The Nobel Prize in Physics 2008: Broken Symmetries

The Nobel Prize in Physics 2008: Broken Symmetries

Johan Bijnens Lund University

bijnens@thep.lu.se

http://www.thep.lu.se/∼bijnens

Overview

The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics for 2008 with one half to

Yoichiro Nambu, Enrico Fermi Institute, University of Chicago, IL, USA

"for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics"

and the other half jointly to

Makoto Kobayashi, High Energy Accelerator Research Organization (KEK), Tsukuba

Toshihide Maskawa,Yukawa Institute for Theoretical Physics (YITP), Kyoto University, Japan

"for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature"

PS: Lots of information: http://nobelprize.org

Yoichiro Nambu

Enrico Fermi Institute, University of Chicago Chicago, IL, USA

b. 1921 (in Tokyo, Japan)

D.Sc. 1952

University of Tokyo, Japan

Makoto Kobayashi

High Energy Accelerator Research Organization (KEK)

Tsukuba, b. 1944

Ph.D. 1972

Nagoya University, Japan

Toshihide Maskawa

Kyoto Sangyo University;

Yukawa Institute for Theo- retical Physics (YITP)

Kyoto University Kyoto, Japan

b. 1940

Ph.D. 1967

Nagoya University, Japan

Papers

Y. Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Phys. Rev. 117 (1960) 648

Y. Nambu, Axial Vector Current Conservation in Weak Interactions, Phys. Rev.

Lett. 4 (1960) 380

Y. Nambu, A ‘Superconductor’ Model of Elementary Particles and its

Consequencies, Talk given at a conference at Purdue (1960), reprinted in Broken Symmetries, Selected Papers by Y. Nambu, ed:s T. Eguchi and K. Nishijima,

World Scientific (1995).

Y. Nambu and G. Jona-Lasinio, A Dynamical Model of Elementary Particles based on an Analogy with Superconductivity I, Phys. Rev. 122 (1961) 345

Y. Nambu and G. Jona-Lasinio, A Dynamical Model of Elementary Particles based on an Analogy with Superconductivity II, Phys. Rev. 124 (1961) 246;

Y. Nambu and D. Lurié, Chirality Conservation and Soft Pion Production, Phys.

Rev. 125 (1962) 1429

Y. Nambu and E. Shrauner, Soft Pion Emission Induced by Electromagnetic and Weak Interactions, Phys. Rev. 128 (1962) 862.

M. Kobayashi and K. Maskawa, CP Violation in the Renormalizable Theory of Weak Interactions, Progr. Theor. Phys. 49 (1973) 652.

Symmetry

no

change if rotated along

vertical axis

Invariant under rotation group

Symmetry

changes if rotated along

vertical axis

But U (1) invariance good approximation

Symmetry explicitly broken by the picture

Spontaneously Broken Symmetry

Pencil balanced on end (on a table) has rotational symmetry about vertical axis.

Symmetry is broken when pencil falls over: lower energy state

Special direction is specified.

But, underlying law of gravity is still symmetrical.

Pencil could have fallen into any available direction

Spontaneously Broken Symmetry

http://nobelprize.org

Symmetry

Symmetries are very well known in physics (and definitely in particle physics)

Heisenberg, Wigner, Eckart, Yang, Mills, Glashow, Weinberg, Salam,. . .

Nambu: Introduced the concept of spontaneously broken symmetries into particle physics

Kobayashi and Maskawa: Proposed the mechanism by which the combined charge-conjugation–parity (CP)

symmetry is broken in the Standard Model

Spontaneous symmetry breaking

Magnets (e.g. in iron):

Symmetry: ≈ ≈

With interactions:

Low energy:

High energy:

Spontaneous symmetry breaking

a generic state

Spontaneous symmetry breaking

the ground state

Spontaneous symmetry breaking

No: A ground state

Spontaneous symmetry breaking

A ground state: choice of vacuum breaks the symmetry Spontaneous symmetry breaking

Spontaneous symmetry breaking

A very low energy excitation: spin wave

Spontaneous symmetry breaking

A very low energy excitation: spin wave A collective excitation: quasiparticle

Nambu and Superconductivity

Superconductivity: conductivity goes to zero below a critical temperature

Discovered by Kamerlingh-Onnes in 1911 Meisnner: expel magnetic fields 1933

Landau-Ginzburg-Abrikosov: Phenomenological Theory (scalar field)

Explained by Bardeen, Cooper and Schrieffer by Cooper pair condensation 1957

Bogolyubov 1958, (Bogolyubov transformation)

Nambu and Superconductivity

Superconductivity: conductivity goes to zero below a critical temperature

Discovered by Kamerlingh-Onnes in 1911 Meisnner: expel magnetic fields 1933

Landau-Ginzburg-Abrikosov: Phenomenological Theory (scalar field)

Explained by Bardeen, Cooper and Schrieffer by Cooper pair condensation 1957

Bogolyubov 1958, (Bogolyubov transformation) Note the many other Nobel Prizes here

Nambu and Superconductivity

Condensed Cooper Pairs: Ground State is Charged What about Gauge Invariance?

Nambu and Superconductivity

Condensed Cooper Pairs: Ground State is Charged What about Gauge Invariance?

Electromagnetic Gauge Invariance

Wave function:_{ψ(x) → e}^−iα(x)ψ(x) does not change _|ψ(x)|²

A(x) → ~~ A(x) − ~∇α(x) and V (x) → V (x) − ^∂α(x)_∂t do not change E(x)^~ and B(x)^~

Minimal coupling: both invariances belong together Requiring gauge invariance determines the form of the interaction

Nambu and Superconductivity

He used the Feynman-Dyson methods for QuantumElectroDynamics (QED)

Reformulated BCS as a generalized Hartree-Fock ground state

When deriving Green functions (which lead to amplitudes) you also need to introduce the loop graphs both in the interaction vertex and in the selfenergies

When you do that all Ward identities are satisfied Hence everything is gauge invariant as should be

Nambu and Superconductivity

He used the Feynman-Dyson methods for QuantumElectroDynamics (QED)

Reformulated BCS as a generalized Hartree-Fock ground state

The photon “mixes” with the available electron/hole states

Longitudinal mode of the photon couples to the collective excitations

In effect produces a mass for the photon ⇒ Meissner effect

Nambu and Superconductivity

He used the Feynman-Dyson methods for QuantumElectroDynamics (QED)

Reformulated BCS as a generalized Hartree-Fock ground state

The photon “mixes” with the available electron/hole states

Longitudinal mode of the photon couples to the collective excitations

In effect produces a mass for the photon

Brout-Englert, Higgs, Guralnik-Hagen-Kibble:

generalize to Yang-Mills

Weinberg, Salam apply to Glashow’s neutral current model: the Standard Model

Nambu: SSB in the Strong Interaction

First an aside: The Weak Interaction and Parity Parity (P ): ~x → −~x

related to mirror symmetry

Lee and Yang 1956: No experiment for P in weak interaction and can solve τ θ puzzle

Suggested ⁶⁰Co and µ-decay: immediately seen

Nambu: SSB in the Strong Interaction

First an aside: The Electromagnetic and Weak Interaction Electromagnetic coupling to electron:

A_µJ^µ = A_µeγ^µe (_{e ≡ ψe})

Current is conserved: ^∂J_∂xµ^µ = 0 ⇒

Matrix element of J^µ at q² = 0 is _{≡ 1}. Weak interaction is (leptonic µ decay)

GF

√2J(µ)_ρJ(e)^ρ = ^G√F

2µγ^ρ (1 − γ⁵) ν_µ ν_eγ^ρ (1 − γ⁵) e Feynman-Gell-Mann, Marshak-Sudarshan: V_ρ − A^ρ V_ρ and A_ρ conserved (CVC and CAC) for hadrons _⇒

≡ 1 at q² = 0.

Nambu: SSB in the Strong Interaction

First an aside: The Electromagnetic and Weak Interaction Electromagnetic coupling to electron:

A_µJ^µ = A_µeγ^µe (_{e ≡ ψe})

Current is conserved: ^∂J_∂xµ^µ = 0 ⇒

Matrix element of J^µ at q² = 0 is _{≡ 1}. Weak interaction is (leptonic µ decay)

GF

√2J(µ)_ρJ(e)^ρ = ^G√F

2µγ^ρ (1 − γ⁵) ν_µ ν_eγ^ρ (1 − γ⁵) e Feynman-Gell-Mann, Marshak-Sudarshan: V_ρ − A^ρ V_ρ and A_ρ conserved (CVC and CAC) for hadrons _⇒

≡ 1 at q² = 0. Neutron decay:

GF

√ 0.97 pγ^ρ (1 − 1.27γ₅) n eγ^ρ (1 − γ⁵) ν_e

Nambu: SSB in the Strong Interaction

g_A = 1.27 6= 1

Solved simultaneously by Gell-Mann-Levy and Nambu PCAC: Partially Conserved Axial Current

Nambu: SSB in the Strong Interaction

g_A = 1.27 6= 1

Solved simultaneously by Gell-Mann-Levy and Nambu PCAC: Partially Conserved Axial Current

A CAC & g_A 6= 0: p g_AF₁(q²)


γ_µγ₅ − ^2M_q^{N2 qµ} γ₅ n But has pseudoscalar coupling and massless pole:

ruled out

Nambu: SSB in the Strong Interaction

g_A = 1.27 6= 1

Solved simultaneously by Gell-Mann-Levy and Nambu PCAC: Partially Conserved Axial Current

A CAC & g_A 6= 0: p g_AF₁(q²)


γ_µγ₅ − ^2M_q^{N2 qµ} γ₅ n But has pseudoscalar coupling and massless pole:

ruled out

p g_AF₁(q²)


γ_µγ₅ − _(q^2M2_+m^Nq2_π^µ₎γ₅ n

Nambu: SSB in the Strong Interaction

Consequences:

PCAC: ^∂A

ρ

∂x^ρ = im²_πF_πφ_π

Partially: conserved when m²_π → 0 G_π =

√2 MNgA

Fπ , the pion coupling to the nucleon is related to the pion decay constant and g_A

Golberger-Treiman relation

Nambu: SSB in the Strong Interaction

Consequences:

PCAC: ^∂A

ρ

∂x^ρ = im²_πF_πφ_π

Partially: conserved when m²_π → 0 G_π =

√2 MNgA

Fπ , the pion coupling to the nucleon is related to the pion decay constant and g_A

Golberger-Treiman relation

Similar suggestion also for the strangeness changing hyperon decays and the pion to Kaon

Both papers also suggested that there should be an axial variant of isospin symmetry: exact for m_π → 0 Isospin: SU (2) symmetry from interchanging p and n

Nambu: SSB in the Strong Interaction

Further developments

He realized that if an exact symmetry is spontaneously broken there must be a massless particle

Also suggested by Goldstone (Nambu-Goldstone Bosons)

Proven by Goldstone-Salam-Weinberg

Goldstone Modes

UNBROKEN: V (φ)

Only massive modes around lowest energy state (=vacuum)

BROKEN: V (φ)

Need to pick a vacuum

hφi 6= 0: Breaks symmetry No parity doublets

Massless mode along ridge

Goldstone Modes

Also here:

low energy excita- tions

Goldstone Modes

Also here:

low energy excita- tions

Overall direction is a symmetry

θ(x) → θ(x) + Θ No Change

Interactions must vanish if θ(x) is constant

Low energy: no interactions

Nambu: SSB in the Strong Interaction

Soft Pions and Current Algebra:

∂A^ρ

∂x^ρ = im²_πF_πφ_π

This means that an axial symmetry rotation connects processes with different numbes of pions

Nambu: SSB in the Strong Interaction

Soft Pions and Current Algebra:

∂A^ρ

∂x^ρ = im²_πF_πφ_π

This means that an axial symmetry rotation connects processes with different numbes of pions

This together with the small interactions at low energies allows for a systematic expansion for processes only

involving the low-energy excitations

Chiral Perturbation Theory is the modern way to use this

How does this fit in with QCD

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: U (3)_V But L_QCD = 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 U (3)_L × U (3)_R.







u_L d_L s_L





 → U_L







u_L d_L s_L





 and







u_R d_R s_R





 → U_R







u_R d_R s_R





 Can also see that left right independent via

v < c, m_q 6= 0 =⇒

v = c, m_q = 0 =⇒/

How does this fit in with QCD

Chiral Symmetry

QCD: 3 light quarks: equal mass: interchange: U (3)_V But L_QCD = 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 U (3)_L × U (3)_R.

Hadrons do not come in parity doublets: symmetry must be broken

A few very light hadrons: π⁰π⁺π⁻ and also K, η Both can be understood from spontaneous Chiral Symmetry Breaking

Kobayashi and Maskawa

Let’s extend now P and

GF

√2 0.97 pγ^ρ (1 − 1.27γ₅) n eγ^ρ (1 − γ⁵) ν_e

Kobayashi and Maskawa

Let’s extend now P and

GF

√2 0.97 pγ^ρ (1 − 1.27γ₅) n eγ^ρ (1 − γ⁵) ν_e P is broken by the weak interaction

CP is not

C: Charge conjugation, replace everyone by their antiparticle

CP conservation still allows to solve the τ θ puzzle

Kobayashi and Maskawa

Let’s extend now P and

GF

√2 0.97 pγ^ρ (1 − 1.27γ₅) n eγ^ρ (1 − γ⁵) ν_e P is broken by the weak interaction

CP is not

C: Charge conjugation, replace everyone by their antiparticle

CP conservation still allows to solve the τ θ puzzle Experimentally CP is violated a little bit in Kaons Cronin-Fitch 1964

K⁰(sd) and K⁰(ds) mix through the weak interaction, eigenstates K_L and K_S

A tiny but well measured mass difference is a consequence

The 0.97

Suggestion: the weak current for hadrons really is the same as for leptons but it mixes ∆S = 1 and ∆S = 0

Gell-Mann-Levy 1960, Cabibbo 1963 J_ρ = cos θ_CJ_ρ^∆S=0 + sin θ_CJ_ρ^∆S=1

∆S = 0 current: with p and n,

∆S = 1 current: with p and Λ

And the same for the coupling of pion to kaon

With sin θ_C ≈ 0.22 this explained the 0.97 and the by 1963 much better measured ∆S = 1 decays of Kaon and Hyper-

Quarks came

In quark language now weak interaction from W^± exchange and the vertex is

W ^+µuγ_µ (1 − γ⁵) (cos θ_Cd + sin θ_Cs)

A mechanism for getting the K_L-K_S mass difference is

K⁰ u

W

W

u K⁰ Result is way too large

Glashow-Iliopoulos-Maiani 1970

Add another quark

W ^+µuγ_µ (1 − γ⁵) (cos θ_Cd + sin θ_Cs) + W ^+µcγ_µ (1 − γ⁵) (− sin θ^Cd + cos θ_Cs) The mechanism is now

K⁰ u, c

W

W

u, c K⁰

Result is zero if m_u = m_c; the vertical lines have sin θ_C cos θ_C − cos θ^C sin θ_C (GIM mechanism) Fits for m_c ≈ 1.5GeV /c² (Gaillard-Lee)

θ _C in the Standard Model

The standard model: interactions with W^µ and the masses:

i, j, k, l = 1, 2, later 1, 2, 3

u₁ = u, u₂ = c, u₃ = t, d₁ = d, d₂ = s, d₃ = b W Interactions: W ^+µ P

i u^W_Liγ_µd^W_Li + h.c.

Mass terms (v is the Higgs vacuum expectation value)

−v P

ij d^W_Liλ^D_ijd^W_Rj − v P

ij u^W_Liλ^U_iju^W_Rj+h.c.

θ _C in the Standard Model

−v P

ij d^W_Liλ^D_ijd^W_Rj − v P

ij u^W_Liλ^U_iju^W_Rj+h.c.

vλ^U and vΛ^D are complex 2 by 2 (3 by 3) matrices Diagonalization is always possible by two complex matrices each

V_L^U vλ^UV_R^{U †} = diag(m_u, m_c, m_t) V_L^Dvλ^DV_R^D† = diag(m_d, m_s, m_b)

θ _C in the Standard Model

−v P

ij d^W_Liλ^D_ijd^W_Rj − v P

ij u^W_Liλ^U_iju^W_Rj+h.c.

vλ^U and vΛ^D are complex 2 by 2 (3 by 3) matrices Diagonalization is always possible by two complex matrices each

V_L^U vλ^UV_R^{U †} = diag(m_u, m_c, m_t) V_L^Dvλ^DV_R^D† = diag(m_d, m_s, m_b) Define: u^W_Li = P

j V_Lij^{U †}u_Lj and u^W_Ri = P

j V_Rij^{U †}u_Rj, d^W_Li = P

j V_Lij^D†d_Lj, d^W_Ri = P

j V_Rij^D†d_Rj mass eigenstates

and substitute

θ _C in the Standard Model

−v X

ijkl

d_LiV_Lij^D λ^D_jkV_Rkl^D†d_Rk − v X

ijkl

u_LiV_Lij^U λ^U_jkV_Rkl^{U †}u_Rk +h.c. = − X

i

d_Lim_did_Ri − X

i

u_Lim_uiu_Ri + h.c.

Phase invariance left:

d_Li → e^iαid_Li, d_Ri → e^−iαid_Ri u_Li → e^iβiu_Li, u_Ri → e^−iβiu_Ri

θ _C in the Standard Model

−v X

ijkl

d_LiV_Lij^D λ^D_jkV_Rkl^D†d_Rk − v X

ijkl

u_LiV_Lij^U λ^U_jkV_Rkl^{U †}u_Rk +h.c. = − X

i

d_Lim_did_Ri − X

i

u_Lim_uiu_Ri + h.c.

The W Interactions: W ^+µ P

i u^W_Liγ_µd^W_Li + h.c. become W ^+µ P

ijkl u_Liγ_µV_Lij^U V_Ljk^D† d_Lk + h.c.

θ _C in the Standard Model

−v X

ijkl

d_LiV_Lij^D λ^D_jkV_Rkl^D†d_Rk − v X

ijkl

u_LiV_Lij^U λ^U_jkV_Rkl^{U †}u_Rk +h.c. = − X

i

d_Lim_did_Ri − X

i

u_Lim_uiu_Ri + h.c.

The W Interactions: W ^+µ P

i u^W_Liγ_µd^W_Li + h.c. become W ^+µ P

ijkl u_Liγ_µV_Lij^U V_Ljk^D† d_Lk + h.c.

V_CKM = V_L^UV_L^D† is a unitary matrix

All other terms in the Lagrangian get V_L^U V_L^{U †} = 1, . . .

θ _C in the Standard Model

The two generation case (V_CKM = V ) Flavours only: ^ _{uL cL} ^ V d_L

s_L

! V is a general 2 by 2 unitary matrix.

θ _C in the Standard Model

The two generation case (V_CKM = V ) Flavours only: ^ _{uL cL} ^ V d_L

s_L

! V is a general 2 by 2 unitary matrix.

Use α_i and β_i to make V₁₁, V₁₂ and V₂₁ real :



e^−iβ1u_L e^−iβ2c_L  V₁₁ V₁₂ V₂₁ V₂₂

! e^iα1d_L e^iα2s_L

!

=

 u_L c_L  e^i(α1−β1)V₁₁ e^i(α2−β1)V₁₂ e^i(α1−β2)V₂₁ e^i(α2−β2)V₂₂

! d_L s_L

!

α₂ − β² = (α₂ − β¹) + (α₁ − β²) − (α¹ − β¹)

θ _C in the Standard Model

The two generation case (V_CKM = V ) Flavours only: ^ _{uL cL} ^ V d_L

s_L

! V is a general 2 by 2 unitary matrix.

Use α_i and β_i to make V₁₁, V₁₂ and V₂₁ real Unitary implies ^P_i V_ik^∗ V_il = δ_il

V = cos θ_C sin θ_C

− sin θ^C cos θ_C

!

This gives exactly what we had before

Kobayashi and Maskawa 1972

Bold suggestion: add a third generation (remember charm not discovered)

Flavours only: ^ _{uL cL tL} ^ V







d_L s_L b_L





 V is a general 3 by 3 unitary matrix.

Kobayashi and Maskawa 1972

Bold suggestion: add a third generation (remember charm not discovered)

Flavours only: ^ _{uL cL tL} ^ V







d_L s_L b_L





 V is a general 3 by 3 unitary matrix.

Use α_i and β_i to make V₁₁, V₁₂, V₁₃, V₂₁ and V₃₁ real Unitary implies ^P_i V_ik^∗ V_il = δ_il

V =







c₁ − s¹c₃ − s¹s₃

s₁c₂ c₁c₂c₃ − s²s₃e^iδ c₁c₂s₃ + s₂c₃e^iδ s₁s₂ c₁s₂c₃ + s₂s₃e^iδ c₁s₂s₃ − c²c₃e^iδ





 c_i = cos θ₁, s_i = sin θ_i

Kobayashi and Maskawa 1972

This extra e^iδ makes the Lagrangian intrinsically complex (i.e. not removable by phase redefinitions) This implies CP violation

can explain CP violation that was seen in kaon decays

via box diagrams ^{K0 u, c, t}

W

W

u, c, t K⁰

Kobayashi and Maskawa 1972

This extra e^iδ makes the Lagrangian intrinsically complex (i.e. not removable by phase redefinitions) This implies CP violation

can explain CP violation that was seen in kaon decays via box diagrams

What has happened since

Third generation discovered

C. Jarlskog: easy way to see from λ^U and λ^D if CP violation is there

Many more predictions have been seen

The predictions: 1

There is another type of CP violation in Kaon decays:

direct CP violation from Penguin diagrams:

K⁰

W

π⁺π⁻, π⁰π⁰ γ, Z, g

The predictions: 1

There is another type of CP violation in Kaon decays:

direct CP violation from Penguin diagrams:

K⁰

W

π⁺π⁻, π⁰π⁰ γ, Z, g

Seen in 1999 by KTeV at Fermilab and NA48 at CERN (earlier at NA31 CERN in 1988 not confirmed by Fermi- lab experiment)

Lund 7/11/2008 The Nobel Prize in Physics 2008: Broken Symmetries Johan Bijnens p.39/44

The origin of penguins

Told by John Ellis:

Mary K. [Gaillard], Dimitri [Nanopoulos], and I first got interested in what are now called penguin diagrams while we were studying CP violation in the Standard Model in 1976. The penguin name came in 1977, as follows.

In the spring of 1977, Mike Chanowitz, Mary K. and I wrote a paper on GUTs [Grand Unified Theories] predicting the b quark mass before it was found. When it was found a few weeks later, Mary K., Dimitri, Serge Rudaz and I immediately started working

on its phenomenology. from symmetry

magazine

That summer, there was a student at CERN, Melissa Franklin, who is now an experimentalist at Harvard. One evening, she, I, and Serge went to a pub, and she and I started a game of darts. We made a bet that if I lost I had to put the word penguin into my next paper. She actually left the darts game before the end, and was replaced by Serge, who beat me.

Nevertheless, I felt obligated to carry out the conditions of the bet.

For some time, it was not clear to me how to get the word into this b quark paper that we were writing at the time. Later, I had a sudden flash that the famous diagrams look like penguins. So we put the name into our paper, and the rest, as they say, is history.

More predictions

In B meson decays you can have all three generations at tree level in a process. CP violations can (and are) much larger

B⁰

b

d

c c s

J/ψ = cc K_S

W

Observed at the predicted level in many processes

More predictions

In B meson decays you can have all three generations at tree level in a process. CP violations can (and are) much larger

B⁰

b

d

c c s

J/ψ = cc K_S

W

Observed at the predicted level in many processes Penguins also contribute and again in many places

has led to great confidence in CKM picture both for the angles and the CP violation part

Results

Parametrization of V_CKM:







V_ud V_us V_ub V_cd V_cs V_cb

V_td V_ts V_tb





 The Unitarity triangle of three complex numbers V_udV_ub^∗ + V_cdV_cb^∗ + V_tdV_tb^∗ = 0 (more exist)

Results

Parametrization of V_CKM:







V_ud V_us V_ub V_cd V_cs V_cb

V_td V_ts V_tb





 The Unitarity triangle of three complex numbers V_udV_ub^∗ + V_cdV_cb^∗ + V_tdV_tb^∗ = 0 (more exist)

|V^us| ≈ 0.2; _{|Vcd| ≈ 0.04} and _{|Vub| ∼ 0.003} Approximate Wolfenstein parametrization:







1 − ^λ₂² λ Aλ³(ρ − iη)

−λ 1 − ^λ₂² Aλ² Aλ³(1 − ρ − iη) −Aλ² 1





 All three side are of order λ³

Results

γ

γ α

α

md

∆ εK

εK

ms

∆ &

md

∆

Vub

β sin 2

(excl. at CL > 0.95) < 0 β sol. w/ cos 2

excluded at CL > 0.95

α γ β

ρ

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

η

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

excluded area has CL > 0.95

ICHEP 08

CK Mf i t t e r

Each num- ber is itself an average of several measure-

ments

Conclusion

I hope I have given you a feeling for what these people have accomplished and what the physics behind is