NLSM $\subset$ Tr$(φ^3)$

Abstract

Scattering amplitudes for the simplest theory of colored scalar particles - the Tr($Φ^3$) theory - have recently been the subject of active investigations. In this letter we describe an unanticipated wider implication of this work: the Tr($Φ^3$) theory secretly contains Non-linear Sigma Model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr$(Φ^3)$ amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in potential, with extrema spontaneously breaking $U(N) \to U(N-k) \times U(k)$. The Goldstone amplitudes for this theory coincide with those of pions in the $U(N) \times U(N) \to U(N)$ chiral Lagrangian to all orders in the planar limit. We also give a purely on-shell understanding of this correspondence, showing integrands defined by the kinematic shifts have the correct residues on poles and appropriately produce the Adler zero. Finally, we discuss how similar kinematic shifts produce certain infinite classes of mixed amplitudes of pions and Tr($Φ^3$) scalars, most of which are not interpretable from the Lagrangian description.

Figures (13)

• Figure 1: Single cut of one-loop 2-point NLSM with $X_{1,z_{1}}=0$ obtained from forward limit of 4-point tree.
• Figure 2: The 2-point 2-loop decorated graph of $\chi,\chi^\dagger$
• Figure 3: Vertices of ${\rm Tr} \phi \chi \chi^\dagger$ and ${\rm Tr} \tilde{\phi} \chi^\dagger \chi$
• Figure 4: The 6-point tree diagram that contains negative shifted $X_{1,3}$, positive shifted $X_{4,6}$, unshifted $X_{3,6}$; and the 4-point 1-loop diagram with odd/even assignment to the loop puncture, where the path of the propagator $X_{1,z}$ goes out and spiral around the puncture with red, red/red, blue arrows, therefore should be shifted by $-\delta$/keep unshifted.
• Figure 5: The shift of $\tilde{\phi}$ propagator in 2-point 2-loop non-planar example. The path of the $\tilde{\phi}$ propagator, denoted by black line with blue dashed or purple dotted arrows on each side inside the fat-graph. Each side extends outward with a corresponding external blue arrow, both of which terminate at the external $\chi^\dagger$. As a result, the corresponding propagator should be shifted by $+\delta$.