• 沒有找到結果。

Hydrodynamics, current algebra, and quantum anomaly

N/A
N/A
Protected

Academic year: 2022

Share "Hydrodynamics, current algebra, and quantum anomaly"

Copied!
26
0
0

加載中.... (立即查看全文)

全文

(1)

Hydrodynamics, current algebra, and quantum anomaly

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

N S

j B

µR 6= µL

立臺灣大學

Masaru Hongo ( RIKEN, iTHES )

Talk by T. Hirano Talk by Y. Hidaka

Talk by D. Kharzeev

(2)

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)?

(3)

- Existence of extremely strong magnetic field

- Chirality drastically affect hydrodynamic transport

y x z

QGP

+

Quark

Gluon

B

e |B| 10

14

T

Photon

QGP

+ + +

+ − +

− −

− −

RightLeft

QGP as Chiral fluid

(4)

Effective theory for macroscopic dynamics

Universal description, not depending on details

Only conserved quantity ~ symmetry of system

Hydro:

{ (x), v(x)}

' '

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

12

cm

T 10

12

K 10

12

kg/cc

10 km

Hydrodynamics is

(5)

◆ 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

µR 6= µL

Macro:Anomalous transport

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

Symmetry breaking and Hydro

(6)

Parity- violating chiral transport

◆Chiral Magnetic Effect (CME)

[Fukushima et al. (2008), Vilenkin (1980)]

N S

j B

µR 6= µL

J = ~ µ 5

2⇡ 2 B ~

◆ Chiral Vortical Effect (CVE)

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

µ

R

6= µ

L

~ j

J = µµ 5

2⇡ 2 ! ~

(7)

Anomaly and chiral transport

Quantum anomaly

N S

j B

µR 6= µL

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

(8)

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

?

B

(9)

Review: Current algebra for QCD

◆ Current algebra for

⇥ ˆ Q

L,a

, ˆ J

L,bµ

(x) ⇤

= if

cab

J ˆ

L,cµ

(x), ⇥ ˆ Q

L,a

, ˆ J

R,bµ

(x) ⇤

= 0

⇥ ˆ Q

R,a

, ˆ J

R,bµ

(x) ⇤

= if

cab

J ˆ

R,cµ

(x), ⇥ ˆ Q

R,a

, ˆ J

L,bµ

(x) ⇤

= 0 SU (N )

R

⇥ SU(N)

L

Low-energy theorem

Goldberger-Treiman relation Soft Pion 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!

(10)

Current algebra and chiral anomaly

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

V

⇥ U(1)

A

⇥ ˆ J

0

(t, x), ˆ J

0

(t, y) ⇤

= ⇥ ˆ J

50

(t, x), ˆ J

50

(t, y) ⇤

= 0

⇥ ˆ J

50

(t, x), ˆ J

0

(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!

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

J ˆ

0

(x) = @ L

@(@ ) i ˆ = iˆ ⇡(x) ˆ(x),

(11)

Current algebra and chiral anomaly

Sketch of Proof.

⇥ ˆ J

0

(t, x), ˆ J

0

(t, y) ⇤

= ⇥ ˆ J

50

(t, x), ˆ J

50

(t, y) ⇤

= 0

Ward-Takahashi identity is not h@µJ5µ(x)iA = 0 but

h@

µ

J

5µ

(x) i

A

= C✏

µ⌫⇢

F

µ⌫

(x)F

(x) ⇠ CdAdA

⇠ CdB

Variation w.r.t gives

A

0

@

µ

hJ

5µ

(x)J

0

(y) i

A

⇠ CddA

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

⇥ ˆ J

50

(t, x), ˆ J

0

(t, y) ⇤

= i

2⇡

2

B

i

(t, y)@

ix

(x y)

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

V

⇥ U(1)

A

(12)

Anomaly and chiral transport

Quantum anomaly

N S

j B

µR 6= µL

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

(13)

◆ EoM given by Mori’s projection operator method

Reversible

@

0

A ˆ

n

(t) = i⌦

nm

A ˆ

m

(t)

Z

t 0

ds

nm

(t s) ˆ A

m

(s, y) + ˆ R

n

(t)

mn (t s) = ˆRn(t s), ˆRm(0) Noise Dissipative

Fluctuation Dissipation relation:

i⌦nm = i

h[ ˆAn(0), ˆAm(0)]i + iµ [ ˆN , ˆAn(0)], ˆAm(0)

{

A~i A~j

B~

P ~ˆB = X

i

aiA~i aj

ai A~?

B~?

A method to write down

A ˆ

n

(t)

Equation of Motion (EoM) only focusing on

Mori’s projection operator method

[Mori (1965)]

(14)

◆ EoM given by Mori’s projection operator method

i⌦nm = i

h[ ˆAn(0), ˆAm(0)]i + iµ [ ˆN , ˆAn(0)], ˆAm(0)

mn (t s) = ˆRn(t s), ˆRm(0) Noise Dissipative

Reversible

@

0

A ˆ

n

(t) = i⌦

nm

A ˆ

m

(t)

Z

t 0

ds

nm

(t s) ˆ A

m

(s, y) + ˆ R

n

(t)

Fluctuation Dissipation relation:

{

A~i A~j

B~

P ~ˆB = X

i

aiA~i aj

ai A~?

B~?

A method to write down

A ˆ

n

(t)

Equation of Motion (EoM) only focusing on

Mori’s projection operator method

[Mori (1965)]

(15)

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

?

B

⇥ ˆJ50(t, x), ˆJ0(t, y)

= i

2⇡2 Bi(t, y)@ix (x y)

A~i A~j

B~

P ~ˆB =X

i

aiA~i

aj

ai A~?

B~?

(16)

Mori's method and current algebra

A ˆ

n

(t)

Choose as conserve charges:

A ˆ

n

(t) = { ˆ T

00

(t, x), ˆ T

0i

(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 0i(t, y)

= i ˆT ji(t, x)@j Tˆ00(t, y)@i (x y)

⇥ ˆT 0i(t, x), ˆT 0j(t, y)

= i ˆT 0j(t, x)@i + ˆT 0i(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

@

0

A ˆ

n

(t) = i

lm

h[ ˆ A

n

(0), ˆ A

m

(0)] i A ˆ

l

(t) + · · ·

(17)

Perfect fluid from Mori's method

⇣ = eq

Z 1

0

dt Z

dd 1x(eQi ˆˆ LtQ ˆp(0, x), ˆˆ Q ˆp(0, 0))

⌘ = eq

(d + 1)(d 2)

Z 1

0

dt Z

dd 1x(eQi ˆˆ LtQ 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

@

0

T ˆ

0i

= ik

i

h

eq ee

T ˆ

00

k

i

k

k

✓ ⇣

h

eq

+ d 3 d 1

⌘ h

eq

+ k

2 ki

h

eq

T ˆ

0k

+ ˆ R

i

◆ Relativistic hydrodynamic from Mori’s method

@

0

T ˆ

00

= ik

i

T ˆ

0i [Minami-Hidaka (2013)]

(18)

CME from anomalous commutation

Choose

A ˆ

n

(t) = { ˆ T

00

(t, x), ˆ T

0i

(t, x), J

0

(t, x), J

50

(t, x) }

@

0

A ˆ

n

(t) = i

lm

h[ ˆ A

n

(0), ˆ A

m

(0)] i A ˆ

l

(t) + · · ·

For EoM:

Current algebra with

anomalous

commutation rel. ⇥ ˆJ0(t, x), ˆJ0(t, y) = ⇥ ˆJ0

5 (t, x), ˆJ50(t, y)

= 0

⇥ ˆT 0i(t, x), ˆJ0(t, y)⇤

= i ˆJ0(t, x)@j (x y)

⇥ ˆT 0i(t, x), ˆJ50(t, y)⇤

= i ˆJ50(t, x)@j (x y)

{ {

⇥ ˆJ50(t, x), ˆJ0(t, y) = 2⇡i 2 Bi(t, y)@ix (x y)

@

0

J ˆ

0

(x) + @

ix

h

nn5

J ˆ

50

(x)

2⇡

2

B

i

(x) i

+ · · · = 0

◆ EoM for Jˆ0(t, x)

(19)

CME from anomalous commutation

= ˆ J

i

(x)

@

0

J ˆ

0

(x) + @

ix

h

nn5

J ˆ

50

(x)

2⇡

2

B

i

(x) i

+ · · · = 0

- Conservation law:

J ˆ

i

(x) =

nn5

J ˆ

50

(x)

2⇡

2

B

i

(x)

@

µ

J ˆ

µ

(x) = 0

- Const. relation:

◆ Summary of result

Chiral Magnetic Effect (CME)

N S

j B

µR 6= µL

Anomalous comm.

(20)

Chiral Magnetic Wave (CMW)

Chiral Magnetic Wave

[Kharzeev, Yee, (2011)]

µ > 0 µ

5

= 0

B ~

J ˆ

i

(x) =

nn5

J ˆ

50

(x)

2⇡

2

B

i

(x) Chiral Magnetic Effect

J ˆ

5i

(x) =

n5n

J ˆ

0

(x)

2⇡

2

B

i

(x) Chiral Separation Effect

Ex.

µ

5

> 0 µ

5

< 0

+

+

Charge

propagation along B

Collective excitation

=

J ~ 5 J ~

J ~

(21)

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

a

, ˆ

i

(x)] i ⌘ Tr ˆ ⇢[i ˆ Q

a

, ˆ

i

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

◆ Spontaneous symmetry breaking & Nambu-Goldstone mode

such that

9 ˆ(x)

Q ˆ

a

For some conserved charge

- Type-A NG modesatisfy

h[i ˆ Q

a

, ˆ Q

b

] i = 0

{

◆ Classification of NG mode

[Hidaka (2012),

Watanabe-Murayama(2012)]

8 ˆ Q

b

- Type-B NG mode

9 ˆ Q

b such that

h[i ˆ Q

a

, ˆ Q

b

] i 6= 0

(22)

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 0i(t, y)⇤

i = i h ˆT ji(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)@ix (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

b such that

h[i ˆ Q

a

, ˆ Q

b

] i 6= 0

@

0

A ˆ

n

(t) = i

lm

h[ ˆ A

n

(0), ˆ A

m

(0)] i A ˆ

l

(t) + · · ·

(23)

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

?

B

⇥ ˆJ50(t, x), ˆJ0(t, y)

= i

2⇡2 Bi(t, y)@ix (x y)

A~i A~j

B~

P ~ˆB =X

i

aiA~i

aj

ai A~?

B~?

i(x) =

nn550(x)

2⇡2 Bi(x)

(24)

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?

µR 6= µL

~

j

J

5

= µ

2⇡

2

B + ~

✓ µ

2

+ µ

25

4⇡

2

+ T

2

12

~

!

◆ Chiral vortical effect

[Crossley et al. (2015)]

MSRJD effective lagrangian w/ ??

Origin of this term?

RR ?? ˜

(25)

◆ 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

µR 6= µL

Macro:Anomalous transport

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

Outlook 2: Chiral superfluid

◆ Symmetry breaking by quantum anomaly

(26)

Back up

參考文獻

相關文件

The CME drastically changes the time evolution of the chiral fluid in a B-field. - Chiral fluid is not stable against a small perturbation on v

Fully quantum many-body systems Quantum Field Theory Interactions are controllable Non-perturbative regime..

These include developments in density functional theory methods and algorithms, nuclear magnetic resonance (NMR) property evaluation, coupled cluster and perturbation theories,

We compare the results of analytical and numerical studies of lattice 2D quantum gravity, where the internal quantum metric is described by random (dynamical)

In x 2 we describe a top-down construction approach for which prototype charge- qubit devices have been successfully fabricated (Dzurak et al. Array sites are de­ ned by

Generalized LSMA theorem: The low-energy states in gapped phases of SU (N ) spin systems cannot be triv- ially gapped in the thermodynamical limit if the total number of

• QCSE and band-bending are induced by polarization field in C-plane InGaN/GaN and create triangular energy barrier in active region, which favors electron overflow. •

– Sonic black hole and Hawking radiation and Unruh effect – Quantum criticality and AdS-CFT correspondance. Discrete