The CEGM NLSM

TL;DR

This work develops a systematic deformation theory for CEGM amplitudes and shows that zero-preserving deformations yield generalized NLSM amplitudes. It proves that the space of pure kinematic shifts has dimension $\gcd(k,n)-1$ and provides an $X$-basis for k-propagators, enabling a transparent description of deformations. The authors establish a residual embedding of the $n$-point NLSM into a mixed $(3,n+2)$ CEGM amplitude, and demonstrate a factorization of an $8$-point NLSM $4$-amplitude into two $m^{(4)}_6$ factors, linking higher-point generalized amplitudes to lower-point structures. Hard/soft limit analysis reveals Adler-zero–like properties for pure generalized NLSM amplitudes, suggesting deep symmetry-driven constraints. Collectively, the results illuminate how NLSM physics can emerge from the CEGM framework and point to rich future directions for higher $k$ and connections to positive tropical geometry and string-inspired integrals.

Abstract

Studying quantum field theories through geometric principles has revealed deep connections between physics and mathematics, including the discovery by Cachazo, Early, Guevara and Mizera (CEGM) of a generalization of biadjoint scalar amplitudes. However, extending this to generalizations of other quantum field theories remains a central challenge. Recently it has been discovered that the nonlinear sigma model (NLSM) emerges after a certain zero-preserving deformation from $\text{tr}(φ^3)$. In this work, we find a much richer story of zero-preserving deformations in the CEGM context, yielding generalized NLSM amplitudes. We prove an explicit formula for the residual embedding of an $n$-point NLSM amplitude in a mixed $n+2$ point generalized NLSM amplitude, which provides a strong consistency check on our generalization. We show that the dimension of the space of pure kinematic deformations is $\gcd(k,n)-1$, we introduce a deformation-compatible modification of the Global Schwinger Parameterization, and we include a new proof, using methods from matroidal blade arrangements, of the linear independence for the set of planar kinematic invariants for CEGM amplitudes. Our framework is compatible with string theory through recent generalizations of the Koba-Nielsen string integral to any positive configuration space $X^+(k,n)$, where the usual Koba-Nielsen string integral corresponds to $X(2,n) = \mathcal{M}_{0,n}$.

Figures (3)

• Figure 1: Directed distance function computation of $X_{13} = \frac{1}{4} \left(s_{12}+3 s_{14}+3 s_{23}+2 s_{24}+s_{34}\right)$. In this case, it simplifies modulo momentum conservation to just $s_{23}$. The length 3 path from $e_{1}+e_3$ to $e_{2}+e_3$ is shown in red, where we abbreviate $e_i+e_j \in \Delta_{2,4}$ as $ij$.
• Figure 2: The mechanics of Theorem \ref{['thm: residual embedding']}. Evaluating the given residue localizes $n$ points in $\mathbb{P}^2$ to the degenerate configuration where points $1,2,\ldots, n-3$ are collinear in a line $L$. The remaining $3$ points project in pairs onto $L$, determining a point in $X(2,n) = \mathcal{M}_{0,n}$.
• Figure 3: Combinatorial interpretation of a compatible collection of inverse propagators involved in the residual embedding of $m^{(2)}_{7}$ amplitude inside $m^{(3)}_{8}$. Deforming this with the mixed kinematic shift $\sigma$ in Equation \ref{['eqn: Xijk mixed deformation']} and taking one more residue where $X_{168}=0$ or $X_{278}=0$ gives a residual embedding of the $6$-point NLSM amplitude into a mixed $8$-point CEGM amplitude.

Theorems & Definitions (21)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Example 3.1
• Proposition 3.2
• proof
• Proposition 3.3: Early:2022mdn
• proof : Sketch of Proof
• Theorem 3.4: Early:2020hap