  We study the problem how to deal with tensor-type two-loop integrals in the
Loop Regularization (LORE) scheme. We use the two-loop photon vacuum
polarization in the massless Quantum Electrodynamics (QED) as the example to
present the general procedure. In the processes, we find a new divergence
structure: the regulated result for each two-loop diagram contains a
gauge-violating quadratic harmful divergent term even combined with their
corresponding counterterm insertion diagrams. Only when we sum up over all the
relevant diagrams do these quadratic harmful divergences cancel, recovering the
gauge invariance and locality.
