Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups

In this paper,linear constraint relations among loop-order five-point color-ordered amplitudes in SO(N)and Sp(2N) gauge theories are derived with the group-theoretic method. These constrains are derived up to four-loop order. It is found that in both theories, there are $n=6,22,34,44,50$ linear constraint relations at $L=0,1,2,3,4$ loop orders. Then the numbers of independent color-ordered five-point amplitudes are $n_{ind.}=6,12,22,34,50$ at each loop order correspondingly.