Infrared Divergences in QED, Revisited

TL;DR

This work reinterprets the IR structure of QED through the lens of vacuum degeneracy arising from large gauge transformations. By incorporating vacuum transitions via dressed charged states (Dirac/Faddeev-Kulish clouds) that shift soft charges, the authors obtain nonzero, IR-finite scattering amplitudes instead of the conventional vanishing results. They show the FK dressings are effectively equivalent at leading IR order to their dressed states, and extend the construction to massive particles with appropriate soft factors on H3, introducing radiative shocks and Coulomb-field considerations. The results unify and generalize the mechanism for IR cancellation, with potential implications for nonabelian gauge theories and gravity, and they address charged states beyond the FK framework, suggesting broad IR finiteness under suitable conditions.

Abstract

Recently it has been shown that the vacuum state in QED is infinitely degenerate. Moreover a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish.

Figures (3)

• Figure 1: IR divergences arise from soft photon exchange between pairs of external charges. When the charges are dressed with apprpropriately correlated clouds of soft photons, these divergences are pairwise cancelled by exchanges involving the soft clouds.
• Figure 2: Dressed massive particles of zero net charge come in from $i^-$, scatter and go out to $i^+$. The Faddeev-Kulish dressing introduces radiative shock waves at $v=0$ and $u=0$ which cancel the asymptotic Coulomb fields of the particles for $v>0$ and $u<0$ respectively. For neutral scattering states the Coulomb field will vanish near spatial infinity while for charged ones it will be an angle-dependent constant.
• Figure 3: In this figure, the outgoing positron in Figure 1 is crossed to an incoming electron, but its associated soft photon cloud remains as part of the out state. The leading IR divergences from the depicted soft photon exchange still cancel, even though the in and out states carry nontrivial $Q^\pm_\varepsilon$ charges and are no longer Faddeev-Kulish states.