Two charged spheres collided and flew apart in different directions. You'd think it was all over — but the light from their farewell still wanders through space. In the ordinary world, this is a well-known effect: an accelerated charge always radiates. But what if there aren't three dimensions, but five, seven, or more?
The authors of the new work took the simplest theory — quantum electrodynamics for massive charged particles — and calculated the radiation of very soft photons (quanta of light with low energy) in a space with dimension $d$ greater than four. It turns out that in such worlds, soft radiation contains an unusual logarithmic factor: $\omega^{d-4}\ln \omega$, where $\omega$ is the photon frequency. This logarithm is born in loop quantum corrections, even though the quantum theory itself is free of infrared divergences there.
The most interesting result awaited at the classical level. When the scientists moved from quantum amplitudes to the shape of the electromagnetic wave, they saw long radiation tails. In even dimensions (starting from six), the main contribution comes from the straight-line motion of particles after scattering — the light arrives as a sharp burst. But acceleration adds universal early and late tails, decaying as $|u|^{-(3d-10)/2}$, where $u$ is retarded time. In odd dimensions (starting from five), even without acceleration there is a power-law tail $u^{-(d-4)/2}$, and acceleration adds a logarithmic correction $\ln u/u^{(3d-10)/2}$.
Thus, light tails become a bridge between the quantum microcosm and the familiar picture of electromagnetic waves. Perhaps someday such tails might be found in data from gravitational-wave observatories or in the cosmic microwave background — as a hint that our space has hidden dimensions.
as written by the authors arXiv:2608.03747v1
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 $$d\geq6$$, 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 $$d\geq5$$, 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.
In even dimensions (6, 8, 10…), light from separating particles arrives at strictly defined intervals, as if on schedule, while in odd dimensions (5, 7, 9…), it stretches into an infinitely long fading trail.
In Carl Sagan's novel 'Contact', the protagonists receive a message encoded in the polarization of radio waves — perhaps similar radiation tails from distant cataclysms carry far more information than we tend to think.