# Transport coefficients of heavy quarkonia comparing with heavy quark
coefficients

Juhee Hong  Department of Physics and Institute of Physics and Applied
Physics, Yonsei University, Seoul 03722, Korea  Su Houng Lee  Department of
Physics and Institute of Physics and Applied Physics, Yonsei University, Seoul
03722, Korea

###### Abstract

We revisit the transport coefficients of heavy quarkonia moving in high-
temperature QCD plasmas. The thermal width and mass shift for heavy quarkonia
are closely related to the momentum diffusion coefficient and its dispersive
counterpart for heavy quarks, respectively. For quarkonium at rest in plasmas
the longitudinal gluon part of the color-singlet self-energy diagram is
sufficient to determine the leading-order thermal width, whereas the momentum
dependence is obtained from the transverse gluon channel. Using the
quarkonium-gluon effective vertex based on the dipole interaction of color
charges, we discuss the damping rate, the effective rest and kinetic mass
shifts of slowly moving quarkonia and compare with the corresponding
coefficients of heavy quarks.

##  I Introduction

The suppression of quarkonium production has been predicted to indicate the
formation of quark-gluon plasmas in relativistic heavy-ion collisions
Matsui:1986dk . In a hot medium, color-screening prevents heavy quark-
antiquark pairs from binding and bound states are dissociated by medium
interactions. The dynamics of heavy quarkonia are characterized by transport
coefficients such as the thermal width and mass shift. These two coefficients
correspond to thermal corrections to the imaginary and real parts of the
static potential for a heavy quark-antiquark pair, respectively Laine:2006ns ;
Brambilla:2008cx .

In high-temperature plasmas, dissociation of a bound state such as
$\Upsilon(1S)$ is described mostly by inelastic parton scattering Hong:2018vgp
. At leading order in a weak coupling expansion, the thermal width is related
to the momentum diffusion coefficient of heavy quark Brambilla:2013dpa ;
Brambilla:2016wgg . Medium interactions of heavy quarks are described by the
diffusion coefficient and its dispersive counterpart Brambilla:2019tpt . In
this work, we investigate the transport coefficients of heavy quarkonia,
especially bound states with finite spatial momentum with respect to plasmas.
Although the transport coefficients for heavy quarks and quarkonia can be
extracted from lattice QCD computations, current lattice data have large
uncertainties and are not practical for measuring the momentum dependence of
the heavy quark-antiquark potential. Furthermore, it would be useful to
identify the main partonic processes contributing to the transport
coefficients.

In previous studies, the momentum dependence of the thermal width has been
considered phenomenologically in a moving medium because most theoretical
investigations have been carried out in the rest frame of a bound state. We
have computed the thermal width by calculating the dissociation cross section
in the quarkonium rest frame and then convoluting parton distribution
functions given by $f(K)=(e^{K\cdot V/T}\pm 1)^{-1}$, where
$V=\frac{1}{\sqrt{1-v^{2}}}(1,\bm{v})$ Hong:2019ade . An effective field
theory approach has also been employed to calculate the thermal width
Escobedo:2013tca . In the kinematical regime $T\gg\frac{1}{r}\gg m_{D}\gg E$,
they have taken into account dynamic screening Weldon:1982aq ; Chu:1988wh
(which depends on velocity) but ignored the energy transfer between a bound
state and plasma. The energy transfer can be neglected at leading order for
quarkonium at rest, but both energy and momentum transfer are required to
evaluate the thermal width of bound states moving in quark-gluon plasmas.

To understand dynamic properties of heavy particles in hot QCD plasmas, there
have been many investigations and they need to be brought together to study
transport processes of heavy quarkonia. The goal of this paper is to compute
the transport coefficients for heavy quarkonium depending on its velocity and
to explain the difference from the heavy quark coefficients which have been
investigated in literature. Our analysis is based on the leading quarkonium-
gluon interaction Oh:2001rm  which has been derived from the Bethe-Salpeter
amplitude using the dipole interaction of color charges Bhanot:1979vb . In
Section II, we calculate the velocity dependence of the thermal width of
$\mathcal{O}(g^{4}T^{3}r^{2})$ through inelastic parton scattering. In Section
III, the rest and kinetic mass shifts for slowly moving quarkonia are
determined at leading order $\mathcal{O}(g^{4}T^{3}r^{2})$ and next-to-leading
order $\mathcal{O}(g^{5}T^{3}r^{2})$. In Section IV, we revisit the damping
rate of nonrelativistic quarkonium by using the spectral density derived from
a Schrödinger-type equation. Finally, we summarize our result in Section V.
The details on the quarkonium-gluon effective interaction are presented in
Appendix A.

##  II Thermal width

In high-temperature QCD plasmas, quarkonium dissociation occurs through
inelastic parton scattering ($\Upsilon+p\rightarrow b+\bar{b}+p$,
$p=g,q,\bar{q}$) exchanging spacelike gluons. The cross section calculated in
hard-thermal-loop (HTL) perturbation theory has been shown to agree with the
result obtained by potential nonrelativistic QCD (pNRQCD) in the kinematical
regime $\frac{1}{r}\gg T\gg m_{D}\gg E$ (where $r$ is the distance between $b$
and $\bar{b}$, and $E$ is the binding energy) Hong:2018vgp . The leading-order
dissociation is described by taking the dipole interaction of color charges
(which is valid at short $r$) and ignoring the interaction between $b$ and
$\bar{b}$ after breakup in the large $N_{c}$ limit Bhanot:1979vb . In the same
approach, we consider the thermal width for moving quarkonium and compare with
the momentum diffusion coefficient of heavy quark.

A bound state of heavy quarks and its interaction with partons can be
described by the Bethe-Salpeter amplitude in the rest frame of quarkonium (see
Appendix A) Oh:2001rm . Using the effective vertex Eq. (27), quarkonium
dissociation through inelastic parton scattering has been calculated
Hong:2018vgp :

$|\mathcal{M}|^{2}=Cg^{4}m^{2}m_{\Upsilon}\,|\nabla\psi(\bm{p})|^{2}\left[\bm{k}^{2}\frac{k_{10}^{2}}{(\bm{k}^{2}+m_{D}^{2})^{2}}\bigg{\\{}\begin{tabular}[]{cc}$(1+\cos\theta_{k_{1}k_{2}})$&\quad($q,\bar{q}$)\\\
$(1+\cos^{2}\theta_{k_{1}k_{2}})$&\quad($g$)\end{tabular}\bigg{.}\right]\,,$
(1)

where $C=16\frac{(N_{c}^{2}-1)N_{f}}{N_{c}}$, $16(N_{c}^{2}-1)$ for
$(q,\bar{q})$, $(g)$, respectively, $\psi(\bm{p})$ is the normalized wave
function for a bound state with the relative momentum,
$\bm{p}=\frac{1}{2}(\bm{p}_{1}-\bm{p}_{2})$, and
$\bm{k}=\bm{k}_{1}-\bm{k}_{2}$ is the momentum transfer. The matrix elements
are similar to those for heavy quark diffusion, $t$-channel gluon exchange
($b+p\rightarrow b+p$) weighted by the squared momentum transfer Moore:2004tg
. The thermal width of quarkonium is obtained by the following phase space
integrations Hong:2018vgp :

$\Gamma=\frac{1}{2d_{\Upsilon}q^{0}}\int\frac{d^{3}\bm{k}_{1}}{(2\pi)^{3}2k_{10}}\int\frac{d^{3}\bm{k}_{2}}{(2\pi)^{3}2k_{20}}\int\frac{d^{3}\bm{p}_{1}}{(2\pi)^{3}2p_{10}}\int\frac{d^{3}\bm{p}_{2}}{(2\pi)^{3}2p_{20}}\\\
\times(2\pi)^{4}\delta^{4}(Q+K_{1}-K_{2}-P_{1}-P_{2})|\mathcal{M}|^{2}n(k_{1})[1\pm
n(k_{2})]\,,$ (2)

where $d_{\Upsilon}$ is the degeneracy factor of quarkonium. For quarkonium at
rest, the interaction rate is proportional to the momentum diffusion
coefficient of heavy quark: $\Gamma^{\rm 1S}=3a_{0}^{2}\kappa$ for a Coulombic
bound state Brambilla:2013dpa ; Brambilla:2016wgg . Note that Eq. (1) involves
only the longitudinal part of the effective gluon propagator as the leading
contribution. Since the transverse gluon propagator couples with the energy
transfer rather than the momentum transfer, the thermal width of quarkonium
with finite momentum will be different from the momentum diffusion coefficient
of heavy quark.

To consider the relative motion between quarkonium and thermal medium, we use
$\cos\theta_{kk_{1}}=\frac{\omega}{k}+\frac{k^{2}-\omega^{2}}{2kk_{1}}$. Then
the square bracket in Eq. (1) becomes

$\left[\frac{k^{2}}{2}[4k_{1}(k_{1}-\omega)-(k^{2}-\omega^{2})]\,|D_{L}(K)|^{2}+\frac{\omega^{2}}{2k^{2}}[(2k_{1}-\omega)^{2}+k^{2}](k^{2}-\omega^{2})\,|D_{T}(K)|^{2}\right]\,$
(3)

at leading order. Due to the long-range interactions mediated by gluon,
infrared divergences arise and HTL resummation is necessary. In Coulomb gauge,
the effective propagators for soft gluon are Braaten:1989mz

$\displaystyle D_{L}(K)$ $\displaystyle=$
$\displaystyle\frac{1}{k^{2}+\Pi_{L}(K)}\,,$ $\displaystyle D_{T}(K)$
$\displaystyle=$ $\displaystyle-\frac{1}{k^{2}-w^{2}+\Pi_{T}(K)}\,,$ (4)

with HTL corrections Weldon:1982aq ; Weldon:1982bn

$\displaystyle\Pi_{L}(K)$ $\displaystyle=$ $\displaystyle
m_{D}^{2}\left[1-\frac{\omega}{2k}\left(\ln\frac{k+\omega}{k-\omega}-i\pi\right)\right]\,,$
$\displaystyle\Pi_{T}(K)$ $\displaystyle=$ $\displaystyle
m_{D}^{2}\left[\frac{\omega^{2}}{2k^{2}}+\frac{\omega(k^{2}-\omega^{2})}{4k^{3}}\left(\ln\frac{k+\omega}{k-\omega}-i\pi\right)\right]\,.$
(5)

The leading-log thermal width is determined by a method similar to that
developed for calculating the screening effect Braaten:1991dd ; Braaten:1991jj
; Braaten:1991we . The effective propagators screen the static Coulomb
interaction, eliminating infrared divergences. Although the purely static
magnetic interaction is not screened, there is dynamic screening at nonzero
frequency Weldon:1982aq : this dynamical screening cuts off infrared
divergences if divergences are only logarithmic. After angular integrations,
the thermal width is given by

$\displaystyle\Gamma(v)=\frac{1}{16(2\pi)^{5}d_{\Upsilon}q_{0}v}\int
dk_{1}\int dk\int_{-vk}^{vk}d\omega\int
dp\,\frac{p^{2}}{p_{10}^{2}}\,|\mathcal{M}|^{2}\,n(k_{1})[1\pm
n(k_{1}-\omega)]\,.$ (6)

For a Coulombic bound state, we have $\langle
r^{2}\rangle=\int\frac{d^{3}\bm{p}}{(2\pi)^{3}}|\nabla\psi_{1S}(\bm{p})|^{2}=3a_{0}^{2}$
and

$\displaystyle\Gamma^{\rm 1S}(v)$ $\displaystyle=$
$\displaystyle\frac{3Cg^{4}a_{0}^{2}}{128\pi^{3}d_{\Upsilon}v}\int
dk_{1}\,k_{1}^{2}\,n(k_{1})[1\pm n(k_{1})]$ (7) $\displaystyle\qquad\times\int
dk\int_{-vk}^{vk}d\omega\left[k^{2}|D_{L}(K)|^{2}+\frac{\omega^{2}(k^{2}-\omega^{2})}{k^{2}}|D_{T}(K)|^{2}\right]\,,$
$\displaystyle\simeq$
$\displaystyle\frac{(N_{c}^{2}-1)g^{4}T^{3}a_{0}^{2}}{8\pi
d_{\Upsilon}v}\left(1+\frac{N_{f}}{2N_{c}}\right)\ln\frac{1+v}{1-v}\,\ln\frac{T}{m_{D}}\,.$

We notice that the velocity-dependence comes from the transverse gluon part:
in Eq. (3) $\mathcal{O}(v^{2})$ terms from the longitudinal part have been
ignored compared to the leading-log term.

Figure 1:  The velocity dependence in a leading-log approximation: comparing
the thermal width of quarkonium and the momentum diffusion coefficient of
heavy quark.

The momentum diffusion coefficient of heavy quark has been computed in a
leading-log approximation Moore:2004tg

$\displaystyle\kappa(v)=\frac{C_{F}g^{4}T^{3}}{12\pi}\left(\frac{N_{f}}{2}+N_{c}\right)\left[1-\frac{1-v^{2}}{6v}\ln\left(\frac{1+v}{1-v}\right)\right]\ln\left(\frac{T}{m_{D}}\right)\,.$
(8)

In Fig. 1, we present the velocity dependence of the thermal width, comparing
with that of the heavy quark diffusion coefficient. The width increases with
quarkonium momentum, in qualitative agreement with the previous investigation
in Ref. Hong:2019ade . For small $v$,
$\frac{\Gamma(v)}{\Gamma(v=0)}=\frac{\kappa(v)}{\kappa(v=0)}=1+\frac{v^{2}}{3}+\mathcal{O}(v^{4})$:
the thermal width of moving quarkonium agrees with the momentum diffusion
coefficient of heavy quark (multiplied by $\langle r^{2}\rangle$) up to
$\mathcal{O}(v^{2})$.

Figure 2:  The self-energy diagram for heavy quark-antiquark color-singlet
with the effective gluon propagator.

Dynamics of heavy quarks and quarkonia is characterized by dissipative
transport coefficients and their dispersive counterparts. To determine the
former such as heavy quark diffusion and damping rate, we consider the
interaction rate of heavy particles. The interaction rate is proportional to
the imaginary part of the self-energy and is related to the spectral density
of soft gluon Weldon:1983jn ; Braaten:1991jj . On the other hand, dispersive
coefficients such as mass shift corresponds to the real part of the gluon
self-energy. Indeed, the thermal width and mass shift of quarkonia are given
by the imaginary and real parts of the color-singlet self-energy (see Fig. 2)
Brambilla:2008cx ; Brambilla:2016wgg .

We reexpress the thermal width in terms of the spectral density of soft gluon,
$\rho_{L,T}(K)=-\frac{1}{\pi}\,{\rm Im}\,D_{L,T}(\omega+i\epsilon,\bm{k})$:

$\Gamma^{\rm
1S}(v)=\frac{3\pi(N_{c}^{2}-1)g^{2}a_{0}^{2}}{N_{c}d_{\Upsilon}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int
d\omega\,[1+n_{B}(\omega)]\,\delta(\omega-\bm{k}\cdot\bm{v})\\\
\times\Big{[}k^{2}\,\rho_{L}(K)+2\omega^{2}\,\rho_{T}(K)\Big{]}\,,$ (9)

where quark and gluon contributions correspond to
$m_{D(q,g)}^{2}=\frac{2N_{f,c}g^{2}}{T}\int\frac{d^{3}k_{1}}{(2\pi)^{3}}\,n(k_{1})[1\pm
n(k_{1})]$ in HTL corrections. For $\omega\ll T$, the bosonic thermal
distribution is expanded as
$1+n_{B}(\omega)=\frac{T}{\omega}+\frac{1}{2}+\cdots$. Since the spectral
density is odd in $\omega$, the first term, $\frac{T}{\omega}$, contributes to
the thermal width. This limit is consistent with the soft momentum transfer in
the thermal width in Eq. (7), and $\Gamma^{\rm 1S}$ for small $v$ will be
reproduced later when discussing the damping rate. Replacing the spectral
density by the corresponding real part in Eq. (9), the second term,
$\frac{1}{2}$, produces the thermal mass shift which will be evaluated in the
next section.

##  III Mass shift

The thermal width $\Gamma\sim g^{4}T^{3}a_{0}^{2}$ is determined through
inelastic parton scattering with soft momentum transfer by using HTL resummed
perturbation theory. The thermal mass shift is considered in the same way, but
the leading contribution $\delta m_{\rm LO}^{\rm
1S}=-\frac{1}{4\pi^{2}}\zeta(3)\left(N_{f}+\frac{4}{3}N_{c}\right)g^{4}T^{3}a_{0}^{2}$
comes from momenta larger than $\mathcal{O}(gT)$ scale Brambilla:2008cx . It
turns out that soft momentum transfer yields $\delta m_{\rm NLO}\sim
g^{5}T^{3}a_{0}^{2}$ which is an $\mathcal{O}(g)$ higher term: the higher
order corrections for the heavy quark diffusion coefficient are known to be
significant at realistic values of the strong coupling constant Caron-
Huot:2007rwy . The mass shift provides thermal corrections to the dispersion
relation of heavy particles, $E_{\bm{p}}=m_{\rm
rest}+\frac{\bm{p}^{2}}{2m_{\rm kin}}$, where $m_{\rm rest}$ and $m_{\rm kin}$
are the rest mass and the kinetic mass, respectively Chesler:2009yg . For
$m\gg T$, the leading corrections to the mass-shell seem to be independent of
the spatial momentum. In this section, we discuss the rest mass shift (which
has been computed in Ref. Brambilla:2008cx ) and the kinetic mass shifts for
slowly moving quarkonia, focusing on smaller momentum transfer than incoming
parton momentum.

The dispersive counterpart of Eq. (9) is given by

$\displaystyle\delta m_{\rm NLO}^{\rm 1S}(v)$ $\displaystyle=$
$\displaystyle\frac{3(N_{c}^{2}-1)g^{2}a_{0}^{2}}{2N_{c}d_{\Upsilon}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int
d\omega\,\frac{1}{2}\,\delta(\omega-\bm{k}\cdot\bm{v})$ (10)
$\displaystyle\qquad\times\,{\rm
Re}\left[\frac{k^{2}}{k^{2}+\Pi_{L}(K)}-\frac{2\omega^{2}}{k^{2}-\omega^{2}+\Pi_{T}(K)}\right]\,.$

For small $v$, we expand the longitudinal gluon propagator:

$\displaystyle\delta m_{\rm NLO}^{\rm 1S}(v)$ $\displaystyle=$
$\displaystyle\frac{3(N_{c}^{2}-1)g^{2}a_{0}^{2}}{2N_{c}d_{\Upsilon}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int
d\omega\,\frac{1}{2}\,\delta(\omega-\bm{k}\cdot\bm{v})$ (11)
$\displaystyle\qquad\times\,k^{2}\left[\frac{1}{k^{2}+m_{D}^{2}}+\frac{m_{D}^{2}\omega^{2}}{(k^{2}+m_{D}^{2})^{3}}+\frac{(4-\pi^{2})m_{D}^{4}\omega^{2}}{4k^{2}(k^{2}+m_{D}^{2})^{3}}+\cdots\right]\,,$
$\displaystyle=$
$\displaystyle\frac{3(N_{c}^{2}-1)g^{2}m_{D}^{3}a_{0}^{2}}{16\pi
N_{c}d_{\Upsilon}}\left[1-\frac{1}{2}\left(1+\frac{\pi^{2}}{16}\right)v^{2}\right]+\mathcal{O}(v^{4})\,,$

which has been evaluated in dimensional regularization. Subtracting from the
integrand the corresponding unresummed propagator
($\frac{1}{k^{2}-\omega^{2}+\Pi_{T}(K)}-\frac{1}{k^{2}-\omega^{2}}$), we can
show that the transverse gluon contributes to $\mathcal{O}(v^{4})$. The first
and second terms in the last line of Eq. (11) correspond to the rest mass
shift and the kinetic mass shift, respectively. We note that the effective
kinetic mass shift is not equal to the rest mass shift. Similarly, the
dispersion relation of heavy quark has been discussed in Ref. Chesler:2009yg :

$\delta
m_{HQ}=-\frac{C_{F}g^{2}m_{D}}{8\pi}+\frac{1}{2}\left(1-\frac{\pi^{2}}{16}\right)\frac{C_{F}g^{2}m_{D}}{24\pi}v^{2}\,.$
(12)

In comparison to heavy quark $\delta m_{HQ}\sim g^{3}T$, the thermal mass
shift of quarkonium is suppressed by a factor $\sim(m_{D}a_{0})^{2}$ in the
case of soft momentum transfer.

The leading-order mass shift of moving quarkonia is

$\delta m_{\rm LO}^{\rm
1S}(v)=-\frac{3(N_{c}^{2}-1)g^{2}a_{0}^{2}}{2N_{c}d_{\Upsilon}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int
d\omega\,\frac{1}{2}\,\delta(\omega-\bm{k}\cdot\bm{v})\left[\frac{{\rm
Re}\,\Pi_{L}(K)}{k^{2}}+\frac{2\omega^{2}{\rm
Re}\,\Pi_{T}}{(k^{2}-\omega^{2})^{2}}\right]\,.$ (13)

Here, we expand the gluon self-energy Kajantie:1982xx ; Heinz:1986kz ;
Kapusta:2006pm  in $\omega$:

$\displaystyle\Pi_{L}(K)$ $\displaystyle=$
$\displaystyle\frac{g^{2}N_{f}}{2\pi^{2}}\int_{0}^{\infty}dk_{1}\,k_{1}n_{F}(k_{1})\left[2+\left(\frac{2k_{1}}{k}-\frac{k}{2k_{1}}\right)\ln\frac{k+2k_{1}}{k-2k_{1}}\right.$
(14)
$\displaystyle\qquad\left.-\frac{\omega^{2}}{k^{2}}\left(\frac{2(k^{2}-8k_{1}^{2})}{k^{2}-4k_{1}^{2}}-\frac{k}{2k_{1}}\ln\frac{k+2k_{1}}{k-2k_{1}}\right)+\cdots\right]$
$\displaystyle+\frac{g^{2}N_{c}}{8\pi^{2}}\int_{0}^{\infty}dk_{1}\,k_{1}n_{B}(k_{1})\left[8-\frac{4k^{2}}{k_{1}^{2}}+\left(\frac{8k_{1}}{k}-\frac{4k}{k_{1}}+\frac{k^{3}}{k_{1}^{3}}\right)\ln\frac{k+2k_{1}}{k-2k_{1}}\right.$
$\displaystyle\qquad\left.-\frac{\omega^{2}}{k^{2}}\left(\frac{3k^{6}-8k^{4}k_{1}^{2}-112k^{2}k_{1}^{4}+256k_{1}^{6}}{k_{1}^{2}(k^{2}-4k_{1}^{2})^{2}}+\frac{5k(k^{2}-k_{1}^{2})^{2}}{2k_{1}^{5}}\ln\frac{k+k_{1}}{|k-k_{1}|}\right.\right.$
$\displaystyle\qquad\left.\left.-\frac{k(5k^{4}-12k^{2}k_{1}^{2}+24k_{1}^{4})}{4k_{1}^{5}}\ln\frac{k+2k_{1}}{k-2k_{1}}\right)+\cdots\right]\,,$

where the first and second integrals correspond to quark and gluon
contributions, respectively. The $\mathcal{O}(\omega^{0})$ and
$\mathcal{O}(\omega^{2})$ terms yield the rest mass shift and the relative
correction, respectively, while the transverse part is subleading. If the
momentum transfer is smaller than the incoming parton momentum, we have

$\displaystyle{\rm Re}\,\Pi_{L}(K)$ $\displaystyle\sim$
$\displaystyle\frac{2g^{2}N_{f}}{\pi^{2}}\int_{0}^{\infty}dk_{1}\,k_{1}n_{F}(k_{1})\left(1-\frac{\omega^{2}}{k^{2}}\right)$
(15)
$\displaystyle\qquad+\frac{2g^{2}N_{c}}{\pi^{2}}\int_{0}^{\infty}dk_{1}\,k_{1}n_{B}(k_{1})\left(1-\frac{\omega^{2}}{k^{2}}\right)+\mathcal{O}\left(\frac{k^{2}}{k_{1}^{2}},\omega^{4}\right)\,,$

and the thermal mass shift is estimated to be $\delta m_{\rm
rest}(1-\frac{v^{2}}{3})$. However, we need to use the resummed propagators
when $k\sim gT$.

##  IV Damping rate

The static potential for a heavy quark-antiquark pair has been computed at
high temperature, $T\gtrsim\frac{1}{r}\sim mg^{2}$ Laine:2006ns ;
Brambilla:2008cx . Employing the potential, the quarkonium contribution to the
spectral function of the electromagnetic current has been numerically computed
Laine:2007gj ; Burnier:2007qm . Especially for $T\gg\frac{1}{r}\sim m_{D}$,
the real part $\mathcal{O}(g^{2}m_{D})$ of the potential is smaller than the
imaginary part $\mathcal{O}(g^{2}T)$ (whose $m_{D}r\ll 1$ expansion yields
$\Gamma\sim g^{2}T(m_{D}r)^{2}$ discussed in Section II), so the heavy quark-
antiquark pair is expected to decay before forming a bound state
Brambilla:2008cx . It has been noticed that the real part correction coincides
with the free energy of static heavy quark-antiquark Gava:1981qd ;
Petreczky:2005bd  (or twice the rest mass shift of heavy quark), while the
imaginary part approaches twice the heavy quark damping rate Pisarski:1988vd ;
Pisarski:1993rf  (with minus sign) in the large $r$ limit Beraudo:2007ky ;
Brambilla:2008cx . The heavy quark propagator can be approximated by its
nonrelativistic particle pole:

$D_{b}\left(P\pm\frac{Q+K}{2}\right)\rightarrow\frac{\pm i(1\pm\not{V})}{2(\pm
p_{0}+\frac{\omega-\bm{k}\cdot\bm{v}}{2}-\frac{E}{2}-\frac{\bm{p}^{2}}{2m}+i\frac{\Gamma_{b}}{2})}\,,$
(16)

where $V=\frac{1}{\sqrt{1-v^{2}}}(1,\bm{v})$ and $\Gamma_{b}$ is the heavy
quark width. For a heavy quark-antiquark pair with the effective vertex
$V_{\rm eff}^{\mu\nu}(K)$ in Fig. 2,

$\int\frac{dp^{0}}{2\pi}D_{b}\left(P+\frac{Q+K}{2}\right)V_{\rm
eff}^{\mu\nu}(K)D_{b}\left(P-\frac{Q+K}{2}\right)V_{\rm
eff}^{\rho\sigma\dagger}(K)\\\ \simeq\frac{1+\not{V}}{2}V_{\rm
eff}^{\mu\nu}(K)\frac{1-\not{V}}{2}V_{\rm
eff}^{\rho\sigma\dagger}(K)\frac{i}{\omega-\bm{k}\cdot\bm{v}-E-\frac{\bm{p}^{2}}{m}+i\Gamma_{b}}\,,$
(17)

where the width of the quarkonium state is twice that of heavy quark
Strassler:1990nw .

In this work, we are interested in the kinematical regime where
$\frac{1}{r}\gg T$ and $m_{D}\gg E$ are satisfied so that a dipole
approximation is appropriate and quarkonium dissociation can be parallel to
heavy quark diffusion. In Eq. (17), without $\Gamma_{b}$ we get
$\delta(\omega-\bm{k}\cdot\bm{v})$ which appears throughout the paper. We have
calculated the thermal width and mass shift of quarkonium moving in quark-
gluon plasmas in the previous two sections. They provide the thermal
corrections, $\delta m-i\frac{\Gamma}{2}$, to the singlet static potential
Brambilla:2008cx ; Brambilla:2016wgg . In the following, we employ the complex
potential to estimate the spectral density of nonrelativistic quarkonium near
the threshold $\omega\sim 2m+\frac{\bm{q}^{2}}{4m}$. The result is then used
to compute the quarkonium damping rate which is comparable with the thermal
width.

As discussed in Refs. Laine:2006ns ; Laine:2007gj ; Burnier:2007qm , a heavy
quark current-current correlator
$C^{>}(X)=\langle\hat{J}^{\mu}(X)\hat{J}_{\mu}(0)\rangle$, where
$\hat{J}^{\mu}(X)=\hat{\bar{\psi}}(X)\gamma^{\mu}\hat{\psi}(X)$, satisfies a
Schrödinger-type equation with a complex potential:

$\left[i\frac{\partial}{\partial
t}-\left(2m+\frac{q^{2}}{4m}-\frac{\nabla_{\bm{r}}^{2}}{m}+V(r)\right)\right]C^{>}(t,r)=0\,.$
(18)

Although one can solve the equation numerically, we consider a simple case in
the static limit: ${\rm Re}\,V(r)\approx-\frac{g^{2}C_{F}}{4\pi r}e^{-m_{D}r}$
(plus the thermal mass shift) and $\int d^{3}\bm{r}\,{\rm
Im}\,V(r)=-\frac{\Gamma}{2}$ at short distance $r$ (we assume that only the
real part depends on $r$). Since the potential is independent of time,
$C^{>}(t,r)\propto e^{-i(E_{\bm{q}}-i\frac{\Gamma}{2})t}$, where
$E_{\bm{q}}=2m+\frac{\bm{q}^{2}}{4m}-E$. The radial part is given by a bound
state for a Debye-screened Coulombic potential,

$\left[\frac{\nabla_{\bm{r}}^{2}}{m}-{\rm Re}\,V(r)\right]\psi(r)=E\psi(r)\,.$
(19)

In the limit $r\rightarrow 0$, we take a Fourier transform with respect to
time:

$\displaystyle\rho(\omega)\propto\int_{0}^{\infty}dt\,e^{-\frac{\Gamma}{2}t}[\cos(E_{\bm{q}}t)\cos(\omega
t)+\sin(E_{\bm{q}}t)\sin(\omega t)]=\frac{\frac{\Gamma}{2}}{(\omega-
E_{\bm{q}})^{2}+\frac{\Gamma^{2}}{4}}\,.$ (20)

Retaining the leading terms in the real and ($\bm{r}$-integrated) imaginary
parts of the potential, we obtain a Breit-Wigner form of the spectral density
for a spherically symmetric wave function: the imaginary part of the potential
provides a finite width of a bound state.

The damping rate is given by the imaginary part of the self-energy at the pole
(which has an imaginary part) in the propagator, while on the mass-shell half
the interaction rate is expected Bellac:2011kqa . Since the spectral density
has been estimated in Eq. (20), we follow Ref. Pisarski:1993rf  for heavy
quark to compute the damping rate of nonrelativistic quarkonium. After
integrating a wave function and averaging over polarization,

$\frac{\Gamma}{2}=\frac{4g^{2}a_{0}^{2}T}{3}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}\int\frac{d\omega}{\omega}\int
d\omega_{\Upsilon}\frac{\frac{\Gamma}{2}}{[(\omega_{\Upsilon}-E_{\bm{q}-\bm{k}})^{2}+\frac{\Gamma^{2}}{4}]}\\\
\times\left[k^{2}\rho_{L}(K)+2\omega^{2}\rho_{T}(K)\right]\delta(\omega+\omega_{\Upsilon}-E_{\bm{q}})\,.$
(21)

We redefine $\omega_{\Upsilon}\rightarrow\omega_{\Upsilon}+E_{\bm{q}-\bm{k}}$
and $E_{\bm{q}-\bm{k}}\approx
m_{\Upsilon}+\frac{\bm{q}^{2}}{2m_{\Upsilon}}-\bm{k}\cdot\bm{v}$. Then the
delta function becomes $\delta(\omega+\omega_{\Upsilon}-\bm{k}\cdot\bm{v})$.
Since $\omega_{\Upsilon}\sim g^{4}T^{3}a_{0}^{2}$ is much smaller than the
scale of $\omega$ or $vk$, we can perform the integrations over
$\omega_{\Upsilon}$ and $\omega$ separately:
$\int_{-vk}^{vk}d\omega_{\Upsilon}\frac{\frac{\Gamma}{2}}{(\omega_{\Upsilon}^{2}+\frac{\Gamma^{2}}{4})}=2\,\arctan\left(\frac{2vk}{\Gamma}\right)$.
Finally, approximating the spectral densities in the limit $\omega\rightarrow
0$,

$\displaystyle\rho_{L}(K)$ $\displaystyle\simeq$ $\displaystyle-{\rm
Im}\,\frac{1}{\pi\big{[}k^{2}+m_{D}^{2}\big{(}1+i\frac{\pi\omega}{2k}\big{)}\big{]}}\,,$
$\displaystyle\rho_{T}(K)$ $\displaystyle\simeq$ $\displaystyle{\rm
Im}\,\frac{1}{\pi\big{(}k^{2}-i\frac{m_{D}^{2}\pi\omega}{4k}\big{)}}\,,$ (22)

we have

$\frac{\Gamma}{2}\simeq\frac{g^{2}a_{0}^{2}}{3\pi}\left(1+\frac{v^{2}}{3}\right)m_{D}^{2}T\ln\frac{T}{m_{D}}\,.$
(23)

This result agrees with the thermal width, Eq. (7), for small $v$.

Due to the quarkonium-gluon effective vertex involving momentum and energy
transfer, the long-range interactions in both longitudinal and transverse
gluon parts are screened by the Debye mass in the leading-log damping rate.
This is in contrast to the heavy quark damping rate where the transverse part
is in need of an infrared regulator such as the magnetic mass ($\sim g^{2}T$)
or the imaginary part of the pole (that is, the damping rate itself)
Thoma:1990fm ; Pisarski:1993rf . The damping rate of slow heavy quark has been
computed for $g^{\frac{2}{3}}\ll v\ll 1$:
$\gamma_{HQ}=\frac{g^{2}C_{F}T}{8\pi}\Big{(}1+\frac{v}{2}\ln\frac{\pi
m_{D}^{2}v^{3}}{4\gamma_{HQ}^{2}}\Big{)}$, where $\gamma_{HQ}$ inside the
logarithm is replaced by its value at $v=0$ Pisarski:1993rf .

##  V Summary

We have presented the velocity dependence of the thermal width and mass shift
for heavy quarkonia slowly moving in hot QCD plasmas. In the kinematical
regime where $\frac{1}{r}\gg T$ and $m_{D}\gg E$, the longitudinal gluon
contribution is same for both the thermal width and the momentum diffusion
coefficient of heavy quarks, but the transverse gluon part is different. For
nonrelativistic quarkonia, we have discussed the thermal width in relation to
the damping rate which is proportional to the imaginary part of the color-
singlet self-energy. We have also discussed the thermal corrections to the
dispersion relation, the effective rest mass shift and its correction (kinetic
mass shift) for small momentum transfer. On the other hand, at a higher
temperature regime $T\gg\frac{1}{r}$ a heavy quark-antiquark pair is likely to
decay before forming a bound state. The real and imaginary part of the static
potential approach the mass shift and the damping rate, respectively, of heavy
quark and antiquark in the large $r$ limit.

For a slowly moving bound state, the thermal width agrees with the momentum
diffusion coefficient of heavy quark up to $\mathcal{O}(v^{2})$. However, the
thermal width of nonrelativistic quarkonium is same as neither the heavy quark
diffusion coefficient nor the damping rate of heavy quark-antiquark pair in
the short $r$ limit. The relative correction to the rest mass shift for
quarkonium is also different from that for heavy quark. In studying quarkonium
dynamics, the transport coefficients of heavy quarks and quarkonia might be
relevant in different kinematical regimes. Eventually the nonperturbative
determination of the transport coefficients is desirable, while our
perturbative analysis of the velocity dependence can supplement current
lattice QCD computations.

##  Appendix A Quarkonium-gluon interaction

In this appendix, we provide the details on the leading-order quarkonium-gluon
interaction which has been derived with the Bethe-Salpeter amplitude Oh:2001rm
through dipole interaction Bhanot:1979vb . The Bethe-Salpeter equation for a
bound state is Salpeter:1951sz

$\Gamma_{\rm BS}^{\mu}(P_{1},-P_{2})\\\
=ig^{2}C_{F}\int\frac{d^{4}K}{(2\pi)^{4}}V(K)\gamma^{\nu}D_{b}(P_{1}+K)\Gamma_{\rm
BS}^{\mu}(P_{1}+K,-P_{2}+K)D_{b}(-P_{2}+K)\gamma_{\nu}\,,$ (24)

where $V(K)\simeq-\frac{1}{\bm{k}^{2}}$ after computing a dominant residue.
For $\bm{q},\bm{p}_{1,2}\gg\bm{k}$ and in the rest frame of quarkonium, the
Bethe-Salpeter amplitude reduces to

$\Gamma_{\rm
BS}^{\mu}\left(P+\frac{Q}{2},P-\frac{Q}{2}\right)=\sqrt{\frac{m_{\Upsilon}}{N_{c}}}\left(E+\frac{\bm{p}^{2}}{m}\right)\psi(\bm{p})\frac{1+\gamma^{0}}{2}\gamma^{i}\delta^{\mu
i}\frac{1-\gamma^{0}}{2}\,,$ (25)

and Eq. (24) becomes the nonrelativistic Schrödinger equation for a Coulombic
bound state,

$\left(E+\frac{\bm{p}^{2}}{m}\right)\psi(\bm{p})=g^{2}C_{F}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}}V(\bm{k})\psi(\bm{p}+\bm{k})\,.$
(26)

As in Eq. (16), the leading expression in the plasma rest frame is expected
with the replacement, $1\pm\gamma^{0}\rightarrow 1\pm\not{V}$.

Inelastic parton scattering is a dominant dissociation process of a bound
state like $\Upsilon(1S)$ which survives in high-temperature plasmas, whereas
at low temperature near the phase transition gluo-dissociation
($\Upsilon+g\rightarrow b+\bar{b}$) becomes effective Hong:2018vgp . Using Eq.
(25), the scattering amplitude of gluon absorption has been obtained as
follows Oh:2001rm :

$\displaystyle\mathcal{M}_{\rm gluo}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-g\sqrt{\frac{m_{\Upsilon}}{N_{c}}}\left[\bm{k}\cdot\frac{\partial\psi(\bm{p})}{\partial\bm{p}}\delta^{\mu
0}+k_{0}\frac{\partial\psi(\bm{p})}{\partial p^{i}}\delta^{\mu
i}\right]\delta^{\nu
j}\bar{u}(P_{1})\frac{1+\gamma^{0}}{2}\gamma^{j}\frac{1-\gamma^{0}}{2}T^{a}v(P_{2})\,,$
(27) $\displaystyle\equiv$ $\displaystyle\bar{u}(P_{1})V_{\rm
eff}^{\mu\nu}(K)v(P_{2})\,,$

where a quarkonium-gluon effective vertex $V_{\rm eff}^{\mu\nu}(K)$ can be
defined. The effective vertex is similar to the pNRQCD vertex Brambilla:2004jw
, with $\frac{\partial\psi(\bm{p})}{\partial\bm{p}}$ corresponding to $\bm{r}$
multiplied by a bound state.

The thermal width by gluo-dissociation is

$\Gamma_{\rm
gluo}=\frac{1}{2d_{\Upsilon}q^{0}}\int\frac{d^{3}\bm{k}}{(2\pi)^{3}2k^{0}}\int\frac{d^{3}\bm{p}_{1}}{(2\pi)^{3}2p_{10}}\int\frac{d^{3}\bm{p}_{2}}{(2\pi)^{3}2p_{20}}\\\
\times(2\pi)^{4}\delta^{4}(Q+K-P_{1}-P_{2})|\mathcal{M}|^{2}_{\rm
gluo}n(k)\,,$ (28)

which corresponds to a different imaginary part of the heavy quark-antiquark
potential from inelastic parton scattering Brambilla:2008cx . Gluo-
dissociation is dominant in the kinematical regime $\frac{1}{r}\gg T\gg E\gg
m_{D}$ Brambilla:2013dpa  and is not directly related to the momentum
diffusion coefficient of heavy quark. In contrast to inelastic parton
scattering, the thermal width by gluo-dissociation decreases with $v$
Hong:2019ade .

## Acknowledgments

This research was supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (No. 2021R1I1A1A01054927).

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