Novel cluster-algebraic letters for 5- and 6-point QCD processes

1. Problem Statement

The calculation of scattering amplitudes in Quantum Chromodynamics (QCD), particularly at higher loops, relies heavily on understanding the "alphabet" of functions (letters) that appear in the symbol of these amplitudes. These letters are algebraic functions of kinematic variables (Mandelstam invariants) that serve as the arguments for transcendental functions (like polylogarithms).

• The Challenge: While the alphabet for massless processes in $\mathcal{N}=4$ Super Yang-Mills (SYM) theory is well-understood via cluster algebras, extending these predictions to QCD processes involving massive external legs (off-shell particles) has been difficult.
• Specific Gap: There was a lack of predictive tools for the alphabets of 6-point processes with one massive leg and 5-point processes with two massive legs in QCD. Existing direct calculation methods often struggle with complex algebraic structures, such as nested square roots, which have recently been observed in massive integrals but were not predicted by standard cluster algebra approaches for massless theories.

2. Methodology

The authors employ a strategy of breaking Dual Conformal Invariance (DCI) to map predictions from $\mathcal{N}=4$ SYM to QCD.

• Starting Point: They begin with the proposed cluster-algebraic alphabet for the 9-particle massless amplitude in planar $\mathcal{N}=4$ SYM theory (based on the $Gr(4,9)$ cluster algebra and its generalizations to handle infinite mutation sequences).
• Reduction to Massive DCI Kinematics: They reduce the 9-point massless alphabet to sub-alphabets describing $(9-k)$-point $k$-mass DCI kinematics. This is achieved by identifying kinematic configurations where certain external legs are formed by sums of massless momenta (e.g., $k_i = p_{i-1} + p_i$). In momentum twistor space, this corresponds to imposing invariance under specific shift symmetries (analogous to BCFW shifts) that render the function independent of specific dual coordinates.
• Breaking DCI to Lorentz Invariance (LI): The core innovation is the "breaking" of DCI. By taking a dual coordinate $x_\infty$ to infinity, the DCI cross-ratios reduce to standard Lorentz-invariant Mandelstam variables.
  • Mechanism: A DCI integral with $n$ points and $k$ massive legs (where the massive legs are adjacent) reduces to an LI integral with $(n-k-1)$ points and $(k-1)$ massive legs.
  • Application:
    • 7-point 2-mass DCI $\to$ 6-point 1-mass LI (QCD).
    • 6-point 3-mass DCI $\to$ 5-point 2-mass LI (QCD).
• Parameterization: The authors utilize momentum twistors to rationalize square roots and simplify the algebraic structure during the reduction process. They then map the resulting letters back to Mandelstam variables using ansatz-based fitting and finite field methods.

3. Key Contributions

1. First Prediction of 6-point 1-mass Letters: The paper provides the first cluster-algebraic candidate letters for planar 6-point Feynman integrals with one massive external leg and all other particles massless.
2. Discovery of Nested Square Roots: The authors discover that the alphabet for these processes contains nested square roots (square roots of expressions that themselves contain square roots).
   • Significance: Previously, nested roots were thought to be exclusive to integrals with massive internal propagators or elliptic sectors. This work proves they appear even in purely polylogarithmic integrals with massless propagators.
3. Validation against 2-loop Massless QCD: By taking the massless limit of their 6-point 1-mass alphabet, they recover the alphabet for 6-point massless QCD processes. They confirm that their prediction essentially contains the alphabet of all known 1-loop and finite 2-loop planar integrals for this kinematics, while also predicting 162 new letters likely to appear at higher loops.
4. 5-point 2-mass Predictions: They generate candidate letters for 5-point 2-mass processes (both "hard" and "easy" configurations) and compare them with existing 2-loop calculations, finding significant overlap and new predictions.

4. Detailed Results

A. 6-point 1-mass Processes

• Total Letters: The reduction yields 246 candidate letters.
  • 119 Rational.
  • 59 Rationalisable (simple square roots that become rational in momentum twistors).
  • 68 Non-rationalisable (containing complex square roots).
• Genuinely 6-point Letters: After removing letters belonging to lower-point sub-kinematics (like 5-point 2-mass), 29 genuinely new letters remain.
  • Nested Roots: 12 of these contain nested square roots of the form:
    $$l_i^\pm = \frac{A_i \pm B_i \epsilon_{2356} + C_i \sqrt{\Delta_\pm}}{A_i \pm B_i \epsilon_{2356} - C_i \sqrt{\Delta_\pm}}$$
    where $\Delta_\pm = F \pm \epsilon_{2356} G$. Here, $\epsilon_{2356}$ is a parity-odd square root, making $\Delta_\pm$ a nested structure.
• Physical Origin: These nested roots are associated with pentagon-triangle integral topologies.

B. 6-point Massless Processes (Limit)

• Taking the limit where the massive leg becomes massless ($p^2 \to 0$) and applying cyclic symmetry yields 374 letters.
• Comparison with Literature:
  • The prediction contains all 167 letters known to contribute to the finite part of 2-loop 6-point massless amplitudes (from refs [42, 43]).
  • It also contains 168 new letters (108 rational, 60 rationalisable) not found in current 2-loop results, suggesting they will appear at 3-loops or higher.
  • The authors note that alternative cluster-algebraic approaches (based on partial flag varieties) miss some of these new letters and specific $O(\epsilon)$ terms.

C. 5-point 2-mass Processes

• Two configurations are analyzed: "Easy" (non-adjacent massive legs) and "Hard" (adjacent massive legs).
• Hard Configuration: The union of two symmetry-breaking maps yields 103 letters (55 rational, 17 rationalisable, 31 non-rationalisable).
• Comparison: When compared to the 2-loop planar alphabet from [44], the authors find a non-trivial overlap.
  • They reproduce most known letters but miss some involving specific non-rationalisable roots ($r_2$) found in [44].
  • Conversely, they predict 495 new letters (arranged in 50 permutation orbits), including 87 new letters that serve as candidates for 3-loop calculations.

5. Significance and Outlook

• Theoretical Breakthrough: The discovery of nested square roots in massless propagator integrals challenges the assumption that such complexity only arises in elliptic or massive-internal-sector integrals. This necessitates updates to algorithms that determine alphabets directly from integrals, which often assume letters are rational or simple square roots.
• Practical Utility: The provided alphabets serve as essential input for the bootstrap program in QCD. By knowing the alphabet in advance, researchers can constrain the form of higher-loop amplitudes without performing the full, computationally expensive integration.
• Future Directions: The method is generalizable to $Gr(4, n)$ for $n > 9$, offering a path to predict alphabets for 6-point 2-mass and 5-point 3-mass processes. The authors suggest that combining their DCI-breaking approach with scattering diagram methods could provide a more robust framework for identifying "trivially recoverable" letters and positivity properties in general gauge theories.

In summary, this paper successfully bridges the gap between the highly symmetric $\mathcal{N}=4$ SYM theory and realistic QCD phenomenology, providing a powerful predictive tool for multi-loop calculations involving massive particles and revealing new algebraic structures (nested roots) in the process.