Unconventional minimal subtraction and Bogoliubov-Parasyuk-Hepp-Zimmermann: massive scalar theory and critical exponents

TL;DR

This paper presents a simplified minimal subtraction renormalization method for massive scalar $ ext{φ}^4$ theory, reducing diagram complexity and connecting it with the BPHZ approach, while calculating critical exponents up to two loops.

Contribution

Introduces an unconventional minimal subtraction scheme for massive scalar theory that simplifies calculations and relates to the BPHZ method.

Findings

Method reduces the number of diagrams needed.

Critical exponents $ ext{η}$ and $ u$ computed up to two loops.

Explicit comparison shows fewer diagrams than BPHZ.

Abstract

We introduce a simpler although unconventional minimal subtraction renormalization procedure in the case of a massive scalar $\lambda \phi^{4}$ theory in Euclidean space using dimensional regularization. We show that this method is very similar to its counterpart in massless field theory. In particular, the choice of using the bare mass at higher perturbative order instead of employing its tree-level counterpart eliminates all tadpole insertions at that order. As an application, we compute diagrammatically the critical exponents $\eta$ and $\nu$ at least up to two loops. We perform an explicit comparison with the Bogoliubov-Parasyuk-Hepp-Zimmermann ($BPHZ$) method at the same loop order, show that the proposed method requires fewer diagrams and establish a connection between the two approaches.