1. Axial symmetry at finite temperature Guido Cossu High Energy Accelerator Research Organization – KEK Lattice Field Theory on multi-PFLOPS computers German-Japanese Seminar 2013 8 Nov 2013, Regensburg, Germany 高エネルギー加速器研究機構

2. U A (1) symmetry at finite temperature ✔ Introduction – chiral symmetry at finite temperature ✔ Simulations with dynamical overlap fermions ✔ Fixing topology ✔ Results: ✔ (test case) Pure gauge ✔ N f =2 case ✔ Other studies on the subject ✔ Conclusions, criticism and future work People involved in the collaboration: JLQCD group: H. Fukaya, S. Hashimoto, S. Aoki, T. Kaneko, H. Matsufuru, J. Noaki See Phys.Rev. D87 (2013) 114514 and previous Lattice proceedings (10-11-12),

3. U A (1) symmetry at finite temperature Pattern of chiral symmetry breaking at low temperature QCD Symmetry of the Lagrangian Symmetries in the (pseudo)real world (N f =3) at zero temperature ● U(1) V the baryon number conservation ● SU(3) V intact (softly broken by quark masses) – 8 Goldstone bosons (GB) ● SU(3) A is broken spontaneously by the non zero e.v. of the quark condensate ● No opposite parity GB, U(1) A is broken, but no 9th GB is found in nature. Axial symmetry is not a symmetry of the quantum theory ('t Hooft - instantons) Witten-Veneziano : mass splitting of the  '(958) from topological charge at large N. topological charge density Finite temperature T > T c

4. U A (1) symmetry at finite temperature At finite temperature, in the chiral limit m q → 0, chiral symmetry is restored ● Phase transition at N f =2 (order depending on U(1) A see e.g. Vicari, Pelissetto) ● Crossover with 2+1 flavors What is the fate of the axial U(1) A symmetry at finite temperature T ≳ T c ? Complete restoration is not possible since it is an anomaly effect. Exact restoration is expected only at infinite T (see instanton-gas models) At most we can observe strong suppression (effective restoration) On the lattice the overlap Dirac operator is the best way to answer this questions since it preserves the maximal amount of chiral symmetry. An operator satisfying the Ginsparg-Wilson relation has several nice properties e.g. ● exact relation between zero eigenmodes (EM) and topological charge ● Negligible additive renormalization of the mass (residual mass) ●...

5. U A (1) symmetry at finite temperature Check the effective restoration of axial U(1) A symmetry by measuring (spatial) meson correlators at finite temperature in full QCD with the Overlap operator Degeneracy of the correlators is the signal that we are looking for (NB: 2 flavors) First of all there are some issues to solve before dealing with the real problem... Dirac operator eigenvalue density is also a relevant observable for chiral symmetry

6. U A (1) symmetry at finite temperature The sign function in the overlap operator gives a delta in the force when HW modes cross the boundary (i.e. topology changes), making very hard for HMC algorithms to change the topological sector In order to avoid expensive methods (e.g. reflection/refraction) to handle the zero modes of the Hermitian Wilson operator JLQCD simulations used (JLQCD 2006): ● Iwasaki action (suppresses Wilson operator near zero modes) ● Extra Wilson fermions and twisted mass ghosts to rule out the zero modes Topology is thus fixed throughout the HMC trajectory. The effect of fixing topology is expected to be a Finite Size Effect (actually O(1/V) ), next slides

7. U A (1) symmetry at finite temperature Partition function at fixed topology Using saddle point expansion around one obtains the Gaussian distribution where the ground state energy can be expanded (T=0)

8. U A (1) symmetry at finite temperature From the previous partition function we can extract the relation between correlators at fixed θ and correlators at fixed Q In particular for the topological susceptibility and using the Axial Ward Identity we obtain a relation involving fermionic quantities: P(x) is the flavor singlet pseudo scalar density operator, see Aoki et al. PRD76,054508 (2007) What is the effect of fixing Q at finite temperature?

9. U A (1) symmetry at finite temperature ✔ Simulation details ✔ Finite temperature quenched SU(3) at fixed topology (proof of concept) ✔ Eigenvalues density distribution ✔ Topological susceptibility ✔ Finite temperature two flavors QCD at fixed topology ✔ Eigenvalues density distribution ✔ Meson correlators BG/L Hitachi SR16K

10. U A (1) symmetry at finite temperature Pure gauge (16 3 x6, 24 3 x6): Iwasaki action + topology fixing term Two flavors QCD (16 3 x8) Iwasaki + Overlap + topology fixing term O(300) trajectories per T, am=0.05, 0.025, 0.01 Pion mass: ~290 MeV @ am=0.015, β =2.30 T c was conventionally fixed to 180, not relevant for the results (supported by Borsanyi et al. results) βa (fm)T (Mev)T/Tc 2.350.132249.10.86 2.400.123268.10.93 2.430.117280.90.97 2.440.115285.70.992 2.4450.1142881.0 2.450.1133290.21.01 2.460.111295.11.02 2.480.107305.61.06 2.500.104316.21.10 2.550.094347.61.20 βa (fm)T (Mev)T/Tc 2.180.1438171.50.95 2.200.1391177.30.985 2.250.12818192.21.06 2.300.1183208.51.15 2.400.1013243.51.35 2.450.0940262.41.45

11. U A (1) symmetry at finite temperature Phase transition

12. U A (1) symmetry at finite temperature Extracting the topological susceptibility: (Spatial) Correlators are always approximated by the first 50 eigenvalues Pure gauge: double pole formula for disconnected diagram Q=0, assume c 4 term is negligible, then check consistency Topological susceptibility estimated by a joint fit of connected and disconnected contribution to maximize info from data Cross check without fixing topology

13. U A (1) symmetry at finite temperature

14. Effect of axial symmetry on the Dirac spectrum If axial symmetry is restored we can obtain constraints on the spectral density Ref: S. Aoki, H. Fukaya, Y. Taniguchi Phys.Rev. D86 (2012) 114512

15. U A (1) symmetry at finite temperature Test (β=2.20, am=0.01): decreasing Zolotarev poles in the approx. of the sign function At N poles = 5 more near zero modes appeared (16 is the number used in the rest of the calculation)

16. U A (1) symmetry at finite temperature Bazavov et al. 2011 Updated 2013 (HotQCD) DWF Low modes Vranas 2000 DWF No restoration Ohno et al. 2011 Updated 2013 HISQ (2+1)

17. U A (1) symmetry at finite temperature Mass Temperature

18. U A (1) symmetry at finite temperature Mass Temperature

19. U A (1) symmetry at finite temperature Mass Temperature

20. U A (1) symmetry at finite temperature Mass Temperature

21. U A (1) symmetry at finite temperature Mass Temperature

22. U A (1) symmetry at finite temperature Mass Temperature

23. U A (1) symmetry at finite temperature Mass Temperature

24. U A (1) symmetry at finite temperature Mass Temperature

25. U A (1) symmetry at finite temperature Mass Temperature

26. U A (1) symmetry at finite temperature Mass Temperature

27. U A (1) symmetry at finite temperature Mass Temperature

28. U A (1) symmetry at finite temperature Disconnected contibution at β=2.30 and several distances Falls faster than exponential

29. U A (1) symmetry at finite temperature Full QCD spectrum shows a gap at high temperature even at pion masses ~250 MeV Statistics: high at T>200 MeV Correlators show degeneracy of all channels when mass is decreased Results support effective restoration of U(1) A symmetry With overlap fermions we have a clear theoretical setup for the analysis of spectral density and control on chirality violation terms. Realistic simulations are possible, but topology must be fixed A check of systematics due to topology fixing at finite temperature is necessary (finite volume corrections are expected) Pure gauge test results show that we can control these errors as in the previous T=0 case. Finite volume effects are small in the SU(3) pure gauge case Criticism, systematics (the usual suspects): Finite volume effects (beside topology fixing): only one volume Lowest mass still quite large (HotQCD addresses this) No continuum limit Future work (ongoing): New action, DWF with scaled Shamir kernel: very low residual mass ~0.5, no topology fix Two volumes(at least), lower masses Systematic check for dependence of results on the sign function approximation Criticism, systematics (the usual suspects): Finite volume effects (beside topology fixing): only one volume Lowest mass still quite large (HotQCD addresses this) No continuum limit Future work (ongoing): New action, DWF with scaled Shamir kernel: very low residual mass ~0.5, no topology fix Two volumes(at least), lower masses Systematic check for dependence of results on the sign function approximation

30. U A (1) symmetry at finite temperature LuxRay Artistic Rendering of Lowest Eigenmode

31. U A (1) symmetry at finite temperature Backup slides

32. U A (1) symmetry at finite temperature Lowest Eigenmode