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, $\Lambda$, and effective cutoff, $\lam$ ($\lam < \Lam$). This generates effective interactions in the renormalized Hamiltonian, $H_{{}_R}$. These effective interactions are derived to order $\alpha$ in this work, with $\alpha \ll 1$. $H_{{}_R}$ 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 $O(\alpha )$ reproduces the exact spectrum of spin splittings, with degeneracies dictated by rotational symmetry, for the ground state through $O({\alpha }^4)$. The entire calculation is performed analytically, and gives the well known singlet-triplet ground state spin splitting of positronium, $7/6{\alpha }^{2}Ryd$. We discuss remaining corrections other than the spin splittings and how they can be treated in calculating the spectrum with higher precision.

46 pages, latex, 3 Postscript figures included, section on remaining corrections added, title changed, error in older version corrected, cutoff placed in a window

References in corpus (2)

• Quarkonia in Hamiltonian Light-Front QCD
• Initial bound state studies in light-front QCD

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