Off-shell BCJ Relation in Nonlinear Sigma Model

TL;DR

This work proves a general off-shell BCJ relation for tree-level currents in the $U(N)$ nonlinear sigma model under Cayley parametrization by introducing a revised BCJ form that expresses a permutation-summed amplitude as a sum over divisions into sub-currents with BCJ-type coefficients. The authors establish a rigorous, recursive proof using Berends-Giele recursion, generalized $U(1)$-decoupling, and division-structure analysis, and confirm the six-point case explicitly. They derive the coefficient matrix $\mathbf{C}(r,s)$ governing the divisions and show the off-shell relation reduces to the standard on-shell BCJ relation in the limit $p_1^2\to0$, thereby establishing a complete off-shell correspondence for tree-level BCJ relations in this theory. The results illuminate the algebraic structure behind amplitude relations beyond YM and point to potential loop-level extensions and broader applicability to theories with the same underlying color-kinematics algebra.

Abstract

We investigate relations among tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, we propose and prove a general revised BCJ relation for even-point currents. Unlike the on-shell BCJ relation, the off-shell one behaves quite differently from Yang-Mills theory although the algebraic structure is the same. After performing the permutation summation in the revised BCJ relation, the sum is non-vanishing, instead, it equals to the sum of sub-current products with the BCJ coefficients under a specific ordering, which is presented by an explicit formula. Taking on-shell limit, this identity is reduced to the on-shell BCJ relation, and thus provides the full off-shell correspondence of tree-level BCJ relation in nonlinear sigma model.

Figures (14)

• Figure 1: Schematic diagram representations to read the coefficients $S_{div\{\mathbf{\alpha_r},\mathbf{\beta_s}\}}$.
• Figure 2: Representation of division $\mathcal{J}(\alpha_1)\mathcal{J}(\alpha_2)\mathcal{J}(\alpha_3)\mathcal{J}(\alpha_4)\mathcal{J}(\alpha_5\alpha_6\alpha_7)$ as an example.
• Figure 3: The Hasse diagram for the different divisions of $\mathcal{J}(\alpha_1\cdots\alpha_7\beta_1\beta_2)$. The divisions in the same row are of same number of parts. We shall use $\mathbf{i}_j$ to denote the $j$-th division in the $i$-th row for simplicity.
• Figure 4: Direct contribution to $\mathbf{3}_1$ from $\mathbf{2}_1$, excluding the contribution from $\mathbf{1}_1$, denoted by $\mathbf{2}_1\rightarrow\mathbf{3}_1$
• Figure 5: $\mathbf{2}_2\rightarrow\mathbf{3}_1$

Theorems & Definitions (14)

• Proposition 3.2
• proof
• Lemma 5.3
• proof
• Definition 5.1
• Definition 5.2
• Lemma 5.4
• proof
• Lemma 5.5
• proof