What is the Simplest Quantum Field Theory? (0808.1446v3)

Abstract: Conventional wisdom says that the simpler the Lagrangian of a theory the simpler its perturbation theory, but an increased understanding of the structure of the S-matrix in gauge theories and gravity has been pointing to the opposite conclusion. In this paper we suggest that N=8 SUGRA has the simplest interacting S-matrix in 4D. Using Grassmann coherent states for external particles shows that amplitudes with maximal SUSY are smooth objects, with the action of SUSY manifest. We show that all tree amplitudes in N=4 SYM and N=8 SUGRA vanish at (supersymmetric) infinite complex momentum, and can thus be determined by recursion relations. We also identify the action of the non-linearly realized E_{7(7)} symmetry of N=8 SUGRA on scattering amplitudes. We give a simple discussion of the structure of 1-loop amplitudes in any QFT, in close parallel to recent work of Forde, showing that the coefficients of scalar "triangle" and "bubble" integrals are determined by the "pole at infinite momentum" of tree amplitude products appearing in cuts. The on-shell superspace for maximal SUSY makes it easy to compute the multiplet sums that arise in these cuts, leading to a simple proof of the absence of triangles and bubbles at 1-loop. We also argue that rational terms are absent. This establishes the recent conjecture that 1-loop amplitudes in N=8 SUGRA have only scalar box integrals, just as N=4 SYM. It is natural to conjecture that with maximal SUSY, amplitudes are completely determined by their leading singularities even beyond tree- and 1-loop level; this would directly imply the perturbative finiteness of N=8 SUGRA. The remarkable properties of scattering amplitudes call for an explanation in terms of a "weak-weak" dual formulation of QFT, a holographic dual of flat space.

• The paper demonstrates that supersymmetry simplifies scattering amplitudes by inducing vanishing behavior at infinite momentum using Grassmann coherent states.
• The paper finds that one-loop amplitudes in N=4 SYM and N=8 SUGRA lack triangles, bubbles, and rational terms through comprehensive supersymmetric recursion analysis.
• The paper proposes that leading singularities may determine every loop order, suggesting inherent finiteness and new insights into quantum gravity.

Overview of "What is the Simplest Quantum Field Theory?"

The paper by Nima Arkani-Hamed, Freddy Cachazo, and Jared Kaplan investigates the simplicity of quantum field theories (QFTs) using the framework of scattering amplitudes. The authors challenge the conventional wisdom that simpler Lagrangians directly lead to simpler perturbation theories. Instead, they propose examining the structure of scattering amplitudes to identify simplicity, particularly focusing on N=8 supergravity (SUGRA) and N=4 super Yang-Mills (SYM).

The focal point of their analysis is the behavior of scattering amplitudes, especially within the context of recursion relations like BCFW (Britto-Cachazo-Feng-Witten) and the absence of higher-order loop divergences. The paper delineates that, contrary to intuition, scattering amplitudes for theories involving massless higher-spin particles such as N=8 SUGRA exhibit profound simplicity due to their favorable UV behavior and their potential perturbative finiteness.

Key Findings and Results

1. Supersymmetric Springboard: The authors explore the rich structure of on-shell superspaces to articulate the power of SUSY (supersymmetry) in simplifying theories at the amplitude level. They utilize the supercharge eigenstates—Grassmann coherent states—to express tree amplitudes in maximally supersymmetric theories as completely determined entities, often exhibiting vanishing behavior at infinite momentum.
2. Vanishing of Rational Terms: A significant result from the paper is the absence of triangles, bubbles, and rational terms in 1-loop amplitudes for N=4 SYM and N=8 SUGRA. These conjectures are verified through comprehensive computations involving supersymmetric recursion relations and complex momentum spaces. The absence of these components indicates a simpler and more predictive framework for these theories.
3. E_7(7) Symmetry: The paper explores the implications of E_7(7) symmetry in N=8 SUGRA, elucidating its presence in the amplitudes and its influence over the moduli space. The single and double soft emission of scalars provides insights into how non-linearly realized symmetries manifest within the on-shell framework by inducing SU(8) rotations in the on-shell particle states.
4. Leading Singularity Hypothesis: The authors speculate that the simplest characterization of theories such as N=8 SUGRA could imply that they are entirely determined by their leading singularities at every loop order. This hypothesis, if substantiated, would suggest intrinsic finiteness and tremendous simplicity for these theories relative to their high-loop structures.

Implications and Future Directions

The insights in this work encourage a deep reconsideration of how QFTs are formulated, especially recognizing the potentially minimalistic amplitude-based approach in contrast to the traditional Lagrangian focus. The fact that N=8 SUGRA may have the simplest analytic properties is promising for further endeavors into a "weak-weak" dual formulation of QFT, potentially leading to revolutionary frameworks or interpretations of space-time and particle interactions.

Moreover, the exploration of maximally supersymmetric theories from an amplitude-centric viewpoint could lead to implications extending towards gravitational theories and search for a consistent theory of quantum gravity. The methodology prompts further investigations into finite theories' consistency within quantum frameworks and their compliance with the principles of locality and causality in physics.

The mathematical implications regarding the role of algebraic and transcendental numbers in scalar integrals also offer an engrossing cross-disciplinary connection worthy of further paper. This work forms a basis for profound exploration both in fundamental theory and mathematical physics, guiding the future trajectory of quantum theory, its calculations, and its interpretations.