PHOTONS and PARTIAL QUENCHING η 3π AT TWO LOOPS: STATUS and PRELIMINARY RESULTS

Transcription

1 lavi net PHOTONS and PARTIAL QUENCHING η 3π AT TWO LOOPS: STATUS and PRELIMINARY RESULTS Johan Bijnens Lund University bijnens Various ChPT: bijnens/chpt.html Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.1/40

2 Overview Chiral Perturbation Theory Photons and Partial Quenching What is partially quenched? PQChPT: what we did earlier Electromagnetism in Lattice QCD Electromagnetism in PQChPT Lagrangians Masses, Decay Constants M 2 and F from quenched photons η 3π What is known Status of p 6 calculation Conclusions Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.2/40

3 Chiral Perturbation Theory Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown Power counting: Dimensional counting Expected breakdown scale: Resonances, so M ρ or higher depending on the channel Chiral Symmetry QCD: 3 light quarks: equal mass: interchange: SU(3) V But L QCD = [i q L D/q L + i q R D/q R m q ( q R q L + q L q R )] q=u,d,s So if m q = 0 then SU(3) L SU(3) R. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.3/40

4 Chiral Perturbation Theory qq = q L q R + q R q L 0 SU(3) L SU(3) R broken spontaneously to SU(3) V 8 generators broken = 8 massless degrees of freedom and interaction vanishes at zero momentum Power counting in momenta: p 2 (p 2 ) 2 (1/p 2 ) 2 p 4 = p 4 1/p 2 d 4 p p 4 (p 2 ) (1/p 2 )p 4 = p 4 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.4/40

5 What is Partially Quenched? In Lattice gauge theory one calculates 0 (uγ 5 d)(x)(dγ 5 u)(0) = [dq][dq][dg](uγ 5 d)(x)(dγ 5 u)(0)e i [dq][dq][dg]e i d 4 yl QCD d 4 yl QCD for Euclidean separations x Integrals performed after rotation to Euclidean (note that I use Minkowski notation throughout) (note that D/ includes also mass term) Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.5/40

6 What is Partially Quenched? [dq][dq][dg](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg]e i R d 4 x( 1/4)G µν G µν (D/ u G ) 1 (x, 0)(D/ d G ) 1 (0,x) det (D/ G ) QCD Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.6/40

7 What is Partially Quenched? [dq][dq][dg](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg]e i R d 4 x( 1/4)G µν G µν (D/ u G ) 1 (x, 0)(D/ d G ) 1 (0,x) det (D/ G ) QCD [dg] done via importance sampling Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.6/40

8 What is Partially Quenched? [dq][dq][dg](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg]e i R d 4 x( 1/4)G µν G µν (D/ u G ) 1 (x, 0)(D/ d G ) 1 (0,x) det (D/ G ) QCD [dg] done via importance sampling Quenched: get distribution from e i R d 4 x( 1/4)G µν G µν only Unquenched: include det (D/ G ) QCD VERY expensive Partially quenched: (D/ u G ) 1 (x, 0)(D/ d G ) 1 (0,x) DIFFERENT Quarks then in det (D/ G ) QCD Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.6/40

9 What is Partially Quenched? Why do this? Is not Quenched: Real QCD is continuous limit from Partially Quenched More handles to turn: Allows more systematic studies by varying parameters Sometimes allows to disentangle things from different observables det (D/ G ) QCD : Sea quarks (D/ u G ) 1 (x, 0)(D/ d G ) 1 (0,x): Valence Quarks Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.7/40

10 What is Partially Quenched? Why not do this? It is not QCD as soon as Valence Sea not a Quantum Field Theory No unitarity No CPT theorem No spin statistics theorem Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.8/40

11 What is Partially Quenched? Do anyway Cheap computationally Allows extra studies Hope it works near real QCD case Is a well defined Euclidean statistical model QCD: m val = m sea m sea PQχPT Lattice m val Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.9/40

12 ChPT and Lattice QCD Mesons Quark Flow Valence Quark Flow Sea = + + They need to be treated separately: i.e. different quark masses Partially Quenched ChPT (PQChPT) Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.10/40

13 PQChPT at Two Loops: General Add ghost quarks: remove the unwanted free valence loops Mesons Quark Flow Valence Quark Flow Valence Quark Flow Sea Quark Flow Ghost = Possible problem: QCD = ChPT relies heavily on unitarity Partially quenched: at least one dynamical sea quark = Φ 0 is heavy: remove from PQChPT Symmetry group becomes SU(n v + n s n v ) SU(n v + n s n v ) (approximately) Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.11/40

14 PQChPT: General Essentially all manipulations from ChPT go through to PQChPT when changing trace to supertrace and adding fermionic variables Exceptions: baryons and Cayley-Hamilton relations So Luckily: can use the n flavour work in ChPT at two loop order to obtain for PQChPT: Lagrangians and infinities Very important note: ChPT is a limit of PQChPT = LECs from ChPT are linear combinations of LECs of PQChPT with the same number of sea quarks. E.g. L r 1 = L r(3pq) 0 /2 + L r(3pq) 1 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.12/40

15 PQChPT: What we did earlier One-loop: Bernard, Golterman, Sharpe, Shoresh, Pallante,... Two loops: Subject started m 2 π simplest mass case: + JB,Danielsson,Lähde, hep-lat/ Other mass combinations: F π +: JB,Lähde, hep-lat/ F π +, m 2 π two sea quarks: + JB,Lähde, hep-lat/ m 2 π : + JB,Danielsson,Lähde, hep-lat/ Neutral masses: JB,Danielsson, hep-lat/ Actual Calculations: heavy use of FORM Vermaseren use PQ without super Φ 0 in supersymmetric formalism Main problem: sheer size of the expressions Iso breaking from lattice data: a and L extrapolations needed Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.13/40

16 Including electromagnetism High precision: need to include electromagnetism Large violations of Dashen s theorem expected Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.14/40

17 Including electromagnetism High precision: need to include electromagnetism Large violations of Dashen s theorem expected Dashen: ( m 2 K + m 2 K 0 ) em = ( m 2 π + m 2 π 0 )em JB,Donoghue,Holstein,Wyler 93: ( m 2 K + m 2 K 0 )em 2( m 2 π + m 2 π 0 ) em Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.14/40

18 Including electromagnetism High precision: need to include electromagnetism Large violations of Dashen s theorem expected Dashen: ( m 2 K + m 2 K 0 ) em = ( m 2 π + m 2 π 0 )em JB,Donoghue,Holstein,Wyler 93: ( m 2 K + m 2 K 0 )em 2( m 2 π + m 2 π 0 ) em Needs checking Lattice: Duncan, Eichten,... Analytical: JB, Prades, Ananthanarayan, Moussallam,... Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.14/40

19 Why so little lattice? [dq][dq][dg][da](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg][da]e i R d 4 x( 1/4)G µν G µν (1/4)F µν F µν (D/ u G,A ) 1 (x, 0)(D/ d G,A ) 1 (0,x) det ( D/ G,A )QCD Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.15/40

20 Why so little lattice? [dq][dq][dg][da](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg][da]e i R d 4 x( 1/4)G µν G µν (1/4)F µν F µν (D/ u G,A ) 1 (x, 0)(D/ d G,A ) 1 (0,x) det ( D/ G,A )QCD [dg][da] done via importance sampling Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.15/40

21 Why so little lattice? [dq][dq][dg][da](uγ 5 d)(x)(dγ 5 u)(0)e i d 4 yl QCD [dg][da]e i R d 4 x( 1/4)G µν G µν (1/4)F µν F µν (D/ u G,A ) 1 (x, 0)(D/ d G,A ) 1 (0,x) det ( D/ G,A )QCD [dg][da] done via importance sampling Quenched: get distribution from e i R d 4 x( 1/4)G µν G µν (1/4)F µν F µν only Unquenched: include det ( D/ G,A )QCD VERY expensive Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.15/40

22 Other possibilities: Include det (D/ G ) QCD, but not det ( D/ G,A )QCD Allows to reuse pure QCD gluon configurations Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.16/40

23 Other possibilities: Include det (D/ G ) QCD, but not det ( D/ G,A )QCD Allows to reuse pure QCD gluon configurations Partially quenched: (D/ u G,A ) 1 (x, 0)(D/ d G,A ) 1 (0,x) DIFFERENT mass and charge Quarks then in det ( D/ G,A )QCD Allows the reuse if we set {q sea } {0} Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.16/40

24 Other possibilities: Include det (D/ G ) QCD, but not det ( D/ G,A )QCD Allows to reuse pure QCD gluon configurations Partially quenched: (D/ u G,A ) 1 (x, 0)(D/ d G,A ) 1 (0,x) DIFFERENT mass and charge Quarks then in det ( D/ G,A )QCD Allows the reuse if we set {q sea } {0} Done to O(e 2,e 2 p 2 ): JB, Danielsson Lagrangian Off-diagonal mesons: masses and decay constants Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.16/40

25 EM PQChPT The whole trace to supertrace goes through again Need to include the spurions with valence, sea and ghost sector ghost: masses and charges equal to valence n F flavour Lagrangian and infinities carry across trace supertrace n F n sea Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.17/40

26 EM PQChPT: Lagrangian r µ = v µ + eq R A µ + a µ l µ = v µ + eq L A µ a µ Q R = Q L = Q = diag(q 1,...,q 9 ) with Q = q 4 +q 5 +q 6 = 0. u = U Q L = uq L u ˆ µ Q L = ud µ Q L u Q R = u Q R u ˆ µ Q R = u D µ Q R u, D µ Q L = µ Q L i [l µ,q L ] D µ Q R = µ Q R i [r µ,q R ] Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.18/40

27 EM PQChPT: Lagrangian L 2 = 1 4 F µνf µν 1 2 λ( µa µ ) 2 + F uµ u µ +χ + +e 2 Z E F 4 0 Q L Q R L S2E2 = e 2 F 2 0 L 4 = L S4 + L S2E2. ( 14 ) Ki E Q s i + K18Q E s 18 + K19Q E s 19 i=1. Q s 18 = Q L u µ Q L u µ + Q R u µ Q R u µ Q s 19 = Q L u µ Q R u µ Ki E i = 1,...,14 terms: same form as Urech, but supertrace,... K18 E and KE 19 new ones. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.19/40

28 EM PQChPT: Lagrangian Q s 1 = 1 2 Q2 L + Q2 R u µu µ Q s 2 = Q LQ R u µ u µ Q s 3 = Q Lu µ Q L u µ Q R u µ Q R u µ Q s 4 = Q Lu µ Q R u µ Q s 5 = (Q2 L + Q2 R )u µu µ Q s 6 = (Q LQ R + Q R Q L )u µ u µ Q s 7 = 2 1 Q2 L + Q2 R χ + Q s 8 = Q LQ R χ + Q s 9 = (Q2 L + Q2 R )χ + Q s 10 = (Q LQ R + Q R Q L )χ + Q s 11 = (Q [ RQ L Q L ] Q R )χ [ ] Q s 12 = i ˆ µ Q R, Q R u µ ˆ µ Q L, Q L u µ Q s 13 = ˆ µ Q L ˆ µ Q R Q s 14 = ˆ µ Q L ˆ µ Q L + ˆ µ Q R ˆ µ Q R Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.20/40

29 EM PQChPT: Lagrangian Relation with unquenched: K 1 = K1 E + KE 18, K 2 = K2 E + 2 1KE 19, K 3 = K1 E KE 18, K 4 = K4 E + KE 19, K 5 = K5 E 2KE 18, K 6 = K6 E KE 19, K i = Ki E; i = 7,...,14 Infinities: taken over from n F -flavour calculation of Knecht- Urech (after putting in minimal basis and correcting misprint) Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.21/40

30 Masses M 2 phys = χ e,ij + δ (4)vs /F 2 0 χ e,ij = χ ij + 2Z E e 2 F 2 0 (q i q j ) 2 δ (4)23 = [48L r 6 24Lr 4 ] χ 1χ 13 + [16L r 8 8Lr 5 ] χ e 2 Z E F0 2Lr 4 q2 13 χ 1 16e 2 Z E F0 2Lr 5 q2 13 χ 13 e 2 F0 2 [ ] 12K Er 1 +12KEr 2 12KEr 7 12KEr 8 Q2 χ 13 e 2 F0 2 [ 4K Er 5 + ] 4KEr 6 q 2 p χ 13 + e 2 F0 2 [ 4K Er 9 + ] 4KEr 10 q 2 p χ p +12e 2 F0 2KEr 8 q2 13 χ 1 + 8e 2 F0 2 [ ] K Er 10 +KEr 11 q 2 13 χ 13 e 2 F0 2 [ 8K Er 18 + ] 4KEr 19 q1 q 3 χ 13 1/3Ā(χ m)rn13 m χ 13 1/3Ā(χ p)rqπηχ p 13 + e 2 F0 2Ā(χ 13)q e 2 Z E F0 2Ā(χ 1s)q 1s q 13 2e 2 Z E F0 2Ā(χ 3s)q 3s q 13 +4e 2 F0 2 B(χ γ,χ 13,χ 13 )q13 2 χ 13 4e 2 F0 2 B 1 (χ γ,χ 13,χ 13 )q13 2 χ 13. Q 2 = (q q2 5 + q2 6 )/3 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.22/40

31 Dashen s theorem again very close to M 2 = M 2 (χ 1,χ 3,q 1,q 3 ) M 2 (χ 1,χ 3,q 3,q 3 ) M 2 (χ 1,χ 1,q 1,q 3 ) + M 2 (χ 1,χ 1,q 3,q 3 ) M 2 D = (m2 K + m 2 K 0 ) (m 2 π + m 2 π 0 ). {q sea } dependence predicted: can be had from quenched photons Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.23/40

32 Decay Constants F phys = F 0 [ 1 + f(4)vs F 2 0 ] + O(p 6,e 2 p 4 ), F LO = F 0 [ ] f (4)23 = +12L r 4 χ 1 + 4L r 5χ e 2 F0 2 K1 Er + K2 Er Q 2 [ ] +2e 2 F0 2 K5 Er + K6 Er qp 2 + 2e 2 F0 2 K12 Er q13 2 [ ] +e 2 F0 2 4K18 Er + 2KEr 19 q 1 q 3 1/12Ā(χ m)rmn13 v +Ā(χ p) [ 1/6Rqπη p 1/12Rp c ] +1/4Ā(χ e,ps) 1/12 B(χ p,χ p, 0)R d p +2e 2 F 2 0 B (χ γ,χ 13,χ 13 )q 2 13 χ 13 e 2 F 2 0 B 1 (χ γ,χ 13,χ 13 )q e 2 F 2 0 B 1(χ γ,χ 13,χ 13 )q 2 13χ 13 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.24/40

33 Decay Constants F = [ F(χ 1,χ 3,q 1,q 3, {q sea }) F(χ 1,χ 3, 0, 0, {0}) ] F(χ 1,χ 1,q 1,q 3, {q sea }) + F(χ 1,χ 1, 0, 0, {0}) /F 0 {q sea } dependence predicted: Actually: no dependence on K Er i can be had from quenched photons Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.25/40

34 Masses/Decay Constants 0.5 M 2, Fit 10, p 4 +e 2 p F, Fit 10, p 4 +e 2 p 2 χ 4 [GeV 2 ] e-04 1e-05 1e-06 χ 4 [GeV 2 ] e-04 1e-05 1e χ 1 [GeV 2 ] χ 1 [GeV 2 ] χ 3 = χ 6 = 0.5 GeV 2 K E i chosen to be same mass combinations as JB,Prades Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.26/40

35 Remember: Eta Physics Handbook Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.27/40

36 η 3π beyond p 4 : Basic Review: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/ ] η pη π+ p π+ π p π π 0 p π 0 s = (p π + + p π ) 2 = (p η p π 0) 2 t = (p π + p π 0) 2 = (p η p π +) 2 u = (p π + + p π 0) 2 = (p η p π ) 2 s + t + u = m 2 η + 2m 2 π + m 2 + π 3s 0 0. π 0 π + π out η = i (2π) 4 δ 4 (p η p π + p π p π 0) A(s,t,u). π 0 π 0 π 0 out η = i (2π) 4 δ 4 (p η p 1 p 2 p 3 ) A(s 1,s 2,s 3 ) A(s 1,s 2,s 3 ) = A(s 1,s 2,s 3 ) + A(s 2,s 3,s 1 ) + A(s 3,s 1,s 2 ), Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.28/40

37 η 3π beyond p 4 : Lowest order Pions are in I = 1 state = A (m u m d ) or α em α em effect is small (but large via m π + m π 0) η π + π π 0 γ needs to be included directly Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.29/40

38 η 3π beyond p 4 : Lowest order Pions are in I = 1 state = A (m u m d ) or α em α em effect is small (but large via m π + m π 0) η π + π π 0 γ needs to be included directly ChPT:Cronin 67: A(s,t,u) = B 0(m u m d ) 3 3F 2 π { 1 + 3(s s 0) m 2 η m 2 π } Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.29/40

39 η 3π beyond p 4 : Lowest order Pions are in I = 1 state = A (m u m d ) or α em α em effect is small (but large via m π + m π 0) η π + π π 0 γ needs to be included directly ChPT:Cronin 67: A(s,t,u) = B 0(m u m d ) 3 3F 2 π or with Q 2 m2 s ˆm 2, ˆm = 1 m 2 d m2 u 2 (m u + m d ) { 1 + 3(s s 0) m 2 η m 2 π } A(s,t,u) = 1 Q 2 m 2 K m 2 π (m 2 π m 2 K ) M(s,t,u) 3 3F 2 π, with at lowest order M(s,t,u) = 3s 4m2 π m 2 η m 2 π. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.29/40

40 η 3π beyond p 4 : p 2 and p 4 Γ(η 3π) A 2 Q 4 allows a PRECISE measurement Q 24 gives lowest order Γ(η π + π π 0 ) 66 ev. Other Source from m 2 K + m 2 K 0 Q 2 = Q = 20.0 ± 1.5 Lowest order prediction Γ(η π + π π 0 ) 140 ev. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.30/40

41 η 3π beyond p 4 : p 2 and p 4 Γ(η 3π) A 2 Q 4 allows a PRECISE measurement Q 24 gives lowest order Γ(η π + π π 0 ) 66 ev. Other Source from m 2 K + m 2 K 0 Q 2 = Q = 20.0 ± 1.5 Lowest order prediction Γ(η π + π π 0 ) 140 ev. dlips A 2 + A 4 2 At order p 4 : = 2.4, dlips A 2 2 (LIPS=Lorentz invariant phase-space) Major source: large S-wave final state rescattering Experiment: Γ(η 3π) = 295 ± 17 ev. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.30/40

42 η 3π beyond p 4 : Dispersive Try to resum the S-wave rescattering: Anisovich-Leutwyler (AL), Kambor,Wiesendanger,Wyler (KWW) Different method but similar approximations Here: simplified version of AL Up to p 8 : No absorptive parts from l 2 = M(s,t,u) = M 0 (s)+(s u)m 1 (t)+(s t)m 1 (t)+m 2 (t)+m 2 (u) 2 3 M 2(s) M I : roughly contributions with isospin 0,1,2 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.31/40

43 η 3π beyond p 4 : Dispersive 3 body dispersive: difficult: keep only 2 body cuts start from πη ππ (m 2 η < 3m 2 π) standard dispersive analysis analytically continue to physical m 2 η. M I (s) = 1 π 4m 2 π ds ImM I (s ) s s iε ImM I (s ) discm I (s) = 1 2i (M I(s + iε) M I (s iε)) Z M 0 (s) = a 0 + b 0 s + c 0 s 2 + s3 ds discm 0 (s ) π s 3 s s iε, Z M 1 (s) = a 1 + b 1 s + s2 ds discm 1 (s ) π s 2 s s iε, Z M 2 (s) = a 2 + b 2 s + c 2 s 2 + s3 ds discm 2 (s ) π s 3 s s iε. Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.32/40

44 η 3π beyond p 4 AL: Lowest order is M(s,t,u) = 3s 4m2 π m 2 η m 2 π zero at s 4 /3m 2 π : remains in the neighbourhood: match position of s A and slope of Adler zero. KWW: fix amplitude at some place(s) in s,t,u plane to be equal to p 4 Both find moderate increases over p 4 : 15% in amplitude Dalitzplot distributions provide a check Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.33/40

45 η 3π beyond p L 2 2 ReM 1 AMP CA 1 0 Adler zero physical region -1 physical region s Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.34/40

46 Two Loop Calculation: why In K l4 dispersive gave about half of p 6 in amplitude Same order in ChPT as masses for consistency check on m u /m d Check size of 3 pion dispersive part Technology exists: Two-loops: Amorós,JB,Dhonte,Talavera,... Dealing with the mixing π 0 -η: Amorós,JB,Dhonte,Talavera 01 Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.35/40

47 Status 3 4 m d m u m s ˆm A(s,t,u) = sin ǫ M(s,t,u) ǫ M(s,t,u) = M 0 (s) + (s u)m 1 (t) + (s t)m 1 (t) + M 2 (t) + M 2 (u) 3 2M 2(s) M (2) = F 1 ( 43 m 2 π 2 π s ) p 4 JB Ghorbani Talavera All agree and with GL p 6 All cancel p 6 Vertex Checked and agree in progress p 6 Rest Comparison in progress p 6 Very Very Numerics preliminary preliminary Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.36/40

48 M 0 : Vertex function contribution p 2 Re Vertex functions Im Vertex functions 5 Re(M 0 ) s [GeV 2 ] Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.37/40

49 M 1 : Vertex function contribution M 0 p Re Vertex functions 0.1 Im Vertex functions 5 Re(M 1 ) s [GeV 2 ] Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.38/40

50 M 2 : Vertex function contribution M 0 p 2 Re Vertex functions Im Vertex functions 5 Re(M 2 ) s [GeV 2 ] Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.39/40

51 Conclusions PQChPT PQChPT at two loops: basic calculations done For formulas: bijnens/chpt.html Numerical programs: ask me (or a collaborator) Photons now also included (but NLO) M 2 and F can be done with quenched photons Lattice: Please use (and cite ) our work η 3π: All three have expressions: do we agree? Euroflavour06, 2-4/11/2006 Photons and Partial Quenching η 3π: status Johan Bijnens p.40/40