The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator

Transcription

1 Stony Brook University Department of Physics and Astronomy The Effect of the Low Energy Constants on the Spectral Properties of the Wilson Dirac Operator Savvas Zafeiropoulos July 29-August 3, 2013 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 1/22

2 Wilson Chiral Perturbation Theory Wilson term breaks χ symmetry explicitly Lattice spacing effects lead to new terms in χ P T Sharpe and Singleton (1998), Rupak and Shoresh (2002), Baer,Rupak and Shoresh (2004) ɛ regime where in the thermodynamic, chiral and continuum limit mv Σ, zv Σ and a 2 V W i kept fixed. At order a 2 it involves three Low Energy Constants (LECs) Z Nf (m, z; a) = du e S[U], where the action is S = m 2 ΣV tr (U + U ) z 2 ΣV tr (U U ) ( +a 2 V W 6 [tr U +U ) ( ] 2 +a 2 V W 7 [tr U U ) ] 2 +a 2 V W 8 tr(u 2 +U 2 ). M Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 2/22

3 Wilson Chiral Perturbation Theory Wilson term breaks χ symmetry explicitly Lattice spacing effects lead to new terms in χ P T Sharpe and Singleton (1998), Rupak and Shoresh (2002), Baer,Rupak and Shoresh (2004) ɛ regime where in the thermodynamic, chiral and continuum limit mv Σ, zv Σ and a 2 V W i kept fixed. At order a 2 it involves three Low Energy Constants (LECs) Z Nf (m, z; a) = du e S[U], where the action is S = m 2 ΣV tr (U + U ) z 2 ΣV tr (U U ) ( +a 2 V W 6 [tr U +U ) ( ] 2 +a 2 V W 7 [tr U U ) ] 2 +a 2 V W 8 tr(u 2 +U 2 ). M Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 2/22

4 Wilson Chiral Perturbation Theory Wilson term breaks χ symmetry explicitly Lattice spacing effects lead to new terms in χ P T Sharpe and Singleton (1998), Rupak and Shoresh (2002), Baer,Rupak and Shoresh (2004) ɛ regime where in the thermodynamic, chiral and continuum limit mv Σ, zv Σ and a 2 V W i kept fixed. At order a 2 it involves three Low Energy Constants (LECs) Z Nf (m, z; a) = du e S[U], where the action is S = m 2 ΣV tr (U + U ) z 2 ΣV tr (U U ) ( +a 2 V W 6 [tr U +U ) ( ] 2 +a 2 V W 7 [tr U U ) ] 2 +a 2 V W 8 tr(u 2 +U 2 ). M Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 2/22

5 The Goal Facilitate simulations in the deep chiral regime by an exact, analytical understanding of the average behavior of the smallest eigenvalues Chiral symmetry breaking from lattice spacing Stability of lattice simulations Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 3/22

6 The Goal Facilitate simulations in the deep chiral regime by an exact, analytical understanding of the average behavior of the smallest eigenvalues Chiral symmetry breaking from lattice spacing Stability of lattice simulations Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 3/22

7 The Goal Facilitate simulations in the deep chiral regime by an exact, analytical understanding of the average behavior of the smallest eigenvalues Chiral symmetry breaking from lattice spacing Stability of lattice simulations Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 3/22

8 Introduction of the Model Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 4/22

9 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

10 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

11 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

12 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

13 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

14 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

15 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

16 Wilson Dirac operator and RMT Partition function of D W with N f flavors : Z RMT,ν N f = dd W det N f (D W + m)p (D W ) P (D W ) is a Gaussian ( ) aa W D W = W + am ab 6 + aλ 7 γ 5 (Damgaard et al (2010),Akemann et al (2010), Kieburg et al (2011,2012) A : n n Hermitian B : (n + ν) (n + ν) Hermitian W : n (n + ν) Complex m 6 and λ 7 scalar random variables At a = 0 : D W has ν generic zero modes At finite a : definition of the index through spectral flow lines or equivalently ν = λ W k R sign( k γ 5 k ) Itoh et al (1987) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 5/22

17 γ 5 - hermiticity D W = 1 2 γ µ( µ + µ) 1 2 a µ µ At a 0 is non-hermitian but retains γ 5 -Hermiticity D W = γ 5D W γ 5 Eigenvalues of D W because of the γ 5 -Hermiticity occur in complex conjugate pairs or are real ONLY eigenvectors corresponding to real eigenvalues have non vanishing chirality k γ 5 k Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 6/22

18 γ 5 - hermiticity D W = 1 2 γ µ( µ + µ) 1 2 a µ µ At a 0 is non-hermitian but retains γ 5 -Hermiticity D W = γ 5D W γ 5 Eigenvalues of D W because of the γ 5 -Hermiticity occur in complex conjugate pairs or are real ONLY eigenvectors corresponding to real eigenvalues have non vanishing chirality k γ 5 k Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 6/22

19 γ 5 - hermiticity D W = 1 2 γ µ( µ + µ) 1 2 a µ µ At a 0 is non-hermitian but retains γ 5 -Hermiticity D W = γ 5D W γ 5 Eigenvalues of D W because of the γ 5 -Hermiticity occur in complex conjugate pairs or are real ONLY eigenvectors corresponding to real eigenvalues have non vanishing chirality k γ 5 k Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 6/22

20 γ 5 - hermiticity D W = 1 2 γ µ( µ + µ) 1 2 a µ µ At a 0 is non-hermitian but retains γ 5 -Hermiticity D W = γ 5D W γ 5 Eigenvalues of D W because of the γ 5 -Hermiticity occur in complex conjugate pairs or are real ONLY eigenvectors corresponding to real eigenvalues have non vanishing chirality k γ 5 k Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 6/22

21 The Eigenvalue Densities Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 7/22

22 Lattice results vs RMT 0.7 Ev_279_20_HYP_Q2_Qm2_m178 m=4.8 a=0.35 ν=2 PREDICTION ρ ν=2 5 5 (λ,m;a) λ ab6 = ab7 = 0.25, ab8 = 0.7 m b = 5.3 ν = 0 (top) and ν = 1 (bottom) m = 4.8, ν = 2 (Damgaard,Heller and Splittorff (2011)) (Deuzeman,Wenger and Wuilloud (2011)) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 8/22

23 Lattice results vs RMT Ev_real_2635_16_HYP_cl_Q1andQm1 Ev_real_2635_16_HYP_cl_Q2andQm2 a 6 =i0.1 a 8 =0 ν=1 a 6 =i0.1 a 8 =0 ν=2 2 ρ real Q (λ w ) λ w ΣV The density of real eigenvalues of D W Cumulative eigenvalue distributions of D 5 with all W 6/7/8 included at ν = 0 (Deuzeman, Wenger and Wuilloud (2011)) Damgaard,Heller and Splittorff (2012)) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 9/22

24 The effects of W 6 and W 7 when W 8 = 0 â 6 and â 7 introduced through the addition of the Gaussian stochastic variable m 6 + λ 7 γ 5 to D W D = D W + (m + m 6 )1 + λ 7 γ 5 When â 8 = 0 D W is anti-hermitian, the eigenvalues of D W ( λ 7, m 6 ) = D m are given by ẑ ± = m 6 ± i λ 2 W λ 2 7 where iλ W is an eigenvalue of D W Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 10/22

25 The effects of W 6 and W 7 when W 8 = 0 â 6 and â 7 introduced through the addition of the Gaussian stochastic variable m 6 + λ 7 γ 5 to D W D = D W + (m + m 6 )1 + λ 7 γ 5 When â 8 = 0 D W is anti-hermitian, the eigenvalues of D W ( λ 7, m 6 ) = D m are given by ẑ ± = m 6 ± i λ 2 W λ 2 7 where iλ W is an eigenvalue of D W Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 10/22

26 The effects of W 6 and W 7 when W 8 = 0 â 6 and â 7 introduced through the addition of the Gaussian stochastic variable m 6 + λ 7 γ 5 to D W D = D W + (m + m 6 )1 + λ 7 γ 5 When â 8 = 0 D W is anti-hermitian, the eigenvalues of D W ( λ 7, m 6 ) = D m are given by ẑ ± = m 6 ± i λ 2 W λ 2 7 where iλ W is an eigenvalue of D W Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 10/22

27 The effects of W 6 and W 7 when W 8 = 0 â 6 and â 7 introduced through the addition of the Gaussian stochastic variable m 6 + λ 7 γ 5 to D W D = D W + (m + m 6 )1 + λ 7 γ 5 When â 8 = 0 D W is anti-hermitian, the eigenvalues of D W ( λ 7, m 6 ) = D m are given by ẑ ± = m 6 ± i λ 2 W λ 2 7 where iλ W is an eigenvalue of D W Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 10/22

28 The effects of W 6 and W 7 Im Im 4 a 6 Re 4 a 7 Re Schematic plots of the effects of W 6 (left plot) and of W 7 (right plot). W 6 broadens the spectrum parallel to the real axis according to a Gaussian with width 4â 6, but does not change the continuum spectrum in a significant way. When W 7 0 and W 6 = 0 the purely imaginary eigenvalues invade the real axis through the origin and only the real (green crosses) are broadened by a Gaussian with width 4â 7 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 11/22

29 Distribution of additional real modes for ν = 1 ς r ς r a 6 0.1, a 7 0.7, a a 6 0.1, a 7 0.7, a 8 0.1, MC a 6 0.1, a 7 0.1, a a 6 0.1, a 7 0.1, a 8 0.7, MC a 6 0.7, a 7 0.7, a a 6 0.7, a 7 0.7, a 8 0.7, MC a 6 0.1, a 7 0.1, a a 6 0.7, a 7 0.1, a x x Notice that the two curves for â 7 = â 8 = 0.1 (right plot) are two orders smaller than the other curves (left plot). Notice the soft repulsion of the additional real modes from the origin at large â 7. The parameter â 6 smooths the distribution. Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 12/22

30 Log-Log plots of additional real modes vs â for ν = 0, 2 10 N add Ν 0 a a a a a a a a a a a a , MC , MC , MC , MC , MC a , MC N add 1 Ν a a 8 Log-log plots of N add as a function of â 8 for ν = 0 (left plot) and ν = 2 (right plot). W 6 has no effect on N add. Saturation around zero due to a non-zero value of â 7. For â 7 = 0 (lowest curves) the average number of additional real modes behaves like â 2ν+2 8. Kieburg, Verbaarschot and SZ (2011) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 13/22

31 Distribution of additional real modes for â >> ς r a a 7 8, a a 7 8, a 8 10 a 7 8, a 8 10, Ν 0, MC 8 10, a 7 12, a Ν 1, MC a 7 12, a , a 8 10, Ν 0, MC 8 10, Ν 1, MC x At â >> 1 ρ r develops square root singularities at the boundaries. Finite matrix size+ finite lattice spacing ρ r has a tail dropping off much faster than the size of the support. The dependence on W 6 and ν is completely lost. Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 14/22

32 Projected distribution of the complex eigenvalues for ν = 1 ς cp a 7 0,a 8 0 a 7 0.1,a a 7 0.1,a 8 0.1, MC a 7 1,a a 7 1,a 8 0.1, MC a 7 0.1,a 8 1 a 7 0.1,a 8 1, MC a 7 1,a 8 1 a 7 1,a 8 1, MC y The distribution of the complex eigenvalues projected onto the imaginary axis for ν = 1. Notice that â 6 does not affect this distribution. The comparison of â 7 = â 8 = 0.1 with the continuum result (black curve) shows that ρ cp is still a good quantity to extract the chiral condensate Σ at small lattice spacing. Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 15/22

33 Chirality distribution for ν = 1 ς Χ a 6 0.1, a 7 0.1, a 8 1 a 6 0.1, a 7 0.1, a 8 1, MC a 6 0.1, a 7 1, a a 6 0.1, a 7 1, a 8 0.1, MC 0.1 a 6 1, a 7 0.1, a a 6 1, a 7 0.1, a 8 0.1, MC a 6 1, a 7 1, a 8 1 a , a 7 1, a 8 1, MC x The distribution is symmetric around the origin. At small â 8 the distributions for (â 6, â 7 ) = (1, 0.1), (0.1, 1) are almost the same Gaussian as the analytical result predicts. At large â 8 the maximum reflects the predicted square root singularity which starts to build up. We have not included the case â 6/7/8 = 0.1 since it exceeds the other curves by a factor of 10 to 100. Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 16/22

34 Extracting the LECs of Wilson chpt Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 17/22

35 Extracting the LECs of Wilson chpt Please do not read this (Figure courtesy of M. Kieburg) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 18/22

36 Extracting the LECs of Wilson chpt (Figure courtesy of M. Kieburg) Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 19/22

37 Conclusions Studied the effect of the three LECs on the spectrum of D W. W 6 and W 7 can be interpreted as collective fluctuations of the spectrum while W 8 induces interactions among all modes. Analytical and numerical results of the eigenvalue densities of D W At small lattice spacing we propose the following quantities for the extraction of LECs ã 2 V W 6 W 7 W 8 = π2 8 4Nadd ν=0 /π2 2σ 2 / 2 x 2 ν=1 ρ χ / 2 x 2 ν=2 ρ χ / 2 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 20/22

38 Conclusions Studied the effect of the three LECs on the spectrum of D W. W 6 and W 7 can be interpreted as collective fluctuations of the spectrum while W 8 induces interactions among all modes. Analytical and numerical results of the eigenvalue densities of D W At small lattice spacing we propose the following quantities for the extraction of LECs ã 2 V W 6 W 7 W 8 = π2 8 4Nadd ν=0 /π2 2σ 2 / 2 x 2 ν=1 ρ χ / 2 x 2 ν=2 ρ χ / 2 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 20/22

39 Conclusions Studied the effect of the three LECs on the spectrum of D W. W 6 and W 7 can be interpreted as collective fluctuations of the spectrum while W 8 induces interactions among all modes. Analytical and numerical results of the eigenvalue densities of D W At small lattice spacing we propose the following quantities for the extraction of LECs ã 2 V W 6 W 7 W 8 = π2 8 4Nadd ν=0 /π2 2σ 2 / 2 x 2 ν=1 ρ χ / 2 x 2 ν=2 ρ χ / 2 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 20/22

40 Conclusions Studied the effect of the three LECs on the spectrum of D W. W 6 and W 7 can be interpreted as collective fluctuations of the spectrum while W 8 induces interactions among all modes. Analytical and numerical results of the eigenvalue densities of D W At small lattice spacing we propose the following quantities for the extraction of LECs ã 2 V W 6 W 7 W 8 = π2 8 4Nadd ν=0 /π2 2σ 2 / 2 x 2 ν=1 ρ χ / 2 x 2 ν=2 ρ χ / 2 Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 20/22

41 for upcoming results... Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 21/22

42 Thank you for your attention! Collaborators: Mario Kieburg 8E A classification of 2 dim Lattice Theory Jacobus Verbaarschot 7D Discretization Effects in the ɛ Domain of QCD Savvas Zafeiropoulos Spectral Properties of the Wilson Dirac Operator 22/22