On-shell recurrence relations for one-loop QCD amplitudes

We present examples of on-shell recurrence relations for determining rational functions appearing in one-loop QCD amplitudes. In particular, we give relations for one-loop QCD amplitudes with all legs of positive helicity, or with one leg of negative helicity and the rest of positive helicity. Our recurrence relations are similar to the tree-level ones described by Britto, Cachazo, Feng, and Witten. A number of new features arise for loop amplitudes in nonsupersymmetric theories like QCD, including boundary terms and double poles. We show how to eliminate the boundary terms, which would interfere with obtaining useful relations. Using the relations we give compact explicit expressions for the $n$-gluon amplitudes with one negative-helicity gluon, up through $n=7$.

Figure 1

Terms in the recursive expression (33) for $A_n^{(1)}(1^{+},2^{+},\dots ,n^{+})$. The vertices labeled by a $T$ are trees, and the ones labeled by an $L$ are loops.

Figure 2

Terms in the recursive expression for $A_5^{(1)}(1^{-},2^{+},3^{+},4^{+},5^{+})$.

Figure 3

The diagrams describing a recursive formula for $A_n^{(1)}(1^{-},2^{+},\dots ,n^{+})$. Diagram (a) vanishes, while diagram (d) needs to be summed over $j$. For $j=3$ diagram (d) contains both single-pole and double-pole contributions.