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Axial symmetry at finite temperature Accelerator Research Organization – KEK Lattice Field Theory on multi-PFLOPS computers German-Japanese Seminar 2013 8 Nov 2013, Regensburg, Germany 高高高高高高高高高高高高高

Axial symmetry at finite temperature

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

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Page 1: Axial  symmetry  at finite temperature

Axial symmetry at finite temperature

Guido CossuHigh Energy Accelerator Research Organization – KEK

Lattice Field Theory on multi-PFLOPS computersGerman-Japanese Seminar 2013

8 Nov 2013, Regensburg, Germany

高エネルギー加速器研究機構

Page 2: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

✔Introduction – chiral symmetry at finite temperature✔Simulations with dynamical overlap fermions✔Fixing topology✔Results:

✔(test case) Pure gauge✔Nf=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),

Summary

Page 3: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Pattern of chiral symmetry breaking at low temperature QCD

Symmetry of the Lagrangian

Symmetries in the (pseudo)real world (Nf=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)

@¹j¹5(x)= 2iN fq(x)

Witten-Veneziano: mass splitting of the '(958) from topological charge at large N.

Introduction – chiral symmetry

topological charge density

Finite temperature T > Tc

Page 4: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

At finite temperature, in the chiral limit mq → 0, chiral symmetry is restored●Phase transition at Nf=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 symmetryat finite temperature T ≳ Tc ?

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)●...

Introduction – finite temperature

Page 5: Axial  symmetry  at finite temperature

UA(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

U (1)A U (1)A

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

Project intent

Page 6: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

The sign function in the overlap operator givesa 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

Simulations with overlap fermions

Page 7: Axial  symmetry  at finite temperature

UA(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)

Fixing topology: zero temperature

Page 8: Axial  symmetry  at finite temperature

UA(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?

Fixing topology

Page 9: Axial  symmetry  at finite temperature

UA(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

Results

Page 10: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Pure gauge (163x6, 243x6):Iwasaki action + topology fixing term

Two flavors QCD (163x8)Iwasaki + Overlap + topology fixing termO(300) trajectories per T, am=0.05, 0.025, 0.01

Pion mass: ~290 MeV @ am=0.015 , β =2.30Tc was conventionally fixed to 180, not relevant for the results

(supported by Borsanyi et al. results)

Simulation details

β a (fm) T (Mev)

T/Tc

2.35 0.132 249.1 0.862.40 0.123 268.1 0.932.43 0.117 280.9 0.972.44 0.115 285.7 0.9922.445 0.114 288 1.02.45 0.1133 290.2 1.012.46 0.111 295.1 1.022.48 0.107 305.6 1.062.50 0.104 316.2 1.102.55 0.094 347.6 1.20

β a (fm) T (Mev)

T/Tc

2.18 0.1438 171.5 0.952.20 0.1391 177.3 0.9852.25 0.1281

8192.2 1.06

2.30 0.1183 208.5 1.152.40 0.1013 243.5 1.352.45 0.0940 262.4 1.45

Page 11: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Phase transition

Overlap eigenvalues distribution, SU(3) pure gauge

Page 12: Axial  symmetry  at finite temperature

UA(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 c4 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

Topological susceptibility

Page 13: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Topological susceptibility, result

Page 14: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Effect of axial symmetry on the Dirac spectrum

If axial symmetry is restored we can obtain constraints on the spectral density

lim̧! 0limm ! 0½m (̧ )¸2 = 0

Ref: S. Aoki, H. Fukaya, Y. TaniguchiPhys.Rev. D86 (2012) 114512

Eigenvalues distribution – QCD Nf=2

Page 15: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Test (β=2.20, am=0.01):decreasing Zolotarevpoles in the approx.of the sign function

At Npoles = 5 morenear zero modes appeared(16 is the number used in the rest of the calculation)

Eigenvalues distribution – Approximation dependence

Page 16: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Bazavov et al. 2011Updated 2013(HotQCD) DWFLow modes

Vranas 2000 DWFNo restoration

Ohno et al. 2011Updated 2013HISQ (2+1)

Other analyses

Page 17: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Mass

Temperature

Meson correlators

Page 18: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 19: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 20: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 21: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 22: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 23: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 24: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 25: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 26: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 27: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Meson correlators

Mass

Temperature

Page 28: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Disconnected diagrams – mass dependence

Disconnectedcontibution at β=2.30 and several distances

Falls faster than exponential

Page 29: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Full QCD spectrum shows a gap at high temperature even at pion masses ~250 MeVStatistics: high at T>200 MeVCorrelators show degeneracy of all channels when mass is decreasedResults 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

Conclusions, criticism, future works

Criticism, systematics (the usual suspects): Finite volume effects (beside topology fixing): only one

volume Lowest mass still quite large (HotQCD addresses this) No continuum limitFuture work (ongoing):New action, DWF with scaled Shamir kernel: very low residual mass ~0.5, no topology fixTwo volumes(at least), lower massesSystematic check for dependence of results on the sign function approximation

Page 30: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Thanks for your attention

LuxRay Artistic Rendering of Lowest Eigenmode

Page 31: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Backup slides

Page 32: Axial  symmetry  at finite temperature

UA(1) symmetry at finite temperature

Lowe

st E

igen

mod

ePure gauge theory