Consistency Conditions on the S-Matrix of Massless Particles (0705.4305v2)

Abstract: We introduce a set of consistency conditions on the S-matrix of theories of massless particles of arbitrary spin in four-dimensional Minkowski space-time. We find that in most cases the constraints, derived from the conditions, can only be satisfied if the S-matrix is trivial. Our conditions apply to theories where four-particle scattering amplitudes can be obtained from three-particle ones via a recent technique called BCFW construction. We call theories in this class constructible. We propose a program for performing a systematic search of constructible theories that can have non-trivial S-matrices. As illustrations, we provide simple proofs of already known facts like the impossibility of spin $s > 2$ non-trivial S-matrices, the impossibility of several spin 2 interacting particles and the uniqueness of a theory with spin 2 and spin 3/2 particles.

• The paper establishes a framework using the BCFW method to reconstruct four-particle amplitudes from three-particle ones, proving non-trivial S-matrices are rare for spins s > 0.
• It demonstrates the uniqueness of three-particle on-shell amplitudes by linking helicity and Lorentz invariance to precise coupling constraints, crucial for models like Yang-Mills and General Relativity.
• The methodology reveals that achieving non-trivial interactions for higher spins requires strict adherence to algebraic structures, setting rigorous conditions for constructible massless particle theories.

Consistency Conditions on the S-Matrix of Massless Particles

The paper under discussion explores the foundational aspects of S-matrix theory for massless particles with arbitrary spin within four-dimensional Minkowski space-time. Its core focus is on establishing a set of consistency conditions for analyzing the possibility of non-trivial S-matrix configurations in such theories. These conditions facilitate the understanding of when the interaction theories of massless particles might yield a trivial S-matrix, implying no scattering or trivial scattering amplitudes.

Key Contributions and Methodological Framework

The authors, Paolo Benincasa and Freddy Cachazo, introduce a methodological framework anchored in the BCFW construction, which allows for reconstructing four-particle scattering amplitudes from three-particle ones. The paper defines a class of theories as "constructible" if such reconstructions lead to non-trivial results devoid of ambiguities. A primary thrust of the work is demonstrating that for spin $s>0$, non-trivial S-matrices are paradoxically rare, underlining that most theories yield trivial results unless the spin and interactions adhere to specific limits.

Principal Findings

1. Three-Particle Amplitudes: Uniqueness - The paper provides a rigorous proof that three-particle on-shell amplitudes for massless particles are determined uniquely up to the choice of coupling constants. This uniqueness is tightly linked with consistency requirements emerging from helicity constraints and Lorentz invariance.
2. Four-Particle Consistency Test - A significant contribution is the "four-particle test," a consistency check that mandates four-particle amplitudes computed through different BCFW deformations converge to a singular, unambiguous result. This test is shown to be stringent, with numerous massless theories failing to yield non-trivial results without consistent alteration of the conventional theory setup.
3. Implications for Higher Spins - The exploration reveals critical constraints for interactions among multiple massless particles of both the same and mixed spins. For instance, for spin 1 particles to interact non-trivially, the structure constants must align with those of a Lie algebra, thereby imposing a constraint that reflects the familiar Yang-Mills theory.
4. Uniqueness of Spin 2 and 3/2 Particles - It is demonstrated that the only permissible self-interaction for a single spin-2 particle coincides with linearized General Relativity. Furthermore, coupling constraints allow for partial theory alignment with $\mathcal{N}=1$ supergravity when associating a spin-2 particle with spin-3/2 particles.

Implications and Future Directions

The findings have profound implications for theoretical physics, especially in the field of higher-dimensional field theories and the quest to harmonize particle interactions governed by quantum mechanics and relativity. The paper delicately uncovers the rigid structures mandated by canonical Poincaré invariance, highlighting the rarity and uniqueness of theories like Yang-Mills and General Relativity within constructible models. Considering future developments, the authors suggest exploring the framework under different symmetry groups and dimensions beyond Minkowski space, like anti-de Sitter space, where known extensions to higher spins exist.

Furthermore, the paper opens avenues to consider anomalies and quantum corrections that may arise when extending the theory beyond tree-level calculations. This signals towards potential integration with supersymmetry and supergravity constructs, aiming to expand theoretical congruence between microscopic particle interactions and cosmological scale phenomena. The challenges identified in achieving constructibility serve as a compelling focal point for further theoretical elaboration and experimental validation in contemporary physics.