Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential

Abstract

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) in the three-loop order for quartic double-well potential. The result for the three-loop contribution coincides in six significant figures with 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.

Figures (3)

• 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 $B_{2loop}$. For the instanton field the effective triple and quartic coupling constants (vertices) 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 $V_4=\frac{1}{2}\,S_0^{-1}$, all marked by (filled) bullets. The tadpole in diagram $c$, which comes from the zero-mode Jacobian rather than from the action, is effectively represented by the vertex (Jacobian source) $V_{tad}=\frac{\sqrt{3}}{4}\frac{{\tanh}(t/2)}{{\cosh}^2(t/2)}\,S_0^{-1/2}$, marked (unfilled) open bullet. The signs of contributions and symmetry factors are indicated.
• Figure 2: Diagrams contributing to the coefficient $B_{2}$. Triple and quartic vertices $V_3, V_4$ are marked by (filled) bullets. The signs of contributions and symmetry factors are indicated.
• Figure 3: Tadpole 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. Tadpole vertex $V_{tad}$ (Jacobian source) is marked by (unfilled) open bullet. The signs of contributions and symmetry factors are indicated.