The soft-energy region in the radiative decay of bound states

Abstract

The orthopositronium decay to three photons is studied in the phase-space region where one of the photons has an energy comparable to the relative three-momentum of the e+e- system (w ~ m alpha). The NRQED computation in this regime shows that the dominant contribution arises from distances ~ 1/(mw)^(1/2), which allows to treat the Coulomb interaction perturbatively. The small-photon energy expansion of the 1-loop decay spectrum from full QED yields the same result as the effective theory. By doing the threshold expansion of the 1-loop QED amplitude we confirm that the leading term arises from a loop-momentum region where q^0 ~ q^2/m ~ w. This corresponds to a new non-relativistic loop-momentum region, which has to be taken into account for the description of a non-relativistic particle-antiparticle system that decays through soft photon emission.Comment: 28 pages, 5 figures; typos corrected, one reference added, published versio

Positronium decay into a photon and neutrinos

“…In three-body channels, the energy of decay products has an extended distribution. Its low-energy tail is sensitive to binding effects; such effects have been determined in the three-photon decay of o-Ps [23][24][25][26]. We find analogous phenomena in the Ps → γν ν decay.…”

“…While normally the dipole approximation is applicable for photons with wavelengths much larger than the spatial extent of the Ps atom, 2/mα (i.e., E γ mα), it has been shown that the dipole approximation of the Coulomb interaction Hamiltonian holds in the enlarged energy region E γ m for the three-body decay o-Ps → 3γ [25,26]. In this energy region, amplitudes where the intermediate states propagate via the Coulomb Green's function, are a series in α m/E γ ∼ √ α rather than integer powers of α.…”

“…where −κ 2 /m = E o − E γ and G C (x, y, κ) is the Coulomb Green's function. The derivative selects the l = 1 partial wave of the Green's function [26] ∂ i x G C (x, y, κ) x=0 = 3y i G 1 (0, y, κ).…”

The $\Upsilon\rightarrow X \gamma$ photon spectrum in the soft-energy region

“…The Non-Relativistic QED (NRQED) framework used in Refs. [1,2] provides a systematic way to compute bound state effects in the the o-Ps decay spectrum when the photon energy is much smaller than the electron mass. Contrary to the usual relativistic approach where the 3photon annihilation is considered to take place at very short distances as compared to the range of the electromagnetic binding force between e + e − , the non-relativistic description takes into account that there is a long-distance part in the o-Ps decay process when one the photons in the final state is not hard (ω m).…”

“…1). The emission of the low energy photon from o-Ps is described by the Coulomb Hamiltonian of the e + e − system in interaction with a quantized electromagnetic field [1,2].…”