Mathematical aspects of the nuclear glory phenomenon: from backward focusing to Chebyshev polynomials

Abstract

The angular dependence of the cumulative particles production off nuclei near the kinematical boundary for multistep process is defined by characteristic polynomials in angular variables $J_N^2(z_N^\theta)$, where $\theta$ is the polar angle defining the momentum of the final (cumulative) particle, $z_N^\theta = cos (\theta/N)$, the integer $N$ being the multiplicity of the process (the number of interactions). Physical argumentation, exploring the small phase space method, leads to the appearance of equations for these polynomials $J_N^2[cos(\pi/N)]=0$. The recurrent relations between polynomials with different $N$ are obtained, and their connection with known in mathematics Chebyshev polynomials of 2-d kind is established. As a result of this equality, differential cross section of the cumulative particle production has characteristic behaviour $d\sigma \sim 1/ \sqrt {\pi - \theta}$ at $\theta \sim \pi$ (the backward focusing effect). Such behaviour takes place for any multiplicity of the interaction, beginning with $n=3$, elastic or inelastic (with resonances excitations in intermediate states), and can be called the nuclear glory phenomenon, or 'Buddha's light' of cumulative particles.