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

TL;DR

This paper applies the Bakamjian-Thomas relativistic quark model to study Isgur-Wise functions for $oldsymbol{ ext{Lambda}_b o ext{Lambda}_c}$ transitions, extending previous meson analyses to baryons with detailed three-body calculations.

Contribution

It provides explicit calculations of Isgur-Wise functions for baryon transitions within the BT model, demonstrating covariance and IW scaling in a complex three-body framework.

Findings

IW functions exhibit expected covariance and scaling properties

Explicit expressions for elastic and inelastic IW functions

Application of BT model to baryons extends previous meson results

Abstract

We study the transitions ${\Lambda_b \to \Lambda_c\left({1 \over 2}^\pm \right)}$ 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 ${\vec \rho}$ (relative space coordinate between the two light spectator quarks) and ${\vec \lambda}$ (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 ${\vec \ell}_\rho$ and ${\vec \ell}_\lambda$, with ${\vec \ell}_\rho + {\vec \ell}_\lambda = {\vec L}$. For the transitions $\Lambda_b \to \Lambda_c\left({1 \over 2}^\pm\right)$, i.e. $L = 0 \to L = 0$ or $L = 0 \to L = 1$, one can see that the moduli $\ell_\rho$ and $\ell_\lambda$ can take an infinite number of values. For $L = 0 \to L = 0$ one has the constraint $\ell_\lambda = \ell_\rho$ and for $L = 0 \to L = 1$ the constraint is $\ell_\lambda = \ell_\rho \pm 1$. We compute explicitly the IW function $\xi_\Lambda (w)$ in the elastic case $\Lambda_b \left({1 \over 2}^+ \right) \to \Lambda_c \left({1 \over 2}^+ \right)$ and the much more involved IW function $\sigma_\Lambda (w)$ in the inelastic case $\Lambda_b \left({1 \over 2}^+ \right) \to \Lambda_c \left({1 \over 2}^- \right)$. These functions exhibit the expected properties of covariance and IW scaling.