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

Transcription

1 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. Aoki, S. Hashimoto, T.H. Hsieh, T. Kaneko, H. Matsufuru, J. Noaki, T. Onogi, N. Yamada (for JLQCD and TWQCD Collaborations)

2 Outline T.W. Chiu, Lattice 2008, July 15, 2008 p.2/30 Introduction Topology with Overlap Dirac Operator Topological Susceptibility in a Fixed Topological Sector Lattice Setup Results using N f = Dynamical Overlap Configurations with Q t = 0 Conclusion and Outlook

3 Introduction T.W. Chiu, Lattice 2008, July 15, 2008 p.3/30 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0), ρ(x) = 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)]

4 Introduction T.W. Chiu, Lattice 2008, July 15, 2008 p.3/30 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0), ρ(x) = 1 32π 2ɛ µνλσtr[f µν (x)f λσ (x)] Veneziano-Witten relation χ t (quenched) = f2 πm 2 η 4N f

5 Introduction T.W. Chiu, Lattice 2008, July 15, 2008 p.3/30 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0), 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) (N f with same mass m q )

6 Introduction T.W. Chiu, Lattice 2008, July 15, 2008 p.3/30 Theoretically, topological susceptibility is defined as χ t = d 4 x ρ(x)ρ(0), 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) (N f with same mass m q ) χ t = Σ ( 1 m u + 1 m d + 1 m s ) 1 + O(m 2 u) (N f = 2 + 1)

7 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.4/30 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. (For application to m π and f π, see J. Noaki s talk) Brower, Chandrasekaran, Negele, Wiese, PLB 560 (2003) 64 Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007)

8 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.4/30 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. (For application to m π and f π, see J. Noaki s talk) 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.

9 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.5/30 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.

10 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.5/30 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

11 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.6/30 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.

12 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.6/30 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.

13 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.7/30 For any topological sector with Q t, using saddle-point expansion, 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)

14 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.7/30 For any topological sector with Q t, using saddle-point expansion, 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)

15 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.8/30 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the correlation function C(t 1 t 2 ) = Q(t 1 )Q(t 2 ) = x 1, x 2 ρ(x 1 )ρ(x 2 )

16 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.8/30 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the 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]

17 Introduction (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.8/30 On a finite lattice, consider two spatial sub-volumes at time slices t 1 and t 2, measure the 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!

18 Topology with Overlap Dirac Operator T.W. Chiu, Lattice 2008, July 15, 2008 p.9/30 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)

19 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.10/30 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)]

20 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.10/30 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.

21 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.11/30 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)

22 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, Lattice 2008, July 15, 2008 p.11/30

23 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.12/30 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.

24 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.12/30 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

25 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.13/30 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 ]

26 Topology with Overlap Dirac Operator (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.13/30 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.

27 T.W. Chiu, Lattice 2008, July 15, 2008 p.14/30 Topological Fluctuations with fixed Q t 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).

28 T.W. Chiu, Lattice 2008, July 15, 2008 p.14/30 Topological Fluctuations with fixed Q t 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)

29 T.W. Chiu, Lattice 2008, July 15, 2008 p.14/30 Topological Fluctuations with fixed Q t 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) But, what is the contribution due to the c 4 term?

30 Topological Fluctuations with fixed Q t (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.15/30 lim x i x j 1 m4 q η (x 1 )η (x 2 )η (x 3 )η (x 4 ) = 3χ2 t Ω 2 ( 1 Q2 t χ t Ω + c ) 2 4 χ 2 tω +O(e m η x i x j ) + O(Ω 4 ) + Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007)

31 Topological Fluctuations with fixed Q t (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.15/30 lim x i x j 1 m4 q η (x 1 )η (x 2 )η (x 3 )η (x 4 ) = 3χ2 t Ω 2 ( 1 Q2 t χ t Ω + c ) 2 4 χ 2 tω +O(e m η x i x j ) + O(Ω 4 ) + Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007) 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 )

32 Topological Fluctuations with fixed Q t (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.15/30 lim x i x j 1 m4 q η (x 1 )η (x 2 )η (x 3 )η (x 4 ) = 3χ2 t Ω 2 ( 1 Q2 t χ t Ω + c ) 2 4 χ 2 tω +O(e m η x i x j ) + O(Ω 4 ) + Aoki, Fukaya, Hashimoto, Onogi, PRD 76 (2007) 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 ) Measure the 2-pt and 4-pt functions of η can determinate both χ t and y c 4 2χ 2 tω

33 Topological Fluctuations with fixed Q t (cont) T.W. Chiu, Lattice 2008, July 15, 2008 p.16/30 Suppose the asymptotic values of 2-pt and 4-pt functions of η are k 2 and k 4 respectively, then χ t and y can be solved as ( χ t = Q2 t Ω + Ω 2k 2 ) k 4 /3 ( k4 ) /3 k 2 y = k4 /3 2k 2 (1 Q2 t χ t Ω )

34 Topological Fluctuations with fixed Q t (cont) Suppose the asymptotic values of 2-pt and 4-pt functions of η are k 2 and k 4 respectively, then χ t and y can be solved as ( χ t = Q2 t Ω + Ω 2k 2 ) k 4 /3 ( k4 ) /3 k 2 y = k4 /3 2k 2 (1 Q2 t χ t Ω If one neglects the y term in 2-pt and 4-pt functions of η, one obtains ) χ t Q2 t Ω + Ωk 2 χ t Q2 t Ω + Ω k 4 /3 which provide another two independent estimates of χ t. If y 1, then all 3 eqs give compatible values of χ t. T.W. Chiu, Lattice 2008, July 15, 2008 p.16/30

35 Lattice Setup (see S. Hashimoto s talk, and H. Matsufuru s poster) T.W. Chiu, Lattice 2008, July 15, 2008 p.17/30 Lattice size: Gluons: Iwasaki gauge action at β = 2.30 Quarks (N f = 2 + 1): 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 u = 0.015, 0.025, 0.035, 0.050, 0.100, each of 500 confs, with m s = 0.100, and Q t = 0. For each configuration, 80 conjugate pairs of low-lying eigenmodes of overlap Dirac operator are projected.

36 Saturation of C η (t) by low-lying eigenmodes T.W. Chiu, Lattice 2008, July 15, 2008 p.18/30 C η (t) = 1 L 3 T T η ( x 2,u + t)η ( x 1,u) u=1 x i m u = 0.015, m s = C η' (t) nev=80 nev=60 nev=40 nev= t

37 Saturation of C 4η (t) by low-lying eigenmodes T.W. Chiu, Lattice 2008, July 15, 2008 p.19/30 C 4η (t) = 1 L 3 T T η ( x 4,u + 3t)η ( x 3,u + 2t)η ( x 2,u + t)η ( x 1,u) u=1 x i 0.03 m u = 0.015, m s = C 4η' (t) nev=80 nev=60 nev=40 nev= t

38 C η (t) T.W. Chiu, Lattice 2008, July 15, 2008 p.20/ m u = 0.015, m s = C η' (t) full disconnected connected t

39 C 4η (t) T.W. Chiu, Lattice 2008, July 15, 2008 p.21/ m u = 0.015, m s = C 4η' (t) full disconnected connected t

40 Results of χ t and y T.W. Chiu, Lattice 2008, July 15, 2008 p.22/30 1e-4 8e-5 χ t 6e-5 a 4 χ t 4e-5 2e-5 0-2e m q 0.2 y m q a

41 Results of χ t and y T.W. Chiu, Lattice 2008, July 15, 2008 p.23/30 1e-4 8e-5 χ t χ t (2 pt) 6e-5 a 4 χ t 4e-5 2e-5 0-2e m q 0.2 y m q a

42 Results of χ t and y T.W. Chiu, Lattice 2008, July 15, 2008 p.24/30 1e-4 8e-5 6e-5 χ t χ t (2 pt) χ t (4 pt) a 4 χ t 4e-5 2e-5 0-2e m q 0.2 y m q a

43 T.W. Chiu, Lattice 2008, July 15, 2008 p.25/ N f = χ t [GeV 4 ] m q [GeV]

44 Realization of the Leutwyler-Smilga relation N f = ChPT fit (N f = 2+1) χ t [GeV 4 ] m q [GeV] ( ) 1 1 Fitting χ t to δ + Σ m u + 1 m d + 1 m s for m u a = 0.015, 0.025, 0.035, 0.050, it gives a 3 Σ = (10) and δ = 4.1(1.1) T.W. Chiu, Lattice 2008, July 15, 2008 p.26/30

45 Realization of the Leutwyler-Smilga relation N f = ChPT fit (N f = 2+1) Nf = 2 χ t [GeV 4 ] m q [GeV] ( ) 1 1 Fitting χ t to δ + Σ m u + 1 m d + 1 m s for m u a = 0.015, 0.025, 0.035, 0.050, it gives a 3 Σ = (10) and δ = 4.1(1.1) T.W. Chiu, Lattice 2008, July 15, 2008 p.27/30

46 Realization of the Leutwyler-Smilga relation N f = ChPT fit (N f = 2+1) Nf = 2 ChPT fit (Nf = 2) χ t [GeV 4 ] m q [GeV] ( ) 1 1 Fitting χ t to δ + Σ m u + 1 m d + 1 m s for m u a = 0.015, 0.025, 0.035, 0.050, it gives a 3 Σ = (10) and δ = 4.1(1.1) T.W. Chiu, Lattice 2008, July 15, 2008 p.28/30

47 Determination of Σ T.W. Chiu, Lattice 2008, July 15, 2008 p.29/30 With a 1 = 1833(12) MeV (see H. Matsufuru s poster), and Zm MS (2 GeV) = 0.826(8) (see J.Noaki s talk), the value of a 3 Σ is transcribed to Σ MS (2 GeV) = (240 ± 5 ± 2 MeV) 3 (N f = 2 + 1)

48 Determination of Σ T.W. Chiu, Lattice 2008, July 15, 2008 p.29/30 With a 1 = 1833(12) MeV (see H. Matsufuru s poster), and Zm MS (2 GeV) = 0.826(8) (see J.Noaki s talk), the value of a 3 Σ is transcribed to Σ MS (2 GeV) = (240 ± 5 ± 2 MeV) 3 (N f = 2 + 1) which is in good agreement with Σ MS (2 GeV) = (242 ± 5 ± 10 MeV) 3 (N f = 2) extracted from χ t measured in N f = 2 QCD. S. Aoki et al. (JLQCD and TWQCD Collaborations) PLB 665 (2008) 294, arxiv: [hep-lat]

49 Conclusion and Outlook T.W. Chiu, Lattice 2008, July 15, 2008 p.30/30 For the topologically-trivial gauge configurations generated with N f = overlap quarks constrainted by extra Wilson and pseudofermions, they possess topologically non-trivial excitations (e.g., instanton and anti-instanton pairs) in sub-volumes.

50 Conclusion and Outlook T.W. Chiu, Lattice 2008, July 15, 2008 p.30/30 For the topologically-trivial gauge configurations generated with N f = 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.

51 Conclusion and Outlook T.W. Chiu, Lattice 2008, July 15, 2008 p.30/30 For the topologically-trivial gauge configurations generated with N f = 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. ( The Leutwyler-Smilga relation χ t = Σ 1 m u + 1 m d + 1 m s ) 1 is realized with Σ MS (2 GeV) = 240(5)(2) MeV, in good agreement with our previous result obtained in N f = 2 QCD.

52 Conclusion and Outlook T.W. Chiu, Lattice 2008, July 15, 2008 p.30/30 For the topologically-trivial gauge configurations generated with N f = 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. ( The Leutwyler-Smilga relation χ t = Σ 1 m u + 1 m d + 1 m s ) 1 is realized with Σ MS (2 GeV) = 240(5)(2) MeV, in good agreement with our previous result obtained in N f = 2 QCD. y = c 4 2χ 2 t Ω < 0.1 for m ua = 0.015, 0.025, 0.035, 0.050

53 Conclusion and Outlook T.W. Chiu, Lattice 2008, July 15, 2008 p.30/30 For the topologically-trivial gauge configurations generated with N f = 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. ( The Leutwyler-Smilga relation χ t = Σ 1 m u + 1 m d + 1 m s ) 1 is realized with Σ MS (2 GeV) = 240(5)(2) MeV, in good agreement with our previous result obtained in N f = 2 QCD. y = c 4 2χ 2 t Ω < 0.1 for m ua = 0.015, 0.025, 0.035, Can we also determine the mass of η?