All one-loop maximally helicity violating gluonic amplitudes in QCD

We use on-shell recursion relations to compute analytically the one-loop corrections to maximally helicity-violating $n$-gluon amplitudes in QCD. The cut-containing parts have been computed previously; our work supplies the remaining rational parts for these amplitudes, which contain two gluons of negative helicity and the rest positive, in an arbitrary color ordering. We also present formulae specific to the six-gluon cases with helicities ($-+-+++$) and ($-++-++$), as well as numerical results for six, seven, and eight gluons. Our construction of the $n$-gluon amplitudes illustrates the relatively modest growth in complexity of the on-shell-recursive calculation as the number of external legs increases. These amplitudes add to the growing body of one-loop amplitudes known for all $n$, which are useful for studies of general properties of amplitudes, including their twistor-space structure.

Figure 1

Schematic representation of tree-level recursive contributions. The labels $T$ refer to tree vertices which are on-shell amplitudes. The momenta $\widehat{j}$ and $\widehat{l}$ are shifted, on-shell momenta, evaluated according to Eqs. (3.14, 3.15).

Figure 2

Schematic representation of one-loop recursive contributions. The labels $T$ and $L$ refer to tree and loop vertices. The factorization-function contribution (c) does not appear for MHV amplitudes.

Figure 3

Diagrammatic representation of overlap contributions. Each overlap diagram corresponds to a physical channel.

Figure 4

Nonvanishing recursive diagrams for the amplitude $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{+},3^{-},4^{+},5^{+})$, using a $[1,3⟩$ shift, as given in Eqs. (4.11, 4.12, 4.13, 4.14).

Figure 5

Channels giving overlap contributions to $A_{5;1}^{\mathcal{N}=0}(1^{-},2^{+},3^{-},4^{+},5^{+})$ with a $[1,3⟩$ shift, at the values of $z$ given in Eq. (4.15).

Figure 6

Nonvanishing recursive diagrams for the amplitude $A_{6;1}^{\mathcal{N}=0}(1,3)$, using a $[1,3⟩$ shift, as given in Eq. (4.23).

Figure 7

Channels that can give overlap contributions to $A_{6;1}^{\mathcal{N}=0}(1,3)$ with a $[1,3⟩$ shift, at the values of $z$ given in Eq. (4.24).

Figure 8

Nonvanishing recursive diagrams for the amplitude $A_{6;1}^{\mathcal{N}=0}(1,4)$, using a $[1,4⟩$ shift, as given in Eq. (4.25).

Figure 9

Channels giving overlap contributions to $A_{6;1}^{\mathcal{N}=0}(1,4)$ with a $[1,4⟩$ shift, at the values of $z$ given in Eq. (4.26).

Figure 10

The propagators corresponding to the sequence of $K_{[i]}$.

Figure 11

Kinematic configuration of the box and triangle functions appearing in the completed-cut contribution given in Eq. (a1).