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Amplitude Relations in Non-linear Sigma Model

In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.