  In this second paper on quantum fluctuations near the classical instanton
configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and
arXiv:1501.03993, we focus on another well studied quantum-mechanical problem,
the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only
the tools from quantum field theory, the Feynman diagrams in the instanton
background, we calculate the tunneling amplitude (the instanton density) to the
three-loop order. The result confirms (to seven significant figures) the one
recently recalculated by G. V. Dunne and M. \"{U}nsal, {\it Phys. Rev. \bf D
89}, 105009 (2014) from the resurgence perspective. As in the double well
potential case, we found that the largest contribution is given by the diagrams
originating from the Jacobian. We again observe that in the three-loop case
individual Feynman diagrams contain irrational contributions, while their sum
does not.
