Abstract
We review a series of unitarization techniques that have been used during the last decades, many of them in connection with the advent and development of current algebra and later of Chiral Perturbation Theory. Several methods are discussed like the generalized effective-range expansion, K-matrix approach, Inverse Amplitude Method, Padé approximants and the N / D method. More details are given for the latter though. We also consider how to implement them in order to correct by final-state interactions. In connection with this some other methods are also introduced like the expansion of the inverse of the form factor, the Omnés solution, generalization to coupled channels and the Khuri-Treiman formalism, among others.
Highlights
The effective chiral Lagrangian formalism has become a well-established methodology to study the interactions of the Goldstone bosons with or without other particle species, like, for example, pions and nucleons, respectively [1,2]
In the early days of partial conservation of the axial-vector current (PCAC), soft pions theorems and realizations based on chiral Lagrangians, it was customary to refer to effective-range expansion (ERE) as a unitarization method based on the identification of a remnant in the inverse of a partial-wave amplitudes (PWAs) free of right-hand cut (RHC) which was expanded in powers of p2
Along the discussion we introduce the way final-state interactions (FSI) are treated in Reference [19], as it is probably the first paper in which next-to-leading order (NLO) Chiral Perturbation Theory (ChPT) is unitarized to account for FSI following the basic notions of unitarity, Watson final-state theorem and use of an Omnès function, which are the basic elements usually employed in the different modern approaches to resum FSI [22,67]
Summary
The effective chiral Lagrangian formalism has become a well-established methodology to study the interactions of the Goldstone bosons with or without other particle species, like, for example, pions and nucleons, respectively [1,2]. It is typically simpler and much more predictive to implement lower-order calculations of ChPT within non-perturbative methods (several examples are given along this review related to meson-meson scattering and spectroscopy, like for example, the ππ phase shifts, scalar and vector pion form factors, impact of the resonances f 0 (500) and ρ(770) in the low-energy phenomenology, η → 3π decays, etc.) Even if such corrections could be computed, the resultant renormalized perturbation series would probably diverge, since the perturbation parameter has the strength characteristic of strong interactions. One close field is infinite nuclear matter, where resummation techniques based on the N/D method, discussed, were applied in References [43,44,45,46] to work out the NN scattering amplitude in the nuclear medium From this result, equations of state for neutron and symmetric nuclear matter were derived each containing only a free parameter, and showing themselves as very successful from the phenomenological point of view. The last section contains our conclusions with extra discussions included
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