$η$ regularisation and the functional measure

TL;DR

This work develops a general operator-valued regularisation of the functional path integral measure to rigorously treat the chiral anomaly. By extending the $η$-regularisation to the spectrum of the Dirac operator, it connects spectral asymmetry with measure transformation and unifies Fujikawa’s analytic regularisation with the Atiyah–Singer index theorem. The authors introduce a scale-dependent regulator $ι_E(Λ)$, derive its Mellin-transform structure, and demonstrate how the regularised measure reproduces the standard chiral anomaly while clarifying regulator dependence. They further bridge different regularisation schemes via a generalized Schwinger proper-time framework and illustrate the connections in the two-dimensional Schwinger model. The framework provides a pathway to robust, regulator-aware anomaly computations and prompts future exploration of higher-dimensional and curved-spacetime applications, with potential implications for UV completion and related conceptual issues.

Abstract

In this paper, we revisit Fujikawa's path integral formulation of the chiral anomaly and develop a generalised framework for systematically defining a regularised functional measure. This construction extends the $η$ regularisation scheme to operator language, making the connection between spectral asymmetry and measure transformation fully explicit. Before recovering Fujikawa's expression for the chiral anomaly from the regularised measure, we explore the deeper number-theoretic structure underlying the ill-defined spectral sum associated with the anomaly, interpreting it through the lens of smoothed asymptotics. Our approach unifies two complementary perspectives: the analytic regularisation of Fujikawa and the topological characterisation given by the Atiyah-Singer index theorem. We further investigate how the measure transforms under changes to the regularisation scale and derive a function $ι_E(Λ)$ that encodes this dependence, showing how its Mellin moments govern the appearance of divergences. Finally, we comment on the conceptual relationship between the regularised measure, $η$ regularisation, and the generalised Schwinger proper-time formalism, with a particular focus on the two-dimensional Schwinger model.