Kaon Decays and Chiral Perturbation Theory

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lavi net KAON DECAYS AND CHIRAL PERTURBATION THEORY

Johan Bijnens Lund University

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

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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.1/64 Overview

Kaon ? A bit of history of Kaons Motivation Effective Field Theory Chiral Perturbation Theory

Kℓ3 Some rare Kaon decays K 3π →

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.2/64 Kaon: a ﬁrst guess

Know All On Nothing

Kill Any Old Nerd

Keep All Original Neanderthals

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.3/64 Try Google

http://www.tech-ex.com/article_images1/324984/DSC02855.JPG

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.4/64 Try Google once more

http://www.mountaindesigns.com.au/ product_images/full/04305_S04.jpg

http://tn3-2.deviantart.com/300W/fs7.deviantart.com/ i/2005/179/d/6/Anti__Kaon_by_Segzhan.jpg

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.5/64 Wikipedia

This article describes the subatomic particle called the kaon. For the ontology infrastructure of the same name, see KAON.

In particle physics, a kaon (also called K-meson and denoted K) is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. In the quark model they are understood to contain a single strange quark (or antiquark).

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.6/64 One of the ﬁrst Kaons

a V particle

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.7/64 One of the ﬁrst Kaons

a τ + π+π+π decay → −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.8/64 Early Kaon results

Produced with strong interaction rates, decay weakly: Introduction of strangeness: Gell-Mann–Pais θ, τ, κ, ...: All the same mass The θ-τ puzzle τ 3π = negative parity, → ⇒ θ 2π = positive parity, → ⇒ Two particles or parity broken 0 K0-K : Two states with very different lifetimes KL and KS are the CP even and odd states CP-violation 0 0 + + 0 ∆I = 1/2 rule: Γ(KS π π ) ≫ Γ(K π π ) → →

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.9/64 More recent Kaon results

Direct CP-violation ε′/ε.

Determination of Vus ππ scattering lengths

· · ·

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.10/64 More recent Kaon results

I will talk about ChPT for:

Kℓ3 A few rare kaon decays K 3π → There are many more ChPT calculations relevant for Kaons Masses and decay constants Rare decays Radiative decays

Kℓ4 Constraints in calculating the nonleptonic matrix elements: ∆I = 1/2 and ε′/ε

Mainz 17/1/2007· · · Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.11/64 The Standard Model

The Standard Model Lagrangian has four parts:

H (φ) + G(W,Z,G) L L Higgs Gauge | {z } | {z } ¯ ¯ ψiD/ψ + gψψ′ ψφψ′

ψ=fermionsX ψ,ψ′=fermionsX gauge-fermion Yukawa | {z } | {z }

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.12/64 The Standard Model

What is tested ? gauge-fermion Very well tested Higgs Limits only, real tests coming up Gauge Well tested, QCD nonperturbative a small E Yukawa Flavour Physics

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.13/64 The Standard Model

What is tested ? gauge-fermion Very well tested Higgs Limits only, real tests coming up Gauge Well tested, QCD nonperturbative a small E Yukawa Flavour Physics Discrete symmetries: C Charge Conjugation P Parity T Time Reversal QCD and QED conserve C,P,T separately, Weak breaks C and P, only Yukawa breaks CP Field theory implies CPT

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.13/64 Weak interaction: quarks to mesons

ENERGY SCALE FIELDS Effective Theory

W,Z,γ,g; MW τ,µ,e,νℓ; Standard Model t, b, c, s, u, d using OPE ⇓ γ,g; µ,e,νℓ; ∆S =1,2 . mc QCD,QED, | | s, d, u Heff ??? ⇓ γ; µ,e,ν ; M ℓ ChPT K π, K, η

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.14/64 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  

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.15/64 Power Counting

➠ gap in the spectrum = separation of scales ⇒ ➠ with the lower degrees of freedom, build the most general effective Lagrangian

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.16/64 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 ?

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.16/64 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 ⇒

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.16/64 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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.16/64 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 ≈ −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.17/64 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: −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.17/64 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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.17/64 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.

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.17/64 Why Field Theory ?

➠ Only known way to combine QM and special relativity ➠ Off-shell effects: there as new free parameters

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.18/64 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 •

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.18/64 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• Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.18/64 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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.19/64 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. ×

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.19/64 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 =/ ⇒ Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.19/64 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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.20/64 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:

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

1/p2 (p2) (1/p2) p4 = p4 d4p p4

Mainz 17/1/2007R Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.20/64 Chiral Perturbation Theory

Large subject:

Steven Weinberg, Physica A96:327,1979: 1662 citations Juerg Gasser and Heiri Leutwyler, Nucl.Phys.B250:465,1985: 2126 citations Juerg Gasser and Heiri Leutwyler, Annals Phys.158:142,1984: 2072 citations Sum: 3509 Checked on 9/1/2007 in SPIRES

For lectures, review articles: see http://www.thep.lu.se/ bijnens/chpt.html ∼

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.21/64 Two Loop: 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 53+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⇒ Note  ➠ 3 flavour is a special case of 3+3 PQ:   Lˆr, Kr Lr,Cr i i → i i Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.22/64  Long Expressions

= ⇒

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.23/64 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 π16 L2 11/3 χηχ4 + χ13 + 13/3¯χ2 + π16 L3 4/9 χηχ4 7/12 χ1χ3 + 11/6 χ13 17/6¯χ1χ13 43/36χ ¯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 χ 13 + 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 − ηη 1 − 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 RpRη13 + A(χη)B(χp,χη; 0) 1/18χ ¯pη1 +1/18χ ¯qη1 +1/18 (Rpp) Rqpη χp − 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) L5 2/3¯χηη1χp 4/3 Rqη χp +4 /3 Rp χp +1/2 Rp χp 1/6 Rp χq − − − ¯ r pp p c ¯ r d 2 + B(χp,χp; 0) L6 4¯χ1χ¯ηη1 +8¯χ1Rqη χp 8¯χ1Rp χp + 4 B(χp,χp; 0) L7 (Rp) − ¯ r pp p 2 c 2 ¯ 2 p d c d + B(χp,χp; 0) L8 4/3¯χηη2 +8/3 Rqη χp 8/3 Rp χp + B(χp,χp; 0) 1/18 RqηRp χp +1/18 RpRp χp − − d 2 ¯ ¯ η d η d + 1/288 (Rp) + 1/18 B(χp,χp; 0)B(χp,χη; 0) RppRp χp R13Rp χp − plus several more pages

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.23/64 Usual ChPT two-loop: A list

Review paper on Two-Loops: JB, LU TP 06-16 hep-ph/0604043 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 Two-Loops Three ﬂavours

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

ΠVVρω Maltman

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

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.24/64 Usual ChPT two-loop: A list

ΠVVK , ΠAAK , FK , mK Amorós, JB, Talavera r r r Kℓ4, hqqi Amorós, JB, Talavera L1,L2,L3

r FM , mM , hqqi (mu 6= md) Amorós, JB, Talavera L5,7,8, mu/md

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 L10

ππ JB,Dhonte,Talavera πK JB,Dhonte,Talavera

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.25/64 Kℓ3

H. Leutwyler and M. Roos, Z.Phys.C25:91,1984. J. Gasser and H. Leutwyler, Nucl.Phys.B250:517-538,1985. J. Bijnens and P. Talavera, hep-ph/0303103, Nucl. Phys. B669 (2003) 341-362 V. Cirigliano et al., hep-ph/0110153, Eur.Phys.J.C23:121-133,2002.

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.26/64 Kℓ3 Deﬁnitions

+ + 0 + K : K (p) π (p′)ℓ (p )ν (pν) ℓ3 → ℓ ℓ 0 0 + K : K (p) π−(p′)ℓ (p )ν (pν) ℓ3 → ℓ ℓ

+ GF ⋆ µ + K : T = Vusℓ Fµ (p′,p) ℓ3 √2 µ µ ℓ =u ¯(pν)γ (1 γ )v(p ) − 5 ℓ + 0 4 i5 + Fµ (p′,p) = <π (p′) V − (0) K (p) > | µ | 1 K+π0 K+π0 = [(p′ + p)µf+ (t)+(p p′)µf (t)] √2 − −

0 + 0 K π− K π Isospin: f+ (t) = f+ (t) = f+(t) 0 + 0 f K π− (t) = f K π (t) = f (t) − Mainz 17/1/2007− Kaon Decays and− Chiral Perturbation Theory Johan Bijnens p.27/64 Kℓ3 Deﬁnitions and Vus

t Scalar formfactor: f0(t) = f+(t) + f (t) m2 m2 − K − π t Usual parametrization: f (t) = f (0) 1 + λ +,0 + +,0 m2 π Vus : Know theoretically f (0) = 1 + | | + · · · Short distance correction to GF from Gµ Marciano-Sirlin Ademollo-Gatto-Behrends-Sirlin theorem: 2 (ms mˆ ) − Isospin Breaking Leutwyler-Roos Radiative corrections: Cirigliano et al

Know experimentally f+(0)

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.28/64 Vus

PDG2002: Vud = 0.9734 0.0008 Vus = 0.2196 0.0026 | |2 2 ± | | ± Vud + Vus = (0.9475 0.0016) + (0.0482 0.0011) = |0.9957| |0.0019| ± ± ± PDG2006: Vud = 0.97377 0.00027 Vus = 0.2257 0.0021 | |2 2 ± | | ± Vud + Vus = (0.94823 0.00054) + (0.05094 0.00095) = |0.99917| | 0.00110| ± ± ± Problems: Ignores ∆(0) = 0.0113 from pure two-loop Conﬂicts between experiments

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.29/64 Kℓ3 Diagrams

¢ ª

• : p2 vertex

¢ ×: p4 vertex ⊗: p6 vertex

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.30/64 f+(t) Theory

(4) (6) f+(t)=1+ f+ (t) + f+ (t)

(4) t r f+ (t) = 2 L9 + loops 2Fπ

(6) 8 r r 2 2 2 t Kπ f+ (t) = 4 (C12 + C34) mK mπ + 4 R+1 −Fπ − Fπ 2 t r r r + 4 ( 4C88 + 4C90) + loops(Li ) Fπ −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.31/64 ChPT ﬁt to f+(t)

1.2

1.18 Kπ ⇒R+1 = 1.16 5 2 −(4.7 ± 0.5) 10− GeV 1.14 CPLEAR ChPT C r=0 1.12 i 4 f (t) ChPT p4 `c+ = 3.2 GeV− ´ + 1.1 ChPT fit λ = 0.245 1.08 + ⇒a+ = 1.009 ± 0.004 1.06

1.04 ⇒λ+ = 0.0170 ± 0.0015 1.02

0.98 0 0.02 0.04 0.06 0.08 0.1 0.12 t [GeV2]

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.32/64 f0(t)

Main Result:

8 r r 2 2 2 f0(t) = 1 4 (C12 + C34) mK mπ − Fπ − t r r 2 2 t +8 4 (2C12 + C34) mK + mπ + 2 2 (FK/Fπ 1) Fπ mK mπ − − 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 ] . Mainz 17/1/2007 Kaon− Decays and Chiral Perturbation± Theory ± Johan Bijnens p.33/64 Experiment

3 3

' ( <>I ".0,..!27 ChPT 1 3σ Q 1 =+

V ( : λ″ ≠0 !σ 3/ + ' ( VV ( % λ = 0 1", 3 1 2 .1 ′ µ3 0 + " ';-G λ 0 A$ ( Κ µ3 , π+

& " ';- < :9 0,..!24 < :9 0,..!27 , π $%& V ( % λ″ ≠0 ,σ + < 8180,..827 − 1 Κ µ3 V ( % λ0 = 0 " 3 ,1 2 . #)*

1 #)*5< 8.!04 W<,..: ,9 7 λ+′3/

( V

0 1 λ″ → − L $# $ Κ µ3 ( % λ0- 0 ,. 3 " 2 . +

→ B.:L (

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.34/64 Experiment

± ± 1 µ1 A 1 1 µ3 λ′+ = 0.02496(79)

! λ′′ = 0.0016(3) > .!.!8092 .,8 "0"2 ".!80 2N . ..:.!101 2 ..11910,"2 +

τ : .0 2 9 :9082 $ ,1980,,2 λ0 = 0.01587(95)

S (B0.2-. 8 092 '+ > Y 5<#+ ,:4 4 9!7

S -. "1""0,"2 A 5<#+> 8 .1,..,4,..87

- .,,! 3 ...

>µ % .. !...1"

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.35/64 Experiment

= Γ(Κ→µν(γ))/Γ(π→µν(γ))

7 π 7 +( +A+(+ , , , , 4 8A/)/91,)5 Γ(Κ→µν(γ))/Γ(π→µν(γ)) ∝ +( + A+(+ π B 6 D π - 90120 −52 ( P $%&6;4- → µ+ν)

' ( A(./!,,0)± 0!//,9

δ.+( +,-+( +,83.8/!//,/2/!///9

#&&='%6& %#H";&=

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.36/64 K γγ S →

Well predicted by CHPT at order p4 from Goity, D’Ambrosio, Espriu

π+, K+ π+, K+ π+, K+

KS KS KS

Prediction was: BR = 2.1 10 6 · − NA48: 2.78(6)(4) 10 6 · −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.37/64 Some other rare decays

KL γγ Needs→ work: main contribution is full of cancellations:

K π0,η,η difﬁcult L ′

0 KL π γγ → 0 KL π γγ OK predicted by CHPT → Main succes: events must be at high mγγ Rate still a problem 0 KS π γγ → Similar problems as in KL γγ → Ecker, Pich, de Rafael, D’Ambrosio,. . .

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.38/64 K 3π → K 3π Decays in Chiral Perturbation Theory, J. Bijnens,→ P. Dhonte and F. Persson, hep-ph/0205341, Nucl. Phys. B648 (2003) 317-344. Isospin Breaking in K 3π Decays I: Strong Isospin Breaking, J. Bijnens and→ F. Borg, hep-ph/0405025, Nuclear Physics B697 (2004) 319-342. Isospin Breaking in K 3π Decays II: Radiative Corrections, J. Bijnens→ and F. Borg, hep-ph/0410333, Eur. Phys. J. C39 (2005) 347-357. Isospin Breaking in K 3π Decays III: Bremsstrahlung and Fit to Experiment,→ J. Bijnens and F. Borg, hep-ph/0501163,Eur. Phys. J. C40 (2005) 383-394

Note: Fredrik Borg = Fredrik Persson

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.39/64 K 3π: Overview →

ChPT in the nonleptonic mesonic sector Lagrangians K 3π kinematics and isospin → Overview of calculations and results Data and Fits

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.40/64 ChPT in the nonleptonic sector

some earlier work: especially on decays with photons Kambor, Missimer, Wyler (KMW) : Constructed and 1990 L ∞ G. Esposito-Farese: Checked and 1991 L ∞ KMW : Calculated K 2π and K 3π 1991 → → Donoghue + Holstein + KMW : clariﬁed the relations between observables 1992 BUT: explicit formulas lost (US Mail) Kambor, Ecker, Wyler : simpliﬁed octet 1993 L K 2π redone: Bijnens, Pallante, Prades 1998 →

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.41/64 ChPT in the nonleptonic sector

First paper: redo K 3π → Ecker Isidori Muller Neufeld Pich: Electromagnetic octet Lagrangian plus 2000 L ∞ applications to K 2π: Several papers → Isospin breaking in K 3π: remaining papers 4 2 2 →2 (p , p (mu m ), e p ). O − d

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.42/64 Lagrangians: p2 and e2

2 F0 µ = S + W + E S = uµu + χ L2 L 2 L 2 L 2 L 2 4 h +i

2 uµ = iu† DµU u† = uµ† , u = U, χ = u†χu† uχ†u , ± ± U contains the Goldstone boson ﬁelds

1 1 + + π3 + η8 π K √ √2 √6 i 2 1 1 0 3 8 U = exp M , M = π− √−2 π + √6 η K . F0 !  0 2  K− K √−6 η8   mu Here χ = 2B m and DµU = ∂µU ie [Q, U] . 0  d  − ms    

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.43/64 Lagrangians: p2 and e2

4 µ ij,kl µ W 2 = CF0 G8 ∆32uµu + G8′ ∆32χ+ + G27t ∆ijuµ ∆klu L " h i h i h ih i#

21,13 13,21 ∆ij uλiju , (λij) δia δ , t = t = 1/3,... ≡ † ab ≡ jb

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.44/64 Lagrangians: p2 and e2

4 µ ij,kl µ W 2 = CF0 G8 ∆32uµu + G8′ ∆32χ+ + G27t ∆ijuµ ∆klu L " h i h i h ih i#

21,13 13,21 ∆ij uλiju , (λij) δia δ , t = t = 1/3,... ≡ † ab ≡ jb

3 GF C = V V : chiral and large Nc limits G = G = 1 − 5 √2 ud us∗ 8 27

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.44/64 Lagrangians: p2 and e2

4 µ ij,kl µ W 2 = CF0 G8 ∆32uµu + G8′ ∆32χ+ + G27t ∆ijuµ ∆klu L " h i h i h ih i#

21,13 13,21 ∆ij uλiju , (λij) δia δ , t = t = 1/3,... ≡ † ab ≡ jb

3 GF C = V V : chiral and large Nc limits G = G = 1 − 5 √2 ud us∗ 8 27 Electromagnetic: 2 4 2 6 E = e F Z L R + e CF GE ∆ R L 2 0 hQ Q i 0 h 32Q i L = uQu , R = u Qu with Q = diag (2/3, 1/3, 1/3) Q † Q † − −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.44/64 Lagrangians: p2 and e2

4 µ ij,kl µ W 2 = CF0 G8 ∆32uµu + G8′ ∆32χ+ + G27t ∆ijuµ ∆klu L " h i h i h ih i#

21,13 13,21 ∆ij uλiju , (λij) δia δ , t = t = 1/3,... ≡ † ab ≡ jb

3 GF C = V V : chiral and large Nc limits G = G = 1 − 5 √2 ud us∗ 8 27 Electromagnetic: 2 4 2 6 E = e F Z L R + e CF GE ∆ R L 2 0 hQ Q i 0 h 32Q i L = uQu , R = u Qu with Q = diag (2/3, 1/3, 1/3) Q † Q † − − 4 Note: F0 not well known, ﬁt CF0 G8,. . .

use numerically F0 = Fπ to quote G8,. . .

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.44/64 Lagrangians: p4 and e2p2

= S + W + S E + W E (G8) . L4 L 4 L 4 L 2 2 L 2 2 r 4 S values of L use Amoros, Bijnens, Talavera p ﬁt L 4 i r r W 4 thirteen Ni (octet) and twelve Di might contribute L r r (two Ni and two Di extra for photon-reducible) r S E eleven K can contribute L 2 2 i r W E fourteen Z can contribute, 27 part not classiﬁed L 2 2 i

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.45/64 Lagrangians: p4 and e2p2

Done here : r r ˜ Isospin conserved: 11 combinations of Ni ,Di relevant: Ki 4 2 2 7 coefﬁcients order mK, 4 order MKmπ 2 (virtually) indistinguishable from G8 and G27. r r r Isospin Broken: 30 combinations of Ni ,Di , Zi relevant r r Results from isospin breaking with Ki = Zi = 0

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.46/64 K 3π Kinematics and Isospin →

0 0 0 L KL(k) π (p )π (p )π (p ) , [A ] , → 1 2 3 000 + 0 L KL(k) π (p1)π−(p2)π (p3) , [A+ 0] , → − + 0 S KS(k) π (p1)π−(p2)π (p3) , [A+ 0] , → − + 0 0 + K (k) π (p1)π (p2)π (p3) , [A00+] , + → + + K (k) π (p1)π (p2)π−(p3) , [A++ ] , → − plus charge conjugate ones S even under p1 p2 except A+ 0 odd ↔ −

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.47/64 K 3π Kinematics and Isospin →

Kinematics: s =(k p )2 , s =(k p )2 , s =(k p )2 . 1 − 1 2 − 2 3 − 3 Dalitz plot variables:

2 2 y =(s s )/m + , x =(s s )/m + , s =(s + s + s ) /3 . 3 − 0 π 2 − 1 π 0 1 2 3 2 Bremsstrahlung problem: si from Ei or mππ different

Decay amplitudes squared expanded as

A(s , s , s ) 2 1 2 3 = 1 + gy + hy2 + kx2 A(s , s , s ) 0 0 0

0 0 0 KL π π π : g = 0 and k = h/3. →

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.48/64 Kinematics and Isospin

Valid also at p6

L A000 = M0(s1)+M0(s2)+M0(s3) , L A+ 0 = M1(s3)+M2(s1)+M2(s2)+M3(s1)(s2 −s3)+M3(s2)(s1 −s3) , − S A+ 0 = M4(s1)−M4(s2)+M5(s1)(s2 −s3)−M5(s2)(s1 −s3)+M6(s3)(s1 −s2) , − A00+ = M7(s3)+M8(s1)+M8(s2)+M9(s1)(s2 −s3)+M9(s2)(s1 −s3) ,

A++ = M10(s3)+M11(s1)+M11(s2)+M12(s1)(s2 −s3)+M12(s2)(s1 −s3) . − 2 polynomial ambiguity from s1 + s2 + s3 = i mi P Isospin:

M0(s) = M1(s) + 2M2(s) ,

2M7(s) + 4M8(s) = M10(s) + 2M11(s) , 1 M4(s) = (M7(s) − M8(s)+ M10(s) − M11(s)) 3 M5(s) − M6(s) = M9(s)+ M12(s) . Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.49/64 Calculations: isospin limit

The diagrams of order p4

Expressions for the Mi(s): see ﬁrst paper

Gamiz, Prades, Scimemi, hep-ph/0305164 Conﬁrmed by Lashin, hep-ph/0308200 (how not clear) Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.50/64 Calculations: Strong Isospin Breaking

Includes:

mu m and the effects of E , E S and W E (G ) • − d L 2 L 2 2 L 2 2 8 π0-η mixing (η via LECs) • ′ Same diagrams as before • Formulas immediately a lot longer: • see http://www.thep.lu.se/ bijnens/chpt.html ∼

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.51/64 Calculations: Strong Isospin Breaking

Includes:

Effects:

F + , F + overall factor • π K Mass difference π+-π0 • Others typically a few % Mainz 17/1/2007• Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.51/64 Phasespace

3 Curves: Phasespace

2 boundaries Dots: ππ pair 1 at rest Lines: x 0 2 000 Show A +-0 | | 00+ along these -1 ++-

-2

-3 -1.5 -1 -0.5 0 0.5 1 1.5 y

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.52/64 K π0π0π0 L →

0.995 2 (0,0)| L 000 A

0.99

⁄ x = 0 y = 0

(x,y) x = √3 y

L 000 isospin limit |A 0.985

0.98 -1.5 -1 -0.5 0 0.5 1 1.5 r

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.53/64 Calculations: Radiative Corrections

Photon Loops •

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.54/64 Calculations: Radiative Corrections

Photon Loops •

Soft Bremsstrahlung •

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.54/64 Calculations: Radiative Corrections Photon reducible diagrams •

Tree Level: •

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.55/64 Calculations: Radiative Corrections Photon reducible diagrams •

Tree Level: •

Extra vanish exactly

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.55/64 Calculations: Radiative Corrections Photon reducible diagrams •

Tree Level: •

Extra vanish exactly

One-loop Level: •

Extra N r,Dr from K πℓ+ℓ i i → − Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.55/64 Calculations: Radiative Corrections Photon reducible diagrams •

Tree Level: •

Extra vanish exactly

One-loop Level: •

Extra N r,Dr from K πℓ+ℓ Numerically Negligible i i → − Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.55/64 + + + K π π π− → 1.3 x = 0 y = 0 1.2 x = √3 y local 2 isospin limit 1.1 (0,0)| − ++ A

⁄ 1 (x,y) −

++ 0.9 |A

0.8

0.7 -1 -0.5 0 0.5 1 r

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.56/64 K+ π0π0π+ → 2

1.8 x = 0 y = 0 1.6 x = √3 y isospin limit 2 1.4 (0,0)| 1.2 00+ A

⁄ 1 (x,y)

00+ 0.8 |A

0.6

0.4

0.2 -1.5 -1 -0.5 0 0.5 1 1.5 r Note: Disagree numerically with Nehme, hep-ph/0406209

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.57/64 Calculations: Hard Bremsstrahlung

Include hard photon Bremsstrahlung to lowest order • Full amplitude from Low’s theorem Do everything also for K 2π to have same treatment • → 2 2 Note: some new pieces, e.g., e p G27

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.58/64 Calculations: Hard Bremsstrahlung

Include hard photon Bremsstrahlung to lowest order • Full amplitude from Low’s theorem Do everything also for K 2π to have same treatment • → 2 2 Note: some new pieces, e.g., e p G27

⊛ Everything done twice indepently  ⊛ UV inﬁnities cancel analytically: no 1/(d 4)  − Checks: ⊛ IR inﬁnities cancel analytically: no mγ    ⊛ soft-hard Bremsstrahlung match: no Eγ  (except for F ,F parts)  π K   Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.58/64 Treatment of Bremsstrahlung

Old experiments: what to do? Typically: Decay rate and distribution measured in part of phasespace and then extrapolated to whole decay region

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.59/64 Treatment of Bremsstrahlung

Old experiments: what to do? Typically: Decay rate and distribution measured in part of phasespace and then extrapolated to whole decay region

Subtract hard Bremsstrahlung from • Decay Rate  From remainder get A(s , s , s ) 2 We adopted: • | 0 0 0 |  using experimental g, h, k   Fit A(s , s , s ) 2, g, h, k with soft • | 0 0 0 |  Bremsstrahlung and loops, ...    Mimicks experimental determination Advantages: • (well, sort of) ( Coulomb singularity avoided •

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.59/64 Results K ππ only: essentially same as Ecker et al. → Tree level: G8 = 10.36 G27 = 0.550

Full: G8 = 5.39 G27 = 0.359

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.60/64 Results K ππ only: essentially same as Ecker et al. → Tree level: G8 = 10.36 G27 = 0.550

Full: G8 = 5.39 G27 = 0.359

µ 0.77 GeV 1.0 GeV 0.6 GeV 0.77 GeV

G8 5.39(1) 4.60(1) 6.43(1) 5.39(1)

G27 0.359(2) 0.301(1) 0.438(2) 0.359(2) o o o o δ2 − δ0 −57.9(1.5) −57.3(1.4) −58.9(1.4) −57.9(1.4) 3 10 K˜1/G8 ≡ 0 ≡ 0 ≡ 0 ≡ 0 3 10 K˜2/G8 48.5(2.4) 56.5(2.4) 41.2(1.9) 46.6(1.6) Full ﬁt: 3 10 K˜3/G8 2.6(1.2) −1.7(1.1) 6.7(1.0) 3.5(0.8) 3 10 K˜4/G27 ≡ 0 ≡ 0 ≡ 0 ≡ 0 3 10 K˜5/G27 −41.2(16.9) −52.0(17.7) −31.1(12.0) −27.0(8.3) 3 10 K˜6/G27 −102(105) −114(105) −93(76) ≡ 0 3 10 K˜7/G27 78.6(33) 78.0(33.5) 79.6(22.7) 50.0(13.0) χ2/DOF 29.3/10 27.2/10 33.0/10 30.5/11

vary K˜1, K˜4: see ﬁrst paper Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.60/64 Octet and µ-variation

octet µ variation µ variation µ 0.77 GeV 1.0 GeV 0.6 GeV Octet: G27 = 0 G8 4.84(1) – – keep also K˜i G27 0.430(1) – – 2 2 o δ2 − δ0 −57.9(0.2) – – with mKmπ factors 3 10 K˜1/G8 2.0(1) −5.88 5.61 3 10 K˜2/G8 63.0(1.5) −2.69 2.57 3 10 K˜3/G8 −6.0(7) 0.159 −0.152 µ variation 3 ˜ 10 K4/G27 ≡ 0 −9.93 9.48 K˜i(µ) K˜i(0.77 GeV ) 3 10 K˜5/G27 ≡ 0 0 0 − 3 10 K˜6/G27 ≡ 0 27.0 −25.8 3 10 K˜7/G27 ≡ 0 −21.5 20.5 Fit about same as 3 10 K˜8/G8 20.4(1) −0.546 0.521 before, some exper- 3 10 K˜9/G8 9.1(1) −2.92 2.79 iments ﬁt better, oth- 3 10 K˜10/G8 ≡ 0 11.6 −11.1 3 ers smaller error 10 K˜11/G8 ≡ 0 −1.66 1.58 χ2/DOF 33.3/10 – –

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.61/64 Models

See Ecker, Kambor, Wyler 1993 for explanations 1 Vector octet dominance = K˜ = K˜ : Not at all • ⇒ 3 −2 2

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.62/64 Models

See Ecker, Kambor, Wyler 1993 for explanations 1 Vector octet dominance = K˜ = K˜ : Not at all • ⇒ 3 −2 2

Weak Deformation and Factorization • r r r r N1 = 2kf (32/3 L1 + 4 L3 + 2/3 L9) , r r r r N2 = 2kf (16/3L1 + 4L3 + 10/3 L9) , r r r N3 = 2kf (8L2 − 2L9) , r r r r N4 = 2kf (−16/3 L1 − 8/3 L3 − 4/3 L9) , r r N5 = 2kf (−L5) , r r N6 = 2kf (2/3 L5) , r r N7 = 2kf (L5) , r r r N8 = 2kf (4 L4 + 2 L5) , r r r r r N9 = N10 = N11 = N12 = N13 = 0

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.62/64 Models

See Ecker, Kambor, Wyler 1993 for explanations 1 Vector octet dominance = K˜ = K˜ : Not at all • ⇒ 3 −2 2

Weak Deformation and Factorization • r r r r µ 0.77 GeV 0.9 GeV 0.842 GeV N1 = 2kf (32/3 L1 + 4 L3 + 2/3 L9) , r r r r G8 4.18(1) 4.42 4.22(1) N2 = 2kf (16/3L1 + 4L3 + 10/3 L9) , r r r G27 0.360(2) 0.326(10) 0.339(10) N3 = 2kf (8L2 − 2L9) , r r r r kF 2.61(1) 4.94(2) 3.60(5) N4 = 2kf (−16/3 L1 − 8/3 L3 − 4/3 L9) , 2 r r χ /DOF 109/14 182/14 60.4/13 N5 = 2kf (−L5) , r r N6 = 2kf (2/3 L5) , WDM or kf = 1/2 No r r N7 = 2kf (L5) , Factorization: surprisingly OK r r r N8 = 2kf (4 L4 + 2 L5) , But not naive one r r r r r kf N9 = N10 = N11 = N12 = N13 = 0

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.62/64 Data and Fits

Decay Width [GeV] ChPT [GeV] Fact. [GeV] + + 0 17 17 17 K → π π (1.1231 ± 0.0078) · 10− 1.123 · 10− 1.127 · 10− 0 0 15 15 15 KS → π π (2.2828 ± 0.0104) · 10− 2.282 · 10− 2.283 · 10− + 15 15 15 KS → π π− (5.0691 ± 0.0108) · 10− 5.069 · 10− 5.069 · 10− 0 0 0 18 18 18 KL → π π π (2.6748 ± 0.0358) · 10− 2.618 · 10− 2.698 · 10− + 0 18 18 18 KL → π π−π (1.5998 ± 0.0271) · 10− 1.658 · 10− 1.711 · 10− + 0 0 + 19 19 19 K → π π π (9.195 ± 0.0213) · 10− 8.934 · 10− 8.816 · 10− + + + 18 18 18 K → π π π− (2.9737 ± 0.0174) · 10− 2.971 · 10− 2.933 · 10− Decay Quantity Experiment ChPT Fact. 0 0 0 KL → π π π h −0.0050 ± 0.0014 -0.0062 -0.0025 + 0 KL → π π−π g 0.678 ± 0.008 0.678 0.654 h 0.076 ± 0.006 0.088 0.083 k 0.0099 ± 0.0015 0.0057 0.0068 + 0 8 8 8 KS → π π−π γS (3.3 ± 0.5) · 10− 3.0 · 10− 2.9 · 10− 0 0 K± → π π π± g 0.638 ± 0.020 0.636 0.648 h 0.051 ± 0.013 0.077 0.080 k 0.004 ± 0.007 0.0047 0.0069 + + + K → π π π− g −0.2154 ± 0.0035 −0.215 −0.226 h 0.012 ± 0.008 0.012 0.019 k −0.0101 ± 0.0034 −0.0034 −0.0033

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.63/64 Conclusions ChPT is a useful tool for Kaon decays

Kℓ3 Predicts λ+′′ Sizable pure two-loop corrections to f+(0) Can combine measurements + dispersive for f0(t) to get fully at f+(0) A very short summary of some rare decays K 3π → Calculated K 3π to ﬁrst nontrivial order in isospin breaking fully → New ﬁt done and tested a few models Fit with/without isospin breaking seems similar CP violation: partly done by Gamiz et al.

Mainz 17/1/2007 Kaon Decays and Chiral Perturbation Theory Johan Bijnens p.64/64