Abstract
We study the symbol and the alphabet for two-loop NMHV amplitudes in planar mathcal{N} = 4 super-Yang-Mills from the overline{Q} equations, which provide a first-principle method for computing multi-loop amplitudes. Starting from one-loop N2MHV ratio functions, we explain in detail how to use overline{Q} equations to obtain the total differential of two-loop n-point NMHV amplitudes, whose symbol contains letters that are algebraic functions of kinematics for n ≥ 8. We present explicit formula with nice patterns for the part of the symbol involving algebraic letters for all multiplicities, and we find 17 − 2m multiplicative-independent letters for a given square root of Gram determinant, with 0 ≤ m ≤ 4 depending on the number of particles involved in the square root. We also observe that these algebraic letters can be found as poles of one-loop four-mass leading singularities with MHV or NMHV trees. As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, particularly simple and free of square roots. As an example, we present the complete symbol for n = 9 whose alphabet contains 59 × 9 rational letters, in addition to the 11 × 9 independent algebraic ones. We also give all-loop NMHV last-entry conditions for all multiplicities.
Highlights
Cases for more general QFT), and it is conceivable that we will have more and more important understandings for both integrands and integrals
We study the symbol and the alphabet for two-loop NMHV amplitudes in planar N = 4 super-Yang-Mills from the Qequations, which provide a first-principle method for computing multi-loop amplitudes
As a byproduct of our algebraic results, we find a large class of components of two-loop NMHV, which can be written as differences of two double-pentagon integrals, simple and free of square roots
Summary
One of the main results of [24] is the following anomaly equation for the Qgenerators: it has been argued based on a Wilson-loop analysis that Qof the amplitude is given in terms of an integral of higher-point one with fermion insertion (which increases k) in the collinear limit, and by taking into account Qof the BDS ansatz we have: QAa Rn,k. For NMHV cases, 2Note that when we take the differential of the usual super-amplitude, it is understood that the differential d only acts on transcendental functions, not on the coefficients which are Yangian ( Q) invariants This is consistent since after we do the replacement χ → dZ for these Yangian invariants, they become differential forms that are closed. Qequations (2.5), with the supplement of dual conformal invariance, can determine the differential of NMHV amplitudes on their own
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