[Paper Review] Note on off-shell relations in nonlinear sigma model

Gang Chen, Yi-Jian Du+2 more|arXiv (Cornell University)|Dec 11, 2014

Black Holes and Theoretical Physics42 references4 citations

TL;DR

This paper proposes and proves a generalized U(1) identity for tree-level off-shell currents in the nonlinear sigma model under Cayley parametrization, showing that all odd-point currents vanish. The generalized identity extends the off-shell U(1) relation of Chen et al. (2013) and reduces to the on-shell Kleiss-Kuijf (KK) relation in the on-shell limit, providing the full off-shell correspondence of the KK relation in this theory.

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.

Motivation & Objective

• To extend the known off-shell U(1) identity in the nonlinear sigma model to a generalized form applicable to even-point currents.
• To establish a complete off-shell correspondence for the on-shell Kleiss-Kuijf (KK) relation in the nonlinear sigma model.
• To resolve the lack of general off-shell relations for KK and BCJ-type identities in the nonlinear sigma model, despite their on-shell validity.
• To provide a systematic framework for off-shell amplitude relations in nonlinear sigma model using Cayley parametrization and current decomposition.

Proposed method

• Utilizes Cayley parametrization to simplify current structures, under which odd-point off-shell currents vanish identically.
• Proposes a generalized U(1) identity for even-point off-shell currents, summing over all ordered permutations of two sets of external legs with fixed internal order.
• Introduces a summation over divisions of the two leg sets into ordered subsets with odd cardinality, constrained by |R_D - S_D| = 1.
• Derives the identity using recursive relations and coefficient matching, solving for structure constants via diagrammatic analysis and momentum conservation.
• Applies momentum conservation and on-shell limits to verify consistency with known on-shell KK relations.
• Validated through explicit computation of structure constants and their dependence on |r - s|, showing δ(|r - s| - 1) behavior.

Experimental results

Research questions

• RQ1 Can a generalized off-shell U(1) identity be formulated for even-point currents in the nonlinear sigma model?
• RQ2 Does this generalized identity reduce to the known on-shell KK relation in the on-shell limit?
• RQ3 Is the off-shell U(1) identity of Chen et al. (2013) a special case of this generalized identity?
• RQ4 Can the full off-shell correspondence of the KK relation be established in the nonlinear sigma model?
• RQ5 What constraints govern the structure constants in the off-shell current decomposition under Cayley parametrization?

Key findings

• All odd-point off-shell currents in the nonlinear sigma model vanish under Cayley parametrization, as established in prior work and confirmed here.
• The generalized U(1) identity for even-point currents is proven, with the sum over all ordered permutations of two leg sets equal to a sum over specific divisions of the sets into odd-sized subsets.
• The identity reduces to the on-shell Kleiss-Kuijf (KK) relation in the on-shell limit, establishing full off-shell correspondence.
• The off-shell U(1) identity of Chen et al. (2013) is shown to be a special case of the generalized identity, with specific constraints on the number of legs.
• Structure constants in the current decomposition are found to be δ(|r - s| - 1), indicating a strict selection rule based on the difference in the number of legs in the two sets.
• The final form of the generalized identity includes a factor of (1/(2F²))^{(r+s-1)/2}, reflecting the energy scale dependence of the amplitude relations.