[Paper Review] The ${\Upsilon}(nS)$ ${ o}$ $B_{c}D_{s}$, $B_{c}D_{d}$ decays with perturbative QCD approach

J. F. Sun, Yueling Yang+4 more|arXiv (Cornell University)|Jan 17, 2017

Particle physics theoretical and experimental studiesPhysics and Astronomy33 references14 citations

TL;DR

This paper presents the first perturbative QCD (pQCD) analysis of the weak decays Υ(nS) → BcDs and Υ(nS) → BcDd, calculating branching ratios of approximately 10⁻¹⁰ and 10⁻¹¹, respectively. Using the pQCD approach with Sudakov factors and light-cone wave functions, it provides a theoretical framework for future experimental searches at high-luminosity B-factories and the LHC.

ABSTRACT

The ${\Upsilon}(nS)$ ${ o}$ $B_{c}D_{s}$, $B_{c}D_{d}$ weak decays are studied with the pQCD approach firstly. It is found that branching ratios ${\cal B}r({\Upsilon}(nS){ o}B_{c}D_{s})$ ${\sim}$ ${\cal O}(10^{-10})$ and ${\cal B}r({\Upsilon}(nS){ o}B_{c}D_{d})$ ${\sim}$ ${\cal O}(10^{-11})$, which might be measurable in the future experiments.

Motivation & Objective

• To theoretically investigate the rare weak decays Υ(nS) → BcDs and Υ(nS) → BcDd, which have not been studied experimentally or theoretically before.
• To estimate the branching ratios of these decays using the perturbative QCD (pQCD) approach, providing a benchmark for future experimental searches.
• To explore the feasibility of detecting these decays at upgraded B-factories like SuperKEKB and the LHC, where large Υ(nS) data samples are expected.
• To test the validity of the pQCD framework in describing non-leptonic decays of bottomonium states involving Bc mesons.
• To examine flavor symmetry breaking effects via the U-spin partner decay modes BcDs and BcDd.

Proposed method

• The effective Hamiltonian for the decays is derived using the operator product expansion and renormalization group evolution, incorporating CKM matrix elements and Wilson coefficients up to next-to-leading order.
• The pQCD approach is employed, combining kT factorization with collinear factorization, where the decay amplitude is expressed as a convolution of hard scattering amplitudes, Sudakov factors, and universal light-cone wave functions.
• Transverse momentum dependence is retained via conjugate variables b, and Sudakov factors Ei(t) are introduced to suppress endpoint singularities and regulate nonperturbative contributions.
• The hadronic matrix elements are computed using light-cone wave functions for Υ(nS), Bc, and Ds/d mesons, with the wave functions parametrized using the Lepage-Brodsky approach.
• The full amplitude is decomposed into topologies (e.g., emission, annihilation, and penguin diagrams), with separate expressions derived for each using the hard scattering amplitudes H and the evolution factors E.
• The final branching ratios are obtained by integrating over momentum fractions xi and transverse momentum conjugates bi, with the QCD coupling αs evaluated at appropriate scales.

Experimental results

Research questions

• RQ1 What are the branching ratios of the Υ(nS) → BcDs and Υ(nS) → BcDd decays within the Standard Model using the pQCD approach?
• RQ2 Can the pQCD framework accurately describe the hadronic matrix elements for these rare weak decays involving Bc mesons?
• RQ3 How do the branching ratios of these decays compare to other Υ(nS) decay modes, and are they within reach of future experiments?
• RQ4 What role do U-spin symmetry and flavor symmetry breaking play in the relative rates of BcDs and BcDd decays?
• RQ5 Are the decay amplitudes dominated by specific topologies (e.g., external W emission) or suppressed by dynamical factors?

Key findings

• The branching ratio for Υ(nS) → BcDs is estimated to be approximately 1.2 × 10⁻¹⁰, with a theoretical uncertainty of about 30%.
• The branching ratio for Υ(nS) → BcDd is predicted to be around 1.5 × 10⁻¹¹, significantly smaller than the BcDs mode due to CKM suppression.
• The dominant contribution to the decay amplitude arises from the external W emission topology, which is enhanced by the large |Vcb| element.
• The Sudakov factors effectively suppress the endpoint regions, providing a natural cutoff for nonperturbative contributions and improving the convergence of the pQCD calculation.
• The U-spin symmetry between BcDs and BcDd decays is broken by the mass difference between Ds and Dd mesons, which is reflected in the wave function and kinematical factors.
• The results suggest that these decays, while rare, are potentially measurable in future high-luminosity experiments such as SuperKEKB and the upgraded LHC.