O(a 2 ) Corrections to the Propagator and Bilinears of Wilson / Clover Fermions

Transcript

1 O(a 2 Corrections to the ropagator and Bilinears of Wilson / Clover Fermions Martha Constantinou Haris anagopoulos Fotos Stylianou hysics Department, University of Cyprus In collaboration with members of the ETM Collaboration (V. Lubicz, V. Giménez, D. alao July 13, 2008

2 OUTLINE 1. Introduction Motivation - Necessity for off-shell improvement Existing wor - Complications with O(a 2 2. Description of the calculation: Wilson/clover/twisted mass fermions, Symanzi improved gluons Results 3. Fermion bilinears improvement The method reliminary results 4. Applications - Future wor

3 Introduction Motivation Necessity for off-shell improvement Space-time discretization leads to systematic errors in simulations Action improvement does not lead to off-shell improvement Improvement of operator matrix elements: minimum discretization errors ahead comparing with continuum results O(a 1 improvement: Automatic in many cases Symanzi s program: irrelevant operators in the action Twisted mass QCD: maximal twist

4 Introduction Existing wor (erturbative evaluation of fermion propagator, bilinears ΨΓΨ O(a 1 improvement only (1-loop: O(g0 2 arbitrary fermion mass (Aoi et al., Capitani et al. O(a 0 to 2-loops (Z Ψ, Z Γ : mass-independent scheme, m = 0 (Souroupathis - anagopoulos New complications with O(a 2 O(a 1 : No new types of IR divergences O(a 2 : Novel IR singularities Non-Lorentz invariant contributions, e.g., µ γµp3 µ p 2

5 Description of the calculation Clover fermions r: Wilson parameter f: flavor index c sw: free parameter S F = 1 X g 2 Tr ˆ1 Π µν(x + X X (4r + m ψ f (xψf(x x, µ, ν f x 1 X X» ψ f (x `r γ µ Uµ(xψf (x + µ + ψ f (x + µ `r + γ µ Uµ(x ψ f (x 2 f x, µ + i 4 c X X SW ψ f (xσ µν ˆFµν(xψ f (x f x, µ, ν We are interested in renormalization constants at the chiral limit Our calculations/results are identical also for the twisted mass action

6 Description of the calculation Clover fermions r: Wilson parameter f: flavor index c sw: free parameter S F = 1 X g 2 Tr ˆ1 Π µν(x + X X (4r + m ψ f (xψf(x x, µ, ν f x 1 X X» ψ f (x `r γ µ Uµ(xψf (x + µ + ψ f (x + µ `r + γ µ Uµ(x ψ f (x 2 f x, µ + i 4 c X X SW ψ f (xσ µν ˆFµν(xψ f (x f x, µ, ν We are interested in renormalization constants at the chiral limit Our calculations/results are identical also for the twisted mass action

7 Symanzi gluons " 2 X X S G = g 2 c 0 ReTr(1 U plaquette + c 1 ReTr(1 U rectangle plaquette rectangle # X X + c 2 ReTr(1 U chair + c 3 ReTr(1 U parallelogram chair parallelogram c 0 + 8c c 2 + 8c 3 = 1, c 2 = 0 Action c 0 c 1 c 3 laquette Symanzi TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = TILW, βc 0 = Iwasai DBW

8 Calculation of Feynman diagrams 1 2 : gluon field : fermion field Technical rocedure Wic contraction of appropriate vertices Simplification of color dependence, Dirac matrices and tensors Exploitation of symmetries of the theory and of the diagrams

9 Isolation of the logarithmic and non-lorentz invariant terms: Subtractions among the propagators ( 1 q 2 = 1ˆq q 2 1ˆq 2 D µν (q = δµν n «ˆqν (1 λ4ˆqµ ˆq 2 (ˆq D µν δµν ˆqν o (q (1 λ4ˆqµ ˆq 2 (ˆq 2 2 ˆq 2 = 4 µ sin2 ( qµ 2 D µν (q : Symanzi propagator q 2 : denominator of fermion propagator All primitive divergent integrals expressed in terms of Wilson gluon propagator 1/ˆq 2

10 Isolation of the logarithmic and non-lorentz invariant terms: Subtractions among the propagators ( 1 q 2 = 1ˆq q 2 1ˆq 2 IR degree of divergence reduced by 2 D µν (q = δµν n «ˆqν (1 λ4ˆqµ ˆq 2 (ˆq D µν δµν ˆqν o (q (1 λ4ˆqµ ˆq 2 (ˆq 2 2 ˆq 2 = 4 µ sin2 ( qµ 2 D µν (q : Symanzi propagator q 2 : denominator of fermion propagator All primitive divergent integrals expressed in terms of Wilson gluon propagator 1/ˆq 2

11 Isolation of the logarithmic and non-lorentz invariant terms: Subtractions among the propagators iteratively ( 1 q 2 = 1ˆq q 2 1ˆq 2 = 1ˆq 2 + ˆq2 q 2 (ˆq (ˆq2 q 2 2 (ˆq 2 2 q 2 =... IR degree of divergence reduced by 2 D µν (q = δµν n «ˆqν (1 λ4ˆqµ ˆq 2 (ˆq D µν δµν ˆqν o (q (1 λ4ˆqµ ˆq 2 (ˆq 2 2 ˆq 2 = 4 µ sin2 ( qµ 2 D µν (q : Symanzi propagator q 2 : denominator of fermion propagator All primitive divergent integrals expressed in terms of Wilson gluon propagator 1/ˆq 2

12 Analytical evaluation of primitive divergent integrals: Non-integer dimensions, D 4 Ultraviolet divergences are isolated à la Zimmermann Z π d 4 1 Example : I 1 = π (2π 4 ˆ 2 + a p 2 neededto O(a 2! Required operations: Z π Z π d 4 (2π 4 = <µ d 4 (2π 4 + Z π π d 4 Z (2π 4 <µ d 4! (2π 4 Z d 4 Z (2π 4 d D (2π D 1 ˆ 2 = ˆ 2 1 «2 repeatedly {z } IR degree of divergence reduced by 2 Z d D Z <µ (2π D = d D Z < (2π D d D µ< < (2π D UV-finite integrands

13 Analytical evaluation of primitive divergent integrals: Non-integer dimensions, D 4 Ultraviolet divergences are isolated à la Zimmermann Z π d 4 1 Example : I 1 = π (2π 4 ˆ 2 + a p 2 neededto O(a 2! Required operations: Z π Z π d 4 (2π 4 = <µ d 4 (2π 4 + Z π π d 4 Z (2π 4 <µ d 4! (2π 4 Z d 4 Z (2π 4 d D (2π D 1 ˆ 2 = ˆ 2 1 «2 repeatedly {z } IR degree of divergence reduced by 2 Z d D Z <µ (2π D = d D Z < (2π D d D µ< < (2π D UV-finite integrands

14 Analytical evaluation of primitive divergent integrals: Non-integer dimensions, D 4 Ultraviolet divergences are isolated à la Zimmermann Z π d 4 1 Example : I 1 = π (2π 4 ˆ 2 + a p 2 neededto O(a 2! Required operations: Z π Z π d 4 (2π 4 = <µ d 4 (2π 4 + Z π π d 4 Z (2π 4 <µ d 4! (2π 4 Z d 4 Z (2π 4 d D (2π D 1 ˆ 2 = ˆ 2 1 «2 repeatedly {z } IR degree of divergence reduced by 2 Z d D Z <µ (2π D = d D Z < (2π D d D µ< < (2π D UV-finite integrands

15 Analytical evaluation of primitive divergent integrals: Non-integer dimensions, D 4 Ultraviolet divergences are isolated à la Zimmermann Z π d 4 1 Example : I 1 = π (2π 4 ˆ 2 + a p 2 neededto O(a 2! Required operations: Z π Z π d 4 (2π 4 = <µ d 4 (2π 4 + Z π π d 4 Z (2π 4 <µ d 4! (2π 4 Z d 4 Z (2π 4 d D (2π D 1 ˆ 2 = ˆ 2 1 «2 repeatedly {z } IR degree of divergence reduced by 2 Z d D Z <µ (2π D = d D Z < (2π D d D µ< < (2π D UV-finite integrands

16 Most divergent piece: Z <µ d 4 1 (2π 4 2 ( + a p 2 = π 2 ln( a2 p 2 «µ 2 D-dimensional lattice integrals with explicit (polynomial external momentum dependence: Bessel functions D-dimensional UV-convergent integrals: evaluated with continuum methods (Chetyrin A host of 4-dimensional finite lattice integrals: numerical integration Z π π d 4 1 (2π 4 ˆ 2 + a p 2 = ln(a2p2 16π 2 + a (3 p 2 + a 2 µ p4 µ 384π 2 p 2 + O(a4 p 4 (Evaluated at 2 further orders in a, beyond the order at which an IR divergence initially sets in D 6

17 Convergent terms: Taylor expansion in the external momentum p and the lattice spacing up to O(a 3 p 3 Numerical integration over the internal momentum lattices with different size L 4 : L sets of the Symanzi parameters (actions: laquette, tree-level improved Symanzi, TILW, Iwasai, DBW2 Extrapolation of results to L combination of 51 functional forms of the type X e i,j L i lnl j i,j accurate estimation of systematic errors

18 Results C F = (N 2 1/(2N S 1 (p = i p + a 2 p2 i a2 6 p 3 p 3 = µ γµp3 µ λ = 1 (0 : Feynman (Landau gauge i p g2 C h F 16 π 2 ε (0, (6 λ + ε (0,2 c SW + ε (0,3 c 2 SW + λ ln(a2 p 2 i a p 2 g2 C F hε (1,1 16 π (2 λ + ε (1,2 c SW + ε (1,3 c 2 SW λ 3 c SW ln(a 2 p 2 i 2 i a 2 p 3 g2 C h F 16 π 2 ε (2, (9 λ + ε (2,2 c SW + ε (2,3 c 2 SW + ε (2,4 1 6 λ ln(a 2 p 2 i i a 2 p 2 p g2 C F 16 π 2 h ε (2, (9 λ + ε (2,6 c SW + ε (2,7 c 2 SW + ε (2, λ + c SW + c 2 SW ln(a 2 p 2 i i a 2 µ p p4 µ p 2 g 2 C h F 16 π 2 ε (2,9 5 i 48 λ

19 Results C F = (N 2 1/(2N S 1 (p = i p + a 2 p2 i a2 6 p 3 p 3 = µ γµp3 µ λ = 1 (0 : Feynman (Landau gauge i p g2 C h F 16 π 2 ε (0, (6 λ + ε (0,2 c SW + ε (0,3 c 2 SW + λ ln(a2 p 2 i a p 2 g2 C F hε (1,1 16 π (2 λ + ε (1,2 c SW + ε (1,3 c 2 SW λ 3 c SW ln(a 2 p 2 i 2 i a 2 p 3 g2 C h F 16 π 2 ε (2, (9 λ + ε (2,2 c SW + ε (2,3 c 2 SW + ε (2,4 1 6 λ ln(a 2 p 2 i i a 2 p 2 p g2 C F 16 π 2 h ε (2, (9 λ + ε (2,6 c SW + ε (2,7 c 2 SW + ε (2, λ + c SW + c 2 SW ln(a 2 p 2 i i a 2 µ p p4 µ p 2 g 2 C h F 16 π 2 ε (2,9 5 i 48 λ

20 Action ε (0,1 ε (0,2 ε (0,3 laquette ( ( (7 Symanzi ( ( (2 TILW, βc 0 = ( ( (2 TILW, βc 0 = ( ( (1 TILW, βc 0 = ( ( (5 TILW, βc 0 = ( ( (2 TILW, βc 0 = ( ( (3 TILW, βc 0 = ( ( (3 Iwasai ( ( (3 DBW ( ( (4 Action ε (1,1 ε (1,2 ε (1,3 laquette ( ( (4 Symanzi ( ( (1 TILW, βc 0 = ( ( (1 TILW, βc 0 = ( ( (1 TILW, βc 0 = ( ( (3 TILW, βc 0 = ( ( (3 TILW, βc 0 = ( ( (2 TILW, βc 0 = ( ( (3 Iwasai ( ( (8 DBW ( ( (1

21 Action ε (2,1 ε (2,2 ε (2,3 ε (2,4 ε (2,5 laquette ( ( (3 101/ (1 Symanzi ( ( ( ( (5 TILW ( ( ( ( ( (7 TILW ( ( ( ( ( (6 TILW ( ( ( ( ( (8 TILW ( ( ( ( ( (2 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 Iwasai ( ( ( ( (1 DBW ( ( ( ( (5 Action ε (2,6 ε (2,7 ε (2,8 ε (2,9 laquette ( (7 59/240-3/80 Symanzi ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 Iwasai ( ( ( (1 DBW ( ( ( (1

22 Fermion bilinears improvement Fermion bilinears improvement The method Scalar, seudoscalar, Vector, Axial, Tensor O Γ = ΨΓΨ Operator Γ Scalar ˆ1 seudoscalar γ 5 Vector γ µ Axial γ µ γ 5 Tensor σ µν γ 5

23 Fermion bilinears improvement O(a 2 improvement: O imp Γ = ΨΓΨ + a! 0 1 nx Γ,1 i Ψ Q i Γ,1 Ψ Xn + a i Ψ Γ,2 Q i Γ,2 Ψ A i=1 i=1 Q i Γ,1 (Qi Γ,2 : same symmetries as Γ, dimension 1 (2 higher include covariant derivatives Ψ ΓΨ: up to O(a 2 ΨQ i Γ,1 Ψ: up to O(a1 ΨQ i Γ,2 Ψ: up to O(a0 i Γ,1, i Γ,2 appropriately chosen to cancel all O(a2 terms

24 Fermion bilinears improvement O(a 2 improvement: O imp Γ = ΨΓΨ + a! 0 1 nx Γ,1 i Ψ Q i Γ,1 Ψ Xn + a i Ψ Γ,2 Q i Γ,2 Ψ A i=1 i=1 "local" operator Q i Γ,1 (Qi Γ,2 : same symmetries as Γ, dimension 1 (2 higher include covariant derivatives Ψ ΓΨ: up to O(a 2 ΨQ i Γ,1 Ψ: up to O(a1 ΨQ i Γ,2 Ψ: up to O(a0 i Γ,1, i Γ,2 appropriately chosen to cancel all O(a2 terms

25 Fermion bilinears improvement O(a 2 improvement: O imp Γ = ΨΓΨ + a! 0 1 nx Γ,1 i Ψ Q i Γ,1 Ψ Xn + a i Ψ Γ,2 Q i Γ,2 Ψ A i=1 i=1 "local" operator extended operators Q i Γ,1 (Qi Γ,2 : same symmetries as Γ, dimension 1 (2 higher include covariant derivatives Ψ ΓΨ: up to O(a 2 ΨQ i Γ,1 Ψ: up to O(a1 ΨQ i Γ,2 Ψ: up to O(a0 i Γ,1, i Γ,2 appropriately chosen to cancel all O(a2 terms

26 Fermion bilinears improvement erturbative calculation of local operators 1 2 : gluon field : fermion field : operator insertion erturbative calculation of extended operators Additional diagrams: 6 7 8

27 Fermion bilinears improvement erturbative calculation of local operators 1 2 : gluon field : fermion field : operator insertion erturbative calculation of extended operators Additional diagrams: 6 7 8

28 Fermion bilinears improvement Scalar: Tr[ Ψˆ1Ψ ](p = g2 C F 4 π 2 h + a 2 p 2 g2 C F 4 π 2 h ε (0,1 S ε (2,1 S reliminary results (8 λ + ε (0,2 S (5 λ + ε (2,2 S + a 2 µ p4 2 µ g CF h p 2 4 π 2 ε (2,4 1 i S 8 λ seudoscalar: Tr[γ 5 Ψγ 5 Ψ ](p = g2 C F 4 π 2 h + a 2 p 2 g2 C F 4 π 2 h ε (2,1 ε (0,1 + a 2 µ p4 2 µ g CF h p 2 4 π 2 ε (2,4 1 i 8 λ c SW + ε (0,3 c 2 S SW ln(a2 p 2 i (3 + λ c SW + ε (2,3 c 2 S SW λ + 3 «2 c SW ln(a 2 p 2 i C F = (N 2 1/(2N λ = 1 (0 : Feynman (Landau gauge (1 λ + ε (0,2 c 2 SW ln(a2 p 2 i (3 + λ (5 λ + ε (2,2 c 2 SW «4 λ ln(a 2 p 2 i

29 Fermion bilinears improvement Action ε (0,1 S ε (0,2 S ε (0,3 S laquette ( ( (6 Symanzi ( ( (5 TILW, βc 0 = ( ( (2 TILW, βc 0 = ( ( (3 TILW, βc 0 = ( ( (4 TILW, βc 0 = ( ( (1 TILW, βc 0 = ( ( (4 TILW, βc 0 = ( ( (3 Iwasai ( ( (4 DBW ( ( (4 Action ε (2,1 S ε (2,2 S ε (2,3 S ε (2,4 S laquette ( ( ( (1 Symanzi ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 TILW, βc 0 = ( ( ( (1 Iwasai ( ( ( (1 DBW ( ( ( (1

30 Fermion bilinears improvement Action ε (0,1 ε (0,2 ε (2,1 ε (2,2 ε (2,3 laquette ( ( ( ( (1 Symanzi ( ( ( ( (1 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 TILW ( ( ( ( ( (1 Iwasai ( ( ( ( (1 DBW ( ( ( ( (1

31 Fermion bilinears improvement Vector: pµ pν Tr[γ ν Ψγ µψ ](p = p 2 + δ µν g 2 C F 4 π 2 h ε (0,1 V + a 2 p 2 µ δµν g 2 C F 4 π 2 h 2 g CF h i 4 π 2 2 λ (4 λ + ε (0,2 V ε (2,1 V λ + ε(2,2 V + a 2 ρ δ p4 2 ρ g CF h µν p 2 4 π 2 ε (2,5 + 5 i V 48 λ + a 2 pµ p3 ν p 2 + a 2 p3 µ pν p 2 2 g CF h 4 π 2 + a 2 p 2 δ µν g 2 C F 4 π 2 h + a 2 p µ p ν g 2 C F 4 π 2 h ε (2,7 + 1 i V 12 λ + + ε (2,9 V c SW + ε (0,3 c 2 V SW λ ln(a2 p 2 i c SW + ε (2,3 c 2 V SW + ε(2,4 ln(a 2 p 2 i V 2 g CF h 4 π 2 + a 2 pµ pν ρ p4 2 ρ g CF h p 2 4 π (1 λ + ε (2,10 V ε (2, V 8 λ 5 12 c SW c2 SW ε (2,13 V (1 λ + ε (2,14 V ε (2, V 4 λ c SW c2 SW ε (2,6 + 1 i V 3 λ ε (2,8 V c SW + ε (2,11 c 2 V SW «ln(a 2 p 2 i c SW + ε (2,15 c 2 V SW «ln(a 2 p 2 i 5 24 λ i

32 Fermion bilinears improvement Axial: Tr[γ 5 γ ν Ψγ 5 pµ pν γ µψ ](p = p 2 δ µν g 2 C F 4 π 2 h ε (0,1 A a 2 p 2 µ δµν g 2 C F 4 π 2 h 2 g CF h i 4 π 2 2 λ (4 λ + ε (0,2 A ε (2,1 A λ + ε(2,2 A a 2 ρ δ p4 2 ρ g CF h µν p 2 4 π 2 ε (2,5 + 5 i A 48 λ a 2 pµ p3 ν p 2 a 2 p3 µ pν p 2 2 g CF h 4 π 2 a 2 p 2 δ µν g 2 C F 4 π 2 h a 2 p µ p ν g 2 C F 4 π 2 h ε (2,7 + 1 i A 12 λ + + ε (2,9 A c SW + ε (0,3 c 2 A SW λ ln(a2 p 2 i c SW + ε (2,3 c 2 A SW + ε(2,4 ln(a 2 p 2 i A 2 g CF h 4 π 2 a 2 pµ pν ρ p4 2 ρ g CF h p 2 4 π (1 λ + ε (2,10 A ε (2, A 8 λ c SW 1 4 c2 SW ε (2,13 A (1 λ + ε (2,14 A ε (2,16 3 A 4 λ 5 6 c SW 1 2 c2 SW ε (2,6 + 1 i A 3 λ ε (2,8 5 i A 24 λ c SW + ε (2,11 c 2 A SW «ln(a 2 p 2 i c SW + ε (2,15 c 2 A SW «ln(a 2 p 2 i

33 Fermion bilinears improvement Tensor: Tr[γ 5 σ ρτ Ψγ 5 σ µνψ ](p = {δ µτ δ νρ} a g 2 C F 4 π 2 h ε (0,1 T (2 λ + ε (0,2 c T SW + ε (0,3 c 2 T SW + (1 λ ln(a2 p 2 i + a 2 p 2 g 2 C h F µ {δµτ δνρ}a 4 π 2 ε (2, (2λ (1 c T SW (1 c 2 i SW + a 2 σ {δ µτ δ p4 2 σ g CF h νρ} a p 2 4 π 2 + a 2 p3 µ {δντ pρ}a 2 g CF h p 2 4 π 2 + a 2 p 2 {δ µτ δ νρ} a g 2 C F 4 π 2 h + a 2 p µ {δ ντ p ρ} a g 2 C F 4 π 2 h + ε (2,2 + 1 i T 3 λ + a 2 pµ {δντ p3 ρ }a 2 g CF h p 2 4 π 2 ε (2,4 + 1 i T 2 λ + a 2 p2 µ pν {δµτ pρ}a 2 g CF h p 2 4 π 2 ε (2,6 T (1 λ + ε (2,7 T ε (2,9 1 T 2 c SW ε (2,10 T «ln(a 2 p 2 i (2 λ + ε (2,11 T + (2 λ c SW ln(a 2 p 2 i c SW + ε (2,8 c 2 T SW ε (2,3 i T ε (2,5 5 i T 24 λ c SW + ε (2,12 c 2 T SW

34 Applications Applications Construction of bilinears with O(a 3 suppressed finite lattice effects O S imp = ΨΨ + a S,1 Ψ D Ψ O imp = Ψγ5 Ψ imp Oµ V = ΨγµΨ + a V,1 ΨD µψ O A µ imp = Ψγµγ 5 Ψ + a i A,1 Ψσµλ γ 5 D λψ imp Oµν T = Ψσµνγ 5 Ψ + a i T,1 Ψ γ µd ν γ νd µ γ 5 Ψ

35 Applications Applications Construction of bilinears with O(a 3 suppressed finite lattice effects O S imp = ΨΨ + a S,1 Ψ D Ψ + a 2 i=1 O imp n! X = Ψγ5 Ψ + a 2,2 i Ψ Q i,2 Ψ! nx S,2 i Ψ Q i S,2 Ψ i=1 imp Oµ V = ΨγµΨ + a V,1 ΨD µψ n! X + a 2 V,2 i Ψ Q i V,2 Ψ i=1 imp Oµ A = Ψγµγ 5 Ψ + a i A,1 Ψσµλ γ 5 D n! X λψ + a 2 A,2 i Ψ Q i A,2 Ψ i=1 imp Oµν T = Ψσµνγ 5 Ψ + a i T,1 Ψ γ µd ν γ n! X νd µ γ 5 Ψ + a 2 T,2 i Ψ Q i T,2 Ψ i=1

36 Future wor Future wor O(a 2 computation of higher dimension operators Ψ(xΓD µψ(x O(a 2 computation of 4-fermi operators Ψ(xΓΨ(x Ψ(xΓ Ψ(x

37 THANK YOU