Isgur-Wise functions and unitary representations of the Lorentz group : the meson case j = 1/2

TL;DR

This paper uses group theory to analyze Isgur-Wise functions for mesons with j=1/2, deriving bounds, integral representations, and criteria for model compatibility, advancing understanding of heavy meson decay form factors.

Contribution

It extends the group theoretical approach to mesons with j=1/2, providing new bounds, integral formulas, and criteria to test Isgur-Wise function models against sum rule constraints.

Findings

Derived integral representation for i(w) with positive measure

Established bounds for slope, curvature, and higher derivatives of i(w)

Identified conditions under which ansatz functions satisfy sum rule constraints

Abstract

We pursue the group theoretical method to study Isgur-Wise functions. We apply the general formalism, formerly applied to the baryon case j^P = 0^+ (for \Lambda_b -> \Lambda_c \ell \nu), to mesons with j^P = 1/2^-, i.e. $\overline{B} -> D(D^{(*)})\ell\nu. In this case, more involved from the angular momentum point of view, only the principal series of unitary representations of the Lorentz group contribute. We obtain an integral representation for the IW function xi(w) with a positive measure, recover the bounds for the slope and the curvature of xi(w) obtained from the Bjorken-Uraltsev sum rule method, and get new bounds for higher derivatives. We demonstrate also that if the lower bound for the slope is saturated, the measure is a delta-function, and xi(w) is given by an explicit elementary function. Inverting the integral formula, we obtain the measure in terms of the IW function, allowing to formulate criteria to decide if a given ansatz for the Isgur-Wise function is compatible or not with the sum rule constraints. Moreover, we have obtained an upper bound on the IW function valid for any value of w. We compare these theoretical constraints to a number of forms for \xi(w) proposed in the literature. The "dipole" function \xi(w) = (2/(w+1))^(2c) satisfies all constraints for c \geq 3/4, while the QCD Sum Rule result including condensates does not satisfy them. Special care is devoted to the Bakamjian-Thomas relativistic quark model in the heavy quark limit and to the description of the Lorentz group representation that underlies this model. Consistently, the IW function satisfies all Lorentz group criteria for any explicit form of the meson Hamiltonian at rest.