How far can you go? Surprises & pitfalls in three-flavour chiral extrapolations

Transcription

1 How far can you go? Surprises & pitfalls in three-flavour chiral extrapolations Sébastien Descotes-Genon Laboratoire de Physique Théorique CNRS & Université Paris-Sud 11, Orsay, France July Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 1 / 26

2 χpt and lattice χpt : structure of π, Kη interactions but not values of couplings Overlap with lattice? Light masses for χpt, but unknown constants Heavier masses for lattice, but extrapolation X chiral polynomial m phys lattice m q In particular, strong impact of chiral logarithms at NLO 4M 2 π log M2 π µ 2 can we see them on the lattice? Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 2 / 26

3 Common lore on χpt χpt good agreement with a large number of observables In the 80 s, with limited accuracy/statistics in experimental data Now, lattice aims at high accuracy, but can χpt deliver it? Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 3 / 26

4 Common lore on χpt χpt good agreement with a large number of observables In the 80 s, with limited accuracy/statistics in experimental data Now, lattice aims at high accuracy, but can χpt deliver it? Converges quickly : NLO is enough Assumption underlying χpt and used to determine LECs NNLO computations H. Bijnens, Prog.Part.Nucl.Phys.58: ,2007 (F 2 π) th = (F 2 π) exp [ ] [LO + NLO + NNLO] (M 2 π) th = (M 2 π) exp [ ] [ Fit 10 ] Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 3 / 26

5 Common lore on χpt χpt good agreement with a large number of observables In the 80 s, with limited accuracy/statistics in experimental data Now, lattice aims at high accuracy, but can χpt deliver it? Converges quickly : NLO is enough Assumption underlying χpt and used to determine LECs NNLO computations H. Bijnens, Prog.Part.Nucl.Phys.58: ,2007 (F 2 π) th = (F 2 π) exp [ ] [LO + NLO + NNLO] (M 2 π) th = (M 2 π) exp [ ] [ Fit 10 ] Proven by E865 data on K l4 decays New preliminary data from NA48 may modify this picture What is exactly determined on χpt through this measurement? Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 3 / 26

6 Common lore on χpt χpt good agreement with a large number of observables In the 80 s, with limited accuracy/statistics in experimental data Now, lattice aims at high accuracy, but can χpt deliver it? Converges quickly : NLO is enough Assumption underlying χpt and used to determine LECs NNLO computations H. Bijnens, Prog.Part.Nucl.Phys.58: ,2007 (F 2 π) th = (F 2 π) exp [ ] [LO + NLO + NNLO] (M 2 π) th = (M 2 π) exp [ ] [ Fit 10 ] Proven by E865 data on K l4 decays New preliminary data from NA48 may modify this picture What is exactly determined on χpt through this measurement? May prove wise not to use χpt blindly but really to test it on the lattice! Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 3 / 26

7 Three chiral limits of interest b,t c c c m u, m d 0 hadronic s u,d N=2 χ f s d u scale N=3 f u,d,s Λ H Λ QCD N f = 3 : m s 0 N f = 2 χ : m s physical N f = 2 lat : no dynamical s Two versions of χpt N f = 2 χ : π only d.o.f (few param. & processes) N f = 3 : π, K, η d.o.f (more param. & processes) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 4 / 26

8 From 2 to 3 massless flavours Σ(2; m s ) = lim 0 ūu 0 m u,m d 0 Σ(3) = Σ(2; 0) Σ(2 χ ) = Σ(2; ms phys ) Σ(2 lat ) = Σ(2; ) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 5 / 26

9 From 2 to 3 massless flavours Σ(2; m s ) = lim 0 ūu 0 m u,m d 0 Σ(3) = Σ(2; 0) Σ(2 χ ) = Σ(2; ms phys ) Σ(2 lat ) = Σ(2; ) Σ(2; m s ) = Σ(2; 0) + m s Σ(2; m s ) m s + O(m 2 s ) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 5 / 26

10 From 2 to 3 massless flavours Σ(2; m s ) = lim 0 ūu 0 m u,m d 0 Σ(3) = Σ(2; 0) Σ(2 χ ) = Σ(2; ms phys ) Σ(2 lat ) = Σ(2; ) Σ(2 χ ) = Σ(3) + m phys s lim i m u,m d 0 d 4 x 0 ūu(x) ss(0) 0 + O(m 2 s ) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 5 / 26

11 From 2 to 3 massless flavours Σ(2; m s ) = lim 0 ūu 0 m u,m d 0 Σ(3) = Σ(2; 0) Σ(2 χ ) = Σ(2; ms phys ) Σ(2 lat ) = Σ(2; ) Σ(2 χ ) = Σ(3) + m phys s lim i m u,m d 0 d 4 x 0 ūu(x) ss(0) 0 + O(m 2 s ) Σ(2 χ ) contains A genuine condensate Σ(3) An induced condensate m s (scalar 1/N c -suppressed) effect from sea s s-pairs u u (similar analysis with F (N f ) 2 = lim Nf F 2 π) s s Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 5 / 26

12 Which scenario for Nf = 2? From ππ scattering data (before preliminary NA48 K`4 data) (mu + md )Σ(2χ ) SDG,L.Girlanda,N.Fuchs,J.Stern = 0.81 ± 0.08 Fπ2 Mπ2... or larger G.Colangelo,J.Gasser,H.Leutwyler NA48 K`4 : increase in a00 = decrease in Σ(2χ ) (down to 0.7?) Se bastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 6 / 26

13 Which scenario for N f = 3? ūu mu,d =0,m s phys }{{} = ūu mu,d,s =0 }{{} sizeable Σ(2 χ ) Σ(3) + m s (ūu)( ss) +... Analysis of fermion det in terms of Dirac eigenvalues: Σ(3) Σ(2 χ ) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 7 / 26

14 Which scenario for N f = 3? ūu mu,d =0,m s phys }{{} = ūu mu,d,s =0 }{{} sizeable Σ(2 χ ) Σ(3) + m s (ūu)( ss) +... Analysis of fermion det in terms of Dirac eigenvalues: Σ(3) Σ(2 χ ) Σ (2 lat) χ Σ (2 ) Σ(3) Σ(m ) s 0 m s phys m s Σ(3) Σ(2 χ ) and (ūu)( ss) small Zweig rule OK for scalars No impact of strange sea quarks or Σ(3) < Σ(2 χ ) and (ūu)( ss) large Large Zweig-rule violation Strange sea quarks important Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 7 / 26

15 Which scenario for N f = 3 (2)? In the scalar sector, Zweig rule and large-n c badly violated described in O(p 4 ) χpt by L 4 and L 6 Large dispersive estimates of (ūu)( ss) L 6 ūu Nf =3 1/2 ūu N f =2 B.Moussallam;SDG Low-energy πk scattering from data > 1 GeV and Roy-Steiner eqs. B.Moussallam,SDG,P.Büttiker Naive comparison with O(p 4 ) series in subthreshold regions yields L 4 (M ρ ) = (0.53 ± 0.39) 10 3 Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 8 / 26

16 Which scenario for N f = 3 (2)? In the scalar sector, Zweig rule and large-n c badly violated described in O(p 4 ) χpt by L 4 and L 6 Large dispersive estimates of (ūu)( ss) L 6 ūu Nf =3 1/2 ūu N f =2 B.Moussallam;SDG Low-energy πk scattering from data > 1 GeV and Roy-Steiner eqs. B.Moussallam,SDG,P.Büttiker Naive comparison with O(p 4 ) series in subthreshold regions yields L 4 (M ρ ) = (0.53 ± 0.39) 10 3 Not a big deal! Really? Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 8 / 26

17 Consequences for three-flavour chiral series Fπ 2 = F (3) (m s + 2m)B 0 L mB 0 L 5 + O(mq) 2 B 0 = lim mu,md,m s 0 ūu /Fπ 2 m = m u = m d L 5 = L 5 (M ρ ) = O(p 4 ) LEC + χ log L 4 = L 4 (M ρ ) = O(p 4 ) LEC + χ log Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 9 / 26

18 Consequences for three-flavour chiral series Fπ 2 = F (3) (m s + 2m)B 0 L mB 0 L 5 + O(mq) 2 B 0 = lim mu,md,m s 0 ūu /Fπ 2 m = m u = m d L 5 = L 5 (M ρ ) = O(p 4 ) LEC + χ log L 4 = L 4 (M ρ ) = O(p 4 ) LEC + χ log L 4 is the awesome guy here Enhanced by m s, related to a Zweig-rule violating correlator... and guesstimated assuming Zweig rule in scalar sector Numerical competition between LO and NLO in chiral series of FP 2? Same (conflictual) relation between L 6 and FP 2M2 P Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 9 / 26

19 Consequences for three-flavour chiral series (2) F 2 π = F (3) (m s + 2m)B 0 L mB 0 L 5 + O(m 2 q) If strong convergence NLO LO, L 4 (M ρ ) = , L 5 (M ρ ) = , F (3) 2 F 2 π = F 2 (3) F 2 (3) + O(p 4 ) 1 8 2M2 K + M2 π Fπ 2 L 4 8 M2 π Fπ 2 L = (s s pairs) 0.04(other) + O(p 4 ) = NLO LO : contradiction! [same with Σ(3), F 2 πm 2 π and L 6 ] Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 10 / 26

20 Consequences for three-flavour chiral series (2) F 2 π = F (3) (m s + 2m)B 0 L mB 0 L 5 + O(m 2 q) If strong convergence NLO LO, L 4 (M ρ ) = , L 5 (M ρ ) = , F (3) 2 F 2 π = F 2 (3) F 2 (3) + O(p 4 ) 1 8 2M2 K + M2 π Fπ 2 L 4 8 M2 π Fπ 2 L = (s s pairs) 0.04(other) + O(p 4 ) = NLO LO : contradiction! [same with Σ(3), F 2 πm 2 π and L 6 ] Positive O(10 3 ) value of L µ 4 (M ρ) yields LO NLO Need to reassess common lore on the usage of chiral expansions Safer to impose only weak convergence NNLO LO+NLO Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 10 / 26

21 Resummed χpt Large effect of s s pairs = weak convergence = 2mB 0 M 2 π Resummed Chiral Perturbation Theory where effects of L 4 and L 6 resummed coping with competition between LO and NLO SDG, hep-ph/ Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 11 / 26

22 Resummed χpt Large effect of s s pairs = weak convergence = 2mB 0 M 2 π Resummed Chiral Perturbation Theory where effects of L 4 and L 6 resummed coping with competition between LO and NLO SDG, hep-ph/ ratios cannot be expanded (perturbatively) with neglect of remainders x 1 x x 1 + x/ r not fixed from M 2 K /M2 π at NLO, but free parameter r 1 = 2 F K M K F π M π 1 8 r r 2 = 2 F 2 K M2 K F 2 πm 2 π L 5 not determined from F K /F π at NLO Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 11 / 26

23 Resummed χpt Large effect of s s pairs = weak convergence = 2mB 0 M 2 π Resummed Chiral Perturbation Theory where effects of L 4 and L 6 resummed coping with competition between LO and NLO SDG, hep-ph/ ratios cannot be expanded (perturbatively) with neglect of remainders x 1 x x 1 + x/ r not fixed from M 2 K /M2 π at NLO, but free parameter r 1 = 2 F K M K F π M π 1 8 r r 2 = 2 F 2 K M2 K F 2 πm 2 π L 5 not determined from F K /F π at NLO Application to ππ and πk scatterings (dispersion relation + frequentist) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 11 / 26

24 Results for CL(r) 1 0,8 CL 0,6 0,4 0,2 pi pi pi K both r = m s m (free, not fixed by M 2 K /M2 π) r For ππ scattering, good agreement with previous work (lower bound on r) SDG, N.Fuchs, L.Girlanda, J.Stern, Eur.Phys.J.C34: ,2004 Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 12 / 26

25 Results for CL(X (3)) 1 CL 0,8 0,6 0,4 0,2 pi pi pi K both 0 0 0,2 0,4 0,6 0,8 1 X(3) X (3) = 2mΣ(3) F 2 πm 2 π = LO(F 2 πm 2 π) F 2 πm 2 π (free, not fixed close to 1 by Zweig rule) X (3) X (2) = 0.81 ± 0.08, fixed by ππ data SDG, N.Fuchs, L.Girlanda, J.Stern, Eur.Phys.J.C24: ,2002 Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 13 / 26

26 Results for CL(Z(3)) 1 CL 0,8 0,6 0,4 0,2 pi pi pi K both 0 0 0,2 0,4 0,6 0,8 1 Z(3) Z(3) = F 2 (3) F 2 π = LO(F 2 π) F 2 π (free, not fixed close to 1 by Zweig rule) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 14 / 26

27 Results for CL(Y (3)) 1 CL 0,8 0,6 0,4 0,2 pi pi pi K both 0 0 0,5 1 1,5 2 Y(3) Y (3) = 2mB 0 M 2 π = LO(M2 π) M 2 π (free, not fixed close to 1 by Zweig rule) No stringent experimental constrain that LO NLO Problems of convergence for three-flavour chiral series? Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 15 / 26

28 Help from lattice QCD Idea: Unquenched lattice to get m s -enhanced Zweig-rule violating effects Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 16 / 26

29 Help from lattice QCD Idea: Unquenched lattice to get m s -enhanced Zweig-rule violating effects Real QCD Lattice 2+1 flavours (m, m, m s ) ( m, m, m s ) m m m s Observables X X Fπ, 2 FK 2, F πm 2 π, 2 FK 2 M2 K Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 16 / 26

30 Help from lattice QCD Idea: Unquenched lattice to get m s -enhanced Zweig-rule violating effects Real QCD Lattice 2+1 flavours (m, m, m s ) ( m, m, m s ) m m m s Observables X X Fπ, 2 FK 2, F πm 2 π, 2 FK 2 M2 K Real QCD: three main parameters in chiral series, X (3) = 2mΣ(3) FπM 2 π 2, Z(3) = F 0 2 Fπ 2, r = m s m Lattice: additional parameter: q = m m s 1/ r Varying q changes the balance between LO, NLO(s s) and NLO(other) = disentangle the contributions to F 2 P and F 2 P M2 P SDG, Eur.Phys.J.C40:81-96,2005 Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 16 / 26

31 Ratios of interest R π = 1 F 2 M π π 2 q FπM 2 π 2 R K = 2 F 2 M K K 2 (q + 1) F 2 K M2 K Rpi z3=0.8 r= Rka z3=0.8 r= q to assess the role of sea s s pairs, negligible (full) or significant (dashed)? q NB : Applying ReχPT to compute (NLO) finite-volume corrections yields FπM 2 π, 2 FK 2 M2 K < 10% for L 2.5 fm Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 17 / 26

32 Conclusions Two chiral limits of interest N f = 3 : m u, m d, m s 0 N f = 2 χ : m u, m d 0, m s physical Σ(2 χ ) = Σ(3) + m s (ūu)( ss) + O(m 2 s ) Role of sea s s-pairs N f -dependence of order parameters Zweig rule violation in scalar sector Numerical competition for LO and NLO in 3-flavour chiral series? Troubles ahead : new systematics in extrapolations Alternative treatment of chiral series : Resummed χpt ππ and πk scatt. data: r 14.8 and 2mΣ(3)/(F 2 πm 2 π) 0.83 Info from lattice simulations with three dynamical flavours e.g., dependence of π, K observables on quark masses Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 18 / 26

33 Conclusions Two chiral limits of interest N f = 3 : m u, m d, m s 0 N f = 2 χ : m u, m d 0, m s physical Σ(2 χ ) = Σ(3) + m s (ūu)( ss) + O(m 2 s ) Role of sea s s-pairs N f -dependence of order parameters Zweig rule violation in scalar sector Numerical competition for LO and NLO in 3-flavour chiral series? Troubles ahead : new systematics in extrapolations Alternative treatment of chiral series : Resummed χpt ππ and πk scatt. data: r 14.8 and 2mΣ(3)/(F 2 πm 2 π) 0.83 Info from lattice simulations with three dynamical flavours e.g., dependence of π, K observables on quark masses Do no trust blindly χpt for extrapolations : Check it! Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 18 / 26

34 Backup Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 19 / 26

35 ππ and πk scatterings : data and dispersive approaches Data on ππ and πk scattering Extrapolation down to unphysical region where χpt converges well = Dispersion relations - Roy and Roy-Steiner equations m π a 3/ CA S-wave scattering lengths ChPT O(p 4 ) Roy-Steiner m π a 1/2 0 B. Ananthanarayan et al. Phys.Rept.353:207,2001 SDG, N.Fuchs, L.Girlanda, J.Stern, Eur.Phys.J.C24:469,2002 P. Buettiker, SDG and B. Moussallam, Eur.Phys.J.C33:409,2004 Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 20 / 26

36 ππ data (1) Analysis of E865 K l4 data Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 21 / 26

37 ππ data (2) E865 and NA48 preliminary K l4 data (e.m. corrections under discussion) Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 22 / 26

38 One-loop resummed χpt (1) All Green functions in generating functional at one loop Z = Z t + Z u + Z A Bare expansion in terms of chiral couplings F 0, B 0, L i... with LO masses 2 M π= Y (3)M 2 π, 2 M K = r Y (3)M2 π r = m s m, Y (3) = 2mB 0 M 2 π Where M 2 P MP 2 in bare expansion? Only if physically supported! Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 23 / 26

39 One-loop resummed χpt (1) All Green functions in generating functional at one loop Z = Z t + Z u + Z A Bare expansion in terms of chiral couplings F 0, B 0, L i... with LO masses 2 M π= Y (3)M 2 π, 2 M K = r Y (3)M2 π r = m s m, Y (3) = 2mB 0 M 2 π Where M 2 P MP 2 in bare expansion? Only if physically supported! Z u one-loop graphs with two O(p 2 ) vertices 2 L 2 L Unitarity cuts at ( MP + MQ) 2 converging to (M P + M Q ) 2 when higher orders into account = Replace M 2 P MP 2 everywhere in Z u Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 23 / 26

40 One-loop resummed χpt (2) Z A purely topological, no chiral couplings L 4 L 2 O(p 2 ) and O(p 4 ) tree graphs : chiral couplings Z t tree and tadpole graphs tadpoles : factors of log modified by higher orders but no physical expectation, so just a change in the log for simplicity 2 2 MP 32π 2 log MP µ 2 2 MP 32π 2 log M2 P µ 2 not M 2 P 32π 2 log M2 P µ 2! Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 24 / 26

41 Why is it a resummation? X (3) = 2mΣ(3) F 2 πm 2 π, Z(3) = F 2 (3) F 2 π, r = m s m F 2 π = F 2 πz(3) + 8Y (3)M 2 π[(r + 2) L 4 + L 5 ] + F 2 πe π F 2 πm 2 π = F 2 πm 2 πx (3) + 16Y 2 (3)M 4 π[(r + 2) L 6 + L 8 ] + F 2 πm 2 πd π Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 25 / 26

42 Why is it a resummation? X (3) = 2mΣ(3) F 2 πm 2 π, Z(3) = F 2 (3) F 2 π, r = m s m F 2 π = F 2 πz(3) + 8Y (3)M 2 π[(r + 2) L 4 + L 5 ] + F 2 πe π F 2 πm 2 π = F 2 πm 2 πx (3) + 16Y 2 (3)M 4 π[(r + 2) L 6 + L 8 ] + F 2 πm 2 πd π Y (3) = 2mB 0 M 2 π = 2[1 ɛ(r) d] [1 η(r) e] + [1 η(r) e] 2 + k [2 L 6 L 4 ] k 32(r + 2) M2 π F 2 π e, d e π,k, d π,k Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 25 / 26

43 Why is it a resummation? X (3) = 2mΣ(3) F 2 πm 2 π, Z(3) = F 2 (3) F 2 π, r = m s m F 2 π = F 2 πz(3) + 8Y (3)M 2 π[(r + 2) L 4 + L 5 ] + F 2 πe π F 2 πm 2 π = F 2 πm 2 πx (3) + 16Y 2 (3)M 4 π[(r + 2) L 6 + L 8 ] + F 2 πm 2 πd π Y (3) = 2mB 0 M 2 π = 2[1 ɛ(r) d] [1 η(r) e] + [1 η(r) e] 2 + k [2 L 6 L 4 ] k 32(r + 2) M2 π F 2 π e, d e π,k, d π,k If small vacuum fluctuations : k [2 L 6 L 4 ] 0 and Y (3) 1 = usual (iterative and perturbative) treatment of chiral series Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 25 / 26

44 Why is it a resummation? X (3) = 2mΣ(3) F 2 πm 2 π, Z(3) = F 2 (3) F 2 π, r = m s m F 2 π = F 2 πz(3) + 8Y (3)M 2 π[(r + 2) L 4 + L 5 ] + F 2 πe π F 2 πm 2 π = F 2 πm 2 πx (3) + 16Y 2 (3)M 4 π[(r + 2) L 6 + L 8 ] + F 2 πm 2 πd π Y (3) = 2mB 0 M 2 π = 2[1 ɛ(r) d] [1 η(r) e] + [1 η(r) e] 2 + k [2 L 6 L 4 ] k 32(r + 2) M2 π F 2 π e, d e π,k, d π,k If small vacuum fluctuations : k [2 L 6 L 4 ] 0 and Y (3) 1 = usual (iterative and perturbative) treatment of chiral series But k 1900 : L 6, L 4 = O(10 3 ) yields shift of Y (3) from 1, = resummation of k [2 L 6 L 4 ] needed Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 25 / 26

45 Variation of pion observables with quark masses Fpi^2Mpi^2 z3= Fpi^2 z3= q F 2 π = F(q = m/m s ) F 2 π M 2 π = F(q = m/m s ) q Infinite volume, continuum limit, no NNLO remainders Zweig-rule violation in 0 ++ : from none (full) to almost maximal (dotted) Varying r = m s /m: 20 (thin) and 30 (thick)... and similar results for kaons Sébastien Descotes-Genon (LPT-Orsay) Lattice and 3-flavour χpt 31/7/7 26 / 26