CHIRAL PERTURBATION THEORY

K _ℓ3 AT TWO LOOPS IN

CHIRAL PERTURBATION THEORY

Johan Bijnens Lund University

lavi

net

Overview

Motivation

Chiral Perturbation Theory K _ℓ3

What is new when including isospin breaking?

First very preliminary results with isospin breaking at

two loops

Motivation

Basically two: Test ChPT and determine V _us

From Moulson, hep-ex/0703013, Flavianet working group result

Mode ∆(SU (2)) ∆(EM ) Decay V _us f ₊ (0) K _e3 ⁰ 0 +0.52% K _Le3 0.21639(55) K _µ3 ⁰ 0 +0.95% K _{Lµ 3} 0.21649(68) K _e3 ^± 2.31% +0.52% K _e3 ^± 0.21844(101) K _µ3 ^± 2.31% +0.95% K _µ3 ^± 0.21809(125)

About 1% discrepancy: is this real or a fluke

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 _{L QCD} = X

q=u,d,s

[i¯ q _L D / q _L + i¯ q _R D / q _R − m ^q (¯ q _R q _L + ¯ q _L q _R )]

So if m _q = 0 then SU (3) _L × SU(3) ^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 _{L QCD} = X

q=u,d,s

[i¯ q _L D / q _L + i¯ q _R D / q _R − m ^q (¯ q _R q _L + ¯ q _L q _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 ^L q _R + ¯ q _R q _L i 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 ^L q _R + ¯ q _R q _L i 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:

p ²

1/p ² R d ⁴ p p ⁴

(p ² ) ² (1/p ² ) ² p ⁴ = p ⁴

(p ² ) (1/p ² ) p ⁴ = p ⁴

Chiral Perturbation Theory

For lectures, review articles: see

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

Review paper on Two-Loops: JB, LU TP 06-16

hep-ph/0604043

Two Loop: Lagrangians

Lagrangian Structure:

2 flavour 3 flavour 3+3 PQChPT p ² F, B 2 F ₀ , B ₀ 2 F ₀ , B ₀ 2 p ⁴ l _i ^r , h ^r _i 7+3 L ^r _i , H _i ^r 10+2 L ^{ˆ r} _i , ˆ H _i ^r 11+2 p ⁶ c ^r _i 53+4 C _i ^r 90+4 K _i ^r 112+3 p ² : Weinberg 1966

p ⁴ : Gasser, Leutwyler 84,85

p ⁶ : 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:

L ˆ ^r _i , K _i ^r → L ^r i , C _i ^r

Long Expressions

=⇒

Long Expressions

=⇒

δ⁽⁶⁾²²_loops= π16L^r₀4/9 χηχ4− 1/2 χ1χ3+ χ²₁₃− 13/3 ¯χ1χ13− 35/18 ¯χ2 − 2 π16L^r₁χ²₁₃

− π16L^r₂11/3 χηχ4+ χ²₁₃+ 13/3 ¯χ2 + π16L^r₃4/9 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^r₀8/3 R^pqηχp+ 2/3 R^cpχp+ 2/3 R^dp + ¯A(χp) L^r₃2/3 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^ppηη1− R^pqηχp

+ 1/3 R^pqηχq+ 1/2 R^cpχp− 1/6 R^cpχq

+ ¯A(χp)²1/16 + 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) π161/8 ¯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) π161/16 ¯χ1R^v_η13− 1/48 R^vη13χη+ 1/16 R_η13^v χ13 + ¯A(χη) L^r₀4R^η₁₃χη+ 2/3 R^v_η13χη − 8 ¯A(χη) L^r₁χη− 2 ¯A(χη) L^r₂χη+ ¯A(χη) L^r₃4R^η₁₃χη+ 5/3 R^v_η13χη

 + ¯A(χη) L^r₄4 χη+ ¯χ1R^v_η13 − ¯A(χη) L^r₅1/6 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^ppηηη1− 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^ηpηpη1+ 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) π161/16 ¯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^ppηη1χp− 4/3 R^pqηχ²p+ 4/3 R^cpχ²p+ 1/2 Rp^dχp− 1/6 Rp^dχq

+ ¯B(χp, χp; 0) L^r64 ¯χ1χ¯^pp_ηη1+ 8 ¯χ1R^pqηχp− 8 ¯χ1R^cpχp

+ 4 ¯B(χp, χp; 0) L^r7(R^dp)²

+ ¯B(χp, χp; 0) L^r84/3 ¯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^ηppR^d_pχp− R^η13R^d_pχp

plus several more pages

Usual ChPT two-loop: A list

Two-Loop Two-Flavour

Bellucci-Gasser-Sainio: γγ → π ⁰ π ⁰ : 1994 Bürgi: γγ → π ⁺ π ⁻ , F π , m π : 1996

JB-Colangelo-Ecker-Gasser-Sainio: ππ, F π , m π : 1996-97 JB-Colangelo-Talavera: F _{V π} (t), F _Sπ : 1998

JB-Talavera: π → ℓνγ : 1997

Gasser-Ivanov-Sainio: γγ → π ⁰ π ⁰ , γγ → π ⁺ π ⁻ : 2005-2006 Two-Loops Three flavours

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

Π _{V V ρω} Maltman

Π _AAπ , Π _AAη , F π , F η , m π , m η Kambor, Golowich; Amorós, JB, Talavera

Π SS Moussallam L ^r ₄ , L ^r ₆

Usual ChPT two-loop: A list

Π V V K , Π AAK , F K , m K Amorós, JB, Talavera K ℓ4 , hqqi Amorós, JB, Talavera L ^r ₁ , L ^r ₂ , L ^r ₃

F _M , m _M , hqqi (m u 6= m _d ) Amorós, JB, Talavera L ^r _5,7,8 , m u /m _d

F _{V π} , F _{V K} ⁺ , F _{V K} ⁰ Post, Schilcher; JB, Talavera L ^r ₉

K _ℓ3 Post, Schilcher; JB, Talavera V us

F Sπ , F SK (includes σ -terms) JB, Dhonte L ^r ₄ , L ^r ₆

K, π → ℓνγ Geng, Ho, Wu L ^r ₁₀

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.

J. Bijnens and K. Ghorbani, to be published.

K _ℓ3 Definitions

K _ℓ3 ⁺ : K ⁺ (p) → π ⁰ (p ^′ )ℓ ⁺ (p _ℓ )ν _ℓ (p _ν ) K _ℓ3 ⁰ : K ⁰ (p) → π ⁻ (p ^′ )ℓ ⁺ (p _ℓ )ν _ℓ (p _ν )

K _ℓ3 ⁺ : T = G _F

√ 2 V _us ^⋆ ℓ ^µ F _µ ⁺ (p ^′ , p) ℓ ^µ = ¯ u(p _ν )γ ^µ (1 − γ ⁵ )v(p _ℓ )

F _µ ⁺ (p ^′ , p) = < π ⁰ (p ^′ ) | V µ ⁴⁻ⁱ⁵ (0) | K ⁺ (p) >

= 1

√ 2 [(p ^′ + p) _µ f ₊ ^{K + π 0} (t) + (p − p ^′ ) _µ f − ^{K + π 0} (t)]

Isospin: f ₊ ^{K 0 π −} (t) = f ₊ ^{K + π 0} (t) = f ₊ (t)

f − ^{K 0 π −} (t) = f − ^{K + π 0} (t) = f ⁻ (t)

K _ℓ3 Definitions and V _us

Scalar formfactor: f ₀ (t) = f ₊ (t) + t

m ² _K − m ² π

f ⁻ (t)

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



1 + λ _+,0 t m ² _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 − ˆ m) ²

Isospin Breaking Leutwyler-Roos : see later

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

Comparison between experiments: we’ll hear more next

week

K _ℓ3 Diagrams

(a)



(b)

( )

(a) (b) ( )

• : p ² vertex

4

(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 _π ² L ^r ₉ + loops f ₊ ⁽⁶⁾ (t) = − 8

F _π ⁴ (C ₁₂ ^r + C ₃₄ ^r ) m ² _K − m ² π

2

+ t

F _π ⁴ R ^Kπ ₊₁ + t ²

F _π ⁴ (−4C 88 ^r + 4C ₉₀ ^r ) + loops (L ^r _i )

ChPT fit to f ₊ (t)

⇒R ₊₁ ^Kπ =

−(4.7 ± 0.5) 10 ⁻⁵ GeV ²

`c + = 3.2 GeV ⁻⁴ ´

⇒a + = 1.009 ± 0.004

⇒λ + = 0.0170 ± 0.0015

0.98 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

0 0.02 0.04 0.06 0.08 0.1 0.12

2

f ₊ (t)

CPLEAR

ChPT C _i ^r =0

ChPT p ⁴

ChPT fit

λ ₊ = 0.245

f ₀ (t)

Main Result:

f ₀ (t) = 1 − 8

F _π ⁴ (C ₁₂ ^r + C ₃₄ ^r ) m ² _K − m ² π

2

+8 t

F _π ⁴ (2C ₁₂ ^r + C ₃₄ ^r ) m ² _K + m ² _π
+ t

m ² _K − m ² π

(F _K /F _π − 1)

− 8

F _π ⁴ t ² C ₁₂ ^r + ∆(t) + ∆(0) .

∆(t) and ∆(0) contain NO C _i ^r and only depend on the L ^r _i at order p ⁶

=⇒

All needed parameters can be determined experimentally

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

Experiment

 ₁

3σ 

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



: λ″

≠0!σ 



λ




 Κ

33 

λ ₊ ′3/ ¹ λ + ′  3 /
1

' (

" ';-G

$%&

#)*

Q      = + 

&

#)*
5



" ';-



λ″

≠0,σ 

Κ

V ( % λ

λ″

<8180,..827

ChPT

New for isospin breaking at two loops

Two-point functions:

i j

+ i j

+ i j

+ i j

+ + i j

The diagrams for _{h 0 | φ i} φ _j | 0i ≡ G ^ij Problem: Mixing

Take LSZ into account properly

Amoros, JB, Talavera, 2001

Isospin Breaking

G = iP + iP (iΠiP ) + iP (iΠiP ) ² + . . . = iP (1 + ΠP ) ^{− 1} . No mixing: Π , P and G all diagonal

det(G ^{− 1} ) = det (1 + ΠP ) P ^{− 1}
= det P ^{− 1} + Π
= 0 .

Zeros give particle masses, residues can also be obtained.

Solve perturbatively for our case of π ⁰ - η mixing.

Lowest order: π ⁰ ≡ cos(ǫ) π ⁰ + sin(ǫ) η

Isospin Breaking

Define lowest order diagonal basis P _ij = δ _ij p ² − m ² _i0 Π starts at next-to-leading order

m ² _phys = m ² ₀ + (m ² ) ⁽⁴⁾ + (m ² ) ⁽⁶⁾ + . . . m ² ₀ = m ² ₃₀ ,

(m ² ) ⁽⁴⁾ = −Π ⁽⁴⁾ ₃₃ (m ² ₃₀ , m ² _i0 , F ₀ ) ,

(m ² ) ⁽⁶⁾ = −Π ⁽⁴⁾ ₃₃ (m ² ₃₀ , m ² _i0 , F ₀ ) − (m ² ) ⁽⁴⁾ ∂

∂p ² Π ⁽⁴⁾ ₃₃ (p ² , m ² _i0 , F ₀ )   

 p ² =m ² ₃₀

+ 1

m ² ₃₀ − m ² ₈₀

 Π ⁽⁴⁾ ₃₈ (m ² ₃₀ , m ² _i0 , F ₀ )  2

Matrix element

Matrix element: j i

A ^{i 1 ...i n} = (−i) ⁿ

pZ _i 1 . . . Z _{i n}

! _n Y

i=1

k _i ² lim → m ² _i (k _i ² −m ² i ) G _{i 1 ...i n} (k ₁ , . . . , k _n )

Isospin Breaking

G ^{− 1} = −iP , _{P = P} ^{− 1} + Π

Z ₃ = 1

∂

∂p ² (det P(p ² ))  

 p ² =m ² _π

P ⁸⁸ (m ² _π ) ,

Leading to _{A 3} = 1

q P ⁸⁸ _∂p ^{∂ 2} P ³³ P ⁸⁸ − P ₃₈ ²
{P ⁸⁸ G ³ − P ³⁸ G ⁸ } .

Now expand in chiral orders and . . .

Isospin Breaking

A ³ = G ₃ ⁽²⁾ +



G ₃ ⁽⁴⁾ − 1

2 Z ₃₃ ⁽⁴⁾ G ₃ ⁽²⁾ − Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽²⁾



+



G ₃ ⁽⁶⁾ − 1

2 Z ₃₃ ⁽⁶⁾ G ₃ ⁽²⁾ − 1

2 Z ₃₃ ⁽⁴⁾ G ₃ ⁽⁴⁾ + 3 8

 Z ₃₃ ⁽⁴⁾  2

G ₃ ⁽²⁾

+ Z ₃₈ ⁽⁴⁾ Π ⁽⁴⁾ ₃₈

∆m ² G ₃ ⁽²⁾ − 1 2

Π ⁽⁴⁾ ₃₈

∆m ²

! 2

G ₃ ⁽²⁾ − Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽⁴⁾

− Π ⁽⁶⁾ ₃₈

∆m ² G ₈ ⁽²⁾ + Π ⁽⁴⁾ ₃₈ Π ⁽⁴⁾ ₈₈

∆m ² G ₈ ⁽²⁾ + 1

2 Z ₃₃ ⁽⁴⁾ Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽²⁾



Isospin Breaking

A ³ = G ₃ ⁽²⁾ +



G ₃ ⁽⁴⁾ − 1

2 Z ₃₃ ⁽⁴⁾ G ₃ ⁽²⁾ − Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽²⁾



+



G ₃ ⁽⁶⁾ − 1

2 Z ₃₃ ⁽⁶⁾ G ₃ ⁽²⁾ − 1

2 Z ₃₃ ⁽⁴⁾ G ₃ ⁽⁴⁾ + 3 8

 Z ₃₃ ⁽⁴⁾  2

G ₃ ⁽²⁾

+ Z ₃₈ ⁽⁴⁾ Π ⁽⁴⁾ ₃₈

∆m ² G ₃ ⁽²⁾ − 1 2

Π ⁽⁴⁾ ₃₈

∆m ²

! 2

G ₃ ⁽²⁾ − Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽⁴⁾

− Π ⁽⁶⁾ ₃₈

∆m ² G ₈ ⁽²⁾ + Π ⁽⁴⁾ ₃₈ Π ⁽⁴⁾ ₈₈

∆m ² G ₈ ⁽²⁾ + 1

2 Z ₃₃ ⁽⁴⁾ Π ⁽⁴⁾ ₃₈

∆m ² G ₈ ⁽²⁾



Compute all diagrams, produce numerical programs, . . .

Numerical results for f ₊ (0)

VERY PRELIMINARY

Decay p ² p ⁴ pure 2-loop L ^r _i at p ⁶ C _i ^r total Iso conserving calculation

K ⁰ 1 −0.02266 0.01130 0.00332 ??? 0.99196 K ⁺ 1 −0.02276 0.01104 0.00320 ??? 0.99154

m u /m _d = 0.45

K ⁰ 1 −0.02310 0.01131 0.00325 ??? 0.99146 K ⁺ 1.02465 −0.01741 0.00379 0.00648 ??? 1.01751

ratio 1.02465 1.0311 1.0262

m u /m d = 0.58

K ⁰ 1 −0.02299 0.01124 0.00325 ??? 0.99150 K ⁺ 1.01702 −0.01897 0.00657 0.00551 ??? 1.01013

ratio 1.0170 1.0215 1.0188

Some Comments

K ⁰ only again on C ₁₂ ^r + C ₃₄ ^r

Ademollo-Gatto + Callan-Treiman as in isoconserving case

but (m _s − m ^u ) ²

Not checked yet whether C _i ^r in K ⁺ decay can b determined

p ⁶ lowers the isospin breaking

p ⁶ fit has m _u /m _d = 0.45 and not 0.58

0.58 → 0.52 from p ⁶ and 0.52 → 0.45 from violation of Dashen’s theorem

Very preliminary numerics: decrease discrepancy by

about 0.5%.