Leading soft theorems on plane wave backgrounds

TL;DR

The paper investigates the fate of leading soft singularities for massless mediators on plane-wave backgrounds in gauge theory and gravity, deriving all-multiplicity tree-level soft theorems that depend on whether the soft particle is ingoing or outgoing and on the background memory. The authors develop exact Feynman rules for plane-wave backgrounds, construct dressed perturbations (scalars, photons, gluons, gravitons), and reveal memory-dependent corrections that modify flat-space soft factors; in particular, self-dual backgrounds simplify certain sectors and allow direct comparisons to self-dual MHV structures. A key finding is that, unlike flat space, the soft theorems on generic backgrounds display background-induced terms (including tail/memory contributions) and require careful treatment of boundary conditions, while negative-helicity states often retain a familiar factorization pattern. The work connects to broader themes in infrared structure, asymptotic symmetries, and the double copy, and sets the stage for future explorations into loops, celestial holography, and perturbations around self-dual sectors on curved backgrounds.

Abstract

The infrared singularities of scattering amplitudes have historically contributed to much development in understanding fundamental structures in physics. However, the fate of the leading soft singularities of amplitudes in non-trivial background fields has remained largely unknown. In this paper, we derive the leading soft theorems for photons, gluons and gravitons on generic plane wave backgrounds in gauge theory and gravity. The results differ from the flat space results through dependence on the initial conditions of the soft mediator. We also consider the special case of self-dual plane wave backgrounds, and match onto the flat space results when the background is treated perturbatively.

Figures (5)

• Figure 1: Possible sources to the leading soft singularity of a scattering process. Here the soft particle is treated with outgoing boundary conditions and all momenta point into the diagram. Whether a hard particle is ingoing or outgoing is determined by the sign of $p_{i \, +}$: if $p_{i \, +}>0$, the particle is considered ingoing, and if $p_{i \, +}< 0$ it is considered outgoing. In (a), the singularity would come from attaching to internal lines or vertices in the scattering amplitude. In (b) the soft outgoing mediator attaches to an outgoing hard particle. In (c) the soft outgoing mediator attached to an ingoing hard particle. The diagrams (b) and (c) are the ones contributing to the flat space leading soft theorem, but diagrams such as (a) can contribute to soft singularities in the case of soft gravitons.
• Figure 2: An ingoing soft photon in an electromagnetic plane wave background remains soft in the outgoing region. Therefore soft singularities occur when the ingoing soft photon attaches to both ingoing and outgoing hard particles.
• Figure 3: A gluon in a Cartan-valued plane wave background that is soft in the ingoing region becomes infinitely hard in the outgoing region. Intuitively, this is why soft singularities arising from an ingoing soft gluon only happen when it couples to an also ingoing hard particle.
• Figure 4: A photon on a gravitational plane wave background that is soft in the ingoing region does not look like a soft Fourier mode in the outgoing region. Instead, it 'spreads' out and the analysis of soft singularities when it couples to outgoing hard particles depends on the fall-off conditions.
• Figure 5: A graviton defined on a gravitational plane wave that is soft in the ingoing region will 'spread' out in the outregion and introduce a large coordinate transformation of the background metric.