Numerical Stochastic Perturbation Theory and The Gradient Flow

Transcription

1 Numerical Stochastic Perturbation Theory and The Gradient Flow Mattia Dalla Brida Trinity College Dublin, Ireland Dirk Hesse Universita degli Studi di Parma, Italia 31st International Symposium on Lattice Field Theory, 29th of July 2013, Mainz, Germany

2 Motivations Goal: The running coupling of QCD Finite-size scaling techniques provide a general solution to scale-dependent renormalization problems. Start: (U. Wolff 86; M. Lüscher, P. Weisz, U. Wolff 91) The finite-volume scheme i.e. the fields boundary conditions Schrödinger functional (K. Symanzik 81; M. Lüscher et. al. 92) The non-perturbative definition of the coupling gradient flow coupling (M. Lüscher 10) We consider pure SU(3) Yang-Mills theory From PT we can obtain important insights into this new tool NSPT is a natural framework for the gradient flow!

3 The gradient flow coupling The gradient flow evolves the gauge field as a function of the flow time parameter t 0 according to, t B µ = D ν G νµ + α 0 D µ ν B ν, B µ t=0 = A µ, where Gµν = µ B ν ν B µ + [B µ, B ν ], D µ = µ + [B µ, ]. Correlation functions of the field B are automatically finite for flow times t > 0, once the theory in 4d is renormalized in the usual way. Energy density E(t) = 1/2 Tr Gµν(t)G µν(t). (M. Lüscher, P. Weisz 11) From flow observables one can define a renormalized coupling, e.g, ḡ 2 (µ) N 1 t 2 E(t), µ = 1/8t, where N is such that ḡ 2 = g0 2 + O(g 0 4). (M. Lüscher 10)

4 The Schrödinger Functional and ḡ GF We consider SF boundary conditions with zero boundary fields, which have to be maintained at all flow times t. To apply finite-volume scaling, one has to run the renormalization scale with the size of the finite volume box given by L, µ = 1/L, and rescale with L all dimensionful parameters, e.g., c = 8t/L, T = L. (Z. Fodor et. al. 12; P. Fritzsch, A. Ramos 13) A c-family of running couplings can be introduced as, ḡ 2 GF(L) N 1 t 2 E(t, T /2) t=c 2 L 2 /8, where N depends on the specific scheme. (P. Fritzsch, A. Ramos 13)

5 The gradient flow on the lattice The gradient flow can be studied on the lattice as, t V xµ (t) = { g 2 0 xµ S G (V (t)) } V xµ (t), V xµ t=0 = U xµ, where is the Lie-derivative on the gauge group, and S G is, e.g., the Wilson gauge action = Wilson flow! (M. Lüscher 10) The SF boundary conditions for zero boundary fields, are realized on the lattice as, V µ (x + ˆkL, t) = V µ (x, t), V k (x, t) x0=0,t = I, t 0. We consider the regularized energy density, Ê(t, x0 ) = 1/2 Tr Ĝµν(t, x 0 )Ĝµν(t, x 0 ), where Ĝ is the clover definition of the field strength tensor corresponding to the lattice flow V.

6 Numerical Stochastic Perturbation Theory (D. Hesse s talk) Stochastic Quantization introduces a stochastic time t s, in which the fundamental fields evolve according to the Langevin equation, ts U xµ (t s ) = { xµ S G (U(t s )) + η xµ (t s ) } U xµ (t s ), where η is a Gaussian distributed noise field. (G. Parisi, Y.S. Wu 81) Considering the formal perturbative expansion, U(t s ) V + k>0 g k 0 U (k) (t s ), in the Langevin equation, one can obtain an approximate solution by solving the resulting hierarchy of equations order-by-order in g 0. Stochastic Perturbation Theory (SPT) [ lim O g0 k U (k) (t s ) ] = g t s η 0 k O k [U] = O[U]. k k NSPT considers a discrete approximation of the Langevin equation, and performs this program numerically! (F. Di Renzo et. al. 94)

7 NSPT and The Gradient Flow In the gradient flow, the noise field is not present and the initial distribution of the fundamental gauge field is taken into account, t V xµ (t) = { g 2 0 xµ S G (V (t)) } V xµ (t), V xµ t=0 = U xµ (t s ). Analogously to the Langevin equation, considering the formal perturbative expansion, V (t; t s ) V + k>0 g k 0 V (k) (t; t s ), V (k) t=0 = U (k) (t s ), k, one can obtain an approximate solution for the gradient flow! SPT for The Gradient Flow [ lim O g0 k V (k) (t; t s ) ] = t s η k k g k 0 O k [V (t)] = O[V (t)]. Using the machinery of NSPT this program can be implemented numerically! No gauge fixing step is needed along the flow.

8 Determination of t 2 Ê(t, T /2) = Ě (0) g Ě(1) g Ě(2) g (P. Fritzsch, A. Ramos 13) 0.009c = c = c = c = c = Ě m (0) L/a = ɛ 2

9 Determination of t 2 Ê(t, T /2) = Ě (0) g Ě(1) g Ě(2) g (P. Fritzsch, A. Ramos 13) 0.009c = c = c = c = c = Ě s (0) L/a = ɛ 2

10 Determination of t 2 Ê(t, T /2) = Ě (0) g Ě(1) g Ě(2) g c = c = c = c = c = Ě s (1) L/a = ɛ 2

11 Determination of t 2 Ê(t, T /2) = Ě (0) g Ě(1) g Ě(2) g Ě s (2) L/a = c = c = c = c = c = ɛ 2

12 Comparison with Monte Carlo data Ěs g 0 2 Ě s (0) c = c = c = c = g0 2

13 Comparison with Monte Carlo data O(g 4 0 ) O(g 6 0 ) c MC NSPT MC NSPT t 2 Ê s (86) (22) (11) (96) (15) (49) (19) (29) (18) (64) (21) (44) (14) (64) (17) (44) t 2 Ê m (90) (23) (11) (10) (17) (52) (21) (31) (22) (66) (27) (46) (21) (63) (25) (47) MC data obtained with a customized MILC code, results are for L/a = 8.

14 Noise to signal ratio vs c ), ɛ = ) / ( E (0) s N meas σ ( E (0) s L/a = 6 L/a = 8 L/a = 10 L/a = 12 L/a = c

15 Autocorrelation time vs c ] /τ meas, ɛ = 0.05, L/a = 8 [Ěm τint O(g 2 0) O(g 4 0) O(g 6 0) c

16 Conclusions Outlook Mild extrapolations, and good statistical behavior for the flow observables we have considered. NPST provides a natural setup for a (numerical) perturbative solution of the gradient flow. The setup is flexible: different action regularizations, boundary conditions, and observables can be implemented easily. Continuum limit extrapolations Cut-off effects in the step-scaling function Λ GF and PT relation to other schemes require bigger lattices (n.b. cost (L/a) 6 ) Inclusion of fermions and QCD

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18 Numerical precision The most expensive simulations were performed at L/a = 12. The results of the extrapolations are, c Ě (0) s δě (0) s Ě (0) s Ě (1) s δě (1) s Ě (1) s (38) 0.5% (37) 0.6% (66) 0.8% (45) 0.8% (78) 1.2% (85) 1.7% (62) 1.5% (11) 3.2% Autocorrelation τ int (τ int /10 LDU) and N eff = N meas /(2 τ int ), ɛ ɛ Ě (0) s τ int c= (94) 4.10(87) 2.07(24) τ int c= (22) 5.1(12) 2.66(33) N eff c= (24) 112(24) 550(63) N eff c= (16) 89(21) 429(53) Ě (1) s τ int c= (14) 3.54(70) 2.54(32) τ int c= (28) 21.3(75) 4.85(78) N eff c= (20) 130(26) 448(56) N eff c= (14) 21.6(76) 235(38)

19 Numerical effort Ě (0) s ɛ L = 4 L/a = 6 L/a = 8 L/a = 12 τ int c= (28) 0.677(19) 0.541(14) τ int c= (48) 0.837(25) 0.567(14) N eff c= (45) N eff c= (38) τ int c= (78) 0.981(30) 0.663(19) τ int c= (17) 1.651(64) 0.842(26) N eff c= (24) N eff c= (18) 7250(28) τ int c= (29) 1.66(12) 0.919(51) τ int c= (54) 2.61(22) 1.390(89) N eff c= (84) 1810(12) 5960 N eff c= (66) 1133(97) 2140(14) τ int c= (94) 4.10(87) 2.07(24) τ int c= (22) 5.1(12) 2.66(33) N eff c= (24) 112(24) 550(63) N eff c= (16) 89(21) 429(53) Ě (1) s ɛ L = 4 L/a = 6 L/a = 8 L/a = 12 τ int c= (54) 0.921(28) 0.636(16) τ int c= (17) 2.030(83) 1.144(37) N eff c= (36) N eff c= (22) 6150(25) 10910(35) τ int c= (10) 1.273(44) 0.762(21) τ int c= (58) 4.20(23) 2.32(10) N eff c= (22) 9400(33) N eff c= (11) 2850(16) 5170(22) τ int c= (46) 1.99(15) 1.143(69) τ int c= (23) 6.00(69) 2.99(26) N eff c= (70) 1490(11) 2610(16) N eff c= (34) 493(57) 995(87) τ int c= (14) 3.54(70) 2.54(32) τ int c= (28) 21.3(75) 4.85(78) N eff c= (20) 130(26) 448(56) N eff c= (14) 21.6(76) 235(38) N eff = N meas 2 τ int

20 Determination of t 2 Ê(t, T /2) = Ě (0) g Ě(1) g Ě(2) g (P. Fritzsch, A. Ramos 13) L = 4 L/a = 6 L/a = 8 L/a = 12 c δě s (0) (23) (26) (31) (30) (28) δě m (0) (23) (25) (30) (28) (26) c δě s (0) (22) (18) (14) (17) (20) δě m (0) (19) (24) (22) (22) (23) c δě s (0) (63) (27) (36) (43) (48) δě m (0) (59) (23) (45) (57) (64) c δě s (0) (68) (68) 0.011(11) 0.014(14) 0.015(16) δě m (0) (80) (87) 0.007(14) 0.009(18) 0.011(21) δě s,m (0) = Ě(0) s,m ˆN s,m 1

21 Autocorrelation time vs c ] /τ meas, ɛ = 0.05 [ E (0) s τint L/a = 8 L/a = 12 L/a = 4 L/a = c

22 Autocorrelation time vs L ] /τmeas, ɛ = 0.05 (a/l) 2 (0) τint [Ě s c = 0.3 c = 0.4 c = L/a