The heat-kernel coefficient a2 and the Casimir energy of a dielectric cylinder

We calculate the heat kernel coefficients for the electromagnetic field in the background of a dielectric cylinder with non equal speeds of light inside and outside. The coefficient a2 whose vanishing makes the vacuum energy of a massless field unique, turns out to be zero in dilute order, i.e., in order (ε-1)2, and nonzero beyond. As a consequence, the vanishing of the vacuum energy in the presence of a dielectric cylinder found by Casimir-Polder summation must take place irrespectively of the methods by which it might be calculated