content_image/322603.jpg content_image/322604.jpg content_image/322605.jpg
are given as 2 

<img src='content_image/322603.jpg'>

<img src='content_image/322604.jpg'>

<img src='content_image/322605.jpg'>

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. 

## III. D$_{2}$ RELATIONS FOR THREE- AND FOUR-POINT TREE AMPLITUDES 

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. 

