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Abstract: The decay rate for a particle in a metastable cubic potential is investigatedin the quantum regime by the Euclidean path integral method in semiclassicalapproximation. The imaginary time formalism allows one to monitor the system asa function of temperature. The family of classical paths, saddle points for theaction, is derived in terms of Jacobian elliptic functions whose periodicitysets the energy-temperature correspondence. The period of the classicaloscillations varies monotonically with the energy up to the sphaleron, pointingto a smooth crossover from the quantum to the activated regime. The softeningof the quantum fluctuation spectrum is evaluated analytically by the theory ofthe functional determinants and computed at low $T$ up to the crossover. Inparticular, the negative eigenvalue, causing an imaginary contribution to thepartition function, is studied in detail by solving the Lam\`{e} equation whichgoverns the fluctuation spectrum. For a heavvy particle mass, the decay rateshows a remarkable temperature dependence mainly ascribable to a low lying softmode and, approaching the crossover, it increases by a factor five over thepredictions of the zero temperature theory. Just beyond the peak value, theclassical Arrhenius behavior takes over. A similar trend is found studying thequartic metastable potential but the lifetime of the latter is longer by afactor ten than in a cubic potential with same parameters. Some formalanalogies with noise-induced transitions in classically activated metastablesystems are discussed.

Autor: Marco Zoli


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