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Note on off-shell relations in nonlinear sigma model

Gang Chen, Yi-Jian Du, Shuyi Li, Hanqing Liu

TL;DR

This work addresses off-shell relations among tree-level currents in the nonlinear sigma model under Cayley parametrization, where odd-point currents vanish. It develops a generalized off-shell $U(1)$ identity for even-point currents and shows that its on-shell limit is equivalent to the Kleiss–Kuijf (KK) relations, establishing a full off-shell KK correspondence. The main technical advance is a BG-recursion-based proof that fixes the division coefficients ${ m cal V}^{(r,s)}$ to ${ig(}1/(2F^2){ig)}^{(r+s-1)/2}oldsymbol{ riangleleft}(|r-s|-1oldsymbol{ riangleright)$, thereby unifying previous off-shell results and their on-shell reductions. The results open avenues for further exploration of off-shell BCJ-type relations and potential loop-level extensions within nonlinear sigma models.

Abstract

In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized $U(1)$ identity for even-point currents. The off-shell $U(1)$ identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.

Note on off-shell relations in nonlinear sigma model

TL;DR

This work addresses off-shell relations among tree-level currents in the nonlinear sigma model under Cayley parametrization, where odd-point currents vanish. It develops a generalized off-shell identity for even-point currents and shows that its on-shell limit is equivalent to the Kleiss–Kuijf (KK) relations, establishing a full off-shell KK correspondence. The main technical advance is a BG-recursion-based proof that fixes the division coefficients to , thereby unifying previous off-shell results and their on-shell reductions. The results open avenues for further exploration of off-shell BCJ-type relations and potential loop-level extensions within nonlinear sigma models.

Abstract

In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized identity for even-point currents. The off-shell identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.

Paper Structure

This paper contains 12 sections, 44 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preparation: Feynman rules, Berends-Giele recursion and $U(1)$ identity
  3. Feynman rules
  4. Berends-Giele recursion
  5. The off-shell versions of $U(1)$ identity
  6. Direct calculation of an eight-point example
  7. Proof of the generalized $U(1)$ identity for off-shell currents
  8. Proof of the validity of \ref{['off-shell-gen-U(1)-1']} with an undetermined coefficient ${\cal V}^{(r,s)}$
  9. Solving ${\cal V}^{(r,s)}$
  10. $r=1$
  11. $r>1$
  12. Conclusions

Figures (11)

  • Figure 1: Two classes of diagrams: (A) diagrams containing substructures of generalized $U(1)$ identity such as $\sigma\in OP(A_i\bigcup B_j)$ where $A_i$ and $B_j$ denote ordered subsets of $\{\alpha\}$ and $\{\beta\}$. (B) diagrams with each subcurrent containing only $\{\alpha\}$ elements or $\{\beta\}$ elements.
  • Figure 2: We redefine the coefficients for $R+S<r+s$ divisions such that they are the right ones as in the general form \ref{['off-shell-gen-U(1)']}. Then we solve the $R=r$, $S=s$ coefficient.
  • Figure 3: Diamgrams contributing to four-point identity.
  • Figure 4: The off-shell extension of four-point identity. Here we absorb the ${1\over p_1^2}$ corresponding to the off-shell leg $1$ into the coefficients for convenience.
  • Figure 5: Redefinition of the coefficient of $(2,3)$ division for the identity with two $\alpha$'s and three $\beta$'s.
  • ...and 6 more figures