CKM Matrix Elements and CKM Angles - Theoretische Physik 1 ...

Introduction 

Flavour and CPV in the Standard Model 

The history of the UT since ∼ 1993 

CKM Matrix Elements 

and 

CKM Angles 

Thomas Mannel 

Theoretische Physik I 

Universität Siegen 

Symposium to celebrate the 

60th aniversary of 

our friend and colleague Hans Kühn 

Thomas Mannel, University of Siegen 

CKM Matrix Elements and CKM Angles

Introduction 



Introduction: What is this talk about? 

Many experts on CKM matrix and CP violation are in 

the audience 

I will be rather general and point out a few 

peculiarities 

Connection to Hans: 

I try to show how our knowledge about CKM and CP 

evolved over the time, since when Hans is active in 

Karlsruhe 

Quite a few of his achivements are 

relevant for CKM / CP 



Introduction 





the audience 


peculiarities 




Karlsruhe 





Introduction 





the audience 


peculiarities 




Karlsruhe 





Introduction 





the audience 


peculiarities 




Karlsruhe 





Introduction 





the audience 


peculiarities 




Karlsruhe 





Introduction 



Contents 

1 Introduction 

2 Flavour and CPV in the Standard Model 

Flavour and CP Basics 

Peculiarities of SM CP / Flavour 

Three Words on Hadronic Matrix Elements 

3 The history of the UT since ∼ 1993 



Introduction 



Flavour in the Standard Model 




Unique origin of flavour in the standard model: 

Yukawa Couplings of the Quarks: Y (u) 

ij 

and Y (d) 

ij 

=⇒ Two Mass Matrices 

for the (up- and down-type) Quarks: 

M (q) 

ij 

= vY (q) 

ij 

(q = u, d) 

Mass Eigenstates: Relative Rotation 

between M (u) and M (d) : CKM Matrix 

Appears in the mass-eigenbasis only in the charged 

current interactions 

Neutral currents remain diagonal: Neutral Higgs 

particles couple with the diagonal mass matrices 



Introduction 






CP violation in the Standard Model 

In Lagrangian Field Theory: 

CP Violation ⇔ complex coupling constants a i : 

L = ∑ i 

a i O i + h.c. 

(CP) O i (CP) † = O † i 

In the (three family) standard model: 

(with a single Higgs doublet and massless neutrinos) 

All couplings can be made real (in the mass eigenbasis) 

except the charged current couplings, encoded in 

V CKM ≠ V ∗ CKM 

CP violation through a single complex phase 



Introduction 









L = ∑ i 


(CP) O i (CP) † = O † i 









Introduction 









L = ∑ i 


(CP) O i (CP) † = O † i 









Introduction 









L = ∑ i 


(CP) O i (CP) † = O † i 









Introduction 



CKM Matrix: Basics 




U 12 = 

Three Euler angles θ ij 

2 

4 c 12 s 12 0 

−s 12 c 12 0 

0 0 1 

3 

5 , U 13 = 

Single phase δ: U δ = 

2 

4 c 3 

13 0 s 13 

0 1 0 

−s 13 0 c 13 

PDG CKM Parametrization: 

2 

4 

5 , U 23 = 

1 0 0 

0 1 0 

0 0 e −iδ 13 

3 

5 . 

2 

1 

0 0 c 23 0 

s 23 

3 

5 

4 

V CKM = U 23 U † δ U 13U δ U 12 

Large Phases in V ub = |V ub |e −iγ = s 13 e −iδ 13 and 

V td = |V td |e iβ 



Introduction 







U 12 = 


2 

4 c 12 s 12 0 

−s 12 c 12 0 

0 0 1 

3 

5 , U 13 = 


2 

4 c 3 

13 0 s 13 

0 1 0 

−s 13 0 c 13 


2 

4 

5 , U 23 = 

1 0 0 

0 1 0 

0 0 e −iδ 13 

3 

5 . 

2 

1 

0 0 c 23 0 

s 23 

3 

5 

4 






Introduction 







U 12 = 


2 

4 c 12 s 12 0 

−s 12 c 12 0 

0 0 1 

3 

5 , U 13 = 


2 

4 c 3 

13 0 s 13 

0 1 0 

−s 13 0 c 13 


2 

4 

5 , U 23 = 

1 0 0 

0 1 0 

0 0 e −iδ 13 

3 

5 . 

2 

1 

0 0 c 23 0 

s 23 

3 

5 

4 






Introduction 







U 12 = 


2 

4 c 12 s 12 0 

−s 12 c 12 0 

0 0 1 

3 

5 , U 13 = 


2 

4 c 3 

13 0 s 13 

0 1 0 

−s 13 0 c 13 


2 

4 

5 , U 23 = 

1 0 0 

0 1 0 

0 0 e −iδ 13 

3 

5 . 

2 

1 

0 0 c 23 0 

s 23 

3 

5 

4 






Introduction 






CP Violation in the Standard model is too small 

The Flavour Physics and the CP Violation of the 

Standard model are peculiar 

However, Flavour and CP violation observed in 

particle physics is (still?) fully compatible with the 

Standard model predictions 

CP Violation (in exclusive non-leptonic decays) is 

hard to compute (precisely) 



Introduction 
















Introduction 
















Introduction 
















Introduction 



Unitarity Triangle(s) 




Out of six Unitarity Triangles only two have sides of 

comparable lengths: 

Invariant measure of CP violation 

= Area of the triangles 



Introduction 



Unitarity Triangle(s) 




Out of six Unitarity Triangles only two have sides of 

comparable lengths: 

Invariant measure of CP violation 

= Area of the triangles 



Introduction 






Invariant measure of CP violation: 

Im∆ = ImV ud V ∗ 

tdV tb V ∗ ub = c 12 s 12 c 2 13s 13 s 23 c 23 sin δ 13 

Maximal possible value δ max = 1 

6 √ 3 ∼ 0.1 

CP Violation is a small effect: 

Measured value δ exp ∼ 0.0001 

CP Violation vanishes in case of degeneracies: (Jarlskog) 

J = Det([M u , M d ]) 

= 2iIm∆(m u − m c )(m u − m t )(m c − m t ) 

×(m d − m s )(m d − m b )(m s − m b ) 



Introduction 










6 √ 3 ∼ 0.1 




J = Det([M u , M d ]) 


×(m d − m s )(m d − m b )(m s − m b ) 



Introduction 










6 √ 3 ∼ 0.1 




J = Det([M u , M d ]) 


×(m d − m s )(m d − m b )(m s − m b ) 



Introduction 










6 √ 3 ∼ 0.1 




J = Det([M u , M d ]) 


×(m d − m s )(m d − m b )(m s − m b ) 



Introduction 






Peculiarities of SM Flavour Mixing 

Hierarchical structure of the CKM matrix 

Quark Mass spectrum ist widely spread 

m u ∼ 10 MeV to m t ∼ 170 GeV 

PMNS Matrix for lepton flavour mixing is not 

hierarchical 

Only the charged lepton masses are hierarchical 

m e ∼ 0.5 MeV to m τ ∼ 1772 MeV 

Up-type leptons ∼ Neutrinos have very small masses 

(?) 

(Enormous) Suppression of Flavour Changing 

Neutral Currents: 

b → s, c → u, τ → µ, µ → e, ν 2 → ν 1 



Introduction 






Peculiarities of SM CP Violation 

Strong CP remains mysterious 

Flavour diagonal CP Violation is well hidden: 

e.g electric dipole moment of the neutron: 

At least three loops (Shabalin) 

d e ∼ e α s GF 

2 mt 

2 Im∆ µ 3 

π (16π 2 ) 2 MW 

2 

∼ 10 −32 e cm with µ ∼ 0.3 GeV 

d exp ≤ 3.0 × 10 −26 e cm 



Introduction 



Semileptonic Decays 




Fairly good control over s.l. decays from the Heavy 

Quark Expansion 

Originally (∼ 1991 ± 2): Best determination of V cb 

(and also V ub ) from exclusive decays: 

Heavy Quark Symmetries 

Later (from about 1995): 

V cb and V ub from inclusive decays 

Requires precise heavy quark masses! 

Recently (∼ 2005): Precise, unquenched lattice data: 

Renaissance of V cb and V ub from exclusive decays ? 



Introduction 






Methods for Hadronic Matrix Elements 

Typical problem: 

Evaluation of a matrix element of the type 

M = 〈f |O|i〉, 

O = Four Quark Operator 

No First Principles Method for exclusive decays 

No precision calculations possible 

Approximate methods 

Flavour Symmetry: Isospin, SU(3) 

QCD Factorization / perturbative QCD 

QCD (Light Cone) Sum rules 



Introduction 









M = 〈f |O|i〉, 










Introduction 









M = 〈f |O|i〉, 










Introduction 









M = 〈f |O|i〉, 










Introduction 









M = 〈f |O|i〉, 










Introduction 









M = 〈f |O|i〉, 










Introduction 









M = 〈f |O|i〉, 










Introduction 



Flavour Symmetries 




I-Spin (u ↔ d), U-Spin (u ↔ d), V-Spin (s ↔ d), 

Full SU(3) 

Group theory supplemented by the study of diagram 

topolgies: 

Color Suppression: ∼ 1/N c 

Annihilation: ∼ f B /M B 

Penguin contractions (of tree operators): 

1 

16π 2 ... 1 

→ needed to identify the contributions carrying weak 

phases 

Uncertainties O(20%), hard to estimate, hard to 

improve 



Introduction 








Full SU(3) 


topolgies: 




1 

16π 2 ... 1 


phases 


improve 



Introduction 








Full SU(3) 


topolgies: 




1 

16π 2 ... 1 


phases 


improve 



Introduction 








Full SU(3) 


topolgies: 




1 

16π 2 ... 1 


phases 


improve 



Introduction 






QCD Factorization, PQCD and SCET 

(Beneke, Buchalla, Neubert, Sachrajda, Bauer, Stewart Pirjol, Feldmann, Li, Keum, ...) 

Starts from the limit m b → ∞ 

Perturbative calculation, systematic expansion (?) 

Strong phases are small (O(α s (m b )) or O(Λ/m b )) 

Sizable uncertainties, subleading terms ? 



Introduction 














Introduction 














Introduction 














Introduction 



QCD (Light Cone) Sum Rules 

(Khodjamirian, Ball, Zwicky, Melcher, ...) 




Relies on Duality and Analyticity 

Estimate of the matrix element based on perturbative 

calculations, → QCD Factorization 

Intrinsic uncertainties in the range (10 - 20)% , hard 

to improve 



Introduction 












to improve 



Introduction 












to improve 



Introduction 






η 

2 

1 

0 

-1 

sin γ 

sin 2β 

sin γ 

sin 2β 

K + →π + νν 

∆m d 

|ε , /ε K 

| , K 0 →π 0 νν 

|ε K 

| 

K 0 →π 0 νν 

|V ub 

/V cb 

| 

sin 2α 

sin 2α 

K + →π + νν 

|ε K 

| 

-2 

-2 -1 0 1 2 

ρ 

Many different 

measurements 

necessary 

A lot ot 

theoretical effort 

over ∼ 15 years 

HQE: Precise 

Quark Masses 

needed! 

Kaon Physics is 

an essential 

input 



Introduction 



Pre-historic Unitarity Triangle 



Introduction 




Situation in 1993: 

HQET was still young (∼ 3 years) 

Hadronic Matrix elements for 

∆m d ∼ fB 2 were quite uncertain 

V ub /V cb was known at the level of ∼ 20% 

The top quark mass was still m t ∼ (140 ± 40) GeV 

No CP violation has been observed except ɛ K 

The UT still could have been “flat” 



Introduction 



Unitarity triangle 1993: f B = 135 ± 25 MeV 



Introduction 



Unitarity triangle 1993: f B = 200 ± 30 MeV 



Introduction 



2001: First observation of “Non-Kaon CPV” 

1 

∆m d 

∆m s /∆m d 

|ε K | 

∆m d 

η 

0 

sin 2β WA 

1 

|V ub /V cb | 

0.8 

∆m d 

∆m s and ∆m d 

|ε K | 

0.6 

-1 

η 

0.4 

|ε K | 

0.2 

|V ub /V cb | 

sin 2β WA 

-1 0 1 2 

ρ 

0 

-1 -0.5 0 0.5 1 

ρ 



Introduction 



Unitarity Triangle 2001 

1 

∆m d 

∆m s /∆m d 

|ε K | 

∆m d 

η 

0 

sin 2β BABAR 

|V ub /V cb | 

|ε K | 

-1 

-1 0 1 2 

ρ 



Introduction 




1.5 

1 

∆m d 

η 

0.5 

0 

ε K 

|V ub /V cb | 

γ 

α 

β 

∆m s & ∆m d 

sin 2β WA 

Some improvement of 

V ub /V cb through the Heavy 

Quark Expansion 

-0.5 

ε K 

More data on 

A CP (B → J/ΨK s ) 

-1 

CKM 

f i t t e r 

FPCP 2003 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 



Introduction 




1.5 

excluded area has < 0.05 CL 

1 

∆m d 

η 

0.5 

0 

-0.5 

-1 

ε K 

|V ub /V cb | 


f i t t e r 

LP 2003 

γ 

α 

∆m s & ∆m d 

sin 2β(WA) 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

ε K 

Slight improvement of f 2 B B B 

from lattice calculations 

Still more data on 


Central value of V ub /V cb 

slightly moved 



Introduction 




1.5 

excluded area has CL < 0.05 

1 

sin 2β 

∆m d 

η 

0.5 

0 

-0.5 

-1 

ε K 

|V ub /V cb | 

α 


f i t t e r 

ICHEP 2004 

γ 

α 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

α 

∆m s & ∆m d 

ε K 

More improvement of f 2 B B B 

from lattice calculations 



First constraints on the 

angle α from B → ρρ 



Introduction 




1.5 

excluded area has CL > 0.95 

excluded at CL > 0.95 

1 

sin 2β 

∆m d 

η 

0.5 

0 

-0.5 

-1 

ε K 

|V ub /V cb | 


f i t t e r 

EPS 2005 

γ 

α 

sol. w/ cos 2β < 0 

(excl. at CL > 0.95) 

∆m s & ∆m d 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

ε K 



Exclusion of the “wrong 

branch” of β 

Dramatic Improvement of 

V ub from the HQE 



Introduction 




1.5 

1 

excluded area has CL > 0.95 

γ 

sin2β 

excluded at CL > 0.95 

∆m d 

& ∆m d 

∆m s 

η 

0.5 

0 

-0.5 

-1 

ε K 

V ub /V cb 

α 


f i t t e r 

γ 

α 

γ 

(excl. at CL > 0.95) 

BEAUTY 2006 

-1.5 

-1 -0.5 0 0.5 1 1.5 2 

ρ 

β 

α 

sol. w/ cos2β 

ε K 

< 0 

TEVATRON measurement 

of ∆m s 

Tighter constraints on α 

First constraints on γ from 

CPV in B → K π 



Introduction 



Unitarity Triangle 2010 (CKM-Fitter illustration) 

1 

0.8 

∆m s 

and ∆m d 

∆m d 

π + vv 

γ 

η-bar 

0.6 

|V ub 

/V cb 

| 

α 

0.4 

γ 

π 0 vv 

0.2 |ε K 

| 

sin 2β 

0 

-1 -0.5 0 0.5 1 

ρ-bar 



Introduction 



Final Remarks ... 

Dramatic improvement of our knowledge of the 

Flavour Sector over the last fifteen years 

The B Factories (and B Physics at colliders) have 

done a similar job in the Flavour Sector as LEP in the 

Gauge Sector 

The story continues: Belle, BaBar, LHC-b, later 

maybe SuperB or SuperKEKB 

However, We have (up to now) confirmed the CKM 

picture 

We still dont know the “grand view” ... 



Introduction 








Gauge Sector 




picture 




Introduction 








Gauge Sector 




picture 




Introduction 








Gauge Sector 




picture 




Introduction 








Gauge Sector 




picture 




Introduction

CKM Matrix Elements and CKM Angles - Theoretische Physik 1 ...

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