Based on the work w/
Riki Oshima (Saga U.)
Hiroaki Kouno (Saga U.)
Kouji Kashiwa (Fukuoka Inst.)
Motoi Tachibana (Saga U.)
2601.16762[hep-ph]
February 11, 2026
Workshop on
particle physics & cosmology: Jeju
@Jeju National University, Korea
Thermodynamic geometry
and Critical Phenomena
-an application into hadron physics-
Summary of this talk
Study of QCD phase diagram provides an interesting connection
between particle physics and cosmology
Especially, dense QCD is a friend of neutron star/gravitational wave
Its theoretical study is hard because of sign problem/strong dynamics
Thermodynamic geometry (TDG)
= Riemannian geometry in thermodynamic space
Application of TDG into hadron resonance gas (HRG) model
Let’s get started
Standard Model of Particle Physics
is so good, but too much standard!
“Missing pieces”
① Dark components
(dark matter/energy)
③ (Quantized) gravity
② Inflation
Vacuum structure
https://www.ivyleagueinsecurities.com/aidan-
donnelley-rowley-2/2010/07/missing-pieces
QCD Lagrangian
(M. –Y. Han and Y. Nambu, 1965)
Just one line, but very rich in physics and math
Testing field for any kind of new ideas
energy scale κ (GeV)
αs(κ)
τ width
Υ decay
DIS
e+e-
Basic properties of QCD
■ Asymptotic freedom (Gross-Politzer-Wilczek, 1973)
“QCD coupling gets weaker as the energy grows”
Coupling
constant
Color
Confinement
Asymptotic
freedom
short distance
long distance
D. Gross
H. Politzer
F. Wilczek
(2004, Nobel Prize)
classical QCD symmetry (m=0)
Quantum QCD vacuum (m=0)
Chiral condensate :
spontaneous mass generation
Axial anomaly :
quantum violation of U(1)A
Chiral basis :
QCD Lagrangian :
■Symmetries of QCD and their breaking patterns
Origin of masses Structure of the vacuum
WMAP, Planck
Cosmological constant
Einstein (1917)
Universe
baryons
RHIC, LHC
“Chiral” condensate
Nambu (1960)
baryon
quark
LHC, ILC
“Higgs” condensate
Anderson (1963)
Englert-Brout, Higgs (1964)
quark
bare
quark
Interesting connection btw
particle physics & cosmology!
A child-like question:
What happens to matter,
as we squeeze it harder and harder,
and/or make it hotter and hotter?
Phases of matter
Solid(ice)
Liquid(water)
Gas(vapor)
T, P
Ex.) H2O
Phases are characterized by “condensates”
chiral condensate
diquark condensate
QCD Phase diagram
“order parameters”
μ
T
Quark-gluon plasma
Hadron
Color superconductor
“Schematic phase diagram”
Big-Bang
LHC
NS
GW
Mass-radius (M-R) relation
Neutron star mass and radius
F. Ozel, D. Psaltis, T. Guver, G. Baym, C. Heinke, S. Guillot, APJ 820 (2016) 28
http://xtreme.as.arizona.edu/NeutronStars/
For example,
EoS has one to one correspondence with the neutron star M-R relation (via TOV equation)
Some EoSs are rejected
Fruitful phase structure of QCD can be discussed
from neutron star properties
GW from tidal force
Restriction: Gravitational wave signal from binary neutron star merger
Maximal mass of neutron star
M. Shibata, S. Fujibayashi, K. Hotokezaka,
K. Kiuchi, K. Kyutoku, Y. Sekiguchi, and
M. Tanaka, Phys. Rev. D 96 (2017)123012
Abotto et al.(LIGO Scientific Collaboration and Virgo Collaboration),
Phys. Rev. Lett. 119 (2017) 161101
There are several constraints
coming from neutron star observations
Taken from T. Kojo, arXiv:1904.05080
Jeju Neutron Star!
Oops!
Our hearts combined like
a neutron star collision
Let me be more serious
-Terra firma, Terra incognita-
Why finite density QCD is hard
□ Sign problem
□ Strong coupling
✅
✅
① Effective field theories (EFTs)
② perturbative QCD (pQCD)
③ Lattice QCD (LQCD)
④ Imaginary chemical potential
⑤ Holographic QCD (HQCD)
How to overcome?
Thermodynamic geometry （TDG）
A new method
Thermodynamic geometry
in hadron resonance gas model
at real & imaginary baryon chemical potential
and
a simple sufficient condition for
quark deconfinement
(Oshima-Kouno-Tachibana-Kashiwa, 2601.16762)
with
Probability density to find in the point
Thermodynamic geometry
Rao, Amari,
Weinhold, Ruppeiner
The entropy can be expanded around the equilibrium point
: an isolated system with large volume (universe)
:an open subsystem of fixed volume
Transformation rules for the Hessian of the entropy
At the equilibrium point, due to the maximum entropy principle, the above becomes the transformation rule for the second rank tensor!
Thus, one can define the metric tensor in thermodynamic space:
Thermodynamic metric
Thermodynamic geometry at criticality
The line element
measures a distance btw 2 different equilibrium states.
A large distance corresponds to a small probability of a fluctuation
from one equilibrium state to another. Moreover, from the metric,
one can obtain the Riemann curvature R that depends on the 2nd
and 3rd moments of thermodynamic variables. Therefore, R contains
information about fluctuations around a phase transition.
Thermodynamic geometry in 2 dimension (T, μ)
Line element in 2d thermodynamic space
Metric
2d Riemann curvature
where for instance,
specific heat
baryon # susceptibility
Einstein meets Boltzmann
boson
fermion
anyon
Mirzba-Mohammadzadeh
0808.0241[cond-mat.stat-mech]
Example: quantum gas
Criteria for Riemann curvature
(associated w/ phase transition)
① divergent R
② vanishing R
Seems to contradict each other... In many cases, however,
the divergence of R is accompanied by the change of its sign.
In the analysis done below, we will use the R=0 criterion.
Castorina, Imbrosciano, Lanteri (2018)
Castorina, Lanteri, Mancani (2018)
Zhang, Wan, Ruggieri (2020)
An application
-hadron resonance gas model-
Essence of hadron resonance gas (HRG) model
Interactions btw hadrons ↔︎ all known resonances
Partition function
HRG does work well at low temperature/density
S. Borsanyi et al, Phys. Lett. B730 (2014) 99
Our main results
w/o excluded volume effect (EVE)
(i.e. hadrons are point-like)
with EVE
(i.e. hadrons w/ finite size)
Study of QCD phase diagram provides an interesting connection
between particle physics and cosmology
Especially, dense QCD is a friend of neutron star/gravitational wave
Its theoretical study is hard because of sign problem/strong dynamics
Thermodynamic geometry (TDG)
= Riemannian geometry in thermodynamic space
Application of TDG into hadron resonance gas (HRG) model
Summary and perspectives
Thank you !
감사합니다 !
感謝！