Three-loop correction to the instanton density. I. the quartic double well potential

This paper deals with quantum fluctuations near the classical instanton configuration. Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude (the instanton density) at three-loop order for a quartic double well potential. The result for the three-loop contribution coincides, to five significant figures, with the one given long ago by J. Zinn-Justin. Unlike the two-loop contribution where all involved Feynman integrals are rational numbers, in the three-loop case Feynman diagrams can contain irrational contributions.