Thermal correlator at null infinity

TL;DR

This work develops a real-time (Schwinger–Keldysh) formalism for thermal Carrollian correlators at null infinity, constructing bulk–to–bulk, bulk–to–boundary, and boundary–to–boundary propagators for a massless scalar in flat spacetime and formulating the Carrollian Feynman rules. The authors show that bulk–to–boundary propagators take an extended Bose–Einstein form in position space, and they translate position-space Carrollian correlators into momentum-space amplitudes via contour-dependent Fourier transforms; at finite temperature new three-to-one and two-to-two scattering structures emerge, with all tree-level four-point functions expressible through Barnes zeta functions. They confirm KMS symmetry and analyze various temperature regimes, revealing vanishing four-point contributions at zero temperature for some types and nontrivial finite-T corrections for others. The results illuminate how thermal effects manifest at null infinity and provide a framework for exploring thermal Carrollian CFTs and flat-space holography, including potential links to magnetic Carrollian sectors and black-hole-related physics.

Abstract

We study the thermal Carrollian correlators at null infinity in the real-time formalism. We derive the Feynman rules to calculate these correlators in the position space. We compute the bulk-to-bulk, bulk-to-boundary and boundary-to-boundary propagators for massless scalar theory. Due to the doubling of the fields degrees of freedom, the number of each propagator is quadrupled. The bulk-to-boundary propagators have the form of (extended) Bose-Einstein distribution in the position space. Utilizing the contour integral of the propagators, we can transform the Feynman rules to momentum space. Interestingly, while the external lines and amplitudes in momentum space depend on the contour, Carrollian correlators in position space are independent of it. We show how to compute four-point correlators at finite temperature. The tree level correlators can be written as the summation of Barnes zeta functions and reduce to the ones in the zero temperature limit.

Figures (15)

• Figure 1: A Feynman diagram for four graviton scattering in a maximally extended Schwarzschild black hole. The dashed lines are event horizons and the wavy lines are bulk-to-boundary propagators for gravitons. The wavy line with a horizontal line represents the singularity. One should integrate out the bulk points, including the black hole and white hole as well as the two asymptotic flat regions I and II to obtain the Carrollian amplitude.
• Figure 2: The time path $C$ in the real-time formalism.
• Figure 3: Feynman diagrams for two-point Green's function up to $\mathcal{O}(\lambda)$.
• Figure 4: Feynman diagrams for four-point Green's function up to $\mathcal{O}(\lambda)$.
• Figure 5: The $n$ point connected Green's function. The black lines are connected to external points. Each external point $x_j$ is connected to a vertex $y_j$ through Feynman propagators. The internal vertices should be integrated out. The shaded part is the connected and amputated correlation function $\mathcal{G}_{a_1a_2\cdots a_n}$ which could be constructed by Feynman rules in the position space.