New Regulators for Quantum Field Theories with Compactified Extra Dimensions. I: Fundamentals

TL;DR

This paper introduces two new regulators, EHC and EDR, for quantum field theories with compactified extra dimensions that preserve higher-dimensional symmetries, enabling more accurate analysis of Kaluza-Klein modes.

Contribution

The paper presents novel regulators that maintain higher-dimensional Lorentz and gauge symmetries in theories with compactified extra dimensions.

Findings

Regulators respect original higher-dimensional symmetries.

Ward-Takahashi identities hold for all Kaluza-Klein photons.

Avoidance of spurious terms in calculations.

Abstract

In this paper, we propose two new regulators for quantum field theories in spacetimes with compactified extra dimensions. We refer to these regulators as the ``extended hard cutoff'' (EHC) and ``extended dimensional regularization'' (EDR). Although based on traditional four-dimensional regulators, the key new feature of these higher-dimensional regulators is that they are specifically designed to handle mixed spacetimes in which some dimensions are infinitely large and others are compactified. Moreover, unlike most other regulators which have been used in the extra-dimension literature, these regulators are designed to respect the original higher-dimensional Lorentz and gauge symmetries that exist prior to compactification, and not merely the four-dimensional symmetries which remain afterward. This distinction is particularly relevant for calculations of the physics of the excited Kaluza-Klein modes themselves, and not merely their radiative effects on zero modes. By respecting the full higher-dimensional symmetries, our regulators avoid the introduction of spurious terms which would not have been easy to disentangle from the physical effects of compactification. As part of our work, we also derive a number of ancillary results. For example, we demonstrate that in a gauge-invariant theory, analogues of the Ward-Takahashi identity hold not only for the usual zero-mode (four-dimensional) photons, but for all excited Kaluza-Klein photons as well.