Group-theoretic relations for partial amplitudes in gauge theories with orthogonal group

It is very important to find some nontrivial relations for color-ordered amplitudes at loop levels. In the last several years, a pure group-theoretic method has been proposed to study loop level relations for color-ordered amplitudes in $\sun$ gauge theories. In this paper, we apply the same method to study the linear relations for loop level four point color-ordered gluon amplitudes in gauge theory with $\son$ group and find that there are $1,6,10,13,16$ constraint relations at $L=0,1,2,3,4$ loop level. It is expected that there are more group-theoretic constraints for high loop levels. The constraint relations should be derived order by order in $\son$ gauge theories. This is unlike the $\sun$ case, where there are only four constraints for higher $(L\geqslant 2)$ loop amplitudes.