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Effective Field Theory for Massive Gravitons and Gravity in Theory Space

Nima Arkani-Hamed, Howard Georgi, Matthew D. Schwartz

TL;DR

This paper develops a covariant, theory-space framework for gravity by introducing link fields that connect multiple sites with independent general coordinate invariances, reproducing massless and massive gravitons within a controlled EFT.Using a Goldstone-boson expansion, it clarifies the longitudinal graviton dynamics, derives the necessary Fierz-Pauli structure to avoid ghosts, and explains the vDVZ discontinuity in flat space versus its absence in AdS backgrounds.The analysis identifies two natural EFT cutoff scales, $\Lambda_5 = (m_g^4 M_{\rm Pl})^{1/5}$ for the simplest FP-like theory and $\Lambda_3 = (m_g^2 M_{\rm Pl})^{1/3}$ in certain non-linear extensions, and provides a systematic power-counting for higher-dimension operators.The work also introduces gravitational analogs of gauge-theory plaquettes, discusses the breakdown of the EFT around heavy sources, and outlines potential applications to gravity in theory space, gravitational dimensions, and UV completions of quantum gravity.

Abstract

We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale Mpl and a mass mg, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Lambda parametrically above mg. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz-Pauli form of the Lagrangian, the presence or absence of the van Dam-Veltman-Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz-Pauli term are easily determined. The cutoff scales as Lambda ~ (mg^4 Mpl)^(1/5) for the Fierz-Pauli theory, but can be raised to Lambda ~ (mg^2 Mpl)^(1/3) in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions.

Effective Field Theory for Massive Gravitons and Gravity in Theory Space

TL;DR

This paper develops a covariant, theory-space framework for gravity by introducing link fields that connect multiple sites with independent general coordinate invariances, reproducing massless and massive gravitons within a controlled EFT.Using a Goldstone-boson expansion, it clarifies the longitudinal graviton dynamics, derives the necessary Fierz-Pauli structure to avoid ghosts, and explains the vDVZ discontinuity in flat space versus its absence in AdS backgrounds.The analysis identifies two natural EFT cutoff scales, $\Lambda_5 = (m_g^4 M_{\rm Pl})^{1/5}$ for the simplest FP-like theory and $\Lambda_3 = (m_g^2 M_{\rm Pl})^{1/3}$ in certain non-linear extensions, and provides a systematic power-counting for higher-dimension operators.The work also introduces gravitational analogs of gauge-theory plaquettes, discusses the breakdown of the EFT around heavy sources, and outlines potential applications to gravity in theory space, gravitational dimensions, and UV completions of quantum gravity.

Abstract

We introduce a technique for restoring general coordinate invariance into theories where it is explicitly broken. This is the analog for gravity of the Callan-Coleman-Wess-Zumino formalism for gauge theories. We use this to elucidate the properties of interacting massless and massive gravitons. For a single graviton with a Planck scale Mpl and a mass mg, we find that there is a sensible effective field theory which is valid up to a high-energy cutoff Lambda parametrically above mg. Our methods allow for a transparent understanding of the many peculiarities associated with massive gravitons, among them the need for the Fierz-Pauli form of the Lagrangian, the presence or absence of the van Dam-Veltman-Zakharov discontinuity in general backgrounds, and the onset of non-linear effects and the breakdown of the effective theory at large distances from heavy sources. The natural sizes of all non-linear corrections beyond the Fierz-Pauli term are easily determined. The cutoff scales as Lambda ~ (mg^4 Mpl)^(1/5) for the Fierz-Pauli theory, but can be raised to Lambda ~ (mg^2 Mpl)^(1/3) in certain non-linear extensions. Having established that these models make sense as effective theories, there are a number of new avenues for exploration, including model building with gravity in theory space and constructing gravitational dimensions.

Paper Structure

This paper contains 14 sections, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of gauge theory
  3. Building blocks for gravity in theory space
  4. Sites and Links
  5. Explicit Goldstone boson expansion
  6. Plaquettes
  7. Massive gravitons
  8. Two site model
  9. Linearized analysis
  10. The vDVZ discontinuity in general backgrounds
  11. Strong coupling scale and power-counting in the effective theory
  12. Adding interactions to raise the cutoff
  13. Breakdown of the effective theory around heavy sources
  14. Summary, Discussion and Outlook

Figures (2)

  • Figure 1: Sites $i$ and $j$ with corresponding GC$_i \times$ GC$_j$ symmetries, connected by a link $Y_{ji}$. Under GC$_i \times$ GC$_j$, $Y_{ji} \to f_j^{-1} \circ Y_{ji} \circ f_i$.
  • Figure 2: Making a plaquette from $Y_{ji} \circ Y_{ik} \circ Y_{kj}$.