Shear viscosity of classical Yang-Mills fieldwith use of scaling invariance
Hidefumi Matsuda(Kyoto Univ.)
Collaborator ： T. Kunihiro（YITP）, B. Müller(Duke Univ.),
A. Ohnishi(YITP), T. T. Takahashi（NIT, Gunma Col.）arXiv:2007.06886 (ver2)
熱場2020• Introduction• Theory• Numerical Results• SummaryInitial stage of relativistic heavy-ion collisions

Flux tube(𝜏 ∼ 0 fm/c)• Strong gluon fields(based on the Color Glass Condensate)

Classical Yang-Mills(CYM) theory

Quark Gluon Plasma(𝜏 ≳ 0.6 fm/c)• Quark and gluon matter• Viscous fluid

Relativistic viscous hydrodynamics

IP-glasma model + Relativistic viscous hydrodynamics
The relativistic viscous hydrodynamics in combination with the initial condition givenby the IP-glasma model successfully reproduces the experimental data at mid-rapidity
IP-glasma model :• The initial condition is given as the CYM fields, taking
account for the event-by-event fluctuations of thenucleon positions in the transverse plane• The time evolution is obtained by solving the classicalequation of motion in the Yang-Mills theory

B. Schenke, C. Shen and P. Tribedy (2018)

Relativistic viscous hydrodynamics :• The initial condition is obtained from the energy-momentum tensor of theCYM fields at a switching time
• The shear viscosity is set as the small value, which is close to the lower boundpredicted by superstring theories with an Einsteinian classical limit, 1/4𝜋

𝜂/𝑠 ∼ 0.1 − 0.2

This talk

The CYM fields is used as the initial condition of the relativistic viscoushydrodynamics in order to reproduce the experimental data. However,
the transport property of the CYM fields hasn’t been clarified.

We investigate the shear viscosity of the CYM
fields by using the Green-Kubo formula• Introduction• Theory• Numerical Results• SummaryGreen-Kubo formula

In the Green- Kubo approach, we can extract the shear viscosity 𝜂 from the thermal
expectation value of the time-correlation function of energy-momentum(EM) tensor,

𝑉𝐶 𝑡 =
3

𝜏𝑖𝑗 𝑡 𝜏𝑖𝑗 (0) ,𝑖≠𝑗by using the Green-Kubo formula,
∞

1𝑖𝜔𝑡
𝜂=lim
𝑑𝑡 𝑒 𝐶 𝑡 .
2𝑇 𝜔→0 −∞1= ∫ 𝑑3 𝑥 𝑇𝑖𝑗 𝑥𝑉𝜏𝑖𝑗 𝑡𝑇𝑖𝑗 𝑥 : EM tensor
𝑉: volume

How to examine thermal expectation value in classical fields
Step1
prepare 𝑁 classical field configurations independently(𝑁 ≫ 1)
• same energy as each other• non-equilibrium

Non-equilibrium

12

Step2
evolve each configuration with the classical equationof motion until it equilibrates

Step3
calculate a observable in each configuration, 𝒪𝑖cl (𝑖 = 1,2, ⋯ 𝑁),and regard the average of them as the expectation value,
1cl
𝑂 eq ∼𝑁𝑁
𝑖=1cl𝑂𝑖N

Equilibrium
12N𝒪𝑁cl

𝒪1cl 𝒪2cl
𝑂 cleqExamination of 𝑂 cleq

CYM theory on lattice
We employ the Hamiltonian formalism to describe the time evolution of the CYM
fields(A. Krasnitz et al.(2003), T. Lappi(2003), P. Romatschke et al.(2006))•Temporal gauge +Coulomb gauge𝑎𝜕
𝐴𝑖 𝑖 𝑖 𝑥 =0

𝐴𝑎0 𝑥 = 0,Hamiltonian1𝐻=
2𝑎𝑎22𝐸𝑥+𝐵𝑥
𝑖𝑎,𝑖,𝑥 𝑖

𝐸𝑖𝑎 𝑥•

Equation of motion

𝜕0 𝐴𝑎𝑖 𝑥

𝜕𝐻= − 𝑎 , 𝜕0 𝐸𝑖𝑎 𝑥
𝜕𝐸𝑖 𝑥

𝜕𝐻= 𝑎
𝜕𝐴𝑖 𝑥

•

Space-averaged energy-momentum tensor (off-diagonal matrix elements, 𝑖 ≠ 𝑗)•𝜏𝑖𝑗 𝑥1=𝑉𝐸1′ (𝑥)𝑎𝑎𝑎𝑎𝐸′(𝑥)𝐸′(𝑥)+𝐵′(𝑥)𝐵′𝑖𝑗𝑖𝑗 (𝑥) ,𝑎,𝑥𝐵3′ (𝑥)𝐸′𝑎𝑖 𝑥== 𝜕0 𝐴𝑎𝑖 𝑥 , 𝐵𝑖𝑎 𝑥

𝐸𝑖𝑎 𝑥 +𝐸𝑖𝑎 𝑥−𝑖2, 𝐵′𝑎𝑖 𝑥

1𝑎= 𝜖𝑖𝑗𝑘 𝐹𝑗𝑘(𝑥)2=

𝐵𝑖𝑎 𝑥 +𝐵𝑖𝑎 𝑥+𝑖
2*The CYM theory is the infrared effectivemodel with a finite ultraviolet cut-off.
*We don’t take a continuum limit anddon’t renormalize.

Scaling property of CYM theory
The dependence of the shear viscosity 𝜂 on the coupling 𝑔 and lattice temperature 𝑇2
2
is represented by a scaling function 𝑓𝜂 (𝑔 𝑇) as 𝜂 = 𝑇𝑓𝜂 (𝑔 𝑇) due to the scaling
property of the CYM theory<Scaling property of CYM theory>
Gibbs ensemble of CYM field𝑒

𝐻 𝐴,𝐸,𝑔𝑇=𝑒

𝐻 𝐴/𝛾,𝐸/𝛾,𝛾𝑔
𝑇/𝛾2

𝐴𝑚 𝐸 𝑛 (𝑔,𝑇) = 𝛾 𝑚+𝑛 𝐴𝑚 𝐸 𝑛 (𝛾𝑔,𝑇/𝛾2)
𝑇𝑚+𝑛−2

𝐴𝑚 𝐸 𝑛 (𝑔,𝑇) is a function of 𝑔2 𝑇
22

𝐶 𝑡; 𝑔, 𝑇 = 𝑇 𝑓𝐶 (𝑡; 𝑔 𝑇)2
𝜂 𝑔, 𝑇 = 𝑇𝑓𝜂 (𝑔 𝑇)

We extract scaling function from numerical calculations over a wide range of 𝑔2 𝑇• Introduction• Theory• Numerical Results• SummarySet up• Number of sites : 163 , 243 , 323• Coupling constant and lattice temperature : g = 0.15 ∼ 20, 𝑇 ∼ 1
g 2 T = 0.0256 ∼ 499• SU(2) Yang-Mills theory

Time-correlation function of EM tensor
𝑡 < 1.5✔ Damped oscillatory behavior✔ Amplitude decreases by around
two orders of magnitude𝑡>5✔ Small remaining correlation decay
2✔𝑔 𝑇 dependence is not weakFourier transformation of 𝐶 𝑡 , 𝜌 𝑡

Solid line: 𝑓𝜌 𝜔; 𝑔2 𝑇 = 𝜌 𝜔; 𝑔, 𝑇 /𝑇 2
Dashed line ： Fourier transformation ofdamped oscillator✔ The broad bump around 𝜔 ∼ 5 reflects the damped oscillatory
behavior of 𝐶(𝑡)✔ With decreasing 𝑔2 𝑇, the height of the sharp peak at 𝜔 ∼ 0 increases
and its width narrows. Thus the 𝜔 ∼ 0 peak may be due to the slow
decay part of 𝐶(𝑡)2Shear viscosity : 𝑔 𝑇 dependence

22

𝑔 𝑇 dependence of 𝑓𝜂 (𝑔 𝑇) is welldescribed by a polynomial function
with parameters 𝛼, 𝛽, 𝛾, 𝛿,
𝐹 𝑥 = 𝛼𝑥

−𝛽/2

+ 𝛾𝑥

−𝛿/2(𝛽 > 𝛿)

1.10−1.88

It turns out that 𝜂 of the CYM field is proportional to 1/𝑔at weak coupling,which has a weaker dependence on 𝑔 than that in the leading-order perturbation
theory 𝜂 ∝ 1/𝑔4 ln𝑔−1 [1] but consistent with that of the "anomalous viscosity"𝜂 ∝ 1/g1−1.5 [2] under the strong disordered field in a turbulent plasma.
[1] P. B. Arnold, G. D. Moore and L. G. Yaffe (2000), (2003)[2] M. Asakawa, S. A. Bass and B. Müller (2006), V. Chandra (2012), J. Hong et al. (2014)

Shear viscosity : physical unit
𝑎−1 = 2 GeV, g = 0.5: same as value in previous work
(T. Epelbaum and F. Gelis (2013))

𝑇phys = 0.2 GeV, 0.4 GeV and 0.8 GeV: temperature at the start of thehydrodynamical simulation(RHIC and LHC)
𝜀lat ∼ 3 𝑁c2 − 1 𝑇lat = 9𝑇lat: harmonic oscillator approximation
within the acuracy of 9%-15%

The obtained shear viscosity is also found to be roughly consistent with that
estimated through the analysis of the near-equilibrium dynamics of the CYM
fields in the boost invariant expanding geometry with recourse to ahydrodynamic equation[3].
[1] T. Epelbaum and F. Gelis (2013))

Summary and Future work
• We calculated the shear viscosity of the classical Yang-Mills (CYM) fields on a lattice byapplying the Green-Kubo formula to the time-correlation function 𝐶(𝑡) of the energymomentum(EM) tensor in equilibrium.
• The correlation function C(t) was found to exhibit damped oscillatory behavior at earlytimes followed by a slow decay which produces a sharp peak at 𝜔 ∼ 0 in the Fouriertransformation of C(t).
• It turns out that 𝜂 of the CYM field is proportional to 1/𝑔1.10−1.88 at weak coupling,which has a weaker dependence on 𝑔 than that in the leading-order perturbation
4−1theory 𝜂 ∝ 1/𝑔 ln𝑔 (P. B. Arnold et al.(2000),(2003)) but consistent with that of the"anomalous viscosity" 𝜂 ∝ 1/g1−1.5 (M. Asakawa et al.(2006),V. Chandra(2012),Hong et al.(2014))
under the strong disordered field in a turbulent plasma.
• The obtained shear viscosity is also found to be roughly consistent with that estimatedthrough the analysis of the near-equilibrium dynamics of the CYM fields in theexpanding geometry with recourse to a hydrodynamic equation(T. Epelbaum et al.(2013)).

We found two interesting behaviors of 𝜂 of the CYM field. However it
is beyond the scope of this work to clarify the physical origins of those.
It is a feature work.