Two-variable longitudinal distribution of the ${\mathrm{\Lambda }}_b$ in a three-quark light-front model

We determine a normalized two-variable longitudinal distribution of the ${\mathrm{\Lambda }}_b=(ud)b$ baryon from a covariant three-quark light-front wave function. The scalar $ud$ channel is selected with the good-component structure $C{\gamma }_5{\gamma }^{+}$, and the two independent transverse momenta are integrated without introducing an elementary diquark. The resulting model-scale valence distribution, ${\mathrm{\Psi }}_2^{\mathrm{L}\mathrm{F}}(x_u,x_d)$, retains both the heavy–light and internal-light momentum scales of the parent wave function. It is symmetric under $x_u\leftrightarrow x_d$, is concentrated where both light-quark fractions are small, and separates the total light-pair momentum from its internal sharing. For the central parameter set we find $⟨x_u⟩=⟨x_d⟩=0.130$, $⟨x_b⟩=0.739$, $⟨{\zeta }^2⟩=0.153$, and $⟨{\zeta }^4⟩=0.055$, where $\zeta =(x_u-x_d)\mathrm{/}(x_u+x_d)$. Parameter variations give $⟨x_u⟩=0.130_{-0.007}^{+0.007}$, $⟨x_b⟩=0.739_{-0.014}^{+0.014}$, $⟨{\zeta }^2⟩=0.153_{-0.016}^{+0.014}$, and $⟨{\zeta }^4⟩=0.055_{-0.009}^{+0.009}$. The heavy–light momentum partition is controlled mainly by ${\beta }_{b,ud}$ and $m_b$, whereas the internal $u$–$d$ sharing is most sensitive to ${\beta }_{ud}$. The result provides a compact finite-mass benchmark for three-quark longitudinal structure, with distinct observables for heavy–light momentum sharing and correlations inside the light pair.