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 {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.

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) 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.

Two charges on plane in a magnetic field: III. $He^+$ ion

The $He^+$ ion on a plane subject to a constant magnetic field $B$ perpendicular to the plane is considered taking into account the finite nuclear mass. Factorization of eigenfunctions permits to reduce the four-dimensional problem to three-dimensional one. The ground state energy of the composite system is calculated in a wide range of magnetic fields from $B=0.01$ up to $B=100$ a.u. and center-of-mass Pseudomomentum $K$ from $0$ to $1000$ a.u. using a variational approach. The accuracy of calculations for $B = 0.1 $ a.u. is cross-checked in Lagrange-mesh method and not less than five significant figures are reproduced in energy. Similarly to the case of moving neutral system on the plane a phenomenon of a sharp change of energy behavior as a function of $K$ for a certain critical $K_c$ but a fixed magnetic field occurs.