Fine structure: higher order contributions

In Chapter 1, the first order contributions to the annihilation rates from the dominant modes of decay of the S-states of both ortho- and para-positronium (for arbitrary principal quantum number nPs) were given as equations (1.5) and (1.6). These contributions are included in the following equations for the rates for the two ground states, which also contain terms of higher order in the fine structure constant, a ... 

Above we have proposed to measure the the small difference frequency df = f(1S-2S) - 3 f(2S-nS). This frequency depends critically on the Lamb shifts of the participating levels, and can provide a stringent test of QED. For n=100, the theoretical uncertainty of df is dominated by the nuclear size effect ( 70 kHz) and by approximations in the computation of electron structure corrections and uncalculated higher order QED corrections (65 kHz). The contribution of the Rydberg constant (1 kHz), the electron mass (0.05 kHz), and the fine structure constant (4 kHz) are negligible by comparison. The QED computations can be improved, and if theory is correct, a precision measurement of df can provide accurate new values for the charge radii of the proton and deuteron. 

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