Gravitational radiation in d>4 from effective field theory

TL;DR

The paper extends the classical EFT approach to gravity (ClEFT) to compute gravitational radiation in arbitrary spacetime dimensions. It derives the Einstein–Infeld–Hoffmann Lagrangian in $d$ dimensions and shows how to obtain the quadrupole formula for energy loss due to gravitational waves in flat $d$-space, including odd dimensions where tails are present in traditional formalisms. By systematically integrating out short-distance, orbital, and radiation scales and applying multipole expansions, the authors provide a unified, gauge-invariant framework with explicit Feynman rules and power counting for classical gravity observables. The work demonstrates that ClEFT yields consistent, dimension-spanning expressions for both the conservative two-body dynamics and the dissipative radiation sector, offering insights for higher-dimensional gravity scenarios and potential phenomenological constraints.

Abstract

Some years ago, a new powerful technique, known as the Classical Effective Field Theory, was proposed to describe classical phenomena in gravitational systems. Here we show how this approach can be useful to investigate theoretically important issues, such as gravitational radiation in any spacetime dimension. In particular, we derive for the first time the Einstein-Infeld-Hoffman Lagrangian and we compute Einstein's quadrupole formula for any number of flat spacetime dimensions.

Figures (4)

• Figure 1: Feynman diagrams that give the leading Newtonian interaction. They are of order $Lv^0$. Actually, the self-energy diagrams b) and c) give a vanishing contribution after doing dimensional regularization. Solid lines represent particle worldlines and dashed lines the potential graviton propagator (\ref{['FR:PgravProp']}). Dots are point particle vertices. In these diagrams they are $V_{\mu\nu}^{(1)}$ in (\ref{['FR:VertexRules']}). Note that loops closed by a "heavy" particle worldline are not quantum and contribute to tree-level results ( i.e., we have no integration over the particle momentum). Quantum loop corrections would correspond to diagrams with extra powers of $\hbar/L\ll 1$, i.e., with graviton loops. (The reader interested in quantum corrections to the gravitational interaction between two bodies can see Donoghue:1994dnBjerrumBohr:2002ktBohr:2001 and references therein.)
• Figure 2: Feynman diagrams for the Einstein-Infeld-Hoffmann correction to the Newton interaction. They are of order $Lv^2$. Only (internal) potential gravitons contribute as mediators of the interaction.
• Figure 3: Feynman diagrams for the coupling between the NR source and the radiation gravitons, up to leading order in $v$ where radiation emission occurs. Diagram a) is of order $\sqrt{L}v^{d-3 \over 2}$, diagram b) is $\mathcal{O}\left( \sqrt{L}v^{d-1 \over 2}\right)$, and c)-e) are order $\sqrt{L}v^{d+1 \over 2}$. Only external radiation gravitons, represented by wavy lines, describe the radiated waves.
• Figure 4: Self-energy diagram whose imaginary contribution gives the quadrupole formula for the radiated power in the form of gravitational waves by a NR system. The double solid lines represent the NR particles. In the low energy ClEFT describing the coupling of the particles to the radiated gravitons, lengthscales smaller than the orbital distance cannot be resolved. This is pictorially emphasized in this diagram by drawing the particle lines close to each other. This diagram is to be understood as a standard self-energy diagram: at a certain point in the past the system radiates a graviton (described by (\ref{['Lrad:final']})) which is then absorbed back in the future. The amplitude for this process is complex. Its imaginary part encodes the information on the radiated energy through gravitational waves that propagates to the asymptotic region. Again, this is a tree-level diagram since the loop is closed by "heavy" particle worldlines whose momenta are not integrated.