High energy physics, e.g., at CERN, promises to discover physics beyond the standard model. However, it is not necessary to expend all this energy and money to probe fundamental physics. One can do so in tabletop experiments on cold atoms, molecules and ions.
Several blog posts have recorded the recent appearance on the arxiv of a review by Ted Haensch of measurements of the fine structure constant by spectroscopy of cold atoms, ions and molecules. The idea is to compare very precise measurements at different times to detect any time dependence of the fine stucture constant, which would show physics beyond the standard model.
This current interest has reminded me of work by Ed Hinds to measure the electron electric dipole moment by observing the change in the energy levels of cold molecules on the application of a magnetic field. A non-zero value of such a quantity would show physics beyond the standard model. The Imperial College website has several pages of lucid explanation for the layperson here. Here is a chart from their website showing how precise they are trying to get.