Three-loop correction to the instanton density. II. The sine-Gordon potential

In this second paper on quantum fluctuations near the classical instanton configurations [see M. A. Escobar-Ruiz, E. Shuryak, and A. V. Turbiner, preceding article, Phys. Rev. D 92, 025046 (2015)], 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 six significant figures) the one recently recalculated by G. V. Dunne and M. Ünsal, Phys. Rev. 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.

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 $B_{2\text{loop}}$. For the instanton field the effective triple, quartic, quintic and sextic vertices are $V_3=4\sqrt{2}\frac{\sinh (t)}{{\cosh }^2(t)}{S}_0^{-1/2}$, $V_4=8(2/{\cosh }^2(t)-1)S_0^{-1}$, $V_5=-32\sqrt{2}\frac{\sinh (t)}{{\cosh }^2(t)}{S}_0^{-3/2}$, $V_6=-64(2/{\cosh }^2(t)-1)S_0^{-2}$, respectively, while for the subtracted vacuum field diagrams we have $V_3=V_5=0$, $V_4=8S_0^{-1}$ and $V_6=64S_0^{-2}$. The tadpole in diagram $c$, which comes from the zero-mode Jacobian rather than from the action, is effectively represented by the vertex $V_{\text{tad}}=\frac{1}{\sqrt{2}}\frac{\sinh (t)}{{\cosh }^2(t)}{S}_0^{-1/2}$ (see text). 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 perturbative vacuum. The signs of the contributions and symmetry factors are indicated.