content_image/325010.jpg content_image/325009.jpg content_image/325008.jpg
## A. D$_{2}$ relations for tree amplitudes with one positive helicity graviton, N − 2 positive helicity gluons and two negative helicity gluons 

The tree amplitudes with two gluons of negative helicity and other legs of positive helicity can be expressed by[28, 29] 

<img src='content_image/325010.jpg'>

where 

<img src='content_image/325009.jpg'>

In M = 1 case where all the legs take positive helicity except 1 and i , the amplitude is reduced to 

<img src='content_image/325008.jpg'>

In the equation above, 〈 1 i 〉 4 〈 12 〉〈 23 〉 ... 〈 N 1 〉 is just the MHV amplitude for N gluons with the per- mutation 1 , ..., N . Using (A5), we have 〈 l, N + 1 〉 [ l, N + 1] = s$_{l,N}$$_{+1}$ . With the eikonal identity(A9), 〈 1 l 〉 〈 1 ,N +1 〉〈 N +1 ,l 〉 and 〈 li 〉 〈 l,N +1 〉〈 N +1 ,i 〉 can split into sums of terms over all the legs 

