Abstract
We generalize soft theorems of the nonlinear sigma model beyond the mathcal{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} (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} (p4), where for at least two specific choices of the mathcal{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} (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} (p2) Lagrangian, while any possible corrections to the subleading part are determined by the mathcal{O} (p4) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.
Highlights
We generalize soft theorems of the nonlinear sigma model beyond the O(p2) amplitudes and the coset of SU(N ) × SU(N )/SU(N )
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 O(p2) Lagrangian, while any possible corrections to the subleading part are determined by the O(p4) Lagrangian alone
It is important to note that in principle, such operation does not work for a general group representation of the NLSM, as it relies on the correct factorization of traces, which is only valid in some cases such as the adjoint of SU(N ) [19]
Summary
As an EFT, the NLSM is valid below a high energy scale Λ, and its Lagrangian admits a derivative expansion of ∂/Λ. The same Lagrangian can be constructed using entirely IR information of the linearly realized group H and its representation R [9, 10], where Ti and Xa are constructed using the generators (T i)ab of R and the structure constants f ijk of H [43]. For Nf = 2, Owzw vanishes while O3 and O4 can be expressed in terms of linear combinations of O1 and O2, so that there are only 2 independent O(p4) operators. For Nf = 3, Owzw is non-vanishing, though O4 can be expressed as a linear combination of O1, O2 and O3, there are 3 independent P-even operators at O(p4). The amplitudes for the NLSM exhibit a derivative expansion, and at tree level one can write.
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