Axial symmetry in the chiral symmetric phase

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1 Axial symmetry in the chiral symmetric phase Swagato Mukherjee June 2014, Stoney Brook, USA

2 Axial symmetry in QCD massless QCD Lagrangian is invariant under U A (1) : ψ (x) e i α ( x) γ 5 ψ(x) μ J 5 μ =0, J 5 μ = ψ γ μ γ 5 ψ at the classical level quantum fluctuations: Adler-Bell-Jackiw anomaly μ J 5 μ = 1 16 π 2 Tr F μ ν F μ ν can be written as a total derivative 't Hooft: global breaking of U A (1) non-trivial topology of the gauge field non-perturbative phenomena generates localized fermionic zero modes index theorem 2

3 Axial symmetry in QCD chiral broken phase: ψ ψ 0 axial symmetry is broken by the vacuum not invariant under U A (1) chiral symmetric phase: mechanism axial symmetry breaking can be studied directly configurations with non-trivial topologies are suppressed due to color screening with increasing T with increasing T, U A (1) breaking becomes smaller can be compared with dilute instanton gas type approximations 3

4 This talk LQCD studies of axial symmetry in the chiral symmetric phase: signatures of U A (1) in two-point correlation functions mechanism of U A (1) breaking through Dirac eigenvalue spectra role of zero & near-zero fermionic modes, chirality, localization mix-&-match of several LQCD studies: staggered (HISQ) fermions: H. Ohno et. al.: arxiv: [hep-lat], arxiv: [hep-lat] domain wall fermions (DWF): P. Hegde, M. Cheng et. al. [HotQCD]: arxiv: [hep-lat]; Z. Lin, H-T Ding et. al. [RBC-LLNL]: arxiv: [hep-lat]; C. Schroeder, Z. Lin et. al. [HotQCD]: arxiv: [hep-lat] overlap fermions: S. Sharma, V. Dick et. al.: arxiv: [hep-lat], unpublished V. Dick: poster sessions 4

5 Lattice trivia staggered (HISQ): emergence of correct axial anomaly is subtle, only in a 0 limit DWF: 5-d formulation exact chiral symmetry & correct anomaly for a=/=0 but in the limit of infinite 5th dimension overlap: 4-d formulation with exact chiral symmetry & correct anomaly for a=/=0 computationally very expensive show results for Dirac eigenvalue spectra measured on gauge configurations generated with HISQ 5

6 Degeneracies in two point correlation functions 2 2 susceptibility: χ i = T V d4 x O i (x)o i (0) con: (quark-line) connected disc: (quark-line) disconnected 6

7 Measure of axial symmetry breaking from 2-pt functions 2 2 axial symmetric: χ π = χ δ, χ σ = χ η chiral symmetric: χ π = χ σ χ δ = χ η χ π χ δ = 2 χ disc χ π χ δ = 2 χ 5, disc topological susceptibility: Q top = m ψ γ 5 ψ χ top = m 2 χ 5, disc measure of axial symmetry breaking: (in the chiral symmetric phase) χ π χ δ =2 χ disc = 2 χ top m 2 7

8 Measure of axial symmetry breaking from 2-pt functions χ disc DWF χ π χ δ DWF chiral crossover: T c =154(9) MeV non-zero at least till T 1.2T c χ π χ δ =2 χ disc = 2 χ top m 2 8

9 Measure of axial symmetry breaking from 2-pt functions χ disc DWF χ π χ δ DWF non-zero at least till T 1.2T c HISQ χ disc χ π χ δ =2 χ disc = 2 χ top m 2 very little volume & quark mass dependence 9

10 Axial symmetry breaking and the Dirac spectra χ SM: ψ ψ = 0 d λ 2mρ(λ) m 2 +λ 2 Banks-Casher: lim m 0 ψ ψ = π sgn(m)ρ(0) U A (1): χ π χ = δ 0 d λ 4m2 ρ(λ) (m 2 +λ 2 ) 2 QCD partition function is non-analytic at m=0 chiral symmetry restoration: axial symmetry breaking: ψ ψ =0 ρ(0)=0 χ π χ δ 0 what is the from of ρ(λ), such that ρ(λ)=0 but χ π χ δ 0? nature of the configurations that give rise to such ρ(λ)? 10

11 Axial symmetry breaking and the Dirac spectra χ SM : ψ ψ = 0 d λ 2mρ(λ) U m 2 +λ 2 A (1): χ π χ = δ 0 d λ 4m2 ρ(λ) (m 2 +λ 2 ) 2 a gap in around ρ(λ) λ=0 : free theory (HISQ) ψ ψ =0, χ π χ δ =0 lowest Matsubara mode 11

12 A gap in Dirac spectra? DWF T 1.3T c overlap T 1.5T c for the interacting theory there is no visible gap in the Dirac spectra till HISQ T 2T c T 2.1T c 12

13 Possible forms of Dirac spectra requirements, m 0: χ π = ψ ψ /m finite = ψ ψ 0 d λ 2mρ(λ) m 2 +λ 2 = 0 χ = δ 0 d λ 2ρ(λ) d dm [ 2m ] m 2 +λ finite 2 χ = disc 0 d λ 2 m(dρ(λ)/dm) = χ m 2 +λ 2 π χ = δ 0 d λ 4 m2 ρ(λ) (m 2 +λ 2 ) 2 0 quark mass dependence of ρ(λ 0) is the key ρ(λ 0) independent of m ruled out 13

14 Possible forms of Dirac spectra chiral: SU L (N f ) SU R (N f ) axial: U A (1) common subgroup: Z (N f ) ψ L e 2i π /Nf ψ L QCD Lagrangian: m - m (for even number of flavors) chiral symmetric phase: also symmetric under this Z (N f ) ρ(m, λ 0) = ρ ( m, λ 0) does not necessarily mean ρ(m, λ 0) is analytic in m 2 remember: lim m 0 ψ ψ = π sgn(m)ρ(0) 14

15 Possible forms of Dirac spectra ρ(m, λ 0) ψ ψ χ π χ δ χ π χ δ 2 χ disc ρ(λ 0) m 2 δ (λ) ρ(λ 0) m a possible choice a possible choice ρ(λ) m 2 : no axial symmetry breaking in 2-pt functions for m 0 not even in 4-pt functions; rigorous proof: S. Aoki et. al, arxiv:

16 LQCD Dirac spectra: DWF zero or near-zero ρ(λ 0) m 2 δ (λ) modes? 16

17 LQCD Dirac spectra: DWF ψ ψ = 0 d λ 2mρ(λ) m 2 +λ 2 + Q top mv T 1.2T c χ π χ = δ 0 d λ 4m2 ρ(λ) (m 2 +λ 2 ) + 2 Q top 2 m 2 V χ top V Q top V V=16 3 V=32 3 ρ (0) 1/ 8 no evidence!! contributions of exact zero modes vanishes in the thermodynamic limit zero modes: ρ(0) 1/ V near-zero ρ(λ 0) m 2 δ (λ) modes? 17

18 Axial symmetry breaking from Dirac spectra: DWF χ π χ = δ 0 d λ 4m2 ρ(λ) (m 2 +λ 2 ) 2 ρ(λ 0) = a 0 + a 1 λ + a 2 m 2 δ (λ) almost the entire contribution to the axial symmetry breaking measure χ π χ δ comes from near-zero modes m 2 δ (λ) for T 1.2T c χ π χ δ = a 0 π/m + 2a 1 + 2a 2 18

19 A cleaner version: overlap bulk zero near-zero T 1.5T c 19

20 A cleaner version: overlap bulk zero near-zero T 1.5T c 20

21 A cleaner version: overlap bulk zero near-zero ρ(λ 0) m 2 δ (λ) dilute instanton gas type picture? chirality wave function 21

22 Closer to the chiral crossover T 1.04T c overlap HISQ no clear m 2 δ (λ) like structure close enough to the chiral crossover O(4) universality driven behavior should dominate T 1.02 T c : ρ(λ 0) m? inconclusive!! 22

23 Summary LQCD answers to the fate of axial symmetry in the chiral symmetric phase of QCD signature of axial symmetry clearly visible in 2-pt correlation function at least for T c T 1.2T c dominant mechanism of axial symmetry appears to be similar to dilute instanton gas type picture for T 1.2T c more interesting scenario for no conclusive answer yet T T c? 23