# One-dimensional complex potentials for quarkonia in a quark-gluon plasma.

Roland Katz  SUBATECH, Université de Nantes, IMT Atlantique, IN2P3/CNRS, 44307
Nantes, France  Stéphane Delorme  SUBATECH, Université de Nantes, IMT
Atlantique, IN2P3/CNRS, 44307 Nantes, France  Pol-Bernard Gossiaux  SUBATECH,
Université de Nantes, IMT Atlantique, IN2P3/CNRS, 44307 Nantes, France

###### Abstract

Master equations of the Lindblad type have recently been derived to describe
the dynamics of quarkonium states in the quark-gluon plasma. Because their
full resolution in three dimensions is very challenging, the equations are
often reduced to one dimension. The main ingredient of these equations is the
complex potential that describes the binding of the heavy quark-antiquark
pairs and their interactions with the medium. In this work, we propose a one-
dimensional complex potential parameterized to reproduce at best two key
properties – the temperature-dependent masses of the eigenstates and their
decay widths – of a three-dimensional lattice QCD inspired potential. Their
spectral decompositions are calculated to check their compatibility with the
positivity of the master equations.

## Introduction

The quark-gluon plasma (QGP) produced in high energy heavy ion collisions is
considered to be the most extreme medium ever created: mainly composed of
deconfined strongly interacting particles, extremely short lived, smallest and
most perfect fluid, highest temperature… Among the particles forming this
medium, the heavy quarks are known to be excellent probes of its peculiar
properties. Being mostly produced at the very beginning of the collision via
well known hard processes, the amount of heavy quarks remains constant
throughout the collision. Thus, the modification of their production as
compared to proton-proton collisions, where no significant QGP production is
expected, is mostly due to their interaction with the rest of the medium
partons.

In this work, we deal with the heavy quark-antiquark systems, whose bound
states are called quarkonia. More specifically, we focus on the key modeling
of the interaction between the heavy quark and antiquark and of their
interactions with the medium partons. Because the heavy quark masses are much
larger than the typical Quantum Chromodynamic (QCD) scale $\Lambda_{\rm
QCD}\sim 200$ MeV, the simplified description of their self interaction via a
(non-relativistic) binding potential is possible. In the vacuum, the basic
model is the so-called Cornell potential [1],

$\displaystyle V(r)=\sigma\,r-\frac{\alpha}{r},$ (1)

where $r$ is the distance between the heavy quark and antiquark, and $\sigma$
and $\alpha$ two parameters. The first term describes the long distance non-
perturbative confinement, whereas the second term corresponds to a short
distance Coulombian like attraction. To take into account the quarkonium
strong decays via fragmentation, the Cornell potential is often saturated to a
certain value $V_{\rm SB}$ or at a certain distance $r_{\rm SB}$, in order to
“free” the heavy quark pairs with higher energies or larger separations. This
simple modeling is close to what is obtained from lattice QCD calculations at
the limit $T\rightarrow 0$ [2, 3, 4, 5, 6]. The hierarchy of scales also
ensures the interactions with the thermal medium to be weak enough to be
described as a temperature dependent modification of the vacuum binding
potential, which can be calculated via the hard-thermal loop (HTL) framework
[7, 8, 9] or evaluated from lattice QCD results [2, 3, 4, 5, 6, 10]. The first
main modification is an exponential damping of the self interaction due to a
Debye-like screening from the color charges in the vicinity of the heavy quark
pair. The second main modification is the addition of an imaginary
contribution which translates the thermal effects – i.e. the direct collisions
between the medium partons and the heavy quarks – and generates finite widths
for the quarkonium states. It is important to note that this imaginary
contribution is derived for a Schrödinger-type equation for a correlator and
not for a wavefunction, i.e. it does not imply a disappearance of the heavy
quarks but instead a decoherence of the pair over time.

In the last decade, a special effort has been made to study the real-time
dynamics of heavy quarks pairs inside the QGP [11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. The purpose is to obtain
a full quantum description of the quarkonium dynamics that can be implemented
in a realistic QGP simulation and compared to experimental data. Inspired by
the open quantum system framework and using the hierarchy of scales at play,
several recent works have derived – from first QCD principles – some quantum
master equations of the Lindblad type for the quantum Brownian motion of the
heavy quarks [15, 17, 18, 20, 23, 21, 22, 26]. Various approaches have been
proposed to resolve these equations at best: reduce them to their semi-
classical limit and obtain a Langevin equation [18, 22, 26], expand them to a
limited number of spherical harmonics [20, 23], turn the equations into a
Schrödinger equation with a stochastic potential [24, 28] or to a stochastic
Schrödinger equation in one or three dimensions [27, 30], or perform a direct
resolution in one dimension [21, 29]. The present authors have also performed
a full resolution in one dimension (in preparation). The resolution of the
full equations in three dimensions remains very challenging and will require a
substantial computational effort. In the meantime, most of the studies
restrict the resolution to one dimension, therefore limiting some aspects of
the underlying physics (e.g. the transitions between states).

For all those approaches the potential is a one of the key ingredients of the
dynamics. Several simple one-dimensional potentials have been proposed [21,
24, 27, 29], mostly based on a single parameter tuned to explore the different
regimes given by the master equations or related. However, these potentials
are disconnected from the in-medium quarkonium physics and the three-
dimensional potentials, making the connection to phenomenology very limited. A
precise quantitative comparison to data would obviously require a three-
dimensional description, but a well calibrated one-dimensional model could
still be used as a good proxy. In this paper, we propose a one-dimensional
potential more suited to quarkonium phenomenology. This potential is built to
reproduce at best two key properties of a three-dimensional potential inspired
by lattice data [6, 10]: the temperature-dependent mass and the decay width of
the eigenstates. The equivalence of the temperature-dependent masses between
the one- and three-dimensional cases ensures to reproduce the quarkonia
spectrum at any temperatures as well as the dissociation temperatures.
Combined with the decay widths equivalence, we ensure the dissociation to free
states and the decoherence of the pair correlations to be approximately
similar in the one- and three-dimensional models.

In Sec. I, we first describe the three-dimensional complex potential obtained
from the perturbative HTL framework. We then derive its bona fide reduction to
one dimension and analyse its properties. In Sec. II, we introduce the three-
dimensional lattice QCD inspired potential developed in [6, 10]. We analyse
the behavior of its real part and the properties of the corresponding
eigenstates. We compare the temperature-dependent masses obtained with the
spectral functions in [6, 10] to the ones obtained with the expectation value
methods. We then study the features of the imaginary part and compare the
thermal decay widths obtained with the spectral function and expectation value
method. We finally calculate the spectral decomposition of the imaginary part
and check its compatibility with the positivity of the Lindblad equation. In
Sec. III, we describe a simple model for the real part of the one-dimensional
potential which reproduces at best the experimental masses of the charmonium
and bottomonium states and the temperature-dependent masses given by the
three-dimensional potential described in Sec. II. The spatial features of the
one- and three-dimensional eigenstates are also compared. In Sec. IV, we
propose an imaginary part for the one-dimensional potential, parameterized
such as to reproduce at best the thermal decay widths of the three-dimensional
potential. We then calculate the spectral decomposition of the one-dimensional
imaginary part and analyse its compatibility with the positivity of the
Lindblad equation. Finally, in Sec. V, the proposed one-dimensional potential
is compared to the reduction of the HTL potential in one dimension given in
Sec. I and to a one-dimensional potential found in the literature [27, 29].

##  I The HTL potential

###  I.1 In three dimensions

A complex in-medium potential, appearing in a Schrödinger equation for the
correlation function of two static heavy quarks, can be derived from the real-
time evolution of the QCD Wilson loop using hard-thermal loop (HTL) resummed
perturbation theory [7, 8]. The imaginary part of this potential originates
from the scattering of the thermal medium particles with the gluons that
mediates interaction between the two heavy quarks, similarly to Landau damping
in electromagnetic plasma. With $r$ the distance between the two heavy quarks
and $T$ the medium temperature, this complex potential reads

$V({\bf r},T)\equiv
g^{2}C_{f}\int\frac{d^{3}\mathbf{q}}{(2\pi)^{3}}\Bigl{(}1-e^{i\mathbf{q}\cdot{\bf
r}}\Bigr{)}\Biggl{[}\frac{1}{\mathbf{q}^{2}+m_{D}^{2}}-i\frac{\pi
m_{D}^{2}T}{|\mathbf{q}|(\mathbf{q}^{2}+m_{D}^{2})^{2}}\Biggr{]}.$ (2)

After integration, this potential yields

$V(r,T)=-\frac{g^{2}C_{f}}{4\pi}\Biggl{[}m_{D}+\frac{e^{-m_{D}r}}{r}\Biggr{]}-i\frac{g^{2}TC_{f}}{4\pi}\phi(m_{D}r),$
(3)

where

$\phi(x)=2\int_{0}^{\infty}\mathrm{d}z\;\frac{z}{\left(z^{2}+1\right)^{2}}\left(1-\frac{\sin(zx)}{zx}\right),$
(4)

and $C_{f}=8/6$. In the framework of effective field theories (EFTs) for non-
relativistic bound states, a complex potential can also be derived, but its
(non-trivial) expression and the underlying physical phenomena depend on the
considered separations of scales [9]. In addition to Landau damping, which
dominates at high temperatures, the transitions between color singlet and
octet states induced by a medium gluon may contribute as well to the imaginary
part. In EFTs, the expression (2) is only recovered when the temperature is
larger than $1/r$ and the Debye mass of the order of $1/r$.

To illustrate the complex potential (3), we use the QCD running coupling
constant $\alpha_{s}$ determined with the one loop contribution and at the
scale $2\pi T$,

$\alpha_{s}(T)=\frac{g^{2}(T)}{4\pi}=\frac{2\pi}{\left(11-\frac{2}{3}\,n_{f}\right)\,\log\left(\frac{2\pi
T}{\Lambda_{\rm QCD}}\right)},$ (5)

with $n_{f}=3$ massless flavors and the QCD scale taken to be $\Lambda_{\rm
QCD}=0.250$ GeV [32, 33]. For the Debye mass one can use its HTL approximation

$m_{D}^{2}(T)=\frac{2\pi}{3}(6+n_{f})\alpha_{s}T^{2}=\frac{3}{2}g^{2}T^{2}.$
(6)

As shown in figures 1 and 2, the real part of (3) is a screened Coulomb
potential with a very small temperature dependence when $T>150$ MeV, and the
imaginary part exhibits a small distance harmonic behavior and a large
distance saturation that grows monotonously with temperature. The second
derivative of the imaginary part at $r=0$ can then be related to the diffusion
coefficients of the heavy quarks within the open quantum system framework [15,
17, 18, 21, 22].

Figure 1: (a) Real part of the potential for different temperatures in GeV.
(b) Values at large $r$ of the real part of the potential as a function of T.

Figure 2: (a) Imaginary part of the potential for different temperatures in
GeV. (b) Values at large r.

The HTL potentials are only valid at high temperatures (from at least few
hundred MeV) where the weak-coupling expansion is applicable. The Coulombic
real part does not include the non-perturbative contributions, such as the
string-like linear rise present at larger $r$ in lattice QCD results [4, 5, 6]
and in the Cornell form [1], which are necessary at lower temperatures to
reproduce the QCD confinement, the quarkonium vacuum spectra and the gradual
“melting” of the various states. Thus, these potentials are not adapted to the
phenomenology of quarkonia in a QGP, which temperature ranges below few
hundred MeV most of its lifetime. Nevertheless, their analytical expressions
can easily be transposed to lower dimensions and it is thus temptating to use
them in one dimension.

###  I.2 A possible reduction of the HTL potential to one dimension

In one dimension, a natural expression for the real part of the potential
inspired by the three-dimensional equation (2) could be taken as111Within the
one dimensional reduction of the HTL framework, there is no screening of the
potential per se.

${\rm Re}V(x,T)=g_{\rm
1D}^{2}C_{f}\int\frac{dq}{2\pi}\frac{1-e^{iq|x|}}{q^{2}+m_{D}^{2}},$ (7)

with $x$ being the distance between the two heavy quarks. After integration,
the expression becomes

${\rm Re}V(x,T)=\frac{g_{\rm
1D}^{2}C_{f}}{2m_{D}}\Bigl{(}1-e^{-m_{D}|x|}\Bigr{)}.$ (8)

As shown in Fig. 3, the real part of this one-dimensional potential is not
Colombic as in three dimensions. At small $m_{D}|x|$, it reduces to a linear
potential $\propto|x|$, and at large $m_{D}|x|$ it asymptotes to a value
strongly dependent on the temperature. In one dimension, $g^{2}$ is not
anymore a dimensionless constant and one would need to derive its correct
expression. It is more convenient to simplify the potential expression (8) to

${\rm Re}V(x,T)=\frac{\sigma}{m_{D}}\Bigl{(}1-e^{-m_{D}|x|}\Bigr{)},$ (9)

by analogy with the linear part of the Cornell potential (1), as the potential
(9) now reduces to $\sigma|x|$ at small distances. One can evaluate the
constant $\sigma$ such as to reproduce, for instance, the quarkonium spectra
(as done in Sec. III.1). Note that the plots in this section are only given on
an indicative basis as we used the expression (6) of the Debye mass $m_{D}$
derived in three dimensions. Nonetheless, the main features of the one-
dimensional potential discussed here should remain valid.

Figure 3: (a) Real part of the one-dimensional potential for different T in
GeV. (b) Values at large $r$ of the real part of the potential as a function
of the temperature. We use $\sigma=1.724$ GeV/fm for the charmonia taken from
Sec. III.1.

In one dimension, the imaginary part of the potential could be taken as

${\rm Im}V(x,T)=-g_{\rm 1D}^{2}C_{f}\pi
m_{D}^{2}T\int_{-\infty}^{\,\infty}\frac{dq}{2\pi}\frac{1-e^{iq|x|}}{|q|(q^{2}+m_{D}^{2})^{2}},$
(10)

which also writes

${\rm Im}V(x,T)=-g_{\rm
1D}^{2}C_{f}m_{D}^{2}T\int_{\,0}^{\,\infty}dq\frac{1-\cos{q|x|}}{q(q^{2}+m_{D}^{2})^{2}}.$
(11)

At small distances $\big{(}|x|\ll\frac{1}{m_{D}}\big{)}$, the imaginary part
reduces to a harmonic potential,

${\rm Im}V(x,T)=-\frac{g_{\rm 1D}^{2}C_{f}T}{4}x^{2},$ (12)

similarly to the three-dimensional case. At large distances
$\big{(}|x|\gg\frac{1}{m_{D}}\big{)}$, the imaginary part diverges
logarithmically,

${\rm Im}V(x,T)\propto-\frac{g_{\rm 1D}^{2}C_{f}T}{m_{D}^{2}}\ln{(m_{D}|x|)},$
(13)

i.e. it does not saturate unlike the three-dimensional case. This logarithmic
growth appears unphysical as a saturation of the imaginary part at large
distances is expected from Landau damping [7, 8, 10]. Additionally, at large
distances, the physics of well separated heavy quarks should not influence the
behaviour of the quarkonium states, which correspond to small distance
correlations.

In accordance with the real part of the potential (9), we can replace $g_{\rm
1D}^{2}C_{f}/2$ by $\sigma$:

${\rm Im}V(x,T)=-2\sigma
m_{D}^{2}T\int_{\,0}^{\,\infty}dq\frac{1-\cos{q|x|}}{q(q^{2}+m_{D}^{2})^{2}}.$
(14)

Using the Debye mass derived in three dimensions as for the real part, the
imaginary part of the potential (14) is shown for different temperatures in
Fig. 4.

Figure 4: Imaginary part of the one-dimensional potential for different
temperatures in GeV. We use $\sigma=1.724$ GeV/fm for the charmonia taken from
Sec. III.1.

Although this one-dimensional potential does not fit our basic physical
expectations - because of the logarithmic growth of the imaginary part - its
properties will be further studied in Sec. V and compared to other models.

##  II The three-dimensional lattice QCD inspired complex potential

To find a three dimension in-medium quarkonium potential valid in a broader
range of temperature and relevant in terms of phenomenology, one can turn to
lattice QCD inspired potentials [2, 3, 4, 6, 10]. The real parts of these
potentials are generally parameterizations of Cornell forms supplemented by
exponential damping factors [2, 3, 4] or parameterizations of more advanced
analytical expressions derived from linear response theory [34, 6, 10]. The
parameters are then generally determined by a fit to lattice QCD numerical
calculations at different distances and temperatures. We focus in this section
on the only complex potential model inspired by lattice QCD results developed
so far [6, 10]. For the sake of completeness, we first sum up the model
further described in [6, 10].

###  II.1 The model from Y. Burnier, O. Kaczmarek, A. Rothkopf and D. Lafferty
[6, 10]

Based on the linear response theory framework and a generalized Gauss law
ansatz, the in-medium complex potential is derived from the vacuum Cornell
potential using the in-medium complex permittivity obtained from a HTL
calculation. Similarly to the Cornell form, the resulting analytic expressions
can be decomposed in a Coulombic (“$V_{C}$”) and a linear string-like part
(“$V_{S}$”) which both depend on a single temperature dependent parameter, the
Debye mass $m_{D}$.

$\mathrm{Re}V=\mathrm{Re}V_{C}+\mathrm{Re}V_{S}+c\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\mathrm{Im}V=\mathrm{Im}V_{C}+\mathrm{Im}V_{S},$ (15)

where $c$ is a constant. The Coulombic part writes:

$\mathrm{Re}V_{C}\\!\left(r\right)=-\alpha_{s}\left[m_{D}+\frac{e^{-m_{D}r}}{r}\right],\quad\mathrm{Im}V_{C}\\!\left(r\right)=\alpha_{s}T\phi(m_{D}r),$
(16)

where

$\phi(x)=2\int_{0}^{\infty}\mathrm{d}z\;\frac{z}{\left(z^{2}+1\right)^{2}}\left(1-\frac{\sin(zx)}{zx}\right).$
(17)

$V_{C}$ is perfectly equivalent to the HTL potential (3) seen in Sec. I.1. The
string part writes:

$\displaystyle\mathrm{Re}V_{S}\\!\left(r\right)=\frac{2\sigma}{m_{D}}-\frac{e^{-m_{D}r}\left(2+m_{D}r\right)\sigma}{m_{D}},\quad\mathrm{Im}V_{S}(r)=\frac{\sigma
T}{m_{D}^{2}}\chi(m_{D}r),$ (18)

where

$\chi(x)=2\int_{0}^{\infty}\mathrm{d}p\;\frac{2-2\cos(px)-px\sin(px)}{\sqrt{p^{2}+\Delta_{D}^{2}}\left(p^{2}+1\right)^{2}},$
(19)

and $\Delta_{D}=\Delta/m_{D}$ $\approx$ 3.0369 with $\Delta$ being a suitably
chosen regularization scale. The latter has been introduced to avoid the
logarithmic divergence of the imaginary string part at large r, which can be
traced back to the absence of string breaking in the initial Cornell
potential. The choice in the formulation of this regularization scale has been
checked to not influence the end results. The set of phenomenological
parameters coming from the vacuum Cornell potential,

$\alpha_{s}=0.513\pm 0.0024,\quad\sqrt{\sigma}=0.412\pm
0.0041~{}\mathrm{GeV},\quad c=-0.161\pm 0.0025~{}\mathrm{GeV},$ (20)

is fixed to reproduce the energy spectrum of the bottomonia with the bottom
quark mass taken to be the so-called renormalon subtracted mass [35],

$m_{b}=m_{b}^{RS}=4.882~{}\mathrm{GeV}.$ (21)

The charm quark mass $m_{c}=m_{c}^{\mathrm{fit}}=1.4692~{}\mathrm{GeV}$ is
then determined to reproduce the energy spectrum of the charmonia while
keeping the same vacuum parameters (20). The Debye mass parameter is obtained
via the next-to-leading order HTL expression supplemented by two terms
(quadratic and cubic in the coupling constant $g$) accounting for non-
perturbative contribution [6, 10, 36]. The two constants appearing in the
latter are fixed using continuum corrected lattice results. The resulting
Debye mass222A good fit to the resulting Debye mass is given by:
$m_{D}/T=1.275\ln{(170.813\,T-19.4086)}-7.593\,T$ for $T_{0}<T<0.2$ GeV and
$m_{D}/T=0.6724\,{\rm
Re}\left[\ln{(95347.824\,T-12617.627)}\right]-7.864\,T^{0.12}+2.4975$ for
$0.2<T<1$ GeV. dependence on temperature is shown in Fig. 5.

Figure 5: Debye mass dependence on the temperature [6, 10].

The zero of the Debye mass defines the “vacuum” temperature $T_{0}\approx
0.126025$ GeV of this potential model. The resulting curves for the real part
of the potential fit nicely the data extracted from the lattice at any
temperature [37, 6]. The curves for the imaginary part also agree quite well
with the tentative lattice QCD results up to $r\approx 1$ fm and down to
$T\sim 160$ MeV (see figure 2 in [10]). At high temperatures (i.e. large
$m_{D}$), the string part $V_{S}$ becomes negligible and the potential reduces
to the HTL result $V_{C}$. At low temperatures (i.e. when $m_{D}\rightarrow 0$
and $T\rightarrow T_{0}$), the model approaches the vacuum Cornell form. To
include string breaking to the real part of the potential, a flat asymptotics
is enforced from $r_{SB}=1.25$ fm at $T=T_{0}$, which corresponds to a
saturation of the potential at $V_{\rm SB}\approx 0.8352$ GeV. The latter is
enforced to be the maximum value of the real part for $T>T_{0}$ as well. The
value of $V_{\rm SB}$ is consistently close to the fragmentation threshold
$2m_{B}-2m_{b}\approx 2m_{D^{0}}-2m_{c}\approx 0.8$ GeV. We do not consider
the complex potential extension to the finite baryon density and finite
velocity regimes developed in [10].

###  II.2 Features of the real part of the potential

In Fig. 6, we show the resulting real part of the potential at different
temperatures and its temperature dependence at large distances. At small
distances $\big{(}r\ll\frac{1}{m_{D}}\big{)}$, the potential becomes purely
Coulombic and is independent on the temperature. At large distances
$\big{(}r\gg\frac{1}{m_{D}}\big{)}$, due to the string part, the potential
decrease with the temperature is much steeper than it was in Fig. 1 with the
HTL potential.

Figure 6: (a) Real part of the potential for different temperatures in GeV.
(b) Values at large distances of the real part of the potential as a function
of the temperature.

Figure 7: (a) Reduced radial wavefunctions $u(r)=rR(r)$ of the charmonium S
states at the vacuum temperature $T_{0}$ (b) Same for bottomonia.

To further analyse the properties of this potential, we use the Hamiltonian of
a non-relativistic two particles system in the $Q{\bar{Q}}$ pair center of
mass frame, given by :

$\displaystyle H=2m_{Q}-\frac{1}{m_{Q}}\nabla^{2}+{\rm Re}V(r,T),$ (22)

where $\nabla$ is the nabla operator and $m_{Q}$ is the heavy quark mass333The
value of ${\rm Re}V(r\rightarrow\infty,T)$ can be interpreted as a temperature
dependent “mean field” contribution to the heavy quark mass [38, 39, 40] :
$m_{Q}(T)=m_{Q}^{0}+{\rm Re}V(r\rightarrow\infty,T)/2$, where $m_{Q}^{0}$ is
the heavy quark bare mass. If one would consider $m_{Q}(T)$ in the
Hamiltonian, then the potential would be rescaled to ${\rm Re}V(r,T)-{\rm
Re}V(r\rightarrow\infty,T)$ and one may use $m_{Q}(T)$ in the kinetic term as
well. To our knowledge, the effect of a temperature dependent inertial mass
has never been studied within the quarkonium framework. In the present work,
this temperature dependence of the heavy quark mass is not considered. (given
by Eq. 21 and beneath). The reduced radial wavefunctions $u(r)=rR(r)$ (with
$R(r)$ the radial part of the wavefunction) for the charmonium and bottomonium
S states at the vacuum temperature $T_{0}$ are shown in Fig. 7. In Tab. 1, the
masses $m^{\rm EV}$ of the different quarkonium S states obtained from the
expectation values of the Hamiltonian at the vacuum temperature $T_{0}$ are
compared to the masses obtained with the spectral functions $m^{\rm SF}$ in
[10] and to the experimental values. The differences between the results of
the two methods are negligible and the values are consistent with the data.
The mean radiuses of the “vacuum” states obtained from the expectation values
$\langle r\rangle$ are also given in Tab. 1.

At $T=T_{0}$ |  $J/\psi$ |  $\psi$(2S) |  $\Upsilon$(1S) |  $\Upsilon$(2S) |  $\Upsilon$(3S)
---|---|---|---|---|---
$m^{\rm PDG}$ [41] |  3.0969 |  3.6861 |  9.4603 |  10.023 |  10.355
$m^{\rm SF}$ [10] |  3.0969 |  3.6632 |  9.4603 |  10.023 |  10.355
$m^{\rm EV}$ |  3.0964 |  3.6642 |  9.4611 |  10.023 |  10.355
$r^{\rm EV}$ |  0.387 |  0.856 |  0.188 |  0.463 |  0.687
Table 1:  Masses (in GeV) and mean radiuses (in fm) for the “vacuum” states
(at $m_{D}=0$ $\Leftrightarrow T_{0}\approx 0.126025$ GeV). $m^{\rm PDG}$ is
the experimental mass given by the particle data group, $m^{\rm SF}$
corresponds to the mass obtained from the spectral functions [10] and $m^{\rm
EV}$ ($r^{\rm EV}$) is the mass (mean radius respectively) calculated from the
expectation values of the Hamiltonian (of the radius operator respectively).

Figure 8: (a) Thermal masses $M(T)=E_{n}(T)+2m_{c}$ of the charmonium states
and (b) Same for bottomonia. Gray areas corresponds to the results available
from spectral functions in [10].

In Fig. 8, the temperature-dependent masses of the states
$M(T)=E_{n}(T)+2m_{Q}$, where $E_{n}$ are the expectation values of the
Hamiltonian, are compared to the temperature-dependent masses obtained in [10]
with the spectral functions via the fit of the spectral peaks with (skewed)
Breit-Wigner distributions. The masses obtained with both methods are similar
and globally decrease with the temperature, but the dissociation temperatures
are observed to be larger with the expectation value method – they correspond
to the crossing of $E_{n}(T)$ and ${\rm Re}V(r\rightarrow\infty)$ – than with
the spectral function method. This difference in behavior close to
dissociation might be explained by the difficulty of fitting large spectral
peaks with skewed Breit-Wigner distributions. As noted in [10], the variation
with temperature is stronger for excited states with smaller binding energies
$E^{\rm bind}_{n}={\rm Re}V(r\rightarrow\infty,T_{0})-E_{n}(T_{0})$, which
corresponds to the intuition that tightly bound states are less sensitive to
medium effects.

###  II.3 Features of the imaginary part of the potential

In Fig. 9, we show the imaginary part of the potential at different
temperatures and its temperature dependence at large distances. As with the
HTL potential (Fig. 2), the imaginary part exhibits a small distance harmonic
behavior with a harmonic coefficient that gets larger with the temperature.
However, the large distance saturation is less trivial than it was in Fig. 2
with the HTL potential: the string part has a strong influence up to at least
T=400 MeV. To evaluate the impact of the imaginary part on the heavy quark
pair, one can compute the thermal decay widths $\Gamma_{n}$ of the bound
states. Within the spectral function method, the $\Gamma_{n}$ correspond to
the widths of the bound state peaks at finite temperature and are calculated
in [10] via a fit of the peaks with (skewed) Breit-Wigner distributions. The
thermal decay width of the state $n$ can also be evaluated via the expectation
value of the imaginary part

$\displaystyle\Gamma_{n}=2{\rm Tr}(\rho_{n}{\rm Im}V)=2\langle{\rm
Im}V\rangle_{n},$ (23)

where $\rho_{n}$ is the density matrix of the state $n$ and
$\langle...\rangle_{n}$ the related expectation value. As shown in Fig. 10,
the thermal decay widths increase with temperature and are larger for excited
states, which corresponds to the intuition that more compact states are less
sensitive to medium effects. The two methods give similar results when the
temperature is much smaller than the dissociation temperature of the
considered state, but increasing discrepancies appear as this temperature is
approached. These differences might originate once again from the difficulty
of fitting large spectral peaks with skewed Breit-Wigner distributions.

Figure 9: (a) Imaginary part of the potential for different temperatures in
GeV. (b) The temperature dependence of the imaginary part at large distances.

Figure 10: (a) $\Gamma_{n}(T)$ of the charmonia and (b) $\Gamma_{n}(T)$ of the
bottomonia. Gray areas corresponds to the results available from spectral
functions in [10].

###  II.4 Spectral decomposition

The spectral decomposition ${\rm Im}{\tilde{V}}({\bm{q}})={\rm
Im}{\tilde{V}}_{c}+{\rm Im}{\tilde{V}}_{s}$ of the imaginary part of the
potential $\mathrm{Im}V({\bm{r}})={\rm Im}V_{c}+{\rm Im}V_{s}$ is given by its
three-dimensional Fourier transform:

${\rm
Im}{\tilde{V}}({\bf{\bm{q}}})=\frac{1}{(2\pi)^{3}}\int_{-\infty}^{\infty}d{\bm{r}}\,\mathrm{Im}V({\bm{r}})\,e^{-i{\bm{q}}{\bm{r}}}.$
(24)

Within the open quantum system framework, it can be shown that the spectral
decomposition of the imaginary part of the potential must be globally positive
(or negative depending on the chosen convention) to ensure the positivity of
the Lindblad equation [15, 17, 18, 21, 22, 26]. In this section, we will check
whether the spectral decomposition of the QCD inspired potential 15 satisfies
this condition.

For the Coulombic part, the angular integration of the Fourier transform yield

$\displaystyle{\rm Im}{\tilde{V}}_{c}=\frac{\alpha_{s}T}{2\pi
q}\int_{0}^{\infty}dr\,r\sin(qr)\big{(}\phi(m_{D}r)-\phi_{\rm asympt}\big{)},$
(25)

where $\phi_{\rm asympt}=1$ is subtracted by hand to remove the asymptotic
value of the complex potential that would lead to a Dirac delta $\delta(q)$.
Similarly, for the string part we get

$\displaystyle{\rm Im}{\tilde{V}}_{s}=\frac{\sigma
T}{2\pi^{2}q\,m_{D}^{2}}\int_{0}^{\infty}dr\,r\sin(qr)\big{(}\chi(m_{D}r)-\chi_{\rm
asympt}\big{)},$ (26)

where $\chi_{\rm asympt}$ is the asymptotic value of the function $\chi$ and
is given by

$\displaystyle\chi_{\rm
asympt}=4\int_{0}^{\infty}dk\frac{1}{(k^{2}+1)^{2}\sqrt{k^{2}+\Delta_{D}^{2}}}.$
(27)

Figure 11: Spectral decomposition of the Coulombic-like imaginary part of the
potential ${\rm Im}V_{c}$

at different temperatures (in GeV).

Figure 12: Spectral decomposition of the String-like imaginary part of the
potential ${\rm Im}V_{s}$

at different temperatures (in GeV). (a) General behaviour. (b) Very small
positive bumps at low temperatures.

In Fig. 11, the spectral decomposition of the Coulombic-like part (equivalent
to the HTL potential of Sec. I.1) is observed to be negative for all $q$,
while in Fig. 12 for the string-like part the spectral decomposition is
negative at small $q$ but exhibits a very small positive bump at larger q and
smaller temperatures.

Figure 13: Spectral decomposition of the imaginary part of the potential ${\rm
Im}V={\rm Im}V_{c}+{\rm Im}V_{s}$

at different temperatures (in GeV). (a) General behaviour. (b) Very small
positive bumps at low temperatures.

Finally, as shown in Fig. 13, when summing the Coulomb and string like parts,
the small positive bump is only observed for temperatures $T\lesssim$ 160 MeV.
We can therefore conclude that the spectral decomposition is globally negative
(with the chosen convention) and that this three-dimensional potential will
globally satisfies the positivity requirement of the Lindblad equations.

##  III A one-dimensional potential: the real part

In this section, we propose a one-dimensional potential parameterized to
reproduce at best the properties of the three-dimensional potential from the
generalized Gauss law model described in Sec. II. We first focus on the real
part of the potential and the corresponding energy spectra. To study the one-
dimensional potential, we use the Hamiltonian :

$\displaystyle H=2m_{Q}-\frac{1}{m_{Q}}\frac{\partial^{2}}{\partial
x^{2}}+{\rm Re}V(x,T).$ (28)

###  III.1 Vacuum potential

In the “vacuum”, a simple form is proposed and tuned to reproduce at best the
experimental masses. Inspired by the Cornell potential and by the one-
dimensional version of the pQCD+HTL potential in Sec. I – which are mainly
linear around the considered eigenenergies –, we approximate the heavy
quark/anti-quark self interaction to a 1D symmetrical linear potential
$1/2\,K|x|+C$ saturated at $V_{\rm SB}\approx 0.8352$ GeV for the string
breaking, i.e.

$\displaystyle\mathrm{Re}V_{\rm
1D}(x,T=T_{0})=\left\\{\begin{array}[]{lcl}1/2\,K|x|+C&\mbox{ when
}&1/2\,K|x|+C<V_{\rm SB}\\\ V_{\rm SB}&\mbox{ when }&1/2\,K|x|+C\geq V_{\rm
SB}.\end{array}\right.$ (31)

We keep the same quark masses as in Sec. II, i.e. $m_{c}=1.4692$ GeV and
$m_{b}=4.882$ GeV. The string parameters $K_{c}$ for the charmonia and $K_{b}$
for the bottomonia are chosen such as to obtain an energy difference between
the first two even eigenstates (subscripted by $n=0$ and $n=2$) given by
$E_{2}-E_{0}=E(\psi^{\prime})-E(J/\psi)=589$ MeV for charmonia and an
intermediate situation between $E_{2}-E_{0}\approx
E(\Upsilon^{\prime})-E(\Upsilon)=563$ MeV and $E_{4}-E_{0}\approx
E(\Upsilon^{\prime\prime})-E(\Upsilon)=895$ MeV for bottomonia (with subscript
$n=4$ for the third even eigenstate); it leads respectively to $K_{c}=1.724$
GeV.fm-1 and $K_{b}=2.692$ GeV.fm-1. The overall constants $C_{c}$ for the
charmonia and $C_{b}$ for the bottomonia are then fixed to get
$E_{0}+2m_{c}=m_{J/\psi}$ for the charmonia and $E_{0}+2m_{b}=m_{\Upsilon}$
for the bottomonia, i.e. $C_{c}=-0.115$ GeV and $C_{b}=-0.55$ GeV
respectively.

Figure 14: Vacuum one-dimensional linear potential for (a) charmonia and (b)
bottomonia.

The corresponding vacuum potentials are shown in Fig. 14 and feature three
charmonium and six bottomonium eigenstates. The masses (and root mean square
radiuses) of the even “S-like” and odd “P-like” states are summarized in Tab.
2 and 3 and compared to the three-dimensional and experimental values. With
errors of maximum $\approx 30$ MeV for the S-like states and $\approx 100$ MeV
for the P-like states, the one dimension linear model reproduces in good
approximation the mass spectra. In Fig. 15, the charmonium and bottomonium
S-like states obtained with the one-dimensional potential are shown, and can
be compared to the states obtained with the three-dimensional potential (Fig.
7). As expected, the shapes of the one- and three-dimensional 1S-states are
different : the radial parts of the three-dimensional wavefunctions are
typically exponential for Coulombic potentials, whereas they are approximately
Gaussian for linear potentials. Despite this difference in nature, some
similar features can be observed : the typical sizes of the states (as
confirmed by the root mean square radiuses in Tab. 2), the large distance
behaviour, the wave nodes and antinodes positions and relative amplitudes…

At $T=T_{0}$ |  $J/\psi$ |  $\psi$(2S) |  $\Upsilon$(1S) |  $\Upsilon$(2S) |  $\Upsilon$(3S)
---|---|---|---|---|---
$m^{\rm PDG}$ [41] |  3.0969 |  3.6861 |  9.4603 |  10.023 |  10.355
$m^{\rm EV}_{\rm 3D}$ |  3.0964 |  3.6642 |  9.4611 |  10.023 |  10.355
$m^{\rm EV}_{\rm 1D}$ |  3.0981 |  3.6858 |  9.4621 |  10.005 |  10.386
$|m^{\rm PDG}-m^{\rm EV}_{\rm 1D}|$ (MeV) |  1.2 |  0.3 |  2.1 |  18 |  31
${\rm 3D}\,\,\,\sqrt{\langle r^{2}\rangle}$ |  0.431 |  0.943 |  0.213 |  0.501 |  0.7449
${\rm 1D}\,\,\,\sqrt{\langle x^{2}\rangle}$ |  0.387 |  0.856 |  0.157 |  0.432 |  0.648
Table 2:  Masses (in GeV) and root mean square radiuses (in fm) for the “vacuum” S-like states (at $T_{0}\approx 0.126025$ GeV). $m^{\rm PDG}$ is the experimental mass given by the particle data group, $m^{\rm EV}_{\rm 3D}$ ($\sqrt{\langle r^{2}\rangle}$) is the mass (root mean square radius respectively) calculated from the expectation values of the Hamiltonian (of the square radius operator respectively) with the three-dimensional potential (see Sec. II). $m^{\rm EV}_{\rm 1D}$ and $\sqrt{\langle x^{2}\rangle}$ are the corresponding values obtained with the one-dimensional potential. At $T=T_{0}$ |  $\chi_{c}$(1P) |  $\chi_{b}$(1P) |  $\chi_{b}$(2P) |  $\chi_{b}$(3P)
---|---|---|---|---
$m^{\rm PDG}$ [41] |  3.494 |  9.888 |  10.252 |  10.534
$m^{\rm EV}_{\rm 3D}$ |  3.509 |  9.932 |  10.273 |  10.540
$m^{\rm EV}_{\rm 1D}$ |  3.453 |  9.783 |  10.209 |  10.544
$|m^{\rm PDG}-m^{\rm EV}_{\rm 1D}|$ (MeV) |  41 |  105 |  43 |  10
Table 3:  Masses (in GeV) for the “vacuum” P-like states (at $T_{0}\approx
0.126025$ GeV). $m^{\rm PDG}$ is the experimental mass given by the particle
data group [41]. $m^{\rm EV}_{\rm 3D}$ is the mass calculated from the
expectation values of the Hamiltonian with the three-dimensional potential and
$m^{\rm EV}_{\rm 1D}$ is the mass obtained with the one-dimensional potential.

Figure 15: (a) Charmonium S-like states obtained with the real part of the
one-dimensional potential in the “vacuum” (at $T_{0}\approx 0.126025$ GeV).
(b) Same but for bottomonia.

###  III.2 Temperature dependence

When $m_{D}>0$ (for $T>T_{0}\approx 0.126025$ GeV), the one-dimensional
potential is parameterized such as to reproduce the temperature dependence of
the mass spectrum obtained from the three-dimensional potential, i.e.

$\displaystyle M^{\rm 1D}(T)=\langle{\hat{H}}^{\rm
1D}(T)\rangle\approx\langle{\hat{H}}^{\rm 3D}(T)\rangle=M^{\rm 3D}(T),$ (32)

where $M^{\rm 1D}$ and ${\hat{H}}^{\rm 1D}$ (respectively $M^{\rm 3D}$ and
${\hat{H}}^{\rm 3D}$) are the mass spectrum and Hamiltonian operator
associated with the real part of the one-dimensional (respectively three-
dimensional) potential. Additionally, the effect of the temperature dependent
Debye screening is implemented by saturating the one-dimensional potential
$\mathrm{Re}V_{\rm 1D}$ to the value of the three-dimensional potential at
large distances,

$\mathrm{Re}V^{x\rightarrow\infty}_{\rm
1D}(T)\equiv\mathrm{Re}V(r\rightarrow\infty,T)={\rm
Minimum}\,[\,2\sigma/m_{D}(T)-\alpha_{s}m_{D}(T)+c,\,V_{\rm SB}].$ (33)

This combination of similar mass spectra and equal large distance values of
the potential implies that the binding energies of the states – which are key
features of the real part of the potential in the open quantum system
framework – are equivalent in both cases. In other words, the energy gaps
between bound and free states are then similar in the one- and three-
dimensional cases.

Inspired by the exponential “$e^{-m_{D}r}$” terms present in
$\mathrm{Re}V_{C}$ and $\mathrm{Re}V_{S}$, we propose the following
parameterization for the one-dimensional potential444Many other
parametrizations were tested, using for instance the the expression of the
string part of the potential 18. Nevertheless, none of those was able to
correctly reproduce the three-dimensional mass spectra (condition 32).:

$\displaystyle\mathrm{Re}V_{\rm
1D}(x,T)=\left\\{\begin{array}[]{lcl}1/2\,K|x|\,e^{-\mu_{1}|x|}+\mu_{2}&\mbox{
when
}&1/2\,K|x|\,e^{-\mu_{1}|x|}+\mu_{2}<\mathrm{Re}V^{x\rightarrow\infty}_{\rm
1D}(T)\\\ \mathrm{Re}V^{x\rightarrow\infty}_{\rm 1D}(T)&\mbox{ when
}&1/2\,K|x|\,e^{-\mu_{1}|x|}+\mu_{2}\geq\mathrm{Re}V^{x\rightarrow\infty}_{\rm
1D}(T),\end{array}\right.$ (36)

where

$\mu_{1}(x,T)=a_{1}\,(T-T_{0})^{b_{1}}\,e^{-\lambda(T-T_{0})\,|x|}\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\mu_{2}(T)=-a_{2}\,(T-T_{0})^{b_{2}}+C.$ (37)

The $\mu_{2}$ term is a global temperature-dependent shift of the potential
leading to a global decrease of the mass spectrum values as the temperature
increases. The $e^{-\mu_{1}|x|}$ term symmetrically curves the shape of the
potential at intermediate distances, resulting in different (temperature-
dependent) variations of the mass spectrum values around the global decrease
given by $\mu_{2}$. The $e^{-\lambda(T-T_{0})\,|x|}$ term present in $\mu_{1}$
is introduced to smoothly cancel $\mu_{1}$ at large distances. Without this
term, the potential would not reach or would not remain at the saturation
value at large distances given by Eq. 33, but would rather tend to the value
$\mu_{2}$. The values of the parameters are summed up in Tab. 4 and were tuned
to reproduce at best the mass spectra of the three-dimensional potential shown
in Fig. 8. We keep the values of the K and C constants determined in the
vacuum case. As $\mu_{1}=0$ and $\mu_{2}=0$ when $T=T_{0}$, the potential
$\mathrm{Re}V_{\rm 1D}(x,T)$ reduces to the vacuum linear potential described
in the previous Sec. III.1.

$\mathrm{Re}V_{\rm 1D}$ parameters |  $K$ |  $C$ |  $a_{1}$ |  $b_{1}$ |  $\lambda$ |  $a_{2}$ |  $b_{2}$
---|---|---|---|---|---|---|---
Charmonia |  1.724 |  -0.115 |  7.5 |  0.98 |  1.5 |  0.96 |  0.97
Bottomonia |  2.692 |  -0.55 |  7.7 |  1.15 |  |  0.58 |  1.15
Table 4:  Parameters for the real part of the 1D potential $\mathrm{Re}V_{\rm
1D}$. $K$ is in GeV.fm-1, $C$ in GeV, $a_{1}$, $b_{1}$ and $\lambda$
dimensions are such as $\mu_{1}$ is in fm-1, $a_{2}$ and $b_{2}$ dimensions
are such as $\mu_{2}$ is in GeV.

Figure 16: (a) One-dimensional potential at different temperatures (in GeV)
for the charmonia and (b) for the bottomonia.

The resulting potentials $\mathrm{Re}V_{\rm 1D}$ for the charmonia and
bottomonia are shown in Fig. 16 at different temperatures. Although the three-
dimensional quarkonium potentials are independent of the heavy quark mass, it
is not the case within this one-dimensional model which rather prioritizes the
phenomenological features. Due to a larger $K$ and a smaller $C$, the
potential well for the bottomonia is narrower and deeper than for the
charmonia. Larger values of $\mu_{1}(x,T)$ in the bottomonium case also lead
to a less curved and narrower potential well close to the asymptotes. The
potentials become completely non-bonding (i.e. constant) at $T\approx 530$ MeV
for the charmonia and $T\approx 800$ MeV for the bottomonia.

In Fig. 17, the temperature dependence of the mass spectra – using the
expectation values of the Hamiltonian – for the charmonium and bottomonium
S-like states obtained from the one-dimensional potential are compared to the
three-dimensional case. The mass spectra are in good agreement at any
temperature for the charmonia and up to $T\approx 550$ MeV for the bottomonia
(which includes the typical range of temperature reached in heavy-ion
collisions). The maximum mass difference between the one- and the three-
dimensional cases is $\approx$ 40 MeV for the S states (mainly for the
$\Upsilon({\rm 1S})$ above $T=550$ MeV and for the $\Upsilon({\rm 3S})$).
Additionally, the dissociation temperature of the $\Upsilon({\rm 1S})$ is
approximately $100$ MeV smaller with the one-dimensional potential ($T_{\rm
diss}^{\Upsilon({\rm 1S})}\approx 740$ MeV).

Figure 17: (a) Charmonium mass spectrum of the S states for the one-
dimensional potential (solid lines) obtained from the expectation values of
the Hamiltonian compared to the three-dimensional case (dashed lines). (b)
Same for the bottomonia.

Figure 18: (a) Bottomonium 1S-like state obtained with the real part of the
one-dimensional potential at various temperatures. (b) Reduced radial part of
the bottomonium 1S state obtained with the three-dimensional potential at
various temperatures.

The bottomonium 1S states obtained with the one- and three-dimensional
potentials are shown at different temperatures in Fig. 18. As in the vacuum
case (see Sec. III.1), the features of the one- and three-dimensional
eigenstates remain roughly similar throughout the temperature range.
Nevertheless, the spatial distributions of the one-dimensional 1S-like states
show more deviations from Gaussianity at temperatures close to the state
dissociation temperature (see for instance $\Upsilon(1S)$ at $T=710$ MeV in
Fig. 18). These deviations can be explained by the one-dimensional potential
being almost flat at these high temperatures (see Fig 16).

In Fig 19, the root-mean-square radiuses of the S states obtained with the
one- and three-dimensional potentials are compared. The 1S states are observed
to be $\approx 35\%$ larger with the three-dimensional potential and $\approx
10-15\%$ larger for the 2S states. For most of the states, the disagreements
increase close to their dissociation temperatures : a large extension of the
state spatial distributions exacerbates the differences between the two
potentials. Nevertheless, as the imaginary part of the potential dominates the
evolution of a states close to its dissociation temperature, these particular
discrepancies are expected to be negligible.

Figure 19: (a) Comparison of the root-mean-square radiuses of the charmonium S
states obtained with the one- and three-dimensional potentials. (b) Same for
the bottomonia.

##  IV A one-dimensional potential: the imaginary part

In this section, we propose an imaginary part for the one-dimensional
potential parameterized to reproduce at best the temperature dependence of the
decay widths of the three-dimensional potential described in Sec. II.

###  IV.1 One dimension model and decay widths

A simple possibility is to extend the radial imaginary part of the three-
dimensional potential $\mathrm{Im}V=\mathrm{Im}V_{C}+\mathrm{Im}V_{S}$ (see
Eqs. 18 and 16) to “$-x$” (i.e. the symmetry to the $x=0$ axis) and to include
by hand two coefficients $\alpha$ and $\beta$ to be tuned,

$\mathrm{Im}V^{\rm
1D}(x,T)=\alpha\,\mathrm{Im}V_{C}\left(|x|,T\right)+\beta\,\mathrm{Im}V_{S}\left(|x|,T\right).$
(38)

Then, like in three dimensions, the one-dimensional imaginary part has a
harmonic behaviour at small distances – that can be related to the diffusion
coefficients of the heavy quarks – and it saturates at large distances – which
is expected from Landau damping –. The values of $\alpha$ and $\beta$ which
lead to the best similarity between the one- and three-dimensional decay
widths are summed up in Tab. 5. The resulting potential $\mathrm{Im}V_{\rm
1D}$ is shown in Fig. 20 (a) at different temperatures. By giving more weights
to the Coulombic part, the coefficients tend to narrow the potential well at
short distances and to uplift the large distance values at high temperatures
as compared to the three-dimensional case (as shown in Fig. 20 (b)).

$\mathrm{Im}V_{\rm 1D}$ coefficients |  $\alpha$ |  $\beta$
---|---|---
Charmonia |  1.7 |  0.8
Bottomonia |  1.4 |  0.9
Table 5:  Coefficients for the imaginary part of the one-dimensional potential
$\mathrm{Im}V_{\rm 1D}$.

Figure 20: (a) Imaginary part of the one-dimensional potential for the
bottomonia. (b) Large distance behaviour of the imaginary part of the one- and
three-dimensional potentials for the bottomonia.

The imaginary part of the one-dimensional potential must reproduce at best the
decay widths of the three-dimensional potential from Sec. II. To compute the
decay widths $\Gamma^{\rm 1D}_{n}$ of the one-dimensional potential we use the
eigenstates of $\mathrm{Re}V_{\rm 1D}$ determined in Sec. III and calculate
the expectation value of $\mathrm{Im}V^{\rm 1D}$. They are then compared to
the decay widths $\Gamma^{\rm 3D}_{n}$ of the three-dimensional potential
calculated with the expectation values of $\mathrm{Im}V$, and we expect that
for a state $n$:

$\displaystyle\Gamma^{\rm 1D}_{n}(T)=2\left\langle{\rm Im}V^{\rm
1D}(T)\right\rangle_{n}\approx 2\left\langle{\rm
Im}V(T)\right\rangle_{n}=\Gamma^{\rm 3D}_{n}(T).$ (39)

The temperature dependent decay widths for the charmonium and bottomonium S
states obtained with the one-dimensional potential (38) and the set of
parameters given in Tab. 5 are shown in Fig. 21. They are in a good agreement
with the three-dimensional decay widths up to $T=600$ MeV, at the exception of
the $J/\psi$ state which is underestimated for $T<350$ MeV and overestimated
for $T>380$ MeV to a maximum of $\approx 30\%$. The large distance saturation
of the imaginary part is observed to be important to obtain decay widths that
are nearly linear with the temperature. At high temperatures, the
discrepancies between the one- and three-dimensional 1S state decay widths can
be traced back to the differences in eigenstate spatial features (see Fig. 18)
and to the differences in complex potentials at large distances (see Fig. 20).

Figure 21: (a) Charmonium decay widths obtained with the one- and three-
dimensional potentials. (b) Same for bottomonia.

###  IV.2 Spectral decomposition

Figure 22: Spectral decomposition of the imaginary part of the potential ${\rm
Im}V^{\rm 1D}=\alpha\,{\rm Im}V_{c}^{\rm 1D}+\beta\,{\rm Im}V_{s}^{\rm 1D}$

at different temperatures (in GeV) for the (a) charmonia and (b) bottomonia.

The spectral decomposition ${\rm Im}{\tilde{V}}^{\rm 1D}(q)=\alpha\,{\rm
Im}{\tilde{V}}_{c}^{\rm 1D}+\beta\,{\rm Im}{\tilde{V}}_{s}^{\rm 1D}$ of this
one-dimensional potential can be calculated analytically. For the Coulombic
part, the Fourier transform is

$\displaystyle{\rm Im}{\tilde{V}}_{c}^{\rm
1D}=\frac{\alpha_{s}T}{2\pi}\int_{-\infty}^{\infty}dx\,\phi\left(m_{D}|x|\right)\cos(qx).$

Integrating by parts, we get:

$\displaystyle{\rm Im}{\tilde{V}}_{c}^{\rm
1D}=\frac{\alpha_{s}T}{2}\left[\frac{q}{q^{2}+m_{D}^{2}}+\frac{1}{m_{D}}\left(\arctan\left(\frac{q}{m_{D}}\right)-\frac{\pi}{2}\right)\right],$

where $-\frac{\pi}{2}$ is added to the primitive to get rid of its asymptotic
value. Similarly, the spectral decomposition of the string part is given by

$\displaystyle{\rm Im}{\tilde{V}}_{s}^{\rm 1D}=\frac{\sigma T}{2\pi
m_{D}^{2}}\int_{-\infty}^{\infty}dx\,\chi\left(m_{D}|x|\right)\cos(qx),$

and the integration by parts yields

$\displaystyle{\rm Im}{\tilde{V}}_{s}^{\rm 1D}=-\frac{\sigma
T}{m_{D}^{3}}\left[\frac{2}{\sqrt{{\Delta_{D}}^{2}+\frac{q^{2}}{m_{D}^{2}}}\left(1+\frac{q^{2}}{m_{D}^{2}}\right)^{2}}+\frac{{\Delta_{D}}^{2}\left(1-3\frac{q^{2}}{m_{D}^{2}}\right)-4\frac{q^{4}}{m_{D}^{4}}}{\left(1+\frac{q^{2}}{m_{D}^{2}}\right)^{3}\left({\Delta_{D}}^{2}+\frac{q^{2}}{m_{D}^{2}}\right)^{3/2}}\right].$

As shown in Fig. 22, the spectral decomposition ${\rm Im}{\tilde{V}}^{\rm 1D}$
is negative for all q and at any temperatures considered here. We can thus
conclude that this one-dimensional potential satisfies the positivity
requirement of the Lindblad equations.

##  V Comparison with other one-dimensional potential models

The one-dimensional potential described in Sec. III and IV (from now on called
“model A”) can be compared to the reduction of the HTL potential proposed in
Sec. I.2 (from now on called “model C”), and to other potentials found in the
literature [21, 24, 27, 29]. Among the latter we choose the potential used in
[27, 29] (“model B”), given by:

$\displaystyle{\rm Re}V^{\rm
1D}(x,T)=-\frac{\alpha}{\sqrt{x^{2}+x_{c}^{2}}}e^{-m_{D}|x|},$ (40)

for the real part, and

$\displaystyle{\rm Im}V^{\rm 1D}(x,T)=\gamma-\gamma\exp(-x^{2}/l_{\rm
corr}^{2}).$ (41)

for the imaginary part. The parameters are taken to be $x_{c}=1/m$, $l_{\rm
corr}=1/T$, $m_{D}=2T$ and $\gamma=T/\pi$. The imaginary part given by Eq. 41
is shown in Fig. 23 for different temperatures. The three models A, B and C
are summed up in Tab. 6.

Figure 23: Imaginary part of the one-dimensional potential used in [27, 29]. It can be compared to the potential proposed in Sec. IV shown in Fig. 20 : their behaviour are qualitatively different for $T<200$ MeV.  1D potential |  Model A |  Model B |  Model C
---|---|---|---
Described in |  Sec. III and IV |  Sec. V [27, 29] |  Sec. I.2 (HTL 1D)
Real part |  $1/2\,K|x|\,e^{-\mu_{1}|x|}+\mu_{2}$ saturated from some $|x|$ |  $-\frac{\alpha}{\sqrt{x^{2}+x_{c}^{2}}}e^{-m_{D}|x|}$ |  $\frac{\sigma}{m_{D}}\bigl{(}1-e^{-m_{D}|x|}\bigr{)}$
Imaginary part |  $\alpha\,\mathrm{Im}V_{C}\left(|x|\right)+\beta\,\mathrm{Im}V_{S}\left(|x|\right)$ (see Eq. 16 and 18) |  $\gamma-\gamma\exp(-x^{2}/l_{\rm corr}^{2})$ |  $-2\sigma m_{D}^{2}T\int_{\,0}^{\,\infty}dq\frac{1-\cos{q|x|}}{q(q^{2}+m_{D}^{2})^{2}}$
Table 6:  The 3 models of one-dimensional potential that are compared in this
section.

As shown in Fig. 24, the energy spectra $E_{n}$ \- and the binding energies
given by ${\rm Re}V(r\rightarrow\infty)-E_{n}$ \- obtained from the real parts
of the models B and C are qualitatively and quantitatively very different from
what is expected from the three-dimensional potential inspired by lattice QCD
(described in Sec. II). Within the range of temperature studied here, these
two models only give one bound S state. At the opposite, the model A was
parameterized to fit at best the energy spectrum given by the three-
dimensional potential (see Fig. 17). In terms of phenomenology, the real parts
of the models B and C are therefore irrelevant.

Figure 24: (a) Charmonium mass spectrum of the S states obtained with the real
part of the one-dimensional model B [27, 29] (solid lines) using the
expectation values of the Hamiltonian compared to the results obtained with
the three-dimensional potential (dashed lines). (b) Same comparison with the
one-dimensional model C (1D HTL).

Figure 25: (a) Charmonium decay widths obtained with the imaginary parts of
the three models and the real part of the model A compared to the results
obtained with the three-dimensional potential (solid lines). (b) Same for
bottomonia.

Because the eigenstates obtained with the three models are very different,
comparing the decay widths given by each full model - i.e. using the
eigenstates given by the real part to compute the expectation value of the
imaginary part - is meaningless. To compare the imaginary parts of the three
models on a common basis, we will only use the eigenstates given by the model
A (see Sec. III). In Fig. 25, the decay widths obtained with the imaginary
parts of the three models and the real part of the model A are compared to the
ones of the three-dimensional potential. The decay widths given by the model C
(1D HTL) overestimate or underestimate quite strongly the three-dimensional
results and show a different qualitative behaviour (exception made of the
$J/\psi$-like state). The decay widths obtained with the model B globally
underestimate the three-dimensional results but seems qualitatively relevant.
As shown in Fig. 26, if one re-scales the imaginary part of the model B, i.e.
multiplies the $\gamma$ parameter by some factor, one can get a much better
quantitative match. Although the correspondence with the three-dimensional
results is slightly less good with the adjusted model B than with the model A
(the $J/\psi$-like state excepted), the adjusted model B can be seen as a
simple alternative to model A (if one considers the decay widths as the only
criterion of choice).

Figure 26: (a) Same as Fig. 25 (a), with an extra global factor for the
imaginary part of the model B, i.e. by multiplying the $\gamma$ parameter by
$4.2$. (b) Same for bottomonia, with a global factor of $3.2$.

## Conclusion

In this paper, we have analysed the behavior and properties of two three-
dimensional complex potentials for in-medium quarkonium physics, derived from
the hard thermal loop perturbation theory or inspired by lattice QCD data [6,
10]. Based on this analysis, we proposed a one-dimensional complex potential
parameterized to reproduce at best two key properties – the temperature-
dependent masses of the eigenstates and their decay widths – of the three-
dimensional lattice QCD inspired potential. The real part of this one-
dimensional potential is given by Eqs. 36 and 37 with the set of parameters
summed up in Tab. 4. The vacuum spectra, the temperature-dependent masses and
the dissociation temperatures of the states correspond in a good approximation
to the ones given by the three-dimensional potential. The features of the one-
and three-dimensional eigenstates are roughly similar at any temperature
relevant for phenomenology, despite their difference of nature (Gaussian-like
for a linear-like potential and exponential-like for the radial part of a
Coulombic-like potential). The imaginary part of this one-dimensional
potential, given by Eq. 38 and the parameters in Tab. 5, is a simple
“symmetrization” of the three-dimensional potential radial component. The
thermal decay widths obtained with the one- and three-dimensional potentials
are similiar up to $T=400$ MeV ($T=600$ MeV) for the charmonia (bottomonia).
Note that within this model, the one-dimensional potential for charmonia and
bottomonia are different. As a sanity check, we calculated the spectral
decompositions of both the one- and three-dimensional potentials and ensured
their compatibility with the positivity of the Lindblad equation. The proposed
potential can thus be used in the resolution of one-dimensional Lindblad
equations, or related stochastic Schrödinger equations, in order to reproduce
in a good approximation the quarkonium phenomenology in heavy ion collisions.
We also compared the proposed potential to a possible reduction of the HTL
potential in one-dimensional and to a potential found in the literature [27,
29]. The behaviour of the real parts of these two potentials is incompatible
with the energy spectra given by the three-dimensional lattice QCD inspired
potential. Nevertheless, combining the real part of the proposed potential and
the imaginary part of the model found in the literature (rescaled by some
factor) reasonably matches the thermal widths of the three-dimensional
potential. This combination can thus be seen as a simple alternative to the
main proposal in this work. We finally emphasize that one-dimensional dynamics
remains inherently limited and does not aim to reproduce the whole physics at
stake. Nevertheless, it can still be used as a good proxy for phenomenology
while being much less demanding from a computational viewpoint.

## Acknowledgments

We wish to thank Thierry Gousset and Alexander Rothkopf for discussions and
help. The authors thank the Région Pays de la Loire and Subatech for support.
R.K. is under contract No. 2015-08473. S.D. is supported by the Centre
national de la recherche scientifique (CNRS) and Région Pays de la Loire.

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# Expressive Text-to-Speech using Style Tag

###### Abstract

As recent text-to-speech (TTS) systems have been rapidly improved in speech
quality and generation speed, many researchers now focus on a more challenging
issue: expressive TTS. To control speaking styles, existing expressive TTS
models use categorical style index or reference speech as style input. In this
work, we propose StyleTagging-TTS (ST-TTS), a novel expressive TTS model that
utilizes a style tag written in natural language. Using a style-tagged TTS
dataset and a pre-trained language model, we modeled the relationship between
linguistic embedding and speaking style domain, which enables our model to
work even with style tags unseen during training. As style tag is written in
natural language, it can control speaking style in a more intuitive,
interpretable, and scalable way compared with style index or reference speech.
In addition, in terms of model architecture, we propose an efficient non-
autoregressive (NAR) TTS architecture with single-stage training. The
experimental result shows that ST-TTS outperforms the existing expressive TTS
model, Tacotron2-GST in speech quality and expressiveness.

Index Terms: speech synthesis, expressive TTS, language model, non-
autoregressive TTS

##  1 Introduction

Recently, end-to-end (e2e) speech synthesis systems have achieved significant
improvement and now show almost human-like speech quality. Among e2e speech
synthesis systems, autoregressive (AR) models such as Tacotron2 [1] and
Transformer TTS [2] first showed state-of-the-art performance by exploiting
the attention mechanism. However, as AR models suffer from slow generation
speed and a lack of stability due to attention failure, non-autoregressive
(NAR) models [3, 4, 5, 6, 7, 8] have been proposed in recent years. By
explicitly modeling the duration of each text unit and generating speech in
parallel, NAR models can synthesize speech much faster than real-time with
comparable speech quality with AR models. Such development enables e2e speech
synthesis systems to be successfully adopted in many real-world applications.

Meanwhile, to imitate the non-phonetic expressions of human utterance,
expressive TTS [9, 10, 11, 12, 13, 14] has been studied in line with the
development of the e2e TTS systems. In expressive TTS, the speaking style is
modeled in a supervised or unsupervised manner and usually combined with an
existing TTS architecture. In terms of style control, there are two approaches
for expressive TTS. Several models exploit the categorical style labels, which
indicate speaking styles such as emotions [9, 10, 11]. Trained with a style-
labeled dataset in a supervised manner, these models can control style using
explicit labels. However, there are limitations in the diversity of
expression, as these models can only express a few pre-defined styles. On the
other hand, some models use reference speech as style input [12, 13, 14]. In
these models, style information is extracted from the reference speech using a
reference encoder and transferred to the generated speech. Although these
methods don’t require a labeled dataset and can express unbounded speaking
styles, the extracted style information is not intuitive and interpretable.
Moreover, choosing a reference speech every time is time-consuming and memory
inefficient, which makes these methods less practical.

In this paper, we introduce a novel style interface for expressive TTS: style
tag. Style tag is a short phrase or word representing the style of an
utterance, such as emotion, intention, and tone of voice. As style is tagged
in the natural language, it is intuitive and interpretable for controlling the
speaking style. In addition, we propose Style Tagging TTS (ST-TTS), which is a
novel non-autoregressive expressive TTS model utilizing style tag as a style
input. Using a pre-trained language model as an interpreter for transforming
natural language to linguistic embedding, ST-TTS can learn the relationship
between linguistic embedding and style embedding space. Due to the
generalization capabilities of the language model, even a style tag unseen in
the training dataset can be expressed by ST-TTS. Besides, ST-TTS also has
several advantages in model architecture. To find alignments between text and
speech, ST-TTS uses Monotonic Alignment Search (MAS) algorithm [5] and
normalizing flow (NF) [15, 16] as an aligner. As the aligner is jointly
optimized with the entire TTS system, ST-TTS can be trained in single-stage,
unlike other NAR models such as FastSpeech [3, 4] and SpeedySpeech [8].
Additionally, at the inference phase, we can optionally use either style tag
or reference speech as a style input by building a shared embedding space of
reference speech and style tag.

This paper is organized as follows. In Section 2, we first describe the
dataset we used. Then we explain the proposed ST-TTS model in section 3, and
experiment results are reported in section 4. Lastly, the conclusion and
discussion are covered in section 5.

##  2 Style Tagging Dataset

As existing expressive TTS datasets [17, 18, 19] have only a few numbers of
speaking styles, a dataset with much more various styles is required for our
proposed method. We introduce a Korean Style Tagging TTS dataset:
FSNR0111FSNR0 is a part of SMART-TTS corpus, a Korean TTS corpus for an
expressive TTS project named SMART-TTS. We are planning to release this corpus
as an open-resource.. FSNR0 consists of {speech, transcript, style tag} tuples
and the entire transcripts and style tags are written in Korean. The
transcripts are recorded by a voice actress expressing given style tags.

In FSNR0, There are 327 style tags including various emotions, intentions,
voice tone and speed. Each style tag appears from one to hundreds of times in
the whole dataset except for the normal reading book style, which takes up
about 25% of the total data. The entire speech data is about 26 hours long and
consists of about 18,700 sentences. Several examples of style tags are shown
in Table 1. In training, the style tags are augmented by several rules such as
switching adverbs and adjectives, and extending keywords to plausible phrase
forms. One of the augmented style tag is sampled every iteration.

Table 1: Examples of style tag. We additionally provide the translated ones to help understanding. style tag | translated | style tag | translated
---|---|---|---
다정하게 | with affection | 매정하게 | heartless
화가난 듯 | seem angry | 기쁜 듯 | pleased
씁쓸한 듯 | bitter | 다급하게 | in a hurry
큰 소리로 | in a loud voice | 속삭이듯 | whispering
졸린 듯 | sleepy | 술취한 듯 | drunken

(a) speech synthesis procedure

(b) modules for duration extraction and prediction

Figure 1: the entire architecture of ST-TTS

##  3 Style Tagging TTS

###  3.1 Model architecture

In this section, we introduce the ST-TTS architecture. Our proposed model
takes grapheme sequence and style tag as input and returns log mel-
spectrogram. Reference speech is additionally used for training and optionally
used for inference. The entire structure is based on non-autoregressive
feedforward TTS models [3, 4, 8], and end-to-end trainable by jointly
optimizing duration information with the entire system. The entire
architecture is shown in Figure 1, and described as follows.

####  3.1.1 Sentence BERT

As style tag is an infinite set, we used a pre-trained language model to embed
a style tag to a meaningful space without Out-Of-Vocabulary (OOV) problem. It
is the core component in ST-TTS that enables generalization of unseen style
tags. We employed Sentence BERT (SBERT) [20] for our proposed model. BERT [21]
is one of the most widely used representation learning for natural language
process (NLP). It is composed of Transformer [22] blocks and trained with
masking and slot filling task. Although BERT performs well in various NLP
tasks such as machine translation, it lacks the ability to extract a sentence-
level embedding. SBERT is a fine-tuned model of BERT optimized for sentence-
level similarity using contrastive learning. SBERT shows better performance in
measuring sentence similarity and clustering sentences than original BERT. We
regard style tags as extremely short sentences and used pre-trained SBERT as a
component for the style encoder.

####  3.1.2 Style encoder

Style encoder is a module that extracts style embedding for the TTS system,
and it consists of a reference encoder and a style tag encoder. Reference
encoder extracts style embedding from reference speech. It has almost the same
architecture with reference encoder of Global Style Tokens (GSTs) [12]. The
difference is that the proposed module does not use style token, and batch
normalization [23] layers are replaced with weight normalization [24]. Style
tag encoder is a module for extracting a style embedding from a style tag. It
consists of pre-trained SBERT and adaptation layers. As mentioned in 3.1.1,
SBERT transforms style tag to semantic embedding. Adaptation layers are made
up of three linear layers with ReLU activation and learn to map from
linguistic semantic space to style embedding space.

In training, style embedding from reference encoder is used for TTS system,
and then the embedding from style tag encoder learns reference embedding by
mean squared error (MSE) loss named style embedding loss. This bi-modal
conditioning method has the following advantages. 1) By building a bi-modal
embedding space of style tag and reference speech, either style tag or
reference speech can be used to extract a style embedding for inference. 2)
Style tag embedding functions as an anchor of reference speeches with the same
style tag, which means the reference encoder learns to form clusters of
reference speeches with the same style tags. As speeches with similar speaking
styles are mapped adjacently in embedding space by the reference encoder,
style tags with similar meanings are also located closely in embedding space.
This property helps to form a well-defined embedding space, and adaptation
layers can easily learn the relationship between style domain and linguistic
domain. While training, we freeze SBERT, and the other components are
optimized for the entire TTS loss and the style embedding loss.

####  3.1.3 Text encoder

Text encoder takes a grapheme sequence and returns a text embedding sequence.
it consists of residual dilated convolution blocks used in parallel
wavegan222referred code : https://github.com/kan-bayashi/ParallelWaveGAN [25].
We used 12 residual blocks with dilation rates (1, 2, 4) 4 times. The kernel
size and hidden dimensions are 5 and 256 respectively.

The output embedding sequence is used for the aligner, duration predictor, and
mel decoder. When used for mel decoder, the text embeddings are copied
multiple times to match text and speech length using duration information from
aligner or duration predictor.

####  3.1.4 Aligner

Aligner calculates alignment between text and log mel-spectrogram and returns
duration of each grapheme. For aligner, we exploit the method devised in Glow-
TTS which uses normalizing flow (NF) and Monotonic Alignment Search (MAS)
algorithm [5]. To calculate alignment, mel-spectrogram $x_{1:T}$ is first
mapped to latent variable $z_{1:T}$ by NF $f^{-1}_{NF}:x\rightarrow z$. Text
encoder output is then projected to the latent space with a linear layer and
regarded as parameter $\mu_{1:N}$ of the latent variable. $T$ and $N$ denote
the length of mel-spectrogram and text tokens respectively. For a possible
alignment $A$, $A(j)=i$ represents $z_{j}$
$\sim\mathcal{N}(\mu_{i},\sigma_{i}$), where $\sigma_{i}$ of the entire frame
is set to 1 for stable training. By change of variables formula used in NF,
the conditional log likelihood of mel-spectrogram given text input $c_{1:N}$
and an alignment $A$ can be expressed as Eq. (1) and Eq. (2):

$\log{P_{X}}(x|c;A)=\log{P_{Z}}(z|c;A)+\log\left|\det\frac{\partial
f_{NF}^{-1}(x)}{\partial x}\right|,$ (1)
$\log{P_{Z}}(z|c;A)=\sum_{j=1}^{T}\log\mathcal{N}(z_{j};\mu_{A(j)},\sigma_{A(j)}).$
(2)

The optimal alignment path $A^{*}$ for maximizing Eq. (1) can be obtained by
the MAS algorithm. In training, NF and text encoder are optimized for
maximizing $log{P_{X}}(x|c;A^{*})$, and duration for length expansion is
calculated by summation $A^{*}$ over mel-spectrogram axis.

In Glow-TTS, NF is itself a generator of mel-spectrogram, but we only exploit
NF as an aligner whose role is just providing an accurate alignment. We used
the same architecture with Glow-TTS333referred code :
https://github.com/jaywalnut310/glow-tts in a much smaller size. The NF is
consists of 6 flow blocks, and the affine coupling layers in each flow block
are a stack of 4 residual convolution layers with kernel size 5 and hidden
dimension 128. Unlike Glow-TTS, we did not squeeze mel-spectrogram for better
alignment.

As this aligning method only considers monotonic alignments between text and
speech, ST-TTS can be trained much faster and robustly compared with the
attention mechanism generally used for TTS alignment. In addition, it does not
require auxiliary methods such as guided attention loss [26] or positional
encoding [22] for attention mechanism.

####  3.1.5 Duration predictor

Duration predictor predicts the number of frames for each grapheme in log
scale. Duration predictor is made up of the same residual convolution blocks
used in the text encoder. As different style yields different speaking speed,
duration predictor also takes a style embedding to predict style dependent
duration. Style embedding is duplicated to the equal length of text embedding
and conditioned to each residual block by the same method used in [25].
Duration predictor is composed of five residual blocks without dilation, and
kernel size and hidden dimension are 5 and 256.

####  3.1.6 Mel decoder

Mel decoder transforms duration-expanded text embedding to log mel-spectrogram
conditioned on a style embedding. It has the same architecture with the
duration predictor in a much larger size. Mel decoder consists of 30 residual
blocks with dilation rate (1, 2, 4, 8, 16) 6 times. The kernel size used in
the mel decoder is 3, and the hidden dimension is 256.

###  3.2 Training and inference

In training, the total loss is the sum of the following four losses.

  * •

Mel reconstruction loss: mean-absolute-error (MAE) between predicted and
target log mel-spectrogram.

  * •

Duration loss: huber loss [27] between output of duration predictor and log
duration acquired by aligner.

  * •

Alignment loss: negative log likelihood loss for training aligner described in
3.1.3

  * •

Style embedding loss : MSE loss between style embeddings from reference
encoder and style tag encoder as described in 3.1.2

These losses are summed with equal weights and jointly optimized for the
entire model except for SBERT in style encoder.

During inference, ST-TTS uses duration from the duration predictor. As the
duration predictor predicts duration in log scale, it is transformed by
exponential operation and rounded to the nearest integer. To prevent skipping
text units, the predicted durations less than one are clamped to one. We can
optionally use style embedding extracted from reference speech or style tag
depend on user convenience, as mentioned in 3.1.2.

##  4 Experiments

###  4.1 Experimental setup

For our experiments, we used the FSNR0 dataset described in section 2. The
audio is downsampled from 48 kHz to 22.05 kHz, and 80-dimensional log mel-
spectrogram was used for the target and reference speech. For input text, we
used Korean grapheme without G2P. Style tags are tokenized by pre-trained
SBERT. We trained our model using the Adam optimizier [28] with learning rate
of 0.02 and the Noam learning scheduler [22]. Total training took about 87
hours for 500k iteration with a single GeForce GTX 2080 GPU. As a vocoder, we
used HiFiGAN [29] which shows state-of-the-art audio quality for speech
synthesis. For the subjective evaluations, 18 native Korean participants
joined the tests described in 4.2.1 and 4.2.2. The audio samples of ST-TTS are
available on the demo website444demo site:
https://gannnn123.github.io/styletaggingtts-demo.

###  4.2 Evaluation

####  4.2.1 Audio quality

We measure the 5 scale Mean Opinion Score (MOS) to estimate the speech quality
of the proposed model. Since ST-TTS is the first TTS model which can control
style with the undefined label or reference speech, we chose Tacotron2-GST
[12] as a baseline. The result of five groups: ground truth (GT), GT with
vocoder reconstruction, Tacotron2-GST, ST-TTS (reference speech), ST-TTS
(style tag) is shown in Table 2. ST-TTS (reference speech) and ST-TTS (style
tag) mean ST-TTS using reference speech and style tag for style input
respectively. The reference speech used for Tacotron2-GST and ST-TTS
(reference speech) is the target speech itself because it shows better
performance for these models. We tested 30 samples each for normal reading
book style samples and expressive samples with style tags, randomly chosen
from the test set. According to Table 2, ST-TTS with either style tag or
reference speech shows better speech quality than the baseline.

Table 2: Mean Opinion Scores with 95% confidence intervals Method | Normal | Expressive
---|---|---
GT |  4.77 $\pm$ 0.04 |  4.80 $\pm$ 0.04
GT (+ HifiGAN) |  4.64 $\pm$ 0.05 |  4.67 $\pm$ 0.05
Tacotron2-GST |  3.37 $\pm$ 0.08 |  3.44 $\pm$ 0.08
ST-TTS (reference) |  4.15 $\pm$ 0.06 |  4.12 $\pm$ 0.06
ST-TTS (style tag) |  4.17 $\pm$ 0.06 |  4.08 $\pm$ 0.06

####  4.2.2 Preference test

To demonstrate that the style tag can control the style of generated speech
intuitively, we tested a preference test with ST-TTS and Tacotron2-GST. In
this test, the participants listened to two samples from ST-TTS and
Tacotron2-GST, and rated which sample reflected the given style tag better in
7 scale555The score is rated in the criterion: A is ( ) than B. Each score is
described below.
-3: much worse, -2: worse, -1: little worse, 0: about same, 1: little better, 2: better, 3: much better. As Tacotron2-GST cannot directly use the style tag, we used the average style token weights of 10 utterances tagged with the given style tag. We regarded these weights as a representation of the given style tag for Tacotron2-GST. We tested 30 sentences with randomly selected style tags from the test set. The transcripts are sampled from another dataset. As a result, ST-TTS got the preference score 1.37 ($\pm$0.14, 95% confidence interval), where the score is ranged in (-3, 3), and a higher score means better performance. This result shows that ST-TTS can express various speaking styles recognized as reflecting style tags.

####  4.2.3 Visualization of style tag embedding

In Figure 2, we visualized the style embeddings of style tags using t-SNE
[30]. There are all the style tags in the dataset and several style tags
unseen in training. This visualization shows that style tags with similar
meanings are embedded adjacently. Particularly, we can see that even the
unseen style tags are properly located in the embedding space, which means the
available style tag is not limited to the training dataset.

Figure 2: Style embedding visualization using t-SNE. The entire t-SNE plot is
at the center, and five local regions are enlarged for closer observation. The
blue points and red points indicate the style tags seen and unseen in
training, respectively. The translations with similar nuance are provided to
help understanding.

##  5 Conclusion and Discussion

In this paper, we proposed a novel expressive TTS model named ST-TTS, which
uses style tag as a style interface. ST-TTS has several advantages in
convenience and performance. 1) As style tag is written in natural language,
users can intuitively control the style of generated speech with ST-TTS. The
language model, SBERT enables ST-TTS to express even the style tags unseen
during training. 2) ST-TTS has a bi-modal embedding space of reference speech
and style tag, so either reference style tag or reference speech can be used
in inference. 3) ST-TTS is a non-autoregressive TTS model which can be trained
in single-stage. For this reason, ST-TTS can be easily trained and has fast
generation speed.

As ST-TTS is the first TTS model that uses style tag, there are still
remaining research topics as our future works. The most important issue in ST-
TTS is the generalization performance for style tags. There are countless
style tags in natural language, and the model should express as many style
tags as possible for human-level expressiveness. This issue can be addressed
in two directions. In terms of data configuration, it is important to use as
many style tags as possible in training. We can augment style tags using
synonyms to expand the covered linguistic embedding region. Next, as style
tags are usually much shorter than common sentences, investigating a language
model which can embed short phrases better can be a solution for better
generalization. Meanwhile, there may be styles that cannot be expressed with a
single style tag. We are planning to consider the methods using multi-style
tags to express various characteristics together.

##  6 Acknowledgements

This work was supported by the Institute of Information & Communications
Technology Planning & Evaluation (IITP) funded by the Korea Government (MSIT)
under Grant 2020-0-00059 (Deep learning multi-speaker prosody and emotion
cloning technology based on a high quality end-to-end model using small amount
of data).

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