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A model of composite gravity with Pauli-Villars regulators

Chris Li, Diana Vaman

TL;DR

This work demonstrates that a massless composite graviton in a non-metric, non-polynomial scalar theory persists when dimensional regularization is replaced by covariant Pauli-Villars regulators. By analyzing zero- and small-momentum transfer in two-to-two scalar scattering and performing a regulator-consistent resummation, the authors show that a graviton-like pole appears only under a fine-tuned choice of V0, with the Planck scale M_Pl depending on the regulator masses and weights. They explore minimal (N_PV=3) and non-minimal (N_PV=4) Pauli-Villars configurations and derive exact or leading-order expressions for the relevant loop integrals, confirming regulator independence of the graviton pole within this covariant framework, while highlighting the regulator-dependence of M_Pl. When the tuning is not imposed, the theory yields a composite massive spin-2 state and a ghost, or no low-energy poles, illustrating a close connection between the graviton pole and higher-derivative/gauge-fixing subtleties in the model, particularly in the large-N limit.

Abstract

We revisit a model of composite gravity, in the form of a reparametrization invariant, non-polynomial, metric-independent action for scalar fields. Previously, the emergence of a composite massless spin 2 particle, the graviton, was demonstrated by analyzing a two-into-two scalar scattering amplitude. Working in the limit of a large number of physical scalars and using dimensional regularization, it was shown that the scattering amplitude had a pole corresponding to a graviton exchange, provided that a certain fine-tuning was implemented; the Planck mass was determined as a function of the dimensional regularization parameter and a mass scale. Here we demonstrate that the presence of the composite graviton is a robust feature of this model and not an artefact of the choice of regulator, by replacing dimensional regularization with Pauli-Villars fields. The presence of the massless graviton is conditioned by a similar fine-tuning as before. This is arguably a more physical regularization, since the Planck mass now depends on the specifics of the Pauli-Villars regulator fields, e.g. their mass as well as their multiplicity.

A model of composite gravity with Pauli-Villars regulators

TL;DR

This work demonstrates that a massless composite graviton in a non-metric, non-polynomial scalar theory persists when dimensional regularization is replaced by covariant Pauli-Villars regulators. By analyzing zero- and small-momentum transfer in two-to-two scalar scattering and performing a regulator-consistent resummation, the authors show that a graviton-like pole appears only under a fine-tuned choice of V0, with the Planck scale M_Pl depending on the regulator masses and weights. They explore minimal (N_PV=3) and non-minimal (N_PV=4) Pauli-Villars configurations and derive exact or leading-order expressions for the relevant loop integrals, confirming regulator independence of the graviton pole within this covariant framework, while highlighting the regulator-dependence of M_Pl. When the tuning is not imposed, the theory yields a composite massive spin-2 state and a ghost, or no low-energy poles, illustrating a close connection between the graviton pole and higher-derivative/gauge-fixing subtleties in the model, particularly in the large-N limit.

Abstract

We revisit a model of composite gravity, in the form of a reparametrization invariant, non-polynomial, metric-independent action for scalar fields. Previously, the emergence of a composite massless spin 2 particle, the graviton, was demonstrated by analyzing a two-into-two scalar scattering amplitude. Working in the limit of a large number of physical scalars and using dimensional regularization, it was shown that the scattering amplitude had a pole corresponding to a graviton exchange, provided that a certain fine-tuning was implemented; the Planck mass was determined as a function of the dimensional regularization parameter and a mass scale. Here we demonstrate that the presence of the composite graviton is a robust feature of this model and not an artefact of the choice of regulator, by replacing dimensional regularization with Pauli-Villars fields. The presence of the massless graviton is conditioned by a similar fine-tuning as before. This is arguably a more physical regularization, since the Planck mass now depends on the specifics of the Pauli-Villars regulator fields, e.g. their mass as well as their multiplicity.

Paper Structure

This paper contains 9 sections, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Regularizing with Pauli-Villars fields
  3. Zero momentum-transfer ($q=0$)
  4. Small momentum-transfer ($q\neq 0$) and the graviton pole
  5. A non-minimal set of Pauli-Villars fields: $N_{PV}=4$
  6. Conclusion
  7. Clock-and-ruler fields, $c_1$ and $c_2$ counterterms
  8. Three Pauli-Villars fields
  9. Four Pauli-Villars fields

Figures (3)

  • Figure 1: Two-into-two scalar scattering
  • Figure 2: Two-into-two scalar scattering recursion relation, with Pauli-Villars fields. The letters $a,b,c$ and P.V. indicate the species and type of the scalar fields. The sum is over all $N$-many intermediate $c$ scalar fields and all $N_{PV}$-many sets of Pauli-Villars fields.
  • Figure 3: Counterterms