Two-Loop All-Plus QCD Amplitude

• The paper demonstrates that the two-loop all-plus-helicity QCD amplitude yields a finite remainder composed solely of rational terms and dilogarithms after infrared subtraction.
• Integrand reduction techniques, including Gröbner bases and unitarity cuts, are used to decompose the amplitude into master integrals and expose its complex IR structure.
• The simplified analytic form of the amplitude enhances precision collider predictions and benchmarks advanced multi-loop computation methods.

The two-loop all-plus-helicity QCD amplitude is the quantum chromodynamics (QCD) scattering amplitude for a process in which all external gluons possess positive helicity and that is computed to order $\alpha_s^2$ (two-loop order) in perturbation theory. Since the tree-level all-plus amplitude vanishes in pure Yang-Mills, such amplitudes provide an arena for explicit, fully analytic multi-loop computations and serve as a theoretical laboratory for exposing the structure of infrared singularities, rational terms, connections to supersymmetric amplitudes, and for benchmarking modern amplitude methodologies.

1. Vanishing at Tree Level and Structure at One and Two Loops

The all-plus helicity amplitude $A_n^{(0)}(1^+, 2^+, \ldots, n^+)$ vanishes identically at tree level, a result of supersymmetric Ward identities and the structure of pure gauge theory (Gehrmann et al., 2015, Mogull, 2017). The first nonzero contribution arises at one-loop, where the amplitude is finite (after UV renormalization) and purely rational in the spinor variables: $$A_n^{(1)}(1^+,2^+,\ldots,n^+) = -\frac{i}{3}\frac{1}{\langle12\rangle\langle23\rangle\cdots\langle n1\rangle}\sum_{1\leq i<j<k<l\leq n} \text{tr}_-[ijkl] \, + O(\epsilon)$$ with $$\text{tr}_-[ijkl]=\langle ij\rangle[jk]\langle kl\rangle[li]$$. At two loops, the amplitude is composed of overlapping infrared (IR) and ultraviolet (UV) singularities (which are entirely determined by universal IR factorization structure, due to the vanishing of the tree) and a finite remainder $F^{(2)}$ which splits into a polylogarithmic (cut-constructible) piece $P^{(2)}$ and a rational term $R^{(2)}$ (Gehrmann et al., 2015, Dunbar et al., 2016, Dunbar et al., 2017, Dunbar et al., 2020): $$A_n^{(2)} = A_n^{(1)} I^{(1)} + F^{(2)} + O(\epsilon)\,,\qquad F^{(2)} = P^{(2)} + R^{(2)}$$ The explicit form for the IR structure $I^{(1)}$ is dictated by Catani's formula.

2. Analytic Results, Infrared Structure, and Finite Remainders

For the two-loop five-gluon all-plus amplitude, the analytic expression after infrared pole subtraction yields a remarkably simple result: the finite remainder is an explicit sum of dilogarithms (weight-two polylogarithms), with all anticipated weight one, three, and four terms canceling identically: $$F_5^{(2)} = \frac{5\pi^2}{12} F_5^{(1)} + \sum_{i=0}^4 \sigma^i \left\{ \frac{v_5\, \mathrm{tr}[(1-\gamma_5)\hat{p}_4\hat{p}_5\hat{p}_1\hat{p}_2]}{v_2 + v_3 - v_5} I_{23,5} + \frac{1}{6} \frac{\left(\mathrm{tr}[(1+\gamma_5)\hat{p}_4\hat{p}_5\hat{p}_1\hat{p}_2]\right)^2}{v_1 v_4} + \frac{10}{3}v_1 v_2 + \frac{2}{3}v_1 v_3 \right\}$$ with $I_{23,5}$ a specific dilogarithm function of the independent invariants [(Gehrmann et al., 2015), Eq. (3)-(4)]. Similar simplification—emergence of rational and dilogarithmic finite remainders—holds at six and seven points, with explicit formulas presented through cyclically symmetric combinations of spinor products (Dunbar et al., 2016, Dunbar et al., 2017).

The IR divergence structure is encoded by universal poles controlled by the factorization theorems: $$A_n^{(2)} = A_n^{(1)}\left[ -\sum_{i=1}^n \frac{1}{\epsilon^2} \left(\frac{\mu^2}{-s_{i,i+1}}\right)^\epsilon + \frac{n\pi^2}{12}\right] + F_n^{(2)} + O(\epsilon)$$ where $s_{i,i+1} = (p_i + p_{i+1})^2$.

3. Integrand and Integrand Reduction Techniques

Multi-loop integrand reduction and algebraic geometry methods are systematically applied to the all-plus sector (Badger et al., 2014, Abreu et al., 2018, Abreu et al., 2018). For two-loop five-point amplitudes, the integrand is decomposed into master integrand topologies, each parametrized by a basis of irreducible scalar products (ISPs). The residue matching and reduction to master integrals is accomplished by solving a system of linear equations arising from unitarity cuts, often using finite field arithmetic for efficiency and exactness.

D-dimensional integrand reduction employs polynomial division by Grӧbner bases and primary decomposition to organize numerator structures and automate the extraction of integrand coefficients, with special adaptation to nonplanar topologies appearing in the all-plus configuration (Badger et al., 2014).

The local integrand representation further simplifies the structure: by constructing loop integrands with manifestly local numerator insertions and eliminating spurious singularities (non-physical poles), both the analytic and numerical properties of the amplitude are improved (Mogull, 2017). Infrared structure is rendered transparent on the integrand, often isolating all divergence into triangle (and lower) topologies.

4. Rational Terms, Recursion, and Singularities

The rational terms, particularly at $n \geq 6$, involve intricate singularity structures, including both simple and double poles in spinor variables. Recursive techniques—specifically augmented BCFW and Risager shifts—are adapted to capture contributions with double poles (Dunbar et al., 2016, Dunbar et al., 2017). These are necessary because, in contrast to tree and one-loop amplitudes, standard recursion does not always sufficiently suppress large-$z$ behavior; a three-line shift is required to expose all poles.

Residues are constructed using on-shell recursion supplemented with off-shell information—aided by axial gauge and nullified momentum prescription—enabling the analytic derivation of terms omitted by four-dimensional unitarity cuts. Leading and sub-leading residues at double poles are systematically isolated, and the full amplitude is reconstructed respecting factorization, cyclic, and reflection symmetries.

The analytic structure of these rational terms is benchmarked through their expected singularities: explicit factorization in multi-particle and collinear limits is verified, and care is taken to remove unphysical coplanar singularities which arise from the analytic continuation of momentum twistors and complex momenta.

5. Color Structure, Symmetries, and Subleading-Traces

The color decomposition at two loops is nontrivial: while the leading-color part is dominant in typical collider observables, subleading single-trace $N_c^0$ color structures can be computed and are expressible in explicit all-$n$ formulas (Dunbar et al., 2020). These partial amplitudes have their origins in string theory as contributions from non-planar two-loop surfaces with a single boundary. The rational part for the single-trace all-plus amplitude admits closed-form expressions in terms of Parke-Taylor factors and spinor contractions, and collinear/decoupling symmetries constrain their functional form and redundancy.

Moreover, connections to $\mathcal{N}=4$ supersymmetric Yang-Mills theory are both direct and quantitative: through dimension-shifting relations in the integrand and color-kinematics duality construction (Mogull, 2017), much of the all-plus multiloop structure inherits properties from maximally supersymmetric gauge theory. At the integrand level, the two-loop all-plus amplitude can be related to the MHV amplitude in $\mathcal{N}=4$ SYM by the replacement $\delta^8(Q) \to (D_s-2)\,\mu^4$ or $F_1(\mu_1,\mu_2)$, with $D_s$ the state-space dimension and $\mu_i$ parametrizing extra-dimensional momenta. This relation extends to integrand representations obeying Jacobi (kinematic) identities.

6. Modern Computational Techniques and Practical Implementations

Practical evaluation of two-loop all-plus amplitudes relies on several algorithmic and technological advances:

• Master integral reduction is performed using integration-by-parts (IBP) identities, implemented via Laporta algorithms, potentially with further optimization using polynomial and syzygy techniques, and is now routinely verified via finite field sampling and rational function reconstruction (Abreu et al., 2018, Abreu et al., 2018, Moodie, 2023).
• Amplitudes are efficiently implemented in publicly available C++ libraries (such as NJet and PentagonFunctions++), employing tiered floating-point precision and numerically optimized pentagon-function evaluations. Reconstruction of rational coefficients is achieved via finite field arithmetic, linear relations among coefficients, denominator structure matching to the pentagon function alphabet, and univariate partial fraction decomposition (Moodie, 2023).
• Stability and performance of these implementations allow precise Monte Carlo integration over physical phase space, with numerically robust behavior even in deep infrared regions.
• Extensions to processes with massive fermions (e.g., $gg \to t\bar{t}$) require master integrals that involve elliptic functions; these are handled via rationalized kinematics, functional reduction to iterated integrals over elliptic curves, and analytic continuation strategies adapted to the resulting singularity structures (Badger et al., 2021, Chaubey, 2021).

7. Significance, Applications, and Further Developments

Two-loop all-plus helicity QCD amplitudes furnish essential ingredients for precision Standard Model predictions, including NNLO and N${}^3$LO corrections in collider processes (such as three-jet, $Z$-boson, $Z\gamma$, and diphoton-plus-jet production). Their compact analytic forms—especially the simplification to dilogarithms in the finite remainders—enable more efficient phenomenological predictions and serve as benchmarks for amplitude reconstruction techniques.

The exposure and removal of spurious singularities, realization of color-kinematics duality in the non-supersymmetric sector, and connections to twistorial and chiral algebra bootstrap approaches (which can reduce genuine two-loop computations to lower-loop ingredients via correlator methods and symmetry constraints) indicate a deepening understanding of gauge theory amplitudes. These structures not only enable novel analytic calculations but also hint at universal underpinnings spreading across QCD, supersymmetric Yang-Mills, and (symbolically) gravity via the double-copy construction.

Ongoing developments include extension to higher multiplicity, more general helicity configurations, full color decomposition at two loops and beyond, analytic continuation to all physical regions using momentum twistor and symbol techniques, and applications to processes at the energy frontier—demonstrating the central role of all-plus two-loop amplitudes in the advancing precision era of collider phenomenology.