Asymptotic Behavior of the \Beta Function in the \Phi^4 Theory: A Scheme Without Complex Parameters

TL;DR

This paper investigates the asymptotic behavior of the eta function in { ext{phi}^4}{ ext{theory}} without relying on complex parameters, clarifying the transition to strong coupling and addressing singularities in the parametric representation.

Contribution

It demonstrates that the parametric representation's singularity at t→0 allows analysis with real g_0, providing insights into the strong coupling regime and supporting renormalizability.

Findings

Singularity at t→0 in the parametric representation

Analysis applicable to real values of g_0

Partial proof of renormalizability in strong coupling

Abstract

The previously obtained analytical asymptotic expressions for the Gell-Mann - Low function \beta(g) and anomalous dimensions of \phi^4 theory in the limit g\to\infty are based on the parametric representation of the form g = f(t), \beta(g) = f1(t) (where t\sim g_0^{-1/2} is the running parameter related to the bare charge g_0), which is simplified in the complex t plane near a zero of one of the functional integrals. In the present paper, it is shown that the parametric representation has a singularity at t\to 0; for this reason, similar results can be obtained for real values of g_0. The problem of the correct transition to the strong coupling regime is simultaneously solved; in particular, the constancy of the bare or renormalized mass is not a correct condition of this transition. A partial proof is given for the theorem of the renormalizability in the strong coupling region.