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Automation of amplitude-level QED resummation with ACRAIC

The paper introduces ACRAIC, an automated tool within the Sherpa framework that implements coherent amplitude-level QED resummation for general scattering processes, enabling precise $\mathcal{O}(\alpha^2)$ predictions with significantly reduced perturbative uncertainties for future $e^+e^-$ colliders.

Original authors: Alan Price

Published 2026-10-21

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CC BY 4.0

Original authors: Alan Price

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-stakes world of particle physics, scientists are building machines to collide electrons and positrons with such precision that they can measure the properties of fundamental particles down to the smallest possible fraction. To do this, they must predict exactly what happens when these particles smash together, accounting for every tiny interaction. One of the most persistent challenges in these predictions is the behavior of light. When charged particles accelerate or change direction during a collision, they inevitably emit photons, the particles of light. Sometimes they emit just one, but often they emit many, creating a complex cloud of radiation that distorts the measurement. If scientists ignore this cloud, their predictions drift wildly from reality. To fix this, they use a mathematical framework that treats these emissions not as a chaotic mess, but as a predictable pattern that can be calculated order by order, ensuring that the theoretical models match the experimental data with extreme accuracy.

For decades, the most successful method for handling this light emission has been a technique that works by squaring the mathematical description of the collision. This approach, while powerful, has a significant limitation: it treats the emissions as if they happen independently, losing the subtle quantum interference that occurs when multiple photons are emitted simultaneously. This loss of information becomes a critical problem when scientists try to predict collisions involving many particles or when they need to reach a level of precision that leaves no room for error. A new tool, developed by Alan Price at the Jagiellonian University and integrated into the SHERPA simulation software, solves this problem by moving the calculation to a deeper level. Instead of squaring the numbers first, this new method, called ACRAIC, performs the complex subtraction of light emissions directly on the quantum amplitudes themselves. This preserves the delicate interference between emissions, allowing for a more stable and accurate prediction, especially in scenarios where the older methods begin to falter.

The core of this achievement lies in how the software organizes the chaos of photon emissions. In the past, scientists had to manually code the rules for how photons attach to specific particles, a process that worked well for simple collisions but became unmanageable for complex ones. ACRAIC automates this entire process. It breaks the collision down into distinct stages of emission, grouping the charged particles into neutral sets that act as coherent sources of light. By ensuring that each stage is electrically neutral, the software maintains the fundamental laws of physics while calculating the probability of every possible way a photon could be emitted. This allows the system to handle any number of final particles, from simple pairs to complex showers of four or more leptons, without needing a new manual code for every single scenario. The result is a system that can generate predictions for general scattering processes automatically, a capability that was previously unavailable for such a broad range of events.

The researchers tested this new tool against existing, highly trusted simulations to ensure it was working correctly. They compared their results for the production of muon pairs at the energy levels expected for future colliders, specifically at 240 GeV and at the mass of the Z boson. The new method agreed with the established results to within a few parts in a thousand, confirming its accuracy. However, the true value of ACRAIC emerged when they looked at more difficult cases. In simulations involving the production of four leptons, the older method produced a small but significant number of "negative weights," a mathematical artifact that occurs when the calculation subtracts more than it adds, leading to statistical noise and inefficiency. In these difficult scenarios, the older method produced negative weights in about 0.3% of events, a fraction that grows with the complexity of the collision. ACRAIC, by contrast, produced zero negative weights at the first order of correction, making it far more efficient for generating the massive datasets required for future experiments.

Beyond simply fixing a calculation, the new tool provides a clearer picture of the uncertainty in these predictions. The researchers used their method to estimate how much their results might change if they included even higher levels of precision that are currently too difficult to calculate. For the production of muon pairs at the Z boson mass, they found that the uncertainty from missing higher-order effects is roughly 0.2% to 0.4%. When they applied the same machinery to the production of a Higgs boson at 240 GeV, the uncertainty was similarly small, around 0.1% to 0.3%. These estimates are crucial because they tell experimentalists how much they can trust their theoretical models. The study also looked at the "recoil mass" of the Higgs boson, a key measurement for future colliders like FCC-ee. They found that the quantum effects of light emission shift the measured mass by only a tiny fraction of a MeV, well within the target precision for measuring the Higgs width. This suggests that the new method can reliably support the next generation of precision measurements.

The paper also highlights where the method still needs work. While it handles the emission of light perfectly, it does not yet include the effects of light-fermion pairs, which are small corrections that must be added from other sources. Furthermore, for the most complex collisions involving four leptons, the researchers encountered some numerical instabilities when trying to push the calculation to the second order of precision, a challenge they plan to address in future work. Despite these remaining hurdles, the tool represents a significant step forward in the automation of high-precision physics. It transforms a process that once required hand-coding for every specific reaction into a general, automated system that can handle the full complexity of future collider experiments. By preserving the quantum interference of light emissions and eliminating the statistical noise of negative weights, ACRAIC provides a robust foundation for the next era of discovery, ensuring that the theoretical predictions are as sharp and reliable as the machines built to test them.

Technical Summary: Automation of Amplitude-Level QED Resummation with ACRAIC

Problem Statement
Precision electroweak predictions for future $e^+e^-$ colliders require theoretical control over QED radiation that matches experimental systematics, necessitating the modeling of multiple-photon emissions and the exponentiation of associated logarithms to the per-mil level. While the Yennie–Frautschi–Suura (YFS) theorem provides a framework for all-order QED by exponentiating infrared (IR) singularities, existing implementations face limitations. The Coherent Exclusive Exponentiation (CEEX) approach, which preserves interference between initial-state radiation (ISR) and final-state radiation (FSR) at the amplitude level, has historically been restricted to fermion-pair production ( $e^+e^- \to f\bar{f}$ ) with hand-coded amplitudes. General-purpose event generators typically rely on squared-level exponentiation (EEX) or universal QED parton showers, which either lose amplitude-level coherence beyond the first emission or lack the precision of fixed-order electroweak corrections. There is a specific need for an automated, amplitude-level resummation scheme that remains stable at high multiplicities, where squared-level matching can become unstable.

Methodology: The ACRAIC Algorithm
The paper presents ACRAIC ("A Coherent Resummation of Amplitudes In C++"), an automated realization of amplitude-level YFS exponentiation integrated into the SHERPA event generator. The method generalizes CEEX to arbitrary scattering processes through three core innovations:

Emission Stages and Partitions: Instead of assigning photons strictly to initial or final states, ACRAIC assigns them to "emission stages" ( $L_g$ ). These stages are sets of charged legs separated by resonant internal propagators. A critical requirement is that each stage must be charge-neutral to ensure gauge invariance of the soft factor terms. This is achieved by deriving stages from the radiating dipoles of the process, treating resonances (like $Z/\gamma^*$ or $W$ pairs) as legs in both their production and decay stages.
Partition-Dependent Propagator Shifts: The coherent sum over photon assignments is constructed by shifting the momenta of internal propagators based on the partition of photons to stages. For a partition $\wp$ , the Born amplitude is evaluated with propagator momenta shifted as if the assigned photons were emitted from that specific stage's legs. This allows the use of standard tree-level helicity amplitudes (computed via the Berends–Giele recursion in COMIX) with modified kinematics to represent the soft limits of specific assignments.
Remainders and Virtual Corrections: The IR-finite remainders ( $\hat{\beta}$ ) are tree-level helicity amplitudes with $N+2$ or $N+3$ legs (for one or two real photons) evaluated with these partition-dependent shifts. The one-loop virtual corrections enter as a per-helicity factor applied to the Born-level coherent amplitude. This factor is derived from the ratio of the loop provider's one-loop amplitude to the tree amplitude, adjusted for the virtual form factor. The method also implements a "massification" technique, allowing massless loop providers to approximate massive amplitudes by applying universal mass-dependent factors, thereby avoiding numerical instabilities associated with small lepton masses.

Key Contributions

Automation: ACRAIC automates the amplitude-level YFS exponentiation for general Standard Model processes, removing the need for hand-coded amplitudes. It utilizes the COMIX recursion for tree-level amplitudes and interfaces with loop providers (OPENLOOPS, RECOLA) for virtual corrections.
Stability at High Multiplicity: The paper demonstrates that the amplitude-level approach remains stable for processes with high final-state multiplicity (e.g., four-lepton production), whereas squared-level matching can suffer from large negative weights and statistical variance due to cancellations.
Positive Weights: At $\mathcal{O}(\alpha)$ , ACRAIC generates strictly positive event weights, unlike squared-level approaches (e.g., YFSNLOEW) which can produce negative weights (approx. 0.3% for $e^+e^- \to \mu^+\mu^-$ at 240 GeV). This improves unweighting efficiency and simplifies detector simulation.
Theoretical Uncertainty Estimation: The framework introduces a method for estimating missing higher-order uncertainties using theory nuisance parameters, constructing bands based on rescaling specific classes of amplitudes (e.g., $\hat{\beta}_0$ , $\hat{\beta}_1$ , $\hat{\beta}_2$ ).

Results and Validation
The implementation was validated against existing codes and theoretical expectations:

Comparison with KKMC: For $e^+e^- \to \mu^+\mu^-$ at photonic $\mathcal{O}(\alpha^2)$ , ACRAIC agrees with the dedicated CEEX code KKMC within $0.5 \times 10^{-3}$ across the Z pole, 161 GeV, and 250 GeV. Differences arise primarily from real-virtual contributions included in ACRAIC but not in the specific KKMC $\mathcal{O}(\alpha^2)$ realization.
Comparison with YFSNLOEW: Across various final states at the Z pole and 240 GeV, ACRAIC and YFSNLOEW agree to within 0.4% at the Z pole. Discrepancies are attributed to differences in how the virtual correction is applied (to the whole $\mathcal{O}(\alpha)$ amplitude in ACRAIC vs. the Born in YFSNLOEW) and the treatment of resonant initial-final logarithms.
Four-Lepton Production: In $e^+e^- \to 4\ell$ at the $W$ -pair threshold (161 GeV), ACRAIC avoids the large statistical tails and negative weights observed in squared-level approaches, providing stable cross-section predictions with statistical errors of 0.3–0.8%.
Perturbative Uncertainty: At $\mathcal{O}(\alpha^2)$ , the estimated perturbative uncertainty shrinks by about an order of magnitude relative to $\mathcal{O}(\alpha)$ . For the Z-pole line shape, the $\mathcal{O}(\alpha^2)$ band is approximately $10^{-4}$ , significantly smaller than the $\mathcal{O}(\alpha)$ band ( $\sim 10^{-3}$ ).

Significance and Claims
The paper claims that ACRAIC provides a robust, automated framework for precision QED resummation essential for future lepton colliders like FCC-ee. By moving the resummation to the amplitude level, it preserves the full interference structure of photon emissions, which is critical for observables sensitive to initial-final interference. The method is particularly significant for its ability to handle complex, high-multiplicity final states without the numerical instabilities inherent in squared-level matching. The author positions ACRAIC as a necessary tool for achieving the per-mil precision required for interpreting data from future colliders, specifically highlighting its application to the Higgs recoil mass measurement and Z-pole precision distributions. The work is presented as a technical realization of a general scattering process resummation, with specific phenomenological applications serving as illustrations of the machinery's capabilities rather than exhaustive phenomenological studies.

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