Dimensional Renormalization in phi^3 theory: ladders and rainbows

TL;DR

This paper solves complex differential equations arising from ladder and rainbow diagrams in phi^3 theory near 6 dimensions, revealing the absence of transcendental renormalization factors and determining anomalous dimensions.

Contribution

It provides exact solutions to the differential equations for ladder and rainbow diagrams in phi^3 theory, demonstrating the non-existence of transcendental renormalization factors.

Findings

No transcendental renormalization factors in the theory.

Explicit expressions for Green functions in terms of hypergeometric functions.

Correct anomalous dimensions obtained in six-dimensional limit.

Abstract

The sum of all ladder and rainbow diagrams in $\phi^3$ theory near 6 dimensions leads to self-consistent higher order differential equations in coordinate space which are not particularly simple for arbitrary dimension D. We have now succeeded in solving these equations, expressing the results in terms of generalized hypergeometric functions; the expansion and representation of these functions can then be used to prove the absence of renormalization factors which are transcendental for this theory and this topology to all orders in perturbation theory. The correct anomalous scaling dimensions of the Green functions are also obtained in the six-dimensional limit.