Extended Metric-Affine $f(R)$ Gravity with Dynamical Connection in Vacuum

We extend the usual vacuum Metric-Affine $f(R)$ Gravity by supplementing it with all parity even quadratic invariants in torsion and non-metricity. As we show explicitly this supplementation drastically changes the status of the Theory which now propagates an additional scalar degree of freedom on top of the graviton. This scalar degree of freedom has a geometric origin as it relates to spacetime torsion and non-metricity. The resulting Theory can be written equivalently as a metric and torsionless Scalar-Tensor Theory whose potential and kinetic term coupling depend on the choice of the function $f(R)$ and the dimensionless parameters of the quadratic invariants respectively.