Particle Physics The Standard Model · Particle Physics The Standard Model Cabibbo Kobayashi Maskawa Frédéric Machefert [email protected] Laboratoire de l’accélérateur linéaire

Particle PhysicsThe Standard Model

Cabibbo Kobayashi Maskawa

Frédéric Machefert

[email protected]

Laboratoire de l’accélérateur linéaire (CNRS)

Cours de l’École Normale Supérieure24, rue Lhomond, Paris

March 15th, 2018

1 / 25

Introduction The Lagrangian K-physics B-physics

Part IX

Cabibbo Kobayashi Maskawa

2 / 25


1 Introduction

2 The Lagrangian

3 K-physics

4 B-physics

3 / 25


• The Particles

• W± couples to SU(2)L doublets

• Z◦: no FCNC (Z◦ cannot change

flavor just like γ)

• AssumptionMassEigenstates=EWEigenstates

• Why?

(

uL

dL

) (

cL

sL

) (

tL

bL

)

(

νeL

eL

) (

νµL

µL

) (

ντL

τ L

)

uR cR tR

dR sR bR

eR µR τR

γg

W±,Z◦

H

4 / 25


• The Particles





• Why?

(

uL

dL

) (

cL

sL

) (

tL

bL

)

(

νeL

eL

) (

νµL

µL

) (

ντL

τ L

)

uR cR tR

dR sR bR

eR µR τR

γg

W±,Z◦

H

4 / 25


• The Particles





• Why?

(

uL

dL

) (

cL

sL

) (

tL

bL

)

(

νeL

eL

) (

νµL

µL

) (

ντL

τ L

)

uR cR tR

dR sR bR

eR µR τR

γg

W±,Z◦

H

4 / 25


• Another perspective: mesonlifetimes

• Weak decays of mesons differ byorders of magnitude?

• How can the s decay weakly?mc > ms?

• Keep leptons untouched

• Introduce the CKM matrix

Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25







Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25







Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25







Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25







Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25







Properties of the c

m0 = 1.27GeV

τ = (1.040 · 10−12)s cd̄

cτ = 311.8µm

Properties of the s

m0 = 100 ± 25MeV

τ = (1.24 · 10−8)s us̄

cτ = 3.7m

5 / 25


Definitiond is the mass Eigenstated′ is the isospin partner of u

V: unitary 3 × 3 matrix V†V = 13

Cannot simplify: masses not equal

LYuk = −umuu − cmcc − tmtt − dmdd − smss − bmbb

= − ( u c t )

mu 0 00 mc 00 0 mt

u

c

t

6 / 25






= − ( u c t )

mu 0 00 mc 00 0 mt

u

c

t

− ( d s b )

md 0 00 ms 00 0 mb

d

s

b

6 / 25






= − ( u c t )

mu 0 00 mc 00 0 mt

u

c

t

− ( d′

s′

b′)V

md 0 00 ms 00 0 mb

V†

d′

s′

b′

6 / 25


Properties of V

• Cabibbo,Kobayashi,Maskawa

• V complex:3 × 3 × 2

• VV† = 13: 9

constraints

• 5 phasesabsorbed

• 3 real mixingangles, 1 complexphase

Cabibbo

(2 generations):(

cos θC sin θC

− sin θC cos θC

)

Wolfenstein

1 − λ2

2 λ Aλ3(ρ− iη)

−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1

• Diagonal entries dominate

7 / 25


Properties of V


• V complex:3 × 3 × 2

• VV† = 13: 9

constraints



Cabibbo

(2 generations):(

cos θC sin θC

− sin θC cos θC

)

Wolfenstein

1 − λ2


−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1


7 / 25


Properties of V


• V complex:3 × 3 × 2

• VV† = 13: 9

constraints



Cabibbo

(2 generations):(

cos θC sin θC

− sin θC cos θC

)

Wolfenstein

1 − λ2


−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1


7 / 25


Properties of V


• V complex:3 × 3 × 2

• VV† = 13: 9

constraints



Cabibbo

(2 generations):(

cos θC sin θC

− sin θC cos θC

)

Wolfenstein

1 − λ2


−λ 1 − λ2

2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1


7 / 25


Charged Current

( uL cL tL )γµ

dL′

sL′

bL′

= ( uL cL tL )γµV

dL

sL

bL

Decays

d → u + W− V11

s → u + W− V12

b → u + W− V13

• V11 = 0.98, V12 = 0.2

• charmed mesons:V 2

11G2 → 0.96G2

• strange mesons:V 2

12G2 → 0.04G2 longer lifetime

8 / 25


Charged Current

( uL cL tL )γµ

dL′

sL′

bL′

= ( uL cL tL )γµV

dL

sL

bL

Decays

d → u + W− V11

s → u + W− V12

b → u + W− V13

• V11 = 0.98, V12 = 0.2


11G2 → 0.96G2



8 / 25


Charged Current

( uL cL tL )γµ

dL′

sL′

bL′

= ( uL cL tL )γµV

dL

sL

bL

Decays

d → u + W− V11

s → u + W− V12

b → u + W− V13

• V11 = 0.98, V12 = 0.2


11G2 → 0.96G2



8 / 25


Charged Current

( uL cL tL )γµ

dL′

sL′

bL′

= ( uL cL tL )γµV

dL

sL

bL

Decays

d → u + W− V11

s → u + W− V12

b → u + W− V13

• V11 = 0.98, V12 = 0.2


11G2 → 0.96G2



8 / 25


Neutral Current

Neutral currents couple to right- and left-handed quarks:

( d′

s′

b′)γµ(− 1

21−γ5

2 + 13 sin2 θW )

d′

s′

b′

= ( d s b )V†γµ(− 12

1−γ52 + 1

3 sin2 θW )V

d

s

b

= ( d s b )V†Vγµ(− 1

21−γ5

2 + 13 sin2 θW )

d

s

b

V†V = 13: does not have a Lorentz index, only family index

9 / 25


Neutral Current


( d′

s′

b′)γµ(− 1

21−γ5

2 + 13 sin2 θW )

d′

s′

b′

= ( d s b )V†γµ(− 12

1−γ52 + 1

3 sin2 θW )V

d

s

b

= ( d s b )V†Vγµ(− 1

21−γ5

2 + 13 sin2 θW )

d

s

b


9 / 25


Neutral Current


( d′

s′

b′)γµ(− 1

21−γ5

2 + 13 sin2 θW )

d′

s′

b′

= ( d s b )V†γµ(− 12

1−γ52 + 1

3 sin2 θW )V

d

s

b

= ( d s b )V†Vγµ(− 1

21−γ5

2 + 13 sin2 θW )

d

s

b


9 / 25


And the up-type sector?

( uL′

cL′

tL

′ )γµ

dL′

sL′

bL′

= ( uL cL tL )V †2 γ

µV

dL

sL

bL

= ( uL cL tL )γµV†2 V

dL

sL

bL

define: V3 = V †2 V

V3V †3 = (V †

2 V)(V †2 V)† = V †

2 VV†V2 = 1

→ 1 matrix sufficient

10 / 25


And the up-type sector?

( uL′

cL′

tL

′ )γµ

dL′

sL′

bL′

= ( uL cL tL )V †2 γ

µV

dL

sL

bL

= ( uL cL tL )γµV†2 V

dL

sL

bL

define: V3 = V †2 V

V3V †3 = (V †

2 V)(V †2 V)† = V †

2 VV†V2 = 1

→ 1 matrix sufficient

10 / 25


• C: transforms particles intoanti-particles

• P: inverts momentum

EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions

µ− → νµLe−νeL

L → LLR

P µ− → νµLe−νeL

R → RRL

C µ+ → νµLe+νeL

L → LLR

CP µ+ → νµLe+νeL

R → RRL

CPT always conserved

11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25




EM Interactions

e− → γe

−

L → −1R

P e− → γe

−

R → +1L

C e+ → γe

+

L → −1R

CP e+ → γe

+

R → +1L

EW Interactions


L → LLR


R → RRL


L → LLR


R → RRL


11 / 25


Neutral Kaons

K◦ = |sd〉

K◦

= −|sd〉-: strong Isospin anti-particle

Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦

not Eigenstates of C

Parity

• (−1)ℓ fromYℓm(π − θ, π + φ) = (−1)ℓY (θ, φ)

• multiplicative:P(p1p2) = P(p1) · P(p2)

• Spinor: γ0ψ (DIRAC equation)• γ

0u(p′) = u(p)• γ

0v(p′) = −v(p)

• relative (−1) between particleand anti-particle

12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


Neutral Kaons

K◦ = |sd〉

K◦


Charge Conjugation

CK◦ = C(|sd〉)

= |sd〉

= −K◦

CK◦

= −|sd〉= −K

◦


Parity




0u(p′) = u(p)• γ

0v(p′) = −v(p)


12 / 25


CP

CPK◦ = (−1) · (−1)ℓ(−K

◦)

= K◦

CPK◦

= (−1) · (−1)ℓ(−K◦)

= K◦

JP(K◦) = JP(K◦) = 0−

CP Eigenstates

(+1) : K1 = 1√2(K◦ + K

◦)

(−1) : K2 = 1√2(K◦ − K

◦)

strong prod, weak decay

π+π−

C(π+π−) = π−π+

P(π+π−) = P(π−)P(π+)= 1

CP(π+π−) = 1

π+π−π

0

C(π0) = C(γ)2 = 1P(π0) = −1CP(π+π−π0) = CP(π+π−)·

CP(π0)= −1

13 / 25


CP

CPK◦ = (−1) · (−1)ℓ(−K

◦)

= K◦

CPK◦

= (−1) · (−1)ℓ(−K◦)

= K◦

JP(K◦) = JP(K◦) = 0−

CP Eigenstates

(+1) : K1 = 1√2(K◦ + K

◦)

(−1) : K2 = 1√2(K◦ − K

◦)


π+π−

C(π+π−) = π−π+

P(π+π−) = P(π−)P(π+)= 1

CP(π+π−) = 1

π+π−π

0


CP(π0)= −1

13 / 25


CP

CPK◦ = (−1) · (−1)ℓ(−K

◦)

= K◦

CPK◦

= (−1) · (−1)ℓ(−K◦)

= K◦

JP(K◦) = JP(K◦) = 0−

CP Eigenstates

(+1) : K1 = 1√2(K◦ + K

◦)

(−1) : K2 = 1√2(K◦ − K

◦)


π+π−

C(π+π−) = π−π+

P(π+π−) = P(π−)P(π+)= 1

CP(π+π−) = 1

π+π−π

0


CP(π0)= −1

13 / 25


CP

CPK◦ = (−1) · (−1)ℓ(−K

◦)

= K◦

CPK◦

= (−1) · (−1)ℓ(−K◦)

= K◦

JP(K◦) = JP(K◦) = 0−

CP Eigenstates

(+1) : K1 = 1√2(K◦ + K

◦)

(−1) : K2 = 1√2(K◦ − K

◦)


π+π−

C(π+π−) = π−π+

P(π+π−) = P(π−)P(π+)= 1

CP(π+π−) = 1

π+π−π

0


CP(π0)= −1

13 / 25


LifetimesKaon mass: 494MeV

K1 → π+π−

τS = 0.9 · 10−10s

K2 → π+π−π0

τL = 5.2 · 10−8s

phase space:

m(π+π−) ≈ 280MeV

m(π+π−π0) ≈ 420MeV

K2 was initially “overlooked”

Time dependence

Decay is described by weakEigenstates with a well-definedlifetime:

|K1(t)〉 = |K1(0)〉exp−iMS t

exp−ΓS t/2

|K2(t)〉 = |K2(0)〉exp−iML t

exp−ΓLt/2

strong as f(weak):

K◦ = 1√

2(|K1〉+ |K2〉)

K◦

= 1√2(|K1〉 − |K2〉)

14 / 25


LifetimesKaon mass: 494MeV

K1 → π+π−

τS = 0.9 · 10−10s

K2 → π+π−π0

τL = 5.2 · 10−8s

phase space:

m(π+π−) ≈ 280MeV

m(π+π−π0) ≈ 420MeV

K2 was initially “overlooked”

Time dependence

Decay is described by weakEigenstates with a well-definedlifetime:

|K1(t)〉 = |K1(0)〉exp−iMS t

exp−ΓS t/2

|K2(t)〉 = |K2(0)〉exp−iML t

exp−ΓLt/2

strong as f(weak):

K◦ = 1√

2(|K1〉+ |K2〉)

K◦

= 1√2(|K1〉 − |K2〉)

14 / 25


Oscillation

AK◦

K◦(t): amplitude to produce at t = 0 a K◦ and find at t a K

◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)

= 12 (exp−iMS t exp−ΓS t/2 − exp−iML t exp−ΓLt/2)

PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)

= 14 (exp−ΓS t +exp−ΓLt −2 cos(∆Mt) exp−(ΓL+ΓS )t/2)

• ΓS ≫ ΓL

• oscillation with frequency ∆M = ML − MS

15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


Oscillation

AK◦


◦

AK◦

K◦(t) = 〈K◦(t)|K◦(t = 0)〉

= 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)


PK◦

K◦(t) = AK◦

K◦(t)A⋆K◦

K◦(t)


• ΓS ≫ ΓL


15 / 25


• AK◦

K◦(t): t = 0 a K◦ and at t a K

◦

• AK◦K

◦(t): t = 0 a K◦

and at t a K◦

AK◦

K◦(t) = 12 (〈K1(t)| − 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)

AK◦K

◦(t) = 12 (〈K1(t)|+ 〈K2(t)|)(|K1(t = 0)〉 − |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉 − 〈K2(t)|K2(t = 0)〉)

= AK◦

K◦((t)

PK◦

K◦(t) = PK◦K

◦(t)

PK◦K

◦(t) = 14 (exp−ΓS t +exp−ΓLt −2 cos(∆Mt) exp−(ΓL+ΓS )t/2)

16 / 25


• AK◦K◦(t): t = 0 a K◦ and at t a K

◦

• AK◦

K◦(t): t = 0 a K

◦and at t a K

◦

• A∆m asymmetry used to express the experimental measurement.Overall normalisation is cancelled in the ratio.

AK◦K◦(t) = 12 (〈K1(t)|+ 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)

AK◦

K◦(t) = 1

2 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)PK◦K◦(t) = P

K◦

K◦(t)

PK◦K◦(t) = 14 (exp−ΓS t +exp−ΓLt +2 cos(∆Mt) exp−(ΓL+ΓS )t/2)

A∆m =(P

K◦K◦ (t)+PK◦

K◦ (t))−(P

K◦

K◦ (t)+PK◦K

◦ (t))

PK◦K◦ (t)+P

K◦

K◦ (t)+P

K◦

K◦ (t)+PK◦K

◦ (t)

= 2 cos(∆Mt) exp−(ΓL+ΓS )t/2

exp−ΓSt + exp−ΓLt

17 / 25



◦

• AK◦

K◦(t): t = 0 a K

◦and at t a K

◦


AK◦K◦(t) = 12 (〈K1(t)|+ 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)

AK◦

K◦(t) = 1

2 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)PK◦K◦(t) = P

K◦

K◦(t)


A∆m =(P

K◦K◦ (t)+PK◦

K◦ (t))−(P

K◦

K◦ (t)+PK◦K

◦ (t))

PK◦K◦ (t)+P

K◦

K◦ (t)+P

K◦

K◦ (t)+PK◦K

◦ (t)



17 / 25



◦

• AK◦

K◦(t): t = 0 a K

◦and at t a K

◦


AK◦K◦(t) = 12 (〈K1(t)|+ 〈K2(t)|)(|K1(t = 0)〉+ |K2(t = 0)〉)

= 12 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)

AK◦

K◦(t) = 1

2 (〈K1(t)|K1(t = 0)〉+ 〈K2(t)|K2(t = 0)〉)PK◦K◦(t) = P

K◦

K◦(t)


A∆m =(P

K◦K◦ (t)+PK◦

K◦ (t))−(P

K◦

K◦ (t)+PK◦K

◦ (t))

PK◦K◦ (t)+P

K◦

K◦ (t)+P

K◦

K◦ (t)+PK◦K

◦ (t)



17 / 25


A recent example (CPLear)

• 200MeV p̄ to rest in Hydrogen

• need to tag the initial state: strange mesons in pairs

• need to tag the final state: sign of electron

p̄p → K−π+

K◦

p̄p → K+π−

K◦

K◦ → π−

e+νeL

s → ue+νeL

K◦ → π+

e−νeL

s → ue−νeL

18 / 25


A recent example (CPLear)

• 200MeV p̄ to rest in Hydrogen

• need to tag the initial state: strange mesons in pairs

• need to tag the final state: sign of electron

p̄p → K−π+

K◦

p̄p → K+π−

K◦

K◦ → π−

e+νeL

s → ue+νeL

K◦ → π+

e−νeL

s → ue−νeL

18 / 25


• at t=0: asymmetry zero

• need interference term!

• A∆m = 2 cos(∆Mt) exp−(ΓL+ΓS )t/2

exp−ΓS t + exp−ΓLt

• ∆M ∼ 3.5 · 10−6eV

d u,c,t s

s u,c,t d

W±

W±

Follow fermion line: transition betweengenerations inevitable!

d

s

s

d

W−

W+

Need CKM non-diagonal: ∼ sin2 θC

19 / 25


All settled?

• BNL AGS 30GeV protons• K1 die out (can be regenerated)• expect no π+π− decays at the K

◦ mass (theoretically)• experimentally: combinatorics• use angle between beam and reconstructed π+π− system• no peak expected

20 / 25


All settled?



20 / 25


All settled?



20 / 25


All settled?



20 / 25


All settled?



20 / 25


All settled?



20 / 25


• PEAK!!!

• level: 10−3

• CP must be violated!

• CKM has a complex phase

21 / 25


• PEAK!!!

• level: 10−3



21 / 25


• PEAK!!!

• level: 10−3



21 / 25


Kaon description

KS = 1√1+|ǫ|2

(|K1〉+ ǫ|K2〉)KL = 1√

1+|ǫ|2(ǫ|K1〉+ |K2〉)

CP violation in mixing

|ǫ| =√

ΓL(π+π−)

ΓS (π+π−)

= 2.268 ± 0.023 · 10−3

CP violation in decay

Re( ǫ′

ǫ) = 1

6 (1−ΓL(π

0π0)ΓS(π+π−)

ΓS(π0π0)ΓL(π

+π−))

(NA31) = 23 ± 6.5 · 10−4

(FNAL) = 7.4 ± 5.9 · 10−4

(FNAL) = 28 ± 4.1 · 10−4

(NA48) = 18.5 ± 7.3 · 10−4

• CP violation discovered

• good for our existence

• αS(K◦)!

22 / 25


Kaon description

KS = 1√1+|ǫ|2

(|K1〉+ ǫ|K2〉)KL = 1√

1+|ǫ|2(ǫ|K1〉+ |K2〉)


|ǫ| =√

ΓL(π+π−)

ΓS (π+π−)

= 2.268 ± 0.023 · 10−3


Re( ǫ′

ǫ) = 1

6 (1−ΓL(π

0π0)ΓS(π+π−)

ΓS(π0π0)ΓL(π

+π−))

(NA31) = 23 ± 6.5 · 10−4

(FNAL) = 7.4 ± 5.9 · 10−4

(FNAL) = 28 ± 4.1 · 10−4

(NA48) = 18.5 ± 7.3 · 10−4



• αS(K◦)!

22 / 25


Kaon description

KS = 1√1+|ǫ|2

(|K1〉+ ǫ|K2〉)KL = 1√

1+|ǫ|2(ǫ|K1〉+ |K2〉)


|ǫ| =√

ΓL(π+π−)

ΓS (π+π−)

= 2.268 ± 0.023 · 10−3


Re( ǫ′

ǫ) = 1

6 (1−ΓL(π

0π0)ΓS(π+π−)

ΓS(π0π0)ΓL(π

+π−))

(NA31) = 23 ± 6.5 · 10−4

(FNAL) = 7.4 ± 5.9 · 10−4

(FNAL) = 28 ± 4.1 · 10−4

(NA48) = 18.5 ± 7.3 · 10−4



• αS(K◦)!

22 / 25


Kaon description

KS = 1√1+|ǫ|2

(|K1〉+ ǫ|K2〉)KL = 1√

1+|ǫ|2(ǫ|K1〉+ |K2〉)


|ǫ| =√

ΓL(π+π−)

ΓS (π+π−)

= 2.268 ± 0.023 · 10−3


Re( ǫ′

ǫ) = 1

6 (1−ΓL(π

0π0)ΓS(π+π−)

ΓS(π0π0)ΓL(π

+π−))

(NA31) = 23 ± 6.5 · 10−4

(FNAL) = 7.4 ± 5.9 · 10−4

(FNAL) = 28 ± 4.1 · 10−4

(NA48) = 18.5 ± 7.3 · 10−4



• αS(K◦)!

22 / 25


B-sector

• Flavour oscillation in all neutralsystems

• mB◦ ∼ 5GeV ≫ mK◦ ∼ 0.5GeV

• lifetime (tag)

Experiments

large production

• dedicated machine: e+

e−

• or pp

• good PID

• BABAR@SLAC PEP-II

• BELLE@KEK-B• asymmetric colliders

• e−: 9.1GeV

• e+: 3.4GeV• Υ4s:

• resonance bb

• decay to B◦d

B◦d and B◦

s B◦s

• 250µm need great vertexdetector

23 / 25


B-sector


• mB◦ ∼ 5GeV ≫ mK◦ ∼ 0.5GeV

• lifetime (tag)

Experiments

large production


e−

• or pp

• good PID



• e−: 9.1GeV

• e+: 3.4GeV• Υ4s:

• resonance bb

• decay to B◦d

B◦d and B◦

s B◦s


23 / 25


B-sector


• mB◦ ∼ 5GeV ≫ mK◦ ∼ 0.5GeV

• lifetime (tag)

Experiments

large production


e−

• or pp

• good PID



• e−: 9.1GeV

• e+: 3.4GeV• Υ4s:

• resonance bb

• decay to B◦d

B◦d and B◦

s B◦s


23 / 25


Back to CKM

• V complex

• V†V = 13: 9 equations

• 6 equations with complex = 0

• 2-coordinate plane: triangle

• α, β, γ

Unitary triangle

(V†V)31 = 0

= V ⋆ubVud + V ⋆

cbVcd + V ⋆tbVtd

Measurements

• ∆ms

• ∆md

• B◦ → π+π−

• B◦ → J/ψKS

• B+ → DK+

• B → τν

• overconstrained

24 / 25


Back to CKM

• V complex




• α, β, γ

Unitary triangle

(V†V)31 = 0


cbVcd + V ⋆tbVtd

Measurements

• ∆ms

• ∆md

• B◦ → π+π−

• B◦ → J/ψKS

• B+ → DK+

• B → τν

• overconstrained

24 / 25


Back to CKM

• V complex




• α, β, γ

Unitary triangle

(V†V)31 = 0


cbVcd + V ⋆tbVtd

Measurements

• ∆ms

• ∆md

• B◦ → π+π−

• B◦ → J/ψKS

• B+ → DK+

• B → τν

• overconstrained

24 / 25


Back to CKM

• V complex




• α, β, γ

Unitary triangle

(V†V)31 = 0


cbVcd + V ⋆tbVtd

Measurements

• ∆ms

• ∆md

• B◦ → π+π−

• B◦ → J/ψKS

• B+ → DK+

• B → τν

• overconstrained

24 / 25


• relationship angles-V

• all measurements inagreement

• no sign of BSM

• impressive progressin 10 years

ρ1 0.5 0 0.5 1

η

1

0.5

0

0.5

1

γ

β

α

)γ+βsin(2

sm∆dm∆ dm∆

Kε

cbVubV

ρ1 0.5 0 0.5 1

η

1

0.5

0

0.5

1

25 / 25

Documents

Particle Physics The Standard Model · Particle Physics The Standard Model Cabibbo Kobayashi Maskawa Frédéric Machefert [email protected] Laboratoire de l’accélérateur linéaire