Grant: Subatomic Physics Envelope - Individual

My goal in the next five years is to finalize an algorithm for finding quantum corrections to properties of bound states. Properties such as energy levels of muonium are determined by the interaction between the electron and the muon, their self-interactions, and their relative motion. Improving the theory of any of these effects requires sifting through contributions arising at various energy scales. At a certain level of precision, complications prohibit the theoretical progress. My group specializes in theoretical predictions to support precise experiments. Multi-loop Feynman amplitudes are evaluated, usually as an expansion in some parameter. The mathematics involved is known as asymptotic expansions, greatly developed in high-energy physics. Resulting large formulas are handled with symbolic computation and we are fortunate to have a cluster of dedicated, large-memory computers in Alberta. We now want to generalize this successful approach to bound states. The challenge is that the amplitudes involve not only small numerical parameters but also operators like the relative momentum or a perturbing potential. They are also small, in the sense of expectation values. The algorithm we envision will generate expressions needed for the desired precision, much like Feynman diagrams in a given loop order can be produced by computer programs. Such an automated approach to bound states is nontrivial. It requires an extension of the known mathematics of expansion of integrals to the situation where an integrand depends on non-commuting operators (momentum and a position-dependent potential). We have not yet been able to solve this problem in all generality, and propose to focus on three specific cases, urgently needed by experiments. We hope that the general method will crystallize out of these efforts. Five years from now, experiments will have progressed on all three fronts. Searches for lepton-flavor violation, COMET and Mu2e, will have taken their first data. Their background will be under control thanks to our results on the bound muon decay. They will map the electron spectrum and test our predictions. Our theory of the bound-electron g-2, together with the anticipated progress in measurements, will result in a more accurate electron mass, and hopefully also a better value of the fine structure constant alpha. Very likely, the measurement of the free-electron g-2 will be improved, and with the new alpha will check whether the current muon g-2 discrepancy is due to new physics. Our improved prediction of the Lamb shift will help interpret the other low-energy conundrum, that of the proton radius. My dream is that by then we will have a fully-fledged algorithm for bound state studies.