### Chiral Matter and Topology in Astrophysics

### Naoki Yamamoto (Keio University)

### “Recent Developments in Chiral Matter and Topology”

### December 8, 2018

### Main topics

### • Core-collapse supernova

### • Chiral hydrodynamics

### • Chiral turbulence in supernovae

### • Photonic chiral vortical effect in pulsars

### Units: ~ = c = k

^{B}

### = e = 1

## Core-collapse supernovae

## explosions

### • One of the most energetic phenomena in the Universe

### • Transition to neutron stars & origin of heavy elements

### • But explosion is difficult in conventional 3D hydrodynamic theory

### Core-collapse supernova explosions

### One of the puzzles in astrophysics

http://www.riken.jp/pr/press/2009/20091211/

### Parity

### Chirality of fermions

**s**

**p**

### left-handed right-handed

mirror

**s**

**p**

### Why is “God” left-handed?

### W. Pauli

### “God is just a weak left-hander. ”

### The laws of physics are left-right symmetric except for the

### weak interaction that acts only on left-handed particles.

### From micro to macro

### Macro Evolution of core-collapse supernovae (giant P violation) Chiral kinetic theory

Son, Yamamoto (2012); Stephanov, Yin (2012); J.

W. Chen, S. Pu, Q. Wang, X. N. Wang (2013), …

### Micro

### Microscopic parity violation is reflected in macroscopic behavior:

### Chirality of fermions (e, ν) in Standard Model

Yamamoto (2016)

### Supernova = Giant Parity Breaker

### p + e ^{L} ! n + ⌫ _{e} ^{L}

### e

L### e

R### e

L### e

Rsupernovae

### ν

L### ν

R### ν

L### ν

ROhnishi, Yamamoto (2014); Grabowska, Kaplan, Reddy (2015); Sigl, Leite (2016), …

supernovae

### m

e### • Neutrino mean free path ~ 1cm at core (ρ

N### ~10

^{15}

### g/cm

^{3}

### ).

### • Neutrino matter = Chiral liquid (μ

ν### ~200 MeV ≫ T~10 MeV) = 3D topological matter

### Neutrino matter in supernovae

### Chiral hydro

### Chiral kinetic theory ν

L### ~ 100 km

### Chiral hydrodynamics

### Chiral magnetic effect

Vilenkin (1980); Nielsen, Ninomiya (1983); Fukushima, Kharzeev, Warringa (2008), …

### j = µ _{R} µ _{L}

### 4⇡ ^{2} B ⌘ µ _{5}

### 2⇡ ^{2} B

### j _{5} = µ _{R} + µ _{L}

### 4⇡ ^{2} B ⌘ µ

### 2⇡ ^{2} B

### Chiral vortical effect

Vilenkin (1979); Erdmenger et al. (2009); Banerjee et al. (2011);

Son, Surowka (2009); Landsteiner et al. (2011)

### j = µµ _{5}

### 2⇡ ^{2} !

### j _{5} =

### ✓ µ ^{2} + µ ^{2} _{5}

### 4⇡ ^{2} + T ^{2} 12

### ◆

### !

### ! ⌘ r ⇥ v

### vorticity

### Lorentz covariant chiral hydro

### @

_{µ}

### T

^{µ⌫}

### = F

^{⌫}

### j

### Energy-momentum conservation:

### Anomaly relation:

Son, Surowka (2009); Sadofyev, Isachenkov (2011); Neiman, Oz (2011)

### T

^{µ⌫}

### = (✏ + P )u

^{µ}

### u

^{⌫}

### P g

^{µ⌫}

### + (di↵usion) j

^{µ}

### = nu

^{µ}

### +⇠

_{B}

### B

^{µ}

### + ⇠!

^{µ}

### + (di↵usion)

### B

^{µ}

### = 1

### 2 ✏

^{µ⌫↵}

### u

_{⌫}

### F

_{↵}

### , !

^{µ}

### = ✏

^{µ⌫↵}

### u

_{⌫}

### @

_{↵}

### u

### @

_{µ}

### j

_{5}

^{µ}

### = CE

^{µ}

### B

_{µ}

### j

_{5}

^{µ}

### = n

_{5}

### u

^{µ}

### +⇠

_{B5}

### B

^{µ}

### + ⇠

_{5}

### !

^{µ}

### + (di↵usion) (dissipation) (dissipation)

### (dissipation)

### Helicity conservation

### @

_{µ}

### j

_{5}

^{µ}

### = CE · B

Yamamoto (2016); see also Avdoshkin et al., (2016)

### Helicity conservation

### d dt

### Z

### d

^{3}

### x

### ✓

### j

_{5}

^{0}

### + C

### 2 A · B

### ◆

### = 0

Yamamoto (2016); see also Avdoshkin et al., (2016)

### CVE CME

### Helicity conservation

### d dt

### Z

### d

^{3}

### x

### ✓

### j

_{5}

^{0}

### + C

### 2 A · B

### ◆

### = 0

### j

_{5}

^{0}

### = n

_{5}

### +⇠

_{5}

### v · ! + ⇠

^{B5}

### v · B

Yamamoto (2016); see also Avdoshkin et al., (2016)

### chiral charge magnetic helicity

### fluid helicity mixed helicity

### CVE CME

### Helicity conservation

### d

### dt Q

_{tot}

### = 0, Q

_{tot}

### ⌘ Q

^{chi}

### + Q

_{mag}

### + Q

_{flu}

### + Q

_{mix}

Q_{mag} =
Z

d^{3}x C

2 A · B

### Q

_{chi}

### = Z

### d

^{3}

### x n

_{5}

### d

### dt Z

### d

^{3}

### x

### ✓

### j

_{5}

^{0}

### + C

### 2 A · B

### ◆

### = 0

Yamamoto (2016); see also Avdoshkin et al., (2016)

Q_{flu} =
Z

d^{3}x ⇠_{5}v · ! Q_{mix} =
Z

d^{3}x ⇠_{B5}v · B

### j

_{5}

^{0}

### = n

_{5}

### +⇠

_{5}

### v · ! + ⇠

^{B5}

### v · B

**B** **B**

**B**

**B**

**v** **v**

**B** **v**

**B**

**v**

### Neutrino chiral hydro

### • Neutrino number + fluid helicity is conserved.

### • Generation of fluid helicity is numerically observed.

### • Chiral hydrodynamic equations for pure neutrino matter:

### CVE CVE

### @

_{t}

### (n + ⇠v · !) + r · j = 0, j = nv + ⇠!

Kobayashi, Okuno, Yamamoto, in preparation

### • When coupled to charged sector, fluid helicity ~ μ

5### for electrons

### j ⇠ (v · !)B

### Chiral MHD turbulence

### in supernovae

### Turbulence and cascade

https://doi.org/10.1515/htmp-2016-0043

### • The structure becomes smaller, and eventually dissipates (direct cascade)

### • Similar in magneto-hydrodynamics (MHD)

### ………

### Direct cascade

### (3D usual matter) Inverse cascade

### (2D usual matter)

### explosion difficult explosion easier

F. Hanke (2014)

### 2D

### 3D

### Cascade and explosion

### What about 3D chiral matter?

### Chiral MHD for supernovae

Masada, Kotake, Takiwaki, Yamamoto, arXiv:1805.10419

### Proto-neutron star (PNS)

### Chiral MHD for supernovae

### “CME”

### chiral anomaly

### • Chiral MHD w/o vorticity at the core (proton, e

R### , e

L### ):

### @

_{t}

### B = r ⇥ (v ⇥ B) + ⌘r

^{2}

### B + ⌘ r ⇥ (⇠

^{B}

### B)

### • Setup for proto-neutron stars (100 MeV = 1) :

Masada, Kotake, Takiwaki, Yamamoto, arXiv:1805.10419

### @

_{t}

### n

_{5}

### = ⌘

### 2⇡

^{2}

### ( r ⇥ B ⇠

_{B}

### B) · B

### ⇢

_{0}

### = 5.0, P

_{0}

### = 1.0, ⇠

_{B0}

### = 4.2 ⇥ 10

^{3}

### , ⌘ = 100.0

### +(di↵usion) (dissipation)

### Movies of 3D simulations are available at:

### http://www.kusastro.kyoto-u.ac.jp/~masada/movie.mp4

Masada, Kotake, Takiwaki, Yamamoto, arXiv:1805.10419

### Energy spectra

### • *As time passes, energy in small-k and large-k regions grows*

### • Eventually, ε

M*~k*

^{-2}

### , ε

K*~k*

^{-5/3}

Masada et al., arXiv:1805.10419; see also Brandenburg et al., arXiv:1707.03385

### ε

M### ε

K### Neutrino chiral radiation hydro

### r ^{↵} (T _{hyd} ^{↵} + T _{⌫} ^{↵} ) = 0

Yamamoto, work in progress

### Stress tensor for ν Stress tensor for N & e

### (Hydro) (Chiral kinetic theory)

### T

_{⌫}

^{ij}

### = Z

p

### |p|

### ✓ ˆ

### p

^{i}

### p ˆ

^{j}

### n

_{⌫}

### 1

### 2 p

^{i}

### ✏

^{jk`}

### ⌦

^{k}

_{p}

### @

_{`}

### n

_{⌫}

### 1

### 2 p

^{j}

### ✏

^{ik`}

### ⌦

^{k}

_{p}

### @

_{`}

### n

_{⌫}

### ◆

### ⌦

_{p}

### = p ˆ 2 |p|

^{2}

### Berry curvature of ν :

### Photonic chiral vortical effect

Avkhadiev-Sadofiev (2017); Yamamoto (2017); V. A. Zyuzin (2017);

Chernodub, Cortijo, Landsteiner (2018), …

### Helicity and Berry curvature

### • Spin-momentum locking ⇄ helicity λ

### • chiral fermions (λ=±1/2)

### • photons (λ=±1)

e.g., Onoda, Murakami, Nagaosa (2004)### • gravitons (λ=±2)

Yamamoto (2018)### • Berry curvature (adiabatic approximation): ⌦

_{p}

### = p ˆ

### |p|

^{2}

### • Semi-classical equations of motion in a rotating frame:

### • Photonic chiral current along a rotation:

### ˙x = ˆ p + p ˙ ⇥ ⌦

^{p}

### p = 2 ˙ |p| ˙x ⇥ ! + O(!

^{2}

### ) Coriolis force

### j

_{CVE}

^{±}

### = 2!

### Z d

^{3}

### p

### (2⇡)

^{3}

### |p|( ˆ p · ⌦

^{p}

### )n

^{±}

_{p}

### = ± T

^{2}

### 6 ! equilibrium non-equilibrium

Yamamoto, arXiv:1702.08886

### Photon gas under rotation

### X-ray pulsars

### Pulsars → Polarized photon flux:

### R

### L

### T ⇠ 10 keV, ! ⇠ 10

^{3}

### Hz

### cf) photon flux from sun: f ⇠ 10

^{17}

### /s · cm

^{2}

### f

^{±}

### ⇠ 10

^{21}

### /s · cm

^{2}