  Cross sections for top quarks provide very interesting physics opportunities,
being both sensitive to new physics and also perturbatively tractable due to
the large top quark mass. Rigorous factorization theorems for top cross
sections can be derived in several kinematic scenarios, including the boosted
regime in the peak region that we consider here. In the context of the
corresponding factorization theorem for $e^+e^-$ collisions we extract the last
missing ingredient that is needed to evaluate the cross section differential in
the jet-mass at two-loop order, namely the matching coefficient at the scale
$\mu\simeq m_t$. Our extraction also yields the final ingredients needed to
carry out logarithmic resummation at next-to-next-to-leading logarithmic order
(or N$^3$LL if we ignore the missing 4-loop cusp anomalous dimension). This
coefficient exhibits an amplitude level rapidity logarithm starting at
$\mathcal{O}(\alpha_s^2)$ due to virtual top quark loops, which we treat using
rapidity renormalization group (RG) evolution. Interestingly, this rapidity RG
evolution appears in the matching coefficient between two effective theories
around the heavy quark mass scale $\mu\simeq m_t$.
