1. Pavel Buividovich (Regensburg)

2. Chiral Magnetic Effect [Kharzeev, Warringa, Fukushima] Chiral Separation Effect [Son, Zhitnitsky] Chiral Vortical Effect [Erdmenger et al., Banerjee et al.]

3. Dissipative transport (conductivity, viscosity) No ground state No ground state T-noninvariant (but CP) T-noninvariant (but CP) Spectral function = anti- Hermitean part of retarded correlator Spectral function = anti- Hermitean part of retarded correlator Work is performed Work is performed Dissipation of energy Dissipation of energy First k → 0, then w → 0 First k → 0, then w → 0 Anomalous transport (CME, CSE, CVE) Ground state Ground state T-invariant (but not CP!!!) T-invariant (but not CP!!!) Spectral function = Hermitean part of retarded correlator Spectral function = Hermitean part of retarded correlator No work is performed No work is performed No dissipation of energy No dissipation of energy First w → 0, then k → 0 First w → 0, then k → 0

4. Folklore on CME & CSE: Transport coefficients are RELATED to anomalies (axial and gravitational) Transport coefficients are RELATED to anomalies (axial and gravitational) and thus protected from: and thus protected from: perturbative corrections perturbative corrections IR effects (mass etc.) IR effects (mass etc.) Check these statements as applied to the lattice. What is measurable? What is measurable? How should one measure? How should one measure? What should one measure? What should one measure? Does it make sense at all? If Does it make sense at all? If

5. ρ = 1.0, m = 0.05

6. ρ = 1.4, m = 0.01

7. ρ = 1.4, m = 0.05

8. Bulk definition of μ 5 !!! Around 20% deviation

9. CLAIM: constant magnetic field in finite volume is NOT a small perturbation “triangle diagram” argument invalid (Flux is quantized, 0 → 1 is not a perturbation, just like an instanton number) More advanced argument: in a finite volume in a finite volume Solution: hide extra flux in the delta-function Fermions don’t note this singularity if Flux quantization!

10. Partition function of Dirac fermions in a finite Euclidean box Partition function of Dirac fermions in a finite Euclidean box Anti-periodic BC in time direction, periodic BC in spatial directions Anti-periodic BC in time direction, periodic BC in spatial directions Gauge field A 3 =θ – source for the current Gauge field A 3 =θ – source for the current Magnetic field in XY plane Magnetic field in XY plane Chiral chemical potential μ 5 in the bulk Chiral chemical potential μ 5 in the bulk Dirac operator:

11. Creation/annihilation operators in magnetic field: Now go to the Landau-level basis: Higher Landau levels (topological) zero modes

12. Dirac operator in the basis of LLL states: Vector current: Prefactor comes from LL degeneracy Only LLL contribution is nonzero!!!

13. Polarization tensor in 2D: [Chen,hep-th/9902199] Value at k 0 =0, k 3 =0: NOT DEFINED Value at k 0 =0, k 3 =0: NOT DEFINED (without IR regulator) (without IR regulator) First k 3 → 0, then k 0 → 0 First k 3 → 0, then k 0 → 0 Otherwise zero Otherwise zero Final answer: Proper regularization (vector current conserved):

14. μ 5 is not a physical quantity, just Lagrange multiplier Chirality n 5 is (in principle) observable Express everything in terms of n 5 Express everything in terms of n 5 To linear order in μ 5 : Singularities of Π 33 cancel !!! Note: no non-renormalization for two loops or higher and no dimensional reduction due to 4D gluons!!!

15. Anomalous current-current correlators: Chiral Separation and Chiral Magnetic Conductivities:

16. Integrate out time-like loop momentum Relation to canonical formalism Virtual photon hits out fermions out of Fermi sphere…

17. Singularity

18. Decomposition of propagators and polarization tensor in terms of in terms of chiral states s,s’–chiralities

19. Individual contributions of chiral states are divergent Individual contributions of chiral states are divergent The total is finite and coincides with CSE (upon μ V → μ A ) The total is finite and coincides with CSE (upon μ V → μ A ) Unusual role of the Dirac mass Unusual role of the Dirac mass non-Fermi-liquid behavior? non-Fermi-liquid behavior? Carefulregularizationrequired!!!

20. Pauli-Villars regularization (not chiral, but simple and works well)

21. Pauli-Villars: zero at small momenta Pauli-Villars: zero at small momenta -μ A k 3 /(2 π 2 ) at large momenta -μ A k 3 /(2 π 2 ) at large momenta Pauli-Villars is not chiral (but anomaly is reproduced!!!) Pauli-Villars is not chiral (but anomaly is reproduced!!!) Might not be a suitable regularization Might not be a suitable regularization Individual regularization in each chiral sector? Individual regularization in each chiral sector? Let us try overlap fermions Advanced development of PV regularization Advanced development of PV regularization = Infinite tower of PV regulators [Frolov, Slavnov, Neuberger, 1993] = Infinite tower of PV regulators [Frolov, Slavnov, Neuberger, 1993] Suitable to define Weyl fermions non-perturbatively Suitable to define Weyl fermions non-perturbatively

22. Lattice chiral symmetry: Lüscher transformations These factors encode the anomaly (nontrivial Jacobian)

23. First consider coupling to axial gauge field: Assume local invariance under modified chiral transformations [Kikukawa, Yamada, hep-lat/9808026]: Require (Integrable) equation for D ov !!!

24. We require that Lüscher transformations generate gauge transformations of A μ (x) Linear equations for “projected” overlap or propagator

25. Dirac operator with axial gauge field Dirac operator with chiral chemical potential Only implicit definition (so far) Only implicit definition (so far) Hermitean kernel (at zero μV) = Sign() of what??? Hermitean kernel (at zero μV) = Sign() of what??? Potentially, no sign problem in simulations Potentially, no sign problem in simulations

26. Axial vertex: Consistent currents!Natural definition:charge transfer

27. In terms of kernel spectrum + derivatives of kernel: Derivatives of GW/inverse GW projection: Numerically impossible for arbitrary background… Krylov subspace methods? Work in progress…

28. Good agreement with continuum expression

29. Small momenta/large size: continuum result

30. At small momenta, agreement with PV regularization

31. As lattice size increases, PV result is approached

32. Let us try “slightly’’ non-conserved current Perfect agreement with “naive” unregularized result

33. Replace D ov with projected operator Lüscher transformations Usual chiral rotations

34. Agreement with PV, but large artifacts They might be crucial in interacting theories

35. Expand anomalous correlators in μ V or μ A : VVA correlator in some special kinematics!!!

36. At fixed k3, the only scale is µ Use asymptotic expressions for k3 >> µ !!! Ward Identities fix large-momentum behavior CME: -1 x classical result, CSE: zero!!! CME: -1 x classical result, CSE: zero!!!

37. 4 independent form-factors 4 independent form-factors Only wL is constrained by axial WIs Only wL is constrained by axial WIs [M. Knecht et al., hep-ph/0311100]

38. CSE: p = (0,0,0,k3), q=0, µ=2, ν=0, ρ=1 ZERO CME: p = (0,0,0,k3), q=(0,0,0,-k3), µ=1, ν=2, ρ=0 IR SINGULARITY Regularization: p = k + ε/2, q = -k+ε/2 ε – “momentum” of chiral chemical potential Time-dependent chemical potential:

39. Spatially modulated chiral chemical potential By virtue of Bose symmetry, only w (+) (k 2,k 2,0) Transverse form-factor Not fixed by the anomaly

40. In addition to anomaly non-renormalization, new (perturbative!!!) non-renormalization theorems [M. Knecht et al., hep-ph/0311100] [A. Vainstein, hep-ph/0212231]: Valid only for massless QCD!!!

41. Special limit: p 2 =q 2 Six equations for four unknowns… Solution: Might be subject to NP corrections due to ChSB!!!

42. In terms of correlators Linear response of currents to “slow” rotation: Subject to PT corrections!!!

43. A naïve method [Yamamoto, 1303.6292]: Analytic continuation of rotating frame metric Analytic continuation of rotating frame metric Lattice simulations with distorted lattice Lattice simulations with distorted lattice Physical interpretation is unclear!!! Physical interpretation is unclear!!! By virtue of Hopf theorem: By virtue of Hopf theorem: only vortex-anti-vortex pairs allowed on torus!!! only vortex-anti-vortex pairs allowed on torus!!! More advanced method [Landsteiner, Chernodub & ITEP Lattice, 1303.6266]: Axial magnetic field = source for axial current Axial magnetic field = source for axial current T 0y = Energy flow along axial m.f. T 0y = Energy flow along axial m.f. Measure energy flow in the background axial magnetic field

44. First lattice calculations: ~ 0.05 x theory prediction Non-conserved energy-momentum tensor Non-conserved energy-momentum tensor Constant axial field not quite a linear response Constant axial field not quite a linear response For CME: completely wrong results!!!... Is this also the case for CVE? Check by a direct calculation, use free overlap

45. No holomorphic structure No holomorphic structure No Landau Levels No Landau Levels Only the Lowest LL exists Only the Lowest LL exists Solution for finite volume? Solution for finite volume?

46. Conserved lattice energy-momentum tensor: not known How the situation can be improved, probably? Momentum from shifted BC [H.Meyer, 1011.2727] We can get total conserved momentum = Momentum density =(?) Energy flow

47. Anomalous transport: very UV and IR sensitive Anomalous transport: very UV and IR sensitive Non-renormalizability only in special kinematics Non-renormalizability only in special kinematics Proofs of exactness: some nontrivial assumptions Proofs of exactness: some nontrivial assumptions  Fermi-surface  Quasi-particle excitations  Finite screening length  Hydro flow of chiral particles is a strange object Field theory: NP corrections might appear, ChSB Field theory: NP corrections might appear, ChSB Even Dirac mass destroys the usual Fermi surface Even Dirac mass destroys the usual Fermi surface What happens to “chiral” Fermi surface and to anomalous transport in confinement regime? What happens to “chiral” Fermi surface and to anomalous transport in confinement regime? What are the consequences of inexact ch.symm? What are the consequences of inexact ch.symm?

48. Backup slides

49. Chemical potential for conserved charge (e.g. Q): In the action In the action Via boundary conditions Via boundary conditions Non-compact Non-compact gauge transform For anomalous charge: General gauge transform BUT the current is not conserved!!! Chern-Simons current Chern-Simons current Topological charge density Topological charge density

50. Simplest method: introduce sources in the action Constant magnetic field Constant magnetic field Constant μ5 [Yamamoto, 1105.0385] Constant μ5 [Yamamoto, 1105.0385] Constant axial magnetic field [ITEP Lattice, 1303.6266] Constant axial magnetic field [ITEP Lattice, 1303.6266] Rotating lattice??? [Yamamoto, 1303.6292] Rotating lattice??? [Yamamoto, 1303.6292] “Advanced” method: Measure spatial correlators Measure spatial correlators No analytic continuation necessary No analytic continuation necessary Just Fourier transforms Just Fourier transforms BUT: More noise!!! BUT: More noise!!! Conserved currents/ Conserved currents/ Energy-momentum tensor Energy-momentum tensor not known for overlap not known for overlap

51. First Lx,Ly →∞ at fixed Lz, Lt, Φ !!!

52. L3 → ∞, Lt fixed: ZERO (full derivative) L3 → ∞, Lt fixed: ZERO (full derivative) Result depends on the ratio Lt/Lz Result depends on the ratio Lt/Lz

53. 2D axial anomaly: Correct polarization tensor: Naive

55. Landau levels for vector magnetic field: Rotational symmetry Rotational symmetry Flux-conserving singularity not visible Flux-conserving singularity not visible Dirac modes in axial magnetic field: Rotational symmetry broken Rotational symmetry broken Wave functions are localized on the boundary Wave functions are localized on the boundary (where gauge field is singular) (where gauge field is singular) “Conservation of complexity”: Constant axial magnetic field in finite volume is pathological

56. CME is related to anomaly (at least) perturbatively in massless QCD CME is related to anomaly (at least) perturbatively in massless QCD Probably not the case at nonzero mass Probably not the case at nonzero mass Nonperturbative contributions could be important (confinement phase)? Nonperturbative contributions could be important (confinement phase)? Interesting to test on the lattice Interesting to test on the lattice Relation valid in linear response approximation Relation valid in linear response approximationHydrodynamics!!! Six equations for four unknowns… Solution:

57. Full phase space is available only at |p|>2|k F |

58. Measure spatial correlators + Fourier transform Measure spatial correlators + Fourier transform External magnetic field: limit k0 → 0 required after k3 → 0, analytic continuation??? External magnetic field: limit k0 → 0 required after k3 → 0, analytic continuation??? External fields/chemical potential are not compatible with perturbative diagrammatics External fields/chemical potential are not compatible with perturbative diagrammatics Static field limit not well defined Static field limit not well defined Result depends on IR regulators Result depends on IR regulators