The Lamb shift contribution of very light millicharged particles

The leading order vacuum polarization contribution of very light millicharged fermions and scalar (spin–0) particles with charge εe and mass μ to the Lamb shift of the hydrogen atom is shown to imply universal, i.e. μ–independent, upper bounds on ε : ε < ∼ 10 for μ < ∼ 1 keV in the case of fermions, and for scalars this bound is increased by a factor of 2. This is in contrast to expectations based on the commonly used approximation to the Uehling potential. The recent observation [1] of an optical rotation of linearly polarized laser light generated in vacuum by a magnetic field may be due [2] to the photon initiated pair production of very light charged fermions with mass μ ≃ 0.1 eV and charge εe where ε ≃ 10. There exist, however, very strong astrophysical and cosmological constraints [3] which exclude the quoted values of ε and μ. Some of these constraints may nevertheless be relaxed in specific paraphoton scenarios [4]. Further constraints on the mass and charge of light charged fermions may be obtained from their leading order vacuum polarization contribution [3, 5, 6] to the Lamb shift of the hydrogen atom δE = E(2S1/2)−E(2P1/2) (1) or from their higher order contribution [5] to the anomalous magnetic moment of the muon. The Lamb shift constraints studied in [3, 5, 6] were restricted to μ > ∼ 1 keV where the commonly used approximate leading order vacuum polarization contribution δEVP ≃ − α5me 30π ( me μ )2 ε (2) is expected to be quite reliable. This, of course, is no longer true for the much lower values of μ considered, for example, in [2] and it is therefore necessary to study the consequences of going beyond this standard approximation relevant only for μ > ∼ αme. The lowest order Coulomb interaction V = −α/r for a point nucleus is modified at the 1-loop level according to V = −α/r + δV where [7, 8] δV (r) = − r 2αε 3π ∫ ∞