Exploring the landscape for soft theorems of nonlinear sigma models

Abstract

We generalize soft theorems of the nonlinear sigma model beyond the $$ \mathcal{O} $$ O (p2) amplitudes and the coset of SU(N) × SU(N)/SU(N). We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known $$ \mathcal{O} $$ O (p2) single soft theorem for SU(N) × SU(N)/SU(N) in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of SO(N), where a special flavor ordering of the “pair basis” is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to $$ \mathcal{O} $$ O (p4), where for at least two specific choices of the $$ \mathcal{O} $$ O (p4) operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at $$ \mathcal{O} $$ O (p2). Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the $$ \mathcal{O} $$ O (p2) Lagrangian, while any possible corrections to the subleading part are determined by the $$ \mathcal{O} $$ O (p4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.

The ChPT: top-down and bottom-up

“…It might be a coincidence, but it is important to study the consequences also for higher orders and define a core of ChPT satisfying KK and/or BCJ relations. Recently, application of the BCJ up to O(p 4 ) was done in [45] and dedicated study to five points in [46]. Both even and odd sectors up to O(p 6 ) and up to 6pt were also studied in [47].…”

Extended DBI and its generalizations from graded soft theorems

“…In this paper we discuss in more detail one of these cases, namely the theory proposed by Cachazo et al [27] which is referred to as extended Dirac-Born-Infeld theory. The CHY representation of this theory was also mentioned in [39], where the double copy structure was established, and in [40], where the soft behavior was discussed.…”

Nonrelativistic effective field theories with enhanced symmetries and soft behavior

“…While considerable effort has been devoted to exploring the properties of relativistic scalar EFTs using state-of-the-art techniques of quantum field theory [21,22,[27][28][29][30][31][32][33][34], nonrelativistic scalar EFTs have received considerably less attention. This has left nonrelativistic EFTs with enhanced symmetries and soft limits a terra incognita.…”