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Register a new userA Fresh Look at the Calculation of Tunneling Actions including Gravitational Effects
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J. R. Espinosa
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Recently, the calculation of tunneling actions, that control the exponential suppression of the decay of metastable vacua, has been reformulated as an elementary variational problem in field space. This paper extends this formalism to include the effect of gravity. Considering tunneling potentials $V_t(phi)$ that go from the false vacuum $phi_+$ to some $phi_0$ on the stable basin of the scalar potential $V(phi)$, the tunneling action is the minimum of the functional $S_E[V_t]=6 pi^2m_P^4int_{phi_+}^{phi_0}(D+V_t)^2/(V_t^2D)dphi $, where $Dequiv [(V_t)^2+6(V-V_t)V_t/m_P^2]^{1/2}$, $V_t=dV_t/dphi$ and $m_P$ is the reduced Planck mass. This one-line simple result applies equally to AdS, Minkowski or dS vacua decays and reproduces the Hawking-Moss action in the appropriate cases. This formalism provides new handles for the theoretical understanding of different features of vacuum decay in the presence of gravity.
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An alternative approach to the calculation of tunneling actions, that control the exponential suppression of the decay of metastable phases, is presented. The new method circumvents the use of bounces in Euclidean space by introducing an auxiliary function, a tunneling potential $V_t$ that connects smoothly the metastable and stable phases of the field potential $V$. The tunneling action is obtained as the integral in field space of an action density that is a simple function of $V_t$ and $V$. This compact expression can be considered as a generalization of the thin-wall action to arbitrary potentials and allows a fast numerical evaluation with a precision below the percent level for typical potentials. The method can also be used to generate potentials with analytic tunneling solutions.
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