Massless scalar Feynman diagrams: five loops and beyond

TL;DR

This paper develops advanced techniques for evaluating massless scalar Feynman diagrams, enabling the calculation of all five-loop diagrams with finite contributions in four-dimensional scalar theories, revealing only a few fundamental constants.

Contribution

It introduces new methods for solving recurrence relations, exploiting symmetry and conformal invariance, to compute complex multi-loop diagrams efficiently and precisely.

Findings

Evaluated all 216 five-loop or fewer diagrams with finite contributions.

Found only five fundamental zeta constants in the results.

Achieved high-precision calculation of the most symmetrical diagram.

Abstract

Several powerful techniques for evaluating massless scalar Feynman diagrams are developed, viz: the solution of recurrence relations to evaluate diagrams with arbitrary numbers of loops in $n=4-2\omega$ dimensions; the discovery and use of symmetry properties to restrict and compute Taylor series in $\omega$; the reduction of triple sums over Chebyshev polynomials to products of Riemann zeta functions; the exploitation of conformal invariance to avoid four-dimensional Racah coefficients. As an example of the power of these techniques we evaluate all of the 216 diagrams, with 5 loops or less, which give finite contributions of order $1/k^2$ or $1/k^4$ to a propagator of momentum $k$ in massless four-dimensional scalar field theories. Remarkably, only 5 basic numbers are encountered: $\zeta(3)$, $\zeta(5)$, $\zeta(7)$, $\zeta(9)$ and the value of the most symmetrical diagram, which is calculated to 14 significant figures. It is conceivable that these are the only irrationals appearing in 6-loop beta functions. En route to these results we uncover and only partially explain many remarkable relations between diagrams.