Radiative corrections in hydrogen-like systems

Abstract

The lowest order term was subsequently calculated relativlstiually, and evaluation of successively higher order terms followed that. To display th?results of these calculations, we express the level shift in the formwhere F(aa) = k k0 * A kl en(za)" 2 + A^za) + A 6n {2afIn Table 1 for all Zot and agrees, by construction, with the small la expan-12 sion.The results of these calculations appear in Fig. 10.1.-it- VW --frThe bar over the wave function denotes the adjoint: 0 ^x) = 0 <(x)rTo keep the terms in the energy shift separately finite, we use the 19 20 covariant regulator method, where in momentum space the following replacement is made for the photon propagator:The limit is to be taken after the integrals in Eq. In order to do the integration over t" -t^ first, we place in the integrand a convergence factor exp(-6Jt -t |), where 6 temporarily satisfies the conditionWe shall eventually let & tend to zero. We now integrate over(2.12) Equation (2.10) can be written -6m Id^tfx)^^) (P.13)whereThe integration over k in T p is done next. It is convenient to integrate over k. firstThen we perform the integration over k in I_ and obtainThe branches of the square roots are determined by the conditionsThe .'-n. •M^'ljl^nlIn order to facilitate the evaluation of (2."9), we change the contour of integration C" to a new contour, and divide the integral The total energy shift is -12-For these contributions, we are interested in the limit as e -» 0+, as 2, and z~ approach zero from below and above the real axis respectively, and as R -*». This limit will be considered first for AEj. and then for /sE".-13-The low-energy part of the energy shift is * |x -x j texp(-bjj 2 -xj) -expC-b'lx^ -xj)) . {3.1)Tne contour CT is shown in Pig. (3.'*)Henceforth we assume tha'. A > E . We then have n X