Classical Effective Field Theory and Caged Black Holes

TL;DR

This work demonstrates that classical matched asymptotic expansion (MAE) is effectively a Classical Effective Field Theory (ClEFT), enabling diagrammatic, renormalizable analyses of systems with widely separated scales. Applying this framework to caged black holes, the authors develop a time-dimensionally reduced effective action, derive a streamlined set of Feynman rules, and reproduce known thermodynamic quantities while enabling higher-order computations; crucially, a single factorizing diagram replaces the prior six two-loop diagrams. The methodology extends to rotating caged black holes, yielding leading mass corrections and showing that angular momentum is not renormalized at leading order, with clear prescriptions for deriving temperature, tension, and entropy corrections. The results generalize to arbitrary compactification manifolds, illustrating that classical renormalization concepts and Feynman-diagram techniques can be fruitfully applied in GR contexts with multiple scales.

Abstract

Matched asymptotic expansion is a useful technique in General Relativity and other fields whenever interaction takes place between physics at two different length scales. Here matched asymptotic expansion is argued to be equivalent quite generally to Classical Effective Field Theory (CLEFT) where one (or more) of the zones is replaced by an effective theory whose terms are organized in order of increasing irrelevancy, as demonstrated by Goldberger and Rothstein in a certain gravitational context. The CLEFT perspective has advantages as the procedure is clearer, it allows a representation via Feynman diagrams, and divergences can be regularized and renormalized in standard field theoretic methods. As a side product we obtain a wide class of classical examples of regularization and renormalization, concepts which are usually associated with Quantum Field Theories. We demonstrate these ideas through the thermodynamics of caged black holes, both simplifying the non-rotating case, and computing the rotating case. In particular we are able to replace the computation of six two-loop diagrams by a single factorizable two-loop diagram, as well as compute certain new three-loop diagrams. The results generalize to arbitrary compactification manifolds. For caged rotating black holes we obtain the leading correction for all thermodynamic quantities. The angular momentum is found to non-renormalize at leading order.

Figures (11)

• Figure 1: A definition of the (asymptotic, renormalized) ADM mass $m$ in terms of a 1-point function as in CGR. The top line is pre-dimensional reduction as in CGR and the wavy lines represent gravitational perturbations (of $g$). The external leg is the asymptotic component $\bar{h}_{00}$ and there is no propagator associated with it. The double solid lines denote the black hole world-line. The bottom line is the translation of the top line into the dimensionally reduced fields which we use. $\bar{\phi}$ is the longest scale ($\gg L$) component of the metric field related to $g_{00}$ through \ref{['def-phi']}. The solid internal lines denote the propagator (\ref{['scalar_prop']}) for the scalar field $\phi$. More details about the Feynman rules will be given later in figure \ref{['Feynman_rules_Sch']}.
• Figure 2: The new definition of $m$ as a 0-point or "vacuum" function, represented in terms of Feynman diagrams. (a) and (b) represent correction to the mass $m$ of different orders (1 and 2 respectively). The order parameter $\lambda$ (related to $r_0/L$) will be defined later in (\ref{['def-lambda']}). More details about the Feynman rules will be given later in figure \ref{['Feynman_rules_Sch']}.
• Figure 3: A Feynman diagram definition of the red-shift factor $R$. (a) and (b) denote contributions to different orders (1 and 2) in the order parameter $\lambda$ which will be defined later in \ref{['def-lambda']}. The $\otimes$ denotes the value of $\phi$ at the location of the black hole, that is $\phi(O)$.
• Figure 4: The Feynman rules for ClEFT are naturally real. As an example we display them for the $\phi^4$ scalar theory.
• Figure 5: Feynman rules obtained from the expansion of (\ref{['eff-Lagrangian']})