QCD Scattering Amplitudes and Prescriptive Unitarity

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.

💡 Research Summary

This paper presents a comprehensive framework for extracting the maximally‑transcendental part of planar, massless QCD six‑gluon MHV amplitudes at two loops, and demonstrates how this part can be completely determined using a symbol‑bootstrap approach combined with prescriptive unitarity. The authors begin by reviewing six‑particle kinematics in spinor‑helicity variables, the color decomposition in the planar limit, and the separation of infrared (IR) and ultraviolet (UV) divergences into a hard function. They then introduce the notion of a maximal‑weight projection, which isolates the highest transcendental weight contributions of the amplitude. Crucially, they show that the prefactors multiplying these weight‑maximal functions are precisely the four‑dimensional leading singularities of the theory. These leading singularities can be computed from on‑shell diagrams, a technique originally developed for N=4 super‑Yang‑Mills (sYM) theory.

The paper classifies all relevant prefactors for two‑loop MHV amplitudes. At one loop, non‑singlet on‑shell diagrams generate the required prefactors. At two loops, the authors identify five distinct maximal‑cut topologies that contribute both in pure Yang‑Mills and in full QCD (including quark loops). Each prefactor is associated with a contour satisfying triangle power‑counting, guaranteeing a complete basis for the maximally‑transcendental sector.

Prescriptive unitarity is then introduced as a refinement of generalized unitarity. By first enumerating all independent L‑loop contour integrals and constructing pure Feynman integrals that have non‑zero leading singularities only on a chosen contour, one obtains a set of “prescriptive‑unitarity integrals”. The two‑loop prescriptive integrals from earlier work form a subset of the function space needed for the present problem. This connection dramatically reduces the number of unknown coefficients in the bootstrap ansatz.

The authors construct a symbol ansatz using the recently completed planar six‑point two‑loop function space, which is spanned by a 137‑letter alphabet. The ansatz is a linear combination of these symbols multiplied by the previously determined prefactors. Physical constraints are then imposed systematically: permutation and cyclic symmetries, cancellation of spurious poles, correct behavior in multi‑collinear and multi‑soft limits, and consistency with lower‑point or lower‑loop results. Because the prescriptive‑unitarity input already fixes many coefficients, the remaining constraints uniquely determine all coefficients, yielding a fully fixed symbol for the maximally‑transcendental part of the two‑loop six‑gluon MHV amplitude.

The resulting symbols are expressed entirely in the reduced 137‑letter alphabet, confirming that this alphabet is complete for two‑loop six‑point massless MHV scattering. From the bootstrap solution the authors also extract two‑loop triple‑collinear splitting functions and double‑soft gluon functions at maximal weight. These functions match the corresponding N=4 sYM results up to the prefactor differences dictated by the QCD color structure, reinforcing the maximal‑transcendentality principle that the most complicated terms in QCD mirror those in the supersymmetric theory.

In the discussion, the paper emphasizes several key achievements: (i) establishing a direct correspondence between leading singularities and prefactors in QCD, (ii) demonstrating that a symbol‑bootstrap augmented by prescriptive unitarity can solve a non‑trivial two‑loop QCD amplitude without any explicit IBP reduction, and (iii) providing new universal splitting and soft functions that are immediately useful for infrared subtraction schemes at higher orders. The authors outline future directions, including extending the method to non‑MHV helicity configurations, amplitudes with external quarks, higher multiplicities, and non‑planar contributions, where new alphabet letters and more intricate prescriptive integrals are expected.

Overall, the work bridges sophisticated on‑shell techniques from N=4 sYM to realistic QCD calculations, offering both a deeper theoretical understanding of the analytic structure of QCD amplitudes and practical tools for precision collider phenomenology.