Study of special functionals under mixed regularization in four dimensions

Investigate the special renormalization functionals that arise in the four-dimensional scalar theory at three-loop order (as listed in the appendix of Ivanov, Nucl. Phys. B, 2024, arXiv:2402.14549) under mixed regularization that combines a coordinate-space cutoff via the quasi-local averaging operator and a momentum-space cutoff, analogous to the analysis of mixed regularization in the two-dimensional case presented in Section 4.3 of this paper.

Background

The paper develops quasi-local probability averaging of fundamental solutions for the Laplace operator and analyzes the resulting cutoff-regularized Green’s functions. It provides general representations and properties (Theorems 1–2) and works through concrete examples relevant to perturbative quantum field theory.

In Section 4.3, the authors introduce a mixed regularization in two dimensions that combines a coordinate-space cutoff (through quasi-local averaging kernels with compact support) and a momentum-space cutoff. There, they define and analyze special functionals (e.g., quantities denoted θj) and show how this mixed scheme yields extra freedom, allowing certain renormalization functionals to vanish in an appropriate limit.

The authors point out that, in four dimensions, sets of analogous special functionals occur in the three-loop analysis of the scalar φ4 model (as documented in the appendix of Ivanov (2024), Nucl. Phys. B). Extending the mixed-regularization study to these four-dimensional functionals is identified as an open direction.