## IV. GENERAL DISCUSSIONS ON D$_{2}$ RELATIONS FOR TREE AMPLITUDES IN MINIMAL COUPLING THEORY OF GAUGE FIELD AND GRAVITY 

In this section, we study general tree amplitudes in minimal coupling theory of gauge field and gravity. D$_{2}$ relations can give amplitudes with N gluons and M gravitons by sum of pure-gluon partial amplitudes with N + 2 M legs except for appropriate factors. If all the gluons and gravitons take positive helicity, the amplitude must vanish. This is because the pure-gluon amplitudes with all legs of positive helicity vanish. 

The tree amplitudes with one gluon of negative helicity and other legs of positive helicity can be expressed as sum of pure-gluon amplitudes with one leg of negative helicity and other legs of positive helicity. Then these amplitudes also vanish. The amplitudes with one graviton of negative helicity and other N + M − 1 legs of positive helicity are expressed by sum of MHV ( N + 2 M )-gluon tree amplitudes where the two gluons corresponding to the negative helicity graviton take negative helicity. The two negative helicity gluons in the ( N + 2 M )-gluon amplitudes take the same momentum. Considering the antisymmetry of the spinor products (A6), these MHV tree amplitudes for N + 2 M gluons vanish. Thus the amplitudes with one graviton of negative helicity and other N + M − 1 legs of positive helicity must vanish. 

The results above given by D$_{2}$ relations are same with those given by KLT relations. Then the D$_{2}$ relations can give the amplitudes with all the legs of positive helicity for gauge- graviton minimal coupling. They also give amplitudes with one leg of negative helicity and other legs of positive helicity for gauge-gravity minimal coupling. 

Though the trivial cases are easy to consider, D$_{2}$ relations for nontrivial helicity con- figurations are not so clear. In the Subsections IV A and IV B, we will give discussions 

