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Vts

tb s

W

γ

• ⇒

• Semileptonic decays

• W ± decays

• CP in kaon (|εK|) and B meson

(sin2β) decays

Measurement of the CKM matrix elements

• deep inelastic neutrino-nucleon scattering

• and mixing

• Penguin decays

µννµ ee−− →

→

+→+→+→+→

++

−−

++

e

e

ee

ud

e

epn

ee

V

νππ

νµννν

µ

0

10101414

,np,

,BC,NO �

Λ→Ξ

→Σ→Λ

→→

−−

−−−

+−++

etc.,

,,

, 00

e

ee

eLe

us

e

enep

eKeK

V

ννν

νπνπ

eecs eKDeKDV νν +++− →→ 00 ,:

→→

→+−+−

��

�

��

�

νρνπν

00 ,

)(

BB

inclusiveXBV

u

ub

( ) 2

csVscW ∝→+��

→→

→∗

��

�

��

�

ννν

DBDB

inclusiveXBV

c

cb,

)(

cscd VV ,⇒

00dd BB − 00

ss BB −

( ) 222tbtdtBBBB VVxSfBm

ddη∝∆

2

2

2

2

td

ts

BB

BB

B

B

V

V

fm

fm

m

m

dd

ss

d

s =∆∆

( ) 22

tstb VVsb ∝→ γ��

FG

W W+

c sd s( )

νµ µ−

µ+V V( )cd cs

νµ

W

W

tu, c,

tu, c,

b d

bdVtd Vts( )

Bd0 Bs

0( )Bd0

s( )

s

B s0( )

( )

Vqq'

W

q

q' lνl

Determination of the CKM matrix elements

Review of Particle Physics (January 1998)

experimental sources

Present analysis: main differences and new results

|Vud| = 0.9740 ± 0.0010 super-allowed nuclear β decays

|Vud| = 0.9743 ± 0.0008 neutron decay

|Vus| = 0.2196 ± 0.0023 kaon semileptonic decays

|Vus| = 0.2200 ± 0.0025

|Vub| = ( ) 3−10⋅80±33 ..

���� νρνπ +−0+−0 →→ BB ,

(CLEO)

|Vub| = ( ) 3105.06.3 −⋅±

updated CLEO measurement; new inclusive analyses

(�

�νuXB → ) by ALEPH and L3

|Vcb| = ( ) 3−10⋅71±539 ..

���� νν DBDB →→ ∗ ,

(ARGUS, CLEO, LEP) inclusive

��νcXB →

|Vcb| = ( ) 3−10⋅71±539 .. new measurement of exclusive

decays (DELPHI)

|Vub/Vcb| = 0.080 ± 0.020 �

�νuXB → inclusive decays (CLEO II)

|Vub/Vcb| = 0.090 ± 0.008 theoretical error reduced on the

basis of |Vub| and |Vcb| measurements; earlier end-point

results by CLEO and ARGUS taken into account;

new inclusive analysis by DELPHI

|Vcd| = 0.224 ± 0.016 deep inelastic neutrino-nucleon

scattering |Vcd| = 0.225 013.0

011.0+−

|Vcs| = 1.04 ± 0.16 D meson semileptonic decays

|Vcs| = 0.996 ± 0.024 deep inelastic neutrino-nucleon scattering; W ± hadronic decays

|Vtb| = 0.99 ± 0.15 if 3×3 unitarity holds top quark decays

|Vtb| = 0.96 16.012.0

+−

CDF update

00 − dd BB mixing

( ) =∝∆ 222tbtdtBBBB VVxSBfm

ddη( ) 1−0180±4720= ps..

( ) 1016.0473.0 −±=∆ psmdB

( )MeVBf thBBd31±207=

( )( )MeV

MeVBf thexpBBd

28220

1722220

±=

±±=

00 − ss BB mixing

( ) 222∝∆ tbtstBBBB VVxSBfmss

η

..%. LCps 95210> 1−

..%953.14 1 LCpsmsB

−>∆

( )MeVBf thBBs37239 ±=

( )( )MeV

MeVBf thexpBBs

31252

1925252

±=

±±=

electromagnetic Penguin b→sγ (CLEO)

|Vts||Vtb|/|Vcb| = 1.10 ± 0.43

|Vts||Vtb|/|Vcb| = 0.96 ± 0.09 new measurements by CLEO and ALEPH; more accurate theoretical

calculation of the inclusive BR CP violation in the

mixing of neutral kaons ( )

( ) 3−10⋅0180±2792=

⋅=

..

,,, tstdcscdKK VVVVfBε

140±870= ..KB 080±940= ..KB

CP violation in

Sdd KJBB ψ→00 decays 410+440−790=2 .

..sin β (CDF)

= 3072.3 ± 0.9 ± 1.1(δC) = 3072.3 ± 2.0average of nine super-allowed decays, from Z’=5 ( ) to Z’=26 ( )

| Vud| = 0.9740 ± 0.00014exp ± 0.00048th = 0.9740 ± 0.0005Further nucleus-dependent corrections needed ? (Wilkinson, Saito-Thomas,...)

Conservative estimate:

|Vud| from super-allowed nuclear β decays

• O+ → O+ (pure vector in the allowed approximation) u ↔ d transitions• ∆T = 0: exact isospin (p-n) symmetry• remaining nucleons behave as spectators

Z-independent

Electromagnetic and nuclear-structure effects

2222 2ud

VG

K

MG

Kft

FfiV

==

( )( )( )RF

CR

udVG

Kftt

∆+=−+≡

1211 22

δδ�δR(Z’), ∆R radiative corrections

δC(Z’) isospin symmetrybreaking corrections

t�

C10 Co54

= 0.974 ± 0.001nuclearudV

|Vud| from

neutron β decay

� counting of β-decays in continuous or pulsed neutron beams

� accumulation of ultra-cold

neutrons : reduced

systematic effects

� polarized neutrons: measurement of the decay asymmetry

� non-polarized neutrons (A=B=D=0): electron-neutrino correlation

deduced from proton spectrum

� rate of muon capture in hydrogenum( )

( ) ( )( ) ( )RRF fG

K

δ++∆+=

1311

2ln/22

s9.17.886n ±=τ0025.02665.1 ±−== VA ggλ

( )

−=−

τt

NNN t exp00

( )eVE 7103 −⋅<

( )ν

ν

ν

ν

ν

νν EE

DE

BE

AEE

aPe

e

e

e

e

ee

pppppppp

×⋅+⋅+⋅+⋅+∝ σσσσ 1),,(

0≠+−⋅= ↓↑

↓↑

NN

NN

v

cA

( ) λλ

λλ ⇒+

+−= 2311

2A

2

2

311

λλ+

−−=a

µνµ +→+− np= 0.9755 ± 0.0019neutronudV

nτVA gg

2

udV

• pure vector transition

• |Vud| value independent of nuclear structure effects

( ) ( )( ) ( ) πτδ

νππ

RRF

eud

fffG

eKV

+∆+→=

++

112

2ln/

212

02 ��

( ) ( ) 80 10034.0025.1 −++ ⋅±=→ ee νππ�� ( ) s80005.06033.2 −±=πτ

|Vud| from pion β decay

= 0.967 ± 0.016pionudV

|Vud| = 0.9743 ± 0.0008Overall average:

pure vector transitions

|Vus|: kaon semileptonic decays

( ) ( )( )∆++=Γ 110192

2

125

3

22

δπ

IfCmVG

KF us

�� νπ +0+ →K

�� νπ +−0 →LK

�� ν+→ us

( )( ) MeV

MeV

e

e

K

K

15

15

10053.0937.4

10033.0560.2

03

3

−

−

±=Γ

±=Γ +(from PDG’98 fitted mean lives and BRs)

• phase-space integrals computed using and p2-dependent decay data

• isospin symmetry breaking corrections from chiral perturbation theory

Consistent but less precise (and theoretically less accurate) |vus| value from hyperon decay data

|Vus| = 0.2200 ± 0.0025

03�K +

3�K

|Vcd| and |Vcs| from deep inelastic neutrino-nucleon scattering

W W+

c sd s( )

νµ µ−

µ+V V( )cd cs

νµ

dimuon production induced by neutrino-nucleon scattering

• neutrino and anti-neutrino dimuon cross-sections measured by CDHS, CCFR (‘95) and CHARM II

• Bc =

determined from measured neutrino production-fractions (E531) and world-average c-hadron semileptonic branching ratios

• κ = relative size of the strange quark sea: obtained by CCFR (‘98)from total neutrino cross-section measurements

( )XN −+→ µµνσ µ ≈ cB ( )νν21 cc +

2

csV2

cdV κ

( )µµνµµν

+−→

→∑ iic

iic

YXXN

f ��

= 1.04 ± 0.16= 0.224 ± 0.014neutrinocdVneutrinocsV

|Vcs| from W decays

• independent of the

hypothesis of 3×3unitarity, provided

that mt’ and mb’ > mW

• phase-space terms oforder (mq /mW)2 neglected

LEP average:

csV from ( )qcW →�� csV from ( )hadronsW →��

DELPHI 0.94 32.026.0

+− stat ± 0.13syst 0.90 ± 0.17stat ± 0.04syst

ALEPH 1.00 ± 0.11stat ± 0.07syst 0.947 ± 0.031stat ± 0.015syst L3 0.98 ± 0.22stat ± 0.08syst 1.032 ± 0.033stat ± 0.018syst

OPAL 0.91 ± 0.07stat ± 0.11syst 1.015 ± 0.029stat ± 0.015syst

= 0.993 ± 0.025WcsV

1)

2)

( )( )hadronsW

hadronsW

→−→+

+

��

��

1 22222

22

cbcdubusud

cbcd

VVVVV

VV

+++++++

= 2

csV

2

csV

22222

cbcdubusud VVVVV +++++∝ 2

csV( )( )hadronsW

qcW bsdq

→→

+=

+

��

�� ,,

|Vcs| from D semileptonic decays

( ) ( ) ( )�10192

2

15

3

22

+=→Γ + IfmVG

eKD DF

ecs

πν

= 1.04 ± 0.16icsemileptoncsV

Best values:

|Vcd| = 0.225

|Vcs| = 0.996 ± 0.024

0.0130.011

+−

|Vub|: b → u semileptonic decays

Different techniques employed to single out the b → u contribution from the large background of b → c events

1) CLEO (1996-): exclusive measurements ofand decays with an end-point approach

�� νπ +−→0B

�� νρ +−0 →B

exclusiveCLEOubV ( ) 325.032.0 1055.025.3 −+

− ⋅±= thexp

2) measurement of the inclusive branching ratio at LEP (ALEPH and L3, 1998-99): study of the invariant-mass distribution of the hadronic products

Average:

HQT (Bigi et al.) ⇒

��νuXb →

|Vub| = (3.6 ± 0.5) ⋅10-3

( ) ( ) 3108.00.2 −⋅±=→�

�νuXb��

inclusiveLEPubV ( ) 3103.09.03.4 −⋅±±= thexp

weighted average:

|Vcb|: b → c semileptonic decays

1) measurement of the differential decay rate of (and ) decays in the limit of zero-recoil

HQET:

with (unknown) universal function,

The experiments fit as

CLEO-LEP average result:

After correcting for the finiteness of masses ( )

��ν∗→ DB

��νDB →

2) inclusive decays: from the world-average ϒ(4�) and Zmeasurements of (corrected for the contribution) HQT calculations lead to

223

2

)(),,(48 cb

VwmmwfG

dw

dDB

F �π

=Γ

)(w� )()1( 22maxqqw === ��

)(w� ])(ˆ)(ˆ)[()( �+1−+1−−11= 22 wcww ρ��

( )cb

VD

⋅1∗� ( ) 3106.10.35 −⋅±=

1=

= 0.91 ± 0.03)(1∗D�

exclusivecbV ( ) 3103.18.15.38 −⋅±±= thexp

inclusivecbV ( ) 3104.28.08.40 −⋅±±= thexp

|Vcb| = (39.5 ± 1.7) ⋅10-3weighted average:

��νub →( )

��νXb →��

KS WSB ACCMM ISGW CLEO

2.2<p"<2.4 GeV 2.4<p"<2.6 GeV

0.095 ± 0.011 0.114 ± 0.018 0.089 ± 0.011 0.148 ± 0.020

ARGUS 2.3<p"<2.6 GeV 0.110 ± 0.012 0.130 ± 0.015 0.110 ± 0.012 0.200 ± 0.023

CLEO II 2.3<p"<2.4 GeV 2.4<p"<2.6 GeV

0.057 ± 0.006 0.075 ± 0.007 0.078 ± 0.008 0.104 ± 0.010

Weighted average 0.073 ± 0.005 0.088 ± 0.006 0.088 ± 0.006 0.124 ± 0.008

χ2/2 → 10.3 6.7 2.5 8.2 |Vub| + |Vcb| +

|Vub/Vcb| DELPHI 0.093 ± 0.011

χ2/3 → 3.1 2.3 0.8 8.0

|Vub / Vcb| from inclusive b → u semileptonic decays

• ARGUS, CLEO (1990) and CLEO II (1993) lepton end pointmeasurements: strongly model-dependent results (see table)

• DELPHI (1999 – invariant mass and whole lepton spectrum):

cbub VV modelsyststat 009.0018.0011.0100.0 ±±±=

cbub VV / 009.0088.0 ±= (s= scale factor applied)5.2

average withDELPHI result: |Vub /Vcb| = 0.090 ± 0.008

Third row: effective FCNC processes

W

W

tu, c,

tu, c,

b d

bdVtd Vts( )

Bd0 Bs

0( )Bd0

s( )

s

B s0( )

( )

• B meson mixing

)(6

222

2

tBBBBWF xSBfmm

Gdddη

π=

dBm∆ 2

tdtbVV

)(6

222

2

tBBBBWF xSBfmm

Gsssη

π=

sBm∆ 2

tstbVV

( ) 1016.0473.0 −±=∆ psmdB

..%953.14 1 LCpsmsB

−>∆

CLEO, ARGUS, CDF, SLD, LEP

CDF, SLD, LEPamplitude method

Am

plit

ude

∆ ( )m psBs

−1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16 18 20

1.645 σ

data ± 1σ

± 1.645 σdata

95% C.L. limit 14.3sensitivity 14.7

ps−1

ps−1

( )tmeBBPs

sB

s

B

t

Bss ∆±1

21=

−00 cos),( �

τ

τ1)3.14(645.1)3.14( 11 ==∆σ+=∆ −− psmpsm

ss BB ��

main theoretical uncertainties: determination of , , and

dBBsBB

dBf sBf

Lattice QCD ⇒

and quite accurate , , determinations

With from decays

15.030.1 ±=== BBB BBBds

sd DB ffss DB ff

ds BB ff

MeVfsD 25254 ±=

�� ν++ →sD

( ) ( )MeVMeVBf thexpBB dd282201722220 ±=±±=

( ) ( )MeVMeVBf thexpBB ss312521925252 ±=±±=

Vts

tb s

W

γ• Electromagnetic penguin decay b→sγ

Using NLO Standard Model calculations (Chetyrkin et al.)

⇒

( ) ( ) 4−10⋅480±143=→ ..γsXb�� CLEO (1995-) and ALEPH (1998) average

17.093.008.014.093.02

22

±=±±= thexp

cb

tbts

V

VV

( ) 2

cbVcb∝

→�

��

( )γsb →��22

tbts VV

• CP violation

( ) 3−10⋅0180±2792= ..Kε

( )12122

2

224 2+

∆212= MMBf

m

mmGe KK

K

KWFi ReIm ξπ

π

080±940= ..KB

%2≈

from Lattice QCD

),(2)()(12 tccttttccc xxSxSxSM ηηη ++= ( )2∗tdtsVV ∗∗

tdtscdcs VVVV( )2∗cdcsVV

Kεkaons:

B mesons: β2sin ∗∗

∗∗

−cbcstbtd

cbcstbtd

VVVV

VVVVIm=

41.044.079.02sin +

−=β (CDF 1999)

Parametrizations

=

=

tbtstd

cbcscd

ubusud

CKM

VVV

VVV

VVV

V

−−−−−−

−

132313231223121323122312

132313231223121323122312

1313121312

1313

1313

13

ccescsscesccss

csesssccessccs

escscc

ii

ii

i

δδ

δδ

δ

≠≠

≠⇔

πγβαη

πδ

,0,,

0

,013

( )Cijijijij cs ϑϑϑϑ === 12cossin

[ [πδ 2,012 ∈

[ ] 3,2,1,2,0 =∈ jiji πϑ

( )

( )

( )

( )6

42222

223

2242

4242

342

21

21

12

12

1

4182

121

1

821

λ

ληλρλλληρλρλ

λλληλρλλ

ηρλλλλ

�+

−

+

−−−

−−+−

+−−

+

−−−

−−−

=

AiAiA

AAiAA

iA

( )1,, �=ηρA

( )ηρλδ iAes i −=− 313

13

223 λAs =

22.012 ≅≅≅= cdus VVsλ

0=++ ∗∗∗tbtdcbcdubud VVVVVV

13arctan δρηγ ==

( )( ) 221

122sin

ηρρηβ+−

−=

( )( ) 2222

2222sinηρρη

ρρηηα+−+

−+=

−≅

21

2ληη

−≅

21

2λρρ

‘canonical’ (PDG)

Wolfenstein

unitarity triangle

CP

Determination of the 3×3 unitary CKM matrix

χ2 minimization with all the constraints expressed in terms of a unitary parametrization (Wolfenstein, canonical)

24 constraints, 16 parameters, 8 degrees of freedom⇒ well determined problem

� constraint:sBm∆

2

∆σ

1−∆)(

)(

s

s

B

B

m

m

�

�=∆

2

sBmχ

� non-constant parameters (δx/x > 1%) entering into the expressions of , and treated as additional constraints. Example:

sBm∆dBm∆ Kε

( )2

2

2 −=

sD

ss

sD

f

DDf

ff

σχ

with ( )�ηρλ ,,, AfmsB =∆

χ2 input values and expressions

χ2 term value

expression

udV 0.9743 ± 0.0008 8

−2

−142 λλ

usV 0.2200 ± 0.0025 λ

ubV ( ) 3105.06.3 −⋅± 223 +ηρλA

cbub VV 0.090 ± 0.008 22 +ηρλ

cdV 013.0011.0225.0 +

− ( )

2−1

2−1

42 ρλλ A

csV 0.996 ± 0.024

2

+81−

2−1

24

2 Aλλ

cbV ( ) 3−10⋅71±539 .. 2λA

tbV 16.012.096.0 +

− 2

−14

2 λA

2

22

cb

tbts

V

VV 0.93 ± 0.17 ( ) ( )[ ]ρρηλρλ −1+−−2−1−1 2242 A

dBm∆ ( ) 1016.0473.0 −± ps ( ) ( ) ( ){

( ) ( )

−++

−+

+−+−+−

=∆

1241

16

22224

22222622

22

2

22

2

ρηρλ

ρηρληρληπ

AA

Am

mSBf

f

fmm

Gm

W

tBBD

D

BBW

FB s

s

d

dd

sBm∆ ..%953.14 1 LCps−> ( ) ( ) ( )[ ]{ }ρρηλρλληπ

−+−−−−

=∆ 1211

6224242

2

22

2

22

2

AAm

mSBf

f

fmm

Gm

W

tBBD

D

BBW

FB s

s

s

ss

Kε ( ) 3−10⋅050±282 ..

( )( )

( )

81+

2−

25−

2−1

+

+

1−4

81+

2+

21−+

21+2−+−−1

+

+

1+2−34

8−

2−1

−

∆26=

242

2

2

2

2

22

24222422

2

242

2

262

2

222

ρλλη

ρηρλρηρλρλη

ρλληηλπ

ε

Am

m

m

mS

AAm

mS

Am

mSAB

m

fmmG

W

t

W

cct

W

ttt

W

cccK

K

KKWFK

,

sin2β 410+440−790 .

.. ( )

( ) 22 +−1−12

ηρρη

Constant and variable parameters of the fit

variables constants

04.076.0 ±=sd DB ff 0040±5740= ..ttη

04.087.0 ±=ss DB ff

( ) 251000001.016639.1 −−⋅±= GeVGF

( )GeVfsD 025.0254.0 ±=

( )GeVmW 10.041.80 ±=

( ) 15.030.1 ±===sd BBB BBB ( )GeVm

dB 0018.02792.5 ±=

010±550= ..Bη ( )GeVmsB 0020.03692.5 ±=

( )GeVmt 5166 ±= ( ) GeVmK1510009.0489.3 −⋅±=∆

( )GeVmc 10.025.1 ±= ( )GeVmK 000031.0497672.0 ±=

530±381= ..ccη ( )GeVfK 00150±15980= ..

040±470= ..ctη

080±940= ..KB

Experimental constraints on the vertex of the unitarity triangle

ρ

η

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

∆mBs

(95% C.L.)

d∆mB

sin2β

εK| || / |V V ub cb

22, ηρ +∝cbubub VVV 007.0090.0 ±=cbub VV

( ) 221 ηρ +−∝∆dBm

( ) 221

1

ηρ +−∝

∆∆

d

s

B

B

m

m

( ) 1016.0473.0 −±=∆ psmdB

..%953.14 1 LCpsmsB

−>∆( )ρηε ⋅−⋅⋅⋅≅ − 75.01101.7 3

K

( ) 310018.0279.2 −⋅±=Kε( )( ) 221

122sin

ηρρηβ+−

−=)(79.02sin 41.0

44.0 CDF+−=β

ρ

η

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

α

ρ

η

0.1

0.2

0.3

0.4

0.5

0.6

-0.2 0 0.2 0.4 0.6 0.8 10

βγ

Present determination of the unitarity triangle

CKM angles:20.022.011.02sin +

−−=α044.0046.0725.02sin +

−=β ( )�3.50.77.63 +

−=γ

Bs oscillations: 10.37.04.15 −+

−=∆ psmsB

046.0034.0175.0 +

−=ρ

031.0032.0354.0 +

−=η

Results of the χ2 minimization

0.176° ÷ 0.230°

2.12° ÷ 2.43°

12.58° ÷ 13.05°(12.82 ± 0.12)°45.4° ÷ 74.4°

0.632 ÷ 0.809sin2β−0.73 ÷ 0.26sin2α0.275 ÷ 0.415

0.103 ÷ 0.288

0.743 ÷ 0.8680.798 ± 0.029A

0.2179 ÷ 0.2258λ

95% C.L.68% C.L.

ρη

13δγ =

12ϑ

23ϑ

13ϑ

ijV( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

+−

+−

+−

+−

+−

+−

4851

1312

3233

1213

2524

4546

999223.00386.000782.0

0393.04697432.0202217.0

00353.0202218.097508.0

÷÷÷÷÷÷÷÷÷

999316.0999118.00418.00363.000846.000701.0

0424.00369.097522.097341.02257.02178.0

00402.000309.02258.02179.097597.097417.0

0020.00021.02219.0 +

−

046.0034.0175.0 +

−031.0032.0354.0 +

−20.022.011.0 +

−−044.0046.0725.0 +

−

( )�3.50.77.63 +

−

( )�074.0071.0250.2 +

−

( )�014.0013.0202.0 +

−

Bayesian primer Probability: a measure of the degree of belief that an event will occur

subjective ⇒ objective probability does not exist Bayes theorem

( ) ( ) ( )ABPAPBAP |⋅=⋅ Our use:

( )AP → initial probability → initial knowledge ( )ABP | → composed probability → experimental results ( )BAP ⋅ → joint probability → final knowledge

Bayes estimators pros:

- force to state all initial assumptions - better internal consistency with respect to other

methods (χ2) - final results are probability distributions (and not

point estimators) cons:

- computationally heavy

Bayes method for CKM estimations Main formula

( ) ( ) ( )( ) ( ) xdixPixe

ixPixeeixP

∫=

||,

||,|,

�

�

with x free parameters of the problem at hand

4 CKM angles + 3 factors Φ parametrizing mainly theoretical uncertainties in Kε ,

dBm∆ and sBm∆

( )ixP | our initial knowledge on the x parameters ( )ixe |,� likelihood ( )eixP |, our final knowledge: it is a probability

distribution of the x parameters

( ) ( ) ( )iPiPixP ||| Φ= ϑ

gaussian uniform in the allowed range [ ] [ ]ππ 2,02,0 3 × � = product of gaussians

{ }βε 2sin,,,,,,,,,,, KBBtbcscdcbcbububusud sdmmVVVVVVVVVe ∆∆=

Probability distribution functions (Bayesian method)

degrees

δ = γ13sin2α

sin2β∆mBs

ps−1

V =CKM| |

d s b

u

c

t

Summary of the constraints on the CKM matrix

udV

udV

ubV

cbub VV

cdV

csV

cbV

2

22

cb

tbts

V

VV

dBm∆

sBm∆

Kε

β2sin

CP violation measurements

⇒ complex matrix

reasonably unaffected by new physics

underlying hypothesis: no contribution to box diagrams

from 4th up-type quark or supersymmetric particles

⇒ 3×3 matrix

3×3 unitarity required explicitlytbV

The vertex of the unitarity after the removal of critical constraints

ρ

η all measurements no CP measurements

ρ

η

ρ

ηonly measurementsVij| |

CKM angles without CP and B-oscillation constraints

p.d.

f.

sin2β

p.d.

f.

sin2α

p.d.

f.

δ = γ13 (degrees)

only V measurements| |ij

no CP measurementsall measurements

HALVED

UNCERTAINTY ↓

( )ρσ ( )ησ ( )βσ 2sin ( )ασ 2sin

(“standard” fit) 0.038 0.032 0.045 0.22

( )udVσ 0.038 0.031 0.045 0.22

( )usVσ 0.038 0.031 0.045 0.21

( )ubVσ 0.036 0.029 0.037 0.22

( )cbub VVσ 0.037 0.024 0.026 0.21

( )cdVσ 0.038 0.032 0.045 0.22

( )csVσ 0.038 0.032 0.045 0.22

( )cbVσ 0.038 0.029 0.043 0.20

( )tbVσ 0.038 0.032 0.045 0.22

( )2

cbtbts VVVσ 0.038 0.032 0.045 0.22

( )dBm∆σ 0.037 0.031 0.045 0.21

( )Kεσ 0.038 0.032 0.045 0.22

( )sDfσ 0.038 0.032 0.045 0.22

( )BBσ 0.038 0.032 0.045 0.22

( )Bησ 0.038 0.032 0.045 0.22

( )tmσ 0.038 0.032 0.045 0.22

( )cmσ 0.038 0.032 0.045 0.22

( )ccησ 0.038 0.032 0.045 0.22

( )ctησ 0.038 0.032 0.045 0.22

( )KBσ 0.038 0.031 0.045 0.21

( )βσ 2sin 0.038 0.031 0.044 0.22

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