Finite volume effects for masses and decay constants

Transcription

1 Finite volume effects for masses and decay constants Christoph Haefeli University of Bern Exploration of Hadron Structure and Spectroscopy using Lattice QCD Seattle, March 27th, 2006 Introduction/Motivation Finite volume effects in Chiral Perturbation Theory Asymptotic formulae Two-loop calculation of M π (L) Summary Work done in collaboration with G. Colangelo, S. Dürr

2 Motivation to study finite volume effects L = 2fm M π = 200MeV (1) Finite volume effect of the pion mass: Finite volume effect is small M π (L) M π M π (1) = 4% But needs to be taken into account for very precise lattice calculations. Extrapolation L is by no means straightforward ChPT provides the proper analytic framework C.Haefeli - Seattle, p.2

3 Introduction: ChPT in finite volume Expansion in m q /Λ and p/λ Quantized momenta in finite volume: p = 2π L n Condition of applicability for ChPT: m q Λ and 2π L Λ Λ 4πF π 2LF π 1 C.Haefeli - Seattle, p.3

4 Introduction: ChPT in finite volume Expansion in m q /Λ and p/λ Quantized momenta in finite volume: p = 2π L n Condition of applicability for ChPT: m q Λ and 2π L Λ Λ 4πF π 2LF π 1 Once this condition is respected, we still have two different physical situations: LM π 1 p regime : M π 1 L O(p) LM π 1 ε regime : M π 1 L 2 O(ε2 ) C.Haefeli - Seattle, p.3

5 p-regime: M π L 1 Calculational rule in ChPT for isotropic finite box with periodic boundary conditions: Lagrangian : Propagator : L L eff = L eff G L (x 0, x) = n Z 3 G(x 0, x + nl) J.Gasser, H.Leutwyler 87 Implication for perturbative calculation: d 4 p L< f(p) = (2π) 4 dp0 2π 1 L 3 p f(p) ( ) = d 4 p (2π) 4 f(p) n Z 3 e i p nl ( ): Poisson summation formula C.Haefeli - Seattle, p.4

6 p-regime: M π L 1 Calculational rule in ChPT for isotropic finite box with periodic boundary conditions: Lagrangian : Propagator : L L eff = L eff G L (x 0, x) = n Z 3 G(x 0, x + nl) J.Gasser, H.Leutwyler 87 Examples: M π (L) = M π ( ξ g(λ) + O(ξ2 ) ), F π (L) = F π ( 1 ξ g(λ) + O(ξ 2 ) ), ξ = M2 π (4πF π ) 2, λ = M π L, g(λ) = n\{0} 0 dx e 1 x x 4 n2 λ 2. C.Haefeli - Seattle, p.5

7 Outline Introduction/Motivation Finite volume effects in Chiral Perturbation Theory Asymptotic formulae Two-loop calculation of M π (L) Summary C.Haefeli - Seattle, p.6

8 Lüscher s formula (applied for M K ) Leading corrections for M π L 1: M K (L) M K = + O(e 2M π L ) = d 4 l ei l nl Mπ 2 + l 2 + O(e 2M π L ), n =1 Integrations over spatial momenta l can be performed within claimed accuracy. It remains the integral over l 0. 3 M K (L) M K = 16π 2 M K L dl 0 e M 2 π +l2 0 L T I=0 πk (il 0 )+O(e 2 M π L ), M. Lüscher 86 C.Haefeli - Seattle, p.7

9 Lüscher s formula: P = π, K, η: 3 M P (L) M P = 16π 2 M P L dy e M 2 π +y2l T I=0 πp (iy) + O(e 2 M π L ). The coupling to the ligthest particle matters, i.e. the pions. Leading corrections of order exp( M π L). C.Haefeli - Seattle, p.8

10 Å Å Lüscher s formula: P = π, K, η: 3 M P (L) M P = 16π 2 M P L dy e M 2 π +y2l T I=0 πp (iy) + O(e 2 M π L ). The coupling to the ligthest particle matters, i.e. the pions. Leading corrections of order exp( M π L). The formula expresses the corrections over a (analytically continued) physical amplitude. Analyticity properties of T I=0 πp (ν): ÁÑ Integration line Ê ÈË Ö Ö ÔÐ Ñ ÒØ C.Haefeli - Seattle, p.8

11 Lüscher s formula or ChPT?: 3 M π,lüscher = 16π 2 M π L dy e M 2 π +y2l T I=0 ππ (iy) + O(e 2M π L ), M π,chpt = ξ M π 4 g(m πl) + O(ξ 2 ). The two formulae give the leading term in two different expansions n one loop ChPT chiral order asymptotic formula C.Haefeli - Seattle, p.9

12 Extension of Lüscher s formula One can extend the formula so that it contains contributions from all n of a single propagator: 1 M π = 32π 2 M π L n =1 m( n ) n dye M 2 π +y2 n L T I=0 ππ (iy)+o(e 2M πl ) The extension does not provide all exponentially subleading terms! n one loop ChPT chiral order asymptotic formula partial resummation C.Haefeli - Seattle, p.10

13 Non-leading exponential terms in M π (L) 0.1 LO, n = 1 NLO, n = 1 NNLO, n = 1 R Mπ 0.01 L = 2 fm L = 3 fm L = 4 fm M π (GeV) R Mπ = M π(l) M π M π C.Haefeli - Seattle, p.11

14 Non-leading exponential terms in M π (L) 0.1 LO, all n NLO, all n NNLO, all n R Mπ 0.01 L = 2 fm L = 3 fm L = 4 fm M π (GeV) R Mπ = M π(l) M π M π C.Haefeli - Seattle, p.12

15 Asymptotic formula for decay constants In Euclidean space: π(p) A µ (0) 0 L = ip µ F π (L). 3 M π = 16π 2 M π L Z dy e M 2 π +y2l T ππ (iy) + O(e 2 M π L ), = + O(e 2M π L ), T ππ (ν) ππ ππ. M. Lüscher 86 F π = 3 8π 2 M π L Z dy e M 2 π +y2l N F (iy) + O(e 2 M π L ), = + O(e 2M π L ), N F (ν) 3π A µ 0 A(τ 3πν τ ). G.Colangelo, C.H. 04 C.Haefeli - Seattle, p.13

16 Definition of the amplitude N F (ν) F π = 3 8π 2 M π L dy e M 2 π +y2l N F (iy) + O(e 2 M π L ), N F (ν) is the subtracted 3π A µ 0 matrix element in the forward kinematic region. N F (ν) = i pµ M π T ( (2π) I=0 ) ππ I=0 π A µ 0 iq µ F π Mπ 2 Q 2 = G.Colangelo and C.H. 04 C.Haefeli - Seattle, p.14

17 Finite volume corrections for F π 0.1 L= 2fm LO, n=1 NLO, n=1 LO, all n NLO, all n -R Fπ 0.01 L = 3fm L = 4fm M π (GeV) R Fπ = F π(l) F π F π C.Haefeli - Seattle, p.15

18 Universality of asymptotic formula Masses (Lüscher): Decay constants: M scattering M π ππ ππ M K πk πk M η πη πη M B πb πb M N πn πn F decays F π τ 3π F K K l4 F η η l4 F B B l4 G. Colangelo, C.H. 04; G. Colangelo, S. Dürr, C.H. 05 G. Colangelo, A. Fuhrer, S. Lanz, work in progress C.Haefeli - Seattle, p.16

19 Outline Introduction/Motivation Finite volume effects in Chiral Perturbation Theory Asymptotic formulae Two-loop calculation of M π (L) Summary C.Haefeli - Seattle, p.17

20 Motivation for two-loop calculation 0.1 NNLO, n = 1, asympt. one-loop NNLO, all n, asympt. M π L = 2 R Mπ 0.01 L = 2 fm R Mπ = M π(l) M π M π M π (GeV) The best estimate goes beyond the one-loop level, but misses terms at the two-loop level Only a full two-loop calculation can clarify, how good the best estimate is C.Haefeli - Seattle, p.18

21 Two-loop calculation L L eff = L eff, G L (x 0, x) = n Z 3 G(x 0, x + nl) M π (L) defined as pole of the connected correlation function G L (p) 1 = M 2 + p 2 Σ L (p 2 ), Σ L (p 2 ) : self-energy in finite volume Motivated from asymptotic formula: split self-energy in terms of number of propagators in finite volume M π (L) 2 = M 2 π Σ (1) Σ (2) C.Haefeli - Seattle, p.19

22 Two-loop calculation Σ (1) = = n d 4 q ( e i q nl M 2 π + q 2 ) Im q 1 Σ (1) = I p + I c + O(ξ 3 ) I p = residue of contour integration asymptotic formula I c = from integration along the cut Re q 1 C.Haefeli - Seattle, p.20

23 Two-loop calculation Σ (1) = = n d 4 q ( e i q nl M 2 π + q 2 ) Σ (1) = I p + I c + O(ξ 3 ) I c = im2 π 32π 3 M π L n=1 m(n) n dy 4 d s e n( s+y 2 )M π L s + 2iy disc [ T ππ ] G.Colangelo and C.H. 06 Numerically, the contributions from I c are very small C.Haefeli - Seattle, p.21

24 Two-loop calculation Σ (2) can be expressed in terms of a three particle scattering amplitude... Σ (2) = 1 8 n,r 0 n r d 4 q (2π) 4 d 4 k (2π) 4 e iqnl (M 2 π + q 2 ) e ikrl (M 2 π + k 2 ) a 1 1 b b }{{} a ˆT πππ C.Haefeli - Seattle, p.22

25 Two-loop calculation Σ (2) can be expressed in terms of a three particle scattering amplitude... Σ (2) = 1 8 n,r 0 n r d 4 q (2π) 4 d 4 k (2π) 4 e iqnl (M 2 π + q 2 ) e ikrl (M 2 π + k 2 ) a 1 1 b b }{{} a ˆT πππ... that again, needs to be subtracted, ˆT πππ = 3π 3π R M 2 π + Q 2 a a a a a a b b 1 1 = b b b b C.Haefeli - Seattle, p.23

26 Two-loop calculation Σ (2) can be expressed in terms of a three particle scattering amplitude... Σ (2) = 1 8 n,r 0 n r d 4 q (2π) 4 d 4 k (2π) 4 e iqnl (M 2 π + q 2 ) e ikrl (M 2 π + k 2 ) a 1 1 b b }{{} a ˆT πππ Σ (2) = M2 πξ 2 8 [ 9g(λπ ) 2 λ π g(λ π )g (λ π ) ] + M 2 πξ 2 + O(ξ 3 ) G.Colangelo and C.H. 06 includes sunset-type contributions that cannot be represented in terms of g(λ π ), but may be evaluated numerically C.Haefeli - Seattle, p.24

27 Numerical evaluation 0.1 LO NLO, asympt. NLO NNLO, asympt. NNLO asympt. + NLO non-asympt. M π L = 2 R Mπ = M π(l) M π M π R Mπ 0.01 L = 3fm L = 2fm L = 4fm M π (GeV) Resummed asymptotic formula is very accurate for M π L > 2 C.Haefeli - Seattle, p.25

28 Summary We discussed: Role of finite volume effects in lattice calculations ChPT as proper framework The need to go beyond leading order The successful results for M π,l,m K,L,M η,l,f π,l,f K,L when combined with an asymptotic formula à la Lüscher First two-loop calculation in L < : M π,l C.Haefeli - Seattle, p.26