Analytic Treatment of Positronium Spin Splittings in Light-Front QED

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, λ (λ < Λ). This generates effective interactions in the renormalized Hamiltonian, H R . These effective interactions are derived to order α in this work, with α ≪ 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, λ, is fixed, and the limit, 0 ←− mα ≪ λ ≪ mα −→ ∞, is taken. This upper bound on λ places the effects of low-energy (energy transfer below λ) emission in the effective interactions in the |ee〉 sector. This lower bound on λ 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(α4). The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, 7 6α Ryd. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision.