# Presentation: 736287

## 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

- Topology in unquenched simulation
- QCD vacuum
-       and Q dependence of the observables
- Topological susceptibility
- Summary

- T. Onogi

- Topology

## Members of Dynamical Overlap projectJLQCD +TWQCD+..

- KEK:        S.Hashimoto, T.Kaneko, H.Matsufuru,
-        J. Noaki, M.Okamoto, N.Yamada
- Riken:       H.Fukaya
- Tsukuba:   S.Aoki, N.Ishizuka, K.Kanaya,
-        Y.Kuramashi, Y.Taniguchi, A.Ukawa,
-                  T.Yoshie
- Hiroshima: K.Ishikawa, M.Okawa
- Kyoto:       T.O.
- Taiwan:     T-W. Chiu, K.Ogawa,...

- T. Onogi

- Topology

## 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

- T. Onogi

- Topology

## 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)

- T. Onogi

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## Our strategy:

-    appearance of low eigenmodes of
-                            Fodor, Katz, Szabo; Cundy et al. ; deGrand, Shafer
- Reflection/Refraction                          Huge numerical cost

-  Fix the topology to avoid the problem

- Low mode spectrum         of Hw

- T. Onogi

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## 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.

- T. Onogi

- Topology

## 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.

- T. Onogi

- Topology

## 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.

- T. Onogi

- Topology

## 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.

- T. Onogi

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##    Chiral Lagrangian for     vacuum 

- Dashen’s phenomena

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## 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)

- T. Onogi

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## Saddle point approximation

- Parameterize the vacuum energy as
- Then, the partition function at fixed topology is
- Changing variables as
- If                         is satisfied

- T. Onogi

- Topology

## 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.

- 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.

- T. Onogi

- Topology

## Example:  Q dependence of the meson mass 

      dependence estimated from ChPT

The correction from fixing the topology is 3%-1%
 for                                              with (2fm)^4

-       dependence estimated from ChPT
- The correction from fixing the topology is 3%-1%
-  for                                              with (2fm)^4

- T. Onogi

- Topology

## 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. 

- ChPT prediction
-           or Q dependent correction only comes  through
-           as subleading corrections.
-                          If pion mass is under control,
-                          other hadronic quantities are safe.

- T. Onogi

- Topology

## 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

- T. Onogi

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## (1) Ward-Takahashi identity
(2) Cluster Property         Q distribution

(1)&(2) 

- (2) Cluster Property         Q distribution
- (1)&(2)

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

- Intuitive proof

- T. Onogi

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

- Consider the topological charge density correlator.
- Using formula
- where
- Using the clustering property

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

Examples

- Examples

- T. Onogi

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## Schwinger model case (fixed topology simulation)

- There is indeed a nonzero constant for

- This constant gives topological susceptibility consisitent with
- direct measurement

- T. Onogi

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## 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..

- T. Onogi

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## Back up slides

- T. Onogi

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- T. Onogi

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## 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

- T. Onogi

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