Theory of Loosely Bound Composite Systems: Multiloop Corrections to Lamb Shift, Hyperfine Splitting, and g-Factors

High precision experiments with two-body bound systems have achieved a new level of accuracy in recent years and further progress is expected. The experimental errors of measurements of energy shifts in hydrogen, muonium and other simple hydrogenlike ions were reduced by orders of magnitude. These recent developments opened new perspectives for precise determination of many fundamental constants (the Rydberg constant, electron-muon mass ratio, electron mass in atomic units, proton charge radius, deuteron structure radius, etc.), and for comparison of the experimental and theoretical results on the Lamb shifts and hyper¯ne splitting. The experimental progress poses new theoretical challenges. Reduction of the theoretical error of the value of the 1S Lamb shift in hydrogen to the level below 1 kHz (and, respectively, of the 2S Lamb shift to below several tenth of kHz) should be considered as a next stage of the theory. The theoretical error of the hyper¯ne splitting in muonium should be reduced to about 10 Hz. The theoretical error of the 2S ¡ 2P Lamb shift in muonic hydrogen should be reduced below 0:001 meV. The problem of the spin dependence of the bound state corrections to g factors should be resolved. The following major problems in the theory of low Z systems will be addressed in the proposed research: ² Three-loop nonrecoil corrections to the Lamb shift and hyper¯ne splitting in hydrogen and muo- nium. ² Three-loop radiative-recoil corrections to hyper¯ne splitting in muonium. ² Nonrecoil high-order corrections to the Lamb shift in muonic hydrogen. ² Binding and recoil corrections to the bound g factor in systems including particles with spins j 6= 1=2. In the proposed research quantum electrodynamics will be applied to loosely bound two-particle systems. The calculations will be done in the framework of perturbation theory and nonrelativistic quantum electrodynamics, and combined expansions in the small parameters ®, Z®, and m=M will be performed. Three-loop nonrecoil corrections of order ®3(Z®)5m to the Lamb shift in hydrogen and of order ®3(Z®)EF to hyper¯ne splitting in muonium will be considered. Calculations of these corrections will be performed in the framework of the skeleton integral approach developed earlier. All three-loop radiative-recoil corrections to hyper¯ne splitting in muonium of order ®3(m=M)EF will be calculated. The two-loop Yennie gauge vertex and the nonrecoil two-loop electron factor ob- tained earlier will be used in these calculations in the framework of the skeleton integral approach. Nonlogarithmic recoil corrections of order (Z®)3(m=M)EF , and ®(Z®)2(m=M)EF to hyper¯ne split- ting in muonium will be considered in the nonrelativistic quantum electrodynamics approach. Methods of nonrelativistic quantum electrodynamics will be employed to obtain nonlogarithmic contribution of order ®2(Z®)4m and light by light scattering contribution of order ®2(Z®)3 to the Lamb shift in muonic hydrogen. Leading relativistic, radiative and recoil corrections to bound state g-factors of particles with ar- bitrary spins will be obtained, and the controversy about the dependence of these corrections on the magnitude of the spins of the constituents will be resolved. As a result of the proposed research the theoretical errors of the Lamb shift in electronic hydrogen, hyper¯ne splitting in muonium, and the Lamb shift in muonic hydrogen will be signi¯cantly reduced. Together with the high precision experimental data and the theoretical results of other authors the results of the proposed research will lead to a more precise determination of a number of important physical constants: electron-muon mass ratio, proton charge radius, the Rydberg constant, etc. The results of the proposed research will be published in refereed journals, will be presented at international conferences and workshops, and used in teaching graduate courses. At least one graduate student will participate in this research.