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.

Figure 1

Diagrams contributing to the two-loop correction $B_1=a+b_1+b_2+c$. They enter into the coefficient $B_2$ via the term $B_{2\hspace{0.17em}\text{loop}}$. For the instanton field the effective triple and quartic coupling constants are $V_3=-\frac{\sqrt{3}}{2}\tanh (t/2)S_0^{-1/2}$ and $V_4=\frac{1}{2}{S}_0^{-1}$, respectively, while for the subtracted anharmonic oscillator we have $V_3=-\frac{\sqrt{3}}{2}{S}_0^{-1/2}$ and $V_4=\frac{1}{2}{S}_0^{-1}$. The tadpole in diagram $c$, which comes from the zero-mode Jacobian rather than from the action, is effectively represented by the vertex $V_{\text{tad}}=\frac{\sqrt{3}}{4}\frac{\tanh (t/2)}{{\cosh }^2(t/2)}{S}_0^{-1/2}$. The signs of the contributions and symmetry factors are indicated.

Figure 2

Diagrams contributing to the coefficient $B_2$. The signs of the contributions and symmetry factors are indicated.

Figure 3

Diagrams contributing to the coefficient $B_2$. They come from the Jacobian of the zero mode and have no analogs in the anharmonic oscillator problem. The signs of the contributions and symmetry factors are indicated.