  We derive the running of the $R^2$ coupling in the Starobinsky Lagrangian
that stems from integrating out quantum torsion fluctuations on a maximally
symmetric Euclidean background. Our analysis is performed in a manifestly
covariant way, exploiting both the recently-introduced spin-parity
decomposition of torsion perturbations and the heat kernel technique. The
Lagrangian we start with is the most general one involving kinetic terms and
couplings to the scalar curvature that is compatible with a gauge-like symmetry
for the torsion. The latter removes the twice-longitudinal vector mode from the
spectrum, and it yields operators of maximum rank four.
