Long-Range Interactions in Celestial CFT

TL;DR

This work analyzes the all-order structure of universal logarithmic soft theorems in massless QED and gravity and their imprint on celestial CFT. By enforcing SL(2,$\mathbb{C}$) covariance, the authors show that the symmetry-governed logarithmic soft factors exponentiate, with the exponent sourced by pairwise long-range exchanges between hard operator insertions, and a gravitational drag that accounts for backreaction. They define conformally soft loop operators and demonstrate that their celestial OPEs vanish in the universal sector, while loop corrections render celestial Ward identities non-local and involve sums over pairs of hard insertions. The paper also derives loop-corrected celestial OPEs and discusses implications for the BMS subalgebra and the $w_{1+\infty}$-type structures in gravity, highlighting the role of infrared scales and Faddeev-Kulish dressings in organizing the soft sector. Overall, the results illuminate how long-range interactions shape the holographic-like structure of flat-space celestial CFT and point to future work on non-universal loop effects and massive-particle extensions.

Abstract

Loop corrections in QED and gravity have recently been conjectured to give rise to an infinite tower of logarithmic soft theorems governing the universal low-energy behavior of photons and gravitons. We explore the implications of this tower for celestial CFT and for the algebra of conformally soft operators. The symmetry-governed part of the tower of logarithmic soft factors is shown to exponentiate, which demonstrates that these loop effects do not represent independent multi-particle interactions, but instead are rooted in the long-range exchange of gauge bosons between pairs of hard operator insertions. We define conformally soft loop operators, and compute their operator product expansions on the celestial sphere. The associated Ward identities exhibit characteristic non-local behaviors, which reflect the pair-wise interactions between hard operator insertions mediated by gauge bosons. We comment on the implications of these results for the soft operator algebra at loop order.

Figures (2)

• Figure 1: Examples of one-loop diagrams with multiple external scalars and one soft graviton, that give rise to terms logarithmic in the soft graviton energy. Straight lines represent scalars and wiggly lines represent gravitons. (a) This diagram contributes a logarithm in the region where the loop momentum lies between the soft energy $\omega$ and the characteristic energy scale $1/L$ of the (hard) scalars. The soft graviton is emitted from a scalar leg. (b) This type of diagram contributes a logarithm in the region where the loop momentum is less than the soft energy $\omega$. The soft graviton is emitted from a graviton loop, so these terms can be interpreted as describing the propagation of a soft graviton through a spacetime that has been backreacted by the external scalars.
• Figure 2: One-loop diagrams that contribute logarithms in the presence of graviton dressings. Straight and wiggly lines represent scalars and gravitons respectively, and the gray ellipse at the end of each external leg represents the gravitational dressing of that leg. Because the dressings carry gravitons of energy less than the energy scale $1/R$ associated with the detector distance $R$, the domain of these loop integrals in figures (b)--(d) are restricted to momentum below the scale $1/R$. The diagram in (a) contributes the same term as the one in figure \ref{['fig:drag']}, while the sum of diagrams (b)--(d) removes the region of loop integral in this contribution below energy scale $1/R$.