Effective Field Theories from Soft Limits (1412.4095v1)

Abstract: We derive scalar effective field theories - Lagrangians, symmetries, and all - from on-shell scattering amplitudes constructed purely from Lorentz invariance, factorization, a fixed power counting order in derivatives, and a fixed order at which amplitudes vanish in the soft limit. These constraints leave free parameters in the amplitude which are the coupling constants of well-known theories: Nambu-Goldstone bosons, Dirac-Born-Infeld scalars, and Galileons. Moreover, soft limits imply conditions on the Noether current which can then be inverted to derive Lagrangians for each theory. We propose a natural classification of all scalar effective field theories according to two numbers which encode the derivative power counting and soft behavior of the corresponding amplitudes. In those cases where there is no consistent amplitude, the corresponding theory does not exist.

• The paper introduces a novel method to derive scalar effective field theories by analyzing soft limits in scattering amplitudes.
• It classifies EFTs using a two-parameter taxonomy, linking derivative power counting with soft behavior to identify theories like DBI and Galileon.
• The approach bypasses traditional Lagrangian formulations, revealing inherent symmetries directly from amplitude constraints and paving the way for further EFT exploration.

Effective Field Theories from Soft Limits: An Analytical Perspective

The paper "Effective Field Theories from Soft Limits" by Cheung et al. advances the theoretical understanding of scalar effective field theories (EFTs) using the novel approach of constructing these theories from the soft limits of on-shell scattering amplitudes. The authors bypass traditional Lagrangian formulations, instead deriving these formulations by imposing constraints such as Lorentz invariance, factorization, and vanishings of amplitudes at specified soft limits.

Overview

The primary contribution of this paper is demonstrating a methodology for deriving EFTs by leveraging the behavior of on-shell scattering amplitudes in the soft limit, where momenta of external legs approach zero. This approach leads to a natural classification of scalar EFTs based on two pivotal parameters: the order of power counting in derivatives and the order at which amplitudes vanish in the soft limit (characterized by the non-negative integer $\nu$). The resulting classification underscores theories such as Nambu-Goldstone bosons, Dirac-Born-Infeld (DBI) scalars, and Galileons, each exhibiting distinct soft limit behaviors.

Strong Numerical Results and Claims

1. Classification and Existence of Theories: The authors identify a systematic approach to classify and enumerate EFTs for massless scalars using a two-number taxonomy, $(\alpha, \nu)$, which denotes the derivative power counting and soft behavior of amplitudes. This classification is pivotal, as it determines whether a consistent theory is possible, such as DBI with $(1,2)$ and Galileon with $(2,3)$.
2. Deriving Lagrangians from Soft Limits: By assuming values for $(\alpha, \nu)$, the authors provide explicit derivations of Lagrangians without direct reliance on Noether's theorem, instead using the soft behavior of scattering amplitudes. This has the remarkable implication that the hidden symmetries of these theories are a direct consequence of their soft limit properties.
3. Analytical Challenges Addressed: The analysis includes rigorous methods for constructing amplitudes that are consistent with symmetry requirements and derivative power counting. The authors overcome challenges related to managing redundant kinematic invariants and ensure the future applicability of their theoretical framework.

Implications and Future Developments

This methodological framework proposes a significant shift in understanding scalar EFTs by focusing on their manifestation through scattering amplitudes instead of conventional Lagrangian mechanics. The implications are manifold:

• Theoretical Implications: This perspective could redefine how symmetries in field theories are understood, emphasizing amplitudes as primary objects that encode the residual symmetries of the theory naturally extending existing paradigms such as the S-matrix approach in particle physics.
• Practical Implications: On a practical level, this approach could influence computational strategies in high-energy physics, providing a more direct path to investigating the implications of EFTs in experimental data without relying heavily on perturbative Lagrangian physics.

In speculation of future developments, this paper opens the avenue for discovering previously unidentified effective field theories by leveraging the unexplored facets of soft limits and non-trivial amplitude structures. Further inquiry is required to ascertain if these results hold through the loop-level amplitudes and in non-trivial spacetime backgrounds. Future work, as hinted by the authors, may extend these methods to fully classify all possible EFTs and unravel additional symmetries and dynamics intrinsic to scalar fields in a quantized setting.