Soft bootstrap and effective field theories

Abstract

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 SU(N ) × SU(N )/SU(N ) nonlinear sigma model (NLSM), where O(p 2 ) amplitudes with an arbitrary multiplicity of external particles can be soft-bootstrapped. We extend the program to 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 softconstructible 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.

UV considerations on scattering amplitudes in a web of theories

“…However, the special theories under consideration do satisfy a symmetry which fixes the higher-point contact terms: the Adler zero, which in this sense can be regarded as a "gauge symmetry" for these scalar theories. Inspired by this observation, the BCFW recursion was extended to EFT's, by modifying the shifts as to include the Adler zero property [18][19][20][21][22][23][24].…”

“…Even if the amplitude does not vanish at large z, the recursion may be generalized to multiline shifts, and complemented by other properties, like the Adler zero for EFT's [9,18,21,23,24]; see also [67,104] for biadjoint scalar amplitudes. In any case, the scaling is crucial but difficult to compute.…”

Higher-order tree-level amplitudes in the nonlinear sigma model

“…Flavour-ordering calculations can be very extensive, but are mathematically trivial and easily automated. Further developments of recursion relations in [15] have offset the algebraic difficulties, but soft recursion retains the fundamental limitation that recursive calculation of an O(p m ) n-point amplitude requires n > m. Therefore, the O(p 6 ) 6-point can not be reached by such means, and must be supplied as a seed amplitude if O(p 6 ) amplitudes are to be calculated for more than 6 particles. For this, our methods seem to be the only viable option other than brute-force Feynman diagrams.…”

“…Using recursion relations, tree-level amplitudes based on the leading-order term in the Lagrangian have been computed with up to 10 external particles [36]. Using more general recursion relations based on soft limits [11], 6-particle tree-level interactions have been computed using the next-to-leading-order Lagrangian [15]. These methods suffer limitations when higher-order Lagrangian terms are used, and can not handle loops.…”

“…Therefore, (4.1) and any statement relying on it can equally well be applied to the stripped amplitudes. The Adler zeroes may be used as a starting point to construct amplitudes through recursion relations [11,15]. For our purposes, however, their main use is in validating the correctness of complicated stripped amplitudes.…”