  We develop a new theoretical framework for the treatment of heavy quark (HQ)
resonances within heavy quark effective theory (HQET). This framework uses
on-shell recursion techniques to express the resonant amplitude as a product of
on-shell subamplitudes, which allows one to employ a form-factor representation
of the hadronic matrix elements and to obtain an HQ expansion, but at the price
of introducing complex momenta. We construct a generalized "holomorphic HQET"
onto which such complex-momentum matrix elements can be matched, and we show
that $PT$ symmetry ensures the Isgur-Wise functions (and the perturbative
corrections) become holomorphic functions of the complex recoil parameter with
real coefficients. They are thus an analytic continuation of the standard HQET
description. This framework admits a HQ hadron (strong decay) width expansion.
At second order, we show it is in good agreement with data for the
$B_{1(2)}^{(*)}$ and $D_{1(2)}^{(*)}$ HQ doublets. Taking the $\bar{B} \to
(D_1^*(1^-) \to D\pi)l\nu$ system as an example, we compute the holomorphic
HQET expansion to first order, as well as the complex-momentum on-shell
subamplitudes. A toy numerical study of the resulting differential rates
demonstrates that this framework generates HQ resonance lineshapes with large
tails, resembling those seen in data.
