Analytic Treatment of Positronium Spin Splittings in Light-Front QED

Abstract

We study the QED bound-state problem in a light-front hamiltonian approach. Starting with a bare cutoff QED Hamiltonian, $H_{_{B}}$, with matrix elements between free states of drastically different energies removed, we perform a similarity transformation that removes the matrix elements between free states with energy differences between the bare cutoff, $Λ$, and effective cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the renormalized Hamiltonian, $H_{_{R}}$. These effective interactions are derived to order $α$ in this work, with $α\ll 1$. $H_{_{R}}$ is renormalized by requiring it to satisfy coupling coherence. A nonrelativistic limit of the theory is taken, and the resulting Hamiltonian is studied using bound-state perturbation theory (BSPT). The effective cutoff, $\lam^2$, is fixed, and the limit, $0 \longleftarrow m^2 α^2\ll \lam^2 \ll m^2 α\longrightarrow \infty$, is taken. This upper bound on $\lam^2$ places the effects of low-energy (energy transfer below $\lam$) emission in the effective interactions in the $| e {\overline e} > $ sector. This lower bound on $\lam^2$ insures that the nonperturbative scale of interest is not removed by the similarity transformation. As an explicit example of the general formalism introduced, we show that the Hamiltonian renormalized to $O(α)$ reproduces the exact spectrum of spin splittings, with degeneracies dictated by rotational symmetry, for the ground state through $O(α^4)$. The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, $7/6 α^2 Ryd$. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision.