⚛️ phenomenology

Two-variable longitudinal distribution of the $\Lambda_b$ in a three-quark light-front model

This paper presents a covariant three-quark light-front model calculation of the normalized two-variable longitudinal distribution for the $\Lambda_b$ baryon, deriving a symmetric valence wave function that quantifies heavy-light momentum partitioning and internal light-quark correlations without relying on an elementary diquark approximation.

Original authors: Shuai Xu, Xiao-Nan Li

Published 2026-10-16✓ Author reviewed ⓘ

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CC BY 4.0

Original authors: Shuai Xu, Xiao-Nan Li

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of every atom lies a world of constant motion, governed by the invisible forces that bind the universe together. At the center of this subatomic realm are protons and neutrons, the building blocks of matter, which are themselves made of even smaller particles called quarks. These quarks do not sit still; they zip around at incredible speeds, carrying energy and momentum. To understand how these particles behave, physicists use a special way of looking at the universe called light-front dynamics. Imagine taking a snapshot of a speeding car not from the side, but from a camera moving alongside it at the same speed. In this view, the car's internal parts seem frozen in a specific arrangement, making it easier to see how the pieces fit together and share the load of the journey. This perspective is crucial for studying heavy particles, like the Lambda-b baryon, which contains a very heavy bottom quark paired with two lighter quarks. Scientists care deeply about how these particles share their momentum because this internal arrangement dictates how they interact with the rest of the universe, influencing everything from the stability of matter to the rare decays that reveal the secrets of fundamental physics.

A team of researchers recently turned their attention to this specific heavy-light particle, the Lambda-b, to map out exactly how its three constituent quarks share the forward motion of the baryon. While many previous studies simplified the picture by treating the two lighter quarks as a single, fused unit, this team chose to keep them distinct. They wanted to see not just how much momentum the heavy bottom quark carries compared to the pair, but also how the two light quarks share the remaining momentum between themselves. Using a sophisticated mathematical model that treats the particle as a system of three moving parts, they calculated a detailed distribution map. This map shows the probability of finding the light quarks with specific amounts of momentum, revealing a structure that is far more nuanced than a simple split between a heavy core and a light shell.

The results of this calculation paint a clear picture of the particle's internal life. The heavy bottom quark dominates the journey, carrying nearly seventy-four percent of the total forward momentum. The two lighter quarks, the up and down quarks, share the remaining twenty-six percent almost equally between them. On average, each light quark holds about thirteen percent of the total momentum. This finding confirms that while the heavy quark is the primary driver, the light quarks are not merely passive passengers; they maintain a dynamic, balanced relationship with each other. The researchers found that the two light quarks prefer to be in a state where they have nearly identical momentum, rather than one hogging the energy while the other lags behind. This internal balance is a key feature of the particle's structure, distinct from the overall split between the heavy and light sectors.

To ensure these numbers were not just a fluke of a single calculation, the team tested how sensitive their results were to changes in the input values, such as the assumed masses of the quarks and the parameters that describe how tightly they are bound. They discovered that the overall split of momentum between the heavy quark and the light pair is primarily controlled by the mass of the heavy quark and the scale of the binding between the heavy and light sectors. In contrast, the way the two light quarks share their small slice of momentum is most sensitive to the specific parameters governing their internal interaction. This separation of influences is a significant insight, showing that the heavy-light relationship and the light-light relationship are governed by different physical levers. The study provides a precise, finite-mass benchmark for how three quarks arrange themselves longitudinally, offering a new standard for comparing different theoretical models.

It is important to note that this work describes a specific model-scale view of the particle, rather than a final, renormalized measurement from a high-energy experiment. The researchers have calculated a valence distribution, which represents the core three-quark structure without the complex cloud of virtual particles that appear at higher energy scales. This distinction is vital because the relationship between this model and the more abstract, high-energy descriptions used in quantum field theory requires further mathematical translation. Nevertheless, the map they have drawn is a robust and self-consistent description of the particle's internal momentum sharing. It offers a compact and detailed reference point for future studies, allowing physicists to test their theories against a clear, three-dimensional picture of how the heavy and light components of the Lambda-b baryon coexist and move together.

Technical Summary: Two-variable longitudinal distribution of the $\Lambda_b$ in a three-quark light-front model

Problem and Motivation
The paper addresses the need for a minimal longitudinal description of genuine three-quark baryons, specifically the heavy-light $\Lambda_b = (ud)b$ system. While light-front dynamics offers a relativistic framework for hadron structure, standard quark-diquark reductions compress the two light constituents into a single effective degree of freedom. This reduction fails to retain the internal momentum sharing between the light quarks as an independent longitudinal variable. Conversely, Heavy Quark Effective Theory (HQET) descriptions of the light sector ( $\omega_u, \omega_d$ ) involve renormalized matrix elements with explicit scale dependence and perturbative matching kernels that differ from finite-mass constituent models. The authors aim to determine a normalized, two-variable longitudinal distribution that explicitly resolves both the heavy-light momentum partition and the internal $u$ - $d$ sharing without introducing an elementary diquark or relying on the renormalization-scale dependence of HQET distribution amplitudes (LCDAs).

Methodology
The authors utilize a finite-mass, covariant three-quark light-front wave function (LFWF) previously developed by Zhao et al. [23]. The methodology proceeds as follows:

Wave Function Construction: The $\Lambda_b$ is treated as a three-quark state with a heavy $b$ quark and a light scalar $ud$ pair. The spin-momentum wave function employs a $C\gamma_5\gamma_+$ projection for the scalar $ud$ channel. The momentum dependence is modeled using two independent Gaussian parameters: $\beta_{b,ud}$ governing the relative motion of the $b$ quark against the $ud$ subsystem, and $\beta_{ud}$ governing the internal relative motion of the $u$ and $d$ quarks.
Projection and Integration: The authors derive the two-variable longitudinal distribution $\Psi^{LF}_2(x_u, x_d)$ by performing a "good-component" projection ( $\gamma_+$ ) of the covariant wave function. Crucially, they integrate over both independent transverse momenta ( $k_{u\perp}, k_{d\perp}$ ) without reducing the system to a diquark.
Normalization: The distribution is normalized such that the integral over the physical triangle ( $x_u > 0, x_d > 0, x_u + x_d < 1$ ) equals unity. This isolates the longitudinal shape from local-current couplings, distinguishing the result as a model-scale valence quantity rather than a renormalized HQET LCDA.
Numerical Implementation: The calculation uses central constituent masses ( $m_q=0.250$ GeV, $m_b=4.800$ GeV) and Gaussian parameters from Ref. [23]. Sensitivity is analyzed by varying these inputs within specified ranges. The internal sharing is characterized by the variable $\zeta = (x_u - x_d)/(x_u + x_d)$ , while the total light-pair momentum is $s = x_u + x_d$ .

Key Results
For the central parameter set, the study yields the following quantitative findings:

Momentum Fractions: The heavy quark carries the majority of the longitudinal momentum, $\langle x_b \rangle = 0.739$ , while the light quarks share the remainder symmetrically, $\langle x_u \rangle = \langle x_d \rangle = 0.130$ . The total light-pair momentum is $\langle s \rangle = 0.261$ .
Internal Sharing: The distribution is symmetric under $x_u \leftrightarrow x_d$ . Moments of the sharing variable $\zeta$ indicate a preference for equal sharing: $\langle \zeta^2 \rangle = 0.153$ and $\langle \zeta^4 \rangle = 0.055$ . These values are lower than those for a uniform distribution, confirming that the light pair concentrates near $x_u \simeq x_d$ .
Factorization: The ratio $\langle s^2 \zeta^2 \rangle / (\langle s^2 \rangle \langle \zeta^2 \rangle) = 0.877$ deviates from unity, demonstrating that the total light-pair momentum and its internal sharing do not factorize exactly.
Parameter Sensitivity:
- The heavy-light momentum partition ( $\langle x_b \rangle, \langle s \rangle$ ) is primarily controlled by the heavy-light Gaussian parameter $\beta_{b,ud}$ and the heavy quark mass $m_b$ .
- The internal $u$ - $d$ sharing (moments of $\zeta$ ) is most sensitive to the internal light-quark parameter $\beta_{ud}$ , with variations in $\beta_{ud}$ changing $\langle \zeta^2 \rangle$ by up to 9.2% and $\langle \zeta^4 \rangle$ by up to 15.1%.
- Constituent light-quark masses ( $m_q$ ) have negligible effects on these observables.

Significance and Claims
The paper claims to provide a "compact finite-mass benchmark" for the three-quark longitudinal structure of the $\Lambda_b$ . Its primary significance lies in:

Resolution of Scales: It successfully separates the heavy-light momentum scale from the internal light-pair scale within a single two-variable distribution, a feature lost in quark-diquark models.
Model-Scale Definition: The authors explicitly state that $\Psi^{LF}_2(x_u, x_d)$ is a model-scale valence projection. It is not identified directly with a renormalized HQET LCDA, as it lacks the ultraviolet operator definition, renormalization-scale dependence, and perturbative radiative tails inherent to QCD LCDAs.
Distinct Observables: The work establishes distinct observables for heavy-light momentum sharing and correlations inside the light pair, offering a basis for comparing different light-front wave functions.

The authors maintain a modest scope, noting that establishing a quantitative link to HQET LCDAs would require separate matching and evolution analyses. The reported uncertainties are presented strictly as model-sensitivity estimates regarding constituent masses and Gaussian parameters, excluding variations in the radial ansatz, higher Fock components, or matching uncertainties.

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