  We investigate the twist four longitudinal structure function
$F_{L}^{\tau=4}$ of deep inelastic scattering and show that the integral of
${F_{L}^{\tau=4} \over x}$ is related to the expectation value of the fermionic
part of the light-front Hamiltonian density at fixed momentum transfer. We show
that the new relation, in addition to providing physical intuition on
$F_{L}^{\tau=4}$, relates the quadratic divergences of $F_{L}^{\tau=4}$ to the
quark mass correction in light-front QCD and hence provides a new pathway for
the renormalization of the corresponding twist four operator. The mixing of
quark and gluon operators in QCD naturally leads to a twist four longitudinal
gluon structure function and to a new sum rule $ \int dx {F_L \over x}= 4 {M^2
\over Q^2}$, which is the first sum rule obtained for a twist four observable.
The validity of the sum rule in a non-perturbative context is explicitly
verified in two-dimensional QCD.
