Energy levels of multiscale bound states from QED energy-momentum trace

Energy levels of QED bound states, which depend on a number of independent mass parameters, can be calculated as matrix elements of the QED energy-momentum tensor trace. As an example of such system we consider muonic hydrogen. The leading one-loop corrections to its energy levels depend on the electron and muon masses. These corrections are calculated as matrix elements of the energy-momentum tensor trace. Respective one-loop trace diagrams are different from the standard Lamb shift diagrams. We explain analytically and diagrammatically why two different sets of diagrams lead to the same results. Similar relationships should also hold beyond the one-loop approximation.