A Systematic Lagrangian Formulation for Quantum and Classical Gravity at High Energies

TL;DR

The paper builds a Glauber SCET-based EFT for gravity in the Regge regime $s\gg M_{pl}^2\gg t$, enabling a double expansion in $\alpha_Q$ and $\alpha_C$ and systematic resummation of $\log(s/t)$ via a rapidity RG. It unveils a soft operator tower whose leading piece yields the graviton Regge trajectory and whose next piece gives a gravitational BFKL equation, with the gravity kernel related to the QCD kernel by a square relation at leading order. The formalism yields an all-orders structure for forward scattering, identifies when classical logs appear (starting at 3PM), and provides a practical method to extract leading logs to any PM order by iterating one-loop RRGE anomalous dimensions. The approach exposes universal features of soft graviton interactions, clarifies the role of Glauber exchanges in gravity, and offers a path toward precise high-energy gravitational predictions relevant to PM expansions and gravitational-wave phenomenology.

Abstract

We derive a systematic Lagrangian approach for quantum gravity in the super-Planckian limit where $s\gg M_{pl}^2\gg t$. The action can be used to calculate to arbitrary accuracy in the quantum and classical expansion parameters $α_Q= \frac{t}{M_{pl}^2}$ and $α_C= \frac{st}{M_{pl}^4}$, respectively, for the scattering of massless particles. The perturbative series contains powers of $\log(s/t)$ which can be resummed using a rapidity renormalization group equation (RRGE) that follows from a factorization theorem which allows us to write the amplitude as a convolution of a soft and collinear jet functions. We prove that the soft function is composed of an infinite tower of operators which do not mix under rapidity renormalization. The running of the leading order (in $G$) operator leads to the graviton Regge trajectory while the next to leading operator running corresponds to the gravitational BFKL equation. For the former, we find agreement with one of the two results previously presented in the literature, while for the ladder our result agrees (up to regulator dependent pieces) with those of Lipatov. We find that the convolutive piece of the gravitational BFKL kernel is the square of that of QCD. The power counting simplifies considerably in the classical limit where we can use our formalism to extract logs at any order in the PM expansion. The leading log at any order in the PM expansion can be calculated without going beyond one loop. The log at $(2N+1)$ order in the Post-Minkowskian expansion follows from calculating the one loop anomalous dimension for the $N+1$th order piece of the soft function and perturbatively solving the RRGE $N-1$ times. The factorization theorem implies that the classical logs which arise alternate between being real and imaginary in nature as $N$ increases.

Figures (10)

• Figure 1: The structure of the perturbative series. The blue circles correspond to classical contributions in the Post-Minkowskian expansion. The pink circles are super classical (box diagrams) while the greens lines indicate quantum corrections from soft and collinear loops. The classical contributions occur at odd orders in the PM expansion. Each soft/collinear loop generates a log while Glauber loops generate $i\pi$.
• Figure 2: The vertical lines are Glaubers while the blob represents all possible soft loops.
• Figure 3: Diagrams needed for the renormalization of $S_{(2,2)}$. The first two diagrams are soft eye insertions into a Glauber rung, while the third diagrams is H graph.
• Figure 4: Prototypical diagrams needed to renormalize $S_{(N+1, N+1)}$. The diagram on the left is the $N+1$-rung Glauber box with a soft eye insertion, and the diagram on the right is the multi-rung H diagram. The soft graviton exchange can be between any two Glauber rungs, and the soft eye can similarly be inserted into any individual rung. The $H$ graph contribution to $S_{(2,2)}$ has no additional Glauber rungs.
• Figure 5: Tree level matching for $n$-$\bar{n}$ Glauber operators. These are the full theory diagrams with a $t$-channel pole. For scalar-scalar scattering this is sufficient to extract the Glauber operator, but for scalar-graviton scattering, one must also include $s$- and $u$-channel graphs, as well as the 4-point contact term. These additional contributions will be automatically accounted for order by order in expansion parameters in the EFT given that we build operators out of gauge invariant building blocks.