We begin by reviewing the fundamentals of renormalisation and its modern formulation via the Exact Renormalisation Group (ERG). Building on this, we explore a recently proposed background-independent ERG framework for gauge theories and gravity, which comes with some advantages: preserves gauge invariance manifestly, avoids introducing unphysical fields, such as ghosts and Pauli-Villars fields, and does not require gauge-fixing. We examine its application in the simple case of SU(N) Yang-Mills, developing the machinery needed to study it rigorously in the process. Despite its appealing features, we show that the longitudinal part of vertex functions is not completely regularised already at one loop, invalidating certain methods for extracting universal components. Moreover we demonstrate a kind of no-go theorem: within the proposed structure, whatever choice is made for covariantisation and cutoff profiles, the two-point vertex flow equation at one loop cannot be both transverse, as required by gauge invariance, and fully regularised.
Based on:
[1] Vlad-Mihai Mandric and Tim R. Morris. Properties of a proposed background independent exact renormalization group. Phys. Rev. D, 107(6):065012, 2023, 2210.00492.