Prescriptive Unitarity Integrals in Planar QCD

Updated 4 February 2026

Prescriptive unitarity integrals are a framework using pure dlog forms that yield integrals of uniform maximal transcendental weight in planar QCD.
They leverage on-shell diagram computations to assign unique residues, bypassing IBP ambiguities and simplifying loop integration.
This method streamlines high-loop amplitude calculations and reveals deep connections between planar QCD and maximally supersymmetric theories.

Prescriptive unitarity integrals are a framework for representing and extracting the maximally transcendental contributions to planar QCD scattering amplitudes. These integrals embed the analytic and algebraic structure of leading singularities—computed via on-shell diagrams—directly into the basis of loop integration, yielding a prescription that bypasses ambiguities of traditional master-integral reduction. The method leverages the uniform transcendental weight structure inherent to four-dimensional pure dlog integrals and connects the assignment of maximally transcendental pieces in QCD to the algebraic machinery developed for $\mathcal{N}=4$ SYM, particularly at multi-loop and multi-leg level (Carrôlo et al., 2 Feb 2026).

1. Definition and Context

Prescriptive unitarity integrals are loop integrals constructed so that each individual basis element evaluates to a pure function of uniform and maximal transcendentality, with four-dimensional leading singularities matching the corresponding residues ("on-shell functions") of the amplitude on chiral maximal-cut contours. The central idea is to avoid the traditional integration-by-parts (IBP) ambiguity of master reductions by fixing canonical dlog forms whose leading coefficients—the "prescriptions"—are fixed uniquely by four-dimensional unitarity cuts (Carrôlo et al., 2 Feb 2026, Henn et al., 2021).

This structure arises in motivated response to the observed universality of maximally transcendental parts of gauge-theory amplitudes, which coincide in planar QCD and $\mathcal{N}=4$ SYM at leading color, modulo a color-factor replacement. Prescriptive unitarity provides the analytic machinery to operationalize this correspondence directly at the integrand level.

2. Uniform Transcendentality and the Maximal-Weight Projector

Transcendental weight $w$ is assigned such that $\mathrm{Li}_n$ and $\zeta_n$ have $w=n$ , $\log^k$ terms have $w=k$ , and each loop order $L$ allows maximal achievable weight $2L$ for $L$ -loop integrals in four dimensions (Henn et al., 2021).

Any $L$ -loop amplitude can be evaluated as a sum of integrals,

$I^{(L)} = \int d^{4L} \ell \; \left( \sum_j R_j \, d\log(\alpha_{j,1}) \wedge \ldots \wedge d\log(\alpha_{j,4L}) + \text{lower-weight terms} \right),$

where the $R_j$ are rational "prefactors" determined by leading singularities.

The maximal weight projector $\mathcal{P}_{\text{max}}$ operates at the integrand level by decomposing the amplitude into a sum over pure dlog forms and discarding all terms (such as those with double poles) that integrate to functions of lower weight. The result is a representation solely in terms of pure, maximally transcendental master integrals (Henn et al., 2021, Carrôlo et al., 2 Feb 2026).

3. Construction of Prescriptive Unitarity Bases

Prescriptive unitarity bases are constructed by imposing that for each four-dimensional, chiral maximal-cut contour, there is a unique basis element that has unit residue on this contour and vanishes on all others. This is accomplished as follows (Carrôlo et al., 2 Feb 2026):

Enumerate all IR-finite pure master integrals (with dlog integrands) compatible with the process at fixed $n$ -points and $L$ -loops.
For each maximal cut of the loop momenta, solve for integrand numerators such that the required residue structure is achieved (e.g., numerators constructed from spinor traces or powers of tree-level Parke-Taylor factors).
The coefficients (prefactors) for each basis element in the amplitude are, by construction, the four-dimensional leading singularities on that cut. These prefactors are computable via on-shell diagrams (bipartite graphs of glued three-point MHV and $\overline{\text{MHV}}$ amplitudes).

Thus, the full amplitude in the maximally transcendental sector is given as a sum over prescriptive integrals,

$\mathcal{A}^{(L)}_{\text{MT}} = \sum_k R^{(L)}_k \, I^{(L)}_k,$

where $I^{(L)}_k$ are canonical dlog integrals and $R^{(L)}_k$ are on-shell prefactors.

4. Leading Singularities and On-Shell Diagram Classification

The algebraic classification of leading singularities appears naturally in the on-shell diagram formalism:

At one loop: For MHV amplitudes, the only IR-finite contour is the two-mass-easy box, with a chiral dlog basis. The numerator is a four-trace in spinor bracket notation, linking directly to the external helicity structure.
At two loops: There are five primary IR-finite topologies for MHV amplitudes—kissing-box, penta-box, double-box, double-pentagon, and hexa-box—each with specifically engineered numerators to enforce unit leading singularities (Carrôlo et al., 2 Feb 2026).
Quark contributions in QCD: In pure Yang-Mills, these are sums over spinor traces of degree four; in QCD with $N_f$ quarks, new non-singlet structures arise, including double-box basis elements accommodating closed quark loops.

Leading singularities $R^{(L)}$ are calculated by evaluating the on-shell functions that result from gluing three-point amplitudes along the maximal-cut solutions. The structure of these singularities firmly determines which dlog integrals can (and cannot) appear in the maximal weight sector.

5. Algorithmic Implementation and Symbol-Level Bootstrap

The application at multi-loop level (e.g., two-loop six-gluon amplitudes) involves:

Building a function space: The set of allowed functions—the "symbol alphabet"—is dictated by the kinematic constraint and maximal cuts. For the two-loop six-point massless sector, a 137-letter alphabet suffices (Carrôlo et al., 2 Feb 2026).
Forming an ansatz: The amplitude is written as a linear combination of prescriptive basis integrals times undetermined weight-$2L$ pure symbols. One- and two-loop "prefactors" (from the on-shell diagram analysis) multiply each functional basis element.
Imposing physical constraints: Spurious-pole cancellation, correct collinear and soft limits, and compatibility with universal IR subtraction relations fix all free coefficients. The final result is unique and manifestly uniform transcendental weight.

6. Universality, Maximal Transcendentality, and Correspondence to $\mathcal{N}=4$ SYM

A key consequence of the prescriptive unitarity approach is the demonstration of the maximal transcendentality principle for planar QCD: after color-factor replacement ( $C_F \to C_A$ ), the maximally transcendental part of the amplitude coincides with the $\mathcal{N}=4$ SYM result, even though the underlying matter content differs. Subleading and matter-dependent integral topologies always enter at lower weight, and all maximally transcendental functions are generated by the uniform-weight, four-dimensional dlog master integrals that are universal to both QCD and $\mathcal{N}=4$ SYM at planar level (Carrôlo et al., 2 Feb 2026, Henn et al., 2021, Dixon et al., 2020).

7. Applications and Physical Implications

Prescriptive unitarity integrals provide a constructive and canonical approach for:

Explicit computation of maximally transcendental weight pieces of multi-loop, multi-leg planar QCD amplitudes (Carrôlo et al., 2 Feb 2026).
Systematic derivation of splitting and soft functions in the multi-collinear and multi-soft limit, matching the known results of $\mathcal{N}=4$ SYM in the extracted sector.
Efficient identification and organization of the function space (symbol alphabet) relevant for a given process, illuminating potential geometric structures (such as cluster-amplituhedron/positivity domains) governing amplitudes (Abreu et al., 2024).
Reducing complexity: all lower-weight corrections (from scalar or quark matter in lesser-supersymmetric theories) are automatically omitted from the MT projection, focusing analytic attention on the most universal and challenging (highest-weight) functions.

The method has been extensively validated in explicit calculations of two-loop Higgs+multi-gluon amplitudes, SL(2) sector form factors, anomalous dimensions, and multi-gluon MHV sectors up to six points (Carrôlo et al., 2 Feb 2026, Dixon et al., 2020, Loebbert et al., 2016, Kotikov, 2010).

Summary Table: Structure of Prescriptive Unitarity Integrals in Planar QCD

Concept	Role in Prescriptive Unitarity	Key Reference
dlog integrals (pure basis)	Provide uniform weight $2L$	(Carrôlo et al., 2 Feb 2026)
Leading singularity prefactors	Classify function multipliers via on-shell diagrams	(Carrôlo et al., 2 Feb 2026)
Maximal-weight projection	Discards lower-weight and double-pole terms	(Henn et al., 2021)
Master function alphabet	Enumerates allowed iterated integrals, e.g., 137 letters at two-loop six-point	(Carrôlo et al., 2 Feb 2026)
$\mathcal{N}=4$ correspondence	Maximally transcendental part matches after $C_F \to C_A$	(Dixon et al., 2020)

All computational steps are justified by the universality of soft/collinear limits, complete polynomial bases for four-dimensional singularities, and consistency with known QCD and $\mathcal{N}=4$ SYM results at maximal weight (Carrôlo et al., 2 Feb 2026).