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 opera­ tor in the EFT of SU(N) x SU(N)/SU(N) nonlinear sigma model (NLSM), where 𝒪(p²) amplitudes with an arbitrary multiplicity of external particles can be soft-bootstrapped. We extend the program to 𝒪(p⁴) 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 SO(N + 1) /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 SU(N) NLSM for N ≥ 3 and the absence of WZW terms in SO(N) NLSM for N ≠ 5.