In a classically scale-invariant quantum field theory, tunneling rates are infrared divergent due to the existence of instantons of any size. While one expects such divergences to be resolved by quantum effects, it has been unclear how higher-loop corrections can resolve a problem appearing already at one loop. With a careful power counting, we uncover a series of loop contributions that dominate over the one-loop result and sum all the necessary terms. We also clarify previously incomplete treatments of related issues pertaining to global symmetries, gauge fixing, and finite mass effects. In addition, we produce exact closedform solutions for the functional determinants over scalars, fermions, and vector bosons around the scaleinvariant bounce, demonstrating manifest gauge invariance in the vector case. With these problems solved, we produce the first complete calculation of the lifetime of our Universe: 10 139 years. With 95% confidence, we expect our Universe to last more than 10 58 years. The uncertainty is part experimental uncertainty on the top quark mass and on α s and part theory uncertainty from electroweak threshold corrections. Using our complete result, we provide phase diagrams in the m t /m h and the m t /α s planes, with uncertainty bands. To rule out absolute stability to 3σ confidence, the uncertainty on the top quark pole mass would have to be pushed below 250 MeV or the uncertainty on α s ðm Z Þ pushed below 0.00025.

The stability of the Standard Model is determined by the true minimum of the effective Higgs potential. We show that the potential at its minimum when computed by the traditional method is strongly dependent on the gauge parameter. It moreover depends on the scale where the potential is calculated. We provide a consistent method for determining absolute stability independent of both gauge and calculation scale, order by order in perturbation theory. This leads to a revised stability bounds m pole h > (129.4 ± 2.3) GeV and m pole t < (171.2 ± 0.3) GeV. We also show how to evaluate the effect of new physics on the stability bound without resorting to unphysical field values.

It is well known that effective potentials can be gauge-dependent while their values at extrema should be gauge-invariant. Unfortunately, establishing this invariance in perturbation theory is not straightforward, since contributions from arbitrarily highorder loops can be of the same size. We show in massless scalar QED that an infinite class of loops can be summed (and must be summed) to give a gauge invariant value for the potential at its minimum. In addition, we show that the exact potential depends on both the scale at which it is calculated and the normalization of the fields, but the vacuum energy does not. Using these insights, we propose a method to extract some physical quantities from effective potentials which is self-consistent order-by-order in perturbation theory, including improvement with the renormalization group.

In applications of machine learning to particle physics, a persistent challenge is how to go beyond discrimination to learn about the underlying physics. To this end, a powerful tool would be a framework for unsupervised learning, where the machine learns the intricate high-dimensional contours of the data upon which it is trained, without reference to pre-established labels. In order to approach such a complex task, an unsupervised network must be structured intelligently, based on a qualitative understanding of the data. In this paper, we scaffold the neural network's architecture around a leading-order model of the physics underlying the data. In addition to making unsupervised learning tractable, this design actually alleviates existing tensions between performance and interpretability. We call the framework Junipr: "Jets from UNsupervised Interpretable PRobabilistic models". In this approach, the set of particle momenta composing a jet are clustered into a binary tree that the neural network examines sequentially. Training is unsupervised and unrestricted: the network could decide that the data bears little correspondence to the chosen tree structure. However, when there is a correspondence, the network's output along the tree has a direct physical interpretation. Junipr models can perform discrimination tasks, through the statistically optimal likelihood-ratio test, and they permit visualizations of discrimination power at each branching in a jet's tree. Additionally, Junipr models provide a probability distribution from which events can be drawn, providing a data-driven Monte Carlo generator. As a third application, Junipr models can reweight events from one (e.g. simulated) data set to agree with distributions from another (e.g. experimental) data set.

Tunneling in quantum field theory is worth understanding properly, not least because it controls the long term fate of our universe. There are however, a number of features of tunneling rate calculations which lack a desirable transparency, such as the necessity of analytic continuation, the appropriateness of using an effective instead of classical potential, and the sensitivity to short-distance physics. This paper attempts to review in pedagogical detail the physical origin of tunneling and its connection to the path integral. Both the traditional potential-deformation method and a recent more direct propagator-based method are discussed. Some new insights from using approximate semi-classical solutions are presented. In addition, we explore the sensitivity of the lifetime of our universe to short distance physics, such as quantum gravity, emphasizing a number of important subtleties.