QCD Scattering Amplitudes and Prescriptive Unitarity

TL;DR

We bootstrap the maximally-transcendental part of planar QCD amplitudes, identifying four-dimensional leading singularities as the source of the rational prefactors that accompany maximal weight functions. By marrying prescriptive unitarity with a symbol-based bootstrap, we compute the two-loop six-gluon MHV QCD amplitudes in symbols, deriving a complete IR-finite basis from prescriptive integrals and constraining the remaining pieces with multi-collinear and soft limits. The resulting six-point, two-loop MHV symbols live in a reduced 137-letter alphabet, and the analysis reveals a close relationship between pure YM results and N=4 sYM through maximal transcendentality, including a precise mapping of hard functions. These results illuminate the analytic structure of QCD amplitudes, enable efficient computation at two loops, and have implications for factorization in collinear and soft regimes as well as for future explorations of NMHV sectors and external fermions.

Abstract

We present a systematic framework for the maximally-transcendental part of planar QCD scattering amplitudes and perform the first bootstrap computation of six-gluon MHV amplitudes in massless QCD at the symbol level. By analyzing the maximal weight projection of amplitudes at the integrand level, we relate their maximally-transcendental parts to prescriptive unitarity integrals. This reveals a novel analytic structure: the prefactors multiplying the functions of maximal transcendentality are identified with the four-dimensional leading singularities of the theory. As a consequence, these prefactors admit a complete classification and can be computed using on-shell diagrams, a formalism originally developed in $\mathcal{N}{=}4$ super Yang-Mills theory. As a concrete application, we determine the two-loop prefactors for planar MHV gluon amplitudes at arbitrary multiplicity. Combining these prefactors with recent advances in the planar two-loop six-point function space and explicit six-point prescriptive-unitarity input, we construct a complete symbol ansatz and uniquely fix the maximally-transcendental part of the two-loop six-gluon MHV QCD amplitudes by imposing physical constraints. The resulting symbols are expressible in a reduced 137-letter alphabet, suggesting that this alphabet is complete for two-loop six-point massless MHV scattering. We also discuss the implications for multi-collinear splitting and multi-soft functions.

Figures (10)

• Figure 1: One-loop MHV contour.
• Figure 2: Five types of two-loop contours for MHV amplitudes.
• Figure 3: Non-singlet two-mass-easy box on-shell diagrams contributing genuinely one-loop prefactors.
• Figure 4: Non-singlet on-shell diagrams contributing to genuinely two-loops prefactors.
• Figure 5: Two-loop on-shell diagrams for the MHV sector $({-}{+}{-}{+}{+}{+})$.