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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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