Cumulants Cn of σ a nonequilibrium chiral Bjorken model
near the critical point
Poramin Saikham
Christoph Herold
School of Physics, Suranaree University of Technology, 111 University Avenue,
Nakhon Ratchasima 30000, Thailand
e-mail: kob_ poramin@hotmail.com
INTRODUCTION
LINEAR SIGMA MODEL
Our universe today we see a hadronic medium rather than Quark-GluonPlasma(QGP). The Hadron evolve to cooling QGP. Evidence for QGP has
been found at ultrarelativistic heavy-ion collision at RHIC and LHC from
STAR Beam Energy Scan (BES) program (STEPHANOV; RAJAGOPAL;
SHURYAK, 1999).
In this work, we concentrate on the dynamical evolution of the σ by using
Langevin equation with linear sigma model. To increase opportunity to find
critical point.
CUMULANTS
The high order moment of the σ field can be calculated from these various
cumulants
C1,N
C2,N
C3,N
C4,N
=
=
=
=
<N >,
< (δN )2 > ,
< (δN )3 > ,
< (δN )4 > −3 < (δN )2 >2 .
Experimentally, the dependence on volume and temperature of these fluctuations is hard to measure directly. We then focus on ratios of cumulants
χ2
χ1
=
χ3
χ2
=
χ4
χ2
=
δN 2
σ2
=
,
hN i
M
δN 3
= Sσ,
hδN 2 i
δN 4
− 3 δN 2 = κσ 2 .
2
hδN i
The Lagrangian density of quark-meson model is defined as
L = q (iγ µ ∂µ − gσ) q +
U (σ) =
1
(∂µ σ)2 − U (σ) ,
2
2
λ2 2
σ − fπ2 − fπ m2π σ + U0 .
4
(1)
(2)
The field q = (u,d) has component of the light quark fields only. The
parameters of this model are chosen as fπ = 93 MeV , mπ = 138 MeV and
U0 , the ground state potential of U (σ). The quark-meson coupling g depend
on the nucleon mass that fixed by gσ equals to around 940 MeV in vacuum.
(HEROLD et al., 2019).
The Langevin equation gives
D
δΩ
σ̈ +
+ η σ̇ +
=ξ ,
(3)
τ
δσ
Under the assumption of the Bjorken model, The damping η and noise ξ (t,x)
term in the above equation satisfy the fluctuation-dissipation theorem :
m σ
hξ(t,~x)ξ(t0 ,~x0 )iξ = δ(~x − ~x0 )δ(t − t0 )mσ η coth
.
(4)
2T
This is equation of motion for a Bjorken expansion similar to what was
found in (NAHRGANG et al., 2011).
RESULT & DISCUSSION
METHODOLOGY
Get the susceptibility of the system computed in theoretical calculations
Initialize Hydro and Fields
Evolve Fluid-dynamics
sigma update
All event
no
Find the sigma average
yes
Get the sigma average
Cumulants
Stop
CONCLUSION
We have the highest fluctuation when the system across the Crossover
boundary. Moreover we see 2 peak of kertosis, so it has to have 2 phase
regions and the QCD CP is the distinguishable tool. It increases opportunity
to find QCD CP. The future, we will add events to accuracy and then compare
with experiment.
ACKNOWLEDGMENT
This work was supported by Suranaree University of Technology(SUT), Development and Promotion
of Science and Technology Talents Project (DPST)
Figure (1) is shown the phase transition of QGP for several initial
conditions. The dashed line corresponds to the phase boundary and the
green dot indicates the position of the CP. Figure (2-5) show high order
moment as a function of time. The last figure (6) is ratio of cumulants
named kertosis. We see that the system across the Crossover boundary have
more fluctuation.
REFERENCE
HEROLD, C. et al. Entropy production and reheating at the chiral phase transition. Physics
Letters B, v. 790, p. 557–562, mar. 2019. DOI: 10.1016/j.physletb.2019.02.004. arXiv:
1810.02504 [hep-ph].
NAHRGANG, M. et al. Nonequilibrium chiral fluid dynamics including dissipation and noise.
Phys. Rev. C, American Physical Society, v. 84, p. 024912, 2 ago. 2011. DOI:
10.1103/PhysRevC.84.024912. Disponível em:
<https://link.aps.org/doi/10.1103/PhysRevC.84.024912>.
STEPHANOV, M.; RAJAGOPAL, K.; SHURYAK, E. Event-by-event fluctuations in heavy ion
collisions and the QCD critical point. prd, v. 60, n. 11, 114028, p. 114028, dez. 1999. DOI:
10.1103/PhysRevD.60.114028. arXiv: hep-ph/9903292 [hep-ph].