Continuity of the four-point function of massive $\phi_4^4$-theory above threshold

TL;DR

This paper proves the continuity of the four-point function in massive ^4-theory above threshold, establishing a rigorous foundation for the physical definition of the renormalized coupling.

Contribution

It provides a rigorous proof of the continuity of the four-point function at on-shell mass, using flow equations and integral representations, closing a longstanding gap in renormalization theory.

Findings

Four-point function is continuous as a function of external momenta.

The proof uses inductive integral representations from flow equations.

Supports a physical, well-defined renormalized coupling.

Abstract

In this paper we prove that the four-point function of massive $\vp_4^4$-theory is continuous as a function of its independent external momenta when posing the renormalization condition for the (physical) mass on-shell. The proof is based on integral representations derived inductively from the perturbative flow equations of the renormalization group. It closes a longstanding loophole in rigorous renormalization theory in so far as it shows the feasibility of a physical definition of the renormalized coupling.