1. condensates and topology fixing action Hidenori Fukaya YITP, Kyoto Univ. Collaboration with T.Onogi (YITP) hep-lat/0403024

2. 1.Introduction 1.Introduction Why topology fixing action ? Why topology fixing action ? An action proposed by Luscher provide us to simulate with An action proposed by Luscher provide us to simulate with a fixed topological charge by suppressing the field strength. a fixed topological charge by suppressing the field strength.  Better statistics of higher topological sectors.  Locality of Overlap D is improved.  Chiral symmetry is also improved.  Theta vacuum if one has a reweighting way. To test the feasibility of this action, we simulate 2-dim QED, as well as developing a reweighting method to treat θ term. An application to QCD is also studied by S.Shcheredin, B.Bietenholz, K.Jansen, K.-I.Nagai,S.Necco and L.Scorzato (hep-lat/0409073). As an example, let us focus on condensates in the massive Schwinger model with a fixed topological charge and in theta vacuum. M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)..

3. 1.Introduction 1.Introduction Overview Overview In QCD or 2-d QED (θ ＝ 0 ), In QCD or 2-d QED (θ ＝ 0 ), but in θ ≠ 0 case, we expect Then the η has a long-range correlation as

4. 1.Introduction 1.Introduction The 2-flavor massive Schwinger model The 2-flavor massive Schwinger model  Confinement  Chiral condensation  U(1) problem (m η ＞ m π )  Bosonization picture ⇒ strong coupling limit We define

5. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum ⇒ We need and.

6. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum We can calculate by generating link variables which satisfy Luscher’s bound variables which satisfy Luscher’s bound ( which are called “admissible”. ); realized by the following gauge action; ⇒ topologcal charge is conserved !!! ※ 0 <ε< π/ 5 → GW Dirac operator is local. ※ 0 <ε< π/ 5 → GW Dirac operator is local. M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999) M.Luscher,Nucl.Phys.B538,515(1999), Nucl.Phys.B549,295(1999)....

7. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum Topological charge is defined as Without Luscher’s bound Without Luscher’s bound ⇒ topological charge can jump ； Q → Q± １... If gauge fields are “admissible” ( ε ＜ 2 ), If gauge fields are “admissible” ( ε ＜ 2 ), ⇒ topological charge is conserved !! ⇒ topological charge is conserved !! In our numerical study, the topological charge is actually conserved !!! Plaquette action(β=2.0) Lusher’s action(β=0.5) (ε ＝ √ ２,Q=2.)..

8. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum Note that the continuum limit is the same as the standard plaquette action at any ε,

9. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum Using the following expression, Using the following expression, we can evaluate normalized by ! we can evaluate normalized by !

10. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum Classical solution ⇒ Constant field strength. Classical solution ⇒ Constant field strength. Moduli (constant potential which affects Polyakov loops ) integral of the determinants Moduli (constant potential which affects Polyakov loops ) integral of the determinants ⇒ Householder and QL method. ⇒ Householder and QL method. Integral of Integral of ⇒ fitting with polynomiyals. ⇒ fitting with polynomiyals. Thus we can calculate the reweighting factor R Q (=Z Q /Z 0 ).

11. 1.Introduction 1.Introduction How to evaluate θ vacuum How to evaluate θ vacuum Now we can evaluate by Luscher’s gauge action and our reweighting method... Details are shown in HF,T.Onogi,Phys.Rev.D68,074503(2003). Our goal is to study : condensates in each sector, : condensates in each sector, : η correlators in each sector, : η correlators in each sector, : θ dependence of condensates, : θ dependence of condensates, : θ dependence of η correlators, : θ dependence of η correlators, The relation between them. The relation between them.

12. 1.Introduction 1.Introduction  Action: Luscher’s action + DW fermion with PV’s.  Algorithm : The hybrid Monte Carlo method.  Gauge coupling : g = 1.0. (β=1/g 2 =1.0.)  Fermion mass : m 1 = m 2 = 0.1, 0.15, 0.2, 0.25, 0.3.  Lattice size : 16 × 16 ( × 6 ).  Admissibility condition : ε= √2  Topological charge : Q = - 4 ～ + 4  50 molecular dynamics steps with the step size Δτ= 0.035 in one trajectory.  300 configurations are generated in each sector.  Admissibility is checked at the Metropolis test... The simulations were done on the Alpha work station at YITP and SX-5 at RCNP. Numerical simulation Numerical simulation

13. 2. Fermion Condensates 2. Fermion Condensates Condensations in each sector Condensations in each sector Integrating the anomaly equation; one obtains the following relation Pseudo scalar condensation in each sector Our data are consistent with the lines (Q/mV) !! Reweighted pseudo scalar condensation ＜ ψγ ５ ψ ＞ Q Z Q /Z 0 Scalar condensation in each sector ＜ ψψ ＞ Q Reweighted scalar codensation ＜ ψψ ＞ Q Z Q /Z 0 Pseudoscalar does condense !!

14. 2. Fermion Condensations 2. Fermion Condensations Condensations in θ vacuum Condensations in θ vacuum We evaluate and Scalar condensation ＜ ψψ ＞ θ The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998) Pseudo scalar condensation ＜ ψγ ５ ψ ＞ θ The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998) m dependence of scalar condensates at θ=0

15. The η correlations in each sector The η correlations in each sector ⇒ We expect theη meson has a long- range correlation in each topological sector. We measure where 2nd part is calculated exactly. 3.The η meson in θ vacuum 3.The η meson in θ vacuum The η correlations in each sector The η correlations in each sector Connected part (pion results) Connected part (pion results) The η correlations in each sector The η correlations in each sector Connected part (pion results) Connected part (pion results) Our numerical data show a good agreement with Our numerical data show a good agreement with the continuum theory at small θ; the continuum theory at small θ; Connected part in each topological sector ＜ π(x)π(0) ＞ Q Pion mass at θ=0Pion mass at m=0.2

16. The η correlations in each sector The η correlations in each sector ⇒ We expect theη meson has a long- range correlation in each topological sector. We measure where 2nd part is calculated exactly. 3.The η meson in θ vacuum 3.The η meson in θ vacuum η meson correlator in each sector Long-range correlation in each sector lim |x|→large ＜ ψγ ５ ψ(x)ψγ ５ ψ( ０ ) ＞ Q Reweighted long-range correlation lim |x|→large ＜ ψγ ５ ψ( ｘ )ψγ ５ ψ( ０ ) ＞ Q Z Q /Z 0

17. 3.The η meson in θ vacuum 3.The η meson in θ vacuum The η correlations in θ vacuum The η correlations in θ vacuum We evaluate ＜ η † (x)η(0) ＞ θ at m=0.1 lim |x|→large ＜ η † (x)η(0) ＞ θ at m=0.1

18. 3.The η meson in θ vacuum 3.The η meson in θ vacuum Clustering decomposition Clustering decomposition Consider the operators put on t = -T/2 and t=T/2 of a large box divided into two parts; We expect the η correlation in each sector is expressed as AB ψγ 5 ψ （ -T/2)ψγ 5 ψ （ T/2)

19. Clustering decomposition Clustering decomposition  Q=0 case 3.The η meson in θ vacuum 3.The η meson in θ vacuum AB ψγ 5 ψ ＜ ψγ 5 ψ ＞ Q

20. Clustering decomposition Clustering decomposition  Q>>0 case We assume Q’ distribution is expressed as Gaussian around Q/2 ; 3.The η meson in θ vacuum 3.The η meson in θ vacuum Long range correlation Consistent with -4Q 2 /m 2 V 2 !!!

21. 4. Summary and Discussion 4. Summary and Discussion The results of 2-d QED The results of 2-d QED We obtain and It is important that vanishing of is non-trivial even in θ= 0 case and ⇒ the fluctuation of disconnected ⇒ the fluctuation of disconnected diagrams !!! diagrams !!!

22. 4. Summary and Discussion 4. Summary and Discussion How about 4-d QCD ? How about 4-d QCD ? We expect and It is important that vanishing of is non-trivial even in θ= 0 case and ⇒ the fluctuation of disconnected ⇒ the fluctuation of disconnected diagrams !!! diagrams !!!

23. How about 4-d QCD ? How about 4-d QCD ? In QCD, one obtains the same anomaly equation; From the clustering decomposition, is also expected, which is consistent with ChPT results in the ε-regime !!!! ChPT results in the ε-regime !!!! (P.H.Damgaard et al., Nucl.Phys B629(2002)445), 4. Summary and Discussion 4. Summary and Discussion Suggestion Suggestion Thus, η correlators should be treated as ; Thus, η correlators should be treated as ; and cancellation of C θ at θ=0 is quite and cancellation of C θ at θ=0 is quite non-trivial. non-trivial. ⇒ An indicator for enough topology ⇒ An indicator for enough topology changes (with standard plaquette action.). changes (with standard plaquette action.). Reweighted long-range correlation lim |x|→large ＜ ψγ ５ ψ( ｘ )ψγ ５ ψ( ０ ) ＞ Q R Q

24. Application to 4-d QCD ? Application to 4-d QCD ? Are R Q =Z Q /Z 0 ’s calculable in 4-d QCD? ⇒ Yes ( in principle. ). ⇒ Yes ( in principle. ).Difficulties Fermion determinants Fermion determinants ⇒ eigenvalue truncation ? Moduli integrals Moduli integrals ⇒ we need all the solutions on 4-d torus. VEV’s of action (ΔS Q ) VEV’s of action (ΔS Q ) ⇒ perturbative analysis ? 4. Summary and Discussion 4. Summary and Discussion It would be difficult but … We should not give up since we already know the answer (Z Q /Z 0 ) is between 0 and 1 !!