[Paper Review] Study of $\Lambda_b ightarrow~ \Lambda l^+l^-$ and $\Lambda_b ightarrow p l \bar{ u}$ decays in the Bethe-Salpeter equation approach

Y. Liu, L. L. Liu+1 more|arXiv (Cornell University)|Mar 24, 2015

Particle physics theoretical and experimental studies4 references3 citations

TL;DR

This study calculates the form factors for Λb → Λl⁺l⁻ and Λb → pl̄ν decays using a quark-diquark model based on the Bethe-Salpeter (BS) equation approach within SU(6) spin-flavor symmetry. The authors find that their predicted branching ratio for Λb → Λμ⁺μ⁻ is consistent with LHCb experimental data, except at high momentum transfer, suggesting room for new physics. The results for Λb → pl̄ν are of the same order as lattice QCD and constituent quark model predictions, but differ significantly from QCD sum rules by up to two orders of magnitude.

ABSTRACT

In our previous work, based on the $SU(6)$ spin-flavor wave function, we regard $\Lambda$ and $p$ as composed of different quark-diquark configurations and established the Bethe-Salpeter (BS) equations of configurations for quark and scalar diquark. In our present work, we apply this model to calculate the form factors of the semileptonic transitions $\Lambda_b ightarrow\Lambda l^+l^-$ $(l=\mu,e, au)$ and $\Lambda_b ightarrow p l\bar{ u}$ within the Standard Model (SM). The decay $\Lambda_b ightarrow\Lambda \mu^+\mu^-$ is especially interesting since it has been measured in CDF and LHCb Collaborations and this rare decay is very sensitive to new physics effects. The decay $\Lambda_b ightarrow p l\bar{ u}$ is a promising mode for the measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$ at the Large Hadron Collider. In our calculations, depending on the ranges of the parameters in the model including the diquark mass and the interaction strength between the quark and the diquark in the kernel of the BS equation, we find that the branching ratio of $\Lambda_b ightarrow\Lambda\mu^+\mu^-$ in our model is consistent with the experimental data and the current experimental results from LHCb agree with the differential branching ratio of $\Lambda_b ightarrow\Lambda\mu^+\mu^-$ from our calculation except at the lager momentum transfer region. This indicates that there is still room for possible new physics effects. We also give comparisions of the total branching ratios of $\Lambda_b ightarrow\Lambda l^+l^-$ and $\Lambda_b ightarrow p l\bar{ u}$ with those given by other phenomenological methods.

Motivation & Objective

• To calculate the form factors for semileptonic Λb decays using a relativistic quark-diquark model based on the Bethe-Salpeter equation.
• To test the model's predictions against experimental data from CDF and LHCb, particularly for the rare decay Λb → Λμ⁺μ⁻.
• To provide a theoretical estimate of the branching ratio for Λb → pl̄ν to aid in the determination of the CKM matrix element |Vub| at the LHC.
• To compare results with other phenomenological methods such as QCD sum rules, lattice QCD, and constituent quark models.

Proposed method

• The study employs a quark-diquark model where Λ and p are described as bound states of a quark and a scalar diquark, using SU(6) spin-flavor wave functions.
• The Bethe-Salpeter (BS) equations for the quark and scalar diquark configurations are solved in the covariant instantaneous approximation with a kernel including scalar confinement and one-gluon exchange.
• The BS wave functions for Λb, Λ, and p are used to compute the hadronic matrix elements required for form factors in Λb → Λl⁺l⁻ and Λb → pl̄ν decays.
• The form factors are derived from the matrix elements of the current ⟨Λ|¯sΓb|Λb⟩, with the decay amplitudes constructed using the weak Hamiltonian and the heavy quark effective theory (HQET) framework.
• Numerical results are obtained by varying model parameters (diquark mass mD and interaction strength κ), and the branching ratios are computed over the physical phase space.

Experimental results

Research questions

• RQ1 How well does the Bethe-Salpeter equation approach in the quark-diquark model reproduce the measured branching ratio of Λb → Λμ⁺μ⁻?
• RQ2 What is the predicted differential branching ratio of Λb → Λμ⁺μ⁻ across different momentum transfer regions, and how does it compare to LHCb data?
• RQ3 How do the predicted branching ratios for Λb → pl̄ν compare with those from QCD sum rules, lattice QCD, and other phenomenological models?
• RQ4 Can the model provide a reliable estimate of |Vub| through Λb → pl̄ν decays, and what are the uncertainties due to model parameters?

Key findings

• The total branching ratio for Λb → Λμ⁺μ⁻ in the model is consistent with the experimental measurement from LHCb, validating the approach.
• The differential branching ratio for Λb → Λμ⁺μ⁻ agrees well with LHCb data across most of the momentum transfer range, except at q² > 15 GeV², where discrepancies suggest possible new physics effects.
• The predicted branching ratio for Λb → pl̄ν ranges from 3.12 to 7.06 ps⁻¹ in units of |Vub|², depending on model parameters mD and κ.
• These values are of the same order as those from lattice QCD in the heavy quark limit and constituent quark models (e.g., HONR/HOSR), but differ by up to two orders of magnitude from QCD sum rule predictions.
• The model's results for Λb → pl̄ν are in better agreement with lattice QCD and quark models than with QCD sum rules, indicating potential limitations in the latter approach for this decay.
• The study concludes that Λb → pl̄ν is a promising channel for future determination of |Vub| at the LHC, with predictions sensitive to the model parameters mD and κ.