A New Physics Study
in B  K p & B  K*r Decays
Sechul OH
(吳世哲)
(오세철)
C.S. Kim, S.O., Y.W. Yoon, PLB665, 231 (2008)
C.S. Kim, S.O., C. Sharma, R. Sinha, Y.W. Yoon, PRD76, 074019 (2007)
National Tsing Hua University, October 23, 2008
1
 The B  K p puzzle
 A model-independent analysis of B  K p
-- reparametrization invariance
-- how to extract New Physics effects
 A model-independent analysis of B  K* r
-- interesting observables sensitive to New Physics
effects
 Summary
2
3
4
5

Cabibbo-Kobayashi-Maskawa (CKM) matrix
Unitarity:
(a)
Unitarity triangle:
(g)
(b)
6

Direct CP violation
Direct CP violation in decay

occurs when
Time-dependent CP violation
= J/y Ks
7
8
 The dominant quark level subprocesses are
loop (penguin) processes

b  s penguin is sensitive to NP
q
q
 The 4 decay channels (& antiparticle decay channels)
9
Conventional Hierarchy in B → Kπ
Large
strong penguin
EW penguin
Small
color-suppressed tree
10
 Branching Ratios
SM :
Fleischer Hep-ph/0701217
=0
March 2007:
Rc = 1.11 ± 0.07
Rn = 0.97 ± 0.07
11
 CP Asymmetries
SM :
12
 Amplitude parameterization
A (B + ® K 0 p + ) = P ¢+ A ¢
¢C - T ¢
A (B 0 ® K + p - ) = - P ¢- PEW
¢ - PEW
¢C - T ¢- C¢- A ¢
2A (B + ® K + p 0 ) = - P ¢- PEW
¢ - C¢
P ¢- PEW
2A (B 0 ® K 0 p 0 ) =
 Hierarchy between the parameters
Pt¢c
1
¢
T ¢, PEW
l
l
2
¢C
C¢, PEW
l
3
A ¢, Puc¢
%¢ + V * V P
%¢
¢
¢
P ¢= V t *bV t s P
tc
ub us uc º Pt c + Puc
l
2
l
4
q
q
13
 Final form
A (B + ® K 0 p + ) º A 0+
= - P¢
= P ¢(1 - rT e i ge i dT¢ )
+0
1
¢
A (B + ®
A + 0e i a =
P ¢(1 - rT e i g e i dT¢ - rC e i g e i dC¢ + r EW e i dEW
2
00
1
¢
A (B 0 ® K 0 p 0 ) º A 00e i a =
P ¢(- 1 - rC e i g e i dC¢ + r EW e i dEW
)
2
A (B ® K p
0
)º
K + p 0) º
+
-
+-
A e
ia + -
P ¢= Pt¢c , rT =





)
T¢
C¢
P¢
, rC =
, r EW = EW
Pt¢c
Pt¢c
Pt¢c
We neglect Puc¢, PEW¢C , A¢
We set the strong phase of P to be zero  all phase is relative to it
¢
We hold 7 unknown parameters P ¢, rT , rC , rEW , δ T¢, δ C¢, δ EW
We use g value given by other analyses
A ij are real and positive, a ij are phases of their amplitude
14
“Reparametrization invariance” of decay amplitudes
Botella and Silva
 NP term is absorbed into SM term:
+0
A ,A
00
1
i g i dC¢
i dE W
N i f N i dN
¢
É
P - rC e e + r EW e
+r e e
2
N
N
æ
ö
N
1
sin
f
sin(
f
- g ) i dN ÷
¢
i
g
i
d
i
d
N
i
d
i
g
N
C
EW
ç
¢
=
P ç- rC e e + r E W e
+r
e e - r
e ÷
÷
÷
2 çè
sin g
sin g
ø
(
)
1
M i g i dCM
=
P ¢ - rC e e
2
(
M
C
r e
i dCM
M
r EMW e i dE W
+ r
M
EW
e
M
i dEW
sin f N i dN
= rC e - r
e
sin g
N
- g ) i dN
¢
i dEW
N sin(f
= r EW e
- r
e
sin g
i dC¢
)
N
15
 Original Form does not change:
= - P¢
A (B + ® K 0 p + ) º A 0+
A (B 0 ® K + p - ) º A + - e i a
A (B
+
® K p
+
0
)º
+ 0 ia +0
A e
A (B 0 ® K 0 p 0 ) º A 00e i a

If there is NP,
+-
00
= P ¢(1 - rT e i ge i dT¢ )
M
1
i g i dT¢
M i g i dCM
M
i dEW
¢
=
P 1 - rT e e - rC e e
+ r EW e
2
M
M
1
=
P ¢ - 1 - rCM e i g e i dC + r EMW e i dEW
2
(
(
)
rCM ¹ (rC )SM , dCM ¹ (dC )SM
M
r EW
¹ (r EW )SM , dEMW ¹ (dEW )SM
16
)
Analytic Solution
P ¢µ
B r 0+
é
sin 2g ê
cot dT¢ =
ê1 ±
(- AC+P- )R ê
êë
rT =
rCM
1 æ
çç
1+
R - 12
cos g ççè
2 öù
æ- A C+ P- R ö
ú
÷÷
çç
÷
÷
ú
÷
÷
÷
÷ø
çè 2 sin g ø
÷ú
ú
û
a
00
R (1 - A C+ P- cot g cot dT¢) - 1
1
=
sin g
2
y + y
2
2
2
- y y cos( h - h )
2
y + y
r
- y y cos(2g + h - h )
2
æ y cos h - y cos h ÷
ö
dCM = ArcT an çç÷
÷
çè y sin h - y sin h ÷
ø
æ y cos( h - g ) - y cos( h + g ) ÷
ö
÷
dEMW = ArcT an çççè y sin( h - g ) - y sin( h + g ) ÷
÷
ø
M
EW
a
00
1
=
sin g
æ2A + 0 2 = z ± ArcCos ççç
çè
2
æ2A + 0 2 = z ± ArcCos ççç
2
èç
2A 00 - P ¢2x 2 ö
÷
÷
÷
00
÷
2A P ¢x
ø
2
2A 00 - P ¢2x 2 ö
÷
÷
÷
00
÷
¢
2A P x
ø
2
A 00 i a 00
-r e e
+r e
= 2
e
+ 1 º y e ih
P¢
M
A 00 i a 00
M - i g i dCM
M
i dEW
- rC e e
+ rEW e
= 2
e
+ 1 º y e ih
P¢
M
C
ig
i dCM
M
EW
i dEMW
17
M
 4 different solutions for rCM , rEWM , δ CM , δ EW
( 4-fold discrete ambiguity )
S K S p 0 = 0.38 ± 0.19(dat a)


We reject “Cases 1 & 3” due to SK π0 predictions different from data
s
The SM estimate rEW = 0.12 > rC = 0.039 , δ C » - 61o , δ EW = 22o


Case 2: Large C
Case 4: Large EW
18
 Find solutions for NP term
M
C
r e
r
M
EW
e
i dCM
i dEMW
= rC e
i dC¢
= r EW e
- r
¢
i dEW
N
sin f N i dN
e
sin g
- r
N
sin(f N - g ) i dN
e
sin g
4 real equations vs 7 unknowns:
¢ , r N ,φN , δ N
rC , rEW , δ C¢, δ EW
Need at least 3 additional inputs to fix NP terms
19
Additional inputs
 Additional inputs from flavor SU(3) symmetry
From B  pp decays
Assuming no NP in B pp
20
B  pp parametrization
2 A (B + ® p + p 0 ) = - (T e i g e i dT + C e i g e i dC )
A (B 0 ® p + p - ) = - (T e i g e i dT + P e - i b )
2A (B + ® p 0p 0 ) = - (C e i g e i dC - P e - i b )
c2-fitting
P
with 5 measurements
3 Br’s, ACP (π+ π- ), Sπ π
with 5 parameters
+
-
δT
T
C
δC
0.33
A CP ( p 0 p 0 )= 0.36-+ 0.31
(dat a)
C ¢=
r EW e
¢
i dEW
V us
C = (3.8 ± 0.4) eV
V ud
3 c 9 + c 10 1
¢
¢
= (rT e i dT + rC e i dC )
2
2 c1 + c2 l R b
Gronau, Pirjol, Yan (1999)
(rC , dC¢) = (0.076 ± 0.008, - 12° ± 15°)
¢ ) = ( 0.14 ± 0.04,
(r EW , dEW
9° ± 10°)
21
 Additional inputs from PQCD result
Li, Mishima, Sanda, PRD72, 114005 (2005)
(rC , dC¢) = (0.039, - 61°)
¢ ) = ( 0.12,
(r EW , dEW
22° )
22
Determining NP parameters
 Solution for NP term with additional inputs
Defining
D rC e
i D dC
º r e
M
C
N
i dCM
- rC e
i dC¢
¢
M
D r EW e i D dEW º r EW
e i dEW - r EW e i dEW
M
D rC e i D dC = - r N
sin f
e id
sin g
D r EW e i D dEW = - r N
With inputs from SU(3) symmetry
N
sin(f N - g ) i dN
e
sin g
d N = D dC or D dC - p
sin f N
D rC
=
N
sin(f - g )
D r EW
sin g
rN =
D rC
sin f N
With inputs from PQCD results
Cases 2 & 4 are suitable and consistent each other between two methods.
23
Dependance on g
24




Due to the Reparametrization Invariance(RI), the NP
terms can be absorbed into the SM terms C & PEW
in pair.
In order to extract NP parameters, we need at least
3 additional inputs.
We could pin down each hadronic parameter under
four-fold discrete ambiguity using analytic method.
And also NP parameter for given additional inputs.
The result shows that there should be quite large NP
contribution with a maximal weak phase ~ p/2.
25
26
B → V V decays
by angular momentum conservation
B ! V1 V2
Spin:
Sz :
0!1+1
)
L = 0, 1, 2 or S, P, D waves
0 ! 1 + (-1)
(-1) + 1
0+0
helicity:
1   2  
hSp
decay amplitudes:
In the B rest frame, the momenta of V1 and V2 are equal and opposite.
 the helicities of both vector mesons are same.
27
♦ The most general covariant amplitude for B  V V


b
ic ab
A ( B  V ( p1 , 1 )V ( p2 ,  2 ) )  1*  2*  ag  
p2 p1 

p1a p2 b 
m1m2
m1m2


• Helicity basis
A 1  a  c x2  1
A 0  ax  b( x2 1)
where x 
p1  p2
m1m2
• Transversity basis
1
A 
( A1  A1 )
2
parallel
A 
1
( A1  A1 )
2
transverse
A0  A0
longitudinal
28

Total decay rate (in the B rest frame)
 ( B  V1V2 ) 
(
2
1
2
p
A

A
 A
V1
0
2
p mB
(
 m  (mV  mV )
1
2
pV1  

2
B
2
)( m
2
B
2
)
Br   B 
)
1/ 2
 (mV1  mV2 ) 
2
2mB
Longitudinal & Transverse polarization fractions
fL 
A0
2
2
A0  A  A
2
2
Standard model estimation:
f 
A
2
2
A0  A  A
2
2
 1 
f L  1 O 2 
 mb 
29

Time dependent measurement
Define A A ( B 0  f1 f 2 ) , A A ( B 0  f1 f 2 )
A 2 A
( B 0 (t )  f1 f 2 )  e t 

2

q
where

p
2
A A
2

2
2

q *
cos(mt )  Im( A A ) sin( mt ) 

p

Hamiltonian Hij  M ij  iij / 2
*
M 12*  i12
M 12  i12
(M ,  : Hermitian)
H11  H22 by CPT invariance
For B  V V decay modes,
A ( B 0  VV
1 2 )  A0 g 0  A g  iA g 
A ( B 0  VV
1 2 )  A0 g 0  A g  iA g 
( g depend purely on angles
describing the kinematics )
A  A0 g0 2  A g 2  A g  2  Re ( A0 A* ) g0 g  Im ( A A0* ) g0 g   Im ( A A* ) g g 
2
2
2
2
A  A0 g0 2  A g 2  A g  2  Re ( A0 A* ) g0 g  Im ( A A0* ) g0 g   Im ( A A* ) g g 
2
2
2
2
30
 Observables
Time dependent measurement:
35 independent observables (18 magnitudes + 17 relative phases)
31
 Observables for B  K p[ Example of B  P P case ]
Only 9 observables
32
An example of New Physics study
beyond the Standard Model
by using B  V V decays
B  K* r
33
 B  K* r is a vector version (B  V V) of B  K p (B  P P)
 The dominant quark level subprocesses are
loop (penguin) processes

b  s penguin is sensitive to NP
q
q
 We expect that NP contribution to B  K* r has the same
nature as that of B  K p
 B  K* r (B  V V) provides enormously many observables
34
Conventional Hierarchy in B  K* r
Large
strong penguin
EW penguin
Small
color-suppressed tree
35
 Parameterization of decay amplitudes
Hierarchy relation in the SM:
Isospin relations:
36
Investigate how much sensitive to possible NP effects each observable
for
decays could be.
Assume that NP contributing via the EW penguins.
For simplicity, further assume that the SM amplitudes
and
are known
(by additional information from somewhere, e.g. from future theoretical
estimates).
Thus, the SM amplitude
is the only one modified by NP.
(SM part)
(NP part)
37
Procedure:
(i) In order to determine the theoretical parameters, adopt the c2
minimization technique
& use the currently available experimental data as constraints
on the parameters.
(HFAG)
fL 
A0
2
2
A0  A  A
2
2
: longitudinal polarization
fraction
38
(ii) [number of data] < [number of parameters]
Try to fit the dominant strong penguins
and their phases
with
, first.
(iii) Assume that the SM amplitudes (
the conventional hierarchy as in
) follows
within the SM:
in PQCD,
(iv) Using the parameters determined, calculate all the 35 observables
in the SM.
(v) To investigate the possible NP effects, consider two different cases.
(SM part)
(NP part)
39
For illustration:
Very sensitive to NP:
40
For illustration:
Very sensitive to NP:
41
For illustration:
Very sensitive to NP:
42
For illustration:
Very sensitive to NP:
43
 B  V V measurements
B factories: Belle (KEK), BaBar (SLAC, closed),
LHC-b (CERN), Tevatron (Fermi Lab), Super-B (?)
 B  K* r decays: useful for New Physics study
certain observables are expected to be very sensitive to NP effects.
44