Subleading soft radiation during scattering of dressed states in QED

We study soft photon emission during scattering of Faddeev-Kulish charged states in QED, at leading order in perturbation theory. The charged asymptotic particles are accompanied by clouds of an infinite number of soft photons of energy less than a characteristic infrared scale $E_d$. When the corresponding ``dressing’’ functions are suitably corrected to subleading order in the soft momentum expansion, as advocated in recent work by Choi and Akhoury, we show explicitly that the emission of additional radiative soft photons with energy less than $E_d$ is completely suppressed. Moreover, the dressing renders the elastic amplitudes infrared-finite, order by order in perturbation theory, regulating the infrared divergences due to virtual soft photons, at the energy scale $E_d$. Therefore, the characteristic energy scale of the soft photons in the clouds provides an effective infrared cutoff, allowing for the formulation of an infrared finite S-matrix.

💡 Research Summary

The paper investigates the infrared (IR) structure of quantum electrodynamics (QED) by focusing on the scattering of dressed charged states, specifically the Faddeev‑Kulish (FK) states, and by extending the dressing functions to subleading order in the soft‑photon momentum expansion. In conventional perturbative QED, amplitudes built on the Fock basis suffer from IR divergences: virtual soft‑photon loops generate logarithmic divergences in the regulator λ, while real soft‑photon emission produces complementary divergences that cancel only after an inclusive sum (Bloch‑Nordsieck theorem). However, the cancellation does not solve the deeper problem that the asymptotic charged states themselves are not eigenstates of the full Hamiltonian because long‑range Coulomb interactions persist at arbitrarily late times.

FK proposed to dress each charged particle with a coherent cloud of infinitely many soft photons whose energies are bounded above by a small scale Ed. The dressing operator (e^{R_f}) creates a coherent photon state characterized by a dressing function (f^\mu(p,k)=e,p^\mu/(p!\cdot!k)-c^\mu e^{-i p\cdot k t_0/p^0}). The cloud contains all modes with λ ≤ |k| ≤ Ed, and the parameter Ed acts as an effective IR cutoff. The dressed states are eigenstates of the asymptotic Hamiltonian, and the corresponding S‑matrix elements are free of IR divergences order‑by‑order.

The novelty of the present work lies in incorporating subleading corrections to the FK dressing, as advocated by Choi and Akhoury. The authors construct a corrected dressing function that includes terms of order k⁰ (the subleading soft factor) in addition to the leading 1/k pole. This refinement is motivated by the subleading soft photon theorem and by large‑gauge‑transformation Ward identities. By explicitly evaluating tree‑level processes—electron–muon scattering, electron–positron scattering, and electron–photon scattering—the authors demonstrate two central results:

1. Infrared‑finite elastic amplitudes. The elastic FK amplitude factorizes as
   (\tilde S_{\beta\alpha}= \langle f_\beta|f_\alpha\rangle, S_{\beta\alpha}),
   where the overlap of the coherent clouds yields (\langle f_\beta|f_\alpha\rangle = \exp