Sum rule for the twist four longitudinal structure function

We investigate the twist four longitudinal structure function FLτ=4 of deep inelastic scattering and show that the integral of FLτ=4x 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 FLτ=4, relates the quadratic divergences of FLτ=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 ∫dxFLx=4M2Q2, 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.