}V_{0}=-125~{}\rm{MeV};\end{split}$

It should be noted that, for harmonic oscillator wave function
$|\Psi(0)|^{2}\propto\frac{1}{b^{3}}$, which is required to estimate the
leptonic and two photon and two gluon decay widths. The $\alpha_{s}$ quoted by
the latest PDG is 0.1182(12)(et al., 2016). The value of $\alpha_{s}$ used in
the present investigation is 0.3.

Table 1:  Mass spectrum (MeV). $n^{2S+1}L_{J}$ | Name | $J^{PC}$ | Present Work | $M_{exp}$ | Godfrey and Isgur (1985) | Ebert  _et al._ (2011) | Radford and Repko (2007) | Cao  _et al._ (2012) | Segovia  _et al._ (2013)
---|---|---|---|---|---|---|---|---|---
|  |  | MeV | MeV | MeV | MeV | MeV | MeV | MeV
$1^{1}S_{0}$ | ${\eta}_{c}{(1S)}$ | $0^{-+}$ | 2984 | 2983.6$\pm$0.7 | 2970 | 2981 | 2981.7 | 2990.4 | 2990
$2^{1}S_{0}$ | ${\eta}_{c}{(2S)}$ | $0^{-+}$ | 3640 | 3639.2$\pm$0.11 | 3620 | 3635 | 3619.2 | 3646.5 | 3643
$3^{1}S_{0}$ | ${\eta}_{c}{(3S)}$ | $0^{-+}$ | 4061 | …. | 4060 | 3989 | 4052.5 | 4071.9 | 4054
$1^{3}S_{1}$ | $J/\psi$ | $1^{--}$ | 3097 | 3096.916$\pm$0.011 | 3100 | 3096 | 3096.92 | 3085.1 | 3096
$2^{3}S_{1}$ | ${\psi}{(2S)}$ | $1^{--}$ | 3687 | 3686.108$\pm$0.018 | 3680 | 3685 | 3686.1 | 3682.1 | 3703
$3^{3}S_{1}$ | ${\psi}{(3S)}$ | $1^{--}$ | 4039 | 4039$\pm$1 | 4100 | 4039 | 4102 | 4100.2 | 4097
$1^{1}P_{1}$ | $h_{c}(1P)$ | $1^{+-}$ | 3525 | 3525.38$\pm$0.11 | 3520 | 3525 | 3523.7 | 3514.6 | 3515
$2^{1}P_{1}$ | $h_{c}(2P)$ | $1^{+-}$ | 3927 | …. | 3960 | 3926 | 3963.2 | 3944.6 | 3956
$3^{1}P_{1}$ | $h_{c}(3P)$ | $1^{+-}$ | 4337 | …. | …. | 4337 | …. | 4333.9 | 4278
$1^{3}P_{0}$ | $\chi_{c0}(1P)$ | $0^{++}$ | 3414 | 3414.75$\pm$0.31 | 3440 | 3413 | 3415.2 | 3351.6 | 3452
$2^{3}P_{0}$ | $\chi_{c0}(2P)$ | $0^{++}$ | 3916 | 3915$\pm$3$\pm$2 | 3920 | 3870 | 3864.3 | 3835.7 | 3909
$3^{3}P_{0}$ | $\chi_{c0}(3P)$ | $0^{++}$ | 4303 | …. | …. | 4301 | …. | 4216.7 | 4242
$1^{3}P_{1}$ | $\chi_{c1}(1P)$ | $1^{++}$ | 3510 | 3510.66$\pm$0.07 | 3510 | 3511 | 3510.6 | 3500.4 | 3452
$2^{3}P_{1}$ | $\chi_{c1}(2P)$ | $1^{++}$ | 3872 | 3872 | 3950 | 3906 | 3950.0 | 3933.5 | 3947
$3^{3}P_{1}$ | ${\chi}_{c1}{(3P)}$ | $1^{++}$ | 4312 | …. | …. | 4319 | …. | 4317.9 | 4272
$1^{3}P_{2}$ | $\chi_{c2}(1P)$ | $2^{++}$ | 3555 | 3556.20$\pm$0.09 | 3550 | 3555 | 3556.2 | 3551.4 | 3532
$2^{3}P_{2}$ | $\chi_{c2}(2P)$ | $2^{++}$ | 3929 | 3927.2$\pm$2.6 | 3980 | 3949 | 3992.3 | 3979.8 | 3969
$3^{3}P_{2}$ | ${\chi}_{c2}{(3P)}$ | $2^{++}$ | 4042 | …. | 4010 | 4041 | …. | 4383.4 | 4043
$1^{1}D_{2}$ | ${\eta}_{c2}{(1D)}$ | $2^{-+}$ | 3812 | …. | 3840 | 3807 | 3822.3 | 3807.3 | 3812
$2^{1}D_{2}$ | ${\eta}_{c2}{(2D)}$ | $2^{-+}$ | 4198 | …. | 4210 | 4196 | 4196.9 | 4173.7 | 4166
$1^{3}D_{1}$ | ${\psi}_{1}{(1D)}$ | $1^{--}$ | 3779 | 3778 | 3820 | 3783 | 3789.4 | 3785.3 | 3796
$2^{3}D_{1}$ | ${\psi}_{1}{(2D)}$ | $1^{--}$ | 4192 | 4191$\pm$5 | 4190 | 4159 | 4159.2 | 4150.4 | 4153
$1^{3}D_{2}$ | ${\psi}_{2}{(1D)}$ | $2^{--}$ | 3823 | 3823 | 3840 | 3795 | 3822.1 | 3807.7 | 3810
$2^{3}D_{2}$ | ${\psi}_{2}{(2D)}$ | $2^{--}$ | 4195 | …. | 4210 | 4190 | 4195.8 | 4173.7 | 4160
$1^{3}D_{3}$ | ${\psi}_{2}{(1D)}$ | $3^{--}$ | 3845 | …. | …. | …. | 3844.8 | 3814.6 | ….
$2^{3}D_{3}$ | ${\psi}_{2}{(2D)}$ | $3^{--}$ | 4220 | …. | …. | …. | 4218.9 | 4182.9 | ….

Table 1 lists the low lying charmonium states in comparison with experimental
data and other theoretical models. Our predictions for the masses agree with
PDG data within a few MeV. The model correctly reproduces the mass spectrum of
charmonium states.The mass of singlet state $\eta_{c}(1S)$ is found to be
$2984$ MeV which is in good agreement with the experimentally measured mass
value $2983.6\pm 0.7$MeV Olive  _et al._ (2014). Lattice QCD calculations
predict a mass of $2985(1)$ MeVKawanai and Sasaki (2015) for $\eta_{c}(1S)$
state. The mass of the spin triplet state $J/\psi(1S)$ calculated in our model
is $3097$ MeV which is in good agreement with the experimental value
$3096.96\pm 0.011$MeVOlive  _et al._ (2014) and the values of other
theoretical models Godfrey and Isgur (1985),Ebert  _et al._ (2011),Radford and
Repko (2007),Cao  _et al._ (2012),Segovia  _et al._ (2013). The lattice QCD
calculations predict a mass of $3099\pm 1$ MeV for $J/\psi(1S)$ state. The
masses of radially excited Charmonium state $\eta_{c}(2S)$ and its triplet
partner $\psi(2S)$ calculated in our model are in good agreement with both
experimental value and with other theoretical models. The lattice QCD
calculations predict a slightly less mass value for $\eta_{c}(2S)$,
($M(\eta_{c}(2S))=3612\pm 9$ MeV) and for $\psi(2S)$($M_{\psi(2S)}=3653\pm 12$
MeV)Kawanai and Sasaki (2015). It is clear from the table, that the mass of
$\eta_{c}(3S)$ and its triplet partner calculated in our model is reasonably
in good agreement with the experimental and with other model values. The
lattice QCD calculations predict a mass of $4074\pm 20$ MeV for $\eta_{c}(3S)$
and $4099\pm 24$ MeV for $\psi(3S)$Kawanai and Sasaki (2015).

The mass of spin-singlet P-wave charmonium $h_{c}(1P)$ is in good agreement
with experiment and other theoretical models. However the lattice QCD predicts
a mass of $3506\pm 6$ MeV for $h_{c}(1P)$ state which is slightly below the
experimental valueKawanai and Sasaki (2015). The spin triplet states
$\chi_{cJ}$ are in good agreement with both experiment and other
phenomenological models. The lattice QCD obtains the masses $20-25$ MeV
smaller than the experimental values for these statesKawanai and Sasaki
(2015). We have also predicted masses of low lying $D$ wave states which is
reasonably in good agreement with available experimental data et al. (2016)and
with other theoretical models Radford and Repko (2007),Cao  _et al._
(2012),Segovia  _et al._ (2013)

The spin-orbit and tensor potentials in eqn(6) are responsible for the
splitting of the charmonium levels. The hyperfine splitting of 1S state
obtained in our model $\Delta M(1^{3}S_{1}-1^{1}S_{0})$ is 113 MeV. The
hyperfine mass splitting calculated in our model agrees with the experimental
value $\Delta M(1^{3}S_{1}-1^{1}S_{0})=113.2\pm 0.7$ MeVOlive  _et al._ (2014)
and lattice QCD results $\Delta M(1^{3}S_{1}-1^{1}S_{0})$ = 114 $\pm$ 1MeV.
The hyperfine mass splitting of 2S states $\Delta M(2^{3}S_{1}-2^{1}S_{0})$=
47 MeV is in good agreement with the $\Delta M(2^{3}S_{1}-2^{1}S_{0})=47\pm 1$
MeVOlive  _et al._ (2014). The lattice QCD predicts a 2S hyperfine splittings
of $57.9\pm 2.0$ MeV Mohler (2012) which is slightly higher than the
experimental value. For $1^{3}P_{1}$ \- $1^{3}P_{0}$ splitting, we obtain
$\Delta M(1^{3}P_{1}-1^{3}P_{0})=96$ MeV which is good agreement with the
experimental value $\Delta M(1^{3}P_{1}-1^{1}P_{0})=95.5\pm 0.8$ MeV Olive
_et al._ (2014). The Lattice QCD calculations predict a 1P splitting of
$68.4\pm 5.0+11.8-3.0$ MeVOkamoto  _et al._ (2002) which is reasonably in good
agreement with our calculations. For $1^{3}P_{2}-1^{3}P_{1}$ splitting, we
obtain $\Delta M(1^{3}P_{2}-1^{3}P_{1})$=45 MeV, which is in good agreement
with the experimental data $\Delta M(1^{3}P_{2}-1^{3}P_{1})=45.7\pm 0.2$ MeV
Olive  _et al._ (2014). The lattice QCD calculations predict $\Delta
M(1^{3}P_{2}-1^{3}P_{1})=31.4\pm 8.4$ MeV Okamoto  _et al._ (2002), which is
rather a low value compared to experimental value.

The ratio between the two hyperfine structures, $\frac{\Delta
M(1^{3}P_{2}-1^{3}P_{1})}{\Delta M(1^{3}P_{1}-1^{3}P_{0})}$ sheds light on the
nature of the confinement. Our estimate of the ratio is $\frac{\Delta
M(1^{3}P_{2}-1^{3}P_{1})}{\Delta M(1^{3}P_{1}-1^{3}P_{0})}$ =0.46, and that
from the experiment is 0.48Okamoto  _et al._ (2002); Lucha and Schöberl
(1996). Another interesting quantity is the P-state hyperfine splitting
$\Delta M(1^{1}P_{1}-1^{3}P)$, where,

$M(1^{3}P)=\frac{5M(1^{3}P_{2})+3M(1^{3}P_{1})+M(1^{3}P_{0})}{9}$ (24)

is the center of gravity of the P-wave system. The P-state hyperfine splitting
should be much smaller than S-state hyperfine splitting, since the P state
wave function is zero at the origin. Our estimate of the P-state hyperfine
splitting is $\Delta M(1^{1}P_{1}-1^{3}P)$ = 0.6 MeV and the experimental
value is $\Delta M(1^{1}P_{1}-1^{3}P)$ = 0.9 MeV Olive  _et al._ (2014). The
lattice QCD calculations obtain P-state hyperfine splitting of -1.4 MeV.

The spin averaged masses are defined by,

$M(n\bar{S})=\frac{3M(n^{3}S_{1})+M(n^{1}S_{0})}{4}$ (25)
$M(n\bar{P})=\frac{3M(n^{1}P_{1})+5M(n^{3}P_{2})+3M(n^{3}P_{1})+M(n^{3}P_{0})}{12}$
(26)

with n=1,2,3,… the radial quantum numbers. The spin averaged masses calculated
in our model are $M(1\bar{S})$=3068.75 MeV, $M(2\bar{S})$=3675.25 MeV,
$M(1\bar{P})$=3524.5 MeV and $M(2\bar{P})$=3913.16 MeV and the experimental
value of spin averaged masses are 2984.3, 3638.5.6 MeV, 3525.3 MeV and 3929
MeV respectively Okamoto  _et al._ (2002) Zhang (2013). The calculated spin
averaged mass splittings are listed in Table 2.

Table 2:  Spin averaged mass splittings (MeV) Mass Splittings | Present Work | Exp | Lattice QCD Okamoto  _et al._ (2002) | GIBarnes  _et al._ (2005) | Deng  _et al._ (2016b) | Li and Chao (2009)
---|---|---|---|---|---|---
$M(1^{3}S_{1}-1^{1}S_{0})$ | 113 | $113.2\pm 0.7$ | $114\pm 1$ | 113 | 114 | 118
$M(2^{3}S_{1}-2^{1}S_{0})$ | 47 | $47\pm 1$ | $57.9\pm 2$ | 53 | 44 | 50
$M(1^{3}P_{2}-1^{3}P_{1})$ | 45 | $45.7\pm 0.2$ | $31.4\pm 8.4$ | 40 | 36 | 44
$M(1^{3}P_{1}-1^{3}P_{0})$ | 96 | $95.5\pm 0.8$ | 68.4 | 65 | 101 | 77
$M(1^{1}P_{1}-1^{3}P)$ | 0.6 | 0.9 | -1.4 |  |  |
$M(1^{1}P_{1}-1\bar{S})$ | 456.25 | 458.5 | $448.8\pm 29$ |  |  |
$M(1^{3}P_{0}-1\bar{S})$ | 345.25 | 347.1 | $419.8\pm 47$ |  |  |
$M(1^{3}P_{1}-1\bar{S})$ | 441.25 | 442.9 | $448.8\pm 34$ |  |  |
$M(1^{3}P_{2}-1\bar{S})$ | 486.25 | 488.6 | $448.8\pm 29$ |  |  |
$M(1\bar{P}-1\bar{S})$ | 455.75 | 457.9 | $457.9$ |  |  |
$M(2\bar{P}-2\bar{S})$ | 237.91 | …. | $462\pm 72$ |  |  |
$M(2\bar{S}-1\bar{S})$ | 606.75 | 595 | $671\pm 21$ |  |  |
$M(2\bar{P}-1\bar{P})$ | 388.66 | …. | $675\pm 76$ |  |  |

Radiative decays of excited charmonium states are powerful tool to study the
internal structure of the mesons. The possible $E1$ decay modes have been
listed in Table 3. Most of the predictions for $E1$ transitions are in
qualitative agreement with other theoretical models. However, there are some
differences in the predictions due to differences in phase space arising from
different mass predictions and also from the wave function effects. We find
our results are compatible with other theoretical model values for most of the
channels.

The M1 transition rates of charmonium states have been calculated using
eqn(11). Allowed M1 transitions correspond to triplet-singlet transition
between S-wave states and between P-state of the same n quantum number, while
hindered M1 transitions are either triplet-singlet or singlet-triplet
transitions between S-wave states of different quantum numbers. In order to
calculate decay rates of hindered transitions, we need to include relativistic
corrections, viz, modification of the nonrelativistic wave functions,
relativistic modification of the electromagnetic transition operator, and
finite-size corrections. In addition to these, there are additional
corrections arising from the quark anomalous magnetic moment.

Corrections to the wave function that give contribution to the transition
amplitude are of two categories: (1) higher order potential corrections, which
are distinguished as (a) the zero recoil effect and (b)recoil effects of the
final state meson, and (2) color octet effects. The color octet effects have
not been included in potential model formulation and are not considered so far
in radiative transitions. The spherical Bessel function $j_{0}(k_{0}r/2)$,
introduced in eqn(11), takes into account the so called finite size
effect(equivalently, re-summing the multipole-expanded magnetic amplitude to
all orders). For small value of $k_{0}$, $j_{0}(k_{0}r/2)\rightarrow 1$, the
transitions with $n^{\prime}=n$ have dominant contribution to the matrix
elements,though the corresponding partial decay widths are suppressed by
smaller $k_{0}^{3}$ factors. For a large value of photon energy (k),
transitions with $n\neq n^{\prime}$ have dominant contribution to the matrix
element, since $j_{0}(k_{0}r/2)$ becomes very small. M1 transition rates are
very sensitive to hyperfine splittings of the levels due to the $k_{0}^{3}$
factor in eqn(11). The resulting M1 radiative transition rates of these states
are presented in Table 4. The M1 transition rates calculated in our model
agree well with the values predicted by other theoretical models.

Using the Van-Royen-Weisskopf relation we have calculated annihilation decay
widths like leptonic decay widths, two -photon and two gluon decay widths with
the inclusion of radiative corrections Royen and Weisskopf (1967). The
resulting leptonic decay widths are listed in Table V. Our predictions for
leptonic decay widths have been compared with experiment and other theoretical
models and are found to be in good agreement. The Tables VI and VII list the
two photon and two gluon decay widths of charmonium states. The two photon and
two-gluon decay widths for charmonium states reasonably agree with the
available experimental data and with other theoretical models.

Table 3: E1 Transition rates E1 Transition | k | Present Work $\Gamma$ | $\Gamma_{Expt}$ |  Barnes  _et al._ (2005)a |  Barnes  _et al._ (2005)b | Brambilla  _et al._ (2004) | Cao  _et al._ (2012) | Radford and Repko (2007)
---|---|---|---|---|---|---|---|---
$i\rightarrow f$ | (MeV) | ($i\rightarrow f$)(keV) | (keV) | (keV) | (keV) | (keV) | (keV) | (keV)
$1^{3}P_{0}\rightarrow 1^{3}S_{1}+\gamma$ | 317 | 114.74 | 119.5$\pm$8 | 152 | 114 | 120 | 97 | 139.3
$1^{3}P_{1}\rightarrow 1^{3}S_{1}+\gamma$ | 413 | 297.61 | 295.8$\pm$13 | 314 | 239 | 241 | 330 | 38.4
$1^{3}P_{1}\rightarrow 1^{1}S_{0}+\gamma$ | 526 | 486.26 |  | 498 | 352 | 482 | 465 | 546.4
$1^{3}P_{2}\rightarrow 1^{3}S_{1}+\gamma$ | 458 | 350.58 | 384.2$\pm$16 | 424 | 313 | 315 | 421 | 319.4
$2^{3}S_{1}\rightarrow 1^{3}P_{0}+\gamma$ | 273 | 55.61 |  | 63 | 26 | 47 | 34 | 25.2
$2^{1}S_{0}\rightarrow 1^{1}P_{1}+\gamma$ | 115 | 40.26 |  | 49 | 36 | 35.1 | 72 | 17.4
$2^{3}S_{1}\rightarrow 1^{3}P_{1}+\gamma$ | 177 | 25.45 | 28.0$\pm$1.2 | 54 | 29 | 42.8 | 48 | 29.1
$2^{3}S_{1}\rightarrow 1^{3}P_{2}+\gamma$ | 132 | 23.9 | 26.6$\pm$1.1 | 38 | 24 | 30.1 | 43 | 26.5
$1^{3}D_{1}\rightarrow 1^{3}P_{0}+\gamma$ | 365 | 200.66 | 199.3$\pm$25 | 403 | 213 | 299 | 367 | 243.9
$1^{3}D_{1}\rightarrow 1^{3}P_{1}+\gamma$ | 269 | 75.34 | 79.2$\pm$16 | 125 | 77 | 99 | 146 | 104.9
$1^{3}D_{1}\rightarrow 1^{3}P_{2}+\gamma$ | 224 | 3.70 | 3.88 | $<24.6$ | 4.9 | 3.3 | 6.8 | 1.9
$1^{3}D_{2}\rightarrow 1^{3}P_{1}+\gamma$ | 313 | 245.35 |  | 307 | 268 | 313 | 321 | 256.7
$1^{3}D_{2}\rightarrow 1^{3}P_{2}+\gamma$ | 268 | 80.64 |  | 64 | 66 | 69.5 | 79 | 61.8
$1^{1}D_{2}\rightarrow 1^{1}P_{1}+\gamma$ | 287 | 310.52 |  | 339 | 344 | 389 | 398 |
$1^{3}D_{3}\rightarrow 1^{3}P_{2}+\gamma$ | 290 | 246.79 |  | 272 | 296 | 402 | 340 |
aNon relativistic quark model |  |  |  |  |  |  |  |
aRelativistic quark model |  |  |  |  |  |  |  |
Table 4: M1 Transition rates $(M_{1})$Transition | k | Present Work $\Gamma$ | $\Gamma_{Expt}$ |  Barnes  _et al._ (2005)c |  Barnes  _et al._ (2005) d |  Cao  _et al._ (2012)c |  Cao  _et al._ (2012)d
---|---|---|---|---|---|---|---
$i\rightarrow f$ | (MeV) | ($i\rightarrow f$)(keV) | (keV) | (keV) | (keV) | (keV) | (keV)
$1^{3}S_{1}\rightarrow 1^{1}S_{0}+\gamma$ | 113 | 2.38 | 1.58$\pm$0.37 | 2.9 | 2.4 | 1.5 | 2.2
$2^{3}S_{1}\rightarrow 2^{1}S_{0}+\gamma$ | 47 | 0.17 | 0.143$\pm$0.027 | 0.21 | 0.17 | 3.1 | 3.8
$2^{3}S_{1}\rightarrow 1^{1}S_{0}+\gamma$ | 703 | 3.96 | 0.97$\pm$0.027 | 4.6 | 9.6 | 6.1 | 6.9
$2^{3}S_{1}\rightarrow 1^{3}S_{1}+\gamma$ | 590 | 5.30 |  | 7.9 | 5.6 | 0.70 | 0.71
$3^{3}S_{1}\rightarrow 2^{1}S_{0}+\gamma$ | 399 | 1.12 |  | 0.61 | 2.6 | 3.2 | 3.7
$3^{1}S_{0}\rightarrow 2^{3}S_{1}+\gamma$ | 374 | 1.42 |  | 1.3 | 0.84 | 1.7 | 1.6
$3^{1}S_{0}\rightarrow 1^{3}S_{1}+\gamma$ | 964 | 6.51 |  | 6.3 | 6.9 | 5.9 | 6.5
$2^{1}P_{1}\rightarrow 1^{3}P_{2}+\gamma$ | 372 | 0.23 |  | 0.071 | 0.11 | 1.3 | 1.2
$2^{1}P_{1}\rightarrow 1^{3}P_{1}+\gamma$ | 417 | 0.18 |  | 0.058 | 0.36 | 0.16 | 0.13
$2^{1}P_{1}\rightarrow 1^{3}P_{0}+\gamma$ | 513 | 1.31 |  | 0.033 | 1.5 | 5.6 | 5.3
$2^{3}P_{2}\rightarrow 1^{1}P_{1}+\gamma$ | 404 | 1.12 |  | 0.67 | 1.3 | 1.0 | 0.89
$2^{3}P_{1}\rightarrow 1^{1}P_{1}+\gamma$ | 347 | 0.15 |  | 0.050 | 0.045 | 0.15 | 0.13
cNon relativistic quark model |  |  |  |  |  |  |
dRelativistic quark model |  |  |  |  |  |  |
Table 5: Leptonic Decay widths (keV) State | Present Work $\Gamma_{l^{+}l^{-}}$ | Exp $\Gamma_{l^{+}l^{-}}$ | Bhaghyesh and Vijyaya Kumar (2012) | Radford and Repko (2007) | Barnes (2004)
---|---|---|---|---|---
$J/\psi$ | 4.65 |  $5.55\pm$0.14 | 3.589 | 4.28 | 12.13
$\psi(2S)$ | 2.25 | $2.33\pm 0.07$ | 1.440 | 2.25 | 5.03
$\psi(3S)$ | 0.98 | $0.86\pm 0.07$ | 0.975 | 1.66 | 3.48
$1^{3}D_{1}$ | 0.35 | $0.242\pm 0.030$ | 0.096 | 0.09 | 0.056
$2^{3}D_{1}$ | 0.68 | $0.83\pm 0.07$ | 0.112 | 0.16 | 0.096
Table 6: Two-Photon Decay widths (keV) State | Present Work $\Gamma$ | Exp$\Gamma$ | Bhaghyesh and Vijyaya Kumar (2012) | Münz (1996) | Lakhina and Swanson (2006)
---|---|---|---|---|---
$\eta_{c}(1S)$ | 6.89 | $7.2\pm 0.7$ | 6.812 | 3.50 | 7.18
$\eta_{c}(2S)$ | 6.68 | $7.0\pm 3.5$ | 2.625 | 1.38 | 1.71
$\eta_{c}(3S)$ | 1.23 |  | 1.760 | 0.94 | 1.21
$\chi_{c0}(1P)$ | 1.96 | $2.36\pm 0.35$ | 2.119 | 1.39 | 3.28
$\chi_{c0}(2P)$ | 1.02 |  | 1.308 | 1.11 |
$\chi_{c2}(1P)$ | 0.82 | $0.66\pm 0.07$ | 0.261 | 0.44 |
$\chi_{c2}(2P)$ | 0.19 |  | 0.168 | 0.48 |
Table 7: Two-Gluon Decay widths (MeV) State | Present Work $\Gamma$ | Exp | Bhaghyesh and Vijyaya Kumar (2012) | Parmar  _et al._ (2010) | Laverty  _et al._ (2009)
---|---|---|---|---|---
$\eta_{c}(1S)$ | 27.61 | $28.6\pm 2.2$ | 22.048 | 15.70 | 32.209
$\eta_{c}(2S)$ | 7.92 | $14\pm 7$ | 8.496 | 8.10 |
$\eta_{c}(3S)$ | 5.67 |  | 5.696 |  |
$\chi_{c0}(1P)$ | 9.67 | $10.3\pm 0.6$ | 6.114 | 4.68 | 10.467
$\chi_{c0}(2P)$ | 3.67 |  | 3.775 |  |
$\chi_{c2}(1P)$ | 2.15 | $1.97\pm 0.11$ | 0.633 | 1.72 | 1.169
$\chi_{c2}(2P)$ | 0.59 |  | 0.401 |  |
$1^{1}D_{2}^{e}$ | 0.068 |  | 0.014 |  |
$2^{1}D_{2}^{e}$ | 0.061 |  | 0.012 |  |
eWithout QCD corrections |  |  |  |  |

##  IV Conclusions and Outlook

Basic aim of the present work is to develop a consistent model which could
reproduce both the spectra and the decay widths with the same set of
parameters and to investigate the effect of instanton potential on masses and
the excited states of charmonium. From our analysis, we infer that the present
model has the right prediction both for the mass spectrum and decay widths.
Since different potentials can reproduce the same spectra the stringent test
for any given model is the calculation of the other observables like leptonic,
the radiative, two-photon and two-gluon decay widths in addition to mass
spectrum. In our earlier work also, we had come to the similar conclusion
while investigating light meson spectrum Vijayakumar  _et al._ (2004)
Bhavyashri  _et al._ (2005). The differences in the prediction for the decay
rates in various theoretical models can be attributed to the differences in
mass predictions and wave function effects. From the study of mass spectra and
decay properties of $c\bar{c}$ states in a phenomenological approach we draw
up the following conclusions.

  1. (1)

Our calculations for the low lying charmonium states are in good agreement
with experimental measurements

  2. (2)

The hyperfine separations are directly related to the spin-spin interaction.
The theoretical predictions of our model are remarkably consistent with well
established experimental data for the conventional charmonium states. Below
the open charm threshold, our theoretical calculations are in agreement with
lattice calculations and experimental results.

  3. (3)

The mass splittings between the radial excitations and the ground states also
provide an important check on the validity of our potential model.

  4. (4)

The calculations demonstrate that the charmonium masses/mass splittings can be
computed with a combination of coulombic, instanton and confinement
potentials. Hence,in heavy quark sector,the instanton potential plays the role
of OGEP. The instanton effects are quite significant in the central part of
the potential.

  5. (5)

They turn out to be rather small in describing the hyperfine mass splittings
of the charmonia. This might be due to the spin-dependent part of the
potential from the instanton vacuum is an order of magnitude smaller than the
central part. The tensor interaction almost does not contribute to the masses.

  6. (6)

Charmonium decays provide a deeper insight on the exact nature of the inter
quark forces and decay mechanisms. Using the predicted masses and the radial
wave function at the origin, leptonic decays,two photon and two gluon decays
are computed using the Van Royen-Weisskopf relation. The calculated values
including the correction factor agree with the experimental values within a
few MeV. From our calculations, we conclude that the inclusion of QCD
correction factors are of importance for obtaining accurate results for decay
rates.

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>}}\end{array}\begin{array}[]{c}\Psi,\,{\footnotesize\texttt{name :
String}}\,\,\vdash\,{\footnotesize\texttt{println!("Hello, \\{\\}!", name);
terminate()}}\,:\\\ {\footnotesize\texttt{PartialSession < (), End
>}}\end{array}$

The first part of derivation tree of hello_client, of how a received channel
is bound to the linear context, is shown as follows:

$\begin{array}[]{c}\Psi\,\,\vdash\,\texttt{receive\\_channel(|a|\\{
send\\_value\\_to(...) \\})}\,:\\\ \texttt{PartialSession < (),
ReceiveChannel< ReceiveValue<String, End>, End > >
>}\end{array}\lx@proof@logical@and\begin{array}[]{c}\Psi\,\,\vdash\,\texttt{|a|\\{
send\\_value\\_to(a, ...) \\}}\,:\\\ \texttt{impl FnOnce(Z) -> PartialSession
< }\\\ \texttt{(ReceiveValue<String, End>, ()), End
>}\end{array}\begin{array}[]{c}\Psi,\,\texttt{a:
Z}\,\,\vdash\,\texttt{send\\_value\\_to(a, ...)}\,:\\\ \texttt{PartialSession
<}\\\ \texttt{(ReceiveValue<String, End>, ()), End
>}\end{array}\mathcal{D}\tiny\begin{array}[]{c}\texttt{()}\,\Rightarrow\,\texttt{AppendContext
<}\\\ \texttt{(ReceiveValue<String, End>, ()),}\\\ \texttt{Appended=}\\\
\texttt{(ReceiveValue<String, End>,())
>}\end{array}\tiny\begin{array}[]{c}\texttt{()}\,\Rightarrow\,\texttt{Context}\\\
\texttt{< Length=Z >}\end{array}$

With the received channel and context lens in the environment, The second part
of derivation tree of hello_client, $\mathcal{D}$, is continued as follows:

$\begin{array}[]{c}\Psi,\,\texttt{a:
Z}\,\,\vdash\,\texttt{send\\_value\\_to(a, "Alice".to\\_string(), wait(a,
...))}\,:\\\ \texttt{PartialSession < (ReceiveValue<String, End>, ()), End
>}\end{array}\lx@proof@logical@and\begin{array}[]{c}\Psi,\,\texttt{a:
Z}\,\,\vdash\,\texttt{wait(a, terminate())}\,:\\\ \texttt{PartialSession
<(End, ()), End
>}\end{array}\lx@proof@logical@and\begin{array}[]{c}\Psi\,\,\vdash\,\texttt{terminate()}\,:\\\
\texttt{PartialSession <}\\\ \texttt{(Empty,()), End
>}\end{array}\tiny\texttt{(Empty,
())}\Rightarrow\texttt{EmptyContext}\tiny\begin{array}[]{c}\texttt{Z}\,\Rightarrow\,\texttt{ContextLens
<}\\\ \texttt{(End,()),}\\\ \texttt{End, Empty,}\\\ \texttt{Target=(Empty,())
>}\end{array}\tiny\begin{array}[]{c}\texttt{Z}\,\Rightarrow\,\texttt{ContextLens
<}\\\ \texttt{(ReceiveValue<String, End>,()),}\\\ \texttt{ReceiveValue<String,
End>,}\\\ \texttt{End,}\\\ \texttt{Target=(End,()) >}\end{array}$

###  B.2. Communication

####  B.2.1. Include Session

Other than the general cut rule, Ferrite also provides a more restricted
version of cut called include:

$\Gamma\,;\,\Delta\,\vdash\,\,x\,\leftarrow\,\mathsf{include}\,a\,;\,b\,::\,B\lx@proof@logical@and\Gamma\,;\,\cdot\,\vdash\,a::A\Gamma\,;\,\Delta,\,x:A\,\vdash\,b::B$

The typing rule T-incl is nearly identical to T-cut, with the additional
restriction that the linear context for $a$ must be empty. This restriction
makes it much easier for the Rust compiler to infer the type for $\Delta$,
since there is no longer a need to split it into two parts for the two
continuations. Without the linear context splitted, the existing context
lenses can also be preserved, making them usable before and after an include.
It is also clear that T-incl is derivable from T-cut, by simply unifiying
$\Delta_{1}$ in T-cut with $\cdot$. As a result, introducing T-incl does not
affect the properties of session types. T-incl is implemented in Ferrite as
the include_session function as follow:

⬇

fn include_session < A: Protocol, B: Protocol, C: Context >

( a: Session < A >,

cont: impl FnOnce ( C::Length )

-> PartialSession < C::Appended, B >

) -> PartialSession < C, B >

where C : AppendContext < ( A, () ) >,

include_session takes a Ferrite program of type Session<A> and appends the
offered channel to the linear context C. Similar to receive_channel,
include_session generates a context lens of type C::Length, which is used by
the continuation closure cont to access the appended channel A to C. cont
returns a PartialSession < C::Appended, B >, indicating that it offers session
type B using the linear context C appended with A. Finally include_session
returns PartialSession < C, B >, which works on the original linear context C
and offers session type B.

The apply_channel construct is implemented using include_session,
send_channel_to, and forward as shown below.

⬇

fn apply_channel < A: Protocol, B: Protocol >

( f: Session < ReceiveChannel<A, B> >, a: Session < A > )

-> Session < B >

{ include_session ( f, | chan_f | {

include_session ( a, | chan_a | {

send_channel_to ( chan_f, chan_a,

forward ( chan_f ) ) }) }) }

We can prove that the typing rule T-app is derivable from T-cut, as shown
below.

$\Gamma\,;\,\cdot\,\vdash\,\mathsf{apply\\_channel}\;f\;a::B\Gamma\,;\,\cdot\,\vdash\,f^{\prime}\,\leftarrow\mathsf{cut}\;f\,;\,a^{\prime}\,\leftarrow\mathsf{cut}\;a\,;\,\mathsf{send\\_channel\\_to}\,f^{\prime}\,a^{\prime};\mathsf{forward}\,f^{\prime};::B\lx@proof@logical@and\Gamma\,;\,f^{\prime}:\,A\multimap
B\,\vdash\,a^{\prime}\,\leftarrow\mathsf{cut}\;a\,;\,\mathsf{send\\_channel\\_to}\,f^{\prime}\,a^{\prime};\,\mathsf{forward}\,f^{\prime}::B\lx@proof@logical@and\Gamma\,;\,f^{\prime}:\,A\multimap
B,\,a:\,A\,\vdash\,\mathsf{send\\_channel\\_to}\,f^{\prime}\,a^{\prime};\,\mathsf{forward}\,f^{\prime}::B\Gamma\,;\,f^{\prime}:B\,\vdash\,\mathsf{forward}\,f^{\prime}::B\Gamma\,;\,\cdot\,\vdash\,a::A...\Gamma\,;\,\cdot\,\vdash\,f::A\multimap
B...$

###  B.3. Recursive Session Types

####  B.3.1. Unrolling Session Types

The function unfix_session_for shown below works with a context lens N
operating on a linear context C to unroll a recursive session type Fix<F>. The
inner protocol F implements TypeApp< Fix <F> >. The target linear context
N::Target is the result of replacing Fix<F> in C with F::Applied, which is
then given as the linear context for the continuation session cont.

⬇

fn unfix_session_for

< N, C: Context, A: Protocol, F: Protocol >

( n : N, cont : PartialSession < N::Target, A > )

-> PartialSession < C, A >

where

F : TypeApp < Fix < F > >,

N : ContextLens < C, Fix < F >, F::Applied, >

Using unfix_session_for, we can define a client stream_client to consume the
Counter channel offered by stream_producer as follows:

⬇

1fn stream_client () -> Session < ReceiveChannel < Counter, End > >

2{ receive_channel ( | stream | {

3 unfix_session_for ( stream,

4 receive_value_from ( stream,

5 async move | count | {

6 println!("Received value: {}", count);

7 include_session ( stream_client (),

8 | next | {

9 send_channel_to ( next, stream,

10 forward ( next ) ) }) }) ) }) }

The function stream_client is a bit more involved than stream_server, so we will go through the steps one line at a time. In the first line, we define stream_client to be a recursive function with no argument that returns a Session< ReceiveChannel<Counter, End> >. In other words stream_client receives a channel of type Counter and then terminates. In the body, receive_channel is used to receive the Counter channel, and the continuation closure binds the stream context lens for accessing the Counter channel in the linear context. Inside the continuation closure (line 3), the expression unfix_session_for (...) have the type PartialSession< (Counter, ()), End >, so it has one channel of session type Counter in the linear context. Using the context lens stream, unfix_session_for(stream, ...) unrolls Counter so that the continuation at line 4 have the type PartialSession< (SendValue<u64, Counter>, ()), End >. With the recursive session type unrolled, receive_value_from(stream, ...) can be used to receive the integer value from SendValue<u64, Counter> using the stream context lens. In line 5 the continuation closure async move | count | {...} is passed to receive_value_from to bind the received integer value to count. Then the next line prints out the received count using println!.

Following that in line 7, the return type for the continuation closure has to
be PartialSession< (Counter, ()), End >, which is the same type as in line 3.
Experienced functional programmers may recognize that we could have done some
inner recursion to get back to line 3, however this is not an option in Rust
as inner recursive closure is not supported. Instead, we need to find some way
of doing recursion back to stream_client, but this would require some way of
converting the Rust type from Session< ReceiveChannel<Counter, End> > to
PartialSession< (Counter, ()), End >. To do this, in line 7 we first call
include_session(stream_client (), ...) to include a new copy of Session<
ReceiveChannel<Counter, End> > into the current continuation by recursively
calling stream_client(). The new channel is then bound to next inside the
continuation closure in line 8. Following that, the continuation at line 9
have the type PartialSession< (Counter, (ReceiveChannel<Counter, End>, ())),
End >, with the context lens stream bound to the first channel Counter and the
context lens next bound to the second channel ReceiveChannel < Counter, End >.
Comparing the session types of stream and next, we can notice that stream can
be sent to next, which can be done using send_channel_to(stream, next, ...).
Finally in the continuation in line 10, we need to produce the Rust type
PartialSession< (Empty, (End, ())), End >. This can be done by forwarding the
channel next using forward ( next ), which then the empty linear context
allows completion of the Ferrite program.


plain arabic

# Meta-Interpolation: Time-Arbitrary Frame Interpolation via Dual Meta-
Learning

Shixing Yu2, Yiyang Ma2, Wenhan Yang2, Wei Xiang3, Jiaying Liu21 3Bigo,
Beijing, China 1Corresponding author.
This work is supported by the National Key Research and Development Program of
China under Grant No. 2018AAA0102702, the National Natural Science Foundation
of China under Contract No.62172020, State Key Laboratory of Media Convergence
Production Technology and Systems, and Key Laboratory of Science, Technology
and Standard in Press Industry (Key Laboratory of Intelligent Press Media
Technology). 2Wangxuan Institute of Computer Technology, Peking University,
Beijing, China

###### Abstract

Existing video frame interpolation methods can only interpolate the frame at a
given intermediate time-step, e.g. 1/2. In this paper, we aim to explore a
more generalized kind of video frame interpolation, that at an arbitrary time-
step. To this end, we consider processing different time-steps with adaptively
generated convolutional kernels in a unified way with the help of meta-
learning. Specifically, we develop a dual meta-learned frame interpolation
framework to synthesize intermediate frames with the guidance of context
information and optical flow as well as taking the time-step as side
information. First, a content-aware meta-learned flow refinement module is
built to improve the accuracy of the optical flow estimation based on the
down-sampled version of the input frames. Second, with the refined optical
flow and the time-step as the input, a motion-aware meta-learned frame
interpolation module generates the convolutional kernels for every pixel used
in the convolution operations on the feature map of the coarse warped version
of the input frames to generate the predicted frame. Extensive qualitative and
quantitative evaluations, as well as ablation studies, demonstrate that, via
introducing meta-learning in our framework in such a well-designed way, our
method not only achieves superior performance to state-of-the-art frame
interpolation approaches but also owns an extended capacity to support the
interpolation at an arbitrary time-step.

##  I Introduction

Video frame interpolation creates non-existent intermediate frames of the
input video and maintains the newly generated video to be continuous spatially
and temporally and to have a pleasing visual effect. The technique has been
studied widely and becomes a hot research topic in the video processing
community. Its applications range from frame rate up conversion [1, 2], novel
view synthesis [3], and inter prediction in video coding [4, 5].

Conventional frame interpolation estimates the optical flow of the input
frames first, then infer the optical flow at the intermediate time-step, and
finally warp the input pixels to the target ones under the guidance of the
optical flow [6, 7]. This kind of method heavily relies on optical flow
estimation, therefore their performance is unstable if there are large motions
as the optical flow estimation is usually inaccurate in this case.
Furthermore, the conventional optical flow estimation might be time-consuming,
therefore the complexity of these methods is usually high.

Nowadays, convolutional neural networks (CNN) have been applied to
synthesizing the intermediate frames, achieving promising performance in
visual quality and time efficiency. All methods can be classified into three
branches: 1) direct generation [8] that takes the original frames as input and
directly predicts the intermediate frames; 2) flow-guided method [9, 10, 11,
12] that simulates the align-and-synthesis paradigm; 3) adaptive kernel based
method [13, 14, 15] that adopts a flow-free pipeline, where the convolution
kernels are learned by passing the original frames through a CNN.

All previous methods come across several neglected issues. 1) It is a dilemma
whether to adopt the optical flow or not. Optical flow estimation is an
effective representation of motion modeling. However, its estimation is not
robust when large motions are included. 2) Due to the fixed model parameters,
all methods can only be applied to handling the interpolation at the given
time-step adopted in the training stage, e.g. 1/2. 3) As all modules adopt
fixed parameters, their adaptivity is not enough to handle different contents
and motion conditions accurately and robustly.

Recently, meta-learning, is introduced to increase the model’s adaptivity via
adjusting the model based on the testing conditions and content of the input
images/videos for many computer vision and image processing tasks [16, 17, 18,
19, 20]. These works motivate us to address the above-mentioned issues via
meta-learning.

In our work, we still follow the paradigm of align-and-synthesis and aim to
realize the time-arbitrary video frame interpolation with accurate and robust
modeling of motions. First, to achieve the time-arbitrary video frame
interpolation, we build a meta-learned frame interpolation module that takes
both the optical flow and time-step as the input for generating the
convolutional kernels used in the convolution operations adaptively to produce
the predicted intermediate frame. Besides, it also makes our model more
adaptive to different motion contexts and contents when meta-learned adaptive
convolutions are introduced. Second, a meta-learned flow refinement module is
introduced to improve the accuracy of the optical flow estimation based on the
down-sampled version of the input frames. As shown in the visual results and
the quantitative results, the proposed method outperforms state-of-the-art
methods and can provide a better solution for the arbitrary-time video frame
interpolation.

Figure 1:  The framework of our proposed method. (a) The content-aware meta-
learned flow refinement module improves the accuracy of the optical flow
estimation based on the down-sampled version of the input frames. (b) With the
refined optical flow and the time-step as the input, the motion-aware meta-
learned frame interpolation module generates the pixel-wise convolutional
kernels, used to fuse the coarse warped version of the input frames for the
frame interpolation.

##  II Meta-Interpolation

###  II-A Motivations

To address the issues mentioned in introduction, we have the following three
motivations in our minds to guide our architecture design:

  * •

Improving robustness and accuracy of motion estimation. Our architecture
inherits the align-and-synthesis paradigm due to the excellent performance
this kind of method offers. Furthermore, we hope to inject the motion
awareness mechanism into the motion estimation.

  * •

Providing feasibility to decide the time-step of the generated intermediate
frame in a flexible way. It is generally preferred if the framework can take
the time-step as the input to support time-arbitrary interpolation under the
control of an input time-step.

  * •

Improving adaptivity by making the developed architecture more aware of
content and motion contexts. Thus, in our architecture, we hope to integrate
time-step utilization, motion estimation and context perception jointly.

###  II-B Arbitrary-Time Interpolation

Conventional video frame interpolation techniques usually take two adjacent
frames $I_{t}$ and $I_{t+1}$ as input and generate the intermediate frame at a
given time-step, usually $I_{t+0.5}$. The process $\Phi_{f}(\cdot)$ can be
formulated as follows,

$\footnotesize I_{t+0.5}=\Phi_{fixed}(I_{t},I_{t+1}|\theta_{f}),$ (1)

where $\theta_{f}$ is the model parameter and $fixed$ means the fixed time-
step. For most of the existing deep-learning video frame interpolation
methods, they can only predict the target frame at that time-step.

For the time-arbitrary video frame interpolation problem, the time-step is
relaxed as a controllable input, instead of a fixed number, which can be
adjusted freely in the testing phase. Eqn. (1) can be extended as follows,

$\footnotesize
I_{t+\alpha}=\Phi_{arbitrary}(I_{t},I_{t+1},\alpha|\theta_{a}),\text{
}0<\alpha<1,$ (2)

where $\theta_{a}$ is the model parameter and $arbitrary$ means the arbitrary
time-step. Many traditional interpolation methods, can be applied to generate
the intermediate frame at an arbitrary time-step. In our work, we hope to give
deep models such capacity via exploiting the power of meta-learning. That is,
the model parameters used for interpolation can be dynamically decided in the
testing phase based on the input context, e.g. frame context, optical flow,
and time-step.

###  II-C Framework Overview

The architecture of our framework is briefly shown in Fig. 1. In general, the
proposed method views video frame interpolation as the result of a
convolutional neural network, adaptive instead of fixed based on the testing
context.

Specifically, we develop a dual meta-learned frame interpolation framework to
synthesize the intermediate frame $I_{t+\alpha}$ with the guidance of optical
flow $f_{t+\alpha,t}$ and $f_{t+\alpha,t+1}$ as well as taking the time-step
$\alpha$ as side information. The total pipeline consists of a content-aware
meta-learned flow refinement module and a motion-aware meta-learned frame
interpolation module. In the next part, these two modules will be further
introduced in detail.

###  II-D Content-Aware Meta-Learned Flow Refinement

1) Initial Optical Flow Estimation. We first aim to obtain a good motion
estimation, fully considering the content information and the time-step. To
this end, we construct a coarse-to-fine optical flow estimation and refinement
pipeline. We first estimate the initial optical flow at both the original
resolution and half resolution spaces. The flow estimation process is denoted
by $E(\cdot)$, then we obtain the optical flow estimations as follows:

$\footnotesize\begin{split}f_{t\to
t+1}^{1/2}=U\left(E\left(I_{t}^{1/2},I_{t+1}^{1/2}\right)\right),\\\ f_{t+1\to
t}^{1/2}=U\left(E\left(I_{t+1}^{1/2},I_{t}^{1/2}\right)\right),\end{split}$
(3) $\footnotesize\begin{split}f_{t\to t+1}=E\left(I_{t},I_{t+1}\right),\\\
f_{t+1\to t}=E\left(I_{t+1},I_{t}\right),\end{split}$ (4)

where $U(\cdot)$ is the up-sampling process that projects the frame/flow back
to the original resolution space.

2) Linear Blend of Flows. Then, based on [21], the flow related to the time-
step $\alpha$ can be inferred by the linear blend:

$\displaystyle\hat{f}_{t+\alpha\to t}$
$\displaystyle=-(1-\alpha)\alpha{{f}_{t\to t+1}}+\alpha^{2}{{f}_{t+1\to t}},$
(5) $\displaystyle\hat{f}_{t+\alpha\to t+1}$
$\displaystyle=(1-\alpha)^{2}{{f}_{t\to t+1}}-\alpha(1-\alpha){{f}_{t+1\to
t}}.$

3) Flow Kernel Generation. After that, we refine the flow estimation based on
these initial estimations. In our method, following [21], we seek to utilize
the multi-scale information to realize the optical flow information
refinement.

The estimated optical flows obtained from $f_{t\to t+1}^{1/2}$ and $f_{t+1\to
t}^{1/2}$ are utilized to provide the context information to infer the refined
version of optical flows at the original resolution space. Through a cascaded
convolutional and fully connected layers, for each location $(i,j)$, input
tensors $s_{t+\alpha\to t}^{r},s_{t+\alpha\to t+1}^{r}\in R^{6\times hw}$ (the
superscript $r$ denoting refinement) with both optical flows and time-step
information is collected in the following way:

$\displaystyle s_{t+\alpha\to t}^{r}(i,j)=s_{t+\alpha\to t+1}^{r}(i,j)=$ (6)
$\displaystyle(f^{1/2}_{t\rightarrow t+1}(i,j),f^{1/2}_{t+1\rightarrow
t}(i,j),\alpha,1-\alpha).$

After that, the refined side information matrices $s^{r}_{t+\alpha\to t}$ and
$s^{r}_{t+\alpha\to t+1}$ are calculated and feed-forwarded to the given
network $K(\cdot)$ to derive the convolutional kernels $k_{t+\alpha\rightarrow
t}^{r}$ and $k_{t+\alpha\rightarrow t+1}^{r}$ as follows,

$\footnotesize k_{t+\alpha\to t}^{r}=K\left(s^{r}_{t+\alpha\to
t}\right),k_{t+\alpha\to t+1}^{r}=K\left(s^{r}_{t+\alpha\to t+1}\right).$ (7)

4) Feature Extraction. Different from previous methods, in our method, the
convolution is not operated directly on the input images. Our adaptive
convolution operations are applied to the extracted features $a_{t+\alpha\to
t}^{r}$, which are generated via the cascaded convolutional layers $A(\cdot)$
as follows,

$\footnotesize a_{t+\alpha\to t}^{r}=A\left(\hat{f}_{t+\alpha\to t}\right).$
(8)

5) Meta-Flow Refinement. Finally, the refined optical flows are inferred as
follows:

$\displaystyle\footnotesize f^{r}_{t+\alpha\rightarrow t}=a_{t+\alpha\to
t}^{r}\otimes k_{t+\alpha\to t}^{r},$ (9)

where $\otimes$ denotes the convolutional operation.

###  II-E Motion-Aware Meta-Learned Frame Prediction

Based on guidance of refined optical flows, we perform the video frame
interpolation.

1) Initial Frame Warping. Based on the initial estimated flows, we can obtain
the coarsely warped results as follows,

$\displaystyle\footnotesize\hat{I}_{t+\alpha,t}$
$\displaystyle=W(I_{t},f^{r}_{t+\alpha\to t}),$ (10)
$\displaystyle\hat{I}_{t+\alpha,t+1}$
$\displaystyle=W(I_{t+1},f^{r}_{t+\alpha\to t+1}),$

where $W(\cdot)$ is backward warping function, which can be implemented by
bilinear interpolation and is differentiable [10].

2) Frame Kernel Generation. We use optical flow information and time stamp as
side information to predict the convolutional filters. For each spatial
location $(i,j)$, we create the input tensor as follows:

$\footnotesize n_{t+\alpha}^{r}(i,j)=(f^{r}_{t+\alpha\to
t}(i,j),f^{r}_{t+\alpha\to t+1}(i,j),\alpha,1-\alpha),$ (11)

where $f^{r}_{t+\alpha\rightarrow t}(i,j)$ is the refined optical flow
extracted from $I_{t+\alpha}$ to $I_{t}$ at location $(i,j)$ including two
flow fields. The superscripted $f^{r}$ which denotes the optical flow is a
refined version generated from the motion-aware meta-learned optical flow
refinement in Sec. 3.4.

After that, the refined side information matrice $n_{t+\alpha}^{r}$ is feed-
forwarded to the given network $V(\cdot)$ to derive the convolutional kernels
$v_{t+\alpha}^{r}$ as follows,

$\footnotesize v_{t+\alpha}^{r}=V\left(n^{r}_{t+\alpha}\right).$ (12)

3) Meta-Frame Regression. To present a coarse-to-fine process, we concatenate
the initial warped frames $\hat{I}_{t+\alpha,t}$ and $\hat{I}_{t+\alpha,t+1}$
together with the original input $I_{t}$, $I_{t+1}$ and extract features
through a cascaded convolutional layer $A_{final}(\cdot)$ similar to the
feature extraction layer in Sec 3.4 and generate feature maps as follows:

$\footnotesize
a_{t+\alpha}^{final}=A_{final}\left(\hat{I}_{t+\alpha,t},\hat{I}_{t+\alpha,t+1},I_{t},I_{t+1}\right).$
(13)

Finally, we infer the video frame interpolation result as follows:

$\footnotesize\hat{I}^{r}_{t+\alpha}=a_{t+\alpha}^{final}\otimes
v_{t+\alpha}^{r}.$ (14)

where $\otimes$ denotes the convolutional operation.

###  II-F Implementation Details

1) Architecture Details. We choose residual dense network (RDN) [22] as the
feature extraction network. For frame kernel generation, the network consists
of 12 residual dense blocks. Each RDB consists of 8 convolution layers and 64
channels. In the flow kernel generation stage, the network’s corresponding
hyper-parameters are 4, 4, 32. The weight prediction networks for both flow
refinement and frame interpolation consist of 2 convolution layers, 2 fully
connected layers. We use the state-of-the-art optical flow prediction
algorithm PWC-Net [23] for optical flow estimation.

2) Loss Function. We denote the ground-truth frame as $I_{t+\alpha}$ and our
prediction as $\hat{I}_{t+\alpha}^{r}$. The reconstruction loss is defined as
follows:

$\footnotesize L_{r}=\left\|\hat{I}_{t+\alpha}^{r}-I_{t+\alpha}\right\|_{1}.$
(15)

For optical flow refinement, we introduce three kinds of losses: flow loss
$L_{f}$, warping loss $L_{w}$, and smooth loss $L_{s}$. The flow loss $L_{f}$
plays a role of regularization for the optical flow refinement by enforcing
the consistency between the refined optical flow and the original optical flow
(estimated directly by a pretrained model). It is denoted as follows:

$\footnotesize L_{f}=\left\|f_{t+\alpha\to t}^{r}-f_{t+\alpha\to
t}^{p}\right\|_{1}+\left\|f_{t+\alpha\to t+1}^{r}-f_{t+\alpha\to
t+1}^{p}\right\|_{1}.$ (16)

The warping loss $L_{w}$ regularizes the consistency between the warped frame
and adjacent frame, implicitly instructing the refinement of the optical flow.
It is represented as follows,

$\displaystyle\footnotesize L_{w}=$
$\displaystyle\left\|W\left(I_{t},f^{p}_{t+\alpha\to
t}\right)-W\left(I_{t},f^{r}_{t+\alpha\to t}\right)\right\|_{1}+$ (17)
$\displaystyle\left\|W\left(I_{t+1},f^{p}_{t+\alpha\to
t+1}\right)-W\left(I_{t},f^{r}_{t+\alpha\to t+1}\right)\right\|_{1}.$

The commonly used smoothness loss $L_{s}$ is also adopted to suppress the
artifacts in the generated optical flow estimations as follows,

$\footnotesize L_{s}=\left\|\Delta\left(f^{r}_{t+\alpha\to
t}\right)\right\|_{1}+\left\|\Delta\left(f^{r}_{t+\alpha\to
t+1}\right)\right\|_{1},$ (18)

where $\Delta(\cdot)$ is the gradient operator.

In summary, our total training loss is given as follows,

$\footnotesize
L=\lambda_{r}L_{r}+\lambda_{f}L_{f}+\lambda_{w}L_{w}+\lambda_{s}L_{s},$ (19)

where $\lambda_{r}$, $\lambda_{f}$, $\lambda_{w}$ and $\lambda_{s}$ are the
weighting parameters that balance the importance of each term. We empirically
set $\lambda_{r}=1.0$ $\lambda_{f}=0.02$, $\lambda_{w}=0.2$ and
$\lambda_{s}=0.5$.

TABLE I: Quantitative evaluation on the UCF-101, Vimeo90K and Middlebury datasets. The best results are denoted in red while the second best results are denoted in blue. Network | Vimeo90K |  | UCF-101 | MiddleBury
---|---|---|---|---
PSNR | SSIM |  | PSNR | SSIM | IE
MIND [24] | 33.50 | 0.9429 |  | 33.93 | 0.9661 | 3.35
DVF[9] | 31.54 | 0.9462 |  | 34.12 | 0.9631 | 7.75
SepConv [15] | 33.79 | 0.9702 |  | 34.69 | 0.9669 | 2.28
SuperSlomo [10] | 33.53 | 0.9653 |  | 34.26 | 0.9664 | 2.47
CyclicGen [25] | 32.10 | 0.9492 |  | 35.11 | 0.9681 | 2.86
CAIN [26] | 34.40 | 0.9715 |  | 34.87 | 0.9685 | 2.28
DAIN [14] | 34.70 | 0.9756 |  | 35.00 | 0.9683 | 2.04
MIN | 34.80 | 0.9761 |  | 35.05 | 0.9687 | 2.08

()

()

()

()

()

()

Figure 2:  Visual result of the video frame interpolation by different
methods. (a) and (b): original inputs $I_{0}$ and $I_{1}$. (c)$\to$(e): the
results produced by SepConv [15], DAIN [14], CAIN [26], respectively. (f): the
result of our method (MIN).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3:  Visual comparison of the proposed method with different components.
(a): Overlapped input. (b): MIN-Base: directly feed-forwarding the input
through an RDN. (c): MIN-UNR: MIN-Base + motion-aware meta-learned model. (d):
SepConv [15]. (e): MIN-Full: our full version. (f): Ground truth.  TABLE II:
Ablation study for
the motion-aware meta-learned frame prediction and
the content-aware meta-learned optical flow refinement.
The best results are denoted in red. Network | Metrics | Vimeo | UCF-101
---|---|---|---
SepConv [15] | PSNR | 33.79 | 34.69
SSIM | 0.9702 | 0.9669
MIN-Base | PSNR | 34.30 | 34.49
SSIM | 0.9735 | 0.9670
MIN-UNR | PSNR | 34.47 | 34.61
SSIM | 0.9746 | 0.9674
MIN-UNC | PSNR | 34.62 | 34.92
SSIM | 0.9750 | 0.9682
MIN-Full | PSNR | 34.80 | 35.05
SSIM | 0.9761 | 0.9687

(a) $f_{t\rightarrow t+1}$

(b) $\hat{f}_{t+\alpha\rightarrow t+1}$

(c) $f^{UNC}_{t+\alpha\rightarrow t+1}$

(d) $f^{r}_{t+\alpha\rightarrow t+1}$

(e) $f_{t+1\rightarrow t}$

(f) $\hat{f}_{t+\alpha\rightarrow t}$

(g) $f^{UNC}_{t+\alpha\rightarrow t}$

(h) $f^{r}_{t+\alpha\rightarrow t}$

Figure 4:  Visualization of the optical flow results of different versions of
our method.

##  III Experiments

1) Datasets and Metrics. We train our proposed video frame interpolation
method on Vimeo90K dataset [12] and validate on UCF101 [27], Vimeo90K [12] and
Middlebury benchmark [6]. Following the setting in [6], We also report
Interpolation Error (IE) on Middlebury benchmark.

2) Training Strategies and Hyper-Parameter Setting. We train the network for
20 epochs with a mini-batch size of 2. The initial learning rate is set to
0.001 and a reduce on plateau strategy. Adam[28] optimzer is used to update
the network parameters, with $\beta_{1}=0.9,\beta_{2}=0.999$.

3) Evaluation. We compare the performance of our approach against several SotA
methods on quantity and quality. The results are shown in Table I and Fig. 2.
Our method is denoted as MIN. As we can see in Table I, our method outperforms
almost all the state-of-the-art methods in all metrics.

4) Ablation Studies. We analyze the effectiveness of each component and
constraint of our method in Table II.

  * •

Motion-Aware Meta-Learned Frame Prediction. We denote MIN-Base as the model
directly feed-forwarding the two concatenated coarse generation results
through the RDN. Meanwhile, we also compare the result of another kernel-based
method, SepConv [15].

  * •

Content-Aware Meta-Learned Flow Refinement. Then, we perform the ablation
study on the content-aware meta-learned flow refinement module. We denote the
model without the proposed optical flow refinement module as MIN-UNR, the
model that only uses the reconstruction loss $L_{r}$ in the training loss as
MIN-UNC. Besides, we visualize the optical flow to show the effectiveness of
this module in Fig. 4. In general, both of our last two versions can well
capture the motions of foreground objects.

  * •

Visual Comparisons. We show qualitative results on one image selected from
UCF101 in Fig. 3. Our method successfully generates spatially consistent
result in Fig. 3 (e).

##  IV Conclusions

In this paper, we develop a dual meta-learned frame interpolation framework
that is capable to synthesize the intermediate frame at an arbitrary
intermediate time-step. First, we create a content-aware meta-learned flow
refinement module that takes the down-sampled version of the input frames as
input to improve the accuracy and robustness of the optical flow estimation.
Second, a motion-aware meta-learned frame interpolation module generates the
convolutional kernels to generate the interpolated frame based on the refined
optical flows and the time-step. Extensive qualitative and quantitative
evaluations demonstrate the superiority of our method and the effectiveness of
each component of our method.

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