Lectures on scattering theory in partial-wave amplitudes

Abstract

These lectures treat scattering theory from a non-perturbative point of view. The course begins with a review of formal aspects in scattering theory, discussing the in/out states and the $S$ matrix that connects them. Unitarity relations, phase space, and the Lippmann-Schwinger equation for the $T$ matrix are discussed. The calculation of cross sections, the optical theorem and Boltzmann $H$-theorem from unitarity of the $S$ matrix are also explained. Special emphasis is given to expansions in partial-wave amplitudes of two-body scattering amplitudes, both for massive and massless particles, and to unitarity in partial-wave amplitudes. In this way, partial-wave expansions in terms of $\ell SJ$ states, and using helicity states with definite total angular momentum $J$ are discussed. Crossing symmetry is also explained, and connected to crossed channels, analyticity and crossed-channel cuts in partial-wave amplitudes. Different non-perturbative techniques and general parameterizations are developed for partial-wave amplitudes that satisfy unitarity and are consistent with the analyticity properties that they must have. Then, the $N/D$ method and additional solutions generated by adding CDD poles are covered in detail. The change to different non-physical Riemann sheets and the search for resonant poles is discussed as well. A differentiation is made between dynamically generated resonances by the degrees of freedom explicitly accounted for versus pre-existing resonances derived from short-distance dynamics. We exemplify it with the cases of the $\sigma/f_0(500)$ and the $\rho(770)$ resonances in $\pi\pi$ scattering. The reader is also introduced to final-state interactions, Watson's theorem, and general parameterizations to take them into account. It ends with an appendix dedicated to the Sugawara-Kanazawa theorem regarding the number of subtractions in dispersion relations.