New recursive construction for tree NLSM and SG amplitudes, and new understanding of enhanced Adler zero

We propose a new bottom up method to construct tree amplitudes of non-linear sigma model (NLSM) and special Galileon theory (SG), based on assuming the universality of soft behaviors and the double copy structure. We extend the on-shell amplitudes to off-shell ones with two off-shell external legs, which allow the numbers of external legs to be odd. Then the $3$-point and $4$-point off-shell amplitudes can be bootstrapped, and the soft behaviors of $4$-point NLSM and SG amplitudes can be derived from them. The universality of soft behaviors allows us to invert the resulted soft theorems to construct higher-point off-shell amplitudes recursively, and express them in the formula of expansions to tree amplitudes of bi-adjoint scalar theory. We emphasize that the exact forms of universal soft behaviors are derived, rather than assumed as the input. Back to the on-shell limit, amplitudes with odd numbers of external legs vanish automatically, and the enhanced Adler zero emerge. From the bottom up perspective without the aid of a Lagrangian, the enhanced Adler zero are understood as that soft behaviors vanish faster than the degree expected from the naive power counting of soft momentum in the formula of expansions. Interestingly, such "zero" have explicit formulas and can be interpreted naturally. For tree amplitudes of Born-Infeld and Dirac-Born-Infeld theories, our method for construction does not make sense, but the enhanced Adler zero can be studied similarly.