⚛️ phenomenology

Helicity amplitudes in massless QED to higher orders in the dimensional regulator

This paper presents an analytical calculation of one- and two-loop helicity amplitudes for four-fermion and Compton scattering in massless QED using a four-dimensional tensor decomposition and an efficient integrand-level algorithm, providing results up to transcendental weight six to support N³LO theoretical predictions.

Original authors: Thomas Dave, William J. Torres Bobadilla

Published 2026-09-16

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Dave, William J. Torres Bobadilla

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the high-energy world of particle physics, scientists act as cosmic detectives, smashing particles together to understand the fundamental rules that govern our universe. When these particles collide, they scatter in complex patterns, and the likelihood of any specific outcome is described by a mathematical object called a scattering amplitude. Think of this amplitude as a precise probability map that tells physicists exactly how often particles will bounce off one another in specific ways. To make these maps accurate enough for modern experiments, researchers must calculate them with extreme precision, often accounting for invisible, fleeting particles that pop in and out of existence during the collision. These calculations are notoriously difficult, requiring the summation of countless possibilities that grow exponentially in complexity as the precision increases. For decades, the focus has been on the strong nuclear force, which binds atomic nuclei together, but the electromagnetic force, which governs how light and charged particles interact, has received less attention at these highest levels of precision, leaving a gap in our theoretical understanding.

A team of researchers at the University of Liverpool has now filled this gap by performing a rigorous, high-precision calculation of how massless electrons and muons scatter off one another, as well as how they produce light, within the framework of quantum electrodynamics. They focused on processes that are essential for reaching the next major milestone in theoretical accuracy, known as the third next-to-leading order. To achieve this, the team developed a new, efficient method for organizing the thousands of intricate diagrams that represent these particle interactions. Instead of getting lost in the sheer volume of possibilities, they created a systematic way to group these diagrams into families, allowing them to solve the underlying mathematics much faster. By treating the particles as if they exist in a four-dimensional space while handling the mathematical complexities of their interactions in a slightly different, higher-dimensional space, they were able to strip away the noise and isolate the core physical signals.

The result is a complete set of analytical formulas that describe these scattering events with unprecedented detail. The researchers expressed their findings in terms of generalized polylogarithms, a sophisticated class of mathematical functions that can describe the subtle, layered dependencies of the particles' energies and angles. Their work covers four distinct scenarios: an electron and a positron turning into a muon and an antimuon, an electron and a muon scattering off each other, an electron and a positron scattering off each other, and an electron and a positron annihilating to produce two photons. For each of these, they calculated the probabilities not just for the simplest interaction, but also for the more complex cases involving one and two loops of virtual particles. These calculations are crucial because they provide the theoretical baseline needed to interpret data from future particle colliders, ensuring that any new discoveries are not mistaken for errors in our current understanding of the electromagnetic force.

To verify their work, the team compared their results against similar calculations made for the strong nuclear force, a field where such high-precision work has already been done. By carefully extracting the parts of the strong-force calculations that behave like the electromagnetic force, they confirmed that their new formulas matched perfectly. This cross-check was vital, as it proved that their new method for organizing the diagrams and their specific mathematical approach were correct. The study also clarified how different mathematical techniques used to handle infinities in the calculations affect the final numbers, showing that while the intermediate steps might look different depending on the method, the final physical predictions remain consistent. This consistency gives physicists confidence that the theoretical tools they are building are robust and ready for the next generation of experiments.

The significance of this work extends beyond just these specific particle collisions. The method the authors developed to group Feynman diagrams—using matrix transformations to identify shared structures—offers a powerful new tool for tackling even more complex problems, including those involving massive particles or three-loop calculations. By demonstrating that these high-order calculations can be done analytically and efficiently for massless quantum electrodynamics, the team has paved the way for similar breakthroughs in other areas of physics. Their results are now available as a resource for the global community, providing the precise ingredients needed to push the boundaries of what we know about the electromagnetic interaction and to prepare for the next leap in our understanding of the subatomic world.

Technical Summary: Helicity Amplitudes in Massless QED to Higher Orders in the Dimensional Regulator

Problem Statement
Precision physics at particle colliders increasingly demands theoretical predictions at higher perturbative orders, specifically aiming for Next-to-Next-to-Leading Order (NNLO) and Next-to-Next-to-Next-to-Leading Order (N3LO). While significant progress has been made in Quantum Chromodynamics (QCD) for multi-loop scattering amplitudes, analogous calculations in Quantum Electrodynamics (QED) have received less attention. Existing massless QED results were largely limited to two-loop calculations performed long ago, with no extensions to higher orders in the dimensional regulator $\epsilon = (4-D)/2$ required for N3LO predictions. Furthermore, recent efforts have focused on massive lepton cases, which introduce additional complexity. This work addresses the gap by analytically calculating one- and two-loop helicity amplitudes for massless QED processes to higher orders in $\epsilon$ , providing the necessary ingredients for N3LO theoretical predictions.

Methodology
The authors employ a diagrammatic approach combined with a four-dimensional tensor decomposition scheme to compute scattering amplitudes for four processes: $e^+e^- \to \mu^+\mu^-$ , $e^+\mu^- \to e^+\mu^-$ , $e^+e^- \to e^+e^-$ , and $e^+e^- \to \gamma\gamma$ .

Tensor Decomposition and Regularization:
The authors adopt the 't Hooft-Veltman dimensional regularization scheme, where internal loop momenta are $D$ -dimensional while external states are treated as four-dimensional. By decomposing amplitudes into form factors multiplied by tensor structures, they reduce the number of independent tensor structures to the number of independent helicity states, regardless of the loop order. This allows the complex spinor and polarization algebra to be encapsulated in four-dimensional tensor structures, while the scalar form factors are computed in $D$ dimensions.
Integrand-Level Grouping Algorithm:
To manage the complexity of multi-loop diagrams, the authors propose a novel algorithm to group Feynman diagrams into integral families. Instead of checking all possible loop momentum shifts (which scales factorially), they represent propagators as matrices. By applying linear algebra transformations (shift matrices) to these representations, they identify momentum shifts that align the propagators of sub-diagrams with parent topologies. This method significantly reduces the computational burden of identifying integral families.
Reduction to Master Integrals:
Using the Laporta algorithm (via LiteRed and FiniteFlow), the form factors are reduced to a minimal basis of Master Integrals (MIs). The authors select a canonical basis of MIs that admit a $d\log$ representation, ensuring they satisfy a system of differential equations in canonical form. These integrals are solved analytically up to transcendental weight six, expressed in terms of Generalized Polylogarithms (GPLs) of a dimensionless variable $x = -t/s$ .
Renormalization and Subtraction:
Ultraviolet (UV) divergences are removed via renormalization in the $\overline{\text{MS}}$ scheme. Infrared (IR) divergences are subtracted using the Soft-Collinear Effective Theory (SCET) formalism, adapted for the Abelian nature of QED by "abelianizing" the QCD subtraction operators (setting $C_A \to 0$ and $C_F \to 1$ ). This yields finite remainders and higher-order terms in $\epsilon$ .

Key Contributions

Analytic Calculation: The paper provides the first analytic calculation of one- and two-loop helicity amplitudes for massless QED processes to higher orders in $\epsilon$ , extending previous results that were limited to lower orders or massive cases.
Algorithmic Innovation: A matrix-based method for grouping Feynman diagrams into integral families is introduced, offering an efficient alternative to traditional momentum-shift enumeration.
Explicit Results: The authors present explicit expressions for the non-vanishing helicity amplitudes of the four processes mentioned above. These are expressed in terms of form factors constructed from Generalized Polylogarithms up to transcendental weight six.
Cross-Process Relations: The work demonstrates how amplitudes for $e^+\mu^- \to e^+\mu^-$ and $e^+e^- \to e^+e^-$ can be derived from the $e^+e^- \to \mu^+\mu^-$ results via specific momentum exchanges and sign changes, streamlining the calculation of related processes.

Results
The authors successfully computed the bare form factors and the subsequent finite remainders for the specified processes.

For four-fermion scattering ( $e^+e^- \to \mu^+\mu^-$ , etc.), the amplitudes are organized into terms proportional to $N_f^0$ , $N_f^1$ , and $N_f^2$ (where $N_f$ is the number of fermion flavors).
For $e^+e^- \to \gamma\gamma$ , the structure involves terms proportional to $N_f^0$ and $N_f^1$ .
The results are presented in terms of Generalized Polylogarithms, with the alphabet reduced to $\{x, 1-x\}$ in the production region.
The paper includes ancillary files containing the full analytic expressions for the form factors and helicity amplitudes.

Significance and Validation
The paper claims that these results are crucial for enabling theoretical predictions at N3LO in QED, a regime where such calculations were previously unavailable. To validate their findings, the authors performed extensive cross-checks against known QCD results (specifically $q\bar{q} \to Q\bar{Q}$ , $q\bar{Q} \to q\bar{Q}$ , $q\bar{q} \to q\bar{q}$ , and $q\bar{q} \to gg$ ). By isolating the Abelian contributions within the QCD amplitudes, they demonstrated complete agreement with QCD results at one and two loops for the finite remainders and higher orders in $\epsilon$ .

The authors note a subtlety regarding the dimensional regularization scheme: while the finite remainders agree perfectly, the bare helicity amplitudes may differ from those in certain QCD references due to scheme-dependent terms in the tensor decomposition. However, by analyzing the iterative IR structure (via the logarithm of the amplitude), they confirm that these discrepancies are resolved and consistent with the expected universal IR behavior.

Conclusion
This work establishes a robust framework for high-precision QED calculations, providing the necessary analytic tools and explicit results for N3LO predictions. The proposed matrix-based grouping algorithm and the four-dimensional tensor decomposition strategy are presented as generalizable techniques that can be applied to other processes, including those with internal massive particles or higher-loop orders (e.g., three-loop calculations).

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