Logarithmic Soft Photon Theorem and Waveform Tails in Higher Dimensions

Abstract

We derive the leading logarithmic soft-photon theorem in $d>4$ spacetime dimensions and its classical radiative counterpart. A direct one-loop analysis in massive scalar quantum electrodynamics (QED) yields a factorizing soft term of order ${\omega }^{d-4}\ln \omega$, although the charged-particle S-matrix is infrared finite. The logarithm is generated by the scale-invariant loop-momentum region $\omega \ll |\ell |\ll \Lambda$, where $\Lambda$ denotes a characteristic hard-particle energy scale. At leading radiative order, the corresponding logarithmic contribution to the classical electromagnetic waveform arises from the long-range acceleration of the asymptotic charged particles. In even $d\ge 6$, the straight-line waveform at this radiative order is distributionally supported in retarded time within an interval whose width is set by the characteristic size of the hard-scattering region, whereas the logarithmic acceleration term produces universal early- and late-time radiative tails proportional to $|u{|}^{-(3d-10)/2}$. In odd $d\ge 5$, straight-line motion already gives a late-time tail at the same radiative order proportional to $u^{-(d-4)/2}$. The acceleration correction adds universal late- and early-time terms proportional, respectively, to $\ln u/u^{(3d-10)/2}$ and $|u{|}^{-(3d-10)/2}$. A comparison of Feynman and retarded boundary conditions separates the classically radiative contribution from the intrinsically quantum part of the logarithmic soft factor.

13 pages + derivation