# Mathematical aspects of the nuclear glory phenomenon; backward focusing and Chebyshev polynomials

## TL;DR

This paper explores the mathematical structure behind the nuclear glory phenomenon, revealing how Chebyshev polynomials describe the angular dependence of cumulative particle production near the backward direction, leading to a universal focusing effect.

## Contribution

It establishes a connection between characteristic polynomials in nuclear physics and Chebyshev polynomials, extending their properties to describe the backward focusing effect in cumulative particle production.

## Key findings

- Differential cross section behaves as 1/√(π - θ) near θ = π
- Characteristic polynomials are related to Chebyshev polynomials of the second kind
- Backward focusing effect occurs for any multiplicity of interactions

## 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, describing spatial momenta of the particles in intermediate and final states. Physical argumentation, exploring the small phase space method, leads to the appearance of equations for these polynomials in $cos (\theta/N)$, where $\theta$ is the polar angle of the momentum of final (cumulative) particle, the integer $N$ being the multiplicity of the process (the number of interactions). It is shown explicitly how these equations appear, and the recurrent relations between polynomials with different $N$ are obtained. Factorization properties of characteristic polynomials found previously, are extended, and their connection with known in mathematics Chebyshev polynomials of 2-d kind is established. As a result, differential cross section of the cumulative particle production has characteristic behaviour $d\sigma \sim 1/ \sqrt {\pi - \theta}$ near the strictly backward direction ($\theta = \pi$, the backward focusing effect). Such behaviour takes place for any multiplicity of the interaction, beginning with $n=3$, elastic or inelastic (with resonance excitations in intermediate states), and can be called the nuclear glory phenomenon, or 'Buddha's light' of cumulative particles.