Two-loop Calculation of Higgs Mass in Gauge-Higgs Unification: 5D Massless QED Compactified on S^1

We calculate the quantum corrections to the mass of the zero mode of the fifth component of the gauge field at two-loop level in a five dimensional massless QED compactified on S^1. We discuss in detail how the divergences are exactly canceled and the mass becomes finite. The key ingredients to obtain the result are the five dimensional gauge invariance and the Ward-Takahashi identity. We also evaluate the finite part of corrections.