New Corrections of Order Α to S-levels of Two-body Systems

New corrections to the energy of S-levels of positronium of order mα which are as large as several hundred kilohertz are obtained. A new recoil correction of order α(Zα)(m/M)m to the Lamb shift in hydrogen is calculated. This correction turns out to be too small from the phenomenological point of view. 1. Recent progress in the spectroscopy of positronium [1, 2, 3, 4, 5] triggered theoretical work on the corrections of order αm to the positronium energy levels. All logarithmic corrections of this order to S-levels were calculated recently in [6, 7]. Complete results for the corrections of order αm to P -levels were obtained in [8]. As emphasized in this last work the large magnitude of the nonlogarithmic corrections to P -levels suggest that calculation of corresponding nonlogarithmic corrections to S-levels is also important. Some of these corrections are already known, e.g., contributions induced by the twoand three-photon annihilation kernels [9, 10, 11]. We present below results of the calculation of nonlogarithmic contributions of order αm to the S-levels of positronium induced by radiative corrections to the Breit potential and by the polarization insertions in the graphs with two-photon exchange. A new radiative-recoil correction of order α(Zα)(m/M)m to the Lamb shift in hydrogen induced by a polarization operator insertion in the twophoton exchange graph is also calculated in this note. Recent experimental achievements in measuring 1S − 2S splitting in hydrogen [12] and the wellknown results on the 2S Lamb shift [13, 14, 15] clearly demonstrate that theoretical calculation of all corrections to the Lamb shift of the order of several kHz for the 1S-state and about 1 kHz for the 2S-state is necessary. Several such contributions were obtained quite recently [16, 17, 18] and the result presented below is one more such contribution (for more detailed description of the current theoretical status of the Lamb shift calculations see, e.g. [19]). 2. Let us consider first corrections of order αm to the S-levels of positronium connected with radiative insertions in the graph with one-photon exchange in Fig 1. As is well known this graph leads to the Breit potential. One may easily obtain the radiatively corrected expression for the Breit potential in the form (see, e.g. [20] and paper in preparation) U(p, r) = −α{ r − π + 8f ′ 1 + 2f2 m2c2 δ(r) + 4πp m2c2 δ(r) (1) + r(rp)p 2m2c2r3 + p 2m2c2r − (3 + 4f2) sl 2m2c2r3 The annihilation diagram contribution is missing in this expression since we do not consider annihilation contributions in this paper.