1 Some Field Theoretical Issues of the Chiral Magnetic Effect Hai-cang Ren The Rockefeller University & CCNU with De-fu Hou, Hui Liu JHEP 05(2011)046 CPODD 2012, BNL

2 The contents An introduction of CME Axial anomaly in QCD General properties of CME One loop calculation Summary Current project

3 (Fukushima, Kharzeev and Warringa) 1. A charged massless quark in a magnetic field Helicity R L charge + — + — Magnetic moment Momentum Current J I. An introduction to CME In a quark matter of net axial charge Color-flavor factor

4 the wind number QCD field strength ii) Magnetic field Generated by an off-central collision 2. RHIC Implementation i) Excess axial change Transition between different topologies of QCD Axial anomaly T=0T≠0 ion

5 iii) May provide a new signal of QCD phase transition. iv) Theoretical approach: ---- Field theory (Fukushima et. al., Kharzeev et. al.) ---- Holographic theory (Yee, Rebhan et. al.) v) There are experimental evidences, remains to be solidified. vi) Complication in RHIC: ＊ Inhomogeneous & time dependent magnetic field ＊ Inhomogeneous temperature and chemical potentials local equilibrium ＊ Beyond thermal equilibrium

6 3. The robustness of under the Infrared limit of i.e.

7 A relativistic quantum field theory at nonzero temperature and/or chemical potentials UV divergence is no worse than vacuum IR is more problematic because: ---- The appearance of the ratio ---- The appearance of the ratios etc Linde’s problem with gluons.

8 Naïve Ward identities: II. Axial anomaly in QCD UV divergence demands regularization (e.g. Pauli Villars) Not all Ward identities can be preserved The ones related to gauge symmetries have to be maintained PV regulators :

9 Ward identities post regularization:

10 ※ The explanation of the rate of ※ The solution of U A (1) problem ※ Link the change of the axial charge and the change of topology. ※ Chiral magnetic effect, chiral vortical effect, etc. Applications of the axial anomaly:

11 III General properties of CME i) Naïve axial charge & conserved axial charge should be used in thermodynamics equilibrium (Rubakov)

12 The usual photon self-energy tensor, subject to higher order corrections ii) Grand partition function: iii) Linear response

13 Chiral magnetic current iv) The Taylor expansion in Normal termAnomaly term

14 to all orders and all T and The limit

15 The limit General tensor structure with Bose symmetry:

16 The electromagnetic gauge invariance: If the infrared limit exists: to all orders

17 III. One loop calculation Continuation of imaginary Matsubara to with real for retarded (advanced) response function after the summation over Matsubara

18 Subtlety of IR limit: IR singularity: Kharzeev & Warringar

19 III. Summary IR limitHigher order 0none none if IR safe yes 0none

20 Son & Surowka Landsteiner et. al. Current project Anomalous transport coefficients

21 Regulated one loopAnomaly termTotal 0

22 Anomaly and thermodynamics Does the anomaly still show up in the regulated ?

23 Thank you!