Anomalies in particle physics and their implications for physics beyond the standard model

The standard model (SM) of particle physics is the mathematical description of the fundamental constituents and interactions of matter. Its last missing particle, the Higgs boson, was observed in 2012. However, there are several phenomena that the SM cannot account for (such as dark-matter particles, or non-vanishing neutrino masses), neither does it describe gravity. There must be more to discover, to extend the SM into a full description of nature. Here we review the hints of new physics, called anomalies, that are seen for various interactions as discrepancies between standard-model predictions and experimental measurements. We consider both direct high-energy searches for new particles at the Large Hadron Collider at CERN and indirect low-energy precision experiments. These anomalies span an energy scale of more than four orders of magnitude: from the mass of the proton, to the electroweak scale (approximately the mass of the Higgs boson), to the teraelectronvolt scale, which is the highest scale directly accessible at the Large Hadron Collider. We discuss the experimental and theoretical status of various anomalies and summarize possible explanations in terms of new particles and new interactions as well as discovery prospects. We suggest, in particular, that new additional Higgs bosons and so-called leptoquarks are promising candidates for extending the standard model.