Topology T. Onogi 1

Should we change the topology at all?

Tetsuya Onogi (YITP, Kyoto Univ.) for JLQCD collaboration

RBRC Workshp: 　“ Domain Wall Fermions at Ten Years”March 16 2007 at BNL

1. Topology in unquenched simulation2. QCD vacuum3. and Q dependence of the observables4. Topological susceptibility5. Summary

Topology T. Onogi 2

Members of Dynamical Overlap projectJLQCD +TWQCD+..

KEK: S.Hashimoto, T.Kaneko, H.Matsufuru, 　　　　　　　 J. Noaki, M.Okamoto, N.YamadaRiken: H.FukayaTsukuba: S.Aoki, N.Ishizuka, K.Kanaya,　　　　　　　 Y.Kuramashi, Y.Taniguchi, A.Ukawa, 　 T.YoshieHiroshima: K.Ishikawa, M.OkawaKyoto: T.O.Taiwan: T-W. Chiu, K.Ogawa,…

Topology T. Onogi 3

The aim of this talk

• Topology change in unquenched QCD is a serious problem. One should carefully think which strategy should be taken: Enforce the topology change ? or fix the topology?

• Review the theoretical understanding of QCD in vacuum and QCD at fixed topology.

• Claim that the fixed Q effect is a finite size effect, which can be removed in large volume or correctly estimated.

• Give a proposal to measure topological susceptibility at fixed topology.

　　　　 Talk by T-W. Chiu

Topology T. Onogi 4

1. Topology in unquenched simulation

Topological charge evolution in HMC becomes slower towards weaker coupling, smaller sea quark mass, and better chirality

Staggered: Bernard et al 2003 Domain wall: RBC, Antonio et al. 2006

Evolution of the topology in 2+1 dynamical domain-wall.

Figiure from RBC hep-lat/0612005

Better chirality: Iwasaki < DBW2 < C (Plaq+Rect)

Topology change: Iwasaki > DBW2 > C (Plaq+Rec)

Topology T. Onogi 5

Additional problem in dynamical overlap fermion: appearance of low eigenmodes of Fodor, Katz, Szabo; Cundy et al. ; deGrand, Shafer

Reflection/Refraction Huge numerical cost

Our strategy:

Fix the topology to avoid the problemLow mode spectrum of Low mode spectrum of

HwHw

Topology T. Onogi 6

How to extract physics from fixed topologies?

• Intuitively, fixing the topology should not affect physics for large enough volume for .

But, finite volume effects should be estimated. If large instanton contributes the vacuum, the finite size effect may be large.

• Is local fluctuation of the topology sufficiently active without the topology change through dislocation?

Measuring topological susceptibility is imporant.

Topology T. Onogi 7

2. 　 QCD vacuum

Witten’s picture

• Solution to U(1) problem leads to the picture that large Nc expansion gives a good approximation to theta dependence of QCD vacuum. E. Witten Nucl.Phys.B156(1979)269

• Instanton (= classical (anti)self-dual configuration) contribtuion is NOT a major component. Local fluctuation of density is the dominant contribution the vacuum. E. Witten Nucl.Phys. B149(1979)

Good news suggesting small finite size effect.

Topology T. Onogi 8

Is there a numerical test of this picture ?• Yes, studies of local chirality of the low mode of Dov. But still controvertial. DeGrand, Hasenfratz : instanton contribution Horvath et al. : no major instanton contribution

• However, it seems that local topological charge density fluctuation dominates.

Correlation length of topological charge is small.

Finite size effect can be small. The vacuum can consist of huge number of relatively independent topological lumps of positive and negative charge.

Topology T. Onogi 9

Local chirality of low modes of Dov for Nf=2

JLQCD nf=2 results for local chirality with

The distribution has peaks

at maximum positive and

negative chirality.

There are local topological

lumps even for Q=0.

Topology T. Onogi 10

Chiral Lagrangian for vacuum

Dashen’s phenomena

Topology T. Onogi 11

3. and Q dependence of the observables

Brower, Chandrasekaran, Negele, Wiese, Phys.Lett.B560(2003)64 +discussions with S.Aoki, H. Fukaya and S. Hashimoto

• : partition function in vacuum

: partition function at fixed Q

• : observable in vacuum

: observable at fixed Q

The partition function and observable at fixed Q

can be obtained from those in vacuum

using saddle point approximation for large V (volume)

Topology T. Onogi 12

Saddle point approximation

Parameterize the vacuum energy as

Then, the partition function at fixed topology is

Changing variables as

If is satisfied

Topology T. Onogi 13

Parameterizing the vacuum energy as

one obtains

“(n)” means

n-th derivative in

• Difference of observables with fixed Q and in vacuum can be estimated as 1/V correction and higher order.

• Topological susceptibility as well as higher moments are the key quantities.

• One can also obtain the dependence of CP-odd observable.

EDM can be obtained.

Topology T. Onogi 14

Example: Q dependence of the meson mass

dependence estimated from ChPT

The correction from fixing the topology is 3%-1%

for with (2fm)^4

Topology T. Onogi 15

Other hadrons (nucleon, )

ChPT prediction

or Q dependent correction only comes through

as subleading corrections.

If pion mass is under control,

other hadronic quantities are safe.

Topology T. Onogi 16

4. Topological susceptibility• Measure the topological susceptibility

– check thermal equilibrium in topology– Useful for estimate the finite size effects

Definitions – Giusti, Rossi, Testa Phys.Lett.B587(2004)157

disconnected loop

– Luescher, Phys.Lett.B(2004)296

n-point function without div.

– Asymptotic value Fukaya, T.O. Phys.Rev.D70(2004)054508

Topology T. Onogi 17

(1) Ward-Takahashi identity

(2) Cluster Property Q distribution

(1)&(2)

Topological suscpetibilitycan be measured indirectly from asymptopic values of Pseudoscalar 2-pt ftn

Intuitive proof

Topology T. Onogi 18

More sytematic proof Aoki, Fukaya, Hashimoto, Onogi in progress

Consider the topological charge density correlator.

Using formula

where

Using the clustering property

Topology T. Onogi 19

One can use arbitrary function to define the topological charge density up to total divergence.

Examples

Topology T. Onogi 20

Schwinger model case (fixed topology simulation)

There is indeed a nonzero constant for

This constant gives topological susceptibility consisitent with direct measurement

Topology T. Onogi 21

5. Summary• The effect of fixing the topology is a finite size effect, which can be removed in large volume or correctly estimated by the topological susceptibility and suitable effective theory.

• Fortunately, the pion mass receives the largest correction but other quantities receives only subleading corrections through pion mass.

• The theta dependence of CP-odd observable can also be extracted from fixed topology simulation.

• Topological susceptibility can be measured by the asymptotic values of single pseudoscalar 2pt function at fixed topology.

Talk by T-W. Chiu• Systematic study of next-leading order (partially quenched ) ChPT is needed..

Topology T. Onogi 22

Back up slides

Topology T. Onogi 23

Topology T. Onogi 24

Low mode and topology change

• Zeros of Hw(m) arise when the topology changes through localized modes.

Edwards, Heller, Narayanan Nucl.Phys.B535(1998)403.

Spectral flow of Hw Localization size of the crossing mode