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Register a new userThree-loop Correction to the Instanton Density. II. The Sine-Gordon potential
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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.
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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.
Extending our previous construction in the sine-Gordon model, we show how to introduce two kinds of fermionic screening operators, in close analogy with conformal field theory with c<1.
Two series of integrable theories are constructed which have soliton solutions and can be thought of as generalizations of the sine-Gordon theory. They exhibit internal symmetries and can be described as gauged WZW theories with a potential term. The spectrum of massive states is determined.
We explore boundary scattering in the sine-Gordon model with a non-integrable family of Robin boundary conditions. The soliton content of the field after collision is analysed using a numerical implementation of the direct scattering problem associated with the inverse scattering method. We find that an antikink may be reflected into various combinations of an antikink, a kink, and one or more breathers, depending on the values of the initial antikink velocity and a parameter associated with the boundary condition. In addition we observe regions with an intricate resonance structure arising from the creation of an intermediate breather whose recollision with the boundary is highly dependent on the breather phase.
The semi-classical spectrum of the Homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepners G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N=2 and N=4 supersymmetric gauge theories.
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