Functional methods for false-vacuum decay in real time
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Springer
Received: May Revised: September Accepted: November Published: December
17, 30, 23, 11,
2019 2019 2019 2019
Wen-Yuan Ai, Bj¨ orn Garbrecht and Carlos Tamarit Physik Department T70, James-Franck-Straße, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany
E-mail: [email protected], [email protected], [email protected] Abstract: We present the calculation of the Feynman path integral in real time for tunneling in quantum mechanics and field theory, including the first quantum corrections. For this purpose, we use the well-known fact that Euclidean saddle points in terms of real fields can be analytically continued to complex saddles of the action in Minkowski space. We also use Picard-Lefschetz theory in order to determine the middle-dimensional steepestdescent surface in the complex field space, constructed from Lefschetz thimbles, on which the path integral is to be performed. As an alternative to extracting the decay rate from the imaginary part of the ground-state energy of the false vacuum, we use the optical theorem in order to derive it from the real-time amplitude for forward scattering. While this amplitude may in principle be obtained by analytic continuation of its Euclidean counterpart, we work out in detail how it can be computed to one-loop order at the level of the path integral, i.e. evaluating the Gaußian integrals of fluctuations about the relevant complex saddle points. To that effect, we show how the eigenvalues and eigenfunctions on a thimble can be obtained by analytic continuation of the Euclidean eigensystem, and we determine the path-integral measure on thimbles. This way, using real-time methods, we recover the one-loop result by Callan and Coleman for the decay rate. We finally demonstrate our real-time methods explicitly, including the construction of the eigensystem of the complex saddle, on the archetypical example of tunneling in a quasi-degenerate quartic potential. Keywords: Nonperturbative Effects, Solitons Monopoles and Instantons ArXiv ePrint: 1905.04236
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)095
JHEP12(2019)095
Functional methods for false-vacuum decay in real time
Contents 1
2 Optical theorem for the decay of the false vacuum 2.1 The optical theorem in scattering theory 2.2 Optical theorem for false-vacuum decay 2.3 Boundary conditions on the path integral, Lefschetz thimbles and the false-vacuum state
5 6 6 7
3 Complex saddle points in the path integral with complex time and the Minkowski case 10 3.1 Complex saddles 11 3.2 Complexified path integral and Gaußian approximation 14 3.3 Flow equations and Jacobian 20 3.4 Generalization to quantum field theory 23 4 Analytic continuation of the fluctuation spectrum 4.1 Eigenmodes and eigenvalues 4.2 Normalization of the eigenmodes 4.3 Completeness of the eigenmodes 4.4 Fluctuation determinant 4.5 Spherical geometry
26 27 30 32 32 36
5 Examples for the analytic continuation of the fluctuation spectrum 5.1 Effective action evaluated at a constant vacuu
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