T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.1/34. The Topology in QCD. Ting-Wai Chiu Physics Department, National Taiwan University

Size: px

Start display at page:

Download "T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.1/34. The Topology in QCD. Ting-Wai Chiu Physics Department, National Taiwan University"

Neal Haynes
5 years ago
Views:

1 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.1/34 The Topology in QCD Ting-Wai Chiu Physics Department, National Taiwan University

2 The vacuum of QCD has a non-trivial topological structure. T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.2/34

3 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.2/34 The vacuum of QCD has a non-trivial topological structure. The cluster property and the gauge invariance require that the ground state must be the θ vacuum, a superposition of gauge configurations in different topological sectors.

4 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.2/34 The vacuum of QCD has a non-trivial topological structure. The cluster property and the gauge invariance require that the ground state must be the θ vacuum, a superposition of gauge configurations in different topological sectors. The topological structure is also essential for the resolution of the U(1) problem, i.e., the flavor singlet pseudo-scalar meson knows the presence of non-trivial topological charge in the QCD vacuum.

5 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.2/34 The vacuum of QCD has a non-trivial topological structure. The cluster property and the gauge invariance require that the ground state must be the θ vacuum, a superposition of gauge configurations in different topological sectors. The topological structure is also essential for the resolution of the U(1) problem, i.e., the flavor singlet pseudo-scalar meson knows the presence of non-trivial topological charge in the QCD vacuum. Therefore, the topological excitations, such as the instantons, plays a central role in understanding the vacuum of QCD.

6 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.2/34 The vacuum of QCD has a non-trivial topological structure. The cluster property and the gauge invariance require that the ground state must be the θ vacuum, a superposition of gauge configurations in different topological sectors. The topological structure is also essential for the resolution of the U(1) problem, i.e., the flavor singlet pseudo-scalar meson knows the presence of non-trivial topological charge in the QCD vacuum. Therefore, the topological excitations, such as the instantons, plays a central role in understanding the vacuum of QCD. Since the topological excitations do not occur in the perturbation theory, theoretical calculations starting from the QCD Lagrangian necessarily involves non-perturbative methods, such as lattice QCD.

7 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.3/34 The main difficulties in lattice QCD are: (i) Definition of the topological charge density using the gauge links causes bad ultraviolet divergences. (the cooling methods devised to tame the short distance fluctuations introduce sizable systematic uncertainties.)

8 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.3/34 The main difficulties in lattice QCD are: (i) Definition of the topological charge density using the gauge links causes bad ultraviolet divergences. (the cooling methods devised to tame the short distance fluctuations introduce sizable systematic uncertainties.) (ii) Unquenched simulations with Wilson/staggered fermion do not respect correct chiral or flavor symmetry at finite lattice spacing, and the definition of the topological charge through the Atiyah-Singer index theorem is ambiguous.

9 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.3/34 The main difficulties in lattice QCD are: (i) Definition of the topological charge density using the gauge links causes bad ultraviolet divergences. (the cooling methods devised to tame the short distance fluctuations introduce sizable systematic uncertainties.) (ii) Unquenched simulations with Wilson/staggered fermion do not respect correct chiral or flavor symmetry at finite lattice spacing, and the definition of the topological charge through the Atiyah-Singer index theorem is ambiguous. (iii) With the HMC algorithm which is based on a continuous evolution of the gauge links, the system is trapped in a fixed topological sector as the continuum limit is approached. Therefore, a proper sampling of different topological sectors cannot be achieved. (Approaching the chiral limit, the suppression of the fermion determinant for Q 0 also makes the tunneling a rare event.)

10 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.4/34 During the last decade, (i) and (ii) have been solved by the realization of exact chiral symmetry on the lattice, with which the topological charge is uniquely defined at any finite lattice spacing by counting the number of fermionic zero-modes.

11 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.4/34 During the last decade, (i) and (ii) have been solved by the realization of exact chiral symmetry on the lattice, with which the topological charge is uniquely defined at any finite lattice spacing by counting the number of fermionic zero-modes. However, (iii) remains insurmountable, since the correct sampling of topology becomes increasingly more difficult towards realistic simulation with lighter quarks and finer lattices.

12 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.4/34 During the last decade, (i) and (ii) have been solved by the realization of exact chiral symmetry on the lattice, with which the topological charge is uniquely defined at any finite lattice spacing by counting the number of fermionic zero-modes. However, (iii) remains insurmountable, since the correct sampling of topology becomes increasingly more difficult towards realistic simulation with lighter quarks and finer lattices. A plausible solution is to perform QCD simulations in a fixed topological sector and to extract χ t from local topological fluctuations. Then any observable measured at a fixed topological charge can be transcribed to its value in the θ vacuum.

13 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.5/34 Topology Susceptibility in 2-flavor QCD with Fixed Topology (JLQCD-TWQCD, arxiv: ) The topological charge density is unambiguously defined on the lattice using the overlap-dirac operator which possesses exact chiral symmetry.

14 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.5/34 Topology Susceptibility in 2-flavor QCD with Fixed Topology (JLQCD-TWQCD, arxiv: ) The topological charge density is unambiguously defined on the lattice using the overlap-dirac operator which possesses exact chiral symmetry. Simulations are performed on a lattice at lattice spacing 0.12 fm at six sea quark masses m q ranging in m s /6 m s.

15 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.5/34 Topology Susceptibility in 2-flavor QCD with Fixed Topology (JLQCD-TWQCD, arxiv: ) The topological charge density is unambiguously defined on the lattice using the overlap-dirac operator which possesses exact chiral symmetry. Simulations are performed on a lattice at lattice spacing 0.12 fm at six sea quark masses m q ranging in m s /6 m s. The χ t is extracted from the constant behavior of the time-correlation of flavor-singlet pseudo-scalar meson two-point function at large distances, which arises from the finite size effect due to the fixed topology.

16 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.5/34 Topology Susceptibility in 2-flavor QCD with Fixed Topology (JLQCD-TWQCD, arxiv: ) The topological charge density is unambiguously defined on the lattice using the overlap-dirac operator which possesses exact chiral symmetry. Simulations are performed on a lattice at lattice spacing 0.12 fm at six sea quark masses m q ranging in m s /6 m s. The χ t is extracted from the constant behavior of the time-correlation of flavor-singlet pseudo-scalar meson two-point function at large distances, which arises from the finite size effect due to the fixed topology. In the small m q regime, our result of χ t is proportional to m q as expected from chiral effective theory. Using the formula χ t = m q Σ/N f by Leutwyler-Smilga, we obtain Σ MS (2 GeV) = [252(5)(10)MeV] 3

17 Outline T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.6/34 Introduction Topology with Overlap Dirac Operator Lattice Setup Results using N f = 2 Dynamical Overlap Configurations with Q t = 0, 2, 4 Conclusion and Outlook

18 Introduction T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.7/34 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0) where ρ(x) = 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)]

19 Introduction T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.7/34 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0) where ρ(x) = 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)] Veneziano-Witten relation χ t (quenched) = f2 πm 2 η 4N f

20 Introduction T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.7/34 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0) where ρ(x) = 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)] Veneziano-Witten relation Leutwyler-Smilga relation χ t (quenched) = f2 πm 2 η 4N f χ t = m qσ N f + O(m 2 q) (in the chiral limit)

21 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.8/34 For lattice QCD with fixed topology in a finite volume, χ t is the most crucial quantity which is used to relate any observable measured in the fixed topology to its physical value. Brower, Chandrasekaran, Negele, Wiese, PLB 560 (2003) 64 Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007)

22 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.8/34 For lattice QCD with fixed topology in a finite volume, χ t is the most crucial quantity which is used to relate any observable measured in the fixed topology to its physical value. Brower, Chandrasekaran, Negele, Wiese, PLB 560 (2003) 64 Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007) In other words, the artifacts due to fixed topology can be removed, provided that χ t has been determined.

23 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.9/34 Since χ t = d 4 x ρ(x)ρ(0) = 1 Ω Q 2 t, Ω = volume where Q t = d 4 x 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)] = integer one can obtain χ t by counting the number of gauge configurations for each topological sector.

24 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.9/34 Since χ t = d 4 x ρ(x)ρ(0) = 1 Ω Q 2 t, Ω = volume where Q t = d 4 x 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)] = integer one can obtain χ t by counting the number of gauge configurations for each topological sector. However, for a set of gauge configurations in the topologically-trivial sector, Q t = 0, it gives χ t = 0

25 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.10/34 Even for a topologically-trivial gauge configuration, it may possess near-zero modes due to excitation of instanton and anti-instanton pairs, which are the origin of spontaneous chiral symmetry breaking in the infinite volume limit.

26 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.10/34 Even for a topologically-trivial gauge configuration, it may possess near-zero modes due to excitation of instanton and anti-instanton pairs, which are the origin of spontaneous chiral symmetry breaking in the infinite volume limit. Thus, one can investigate whether there are topological excitations within any sub-volumes, and to measure the topological susceptibility using the correlation of the topological charges of two sub-volumes.

27 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.11/34 For any topological sector with Q t, using χpt, it can be shown that lim ρ(x)ρ(y) = 1 ( Q 2 t x y Ω Ω χ t c ) 4 + O(Ω 3 ) 2χ t Ω Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007)

28 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.11/34 For any topological sector with Q t, using χpt, it can be shown that lim ρ(x)ρ(y) = 1 ( Q 2 t x y Ω Ω χ t c ) 4 + O(Ω 3 ) 2χ t Ω Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007) Thus, in the trivial sector with Q t = 0, for any two widely separated sub-volumes Ω 1 and Ω 2, the correlation of their topological charges would behave as Q 1 Q 2 Ω 1Ω 2 Ω ( χ t + c 4 2χ t Ω ) Q i = Ω i d 4 x ρ(x)

29 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.12/34 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the time-correlation function C(t 1 t 2 ) = Q(t 1 )Q(t 2 ) = x 1, x 2 ρ(x 1 )ρ(x 2 )

30 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.12/34 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the time-correlation function C(t 1 t 2 ) = Q(t 1 )Q(t 2 ) = x 1, x 2 ρ(x 1 )ρ(x 2 ) Then its plateau at large t 1 t 2 can be used to extract χ t, provided that c 4 2χ 2 tω, c 4 = 1 Ω [ Q 4 t θ=0 3 Q 2 t 2 θ=0]

31 Introduction (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.12/34 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the time-correlation function C(t 1 t 2 ) = Q(t 1 )Q(t 2 ) = x 1, x 2 ρ(x 1 )ρ(x 2 ) Then its plateau at large t 1 t 2 can be used to extract χ t, provided that c 4 2χ 2 tω, c 4 = 1 Ω [ Q 4 t θ=0 3 Q 2 t 2 θ=0] However, on a lattice, it is difficult to extract ρ(x) unambiguously from the link variables!

32 Topology with Overlap Dirac Operator T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.13/34 It is well known that the topological charge density can be defined via the overlap Dirac operator as ρ(x) = tr[γ 5 (1 rd) x,x ], r = 1 2m 0 where D is the overlap Dirac operator D = m 0 (1 + V ), V = γ 5 H w H 2 w, H w = γ 5 ( m 0 + γ µ t µ + W)

33 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.14/34 Here ρ(x) = tr[γ 5 (1 rd) x,x ] is justified to be a definition of topological charge density since it has been asserted (Kikukawa & Yamada, 1998) ρ(x) a π 2ɛ µνλσtr[f µν (x)f λσ (x)]

34 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.14/34 Here ρ(x) = tr[γ 5 (1 rd) x,x ] is justified to be a definition of topological charge density since it has been asserted (Kikukawa & Yamada, 1998) ρ(x) a π 2ɛ µνλσtr[f µν (x)f λσ (x)] Note that the index theorem on the lattice index(d) = n + n = x ρ(x) = Q t had been observed by Narayanan and Neuberger in 1995, using the spectral flow of H w (m 0 ), before the Ginsparg-Wilson relation was rejuvenated in 1998.

35 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.15/34 It seems natural to use ρ(x) = tr[γ 5 (1 rd) x,x ] to compute the topological susceptibility χ t = 1 Ω Q2 t = 1 Ω ρ(x)ρ(y) = x,y x ρ(x)ρ(0)

36 Topology with Overlap Dirac Operator (cont) It seems natural to use ρ(x) = tr[γ 5 (1 rd) x,x ] to compute the topological susceptibility χ t = 1 Ω Q2 t = 1 Ω ρ(x)ρ(y) = x,y x ρ(x)ρ(0) On the other hand, one can derive the relation index(d) = m x tr[γ 5 (D c + m) 1 x,x] = m Tr[γ 5 (D c + m) 1 ] where D c = D(1 rd) 1 = 2m 0 (1 + V )(1 V ) 1 is chirally symmetric but non-local (Chiu & Zenkin, 1998). Note that for the topologically-trivial configurations, D c is well-defined (without any poles). T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.15/34

37 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.16/34 Thus one can regard ρ 1 (x) = m tr[γ 5 (D c + m) 1 x,x] as a definition of topological charge density, for any valence quark mass m.

38 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.16/34 Thus one can regard ρ 1 (x) = m tr[γ 5 (D c + m) 1 x,x] as a definition of topological charge density, for any valence quark mass m. Obviously, the identity index(d) = m Tr[γ 5 (D c + m) 1 ] can be generalized to index(d) = m 1 m 2 m k Tr[γ 5 (D c + m 1 ) 1 (D c + m 2 ) 1 (D c + m k ) 1 ] with the generalized topological charge density ρ k (x) = m 1 m 2 m k tr[γ 5 (D c + m 1 ) 1 (D c + m 2 ) 1 (D c + m k ) 1 ] x,x

39 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.17/34 Presumably, any ρ k can be used to compute χ t. In general, χ t = m 1 m k m k+1 m l Tr[γ 5 (D c + m 1 ) 1 (D c + m k ) 1 ] Ω Tr[γ 5 (D c + m k+1 ) 1 (D c + m l ) 1 ]

40 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.17/34 Presumably, any ρ k can be used to compute χ t. In general, χ t = m 1 m k m k+1 m l Tr[γ 5 (D c + m 1 ) 1 (D c + m k ) 1 ] Ω Tr[γ 5 (D c + m k+1 ) 1 (D c + m l ) 1 ] It has been pointed out by Lüscher, for k 2 and l 5, χ t avoids the short-distance singularities in the continuum limit.

41 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.18/34 However, on a finite lattice, lim x y 1 ρ 1 (x)ρ 1 (y) 1 Ω ( Q 2 t Ω χ t c ) 4 2χ t Ω + O(e m π x y ) +O(e m η x y ) + O(Ω 3 ) + is contaminated by m π, m η,, which can couple to ρ 1 (x)ρ 1 (y).

42 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.18/34 However, on a finite lattice, lim x y 1 ρ 1 (x)ρ 1 (y) 1 Ω ( Q 2 t Ω χ t c ) 4 2χ t Ω + O(e m π x y ) +O(e m η x y ) + O(Ω 3 ) + is contaminated by m π, m η,, which can couple to ρ 1 (x)ρ 1 (y). A better alternative is to compute the correlator of flavor-singlet η, which behaves as lim x y 1 m2 q η (x)η (y) 1 Ω ( Q 2 t Ω χ t c ) 4 2χ t Ω +O(Ω 3 ) + + O(e m η x y ) Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007)

43 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.19/34 Time-Correlation Function of η' 1 N f x xcη'(t) = Σ X 0 Σ x x X 0

44 Lattice Setup T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.20/34 Lattice size: Gluons: Iwasaki gauge action at β = 2.30 Quarks (N f = 2): overlap Dirac operator with m 0 = 1.6 Add extra Wilson fermions and pseudofermions det(hov) 2 det(hov) 2 det(hw) 2 det(hw 2 + µ 2 ), µ = 0.2 to forbid λ(h w ) crossing zero, thus Q t is invariant. Quark masses: m sea = 0.015, 0.025, 0.035, 0.050, 0.070, 0.100, each of 500 confs with Q t = 0. For m sea = 0.05, 250 confs with Q t = 2, 4 respectively. For each configuration, 50 conjugate pairs of low-lying eigenmodes of overlap Dirac operator are projected.

45 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.21/ x32, β=2.30, m sea =m val =0.025 no. of confs = disconnected (hairpin) C(t) t

46 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.22/ x32, β=2.30, m sea =m val =0.025 no. of confs = disconnected (hairpin) connected C(t) t

47 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.23/ x32, β=2.30, m sea =m val =0.025 no. of confs = disconnected (hairpin) connected η' = hairpin - connected / N f C(t) t

48 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.24/ x32, β=2.30, m sea =m val =0.025 no. of confs = disconnected (hairpin) connected η' = hairpin - connected / N f cosh + c η' C(t) t

49 T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.25/ x32, β=2.30, m sea =m val =0.025 no. of confs = disconnected (hairpin) connected η' = hairpin - connected / N f C(t) cosh + c η' c η' t

50 Saturation of η by low-lying eigenmodes T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.26/ x32, β=2.30, m sea =m val = e-3 no. of configurations = e-3 C η' (t) -3.0e-3 nev = e-3-5.0e t

51 Saturation of η by low-lying eigenmodes T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.27/ x32, β=2.30, m sea =m val = e-3 no. of configurations = e-3 C η' (t) -3.0e-3 nev = nev = e-3-5.0e t

52 Saturation of η by low-lying eigenmodes T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.28/ x32, β=2.30, m sea =m val = e-3 no. of configurations = e-3 C η' (t) -3.0e-3 nev = nev = nev = e-3-5.0e t

53 Saturation of η by low-lying eigenmodes T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.29/ x32, β=2.30, m sea =m val = e-3 no. of configurations = e-3 C η' (t) -3.0e-3 nev = nev = nev = nev = e-3-5.0e t

54 Saturation of η by low-lying eigenmodes T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.30/ x32, β=2.30, m sea =m val = e-3 no. of configurations = e-3 C η' (t) -3.0e-3-4.0e-3 nev = nev = nev = nev = nev = e t

55 Realization of Leutwyler-Smilga relation T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.31/34 2.0e-4 1.5e-4 _ ChPT χ t = mσ/n f Σ MS (2 GeV) = [252(5)(6) MeV] 3 a 4 χ t 1.0e-4 5.0e m q In the limit m 0, χ t mσ/n f, in agreement with ChPT.

56 Determination of Σ T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.32/34 From the slope of the linear fit of χ t vs. m q for m q a = 0.015, 0.025, and 0.035, it gives a 3 Σ = (10) With a 1 = 1670(20)(20) MeV, and Z MS m (2 GeV) = 0.742(12), the value of a 3 Σ is transcribed to Σ MS (2 GeV) = (252 ± 5 ± 10 MeV) 3 in good agreement with our previous result 251(7)(11) MeV obtained in the ɛ-regime. H. Fukaya et al. (JLQCD-TWQCD) PRL 98 (2007) ; PRD 76 (2007)

57 Universality of χ t for different Topological Sectors T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.33/34 3.0e x32, β=2.30, m sea =m val = e-4 Q t = 0, 500 confs. Q t =-2, 250 confs. Q t =-4, 250 confs. a 4 χ t 1.0e Q top

58 Conclusion and Outlook T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.34/34 For the topologically-trivial gauge configurations generated with N f = 2 dynamical overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes.

59 Conclusion and Outlook T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.34/34 For the topologically-trivial gauge configurations generated with N f = 2 dynamical overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes. These near-zero modes allow us to determine χ t and Σ.

60 Conclusion and Outlook T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.34/34 For the topologically-trivial gauge configurations generated with N f = 2 dynamical overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes. These near-zero modes allow us to determine χ t and Σ. In the chiral limit, χ t = mσ/n f is realized, with Σ MS (2 GeV) = 252(5)(10) MeV, in good agreement with our result in ɛ-regime.

61 Conclusion and Outlook T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.34/34 For the topologically-trivial gauge configurations generated with N f = 2 dynamical overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes. These near-zero modes allow us to determine χ t and Σ. In the chiral limit, χ t = mσ/n f is realized, with Σ MS (2 GeV) = 252(5)(10) MeV, in good agreement with our result in ɛ-regime. For m sea = 0.05, χ t extracted from different topological sectors (Q t = 0, 2, 4) are consistent with each other.

62 Conclusion and Outlook T.W. Chiu, Chung-Yuan Christian Univ, May 13, 2008 p.34/34 For the topologically-trivial gauge configurations generated with N f = 2 dynamical overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes. These near-zero modes allow us to determine χ t and Σ. In the chiral limit, χ t = mσ/n f is realized, with Σ MS (2 GeV) = 252(5)(10) MeV, in good agreement with our result in ɛ-regime. For m sea = 0.05, χ t extracted from different topological sectors (Q t = 0, 2, 4) are consistent with each other. It remains to obtain an upper bound of c 4 (from 2-pt and 4-pt correl. fn.) to see whether c 4 2χ 2 tω is satisfied.

Topological susceptibility in (2+1)-flavor lattice QCD with overlap fermion

T.W. Chiu, Lattice 2008, July 15, 2008 p.1/30 Topological susceptibility in (2+1)-flavor lattice QCD with overlap fermion Ting-Wai Chiu Physics Department, National Taiwan University Collaborators: S.