The decay rate of the electroweak (EW) vacuum is calculated in the framework of the standard model (SM) of particle physics, using the recent progresses in the understanding of the decay rate of metastable vacuum in gauge theories. We give a manifestly gauge-invariant expression of the decay rate. We also perform a detailed numerical calculation of the decay rate. With the best-fit values of the SM parameters, we find that the decay rate of the EW vacuum per unit volume is about 10 −554 Gyr −1 Gpc −3 ; with the uncertainty in the top mass, the decay rate is estimated as 10Introduction: It is highly non-trivial whether the vacuum we are living in, which we call electroweak (EW) vacuum, is absolutely stable or not. If there exists a vacuum which has lower energy density than that of the EW vacuum, which is the case in a large class of particle-physics models, the EW vacuum decays via the quantum tunneling effect. If the decay rate is too large, the universe should have been experienced a phase transition before the present epoch, with which the universe would show completely different aspects than the present one. From the particle-physics and cosmology points of view, the stability of the EW vacuum is of particular interest to have deep insight into particle-physics models and the nature of the universe. Even in the standard model (SM) of particle physics, which is extremely successful to explain particle interactions, the EW vacuum may be metastable [1][2][3][4][5][6][7]. In particular, the discovery of the Higgs boson by the LHC experiments [8,9] shed light on the stability of the EW vacuum. The observed value of the Higgs mass suggests that the Higgs quartic coupling becomes negative via the renormalization group (RG) effects at energy scale much higher than the EW scale. This fact implies that the Higgs potential becomes negative and that the EW vacuum is not absolutely stable if the SM is valid up to a scale much higher than the EW scale.The decay rate of the EW vacuum has been estimated in the past, mostly using the method given in [10][11][12]. The decay rate of the metastable vacuum (i.e., false vacuum) per unit volume, which we call γ, is given in the following form:

We perform a precise calculation of the decay rate of the electroweak vacuum in the standard model as well as in models beyond the standard model. We use a recently developed technique to calculate the decay rate of a false vacuum, which provides a gauge invariant calculation of the decay rate at the oneloop level. We give a prescription to take into account the zero modes in association with translational, dilatational, and gauge symmetries. We calculate the decay rate per unit volume, γ, by using an analytic formula. The decay rate of the electroweak vacuum in the standard model is estimated to be log 10 γ × Gyr Gpc 3 ¼ −582 þ40þ184þ144þ2 −45−329−218−1 , where the first, second, third, and fourth errors are due to the uncertainties of the Higgs mass, the top quark mass, the strong coupling constant and the choice of the renormalization scale, respectively. The analytic formula of the decay rate, as well as its fitting formula given in this paper, is also applicable to models that exhibit a classical scale invariance at a high energy scale. As an example, we consider extra fermions that couple to the standard model Higgs boson, and discuss their effects on the decay rate of the electroweak vacuum.

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Based on the gradient flow, we propose a new method to determine the bounce configuration for false vacuum decay. Our method is applicable to a large class of models with multiple fields. Since the bounce configuration is a saddle point of an action, a naive gradient flow method does not work. We point out that a simple modification of the flow equation can make the bounce configuration its stable fixed point while the false vacuum configuration an unstable one. Consequently, the bounce configuration can be obtained simply by following the flow without a careful choice of an initial configuration. With numerical analysis, we confirm the validity of our claim, checking that the flow equation we propose indeed has solutions that flow into the bounce configuration.Introduction: Study of false vacua (and metastable states) has been important in various fields, like particle physics, cosmology, nuclear physics, condensed matter physics, and so on. For example, in the field of particle physics and cosmology, the stability of the electroweak vacuum has been attracted much attention. In particular, taking the best-fit values of the observed top-quark and Higgs-boson masses, and assuming that the standard model of particle physics is valid up to a very high scale (like the Planck scale), the electroweak vacuum is metastable [1,2]. It is because the Higgs quartic coupling constant becomes negative at a high energy scale due to the renormalization group effects. Performing precise calculation based on relativistic quantum field theory, the decay rate of the electroweak vacuum per unit volume is known to be ∼ 10 −582 Gyr −1 Gpc −3 [3-5], with which the stability of our universe looks plausible for the present cosmic time scale. However, this conclusion may be altered with the introduction of new physics beyond the standard model. The studies of the stability of the electroweak vacuum in such new physics models remain important.In relativistic quantum field theory, the decay of the false vacuum is mainly induced by the field configuration called "bounce" [6][7][8]. Bounce is a configuration obeying the classical equation of motion (EOM) derived from the Euclidean action. With the bounce, which we denote as φ, the decay rate of the false vacuum per unit volume is given in the following form:

There are many models beyond the standard model which include electroweakly interacting massive particles (EWIMPs), often in the context of the dark matter. In this paper, we study the indirect search of EWIMPs using a precise measurement of the Drell-Yan cross sections at future 100 TeV hadron colliders. It is revealed that this search strategy is suitable in particular for Higgsino and that the Higgsino mass up to about 1.3 TeV will be covered at 95 % C.L. irrespective of the chargino and neutralino mass difference. We also show that the study of the Drell-Yan process provides important and independent information about every kind of EWIMP in addition to Higgsino.