transport coefficients near qcd critical point by dynamic rg · 2012-07-08 · transport...

Transport coefficients near QCD critical point by dynamic RG

Yuki Minami

Math. Phys. Lab., RIKEN

YM., Phys. Rev. D 83, 094019 (2011). YM., arXiv:1201.6408 [hep-ph].

プレゼンター

プレゼンテーションのノート

Thank you. Today, I will be taking about transport coefficients near QCD critical point. My collaborator is Kunihiro san.

Bulk viscosity:

Thermal conductivity:

Correlation length:

Not perfect fluid

by dynamic RG

Outline • Introduction • QCD critical point • Short review of critical dynamics

• Transport coefficients near QCD critical point • Nonlinear Langevin equation • Dynamic renormalization group

• Summary

プレゼンター


In this presentation, first I will provide some of the earlier studies on the transport coefficients near QCD critical point. Next, I will shortly review the theory to describe critical dynamics in condensed matter physics and explain why we consider nonlinear Langevin equation. After that, I will construct nonlinear Langevin equation for the QCD critical point based on the generalized Langevin equation and the relativistic fluid dynamics. Then, I will apply the dynamic renormalization theory to the constructed equation and analyze the critical behavior of the transport coefficients. Finally, I will make a brief summary.

QCD phase diagram

プレゼンター


Let me refresh your memory on QCD critical point. This is a typical QCD phase diagram. An interesting feature of this phase diagram is the critical point, which is predicted by various effective models of QCD. The critical point is expected as the end point of the first order phase transition line. The significance of this QCD Critical point is that the phase transition at this point is of second order, and thereby we can expect critical phenomena due to the divergence of correlation length.

QCD phase diagram (2) Lattice QCD

Perturbative QCD

?

Only by effective models

Lowering

Beam energy scan program Experimental probe

QCD critical point (CP) 2nd order transition

Correlation length

Singularity

Singularity at a CP Free energy

Curvature along order parameter ~ 0

1 / (Curvature)

: an Order parameter

Order parameter for QCD CP Free energy

by NJL model Flat direction

linear combination

For

H. Fujii and M. Ohtani, Phys. Rev. D 70, 014016 (2004)

: Baryon num

ber density

Bulk viscosity near QCD CP

・Counter arguments P. Romatschke and D. T. Son, Phys. Rev. D 80, 065021(2009)

プレゼンター


As one of such critical phenomena, some authors suggested a divergent behavior of bulk viscosity at the QCD CP. However, their validity of their argument is very much controversial. For instance, these authors point out that the ansatz for the spectral function adopted in this paper is not necessarily true, and a microscopic calculation based on the relativistic Boltzmann equation shows that the bulk viscosity is finite at QCD critical point. Therefor, it is still uncertain whether the transport coefficients will show a divergent behavior near the QCD CP.

Shear viscosity and Thermal conductivity

solid

gas

liquid Critical point

D. T. Son and M. A. Stephanov, Phys. Rev. D 70 (2004), 056001.

Dynamic universality class: H ? P.C. Hohenberg

and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).

Outline • Introduction • QCD critical point

• Short review of critical dynamics

• Transport coefficients near QCD critical point • Nonlinear Langevin equation • Dynamic renormalization group

• Summary

プレゼンター


The divergent behavior of transport coefficients is common to several critical points in condensed matter, for instance, the critical point of liquid-gas or ferromagnetic phase transitions and so on. Then, I will review the theory to describe critical dynamics developed in condensed matter physics.

ex. thermal conductivity near liquid-gas CP

:Heat current Microscopic processes, usually

can contain nonlinear fluctuations of macroscopic variables.

Dominant part near the CP

H. Mori, Prog. Theor. Phys. 33, 423 (1965)

Critical divergence of transport coefficients

プレゼンター


As well known in condensed matter physics, the divergent behavior of transport coefficients is caused by the universal mechanism. I would like to now illustrate how the mechanism cause the critical divergence. For example, let us consider the thermal conductivity near liquid-gas critical point. The thermal conductivity is given by Kubo formula as the correlation function of the heat current. This heat current usually represents microscopic processes However, it can contain nonlinear fluctuation of macroscopic variables. Then, the heat current consists of microscopic and macroscopic processes. We may write the macroscopic heat current as this. Here, s is the entropy fluctuation and v is the fluid velocity fluctuation. This heat current represents is nonlinear in fluctuations, so we can neglect this current in a normal region. However, near the critical point, this term becomes dominant part because fluctuations are enhanced near the critical point.

Critical divergence

Nonlinear Langevin equations

+ Renormalization group

Macroscopic processes cause the divergence

Macroscopic

プレゼンター


Therefore, the thermal conductivity also consists of the two parts. The first term represents the thermal conductivity coming from microscopic processes while the second term represents the contribution from the macroscopic nonlinear fluctuation. Because the entropy fluctuation is the soft mode near the liquid-gas critical point, this term diverges at the critical point. Let us call the transport coefficients, such as \lambda_{micro}, bare transport coefficients, while those including macroscopic processes renormalized transport coefficients. The important point is that the macroscopic nonlinear fluctuation causes the critical divergence of the transport coefficients. In other words, microscopic processes would give a only minor contribution to the critical divergence of these quantities, if any. Although the earlier studies on the bulk viscosity near QCD critical point are based on microscopic theories, we may analyze the critical divergence with a macroscopic theory. The dynamic renormalization group theory is a standard technique used in critical dynamics, which systematically incorporate the macroscopic fluctuations causing the divergent behavior of transport coefficients. In this theory, the nonlinear Langevin equation is used as an infrared effective theory. On the next slide, I will provide the formalism of Langevin equation.

Langevin equation The simplest Langevin equation EOM of Brownian motion

Brownian particle

:velocity of Brownian particle

:relaxation rate

:random noise

Systematic part Random part

Systematic part only macroscopic variables.

Random part microscopic processes.

Small particle

プレゼンター


I think that some people are not familiar with Langevin equations, so let me start with the simplest Langevin equation, namely, Brownian motion. Brownian motion is a zig-zag motion of a relatively large particle called Brownian particle in a fluid and its equation of motion is given by this Langevin equation. u is the velocity of the Brownian particle , gamma is its relaxation rate and theta is the random noise caused by a large number of collisions with the surrounding particle. The important point is that Langevin equations generally consist of two parts; a systematic part and a random part. The systematic part is the part described by only macroscopic variables and the random part comes from microscopic processes. Although this Langevin equation is just phenomenological equation, we can formally derive the Langevin equation from microscopic equations of motion.

Generalized Langevin equation EOM of arbitrary slowly varying variables.

Liouville equation

Heisenberg equation

： arbitrary slowly varying variable

：microscopic Hamiltonian

Systematic part and Random part

Generalized Langevin equation H. Mori and H.Fujisaka, Prog. Theor. Phys.49, 764 (1973). H. Mori, Prog. Theor. Phys. 33, 423 (1965).

Microscopic EOM

Formal decomposition

プレゼンター


The starting microscopic equation is given as the Liouville or Heisenberg equation for classical and quantum cases, respectively. Here A is an arbitrary slowly varying variables, H is a microscopic Hamiltonian, this bracket is the Poisson bracket and this is the commutation relation. We may formally decompose this microscopic equation into a systematic part and a random part using the projection operator method. The decomposed equation is called the generalized Langevin equation. The decomposition is just formal one, so we may consider the generalized Langevin equation even for the QCD critical point.

Generalized Nonlinear Langevin equation

:Slow variables = Order parameters + Conserved densities

:Kinetic coefficients :Noise terms

:Free energy

:Streaming terms

H. Mori and H.Fujisaka, Prog. Theor. Phys.49, 764 (1973)

time-reversible irreversible

プレゼンター


This is the generalized nonlinear Langevin equation. Here, A is slow variables and chosen as soft modes and conserved densities. This part is the systematic part and nonlinear in slow variables while this part is the random part. v is called streaming terms, which give time-reversible change rates and dynamic nonlinear interactions among slow variables. This term represents dissipative effects. H is a thermodynamic potential and L is bare kinetic coefficients. How do we construct this Langevin equation ?

Dynamic renormalization group Average over the short wavelength component of

from nonlinear Langevin equation.

Renormalization group (RG) equations

:ultraviolet cutoff

プレゼンター


I would like to next move on dynamic renormalization group theory. First of all, I note that the Langevin equation is an infrared effective theory and has the ultra violet cutoff. As dynamic RG transformation, we average over the short wavelength component of the slow variables in the shell from the nonlinear Langevin equation. In other words, we slightly coarse grain the nonlinear Langevin equation. From inspecting on the coarse-grained equation, we can know RG equations for transport coefficients.

Purpose

Dynamic RG

:thermal conductivity :shear viscosity

Nonlinear Langevin eq.

:bulk viscosity

プレゼンター


Therefor, I will construct the nonlinear Langevin equation for the QCD critical point based on the general form. After that, I will apply the dynamic RG theory to the obtained equation and analyze the transport coefficients near QCD critical point.


• Transport coefficients near QCD critical point

• Nonlinear Langevin equation • Dynamic renormalization group

• Summary

プレゼンター


I would like to next move on the construction of the nonlinear Langevin equation for QCD critical point.

Slow variables for QCD CP (1)

Conserved densities :Energy-momentum

Order parameter

Long-time behavior is determined only by .

is needless for the slow dynamics. D. T. Son and M. A. Stephanov, Phys. Rev. D 70 (2004), 056001.

Slow variables for QCD CP (2)

Note: Near QCD CP Relativistic critical fluid

Slow variables:

プレゼンター


Frist, let us specify the slow variables near the QCD critical point. What are the soft modes near the QCD critical point? The answer is given by these authors, and the soft modes are the long wavelength fluctuations of conserved densities , the baryon number density and the energy momentum tensor. Because the soft modes are the conserved densities, we may choose only the conserved densities as the slow variables, namely, the density , the energy density, and the momentum density. The relevant variables are only the conserved density, so we may describe the system near the QCD critical point as a relativistic critical fluid.

Hydrodynamic and critical regimes

normal fluid

critical fluid:

Far from CP

hydrodynamics

Near CP

・hydrodynamic regime

・critical regime

normal fluid + nonlinear effects

mean free path

Linearised relativistic hydrodynamics +

Static Scaling laws

YM. et al., Prog. Theor. Phys. 122, 881 (2009).

Tendency

＆ Enhanced: Not enhanced:

Not all fluctuations are enhanced

プレゼンター


Because the momentum density is not a thermodynamic quantity. we must consider the linearized relativistic hydrodynamics in addition to static scaling laws for the slow variables. Such study is given by my this paper, I will provide a part of this paper associated with this point.

＆

Free energy for QCD CP

We may use the free energy for 3d Ising system.

:Gaussian form

:Gaussian form

Static universality class:

Ising CP

プレゼンター


Next, I would like to construct the thermodynamic potential in terms of the slow variables. For the momentum density, we may take a Gaussian form, because the momentum density is not enhanced. For the density and the energy density, we can not take Gaussian forms. Alternatively, we may use the thermodynamic potential for 3d Ising system. because the thermodynamic potential is the quantity that determines the static property and the static universality class is Z2, which is of the Ising critical point.

Free energy for Ising CP

Spin density (order parameter) Energy density

B.I. Halperin, P.C. Hohenberg, and S. Ma, Phys. Rev. B13, 4119 (1976)

We need mapping relation.

プレゼンター


This is the thermodynamic potential for 3d Ising system called GLW Hamiltonian. Here, psi is the spin density, m is the exchange energy, tau is the reduced temperature, h is the magnetic field. Because this potential is written by the Ising variables, we need the mapping relation between the two systems.

solid

gas


Mapping relation Ising systems Grand canonical ensembles

A. Onuki, Phys. Rev.E 55,403 (1997)

Mapping relation

プレゼンター


The mapping relation between a Ising system and a grand canonical ensemble in Z2 class is developed by Onuki, fortunately. I will now provide the mapping relation. First, we assume this linear relation. This relation represents that which directions in QCD phase diagram corresponds to which directions in Ising phase diagram near the critical points. Then, we may assume that the coefficients, alpha and beta, are no singular. The important point is that we need not to specify this coefficients for the RG theory because they have not singularity.

The rest parts

Continuity equations +

General conditions

Relativistic hydrodynamics (Landau equation)

Nonlinear Langevin equation for QCD CP

Nonlinear force near the CP

no dissipation

cf. for normal fluid

プレゼンター


We have determined the all components of the Langevin equation at last. This is the constructed equation. These terms represents the convection effects due to the fluid velocity fluctuation. This term represents the intrinsic force near the QCD CP. This intrinsic force is absent in a normal relativistic fluid. The important point is that there are four nonlinear term. Namely, these reversible terms and this irreversible term are nonlinear.


• Transport coefficients near QCD critical point • Nonlinear Langevin equation

• Dynamic renormalization group

• Summary

プレゼンター


I would like to next move on the Dynamic RG analysis.

Diagramatic representation

プレゼンター


Let us now represent the slow variables with these diagram. The solid line is the density, and the dashed and wavy lines are the transvers and longitudinal momentum densities, respectively. With these diagrams, we can represent VN as this two vertexes. Although VJ contains many nonlinear terms, dominant terms are only two terms and given these vertexes. An important point is that these vertex functions are different. This difference results in the difference between the bulk and shear viscosities.

Renormalization of propagators

プレゼンター


From a similar calculation of a beta function in a quantum field theory, we can obtain the RG equation for the transport coefficients. Specifically, from the renormalization of the density propagator, we can obtain the RG equation for the thermal conductivity. And, from the renormalization of the transvers and longitudinal momentum density propagators we can obtain for the shear and bulk viscosities.

Dynamic RG equations near QCD CP

unimportant constants a static parameter

space dimension

プレゼンター


This is the obtained RG equation. Here, I have introduced F for convenience, and F is given as this. epsilon is four minus the space dimension gamma is a static parameters in the thermodynamic potential. A and B are unimportant constants. From these equations, we can extract the critical exponents of the transport coefficients.

Transport coefficients near QCD CP

Thermal conductivity:

Shear viscosity:

Bulk viscosity:

プレゼンター


This is the result at three dimensions. We see that the bulk viscosity and the thermal conductivity strongly diverge while the shear viscosity has a very weak divergent behavior. Therefor, these transport coefficients can become more important than the shear viscosity near QCD critical point, although these quantities are usually neglected in heavy ion physics. Moreover, from these divergent behaviors, we can see that the description as a perfect fluid is not valid near the critical point. If I compare with the result by Karsh, which I mentioned in the Introduction, the divergent behavior of the bulk viscosity is the same, but, the critical exponent is different. The critical exponent by this paper is small contrary to our result.

Comparison with liquid-gas CP

solid

gas



33

プレゼンター


Let me compare with the liquid-gas critical point, namely, the non-relativistic critical fluid. Some authors conjectured that the QCD critical point has the same critical behaviors as the liquid-gas critical point has. Does the QCD critical point actually have the same critical behaviors? I will answer such question.

{

{

For QCD CP

For liquid-gas CP A. Onuki, Phys. Rev.E 55,403 (1997)

The dissipative terms are different.

No dissipation

No dissipation

34

プレゼンター


The nonlinear Langevin equations for the QCD and liquid-gas critical points are given by like this. From this, we see that the dissipative terms are different. For the QCD critical point, the density has the dissipative term, but the energy density has no dissipation. On the other hand, for the liquid-gas critical point, the density has no dissipation, but the energy has the dissipation term. How does this difference result in the dynamic critical behaviors?

Relativistic effects in dynamic RG

Relativistic effects appear in only unimportant constants.

・At one loop calculation

プレゼンター


The answer is that the relativistic effects appear in only unimportant constants, fortunately or unfortunately. For me, it is unfortunate.

solid

gas


The same

Summary

:thermal conductivity :shear viscosity

Nonlinear Langevin eq.

Relativistic critical fluid

:bulk viscosity

Dynamic RG

プレゼンター


Let me summarize my talk. I have constructed the nonlinear Langevin equation for the QCD critical point based on the general form of Langevin equations and the description as a relativistic critical fluid. After that, I have applied the dynamic RG theory and analyzed the transport coefficients.

Future work

Observables in RHIC

Thank you!

Kinetic coefficients

:bare thermal conductivity

:bare bulk viscosity :bare shear viscosity

From Landau equation

The other coefficients are zero.

for small

プレゼンター


Next, I would like to determine the kinetic coefficients. The kinetic coefficients are only phenomenologically determined. So let us determine the kinetic coefficients from a relativistic hydrodynamic equation. Here, let us take the Landau equation. From the Landau equation, the kinetic coefficients for the small fluid velocity fluctuation read like this, and the other coefficients are zero.

Signal for QCD CP M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009).

: Baryon number

Construction method 1. Specify slow variables

2. Construct free energy

3. Determine streaming terms

or continuity eq.

Conditional average

プレゼンター


I will now provide a general prescription for construction. First, we must specify the slow variables, namely, soft modes and conserved densities. Next, we must construct the thermodynamic potential in terms of the slow variables. After that, we must calculate the streaming terms from this relation in general. Here, Q is the conditional average of Poisson brackets or commutation relations. In the conditional average, the slow variables are fixed at this values. This is a kind of coarse graining. In this average, microscopic degree of freedom is integrated out and eliminated while slow variables are retained. If the slow variables are conserved, we may also determine the streaming terms from continuity equations.

Static and Dynamic RG

Static

Dynamic

Infrared effective theory

Keys for construction

Free energy Nonlinear Langevin eq.

Space dimension

Symmetry

Space dimension Symmetry

Poisson bracket relations

Relevant variables

Order parameters

Order parameters

Conserved densities +

+

プレゼンター


Let me compare with static RG. As I have shown in the previous slide, the dynamic RG transformation is almost the same as static RG one. However, the considered infrared effective theory is fairly different. In the static case, we consider the thermodynamic potential as the infrared effective theory. and relevant variables are given by only order parameters. Key components for construction the thermodynamic potential are only the space dimension and symmetry among the order parameters. So the static universality class is universal as its name means. While, in the dynamic case, we consider nonlinear Langevin equation as the infrared effective theory. As relevant variables, we must consider the conserved densities in addition to order parameters because the nonlinear couplings among them causes the critical divergence of the transport coefficients. And such nonlinear couplings, namely, streaming terms are determined from the Poisson bracket relations in general. Threfor the Poisson brackets relations come into as a key component.

Dynamic universality class Poisson bracket relations depend on microscopic expressions

Not so universal


QCD CP Class H ?

Model H is based on nonrelativistic relations. ex. Microscopic expression for energy density

K. Kawasaki, Ann. Phys. (N.Y.) 61,1 (1970).

プレゼンター


However, the Poisson bracket relations depend on microscopic expressions, so the concept of dynamic universality class is not so universal compared to static one. Some authors claim that the dynamic universality class of QCD critical point is class H, which is of the liquid-gas critical point, and the dynamic critical behavior is of class H. However, the model H, which is the nonlinear Langevin equation for class H is base on nonrelativistic relations. For example, the microscopic expression of the energy density is assumed as this form Obviously, we can not adopt this relation for quark and gluon. So it is not trivial how the transport coefficients behave near the QCD critical point.

Relativistic effects

Linearized Landau equation

No dissipation

:enthalpy density :fluid velocity

EOM for the slow variables at linear level

With suffix c → equilibrium values With → fluctuations,

プレゼンター


What you see here is the linearized Landau equation, which is a relativistic hydrodynamic equation. This equation is the equation of motion for the slow variables at linear level. Here, the variables with delta are fluctuations and with the suffix c are the equilibrium values. We can see that the relativistic effect comes into the density conservation law and the energy density has no dissipation. This relativistic effects come from that the fluid velocity represents the energy flow. In the nonrelativistic case, the fluid velocity represents the particle flow.

Hydrodynamic modes and these behaviors

Thermal mode

Sound mode

Viscous mode

＆ Enhanced: Not enhanced:

＆

Static scaling law

プレゼンター


The Linearized Landau equation has the three modes, namely, the thermal mode, the sound mode and the viscous mode. These modes correspond to the slow variables. Here, JL is the longitudinal component of the momentum density and JT is the transverse component. Using, additionally, static scaling laws, we can know the behaviors of these modes. As a result, we obtain that the density and the energy density are enhanced while the momentum density is not enhanced. The important point is that also the fluid velocity fluctuation is not enhanced.

Hydrodynamic and critical regimes (2) Initial cutoff :

RG transformation

Only hydrodynamic regime

cf.

Behavior of fluctuations around QCD CP

Not all fluctuations are enhanced near QCD CP.

ex. from static scaling laws specific heat and compressibility

For grand canonical ensemble

: enhanced : not enhanced

プレゼンター


For the following construction, we must take care that not all fluctuations are enhanced near the QCD critical point. For example, static scaling laws tell us behavior of some fluctuations. From static scaling laws, we can know that the specific heat at constant pressure and the adiabatic compressibility diverge at the critical point. On the other hands, we can relate the thermodynamic quantities, and the susceptibility of the entropy and pressure fluctuations. Therefor, the entropy susceptibility diverges at the critical point the pressure susceptibility is suppressed. In other words, we can see that the entropy fluctuation is enhanced while the pressure fluctuation is not enhanced.

Streaming terms

＆

From continuity eq.

is the Lorentz factor of

We may approximate

Note:

:reversible change rates

Reversible currents

プレゼンター


Next, let us determine the streaming terms from continuity equations. From continuity equations, we can know that the streaming terms for the density and the energy density are written as a divergence of the reversible current. The reversible currents in the relativistic fluid are almost trivial, and given as this. This currents simply represents the effect of the convection due to the fluid velocity fluctuation. Here, gamma is the Lorentz factor of the fluid velocity fluctuation and we may approximate one, because the fluid velocity fluctuation is not enhanced. Therefor, we obtain these streaming terms.

Potential condition

for a continuum system

A. Onuki (2002)

From continuity eq.

General condition

プレゼンター


For the momentum density, the continuity equation tell us that the streaming term is a divergence of the reversible stress tensor. However, the reversible stress tensor is not trivial. Then, We need another prescription. Let me take a tricky method developed Onuki. Namely, I determine the streaming term from the potential condition, which is a general condition. The potential condition is given as this. The important point that the right hand side generally vanishes for a continuum system. Then, we have this condition. Because the unknown variable in this condition is only the VJ, we can determine VJ from this condition. As a result, we obtain this expression.

Behavior of fluctuations around QCD CP

Not all fluctuations are enhanced near QCD CP.

ex. from static scaling laws specific heat and compressibility

For grand canonical ensemble

: enhanced : not enhanced

プレゼンター


For the following construction, we must take care that not all fluctuations are enhanced near the QCD critical point. For example, static scaling laws tell us behavior of some fluctuations. From static scaling laws, we can know that the specific heat at constant pressure and the adiabatic compressibility diverge at the critical point. On the other hands, we can relate the thermodynamic quantities, and the susceptibility of the entropy and pressure fluctuations. Therefor, the entropy susceptibility diverges at the critical point the pressure susceptibility is suppressed. In other words, we can see that the entropy fluctuation is enhanced while the pressure fluctuation is not enhanced.

Perturbative treatment (1)

Linear terms = Nonlinear + Noise terms

Linear operator

Rewrite

Langevin equation

Bare propagator:

Iteration gives a perturbative expansion of

Perturbative treatment (2)

Self-consistent equation: cf.

Dynamic RG near QCD CP

in

Static RG No renormalization

＆

Nonlinear terms:

At a one loop calculation

only

プレゼンター


As I have shown, we have the four nonlinear terms. However, we need not to renomalize all nonlinear terms. At a one loop calculation, irreversible nonlinear terms generally renormalize only static parameters, and we can know such static critical behaviors from static RG. Then, we can effectively neglect this irreversible nonlinear term. In addition, we can neglect this nonlinear term because this term does not renomalize. Therefor, we must reomalize only this two term.

Message

Not Perfect fluid

プレゼンター


Let me summarize my talk. I have constructed the nonlinear Langevin equation for the QCD critical point based on the general form of Langevin equations and the description as a relativistic critical fluid. After that, I have applied the dynamic RG theory and analyzed the transport coefficients.

transport coefficients near qcd critical point by dynamic rg · 2012-07-08 · transport...

Documents