All next-to-maximally-helicity-violating one-loop gluon amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory

We compute the next-to-MHV one-loop $n$-gluon amplitudes in $\mathcal{N}=4$ super-Yang–Mills theory. These amplitudes contain three negative-helicity gluons and an arbitrary number of positive-helicity gluons, and are the first infinite series of amplitudes beyond the simplest, MHV, amplitudes. We also discuss some aspects of their twistor-space structure.

Figure 1

Examples of box integral functions $B(i,j,k,l)$ appearing in seven-point amplitudes; the arguments $i,j,k,l$ are the uncanceled propagators: (a) the one-mass box $B(3,4,5,6)=F^{1\mathrm{m}}(s_{34},s_{45},s_{345})$, (b) the “easy” two-mass box $B(3,4,6,7)=F^{2\mathrm{m}e}(s_{345},s_{456},s_{45},s_{712})$, (c) the “hard” two-mass box $B(3,5,6,7)=F^{2\mathrm{m}h}(s_{56},s_{345},s_{712},s_{34})$, and (d) the three-mass box $B(2,4,6,7)=F^{3\mathrm{m}}(s_{671},s_{456},s_{71},s_{23},s_{45})$.

Figure 2

A generalized triple cut. The three propagators cut by the dashed lines are required to be “open”.

Figure 3

The box integral functions labeled by the clusters of masses.

Figure 4

Schematic depiction of the easy two-mass box coefficients, expressed as a double sum over three-mass box coefficients.

Figure 5

An easy two-mass box where positive-helicity legs $a$ and $b$ are buried inside a cluster.

Figure 6

The three-mass box integrals that may be used to verify all hard two-mass coefficients. Leg $s$ becomes soft.

Figure 7

The boxes whose coefficients combine to give one of the new representations of any one-loop $n$-point tree amplitude. These boxes are the ones with infrared singularities of the form $\ln (-s_{12})/ϵ$.

Figure 8

The twistor-space configuration for the three-mass-box coefficients $c^{3\mathrm{m}}$ given in Eq. (15) is depicted on the right. The corresponding three-mass integral is shown on the left. All points lie in a plane.

Figure 9

The twistor-space configuration for the four-mass-box coefficient of an ${\mathrm{N}}^2\mathrm{MH}V$ amplitude is depicted on the right. The corresponding four-mass integral is shown on the left. The points need not lie in a plane, but lie in the intersection of two planes.