Isgur-Wise functions and unitary representations of the Lorentz group : the baryon case j = 0

Abstract

We propose a group theoretical method to study Isgur-Wise functions. A current matrix element splits into a heavy quark matrix element and an overlap of the initial and final clouds, related to the IW functions, that contain the long distance physics. The light cloud belongs to the Hilbert space of a unitary representation of the Lorentz group. Decomposing into irreducible representations one obtains the IW function as an integral formula, superposition of irreducible IW functions with positive measures, providing positivity bounds on its derivatives. Our method is equivalent to the sum rule approach, but sheds another light on the physics and summarizes and gives all its possible constraints. We expose the general formalism, thoroughly applying it to the case j = 0 for the light cloud, relevant to the semileptonic decay Lambda_b -> Lambda_c + l + nu. In this case, the principal series of the representations contribute, and also the supplementary series. We recover the bound for the curvature of the j = 0 IW function xi_Lambda (w) that we did obtain from the sum rule method, and we get new bounds for higher derivatives. We demonstrate also that if the lower bound for the curvature is saturated, then xi_Lambda (w) is completely determined, given by an explicit elementary function. We give criteria to decide if any ansatz for the Isgur-Wise function is compatible or not with the sum rules. We apply the method to some simple model forms proposed in the literature. Dealing with a Hilbert space, the sum rules are convergent, but this feature does not survive hard gluon radiative corrections.

70 pages

References in corpus (12)

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• Lower bounds on the curvature of the Isgur-Wise function
• Sum rules in the heavy quark limit of QCD
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• A Few Aspects of Heavy Quark Expansion
• Bound on the curvature of the Isgur-Wise function of the baryon semileptonic decay Lambda_b -> Lambda_c + l + nu
• Perturbative corrections to curvature sum rules
• Bounds on the derivatives of the Isgur-Wise function with a non-relativistic light quark

Cited by in corpus (9)

• Measurement of the shape of the ${\Lambda }_b^0\to {\Lambda }_c^{+}{\mu }^{-}\overline{\nu }$ differential decay rate
• Heavy baryon wave functions, Bakamjian-Thomas approach to form factors, and observables in ${\Lambda }_b\to {\Lambda }_c\left({\frac{1}{2}}^{\pm }\right)\ell \overline{\nu }$ transitions
• Sum rules of Bjorken-Uraltsev type in the Bakamjian-Thomas relativistic quark model
• New results on the baryon decay Lambda_b -> Lambda_c ell nu in Heavy Quark Effective Theory
• Isgur-Wise functions and unitary representations of the Lorentz group : the meson case j = 1/2
• Unexpected pattern of transitions to radial excitations by heavy quark current
• Simple operator formulation of the Bakamjian-Thomas approach to heavy quark current , with generalisation to HQET, and with applications to transitions of Lambda-b
• Isgur-Wise functions, Bjorken-Uraltsev Sum Rules and their Lorentz group interpretation, In Memoriam Nikolai Uraltsev
• On the form factors of semileptonic baryon decays in Heavy Quark Effective Theory