UNDERSTANDING THE DYNAMICS OF THE CHIRAL VORTICAL …

F. Peña-Benítez

Crete Center for theoretical physics

Kolymbari, sept 2014

UNDERSTANDING THE DYNAMICS OF THE CHIRAL VORTICAL EFFECT

based on: arXiv:1312.1204

K. Landsteiner, E. Megías, F. P-B

OUTLINE• the chiral magnetic effect (CME) and quantum anomalies

• time evolution

• the chiral vortical effect (CVE)

• holographic model

• time evolution

• consistency checks

• physical implications

2

MOTIVATION• quantum anomalies are interesting

• mixed gravitational anomaly could be measure for the first time

• non dissipative transport

• the non dissipative nature is not destroyed at high temperature (could be powered in some cases)

• macroscopical effect of a purely quantum property

• origin of the charge separation observed in the QGP???

• possibly realized in Weyl semi-metals

• non renormalization theorems

• …

TA

TB

TC

Aµ

Aν

Aρ

+++++

In a field theory global symmetries can be violated at the quantum level

Theories with massless fermions have axial and vector global symmetries

It is not possible to preserve in the QFT (even dimensions) both symmetries

In (3+1)d there are two types of axial anomalies

what do we know about quantum anomalies?

ANOMALIES

4

~B

electric charge flow

axial charge flow

momentum flow

~!electric charge flow

axial charge flow

momentum flow

~B

electric charge flow

axial charge flow

momentum flow

~ja =

0

@~je~j5~j"

1

Aji" = T ticharge transportchirality transportenergy transport

Kubo formulas

Kubo formulas

~ja =

0

@~je~j5~j"

1

Aji" = T ticharge transportchirality transportenergy transport

Ba = lim

qz

!0

i

qzGR

jxa

jy (0, qz)

~ja = Ba~k ~A

t,y

x

~k

z

Kubo formulas for anomaly induced transport~J = V r ~v

uµ = (1,~0) ! uµ = gtµ

Agi gti

~J = V r ~Ag

ds

2 = (µ + hµ)dxµdx

V = limqz

!0

i

qzGR

jx T ty

(0, qz)

[Amado, Landsteiner & F. P-B, JHEP.1105 (2011)]

= limk!0

1

k

~jc = Tc~P+

J iA T 0j

q

q + k

T 0i =i

4 (0@i + i@0)P+

Va =

1

162

2

4X

b,c

Tr(TaHb, Hc)µbµc +22

3T 2Tr(Ta)

3

5

Bab =

1

42

X

c

Tr(TaTb, Hc)µc

[Landsteiner, Megías & F. P-B, PRL. 107 (2011)]

[Kharzeev & Warringa, PRD.80 (2009)]

free fermions &

anomaly induce transport

µf =X

a

qfaµa , Ha = qfafg

TA

TB

TC

Aµ

Aν

Aρ

TA

hµν

hλβ

Aρ

ba Tr[Ta]cabc Tr[TaTb, Tc]

in the case of interest for QCD U(1)VxU(1)A

rµjµ = 0

rµjµ5 =

1

162Fµ F

µ +1

3842µR↵

µR

↵

Solutions for U(1)VxU(1)A

free fermions conductivities

BA,(0) =

1

22

8<

:

µ5

µµµ5

, A = e, 5,

B5

A,(0) =1

22

8<

:

µµ5µ2+µ2

52 + 2T 2

6

τ = 0.1/T

τ = 0.2/T

τ = 1/T

t/τ

j(t)σ0B0

151050-5

0.4

0.3

0.2

0.1

0

σ′′χ

σ′χ

ω/T

σχ(ω)σ0

3020100

1

0.5

0

5 10 15 20 25 30

Ω

T

"0.5

0.5

1.0

Σ

Σ0

Μ!T%0.1

free fermions magnetic

conductivity

holographic magnetic

conductivity

B(t) =1

[1 + (t/)2]3/2B0

[Kharzeev & Warringa, arXiv:0907.5007]

[Ho-Ung Yee, arXiv:0908.4189]


[Vilenkin, Phys. Rev. D20, 1807 (1979)]

VA,(0) =

1

22

8><

>:

µµ5µ2+µ2

52 + 2T 2

6

µ5

µ2+µ2

53 + 2T 2

3


[Vilenkin, Phys. Rev. D20, 1807 (1979)]

free fermions conductivities

V (!) = V0 (!,0 + i!(!))

VA,(0) =

1

22

8><

>:

µµ5µ2+µ2

52 + 2T 2

6

µ5

µ2+µ2

53 + 2T 2

3

[Landsteiner, Megías & P-B. 1312.1204]

holographic modelwe need to build a model with a U(1)VxU(1)A global symmetry the vector current must be conserved the axial current must be anomalous with the mixed gauge-gravitational anomaly included

S =1

16G

Zd

5x

pg

R+

12

L

2 1

4

FMNF

MN + F

(5)MNF

(5)MN

is a vector gauge field in the bulk

is an axial gauge field in the bulk

the boundary value corresponds to a background vector source

the boundary value corresponds to a background axial source

AM (r, x)

A

(5)M (r, x)

A

(5)µ (1, x)

Aµ(1, x)

is the bulk metric the boundary background metric hµ(1, x)gMN (r, x)


is a vector gauge field in the bulk

is an axial gauge field in the bulk

the boundary value corresponds to a background vector source

the boundary value corresponds to a background axial source

AM (r, x)

A

(5)M (r, x)

A

(5)µ (1, x)

Aµ(1, x)

is the bulk metric the boundary background metric hµ(1, x)gMN (r, x)

S

ano

=

Zd

5x

pg

h

MNPQR

A

(5)M

3F

(5)NP

F

(5)QR

+ F

NP

F

QR

+ R

A

BNP

R

B

AQR

i


S

ano

=

Zd

5x

pg

h

MNPQR

A

(5)M

3F

(5)NP

F

(5)QR

+ F

NP

F

QR

+ R

A

BNP

R

B

AQR

i

5(S + S

ano

+ S

bound

) /Z

@

d

4x

ph5

µ

3F

(5)µ

F

(5)

+ F

µ

F

+ R

↵

µ

R

↵


S

ano

=

Zd

5x

pg

h

MNPQR

A

(5)M

3F

(5)NP

F

(5)QR

+ F

NP

F

QR

+ R

A

BNP

R

B

AQR

i

5(S + S

ano

+ S

bound

) /Z

@

d

4x

ph5

µ

3F

(5)µ

F

(5)

+ F

µ

F

+ R

↵

µ

R

↵

rµjµ5 =

1

162Fµ F

µ +1

3842µR↵

µR

↵

++

ds

2 =r

2

L

2

f(r)dt2 + d~x

2+

L

2

r

2f(r)

dr

2

A =

µ r2H

r2

dt

A(5) =

µ5 r2H

r2

dt

Q =r2Hp3µ Q5 =

r2Hp3µ5

AdS Reissner-Nordström blackhole

Numerical vortical conductivity seems to agree with the free case

V (!) = V0 (!,0 + i!(!))

vortical conductivities

0 7 14-3. ¥ 10-6

0

6. ¥ 10-6

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

w

T

s5VHw, 0L

k

T!

2 Π

5

k

T!Π

25

k

T!Π

250

Hydro prediction

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

4ΠTΩ

k2

Σe!!Ω, 0"

Numerical vortical conductivity seems to agree with the free case

Notice the peak seems to be at

! 1

4Tk2

this coefficient coincides with the SYM diffusion

constant

V (!) = V0 (!,0 + i!(!))

vortical conductivities

HYDRODYNAMICS

Tµ = (+ P )uµu + Pgµ µ + B (Bµu +Buµ) + V

(!µu + !uµ) ,

Jµ = uµ + BBµ + V !µ

rµTµ = F µJµ

T ti = (+ P )vi + Phti ikijV (vj + htj) ikij

B Aj

J i = vi ikijV (vj + htj) ikij

BAj

ij

B

!kAj

+ (i!(+ P ) + k2)vi

i!(P + )hti

ij

V

!k(htj

+ vj

) !khxi

i!Ai

= 0

k

T!

2 Π

5

k

T!Π

25

k

T!Π

250

Hydro prediction

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

4ΠTΩ

k2

Σe!!Ω, 0"

Hydro prediction

0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

4ΠTΩ

k2

Σ!

!!Ω, 0"

< T tiT tj > = ikijV

D2k4

(! + iDk2)2

< T tiJj > = < J iT tj >= ikijB

iDk2

(! + iDk2)

!M = ± 1p3Dk2 , < T tiT tj >

!M = ±Dk2 , < T tiJj >

ENERGY CONSERVATION

W =

Zd

4x

W

gµ(x)gµ(x) +

W

Aµ(x)Aµ(x)

= 0

rµ

2pg

W

gµ(x)

+rµ

1pg

W

Aµ(x)

A

(x) +1pg

W

Aµ(x)Fµ

(x) = 0

Euclidean effective action

ENERGY CONSERVATION

i(! + i)0x,0y(i!

n

= ! + i, kz

)

ikz

= 0x,zy(i!n

= ! + i, kz

)

! V (!) = 0

i(! + i)Gx,0y(i!

n

= ! + i, kz

)

ikz

kz!0

= Gx,zy(i!n

= ! + i,~k = 0)

! V (!) = 0

vanishing at zero momentum because of rotational invariance

to understand the physical implication of these result, let's play a game

0

!(s) =ik

s+ i

V (s) = V0

iDk2

s+ iDk2

J(t) = V0 k(1 eDk2t)(t)

k

TA

TB

TC

Aµ

Aν

Aρ

+++++

It is a temptation to play with this result and the characteristic numbers of the QGP

for the holographic plasma

the lifetime of the QGP is of order

J(t) = V0 k(1 eDk2t)(t) =

1

Dk2

D =1

4T

2100fm

L 10fm

T 350Mev

10fm

It is a temptation to play with this result and the characteristic numbers of the QGP

for the holographic plasma

the lifetime of the QGP is of order

However in the QGP the vorticity is not an external source and the real problem is much more complicated because the vorticity is a dynamical variable!

J(t) = V0 k(1 eDk2t)(t) =

1

Dk2

D =1

4T

2100fm

L 10fm

T 350Mev

10fm0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

t

tcCVE

J!t"#!Σ

V"

k" " !t" # "k $ !t"" !t" # "k $$ !t" % $ %t%t f &'

& tcCVEΤQGP &

%%%%%

• we need to study more realistic situations because the vorticity is a dynamical quantity

• looking forward for Weyl semimetals!

• are we close to see experimental evidence of the presence of the mixed gravitational anomaly in nature?

• the existence of the anomalous conductivities is well understood already.

• now is necessary to understand the renormalization issues in presence of dynamical gauge fields

• generation of axial chemical potential mechanism in order to be able to do good predictions for QGP

BA,(0) =

1

22

8<

:

µ5

µµµ5

, A = e, 5,

B5

A,(0) =1

22

8<

:

µµ5µ2+µ2

52 + 2T 2

6

VA,(0) =

1

22

8><

>:

µµ5µ2+µ2

52 + 2T 2

6

µ5

µ2+µ2

53 + 2T 2

3

This research has been co-financed by the European Union (European Social Fund, ESF) and!Greek national funds through the Operational Program "Education and Lifelong Learning"!of the National Strategic Reference Framework (NSRF), under the grants schemes "Funding!of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC Grant Schemes"!and the program "Thales"

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