Random Matrix Theory for the Wilson-Dirac operator

Transcription

1 Random Matrix Theory for the Wilson-Dirac operator Mario Kieburg Department of Physics and Astronomy SUNY Stony Brook (NY, USA) Bielefeld, December 14th, 2011

2 Outline Introduction in Lattice QCD and in the Random Matrix Model The Joint Probability Densities The Eigenvalue Densities Conclusions and Outlook

3 Introduction in Lattice QCD and in the Random Matrix Model

4 Action of continuums QCD The partition function of N f fermionic flavors N f Z = exp ıs YM (A) ı ψ j (ıd(a) m j )ψ j d 4 x D[A, ψ] j=1 The Yang-Mills action of SU(3) S YM (A) = 1 4g 2 tr F µν F µν d 4 x with the field strength tensor F µν = µ A ν ν A µ + g [A µ, A ν ] The four components of the vector potential A µ su(3) are 3 3 matrix valued functions.

5 The continuum Dirac-operator Fermionic fields ψ j are Grassmann variables Z = N f det ( ) ıd(a) m j exp [ ısym (A)] D[A] j=1 The Dirac operator D(A) = γ µ ( 1 ı µ + ga µ ) Index-theorem: number of zero modes (index)=topological charge ν = 1 32π 2 ε µνλκ tr F µν F λκ d 4 x

6 Lattice QCD space-time becomes discrete with lattice spacing ã vector field A µ su(3) replaced by U µ SU(3) Minkowski space Euclidean space (Wick-rotation t ıt) The Dirac matrices in the chiral basis ( ) ( ) γ = γ = ( ) ( ) γ k 0 σ k = σ k γ k 0 ıσ k = 0 ıσ k 0 ( ) ( ) γ = γ =

7 Fundamental problem on the lattice Energy in continuum: Energy on lattice: Doubler Problem: E 2 = k µ k µ + M 2 0 E 2 = sin2 (k µ ã) ã 2 + M 2 0. k µ { kµ π ã k µ one momentum=(2 4 = 16) particles

8 The Wilson-Dirac-operator Main idea: Make 15 particles in the continuum limit (ã 0) infinitely heavy. too inertial, decouple from the system Wilson-Dirac operator D W = D(A) + ã γ µ sin(k µ ã) + sin2 (k µ ã/2) ã Laplace operator additional effective mass explicitly breaks chiral symmetry Dirac operator not anymore anti-hermitian, but γ 5 -Hermitian D W = γ 5D W γ 5

9 The ϵ-regime of QCD infrared limit of QCD lattice volume (space-time volume) V large Compton wavelength of Goldstone bosons box size V 1/4 Saddlepoint approximation spontaneous breaking of chiral symmetry global Goldstone bosons = Mesons e.g. N f = 2, then SU(2)-integral = zero momentum modes of the three pions

10 Partition function in ϵ-regime for N f flavors Z exp[l(u)]dµ(u) SU(N f ) ν Z U (N f ) Lagrangian of the Goldstone bosons: L(U) = ΣV 2 tr ( M R U + U ML ) exp[l(u)]det ν Udµ(U) VW 6 ã 2 [tr (U + U )] 2 VW 7 ã 2 [tr (U U )] 2 VW 8 ã 2 tr (U 2 + U 2 ) index of the Dirac operator: ν masses for right- and left-handed particles: M R/L = M ± Λ lattice spacing: ã low energy constants: Σ, W 6, W 7, W 8 spacetime volume: V

11 a 2 -terms of the potential: VW 6 ã 2 [tr (U + U )] 2 + VW 7 ã 2 [tr (U U )] 2 + VW 8 ã 2 tr (U 2 + U 2 ) For SU(2): Sharpe and Singleton (1998) Bär, Necco and Schaefer (2009) For general number of flavors: Bär, Rupak and Shoresh (2004) Sharpe (2006)

12 Simplification exp [VW 6 ã 2 [tr (U + U )] 2 + VW 7 ã 2 [tr (U U )] 2] = 1 d[m 6, λ 7 ] exp [ m 6 2 ] + λ 2 7 2π 2 [ exp VW6 ãm 6 tr (U + U ) + ] VW 7 ãλ 7 tr (U U ) m 6, λ 7 can be considered as additional masses omitting the squared trace terms can be introduced later on

13 Random Matrix Ensemble ( aa W D W = W ab ) distributed by P(D W ) exp [ n 2 (tr A2 + tr B 2 ) n tr WW ] Hermitian matrices A (n n) and B ((n + ν) (n + ν)) are the Wilson-terms breaking of chiral symmetry complex W (n (n + ν)) matrix at a = 0: chgue describing continuum QCD (Shuryak, Verbaarschot; 1993) corresponds to W 8 > 0 Damgaard, Splittorff, Verbaarschot (2010)

14 Microscopic Limit Partition function for N f flavors Z N f d[d W ]P(D W ) det(d W + m j 1 2n+ν λ j γ 5 ) j=1 λ: eigenvalues for D 5 = D W γ 5 with γ 5 = diag (1 n, 1 n+ν ) spacetime volume V / matrix dimension n fixed parameters: ΣV diag ( MR, M L ) = 2n diag (m + λ, m λ) = diag ( M R, M L ) V W8 ã = n/2a = â

15 Outcome Z U (N f ) exp[l(u)]det ν Udµ(U) Lagrangian of the Goldstone bosons: L(U) = tr ( M R U + U ML ) â 2 tr (U 2 + U 2 ) Damgaard, Splittorff, Verbaarschot (2010)

16 The Joint Probability Densities

17 Properties of D W and D 5 D W is γ 5 -Hermitian D 5 = D W γ 5 is Hermitian form invariance: D 5 : V U (2n + ν), V 1 = V, compact D W : U U (n, n + ν), U 1 = γ 5 U γ 5, non-compact diagonalization: D 5 = VxV 1 D W = U x, x j, y 2 are real diagonal x x 2 y y 2 x x 3 U 1

18 Diagonalization of D W and D 5 D 5 = VxV 1 x: 2n + ν dim pure real spectrum Let 0 l n D W = U x 1 : l dim real spectrum x x 2 y y 2 x x 3 U 1 x 2, y 2 : n l dim complex conj. eigenvalue pairs x 2 ± ıy 2 x 3 : ν + l dim real spectrum

19 Spectral flow of D 5 (Example for n = 1 and ν = 2)

20 Definition of the Joint Probability Density Let f be arbitrary integrable function invariant under Gl(2n + ν, C): f (D 5 )P(D W )d[d W ] = f (x)p 5 (x)d[x] f (D W )P(D W )d[d W ] = = n l=0 f (z (l) )p (l) W (z(l) )d[z (l) ] f (z)p W (z)d[z] where p W (z) = n l=0 ( p (l) W (z(l) )δ z z (l)) d[z (l) ], z (l) = diag (x 1, x 2 + ıy 2, x 2 ıy 2, x 3 )

21 Results For D 5 a 2(n + ν) dim Pfaffian: [ p 5 (x) 2n+ν (x)pf degenerated quark mass m Akemann, Nagao (2011) For D W a n + ν dim determinant: g 2 (x a, x b ) xa b 1 g 1 (x a ) x a 1 b g 1 (x b ) 0 ] p W (z) 2n+ν (z) [ gc (z det ar )δ (2) (z ar zbl ) + g r(x ar, x bl )δ(y ar )δ(y bl ) x a 1 bl g 1(x bl )δ(y bl ) ] degenerated source term λγ 5 Kieburg, Verbaarschot, Zafeiropoulos (2011) Remark: Vandermonde determinant 2n+ν (x) = i<j(x i x j )

22 Side remark: Orthogonal Polynomial Theory A novel mixing of orthogonal and skew-orthogonal polynomials orthogonal polynomials from order 0 to ν 1 p i p j δ ij skew-orthogonal polynomials from order ν additional relations (q ν+2i q ν+2j+1 ) δ ij (q ν+2i q ν+2j ) = (q ν+2i+1 q ν+2j+1 ) = 0 (p i p j ) = (p i q ν+j ) = p i q ν+j = 0 with f 1 f 2 = (f 1 f 2 ) = f 1 (x)f 2 (x)g 1 (x)dx (f 1 (x 1 )f 2 (x 2 ) f 1 (x 2 )f 2 (x 1 ))G 2 (x 1, x 2 )d[x] essentially the same system of equations for D W

23 Side remark: Orthogonal Polynomial Theory Wish list: relation to Hermite polynomials recursion relation Christoffel Darboux-like formula representation as a matrix integral Rodrigues formula asymptotics in the microscopic limit

24 The Eigenvalue Densities

25 Definition For D 5 ρ 5 (x 1 ) = p 5 (x)d[x 1 ] one level density ρ 5 For D W ρ R (x 1R )δ(y 1R ) ρ c(z 1R ) = ρ L (x 1L )δ(y 1L ) ρ c(z 1L ) = p W (z)d[z 1R ] p W (z)d[z 1L ] three level densities ρ c and ρ r = 2ρ R = ρ R + ρ L ρ χ ρ χ = ρ R ρ L

26 In the Microscopic Limit For â = 0: ρ(y) = νδ(y) + y 2 (J2 ν (y) J ν 1 (y)j ν 2 (y)) = νδ(y) + ρ (ν) i (y) For â 0: two-fold integrals for ρ 5, ρ c, ρ r, ρ χ Akemann, Damgaard, Kieburg, Nagao, Splittorff, Verbaarschot, Zafeiropoulos (2010/11)

27 The Density ρ c ρ c (z) = exp[ x 2 /8â 2 ] y 8πâ Θ(8â 2 x ) 16πâ 2 erf z ρ(ν) i ( z ), â 1 ], â 1 [ y 8â Kieburg, Verbaarschot, Zafeiropoulos (2011)

28 The Density ρ c ρ c (z) = along y = 5â axis exp[ x 2 /8â 2 ] y 8πâ Θ(8â 2 x ) 16πâ 2 erf z ρ(ν) i ( z ), â 1 ], â 1 [ y 8â along imaginary axis Kieburg, Verbaarschot, Zafeiropoulos (2011)

29 The Density ρ χ ρ χ (x) = ( ρ (ν) x GUE, 4â) â 1 νθ(8â 2 x ) π 64â 4 x, 2 â 1 Akemann, Damgaard, Splittorff, Verbaarschot (2010/11)

30 Number of additional real modes N add N add = ρ r (x)dx = (â2 ) ν+1 [2(ν + 1)]! [(ν + 1)!] 2, â 1 2 ( ) 2 3/2 â, â 1 π Kieburg, Verbaarschot, Zafeiropoulos (2011) Deuzeman, Wenger, Wuilloud (2011)

31 ρ 5 at small Lattice Spacing ( x m 4â = ρ χ (x m) + 8πâ exp [ m 2 8â 2 ρ 5 (x) = ρ (ν) GUE ) x + x 2 m 2 ρ(ν) i ( x 2 m 2 ) ] ρ c (Re = ı m, Im = x) m = 3, â = 0.2 ν {0, 1, 2, 3, 4} Akemann, Damgaard, Splittorff, Verbaarschot (2010/11)

32 Lattice Spacing Dependence of ρ 5 ρ 5 (x) = ρ (ν) GUE ( ) x m x + 4â x 2 m 2 ρ(ν) i ( x 2 m 2 ), â 1 [ ] cos2 x π 8â 2, â 1 m = 3, ν = 1 â { 1 8, 1 4, 1 2, 1, 2}

33 Mean Field Limit of ρ 5 Scaling: m s 2 m x s 2 x â sâ m = 3, ν = 1, â = 0.25 s {1, 2, 3, 4}

34 Mean Field Limit of ρ 5 Scaling: m s 2 m x s 2 x â sâ m = 3, ν = 1, â = 0.5 s {1, 2, 3, 4}

35 Mean Field Limit of ρ 5 Scaling: m s 2 m x s 2 x â sâ m = 3, ν = 1, â = 1 s {1, 2, 3, 4}

36 ν Dependence of ρ 5 at large Lattice Spacing Scaling: m s 2 m x s 2 x â sâ m = 3, â {0.5, 1}, s = 4 gap in x ( m 2/3 4â 4/3) 3/2 gap only if m > 8â 2 not Aoki Phase Aoki Phase Damgaard, Splittorff, Verbaarschot (2010)

37 Conclusions and Outlook

38 Summary joint probability densities of D W and D 5 small â 1: broadening of ν formerly zero modes by GUE Gaussian broadening of ρi (y) for D W additional real modes are strongly suppressed with increasing ν large â 1: finite support of size â 2 along and parrallel to the real axis for D W ν independence of D W and D 5 mean field limit: gap for ρ5 if m > 8â 2

39 Outlook non-degenerated masses higher correlation functions comparison with lattice data Damgaard, Heller, Splittorff (2011) single eigenvalue distributions

40 Thank you for your attention! Collaborators: Gernot Akemann Poul H. Damgaard Kim Splittorff Jacobus J. M. Verbaarschot Savvas Zafeiropoulos

41 Appendix

42 The Density ρ r â 2ν+1 2 ( x 2 p ν ρ r (x) = 1 ) â 2, λ2 1 exp [ x 2 16â 2 λ2 ] dλ, â 1 Θ(8â 2 x ) (2π) 3/2 2â, â 1 Kieburg, Verbaarschot, Zafeiropoulos (2011)