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Nonlinear Properties of Vielbein Massive Gravity

Stefan Groot Nibbelink, Marco Peloso, Matthew Sexton

TL;DR

This work develops a nonlinear, Lorentz-invariant massive gravity model in the vielbein formalism with an external Minkowski background, motivated by the need for viable large-scale modifications of gravity. Through a bi-gravity origin and a Stueckelberg decomposition, it yields a massive graviton whose coupling to matter can be universal when the massless mode decouples. Using a 't Hooft–Feynman-like gauge, the authors derive decoupled propagators for tensor, vector, and scalar modes and show the leading nonlinear interactions saturate at the scale $\Lambda_3 = (m_g^2 M_P)^{1/3}$, i.e., the softest UV behavior expected for Lorentz-invariant massive gravity. They further demonstrate the absence of dangerous scalar-only and single-vector interactions below $\Lambda_3$ and identify a parameter region where tree-level $\phi\phi \to \phi\phi$ scattering can vanish at $\Lambda_3$, illustrating tunable high-energy behavior. Overall, the paper presents a minimal, well-behaved massive gravity prototype in the vielbein language and highlights the potential advantages of the vielbein formulation for addressing nonlinear issues in massive gravity.

Abstract

We propose a non-linear extension of the Fierz-Pauli mass for the graviton through a functional of the vielbein and an external Minkowski background. The functional generalizes the notion of the measure, since it reduces to a cosmological constant if the external background is formally sent to zero. Such a term and the explicit external background, emerge dynamically from a bi--gravity theory, having both a massless and a massive graviton in its spectrum, in a specific limit in which the massless mode decouples, while the massive one couples universally to matter. We investigate the massive theory using the Stueckelberg method and providing a 't Hooft-Feynman gauge fixing in which the tensor, vector and scalar Stueckelberg fields decouple. We show that this model has the softest possible ultraviolet behavior which can be expected from any generic (Lorentz invariant) theory of massive gravity, namely that it becomes strong only at the scale Lambda_3 = (m_g^2 M_P)^{1/3}.

Nonlinear Properties of Vielbein Massive Gravity

TL;DR

This work develops a nonlinear, Lorentz-invariant massive gravity model in the vielbein formalism with an external Minkowski background, motivated by the need for viable large-scale modifications of gravity. Through a bi-gravity origin and a Stueckelberg decomposition, it yields a massive graviton whose coupling to matter can be universal when the massless mode decouples. Using a 't Hooft–Feynman-like gauge, the authors derive decoupled propagators for tensor, vector, and scalar modes and show the leading nonlinear interactions saturate at the scale , i.e., the softest UV behavior expected for Lorentz-invariant massive gravity. They further demonstrate the absence of dangerous scalar-only and single-vector interactions below and identify a parameter region where tree-level scattering can vanish at , illustrating tunable high-energy behavior. Overall, the paper presents a minimal, well-behaved massive gravity prototype in the vielbein language and highlights the potential advantages of the vielbein formulation for addressing nonlinear issues in massive gravity.

Abstract

We propose a non-linear extension of the Fierz-Pauli mass for the graviton through a functional of the vielbein and an external Minkowski background. The functional generalizes the notion of the measure, since it reduces to a cosmological constant if the external background is formally sent to zero. Such a term and the explicit external background, emerge dynamically from a bi--gravity theory, having both a massless and a massive graviton in its spectrum, in a specific limit in which the massless mode decouples, while the massive one couples universally to matter. We investigate the massive theory using the Stueckelberg method and providing a 't Hooft-Feynman gauge fixing in which the tensor, vector and scalar Stueckelberg fields decouple. We show that this model has the softest possible ultraviolet behavior which can be expected from any generic (Lorentz invariant) theory of massive gravity, namely that it becomes strong only at the scale Lambda_3 = (m_g^2 M_P)^{1/3}.

Paper Structure

This paper contains 13 sections, 64 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Discussion
  2. A massive gravity theory described in the vielbein formalism
  3. Stückelberg fields
  4. Graviton and Stückelberg propagators
  5. Dominant interactions at high energies
  6. Absence of interactions with only scalars or at most one vector
  7. All interactions become strong at scale $\boldsymbol{\Lambda_3}$ or above
  8. Can the $\boldsymbol{\phi\phi \rightarrow \phi\phi}$ scattering amplitude vanish?
  9. Conclusions
  10. Derivation from a bi--gravity theory
  11. Properties of $\boldsymbol{\langle ABCD \rangle}$
  12. Perturbative expansions
  13. Exact expressions

Figures (1)

  • Figure 1: This diagram displays the three diagrams that contribute to the four scalar scattering at the scale $\Lambda_3$, which result from \ref{['intergL3']}. In the first diagram the scalar $\phi$ is exchanged, while in the second diagram the graviton $\hat{f}_{\mu\nu}$ is the mediating particle. The last diagram results from the four point interactions of scalars.