Ⅱ-5. Hadron Physics (Theory)

RIKEN Accel. Prog. Rep. 49 (2016)

P-odd spectral density of quark-gluon plasma at weak coupling
H. -U. Yee∗1,∗2Topological fluctuations of QCD is one of the fundamental aspects of QCD dynamics in the plasma ofquarks and gluons. They lead to thermal fluctuationsof axial charge density that has distinct P- and CP-oddparities, which induces several unique phenomena suchas chiral magnetic effect1) . At weak coupling regime inhigh enough temperature, the rate of topological fluctuations is Γs ∼ αs5 log(1/αs )T 4 , and this dependence
on the coupling constant also governs how fast the axial charge, once created, decays in time. If the timescale one is interested in is faster than this, the axial
charge is approximately conserved. The boundary ofvalidity regime of hydrodynamics is αs2 T , the scale ofhard electromagnetic probes such as photon emission
rates is dΓ/d3 k ∼ αEM αs T , the scale of heavy quark
momentum diffusion is κ ∼ αs2 T 3 , and the jet quenching parameter is q̂ ∼ αs2 T 3 . Therefore, for many interesting dynamics in the quark-gluon plasma at weakcoupling regime, the axial charge can be considered asa conserved charge.
Due to its unique P- and CP-odd parities, the axial charge leads to an interesting P-odd structure in
the current-current thermal correlation functions ofthe plasma. The retarded current-current correlationij
ij
function Gij
R (k) ≡ (−i)Gra (k) = (−i)Jr (k)Ja (−k)
written in the ra-basis of Schwinger-Keldysh formalism, encodes “chiral magnetic conductivity”2) σχ (k) byijl l
Gij
R (k) ∼ iσχ (k) k , that is responsible for the chiral magnetic effect with arbitrary momentum of theIt has been wellmagnetic field: J = σχ (k)B(k).
established that the zero momentum limit of σχ (k)is topologically protected by chiral anomaly coefficient: limk→0 σχ (k) = Nc NF µA /(2π 2 ) where µA is axial chemical potential.
To get a better picture of full real-time P-odd correlation functions, one starts with the fluctuationdissipation relation
 ij0(1)Gijrr (k) = 1/2 + nB (k ) ρ (k) ,ijwhere ρij (k) ≡ Gijra (k) − Gar (k) is the spectral density that governs the strength of thermal fluctuations
represented by the left-hand side. From the hermiticity
of current operator, one can show that ρij (k) is a hermitian matrix in terms of spatial indices i, j: with rotational invariance, it allows to have a P-odd structureρij (k) ∼ iρodd (k)ijl k l with a real function ρodd (k).
We call ρodd (k) “P-odd spectral density”3) . It is easy
to show that it is the imaginary part of the chiral magnetic conductivity: ρodd (k) = −2 Im(σχ (k)).
Contrary to the real part of σχ (k) in k → 0 limit, the∗1∗2

Department of Physics, University of Illinois
RIKEN Nishina Center- 115 -

P-odd spectral density is sensitive to real-time thermal
dynamics of quark-gluon plasma as it governs the Podd part of thermal current-current fluctuations, andit shares many common features with the P-even spectral density that is responsible for dissipative transportcoefficients in hydrodynamic regime. Like P-even spectral densities, ρodd (k) is an odd function of k 0 ≡ ω witha small ω expansion ρodd (k) ∼ 2ξ5 ω + · · ·, where ξ5 is
one of the second order transport coefficients in chi
 + ξ5 dB/dt.
ral anomalous hydrodynamics4) : J = σ0 B
Like the ordinary electric conductivity, the ξ5 turns
out to be non-analytic in coupling constant, and theleading log computation requires a re-summation of allladder diagrams with HTL re-summation of exchanged
gluons, that eventually leads to solving second orderdifferential equations in momentum space. For 2-flavor
QCD, we have obtained3)µA
2.003.
ξ5 = − 4gs log(1/gs ) T(2)For other kinematics of momentum k, the P-odd
spectral density generally gives P- and CP-odd emission rates of spin-polarized (that is, circular polarized) virtual photons with momentum k. To see this,the unique P- and CP-odd observable in photon emissions is given by the spin polarization5) : Γodd =Γ(+ ) − Γ(− ) where ± are circular polarization vectors of photons. Using the results in Ref.5) that relates
Γodd with the P-odd part of current-current correlation
functions, one can show thatdΓodde2
odd=−nB (ω)ρ (k)
.
(3)
d3 k(2π)3ω=|k|

We have computed dΓodd /d3 k in complete leading order at weak coupling6) , by computing 1) hard Compton/pair annihilation rates, 2) soft t- or u-channel
exchange contributions with HTL re-summation, and
3) Landau-Pomeranchuk-Migdal re-summation of soft
collinear Bremstrahlung and inelastic pair annihilation
processes.
References
1) K. Fukushima, D. E. Kharzeev and H. J. Warringa,Phys. Rev. D 78, 074033 (2008).
2) D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 80,034028 (2009).
3) A. Jimenez-Alba and H. U. Yee, Phys. Rev. D 92, no.
1, 014023 (2015).
4) D. E. Kharzeev and H. U. Yee, Phys. Rev. D 84, 045025(2011).
5) K. A. Mamo and H. U. Yee, Phys. Rev. D 88, no. 11,114029 (2013).
6) K. A. Mamo and H. U. Yee, arXiv:1512.01316 [hep-ph].