⚛️ phenomenology

Towards the inclusion of NLO EW corrections in the MiNLO method in Drell-Yan processes

This paper presents the first application of the MiNLO method to QED NLO corrections in Drell-Yan processes, specifically addressing initial-state radiation for Z boson decays into neutrinos, proposing a modified formula to handle QED-specific challenges, and quantifying uncertainties as a foundational step toward integrating full electroweak effects into the MiNNLOPS framework.

Original authors: Filippo Belloni, Mauro Chiesa, Carlo Oleari, Emanuele Re

Published 2026-02-19

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CC BY 4.0

Original authors: Filippo Belloni, Mauro Chiesa, Carlo Oleari, Emanuele Re

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

Imagine you are trying to predict the exact path of a billiard ball rolling across a table. In the world of particle physics, the "billiard balls" are subatomic particles like protons, and the "table" is the massive Large Hadron Collider (LHC).

For decades, physicists have been incredibly good at predicting how these particles behave when they crash into each other, thanks to a set of rules called QCD (Quantum Chromodynamics). Think of QCD as the "heavy gravity" of the particle world—it's strong, messy, and dominates the action.

However, there's a weaker force called Electroweak (specifically QED, or Quantum Electrodynamics), which is like a gentle breeze. For a long time, scientists ignored this breeze because the gravity (QCD) was so loud. But now, with the LHC running at super-high speeds, that gentle breeze is starting to push the billiard ball off its predicted path just enough to ruin our precision measurements.

The Problem:
We have a very sophisticated GPS for the "heavy gravity" (QCD) that can predict the ball's path perfectly, even when it's wobbling. But we don't have a GPS that can handle the "gentle breeze" (QED) while the ball is wobbling. Current tools either ignore the breeze or try to add it in a clumsy way that doesn't match the GPS data.

The Solution (The MiNLO Method):
The authors of this paper are trying to build a new, super-accurate GPS that handles both the heavy gravity and the gentle breeze simultaneously. They are adapting a famous existing method called MiNLO (which is great for QCD) to work for QED.

Here is how they did it, using some creative analogies:

1. The "Abelianization" (Changing the Rules)

In the QCD world, particles carry "color charges" (like red, blue, green). In the QED world, they carry "electric charges" (positive or negative).

The Analogy: Imagine the QCD method is a recipe for baking a spicy chili. The authors realized they can't just use the chili recipe to make a sweet cake (QED). They had to take the chili recipe and swap out the chili peppers for sugar, and the cumin for vanilla.
In the paper: They took the complex mathematical formulas for QCD and systematically replaced the "color factors" with "electric charge factors." This is called abelianization. It's like translating a book from a language with complex grammar (QCD) into a simpler language (QED) while keeping the story the same.

2. The "Sudakov Peak" (The Impossible Hill)

When particles emit radiation (like a photon), there is a mathematical "hump" or peak in the probability of where that radiation goes.

The Analogy: In the QCD world, this peak is like a mountain you can climb. It's at a reasonable height (a few GeV). But in the QED world, because the electric force is so much weaker, this "mountain" is actually a microscopic hill located at the bottom of an ocean. It's so low (trillions of times smaller than a proton) that if you try to measure it with a ruler, your ruler breaks before you get there.
The Problem: The computer code crashes because it tries to calculate numbers that are too small to exist in the real world.
The Fix: The authors realized they couldn't measure the bottom of the ocean. So, they drew a line (a "technical cutoff") a little higher up.
- Above the line: They use the standard, complex MiNLO GPS.
- Below the line: They use a clever mathematical trick. Since the area below the line is so small and smooth, they can just calculate the total area under the curve without worrying about the tiny details. It's like saying, "We don't need to count every grain of sand on the beach; we just know the beach is flat and small, so we can estimate the total sand easily."

3. The "Fake" Experiment

To prove their new GPS works, they needed to test it. But testing it with real physics (where the electric force is tiny) would take forever because the "breeze" is so weak, the errors would be invisible.

The Analogy: Imagine you are testing a new windmill design. To see if it works, you don't wait for a gentle breeze; you put it in a hurricane. If it survives the hurricane, you know it will work in a gentle breeze.
In the paper: They artificially made the electric force 5 times stronger than it actually is. This amplified the "breeze" so they could see if their math was breaking. They also created a fake set of "parton distribution functions" (which are like maps of where the particles are inside the proton) just for this test.
The Result: Even with the hurricane-force wind, their new GPS worked perfectly. The predictions matched the "true" answer to within 0.01%.

Why Does This Matter?

The Large Hadron Collider is now taking data at a level of precision where even the tiniest "breeze" matters. If we want to measure the mass of the W-boson (a fundamental particle) or test if the Standard Model of physics is perfect, we cannot ignore the QED effects.

This paper is the blueprint for the next generation of particle physics simulations. It proves that we can finally combine the "heavy gravity" (QCD) and the "gentle breeze" (QED) into one unified, ultra-precise prediction tool.

In summary: The authors took a complex tool designed for strong forces, rewrote its rules for weak forces, figured out how to handle a mathematical glitch where the numbers get too small, and proved it works by testing it in a "super-storm." This paves the way for the next era of precision physics at the LHC.

1. Problem Statement

The precision physics program at the High-Luminosity LHC (HL-LHC) requires theoretical predictions that combine high-order Quantum Chromodynamics (QCD) and Electroweak (EW) corrections. While Next-to-Leading Order (NLO) QCD corrections matched to Parton Showers (PS) are standard, the consistent matching of NLO EW corrections (specifically QED) to parton showers remains a significant challenge.

Current simulation strategies often suffer from a mismatch:

Inclusive observables are modeled with high-order QCD but often only leading-order (LO) or approximate EW accuracy.
Radiation-sensitive observables (like the vector boson transverse momentum, $p_T$ ) are modeled with QCD but lack consistent NLO EW treatment.
There is a lack of a unified framework that consistently describes both unresolved (inclusive) and resolved (radiative) QED emissions at NLO accuracy.

The authors aim to bridge this gap by adapting the MiNLO (Multi-scale improved NLO) method, originally designed for QCD, to handle NLO QED corrections in Drell-Yan (DY) processes.

2. Methodology

The authors apply the MiNLO formalism to the process $pp \to Z A$ , where $Z$ decays into neutrinos ( $\nu\bar{\nu}$ ) to isolate Initial State Radiation (ISR) and avoid complications from Final State Radiation (FSR). The radiation $A$ can be a photon ( $\gamma$ ) or a quark/antiquark ( $q\bar{q}$ ) arising from QED splitting.

Key methodological adaptations include:

Abelianization of MiNLO Formulae:
The standard QCD MiNLO formulas rely on non-abelian color factors ( $C_A, C_F$ ). For QED, these are replaced with abelian counterparts:
- $C_A \to 0$
- $C_F \to Q_f^2$ (fermion electric charge squared)
- $T_F \to N_c Q_f^2$
- The $\beta$ -function is updated to the QED version ( $\beta_0^{QED}$ ).
Running Coupling and Renormalization:
The electromagnetic coupling $\alpha(\mu)$ is treated as a running coupling in the $\overline{\text{MS}}$ scheme. The authors evaluate the coupling at the scale of the transverse momentum ( $p_T$ ) for the radiation vertex, while keeping the Born-level couplings fixed. They utilize a simplified parton model with only $u$ and $d$ quarks to isolate QED effects.
The Sudakov Peak Problem:
In QCD, the Sudakov peak (where the $p_T$ distribution is maximal) occurs at a few GeV. In QED, due to the finite value of $\alpha$ at low scales (unlike the divergent $\alpha_S$ ) and smaller coefficients, the peak position is estimated to be extremely low ( $p_T \sim 10^{-104}$ GeV). This makes direct numerical evaluation of the MiNLO formula unstable in the low- $p_T$ region.
Proposed Solution: The Technical Cutoff ( $p_T^c$ ):
To circumvent the numerical instability of the Sudakov peak, the authors split the phase space at a technical cutoff $p_T^c$ :
1. Region $p_T > p_T^c$ : The standard (abelianized) MiNLO formula is applied.
2. Region $p_T < p_T^c$ : The non-singular term $R_f$ is neglected. The cross-section is treated as an exact differential, allowing analytical integration to recover the NLO inclusive cross-section without evaluating unstable amplitudes at extremely low $p_T$ .
PDF Evolution:
Since the calculation involves QED radiation, the authors constructed an ad-hoc Parton Distribution Function (PDF) set evolved exclusively with QED kernels (using the HOPPET library). This ensures consistency between the factorization scale and the QED evolution, avoiding large residual scale dependencies.

3. Key Contributions

First Application of MiNLO to QED: This is the first study applying the MiNLO method to calculate NLO QED corrections for Drell-Yan processes, merging inclusive and radiative descriptions.
Handling the QED Sudakov Peak: The authors propose and validate a novel strategy using a technical cutoff to separate the singular and non-singular regions, solving the numerical instability caused by the QED Sudakov peak's position.
Abelianization Framework: They provide a systematic procedure for converting QCD MiNLO coefficients and splitting functions into their QED equivalents.
Validation of NLO Accuracy: They demonstrate that their method reproduces the NLO inclusive cross-section for $pp \to Z$ within a few per mille, confirming that the method correctly merges the resolved radiation ( $Z+\gamma$ ) with the inclusive process.

4. Results

The authors performed numerical validations using an enhanced electromagnetic coupling ( $\bar{\alpha}_0 \approx 0.04$ , roughly 5 times the physical value) to amplify effects and test the formalism's robustness.

Inclusive Cross-Section Agreement:
- When summing the contributions from $p_T < p_T^c$ and $p_T > p_T^c$ , the total cross-section matches the fixed-order NLO Drell-Yan prediction.
- The relative difference ( $\delta$ ) between the MiNLO prediction and the NLO target is found to be negligible (order of $10^{-3}$ to $10^{-4}$ ) for various cutoff values ( $p_T^c$ ranging from $M_Z$ down to $M_Z/200$ ).
- This confirms that neglecting the non-singular term $R_f$ in the low- $p_T$ region is a valid approximation.
Differential Distributions:
- Differential distributions in rapidity ( $y_Z$ ) and invariant mass ( $M_{\nu\bar{\nu}}$ ) show agreement with the NLO target at the level of a few per mille.
- The NLO corrections to the LO cross-section are positive and approximately 2% (for the enhanced coupling).
Physical Coupling Test:
- Using the physical value of $\alpha$ ( $\bar{\alpha}_0 \approx 0.008$ ), the NLO corrections reduce to $\sim 3.7$ per mille.
- The discrepancy between the MiNLO prediction and the NLO target scales as $\alpha^2$ , resulting in a relative uncertainty of approximately 0.01% (0.1 per mille) for realistic parameters and a cutoff $p_T^c \sim 1$ GeV.

5. Significance

This work represents a critical first step toward the MiNNLOPS framework for Electroweak physics.

Foundation for Future Work: It lays the groundwork for combining NLO QCD and NLO EW corrections with parton showers, which is essential for the precision era of the HL-LHC.
Unified Framework: It addresses the current lack of tools that can simultaneously describe inclusive observables and radiation-sensitive variables (like $p_T$ ) with consistent NLO EW accuracy.
Scalability: The methodology developed here (abelianization, handling the Sudakov peak, and PDF evolution) can be extended to include Final State Radiation (FSR) and full electroweak corrections (including weak bosons), eventually leading to a generator capable of NNLO QCD + NLO EW precision.

In conclusion, the paper successfully demonstrates that the MiNLO method can be adapted to QED, providing a robust, accurate, and numerically stable framework for simulating Drell-Yan processes with NLO QED corrections.