# Functional Renormalization Group Study of Thermodynamic Geometry Around the
Phase Transition of Quantum Chromodynamics

Fabrizio Murgana  Department of Physics and Astronomy, University of Catania,
Via S. Sofia 64, I-95125 Catania, Italy  INFN-Sezione di Catania, Via S. Sofia
64, I-95123 Catania, Italy  Institut für Theoretische Physik, Goethe-
Universität, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany
Vincenzo Greco  Department of Physics and Astronomy, University of Catania,
Via S. Sofia 64, I-95125 Catania, Italy  Laboratori Nazionali del Sud, INFN-
LNS, Via S. Sofia 62, I-95123 Catania, Italy  Marco Ruggieri  Department of
Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125
Catania, Italy  INFN-Sezione di Catania, Via S. Sofia 64, I-95123 Catania,
Italy  Dario Zappalà  INFN-Sezione di Catania, Via S. Sofia 64, I-95123
Catania, Italy  Centro Siciliano di Fisica Nucleare e Struttura delle Materia,
Via S. Sofia 64, I-95123 Catania, Italy

###### Abstract

We investigate the thermodynamic geometry of the quark-meson model at finite
temperature, $T$, and quark number chemical potential, $\mu$. We extend
previous works by the inclusion of fluctuations exploiting the functional
renormalization group approach. We use recent developments to recast the flow
equation into the form of an advection-diffusion equation. We adopt the local
potential approximation for the effective average action. We focus on the
thermodynamic curvature, $R$, in the $(\mu,T)$ plane, in proximity of the
chiral crossover, up to the critical point of the phase diagram. We find that
the inclusion of fluctuations results in a smoother behavior of $R$ near the
chiral crossover. Moreover, for small $\mu$, $R$ remains negative, signaling
the fact that bosonic fluctuations reduce the capability of the system to
completely overcome the fermionic statistical repulsion of the quarks. We
investigate in more detail the small $\mu$ region by analyzing a system in
which we artificially lower the pion mass, thus approaching the chiral limit
in which the crossover is actually a second order phase transition. On the
other hand, as $\mu$ is increased and the critical point is approached, we
find that $R$ is enhanced and a sign change occurs, in agreement with mean
field studies. Hence, we completely support the picture that $R$ is sensitive
to a crossover and a phase transition, and provides information about the
effective behavior of the system at the phase transition.

Functional renormalization group, thermodynamic geometry, quark-Meson model,
QCD phase diagram, chiral phase transition, critical endpoint, quark-gluon
plasma

###### pacs:

12.38.Aw,12.38.Mh

##  I Introduction

Within the theory of fluctuations among equilibrium states, one of the most
interesting ideas is given by the thermodynamic curvature [1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26],
which represents an innovative perspective in the field of thermodynamics.
This employs non-euclidean geometry to represent fluctuations and interactions
among thermodynamic variables. The geometric approach sheds new light on
understanding phase transitions and emergent properties in complex systems,
providing an intriguing connection between geometry and thermodynamics. In
fact, in the grand-canonical ensemble one can consider any pair of intensive
variables $(\beta^{1},\beta^{2})$: given these, the probability of a
fluctuation from the state $S_{1}=(\beta^{1},\beta^{2})$ to
$S_{2}=(\beta^{1}+\delta\beta^{1},\beta^{2}+\delta\beta^{2})$ is proportional
to

$\sqrt{g}\exp\left({-\frac{d\ell^{2}}{2}}\right).$ (1)

Here,

$\displaystyle
d\ell^{2}=g_{\beta^{1}\beta^{1}}d\beta^{1}d\beta^{1}+2g_{\beta^{1}\beta^{2}}d\beta^{1}d\beta^{2}+g_{\beta^{2}\beta^{2}}d\beta^{2}d\beta^{2},$
(2)

where

$\displaystyle
g_{ij}=\frac{\partial^{2}\log\mathcal{Z}}{\partial\beta^{i}\partial\beta^{j}}$
(3)

is the metric tensor in the 2-dimensional manifold; $\mathcal{Z}$ is the
grand-canonical partition function. Finally, $g$ is the determinant of the
metric tensor. $d\ell^{2}$ measures the distance between the states $S_{1}$
and $S_{2}$. Equipped with the metric, we introduce the thermodynamic
curvature, $R=2R_{1212}/g$, where $R_{1212}$ corresponds to the only
independent component of the Riemann tensor for a twodimensional manifold. $R$
depends on the second and third order moments of the thermodynamic variables
that are conjugated to $(\beta^{1},\beta^{2})$: it thus carries information
about the fluctuation of the physical quantities.

Within the context of thermodynamic geometry, interesting and recent
developments regard its applications to effective models, in particular to the
quark-meson (QM) model [27, 28]. The QM model is a well known low energy
effective model of Quantum Cromodynamics (QCD) [29, 30, 31, 32]; it has mostly
been used because it could qualitatively capture the chiral phase transition
of QCD, i.e the transition from chiral symmetry broken phase in QCD vacuum, to
a chiral symmetry restored phase at finite temperature, $T$, and quark number
chemical potential, $\mu$.

Figure 1: Phase diagram of the quark-meson model at finite temperature T and
chemical potential $\mu$. The chirally spontaneously (and explicitly) broken
phase and the (approximately) chirally symmetric phase are indicated in the
plot. The dashed line, indicating a crossover, and the 1st order phase
transition lines merge at the critical end point, labeled CP.

In Fig. 1 we show a cartoon phase diagram of the QM model in the $(\mu,T)$
plane; the diagram is drawn assuming a finite quark mass which explicitly
breaks the chiral symmetry. In the figure, the dashed line corresponds to a
smooth crossover, while the solid line to a first order transition; in both
cases, chiral symmetry is spontaneously broken in the low temperature phase
while it is (approximately) restored in the high temperature phase. The two
lines meet at a point, known as the critical endpoint (CP). At this point, the
crossover becomes a real second order phase transition with a divergent
correlation length. The purpose of the present work is to study the
thermodynamic curvature around the crossover, both at small $\mu$ and in the
proximity of CP.

One of the most viable uses of the QM model is related to the study of the
strongly interacting matter at finite $T$ and/or $\mu$. Similarly to the
already discussed phase diagram of the QM model, first principle calculations
performed in Lattice-QCD (LQCD) [33, 34, 35] show that at $\mu=0$, a smooth
crossover happens between a low-temperature phase, in which chiral symmetry is
spontaneously broken and the relevant degrees of freedom are hadrons, to a
quark-gluon plasma phase at high temperature where chiral symmetry is
approximately restored and the relevant degrees of freedom are quarks and
gluons. In addition to this, it is commonly believed that there should exist a
critical endpoint in the $(T,\mu)$ plane in full QCD, similar to the one found
in the QM model, see [36, 37] for reviews. However, first principle
calculations are not feasible at finite $\mu$ due to the (in)famous sign
problem [33, 34, 35]. This is where effective models, like the QM model, can
help.

In this work, we study the QM model at finite $T$ and $\mu$, focusing of its
thermodynamic geometry; this is the natural continuation of previous works on
the same subject [27, 28, 38]. The aspect of novelty of the study is the
inclusion of the quantum fluctuations via the functional renormalization group
approach (FRG) [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62]; this is based on the ideas introduced by
Wilson [44, 45, 46, 47, 48] and others, see, e.g., Refs. [49, 50]. Since the
FRG method belongs to the class of non-perturbative approaches to quantum
field theories and it is built in such a way to link different energy scales
and the associated degrees of freedom, it is a suitable tool to deal with
second order phase transitions and critical long wave length phenomena in
general, whose nature is highly non-perturbative.

The reason why the investigation of the critical behaviour of the QM model via
the thermodynamic geometry is worthy lies in the information one can extract
from $R$ on the system. Particularly, it has been argued that the sign of $R$
is connected to the dominance of a fermionic or bosonic-like behavior of the
system [14, 15, 9, 63]: a positive $R$ corresponds to a boson-like behavior
while a negative $R$ to a fermion-like behavior. This behavior has been
understood as attraction or repulsion in the phase space [14, 15, 9, 63].
Therefore, the study of the sign of $R$ near a second order phase transition
(and to some extent, around a crossover [27, 28, 38]) can reveal information
about the effective dynamics which develops in the proximity of this
transition. Another interesting feature of $R$, anticipated in [27, 28, 38],
is linked to the presence of a peak structure close to the crossover
temperature for small values of $\mu$, which eventually should turn into a
divergence when the critical point is reached. We complete the studies of [27,
28, 38] by analyzing these aspects of $R$ within the FRG approach.

This article is structured as follows: In Sec. II, we briefly introduce the
thermodynamic geometry. In Sec. III, we discuss the basic features of the
quark-meson model. In Sec. IV, we present a short introduction to the
functional renormalization group and how this can be specifically applied to
the quark-meson model. In Sec. V, we discuss the results on the thermodynamic
geometry for the quark-meson model. Finally, in Sec. VI we draw our
conclusions and present an outlook. Within this article we use natural units
$\hbar=c=k_{B}=1$.

##  II Thermodynamic Curvature

We consider a thermodynamic system in the grand-canonical ensemble whose
equilibrium state is characterized by the pair $(\mu,T)$, where $T$ is the
temperature and $\mu$ is the quark number chemical potential (conjugated to
quark number density). Thermodynamic geometry is more conveniently defined in
terms of the variables $(\beta=1/T\,,\,\gamma=-\mu/T)$. A thermodynamic system
at equilibrium at the point $(\beta,\gamma)$ can fluctuate to another
equilibrium state $(\beta^{\prime},\gamma^{\prime})$, and the probability of
this fluctuation can be computed within the standard thermodynamic fluctuation
theory. In fact, as already mentioned in the introduction, we firstly define a
distance in the two-dimensional manifold spanned by $(\beta,\gamma)$,

$\displaystyle d\ell^{2}=g_{\beta\beta}d\beta d\beta+2g_{\beta\gamma}d\beta
d\gamma+g_{\gamma\gamma}d\gamma d\gamma$ (4)

where the metric tensor is

$\displaystyle
g_{ij}=\frac{\partial^{2}\log\mathcal{Z}}{\partial\beta^{i}\partial\beta^{j}}=\frac{\partial^{2}\phi}{\partial\beta^{i}\partial\beta^{j}}\equiv\phi_{,ij},$
(5)

with $\phi=\beta P$, $P=-\Omega$ and $\Omega$ denotes the thermodynamic
potential density; moreover, we used the standard notation $\beta^{1}=\beta$
and $\beta^{2}=\gamma$. Given these, the fluctuation probability is

$\frac{dp}{d\beta
d\gamma}\propto\sqrt{g}\exp\left({-\frac{d\ell^{2}}{2}}\right).$ (6)

where

$\displaystyle g=g_{\beta\beta}g_{\gamma\gamma}-g_{\beta\gamma}^{2}$ (7)

is the determinant of the metric. Large probability of a fluctuation
corresponds to small $d\ell^{2}$. Therefore, a large thermodynamic distance
between two equilibrium states means a small probability to fluctuate between
the two states. According to these considerations, Eq. (4) measures the
distance in the $(\beta,\gamma)$ plane between two thermodynamic states in
equilibrium.

Thermodynamic stability requires that $g_{\beta\beta}>0$ and $g>0$, while
$g=0$ corresponds to a phase boundary and regions with $g<0$ are
thermodynamically unstable: hence, the stability conditions ensure that
$d\ell^{2}>0$. Furthermore,

$\displaystyle V\phi_{,\beta\beta}=\langle(U-\langle U\rangle)^{2}\rangle,$
(8) $\displaystyle V\phi_{,\beta\gamma}=\langle(U-\langle
U\rangle)\rangle\langle(N-\langle N\rangle)\rangle,$ (9) $\displaystyle
V\phi_{,\gamma\gamma}=\langle(N-\langle N\rangle)^{2}\rangle.$ (10)

where $U$ and $N$ denote the internal energy and the particle number,
respectively, while $V$ stands for the volume of the system.

Once the manifold has been provided with the metric tensor, one can define the
Riemann tensor as

$R^{i}_{klm}=\frac{\partial\Gamma^{i}_{km}}{\partial
x^{l}}-\frac{\partial\Gamma^{i}_{kl}}{\partial
x^{m}}+\Gamma^{i}_{nl}\Gamma^{n}_{km}-\Gamma^{i}_{nm}\Gamma^{n}_{kl},$ (11)

with the Christoffel symbols

$\Gamma^{i}_{kl}=\frac{1}{2}g^{im}\left(\frac{\partial g_{mk}}{\partial
x^{l}}+\frac{\partial g_{il}}{\partial x^{k}}-\frac{\partial g_{kl}}{\partial
x^{m}}\right).$ (12)

The standard contraction procedure gives the Ricci tensor
$R_{ij}=R^{k}_{ikj}$, and the scalar curvature $R=R^{i}_{i}$: within
thermodynamic geometry, $R$ is called the thermodynamic curvature. For the
two-dimensional manifold that we consider in this study the expression of $R$
considerably simplifies, namely [12]

$\displaystyle
R=-\frac{1}{2\,g^{2}}\begin{vmatrix}\phi_{,\beta\beta}&\phi_{,\beta\gamma}&\phi_{,\gamma\gamma}\\\
\phi_{,\beta\beta\beta}&\phi_{,\beta\beta\gamma}&\phi_{,\beta\gamma\gamma}\\\
\phi_{,\beta\beta\gamma}&\phi_{,\beta\gamma\gamma}&\phi_{,\gamma\gamma\gamma}\end{vmatrix},$
(13)

where $||$ indicates the determinant of the matrix. The curvature diverges for
$g\rightarrow 0$, which corresponds to a phase boundary, unless the numerator
of Eq. (13) vanishes on the same boundary.

Let $\xi$ denote the correlation length of the order parameter: then,
$|R|\propto\xi^{3}$ near a second-order phase transition [1], which naturally
results from hyperscaling. Theoretical calculations based on different models
confirm this hypothesis [1, 12, 64, 65, 66]; therefore, the study of $R$ in
the $(\mu,T)$ plane allows to estimate the correlation volume based only on
the thermodynamic potential: this is one of the merits of the thermodynamic
geometry. It has also been suggested that the sign of $R$ conveys details
about the nature of the interaction, attractive or repulsive, at a mesoscopic
level in proximity of the phase transition.

Within our sign convention, $R>0$ indicates an attractive interaction while
$R<0$ corresponds to a repulsive one. These interactions include not only real
interactions [19, 24, 17, 67, 68], but also the statistical attraction and
repulsion that ideal quantum gases feel in phase space [69, 70, 71, 72, 73]:
an ideal fermion gas has $R<0$ due to the statistical repulsion, while an
ideal boson gas has $R>0$ due to the statistical attraction. The thermodynamic
curvature is known to be identically zero only for the ideal classical gas.
Other fields of application of thermodynamic geometry include Lennard-Jones
fluids [17, 67, 68], ferromagnetic systems [74], black holes [75, 76, 77, 78,
79, 80, 17, 81, 82, 83, 84, 85, 86, 63, 87, 88, 89], strongly interacting
matter [90, 91, 27] and others [92, 93].

##  III The Quark-Meson Model

The Quark-Meson model (QM) can be understood as arising from the well-known
linear-sigma model coupled to fermions ([29, 30, 31, 32]). In particular, the
$N_{f}=2$ QM model uses as fundamental degrees of freedom four mesons, an
isotriplet of pion fields $\vec{\pi}=(\pi^{1},\pi^{2},\pi^{3})$ and an
isosinglet field $\sigma$, coupled to a massless isodoublet fermionic field
$\psi$ representing up and down quarks. The euclidean lagrangian density of
the model then reads

$\displaystyle\mathcal{L}^{E}_{QM}$ $\displaystyle=$
$\displaystyle\bar{\psi}(\gamma_{\mu}\partial^{\mu}+h(\sigma+i\gamma^{5}\vec{\tau}\vec{\pi}))\psi$
$\displaystyle+\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\vec{\pi})^{2}+U(\sigma^{2}+\vec{\pi}^{2})-c\sigma.$

in which $\gamma_{\mu}$ and $\gamma^{5}$ are the standard euclidean Dirac
matrices in Dirac space, $h$ is the strength of the Yukawa coupling between
quarks and mesons and $\vec{\tau}=(\tau^{1},\tau^{2},\tau^{3})$ represents the
vector of Pauli matrices in flavor space. $U(\sigma^{2}+\vec{\pi}^{2})$
indicates a generic interaction potential among the mesons, and it constructed
in order to be O$(4)$ symmetric, since it depends on the O$(4)$ invariant
mesonic combination $\sigma^{2}+\vec{\pi}^{2}$. Anyway, if the potential
develops a finite minimum along one radial direction, the O$(4)$ symmetry is
spontaneously broken into a O$(3)$ symmetry. Due to this residual symmetry,
one is free to choose the ground state in the vacuum as

$\langle\vec{\pi}\rangle=0,~{}~{}~{}\langle\sigma\rangle=f_{\pi}\neq 0,$ (15)

where $f_{\pi}=0.093$ GeV is the pion decay constant.

Even though the lagrangian does not contain an explicit mass term for the
fermions, when the symmetry is spontaneously broken they acquire a dynamical
constituent mass given by $M=h\langle\sigma\rangle$. Furthermore, the O$(4)$
symmetry is also explicitly broken by the term $-c\sigma$, which mimics the
presence of a finite current mass for quarks. In this way, the pions turn into
massive pseudo-Goldstone mesons, since the spontaneous symmetry breaking
pattern is not exact, acquiring a finite mass $M^{2}_{\pi}=c/f_{\pi}$.

##  IV The Functional Renormalization Group

In this section, we provide a brief introduction to the FRG, following the
interpretation provided by Wetterich et al. [51, 52, 53, 54], see [39, 40, 41,
42, 43] for reviews. When performing an FRG study, one changes the focus from
the usual generating functional of the theory, or the partition function, to
the effective action $\Gamma[\Phi]$, which serves as the generating functional
of 1PI vertex functions. Within the FRG formalism, similarly to Wilson’s
approach to renormalization, for the computation of $\Gamma[\Phi]$ one
integrates out the fluctuations by successive momentum shells; this is
achieved by introducing the effective average action, $\Gamma_{k}[\Phi]$,
which depends on $k$ that represents the momentum up to which fluctuations
have been effectively integrated out, thus serving as a coarse-graining scale.
In particular, the effective average action interpolates between the bare
classical action, $S_{\textrm{bare}}[\Phi]$, in the UV, i.e, at the cutoff
scale $k=\Lambda$ when no fluctuations have been taken into account, and the
full quantum effective action $\Gamma[\Phi]$ in the IR, i.e. $k=0$, when all
quantum fluctuations have been integrated out, namely

$\Gamma_{k\to\infty}[\Phi]=S_{\textrm{bare}}[\Phi],~{}~{}~{}\Gamma_{k\to
0}[\Phi]=\Gamma[\Phi].$ (16)

The evolution of the effective average action from the UV to the IR is
described by the FRG flow equation, that is [55, 52, 56]

$\partial_{k}\Gamma_{k}[\Phi]=\mathrm{Tr}\Big{[}\big{(}\tfrac{1}{2}\,\partial_{k}R_{k}\big{)}\big{(}\Gamma^{(2)}_{k}[\Phi]+R_{k}\big{)}^{-1}\Big{]}$
(17)

in which the trace is understood as the sum over internal degrees of freedom
of the theory, such as color, flavor, spin etc, as well as an integral over
Fourier momenta. $R_{k}$ in Eq. 17 is a regulator which effectively defines
the support of the momentum integral in the UV and acts as a screening term in
the IR. $R_{k}$ must be chosen in order for $\Gamma_{k}$ to fulfill the
conditions (16). In particular, in order to recover the full quantum effective
action in the IR, it has to vanish for $k\to 0$. Moreover, $R_{k}$ has to
diverge for large $k$ in order to ensure that the bare action represents a
stationary point for the path integral through which $\Gamma_{k}$ is
introduced.

$G_{k}[\Phi]\equiv({\Gamma}^{(2)}_{k}[\Phi]+R_{k})^{-1}$ in Eq. 17 indicates
the exact, scale dependent propagator, thus the FRG flow equation has a one-
loop structure. Despite this apparent simplicity, it still consists of a hard-
to-solve functional integro-differential equation; hence, except for very
special cases (see for example [57, 58, 59]), one has to rely on
approximations in order to solve it. A first possibility is called the vertex-
expansion [53, 60], in which $\Gamma_{k}[\Phi]$ is expanded in powers of the
fields $\Phi$ around a certain field configuration $\Phi_{0}$, and the
expansion coefficients are the vertex functions
$\Gamma_{k}^{(n)}(x_{1},\ldots,x_{n})=\delta^{n}\Gamma_{k}[\Phi]/\delta\Phi(x_{1})\cdots\delta\Phi(x_{n})\big{|}_{\Phi=\Phi_{0}}$.
Plugging the vertex expansion into the FRG flow equation, one obtains a system
of infinite coupled integro-differential equations which are then truncated at
certain finite order.

Another approximation scheme, which is the one we adopt in the present work,
is called the derivative expansion [43, 62], which approximates
$\Gamma_{k}[\Phi]$ in terms of powers of space-time derivatives of the fields.
Also in this case one would obtain an infinite system of coupled equations,
one per each operator compatible with the symmetries of the theory, and thus
for a practical solution one has to choose a order of space-time derivatives
of the fields to which the action is truncated. One of the main advantages of
the derivative expansion is that one does not need to assume an analytic
behaviour in field space of the effective action during the flow evolution,
which on the other hand is required by the vertex expansion.

In this work, we focus on critical phenomena which involve second order phase
transitions (in the chiral limit), crossovers and critical endpoints in the
phase diagram. It is known that, under these conditions, the free energy is
not arbitrarily differentiable (see for example [94, 95, 96, 41]);
furthermore, the effective action can develop discontinuities, or in general
singular points, when dealing with such a kind of phenomena [97, 98, 99, 100].
This suggests us to take advantage of the properties of the derivative
expansion, also at the lowest order called local potential approximation (LPA)
[100]. Particularly, we adopt the LPA ansatz for the QM model at finite
temperature and quark number density

$\displaystyle\Gamma_{k}[\bar{\Psi},\Psi,\phi]$ $\displaystyle=$
$\displaystyle\int_{0}^{1/T}\\!\\!dx_{4}\int\\!\\!d^{3}x~{}\left\\{\bar{\psi}\big{(}\gamma_{\mu}\partial^{\mu}+h(\sigma+i\gamma^{5}\vec{\tau}\vec{\pi})-\mu\gamma_{0}\big{)}\psi+\frac{1}{2}(\partial_{\mu}\sigma)^{2}+\frac{1}{2}(\partial_{\mu}\vec{\pi})^{2}+U_{k}(\sigma^{2}+\vec{\pi}^{2})-c\sigma\right\\}.$

Using the three-dimensional Litim regulator [101, 102] for both fermions and
bosons,

$\displaystyle R_{k,B}(p)$
$\displaystyle=(k^{2}-\bm{p}^{2})\Theta(k^{2}-\bm{p}^{2})$ (19) $\displaystyle
R_{k,F}(p)$
$\displaystyle=i\not{p}\left(\sqrt{\frac{k^{2}}{\bm{p}^{2}}}-1\right)\Theta\left(k^{2}-\bm{p}^{2}\right),$
(20)

we get the flow equation for the effective potential, that is

$\partial_{t}U_{k}(\sigma)=-\frac{k^{5}}{12\pi^{2}}\left\\{\left[\frac{1}{E_{k,\sigma}}\coth\left(\frac{E_{k,\sigma}}{2T}\right)+\frac{3}{E_{k,\pi}}\coth\left(\frac{E_{k,\pi}}{2T}\right)\right]-4N_{c}\frac{1}{E_{\psi}}\left[\tanh\left(\frac{E_{\psi}-\mu}{2T}\right)+\tanh\left(\frac{E_{\psi}+\mu}{2T}\right)\right]\right\\},$
(21)

with $t=-\ln{k/\Lambda}$ and

$\displaystyle
E_{k,\sigma}=\sqrt{k^{2}+\partial^{2}_{\sigma}U_{k}(\sigma)},~{}~{}~{}E_{k,\pi}=\sqrt{k^{2}+\frac{\partial_{\sigma}U_{k}(\sigma)}{\sigma}}.$
(22)

Moreover,

$E_{\psi}=\sqrt{\bm{p}^{2}+M^{2}},$ (23)

with $M=h\langle\sigma\rangle$ denoting the previously defined constituent
quark mass.

Analogously to e.g. [100, 103, 104], we introduce the following variables

$u_{k}(\sigma)=\partial_{\sigma}U_{k}(\sigma),~{}~{}~{}u^{\prime}_{k}(\sigma)=\partial_{\sigma}u_{k}(\sigma).$
(24)

Taking the derivative of equation (21) with respect to $\sigma$ we are able to
cast the FRG flow equation into an advection-diffusion equation with a source
term [98, 99], thus obtaining

$\partial_{t}u_{k}(\sigma)+\partial_{\sigma}f_{k}(\sigma,u_{k}(\sigma))=\partial_{\sigma}g_{k}(u^{\prime}_{k}(\sigma))+N_{c}\partial_{\sigma}S_{k}(\sigma)$
(25)

where we defined the advection flux as

$f_{k}(\sigma,u_{k}(\sigma))=\frac{k^{5}}{4\pi^{2}}\frac{1}{E_{k,\pi}}\coth\left(\frac{E_{k,\pi}}{2T}\right),$
(26)

the diffusion flux, namely

$g_{k}(u^{\prime}_{k}(\sigma))=-\frac{k^{5}}{12\pi^{2}}\frac{1}{E_{k,\sigma}}\coth\left(\frac{E_{k,\sigma}}{2T}\right),$
(27)

and finally the source term

$N_{c}S_{k}(\sigma)=\frac{N_{c}\,k^{5}}{3\pi^{2}}\frac{1}{E_{\psi}}\left[\tanh\left(\frac{E_{\psi}-\mu}{2T}\right)+\tanh\left(\frac{E_{\psi}+\mu}{2T}\right)\right].$
(28)

In this framework, the derivative of the potential $u_{k}(\sigma)$ plays the
role of a conserved quantity, in the sense that it satisfies a generalized
conservation law [97, 57, 105]. Each of the previously defined contributions
arises from the various particle species involved in the model. The advection
flux, which is responsible of the bulk motion of the conserved quantity $u$,
is originated from the pions. Indeed, there is a factor 3 that appears in Eq.
21 and multiplying the advection term. Furthermore, as it can be seen from the
definition of the energy $E_{k,\pi}$, the mass term for the pions,
$u_{k}(\sigma)/\sigma$, vanishes at the minimum of the effective potential, in
agreement with the nature of the pions as Goldstone bosons (since the explicit
symmetry breaking term is linear in the sigma field, it does not contribute to
the flow equation and is just added to the IR potential). One can verify that
the speed of characteristics $\partial_{u}f_{k}(\sigma,u_{k}(\sigma))$ is
positive if $\sigma<0$ and negative if $\sigma>0$, implying that the conserved
quantity $u_{k}(\sigma)$ and the minimum of the potential are always
transported towards smaller values of $\sigma$ by the advection.

On the other hand, the one radial sigma mode produces the diffusion term,
which depends on the curvature mass $u^{\prime}_{k}(\sigma)$. The diffusion
has no specific direction since it depends on the local gradients of the
conserved quantity, meaning that smears out peaks and discontinuities.

The fermionic loop gives rise to a time and $\sigma$ dependent source term,
which we identify as such since it is independent of the conserved quantity
$u_{k}(\sigma)$.

In order to compute thermodynamic quantities we need the effective potential,
but solving the flow equation (25) we obtain its derivative w.r.t. $\sigma$.
This means that we need to integrate the solution in $\sigma$, and so the
effective potential would be defined up to an arbitrary integration constant,
which is $\sigma$-independent but in principle $T$ and $\mu$-dependent. Thus
to obtain the correct thermodynamic properties we need to calculate this
constant using directly the flow equation (21), evaluated in a generic point
(that for us is $\sigma=0$). Thus, together with Eq. 25 we also solve

$\displaystyle\partial_{t}U_{k}(0)=-\frac{k^{5}}{12\pi^{2}}\left\\{\left[\frac{4}{E_{k,\sigma}(\sigma=0)}\coth\left(\frac{E_{k,\sigma}(\sigma=0)}{2T}\right)\right]-4N_{c}\frac{1}{k}\left[\tanh\left(\frac{k-\mu}{2T}\right)+\tanh\left(\frac{k+\mu}{2T}\right)\right]\right\\}.$

##  V Results

Throughout this section we compare the results obtained within the mean field
approximation, obtained by neglecting the bosonic fluctuations, with the
calculation that takes into account the fluctuations by solving the full FRG
flow equation for the average effective action. In the mean field
approximation we firstly introduce the rescaling
$\sigma\to\sqrt{N_{c}}\sigma$, $U_{k}(\sigma)\to N_{c}U_{k}(\sigma)$ and
$u_{k}(\sigma)\to\sqrt{N_{c}}u_{k}(\sigma)$, then we get the rescaled flow
equation

$\displaystyle\partial_{t}u_{k}(\sigma)+\frac{1}{\sqrt{N_{c}}}\partial_{u_{k}}f_{k}(\sigma,u_{k}(\sigma))u^{\prime}_{k}(\sigma)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{\sqrt{N_{c}}}\partial_{\sigma}g_{k}(u^{\prime}_{k}(\sigma))+\partial_{\sigma}S_{k}(\sigma).$
(29)

The mean field flow equation is obtained in the $N_{c}\to\infty$ limit, that
is

$\partial_{t}u_{k}(\sigma)=\partial_{\sigma}S_{k}(\sigma).$ (30)

As initial condition in the UV ($k=\Lambda$ or $t=0$) for the flow equation we
choose a quartic potential

$U_{\Lambda}(\sigma)=\frac{m^{2}_{UV}}{2}\sigma^{2}+\frac{\lambda_{UV}}{4}\sigma^{4}.$
(31)

Due to the presence of a finite cutoff $\Lambda$, when performing the
Matsubara sum, thermal modes with $2\pi T>\Lambda$ are factually excluded.
This is a serious problem especially for the calculation of thermodynamic
quantities at high temperatures. This issue can be fixed including the missing
high-momentum modes in the effective potential. Since one expects the
fermionic degrees of freedom to be relevant at higher temperature, a standard
procedure consists of integrating the fermionic part of Eq. (21) from
$k\to\infty$ to $k=\Lambda$ and add it to the effective potential. So we
calculate

$U^{\infty}_{\Lambda}(\sigma)=\int_{\infty}^{\Lambda}\,\,S_{k}(\sigma)dk$ (32)

and then add it to the effective potential at the UV scale $\Lambda$

$U_{\Lambda}(\sigma)\,\to\,U_{\Lambda}(\sigma)+U^{\infty}_{\Lambda}(\sigma)$
(33)

Our set of parameters is $\Lambda=1.0$ GeV, $f_{\pi}=0.093$ GeV, $h=3.6$,
$c=1.78\times 10^{-3}$ GeV3. The initial condition parameters are
$m_{UV}=0.762$ GeV and $\lambda_{UV}=1.05$ for MF calculations, and
$m_{UV}=0.812$ GeV and $\lambda_{UV}=3.08$ in the FRG case. These parameters
are chosen such that in the vacuum we get $\langle\sigma\rangle=f_{\pi}$ and
$\partial^{2}_{\sigma\sigma}U(\langle\sigma\rangle)=M^{2}_{\sigma}=0.36$ GeV2.

Figure 2: Constituent quark mass, $M=h\langle\sigma\rangle$, versus
temperature, for $\mu=0$ and $\mu=0.3$ GeV. MF and FRG stand for mean field
and functional renormalization group respectively.

In Fig. 2 we show the constituent quark mass, $M=h\langle\sigma\rangle$,
versus temperature, for $\mu=0$ and $\mu=0.3$ GeV; $\langle\sigma\rangle$
corresponds to the value of $\sigma$ that minimizes the effective action. We
note that there is a range of temperatures in which the condensate decreases
from its zero temperature value to a smaller one, signaling the crossover from
the low temperature phase in which chiral symmetry is spontaneously broken to
the high temperature phase in which the symmetry is (approximately) restored.
The picture remains qualitatively the same also after fluctuations are
included; quantitatively, fluctuations lower the temperature range in which
the chiral crossover takes place. We also note that increasing the chemical
potential results in the hardening of the crossover, since changes in
$\langle\sigma\rangle$ occur in a smaller range of temperature.

Figure 3:  Critical temperature, $T_{c}$, versus $\mu$, for the mean field
(dashed line) and the full FRG (solid line) calculations.

The results shown in Fig. 2 allow us to define a (pseudo-)critical
temperature, $T_{c}$, as the temperature at which the highest change of
$\langle\sigma\rangle$ occurs. In Fig. 3 we plot $T_{c}$ versus $\mu$ for both
the mean field case and the full FRG calculation. Below the critical lines the
chiral symmetry is spontaneously broken, while above the lines chiral symmetry
is approximately restored. For both cases the lines are stopped at the
critical endpoint, where the crossover changes into a real second order phase
transition where the chiral susceptibility diverges; for larger values of
$\mu$ the phase transition is of the first order: in this case the effective
potential exhibits two separate finite minima which become degenerate at the
phase transition.

Figure 4: Determinant of the thermodynamic metric, $g$, versus $\mu$ computed
at $T=T_{c}$, within the mean field approximation (dashed line) and FRG (solid
line).

Next we turn to the main focus of this work, namely the thermodynamic
geometry. Firstly, we show in Fig. 4 the determinant of the thermodynamic
metric, $g$, versus $\mu$, computed at $T=T_{c}$. We present the results
obtained within the mean field approximation and within the FRG. The results
are in qualitative agreement with [28], where fluctuations were introduced
within a gaussian approximation. $g=0$ corresponds to the thermodynamic
instability of the system, that is to a phase transition. We do not find $g=0$
because in this model chiral symmetry is explicitly, albeit softly, broken by
the finite quark mass, hence a phase transition is replaced by a smooth
crossover. However, $g(T_{c})$ decreases with $\mu$, signaling that the system
is approaching criticality, that is the critical endpoint. Moreover, we note
that including fluctuations results in the lowering of $g$, in agreement with
previous findings [28].

Figure 5: Thermodynamic curvature, $R$, versus $T$, for several values of $c$.
Upper panel corresponds to the mean field approximation, while the lower panel
to the calculations within the FRG. $c=M_{\pi}^{2}f_{\pi}$ denotes the
physical value of the symmetry breaking parameter in Eq.
(LABEL:eq:qm_lagr_111).

We now discuss the thermodynamic curvature, $R$. It is expected that $R$
diverges at a second order phase transition, while it is not obvious the
behavior of $R$ near a smooth crossover. In order to better understand the
results on $R$ obtained within the FRG, we preliminarily analyze the curvature
versus $T$ at $\mu=0$, with and without fluctuations, for several values of
the parameter $c$ that regulates the explicit breaking of chiral symmetry. In
fact, in the limit $c=0$, chiral symmetry is not explicitly broken and
restoration of chiral symmetry at $\mu=0$ happens by a second order phase
transition. Performing calculations of $R$ in the chiral limit is numerically
demanding near the phase transition, hence we limit ourselves to analyze cases
in which $c$ is small but nonzero. In order to avoid confusion, from now on
with $c$ we denote solely the value of the parameter at the physical point,
namely $c=M_{\pi}^{2}f_{\pi}$; we then artificially lower the value of this
parameter.

In Fig. 5, we plot $R$ versus $T$ at $\mu=0$ within the mean field
approximation (upper panel) and with fluctuations included (lower panel); we
show results for several values of the symmetry breaking parameter in Eq.
(LABEL:eq:qm_lagr_111), namely the physical value, then $c/40$ and $c/200$. We
note in both panels of Fig. 5 that as the chiral limit is approached, the
curvature is enhanced in the pseudo-critical region, while the peak of $R$
becomes smoother as $c$ approaches the one in the physical limit. Moreover,
including fluctuations results in the lowering of the peaks of $R$ in the
pseudo-critical region. Within the mean field approximation, $R$ changes sign
around $T_{c}$, in agreement with previous results [38, 28, 27]: this was
interpreted as the emergence of a boson-like behavior of the system around
$T_{c}$, namely of a statistical attraction in phase space due to long range
correlations that develop around $T_{c}$ that overcomes the statistical
fermionic repulsion. This behavior of $R$ is partially found when fluctuations
are included (lower panel of Fig. 5). However, we note that fluctuations lower
the overall magnitude of $R$, in agreement with [28]. We also note that
despite the peaks of $R$ become more prominent as the chiral limit is
approached, it does not change sign as it does in the mean field calculations.
Hence, it is likely that although the qualitative behavior of $R$ is
independent on the approximation used (mean field versus FRG), at the physical
point, the change of nature from fermion-like to boson-like depends on the
calculation scheme adopted in proximity of the crossover. The results shown in
Fig. 5 will be useful to interpret the behavior of $R$ we discuss below. As a
final remark, we note that the peak of $R$ moves towards smaller temperatures
as the chiral limit is approached. This is in agreement with the fact that the
critical temperature is lower in the chiral limit. We will see that as the
critical point is approached, the change of sign of $R$ appears both in the
mean field and in the FRG cases.

Figure 6: Thermodynamic curvature, $R$, versus $T$, for several values of
$\mu$. Calculations correspond to the mean field approximation (upper panel)
and the FRG scheme (lower panel).

In Fig. 6 we plot $R$ versus $T$ for several values of $\mu$, obtained within
the mean field approximation and the full FRG calculation at the physical
point $c=M_{\pi}^{2}f_{\pi}$. In both the MF and FRG panels of Fig. 6, we
multiplied the results by $100$ in all but the $\mu=0.3$ GeV cases to make the
results more readable. Firstly, we note that the trend of $R$ is qualitatively
similar in both calculations. Within the mean field approximation, at $\mu=0$
the curvature locally develops a peak in correspondence of the chiral
crossover, signaling that $R$ is capable to capture the pseudo-critical
behavior of the quark condensate. Increasing the chemical potential, $R$
maintains its local peak structure, but as the critical endpoint is
approached, the peaks become more pronounced. This is in agreement with the
general understanding on $R$ which is expected to diverge at a second order
phase transition. Moreover, we note that at large $\mu$ the thermodynamic
curvature develops several peaks in the temperature range of the chiral
crossover, although the most pronounced peak does not necessarily show up at
the critical temperature. This behavior, already noticed in [27], shows that
$R$ is not necessarily as sensitive as other quantities, like the chiral
susceptibility or $|dM/dT|$, at the changes of the quark condensate at
$T=T_{c}$, but it is still capable to measure sensible deviations in the
pressure around the chiral crossover.

Including fluctuations does not change the qualitative behavior of $R$.
Therefore, we conclude that the fact that $R$ is sensitive to the chiral
crossover is not an artifact of the mean field approximation, rather it is a
quite solid statement. However, as already remarked in Fig. 5, the inclusion
of fluctuations lowers the value of $R$ around the chiral crossover;
particularly, when $\mu$ is small, $R$ remains negative also around the
crossover, while in the mean field approximation it changes sign. Therefore,
it is likely that the change of nature of the interaction at the mesoscopic
level, from fermion-like to boson-like, depends on the approximation used in
the calculation when the system is far from criticality. Hence, this implies
that, at small $\mu$, the fluctuations substantially change the geometry of
the manifold.

On the other hand, when the critical endpoint is approached at large $\mu$, we
find that $R$ changes sign also in the FRG calculation, and the mean field
results do not qualitatively differ from the those obtained within the FRG.
Hence, we conclude that when this system approaches criticality, $R$ changes
sign and locally develops a marked peak: this conclusion was anticipated in
previous mean field calculations [38, 27] and stands also in case fluctuations
are taken into account via FRG.

##  VI Conclusions and outlook

We studied the thermodynamic geometry, and in particular computed the
thermodynamic curvature $R$, of the chiral phase transition of Quantum
Chromodynamics, within the quark-meson model and the functional
renormalization group method. The advantage of this method is that it allows
to exactly include fluctuations, differently from previous approaches [28] in
which fluctuations were introduced only within a gaussian approximation. As a
matter of fact, the inclusion of quantum fluctuations via the functional
renormalization group represents a remarkable improvement compared to previous
mean field calculations. We found that the qualitative behavior of $R$ is not
very different from the one previously computed within mean field
calculations, as well as within calculation schemes that include fluctuations
by a gaussian approximation. In particular, $R$ seems to keep its local peak
structure in proximity of the chiral crossover at small chemical potential;
moreover, it is enhanced at the critical point, signaling that when the system
approaches criticality $R$ could diverge, supporting the arguments of
hyperscaling [1].

We also found that the change of sign of $R$ near the chiral crossover,
discussed previously in the literature [27, 90, 91, 38, 28], does not always
happen when fluctuations are taken into account within the functional
renormalization group approach; however, as the system approaches criticality,
the change of sign from negative (fermion-like behavior) to positive (boson-
like behavior) takes place. Hence, we conclude that the change of sign of $R$
near the critical endpoint seems to be quite a robust prediction of the chiral
effective models.

It will be interesting to analyze if the behavior of $R$ we highlighted in
this article does not change when the truncation adopted in the functional
renormalization group approach is improved; for example, the inclusion of the
scale-dependent wave function renormalization factors of the boson and the
quark fields is worth of further investigation, due to the possible link of
this to the formation of inhomogeneous phases at large chemical potential.
Another possible improvement is the introduction of other condensation
channels, which could include diquarks or meson condensates. A potential
application would be the QM model with a finite isospin chemical potential,
$\mu_{I}$, at vanishing (or small) $\mu$. This would be extremely interesting,
due to the opportunity to directly compare the obtained results with lattice
QCD calculations, since a finite $\mu_{I}$ does not lead to a sign problem,
and would give the opportunity to study $R$ in presence of potentially two
condensates, namely a pion condensate beside the chiral condensate. We plan to
address these issues in the near future.

###### Acknowledgements.

M. R. acknowledges John Petrucci for inspiration. M. R. and F. M. acknowledges
Gabriele Parisi for numerous discussions. F. M. acknowledges discussions with
D. Rischke. This work has been partly funded by the European Union – Next
Generation EU through the research grant number P2022Z4P4B “SOPHYA -
Sustainable Optimised PHYsics Algorithms: fundamental physics to build an
advanced society” under the program PRIN 2022 PNRR of the Italian Ministero
dell’Università e Ricerca (MUR).

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# FEANet: Feature-Enhanced Attention Network for RGB-Thermal Real-time
Semantic Segmentation

Fuqin Deng1,3,4∗, Hua Feng1∗, Mingjian Liang1, Hongmin Wang1, Yong Yang4, Yuan
Gao3,
Junfeng Chen3, Junjie Hu3, Xiyue Guo3, and Tin Lun Lam2,3,† This work was
supported in part by the National Key R$\&$D Program of China
(2020YFB1313300), the funding (AC01202101025, AC01202101026) from the Shenzhen
Institute of Artificial Intelligence and Robotics for Society, the special
projects in key fields of Guangdong Provincial Department of Education
(2019KZDZX1025), Innovative Program for Graduate Education (503170060259) from
the Wuyi University, and Shenzhen Peacock Plan of Shenzhen Science and
Technology Program (KQTD2016113010470345). We would also like to thank Dr. Li
Nan Nan from Macau University of Science and Technology for his valuable
discussion on this paper.∗Authors contributed equally1School of Intelligent
Manufacturing, the Wuyi University, Jiangmen, China. 2School of Science and
Engineering, the Chinese University of Hong Kong, Shenzhen, China. 3Shenzhen
Institute of Artificial Intelligence and Robotics for Society, the Chinese
University of Hong Kong, Shenzhen, China. 43irobotix Co.,Ltd, Shenzhen, China.
†Corresponding author is Tin Lun Lam<EMAIL_ADDRESS>
###### Abstract

The RGB-Thermal (RGB-T) information for semantic segmentation has been
extensively explored in recent years. However, most existing RGB-T semantic
segmentation usually compromises spatial resolution to achieve real-time
inference speed, which leads to poor performance. To better extract detail
spatial information, we propose a two-stage Feature-Enhanced Attention Network
(FEANet) for the RGB-T semantic segmentation task. Specifically, we introduce
a Feature-Enhanced Attention Module (FEAM) to excavate and enhance multi-level
features from both the channel and spatial views. Benefited from the proposed
FEAM module, our FEANet can preserve the spatial information and shift more
attention to high-resolution features from the fused RGB-T images. Extensive
experiments on the urban scene dataset demonstrate that our FEANet outperforms
other state-of-the-art (SOTA) RGB-T methods in terms of objective metrics and
subjective visual comparison (+2.6% in global mAcc and +0.8% in global mIoU).
For the 480 × 640 RGB-T test images, our FEANet can run with a real-time speed
on an NVIDIA GeForce RTX 2080 Ti card.

##  I INTRODUCTION

As a fundamental but challenging task in computer vision, semantic
segmentation can be broadly applied to the fields of path planning, autonomous
driving, and video surveillance [1] [2] [3]. Most of the existing deep
learning-based semantic segmentation networks [4] [5] mainly deal with RGB
images. However, RGB images could provide less information for the model
training and produce inaccurate prediction results on the scenarios of similar
texture, complex background with dim light, or total darkness. With the
popularity of thermal imaging cameras, some

Figure 1: Qualitative comparison with two latest networks in daytime and
nighttime urban street scenes. The color cones (the objects marked in the red
frame) are too small to detect and segment. We can see that the color cone
boundaries segmented by the RTFNet are not so sharp or fail to segment them
correctly (e.g., (d), (e), (j), (k)), whereas our FEANet gives a more ideal
segmentation result (e.g., (f), (l)).

researchers found that thermal information is robust and effective for
reducing the ambiguity in challenging lighting conditions [6], such as in
urban street scenes. Therefore, the thermal images created by the thermal
imaging cameras can be added as important supplements to improve the
performance of RGB-T semantic segmentation.

Recently, RGB-T semantic segmentation has received increasing research
attention. Various RGB-T models [7] [8] [9] have been proposed to improve the
segmentation performance by combining RGB and thermal information. However,
the performance of existing RGB-T models tends to drastically decrease when
faced with certain complex scenarios (e.g., cluttered backgrounds, varying
illuminations). Therefore, the existing RGB-T methods still need to solve the
following challenges in order to make further progress.

The first challenge is to effectively extract multi-level features from RGB-T
fused data. Generally, high-level features contain rich semantic information
which can be used for object location, while low-level features provide
plentiful micro details that are useful for reducing glitch noise and refining
segmentation boundaries. Therefore, current RGB-T semantic segmentation
methods (e.g., MFNet [6], RTFNet [7]) use either a direct feature extracting
strategy or a progressive multi-data fusing process to leverage multi-level
features. However, due to the direct multi-level features extracting and
merging strategy without considering differences between levels, these
processes suffer from the incomplete extraction problem of noisy low-level
features. As shown in Fig. 1, the segmented object boundaries in the RTFNet
prediction maps are not sharp (e.g., (d), (e), (j), (k)).

The second challenge is to excavate informative features from the thermal
modality. Thermal images are of low quality, which leads to unpredictable
noise during the data fusion process. Previous RGB-T models usually treat the
extra thermal images as a fourth-channel input without the modification of
three-channel RGB encoder stream or fuse RGB and thermal features by simple
summation and multiplication. These methods treat thermal and RGB information
from the same perspective and ignore the fact that RGB images contain color
and texture, whereas thermal maps contain the spatial relations among objects.
Due to this modality difference, the above-mentioned simple combination
methods [7] [9] are not effective. As shown in Fig. 1 (d), (e), (j), (k), the
RTFNet fails to detect and segment the small target objects (e.g., color
cones).

To address the above issues, we propose a two-stage FEANet for better RGB-T
semantic segmentation performance. As shown in Fig. 2, our FEANet contains a
two-stages process for feature extraction and fusion. In stage 1, we introduce
a FEAM module, which exploits the inter-channel and spatial relations for the
detail information. The proposed FEAM exploits multi-level features in a
progressive refinement way to suppress distractors in the encoder stream. This
strategy is based on the observation that low-level features provide
discriminative semantic information with micro details, which may contribute
significantly to eliminating the background distractors. In stage 2, to
improve the compatibility of RGB and thermal features, the corresponding RGB
and thermal feature maps are aggregated through elementwise summation into the
RGB encoder stream. Our main contributions are summarized as follows:

  * •

We design a two-stage FEANet to deal with the object boundaries and the small
target object for RGB-T semantic segmentation in urban scenes.

  * •

We introduce a FEAM module to enhance multi-level features and fuse RGB and
thermal information in a complementary way.

The remainder of this letter is structured as follows. In section II, related
works have been reviewed. In section III, we describe our network in detail.
In section IV, experimental results and discussions are presented. Conclusions
and future work are drawn in the last section.

##  II RELATED WORKS

###  II-A Semantic Segmentation

Over the past few years, semantic segmentation is a great challenge for
detecting and locating target objects in computer vision. The Convolutional
Neural Networks (CNNs) [10] had been applied to improve the accuracy in the
image classification and semantic segmentation tasks since 2012. In 2015,
Fully Convolutional Networks (FCN) [11] had been proposed for the semantic
segmentation, which had an end-to-end network architecture and outperformed
the traditional methods that rely on the hand-crafted features extraction
mechanism.

Similar to the FCN, the SegNet [12] adopted the Encoder-Decoder network
architecture as the backbone for semantic segmentation tasks. The SegNet used
a pre-trained VGG16 architecture as its encoder, and then applied the output
of the encoder as the input of the up-sampling decoder. SegNet achieved state-
of-the-art accuracy while getting the low inference speed. In subsequent
years, the Encoder-Decoder network structure is widely used in semantic
segmentation methods. UNet [13] and DeepLabv3 [14] have the large Encoder-
Decoder architecture, specially, the decoder can restore the high-resolution
feature maps from the low layers through the short-cut connections.

Real-time semantic segmentation methods aim to generate high-quality
segmentation results in real-time. ENet [15] also followed the Encoder-Decoder
architecture to achieve real-time semantic segmentation, but it was optimized
for fast inference and high accuracy. Due to the efficiency of ENet, it can
effectively process images (480×640 RGB) in the requiring high-speed inference
situations. However, ENet failed to perform as well as SegNet on spectral
images datasets, such as SUN RGB-D [16].

To improve the accuracy and speed of semantic segmentation, the BiSeNet [17]
had the spatial path that preserves the spatial information and the semantic
path that obtains the sufficient receptive field. Based on these two paths, a
new Feature Fusion Module was developed to combine the features efficiently.
However, the BiSeNet just captures the information from the lower layer to
sharpen the boundaries with a slow inference speed.

###  II-B RGB-T Semantic Segmentation

Some methods adopting CNNs were designed for the RGB-D dataset, which contains
images that were acquired by multispectral cameras. In these works, we found
that some ideas were useful for designing our method. Hazirbas et al. [18]
proposed a new CNN network named FuseNet, which contained an Encoder-Decoder
structure that simultaneously extracts features from RGB and depth images. In
[19], RGB and spectral images feature maps were not only processed separately
in the encoder stream but also in the decoder stream.

The existing urban scenario image segmentation datasets are based on visible
spectral images (RGB images), such as Cityscapes [20] and Daimler Urban
dataset [21]. Naturally, semantic segmentation methods based on these datasets
can only be used to process RGB images. Furthermore, most of these methods
focused only on improving the segmentation accuracy while neglecting the
inference speed. For RGB-T semantic segmentation of urban scenes, MFNet,
RTFNet, and FuseSeg-161 were proposed to fuse RGB and thermal data in a novel
Encoder–Decoder structure. In this structure, two identical encoders were
employed to extract features from RGB and thermal data, respectively, and one
decoder was designed to gradually restore the resolution. In addition to the
above methods, recently, other RGB-T fusion methods [22] [23] utilized the
combination of omnidirectional (O-D) infrared sensors and O-D visual RGB
sensor for semantic segmentation in autonomous robotic systems.

Figure 2: The overall architecture of the proposed FEANet. From left to right
are Thermal Stream, RGB Stream, and Output Stream. The encoder in Thermal
Stream and RGB Stream contains two extracting stages. In stage 1, Thermal
Stream and RGB Stream use ResNet [24] as the feature extractor layer. The
output part of each layer is weighted through the FEAM. In stage 2, the output
map of Thermal Stream is fused into the RGB Stream. The decoder in Output
Stream is composed of Transposed blocks A and B.

##  III PROPOSED METHOD

In this section, we first introduce the overall architecture of our FEANet,
which contains two extracting encoder streams and an output decoder stream. As
the Encoder-Decoder structure has been confirmed as an effective architecture
in many semantic segmentation networks, our FEANet also adopts this structure.
Then, to fully excavate informative cues from both the RGB and thermal feature
maps, we present the FEAM to enhance the multi-level features for superior
segmentation performance.

###  III-A Overall Architecture

As shown in Fig. 2, our FEANet contains two main steps: feature extracting and
resolution restoring. Two encoder streams and one decoder stream are designed
for the feature extraction and recovery, respectively. In the feature
extraction progress, two encoders extract the multi-level features from three-
channel RGB and one-channel thermal images, respectively. With increasing
encoder stream depth, the high-level features (e.g., L3, L4 in Fig. 2) will be
more useful for capturing global context, while they lose the object details.
When we up-sample the high-level feature maps, the output prediction will be
blurred and object boundaries will become unclear. Instead, the proposed FEAM
can distinguish object regions which are too small to detect. In the
resolution restoration progress, the decoder gets dense output predictions. At
the end of FEANet, the final softmax layer is adapted to get the prediction
output map for the RGB-T semantic segmentation results.

Figure 3: Architecture of the Feature-Enhanced Attention Module (FEAM) Figure
4:  Architecture of the Transposed block. Conv, TransConv and BN refer to the
convolutional layers, transposed convolutional layers and the batch
normalization layers, respectively. The detailed channel numbers for Conv and
TransConv layers are listed in Tab. I.

The proposed FEANet explores the two-stages cross-modal fusion methods. In the
first stage, we first extract both the RGB and thermal feature maps by the
ResNet block and then refine the detail features through the FEAM module. In
the second stage, the corresponding RGB and thermal feature maps are
aggregated through elementwise summation into the RGB encoder stream. At the
end of two encoders, the final refined feature maps are transmitted to the
decoder. With the two-stage feature extracting strategy, the loss of rich
semantic information through the intensive feature extracting could be
recovered.

###  III-B Encoder-Feature Extracting

In the proposed FEANet, both the RGB and thermal features are extracted in two
encoder streams. Specifically, both the RGB and thermal encoder streams employ
five convolutional blocks from ResNet [24] as the standard backbone and attach
an additional FEAM after every single convolutional block, respectively.
Usually, the existing ResNet is designed for the three-channel RGB images
extracting, which is not suitable for the single-channel images, then we
modify the number of the first convolutional layer to be one to extend it to
the thermal image.

TABLE I: Configuration for the convolution (Conv) and transposed convolution (TransConv) layers in the individual module of the decoder. |  | Name | Kernel Size | Stride | Padding
---|---|---|---|---|---
Block A | Conv 1 | 3x3 | 1 | 1
Conv 2 | 3x3 | 1 | 1
Block B | Conv 1 | 3x3 | 1 | 1
TransConv 1 | 2x2 | 2 | 0
TransConv 2 | 2x2 | 2 | 0

To effectively extract features from both RGB and thermal images is the focus
of this paper. When in the nighttime, some colorful objects in RGB maps are
invisible but can be clearly seen in the thermal maps. Considering the
modality difference, RGB and thermal features need to be enhanced. Inspired by
[25], we design a FEAM module using an attention component to learn features
from the fused data and then refine the prediction. In Fig. 2, the FEAM is
added after each convolutional layer in two encoder streams, which can enhance
the compatibility of the features. This extraction process improves the
representation of image features and preserves the multi-level information. To
better understand the working mechanism of the FEAM module, the channel-wise
feature maps from FEAM are visualized at different levels.

As illustrated in Fig. 3, the FEAM contains a sequential channel attention
operation and a spatial attention operation. Channel attention operation
shifts attention to the feature that extracted from the convolutional layer
and then explores foreground cues. Complementarily, spatial attention
operation focuses on the global area to explore the informative cues, looking
for possible small target objects within it. To the best of our knowledge, we
are the first to introduce the attention mechanism to excavate informative
cues from both RGB and thermal multi-level features. Our experiments in Tab.
II and Fig. 5 demonstrate the effectiveness of our approach in improving RGB-T
semantic segmentation performance.

###  III-C Decoder-Resolution Restoring

After computing multi-level features from two encoder streams, which are the
final map of the RGB and thermal features. The decoder is mainly designed to
efficiently leverage the multi-level information to carry out the detail
pixels refinement. Our decoder architecture is refined from the RTFNet decoder
and then restores the feature map to the original images. Different from
RTFNet, we delete two sequential 1 × 1 convolutions of the original block,
which avoids the complicated up-sample process in the decoder. As illustrated
in Fig. 4, the decoder consists of two blocks (Transposed blocks A and B).
Specifically, Transposed block B contains an additional branch to enlarge the
receptive field and a residual connection to preserve the information.

Detailed configurations for the neural network layers in the Transposed blocks
are displayed in Tab. I. In block A, there is a batch normalization (BN) layer
[26] and a ReLu activation layer [27] followed by the convolutional layer as
the feature resolution. The short cut from the input and the output of the
final BN layer is element-wisely added up. In block B, it consists of Conv 1
and two TransConv layers. Each residual-based transposed block contains a 3×3
convolution and a residual-based transposed convolution. Through Conv 1, the
resolution of the map is the same as the original one, however, the number of
feature channels is decreased by a factor of 2. And the TransConv 1 keeps the
number of channels unchanged and increases the resolution by a factor of 2.
Different from the TransConv 1, the TransConv 2 needs to increase the
resolution and decrease the number of feature channels. Finally, the decoder
will get more details to generate the final predicted map in a progressive
upsampling way.

TABLE II: Comparison result on the test set (%). 3c and 4c represent that the networks are tested with the three-channel RGB data and four-channel RGB-Thermal data, respectively. Note that mAcc and mIoU are calculated with the unlabeled classes, but the results for the unlabeled classes are not displayed. The bold font highlights the best result in each column. Methods | Car | Person | Bike | Curve | Car Stop | Guardrail | Color Cone | Bump | mAcc | mIoU
---|---|---|---|---|---|---|---|---|---|---
Acc |  IoU | Acc |  IoU | Acc |  IoU | Acc |  IoU | Acc |  IoU | Acc |  IoU | Acc |  IoU | Acc |  IoU
FRRN(4c) | 81.9 | 74.7 | 66.2 | 60.8 | 62.8 | 50.3 | 41.2 | 35.0 | 12.5 | 11.5 | 0.0 | 0.0 | 37.2 | 34.0 | 35.2 | 34.6 | 48.5 | 44.2
FRRN(3c) | 80.0 | 71.2 | 53.0 | 46.1 | 65.1 | 53.0 | 34.0 | 27.1 | 21.6 | 19.1 | 0.0 | 0.0 | 34.7 | 32.5 | 36.2 | 30.5 | 47.1 | 41.8
BiSeNet(4c) | 89.7 | 84.1 | 72.0 | 63.2 | 74.1 | 60.1 | 45.1 | 36.7 | 34.2 | 25.3 | 18.2 | 5.0 | 47.4 | 42.2 | 39.8 | 35.9 | 57.7 | 50.0
BiSeNet(3c) | 90.0 | 84.5 | 65.0 | 54.3 | 75.0 | 61.4 | 32.1 | 25.7 | 32.3 | 26.2 | 3.2 | 0.9 | 49.6 | 43.3 | 48.1 | 40.5 | 54.9 | 48.2
DFN(4c) | 90.0 | 84.4 | 73.2 | 65.0 | 75.5 | 60.9 | 54.0 | 40.4 | 38.9 | 25.7 | 10.2 | 2.7 | 48.3 | 42.5 | 55.8 | 47.4 | 60.5 | 52.0
DFN(3c) | 90.7 | 81.4 | 67.7 | 52.8 | 71.5 | 57.5 | 49.2 | 34.9 | 35.1 | 23.8 | 4.1 | 1.4 | 44.2 | 31.0 | 54.6 | 47.5 | 57.3 | 47.5
SegHRNet(4c) | 92.8 | 87.6 | 79.3 | 71.0 | 78.3 | 63.4 | 59.8 | 42.5 | 25.7 | 19.1 | 18.8 | 0.0 | 56.5 | 49.8 | 63.5 | 44.5 | 63.7 | 53.2
SegHRNet(3c) | 92.2 | 86.6 | 73.1 | 59.8 | 74.9 | 61.3 | 47.0 | 33.2 | 23.8 | 28.7 | 7.3 | 0.0 | 54.6 | 47.2 | 61.5 | 46.2 | 60.9 | 51.3
MFNet | 77.2 | 65.9 | 67.0 | 58.9 | 53.9 | 42.9 | 36.2 | 29.9 | 19.1 | 9.9 | 0.1 | 8.5 | 30.3 | 25.2 | 30.0 | 27.7 | 45.1 | 39.7
FuseNet | 81.0 | 75.6 | 75.2 | 66.3 | 64.5 | 51.9 | 51.0 | 37.8 | 28.7 | 15.0 | 0.0 | 0.0 | 31.1 | 21.4 | 51.9 | 45.0 | 52.4 | 45.6
DepthAwareCNN | 85.2 | 77.0 | 61.7 | 53.4 | 76.0 | 56.5 | 40.2 | 30.9 | 9.9 | 29.3 | 22.8 | 6.4 | 32.9 | 30.1 | 36.5 | 32.3 | 55.1 | 46.1
RTFNet-50 | 91.3 | 86.3 | 78.2 | 67.8 | 71.5 | 58.2 | 69.8 | 43.7 | 32.1 | 24.3 | 13.4 | 3.6 | 40.4 | 26.0 | 73.5 | 57.2 | 62.2 | 51.7
RTFNet-152 | 93.0 | 87.4 | 79.3 | 70.3 | 76.8 | 62.7 | 60.7 | 45.3 | 38.5 | 29.8 | 0.0 | 0.0 | 45.5 | 29.1 | 74.7 | 55.7 | 63.1 | 53.2
FuseSeg-161 | 93.1 | 87.9 | 81.4 | 71.7 | 78.5 | 64.6 | 68.4 | 44.8 | 29.1 | 22.7 | 63.7 | 6.4 | 55.8 | 46.9 | 66.4 | 47.9 | 70.6 | 54.5
FEANet(Ours) | 93.3 | 87.8 | 82.7 | 71.1 | 76.7 | 61.1 | 65.5 | 46.5 | 26.6 | 22.1 | 70.8 | 6.6 | 66.6 | 55.3 | 77.3 | 48.9 | 73.2 | 55.3
Figure 5: Qualitative demonstrations for the fusion networks in daytime or
nighttime. We can see that our FEANet can provide acceptable results in
various lighting conditions. The comparative results demonstrate our
superiority.

##  IV EXPERIMENTS AND RESULTS

###  IV-A The RGB-T Dataset

We use the public data set released by the MFNet [6]. It was recorded in urban
street scenes, which contains eight hand-labeled object classes and one
unlabelled background class. This dataset contains 1569 pairs of RGB and
thermal images, in which 820 taken at daytime and 749 taken at nighttime. We
follow the dataset splitting scheme proposed in [6]. The training set consists
of 784 pairs of images. The validation set consists of 392 pairs of images.
The other images are used for testing.

###  IV-B Training Details

We use the Stochastic Gradient Descent (SGD) optimization solver for training.
The momentum and weight decay are set to 0.9 and 0.0005, respectively. The
initial learning rate is set to 0.03. We adapt the CosineAnnealingWarmRestarts
[28] to gradually decrease the learning rate. The loss function uses the
DiceLoss [29] and the SoftCrossEntropy [30] for training. We give the DiceLoss
and the SoftCrossEntropy a weight of 0.5 and add them to get the loss
function:

$\textup{DiceLoss}=1-\frac{2\sum_{i}^{N}p_{i}g_{i}}{\sum_{i}^{N}p_{i}^{2}+\sum_{i}^{N}g_{i}^{2}}$
$None$

where the sums run over the $N$ voxels, of the predicted binary segmentation
volume $p_{i}\in P$ and the ground truth binary volume $g_{i}\in G$. This
formulation of DiceLoss can be differentiated yielding the gradient.

$\textup{SoftCrossEntropyloss
}=-\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{c}\hat{y}_{ij}\log(y_{ij}^{d})$ $None$

where the $n$ is the number of the batch. In this work $n=5$, $\hat{y}_{ij}$
is is binary indicator if class label $c$ is the correct classification for
pixel $i$, and $y_{ij}^{d}$ is the corresponding predicted probability be
normalized to a probability distribution.

###  IV-C Evaluation Metrics

The Accuracy (Acc) and the Intersection-over-Union (IoU) are used as the
evaluation indicators of our model. The Acc defines the overall accuracy as
the probability of correspondence between a positive decision and true
condition. The IoU calculates the intersection of the true label and the
predicted result separately for each class. Both mAcc and mIoU are the average
values across all the classes for the Acc and the IoU.

$\textup{{m}{A}{c}{c}}=\frac{1}{k+1}{\sum_{i=0}^{k}}\frac{p_{ii}}{\sum_{j=0}^{k}p_{ij}}$
$None$

$\textup{{m}{I}{o}{U}}=\frac{1}{k+1}\sum_{i=0}^{k}\frac{p_{ii}}{\sum_{j=0}^{k}p_{ij}+\sum_{j=0}^{k}p_{ji}-p_{ii}}$
$None$

where the $k$ is the number of the hand-labeled object classes, in this work,
$k=8$. $p_{ii}$ is the number of the pixels of class $i$ that are correctly
classified as class $i$, $p_{ij}$ is the number of pixels of class $i$ that
are wrongly classified as class $j$, $p_{ji}$ is the number of pixels of class
$j$ that are wrongly classified as class $i$.

###  IV-D Results And Analysis

The complete quantitative evaluation results of the networks are listed in
Tab. II. Our FEANet achieves some remarkable advantages over the comparison
methods in terms of both mAcc and mIoU indicators. Compared to the SOTA RGB-T
semantic segmentation methods, our proposed FEANet has a great improvement on
small target object detection and segmentation, especially in the Guardrail
class. Compared to the SOTA FuseSeg-161, We find the Guardrail class has the
+7.1 % Acc and +0.2 % IoU results in improvement. At the same time, the Color
Cone class has the +5.4 % Acc and +8.4 % IoU results in improvement. And the
other objects also have good segmentation performance. This indicates that our
FEANet can make more efficient detection and segmentation on small target
objects. To further demonstrate the effectiveness of our FEANet, we visualize
the prediction maps of our FEANet and other top 2 methods in Fig. 5.
Experiments show that Our FEANet effectively utilizes the RGB and thermal
information for sharp object boundaries, while the others are disturbed by the
background.

TABLE III: Inference speed of SOTA networks. ms and FPS represent the time of milliseconds and the speed of Frames Per Second, respectively. Methods | RTX 2080 Ti
---|---
ms |  | FPS
RTFNet-50 | 11.25 |  | 88.87
RTFNet-152 | 30.47 |  | 32.81
FuseSeg-161 | 33.32 |  | 30.01
FEANet(ours) | 28.52 |  | 35.06

Although our FEANet can get prior results in the tiny object classes, there
are some limitations in our network. According to Tab. II, FuseSeg-161 gets
the best results in the Person and Bike classes. This proves the effectiveness
of the DenseNet, which can keep the feature map resolution unchanged in the
encoder stream. And for the RTFNet, our results are better than those of the
RTFNet in most object classes, however, the RTFNet152 gets the best indicators
in the Car Stop class. This proves network can get the dense feature in a
deeper layer to improve the segmentation performance. As indicated in Tab.
III, our FEANet achieves real-time inference speed (approximately 35 images/s)
on a single NVIDIA Geforce RTX 2080 Ti GPU.

###  IV-E Ablation Study

TABLE IV: The comparison of mAcc ($\%$) and mIoU ($\%$) on the test-sets for the NFRTS, NFRS, NFTS and FRTS(ours) variants. The bold font highlights the better results in each scenario. Variants | Test-set
---|---
mAcc |  | mIoU
NFRTS | 63.9 |  | 50.0
NFRS | 69.5 |  | 54.5
NFTS | 65.3 |  | 50.6
FRTS(ours) | 73.2 |  | 55.3

In order to verify that our FEAM module is effective at each feature level, we
removed the FEAM module from the RGB stream and thermal stream, respectively,
then we can see the performance without using the attention module FEAM.
Therefore, we call no FEAM in a thermal stream as NFTS. Similarly, no FEAM in
RGB stream as NFRS and no FEAM in RGB and thermal stream is named as NFRTS,
respectively. FRTS means that the FEAM is both in RGB and thermal stream. Tab.
IV shows the quantitative comparison test results. By comparing the results of
NFRTS, NFRS, NFTS, and FRTS, we find that FRTS usually provides better
performance than NFRS, NFRTS and NFTS in the RGB-T semantic segmentation task.
The performance in FRTS can also prove that FEAM can enhance the fusion effect
of RGB image information and thermal image information. In this experiment,
the FEAM in every layer facilitates a universal improvement in detection
performance. In addition, we find that FEAM applied in the thermal stream
contributes more to the results.

##  V CONCLUSIONS

We proposed a novel two-stage FEANet to excavate informative thermal cues from
both RGB and thermal images for the semantic segmentation of urban scenes.
Specifically, we introduce a FEAM to excavate and enhance informative features
from both the channel and spatial views. The experimental results demonstrate
that FEANet performs better on small target object segmentation and produces
sharp object boundaries. The proposed FEANet runs at real-time speed on a
single GPU, making it a potential solution for autonomous driving
applications. In the future, we would like to fuse more different modalities
of information (e. g., depth, audio) into a network for segmentation
improvement.

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