Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 48 (2015)

Chiral Magnetic and Vortical Effects at Weak Coupling
H. -U. Yee∗1,∗2
Chiral magnetic and vortical effects are parityodd transport phenomena in the hydrodynamics of aplasma of chiral massless fermions, which are macroscopic manifestations of the underlying microscopicchiral anomaly. For the plasma of a single right-handed
Weyl fermion, the chiral magnetic effect dictates a cur as
rent along an applied external magnetic field B where µ is the chemical potential, and inJ = 4πµ2 Bthe case of chiral vortical effect, the vorticity of fluid × v plays a role similar to the magneticω = (1/2)∇
ω . These effects are robust
field, J = (µ2 + π 2 T 2 /3)due to the topological nature of chiral anomaly, andshould persist both in weak and strong coupling limits. Demonstrating the expected universality of these
phenomena in weak and strong coupling limits is aninteresting and non-trivial test of the topological nature of chiral anomaly. While the strong coupling limit
provided by AdS/CFT correspondence has successfully
confirmed the universality of these effects, the weakcoupling picture, albeit more intuitive, contains more
subtleties that need to be carefully taken into accounts.
In general, one can define the chiral magnetic
k)conductivity1) σχ (ω, k) by J = σχ (ω, k)B(ω,where (ω, k) are frequency and longitudinal momentum of the in-homogeneous time-dependent magnetic field B(ω,k). One expects the zero frequencymomentum limit of σχ (ω, k) to reproduce the topologically robust result of chiral magnetic effect,limk→0 limω→0 σχ (ω, k) = 4πµ2 . In the interactionfree limit, both diagrammatic1) and kinetic2) approaches give a result for σχ (ω, k) such thatlimk→0 limω→0 σχ (ω, k) = limω→0 limk→0 σχ (ω, k) =µ1
3 · 4π 2 , while a hydrodynamic argument indicates that
there should not be such a difference between the twolimits3) . Since hydrodynamic regime exists only in aninteracting theory, it is natural to study this issue in
an interacting theory going beyond the non-interacting
limit. In Ref.3) we showed in both kinetic and diagrammatic approaches that the above difference between two limits disappears in the presence of relaxation dynamics caused by interactions, confirming theexpectation from the hydrodynamics argument.
It is interesting to understand how chiral magnetic
and vortical effects arise in the quasi-particle pictureof kinetic theory of chiral fermions. The essential
ingredient that is responsible for chiral anomaly is
the Berry’s curvature in momentum space4,5) , b =
 p × Ap = p/2|p|3 , where Ap is the Berry’s connection∇of momentum-dependent chiral spinors. We showed inRef.6) that the quasi-particle energy in the presence of∗1
∗2

Department of Physics, University of Illinois, Chicago
RIKEN Nishina Center · p/2|magnetic field is dictated to be (p) = |p| − Bp|2by Lorentz invariance where the second term is a spinmagnetic moment interaction. This term brings about
several interesting consequences. Since the equilibriump) = 1/(exp[β(p)] + 1) the distridistribution is f eq (bution becomes distorted by the magnetic field, whichcauses a net current along the magnetic field. This effect turns out to explain 1/3 of the full chiral magnetic
effect. On the other hand, the equation of motions of5)
quasi-particles with the Berry’s curvature√ is given by√ where G = 1 + b · BGx˙ = ∂/∂p + (b · ∂/∂p)Bis the modified phase space density. The second term induces a net current along the magproportional to B
netic field even with spherically symmetric equilibrium
distribution. This contribution gives the rest 2/3 of thefull chiral magnetic effect.
Interestingly, a similar feature also exists in the chi , a deral vortical effect6) . With the fluid vorticity ω
tailed valance in the Boltzmann equation with conser = x × p + (1/2)
vation of angular momentum Lp/|p|,where the second term is the spin angular momentum, dictates the equilibrium distribution to be f eq =ω ·p1/(exp[β( − 2|p| )] + 1) , which induces a net current
proportional to ω : the result comprises 1/3 of thefull chiral vortical effect. From the spin-magnetic mo · p/2|ment interaction term in the energy, −B
p|2 , onecan obtain a contribution to the current by the variation of the quasi-particle action with respect to eternal gauge
which takes a form
 potential, the result of × p/(2|p|2 )). With distribution∆J = d3 p/(2π)3 (∇ffunction f = f eq ( − v · p), the resulting current alongω constitutes the rest 2/3 of the chiral vortical effect.
In space-time dimensions higher than four, there aregeneralizations of chiral magnetic and vortical effects.
A weak coupling computation of them in real-timeformalism was performed in Ref.7) , which successfully
confirmed the hydrodynamics prediction in Ref.8)
References
1) D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 80,034028 (2009).
2) D. T. Son and N. Yamamoto, Phys. Rev. D 87, no. 8,085016 (2013).
3) D. Satow and H. U. Yee, Phys. Rev. D 90, 014027(2014).
4) D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109,181602 (2012).
5) M. A. Stephanov and Y. Yin, Phys. Rev. Lett. 109,162001 (2012).
6) J. Y. Chen, D. T. Son, M. A. Stephanov, H. U. Yee andY. Yin, Phys. Rev. Lett. 113, no. 18, 182302 (2014).
7) H. U. Yee, Phys. Rev. D 90, 065021 (2014).
8) D. E. Kharzeev and H. U. Yee, Phys. Rev. D 84, 045025(2011).

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