Mass gap in non-perturbative quadratic $\mathcal{R}^2$ gravity via Dyson-Schwinger

We apply in a simple model derived from quadratic $\mathcal{R}^2$ gravity the technique of Dyson-Schwinger equations to solve for its corresponding quantum theory. Particularly, we solve the classical equations of motion to get a solution to the hierarchy of Dyson-Schwinger equations in the limit of large Ricci scalar, assumed to be constant and larger than the square of the Starobinsky mass. Moving to the Einstein frame, the model admits Higgs-like solutions with a single particle having a finite mass. We quantize the scalar field showing the appearing of a mass gap through a Higgs-like solution. The presence of the mass gap, that increases with the square root of the Ricci scalar, shows how the effect of the scalar sector at low-energy becomes ineffective, making it relevant only at short distances.