Three-Loop Slope of the Dirac Form Factor and the1SLamb Shift in Hydrogen

Abstract

The last unknown contribution to hydrogen energy levels at order mα 7 , due to the slope of the Dirac form factor at three loops, is evaluated in a closed analytical form. The resulting shift of the hydrogen nS energy level is found to be 3.016/n 3 kHz. Using the QED calculations of the 1S Lamb shift, we extract a precise value of the proton charge radius rp = 0.883 ± 0.014 fm.Precision experiments with hydrogen and, more generally, with hydrogen-like atoms serve as an excellent laboratory to test theoretical approaches to bound state QED (for a recent review see, e.g. [1]). These experiments address a number of features of the simplest atoms, such as the energy levels of the ground and excited states, and the corresponding lifetimes.In recent years we have seen remarkable progress in the experimental study of the hydrogen atom. In particular the accuracy of the 1S Lamb shift measurements has increased dramatically over the years [2][3][4][5][6][7]. All measurements are consistent with each other and the most accurate value so far was determined in Ref. [7]:This result was obtained by analysing the most precise measurements for the transition frequencies in hydrogen (see [7] for details). Theoretically, since the ratio of the electron mass m to the proton mass M is very small, it is convenient to write hydrogen energy levels as a double expansion in α and m/M . The corrections which survive in the limit M → ∞ are known as non-recoil corrections, the other corrections are called recoil ones. It is further convenient to organize non-recoil corrections in powers of α n (Zα) l , assigning an auxiliary notation Z for the proton charge. In this case the correction α n (Zα) l describes a contribution of all diagrams with l − 3 Coulomb photons exchanged between the electron and the proton and with * e-mail: melnikov@slac.stanford.edu † e-mail: timo@particle.physik.uni-karlsruhe.de n photons emitted and absorbed by the electron. The general expression for the nS-level shift can be written as:where we have also added a contribution of muons and hadrons to photon vacuum polarization and a contribution due to the proton structure (see a discussion below). We begin with non-recoil corrections. These corrections can be parameterized as (see e.g. [8][9][10]):where m r = mM/(m + M ) is the reduced mass of the electron and L = log m/(m r (Zα) 2 ) and the function A(Zα) contains all higher order terms in the expansion in Zα. We have also indicated that the higher order corrections contain a logarithm of the fine structure constant in the third power [11]. All the terms in the above equation, with the exception for C 40 , are currently known. It is the purpose of this paper to report on the calculation of the last missing ingredient in C 40 , the slope of the Dirac form factor at zero momentum transfer.The hydrogen atom is formed because of a Coulomb interaction of the proton with the electron. The interaction of the virtual photon with the electron on its mass shell can be parameterized by the so-called Dirac and Pauli form facto...

Theory of light hydrogenlike atoms

“…The three-loop contribution to the slope of the Dirac form factor (Fig. 14) was recently calculated analytically [56] …”

Measuring the running top-quark mass

“…(1)) and the scale dependent MS mass m(µ r ) we exploit well-known relations to NNLO [23] (see also Refs. [24,25]). We investigate the apparent convergence of both definitions, m t and m(µ r ), in perturbation theory through NNLO.…”

Two-Loop Bethe-Logarithm Correction in Hydrogenlike Atoms

“…(3), and the three-loop binding correction C 50 (for a recent evaluation of C 40 see Ref. [22]). Theoretical predictions for the 1S state agree well with the experimental value of the combined result from the Garching and the Paris groups [3,4,23]:…”