content_image/305647.jpg content_image/305648.jpg content_image/305649.jpg
are inserted at the positions between 1, l and l , i respectively, the relation (21) depends on the relative positions of the two negative helicity gluons. Then we expect the relation (21) can be extended to that independent of the relative positions of the two negative helicity legs. In fact, in the expression (17), we can sum over l (1 < l ≤ N ). This is because 〈 li 〉 vanishes for l = i . Using the eikonal identity (A9) and the identity (A10) implied by momentum conservation, the amplitude becomes 

<img src='content_image/305647.jpg'>

Then we repeat the discussions above. The amplitude satisfy the relation 

<img src='content_image/305648.jpg'>

where we sum over all the external gluons. For any given l , P ′ in this relation denote all the permutations where one of the two gluons corresponding to the graviton is inserted at positions to the left of l and right of 1, the other one is inserted to the left of 1 and right of l (See Fig. 1(c)). Since this expression of the amplitude does not depend on the helicity configuration of the legs. We suggest that for any helicity configuration, the tree amplitude with N gluons minimally coupled to one graviton satisfy the relation 

<img src='content_image/305649.jpg'>

Though this extension will be more complicated, we expect there must be such extensions to the relations for arbitrary helicity configurations. 

