1. YongPyong Winter Conference on Particle Physics
Branching Ratios of Bc Meson Decaying to Axial-Vector Mesons in the Final State
Rohit Dhir
Yonsei University, Seoul, Korea
YongPyong-2013
YongPyong Winter Conference on Particle Physics
Rohit Dhir and C.S. Kim, PRD 87, (2013) arXiv:
2/25/2013

2. 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.

3. 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, 2012] 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.

4. VARIOUS QUARK LEVEL PROCESSES THAT CONTIBUTE TO THE NONLEPTONIC DECAYS

5. FACTORIZATION SCHEME (Preliminary Estimates of BRs)
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:
Class I transition (caused by color favored),
Class II transition (caused by color suppressed) and
Class III transition (caused by both color favored and color suppressed diagrams).

6. WEAK HAMILTONIAN
BOTTOM CHANGING DECAYS

7. Some More Details
Sandwiching the weak Hamiltonian between the initial and the final states, the decay amplitudes for various Bc → MA decay modes (M = V or A) can be obtained for the following three categories :
Class I transitions: decay amplitudes are proportional to a1.
Class II transitions: decay amplitudes in this class are proportional to a2.
Class III transitions: these decays are caused by the interference of color singlet and color neutral currents i.e. the amplitudes a1 and a2 interfere.
where a1,2 (μ) = c1,2 + (1/Nc ) c2,1 (μ), and Nc is the number of colors.
It may be noted that Nc, number of color degrees of freedom, may be treated as a phenomenological parameter in weak meson decays, which account for non-factorizable contributions. It implies that the effective expansion parameter is something like, 1/(4π) Nc, (1/Nc)2... or non-leading 1/ Nc terms are suppressed by some reason.
In order to study the variation in decay rates and branching ratios, we effectively vary the parameter Nc from 3 to 10. The obtained results are thus presented as an average with uncertainties between branching ratios at Nc = 3 to Nc = 10.
Taking in to account the constructive interference observed for B meson decays involving both the color favored and color suppressed diagrams. We use the ratio a2/a1 to be positive in the present calculations.

8. For Isovector : Isoscalars: where
AXIAL-VECTOR MESON SPECTROSCOPY Experimentally, two types of the axial-vector mesons exist i.e and
For
Isovector :
Isoscalars:
where

9. For
Isovector :
Isoscalars:
where
with

10. Mixing of Charmed and Strange Charmed states
MIXING IN STARNGE AND CHARM AXIAL-VECTOR MESONS and Mixing of Strange states
Mixing of Charmed and Strange Charmed states
&

11. 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
Mixing of Charmed states
Mixing of strange-Charmed states
with

12. DECAY CONSTANTS (in GeV) OF THE AXIAL-VECTOR MESONS
The decay constants for axial-vector mesons are defined by the matrix elements given in the previous section. It may be pointed out that the axial-vector meson states are represented by 3×3
matrix and they transform under the charge conjugation as
Since the weak axial-vector current transfers as under charge conjugation, the decay constant of the 1P1 meson should vanish in the SU(3) flavor limit [M. Suzuki PRD 47, 1252 (1993); 55, 2840 (1997)].
Experimental information based on τ decays gives decay constant fK1 (1270) = ± GeV [H. Y. Cheng, PRD 67, (2003)], while decay constant for K1(1.400) can be obtained from relation fK1(1.400)/fK1(1.270) = cot θ1 i.e. fK1 (1.400) = (−0.109 ± 0.12) GeV, for θ1 = −37◦ used in the present work .
Numerous analysis based on phenomenological studies indicate that strange axial vector meson states mixing angle θK lies in the vicinity of ∼ 35◦ and ∼ 55◦. Recently, it has been pointed out [H.Y. Cheng, PLB 707, 116 (2012)] that mixing angle θ1 ∼ 35◦ is preferred over ∼ 55◦, we use θ1 = −37◦ in our numerical calculations. It is based on the observation that choice of angle for f − f′ and h − h′ mixing schemes (which are close to ideal mixing) are intimately related to choice of mixing angle θ1.

13. In case of non-strange axial vector mesons, Nardulli and Pham [36] used mixing angle for strange axial vector mesons and SU(3) symmetry to determine fa1 = GeV. Since, a1 and f1 lies in the same nonet we assume ff1 ≈ fa1 under SU(3) symmetry. Due to charge conjugation invariance decay constants for 1P1 nonstrange neutral mesons b0, h1(1.235), h1(1.170), and h′1(1.380) vanish. Also, owing to G-parity conservation in the isospin limit decay constant fb1 = 0.
SUMMARY OF DECAY CONSTANTS (in GeV)

14. 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

15. CALCULATION OF THE FORM FACTORS IN ISGW II MODEL
Expressions for Bc A Transition Form Factors
where

16. Bc A’ Transition Form Factors
with appropriate

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

19. Form factors of Bc A transitions at q2 = tm
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

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

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

22. DECAY AMPLITUDES AND RATES
The factorization Scheme expresses the decay amplitudes as a product of matrix element of the weak currents
M = V or A.
The matrix element of current between mesons states are expressed as
DECAY AMPLITUDES AND RATES

23. Since, final states of BcVA/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 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.

24. 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

25. Results for Bc AA Decays

30. Comparison with other works
For references [13, 15, 13, 25] plz see arXiv:

31. Summary of Results

32. THANK YOU