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Soft Bootstrap and Effective Field Theories

The soft bootstrap program aims to construct consistent effective field theories (EFT's) by recursively imposing the desired soft limit on tree-level scattering amplitudes through on-shell recursion relations. A prime example is the leading two-derivative operator in the EFT of $\text{SU} (N)\times \text{SU} (N)/\text{SU} (N)$ nonlinear sigma model (NLSM), where $ \mathcal{O} (p^2)$ amplitudes with an arbitrary multiplicity of external particles can be soft-bootstrapped. We extend the program to $ \mathcal{O} (p^4)$ operators and introduce the "soft blocks," which are the seeds for soft bootstrap. The number of soft blocks coincides with the number of independent operators at a given order in the derivative expansion and the incalculable Wilson coefficient emerges naturally. We also uncover a new soft-constructible EFT involving the "multi-trace" operator at the leading two-derivative order, which is matched to $\text{SO} (N+1)/ \text{SO} (N)$ NLSM. In addition, we consider Wess-Zumino-Witten (WZW) terms, the existence of which, or the lack thereof, depends on the number of flavors in the EFT, after a novel application of Bose symmetry. Remarkably, we find agreements with group-theoretic considerations on the existence of WZW terms in $\text{SU} (N)$ NLSM for $N\ge 3$ and the absence of WZW terms in $\text{SO} (N)$ NLSM for $N\neq 5$.