  The soft bootstrap is an on-shell method to constrain the landscape of
effective field theories (EFTs) of massless particles via the consistency of
the low-energy S-matrix. Given assumptions on the on-shell data (particle
spectra, linear symmetries, and low-energy theorems), the soft bootstrap is an
efficient algorithm for determining the possible consistency of an EFT with
those properties. The implementation of the soft bootstrap uses the recently
discovered method of soft subtracted recursion. We derive a precise criterion
for the validity of these recursion relations and show that they fail exactly
when the assumed symmetries can be trivially realized by independent operators
in the effective action. We use this to show that the possible pure (real and
complex) scalar, fermion, and vector exceptional EFTs are highly constrained.
Next, we prove how the soft behavior of states in a supermultiplet must be
related and illustrate the results in extended supergravity. We demonstrate the
power of the soft bootstrap in two applications. First, for the N= 1 and N=2
CP^1 nonlinear sigma models, we show that on-shell constructibility establishes
the emergence of accidental IR symmetries. This includes a new on-shell
perspective on the interplay between N=2 supersymmetry, low-energy theorems,
and electromagnetic duality. We also show that N=2 supersymmetry requires
3-point interactions with the photon that make the soft behavior of the scalar
O(1) instead of vanishing, despite the underlying symmetric coset. Second, we
study Galileon theories, including aspects of supersymmetrization, the
possibility of a vector-scalar Galileon EFT, and the existence of
higher-derivative corrections preserving the enhanced special Galileon
symmetry. This is addressed by soft bootstrap and by application of
double-copy/KLT relations applied to higher-derivative corrections of chiral
perturbation theory.
