  Jet calculus offers a unique mathematical technique to bridge the area of QCD
resummation with Monte Carlo parton showers. With the ultimate goal of
constructing next-to-next-to-leading logarithmic (NNLL) parton showers we
study, using the language of generating functionals, the collinear
fragmentation of final-state partons. In particular, we focus on the definition
and calculation of the Sudakov form factor, which physically describes the
no-emission probability in an ordered branching process. We review recent
results for quark jets and compute the Sudakov form factor for the collinear
fragmentation of gluon jets at NNLL. The NNLL corrections are encoded in a $z$
dependent two-loop anomalous dimension $B_2(z)$, with $z$ being a suitably
defined longitudinal momentum fraction. This is obtained from the integration
of the relevant $1\to 3$ collinear splitting kernels combined with the one-loop
corrections to the $1\to 2$ counterpart. This work provides the missing
ingredients to extend the methods of jet calculus in the collinear limit to
NNLL and gives an important element of the next generation of NNLL parton
shower algorithms. As an application we derive new NNLL results for both the
fractional moments of energy-energy correlation $FC_x$ and the angularities
$\lambda_x$ measured on mMDT/Soft-Drop ($\beta=0$) groomed jets.
