**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

### +

^{Quark}

Gluon

### B

### e |B| 10

^{14}

### T

^{Photon}

### QGP

### + + +

### + − +

### − −

### − −

^{Right}

_{Left}

**QGP as Chiral fluid**

**•**

**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**

**◆ 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：π**^{0}** 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)**

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

### j B

µ_{R} 6= µ^{L}

### 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} ! ~

**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}

µ_{5} = 0
j

### ?

B**Review: Current algebra for QCD**

**◆ Current algebra for **

### ⇥ ˆ Q

_{L,a}

### , ˆ J

_{L,b}

^{µ}

### (x) ⇤

### = if

^{c}

_{ab}

### J ˆ

_{L,c}

^{µ}

### (x), ⇥ ˆ Q

_{L,a}

### , ˆ J

_{R,b}

^{µ}

### (x) ⇤

### = 0

### ⇥ ˆ Q

_{R,a}

### , ˆ J

_{R,b}

^{µ}

### (x) ⇤

### = if

^{c}

_{ab}

### J ˆ

_{R,c}

^{µ}

### (x), ⇥ ˆ Q

_{R,a}

### , ˆ J

_{L,b}

^{µ}

### (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!**

**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

_{5}

^{0}

### (t, x), ˆ J

_{5}

^{0}

### (t, y) ⇤

### = 0

### ⇥ ˆ J

_{5}

^{0}

### (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!

Jˆ_{5}^{0}(x) = iˆ⇡(x) _{5} ˆ(x)
Definition of Noether current gives

### J ˆ

^{0}

### (x) = @ L

### @(@ ) i ˆ = iˆ ⇡(x) ˆ(x),

**Current algebra and chiral anomaly**

**Sketch of Proof.**

### ⇥ ˆ J

^{0}

### (t, x), ˆ J

^{0}

### (t, y) ⇤

### = ⇥ ˆ J

_{5}

^{0}

### (t, x), ˆ J

_{5}

^{0}

### (t, y) ⇤

### = 0

**Ward-Takahashi identity is not ** h@^{µ}J_{5}^{µ}(x)i^{A} = 0 but

### h@

^{µ}

### J

_{5}

^{µ}

### (x) i

^{A}

### = C✏

^{µ⌫⇢}

### F

_{µ⌫}

### (x)F

_{⇢}

### (x) ⇠ CdAdA

### ⇠ CdB

**Variation w.r.t gives**

### A

_{0}

### @

_{µ}

### hJ

_{5}

^{µ}

### (x)J

^{0}

### (y) i

^{A}

### ⇠ CddA

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

### ⇥ ˆ J

_{5}

^{0}

### (t, x), ˆ J

^{0}

### (t, y) ⇤

### = i

### 2⇡

^{2}

### B

^{i}

### (t, y)@

_{i}

^{x}

### (x y)

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

_{V}

### ⇥ U(1)

^{A}

**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**

### @

_{0}

### A ˆ

_{n}

### (t) = i⌦

_{n}

^{m}

### A ˆ

_{m}

### (t)

### Z

t 0### ds

_{n}

^{m}

### (t s) ˆ A

_{m}

### (s, y) + ˆ R

_{n}

### (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

a^{i}A~_{i}
a^{j}

a^{i}
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)]

**◆ 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**

### @

_{0}

### A ˆ

_{n}

### (t) = i⌦

_{n}

^{m}

### A ˆ

_{m}

### (t)

### Z

t 0### ds

_{n}

^{m}

### (t s) ˆ A

_{m}

### (s, y) + ˆ R

_{n}

### (t)

**Fluctuation Dissipation relation: **

### {

A~_{i}
A~_{j}

B~

P ~ˆB = X

i

a^{i}A~_{i}
a^{j}

a^{i}
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)]

**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

µ_{5} = 0
j

### ?

B⇥ ˆJ_{5}^{0}(t, x), ˆJ^{0}(t, y)⇤

= i

2⇡^{2} B^{i}(t, y)@_{i}^{x} (x y)

A~_{i}
A~j

B~

P ~ˆB =X

i

a^{i}A~i

a^{j}

a^{i}
A~_{?}

B~_{?}

**Mori's method and current algebra**

### A ˆ

_{n}

### (t)

**Choose as conserve charges:**

### A ˆ

_{n}

### (t) = { ˆ T

^{0}

_{0}

### (t, x), ˆ T

^{0}

_{i}

### (t, x) }

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

⇥ ˆT ^{0}_{0}(t, x), ˆT ^{0}_{0}(t, y)⇤

= i ˆT ^{k}_{0}(t, x) + ˆT ^{k}_{0}(t, y) @_{k} (x y)

⇥ ˆT ^{0}_{0}(t, x), ˆT ^{0}_{i}(t, y)⇤

= i ˆT ^{j}_{i}(t, x)@_{j} Tˆ^{0}_{0}(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**

### @

_{0}

### A ˆ

_{n}

### (t) = i

^{lm}

### h[ ˆ A

_{n}

### (0), ˆ A

_{m}

### (0)] i A ˆ

_{l}

### (t) + · · ·

**Perfect fluid from Mori's method**

⇣ = _{eq}

Z _{1}

0

dt Z

d^{d 1}x(e^{Qi ˆ}^{ˆ} ^{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 ˆ}^{ˆ} ^{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**

### @

_{0}

### T ˆ

^{0}

_{i}

### = ik

_{i}

### h

_{eq}

^{ee}

### T ˆ

^{0}

_{0}

###

### k

_{i}

### k

^{k}

### ✓ ⇣

### h

_{eq}

### + d 3 d 1

### ⌘ h

_{eq}

### ◆

### + k

^{2 k}

_{i}

### ⌘

### h

_{eq}

### T ˆ

^{0}

_{k}

### + ˆ R

_{⇡}

_{i}

**◆ Relativistic hydrodynamic from Mori’s method**

### @

_{0}

### T ˆ

^{0}

_{0}

### = ik

^{i}

### T ˆ

^{0}

_{i}[Minami-Hidaka (2013)]

**CME from anomalous commutation**

**Choose**

### A ˆ

_{n}

### (t) = { ˆ T

^{0}

_{0}

### (t, x), ˆ T

^{0}

_{i}

### (t, x), J

^{0}

### (t, x), J

_{5}

^{0}

### (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.** _{⇥ ˆ}_{J}^{0}_{(t, x), ˆ}_{J}^{0}_{(t, y)}^{⇤} _{=} _{⇥ ˆ}_{J}^{0}

5 (t, x), ˆJ_{5}^{0}(t, y)⇤

= 0

⇥ ˆT ^{0}_{i}(t, x), ˆJ^{0}(t, y)⇤

= i ˆJ^{0}(t, x)@_{j} (x y)

⇥ ˆT ^{0}_{i}(t, x), ˆJ_{5}^{0}(t, y)⇤

= i ˆJ_{5}^{0}(t, x)@_{j} (x y)

## { ^{{}

^{⇥ ˆ}

^{J}

^{5}

^{0}

^{(t, x), ˆ}

^{J}

^{0}

^{(t, y)}

^{⇤}

^{=}

^{2⇡}

^{i}

^{2}

^{B}

^{i}

^{(t, y)@}

^{i}

^{x}

^{(x}

^{y)}

### @

_{0}

### J ˆ

^{0}

### (x) + @

_{i}

^{x}

### h

^{nn}

^{5}

### J ˆ

_{5}

^{0}

### (x)

### 2⇡

^{2}

### B

^{i}

### (x) i

### + · · · = 0

**◆ EoM for** J^{ˆ}^{0}(t, x)

**CME from anomalous commutation**

### = ˆ J

^{i}

### (x)

### @

_{0}

### J ˆ

^{0}

### (x) + @

_{i}

^{x}

### h

^{nn}

^{5}

### J ˆ

_{5}

^{0}

### (x)

### 2⇡

^{2}

### B

^{i}

### (x) i

### + · · · = 0

** - Conservation law:**

### J ˆ

^{i}

### (x) =

nn_{5}

### J ˆ

_{5}

^{0}

### (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

µ_{R} 6= µ^{L}

**Anomalous comm.**

**Chiral Magnetic Wave (CMW)**

**Chiral Magnetic Wave**

*[Kharzeev, Yee, (2011)]*

### µ > 0 µ

_{5}

### = 0

### B ~

### J ˆ

^{i}

### (x) =

nn_{5}

### J ˆ

_{5}

^{0}

### (x)

### 2⇡

^{2}

### B

^{i}

### (x) **Chiral Magnetic Eﬀect**

### J ˆ

_{5}

^{i}

### (x) =

n_{5}n

### 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 ~

**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 mode**： **satisfy**

### 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

**CMW ≒ Type-B NG mode?**

**Hydrodynamics of chiral plasma contains **

**- Sound wave**

**- Chiral Magnetic Wave (CMW)**

### {

**massless collective excitation known as **

h⇥ ˆT ^{0}_{0}(t, x), ˆT ^{0}_{i}(t, y)⇤

i = i h ˆT ^{j}_{i}(t, x)i@^{j} h ˆT ^{0}_{0}(t, y)i @^{k} (x y)6= 0
h⇥ ˆJ_{5}^{0}(t, x), ˆJ^{0}(t, y)⇤

i = i

2⇡^{2} B^{i}(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

_{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) + · · ·

**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

µ_{5} = 0
j

### ?

B⇥ ˆJ_{5}^{0}(t, x), ˆJ^{0}(t, y)⇤

= i

2⇡^{2} B^{i}(t, y)@_{i}^{x} (x y)

A~_{i}
A~j

B~

P ~ˆB =X

i

a^{i}A~i

a^{j}

a^{i}
A~_{?}

B~_{?}

Jˆ^{i}(x) =

nn_{5}Jˆ_{5}^{0}(x)

2⇡^{2} B^{i}(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### J

_{5}

### = µ

### 2⇡

^{2}

### B + ~

### ✓ µ

^{2}

### + µ

^{2}

_{5}

### 4⇡

^{2}

### + T

^{2}

### 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：π**^{0}** 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)]