Hydrodynamics, current algebra, and quantum anomaly

Hydrodynamics, current algebra, and quantum anomaly

Workshop of Recent developments in QCD and Quantum field theories, 2017 Nov. 11th

N S

j B

µ_R 6= µ^L

國 立臺灣大學

Masaru Hongo ( RIKEN, iTHES )

Talk by T. Hirano Talk by Y. Hidaka

Talk by D. Kharzeev

Outline

Motivation:

Approach:

Result:

Anomalous commutation:

Mori’s method as a generalization of current algebra

Chiral Magnetic Effect in operator formalism:

Origin of chiral transport (Chiral Magnetic Effect)?

- Existence of extremely strong magnetic field

- Chirality drastically affect hydrodynamic transport

y x z

QGP

+

Gluon

B

e |B| 10

T

QGP

+ + +

+ − +

− −

− −

QGP as Chiral fluid

•

Effective theory for macroscopic dynamics

•

Universal description, not depending on details

•

Only conserved quantity ~ symmetry of system

Hydro:

' '

http://www.bnl.gov/rhic/news2/news.asp?a=1403&t=pr

Quark-Gluon Plasma

http://newsoffice.mjitugenn.edu/2012/model-bursting-star-0302

Neutron Star

10

cm

T 10

K 10

kg/cc

10 km

Hydrodynamics is

◆ Spontaneous symmetry breaking

◆ Symmetry breaking by quantum anomaly

https://en.wikipedia.org/wiki/Superfluidity#/

media/File:Liquid_helium_Rollin_film.jpg

Macro：Superfluid Micro：Selecting vacuum

0

Micro：π⁰ decay

[Adller (1969), Bell-Jackiw (1969)]

N S

j B

µ_R 6= µ^L

Macro：Anomalous transport

[Erdmenger et al. (2008), Son-Surowka (2009)]

Symmetry breaking and Hydro

Parity- violating chiral transport

◆Chiral Magnetic Effect (CME)

N S

j B

µ_R 6= µ^L

J = ~ µ ₅

2⇡ ² B ~

◆ Chiral Vortical Effect (CVE)

µ

6= µ

~ j

J = µµ ₅

2⇡ ² ! ~

Anomaly and chiral transport

Quantum anomaly

N S

j B

µ_R 6= µ^L

CME

Can we understand this based on current algebra?

Not vacuum physics as is the case for QCD!

→ We have to generalize current algebra for

Problem.

T 6= 0, µ 6= 0

Outline

Motivation:

Approach:

Result:

Anomalous commutation:

Mori’s method as a generalization of current algebra

Chiral Magnetic Effect in operator formalism:

Origin of chiral transport (Chiral Magnetic Effect)? ^{S N}

µ₅ = 0 j

?

Review: Current algebra for QCD

◆ Current algebra for

⇥ ˆ Q

, ˆ J

(x) ⇤

= if

J ˆ

(x), ⇥ ˆ Q

, ˆ J

(x) ⇤

= 0

⇥ ˆ Q

, ˆ J

(x) ⇤

= if

J ˆ

(x), ⇥ ˆ Q

, ˆ J

(x) ⇤

= 0 SU (N )

⇥ SU(N)

Low-energy theorem

{

Universal results for process with low-energy pion scattering!

If current algebra satisfies the above relations,

it does not matter whether UV theory is QCD, NJL model, or anything!

Current algebra and chiral anomaly

◆ Current algebra in external EM fields for U (1)

⇥ U(1)

⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= ⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= 0

⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= 0

Proof.

⇥ ˆ(t, x), ˆ⇡(t, y)⇤

= i (x y) Using canonical commutation relation

we can directly show the above current algebraic structure!

Jˆ₅⁰(x) = iˆ⇡(x) ₅ ˆ(x) Definition of Noether current gives

J ˆ

(x) = @ L

@(@ ) i ˆ = iˆ ⇡(x) ˆ(x),

Current algebra and chiral anomaly

Sketch of Proof.

⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= ⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= 0

Ward-Takahashi identity is not h@^µJ₅^µ(x)i^A = 0 but

h@

J

(x) i

= C✏

F

(x)F

(x) ⇠ CdAdA

⇠ CdB

Variation w.r.t gives

A

@

hJ

(x)J

(y) i

⇠ CddA

“Corr. function = T-product in operator formalism” gives the above

⇥ ˆ J

(t, x), ˆ J

(t, y) ⇤

= i

2⇡

B

(t, y)@

(x y)

◆ Current algebra in external EM fields for U (1)

⇥ U(1)

Anomaly and chiral transport

Quantum anomaly

N S

j B

µ_R 6= µ^L

CME

Can we understand this based on current algebra?

Not vacuum physics as is the case for QCD!

→ We have to generalize current algebra for

Problem.

T 6= 0, µ 6= 0

◆ EoM given by Mori’s projection operator method

Reversible

@

A ˆ

(t) = i⌦

A ˆ

(t)

Z

ds

(t s) ˆ A

(s, y) + ˆ R

(t)

mn (t s) = ˆR_n(t s), ˆR^m(0) Noise Dissipative

Fluctuation Dissipation relation:

i⌦_n^m = i

h[ ˆA_n(0), ˆA^m(0)]i + iµ [ ˆN , ˆA_n(0)], ˆA^m(0)

{

A~_i A~_j

B~

P ~ˆB = X

i

aⁱA~_i a^j

aⁱ A~_?

B~_?

A method to write down

A ˆ

(t)

Equation of Motion (EoM) only focusing on

Mori’s projection operator method

[Mori (1965)]

◆ EoM given by Mori’s projection operator method

i⌦_n^m = i

h[ ˆA_n(0), ˆA^m(0)]i + iµ [ ˆN , ˆA_n(0)], ˆA^m(0)

mn (t s) = ˆR_n(t s), ˆR^m(0) Noise Dissipative

Reversible

@

A ˆ

(t) = i⌦

A ˆ

(t)

Z

ds

(t s) ˆ A

(s, y) + ˆ R

(t)

Fluctuation Dissipation relation:

{

A~_i A~_j

B~

P ~ˆB = X

i

aⁱA~_i a^j

aⁱ A~_?

B~_?

A method to write down

A ˆ

(t)

Equation of Motion (EoM) only focusing on

Mori’s projection operator method

[Mori (1965)]

Outline

Motivation:

Approach:

Result:

Anomalous commutation:

Mori’s method as a generalization of current algebra

Chiral Magnetic Effect in operator formalism:

Origin of chiral transport (Chiral Magnetic Effect)?

Sound and Chiral Magnetic Wave as a family of Type-B NG mode

S N

µ₅ = 0 j

?

⇥ ˆJ₅⁰(t, x), ˆJ⁰(t, y)⇤

= i

2⇡² Bⁱ(t, y)@_i^x (x y)

A~_i A~j

B~

P ~ˆB =X

i

aⁱA~i

a^j

aⁱ A~_?

B~_?

Mori's method and current algebra

A ˆ

(t)

Choose as conserve charges:

A ˆ

(t) = { ˆ T

(t, x), ˆ T

(t, x) }

EoM(LO) is controlled by energy-momentum density algebra!

⇥ ˆT ⁰₀(t, x), ˆT ⁰₀(t, y)⇤

= i ˆT ^k₀(t, x) + ˆT ^k₀(t, y) @_k (x y)

⇥ ˆT ⁰₀(t, x), ˆT ⁰_i(t, y)⇤

= i ˆT ^j_i(t, x)@_j Tˆ⁰₀(t, y)@_i (x y)

⇥ ˆT ⁰_i(t, x), ˆT ⁰_j(t, y)⇤

= i ˆT ⁰_j(t, x)@_i + ˆT ⁰_i(t, y)@_j (x y)

{

Current algebra

EoM for perfect fluid (Sound wave) is derived!!

◆ Current algebra related to relativistic hydrodynamics

(  : inv. suscep.)lm

◆ Leading Order term in EOM

@

A ˆ

(t) = i

h[ ˆ A

(0), ˆ A

(0)] i A ˆ

(t) + · · ·

Perfect fluid from Mori's method

⇣ = _eq

Z ₁

0

dt Z

d^{d 1}x(e^{Qi ˆˆ Lt}Q ˆp(0, x), ˆˆ Q ˆp(0, 0))

⌘ = ^eq

(d + 1)(d 2)

Z ₁

0

dt Z

d^{d 1}x(e^{Qi ˆˆ Lt}Q ˆ⇡ˆ ^ik(0, x), ˆQ ˆ⇡^jl(0, 0)) ^{ij kl}

◆ Green-Kubo formula for transport coefficients (viscosity)

Reversible → Sound wave / Dissipative → Diffusion mode

@

T ˆ

= ik

h

T ˆ



k

k

✓ ⇣

h

+ d 3 d 1

⌘ h

◆

+ k

⌘

h

T ˆ

+ ˆ R

◆ Relativistic hydrodynamic from Mori’s method

@

T ˆ

= ik

T ˆ

CME from anomalous commutation

Choose

A ˆ

(t) = { ˆ T

(t, x), ˆ T

(t, x), J

(t, x), J

(t, x) }

@

A ˆ

(t) = i

h[ ˆ A

(0), ˆ A

(0)] i A ˆ

(t) + · · ·

For EoM:

Current algebra with

anomalous

commutation rel. _{⇥ ˆJ}⁰_{(t, x), ˆJ}⁰_{(t, y)}^⇤ _{= ⇥ ˆJ}⁰

5 (t, x), ˆJ₅⁰(t, y)⇤

= 0

⇥ ˆT ⁰_i(t, x), ˆJ⁰(t, y)⇤

= i ˆJ⁰(t, x)@_j (x y)

⇥ ˆT ⁰_i(t, x), ˆJ₅⁰(t, y)⇤

= i ˆJ₅⁰(t, x)@_j (x y)

{ ^{

@

J ˆ

(x) + @

h

J ˆ

(x)

2⇡

B

(x) i

+ · · · = 0

◆ EoM for J^ˆ0(t, x)

CME from anomalous commutation

= ˆ J

(x)

@

J ˆ

(x) + @

h

J ˆ

(x)

2⇡

B

(x) i

+ · · · = 0

- Conservation law:

J ˆ

(x) =

nn₅

J ˆ

(x)

2⇡

B

(x)

@

J ˆ

(x) = 0

- Const. relation:

◆ Summary of result

Chiral Magnetic Effect (CME)

N S

j B

µ_R 6= µ^L

Anomalous comm.

Chiral Magnetic Wave (CMW)

Chiral Magnetic Wave

µ > 0 µ

= 0

B ~

J ˆ

(x) =

nn₅

J ˆ

(x)

2⇡

B

(x) Chiral Magnetic Effect

J ˆ

(x) =

n₅n

J ˆ

(x)

2⇡

B

(x) Chiral Separation Effect

Ex.

µ

> 0 µ

< 0

+

+

Charge

propagation along B

Collective excitation

=

J ~ _{5 J} ^~

J ~

Interpretation as Type-B NG mode

Massless mode = Nambu-Goldstone (NG) mode appears!

Generalization of Nambu-Goldstone’s theorem for type-B NG mode

h[i ˆ Q

, ˆ

(x)] i ⌘ Tr ˆ ⇢[i ˆ Q

, ˆ

(x)] 6= 0 Spontaneous Symmetry Breaking (SSB)

◆ Spontaneous symmetry breaking & Nambu-Goldstone mode

such that

9 ˆ(x)

Q ˆ

For some conserved charge

- Type-A NG mode： satisfy

h[i ˆ Q

, ˆ Q

] i = 0

{

◆ Classification of NG mode

Watanabe-Murayama(2012)]

8 ˆ Q

- Type-B NG mode：

9 ˆ Q

h[i ˆ Q

, ˆ Q

] i 6= 0

CMW ≒ Type-B NG mode?

Hydrodynamics of chiral plasma contains

- Sound wave

- Chiral Magnetic Wave (CMW)

{

massless collective excitation known as

h⇥ ˆT ⁰₀(t, x), ˆT ⁰_i(t, y)⇤

i = i h ˆT ^j_i(t, x)i@^j h ˆT ⁰₀(t, y)i @^k (x y)6= 0 h⇥ ˆJ₅⁰(t, x), ˆJ⁰(t, y)⇤

i = i

2⇡² Bⁱ(t, y)@_i^x (x y)6= 0

◆ Origin of Sound wave and CMW

The above definition states they are a friend of Type-B NG mode!

- Type-B NG mode：

9 ˆ Q

h[i ˆ Q

, ˆ Q

] i 6= 0

@

A ˆ

(t) = i

h[ ˆ A

(0), ˆ A

(0)] i A ˆ

(t) + · · ·

Summary

Motivation:

Approach:

Result:

Anomalous commutation:

Mori’s method as a generalization of current algebra

Chiral Magnetic Effect in operator formalism:

Origin of chiral transport (Chiral Magnetic Effect)?

Sound and Chiral Magnetic Wave as a friend of Type-B NG mode

S N

µ₅ = 0 j

?

⇥ ˆJ₅⁰(t, x), ˆJ⁰(t, y)⇤

= i

2⇡² Bⁱ(t, y)@_i^x (x y)

A~_i A~j

B~

P ~ˆB =X

i

aⁱA~i

a^j

aⁱ A~_?

B~_?

Jˆⁱ(x) =

nn₅Jˆ₅⁰(x)

2⇡² Bⁱ(x)

Outlook 1

Originated from chiral anomaly

Path-integral formalism

Chiral pert. w/ Wess-Zumino term CA w/ Anomalous CR

Operator formalism QCD

Hydro Mori’s projection w/

Anomalous CR

◆ Path-integral treatment?

µ_R 6= µ^L

~

J

= µ

2⇡

B + ~

✓ µ

+ µ

4⇡

+ T

12

◆

~

!

◆ Chiral vortical effect

[Crossley et al. (2015)]

MSRJD effective lagrangian w/ ？？

Origin of this term?

RR ?? ˜

◆ Spontaneous symmetry breaking

◆ Symmetry breaking by quantum anomaly

https://en.wikipedia.org/wiki/Superfluidity#/

media/File:Liquid_helium_Rollin_film.jpg

Macro：Superfluid Micro：Selecting vacuum

0

Micro：π⁰ decay

[Adller (1969), Bell-Jackiw (1969)]

N S

j B

µ_R 6= µ^L

Macro：Anomalous transport

[Erdmenger et al. (2008), Son-Surowka (2009)]

Outlook 2: Chiral superfluid

◆ Symmetry breaking by quantum anomaly

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