Isgur-Wise functions for $\boldsymbol{Λ_b \to Λ_c\left({1 \over 2}^\pm \right)}$ transitions in the Bakamjian-Thomas Relativistic Quark Model

Abstract

We study the transitions ${Λ_b \to Λ_c\left({1 \over 2}^\pm \right)}$ in the Bakamjian-Thomas (BT) relativistic quark model formalism, which describes hadrons with a fixed number of constituents. In the heavy quark limit, the BT model yields covariant form factors and Isgur-Wise (IW) scaling, regardless of the spectroscopic model used to describe the bound states. It has been extensively applied to heavy mesons where subtle properties of the IW limit of QCD, including the Bjorken-Uraltsev sum rules, have been shown to be satisfied. The present paper, where the BT construction is applied to baryons, is unavoidably technical because one is dealing with a three-body problem. The complications originate from the natural choice of the Jacobi coordinates ${\vec ρ}$ (relative space coordinate between the two light spectator quarks) and ${\vec λ}$ (relative space coordinate between the center-of-mass of the two light quarks and the heavy quark). The corresponding orbital angular momenta are denoted by ${\vec \ell}_ρ$ and ${\vec \ell}_λ$, with ${\vec \ell}_ρ+ {\vec \ell}_λ= {\vec L}$. For the transitions $Λ_b \to Λ_c\left({1 \over 2}^\pm\right)$, i.e. $L = 0 \to L = 0$ or $L = 0 \to L = 1$, one can see that the moduli $\ell_ρ$ and $\ell_λ$ can take an infinite number of values. For $L = 0 \to L = 0$ one has the constraint $\ell_λ= \ell_ρ$ and for $L = 0 \to L = 1$ the constraint is $\ell_λ= \ell_ρ\pm 1$. We compute explicitly the IW function $ξ_Λ(w)$ in the elastic case $Λ_b \left({1 \over 2}^+ \right) \to Λ_c \left({1 \over 2}^+ \right)$ and the much more involved IW function $σ_Λ(w)$ in the inelastic case $Λ_b \left({1 \over 2}^+ \right) \to Λ_c \left({1 \over 2}^- \right)$. These functions exhibit the expected properties of covariance and IW scaling.