⚛️ phenomenology

Weak Decays of the $\Lambda_{b}$ Baryon in Light-Front Dynamics

This paper investigates exclusive semileptonic and nonleptonic $\Lambda_{b} \to \Lambda_{c}(p)$ decays within the Standard Model using a light-front quark model enhanced by Bethe-Salpeter formalism to account for nonvalence contributions, yielding branching fractions and asymmetry ratios that are consistent with current experimental measurements.

Original authors: Chong-Chung Lih, Cong-Yu Li, Chao-Qiang Geng

Published 2026-09-16

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CC BY 4.0

Original authors: Chong-Chung Lih, Cong-Yu Li, Chao-Qiang Geng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the heart of matter, protons and neutrons are not solid, indivisible spheres but rather bustling cities of smaller particles called quarks. These quarks are bound together by a powerful force known as the strong interaction, which acts like an invisible glue holding the atomic nucleus together. While the rules governing this glue are well understood in theory, calculating exactly how these quarks move and interact inside a particle is one of the most difficult challenges in modern physics. This is because the force is so strong at these tiny scales that standard mathematical tools often fail, requiring scientists to use complex simulations to peek inside. Among the most interesting of these particles are heavy baryons, which are made of three quarks, including a very heavy one called a bottom quark. When these heavy particles decay, or break apart, they transform into lighter particles, a process that offers a rare glimpse into the fundamental laws of nature and helps physicists test whether our current understanding of the universe is complete.

A team of researchers recently turned their attention to a specific heavy baryon known as the Lambda-b, which contains a bottom quark. They wanted to understand exactly how this particle decays into two different lighter baryons: one containing a charm quark and another made of just up and down quarks. To do this, they focused on the "weak force," the mechanism responsible for these transformations, which allows quarks to change their identity. The scientists used a sophisticated computational approach called light-front dynamics, a method that treats the particle as a snapshot in time to calculate the probabilities of these changes. A key part of their work involved accounting for the complex internal structure of the baryon, specifically looking at how the quarks and a temporary pair of quarks, known as a diquark, interact during the decay. They paid special attention to a subtle effect where the internal structure of the particle might contribute to the decay in ways that are usually ignored, a phenomenon they termed "nonvalence" contributions.

The researchers found that when they included these subtle internal contributions in their calculations, the predicted rates at which the Lambda-b baryon decays matched experimental measurements very closely. For the decay into a charm-containing baryon and a lepton pair, their model predicted a branching fraction of approximately 5.39 percent, which aligns well with the range observed in real-world experiments. Similarly, for the decay into a proton and a lepton pair, they calculated a rate of about 0.033 percent, again consistent with what has been seen in particle accelerators. The study also looked at decays involving a heavy tau lepton, a heavier cousin of the electron, finding that the ratio of tau decays to lighter lepton decays was roughly 0.26 for the charm baryon and 0.62 for the proton. These numbers are crucial because they help physicists check a principle called lepton flavor universality, which states that the weak force should treat all three types of charged leptons equally, aside from differences caused by their mass. The results suggest that the Standard Model of physics holds up well in these heavy baryon decays.

Beyond the rates of decay, the team also investigated the direction in which the particles fly after the decay, a measurement known as forward-backward asymmetry. They discovered that the subtle internal contributions they had included made almost no difference to these directional predictions. This finding is significant because it suggests that the uncertainties in the measurements come more from the known properties of the particles themselves rather than from the complex, hard-to-calculate internal dynamics. The researchers also applied their method to non-leptonic decays, where the baryon transforms into another baryon and a meson, a type of particle made of a quark and an antiquark. In most cases, their predictions for these decay rates matched existing data, though they noted one specific case involving a proton and a strange meson where their calculated rate was about 50 percent higher than the current experimental value. This discrepancy highlights an area where further study is needed to fully understand the interplay of forces at work.

Ultimately, this work provides a unified and detailed picture of how heavy bottom baryons transform into lighter particles. By using a framework that directly handles the time-like nature of these decays and carefully separating the main particle components from the more complex internal fluctuations, the scientists were able to produce predictions that are robust and consistent with observation. Their results confirm that the light-front quark model is a powerful tool for describing these complex interactions, offering a reliable way to calculate the behavior of heavy baryons without needing to rely on approximations that might miss important details. The study reinforces the current understanding of the weak force in the realm of heavy baryons while pinpointing specific areas, like the decay into a proton and a strange meson, where future experiments could help refine our knowledge of the subatomic world.

Technical Summary: Weak Decays of the $\Lambda_b$ Baryon in Light-Front Dynamics

Problem Statement
Weak decays of bottom baryons, particularly the $\Lambda_b$ , serve as critical testing grounds for the Standard Model (SM) and nonperturbative Quantum Chromodynamics (QCD). While semileptonic decays offer direct access to Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and tests of lepton flavor universality (LFU), and nonleptonic decays probe hadronic weak mechanisms and CP violation, theoretical predictions are hindered by uncertainties in hadronic transition form factors. Unlike mesons, baryons possess a three-quark structure that introduces complex nonperturbative dynamics. Existing theoretical approaches, including Lattice QCD (LQCD), QCD sum rules, and perturbative QCD, often exhibit discrepancies, particularly in regions dominated by nonperturbative effects. A specific challenge in Light-Front Quantization (LFQ) is the treatment of higher-Fock state contributions (nonvalence configurations) in the timelike region ( $q^2 > 0$ ), where standard light-front quark models (LFQM) often rely on spacelike extrapolations or neglect these contributions.

Methodology
The authors employ the Light-Front Quark Model (LFQM) within the framework of light-front quantization to calculate hadronic transition form factors for $\Lambda_b \to \Lambda_c$ and $\Lambda_b \to p$ transitions directly in the physical timelike region.

Diquark Picture: The baryon is modeled as a bound state of an active quark and a spectator diquark, reducing the three-body problem to an effective two-body system.
Bethe-Salpeter Formalism: To address the contributions of nonvalence quarks in frames where $q^+ \neq 0$ , the authors utilize a Bethe-Salpeter (B-S) based light-front framework. This allows for the systematic inclusion of nonvalence configurations (where the spectator diquark is off-shell) alongside valence contributions.
Form Factor Calculation: The transition matrix elements are decomposed into valence and nonvalence regions based on the longitudinal momentum fraction $x$ . The valence region ( $0 < x < \alpha$ ) is calculated using standard LF wavefunctions, while the nonvalence region ( $\alpha < x < 1$ ) is treated via analytic continuation of the B-S amplitude. The kernel connecting these regions is approximated as a constant ( $G_{B_i B_f}$ ) due to the strong suppression of the integrand by the incoming baryon wavefunction.
Parametrization: The $q^2$ dependence of the six invariant form factors ( $f_{1,2,3}$ and $g_{1,2,3}$ ) is described using a double-pole parametrization.
Observables: Using these form factors, the authors compute branching fractions for semileptonic decays ( $\Lambda_b \to \Lambda_c(p) \ell \bar{\nu}_\ell$ ) and two-body nonleptonic decays ( $\Lambda_b \to \Lambda_c(p) M$ , where $M$ is a pseudoscalar or vector meson). They also calculate the integrated forward-backward asymmetries ( $\langle A_{FB} \rangle$ ) and the ratios $R_{\ell\tau}$ .

Key Contributions

Unified Framework: The paper provides a unified description of $\Lambda_b$ weak decays within the LFQM, explicitly incorporating nonvalence contributions derived from the B-S formalism in the timelike region.
Treatment of $f_3$ and $g_3$ : The study highlights the necessity of including the form factors $f_3$ and $g_3$ , which are often neglected in spacelike calculations ( $q^+=0$ ). The authors demonstrate that while negligible for light leptons ( $e, \mu$ ), these form factors significantly impact $\tau$ -lepton modes and temporal helicity amplitudes.
Quantification of Nonvalence Effects: The work systematically compares results with and without nonvalence contributions to isolate their impact on physical observables.

Results

Branching Fractions:
- Semileptonic: The calculated branching fractions for $\Lambda_b \to \Lambda_c \ell \bar{\nu}_\ell$ and $\Lambda_b \to \Lambda_c \tau \bar{\nu}_\tau$ are approximately $5.39\%$ and $1.41\%$ , respectively. For $\Lambda_b \to p \ell \bar{\nu}_\ell$ and $\Lambda_b \to p \tau \bar{\nu}_\tau$ , the values are $3.33 \times 10^{-4}$ and $2.08 \times 10^{-4}$ . These results are consistent with experimental measurements (PDG, LHCb).
- Nonleptonic: Predictions for $\Lambda_b \to \Lambda_c M$ and $\Lambda_b \to p M$ generally agree with current experimental data. However, the prediction for $\Lambda_b \to p D_s$ is approximately 58% higher than the PDG value, though the uncertainty ranges overlap.
Ratios ( $R_{\ell\tau}$ ): The ratios $R_{\ell\tau}(\Lambda_c) = 0.261^{+0.097}_{-0.117}$ and $R_{\ell\tau}(p) = 0.624^{+0.119}_{-0.129}$ are found to be consistent with LHCb measurements.
Forward-Backward Asymmetry: The integrated forward-backward asymmetries are calculated. The study finds that nonvalence contributions to these asymmetries are negligible compared to uncertainties induced by the shape parameter $\beta$ . A sign reversal in the asymmetry is observed for $\tau$ modes due to the finite lepton mass.
Sensitivity: The results indicate that the inclusion of nonvalence contributions has a negligible effect on semileptonic branching fractions (differences are less than 1%), except for a slight decrease in the $\Lambda_b \to p$ rate. The dominant uncertainties arise from the baryon shape parameters ( $\beta$ ).

Significance
The paper claims that its primary significance lies in providing a systematic, relativistic, and nonperturbative framework for calculating $\Lambda_b$ transition form factors directly in the physical timelike region without the need for spacelike extrapolation. By successfully incorporating nonvalence contributions via the Bethe-Salpeter formalism, the authors demonstrate that the light-front framework yields results consistent with experimental data for both semileptonic and nonleptonic decays. The work offers a compact set of theoretical inputs and observables (branching ratios, $R_{\ell\tau}$ , and asymmetries) that serve as benchmarks for testing form factor treatments and guiding future experimental measurements of angular distributions in bottom baryon decays. The study confirms that while nonvalence effects are theoretically necessary for a complete description, their numerical impact on integrated rates is small compared to other model uncertainties, validating the robustness of the LFQM approach for these processes.

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