are given as 2 where s$_{ij}$ = 2 k$_{i}$ · k$_{j}$ , 3$_{g}$ and 4$_{g}$ have momentum 1 $_{2}$k$_{3}$ while 5$_{g}$ and 6$_{g}$ have momentum 1 $_{2}$k$_{4}$ . The total amplitude is derived by substitute the relations (6) into Eq. (4). So far, we have seen, though KLT and D$_{2}$ relations hold on worldsheets with different topologies in string theory. They can both give the amplitudes for gauge-gravity coupling. KLT relations can given the amplitudes for gauge-gravity minimal coupling. Then we should consider the question: Can the field theory limits of the two different relations in string theory hold in a same theory for gauge-gravity coupling? In the next two sections, we will show the D$_{2}$ relations also hold in minimal coupling theory of gauge field and gravity. In this section, we use the D$_{2}$ relations (6) to give the three- and four-point tree ampli- tudes for gauge-gravity coupling. These results are same with those given by using KLT relations[17]. Thus D$_{2}$ relations also hold in minimal coupling theory of gauge field and gravity. 2 In this paper, we use i$_{g}$ and j$_{h}$ to denote gluons and gravitons respectively. We do not consider the amplitudes where all the external legs are gravitons. We also do not consider the amplitudes where graviton exchanges between gluons, which contribute to higher order process. 7 III. D$_{2}$ RELATIONS FOR THREE- AND FOUR-POINT TREE AMPLITUDES A (1$_{g}$ , 2$_{g}$ , 3$_{h}$) ∼ κs$_{12}$ A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{g}$) , (6a) A (1$_{g}$ , 2$_{g}$ , 3$_{g}$ , 4$_{h}$) ∼ gκs$_{13}$ A (1$_{g}$ , 5$_{g}$ , 2$_{g}$ , 4$_{g}$ , 3$_{g}$) , (6b) A (1$_{g}$ , 2$_{g}$ , 3$_{h}$ , 4$_{h}$) ∼ κ $^{2}$[ s 2 12 A (1$_{g}$ , 6$_{g}$ , 3$_{g}$ , 5$_{g}$ , 4$_{g}$ , 2$_{g}$) − s$_{12}$s$_{13}$ A (1$_{g}$ , 3$_{g}$ , 5$_{g}$ , 4$_{g}$ , 2$_{g}$ , 6$_{g}$)] , (6c)