Dimensionally continued multi-loop gauge theory

Abstract

A dimensionally continued background-field method makes the rationality of the 4-loop quenched QED beta function far more reasonable than had previously appeared. After 33 years of quest, dating from Rosner's discovery of 3-loop rationality, one finally sees cancellation of zeta values by the trace structure of individual diagrams. At 4-loops, diagram-by-diagram cancellation of $ζ(5)$ does not even rely on the values of integrals at d=4. Rather, it is a property of the rational functions of $d$ that multiply elements of the full d-dimensional basis. We prove a lemma: the basis consists of slices of wheels. We explain the previously mysterious suppression of $π^4$ in massless gauge theory. The 4-loop QED result $β_4=-46$ is obtained by setting d=4 in a precisely defined rational polynomial of d, with degree 11. The other 5 rational functions vanish at d=4.