Resonances parameterization by Jost function zeros

R. O. BARRACHINA; P. A. MACRI

Lanzhou

Conferencia; XXVII International Conference on the Physics of Electronic and Atomic Collisions; 2013

One of the main goals of Scattering Theory is to predict the behavior of cross sections as a function of the parameters which define the interaction. Frequently, full knowledge of these parameters is not accessible but instead some spectral information of the interaction is available as, for instance, the position of bound, virtual and resonant states.  In fact, most of the striking phenomena observed in scattering experiments~cite{Taylor1972} can be described in terms of these spectral characteristics of the interaction.  In particular, resonances which display a sudden increase of the cross section at an energy $E_o$ are considered to be the manifestation of a transitorily bound state with energy $E_o$ and half life $1/Gamma$, and described by the celebrated Breit-Wigner formula ~cite{Breit1936} $sigma propto  (Gamma^2 / 4) / ((E - E_o)^2 + Gamma^2 / 4)$. This formula provided one of the first descriptions of a specific feature of a cross section in terms of the spectral properties of the interaction. In spite of its enormous success in nuclear, atomic and molecular physics, this formula presents severe shortcomings if (i) the energy $E_o$ lies close to threshold, (ii) the half life $1/Gamma$ is not large enough, (iii) the interaction produces a noticeable background phase shift before the onset of the resonance or (iv) there is another proximate resonant state.In this work, we propose a new spectral formula based on one or more $ell-$wave Jost function´s zeros on the complex momentum plane~cite{Jost1951,Macri2013}. We demonstrate that it is possible to solve difficult (i) by considering the correct low energy expansion of the Jost function. In this way, a momentum-dependent half life is introduced allowing to find a formula which satisfies the well-known partial-wave threshold law $sigma propto E^{2ell}$~cite{Wigner1948}. Difficulty (ii) appears when the net repulsive part of the potential barrier is weaker than a centrifugal-like term. It can show up even for $s-$waves in systems with "negative" dipole polarizabilities as observed in the $5^2G$ partial photodetachment of K$^-$ ~cite{Lindah2012}. This difficulty can be solved by considering the contribution of the imaginary part of the Jost zero to the resonant position which is ignored in the Breit-Wigner formula. Difficulty (iii) can be solved by means of a non-spectral contribution to the background Jost function which plays the role of the well-known Fanos´s shape parameter $q$~cite{Fano1961}. Finally, shortcoming (iv) can be overcame simply by considering the contribution of two or more relevant zeros.We finally demonstrate that the obtained formula can be extended to inelastic processes.