Test of the Schrödinger functional with chiral fermions in the Gross-Neveu model. B. Leder. Humboldt Universität zu Berlin

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1 Test of the Schrödinger functional with chiral fermions in the Gross-Neveu model B. Leder Humboldt Universität zu Berlin XXV International Symposium on Lattice Field Theory Regensburg, July 30, 2007 LPHA ACollaboration B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

Leder Humboldt Universität zu Berlin XXV International Symposium on Lattice

2 The Schrödinger functional Z = DψDψ e S F, S F = T 0 d x 0 d d x ψ(x) {γ µ D µ + m} ψ(x) P + ψ = 0 ψp = 0 L d x 0 = 0 ψγ 0 γ 5 ψ x 0 = T P ψ = 0 ψp + = 0 ψ(x + L) = e iθ ψ, θ [0, 2π) SF b.c. need no tuning (SF universality class) P ± = 1 2 (1 ± γ 0) finite number of boundary counterterms needed finite size effects are reinterpreted as renormalisation effect: µ = 1/L used to determine scale dependence of the fundamental parameters of LPHA QCD (pioneered by ACollaboration) B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

need no tuning (SF universality class) P ± = 1 2 (1 ± γ 0) finite number of boundary counterterms needed finite size effects are

3 The Schrödinger functional and chiral fermions P + ψ = 0 ψp = 0 L d x 0 = 0 ψγ 0 γ 5 ψ x 0 = T P ψ = 0 ψp + = 0 γ 5 D + D γ 5 = a Dγ 5 D But: SF b.c. break chiral symmetry B supported at the boundary with exponentially decaying tails Solution: [Lüscher, 2006] { D N = 1 ā (U + γ 5U γ 5 ) } A U=, ā= a A A+caP 1+s, A=1+s ad W Pψ(x)= 1 a {δx 0,aP ψ(x) x 0 =a+δx 0,T ap+ψ(x) x 0 =T a} B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

break chiral symmetry B supported at the boundary with exponentially decaying tails Solution: [Lüscher, 2006] { D N = 1

4 The Schrödinger functional and chiral fermions P + ψ = 0 ψp = 0 L d x 0 = 0 ψγ 0 γ 5 ψ x 0 = T P ψ = 0 ψp + = 0 γ 5 D + D γ 5 = a Dγ 5 D + B But: SF b.c. break chiral symmetry B supported at the boundary with exponentially decaying tails Solution: [Lüscher, 2006] { D N = 1 ā (U + γ 5U γ 5 ) } A U=, ā= a A A+caP 1+s, A=1+s ad W Pψ(x)= 1 a {δx 0,aP ψ(x) x 0 =a+δx 0,T ap+ψ(x) x 0 =T a} B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

5 Free fermions D N = 1 ā { (U + γ 5U γ 5 ) } A U=, ā= a A A+caP 1+s, A=1+s ad W A A + cap bounded by (1 s ) 2 for s < 1 and c 1 spatial part of D N can be diagonalised (periodic b.c.) remaining (d + 1)(T 1) (d + 1)(T 1) matrix must be computed numerically modification to GW relation γ 5 D N + D N γ 5 = a D N γ 5 D N + B 0 2 x0 y0 = 0 x0 y0 = T/4 x0 y0 = T/2 log 10 ( a B(x, y) ) distance from boundary at x0 = 0 B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

6 Chiral Gross-Neveu model S c CGN = dx 2 { ψ γ µ µ ψ 1 2 g2 (O SS O PP ) 1 2 g2 VO VV } [Thirring, 1958, Gross, Neveu, 1974] O SS = (ψψ) 2, O PP = (ψγ 5 ψ) 2, O VV = µ (ψγ µ ψ) 2 continuous chiral U(1) symmetry asymptotically free for N 2 (coupling g 2 ) Wilson fermions: S CGN,W = a 2 x { ψ (DW + m 0 ) ψ 1 2 g2 (O SS O PP ) 1 2 δ2 PO PP 1 2 g2 VO VV } Ginsparg-Wilson fermions: S CGN,GW = a 2 x { } ψ D N ψ 1 2 g2 (Ô SS Ô PP ) 1 2 g2 VÔVV ˆψ = (1 a 2 D N)ψ B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

CGN,W = a 2 x { ψ (DW + m 0 ) ψ 1 2 g2 (O SS O PP ) 1 2 δ2 PO PP 1 2 g2 VO VV } Ginsparg-Wilson fermions: S CGN,GW = a 2 x { } ψ D N ψ 1

7 Chiral Ward identity 1 Continuum: µ A µ (x) O(y) = 2m P(x) O(y), x y Wilson fermions (m R = 0): [Leder, 2005] (O) R µ (A µ ) R (x) = O(a) critical mass m c and symmetric coupling δ P,s to second order in PT Correlation functions: f A (x 0 ) = a2 2N and similar for pseudo scalar density f P (x 0 ) Axial Current Insertion { }} { ψ(x) γ 0 γ 5 ψ(x) (y 1 ) γ 5 (z 1 ) } {{ } y 1,z 1 PS Boundary State B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

2N and similar for pseudo scalar density f P (x 0 ) Axial Current Insertion { }} { ψ(x) γ 0 γ 5 ψ(x) (y 1 ) γ 5 (z 1 ) }

8 Chiral Ward identity 2 Chiral fermoins: Compute in lattice PT r(x 0 ) = a 0 f A (x 0 ) 2f P (x 0 ) = r 0 (x 0 ) + r 1 (x 0 )g 2 + O(g 4 ) as measure of the deviation from continuum form r 0 vanishes identically, r 1 see below Wilson chiral 10 2 chiral r1(x0) r1(x0) x0/a x0/a B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

05 0.04 Wilson chiral 10 2 chiral 0.03 10 3 r1(x0) 0.02 0.01 r1(x0) 10 4 0 10 5 0.01 10 6 0.

9 Renormalised coupling Correlation functions: f 4 = a4 2(N 2 1)L 2 u 1v 1y 1z 1 (u1 ) Γ Bλ a (v 1 ) (y 1 ) Γ B λ a (z 1 ) f 2 = a2 NL (u1 ) (z 1 ), (f 2 ) R = Z 2 f 2 and (f 4 ) R = Z 4 f 4 u 1z 1 Wilson: R(θ) = (f 4) R (f 2 ) 2 1 = f 4 R (f 2 ) 2 1 at m R = 0 b 0 = N/π g 2 a 0 = g 2 w + g 4 w b 0 ln(a/l) + c w g + O(g 6 ), δ2 V a 0 = δ 2 V,w + c w V c w g /N + O(g 6 ) Ginsparg-Wilson: b 0 = N/π g 2 a 0 = g 2 gw + g 4 gw b 0 ln(a/l) + c gw g + O(g 6 a 0 ), δ2 V = δv,gw 2 + c gw V cgw g /N + O(g 6 ) B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

0 ln(a/l) + c w g + O(g 6 ), δ2 V a 0 = δ 2 V,w + c w V c w g /N + O(g 6 ) Ginsparg-Wilson: b 0 = N/π g 2 a 0 = g 2 gw + g 4 gw b 0 ln(a/l) + c gw g

10 A universal quantity Expansion: F RGI (θ, a/l) = g 2 (θ, a/l) g 2 (θ 0, a/l), θ 0 = 1 F RGI (θ, a/l) = F (1) 1 (θ, a/l) g 4 + F (1) 2 (θ, a/l) g 2 g 2 V + F (1) 3 (θ, a/l) g 4 V + O(g 6 ) Coefficients given by finite parts: F (1) i (θ, a/l) = c i (θ, a/l) c i (θ 0, a/l) Universal quantity: F (1) i,w (θ, a/l)! F (1) i,gw (θ, a/l) θ F (1) 1,w (6) (5) (3) (1) (1) F (1) 1,gw 5.078(2) 4.469(2) 3.638(2) 2.762(2) 1.966(2) B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

Universal quantity: F (1) i,w (θ, a/l)! F (1) i,gw (θ, a/l) θ 0.1 0.2 0.3 0.4 0.5 F (1) 1,w 5.0797(6) 4.4704(5) 3.6395(3) 2.7629(1) 1.

11 Summary Formulation of the SF for fermionic models of the Gross-Neveu type. In 1-loop lattice PT the theory is renormalisable with Wilson and Ginsparg-Wilson fermions. First check of the recently proposed chiral SF Dirac operator beyond the free theory. Chiral Ward identities take their continuum value up to exponentially decaying tails. Outlook: O(a) improvement non-perturbative tests of universality: staggered fermions, mixed actions QCD B. Leder (HU Berlin) Test of the SF with chiral fermions in the GN model Lattice / 10

First check of the recently proposed chiral SF Dirac operator beyond the free theory.

Standard Model of Particle Physics

Standard Model of Particle Physics Chris Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ UK SUSSP61, St Andrews August 8th 3rd 006 Contents 1. Spontaneous Symmetry