Hadronic B decays involving tensor mesons
Hai-Yang Cheng (鄭海揚)
Academia Sinica
 Properties of tensor mesons
 QCD factorization
 Comparison with experiment
in collaboration with Kwei-Chou Yang
2011 Cross Strait Meeting on Particle Physics and Cosmology
April 5, 2011
Even-parity mesons
Scalar mesons (JPC= 0++)
0
 1 GeV
a00 , f 0 , 
a0

K 0*
K 0*0

a0
a0
K 0*
0
a0
a00 , f 0 , f 0
> 1 GeV
K 0*0
Axial-vector mesons
K1A
K10A
3P
1
a1
(JPC=1++)
a1
a10 , f1 , f '1
K1A
K10A
Kwei-Chou Yang,
K1B
K10B
b1
b1
b10 , h1 , h'1
K1B
1P
1
(JPC=1+-)
K10B
Nucl. Phys. B776, 187-257 (2007).
2
Tensor mesons
For JP=2+ tensor mesons
3P
2
nonet:
I=0:
f2(1270), f’2(1525),
K 2*
K 2*0
a2
a2
a20 , f 2 , f '2
I=1/2: K2*(1430)
I=1:
a2(1320)
K 2*
K 2*0
close to ideal mixing, f2  5.8o
3
B SM (M=P,V):
HYC, Chua, Yang in QCD factorization (’06, ’08)
C.D. Lu et al. in pQCD (’06, ’07, ’09)
Delepine et al. (’08)
Z. J. Xiao et al. in pQCD (’08 - ’10)
B AM:
HYC, Yang in QCDF (’07)
C.D. Lu et al. in pQCD (’07)
B TM:
last enterprise
4
To study B → TM (M=P,V) decays, we need to know
 mixing angles
 decay constants
 light-cone distribution amplitudes
 form factors for B → T transition
Aliev & Shifman (’82)
HYC, Koike, Yang (’10)
Braun & Kivel (’01)
W.
Wang
(’10), Yang
ISGW
(’89,’95),
CCH(’10),
(’01)
Z.G. Wang (’10)
 vertex corrections, spectator interactions, annihilation for decay
HYC, Yang (’10)
amplitudes
5
Decay constants
 Tensor meson cannot be produced from local V-A current owing
to p=0
T ( p,  ) | V , A | 0  0
 Can be created from local current involving covariant derivatives
with
Previous estimates: Aliev & Shifman (’82); Aliev, Azizi, Bashiry (’10)
Based on QCD sum rules we obtain (HYC, Koike, Yang, arXiv:1007.3526)
6
Form factors for B → T
 ISGW (Isgur-Scora-Grinstein-Wise) non-relativistic quark model (’89,’95)
 Covariant light-front quark model (Chua, Hwang, HYC, ’04)
Relativistic effects in B-to-light transitions at q2=0 are important
 Large energy effective theory (LEET) (Charles et al. ’99)
 pQCD approach (W. Wang, arXiv:1008.5326)
 QCD sum rules (K.C. Yang, arXiv:1010.2144; Z.G. Wang, arXiv:1011.3200)
7
Light-cone distribution amplitudes (LCDAs)
first studied by
Braun & Kivel (‘01)
twist-2: ∥,
twist-3: gv, ga, ht, hs twist-4: g3, h3
Due to even G-parity, these LCDAs are anti-symmetric under the replacement
u→1-u in SU(3) limit
Ci3/2: Gegenbauer polynomial
8
 Longitudinal & transverse helicity projectors for tensor mesons:
Transverse momentum derivative terms should be included before
taking collinear approximation
 Helicity projectors for vector mesons:
9
B→ TM in QCDF
Apply QCD factorization to B→TM (Beneke, Buchalla, Neubert, Sachrajda)
vertex & penguin
spectator int.
annihilation
10
Data
dominated by BaBar, f2K modes are due to Belle
Previous studies based on naïve or generalized factorization predict rates typically
too small by 1-2 orders of magnitude compared to experiment
Penguin-dominated B TP
12
B- K2*0 vanishes in naïve factorization,
while its BR is measured to be ~ 5.610-6
 importance of nonfactorizble effects
 Beyond naïve factorization, contributions  fT defined from local
currents involving covariant derivatives can be produced from
nonfactorizable contributions such as vertex, penguin and hard
spectator corrections
A( B   K 2*0  )  2if T mB pc F1B ( mK2 * )
2
 Penguin annihilation is needed in QCDF to account for rates & CP
asymmetries
1
dy
m
X A    ln B 1   Aei A 
y
h
0
ATP=0.83, ATP = -70o
APT=0.75,
APT = -30o
similar to the parameters
for B PP
13
Penguin-dominated B TP
14
B K2*, K2*’
  q cos  s sin  ,  '  q sin   s cos with   42o
Interference between (b) & (c) is constructive for K2*’ and
destructive for K2*  large rate of K2*’ than K2*
C.S. Kim et al. obtained Br(B K2*’)/Br(B K2*) ~ 45, while it is ~ 2
experimentally. This is because the matrix elements
m2' s
 ' | s  5 s | 0  i
f ' ,
2ms
m2 s
 | s  5 s | 0  i
f
2ms
do not have correct chiral limit behavior due to anomaly and should be
replaced by
m2'
m2
1 q
1 q
s
 ' | s  5 s | 0  i
( f ' f ' ),  | s  5 s | 0  i
( fs f )
2ms
2ms
2
2
15
Tree-dominated B TP
16
Penguin-dominated B TV
17
Rate puzzle in B K2* decays
1
( a4  r a6   3 ) X ( B , K 2* ),
2
A( B   K 2* )  ( a4  r a6   3 ) X ( BK 2* , )
A( B   K 2* ) 
It is naively expected that
2
Br ( B   K 2* ) 1 X ( B , K 2* )

 0.15
Br ( B   K 2* ) 2 X ( BK 2* , )
just as
Br ( B   K * )
Br ( B   K * )
 0.30
Experimentally, Br(B K2*)  Br(B K2*). This can be
accommodated by having penguin annihilation such that 3(K2*)
>> 3(K2*). But why ? What is the dynamical origin ?
18
Polarization puzzle in charmless B→VV decays
 QCD   QCD 

A0 : A : A  1 :
: 
mb  mb 
2
A00 >> A-- >> A++
In transversity basis
A  ( A  A ) / 2 ,
fT  f||  f   1  f L  O(mV2 / mB2 ),
A||  ( A  A ) / 2
f|| / f   1  O(mV / mB )
Why is fT so sizable ~ 0.5 in penguin-dominated
B K*, K*, K*00 decays ?
19
 NLO corrections alone can lower fL and enhance fT significantly !
Beneke,Rohere,Yang
HYC,Yang
constructive (destructive) interference in A- (A0)
⇒ fL  0.58
 Although fL is reduced to 60% level, polarization puzzle is not
completely resolved as the predicted rate, BR  4.310-6, is too small
compared to the data, ~ 1010-6 for B →K*
(S-P)(S+P)
Kagan
(S-P)(S+P) penguin annihilation
contributes to A-- & A00 with similar
amount
2
2
  QCD mb    QCD mb    QCD 
PA
PA
PA
 : 
 : 

A0 : A : A  
ln
ln
 h   mb
 h   mb 
 mb
4
20
Polarization puzzle in B  K2*
fL(K2*+) = 0.560.11, fL(K2*0) = 0.450.12,
BaBar
fL(K2*+) = 0.800.10, fL(K2*0) = 0.901+0.059-0.069
Why is fT/ fL <<1 for B K2* and fT /fL 1 for B K2* ?
Why is that fT behaves differently in K2* and K* ?
In QCDF, fL is very sensitive to the phase ATV for B K2*, but not so
sensitive to AVT for B K2*
fL(K2*) = 0.88, 0.72, 0.48 for ATV = -30o, -45o, -60o,
fL(K2*)= 0.68, 0.66, 0.64 for AVT = -30o, -45o, -60o
Rates & polarization fractions can be accommodated in QCDF
but no dynamical explanation is offered
21
Conclusions
 Tensor meson cannot be created from local V-A current, but its decay
constant can be defined through non-local current or local current with
covariant derivative.
 Some decays e.g. B- K2*0- prohibited in naïve factorization receive
sizable nonfactorizable corrections
 Predictions of QCD factorization in general agree with experiment for
B TM (M=P,V), but there remains puzzles to be resolved: rate of K2* and
polarization of K2*
22