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Theoretical description of pygmy (dipole) resonances

Stable and unstable nuclei with neutron excess ($N>Z$) show - in the isovector dipole transition strength distribution - a small hump around the neutron emission threshold energy known as Pygmy Dipole Resonance (PDR). One of its main features is the isospin mixing allowing the experimental studies with both isovector and isoscalar probes. Different theoretical approaches and methodologies are used to deduce the characteristics of the PDR. In this Chapter, the various mean-field theories and their extensions, devoted to understand and reproduce the strength distribution of these low-lying dipole states, are summarised. Special attention is dedicated to the calculations of the inelastic cross section, aspect that is particularly important in the investigation with isoscalar probes, such as $α$-particles or $^{17}$O. The relevance of the radial form factors is presented in relation to the inelastic cross-section calculations.