  We compute two-loop form factors of operators in the $SU(2|3)$ closed
subsector of $N=4$ supersymmetric Yang-Mills. In particular, we focus on the
non-protected, dimension-three operators $\mathrm{Tr} (X[Y,Z])$ and
$\mathrm{Tr} ( \psi \psi)$ for which we compute the four possible two-loop form
factors, and corresponding remainder functions, with external states $\langle
\bar{X} \bar{Y} \bar{Z}|$ and $\langle \bar{\psi} \bar{\psi}|$. Interestingly,
the maximally transcendental part of the two-loop remainder of $\langle\bar{X}
\bar{Y} \bar{Z}| \mathrm{Tr} (X[Y,Z]) |0\rangle$ turns out to be identical to
that of the corresponding known quantity for the half-BPS operator $\mathrm{Tr}
(X^3)$. We also find a surprising connection between the terms subleading in
transcendentality and certain a priori unrelated remainder densities introduced
in the study of the spin chain Hamiltonian in the SU(2) sector. Next, we use
our calculation to resolve the mixing, recovering anomalous dimensions and
eigenstates of the dilatation operator in the SU(2|3) sector at two loops. We
also speculate on potential connections between our calculations in $N=4$ super
Yang-Mills and Higgs + multi-gluon amplitudes in QCD in an effective Lagrangian
approach.
