Chiral Magnetic and Vortical Effects at Weak Coupling

Chiral Magnetic and Vortical Effects at Weak Coupling University of Illinois at Chicago and RIKEN-BNL Research Center June 19, 2014 XQCD 2014 Stonybrook University, June 19-20, 2014

Chiral Magnetic and Vortical Effects of Chiral Weyl Fermion Chiral Magnetic Effect (Fukushima-Kharzeev-Warringa, Son-Zhitnitsky, Vilenkin) J = 1 4π 2 µ B Chiral Vortical Effect (Erdmenger.et al, Banerjee.et al, Vilenkin) J = 1 4π 2 ( µ 2 + π2 3 T 2 ) ω, ω = 1 2 v

They are robust and protected by triangle anomaly They have been checked at strong coupling (HUY, Rebhan-Schmitt-Stricker, Gynther.et al) in hydrodynamics (Son-Surowka) on lattices (Buividovich.et al, Abramczyk.et al, Yamamoto, Bali.et al)

We will see that the weak coupling picture is a bit more subtle

Quasi-particle picture of CME (Kharzeev-Warringa) Quantized Weyl particles (p) and anti-particles ( p) S = ± 1 2 p, p µ M = ± S = 1 p p 2 p 2

Quasi-particle picture of CME (Kharzeev-Warringa) Energy shift in a magnetic field: E = µ M B = 1 p B 2 p 2 It gives rise to a tendency to align the momentum along the magnetic field direction

Let s try to be more quantitative The energy shift E = 1 p B will modify the 2 p 2 equilibrium distribution of particles (f eq + ) and anti-particles (f eq f eq ± = from ) f (0) ± ( exp[β( p µ)] + 1 ) 1 ( to exp[β( p 1 2 p B ) 1 p µ)] + 1 2 f (0) ± + βf (0) ± (1 f (0) ± ) p B 2 p 2 + O(B2 )

The net current is d 3 p d 3 J = x (f eq (2π) 3 + f eq p p ) = (f eq (2π) 3 + f eq ) p = β d 3 p p p B ( ) f (0) 2 (2π) 3 p p 2 + (1 f (0) + ) f (0) (1 f (0) ) = 1 3 1 ( ) B β dp p f (0) 4π 2 + (1 f (0) + ) f (0) (1 f (0) ) 0 = 1 3 µ B 4π 2 where β ( ) dp p f (0) 0 + (1 f (0) + ) f (0) (1 f (0) ) = µ independent of temperature

This contribution from the energy shift explains only of the full result 1 3 Identifying the remaining 2 contribution to the CME 3 needs a complete picture of microscopic motions of fermions under a magnetic field

Berry Phase in Momentum Space (Son-Yamamoto, Stephanov-Yin, Zahed) The motion of Weyl particle is described by the action ( S + = dt p x + A x + A 0 E A p p ) where the last term is the Berry s connection coming from the chiral spinor wave-function A p = iψ ( p) p ψ( p), ( σ p)ψ( p) = p ψ( p) whose curvature is of the monopole form b Ap = p 2 p 3

Picture of Berry s Phase in Momentum Space

Picture of Berry s Phase in Momentum Space

Motion of Weyl Particle in a Magnetic Field Using the relativistic energy E = p 1 2 p B p 2 the equation of motion from the action gives G x = E ( ) p + B E p b = p p + p( p B) + O(B 2 ) p 4 where G = (1 + B b) is the modified phase space measure The second term is the new velocity from triangle anomaly (Stephanov-Yin)

The current from this new velocity is d 3 p ( ) J = G x f (0) (2π) 3 + f (0) d 3 p p( p = B) ( ) f (0) (2π) 3 p 4 + f (0) = 2 3 1 ( ) B dp f (0) 4π 2 + f (0) 0 = 2 3 µ B 4π 2 where ( ) dp f (0) 0 + f (0) = µ independent of temperature

Breaking Up The Equilibrium CME Value 2 3 1 3 comes from the modification of equilibrium distribution due to energy shift Let s call it energetic contribution comes from a new component of velocity due to anomaly, which is more kinematic Let s call it kinematic contribution The point: In out-of-equilibrium conditions, the energetic contribution is expected to be lost, while the kinematic contribution always exists Prediction: At weak coupling, the out-of-equilibrium CME should be roughly 2 of equilibrium CME 3

A Similar Story for Chiral Vortical Effect (Chen-Son-Stephanov-HUY-Yin)

Derivation of equilibrium distribution in the presence of vorticity ω = 1 2 v Detailed balance in the Boltzmann equation dictates M i f 2 f eq ( x i, p i ) ( ( )) 1 f eq x f, p f i f = M f i 2 f f eq ( x f, p f ) i ( 1 f eq ( x i, p i ))

In a time-reversal invariant theory, we have M i f 2 = M f i 2, and i f eq ( x i, p i ) ( 1 f eq ( x i, p i )) = f f eq ( x f, p f ) ( 1 f eq ( x f, p f )) This is satisfied if [ f eq ( x, p) ( ( 1 f eq x, p )) = exp ] β l Q l ( x, p) l where Q l ( x, p) are additive conserved charges carried by a particle at ( x, p), and β l are arbitrary constants Possible Q l ( x, p) are: Energy E( p), Momentum p, and Angular momentum L = x p + 1 p 2 p

We have [ ( exp β E v 0 p α L )] [ ( ( = exp β E v 0 p α x p + 1 ))] p 2 p [ ( = exp β E ( v 0 + α x ) p 1 )] 2 α p p [ ( = exp β E v( x) p 1 )] 2 α p p where v( x) v 0 + α x can be interpreted as the fluid velocity, now depending on the position x. Computing 1 2 v = α, we see that α is in fact the vorticity ω of the fluid.

Local Equilibrium Distribution With a Vorticity In the presence of vorticity ω, the local equilibrium distribution at rest is shifted from f (0) ± = (exp[β(u µ P µ µ)] + 1) 1 = ( exp[β( p µ)] + 1 ) 1 f eq ± = to ( exp[β( p µ 1 p ω 2 p )] + 1 ) 1 f (0) ± ± βf (0) ± (1 f (0) ± ) p ω 2 p + O(ω2 )

The net current from equilibrium distribution shift is d 3 p d 3 J = x (f eq (2π) 3 + f eq p p ) = (f eq (2π) 3 + f eq ) p = β d 3 p p p ω ( ) f (0) 2 (2π) 3 + (1 f (0) + ) + f (0) (1 f (0) ) p p = 1 3 1 ( ) 4π ω β dp p 2 f (0) 2 + (1 f (0) + ) + f (0) (1 f (0) ) 0 = 1 ) 3 1 (µ 2 + π2 4π 2 3 T 2 ω where β ( ) dp 0 p2 f (0) + (1 f (0) + ) + f (0) (1 f (0) ) = µ 2 + π2 3 T 2 Again, this is precisely 1 3 of the full result

Kinematic Contribution to CVE There exists a kinematic term in the current (Son-Yamamoto, Chen-Son-Stephanov-HUY-Yin) J d 3 p (2π) 3 ( ) p 2 p (f 2 + + f ) Using f ± = ( exp[β( p v( x) p µ)] + 1 ) 1 with 1 v = ω, 2 a short computation shows that this kinematic contribution gives the remaining 2 of the full CVE 3 Prediction: Out-of equilibrium CVE should be roughly of the equilibrium CVE 2 3

Let s Change the Topic to Chiral Magnetic Effect at Weak Coupling with Relaxation Dynamics (Satow-HUY)

Chiral Magnetic Conductivity at Finite (ω, k) (Kharzeev-Warringa) Chiral Magnetic Effect is a response of the current to an external magnetic field J(ω, k) = σχ (ω, k) B(ω, k) The chiral magnetic conductivity σ χ (ω, k) is computed from the P-odd retarded function G ij R = iθ(t t ) [ J i (t), J j (t ) ] iσ χ (ω, k)ɛ ijl k l We would expect lim k 0 lim ω 0 σ χ (ω, k) = µ 4π 2

A Puzzle in Free Fermion Computations (Kharzeev-Warringa) One loop computation with free fermion

A Puzzle in Free Fermion Computations (Kharzeev-Warringa) lim lim σ χ(ω, k) lim lim σ χ (ω, k) k 0 ω 0 ω 0 k 0 lim lim σ χ(ω, k) = µ k 0 ω 0 4π 2 but lim lim σ χ(ω, k) = 1 ω 0 k 0 3 µ 4π 2

We show that this issue disappears with finite interactions (as originally suggested by Kharzeev-Warringa) in the kinetic theory with finite collision term in the diagrammatic computation with resummed damping rate

Result from Chiral Kinetic Theory with a Finite Relaxation Time τ F(ω, k) = ω k σ χ (ω, k) = µ (1 F(ω, k)) 4π2 (( ) ω + i τ k where ( k 2 ( ω + i τ 2k 2 ) 2 ) log ( )) ω k + i τ ω + k + i τ lim lim σ χ(ω, k) = lim lim σ χ (ω, k) = µ k 0 ω 0 ω 0 k 0 4π 2 No Non-Commutativity

1 F Ω,k 0 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ω τ = 1 (dashed), τ = 10 (dotted), τ = 100 (solid)

Diagrammatic Computation We compute the P-odd part of the real-time retarded current-current correlation function, G ra, in the Swinger-Keldysh contour j µ (k) = ( 1)ie 2 A ν(k) d 4 p (2π) 4 tr [σµ S ra(p + k)σ ν S rr (p) + σ µ S rr (p)σ ν S ar (p k)]

Fermion Propagators with Damping Rate Included S ra(p) θ(x 0 y 0 ) {ψ(x), ψ (y)} = s=± i p 0 s p + iζ/2 Ps( p) S ar (p) θ(y 0 x 0 ) {ψ(x), ψ (y)} = i p 0 s p iζ/2 Ps( p) s=± S rr (p) 1 ( ) 1 2 [ψ(x), ψ (y)] = 2 n+(p0 ) ρ(p) where the spectral density ρ is ρ(p) = t=± ζ (p 0 t p ) 2 + (ζ/2) 2 Pt( p) The projection operators P ±( p) are defined as P ±( p) 1 ( ) σ p 1 ± 2 p

Absence of Pinch Singularity in P-odd Part In the longitudinal P-even part responsible for electric conductivity, we see the Pinch singularity as expected j 3 (k) iω e2 χ 3ζ A 3(k) as (ω, k) 0, where χ susceptibility ( T 2 6 + µ2 2π 2 ) is the Since ζ g 2 log(1/g), this changes the normal power counting scheme, and a re-summation of infinitely many ladder diagrams is necessary (Jeon-Yaffe)

Absence of Pinch Singularity in P-odd Part However, in the transverse P-odd part responsible for CME, the Pinch singularity cancels between the two Feynman diagrams The first diagram gives ( ) J i µ 4ik 0 (p) = i 8π 2 ɛijk A j (p)p k 3ζ + 1 + O ( p 0, p 3) The second diagram gives J i µ (p) = i 8π 2 ɛijk A j (p)p k ( 4ik ) 0 3ζ + 1 + O ( p 0, p 3)

Absence of Pinch Singularity in P-odd Part The sum of the two gives the right magnitude of the Chiral Magnetic Effect J i µ (p) = i 4π 2 ɛijk A j (p)p k Questions: Does this feature persist in higher loops? There may appear a diffusive pole if we include the coupling to the energy-momentum sector (Chiral Shear Wave) (Sahoo-HUY, Matsuo-Sin-Takeuchi-Tsukioka). How do we see it diagrammatically?

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