1. Towards θvacuum simulation in lattice QCD Hidenori Fukaya YITP, Kyoto Univ. Collaboration with S.Hashimoto (KEK), T.Hirohashi (Kyoto Univ.), K.Ogawa(Sokendai), T.Onogi(YITP) K.Ogawa(Sokendai), T.Onogi(YITP)

2. Contents 1. Introduction 2. θ vacuum simulation of 2D QED 3. How to fix topology in 4D QCD 4. Tempering for QCD θ vacuum 5. Conclusion

3. 1.Introduction 1.Introduction Chiral symmetry and topology are very important and related each other... Chiral symmetry breaking Chiral symmetry breaking Anomaly Anomaly Construction of chiral gauge theory Construction of chiral gauge theory 4D N=1 SUSY (Chiral super fields) 4D N=1 SUSY (Chiral super fields) Our aim is to keep chiral symmetry and Our aim is to keep chiral symmetry and topological properties in 4D lattice QCD. topological properties in 4D lattice QCD. ⇒ θ vacuum simulation ⇒ θ vacuum simulation

4. 1.Introduction 1.Introduction Ginsparg-Wilson relation Ginsparg-Wilson relation ⇒ exact chiral symmetry at classical level. Luscher’s “admissibility” condition Luscher’s “admissibility” condition ⇒ locality of Dirac operator ⇒ Stability of the index (topological charge) Phys.Rev.D25,2649(‘82) Phys.Lett.B428,342(‘98),Nucl.Phys.B549,295(‘99)

5. 1.Introduction 1.Introduction Our goal is to calculate path integrals with θterm; ⇒ We need and.

6. The 2-flavor massive Schwinger model The 2-flavor massive Schwinger model We have a geometrical definition of the topological charge; We have a geometrical definition of the topological charge; 2. θvacuum simulation of 2D QED 2. θvacuum simulation of 2D QED 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 !! The “admissibility” condition is realized by a gauge action proposed by Luscher; Note : the effect of ε is only O(a 4 ). In our numerical study ( using HMC ), the topological charge is actually conserved !!! Plaquette action(β=2.0) Luscher’s action(β=0.5) (ε ＝ √ ２,Q=2.)

7. We also developed a method to calculate 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 ). 2. θvacuum simulation of 2D QED 2. θvacuum simulation of 2D QED

8. Thus, we could evaluate by Luscher’s gauge action and our reweighting method and the results were consistent with the continuum theory... Details are shown in HF,T.Onogi,Phys.Rev.D68,074503(2003).

9. For example, we calculated the pseudoscalar condensates, which can be obtained from the anomaly equation, Pseudo scalar condensates in each sector Our data (using DWF.) are consistent with Q/mV !! 2. θvacuum simulation of 2D QED 2. θvacuum simulation of 2D QED

10. We evaluate the total expectation value in θvacuum; θvacuum; Pseudo scalar condensates ＜ iψγ ５ ψ ＞ θ The dashed line:Y.Hosotani and R.Rodoriguez,J.Phys.A31,9925(1998) 2. θvacuum simulation of 2D QED 2. θvacuum simulation of 2D QED does condense !! does condense !! Details are in HF,T.Onogi, Phys.Rev.D70,054508(2004).

11. Also in 4D QCD, we expect that can be obtained with the action; but … No geometrical (and practical ) definition of Q. No geometrical (and practical ) definition of Q. The index of D ⇒ a lot of computational costs. The index of D ⇒ a lot of computational costs. 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD

12. (New) Cooling method (New) Cooling method We “cool” the configuration smoothly by performing the hybrid Monre Carlo steps with decreasing g 2 (admissibility is always satisfied.). ⇒ We obtain a “cooled ” configuration close to the classical background at close to the classical background at very weak g 2 ～ 0.0001 then, very weak g 2 ～ 0.0001 then, gives an integer with 10% accuracy. gives an integer with 10% accuracy. 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD

13. How can ε be determined ? How can ε be determined ? The stability of Q is proved only when ε ＜ ε max ～ 1/20.49. ⇒ configuration space is too narrow on 10 4 ～ 20 4 lattice. on 10 4 ～ 20 4 lattice. 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD S G ε< 1/20.49 Q=0 ε=∞ Q=1 S G ε= 1.0 Q=0 Q=1 If the barrier is high enough, Q may be fixed. Let us search the parameters; β, ε, V, Δτ … which can fix Q. Anothor group is also studying this action. S.Shcheredin, W.Bietenholz, K.Jansen, K.I.Nagai, S.Necco and L.Scorzato, hep-lat/0409073

14.  Action: Luscher action (quenched)  Algorithm : The hybrid Monte Carlo method.  Gauge coupling : β= (6/g 2 ) = 1.0 ～ 2.8  Lattice size : 10 4,14 4,16 4  Admissibility condition : 1/ε= 2/3 ～ 1.0  Topological charge : Q = - ３ ～ + ３  10 ～ 40 molecular dynamics steps with the step size Δτ= 0.001-0.02 in one trajectory. size Δτ= 0.001-0.02 in one trajectory... The simulations were done on the Alpha work station at YITP and SX-5 at RCNP. Numerical simulations Numerical simulations 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD

15. Initial configuration Initial configuration For topologically non-trivial initial configuration, we use a discretized version of instanton solution on 4D torus; which gives constant field strength with which gives constant field strength with topological charge Q. topological charge Q. 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD A.Gonzalez-Arroyo,hep-th/9807108, M.Hamanaka,H.Kajiura,Phys.Lett.B551,360(‘03)

16. Results Results 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD β=1.25, 1/ε=1.0β=1.5, 1/ε=1.0 1000 trajectory length =100 uncorrelated confs β=2.4, 1/ε=2.0/3.0β=2.6, 1/ε=2.0/3.0 ×Δτ=0.001 V = 10 4 V = 10 4 β= 6.0 1/ε=0.0 ○Δτ=0.003 V = 10 4 V = 10 4 β= 2.7 ○Δτ=0.003 V = 10 4 V = 10 4 β= 2.6 ×Δτ=0.008 V = 10 4 V = 10 4 β= 2.6 ×Δτ=0.003 V = 10 4 V = 10 4 β= 2.5 ×Δτ=0.003 V = 10 4 V = 10 4 β= 2.4 1/ε=2/3 ○Δτ=0.004 V = 14 4 V = 14 4 β= 1.5 ○Δτ=0.004 V = 10 4 V = 10 4 β= 1.55 ○Δτ=0.004 V = 10 4 V = 10 4 β= 1.5 ○Δτ=0.004 V = 10 4 V = 10 4 β= 1.4 ×Δτ=0.001 V = 10 4 V = 10 4 β= 1.25 ×Δτ=0.001 V = 10 4 V = 10 4 β= 1.0 1/ε=1.0 Topology fixing for 2000 trajectory length

17. Physical scale of the lattice Physical scale of the lattice 3. How to fix topology in 4D QCD 3. How to fix topology in 4D QCD Wilson loops Data of plaquette action are from H.Matsufuru’s web page. Both 1/ε=1.0, β ≧ 1.4 and 1/ε=2/3, β ≧ 2.6 are corresponding β ≧ 6.0 （ 1/a ～ 2.0 GeV ） for plaquette action. ⇒ Q can be fixed on lattices finer than (2.0GeV) -1.

18. Our method to calculate the partition function used in 2D QED; is not applicable to 4D QCD due to its large numerical costs. 4. Tempering for QCD θvacuum 4. Tempering for QCD θvacuum

19. Tempering method Tempering method Let us extend the configuration space; 4. Tempering for QCD θvacuum 4. Tempering for QCD θvacuum Extended configuration space Luscher action plaquette action ： Q = 0 Q = 1 Q = -1 Q is not conserved. If we obtain ↓ ↓↓N0,N0,↓↓N0,N0,↓ ↓↓N1,N1,↓↓N1,N1,↓ N -1, ・・・ samples,

20. There are several tempering methods... There are several tempering methods... Simulated tempering G.Parisi et al,Europhys.Lett.19,451(‘92) Simulated tempering G.Parisi et al,Europhys.Lett.19,451(‘92) Parallel tempering E.Marianari et al,cond-mat/9612010, Parallel tempering E.Marianari et al,cond-mat/9612010, K.Hukushima et al, cond-mat/9512035 K.Hukushima et al, cond-mat/9512035 … If the tempering is applicable for Luscher action, we may able to treat θvacuum, 4. Tempering for QCD θvacuum 4. Tempering for QCD θvacuum

21. We find at V=10 4, 14 4, We find at V=10 4, 14 4, Well- controlled topological charge can be measured Well- controlled topological charge can be measured by our cooling method. by our cooling method. Q can be fixed for more than 2000 trajectory Q can be fixed for more than 2000 trajectory length (200 uncorrelated samples.) if the lattice is fine enough (1/a ～ 2.0GeV) even when we set fine enough (1/a ～ 2.0GeV) even when we set 1/ε=2/3. Next we will try Next we will try more quantitative studies more quantitative studies large volume large volume The index of GW fermion The index of GW fermion tempering for θvacuum tempering for θvacuum full QCD full QCD 5. Conclusion 5. Conclusion

22. Luscher’s ‘admissibility’ condition will be Luscher’s ‘admissibility’ condition will be important for … Locality of Dirac operators Locality of Dirac operators ε-regime ε-regime Finite temperatures and Chiral transition Finite temperatures and Chiral transition SUSY SUSY Majorana fermions? Majorana fermions? Non commutative lattice ?? Non commutative lattice ?? Matrix model ??? Matrix model ??? … 6. Outlook 6. Outlook