We calculate the perifocus precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Our result is in the form of a one-dimensional integral that is straightforward to evaluate numerically. We demonstrate the effectiveness of our formula for the case of the Yukawa potential. We obtain analytic results for potentials of the form $V(r)=\alpha r^n$ and $V(r)=\alpha \ln (r/\lambda )$ in terms of the hypergeometric function ${}_{2}F_1(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2;e^2)$, where $e$ is the eccentricity. Our results reproduce the known general relativistic ($n=-3$), constant force ($n=1$), and cosmological constant ($n=2$) precession formulas. Planetary perihelion precessions are often used to constrain the sizes of hypothetical new weak forces—our results allow for more precise, and often stronger, constraints on such proposed new forces.

We study the effect of cosmological expansion on orbits—galactic, planetary, or atomic—subject to an inverse-square force law. We obtain the laws of motion for gravitational or electrical interactions from general relativity—in particular, we find the gravitational field of a mass distribution in an expanding universe by applying perturbation theory to the Robertson-Walker metric. Cosmological expansion induces an ($\ddot{a}/a)\overrightarrow{r}$ force where $a(t)$ is the cosmological scale factor. In a locally Newtonian framework, we show that the $(\ddot{a}/a)\overrightarrow{r}$ term represents the effect of a continuous distribution of cosmological material in Hubble flow, and that the total force on an object, due to the cosmological material plus the matter perturbation, can be represented as the negative gradient of a gravitational potential whose source is the material actually present. We also consider the effect on local dynamics of the cosmological constant. We calculate the perihelion precession of elliptical orbits due to the cosmological constant induced force, and work out generalizations to the rotation curve and virial relation applicable to clusters of galaxies.

A recently introduced Bethe-Salpeter formalism is applied to the calculation of recoil corrections to the energy levels of all $n=1$, 2, and 3 states of hydrogenic ions. Finite basis set techniques are shown to allow the accurate evaluation of expressions that sum all orders of $Z\alpha$, which are rederived in the formalism using combinatoric techniques. A comparison of the all-order results with one-loop calculations and known terms of the perturbation expansion in $Z\alpha$ is made. Good agreement of the results of the present work with previous calculations is shown, and a discussion of issues that will have to be treated for the many-electron case, where highly accurate experiments have been carried out, is given.

Positronium spectroscopy ($n=1$ hyperfine splitting, $n=2$ fine structure, and the $2S\text{−}1S$ interval) has reached a precision of order 1 MHz. Vigorous ongoing efforts to improve the experimental results motivate the calculation of the positronium energy levels at order $m{\alpha }^7$. In this Letter, we present the result for a complete class of such contributions—those involving virtual annihilation of positronium to three photons in an intermediate state. We find an energy shift of $2.6216(11)m{\alpha }^7/(n\pi {)}^3=11.5/n^3\mathrm{kHz}$ for the spin-triplet $S$ state with principal quantum number $n$. The corresponding energy shift for true muonium (the ${\mu }^{+}{\mu }^{-}$ bound state) is $2.38/n^3\mathrm{MHz}$ with an additional $-5.33/n^3\mathrm{MHz}$ coming from electronic vacuum polarization.

Positronium plays a unique role in the continuing development of the techniques of bound-state QED and high-precision tests of the standard model as a Coulombic bound system that is almost free of strong and weak interaction contaminations and that exhibits large recoil and annihilation (virtual and real) effects. The present frontier is at the level of three-loop corrections to energy levels and decay rates. We have calculated the three-loop correction to the positronium hyperfine splitting due to light-by-light scattering in the exchange of two photons between the electron and positron. Our result is $\Delta E=-0.235355\left(8\right)m{\alpha }^4{\left(\frac{\alpha }{\pi }\right)}^3=-1.033\mathrm{kHz}$. As a check of our approach, we confirm earlier evaluations of the analogous correction for a bound system (such as muonium) with unequal masses. DOI: http://dx.doi.org/10.1103/PhysRevA.89.052518 ©2014 American Physical Society