Branching Ratios of Bc Meson Decaying to

Vector and Axial-Vector Mesons

Rohit Dhir

Department of Physics,Yonsei University,

Seoul, Korea.Dated:21-Sept-2012

INTRODUCTION

• Weak currents in the Standard Model generate leptonic, semileptonic and hadronic decays of the heavy flavor hadrons.

• Since the quarks are confined inside the colorless hadrons, matching between theory and experiment requires an exact knowledge of the low energy strong interactions.

• The weak decays of heavy quark hadrons provide a unique opportunity to learn more about QCD particularly on the interface between the perturbative and nonperturbative regimes, to determine SM parameters and finally to search for the physics lying beyond the model.

• In the present work, we study Axial Vector emitting decays of Bc meson in the Standard Model Framework. We present preliminary estimates of Branching Ratios for these decays.

Bc MESON

• The Bc meson discovered at Fermilab, is a unique quark-antiquark bound state composed of two heavy quarks with different flavors and are thus flavor asymmetric.

• A peculiarity of Bc decays, w.r.t. the B and Bs Decays, is that both quarks (b, c) can involve in weak decays.

• The decay processes of the Bc meson can be broadly divided into three classes:

i) involving the decay of b quark with c being spectator,

ii) involving the decay of c quark with b being spectator and

iii) the two component annihilate, b and c, weakly.

Processes i) and ii), as mentioned above, can contribute to semileptonic and nonleptonic weak decays. Though we ignore contributions from third process at present.

• There have been many theoretical efforts to study the bottom meson emitting decays involving s-wave mesons (Bc PP/PV/VV) i.e. pseudoscalar (P) and vector (V) mesons using the factorization scheme.

• Bc mesons being heavy, can also emit p-wave mesons i.e. axial-vector (A), tensor (T) and scalar (S) mesons, so far, relatively less attention is paid to these decays. However, we restrict our self only to emission of axial vector meson in the final state.

• Experimental study [PDG, 2011] of the Bc mesons are in plan for B-Physics both at the TEVATRON and Large Hadron Collider (LHC). These experimental efforts have opened up new investigation concerning the structure of strong and weak interactions for heavy flavor sector. Also, Bc meson attracts the interest of experimentalists for testing the predictions of various theoretical efforts in the laboratory.

VARIOUS QUARK LEVEL PROCESSES THAT CONTIBUTE TO THE NONLEPTONIC DECAYS

FACTORIZATION SCHEME (Preliminary Estimates of BRs)

Factorization is the assumption that the two-body hadronicdecays 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

BOTTOM CHANGING DECAYS

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

1 1

1 1

1 1 1

1 1 1

(2.427) sin cos ,

(2.422) cos sin ,

A D A D

A D A D

D D D

D D D

1 1

1 1

1 1 1

1 1 1

(2.460) sin cos ,

(2.535) cos sin ,s s

s s

s s A D s A D

s s A D s A D

D D D

D D D

Mixing of Charmed and Strange Charmed states

MIXING IN STARNGE AND CHARM AXIAL-VECTOR MESONS

and

Mixing of Strange states

(1 )A (1 )A

0 01 58 ( 27 )

&

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

Mixing of Charmed states

Mixing of strange-Charmed states

with

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

1 2 1 31 1 1

3 2 1 31 1 1

1 2| | | ,

3 3

2 1| | | .

3 3

P P P

P P P

However, in the heavy quark limit, the physical mass eigenstates with are and rather than and states as the heavy quark spin decouples from the other degrees of freedom so that

1PJ 3 2

1P 1/21P 3

1P 11P

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

, ,

, ,

.

SUMMARY OF DECAY CONSTANTS (in GeV)

DECAY CONSTANTS (in GeV) OF THE VECTOR MESONS

0.221f * 0.220Kf * 0.245Df

* 0.273sD

f / 0.411Jf

ISGW II MODEL

The new features are:• Heavy quark symmetry constraints on the relations between form

factor from zero-recoil are respected and slopes of form factors near zero-recoil are built

• The naive currents of the quark model are related to the full weak currents via the matching conditions of heavy quark effective theory

• Heavy-quark-symmetry- breaking color magnetic interactions are included, whereas ISGW only included the symmetry breaking due to the heavy quark kinetic energy,

• The ISGW prescription for connecting its quark model form factor to physical form factors is modified to the consistent with the constraints of heavy quark symmetry breaking at order 1/mQ

• Relativistic corrections to the axial vector coupling constants are taken into account

• More realistic form factor shapes based on the measured pion form factor, are employed

Expressions for Bc A Transition Form Factors

where

CALCULATION OF THE FORM FACTORS IN ISGW II MODEL

with appropriate

Bc A’ Transition Form Factors

The values of parameter for s-wave and p-wave mesons in the ISGW II quark model

Form factors of Bc A transitions at q2 = tm

Form factors of Bc A transitions at q2 = tm (in BSW model type notations)

Form factors of Bc A’ transitions at q2 = tm

Form factors of Bc A’ transitions at q2 = tm (in BSW model type notations)

DECAY AMPLITUDES AND RATES

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

0 0 ,wVA H B V J A J B A J V J B

*( , ) 0A A AA k A m f

*( , ) 0A A AA k A m f

* * *( , ) ( ) ( )( ) ( )( ) ,c c c c cA c B B B A B B AA k V B k l c k k k c k k k

* * *( , ) ( ) ( )( ) ( )( )c c c c cA c B B B A B B AA k V B k r s k k k s k k k

( , ) ( ) ' ( ) ( ) ,c c cA c B B A B AA k A B k iq k k k k

' ''( , ) ( ) ( ) ( )c c cA c B B A B AA k A B k i v k k k k

The matrix element for various Bc V transition are given by

It may be noted that Bc A/A’ transition form factors in ISGW2 framework are related to BSW type form factor notations

Since, final states of BcVA/AA carry spin degrees of freedom, the decay

amplitudes in terms of helicities, like those in the Bc VV decays, can

be generally described by

with M = V or A.

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

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

2 c c

c

c B P A B P AB

p m m m m m mm

ResultsBc VA Decays

Bc AA Decays

Comparison with other works

Summary of Results

THANK YOU