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

We compute the non-MHV (non-maximally-helicity-violating) one-loop seven-gluon amplitudes in $\mathcal{N}=4$ super-Yang-Mills theory, which contain three negative-helicity gluons and four positive-helicity gluons. There are four independent color-ordered amplitudes, $(---++++)$, $(--+-+++)$, $(--++-++)$ and $(-+-+-++)$. The MHV amplitudes containing two negative-helicity and five positive-helicity gluons were computed previously, so all independent one-loop seven-gluon helicity amplitudes are now known for this theory. We present partial information about an infinite sequence of next-to-MHV one-loop helicity amplitudes, with three negative-helicity and $n-3$ positive-helicity gluons, and the color ordering $(---++\cdots ++)$; we give a new coefficient of one class of integral functions entering this amplitude. We discuss the twistor-space properties of the box-integral-function coefficients in the amplitudes, which are quite simple and suggestive.

Figure 1

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

Figure 2

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

Figure 3

A schematic depiction of multiparticle factorization at one loop.

Figure 4

The class of three-mass box functions whose coefficient is given in Eq. (199).

Figure 5

Examples of twistor-space configurations for single-term box coefficients in the helicity amplitude $A_{7;1}^{\mathcal{N}=4}(1^{-},2^{-},3^{+},4^{-},5^{+},6^{+},7^{+})$. In every case, all the points lie in a plane. (a) The easy two-mass box coefficient $c_{125}$, (b) the hard two-mass box coefficient $c_{237}$, (c) the three-mass box coefficient $c_{135}$.

Figure 6

The twistor-space configuration for the class of three-mass-box coefficients in the helicity amplitude $(---++\cdots ++)$ discussed in the text. All points lie in a plane.