The CKM Angles a and b
Introduction
Theory overview
Experiments at B-factories
a/f2
g/f3
Richard Kass
Measuring b/f1
Measuring a/f2
Summary
b/f1
1
The Cabibbo-Kobayashi-Maskawa Matrix
• The weak interaction can change
the favor of quarks and lepton
• Quarks couple across generation
boundaries
Vcb
Vub
d
• Mass eigenstates are not the
weak eigenstates
• The CKM Matrix rotates the
quarks from one basis to the
other
Richard Kass
d’
s’
b’
u
s
b
Vud Vus
Vubl d
l
3
= c Vcdl Vcs Vcbl
2
t
Vtdl Vltd Vtb
3
2
l=sin(qc)=0.22
s
b
2
Visualizing CKM information from Bd decays
The Unitarity Triangle
The CKM matrix Vij is unitary with 4
independent fundamental parameters
Unitarity constraint from 1st and 3rd columns:
i V*i3Vi1=0
d
s
b
u
Vud
Vus
Vub
c
Vcd
Vcs
Vcb
t
Vtd
Vts
Vtb
CKM phases
(in Wolfenstein convention)
 1 1 e-iγ 


 1 1 1 
 e-iβ 1 1 


Testing the Standard Model
Measure angles, sides in as many ways possible
Area of triangle proportional to amount of CP violation
Richard Kass
3
Three Types of CP Violation
I) Indirect CP violation/CP violation in mixing
KKlnexpected to be small (SM: 10-3) for B0’s
II) Direct CP violation: Prob(Bf) Prob(Bf)
Only CP violation possible for
/ in K
charged B’s
Br(B0-+) Br(B0+-)
III) Interference of mixing & decay: Prob(B(t)fCP) Prob(B(t)fCP)
B0s
B0+-
(CKM angle b)
(CKM angle a)
B
B
0
0
f CP
Due to quantum numbers of
Y(4S) and B meson we must
measure time dependant
quantities to see this CP violation
In this talk we will be discussing type III) CP violation
Richard Kass
4
CP Violation at the Y(4S)
CP violation from the interference between two paths, decay with and without mixing
AfC P
mixing
|BL>=p|B0>+q|B0>
|BH>=p|B0>- q|B0>
B0
q/p
B
t
fCP
AfCP
0
Measure time dependent decay rates
&
m from B0B0 mixing
t 0
ACP (t ) 
 ( B 0 (t )  f ) - ( B 0 (t )  f )
 ( B 0 (t )  f ) + ( B 0 (t )  f )
 S f sin (mt ) - C f cos (m t )
Cf 
Sf 
Richard Kass
1- | l f |
2
1+ | l f |2
- 2 Im l f
1+ | l f |2
q Af
lf  
p Af
Direct CP Violation: C
|Af/Af|≠1→ direct CP violation
|q/p|≠1→ CP violation in mixing
Sf and Cf depend on CKM angles
5
Getting the Data Sample
Use e+e- annihilations at Y(4S) to get a clean sample of B mesons
At Y(4S) produce B-/B+ (bu/bu) and B0B0 (bd/bd) mesons
BB Threshold
mB0 ~ mB- ~ 5.28 GeV
 
 bb
 0.28
 hadr 
The Y(4S) - a copious, clean source of B meson pairs
1 of every 4 hadronic events is a BB pair
No other particles produced in Y(4S) decay
Equal amounts of matter and anti-matter
Richard Kass
6
B Factories
To get the large data set necessary to measure CP-violation with B’s use B-factories
SLAC and KEK
Both factories have attained unprecedented high luminosities: >1034/cm/s2
BaBar has 352 fb-1 and Belle has 610fb-1 of data
Note: 1fb-1 ~ 1.1 million BB pairs
Richard Kass
7
Asymmetric e+e- Colliders
KEKII
PEPII
KEK/SLAC are asymmetric e+e− colliders
KEK: 8 GeV (e-)/3.5 GeV (e+)
SLAC: 9 GeV (e-)/3.1 GeV (e+)
B travels a measurable distance before decay:
SLAC: bg=0.56 → bgct~260mm
KEK: bg=0.42 → bgct~193mm
Richard Kass
8
Detectors at Asymmetric e+e- Colliders
Both detectors feature:
Charged particle tracking (silicon+drift chambers + 1.5T B-field)
Electromagnetic calorimetry (CsI)  g and electron ID
/K/p separation up to the kinematic limit
BABAR: dE/dx+DIRC
Belle: dE/dx+aerogel+ToF
Muon/KL identification
Richard Kass
9
Key Analysis Techniques
Threshold kinematics: we know the initial energy of the Y(4S) system
Therefore we know the energy and magnitude of momentum of each B
*2
mES  Ebeam
- pB*2
Signal
*
E  EB* - Ebeam
Event topology
Signal
(spherical)
Background
Background
(jet-structure)
Most analyses use an unbinned maximum likelihood fit to extract parameters of interest
Richard Kass
10
How to Measure Time Dependent Decay Rates
t =0
We need to know the flavour of the B at a reference t=0.
z = t gbc
0
At t=0 we
B0
know this
meson is B0
B
rec
K s
(4S)
bg =0.56
B0
The two mesons oscillate
coherently : at any given
time, if one is a B0 the
other is necessarily a B0
Richard Kass
tag
W
l - (e-, m-)
In this example, the tagside meson decays first.
It decays semi-leptonically
and the charge of the
lepton gives the flavour of
the tag-side meson :
l -= B0
l += B 0.
Kaon tags also used.
B0
-
l-
nl
b
d
t picoseconds
later, the B 0 (or
perhaps it is
now a B 0)
decays.
11
The Many Ways to Measure sin2b
Can use 3 different categories of B0 decays to measure b:
b) b  cc d charm
(and charmonium )
a) b  cc s
(charmoniu m)
J /K S0
golden mode
 (2S ) K S0 ,  c1 K S0 , c K S0
J /K ( K
Richard Kass
-
*0
K  )
0
S
0
+
D D ,D D
*+
fK 0 , K + K - K S0 ,
-
J / , D D
0
J /K L0
*0
*+
c) Penguin - dominated
b  dd s, b  ss s
*-
K S0 K S0 K S0 , K 0 , K S0 0 ,
K S0 , f 0 (980) K S0
12
Precise Measurement of sin2b from B0charmonium K0
Theoretically very clean:
ACP(t)=Sfsin(mt)-Cfcos(mt)
The dominant penguin amplitude (suppressed by l2Cab) has same phase as tree
SM prediction: Cf=0 ACP(t)=Sfsin(mt)
confirmed by recent model-independent analyses [e.g. PRL 95 221804 (2005)]
S=0.000±0.012
Experimentally very clean:
Many accessible decay modes
with (relatively) large BFs
CP odd
CP even
B→ψK0~8.5x10-4
B→ψ(2S)K0~6.2x10-4
B→χc1K0~4x10-4
B→ηcK0~1.2x10-3
Richard Kass
13
Precise Measurement of sin2b from B0charmonium K0
hep-ex/0507037
386x106 BB
sin2b=0.652±0.039±0.020
(was 0.728±0.056±0.023, Nov. 2004
PR D71, 072003, 152x106 BB)
PRL94, 161803 (2005)
227x106 BB
sin2b=0.722±0.040±0.023
(cc) KS (CP odd) modes
Richard Kass
J/ KL (CP even) mode
14
Brief history of sin2b from B0charmonium K0
1 CKM fit
2
World Average
sin2b[WA]=0.687±0.032
From external constraints
sin2bUTFit= 0.793±0.033 (sides)
sin2bUTFit=0.734±0.024 (all)
Great success for Standard Model
Great success for all of us
theorists, experimentalists, accelerator physicists
Richard Kass
15
Resolving the sin(2b) Ambiguity
sin(2b) is the same for b/2-b+b3/2-b
Belle: Use bcud [B 0DCP(Ks+-) h0] decays
[A.Bondar, T.Gershon, P.Krokovny, PL B624 1 (2005)]
h0=,,
Theoretically clean (no penguins), Neglect DCS B0DCPh0 decay
Interference of Dalitz amplitudes sensitive to cos2b
| f  |  | f (mK2   , mK2   ) |2
S
S
Dalitz model fitted in D*-tagged D0 decays
f116±21±12o rules out f1=68o @ 97% CL
[Belle. hep-ex/0507065]
BABAR: B0J/K*0(K*0Ks0)
Extract cos2b from interference of CP-even and CP-odd
states (L=0,1,2) in time-dependent transversity analysis
cos2b<0 excluded at 86% C.L.
[BABAR, PRD 71, 032005 (2005)]
Richard Kass
16
b  cc d decays : B  J /
0
0
These decays suffer from potential penguin-pollution:
bd penguin amplitude
has different weak & strong
phases with respect to tree.
S  sin 2b , C  0
BABAR: B0 J/0 updated measurements [hep-ex/0603012, submitted PRD-RC]:
Br(B0J/0)=(1.94±0.22±0.17)x10-5
SJ/0=-0.68±0.30±0.04
CJ/0=-0.21±0.26±0.09
Consistent with previous Belle results:
PRL93, 261801 (2004)
SJ/0=-0.72±0.42±0.09
CJ/0=-0.01±0.29±0.03
Richard Kass
17
b  cc d decays : B  D
0
(*)+
D
(*)-
D*+D*-: [PRL 95, 151804 (2005)]
VV decay: both CP-odd and CP-even components.
CP-odd fraction extracted with transversity analysis:
fodd=0.125±0.044±0.070
S+=-0.75±0.25±0.03
C+=+0.06±0.17±0.03
D(*)+D- [PRL 95, 131802 (2005)]:
SDD =-0.29±0.63±0.06
CDD =+0.11±0.35±0.06
SD*+D-=-0.54±0.35±0.07
CD*+D-=+0.09±0.25±0.06
SD*-D+=-0.29±0.33±0.07
CD*-D+=+0.17±0.24±0.04
Richard Kass
D*+D-
D*-D+
D+D-
18
b  cc d decays : summary
All results consistent with SM expectation of tree dominance
SDD≡SDD-sin2b~.2-.5[Z-Z. Xing, PR D61 014010 (2000)]
Still below current experimental sensitivity
Richard Kass
19
Sin2beff in b → s Penguins
Decays dominated by gluonic penguin diagrams
Golden example: B0→fKS
No tree level contributions: theoretically clean
SM predicts: ACP(t) = sin2bsin(mt)
NP
SM
d
d
Impact of New Physics could be significant
New particles could participate in the loop → new CPV phases
Measure ACP in as many b→sqq penguins as possible!
φK0, K+ K− KS, η′ KS, KS π0, KS KS KS, ω KS, f0(980) KS
BUT there are complications:
Richard Kass
b
u
Low branching fractions (10-5)
B
non-penguin processes can pollute d
u
s
d
20
All “sin2b” Results Compared
Will be discussed in O. Long’s talk “Rare decays and new physics studies”
Richard Kass
21
The Unitarity Triangle
(r,)
Vub* Vud
Vcd Vcb*
(0,0)
Richard Kass
a
g
Vtd Vtb*
Vcd Vcb*
o
(0,1)
[21.7 ± 1.3]
22
The CKM angle a
In an ideal world we could access a from the interference of
a b→u decay (g) with B0B0 mixing (b):
Tree decay
B0B0 mixing
b
B
0
d
Vtb*
Vtd*
t
t
Vud*
g
d
B
b
Vtb
Vtd
q / p  Vtb*Vtd / VtbVtd*
0
B
0 b
d
Vub
d
u
u
d
-
+
A  Vud* Vub
q A
l
 e -i 2 b e -i 2g  ei 2a
p A
But we do not live in the ideal world.
There are penguins...
Richard Kass
a- b- g
B0→K+- large
Br~2x10-5
~Pure Penguin
23
sin(2a): Overcoming Penguin Pollution
Access to a from the interference of a b→u decay (g) with B0B0 mixing (b)
complicated by Penguin diagram
Tree decay
B0B0 mixing
b
B
0
d
Vtb*
Vtd*
t
t
Vud*
g
d
B
0
B
b
Vtb
Vtd
q / p  Vtb*Vtd / VtbVtd*
Penguin decay
Vub
0 b
d
d
u
u
d
-

+

B
b
0 u,c,t
d
lCP
lCP  e
i 2a
Inc. penguin contribution
S  sin( 2a )
C 0
Richard Kass
+
T + P e + ig ei
T + P e -ig ei
S  1 - C 2 sin( 2a eff )
C  sin 
T = "tree" amplitude P = "penguin" amplitude =strong phase
Time-dep. asymmetry :
-
A  Vtd*Vtb
A  Vud* Vub
q A

 e -i 2 b e -i 2g  ei 2a
p A
g
d
u
u
d
A(t )  S sin( md t ) - C cos(md t )
How can we
obtain α
from αeff ?
24
How to estimate |a-aeff|: Isospin analysis
Use SU(2) to relate decay rates of different  final states
+-+
Important point is that  can have I=0 or 2 but gluonic penguins only
contribute to I=0 (by I=1/2 rule) &EW penguins are negligible
Need to measure several B.F.s:
a2|a -a|
eff
B 0   + - B 0   + B 0   0 0 B 0   0 0
B -   - 0 B +   + 0
1
2
+-
AB->
BF(B++)=BF(B--) since
+ is pure I=2, only tree amplitude
1
2
~
+AB->
Richard Kass
~

AB->
f
However, for this technique to work
 amplitudes must be very small
or very large!
~

AB->
-
- 
++
AB-> AB->
Gronau-London: PRL65, 3381 (1990)
25
B0→+275×106
227×106 B pairs
B pairs
666±43 signal events
467±33 signal events
B0
B0
Phys.Rev.Lett. 95 (2005) 101801
Richard Kass
Phys.Rev.Lett. 95 (2005) 151803
26
B0→+B0
B0
S = −0.67±0.16±0.06
B0
B0
S = −0.30±0.17±0.03
C = −0.56±0.12±0.06
C = −0.09±0.15±0.04
So two comparable measurements of S = sin(2aeff)
Measurements of direct CP asymmetry less compatible
Richard Kass
27
B0→
B0
B0
ACP  -0.44 +-00..53
52  0.17
ACP 
Richard Kass
227 x 106 BB pairs
PRL 94, 181802 (2005)
PRL 94, 181803(2005)
275 x 106 BB pairs
ACP  -0.12  0.56  0.06
| AB 0  0 0 |2 - | AB 0  0 0 |2
| AB 0  0 0 |2 + | AB 0  0 0 |2
28
Using isospin in  system

++
a - a eff  35 at 90% CL
PRL 94, 181802 (2005)
Precision measurement of a not possible with current stats using 
Richard Kass
29
B → rrto the Rescue
Pseudoscalar→ Vector Vector
3 possible ang. mom. states:
S wave (L=0, CP even)
P wave (L=1, CP odd)
D wave (L=2, CP even)
d 2N
 f L cos 2 q1 cos 2 q 2 + 14 (1 - f L ) sin 2 q1 sin 2 q 2
d cosq1d cosq 2
Nature is KIND!
PRL 93 (2004) 231801
B0r+r-~100% longitudinally polarized!
Transverse component taken as 0 in analysis,
essentially all CP even
Large Branching Fraction!
Br(B0r+r-)=(30±4±5)x10-6
Br(B0r+r-)~6xBr(B0+-)
Richard Kass
r helicity angle
signal
bkg
30
B0 → r + r BaBar (227 x 106 BB)
PRL 95, 041805 (2005)
Richard Kass
Belle (275 x 106
PRL 96, 171801 (2006)
BB)
08
S rr  -0.33  0.24 +- 00..14
S rr  -0.08  0.41  0.09
C rr  -0.03  0.18  0.09
Crr  0.00  0.30  0.09
31
But How large is B0→rr ?
Phys.Rev.Lett. 94 (2005) 131801
In rr system the amount of neutral
decays is small
22
0
 
33

Measure:
20 12 B →r r
events in 227 x 106 BB events
Br < 1.1 x 10-6 at 90% CL.
Isospin triangle for rr is flattened compared to 
k2|aeff -a|
1
2
1
2
+-
AB->rr
~
-
- 
~
+AB->rr

Penguin Pollution Defeated!
AB->rr
~

AB->rr
++
AB->rr AB->rr
Richard Kass
32
B±→r±r
New result with 231 x 106 BB events
Br  (17.2  2.5  2.8)  10- 6
Moriond QCD 2006
f L  0.96  0.04  0.05
ACP  0.10  0.14  0.09
Previous BaBar/Belle HFAG average


Br  26.4+-66..14 10-6
hep−ex/0603003
New result is better match to isospin model
Smaller uncertainty on |aeff−a|
compared to  mode
a - aeff  11 @ 68% C.L.
Richard Kass
33
B0 → r Analysis
B0 → r→+  is not a CP eigenstate
– 6 decays to disentangle: B 0 B 0  r   , r 0 0
– Tried by BaBar and Belle for just r± phase space
– Did not set limits on a
– Can use a Dalitz plot analysis to get a from decays
 
Snyder & Quinn: Phys. Rev. D48, 2139 (1993)
r+-
MC
Convert to a square Dalitz plot
Mostly resonant decays
Move signal away from edges
Simplifies analysis
q 
m 
r-+
Richard Kass
q0

1

q0=r helicty angle
cos -1 (2
m0 - 2m +
mB0 - m 0 - 2m +
- 1)
m0=invariant mass of charged tracks
34
B0 → (r)0 Dalitz plot analysis
Dalitz plot analysis yields CP asymmetries and strong phases of decays
Using 213 x 106 BB events hep-ex/0408099
1184±58 B→+-
Signal events
S  -0.10  0.14  0.04
C  0.34  0.11  0.05
Blue histos are
different types of
backgrounds
m’ and q’ are variables
for a square Daltiz plot
Analysis provides a weak
determination of a:
27
a  (113  +-17
6)
However, useful for
resolving ambiguities…..
Richard Kass
35
Combined constraints on a
rr gives single best measurement
World Average:
r resolves 2-fold ambiguity from rr
a WA  99+-12
9 (HFAG)
Global CKM Fit (w/o a):
a  97
Richard Kass
+5
-16
ms
36
The Unitarity Triangle
[99(r,)
± 11]o
Vub* Vud
Vcd Vcb*
(0,0)
Richard Kass
g
Vtd Vtb*
Vcd Vcb*
b
[21.7 ±(0,1)
1.3]o
37
Summary and Outlook
BABAR & Belle measure sin2b in ccK0 modes to 5% precision
sin2bcharmonium=0.687±0.032
Comparison with sin2beff in b s penguin modes could reveal new physics effects
BUT need to carefully evaluate SM contributions
sin2beff measurements are statistically limited, can add new modes, beat 1/√L scaling
Extraction of a depends crucially on penguin contributions
B→rr/r+r
Theory  experimental feedback is helpful
Expected precision Vs time
sin2b in
penguins
reference +1
reference (current r0r0 Br)
reference -1
a from rr
Richard Kass
Luminosity (ab-1)
38
Putting it all together
As of today the complex phase in the CKM matrix correctly
describes CP Violation in the B meson system!

r
More to come from BABAR/Belle, CDF/D0, and LHCb
Will they find CKM violation????
Richard Kass
39
Extra Slides
Richard Kass
40
Adding Theoretical Uncertainties
•
size of possible discrepancies
Δsin2β have been evaluated for
some modes:
– estimates of deviations based on
QCD-motivated specific models;
some have difficulties to reconcile
with measured B.R.
•
•
•
•
•
Beneke at al, NPB675
Ciuchini at al, hep-ph/0407073
Cheng et al, hep-ph/0502235
Buras et al, NPB697
Charles et al, hep-ph/0406184
2xΔsin2β
– model independent upper limits
based on SU(3) flavor symmetry
and measured b d,sqq B.R.
• [Grossman et al, PRD58;
Grossman et al, PRD68; Gronau,
Rosner, PLB564; Gronau et al,
PLB579; Gronau et al, PLB596;
Chiang et al, PRD70]
‘naive’ upper limit based on final state quark content,
CKM (λ2) and loop/tree (= 0.2-0.3) suppression factors
[Kirkby,Nir, PLB592; Hoecker, hep-ex/0410069]
Richard Kass
41
There is a problem


B0  +K
K
K
B0  K+-
q
q

B0+-
157  19
(4.7  0.6  0.2) x 10-6
B0K+-
589  30
(17.90.9 0.7) x
Richard Kass
10-6
Penguin/Tree ~ 30%
42