INTRODUCTION AND APPLICATIONS. Johan Bijnens Lund University. bijnens

Transcription

1 EFFECTIVE FIELD THEORY: lavi net INTRODUCTION AND APPLICATIONS Johan Bijnens Lund University bijnens Various ChPT: bijnens/chpt.html Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.1/103

2 Overview What is effective field theory? Introduction to ChPT including possible problems Results for two-flavour ChPT Results for three flavour ChPT Partial quenching and ChPT η 3π Some comments about Higgs sector and ChPT Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.2/103

3 Wikipedia Effective_field_theory In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.3/103

4 Wikipedia Chiral_perturbation_theory Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD. As QCD becomes non-perturbative at low energy, it is impossible to use perturbative methods to extract information from the partition function of QCD. Lattice QCD is one alternative method that has proved successful in extracting non-perturbative information. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.4/103

5 Effective Field Theory Main Ideas: Use right degrees of freedom : essence of (most) physics If mass-gap in the excitation spectrum: neglect degrees of freedom above the gap. Examples: Solid state physics: conductors: neglect the empty bands above the partially filled one Atomic physics: Blue sky: neglect atomic structure Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.5/103

6 Power Counting gap in the spectrum = separation of scales with the lower degrees of freedom, build the most general effective Lagrangian Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103

7 Power Counting gap in the spectrum = separation of scales with the lower degrees of freedom, build the most general effective Lagrangian # parameters Where did my predictivity go? Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103

8 Power Counting gap in the spectrum = separation of scales with the lower degrees of freedom, build the most general effective Lagrangian # parameters Where did my predictivity go? = Need some ordering principle: power counting Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103

9 Power Counting gap in the spectrum = separation of scales with the lower degrees of freedom, build the most general effective Lagrangian # parameters Where did my predictivity go? = Need some ordering principle: power counting Taylor series expansion does not work (convergence radius is zero) Continuum of excitation states need to be taken into account Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.6/103

10 Why is the sky blue? System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å Atomic excitations suppressed by 10 3 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103

11 Why is the sky blue? System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å Atomic excitations suppressed by 10 3 L A = Φ v t Φ v +... L γa = GF 2 µνφ vφ v +... Units with h/ = c = 1: G energy dimension 3: Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103

12 Why is the sky blue? System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å Atomic excitations suppressed by 10 3 L A = Φ v t Φ v +... L γa = GF 2 µνφ vφ v +... Units with h/ = c = 1: G energy dimension 3: σ G 2 E 4 γ Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103

13 Why is the sky blue? System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å Atomic excitations suppressed by 10 3 L A = Φ v t Φ v +... L γa = GF 2 µνφ vφ v +... Units with h/ = c = 1: G energy dimension 3: σ G 2 E 4 γ blue light scatters a lot more than red Higher orders suppressed by 1 Å/λ γ. { = red sunsets = blue sky Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103

14 Why Field Theory? Only known way to combine QM and special relativity Off-shell effects: there as new free parameters Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.8/103

15 Why Field Theory? Only known way to combine QM and special relativity Off-shell effects: there as new free parameters Drawbacks Many parameters (but finite number at any order) any model has few parameters but model-space is large expansion: it might not converge or only badly Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.8/103

16 Why Field Theory? Only known way to combine QM and special relativity Off-shell effects: there as new free parameters Drawbacks Many parameters (but finite number at any order) any model has few parameters but model-space is large expansion: it might not converge or only badly Advantages Calculations are (relatively) simple It is general: model-independent Theory = errors can be estimated Systematic: ALL effects at a given order can be included Even if no convergence: classification of models often useful Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.8/103

17 Examples of EFT Fermi theory of the weak interaction Chiral Perturbation Theory: hadronic physics NRQCD SCET General relativity as an EFT 2,3,4 nucleon systems from EFT point of view Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.9/103

18 references A. Manohar, Effective Field Theories (Schladming lectures), hep-ph/ I. Rothstein, Lectures on Effective Field Theories (TASI lectures), hep-ph/ G. Ecker, Effective field theories, Encyclopedia of Mathematical Physics, hep-ph/ D.B. Kaplan, Five lectures on effective field theory, nucl-th/ A. Pich, Les Houches Lectures, hep-ph/ S. Scherer, Introduction to chiral perturbation theory, hep-ph/ J. Donoghue, Introduction to the Effective Field Theory Description of Gravity, gr-qc/ Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.10/103

19 School PSI Zuoz Summer School on Particle Physics Effective Theories in Particle Physics, July 16-22, previous_summerschools/zuoz2006/index.html R. Barbieri: Effective Theories for Physics beyond the Standard Model M. Beneke: Concept of Effective Theories, Heavy Quark Effective Theory and Soft-Collinear Effective Theory G. Colangelo: Chiral Perturbation Theory U. Langenegger: B Physics and Quarkonia H. Leutwyler: Historical and Other Remarks on Effective Theories A. Manohar: Nonrelativistic QCD L. Simons: Pion-Nucleon Interaction M. Sozzi: Kaon Physics Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.11/103

20 Chiral Perturbation Theory Exploring the consequences of the chiral symmetry of QCD and its spontaneous breaking using effective field theory techniques Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.12/103

21 Chiral Perturbation Theory Exploring the consequences of the chiral symmetry of QCD and its spontaneous breaking using effective field theory techniques Derivation from QCD: H. Leutwyler, On The Foundations Of Chiral Perturbation Theory, Ann. Phys. 235 (1994) 165 [hep-ph/ ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.12/103

22 The mass gap: Goldstone Modes UNBROKEN: V (φ) BROKEN: V (φ) Only massive modes around lowest energy state (=vacuum) Need to pick a vacuum φ 0: Breaks symmetry No parity doublets Massless mode along bottom For more complicated symmetries: need to describe the bottom mathematically: G H = G/H (explain) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.13/103

23 The two symmetry modes compared Wigner-Eckart mode Nambu-Goldstone mode Symmetry group G Vacuum state unique Massive Excitations States fall in multiplets of G Wigner Eckart theorem for G G spontaneously broken to subgroup H Vacuum state degenerate Existence of a massless mode States fall in multiplets of H Wigner Eckart theorem for H Broken part leads to low-energy theorems Symmetry linearly realized Full Symmetry, G, nonlinearly realized unbroken part, H, linearly realized Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.14/103

24 Some clarifications φ(x): orientation of vacuum in every space-time point Examples: spin waves, phonons Nonlinear: acting by a broken symmetry operator changes the vacuum, φ(x) φ(x) + α The precise form of φ is not important but it must describe the space of vacua (field transformations possible) In gauge theories: the local symmetry allows the vacua to be different in every point, hence the Goldstone Boson must not be observable as a massless degree of freedom. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.15/103

25 The power counting Very important: Low energy theorems: Goldstone bosons do not interact at zero momentum Heuristic proof: Which vacuum does not matter, choices related by symmetry φ(x) φ(x) + α should not matter Each term in L must contain at least one µ φ Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.16/103

26 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103

27 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. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103

28 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. Can also see that via v < c, m q 0 = v = c, m q = 0 = / Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103

29 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.18/103

30 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: (explain) 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.18/103

31 Chiral Perturbation Theory Large subject: Steven Weinberg, Physica A96:327,1979: 1789 citations Juerg Gasser and Heiri Leutwyler, Nucl.Phys.B250:465,1985: 2290 citations Juerg Gasser and Heiri Leutwyler, Annals Phys.158:142,1984: 2254 citations Sum: 3777 Checked on 18/2/2007 in SPIRES For lectures, review articles: see bijnens/chpt.html Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.19/103

32 Chiral Perturbation Theories Baryons Heavy Quarks Vector Mesons (and other resonances) Structure Functions and Related Quantities Light Pseudoscalar Mesons Two or Three (or even more) Flavours Strong interaction and couplings to external currents/densities Including electromagnetism Including weak nonleptonic interactions Treating kaon as heavy Many similarities with strongly interacting Higgs Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.20/103

33 Lagrangians U(φ) = exp(i 2Φ/F 0 ) parametrizes Goldstone Bosons Φ(x) = 0 π η 8 6 π + K + π π0 2 + η 8 6 K 0 K K 0 2 η C A LO Lagrangian: L 2 = F { D µ U D µ U + χ U + χu }, D µ U = µ U ir µ U + iul µ, left and right external currents: r(l) µ = v µ + ( )a µ Scalar and pseudoscalar external densities: χ = 2B 0 (s + ip) quark masses via scalar density: s = M + A = Tr F (A) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.21/103

34 External currents? in QCD Green functions derived as functional derivatives w.r.t. external fields Green functions are the objects that satisfy Ward identities By introducing a local Chiral Symmetry, Ward identities are automatically satisfied (great improvement over current algebra) QCD Green functions form a connection QCD ChPT [dgdqdq]e i R d 4 xl QCD (q,q,g,l,r,s,p) [du]e i R d 4 xl ChP T (U,l,r,s,p) so also functional derivatives are equal Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.22/103

35 Lagrangians L 4 = L 1 D µ U D µ U 2 + L 2 D µ U D ν U D µ U D ν U +L 3 D µ U D µ UD ν U D ν U + L 4 D µ U D µ U χ U + χu +L 5 D µ U D µ U(χ U + U χ) + L 6 χ U + χu 2 +L 7 χ U χu 2 + L 8 χ Uχ U + χu χu il 9 F R µν Dµ UD ν U + F L µν Dµ U D ν U +L 10 U F R µνuf Lµν + H 1 F R µνf Rµν + F L µνf Lµν + H 2 χ χ L i : Low-energy-constants (LECs) H i : Values depend on definition of currents/densities These absorb the divergences of loop diagrams: L i L r i Renormalization: order by order in the powercounting Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.23/103

36 Lagrangians Lagrangian Structure: 2 flavour 3 flavour 3+3 PQChPT p 2 F,B 2 F 0,B 0 2 F 0,B 0 2 p 4 li r,hr i 7+3 L r i,hr i 10+2 ˆL r i,ĥr i 11+2 p 6 c r i 52+4 Ci r 90+4 Ki r p 2 : Weinberg 1966 p 4 : Gasser, Leutwyler 84,85 p 6 : JB, Colangelo, Ecker 99,00 replica method = PQ obtained from N F flavour All infinities known 3 flavour special case of 3+3 PQ: ˆL r i,kr i Lr i,cr i arxiv: [hep-ph] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.24/103

37 Chiral Logarithms The main predictions of ChPT: Relates processes with different numbers of pseudoscalars Chiral logarithms m 2 π = 2B ˆm + ( 2B ˆm F ) 2 [ ] 1 (2B ˆm) 32π2log µ 2 + 2l3(µ) r + M 2 = 2B ˆm B B 0, F F 0 (two versus three-flavour) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.25/103

38 LECs and µ l r 3(µ) li = 32π2 li r (µ) log M2 π γ i µ 2. Independent of the scale µ. For 3 and more flavours, some of the γ i = 0: L r i (µ) µ : m π, m K : chiral logs vanish pick larger scale 1 GeV then L r 5 (µ) 0 large N c arguments???? compromise: µ = m ρ = 0.77 GeV Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.26/103

39 Expand in what quantities? Expansion is in momenta and masses But is not unique: relations between masses (Gell-Mann Okubo) exists Express orders in terms of physical masses and quantities (F π, F K )? Express orders in terms of lowest order masses? E.g. s + t + u = 2m 2 π + 2m 2 K See e.g. Descotes-Genon talk I prefer physical masses Thresholds correct in πk scattering Chiral logs are from physical particles propagating Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.27/103

40 An example m π = m a m0 f0 f π = f b m0 f0 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103

41 An example m π = m a m0 f0 f π = f b m0 f0 m π = m 0 a m2 0 + a 2m3 0 f 0 f0 2 + ( f π = f 0 1 b m 0 + b 2m2 0 f 0 f0 2 + ) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103

42 An example m π = m a m0 f0 f π = f b m0 f0 m π = m 0 a m2 0 + a 2m3 0 f 0 f0 2 + ( f π = f 0 1 b m 0 + b 2m2 0 f 0 f0 2 + ) m π = m 0 a m2 π f π + a(b a) m3 π f 2 π m π = m 0 ( 1 a m π f π + ab m2 π f 2 π + ) + f π = f 0 ( 1 b m π f π + b(2b a) m2 π f 2 π + ) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103

43 An example m π = m a m0 f0 f π = f b m0 f0 m π = m 0 a m2 0 + a 2m3 0 f 0 f0 2 + ( f π = f 0 1 b m 0 + b 2m2 0 f 0 f0 2 + ) m π = m 0 a m2 π f π + a(b a) m3 π f 2 π m π = m 0 ( 1 a m π f π + ab m2 π f 2 π + ) + f π = f 0 ( 1 b m π f π + b(2b a) m2 π f 2 π a = 1 b = 0.5 f 0 = 1 + ) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103

44 An example: m 0 /f m π LO NLO NNLO m π m 0 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.29/103

45 An example: m π /f π m π LO NLOp NNLOp m π m 0 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.30/103

46 Two-loop Two-flavour Review paper on Two-Loops: JB, hep-ph/ Prog. Part. Nucl. Phys. 58 (2007) 521 Dispersive Calculation of the nonpolynomial part in q 2,s,t,u Gasser-Meißner: F V, F S : 1991 numerical Knecht-Moussallam-Stern-Fuchs: ππ: 1995 analytical Colangelo-Finkemeier-Urech: F V, F S : 1996 analytical Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.31/103

47 Two-Loop Two-flavour Bellucci-Gasser-Sainio: γγ π 0 π 0 : 1994 Bürgi: γγ π + π, F π, m π : 1996 JB-Colangelo-Ecker-Gasser-Sainio: ππ, F π, m π : JB-Colangelo-Talavera: F V π (t), F Sπ : 1998 JB-Talavera: π lνγ: 1997 Gasser-Ivanov-Sainio: γγ π 0 π 0, γγ π + π : m π, F π, F V, F S, ππ: simple analytical forms Colangelo-(Dürr-)Haefeli: Finite volume F π,m π Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.32/103

48 LECs l1 to l 4 : ChPT at order p 6 and the Roy equation analysis in ππ and F S Colangelo, Gasser and Leutwyler, Nucl. Phys. B 603 (2001) 125 [hep-ph/ ] l5 and l 6 : from F V and π lνγ JB,(Colangelo,)Talavera l1 = 0.4 ± 0.6, l2 = 4.3 ± 0.1, l3 = 2.9 ± 2.4, l4 = 4.4 ± 0.2, l6 l 5 = 3.0 ± 0.3, l6 = 16.0 ± 0.5 ± 0.7. l from π 0 -η mixing Gasser, Leutwyler 1984 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.33/103

49 LECs Some combinations of order p 6 LECs are known as well: curvature of the scalar and vector formfactor, two more combinations from ππ scattering (implicit in b 5 and b 6 ) Note: c r i for m π, f π, ππ: small effect c r i (770MeV ) = 0 for plots shown expansion in m 2 π/f 2 π shown General observation: Obtainable from kinematical dependences: known Only via quark-mass dependence: poorely known Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.34/103

50 m 2 π LO NLO NNLO m π M 2 [GeV 2 ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.35/103

51 m 2 π ( l 3 = 0) m π LO NLO NNLO M 2 [GeV 2 ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.36/103

52 F π F π [GeV] LO NLO NNLO M 2 [GeV 2 ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.37/103

53 Two-loop Three-flavour, 2001 Π V V π, Π V V η, Π V V K Kambor, Golowich; Kambor, Dürr; Amorós, JB, Talavera Π V V ρω Π AAπ, Π AAη, F π, F η, m π, m η Maltman Kambor, Golowich; Amorós, JB, Talavera Π SS Moussallam L r 4,Lr 6 Π V V K, Π AAK, F K, m K Amorós, JB, Talavera K l4, qq Amorós, JB, Talavera L r 1,Lr 2,Lr 3 F M, m M, qq (m u m d ) Amorós, JB, Talavera L r 5,7,8,m u/m d Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.38/103

54 Two-loop Three-flavour, 2001 F V π, F V K +, F V K 0 Post, Schilcher; JB, Talavera L r 9 K l3 Post, Schilcher; JB, Talavera V us F Sπ, F SK (includes σ-terms) JB, Dhonte L r 4,Lr 6 K,π lνγ Geng, Ho, Wu L r 10 ππ πk relation li r and Lr i,cr i Finite volume qq JB,Dhonte,Talavera JB,Dhonte,Talavera Gasser,Haefeli,Ivanov,Schmid JB,Ghorbani Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.39/103

55 Two-loop Three-flavour Known to be in progress η 3π: being written up K l3 iso: preliminary results available JB,Ghorbani JB,Ghorbani Finite Volume: sunsetintegrals being written up JB,Lähde relation c r i and Lr i,cr i Gasser,Haefeli,Ivanov,Schmid Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.40/103

56 C r i Most analysis use: Ci r from (single) resonance approximation π π ρ,s q 2 π π q 2 << m 2 ρ, m 2 = S C r i Motivated by large N c : large effort goes in this Ananthanarayan, JB, Cirigliano, Donoghue, Ecker, Gamiz, Golterman, Kaiser, Knecht, Peris, Pich, Prades, Portoles, de Rafael,... Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.41/103

57 C r i L V = 1 4 V µνv µν m2 V V µv µ f V 2 2 V µνf µν + ig V 2 2 V µν[u µ, u ν ] + f χ V µ [u µ, χ ] L A = 1 4 A µνa µν m2 A A µa µ f A 2 2 A µνf µν L S = 1 2 µ S µ S M 2 S S2 + c d Su µ u µ + c m Sχ + L η = 1 2 µp 1 µ P M2 η P i d m P 1 χ. f V = 0.20, f χ = 0.025, g V = 0.09, c m = 42 MeV, c d = 32 MeV, dm = 20 MeV, m V = m ρ = 0.77 GeV, m A = m a1 = 1.23 GeV, m S = 0.98 GeV, m P1 = GeV f V, g V, f χ, f A : experiment c m and c d from resonance saturation at O(p 4 ) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.42/103

58 C r i Problems: Weakest point in the numerics However not all results presented depend on this Unknown so far: Ci r in the masses/decay constants and how these effects correlate into the rest No µ dependence: obviously only estimate Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.43/103

59 C r i Problems: Weakest point in the numerics However not all results presented depend on this Unknown so far: Ci r in the masses/decay constants and how these effects correlate into the rest No µ dependence: obviously only estimate What we did about it: Vary resonance estimate by factor of two Vary the scale µ at which it applies: MeV Check the estimates for the measured ones Again: kinematic can be had, quark-mass dependence difficult Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.43/103

60 Inputs K l4 : F(0), G(0), λ m 2 π 0, m 2 η, m 2 K +, m 2 K 0 E865 BNL em with Dashen violation F π + F K +/F π + m s / ˆm 24 (26) ˆm = (m u + m d )/2 L r 4,Lr 6 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.44/103

61 Outputs: I fit 10 same p 4 fit B fit D 10 3 L r ± L r ± L r ± L r L r ± L r L r ± L r ± errors are very correlated µ = 770 MeV; 550 or 1000 within errors varying Ci r factor 2 about errors L r 4, L r 6 0.3,..., OK fit B: small corrections to pion sigma term, fit scalar radius fit D: fit ππ and πk thresholds Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.45/103

62 Correlations L 2 r (older fit) 10 3 L r 1 = 0.52 ± L r 2 = 0.72 ± L r 3 = 2.70 ± L 3 r L 1 r Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.46/103

63 Outputs: II fit 10 same p 4 fit B fit D 2B 0 ˆm/m 2 π m 2 π: p 4, p , , , ,0.133 m 2 K : p4, p , , , ,0.201 m 2 η: p 4, p , , , ,0.197 m u /m d 0.45± F 0 [MeV] F K Fπ : p 4, p , , , ,0.061 m u = 0 always very far from the fits F 0 : pion decay constant in the chiral limit Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.47/103

64 ππ p 4 p 6 p 4 p 6 a 0 0 a r L r L r L r L a 0 0 = ± 0.005, a2 0 Colangelo, Gasser, Leutwyler = ± a 0 0 = a2 0 = at order p2 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.48/103

65 ππ and πk 0.6 ππ constraints 0.6 πk constraints C + 10 a 0 3/2 C 1 a L 6 r L 6 r a 2 0 C 1 a 0 3/2 C r L r L 4 preferred region: fit D: 10 3 L r 4 0.2, 103 L r General fitting needs more work and systematic studies Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.49/103

66 Quark mass dependences Updates of plots in Amorós, JB and Talavera, hep-ph/ , Nucl. Phys. B585 (2000) 293 Some new ones Procedure: calculate a consistent set of m π,m K,m η,f π with the given input values (done iteratively) vary m s /(m s ) phys, keep m s / ˆm = 24 m 2 π, m 2 K, F π, F K vary m s /(m s ) phys keep ˆm fixed m 2 π, F π vary m π, keep m K fixed f + (0): the formfactor in K l3 decays f + (0),f + (0)/ ( m 2 K m2 π ) 2 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.50/103

67 m 2 π fit 10 m π 2 [GeV 2 ] LO NLO NNLO m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.51/103

68 m 2 π fit D m π 2 [GeV 2 ] LO NLO NNLO m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.52/103

69 m 2 K fit LO NLO NNLO m K 2 [GeV 2 ] m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.53/103

70 m 2 K fit D LO NLO NNLO m K 2 [GeV 2 ] m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.54/103

71 F π fit LO NLO NNLO F π [GeV] m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.55/103

72 F K fit F K [GeV] LO NLO NNLO m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.56/103

73 F K /F π fit F K /F π NLO NNLO m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.57/103

74 m 2 π fit 10, fixed ˆm m π 2 [GeV 2 ] LO NLO NNLO m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.58/103

75 F π fit 10, fixed ˆm LO NLO NNLO F π [GeV] m s /(m s ) phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.59/103

76 f + (0) fit 10, fixed m K correction to f + (0) sum p 4 p 6 2-loops p 6 L i r m π 2 /(mk 2 )phys Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.60/103

77 (f + (0))/ ( m 2 K m2 π) 2 fit 10, fixed mk 0.3 (correction to f + (0))/(m K 2 mπ 2 ) 2 [GeV 4 ] m π 2 /(mk 2 )phys sum p 4 p 6 2-loops p 6 L i r Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.61/103

78 f 0 (t) in K l3 Main Result: JB,Talavera f 0 (t) = 1 8 F 4 π +8 t F 4 π 8 F 4 π (C12 r + C34) r ( m 2 K ) 2 m2 π (2C12 r + C34) r ( m 2 K + ) t m2 π + m 2 K m2 π t 2 C r 12 + (t) + (0). (F K /F π 1) (t) and (0) contain NO C r i and only depend on the Lr i at order p 6 = All needed parameters can be determined experimentally (0) = ± [loops] ± [L r i]. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.62/103

79 3-flavour: PQChPT 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.63/103

80 PQChPT One-loop: Bernard, Golterman, Sharpe, Shoresh, Pallante,... with electromagnetism: JB,Danielsson, hep-lat/ Two loops: m 2 π + simplest mass case: JB,Danielsson,Lähde, hep-lat/ 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/ Lattice data: a and L extrapolations needed Programs available from me (Fortran) Formulas: bijnens/chpt.html Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.64/103

81 Partial Quenching and ChPT Mesons Quark Flow Valence Quark Flow Sea = + + Lattice QCD: Valence is easy to deal with, Sea very difficult They can be treated separately: i.e. different quark masses Partially Quenched QCD and ChPT (PQChPT) One Loop or p 4 : Bernard, Golterman, Pallante, Sharpe, Shoresh,... Two Loops or p 6 : This talk JB, Niclas Danielsson, Timo Lähde Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.65/103

82 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) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.66/103

83 PQChPT at Two Loops: 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.67/103

84 PQChPT at Two Loop: Papers valence equal mass, 3 sea equal mass: m 2 π + : JB,Danielsson,Lähde, hep-lat/ Other mass combinations: F 2 π + : JB,Lähde, hep-lat/ F 2 π +, m 2 π + two sea quarks: JB,Lähde, hep-lat/ In progress: the other charged masses Actual Calculations: heavy use of FORM Vermaseren Main problem: sheer size of the expressions No fits to lattice data (yet): a and L extrapolations needed Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.68/103

85 Long Expressions = Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.69/103

86 Long Expressions δ (6)22 loops = π16 L r [ 0 4/9 χηχ4 1/2 χ1χ3 + χ 2 ] 13 13/3 χ1χ13 35/18 χ2 2 π16 L r 1 χ 2 13 π16 L r [ 2 11/3 χηχ4 + χ /3 ] χ2 + π16 L r [ 3 4/9 χηχ4 7/12 χ1χ3 + 11/6 χ /6 χ1χ13 43/36 ] χ2 + π16 2 [ 15/64 χηχ4 59/384 χ1χ3 + 65/384 χ /2 χ1χ13 43/128 ] χ2 48 L r 4 L r 5 χ1χ13 72 Lr2 4 χ2 1 8 L r2 5 χ [ Ā(χp)π16 1/24 χp + 1/48 χ1 1/8 χ1 Rqη p + 1/16 χ1 Rc p 1/48 Rp qη χp 1/16 Rp qη χq + 1/48 Rpp η χη + 1/16 Rp c χ13] + Ā(χp)L r [ 0 8/3 R p qη χp + 2/3 Rp c χp + 2/3 Rp d ] + Ā(χp)L r [ 3 2/3 R p qη χp + 5/3 Rp c χp + 5/3 p] Rd + Ā(χp)L r [ 4 2 χ1 χ pp ηη0 2 χ1 Rp qη + 3 p] χ1 Rc + Ā(χp)L r [ 5 2/3 χ pp ηη1 Rp qη χp + 1/3 Rqη p χq + 1/2 Rp c χp 1/6 Rp c ] χq + Ā(χp) 2 [ 1/16 + 1/72 (Rqη) p 2 1/72 RqηR p p c + 1/288 (Rp) c 2] + Ā(χp)Ā(χps) [ 1/36 Rqη p 5/72 Rp sη + 7/144 p] Rc Ā(χp)Ā(χqs) [ 1/36 Rqη p + 1/24 Rp sη + 1/48 ] Rc p + Ā(χp)Ā(χη) [ 1/72 Rqη p Rv η13 + 1/144 Rc p η13] Rv + 1/8 Ā(χp)Ā(χ13) + 1/12 Ā(χp)Ā(χ46)Rη pp + Ā(χp) B(χp, χp; 0) [ 1/4 χp 1/18 RqηR p p c χp 1/72 RqηR p p d + 1/18 (Rp) c 2 χp + 1/144 RpR c p d ] + Ā(χp) B(χp, χη; 0) [ 1/18 Rpp η Rc p χp 1/18 ] Rη 13 Rc p χp + Ā(χp) B(χq, χq; 0) [ 1/72 Rqη p Rd q + 1/144 ] Rc p Rd q = 1/12 Ā(χp) B(χps, χps; 0)Rsη p χps 1/18 Ā(χp) B(χ1, χ3; 0)RpηR q p c χp + 1/18 Ā(χp) C(χp, χp, χp; 0)Rp c Rd p χp + Ā(χp; [ ε)π16 1/8 χ1r qη p 1/16 χ1 Rc p 1/16 Rc p χp 1/16 ] Rd p + Ā(χps)π16 [1/16 χps 3/16 χqs 3/16 χ1] 2 Ā(χps)Lr 0 χps 5 Ā(χps)Lr 3 χps 3 Ā(χps)Lr 4 χ1 + Ā(χps)Lr 5 χ13 + Ā(χps)Ā(χη) [ 7/144 Rpp η 5/72 Rps η 1/48 Rqq η + 5/72 Rqs η 1/36 R η ] 13 + Ā(χps) B(χp, χp; 0) [ 1/24 R p sη χp 5/24 Rp sη χps ] + Ā(χps) B(χp, χη; 0) [ 1/18 R η ps Rz qpη χp 1/9 R η ps Rz qpη χps ] 1/48 Ā(χps) B(χq, χq; 0)R d q + 1/18 Ā(χps) B(χ1, χ3; 0)R q sη χs + 1/9 Ā(χps) B(χ1, χ3; 0, k)rsη q + 3/16 Ā(χps; ε)π16 [χs + χ1] 1/8 Ā(χp4)2 1/8 Ā(χp4)Ā(χp6) + 1/8 Ā(χp4)Ā(χq6) 1/32 [ Ā(χp6)2 + Ā(χη)π16 1/16 χ1 Rη13 v 1/48 Rη13 v χη + 1/16 Rη13 v ] χ13 [ + Ā(χη)Lr 0 4R η 13 χη + 2/3 ] Rv η13 χη 8 Ā(χη)L r 1 χη 2 Ā(χη)Lr 2 χη + [ Ā(χη)Lr 3 4R η 13 χη + 5/3 ] Rv η13 [ χη + Ā(χη)Lr 4 4 χη + χ1 Rη13 v ] Ā(χη)L r [ 5 1/6 R η pp χq + R η 13 χ13 + 1/6 Rv η13 χη] + 1/288 Ā(χη) 2 (Rη13) v 2 + 1/12 Ā(χη)Ā(χ46)Rv η13 + Ā(χη) B(χp, χp; 0) [ 1/36 χ ppη ηη1 1/18 Rp qηrpp η χp + 1/18 RppR η p c χp + 1/144 RpR d η13 v ] + Ā(χη) B(χp, χη; 0) [ 1/18 χ ηpη pη1 + 1/18 χηpη qη1 + 1/18 (Rη pp) 2 Rqpη z ] χp 1/12 Ā(χη) B(χps, χps; 0)Rps η χps Ā(χη) B(χη, χη; 0) [ 1/216 Rη13 v χ4 + 1/27 ] Rv η13 χ6 1/18 Ā(χη) B(χ1, χ3; 0)RηηR 1 ηη 3 χη + 1/18 Ā(χη) C(χp, χp, χp; 0)RppR η p d χp + Ā(χη; ε)π16 [1/8 χη 1/16 χ1 Rη13 v 1/8 Rη 13 χη 1/16 ] Rv η13 χη + Ā(χ1)Ā(χ3)[ 1/72 Rqη p Rc q + 1/36 R1 3η R3 1η + 1/144 ] Rc 1 Rc 3 4 Ā(χ13)Lr 1 χ13 10 Ā(χ13)Lr 2 χ13 + 1/8 Ā(χ13)2 1/2 Ā(χ13) B(χ1, χ3; 0, k) + 1/4 Ā(χ13; ε)π16 χ13 + 1/4 Ā(χ14)Ā(χ34) + 1/16 Ā(χ16)Ā(χ36) 24 Ā(χ4)Lr 1 χ4 6 Ā(χ4)Lr 2 χ Ā(χ4)Lr 4 χ4 + 1/12 Ā(χ4) B(χp, χp; 0)(R p 4η )2 χ4 + 1/6 Ā(χ4) B(χp, χη; 0) [ R p 4η Rη p4 χ4 Rp 4η Rη q4 χ4 ] 1/24 Ā(χ4) B(χη, χη; 0)Rη13 v χ4 1/6 Ā(χ4) B(χ1, χ3; 0)R4η 1 R3 4η χ4 + 3/8 Ā(χ4; ε)π16 χ4 32 Ā(χ46)Lr 1 χ46 8 Ā(χ46)Lr 2 χ Ā(χ46)Lr 4 χ46 + Ā(χ46) B(χp, χp; 0) [ 1/9 χ46 + 1/12 R η pp χp + 1/36 R η pp χ4 + 1/9 R η p4 χ6 ] + Ā(χ46) B(χp, χη; 0) [ 1/18 R η pp χ4 1/9 R η p4 χ6 + 1/9 Rη q4 χ6 + 1/18 Rη 13 χ4 ] 1/6 Ā(χ46) B(χp, χη; 0, k) [ R η pp R η 13] + 1/9 Ā(χ46) B(χη, χη; 0)R v η13 χ46 Ā(χ46) B(χ1, χ3; 0) [2/9 χ46 + 1/9 R η p4 χ6 + 1/18 Rη 13 χ4] 1/6 Ā(χ46) B(χ1, χ3; 0, k)r η /2 Ā(χ46; ε)π16 χ46 + B(χp, [ χp; 0)π16 1/16 χ1 Rp d + 1/96 Rd p χp + 1/32 ] Rd p χq + 2/3 B(χp, χp; 0)L r 0 Rd p χp + 5/3 B(χp, χp; 0)L r 3 Rp d χp + B(χp, χp; 0)L r [ 4 2 χ1 χ pp ηη0 χp 4 χ1rp qη χp + 4 χ1r p c χp + 3 χ1r p d ] + B(χp, χp; 0)L r [ 5 2/3 χ pp ηη1 χp 4/3 Rp qη χ2 p + 4/3 Rc p χ2 p + 1/2 Rd p χp 1/6 ] Rd p χq + B(χp, χp; 0)L r [ 6 4 χ1 χ pp ηη1 + 8 χ1rp qη χp 8 ] χ1rc p χp + 4 B(χp, χp; 0)L r 7 (Rd p )2 + B(χp, χp; 0)L r [ 8 4/3 χ pp ηη2 + 8/3 Rp qη χ 2 p 8/3 Rp c χ 2 p] + B(χp, χp; 0) 2 [ 1/18 RqηR p p d χp + 1/18 RpR c p d χp + 1/288 (Rp d )2] + 1/18 B(χp, χp; 0) B(χp, χη; 0) [ Rpp η Rd p χp ] Rη 13 Rd p χp plus several more pages Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.69/103

87 Why so long expressions Many different quark and meson masses (χ ij ) Charged propagators: ig c ij (k) = ǫ j k 2 χ ij +iε (i j) Neutral propagators: G n ij (k) = Gc ij (k)δ ij 1 n sea G q ij (k) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103

88 Why so long expressions Many different quark and meson masses (χ ij ) Charged propagators: ig c ij (k) = ǫ j k 2 χ ij +iε (i j) Neutral propagators: G n ij (k) = Gc ij (k)δ ij 1 n sea G q ij (k) ig q ii (k) = R d i (k 2 χ i +iε) 2 + R c i k 2 χ i +iε + Rπ ηii k 2 χ π +iε + Rη πii k 2 χ η +iε R i jkl = Rz i456jkl, Rd i = Rz i456πη, R c i = Ri 4πη + Ri 5πη + Ri 6πη Ri πηη R i ππη R z ab = χ a χ b, R z abcdefg = (χ a χ b )(χ a χ c )(χ a χ d ) (χ a χ e )(χ a χ f )(χ a χ g ) R z abc = χ a χ b χ a χ c, R z abcd = (χ a χ b )(χ a χ c ) χ a χ d Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103

89 Why so long expressions Many different quark and meson masses (χ ij ) Charged propagators: ig c ij (k) = ǫ j k 2 χ ij +iε (i j) Neutral propagators: G n ij (k) = Gc ij (k)δ ij 1 n sea G q ij (k) ig q ii (k) = R d i (k 2 χ i +iε) 2 + R c i k 2 χ i +iε + Rπ ηii k 2 χ π +iε + Rη πii k 2 χ η +iε R i jkl = Rz i456jkl, Rd i = Rz i456πη, R c i = Ri 4πη + Ri 5πη + Ri 6πη Ri πηη R i ππη R z ab = χ a χ b, R z abcdefg = (χ a χ b )(χ a χ c )(χ a χ d ) (χ a χ e )(χ a χ f )(χ a χ g ) R z abc = χ a χ b χ a χ c, R z abcd = (χ a χ b )(χ a χ c ) χ a χ d Relations = order of magnitude smaller Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103

90 PQChPT at Two Loop Problem: Plotting with many input parameters Plot masses as a function of lowest order mass squared I.e. of quark mass: χ i = 2B 0 m i = m 2(0) M Remember: χ i 0.3 GeV 2 (550 MeV ) 2 border ChPT Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103

91 PQChPT at Two Loop Problem: Plotting with many input parameters Plot masses as a function of lowest order mass squared I.e. of quark mass: χ i = 2B 0 m i = m 2(0) M Remember: χ i 0.3 GeV 2 (550 MeV ) 2 border ChPT 1+1 case: Valence: χ 1 = χ 2 = χ 3 Sea: χ 4 = χ 5 = χ 6 Plot along curves: χ 4 = tanθ χ 1 or χ 4 constant Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103

92 PQChPT: 1+1 case, 3 sea quarks Sea χ 4 [GeV 2 ] o 60 o 45 o 30 o A o Valence χ 1 [GeV 2 ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103

93 PQChPT: 1+1 case, 3 sea quarks A Sea χ 4 [GeV 2 ] o o 45 o o A o Valence χ 1 [GeV 2 ] δ (4) /F o 0 60 o o 30 o o p 4 mass χ 1 [GeV 2 ] Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103

94 PQChPT: 1+1 case, 3 sea quarks A Sea χ 4 [GeV 2 ] o o 45 o o A o Valence χ 1 [GeV 2 ] δ (4) /F o 0 60 o o 30 o o p 4 mass χ 1 [GeV 2 ] Notice the Quenched Chiral Logs: m2 π χ 1 = 1+ α F 2χ 4 log χ 1 + Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103

95 PQChPT: 1+1 case, 3 sea quarks o 60 o 45 o δ (4) /F 0 2 +δ (6) /F A 30 o o χ 1 [GeV 2 ] p 4 + p 6 relative correction mass Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103

96 PQChPT: 1+1 case, 3 sea quarks o 60 o 45 o o 15 o A δ (4) /F 0 2 +δ (6) /F A 30 o 15 o δ (4) /F 0 2 +δ (6) /F o 60 o 45 o χ 1 [GeV 2 ] χ 1 [GeV 2 ] p 4 + p 6 relative correction mass decay constant Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103

97 PQChPT: 1+1 case, 2 sea quarks 0.4 M χ 3 [GeV 2 ] χ 1 [GeV 2 ] Relative Correction: Mass χ 1 : valence mass, χ 3 : sea mass Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103

98 PQChPT: 1+1 case, 2 sea quarks 0.4 M 0.4 F χ 3 [GeV 2 ] χ 3 [GeV 2 ] χ 1 [GeV 2 ] χ 1 [GeV 2 ] Relative Correction: Mass Decay Constant χ 1 : valence mass, χ 3 : sea mass Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103

99 PQChPT: 2 sea quarks χ 3 = χ 1 tan(θ) χ 4 = z χ 3 θ = 70 o z = 0.4 z = M n f = 2 d val = 1 d sea = 2 z = 0.8 z = 1.0 z = 1.2 z = χ 1 [GeV 2 ] Relative Correction: Mass 1+2 case Valence: χ 1 χ 2 Sea: χ 3 = χ 4 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103

100 PQChPT: 2 sea quarks χ 3 = χ 1 tan(θ) χ 4 = z χ 3 θ = 70 o z = 0.4 z = χ 3 = χ 1 tan(θ) χ 2 = x χ 1 θ = 70 o x = 1.8 x = M 0 M n f = 2 d val = 1 d sea = 2 z = 0.8 z = 1.0 z = 1.2 z = χ 1 [GeV 2 ] Relative Correction: Mass 1+2 case Valence: χ 1 χ 2 Sea: χ 3 = χ n f = 2 d val = 2 d sea = 1 x = 1.4 x = 1.2 x = 1.0 x = χ 1 [GeV 2 ] Mass 2+1 case Valence: χ 1 = χ 2 Sea: χ 3 χ 4 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103

101 PQChPT: Fitting Strategy For masses and Decay constants At order p 4 : L r 4,Lr 5,Lr 6,Lr 8 At order p 6 : all allowed quadratic quark mass combinations show up Problem: Need L r 0,Lr 1,Lr 2,Lr 3 only to order p 4 from ππ,πk scattering but Nonanalytic structure at p 6 given (only one included in numerics shown) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.76/103

102 Conclusions PQChPT Part All relevant mass combinations for masses and decay constants for charged pseudoscalar mesons now known to two loops Decay constants and masses for two sea quarks converge nicely Masses for three sea quarks convergence slower Looking forward to getting lattice data to fit Expressions available from bijnens/chpt.html For the numerical programs contact the authors Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.77/103

103 Wishing list General: Quark-mass dependences everywhere Not only fits but also continuum infinite volume results at a given quark-mass (so we can also fit ourselves for studying other inputs) More use of the existing two-loop calculations Analytical NNLO only: not really fewer parameters Only more in LECs: p 4 scattering LECs also in masses/decay constants I.e. l1 r,lr 2 (n F = 2), L r 1,Lr 2,Lr 3 (n F = 3), ˆL r i, i = 0, 1, 2, 3 (PQChPT) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.78/103

104 Wishing list: Two-flavour (with input from Gasser et al.) l3 and errors l4 : from F π (m q ) and scalar radius: can lattice check this relation a 2 0 accurately predicted in terms of scalar radius: can lattice check this Isospin breaking in ππ scattering (important for CP violation in K ππ) l5 l 6 Needed for π lνγ, can be had from Π AA Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.79/103

105 Wishing list: 3-flavour Ideas on how to make all those calculations usable for you Three and more flavour: typically slow numerically large N c suppressed couplings: i.e. m s dependence of m π,f π L r 4, Lr 6 sigma terms and scalar radii Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.80/103

106 Conclusions and final comments Lots of analytical work done in ChPT Use the correct ChPT 2-flavour for varying ˆm and possible for N f = 2 and N f = at fixed m s (but have different LECs) otherwise 3-flavour the various partially quenched versions Remember at which order in ChPT you compare things Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.81/103

107 Conclusions and final comments Lots of analytical work done in ChPT Use the correct ChPT 2-flavour for varying ˆm and possible for N f = 2 and N f = at fixed m s (but have different LECs) otherwise 3-flavour the various partially quenched versions Remember at which order in ChPT you compare things Seen a lot of lattice work, looking forward to seeing more LECs in many talks: Boyle, Matsufuru, Urbach, Kuramashi and many parallel session talks Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.81/103

108 η 3π Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/ ] JB, Acta Phys. Slov. 56(2005)305 [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 ), Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.82/103

109 η 3π: Lowest order (LO) 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103

110 η 3π: Lowest order (LO) Pions are in I = 1 state = A (m u m d ) or α em 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103

111 η 3π: Lowest order (LO) Pions are in I = 1 state = A (m u m d ) or α em 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 with Q 2 m2 s ˆm 2 m 2 d m2 u or R m s ˆm m d m u ˆm = 1 2 (m u + m d ) 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, A(s,t,u) = 3 4R M(s,t,u) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103

112 η 3π: Lowest order (LO) Pions are in I = 1 state = A (m u m d ) or α em 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 with Q 2 m2 s ˆm 2 m 2 d m2 u or R m s ˆm m d m u ˆm = 1 2 (m u + m d ) 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, A(s,t,u) = 3 4R M(s,t,u) LO: M(s,t,u) = 3s 4m2 π m 2 η m 2 π M(s,t,u) = 1 F 2 π ( ) 4 3 m2 π s Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103

113 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. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103

114 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. Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103

115 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 Gasser-Leutwyler 1985: = 2.4, dlips A 2 2 (LIPS=Lorentz invariant phase-space) Major source: large S-wave final state rescattering Experiment: 295 ± 17 ev (PDG 2006) Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103

116 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 Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.85/103