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) and

3. 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 rules

1, = 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 SPECTROSCOPY

Experimentally, 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 ( )sin

21

(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 ( )sin

21

(1.380) ( )sin ( )cos2

A A

A A

h uu dd ss

h uu dd 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 K

K 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( ) ( )

8c

A

pB P A A B P A

m

2 2 2 2 1/21{[ ( ) ][ ( ) ]}

2c B P A B P AB

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 2lA B

B BB BA

m m ml m F

m

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 F

m m m

m m m mc c F

m m m

B A

2 ( )252

1

1[ ( 1) ] ,

32rB B A

B

m m mr F

m

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 F

m m

m m ms s F

m 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 F

m m

m mF F F

m m

m mF F F

m m

2 2 21

2BX B X

1

1 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 transverse

perpendicular components, respectively. In sum, the decay rate expressed by polarization amplitudes is given by

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