KAON DECAYS AND CHIRAL PERTURBATION THEORY

KAON DECAYS AND CHIRAL PERTURBATION THEORY

Johan Bijnens Lund University

bijnens@thep.lu.se

http://www.thep.lu.se/∼bijnens

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

lavi

net

Overview

Kaon ?

A bit of history of Kaons Motivation

Effective Field Theory

Chiral Perturbation Theory K_ℓ3

Some rare Kaon decays K → 3π

Kaon: a first guess

Know All On Nothing Kill Any Old Nerd

Keep All Original Neanderthals

Try Google

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

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).

One of the first Kaons

a V particle

One of the first Kaons

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

K⁰-K⁰: Two states with very different lifetimes K_L and K_S are the CP even and odd states CP-violation

∆I = 1/2 rule: Γ(K_S → π⁰π⁰) ≫ Γ(K⁺ → π⁺π⁰)

More recent Kaon results

Direct CP-violation ε^′/ε. Determination of V_us

ππ scattering lengths

· · ·

More recent Kaon results

I will talk about ChPT for:

K_ℓ3

A few rare kaon decays K → 3π

There are many more ChPT calculations relevant for Kaons Masses and decay constants

Rare decays

Radiative decays K_ℓ4

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

The Standard Model

The Standard Model Lagrangian has four parts:

L^H(φ)

| {z } Higgs

+ L^G(W, Z, G)

| {z }

Gauge

X

ψ=fermions

ψiD¯ / ψ

| {z }

gauge-fermion

+ X

ψ,ψ^′=fermions

g_ψψ^′ψφψ¯ ^′

| {z }

Yukawa

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

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

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

Weak interaction: quarks to mesons

ENERGY SCALE FIELDS Effective Theory

M_W

W, Z, γ, g; τ, µ, e, ν_ℓ; t, b, c, s, u, d

Standard Model

⇓ using OPE . m_c γ, g; µ, e, ν_ℓ;

s, d, u QCD,QED,_H^|∆S|=1,2 eff

⇓ ???

Effective Field Theory

Main Ideas:

Use right degrees of freedom : essence of (most) physics

If mass-gap in the excitation spectrum: neglect degrees of freedom above the gap.

Examples:











Solid state physics: conductors: neglect the empty bands above the partially filled one

Atomic physics: Blue sky: neglect atomic structure

Power Counting

➠ gap in the spectrum _=⇒ separation of scales

➠ with the lower degrees of freedom, build the most general effective Lagrangian

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 ?

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 ?

= ⇒

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 ?

= ⇒

➠ Taylor series expansion does not work (convergence radius is zero)

➠ Continuum of excitation states need to be taken into account

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

Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

σ ≈ G²E_γ⁴

Why is the sky blue ?

System: Photons of visible light and neutral atoms Length scales: a few 1000 Å versus 1 Å

Atomic excitations suppressed by _{≈ 10}⁻³

L^A = Φ^†_v∂_tΦ_v + . . . L^γA = GF_µν² Φ^†_vΦ_v + . . . Units with h/ = c = 1: G energy dimension ₋₃:

σ ≈ G²E_γ⁴ blue light scatters a lot more than red

=⇒ red sunsets

=⇒ blue sky Higher orders suppressed by 1 Å/λ_γ.

Why Field Theory ?

➠ Only known way to combine QM and special relativity

➠ Off-shell effects: there as new free parameters

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

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

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

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

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

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

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU (3)_V But _LQCD = X

q=u,d,s

[i¯q_LD/ q_L + i¯q_RD/ q_R − m^q (¯q_Rq_L + ¯q_Lq_R)]

Chiral Perturbation Theory

Degrees of freedom: Goldstone Bosons from Chiral Symmetry Spontaneous Breakdown

Power counting: Dimensional counting

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

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: SU (3)_V But _LQCD = X

q=u,d,s

[i¯q_LD/ q_L + i¯q_RD/ q_R − m^q (¯q_Rq_L + ¯q_Lq_R)]

So if m_q = 0 then SU (3)_L × SU(3)^R.

Can also see that via ^{v < c}, m_q 6= 0 =⇒

v = c, m_q = 0 =⇒/

Chiral Perturbation Theory

h¯qqi = h¯q^Lq_R + ¯q_Rq_Li 6= 0

SU (3)_L × SU(3)^R broken spontaneously to SU (3)_V

8 generators broken _=⇒ 8 massless degrees of freedom and interaction vanishes at zero momentum

Chiral Perturbation Theory

h¯qqi = h¯q^Lq_R + ¯q_Rq_Li 6= 0

SU (3)_L × SU(3)^R broken spontaneously to SU (3)_V

8 generators broken _=⇒ 8 massless degrees of freedom and interaction vanishes at zero momentum

Power counting in momenta:

p²

1/p² R d⁴p p⁴

(p²)² (1/p²)² p⁴ = p⁴

(p²) (1/p²) p⁴ = p⁴

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

Two Loop: Lagrangians

Lagrangian Structure:

2 flavour 3 flavour 3+3 PQChPT p² F, B 2 F₀, B₀ 2 F₀, B₀ 2 p⁴ l_i^r, h^r_i 7+3 L^r_i, H_i^r 10+2 L^ˆr_i, ˆH_i^r 11+2 p⁶ c^r_i 53+4 C_i^r 90+4 K_i^r 112+3 p²: Weinberg 1966

p⁴: Gasser, Leutwyler 84,85

p⁶: JB, Colangelo, Ecker 99,00

Note











➠ replica method _=⇒ PQ obtained from N_F flavour

➠ All infinities known

➠ 3 flavour is a special case of 3+3 PQ:

Lˆ^r, K^r → L^r, C^r

Long Expressions

=⇒

Long Expressions

=⇒

δ⁽⁶⁾²²_loops= π16L^r₀

4/9 χηχ4− 1/2 χ1χ3+ χ²₁₃− 13/3 ¯χ1χ13− 35/18 ¯χ2

− 2 π16L^r₁χ²₁₃

− π16L^r₂

11/3 χηχ4+ χ²₁₃+ 13/3 ¯χ2 + π16L^r₃

4/9 χηχ4− 7/12 χ1χ3+ 11/6 χ²₁₃− 17/6 ¯χ1χ13− 43/36 ¯χ2 + π₁₆² 

−15/64 χηχ4− 59/384 χ1χ3+ 65/384 χ²₁₃− 1/2 ¯χ1χ13− 43/128 ¯χ2

− 48 L^r4L^r₅χ¯1χ13− 72 L^r24χ¯²₁

− 8 L^r25χ²₁₃+ ¯A(χp) π16

−1/24 χp+ 1/48 ¯χ1− 1/8 ¯χ1R^p_qη+ 1/16 ¯χ1R^c_p− 1/48 R^pqηχp− 1/16 R^pqηχq

+ 1/48 R^ηppχη+ 1/16 R^cpχ13

+ ¯A(χp) L^r₀

8/3 R^pqηχp+ 2/3 R^cpχp+ 2/3 R^dp

+ ¯A(χp) L^r₃ 2/3 R^pqηχp

+ 5/3 R^cpχp+ 5/3 R^dp

+ ¯A(χp) L^r₄

−2 ¯χ1χ¯^pp_ηη0− 2 ¯χ1R^pqη+ 3 ¯χ1R^cp

+ ¯A(χp) L^r₅

−2/3 ¯χ^pp_ηη1− R^pqηχp

+ 1/3 R^pqηχq+ 1/2 R^cpχp− 1/6 R^cpχq

+ ¯A(χp)²

1/16 + 1/72 (R^pqη)²− 1/72 R^pqηR^cp+ 1/288 (R^cp)² + ¯A(χp) ¯A(χps)

−1/36 Rqη^p− 5/72 Rsη^p+ 7/144 R^cp

− ¯A(χp) ¯A(χqs)

1/36 R^pqη+ 1/24 R^psη+ 1/48 R^cp

 + ¯A(χp) ¯A(χη)

−1/72 Rqη^pR^v_η13+ 1/144 R_p^cR^v_η13

+ 1/8 ¯A(χp) ¯A(χ13) + 1/12 ¯A(χp) ¯A(χ46) R^η_pp + ¯A(χp) ¯B(χp, χp; 0)

1/4 χp− 1/18 R^pqηRp^cχp− 1/72 R^pqηR^dp+ 1/18 (R^cp)²χp+ 1/144 R^cpR^dp

 + ¯A(χp) ¯B(χp, χη; 0)

1/18 R^ηppR^cpχp− 1/18 R^η13R^cpχp

+ ¯A(χp) ¯B(χq, χq; 0)

−1/72 R^pqηR^dq+ 1/144 R^cpRq^d



− 1/12 ¯A(χp) ¯B(χps, χps; 0) R^psηχps− 1/18 ¯A(χp) ¯B(χ1, χ3; 0) R^qpηR^cpχp

+ 1/18 ¯A(χp) ¯C(χp, χp, χp; 0) R^cpR^dpχp+ ¯A(χp; ε) π16

1/8 ¯χ1R^pqη− 1/16 ¯χ1R^cp− 1/16 R^cpχp− 1/16 R^dp

 + ¯A(χps) π16[1/16 χps− 3/16 χqs− 3/16 ¯χ1] − 2 ¯A(χps) L^r₀χps− 5 ¯A(χps) L^r₃χps− 3 ¯A(χps) L^r₄χ¯1

+ ¯A(χps) L^r₅χ13+ ¯A(χps) ¯A(χη)

7/144 R^ηpp− 5/72 R^ηps− 1/48 R^ηqq+ 5/72 R^ηqs− 1/36 R^η13

 + ¯A(χps) ¯B(χp, χp; 0)

1/24 R^psηχp− 5/24 R^psηχps

+ ¯A(χps) ¯B(χp, χη; 0)

−1/18 R^ηpsR^zqpηχp

− 1/9 R^ηpsR^zqpηχps

− 1/48 ¯A(χps) ¯B(χq, χq; 0) R^dq+ 1/18 ¯A(χps) ¯B(χ1, χ3; 0) R^qsηχs

+ 1/9 ¯A(χps) ¯B(χ1, χ3; 0, k) R^q_sη+ 3/16 ¯A(χps; ε) π16[χs+ ¯χ1] − 1/8 ¯A(χp4)²− 1/8 ¯A(χp4) ¯A(χp6) + 1/8 ¯A(χp4) ¯A(χq6) − 1/32 ¯A(χp6)²+ ¯A(χη) π16

1/16 ¯χ1R^v_η13− 1/48 R^vη13χη+ 1/16 R_η13^v χ13 + ¯A(χη) L^r₀

4R^η₁₃χη+ 2/3 R^v_η13χη

− 8 ¯A(χη) L^r₁χη− 2 ¯A(χη) L^r₂χη+ ¯A(χη) L^r₃

4R^η₁₃χη+ 5/3 R^v_η13χη

 + ¯A(χη) L^r₄

4 χη+ ¯χ1R^v_η13

− ¯A(χη) L^r₅

1/6 R^ηppχq+ R^η₁₃χ13+ 1/6 R^v_η13χη

+ 1/288 ¯A(χη)²(R_η13^v )² + 1/12 ¯A(χη) ¯A(χ46) R^v_η13+ ¯A(χη) ¯B(χp, χp; 0)

−1/36 ¯χ^ppη_ηη1− 1/18 R^pqηR^ηppχp+ 1/18 R^ηppR^cpχp

+ 1/144 R^dpR^v_η13

+ ¯A(χη) ¯B(χp, χη; 0)

−1/18 ¯χ^ηpη_pη1+ 1/18 ¯χ^ηpη_qη1+ 1/18 (Rpp^η)²Rqpη^z χp

− 1/12 ¯A(χη) ¯B(χps, χps; 0) R^ηpsχps− ¯A(χη) ¯B(χη, χη; 0)

1/216 R^vη13χ4+ 1/27 R^vη13χ6



− 1/18 ¯A(χη) ¯B(χ1, χ3; 0) R¹ηηRηη³χη+ 1/18 ¯A(χη) ¯C(χp, χp, χp; 0) R^ηppRp^dχp+ ¯A(χη; ε) π16[1/8 χη

− 1/16 ¯χ1R^v_η13− 1/8 R13^ηχη− 1/16 R^vη13χη

+ ¯A(χ1) ¯A(χ3)

−1/72 R^pqηR^cq+ 1/36 R¹_3ηR³_1η+ 1/144 R^c₁R₃^c

− 4 ¯A(χ13) L^r₁χ13− 10 ¯A(χ13) L^r₂χ13+ 1/8 ¯A(χ13)²− 1/2 ¯A(χ13) ¯B(χ1, χ3; 0, k)

+ 1/4 ¯A(χ13; ε) π16χ13+ 1/4 ¯A(χ14) ¯A(χ34) + 1/16 ¯A(χ16) ¯A(χ36) − 24 ¯A(χ4) L^r₁χ4− 6 ¯A(χ4) L^r₂χ4

+ 12 ¯A(χ4) L^r₄χ4+ 1/12 ¯A(χ4) ¯B(χp, χp; 0) (R^p_4η)²χ4+ 1/6 ¯A(χ4) ¯B(χp, χη; 0)

R^p_4ηR^η_p4χ4− R^p4ηR^η_q4χ4

− 1/24 ¯A(χ4) ¯B(χη, χη; 0) R^v_η13χ4− 1/6 ¯A(χ4) ¯B(χ1, χ3; 0) R¹_4ηR³_4ηχ4+ 3/8 ¯A(χ4; ε) π16χ4

− 32 ¯A(χ46) L^r₁χ46− 8 ¯A(χ46) L^r₂χ46+ 16 ¯A(χ46) L^r₄χ46+ ¯A(χ46) ¯B(χp, χp; 0)

1/9 χ46+ 1/12 Rpp^ηχp

+ 1/36 R^η_ppχ4+ 1/9 R^η_p4χ6

+ ¯A(χ46) ¯B(χp, χη; 0)

−1/18 R^ηppχ4− 1/9 Rp4^ηχ6+ 1/9 R_q4^ηχ6+ 1/18 R^η₁₃χ4

− 1/6 ¯A(χ46) ¯B(χp, χη; 0, k) R^ηpp− R^η13

+ 1/9 ¯A(χ46) ¯B(χη, χη; 0) R^v_η13χ46− ¯A(χ46) ¯B(χ1, χ3; 0) [2/9 χ46

+ 1/9 R^η_p4χ6+ 1/18 R^η₁₃χ4] − 1/6 ¯A(χ46) ¯B(χ1, χ3; 0, k) R^η₁₃+ 1/2 ¯A(χ46; ε) π16χ46

+ ¯B(χp, χp; 0) π16

1/16 ¯χ1R^dp+ 1/96 R^dpχp+ 1/32 R^dpχq

+ 2/3 ¯B(χp, χp; 0) L^r₀R^dpχp

+ 5/3 ¯B(χp, χp; 0) L^r₃R^dpχp+ ¯B(χp, χp; 0) L^r₄

−2 ¯χ1χ¯^pp_ηη0χp− 4 ¯χ1R^pqηχp+ 4 ¯χ1R^cpχp+ 3 ¯χ1Rp^d

 + ¯B(χp, χp; 0) L^r₅

−2/3 ¯χ^pp_ηη1χp− 4/3 R^pqηχ²p+ 4/3 R^cpχ²p+ 1/2 Rp^dχp− 1/6 Rp^dχq

 + ¯B(χp, χp; 0) L^r6

4 ¯χ1χ¯^pp_ηη1+ 8 ¯χ1R^pqηχp− 8 ¯χ1R^cpχp

+ 4 ¯B(χp, χp; 0) L^r7(R^dp)² + ¯B(χp, χp; 0) L^r8

4/3 ¯χ^pp_ηη2+ 8/3 R^pqηχ²p− 8/3 R^cpχ²p

+ ¯B(χp, χp; 0)²

−1/18 R^pqηR^dpχp+ 1/18 R^cpR^dpχp

+ 1/288 (R^d_p)²

+ 1/18 ¯B(χp, χp; 0) ¯B(χp, χη; 0)

R^η_ppR^d_pχp− R^η13R^d_pχp



plus several more pages

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: F_{V π}(t), F_Sπ: 1998

JB-Talavera: π → ℓνγ: 1997

Gasser-Ivanov-Sainio: γγ → π⁰π⁰, γγ → π⁺π⁻: 2005-2006 Two-Loops Three flavours

ΠV V π, ΠV V η, ΠV V K Kambor, Golowich; Kambor, Dürr; Amorós, JB, Talavera

Usual ChPT two-loop: A list

ΠV V K, ΠAAK, FK, mK Amorós, JB, Talavera

Kℓ4, hqqi Amorós, JB, Talavera L^r₁, L^r₂, L^r₃ F_M, m_M, hqqi (mu 6= m_d) Amorós, JB, Talavera L^r_5,7,8, mu/m_d

F_{V π}, F_{V K}⁺, F_{V K}⁰ Post, Schilcher; JB, Talavera L^r₉

K_ℓ3 Post, Schilcher; JB, Talavera Vus

FSπ, FSK (includes σ-terms) JB, Dhonte L^r₄, L^r₆

K, π → ℓνγ Geng, Ho, Wu L^r₁₀

ππ JB,Dhonte,Talavera

πK JB,Dhonte,Talavera

K _ℓ3

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.

K _ℓ3 Definitions

K_ℓ3⁺ : K⁺(p) → π⁰(p^′)ℓ⁺(p_ℓ)ν_ℓ(p_ν) K_ℓ3⁰ : K⁰(p) → π⁻(p^′)ℓ⁺(p_ℓ)ν_ℓ(p_ν)

K_ℓ3⁺ : T = G_F

√2V_us^⋆ ℓ^µF_µ⁺(p^′, p) ℓ^µ = ¯u(p_ν)γ^µ(1 − γ⁵)v(p_ℓ)

F_µ⁺(p^′, p) = < π⁰(p^′) | Vµ⁴⁻ⁱ⁵(0) | K⁺(p) >

= 1

√2[(p^′ + p)_µf₊^K+π0(t) + (p − p^′)_µf₋^K+π0(t)]

Isospin: f₊^K0π−(t) = f₊^K+π0(t) = f₊(t) f^K0π−(t) = f^K+π0(t) = f (t)

K _ℓ3 Definitions and V _us

Scalar formfactor: f₀(t) = f₊(t) + t

m²_K − m²π

f₋(t)

Usual parametrization: f_+,0(t) = f₊(0)



1 + λ_+,0 t m²_π



|V^us|: Know theoretically f₊(0) = 1 + · · ·

Short distance correction to G_F from G_µ

Marciano-Sirlin

Ademollo-Gatto-Behrends-Sirlin theorem:

(m_s − ˆm)²

V _us

PDG2002:

|V^ud| = 0.9734 ± 0.0008 |V^us| = 0.2196 ± 0.0026

|V^ud|² + |V^us|² = (0.9475 ± 0.0016) + (0.0482 ± 0.0011) = 0.9957 ± 0.0019

PDG2006:

|V^ud| = 0.97377 ± 0.00027 |V^us| = 0.2257 ± 0.0021

|V^ud|² + |V^us|² = (0.94823 ± 0.00054) + (0.05094 ± 0.00095) = 0.99917 ± 0.00110

Problems:

Ignores ∆(0) = 0.0113 from pure two-loop Conflicts between experiments

K _ℓ3 Diagrams

(a)



(b)

( )

(a) (b) ( )

(a)



(b) ( )



(d)



(e) (f) (g) (h)

(i) (j)



(k) (l)

f ₊ (t) Theory

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

2F_π² L^r₉ + loops f₊⁽⁶⁾(t) = − 8

F_π⁴ (C₁₂^r + C₃₄^r ) m²_K − m²π

2

+ t

F_π⁴R^Kπ₊₁ + t²

F_π⁴(−4C88^r + 4C₉₀^r ) + loops(L^r_i)

ChPT fit to f ₊ (t)

⇒R₊₁^Kπ =

−(4.7 ± 0.5) 10⁻⁵ GeV²

`c+ = 3.2 GeV⁻⁴´

⇒a+ = 1.009 ± 0.004

⇒λ+ = 0.0170 ± 0.0015

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

f₊(t)

CPLEAR ChPT C_i^r=0 ChPT p⁴ ChPT fit λ₊ = 0.245

f ₀ (t)

Main Result:

f₀(t) = 1 − 8

F_π⁴ (C₁₂^r + C₃₄^r ) m²_K − m²π

2

+8 t

F_π⁴(2C₁₂^r + C₃₄^r ) m²_K + m²_π

+ t

m²_K − m²π

(F_K/F_π − 1)

− 8

F_π⁴ t²C₁₂^r + ∆(t) + ∆(0) .

∆(t) and ∆(0) contain NO C_i^r and only depend on the L^r_i at order p⁶

=⇒

All needed parameters can be determined experimentally

∆(0) = −0.0080 ± 0.0057[loops] ± 0.0028[L^r] .

Experiment

 ₁

3σ 

' (
<>I".0,..!27

^
_ : λ″₊ ≠0!σ 

^_µ3 λ₀ ^1


 Κ⁰_µ3

33 

λ₊′3/¹ λ +′3/
1

' (

" ';-G

$%&

#)*

Q=+ ,π+& ,π

#)*
5

" ';-

^
_ λ″₊ ≠0,σ 

Κ⁻_µ3 V ( % λ₀ ^1

<8180,..827

ChPT

Experiment



    

λ′₊ = 0.02496(79) λ′′₊ = 0.0016(3) λ₀ = 0.01587(95)

S(_B0.2-. 8092

S_-. "1""0,"2



 ..!...1"

>_µ%

,1980,,2

:.0 2

τ

..11910,"2 ..:.!1012

".!80 2N.^!

.,8 "0"2 .!.!8092

>

^± µ3

 ^± 1

_A1

 µ1

1

_

Experiment



  

 Γ(Κ→µν(γ))/Γ(π→µν(γ))

7


__π
+(_+A+(_+

4
 8A/)/91,)5 Γ(Κ→µν(γ))/Γ(π→µν(γ)) ∝ +(_+^,A+(_+^,_^,_π^,

_
_π - 90120^B6₋₅
P

$%&
6;4^- → µ⁺ν)

(_A(_
.
/!,,0)
± 0!//,9



=



δ.+(+^,-+(+^,83.8/!//,/2/!///9

#&&=

'%
6&
%#H";&=

K _S → γγ

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

K_S

π⁺, K⁺

K_S

π⁺, K⁺

K_S

π⁺, K⁺

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

Some other rare decays

K_L → γγ

Needs work: main contribution is full of cancellations:

difficult ^{KL π}

0, η, η^′

K_L → π⁰γγ

K_L → π⁰γγ OK predicted by CHPT

Main succes: events must be at high m_γγ Rate still a problem

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

K → 3π: Overview

ChPT in the nonleptonic mesonic sector Lagrangians

K → 3π kinematics and isospin

Overview of calculations and results Data and Fits

ChPT in the nonleptonic sector

some earlier work: especially on decays with photons Kambor, Missimer, Wyler (KMW) : Constructed _L and

∞ 1990

G. Esposito-Farese: Checked _L and _∞ 1991 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 _L 1993 K → 2π redone: Bijnens, Pallante, Prades 1998

ChPT in the nonleptonic sector

First paper: redo _{K → 3π}

Ecker Isidori Muller Neufeld Pich: Electromagnetic octet L Lagrangian plus _∞ 2000

applications to _{K → 2π}: Several papers

Isospin breaking in _{K → 3π}: remaining papers O (p⁴, p²(m_u − m^d), e²p²).

Lagrangians: p ² and e ²

L² = L^S2 + L^W2 + L^E2 L^S2 = F₀²

4 hu^µu^µ + χ₊i

u_µ = iu^† D_µU u^† = u^†_µ , u² = U , χ_± = u^†χu^† ± uχ^†u , U contains the Goldstone boson fields

U = exp i√ 2 F₀ M

!

, M =





√1

2π3 + ^√1₆η8 π⁺ K⁺ π^{− √−1}

2π3 + ^√1

6η8 K⁰

K⁻ K^{0 −2√}

6η8



 .

Here χ = 2B₀





m_u

m_d

m_s



 and D_µU = ∂_µU −ie [Q, U] .

Lagrangians: p ² and e ²

L^W2 = CF₀⁴

"

G₈h∆³²u_µu^µi + G^′₈h∆³²χ₊i + G²⁷t^ij,kl h∆^iju_µih∆^klu^µi

#

∆_ij ≡ uλ^iju^† , (λ_ij)_ab ≡ δ^ia δ_jb , t^21,13 = t^13,21 = 1/3, . . .