Unitarization of $R + αR^2$ gravity

TL;DR

This paper advances the unitarization of gravity in the Starobinsky model $R+\alpha R^2$ by employing an improved K-matrix (IK) approach in a coupled-channel framework that includes graviton and scalaron states. It reveals a genuine scalaron resonance on the 22 channel at $s= m^2$, while identifying the graviball as an infrared artifact that moves toward the origin as the regulator is removed; it also uncovers subthreshold structures that may correspond to bound states of scalarons. The analysis demonstrates that unitarization tames high-energy behavior and that the scalar sector exhibits non-decoupling behavior as $\alpha$ varies, with the subplanckian regime showing universal asymptotics akin to pure Einstein–Hilbert gravity. The results provide qualitative predictions for the resonance spectrum of $R+\alpha R^2$ gravity and motivate further checks with different unitarization schemes.

Abstract

We make use of the improved K-matrix algorithm to obtain unitarized amplitudes in $R+αR^2$ gravity (the so-called Starobisnsky model, of cosmological relevance). The procedure is of some complexity because infrared divergences are present and need to be properly regulated. We focus on the behaviour of a bona fide scalar resonance, known to exist in this model, and compare it to an apparent resonance detected in previous studies, thus confirming that the latter seems to be an artifact due to the introduction of the infrared regulator. We analyze the existence of other dynamical resonances and dwell on the amplitudes made unitary by this procedure.

Figures (13)

• Figure 1: Diagrams contributing to the elastic scattering of gravitons in the helicity states $\lambda_i=\pm 2$.
• Figure 2: Diagrams contributing to the crossed process $hh\to \varphi\varphi$ where the gravitons are in the helicity states $\lambda_i=\pm 2$.
• Figure 3: Diagrams contributing to the elastic scattering of scalarons.
• Figure 4: Both sides of the full unitarity condition in Eq. (\ref{['eq_g_unitcondition']}) for the values $m=0.3M_{\text{pl}}$ and $\mu=10^{-10}M_{\text{pl}}$. Both lines coincide meaning that the scalar waves fulfill unitarity as expected
• Figure 5: Position of the scalaron and the $3m^2$ logarithmic branch point for various values of $\overline{m}$ and the indrared cut-off $\overline{\mu}$. In each panel, the values for the dimensionless IR regulator are $\overline{\mu}=10^{-10}, 10^{-15}, 10^{-20}$ and are identified with different pointsizes---bigger pointsizes represent bigger values of $\overline{\mu}$. In the last two panels, the position of the logarithmic branch point is unchanged by the regulator.