Measuring the angle /3 at Belle
Sanjay Kumar Swain
University of Hawaii
• Accelerator and Detector(& recent upgrades)
• CP violation and Unitarity of CKM matrix
(What is the angle  ?)
• DCPK(*)- mode
• Sensitivity for 
• Related decay mode(s)
• Conclusion
KEK-B accelerator
Two separate rings: 8GeV(e-)/3.5GeV(e+)
Finite crossing angle
Design luminosity = 1034 cm-2sec-1
Achieved lum =1.06  1034 cm-2sec-1
Belle logged: 158 fb-1
KEKB Collider
Future upgrade planned for 2005 [luminosity:factor of 2 ]
For a finite crossing angle  Geometrical luminosity loss
 Beam instability
Without crab cavities:
22mrad
With crab cavities:
Complete overlap of beams
Head-on collision
Crab cavities
Belle Detector
SC solenoid
1.5T
Aerogel Cherenkov cnt.
n=1.015~1.030
3.5GeV e+
CsI(Tl)
TOF counter
8GeV eTracking + dE/dx
Si vtx. det.
3 lyr. DSSD
m / KL detection
14/15 lyr. RPC+Fe
Particle identification
•Particle identification: ACC, TOF , dE/dx from CDC
L (K )
PID(K) =
L (K ) + L ( )
L (K ) = pKdE/dx
• Very good in a wide momentum range

pKTOF

pKACC
Upgrade (fall-2003): New beam-pipe
SR mask
SR mask
No SR mask
New beam-pipe is longer , 16cm  24cm
Upgrade (fall-2003): New beam-pipe
Double walled Be beampipe
HER
QCSLE
QC2
BC3
QC1
LER mask
QCSRP
LER
Removed for new beam-pipe
Upgrade (fall-2003): New beam-pipe
SR mask
SR mask
No SR mask
New beam-pipe is longer , 16cm  24cm
Upgrade (fall-2003): cont…
• More synchrotron radiation protection
LER
HER
BC3
QC2
QC1
LER mask
Saw tooth shape
No SR hits
Beam-pipe
Upgrade (fall-2003): New beam-pipe
SR mask
No SR mask
New beam-pipe is longer , 16cm  24cm
Upgrade (fall-2003): SVD2
• Increase the number of layers , 3 layers  4 layers
• smaller radius for inner-most layer
• Better vertex resolution (  1/distance 1st detection layer)
Upgrade (fall-2003): New Vertex Detector(SVD2)
An event with new Vertex Detector
Standard Model lagrangian for q-W interaction
Lint(t) =  d3x (LqW(x) + L†qW(x))
LqW(x)=
Ui (x )
V
ij Ui
m ( 1- 5) Dj Wm
 u( x ) 


c
(
x
)

=
 t( x) 


V=
Dj (x ) =
 Vud

 Vcd
 Vtd

Vub 

Vcb 
Vtb 
Vus
Vcs
Vts
 d( x ) 


s
(
x
)


 b( x ) 


(CKM matrix)
Experimentally, V has a hierarchical structure.
1




3

1

2



1 
3
2
( = 0.22)
Transformation of Lint under CP
exchanges particle (n)  antiparticle ( n )


CP: flips momentum sign ( p  -p )
keeps the spin z-component () the same
under CP, LqW transforms as
CP LqW(x) P†C†
= g  u*d Vij (Ui (x’) m ( 1- 5) Dj(x’) Wm (x’))†
8

x ’ = ( t , -x
)
If u*d can be chosen such that
u*d Vij = Vij*
Then, Lint(t) becomes invariant under CP:
CP LqW(x) P†C† = L†qW(x)
CP Lint(t) P†C† =  d3x CP (LqW(x) + L+qW(x)) P†C†
=  d3x (L+qW(x) + LqW(x) ) = Lint(t)
(1)
Condition for CP invariance
u*d Vij = Vij*
Condition (1) is equivalent to rotating the quark phases
to make Vij all real
In general, there are 5 phase differences for 6 quarks
 5 elements of VCKM can be set to real always
+
3 phases can be related to Euler angles
There is one phase which cannot be removed
 CP violation
Wolfenstein parameterization
 d '  VudVusVub  d 
 
  
 s'   VcdVcsVcb  s 
 b'  V V V  b 
   td ts tb  
Wolfenstein parameterization



V
ud
V
us
V
ub


/
2

A

(  - i ) 




 

2
-
-  / 2
A

 Vcd Vcs Vcb   

 Vtd Vts Vtb   A (1 -  - i ) - A2


 

What is the angle  ?
Orthogonality of d-colmn and b-column:
* + V V* + V V * = 0
VudVub
cd cb
td tb
*
VudVub
*
Vub
 Vcb
VtdVtb*



*
VcdVcb
a

b
-b
 = arg
a
-b
*
Vud Vub


 = arg 
* 
V
cd
V
cb 

CKM fitter: 39°    80°
@ 95% C.L
Methods to extract the angle 
s
b
Bu
Vcb
u
K-
c
o
b
u
Vub
B-
D
-
u
-
-o
-c
D
s
K-
-
u
u
• One needs interference between D0K- and D0K-
D0 and D0 decay to common final state
Example:
K+K-, KS0 etc ( CP eigenstates)
KS+- Dalitz analysis
K*K (singly Cabibbo suppressed mode)
K+- ( doubly Cabibbo suppressed mode)
I will discuss
(Gronau , London and Wyler)
PLB 253(1991)483
PLB 265(1991)172
Gronau-London-Wyler method to extract 
-0 )
B-  DCPK- where Dcp  (D0 ± D
-
• Amp(B-  DCPK-) = Amp(B-  D0K-) + Amp(B-  D0K-)
-
K+K-, +-
 2 diagrams
CP + modes
s

b
-
Vcb
KS0, KS, KS, KS, KS’
D0 and D0
u
K-
b
-
u
Color-favored
~  Vcb
Vub
*)
=arg(Vub
D
u
u
B-
c
o
B-
CP - modes
-
-o
-c
D
s
K-
-
u
u
Color-suppressed
~Vub
GLW method cont…
Strong_final-state-interaction phase:
B-  D0K- relative to B-  D0K- is ei
_
• Amp(B-  DCPK-) = |Amp(B-  D0K-)| + |Amp(B-  D0K-)| ei(+)
_
• Amp(B+ DCPK+) = |Amp(B+ D0K+)| + |Amp(B- D0K+)| ei(-)
—
—
A(B- D0K-) = A(B+ D0K+)

-
GLW method cont…
_
• Amp(B-  DCPK-) = |Amp(B-  D0K-)| + |Amp(B-  D0K-)| ei(+)
_
• Amp(B+ DCPK+) = |Amp(B+ D0K+)| + |Amp(B- D0K+)| ei(-)
—


—
A(B- D0K-) = A(B+ D0K+)
-

Reconstruct the two triangles 

Non-vanishing strong phase (  0)  Direct CP violation
GLW method cont…
One can measure  even if =0( No strong phase)
no direct CPV
One needs to measure the sides and reconstruct triangle 
—
—
A(B- D0K-) = A(B+ D0K+)

-
Problem: Color-suppressed mode
B-  D0K
K+-
and
B-  D0K
K+-
—

How to measure ?
-
Ratio of amplitudes ~ 1  Good or Bad ?
Method of Atwood, Dunietz and Soni
Theoretical solution
One can instead measure:
R1,2 = RDCP /R Dnon-CP = 1 + r2  2r cos()cos()
where R
R1 + R2
2
DCP
=
B (B-  D1,2K-) + C.C
B (B-  D1,2-) + C.C
= 2( 1 + r2)
r = |BKD|/|BKD|
In principle one can obtain r
Useful inequality:
Sin2 < R 1,2  R1 or R2 < 1.0 (assuming small r )
Solution
B
(B-  D1,2K-) - B (B+  D1,2K+)
A1,2 =
=
B (B-  D1,2K-)+ B (B+  D1,2K+)
 2r sin()sin()
R1,2
4 measurements : A1,2 and R1,2
But A1R1 = - A2R 2
r = |BKD|/|BKD|
3 independent measurements  3 unknowns ( r , , )
Solve !
Variables to identify signal
B  f1….. fn
Energy and absolute value of momentum is known:
E
EB = Ebeam = 5.29 GeV

PB = E 2 - M 2 = 0.34 GeV/c
beam
B
Requires that the candidates satisfy

n
n

EB =  E i |PB| =| Pi|
i 1
i 1
Peaks at: 0.0 GeV
5.279 GeV
Mbc

Instead of EB and PB , we historically use
Energy difference E  EB - Ebeam

2
- | PB |2
Beam constrained mass M bc  Ebeam
Mbc ~ 2.5 MeV
~  10 better
inv mass
Hadronic cross sections @(4S) peak energy
channel
(4S)
(nb)
uu
dd
ss
cc
1.05
1.39
0.35
0.35
1.30
Hadronic total
4.44
~ 76% is qq 2 jet-type (“ continuum events”)
KEK-B
operates here
The continuum is monitored
by taking data just below the
(4S) resonance (60 MeV)
off (4S) 
1
x on (4S) (Belle)
10
Rare decay background
is usually dominated by
continuum
Suppressing continuum events
•Variables to distinguish signal from continuum events
CosB
signal ~ sin2
continuum ~ flat
B
e+
B
e-
Continuum
BB Signal
Suppressing continuum
signal

• Fisher discriminant
of variables x = (x1…….xn)

F =  .x

: constants to be chosen to maximize
separation(S) between signal and
background , S = func (FS , FB )
S
=0
 i
Xi =
Rl 
Fox-Wolfram moments
 | Pi || Pj | p l (cos  ij )
i,j
 | Pi || Pj |
i,j
R l  R so [l] + R oo [l]
4
  i R so [l ] +   m R oo [m]
i 1
m 1, 3
Get values of ’s
continuum
continuum
signal
Suppressing continuum
Most effective way to suppress the continuum events
 Combine Fisher discriminant(F) and cosB
Likelihood ratio (LR)
_
_
L(BB)
LR(BB) =
_
L(BB) + L(qq)
_
_
_
L(BB) = L(BB)(F) x L(BB)(cosB)
continuum
signal
_
0 for continuum
_ events
LR(BB) peak at: { 1 for signal(BB)
events
Performance( B- D0[K-+] - ) :
LR > 0.4 keeps 87.5% signal
removes 73% continuum
LR
Results for calibration mode B-  D0K-(-)
@78fb-1
/K separation by Aerogel Cherenkov Counter
( with dE/dx, and TOF)
Prompt Kaon is reconstructed
with pion mass assumption
Allows simple cuts/fits :
shifts E by -49 MeV
B-  D0 K-
B-  D0 -
347.5±21
6058±88
B-  D0 134.4±14.7
B- D0*-, D0D00, D0
B- D0*K-, D0K*-
continuum
D00, D0
Br (B -  D 0 K - )
R=
0 - = 0.077 ± 0.005(stat) ± 0.006(sys)
Br (B  D  )
Results for B-  D1K-(-) mode
For
B-  D0[K+K-]K-
background
CP-even
B-  D0[+-]K-
 23
B-  K+K-K-
 100
B-  K-+-
Can be estimated from
D0 sideband data
 included in systematic error
683.4±32.8
Br (B -  D1 K - )
=0.093±0.018±0.008
R=
Br (B  D1  )
Double ratio
R1 = 1.21 ±0.25 (stat)±0.14 (sys)
CP asymmetry
A1 = +0.06 ± 0.19 (stat)±0.04(sys)
47.3±8.9
15.6±6.4
25±6.5
22.1±6.1
Results for B-  D2K-(-) mode
CP-odd
B-  D0[KS]K- where +-0
For
background
For
B-  D0[K*-+]K- where K*-KS- and + +0
KS- invariant mass difference> 75 MeV
-
R=
-
Br (B  D 2 K )
=0.108±0.019±0.007
Br (B  D 2  )
648.3±31.0
52.4±9.0
6.3±5.0
Double ratio
R2 = 1.41 ±0.27 (stat)±0.145(sys)
CP asymmetry
A2 = - 0.19 ± 0.17 (stat)±0.05(sys)
20.5±5.6
A1,2 and R1,2 are useful quantities for determining 
29.9±6.5
Measuring  using B-  DCPK- mode
Using the measured value of R1,2
and
A1,2
We find:
r2
=
R1 + R 2
= 0.31  0.21
r = |BKD|/|BKD|
2
Just 1.5 away from physical boundary: r2 =0
otherwise r would be imaginary
A lot more statistics needed for this method to be useful
This corresponds to r = 0.57 0.19  very unlikely (theory: 0.1 0.2)
Measurement of “r”
s
b
B
u
Vcb
u
K-
c
o
b
B-
D
-
-
u
u
-
0
Amp(B  D K - )
r=
Amp(B -  D 0 K - )
Vub
u
-o
-c
D
s
K-
-
u
Measurement of “r” using B0  D0K0 mode
s
B-
-u
Vcb
b
c
-u
-u
b
K-
B
D
s
-d
-d
BF(B0D0K0)=(5.0±1.3)  10-5
-
0
Amp(B  D K - )
r=
Amp(B -  D 0 K - )
 (CKM factor)(color factor)
0.45
 r  0.2
??
0
0
Amp(B  D K )
Color factor=
~
0
Amp(B  D K )
0

0
o
-u D
Vcb
o
o
c
0
BF (B  D 0 K )
BF (B -  D 0 K - )
~ 0.4
K
o
Measuring  using B-  DCPK- mode
Assume we measured r = 0.2 , what would be  ?
R1,2 =
1.4
1 + r2  2r cos()cos()
1.2
A1,2 =
 2r sin()sin()
1.0
Taken from
Gronau
R1,2
0.8
0.6
 (degrees)
At 1, the angle  <33 or  >
Excluded by CKM fit
  >
*-
Additional modes: B-  DCPK
mode
Same principle as B- -> DCPK- decay: K- to be replaced by K*First step:
Flavor specific modes
D0  K-+ , K-+0 , K-+-+
169.5±15.4
16 
Only KS- is used
(K-0 could be included
worry: handling background)
Consistency check:
B-

D0K
*-
mode
Yields in KS- mass and helicity bins
• Points with error bars  data

Fit to E for each bin
• Hatched histogram  Signal Monte Carlo
B-
*-
 DCPK mode
13.1 ± 4.3
4.3
B-  D1K*-
7.2 ± 3.6
2.4
B-  D2K*-
CP asymmetries :
A1 = -0.02 ± 0.33(stat) ± 0.07(sys)
A2 = 0.19± 0.50(stat) ± 0.04(sys)
Cannot now constrain  -> need more data
Additional modes: Atwood, Dunietz and Soni method
K-
Doubly Cabibbo
Cabibbo Allowed
Suppressed
+
K 
K-
D0
D0
1
B-

Maximum
Interference
B-
Measure B-  DK- in two decay modes of D:
e.g K+- and KS0 ( their CP conjugates)
[B- (K+-)K-]
[B+ (K-+)K+]
[B- (KS0)K-]
[B+ (KS0)K+]
Solve for   K  , K  ,  and r
+ -
S
0
Additional modes: ADS method
@78fb-1
Only ~ 10-12 events,
Cabibbo-suppressed
D0K down by ~1/15
E
E
Promising method but requires lots & lots of data
B- D0(KS+-)K- Dalitz analysis
Previously: B-  DCPK- where DCP =(D0  D0 )
both D0 and D0 decays to CP eigenstates ( K+K-..)
D0K0 +D0K0 +-
DKS +-
Amp(B+ ->DK+) = f(m+2,m-2 ) + r. ei( + ) f(m-2 , m+2 )
where m+/-2= M2(KS+/-)
r = |BKD|/|BKD|
f( m+2,m-2) =  ak. ei Ak(m+2,m-2) + b ei
-> both 2-body resonances and non-res component
Simple example
Suppose all DKS+- decays are via K*
D0K*-+
D0K*+-
-
KS  +
M(KS +)2
Dalitz plot
KS
interference
M(KS -)2
Reality is more complex ( & better)
D0KS +K*
many amplitudes &
strong phases(13)
lots of interference
K S
K Sf 2
K S
CLEO model of D0 decay
Resonance
K*-(892)+
KS0
K*+(892)KS
KSf0(980)
KSf0(1370)
KSf2(1270)
K*0-(1430)+
K*2-(1430)+
K*-(1680)+
KS1(M=535±6 MeV, =460±15 MeV)
KS2(M=1063±7 MeV, =101±12 MeV)
Non-resonant
Amplitude
Phase
1.706 ± 0.015
1.0(fixed)
0.136 ± 0.008
0.032 ± 0.002
0.385 ± 0.011
0.49 ± 0.04
1.66 ± 0.05
2.09 ± 0.05
1.2 ± 0.05
1.62 ± 0.024
1.66 ± 0.09
0.31 ± 0.04
6.51 ± 0.22
138 ± 0.9
0 (fixed)
330 ± 3
114 ± 3
214.2 ± 2.3
311 ± 6
341.3 ± 2.3
353.6 ± 1.8
316.9 ± 2.1
84 ± 10
217.3 ± 1.4
257 ± 11
149 ± 1.6
Result with 140 fb-1
B-



B+
Use D0KS +- to make Dalitz-plot model (D*+ sample)
 fit 58K events with 13 amplitudes
Select B±K± D0(KS +-) events
-1 Belle data
 107 ± 12 events in 140 fb
Form Dalitz plots for B+ & B-
Fit Dalitz distributions for B+ and B- decay simultaneously
-> r ,  ,  as free parameters
Measuring  using B- D0(KS+-)K- Dalitz analysis


Weak phase  = 950 ±250(stat) ±130 (sys)±100
strong phase  = 1620 ±250(stat) ±120(sys) ±240
(3rd error is model uncertainty)
_
r = |BKD|/|BKD|
r = 0.33 ± 0.10(stat)
@90% C.L :
0.15<r<0.5 ,610<<1420, 1040<<2140
Summary
• There are many methods to constrain  ( couldn’t discuss
all of them)
e.g: B D* (small asymmetry), measures sin(2+)
B  K,  , large theoretical uncertainty
• B- DCPK(*)- needs value of “r” as input.
r2 = 0.31 ±0.21 (assuming r =0.2  r2 = 0.04, r2 = ? )
• B-  DdcsdK- is promising but need lots of data (ADS method)
•All methods have discrete ambiguities.
decay modes with large asymmetry have small BF’s
decay modes with large BF’s have small asymmetry
• Either way, one needs large data sample to constrain 