The Invisible Renormalon
G. Martinelli, M. Neubert, C. T. Sachrajda
TL;DR
This paper demonstrates that the appearance and interpretation of renormalons in HQET depend on the chosen regularisation scheme, linking UV renormalons to power divergences in hard-cutoff schemes and showing that dimensional regularisation hides such divergences while revealing their structure via the Borel plane. By expanding the heavy-quark propagator to order $1/m_Q$ and analyzing the corresponding operator matrix elements, the authors map renormalon poles to the mixing of higher-dimensional operators, and show cancellations between IR and UV renormalons in the HQET expansion. A central finding is the so-called invisible renormalon: in dimensional and certain cut-off schemes the anticipated quadratic UV divergence of the kinetic energy operator is absent, while lattice regularisation exhibits a genuine quadratic divergence and operator mixing, raising questions about underlying symmetries or lattice artefacts. The results highlight important subtleties in EFT renormalon structure and have implications for precise HQET calculations of heavy-hadron properties and decays.
Abstract
We study the structure of renormalons in the Heavy Quark Effective Theory, by expanding the heavy quark propagator in powers of $1/m_Q$. We demonstrate that the way in which renormalons appear depends on the regularisation scheme used to define the effective theory. In order to investigate the relation between ultraviolet renormalons and power divergences of matrix elements of higher-dimensional operators in the heavy quark expansion, we perform calculations in dimensional regularisation and in three different cut-off regularisation schemes. In the case of the kinetic energy operator, we find that the leading ultraviolet renormalon which corresponds to a quadratic divergence, is absent in all but one (the lattice) regularisation scheme. The nature of this ``invisible renormalon'' remains unclear.