Bootstrapping Gravity with Crossing Symmetric Dispersion Relations

This paper derives new bounds on Wilson coefficients in gravitational effective field theories by employing fully crossing symmetric dispersion relations that isolate low-energy couplings without relying on the forward limit, successfully validating the method against known scalar and graviton scattering results while establishing novel constraints on graviton interactions with massive spin-4 states.

The Gist

Imagine you are trying to figure out the rules of a game, but you can only see the players when they are far away from each other. You can't see the tiny, fast interactions happening right in the middle of the field because the "camera" (your mathematical tools) gets blurry or breaks down when things get too close.

This is the challenge physicists face when studying gravity at very small scales. They use "Effective Field Theories" (EFTs) to describe how gravity works at low energies, but these theories have "knobs" (called Wilson coefficients) that need to be set correctly. The problem is, we don't know the ultimate "UV completion"—the true, high-energy theory of everything that sets those knobs.

This paper introduces a new, clever way to figure out the limits of those knobs without needing to know the full high-energy theory. Here is how the author, Celina Pasiecznik, does it, using simple analogies:

1. The Problem: The "Forward Limit" Trap

Traditionally, physicists tried to solve this by looking at particles bouncing straight back at each other (the "forward limit"). It's like trying to judge a car's engine by listening to it drive straight toward you.

• The Issue: In gravity, this "straight-on" view is broken. The math explodes (diverges) because gravity has a "pole" (a singularity) right there. It's like trying to listen to a whisper while standing next to a jet engine; the noise drowns out the signal.
• The Old Fix: Scientists had to use complicated "smearing" techniques (averaging over a range) and mix different equations together to cancel out the noise. It worked, but it was messy and required many steps.

2. The New Solution: The "Symmetric Mirror"

The author proposes using Crossing Symmetric Dispersion Relations.

• The Analogy: Imagine you have a magical mirror that shows you the same scene from three different angles simultaneously (Left, Right, and Center). In physics, this is called "crossing symmetry." It means the rules of the game look the same whether you swap the roles of the particles (like swapping who is throwing the ball and who is catching it).
• How it helps: Instead of looking at just one angle (the broken forward view), this new method looks at the "whole room" at once. By using a special mathematical variable (called $z$) that treats all angles equally, the method naturally filters out the noise.
• The Result: It isolates the specific "knobs" (couplings) we care about automatically. We don't need to manually mix equations to cancel out the noise; the symmetry does it for us. It's like having a filter that only lets the specific color you want through, blocking everything else instantly.

3. Testing the New Tool

The author didn't just invent a new tool; they tested it to make sure it works.

• The Test: They applied this new "Symmetric Mirror" method to two known scenarios:
  1. Scalar particles (simple, point-like particles) interacting with gravity.
  2. Gravitons (particles of gravity) scattering off each other.
• The Outcome: The results matched perfectly with the best previous calculations. This proves the new method is just as accurate as the old, complicated ways, but it's much more direct and elegant.

4. Adding "Heavy" Guests to the Party

The paper also explores what happens if we assume there are specific, heavy, invisible particles (like a massive "spin-4" state) running around in the background.

• The Analogy: Imagine you are trying to figure out the rules of a dance, but you suspect a giant, invisible dancer is occasionally stepping in. The author's method allows them to calculate exactly how strong the connection (coupling) between the visible dancers and this invisible giant can be, depending on how heavy the giant is.
• The Discovery: They found a "tipping point." If the invisible giant is too heavy compared to the energy limit of the theory, the connection must be zero. It's like a bridge that can only hold a certain weight; if the truck (the heavy particle) is too heavy, the bridge (the theory) collapses unless the truck isn't there at all.

5. Why This Matters

The main takeaway is that this new method is a powerful, cleaner tool for the "S-matrix bootstrap" (a program to figure out the laws of physics using only basic rules like cause-and-effect and energy conservation).

• It avoids the "broken camera" problem of the forward limit.
• It works naturally for particles that spin (like gravitons), which is much harder to do with old methods.
• It sets strict boundaries on what is possible in our universe's gravitational theories, telling us which combinations of "knobs" are allowed and which are forbidden by the laws of physics.

In short, the author built a new mathematical lens that lets us see the rules of gravity clearly, even when the view is usually blurry, and confirmed that it sees exactly what we expect to see.

Technical Summary

Technical Summary: Bootstrapping Gravity with Crossing Symmetric Dispersion Relations

Problem Statement
The S-matrix bootstrap program aims to constrain the Wilson coefficients of gravitational Effective Field Theories (EFTs) using fundamental principles: analyticity, unitarity, and crossing symmetry. In theories without gravity, the forward limit ($t \to 0$) has been successfully used to isolate finite subsets of couplings and derive positivity bounds. However, in gravitational theories, the presence of the graviton pole obstructs the forward limit, necessitating the use of "improved sum rules." These improved rules rely on carefully constructed linear combinations of sum rules to isolate specific couplings while avoiding the singularity. Furthermore, existing methods often struggle with loop-level calculations where light states in loops create singularities in the forward limit, and they can be computationally cumbersome when incorporating specific spectral assumptions about massive higher-spin states.

Methodology
This paper proposes a direct alternative to improved sum rules by utilizing fully crossing symmetric dispersion relations. The methodology involves the following key components:

1. Crossing Symmetric Variables: The authors employ a complex variable $z$ and an auxiliary momentum variable $p$, related to the Mandelstam invariants ($s, t, u$) via cubic roots of unity. This formulation allows the dispersion relations to be written in a way that is manifestly crossing symmetric without taking the forward limit.
2. Kernel Construction: The dispersion relations utilize a crossing symmetric kernel $K_k(z, p^2)$, where $k$ labels the subtraction order. By deforming the integration contour around high-energy branch cuts and low-energy poles, the authors derive sum rules that relate low-energy EFT couplings to high-energy spectral densities.
3. Natural Isolation of Couplings: A primary advantage of this approach is that for a chosen $k$, the low-energy side of the sum rule naturally isolates a finite subset of Wilson coefficients. This eliminates the need for the linear combinations of sum rules required by improved sum rules.
4. Null Constraints: Crossing symmetry naturally generates null constraints (relations between couplings that must vanish) as higher-$k$ sum rules, which are used to strengthen the bounds.
5. Smeared Functionals and SDPB: To handle the graviton pole and ensure convergence, the functionals are smeared against wave-packets in momentum space. The resulting optimization problem is solved using Semidefinite Programming (SDPB), imposing unitarity constraints on the spectral density (positivity of partial wave coefficients).
6. Extension to Spinning States: For graviton scattering, the authors construct fully crossing symmetric combinations of Maximally Helicity Violating (MHV) amplitudes. They utilize Wigner-$d$ functions for the partial wave decomposition of spinning external states, adapting the formalism previously used for scalar scattering.

Key Contributions and Results

• Validation on Scalar Scattering: The authors applied the crossing symmetric sum rules to scalar scattering with gravity. By choosing a smearing range of $p_{\text{max}} = M$ (where $M$ is the EFT cutoff), they exactly reproduced the bounds previously derived using improved sum rules in Ref. [36]. They demonstrated that a smaller smearing range ($p_{\text{max}} = M/\sqrt{3}$) yields weaker bounds, confirming that the full range is necessary to recover optimal constraints.
• Maximal Supergravity: The method was applied to $D=10$ maximal supergravity. The authors reproduced the known bounds on the coupling $g_0$ (Ref. [37]) and the coupling of external gravitons to a heavy spin-4 state (effective spin-0 in the auxiliary scalar amplitude). This validated the approach for theories with supersymmetric Ward identities.
• Graviton Scattering in $D=4$: The paper constructs crossing symmetric combinations for MHV graviton amplitudes. Using these, they computed bounds on the Wilson coefficients $g_3, g_4,$ and $g_5$ for Einstein gravity with higher-derivative corrections. These results matched the bounds found in Ref. [38] using improved sum rules, providing a non-trivial check of the formalism for spinning external particles.
• Bounds on Massive Higher-Spin Exchange: A novel application involved "integrating in" a massive spin-4 state below the cutoff. The authors derived bounds on the coupling of external gravitons to this spin-4 state as a function of the mass ratio $M^2/m_{J=4}^2$.
  • The results showed a sharp drop in the allowed coupling when the mass ratio $M^2/m_{J=4}^2 \approx 2.2$, beyond which a non-zero coupling is incompatible with the assumed spectral gap.
  • The bounds were found to depend on the assumed spectrum of light low-spin states (spin-0 and spin-2), with the inclusion of light spin-0 states generally weakening the bounds.

Significance
The paper claims that crossing symmetric dispersion relations offer a more direct and systematic framework for the gravitational S-matrix bootstrap compared to improved sum rules. Its primary significance lies in:

1. Elimination of the Forward Limit: The method naturally isolates finite sets of couplings without relying on the forward limit, thereby avoiding the graviton pole obstruction entirely.
2. Natural Null Constraints: Null constraints arise naturally from the structure of the crossing symmetric sum rules rather than being artificially constructed.
3. Loop-Level Potential: The authors argue that this approach is particularly advantageous for loop-level calculations. In loop processes, light states can induce singularities in the forward limit; the crossing symmetric approach, working away from the forward limit, provides a natural framework to handle these cases where improved sum rules may be unstable.
4. Spectral Flexibility: The formalism effectively incorporates explicit assumptions about the spectrum of massive states (e.g., integrating in specific higher-spin particles), allowing for the bounding of couplings to these states in a controlled manner.

The work demonstrates that crossing symmetric sum rules are a robust tool for deriving bounds in gravitational EFTs, successfully reproducing established results while offering a streamlined path for future investigations into spinning amplitudes and loop corrections.