Axial Vector Meson Emitting Decays of Bc
Dated: 12 JUNE, 2012
VARIOUS QUARK LEVEL PROCESSES THAT CONTIBUTE TO THE NONLEPTONIC DECAYS
These Processes are Classified as:
FACTORIZATION SCHEME Factorization is the assumption that the two-body
hadronic decays of B mesons can be expressed as the product of two independent hadronic currents.
The decay amplitude is given by
Three classes of the decays:1. Class I transition (caused by color favored),2. Class II transition (caused by color suppressed) and3. Class III transition (caused by both color favored and
color suppressed diagrams).
† †1 2 1 2< | | > < | | 0 >< | | >M M J J B M J M J B
1 2
† †1 2 2 1
( )2
0 0 .
FGB M M Cabibbo factors QCD factors
M J M J B M J M J B
WEAK HAMILTONIAN
Selection rules1, = 1, = 0,b C S
1, = 0, = 0,b C S
1, = 0, = 1,b C S
1, = 1, = 1,b C S
1, = 1, = 1,b C S
1, = 1, = 0,b C S
CKM favored decays
1. involving bc transition
2. involving bu transition
11 2
1 2
[ [ ( )( ) ( )( )]2
[ ( )( ) ( )( )]],
b * Fw cb ud
* cb cs
GH V V a cb du a db cu
V V a cb sc a sb cc
1, = 1/ 0, = 0 / 1b C S
11 2
1 2
[ [ ( )( ) ( )( )]2
[ ( )( ) ( )( )]],
b * Fw ub cs
* ub ud
GH V V a ub sc a sb uc
V V a ub du a db uu
1, = 1/ 0, = 1/ 0b C S
CKM suppressed decays
1. involving bc transition
2. involving bu transition
where
11 2
1 2
[ [ ( )( ) ( )( )]2
[ ( )( ) ( )( )]],
b * Fw cb us
* cb cd
GH V V a cb su a sb cu
V V a cb dc a db cc
11 2
1 2
[ [ ( )( ) ( )( )]2
[ ( )( ) ( )( )]],
b * Fw ub us
* ub cd
GH V V a ub su a sb uu
V V a ub dc a db uc
1 1.12a 2 0.26a ,
There have been many theoretical efforts to study the bottom meson emitting decays involving s-wave mesons (B PP/PV/VV) i.e. pseudoscalar (P) and vector (V) mesons using the factorization scheme.
However, B mesons being heavy, can also emit p-wave mesons i.e. axial-vector (A), tensor (T) and scalar (S) mesons, which we have studied in the next chapters.
AXIAL-VECTOR MESON SPECTROSCOPYExperimentally, two types of the axial-vector mesons exist i.e. and
For
Isovector :
Isoscalars:
where
31( 1 )PCP J 1
1( 1 )PCP J
1
01 1 1 1(1.230) : , ,a a a a
1
1
1(1.285) ( ) cos ( )sin2
1(1.512) ( )sin ( ) cos2
A A
A A
f uu dd ss
f uu dd ss
1(3.511) ( )c cc
)()( physicalideal AA
For
Isovector :
Isoscalars:
where
with
1
01 1 1 1(1.229) : , ,b b b b
1
1
1(1.170) ( )cos ( )sin2
1(1.380) ( )sin ( ) cos2
A A
A A
h uu d d ss
h uu d d ss
)()526.3(1 cchc
( ) ( )A Aideal physical
0 AA
MIXING IN STARNGE AND CHARM AXIAL-VECTOR MESONS
Strange and charm mesons are the mixing of and
Mixing of Strange states(1 )A (1 )A
Mixing of Charmed states
Mixing of strange-Charmed states
with
1 1 1 1 1
1 1 1 1 1
(1.270) sin cos ,(1.400) cos sin .
A A
A A
K K KK K K
01 58
1 2 3 21 1 2 1 2
1 2 3 21 1 2 1 2
(2.427) cos sin ,
(2.422) sin cos .
D D D
D D D
1 2 3 21 1 3 1 3
1 2 3 21 1 3 1 3
(2.460) cos sin ,
(2.535) sin cos .s s s
s s s
D D D
D D D
2 ( 5.7 2.4) 3 7
DECAY AMPLITUDES AND RATES
where
The factorization Scheme expresses the decay amplitudes as a product of matrix element of the weak currents
The matrix element of current between mesons states are expressed as
32
2( ) ( )8
c
A
pB P A A B P Am
2 2 2 2 1/21 {[ ( ) ][ ( ) ]}2c B P A B P A
B
p m m m m m mm
0 0 ,
0 0 .w
w
PA H B P J A J B A J P J B
PA H B P J A J B A J P J B
*( , ) 0A A AA k A m f
*( , ) 0A A AA k A m f
* * *( , ) ( ) ( )( ) ( )( ) ,A B B B A B B AA k V B k l c k k k c k k k
* * *( , ) ( ) ( )( ) ( )( )A B B B A B B AA k V B k r s k k k s k k k
Finally the decay amplitude becomes
where
2 21( ) ( 2 ( ) ( ))B P B A
A A A P PA B PA m f F m f F m
2 21( ) ( 2 ( ) ( )) ,B P B A
A A A P PA B PA m f F m f F m
2 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) ,B AP P B A P P PF m l m m m c m m c m
2 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ).B AP P B A P P PF m r m m m s m m s m
ISGW II MODEL
CALCULATION OF THE FORM FACTORS IN ISGW II MODEL
For transition form factors have the following expressions
For transition form factors have the following expressions
B A
2( )2 2
52 21
( 1)1 5[ ( )] ,6 2
lA BB B
B BA
m m ml m Fm
2( )2 1 2
21 2
2( )2 1 2
21 2
1 ,2 2
2 ,2 3 2
c cA B
B A BA
c cA B
B A BA
m m m mc c Fm m m
m m m mc c Fm m m
B A
2 ( )252
1
1[ ( 1) ] ,32
rB B A
B
m m mr Fm
2( )2 2 2
21
2( )2 1 2
21
1 ,2 2
4 ,2 3 2
s sB
B B BA
s sB
B A BA
m m ms s Fm m
m m ms s Fm m
where
The value of parameter for s-wave and p-wave are
and
1 1( ) ( ) 2 25 5 5
3 1( ) ( ) 2 25 5 5
1 1( ) ( ) 2 25 5 5
( ) ( ) ,
( ) ( ) ,
( ) ( ) .
l r B A
B A
c c s s B A
B A
c c s s B A
B A
m mF F Fm m
m mF F Fm mm mF F Fm m
2 2 212BX B X
11 1
q bm m
DECAY CONSTANTS (in GeV) OF THE AXIAL-VECTOR MESONS
1 (1270) 0.175,Kf 1 (1.400) 0.087 ,Kf
10.203,af
1 1f af f
10.127,
ADf
10.045,
BDf
10.121,
s ADf
10.038,
s BDf
10.160.
cf
COMPARISION WITH THE EXPERIMENTAL DATA
HADRONIC WEAK DECAYS OF Bc MESON:
NAKED BOTTOM-CHARM STATE TO
A PSEUDOSCALAR AND A P-WAVE MESONS
UNIQUELY OBSERVED BOTTOM CHARM (Bc) MESON
In this chapter, we studied the weak hadronic decays of Bc meson emitting pseudoscalar and one p-wave meson in the final state.
Bc MESON EMITTING DECAYS OF PSEUDOSCALAR AND AXIAL-VECTOR MESONS
BOTTOM MESON SPECTROSCOPY
WEAK HAMILTONIAN
CALCULATION OF THE FORM FACTORS IN ISGW II MODEL
So far, theoretical focus has also been on the s-wave mesons i.e., pseudoscalar and vector mesons emitting decays. However, the bottom mesons and uniquely observed bottom-charm mesons, being heavy, can also emit p-wave mesons i.e., axial-vector, tensor and scalar mesons.
The hadronic weak currents are expressed in the terms of the form factors which are usually calculated from the phenomenological models, we have employed BSW model to calculate the BP form factors which match well with the experimental information.
We have also studied the hadronic weak decays of uniquely observed bottom-charm (Bc) meson made up of both heavy quarks
For the BcA transition form factors appearing in the decay matrix elements, we employ ISGW II model because it provide the more reliable form factors. We obtained the decay amplitude and consequently predicted the branching ratios for BcPA decays.
In case of Bc meson, one naively expects the bottom conserving modes (c -> u, s transitions) to be kinematically suppressed in comparison to the bottom changing ones. However, the large CKM angle involved in the charm changing modes overcomes the kinematics suppression.Consequently, bottom changing decays get suppressed in comparison to bottom conserving decays.
Measurements of their branching ratios provide a useful test of our model.
We look to extend the present approach to calculate Bc VA decays.
Here also, we look forward to calculate to use the ISGW II model form factors to calculate Bc V transition form factors.
It may be pointed out that so for these transition formfactors have not been used by any one.
The matrix element for various Bc A and Bc V transition are given by
Since, final states of BcVA carry spin degrees of freedom, the decay amplitudes in terms of helicities, like those in the BcVV decays, can be generally described by
Because, Bc is a pseudoscalar, the two outgoing vector mesons A and V have to carry the same helicity. Consequently, the amplitudes with different helicities can be decomposed as
where p is the magnitude of vector momenta of vector mesons.
In addition, we can also write the amplitudes in terms of polarizations as
Accordingly, the polarization fractions can be defined to be
representing longitudinal, transverse parallel and transverseperpendicular components, respectively. In sum, the decay
rate expressed by polarization amplitudes is given by
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