Tree-level Amplitudes in the Nonlinear Sigma Model

TL;DR

This work develops a comprehensive on-shell framework for tree-level Goldstone boson amplitudes in the SU(N) nonlinear sigma model by formulating flavor-ordered Feynman rules, introducing semi-on-shell Berends-Giele currents, and constructing BCFW-like recursions via Cayley parametrization. It demonstrates parametrization independence of stripped amplitudes, derives explicit low-point results, and proves scaling properties that enable a one-subtraction BCFW reconstruction for semi-on-shell amplitudes beyond four dimensions. The authors establish Adler zero behavior for odd momenta and derive a universal double-soft limit formula, highlighting the potential for on-shell methods to extend to effective field theories. The approach provides efficient computational tools for Goldstone boson scattering and suggests avenues for purely on-shell formulations and loop extensions.

Abstract

We study in detail the general structure and further properties of the tree-level amplitudes in the SU(N) nonlinear sigma model. We construct the flavor-ordered Feynman rules for various parameterizations of the SU(N) fields U(x), write down the Berends-Giele relations for the semi-on-shell currents and discuss their efficiency for the amplitude calculation in comparison with those of renormalizable theories. We also present an explicit form of the partial amplitudes up to ten external particles. It is well known that the standard BCFW recursive relations cannot be used for reconstruction of the the on-shell amplitudes of effective theories like the SU(N) nonlinear sigma model because of the inappropriate behavior of the deformed on-shell amplitudes at infinity. We discuss possible generalization of the BCFW approach introducing "BCFW formula with subtractions" and with help of Berends-Giele relations we prove particular scaling properties of the semi-on-shell amplitudes of the SU(N) nonlinear sigma model under specific shifts of the external momenta. These results allow us to define alternative deformation of the semi-on-shell amplitudes and derive BCFW-like recursion relations. These provide a systematic and effective tool for calculation of Goldstone bosons scattering amplitudes and it also shows the possible applicability of on-shell methods to effective field theories. We also use these BCFW-like relations for the investigation of the Adler zeroes and double soft limit of the semi-on-shell amplitudes.

Figures (14)

• Figure 1: Graphical representation of the 6-point amplitude (\ref{['6pt_amplitude']}) with cycling tacitly assumed.
• Figure 2: Illustration of the contour used for the derivation of the subtracted Cauchy formula (\ref{['subtracred_Cauchy']}) with $k=1$ and $n_{C(R)}=3$.
• Figure 3: Graphical representation of the Berends-Giele recursive relations
• Figure 4: Scaling of the building blocks on the right hand hand of the Berends-Giele recursion relation according to the induction hypothesis when the odd momenta are scaled.
• Figure 5: The terms on the right hand hand of the Berends-Giele recursion relation which are automatically $O(t^2)$ using the induction hypothesis when the odd momenta are scaled.