1. 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. 2 The contents An introduction of CME Axial anomaly in QCD General properties of CME One loop calculation Summary Current project

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

7. 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. 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. 9 Ward identities post regularization:

10. 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. 11 III General properties of CME i) Naïve axial charge & conserved axial charge should be used in thermodynamics equilibrium (Rubakov)

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

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

14. 14 to all orders and all T and The limit

15. 15 The limit General tensor structure with Bose symmetry:

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

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

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

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

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

21. 21 Regulated one loopAnomaly termTotal 0

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

23. 23 Thank you!