Effective Field Theory: Introduction and Applications

lavi net EFFECTIVE FIELD THEORY: INTRODUCTION AND APPLICATIONS

Johan Bijnens Lund University

[email protected] http://www.thep.lu.se/∼bijnens

Various ChPT: http://www.thep.lu.se/∼bijnens/chpt.html

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.1/103 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 Wikipedia

http://en.wikipedia.org/wiki/ 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 Wikipedia

http://en.wikipedia.org/wiki/ 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 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.

Solid state physics: conductors: neglect  the empty bands above the partially ﬁlled Examples:  one   Atomic physics: Blue sky: neglect atomic structure  

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.5/103 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 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 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 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 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 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 ≈ −

2 A = Φ†∂tΦv + . . . γA = GF Φ†Φv + . . . L v L µν 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 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 ≈ −

2 A = Φ†∂tΦv + . . . γA = GF Φ†Φv + . . . L v L µν v Units with h/ = c = 1: G energy dimension 3: − σ G2E4 ≈ γ

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103 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 ≈ −

2 A = Φ†∂tΦv + . . . γA = GF Φ†Φv + . . . L v L µν v Units with h/ = c = 1: G energy dimension 3: − σ G2E4 ≈ γ = red sunsets blue light scatters a lot more than red ⇒ = blue sky ⇒ Higher orders suppressed by 1 Å/λγ.

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.7/103 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 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 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 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 references

A. Manohar, Effective Field Theories (Schladming lectures), hep-ph/9606222 I. Rothstein, Lectures on Effective Field Theories (TASI lectures), hep-ph/0308266 G. Ecker, Effective ﬁeld theories, Encyclopedia of Mathematical Physics, hep-ph/0507056 D.B. Kaplan, Five lectures on effective ﬁeld theory, nucl-th/0510023 A. Pich, Les Houches Lectures, hep-ph/9806303 S. Scherer, Introduction to chiral perturbation theory, hep-ph/0210398 J. Donoghue, Introduction to the Effective Field Theory Description of Gravity, gr-qc/9512024

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.10/103 School

PSI Zuoz Summer School on Particle Physics Effective Theories in Particle Physics, July 16 - 22, 2006

http://ltpth.web.psi.ch/zuoz_school/ 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 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 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/9311274]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.12/103 The mass gap: Goldstone Modes

UNBROKEN: V (φ) BROKEN: V (φ)

Only massive modes Need to pick a vacuum φ = 0: Breaks symmetry around lowest energy h i 6 state (=vacuum) 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 The two symmetry modes compared

Wigner-Eckart mode Nambu-Goldstone mode

Symmetry group G G spontaneously broken to subgroup H

Vacuum state unique Vacuum state degenerate

Massive Excitations Existence of a massless mode

States fall in multiplets of G States fall in multiplets of H

Wigner Eckart theorem for G 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 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 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 must contain at least one ∂µφ L

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.16/103 Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown Power counting: Dimensional counting

Expected breakdown scale: Resonances, so Mρ or higher depending on the channel

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

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown Power counting: Dimensional counting

Expected breakdown scale: Resonances, so Mρ or higher depending on the channel

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU(3)V

But QCD = [iq¯LD/qL + iq¯RD/qR mq (¯qRqL +q ¯LqR)] L − q=Xu,d,s

So if mq = 0 then SU(3)L SU(3)R. ×

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

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown Power counting: Dimensional counting

Expected breakdown scale: Resonances, so Mρ or higher depending on the channel

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU(3)V

But QCD = [iq¯LD/qL + iq¯RD/qR mq (¯qRqL +q ¯LqR)] L − q=Xu,d,s

So if mq = 0 then SU(3)L SU(3)R. × v < c, mq = 0 = Can also see that via 6 ⇒ v = c, mq = 0 =/ ⇒ Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.17/103 Chiral Perturbation Theory

qq¯ = q¯LqR +q ¯RqL = 0 h i h i 6 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 Chiral Perturbation Theory

qq¯ = q¯LqR +q ¯RqL = 0 h i h i 6 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)

p2 (p2)2 (1/p2)2 p4 = p4

1/p2 (p2) (1/p2) p4 = p4 d4p p4 R Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.18/103 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 http://www.thep.lu.se/ bijnens/chpt.html ∼

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

U(φ) = exp(i√2Φ/F0) parametrizes Goldstone Bosons

0 π η8 + π+ K+ 0 √2 √6 1 0 π η8 Φ(x)= B π− + K0 C . B C B − √2 √6 C B − 2 η8 C B K K¯ 0 C B − √ C @ 6 A

2 F0 µ LO Lagrangian: = DµU D U + χ U + χU , L2 4 {h † i h † †i}

DµU = ∂µU irµU + iUlµ , − left and right external currents: r(l)µ = vµ +( )aµ −

Scalar and pseudoscalar external densities: χ = 2B0(s + ip) quark masses via scalar density: s = + M · · · A = T rF (A) Odenseh 19-20/2/2008i Effective Field Theory: Introduction and Applications Johan Bijnens p.21/103 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 ⇔ 4 i R d x QCD(q,q,G,l,r,s,p) [dGdqdq]e L Z ≈ 4 i R d x ChPT (U,l,r,s,p) [dU]e L Z so also functional derivatives are equal

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.22/103 Lagrangians

µ 2 µ ν = L DµU D U + L DµU DνU D U D U L4 1h † i 2h † ih † i µ ν µ +L D U DµUD U DνU + L D U DµU χ U + χU 3h † † i 4h † ih † †i µ 2 +L D U DµU(χ U + U χ) + L χ U + χU 5h † † † i 6h † †i +L χ U χU 2 + L χ Uχ U + χU χU 7h † − †i 8h † † † †i iL F R DµUDνU + F L DµU DνU − 9h µν † µν † i +L U F R UF Lµν + H F R F Rµν + F L F Lµν + H χ χ 10h † µν i 1h µν µν i 2h † i

Li: Low-energy-constants (LECs) Hi: Values depend on deﬁnition of currents/densities

r These absorb the divergences of loop diagrams: Li Li Renormalization: order by order in the powercounting→

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.23/103 Lagrangians

Lagrangian Structure: 2 ﬂavour 3 ﬂavour 3+3 PQChPT 2 p F,B 2 F0,B0 2 F0,B0 2 4 r r r r ˆr ˆ r p li , hi 7+3 Li ,Hi 10+2 Li , Hi 11+2 6 r r r p ci 52+4 Ci 90+4 Ki 112+3

p2: Weinberg 1966 p4: Gasser, Leutwyler 84,85 p6: JB, Colangelo, Ecker 99,00

➠ replica method = PQ obtained from NF flavour  ➠ All infinities known⇒  ➠ 3 flavour special case of 3+3 PQ: Lˆr, Kr Lr,Cr  i i → i i ➠ 53 52 arXiv:0705.0576 [hep-ph]  → Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.24/103 Chiral Logarithms

The main predictions of ChPT:

Relates processes with different numbers of pseudoscalars Chiral logarithms

2Bmˆ 2 1 (2Bmˆ ) m2 = 2Bmˆ + log + 2lr(µ) + π F 32π2 µ2 3 · · ·

M 2 = 2Bmˆ B = B , F = F (two versus three-ﬂavour) 6 0 6 0

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.25/103 LECs and µ

r l3(µ)

2 2 ¯ 32π r Mπ li = li (µ) log 2 . γi − µ Independent of the scale µ.

r For 3 and more ﬂavours, some of the γi = 0: Li (µ) µ :

mπ, mK: chiral logs vanish pick larger scale r 1 GeV then L (µ) 0 large Nc arguments???? 5 ≈ compromise: µ = mρ = 0.77 GeV

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.26/103 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 (Fπ, FK)? Express orders in terms of lowest order masses? 2 2 E.g. s + t + u = 2mπ + 2mK in πK scattering See e.g. Descotes-Genon talk

I prefer physical masses Thresholds correct Chiral logs are from physical particles propagating

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

m0 f0 mπ = m0 fπ = m0 1 + a f0 1 + b f0

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103 An example

m0 f0 mπ = m0 fπ = m0 1 + a f0 1 + b f0 2 3 m0 2 m0 mπ = m0 a + a 2 + − f0 f0 · · · 2 m0 2 m0 fπ = f0 1 b + b 2 + − f0 f0 · · ·

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103 An example

m0 f0 mπ = m0 fπ = m0 1 + a f0 1 + b f0 2 3 m0 2 m0 mπ = m0 a + a 2 + − f0 f0 · · · 2 m0 2 m0 fπ = f0 1 b + b 2 + − f0 f0 · · · 2 3 mπ mπ mπ = m0 a + a(b a) 2 + − fπ − fπ · · · 2 mπ mπ mπ = m0 1 a + ab 2 + − fπ fπ · · · 2 mπ mπ fπ = f0 1 b + b(2b a) 2 + − fπ − fπ · · ·

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103 An example

a = 1 b = 0.5 f0 = 1

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.28/103 An example: m0/f0

0.5

0.4 mπ LO NLO NNLO 0.3 π m

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5

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

0.5

0.4 mπ LO NLOp NNLOp 0.3 π m

0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.30/103 Two-loop Two-ﬂavour

Review paper on Two-Loops: JB, hep-ph/0604043 Prog. Part. Nucl. Phys. 58 (2007) 521

Dispersive Calculation of the nonpolynomial part in q2, s, t, u

Gasser-Meißner: FV , FS: 1991 numerical Knecht-Moussallam-Stern-Fuchs: ππ: 1995 analytical

Colangelo-Finkemeier-Urech: FV , FS: 1996 analytical

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.31/103 Two-Loop Two-ﬂavour

Bellucci-Gasser-Sainio: γγ π0π0: 1994 → + Bürgi: γγ π π , Fπ, mπ: 1996 → − JB-Colangelo-Ecker-Gasser-Sainio: ππ, Fπ, mπ: 1996-97

JB-Colangelo-Talavera: FV π(t), FSπ: 1998 JB-Talavera: π ℓνγ: 1997 → 0 0 + Gasser-Ivanov-Sainio: γγ π π , γγ π π−: 2005-2006 → →

mπ, Fπ, FV , FS, ππ: simple analytical forms

Colangelo-(Dürr-)Haefeli: Finite volume Fπ,mπ 2005-2006

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.32/103 LECs

6 ¯l1 to ¯l4: ChPT at order p and the Roy equation analysis in ππ and FS Colangelo, Gasser and Leutwyler, Nucl. Phys. B 603 (2001) 125 [hep-ph/0103088]

¯l and ¯l : from FV and π ℓνγ JB,(Colangelo,)Talavera 5 6 → ¯l = 0.4 0.6 , 1 − ± ¯l = 4.3 0.1 , 2 ± ¯l = 2.9 2.4 , 3 ± ¯l = 4.4 0.2 , 4 ± ¯l ¯l = 3.0 0.3 , 6 − 5 ± ¯l = 16.0 0.5 0.7 . 6 ± ±

l 5 10 3 from π0-η mixing Gasser, Leutwyler 1984 7 ∼ · − Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.33/103 LECs

Some combinations of order p6 LECs are known as well: curvature of the scalar and vector formfactor, two more combinations from ππ scattering (implicit in b5 and b6)

r Note: ci for mπ, fπ, ππ: small effect

r ci (770MeV ) = 0 for plots shown

2 2 expansion in mπ/Fπ 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 2 mπ

0.25

0.2 LO NLO NNLO 0.15 2 π m 0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 M2 [GeV2]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.35/103 2 ¯ mπ (l3 =0)

0.35

0.3

0.25

0.2 LO 2

π NLO

m NNLO 0.15

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 M2 [GeV2]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.36/103 Fπ

0.12

0.1

0.08

0.06 LO [GeV]

π NLO F NNLO 0.04

0.02

0 0 0.05 0.1 0.15 0.2 0.25 M2 [GeV2]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.37/103 Two-loop Three-ﬂavour, 2001 ≤

ΠV V π, ΠVVη, ΠVVK Kambor, Golowich; Kambor, Dürr; Amorós, JB, Talavera

ΠV V ρω Maltman

ΠAAπ, ΠAAη, Fπ, Fη, mπ, mη Kambor, Golowich; Amorós, JB, Talavera r r ΠSS Moussallam L4,L6

ΠVVK, ΠAAK, FK, mK Amorós, JB, Talavera r r r K , qq Amorós, JB, Talavera L ,L ,L ℓ4 h i 1 2 3 r FM , mM , qq (mu = m ) Amorós, JB, Talavera L ,mu/m h i 6 d 5,7,8 d

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.38/103 Two-loop Three-ﬂavour, 2001 ≥

r FV π, FVK+ , FVK0 Post, Schilcher; JB, Talavera L9

Kℓ3 Post, Schilcher; JB, Talavera Vus r r FSπ, FSK (includes σ-terms) JB, Dhonte L4,L6 r K,π ℓνγ Geng, Ho, Wu L → 10 ππ JB,Dhonte,Talavera

πK JB,Dhonte,Talavera r r r relation li and Li ,Ci Gasser,Haefeli,Ivanov,Schmid Finite volume qq JB,Ghorbani h i

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.39/103 Two-loop Three-ﬂavour

Known to be in progress

η 3π: being written up JB,Ghorbani → Kℓ3 iso: preliminary results available JB,Ghorbani Finite Volume: sunsetintegrals being written up JB,Lähde r r r relation ci and Li ,Ci Gasser,Haefeli,Ivanov,Schmid

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.40/103 r Ci

Most analysis use: r Ci from (single) resonance approximation

π π q2 << m2, m2 Cr ρ, S | |= ρ S i q2 π → π ⇒

Motivated by large Nc: 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 r Ci

1 µν 1 2 µ fV µν V = Vµν V + mV VµV Vµν f+ L − 4 h i 2 h i− 2√2 h i

igV µ ν µ Vµν [u ,u ] + fχ Vµ[u ,χ−] − 2√2 h i h i

1 µν 1 2 µ fA µν A = Aµν A + mA AµA Aµν f− L − 4 h i 2 h i− 2√2 h i

1 µ 2 2 µ S = S µS M S + cd Su uµ + cm Sχ+ L 2 h∇ ∇ − S i h i h i

1 µ 1 2 2 ′ = ∂µP1∂ P1 M ′ P1 + id˜mP1 χ− . Lη 2 − 2 η h i

fV = 0.20, fχ = 0.025, gV = 0.09, cm = 42 MeV, c = 32 MeV, d˜m = 20 MeV, − d

mV = mρ = 0.77 GeV, mA = ma1 = 1.23 GeV, mS = 0.98 GeV, mP1 = 0.958 GeV

fV , gV , fχ, fA: experiment 4 cm and c from resonance saturation at (p ) d O

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.42/103 r Ci

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.43/103 r Ci

Problems: Weakest point in the numerics However not all results presented depend on this r Unknown so far: Ci 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: 600-900 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 Inputs

Kℓ4: F (0), G(0), λ E865 BNL 2 2 2 2 mπ0 , mη, mK+ , mK0 em with Dashen violation

Fπ+

FK+ /Fπ+

ms/mˆ 24 (26) mˆ =(mu + md)/2 r r L4,L6

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.44/103 Outputs: I

fit 10 same p4 fit B fit D 103Lr 0.43 0.12 0.38 0.44 0.44 1 ± 103Lr 0.73 0.12 1.59 0.60 0.69 2 ± 103Lr 2.35 0.37 2.91 2.31 2.33 3 − ± − − − 103Lr 0 0 0.5 0.2 4 ≡ ≡ ≡ ≡ 103Lr 0.97 0.11 1.46 0.82 0.88 5 ± 103Lr 0 0 0.1 0 6 ≡ ≡ ≡ ≡ 103Lr 0.31 0.14 0.49 0.26 0.28 7 − ± − − − 103Lr 0.60 0.18 1.00 0.50 0.54 8 ± ➠ errors are very correlated ➠ µ = 770 MeV; 550 or 1000 within errors ➠ r varying Ci factor 2 about errors ➠ r r 0 3 0 6 10−3 L4, L6 . ,..., . 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 Correlations

(older ﬁt) 3 r 10 L1 = 0.52 0.23 3 r ± 10 L2 = 0.72 0.24 Lr 3 r ± 2 10 L = 2.70 0.99 3 − ±

1.4

1.2

0.8

0.6

0.4

0.2 1 0.8 0 -5 0.6 r -4 0.4 L -3 1 r -2 0.2 L3 -1 0 0

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.46/103 Outputs: II

fit 10 same p4 fit B fit D 2 ˆ 2 B0m/mπ 0.736 0.991 1.129 0.958 m2 : p4,p6 0.006,0.258 0.009, 0 0.138,0.009 0.091,0.133 π ≡ − − m2 : p4,p6 0.007,0.306 0.075, 0 0.149,0.094 0.096,0.201 K ≡ − − m2: p4,p6 0.052,0.318 0.013, 0 0.197,0.073 0.151,0.197 η − ≡ − − mu/md 0.45 0.05 0.52 0.52 0.50 ± F0 [MeV] 87.7 81.1 70.4 80.4 FK : p4,p6 0.169,0.051 0.22, 0 0.153,0.067 0.159,0.061 Fπ ≡

➠ mu = 0 always very far from the ﬁts

➠ F0: pion decay constant in the chiral limit

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.47/103 ππ

p4 p4 p6 p6

0 2 a0 a0

0.225 -0.039 -0.04 0.22 -0.041 0.215 -0.042 -0.043 0.21 -0.044 0.205 0.6 -0.045 0.6 0.5 0.5 0.4 -0.046 0.4 0.2 0.3 0.3 -0.047 0.2 3 r 0.2 3 r 0.195 0.1 10 L6 -0.048 0.1 10 L6 -0.4 0 -0.4 0 -0.2 -0.1 -0.2 -0.1 0 0 0.2 -0.2 0.2 -0.2 3 r 0.4 -0.3 3 r 0.4 -0.3 10 L4 0.6 10 L4 0.6

0 2 a0 = 0.220 0.005, a0 = 0.0444 0.0010 Colangelo, Gasser,± Leutwyler− ± a0 = 0.159 a2 = 0.0454 at order p2 0 0 −

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.48/103 ππ and πK

ππ constraints πK constraints 0.6 0.6

+ 0.5 0.5 C10 3/2 a0 0.4 0.4 C1 2 a0 0.3 0.3 r 6 r 6 0.2 0.2 L L 3 3 10 10 0.1 0.1

2 0 a0 0 C1 -0.1 3/2 -0.1 a0 + C10 -0.2 -0.2

-0.3 -0.3 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 3 r 3 r 10 L4 10 L4 preferred region: ﬁt D: 103Lr 0.2, 103Lr 0.0 4 ≈ 6 ≈ General ﬁtting needs more work and systematic studies

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.49/103 Quark mass dependences

Updates of plots in Amorós, JB and Talavera, hep-ph/0003258, Nucl. Phys. B585 (2000) 293 Some new ones Procedure: calculate a consistent set of mπ,mK,mη,fπ with the given input values (done iteratively)

vary ms/(ms)phys, keep ms/mˆ = 24 2 2 mπ, mK, Fπ, FK

vary ms/(ms)phys keep mˆ ﬁxed 2 mπ, Fπ

vary mπ, keep mK ﬁxed f+(0): the formfactor in Kℓ3 decays 2 f (0),f (0)/ m2 m2 + + K − π Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.50/103 2 mπ ﬁt 10

0.02

0.018

0.016

0.014 LO NLO ] 2 0.012 NNLO 0.01 [GeV 2 π

m 0.008

0.006

0.004

0.002

0 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.51/103 2 mπ ﬁt D

0.02

0.018

0.016

0.014 LO NLO ] 2 0.012 NNLO 0.01 [GeV 2 π

m 0.008

0.006

0.004

0.002

0 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.52/103 2 mK ﬁt 10

0.25

0.2 LO NLO NNLO ]

2 0.15 [GeV 2 K

m 0.1

0.05

0 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.53/103 2 mK ﬁt D

0.25

0.2 LO NLO NNLO ]

2 0.15 [GeV 2 K

m 0.1

0.05

0 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.54/103 Fπ ﬁt 10

0.1

0.098 LO NLO NNLO 0.096

0.094 [GeV] π

F 0.092

0.09

0.088

0.086 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.55/103 FK ﬁt 10

0.115

0.11

0.105

0.1 [GeV] K LO F NLO 0.095 NNLO

0.09

0.085 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.56/103 FK/Fπ ﬁt 10

1.25

1.2

1.15 NLO

π NNLO /F K F 1.1

1.05

1 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.57/103 2 mπ ﬁt 10, ﬁxed mˆ

0.02

0.018

0.016

0.014 ]

2 0.012

0.01 LO [GeV 2

π NLO

m 0.008 NNLO 0.006

0.004

0.002

0 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.58/103 Fπ ﬁt 10, ﬁxed mˆ

0.1

0.098 LO NLO NNLO 0.096

0.094 [GeV] π F 0.092

0.09

0.088

0.086 0 0.2 0.4 0.6 0.8 1

ms/(ms)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.59/103 f+(0) ﬁt 10, ﬁxed mK

0.015

0.01

0.005 (0)

+ 0

-0.005

-0.01 sum

correction to f p4 p6 2-loops -0.015 6 r p Li -0.02

-0.025 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2 mπ/(mK)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.60/103 2 (f (0))/ m2 m2 ﬁt 10, ﬁxed m + K − π K 0.3 ]

4 0.2 −

0.1 [GeV 2 ) 2 π 0 m −

2 K -0.1 (0))/(m

+ -0.2

-0.3 sum p4 p6 2-loops -0.4 6 r (correction to f p Li -0.5 0 0.2 0.4 0.6 0.8 1 2 2 mπ/(mK)phys

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.61/103 f0(t) in Kℓ3

Main Result: JB,Talavera

8 2 f (t) = 1 (Cr + Cr ) m2 m2 0 − F 4 12 34 K − π π t r r 2 2 t +8 (2C + C ) m + m + (FK/Fπ 1) F 4 12 34 K π m2 m2 − π K − π 8 2 r 4 t C12 + ∆(t) + ∆(0) . −Fπ

r r ∆(t) and ∆(0) contain NO Ci and only depend on the Li at order p6 = All⇒ needed parameters can be determined experimentally

r ∆(0) = 0.0080 0.0057[loops] 0.0028[Li ] . Odense 19-20/2/2008 Effective− Field Theory: Introduction± and Applications± Johan Bijnens p.62/103 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.

r r(3pq) r(3pq) E.g. L1 = L0 /2 + L1

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

One-loop: Bernard, Golterman, Sharpe, Shoresh, Pallante,. . . with electromagnetism: JB,Danielsson, hep-lat/0610127 Two loops: 2 mπ+ simplest mass case: JB,Danielsson,Lähde, hep-lat/0406017 Fπ+ : JB,Lähde, hep-lat/0501014 2 Fπ+ , mπ+ two sea quarks: JB,Lähde, hep-lat/0506004 2 mπ+ : JB,Danielsson,Lähde, hep-lat/0602003 Neutral masses: JB,Danielsson, hep-lat/0606017 Lattice data: a and L extrapolations needed Programs available from me (Fortran) Formulas: http://www.thep.lu.se/ bijnens/chpt.html ∼

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.64/103 Partial Quenching and ChPT

Mesons Quark Flow Quark Flow Valence 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 p4: Bernard, Golterman, Pallante, Sharpe, Shoresh,. . . Two Loops or p6: This talk JB, Niclas Danielsson, Timo Lähde

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.65/103 PQChPT at Two Loops: General

Add ghost quarks: remove the unwanted free valence loops

Mesons Quark Flow Quark Flow Quark Flow Quark Flow Valence Valence Sea Ghost = + + +

Possible problem: QCD = ChPT relies heavily on unitarity ⇒ Partially quenched: at least one dynamical sea quark = Φ is heavy: remove from PQChPT ⇒ 0

Symmetry group becomes SU(nv + ns nv) SU(nv + ns nv) (approximately) | × |

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

r r(3pq) r(3pq) E.g. L1 = L0 /2 + L1

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

valence equal mass, 3 sea equal mass: 2 mπ+ : JB,Danielsson,Lähde, hep-lat/0406017 Other mass combinations: 2 Fπ+ : JB,Lähde, hep-lat/0501014 2 2 Fπ+ , mπ+ two sea quarks: JB,Lähde, hep-lat/0506004 In progress: the other charged masses

➠ heavy use of FORM Vermaseren Actual Calculations:  ➠ 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 Long Expressions

= ⇒

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

(6)22 r 2 r 2 δ = π L 4/9 χηχ 1/2 χ χ + χ 13/3¯χ χ 35/18χ ¯ 2 π L χ loops 16 0 4 − 1 3 13 − 1 13 − 2 − 16 1 13 r 2 r 2 π L 11/3 χηχ + χ + 13/3¯χ + π L 4/9 χηχ 7/12 χ χ + 11/6 χ 17/6¯χ χ 43/36χ ¯ − 16 2 4 13 2 16 3 4 − 1 3 13 − 1 13 − 2 2 2 r r r2 2 + π 15/64 χηχ 59/384 χ χ + 65/384 χ 1/2¯χ χ 43/128χ ¯ 48 L L χ¯ χ 72 L χ¯ 16 − 4 − 1 3 13 − 1 13 − 2 − 4 5 1 13 − 4 1 r2 2 p c p p 8 L χ + A¯(χp) π 1/24 χp +1/48χ ¯ 1/8¯χ R +1/16χ ¯ R 1/48 R χp 1/16 R χq − 5 13 16 − 1 − 1 qη 1 p − qη − qη η c ¯ r p c d ¯ r p + 1/48 Rpp χη +1/16 Rp χ13 + A(χp) L0 8/3 Rqη χp +2/3 Rp χp +2/3 Rp + A(χp) L3 2/3 Rqη χp c d r pp p c r pp p + 5/3 R χp +5/3 R + A¯(χp) L 2¯χ χ¯ 2¯χ R +3¯χ R + A¯(χp) L 2/3¯χ R χp p p 4 − 1 ηη0 − 1 qη 1 p 5 − ηη1 − qη p c c 2 p 2 p c c 2 + 1/3 R χq +1/2 R χp 1/6 R χq + A¯(χp) 1/16+1/72 (R ) 1/72 R R +1/288(R ) qη p − p qη − qη p p p p c p p c + A¯(χp)A¯(χps) 1/36 R 5/72 R +7/144 R A¯(χp)A¯(χqs) 1/36 R +1/24 R +1/48 R − qη − sη p − qη sη p p v c v η + A¯(χp)A¯(χη) 1/72 R R +1/144 R R + 1/8 A¯(χp)A¯(χ )+1/12 A¯(χp)A¯(χ ) R − qη η13 p η13 13 46 pp p c p d c 2 c d + A¯(χp)B¯(χp,χp; 0) 1/4 χp 1/18 R R χp 1/72 R R +1/18 (R ) χp +1/144 R R − qη p − qη p p p p η c η c p d c d + A¯(χp)B¯(χp,χη; 0) 1/18 R R χp 1/18 R R χp + A¯(χp)B¯(χq,χq; 0) 1/72 R R +1/144 R R pp p − 13 p − qη q p q p q c 1/12 A¯(χp)B¯(χps,χps; 0) R χps 1/18 A¯(χp)B¯(χ ,χ ; 0) R R χp − sη − 1 3 pη p c d p c c d + 1/18 A¯(χp)C¯(χp,χp,χp; 0) R R χp + A¯(χp; ε) π 1/8¯χ R 1/16¯χ R 1/16 R χp 1/16 R p p 16 1 qη − 1 p − p − p ¯ ¯ r ¯ r ¯ r + A(χps) π16 [1/16 χps 3/16 χqs 3/16χ ¯1] 2 A(χps) L0 χps 5 A(χps) L3 χps 3 A(χps) L4 χ¯1 r − − η − η −η η − η + A¯(χps) L χ + A¯(χps)A¯(χη) 7/144 R 5/72 R 1/48 R +5/72 R 1/36 R 5 13 pp − ps − qq qs − 13 p p η z + A¯(χps)B¯(χp,χp; 0) 1/24 R χp 5/24 R χps + A¯(χps)B¯(χp,χη; 0) 1/18 R R χp sη − sη − ps qpη η z d q 1/9 R R χps 1/48 A¯(χps)B¯(χq,χq; 0) R + 1/18 A¯(χps)B¯(χ ,χ ; 0) R χs − ps qpη − q 1 3 sη q 2 + 1/9 A¯(χps)B¯(χ ,χ ;0, k) R + 3/16 A¯(χps; ε) π [χs +χ ¯ ] 1/8 A¯(χp ) 1/8 A¯(χp )A¯(χp ) 1 3 sη 16 1 − 4 − 4 6 = 2 v v v + 1/8 A¯(χp )A¯(χq ) 1/32 A¯(χp ) + A¯(χη) π 1/16χ ¯ R 1/48 R χη +1/16 R χ 4 6 − 6 16 1 η13 − η13 η13 13 r η v r r r η v + A¯(χη) L 4R χη +2/3 R χη 8 A¯(χη) L χη 2 A¯(χη) L χη + A¯(χη) L 4R χη +5/3 R χη 0 13 η13 − 1 − 2 3 13 η13 ⇒ r v r η η v 2 v 2 + A¯(χη) L 4 χη +χ ¯ R A¯(χη) L 1/6 R χq + R χ +1/6 R χη + 1/288 A¯(χη) (R ) 4 1 η13 − 5 pp 13 13 η13 η13 v ppη p η η c + 1/12 A¯(χη)A¯(χ ) R + A¯(χη)B¯(χp,χp; 0) 1/36χ ¯ 1/18 R R χp +1/18 R R χp 46 η13 − ηη1 − qη pp pp p d v ηpη ηpη η 2 z + 1/144 R R + A¯(χη)B¯(χp,χη; 0) 1/18χ ¯ +1/18χ ¯ +1/18 (R ) R χp p η13 − pη1 qη1 pp qpη η v v 1/12 A¯(χη)B¯(χps,χps; 0) R χps A¯(χη)B¯(χη,χη; 0) 1/216 R χ +1/27 R χ − ps − η13 4 η13 6 1 3 η d 1/18 A¯(χη)B¯(χ ,χ ; 0) R R χη + 1/18 A¯(χη)C¯(χp,χp,χp; 0) R R χp + A¯(χη; ε) π [1/8 χη − 1 3 ηη ηη pp p 16 v η v p c 1 3 c c 1/16χ ¯ R 1/8 R χη 1/16 R χη + A¯(χ )A¯(χ ) 1/72 R R +1/36 R R +1/144 R R − 1 η13 − 13 − η13 1 3 − qη q 3η 1η 1 3 4 A¯(χ ) Lr χ 10 A¯(χ ) Lr χ + 1/8 A¯(χ )2 1/2 A¯(χ )B¯(χ ,χ ;0, k) − 13 1 13 − 13 2 13 13 − 13 1 3 ¯ ¯ ¯ ¯ ¯ ¯ r ¯ r + 1/4 A(χ13; ε) π16 χ13 + 1/4 A(χ14)A(χ34) +1/16 A(χ16)A(χ36) 24 A(χ4) L1 χ4 6 A(χ4) L2 χ4 r p 2 − p −η p η + 12 A¯(χ ) L χ + 1/12 A¯(χ )B¯(χp,χp;0)(R ) χ + 1/6 A¯(χ )B¯(χp,χη; 0) R R χ R R χ 4 4 4 4 4η 4 4 4η p4 4 − 4η q4 4 v 1 3 1/24 A¯(χ )B¯(χη,χη; 0) R χ 1/6 A¯(χ )B¯(χ ,χ ; 0) R R χ + 3/8 A¯(χ ; ε) π χ − 4 η13 4 − 4 1 3 4η 4η 4 4 16 4 r r r η 32 A¯(χ ) L χ 8 A¯(χ ) L χ + 16 A¯(χ ) L χ + A¯(χ )B¯(χp,χp; 0) 1/9 χ +1/12 R χp − 46 1 46 − 46 2 46 46 4 46 46 46 pp η η η η η η + 1/36 R χ + 1/9 R χ + A¯(χ )B¯(χp,χη; 0) 1/18 R χ 1/9 R χ +1/9 R χ +1/18 R χ pp 4 p4 6 46 − pp 4 − p4 6 q4 6 13 4 η η v 1/6 A¯(χ )B¯(χp,χη;0, k) R R + 1/9 A¯(χ )B¯(χη,χη; 0) R χ A¯(χ )B¯(χ ,χ ;0) [2/9 χ − 46 pp − 13 46 η13 46 − 46 1 3 46 + 1/9 Rη χ + 1/18 Rη χ ] 1/6 A¯(χ )B¯(χ ,χ ;0, k) Rη + 1/2 A¯(χ ; ε) π χ p4 6 13 4 − 46 1 3 13 46 16 46 ¯ d d d ¯ r d + B(χp,χp; 0) π16 1/16χ ¯1 Rp +1/96 Rp χp +1/32 Rp χq + 2/3 B(χp,χp; 0) L0 Rp χp r d r pp p c d + 5/3 B¯(χp,χp; 0) L R χp + B¯(χp,χp; 0) L 2¯χ χ¯ χp 4¯χ R χp +4¯χ R χp +3¯χ R 3 p 4 − 1 ηη0 − 1 qη 1 p 1 p r pp p 2 c 2 d d + B¯(χp,χp; 0) L 2/3¯χ χp 4/3 R χ +4/3 R χ +1/2 R χp 1/6 R χq 5 − ηη1 − qη p p p p − p r pp p c r d 2 + B¯(χp,χp; 0) L 4¯χ χ¯ +8¯χ R χp 8¯χ R χp + 4 B¯(χp,χp; 0) L (R ) 6 1 ηη1 1 qη − 1 p 7 p r pp p 2 c 2 2 p d c d + B¯(χp,χp; 0) L 4/3¯χ +8/3 R χ 8/3 R χ + B¯(χp,χp; 0) 1/18 R R χp +1/18 R R χp 8 ηη2 qη p − p p − qη p p p d 2 η d η d + 1/288 (R ) + 1/18 B¯(χp,χp; 0)B¯(χp,χη; 0) R R χp R R χp p pp p − 13 p plus several more pages

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.69/103 Why so long expressions

Many different quark and meson masses (χij)

c ǫj Charged propagators: iG (k) = 2 (i = j) ij k χij +iε − − 6 n c 1 q Neutral propagators: G (k) = G (k) δij G (k) ij ij − nsea ij

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103 Why so long expressions

Many different quark and meson masses (χij)

c ǫj Charged propagators: iG (k) = 2 (i = j) ij k χij +iε − − 6 n c 1 q Neutral propagators: G (k) = G (k) δij G (k) ij ij − nsea ij d c π η q Ri Ri Rηii Rπii iG (k) = 2 2 + 2 + 2 + 2 ii (k χi+iε) k χi+iε k χπ+iε k χη+iε − − − − − i z d z Rjkl = Ri456jkl, Ri = Ri456πη, Rc = Ri + Ri + Ri Ri Ri i 4πη 5πη 6πη − πηη − ππη z z χa χb z (χa χb)(χa χc) R = χ χ , R = − , R = − − ab a b abc χa χc abcd χa χd − − − z (χa χb)(χa χc)(χa χd) R = − − − abcdefg (χa χe)(χa χf )(χa χg) − − −

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.70/103 Why so long expressions

Many different quark and meson masses (χij)

Problem: Plotting with many input parameters

Plot masses as a function of lowest order mass squared

2(0) I.e. of quark mass: χi = 2B0mi = mM

2 2 Remember: χi 0.3 GeV (550 MeV ) border ChPT ≈ ≈ ∼

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103 PQChPT at Two Loop

Problem: Plotting with many input parameters

Plot masses as a function of lowest order mass squared

2(0) I.e. of quark mass: χi = 2B0mi = mM

2 2 Remember: χi 0.3 GeV (550 MeV ) border ChPT ≈ ≈ ∼

Valence: 1+1 case: χ1 = χ2 = χ3 Sea: χ4 = χ5 = χ6

Plot along curves: χ4 = tan θ χ1 or χ4 constant Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.71/103 PQChPT: 1+1 case, 3 sea quarks

0.5

0.4

75o

] 0.3 Sea 2 60o

o [GeV

4 45 χ 0.2 30o A 0.1

15o

0 0 0.1 0.2 0.3 0.4 0.5 χ 2 Valence 1 [GeV ]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103 PQChPT: 1+1 case, 3 sea quarks

0.5 0.06 A

0.4 0.04

75o

] 0.3 Sea 0.02 o 2 60o 75 2 0 /F o [GeV (4)

4 45 δ χ 0.2 0 o 60 30o A o -0.02 45 0.1 30o 15o -0.04 15o 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 χ 2 4 χ 2 Valence 1 [GeV ] p mass 1 [GeV ]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103 PQChPT: 1+1 case, 3 sea quarks

0.5 0.06 A

0.4 0.04

75o

] 0.3 Sea 0.02 o 2 60o 75 2 0 /F o [GeV (4)

4 45 δ χ 0.2 0 o 60 30o A o -0.02 45 0.1 30o 15o -0.04 15o 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 χ 2 4 χ 2 Valence 1 [GeV ] p mass 1 [GeV ]

2 mπ α Notice the Quenched Chiral Logs: = 1+ 2 χ4 log χ1 + χ1 F · · ·

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.72/103 PQChPT: 1+1 case, 3 sea quarks

0.7

0.6 60o 75o 0.5 45o

4 0 0.4 /F A o

(6) 30 δ +

2 0 0.3 /F (4) δ 0.2

15o 0.1

0 0.1 0.2 0.3 0.4 0.5 χ 2 1 [GeV ] p4 + p6 relative correction mass

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103 PQChPT: 1+1 case, 3 sea quarks

0.7 0.5

75o 15o A 0.6 60o 75o 0.4 0.5 45o

30o 4 0 0.4 4 0 0.3 /F A o /F o

(6) 60 30 (6) δ δ + + 2 0 0.3 2 0 /F /F 0.2 (4) (4) δ 0.2 δ 45o 15o 0.1 0.1

0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 χ 2 χ 2 1 [GeV ] 1 [GeV ] p4 + p6 relative correction mass decay constant

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.73/103 PQChPT: 1+1 case, 2 sea quarks

∆ M 0.4

0.35 0.4 0.3 0.3 0.2

0.25 0.1 ] 2 0 0.2

[GeV -0.1 3 χ 0.15 -0.2

0.1

0.05

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 χ 2 1 [GeV ]

Relative Correction: Mass

χ1: valence mass, χ3: sea mass

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103 PQChPT: 1+1 case, 2 sea quarks

∆ ∆ M F 0.4 0.4

0.35 0.4 0.35 0.4 0.35 0.3 0.3 0.3 0.3 0.2 0.25 0.2 0.25 0.1 0.25 ] ] 2

2 0.15 0 0.1 0.2 0.2 [GeV

[GeV -0.1

3 0.05 3 χ χ 0 0.15 -0.2 0.15

0.1 0.1

0.05 0.05

0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 χ 2 χ [GeV2] 1 [GeV ] 1

Relative Correction: Mass Decay Constant

χ1: valence mass, χ3: sea mass

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.74/103 PQChPT: 2 sea quarks

0.2 χ χ θ 3 = 1 tan( ) z = 0.4 0.15 χ χ z = 0.6 4 = z 3 θ o 0.1 = 70

0.05

M 0 ∆

-0.05

-0.1 n = 2 z = 0.8 f z = 1.0 -0.15 dval = 1 z = 1.2 dsea = 2 z = 1.4 -0.2 0 0.05 0.1 0.15 0.2 χ 2 1 [GeV ] Relative Correction: Mass 1+2 case Valence: χ = χ 1 6 2 Sea: χ3 = χ4

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103 PQChPT: 2 sea quarks

0.2 0.2 χ χ θ χ χ θ 3 = 1 tan( ) z = 0.4 3 = 1 tan( ) 0.15 χ χ z = 0.6 0.15 χ χ 4 = z 3 2 = x 1 θ o θ o x = 1.8 0.1 = 70 0.1 = 70 x = 1.6

0.05 0.05

M 0 M 0 ∆ ∆

-0.05 -0.05

-0.1 -0.1 n = 2 z = 0.8 n = 2 x = 1.4 f z = 1.0 f x = 1.2 -0.15 dval = 1 z = 1.2 -0.15 dval = 2 x = 1.0 dsea = 2 z = 1.4 dsea = 1 x = 0.8 -0.2 -0.2 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 χ 2 χ 2 1 [GeV ] 1 [GeV ] Relative Correction: Mass Mass 1+2 case 2+1 case Valence: χ1 = χ2 Valence: χ1 = χ2 Sea: χ = χ 6 Sea: χ = χ 3 4 3 6 4

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.75/103 PQChPT: Fitting Strategy

For masses and Decay constants 4 r r r r At order p : L4,L5,L6,L8 At order p6: all allowed quadratic quark mass combinations show up r r r r Problem: Need L0,L1,L2,L3 from ππ,πK scattering but only to order p4 Nonanalytic structure at p6 given (only one included in numerics shown)

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.76/103 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 http://www.thep.lu.se/ 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 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: p4 scattering LECs also in masses/decay constants r r r r r I.e. l1, l2 (nF = 2), L1,L2,L3 (nF = 3), ˆr Li , i = 0, 1, 2, 3 (PQChPT)

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.78/103 Wishing list: Two-ﬂavour

(with input from Gasser et al.)

¯l3 and errors

¯l4: from Fπ(mq) and scalar radius: can lattice check this relation 2 a0 accurately predicted in terms of scalar radius: can lattice check this Isospin breaking in ππ scattering (important for CP violation in K ππ) → ¯l ¯l Needed for π ℓνγ, can be had from ΠAA 5 − 6 →

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.79/103 Wishing list: 3-ﬂavour ≥ Ideas on how to make all those calculations usable for you Three and more ﬂavour: typically slow numerically

large Nc suppressed couplings: i.e. ms dependence of mπ,Fπ r r L4, L6 sigma terms and scalar radii

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.80/103 Conclusions and ﬁnal comments

Lots of analytical work done in ChPT Use the correct ChPT

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.81/103 Conclusions and ﬁnal comments

Lots of analytical work done in ChPT Use the correct ChPT

2-flavour for varying mˆ and possible for Nf = 2 and Nf =2+1 at fixed ms (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 η 3π →

Reviews: JB, Gasser, Phys.Scripta T99(2002)34 [hep-ph/0202242] JB, Acta Phys. Slov. 56(2005)305 [hep-ph/0511076]

+ π pπ+ 2 2 s = (p + + p − ) =(pη p 0 ) π π − π η π−pπ− 2 2 pη t = (p − + p 0 ) =(pη p + ) π π − π 0 2 2 π pπ0 u = (p + + p 0 ) =(pη p − ) π π − π 2 2 2 s + t + u = m + 2m + + m 0 3s . η π π ≡ 0 0 + 4 4 π π π−out η = i (2π) δ (pη p + p − p 0 ) A(s, t, u) . h | i − π − π − π 0 0 0 4 4 π π π out η = i (2π) δ (pη p p p ) A(s , s , s ) h | i − 1 − 2 − 3 1 2 3

A(s1, s2, s3) = A(s1, s2, s3) + A(s2, s3, s1) + A(s3, s1, s2) ,

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.82/103 η 3π: Lowest order (LO) →

Pions are in I = 1 state = A (mu m ) or αem ⇒ ∼ − d αem effect is small (but large via m + m 0 ) π − π η π+π π0γ needs to be included directly → −

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

Pions are in I = 1 state = A (mu m ) or αem ⇒ ∼ − d

B0(mu md) 3(s s0) ChPT:Cronin 67: A(s, t, u) = −2 1 + 2 − 2 3√3F mη mπ π −

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

Pions are in I = 1 state = A (mu m ) or αem ⇒ ∼ − d

B0(mu md) 3(s s0) ChPT:Cronin 67: A(s, t, u) = −2 1 + 2 − 2 3√3F mη mπ π − 2 2 2 ms mˆ ms mˆ 1 with Q 2 − 2 or R mˆ = (m + m ) m m md −mu 2 u d ≡ d− u ≡ − 1 m2 (s, t, u) A(s, t, u) = K (m2 m2 ) M , 2 2 π K 2 Q mπ − 3√3Fπ √3 A(s, t, u) = M(s, t, u) 4R

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

Pions are in I = 1 state = A (mu m ) or αem ⇒ ∼ − d

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.83/103 η 3π beyond p4: p2 and p4 → 2 4 Γ(η 3π) A Q− allows a PRECISE measurement → ∝ | | ∝ Q 24 gives lowest order Γ(η π+π π0) 66 eV . ≈ → − ≈

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.84/103 η 3π beyond p4: p2 and p4 → 2 4 Γ(η 3π) A Q− allows a PRECISE measurement → ∝ | | ∝ Q 24 gives lowest order Γ(η π+π π0) 66 eV . ≈ → − ≈ 2 2 2 Other Source from m + m 0 Q = Q = 20.0 1.5 K − K ∼ − ⇒ ± Lowest order prediction Γ(η π+π π0) 140 eV . → − ≈

2 dLIP S A2 + A4 Z | | At order p4 Gasser-Leutwyler 1985: = 2.4 , 2 dLIP S A2 Z | | (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 η 3π beyond p4: 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 p8: No absorptive parts from ℓ 2 ≥ = M(s, t, u) = ⇒ 2 M (s)+(s u)M (t)+(s t)M (t)+M (t)+M (u) M (s) 0 − 1 − 1 2 2 − 3 2

MI: “roughly” contributions with isospin 0,1,2

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.85/103 η 3π beyond p4: Dispersive → 3 body dispersive: difﬁcult: keep only 2 body cuts start from πη ππ (m2 < 3m2 ) standard dispersive analysis → η π 2 analytically continue to physical mη.

1 ∞ ImMI(s′) MI(s) = ds′ π Z 2 s s iε 4mπ ′ − − ′ 1 ImMI (s ) discMI (s)= (MI (s + iε) MI (s iε)) −→ 2i − −

3 ′ ′ 2 s ds discM0(s ) M0(s) = a0 + b0s + c0s + , π Z s′3 s′ s iε 2 ′ ′− − s ds discM1(s ) M1(s) = a1 + b1s + , π Z s′2 s′ s iε 3 − ′ − ′ 2 s ds discM2(s ) M2(s) = a2 + b2s + c2s + . π Z s′3 s′ s iε − − Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.86/103 η 3π beyond p4 → Technical complications in solving • Only 4 relevant constants: • M(s, t, u) = a + bs + cs2 d(s2 + tu) − 4 4L 1/(64π2) M (s) + M (s) sM (s) + M (s) + s2 3 − 0 3 2 1 2 F 2(m2 m2 ) π η − π converge better

′ 4 1 ds ′ 4 ′ c = c0 + c2 = discM0(s )+ discM2(s ) , 3 π Z s′3  3 ﬀ 2 ′ 4L3 1/(64π ) 1 ds ′ ′ ′ d = − + s discM1(s ) + discM2(s ) − F 2(m2 m2 ) π Z s′3 π η − π ˘ ¯

Fix a, b by matching to tree level or p4 amplitude

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.87/103 η 3π beyond p4 →

2.5 2.5 tree tree 2 p4 2 p4 dispersive dispersive 1.5 1.5

1 1

Re M (AL) Adler zero Re M (KWW) 0.5 0.5

0 0

-0.5 -0.5 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 2 2 s/mπ s/mπ Along s = u KWW Along s = u AL

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.88/103 η 3π beyond p4 →

2.5 p2 2 dispersive (match p2) dispersive (match (|M|) 1.5 dispersive (match Re M)

1 Very simpliﬁed analysis Re M JB, Gasser 2002 0.5 looks more like AL 0

-0.5 1 2 3 4 5 6 7 8 2 s/mπ Along s = u BG

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.89/103 Two Loop Calculation: why

6 In Kℓ4 dispersive gave about half of p in amplitude Same order in ChPT as masses for consistency check on mu/md Check size of 3 pion dispersive part At order p4 unitarity about half of correction Technology exists: Two-loops: Amorós,JB,Dhonte,Talavera,. . . Dealing with the mixing π0-η: Amorós,JB,Dhonte,Talavera 01

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.90/103 Two Loop Calculation: why

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.90/103 Diagrams

(a) (b) (d) (c)

(a) (b) (c) (d) (e) (f) (g) (h)

(i) (j) (k) (l)

√3 Include mixing, renormalize, pull out factor 4R ,... Two independent calculations (comparison major amount of work)

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.91/103 Dalitzplot

mpiav mpidiff 0.16 u=t s=u t-threshold u theshold s threshold x=y=0 0.14 ] 2 x variation: 0.12

t [GeV vertical y variation: 0.1 parallel to t = u

0.08

0.08 0.1 0.12 0.14 0.16 s [GeV2]

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.92/103 η 3π: M(s, t = u) →

Li fit 10 and Ci Li fit 10 and Ci 40 8

Re p6 pure loops 6 r Re p Li 2 6 r 30 Re p 6 Re p Ci Re p2+p4 sum p6 Re p2+p4+p6 Im p4 20 Im p4+p6 4 M(s,t,u=t) M(s,t,u=t) − − 10 2

0 0

-10 -2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 s [GeV2] s [GeV2] Along t = u Along t = u parts

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.93/103 η 3π: M(s, t = u) →

Li = Ci = 0 Li fit 10 and Ci 40 40

30 Re p2 30 Re p2+p4 Re p2+p4 µ= 0.6 GeV Re p2+p4+p6 µ= 0.9 GeV Im p4 Re p2+p4+p6 20 Im p6 20 µ= 0.6 GeV µ= 0.9 GeV M(s,t,u=t) M(s,t,u=t) − 10 10

0 0

-10 -10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 s [GeV2] s [GeV2] Along t = u Along t = u: µ dependence r r r Li = Ci = 0 I.e. where Ci (µ) estimated

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.94/103 η 3π: M(s = u, t) →

30 Re p2 Re p2+p4 Re p2+p4+p6 Im p4 4 6 20 Im p +p Shape agrees with AL M(s,t,u=s) − 10 Correction larger: 20-30% in amplitude 0

-10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 s [GeV2] Along s = u

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.95/103 Dalitz plot

T+ T √3 x = √3 − − = (u t) Qη 2mηQη − 2 3T0 3((mη mπo ) s) iso 3 y = 1 = − − 1 = (s0 s) Qη − 2mηQη − 2mηQη − Qη = mη 2m + m 0 − π − π T i is the kinetic energy of pion πi

2 2 3Ei mη i z = − 0 Ei is the energy of pion π 3 mη 3m iX=1,3 − π

M 2 = A2 1 + ay + by2 + dx2 + fy3 + gx2y + | | 0 · · · 2 M 2 = A (1+2αz + ) | | 0 · · ·

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.96/103 Experiment: charged

Exp. a b d

+0.008 +0.007 KLOE 1.090 0.005− 0.124 0.006 0.010 0.057 0.006− − ± 0.019 ± ± ± 0.016 Crystal Barrel 1.22 0.07 0.22 0.11 0.06 0.04 (input) − ± ± ± Layter et al. 1.08 0.014 0.034 0.027 0.046 0.031 − ± ± ± Gormley et al. 1.17 0.02 0.21 0.03 0.06 0.04 − ± ± ± KLOE has: f = 0.14 0.01 0.02. ± ± Crystal Barrel: d input, but a and b insensitive to d

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.97/103 Theory: charged

2 A0 a b d f LO 120 1.039 0.270 0.000 0.000 − NLO 314 1.371 0.452 0.053 0.027 − NLO to NLO (Lr = 0) 235 1.263 0.407 0.050 0.015 i − NNLO: NNLO 538 1.271 0.394 0.055 0.025 − Little NNLOp (y from T 0) 574 1.229 0.366 0.052 0.023 − change NNLOq (incl (x, y)4) 535 1.257 0.397 0.076 0.004 − NNLO (µ = 0.6 GeV) 543 1.300 0.415 0.055 0.024 − NNLO (µ = 0.9 GeV) 548 1.241 0.374 0.054 0.025 − NNLO (Cr = 0) 465 1.297 0.404 0.058 0.032 Error on i − 2 NNLO (Lr = Cr = 0) 251 1.241 0.424 0.050 0.007 M(s, t, u) : i i − | | dispersive (KWW) — 1.33 0.26 0.10 — − M (6) M(s, t, u) tree dispersive — 1.10 0.33 0.001 — − | | absolute dispersive — 1.21 0.33 0.04 — − error 18 0.075 0.102 0.057 0.160

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.98/103 Experiment: neutral

2 A0 α LO 1090 0.000 NLO 2810 0.013 Exp. α NLO (Lr = 0) 2100 0.016 +0.004 i KLOE 2007 0.027 0.004− − ± 0.006 NNLO 4790 0.013 KLOE (prel) 0.014 0.005 0.004 − ± ± NNLOq 4790 0.014 Crystal Ball 0.031 0.004 r − ± NNLO (Ci = 0) 4140 0.011 WASA/CELSIUS 0.026 0.010 0.010 r r − ± ± NNLO (Li = Ci = 0) 2220 0.016 Crystal Barrel 0.052 0.017 0.010 − ± ± dispersive (KWW) — (0.007—0.014) GAMS2000 0.022 0.023 − − ± tree dispersive — 0.0065 SND 0.010 0.021 0.010 − − ± ± absolute dispersive — 0.007 − Borasoy — 0.031 − error 160 0.032 0 Note: NNLO ChPT gets a0 in ππ correct

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.99/103 α is difﬁcult

Expand amplitudes and isospin:

2 2 M(s, t, u) = A 1 +a ˜(s s0) + ˜b(s s0) + d˜(u t) + − − − · · · 2 2 2 M(s, t, u) = A 3 + ˜b + 3d˜ (s s0) +(t s0) +(u s0) + − − − · ·

Gives relations (Rη = (2mηQη)/3)

2 2 ˜ 2 ˜ a = 2Rη Re(˜a) , b = Rη a˜ + 2Re(b) , d = 6Rη Re(d) . − | | 1 1 1 1 α = R2 Re ˜b + 3d˜ = d + b R2 a˜ 2 d + b a2 2 η 4 − η| | ≤ 4 − 4 equality if Im(˜a) = 0 Large cancellation in α, overestimate of b likely the problem

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.100/103 r and decay rates

sin ǫ = √3 + (ǫ2) 4R O

+ 0 2 Γ(η π π−π ) = sin ǫ 0.572 MeV LO , → · sin2 ǫ 1.59 MeV NLO , · sin2 ǫ 2.68 MeV NNLO , · sin2 ǫ 2.33 MeV NNLO Cr = 0 , · i Γ(η π0π0π0) = sin2 ǫ 0.884 MeV LO , → · sin2 ǫ 2.31 MeV NLO , · sin2 ǫ 3.94 MeV NNLO , · sin2 ǫ 3.40 MeV NNLO Cr = 0 . · i

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.101/103 r and decay rates

Γ(η π0π0π0) r → ≡ Γ(η π+π π0) → −

rLO = 1.54 rNLO = 1.46 rNNLO = 1.47

r rNNLO Ci =0 = 1.46

PDG 2006

r = 1.49 0.06 our average . ± r = 1.43 0.04 our ﬁt , ± Good agreement

Odense 19-20/2/2008 Effective Field Theory: Introduction and Applications Johan Bijnens p.102/103 R and Q

r LO NLO NNLO NNLO (Ci = 0) R (η) 19.1 31.8 42.2 38.7 R (Dashen) 44 44 37 — R (Dashen-violation) 36 37 32 — Q (η) 15.6 20.1 23.2 22.2 Q (Dashen) 24 24 22 — Q (Dashen-violation) 22 22 20 —

2 2 2 mK0 + mK+ 2mπ0 LO from R = 2 −2 (QCD part only) 2 m 0 m + K − K NLO and NNLO from masses: Amorós, JB, Talavera 2001 m +m ˆ Q2 = s R = 12.7R (m /mˆ = 24.4) 2m ˆ s

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