Loop functions of sunset diagrams in 2+1 space-time dimensions

Abstract

In these notes the relativistic $n$-body phase-phase is calculated iteratively in $2+1$ space-time dimensions for all $n$. The obtained result shows a simple power-law behavior $\alpha_n (\mu-M)^{n-2}/\mu$ with a dependence only on the total mass $M=m_1+\dots + m_n$. As a consequence of this feature, the $(n-1)$-loop integrals $J_n(-q^2)$ associated to sunset diagrams with $n$ internal lines can be expressed through of elementary (arctangent and logarithmic) functions, modulo polynomial terms in $q^2$ with regularization-dependent coefficients. An outlook to the analogous situation in $4+1$ space-time dimensions is given by computing the $n$-body phase-phases for $n=2,3,4,5$ with their totally symmetric dependence on the involved masses. Moreover, a digression to $1+1$ space-time dimensions reveals that there the three-body phase-space is already proportional to a complete elliptic integral.