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

TL;DR

The paper analyzes sunset diagrams in $2+1$ space–time dimensions by first deriving a remarkably simple $n$-body phase-space form $\Gamma_n(\mu;M)=\alpha_n (\mu-M)^{n-2}/\mu$ that depends only on the total mass $M$. Using this, it expresses the sunset-loop functions $J_n(-q^2)$ in terms of elementary $\arctan$ and $\ln$ functions up to regularization-dependent polynomials, providing explicit results for small $n$ and a general decomposition in terms of $A_n$ and $L_n$-type coefficients. The work then extends the construction to $4+1$ dimensions, where the $n$-body phase-space becomes a symmetric polynomial in the masses, and derives analogous loop-function structures with polynomial prefactors. A digression to $1+1$ dimensions reveals that the three-body phase-space is elliptic, connected to complete elliptic integrals and Weierstraß invariants, highlighting dimensional parallels and the complexity of master integrals across 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 $α_n (μ-M)^{n-2}/μ$ 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.

Figures (1)

• Figure 1: Sunset diagram with $n$ internal lines carrying masses $m_1, m_2, \dots , m_n$. The vertical dotted line symbolizes the cut, which gives rise to the integrated $n$-body phase-space.