⚛️ phenomenology

Improving predictions for the $pp \to t\bar{t}W^+$ process at the LHC with the MINLO method

This paper compares standard NLO and MiNLO approaches for the full off-shell $pp \to t\bar{t}W^+ + X$ process at the LHC, demonstrating how the MiNLO method's dynamic scale setting and Sudakov form factors improve predictions, particularly for $t\bar{t}W^+j$ production and merged multi-jet modeling.

Original authors: Nikolaos Dimitrakopoulos

Published 2026-09-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Nikolaos Dimitrakopoulos

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

Inside the massive, ring-shaped tunnels of the Large Hadron Collider, scientists smash protons together at speeds approaching that of light. These collisions recreate the intense energy conditions that existed just moments after the universe began, allowing researchers to observe particles that are usually too heavy or unstable to exist on their own. Among the most significant of these particles is the top quark, the heaviest known elementary particle, which decays almost instantly into other particles. When a top quark is produced alongside its antimatter twin and a W boson—a carrier of the weak nuclear force—the resulting event is a complex, chaotic burst of energy. Physicists study this specific combination of particles, known as top-antitop-W production, because it serves as a precise test of the Standard Model, the theory that describes how fundamental particles interact. Any small deviation between what theory predicts and what detectors actually see could hint at new, undiscovered laws of physics. However, predicting exactly how these particles behave is incredibly difficult because the calculations involve countless possibilities for how the particles might scatter and emit additional jets of energy.

To tackle this challenge, a researcher led by Nikolaos Dimitrakopoulos at RWTH Aachen University in Germany has refined a sophisticated method for simulating these collisions. The researcher focused on a process where a top quark, an antitop quark, and a W boson are produced together, along with any number of additional jets of particles. In the past, scientists relied on standard calculation methods that required them to manually choose specific values for the energy scales used in their equations. This manual choice introduced a degree of uncertainty, much like trying to measure a room with a ruler that you are allowed to stretch or shrink slightly depending on your preference. The researcher compared these traditional methods with a newer approach called MiNLO, which automatically determines the most appropriate energy scales based on the specific details of each simulated collision. Instead of relying on a fixed, pre-set value, the MiNLO method looks at the actual motion and energy of the particles in the event and adjusts its calculations dynamically. This allows the method to account for complex effects, such as the suppression of unlikely particle configurations, which helps to smooth out the jagged edges often found in standard predictions.

The study, presented at the DIS2026 workshop in Bologna, reveals that when the researcher uses a flexible, dynamic way to set their energy scales, the new MiNLO method and the traditional standard method produce nearly identical results. Both approaches predict the total number of these events with a high degree of agreement, differing by only about three percent, a margin that is well within the expected range of uncertainty. However, the real advantage of the MiNLO method becomes clear when scientists are forced to use a fixed, unchanging energy scale. In these scenarios, the traditional method often produces large, unpredictable swings in its predictions, particularly when looking at the high-energy tails of the data distributions. The MiNLO method, by contrast, significantly reduces these swings. For instance, when analyzing the momentum of the hardest jet produced in the collision, the uncertainty bands in the MiNLO predictions were nearly half the size of those from the standard method when a fixed scale was used. This happens because the MiNLO algorithm naturally adapts to the event's geometry, effectively removing the problematic scale choices that cause the standard method to stumble.

Beyond improving single-event predictions, the researcher also explored how to combine data from collisions with different numbers of jets to create a more complete picture of the physical process. They developed a merging procedure that stitches together simulations of events with zero, one, and two additional jets, ensuring that no part of the data is double-counted or missed. By carefully selecting the threshold at which these different samples are combined, they found that the overall uncertainty in the final prediction could be reduced. The study showed that merging samples up to two jets provided a more realistic description of the full collision environment, although it required careful handling to maintain accuracy. The results indicate that while the new method does not overturn existing theories, it provides a much sharper tool for measuring them. By reducing the ambiguity associated with how calculations are set up, the MiNLO approach allows physicists to trust their theoretical models more deeply, ensuring that any future discrepancies they find with experimental data are truly signs of new physics rather than just artifacts of the calculation method.

Technical Summary: Improving predictions for the $pp \to t\bar{t}W^+$ process at the LHC with the MiNLO method

Problem Statement
The production of a top-antitop pair in association with an electroweak $W^\pm$ boson ( $pp \to t\bar{t}W^\pm$ ) is a critical process for Standard Model measurements and new physics searches at the LHC, particularly in channels involving same-sign leptons. While recent Next-to-Leading Order (NLO) QCD calculations and even NNLO results for on-shell production exist, a realistic description of the fiducial phase space requires full off-shell predictions that account for decay products and additional jet activity. Previous studies have shown that the fiducial cross-section for $pp \to t\bar{t}W^+ j$ constitutes a substantial fraction of the inclusive $t\bar{t}W^+$ cross-section, and significant distortions appear in differential distributions. However, theoretical predictions for these processes suffer from ambiguities associated with the choice of renormalization ( $\mu_R$ ) and factorization ( $\mu_F$ ) scales, particularly when fixed scale choices are employed. These scale ambiguities can lead to large theoretical uncertainties, especially in the tails of differential distributions.

Methodology
This work presents a comparison between the standard NLO approach and the Multi-scale Improved NLO (MiNLO) method for the full off-shell process $pp \to t\bar{t}W^+ + X$ in the multilepton channel ( $pp \to e^+\nu_e \mu^-\bar{\nu}_\mu \tau^+\nu_\tau b\bar{b}$ ). The study utilizes the Helac-Nlo framework, where the MiNLO method has been recently implemented.

Key methodological features include:

Dynamic Scale Determination: Unlike standard NLO where scales are user-defined, MiNLO dynamically extracts $\mu_R$ and $\mu_F$ from event kinematics. This is achieved by identifying a core system (the decay products of the $t\bar{t}W^+$ system) and using an inverse- $k_T$ clustering algorithm to reconstruct the most probable branching history.
Scale Definitions: The core scale ( $q_{core}$ ) is associated with the hard process, while nodal scales ( $q_k$ ) are derived from the $k_T$ measures of reconstructed branchings. The study employs two scale choices for comparison: a fixed scale ( $\mu_0 = m_t + m_W/2$ ) and a dynamical scale ( $\mu_0 = E_T/2$ ), where $E_T$ is defined by the transverse energies of the top quarks, $W$ boson, and the hardest light jet.
Sudakov Form Factors: The method incorporates Sudakov form factors to resum large logarithms arising from scale hierarchies. A specific modification in this implementation requires probabilistic Sudakov form factors that do not exceed unity, and consistent variation of incoming line factors to match PDF evolution.
Merging: To improve modeling, the author discusses merging samples with different jet multiplicities (up to one and two jets) using a merging-scale parameter ( $p_{T, merging}$ ). Due to the complexity of the full off-shell phase space, a "Direct sum" approach is used alongside MiNLO, as the standard MiNLO merging requires process-dependent modifications to Sudakov factors that are not fully known for this specific process.

Key Results
The comparison between standard NLO and MiNLO yields the following findings:

Integrated Cross Sections: For the full off-shell $pp \to t\bar{t}W^+ j + X$ process, the integrated fiducial cross sections calculated via NLO and MiNLO differ by only approximately 3%, a discrepancy fully covered by scale uncertainties.
Scale Uncertainties:
- With the dynamical scale choice ( $\mu_0 = E_T/2$ ), MiNLO and standard NLO behave similarly, producing theory uncertainties of comparable size (approximately 10% for both).
- With the fixed scale choice ( $\mu_0 = m_t + m_W/2$ ), MiNLO significantly reduces theoretical uncertainties compared to standard NLO. At the integrated level, uncertainties drop from 14% (NLO) to 11% (MiNLO). In differential distributions (specifically the transverse momentum of the hardest light jet, $p_{T, j1}$ ), the uncertainty bands in the high- $p_T$ tail are reduced by nearly half compared to standard NLO. This improvement is attributed to the MiNLO procedure forcing the core scale to equal the nodal scale ( $q_{core} = q_1$ ) in cases of unordered clustering, thereby eliminating problematic fixed scale choices in that regime.
Differential Observables: The reduction in uncertainty is most pronounced for observables involving light jets. Dimensionless observables or those not involving light jets (e.g., purely leptonic) show less improvement, as the MiNLO procedure primarily affects jet dynamics.
Multi-jet Merging: Merging samples up to two jets (using a LO $t\bar{t}W^+ jj$ sample) with appropriate merging scales ( $p_{T, merging} > 55$ GeV) results in predictions with reduced uncertainties compared to the inclusive $t\bar{t}W^+$ cross-section. While merging up to two jets initially shows larger uncertainties for small merging scales, the procedure offers a path to describe the second leading jet with NLO accuracy (if an NLO $t\bar{t}W^+ jj$ sample were included), potentially exceeding formal NNLO accuracy.

Significance and Claims
The paper claims that the MiNLO method, now available within the Helac-Nlo framework for full off-shell processes, offers a robust alternative to standard NLO calculations. Its primary significance lies in its ability to dynamically determine scales, which substantially reduces theoretical uncertainties associated with fixed scale choices, particularly in the tails of differential distributions where standard NLO predictions often struggle. While the method does not drastically alter results when well-behaved dynamical scales are already used, it provides a more stable framework for fixed-scale analyses. Furthermore, the study demonstrates that merging samples with different jet multiplicities using MiNLO improves the overall modeling of the $pp \to t\bar{t}W^+ + X$ process, providing a more complete description of the fiducial phase space with reduced theoretical uncertainties. The author concludes that these improvements are essential for drawing precise conclusions regarding tensions between theory and experimental data in $t\bar{t}W^\pm$ production.

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