Parametric instabilities of

color magnetic field

04/19/2023lunch seminar @ BNL

Shoichiro Tsutsui (Kyoto)

In collaboration with Hideaki Iida (Kyoto), Teiji Kunihiro (Kyoto), Akira Ohnishi (YITP)

Outlines

Introduction Early thermalization problem

Instabilities triggered by color magnetic field

Formalism Floquet theory

Application to CYM

Numerical results

Summary

04/19/2023lunch seminar @ BNL

Early thermalization problem

QGP

time

？

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hydrodynamics is applicable

Heinz (2005)

How can we understand the considerably short time ?

Instabilities in HIC

the earliest stage of HIC…

(under strong EM fields)

chaos

Kunihiro, Muller, Ohnishi, Schafer (2009)Kunihiro et. al. (2010)Iida et. al. (2013)

• non-Abelian Weibel instability• Nielsen-Olesen instability• parametric resonance

instability has been expected to be the triggers leading to early thermalization

turbulence

Berges, Boguslavski, Schlichting, Venugopalan (2014)

Micha, Tkachev (2004)

isotropization

Epelbaum, Gelis(2014)

Arnold, Lenaghan, Moore, Yaffe (2005)

Instabilities triggered by color magnetic field

homogeneous

color magnetic field

gauge fields are also homogeneous

“abelian” configuration

“non-abelian” configuration

lunch seminar @ BNL

Nielsen-Olesen instability

parametric resonance

Fujii, Itakura(2008) Iwazaki(2009)Tanji, Itakura (2012)

Berges, Scheffler, Schlichting, Sexty(2012)

lunch seminar @ BNL

Previous work

performed by classical statistical simulation

Berges, Scheffler, Schlichting, Sexty(2012)

time evolution of gauge field

unstable behavior is observed

time evolution under the “non-abelian” configuration

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lunch seminar @ BNL

Growth rate

growth rate is given by

sub-dominant instability bandis also discussed

parametric resonance

Berges, Scheffler, Schlichting, Sexty(2012)

time averageNO-like instability

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lunch seminar @ BNL

What is the origin ?

system is spatially homogeneous

however…

magnetic field

Landau quantization ?

“non-abelian” configurationdoes not form Landau orbit !

growth rate is still unknown

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lunch seminar @ BNL

Our work

• SU(2) pure Yang-Mills• temporal gauge• non-expanding geometry• under homogenous and time

dependent magnetic field

• linear analysis

We show that instability under non-abelian conf. is completely different from Nielsen-Olesen instability

but enable to determine instability bands precisely

same as Berges, Scheffler, Schlichting, Sexty

Outlines

Introduction Early thermalization problem

Instabilities triggered by color magnetic field

Formalism Floquet theory

Application to CYM

Numerical results

Summary

04/19/2023lunch seminar @ BNL

ODE with periodic function

n- independent solutions

differential eq. with periodic coefficient

solution matrix

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periodic function

Monodromy matrix

follow same equation

solution matrix

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monodromy matrix is given by

periodic function

Floquet’s theorem

Floquet’s theorem

solution matrix is given by

stability of the solution

unstable (exponential growth)

stable

(anti-)periodic or polynomial growth

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eigenvalue of M

Algorithm

set initial conditions

unstable mode

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EOM

solve the EOM for a period

get the monodromy matrix

calculate eigenvalues (characteristic

exponent)

fixed parameter

Merit of Floquet analysis

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Floquet analysis

Numerical calc.

non-linearity

cost

how to find instability bands

linear

non-linear

low(solve EOM for only one period)

high

fittingcheck

Floquet analysis is economicalcriterion of instability is clear

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lunch seminar @ BNL

instability bands

Example: Lame eq.

Lame eq.

unstable mode

the simplest case (n=2)

Outlines

Introduction Early thermalization problem

Instabilities triggered by color magnetic field

Formalism Floquet theory

Application to CYM

Numerical results

Summary

04/19/2023lunch seminar @ BNL

Time evolution of background

Classical Yang-Mills eq. (temporal gauge)

lunch seminar @ BNL

solution is periodic

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“non-abelian” configuration

analytic solution is available

period

Time evolution of fluctuations

Classical Yang-Mills eq.

lunch seminar @ BNL

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EOM of fluctuation(multi component Hill’s

eq. )

2nd order ODE with periodic function

without loss of generality

Outlines

Introduction Early thermalization problem

Instabilities triggered by color magnetic field

Formalism Floquet theory

Application to CYM

Numerical results

Summary

04/19/2023lunch seminar @ BNL

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lunch seminar @ BNL

Complete band structure

map view

characteristic exponent

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Complete band structure

agree with Berges et. al. (2012)

NO-like band

Complete band structure

broad instability region in pT direction

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Complete band structure

broad instability region in pT direction

many bands around

Lowest Landau mode (n=0) is only unstable in NO instability

quantized transverse mom

NO-like band at pT=0 plane should be regarded as a part of parametric resonance bands

NO-like band

parametric resonance bands

lunch seminar @ BNL

Summary

We investigate parametric resonance in Yang-Mills theory

Floquet theory enable us to perform in a systematic way

Instability considered here is completely different from Nielsen-Olesen instability

(there exists broad instability region in pT direction)

Can we see various unstable modes in full numerical simulation?

(work in progress)

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outlook

back up

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CYM under color magnetic field

EOM of fluctuation(multi component Hill’s

eq. )

without loss of generality

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CYM under color magnetic field

lunch seminar @ BNL

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notation

lunch seminar @ BNL

color (a=1,2,3)Lorentz(i=x,y,z)

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